evaluation of the optical design and performance of

evaluation of the optical design and performance of
EVALUATION OF THE OPTICAL DESIGN AND
PERFORMANCE OF TRANSLATING-OPTICS
ACCOMMODATING INTRAOCULAR LENSES
Jit Bahadur Ale Magar
A thesis submitted in partial fulfilment of the requirement for
Doctor of Philosophy
September, 2010
Vision Cooperative Research Centre, Brien Holden Vision Institute
School of Optometry and Vision Science
University of New South Wales, Australia
Surname or Family name:
THE UNIVERSITY OF NEW SOUTH WALES
Thesis/Dissertation Sheet
ALE MAGAR
First name: Jit
Other name/s: Bahadur
Abbreviation for degree as given in the University calendar:
PhD
School: School of Optometry and Vision Science
Faculty: Science
Title: Evaluation of Optical Design and Performance of Translating-optics Accommodating Intraocular Lens
Abstract
The quest for restoration of accommodative ability in pseudophakic eyes have led to number of advancement in the
technology including the development of accommodating intraocular lenses. Translating-optics accommodating
intraocular lenses is a rapidly emerging class of implant devices which has debatably gained increasing popularity among
scientists and practitioners within a short period of time. A large variety of accommodating intraocular lenses,
prominently represented by single-element and two-element designs, has been proposed. These devices work under the
principle of changing effective power by shifting axial position of the optical element within the eye which is brought
about by complex opto-mechanical arrangements.
A comprehensive theoretical design analysis and evaluation of their performance is lacking. Virtually no report exists
investigating other important aspects of their optical performance such as magnification, aberrations and optimization of
retinal image quality. This thesis, using paraxial optical principles, undertakes a systematic development of the optical
principles involved in the evaluation of design and performance of translating optics accommodating intraocular lenses
and applies the results to seek superior performance.
It is demonstrated in this thesis that an appropriate design of accommodating intraocular lens may solve a long-standing
unmet visual need of pseudophakic presbyopia. A two-element accommodating intraocular lens has potential to afford
greater accommodative amplitude compared to a single element lens. It is shown that a wide range of accommodating
intraocular lens designs may effectively eliminate or control several on and off-axis aberrations and optimize the retinal
image quality. Dynamic magnification induced due to pseudophakic accommodation is one of the important issues
identified. The power combination of lens elements in two-element accommodating intraocular lens designs requires
special attention prior to implantation to avoid potential dynamic anisometropia and dynamic aniseikonia.
Finally, while two-element accommodating intraocular lenses offer a potentially promising answer to post-cataract
surgical presbyopia, further work is recommended particularly in understanding the mechanics of accommodation, the
causes of presbyopia and to develop improved optical materials and mechanical designs to maximize translation and
provide robust peripheral structures that may withstand mechanical strain arising from the numerous continuous
accommodative cycle and also to resist long-term biological changes.
Declaration relating to disposition of project thesis/dissertation
I hereby grant to the University of New South Wales or its agents the right to archive and to make available my thesis or
dissertation in whole or in part in the University libraries in all forms of media, now or here after known, subject to the
provisions of the Copyright Act 1968. I retain all property rights, such as patent rights. I also retain the right to use in
future works (such as articles or books) all or part of this thesis or dissertation.
I also authorize University Microfilms to use the 350 word abstract of my thesis in Dissertation Abstracts International.
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The University recognises that there may be exceptional circumstances requiring restrictions on copying or conditions
use. Requests for restriction for a period of up to 2 years must be made in writing. Requests for a longer period of
restriction may be considered in exceptional circumstances and require the approval of the Dean of Graduate Research.
FOR OFFICE USE ONLY
Date of completion of requirements for Award:
[ii]
TABLE OF CONTENT
Copyright Statement…………………………………………………. iii
Authenticity Statement………………………………………………….iii
Originality Statement …………………………………………………. iv
Acknowledgement ………………………………………………………vi
Abstract ……………………………………………………………... viii
List of Figures …………………………………………………………. x
List of Tables …………………………………………………………xvii
List of Abbreviations ………………………………………………. xix
Chapter 1: General Introduction ……………………………………….. 1
Chapter 2: Literature Survey …………………………………………..14
Chapter 3: Accommodative Performance ……………………………. 51
Chapter 4: Magnification ……………………………………………... 73
Chapter 5: Depth of Field …………………………………………….. 99
Chapter 6: Optical Design ……………………………………………125
Chapter 7: Performance in Presence of Misalignment ……………… 205
Chapter 8: Chromatic Aberration …………………………………… 264
Chapter 9: Summary and Recommendation ………………………… 294
Appendix ………………………………………….. ……………….. 304
[iii]
COPYRIGHT STATEMENT
‘I hereby grant to the University of New South Wales or its agents the right to archive and to
make available my thesis or dissertation in whole or in part in the University libraries in all
forms of media, now or here after known, subject to the provisions of the Copyright Act 1968. I
retain all property rights, such as patent rights. I also retain the right to use in future works (such
as articles or books) all or part of this thesis or dissertation.
I also authorize University Microfilms to use the 350 word abstract of my thesis in Dissertation
Abstracts International.
I have either used no substantial portions of copyright material in my thesis or I have obtained
permission to use copyright material; where permission has not been granted I have applied/will
apply for a partial restriction of the digital copy of my thesis dissertation.’
Signed: ........................................................................
Date: ............................................................................
AUTHENTICITY STATEMENT
‘I certify that the Library deposit digital copy is a direct equivalent of the final officially
approved version of my thesis. No emendation of content has occurred and if there are any minor
variations in formatting, they are the result of the conversion to digital format.’
Signed: ........................................................................
Date: ............................................................................
[iv]
ORIGINALITY STATEMENT
‘I hereby declare that this submission is my own work and to the best of my knowledge it
contains no materials previously published or written by another person, or substantial
proportions of material which have been accepted for the award of any other degree or diploma
at UNSW or any other educational institution, except where due acknowledgement is made in
the thesis. Any contribution made to the research by others, with whom I have worked at UNSW
or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content
of this thesis is the product of my own work, except to the extent that assistance from others in
the project's design and conception or in style, presentation and linguistic expression is
acknowledged.’
Signed ……………………………………………..............
Date:
[v]
to my parents, wife, daughter and son
[vi]
ACKNOWLEDGEMENTS
During the course of my PhD, a large number of people contributed in assorted ways. It is a
pleasure to convey my gratitude to them all in this, my humble acknowledgment.
In the first place, my deepest gratitude is to Prof. Arthur Ho for his magnificent supervision,
guidance and advice. Above all and the most needed, he provided me with extraordinary
encouragement, experiences and if required, ‘spoon-feeding’ support from every dimension. He
gave me the freedom to explore on my own, and at the same time guided me to recover when my
steps faltered. His truly scientific intuition and passion for science always inspired me to grow as
a student and a researcher.
I am equally indebted to A/Prof. Fabrice Manns. His exceptional supervision, advice and
contribution are truly a backbone of this research and so to this thesis. His involvement with his
originality has triggered and nourished my intellectual maturity from which I will benefit for a
long time to come. I will always admire his amazing critical assessment skill.
In accordance, I honour the brief but incomparable guidance provided by the late and great Prof.
George Smith in the early stage of my candidature. He passed away, and I am one of his
apprentices unfortunate to lose him before completion of the project.
Many thanks go to Dr. Judith Flanagan for her crucial assistance in preparing manuscripts and
transforming my ‘science’ into English. Generous support from the IT, administrative, corporate
and graphics departments of Brien Holden Vision Institute are highly appreciated. Collective and
individual acknowledgments are also owed to my fellow postgraduates; I value their friendship
and support throughout these difficult years.
[vii]
I gratefully acknowledge the generous financial support from the University of New South
Wales Postgraduate Research Award (UIPA) and the Australian Government Vision CRC
Scheme (top-up scholarship), without which this research could not have been carried out.
Most importantly, none of these would have been possible without my family. My parents
deserve a special mention for their consistent encouragement and inseparable support. Words fail
me to express my appreciation to my wife Anita’s unselfish dedication, love and persistent
confidence in me which took the load off my shoulders. I adore my daughter Kreity and son
Anshu who made me cheerful even in the most difficult times. I dedicate this dissertation to my
family.
[viii]
ABSTRACT
Implantation of an intraocular lens (IOL) has become a standard procedure for vision correction
following cataract surgery. Continually rising demand for perfect long-term post-operative vision
has led to a proliferation of novel IOL designs. Though a standard monofocal or single-vision
IOL affords near perfect vision for a fixed viewing distance, a patient may yet require
supplementary optical devices for comfortable near vision.
The quest for restoration of accommodative ability in pseudophakic eyes have led to number of
advancement in the technology including the development of accommodating intraocular lenses.
Translating-optics accommodating intraocular lenses is a rapidly emerging class of implant
devices which has debatably gained increasing popularity among scientists and practitioners
within a short period of time. A large variety of accommodating intraocular lenses, prominently
represented by single-element and two-element designs, has been proposed. These devices work
under the principle of changing effective power by shifting axial position of the optical element
within the eye which is brought about by complex opto-mechanical arrangements.
Earlier studies on accommodating intraocular lenses are primarily focused on their
accommodative capacity. A comprehensive theoretical design analysis and evaluation of their
performance is lacking. Virtually no report exists investigating other important aspects of their
optical performance such as magnification, aberrations and optimization of retinal image quality.
This thesis, using paraxial optical principles, undertakes a systematic development of the optical
principles involved in the evaluation of design and performance of translating optics
accommodating intraocular lenses and applies the results to seek superior performance.
In the thesis, it is demonstrated that an appropriate design of accommodating intraocular lens
may enhance performance overcoming a long-standing unmet visual need of pseudophakic
presbyopia. A two-element accommodating intraocular lens has been found to afford greater
[ix]
accommodative amplitude compared to a single element accommodating intraocular lens. It is
also shown that a wide range of accommodating intraocular lens designs may effectively
eliminate or control several on and off-axis aberrations and optimize the retinal image quality.
Dynamic magnification induced due to pseudophakic accommodation is one of the important
issues identified. The power combination of lens elements in two-element accommodating
intraocular lens designs requires special attention prior to implantation to avoid potential
dynamic anisometropia and dynamic aniseikonia.
Finally, while two-element accommodating intraocular lenses offer a potentially promising
answer to post-cataract surgical presbyopia, further work is recommended particularly in
understanding the mechanics of accommodation, the causes of presbyopia and to develop
improved optical materials and mechanical designs to maximize translation and provide robust
peripheral structures that may withstand mechanical strain arising from the numerous continuous
accommodative cycle and also to resist long-term biological changes.
[x]
List of Figures
Figure 2.1:
Helmholtz’ drawing demonstrating his theory of accommodation ....................... 18
Figure 2.2:
Schachar theory accommodated ........................................................................... 19
Figure 2.3:
Various methods of treating presbyopia ............................................................... 23
Figure 2.4:
Accommodative function of the dual optic IOL ................................................. 34
Figure 2.5:
A design of 2E-AIOL ........................................................................................... 35
Figure 2.6:
Accommodative performance of AIOLs (literature) ............................................ 37
Figure 2.7:
Defocus curves for monofocal and dual-optic IOLs (literature) .......................... 37
Figure 3.1:
Basic schema of the model eye implanted with AIOL ......................................... 55
Figure 3.2:
Accommodative performance versus front element power for a 2E-AIOL ......... 64
Figure 3.3:
Accommodative performance versus back element power for a 2E-AIOL ......... 64
Figure 3.4:
Paraxial optics prediction of change in accommodation ...................................... 65
Figure 3.5:
Accommodative performance versus total IOL power for 1E-AIOL .................. 67
Figure 4.1:
Layout of the eye model with AIOL .................................................................... 80
Figure 4.2:
Relative Lateral Magnifications of 1E-AIOL ...................................................... 86
[xi]
Figure 4.3:
Relative Lateral Magnifications of 2E-AIOLs ..................................................... 87
Figure 4.4:
Absolute values of Angular Magnification .......................................................... 88
Figure 4.5:
Change in distance between nodal point and retina .............................................. 89
Figure 4.6:
Relative Lateral Magnification for various models of AIOL ............................... 90
Figure 4.7:
Absolute values of angular magnification as a function of translation ................ 91
Figure 5.1:
Geometrical representation of the eye used to calculate the position and size
of the exit pupil, limit of blur circle size and depth of field ............................... 105
Figure 5.2:
Basic schema of the ray tracing .......................................................................... 110
Figure 5.3:
Rate of change in depth of field ......................................................................... 115
Figure 5.4:
Effect of front lens element power on depth of field ......................................... 116
Figure 5.5:
Effect of implantation depth on depth of field ................................................... 117
Figure 5.6:
Effect of implantation depth on effective depth of field .................................... 118
Figure 6.1:
Corneal, Internal and Ocular coefficients C(3, 1) and C(4, 0) ............................ 134
Figure 6.2:
Change in RMS wavefront error with accommodation ...................................... 136
Figure 6.3:
Spherical aberration as a function of accommodation (He et al, 2000) ............. 137
Figure 6.4:
Basic schema of a 2E-AIOL ............................................................................... 142
Figure 6.5:
Illustration of the spacing between elements ..................................................... 144
[xii]
Figure 6.6:
Possible forms elements of 2E-AIOL and their combinations ........................... 145
Figure 6.7:
Xmin of an 1E-AIOL for near and distance vision ............................................ 152
Figure 6.8:
Minimum spherical aberration of 1E-AIOL ....................................................... 153
Figure 6.9:
Seidel Spherical aberration produced by 1E-AIOL ........................................... 154
Figure 6.10:
Seidel spherical aberration of the model eye with 1E-AIOL ............................. 155
Figure 6.11:
Seidel spherical aberration of an AIOL .............................................................. 156
Figure 6.12:
Seidel primary coma of 1E-AIOL ...................................................................... 160
Figure 6.13:
Spherical aberration and coma of 1E-AIOL ...................................................... 161
Figure 6.14:
Design of spherical 2E-AIOL for criterion 1(i) ................................................. 166
Figure 6.15:
Design of spherical 2E-AIOL for criterion 1(ii) ................................................ 168
Figure 6.16:
Design combination of refractive indices for 2E-AIOL ..................................... 169
Figure 6.17:
Design of spherical 2E-AIOL for criterion 2(i) ................................................. 170
Figure 6.18:
Spherical design of 2E-AIOL for criteria 2(ii) ................................................... 171
Figure 6.19:
Design of 2E-AIOL for coma: criterion 1 .......................................................... 174
Figure 6.20:
Design of 2E-AIOL for coma: criterion 2........................................................... 175
Figure 6.21:
Designs of 2E-AIOL for zero coma and spherical aberration ............................ 176
Figure 6.22:
Spherical aberration of the 2E-AIOL as a function of bending factors ............. 177
[xiii]
Figure 6.23:
Two-dimensional conic sections ......................................................................... 181
Figure 6.24:
Design of 1E-AIOL with aspheric front surface ................................................. 185
Figure 6.25:
Design of 1E-AIOL with front aspheric surface ................................................. 186
Figure 6.26:
Design of 2E-AIOL for coma and spherical aberration ...................................... 187
Figure 6.27:
Aberrations of the aspheric 1E-AIOL ................................................................ 188
Figure 6.28:
Asphericities of the front element and rear element for various criteria............. 191
Figure 7.1:
Aligned and misaligned elements in 2E-AIOL ................................................... 219
Figure 7.2:
Change in spherical, coma and astigmatism with various amount of tilt ........... 223
Figure 7.3:
Change in spherical, coma and astigmatism due decentration ........................... 224
Figure 7.4:
Change in spherical aberration with bending factor of 1E-AIOL....................... 225
Figure 7.5:
Change in vertical coma as a function of bending factor of 1E-AIOL .............. 226
Figure 7.6:
Change in astigmatism as a function of bending factor of 1E-AIOL ................. 227
Figure 7.7:
Changes in Zernike coefficient for spherical aberration due to tilt and
decentration of 2E-AIOL .................................................................................... 228
Figure 7.8:
Changes in Zernike coefficient for coma due to tilt and decentration of 2EAIOL .................................................................................................................. 229
Figure 7.9:
Changes in astigmatism due to tilt and decentration of 2E-AIOL ..................... 229
[xiv]
Figure 7.10:
Spherical aberration when front element alone is misaligned............................. 231
Figure 7.11:
Spherical aberration when rear element alone is misaligned .............................. 231
Figure 7.12:
Spherical aberration in waves when system is misaligned ................................. 232
Figure 7.13:
Coma aberration when front element alone is misaligned .................................. 233
Figure 7.14:
Coma aberration when rear element alone is misaligned.................................... 233
Figure 7.15:
Coma aberration when system is misaligned ...................................................... 234
Figure 7.16:
Astigmatism when front element alone is misaligned ........................................ 235
Figure 7.17:
Astigmatism when rear element alone is misaligned .......................................... 235
Figure 7.18:
Astigmatism when system in a group is misaligned ........................................... 236
Figure 7.19:
An example of polynomial fit of the spherical aberration induced by
extreme misalignment as a function of the bending factor ................................. 238
Figure 7.20:
Design of misaligned 2E-AIOL to eliminate spherical aberration...................... 240
Figure 7.21:
Design of misaligned 2E-AIOL to eliminate spherical aberration for near
and distance vision .............................................................................................. 241
Figure 7.22:
Design of 2E-AIOL to eliminate coma ............................................................... 243
Figure 7.23:
Design of 2E-AIOL to eliminate astigmatism..................................................... 246
Figure 8.1:
Chromatic Dispersion of the ocular media.......................................................... 271
[xv]
Figure 8.2:
Chromatic Dispersion of IOL materials .............................................................. 273
Figure 8.3:
Chromatic focal shift of phakic and pseudophakic model eyes .......................... 277
Figure 8.4:
Lateral chromatic aberration of 1E-AIOL........................................................... 278
Figure 8.5:
TCA of the eye implanted with 1E-AIOL........................................................... 279
Figure 8.6:
Chromatic difference of refraction of 2E-AIOL ................................................. 280
Figure 8.7:
LCA of the eye with 2E-AIOL as a function of accommodation ....................... 281
Figure 8.8:
TCA of the eye with 2E-AIOL as a function of accommodation ....................... 281
Figure 8.9:
Refractive index and V-number of the 1E-AIOL material for zero LCA ........... 284
Figure 8.10:
V-numbers of the front and rear elements required for achromatisation ............ 286
Figure 8.11:
Residual LCA of the eye after achromatisation .................................................. 287
Figure A.1:
Two-element Alvarez varifocal transverse translating AIOL ............................. 309
Figure A.2:
Dynamic optic AIOL of Cumming .................................................................... 310
Figure A.3:
Design of AIOL by Skottun ............................................................................... 310
Figure A.4:
Design of a dynamic AIOL ................................................................................ 311
Figure A.5:
NuLens design..................................................................................................... 312
Figure A.6:
BioComFold (Morcher, Germany) 1E-AIOL ..................................................... 312
Figure A.7:
1 CU and AT-45 1E-AIOL.................................................................................. 313
[xvi]
Figure A.8:
A design of 2E-AIOL ......................................................................................... 314
Figure A.9:
A design of 2E-AIOL where one element is implanted in anterior chamber...... 315
Figure A.10: Pyson’s magnet driven design of a 2E-AIOL .................................................... 315
Figure A.11: A design of 2E-AIOL ......................................................................................... 316
Figure A.12: A design of 2E-AIOL ......................................................................................... 317
Figure A.13: Zhang et al design of a 2E-AIOL ....................................................................... 317
Figure A.14: A design of 2E-AIOL as described by Magnante .............................................. 318
Figure B.1:
Diagram illustrating the sign convention ........................................................... 320
Figure B.2:
Symbols used in ray-tracing equations................................................................ 321
Figure B.3:
Diagrammatic illustration of ray transfer ........................................................... 323
Figure B4:
A schema of off-axis ray tracing ........................................................................ 334
Figure C1:
Navarro finite model eye implanted with AIOL ................................................. 337
[xvii]
List of Tables
Table 1.1:
Important milestones in cataract and IOL implantation.......................................... 4
Table 2.1:
Theoretically predicted accommodation ............................................................... 29
Table 2.2:
Accommodation with 1E-AIOL............................................................................ 30
Table 2.3:
Translation of 1E-AIOL ........................................................................................ 31
Table 2.4:
Near vision obtained with AIOLs ......................................................................... 32
Table 2.5:
Accommodative performance of 2E-AIOL........................................................... 36
Table 4.1:
Changes in relative magnifications as a function of accommodation ................. 92
Table 5.1:
Values of depth of focus in the eyes ................................................................... 102
Table 5.2:
Summary of the change in depth of focus at various conditions ........................ 114
Table 5.3:
Rate of change in effective depth of focus.......................................................... 114
Table 6.1:
Zernike coefficients for Spherical aberration reported in the literatures …… ... 132
Table 6.2:
Published bending factors of conventional IOL.................................................. 139
Table 6.3:
Spherical aberration and coma of various model eyes ........................................ 162
Table 6.4:
Designs to satisfy specific criteria for distance and near foci simultaneously ... 179
Table 6.5:
Asphericities representing various aspheric surfaces.......................................... 182
[xviii]
Table 7.1:
Effect of capsulotomy type on tilt and decentration of IOLs .............................. 212
Table 7.2:
Effect of fixation site/haptics position in the bag on tilt and decentration.......... 213
Table 7.3:
Effect of ocular pathology on the alignment of IOL........................................... 214
Table 7.4:
Tilt and decentration of the crystalline lens and IOL position............................ 217
Table 7.5:
Misalignment induced spherical aberrations of a 2E-AIOL ............................... 237
Table 7.6:
Polynomial fitting for misalignment induced spherical aberration ..................... 239
Table 7.7:
Optimal bending factors for coma in presence of misalignment ........................ 242
Table 7.8:
Roots and polynomial equations obtained for the design of 2E-AIOL............... 243
Table 7.9:
Optimal design of misaligned 2E-AIOL to produce minimum astigmatism ..... 244
Table 7.10:
Roots and polynomial equations to eliminate astigmatism................................. 245
Table 8.1:
Coefficients for the Cauchy equation for individual ocular media ..................... 270
Table B.1
Values of p and q for various monochromatic aberrations ................................ 334
[xix]
List of Abbreviation
1E
Single-element (mono element)
2E
Two-element (dual element)
AIOL
Accommodating Intraocular Lens
AM
Angular Magnification
DoF
Depth of Field/Depth of Focus
ECCE
Extracapsular Cataract Extraction
FDA
Food and Drug Association
GRIN
Gradient Refractive Index
ICCE
Intracapsular Cataract Extraction
IOL
Intraocular Lens
LCA
Longitudinal Chromatic Aberration
LEC
Lens Epithelial Cell
LM
Lateral Magnification
MRI
Magnetic Resonance Imaging
NA
Not Available
PCIOL
Posterior Chamber Intraocular Lens
PCO
Posterior Capsular Opacification
PMMA
Poly(methyl)methacrylate
RSM
Relative Spectacle Magnification
SA
Spherical Aberration
TCA
Transverse Chromatic Aberration weird
General Introduction
Chapter 1
General Introduction
1
General Introduction
TABLE OF CONTENT
1.1 HISTORY OF CATARACT SURGERY AND IOL IMPLANTATION ............................. 3
1.2 IOLS FOR NEAR VISION ................................................................................................... 6
1.3 OBJECTIVE OF THE RESEARCH .................................................................................... 8
1.4 OVERVIEW OF METHODS ............................................................................................... 9
1.5 STRUCTURE OF THIS THESIS ....................................................................................... 10
1.6 REFERENCES .................................................................................................................... 11
2
General Introduction
1.1 History of Cataract Surgery and IOL Implantation
Cataract is a general loss of transparency in the crystalline lens, primary effect being a gradual
but progressive loss of optical clarity of the crystalline lens bringing about a reduction of vision
which eventually leads to blindness. Fortunately, the blindness caused by cataracts can be
reversed with relatively uncomplicated surgical intervention albeit the only available treatment
for centuries. In early days, surgical removal of cataractous lens left the patient without the
optical benefit of the natural lens (aphakia), requiring thick plus lenses (about +10.0D or more)
for post-operative hyperopic correction.
In l949, a breakthrough occurred in medical treatment when Sir Harold Ridley implanted the first
artificial intraocular lens (IOL). This first ever IOL (manufactured by the Rayner Company in
England) did not find widespread acceptance until the 1970s, following further developments in
the lens design and surgical techniques. While far from perfect, the procedure worked well
enough to encourage further refinement, and today, virtually all cataract patients have the benefit
of the device. Improvements in design and manufacture are now providing almost perfect IOLs
which can be customized to an exact power and size for each patient. IOL implantation has been
clinically successful since the mid-1960s, though the first FDA approval occurred in 1977
(Milton Roy Co. Analytical). Since its introduction, the IOL has undergone a wide variety of
changes. Now, implantation of the IOL is a standard procedure of visual rehabilitation following
cataract removal. A range of novel lens designs have been developed to meet the constantly
rising demands for perfect vision. Important milestones of cataract surgery and IOL implantation
are summarized in Table 1.1.
3
General Introduction
Table 1.1: Important milestones in cataract surgery and IOL implantation (Jaffe, 1998; Buznego
& Trattler, 2008)
Year
Event
Place
Researchers/Lens
800
Couching
India
Unknown
1015
Needle aspiration
Iraq
Unknown
1500
Couching
Europe
Unknown
1745
ECCE
France
Daviel
1753
ICCE
England
Sharp
1949
ECCE with PCIOL
England
Ridley
1967
ECCE by phacoemulsification
USA
Kelman
1984
Foldable IOLs
USA & SA
Mazzocco, Epstein
1997
Accommodating IOL prototype
Europe
Cumming, Kamman
1997
First FDA Approved multifocal
USA
AMO Array lens
2003
First FDA Approved AIOL
USA
Crystalens AT-45
ICCE = intracapsular cataract extraction, ECCE = extracapsular cataract
extraction, PCIOL = posterior chamber intraocular lens, FDA = US Food and Drug
Administration, AIOL = accommodating intraocular lens.
Much research has been invested in optimizing the performance of traditional designs of
monofocal (or single-vision) IOLs. The primary focus of improvement has been mostly in the
geometrical/optical surface design. Production of monofocal IOL of various lens forms (e.g.
plano-convex, biconvex) and introduction of aspheric surface are some of the improvements.
AcrySof (Alcon) and Tecnis (Abbott Medical Optics) are some other examples of commercially
available aspheric IOLs. These lenses are primarily intended to control spherical aberration
which thereby improves retinal image quality. The effect of aspheric designs on optical
performance will be discussed in Chapter 6.
4
General Introduction
Posterior capsular opacification (PCO) is one of the most common post-operative complications
of the cataract surgery. An attempt to eliminate this complication through improved physical
design of IOLs, especially edge design, is another important area of development. There has
been continuous improvement in edge design for the prevention of capsular epithelial migration
and subsequent formation of PCO. Square and truncated edge designs have been found to be
effective. Maximal optics contact to the posterior capsule has also been credited for minimizing
occurrence of PCO.
Another improvement in the implant technology is the advent of newer materials for IOLs.
Material choice must take into account:
1) stability and inertness when implanted
2) biocompatibility and safety inside the eye
3) a high level of light transmissibility
4) preferably high refractive index to reduce the overall thickness and bulk of the lens
5) low mass, and
6) soft and flexible material to allow folding and insertion through small incision.
Most of the olden days IOLs were made of glass material though the introducing of the IOL was
made of Perspex. Plastics were used later, after evidence of their inertness following an incident
in which a fragment of shattered windshields from a bullet was lodged in the eye of a pilot
during War II. It was noted that this fragment did not produce inflammation. This
polymethylmethacrylate (PMMA, chemical formula [C5O2H8]n) was the most popularly used
5
General Introduction
material for IOL fabrication (Colenbrander et al., 1988). For a long period, PMMA remained as
the only IOL material for its high impact resistant to aging and climatic changes, excellent
biocompatibility and cost effectiveness. However, PMMA, being relatively rigid, requires a
surgical wound that is sufficiently wide for the implantation of the IOL. With the increasing
popularity of small incision cataract surgery for its advantages such as the acceleration of visual
rehabilitation and minimization of post-operative astigmatism, people sought for more flexible
(soft) material that can be squeezed through a small surgical port (Linebarger et al, 1999).
Today, silicone-based polymer is one of the most frequently used materials in fabricating modern
foldable IOL primarily for its better flexibility. More accurately known as polymerized siloxanes
or polysiloxanes, silicone material is a mixture of inorganic-organic polymers with the general
chemical formula [R2SiO] where R represents organic groups such as methyl, ethyl, and phenyl.
This material consists of an inorganic silicon-oxygen backbone (…-Si-O-Si-O-Si-O-…) with
organic side groups attached to the silicon atoms. In some cases, organic side groups can be used
to link two or more of these -Si-O- backbones together. By varying the -Si-O- chain lengths, side
groups, and cross-linking, silicones can be synthesized with a wide variety of properties and
compositions. They can vary in consistency from liquid to gel to rubber to hard plastic.
Phacoflex (Advanced Medical optics, Inc.), Acrysof (Alcon Laboratories), NaturalLensTM
(Eyekon Medical), Matrix Acrylic (Medennium, Inc.) are some of the commercially available
foldable IOLs fabricated from flexible silicones. Hydrophillic material is another addition.
1.2 IOLs for Near Vision
Traditional IOLs are monofocal optical devices which provide clear vision at a fixed distance.
Typically surgeons implant an IOL aiming to provide clear distance vision. As a result, a patient
often requires an additional optical device such as spectacle or contact lens for a comfortable
near vision. The demand for optimum near vision among pseudophakic patients and a concurrent
preference for good cosmesis is driving the development of innovative IOL designs that also
6
General Introduction
support good near vision. Developments in multifocal IOLs represent the early attempts to
improve near vision for pseudophakic presbyopes. These IOLs employ multiple optical zones or
power progressions (e.g. ReZoom, Advanced Medical Optics) or zones with diffractive optics
(e.g. ReStor, Alcon) to simultaneously provide near and distance images. Due to the
simultaneous presentation of images on the retina for a range of viewing distances to achieve
near and distance focus, these devices typically compromise on some aspects of visual
performance including subjective vision complaints such as glare, halos, poor contrast and
ghosting (Artal et al., 1995, Dick et al., 1999, Manns et al., 2004). But a major weakness of such
devices is that they do not provide a continuously variable change in focus afforded by natural
accommodation and enjoyed by the pre-presbyopic eye. Some surgeon attempted intentional
myopization of one eye by implanting the IOL power calculated for near vision (Baikoff, 2004);
also called the monovision technique. Though this technique is cosmetically acceptable, due to
the loss of binocularity and depth perception, monovision is not tolerated by about one quarter of
the population (Evans, 2007, Jain et al., 1996, Erickson & Schor, 1990).
Ahead of these developments, accommodating IOLs (AIOL) have emerged with rapid progress
in development. While monofocal IOLs remain the most frequently implanted post-cataract
devices by far, AIOLs are becoming a preferred device family for restoring accommodation to
pseudophakic presbyopes (Leaming, 2004). Within the family of AIOLs, a range of designs and
configurations have been proposed and developed (Assia, 1997, Dick, 2005, Doane and Jackson,
2007). The most prominent of the designs and configurations is represented by the singleelement AIOL (1E-AIOL) and two-element AIOL (2E-AIOL) (Assia, 1997, Waring, 1992).
These AIOL designs operate on the principle of changing effective power by shifting the axial
position (a process often called “translation”) of one or more lens elements of the AIOL within
the eye under the influence of physiological mechanism accommodation. Various designs of
AIOL potentially able to restore accommodation in the pseudophakic eye have been proposed
and some have been developed. Some of these designs will be discussed in Chapter 2 and
Appendix A.
7
General Introduction
In addition to AIOLs, a range of ‘lens-centric’ strategies for restoring accommodation, such as
lens refilling (Kessler, 1964, Koopmans et al., 2006, Nishi and Nishi, 1998, Nishi et al., 1998,
Nishi et al., 2008, Parel et al., 1986) and photomodulation (Krueger et al., 2001, Myers and
Krueger, 1998, Krueger et al., 2005, Ripken et al., 2008)
are currently being developed.
Detailed discussion of these procedures appears in the literature (Manns et al., 2004) and is
beyond the scope of this thesis.
1.3 Objective of the Research
The quest of presbyopia reversal and restoration of accommodative ability in a pseudophakic eye
has led to a number of advancements in ophthalmic implant technology including the
development of accommodating IOLs (AIOL). AIOLs are very much in their infancy;
nonetheless a number of variants have already been tested in the clinical setting. Studies carried
out with respect to AIOLs so far have focused exclusively on the accommodative performance.
Virtually no reports exist that investigate other important aspects of AIOL optical performance
such as magnification, and aberration. Moreover, designs to optimize retinal image quality have
yet to be explored. A broad objective of this project therefore is to investigate the principles of
design for translating-optics AIOLs (1E & and 2E-AIOL) with a view to optimizing retinal
image quality in a pseudophakic eye and thereby improving optical performance. In lieu of
available prototypes, as well as embracing the broad generalisability of analytical approachs for
performance evaluation, this thesis will focus on the theoretical analysis and design evaluation of
the optical performance of AIOLs.
Specifically, the project aims to:
1)
Evaluate the accommodative performance of 1E & 2E translating-optics AIOLs to
determine optimal designs with respect to accommodative performance
8
General Introduction
2)
Estimate magnification changes introduced by AIOL and its potential effects on visual
performance
3)
Within paraxial optical theory, derive and apply equations of Seidel primary aberrations
pertaining to AIOL designs to control on and off-axis aberrations
4)
Assess optical performance in terms of chromatic aberration and identify method for
achromatization of AIOL
5)
Evaluate optical performance when the device is misaligned in an idealized eye
1.4 Overview of Methods
The following two approaches are predominantly adopted in the study:
1)
Analytical Approach and
2)
Computational Approach
Analytical approach forms a fundamental method in optical design as it can provide closed-form
solutions that encompass an entire AIOL design space. In this thesis, where possible, we will
review and derive analytical equations based on geometrical optics to evaluate the performance
of AIOL in terms of accommodation, aberrations, magnification and depth of focus.
Computational approach is another important method widely applied throughout this study.
Computer-based optical design (Zemax) and mathematical (Matlab) softwares are used in the
following conditions:
9
General Introduction
1)
When the analytical approach is impractical; for example when the equations are too
complex, tedious and extremely protracted.
2)
When exact ray-tracing is desired, e.g., when Gaussian or third-order analysis is not
adequate
3)
When a design evaluation is too intricate for the analytical approach, e.g. for
tolerancing.
1.5 Structure of this Thesis
In general, this thesis proceeds with a brief review of the literature pertinent to theories of
accommodation, presbyopia and design and performance of accommodating IOLs (AIOLs)
which will help in identifying key problems associated with the existing concept of the
translating-optics AIOLs. This is followed by designing AIOLs and evaluating the performance.
More specifically, Chapter 2 reviews various principles of the accommodative mechanism and
theories of the development of presbyopia in brief. The performance of AIOLs in terms of
accommodation, is investigated and summarized in this chapter. Extending on the work
established in Chapter 2, the accommodative performance of translating AIOLs is evaluated in
Chapter 3 using paraxial analytical and computational approaches. In Chapter 4, magnifications
resulting from the AIOL are studied. In Chapter 5, depth of field of various models of
translating-optics AIOLs is studied. Optimal design of spherical and aspheric AIOLs to eliminate
or control on-axis (spherical) and off-axis (coma) aberrations is investigated in Chapter 6. In this
Chapter, Seidel primary aberration theories are employed to evaluate aberrations and designs.
The optical performances of AIOLs in the presence of misalignment are evaluated in Chapter 7
with a focus on induced spherical, coma and astigmatic aberrations. In this Chapter, optimal
designs taking into account tilt & decentration is proposed. In Chapter 8, the chromatic
10
General Introduction
aberration of AIOLs is evaluated and a method of achromatization is proposed. Chapter 9
summarizes and discusses the overall study including suggestions for further work.
As part of the literature survey, various designs of translating-optics AIOL found from the patent
search are described briefly in Appendix A. The optical principles such as sign convention, ray
tracing methods and Seidel Primary aberrations are defined in Appendix B. A description of the
model eye used in the experiments is given in Appendix C.
1.6 References
Artal, P., Marcos, S., Navarro, R., Miranda, I. and Ferro, M. (1995). Through focus image
quality of eyes implanted with monofocal and multifocal intraocular lenses. Optical Eng. 34,
772:779.
Assia, E. I. (1997). Accommodative intraocular lens: a challenge for future development. J
Cataract Refract Surg. 23, 458-460.
Baikoff, G. (2004). Surgical treatment of presbyopia: scleral, corneal, and lenticular. Curr Opin
Ophthalmol. 15, 365-369.
Buznego, C. & Trattler, W.B. (2008). Presbyopia-correcting intraocular lenses. Curr Opin
Ophthalmol 2008; 20: 6.
Colenbrander, A., Woods, L. V. and Stamper, R. L. (1988). Intraocular Lens data.
Ophthalmology, Instrument and Book Suplement, 9.
Dick, H. B. (2005). Accommodative intraocular lenses: current status. Curr Opin Ophthalmol.
16, 8-26.
Dick, H. B., Krummenauer, F., Schwenn, O., Krist, R. and Pfeiffer, N. (1999). Objective and
subjective evaluation of photic phenomena after monofocal and multifocal intraocular lens
implantation. Ophthalmol. 106, 1878-1886.
11
General Introduction
Doane, J. F. and Jackson, R. T. (2007). Accommodative intraocular lenses: considerations on
use, function and design. Curr Opin Ophthalmol. 18, 318-324.
Erickson P. and Schor C. (1990). Visual function with presbyopic contact lens correction. Optom
Vis Sci 67, 22-28
Evans, B. J. W. (2007). Monovision: a review. Ophthalmic Physiol Opt 27, 417-429.
Jaffe, N. S. (1998). History of Cataract Surgery. In: The History of Modern Cataract Surgery
(eds M. L. Kwitko and C. D. Kelman), Kugler Publication, New York.
Jain, S., Arora, I. and Azar, D. T. (1996). Success of monovision in presbyopes: Review of the
literature and potential applications to refractive surgery. Sur Ophthalmol 40, 491-499.
Kessler, J. (1964). Experiments in Refilling the Lens. Archives of Ophthalmol. 71, 412-417.
Koopmans, S. A., Terwee, T., Glasser, A., et al. (2006). Accommodative lens refilling in rhesus
monkeys. Invest Ophthalmol Vis Sci. 47, 2976-2984.
Krueger, R. R., Kuszak, J., Lubatschowski, H., Myers, R. I., Ripken, T. and Heisterkamp, A.
(2005). First safety study of femtosecond laser photodisruption in animal lenses: tissue
morphology and cataractogenesis. J Cataract Refract Surg. 31, 2386-2394.
Krueger, R. R., Sun, X. K., Stroh, J. and Myers, R. (2001). Experimental increase in
accommodative potential after neodymium: yttrium-aluminum-garnet laser photodisruption
of paired cadaver lenses. Ophthalmol. 108, 2122-2129.
Leaming, D. V. (2004). Practice styles and preferences of ASCRS members--2003 survey. J
Cataract Refract Surg. 30, 892-900.
Linebarger, E.J; Hardten, D.R.; Shah, G.K.; and Lindstrom, R.L. (1999). Phacoemulsification
and modern cataract surgery. Surv Ophthalmol 44, 123-47.
Manns, F., Ho, A. and Kruger, R. (2004). Customized Visual Correction of Presbyopia. In:
Wavefront-guided Visual Corrections: the Quest for Super Vision II (eds R. Kruger, H.
Helmholtz and S. Macrae), SLACK Inc., Thorofare, NJ, pp 353 - 362.
12
General Introduction
Myers, R. I. and Krueger, R. R. (1998). Novel approaches to correction of presbyopia with laser
modification of the crystalline lens. J Refracte Surg. 14, 136-139.
Nishi, O. and Nishi, K. (1998). Accommodation amplitude after lens refilling with injectable
silicone by sealing the capsule with a plug in primates. Archives Ophthalmol. 116, 13581361.
Nishi, O., Nishi, K., Mano, C., Ichihara, M. and Honda, T. (1998). Lens refilling with injectable
silicone in rabbit eyes. J Cataract Refract Surg. 24, 975-982.
Nishi, O., Nishi, K., Nishi, Y. and Chang, S. (2008). Capsular bag refilling using a new
accommodating intraocular lens. J Cataract Refract Surg. 34, 302-309.
Parel, J. M., Gelender, H., Treffers, W. F. and Norton, E. W. (1986). Phaco-Ersatz: cataract
surgery designed to preserve accommodation. J Optic Soc Am. 224, 165-173.
Ripken, T., Oberheide, U., Fromm, M., Schumacher, S., Gerten, G. and Lubatschowski, H.
(2008). fs-Laser induced elasticity changes to improve presbyopic lens accommodation.
Graef Arch Clin Exp Ophthalmol. 246, 897-906.
Waring, G. O., 3rd (1992). Presbyopia and accommodative intraocular lenses--the next frontier
in refractive surgery? Refract Corneal Surg. 8, 421-423.
13
Literature Survey
Chapter 2
Literature Survey
14
Literature Survey
TABLE OF CONTENT
2.1 PHYSIOLOGY OF ACCOMMODATION AND PRESBYOPIA ..................................... 16
2.2 ACCOMMODATION ......................................................................................................... 16
2.3 PRESBYOPIA ..................................................................................................................... 19
2.4 TREATMENT OF PRESBYOPIA ..................................................................................... 20
2.5 TREATMENT OPTIONS FOR PSEUDOPHAKIC PRESBYOPIA................................. 23
2.6 TRANSLATING-OPTICS AIOL........................................................................................ 25
2.6.1 Single-element AIOL (1E-AIOL) ............................................................................25
2.6.2 Dual-element AIOL (2E-AIOL) ..............................................................................32
2.7 SUMMARY ......................................................................................................................... 37
2.8 REFERENCES .................................................................................................................... 39
15
Literature Survey
2.1 Physiology of Accommodation and Presbyopia
Accommodation is a physiological process which facilitates the provision of clear vision of
objects located at a range of distances. This process is accomplished through a complex
coordination of the neuromuscular action of the visual system to effect precisely controlled
alteration in crystalline lens power. The accommodative ability of the eye decreases with age,
eventually leading to the loss of near-visual ability, a condition known as presbyopia. Presbyopia
has been a topic of intense investigation from the early days of physiological optics (Duane,
1922, Duane, 1912, Donders, 1864). Yet, its precise mechanism and the functional relationship
between the crystalline lens and other associated ocular structures are still debated today.
Over one billion individuals are estimated to live with presbyopia globally (Holden et al., 2008).
Several optical devices such as spectacles and contact lenses are readily available for the
correction of this visual disorder. These devices compensate for the loss of accommodative
ability but do not restore the full accommodative response. Some innovative surgical and implant
technologies have been proposed to reverse presbyopia. The success of these innovations, some
of which have become controversial, require a complete and accurate understanding of the
normal physiological mechanism of accommodation and presbyopia. We will briefly discuss
some of the more important attempts to reverse presbyopia in the following section. Before this,
we will briefly review the main theories of the mechanism of accommodation and
pathophysiology of presbyopia.
2.2 Accommodation
Historical aspects of the theories of accommodation can be found in the literature (Strenk et al.,
2005, Weeber, 2008). Descartes was the first to report changes in crystalline lens shape during
accommodation back in 1677. These observations were confirmed by Cramer (1853) by
examining the Purkinje images in which he reported an increase in the anterior curvature of the
16
Literature Survey
crystalline lens. Cramer assumed that contraction of the ciliary muscle and the iris together
brings about accommodative changes. A group of scientists (Fincham, 1937, Gullstrand, 1909,
Helmholtz, 1924, Coleman, 1970) proposed that during accommodation, the action of the ciliary
muscle and choroid create pressure in the vitreous cavity which causes the changes in surface
curvature of the crystalline lens. Today there is general consensus that accommodation results
from changes in the shape of the lens. Yet, controversy exists regarding the mechanism by which
lens alters its shape. Currently, there are two contradicting major theories of the mechanism of
accommodation.
According to classic Helmholtz theory (Helmholtz, 1855, Fisher, 1977), during accommodation
contraction of the ciliary muscle releases the tension on the zonules, which in turn allows the
crystalline lens to increase its curvature owing to its inherent visco-elastic properties.
Simultaneously, the equatorial diameter of the lens decreases whereas its central thickness
increases. In the resting position, the ciliary muscle is relaxed and the eye is focused at distance,
when the eye returns to its resting position, the size of the ciliary aperture is increased.
Consequently the tension on the zonules increases, which leads to a flattening of the lens. Later
Gullstrand
expanded
this
theory
introducing
the
now-adopted
term
intracapsular
“accommodation” (Gullstrand, 1911) and proposed that there is a forward translation of the
anterior lens surface, the refractive gradient index changes within the lens and an elastic force of
the choroid helps restoring the ciliary muscle contraction (Helmholtz, 1924). The basic principles
of the Helmholtz theory of accommodation are overwhelmingly supported by the vast majority
of recent in-vivo (Baikoff, 2004, Fincham, 1937, Strenk et al., 1999, Strenk et al., 2004, Wilson,
1997) and ex-vivo (Ehrmann et al., 2008, Manns et al., 2007, Glasser et al., 2001, Glasser and
Kaufmann, 1999) experiments.
On the other hand, Tscherning believed that accommodation is brought about by an increase in
the contraction of the ciliary muscle would increase, rather than decrease, the zonular tension
causing the zonular force to flatten the lens peripherally with resultant forward bulging of the
central region (Tscherning, 1924). He assumed that since the nucleus is more resistant to
17
Literature Survey
mechanical force, the flattening of the peripheral part helps to increase the central curvature of
the lens. According to this theory, the equatorial diameter of the lens increases during
accommodation (it decreases in Helmholtz theory). Recent variants of this theory have been
proposed by Schachar (Schachar, 2001a, Chien et al., 2003, Schachar, 1992).
Figure 2.1: Helmholtz’ drawing demonstrating his theory of accommodation. According to this
theory the ciliary muscle contracts, all the zonules relax, the lens rounds up as a result of its own
elasticity, and the equatorial diameter of the lens decreases. The left half of the image shows
relaxed accommodation. The right half shows the increase in lens thickness and decrease in
equatorial diameter after ciliary muscle contraction.
More recently, two-dimensional ultrasound biomicroscopy and real time videography techniques
have been widely used to visualize the changes in the ciliary body, zonules and lens structures
(Bacskulin et al., 2000, Ludwig et al., 1999, Kano et al., 1999, Neider et al., 1990, Croft et al.,
1998b) and evidence from these studies does not support the Schachar’s postulation (Schachar,
2006, Schachar, 2001a) that the ciliary processes move away from the lens during
accommodation. Using three-dimensional ultrasound Stachs et al (Kirchhoff et al., 2001, Stachs
et al., 2002) found the ciliary processes shifting towards the lens by 0.36 mm in young eyes and
0.18 mm for older eyes during accommodation.
18
Literature Survey
Figure 2.2: (A) In the unaccommodated state, all the zonules are under tension. (B) According to
the Schachar theory, in the accommodated state, the equatorial zonules are under increased
tension and the anterior and posterior zonules are relaxed.
Strenk et al (Strenk et al., 2006, Strenk et al., 2005, Strenk et al., 2004) using magnetic
resonance imaging (MRI) found a contraction of the ciliary body diameter during
accommodation. These evidence from other highly sophisticated studies (Croft et al., 1998a,
Fincham, 1937, Glasser et al., 2001, Kano et al., 1999, Kirchhoff et al., 2001, Ludwig et al.,
1999, Marchini et al., 2004, Neider et al., 1990, Strenk et al., 1999) also support that the ciliary
muscle contracts and shifts anteriorly and inwards towards the lens equator during
accommodation validating the classic Helmholtz theory of accommodation.
2.3 Presbyopia
While the Helmholtz theory of accommodation is almost universally accepted today, there is less
agreement regarding the mechanism of presbyopia. A number of age-related changes occur in
both the crystalline lens and extra-lenticular structures, which all could contribute to the
development of presbyopia. On this basis, there are three broad theories of presbyopia: the
Lenticular or Duane-Fincham theory, Extra-lenticular or Hess-Gullstrand theory and the
Geometric theory.
19
Literature Survey
The lenticular theory, supported by various scientists (Fincham, 1937, Fisher, 1971, Pau and
Kranz, 1991), relies on the assumption that the lens material and the capsule lose their
deformability with age. The crystalline lens consequently becomes harder and less deformable
with age. A constant amount of ciliary muscle force will therefore produce a reduced amount of
(dis)accommodative response. In contrast to the lenticular theory, extra-lenticular explanations
purport that the ciliary muscle becomes less efficient with age. This concept was supported by
studies on rhesus monkeys where the ability of the ciliary muscle to respond to cholinergic drugs
and stimulation to the Westphal nucleus declined with age (Croft et al., 1998a, Lutjen-Drecoll et
al., 1988, Neider et al., 1990, Croft and Kaufman, 2006, Croft et al., 2006). Geometric theory
attempts to explain presbyopia in terms of age-related changes in the geometry of the
accommodative structures, including the lens size, circumlental space and zonular insertion
points on the lens capsule and ciliary body architecture (Strenk et al., 2005, Atchison, 2000,
Smith and Atchison, 1997).
In addition, the neural control of the accommodation is necessary to complete our understanding
of the process of accommodation. This is also fundamental to the functioning of accommodative
devices ion that the neural control remains unaltered with presbyopia as the loss of
accommodation is primarily due to the changes in the biomechanical plant.
2.4 Treatment of Presbyopia
Given our current understanding of accommodation and presbyopia as outlined in the previous
section, it is clear that in order to treat presbyopia and restore ‘true’ accommodation; in the sense
that continuously variable, as-required change in focus is restored to the eye, the effective
solution must address the lens.
Attempts of correcting presbyopia can grossly be categorized into two groups: surgical and nonsurgical procedures. Non surgical methods, which are the traditional techniques, use external
20
Literature Survey
optical devices such as spectacles and contact lenses. While these methods are the easiest to
implement, they do not restore accommodation. Surgical methods for restoring accommodation
are technically more sophisticated and consequently complex, but they have the potential to
restore the normal accommodative function. A number of presbyopia reversing surgical
techniques has been proposed; most of which have been reviewed and summarized in earlier
publications (Baikoff, 2004, Glasser, 2008). These techniques can be divided into two broad
classes (Figure 2.5): 1) surgical techniques that attempt to restore accommodation by altering the
accommodative structures of the phakic eye, and 2) accommodating intraocular lens implants
(AIOLs) aiming to alleviate presbyopia in pseudophakic eyes.
Though there is no direct role of the cornea in visual accommodation, some corneal surgical
procedures such as LASIK (Anschutz, 1994, Bauerberg, 1999, Epstein et al., 2001, Telandro,
2004) and implantation of intra-corneal inlay (Yilmaz et al., 2008) have been suggested.
Scleral approach of the presbyopia reversing surgical procedure is the most debated topic in
ophthalmology which is influenced by the Schachar theory of accommodation (Schachar, 1992,
Schachar, 2006). This surgical procedure involves implanting scleral expansion band
surrounding the ciliary body (Malecaze et al., 2001, Qazi et al., 2002, Schachar, 2001a) or
creating anterior ciliary sclerotomy (Hamilton et al., 2002, Thornton, 1997) which facilitates the
ciliary body to expand externally. Though the subjective amplitude of accommodation is found
to improve following the surgery (Baikoff, 2004, Lin and Mallo, 2003, Qazi et al., 2002), the
amount of true accommodative amplitude restored has been questioned (Elander, 1999,
Mathews, 1999). Even assuming that the Schachars’ theory of accommodation is true and the
surgical technique based on this principle can be made effective, we may anticipate that
presbyopia will recur over time because the slackening of the zonules continues as the lens keeps
growing. Disruption of the lens using femto-second lasers (Blum et al., 2006, Krueger et al.,
2001, Myers and Krueger, 1998, Glasser, 2008) is another novel approach being tested. Other
categories and methods of treatment of presbyopia is summarized in the following diagram
(Figure 2.3).
21
Literature Survey
Presbyopia Reversal in Phakic Eye
Restoration of accommodation in
Pseudophakic Eye
Extra-lenticular Procedure
Lenticular Procedure
Pharmacological
Corneal Surgery
Phaco-Photo disruption
Scleral Expansion Band
External Device
Intraocular Implant
Spectacles
Pseudo-accommodative
Accommodating
Contact Lens
Conventional IOL
Dynamic Optics
e.g. NuLens
Endo-capsular Balloon
Transverse Translating
Axially Translating
Polymer Injection
e.g. Phaco-Ersatz
Figure 2.3: Various methods of treating presbyopia and taxonomy of the implant devices.
Techniques or devices in bold fonts are the main concern of this thesis.
22
Literature Survey
2.5 Treatment Options for Pseudophakic Presbyopia
Conventionally, spectacles and contact lenses are used to treat presbyopia in the pseudophakic
eye. Other methods applied include the monovision method which is a technique of correcting
pseudophakic presbyopia using a traditional IOL where the power of the lens to be implanted in
one of the eyes is more than what is required for emmetropia (Baikoff, 2004), also called
monocular myopization. Though this technique is cosmetically acceptable and can be achieved
with a standard cataract surgery procedure, quality of the vision may be compromised due to loss
of binocularity and stereopsis (Erickson & Shcor, 1990). Therefore, this technique is not suitable
for people who require excellent binocular vision. However, some practitioners prefer the
monovision technique over implanting multifocal and currently available accommodating IOLs
(Olson, 2008).
Another, implant technology which has been developed is the multifocal intraocular lenses. With
this device at least two foci in the eye is achieved by utilizing various optical strategies such as
zonal refraction, diffraction and combined refraction/diffraction (Freeman, 1984, Simpson, 1989)
which enables simultaneous vision at different object distances. Though various designs of
multifocal IOLs are available in the market and have been found beneficial in terms of spectacle
independence for near work (Blum et al., 2006, Elander, 1999), their performance is
compromised due to several undesirable optical phenomena such as glare and reduced contrast
(Artal et al., 1995, Dick et al., 1999, Montes-Mico et al., 2004).
Accommodative IOLs are a family of innovative devices that have the ability to change power
directly either by changing the curvature (dynamic optics AIOL) or by altering their (or some of
their components’) relative axial position within the eye (translating-optics AIOL). We will be
discussing this type of AIOL designs in more details in subsequent chapters.
As described above, accommodative IOLs that were originally designed to function by shifting
their optic forward have also taken advantage of the deformation of the central optic to create
23
Literature Survey
more power. Several newer IOL designs were more explicitly designed to take advantage of
curvature changes of the optic. The FlexOptic IOL (Advanced Medical Optics Inc, Irvine, CA)
uses direct radial compression of the optic to flex the optic, as well as translating the optic
forward. The Nulens (NuLens Ltd) is sulcus-based, and is implanted in front of a collapsed
capsular bag that acts as a diaphragm (Ben-Nun, 2006). When the accommodation is relaxed, the
capsular diaphragm tightens and pushes a flexible optic against a rigid aperture, creating a
central bulge (accommodation mechanism of this design is in the contrary direction to the
conventional mechanism of accommodation). The author claimed that this approach can generate
50 to 70 D of accommodation in birds. The lens has been implanted in a small number of
partially sighted eyes with macular degeneration, and may be capable of higher accommodative
amplitudes than has been seen in the current generation of accommodative IOLs (Pepose, 2009).
FluidVision lens (PowerVision Inc, Belmont, CA), is another variation of the curvature-change
strategy, in which the ciliary muscle motion squeezes fluid from the haptics into a deformable
optic, changing its power.
Though the initial concept dates back to 1964 (Kessler, 1964), lens refilling strategies, also
called “phaco-ersatz” (Parel et al., 1986), is arguably the technically most advanced and
physiologically the most promising method for the restoration of accommodation as it seeks to
replicate the mechanics of the pre-presbyopic eye. It is expected that the lens shape and optical
performance of the polymer gel-replaced lens will not depart far from the natural one (Nishi et
al., 1997). Being at relatively early stages of development, there are several issues to be
addressed before it can be applied in living human eyes. Clouding of the polymer with time,
perhaps due to epithelial proliferation is the most challenging complication (Nishi et al., 2009).
Development of materials that do not lose transparency and elasticity (deformability) over a
period of time may be a challenge for this technology.
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2.6 Translating-optics AIOL
The Accommodating IOLs in this category increase the overall power of the eye by altering the
relative position of the optical element(s) within the eye, a process often called ‘translation of the
optics’. As reviewed earlier, the normal accommodative response brings about several changes in
the accommodative structures, which can be categorized as primary and secondary changes.
Primary changes include contraction of the ciliary muscle, forward translation of the ciliary body
and constriction of the pupil. Secondary changes include decreased ciliary aperture diameter,
relaxation of the zonules, relaxation and stretching of the capsular bag followed by changes in
shape of the crystalline lens, increased vitreous pressure, and forward movement of the lens.
Principally, any anatomical changes occurring on accommodation may be used as a driving force
for an AIOL.
AIOLs in this category may be divided into two types according to the number of optical
elements in the system: single element AIOLs (1E-AIOLs) and dual element AIOLs (2EAIOLs). Throughout this thesis, the word “element” is used to refer to the optical component of
an AIOL which effects some refractive power and may undergo movement or translation. It
should be noted that an element in this context is closer in definition to optical ‘groups’ in
traditional optical (e.g. camera lens) design parlance in that the element of an AIOL may consist
of multiple lenses all moving together. The number of elements favoured in AIOL design
currently is either one or two.
2.6.1 Single-element AIOL (1E-AIOL)
The invention of the single optic AIOL (1E-AIOL) comes from the innovative observations that
a pseudophakic eyes has ability to accommodate (Niessen et al, 1992) as some of the patients
were able to read exceptionally well through their distance corrected IOL (Cumming &
Kammann, 1996; Cumming, 2010). From the investigation of intraocular axial biometry he
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found that the IOL was moving forward (Doane, 2004). He developed a series of accommodating
IOLs designs that were evaluated in Germany with Professor Jochen Kammann. The seventh
generation of his design underwent Food and Drug Administration trials. In 2003, it became the
first AIOL approved for commercial usage in the US under the name Crystalens AT-45, which is
currently commercialized by Eyeonics, Inc. (Eyeonics is now owned by Bausch & Lomb Inc.)
CA (Cumming et al., 2006, Cumming et al., 2001, Buznego and Trattler, 2009). Other
commercially available 1E-AIOLs include new generations of the Crystalens (AT-50, AT-52),
the HumanOptics 1-CU (Erlangen, Germany) (Cumming et al., 2006, Cumming et al., 2001,
Kuchle et al., 2002), the Kellan Tetraflex (KH_3500, Lenstec, FL, USA), the OPAL (Bausch &
Lomb, Rochester, New York), the C-Well (Acuity Ltd. Or-Yehuda, Israel), the BioComFold
(Morcher) and the TekClear (Irvine, California, USA) (Doane and Jackson, 2007). All 1E-AIOLs
are designed to increase the effective power of the eye by relying on a flexible haptic design
(except of Tetraflex which brings about accommodation in virtue of flexure of optics) which
produces an anterior shift of the optic element under the influence of mechanical forces produced
by contraction of the ciliary muscle during accommodation (Dick, 2004). There are a number of
mechanical (haptic) designs proposed for 1E-AIOLs some of which are described in Appendix
A.3.
2.6.1.1 Performance of 1E-AIOL
The success of 1E-AIOL is limited. Some studies have reported promising results whereas others
found effectively no accommodation. Most of the studies rely on subjective measures of near
visual function and amplitude of accommodation. Most of the reports suggest that the objective
amplitude of accommodation obtainable from this type of AIOL is about 1 D (Buratto and Di
Meglio, 2006, Kuchle et al., 2004, Marchini et al., 2007, Wolffsohn et al., 2006a, Wolffsohn et
al., 2006b) which agrees with theoretical predictions (Ho et al., 2006, Langenbucher et al., 2004,
Nawa et al., 2003, Sauder et al., 2005). In contrast a subjective accommodation amplitude of
about 2.5 D has been reported with some designs (Kuchle et al., 2004, Wolffsohn et al., 2006a,
Wolffsohn et al., 2006b). This discrepancy lies the fact that the subjective methods includes the
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effect of depth of focus and therefore significantly overestimates the accommodation amplitude
(Win-Hall and Glasser, 2009), hence a major portion of the accommodation obtained with this
method essentially represents depth of focus.
The validity of reported values of accommodative amplitude (Menapace et al., 2007, Glasser,
2006) and the stability of the outcome over a period of time (Mastropasqua et al., 2007) have
also been questioned. It is evident from published studies that 1E-AIOLs are not equally
effective in all eyes (Heatley et al., 2005; Wolffsohn et al., 2006b). The limited movement of
the IOL in the eye (<1 mm) (Rana et al., 2003), the necessarily, relatively low power of the lens
(Ho et al., 2006, Nawa et al., 2003), and some physiological aspects such as post-operative
capsular opacification and fibrosis (Mastropasqua et al., 2007) are some of the reasons for the
limited success of these devices. Other factors include the variability of the capsular bag size
among individuals, the failure to implant the lens precisely in the bag and the inefficiency of the
haptic design to respond to the dynamics of the accommodative mechanism.
An analytical report (Menapace et al., 2007) on 1E-AIOLs, though arguably overly critical, has
summarized the potential causes of failure of 1E-AIOLs. Findl and Leytdolt (Findl and Leydolt,
2007) meta-analyzed clinical reports published in the literature investigating the accommodative
performance of three types of commercially available 1E-AIOLs [1CU (HumanOptics),
BioComFold (Morcher) and AT-45 (Eyeonics, Inc.)]. They found variable results. The tables
below summarize the performance of 1E-AIOLs reported in the literature in terms of
theoretically predicted values (Table 2.1), measured achieved accommodation (Table 2.2),
translation obtained in the eye (Table 2.3) and near vision outcome (Table 2.4).
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Table 2.1: Theoretically predicted accommodation
Author
Nawa et al., 2003
Missotten et al., 2004
Langenbucher et al.,
2004
Langenbucher et al.,
2005
Ho et al., 2006
Method
Accommodation ( D/mm), comment
Ray-tracing,
Wolfram
Vergence
calculation
Vergence
Calculation
1.3 & 2.3 for 20D & 30D AIOL respectively,
varies with axial length and IOL power
Purkinje image
1.10, parameters of AIOL as 1 CU
Ray tracing, Zemax
1.20 (power = 21D), depends power
1.46 for 24mm AL, 3mm ACD & K = 38D
1.4-1.5 for normal eyes, more for hyperopic
There are some serious issues reported relating to the 1E-AIOL. AT-45 optic was intended to
vault posteriorly to theoretically maximize its movement. Instead, Stachs et al found that the
lenses vaulted anteriorly against the iris (Stachs et al., 2006). Studies have even reported
paradoxical backward translation of the optics resulting in a dis-accommodation effect (Koeppl
et al., 2005). Gradual loss of accommodative performance over time has been reported (Dogru et
al., 2005, Heatley et al., 2005, Mastropasqua et al., 2007, Wolfsohn et al, 2006a) which marks
an important issue to be solved for sustained performance over the lifespan of the patient.
Frequently reported ‘z-syndrome’ (vaulting or tilting) of the single optic AIOL cannot be
disregarded, which results mostly due to delicate haptics design (Cazal et al., 2005, Yuen et al.,
2008, Daniela et al., 2006). Performance of AIOL is significantly associated with the posterior
capsular opacification. Incidence rate of PCO in 1-CU is reported to be consistently higher than
other IOLs (Mastropasqua et al., 2003, Harman et al., 2008) which went up to 86% in 12
months post-operatively (Dogru et al., 2005).
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Table 2.2: Accommodation with 1E-AIOL
Author
AIOL
2003a, Langenbucher 1 CU
Subjective
Method
1 CU
2006b
c
2007
ex
Autorefractor
0.99±0.48
Retinoscopy
RAF rule
0.72±0.38
Autorefractor
RAF rule
0.39±0.53
Autorefractor
Defocus
2.24±0.42
3.1±1.6
2003
2009
>1.0
Method
1.0±0.44
1.6±0.55
et al., 2003b
2006a
Objective
up)
NA
1 CU
1.9±0.77
Defocus
NA
(*IOL)
3.64 ±1.38
Push up
0.11±0.50
Autorefrator
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Table 2.3: Translation of 1E-AIOL
Author
AIOL
Translation (mm)
Method used
Findl et al., 2004
1 CU
0.37±0.29
Interferometry
Sauder et al., 2005
1 CU
0.82±0.30
IOL master
Hancox et al., 2006
1 CU
0.22±0.17
Interferometry
Legeais et al., 1999
BioComFold 0.71±0.55
Langenbucher et al., 2003b
1 CU
Paraxial A-Scan
0.78±0.12
IOL master
0.63±0.15
Immersion UBM
Schneider et al., 2006
1 CU
0.30±0.32
Slit Lamp
Kuchle et al., 2002
1 CU
0.63±0.16
IOL Master
Stachs et al., 2005
1 CU
0.35±0.11
3D Ultrasound
1 CU
0.49±1.26
IOL Master
0.34±1.12
US Biometry
Langenbucher et al., 2003a
Findl et al., 2003
BioComFold 0.12±0.11
Interferometry
Koeppl et al., 2005
AT-45
-0.15±0.08
Interferometry
Marchini et al., 2004
AT-45
0.33±0.25
UBM
Stachs et al., 2006
AT-45
0.13±0.08
3D Ultrasound
Kuchle et al., 2004
1 CU
0.42±0.18
IOL Master
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Table 2.4 (a): Near vision obtained
Findl et al., 2004
1 CU
18
20/25
No statistical difference with control
Pepose et al., 2007
AT-45
49
20/41
MN read card held at 16 inches
Mastropasqua et al., 2003
1 CU
42
J3.7±2.1
Range J2 – J7 at six months
Heatley et al., 2005
1 CU
60
J9.3±0.7
Required 2.3±1.9D add to read J1
Sauder et al., 2005
1 CU
38
Good
Landolt C held at 40 cm
Sanders and Sanders, 2007
Tetraflex
95
J8.5±1.2
63% achieved •20/40
Hancox et al., 2006
1 CU
20
J1
Range J1 – J7, near target at 35cm
Dogru et al., 2005
1 CU
22
20/40
Benefit of AIOL disappeared in 12 months
Buratto and Di Meglio, 2006
1 CU
108
J3
1CU slightly better than AT-45
Koeppl et al., 2005
AT-45
54
J4
Backward movement noticed
Cumming et al., 2001
AT-45
62
J4
97% had •20/30, near target at 14 inches
Marchini et al., 2004
AT-45
20
J7.3±2.1
Anterior displacement of AIOL confirmed
Buratto and Di Meglio, 2006
AT-45
69
J3 or better
in 55% after 1 year
Cumming et al., 2006
1 CU
39
J3 or better
in 72% after 1 year
Macsai et al., 2006
AT-45
28
J3 (median)
Excellent near vision
Table 2.4 (b): Accommodation Obtained
Schneider et al., 2006
1 CU
30
0.30±0.11
No statistical difference with control
Kuchle et al., 2002
1 CU
12
0.34±0.17
Value stable till 1 year
Langenbucher et al., 2003a
1 CU
23
0.32±0.11
Objective&subjective values do not agree
Vargas et al., 2005
1 CU
19
0.5
Range: 0.3 – 0.94
Kuchle et al., 2003
1 CU
20
0.37±0.12
Significantly better than control
Kuchle et al., 2004
1 CU
20
0.36±0.10
Control had 0.16±0.06
Langenbucher et al., 2003b
1 CU
15
0.32±0.11
Range 0.2-0.6, near target at 35cm
Alio et al., 2004
AT-45
12
0.8±0.10
Better performance than multifocal
Wolffsohn et al., 2006a
1 CU
20
0.60±0.09
Best corrected, 4 months post-op
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2.6.2 Dual-element AIOL (2E-AIOL)
2.6.2.1 Optics of 2E-AIOL
The first published report of this type of AIOL dates back to 1990 (Hara et al., 1990). Typically
a 2E-AIOL consists of a front lens with a high positive power and a rear lens with a
complementary, usually negative, power. The elements are connected with specially designed
haptics which are intended to hold the optics in place and also effect translation of the elements
during accommodation. Figure 2.4 shows the basic schema of a dual optic AIOL implanted in
the eye.
The Synchrony Dual Optic AIOL (Visiogen, Irvine, CA) (Glasser, 2008, McLeod et al., 2003) is
the first 2E-AIOL lens to be tested clinically. According to the developers (McLeod et al., 2007),
the lens is made of a silicone material with 5.5 mm optics diameter. It is implanted through a 3.6
– 3.8 mm corneal incision. The power of the front element in is about +32 D. The front element
is connected to a 6.0 mm diameter rear element having negative power. The haptics is designed
to store and release energy in response to ciliary body contraction, following the Helmholtz
theory of accommodation. Accommodation is accomplished by axial translation of the front or
back element of the system. In the resting position (accommodation relaxed), the elements are at
minimum separation, and the eye is focused at distance. The total thickness of the assembly
when compressed is 2.2 mm. During accommodation, zonular relaxation reduces the tension on
the capsule, which produces a separation of the lens elements through the action of the haptics.
The increased distance between the elements results in an increased effective power of the
assembly. An initial clinical study (Ossma et al., 2007) reported a mean subjective amplitude of
accommodation (obtained from defocus curve analysis) of 3.22±0.88 D after six months of
implantation. The defocus curve in this report is symmetrical along zero (Figure 2.7) suggesting
that the AIOL has greater depth of focus which may infer less accommodation.
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Sarfarazi patented a series of similar designs [US patents: 4946469 (1990); 5,275,623 (1994);
6,488,708 (2002); 6423094 (2002); 6884265 (2005) to Sarfarazi] with different haptics. All the
systems disclosed in these patents are to be implanted in the capsular bag. As shown in Figure 6,
the two elements of the Sarfarazi 2E-AIOL are connected with three spring-loaded haptics which
have natural shape memory characteristics and are elliptically curved. In the unaccommodated
state, the capsular bag is pulled radially outward in the stretched position, which squeezes the
spring loaded haptics and bringing the two elements close together. When accommodated, the
zonules are relaxed, which releases the capsular tension and the pressure on the spring-loaded
haptics, as a result of which the lenses move apart.
Figure 2.4: McLeod et al, JCRS 2007 - A schema of the accommodative function of the dual
optic IOL. At ciliary body rest, the zonules are put on stretch producing axial shorting of the
capsular bag, thus pulling the optics together and loading the haptic springs. With
accommodative effort, zonular tension is released, compressive capsular tension on the optics
and spring haptics is released and thus the anterior optic moves forward. Two variants of 2EAIOL are already in the stage of clinical trial.
This design of 2E-AIOL, also known as the Sarfarazi twin-optic Elliptical AIOL (Bausch &
Lomb Surgical, Rochester, NY) had been tested clinically (Sarfarazi, 2006). (The project appears
to be suspended or abandoned as no report has been made since that publication.) The design
aspect of this lens and its performance are included here as this forms one of the recourses in this
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category of AIOL. Optical diameter is 5 mm and the overall diameter is 9.5 mm. When the lens
is in the unaccommodated sate, the antero-posterior thickness is about 4.5 mm. Monkey eyes
implanted with this AIOL returned up to 7 D accommodation (McDonald, 2003, Nataloni, 2003).
The essential difference of this IOL compared to the Synchrony 2E-AIOL is in the mechanical
design of the haptics. A number of other designs of 2E-AIOL has been proposed and patented.
These are described in Appendix A.4.
Figure 2.5: A design of 2E-AIOL (US patent # 0,015,236 to Sarfarazi)
The mechanical design of the haptics plays a major role in the performance of an AIOL. The
majority of the proposed haptics in a two-element AIOL utilize the force transferred on to the
capsular bag. For instance, hinged or spring-loaded haptics are operated by the pressure exerted
by the ‘walls’ of the capsular bag. A US patent application for a design was lodged which is
claimed to works under the direct influence of the forces exerted by the ciliary muscle (US patent
application 10/738,271 to Magnante), however the mechanism by which the haptics are
connected to the ciliary muscle is unclear. Another proposed design works under the influence of
vitreous pressure changes during accommodation (US patent no 7,238,201 to Portney#), and yet
another design relies on the movement of the ciliary body to initiate the movement of magnets
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incorporated within the lenses (US patent application no. 20070118216 to Pynson). Several
designs of this category are discussed in Appendix A4.
In summary, the most common forms of 2E-AIOLs rely on the tension created on the capsular
bag during accommodation. Therefore the vast majority of designs require the capsular bag to be
intact. There is no literature evidence that any of the other designs have been tested clinically.
2.6.2.2 Performance of 2E-AIOLs
There are limited reports in the literature investigating the performance of 2E-AIOL. A few
theoretical studies predict the accommodative performance (Ho et al., 2006, Langenbucher et al.,
2004, McLeod et al., 2003); one study reported on clinical outcomes (McLeod et al., 2007) and
another study reported on the results from non-human primate studies (McDonald, 2003).
Theoretically calculated and clinically determined accommodative performances of various types
and designs of AIOLs are given in Table 2.5 and Figures 2.6 and 2.7.
Table 2.5: Accommodative performance of 2E-AIOL
Authors
McLeod et al., 2003
Method
Ray Tracing
Accommodation
(mm-1 translation)
Comment
2.20D
+32D front power
Langenbucher et al., 2004 Vergence
2.50D
+32D front power
Ho et al., 2006
Ray Tracing
3.0-4.0D
Depends on combination
Ossma et al., 2007
Defocus curve
3.2D
Subjective
McDonald, 2003
Auto -refraction
4.0 -7.0D
average value
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Figure 2.6: Ho et al (JCRS, 2006). Ray-tracing prediction of change in near point vergence
(accommodative state referenced at the anterior corneal plane) of a 2E-IOL with mobile front
element (a) and mobile back element (b). The solid line represents the variation corresponding
to a 1E-IOL. The axial position of 0.00 mm is when the IOL elements are at their greatest
separation (4.00 mm).
Figure 2.7: From Ossma et al (JCRS; 2007). Defocus curves generated in 10 eyes with a
monofocal IOL compared to 24 eyes with the dual-optic IOL. The x-axis indicates the power in
diopters of the defocusing lens placed in front of the tested eye through which the visual acuity
(y-axis) was measured.
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2.7 Summary
The mechanism of accommodation and presbyopia has been long debated; nevertheless, recent
evidences strongly support the Helmholtz theory of accommodation and lenticular theories of
presbyopia development. Since the ciliary muscle remains anatomically and functionally intact
even in aging population (Croft et al., 1998a, Kirchhoff et al., 2001, Stachs et al., 2002, Strenk et
al., 1999), a presbyopia-reversing procedure targeting the crystalline lens should have a high
likelihood of success. On the other hand, procedures that target extra-lenticular structures are
questionable.
Despite the fact that ciliary muscle is still functional after cataract surgery, most patient
undergoing cataract surgery are left with uncorrected refractive presbyopia, which currently has
no solution that approaches the elegance or effectiveness of the pre-presbyopic crystalline lens
and its accommodative system. Several optical devices have been proposed and some have been
developed to alleviate the debilitation of pseudophakic presbyopia. The traditional method of
prescribing spectacles and contact lenses for near vision, though non-surgical, work only for
discrete, fixed viewing distances. Monocular myopization (monovision) with IOLs is not an ideal
method as it compromises binocular vision. Pseudo-accommodative (multifocal) IOLs can
improve near and intermediate vision, but they do not replicate the continuous change in focus
afforded by normal accommodation. They also produce subjective photic phenomena including
loss of contrast (Pepose et al., 2007). Accommodating IOLs provide a promising alternative. 1EAIOLs have already been used clinically for about a decade, albeit with mixed results. 2EAIOLs, which are currently being tested clinically, seem to afford better accommodative
amplitude than 1E-AIOLs. Other designs of AIOLs, some of which appear promising are still
under development. Importantly, the optical performance of AIOLs is a cornerstone for their
widespread acceptance among practitioners and patients alike. Considering an average working
distance for near is about 40 cm (certainly this varies among individuals depending on several
factors such as profession, habitual working distance and length of the arms; and with the
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increasing use of small screen mobile phone and personal digital assistance, the modern habitual
near point may well be much closer), a minimum of 2.5 D accommodation would be desirable.
From the foregoing survey, 1E-AIOL may not offer this amount of accommodation. On the other
hand, theoretically a 2E-AIOL could reach the minimum amplitude target assuming that the
mechanical design of haptics is sufficient to translate the optical elements effectively.
AIOLs provide an opportunity to restore the native ‘youthful’ accommodation of the eye without
the limitations of multifocality and monovision approach. These devices currently receive the
most attention in terms of development – perhaps spurred by their relative ease to bring to
market because they are based on traditional IOL implantation and require relatively less training
on the part of the practitioner compared to other AIOL options. As such IOL companies are
being seen to be investing significant effort and resources to develop these AIOLs. Given that
many designs are anticipated in the near future, it is important to understand the optics behind
these devices, so that optimal designs can be achieved.
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Dick, H. B., Krummenauer, F., Schwenn, O., Krist, R. and Pfeiffer, N. (1999). Objective and
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Doane, J. F. (2004). Accommodating intraocular lenses. Cur Opin Ophthalmol 15, 16-21.
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function and design. Cur Opin Ophthalmol 18, 318-324.
Dogru, M., Honda, R., Omoto, M., et al. (2005). Early visual results with the 1CU accommodating
intraocular lens. J cataract refract surg. 31, 895-902.
Donders, F. C. (1864). On the Anomalies of Accommodation and refraction of the eye. The New
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49
Accommodative Performance
Chapter 3
Accommodative Performance
A portion of this chapter has been published in the following peer reviewed journal article
Ale J, Manns F & Ho A (2010). Evaluation of the performance of accommodating IOLs using a
paraxial optics analysis. Ophthalmic Physiol Opt 2010; 30: 132 - 142.
51
Accommodative Performance
TABLE OF CONTENT
3.1 INTRODUCTION..................................................................................................................53
3.2 MATERIALS AND METHODS ..........................................................................................54
3.2.1 Analytical Approach .....................................................................................................56
3.2.2 Computational Approach..............................................................................................59
3.3 RESULTS ...............................................................................................................................59
3.3.1 Performance of 2E-AIOL..............................................................................................59
3.3.2 Performance of 1E-AIOL..............................................................................................66
3.4 DISCUSSION .........................................................................................................................67
3.5 REFERENCES.......................................................................................................................71
52
Accommodative Performance
3.1 Introduction
The accommodative performance of the translating-optics accommodating intraocular lens
(AIOL) is based on increasing its effective power by altering the axial position of some or all of
its optical elements within the eye. From the optical design standpoint, the empirical and clinical
studies reported in the literature provide incomplete information regarding the optical
performance of AIOL (Langenbucher et al., 2003a, Lesiewska-Junk and Kaluzny, 2000).
Limited literature exists addressing the objective measurement of the performance of AIOL
based on direct quantification of accommodation amplitude. The majority of studies rely on
subjective vision testing while some have employed indirect measurements of accommodation
such as the amount of translation effected in an AIOL using special observational microscopy or
ultrasonographic techniques (Cumming et al., 2006, Cumming et al., 2001, Macsai et al., 2006,
Pepose et al., 2007).
A few studies have employed theoretical analytical approach to model or predict the
accommodative performance of AIOLs. Langenbucher et al (2003b) in one of their studies used
the matrix method to calculate changes in anterior and posterior focal points of the eye with
change in axial position of commercially available 1E-AIOL (Langenbucher et al., 2003c). This
calculation was used in determining the amount of accommodative amplitude achievable. The
study however, did not analyse the effect of corneal power or axial length on accommodative
performance. Another study (Langenbucher et al., 2004), again using the matrix method,
evaluated and compared the accommodative performance of 1E and 2E-AIOLs. In this study
they considered one specific power combination (comprising a +32 D front element) of 2EAIOL with a fixed amount of translation (1 mm). While they effectively compared the
performance of two types of AIOLs, the study was not intended to identify the optimal design in
terms of all power combinations and directions of translation of the AIOL. Similar to the earlier
study, the effect of corneal power or axial length was not extensively examined. Another study
(Missotten et al., 2004) used vergence equations to predict the accommodative performance of
53
Accommodative Performance
1E-AIOL. They explained the effect of corneal power, anterior chamber depth and axial length
on accommodative performance but the study did not evaluate 2E-AIOLs or attempted to
identify optimal designs of AIOLs.
To understand the performance of AIOL especially with a view to optimising their designs,
approaches that can consider all possible design configurations within the entire design space
(that is, the collection of every possible combination of values for all relevant design variables)
in a comprehensive manner should be preferred over those employing modelling of certain
select, specific design configurations. Studies which only consider existing or available designs
would be limited in their scope of evaluation and may not identify the ultimate optimum design
configuration. Similarly, studies which employ finite ray-tracing techniques (Ho et al., 2006,
Nawa et al., 2003) would in practice be limited to evaluation of selected design examples.
Extrapolation of the findings from such studies risks compromising validity. Thus, to explore the
entire design space of AIOLs and to identify an optimum design, a mathematically closed-form
analytical approach is required. In this chapter, we extend on the knowledge with an objective to
understand the control variables relevant to the design optimisation of AIOL in terms of their
accommodative performance.
3.2 Materials and Methods
So far, the conceptualized operation of a 2E-AIOL is based on both the shifting of axial position
and separation of the two lens elements. In practice, both elements of the AIOL are likely to
translate simultaneously during accommodation due to the various forces exerted by the ciliary
body, zonules and capsule, as well as the hydraulic and mechanical resistance from the vitreous
and iris (Crawford et al., 1990, Heatley et al., 2004). However, for simplicity, we employed only
two simplified design operational configuration similar to that proposed elsewhere (Ho et al.,
2006):
54
Accommodative Performance
Configuration 1: an AIOL with a mobile front element, where only the front element moved in
the forward direction; the back element remained stationary; and
Configuration 2: an AIOL with a mobile back element, where only the rear element moved in
the backward direction; the front element remained fixed.
Back Lens
Front Lens
Cornea
Retina
t
t
t
t
D
F
F
Back
Reference
Figure 3.1: Basic schema of the model eye implanted with AIOL. F1 and F2 are the powers of the
front and back lens elements respectively (specified by Ff and Fb, respectively, in the text); Tf is
the distance from pupil to front lens element; ts the distance between AIOL lens elements; tb the
distance from back lens element to vitreous (labelled as the back reference plane); tr is the
vitreous depth from back reference plane to retina; and D is the total maximum space available
for the AIOL including thickness and translation distance (4 mm).
Two approaches were considered in evaluating the performance of AIOLs under the two design
configurations mentioned (1E-AIOL and 2E-AIOL): an analytical approach and a computational
approach. The aspheric ocular surfaces of the model eye were modified and simplified to
spherical paraxial refracting surfaces. Other settings are identical to those described for the
model eye in Appendix C and portrayed in Figure 3.1.
55
Accommodative Performance
3.2.1 Analytical Approach
In the analytical approach, equations were developed by employing a paraxial, first-order theory
model of the eye and AIOL. A vergence equation of paraxial optics could have been employed in
this optical model. However, due to potentially tedious expressions requiring involved algebraic
manipulation that may be encountered, we employed the matrix method of the paraxial optics
(Gerrard and Burch, 1994, Halbach, 1964, Langenbucher et al., 2003b, Long, 1976) in the
derivation of equations depicting the models. Derivation of the matrix method in the paraxial
optics has been discussed in Appendix B3. In the matrix method, a refraction matrix R is given
by:
ª1 − F º
R=«
»
¬0 1 ¼ .............................................................................................................. (3.1)
where F is the dioptric power of the surface. Similarly, a translation matrix T is given by:
ª 1 0º
T =«
»
¬t / n 1¼ ................................................................................................................ (3.2)
where t/n represents the reduced distance between two consecutives surfaces.
Under the matrix method, a backward ray-trace from the retina to the cornea of an optical system
comprising the eye and two elements of AIOL can be described by the matrix system
M = K ⋅ Ta ⋅ Tf ⋅ R f ⋅ Ts ⋅ R b ⋅ Tb ⋅ Tr .............................................................................. (3.3)
where
56
Accommodative Performance
denotes the ray-transfer matrix representing the vergence of light at the posterior
corneal surface,
denotes ray-transfer matrices,
denotes refraction matrices,
and suffixes
b refers to the back reference plane,
a refers to the anterior chamber,
f refers to the front reference plane,
s refers to the separation between A-IOL elements, and
r refers to the vitreous chamber.
or more explicitly in terms of matrix elements Mi.j,
ª S11
«S
¬ 21
S12 º ª 1
=
S 22 »¼ «¬1 / K
0º ª 1
.
1 »¼ «¬t a
0º ª 1
.«
1 »¼ ¬t f
0 º ª1 − F f º ª 1
.
.
1 »¼ «¬0
1 »¼ «¬t s
0º ª1 − Fb º ª 1
.
.
1 »¼ «¬0
1 »¼ «¬t b
0º ª 1
.
1»¼ «¬t r
0º
1»¼ ............ (3.4)
where, suffixes correspond to foregoing usage, and
t
denote optical thicknesses (t/n) between surfaces measured in meters where n is the
refractive index of aqueous and vitreous,
57
Accommodative Performance
F denotes refractive powers of components or surfaces measured in dioptres, and
K is the relative accommodative state in dioptres.
Since refraction at the cornea only serves to add a fixed vergence to the relative accommodative
state, K, and we are only concerned with changes in vergence (i.e. amplitude of accommodation)
relative to the unaccommodated state, we can omit the corneal refraction matrix from the system
matrix to simplify the derivation. The actual amount of the accommodation will be calculated
from the difference between the value of K in the system matrix (Equation 3.4) without
translating and after translating the A-IOL element.
As employed in an earlier study (Ho et al., 2006), the introduction of the fixed-position reference
planes to this model simplifies derivation of the paraxial formulae as it provides a fixed point of
reference from which the position of either the front or back element of the 2E-AIOL can be
defined as a function of the lens element translation.
Since the system is operating principally at finite conjugate ratio, the ray vergence, K at the plane
of the posterior corneal surface can be obtained by setting element S12 of matrix S to zero and
solving for K.
Accommodative performance (Pk) is defined as the rate of change in vergence of the rays with
translation of the mobile lens element. A large value represents greater amount of
accommodation induced by a given amount of translation. Mathematically, accommodative
performance is the first derivative of the ray vergence (differentiating the resulting equation
defining the vergence of ray exiting the corneal plane) at the corneal plane with respect to the
amount of translation undertaken by the mobile element.
58
Accommodative Performance
3.2.2 Computational Approach
In addition to derivation and analysis of paraxial equations describing the performance of the two
design configurations of AIOLs, by way of illustration of relationships between control
parameters and performance, the accommodative performance was computed for specific
configurations using the derived paraxial equations. The computation included determining the
change in ray vergence at the corneal plane K, as defined by equations 3.7 and 3.12 (below),
when the front or back element of the 2E-AIOL was translated from its initial position. Different
variants of the design configurations were tested by varying the power combinations of the
elements. In this series of computation, the refractive powers of both lens elements were limited
to less than approximately ± 40 D.
3.3 Results
3.3.1 Performance of 2E-AIOL
For Configuration 1, the back element is fixed in position at the back reference plane. Hence, tb
= 0, and matrix Tb may be ignored.
In addition, given the mobile front element in this configuration, ts the separation between the
two elements and ts, the distance of the front element from the front reference plane, may be
defined in terms of the amount of axial shift of the front element Zf, as well as the maximum
separation of the two elements D. Therefore;
tf
D Z f ................................................................................................................... (3.5)
and
59
Accommodative Performance
t s = Z f ........................................................................................................................... (3.6)
Since the system is operating principally at finite conjugate ratios (as vergence produced by
corneal refraction has been ignored), the accommodative state, K, may be obtained by setting m12
= 0 (Gerrard et al., 1994). On evaluating (3.4) and substituting (3.5) and (3.6), the solution for K
becomes
K = ( n ( nF f (t r + Z f ) + Fb ( n − F f Z f )t r − n 2 )).( n 2 (t a + D + t r ) − nFb (t a + D )t r …
+ F f Fb ( t a + D − Z f ) Z f t r − nF f ( Z f + t r )( t a + D − Z f )) − 1 ............................................ (3.7)
where suffixes correspond to foregoing usage, and
n
is the refractive index of the aqueous and vitreous (considered equal for simplicity)
D
is the maximum total thickness expressed in metres of the 2E-AIOL (set to 4.0 mm),
Zf
is the translation expressed in metres of the mobile element (varying from 0.0 mm being
the initial position, to 1.5 mm being at extreme position),
As defined earlier accommodation performance Pk is the rate of change of ray vergence K with
respect to axial shift Zf of the mobile element. Thus Pk has units of D/mm and mathematically, is
the first derivative of K with respect to Zf. The exact formula for Pk is a rather protracted
equation. However, the equation may be simplified by approximation as follows. In the optical
system of the combined eye and AIOL described, terms relating to refractive power are typically
of the order of many tens of dioptres (i.e. 102 m-1), while terms relating to distances are typically
of the order of a few millimetres (10-3 m). Thus, in the expansion of the equation, groupings
60
Accommodative Performance
involving distance terms tc, D and Zf, may be considered negligible. Subsequently, an
approximate formula for Pk may be derived as
Pk ≈ D → 0 ,lim
Z→0, t
c →0
dK
dZ f ...................................................................................................... (3.8)
This approximation yields the following simplified formula
Pk ≈
º
1 ª Ff
− 2 F f Fb − F f2 » ....................................................................................... (3.9)
«2n
n ¬ tr
¼
A similar approach can be employed for Configuration 2 AIOL where the back element is
mobile and the front element is fixed in position. Here, tf = 0, and matrix Tf may be ignored.
With this configuration, ts, the separation between the two elements and tb, the distance of the
back element from the back reference plane, may be defined in terms of the amount of
translation of the back element Zb, as well as the maximum separation of the two elements D
such that
tb = D − Z b ................................................................................................................... (3.10)
t s = Z b .......................................................................................................................... (3.11)
By substituting (310) and (3.11) into (3.4), the ray vergence at the plane of the corneal back
surface now equates to
61
Accommodative Performance
K = ( n ( nF f ( D + t r ) − F f Fb ( D − Z b + t r ) Z b + nFb ( D − Z b + t r ) − n 2 )).( F f Fb (t r + D − Z b ) Z b t a
+ n 2 (t a + D + t r ) − nF f (t r − D )t a − nFb ( Z 2 − t a D − t r Z + t a Z − DZ − t a t r )) −1
.............. (3.12)
An approximation for Pk derived from the rate of change of K with respect to Zb can be found by
employing the same approach for approximation used for the Configuration 1 equations. A
simplified equation for accommodative performance for Configuration 2 is found as
Pk ≈ D →0 ,lim
Z →0 , t →0
b
dK
dZ b ....................................................................................................... (3.13)
which yields the following simplified formula
Pk ≈
Fb º
1ª 2
« Fb − 2n » .................................................................................................. (3.14)
n¬
tr ¼
From equations (3.9) and (3.14), more clearly in (3.7) and (3.12), it can be observed that some
design variants increase in power (i.e. accommodate) when the mobile element translates
anteriorly (towards the cornea) while other design variants accommodate when the mobile
element translates posteriorly. This provides some flexibility in the mechanical design of the
AIOL supporting mechanical features for effecting accommodation relative to the dynamics of
the ciliary body, zonule and lens capsule. Accommodation effects with axial translation of
selected design variants of AIOL are shown in Figures 3.2 to 3.4 to illustrate these relationships.
Equation (3.9), corresponding to Configuration 1 AIOLs, is a second-order function that
concaves upward and has zero values at Ff = 2(n/tr-Fb) and Ff = 0 (Figure 3.2). Any value of Ff
bracketed by these two points produces ‘negative’ accommodation (i.e. with ciliary body
constriction and zonular relaxation – actions that are normally, physiologically associated with
62
Accommodative Performance
accommodation – the power of the eye becomes less positive or more negative. Throughout this
thesis, this will be called “de-accommodation”. Note that de-accommodation differs from disaccommodation. The latter refers to the normal physiological process of reducing power of the
lens by relaxation of the ciliary muscle and applying tension to the zonules. The former is a loss
in lens power while the accommodative system undergoes the normal physiological
accommodative process of attempting to increase lens power) whereas values outside these
points produce positive accommodation (i.e. normal accommodative behaviour with
physiological accommodative process). However, values of Ff beyond the point defined by Ff =
2(n/tr-Fb) represent extremely high negative powers which, from the device thickness and
mechanics standpoints, would not be advantageous for an AIOL.
dK/dZ
2(n/tr-Fb)
-F
0
f
0
+F
f
Figure 3.2: Accommodative performance (dK/dZ) versus front element power (Ff) for a 2E-AIOL
of Configuration 1. Practically, good accommodative performance can be achieved more
efficiently with positive power front elements. In contrast, an extremely high negative power
front element is required to produce matching amount of accommodation. Negative power of the
element between the two zero-points results in de-accommodation on forward translation.
63
Accommodative Performance
As the fixed equivalent power of the AIOL (Fiol) for distance focus (i.e. with accommodation
relaxed) is approximately the sum of Ff and Fb, the first zero of equation (3.9) can be rewritten
as Ff § 2Fiol-2n/tr. Since refraction at the cornea contributes to the focusing of light to the retina,
Fiol has a less positive value than n/tr. Thus 2Fiol – 2n/tr is negative and any positive Ff gives
greater accommodative performance Pk (Figure 3.2) than a negative Ff of the same absolute
value. Therefore, selecting a design based on the practical range of power as discussed in the
previous paragraph, greater accommodative performance can be realised when a positivepowered front element is used rather than a negative-powered one.
dK/dZ
0
-F
b
2n/tr
+F
b
Figure 3.3: Accommodative performance (dK/dZ) versus back element power (Fb) for a 2E-AIOL
of Configuration 2. Practically, a good accommodative performance can be achieved more
efficiently with negative power back elements. In contrast, an extremely high positive power back
element is required to produce matching amount of accommodation. Positive power of the
element within the zero-points results in de-accommodation on backward translation.
Similarly, the curve described by Equation (3.14) for Configuration 2 AIOL is concave upward
and is disposed approximately symmetrically about Fb = n/tr with zero values at Fb = 2n/tr and Fb
64
Accommodative Performance
= 0. Following the same considerations as for equation (3.9), that Ff + Fb = Fiol, Equation (3.14)
indicates that greater accommodative performance can be realised with a negative value of Fb.
Accommodation (D)
5
4
+40
3
+30
1E-AIOL
2
+20
1
+10
0
+0
-10
-1
0.0
Figure 3.4(a)
0.5
1.0
1.5
Translation (mm)
Accommodation (D)
4
3
+40
2
+30
1
+20
~+17
0
+10
-1
-2
0.0
Figure 3.4 (b)
1E-AIOL
0.5
1.0
1.5
Translation (mm)
Figure 3.4: Paraxial optics prediction of change in accommodation (D) with axial position of the
mobile element for seven design variants of a two-element 2E-AIOL with (a) Configuration 1 and
(b) Configuration 2. Power of the front element is indicated by the labels. Solid line represents
the variant corresponding to a 1E-AIOL and long dashed line represents the variant portraying
an immobile (single-vision) IOL. Negative values in y-axis represent de-accommodation.
Figure 3.4a and 3.4b are the computational results of accommodative performance for two
configurations respectively. These figures concord with the figures stated in the analytical
65
Accommodative Performance
results. In Configuration 1, higher accommodation is obtained for the greater convex power of
the front element and similarly the larger negative power of the rear element have produced
greater accommodation in Configuration 2.
3.3.2 Performance of 1E-AIOL
When the power of the immobile element for each configuration (i.e. Fb for Configuration 1 and
Ff for Configuration 2) is set to zero in equations (3.9) and (3.14) respectively, a 1E-AIOL is
modeled. In these design variants, accommodation performance is equal to
Pk ≈ −
1
Fiol2 − 2 n Fiol t r ........................................................................................... (3.15)
where Fiol is the power of the single mobile lens of the 1E-AIOL.
Equation (3.15) describes a curve that concave downward and has zero values at Fiol = 0 and Fiol
= 2n/tr . From this equation and corresponding Figure 3.5, it can be seen that any power of the
1E-AIOL beyond these points on either side produces a negative accommodation effect with
translation. The global maximum of the curve is at the coordinate (n/tr, n/tr2) suggesting that the
maximum accommodation is obtained when Fiol = n/tr (Figure 3). However, this is an impractical
case as this power would require the cornea to be effectively plano for the eye to be emmetropic.
Since Fiol must be fixed to a value that renders an eye emmetropic, the accommodation
performance of a 1E-AIOL is also fixed for a given eye. Consequently, the relationship portrayed
by equation (3.15) shows that the lower the corneal power (e.g. hyperopes), the greater the
amount of accommodation that can be derived with 1E-AIOL.
To be exhaustive, when the refractive power of the mobile element is set to zero (dashed
horizontal line in Figure 3.4a we obtain the accommodation performance for the trivial design
66
Accommodative Performance
variant of a conventional (single-vision) IOL without translation, which does not produce any
accommodation effect.
2
(n/tr, n/tr )
dK/dZ
-F
+F
iol
0
iol
2n/tr
Figure 3.5: A plot of accommodative performance (dK/dZ) versus total IOL power (Fiol) for 1EAIOL. Good performance can be achieved more efficiently for higher positive power of the
AIOL reaching a maximum when it is equal to the equivalent dioptric value of the vitreous
chamber depth.
3.4 Discussion
Results of the analysis under paraxial approximations are in good agreement with those from
finite ray-tracing. For an emmetropic eye with typical dimensions, Nawa et al (2003) predicted
that 1 mm translation of a 20 D 1E-AIOL brings approximately 1.30 D of accommodation. Ho
and colleagues (Ho et al., 2006) employing a ray-tracing model also predicted a similar value
(1.25 D/mm). The small differences in predicted accommodation can be readily accounted for by
the differences in model eye parameters as well as the starting axial position of the 1E-AIOL. In
addition, these predicted values are in close agreement with the values reported from a clinical
study where accommodation of 1.06 D with 0.88 mm of translation (equivalent to 1.2 D/mm)
67
Accommodative Performance
was reported (Langenbucher et al., 2003c). The analysis from this study predicted a value of
1.25 D/mm which again is in good agreement.
The model of 1E-AIOL predicts that individuals with shorter axial length, i.e. typically
hyperopes, would enjoy greater accommodation amplitude for the same translation in 1E-AIOL.
More precisely, the hyperopic eye, which is typically associated with a shorter axial length,
demands a higher power of the IOL for distance refractive correction; and it is this greater power
of the AIOL that brings greater accommodation. However, the power of the IOL and the
accommodative performance does not depend merely on the vitreous chamber depth but also on
the corneal power and the anterior chamber depth (equations 3.2 and 3.3). For example, an eye
with high corneal power and low natural lens power would require a correspondingly lower
power for a 1E-AIOL resulting in lesser amount of accommodation. Conversely a myopic eye
with flat, low-powered cornea but higher power natural lens would require relatively higher 1EAIOL power resulting in greater amount of accommodation.
From the analysis, a larger range of accommodation amplitude (up to 4 D) has been predicted for
2E-AIOL depending on the design. Performance of 2E-AIOL has been previously investigated
(Hara et al., 1992, Hara et al., 1990, McDonald, 2003, MSC Software Corporation, 2001,
Nataloni, 2003). One 2E-AIOL tested in vivo using a primate model (McDonald, 2003) achieved
as high as 6 D of accommodation. Computer-assisted modelling (MSC Software Corporation,
2001, Nataloni, 2003) predicted that such a 2E-AIOL could deliver 4 D of accommodation
assuming translation of 1.9 mm. Predictions from the present study agrees with these results. For
example, according to the analytic results of this study, 4 D of accommodation can be achieved
by a 2E-AIOL of Configuration 1 with approximately a +35 D front element and a –15.50 D
back element when translated by 1.7 mm. Calculations based on another design variant for a 2EAIOL suggest that accommodation amplitudes of 2.2 D to 2.4 D may be achieved with a +30 D
to +35 D front lens element (McLeod et al., 2003).
68
Accommodative Performance
It has been noted that both elements in a 2E-AIOL are likely to translate simultaneously with the
effort of accommodation in the eye (Ho et al., 2006, Lin, 2006). Though simultaneous translation
of both elements was not modelled explicitly, the effect can be extrapolated by considering the
analytical results of models for Configurations 1 and 2. It is clear from our 2E-AIOL model that
the optimal performance is achieved when a plus lens moves forward (towards the cornea) or
when a minus lens moves backward (towards the retina). Therefore, relative to normal
physiological behaviour, maximum accommodation would be obtained when a 2E-AIOL
contains a positive front element that moves in the forward direction and a negative back element
that moves in the backward direction.
Mathematically, a desired equivalent power of the 2E-AIOL can be obtained from any of an
infinite number of combinations of front and rear element powers. From consideration of
Equations 3.12 and 3.14 it is observed that the accommodative performance of the 2E-AIOL is
similarly dictated by the combination of power of the elements. Thus, unlike for 1E-AIOL,
accommodative performance of a 2E-AIOL does not depend directly on the refractive state of the
eye. For example, a myopic eye may be corrected by using a less positive front element power or
a more negative back element power. The first configuration would produce a lower
accommodative performance while the latter a higher performance. Therefore, the choice of
element power combination is of greater consideration to accommodative performance than the
actual refractive state to be corrected. However, from an implementation and optical engineering
standpoint, the refractive state of the eye (and hence the individual element powers) may have a
slight effect on the thicknesses of the AIOL elements which consequently may somewhat affect
the final performance. For example, for the same accommodative performance, a hyperopic eye
would require a thicker front element (plus lens) of 2E-AIOL (compared to the one required for
myopic eye) for a given power of the rear element necessitating greater axial space. Such an
embodiment in turn reduces the inter-element space available for translation and consequently,
would reduce the accommodative performance.
69
Accommodative Performance
In conclusion, in this study, we have employed a paraxial optics approach to evaluating the
accommodative performance of 1E-AIOL and especially 2E-AIOL. We developed equations
suitable for the analytical description of the efficiency of an AIOL, its accommodative
performance, as a function of lens element power, and position, separation and translation of
elements for mobile front or back element AIOL. Consistent with the previous theoretical results,
our model suggests that 2E-AIOL produces higher amplitude of accommodation compared to
1E-AIOL. By mathematical approximation, closed-forms of the predictive equations for
accommodative performance were derived. These derivations provide the basis for optical design
optimisation of AIOL from the point of view of maximizing accommodation amplitude. One
potential issue raised with respect to the designs of 2E-AIOL is aniso-accommodation and
associated dynamic aniseikonia when two eyes are implanted with dissimilar designs. This issue
will be evaluated in Chapter 4 with greater details.
70
Accommodative Performance
3.5 References
Crawford, K. S., Kaufman, P. L. and Bito, L. Z. (1990). The role of the iris in accommodation of rhesus
monkeys. Invest Ophthalmol Vis Sci 31, 2185-2190.
Cumming, J. S., Colvard, D. M., Dell, S. J., et al. (2006). Clinical evaluation of the Crystalens AT-45
accommodating intraocular lens: results of the U.S. Food and Drug Administration clinical trial. J
Cataract Refract Surg 32, 812-825.
Cumming, J. S., Slade, S. G. and Chayet, A. (2001). Clinical evaluation of the model AT-45 silicone
accommodating intraocular lens: results of feasibility and the initial phase of a Food and Drug
Administration clinical trial. Ophthalmol. 108, 2005-2009;
Gerrard, A. and Burch, J. (1994). Matrix methods in Paraxial Optics. In: Introduction to Matrix Methods
in Optics, Dover Publications, New York, pp 24-42.
Halbach, K. (1964). Matrix representation of Gaussian optics. Am J Phys. 32, 90-108.
Hara, T., Hara, T., Yasuda, A., Mizumoto, Y. and Yamada, Y. (1992). Accommodative intraocular lens
with spring action--Part 2. Fixation in the living rabbit. Ophthalmic Surg 23, 632-635.
Hara, T., Hara, T., Yasuda, A. and Yamada, Y. (1990). Accommodative intraocular lens with spring
action. Part 1. Design and placement in an excised animal eye. Ophthalmic Surg 21, 128-133.
Heatley, C. J., Spalton, D. J., Boyce, J. F. and Marshall, J. (2004). A mathematical model of factors that
influence the performance of accommodative intraocular lenses. Ophthalmic Physiol Opt 24, 111118.
Ho, A., Manns, F., Therese, P. and Parel, J. M. (2006). Predicting the performance of accommodating
intraocular lenses using ray tracing. J Cataract Refract Surg 32, 129-136.
Langenbucher, A., Huber, S., Nguyen, N. X., Seitz, B., Gusek-Schneider, G. C. and Kuchle, M. (2003a).
Measurement of accommodation after implantation of an accommodating posterior chamber
intraocular lens. J Cataract Refract Surg 29, 677-685.
Langenbucher, A., Huber, S., Nhung, X. N., Seitz, B. and Kuchle, M. (2003b). Cardinal points and imagobject magnification with an accommodative lens implant (1 CU). Ophalmic Physiol Opt 23, 61 - 70.
71
Accommodative Performance
Langenbucher, A., Reese, S., Jakob, C. and Seitz, B. (2004). Pseudophakic accommodation with
translation lenses--dual optic vs mono optic. Ophthalmic Physiol Opt 24, 450-457.
Langenbucher, A., Seitz, B., Huber, S., Nguyen, N. X. and Kuchle, M. (2003c). Theoretical and measured
pseudophakic accommodation after implantation of a new accommodative posterior chamber
intraocular lens. Arch Ophthalmol. 121, 1722-1727.
Lesiewska-Junk, H. and Kaluzny, J. (2000). Intraocular lens movement and accommodation in eyes of
young patients. J Cataract Refract Surg 26, 562-565.
Lin, J. T. (2006). Efficiency analysis of the dual-optics accommodating IOL. J Cataract Refract Surg 32,
1986.
Long, W. F. (1976). A matrix formalism for decentration problems. Am J Optom Physiol Opt 53, 27-33.
Macsai, M. S., Padnick-Silver, L. and Fontes, B. M. (2006). Visual outcomes after accommodating
intraocular lens implantation. J Cataract Refract Surg 32, 628-633.
Mcdonald, J. P. C., M.A; Vinje, E; Glasser, a; Heatley, G.A; Kaufman, P;
Sarfarazi, F.M. (2003).
Sarfarazi Elliptical Accommodating IntraOcular Lens (EAIOL) in Rhesus Monkey Eyes In Vitro and
In Vivo. ARVO E-Abstract 256 44, 1.
Mcleod, S. D., Portney, V. and Ting, A. (2003). A dual optic accommodating foldable intraocular lens. B
J Ophthalmol 87, 1083-1085.
Missotten, T., Verhamme, T., Blanckaert, J. and Missotten, G. (2004). Optical formula to predict
outcomes after implantation of accommodating intraocular lenses. J Cataract Refract Surg 30, 20842087.
Msc Software Corporation (2001). Innovative Intraocular Lens design proven with simulation,
http://www.mscsoftware.com/assets/2702_Patran_Optics.pdf.
Nataloni,
R.
(2003).
Twin-Optic
Elliptical
IOL
emulates
natural
accommodation,
http://www.eyeworld.org/sep03/0903p50.html.
Nawa, Y., Ueda, T., Nakatsuka, M., et al. (2003). Accommodation obtained per 1.0 mm forward
movement of a posterior chamber intraocular lens. J Cataract Refract Surg 29, 2069-2072.
Pepose, J. S., Qazi, M. A., Davies, J., et al. (2007). Visual performance of patients with bilateral vs
combination Crystalens, ReZoom, and ReSTOR intraocular lens implants. Am J Ophthalmol 144,
347-357.
72
Magnification of AIOL
Chapter 4
Magnification of AIOL
A portion of this chapter has been published in the following peer reviewed journal
Ale J, Manns F & Ho A (2010). Magnification of single element and dual element
accommodating intraocular lens: Paraxial analysis. Ophthalmic Physiol Opt. 2011, 31, 7-16.
73
Magnification of AIOL
TABLE OF CONTENT
4.1 INTRODUCTION..................................................................................................................75
4.2 METHODS .............................................................................................................................76
4.2.1 Lateral magnification
77
4.2.2 Relative lateral magnification
78
4.2.3 Angular magnification
79
4.2.4 Image nodal point position
81
4.2.5 Computations
81
4.3 RESULTS ...............................................................................................................................82
4.3.1 Paraxial Analysis
82
4.3.2 Computational Results
85
4.4 DISCUSSION .........................................................................................................................91
4.5 REFERENCES.......................................................................................................................97
74
Magnification of AIOL
4.1 Introduction
Magnification is one of the important optical performance parameters that require special
attention in relation to design because it may significantly influence visual performance. The
change in magnification has been considered to be due to a continuous axial shifting of the optics
that leads to dynamic changes (i.e. changes associated with the action of accommodation) in the
positions of the cardinal points (e.g. the nodal points) in the pseudophakic eye (Langenbucher et
al., 2003) in accompaniment with the accommodation. This in turn may result a dynamic change
in ocular magnification. The magnitude of the effect and its potential implications remain to be
fully investigated, though the issue has been identified previously (Langenbucher et al., 2003,
Ho et al., 2006). Several design parameters including power of the AIOL elements, distance and
direction of the translation and anatomical factors such as corneal power, anterior chamber depth
and axial length may impact the ocular magnification. A theoretical study (Langenbucher et al.,
2003) computed retinal image sizes of an eye implanted with 1E-AIOL at near focus and
compared it to that obtained with near vision spectacle correction. It was concluded that the
lateral magnification does not differ significantly between pseudophakic accommodation and
near correction with spectacles, as the difference was <1%. However, the authors did not
examine the dynamic effect associated with accommodative change. To my knowledge,
magnification of 2E-AIOL is yet to be investigated.
Magnification of an optical system may be characterised in angular and linear (lateral) terms.
Though both types may be used interchangeably, they are frequently treated as different entities
and one may become more meaningful over another in certain circumstances. For instance,
angular magnification is more meaningful for a distance object where the lateral magnification
becomes undefined (Keating, 1980). In a clinical setting, expressing magnification in terms of
change in the retinal image size, a measure of lateral magnification, may be more relevant.
75
Magnification of AIOL
In this chapter both angular and lateral magnifications of the pseudophakic model eye implanted
with 1E-AIOL and 2E-AIOL are investigated. The effects of translation of optics, and types and
designs of AIOLs are evaluated using the matrix method of paraxial optics. The analytical
approach employed in this experiment is primarily aimed at understanding the key control
variables relevant to the design optimisation of the AIOLs and also attempts to determine the
potential clinical significance associated with the performance of such devices.
4.2 Methods
Recalling the introduction of matrix method in Appendix B3 extensively used in Chapter 3, the
system matrix of the optical system representing the eye with AIOL is given by:
S Eye = Tr .Tb .Rb .Ts .R f .T f .TAC .RK .To
................................................................................ (4.1)
where Tr, Tb, Ts, Tf, TAC , and To are translation matrices for the vitreous, space between vitreous
face and rear element, space between the two elements, space between the front element and the
pupil, anterior chamber and object space respectively, and Rb, Rf and RK are the refraction
matrices for the back and front elements of the AIOL and the cornea respectively. System matrix
S has the following structure:
ªS
S = « 11
¬ S 21
S12 º
S 22 »¼ ............................................................................................................. (4.2)
Input ray (Ը) parameters on the first surface of an optical system may be represented by:
ªn ⋅ α º
ℜ=«
» .................................................................................................................. (4.3)
¬ h ¼
76
Magnification of AIOL
where n is the refractive index of the medium of incidence, Į is the angle between the ray and the
optical axis, and h is the height on the surface. The emergent ray leaving the optical system has
parameters:
ªn'⋅α 'º
ªn ⋅ α º
« h' » = S .« h »
¬
¼
¬
¼ ......................................................................................................... (4.4)
where nƍ is the refractive index of the medium in image space
4.2.1 Lateral magnification
Lateral magnification (LM) is defined as the ratio of the object size to image size (Keating,
1988b). The relation between the lateral magnification and the coefficients of the matrix S of an
optical system can be found by calculating the matrix that relates the ray parameters for a set of
conjugate object and image points. Denoting l and lƍ as reduced object and image distances
respectively, the ray parameter at the object point is related to the ray parameter at the image
point by:
ªn'.α 'º ª1 0º ª S11
« h' » = «l ' 1».« S
¼ ¬ 21
¬
¼ ¬
S12 º ª1 0º ªn.α º
.
.
S 22 »¼ «¬l 1»¼ «¬ h »¼
..................................................................... (4.5)
Expanding Equation (4.5) we obtain
S11 + S12l
S12 º ªn.α º
ªn'.α 'º ª
« h' » = « S + S l + ( S + S l )l ' S + S l '».« h »
¬
¼ ¬ 21
¼ ................................................ (4.6)
22
11
12
22
12 ¼ ¬
When the first surface is the object plane and the last surface the image plane, and since the
object and image points are conjugate to each other, by definition, all rays passing through the
77
Magnification of AIOL
object point of height h also pass through the same image point of height h', independent of the
initial value of Į. In other words, h' must be independent of a ray angle. This condition can only
be satisfied when the term at position S21 in the system matrix equals zero, i.e.:
S 21 + S 22l + ( S11 + S12l )l ' = 0
With this, the lateral magnification is therefore obtained by:
LM =
h'
= S 22 + S12 l '
.................................................................................................... (4.7)
4.2.2 Relative lateral magnification
In addition to absolute lateral magnification, relative lateral magnification, equivalent to relative
spectacle magnification (RSM), was calculated using two different approaches. Firstly, the
relative lateral magnification (rLM1), defined as the ratio of the lateral magnification of the
pseudophakic eye at various levels of accommodation to the lateral magnification of the phakic
model eye in unaccommodated state, was calculated. This index provides a measure of the
change in lateral magnification as a function of pseudophakic accommodation with respect to the
magnification of an average phakic eye focused at distance. Mathematically, it is given by:
rLM 1 =
LM a
LM eye
.............................................................................................................. (4.8)
where, LMeye refers to the lateral magnification of the phakic model eye at distance focus state.
Secondly, relative lateral magnification (rLM2), defined as the ratio of the lateral magnification
of the pseudophakic eye at various levels of accommodation to the lateral magnification of the
78
Magnification of AIOL
same eye at distance focus (unaccommodated state), was calculated. This index provides a
measure of changes in apparent retinal image size as a function of pseudophakic accommodation
in the same eye. Mathematically, it is given by:
rLM 2 =
LM a
LM d .............................................................................................................. (4.9)
where, LMd and LMa are, respectively, the lateral magnifications at distance and accommodated
states of the pseudophakic eye.
4.2.3 Angular magnification
Various definitions for angular magnification exist in the literature (Keating, 1988a).
Conventionally, angular magnification (AM) is the ratio of the image space ray angle (Į') to the
object space ray angle (Į) for a ray passing through conjugate points lying on the optical axis
(angles shown illustrated in Figure 4.1). Reference rays are the rays that pass either through the
principal points (Born and Wolf, 1999) or through the centres of the entrance and exit pupils
(Keating, 1988a). In the present study, the ray passing through the centre of the entrance pupil
(chief ray) was used as the reference for angular magnification since positions of the principal
planes vary with translation of the AIOL and with the AIOL design, whereas the position of the
entrance pupil (the image of the real pupil formed by the cornea) is independent of the amplitude
of pseudophakic accommodation and the design of the AIOL. For a given accommodation
amplitude the reference angle α, will therefore be the same for all AIOL designs.
79
Magnification of AIOL
K
H1 H2
P
E1
F1 F2
R
E2
ɲ
ƍ
AC
f
s
b
V
Figure 4.1: Layout of the eye model with AIOL in which K refers to the cornea, H1 and H2 refer
to the primary and secondary principal planes, P refers to the pupil, F1 and F2 refer to the front
and rear lens elements of the AIOL respectively and R refer to the retina (these elements are
referred as Ff and Fb respectively when according to make contextual elsewhere in the thesis).
The cornea was assumed to be a thin lens positioned in the plane of the anterior corneal surface.
Distance AC is the anterior chamber depth, f is the space between the front element and pupil, s
is the space between the two elements, b is the space between the anterior vitreous face and the
rear element and V is the distance from the anterior vitreous face to the retina (the distances are
exaggerated for clarity). Angle Į is the incident chief ray angle at the centre of the entrance
pupil E1 and Įƍ is the emergent chief ray angle at the centre of the exit pupil E2.
To find the relation between angular magnification and the coefficients of the matrix of an
optical system, the same method as for lateral magnification was used. This time, the object point
is the centre of the entrance pupil and the image point is the centre of the exit pupil. Denoting x
as a reduced distance of the entrance pupil from the corneal surface and xƍ as the reduced
distance of the exit pupil from the last surface of the AIOL, ray parameters at the exit pupil plane
is given by:
ªn'.α p 'º ª 1 0º ª S11
« h ' »=«
».«
¬ p ¼ ¬ x ' 1¼ ¬ S 21
S12 º ª1 0º ªn.α p º
.«
.
»
S 22 »¼ «¬ x 1»¼ ¬ h p ¼
.............................................................. (4.10)
80
Magnification of AIOL
where subscript p denotes a quantity measured in the pupil planes. For the chief ray, hp=hp’=0.
Solving Equation (4.10) for angular magnification ( α p ' / α p ), we obtain:
AM =
p
αp
'
=
n
( S11 + S 22 .x)
n'
...................................................................................... (4.11)
4.2.4 Image nodal point position
During pseudophakic accommodation, positions of the cardinal points of the eye change due to
movement of the AIOL. A change in the position of the image space (second) nodal point (N2)
with respect to the retina produces a change in retinal image size (Smith et al., 1992). Therefore
the positional shift of the image nodal point was also calculated. In the matrix method, the
position of the image nodal point N2 with respect to the retina R can be obtained from
(Langenbucher et al., 2003, Raasch and Lakshminarayanan, 1989, Rosenblum and Christensen,
1974):
N2R = −
nn'
S12 ( S 22 − n' ) .................................................................................................. (4.12)
4.2.5 Computations
Accommodation-dependent parameters of the Navarro et al (1985) model were used. Again, the
crystalline lens in the model eye was replaced with an idealized thin-lens model of a 2E-AIOL.
AIOL elements and ocular spaces are portrayed in Figure 4.1. The system matrix of the model
pseudophakic eye from the cornea to the anterior vitreous boundary is given by:
S Eye = Tb .RF 2 .Ts .RF 1 .T f .TAC .RK
................................................................ 4.13a
81
Magnification of AIOL
where TAC, Tf, Ts and Tb are translation matrices for the anterior chamber, space between the
front element and the pupil, space between the two elements and space between anterior vitreous
face and rear element respectively, and RF2, RF1 and RK are the refraction matrices for the back
and front elements of the AIOL and the cornea respectively.
To evaluate the effect of power combinations of the 2E-AIOL elements, front element power
ranging from +20 D to +40 D in 5 D steps were used where +20 D approximately represents a
model of the 1E-AIOL. To simulate pseudophakic accommodation, two types of translation
models were configured. Magnification for both configurations (Configuration 1 comprised an
AIOL with a mobile front element in which the back element is maintained in a fixed position;
and Configuration 2 comprised an AIOL with a mobile back element where the front element
remained fixed) were studied. Maximum translation was constrained to 2 mm in either
configuration. All AIOL elements are thin-lenses in contact at their distance focus state (s = 0)
implanted 2 mm behind the pupil plane.
4.3 Results
4.3.1 Paraxial Analysis
4.3.1.1 Lateral Magnification
According to the Equation (4.7), the lateral magnification of the eye from Equation 4.13a is
given by:
LM =
­ s ( F2b − 1) − b + σ 1 ( AC + f ) − F2b − F1 b − s ( F2b − 1) ½
h'
= K®
¾
h
¯+ l ' [K ( F2 s + σ 2 ( AC + f ) − 1) − F2 + F1 ( F2 s − 1)] + 1 ¿
.................... (4.13)
82
Magnification of AIOL
where
σ 1 = F2 b + F1 b − s ( F2 b − 1) − 1
σ 2 = F2 b + F1 [b − s ( F2 b − 1)] − 1
As in Chapter 3, section 3.3.1, defining separation between the elements (s) and the space
between the front element and the front reference plane (f) in terms of translation of the front
element:
D
s
f, f
D
Zf and s = Z f
Replacing with these in Equation 4.13 and differentiating with respect to Zf, yields the rate of
change in lateral magnification with respect to the forward translation of the optics.
Mathematically, it is given by:
LM
= F1 ( F2 b − 1) − K F1 (b − Z f ( F2 b − 1)) − F1 ( Z f − D )( F2 b − 1) + ACF1 ( F2 b − 1) ...
δZ f
+ l ' K ( F1 ( F2 Z f − 1) + F1 F2 ( Z f − D ) − ACF f F2 + F f F
................................. (4.14)
Except for image distance (l’) all other distances appearing in the equation may be ignored as
these are small quantities in the order of
−3
. With this simplification, the rate of change in
magnification with translation of the optics may be given by:
LM
lim
≅
≅ F1 [l ' ( K − F2 ) + 1]
D, Z f , AC , b → 0
δZ f
.......................................................... (4.15)
83
Magnification of AIOL
Therefore for configuration 1, the lateral magnification is linearly and proportionally related to
the power of the front element. It also depends on the corneal power (K) and image distance (l)
or equivalently the vitreous chamber depth.
Similar approach can be applied to derive the equations for AIOL operating under configuration
2 (rear element moving backward). The exact rate of change in magnification is given by:
LM
δZ b
=
h'
= F2 − K {F2 Z f + AC F2 − F1 ( F2 Z f + F2 ( Z f − D) + f F2 − F1 ( F2 Z f + F2 ( Z f − D) + F2 ( Z f − D)
h
l ' K ( ACF1F2 F2
F1F2 f ) F1 ( F2 Z f
F2 ( Z f
D)
........................................ (4.16)
Approximate rate of change in the magnification is given by:
lim
LM
≅
≅ F2 [1 + l ' ( F1 + K )]
δZ b
D, Z b , AC , f → 0
.......................................................... (4.17)
Again, the magnification is linearly and proportionally related to the powers of the elements,
corneal power and the image distance. An important observation to be made is that the negative
power of the rear element may contribute a minification effect.
4.3.1.2 Angular Magnification
From Equation (4.11), angular magnification of the eye is given by:
AM =
'
α
= x{K [σ 1 ( AC + f ) − s ] + σ 1 )} − ACσ 2 − F2 s − fσ 2 + 1
............................. (4.18)
where x is the distance of entrance pupil from the corneal surface and
84
Magnification of AIOL
σ 1 = F1 s − 1
σ 2 = F2 − F1 ( F2 s − 1)
Angular magnification is independent of the translation matrices beyond the last refracting
surface in the system. Therefore the Equation (4.18) is valid for both configurations.
The rate of change in angular magnification may be obtained differentiating the equation with
respect to translation (Zf) which is given by:
AM
δ
= ACF1 F2 − x F1 − K F1 ( D − Z f ) + ACF1 − 1 − F2 + F1 F2 f
............................. (4.19)
And approximate equation is given by:
AM
lim
≅
≅ − F2 − x ( F1 + K )
D, Z , AC , f → 0
δZ
............................................................ (4.20)
4.3.2 Computational Results
4.3.2.1 Effect of Accommodation
Both lateral and angular magnifications increased with accommodation in the pseudophakic eye
whereas they decreased marginally in the phakic model eye (Figure 4.2 to 4.4). For a 2E-AIOL
with +35.0 D front element operating under configuration 1 (front element moving), the retinal
image size increased by about 25.8% at 4.0 D accommodation compared to the retinal image size
of the same eye at distance focus (unaccommodated) state (or rLM2), and by 37.48% compared
to the retinal image size of the unaccommodated phakic model eye (rLM1). Rates of increment in
85
Magnification of AIOL
the apparent retinal image size in rLM1 and rLM2 were 9.37% and 6.46% per dioptre
accommodation respectively (Table 4.1).
Actual changes in the relative lateral magnification for various models and configurations of the
AIOLs as a function of pseudophakic accommodation can be seen in Figures 4.2 and 4.3. In
these figures it is evident that relative lateral magnification increased at higher rates in
configuration 2 (rear element moving) than in configuration 1 of the 2E-AIOL (Figure 4.4). The
rate of change in magnification with accommodation increased with increased power of the front
element in the 2E-AIOL (Table 4.1).
ZĞůĂƚŝǀĞ>ĂƚĂƌĂůDĂŐŶŝĨŝĐĂƚŝŽŶ
ϭ͘ϯϱ
Ϯ;нϰϬ͘ϬͿ
Ϯ;нϯϬ͘ϬͿ
Ϯ;нϮϱ͘ϬͿ
ϭ͘Ϯϱ
DŽĚĞůLJĞ
ϭ͘ϭϱ
ϭ͘Ϭϱ
Ϭ͘ϵϱ
Ϭ͘ϬϬ
ϭ͘ϬϬ
Ϯ͘ϬϬ
ϯ͘ϬϬ
ϰ͘ϬϬ
ĐĐŽŵŵŽĚĂƚŝŽŶ;Ϳ
Figure 4.2: Relative Lateral Magnifications of 1E-AIOL and configuration 1 (front-moving) of
the various models of 2E-AIOL compared with that of phakic model eye. Solid plots represent
rLM1 and dashed plots represent rLM2. Relative magnification of the phakic model eye (dotted
line) is included for comparison.
For a 1E-AIOL with 2.5 D accommodation, which may be considered the maximum
accommodation obtainable, rLM1 and rLM2 increased by 23.7% and 16.8% respectively. The 1E86
Magnification of AIOL
AIOL produced a higher rate of change in magnification with accommodation than any model of
the 2E-AIOL. It should be noted that a 1E-AIOL requires translation over a larger distance than a
2E-AIOL to produce an equal amount of accommodation. The lateral and angular magnifications
of the phakic model eye decreased with accommodation (Figure 4.2 to 4.4).
The angular magnification of the 2E-AIOL with +35 D front element operating under
configuration 1 increased at a rate of 0.013/D of accommodation. The rate of change increased
with a decrease in the front element power of the AIOL. Slightly lower rates of change were
observed for 2E-AIOLs in configuration 2 (Figure 4.4). Angular magnification of the phakic
model eye decreased at a rate of -0.002/D of accommodation (Table 4.1).
ϭ͘ϴϬ
Ϯ;нϰϬ͘ϬͿ
ZĞůĂƚŝǀĞ>ĂƚĞƌĂůDĂŐŶŝĨŝĐĂƚŝŽŶ
Ϯ;нϯϱ͘ϬͿ
ϭ͘ϲϬ
Ϯ;нϯϬ͘ϬͿ
Ϯ;нϮϱ͘ϬͿ
DŽĚĞůLJĞ
ϭ͘ϰϬ
ϭ͘ϮϬ
ϭ͘ϬϬ
Ϭ͘ϴϬ
Ϭ͘ϬϬ
ϭ͘ϬϬ
Ϯ͘ϬϬ
ϯ͘ϬϬ
ϰ͘ϬϬ
ĐĐŽŵŵŽĚĂƚŝŽŶ;Ϳ
Figure 4.3: Relative Lateral Magnifications of 2E-AIOLs with configuration 2 (rear-moving)
compared with that of the phakic model eye magnification (dashed line). Solid plots represent
rLM1 and dashed plots represent rLM2. Relative magnification of the phakic model eye (dotted
line) is included for comparison.
87
Magnification of AIOL
ϭ͘ϬϬ
ŶŐƵůĂƌDĂŐŶŝĨŝĐĂƚŝŽŶ
Ϭ͘ϵϲ
Ϯ;нϰϬ͘ϬͿ
Ϭ͘ϵϮ
Ϯ;нϯϱ͘ϬͿ
Ϭ͘ϴϴ
Ϯ;нϮϱ͘ϬͿ
ϭ;нϮϬ͘ϳͿ
DŽĚĞůLJĞ
Ϭ͘ϴϰ
Ϭ͘ϴϬ
Ϭ͘ϬϬ
ϭ͘ϬϬ
Ϯ͘ϬϬ
ϯ͘ϬϬ
ϰ͘ϬϬ
ĐĐŽŵŵŽĚĂƚŝŽŶ;Ϳ
Figure 4.4: Absolute values of angular magnification as a function of accommodation for
various models of 2E-AIOL, 1E-AIOL and phakic model eyes magnification (dashed line). Solid
lines represent configuration 1 (front-moving) and dashed lines represent configuration 2 (rearmoving). Angular magnification of the phakic model eye (dotted line) is included for comparison.
The nodal point shifted towards the cornea (moved away from the retina) with pseudophakic
accommodation (Figure 4.5). At distance focus state, the image space nodal point (N2) remained
behind the optical system (about 17.34 mm in front of the retina); no significant differences in
the position were observed between types and models of AIOLs. With each dioptre of
accommodation, the nodal point shifted anteriorly at a rate ranging between 0.58 mm and
0.86 mm in Configuration 1 (front element moving) depending on the power combinations of the
elements; the rate of shift decreases as the power of the AIOL front element increases. In the
phakic model eye, the nodal point moves away from the retina at a rate of 0.30 mm/D of
accommodation.
88
Magnification of AIOL
Ϯϭ͘Ϭ
ŝƐƚĂŶĐĞďĞƚǁĞĞŶEϮ ΘZĞƚŝŶĂ;ŵŵͿ
Ϯ;нϰϬ͘ϬͿ
Ϯ;нϯϱ͘ϬͿ
Ϯ;нϯϬ͘ϬͿ
ϮϬ͘Ϭ
Ϯ;нϮϱ͘ϬͿ
ϭ;нϮϬ͘ϳͿ
ϭϵ͘Ϭ
DŽĚĞůLJĞ
ϭϴ͘Ϭ
ϭϳ͘Ϭ
ϭϲ͘Ϭ
Ϭ͘ϬϬ
ϭ͘ϬϬ
Ϯ͘ϬϬ
ϯ͘ϬϬ
ϰ͘ϬϬ
ĐĐŽŵŵŽĚĂƚŝŽŶ;Ϳ
Figure 4.5: Distance between the image space nodal point (N2) and the retina as a function of
accommodation for the phakic model eye magnification (dotted line), 1E-AIOL and 2E-AIOL
operating under configuration 1 (front-moving).
4.3.2.2 Effect of Translation
The rate of change in relative lateral magnifications (rLM1 and rLM2) ranged between 11.0% and
32.2% relative to the power of the front element and the configuration of the 2E-AIOL (Table
4.1). A 2E-AIOL in configuration 2 produces a higher rate of change in lateral magnification
compared to a 2E_AIOL in configuration 1 (Figure 4.6). The rate of change in the lateral
magnification per mm translation was less for a 1E-AIOL than a 2E-AIOL. The opposite was
found when the change in magnification was quantified in terms of accommodation instead of
translation.
89
Magnification of AIOL
ϭ͘ϴϬ
ZĞůĂƚŝǀĞ>ĂƚĞƌĂůDĂŐŶŝĨŝĐĂƚŝŽŶ;ƌ>DϮͿ
Ϯ;нϰϬ͘ϬͿ
Ϯ;нϯϱ͘ϬͿ
ϭ͘ϲϬ
Ϯ;нϯϬ͘ϬͿ
ϭ͘ϰϬ
ϭ;нϮϬ͘ϳͿ
ϭ͘ϮϬ
ϭ͘ϬϬ
Ϭ͘ϴϬ
Ϭ͘ϬϬ
Ϭ͘ϱϬ
ϭ͘ϬϬ
ϭ͘ϱϬ
Ϯ͘ϬϬ
dƌĂŶƐůĂƚŝŽŶ;ŵŵͿ
Figure 4.6: Relative Lateral Magnification (rLM2) for various models of 2E-AIOL and 1E-AIOL
as a function of translation of the optics. Solid lines represent configuration 1 (front-moving)
and dashed lines represent configuration 2 (rear-moving). Relative Lateral Magnification is
defined as the ratio of the magnification of the pseudophakic eye at various stages of translation
to the magnification of the same eye at distance focus (unaccommodated state).
Angular magnification increased at a rate of 0.03 times /mm translation for a configuration 1
(front-moving) 2E-AIOL with +35.0 D front element. The change in angular magnification was
less for a Configuration 2 (rear element moving) 2E-AIOL than a Configuration 1 (front element
moving) 2E-AIOL (Figure 4.7).
90
Magnification of AIOL
ϭ͘ϬϬ
ŶŐƵůĂƌDĂŐŶŝĨŝĐĂƚŝŽŶ
Ϭ͘ϵϲ
Ϭ͘ϵϮ
Ϭ͘ϴϴ
Ϯ;нϰϬ͘ϬͿ
Ϯ;нϯϱ͘ϬͿ
Ϯ;нϯϬ͘ϬͿ
Ϭ͘ϴϰ
Ϯ;нϮϱ͘ϬͿ
ϭ;нϮϬ͘ϳͿ
Ϭ͘ϴϬ
Ϭ͘ϬϬ
Ϭ͘ϱϬ
ϭ͘ϬϬ
ϭ͘ϱϬ
Ϯ͘ϬϬ
dƌĂŶƐůĂƚŝŽŶ;ŵŵͿ
Figure 4.7: Absolute values of angular magnification as a function of translation of the optics for
1E-AIOL and various models of 2E-AIOL. Solid lines represent configuration 1 (front-moving)
and dashed lines represent configuration 2 (rear-moving).
4.4 Discussion
Changes in both angular and lateral magnification of AIOLs were quantified. Lateral
magnification provides a direct measure of the change in retinal image size for an object at finite
distance. While clinical interpretation of lateral magnification is relatively intuitive, it cannot be
used for a distance object (object at infinity) as the magnification becomes undefined. For this
reason, angular magnification was also quantified. Angular magnification is independent of the
object distance and remains finite for all conjugate distances, including distance vision.
91
Magnification of AIOL
Table 4.1: Changes in relative magnifications, angular magnification, and image space nodal
point (N2) positions as a function of accommodation and translation in phakic and pseudophakic
model eyes. In configuration 1 of 2E-AIOL (Conf. 1), the front element moved in the forward
direction and in configuration 2 (Conf. 2), the rear element moved in the backward direction for
accommodation. rLM1
-
the ratio of lateral magnification of AIOL at various levels of
accommodation to the magnification of the phakic model eye at distance focus
(unaccommodated) state; rLM2 - the ratio of magnifications at accommodated state to the
magnification at distance focus state for the same eye; AM - Angular magnification.
Change in
rLM1 (% per D of
accommodation)
rLM2 (% per D of
accommodation)
rLM1 (% per mm
of translation)
Conf. 1
Conf. 2
Conf. 1
Conf. 2
Conf. 1
Conf. 2
2E-AIOL Front Element Power
+40 D +35 D +30 D
+25 D
9.27
9.37
9.51
9.49
21.57 27.36 39.85
81.03
6.36
6.46
6.53
6.59
20.63 26.17 38.29
77.97
19.98 17.04 14.23
11.54
32.16 29.45 26.85
24.37
1EAIOL
9.62
6.7
9.36
-
Model
Eye
-0.25
-
rLM2 (% per mm Conf. 1
of translation)
Conf. 2
AM (per D of Conf. 1
accommodation)
Conf. 2
19.11
16.30
13.61
11.04
8.95
30.76
0.013
28.17
0.013
25.69
0.014
23.31
0.015
0.016
0.010
0.010
0.011
0.012
-
AM (per mm of Conf. 1
translation)
Conf. 2
0.037
0.015
0.033
0.011
0.029
0.007
0.025
0.003
0.021
-
-
Forward shift of Conf. 1
N2 (mm per D of
Conf. 2
accommodation)
0.58
0.63
0.71
0.79
0.86
0.30
1.07
1.43
2.19
4.66
-
-0.002
In addition, in the present case, angular magnification is advantageous because it can be referred
to a reference input ray which is independent of the design of the AIOL and axial length of the
eye if the entrance and exit pupil planes are used as the reference planes to define angular
magnification. Since the entrance pupil of the eye remains unchanged with accommodation and
with IOL implantation, the input reference ray is an ‘invariant’ ray. A further advantage of the
92
Magnification of AIOL
angular magnification is that it is independent of the axial length of the eye. Lateral
magnification can be readily determined from the angular magnification for any value of the
axial length.
The magnification of an eye implanted with an AIOL changes with the change in position of the
lens elements. Results in this chapter suggest that both the angular and the lateral magnifications
of the eyes proportionally increase with translation of the lens elements during pseudophakic
accommodation. This result is in contrast to the phakic eye where magnifications decrease with
accommodation (Figures 4.2 to 4.4). These results support earlier reports that phakic eye
accommodation causes a decrease in apparent retinal image size (Biersdorf and Baird, 1966); a
phenomenon often referred to as “accommodative micropsia”. Two potential bases, neural
(Hochberg, 1972) and optical (Biersdorf and Baird, 1966), of this effect have been proposed. A
computational study demonstrated the optical minification effect of accommodation in a range of
schematic eyes for which the retinal image size decreased by as much as 1.5% at 10.0 D
accommodation (Smith et al., 1992).
The amount and direction of translation and powers of the lens elements are major design
parameters significantly associated with magnification.
These design parameters are also
relevant to the accommodative performance of the AIOL (Chapter 3). To achieve a given amount
of accommodation, an AIOL with higher front element power requires less translation than an
AIOL with lower front element powers (Ale et al., 2010, Ho et al., 2006). With an equal amount
of translation, a higher power of the front element produces higher amount of magnification.
However, because accommodation is also proportional to the distance of translation of the optics,
an AIOL with a higher power of the front element would produce a lesser amount of
magnification at a given accommodation amplitude.
An optical explanation for the change in magnifications in translating-optics AIOLs can be
founded on the relative position of the image space nodal point. Retinal image height is
dependent on the distance between the retina and the image plan; for a given nodal ray angle, the
93
Magnification of AIOL
image height increases as the nodal point to retina distance increases. In our model, distance
between the retina and the image space nodal point increased with translation of the optics
elements (Figure 4.5) which led to an increase in magnification. The rate of change in distance
between the image space nodal point and the retina per millimetre translation was greater for a
2E-AIOL with higher power of the front element. Also a higher rate was observed for a 2EAIOL operating under configuration 2 (rear element moving) than a 2E-AIOL operating in
configuration 1.
It is possible that the magnification arising from the front element (positive lens) is partly
counteracted by the minification effect from the rear element (negative lens). For an AIOL
operating under configuration 2, the magnification from the rear element decreases as it
translates posteriorly (moving towards the nodal point); since the front element magnification
effect remains constant, this gives rise to a higher rate of increment and amount of magnification.
The results in this chapter raise a potential clinical issue relating to implantation of AIOL: that of
dynamic aniseikonia. Dynamic aniseikonia may arise in two ways; either directly as a result of
inter-ocular differences in magnification, or indirectly due to aniso-accommodation of AIOL
between eyes.
Although a study found that motor fusion may be obtained for up to 4D anisometropia and 11%
aniseikonia (Bhardwaj & Candy, 2011), others found that when the difference in magnification
between the eyes exceeds 4-5%, the two retinal images cannot be binocularly fused to provide
the percept of a single image (Achiron et al., 1997). Unlike motor fusion, stereopsis is extremely
fragile. As little as 1% imbalance in retinal image size has been reported to impair stereopsis
(Reading and Tanlami, 1980). Assuming a nominal 2.50 D accommodation obtainable with a 2EAIOL operating under configuration 1 and consisting of a 35.0 D front element, the relative
retinal image size of the eye increases by an alarming 15.7% compared to that in the
unaccommodated phakic model eye. Taking into account the accommodative minification of the
94
Magnification of AIOL
normal phakic eye, this suggests monocular implantation of an AIOL may severely deteriorate
binocular visual function.
Even when AIOLs are implanted binocularly, care must be taken to ensure that bilateral
magnifications are matched along the entire range of accommodation in order to avoid dynamic
aniseikonia-induced binocular imbalance. Though the difference in magnification between
various design combinations of 2E-AIOL element powers are reasonably similar for matching
accommodation levels up to 4 D, dynamic aniseikonia may rise due to differences in
accommodation between the two AIOLs (i.e. aniso-accommodation). The results in this study
indicate that aniso-accommodation of about 1 D would induce a retinal image size disparity of
about 6% which is sufficient to severely compromise binocular vision. It must be appreciated
that this scenario differs from the results of a study where 1E-AIOLs were implanted in one eye,
and a traditional monofocal non-accommodating IOL was implanted in the other eye. When the
eye implanted with the traditional IOL was corrected with near-vision spectacles and the other
eye relied on the AIOL for near vision, only 1% difference in retinal image size was reported
between the two eyes (Langenbucher et al., 2003). It may be inferred that, while the difference in
magnification induced by small aniso-accommodation (<1 D) may not induce diplopia, it may be
at a level sufficient to impair stereopsis.
In order to facilitate modelling, some assumptions and approximations were used. Hence, there
may be aspects which have not been considered. Since the thin-lens approximation was used,
magnification produced by changes in the shape-factor of lens elements could not be evaluated.
However, a brief calculation shows that the effect is likely to be minimal contributing
approximately 1% (for 1 mm thick biconvex lens of 40 D lens) of the total magnification
(Keating, 1988b). Departure from constant magnification across the field (distortion) can be
brought about by the implementation of specific individual and combinations of lens forms
(bending factors) for the lens elements. Thus, the present results are theoretically only valid
within the domain of paraxial optics approximations. However, since visual performance is
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Magnification of AIOL
typically primarily concerned with high visual acuity, i.e. within the central retinal field, use of
the paraxial approximation is reasonable.
The refractive status of the eye (myopic or hyperopic) and other ocular parameters such as
corneal power and axial length, have also not been considered, but presumably have some effect
on ocular magnification (Felipe et al., 2007). For more precise prediction of the magnification
effect for different refractive error and ocular anatomical dimensions, analyses incorporating
additional parameters would be required which is beyond the scope of the present work.
In conclusion, from the optical design perspective of AIOLs, the power combination of the AIOL
elements and the amount of translation are shown to be two key design parameters governing
dynamic accommodation in the eye implanted with AIOL although implant position, refractive
status and ocular dimensions should not be ignored. The power combination of 2E-AIOL lens
elements requires special attention prior to implantation to avoid potential dynamic
anisometropia and dynamic aniseikonia. In some cases, customised selection of designs and
element combinations may be required to match their performance in a binocularly balanced
manner.
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Magnification of AIOL
4.5 References
Achiron, L. R., Witkin, N., Primo, S. and Broocker, G. (1997). Contemporary management of
aniseikonia. Surv Ophthalmol 41, 321-330.
Ale, J., Manns, F. and Ho, A. (2010). Evaluation of the performance of accommodating IOLs
using a paraxial optics analysis. Oph Physiol Opt 30, 132 - 142.
Bhardwaj S.R. and Candy T.R (2011). The effect of lens-induced anisometropia on
accommodation and vergence during human visual development. Invest ophthalmol Vis Sci
10, 6214
Biersdorf, W. R. and Baird, J. C. (1966). Effect of an artificial pupil and accommodation on
retinal image size. J Opt Soc Am 56, 1123-1129.
Born, M. and Wolf, E. (1999). Geometrical Theory of Optical Imaging. In: Principles of Optics,
Cambridge University Press, New York, pp 142-227.
Felipe, A., Díaz-Llopis, M., Navea, A. and Artigas, J. (2007). Optical analysis to predict
outcomes after implantation of a double intraocular lens magnification device. J Cataract
Refract Surg 33, 1781-1789.
Ho, A., Manns, F., Therese and Parel, J. M. (2006). Predicting the performance of
accommodating intraocular lenses using ray tracing. J Cataract Refract Surg 32, 129-136.
Hochberg, J. (1972). Perception II: Space and Movement. Methuen, London.
Keating, M. P. (1980). Lateral Magnification-Angular Magnification Relationship for a Simple
Magnifier. Am J Physic. 48, 214-217.
Keating, M. P. (1988a). Angular Magnification. In: Geometric, Physical and Visual Optics (ed
M. P. Keating), Butterworth, Stoneham, pp 247-266.
Keating, M. P. (1988b). Spectacle magnification and relative spectacle magnigication. In:
Geometric, Physical and Visual Optics (ed M. P. Keating), Butterworth, Boston, pp 267293.
97
Magnification of AIOL
Langenbucher, A., Huber, S., Nhung, X. N., Seitz, B. and Kuchle, M. (2003). Cardinal points
and imag-object magnification with an accommodative lens implant (1 CU). Ophthalmic
Physiol Opt 23, 61 - 70.
Navarro, R., Santamaria, J. and Bescos, J. (1985). Accommodation-dependent model of the
human eye with aspherics. Graef Arch Clin Exp Ophthalmol. 2, 1273-1281.
Raasch, T. and Lakshminarayanan, V. (1989). Optical matrices of lenticular polyindicial
schematic eyes. Ophthalmic Physiol Opt 9, 61-65.
Reading, R. W. and Tanlami, T. (1980). The threshold of stereopsis in the presence of difference
in magnification of the ocular images. J Am Optom Assoc. 52, 593-595.
Rosenblum, W. M. and Christensen, J. L. (1974). optical matrix method: optometeric
applications. Am J Optom Physiol Opt 51, 961-968.
Smith, G., Meehan, J. W. and Day, R. H. (1992). The effect of accommodation on retinal image
size. Human Factors 34, 289-301.
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Depth of Field
Chapter 5
Depth of Field
Part of this chapter was published in the following peer reviewed journal
Ale J, Ho A, Manns F: Paraxial analysis of the Depth-of-Field of a pseudophakic eye with
Accommodating intraocular lens. Optom. Vis Scie 2011, Accepted in 22 Feb, (Reference - Ms.
No. OVS10357R1)
Part of this chapter was presented in the following conference
Ho A, Manns F, Ale J, Lee Y, Parel J-M: Depth of Field of Accommodating IOL: paraxial optics
analysis. Oral presentation by Arthur Ho. Ophthalmic Technologies conference XVIII at
Biomedical Optics Session, Photonics West Meeting, SPIE, San Jose, 2008.
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Depth of Field
TABLE OF CONTENT
5.1 INTRODUCTION................................................................................................................101
5.2 OBJECTIVE ........................................................................................................................103
5.3 METHODS ...........................................................................................................................104
5.3.1 Analytical: General Equation for DoF
104
5.3.2 Computation Method
107
5.3.3 Computation of DoF
107
5.3.4 Computation of Effective DoF (eDoF)
109
5.4 RESULTS .............................................................................................................................111
5.4.1 Analytical Results
111
5.4.2 Computational Results
113
5.4.2.1 Effect of Implant Depths and Power Combinations
115
5.4.2.2 Effect of Accommodation
115
5.5 DISCUSSION .......................................................................................................................119
5.6 REFERENCES.....................................................................................................................121
100
Depth of Field
5.1 Introduction
Other than psychological and neural factors which is an complex component to determine, an
individual’s perception of blur definably depends on two main factors: first, the physiological
factor which is determined by the size and distribution of photoreceptors on the retina (Geisler,
1984, Legge et al., 1987, Wang and Ciuffreda, 2005, Williams and Coletta, 1987) and second,
the optical factor which is responsible for bringing light from an object into focus. Foveally,
defocus is the more dominant factor responsible for visual blur. The human eye often suffers
from error of refraction and other optical aberrations leading to certain degrees of blur. Perhaps
conveniently, the visual system does not detect blur until the magnitude of defocus exceeds a
perceptual tolerable range termed the depth of focus (whose conjugate in object space is termed
the depth of field). While the term “depth of focus” is commonly used in the literature, more
correctly, the range of acceptably clear vision in the object space should be termed the “depth of
field”. Though, when measured as vergence in dioptric units, the depth of field and depth of
focus should be quantitatively the same, we use the term “depth of field” (abbreviated to “DoF”)
throughout this chapter, except where referenced sources employ otherwise.
Many studies have investigated the factors influencing the depth of field of the human eye. A
key determinant of the depth of focus is the maximum size of a blur circle (assuming a circular
pupil) projected from a point object on to the retina that does not elicit significant detectable
visual blur. This in turn depends on the size and density of photoreceptors, along with some
other physiological factors such as the Stiles-Crawford effect (Campbell, 1957, Tucker and
Charman, 1975). For example, depth of focus is greater in the peripheral retina, where rods and
cones are more sparsely distributed, than in the foveal region (Wang and Ciuffreda, 2004). It has
been well established that the primary factors influencing depth of focus of the human eye
include the refractive power of the eye (Jiang and Morse, 1999, Rosenfield and Abraham-Cohen,
1999), axial length (Green et al., 1980) and pupil size (Atchison et al., 1997b, Campbell, 1957,
Charman and Whitefoot, 1977, Legge et al., 1987, Tucker and Charman, 1975). Some other
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Depth of Field
factors that are associated include luminance (Campbell, 1957), contrast (Atchison et al., 1997b,
Campbell, 1957), colour (Campbell, 1957, Marcos et al., 1999), size of target (Atchison et al.,
1997a, Jacobs et al., 1989), visual acuity (Green et al., 1980), refractive error (Rosenfield and
Abraham-Cohen, 1999), retinal eccentricity (Wang and Ciuffreda, 2004), accommodation,
diffraction and aberrations (Green et al., 1980, Westheimer, 1953). The above-mentioned optical
and physiological factors differ from individual to individual. Therefore a range of DoF is
expected in the population. Table 5.1 summarizes the DoF in the population as reported in the
literature.
Table 5.1: Values of DoF of eyes reported by various authors in the literature.
Investigators
Year
DoF (D)
Comment
Bohr G von
Campbell
1952
1957
±0.15 – 1.5
±0.34 – 0.56
Edged fixed target
Effect of luminance, pupil size
Campbell & Westheimer
Ogle & Schwartz
Tucker & Charman
Charman & Whitefoot
Green et al.
Jacobs et al.
Elder et al.
Atchison
1958
1959
1975
1977
1980
1989
1996
1997
±0.35
±0.30 – 0.45
±0.70 – 1.50
±0.15 – 1.80
±0.04 – 0.50
±0.20
±0.85
±0.28 – 0.43
Fixed stimulus parameters
Effect of target details and pupil size
Fixed stimulus parameters
Effect of pupil size
In terms of visual acuity
Blur threshold method
For near, pseudophakia
Variable target detail, pupil size
Marcos et al.
Nio et al.
1999
2003
±0.11 – 0.27
1.82 ± 0.34
Effect of chrom. aberration
Pseudophakic eye, based on contrast
sensitivity
Wang & Ciufreda
2004
±0.45 – 1.76
Effect of eccentricity
Ciuffreda et al.
2005
±0.73 – 1.15
Naturalistic pictorial stimuli used,
central and peripheral retina evaluated
Studies have been reported in which pseudophakic individuals implanted with a rigid monofocal
IOL can achieve better than predicted near visual acuity without optical aid (Bradbury et al.,
1992, Huber, 1981, Lesiewska-Junk and Kaluzny, 2000) despite the assumed complete loss of
accommodative ability. This is a common observation among clinicians. In addition to this
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Depth of Field
apparent accommodation or ‘pseudo-accommodation’ (Nakazawa and Ohtsuki, 1983, Nakazawa
and Ohtsuki, 1984), improved depth of focus (Kamlesh et al., 2001, Nio et al., 2003, Tucker and
Rabie, 1980) is often credited as a factor associated with spectacle independence in these eyes
though a study (Elder et al., 1996) reported no difference between phakic and pseudophakic
groups. Other factors suggested for the improved depth of focus in the pseudophakic eye include
smaller pupil size (Tucker and Rabie, 1980), induced spherical aberration (Marcos et al., 2005),
and a combination of astigmatism and myopia (Huber, 1981).
5.2 Objective
The importance of DoF in extending the range of near vision is well established. Its effect is
more evident and arguably of greater importance in pseudophakic eyes, implanted with
conventional intra-ocular lenses (IOL), as no accommodation is available due to the replacement
of the natural crystalline lens with a fixed-focus device. Whether theoretically predicted or
clinically measured, DoF is insufficient for practical near vision. When the target near acuity is
approximately N5 point at 40 cm (Wolffsohn et al., 2006), a post-operatively emmetropic
pseudophakic eye is typically expected to require about +2.50 D of near correction for clinically
acceptable near vision.
Near vision outcomes after implantation of a single-element AIOL (1E-AIOL) (Claoue, 2004,
Cumming et al., 2001, Kuchle et al., 2004, Mastropasqua et al., 2003) are reported to be higher
than expected from theoretically predicted accommodation (Ho et al., 2006, Langenbucher et al.,
2004, Nawa et al., 2003). Clinical intuition would attribute such improved performance to an
extension of the range of near vision due to increased DoF. A press article have attributed the
improved near performance of one AIOL (Crystalens) to an improvement in depth of focus
caused by a posterior shift of the nodal point of the eye relative to its position in the phakic eye
or in an eye implanted with a standard IOL (Coombes, 2006). The shift of the nodal point is
assumed to be caused by the more posterior position of the AIOL. However, to date, no scientific
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Depth of Field
literature exists to evaluate or confirm the influence of various design and implementation
factors on the DoF of the AIOLs. Such knowledge would also be useful for optimising the near
vision performance of AIOLs. With an aim to further enhance near vision in the pseudophakic
eye, this chapter investigates the scope of improving the DoF of the AIOLs.
The major objectives of this study are to find:
1) if pseudophakic accommodation has any influence on the DoF
2) if DoF may be improved with a design modification of AIOLs that would enhance near
vision by supplementing the accommodative response in a pseudophakic eye, and
3) if the surgical implant position of the AIOL affects the DoF of the eye
Analytical and computational methods are employed to understand the phenomena and quantify
the DoF for various conditions including types of AIOLs (single or two lens elements), amount
of lens element translation, power combinations of lens elements and implantation depth (anteroposterior positioning relative to the pupil).
5.3 Methods
5.3.1 Analytical: General Equation for DoF
The derivation of equations and determination of the limit of the retinal blur circle diameter and
of the size and location of exit pupil are based on the diagram illustrated in Figure 5.1. From
similar triangles ABO and XYO
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Depth of Field
P
=
0=
(l o)
and
dl
− .................................................................................................................... (5.1)
here d is the diameter limit of the retinal blur circle, l is the distance from the exit pupil to the
retina, o is the distance from the outside focus to the retina and P is the exit pupil diameter given
by the equation P = pm where p is the real pupil diameter and M is the pupil magnification.
Similarly from congruent triangles ABI and YXI we get
i=
dl
+
..................................................................................................................... (5.2)
where i is the distance of inside focus from the retina
Figure 5.1: Geometrical representation used to calculate the position and size of the exit pupil,
limit of blur circle size and DoF. I & O represent the inside and outside foci respectively, the
distance IO is the linear extent of depth of focus. XY represents limit diameter of retinal blur
circle.
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Depth of Field
From Equations 5.1 and 5.2, it is clear that the distances from the retina to the inside and outside
foci are not equal. The total depth of focus (į) is obtained by summing these two distances i.e.:
i o .
On substitution we get:
δ=
2Pdl
2
− 2 ................................................................................................................ (5.3)
Equation (5.3) gives the linear range of clear vision straddling the image plane i.e. the depth of
focus. From the Figure 5.1, the dioptric depth of focus can be derived as:
DoF =
n
n
−
−
+
where n is the refractive index of aqueous.
Substituting i & o from Equations 5.1 & 5.2, we get:
DoF =
2nd
................................................................................................................. (5.4)
It is clear from this equation that depth of focus is directly proportional to the retinal blur circle
diameter (d) and inversely proportional to the distance (l) from the exit pupil to the retina
(effectively, approximately the vitreous chamber depth), and to the exit pupil diameter (P). The
position and size of the exit pupil is related to the magnification of the AIOL which is a function
106
Depth of Field
of the power combination of the AIOL elements and translation of the optics (pseudophakic
accommodation). In the following section, we will examine the effect of these factors.
5.3.2 Computation Method
The eye-AIOL model (i.e. eye implanted with an AIOL) used here resembles the one described
in Chapter 3. Briefly, a single or two thin-lens elements replace the crystalline lens of the
Navarro model eye to simulate a single-element (1E) or two-element (2E) AIOL. For the 1EAIOL, the power of the AIOL lens element was selected so that at the initial implantation
position the eye is rendered emmetropic. The power of the front element in 2E-AIOL was varied
from +20.0 D and +40.0 D in 5 D steps with the power of the back element calculated to return
the model eye to emmetropia at the initial position and implantation depth of the AIOL. This
allows the effect of the power combination of the lens elements to be tested.
The effect of implantation depth, defined as the distance from the pupil plane to the front
element of the AIOL, was studied by varying the position of the front lens element from
0.00 mm (minimum) to 2.00 mm (maximum) behind the pupil. The initial (minimum) and final
(maximum) separations between the lens elements of 2E-AIOL were 0.00 mm and 2.00 mm
respectively. The distance from the back lens element of the AIOL to the back reference plane
(see Figure 3.1) was varied to maintain a constant total eye axial length of 23.94 mm, which is
the axial length of the emmetropic model eye with natural crystalline lens.
5.3.3 Computation of DoF
In this calculation, the DoF is defined as the dioptric difference between the vergences,
calculated at the anterior corneal surface, of rays emanating from the outside and inside focal
points. Mathematically it is expressed as:
107
Depth of Field
DoF = Lo − Li .............................................................................................................. (5.5)
where Lo and Li are the output vergences at the anterior corneal plane for the rays originating
from the outside and inside focal points respectively.
Vergence calculations along each interface in the reference eye and the eye-AIOL model were
carried out in Excel (MS Office 2007). In this process, the diameter of the retinal blur circle (d)
of the reference eye corresponding to a DoF of 0.5 D was first determined using the “Solver”
add-in analysis tool. Though a wide range (± 0.01 D to ± 1.85 D) of DoFs for normal phakic eyes
has been reported (Wang and Ciuffreda, 2006, Bahr, 1952), 0.5 D represents approximately an
average DoF for a 4 mm diameter pupil (Campbell, 1957, Ogle and Schwartz, 1959, Wang and
Ciuffreda, 2004). The blur circle diameter and the positions of the inside and outside foci
respective to the back reference plane of the reference eye were then calculated from equations:
InsideFocus( I ) = Tv +
OutsideFocus(O) = Tv −
dl
+
................................................................................. (5.6) and
dl
−
.................................................................................... (5.7)
where Tv is the vitreous chamber depth (distance from the back reference plane to the retina), d is
the diameter of the retinal blur circle, l is the distance from the exit pupil to the retina and D is
the exit pupil diameter.
Starting from the inside and outside foci, vergence was calculated using a step-wise convergence
method within Solver (with tangent estimates, forward derivatives and Newton search algorithm)
for various accommodation levels of the reference eye and of the eye-AIOL model with the
experimental settings described above (which includes types of AIOL, power combination of the
lens elements, implantation positions and the accommodated states of the AIOL). From these
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Depth of Field
values, the output vergence of the ray emanating from the anterior corneal plane was determined
and DoF was calculated using Equation 5.5.
5.3.4 Computation of Effective DoF (eDoF)
In the vision science laboratory environment, DoF measurements are carried out by taking into
consideration additional factors such as object and retina image size of the test target to arrive at
the true value of DoF. However, in the clinic, a more typical method for establishing the range
of near vision for a patient is to direct the patient to view a small target on a near-point chart. The
chart is then moved away from and towards the patient’s eye until the chosen target letter is no
longer legible to establish the range of clear near vision. This method, also called push-up test, is
influenced, among other factors, by the magnification of the eye and any optical devices
involved.
Assuming all other parameters remain constant, an AIOL that produces a higher image
magnification would produce a smaller blur size relative to the overall size of the object. Thus,
for the same target letter viewed, the AIOL with the higher magnification would exhibit an
apparently greater range of near vision since the higher magnification means the consequently
larger retinal image would need to be blurred to a greater amount before the target letter is no
longer legible.
In order to portray this aspect of clinical estimations of a range of near vision, we derived a
parameter called the “effective DoF” (eDoF) which is calculated using an adjusted retina blur
size limit that takes into account the magnification of the AIOL. Since the retina blur size limit
required to affect image resolution would be directly proportional to the retinal image
magnification; the adjusted retina blur size limit is calculated as:
d' dM ....................................................................................................................... (5.8)
109
Depth of Field
Where d’ is the adjusted blur circle diameter; d is the actual diameter of the blur circle; and M is
the magnification of the eye given by:
M=
L' M
LM
................................................................................................................... (5.9)
where LƍMis the product of all the vergences after transmission through the optical interfaces of
the system and LM is the product of all the vergences of rays incident at the interfaces of the
system
Figure 5.2: Basic schema of the backward ray trace starting from inside (solid line) and outside
(dotted line) retinal focal positions through the two AIOL (F1 and F2) elements and finally
emanating from the corneal plane. DoF is defined as the difference between the vergence
starting from the outside retinal focus to that starting from the inside retinal focus at the corneal
plane. The linear difference between inside and outside retinal foci is the total range of linear
depth of focus.
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Depth of Field
5.4 Results
5.4.1 Analytical Results
As discussed in Chapter 4, we can apply the matrix method of paraxial optics to calculate the
magnification (exit pupil diameter) and position of the exit pupil. The translation matrices
involved are the space between the pupil (Tf) and the front element and the space between the
lens elements (Ts). The refraction matrices involved are those representing the front (RF1) and
rear (RF2) elements. If the front element is located at distance f from the pupil, the ray exiting
from the rear element is given by:
0º ªθ / nº
ªθ ' / n'º
ª 1
« h' » = S .« f / n 1».« h » ..................................................................................... (5.10)
¬
¼
¬
¼¬
¼
where; ș and h are the ray angle and height of the ray from the edge of the pupil, n is the
refractive index of the media and S is the system matrix given by:
S = RF 2 .Ts .RF1
From Chapter 4 Equation (4.7), the lateral magnification of the pupil is given by:
LM = 1 + Tv [ S − f ( F1 S − 1)] − F1S .............................................................................. (5.11)
where Tv is the distance between the rear element and the retina (vitreous chamber depth).
Now, the position of the exit pupil is given by:
l
tan .h
111
Depth of Field
The tangent can be approximated by the angle itself for small angles (paraxial case). With this
simplification and replacing h = p/2 (Ray height h here is the semi diameter of the pupil) we
obtain:
l=
p2
{( F1 S − 1)[ F2 ( F1 S − 1)]}
.................................................................................... (5.12)
Finally, Equation (5.3) for the linear depth of focus (į) becomes:
δ =−
d
( F2 F1 1 ){Tv [ S
1)]}
1( f
2
2
2
2{d − p [Tv ( S − fσ 1 ) − σ 1 ] }
................................................................... (5.13)
1
where σ 1 = ( F1 s − 1))
For simplicity, the angle of incidence ray with the optical axis is considered zero.
Similarly, replacing l and P in Equation (5.4) we obtain
DoF ( D) = −
8dn
p σ 1 ( F2 − F1σ 1 ){Tv [ s − σ 1 ( f + 1)]} ....................................................... (5.14)
2
For pseudophakic accommodation, the space between the elements (s) increases. Hence,
differentiation of Equation (5.14) with respect to s gives an expression which corresponds to the
rate of change in DoF as a function of the translation (accommodation). The exact expression is
rather protracted, but we may ignore the distances s and f (these quantities being sufficiently
small) to get an approximate expression which is:
112
Depth of Field
8F dn ª
F2 º
∂DoF
≅ lim s → 0 ≅ − 3 1
«2 +
»
∂s
p ( F1 + F2 ) ¬ ( F1 + F2 ) ¼
.................................................... (5.15)
Similarly, when Equation (5.14) is differentiated with respect to F1, it gives the relation of DoF
with power of the AIOL elements. An approximate equation for the rate of change in DoF with
the change in power of the front element is given by:
DoF
8dn
≅ lim s → 0 ≅ − 3
∂F1
p ( F1 + F2 ) 2 .......................................................................... (5.16)
5.4.2 Computational Results
The results for DoF, angular magnification and effective DoF with different depths of
implantation, translations and combinations of lens element power are shown in Figures 5.4 to
5.6. The rates of change of DoF with implant depth and accommodation are summarised in
Tables 5.2 and 5.3. In all cases, the changes in DoF were very small. For all practical purposes,
DoF can be assumed to be constant.
It can be observed from these equations that for certain combinations of AIOL elements power,
DoF decreases with accommodation (translation of the optics) whereas it increases for others.
For example, when the front element power is +40.0 D, the rear element power is approximately
-20.0 D; DoF clearly decreases with accommodation for this combination with an approximate
rate of 0.0013 D per mm translation of the optics. Similarly, when the front element power is 20.0 D, the approximate power of the rear element is +40.0 D; in this case DoF increases with an
approximate rate of 0.003 D per mm translation of the optics. In both cases, the magnitude of the
change in DoF is extremely small. For all practical purposes, DoF can be assumed to be constant.
The rate of change in DoF for various power combinations are shown in Figure 5.3.
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Depth of Field
Table 5.2: Summary of the change in depth of focus with implant depth of AIOL, accommodation
and power combinations of the elements for 2E-AIOL. mD = millidioptre which is one
thousandth of a dioptre. 1E-AIOL- single element accommodating IOL, 2E-AIOL-dual element
accommodation AIOL.
Change in Depth of Focus with
Change in
Implant Depth (mD/mm)
Accommodation
(mD/mm)
Front element moving
Back element moving
2E-AIOL
Front
Front
Front
+40D
+30D
+20D
0.099
0.101
0.103
-0.052
-0.059
-0.067
-0.042
-0.044
1E-
Ref.
AIOL
Eye
0.176
-
-0.14
-0.018
-0.035
Table 5.3: Rate of change in effective depth of focus with implant depth, and accommodation and
power combination of the elements for 2E-AIOL. mD = millidioptre which is one thousandth of a
dioptre. 1E-AIOL- single element accommodating IOL, 2E-AIOL-dual element accommodation
AIOL.
Change in effective Depth of Focus
2E-AIOL
Front
Front
Front
+40D
+30D
+20D
0.00
0.00
0.00
Accommodation Front element moving -0.052
-0.055
-0.067
(mD/mm)
-0.045
-0.027
with Change in
Implant Depth (mD/mm)
Back element moving
-0.043
1E-
Ref.
AIOL
Eye
0.097
-
-0.093
-0.024
114
Depth of Field
0.007
Rate of change in DoF (D/mm)
0.006
0.005
0.004
0.003
0.002
0.001
0.00
-20
-10
0.0
10
F1 (D)
20
30
40
Figure 5.3: Approximate rate of change in DoF as a function of power combinations of the AIOL
elements. F1 – power of the front element.
5.4.2.1 Effect of Implant Depths and Power Combinations
DoF increased slightly with more posterior implantation depth for both types AIOL. However,
the effect on effective DoF is less, particularly in the 2E-AIOLs (Figures 5.4 and 5.5). The
effective depth of focus was virtually not affected by the implantation depth of the 2E-AIOL and
the effect was minimal for the 1E-AIOL (Figure 5.5).
5.4.2.2 Effect of Accommodation
Accommodation had a noticeable effect on eDoF for both types of AIOLs (Figure 5.6).
Translation of the optics (accommodation) decreased the DoF in all types of AIOLs. The effect is
reversed in the reference eye. At 3 mm implant depth, i.e. when the front element of the AIOL is
located 3 mm behind the pupil plane, the DoF of a 2E-AIOL with a front element power of
+30 D was approximately identical to that of the reference eye (with natural crystalline lens) at
115
Depth of Field
distance focus. DoF in the accommodated state decreases as the power of the front lens element
of the 2E-AIOL increases. The rates of change in DoF and eDoF with accommodation were
similar for both types of AIOL (Table 5.2). However, the rate of change for configuration 2 of
the 2E-AIOL is marginally slower. For all power combinations of the 2E-AIOL, DoF approaches
that of the reference eye as the accommodation amplitude increases (Figure 5.6).
Depth of Field for Dual-Element AIOL (Front Translating)
0.51
F1=+21D
Depth of Field (D)
F1=+25D
F1=+30D
0.50
F1=+35D
F1=+40D
Reference Eye
Distance Focus
0.49
0.5 mm Translation
1.0 mm Translation
0.48
0.00
1.00
2.00
3.00
Accommodation Power (D)
(A)
Depth of Field for Dual-Element AIOL (Back Translating)
0.51
F1=+21D
Depth of Field (D)
F1=+25D
F1=+30D
0.50
F1=+35D
F1=+40D
Reference Eye
Distance Focus
0.49
0.5 mm Translation
1.0 mm Translation
0.48
0.00
(B)
1.00
2.00
3.00
Accommodation Power (D)
Figure 5.4: Effect of front lens element power on depth of field for 2E-AIOL with translating
front (A) or translating back (B) element. Each line represents a different front lens element
power. DoF is plotted along vertical axis. Amount of accommodation power achieved with
translation is plotted along horizontal axis.
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Depth of Field
Depth of Field for Single-Element AIOL
0.51
1.0 mm Depth
Depth of Field (D)
1.5 mm Depth
2.0 mm Depth
0.50
2.5 mm Depth
3.0 mm Depth
Reference Eye
0.49
0.48
0.00
1.00
2.00
3.00
Accommodation Power (D)
(A)
Depth of Field for Dual-Element AIOL (Front Translating)
0.51
1.0 mm Depth
Depth of Field (D)
1.5 mm Depth
2.0 mm Depth
0.50
2.5 mm Depth
3.0 mm Depth
Reference Eye
0.49
0.48
0.00
(B)
1.00
2.00
3.00
Accommodation Power (D)
Depth of Field for Dual-Element AIOL (Back Translating)
0.51
1.0 mm Depth
Depthof Field(D)
1.5 mm Depth
2.0 mm Depth
0.50
2.5 mm Depth
3.0 mm Depth
Reference Eye
Distance Focus
0.49
0.5 mm Translation
1.0 mm Translation
0.48
0.00
(C)
1.00
2.00
3.00
Accommodation Power (D)
Figure 5.5: Effect of implantation depth (tf) on DoF for: 1E-AIOL (A) and 2E-AIOL with
translating front (B) or translating back (C) element. Each line represents a different
implantation depth. DoF is plotted along vertical axis. Amount of accommodation power
achieved with translation is plotted along horizontal axis.
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Depth of Field
Effective Depth of Field for Single-Element AIOL
0.51
EffectiveDepthof Field(D)
1.0 mm Depth
1.5 mm Depth
2.0 mm Depth
2.5 mm Depth
3.0 mm Depth
0.50
Reference Eye
Distance Focus
0.5 mm Translation
1.0 mm Translation
0.49
0.00
(A)
1.00
2.00
3.00
Accommodation Power (D)
Effective Depth of Field for Dual-Element AIOL (Front Translating)
0.51
Effective Depthof Field(D)
1.0 mm Depth
1.5 mm Depth
2.0 mm Depth
2.5 mm Depth
3.0 mm Depth
0.50
0.49
0.00
Reference Eye
1.00
2.00
3.00
Accommodation Power (D)
(B)
Effective Depth of Field for Dual-Element AIOL (Back Translating)
0.51
EffectiveDepthof Field(D)
1.0 mm Depth
1.5 mm Depth
2.0 mm Depth
2.5 mm Depth
3.0 mm Depth
0.50
0.5 mm Translation
1.0 mm Translation
0.49
0.00
(C)
Reference Eye
Distance Focus
1.00
2.00
3.00
Accommodation Power (D)
Figure 5.6: Effect of implantation depth (tf) on effective DoF for 1E-AIOL (A) and 2E-AIOL with
translating front (B) or translating back (C) element. Each line represents a different
implantation depth. Effective DoF is plotted along vertical axis. Amount of accommodation
power achieved with translation is plotted along horizontal axis.
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Depth of Field
5.5 Discussion
In this chapter, we evaluated the DoF of the eye-AIOL system in relation to types, implant
position and power combination using analyses based on paraxial optics. Here, DoF refers to a
static state of the eye implanted with the AIOL (i.e. for a given focal position of the AIOL with
its lens element in a fixed position) and is not to be confused with (although is often misused
thusly), for example, “range of clear vision”, which includes the effect provided by translation of
the AIOL. The results in this study are in good agreement with the values reported by Campbell
(Campbell, 1957) and Charman (Charman and Whitefoot, 1977) for normal eyes. Both authors
estimated the minimum depth of focus to be ±0.3 D under optimal conditions for a pupil
diameter of 3 mm in normal eyes. A slightly higher mean value of objective depth of focus (0.59
± 0.1 D) for a 5 mm pupil diameter was reported recently (Vasudevan et al., 2007). In the present
study, the DoF of the pseudophakic eye is consistently lower than that of the reference eye.
Subjective and objective values of the depth of focus of pseudophakic eyes reported in the
literatures are consistently higher than the values predicted in this study. A study reported a
subjective depth of focus of ±0.85 D for a 2.5 mm pupil diameter (Elder et al., 1996) in
pseudophakic eyes implanted with a spherical IOL. Another study found a value of 1.65 D for a
monofocal spherical IOL (Kamlesh et al., 2001), which is identical to that reported in another
similar study (Post, 1992). Nio et al (Nio et al., 2003) reported a mean depth of focus of 1.82 ±
0.34 D measured on the basis of contrast sensitivity as a function of defocus. One immediately
identifiable reason for the smaller values in the current study is that the simulations are based on
paraxial geometrical optics and, unlike in real eye conditions, the results are free from the
influence of other optical and physiological factors which may affect the DoF, such as the effect
of optical aberrations and magnification.
For the AIOL and eye combinations exhibiting
aberrations that depart significantly from normal values, it would be worthwhile repeating this
study by incorporating factors including point-spread function and Stiles-Crawford effect in the
analyses. In this study, the DoF also decreased with a more anterior placement of the AIOL.
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Depth of Field
In the clinical setting, comparison of the DoF of different AIOLs would most likely employ testletter types of the same size. Thus, an AIOL that provide greater magnification of the object may
give apparently a greater DoF for the same actual DoF. The concept of ‘effective’ DoF
calculated in this study attempts to take into account this factor and shows that implant position
affects DoF by only a very small amount (less than 10 mD). This ‘compensatory’ effect of
magnification seems also to apply to near vision.
Although the analytical results suggest that DoF may be improved with certain designs of
AIOLs, the computational results suggest that the DoF of an eye-AIOL combination is linked to
the magnification. However, the effects of all the tested variables are miniscule and far below
those observable clinically. Hence they are expected to bear no practical significance. The
change in DoF due to changes in implantation depth, element translation (accommodation) and
combination of lens element power within practical limits never exceeds 0.02 D. Hence the
notion of improved near visual acuity in pseudophakic eyes due to improved DoF is not justified.
The results of this study provide no support for the popular statement that DoF improves with a
more posterior implant position.
120
Depth of Field
5.6 References
Atchison, D. A., Charman, N. and Rl, W. (1997a). Subjective Depth-of-Focus of the Eye. Optom
Vis Sci 74, 10.
Atchison, D. A., Charman, W. N. and Woods, R. L. (1997b). Subjective depth-of-focus of the
eye. Optom Vis Sci 74, 511-520.
Bahr, G. von. (1952). Studies on the depth of focus of the eye. Acta Ophthalmol (Copenh) 30,
39-44.
Bradbury, J. A., Hillman, J. S. and Cassells-Brown, A. (1992). Optimal postoperative refraction
for good unaided near and distance vision with monofocal intraocular lenses. B J
Ophthalmol 76, 300-302.
Campbell, F. W. (1957). The Depth of Field of the Human Eye. J Modern Opt 4, 8.
Charman, W. N. and Whitefoot, H. (1977). Pupil diameter and the depth of field of the human
eye as measured by laser speckle. J Modern Opt 24, 6.
Claoue, C. (2004). Functional vision after cataract removal with multifocal and accommodating
intraocular lens implantation: prospective comparative evaluation of Array multifocal and
1CU accommodating lenses. J Cataract Refract Surg 30, 2088-2091.
Coombes, S. (2006). Baby Boomer Tsunami and IOL for Presbyopia, Online Press:
http://www.vision.io.csic.es/papers_pdfs/JRS_asph_sph_IOLs_preprint.pdf.
Cumming, J. S., Slade, S. G. and Chayet, A. (2001). Clinical evaluation of the model AT-45
silicone accommodating intraocular lens: results of feasibility and the initial phase of a Food
and Drug Administration clinical trial. Ophthalmol. 108, 2005-2009; discussion 2010.
Elder, M. J., Murphy, C. and Sanderson, G. F. (1996). Apparent accommodation and depth of
field in pseudophakia. J Cataract Refract Surg 22, 615-619.
Geisler, W. S. (1984). Physical limits of acuity and hyperacuity. J Opt Soc Am A 1, 775-782.
121
Depth of Field
Green, D. G., Powers, M. K. and Banks, M. S. (1980). Depth of focus, eye size and visual acuity.
Vision research 20, 827-835.
Ho, A., Manns, F., Therese and Parel, J. M. (2006). Predicting the performance of
accommodating intraocular lenses using ray tracing. J Cataract Refract Surg 32, 129-136.
Huber, C. (1981). Planned myopic astigmatism as a substitute for accommodation in
pseudophakia. J Am Intraocular Imp Soc 7, 6.
Jacobs, R. J., Smith, G. and Chan, C. D. (1989). Effect of defocus on blur thresholds and on
thresholds of perceived change in blur: comparison of source and observer methods. Optom
Vis Sci 66, 545-553.
Jiang, B. C. and Morse, S. E. (1999). Oculomotor functions and late-onset myopia. Ophthalmic
Physiol Opt 19, 165-172.
Kamlesh, Dadeya, S. and Kaushik, S. (2001). Contrast sensitivity and depth of focus with
aspheric multifocal versus monofocal intraocular lens. Can J Ophthalmol 36, 197-201.
Kuchle, M., Seitz, B., Langenbucher, A., Gusek-Schneider, G. C., Martus, P. and Nguyen, N. X.
(2004). Comparison of 6-month results of implantation of the 1CU accommodative
intraocular lens with conventional intraocular lenses. Ophthalmol 111, 318-324.
Langenbucher, A., Reese, S., Jakob, C. and Seitz, B. (2004). Pseudophakic accommodation with
translation lenses--dual optic vs mono optic. Ophthalmic Physiol Opt 24, 450-457.
Legge, G. E., Mullen, K. T., Woo, G. C. and Campbell, F. W. (1987). Tolerance to visual
defocus. J Opt Soc Am A 4, 851-863.
Lesiewska-Junk, H. and Kaluzny, J. (2000). Intraocular lens movement and accommodation in
eyes of young patients. J Cataract Refract Surg 26, 562-565.
Marcos, S., Barbero, S. and Jimenez-Alfaro, I. (2005). Optical quality and depth-of-field of eyes
implanted with spherical and aspheric intraocular lenses. J Refract Surg 21, 223-235.
Marcos, S., Moreno, E. and Navarro, R. (1999). The depth-of-field of the human eye from
objective and subjective measurements. Vision research 39, 2039-2049.
122
Depth of Field
Mastropasqua, L., Toto, L., Nubile, M., Falconio, G. and Ballone, E. (2003). Clinical study of
the 1CU accommodating intraocular lens. J Cataract Refract Surg 29, 1307-1312.
Nakazawa, M. and Ohtsuki, K. (1983). Apparent accommodation in pseudophakic eyes after
implantation of posterior chamber intraocular lenses. Am J Ophthalmol 96, 435-438.
Nakazawa, M. and Ohtsuki, K. (1984). Apparent accommodation in pseudophakic eyes after
implantation of posterior chamber intraocular lenses: optical analysis. Invest Ophthalmol Vis
Sci 25, 1458-1460.
Nawa, Y., Ueda, T., Nakatsuka, M., et al. (2003). Accommodation obtained per 1.0 mm forward
movement of a posterior chamber intraocular lens. J Cataract Refract Surg 29, 2069-2072.
Nio, Y. K., Jansonius, N. M., Geraghty, E., Norrby, S. and Kooijman, A. C. (2003). Effect of
intraocular lens implantation on visual acuity, contrast sensitivity, and depth of focus. J
Cataract Refract Surg 29, 2073-2081.
Ogle, K. N. and Schwartz, J. T. (1959). Depth of focus of the human eye. J Opt Soc Am 49, 273280.
Post, C. T. J. (1992). Comparison of depth of focus and low-contrast acuities for monofocal
versus multifocal intraocular lens patients at 1 year. Ophthalmol. 99, 1658-1663; discussion
1663-1654.
Rosenfield, M. and Abraham-Cohen, J. A. (1999). Blur sensitivity in myopes. Optom Vis Sci 76,
303-307.
Tucker, J. and Charman, W. N. (1975). The depth-of-focus of the human eye for Snellen letters.
Am J Optom Pphysiol Opt 52, 3-21.
Tucker, J. and Rabie, E. P. (1980). Depth-of-focus of the pseudophakic eye. B J Physiol Opt 34,
12-21.
Vasudevan, B., Cuiffreda, K. J. and Wang, B. (2007). Subjective and objective Depth of Focus. J
Modern Opt 54, 10.
Wang, B. and Ciuffreda, K. J. (2004). Depth-of-focus of the human eye in the near retinal
periphery. Vis Res 44, 1115-1125.
123
Depth of Field
Wang, B. and Ciuffreda, K. J. (2005). Blur discrimination of the human eye in the near retinal
periphery. Optom Vis Sci 82, 52-58.
Wang, B. and Ciuffreda, K. J. (2006). Depth-of-focus of the human eye: theory and clinical
implications. Surv Ophthalmol 51, 75-85.
Westheimer, G. (1953). The effect of spectacle lenses and accommodation on the depth of focus
of the eye. Am J Optom Arch Am Acad Optom 30, 513-519.
Williams, D. R. and Coletta, N. J. (1987). Cone spacing and the visual resolution limit. J Opt Soc
Am A 4, 1514-1523.
Wolffsohn, J. S., Hunt, O. A., Naroo, S., et al. (2006). Objective accommodative amplitude and
dynamics with the 1CU accommodative intraocular lens. Invest Ophthalmol Vis Sci 47,
1230-1235.
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Optical Design
Chapter 6
Optical Design of AIOL
125
Optical Design
TABLE OF CONTENT
6.1 INTRODUCTION ..................................................................................................... 128
6.2 LITERATURE SURVEY ......................................................................................... 130
6.2.1 Aberration in the Population............................................................................ 130
6.2.2 Effect of age on Aberration .............................................................................. 133
6.2.3 Effect of Accommodation on Aberration .......................................................... 135
6.2.4 IOL design and Posterior Capsular Opacification .......................................... 135
6.2.5 Optimum Shape of a Conventional IOL ........................................................... 137
6.3 SETTING DESIGN CRITERIA FOR AIOLS ....................................................... 139
6.3.1 Biologically acceptable AIOL shape ................................................................ 141
6.3.2 Minimum overall thickness of the AIOL system ............................................... 142
6.3.3 Design Criteria to Control Aberrations ........................................................... 146
6.4 ABERRATION GUIDED DESIGN OF SPHERICAL 1E-AIOL......................... 148
6.4.1 Spherical Aberration ........................................................................................ 149
6.4.2 Coma Aberration .............................................................................................. 156
6.4.3 Design to Control Coma and Spherical Aberration......................................... 159
6.4.4 Summary of Spherical Design of 1E-AIOL ...................................................... 161
6.5 DESIGN OF SPHERICAL 2E-AIOL...................................................................... 162
6.5.1 Spherical Aberration ........................................................................................ 163
6.5.2 Off-axis aberration: Coma ............................................................................... 171
6.5.3 Design to control coma and spherical aberration ........................................... 174
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Optical Design
6.5.4. Summary of the Spherical Design of AIOL ..................................................... 178
6.6 DESIGN OF ASPHERIC AIOL .............................................................................. 179
6.6.1 Introduction ...................................................................................................... 179
6.6.2 Aspheric Surface............................................................................................... 180
6.6.3. Contribution of Asphericity to Aberrations..................................................... 182
6.6.4 Aspheric Design of 1E-AIOLs .......................................................................... 183
6.6.5 Aspheric Design of 2E-AIOL............................................................................ 189
6.7 DISCUSSION............................................................................................................. 192
6.8 REFERENCES .......................................................................................................... 195
127
Optical Design
6.1 Introduction
The eye is a wide-aperture optical instrument and hence the design of an IOL must consider
image blurring due to aberrations (Jalie, 1978). Amid some speculations that reduction of
aberration in IOL is impractical (Holladay, 1986, Simpson, 1992) since pre- and postoperative ocular optical parameters vary greatly between individuals. Nonetheless, vision
scientists and vision care practitioners are increasingly directing their interest to methods of
reducing optical aberrations in the pseudophakic eyes (Guirao et al., 2002, Liang and
Williams, 1997). As a result a variety of IOL designs such as customised IOLs (also called
aberration-correcting IOL (Holladay et al., 2002)) and aberration-free IOLs are being
introduced.
Several design parameters may be used to control or eliminate the aberrations in the lens.
Bending factor (also called Coddington’s shape factor) which is determined by combination
of the surface curvatures, surface designs (e.g. aspheric surface) and refractive index of the
material are some of the important design parameters being employed. The bending factor
as a design criterion for conventional IOLs has been widely investigated (Atchison, 1989a,
Atchison, 1989b, Lu and Smith, 1990, Pomerantzeff et al., 1985, Wang and Pomerantzeff,
1982).
The design of an AIOL should differ from the design of a conventional monofocal IOL.
The translational AIOLs are designed to change in axial position continuously with the
dynamics of visual accommodation which leads to a continuous change in object-image
conjugate. Therefore, unlike in the monofocal IOL design which typically targets aberration
control for one conjugate ratio, the design of the AIOL need to also include maintaining
satisfactory optical performance at as many object-image conjugates as possible and
especially at the distance and near visual points.
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Optical Design
In addition to improving retinal image quality by controlling aberrations, the design of an
AIOL is also important in terms of its biological performance. Surface design, material
property and shape of an IOL have often been associated with proliferation of lens
epithelial cells to form one of the most common postoperative complications known as
posterior capsule opacification (PCO), (Born and Ryan, 1990, Oshika et al., 1998a,
Sellman and Lindstrom, 1988). Hence, the design of an AIOL need to take into
consideration both optical/visual and non-optical requirements. In the present work, while
the focus will primarily be on achieving optimum optical performance, some consideration
will also be given to designs that could address the biological requirements.
This chapter aims to investigate the spherical and aspheric designs of AIOLs that:
1)
Are effective in controlling or eliminating aberrations from the AIOL and/or from
the pseudophakic eye to enhance retinal image quality for at least two object-image
conjugates (distance and near vision) simultaneously
2)
Have a clinically acceptable form that could also impede the migration of
equatorial epithelial cells thereby retarding the formation of PCO
Though aberration coefficients and their effects interact (Lia et al., 2010, McLellan et al.,
2006), spherical and coma aberrations are the most important ocular aberrations in
determining the retinal image quality (Jalie, 1978, Smith and Lu, 1988, Applegate et al.,
2003). Therefore, this chapter focuses on improving AIOL designs to control these two
aberrations. Seidel aberration theory was employed in calculating the aberrations.
With a ‘conservative’ notion that the aberration profile of a pseudophakic eye should
correspond to that of the phakic eye profile preoperatively (Guirao et al., 2002, Liang and
Williams, 1997), understanding of the aberration profile in a population (and in an average
eye) is crucial. This underpins criteria for a desired lens design. Furthermore, design
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Optical Design
parameters that are reported to be associated with PCO development will also be reviewed
which can be set as constraints in the study.
Therefore, this chapter first briefly reviews the literature reporting ocular aberrations in a
population and lens designs found to be effective in controlling PCO. With these criteria in
place, the chapter proceeds by investigating the potential designs for optimum optical
performance. Throughout the chapter, unless specified, a 4 mm pupil diameter is used in
calculating the aberrations. Aberrations are presented in unit of waves with a reference
wavelength of 5893 microns (Fraunhofer D-line). Aberration data obtained from the
literatures are also converted to the unit waves when required for consistency.
6.2 Literature Survey
6.2.1 Aberration in the Population
The majority of normal eyes exhibit with positive spherical aberration (Applegate et al.,
2007, He et al., 2002, Liang and Williams, 1997) mostly originating from the anterior
surface of the cornea (Artal et al., 2002a, Artal and Guirao, 1998, Guirao et al., 2002). Due
to its gradient distribution of the refractive index (GRIN) and aspheric surfaces (Smith,
2003, Atchison and Smith, 1995, Nakao et al., 1969), the crystalline lens produces negative
spherical aberration which partially compensates for the positive aberration from the cornea
(Artal and Guirao, 1998, Smith et al., 2001, Marcos et al., 2008) to result in a lower total
but still positive ocular spherical aberration. In a pseudophakic eye, the advantage of this
compensating effect becomes unavailable due to the loss of the natural crystalline lens and
its refractive index gradient. Additionally, a conventional IOL also produces positive
spherical aberration which adds to the aberration produced by the cornea (Atchison, 1989a,
Smith and Lu, 1988) to result in an elevated total ocular spherical aberration.
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Optical Design
A number of studies have been published reporting on ocular aberrations (Charman, 1991,
He et al., 1998, Liang et al., 1994, Mrochen et al., 2000, Walsh et al., 1984). These reports
suggest that a large inter-subject variability of the aberrations exists in the population
(Guirao et al., 1999, Howland and Howland, 1977, Porter et al., 2001, Smirnov, 1961,
Walsh et al., 1984). Some population studies have demonstrated that young children (age 5
to 15) exhibit negative total spherical aberration (Martinez, PhD thesis; Phillip, PhD thesis
– University of New South Wales); whereas the aberration shifts towards the positive
direction with age (Atchison and Smith, 1995). Anatomical inconsistency of optical
components of the eye is thought to be a main source of this variability. Population
aberration data obtained from a few literatures are presented in Table 6.1.
Understanding optical aberrations of a normal eye and its individual components,
particularly of the crystalline lens, is one of the important requirements for designing
aberration-correcting IOLs. The contributions to aberrations from each ocular component
are summarised in the following section.
6.2.1.1 Aberration from the Cornea
Contribution of the cornea in ocular aberrations has been widely investigated (summarized
in Norrby et al, 2007). Normal corneal aberrations are largely based on its surface profile.
Corneal shapes can vary widely among individuals, but based on the available data in the
literature (Liou and Brennan, 1997), an average corneal spherical aberrations range
between 0.25 to 8 waves at 6 mm pupil diameter. The majority of normal eyes are affected
with positive spherical aberration (Applegate et al., 2007, Artal et al., 2002a, Brunette et
al., 2003, He et al., 1998, He et al., 2002, Liang and Williams, 1997, Liou and Brennan,
1997, Tabernero et al., 2007a) mostly originating from the anterior surface (Artal et al.,
2002a, Artal and Guirao, 1998, Brunette et al., 2003, Guirao et al., 2002, Millodot and
Sivak, 1979; ). The posterior surface produces negligible negative spherical aberration
(Rengstorff, 1985). The differences in refractive index at the cornea-air interface are
significantly higher than at the cornea-aqueous, lens-aqueous, and lens-vitreous interfaces
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Optical Design
which optically establish the anterior corneal surface as the most significant contributor for
total ocular aberration.
6.2.1.2 Aberration from the Crystalline Lens
The complete optical properties of the human crystalline lens are not yet fully known;
therefore prediction of an exact contribution of the lens to total ocular aberration is
difficult.
One method of elucidating the crystalline lens contribution is to find the
difference between the corneal and total aberrations of an eye. Another method is to
physically or optically eliminate the effect of the corneal contribution, by immersing it in
water and measuring the residual aberrations (Millodot and Sivak, 1979).
Clinical reports suggest that the spherical aberration of the aphakic eye is identical to the
corneal aberration alone (Barbero et al., 2002) indicating that the crystalline lens is the
major source of internal aberration (Cheng et al., 2004, el-Hage and Berny, 1973). As the
posterior surface of the cornea contributes only a small amount of internal aberration, it
would be reasonable to consider the whole internal aberration as a near estimate of the
lenticular aberration.
The crystalline lens, by virtue of its gradient distribution of the refractive index (GRIN) and
curvature profile (aspheric surfaces) (Smith, 2003, Atchison and Smith, 1995, Nakao et al.,
1969), produces negative spherical aberration which partially compensates the positive
aberration from the cornea (Artal and Guirao, 1998, Smith et al., 2001). The compensated
amount of aberration is reported to range from 30% to 85% of the corneal aberration
(Barbero et al., 2002, Smith et al., 2001, Kelly et al., 2004) and the amount of
compensation reduces with age (Artal et al., 1993, Artal et al., 2003, Calver et al., 1999,
Lopez-Gil et al., 2008, Oshika et al., 1999).
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Optical Design
Table 6.1: Zernike coefficients for Spherical aberration of the population reported in some
earlier studies. The values should be viewed with caution as the methods of measurement
may vary. The age range of the subjects is also not consistent.
Sample
(n)
75
Age of Subjects
(Yrs)
18 – 69
(mean=43.5)
Pupil
(mm)
6
15 young
24.2±3
6
15 older
68±5
Cheng et al.,
2004
76
21 – 40
(mean = 24.8±4)
5
He et al., 2003
45
9 - 29
6
Author
Amano et al.,
2004
Calver et al.,
1999
S.A. (ȝm)
Conclusion/Comment
0.175
Coma and SA increased with
age. No change in SA with
age but did coma.
Wavefront aberration
increased with age
0.095±0.13
0.175±0.13
0.065±0.083 Spherical aberration changed
by -0.044ȝm/D of
accommodation
0.06±0.22 Corneal SA compensated by
internal aberration, coma and
HOA are additive, coma
increased with accommodation.
Kelly et al., 2004 30
20.5 (mean years)
Llorente et al 24 myope
23 – 40
2004
22 hyperope
Porter
2001
et
0.132 ± 0.017
0.10 ± 0.13
0.22 ± 0.17
21 – 65 (mean=41) 5.7
0.138±0.103
Salmon and van 2560
de Pol, 2006
Mean age between 6
24±5 and 47±14
0.128±0.096
Thibos
2002
22 – 35
0.150
et
al., 109
6
6
al., 200
6
Inter-eye
aberrations,
particularly spherical, are
highly correlated
Spherical aberration increased
with age with correlation
coefficient 0.25
Population average of HOA
nearly zero except for
spherical aberration
6.2.2 Effect of age on Aberration
The profile of ocular aberrations changes with age (Artal et al., 1993, Brunette et al., 2003,
Calver et al., 1999, Guirao et al., 1999, He et al., 2002, McLellan et al., 2001, Artal et al.,
2002a, Oshika et al., 1999, Amano et al., 2004, Fujikado et al., 2004, Wang and Koch,
2004). The changes in the eye’s optical parameters such as increased central thickness of
the lens and associated decrease in anterior chamber depth (Dubbelman et al., 2001),
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Optical Design
decreased radii of curvatures of the unaccommodated lens (Brown, 1974, Dubbelman and
Van der Heijde, 2001), alterations in distribution of gradient index of the lens (Jones et al.,
2005) and decreased corneal curvature (Hayashi et al., 1995) contribute significantly. In
general, age influences all types of aberrations to some degree (Amano et al., 2004,
Atchison and Markwell, 2008). While the corneal spherical aberration is minimally affected
by age (Oshika et al., 1999, Artal et al., 2002a), the crystalline lens loses its compensatory
property (Artal et al., 2001, Artal et al., 2002a, Amano et al., 2004). Consequently the
corneal aberration becomes progressively more under-compensated and hence the total
spherical aberration increases towards the positive direction with age. One study found
slopes of linear regression for corneal and total eye aberration of 0.0013 and 0.011 ȝm per
year (Artal et al., 2002a) respectively. Amano and co-authors (Amano et al., 2004) found
an increase in coma with age due to an increase in corneal coma. Atchison and Markwell
(Atchison and Markwell, 2008) in their emmetropic subjects found significant changes in
coefficients for horizontal coma with age. Figure 6.1 shows the aberration as a function of
age reported by these Authors.
Figure 6.1: Atchison & Markwell (2008): Corneal, Internal and Ocular coefficients C(3, 1)
and C(4, 0) as a function of age (5 mm pupils). (I am most grateful and acknowledge these
authors for giving me their written permission to reproduce this figure here)
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Optical Design
6.2.3 Effect of Accommodation on Aberration
The aberrations of an optical system can be expected to change with misalignment or
change in position of the optical components within the system. During accommodation,
the position and shape of the crystalline lens changes; simultaneously anterior chamber
depth decreases.
In spite of significant attempts to understand optical performance of the eye in relation to
accommodation (Cheng et al., 2004, Atchison et al., 1995, He et al., 2000, Lopez-Gil et al.,
2008, Ninomiya et al., 2002, Dubbleman et al, 2001) it is still not well characterised. While
some studies (He et al., 2000, Atchison et al., 1995) did not find a clear trend in the amount
or direction of change in aberrations with accommodation, others found a negative shift
(Cheng et al., 2004, Lopez-Gil et al., 2008). As there is little alteration in the corneal
aberrations during accommodation (Artal et al., 2002b, Buehren et al., 2003, Amano et al.,
2004), any major change arises solely from the crystalline lens. One study (Cheng et al.,
2004) reported decreased spherical aberration with accommodation with a linear regression
slope of -0.0435 micron per dioptre of accommodation. This study concluded that given the
aberration value for the resting state of the eye, the spherical aberration for any level of
accommodation can be predicted within an accuracy of ±0.085 ȝm 95% confidence
interval. Figures 6.2 and 6.3 show the change in aberration with accommodation in phakic
eyes reported in the literature.
6.2.4 IOL design and Posterior Capsular Opacification
Though, beyond the scope of this thesis, the physiological effect of the shape of an IOL
cannot be ignored as this has been associated with post-operative posterior capsule
opacification (PCO). PCO occurs in up to 50% of eyes following cataract surgery (Apple et
al., 1992). A plethora of techniques have been proposed to control formation of PCO; and a
variety of IOL designs have been recognised as promising (Coombes and Seward, 1999,
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Optical Design
Nishi et al., 2007). A lower incidence of PCO has been reported in eyes with an IOL made
of acrylic materials, compared to silicone and PMMA lenses (Ursell et al., 1998); however
others found no significant difference in PCO implanted with various IOLs (Mester et al,
2003). A convex posterior surface affords a larger area of contact with the posterior capsule
which may minimise the rate of equatorial lens epithelial cell (LEC) migration (Born and
Ryan, 1990, Sellman and Lindstrom, 1988, Sterling and Wood, 1986). Similarly, a sharp
posterior edge of the lens has also been found to be effective in controlling LEC migration
(Nishi et al., 1998, Oshika et al., 1998b). A laser ridge, also known as “Hoffer ridge” in an
IOL placed on the optic against the posterior capsule was also found beneficial in reducing
PCO (Maltzman et al., 1989) and a plano-convex IOL with plane surface facing posteriorly
has been reported to have a similar effect (Yamada et al., 1995). In summary the lens optics
with squared, truncated and thick edges are found to effective in controlling the LEC
migration compared to the round or tapered edge optics (Peng et al., 2000, Gerd U.
Auffarth et al., 2003, Nishi et al., 2004, Nishi et al., 2007). These observations can inform
the design of AIOLs.
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Optical Design
Figure 6.2: Lopez-Gil et al (2008) Change in RMS wavefront error for spherical
aberration as a function of accommodation. (A) 19–29yrs (B) 30–39yrs, (C) 40–49yrs, (D)
50–60yrs. (I am most grateful and acknowledge these authors for giving me their written
permission to reproduce the figure)
6.2.5 Optimum Shape of a Conventional IOL
Optically, shape as surface profiles and lens forms, is an important design parameter in
controlling aberrations. For a given power of a lens, the aberrations depend upon the
distribution of curvature between the two surfaces (Smith and Atchison, 1997). Spherical
and coma aberrations are largely dependent on the shape of the lens (Atchison, 1989b).
Though practically insignificant, except with very steeply shaped surfaces, the lens shape
also influences the amount of light loss due to reflection on lens surfaces (Lu and Smith,
1990).
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Optical Design
Figure 6.3: He et al (2000): Spherical aberration as a function of accommodation for eight
subjects (a) and the average (b). Different symbols represent different subjects, the error
bar in panel b shows the ±1 Standard Error of Mean (I am most grateful and acknowledge
these authors for giving me their written permission to reproduce the figure)
Most of the conventional IOLs are made in one of three forms (Atchison, 1989a, Tabernero
et al., 2007b): convex-plano (curved surface towards the cornea), plano-convex (convex
surface towards the retina) and biconvex. The shape of the lens has been one of the
important components investigated by IOL designers in attempts to optimise the optical
image performance. Various design approaches have been proposed (See Table 6.2).
However, it could be argued that optimising the post-operative refractive state with the high
level of precision required to control aberrations is currently not feasible. (More about this
in Chapter 7 on misalignment) Importantly, model eyes used by the designers in their
experiment differ from each other as well as from actual individuals.
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Optical Design
From a biological point of view, though the importance of IOL geometrical designs in
preventing PCO has been widely accepted, there is no consensus attained regarding a
specific range of designs or even design features to be used. However, plano or convex
posterior surfaces are frequently reported to be beneficial. A convex posterior surface
affords larger area of contact with posterior capsule which may minimise the rate of
equatorial epithelial cell migration through mechanical contact (Born and Ryan, 1990,
Sellman and Lindstrom, 1988, Sterling and Wood, 1986, Peng et al., 2000, Gerd U.
Auffarth et al., 2003, Mester et al, 2003).
In recent days, interest has grown in eliminating or correcting aberrations in pseudophakic
eyes (Altmann, 2004, Bellucci and Morselli, 2007). Studies have shown that the aberration
in a pseudophakic eye cannot be eliminated with spherical surface of an IOL. Aspherising a
surface of an AIOL provides an additional degree of design freedom (Lu and Smith, 1990,
Atchison, 1991) which has been the only mean to achieve the goal. Aspherising one or both
surfaces of an AIOL may produce negative or positive aberrations as desired. Optimum
aspheric values to eliminate spherical aberration in the pseudophakic eye implanted with a
conventional IOL have been proposed (Atchison, 1991, Lu and Smith, 1990, Smith and Lu,
1988, Holladay et al., 2002). Although improved retinal image quality has been reported
(Altmann, 2004, Bellucci and Morselli, 2007), the popularity of aspheric IOLs is limited
only to the developed countries due to technical difficulties in the manufacturing processes
and the relatively higher cost. But with the rapid improvement and reduction in cost of
modern manufacturing technology, aspheric designs may be expected to be the norm in
future devices.
6.3 Setting Design Criteria for AIOLs
In many aspects, both conventional and accommodating IOLs share some similar design
principles. For instance, a design of AIOL should also aim to minimise LEC migration and
controlling aberrations. In addition, there are some constraints to be considered in designing
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Optical Design
AIOL which are practically irrelevant in a conventional IOL; maintaining optimal image
quality for more than one viewing distance is a main one. Since, no studies have been
reported investigating the optimum design of AIOLs, some criteria have to be defined prior
to the actual design process. In this section, we will first set some important criteria that
need to be satisfied.
Table 6.2: Published bending factors of conventional IOL. Bending factor is a common
design parameter of a lens which determines the distribution of the curvatures between two
surfaces. Optical importance of the bending factor is discussed later in this chapter in more
details.
Designer
Shape factor
Jalie 1978
-1.0
Corneal
Asphericity
0.000
Rosenblum and Shealy 1979
-0.5
0.000
Wang and Pomerantzeff, 1982
+1.1 and -0.6
-0.659
Pomerantzeff et al 1985
+1.1
-0.338
Smith and Lu, 1988
>+1.1 and <+1.1
-0.260
Atchison, 1989a
+1.1 – (-0.9 - -3.4)
-0.260
Atchison, 1989b
+0.5
-0.260
Tabernero, 2007b
-1.0
-0.260
Three basic criteria for the optimum design of an AIOL in terms of physical shape need to
be considered:
1) A biologically acceptable form that is effective in preventing PCO formation and
that does not cause constant iris irritation, hence avoids persistent intraocular
inflammation.
2) A low overall thickness of the AIOL system is desirable such that sufficient space
(axial distance) is available within the capsular bag for translation of the optics.
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Optical Design
3) An appropriate design (shape, refractive index, etc) that can effectively control or
eliminate select aberrations. A key design tool in this pursuit is the manipulation of
the bending factor of the optical elements. The bending factor is defined as the
distribution of curvatures between two surfaces of a lens that determines physical
shape and several optical characteristics. This chapter focuses on this, the optical
performance aspect of design.
These three criteria are discussed in the following sections.
6.3.1 Biologically acceptable AIOL shape
From the biological perspective, the design of an AIOL should not differ greatly from the
design of a conventional IOL except that for the AIOL, the anterior-most surface (front
surface of the front element in a 2E-AIOL) also needs special attention. This is because this
element must be free to move continuously with the effort of accommodation and may
come into contact (indirectly via the capsule, or directly through the capsulorhexis) with the
posterior surface of the iris. If the anterior edge of the front element is square or sharp, it
may represent a constant irritant to the iris causing persistent intraocular inflammation. For
this reason, a convex anterior-most surface with smoothly convexly curved edge is
preferred. The posterior-most surface (back surface of the rear element) should also
preferably be a convex surface to comply with the concept of “no space, no cells” which
states that close proximity between the lens surface and the posterior capsule is effective in
controlling the proliferation of equatorial epithelial cells and formation of PCO (Apple et
al., 1992). However, as discussed in Section 6.2.4 (biological effect), a plano posterior
surface may be a good alternative.
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Optical Design
6.3.2 Minimum overall thickness of the AIOL system
From the accommodative performance standpoint, the elements should be free to translate
as much as possible to maximise accommodative amplitude (Ale et al., 2010, Ho et al.,
2006). The amount of translation is predominantly governed by the degree to which the
accommodative apparatus in the eye remains intact post-operatively, the mechanical design
of the translating components (haptics) of the AIOL, and the space available for the
movement of the lens elements. Thickness of a 1E-AIOL should not be an issue as space
available in the posterior chamber is normally sufficient for a single lens (1E-AIOL).
However, antero-posterior space may become a constraining factor in relation to a 2EAIOL. Therefore we will concentrate our discussion on the 2E-AIOL.
Given a fixed space available in the capsular bag, the space available for translation within
the capsular bag is dependent on the total thickness of the 2E-AIOL system which is
governed by: a) thickness of the AIOL elements and b) the space between the elements.
There are three important aspects that determine the minimum thickness of a 2E-AIOL:
1) A 2E-AIOL comprises of two lenses; hence more space is occupied by the lens
elements themselves, as compared to a 1E-AIOL.
2) Because of their relatively high power, the elements are likely to be relatively thick
to accommodate the curvature requirements.
3) The extent to which the two facing surfaces of the two elements ‘couple’ when in
their nearest proximity position. e.g. if the two elements have concave surfaces
facing, then there will be a ‘dead’ space between the two elements that adds to the
total device thickness but which cannot contribute to translation.
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Optical Design
Figure 6.4: Basic schema of a 2E-AIOL.C11- front surface curvature of front element, C12back surface curvature of front element, C21 front surface curvature of rear element and
C22 back surface curvature of rear element.
The thicknesses of the lens elements are mainly controlled by their dioptric powers,
refractive index, diameter and surface geometry (asphericity). The space between the
elements is exclusively governed by the curvatures of the adjacent (facing) surfaces i.e. the
back surface of the front element (C12) and the front surface of the rear element (C21).
Optimum combination in terms of useful translation space is obtained when these
curvatures are equal (C12=C21), that is the sagittal profiles of both surfaces are equal and
opposite in sign (assuming that the two elements have equal meridional diameters) enabling
perfect coupling.
When C12 is steeper than C21, there will be contact at the centre of the two surfaces creating
an undesirable space in the periphery. The reverse combination creates a space at the centre
(peripheral contact). Using the sag formula, one can easily calculate the space between the
elements at the centre and periphery for a given curvature of the surface.
Using algebraic distances, according to the figure:
p = PP ' = PC + CC ' + C ' P ' = − s1 + d + s 2 .....................................................................(6.1)
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Optical Design
where s1 and s2 are sags of the front and the back curves respectively, and d is the axial
distance between the two surfaces
The sag (s) of a surface is defined by s = r − r 2 − h 2 for a convex surface (r>0) and
s = r + r 2 − h 2 for a concave surface. Substituting s1 and s2 we get, assuming two convex
surfaces:
p = (r2 − r1 − d ) + r12 − h 2 − r22 − h 2 ..........................................................................(6.2)
and, assuming two concave surfaces:
p = (r2 − r1 − d ) + r2 − h 2 − r1 − h 2 .........................................................................(6.3)
2
2
For a given radius of one of the surfaces, Equation 6.2 or 6.3 can be used to find the radius
of curvature of the other surfaces for any desired amount of central and peripheral spacing.
Figure 6.5: Illustration of the spacing between elements. s1 and s2 are the sags of the back
surface of the front element and front surface of the rear element respectively, PP’ is the
distance between two points at the periphery, CC’=d is the central spacing and h is the half
diameter of the AIOL elements.
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Optical Design
Figure 6.6 outlines possible combinations of the front and the rear elements of a 2E-AIOL.
Designs that are most desirable are those that maintain minimum space between the
elements and at the same time maintain the convex anterior surface of the front element to
minimise detrimental biological effects. These designs can be summarised as follows in
terms of the bending factor:
a) For optimal biological effect:
Greater than -1 for front element (given no rounding of the edges). In this range, the
anterior surface of the front element (anterior-most surface for the AIOL) is convex and
entire bending factors for rear element. The posterior-most surface can have plano, convex
or concave configurations as all are likely to be effective in controlling LEC migration.
b) To minimise spacing between the element (maximise space available for
translation):
Between > -1 and < + 1 for the front element. Lenses outside this range will result in an
extreme form, which will minimise the lens diameter and maximise lens thickness
particularly in the more likely designs for which the power of this element is high
(> +30 D). A bending factor > +1 produces an extremely curved front surface and < -1
produces an extremely curved rear surface. Preferably between > -1 and < + 2 for the rear
element. There is greater freedom for the rear element compared to the front element as the
power is lower (typically < -20 D). However, extreme bending factors may lead to a
smaller diameter of the element.
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Optical Design
Figure 6.6: Possible forms of front and rear elements of 2E-AIOL and their combinations.
The first row represents the bending factor for the front element and first column
represents the bending factor of the rear element. Only plus lenses for the front element
and minus lenses for the rear element are considered as this is the preferred combination of
powers. This combination returns higher accommodative amplitudes.
6.3.3 Design Criteria to Control Aberrations
The presence of spherical aberration in the eye, to some limit, is beneficial from a
psychophysical point of view (Franchini, 2007, Marcos et al., 2005) as this play a number
of important roles in the process of accommodation. First it has been shown to aid the
determination of the polarity of defocus (myopic or hyperopic) thereby driving the lens to
either accommodate or disaccommodate (Kruger and Pola, 1986). Secondly, higher order
aberrations tend to interact with the defocus term to determine the final steady-state of
accommodation (Plainis et al, 2005). Therefore an optimal design may arguably be
obtained not with an aberration-free IOL but with one that partially compensates corneal
aberrations (Guirao et al., 2002, Atchison, 1991). However, from the optical performance
standpoint irrespective of the visual benefits, the best retinal image quality is obtained when
the ocular (eye-AIOL combined) aberrations are completely eliminated (Piers et al, 2007).
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Optical Design
To date, studies on optical analysis have focused on conventional IOL and testing standards
are set for the same (International Standard Organization, 1999). Since this work is not
limited only to distance vision and a single element of IOL, the design parameters and
optical performances for intermediate and near vision are also to be evaluated. A major
criterion in designing AIOLs is to eliminate or control the aberrations within a minimum
acceptable range for all viewing distances along a continuum. In this study, the following
design criteria are formulated:
1)
Elimination of Seidel aberration
i.
of AIOL only and
ii.
of whole eye (cornea + AIOL)
2)
Obtain a target Seidel aberration value
i.
for AIOL only and
ii.
for whole eye (cornea + AIOL)
For criterion 2, the target values of aberrations are based on the literature as discussed
earlier in Section 6.2. Briefly, the population mean of the spherical aberration (0.15 ȝm,
0.088 waves at 587 nm) for a normal eye (Thibos et al., 2002) was considered as a target
value for distance vision. The target aberration for near vision was calculated from the
relation between accommodation and aberration reported previously (Cheng et al., 2004).
Although some investigators could not identify any clear trend (He et al., 2000, Atchison et
al., 1995), Cheng et al reported that each dioptre of accommodation reduces spherical
aberration by -0.0435 ȝm. Thus, considering 2.5 D as the accommodation for near vision,
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Optical Design
spherical aberration would be reduced by 0.109 ȝm. Hence, throughout the study, 0.04 ȝm
(0.024 waves at 587 nm) was set as the performance target for spherical aberration at near.
6.4 Aberration guided design of spherical 1E-AIOL
Several optimum designs of conventional monofocal IOLs reported in the literature are
reviewed in Section 6.2.5 (Atchison, 1989a, Atchison, 1989b, Lu and Smith, 1990,
Pomerantzeff et al., 1985, Wang and Pomerantzeff, 1982, Smith and Lu, 1988, Tabernero
et al., 2007b). These studies have shown that single-focus spherical IOLs cannot eliminate
or reduce spherical aberration, irrespective of the optimum lens bending. The use of
aspheric surfaces is the only way of eliminating or controlling spherical aberration in the
pseudophakic eye (Atchison, 1991, Lu and Smith, 1990). Although improved retinal image
quality is reported (Altmann, 2004, Bellucci and Morselli, 2007), some theoretical studies
have demonstrated suboptimal performance of aspheric IOLs in the presence of
misalignment (Altmann et al., 2005, Eppig et al., 2009, Pieh et al., 2009).
As indicated earlier, there is a basic difference between the optimum design of a single
focus IOL and a translating-optics AIOL. The former is designed to optimise vision for a
static axial position of the lens whereas in the latter, there is a continuous shifting of the
axial position of the lens with a concomitant change in viewing distance (dynamic changes
in object-image conjugates). Optimum performance must be maintained throughout the
range (near to distance). In this section, Seidel first-order aberration theory is used to
investigate the optimum design of a spherical 1E-AIOL. Given that the spherical aberration
of the 1E-AIOL may not be eliminated, the objective of this section is to find optimum
design of 1E-AIOL to optimise (i.e. to minimise) aberrations at distance (unaccommodated)
and near (accommodated) focus conditions.
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Optical Design
6.4.1 Spherical Aberration
From Appendix B4.3, Seidel Primary spherical aberration (SI) for a thin lens surrounded by
homogenous media is given by:
h4F 3
SI =
4n 2
ª n 2 (n'+2n) 2 4n(n'+ n)
(3n'+2n) 2
n' 2 º
Y +
« n' (n'− n) 2 X − n' (n'− n) XY +
n'
(n'− n) 2 »¼ ...........................(6.5)
¬
where, h is the ray height, F is the power of the lens, n and nƍ are the refractive indices in
object and image spaces respectively, X is the normalised bending factor given by:
X =
(n' n)c1
( n ' n ' ' )c 2
..............................................................................................(6.6)
Fe
c1 and c2 are the curvatures of the first and the second surfaces of a thin lens respectively
(c
1 / R ) . In terms of equivalent power and shape factor, the curvatures are given by:
c1 =
Fe ( X + 1)
F ( X − 1)
c2 = e
and
2( n'− n)
2(n'− n' ' )
and Y is the conjugate factor given by:
Y=
n ' ' u ' ' nu
=
nl
n' ' l '
=
L' L
=
1 M
.....................................................................(6.7)
where u and u” are the paraxial object and image ray angle on the surface respectively, l
and l” are the object and image distances from the surface respectively. L and Lƍ are the
incident and emergent vergences from the surface respectively and M is the magnification
of the image.
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Optical Design
The goal of the 1E-AIOL design is to find the values of the bending factor (X) that
produces at most the spherical aberration performance target for values of the conjugate
factor (Y) corresponding to far and near distance.
We will express the spherical aberration of a thin 1E-AIOL focused at distance as:
SI
K ..........................................................................................................................(6.8)
Where, from Eq. (6.5) K =
h4F 3
2
and ı is given by
σ = AX 2 − BXY + CY 2 + D ......................................................................................... (6.9)
where A =
(3n' 2n)
4n( n' n)
n' 2
n 2 (n'+2n)
,
B
=
−
,
C
=
,
D
=
n' (n'−n) 2
(n'−n) 2
n' (n'−n)
When the 1E-AIOL is translated axially (for accommodation) the marginal ray height (h)
and conjugate factor (Y) change. The spherical aberration of the IOL at near focus condition
(accommodated state) may be written as;
S 'I
K ' ' ....................................................................................................................(6.10)
where Kƍ and σƍ have similar expressions as K and σ, except that hƍ replaces h, and Yƍ
replaces Y. All other parameters remain unchanged.
Equations 6.8 and 6.10 are quadratic equations in terms of the bending factor X which are
n 2 (n'+2n)
concave upward as
> 0, and have no roots indicating that neither S1 nor S '1 can
n' (n'− n) 2
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Optical Design
be zero. This rules out achieving both criterion 1 (elimination of aberration) and criterion 2
within clinically meaningful range i.e. obtaining aberration less than the corneal spherical
aberration. In Equation 6.8, the term K is positive. Also, normally nƍ > n which produces
positive ı. Therefore, the spherical aberration produced by 1E-AIOL is positive which adds
to the corneal aberration resulting in a higher positive spherical aberration.
The design goal will therefore be to find the bending factor, Xmin, that minimises spherical
aberration. From Equations 6.8 and 6.10, Xmin satisfies when 2 AX min − BY = 0 which
gives: For distance focus:
X min =
2(n'2 −n 2 )
Y
n(n'+2n) ......................................................................................................(6.11)
and for near focus:
X ' min =
2(n'2 −n 2 )
Y'
n(n'+2n)
............……………………………………………………….. (6.12)
It is seen that Xmin is a function of refractive index and the conjugate factor Y. A higher
index of lens corresponds to a larger Xmin value for the same conjugate ratio. For an AIOL
made out of PMMA ( n' 1.492) , we obtain: X min
0.157Y . Similarly the conjugate factor
decreases with accommodation with corresponds to a decrease in the value of Xmin.
It can be observed from Equations 6.11 and 6.12 that the optimal bending factor for near
and distance are not the same, since the conjugate factor changes. Since the Xmin value for a
given implant position is unique, the optimum bending factor at distance vision is not
optimal for near vision. The values of Xmin for distance and near focus states are plotted in
Figure 6.7 as a function of refractive index, for two values of pupil position, where the
pupil diameter is 4 mm.
151
Optical Design
The graph shows that the optimal bending factor for a 1E-AIOL made of PMMA to be
implanted between 1 and 2 mm behind the pupil is X = +1, plano-convex with curved
surface facing the retina. This is consistent with an earlier report on conventional IOLs
(Atchison, 1989a). The graph also suggests that accommodation slightly decreases the Xmin
value. However, accommodation and implant position only have a small effect on the
optimal lens shape. For example, an AIOL made up of PMMA and implanted 2 mm behind
the pupil has Xmin values of 0.963 and 0.902 for distance and near respectively (2 mm
forward translation for accommodation). The values become 1.01 and 0.945 respectively
for a lens implanted 1 mm behind the pupil. It should be noted that when Xmin is greater
than +1, the lens becomes a convex meniscus with the concave surface facing the retina and
when it is less than +1, the lens is biconvex with the more curved surface facing the cornea.
Xmin vs Refractive Index of 1E-AIOL
4.0
3.5
2.5
2.0
X
min
Value
3.0
1.5
1.0
2mm
2mm
1mm
1mm
0.5
0.0
1.4
1.45
1.5
1.55
1.6
1.65
1.7
post pupil - Distance
postpupil-Near
post pupil-Distance
post pupil-Near
1.75
1.8
1.85
1.9
Refractive Index
Figure 6.7: Xmin of an 1E-AIOL for near and distance vision versus refractive index for two
implant positions. With an increase in refractive index, the bending factor that gives
minimum Seidel spherical aberration increases. The position of the implant and the
accommodative state has a small effect on the optimal lens shape. Extended refractive
index (x-axis) up to 1.9 is shown for reference though they are practically unavailable for
manufacturing an IOL.
152
Optical Design
Replacing X in Equations 6.8 and 6.10 with the expression of Xmin returns the expression for
minimum spherical aberration (SImin):
S I min =
º
h 4 F 3 ª n' 2
n'
−
Y 2 » ............................................................................(6.13a)
2 «
4n ¬ ( n'− n) n'+2n ¼
and
S I 'min =
h' 4 F 3
4n 2
ª n' 2
n'
2º
« (n'− n) − n'+2n Y ' »
¬
¼ ..........................................................................(6.13b)
Smin as a function of Refractive Index
1.2
Eye: Distance
Eye: Near
AIOL only: Distance
AIOL only: Near
0.8
0.6
040
W
(waves)
040
1.0
0.4
0.2
0.0
1.4
1.5
1.6
1.7
1.8
1.9
2.0
Refractive Index
Figure 6.8: Minimum spherical aberration in waves (Ȝ = .587 ȝm)as a function of
refractive index of AIOL.
153
Optical Design
The minimum spherical aberration (SImin) of the 1E-AIOL and of the whole eye (Navarro
model eye) for near and distance vision at 4 mm pupil diameter are plotted in Figure 6.8 as
a function of refractive index.
The spherical aberration of a 1E-AIOL made out of PMMA and the total spherical
aberration of the whole eye for unaccommodated state as a function of the bending factor of
the AIOL can be seen in Figures 6.9 and 6.10 respectively. The spherical aberration of the
1E-AIOL is consistently positive for all values of the bending factor. The additive effect of
the aberrations from the cornea and lens may be clearly seen in Figure 6.10.
W040 vs Bending Factor
8
3mm post-pupil
2mm post-pupil
1mm post-pupil
on pupil plane
7
W 0 4 0 (Waves)
6
5
4
3
2
1
0
-3
-2
-1
0
1
2
3
4
5
Bending Factor
Figure 6.9: Seidel spherical aberration (in waves, Ȝ = .587 ȝm) produced by 1E-AIOL only
for various axial positions (implant depths) in the eye. The refractive index of PMMA (n =
1.492) was used for the calculation. The aberration on translation (accommodation) is not
shown here.
Similar findings are obtained when the AIOL is translated in the axially forward direction
for near vision (pseudophakic accommodation) (Figure 6.11). An important observation to
154
Optical Design
be made from Figure 6.11 is that the spherical aberration increases with pseudophakic
accommodation which is in contrast to reports for phakic eye accommodation where the
aberration decreases.
W040 vs Bending Factor
8
3mm post-pupil
2mm post-pupil
1mm post-pupil
on pupil plane
7
W 0 4 0 (Waves)
6
5
4
3
2
1
0
-3
-2
-1
0
1
2
3
4
5
Bending Factor
Figure 6.10: Seidel spherical aberration (in waves, Ȝ = .587 ȝm) of the model eye with 1EAIOL for various axial positions (implant depths) at distance focus. The IOL is made of
PMMA (n = 1.492). Higher aberrations compared to that in Figure 6.9 can be observed
which shows the additive effect of the corneal aberration.
All of the above results show that accommodation and implant position have a minimal
effect on the optimal shape of the 1E-AIOL. The optimum shapes of 1E-AIOL for all
implant positions and accommodated state are consistent to that reported for a conventional
IOL. The optimal design of the 1E-AIOL is therefore expected to be similar to that of a
traditional non-accommodating IOL.
155
Optical Design
Spherical Aberration with Accommodation
3.0
Distance
1.0 D Accommodation
2.5 D Accommodation
2.5
W 04 0 (waves)
2.0
1.5
1.0
0.5
0.0
-3
-2
-1
0
1
2
3
4
5
Bending Factor
Figure 6.11: Seidel spherical aberration of an A-IOL made of PMMA for three
accommodative states in waves (Ȝ = .587 ȝm) as a function the bending factor. The optimal
bending factor (Xmin) is approximately equal to +1 for all three viewing distances.
6.4.2 Coma Aberration
Seidel primary coma (SII) aberration for a thin lens surrounded by homogenous media is
given by:
S II =
(2n'+ n) º
Hh 2 F 2 ª n(n'+ n)
« n' (n'−n) X − n' Y » ......................................................................(6.14)
2
2n ¬
¼
where, h is the ray height, F is the power of the lens, n and nƍ are the refractive indices on
object and image spaces respectively, X is the normalised bending factor given by:
X =
( n' n)c1
( n' n' ' )c2
Fe
156
Optical Design
c1 and c2 are the curvatures of the first and the second surfaces of a thin lens respectively
( c 1 / R ) . In terms of equivalent power and shape factor, the curvatures are given by:
c1 =
Fe ( X + 1)
F ( X − 1)
c2 = e
and
2(n'− n)
2(n'− n' ' )
and Y is the conjugate factor given by:
Y=
n ' ' u ' ' nu
=
nl
n' ' l '
=
L' L
=
1 M
Where u and uƍƍ are the paraxial object and image ray angle on the surface respectively, l
and /ƍ are the object and image distances from the surface respectively. L and Lƍ are the
incident and emergent vergences from the surface respectively and M is the magnification
of the image.
H is the Smith-Helmhotz-Lagrange invariant given by;
H = ni (u i hi − u i hi ) = n'i (u 'i hi − u ' i hi )
where hi & u i are paraxial pupil (chief) ray height and angles at the i th surface respectively
and ui and hi are paraxial marginal ray angles and heights respectively.
Equation 6.14 can be represented as:
S II = −
h2 F 2 H
2
( BX + GY ) ...........................................................................................(6.15)
157
Optical Design
4( 2 n ' n )
4 n( n' n )
Where B = − n ' ( n'− n ) and G =
.
From Equation 6.14, it is clear that when an AIOL is implanted in the plane of the aperture
stop, coma varies linearly with the bending factor (X) and conjugate factor (Y). However,
the condition is different in a real situation where the lens is implanted behind the pupil
(Equation 6.13). When the aperture-shift factor is taken into account, the coma of a 1EAIOL is given by;
S II* = S II + εS I ..............................................................................................................(6.16)
where
is the stop shift factor given by ε =
h
Therefore, from Eq. (6.8) and (6.15):
S
*
II − IOL
h2F 2
=
4n 2
H
ª
º
2
2
«¬h hF AX − BXY + CY + D − 2 ( BX + GY )»¼ .................................(6.16)
(
)
Using the same methods as for the study of spherical aberrations, we find that coma is
minimised when the bending factor takes the following optimal value Xmin:
X min = −
B(2h hFY − H )
..............................................................................................(6.17)
From the superposition principle, the total primary coma of the eye is given by:
S II − eye
S II − cornea
S II − IOL
.......................................................................................(6.18)
158
Optical Design
The primary coma of the eye at 10° field angle as a function of the bending factor of a 1EAIOL at distance and near focus is plotted in Figure 6.14. The figure shows that coma
becomes more sensitive to the bending factor as the implant depth increases. During
pseudophakic accommodation the lens shifts towards the pupil (aperture stop) minimizing
the effect of pupil shift factor. This leads to a decrease in the aberration. Therefore, coma is
less for near focus than distance focus. The difference between distance focus and near
focus is larger than for spherical aberration. The graph also shows that the bending factor
that minimises coma is approximately Xmin = -1, for both distance and near focus..
6.4.3 Design to Control Coma and Spherical Aberration
The ideal 1E-AIOL design would have a lens bending factor that would simultaneously
minimise coma and spherical aberration. However as mentioned earlier, the optimum
bending factor for spherical aberration falls around +1 whereas it is around -1.0 for coma.
This indicates that a single AIOL cannot optimally control both aberrations simultaneously.
Therefore a compromise between the aberrations is required. A major issue is to determine
which of the aberrations should guide the design. Coma was found to be more sensitive to
the bending factor than spherical aberration. In addition, considering that coma has greater
impact on visual quality (Thibos et al, 2002), it is preferable to select the bending factor
which minimises coma.
A counter-argument, in favour of designing for spherical aberration, is that in an idealised
system, it is central vision that should be optimised which by definition implies spherical
aberration instead of coma. In this study, the eye is assumed to be axially symmetrical. As
is well known, the natural eye possesses coma aberration even along the visual axis due to
the misalignment of the ocular components. In this ‘real world’ situation, the control or
elimination of such ‘on-axis’ coma would require the AIOL to embrace asymmetric designs
or be intentionally misaligned on implantation, the study of which is beyond the scope of
the present work.
159
Optical Design
6
ID=1mm:
ID=1mm:
ID=2mm:
ID=2mm:
ID=3mm:
ID=3mm:
Aberration (Waves)
5
4
Distance
Accommodated
Distance
Accommodated
Distance
Accommodated
3
2
1
0
-4
-3
-2
-1
0
1
2
3
Bending Factor
Figure 6.12: Seidel primary coma of an eye at 10° field implanted with a 1E-AIOL made of
PMMA (n = 1.492) as a function of bending factor, for various implant depths. The
accommodative amplitude used in this example is 2.5 D. Minimum coma obtained is 0.46,
0.53 and 1.21 waves for distance and 0.42, 0.45 and 0.55 waves for near for 1 mm, 2 mm
and 3 mm implant depths respectively (Ȝ =.5876 ȝm). The values of X min increased with
more posterior position of the AIOL.
Coma and spherical aberration of an eye implanted with an 1E-AIOL made of PMMA are
shown in Figures 6.13, as a function of the bending factor, for near and distance vision.
Table 6.3 summarises the values of aberrations obtained with various design of the lens in
the model eye. The best lens for coma has X = -1.01, which effectively maintains a low
spherical aberration as well. On the other hand, the optimal bending factor of the lens for
controlling spherical aberration (X = 0.96) produces significant amounts of residual coma.
A bending factor X = -0.5 may be a good compromise.
160
Optical Design
Seidel Primary Spherical Aberration & Coma of 1E-AIOL
0.8
S II: Distance Vision
0.7
S II: Near Vision
Aberration (Waves)
0.6
S I: Distance Vision
S I: Near Vision
0.5
0.4
0.3
0.2
0.1
0.0
-3
-2
-1
0
1
2
3
Bending Factor
Figure 6.13: Spherical aberration and coma of the model eye implanted with 1E-AIOL
made of PMMA implanted 2 mm behind the pupil. The pupil diameter is 4 mm pupil. The
optimum bending factor is X = +1 for spherical aberration and X = -1 for coma. Solid lines
represent the aberration at distance focus and dotted lines represent the aberrations at
near focus.
6.4.4 Summary of Spherical Design of 1E-AIOL
A spherical design 1E-AIOL cannot eliminate spherical aberration or coma for either
distance or near focus. Hence the approach is to select a design that minimises spherical
aberration and coma. While the optimum bending factor is approximately X = +1 for
spherical aberration, it is about X = -1 for coma. Both these shapes are biologically
acceptable according to the criteria set in Section 6.3.1. The optimum design in terms of
controlling both aberrations simultaneously is one with a bending factor of approximately
-0.5 which is a biconvex design with the more curved surface facing the retina. The study
shows that the optimal bending factor does not depend significantly on the accommodative
state or implant position. A design that minimises aberrations at distance will also be close
161
Optical Design
to optimal for near vision. For a 1E-AIOL with an optimal shape, spherical aberration and
coma do not change significantly with accommodation.
6.5 Design of Spherical 2E-AIOL
With the additional degrees of design freedom provided by 2E-AIOLs, i.e. bending factors
and refractive indices for two elements, it may be possible to eliminate or control to a
desired level the aberrations of the eye implanted with a 2E-AIOL. As stated earlier, an
ideal AIOL is one which eliminates or controls aberrations for the entire range of viewing
distances. However, practically it may not be possible to design such a lens. Therefore, I
have limited the analysis only to two object-image conjugates: distance (infinite object) and
near (finite conjugate with object at 40 cm which is equivalent to 2.5 D accommodation)
focus.
Table 6.3: Spherical aberration and coma in waves (Ȝ = 587.6 nm) produced in the model
eye by various bending factors of 1E-AIOL.
Bending
Factor
-1.01
-0.5
0.963
Aberration Distance
Near
Spherical
0.129
0.154
Coma
0.083
0.035
Spherical
0.094
0.109
Coma
0.124
0.043
Spherical
0.079
0.055
Coma
0.717
0.275
162
Optical Design
6.5.1 Spherical Aberration
6.5.1.1. General Equations and Methods
The total spherical aberration of a 2E-AIOL is the sum of the aberrations produced from the
individual elements. Hence using the same notation as in the previous section:
S I −total = S I − front + S I −rear = K1σ 1 + K 2σ 2 ....................................................................(6.19a)
where subscripts 1 and 2 refer to the front and rear elements respectively
K1 =
h14 F13
2
and K 2 =
h24 F23
2
σ 1 = A1 X 12 + B1 X 1Y1 + C1Y12 + D1
σ 2 = A2 X 22 + B2 X 2Y2 + C 2Y22 + D2
In Equation 6.19a, ray height (h) and conjugate factor (Y) are accommodation-dependent
variables that change with translation of the optics; all other factors are constant. Denoting
accommodation-induced changes with primes (h’ and Y’), the spherical aberration of the
2E-AIOL for near focus is given by:
S ' I −total = S ' I − front + S ' I −rear = K '1 σ '1 + K '2 σ '2 ...............................................................(6.19b)
Similarly, the spherical aberration of the eye (SI-eye) is the sum of contributions from the
cornea (SI-cornea) and the 2E-IOL (SI-total). Therefore the spherical aberration of the whole
eye is given by:
163
Optical Design
For distance: S eye = S I −cornea + S I − front + S I − rear ...........................................................(6.20a)
For near: S 'eye = S ' I −cornea + S ' I − front + S ' I −rear ...............................................................(6.20b)
Equations 6.19a can be expressed in terms of desired total aberration of the 2E-AIOL.
Arbitrarily setting the aberration of the front element, the rear element must satisfy:
S I −rear = S I −total − S I − front .............................................................................................(6.21a)
On expansion with the coefficients for the rear element, we obtain:
K 2 A2 X 2 − K 2 B2 X 2Y2 + K 2 C 2Y2 + K 2 D2 = S I −total − S I − front .....................................(6.21b)
2
2
This is a second-degree equation in X2 which has roots:
K 2 B2Y2 ±
(K 2 B2Y2 )2 − 4 K 2 A2
K 2 C 2Y2 + K 2 D2 − S I −total + S I − front
2
2 K 2 A2
........................(6.21c)
The determinant in Equation 6.21c which needs to be satisfied to result in a real root is:
K 2 B2Y2 ≥ 2 K 2 A2 K 2 C 2Y2 + K 2 D2 − S I −total + S I − front ............................................(6.21d)
2
Again from the Equation 6.20a, for a known total ocular spherical aberration and arbitrary
front element aberration, the equation can be written as:
S I − rear = S I −eye − S I −cornea + S I − front
...........................................................................(6.22a)
164
Optical Design
On expanding SI-rear we get:
2
2
K 2 A2 X 2 − K 2 B2 X 2Y2 + K 2 C 2Y2 + K 2 D2 = S I − eye − S I − cornea + S I − front ................(6.22b)
Equation 6.22b has roots:
K 2 B2Y2 ±
(K 2 B2Y2 )2 − 4 K 2 A2
2
K 2 C 2Y2 + K 2 D2 + S I − eye − S I − cornea + S I − front
2 K 2 A2
............(6.22c)
Existence of these roots (Equation 6.21c and 6.22c) indicates that the spherical aberration
of the 2E-AIOL and the eye can be eliminated under the circumstances defined by the
conditions for the root(s) to the equations. Similar expressions can be derived for the
accommodated states (Equation 6.19b and 6.20b). Now, these equations will be employed
to find designs in each of the optical criteria stated in Section 6.3.2. Equations depicting the
criteria for optimal designs are rather protracted, because of the large number of
permutations of the design. However, it can be observed in Equations 6.19 to 6.22 that the
aberration from the front element, the conjugate factors, power of the element and the
refractive indices of both elements determine the optimum design of a 2E-AIOL. These will
be illustrated for selected designs in the following sections.
6.5.1.2 Criterion 1 (i): Elimination of aberration from AIOL only
Solving Equation 6.21b for X2, we obtain the bending factor of the rear element for a given
bending factor of the front element (X1) satisfying the criterion 1(i) i.e. to produce zero
spherical aberration of the combination of two elements. Equations 6.19a and 6.19b each
produces a pair of hyperbolic curves, where only the upper branches correspond to practical
design solutions. The result obtained for a 2E-AIOL with a +35 D front element is shown in
Figure 6.17, for distance and near vision.
165
Optical Design
A combination of bending factors for the front and rear elements producing zero aberration
at both object-image conjugates is represented by the intersection (henceforth referred as
‘optimal solution’) between the two design curves (near and distance). In Figure 6.14, it is
observed that this optimal solution may not exist when the elements are made from
materials such as silicone (n = 1.45) and PMMA (n = 1.492) which are commonly used in
fabricating IOLs. It exists only when the refractive index of the material is roughly around
1.70 or more. However, even when the optimal solution does not exist, the difference in
optimal bending factors of the near and distance vision solutions is relatively small.
2E-AIOL: Criteria 1(i)
Bending Factor (Back Element)
3.0
1.45
2.0
1.55
1.65
1.70
1.0
0.0
-1.0
-2.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Bending Factor (Front Element)
Figure 6.14: Design of spherical 2E-AIOL for criterion 1(i). Refractive indices of the
elements are indicated by labels. Solid lines represent the design for distance and dashed
lines represent the design for near. The power combination of the AIOL elements is
constant (+35 D front element)
166
Optical Design
6.5.1.3 Criterion 1(ii): Elimination of the spherical aberration in the eye (eye+AIOL)
Solving Equation 6.22b for X2, we obtain the bending factor of the rear element for a given
bending factor of the front element (X1) satisfying criterion 1(ii) i.e. to produce zero
resultant spherical aberration of the whole eye. Plots obtained from Equation 6.21a
designed to eliminate the ocular spherical aberration (SIeye=0) are shown in Figure 6.15.
One or two optimal solutions for simultaneously near and distance designs are available
when the refractive index of the elements is approximately 1.55 or more. Increasing the
refractive index of the material shifts the curves towards a more positive bending factor of
the front element. The vertices of the curves representing distance (solid lines) and near
(dashed lines) vision approach each other, until they eventually intersect at one or two
points, producing one or two optimal solutions.
Figure 6.16 gives an example of a design approach using different refractive indices for the
front and rear elements to produce an optimal solution according to criterion 1(ii) for a
range of bending factors for the elements. In this example, the front element power is
+35 D, which is implanted 2 mm behind the pupil. In the figure, points A, B, C and D
represent the coordinates for optimal solutions (intersection of designs for near and distance
focus). When fitted with polynomial function along these points, the curve represents a
design in terms of refractive indices for two elements to satisfy the criterion. In this
example, a second-order polynomial provided a near perfect fit (r2 = 0.999) with equation:
y = 8.33x 2 − 26.44 x + 22.8 where x and y are the refractive indices of the front and rear
elements respectively. This second order equation represents a curve concaving upward
(coefficient along x2 is positive) and has minima at x = 1.587 which shows that 1.587 is the
minimum refractive index of the front element required to produce a real root.
Corresponding y is 1.528 which represents the refractive index of the rear element for the
optimum solution.
167
Optical Design
2E-AIOL: Criteria 1(ii)
Bending Factor (Rear Element)
4.0
3.5
1.45
1.50 1.55
1.60
3.0
2.5
2.0
1.5
1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
Bending Factor (Front Element)
Figure 6.15: Design of spherical 2E-AIOL for criterion 1(ii). The refractive index of the
elements is indicated by labels. Solid lines represent the design for distance and dashed
lines represent the design for near. In this criterion, solutions for near and distance vision
simultaneously occur when the refractive index of the material is approximately 1.55 or
more.
6.5.1.4 Criteria 2 (i): Obtain a Target aberration for the AIOL only
When we assign a value for SI-total in Equation 6.19a, it represents the equation for this
criterion and the assigned value is the performance target for that aberration. With target
aberration of 0.088 waves (Ȝ = 0.587 ȝm) design curves obtained from Equations 6.19a and
b are shown in Figure 6.17. The front element power is +35 D in this example which is
implanted 2 mm behind the pupil. Again the design solutions were found only for the
materials with refractive index near to or beyond 1.70.
168
Optical Design
Spherical Design of 2E-AIOL
1.60
X2 = +1.5
X1 =
1.55
A = +2.0
B = +1.5
C = +1.0
D = +0.5
E = 0.0
F = -0.5
C
B
A
D
1.50
E
1.45
F
1.40
1.35
1.35
1.4
1.45
1.5
1.55
1.6
1.65
1.7
Index (Front Element)
Figure 6.16: Combination of refractive indices of the two elements of a 2E-AIOL to
produce zero total ocular spherical aberration for various bending factors. Solid lines
represent designs for distance and dashed lines represent designs for near. Solutions were
obtained for all positive bending factors of the front element.
169
Optical Design
Criteria 2(i)
Bending Factor (Rear Element)
3.0
2.5
1.45
1.50
2.0
1.55
1.5
1.60
1.0
1.70
0.5
0.0
-0.5
-1.0
-1.5
0.0
0.5
1.0
1.5
2.0
2.5
Bending Factor (Front Element)
Figure 6.17: Design of spherical 2E-AIOL for criterion 2(i) with target aberration of 0.088
waves for unaccommodated and 0.024 waves (Ȝ = 0.587 ȝm) for accommodated states.
Refractive indices of the elements are indicated by labels. Solid lines represent the design
for distance and dashed lines represent the design for near.
6.5.1.5 Criteria 2 (ii): Obtain Target aberration for Eye (Eye + AIOL)
When we assign a value for SI-eye in Equation 6.19a, it represents the equation for this
criterion and the assigned value is the target aberration. With target aberration of 0.088
waves (Ȝ=0.587 ȝm) for unaccommodated and 0.024 waves for accommodated states,
design curves of criterion 2(ii) for different refractive indices of the front and rear elements
are illustrated in Figure 6.18. Front element had +35 D power in this example. In this
calculation, the optimal solutions were produced when the elements had refractive index of
approximately 1.60.
170
Optical Design
2E-AIOL: Criteria 2(ii)
Bending Factor (Back Element)
5.0
4.5
1.45
4.0
1.50
3.5
1.55
3.0
1.60
2.5
2.0
1.5
1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Bending Factor (Front Element)
Figure 6.18: Spherical design of 2E-AIOL for criteria 2(ii). Refractive indices of the
elements are indicated by labels. Solid lines represent the design for distance and dashed
lines represent the design for near.
6.5.2 Off-axis aberration: Coma
The Seidel primary coma of an eye implanted with a 2E-AIOL is the sum of the aberrations
from the cornea and two elements of the 2E-AIOL:
S II − eye
S II − cornea
S II − front
S II − rear ...........................................................................(6.23)
Adapting the equations and notation derived for the coma of 1E-AIOLs gives:
2
h12 F1 ª
H1
H
º h2 F2 ª
º
−
+
+
h
h
F
σ
B
X
G
Y
h h F σ − 2 ( B2 X 2 + G2Y2 )»
(
)
1 1
1 1 »
2 « 1 1 1 1
2 « 2 2 2 2
2
2
4n ¬
¼ 4n ¬
¼
2
S II −eye = S II − cornea +
2
171
Optical Design
The following designs for coma correction are evaluated:
Criteria 1(i) Zero coma of the 2E-AIOL
for distance:
for near:
S II − front + S II − rear = 0
S ' II − front S ' II − rear
0
............................................................................ (6.24a)
.................................................................................. (6.24b)
Criteria 1(ii) Zero coma of the eye with 2E-AIOL
for distance:
for near:
S Cornea
S II − front
S II − rear
0
S 'Cornea + S ' II − front + S ' II − rear = 0
....……………………………………… (6.25a)
..........……………………………………… (6.25b)
Criteria 2(i) Target coma of the 2E-AIOL
for distance:
for near:
S II − front
S II − rear
S ' II − front S ' II − rear
...……………………………………………….. (6.26a)
'
........……………………………………………….. (6.26b)
Criteria 2(ii) Target coma of the eye with 2E-AIOL
for distance:
for near:
S Cornea
S II − front
S II − rear
S 'Cornea S ' II − front S ' II − rear
'
...………………………………………(6.27a)
..........……………………………………… (6.27b)
172
Optical Design
Each of these equations can be expanded and solved for X2 (or X1) to obtain the
corresponding design curves and solutions. For example, solving Equation 24a for X2 we
obtain:
X2 = −
J 2 B 2 ( 2 P2 Y 2 − H 2 ) ±
4 J 22 P22 ( B 22 Y 22 − 4 A 2 C 2 Y 22 − 4 A 2 D 2 ) + J 22 B 22 H 2 ( H − 4 P2 Y 2 ) − 16 J 2 P2 A 2 S II − front
4 P2 J 2 A 2
(6.28)
where J 2 =
h22 F2
2
2
and P2 = h2 h2 F2
Figures 6.19(a) and (b) and 6.20(a) and (b) represent the corresponding designs of the 2EAIOL elements for the various criteria. In all cases, a field anlge of 10° was used in the
calculations. While an optimal solution does not exist for criteria 1 (i) and (ii) and criterion
2(i) indicating that coma cannot be eliminated for near and distance simultaneously,
designs for the accommodated and unaccommodated states are very close which may
sufficiently satisfy the criteria in practice. In the example, target coma for accommodated
and unaccommodated states are 0.062 waves and 0.011 waves (Ȝ=0.587 ȝm) respectively.
Power of the front element is +35 D.
173
Optical Design
2E-AIOL Criteria 1(i)
Bending Factor (Rear Element)
(a)
5
0
Distance
NEar
-5
-10
-3
-2
-1
0
1
2
3
4
Bending Factor (Front Element)
2E-AIOL criteria 1(ii)
Bending Factor (Rear Element)
(b)
5
Distance
NEar
0
-5
-10
-3
-2
-1
0
1
2
3
4
Bending Factor (Front Element)
Figure 6.19: Design of 2E-AIOL for coma at 10º off-axis position of object: criterion
1(i)(a) and criterion 1(ii) (b). Both of the elements were made up in silicone materials (n =
1.45) implanted 2 mm behind the pupil. The front element (power = +35 D) was translated
in the forward direction to simulate pseudophakic accommodation of about 2.5 D. No
solution was found that satisfies the criterion for both distance and near, but the difference
in bending factor for the near and distance solutions is small.
6.5.3 Design to control coma and spherical aberration
Although individual aberrations can be controlled for both near and distance, analytical
results suggest that spherical design of 2E-AIOL cannot meet the criteria simultaneously
for spherical aberration and coma. A solution could not be found for any practical
combination of bending factors and refractive indices. Figure 6.21 summarises the designs
of two elements for spherical (second row) and coma (first row) aberrations under
criterion 1 (i) (first column) and criterion 1 (ii) (second column).
174
Bending Factor (Rear Element)
Optical Design
2E-AIOL: Criteria 2(i)
(a)
4
2
0
Distance
Near
-2
-4
-2
-1
0
1
2
3
4
Bending Factor (Rear Element)
Bending Factor (Front Element)
2E-AIOL: criteria 2(ii)
(b)
4
2
0
Distance
Near
-2
-4
-2
-1
0
1
2
3
4
Bending Factor (Front Element)
Figure 6.20: Design of 2E-AIOL for coma at 10º off-axis position of object: criterion
2(i)(a) and criterion 2(ii) (b). Both of the elements were made in silicone materials (n =
1.45) implanted 2 mm behind the pupil. The front element was translated in the forward
direction to simulate pseudophakic accommodation of about 2.5 D. A solution that cancels
coma at both distance and near was found for criterion (2(ii).
The spherical aberration of the model pseudophakic eye implanted with a 2E-AIOL with
various combinations of elements is given in a three-dimensional plot in Figure 6.22.
Consistent to the eye implanted with 1E-AIOL and unlike in the phakic eye where the
natural accommodation induces aberration towards negative direction, pseudophakic
accommodation with translation of the front element produces more positive spherical
aberration. The figure also shows that the spherical aberration is more sensitive to the
bending factor of the front element compared to that of the rear element. The rate of change
in spherical aberration as a function of the bending factor is steeper for designs with higher
power front element. Certain combinations do produce zero aberration.
175
Optical Design
(b)
Bending Factor (Rear Element)
II
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
-2
-1
Bending Factor (Rear Element)
(c)
0
1
2
3
-2
S : 2E-AIOL criteria 1(i)
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
-1
0
1
II
-1
2
Bending Factor (Front Element)
3
-2
0
1
2
3
2
3
S : 2E-AIOL criteria 1(ii)
(d)
I
6
-2
S : 2E-AIOL criteria 1(ii)
I
-1
0
1
Bending Factor (Front Element)
Figure 6.21: Designs of 2E-AIOL for zero coma at 10° field and spherical aberration of the
implant and eye with implant. The refractive indices of the elements were 1.45 and
accommodation was 2.5 D. Design curves for criteria (i) are close to the optimal solutions
but not for criteria (ii). Power of the front element is +35 D. Solid red curves represent the
design for distance and blue dotted lines are the design for near (2.5 D accommodation)
In the figure, it may be clearly observed that the aberration for near and distance are not
greatly different when the front element bending factor is between zero and +1. The
bending factors of the elements producing minimal aberration are approximately +1 (planoconvex) for the front element and between -1 to +1 for the rear element and this finding
concords with the analytical results discussed in the preceding section. For example,
Figure 6.15 shows that +0.5 bending factor of the front element corresponds to a bending
176
Optical Design
factor +1.5 for the rear element to satisfy criterion 1(ii). In Figure 6.22, this combination is
approximately producing close to zero aberration for both power combinations.
(a)
3
W 0 4 0 (waves)
2
1
0
-1
2
Distance
Near
1
X2
2
0
1
-1
0
-1
X1
(b)
3
W 04 0 (waves)
2
1
0
-1
2
Distance
Near
1
X2
2
0
1
-1
0
-1
X1
Figure 6.22: Spherical aberration of the 2E-AIOL in waves (Ȝ = .587 ȝm) as a function of
bending factors of the elements, front element powers are (a) +40 D and (b) +35 D.
Refractive index of the lens material is 1.492 for both elements. The horizontal coordinates
represent the bending factors of the front (X1) and rear (X2) elements. Note that the
aberration is more sensitive to the bending factor of the front element.
177
Optical Design
6.5.4. Summary of the Spherical Design of AIOL
A wide range of 2E-AIOL designs satisfy the design criteria for distance and near foci for
individual aberrations. Combinations of the bending factors and refractive indices for the
two elements in a 2E-AIOL can readily be exploited as degrees of freedom in design. These
designs fit equally well in terms of biological criteria set in Section 6.3 of this chapter.
However, there are some concerns regarding the feasibility of designs performing optimally
for distance and near foci simultaneously. In most cases, such optimal solutions require
refractive indices that are beyond the ones commonly available for IOL fabrication. For
example, one of the solutions obtained for spherical aberration correction with
criterion 1(ii) is X1 = 0.72, X2 = 1.51. These correspond to a biologically acceptable shape
but the refractive index of the material is about 1.55 which limits the possibilities relative to
other desirable properties such as chromatic dispersion, mechanical properties and
biocompatibility.
Table 6.4 summarises the optimal designs or solutions and refractive index for correction of
spherical and coma aberration. As can be observed, the bending factors between zero and
+1 for front element and greater than +1 for rear element may be a good compromise to
balance these two aberrations and such designs also represent biologically compatible
shapes.
178
Optical Design
Table 6.4 a: Pair of designs to satisfy specific criteria for distance and near foci
simultaneously for spherical aberration (SI). Values marked with asterisk are either
biologically compromised shape or occupy space which may reduce accommodation
amplitude. X1 – front element bending factor X2 –rear element. bending factor
Criteria
Solution 1
Solution 2
Refractive
X1
X2
X1
X2
Index
SI-1(i)
1.40*
-1.48
2.0*
-0.92
1.70
SI-1(ii)
0.70
1.53
1.20*
1.67
1.55
SI-2(i)
1.40*
-1.15
2.0*
-0.63
1.70
0.80
1.54
1.60*
1.85
1.60
0.80
0.54
1.60*
0.67
1.60/1.45
SI-2(ii)
Criteria
Solution 1
Solution 2
Refractive
X1
X2
X1
X2
Index
SII-1(i)
0.52
0.66
0.44
-2.71*
1.49
SII-1(ii)
0.52
1.36
0.52
-3.64*
1.49
SII-2(i)
1.33*
-0.85
1.33*
-1.66
1.49
SII-2(ii)
0.84
1.06
1.25*
-3.57*
1.49
6.6 Design of Aspheric AIOL
6.6.1 Introduction
In the previous section, we found that a spherical 1E-AIOL cannot eliminate spherical
aberration or coma in the pseudophakic eye. Spherical 2E-AIOLs can eliminate these
aberrations but designs producing solutions simultaneously for near and distance require
179
Optical Design
refractive indices that are not readily available. Aspherising a surface of an AIOL provides
an additional degree of design freedom and further improves optical performance beyond
that achievable with spherical surfaces (Lu and Smith, 1990, Atchison, 1991). Aspherising
one or both surfaces or elements of an AIOL may produce negative or positive aberrations
as desired. Optimum aspheric values to eliminate spherical aberration in the pseudophakic
eye implanted with a conventional IOL have been proposed (Atchison, 1991, Lu and Smith,
1990, Smith and Lu, 1988). In the current section we will investigate optimum asphericities
for AIOLs.
6.6.2 Aspheric Surface
The most common aspheric surface is a three-dimensional conicoid, or more correctly as
we are considering rotationally symmetrical elements, conicoids (i.e. ellipsoid, paraboloid
or hyperboloid) of rotation. In a rectangular (Cartesian) coordinate system a conicoid of
rotation with its axis of symmetric about the z-axis can be expressed as
c[ ρ 2 + Z 2 (1 + Q)] − 2 Z = 0 ..........................................................................................(6.29)
where, Z is the axial coordinate, c is the curvature at the vertex of the surface and
ρ = x 2 + y 2 (x and y are the xy coordinates)
Q is the asphericity parameter (sometimes called “conic constant”) specifying the type of
conicoid which has interpretations as described in Table 6.5. Equation 6.29 is quadratic in Z
which has two roots. Solving the equation for Z we find (Atchison, 2000)
Z =
cρ 2
1 + [1 − c 2 ρ 2 (1 + Q )
............................................................................................(6.30)
180
Optical Design
The square root term in the denominator of Equation 6.30 needs to be greater or equal to
zero, i.e. 1 − c 2 ρ 2 (1 + Q) ≥ 0 or 1 ≥ c 2 ρ 2 (1 + Q)
This constraint limits the value of Q for a given maximum ray height or aperture radius. For
example, when the aperture radius is 3 mm (6 mm diameter), the limit of asphericity may
be given by:
Q ≤ (3c) −2 − 1 ...............................................................................................................(6.31)
Quadratic Curves with Q
Q = -1
Q =0
Q = +1
Z - Axis
Q= -2
Q =-2
Y - Axis
Figure 6.23: Two-dimensional conic section (x = 0) as produced by Equation 6.30. The
section is a circle when the asphericity is zero (Q=0). When Q<-1 the curve is hyperbolic
and when Q=-1 it is a parabola. Q lying between -1 and 0 describes prolate ellipses, and
Q>0 describes oblate ellipses. Q=0 represents a circle.
181
Optical Design
Table
6.5:
Different
Q
values
representing
corresponding
aspheric
surfaces.
P = 1+Q is the shape factor
Surface Type
Conic Constant (Q)
P=1+Q
Circle
0
1
Parabola
-1
0
Hyperbola
<-1
<0
Prolate Ellipse
-1< Q < 0
0< P < 1
Oblate Ellipse
>0
>1
6.6.3. Contribution of Asphericity to Aberrations
The Seidel aberration contribution of the asphericity Q of a surface can be represented by
(Hopkins & Welford, 1950);
α = c 3 h 4 Q(n'−n) ..........................................................................................................(6.32)
Where n and n’ are the refractive indices of the media in the object and image spaces
respectively, h is the ray height and c is the apex curvature. If more than one surface in the
system is aspherised, the effects are additive.
For a thin lens with one aspheric surface, we can express this equation in terms of
equivalent power and bending factor of the lens by replacing the surface curvature as;
if the front surface is aspheric:
α=
Fe3 ( X + 1) 3 h 4 Q
8(n'− n) 2
.................................................................................................. (6.33a)
182
Optical Design
if the back surface is aspheric
α=
Fe3 ( X − 1) 3 h 4 Q
8(n'− n) 2
..................................................................................................(6.33b)
The aberration of an AIOL with aspheric surfaces may thus be represented by:
S I −asph = S I + ¦ α ........................................................................................................(6.34)
S II −asph = S II + ε ¦ α
where
....................................................................................................(6.35)
indicates the sum of the aspheric contributions of all surfaces.
6.6.4 Aspheric Design of 1E-AIOLs
Seidel spherical aberration of the model eye implanted with a 1E-AIOL exhibiting a front
aspheric surface can be determined by;
for distance S I −eye = S I − IOL + α ..................................................................................(6.36a)
and for near S ' I −eye = S ' I − IOL +α ' .................................................................................(6.36b)
Solving these Equations 6.36a and 6.36b analytically for Q, the value for the asphericity
required to produce zero or any target value of the aberration can be determined. For a thin
lens implanted 2 mm behind the pupil and 2.5 D of accommodation in the model eye, the
asphericities for a series of bending factors are plotted below according to the various
design criteria (Figures 6.24 to 6.26).
183
Optical Design
Optimal solutions correcting aberrations for both distance and near vision exist for
spherical aberration but the asphericity values and corresponding bending factor are
questionable in terms of precision manufacturability due to their large values. No solution
was found for simultaneous correction of coma and spherical aberration.
Figure 6.27(a) shows spherical and coma aberrations induced when the front surface of the
AIOL is aspherised with an optimum asphericity value to eliminate spherical aberration (Q
= -1.72). In the figure, spherical aberration of the whole eye at bending factor about +1 is
close to zero (0.09 waves) but the spherical aberrations for AIOL alone and for coma are
over-corrected. Similarly, in Figure 6.27(b) which represents the aberration of an
aspherised AIOL with an optimum asphericity to eliminate coma (Q = -0.18), coma
aberration of the whole eye is well corrected at bending factor -0.95 but it is over-corrected
for AIOL alone and under-corrected for spherical aberration. In both of these figures, and
consistent with the spherical designs of 1E-AIOL, coma aberration is more sensitive to the
bending factors, the spherical aberration has become relatively insensitive to the bending
factor once it is aspherised.
184
Optical Design
Bending Factor
4
2
SA
0
Coma
Coma
-2
-4
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
1.0
1.5
2.0
Criteria 1(ii)
Bending Factor
4
2
SA
0
Coma
Coma
-2
-4
-2.0
-1.5
-1.0
-0.5
0.0
Asphericity
0.5
Figure 6.24: Design of 1E-AIOL with aspheric front surface: Designs eliminating spherical
aberration and coma at 10° field for criteria 1(i) and (ii). The AIOL was made up of
PMMA (n = 1.492) and implanted 2 mm behind the pupil. Solid lines represent distance
and dashed lines represent near focus. 2.5 D accommodation was used for near focus.
Power of the front element is +35 D implanted 2 mm behind the 4 mm diameter pupil.
185
Optical Design
Criteria 2(i)
Bending Factor
4
2
SA
Coma
0
-2
-4
Coma
-2
-1
0
1
2
1
2
Criteria 2(ii)
Bending Factor
4
SA
2
Coma
0
Coma
-2
-4
-2
-1
0
Asphericity
Figure 6.25: Design of 1E-AIOL with front aspheric surface. Designs eliminating spherical
aberration and coma at 10° field for criteria 2(i) and (ii). The AIOL was made up of
PMMA (n = 1.492) and implanted 2 mm behind the pupil. Solid lines represent distance
and dashed lines represent near focus. 2.5 D accommodation was used for near focus.
Power of the front element is +35 D implanted 2 mm behind the 4 mm diameter pupil.
186
Optical Design
1E-AIOL: Solution for SI & SII
3
Bending Factor
2
1
Coma
0
Solutions
[-0.104, -0.471]
&
[-0.103, -1.46]
Spherical Aberration:
Solutions at extreme
values
-1
-2
-3
-2.0
-1.5
-1.0
-0.5
0.0
Asphericity
Figure 6.26: Design of 1E-AIOL for coma at 10° field and spherical aberration – target
spherical aberrations were 0.088 and 0.024 waves (Ȝ=0.587 ȝm) for distance and near
respectively; target coma 0.04 waves for both near and distance. Solutions are (X,Q) =
(-0.104, -0.471) and (-0.103, -1.46). Refractive index of the material is 1.492. The
asphericity to be provided on the front surface. Solid and dashed lines represent the design
for unaccommodated and accommodated states respectively.
Designs obtained in this example for the distance focus state closely resembles the
asphericities reported earlier for conventional monofocal IOL (Lu and Smith, 1990). In the
study of Lu and Smith, for zero spherical aberration of a model eye (corneal asphericity =
-0.26) with the IOL implanted 1 mm behind the pupil, made in PMMA material, bending
factor +1.0 (optimum for spherical aberration) and a power of +17 D required an
asphericity value of -1.72 which is in close agreement with our results (-1.41) illustrated in
Figure 6.26. The small difference might have originated in the difference in implant depth
(2 mm behind the pupil in this example) and different power of the lens (+ 19.85 D).
187
Optical Design
0.6
S I: Eye
S I: AIOL only
Aberrations (waves)
0.4
S II: Eye
S II: AIOL only
0.2
0.0
-0.2
-0.4
(a)
-0.6
-6
-4
-2
0
2
Bending Factor
4
6
0.6
Aberrations (waves)
0.4
0.2
0.0
-0.2
S I: Eye
S I: AIOL only
-0.4
S II: Eye
S II: AIOL only
(b)
-0.6
-6
-4
-2
0
2
Bending Factor
4
6
Figure 6.27:Aberrations of the aspheric 1E-AIOL implanted 2 mm behind the pupil with (a)
asphericity to eliminate spherical aberration (Q = -1.72) and (b) asphericity to eliminate
coma (Q = +0.18). The aberrations are calculated only for unaccommodated states. Pupil
diameter used is 4 mm, the lens material has refractive index 1.45. Solid and dotted lines
represent data for the whole eye and AIOL only, respectively.
In summary, aspherising the surface of 1E-AIOL may satisfy the design criteria in terms of
eliminating or controlling spherical or coma aberration but not both simultaneously. An
elliptical lens surface is required for controlling spherical aberration whereas elliptical as
188
Optical Design
well as hyperbolic surfaces may also be used in controlling coma within the field angle of
10º depending on the bending factor of the lens. It should be noted that the asphericity of an
AIOL is influenced by the power of the lens and position relative to the pupil in addition to
the corneal asphericity.
Again, an extreme bending factor and asphericity (which may be practically infeasible in
terms of manufacturing tolerance) are required to provide optimal solutions for near and
distance foci simultaneously (Figure 6.26). However, coma within a small field angle may
be readily corrected simultaneously for both near and distance vision.
6.6.5 Aspheric Design of 2E-AIOL
It has been demonstrated in the present chapter that a wide range of spherical 2E-AIOL
designs may satisfy the optical design criteria. Therefore, it may not be required to
aspherise a 2E-AIOL just to control or eliminate the aberration. However, we found that the
optimal solutions of the spherical designs either had clinically unacceptable forms or
required refractive indices that may not be readily available. Therefore a major advantage
of aspherising 2E-AIOLs is to obtain practical solutions which preserve a clinically useful
shape and practical values of the refractive index with as small an asphericity as possible.
Extending the derivations and notation for the aspheric 1E-AIOL, the spherical aberration
of the aspheric 2E-AIOL can be written as:
For 2E-AIOL only:
For distance:
For near:
S I − IOL = S I − front + α1 + S I −rear + α 2
S ' I −IOL = S 'I − front +α '1 + S 'I −rear +α '2
........................................................(6.37a)
...............................................................(6.38b)
189
Optical Design
For whole eye:
For distance:
For near:
S I −eye = S I −cornea + S I − front + α1 + S I −rear + α 2
S 'I −eye = S 'I −cornea + S 'I − front +α '1 + S 'I −rear +α '2
.........................................(6.39a)
...............................................(6.39b)
In Section 6.3.2 biologically optimum shapes of the elements in which bending factor for
the front element ranged between >-1 and <+1 were determined, whereas the rear element
had greater freedom but preferably ranging between >-1 and <+2.
Since not all the
solutions obtained in the spherical designs satisfied the biological criteria, the required
asphericities of the elements such that the elements fall within the defined shapes for a
commonly used lens material (n = 1.45) will be investigated. The front element power is
+35.0 D for all simulations. Front element bending factors tested in this simulation was
-0.5, 0.0 or +0.5, and the rear element bending factor was 0, 1 or 2. Figure 6.28 shows the
asphericity values for the selected optical criteria.
Briefly, it is observed in Figure 6.28 that elliptical and hyperbolic surfaces of the front
element and hyperbolic surface of the rear element is the most common form of aspheric
surface required for an optimum design of the AIOLs. When the front element possesses an
oblate ellipse (positive asphericity), the rear element is also required to be an oblate ellipse.
190
Optical Design
Criterion 1(i)
Criterion 1(ii)
4
4
2
2
I
Ql 2
0
D
H
0
H
C
G
C
-2
F
A
E
-6
-1.5
G
-2
B
F
-4
D
I
-4
-1.0
-0.5
Ql1
0.0
0.5
B
E
A
-6
-1.5
-1.0
Criterion 2(i)
-0.5
Ql
0.0
0.5
Criterion 2(ii)
4
4
2
2
I
D
Q l2
0
D
I
0
H
-2
G
H
-2
C
G
C
-4
B
F
F
B
E
-4
E
A
A
-6
-1.5
-1.0
-0.5
Ql1
0.0
0.5
-6
-1.5
-1.0
-0.5
Ql1
0.0
Figure 6.28: Asphericities of the front element (Ql1) and rear element (Ql2) for various
criteria. Q values differ slightly for each criterion. Solid lines represent design for distance
and dotted lines represent the design for near. Power of the front element (+35 D) and
refractive index (n = 1.45) are constant throughout. Labelled plots are for various
combinations of bending factors for the elements: (A) +0.5/+2.0; (B) +0.5/+1.0; (C)
+0.5/0; (D) 0/+2.0; (E) 0/+1.0; (F) 0/0.0; (G) -0.5/+2.0; (H) -0.5/+1.0; (I) -0.5/0; where
numbers in the numerator and denominator are bending factors of the front and rear
elements respectively.
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6.7 Discussion
The design philosophy of the AIOL is an extension of the standard IOL but need to be more
stringent because, while the latter is fixed in position, the former continuously changes its
axial position within the eye. This brings about changes in its aberration profile. An
optimum design of AIOL should maintain satisfactory performance in all positions. It is
very important, but difficult to define an “optimum design” particularly as differences in
philosophy exist concerning whether to totally eliminate aberrations from the eye or to
preserve some residual aberrations that match the aberration profile of the natural eye.
Aberration profiles in the population vary widely and inconsistently which makes the task
of setting criteria for the optimum level of aberration to be controlled even more difficult.
In the pseudophakic eye, spherical aberration increased with the translation of the optics in
both types of AIOL, whereas it decreases in the phakic eye with accommodation (Cheng et
al., 2004). Therefore, correcting or controlling spherical aberration in an AIOL is essential
to preserve or improve retinal image quality. Use of the model eye in the present study
indicated that, as is the case of conventional IOL, the spherical aberration of the eye cannot
be eliminated with a 1E-AIOL. An optimum 1E-AIOL design will only minimise
aberrations. Coma may be eliminated theoretically when the lens is implanted in the pupil
plane; however, this would be an unrealistic position as the AIOL is typically implanted in
the posterior chamber to create some space between the lens and the iris. The bending
factor and refractive index are two important parameters that are commonly used as
variables in the design process but both of these parameters also have limits. The
availability of material for fabrication of an AIOL and physical shapes that are biologically
acceptable are some of the constraints.
It was determined that the aberration outcomes of bending a 1E-AIOL to produce minimum
spherical aberration and coma are essentially similar to those obtained with a standard IOL.
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Optical Design
Optimal bending factors of an 1E-AIOL to be implanted 1 mm behind the pupil for
minimum spherical aberration and coma are opposite in sign: +1.0 or plano-convex with
curved surface facing the cornea for minimum spherical aberration and -1 or plano convex
with curved surface facing the retina for minimum coma. This indicates that a compromise
may be required to balance these aberrations to an ‘acceptable’ level. The optimum bending
of the 1E-AIOL does not change significantly with accommodation, though slightly higher
spherical aberration and lower coma are predicted when the 1E-AIOL translates. Within the
expected range of pseudophakic accommodation, spherical aberration increases and the
coma decreases with accommodation. Hence, an AIOL that controls, for example, spherical
aberration for both distance and near and coma only for distance may offer satisfactory
performance and realistic design solutions.
Selecting the shape which produces minimum possible coma and subsequently aspherising
it to control the spherical aberration is potentially a better option, since coma is more
sensitive to the bending factor whereas spherical aberration is more sensitive to the
asphericity. It was determined that eliminating spherical aberration and coma from the eye
requires an elliptical or hyperbolic surface of the 1E-AIOL with asphericity values < 0.
Required asphericity also depends on the power of the lens and implant position. It was
shown that an asphericity value effective in controlling aberration at distance focus state
may not be effective at near focus state; the optimal solution requires an extremely elliptical
surface.
The 2E-AIOL has additional degrees of design freedoms such as distribution of power,
bending factors and refractive indices between two elements. However, the results in our
study indicate that bending factor is the parameter that can be most readily exploited.
Therefore, in contrast to a spherical design 1E-AIOL, a wide range of 2E-AIOLs may be
designed to eliminate or control the aberrations to a desired level. However, only a few
combinations afford optimum performance for near and distance focus conditions
simultaneously. Further this requires a material of high refractive index; particularly for the
front element. Our results suggest that a 2E-AIOL with a lower power front element has a
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Optical Design
flatter rate of change in spherical aberration with the change in bending factor, which is
advantageous. However, a 2E-AIOL with a lower power front element returns less
accommodative amplitude.
This disadvantage may be overcome by imposing asphericities on the elements.
Appropriate combination of the asphericities between the elements may be imposed to
achieve all optical and biological criteria. The surface may be oblate ellipse (Q>0), prolate
ellipse (-1<Q<0) or hyperbolic (Q <-1).
194
Optical Design
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Chapter 7
Performance in the Presence of Misalignment
Part of this chapter was presented in the following conference
Ale J, Manns F, Ho A “The Optical Performance of Accommodating IOL in presence of
Misalignment”. Association for Research in Vision and Ophthalmology (ARVO) May 2-6
2010 (Poster 5744/D950), annual meeting, Fort Lauderdale, FL.
205
Misalignment of AIOL
TABLE OF CONTENT
7.1 INTRODUCTION ...................................................................................................... 208
7.2 LITERATURE: MISALIGNMENT OF CONVENTIONAL IOL........................ 210
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7.3 METHODS.................................................................................................................. 217
7.4 TILT AND DECENTRATION RESULTS .............................................................. 223
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7.5 DESIGN OF 2E-AIOL IN THE PRESENCE OF MISALIGNMENT ................. 236
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206
Misalignment of AIOL
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7.6 DISCUSSION.............................................................................................................. 246
7.7 REFERENCES ........................................................................................................... 250
7.8 APPENDIX ................................................................................................................. 258
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207
Misalignment of AIOL
7.1 Introduction
Misalignment of an IOL, including tilt and decentration, is one of the most persistent and
frequently reported post-operative complications (Apple et al., 1984, Bush, 1983, Mamalis
et al., 2008) which occurs even with uneventful implantation in the capsular bag. Several
reports investigating tilt and decentration have proposed a number of factors associated
with these complications. The rate and extent of the complication has substantially
decreased with improved IOL designs and surgical techniques (Linnola and Holst, 1998);
yet, extreme misalignments requiring explantation are being reported in some cases
(Gimbel et al., 2005, Mamalis et al., 2008). The rate of explantation is 0.77 – 2.7%, of
which, 10 to 40% are due to misalignment alone, depending on the type of IOL and
surgical procedure (Auffarth et al., 1995, Lyle and Jin, 1992, Mamalis et al., 2008).
For a conventional monofocal IOL, a certain degree of tilt and decentration are not usually
regarded as of clinical importance (Baumeister et al., 2009, Mester et al., 2009, Norrby et
al, 2007). However, theoretical studies have shown that even a small misalignment of
modern IOL designs including multifocal and customized aberration-correcting IOLs, leads
to significantly reduced visual performance (Altmann et al., 2005). Theoretical reports
suggest that aspheric IOLs (aberration-correcting or customized wavefront-correcting and
aberration-free) are extremely sensitive to misalignment (Eppig et al., 2009, Montes-Mico
et al., 2009, Pieh et al., 2009, Piers et al, 2007). Holladay et al. (2002) indicated that when
decentred by more than 0.4 mm and tilted by more than 7º, the performance of aspheric
IOL is worse compared to that of spherical IOL. Altman and colleagues (Altmann et al.,
2005) warned the advantage of aspheric IOL is lost when it is decentred by more than 0.5
mm. Pieh et al. (2009) reported that the Strehl ratio decreases more rapidly for wavefront
customized IOL compared to other types of IOL. Therefore, increasing interest in
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Misalignment of AIOL
correcting aberrations in a pseudophakic eye, by means of IOL technology, demands
additional precision in IOL alignment and centration.
This Chapter analytically evaluates the consequences of tilt and decentration of AIOLs.
Various degrees of these misalignments are imposed on various designs of 1E and 2E
AIOLs to determine the effect on aberrations.
While intuitively, it may be anticipated that the performance of misaligned 1E-AIOLs and
monofocal IOLs should not differ greatly; significantly deteriorated optical performance
might be expected when a 2E-AIOL is misaligned. Further, a small tilt and decentration
may perhaps induce substantial levels of refractive errors and aberrations. This notion may
be formed primarily because of the high dioptric power of the lenses used in 2E-AIOL.
Therefore, we may hypothesise that implantation of such AIOLs demands additional
precision in surgical and implant technologies.
Tilt and decentration of a lens alter the aberration profile of the pseudophakic eye,
primarily altering the angle and height of an incoming ray on the lens surface which is
analogous to off-axis ray incidence. Hence, off-axis aberrations such as coma and
astigmatism are potentially affected. Several clinical studies on misalignment of monofocal
IOLs have been reported which provide important information on how higher-order
aberrations are affected. No studies have reported the effect of misalignment of the 2EAIOL. In this chapter, using exact ray tracing methods, the potential effect of tilt and
decentration of AIOLs on optical performance is investigated. The effect of tilt and
decentration of 2E-AIOLs, both for individual elements and as a group, on higher-order
aberrations up to 4th order, spherical (C40), coma (C±31), and astigmatism (J180) is
evaluated, and the effect of the element shapes and select proposed optimal designs is also
analysed.
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Misalignment of AIOL
Before describing the study, the following section reviews the literature on tilt and
decentration of monofocal and multifocal IOLs. The published data will help in estimating
the typical and boundary values of individual misalignments to be tested in the simulations.
7.2 Literature: Misalignment of Conventional IOL
7.2.1 Methods used for Measuring Tilt and Decentration
Purkinje and Scheimpflug image analyses are two methods that have been used to measure
tilt and decentration of IOL in vivo. Purkinje images are the images formed by reflected
light from the corneal and lens surfaces. The size and relative position of Purkinje III (from
the anterior lens surface) and Purkinje IV images (from the posterior lens surface) can be
used in determining the tilt and decentration of an IOL. A systematic approach of using this
technique was described by Phillips et al. (1988), who used a camera to photograph
Purkinje images. Using this method, Phillips and colleagues found an average 7.8º tilt and
0.7 mm decentration in 13 subjects. The method was subsequently improved and is widely
used to measure tilt and decentration of the crystalline lens (Barry et al., 2001) and IOL
(Auran et al., 1990, Akkin et al., 1994).
Using a slit lamp, Scheimpflug imaging provides a sharp image of the entire anterior
segment of the eye (Sasaki et al., 1989, Wolffsohn and Davis, 2007). A narrow slit beam
delivered perpendicularly to the cornea illuminates the cornea and lens. The light
backscattered by the ocular media is imaged by an objective onto a camera. To satisfy the
Scheimpflug condition, the optical axis of the objective is typically oriented at an angle of
45º from the slit beam and the camera is mounted perpendicularly to the slit beam (Koretz
et al., 2004). Today, there are several commercially available Scheimpflug instruments
designed for anterior segment imaging such as EAS-1000 (NIDEK Co. Ltd, Japan),
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Misalignment of AIOL
Pentacam (Oculus Optikgeräte GmbH, Wetzlar, Germany) and Galilie (Ziemer Group,
Switzerland).
7.2.2 Incidence and Extent of the Misalignments
The rate and extent of tilt and decentration varies across the studies depending on factors
such as surgical techniques, IOL type, fixation site, pathology, post implant duration and
methods of measurement. Brems et al. (1986) reported the presence of a positioning hole of
the IOL within the papillary area (pupil diameter = 3.45 mm) in 71% of 75 autopsied eyes,
indicating a high rate of decentration. The degree of misalignment also varied among
reports. Highest tilt and decentration was reported in envelope and continuous curvilinear
(circular) capsulotomy (CCC) groups (Akkin et al., 1994). Maximum tilts in these groups
were 22º and 13.8º and maximum decentrations were 2.7 mm and 1.3 mm respectively.
More recent studies demonstrated a reduced amount of tilt and decentration (Baumeister et
al., 2005).
7.2.3 Factors affecting the Alignment of IOL
7.2.3.1 Fixation Site
Haptics position can be categorized as symmetric (both haptics in the bag or bag-bag
fixation and both in the ciliary sulcus or sulcus-sulcus fixation) or asymmetric (one haptic
in the bag and another in the ciliary sulcus or bag-sulcus fixation). Hansen and colleagues
(Hansen et al., 1988) reported that 60% of the asymmetrically fixated IOLs (bag-sulcus)
decentred more than 0.8 mm whereas 75% of the symmetrically fixated IOLs (bag-bag or
sulcus-sulcus) decentred less than 0.6 mm. For the symmetrical fixations, 13% of the
subjects with bag-bag fixation and 5% of the subjects with sulcus-sulcus fixation had
decentration ≤ 0.2 mm. Several studies have been published reporting the effect of haptics
position on the misalignment of the IOL (Table 7.2)
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Misalignment of AIOL
Table 7.1: Effect of capsulotomy on tilt and decentration of IOLs. CCC – continuous
curvilinear capsulorhexis
Author
Type
Tilt (degrees)
Dec (mm)
Comment
Envelop
5.66 (max=22)
0.65 (max=2.7)
Incidence rate = 74%
CCC
1.13 (max=13.8)
0.15 (max=1.3)
Incidence rate = 31%, p = <0.01
No tear
1.1
0.09
With tear
4.16
0.34
Caballero et al.,
Envelop
-
0.42±0.02
1995
CCC
-
0.27±0.01
p =<0.01for CCC with tear vs.
No tear
--
0.23±0.02
no tear
With tear
--
0.42±0.06
No statistical difference
CanOpener
6.95±2.46
0.66±0.28
Can Opener
8.49±4.86
0.89±0.39
Envelope
7.30±3.81
0.78±0.22
CCC no tear
3.80±1.74
0.26±0.08
CCC 1 tear
5.66±2.33
0.48±0.25
CCC 2 tears
6.35±3.48
0.67±0.34
CCC with tear
--
0.35±0.25
No tear
--
0.18±0.09
no tear
1.25±0.57
0.21±0.17
The tears (incision) created
2tear
1.22±0.77
0.27±0.33
using YAG laser
3tear
1.41±0.62
0.24±0.14
no tear
--
0.20±0.05
1 tear
--
0.22±0.08
2 tear
--
0.46±0.25
Akkin et al., 1994
Lu
and
Shen,
Identical to envelope technique
1999
Oner et al., 2001
Assia et al., 1993
Hayashi
et
al.,
2008
Legler
1992
et
al.,
CCC with no tear and 1 tear had
significantly
less
tilt
and
decentration compared to other
groups (p = <0.05), but there
was no difference between these
two
IOL position in the bag
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Misalignment of AIOL
Table 7.2: Effect of fixation site/haptics position in the bag on tilt and decentration
Fixation site (Tilt /Decentration)
Author
Akkin
Bag-Bag
et
al.,
et
al.,
1994
Assia
1993
5.43/0.43
Bag-Sulcus
8.8/1.24
Sulcus-Sulcus
--
Other note
Envelope capsulotomy
Difference significant for
-- /0.31
--/0.66
-- /0.39
position (p = <0.01) but not for
-- /0.25
-- /0.63
-- /0.54
loop type
Hayashi et al.,
3.18±1.66/0.2
2.93±1.81/0.47±0.4
1999
9±0.21
7
-- /0.65
Bag-sulcus had capsule tear
-- /0.46
-- /0.91
--
-- /0.20±0.05
-- /0.68±0.28
--
-- /0.40
-- /0.80
J loop
C loop
Caballero et al.,
1991
Legler
et
al.,
1992
Hansen et al.,
1988
--/0.60
(max 2.31)
Postmortem eyes, p = <0.05
7.2.3.2 Capsulotomy Type and Integrity
The can-opener technique was reported to be the least effective capsulotomy type for IOL
centration compared to the envelope and CCC techniques. While the intact CCC was the
best method, its effectiveness decreased in the presence of tear, to a level similar to the
other two methods. A summary of results from various studies reporting decentration of
IOLs as a function of capsulotomy types is presented in Table 7.1.
7.2.3.3 Pathology
Due to complex cellular reactions and biological changes, shrinking of the capsular bag is
reported to be marked when the eye is predisposed to some pathologies such as
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Misalignment of AIOL
pseudoexfoliation (Davison, 1993, Hayashi et al., 1998a), diabetes (Kato et al., 2001),
glaucoma (Hayashi et al., 1999a) and retinitis pigmentosa (Nishi and Nishi, 1993, Hayashi
et al., 1998c). Tilt and decentration in various pathologies are shown in Table 7.3.
Table 7.3: Effect of ocular pathology on the alignment of IOL. PE – pseudoexfoliation, RP
– retinitis pigmentosa, CAG – closed angle glaucoma, OAG –open angle glaucoma.
Author
Pathology (n)
IOL description
Tilt (degrees)
/Decentration (mm)
Hayashi et al.,
1998a
PE (53)
Control (53)
Both group had Soft
acrylic IOL 6 mm optic
3.66±2.12/0.28±0.18
2.77±1.97/0.30±0.15
p = 0.024/p = 0.36
Hayashi et al.,
1998c
RP (47)
Control (47)
Both group had 1pc
PMMA IOL 6 mm optic
3.54±2.78/0.40±0.32
2.46±1.23/0.26±0.14
p=0.048/0.002
Hayashi et al.,
1999b
ACG (29)
OAG (23)
Control (52)
All received soft acrylic
foldable IOL
3.99±2.07/0.33±0.26
3.38±2.33/0.30±0.16
2.71±1.93/0.27±0.10
p = tilt/decentration
p=0.03/0.32
7.2.3.4 Other Factors
The effect of other factors such as optics/haptics material, size, configuration of the loop
and optics design on IOL alignment has been debated. Caballero et al. (1995) found
significantly less decentration of IOLs with total diameter of ≤11.0 mm than those which
had overall diameter • 13.5 mm. The authors suggested that the C or J loops comprise short
contact arches resting against the bag equator, and hence asymmetric fibrosis may easily
displace the lenses in one direction. In contrast, Legler et al. (1992) found no difference
between IOLs with various loop diameters (12-14 mm) implanted in three human eyes
obtained postmortem. The study also involved IOLs with various constructions (1-piece
and 3-piece), and haptic configurations (J and C loops). Nejima et al. (2006) also did not
find any difference with haptic angulations (0º and 10º) and materials (acrylic and PMMA).
Several studies compared tilt and decentration for various designs of IOL including optic
diameter (Mutlu et al., 1998, Taketani et al., 2004a), surface designs (Ohtani et al., 2009,
Taketani et al., 2005), monofocal and multifocal (Hayashi et al., 2001, Jung et al., 2000)
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Misalignment of AIOL
and optics material (Baumeister et al., 2005, Hayashi et al., 1997); no difference was
observed.
7.2.4 Consequences of Tilt and Decentration of IOL
7.2.4.1 Refractive Error and Visual Acuity
Any amount of tilt and decentration may induce higher-order aberrations, leading to
reduced retinal image quality. However, sufficiently large misalignment is required to
produce clinically observable spherical and astigmatic refractive errors and associated
diminution of vision. Theoretical and clinical studies suggest that minor tilt and
decentration of the standard IOL induce very small amounts of refractive error which are
not sufficient to decrease the clinically measureable visual acuity. A study in a model eye
found 0.42 D and 0.58 D defocus with 1 mm decentration and 10º tilt respectively in
average (Turuwhenua, 2005). Atchison found 1.5 D astigmatism with 20º tilt (Atchison,
1989b). One clinical study found unaffected contrast sensitivity and stereopsis in the eyes
with average tilt and decentration (Hayashi and Hayashi, 2004).
7.2.4.2 Higher-Order Aberrations
Tilt and decentration of an IOL are identified as factors associated with higher-order
aberrations (HOA) in a pseudophakic eye (Oshika et al., 2005, Taketani et al., 2004b).
Clinical studies have shown that both coma-like aberration and astigmatism are more
sensitive to tilt and decentration (Oshika et al., 2005, Taketani et al., 2005) of an IOL. In an
extreme form of tilt (28.7º) and decentration (1.78 mm), Oshika et al. (2005) reported very
high coma-like aberration (0.45 μm) whereas spherical aberration was not affected (0.08
μm). Theoretical studies using model eyes (Atchison, 1989a, Turuwhenua, 2005) have
found depressed modulation transfer functions (MTF) when the IOL is tilted and displaced.
Barbero et al., (2003) in their theoretical investigation, found significant aberration (0.15
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Misalignment of AIOL
μm) with 4º tilt of a spherical IOL, which is sufficiently large to degrade retinal image
quality.
Some investigators have indicated that the aberration-compensating effect of the “lens
position factor” is inherent to pseudophakic eyes (Artal et al., 2001, Kelly et al., 2004,
Marcos et al., 2008). In pseudophakic eyes 36% to 70% spherical and 51% to 87% coma
aberrations were compensated. This compensation was attributed in part to tilt and
decentration of the IOL. However, complete data on the pre-operative position of the
crystalline lens and post-operative position of the IOL, which were lacking in the studies, is
required to validate this assumption.
7.2.5 Direction of Tilt and Decentration
The direction of tilt and decentration of an IOL has a significant effect on clinical outcome.
Moreover, pre-operative position of the crystalline lens (capsular bag) has an important role
in determining the position of the IOL after implantation. The crystalline lens is not
perfectly centred with respect to the visual axis and the pupil centre (Artal & Tabernero,
2008). A few studies have attempted to investigate the exact position of the crystalline lens.
Mester et al. (2009) investigated the direction of tilt and decentration of posterior chamber
IOLs (PCIOLs) and compared it with the alignment of the young crystalline lens. All
crystalline lenses displaced temporally and downward whereas the IOLs displaced nasally
and marginally upward. Table 10 summarizes and compares the tilt and decentration of the
crystalline lens and IOLs.
7.2.6 Alignment of AIOL
Except for two case reports of extreme misalignment of 1E-AIOLs (Cazal et al., 2005,
Oshika et al., 2005) (one of which was repositioned and another explanted eventually), no
studies have investigated the pattern of tilt and decentration of AIOLs. Due to the complex
216
Misalignment of AIOL
mechanical design of the haptics, any pattern in AIOL misalignment is of significant
interest. Typically, the AIOL haptics are hinged or spring-loaded such that they flex under
the influence of accommodative force to facilitate translation (Dick, 2004, McLeod et al.,
2003). Capsular contraction and asymmetric fibrosis may lead to significantly higher levels
of misalignment of such devices.
Table 7.4: Tilt and decentration of the crystalline lens compared with IOL position. Under
the columns covered by tilt; (I) denotes the forward slanting of the inferior edge (S) denotes
forward shifting of superior edge, (N) denotes forward shifting of nasal edge, (T) denotes
forward shifting of temporal edge. Similarly, under the columns covered by decentration,
(D) denotes downward displacement, (U) denotes upward displacement, (T) denotes
displacement towards temporal side and (N) denotes displacement towards the nasal side.
Author
Tilt (deg, Mean)
Decentration (mm, Mean)
Vertical
2.2 (I)
Horizontal
3.1 (N)
Vertical
0.16 (D)
Horizontal
0.07 (T)
2.5 (I)
2.6 (N)
0.02 (U)
0.06 (N)
Crystalline
0.77 (I)
1.05 (T)
0.06 (D)
0.28 (T)
IOL
2.30 (I)
0.87 (T)
0.41 (D)
0.25 (T)
Lens Type
Crystalline
Mester et al., 2009
Rosales and Marcos,
2006
IOL
Kirschkamp et al.,
2004
Crystalline
0.1 (I)
--
--
0.07 (T)
Dunne et al., 2005
Crystalline
0.1 (S)
--
--
0.06 (T)
Auran et al., 1990
IOL
Tscherning, 1924
Crystalline
de Castro et al., 2007
IOL
6.75 (S/N)
0.64 (U/T)
4-7 (S/N)
0.25 (U)
1.89 (S)
2.34 (N)
0.17 (U)
0.17 (N)
7.3 Methods
Ray tracing through a finite eye model using AIOL and computation of the aberrations,
were conducted in the optical design software Zemax (August 2009 to September 2010,
Zemax Development Corporation). The crystalline lens of the model eye was replaced with
217
Misalignment of AIOL
a single-element (1E) or two-element (2E) AIOL. A model of the 1E-AIOL included in this
experiment possessed bending factor 0.98 which is the optimum shape of the spherical
design in terms of controlling spherical aberration (Chapter 6, Table 6.3). 2E-AIOL had
bending factors 0.45 and 1.51 for the front element and rear elements, respectively, which
represents an optimum design combination of the elements to eliminate spherical aberration
from the eye (Criteria 1(ii) in Chapter 6). The front element power was +35.0 D; both
elements had identical refractive indices (n = 1.45). The lenses had spherical surfaces.
The effects of mispositioning on the Zernike coefficients for spherical aberration (C40),
coma (C±31), and astigmatism were calculated. Astigmatism was converted into the power
vector component J180 in dioptres (Atchison, 2004b) using equation:
J 180 =
− ( 2 6C 22 − 6 10 C 42 )
2
......................................................................................(7.1)
where C 22 and C42 are the Zernike coefficients for second and fourth-order astigmatism
respectively and r is the pupil radius.
218
Misalignment of AIOL
Aligned elements
Front tilt
Front decentration
Rear tilt
Rear decentration
System tilt
System decentration
Figure 7.1: Diagrams to illustrate the aligned and misaligned elements in 2E-AIOL
iterated in the simulations.
219
Misalignment of AIOL
The following steps were followed:
1) To investigate the effect of the lens form, aberrations were calculated for various
bending factors (X) of the 1E-AIOL and both elements in 2E-AIOL ranging
between -2.0 and +2.0 in the step of 0.5 in presence of various combinations of tilt
and decentration.
2) From the literature survey, 10º tilt and 1 mm decentration are the limits of
misalignment. Hence the lenses were decentred up to 1 mm in 0.02 mm steps along
the y-axis and tilted up to 10º in 2º steps about the x-axis. Parameters 2
(decentration) and 3 (tilt) in the multi-configuration setting of the software (Zemax)
were used to achieve the desired misalignment.
3) Tilt and decentration can be uni- or multi-axial. We considered misalignment only
in one direction: tilt about the x-axis and decentration along the y-axis. These
settings produce a zero horizontal coma and zero oblique component of the
astigmatism, which simplified the simulation and analysis. In practice, tilt and
decentration are likely to occur simultaneously in real conditions. To determine the
combined effect of various degrees of tilt and decentration (in addition to each
misalignment individually), the aberrations were calculated for the following
combinations:
i. Moderate tilt (5º) + moderate decentration (0.5 mm), denoted as 5/0.5
ii. Extreme tilt (10º) + extreme decentration (1 mm), denoted as 10/1.0
4)
Misalignment of 2E-AIOLs can be grossly classified as misalignment of the front
element alone, rear element alone and of the two elements together as a group.
220
Misalignment of AIOL
Similarly misalignment can be tilt only, decentration only and combined (tilt +
decentration). This leads to the following simulation settings for 2E-AIOLs:
i. Effect of tilt
o When the front element alone is tilted
o When the rear element alone is tilted
o When tilted as a group (equal amount of tilt of front and rear element)
ii. Effect of decentration
o When the front element alone is decentred
o When the rear element alone is decentred
o When decentred as a group (equal amount of front and rear element
decentration)
iii. Effect of tilt + decentration
o When only the front element is tilted and decentred
o When only the rear element is tilted and decentred
o When the system is tilted and decentred as a group (equal amount for
front and rear element)
To calculate the effect of misalignment, the following difference was quantified:
221
Misalignment of AIOL
ǻ aberration = aberration with misalignment – aberration without misalignment
With the on-axis object and no misalignment, coma and astigmatic aberrations are zero;
hence any coma and astigmatic aberrations present with misalignment, is due only to the
misalignment and no adjustment is required for these aberrations. Two axial positions of
the lens elements were considered: distance focus and near focus.
Image quality is a derivative of the interaction between the aberrations. Understanding the
tolerance level of the aberrations in the eye is complex as the neural factor also comes into
play. Therefore, to predict the retinal image quality with useful accuracy on the basis of one
or two aberration coefficients may be exceeding the bounds of validity. However, a rough
estimate can be made by converting the wave aberrations into the spherical error (S.E.)
form using the following equation (Atchison, 2004a):
SE =
4 3C nm
2
............................................................................................................... (7.1)
m
where, Cn is the Zernike coefficient for the specific aberration.
Considering that 0.25 D of spherical equivalent may be sufficient to drop visual acuity to a
clinically observable extent, 0.082 waves (or 0.14 micron; Ȝ= 0.5876 ȝm) with pupil
diameter 4mm (r = 2) may be assumed as the clinical limit of tolerance.
222
Misalignment of AIOL
7.4 Tilt and Decentration Results
7.4.1 Tilt and Decentration of 1E-AIOL
Changes in spherical aberration (C40) with up to 10º tilt and up to 1 mm decentration were
extremely small. Coma and astigmatism (J180) increased with increase in tilt. The bending
factors between -0.5 and + 1.5 showed minimum changes in aberrations except for
astigmatism with maximum tilt (Figure 7.2). This is consistent with the optimum bending
factor for the aligned optical system (Chapter 6).
Ϭ͘ϭϬ
Ϭ͘Ϭϱ
Ϭ͘ϬϬ
ͲϬ͘Ϭϱ
;ϯ͕ϭͿ
:ϭϴϬ
;ϰ͕ϬͿ
ͲϬ͘ϭϬ
ͲϮ
Ͳϭ
Ϭ
ϭ
Ϯ
ĞŶĚŝŶŐ&ĂĐƚŽƌ
Figure 7.2: change in spherical, coma and astigmatism due to 10° (red lines) and 5 degrees
(blue lines) tilting of 1E-AIOL as a function of shape factors. Solid and dotted lines
represent data for unaccommodated and accommodated states. Differences of change in
(C40) between the extents of tilt are extremely small, hence the plots are overlapped.
223
Misalignment of AIOL
ǻ ďĞƌƌĂƚŝŽŶƐ;ǁĂǀĞƐŽƌͿ
Ϭ͘ϯϬ
Ϭ͘ϮϬ
Ϭ͘ϭϬ
Ϭ͘ϬϬ
ͲϬ͘ϭϬ
;ϯ͕ϭͿ
:ϭϴϬ
;ϰ͕ϬͿ
ͲϬ͘ϮϬ
ͲϮ
Ͳϭ
Ϭ
ϭ
Ϯ
ĞŶĚŝŶŐ&ĂĐƚŽƌ
Figure 7.3: change in spherical (in wave), coma (in wave) and astigmatism (in dioptre) due
to 1 mm (represented by red lines) and 0.5 mm (represented by blue lines) decentration of
1E-AIOL as a function of shape factors. Solid and dotted lines represent the aberrations for
unaccommodated and accommodated states respectively. Differences of change in
spherical aberration (C40) between the extents of tilt are extremely small, hence the plots
are overlapped.
Spherical aberration is more sensitive to tilt than to decentration while coma aberration is
comparatively more sensitive to decentration (Figure 7.2 vs 7.3). In the presence of
decentration alone, changes in all types of aberrations were minimal for bending factors
ranging between zero and +1.0 (Figure 7.3). When the bending factor ranged between -1
and +1.5, the maximum tilt and decentration tested in this computation produced a change
in aberration less than 0.08 waves, the limit above which we assumed the aberration affects
visual acuity.
The combined effect of misalignment, tilt and decentration of 1E-AIOL on various
aberrations can be seen in Figures 7.4 to 7.6. Combined misalignments exhibited significant
changes compared to the individual misalignments.
224
Misalignment of AIOL
ϱͬϬ͘ϱ
ϭϬͬϭ͘Ϭ
ǻ ;ϰ͕ϬͿǁĂǀĞƐ
Ϭ͘ϲϬ
Ϭ͘ϰϬ
Ϭ͘ϮϬ
Ϭ͘ϬϬ
ͲϮ͘ϬϬ
Ͳϭ͘ϬϬ
Ϭ͘ϬϬ
ϭ͘ϬϬ
Ϯ͘ϬϬ
ĞŶĚŝŶŐ&ĂĐƚŽƌ
Figure 7.4: Change in spherical aberration (C4,0) as a function of bending factor of
spherical 1E-AIOL and combinations of tilt and decentration. The combinations of
misalignments imposed are: 5º tilt and 0.5 mm decentration (represented by red plots and
10º tilt with 1 mm decentration represented by blue plots. Solid lines represent data for
unaccommodated state and the dotted lines data for accommodated state.
Bending factors between zero and +1.5 are sufficiently resistant to misalignments (up to 10º
tilt and 1 mm decentration) limiting the change in aberration to the clinically insignificant
level (changes less than 0.08 waves). However, coma and astigmatism increased sharply
with the change in lens shape. Change in coma is within the clinically tolerable level when
the bending factor is about -1. Even a small change in the bending factor produced
significantly high aberration (Figure 7.5). Astigmatism is also extremely sensitive to the
bending factor. Bending factors between -1 and zero are effective in controlling
astigmatism in the presence of combined misalignments (Figure 7.6).
225
Misalignment of AIOL
ȴ ;ϯ͕ϭͿǁĂǀĞƐ
Ϭ͘ϱϬ
Ϭ͘ϬϬ
ͲϬ͘ϱϬ
Ͳϭ͘ϬϬ
Ͳϭ͘ϱϬ
ͲϮ͘ϬϬ
ͲϮ͘ϬϬ
Ͳϭ͘ϬϬ
Ϭ͘ϬϬ
ϭ͘ϬϬ
Ϯ͘ϬϬ
Figure 7.5: Change in vertical coma (C3-1) of a spherical 1E-AIOL as a function of
bending factor and combinations of tilt and decentration. The combinations of
misalignments imposed are: 5º tilt and 0.5 mm decentration (represented by red plots and
10º tilt with 1 mm decentration represented by blue plots. Solid lines represent data for the
unaccommodated state and the dotted lines data for the accommodated state
226
Misalignment of AIOL
ϱͬϭ͘Ϭ
ϭ͘ϱϬ
ϭϬͬϭ͘Ϭ
ϭ͘ϬϬ
Ϭ͘ϱϬ
Ϭ͘ϬϬ
ͲϬ͘ϱϬ
Ͳϭ͘ϬϬ
ͲϮ͘ϬϬ
Ͳϭ͘ϬϬ
Ϭ͘ϬϬ
ϭ͘ϬϬ
Ϯ͘ϬϬ
ĞŶĚŝŶŐ&ĂĐƚŽƌ
Figure 7.6: Change in astigmatism (J180) as a function of bending factor for a spherical
1E-AIOL and various combinations of tilt and decentration. The combinations of
misalignments imposed are: 5º tilt and 0.5 mm decentration (represented by red plots and
10º tilt with 1 mm decentration represented by blue plots. Solid lines represent data for the
unaccommodated state and the dotted lines data for the accommodated state.
7.4.2 Tilt and Decentration of 2E-AIOL
7.4.2.1 Effect of Tilt
Effect of tilt on spherical aberration was negligible for both accommodated and
unaccommodated states (Figure 7.7a). Coma was linearly related to the angle of tilt. Tilting
of the front element and the system as a group induced negative change in coma whereas
tilting of the rear element induced positive changes (Figure 7.8a). The changes increased
with the pseudophakic accommodation. Astigmatic aberration (J180) increased with the
angle of tilt in a quadratic fashion (Figure 7.9a), conforming to virtually a perfect fit with a
second order polynomial (R2 = 1.00).
227
Misalignment of AIOL
ȴ ;ϰ͕ϬͿǁĂǀĞƐ
Ϭ͘ϬϭϮ
&ƌŽŶƚ
ZĞĂƌ
Ϭ͘ϬϬϴ
Ϭ͘ϬϬϴ
Ϭ͘ϬϬϰ
Ϭ͘ϬϬϰ
Ϭ͘ϬϬϬ
Ϭ͘ϬϬϬ
ͲϬ͘ϬϬϰ
ZĞĂƌ
/Ŷ'ƌŽƵƉ
ͲϬ͘ϬϬϰ
Ϭ
Ϯ
(a)
ϰ
ϲ
dŝůƚ;ĞŐƌĞĞƐͿ
ϴ
ϭϬ
(b)
ĞĐĞŶƚƌĂƚŝŽŶ;ŵŵͿ
0
Figure 7.7: Changes in Zernike coefficient for spherical aberrations C 4 in waves (Ȝ =
0.587 ȝm) due to (a) tilt and (b) decentration of the individual elements and the system as a
group. Solid and dotted lines represent data for unaccommodated and accommodated
states, respectively.
7.4.2.2 Effect of Decentration
For ”1 mm decentration of the individual elements or the system as a group, increase in
spherical aberration did not exceed 0.01 waves (Figure 7.7b). Change in coma was again
linearly related to the amount of decentration (Figure 7.8b). Astigmatic aberration (J180)
also increased with decentration; the pattern of the increment was quadratic in nature
(Figure 7.9b) conforming to a perfect fit with a second order polynomial (R2 = 1.00). All
these aberrations were higher at the accommodated state, except when the rear element
alone was decentred.
228
Misalignment of AIOL
Ϭ͘Ϯ
Ϭ͘ϭ
Ϭ͘ϭ
Ϭ͘Ϭ
Ϭ͘Ϭ
ͲϬ͘ϭ
ͲϬ͘ϭ
&ƌŽŶƚ
ZĞĂƌ
/Ŷ'ƌŽƵƉ
ͲϬ͘Ϯ
&ƌŽŶƚ
ZĞĂƌ
ͲϬ͘Ϯ
/Ŷ'ƌŽƵƉ
ͲϬ͘ϯ
ͲϬ͘ϯ
Ϭ
Ϯ
(a)
ϰ
ϲ
ϴ
ϭϬ
Ϭ
Ϭ͘Ϯ
Ϭ͘ϰ
Ϭ͘ϲ
Ϭ͘ϴ
ϭ
(b)
dŝůƚ;ĞŐƌĞĞƐͿ
1
Figure 7.8: Changes in Zernike coefficient for coma C3 in waves (Ȝ = 0.587 ȝm) due to
(a) tilt and (b) decentration of the individual elements and the system in a group. Solid and
dotted lines represent data for unaccommodated and accommodated states, respectively.
Ϭ͘ϰ
&ƌŽŶƚ
&ƌŽŶƚ
ZĞĂƌ
ZĞĂƌ
/Ŷ'ƌŽƵƉ
Ϭ͘Ϯ
Ϭ͘Ϭ
Ϭ͘Ϭ
ͲϬ͘Ϯ
ͲϬ͘Ϯ
Ϭ
(a)
/Ŷ'ƌŽƵƉ
Ϭ͘Ϯ
Ϯ
ϰ
ϲ
ϴ
Ϭ
ϭϬ
(b)
Ϭ͘Ϯ
Ϭ͘ϰ
Ϭ͘ϲ
Ϭ͘ϴ
ϭ
ĞĐĞŶƚƌĂƚŝŽŶ;ŵŵͿ
Figure 7.9: Changes in astigmatism (J180) due to (a) tilt and (b) decentration of the
individual elements and the system in a group. Solid and dotted lines represent data for
unaccommodated and accommodated states respectively.
Front element misalignment of a 2E-AIOL induced the most significant effects on
aberrations; yet, spherical aberration remained relatively unaffected. Even approaching the
maximal extent of tilt (10º) and decentration (1 mm), change in spherical aberration did not
reach clinical significance (Figure 7.7). However, the change in coma exceeded the
tolerable limit when the tilt and decentration exceeded 3º and 0.2 mm respectively. The rear
element and the system as a group could be tilted up to 4º and 6º, respectively, before visual
acuity would be affected. Decentration of the rear element up to 1 mm (maximum extent in
229
Misalignment of AIOL
the experiment) did not affect coma significantly. The change remained well below the
clinically significant level (<0.08 waves). Upon induction of maximum tilt and decentration
of the rear element and the system as a group, the astigmatism induced was below visually
significant levels. The front element could also be tilted or decentred up to 8º and 0.8 mm
without inducing any visually significant astigmatism.
7.4.2.3 Effect of combined misalignment (Tilt + decentration)
The effect of tilt and decentration in combination with higher-order aberrations were
evaluated for various shapes of the 2E-AIOL elements. Two combinations of tilt and
decentration tested were: moderate (5/0.5) and extreme (10/1.0). Effects on individual
aberrations are discussed separately.
7.4.2.4 Effect on Spherical Aberration
Spherical aberration was least affected when the rear element alone was misaligned.
Misalignment of the front element alone, and as a group, produced similar aberration.
Spherical aberration varied quadratically with the bending factor of the elements in both
aligned and misaligned conditions. Aberration was relatively insensitive to the shape of the
rear element. Varying the front element bending factor from 0 to +1 appears to be the most
effective method for controlling spherical aberration. A negative bending factor of the front
element produced the worst aberration. Spherical aberration was reduced for near focus,
which is consistent with the results of Chapter 6. Negative aberration was produced at near
vision when the rear element alone was misaligned. Spherical aberration produced by
misalignment of the individual element, and as a group, as a function of the bending factors
of the elements is plotted in Figures 7.10 to 7.12. Examples shown in these figures
correspond to a +35 D front element made with material of refractive index 1.45.
230
Misalignment of AIOL
4
4
3
ΔZ(4,0) waves
3
2
2
1
1
0
0
-1
-2
-2
-1
-1
0
X1
+1
+2
0
+1 X2
+2
-1
-2
-1
0
X1
+1
+2
-2
-1
0
+1 X
+2
2
Figure 7.10: Spherical aberration in waves (Ȝ=0.587 ȝm) when front element alone was
misaligned. The figure on the left represents extreme misalignment (10º tilt + 1 mm
decentration) and on the right represents moderate misalignment (5º tilt + 0.5 mm
decentration). Blue and red curves represent data for distance and near focus states,
respectively.
0.25
0.25
0
0.0
-0.25
-0.25
-0.5
-2
-2
-1
-0.50
0
-2
+1 X
+2
2
-1
0
X1
+1 +2
-1
0 +1
+2
X1
-1
0
+1
+2
X
-2
2
Figure 7.11: Spherical aberration in waves (Ȝ=0.587 ȝm) when rear element alone was
misaligned. The figure on the left represents extreme misalignment (10º tilt + 1 mm
decentration) and on the right represents moderate misalignment (5º tilt + 0.5 mm
decentration). Blue and red curves represent data for distance and near focus states,
respectively.
231
Misalignment of AIOL
7.4.2.5 Effect on Coma
When the front element was misaligned, the bending factor of the rear element had no
significant effect on change in coma. Similarly, the bending factor of the front element had
no significant effect when the rear element alone was misaligned. An important finding is
that coma cannot be eliminated when individual elements are misaligned. Upon
misalignment of the front element alone, minimum coma was observed when the bending
factor of the front element was 0.74. The optimum bending factor of the front element
shifted to -1.55 when the rear element alone was misaligned. Coma produced by
misalignment of the individual elements, and of the 2E-AIOL as a group, is plotted in
Figures 7.13 to 7.15.
4
Z(4,0) waves
4
3
3
2
2
1
1
0
0
-2
-1
-2
-1
-1
0
+1 X
0
2
X1 +1 +2 +2
-1
-2
-1
0
-1
+1 X
0 +1
2
+2 +2
X1
-2
Figure 7.12: Spherical aberration in waves (Ȝ=0.587 ȝm) when system was misaligned.
The figure on the left represents extreme misalignment (10º tilt + 1 mm decentration) and
on the right represents moderate misalignment (5º tilt + 0.5 mm decentration). Blue and
red curves represent data for distance and near focus states, respectively.
232
Misalignment of AIOL
0.0
0.0
-1.0
-1.0
-2.0
-2.0
-3.0
-2 -1
0 +1
X1
+2
+2
+1
0
-1
-2 -3.0
-2 -1
X2
0 +1
+2
X
1
+2
+1
0
-1
-2
X2
Figure 7.13: Coma aberration in waves (Ȝ=0.587 ȝm) when front element alone was
misaligned. The figure on the left represents extreme misalignment (10º tilt + 1 mm
decentration) and on the right represents moderate misalignment (5º tilt + 0.5 mm
decentration). Blue and red curves represent data for distance and near focus states,
respectively.
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0.0
-2
0.0
-1
0
X1
+1
+2
+2
+1
0
-1
X2
-2 -2
-1
0
-1
-2
0
+1 X
+1
X1
+2 +2
2
Figure 7.14: Coma aberration in waves (Ȝ=0.587 ȝm) when rear element alone was
misaligned. The figure on the left represents extreme misalignment (10º tilt + 1 mm
decentration) and on the right represents moderate misalignment (5º tilt + 0.5 mm
decentration). Blue and red curves represent data for distance and near focus states,
respectively.
233
Misalignment of AIOL
1.0
1.0
0.0
0.0
-1.0
-1.0
-2.0
-2
-1
0
X1
+1
0
+2
+2
-1
+1 X
2
-2
-2.0
-2
-1
0
X1 +1 +2
-1
0
+1 X
2
+2
-2
Figure 7.15: Coma aberration in waves (Ȝ=0.587 ȝm) when system (group) was
misaligned. The left figure represents extreme misalignment (10º tilt + 1 mm decentration)
and on the right represents moderate misalignment (5º tilt + 0.5 mm decentration). Blue
and red curves represent data for distance and near focus states respectively.
7.4.2.6 Effect on astigmatism (J180)
Similarly to the coma aberration, J180 was not affected by the bending factor of the aligned
element. It was only the bending factor of the misaligned element that induced astigmatism.
J180 could not be eliminated when individual elements were misaligned.
234
ΔJ180(D)
Misalignment of AIOL
3.0
3.0
2.0
2.0
1.0
1.0
0.0
-2
-2
-1
-1
0
X1
0
+1
+1
+2
+2
-2
0.0
-2
-1
-1
X2
0
X1
0
+1
+1
+2
+2
X2
Figure 7.16: Astigmatism when front element alone was misaligned. The figure on the left
represents extreme misalignment (10º tilt + 1 mm decentration) and on the right represents
moderate misalignment (5º tilt + 0.5 mm decentration). Blue and red curves represent data
ΔJ180 (D)
for distance and near focus states, respectively.
1
1.0
0.0
0.0
-1.0
-1.0
-2.0
-2 -1
0 +1
+2
X1
-1
0
+2 +1
X2
-2
-2.0
-2
-1
0
+1
+2
X1
-1
0
+2 +1
X2
-2
Figure 7.17: Astigmatism when rear element alone was misaligned. The figure on the left
represents extreme misalignment (10º tilt + 1 mm decentration) and on the right represents
moderate misalignment (5º tilt + 0.5 mm decentration). Blue and red curves represent data
for distance and near focus states, respectively.
235
J180 (D)
Misalignment of AIOL
1.0
1.0
0.0
0.0
-1
-1.0
1
1
-1.0
-1
0
0
X1
-1
-1 -2
X2
-2
X1
-2
X2
Figure 7.18: Astigmatism when system in a group was misaligned. The figure on the left
represents extreme misalignment (10º tilt + 1 mm decentration) and on the right represents
moderate misalignment (5º tilt + 0.5 mm decentration). Blue and red curves represent data
for distance and near focus states, respectively.
7.5 Design of 2E-AIOL in the Presence of Misalignment
7.5.1 Optimum Design for Spherical Aberration
The combination of shapes inducing minimum spherical aberration range between -1 to +1
for the front element, and -2 to +1 for the rear element. When individual elements are
misaligned, the induced aberration is mostly dictated by the bending factor of the
misaligned element. The bending factor of the aligned element has minimal influence.
From the results of spherical aberration induced by misalignment as a function of the
bending factors (section 7.4.2.4), the preferred bending factors of the front and rear
elements in presence of misalignment can be summarized as:
Front element misalignment: X1 = 0 to +1
and
X2 = entire range
236
Misalignment of AIOL
Rear element misalignment: X1 = -1 to +1 and
X1 = 0 to +1
Group misalignment:
X2 = 0 to -2
X2 = ≤+1
and
The results also indicate that elimination of spherical aberration in the presence of
misalignment is possible when the bending factors of the elements are properly selected. A
potential optimal design for 2E-AIOLs was evaluated through several steps of the
theoretical experiment as described below.
1) The aberration was calculated for various combinations of bending factors of the
elements. Table 7.7 presents Z (40) data for each combination. In this example, the
power of the front element is +35 D, and the AIOL is located 2 mm behind the pupil
in the unaccommodated state. The refractive index of both elements is 1.45.
Table 7.5: Misalignment induced spherical aberrations (ΔC(40)) in waves (Ȝ=0.587 ȝm) of
a spherical 2E-AIOL for extreme misalignment (10º tilt and 1 mm decentration) of the front
element alone at distance. X1 and X2 are bending factors of the front and rear elements
respectively.
X2
X1
-2
-1
0
1
2
-2
5.56699
5.54197
5.50317
5.44746
5.36055
-1
0.94462
0.94591
0.92273
0.87178
0.7775
0
0.1537
0.15975
0.13918
0.08882
-0.00885
1
0.13892
0.14469
0.12294
0.07034
-0.03302
2
0.94917
0.94998
0.91932
0.85279
0.72036
237
Misalignment of AIOL
2) Data in Table 7.5 were plotted on a scatter graph exhibiting induced spherical
aberration as a function of the bending factor of the front element (X1). There are a
total of five scatter plots, one for each value of X2 in Table 7.5. Each plot has five
data points corresponding to the five values of X1 in Table 7.5. Each plot was fit
with a second degree polynomial curve. Examples of plots are shown in Figure
7.22.
Ϭ͘ϴϬ
ǻ ;ϰ͕ϬͿ
Ϭ͘ϲϬ
&ŽƌWůŽƚϭ͗LJϭсϬ͘ϰϬdžϮ Ͳ Ϭ͘ϰϬdžнϬ͘ϭϱ
Zϸсϭ͘ϬϬ
&ŽƌWůŽƚϮ͗LJсϬ͘ϯϴdžϮͲ Ϭ͘ϰϬdžͲ Ϭ͘Ϭϭ
Zϸсϭ
Ϭ͘ϰϬ
Ϭ͘ϮϬ
Ϭ͘ϬϬ
ͲϬ͘ϮϬ
Ͳϭ͘ϬϬ
ͲϬ͘ϱϬ
Ϭ͘ϬϬ
Ϭ͘ϱϬ
ϭ͘ϬϬ
ϭ͘ϱϬ
ĞŶĚŝŶŐ&ĂĐƚŽƌ;&ƌŽŶƚůĞŵĞŶƚͿ
Ϯ͘ϬϬ
Figure 7.19: An example of polynomial fit of the spherical aberration induced by extreme
misalignment (10º tilt + 1 mm decentration) as a function of the bending factor. The dotted
plot corresponds to X2 = -1 and the solid plot corresponds to X2 = +1. Note that the solid
plot has two roots (indicated by circles) whereas the dotted plot has no root as it does not
cross the line representing zero spherical aberration.
3) The roots of each polynomial were determined when they existed (see Table 7.8).
The roots represent the bending factor of the front element which produces zero
aberration when combined with the corresponding bending factor of the rear
element.
238
Misalignment of AIOL
4)
The roots were defined as a function with independent variable X1 and dependent
variable X2. This function was fit with a polynomial. The order of the polynomial
was the smallest order that produced a ‘good’ fit (R2 > 0.99). In Table 7.8, the
points indicated by real roots have coordinates (1.0779, 2), (-0.027, 2), (0.7254, 1)
and (0.2937, 1) where the x-coordinate is X1 and y-coordinate is X2. Polynomial
fitting produces the regression equation:
X2 = 3.8649X12 – 0.0453X1+1.8783 (R2 = 0.9991)
Table 7.6: Second-degree polynomial equations representing the spherical aberration
induced due to extreme misalignment of the front element for each combination of the
bending factors. Where there is no root, the combination of bending factors cannot
eliminate spherical aberration.
Plot #
Polynomial
R2
Root1
Root 2 Comment
1
0.3849x – 0.0405x-0.0112
1
1.0779
-0.027
X2 = 2
2
0.3914x – 0.3989x – 0.0834
0.9999
0.7254
0.2937
X2 = 1
3
0.395x – 0.3976x – 0.1324
0.9998
No real root
No real root
4
0.3979x – 0.3982x + 0.1524
0.9998
No real root
No real root
5
0.4003x – 0.4004x + 0.1464
0.9998
No real root
No real root
Similar tables for other conditions (near and distance, extreme and moderate misalignment
and front, rear or group misalignment) are given in Appendix at the end of this chapter.
5) Using this regression equation, we can determine the optimal bending factor for the
rear element for any value of the bending factor of the front element. The
polynomial represents the locus of all sets of bending factors which produces zero
239
Misalignment of AIOL
spherical aberration. Polynomials corresponding to all simulations are given in the
Appendix.
Figure 7.20: Design of 2E-AIOL to eliminate spherical aberration in presence of various
degrees of misalignment. Red and blue plots represent extreme and moderate
misalignments, respectively, as defined in the text. Solid plots represent the design for
distance, whereas the dashed plots represent designs for near. The points of intersection
are representative of a design that eliminates spherical aberration for near and distance
foci (also called ‘optimal solution’). X1 – bending factor of the front element, X2 – bending
factor of rear element. Curves that are missing correspond to curves for which there is no
root (i.e. spherical aberration cannot be eliminated) .
240
Misalignment of AIOL
6) These polynomials are then plotted in sets of graphs that present the optimal designs
for near and distance vision, and the different types of misalignment, as shown in
Figure 7.23.
Figure 7.21: Design of 2E-AIOL to eliminate spherical aberration for near and distance
vision and various degrees of misalignment (tilt+decentration). X1 – bending factor of the
front element, X2 – bending factor of rear element. Moderate misalignment refers to the
combination of 5º tilt and 0.5 mm decentration . Extreme misalignment refers to the
combination of 10º tilt and 1 mm decentration.
Comparing the subplots in Figure 7.23, it can be observed that the difference between near
and distance vision is much larger for the misaligned 2E-AIOL (Subplots b to d) than the
design for aligned 2E-AIOL (Subplot a). Interestingly, the combination X1 § 0 and X2 § +1
241
Misalignment of AIOL
represents the best choice of design for optimal performance for both aligned and
misaligned conditions. This needs to be experimentally verified in future works.
7.5.2 Optimum Design for Coma
The same approach as for spherical aberration was used to evaluate designs optimized for
correction of coma. The polynomials representing coma aberration for the different designs
and misalignments produced no roots, indicating that there exists no combination of the
bending factors that can eliminate coma. However, global minima or maxima of the
polynomial representing design points that produces minimum aberration can be identified
(Table 7.10). As discussed in the results (Section 7.4), coma aberration is sensitive only to
the bending factor of the misaligned element. Therefore, the bending factors of the aligned
element in the table are not shown. In other words, there is a degree of freedom in design
for the aligned element that may be used to address other design requirements.
Table 7.7: Optimal bending factors of the elements to produce minimum coma-like
aberration in presence of various degrees of misalignment. Designs of the aligned elements
are not shown as they have no effect on coma, hence entire range (+2 to -2) may be used.
Unaccommodated: Extreme
Front
Misalignment
X1 = 0.740
Rear
Misalignment
X2 = -1.550
Unaccommodated: Moderate
X1 = 0.758
X2 = -1.982
Accommodated: Extreme
X1 = 0.741
X2 = -1.991
Accommodated: Moderate
X1 = 0.709
X2 = -1.975
State/Degree of Misalignment
Unlike the misalignments of individual elements, coma could be eliminated when the 2EAIOL system was misaligned as a group. Roots and regression obtained by polynomial
fitting for various degrees of misalignment as a group are given in Table 7.10.
Corresponding designs are plotted in Figure 7.25.
242
Misalignment of AIOL
Table 7.8: Roots and polynomial equations obtained for the design of 2E-AIOL to eliminate
coma-like aberration in the presence of group misalignment. Scatter plots of these roots
can be closely fitted with a 4th order polynomial (see R2 values).
X 2 = − 0 .8335 X 14 − 0 .6843 X 13 ...
Accommodated: (-0.944, 1); (0.5588, 1)
Extreme
(-0.6055, 0); (0.1261, 0)
+ 2.6457 X 12 + 1.1531 X 1 − 0.2393
X 2 = − 0 .9323 X 14 + 1 .2052 X 13 ...
Accommodated: (-0.0574, 1); (1.6208, 1)
Moderate
(0.5392, 0); (1.0253, 0)
+ 0.4621X 12 − 2.1688 X 1 0.8758
X 2 = − 0 .8336 X 14 + 2 .6503 X 13 ...
Extreme
(0.0561, 1); (1.5588, 1)
− 0.3028 X 12 − 2.8576 X 1 + 1.1044
(0.3945, 0); (1.1261, 0)
X 2 = −0 .9148 X 14 + 2 .6344 X 13 ...
Moderate
(0.0212, 1); (1.4398, 1)
+ 0.0251 X 12 − 2.7957 X 1 + 1.0496
(0.4392, 0); (0.9912, 0)
Design to Eliminate Coma
3.0
Unaccommodated: Extreme
Accommodated: Extreme
Unaccommodated: Moderate
Accommodated: Moderate
2.5
2.0
X
2
1.5
1.0
0.5
0.0
-0.5
-1.0
-2.0
-1.5
-1.0
-0.5
0.0
X1
0.5
1.0
1.5
2.0
Figure 7.22: Design curves to eliminate coma. Curves correspond to the polynomials given
in the Table 7.10.
243
Misalignment of AIOL
7.5.3 Optimum Design for Astigmatic Aberration
In the same manner of determination as the analysis for coma aberration, the polynomials
for astigmatism (J180) produced no roots when individual elements were misaligned which
indicates that astigmatism cannot be eliminated in these conditions. Additionally, a change
in astigmatism is entirely governed by the bending factor of the misaligned element. The
aligned element remained without influence on astigmatism. Optimum designs (global
minima or maxima of the polynomial) of the misaligned element producing minimum
astigmatism are detailed in Table 7.11.
In results that show some similarity with those for coma, when an individual element is
misaligned, astigmatism could not be eliminated, however, when the 2E-AIOL is
misaligned as a group, elimination of astigmatic aberration is possible.
The possible
solutions are shown in Table 7.12 and the corresponding designs plotted in Figure 7.26.
Table 7.9: Optimal bending factors of the misaligned element to produce minimum J180 in
presence of various misalignments. The design of the misaligned element shown, is effective
for the entire range of bending factors tested (+2 to -2) of the aligned element.
State/Degree of Misalignment
Front
Rear
Misalignment
Misalignment
Unaccommodated: Extreme
X1 = 0.343
X2 = -1.966
Unaccommodated: Moderate
X1 = 1.333
X2 = -1.973
Accommodated: Extreme
X1 = 1.361
X2 = -1.631
Accommodated: Moderate
X1 = 1.337
X2 = -1.975
244
Misalignment of AIOL
Table 7.10: Roots and polynomial equations obtained for the design to eliminate J180
when a 2E-AIOL is misaligned as a group. Scatter plots of these roots were closely fitted
with 4th order polynomials.
Misalignment
Real Roots
Regression Formula
R2
X 2 = − 0 .244 X 14 + 0 .4335 X 13 ...
Accommodated:
(1.3902, 1); (-0.9514, 1)
Extreme
(1.1549, 0); (-0.4765, 0)
+ 1.7162 X 12 − 1.2321X 1 − 1.0389
X 2 = − 0 .0371 X 14 + 0 .3062 X 13 ...
Accommodated:
(2.6279, 1); (-0.1156, 1)
Moderate
(2.1574, 0); (0.477, 0)
+ 0.1568 X 12 − 1.7623 X 1 + 0.7845
X 2 = −0 .2441 X 14 + 1 .4099 X 13 ...
Extreme
(2.39, 1); (0.0486, 1)
(2.1549, 0); (0.5235, 0)
− 1.0484 X 12 − 2.3883 X 1 + 1.2321
X 2 = − 0.0075 X 14 + 0.2066 X 13 ...
Moderate
(2.6359, 1); (0.0302, 1)
(2.0975, 0); (0.566, 0)
+ 0.2175 X 12 − 1.8102 X 1 + 0.9893
245
Misalignment of AIOL
Design to Eliminate J180
4
Unaccommodated: Extreme
Accommodated: Extreme
Unaccommodated: Moderate
Accommodated: Moderate
3
X2
2
1
0
-1
-2
-3
-2
-1
0
X1
1
2
3
Figure 7.23: Design curves produced from polynomials given in the Table 7.10.
7.6 Discussion
Misalignment of an IOL through tilt and decentration, in a pseudophakic eye is one of the
most frequent postoperative complications of modern cataract surgery. Though the average
amount of misalignment of conventional IOLs is reported to be clinically irrelevant, image
quality may be significantly degraded when misalignment exceeds a certain threshold.
About 10% of post-operative eyes suffer > 5º tilt and > 0.5 mm decentration following
uneventful implantation of a conventional IOL (Hayashi et al., 1999b). This is well above
the tolerable limit considering that about 3° tilt and 0.2 mm decentration induce clinically
observable aberrations. Assuming that an AIOL follows similar patterns of misalignment in
the eye to that of the conventional IOL, there are obvious potential consequences.
Using ray tracing, we evaluated theoretically the performance of spherically surfaced
translating optics AIOLs in the presence of misalignment. For each type of misalignment,
spherical, coma and astigmatic aberrations were calculated. We observed that astigmatism
246
Misalignment of AIOL
and coma are more sensitive to misalignment than is spherical aberration, which remained
virtually unaffected, regardless of the type and amount of misalignment, type of AIOL and
the range of pseudophakic accommodation. Aberrations as a consequence of misalignment
were slightly more evident in the accommodated state than in the unaccommodated state.
A spherical 1E-AIOL design showed considerable tolerance to misalignment which is
clinically beneficial. Changes in aberrations were minimal (<0.01 waves) for bending
factors ranging between -0.5 and +1.5. Assuming that 0.25 D is the tolerable level of
spherical refractive error, a change in aberration by 0.08 waves (Ȝ = 0.5876) at 4 mm pupil
diameter (Atchison, 2005) would comprise the aberration tolerance for clinically relevant
deterioration of visual acuity. The change in aberration due to the imposed misalignment up
to 1 mm decentration and 10° tilt for a 1E-AIOL with a bending factor within the range
specified above, remained well below the limit of aberration tolerance. However, the
change in coma aberration exceeded the significance level when the bending factor of the
1E-AIOL departed from the optimum values. The results for spherical 1E-AIOL accord
with previous reports that spherical IOLs are relatively insensitive to tilt and decentration
(Altmann et al., 2005, Eppig et al., 2009, Holladay et al., 2002).
The patterns of change in aberration due to misalignment of a 2E-AIOL differed slightly
from the patterns observed with misalignment of a 1E-AIOL. Misalignment of the front
element alone affected aberrations more than misalignment of the rear element alone, or of
the system as a group. Decentration of the system as a group produced the least change in
aberration. Aberrations are worse when the front element has a negative bending factor,
indicating that positive bending of the element may afford better performance.
The change in spherical aberration again remained relatively unaffected under
misalignment of a 2E-AIOL. It did not attain clinically significant levels even with
maximum tilt (10°) and decentration (1 mm) (Figure 7.7) of individual elements, or of the
247
Misalignment of AIOL
system as a group. Combined misalignment of the front element alone, and of the system as
a group, produced identical effects on spherical aberration.
Coma was the aberration most vulnerable to misalignment. The change in coma exceeded
the tolerable limit when the tilt and decentration of the front element exceeded 3° and
0.2 mm, respectively. The rear element, and the system as a group, may be tilted up to 4°
and 6°, respectively, before visual acuity is affected. Decentration of the rear element up to
1 mm (maximum extent in the experiment) did not affect coma significantly, remaining
well below the clinically significant level (< 0.08 waves). It was found that the system may
be decentred up to 0.3 mm before visual acuity is affected. The entire test range of tilt and
decentration of the rear element, and the system as a group, did not induce clinically
relevant astigmatism. The front element was also able to be tilted or decentred up to 8° and
0.8 mm without inducing visually significant astigmatism.
It must be borne in mind that this study is based on a theoretical model eye that may not be
an exact representation of actual eyes or misalignment in the clinical setting. However,
these findings do provide an understanding of the qualitative and general quantitative
consequences of AIOL misalignment. Throughout, there is the underlying assumption that
AIOLs follow a pattern of misalignment similar to conventional IOLs, which may not be
the case since the mechanical design of haptics in AIOLs and their total device thickness,
especially for 2E-AIOLs, are different. AIOLs incorporate more complex haptics designs to
facilitate translation of the optical components.
To conclude, an optimal design of the AIOL is alignment specific. The optimum bending
factor of the 1E-AIOL for misalignment is consistent with the optimum design for the
aligned position (Chapter 6) and that for conventional IOLs, which is about +1 for minimal
spherical aberration and about -1 for minimal coma. An appropriate compromise to provide
the best tolerance to misalignment may lie in the range -0.5. For 2E-AIOLs, the optimal
front element bending factors lie between -1 and +1, with a wide range of workable rear
248
Misalignment of AIOL
element bending factors (between -2 to +2). Given that an AIOL is designed for optimal
performance in the aligned position, minor tilt and decentration are optically well tolerated
and do not affect visual acuity.
One recommendation to the industry from this study can be identified from the analysis of
designs to eliminate coma or astigmatism in the presence of misalignment. It was
demonstrated through select examples that while no solution exists for elimination of either
coma or astigmatism when an individual element is misaligned in a 2E-AIOL, design
solutions do exist for misalignment that involve the entire 2E-AIOL as a group. The
implication of these findings may be expressed as follows. Provided well-designed and
constructed mechanical supports and haptics are built into a 2E-AIOL, then during
implantation, the only likely misalignment would primarily be with the device as a group.
Therefore, the AIOL designer and manufacturer can significantly reduce the optical/visual
impact of misalignment errors introduced at the implantation stage, and thereby increasing
the tolerance afforded to the surgeon by (1) employing an optimum optical design that
eliminates or minimises coma or astigmatism in the presence of misalignment, (2)
incorporating haptics design that ensures maintenance of relative alignment of the two
optical elements, for example by employing robust mechanical coupling, and (3) ensuring
manufacturing tolerances keep inter-element misalignments below critical values. This
strategy would effectively neutralise the effect of AIOL misalignment.
It should be emphasised that the effects demonstrated in this study are valid only for
designs utilising spherical surfaces. Given the greater degrees of freedom available in
design, outcomes are anticipated to differ for aspheric designs of AIOL, with potential
optimum designs for misalignment cases for which no solution exists using spherical
optical elements.
249
Misalignment of AIOL
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Misalignment of AIOL
7.8 Appendix
7.8.1 Polynomials for spherical Aberration
Polynomial equations, R2 and roots produced from the spherical aberration data which were
used in finding the optimum design of 2E-AIOL in presence of misalignment.
Unaccommodated, Extreme misalignment of Front Element
Plot #
Polynomial
R2
Root1
Root 2
Comment
1
0.3849x – 0.0405x – 0.0112
1.000
1.0779
-0.027
X2 = 2
2
0.3914x – 0.3989x – 0.0834
0.9999 0.7254
0.2937
X2 = 1
3
0.395x – 0.3976x – 0.1324
0.9998 No real root
No real root
4
0.3979x – 0.3982x + 0.1524
0.9998 No real root
No real root
5
0.4003x – 0.4004x + 0.1464
0.9998 No real root
No real root
Unaccommodated, Extreme misalignment of Rear element
Plot #
Polynomial
R2
Root1
Root 2
Comment
1
0.1887x – 0.2092x – 0.2598
0.997
0.7434
-1.8520 X2 = 2
2
0.1995x + 0.2346x + 0.0238
0.997
-0.1121
-1.0638 X2 = 1
3
0.2045x + 0.2469x + 0.112
0.996
No real root
No real root
4
0.2069x + 0.2533x + 0.1538
0.996
No real root
No real root
5
0.2077x + 0.2556x + 0.1487
0.996
No real root
No real root
258
Misalignment of AIOL
Unaccommodated, Extreme misalignment in Group
Plot #
Polynomial
R2
Root1
Root 2
Comment
1
0.3851x + 0.3022x – 0.2958
1.000
0.5679
-1.3526
X2 = 2
2
0.3646x + 0.3046x – 0.0018
0.9998 0.0059
-0.8413
X2 = 1
3
0.3868x + 0.3683x + 0.1134
1.000
No real root
No real root
4
0.3963x + 0.3917x + 0.1495
0.9999 No real root
No real root
5
0.4028x + 0.4006x + 0.1453
0.9998 No real root
No real root
Unaccommodated, Moderate misalignment of Front element
Plot #
Polynomial
R2
1
0.1832x + 0.1689x – 0.1442
2
Root1
Root 2
Comment
0.9998 0.5388
-1.4608
X2 = 2
0.1966x + 0.1953x + 0.0168
0.9994 -0.095
-0.8983
X2 = 1
3
0.2027x + 0.2069x + 0.0886
0.9993 No real root
No real root
4
0.2059x + 0.2129x + 0.1161
0.9992 No real root
No real root
5
0.207x + 0.2146x + 0.1099
0.9992 No real root
No real root
Unaccommodated, Moderate misalignment of Rear element
Plot #
Polynomial
R2
1
0.1573x – 0.1442x – 0.145
2
Root1
Root 2
Comment
0.9972 1.5241
-0.6048
X2 = 2
0.1652x – 0.1456x + 0.0039
0.997
0.0277
X2 = 1
3
0.1686x – 0.1461x + 0.0707
0.9969 No real root
No real root
4
0.1703x – 0.1461x + 0.0958
0.9968 No real root
No real root
5
0.1707x – 0.1459x + 0.0889
0.9968 No real root
No real root
0.8537
259
Misalignment of AIOL
Unaccommodated, Moderate misalignment in Group
Plot #
Polynomial
R2
1
0.1928x – 0.1966x – 0.0901
2
Root1
Root 2
Comment
0.9995 1.3627
-0.3429
X2 = 2
0.1998x – 0.1978x + 0.0263
0.9993 0.8317
0.1583
X2 = 1
3
0.2031x – 0.1989x + 0.0856
0.9993 No real root
No real root
4
0.205x – 0.1995x + 0.1096
0.9992 No real root
No real root
5
0.2057x – 0.2002x + 0.1027
0.9993 No real root
No real root
Accommodated, Extreme misalignment of Front element
Plot #
Polynomial
R2
Root1
Root 2
Comment
1
0.3849x – 0.0405x – 0.0122
1
1.0803
-0.0293
X2 = 2
2
0.3914x – 0.3989x + 0.0834
0.9999 0.7254
0.2937
X2 = 1
3
0.395x – 0.3976x + 0.1324
0.9998 No real root
No real root
4
0.3979x – 0.3982x + 0.1524
0.9998 No real root
No real root
5
0.4003x – 0.4004x + 0.1464
0.9998 No real root
No real root
Accommodated, Extreme misalignment of Rear element
Plot #
Polynomial
R2
Root1
Root 2
Comment
1
0.1887x – 0.1683x – 0.2802
0.997
1.7435
-0.8517 X2 = 2
2
0.1995x – 0.2346x + 0.0238
0.996
0.8874
-0.0638 X2 = 1
3
0.2045x – 0.1621x + 0.0796
0.993
No real root
No real root
4
0.2069x – 0.1606x + 0.1074
0.992
No real root
No real root
5
0.2077x – 0.1599x + 0.1008
0.991
No real root
No real root
260
Misalignment of AIOL
Accommodated, Extreme misalignment in Group
Plot #
Polynomial
R2
Root1
Root 2
Comment
1
0.3327x – 0.4785x – 0.1815
0.996
1.75
-0.3117
X2 = 2
2
0.3646x – 0.4245x + 0.0582
0.9998 1.0055
0.1587
X2 = 1
3
0.3868x – 0.4052x + 0.1318
1
No real root
No real root
4
0.3963x – 0.4009x + 0.1541
0.9999 No real root
No real root
5
0.4028x – 0. 405x + 0.1475
0.9998 No real root
No real root
Accommodated, Moderate misalignment of Front element
Plot #
Polynomial
R2
1
0.239x – 0.214x – 0.0527
2
Root1
Root 2
Comment
0.9985 1.0965
-0.2011
X2 = 2
0.2433x – 0.212x + 0.0468
0.9983 0.7568
0.1423
X2 = 1
3
0.2455x – 0.2117x + 0.0979
0.9982 No real root
No real root
4
0.247x – 0.2118x + 0.1187
0.9981 No real root
No real root
5
0.2479x – 0.2123x + 0.1125
0.9981 No real root
No real root
Accommodated, Moderate misalignment of Rear element
Plot #
Polynomial
R2
1
0.1984x – 0.1631x – 0.1075
2
Root1
Root 2
Comment
0.9967 1.2541
-0.432
X2 = 2
0.2036x – 0.1611x + 0.0251
0.9964 0.578
0.2133
X2 = 1
3
0.206x – 0.1604x + 0.085
0.9963 No real root
No real root
4
0.2073x – 0.1599x + 0.1074
0.9962 No real root
No real root
5
0.2079x – 0.1597x + 0.1011
0.9961 No real root
No real root
261
Misalignment of AIOL
Accommodated, Moderate misalignment in Group
Plot #
Polynomial
R2
Root1
Root 2
Comment
1
0.232x – 0.2219x – 0.0894
0.999
1.2018
-0.3054
X2 = 2
2
0.2403x – 0.2155x + 0.0391
0.9985 0.6642
0.2526
X2 = 1
3
0.2442x – 0.2131x + 0.0972
0.9982 No real root
No real root
4
0.2467x – 0.2123x + 0.1189
0.9981 No real root
No real root
5
0.2482x – 0.2129x + 0.1126
0.9981 No real root
No real root
7.8.2 Optimal 2E-AIOL designs
Polynomial equations and corresponding R2 representing the optimum design of 2E-AIOL
to produce zero spherical aberration in presence of misalignment. Plots from these
polynomials are included in the main text (Figure 7.24).
Unaccommodated, extreme misalignment
Misalign
Front only
Real Roots
(-0.27, 2); (0.2937, 1);
Regression Formula
R2
Min X1
y = 3.8649x –0.0453x+1.8783 0.9991 0.5233
(0.7254, 1); (1.0779, 2)
Rear only
(-0.7434, 2); (-0.1121, 1); y = 0.6859x +0.7659x+1.0583 0.9994 -0.5583
(-1.0638, 1); (-1.852, 2)
System
(-1.3526,2); (0.0059, 1);
y = 1.3454x +1.0669x+0.9702 0.9986 -0.3946
(-0.8413, 1); (0.5679, 2)
262
Misalignment of AIOL
Unaccommodated, moderate misalignment
Misalign
Front only
Rear only
System
Real Roots
(0.5388, 2); (-1.4602, 2);
(-0.0951, 1); (-0.8983, 1)
(1.5241, 2); (-0.6048, 2);
0.8537, 1); (0.0277, 1)
(1.3627,2); (-0.3429, 2);
(0.8317, 1); (0.1583, 1)
Regression Formula
y = 1.1918x +1.1107x+1.0664
R2
0.998
Min X1
-0.466
y = 1.0388x –0.9498x+1.0399
0.9995 0.4572
y = 1.6286x –1.6543x+1.2354
0.9995 0.5079
Regression Formula
y = 3.8264x –0.4006x+1.8698
R2
Min X1
0.9988 0.5233
y = 0.6859x –0.6061x+0.9786
0.9991 0.4418
y = 1.1152x –1.5582x+1.344
0.9705 0.6986
Regression Formula
y = 3.0623x –2.7441x+1.3257
R2
Min X1
0.9998 0.451
y = 1.4763x –1.2117x+1.1993
0.9999 0.4104
y = 1.7381x –1.6562x+1.3269
0.9992 0.4764
Accommodated, extreme misalignment
Misalign
Front only
Rear only
System
Real Roots
(1.0803, 2); (0.7254, 1);
(-0.0293, 2); (0.2937, 1)
(1.7435, 2); (0.8874, 1);
(-0.8517, 2); (-0.0638, 1)
(1.75, 2); (1.0055, 1);
(-0.3117, 2); (0.1587, 1)
Accommodated, moderate misalignment
Misalign
Front only
Rear only
System
Real Roots
(1.0965, 2); (0.7568, 1);
(-0.2011, 2); (0.1423, 1)
(1.2541, 2); (0.578, 1);
(-0.432, 2); (0.2133, 1)
(1.2618, 2); (0.6442, 1);
(-0.3054, 2); (0.2526, 1)
263
Chromatic Aberration
Chapter 8
Chromatic Aberration
Part of this chapter was presented in the following conference
Ale J, Manns F, Ho A “Chromatic Aberration of AIOL and a Method of Achromatization:
Paraxial and Finite ray-Tracing Analyses” (Poster # 5596). Association for Research in Vision
and Ophthalmology (ARVO) May 3 – 7, 2009, Fort Lauderdale, FL.
264
Chromatic Aberration
TABLE OF CONTENT
8.1 INTRODUCTION................................................................................................................266
8.2 METHODS ...........................................................................................................................269
8.2.1 Chromatic Dispersion................................................................................................ 269
8.2.2 Computation of Chromatic Aberrations .................................................................... 273
8.3 RESULTS .............................................................................................................................277
8.3.1 Chromatic Aberration of 1E-AIOL ............................................................................ 277
8.3.2 Chromatic Aberration of 2E-AIOL ............................................................................ 279
8.4 ACHROMATIZATION OF THE MODEL EYE WITH AIOLS...................................282
8.4.1 Achromatization with 1E-AIOL ................................................................................. 282
8.4.2 Achromatization with 2E-AIOL ................................................................................. 285
8.4 DISCUSSION AND CONCLUSION .................................................................................286
8.5 REFERENCES.....................................................................................................................290
265
Chromatic Aberration
8.1 Introduction
Refraction of the eye is not constant across the spectrum due to chromatic dispersion of light, i.e.
the wavelength dependency of refractive index, first reported by Newton (Cantor, 1984). The eye
that is emmetropic for a given wavelength is myopic for shorter wavelengths and hyperopic for
longer wavelengths with a total difference in refraction across the visible spectrum amounting
about 2 D (Bedford & Wyszecki, 1957, Wald & Donald, 1947).
This manifestation of dispersion; the chromatic aberration of the eye, has significant effect on
retinal image quality. Considering the considerable values (about 2D) of longitudinal chromatic
aberration (LCA) in the eye, it is logical to assume that its correction would have significant
impact on spatial vision. However, the visual benefit of the correction has been debated.
Campbell and Gubisch (1967) found a small improvement in contrast sensitivity measured with
monochromatic light for a small pupil diameter whereas no effect was observed in 4.5 mm pupil.
Others argue that the higher-order monochromatic aberrations may attenuate the negative effects
of LCA in larger pupil (McLellan et al, 2002; Ravikumar et al, 2008) and the possible benefits of
its correction may be unimportant (Yoon & Williams, 2002). Presumably, benefit of correcting
LCA may also be eased by neural mechanisms of the visual system as the eye is less sensitive to
the extreme wavelengths of the visible spectrum. Lapicque (1937) calculated the combined effect
of diffraction, SA, and CA on the retinal image. He concluded that LCA this is the most
significant aberration in white light when the spherical aberration is removed. A study found not
effect on visual acuity with correction of LCA (Van Heel, 1946). Yoon and Williams (2002)
performed an experiment using adaptive optics to correct all higher-order monochromatic
aberrations and monochromatic light to remove the effects of LCA in which a clear improvement
in visual quality was reported. More recently, correction of spherical aberration along with
chromatic aberration improved the visual acuity in laboratory setting (Artal et al, 2010) which
266
Chromatic Aberration
concluded that the advantage of the combined correction of spherical and chromatic aberration is
to provide superior performance for a larger range of pupil diameters. Authors claim that, for the
smaller pupil, LCA will have greater effect on spatial vision (SA would have a smaller impact)
and for larger pupils the opposite situation would occur. Similarly, LCA has been found to affect
the modulation transfer function of the eye (Williams et al., 2000, Negishi et al., 2001). In
addition, LCA also affects some important visual functions such as accommodation (Kruger and
Pola, 1986), resolution and depth of focus (Kruger et al., 1993).
Transverse chromatic aberration (TCA), the variation of image point position across an image for
different wavelengths, is potentially a major factor for limiting image quality when the
polychromatic light is incident obliquely in the eye (Howarth, 1984, Thibos, 1987). Therefore a
good estimate of chromatic aberration is an important precondition in the design of any optical
device.
In the phakic eye, the cornea and the crystalline lens are primarily responsible for the total
chromatic aberration (Hartridge, 1950, Millodot, 1976). In a pseudophakic eye, total chromatic
aberration is mostly dependent on the dispersion of the intraocular lens and reducing LCA of an
IOL by using a material with a high Abbe number also improves overall pseudophakic optical
performance (Zhao and Mainster, 2007). Thus, it is important to make a correct choice of IOL
materials for optimum optical performance. While reliance is placed on the designers and
manufacturers to make the appropriate choice in delivering their devices, clinicians should
prescribe with consideration of optical and visual performance; including those that can be
influenced by chromatic aberrations. Unfortunately, the latter is made somewhat difficult as
typically IOL manufacturers do not include the dispersion data in their product guides. One study
(Nagata et al., 1999) reported that longitudinal chromatic aberration was lesser in pseudophakic
eyes implanted with PMMA IOL (0.64D) than in phakic eye (0.74D). Highest aberration was
found in eyes implanted with Acrysof IOL (0.98D). This result has been replicated by another
study (Rog et al., 1986). In contrast, Siedlecki and Ginis (Siedlecki et al., 2006), in their
267
Chromatic Aberration
theoretical calculations, found three times higher LCA in IOLs than their model of the crystalline
lens – which was based on the Le Grand eye model. In arriving at this conclusion, the authors did
not consider the influence on chromatic aberration that may be produced by the gradient
refractive index, and presumably ‘gradient dispersion’ of the natural crystalline lens.
Several efforts have been made to correct chromatic aberration. Achromatizing lenses consisting
of a triplet (Lewis et al., 1982) and a triplet coupled with an air-spaced doublet (Powell, 1982)
have been proposed for correcting phakic eyes. Though achromatism may be achieved, the
practicality of such a device (e.g. cosmesis) has not been discussed. More problematically, any
potential improvement in the visual function following LCA correction is lost when the device
suffers alignment error with respect to the eye brought about by chromatic parallax (Bradley et
al., 1991, Zhang et al., 1991). An attempt to the achromatizing pseudophakic eye with
achromatic IOL has also drawn some interest. Diaz et al (2004) proposed an achromatizing
hybrid doublet IOL. The design, however, did not completely eliminate the chromatic aberration;
secondary defocus of 0.35 D for the visible spectrum was found. Similarly a theoretical study
(Lopez-Gil and Montes-Mico, 2007) proposed a refractive-diffractive hybrid IOL to correct
longitudinal chromatic and spherical aberrations. These devices may be effective in controlling
the aberrations, but commonly reported optical adverse effects such as glare and haloes
associated with the diffractive echelettes may counteract the benefit. Hong et al (2008) patented
a design of achromatizing IOL consisting of two elements of different dispersion properties
which is yet to be introduced in practice.
The chromatic aberration of a pseudophakic eye implanted with AIOL is of significant interest
due to its dynamically changing axial position. Chromatic focal shift of the 1E-AIOL for a given
fixed axial position in the eye should be identical to that of a conventional monofocal IOL;
however, different behaviour for a 2E-AIOL may be expected because of the involvement of its
multiple lens elements. The aims of this chapter include theoretical investigation into chromatic
aberration of AIOLs. Effect of pseudophakic accommodation on chromatic aberration is
268
Chromatic Aberration
examined. And finally, design strategies for achromatization, where possible, are also discussed.
8.2 Methods
8.2.1 Chromatic Dispersion
Wavelength dependency of the refractive index of a material is called chromatic dispersion.
Refractive index is higher for the shorter wavelength and lower for the longer wavelengths. In
practice, optical calculation is made for a certain wavelength (monochromatic) of the light,
however, evaluation of the chromatic aberration requires calculating refractive index for a range
of wavelengths (polychromatic). A number of equations have been purposed to calculate the
dispersion of a material. In this study, the dispersion of the ocular media and lens materials are
described by different dispersion equations.
8.2.1.1 Dispersion of the Ocular Media
Couchy’s dispersion formula was used to calculate the refractive indices of the cornea, aqueous,
crystalline lens and vitreous. The equation is given by:
n (λ ) = A +
B
λ
2
+
C
λ
4
+
D
λ6
+ ....
....................................................................................... (8.1)
where Ȝ is the wavelength in nanometres and A, B, C and D are coefficients. The practical
advantages in using this formula are that the equation is linear in coefficients and, more
importantly, the predicted dispersion are close to most of the reported values for the ocular
media. Coefficients of the equations are adopted from the literature (Atchison and Smith, 2005)
(Table 8.1). Figure 8.1 shows the chromatic dispersion of the ocular media.
269
Chromatic Aberration
Table 8.1: Coefficients for the Couchy’s equation for individual ocular media suggested by
Atchison and Smith (Atchison and Smith, 2005).
Ocular Media
A
B
C
D
Cornea
1.361594
6.009687E+3
-6.760760E+8 5.908450E+13
Aqueous
1.321631
6.070796E+3
-7.062305E+8 6.147861E+13
Lens
1.389248
6.521218E+3
-6.110661E+8 5.908191E+13
Vitreous
1.322357
6.560240E+3
-5.817391E+8 5.036810E+13
8.2.1.1 Dispersion of AIOL Materials
Polymethylmethacrylate (PMMA), acrylic and silicone are the most common materials used in
fabricating IOLs. Dispersion of these materials would be expected to be different from their
‘pure’ chemical forms due to chemical additives (e.g. metallic oxides to absorb UV). Since IOL
manufacturers usually do not provide dispersion data of the material in the product guide, it is
extremely difficult to determine the exact dispersion coefficients for optical materials used in
IOLs. Based on information available in the literature, we used the Conrady dispersion formula
to describe the dispersion of PMMA and acrylic materials which have previously been used in
calculating the chromatic aberration of specific IOLs (Siedlecki and Ginis, 2007). The equations
are given as:
270
Chromatic Aberration
1.44
Refractive Index
Cornea (V # 55.5)
Aqueous (V # 50.4)
Lens (V # 47.3)
Vitreous (V # 51.0)
1.40
F-line
D-line
C-line
1.36
1.32
0.35
0.45
0.55
Wavelength ( μ)
0.65
0.75
Figure 8.1: Chromatic dispersion of the ocular media calculated from the four coefficient
Couchy’s equation with the coefficients suggested by Atchison and Smith as given in Table 8.1.
Fraunhofer F-line, D-line and C-line (dashed vertical lines) are shown for reference.
PMMA n(λ) = 1.46876 +
7.138594× 10−3 1.143317 ×10−3
+
.................................... (8.2)
λ
λ3.5
Acrylic n(λ) = 1.52006 +
7.365776× 10−3 1.97803× 10−3
+
...................................... (8.3)
λ
λ3.5
To our knowledge, no literature exists reporting the exact dispersion characteristics of silicone
materials as used in modern IOLs. One publication (Siedlecki et al, 2007) tabulated the refractive
index and Abbe number of two silicone IOL materials. Of particular interest is the SofPort AO
IOL from Bausch & Lomb as this silicone material has a low Abbe number of 24.7. While an
exact dispersion formula is not available, from the known refractive index and Abbe number, an
approximation of chromatic dispersion of a silicone material may be established by assuming
271
Chromatic Aberration
that progression of refractive index with wavelength (i.e. the dispersion curve) is similar between
the silicone material and, for example, the PMMA material. Equation 8.2 may be linearly scaled
and offset to produce a dispersion formula that will return matching refractive index and Abbe
number according to the following equation:
nsilicone(λ) = SC ⋅ nPMMA(λ) + OF
where SC is the scaling factor and OF the offset.
By solving for SC and OF and applying those values to the original coefficients of Equation 8.2,
an approximate model dispersion equation for silicone was obtained. The dispersion of the
silicone material is given by:
n(λ ) = 1.40138748 +
9.425717 × 10−3
λ
+
1.509623× 10−3
λ3.5
............................................. (8.4)
It should be emphasised that the Equation (8.4) is only a model based on assumptions discussed
above. But its precision should suffice for illustrative purposes throughout this chapter.
272
Chromatic Aberration
1.65
PMMA (V# 37.3)
Acrylic (V# 32.1)
Silicone (V# 24.7)
1.60
Refractive Index
F-line
D-line
C-line
1.55
1.50
1.45
1.40
0.35
0.45
0.55
Wavelength( μ)
0.65
0.75
Figure 8.2: Chromatic dispersion of three materials commonly used for fabricating IOLs.
Conrady formula was used for PMMA and Acrylic as reported in the literature and an
approximation based on scaling of the PMMA curve to match published values for refractive
index and Abbe number was used for Silicone. Fraunhofer F-line, D-line and C-line (dashed
vertical lines) are shown for reference. V-numbers of the materials are shown on the legend.
8.2.2 Computation of Chromatic Aberrations
In this experiment, the matrix method of the paraxial optics was employed to trace rays with
various wavelengths. One model each for 2E and 1E-AIOL containing finite thicknesses (front
element 1.2 mm and rear element 0.3 mm for 2E-AIOL and 0.9mm for 1E-AIOL) was tested;
equivalent power of the front element in the 2E-AIOL was +35 D and that of the 1E-AIOL was
+20.7 D. The front surface of the front element was positioned 2 mm behind the pupil and the
two elements were separated by 0.3 mm. Only one configuration of translation of optics for
accommodation, front element translated in the forward direction, was studied. To return the
target accommodation of 2.5 D, the required translations were 1.014 mm and 1.98 mm for 2E
273
Chromatic Aberration
and 1E-AIOL respectively.
AIOLs used in the experiment consisted of optimum designs to eliminate spherical aberration of
the model eye as found in Chapter 6. Unless specified otherwise, 2E-AIOL elements tested in
this experiment had bending factors of -0.5 for the front element and +1.5 for the rear element.
Similarly, bending factor +0.95 was used for the 1E-AIOL.
Since power of the lenses and intraocular spaces are wavelength dependent, paraxial ray tracing
was conducted for wavelengths ranging between 0.40 and 0.70 ȝm using the matrix method. For
this, refraction and translation matrices were constructed representing each optical element in the
model eye-AIOL system, where a refraction matrix RȜ for a wavelength is given by:
ª (nλ' − nλ ) º
»
Rλ = «1
R »
«0
1
¬
¼
...................................................................................................... (8.5)
where nλ' and nλ are the refractive indices of the lens and incident medium respectively for a
given wavelength (Ȝ) and R is the radius of curvature of the surface.
Similarly, a translation matrix TȜ for a given wavelength is given by:
ª 1
Tλ = «
¬t / nλ
0º
1»¼
............................................................................................................ (8.6)
A system matrix of the eye implanted with a 2E-AIOL is given by:
Sλ
RλL 22 .TλL 2 .RλL 21 .Tλs .RλL12 .TλL1.RλL11.Tλf .TλAC .RλC 2 .TλC .RλC1
....................................... (8.7)
274
Chromatic Aberration
where,
RλC1 – refraction matrix of the anterior corneal surface
RλC 2 – refraction matrix of the posterior corneal surface
RλL11
– refraction matrix of the anterior surface of the front element
RλL12
– refraction matrix of the posterior surface of the front element
RλL 21 – refraction matrix of the anterior surface of the rear element
RλL 22 – refraction matrix of the posterior surface of the rear element
TλC – translation matrix of the corneal thickness
TλAC – translation matrix of the anterior chamber
Tλf
– translation matrix of the space between the pupil and the front element
TλL1 – translation matrix of the front lens thickness
Tλs – translation matrix of the space between the elements
TλL 2 – translation matrix of the rear element thickness
The system matrix SȜ can be represented as:
S λ12 º
ªS
S λ = « λ11
»
¬ Sλ 21 S λ 22 ¼ ........................................................................................................ (8.8)
Element SȜ12 represents the equivalent power (Fe) of the optical system for a given wavelength.
The longitudinal chromatic aberration, expressed in dioptre, is the difference in equivalent power
of the system for two select ‘extreme’ (representing blue and red) wavelengths. Therefore,
assuming FȜ1 and FȜ2 as the equivalent powers of the system for two extreme wavelengths, the
275
Chromatic Aberration
LCA is given by:
LCA = Fλ1 − Fλ 2 ........................................................................................................... (8.9)
For example, when the wavelength range is limited to the Fraunhofer F-line (Ȝ = 0.486 ȝm) and
C-line (Ȝ = 0.656 ȝm), equation of LCA is given by:
LCA = S F 12 − SC12 = FF − FC
Similarly, TCA or chromatic difference in magnification (įM) may be defined as the difference
in image magnifications for the two given wavelengths. From the definition of lateral
magnification in Chapter 4 (Equation 4.7), at the Fraunhofer F-line;
M F = S F 22 + S F12 .t Fv ................................................................................................... (8.10)
where, tFv is the reduced distance between the image space nodal point and the retina (effectively
the vitreous chamber depth).
Similarly the magnification at the Fraunhofer C-line is given by:
M C = SC 22 + SC12 .tCv ................................................................................................... (8.11)
Therefore the TCA is:
TCA = δM = M F − M C ................................................................................................ (8.12)
276
Chromatic Aberration
8.3 Results
8.3.1 Chromatic Aberration of 1E-AIOL
Focal shift of the 1E-AIOL as a function of wavelength is given in Figure 8.3. Total difference in
the aberration is about 4 D for the material with V-number 32.1 (Acrylic) and just below 6 D for
the material with V-number 24.7 (Silicone). As a comparison, the phakic model eye produced
about 2.6 D differences in total focal shift. Chromatic aberration is slightly higher in the
accommodated state.
5
Phakic Eye
PMMA
Acrylic
Silicone
Phakic Eye
PMMA
Acrylic
Silicone
Focal Shift (D)
4
3
2
1
0
-1
0.40
0.50
Wavelength ( μ)
0.60
0.70
Figure 8.3: Chromatic focal shift (LCA) of phakic and pseudophakic model eyes implanted with
1E-AIOL. Solid and dotted lines represent the focal shifts for unaccommodated and
accommodated (2.0D) states, respectively. Aberration of the phakic model eye (black plots) is
included for comparison. The reference wavelength is 0.5893 ȝm where the shifts are zero.
When the radii of curvatures are expressed in terms of the bending factors and equivalent power,
277
Chromatic Aberration
effect of the physical form of the lens on aberrations can be visualized. This has been portrayed
in Figure 8.4 where it is clear that chromatic aberration is less for negative shape of the AIOL.
3.0
LCA (D)
2.5
2.0
Phakic
PMMA
Acrylic
Silicone
1.5
1.0
-2
-1
0
Bending Factor
1
2
Figure 8.4: Longitudinal chromatic aberration of 1E-AIOL made in various material as a
function of the bending factor and pseudophakic accommodation (2.0 D). Solid and dotted lines
represent the data for unaccommodated and accommodated states respectively. Aberration of the
phakic model eye (black plots) is included for comparison.
For a lens with bending factor 0.95, the TCAs within the spectral range of Fraunhofer F and Clines were 0.55% for PMMA, 0.59% for acrylic and 0.64% for silicone materials which slightly
increased with accommodation for each material. Similar to LCA, the negative shape factors of
the AIOL produced lesser but negligible chromatic difference of magnification (Figure 8.5). For
phakic eye TCA was 0.82%. It is evident that accommodation and shape factor has clinically
insignificant effect on the TCA.
278
Chromatic Aberration
1.0
TCA(%)
0.8
0.6
0.4
Phakic
PMMA
Acrylic
Silicone
0.2
0.0
-4
-2
0
Bending Factor
2
Figure 8.5: TCA of the model pseudophakic eye implanted with 1E-AIOL made in various
materials. Solid and dotted lines represent the data for unaccommodated and accommodated
states respectively. Aberration of the phakic model eye (dashed black line) is included for
comparison.
8.3.2 Chromatic Aberration of 2E-AIOL
Figure 8.6 shows the LCA of model eye implanted with 2E-AIOL. As can be observed, the
aberration was essentially similar to that found in 1E-AIOL. Again, the aberration is greater
when the lenses are made up of material with high dispersion (low V-number). Accommodation
increased the aberration in both phakic and pseudophakic model eyes, however, the changes are
miniscule.
279
Chromatic Aberration
5
Phakic Eye
PMMA
Acrylic
Silicone
Phakic Eye
PMMA
Acrylic
Silicone
Focal Shift (D)
4
3
2
1
0
-1
0.4
0.5
Wavelength ( μ)
0.6
0.7
Figure 8.6: Chromatic difference of refraction of 2E-AIOL with various materials. Solid and
dotted lines represent the focal shifts for unaccommodated and accommodated states
respectively. Aberration of the phakic model eye (black plots) is included for comparison. The
reference wavelength is 0.5893 ȝm where the shifts are zero.
Figure 8.7 shows the LCA of the eye as a function of bending factors of the front and rear
elements and accommodation. Similar to the 1E-AIOL only subtle effects from the AIOL design
can be observed.
Figure 8.8 shows the TCA of 2E-AIOL as a function of accommodation and bending factors of
the elements. Pseudophakic accommodation led an increase in TCA, though the changes are
small. The negative bending factor of the front element is effective in controlling the aberration
whereas the bending factor of the rear element had virtually no influence. There is about 0.2%
change in magnification with 2.5 D accommodation.
280
Chromatic Aberration
1.15
LCA (D)
1.10
1.05
1.00
2
1
0
X2
-1
-2
-2
-1
0
X1
2
1
Figure 8.7: LCA of the eye with 2E-AIOL as a function of accommodation and bending factors of
the front (X1) and rear (X2) elements. Both the elements are made in PMMA (nD = 1.49, V =
37.3). Solid (red) and dotted (black) lines represent the focal shifts for unaccommodated and
accommodated states respectively.
1.8
TCA (%)
1.6
1.4
1.2
1.0
2
2
1
1
0
0
X2
-1
-1
-2
-2
X1
Figure 8.8: TCA of the eye with 2E-AIOL as a function of accommodation and bending factors of
the front (X1) and rear (X2) elements. Both the elements are made in PMMA (nD = 1.49, V =
37.3). Solid (red) and dotted (black) lines represent the data for unaccommodated and
accommodated states respectively.
281
Chromatic Aberration
8.4 Achromatization of the Model Eye with AIOLs
Within the paraxial domain, the chromatic difference in magnification (TCA) was demonstrated
to be extremely small which is negligible in practical sense. This supports the earlier notion that
the TCA becomes clinically problematic only for the large off-axis position of the object (Thibos
et al., 1991). Since these analyses have been based on paraxial optics approximations, the focus
in this section will concern only elimination of the LCA.
Achromatic lenses eliminate the chromatic aberration for two colours. For simplicity, we limit
our analysis for achromatizing to the two conventional wavelengths (i.e. Fraounhofer F and C
lines). With this approach, dispersion property of a material can be expressed in terms of the
Abbe number (V) given by:
V=
nD 1
nF − nC ............................................................................................................... (8.13)
where nD, nF and nC are the refractive indices of a material at the Fraunhofer D, F and C lines.
LCA of a lens with equivalent power (Fe) is given by (Hazra and Delisle, 1998, Smith and
Atchison, 1997):
LCA =
Fe ª n' D −1 nD − 1º
−
nD '− nD «¬ V '
V »¼ ................................................................................ (8.14)
8.4.1 Achromatization with 1E-AIOL
Equivalent to Equation 8.14 for a 1E-AIOL, the LCA of the IOL is given by:
282
Chromatic Aberration
IOLLCA =
FIOL ª nIOL − 1 na − 1º
−
«
»
nIOL − na ¬ VIOL
Va ¼
........................................................................ (8.15)
Achromatization of the whole eye requires consideration of the corneal chromatic aberration.
Chromatic aberration of the eye is the sum of the aberrations from the cornea (CLCA) and the
AIOL (IOLLCA) (Born and Wolf, 1999, Miks et al., 2008). Therefore:
EyeLCA = CLCA + IOLLCA
.............................................................................................. (8.16)
Achromatization is obtained when EyeLCA = 0.
Substituting this in Equation 8.16 and solving for VIOL, the following is obtained:
VIOL =
FIOLV A (nIOL 1)
C LCAV A (na − nIOL ) + FIOL (na − 1) ..................................................................... (8.17)
VIOL in the Equation 8.17 is the dispersive power of the 1E-AIOL to produce zero LCA of the
eye. An important observation to be made in the equation is that VIOL may be negative when
CLCAVA(na-nIOL) in the denominator is greater than FIOL(na-1). Usually (nIOL > na) and FIOL and
VA are positives; hence the numerator of the equation is positive. For a standard material, unless
the surface is modified (diffractive optics for example), the Abbe number cannot be negative.
Hence this equation sets a boundary for the refractive index of the AIOL. For positive VIOL, the
condition is:
nIOL < na +
FIOL (na 1)
Va C LCA .............................................................................................. (8.18)
283
Chromatic Aberration
Achromatization with 1E-AIOL
1000
500
V-number
achromatization
possible zone
achromatization
not possible
zone
boundary
0
-500
1.40
1.44
1.48
1.52
1.56
1.60
Refractive Index
Figure 8.9: Refractive index and V-number of the 1E-AIOL material for zero LCA. The plot also
illustrates the boundary value of the refractive index. Refractive index of the AIOL, as calculated
from the Equation (8.33), must be approximately < 1.493 to obtain positive V-number.
When the corneal LCA is accounted for, minimum V-number of the 1E-AIOL required to
achromatize the model eye is about 252 for a material with refractive index of 1.450. When
Equation 8.17 is plotted for various V-numbers with values extending into the (hypothetical)
negative values, it produces a hyperbolic curve as illustrated in Figure 8.9 which asymptotes at
nIOL = 1.493, close to that of PMMA.
This observation is supported by the results in the previous section. In Figures 8.3 and 8.6, it can
be observed that the aberration is small for material with higher V-number whereas the
aberration is significantly greater for low V-number materials.
284
Chromatic Aberration
8.4.2 Achromatization with 2E-AIOL
LCA of the eye implanted with thin-lens 2E-AIOL may be given by:
EyeLCA =
Subscripts
FL1 ª nL1 − 1 na − 1º
FL 2 ª nL 2 − 1 na − 1º
−
+
−
+ C LCA
«
»
nL1 − na ¬ VL1
VA ¼ nL 2 − na «¬ VL 2
VA »¼
.................... (8.19)
L1
and
L2
denote the front and rear elements respectively and a refers to aqueous.
Again, achromatism is obtained when EyeLCA = 0. With known VL1, when Equation 8.19 is
solved for VL2 (V-number of the rear element) we obtain:
VL 2 = −
VL1
2
ℜ1η1 + VL1[C LCA −
2
η
VA
(ℜ1 + ℜ 2 )]
.................................................................... (8.20)
η = na − 1 , ℜ1 = FL1 and ℜ 2 = FL 2
where η1 = nL1 − 1 , η 2 = nL 2 − 1 ,
nL1 − na
n L 2 − na
Figure 8.10 shows the V-numbers required for the elements of 2E-AIOL to achromatize the
model eye.
It is observed in Figure 8.10 that V-number of the front element needs to be larger than the Vnumber of the rear element. For example, when VL1 = 37.2 (PMMA), VL2 equals 14.01, 17.87
and 26.55 when the refractive index for the back element is 1.55, 1.50 and 1.45 respectively.
With the combination of VL1 = 37.31 (PMMA, n = 1.492) and VL2 = 17.52 (a hypothetical
material with nD = 1.55), we obtain residual chromatic aberration as shown in Figure 8.11. In the
285
Chromatic Aberration
figure, since dioptric equivalent is too small to be displayed (<0.1 D), the focal shifts are shown
in distance measured from the retina.
55
V-number (Rear Element)
constant nl1 (1.492)
45
35
25
nl2 = 1.45
nl2 = 1.50
15
nl2 = 1.55
20
30
40
50
60
70
80
V-number (Front Element)
90
100
110
Figure 8.10: V-numbers of the front and rear elements required for achromatizing the model eye
with 2E-AIOL. Refractive index for the front element is considered constant (n=1.492) in this
example. Results are plotted for three select refractive indices for the back element material.
8.4 Discussion and Conclusion
We used the matrix method for paraxial optics to calculate the longitudinal and transverse
chromatic aberrations of model eye implanted with AIOLs made up of various materials.
Predicted chromatic aberrations in the current study are well within the range of clinically
measured aberrations reported in the literatures. The majority of published studies have found
that the chromatic difference of refraction of a normal eye is approximately 2 D (Bedford and
Wyszecki, 1957, Charman and Jennings, 1976, Howarth and Bradley, 1986, Thibos et al., 1992,
286
Chromatic Aberration
Wald and Donald, 1947, Wang et al., 2008, Ware, 1982) within the visible spectrum (400 to 700
nm) while as high as 3.2 D has been reported for an extended range of wavelength (365 to 750
nm) (Wald and Donald, 1947). In the current study, 2.6 D LCA is predicted for the phakic model
eye (spectrum range 400 to 700 nm) and this value is close (2.2 D) to that reported by Atchison
and Smith for the same model eye (Atchison and Smith, 2005).
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hŶĂĐĐŽŵŵŽĚĂƚĞĚ
ĐĐŽŵŵŽĚĂƚĞĚ
Ϭ͘ϬϬ
>;Ϳ
ͲϬ͘Ϭϭ
ͲϬ͘ϬϮ
ͲϬ͘Ϭϯ
ͲϬ͘Ϭϰ
ͲϬ͘Ϭϱ
Ϭ͘ϰϴϲ
Ϭ͘ϱϭϰ
Ϭ͘ϱϰϯ
Ϭ͘ϱϳϭ
Ϭ͘ϲϬϬ
tĂǀĞůĞŶŐƚŚ;ʅͿ
Ϭ͘ϲϮϴ
Ϭ͘ϲϱϲ
Figure 8.11: An example of secondary longitudinal chromatic aberration of the eye following
achromatization with 2E-AIOL. In this example the front element was manufactured in PMMA
material (nD = 1.492, V = 37.3). The rear element consist of a hypothetical material with nD =
1.55, and V = 17.52.
In a pseudophakic eye, the chromatic aberration is greatly dependent on the dispersion of the
intraocular lens (Zhao and Mainster, 2007). One study (Nagata et al., 1999) reported the
aberration of pseudophakic model eye implanted with various IOLs (the model eye used in that
study was different from the one used in the current study) where LCA for wavelength range of
380 to 700 nm were 2.8, 1.9 and 0.2 D for IOLs with V-numbers 35, 47 and 200 respectively.
287
Chromatic Aberration
Similarly another study reported clinically measured LCA of 3.5 D, 2.8 D, 2.4 D and 2.0 D LCA
for IOL materials with V-numbers 27.3, 32.05, 37.29 and 44.01 respectively (Siedlecki and
Ginis, 2007). These values are in good agreement to the predictions in the current study for both
types of AIOLs in which LCA are 3.6, 2.4 and 5.5 D for materials with V-numbers 32.2, 37.3
and 24.7 respectively. Small discrepancies are expected due to the difference in the V-numbers
of the materials. The different model eyes used in the studies is another possible source of minor
differences. For the same model eye as used in this study, Norberto and Montes-Mico (2007)
reported 0.9 D LCA for PMMA material within the spectral range of 470 to 650 nm which
closely agrees with the prediction in the present study for this spectral range (0.92 D).
Clinical studies reporting the transverse chromatic aberration are very limited. A laboratory
study reported 0.8 radians per metre TCA for wavelength range between 0.486 and 0.656 ȝm
measured with a Vernier adjustment method (Simonet and Campbell, 1990). Similarly, Thibos et
al (1991) found 0.8% chromatic difference in magnification within the same spectral range. Our
study demonstrated slightly lesser TCA (0.71%) for the phakic model eye. TCA of pseudophakic
eyes is slightly higher (1.1%) for 2E-AIOL and smaller (0.4%) for 1E-AIOL which slightly
increased with accommodation.
The paraxial analysis in this study suggests that the material for AIOL is probably the only
degree of freedom that can be practically utilised to eliminate the chromatic aberration in the
pseudophakic eye. Designs of AIOLs to manipulate chromatic aberration by modifying bending
factors produce insignificant changes. Achromatization of the model eye with 1E-AIOL required
extreme values of V-number which, practically is probably unattainable. However, it is
demonstrated that a wide range of 2E-AIOL can be designed which may eliminate or control
chromatic aberration. From this study, it is found that the amount of residual (secondary)
chromatic error that can be achieved can be far lower than a previous attempt (Diaz, 2005). In
practice though, the availability of suitable material that has the appropriate mechanical
properties and is also biocompatible might pose a challenge.
288
Chromatic Aberration
Chromatic aberration, to some extent, is clinically beneficial for a pseudophakic eye because it
enhances near vision by improving depth of focus (Campbell and Weir, 1953, Marcos et al.,
1999). However, the limit that can be tolerated physiologically and remain beneficial is not fully
known. Therefore, debate will continue regarding need for correction of the chromatic aberration
in a pseudophakic eye. In relation to AIOL, since the implant delivers sufficient amount of
accommodation, the magnitude of depth of focus offered by the aberration might be relatively
insignificant.
In conclusion, chromatic aberrations of the AIOLs are similar to that observed in conventional
monofocal IOLs. While achromatisation with 1E-AIOL made in commonly available IOL
material and with standard surface design is not possible (unless surface design is modified to
refractive-diffractive system), a 2E-AIOL could be designed to act as a spaced achromatic
doublet though availability of a suitable material might prove challenging to the industry.
Nonetheless, if achieved, the same combination of materials of 2E-AIOL elements is effective in
achromatizing the eye for both unaccommodated (distance focus) and accommodated (near
focus) states.
289
Chromatic Aberration
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Bedford, R. E. and Wyszecki, G. (1957). Axial chromatic aberration of the human eye. J Opt Soc
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Campbell F. W., and Gubisch R. W. (1967). Effect of chromatic aberration on visual acuity. J
Physiol-London 192, 345.
Campbell, F. W. and Weir, J. B. (1953). The depth of focus of the human eye. J Physiol 120,
59P-60P.
Cantor, G. N. (1984). Newton on Optics: The Optical Papers of Isaac Newton. Science 224, 724725.
Charman, W. N. and Jennings, J. A. M. (1976). Objective measurements of the longitudinal
chromatic aberration of the human eye. Vis Res. 16, 999-1005.
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achromatize the human eye. J Modern Opt 51, 12.
Hartridge, H. (1950). The chromatic aberration of the human eye and its physiological
correction. Experientia 6, 1-10.
Hazra, L. N. and Delisle, C. A. (1998). Primary aberrations of a thin lens with different object
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Chromatic Aberration
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Hong X, K. M., Zhang X, Weinschenk J, Carson Dr (2008). Correction of Chromatic
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223-226.
Howarth, P. A. and Bradley, A. (1986). The longitudinal chromatic aberration of the human eye,
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Kruger, P. B., Mathews, S., Aggarwala, K. R. and Sanchez, N. (1993). Chromatic aberration and
ocular focus: Fincham revisited. Vision Res 33, 1397-1411.
Kruger, P. B. and Pola, J. (1986). Stimuli for accommodation: blur, chromatic aberration and
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Marcos, S., Moreno, E. and Navarro, R. (1999). The depth-of-field of the human eye from
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Nagata, T., Kubota, S., Watanabe, I. and Aoshima, S. (1999). Chromatic aberration in
pseudophakic eyes. Nippon Ganka Gakkai Zasshi 103, 237-242.
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contrast sensitivity in pseudophakic eyes. Arch Ophthalmol 119, 1154-1158.
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Ravikumar S., Thibos L. N., and Bradley A. (2008). Calculation of retinal image quality for
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pseudophakia. Am J Optom Physiol Opt 63, 867-872.
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Chromatic Aberration
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correcting higher order aberrations of the eye. J Refract Surg 16, S554-559.
Yoon G. Y., and Williams D. R. (2002). Visual performance after correcting the monochromatic
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Zhang, X. X., Bradley, A. and Thibos, L. N. (1991). Achromatizing the human eye: the problem
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293
Summary and Recommendation
Chapter 9
Summary & Recommendations
294
Summary and Recommendation
TABLE OF CONTENT
9.1 SUMMARY................................................................................................................. 296
9.2 OPTICAL PERFORMANCE OF AIOLS ............................................................... 298
9.2.1 Accommodative Performance............................................................................ 298
9.2.2 Magnification and Depth of Field ..................................................................... 299
9.2.3 Performance in Presence of Misalignment ....................................................... 300
9.2.3 Aberrations ........................................................................................................ 301
9.3 DESIGN OF AIOLS................................................................................................... 301
9.6 RECOMMENDATIONS ........................................................................................... 303
295
Summary and Recommendation
9.1 Summary
Restoring accommodation in the pseudophakic eye is definitely one of the most challenging
tasks of modern vision treatment; but also of great importance and potentially great
rewards. Not only will this area of vision correction be of benefit to the individual patients
in terms of excellence of vision and quality of life, but improvements will be of importance
to productivity, economics and socio-political well-being of nations. And this can only
increase in importance and value with the ‘ageing’ of the world’s population, the likelihood
of increase of retirement age and the pace of technology development making near vision
tasks such as computers, mobile phone and personal digital assistants (PDAs) a vocational,
even day-to-day survival, requirement for the individual and the population.
Despite the fact that ciliary muscle and other components of the accommodative apparatus
remain functional after cataract surgery, most patients undergoing cataract surgery are left
with uncorrected presbyopia. In practice, these people live with their presbyopia
uncorrected other than with spectacles or contact lenses, compromising the visual
performance and impacting convenience and quality of life, to some extent.
Among the several fundamental approaches to treat pseudophakic presbyopia,
accommodating IOLs have considerable potential to be successful in addressing the longstanding visual need. At present, 1E-AIOL is the only theoretically true accommodating
device commercially available for the public. Unfortunately, clinical experience with such
devices are at least equivocal and at worst controversial – the sufficiency of near vision
afforded by such devices being debatable. 2E-AIOL has been gaining significant interest as
this type of accommodating device potentially offers a greater amount of accommodation
and initial trials of Synchrony 2E-AIOL have been promising. To advance development in
296
Summary and Recommendation
this class of devices, what has been needed is a comprehensive, systematic investigation of
their design options and associated performance.
This thesis evaluated, through theoretical treatment, the optical performance and set out the
design principles of AIOLs – both 1E-AIOL and 2E-AIOL – with a view to identifying
strategies for optimizing accommodative efficiency and visual performance in the
pseudophakic eye. Specifically, the thesis:
•
surveyed the literature relating to accommodation, presbyopia and some conceptualized
methods of alleviating the persisting problem of pseudophakic presbyopia
•
evaluated and compared the accommodative performance of two types of translatingoptics AIOL and proposed design approaches and some design options for optimising
their accommodative efficiency
•
examined the effect of translation of AIOL element/s on magnification and depth of
field
•
studied the ocular aberrations (spherical, coma, astigmatism and chromatic) produced
by various types and designs at distance and near focus states and proposed design
approaches and some design options to optimise performance or control and manage
aberrations
•
analysed the effect of misalignment of AIOLs on aberrations and identified some design
options to maintain or optimise performance in such conditions.
The following major conclusions and key points of relevance can be drawn from the studies
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Summary and Recommendation
in this thesis.
9.2 Optical Performance of AIOLs
9.2.1 Accommodative Performance
The magnitude of accommodation is one of the most important performance indicators
pertaining to an AIOL and this was evaluated employing analytical and computational
methods. The analytical method helped in visualizing the design parameters that can be
used for optimum performance whereas the computational method offered exact prediction
of the accommodation obtained with various select models. Consistent with earlier
theoretical results and most results from clinical studies which employed objective
methods, 1E-AIOL affords about 1 D accommodation with each millimetre of forward
translation of its optics. De-accommodation (i.e. ‘reverse’ accommodation) may result
when the optics is translated in the backward direction. While the accommodative
performance of 1E-AIOL is dictated by the refractive state of the eye (which ultimately
determines the power of the lens to be implanted), the performance of 2E-AIOL is
determined by the choice of lens element powers selected by the manufacturers in their
design. Certain designs of 2E-AIOL deliver higher amounts of accommodative amplitude; a
design consisting of a high power for the front element returns a greater amount of
accommodation.
With consideration of the overall thickness and translation distances that are practical, the
accommodative amplitude available from a 2E-AIOL should be around 2.5 D to 4 D.
Due to the potential for mixing designs (for the same refractive error) or the need for
different distance refractive power (for anisometropia) between eyes in the one patient,
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Summary and Recommendation
the issue of aniso-accommodation causing refractive imbalance at near need to be carefully
considered in the designing and prescribing of these devices. Custom design of AIOL may
be required for post operative visual satisfaction.
9.2.2 Magnification and Depth of Field
One of the issues that require considerable attention is the ocular magnification produced
due to translation of the optics which is more significant in 2E-AIOL. Forward translation
of the AIOL leads to an increase in retinal image size. The magnitude of change is
dependent greatly on the power of the elements and the amount of shift. Considering about
2.5 D accommodation obtainable in a 2E-AIOL, more than 10% dynamic (changes with
accommodation) increase in retinal image size may be expected depending on the power of
the front element incorporated in the design. This dynamic magnification is in contrast to
the dynamic ‘minification’ of the natural phakic eye. This dynamic magnification effect
may even be advantageously exploited by the user for enhancing near vision.
The potential problem of dynamic aniseikonia must be recognised by the clinician and the
designer alike. When both eyes are implanted with identical design, type and power of
AIOLs, the difference in magnification between eyes may be clinically insignificant.
However, dynamic aniseikonia may arise from either the monocular implantation of AIOL,
the use of non-matching types or designs of AIOL of the same power between isometropic
eyes, or when prescribing for anisometropia without due consideration to magnification
effects. Disregarding this effect may severely deteriorate binocular visual function.
Depth of field has considerable effect in aiding near vision. The performance of AIOLs was
evaluated with an objective to identify design parameters that may be of assistance in
improving the depth of field. Various factors such as implant position, power combination
of the elements, amount of translation and types of AIOL were examined. Though all
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Summary and Recommendation
the factors showed some amount of changes in depth of field, none influenced sufficiently
to a clinically observable level concluding that design parameters and pseudophakic
accommodation are irrelevant in practical sense of depth of field. This finding refutes many
of the popular beliefs around the relationship between the depth of focus available with an
AIOL and parameters such as depth or position of the implant.
9.2.3 Performance in Presence of Misalignment
Misalignments of an IOL, i.e. tilt and decentration, have significant effect on off-axis
aberrations; particularly coma and astigmatism. The performance of AIOLs was evaluated
in terms of induced spherical aberration, coma and astigmatism in the presence of
misalignments. Tilt and decentration were found to have significant effects on coma and
astigmatism of both types of AIOLs while spherical aberration remained practically
unaffected. The effects were more pronounced in the accommodated (near focus) state
compared to unaccommodated (distance focus) state. In a 2E-AIOL, the design of the
aligned elements had no effect on the induced aberrations.
Though virtually no studies have reported misalignment of AIOL, due to the complex
mechanical arrangement in the haptics design, the rate and extent of AIOL misalignment
may be assumed to be similar to (or perhaps greater than) that for conventional IOLs.
Assuming equivalent amounts of misalignment to conventional IOL, the effect on
aberrations of misalignments of AIOLs is not expected to present a clinical problem using
visual acuity as a criterion. However, biological consequences are yet to be investigated.
Of interest to designers and manufacturers may be the finding that while the aberrations
caused by the misalignment of a single element in a 2E-AIOL cannot be eliminated or
controlled by the design of either elements, misalignment of the entire 2E-AIOL as a group
can. The implication is that through careful design of robust coupling mechanisms to
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Summary and Recommendation
ensure co-alignment of the elements and incorporation of appropriate optical design, the
manufacturer can play a role in reducing the impact of misalignment introduced at the
surgical implantation stage on vision. This increase of tolerance for the surgeon can only
improve the acceptability and wider utilisation of AIOLs.
9.2.3 Aberrations
Seidel primary aberration theory was used to evaluate the monochromatic aberrations
(spherical aberration and coma) of AIOLs. It was found that bending factor of the lens and
its refractive index are the two more important parameters in governing resultant
aberrations. With pseudophakic accommodation (translation of the optics), aberrations
changed significantly; spherical aberration increased and the coma decreased in stark
contrast to phakic accommodation of the natural eye which has the opposite trend. Coma
aberration is more sensitive to bending factor and translation of the AIOLs compared to
spherical aberration.
Chromatic aberrations of the AIOLs are similar to that reported for pseudophakic eyes
implanted with conventional monofocal IOLs. Pseudophakic accommodation and the shape
of the AIOLs have insignificant effect on chromatic aberration While pseudophakic
accommodation leads to an increase in the chromatic aberration, it is clinically
insignificant. The dispersion property of the lens material is an extremely important
parameter in determining the chromatic aberration in the pseudophakic eye with AIOL.
9.3 Design of AIOLs
In terms of controlling monochromatic aberrations, optimum bending factor of the 1EAIOL is approximately +1 (plano-convex) which is consistent with the optimum shape
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Summary and Recommendation
of the conventional IOL. Elimination of spherical aberration is not possible with spherical
1E-AIOL even with the optimum shape. The coma aberration may be eliminated with a
lens of -1 bending factor for the distance focus state (which is consistent with the optimum
shape for conventional IOL).
Optimum shape of the 1E-AIOL for minimising coma and spherical aberrations are not the
same. Therefore a compromise may be required to manage these two aberrations to
acceptable levels simultaneously. Computational results suggested that a bending factor of
-0.5 provides an ideal intermediate choice. However, the optimum bending factor for
controlling coma may also be a good choice as coma is more sensitive to the change in
shape factor. Elimination of spherical aberration is possible only with aspheric surface of
the AIOL.
A wide range of spherical 2E-AIOL designs is possible to eliminate or control the
aberration to a desirable level using the additional degrees of design freedom available for
two elements, including combinations of the shape factors and refractive indices. Therefore,
technically complex and consequently more expensive manufacturing methods associated
with aspherised surfaces may not be required in 2E-AIOL. Importantly, biologically
attractive (with the potential to limit PCO) designs are also possible without imposing
asphericity.
The optimum shape of the AIOL in the aligned position remains the same when the AIOL
takes on misaligned positions. The optimum shapes to perform optimally in the presence of
the misalignments differ for various degrees of misalignments imposed; nonetheless,
optimum shapes for coma and astigmatic aberrations were nearly identical.
In terms of controlling chromatic aberration for the pseudophakic eye, dispersion property
of the lens material is only the design freedom. 1E-AIOL may not eliminate the aberration
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Summary and Recommendation
unless its surface design is modified (diffractive design). However, 1E-AIOL with less
dispersive power (High V-number) may effectively control the aberration within the
magnitude reported for the phakic eye. Proper combination of the materials for 2E-AIOL
elements may eliminate the aberration. Since the effect of optical shift on the aberration is
negligible, same combination of the materials of two elements is effective in achromatizing
the eye for both unaccommodated (distance focus) and accommodated (near focus) states.
If there is to be only a single summary point, then the revelation of the analyses would have
to be that practically workable, optimum AIOL designs are possible that can
simultaneously address optical and other (e.g. PCO) requirements. However, advances in
material developments to realise a greater range of refractive indices and dispersion can
broaden the range of optimum designs that can be made available to the clinician and
patient.
9.6 Recommendations
Technology relies on scientific knowledge. So the improvement of AIOL technology can
only advance with detailed understanding of the accommodative system and a full
comprehension of presbyopia. Though the classic Helmholtz theory of accommodation has
gained widespread acceptance and can explain accommodation at the general level, the
persistence of contrary theories (such as those of Schachar) in the scientific literature
signals that the full working details of the accommodative system and its physiological
apparatus is still yet to be acquired. Certainly a consensus of the science of accommodation
and presbyopia has not been reached. Unless the mechanism is fully understood,
development of the perfect AIOL cannot be forthcoming. New knowledge will facilitate the
designing of accurate and more effective mechanical arrangements to facilitate translation
of the optical elements in AIOLs to enhance the accommodative performance.
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Summary and Recommendation
Therefore, continuation in the scientific investigation into the crystalline lens, the
accommodative mechanism and the causes of presbyopia is a prerequisite to further
advances in AIOL technology.
The greatest potential ‘killer’ of optical performance for AIOL has a non-optical origin.
Performance of AIOL is significantly associated with the presence of posterior capsular
opacification. PCO, which potentially impedes the efficiency of AIOL substantially, is
reported to occur in all patients following cataract surgery over the period of time. The ‘zsyndrome’ (vaulting or tilting) of the single-element AIOL is reported to be a result of PCO
perhaps exacerbated by delicate haptics design. In addition, the gradual breakdown of
accommodative performance over time is reported which marks an important issue to be
solved in order to achieve sustained, useful performance over the lifespan. Though
clinically sufficient amount of accommodative amplitude may be obtainable with some
designs of AIOL in the ideal setting, consideration of the sustainability of the performance
over time is beyond the scope of this thesis. Studies to evaluate the long-term clinical
experience of such AIOLs may be helpful in understanding the key drivers for long-term
success. Development of materials for peripheral mechanical structures (haptics) that can
withstand the physical strain arising from the enormous number of continuous
accommodation – disaccommodation cycle over the working life of the device and
resistance to biological changes (e.g. cellular reaction, encapsulation, etc) are also critical
for long-term success.
Assuming 4 D accommodation is required for a comfortable near vision and given an eye
generally has about 1 to 2 D pseudo-accommodation, an AIOL should provide at least
another 2 D of accommodation to be euphemistically called a ‘success’. The existing
examples of 1E-AIOLs are well below this criterion for success. Improvements in design
and engineering may further enhance the performance. 2E-AIOL can be a good alternative
as, at least theoretically, it has the potential to afford much higher accommodation.
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Summary and Recommendation
Designs optimised for optical performance can be achieved. However, more work needs to
be done to identify specific design variants and submit these through more exhaustive
analyses than can be undertaken within the limited scope of the present work.
Improvements through further research can be made in both design configurations and
material properties.
Accommodative performance of a 2E-AIOL is linked to its element powers. In order for
clinicians and end-users to be able to make informed decisions about their prescription and
use, the manufacturers should be encouraged or perhaps induced by regulations or
standards to provide detailed design data with the devices. Further, it may be of assistance
for the practitioners advising their patients to provide information about the accommodative
efficiency (i.e. the amount of accommodation achievable for a unit amount of translation)
and other optical performance parameters.
The issue of binocular vision associated with bilateral implantation of AIOL needs to be
investigated. The limits for design and type differences in AIOL as well as acceptable
limits of dynamic anisometropia and dynamic aniseikonia are important parameters that
must be understood for AIOL to be truly safe and functional.
305
Appendix
Appendix
306
Appendix
TABLE OF CONTENT
APPENDIX A: PATENT OF AIOL DESIGNS ........................................................................... 308
A1: Transverse Translating-optics AIOL .........................................................................308
A2: Curvature changing AIOL .........................................................................................309
A3: Single-Element translating-optics AIOL ....................................................................312
A4: Two-element Translating-optics AIOL.......................................................................313
APPENDIX B: REVIEW OF THE OPTICAL PRINCIPLES .................................................... 319
B1: Sign Convention .........................................................................................................319
B2: Optical Ray Tracing...................................................................................................320
Appendix B3: Matrix method in Paraxial optics ..............................................................324
APPENDIX B4: ABERRATION THEORY ................................................................................ 327
B4.1 Equation for Seidel Primary Aberrations ................................................................327
B4.2 Primary Aberration of a Thin Lens ..........................................................................329
B4.3 Central Monochromatic Aberrations of AIOL .........................................................332
B4.4 Non-central Monochromatic Aberrations of AIOL ..................................................333
APPENDIX C: MODEL EYE ...................................................................................................... 335
REFERENCES.............................................................................................................................. 338
307
Appendix
Appendix A: Patent of AIOL Designs
Unlike pseudo-accommodative IOLs (pIOLs) such as multifocal IOLs, accommodating IOLs
(AIOLs) are capable of changing their refractive power with two principal mechanisms: 1) by
changing surface curvature and 2) by shifting axial position within the eye also called translating
optics AIOL. Depending on their mechanism of action, AIOLs may be categorized as illustrated
in Chapter 2 (Table 2.1). Plethora of AIOL designs are patented over last couple of decades. In
this appendix, some of the important patented designs of AIOLs are briefly discussed. All the
designs included here are obtained by performing patent search in one of the following websites:
1)
http://www.uspto.gov/patents
2)
http://www.wipo.int/pctdb/en/
3)
http://www.epo.org/
A1: Transverse Translating-optics AIOL
The transverse translating IOL design consists of two optical elements translating radially,
(transverse translations) (Simonov, Vdovin et al. 2006). The elements are based on a twoelement Varifocal Alvarez lens. The front element consists of a combined spherical lens with a
cubic surface for the varifocal effect superimposed with the posterior element which has only a
cubic surface. The power changes when these elements shift in opposite directions in a plane
perpendicular to the optical axis. About 4.0D of accommodation and 0.75mm translation has
been reported.
308
Appendix
Figure A.1: Two-element Alvarez verifocal transverse translating AIOL
A2: Curvature changing AIOL
A number of designs have been proposed within this category. One design that is patented by
Cumming (US patent # 0,129,801) comprises a combination of solid and liquid silicone
materials. The anterior optic has a central area with an elastic membrane that can vary in radius
in response to accommodative effort of the eye. The designer believes that the accommodation
induced contraction of the ciliary body and the consequential increase in the vitreous pressure
push the posterior surface of the solid silicone material in the forward direction. This in turn
causes bulging of the thinner anterior surface, which is made with an elastic membrane thereby
increasing the power of the lens assembly. (Figure A.2)
309
Appendix
Figure A.2: A design of dynamic optic AIOL patented by Cumming (US patent # 0,129,801)
Skottun patented two designs of deformable AIOLs. The first one patented in 1996 (US patents
5,489,302) consists of a medium having a refractive index less than the surrounding media
(aqueous and vitreous) such as air, enclosed between two transparent membranes. At least one of
the membranes is created to have a concave surface that results in plus refractive power. He
claimed that if the surface is made of a resilient material it is possible to make the lens alter its
shape. He believed that a change in power is accomplished by manipulating the pressure in the
interior of the lens which is brought about by varying the volumes of the liquid in the chamber.
The accommodation generated force in the capsular bag affects a specially designed haptics
which in turn control the volume in the interior of the lens.
Figure A.3: Design patented by Skottun (US patent # 5,489,302)
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Appendix
Another patented design of Cumming (US 6,117,171) has a flexible interior membrane dividing
a bulk lens into two compartments. The compartments have fluid media with different refractive
indices. Cumming proposed that the overall optical power can be altered by changing the
curvature of the transparent membrane in the chambers. The front chamber has an extension to
the fluid filled auxiliary space which is used to control the volume and pressure in the chamber
by means of interior extensions of the haptics. The haptics are attached to the encapsulated rigid
shell in such a manner so as to be able to rotate around under the influence of the force generated
during accommodation and control the movement of fluid in auxiliary space thereby changing
the shape of the membrane.
Figure A.4: Design of a dynamic AIOL patented by Cumming (US patent # 6,117,171)
The NuLens, designed in Herzliya, Israel, is another innovative design of curvature changing
AIOL which consists of a piston-like configuration that, with an accommodative effort,
compresses a central cylinder containing a flexible Silicone gel encased within a deformable
membrane (Ben-Nun and Alio 2005). A button-hole in the front plate compresses the anterior
surface of the membrane in the periphery, and as a result the central part bulges out to increase
the curvature. The authors reported that the NuLens produced more than 40 D accommodative in
monkey eyes.
311
Appendix
Figure A.5: Dynamic optic AIOL designed by Ben-Nun in Harzliya, Israel
A3: Single-Element translating-optics AIOL
A single optics translating-optics AIOL named BioComFold (Morcher, Germany) has three
anteriorly angulated haptics with holes, which are supported by a segmented ring. The number of
holes differs in each model. On contraction of the ciliary muscle, there is a compression on
circumferential ring which results an anterior vaulting of the optics through an action of the
translation zone.
Figure A.6: BioComFold (Morcher, Germany) 1E-AIOL
312
Appendix
The 1CU AIOL (HumanOptics, Germany) has four flexible haptics with a hinge at the
connection point with the optics. The end of the haptics is bent upward. On accommodation, the
optics translates anteriorly due to the compressive force exerted by the capsular bag on the
haptics.
Figure A.7: Schematic diagram of design of 1 CU (HumanOptics, Germany) and AT-45
(Crystalens) 1E-AIOL
The AT-45 (Eyeonic Inc, CA) consists of two plate haptics which are hinged (groove) at junction
to the optics. The other end of each haptic has a pair of fixating extensions. This AIOL is
assumed to be pushed forward due to increased vitreous pressure during accommodation.
There are many other patented designs of translating-optics 1E-AIOL; for example, US
6,494,911 (Cumming, 2002), US 6,551,355 (Ghazizadeh et al, 2003), US 0,111,152 (Kelman,
2004), US 7,125,422 (Woods, 2006), US 7,018,410 (Vazeen, 2006), US 0,089,712 (Malecaze,
2006), US 0,239,274 (Kellan, 2007), US 0,142,913 (Philips, 2007), US 7,435,258 (Blake, 2008),
US 7,354,451 (Koch, 2008).
A4: Two-element Translating-optics AIOL
An intelligent concept of AIOL using two lenses has drawn plenty of interest. These designs
typically comprise of two lenses or elements altering inter-element distance relative to each other
313
Appendix
to accommodate. Translation of the optics are facilitated by often complex but carefully designed
spring loaded or hinged haptics. Various designs of such AIOLs have been patented; however
they share more or less the similar principles of optics but differ in their mechanical
arrangement.
Figure A.8: A design of 2E-AIOL as described in US patent # 4,994,082
Richards et al [US patent 4,994,082 (1991)] proposed a design to be implanted in the ciliary
sulcus which works under the influence of the force from the ciliary body. In this design, when
the ciliary muscle contracts during accommodation, the haptics (70 & 72 in the Figure) move
towards each other (52 and 54 in the Figure) separating two lenses apart.
A design patented by Zhang (US 7,150,760) has a fixed front optic to be implanted in the
anterior chamber of the eye and a mobile back element to be implanted in the capsular bag.
314
Appendix
Figure A.9: A design of 2E-AIOL where one element is implanted in anterior chamber and
another in posterior chamber as described in US patent # 7,150,760
A design patented by Pynson (US 0,118,216) is quite different from others in that the optics is
translated by means of two permanent magnets implanted: one in the ciliary body or zonules and
another coupled with the lens optics. The working principle of this design is that the
accommodation induced movement of the magnetic medium in the ciliary body leads to a
synchronous movement of the magnet coupled optics of the IOL, thus producing
accommodation. It is claimed that the pseudophakic accommodation produced with this AIOL is
independent of zonular movement and that this design has the advantage that the post operative
capsular fibrosis will not deteriorate the accommodation efficiency.
Figure A.10: Pyson’s magnet driven design of a 2E-AIOL (US patent#0,118,216)
Portney et al patented a design which works under the influence of pressure from the vitreous
body [US patents 0,162,612 (2004) & 7,238,201 (2007)]. As claimed the system can be
implanted in the bag or ciliary sulcus. It rests against the vitreous face and does not require the
posterior capsule. The system works under the belief that, during accommodation, the vitreous
315
Appendix
pressure increases and the vitreous moves in the forward direction. The mechanical design of the
haptics is such that a small forward movement of the back element due to vitreous pressure
amplifies the forward movement of the front element.
Woods’ designs [US patents 6,299,641 (2001) & 6,443,985 (2002)] have principally similar
haptics however, this patent contains only one optic.
The system is again to be implanted in the intact capsular bag. In the unaccommodated state, the
capsular bag is pulled radially outward in the stretched position, which squeezes the spring
loaded (resilient) haptics and drives the lens in a more posterior position. When accommodated,
the zonules are relaxed which release capsular force on the spring loaded haptics, as a result of
which the optics move forward.
Figure A.11: A design of 2E-AIOL as described in US patent 6,443,985
Bandhauer et al [US patent 6,767,363 (2004)] is also to be implanted in the capsular bag. The
system has locked haptics engaged in the equator.
316
Appendix
Figure A.12: A design of 2E-AIOL described in US patent 6,767,363
The driving force is again the accommodation induced tension on the capsular bag. When
accommodated, the equatorial diameter of the bag is decreased which pushes the haptics inward
radially propelling the lenses apart.
Zhang et al proposed another design with slightly different haptics design [US patents 0,236,422
(2004) & 7,223,288 (2007)].
Figure A.13: Zhang et al design of a 2E-AIOL (US patent# 7,223,288)
In this patent the driving force is again from the capsular bag. Here the posterior element is
implanted first in the bag followed by the anterior element. The posterior element has haptics
317
Appendix
with notches (18 as seen in the centre Figure) in which the anterior haptics is engaged through
projection (28 as seen in the first Figure).
Magnante et al proposed a simple design [US patent 0,060,032 (2005)] that works under the
direct influence of the ciliary body. In this design, the haptic from the front and back elements
are directly and individually connected to the ciliary muscle. It is claimed that the contraction of
the ciliary muscle drives the lenses in opposite directions separating them apart.
Many other 2E-AIOL designs have been patented which are to be implanted in the bag and work
under the influence of capsular tension. A few examples are: US patents 0,148,023 (Shu, 2004);
6,767,363 (Bandhauer et al, 2004); 0,204,255 (Peng et al, 2003); 6,926,736 (Peng et al, 2005);
6,969,403 (Peng et al, 2005); 0,113,914 (Miller et al, 2005); 7,118,597 (Miller et al, 2006);
0,259,139 (Azndo-Azizi et al, 2006); 7,198,640 (Nguyen, 2007); 0,016,293 (Tran ST, 2007);
0,260,309 & 0,260,310 (Richardson, 2007); 0,154,364 (Richardson, 2008); and 7,316,713
(Zhang, 2008).
Figure A.14: A design of 2E-AIOL as described by Magnante in US patent # 0,060,032
318
Appendix
Appendix B: Review of the Optical Principles
B1: Sign Convention
The convention of sign must be defined and has to be consistent with textbooks and commercial
design programs. Unfortunately various notations are used. In this thesis, a right-handed real
space (x,y,z) coordinate system with z-axis along the optical axis will be considered as defined
by Conrady (1957).
1) The light is considered to be incident from the left
2) The radius (R) is positive if the centre of curvature is to the right of the vertex and
negative otherwise. The curvature ‘c’ is inverse of the radius of curvature (c=1/r).
3) An angle of ray (ș) is considered positive if the ray has to be rotated anti-clockwise into
the optical axis; otherwise it is considered negative.
4) The distance (l) (e.g. thickness, height) is positive if it is measured to the right and up
from the lens pole or optical axis; distances measured to the left and down will be
considered negative.
5) The z-axis of the coordinate system is considered as the optical axis
With this convention for any particular ray, not all three parameters l, h and ș can be positive at
the same time. These sign conventions are illustrated in Figure B1.
319
Appendix
Figure B.1: Diagram illustrating the sign convention followed in this thesis
B2: Optical Ray Tracing
Ray tracing is a technique to find the exact path of a ray along the optical system. Exact (finite)
ray tracing of a meridional ray can be conveniently traced by applying Snell’s law however,
tracing a skewed ray and tracing through higher order surface (e.g. conicoids) is much more
complex. In this thesis, most of the complex ray tracing is performed using optical design
software Zemax (Chapters 6 and 7). In simple instances such as calculation of magnification
(Chapter 4), depth of focus (Chapter 5), and calculation of chromatic aberrations (Chapter 8) are
performed using one of the following methods of paraxial ray tracing:
6) Algebraic equation of the Gaussian optics which is based on Snellen’s principle of
refraction
7) Matrix method of paraxial optics
These two methods of tracing rays are described in this section.
320
Appendix
B2.1 Algebraic equation of ray-tracing
Following procedure and mathematical derivations were performed to arrive at convenient raytracing equations.
O
i
P
iƍ
R
u
uƍ
A
C
B
Bƍ
lƍ
l
Figure B.2: Symbols defined in the diagram are used in deriving ray-tracing equations
Derivation of paraxial refraction equation
According to Snellen’s law of refraction
............................................................................................... (B2.1)
Where n and n’ are the refractive indices in the object and images spaces respectively and I & iƍ
are angle of incidence and refraction respectively.
For small angle approximation
ni
n'i ' ....................................................................................................................... (B2.2)
321
Appendix
Using the equations described above and to eliminate i
n ' (u ' g )
n(u
g ) Or n' u ' nu
g ( n ' n)
As the angle g is related to ray height h and the radius of curvature r, for paraxial approximation
g
h/r
g = − hC where c is the curvature (1/r)
Hence
n' u ' nu
hC (n' n)
n' u '−nu = −hF
................................................................................................. (B2.3)
These are the equations for paraxial refraction.
From the figure above it can be shown that
tan(u ) h /(l
h = −u (l − z )
z)
This equation can also be expressed simply as
h
ul
h = −u 'l '
Paraxial equation can also be independent of u, u’ and h which is given by
322
Appendix
n'
−
n
= F ................................................................................................................. (B2.4)
this is equivalent to vergence equation.
uƍ
hƍ
h
z1
uƍƍ
z2
d
Figure B.3: Diagrammatic illustration of ray transfer
From the figure it can be shown that
h' h [d ( z1
z 2 )] tan(u' )
For paraxial approximation, z1 and z2 both are zero and tan (uƍ) = uƍ, such that:
h' h u ' d ..................................................................................................................(B2.5)
This is the paraxial transfer equation.
323
Appendix
Paraxial ray-tracing through a system with multiple refracting surfaces
For a ray h1
u1l1 .................................................................................................... (B2.6)
For a system with k surfaces, if u’k is the final angle in image space and hk is the ray height at last
surface, then the distance lk of the image from last surface is given by l 'k
hk / u 'k
For jth surface, the refraction equation can be written as
n' j u ' j n j u j
hj Fj
And the paraxial transfer equation can be represented as
h j +1
hj
u' j d j
....................................................................................................... (B2.7)
Appendix B3: Matrix method in Paraxial optics
The matrix method has been widely used in visual optics to calculate retinal image size
(Langenbucher, Huber et al. 2003; Langenbucher, Seitz et al. 2007), spectacle and relative
spectacle magnification (Keating 1982; Garcia, Gonzalez et al. 1995) and IOL power (Colliac
1990). It has also become a popular tool in ray-tracing analyses of astigmatic (Harris 1993;
Langenbucher, Reese et al. 2004) and aspheric surfaces (Langenbucher, Viestenz et al. 2006).
The matrix method of paraxial optics bypasses tedious algebraic derivations and therefore saves
time, and avoids potential errors that may arise from protracted arithmetic derivation. In Chapter
3 (accommodative performance), 4 (magnification) and 8 (chromatic aberration) of the thesis, the
matrix method of ray tracing and optical calculations has been extensively used.
324
Appendix
Incident rays arriving at an optical system consisting of multiple surfaces undergo a number of
successive refractions and translations before they finally emanate from the last surface. In a
matrix convention, for a rotationally symmetric optical system, refraction and translation of a ray
can be represented by a set of 2×2 refraction and translation matrices (Langenbucher, Huber et
al. 2003; Haigis 2009). It should be noted that while other conventions of matrical treatment are
also in use (Gerrard and Burch 1994; Garcia, Gonzalez et al. 1995) we employed the one which
appears to have been applied to the study of accommodating IOLs first (Langenbucher, Huber et
al. 2003; Ale, Manns et al. 2010).
A ray is defined by two parameters: angle and its height on a surface. Input ray (Ը) parameters
on the first surface of an optical system may be represented by:
ªn ⋅ α º
ℜ=«
»
¬ h ¼ ..................................................................................................................(B3.1)
where n is the refractive index of the medium of incidence, Į is the angle between the ray and the
optical axis, and h is the ray height on the surface.
A refraction matrix R s given by:
ª1 − F º
R=«
»
¬0 1 ¼ .............................................................................................................(B3.2)
where F is the dioptric power of the surface. Similarly, a translation matrix T is given by:
ª 1 0º
T =«
»
¬t / n 1¼ ..............................................................................................................(B3.3)
325
Appendix
where t/n represents the reduced distance between two consecutives surfaces.
For a system containing m surfaces, where suffix 1 represents the first and m represents the last
surface with dioptric powers F1 and Fm, the system matrix S (sometimes referred to as the
transformation (Dragt 1982) or transference matrix(Gerrard and Burch 1994)) reads:
S
Rm .Tm−1 .Rm−1 .Tm−2 ..........R2 .T1 .R1 ...........................................................................(B3.4)
The system matrix S is the 2×2 matrix, with standard notational representation:
ªS
S = « 11
¬ S 21
S12 º
S 22 »¼
............................................................................................................(B3.5)
Having defined the incident ray on the first surface of the system and having determined the
system matrix, the emergent ray leaving the optical system has parameters:
ªn'⋅α 'º
ªn ⋅ α º
« h' » = S .« h »
¬
¼
¬
¼ ........................................................................................................(B3.6)
When the object and image distance is included in the calculation, the matrix is called as objectimage system matrix in which the first surface of the system matrix is represented by object
plane and the last surface by image plane.
326
Appendix
Appendix B4: Aberration Theory
B4.1 Equation for Seidel Primary Aberrations
The Seidel aberration theory, the wave aberration theory, is widely used by many vision
scientists to investigate the spherical aberration of IOLs in the pseudophakic eyes. Convenience
of this theory as an excellent starting point in a general design of any optical system can be
explained on the ground that:
8) The aberration of the whole optical system is the sum of the surface contribution hence
easier to understand and manipulate each surface parameter individually
9) Simple mathematical expressions of aberrations exist
10) The shape of the lens for zero or minimum aberrations is independent of pupil size
11) The effect of aspherizing can easily be quantified using simple equations
The paraxial marginal and pupil rays are utilized to formulate equations for the primary Seidel
aberrations. The following equations, derived by Hopkins (Hopkins 1950) and Welford (Welford
1986), are extensively used to calculate the contribution of the ith surface of an optical system to
Seidel monochromatic aberrations:
u
S I = Ai2 hi Δ ( ) i
...........................................................................................(B4.1 ) Spherical
u
S II = Ai Bi hi Δ ( ) i
............................................................................................ (B4.2) Coma
327
Appendix
u
S III = Bi2 hi Δ ( ) i
S IV = H 2 Pi
....................................................................................... (B4.3) stigmatism
............................................................................................. (B4.4) Petzval sum
u
Bi [ H 2 Pi + Bi2 hi Δ( )i ]
n
SV =
Ai
...................................................................... (B4.5) Distortion
Where;
ni (hi Ci
Ai = ni (hi Ci + ui ) = n'i (hi Ci + u 'i )
ui )
n
and
n'i (hi Ci
u'i )
n' '
ij and ij’ are angle of incidence and refraction respectively
and n ij = n’ ij’ is the paraxial version of Snell’s law,
Ci is curvature of the ith surface
Bi = n i ( hi C i + u i ) = n ' i ( hi C i + u ' i )
Where hi & ui are paraxial pupil (chief) ray height and angles on i th surface respectively and
ui and hi are paraxial marginal ray angles and heights respectively.
328
Appendix
Pi = −Ci [(
1
1
) − ( )]
n' i
ni
Δ(u / n) i = (−
ui '
u
)+( i)
ni '
ni
H is the Smith-Helmhotz-Lagrange invariant given by;
H = ni (u i hi − u i hi ) = n 'i (u 'i hi − u ' i hi ) ......................................................................(B4.6)
Once the individual surface contribution is determined, we can sum them up to find the total
system aberration, called the Seidel sum which is given by the equation:
k
S j = ¦ S j ,i
i =1
............................................................................................................…(B4.7)
where k is the number of surfaces and Sj,i is the jth Seidel aberration at the ith surface.
B4.2 Primary Aberration of a Thin Lens
Hazra and Delisle (Hazra and Delisle 1998), based on the third order equations derived by
Hoppkins (Hopkins 1950) and Welford (Welford 1986), proposed a set of simple equations to
calculate monochromatic Seidel aberrations for a thin lens of refractive index n' immersed in
inhomogeneous media with objet space refractive index
and image space refractive index n' ' .
Seidel spherical aberration (SI), Seidel Coma (SII) and Seidel Astigmatism (SIII) are the most
important ocular aberrations that have been frequently evaluated in visual optics; hence, only
these aberrations are evaluated in this thesis.
329
Appendix
B4.2.1 Primary Spherical Aberration
1
3
S I = h4 Fe (a1 X 3 + a2Y 3 + a3 X 2Y + a4 XY 2 + a5 X 2 + a6Y 2 + a7 XY + a8 X + a9Y + a10 )
...........(B4.8)
where,
a1 =
1
1 § n2
n ' '2 ·
¸ a2 = − 2
¨
−
2 ¨
2
2 ¸
n' © (n'−n) (n'−n' ' ) ¹
n'
,
a3 = −
a5 =
§ n' 2 − n 2 n' 2 − n' ' 2 ·
¸
¨¨
−
2
n' ' 2 ¸¹
© n
,
1 § n'+3n n'+3n' ' ·
1 § 2n'+3n 2n'+3n' ' ·
−
−
¸ a4 = 2 ¨
¸
2 ¨
n' © n'− n
n'− n' ' ¹ ,
n' © n
n' ' ¹ ,
1 § n'+2n
n'+2n' ' ·
¨¨
¸
+
1 3n'+2n 3n'+2n' ' ·
2
n' © (n'−n)
(n'−n' ' ) 2 ¸¹ , a6 = §¨
+
¸
n' © n 2
n ' '2 ¹ ,
a7 = −
§ 2n'+ n
4 § n'+n
n'+ n' ' ·
2n'+ n' ' ·
¨¨
¸¸ a8 = ¨¨
¸
+
−
2
n' © n(n'−n) n' ' (n'−n' ' ) ¹ ,
n' ' (n'−n' ' ) 2 ¸¹ ,
© n(n'−n)
§ 3n'+n
§
·
3n'+ n' ' ·
1
1
¸¸ a10 = n'2 ¨¨ 2
¸
− 2
+ 2
a9 = −¨¨ 2
2
2 ¸
n' ' (n'−n' ' ) ¹
© n (n'−n) n' ' (n'−n' ' ) ¹ ,
© n (n'−n)
where h is the object ray height on the surface, Fe is the equivalent power of the lens, X is the
normalized shape factor given by
330
Appendix
( n' n)c1
X =
( n' n' ' )c2
Fe
...........................................................................................(B4.9)
c1 and c2 are the curvatures of the first and the second surfaces of a thin lens respectively
(c
1 / R ) . In terms of equivalent power and shape factor, the curvatures are given by
c1 =
Fe ( X 1)
F ( X 1)
c2 = e
2(n'− n) and
2(n'−n' ' ) ........................................................................( B4.10)
It is noticeable that the change in shape factor also leads to the change in curvature for a constant
equivalent power of the lens. Y is called conjugate factor given by
Y =
n ' ' u ' ' nu nl n ' ' l ' L ' L 1 M
=
=
=
−
−
−
−
.............................................................( B4.11)
where
and u ' ' are the paraxial object and image ray angle on the surface respectively, l and l'
are the object and image distances from the surface respectively. L and Lƍ are the incoming and
outgoing vergences from the surface respectively and M is the magnification of the image.
B4.2.1 Primary Coma
S II =
1
h 2 F 2 H ( p1 X 2 + p 2Y 2 + p3 XY + p 4 X + p5Y + p6 )
.......................................( B4.12)
Where p1 through p6 are refractive index related coefficients given by
1 § n
n' ' · p = 1
−
p1 = 2 ¨
2
¸
n' 2
n' © n'− n n'− n' ' ¹ ,
§ n' 2 − n 2 n' 2 − n' ' 2 ·
¨¨
¸
−
2
n' '2 ¸¹
© n
331
Appendix
p3 =
− 1 § n'+2n n'+2n' ' · p = 1 §¨ n'+ n + n'+ n' ' ·¸
−
¨
¸
4
n' ¨© n( n'−n) n' ' (n'−n' ' ) ¸¹
n '2 © n
n' ' ¹ ,
p5 =
·
§
1
1
− 1 § 2 n ' + n 2 n '+ n ' ' ·
¸¸
− 2
¨ 2 +
¸ p6 = n' ¨¨ 2
2
n' © n
n' ' ¹ ,
© n (n'−n) n' ' (n'−n' ' ) ¹
B4.2.3 Primary Astigmatism
S III =
1 · § 1
1 · º
H 2 F ª§ 1
¨ 2 + 2 ¸ − ¨ 2 − 2 ¸Y »
«
2 © n
n' ' ¹ © n
n' ' ¹ ¼ ................................................................(B4.13)
B4.3 Central Monochromatic Aberrations of AIOL
For simplicity, the refractive indices of aqueous and vitreous in the eye can reasonably be
considered equal. This assumption follows that the indices of object and image spaces are the
same i.e. n
n' ' . Hence the equations of monochromatic aberrations of the IOL can be reduced
to:
h4 F 3
SI =
4n 2
S II =
S III =
ª n 2 (n'+2n) 2 4n(n'+ n)
n' 2
(3n'+2n) 2 º
Y »
« n' (n'−n) 2 X − n' (n'−n) XY + (n'−n) 2 +
n'
¼ ......................(B4.14)
¬
Hh 2 F 2 ª n(n'+ n)
(2n'+ n) º
X−
Y » .................................................................(B4.15)
«
2
n'
2n ¬ n' (n'−n)
¼
H 2F
2
................................................................................................................(B4.16)
332
Appendix
The equations for aberrations described so far give the magnitude of the aberrations for rays
traced from the edge of the field through the edge of the aperture. In general, the aberration
increases with the distance from the axis. Another important advantage of Seidel aberration
coefficients is that, once the aberration for particular aperture ( ) and field size ( ) is known,
they can easily be calculated for any other aperture and field size by multiplying the calculated
values with the scaling factor ( ) given by the equation (Smith and Atchison 1997):
p
q
ªφ º ª ∃ º
ϑ = « new » × « new » ...............................................................................................(B4.17)
¬ φ old ¼ ¬ ∃old ¼
In this equation, the first term on right hand side is called the aperture value and second term is
called the field value. The values of the coefficients p and q are given in the Table 1.
B4.4 Non-central Monochromatic Aberrations of AIOL
When the aperture stop is not located in the plane of the lens, i.e. when there is a space between
the lens and the aperture stop, the magnitude of aberration is changed. To calculate the effect of
the shift of the aperture stop on the Seidel aberration, a stop-shift factor is introduced:
ε=
h
.......................................................................................................................( B4.18)
333
Appendix
Table B.1: Values of p and q for various monochromatic aberrations.
Aberration Wave
Transverse
Longitudinal
Name
p
q
p
q
p
q
SI
4
0
3
0
2
0
S II
3
1
2
1
1
1
S III
2
2
1
2
0
2
where the notation and other stop-shift parameters are defined in the Figure B4.1.
Figure B4: A schema of off-axis ray tracing through a lens to describe paraxial marginal and
paraxial chief ray. The system stop is on lens. Various symbols which are used in equations are
illustrated.
The new Seidel aberration for altered stop position can then be calculated using the following
equations:
S I' = S I ...................................................................................................................(B4.19 a)
S II' = S II + εS I .........................................................................................................(B4.20 b)
334
Appendix
'
S III
= S III + 2εS II + ε 2 S I .........................................................................................(B4.21c)
The quantities with primes are non-central aberrations. It is clear from the above equations that
spherical aberration is not affected by the shift of the aperture.
Appendix C: Model Eye
The optical design of visual optical devices requires that a satisfactory image be formed on the
retina. Unlike spectacles and contact lenses, IOL cannot be tested in a trial-and-error basis which
relies on the subjective feedback from the patient; this in turn demands the need of model eye in
the process of evaluating the performances. The suitability and adequacy of the model eye
employed will have a profound influence in replicating the real-eye condition of the lens design
problems and the quality of the results.
Generally, a model eye functions as an “average observer”. The selection and use of a model eye
raises many issues about what is an average or normal for the human population. A suitable
model eye is the one which replicates optical and anatomical properties closest to an average
human eye. Wide range of model eyes is commonly used in evaluating the several optical
phenomena in the eye including the process of ophthalmic lens and intraocular lens designing,
simulation of the performance of optical visual devices and predicting performance of the
corneal-refractive surgeries.
Several models of eyes have appeared over the last century with various levels of complexity in
their construction which ranges from simple single-surfaced reduced eyes to complicated models
containing aspheric surfaces and gradient distribution of the refractive index of the crystalline
lens. Differences in various models rest on the structure of the cornea and the crystalline lens
which are the major refractive components of the eye. The advent of ray tracing capability
through gradient index media has led to shell structured lenses being replaced by gradient
335
Appendix
refractive indices bounded by two aspheric surfaces. Several such finite, or wide angle, designs
have been suggested as offering good predictions for both on, and off, axis aberrations.
Finite model eyes, which closely replicate the anatomical and optical properties of the real
include, but not limited to, the models proposed by Navarro et al (Navarro, Santamaria et al.
1985), Liou & Brennan (Liou and Brennan 1997) and Kooijman (Kooijman 1983) which are
more commonly used in the recent days (Bakaraju, Ehrmann et al. 2008). In this research, we
selected the finite model eye of Navarro et al (Navarro et al., 1985) which incorporates
asphericities in the anterior corneal surface and both surfaces of the lens. More importantly the
crystalline lens parameters (radii of curvatures of the front and real surfaces, thickness and
refractive index) and the anterior chamber depth are accommodation dependent which allows
efficient comparison of some accommodation dependent optical phenomena with that of AIOLs.
Parameters of the model eye are given in the following table.
Surface
Radius
Asphericity Thickness
Index
Front Cornea
7.72 mm
-0.26
0.55 mm
1.376
Back Cornea
6.50 mm
0
3.05
1.3374
Pupil (stop)
Infinity
0
0 mm
1.3374
Lens Anterior
10.20*
-3.131
4.0
1.42*
Lens Posterior
-6.00*
-1.00
16.40
1.336
Retina
-
Parameters marked with an asterisk are accommodation dependent variables given by:
Anterior chamber depth:
AC = 3.05 – 0.05 log (A+1)
Posterior surface radius:
R1 = 10.2 – 1.75 log (A+1)
Anterior surface radius:
R2 = -6 + 0.229 log (A+1)
336
Appendix
Crystalline lens thickness:
TL = 4 + 0.1 log (A+1)
Refractive Index:
nL = 1.42 + 9× 10-5 (10A + A2)
In this thesis, the crystalline lens was replaced with idealized thin-lens model of AIOLs in all
analytical calculations. The schema of the model eye, with the crystalline replaced with an
AIOL, is portrayed in the following Figure C.1.
Retina
F1 F2
R1
R2
Figure C1: Schematic illustration of the Navarro finite model eye implanted with two thin lens
model of accommodating intraocular lens. AC – anterior chamber, R1 and R2 - the reference
planes corresponding pupil plane and anterior vitreous face respectively, F1 and F2 – front and
rear elements of the AIOL.
337
Appendix
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339
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