Compact Fiber-Optic Diffuse Reflection Probes for Medical Diagnostics

Compact Fiber-Optic Diffuse Reflection Probes for Medical Diagnostics
Compact Fiber-Optic Diffuse Reflection
Probes for Medical Diagnostics
Theodore Paul Moffitt
B.S., Physics, Western Washington University (1997)
M.S., Electrical and Computer Engineering, Oregon Graduate Institute
of Science and Technology (2000)
A dissertation submitted to the faculty of the
OGI School of Science & Engineering
at Oregon Health & Science University
in partial fulfillment of the
requirements for the degree
Doctor of Philosophy
in
Electrical and Computer Engineering
July 2007
The dissertation “Compact Fiber-Optic Diffuse Reflection Probes for Medical Diagnostics” by Theodore Paul Moffitt has been examined and approved by the following
Examination Committee:
Scott A. Prahl
Assistant Professor
Thesis Research Advisor
Steven L. Jacques
Professor
Sean J. Kirkpatrick
Associate Professor
James W. Whittaker
Associate Professor
ii
Acknowledgements
I would like to express thanks to the following people in their many capacities in which
they have helped me to achieve the accomplishment of this thesis. First and foremost, I
would like to thank my research advisor, Dr. Scott Prahl, who encouraged, supported and
mentored my development in scientific research. Next, I express my gratitude to Dr. Steve
Jacques, who among many other things showed me that working hard and playing hard
can be one and the same task. I give my thanks to Dr. Sean Kirkpatrick who taught me
the value of speckle and shared the joys of mechanical testing. I thank Dr. Jim Whittaker
for teaching me the many modes of spectroscopy. I am most grateful for the opportunity
to work with these four individuals who make up my advisory committee. In addition, I
express my gratitude to Dr. Ken Gregory who supported my graduate studies not only
financially, but also in creating a unique multidisciplinary environment at OMLC which
fostered my growth. I thank Dr. Jim Kratochvil for the opprotunity to work with him
in the OH&SU Dental School and to Sean who brought Jim’s intriguing problem to my
attention. To Dr. Theresa Goodell, I give thanks for nursing me through the world of
statistics.
I give thanks to all the people with whom I have had the pleasure and pain of insightful
discussions. I thank Dr. Jessica Ramella-Roman, Dr. Paulo Bargo and Dr. John Viator
who shared my early years. I am also thankful to other various students Dr. Dan Gareau,
Kirstin Engelking, Dr. Yin-chu Chen, Jon Adams, Yongji Fu, Ravi, Patrick Roman, Deb
Baker, Abe Janis, and Laurel Jones. I am thankful to Lisa Buckley for her encouragement.
I must thank all the wonderful people who have come and gone through OMLC; for they
have been a very supportive and wonderful group to work with. I also thank Rob Kalke,
the original OMLC groupie for sharing great discussions over beer. And a hearty thanks
goes to Tim Thomas, who not only provided complete SEM support, but also opened a
iii
door towards employment.
Finally, I would like to thank my family who allowed me to grow to be what I am today.
My parents, Chuck Moffitt and Cathy Kornmeyer taught me the value of an education
and I am very thankful to all they have given me. I thank my brother John, who inspired
me to always achieve more in life. I give thanks to my in-laws who supported and believed
in my success. Last but not least, I am eternally grateful to my wife, Supriya, who did
all the little things, in addition to encouragement and support while prodding me to the
finish-line with this thesis.
iv
Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Fiber optic devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Tissue optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Preparation and Characterization of
Polyurethane Optical Phantoms . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . .
2.2 Materials and Methods . . . . . . . . . . . . . .
2.2.1 Component material selection . . . . . .
2.2.2 Initial studies . . . . . . . . . . . . . . .
2.2.3 Phantom design . . . . . . . . . . . . .
2.2.4 Fabrication method of tissue phantoms
2.2.5 Testing of phantom optical properties .
2.3 Results and Discussion . . . . . . . . . . . . . .
2.3.1 Absorption characterization . . . . . . .
2.3.2 Scatterer characterization . . . . . . . .
2.3.3 Phantom optical properties . . . . . . .
2.4 Conclusions . . . . . . . . . . . . . . . . . . . .
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3 A Combined Inverse Adding-doubling and Monte Carlo Model with Experimental Correction Factors for Optical Property Determination using
Single and Double Integrating Sphere Measurements . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Single integrating sphere . . . . . . . . . . . . . . . . . . . . . . . . .
v
1
1
4
11
11
13
13
14
17
18
20
22
22
23
24
39
41
41
43
43
3.3
3.4
3.5
3.6
3.2.2 Double spheres . . . . . . . . . . .
3.2.3 Sphere parameters . . . . . . . . .
Materials and Methods . . . . . . . . . . .
3.3.1 Test samples . . . . . . . . . . . .
3.3.2 Integrating sphere measurements .
3.3.3 Measuring the Sphere Parameters
3.3.4 Optical property inversion . . . . .
3.3.5 Error analysis . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . .
3.4.1 Optical properties . . . . . . . . .
3.4.2 Error analysis . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . .
3.5.1 Sources of error . . . . . . . . . . .
3.5.2 The inversion model . . . . . . . .
3.5.3 Best practices . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . .
4 Sized-Fiber Reflectometry
for Measuring Local Optical Properties
4.1 Introduction . . . . . . . . . . . . . . . .
4.2 Materials and Methods . . . . . . . . . .
4.2.1 Sized-Fiber Device . . . . . . . .
4.2.2 Tissue Model . . . . . . . . . . .
4.2.3 Chicken Breast Experiment . . .
4.2.4 Planar Absorber Effect on Signal
4.2.5 Monte Carlo Simulations . . . .
4.3 Results . . . . . . . . . . . . . . . . . . .
4.3.1 Tissue Model . . . . . . . . . . .
4.3.2 Monte Carlo Simulations . . . .
4.4 Discussion . . . . . . . . . . . . . . . . .
5 Specular Reflection Problem with
lection . . . . . . . . . . . . . . . . .
5.1 Introduction . . . . . . . . . . . .
5.2 Materials and Methods . . . . . .
5.3 Results . . . . . . . . . . . . . . .
5.4 Discussion . . . . . . . . . . . . .
5.5 Beveled optical fibers . . . . . . .
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47
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68
70
73
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Experiment
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81
81
83
83
85
86
86
87
90
90
94
95
a Single Fiber for
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vi
Emission and
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Col. . . 99
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. . . 101
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. . . 110
. . . 112
5.5.1
5.5.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6 Interpretation of Two Fiber Reflectometer Measurements
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Device Design . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Monte Carlo Model . . . . . . . . . . . . . . . . . . .
6.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Materials and Methods . . . . . . . . . . . . . . . . .
6.3.2 Results and Discussion . . . . . . . . . . . . . . . . . .
6.4 Dual-fiber probe with single emission fiber . . . . . . . . . . .
6.4.1 Monte Carlo simulations . . . . . . . . . . . . . . . . .
6.4.2 Results and Disucssion . . . . . . . . . . . . . . . . . .
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Non-invasive Discrimination of Amalgam and
Oral Lesions using a Dual Optical-fiber Probe .
7.1 Introduction . . . . . . . . . . . . . . . . . . . . .
7.2 Materials and methods . . . . . . . . . . . . . . .
7.2.1 Study population . . . . . . . . . . . . . .
7.2.2 Device . . . . . . . . . . . . . . . . . . . .
7.2.3 Amalgam characterization . . . . . . . . .
7.2.4 Data Analysis . . . . . . . . . . . . . . . .
7.3 Results . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Discussion . . . . . . . . . . . . . . . . . . . . . .
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . .
Melanin
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in
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. 116
. 116
. 117
. 117
. 126
. 126
. 132
. 134
. 135
. 137
. 147
Pigmented
. . . . . . . 148
. . . . . . . 148
. . . . . . . 153
. . . . . . . 153
. . . . . . . 153
. . . . . . . 154
. . . . . . . 157
. . . . . . . 164
. . . . . . . 182
. . . . . . . 184
8 General Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . .
8.1 Fabrication of optically stable phantoms . . . . . . . . . . . . . . . . . . .
8.2 Combined Monte Carlo and inverse adding-doubling model for use with
integrating sphere measurements. . . . . . . . . . . . . . . . . . . . . . . .
8.3 Sized-fiber reflectometry . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4 Specular reflectance noise with single fiber measurements . . . . . . . . .
8.5 Two dual fiber reflectance probe designs . . . . . . . . . . . . . . . . . . .
8.6 Clinical discrimination of amalgam tattoos from melanin pigmented oral
lesions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 187
. 188
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189
189
190
190
. 191
A Angled-Fiber Monte Carlo Program . . . . . . . . . . . . . . . . . . . . . . 193
vii
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
viii
List of Tables
5.1
The mean and standard deviation (listed as a percentage of the mean) of
the distribution of repeated measurements. A value of 1.00 in the mean
implies 100% of the emitted light is collected. . . . . . . . . . . . . . . . . . 115
7.1
Classification table of measurements from the test group as predicted using
discriminant function analysis on an independent classification group of
measurements. Overall, 82.5% of measurements are correctly classified. . . . 165
ix
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
The absorption coefficient spectra of 10 µg/mL Epolight 6084 in polyurethane
and 17 µg/mL Epolight 4148 in polyurethane. . . . . . . . . . . . . . . . .
Process for maintaining the four desired optical properties with only three
independent variables. The reduced scattering of the first wavelength establishes the reduced scattering at the second wavelength. A new absorption
coefficient is calculated at the second wavelength that holds the desired reduced albedo at the second wavelength constant relative to the new optical
properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Diagrams of various integrating sphere reflectance measurements that are
needed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Diagrams of various measurements needed to determine the fraction of diffuse illumination f and the sphere parameter γ. . . . . . . . . . . . . . . . .
Diagrams of various integrating sphere transmission measurements that are
needed. The empty port on the bottom of the spheres should be identical
to the size of the entrance port in the sphere reflection measurements. . . .
The absorption coefficient at 690 nm as a function of dye concentration of
Epolight 6084 in part A of the polyurethane (circles) and after curing in
polyurethane (squares). The fitting line is µ690
= a1 C6084 + µ690
a
a0 , where a1
−1 is the absorption coefficient of
= 0.16 cm−1 mL/µg and µ690
a0 = 0.046 cm
polyurethane at 690 nm one week after casting. . . . . . . . . . . . . . . . .
The absorption coefficient at 830 nm as a function of dye concentration of
Epolight 4148 in part B of the polyurethane (circles) and after curing in
polyurethane (squares). The fitting line is µ830
= a1 C4148 + µ830
a
a0 , where a2
−1
830
−1
= 0.065 cm mL/µg and µa0 = 0.042 cm is the absorption coefficient of
polyurethane at 830 nm one week after casting. . . . . . . . . . . . . . . . .
The absorption coefficient spectra of an epoxy resin, a polymer resin, and
the polyurethane 13 months after being cast in addition to the polyurethane
1 week after casting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Typical reflectance and transmission measurements used to derive optical
properties using the inverse adding-doubling method. . . . . . . . . . . . . .
x
15
26
27
28
29
30
31
32
33
2.10 The reduced scattering and absorption coefficients spectra for the reflection and transmission data in figure ?? is shown in comparison to the
spectrometer measurement of absorption for the same phantom made without added scatterer. The error bars are the standard deviation of three
spectrometer measurements through the clear phantom. Above 830 nm,
the sensitivity of the detector drops precipitously causing error in the resultant optical properties. The reduced scattering is fit by the relation
µ0s = (5.2 cm−1 )( 1000λ nm )−0.8 . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 The size distribution of diameters of 112 titanium dioxide particles has a
mean of 340± 90 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12 The variation in reduced scattering coefficient for ten different phantoms
at 690 nm (circles) and 830 nm(squares) measured by the IAD method. All
samples were designed to have the same reduced scattering properties. The
error bars are the standard deviation of four sets or reflection and transmission measurements for each sample. . . . . . . . . . . . . . . . . . . . .
2.13 The predicted absorption coefficient using equations 1 and 2 are shown relative the spectrometer measured absorption coefficient for the ten different
phantoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.14 The absorption and reduced scattering coefficient spectra for twelve of the
20 identical phantoms. The error bars show the standard deviation for each
optical property among the twelve phantoms and are approximately ± 3%
between 600 and 800 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
3.2
3.3
. 34
. 35
. 36
. 37
. 38
The gain (relative to a black sphere) for a 203 mm (8 in.) integrating sphere.
The wall reflectivity is assumed to be 0.975 and the detector reflectivity to
be zero. In both curves, the entrance port (6.35 mm) and detector port areas
(3.18 mm) are identical. The increase in the gain as the sample reflection
increases is the primary reason that sphere corrections are needed. . . . . . 55
Diagrams of various integrating sphere reflectance measurements needed to
make a reflectance measurement (eqn. ??) using a single sphere set-up. . . . 56
Diagrams of various integrating sphere transmission measurements needed
to make a transmittance measurement (eqn. ??) using a single sphere set-up. 57
xi
3.4
3.5
3.6
3.7
3.8
3.9
The difference between measured and true sample reflectance and transmittance are shown in comparison to unity slope. The relationship is determined by combining eq. ?? with eq. ?? to describe reflectance and transmittance by combinging eq. ?? with eq. ?? for a 203.2 mm diameter sphere
with a wall reflectivity of 0.975, a 50.8 mm diameter sample port, a 6.35 mm
diameter entrance port, and a 3.18 mm diameter detector port. . . . . . . .
Diagram of a double-integrating sphere experiments needed for the simultaneous measurement of reflectance (eqn. ??) and transmittance (eqn. ??).
Diagrams of various integrating sphere measurements needed to determine
the fraction of diffuse illumination f and the average sphere wall reflectance.
The logic flow of inverse adding-doubling program with a Monte Carlo
model for calculation of light lost through the sample edges. . . . . . . . . .
Light incident upon the sample from the initial irradiating beam that scatters out the side of the sample so that it neither enters the reflectance sphere
or a transmission sphere and leads to larger value than the true absorption
of the sample. This error is referred to as direct light loss. Light reflected
by the sample back into the integrating sphere becomes diffuse in nature
after reflecting off the sphere wall. Diffuse light incident upon the sample
has a greater likely-hood of scattering out the side of the sample. Light lost
in this manner is referred to as diffuse light loss. . . . . . . . . . . . . . . .
The calculated mean absorption and reduced scattering coefficient for a
6.67 mm thick polyurethane phantom measured with sample port sizes of
6.35, 12.7, 19.05, 25.4, 31.75, 44.45 and 50.8 mm in diameter for a singlesphere measurements is shown (solid thin line). Error bars indicate the
standard deviation. The calculated mean absorption and reduced scattering coefficient for a 4.9 and 6.7 mm thick identical polyurethane phantoms
measured with sample port sizes of 19.05, 31.75, 44.45 and 50.8 mm in diameter for a double-sphere measurements is shown as the dashed line. The
illumination beam was 4 mm in diameter on the sample in all measurements.
The bold line is the absorption coefficient determined by the spectrometer
for a phantom without scatterer. . . . . . . . . . . . . . . . . . . . . . . . .
xii
58
59
60
63
64
75
3.10 The calculated absorption coefficient for a 6.67 mm thick polyurethane
phantom measured with a 6.35 mm diameter sample port and 4 mm illumination beam. The dashed curve is the absorption coefficient measured
for a phantom without added scatterer. After 5 iterations of a Monte Carlo
simulation to correct for light lost through the sample edge, the absorption
was over-predicted at coefficients less than 0.1 mm−1 , but by less than a
factor of 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11 The absorption and reduced scattering coefficient for two thicknesses (4.93
and 6.67 mm) of a polyurethane phantom. The intrinsic absorption of the
polyurethane at 450 nm is 0.004 mm− 1 [103]. For comparison, the absorption coefficient measured for a phantom without added scatterer is shown
as a bold line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.12 The reduced scattering and absorption coefficients determined with the inverse adding-doubling algorithm for single sphere experiments. The absorption coefficient for the 0.15 mm sample is shown from 1350–1700 nm. For
comparison, the absorption coefficient of water compiled by Segelstein [104]
is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.13 The percent change for the absorption coefficient and reduced scattering coefficient for no corrections (top), for sphere corrections only (middle), and
for lost light corrections (bottom). Sphere corrections account for the nonlinear relationship between the reflectance (transmittance) and the sample
reflectivity (transmissivity). Lost light is predicted using Monte Carlo simulations of the sphere experiment. . . . . . . . . . . . . . . . . . . . . . .
3.14 The percent change for the absorption coefficient and reduced scattering coefficient for a 1% underestimate of the reflectance (top), transmittance (middle), and the sphere wall reflectivity (bottom). The perturbations shown
correspond to single sphere experiments using a 25.4 mm diameter sample
port with the 6.67 mm thick polyurethane phantom. . . . . . . . . . . . .
4.1
. 76
. 77
. 78
. 79
. 80
This is a diagram of the sized-fiber device. A pair of bifurcated fibers emit
and collect light simultaneously by chopping at two distinct frequencies
exceeding 690 Hz. The fibers are submerged at least 1 cm into the Intralipid
to eliminate boundary effects. . . . . . . . . . . . . . . . . . . . . . . . . . . 84
xiii
4.2
4.3
4.4
4.5
4.6
4.7
4.8
The 200 and 600 µm fibers are joined with a medical grade epoxy so that
they may be simultaneously inserted through a needle for measurement of
tissues. The two fibers can easily fit through a 16 gauge needle. When
measuring tissues, water is placed at the tip of the fiber to reduce the index
mismatch between the fiber cores and the tissue. . . . . . . . . . . . . . . .
The optical properties can be determined from a mapping of the reflectance
for each fiber. The open circles are the experimental measurements of each
sample. The solid squares are the Monte Carlo simulation results. For
samples with µ0s = 5.2 and 10.4 cm−1 , Monte Carlo results are only shown
for µa = 0.1 and 2.0 cm−1 for clarity. . . . . . . . . . . . . . . . . . . . . . .
Measurements of the twenty Intralipid mixtures and repeated measurements
of chicken muscle with two distinct fiber orientations relative to the muscle
grain. Each cluster is a constant Intralipid concentration with five different
absorption coefficients which decrease from left to right. . . . . . . . . . . .
Independence of sized-fiber measurements for two scattering samples with
three different absorption coefficients. The filled and unfilled circles correspond to µ0s = 47 and 52 cm−1 respectively. Each joined cluster of three
points represents three measurements on a single sample. . . . . . . . . . .
The signal attenuation for 200 and 600 µm fibers is shown as the fibers
approach an absorbing plane of India ink in acrylamide gelatin. The signal
is normalized to the diffuse reflectance measured in the intralipid solution
far from boundaries (µ0s = 10.4 cm−1 , µa = 0.5 cm−1 , g = 0.83). The curves
represent Monte Carlo simulations of the experiment with standard errors
of the mean and the points are the experimental data. . . . . . . . . . . . .
Monte Carlo simulations show the separation between a fiber and an absorbing plane where a 50% drop in signal occurs for variations in the absorption
and scattering properties. The anisotropy is set to 0.83 to simulate Intralipid.
For a fixed acceptance angle of the fiber, the maximum angle of launch from
a gaussian distribution does not influence the diffuse reflectance collected.
This suggests that complete mode filling of the fibers is unnecessary to
obtain a consistent result; and by extension, modal variation of the emission
does not have any effect upon the measurement. The optical properties used
are µ0s = 10.4 cm−1 , µa = 0.5 cm−1 , and g = 0.83. . . . . . . . . . . . . . .
xiv
85
89
90
91
92
93
94
4.9
The fraction of diffuse light collected by each fiber is dependent on the acceptance angle of the fiber θ in air, increasing with a 1 − cos θ dependence
which follows from a geometrical result if light returns to the fibers uniformly from all directions. A is a scaling parameter to fit the relation. The
photon launch angle distribution is held constant and only the acceptance
angle of the fiber is varied. The optical properties used are µ0s = 10.4 cm−1 ,
µa = 0.5 cm−1 , and g = 0.83. . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1
The experimental set-up consists of two 100 micron fibers that are endcoupled to a 200 micron fiber to separate illumination from collection light.
The photo on the right shows the fiber mounted in the positioning stages
over a first surface mirror. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Diagrams of the geometry used for the angular rotation of the fiber (left)
and the displacement of a fiber form the surface (right). . . . . . . . . . . . 101
Distribution of 32 measurement of a diffusely scattering epoxy resin block
with a 200 micron fiber when the fiber is held by hand (Top) and when the
fiber is mounted to a translation stage and positioned in contact with the
block (Bottom). The fiber is lifted from the surface of the block between
every measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Distribution of 32 measurement on a forearm with a 200 micron fiber that
is held by hand. The fiber is lifted from the arm between every measurement.105
The specular reflectance collected from a 200 micron fiber in contact with a
first surface mirror. The fiber’s optical axis is normal to the mirror at zero
degrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
The reflectance (specular and diffuse) collected by a 200 micron fiber in
contact with an optically diffusing epoxy block. . . . . . . . . . . . . . . . . 107
The specular reflectance collected with a 200 micron fiber displaced from the
surface of a first surface mirror(points) and the expected geometric result
(curve). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
The reflectance (specular and diffuse) collected with a 200 micron fiber displaced from the surface of a first surface mirror. . . . . . . . . . . . . . . . . 109
Transmitted specular reflectance from the fiber face boundary is reduced
for the fibers polished at a 65 degree angle as shown by the ratio of the
specular reflection for each media divided by the signal from a epoxy resin
block standard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
xv
5.10 Repeated measurements of a resin block tissue phantom (N= 12) and skin
(N=13) from the dorsal side of the forearm. Open points represent the
angled polish probe and filled points are for the perpendicular polished
probe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.1
6.2
6.3
In the Monte Carlo model, photon bundles are launched from the last complete plane within the fiber perpendicular to the optical axis of the fiber.
Photon bundles are launched with a truncated gaussian angular distribution with a cut-off at the angle with an e−2 likely-hood to the central angle.
The cut-off angle corresponds to the fiber numerical aperture, which is 0.22
in this model. Photons are tracked to the face of the fiber and the refracted
into the tissue as shown for paths A and B. In the case of photon bundle
C, the path intersects the fiber wall before reaching the fiber face and is
therefore undergoes total internal reflection at the intersection. Paths A0 ,
B0 and C0 (dashed rays) show the specular reflection path from the fiber
face which leaks out the side of the fiber. . . . . . . . . . . . . . . . . . . . 118
Bundles of photons are propagated using a propagational vector given by
the direction cosines defined as the cosine of the propagation vector with
respect to the x, y and z axes. The initial propagation vector is described a
gaussian distribution with a e−2 width set by the numerical aperture of the
fiber and centered on the optical axis of the fiber. For a 65 degree beveled
fiber, the optical axis has direction cosine vector of (0.423, 0, 0.906). The
top figure shows the distribution of angle for launch photons and collected
photons by the direction cosine along the x-axis. The middle figure shows
the collected distribution direction cosine for the y-axis with the launched
photons omitted as the overlay the collected photons and has a similar angular spread as for the x-axis. The bottom figure gives the launched and
collected photon bundle distributions along the z-axis which is perpendicular to the semi-infinite medium boundary. . . . . . . . . . . . . . . . . . . 120
A reflectance map of optical properties in the range of tissues. Each cluster
of points represent a single reduced scattering coefficient where scattering
increases for each cluster as the reflectance increases. The error bars represent the standard deviation of five runs of the Monte Carlo with a different
starting seeds. Within each cluster absorption in creases as the reflectance
decreases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
xvi
6.4
The diffuse reflectance collected by the 200 and 1000 micron fibers decreases
as the light becomes more forwardly scattered (higher g). Anisotropy values
range from g= 0–0.99. The reduced scattering coefficient is held constant
at 30 cm−1 and the absorption constant at 0.2 cm−1 . . . . . . . . . . . . .
6.5 The mean optical path through which collected light travels is presented
as a function of optical properties in terms of the number of mean free
paths traveled (mfp = 1/(µa + µs ). At any particular absorption coefficient, the scattering coefficient increases with the mean number of mean
free paths traveled by collected photons for values of 100, 200, 300, 500,
750, 1000 cm−1 . For all simulations the anisotropy was fixed to g = 0.90. .
6.6 A contour plot of the absolute relative error in the absorption coefficient
is evaluated with a linear approximation of the Monte Carlo simulations in
conjunction with experimentally determined variation. . . . . . . . . . . .
6.7 This is a diagram of the sized-fiber device. A pair of bifurcated fibers emit
and collect light simultaneously by chopping at two distinct frequencies.
The fibers are mounted in tubing and have SMA connectors to connects
the fibers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8 This diagram shows a bisecting view of the 200 and 1000 µm fibers mounted
together in a black plastic SMA connector. A fiber support 1.25 cm in
diameter connects to the fibers via the SMA keeping the fiber concentrically
positioned flush within the support. . . . . . . . . . . . . . . . . . . . . .
6.9 This is a picture of the the 200 and 1000 µm fibers attached to the fiber
support that aids in placement of the fibers. An end-view of the fibers is
pictured in the inset looking down the axis of the fibers to show the emitted
light from the two fibers. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Measurements of an array of semi-infinite tissue phantoms are compared
to Monte Carlo simulations with absorption coefficients of 0.1, 0.4, 0.9 and
1.6 cm−1 . The error bars represent the standard deviations of five Monte
Carlo simulations with different starting seeds. Significant changes in the
tissue phantoms scattering coefficient occur with changes in hydration of the
gelatin. Stability of the phantoms scattering coefficient was compromised
due to condensation on the phantoms while stored under refrigeration resulting in the observed difference between the phantoms and the Monte
Carlo model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvii
. 123
. 124
. 125
. 128
. 129
. 130
. 133
6.11 Two 400 micron diameter fibers are mounted in contact and stripped of the
jackets down to the cladding. The fibers are polished at a 65 degree angle
making the the fiber faces elliptical. The fibers are aligned along the minorelliptical axis of their faces. In this probe design, only one fiber emits light
while both fiber collect back-scattered light. . . . . . . . . . . . . . . . . .
6.12 The mean path traveled for light collected by two side by side 400 micron
diameter fibers is shown (top) in comparison to the mean path travel for the
independent emission and collection of 200 and 1000 micron diameter fibers
(bottom) as determined through Monte Carlo simulations. Three reduced
scattering coefficients are shown 20, 50 and 100 cm−1 with the anisotropy
fixed at 0.9. Error bars show the standard deviation of the mean for five
runs at each combination of optical properties. The collection fiber has
a 0.5 mm greater mean path-length for collected light than a 1000 micron
fiber collecting its own emitted light. . . . . . . . . . . . . . . . . . . . . .
6.13 A mapping of the reflectance for two 400 micron fibers where one fiber emits
and collects lights (Remission ) and the other fiber (Rcollection ) only collects
light emitted by the first fiber. Each cluster represents a single scattering
coefficient and absorption increases as the reflectance decreases within each
cluster. Error bars are given by the standard deviation of five runs of the
Monte Carlo model for each pair of optical coefficients. The anisotropy is
0.9 for all simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.14 The base ten logarithm of the diffuse reflectance for the emission fiber as
determined by a Monte Carlo model is modeled by equation ?? (lines). . .
6.15 The base ten logarithm of the diffuse reflectance for the collection fiber as
determined by a Monte Carlo model. The lines represent a linear fitting
lines similar to equation ?? for the emission fiber, however the slope and offset do not have a simple functional relationship to the scattering coefficient
as does the emission fiber. . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.16 The relative residual error of the predicted reflectance to the reflectance
established by the Monte Carlo model is given by (Rpredicted − Rtrue )/Rtrue .
The predicted reflectance for the emission fiber is given by equation ??. .
6.17 The predicted scattering coefficient plotted against the true scattering coefficient (top) for the reflectance generated from Monte Carlo model using
equation ??. The relative residual error (bottom) of the predicted scattering
coefficient is given by (µ0predicted
− µ0true
)/µ0true
. . . . . . . . . . . . . . . .
s
s
s
xviii
. 136
. 138
. 139
. 140
. 141
. 142
. 144
6.18 The predicted absorption coefficient plotted against the true absorption
coefficient (top) for the reflectance generated from Monte Carlo model using
equation ?? where the scattering coefficient is known exactly. The bottom
plot is similar, except that the scattering coefficient is predicted as shown
in figure ??. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.19 The measured optical properties of a polyurethane phantom using the dual
400 micron diameter fiber probe and eqs. ?? and ?? is shown in relation to
the measured optical properties using inverse adding-doubling (IAD) with
the single integrating sphere method presented in chapter 3. . . . . . . . . . 146
7.1
7.2
7.3
7.4
Above: An amalgam tattoo on the alveolar ridge in one subject is found
where a tooth has been extracted leaving a bluish-black lesion as indicated
by the arrow. Below: The radiograph of the above subject clearly showing
a large cluster of amalgam particles emedded in the soft tissue. Large
amalgam particles may be seen in radiographs whereas fine particles are
not always observable, therefore the dimensions of the tattoo in the picture
do not match the radiograph image. . . . . . . . . . . . . . . . . . . . . .
Measurements of melanin sites were performed on subjects with obvious
racial pigmentation. This typical subject had melanin along the gum line
and highly delineated non-pigmented area nearby which is measured as a
control site. The darkness of the melanin pigmentation varied widely among
subjects as well as the intra-subject variation shown here. The melanin is
visibly more concentrated at the point of the arrow for the melanin site
than to the left of the arrow. . . . . . . . . . . . . . . . . . . . . . . . . .
The device consists of two 400 micron fibers that are held together in an
acrylic housing. One of the the fibers is bifurcated by end-coupling two
400 micron diameter fibers to a single 400 micron diameter for illumination and collection (the emission fiber) while the second fiber (collection
fiber) collects light emitted by the emission. The fibers are switched into
a spectrometer to record diffuse reflection for both fibers sequentially with
a 0.6 s integration time. All optical fibers are encased in teflon tubing for
robustness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The absorption coefficients for the chromophores in tissue are used to fit
to the spectral shape of the fiber measurements made with the collection
fiber to determine if other optical characteristics of the measured spectra
beyond melanin absorption can discriminate among sites. . . . . . . . . .
xix
. 151
. 152
. 155
. 157
7.5
Each fiber measurements was multiplied by the 11 gaussians shown here to
simulate the collection of light using spectral filters. Each filter is a gaussian
with a 12 nm at FWHM and normalized to have an area under the curve of
one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.6 A scatterplot of the measurements in the classification group in the discriminant function space. Points indicated by diamonds represent measurements
from melanin sites, circles indicate amalgam sites, and triangles indicate the
non-pigmented sites. The centroids of each group is shown as a filled square.
Future cases are determined by the nearest group centroid. . . . . . . . . . 166
7.7 The fraction of light absorbed of three commercial alloys of dental amalgams
determined from single integrating sphere measurements. In the visible
spectrum amalgam absorbs light uniformly acting like a neutral density
filter. The absorption coefficient was not determinable due to the unknown
concentration of particles in the light-path for each alloy. In this regard, it
is not determinable whether one ally absorbs more strongly than any other. 167
7.8 The normalized measured reflectance spectra for all sites is shown with
the emission fiber indicated as probe 1 and the collection fiber as probe
2. Measurement variation necessitates the use of a baseline adjustment and
instrument scaling factor in order to fit the measurements using the spectral
shape of the known components. . . . . . . . . . . . . . . . . . . . . . . . . 168
7.9 The mean reflectance collected by the emission fiber for all amalgam, melanin
and non-pigmented sites is shown. The error bars represent the standard
deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.10 The mean reflectance collected by the collection fiber for all amalgam,
melanin and non-pigmented sites is shown. The error bars represent the
standard deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.11 The measured reflectance spectra with the collection fiber are fit using the
known absorption spectra for blood, water and melanin, to evaluate concentrations and scattering properties. Typical fits are shown over-layed on
the measured reflectance for an amalgam site (upper left) and a nearby nonpigmented site on the same patient (lower left) and a melanin site (upper
right) and its nearby non-pigment site (lower right). Five parameters were
varied in the fitting: the blood fraction, the oxygen saturation, a reflectance
scaling factor, a scattering scaling factor, and a wavelength invariant baseline.171
xx
7.12 Histograms of the fraction of blood in the tissue and oxygen saturation
for all measurements with the collection fiber are shown separated by type
of site. The blood content is similar for all measurements but the oxygen
saturation is given a low value when the blood content is relatively low. .
7.13 The melanin concentration was proportionally fixed to the negative slope to
the logarithm of the reflectance between 650-750nm for the collection fiber
measurements. This prevented cross-talk between melanin concentration
and the scattering scalar. An increase in melanin has a similar overall
effect to the reflectance measurement as does a decrease in scattering in
the fitting model. In establishing the melanin concentration linearly to the
slope, an offset slope equal to the mean of sites without melanin is used to
account for the slope with the fiber without melanin absorption. However,
this approach forces the non -pigmented sites to register a small negative
melanin content. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.14 Histograms of the scaler to the scattering coefficient that has a wavelength
dependence the follows a linear combination of Rayleigh and Mie scattering
terms. Melanin sites show a wider variability in scattering but the values
above 4 were poorly fits with the model. . . . . . . . . . . . . . . . . . . .
7.15 The overall predicted measurement is scaled by a wavelength independent
constant to correct for differences in fiber contact to the tissue among measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.16 The baseline for each measured site was allowed to differ to account for
contact variability. It is expected that the offset would be similar for all
measurements with the exception of the amalgam sites due to absorption
by the amalgam which is spectrally flat. however, no discernible difference
is seen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.17 TThe fraction of blood versus the fraction of melanin is shown for all measurements with the collection fiber. Melanin sites are shown as squares,
amalgam sites by circles, and non-pigmented sites nearby amalgam and
melanin sites are respectively diamonds and triangles. The blood fraction
does not distinguish among site types as expected, while the melanin sites
are clearly differentiated. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xxi
. 172
. 173
. 174
. 175
. 176
. 177
7.18 This is a box plot of the melanin slope m with the emission fiber (top) and
collection fiber (bottom). The control sites refer to the non-pigmented sites
adjacent to a type of pigmented site. The control sites and amalgam sites
show similar distributions while the melanin sites generally have a greater
magnitude of slope which has a wide distribution that mirrors the skin color
range of the subjects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.19 ROC curve for sites containing melanin as determined by a threshold value
for the melanin slope m by the emission fiber. As the threshold value increases (values labeled), the sensitivity increases at the sacrifice of a decrease
in specificity (shown as 1 − specificity). . . . . . . . . . . . . . . . . . . . .
7.20 ROC curve for sites containing melanin as determined by a threshold value
for the melanin slope m by the collection fiber. For clarity, the domain and
range are truncated to show detail. As the threshold value increases (values
labeled), the sensitivity increases at the sacrifice of a decrease in specificity
(shown as 1 − specificity). . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.21 The discriminant functions scores of measurements in the test case group
are shown in a scatterplot. Measurements are classified according to the
shortest distance to one of the group centroids (shown as filled squares) established from the the classification group measurements. Points indicated
by diamonds represent measurements from melanin sites, circles indicate
amalgam sites, and triangles indicate the non-pigmented sites. . . . . . . .
7.22 The top image shows normal oral buccal mucosa with melanin pigmentation. The melanin is formed along the epithelium border to the underlying
connective tissue. The bottom image is from a biopsy of an amalgam tattoo showing both large and fine amalgam particles. The fine particles are
digested by macrophages and giant cells resulting in the progressive loss
of tin and mercury leaving behind fine particles containing silver and sulphur [150, 151]. Amalgam is not found in the epithelium as it is typically
shed every 5-6 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 178
. 179
. 180
. 181
. 186
A.1 Sample output of five runs with the same optical parameters using a 65 degree
beveled fiber 400 micron in diameter. . . . . . . . . . . . . . . . . . . . . . . 194
xxii
Abstract
Fiber optics can facilitate a non-invasive means of tissue biopsy through optical inspection
in small volumes to differentiate between normal and diseased tissue. Optical fibers provide
a non-complex means to deliver and collect reflected light allowing for the determination
of the absorption and scattering properties. These optical properties correlate with the
chemical and structural state of the tissue. The arrangement of the emission and collection
fibers dictates the volume of tissue that is optically sampled. Fiber-optic probes that
sample the smallest possible volume by emitting and collecting their own backscattered
light are investigated.
Two complimentary studies are presented that are necessary to provide the tools to
evaluate the behavior of fiber-optic probes. First, a method to fabricate optically stable
phantoms is investigated. The optical properties of the phantoms are defined at two
wavelengths. Linear relations are given for the concentration of dyes and TiO2 scattering
agent that predict the absorption and scattering properties of the finished phantoms with
less than 4% error. The phantoms are demonstrated to be stable over a period exceeding
one year. Second, a combined inverse adding-doubling and Monte Carlo model is presented
to evaluate the optical properties of the phantoms using integrating sphere measurements.
The combined IAD/MC model is demonstrated to accurately determine optical properties
of homogenous optically turbid samples with a reasonable precision using multiple sample
thicknesses and sample port sizes for both single and double sphere experiments. The
scattering is predicted with 1% error and absorption error is 2–4%. The nomenclature for
integrating sphere measurements is simplified and rationalized using the concept of sphere
gain to express the results. Explicit directions for determining sphere parameters were
shown. Formulas were given that work for diffuse incidence or collimated incidence or any
combination thereof.
A new kind of fiber optic probe, a sized-fiber reflectometry device is presented and
investigated. Experimental studies are performed using phantoms with known absorption
and scattering properties. A Monte Carlo model is developed to simulate the device
behavior to evaluate effects due to absorption scattering, scattering anisotropy, and optical
sampling volume. The model is validated by comparison to experimental results. Both
experiments and Monte Carlo simulations of the sized-fiber device indicate that 50% of
the signal arises from roughly 1.2 and 1.9 reduced mean free paths for the 200 and 600 µm
fibers respectively and that in general larger fibers sample deeper optically. Specular
reflectance is shown to act as a noise source comparable in magnitude to the diffuse
reflection signal for perpendicularly polished fibers and can be rejected with bevel-tipped
fibers. The measurement variability decreases 6 fold to 4.5% on in vivo skin with a
200 micron fiber with the beveled-tip fiber and to 2.2% for a 1000 micron fiber with the
beveled-tip fiber. The absorption and scattering sensitivity is presented for a bevel- tipped
sized fiber device using a Monte Carlo generated grid to invert optical properties from the
measured diffuse reflectance. The scattering coefficient could be predicted with an error
of 1.5 ± 0.2% over the entire range of absorption and scattering properties. A second
two-fiber probe design is investigated that uses two identical diameter fibers with only a
single source fiber and both fibers collecting reflected light. The inversion of absorption
and the reduced scattering is investigated using a heuristically determined closed form
relationship from device simulations. The scattering coefficient could be predicted with a
mean error of ±4.3%. Typical error for absorption coefficient determination is shown to
be between 50-100% for absorption less than 2 cm−1 . The poor resolution of absorption is
related to the mean optical path for collected light which typically is less than 2 mm for
tissue.
A clinical study is presented using a dual 400 micron fiber probe to distinguish oral
pigmented lesions caused by either melanin or dental amalgam. Two methods of discrimination were investigated. The first method used the spectral features of melanin in
the 640–720 nm wavelength band. The pigmented lesions containing melanin exhibited
a higher change in reflectance with respect to wavelength over this band in comparison
to amalgam tattoos or non-pigmented sites. The sensitivity and specificity for identifying melanotic lesions from amalgam tattoo was 98% and 92% respectively. The second
method of discriminating amalgam tattoo, melanin pigment and non-pigmented sites uses
discriminant function analysis on uniformly spaced wavelength bands of reflectance to
ii
simulate spectrally filtered reflected light. The sensitivity and specificity was 94% and
100% respectively for classifying melanin pigmented sites.
iii
Chapter 1
Introduction
The purpose for this body of work is to develop a non-invasive and highly-localized
method of performing a biopsy optically with fiber-optics. In order to achieve this goal,
two parallel studies are presented that are necessary to characterize the fiber optic devices.
First, a method for fabricating optically stable phantoms that mimic tissue with engineered
optical properties is presented. Second, a technique that uses the integrating spheres in
conjunction with inverse adding doubling for determining optical properties of the tissuephantoms or any other homogeneous material in a slab geometry is presented for use as
a gold standard method of establishing optical properties. An overview of the sized-fiber
device from its original conception [1] is given in conjunction with basic performance.
Corrections in the fiber design compensate for specular reflectance noise in single fiber
measurements and increase sensitivity relating to absorption. Finally, a clinical study
using the fiber device is presented showing the ability to distinguish pigmented oral lesions.
1.1
Fiber optic devices
The optical fiber provides a highly adaptable and low-cost means to interrogate tissue in situ using optical spectroscopy. The flexible nature of optical fibers provides a
robust and convenient means to deliver and collect light. The versatility of fiber optic
probes allows a consistent optical interface between the tissue and the instrumentation
in numerous configurations. Fiber-probes have been developed using direct perpendicular
contact with the tissue [2,3], offset from the tissue [3], and in side-firing (i.e. side-viewing)
configurations [4, 5]. In addition to topical probe designs, the compact dimensions and
1
2
rigid structure of optical-fibers allows for insertion of fiber-probes through narrow tubular channels in devices such as endoscopes and catheters or through needles to penetrate
into tissue structures for minimally invasive applications [5]. Finally, the optical sampling
volume in tissue may be controlled by the selection of spacing between the emission and
collection fibers. A goal of this work is to elucidate the design implications where the
same fiber is used for both light delivery and collection giving the smallest possible optical
sampling volume.
Due to the adaptability of fiber optics, they are widely employed in application of a variety of spectroscopic techniques. Optical biopsy (tissue diagnosis by optical spectroscopy)
has been extensively studied over the last 15 years [3,6,7]. Several recent studies employed
fiber optic devices to assess differences in optical properties derived from diffuse reflectance
spectroscopy to discriminate normal and abnormal tissue in the esophagus [8, 9], oral cavity [10], kidney [11], ovaries [12], colon [13], breast [14] and skin [15]. Diffuse reflectance
spectroscopy has three modes of application: steady-state reflectance [16,17], time-resolved
reflectance [18–20], and frequency modulated reflectance [21–23]. Other spectroscopic
modalities include polarized reflectance [24–26], fluorescence spectroscopy [27, 28], and
Raman spectroscopy (inelastic light scattering) [29, 30]. Though generally not used spectroscopically, imaging fiber bundles are commonly used in medicine (e.g., endoscopy or
laparoscopy).
Though many modes are possible, I will limit the discussion here to diffuse reflectance
spectroscopy, the mode used in this work. For each of the temporal methods of diffuse
reflectance, the absorption and scattering properties of diseased and healthy tissue must be
differentiated for diagnosis. Time-resolved measurements provide a robust determination
of scattering properties given by time of delay between the input and collection of light
and the fraction of light collected relates to the absorption [18, 31]. In the time domain,
laser sources provide pulse lengths of less than 100 ps that provide adequate temporal
resolution. Two options have been employed for detectors, streak cameras [32] and time
correlated single photon counting [19]. The major drawback of time-domain measurements
is the high cost of the light sources and detectors.
3
Frequency domain measurements contain the same information as time-domain measurements. In the frequency domain, the phase delay in the reflected pulse and amplitude
reduction (modulation) are related to the scattering and absorption properties [21,33]. At
frequencies below 300Mhz, the cost of instrumentation is considerably less expensive than
time-domain techniques. Just as with time-domain systems, the sensitivity of frequency
domain measurements increase with increasing source-detector separation; the phase shift
increases with distance and the modulation falls below unity. The optimal source-detector
separation is in the range of 1-2 cm [34]. Shorter source detector separation requires an
increase in the frequency in order to resolve phase shifts in the collected light relative to
the source, increasing the cost of instrumentation. When a single fiber is used for both
emission and collection, the cost benefit of the frequency domain measurements versus
time domain measurements is nullified at the frequencies necessary to resolve a phase
shift.
Steady-state measurements convey less information than the time and frequency domain measurements since the measured attenuation contains no temporal information.
The determination of scattering and absorption properties using steady-state measurements has historically relied upon spatially resolved reflectance measurements. Data is
fitted into an analytical expression based upon diffusion theory [17, 20, 35–37]. Like frequency domain, the spatial distribution of measurements is optimal for source-detector
separations greater than 1 cm. In the last decade, work has begun to determine optical
properties from steady-state diffuse reflectance probes with source-detector fiber separations less than 1 cm with the drive to minimize the optical sampling volume.
Though many studies exist using spatially resolved diffuse reflectance, far less work
exists that investigates diffuse reflectance probes with emission and collection fiber separations where the assumptions of diffusion theory no longer hold. In one of the first
studies, Mourant et al. show that measurements made with separations between emission
and collection fibers less than one mean free path-length are more sensitive to scatterer
size (spheres) than at larger separations [38]. This is followed by a study showing the
existence of a fiber separation that was nearly independent of scattering properties for
optical properties in the range for tissues [39] which occurs with small fiber separation
4
distance. In the following year, the first description of a fiber device is proposed that used
a single fiber both emission and collection of light for reflectance [1] and fluorescence [27].
Bevilacqua et al. present a fiber probe design to optically characterize a few cubic millimeters of tissue using spatially resolved measurements with source detector separation
in the range of 0.3-1.4 mm in conjunction with a Monte Carlo model of light transport to
extract optical properties. Two other significant fiber probe designs using small fiber separations have either several fibers [40] or two fibers [41, 42] and use a priori chromophore
absorption spectra to establish relative chromophore content and the scattering properties
within a small tissue volume.
1.2
Tissue optics
The optical characteristics of tissues are generally described by three parameters: the
absorption coefficient (µa ), scattering coefficient (µs ), the scattering anisotropy (g). The
refractive index (n) is a fourth parameter that does not characterize tissues since it is
essentially a constant value [43].
The absorption coefficient describes the exponential decrease in irradiance I of a collimated beam of light through a homogenous medium over the distance d by
I = I0 e−µa d
where the incident irradiance is given by I0 . The Beer-Lambert law defines this relationship, but also states that the absorption coefficient is proportional to chromophore
concentration C,
µa = Cσa
where σa is the absorption cross-section.
The scattering mean free path between describes the average distance that a packet
of photons will travel between scattering events. The scattering mean free path is the
reciprocal to the scattering coefficient (1/µs ), the parameter of characterization. Ballistic
light (unscattered) is described by
I = I0 e−µs d
5
for a collimated source in a non-absorbing medium. Both the scattering and absorption
coefficients have units of inverse distance (e. g., cm−1 or mm−1 ). Scattering is also described by another parameter, the anisotropy, g that is the mean cosine of the scattering
angle
Z
2π
cos θf (cos θ) d cos θ
g=
0
where f (θ) is the scattering phase distribution function. The scattering anisotropy at
g = 0 is isotropically scattered and at g = 1 light is completely forward scattered. In
the visible and NIR spectrum, tissue has been shown to be strongly forward scattering
with anisotropy in the range of 0.69 ≤ g ≤ 0.99 [44]. Often, the scattering anisotropy
is combined with the scattering coefficient defined as the reduced scattering coefficient
marked with a prime
µ0s = (1 − g)µs
that describes the effective reduced mean free path (mfp 0 ) between scattering events as if
scattering is isotropic. Macroscopically, similar light distributions are obtained with the
reduced scattering approximation in comparison to models that distinguish the anisotropy from the scattering coefficient, but the reduced mean free path has no real physical
meaning.
An optical biopsy characterizes the optical properties of the tissue to determine the
chemical or structural content of the volume of tissue being measured. The absorption
coefficient is a composite of absorption by many individual chromophores. The major endogenous absorbers in the visible spectrum in tissue are oxy and deoxygenated hemoglobin,
myoglobin, melanin, and bilirubin. In the NIR spectral region water and fat also significantly contribute to absorption [45]. The “therapeutic window” is considered to be
between 600–1000 nm where absorption is low and optical penetration can be centimeters [46]. Below 600 nm, optical penetration is limited to a few millimeters before absorption by hemoglobin [47]. Much work has been done to quantify exogenous chromophores
such as photodynamic agents to measure in vivo drug concentrations [48–50]. Information
regarding the structural composition of tissue is imparted by the size, shape and concentration of the scattering particles. The average effective scatterer size can be determined
6
by the wavelength dependence of the scattering coefficient [51] or by single fiber reflectance
measurements [52]. Spatially resolved reflectance can establish collagen fiber orientation
by a two-fold change in the scattering coefficient with directional orientation of the fiber
probe [53].
Complete optical characterization of tissue is rarely performed with a single device as a
minimum of four measurements are necessitated to determine parameters. The refractive
index is generally ignored as it has been shown to be relatively constant for many tissues
with a value between 1.38-1.41 [43]. Microscopic variations of the refractive index account
for scattering but a value of 1.40 is used for the average macroscopic refractive index of
tissue throughout this thesis unless otherwise noted. To reduce the independent measures
needed further, the reduced scattering coefficient is determined rather than separating the
scattering anisotropy from the scattering coefficient. In total two parameters are determined, the absorption and reduced scattering coefficient from two independent measures
of diffuse reflectance.
The main goal in this thesis is to develop an optical fiber probe to perform diagnostics
based on the light absorption and scattering properties in the smallest possible volume of
tissue. By focusing on a compact probe design, the range of minimally-invasive applications increase due to the ease that the device may be employed such as device insertion
through a needle. Furthermore, it is desired that the optically sampled volume is kept
smaller than the dimension of the features in the tissue of interest (e. g., lesions). The
contents of this thesis is organized as follows.
In chapter 2, I describe a method for the preparation of a polyurethane phantom
to simulate the optical properties of biological tissues at two wavelengths in the visible
and near infrared spectral range. Molecular absorbers with relatively narrow absorption
bands (FWHM 32 and 76 nm for Epolight 6084 and 4148 respectively) provide independent absorption at 690 nm for absorption up to 5 cm−1 and 830 nm for absorption up to
3 cm−1 . Absorption by both dyes is linear in these respective regions and is consistent in
polyurethane both before and after curing. The dyes are stable over long durations with
no more than 4% change. The absorption of visible light by polyurethane decreases with
time and is stable in less than one year with a drop of 0.03 ± 0.003 cm−1 from 500–830 nm.
7
The scattering properties arise from the addition of TiO2 particles to the polyurethane.
These properties at 690 and 830 nm are related to the weight per volume. The variation
in absorption and scattering properties for large batch fabrication (12 samples) is demonstrated to be ±3%. The optical properties of the phantoms did not significantly change
over a period of more than one year which makes them suitable for use as a reference
standard.
Chapter 3 describes a combined inverse adding-doubling and Monte Carlo model to
extract the optical properties from measurements made using integrating spheres. Models
used to extract optical properties of a sample from total diffuse reflection and transmission measurements made with integrating spheres overlook two effects that occur when
integrating spheres are used. The first effect is that the sphere itself alters the measurements. Second is the effect of light that leaks out through the edges of the sample. An
expression for the light collected for both single and double-integrating sphere experiments is derived. Correcting for the integrating sphere influence requires knowledge of
the sphere wall reflectance; methods for measuring this are given. These equations may
be used with any one-dimensional light transport model including Monte Carlo models.
I present a theoretical framework for single and double integrating sphere measurements.
These equations are the basis for a combined inverse adding-doubling and Monte Carlo
model to invert optical properties from integrating sphere measurements. The method is
tested for accuracy and precision using multiple sample thicknesses, multiple sample port
dimensions for both single and double sphere arrangements. The standard deviation of
optical properties for measurements with two sample thicknesses and 4 sample port sizes is
7% for absorption and 4% for scattering with double-sphere experiment. For single-sphere
results the 6 sample ports sizes were measured for one sample giving a standard deviation
of 5% for absorption and 2% for scattering. Absorption is shown to be resolvable down to
0.005 mm−1 with sample thickness of 10 mm. The absorption and reduced scattering coefficients are measured for a suspension of titanium dioxide in water over the wavelengths
400–1700 nm.
In chapter 4, the sized-fiber probe is introduced. Sized-fiber spectroscopy describes
a device and method for measuring absorption and reduced scattering of tissue using
8
optical fibers with different diameters. The device used in this paper consists of two fibers
with diameters of 200 and 600 µm. Each fiber emits and collects its own backscattered
light. Backscattered light measurements for solutions with absorption coefficients of 0.1–
2.0 cm−1 and reduced scattering coefficients of 5–50 cm−1 demonstrate that the device
is most sensitive for the highest scattering materials. Monte Carlo simulations suggest
the device is insensitive to the fiber illumination distribution and that the light returning
to the fiber is nearly uniform over all directions. Finally, experiments and Monte Carlo
simulations of the sized-fiber device indicate that 50% of the signal arises from roughly
1.2 and 1.9 reduced mean free paths for the 200 and 600 µm fibers respectively.
Chapter 5 considers the effects of specular reflections when a single fiber is employed
to emit and collect light from an optically diffusing medium such as biological tissues.
The light collected by the fiber consists of two components: diffusely scattered light from
within the tissue and specularly reflected light from the surface. Only the diffuse reflection
contains the desired information regarding the optical absorption and scattering properties of the tissue, but the specular component is comparable in magnitude to the diffuse
reflection with visible light. The refractive index mismatch between the fiber and tissue
account for a portion of the specular reflection. However, imperfect contact of the fiber
with the surface of tissue creates additional boundaries and thus additional specular reflections. Experiments performed with a 200 micron diameter fiber and a 632.8 nm He-Ne
source characterize the collection of specular reflections through the same fiber using water
as a coupling medium. The collection efficiency measured for a fiber in imperfect contact
shows the sensitivity of fiber angular tilt with the surface on a glass substrate (specular
reflection only) and an epoxy resin tissue phantom (specular and diffuse reflection components). Next, the collection efficiency is measured for a separation between the fiber and
the samples for perpendicular illumination to the surface, 14 degrees, and 25 degrees from
the normal. Imperfect contact is demonstrated to vary the amount of specular reflection
collected using a single fiber where changes in angle greater than 4 degrees or a separation
between the fiber and the surface in excess of 400 micron caused a minimum of 7 percent
reduction of the collected specular reflection.
In chapter 6, two dual fiber probe designs are tested for the purpose of determining the
9
absorption and reduced scattering properties of highly turbid media using bevel-tipped
fibers. The first device is a sized-fiber probe with 200 and 1000 microns diameter fibers.
Optical properties are determined using a Monte Carlo grid of diffuse reflectance for absorption coefficients between 0.1–5.0 cm−1 , scattering between 100–1000 cm−1 and the
anisotropy constant at 0.9. The expected error for predicting scattering is less than 2%
using linear interpolation between points on the grid. The error for predicted absorption
is greater than 50% for absorptions less than 1 cm−1 due to the small mean optical pathlength for collected light. A second probe is shown using dual 400 micron diameter fibers
with one emitting fiber and both fibers collecting reflected light. Only minor difference
exists in performance between the probes. The single emitting fiber fiber probe has more
compact physical dimensions and a slightly longer optical sampling path-length (a 25-30%
increase) relative to the sized-fiber probe. The Monte Carlo calculated mean optical pathlength is 2.5 mm or less for the single emitter fiber probe. Relations are presented between
the collected light and the absorption and scattering based on Monte Carlo calculations.
However, the inversion of optical properties using linear interpolation of optical properties
from a Monte Carlo generated grid demonstrated better performance than the empirically
derived relations for the single emitting fiber probe, despite its greater mean sampling
optical path-length. The optical properties of a polyurethane phantom are measured with
the single emitting fiber probe and differ by a factor of 2 with the results measured using
an integrating sphere.
Chapter 7 presents a clinical application of the dual 400 micron fiber probe to differentiate specific oral pigmented lesions in a pilot study as a non-invasive screening device.
Pigmented oral lesions are biopsied to rule out the possibility of oral melanoma, a rare
condition with poor prognosis. Pigmented oral lesions caused by implantation of dental
amalgam (amalgam tattoo) or by natural racial melanin pigmentation are measured in
thirty subjects. Two methods of discrimination are presented. In the first method, the
content of melanin is related to the spectral slope in the 640–720 nm wavelength band
and a threshold is used to differentiate pigments. The receiver operator characteristic
(ROC) curve for various threshold values is reported independently for each optical fiber
measurement. A sensitivity of 98% was obtained with a specificity of 92% with a slope
10
threshold of −0.0017 for the collection fiber to identify melanin sites. The second method
of analysis used the reflectance spectra obtained in the clinic to simulate data collected
using either spectral filters or LED sources centered at 11 wavelengths from 450-700 nm.
Separation of three groups: amalgam tattoo, melanin pigment and non-pigmented sites
is performed using discriminant function analysis. The discriminant functions are created using two-thirds of the measurements to determine the classification functions and
the remaining third of the measurements to evaluate the correct determination of the
groups by the these functions. 94% of melanin pigmented lesion were correctly classified
with 6% (one measurement) falsely attributed as amalgam. 53% of melanin sites were
correctly classified with the remaining classified falsely as non-pigmented sites. 90% of
non-pigmented sites were correctly classified as non-pigmented with all false attributions
being classified as amalgam tattoo.
Chapter 2
Preparation and Characterization of
Polyurethane Optical Phantoms
2.1
Introduction
Phantoms that simulate the optical characteristics of tissues are commonly used to
mimic light distributions in living tissue. Tissue phantoms are often designed and utilized
for three purposes: to simulate light distributions with a geometry of physical tissue [54],
for the calibration of optical devices [13, 55], and for recording a reference measurement
with an optical measurement device [56]. For all three uses, the absorption and scattering
properties of the created tissue phantom is the primary design factor. Optical tissue
phantoms are necessary to calibrate any optical measurements on real tissues to establish
quantitative information [13]. Optical device calibration requires using several phantoms
with the optical properties that span the typical range of the tissue to be simulated.
Furthermore, phantoms that are optically stable are also suitable for use as a reference
measurement taken in conjunction with measurements on living tissues.
Several types of phantoms to mimic the optics of human tissue have been described
in the literature including homogenized milk [57], non-dairy creamer [58], wax [59], and a
blood and yeast suspension [60]. Suspensions of oils/fats in an aqueous solution such as
Intralipid [61] and a water soluble dye (India ink [62]) appear to be the most commonly
This chapter was originally published under the title, “Preparation and characterization of
polyurethane optical phantoms” by T. P. Moffitt, Y. Chen and S. A. Prahl in Journal of Biomedical Optics,
Vol. 11(4), 041103, 2006.
11
12
used phantom materials. Difficulties arise in creating inhomogeneities using liquid phantoms such as layered structure. A desire for solid phantoms led to the development of
suspensions of Intralipid and India ink in agar [63], agarose [64] and polyacrylamide [65]
that could be cut and shaped to provide inhomogeneities in the phantoms. The inclusion
of inhomogeneities limited the selection of dyes used to waterproof inks to prevent diffusion of dyes between regions [63]. However, these phantoms mostly suffer from relatively
short useable lifetimes which are usually limited to no more than two months.
More robust phantoms made of rubbers or plastics have been described that give longterm optical stability and greater shaping flexibility. Phantoms compositions of silicone
[54, 66], polyester [67, 68], polyurethane [69], and epoxy resin [70] have been described.
There may be no great advantage of one material over another. For example, silicone
more closely matches the mechanical properties of tissue and may be cast into arbitrary
shapes, whereas epoxy, polyurethane, and polyesters are easier to machine after casting.
Typically, the selection of a material is determined by the choice of absorbers and their
stability in that medium.
The choice of scattering agents in solids is usually limited to aluminum oxide (Al2 O3 )
[54, 66], titanium dioxide [67], and silicon dioxide [68, 70], polyester [54], polystyrene,
or latex microspheres. Mie theory can be used to predict the scattering properties of
microspheres with knowledge of the relative refractive index, size distribution and number
density. Unfortunately, microspheres tend to be much more expensive than aluminum
oxide or titanium dioxide particles. One primary difference between aluminum oxide and
titanium dioxide particles may be the maximum attainable value for g (the cosine for the
mean angle of scattering) which describes the scattering anisotropy; Firbank and Deply
found that TiO2 is limited to g = 0.7 whereas Al2 O3 can reach g = 0.97 in polyester
resin [67]. Arridge et al. [71] has shown that any arbitrary scattering angle distribution
can be theoretically matched using an appropriate distribution of particle sizes.
Our goal was to make durable stable optical phantoms with predefined optical properties at two wavelengths, 690 and 830 nm that are commonly used for oximetry [72,73]. To
our knowledge, previous phantom designs have been developed for a single wavelength or
for use with a single broadband absorbing dye that determines the absorption spectrum. I
13
wanted to use component materials that would be photostable. Moreover, the dyes needed
to have absorption spectra with minimal overlap. Two problems during initial material
selection experiments were encountered . First, dyes that absorb strongly at 830 nm also
exhibited comparable absorption at 690 nm and second, dyes that bleached during the
curing process of the binding medium. The dyes selected for our phantoms determined
the bulk material used to suspend the dyes and scatterers.
A method is needed to determine the scattering characteristics of the phantoms. Inverse adding-doubling (IAD) was used to find the scattering and absorption of a slab of
turbid material using total reflection and total transmission measurements. This method
is applicable to homogeneous turbid slabs with any optical thickness, albedo, or phase
function. The slab may have a different index of refraction from its surroundings and
may or may not be bounded by glass. The optical properties are obtained by iterating an
adding-doubling solution of the radiative transport equation until the calculated values of
the reflection and transmission match the measured ones. Alternatively, spatially [4,17,37]
and/or temporally [23,74,75] resolved measurements of diffuse light distributions can also
be used to quantitatively determine absorption and reduced scattering coefficients.
A method for fabricating solid optical tissue phantoms that are castable and photostable is presented. These optical phantoms are designed to be suitable reference standards
at two distinct wavelengths and whose optical properties have been carefully measured.
The final materials were verified by making several optical phantoms with differing quantities of dye and scattering particles. Phantoms made without added scatterers were made
to verify the stability of the dyes’ absorption properties before and after the curing process
and over a duration of 14 months.
2.2
2.2.1
Materials and Methods
Component material selection
The optical phantoms consist of three components: the polyurethane, the absorbing
chromophores, and the scattering agent. Selection of the three primary components were
driven by our choice of absorbing dyes.
14
A phthalocyanine dye (Epolight 6084, Epolin Inc., Newark, NJ) was used as the absorber at 690 nm. The absorption at 830 nm was achieved using a palladium dye (Epolight
4148). The dye manufacturer listed these dyes for use in optical filters and laser goggle applications and suggested polyurethane as a casting material. I verified that both dyes lost
their absorption during the curing process in epoxy resin (Marine grade epoxy resin, Side
A 314 resin and Side B 109 hardener, Tap Plastics, Inc., Dublin CA). Both dyes exhibit
narrow absorption bands (FWHM 32 and 76 nm for Epolight 6084 and 4148 respectively)
that allow nearly independent absorption at the wavelengths 690 nm by Epolight 6084
and 830 nm by Epolight 4148 when cured in polyurethane (Figure 2.1). Titanium dioxide
(Ti-Pure R-900, Dupont Chemicals, Wilmington, DE) was assumed to behave as a pure
scatterer with negligible absorption between 650 and 850 nm. Finally, a two part polyurethane system (WC-781 A/B, BJB Enterprises Inc., Tustin, CA) that had a 30 minute pot
life was used to suspend the titanium dioxide and absorbing dyes. This polyurethane is
a clear rigid casting resin that was designed to be tintable and pigmentable. This polyurethane was formed by mixing part “A” with part “B” in a ratio of 100/85 by weight
(100/88 by volume). The pot life for the polyurethane was a convenient duration because
it allowed sufficient time to mix and de-gas but still cured quickly enough that no apparent
settling of the scatterers took place. The viscosity of the mixed polyurethane was 650 cps.
The demolding time was 24 hours. Optical measurements could be made at 24 hours.
However, complete curing of the polyurethane required 5-7 days at room temperature or
16 hours at 71–82 degrees C.
2.2.2
Initial studies
Absorber characterization
The absorption coefficients as a function of dye concentration were measured in cured
polyurethane (WC-781 A/B, BJB Enterprises Inc., Tustin, CA). Stock solutions of the
two powdered dyes were prepared in xylene with concentrations of 6.64 mg/mL of Epolight
6084 and 1.56 mg/mL of Epolight 4148. Stock solutions of the dyes were pipetted and
mixed in 5 quantities of 15, 25, 35, 40, and 45 µl of Epolight 6084 stock into 10 ml of
polyurethane part A and 5 quantities ranging 50 to 250 µl in 50 µl increments of Epolight
15
2
6084
-1
! [cm ]
1.5
1
a
4148
0.5
0
400
500
600
700
800
900
Wavelength [nm]
Figure 2.1: The absorption coefficient spectra of 10 µg/mL Epolight 6084 in polyurethane
and 17 µg/mL Epolight 4148 in polyurethane.
16
4148 stock into 8.8 ml of polyurethane part B. The absorbance of each concentration of
Epolight 6084 in part A was measured using a dual beam Cary 100 Bio Spectrophotometer
(Varian Scientific Instruments Inc., Walnut Creek, CA) with a cuvette of polyurethane
part A in the path of the reference beam. Likewise, the absorbance of Epolight 4148 for
each concentration was measured with a cuvette of polyurethane part B in the reference
beam path. Then the two parts of the polyurethane were randomly combined with the
concentration of dye in part A randomly assigned to each concentration of dye in part
B. The samples are thoroughly mixed together and poured into 1 cm cuvettes. Degassing
of the mixtures was performed by placing the samples into a desiccator/vacuum chamber
(Nalgene 5311 desiccator, Nalge Nunc International, Rochester, NY) that was connected
to a vacuum pump (Speedivac 2, Edwards High Vacuum Int., England) capable of reducing
the pressure to less than 1 mbar. After full curing, the absorbance of the cuvettes was
measured on the Cary spectrometer. These measurements used polyurethane without
added absorbers cured in an identical cuvette in the reference beam path.
A difference of absorbance measurements with the Cary spectrophotometer were used
to determine the absorption coefficient of polyurethane. Two samples of polyurethane
were cast without any added absorbers, one as a thin slab and another as a tall cylinder.
After 48 hours of curing, the samples were removed from their respective molds. The
surfaces of each sample was hand polished starting with 9 micron grit aluminum oxide
sandpaper (AngstromLap, Fiberoptic Center, Inc., New Bedford, MA) and finishing with
0.3 micron grit paper to create an optically smooth surface. The final thickness of the slab
and cylinder of polyurethane were respectively 0.41 and 3.44 cm. The difference between
the two samples relative to air (i.e. nothing in the reference beam path) produced a lower
noise signal than when the thin slab was placed in the reference arm of the spectrometer
and the cylinder in the sample arm.
Titanium dioxide scattering
A stock solution of titanium dioxide consisted of 20.0 grams of titanium dioxide suspended in 60 mL ethanol. Before dilution into the polyurethane, the titanium dioxide
stock solution was shaken by hand then sonicated for at least five minutes to ensure
17
complete suspension of TiO2 particles. To test the scattering coefficients of different concentrations of titanium dioxide in polyurethane, different amounts of TiO2 stock solution
were added into polyurethane in two separate batches. The first batch had two samples
with a TiO2 concentrations of 2.5 and 5 mg/ml; I added 3.84 µg/ml of Epolight 6084 dye
and 3.11 µg/ml of Epolight 4148 dye for final absorption coefficients, µ690
= 0.74 cm−1
a
and µ830
= 0.19 cm−1 to both samples. For the second batch consisting of two samples
a
with TiO2 concentrations of 0.5 and 1 mg/ml, I added 7.69 µg/ml of Epolight 6084 dye
and 1.56 µg/ml of Epolight 4148 dye for final absorption coefficients, µ690
= 1.17 cm−1
a
and µ830
= 0.11 cm−1 to each sample. From each sample in both batches, a small pora
tion of the final mixture was poured into a weighing boat, degassed for about 8 minutes.
This provided a thin disk used for the inverse adding-doubling method. The inverse
adding-doubling (IAD) method (described in a later section) was used to characterize the
scattering coefficient for each of the above samples.
Scanning electron microscopy was used to evaluate the size distribution of the TiO2
particles. Three samples were prepared by diluting the stock solution of TiO2 with additional ethanol. A 5 µl drop of the diluted stock was pipetted onto a clean microscope cover
slip and allowed to evaporate leaving a film of TiO2 particles. The horizontal width of
112 randomly selected particles was measured from 5 separate SEM images using ImageJ
software. The scale bar was used to establish the width of a pixel within each image.
2.2.3
Phantom design
Our two wavelength phantoms had four variables to specify (µa,λ1 , µa,λ2 , µ0s,λ1 , and
µ0s,λ2 ) but only three free parameters since the reduced scattering coefficients at the two
wavelengths are related by the use of a single scattering agent. In other words, the
optical properties at the first wavelength could be selected but, the reduced scattering at
the second wavelength has a constant proportional relationship to the reduced scattering
at first wavelength with respect to scatterer concentration. To compensate, the desired
optical properties were condensed into three parameters µa,λ1 , µa,λ2 , and the reduced
18
albedo at the second wavelength (a0λ2 ). The reduced albedo was defined by
a0λ2
µ0s,λ2
=
µa,λ2 + µ0s,λ2
The desired absorption and reduced scattering properties at the second wavelength (830 nm)
would be modified by the following method which kept the reduced albedo constant for λ2 .
A flow chart of this process was diagrammed in figure 2.2. The wavelength dependence of
the reduced scattering coefficient for our TiO2 particles was determined using IAD and fit
by the equation µ0s = (5.2 cm−1 )( 1000λ nm )−0.8 . The effective reduced scattering coefficient
at 830 nm was calculated given the desired reduced scattering coefficient at 690 nm. The
modified absorption coefficient was calculated using the desired reduced albedo and the
dependent reduced scattering coefficient at 830 nm.
2.2.4
Fabrication method of tissue phantoms
The optical properties of the tissue phantoms were chosen to represent a range of
different optical properties at the 690 and 830 nm wavelengths. The absorption coefficient
at these two wavelengths would be varied to correspond to physiologic relevant values for
the sum total of contributions by a fractions of water, oxygenated and deoxyhemoglobin
and a wavelength constant background absorption. For all the phantoms, the scattering
coefficient would be held constant. However, the wavelength dependence of the scattering
from TiO2 differs from tissue in that the ratio of scattering coefficients at 690 nm to
830 nm for TiO2 does not equal the target ratio for tissue. To account for this effect, the
target absorption coefficients at 830 nm were adjusted as described in the previous section.
Finally, a single set of optical properties was used to make twenty identical phantoms.
The sequence for phantoms preparation was as follows using the stock solutions of dye
and TiO2 described above. Cylindrically-shaped phantoms 7 cm in diameter and about
5 cm in height were cast. This corresponded to ∼200 mL polyurethane by mixing 113.4 g
of part A and 97 g of part B. In addition, an extra 5.3 g of part A and 4.5 g of part B
was prepared for making thin slab for optical property characterization. The proportions
of each part of polyurethane was determined by weight and the addition of either dye or
TiO2 were neglected since their contribution was less than 1 part in 200 for all phantoms.
19
Absorbers were added to the polyurethane in part A while the scatterers were mixed to
part B. Since the scattering of all phantoms were to be identical, 11 ml of TiO2 stock was
mixed into 2628 g of part B polyurethane which was sufficient quantity for all phantoms.
Part A of the polyurethane (118.8 g) was weighed out for a single phantom into a
polyethylene container. Stock of Epolight 4148, the near-infrared dye was first pipetted
into part A and stirred in with a glass rod until visibly homogenous. This dye had a pale
violet tint when mixed into the polyurethane but was not noticeable after the addition of
the Epolight 6084 stock which produces a strong turquoise blue tint. When both dyes are
homogeneously stirred into part A, about 2 ml was placed in a cuvette for an absorbance
measurement on the Cary Spectrometer using a cuvette of part A polyurethane without
dye in the reference arm to verify proper absorption attenuation.
At this point, 5.3 g of the part A with dye was set aside in a plastic weigh boat dish.
The remaining portion of polyurethane part A with dye was weighed again to calculate the
appropriate quantity of the part B with TiO2 to be added by weight. While on the scale,
part B was poured into part A. The final amount of Part B was added drop by drop using
a wooden tongue depressor. The large flat area of the tongue depressor allowed drops to be
controlled so that weight of any drops could be controlled with precision of about 0.03 g.
The polyurethane is thoroughly mixed using a tongue depressor to scrape the container
sides/bottom and stir for about 2 minutes. Most of the mixture was then poured into
16 oz. HDPE molds (16 oz specimen containers, Fisher Scientific International, Inc., Fair
Lawn, NJ). The remaining material was poured into two plastic weighing-boats to cast thin
disks of differing thickness and about 6 cm in diameter. The set aside 5.3 g of polyurethane
part A and dye was treated identical as above, except it had 4.5 g of transparent part B
added to it. Also these samples were mixed then poured into 1 cm pathlength cuvettes for
casting.
It was necessary to remove entrapped bubbles in the polyurethane introduced by the
mixing of components together. Up to five phantoms or weigh boat slabs could be degassed
at a time. This restricted us to mixing no more than two sets of optical properties together
at a single time since for each set, a phantom and two slabs were made. The second second
set was mixed while the first set was degassing. Degassing was done by placing the samples
20
in a desiccator/vacuum chamber (Nalgene 5311 desiccator, Nalge Nunc International,
Rochester, NY). The chamber was connected to a vacuum pump (Speedivac 2, Edwards
High Vacuum Int., England) capable of reducing the pressure to less than 1 mbar. Samples
were degassed for about 8 minutes at which time bubbles stopped forming.
After degassing, the phantoms were set onto a level surface and covered. The phantoms
were solid after 24 hours and could be removed from the mold. However at 24 hours, the
polyurethane is relatively soft, such that fingerprints can be embedded on the surface with
moderate pressure. In general, a period of 48 hours passed before removing the samples
from the casting mold. The phantoms in the specimen container molds were removed
by flexing the walls of the container and then pushing on the bottom while holding the
phantom upside down. The samples cast in weigh boats were removed by peeling the
weigh boat which tore away from the polyurethane.
The phantoms were finished by leveling the top surface of the polyurethane block. A
lathe was used to mill the meniscus lip from the top of each phantom until the shiny
surface was completely removed. Then each phantom was sanded by hand in two steps:
rough finishing with 200 grit wet sandpaper and fine finishing with 600 grit wet sandpaper.
In each sanding step, the paper was wet with water and placed on a glass surface while
the polyurethane block was sanded in a circular motion. Elimination of visible surface
scratches were used as a completion condition.
2.2.5
Testing of phantom optical properties
Absorption of each phantom was measured using the Cary spectrometer 48 hours
after casting. The clear samples cured in cuvettes for each phantom were measured referenced to cured polyurethane without dyes in a cuvette. Four absorbance measurements of
each sample were recorded. Additional measurements of the same cuvette samples were
recorded 13 and 14 months later to establish long term stability.
Inverse Adding-Doubling (IAD) was used to find the scattering and absorption of a
slab of turbid material using total reflection and total transmission measurements. Two
measurements on each of the two thin slabs for each phantom were made. Total diffuse
21
reflection measurements were made using an 203.2 mm diameter integrating sphere (IS080-SF, Labsphere, Inc., North Sutton, NH) with a 12.7 mm diameter detector port and
a 31.75 mm diameter sample port with a baffle between ports. Light was guided to the
phantom disk through a 400 micron diameter optical fiber (0.22 numerical aperture) placed
about 4 mm above the sample centered with the port hole. A 4 mm diameter steel tube
coated with multiple coats of flat white paint was used to sleave the fiber through a 6.35 mm
top port down to the phantom disk. An Oriel mercury lamp was used for the source and a
600 micron diameter (0.38 numerical aperture) delivered light from the integrating sphere
to a scanning grating detector system of a Fluorolog-3 spectrofluorometer (Instruments
S.A., Inc., Edison, NJ). Signals above 850 nm were often poor as a consequence of lower
power from the lamp at longer wavelengths and lesser efficiency of the photomultiplier
tube above 800 nm. The detector fiber was connected to the integrating sphere using an
SMA connector surrounded by white shielding in the 12.7 mm port. Measurements were
referenced to a 99% Spectralon reflectance standard (Labsphere, Inc., North Sutton, NH)
and a dark measurement where the sample port was open and the fiber illuminating black
felt on the optical table 254 mm below the port. Four measurements were made on each
disk; two measurements on each side. The phantom disks were placed flush with the
sample port. Figure 2.3 shows the sphere measurements for total reflection and figure 2.4
shows the reference sphere calibration measurements.
Total diffuse transmission measurements were made with another identical integrating
sphere. In this set-up only two ports were open, the 25.4 mm diameter sample port and
12.7 mm diameter detector port with a baffle between ports. For this arrangement, the
phantom disk was placed on the top port and light was collected with the same detector
600 micron diameter fiber described previously. Light was delivered to the phantom disk
with the 400 micron fiber. The end of the fiber was positioned just above the phantom
disk (< 1 mm) centered with the port hole. Again, four measurements were recorded on
each disk; two measurements on each side. Measurements were referenced to 100% with
the fiber illuminating the open port hole and a dark measurement with an open port but
the fiber removed from the sample porthole. Figure 2.5 shows the sphere arrangement for
total diffuse transmission measurements.
22
Two other values were needed to calculate the scattering properties. The thickness
of each phantom disk was measured using a depth gage (ID-C112EB, Mitutoyo Corp.
Japan). The refractive index of the cured polyurethane was measured at 670 nm using an
Abbé refractometer as 1.468. A negligible change in the refractive index over the spectral
range of 500 to 900 nm was assumed.
The total diffuse reflectance and transmittance measurements in terms of percentage
at all wavelengths were input into the IAD program. The total diffuse reflection was
calculated using
rsample =
Psample − P0
Pstd − P0
The total diffuse transmission was calculated using
tsample =
0
− P00
Psample
0 − P0
Pstd
0
The batch variability was measured on twelve out of twenty phantoms from a single
batch to establish the variability in our fabrication method. A thin slab was cut and
sanded flat from twelve of the phantoms. The thickness of each slab was measured with a
depth gage. Total diffuse reflection and transmission measurements were recorded for each
of the twelve slabs along with reference measurements as previously described. Inverse
adding-doubling was used to determine the absorption and reduced scattering coefficient
for each slab over the wavelength range of 650–850 nm.
2.3
2.3.1
Results and Discussion
Absorption characterization
Both dyes exhibit stable absorption properties in polyurethane after a period of 14
months from casting. The visible/NIR spectrum for the absorption coefficient of Epolight
6084 and 4148 is shown in figure 2.1. The absorption coefficient at 690 nm relates only to
the concentration of Epolight 6084 (C6084 ) by the relation
690
µ690
a = a1 C6084 + µa0
(2.1)
−1 is the absorption coefficient for polywhere a1 = 0.16 cm−1 mL/µg and µ690
a0 = 0.019 cm
urethane at 690 nm after 13 months when the Epolight 4148 is present at a concentration
23
less than 21 µg/mL. Likewise, when Epolight 6084 is less than 16 µg/mL, the absorption
coefficient at 830 nm depends solely on the concentration of Epolight 4148 (C4148 ) by the
linear relation
830
µ830
a = a2 C4148 + µa0
(2.2)
−1 is the absorption coefficient for
where a2 = 0.065 cm−1 mL/µg and µ830
a0 = 0.013 cm
polyurethane at 830 nm after 13 months. The linear relations for the absorption coefficient
are applicable for Epolight 6084 up to a concentration of at least 32 µg/mL and for Epolight
4148 up to 45 µg/mL. A comparison of the absorption from each dye in solution in one
part of the polyurethane and then after casting is shown in figure 2.6 for Epolight 6084
and figure 2.7 for Epolight 4148. The absorption spectrum of both dyes was unaffected
by curing.
The tissue phantoms are limited by a minimum intrinsic absorption coefficient that is
due to the absorption the of the binding medium. The absorption coefficient of polyurethane at a time point one week after casting is greater than after 13 months from casting
as shown in figure 2.8. The change of absorption at these time points is 0.03 ± 0.003 cm−1
over the wavelength range of 500–835 nm. The polyurethane absorption is stable between
13 and 14 month time points, but it is not known how gradual or when exactly the drop
in absorption occurred between the one week and 13 month time points. The polyurethane exhibited less absorption than two other resins (an epoxy resin and a polymer resin)
which showed significant absorption occurring below 500 nm thus giving each a yellow tint
as shown in figure 2.8.
2.3.2
Scatterer characterization
The scattering coefficients were determined from diffuse reflection and transmission
measurements. A typical spectra is shown in figure 2.9. This illustrates the wavelength
separation in the absorption of the two dyes in the reflected and transmitted light. The
reduced scattering coefficient as a function of wavelength (figure 2.10) was obtained using
inverse adding-doubling using the total diffuse reflection and transmission measurements
shown in figure 2.9. For comparison, the absorption coefficient obtained from a spectrometer measurement of a sample without titanium dioxide is also shown in figure 2.10
24
with the absorption coefficient calculated by IAD. The reduced scattering coefficient, µ0s
relates to the scattering coefficient (µs ) and the cosine of the mean angle of scattering
(g) from the incident direction of light by the equation µ0s = (1 − g)µs . The reduced
scattering coefficient is an equivalent optical parameter that is one over the mean free
path between scattering events when the scattering phase function is isotropic. Since, two
measurements are recorded, only absorption and the reduced scattering coefficient can be
determined. The anisotropy and the scattering coefficient, µs were not determinable separately with another measurement of ballistic photons. Our slabs (ranging from 2–7 mm)
were sufficiently thick that all transmitted light was completely diffuse.
The derived scattering properties for the titanium dioxide follow the relations
µ0s,690 = a3 C
(2.3)
µ0s,830 = a4 C
(2.4)
where a3 = 8.0 cm−1 mL/mg, a4 = 6.6 cm−1 mL/mg and C is the concentration of the
titanium dioxide. The different slopes correspond to the altered scattering efficiencies of
the titanium dioxide particles at 690 and 830 nm. The size distribution of the titanium
dioxide particles is shown in figure 2.11 having a mean of 340± 90 nm. By approximating
the particles as spheres, Mie theory was used to predict the scattering anisotropy for
the average particle size. Three-fourths of all particles were between 250 and 400 nm in
diameter. An average of the g-value calculated in 10 nm increments across the range 250
and 400 nm in particle diameter is g = 0.51 at the 690 nm wavelength and g = 0.52 at the
830 nm wavelength.
2.3.3
Phantom optical properties
A set of ten optical phantoms were made with differing levels of absorption but with
the same scattering properties. The variation among the ten phantoms in the scattering
coefficient at 690 and 830 nm is shown in figure 2.12. The measured absorption coefficients
at 690 and 830 nm of the ten samples are shown relative to the predicted absorption from
equations 1 and 2 in figure 2.13. The difference between predicted and the measured
25
absorption coefficients for the ten phantoms was plus or minus 3 percent at 690 nm and 6
percent at 830 nm.
The reproducibility of a phantom with a single set of absorption and scattering properties is shown in figures 2.14 for twelve phantoms made from a single batch. Both absorption
and scattering varied by plus or minus 3% between 600 and 800 nm. Between 500 and
600 nm, the percentage absorption variation increases but the absolute error remains constant. At wavelengths longer than 800 nm, the error increased (as previously stated) due
to lamp power and decreased detector efficiency. Since both the Epolight 6084 and 4148
were mixed together in one part of the polyurethane and titanium dioxide to the other part
before combining the two parts, the actual percentage variation in absorption at 830 nm
should be comparable to that at 690 nm. Since all twenty phantoms come from a single
large batch where all the dye was added to Part A of the polyurethane and the TiO2 was
added to part B for all twenty samples, then the variation in the samples can only come
from 2 sources: weighing errors for the combination to the two parts of polyurethane or
inhomogeneities in our large solutions of components. The largest weighing error was less
than 2% for all phantoms including the ten with different absorption properties.
To minimize error due to inhomogeneities, the part of the polyurethane with TiO2 was
regularly stirred to prevent settling of the scatterers before mixing with the part with the
added absorbers. However after combining the two parts of the polyurethane together,
the phantom cannot be stirred once the degassing step begins. Samples were held under
vacuum at a minimum of eight minutes but may be degassed for 10-12 minutes. Though
the pot life of the polyurethane was a half-hour, by 30 to 45 minutes the polyurethane was
still liquid but warming and noticeably beginning to gel. It is doubtful that full degassing
of phantoms can be performed with a pot-life less than half an hour since gelling of the
polyurethane will inhibit gas from escaping. It took about 5 hours to make all twenty
phantoms with two people working. The timeliest part of the process was degassing due
to a space limitation of only being able to place 4 or 5 phantoms in a vacuum chamber.
26
Desired optical properties
!a,!1
!s',!1
!a,!2
Use measured
TiO2
!s'(!) = a!"b
Determine effective
!s',!2
!s' ,!2
Evaluate
albedo at !2
Calculate
modified
!a,!2
!a,!1
!s',!1
!s' ,!2
!a,!2
Modified optical properties of phantom
Figure 2.2: Process for maintaining the four desired optical properties with only three
independent variables. The reduced scattering of the first wavelength establishes the
reduced scattering at the second wavelength. A new absorption coefficient is calculated
at the second wavelength that holds the desired reduced albedo at the second wavelength
constant relative to the new optical properties.
27
Pstd
entrance port
detector
baffle
standard
Psample
entrance port
detector
sample
P0
entrance port
detector
empty port
Figure 2.3: Diagrams of various integrating sphere reflectance measurements that are
needed.
28
diffuse
Pstd
entrance port
detector
baffle
standard
P0diffuse
entrance port
detector
empty port
Figure 2.4: Diagrams of various measurements needed to determine the fraction of diffuse
illumination f and the sphere parameter γ.
29
P'std
empty port
baffle
empty port
P'sample
detector
sample
detector
empty port
P'0
blocked beam
detector
empty port
Figure 2.5: Diagrams of various integrating sphere transmission measurements that are
needed. The empty port on the bottom of the spheres should be identical to the size of
the entrance port in the sphere reflection measurements.
30
6
4
3
pre-cure
-1
! [cm ] at 690nm
5
post-cure
a
2
1
0
0
5
10
15
20
25
30
35
40
Concentration [!g/ml]
Figure 2.6: The absorption coefficient at 690 nm as a function of dye concentration of
Epolight 6084 in part A of the polyurethane (circles) and after curing in polyurethane
−1
(squares). The fitting line is µ690
= a1 C6084 + µ690
a
a0 , where a1 = 0.16 cm mL/µg and
690
−1
µa0 = 0.046 cm is the absorption coefficient of polyurethane at 690 nm one week after
casting.
31
3.5
a
-1
! [cm ] at 830nm
3
2.5
pre-cure
2
1.5
1
post-cure
0.5
0
0
10
20
30
40
50
Concentration [!g/ml]
Figure 2.7: The absorption coefficient at 830 nm as a function of dye concentration of
Epolight 4148 in part B of the polyurethane (circles) and after curing in polyurethane
−1
(squares). The fitting line is µ830
= a1 C4148 + µ830
a
a0 , where a2 = 0.065 cm mL/µg and
830
−1
µa0 = 0.042 cm is the absorption coefficient of polyurethane at 830 nm one week after
casting.
.
32
Polymer
-1
!a [cm ]
1
Polyurethane
1 week
0.1
Epoxy
Polyurethane
13 months
0.01
400
500
600
700
800
900
Wavelength [nm]
Figure 2.8: The absorption coefficient spectra of an epoxy resin, a polymer resin, and
the polyurethane 13 months after being cast in addition to the polyurethane 1 week after
casting.
33
Reflection [%]
Transmission [%]
0.6
0.5
R
0.4
0.3
T
0.2
0.1
650
700
750
800
850
Wavelength [nm]
Figure 2.9: Typical reflectance and transmission measurements used to derive optical
properties using the inverse adding-doubling method.
34
4
8
µs'
3
6
2
4
2
0
650
1
µa
absorption coefficient [cm-1]
scattering coefficient [cm-1]
10
0
700
750
800
850
Wavelength [nm]
Figure 2.10: The reduced scattering and absorption coefficients spectra for the reflection and transmission data in figure 2.9 is shown in comparison to the spectrometer
measurement of absorption for the same phantom made without added scatterer. The
error bars are the standard deviation of three spectrometer measurements through the
clear phantom. Above 830 nm, the sensitivity of the detector drops precipitously causing
error in the resultant optical properties. The reduced scattering is fit by the relation
µ0s = (5.2 cm−1 )( 1000λ nm )−0.8 .
35
40
Number of Partilces
35
30
25
20
15
10
5
0
100
200
300
400
500
600
700
800
Particle Diameter [nm]
Figure 2.11: The size distribution of diameters of 112 titanium dioxide particles has a
mean of 340± 90 nm.
36
7
s
! ' [cm-1]
8
6
5
0
1
2
3
4
5
6
7
8
9
10
Sample #
Figure 2.12: The variation in reduced scattering coefficient for ten different phantoms at
690 nm (circles) and 830 nm(squares) measured by the IAD method. All samples were
designed to have the same reduced scattering properties. The error bars are the standard
deviation of four sets or reflection and transmission measurements for each sample.
37
3
-1
Predicted ! [cm ]
2.5
2
a
690 nm
1.5
1
0.5
830 nm
0
0
0.5
1
1.5
2
-1
2.5
3
Measured ! [cm ]
a
Figure 2.13: The predicted absorption coefficient using equations 1 and 2 are shown
relative the spectrometer measured absorption coefficient for the ten different phantoms.
38
!'
10
s
-1
! ' [cm ]
12
6
4
-1
! [cm ]
s
8
!
a
a
2
0
500
550
600
650
700
750
800
Wavelegnth [nm]
Figure 2.14: The absorption and reduced scattering coefficient spectra for twelve of the 20
identical phantoms. The error bars show the standard deviation for each optical property
among the twelve phantoms and are approximately ± 3% between 600 and 800 nm.
39
2.4
Conclusions
I presented a method to fabricate tissue phantoms that simulate the optical properties
of living tissue at the 690 and 830 nm wavelengths. The optical properties of the phantoms
has been consistent in a period of 14 months from the date of casting. The two molecular
dyes provided independent absorption between our chosen wavelengths for absorption
coefficients up to 5 and 3 cm−1 at 690 and 830 nm respectively. These dyes exhibited stable
absorption through the curing process of polyurethane. In addition, the high molecular
extinction coefficient of each dye allowed us to keep the addition of dye stock to less
than 0.1 percent of the volume of the polyurethane. The polyurethane itself was the
only component that changed optical properties after casting. The absorption due to
the polyurethane decreased approximately by a factor of 2 over the visible spectrum.
This change is assumed to occur gradually though I don’t have evidence to support this
belief. Though polyurethane slowly changes absorption properties, the effect is to make the
material more colorless which help to make any added dyes the more dominant absorber.
Polyurethane as a phantom material was easy to work with and handle. The material
becomes solid by 24 hours after mixing but is not fully cured. I found that the polyurethane
could not be machined for at least 48 hours. Even sanding of the material before 48 hours
is futile because ripples would form on the surface. This polyurethane may be cast into any
shape but the material of the mold determines how easy it is to extract the polyurethane
after curing. Polyethylene containers readily released after curing. Attempts to cast the
polyurethane between glass slides proved more problematic. Spray on mold release (Ease
Release 400, Mann Formulated products, Easton, PN) allowed the polyurethane to release
from glass but the thin film of mold release caused surface roughness to the polyurethane
making it like frosted glass. Polycarbonate molds behaved more like glass in terms of
demolding. The mold release allowed polyvinylchloride (pvc) pipe to act as a mold too,
but only the polyethylene worked well without the mold release.
Degassing of the polyurethane was a necessary though time-consuming step in the
phantom preparation. Significant bubble formation occurs during each mixing step. For
our phantoms at least three mixing steps occurred for each phantom, since two dyes and
40
the two components of polyurethane were combined. Bubbles that were not removed
provide many inhomogeneities, and may act as large scatterers. Removal of the bubbles
allowed for consistency in the finished product that otherwise would not be obtainable.
Four linear relationships (eqs. 1–4) that describe the the absorption coefficient as a
function of dye concentration and the reduced scattering coefficient as a function to the
titanium dioxide concentration were determined. These relationships predicted the measured absorbing and scattering properties with less than 4% error. The optical properties
of the phantoms have shown to be stable over a period of 14 months making the phantoms
suitable for use as reference standards.
Chapter 3
A Combined Inverse Adding-doubling
and Monte Carlo Model with
Experimental Correction Factors for
Optical Property Determination using
Single and Double Integrating Sphere
Measurements
3.1
Introduction
Numerous instances have been presented using the integrating sphere as a tool to
measure the total diffuse reflection and transmission of optically turbid and homogeneous
materials, including several biological tissues [76–81] since it was first described by Ulbricht
[82] in 1900. Using a light propagation model, the integrating sphere measurements may
be used to calculate the absorption and scattering properties for a sample with a slab
geometry. The inverse adding-doubling (IAD) algorithm is a fast and accurate method
to solve for the absorption and scattering coefficients of in vitro tissues or other optically
turbid materials from measurements of the total diffuse reflectance and transmittance
[83]. Monte Carlo (MC) models of light transport have been used with integrating sphere
measurements to determine optical properties [76–79, 84–86]. Like the inverse addingdoubling method, an optimization routine finds the absorption and scattering coefficients
by iteratively calculating the reflectance and transmittance. The IAD and MC models
41
42
employed often overlook important effects that occur when integrating spheres are used to
measure the reflection and transmission Many MC models ignore the fact that the sphere
itself alters the measurements. The IAD model ignores the light that escapes through the
edges of the sample.
The effect of the integrating sphere on the measured reflectance has been documented
extensively [87–94]. An experimental method to account for the sphere’s perturbation of
the measurements was attempted by Pickering [95]. Unfortunately, inconsistencies in the
experimental and theoretical parameters make this method difficult to use. Pickering also
gave the first description of a method using two spheres for simultaneous measurement of
both the reflectance and transmittance [94]. This was novel in that the dynamic processes
caused by external stimuli such as heating [96,97] and dehydration [98] could be monitored.
However, double sphere measurements are made when single sphere measurements would
suffice, often due to the convenience of simultaneous measurements of reflectance and
transmittance [99–101].
Light lost through the edges of the sample is a common source of experimental error
[83]. A discussion of the consequences for light loss errors is given by Pickering, where
the effect leads to an over-estimation of absorption by the sample [95]. The light loss
(which may be as high as 30%) is a function of the sample dimensions, the sample port
size, the illuminating beam diameter, and the optical properties of the sample [102]. Lost
light must be corrected for each of these variables making it impractical to pre-calculate.
Therefore, a three dimensional model such as a freely adaptable MC model is commonly
used for inverting optical properties and calculating light loss.
The purpose of this work is to improve the accuracy of integrating sphere measurements. A mathematical model of integrating sphere measurements for both single sphere
and double-sphere arrangements is presented. This derivation is consistent with prior
integrating sphere theory work [87, 88, 92, 94]. A clear and simple characterization of
the effect of the integrating spheres upon measurements is given as a single function for
gain. Practical correction factors for single and double integrating sphere arrangements
are presented. These factors are included into a combined IAD and MC model to calculate
optical properties from sphere measurements. I demonstrate the robustness of the model
43
to produce consistent results for an optically stable polyurethane phantom under a wide
range of experimental sphere conditions (eg. multiple sample thicknesses and sample port
sizes). I measure the optical properties of a water suspension of titanium dioxide particles
over the wavelength range of 400-1700 nm. This demonstrates the ability to resolve absorption for more than three orders of magnitude. An analysis of potential errors in sphere
measurements shows that experimental error is comparable to optical property error with
appropriately matched experiments and an inversion model. Finally, a set of guidelines
for obtaining the best results using integrating spheres is given.
3.2
Theory
Quite a few variables are required to describe the light within an integrating sphere.
There are variables describing areas within the sphere: the inside area of the sphere A,
the area of the sample port As , the detector port Ad , the entrance port Ae , and the area
of the sphere wall Aw . For simplicity, these are normalized to as = As /A, ad = Ad /A,
ae = Ae /A, and aw = 1 − as − ae − ad so that they describe the fractional area of each
region in the sphere. The average reflection of the sphere wall, the detector, and the
sample under diffuse illumination are denoted rw , rd , and rs respectively. The reflection
of the sample by direct or collimated illumination is given by rsdirect . The term rstd is
used for the sample reflection of a reference standard. The transmission of the sample for
, while the transmission of the sample
direct or collimated illumination is denoted tdirect
s
with diffuse illumination is ts . The illumination light power is P . The power incident
upon a region of the sphere is denoted by a subscript with the regions being the sphere
wall Pw , the sample Ps , and the detector Pd .
3.2.1
Single integrating sphere
This section describes the interaction of reflected (or transmitted) light with a single
integrating sphere. The greatest advantage of the integrating sphere is that it collects
nearly all the light reflected by (or transmitted through) a sample. A significant fraction
of this light is collected and redirected back to the detector. The exact amount depends on
44
geometric sphere factors and the reflectivity of the sphere walls and sample. The concept
of the sphere gain to describe the increase in the irradiance on the detector due to the
reflective sphere walls is introduced. Unfortunately, the sphere gain is not an experimental
constant, but instead depends on the reflectance of the sample rs .
Single Sphere Gain
Assume that a sphere is illuminated with diffuse light having a power P and that this
light reaches all parts of the sphere — specifically, light from this source is not blocked
by a baffle. Subsequent reflections however are restricted by a baffle located between
the sample and detector ports. Multiple reflections in the sphere will increase the power
falling on non-white areas in the sphere (e.g., the sample, detector, and entrance). To
find the total light falling on each area of the sphere, the total power is summed for each
successive incidence (followed by a reflection). A superscript denotes the number of times
light is incident upon each surface within the sphere and pertains only to the contribution
of power for that incidence. The first incidence for the diffuse light is
Pw(1) = aw P,
(1)
Ps(1) = as P,
Pd
= ad P
The second incidence on the wall is
(1)
Pw(2) = aw rw Pw(1) + (1 − ae )rd Pd + (1 − ae )rs Ps(1)
The light from the detector and sample is multiplied by (1−ae ) and not by aw because the
light from the detector (and sample) is not allowed to hit either the detector or sample.
The light that hits the walls on the kth incidence has the same form as above
(k−1)
Pw(k) = aw rw Pw(k−1) + (1 − ae )rd Pd
+ (1 − ae )rs Ps(k−1)
Since the light falling on the sample and detector can only arrive from the wall due to a
baffle,
Ps(k) = as rw Pw(k−1)
and
(k)
Pd
= ad rw Pw(k−1) ,
Therefore, the incident light on the wall for the kth incidence becomes
Pw(k) = aw rw Pw(k−1) + (1 − ae )rw (ad rd + as rs )Pw(k−2)
45
The total power falling the detector is
Pd = ad P +
∞
X
ad rw Pw(k−1) = ad P + ad rw
k=2
aw + (1 − ae )(ad rd + as rs )
P
1 − aw rw − (1 − ae )rw (ad rd + as rs )
This result is in agreement with other investigators [88, 92, 94].
The gain G(rs ) is defined as the irradiance on the detector relative to a black sphere
as
G(rs ) ≡
Pd /Ad
P/A
where Ad and A are respectively the area of the detector port and the entire area of the
sphere. The gain is
G(rs ) = 1 +
rw
aw + (1 − ae )(ad rd + as rs )
·
aw 1 − aw rw − (1 − ae )rw (ad rd + as rs )
(3.1)
The gain for a detector in a transmission sphere is similar [94], except that primed parameters are used to designate the use of a potential second sphere. For a black sphere the
gain G(0) = 1, which is easily verified by setting rw = 0, rs = 0, and rd = 0. Conversely,
when the sphere walls are perfectly white, the irradiance at the entrance port, the sample
port, and the detector port must increase so that the total power leaving via these ports
is equal to the incident diffuse power P .
Single Sphere Measurements
The power falling on the detector in a single reflectance sphere will be
Pd = ad · (initial uniform diffuse light) · G(rs )
In a typical measurement, the initial diffuse light in the reflection sphere will arise from
(1) incident light that hits the sphere wall before the sample and (2) incident light that
hits the sample first. Since the fraction of light that first hits the wall is defined as f , the
first reflection is rw f P . However this is not uniformly diffuse (since it cannot hit either
the sample or the detector because of the presence of a baffle) so the light must strike the
walls one more time before becoming uniform. This means that the first portion is given
2 (1 − a )P .
by f rw
e
46
The light reflected by the sample is (1 − f )rsdirect P . Since baffles ensure that the light
cannot reach the detector from the sample, the light must bounce off the sphere walls to
become a uniform diffuse source. The contribution will then be (1 − f )rsdirect (1 − ae )rw P .
The measured reflection will be
R(rsdirect , rs ) = ad (1 − ae )rw [(1 − f )rsdirect + f rw ]P · G(rs )
(3.2)
Similarly the power falling on the detector measuring transmitted light is
0
T (rsdirect , rs ) = a0d tdirect
rw
(1 − a0e )P · G0 (rs ).
s
(3.3)
These expressions for R and T also work for entirely diffuse illumination where the incident
light initially strikes the sphere wall in which case f = 1. However, this is generally
undesirable because of the small difference between R(rstd , rstd ) and R(0, 0).
To remove the source power(P ) dependence, I define the sample reflectance MR as
MR = rstd ·
R(rsdirect , rs ) − R(0, 0)
.
R(rstd , rstd ) − R(0, 0)
(3.4)
MR is the ratio of the difference in power measured between a sample and an open sample
port to the difference in power measured between a reflection standard and an open sample
port, where the experimental setups for R(rsdirect , rs ), R(0, 0), and R(rstd , rstd ) are specified
in figure 3.2. Subtraction of a dark signal measurement provides two significant benefits
over normalization by a ratio R(rsdirect , rs )/R(rstd , rstd ). First, error is reduced especially
for low reflection samples caused by a DC offset in the signal due to dark current in the
detector. Second, the measurement R(0, 0) is largely a measure of the fraction of light
that is initially incident upon the sphere wall f , especially when white light sources are
employed. By subtracting R(0, 0), the effect of f may be accounted for without a direct
method of determining its value. Similarly, the measured transmittance is defined as
MT = tstd ·
T (tdirect
, rs ) − Tdark
s
T (0, 0) − Tdark
(3.5)
where the experimental setups for T (tdirect
, rs ), T (0, 0), and Tdark are shown in figure 3.3.
s
Additionally, reflectance and transmittance values determined using dark subtraction
have a range between 0 and 1, as opposed to a constant lower limit greater than zero that
47
depends upon the experimental apparatus. The measured reflectance with a sphere and
the true reflectance are exact at values of zero and rstd and are relative at intermediate
values as shown in figure 3.4.
3.2.2
Double spheres
When two integrating spheres are used (fig. 3.5), all light must be absorbed by the
sample, the sphere walls, the detector, or lost through the entrance port just as for single
sphere measurements. However, the power of light falling on the detector differs from single
sphere measurements because light may repeatedly pass back and forth through the sample
between the two spheres. Thus, the gain becomes more complicated and now depends on
both the reflectance and transmittance of the sample. In the following derivation, the
subscript 1 refers to the reflection sphere and the subscript 2 to the transmission sphere
and arrows are used to denote the direction of the source to the destination sphere.
Two sphere gains
The double-sphere gain may be derived from the single-sphere gain by considering the
light passing through the sample between spheres as a new source that sums ad infinitum.
Here a superscript denotes the number of times light passes through the sample between
spheres and pertains only to the contribution of power for that number of passes. If P is
uniform diffuse light in the first (reflection) sphere, then the light falling on the detector
(assuming that no light has passed back and forth through the sample) is
(0)
P1→1 (rs , ts ) = ad G(rs )P
The attenuation factor X1→2 for light crossing from the first (reflection) sphere to the
second (transmission) is
0
X1→2 = as ts (1 − a0e )rw
For light crossing the other direction, the attenuation factor is
X2→1 = a0s ts (1 − ae )rw
48
It follows that the light reaching the detector in the first sphere, after the light has crossed
back and forth through the sample k times is
(k)
k
P1→1 (rs , ts ) = ad G(rs )P · G0 (rs )X1→2 X2→1 G(rs )
The total light on the detector of the first sphere for diffuse illumination originating in
the first sphere is
P1→1 (rs , ts ) =
1−
G0 (r
ad G(rs )P
s )X1→2 X2→1 G(rs )
The corresponding gain is defined as
G1→1 (rs , ts ) ≡
G(rs )
P1→1 (rs , ts )/Ad
=
0
0
P/A
1 − as as rw rw (1 − ae )(1 − a0e )G(rs )G0 (rs )t2s
The gain for light on a detector in the second sphere for diffuse light P arising in the first
sphere is identical except for an extra factor of X1→2 G0 (rs ). Thus
0
G1→2 (rs , ts ) = as (1 − a0e )rw
ts G0 (rs )G1→1 (rs , ts )
Similarly, when the light starts in the second sphere, the gain for light on the detector
in the second sphere G2→2 is found by switching all primed variables to unprimed. Thus
G2→2 (rs , ts ) is
G2→2 (rs , ts ) =
G0 (rs )
0 (1 − a )(1 − a0 )G(r )G0 (r )t2
1 − as a0s rw rw
e
s
s s
e
and
G2→1 (rs , ts ) = a0s ts (1 − ae )rw G(rs )G2→2 (rs , ts )
Double integrating sphere measurements
The expressions for direct illumination of a sample between two integrating spheres
follow from the above formulas for diffuse illumination. In this case, the light on the
detector in the reflection (first) sphere arises from three sources: the fraction of light
2 (1 − a )P , the fraction of light reflected by the
directly reflected off the sphere wall f rw
e
2 (1 − a )P , and the light transmitted through the sample (1 −
sample (1 − f )rsdirect rw
e
49
0 (1 − a0 )P ,
f )tdirect
rw
s
e
2
R2 (rsdirect , rs , tdirect
, ts ) = G1→1 (rs , ts ) · ad (1 − ae )rw
fP
s
+ G1→1 (rs , ts ) · ad (1 − ae )rw (1 − f )rsdirect P
0
+ G2→1 (rs , ts ) · ad (1 − a0e )rw
(1 − f )tdirect
P
s
which simplifies to
R2 (rsdirect , rs , tdirect
, ts ) = ad (1 − ae )rw P G1→1 (rs , ts )
s
0
(1 − f )rsdirect + f rw + (1 − f )a0s rw
(1 − a0e )tdirect
ts G0 (rs )
s
×
For the power on the detector in the transmission (second) sphere one has the same three
sources. After a bit of algebra,
0
T2 (rsdirect , rs , tdirect
, ts ) = a0d (1 − a0e )rw
P G2→2 (rs , ts )
s
×
(1 − f )tdirect
+ (1 − ae )rw as ts (f rw + (1 − f )rsdirect )G(rs )
s
Double-sphere normalization differs slightly from the single-sphere method. The experimental arrangement for the spheres in double-sphere measurements is shown in figure
3.5. The normalized reflectance is then
MR = rstd ·
R2 (rsdirect , rs , tdirect
, ts ) − R2 (0, 0, 0, 0)
s
R2 (rstd , rstd , 0, 0) − R2 (0, 0, 0, 0)
(3.6)
and transmission by
MT =
3.2.3
T2 (rsdirect , rs , tdirect
, ts ) − T2 (0, 0, 0, 0)
s
.
T2 (0, 0, 1, 1) − T2 (0, 0, 0, 0)
(3.7)
Sphere parameters
Two experimental parameters are needed to fully evaluate equations 3.2–3.7. These
are the fraction of diffuse illumination (f ) and average sphere wall reflectance (rw ). Fortunately as is shown later, it is not necessary to measure f as long as it is less than 40%.
If needed, f is
f = R(0, 0)/R0diffuse
as shown in figures 3.2 and 3.6.
50
The average reflectance of the sphere wall needs to be measured to minimize error. It
may also be determined with two measurements. Both measurements use diffuse illumination, but one has an open sample port (R0diffuse ) and the other a reflectance standard in
diffuse
the sample port (Rstd
). To make the light diffuse, all the light is directly incident on
the sphere wall behind the baffle in relation to the detector port. The ratio of the light
when no sample is present to that when a reflectance standard may be represented as
R0diffuse
diffuse
Rstd
=
1 − aw rw − (1 − ae )rw (ad rd + as rstd )
.
1 − aw rw − (1 − ae )rw ad rd
A solution for the wall reflectance is then
1−
rw = 1−
R0diffuse
diffuse
Rstd
R0diffuse
diffuse
Rstd
.
(3.8)
(aw + ad rd (1 − ae )) + as (1 − ae )rstd
The reciprocal of the wall reflectance gives a cleaner relationship,
Rdiffuse
1
= aw + ad rd (1 − ae ) + as (1 − ae )rstd diffusestd diffuse .
rw
Rstd
− R0
(3.9)
Equation 3.8 is consistent with other investigators method for determining the wall
reflectance [92, 93] and illustrates the difficulty in making accurate measurements of the
diffuse
and R0diffuse will only differ by
sphere wall reflectance. The two diffuse reflectances Rstd
the amount of diffuse light leaking from the sphere when the port is empty. Consequently,
the difference will be small and any errors in the measurements will be magnified when
the division is done.
It is possible to gain insight into the relation between measurements and rw by making
a few assumptions. For example, the relative detector area ad and the entrance port area
ae are usually negligible and can be ignored. The equation for rw can be expanded in
terms of as
rw = 1 − as
diffuse
R0diffuse − (1 − rstd )Rstd
diffuse
Rstd
−
R0diffuse
+ O(a2s ).
(3.10)
Since rstd is often close to one, the second term in the numerator is small which leads to
rw ≈ 1 − as
R0diffuse
diffuse
Rstd
− R0diffuse
if
rstd = 0
(3.11)
51
diffuse
Ultimately, only two calibration measurements (per integrating sphere) Rstd
and
R0diffuse are necessary to perform sphere corrections. There are six other constants to be
calculated: the fractional area of the sphere wall aw , the fractional area of the sample port
as , the fractional area of the detector ad , the fractional area of the entrance port ae , the
reflectance of the detector rd ; and the reflectance of the standard rstd . These parameters in
combination with the five additional measurements form a complete set of data necessary
to determine optical properties using an integrating sphere, two reflection measurements
R(rsdirect , rs ) and R(rstd , rstd ) and three transmission measurements T (rsdirect , rs ), Tdark ,
and T (0, 0).
3.3
Materials and Methods
The aim is to provide two methods using integrating spheres to indirectly determine
the optical properties of a material and show that the process is accurate and consistent
under a wide variety of experimental conditions.
3.3.1
Test samples
To show that our experimental method is robust, a sample is required that is simple, stable and whose absorption properties are known by independent means from the
method presented here. The majority of experiments used a solid polyurethane phantom. An advantage of the solid phantom was that it could be placed over the port of
the integrating sphere without a special holder or glass covers. Construction of the polyurethane phantom was shown previously and demonstrated to be optically stable [103].
Two slabs roughly 60 mm in diameter of the same batch were produced with thicknesses
of 4.93 ± 0.01 and 6.67 ± 0.01 mm. Additionally, a sample was made from the same batch
cast in a 10 mm pathlength cuvette without added scatterer (TiO2 particles) for independent absorption coefficient determination using an absorbance spectrometer (Cary 100 Bio
Spectrophotometer, Varian Scientific Instruments Inc., Walnut Creek, CA). The refractive
index of the polyurethane (1.468) was measured using an Abbé refractometer at visible
wavelengths.
52
A second sample was included that provided a greater range of absorption over the
visible and near-infrared spectrum. A solution of titanium dioxide particles (Ti-pure R900, Dupont Chemicals, Wilmington, DE) suspended in ultra-filtered water was prepared
at a concentration of 0.63 mg/ml. The TiO2 particles are the same as in the solid phantoms
and were shown to have a mean diameter of 340 ± 90 nm using SEM imaging [103] . The
particle suspension was sonicated for an hour to prevent settling and then pipetted into
holders and measured within the next hour. During that time, notable settling did not
occur. Two sample holders were used with optical pathlengths of 0.15 mm and 11.3 mm.
Water was assumed to be the dominant absorber especially at infrared wavelengths.
3.3.2
Integrating sphere measurements
Single sphere experiments were performed as diagrammed in figures 3.2 and 3.3. A
203.2 mm diameter integrating sphere (IS-080-SF, Labsphere, Inc., North Sutton, NH)
with a Spectraflect coating was used for all measurements. The sphere has three 50.8 mm
ports and a 12.7 mm port adjacent to a baffle. The ports are spaced equally along the
equator of the sphere. The port opposite the 12.7 mm detector port was capped to close
the port. The detector port had a custom fabricated cap with an SMA optical fiber connector that reduced the detector port to a 3.2 mm diameter. This custom cap was coated
on the interior surface with barium sulfate (white reflectance coating #6080, Eastman
Kodak, Rochester, NY). A 1 mm diameter fiber (PWF1000T, CeramOptec Industries,
East Longmeadow, MA) guided light from the detector port to one of two scanning grating detectors in a spectrofluorometer (Fluorolog-3, Instruments S. A.,Inc., Edison, NJ); a
photo-multiplier tube (PMT) detector for visible wavelengths to NIR (400–850 nm) and a
liquid nitrogen cooled germanium detector for the wavelength range of 800–1700 nm.
Slightly different integrating sphere arrangements were used for the two types of samples. For the polyurethane phantom, a 1 mm optical fiber was used to deliver light to
the sample from a tungsten lamp, (Fiber-lite, Dolan-jenner Industries, Boxborough, MA).
The optical fiber was sleeved through a steel tube (4 mm diameter) holding the fiber face
centered with the sample port at 10 mm from the sample giving a 4 mm diameter spot
53
size. The steel tube was coated with multiple layers of flat white paint. The fiber extended beyond the steel tube about 5 mm. The entrance port was 6.35 mm in diameter
for reflection (fig. 3.2) measurements but closed for transmission (fig. 3.3). The 6.67 mm
thick phantom was measured with multiple sample port diameters: 6.35, 12.7, 19.1, 25.4,
31.8, 44.5, and 50.8 mm for single-sphere experiments. Only the 25.4 mm sample port was
used to measure the 4.93 mm thick phantom in a single sphere experiment.
However, additional measurements were recorded with a 25.4 mm sample port from
400–1000 nm using a silicon diode array grating spectrometer detector (SD2000, Ocean
Optics inc., Dunedin, FL) for both of the solid phantom thicknesses. Due to the limited dynamic range of the silicon detector, full spectrum results could not be obtained
from a single set of measurements. And so, multiple reflectance and transmittance were
stitched together (400–500 nm, 500–600 nm, 700–800 nm and 800–1000 nm) such that the
measurement of the reference standard was at least 50% detector saturation limit (but
not saturated) in each spectral section. Neutral density filters were used to attenuate the
light source and the integration time was set to 0.5, 0.7 or 1.0 s to meet that criteria. To
further reduce noise, three acquisitions were averaged during collection.
Measurements of the titanium particle suspension used a single-sphere arrangement.
For the light source, a tungsten lamp (6333, Oriel Instruments, Stratford, CT) had an
image of the filament focused in the plane of the sample port with a spot size of 10 mm for
the largest dimension. For reflection measurements, an entrance port cap reduced the port
diameter to 19.1 mm and the sample port diameter was the maximum possible 50.8 mm.
Transmission measurements were made with a sample port of 50.8 mm in diameter. The
entrance port cap (19.1 mm) from the reflection measurements was placed opposite to
the detector, keeping the fractional sphere wall area, aw constant for both reflection and
transmission measurements.
Double sphere experiments were performed as diagrammed in figure 3.5 using two identical spheres, the same as described for the single sphere experiments. Measurements were
made for four sample port sizes (19.7, 31.8, 44.5, and 50.8 mm) for each of the polyurethane phantom sample thicknesses (4.93 and 6.67 mm). The two spheres were mounted to
54
an optical rail to allow for the sandwiching of the sample between spheres. Only the transmission sphere was moved to keep the reflection sphere fixed with respect the illumination
fiber (4 mm beam-diameter on the sample). The PMT detector of the spectrofluorometer
was used as described above for the single-sphere using a 1 mm collection fiber. Reflection and transmission were recorded by serially alternating the connection of the detector
collection fiber between spheres without otherwise changing the sphere arrangement.
For all experiments, the reference was a 50.8 mm diameter 99% Spectralon reflection standard (Labsphere, Inc., North Sutton, NH) for reflection or an open port as the
reference in transmission. The optical properties were determined from the measured
reflectance (eqn. 3.4 for single-sphere experiments or eqn. 3.6 for double-spheres) and
transmittance (eqn. 3.5 for single-sphere experiments or eqn. 3.7 for double-spheres) using
an IAD program [83].
3.3.3
Measuring the Sphere Parameters
Though values of the sphere wall reflectivity usually accompany vendor supplied integrating spheres, the reflectivity may change over time as dust settles on the sphere surface
over time. The experimental determination of the average sphere wall reflectance follows
diffuse
and R0diffuse were used to determine the sphere
section 2.3. The measurements of Rstd
wall reflectance rw (eqn. 3.8) as shown in figure 3.6. The integrating sphere was rotated
with respect to the collimated lamp beam so that light was directly incident upon the
sphere wall between the sample port and the baffle. The fractional area of the sphere wall
aw , the sample port as and the detector port ad was calculated using a disc geometry for
the ports rather than the area of a spherical cap. Goebel showed that flat surfaces within
an integrating sphere reduce the overall efficiency of the sphere introducing an error equivalent the the difference in area between a spherical cap and a disc of the same radius [92].
Finally, I assumed a detector reflectance, rd of 0.05 to account for the reflectance by the
fiber in addition to the steel SMA connector which made up 90% of the area of the detector
port.
55
45
40
25.4mm sample port
G(rs)
35
30
50.8mm sample port
25
20
0
0.2
0.4
0.6
0.8
1
rs
Figure 3.1: The gain (relative to a black sphere) for a 203 mm (8 in.) integrating sphere.
The wall reflectivity is assumed to be 0.975 and the detector reflectivity to be zero. In both
curves, the entrance port (6.35 mm) and detector port areas (3.18 mm) are identical. The
increase in the gain as the sample reflection increases is the primary reason that sphere
corrections are needed.
56
detector
R (r
std
,rstd )
standard
R (r
detector
,r )
direct
s
s
sample
detector
R (0,0)
empty port
Figure 3.2: Diagrams of various integrating sphere reflectance measurements needed to
make a reflectance measurement (eqn. 3.4) using a single sphere set-up.
57
detector
T (0,0)
empty port
detector
T (r
,r )
direct
s
s
sample
detector
Tdark
blocked
beam
Figure 3.3: Diagrams of various integrating sphere transmission measurements needed to
make a transmittance measurement (eqn. 3.5) using a single sphere set-up.
58
True Reflectance or Transmittance
1
0.8
0.6
R
0.4
T
0.2
0
0
0.2
0.4
0.6
0.8
1
Measured Reflectance or Transmittance
Figure 3.4: The difference between measured and true sample reflectance and transmittance are shown in comparison to unity slope. The relationship is determined by combining
eq. 3.2 with eq. 3.4 to describe reflectance and transmittance by combinging eq. 3.3 with
eq. 3.5 for a 203.2 mm diameter sphere with a wall reflectivity of 0.975, a 50.8 mm diameter
sample port, a 6.35 mm diameter entrance port, and a 3.18 mm diameter detector port.
59
R2 (r
,r ,t sdirect ,ts)
direct
s
s
T 2 (r
detector
,r ,t sdirect ,ts)
direct
s
s
detector
incident
light
sample
R 2(r
std
T2(0,0,1,1)
,rstd ,0,0 )
detector
detector
standard
detector
incident
light
open port
R2(0,0,0,0)
T 2 (0,0,0,0)
detector
detector
open port
detector
blocked
beam
open port
Figure 3.5: Diagram of a double-integrating sphere experiments needed for the simultaneous measurement of reflectance (eqn. 3.6) and transmittance (eqn. 3.7).
60
detector
R
diffuse
std
standard
detector
R
diffuse
0
empty port
Figure 3.6: Diagrams of various integrating sphere measurements needed to determine the
fraction of diffuse illumination f and the average sphere wall reflectance.
61
3.3.4
Optical property inversion
A new IAD program was made that follows the experimental method described in the
above single and double sphere experiments and follows the mathematical model of the
Theory section. Monte Carlo (MC) simulations of light transport were included in the
IAD program to evaluate the fraction of light lost out of the edges of the sample. The
measured reflectance and transmittance were input to the program along with experiment
specific values. These values included the sample thickness, the refractive index of the
sample and any cover slides, the number of spheres used, the port diameters for each
sphere, the reflectivity of the detector and the sphere wall for each sphere and the value
of the reference reflection and transmission.
A flow chart of the program logic is shown in figure 3.7. The program initializes by
calculating a course grid for measured reflectance and transmittance values with preset
absorption (µa ) and scattering coefficients (µs ) and anisotropy (g) (when collimated transmission is input) with the adding-doubling algorithm. The light lost through the sample
edges is initialized to zero. The adding-doubling algorithm then loops to calculate the
diffuse and direct reflectance and transmittance for a set of optical coefficients (µa , µs and
g), subtracts the light lost proportionally for the diffuse and direct fractions, and calculates the total reflectance and transmittance. The process is repeated changing the optical
coefficients until the calculated values equal the measured values. When they match, a
MC simulation is performed to calculate the fraction of lost light. The adding-doubling
loop is repeated to find new optical coefficients with the light loss correction. The MC
simulations are repeated until both the change in the predicted absorption and scattering
coefficients is less than 0.1%.
3.3.5
Error analysis
Integrating sphere measurements are generally performed to determine the optical
absorption and scattering properties of a sample. Since the optical properties are the
desired result, the percent change of the inverted optical properties are calculated from
independent perturbations to three input measurements (MR , MT and rw ). For our data
62
set, the reflectance and transmittance of the 6.67 mm thick polyurethane phantom is used
with a single sphere arrangement and a 25.4 mm diameter sample port. The analysis
was divided into two areas, the model corrections of the measurements and experimental
measurements errors.
There were two types of corrections made with our model of the sphere measurements.
The first type was sphere corrections that accounted for the corresponding differences
between the measured values of reflectance and transmission and the true reflectance
rsdirect and transmission tdirect
. Second, light loss corrections were made using a Monte
s
Carlo simulation to calculate the light lost through the sample edges. The light loss was
separated into two categories, direct loss and diffuse loss which are shown respectively in
figure 3.8. The optical properties using sphere and light loss corrections were determined
using IAD. Next, the percent change to the absorption and reduced scattering coefficients
with no corrections, sphere corrections alone, and light loss corrections alone was calculated
using the relation
% change =
µlimited
a,s
corrections
µsphere
a,s
− µsphere
a,s
and MC corrections
and MC corrections
The sphere corrections accounted for the corresponding differences between the measured
values of reflectance and transmission and the true reflectance rsdirect and transmission
tdirect
.
s
Experimental error was produced by a perturbation of the data set to run with IAD
to determine the percent change of the output optical properties. I ran the same set of
data through the IAD program making independent 1% reductions (a multiplicative factor
of 0.99) to the reflectance, transmittance, and the sphere wall reflectance. The percent
change in optical properties was calculated by the equation
% change =
µnormal
− µperturbed
a,s
a,s
µnormal
a,s
63
MR M T
Initialize coarse grid of
µ a , µ s, g
lost light = 0
Find diffuse and direct
reflectance using
adding-doubling
Subtract MC loss
Calculate the
values M R, M T
from R and T
Change µa , µ s, g
No
calculated = measured
Yes
Calculate new
diffuse and direct
loss using MC and
reiterate above loop
Change in µ a and µs
is less than 1% for
new MC loss
Yes
No
Done
Figure 3.7: The logic flow of inverse adding-doubling program with a Monte Carlo model
for calculation of light lost through the sample edges.
64
fiber sleeve
diffuse light
from sphere wall
optical fiber
direct loss
sample
diffuse loss
Figure 3.8: Light incident upon the sample from the initial irradiating beam that scatters
out the side of the sample so that it neither enters the reflectance sphere or a transmission
sphere and leads to larger value than the true absorption of the sample. This error is
referred to as direct light loss. Light reflected by the sample back into the integrating
sphere becomes diffuse in nature after reflecting off the sphere wall. Diffuse light incident
upon the sample has a greater likely-hood of scattering out the side of the sample. Light
lost in this manner is referred to as diffuse light loss.
65
3.4
3.4.1
Results
Optical properties
The mean absorption and reduced scattering coefficients of the polyurethane phantom
are shown in figure 3.9 for seven different single sphere experiments and eight different
double sphere experiments. The variation among measured optical properties made with
differing sample port sizes exceeded the variation in measurements for a single experimental set-up. The standard deviation for the single sphere arrangements is 4% for absorption
and 3% for scattering. The double sphere arrangements have a standard deviation of the
mean of 6.5% for absorption and 4% for scattering. The measured absorption of the
non-scattering sample differed by as much as 13% from the sphere measurements. Fixed
numbers of iterations of the Monte Carlo simulations were performed on a single sample to
evaluate for absorption convergence for multiple corrections for light lost through the sample edges. The result for absorption is shown in figure 3.10. A comparison of the optical
properties is shown figure 3.11 using different detectors over a greater spectral range. Figure 3.12 shows the calculated optical properties for a 0.63 mg/ml titanium dioxide particle
suspension in water in a 11.3 mm thick sample holder. The fraction of transmitted light
drops below the noise level of the detector above 1350 nm due to absorption by water. The
calculated absorption of the 0.15 mm thick sample holder is also shown for wavelengths
ranging 1350–1700 nm.
3.4.2
Error analysis
A single data set of total diffuse reflection and transmission measurements are used
for the error analysis with the polyurethane phantom. The optical property coefficients
for absorption and scattering differ from the mean optical properties shown in figure 3.9
by less than 2%. The change in the optical properties calculated without corrections,
with light loss only and sphere corrections only relative to optical properties with both
light loss and sphere corrections are shown in figure 3.13. The effect of a 1% decrease
to the sphere wall reflectance upon the IAD determined optical coefficients is shown in
figure 3.14(bottom). The effect of a change in the fraction of initial diffuse light (f )(data
66
not shown) is spectrally flat for both absorption and scattering. This partial derivative
is at least three orders of magnitude smaller than the sphere wall reflectance, the total
diffuse reflection, or total diffuse transmission. The change in optical properties for a
perturbation of a 1% decrease to the reflectance and transmittance are respectively shown
in figures 3.14 top and middle.
3.5
Discussion
I demonstrate in figure 3.9 that our method of calculating absorption and the reduced
scattering coefficient gives consistent results regardless of the experimental conditions. The
optical properties determined using a single sphere is indistinguishable from two sphere
experiments. The maximum deviation for any single experiment is 8% from the mean
for the reduced scattering coefficient. The over-prediction of small absorption coefficients
had been widely discussed as a concern with IAD [80, 81, 95] and integrating sphere measurements in general [84–86]. Monte Carlo simulations were shown to increase the lower
resolution of absorption down to 0.05 mm−1 with loss corrections [85]. In figure 3.11, the
absorption coefficient is in good agreement with the spectrometer absorbance measurement down to 0.01 mm−1 for samples roughly 5 and 7 mm thick. The titanium particle
suspension in water follows the absorption spectrum of water down to 0.006 mm−1 with a
sample thickness of about 11 mm.
A suspension of TiO2 particles in water was chosen expecting that water would be the
dominant absorber which is well characterized [104]. In the visible to near infrared wavelengths water absorption spans several orders of magnitude. Our results showed (fig. 3.12)
that water absorption is dominant in the NIR. However in the visible portion of the spectrum, our measurement of absorption deviates dramatically from water absorption alone.
Cabrera et al. [105] measured the absorption coefficient for rutile TiO2 and water suspensions for five manufacturers’ particles over the wavelengths 275- 405 nm. They show that
properties vary significantly between manufacturers. Our measured absorption between
390-405 nm is within a factor of two of Cabrera et al. To our knowledge, no studies have
shown absorption properties in the visible spectrum of titanium dioxide particles which
67
may be quite different than the properties of films. Therefore, I believe that absorption
between 400-800 nm is dominated by the TiO2 particles rather than water. Additional
studies are needed to validate this result.
The combined IAD and MC model to determine optical properties utilizes the advantage of both techniques. IAD is a one dimensional model and therefore gives a fast and
accurate method of calculating optical properties. The MC model is three dimensional
and so it accounts for effects of light lost out the sample’s edges which can not be accounted for with IAD alone. Since the MC model is used in a limited role, fewer photon
bundles are needed to solve the light loss problem alone than are needed to solve both
the inversion and light loss problems. The iterative use of the MC model to calculate
the light loss hides the statistical error of MC models. It also removes the guess work in
deciding the number of MC corrections needed to reach a stable set of optical properties.
The drawback to the iterative approach is that a minimum of two MC simulations must
be performed for each data point even when only a single iteration is sufficient to meet
the tolerance for stability.
3.5.1
Sources of error
The error is separated into two classes, model errors and experimental errors. The
model presented for both single and double sphere experiments is specific to the experimental method. Total reflectance and transmittance measurements using integrating
spheres need to account for the effect the sphere has upon the measurement. The problem is that the measured reflectance and transmittance are not linearly related to the
sample reflectivity and transmissivity as shown in figure 3.4. Inexact correction for this
problem is referred to as a sphere correction error. I also refer to improper correction for
light lost through the sample edges as a modeling error which is solved for with a MC
model. Figure 3.13 shows the percent change in optical properties with either one or both
model corrections not performed. Without either of the model corrections the absorption
is over-predicted by 10–160% whereas the reduced scattering coefficient is under-predicted
by 3–4%. Adding sphere corrections reduces the scattering error to 1%. MC simulations
correct most of the error in the predicted absorption, but sphere corrections still account
68
for 2–4% error. In all cases, the magnitude of the error for the predicted absorption
decreases with higher absorption.
Experimental error in which one of the measured values (e.g. reflectance, transmittance and sphere wall reflectivity) is imprecisely determined is shown in figure 3.14. The
magnitude of the error for the predicted optical properties is generally equal to the error in the measurement. At low absorption, the error in the predicted absorption does
increase to about 2% for a 1% perturbation in the reflectance and transmittance. The
difference between modeling error and experimental error is stark. It implies that the
greatest error occurs when the experimental method does not match the method assumed
by the inversion model. When the inversion program accurately models the experiment,
the uncertainties in the predicted optical properties will be approximately equal to the
experimental uncertainties.
3.5.2
The inversion model
Since accurate determination of optical properties relies upon the model matching the
experimental method, the implicit assumptions of combined IAD and MC inversion model
are discussed in detail. Four assumptions are made. 1. The integrating sphere measurements are performed using the substitution method (defined below). 2. The reflectance
and transmittance are normalized with a dark measurement subtraction. 3. The specular
reflection from the sample is collected in the reflectance measurement. 4. There is a baffle
inside the sphere blocking direct reflections between the sample and detector ports.
Two modes of integrating sphere experiments are referred to as substitution and comparison methods [88, 92]. The substitution method implies that the integrating sphere
has a single sample port. The sample is substituted by the reflectance standard in a
subsequent measurement to normalize the reflectance and transmittance. An alternative
method used by investigators [79] is to make comparison measurements using two samples
ports. With comparison measurements, the sample is placed in one sample port while
the standard is placed in the other identical size sample port. Light is incident upon
the sample for the sample measurement and then made incident upon the standard for
the reference measurement. Since both the sample and the reference are present on both
69
measurements, the sphere gain is identical on both measurements. The advantage of the
comparison method is that the reflectance and transmittance equal the sample reflectivity
and transmissivity, but only when no light is initially incident upon the sphere wall.
I considered two experimental parameters in the IAD model, the fraction of light
directly incident upon the sphere wall f and the sphere gain G(rs ). Subtraction of a
dark-signal measurement from both the sample and reference measurements (eqs. 3.4,
3.5, 3.6, and 3.7) nearly eliminates the effect of f . Subtraction of a dark signal containing
the light initially incident upon the sphere wall removes the large background signal of
diffuse light from light directly incident on the sphere wall. Dark measurement subtraction
also removes any DC offset due to dark current in the detector. The greatest difference
between optical properties determined with IAD that includes corrections for f versus
without correction is 0.003 mm−1 for the reduced scattering and 0.001 mm−1 for absorption
measurements on the polyurethane phantom (single-sphere data from (fig. 3.9) for the
6.35, 12.7, 19.05 and 25.4 mm sample ports and a 4 mm beam diameter. In these data
sets, f varies from 0.07–0.15. There are two cases where corrections for f are substantial
(1-5% error): when more than half of the light is initially incident upon the sphere wall or
when the sample reflectance is close to zero. Both cases are the result of a poorly designed
experiment and are best avoided. Therefore, I believe that it is unnecessary to make a
measurement of f to accurately determine optical properties using the presented method
of signal normalization.
Our model assumes that specularly reflected light is collected inside the integrating
sphere. Some integrating spheres are designed to trap the reflected specular light [79].
A light trap acts as an additional port with zero reflectivity. Whether or not specular
collection is collected makes little difference, provided the inversion model is consistent
with the experiments.
The foundation of the presented IAD/MC inversion model is based on eqs. 3.2 and
3.3 for single sphere experiments and eqs. 3.6 and 3.6 for double sphere experiments. The
lone restriction for these equations is that a baffle is present between the sample and the
detector blocking direct reflections between the ports. These equations are consistent with
70
previous integrating sphere models [88, 92, 94]. The complexity in the equations is contained by the sphere gain for substitution mode experiments. The gain terms quantify the
effect of multiple reflections caused by the presence of the sphere relating to the efficiency
of light incident on the detector port. The gain term for single sphere experiments is
functionally related to the sample reflectance. However, double sphere gain also depends
on the transmission of the sample because it also includes the cross-talk of light traveling
between the spheres through the sample. It is possible to create an inversion model for
any method of signal normalization based on the equations 3.2, 3.3, 3.6, and 3.6, as they
are universal in describing an individual measurement recorded with an integrating sphere
with a baffle.
3.5.3
Best practices
There are many ways that integrating spheres may be employed to measure reflectance
and transmittance. Therefore, I address the variables involved with spheres to elucidate
how to obtain the best results. To begin, I consider the difference between single or double
sphere measurements. The use of two spheres allows for the simultaneous measurement
of reflectance and transmittance which is necessary for monitoring dynamic processes.
However, the use of two spheres is more involved than a single sphere arrangement .
For example with two spheres, the sample ports must be aligned and the sample held
between the ports without gaps for light to escape. Single sphere arrangements are more
versatile in that either comparison mode or substitution mode can be made whereas only
the latter mode can be performed using two spheres. Substitution mode is better suited
for measurements using white light sources which inevitably produce some light that is
initially incident upon the sphere wall. Comparison measurements have the advantage
that no sphere corrections are necessary only if absolutely all light is initially incident
upon the sample (or standard). Therefore, the comparison mode is best suited for sources
which are easily collimated such as lasers.
The effect of the sphere upon measurements in substitution mode is related to the
choice of the reflection and transmission reference. The magnitude of the correction for
transmittance is nearly equal to the correction reflectance, but never greater than the
71
correction for reflectance when a 99% reflectance standard is used as the reference. It
should also be noted that in reflectance, the smaller the difference between the reflectance
of the standard and the sample, the smaller the correction is for the sphere. Therefore
when a sample is expected to have low reflectivity, the extrapolation between the true and
measured values for a comparable reflectance standard would be much less than for a 99%
standard. Thus, the likelihood for error due to the sphere corrections would be reduced
for inexact values of the sphere parameters (e.g. sphere wall reflectance).
Without light loss corrections, the choice of a sample port size is restricted by the ratio
of beam diameter to the sample port [95,106]. The physical dimensions of the sample, the
integrating sphere port, the optical properties of the sample, and the size of the incident
beam all contribute to the amount of light lost out through the side of the sample. For
a fixed port diameter, the light loss is smaller for thinner samples, samples with higher
absorption and scattering coefficients, and for smaller diameter illumination beams. The
amount of light escaping the sample edges decreases with larger diameter sample ports
for both direct and diffuse illumination. The combined IAD/MC inversion model gives
consistent absorption results regardless of port size. The advantage of using larger ports
is that generally fewer MC simulations will be performed in calculating light loss before
convergence thereby reducing the calculation time. In comparing sample port sizes with
the polyurethane phantom (6.67 mm thick), a 25.4 mm port needed an average of 2.4
iterations. The 19.05 mm sample port averages 3.4 iterations, the 12.7 mm port averages
5.8 iterations and the 6.35 mm sample port averages 13.5 iterations. Each iteration adds
about 4 minutes per 100 data points. However, the sample should not be smaller in
dimension than the port. An under-filled sample port can be accounted for by increasing
the area of the entrance port by the same area by which the sample does not cover the
port. This practice is difficult to quantify and is likely to increase error in the predicted
absorption.
Erroneous determination of optical properties occurs when information is lacking in the
reflectance and transmittance. When the transmission measurement has a value of zero,
it becomes impossible to obtain both the absorption and reduced scattering for the sample. In this case, only the ratio of absorption to scattering (or rather the reduced albedo,
72
µ0s
µ0s +µa )
is uniquely determinable. Less obvious is the limited information when the sum of
the reflectance and transmittance is close to one. The sum of absorption, reflection and
transmission must equal the incident power. When the absorption is small, the accuracy
is limited by the precision of both the reflectance and transmittance measurements. An
example of this condition occurs in the visible region (450–550 nm) of figure 3.11. In that
region, the measured absorption coefficient is 0.001–0.004 mm−1 greater than the polyurethane matrix absorption alone [103]. The limit of absorption resolution is 0.001 mm−1 for
samples on the order of 10 mm thick for a reduced scattering of 0.9 mm−1 .
The use of an optical fiber to illuminate the sample alters the effect of the entrance
port. In addition, the use of a sleeve [5] to direct and hold the optical fiber near the sample
in a reflection measurement deserves discussion. The effect of a sleeve is to close a portion
of the area of the entrance port and translate that area to a position inside the sphere. It
is assumed that the sleeve is coated so that it has the same reflectance as the sphere wall
or is inside the sphere when measuring the average sphere wall reflectance. The sleeve
then behaves as a baffle inside the sphere. To verify that the sleeve had minimal impact, I
measured diffuse light inside a 200 mm diameter sphere with and without a sleeve inserted
approximately 170 mm (data not shown). The two measurements were indistinguishable.
The combined IAD/MC model may also be used to evaluate the scattering anisotropy
(average cosine of the scattering angle) with a third measurement of the fraction of unscattered transmission. A port may be open opposite the sample in a transmission measurement to let the unscattered light pass to a detector far from the sphere [96,99,101]. Though
double spheres arrangements are necessary to measure dynamic processes, an experimental
error is introduced in the transmittance [107]. Roos shows that a fraction of highly forward
scatttered light also escapes along with the unscattered transmission through the port.
Additional measurements can be made to estimate the fraction escaped [107]. However, a
measurement of the unscattered transmission is simple to measure independently without
the integrating sphere. A good unscattered transmission measurement requires that a
sample be optically thin, preferably having an optical thickness τ < 5 (τ = (µa + µs )d
where d is the sample thickness) [95]. High albedo samples often will need large optical thickness to resolve small absorption coefficients relative to the scattering coefficient
73
which is contrary to the unscattered transmission preference. Separate independent measurements of the unscattered transmission prevent conflict by using two sample thickness
which are appropriate for each measurement. The unscattered transmission should then
be included in the integrating sphere transmittance.
Sample handling
Biological tissue samples provide a unique challenge to measure optically. The method
of handling of a sample may cause dramatic differences in the optical properties. Tissue
dehydration effects the refractive index of the tissue [108], its scattering characteristics
[109], and even absorption [110]. Sandwiching a tissue between glass or quartz plates limits
the rate at which a tissue dehydrates and provides support and well defined boundary
conditions for soft tissues and fluids. Additionally saline may be sandwiched with the tissue
to clamp the hydration level of the tissue. Compression of tissue can lead to an increase
in scattering and absorption due to a reduction in tissue volume increasing chromophore
and scatterer concentrations [111] The inverse adding-doubling algorithm calculates the
Fresnel reflections of incident light from the sample [83]. A flat surface to the sample
simplifies the accounting for Fresnel reflections whereas a rough sample will specularly
reflect light in all directions in the integrating sphere.
3.6
Conclusions
The combined IAD/MC model was demonstrated to accurately determine optical properties of homogenous optically turbid samples with a reasonable precision using multiple
sample thicknesses and sample port sizes. Inversion of optical properties from integrating
sphere measurements requires a theoretically compatible inversion model and experimental
methods otherwise gross error occurs. With substitution method sphere measurements,
the inversion model must correct for the sphere gain which increases the measured reflectance and decreases transmittance. A 3D model such as Monte Carlo is essential to
correct for light lost through sample edges which improves the resolution and accuracy
of samples with small absorption coefficients. With these two corrections, experimental
74
error becomes comparable in magnitude to the error in the predicted optical properties
for the presented IAD/MC model.
75
1
Optical coefficients [mm-1]
µs'
0.1
µa
0.01
650
700
750
800
Wavelength [nm]
Figure 3.9: The calculated mean absorption and reduced scattering coefficient for a
6.67 mm thick polyurethane phantom measured with sample port sizes of 6.35, 12.7, 19.05,
25.4, 31.75, 44.45 and 50.8 mm in diameter for a single-sphere measurements is shown (solid
thin line). Error bars indicate the standard deviation. The calculated mean absorption
and reduced scattering coefficient for a 4.9 and 6.7 mm thick identical polyurethane phantoms measured with sample port sizes of 19.05, 31.75, 44.45 and 50.8 mm in diameter for
a double-sphere measurements is shown as the dashed line. The illumination beam was
4 mm in diameter on the sample in all measurements. The bold line is the absorption
coefficient determined by the spectrometer for a phantom without scatterer.
76
0.3
Sphere corrections only
0.25
1 MC iteration
2 MC iterations
µa[mm-1]
0.2
5 MC iterations
0.15
10 MC iterations
0.1
20 MC iterations
0.05
spectrometer absorption
0
650
700
750
800
Wavelength [nm]
Figure 3.10: The calculated absorption coefficient for a 6.67 mm thick polyurethane phantom measured with a 6.35 mm diameter sample port and 4 mm illumination beam. The
dashed curve is the absorption coefficient measured for a phantom without added scatterer. After 5 iterations of a Monte Carlo simulation to correct for light lost through the
sample edge, the absorption was over-predicted at coefficients less than 0.1 mm−1 , but by
less than a factor of 2.
77
1
Optical coefficient [mm-1]
µs'
µa
0.1
0.01
spectrometer absorption
0.001
400
500
600
700
800
900
1000
900
1000
Wavelength [nm]
µa residual [mm-1]
0.02
0.01
0
-0.01
-0.02
400
500
600
700
800
Wavelength [nm]
Figure 3.11: The absorption and reduced scattering coefficient for two thicknesses (4.93
and 6.67 mm) of a polyurethane phantom. The intrinsic absorption of the polyurethane
at 450 nm is 0.004 mm− 1 [103]. For comparison, the absorption coefficient measured for a
phantom without added scatterer is shown as a bold line.
78
10
0.15 mm sample
µs'
-1
Optical coefficient [cm ]
1
0.1
0.01
µa
0.001
Segelstein
0.0001
10-5
400
600
800
1000
1200
1400
1600
Wavelength [nm]
Figure 3.12: The reduced scattering and absorption coefficients determined with the inverse adding-doubling algorithm for single sphere experiments. The absorption coefficient
for the 0.15 mm sample is shown from 1350–1700 nm. For comparison, the absorption
coefficient of water compiled by Segelstein [104] is shown.
79
50
800
200
150
100
50
0
650
700
750
800
10
8
6
4
2
0
650
700
750
wavelength [nm]
800
4
2
0
-2
-4
650
(sphere corrections only) (sphere corrections only)
750
µs' error [%]
(sphere corrections only)
(MC corrections only)
µa error [%]
700
(no corrections)
100
0
650
µa error [%]
µs' error [%]
150
µs' error [%]
(no corrections)
µa error [%]
200
700
750
800
700
750
800
700
750
800
4
2
0
-2
-4
650
4
2
0
-2
-4
650
wavelength [nm]
Figure 3.13: The percent change for the absorption coefficient and reduced scattering
coefficient for no corrections (top), for sphere corrections only (middle), and for lost light
corrections (bottom). Sphere corrections account for the non-linear relationship between
the reflectance (transmittance) and the sample reflectivity (transmissivity). Lost light is
predicted using Monte Carlo simulations of the sphere experiment.
0
-2
-4
2
0
-2
-4
750
800
4
2
0
-2
-4
650
700
750
wavelength [nm]
800
2
0
-2
-4
700
750
800
700
750
800
700
750
800
4
2
0
-2
-4
650
(for Rw = 0.99Rw)
700
4
650
(for MT = 0.99MT)
800
4
650
(for Rw = 0.99Rw)
750
µs' error [%]
(for MT = 0.99MT)
µa error [%]
700
(for MR = 0.99MR)
2
650
µa error [%]
µs' error [%]
4
µs' error [%]
(for MR = 0.99MR)
µa error [%]
80
4
2
0
-2
-4
650
wavelength [nm]
Figure 3.14: The percent change for the absorption coefficient and reduced scattering
coefficient for a 1% underestimate of the reflectance (top), transmittance (middle), and
the sphere wall reflectivity (bottom). The perturbations shown correspond to single sphere
experiments using a 25.4 mm diameter sample port with the 6.67 mm thick polyurethane
phantom.
Chapter 4
Sized-Fiber Reflectometry
for Measuring Local Optical Properties
4.1
Introduction
Determining the optical scattering and absorption properties of tissue for diagnostic
and therapeutic applications is of interest in medicine. For example, the dosimetry of
photodynamic therapy is greatly dependent upon the scattering and absorption properties [48,112] for both light delivery and measurement of drug concentrations. Optical properties have been used to estimate exogenous [113, 114] and endogenous [115] chromophore
concentrations. Moreover, light scattering and absorption can provide information about
both chromophore content and structure, which might be used to distinguish normal and
malignant tissues as well as chemical state information [116, 117] (e.g. oxy- versus deoxyhemoglobin). In this regard, an optical biopsy can be performed using optical information
of hemoglobin and water content differences in normal and malignant tissue [23, 74].
Several techniques and algorithms have been developed to measure the optical properties. Spatially [16, 37, 39] and temporally [22, 31, 118] resolved measurements have been
proposed to extract the optical properties using light distributions based on diffusion theory [17, 119] and/or Monte Carlo simulations [120]. A wide variety of algorithms have
been explored to extract optical properties including neural network [121] and multiple
polynomial regression methods [122]. However, most reflectance techniques rely on light
c
[2001]
IEEE. Reprinted with permission from IEEE Journal of Selected Topics in Quantum Electronics, Lasers in Medicine and Biology, November/December 2001, Vol. 7, No. 6., pp. 952–958
81
82
distribution information over a large area (>1 cm2 ) or have separate illumination and
collection fibers with a separation distance on the order of 1 cm.
Much work has been compiled on devices with separate source and detector fibers.
Two studies show that the mean photon penetration depth increases as the square root of
the separation between source and detector fibers either spatially [123] or temporally [124].
Source-detector fiber devices sample relatively large volumes, because of the separation
between the fibers. Though effort has been made to minimize the sampling volume using
small source-detector separations [125], little work has been done on photon penetration
depths when the source fiber is also used to collect backscattered photons [1]. Since large
sampling volumes are less likely to be homogenous, we developed a device that acquires
information from small volumes of tissue (< 1 mm3 ) by using the same fiber for both source
and detector to minimize the sampling volume.
We propose a compact dual-fiber device to make simple and rapid measurements of the
absorption and reduced scattering properties of tissue. Each fiber illuminates and collects
light independently of the other fiber. The device is based on the fact, that in general,
tissues with different scattering and absorption properties will scatter different numbers of
photons back into a fiber. If only a single fiber is used, two samples with different optical
properties could backscatter the same number of photons. This paper proposes a device
containing a second, different size fiber to make a second measurement. This second fiber
collects information from a different effective volume of the sample than the first fiber.
In this study, we present experiments on an Intralipid tissue model and Monte Carlo
simulations to elucidate the details of light emission and collection with a single fiber. As
a first step, we conduct single wavelength experiments at 632.8 nm with a device using 200
and 600 µm fibers to assess the feasibility of extracting optical properties for these fiber
sizes. We have adapted the sized-fiber system for use with white light (500–800 nm) to
measure the wavelength dependent absorption and reduced scattering coefficients (herein
referred to as µa and µ0s = (1 − g)µs , respectively). Experimental results are presented for
an array of Intralipid solutions of known absorption, 0.1–2.0 cm−1 , and reduced scattering,
5.2–52 cm−1 . Monte Carlo simulations of a 200 and 600 µm fiber device are presented
for comparison. Also presented are experiments and Monte Carlo simulations of the
83
penetration depth of photons collected by two fiber sizes, 200 and 600 µm in diameter.
Finally, Monte Carlo simulations are used to determine the effect numerical aperture of a
fiber and the influence of mode filling of the fiber on reflectance measurements.
4.2
4.2.1
Materials and Methods
Sized-Fiber Device
The sized-fiber device uses a pair of bifurcated fibers to separate the illumination light
from the backscattered light. The first of these is made by end-coupling two 300 µm
diameter fibers onto the face of a 600 µm diameter fiber with SMA connectors. The
second bifurcated fiber consists of two 100 µm diameter fibers end-coupled to a 200 µm
fiber also using SMA connectors. One each of the 100 and 300 µm fibers have He-Ne
laser light (632.8 nm) focused onto their face. The remaining 100 and 300 µm fibers are
coupled to UDT silicon photodiodes. The ends of the 200 and 600 µm fibers are epoxied
(120387, Dymax Corp., Torrington, CT) together by UV curing and then polished (Fig.
4.2). All fibers are fused silica glass/glass from Polymicro Technologies LLC, Phoenix, AZ.
The manufacturer cites a numerical aperture of 0.22 which corresponds to a maximum
exit angle of 12.7 degrees in air. We measured 13 ± 1 degrees for both emission and
transmission with the 200 and 600 µm fibers for the maximum acceptance angle. The
maximum half-angle for acceptance, θ depends on the numerical aperture of the fiber, NA
= n sin θ, where n is the index of refraction of the surrounding medium. This corresponds
to an angle of 9.5 degrees for our fibers in water. The distal ends of the 200 and 600 µm
fibers are sleeved through one meter of polycarbonate tubing (6.2 mm O.D., 4.7 mm I.D.)
to inhibit tight bending of the fibers. The 100 and 300 µm fibers outside the polycarbonate
tubing are kept fixed throughout the experiments so that losses through the fiber cladding
remain relatively constant from measurement to measurement in those sections of fiber.
Two choppers (MC 1000, Thorlabs Inc., Newton, NJ) operating at 690 Hz and 700 Hz
eliminate cross-talk between fibers. The reflectance signal from each fiber is detected by
a lock-in amplifier (LIA 100, Thorlabs Inc., Newton, NJ). The fibers are rinsed in a water
bath then washed with ethyl alcohol between measurements. Washing the fibers is critical
84
50/50
chopper, 690 Hz
He-Ne laser
! = 632.8 nm
detector
detector
chopper, 700 Hz
600 µm
fiber
200 µm
fiber
IL & ink
sample
Figure 4.1: This is a diagram of the sized-fiber device. A pair of bifurcated fibers emit
and collect light simultaneously by chopping at two distinct frequencies exceeding 690 Hz.
The fibers are submerged at least 1 cm into the Intralipid to eliminate boundary effects.
for reproducibility with the Intralipid tissue model which can leave an oily residue on the
fiber that affects later measurements.
The sized-fiber device also works with white light reflectometry [126]. A tungstenhalogen white lamp (LS-1, Ocean Optics,Inc., Dunedin, FL) couples to the source fibers via
the SMA connectors. The collection fibers are coupled into a spectrometer (s2000, Ocean
Optics,Inc., Dunedin, FL). The spectrum is processed in near real-time using LabView
(National Instruments, Austin, TX). A baffle must be inserted so that only one fiber emits
and collects at a time, since cross-talk will not be rejected with this method.
All data points are normalized by using Fresnel reflection from the fiber face in air and
water. The voltage returned from the photodiode is converted by
Reflectance =
(Vsample − Vwater )
∗ (3.46% − 0.19%) + 0.19%
(Vair − Vwater )
where 3.46% is the Fresnel reflectance for the fiber core/air junction and 0.19% is the
Fresnel reflectance for the fiber core/water junction at normal incidence. Measurements
of the air and water Fresnel reflectance are taken periodically throughout the experiments
85
200 µm fiber
epoxy
16 Gauge Needle
600 µm fiber
Figure 4.2: The 200 and 600 µm fibers are joined with a medical grade epoxy so that they
may be simultaneously inserted through a needle for measurement of tissues. The two
fibers can easily fit through a 16 gauge needle. When measuring tissues, water is placed
at the tip of the fiber to reduce the index mismatch between the fiber cores and the tissue.
to correct for any variations in laser power.
4.2.2
Tissue Model
For tissue phantoms with known absorption and reduced scattering properties, an
array of Intralipid (Liposyn II, Abbott Laboratories, North Chicago, IL) and India ink
mixtures with optical properties in the range of tissues were constructed. Five dilutions
of the Intralipid are made from a single bottle of Intralipid making concentrations of 1,
2, 4, 9 and 10% (10% lipid indicates 10 g of lipid per 100 ml of suspension) to be used as
stock solutions. Also five stock solutions of India ink (No. 4415, Higgins, Lewisburg, TN)
and deionized/distilled water are made with absorption of 0.2, 0.4, 1.0, 2.0, and 4.0 cm−1
at 632 nm as measured using a spectrophotometer (HP 8452A). The absorption of light
at 632 nm by Intralipid [61] is less than 0.01 cm−1 at the highest concentration and thus
is negligible in comparison to absorption by the India ink. The phantoms are made by
mixing one part of each Intralipid concentration to one part of each ink solution for a
total of 25 samples. The resultant solutions have all possible combinations of Intralipid
concentrations of 0.5, 1, 2, 4.5, and 5% corresponding to reduced scattering coefficients
of 5.2, 10.4, 20.8, 46.8, and 52.0 cm−1 and absorption coefficients of 0.1, 0.2, 0.5, 1.0, and
2.0 cm−1 at 632 nm. The optical properties are determined by R(r) measurements [17] and
cross-checked using total diffuse reflectance measurements using an integrating sphere [95]
in conjunction with inverse adding-doubling [127]. We measured a reduced scattering
86
coefficients of 104 cm−1 , which is between those of and Flock et. al [61] and van Staveren
et. al. [128] who measured of 65 and 130 cm−1 respectively for 10% Intralipid.
4.2.3
Chicken Breast Experiment
Chicken breast, purchased from the supermarket, was measured using the sized-fiber
device inside a 16 gauge needle. The needle was filled with water before inserting the
optical fibers to aid the coupling of light between the fibers and the tissue. The needle
was inserted into the muscle along the muscle grain in three locations. The needle was
withdrawn and inserted so that the needle tip was in the same three locations but oriented
perpendicular to the muscle grain for the second set of measurements. All the measurements were performed within 5 minutes to minimize changes in the optical properties due
to dehydration of the tissue. Additionally, paper towels saturated with phosphate buffered
saline covered the chicken between measurements.
4.2.4
Planar Absorber Effect on Signal Experiment
To measure the effect of an absorbing boundary on the signal detection, the Thorlabs
lock-in amplifiers were replaced by SR830 lock-in amplifiers (Stanford Research Systems,
Sunyvale, CA) for their phase adjustment capability and improved signal to noise ratio.
Optically thick India ink mixed in acrylamide was used for an absorbing plane. The 1 cm
thick acrylamide was set in the bottom of a 50 ml beaker. The Intralipid phantom described above with an absorption coefficient of 0.5 cm−1 and reduced scattering coefficient
of 20.8 cm−1 covered the black acrylamide layer. The fibers were brought into contact
with the gelatin before the Intralipid was added to obtain a zero position. The fibers were
backed away from the surface for a distance of 1 cm. Measurements were then recorded
for the 200 and 600 µm fiber simultaneously in 0.1 mm increments until the fibers were
submerged into the gelatin. At this point the signal was constant and equal to the Fresnel
reflection of the fiber in water. Measurements of the reflectance in air and water were also
recorded for calibration.
87
4.2.5
Monte Carlo Simulations
A Monte Carlo program was adapted to simulate the light collected by a fiber irradiating an infinite homogeneous scattering and absorbing medium; the fiber face was the only
boundary included into the simulations, thus neglecting the sides of the fiber (e.g. the
fiber jacket, etc.). Experimental results confirm the validity of the computer simulations
(Fig. 4.3). Also, the photon penetration depth was tracked to give insight as to the volume
being sampled for each size fiber. Additional simulations were performed to examine the
effect of photon launching angle and the effect of varying the numerical aperture.
The geometry of the Monte Carlo simulation is set up as follows. Let the z-axis be
parallel with that of the fiber so that the face of the fiber is in the z = 0 plane. Photons
are launched with equal probability over the entire face of the fiber. The direction by
which photons are launched is specified by the direction cosines (νx , νy , νz ). The angle
νz = cos θz is given a Gaussian distribution that depends on the acceptance angle of the
fiber while the other direction cosines are uniformly distributed. The angular distribution
chosen approximates the emission from the fibers but due to under-filled modes, the experimentally measured distribution is not a true Gaussian but only deviates by 12%. The
distribution of angles that the photon might take is given by the function
!
1
θ2
p(θz )dθz = √ exp − z2 dθz
2θa
2π
where θa is the acceptance angle for the fiber. Moreover, the angles were limited such that
θz ≤ θa .
The primary statistic collected by the Monte Carlo program was the fraction of light
backscattered into the fiber. Photons which were collected must pass back through the
fiber face. Only photons that had an angle less than or equal to the maximum acceptance
angle for the fiber were counted into the diffuse reflectance. The maximum acceptance
angle for the fiber was input into the program for the fiber in air, then converted using
Snell’s law for the index of refraction for the incident medium. Photons incident on the
fiber face but outside the cone of acceptance were attributed to light lost through the
cladding. The acceptance angle was corrected for the index of refraction change of the
medium. The index of refraction for the core of the fiber was assumed to be 1.457 as
88
given by the manufacturer’s specifications; 1.333 was used for the index of the medium. A
returning photon’s weight was attenuated due to an index mismatch for both launching and
collection using the Fresnel reflection for normal incidence introducing an error ≈ 0.02%.
The anisotropy was chosen to be 0.83 in accordance to Flock et. al. [61]. In a single
simulation, a minimum of 20,000 photons were launched ten times for each simulation to
accumulate statistical error of the mean. Specular reflectance due to the index mismatch
at the fiber face while launching was added to the diffuse reflectance to get the total
reflectance.
In the depth profiling absorbing plane simulations, 400,000 photons were launched.
A perfect absorbing planar boundary was added parallel to the x-y plane. The collected
diffuse reflectance was related to the fiber-plane separation distance, simulating the experiment in section 2.4. Additional simulations were performed with the absorption held
fixed at 0.5 cm−1 while the scattering was varied (µ0s = 8.5–34 cm−1 ) and with the reduced scattering fixed at 21 cm−1 while absorption was varied (µa = 0.05–2.0 cm−1 ). In
the Monte Carlo code, the photon weight became zero when it crossed into the absorbing
plane. The location of the boundary, determined by a z-position was varied for each set
of optical properties.
Finally, the maximum angles for launching and collection were varied independently.
The maximum angle (measured from the normal of the fiber face) for the launching photons
was varied independent of the collection angle, to simulate under-filling of fiber modes.
The maximum launch angle was stepped in two degree increments while the collection
angle was held fixed at 12 degrees for the fiber in air. Next, the maximum launch angle
was held fixed while the maximum acceptance angle of the fiber was varied to evaluate
the sensitivity with respect to numerical aperture. The acceptance angles chosen were
concentrated around 13 degrees for the fiber in air (for a NA ≈ 0.2) and then extended
to include very small and large NA fibers (0.02–0.65). For all simulations the optical
properties were held constant with µs = 61.18 cm−1 , µa = 0.5 cm−1 , and g = 0.83.
89
µs' = 52 cm
200 µm fiber
Total Reflectance
0.005
0.004
-1
decreasing µa
in all clusters
-1
µs ' = 20.8 cm
0.003
-1
µs' = 10.4 cm
0.002
0.002
µs ' = 5.2 cm-1
0.004
0.006
0.008
0.01
0.012
600 µm fiber
Total Reflectance
Figure 4.3: The optical properties can be determined from a mapping of the reflectance for
each fiber. The open circles are the experimental measurements of each sample. The solid
squares are the Monte Carlo simulation results. For samples with µ0s = 5.2 and 10.4 cm−1 ,
Monte Carlo results are only shown for µa = 0.1 and 2.0 cm−1 for clarity.
200 µm fiber Reflectace
90
0.005
-1
µs' = 52 cm
0.004
-1
µs ' = 20.8 cm
0.003
decreasing µa
in all clusters
-1
µs' = 10.4 cm
-1
µs ' = 5.2 cm
0.002
chicken breast perpendicular to muscle orientation
chicken breast parallel to muscle orientation
0.001
0.000
0.000
0.002
0.004
0.006
0.008
0.010
0.012
600 µm fiber Reflectance
Figure 4.4: Measurements of the twenty Intralipid mixtures and repeated measurements
of chicken muscle with two distinct fiber orientations relative to the muscle grain. Each
cluster is a constant Intralipid concentration with five different absorption coefficients
which decrease from left to right.
4.3
4.3.1
Results
Tissue Model
Fig. 4.3 shows measurements of the Intralipid array as well as the results of the Monte
Carlo simulations. The experimental variation in measurement was smaller than that
returned by the simulations. Typical experimental variation are shown in Fig. 4.5.
Fig. 4.4 also shows twenty measurements of the samples (the 4.5% Intralipid are omitted for clarity) along with in vitro chicken breast. All concentrations of ink are shown at
each Intralipid concentration. Each cluster represents solutions with the same scattering
coefficient. Within each cluster, the higher absorbing solution has a lower reflection with
respect to the 600 µm fiber reflectance. Marijnissen [129] et. al. reported a scattering
coefficient for chicken muscle (µ0s = 3.3 cm−1 and µa = 0.17 cm−1 ) which agrees with the
91
200 µm fiber
Total Reflectance
0.0048
µa = 0.1 cm-1
-1
0.0046
µa = 0.5 cm
-1
µa = 2 cm
0.0044
0.0042
µa = 0.1 cm-1
0.0040
µ = 2 cm-1
a
0.0038
0.008
0.009
µ = 0.5 cm-1
a
0.010
0.011
600 µm fiber
Total Reflectance
Figure 4.5: Independence of sized-fiber measurements for two scattering samples with three
different absorption coefficients. The filled and unfilled circles correspond to µ0s = 47 and
52 cm−1 respectively. Each joined cluster of three points represents three measurements
on a single sample.
Diffuse Reflectance (!)
Diffuse Reflectance (z)
92
1.0
0.8
200 µm fiber
0.6
600 µm fiber
0.4
0.2
0.0
0.01
0.1
1
10
Fiber - Absorbing Plane Separation, z [mm]
Figure 4.6: The signal attenuation for 200 and 600 µm fibers is shown as the fibers approach
an absorbing plane of India ink in acrylamide gelatin. The signal is normalized to the
diffuse reflectance measured in the intralipid solution far from boundaries (µ0s = 10.4 cm−1 ,
µa = 0.5 cm−1 , g = 0.83). The curves represent Monte Carlo simulations of the experiment
with standard errors of the mean and the points are the experimental data.
data presented here, but the absorption coefficient of chicken is unresolvable with the
sized-fiber device for such a low scattering coefficient. Fig. 4.5 shows a comparison of
the 5% Intralipid samples with absorption coefficients of 0.1, 0.5, and 2.0 cm−1 with 4.5%
Intralipid solutions with the same absorption coefficients. A slight decrease of scattering
(∆µ0s = 5.2 cm−1 ) in one sample relative to another is distinguishable by a drop in the
200 µm fiber reflectance. A drop in the reflectance also occurs as the absorption increases
but the effect is more pronounced with the 600 µm fiber reflectance measurement.
93
Depth for 50% Signal [µm]
400
-1
µs' = 21 cm
350
300
600 µm fiber
250
200
150
200 µm fiber
100
50
0
0
0.5
1
-1
1.5
2
2.5
µ a [cm ]
Depth for 50% Signal [µm]
400
-1
µa = 0.5 cm
350
300
600 µm fiber
250
200
150
200 µm fiber
100
50
0
5
10
15
20
-1
25
30
35
µs' [cm ]
Figure 4.7: Monte Carlo simulations show the separation between a fiber and an absorbing
plane where a 50% drop in signal occurs for variations in the absorption and scattering
properties. The anisotropy is set to 0.83 to simulate Intralipid.
94
0.0020
Diffuse Reflectance
600 µm fiber
0.0015
0.0010
0.0005
0.0000
0
200 µm fiber
5
10
15
20
25
30
35
Maximum Launch Angle [Degrees]
Figure 4.8: For a fixed acceptance angle of the fiber, the maximum angle of launch from
a gaussian distribution does not influence the diffuse reflectance collected. This suggests
that complete mode filling of the fibers is unnecessary to obtain a consistent result; and by
extension, modal variation of the emission does not have any effect upon the measurement.
The optical properties used are µ0s = 10.4 cm−1 , µa = 0.5 cm−1 , and g = 0.83.
4.3.2
Monte Carlo Simulations
The effect of attenuation near an absorbing plane is shown experimentally in Fig.
4.6. Monte Carlo simulations are also shown which are confirmed by the experimental
results. All reflectances are normalized by dividing the perturbed signal by the reflectance
that would be measured in the same turbid medium when the fibers are far from any
boundaries. Fig. 4.7 shows Monte Carlo results of the 200 and 600 µm fibers indicating at
what distance the diffuse reflection drops in half due to a perfectly absorbing plane with
respect to independent variation of the absorption and scattering properties.
Monte Carlo simulations also show that the photon launch angle has no effect upon
the reflectance collected by the fiber as shown in Fig. 4.8. Thus, the photon launching
distribution is inconsequential to the amount of backscattered light collected by the fiber.
95
Diffuse Reflectance
0.020
Fit: A(1 - cos !)
0.015
0.010
600 µm fiber
A= 6.0
0.005
200 µm fiber
A= 2.0
0.000
0
10
20
30
40
50
Acceptance Angle [degrees]
Figure 4.9: The fraction of diffuse light collected by each fiber is dependent on the acceptance angle of the fiber θ in air, increasing with a 1 − cos θ dependence which follows
from a geometrical result if light returns to the fibers uniformly from all directions. A
is a scaling parameter to fit the relation. The photon launch angle distribution is held
constant and only the acceptance angle of the fiber is varied. The optical properties used
are µ0s = 10.4 cm−1 , µa = 0.5 cm−1 , and g = 0.83.
On the other hand, the numerical aperture of the fiber affects the amount of light
collected from the tissue. Figure 4.9 shows Monte carlo simulations which demonstrate
how the collected diffuse reflectance varies with the maximum acceptance angle dictated
by the fiber numerical aperture. The change in maximum half-angle of acceptance has a
1 − cos θ relation to the amount of light collected where θ is the half-angle of the maximum
cone of light propagated by the fiber.
4.4
Discussion
Monte Carlo simulations give information about the photons backscattered into the
illumination fiber; in particular, the fraction of light collected by the fiber increases with
96
the maximum angle of acceptance for the fiber following 1−cos θ. The 1−cos θ dependence
is the expected geometric result if the light returning to the fiber is uniform in all directions
at the fiber face. The fraction of light collected for this case is the ratio of the area of a
spherical cap bounded by the numerical aperture cone, divided by the area of a hemisphere,
2πr2 (1−cos θ)
2πr2
= (1 − cos θ). Furthermore, the angular distribution for which photons are
emitted from the fiber is independent of the fraction of light collected by the fibers as
shown in Fig. 4.8. This is a significant detail which indicates that complete mode filling
of a fiber is unnecessary to obtain consistent results, and implies that modal variation in
emission does not have any effect upon the measurement. Finally, the angular dependence
of both the emission and the collection of photons indicate that light returning to the fiber
is uniformly distributed over all directions.
Smaller fibers collect less light and are less sensitive to optical property changes. The
relationship between fiber size and sensitivity can be explained by crudely approximating
the light returning to the fiber by an isotropic point source, positioned one reduced mean
free path from the fiber face. If ρfiber = (µa + µ0s )Rfiber is the radius of the fiber in reduced
mean free paths, then the half-angle of the cone formed by the point source and the face of
the fiber is θc = tan−1 ρfiber . The fraction of light returning to the fiber can be estimated
by the solid angle subtended the face of the fiber relative to the point source divided by
4π steradians (all directions). This fraction reduces to the form,
2πmfp0 2 (1 − cos θc )
1
1
ρfiber
= (1 − q
)≈
2
0
2
4
4πmfp
ρ2fiber + 1
where mfp0 =
ρfiber
4
1
µa +µ0s
is the reduced mean free path. For this crude approximation when
drops below 1/12 then our system cannot resolve absorption changes less than 2 cm−1 .
A diffuse point source [17] cannot be used to estimate the amount of the returning light
to the fiber because the diffusion approximation is invalid at this distance. In general, as
the reduced mean free path increases, the signal collected decreases such that, for reduced
mean free paths greater than 500 µm, the 200 µm fiber can no longer resolve changes in
absorption smaller than a factor of two (e.g., absorption coefficients of 0.5 and 2.0 cm−1
are clearly distinct from 1.0 cm−1 ). The 600 µm fiber can resolve changes in absorption
by a factor of two for reduced mean free paths up to 1900 µm.
97
We found that the 200 and 600 µm fiber system gives optimal resolution when the reduced scattering coefficient is in the range 20–52 cm−1 , corresponding to a reduced mean
free path range of 190–500 µm. In this range, absorption coefficients of 0.1, 0.2, 0.5, 1.0
and 2.0 cm−1 are distinct and changes of 5 cm−1 in the reduced scattering coefficient are
resolved by this system. Tissues within this range at visible wavelengths include but are
not limited to aorta, brain, liver, lung and skin [44]. Differentiation and characterization
of tissue is ideally suited for UV/visible wavelengths rather than NIR/IR due to shorter
scattering pathlengths in the UV/visible range. When the reduced mean free path for
scattering is 500–1900 µm, only absorption coefficients of 0.1, 0.5, and 2.0 cm−1 are distinct. For reduced scattering mean free paths above 1900 µm, the absorption coefficient
resolution becomes larger than 2 cm−1 .
Additionally, we determined that larger fibers collect light from deeper in tissue than
smaller fibers. Fig. 4.6 shows that more than half the signal is from photons traveling 1.2
mean free paths into the medium for a 200 µm fiber and 1.9 mean free paths for a 600 µm
fiber. The depth for which half of returning photons travel is fairly insensitive to changes
in the absorption and the scattering coefficient as shown in Fig. 4.7. Pogue and Burke [27]
suggest that light being collected by a 200 µm fiber has an average of 1.2 scattering events
while the 600 µm fiber has 1.5 scattering events for µ0s = 10 cm−1 and µa = 0.1 cm−1 ,
which agrees well with our results, though our results are for an absorption coefficient of
0.5 cm−1 . Fig. 4.7 shows that the mean depth of photon travel does not change significantly
for absorption in the range of 0.05–2.0 cm−1 . Also, nearly 90% of all the collected photons
travel less than 800 µm (approximately 1 reduced mean free path) into the medium for
both fiber sizes. Effectively, both fibers are sampling the same depth of tissue; however,
a greater fraction of photons undergo a single scattering event before returning to the
200 µm fiber than for the 600 µm fiber. In other words, larger fiber diameters collect a
greater proportion of multiply scattered photons than smaller fibers.
This study demonstrates feasibility of a sized-fiber device using 200 and 600 µm fibers.
We have shown that measurements taken with this device on a Intralipid tissue model agree
with our Monte Carlo simulations over a range of absorption 0.1–2.0 cm−1 , and reduced
scattering, 5.2–52 cm−1 . However, further studies are needed. An inversion technique
98
relating optical properties to fiber reflectance has yet to be implemented. An empirical
relation between reflectance and optical properties would be preferable, yet the 200 µm
fiber signal is nearly independent of the absorption properties in the range of this study
and may be directly related to the reduced scattering coefficient. The 600 µm signal could
then be corrected for scattering to extract the absorption coefficient. With an inversion
technique, an in vivo study measuring optical properties with the sized-fiber device can
be performed. Furthermore, studies on anisotropy need to be addressed to determine if
the system can resolve tissues with the same reduced scattering coefficient but differ in
anisotropy and scattering coefficients.
Sized-fiber measurements have several advantages. The device is compact and measurements can be taken through a needle or endoscope. Real-time monitoring can be performed with the device using a broad spectral source which is best suited in the UV/visible.
System calibration only requires measurement of the Fresnel reflectance of air and water.
Finally, the device acquires localized information from a volume less than a cubic millimeter extending no more than a millimeter from the fiber face when absorption is resolved.
Chapter 5
Specular Reflection Problem with a
Single Fiber for Emission and Collection
5.1
Introduction
A single fiber may be employed to emit and collect light from an optically diffusing
medium such as a biological tissue. However, the light returned by the fiber consists
of three components: Illumination light, specularly reflected light from the surface, and
diffusely scattered light from within the tissue. The first component can be eliminated
by using a bifurcated fiber design. Of the remaining signal, only the diffuse reflection
contains the desired information regarding the optical absorption and scattering properties
of the tissue. Unfortunately, the specular component is comparable in magnitude to the
diffuse reflection for visible light. The refractive index mismatch between the fiber and
tissue account for a portion of the specular reflection. However, imperfect contact of the
fiber with the surface of tissue creates additional boundaries and thus additional specular
reflections. In this chapter, the effects of surface contact of the fiber face to the medium
are examined to reduce inter-measurement variation.
Sized-fiber reflectometry is founded upon the idea of using the same fiber for emission
and collection of light. The collected light is related to the absorption and reduced scattering of tissue. A sized-fiber reflectometer consists of two fibers with diameters of 200
and 1000 microns. Each fiber emits and collects its own backscattered light but the larger
Portions of this chapter were originally published under the title “The specular reflection problem with
a single fiber for emission and collection.” by T. P. Moffitt and S. A. Prahl in SPIE Proceedings of Saratov
Fall Meeting: Optical Technologies in Biophysics and Medicine, Vol. 5068, 2002.
99
100
Chopper
335 Hz
He-Ne Laser
632.8 nm
focusing
lens
detector
Lock- in Amp.
200 µm
fiber
Translation &
rotation stage mount
Sample
Figure 5.1: The experimental set-up consists of two 100 micron fibers that are end-coupled
to a 200 micron fiber to separate illumination from collection light. The photo on the right
shows the fiber mounted in the positioning stages over a first surface mirror.
fiber collects light that on average travels deeper into the tissue [130]. The absorption
and reduced scattering properties can be determined using a lookup table based on Monte
Carlo simulations of the diffuse reflectance collected by these fiber sizes.
The goal of this study was to determine the contribution of the specular reflection to
variation between measurements. We examine the effect of contact by a fiber with the
surface of a sample to elucidate its influence on the specular reflectance collected by the
fiber. Measurements of pure specular reflection are measured by independently varying the
angle of contact a fiber makes with a front-surface mirror. Next, the fiber was positioned
normal to the surface of the mirror to measure the effect of displacement between fiber
face and the surface. These two experiments were then repeated on a epoxy resin block
with diffuse scattering properties. Finally, a comparison is shown for the effects of specular
reflection collected using perpendicular polished fiber and a fiber with a beveled tip.
101
Rotational Experiment
Displacement Experiment
200
micron
fiber
200
micron
fiber
d0
9.5º
water
Rotation of
fiber angle,
θ
Surface of Sample
d
water
Surface of Sample
Figure 5.2: Diagrams of the geometry used for the angular rotation of the fiber (left) and
the displacement of a fiber form the surface (right).
5.2
Materials and Methods
A single fiber was bifurcated to separate the illumination light from the back-scattered
light [130]. A 200 micron fiber was used for emission and collection by being end-coupled
to a pair of 100 micron fibers mounted together in an SMA connector. The light source
consisted of a 632.8 nm He-Ne laser that was chopped (MC1000, Thorlabs Inc., Newton,
NJ) at 335 Hz then focused into one 100 micron fiber. The back-scattered light was
channeled through the other 100 micron fiber to a UDT silicon photodiode. A lockin amplifier (SR830, Stanford Research Systems, Sunyvale, CA) was used to display the
voltage produced by the photodiode. All fibers were fused silica glass/glass from Polymicro
Technologies LLC, Phoenix, AZ. The manufacturer cited a numerical aperture of 0.22 that
corresponds to a maximum exit angle of 12.7 degrees in air. The maximum acceptance
angle in air was measured to be 13 ± 1 degrees for the 200 micron fiber. All fibers were
encased in Teflon tubing and polished flush in SMA connectors.
Fiber positioning relative to the sample surface is achieved by mounting the 200 micron
fiber simultaneously to both a rotation and a translation stage. The sample is set onto
102
a lab-jack to allow coarse movement of the sample relative to the fiber. To test angular
effects, the angle is set with the rotation stage, then the fiber is lowered until the fiber just
touches the sample surface using a vernier translator (figure 5.2 left). To test displacement,
the fiber angle is aligned perpendicular to the sample and the fiber is brought into contact
with the surface using the vernier translator (figure 5.2 right). In all measurements, a
drop of water is placed on the sample to act as a coupling agent between the fiber and
the surface of the sample.
Two different samples were used for all measurements: a first surface mirror and an
epoxy resin block. The mirror returns only specularly reflected light. The resin block is a
solid sample with added absorber (India ink) and a scattering agent (Titanium dioxide).
The resin block has a large flat surface area that is smoothed with 400 grit sandpaper.
Thus the block has some specular reflection in addition to diffusely reflected light. The
optical properties of the block are similar to those found in tissue: µ0s ≈ 10/cm and
µa ≈ 1.0/cm at 632.8 nm.
Data points are normalized using Fresnel reflection from the fiber face in air and water.
The voltage returned from the photodiode is converted by
Total Reflectance =
Vsample − Vwater
[Rair − Rwater ] + Rwater
Vair − Vwater
where Rair = 0.03465 is the Fresnel reflectance for the fiber core/air junction, Rwater =
0.00204 is the Fresnel reflectance for the fiber core/water junction at normal incidence.
Vsample , Vwater , and Vair are the measured voltages for the fiber incident upon the sample,
water, and air respectively. The fiber core has an index of refraction of 1.457 at the
632.8 nm wavelength. Measurements of the air and water Fresnel reflectance are taken
periodically throughout the experiments to correct for any variations in light output.
Comparison measurements are made with fibers that are polished at a 65 degree angle
tip. The beveled fiber face reduces specular reflections therefore normalization is given by
Reflectance =
(Vsample − Vwater )
∗ (% reflection of standard)
(Vstandard − Vwater )
where Vsample , Vstandard , and Vwater are the detector voltages returned from the tissue
sample, a resin block standard, and the specular reflection for the fiber in water. Repeated
103
measurements were recorded for both probe designs on a resin block tissue phantom and a
single spot of skin on the dorsal side of the forearm. Water was used to couple light between
the fiber and the samples, though for measurements on the resin block no measurable
difference was observed with or without water. The fibers were held by hand in the
support mount during each measurement and the fiber was completely removed from the
surface of the sample and replaced between all measurements. The skin was marked with
a ball point pen by outlining the plastic fiber support to ensure proper alignment. A
maximum of five measurements were recorded in succession on the forearm to prevent
pressure induced erythema. Measurements were recorded for specular reflections from the
fiber face in air and in water. Specular reflection measurements were also made with the
fiber faces in contact to a first surface mirror using translation and rotation stages to
position the fibers.
5.3
Results
Measurement to measurement variation is demonstrated in figure 5.3 showing the
distribution of the total reflectance measured on an epoxy resin standard when the fiber
was held by hand and when the fiber was mounted (stationary). Typical measurement
variation for the forearm is shown in figure 5.4. The fiber was mounted in an SMA
connector to achieve measurement consistency. Direct pressure from a bare fiber caused
the signal to increase steadily in excess of 200 percent of the initial value over a period 30
seconds.
Figure 5.5 shows the contribution due to specular reflection when the central-axis of
the 200 micron fiber is not normal to a first surface mirror but remains in contact with
the mirror. Figure 5.6 shows the deviation in total reflectance when the central-axis of
a 200 micron fiber is not normal to a surface on an epoxy resin block, but the fiber still
maintains contact with the surface of the block.
Figure 5.7 shows contributions due to the specular reflection when the fiber is displaced
from the surface while remaining normal to the surface. Figure 5.8 shows the effect of
the specular reflection change due to a displacement of the fiber from the surface of an
104
20
Hand-held 200 micron fiber
Count
15
10
5
0
0
0.005
0.01
0.015
Total Reflectance
20
Fixed 200 micron fiber
Count
15
10
5
0
0
0.005
0.01
0.015
Total Reflectance
Figure 5.3: Distribution of 32 measurement of a diffusely scattering epoxy resin block with
a 200 micron fiber when the fiber is held by hand (Top) and when the fiber is mounted to
a translation stage and positioned in contact with the block (Bottom). The fiber is lifted
from the surface of the block between every measurement.
105
20
Hand-held 200 micron fiber
Count
15
10
5
0
0
0.005
0.01
0.015
Total Reflectance
Figure 5.4: Distribution of 32 measurement on a forearm with a 200 micron fiber that is
held by hand. The fiber is lifted from the arm between every measurement.
106
0.2
Specular Reflectance
Mirror
0.15
0.1
0.05
0
-16
-12
-8
-4
0
4
8
12
16
Angle [degrees]
Figure 5.5: The specular reflectance collected from a 200 micron fiber in contact with a
first surface mirror. The fiber’s optical axis is normal to the mirror at zero degrees.
107
0.015
Total Reflectance
Epoxy
0.01
0.005
0
-15
-10
-5
0
5
10
15
Angle [degrees]
Figure 5.6: The reflectance (specular and diffuse) collected by a 200 micron fiber in contact
with an optically diffusing epoxy block.
108
Specular Reflection
0.2
0.15
Mirror
0.1
0.05
! r2
fiber
!((2d +d )tan(9.5))2
0
0
0
0.5
1
1.5
2
2.5
3
Fiber-Mirror Separation [mm]
Figure 5.7: The specular reflectance collected with a 200 micron fiber displaced from the
surface of a first surface mirror(points) and the expected geometric result (curve).
109
0.01
Total Reflecatance
0.008
Normal to surface
0.006
0.004
14 degrees
25 degrees
0.002
Fresnel reflection for water/glass interface
0
0
0.5
1
1.5
2
Separation [mm]
Figure 5.8: The reflectance (specular and diffuse) collected with a 200 micron fiber displaced from the surface of a first surface mirror.
110
epoxy resin block while the fiber is normal to the surface, and at 65 and 76 degrees to the
surface.
5.4
Discussion
When making any type of measurement, the resulting measurement must be reproducible. Single fiber reflectance measurements are prone to significant variability if fiber
placement on the material is not carefully controlled. Figure 5.3 exemplifies this point.
When a fiber is carefully and precisely aligned (figure 5.3 bottom), the standard deviation
is less than 1.8 percent (mean = 0.011); less precise hand-held measurements (figure 5.3
top) have a standard deviation of 12 percent and an average value that is 17 percent
lower. The solid block is more sensitive to fiber holding than soft tissue (comparing figure
5.3(upper) to figure 5.4), because the soft tissue deforms and make better contact with
the fiber.
Tissue deformation is also a hindrance. Pressure induced by the contact of the optical
fiber causes skin to visibly blanch by driving the blood out of the immediate tissue. I am
unable to achieve a steady signal with a bare 200 micron fiber on skin. The fiber must be
mounted so that a larger area (≈ 7 mm2 ) of support is given around the face of the fiber.
Ideally, a fiber may be positioned such that the fiber face will be in perfect contact
with negligible pressure and any pressure fluctuations would have minimal effect. In less
than ideal circumstances, two problems affect measurements: displacement of the fiber
from the surface and a non-normal fiber angle to the surface. By using a first surface
mirror, these two effects can be described for the extreme case of pure specular reflection.
First, a small change in the angle less than 2 degrees from the normal causes a 2 percent or
less drop in collected light. At 4 degrees, the signal drops by 6.5 percent and at 6 degrees,
the signal drops to 83 percent of the value at normal incidence. Beyond 10 degrees, the
reflected light incident upon the fiber is outside the numerical aperture (0.22) of the fiber
and the collected signal is nearly equal to the Fresnel reflection due to the index mismatch
at the fiber face.
Displacement between the fiber and the surface of the sample has a similar effect. A
111
200 micron displacement causes a 3 percent loss in signal, 400 micron an 8 percent loss,
and 600 micron a 17 percent loss of signal. However, the signal does not drop off as
rapidly as a pure geometric solution would suggest. In a first approximation that assumes
that all modes are evenly filled (the modes were not perfectly even), the collected specular
reflection can be described by the ratio of the area of the fiber face relative to the spot
size reflected back in the plane of the fiber face (figure 5.2 right) giving the equation
Specular Reflection =
2
Cπrfiber
π((2d + d0 ) tan 9.5)2
where C is the maximal reflection from the mirror (C = 1), rfiber is the radius of the fiber,
d is the distance between the fiber face and the surface of the sample, and d0 is the height
of the triangle (600 micron) that forms the cone of light emitted from the fiber (figure 5.2
right). In water, the maximum half angle of emission of the fiber would be 9.5 degrees.
The discrepancy may be due to additional reflections off the meniscus of water back onto
the mirror being captured by the fiber. Uneven annular modes of propagation through
the fiber may also attribute to the discrepancy especially if the central modes carries
substantially less light than outer annular modes. Under-filling of modes will change the
fiber exit angle (to less than 9.5 degrees) and not change the general shape of the curve
and so this cannot be the sole explanation of the difference between the experiment and
the geometric model.
On measurements of the epoxy resin block, the results are similar to the mirror though
the deviations were not as pronounced. This was the expected result since the block
produces much less specular reflection than a mirror. The collected reflectance did not
drop to the surface Fresnel reflection value due to the presence of diffuse reflection. Figure
5.6 shows that the influence of the specular reflection is about the same magnitude as
the diffuse reflection. In figure 5.8, the change in specular reflection at normal incidence
almost equally offsets any changes in the collected diffuse reflection for separations less
than 400 microns. At large fiber angles (≥ 14 degrees), the specular reflection is not
collected and the diffuse reflection collected slowly decreases with increasing separation
between the fiber and the surface of the epoxy block.
In this study, we examined the effect of fiber contact on the collection of specular
112
reflection. When fiber contact between the face of the fiber and the surface of the sample
is imperfect, the collected specular reflection decreases. Changes in angle greater than 4
degrees or a separation between the fiber and the surface in excess of 400 micron causes at
least a 7 percent attenuation of the collected specular reflection. On diffusely scattering
media, the change in the specular reflection is nearly identical to the change in diffusely
collected light for fiber separations less than 400 micron. Any change in the fiber’s optical
axis from normal to the surface caused signals to decrease. Imperfect contact of any
sort between a fiber and the material caused the light collected to decrease. Therefore,
reproducible measurements using a fiber in this manner requires precise control of fiber
placement to avoid artificially low signals that may be falsely attributed to increased
absorption or a decrease in scattering.
5.5
Beveled optical fibers
The tip of an optical fiber may be angled to prevent the propagation of specular
reflectance back through the fiber. A set of experiments is presented to compare the
degree of specular reflection collected between fibers with a right-angle tip and a tip
polished to a 65 degree angle for both 200 and 1000 micron diameter fibers. In the first
experiment, the specular reflectance of each fiber is measured in air, water, and flush
against a first-surface silvered mirror. The specular reflectance is normalized by dividing by
the reflectance collected from the epoxy block as described above. In another experiment,
the measurement variation was evaluated on the epoxy phantom and on skin from the
dorsal side of the forearm using hand placement of all fibers.
5.5.1
Results
The relative specular reflectance collected by the 200 and 1000 micron fibers for a
perpendicular polished and beveled fiber polish is shown in figure 5.9. To make for a
better comparison, the specular reflectance signal from each material is divided by the
signal collected by the same fiber on a resin block standard that is roughly the same for
both fibers.
113
Perpendicular polish probe
10
1000!m
Fiber
200!m
Fiber
1
1000!m
Fiber
0.1
200!m
Fiber
Water
Air
Mirror
Water
Air
Mirror
Water
Air
Mirror
Water
Air
0.01
Mirror
Specular Signal / Standard
[A.U.]
Angled polish probe
Figure 5.9: Transmitted specular reflectance from the fiber face boundary is reduced for
the fibers polished at a 65 degree angle as shown by the ratio of the specular reflection for
each media divided by the signal from a epoxy resin block standard.
114
1000µm Fiber Reflectance
0.005
0.004
0.003
in vivo skin
0.002
tissue phantom
0.001
0
0
0.0005
0.001
0.0015
200µm Fiber Reflectance
Figure 5.10: Repeated measurements of a resin block tissue phantom (N= 12) and skin
(N=13) from the dorsal side of the forearm. Open points represent the angled polish probe
and filled points are for the perpendicular polished probe.
Typical distributions of repeated measurements of reflectance collected by the 200 micron
diameter plotted against the reflectance collected by the 1000 micron diameter fiber are
shown in figure 5.10 on an epoxy resin block tissue phantom and for in vivo skin. Though
the signal correction does not correct for the specular reflection offset, the relationship between measurements in a cluster does not change since the specular reflection is a constant
for each cluster. Clearly the cluster of 12 measurements for the angled polish probe pack
tighter than for the perpendicularly polished probe. The mean and standard deviations
are presented in table 5.1.
5.5.2
Discussion
Specularly reflected light behaves as a noise source since it does not contain any information about the absorption or scattering properties of the tissue sample being measured.
115
Fiber
diameter
Angled polish probe
Tissue phantom!
Skin
N = 12
N = 13
Perpedicular polish probe
Tissue phantom!
Skin
N = 12
N = 13
200 micron
0.00044 ±2.2%
0.00078 ±1.2%
0.00054 ±16%
0.00113 ±7.2%
1000 micron
0.00214 ±0.9%
0.00343 ±0.6%
0.00068 ±5.4%
0.00129 ±1.4%
Table 5.1: The mean and standard deviation (listed as a percentage of the mean) of the
distribution of repeated measurements. A value of 1.00 in the mean implies 100% of the
emitted light is collected.
Elimination of the specular light from the fiber face removes a portion of the signal variation. The specular reflectance is related to the contact between a fiber and the surface. It
was shown in figure 5.5 that the specular reflection signal from a perpendicular polished
fiber changes by 6.5% with only a 4 degree rotation of the fiber relative to the normal of
the surface being measured. The perpendicularly polished fiber probe collects a Fresnel
reflection signal in air of approximately 3.5%, 0.24% in water, and 92% from the first
surface mirror. This specular component is reduced 5 times with the beveled fiber-tip in
water and by an order of magnitude in air or on the mirror.
Inter-measurement variation with fibers polished at a 65 degree angle is also reduced
in comparison to perpendicularly polished fibers. We find that measurement variability
decreases 6 fold on in vivo skin and nearly 8 fold on a resin tissue phantom with the
200 micron fiber. On a solid resin tissue phantom, the difference between the highest
and lowest of 12 measurements is 6.5 and 2.4 percent for the 200 and 1000 micron fiber
respectively. On in vivo skin where the surface is compliant, the difference between the
highest and lowest of 13 measurements drops to 4.5 percent difference with a 200 micron
fiber while the 1000 micron fiber is 2.2 percent difference. The greatest contributing factor
for the decrease in variation is the elimination of specularly reflected light in the collected
signal.
Chapter 6
Interpretation of Two Fiber
Reflectometer Measurements
6.1
Introduction
Though measurements using a single emission and collection fiber minimize sampling
volume, the collected light also includes a specular reflection component due to a refractive
index mismatch between the fiber and the tissue. The specular reflections act as a noise
source that are similar in magnitude to the collected diffusely reflected light. Imperfect
contact between the fiber and tissue causes measurement variability, in part owing to
the specularly reflected light component leading to an inability to predict the optical
properties [131]. In this chapter, two dual-fiber probe designs are presented and evaluated
taking into consideration the results of the previous chapters. Though specular reflections
provide a simple and reliable way to calibrate the measured reflectance, the unpredictable
contact of the fiber to tissue leads to imprecise determination of the specular light collected
for each fiber. The designs of both probes remove the specular component by beveling
the face of the fiber. Given a sufficiently large bevel angle, light reflected internally at
the fiber face reflects to a new angular distribution that exceeds the numerical aperture
of the fiber. The specular light is then lost through the fiber cladding and is therefore not
propagated back to the detector.
Two dual fiber probe designs are presented that optimally collect diffusely scattered
light while rejecting specularly reflected light. Monte Carlo models for both probe designs
are described that are used to create grids of the collected diffuse reflectance for a range
116
117
of optical properties typical for tissues (absorption between 0.1–5 cm−1 and reduced scattering between 10–100 cm−1 ). The Monte Carlo models are used to evaluate the mean
optical path-length sample for each fiber in the two designs. Lastly, the effects of changes
in anisotropy with constant reduced scattering is evaluated using the models.
The first probe is a modified sized-fiber probe as described in chapter 4. The difference in diameter of the two optical fibers is increased to improve contrast between the
reflectance of each fiber. In addition, the original design used the specular reflectance of
water/glass and air/glass interface of the fiber to normalize the reflectance. This design
requires signal normalization using a diffuse reflection standard since specular Fresnel reflections are greatly reduced. The diffuse reflectance from a 4 by 4 array of tissue phantom
array is measured with the device to check the validity of a Monte Carlo model. Finally,
the absolute relative error distributions are calculated for the absorption and scattering
coefficients presented as a function of the absorption and scattering coefficients using a
linear differential approximation of the diffuse reflection collected by the 200 and 1000 µm
fibers.
The second probe consists of two 400 µm diameter fibers, one of which emits light while
both fiber collect diffusely reflected light. In contrast to the Sized -fiber device, empirically
determined relationships between the absorption and reduced scattering coefficients and
the collected light for the probe is presented based on the the results of the Monte Carlo
model. The accuracy of the empirical method is evaluated using measurements with the
probe on a polyurethane phantom characterized using the inverse adding-doubling method
from integrating sphere measurements.
6.2
6.2.1
Device Design
Monte Carlo Model
Description
A Monte Carlo program was adapted to simulate the light collected by a fiber irradiating a semi-infinite homogeneous scattering and absorbing medium. The geometry of
the fiber includes a beveled fiber face that affects the emission profile of light originating
118
B'
optical fiber
A'
Ph
oto
C'
Air
Tissue
n= 1.4
nl
au
nc
hin
gp
lan
e
65º
A
C
B
Figure 6.1: In the Monte Carlo model, photon bundles are launched from the last complete
plane within the fiber perpendicular to the optical axis of the fiber. Photon bundles are
launched with a truncated gaussian angular distribution with a cut-off at the angle with an
e−2 likely-hood to the central angle. The cut-off angle corresponds to the fiber numerical
aperture, which is 0.22 in this model. Photons are tracked to the face of the fiber and
the refracted into the tissue as shown for paths A and B. In the case of photon bundle C,
the path intersects the fiber wall before reaching the fiber face and is therefore undergoes
total internal reflection at the intersection. Paths A0 , B0 and C0 (dashed rays) show the
specular reflection path from the fiber face which leaks out the side of the fiber.
119
from the fiber and the collection of reflected photons back into the fiber. The face of the
fiber is beveled to a 65 degree angle. The face of the fiber is modeled in perfect contact
at the boundary of the medium with air surrounding the fiber as shown in figure 6.1.
To account for the asymmetry of the beveled fiber, photon packets are launched inside
the optical fiber at the last plane that is a complete cross-section of the fiber due to the
beveling of the fiber tip begins (see figure 6.1). In this plane, the launching of photons
uses the same spatial and angular launching distributions as for a perpendicularly polished
fiber. The launching position was spatially uniform across the cross-section of the fiberoptic of the initial launching plane. Each photon bundle is assigned a propagation vector
with a gaussian angular distribution centered about the fiber’s optic axis with the extreme
angle cut-off at the e−2 level of the gaussian. The width of the gaussian distribution is
defined using the maximum propagation angle within the fiber as dictated by the numerical
aperture (θmax = sin−1 (N.A./nfiber )) such that θmax corresponds to the e−2 level. The
initial angular distribution of trajectories for photon bundles in the x and z axes is shown
in figure 6.2. The photon packets travel along the remainder of the fiber tip until reaching
the fiber face and reflecting of the fiber wall whenever the path intercepts the fiber wall.
At the fiber face, the propagation vector is refracted into the semi-infinite medium (n
= 1.4 for tissue and nfiber = 1.457). Each photon bundle is initialized with a weight of
unity that corresponds to the energy in each bundle. The weight of each photon packet
is decreased by an amount equal to a Fresnel reflection at normal incidence as it crosses
from the fiber into the medium, introducing error that is less than 0.1%.
The tracking of discrete photon packets follows the sequence: photon packets are
initiated inside the fiber, undergo refraction as they enter the medium, then propagate
through the medium until either scattering back into the fiber, lost through emission
through the free space boundary of the medium, or are absorbed by the medium. Photon
bundles returning to the fiber face are refracted by the fiber giving the propagation vector
inside the fiber. The propagation vector is described by the cosine of the vector along
the x, y and z axes (direction cosines). The axis are defined so that the fiber face lies
is centered in the x-y plane at z=0 and z is positive going into the medium from the
fiber. The dot product of the photon’s propagation vector and a vector corresponding to
120
400
1000 photons launched
mean = -0.44 ± 0.10
Count
300
200
358/100000 photons collected
mean = 0.42 ± 0.07
100
0
-1
-0.5
0
0.5
1
Direction Cosine, x
400
Count
300
200
358/100000 photons collected
mean = -0.00 ± 0.08
100
0
-1
-0.5
0
0.5
1
Direction Cosine, y
400
1000 photons launched
mean = 0.88 ± 0.05
Count
300
200
358/100000 photons collected
mean = -0.90 ± 0.03
100
0
-1
-0.5
0
0.5
1
Direction Cosine, z
Figure 6.2: Bundles of photons are propagated using a propagational vector given by the
direction cosines defined as the cosine of the propagation vector with respect to the x,
y and z axes. The initial propagation vector is described a gaussian distribution with a
e−2 width set by the numerical aperture of the fiber and centered on the optical axis of
the fiber. For a 65 degree beveled fiber, the optical axis has direction cosine vector of
(0.423, 0, 0.906). The top figure shows the distribution of angle for launch photons and
collected photons by the direction cosine along the x-axis. The middle figure shows the
collected distribution direction cosine for the y-axis with the launched photons omitted
as the overlay the collected photons and has a similar angular spread as for the x-axis.
The bottom figure gives the launched and collected photon bundle distributions along the
z-axis which is perpendicular to the semi-infinite medium boundary.
121
the fiber’s central axis are compared to the cosine of θmax within the fiber to determine
whether the photon path was inside the numerical aperture and is transmitted through the
fiber. The angular distribution of trajectories for photon bundles collected by the fiber in
the x, y and z axes is shown in figure 6.2. Photons with trajectories outside the numerical
aperture count as light lost through the cladding of the fiber. Light incident upon the
boundary of the semi-infinite medium but outside the area of the fiber face is lost to
free space emission and no longer tracked, excepting for the fraction of energy specularly
reflected back into the medium due to refractive index mismatch at the boundary and the
air.
In each simulation of the Monte Carlo model, two million photons are launched for each
combination of optical properties giving adequate balance of accuracy and computation
time. Mean diffuse reflections and the standard deviations were calculated from five runs
of the model using different starting seeds for each combination of optical properties.
The absorption coefficient was varied from 0.1 to 5.0 cm−1 and the scattering coefficient
varied from 100 to 1000 cm−1 with an anisotropy of 0.9 using a Henyey-Greenstien phase
function. Additional details regarding the collected light are examined using the Monte
Carlo model. First, the weighted average path-length of the collected diffuse light by
Σ(photon weight ∗ pathlength)
Σphoton weight
is recorded for each pair of absorption and scattering coefficients. Second, several simulations examine the change in collected diffuse reflectance by varying the anisotropy while
keeping the reduced scattering and absorption coefficients constant (µ0s = 30 cm−1 and
µa = 0.2 cm−1 ).
Results
Monte Carlo simulation results for the angled polish probe design are shown in figure
6.3 for an array of optical properties with absorption coefficient values of 0.1, 0.2, 0.5,
1.0, 2.0, and 5.0 cm−1 and reduced scattering coefficient values of 10, 20, 30, 50, 75, and
100 cm−1 for 200 and 1000 micron diameter fibers. Each cluster represents a single reduced
scattering coefficient and reflectance decreases within each cluster as absorption increases.
0.004
µ ' = 100 cm-1
s
s
µ ' = 75 cm-1
s
0.006
µs ' = 20 cm
-1
-1
0.008
µ ' = 30 cm-1
0.01
µs ' = 10 cm
1000 µm Fiber
Diffuse Reflectance
0.012
µs ' = 50 cm-1
122
0.002
0
0
0.001
0.002
0.003
0.004
200 µm Fiber
Diffuse Reflectance
Figure 6.3: A reflectance map of optical properties in the range of tissues. Each cluster of
points represent a single reduced scattering coefficient where scattering increases for each
cluster as the reflectance increases. The error bars represent the standard deviation of five
runs of the Monte Carlo with a different starting seeds. Within each cluster absorption in
creases as the reflectance decreases.
An estimation of the optical properties using the Monte Carlo results of the in vivo skin
from figure 5.10 for the angled polish probe result in an absorption coefficient of about
0.2 cm−1 and a reduced scattering coefficient of 18 cm−1 at 633 nm which agrees with
values listed by Cheong [44].
Although I assume that in tissue the anisotropy is constant, figure 6.4 shows that the
collected reflectance changes with the anisotropy, g, when the reduced scattering coefficient
and absorption remain constant. The volume of the tissue (or other turbid medium) being
sampled is influenced by the optical properties. The weighted mean path-length for which
123
0.008
Diffuse Reflectance
Monte Carlo
0.007
1000!m Fiber
0.006
0.005
0.004
0.003
200!m Fiber
0.002
0.001
0
0
0.2
0.4
0.6
0.8
1
anisotropy, g
Figure 6.4: The diffuse reflectance collected by the 200 and 1000 micron fibers decreases as
the light becomes more forwardly scattered (higher g). Anisotropy values range from g=
0–0.99. The reduced scattering coefficient is held constant at 30 cm−1 and the absorption
constant at 0.2 cm−1 .
collected light travels is shown in figure 6.5.
Fitting Technique
To evaluate the absorption and reduced scattering coefficients a linear interpolation is
made using the grid of optical properties as shown in figure 6.3 for the diffuse reflectance
collected by 200 and 1000 micron diameter fibers. The change in absorption coefficient
(µa ) are related to a change in the reflectance measured by a 200 micron diameter fiber
(R200 ) and 1000 micron diameter fiber (R1000 ) given by
2
(∆µa ) =
∂µa
∆R200
∂R200
2
+
∂µa
∆R1000
∂R1000
2
.
Mean weighted Photon Path [mfp]
124
50
200 µm Fiber
40
30
-1
µs ' = 10 cm
-1
µs ' = 20 cm
20
µs ' = 30 cm-1
µ ' = 50 cm-1
s
µs ' = 75 cm-1
µs ' = 100 cm-1
10
0
0.1
1
-1
10
Mean Weight Photon Path [m.f.p.]
µ [cm ]
a
140
1000 µm Fiber
120
100
-1
µs ' = 20 cm
80
µs ' = 10 cm-1
µ ' = 30 cm-1
s
60
-1
µs ' = 50 cm
40
20
µs ' = 75 cm-1-1
µs ' = 100 cm
0
0.1
1
-1
10
µa[cm ]
Figure 6.5: The mean optical path through which collected light travels is presented as a
function of optical properties in terms of the number of mean free paths traveled (mfp =
1/(µa + µs ). At any particular absorption coefficient, the scattering coefficient increases
with the mean number of mean free paths traveled by collected photons for values of 100,
200, 300, 500, 750, 1000 cm−1 . For all simulations the anisotropy was fixed to g = 0.90.
125
1000
75-100%
900
800
>100%
µ s' [cm-1]
700
50-75%
600
25-50%
500
400
300
< 25%
200
100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
µ a [cm-1]
Figure 6.6: A contour plot of the absolute relative error in the absorption coefficient is
evaluated with a linear approximation of the Monte Carlo simulations in conjunction with
experimentally determined variation.
126
The partial of the absorption coefficient with respect to the reflectance for each fiber is
linearly approximated by dividing the difference of two adjacent absorption coefficients in
the Monte Carlo grid (µa,1 −µa,2 ) by the difference in by the respective reflectance collected
for each absorption coefficient by that fiber (e. g. R200,1 − R200,2 for the 200 micron fiber).
An absolute relative error map of absorption is presented in figures 6.6. The percent
relative error for absorption is calculated by
∆µa
µa
2
!2 =
µa,1 − µa,2
R200,1 − R200,2
+
µa,1 − µa,2
R1000,1 − R1000,2
∆R200
R200
!2 2 ∆R1000
R1000
R200
µa
2 2
R1000
µa
2
where the standard deviations of measurements on skin in table 5.1 as our relative error in
the 200 and 1000 µm fiber reflectance for the respective terms of
Finally, the terms
R200
µa
and
R1000
µa
∆R200
R200
and
∆R1000
R1000
.
scales the error into relative error. Similarly, the
percent relative error for scattering is calculated by
∆µs
µs
2
µs,1 − µs,2
R200,1 − R200,2
µs,1 − µs,2
R1000,1 − R1000,2
=
+
2 ∆R200
R200
2 2 ∆R1000
R1000
R200
µs
2 2
R1000
µs
2
with the analogous terms as for absorption but relating to scattering (not reduced scattering). The error contours are generated using MatLab (Mathworks, Inc., Natick, MA)
with a base 10 logarithmic contour spacing. The absolute relative error for the scattering
coefficient was 1.5 ± 0.2% over the entire range of absorption and scattering.
6.3
6.3.1
Measurements
Materials and Methods
The basic probe design (shown in figure 6.7) has previously been described [130] but,
a few key modifications are now introduced. The sized-fiber devices consist of 200 and
1000 micron diameter fibers bundled together. Emission and collection light is bifurcated
from the 200 and 1000 micron diameter fibers by end-coupling two 100 µm diameter fibers
onto the face of a 200 micron diameter fiber and 400 µm diameter fibers onto the face
127
of a 1000 micron diameter fiber using SMA connectors. The new probe has the 200 and
1000 micron diameter fibers polished at an angle of 65 degrees relative to the fiber’s central
axis (herein referred to as angled polish) rather than a 90 degree angle used by the previous
design (herein referred to as perpendicular polish). The 200 micron fiber with its core,
cladding and jacket intact is sleeved with the 1000 micron fiber with the jacket removed
but cladding intact into a snug fitting brass tube housing for protective strength. The
sleeved fibers are mounted into an SMA type fiber connector made from black Delrin
plastic. The fibers are polished within the SMA so that the SMA connector and the fibers
are polished to a 65 degree angle. A fiber support fabricated from Delrin plastic that is
1.25 cm in diameter concentrically fastens to the probe end of the two fibers via the SMA
connector. A diagram of the fiber tip and the fiber support is shown in figure 6.8 and a
picture of the two fibers mounted in the support is shown in figure 6.9). All fibers have
a 0.22 a numerical aperture and are fused silica glass/glass from CeramOptec industries,
Inc., San Jose, CA.
A He-Ne laser beam (632.8 nm) is split and focused into a 100 and 400 micron fiber that
respectively feed to 200 and 1000 micron fibers. Each beam is chopped (MC1000, Thorlabs
Inc., Newton, NJ) at differing frequencies (335 and 500 Hz) before entering the fibers.
The return leg of the bifurcation fibers are coupled to UDT silicon photodiodes (unbiased,
100 mm2 area). Lock-in amplifiers (SR830, Stanford Research Systems, Sunyvale, CA) are
used to record the signals using a 100 ms time constant.
Signal normalization is performed using measurements of a dark signal subtraction
and reference signal. The dark signal is a measure of the specular reflectance in water
obtained by placing the fibers into a beaker of water with walls painted flat-black. The
reference measurement is taken on a block or resin. Measurements are converted to percent
reflectance and corrected for a small degree of specular reflection by the equation
Reflectance =
(Vsample − Vwater )
∗ (% reflection of standard)
(Vstandard − Vwater )
where Vsample , Vstandard , and Vwater are the detector voltages returned from the tissue
sample, a resin block standard, and for the specular reflection for the respective fibers
in water. The absorption and reduced scattering properties of the resin block standard
128
50/50
B.S.
chopper
at 335 Hz
He-Ne laser
λ = 632.8nm
detector
chopper
at 500 Hz
detector
1000 µm
fiber
200 µm
fiber
Sample
Figure 6.7: This is a diagram of the sized-fiber device. A pair of bifurcated fibers emit
and collect light simultaneously by chopping at two distinct frequencies. The fibers are
mounted in tubing and have SMA connectors to connects the fibers.
129
Fibers in SMA Connector
200µm fiber
1000µm fiber
threaded connector
black plastic
Fibers support
with male
connector
65º
1.25 cm
Figure 6.8: This diagram shows a bisecting view of the 200 and 1000 µm fibers mounted
together in a black plastic SMA connector. A fiber support 1.25 cm in diameter connects
to the fibers via the SMA keeping the fiber concentrically positioned flush within the
support.
130
Figure 6.9: This is a picture of the the 200 and 1000 µm fibers attached to the fiber
support that aids in placement of the fibers. An end-view of the fibers is pictured in the
inset looking down the axis of the fibers to show the emitted light from the two fibers.
131
were determined by two methods, semi-infinite R(r) measurements as described by Farrell
et al. [17] and by total diffuse transmittance and reflectance integrating sphere measurements [83]. Both techniques resulted in an absorption coefficient of 0.1 cm−1 and a reduced
scattering coefficient of 10 cm−1 at 632.8 nm. The Monte Carlo model was used to determine the percent reflectance collected with the 200 and 1000 micron fibers for the optical
properties of the standard which are 0.00048 ± 0.00002 and 0.00192 ± 0.00002 respectively
(1.0 = 100%)
An array of tissue phantoms were measured with the angled fiber probe design to
check the validity of Monte Carlo simulations. The phantoms were constructed using
20% (20 g/100 ml) Intralipid (Liposyn II, Abbott Laboratories, North Chicago, IL) as a
scatterer and India ink (No. 4415, Higgs, Lewisburg, TN) as an absorber gelled in a 18%
(g/100 ml) acrylamide gelatin. The array was prepared using stock solutions of Intralipid,
ink and acrylamide with added water to make each phantom with a final volume of 100 ml
and dimensions approximately 4 cm deep and 5 cm in diameter. Aliquots of Intralipid were
used to make concentrations of 7, 5, 3.5 and 2.5%. The absorption coefficient of the India
ink stock solution was measured by a spectrophotometer (8452A, Hewlett-Packard, Palo
Alto, CA) and added to the Intralipid to give final absorption coefficients of 0.1, 0.4, 0.9
and 1.6 cm−1 at 632 nm. The acrylamide stock composed of 1.4 kg of acrylamide acid (99%,
electrophoresis grade, Fisher) and 35 g of bis-acrylamide (1:40 ratio, Fisher) in water to
create a final volume of 3.5 liters (40% concentration). The gelatin was set by adding 0.4 g
of ammonium persulfate (Fisher) and 100 µl of TEMED (Fisher) to each 100 ml sample
within the array. Scattering properties were determined after all gelatin samples had
set using total diffuse reflectance measurements with an 8 inch integrating sphere(IS-080,
Labsphere Inc., North Sutton, NH). The experimental and calculation details to determine
the optical properties of the phantom array were elaborated by Bargo [50]. Unlike the
absorption coefficient, the scattering properties were found not to be linear with respect
to the Intralipid concentration with final reduced scattering coefficients of 13, 18, 23 and
27 cm−1 at 632 nm in order of lowest to highest Intralipid concentration.
132
6.3.2
Results and Discussion
Measurements of an array of Intralipid and India ink gelatin phantoms are shown in
comparison to Monte Carlo simulations in figure 6.10 for the expected optical properties
of the array. The absorption coefficient of the array had the values of 0.1, 0.4, 0.9, and
1.6 cm−1 and the reduced scattering coefficient had the range of 13, 18, 23, 27 cm−1 at
632 nm where the anisotropy was modeled with g = 0.83 according to van Staveren et
al. [128].
The typical absolute relative error for predicting the absorption coefficient is roughly
50% (0.5 cm−1 ) for absorption coefficients of 1 cm−1 for reduced scattering exceeding
20 cm−1 . Below 20 cm−1 of reduced scattering the absorption error decreases since the
optical path-lengths increase, increasing the effect of absorption. Since the error is fractionally related to the absorption coefficient, the expected result is for the relative error to
increase dramatically especially as absorption coefficient drops below 1 cm−1 . The resolution of small changes in absorption is poor but major absorption changes will be resolvable
such as the presence of a blood vessel near to the face of the probe.
The precision of predicting small changes of absorption is not surprising when considering the mean path-length that the collected diffuse light travels. The longer mean
path-lengths occur with smaller scattering coefficients giving absorption a greater chance
to attenuate the collected signal. For absorption at 1 cm−1 and below the collected diffuse
reflection can be simplified using a Beer’s law approach where
Rcollected = R0 e−µa d .
In this absorption range, R0 varies less than 3% for reduced scattering coefficients between
10–100 cm−1 for the 1000 µm fiber. The longest mean pathlength of 2.24 mm occurs with
µa = 0.1 cm−1 and µ0s = 10 cm−1 which leads to a 2.2% attenuation of R0 . In contrast for
the optical properties with greatest absorption impact, µa = 1.0 cm−1 and µ0s = 10 cm−1
the absorption attenuation is 14% of R0 with a 1.55 mm path-length. The change in the
optical path-length is inversely related to the absorption coefficient making the effect of
absorption less pronounced in the collected signal.
The collected diffuse light is mostly influenced by scattering. Scattering coefficients
1000µm Diffuse Reflectance
133
0.005
0.004
0.003
error bars represent MC simulations
open points represent experimental data
0.002
0.0005
0.0007
0.0009
0.0011
200µm Diffuse Reflectance
Figure 6.10: Measurements of an array of semi-infinite tissue phantoms are compared to
Monte Carlo simulations with absorption coefficients of 0.1, 0.4, 0.9 and 1.6 cm−1 . The
error bars represent the standard deviations of five Monte Carlo simulations with different
starting seeds. Significant changes in the tissue phantoms scattering coefficient occur
with changes in hydration of the gelatin. Stability of the phantoms scattering coefficient
was compromised due to condensation on the phantoms while stored under refrigeration
resulting in the observed difference between the phantoms and the Monte Carlo model.
134
may be resolved with less than 2% error for reduced scattering coefficients between 10 and
75 cm−1 assuming that anisotropy is constant and known a priori. Distinctions between
two media with identical reduced scattering coefficients but differing anisotropy will be
observed. Without knowledge of the anisotropy error will be introduced to predicted
reduced scattering coefficient. Therefore, the sized-fiber device is sensitive to scattering
mean free path, and not the approximation of the reduced scattering coefficient but since
only two measurements are made the reduced scattering coefficient is utilized.
Sized-fiber reflectometry is not possible without controlling for specularly reflected
light at the probe’s fiber faces. The specular light was reduced by 50 and 15 times for the
200 and 1000 µm fibers respectively using angled fiber tip at 65 degrees. The experimental
variation of measurements on skin of 2.2% is obtained as shown in the previous chapter.
The improved signal reproduction gives the ability to determine scattering coefficients
with a relative error less than 2% assuming that the anisotropy is known for reduced
scattering coefficients between 10 and 75 cm−1 . The system resolves absorption with a
50% error for absorption above 1 cm−1 . Generally, the device is best suited for detecting
gross changes in absorption and ideally suited for the determination of highly localized
changes in scattering properties.
6.4
Dual-fiber probe with single emission fiber
In chapter 4, it was shown that single fiber emission and collection of its own backscattered light is predominated by singly scattered photons. Another dual fiber probe
design is now considered that uses a single fiber that emits and collects its back-scattered
light (emission fiber) and a second fiber (collection fiber) which is in contact with the
first fiber and collects light emitted by the first fiber. The collection fiber is expected to
collect light that travels over a longer average path-length than the 1000 micron diameter
fiber since light must travel laterally to get collected by the emitting fiber as confirmed
in figure 6.12. A longer path-length for collected light gives an increase in sensitivity and
resolution of absorption without increasing the diameter of the probe relative to the dual
sized-fiber design. The probe consists of a 400 micron diameter fiber to emit and collect
135
its back-scattered light. The second fiber only collects light emitted by the first fiber and
is also 400 microns in diameter. Since the fibers are polished at a 65 degree angle, the
arrangement of the fibers influences the light collection behavior by the collection only
fiber, but only one arrangement is investigated. The fibers are fixed in contact along the
short axis of the elliptical faces of both fibers. The geometry of the fiber faces is shown
in figure 6.11.
The design of the probe takes into account the findings in the previous sections of this
chapter. The two optical fibers are mounted in a clear acrylic rod for support, robustness
and reproducibility of signals. The relatively large area of the acrylic rod (6 by 7 mm
for the polished face) to the optical fiber faces reduces measurement variability due to
pressure effects during placement on tissues. The rod (6 by 6 mm) has two 430 micron
holes into the center of the face of the rod so that two holes barely overlap and are about
4 mm deep. From the other side, a single channel 1 mm in diameter is drilled down to
the two smaller holes. A counter bore hole is drilled for the teflon tubing that encases
the fibers between the probe tip and the SMA connectors at the other end of the fibers.
The fiber holes in the acrylic rod are filled with a light-cured medical grade epoxy (3211,
Dymax Corp., Torrington, CT). The polyimide jacket is burned off the ends of the fiber
with on open flame. The fibers are inserted through the uncured epoxy filled channels
until they extend out through each of the 430 micron holes. The fibers are then fixed by
curing the epoxy for 30 s with a fluid core light guide UV curing lamp (Dymax Corp.,
Torrington, CT) with the protective Teflon tubing set in the counter bore hole. Epoxy
is placed on the outside of the tubing and on the top of the rod and cured as before for
added strength. The acrylic rod and fibers are then polished together to a 65 degree angle
starting with 400 grit wet sand paper to remove the bulk material followed by 9 micron
and 0.3 micron wet lap paper to finish.
6.4.1
Monte Carlo simulations
The above Monte Carlo model is adapted to reflect this fiber probe. The changes reflect
the fact that only one fiber emits light, while the second fiber collects light originating from
the emission fiber. The launching of photons is therefore unchanged. A second boundary
136
Figure 6.11: Two 400 micron diameter fibers are mounted in contact and stripped of the
jackets down to the cladding. The fibers are polished at a 65 degree angle making the the
fiber faces elliptical. The fibers are aligned along the minor-elliptical axis of their faces. In
this probe design, only one fiber emits light while both fiber collect back-scattered light.
137
condition is added for photons incident upon the face of the second (collection) fiber. The
fibers are modeled to be in contact along the short axis of the elliptical face of the two
fibers with the fiber center to center spacing 430 microns apart to reflect the 15 micron
thickness of the cladding on each fiber. Collection of photons was unaltered except for
the addition of the second fiber. One million photons are launched for each combination
of optical properties. The absorption coefficients were 0.05, 0.1, 0.2, 0.5, 1.0, 2.0 and
5.0 cm−1 and scattering coefficients of 100, 200, 500 and 1000 cm−1 with anisotropy of
0.9 using a Henyey-Greenstien phase function. Mean diffuse reflectance and the standard
deviations were calculated from five runs of the model using different starting seeds for
each combination of optical properties. Finally, the weighted average path-length of the
collected diffuse light by emission and collection fibers was calculated.
6.4.2
Results and Disucssion
A common approach to derive the optical properties from measurements is to use interpolate values from maps generated by simulations (such as Fig. 6.13) or from maps
of measurements on phantoms with known optical properties [5]. However, here a different empirical model is presented that relates the collected diffuse reflectance for the
emission and collection fibers to the absorption and reduced scattering coefficient where
the anisotropy is 0.9. The reflectance for a 400 micron diameter emission fiber may be
approximated
− log10 (Remission ) = µa F (µs )/100 + F (µs )
(6.1)
where
F (µs ) = 4.91 − 0.96 log10 (µs ).
The base ten logarithm of the diffuse reflectance is shown for the emission fiber in figure
6.14 and the collection fiber in figure 6.15. The relative residual for this approximation is
shown in figure 6.16. A relationship of this form for the reflectance by collection fiber was
not satisfactory, producing residuals well in excess of 10% and continued to exhibit dependence upon either the absorption or scattering coefficient. This relation was determined
by noticing that for a single scattering coefficient, the data points could be collapsed onto
Mean Optical Pathlength [mm]
Mean Optical Pathlength [mm]
138
3
2.5
400µm collection fiber
2
µs ' = 20 cm-1
1.5
µs ' = 20 cm-1
1
0.5
0
µs ' = 100 cm-1
400µm emission fiber
µ ' = 100 cm-1
s
0.1
1
-1
µa [cm ]
3
2.5
2
1000 µm fiber
1.5
µ ' = 20 cm-1
s
µs ' = 100 cm-1
1
µs ' = 20 cm-1
0.5
200 µm fiber
0
0.1
µs ' = 100 cm-1
1
-1
µa [cm ]
Figure 6.12: The mean path traveled for light collected by two side by side 400 micron
diameter fibers is shown (top) in comparison to the mean path travel for the independent
emission and collection of 200 and 1000 micron diameter fibers (bottom) as determined
through Monte Carlo simulations. Three reduced scattering coefficients are shown 20, 50
and 100 cm−1 with the anisotropy fixed at 0.9. Error bars show the standard deviation of
the mean for five runs at each combination of optical properties. The collection fiber has
a 0.5 mm greater mean path-length for collected light than a 1000 micron fiber collecting
its own emitted light.
139
µs = 1000 [cm-1]
0.0015
µs = 500 [cm-1]
increasing µ
Rcollection
0.001
a
-1
µs = 200 [cm ]
0.0005
µ = 100 [cm-1]
s
0
0
0.002
0.004
0.006
0.008
0.01
Remission
Figure 6.13: A mapping of the reflectance for two 400 micron fibers where one fiber emits
and collects lights (Remission ) and the other fiber (Rcollection ) only collects light emitted
by the first fiber. Each cluster represents a single scattering coefficient and absorption
increases as the reflectance decreases within each cluster. Error bars are given by the
standard deviation of five runs of the Monte Carlo model for each pair of optical coefficients. The anisotropy is 0.9 for all simulations.
140
3.4
µs' = 100 [cm-1]
3.2
-log(Remission)
3
200 [cm-1]
2.8
2.6
500 [cm-1]
2.4
1000 [cm-1]
2.2
2
0
1
2
3
4
5
6
µ [cm-1]
a
Figure 6.14: The base ten logarithm of the diffuse reflectance for the emission fiber as
determined by a Monte Carlo model is modeled by equation 6.1 (lines).
141
4
-1
µs' = 100 [cm ]
3.8
200 [cm-1]
-log(Rcollection)
3.6
3.4
500 [cm-1]
3.2
3
1000 [cm-1]
2.8
2.6
0
1
2
3
4
5
6
µa[cm-1]
Figure 6.15: The base ten logarithm of the diffuse reflectance for the collection fiber as
determined by a Monte Carlo model. The lines represent a linear fitting lines similar to
equation 6.1 for the emission fiber, however the slope and offset do not have a simple
functional relationship to the scattering coefficient as does the emission fiber.
142
predicted Remission
0.01
0.008
0.006
0.004
0.002
0
0
1
0
0.002
2
3
-1
µ a [cm ]
4
5
6
2
Residual [%]
1.5
1
0.5
0
-0.5
0.004
R
0.006
0.008
0.01
emission
Figure 6.16: The relative residual error of the predicted reflectance to the reflectance
established by the Monte Carlo model is given by (Rpredicted − Rtrue )/Rtrue . The predicted
reflectance for the emission fiber is given by equation 6.1.
143
each other by the log-linear relation in eq. 6.1. F (µs ) was determined by fitting for the
slope and intercept coefficients of eq. 6.1 for each scattering coefficient. It was purely
coincidental that the slope and intercept were proportional. The predictive nature eq.
6.1 is limited at this point, in the sense that it has not been tested for other scattering
anisotropies (other than 0.9) or for other fiber diameters. This factor might be explored
as a logical next step.
A closed-form relationship for single fiber measurements to the absorption and scattering properties would provide a similar tool as the diffusion approximation does for large
fiber separations [17]. Equation 6.1 is used to determine the absorption coefficient given
the scattering coefficient and the emission fiber reflectance. The scattering coefficient was
determined from the mapping shown in figure 6.13 where the slope of a linear fit is related
to the scattering by
Rcollection =
(Remission + 0.006)
− 0.005
0.0012(µs + 1000))
(6.2)
where Rcollection and Remission are the diffuse reflectance collected by the collection and
emission fibers respectively for absorption coefficients less than 5 cm−1 . Figure 6.17 shows
the predicted versus true scattering coefficient as determined by eq. 6.2 and the residual
error. FIgure 6.18 shows the predicted absorption versus the true absorption for the case
where the scattering is known a priori and when predicted using eq.6.2.
The accuracy of equations 6.1 and 6.2 were further evaluated experimentally using a
polyurethane phantom created as described in chapter 2 and characterized using the single
sphere integrating sphere method described in chapter 3. Reflectance measurements were
normalized using a teflon block as a reference and a dark measurement subtraction given
by
Rphantom = Rref erence
Mphantom − Mdark
Mreference − Mdark
where Mphantom is the measured signal for the phantom, Mdark is the measured signal for
the probe in a black box and Mreference is the measured signal for the teflon block. The
teflon block’s absorption and scattering were also characterized using the same integrating
sphere method as the phantom. Monte Carlo simulations were then used to determine
the reflectance of the teflon block (Rreference ) for the emission and collections fibers based
144
-1
predicted µs [cm ]
1200
1000
800
600
400
200
0
0
200
400
600
800
1000
1200
1000
1200
true µs [cm-1]
20
Residual [%]
15
10
5
0
-5
-10
0
200
400
600
800
-1
µs [cm ]
Figure 6.17: The predicted scattering coefficient plotted against the true scattering coefficient (top) for the reflectance generated from Monte Carlo model using equation 6.2.
The relative residual error (bottom) of the predicted scattering coefficient is given by
(µ0predicted
− µ0true
)/µ0true
.
s
s
s
145
4
with µs known
-1
predicted µa [cm ]
5
3
2
1
0
-1
-2
0
1
3
4
5
4
5
true µa [cm-1]
5
4
with µs predicted
-1
predicted µa [cm ]
2
3
2
1
0
-1
-2
0
1
2
3
true µa [cm-1]
Figure 6.18: The predicted absorption coefficient plotted against the true absorption coefficient (top) for the reflectance generated from Monte Carlo model using equation 6.1
where the scattering coefficient is known exactly. The bottom plot is similar, except that
the scattering coefficient is predicted as shown in figure 6.17.
146
Optical Coefficients [cm-1]
20
15
10
Measured µs '
IAD determined µs '
5
IAD determined µa
0
600
650
Measured µa
700
750
800
Wavelength [nm]
Figure 6.19: The measured optical properties of a polyurethane phantom using the dual
400 micron diameter fiber probe and eqs. 6.1 and 6.2 is shown in relation to the measured
optical properties using inverse adding-doubling (IAD) with the single integrating sphere
method presented in chapter 3.
147
on those optical properties. A comparison of the predicted optical properties in relation
to the optical properties measured using the integrating sphere method is shown in figure
6.19. This method gives considerable cross-talk between absorption and scattering. In
addition, the error in the predicted scattering leads to a doubling of the error in the
predicted absorption as shown in figure 6.18. The error in the predicted absorption from
the measurements of the optical phantom is consistent with figure 6.18. Moreover, the
performance in absorption resolution is on par with the sized-fiber method that was shown
to optically sample roughly a 25% shorter optical path.
6.5
Conclusions
Two dual-fiber devices were presented with corresponding methods to invert optical
properties from diffuse reflection measurements, a sized-fiber probe that uses a 200 and
1000 micron diameter fibers both emitting and collecting light independently and a dual
400 micron diameter fiber probe with one emitting fiber and both fibers collecting reflected
light. Only minor differences exists in performance between the probes. The single emission fiber probe has more compact physical dimensions and has a slightly longer optical
sampling path-length (a 25-30% increase) relative to the sized-fiber probe. In the expected
range of tissue optical properties for the visible spectrum, the greatest mean optical pathlengths measured are 2.5 mm or less, restricting the resolution of absorption where 100%
error is possible. Absolute determination of absorption with either probe is poor for such
short path-lengths, yet the relative absorption spectral shape was demonstrated on a measurement of an optical phantom, even with cross-talk between the measured absorption
and scattering properties. The dual 400 micron fiber probe is a better design than the
sized-fiber probe for two reasons. First, it is a less complex design with a single source
fiber that is bifurcated for simultaneous emission and collection of light. Second, it has a
slightly larger optical sampling path and a more narrow overall diameter. However for the
inversion of optical properties, the linear interpolation of optical properties from a Monte
Carlo generated grid gives the ability to determine scattering coefficients with a relative
error less than 2% when the anisotropy is known a priori.
Chapter 7
Non-invasive Discrimination of Amalgam
and Melanin in Pigmented Oral Lesions
using a Dual Optical-fiber Probe
7.1
Introduction
Oral malignant melanoma has an estimated lifetime risk of 1/2,000,000 [132], but has a
5–year survival rate in the range 10–25% with a median survival of less than 2 years [133].
The poor survival of cases involving oral melanoma is complicated by several factors. Oral
melanoma generally exhibits few symptoms; pain, ulceration, or bleeding are rare until late
in the disease [134]. Though oral melanoma is rare, common benign pigmented lesions can
easily be confused with melanoma [135]. Pigmented lesions not recognized as a specific
entity such as an amalgam tattoo should be diagnosed by biopsy [136]. Buchner and
Hansen stressed that histologic diagnosis is required for most pigmented oral lesions [137].
One of the more common oral benign pigmented lesions is the dental amalgam tattoo.
A mass screening study of 23,616 predominantly white adults over 35 found melanotic
macules and oral nevi (moles) with incidences of 0.38 and 0.47 lesions respectively per 1000
persons [138]. Amalgam tattooing occurs when small grains of amalgam are embedded
into the mucosa. This is possible during restorative work when previous fillings are drilled
out or even when flossing between teeth with amalgam fillings. These tattoos may exhibit
a similar clinical coloration to melanin pigmentation and may also exhibit an uneven
border characteristic of melanomas. X-ray images are the primary method to confirm the
presence of amalgam in suspected tattoos. However, approximately only one quarter of
148
149
amalgam tattoos are visible by radiograph [135] and so the remainder require a biopsy
to confirm their identity and rule out melanoma. Amalgam tattoos are 2-10 times more
common than melanotic macules or oral nevi [139] but the true incidence is unknown [140].
The standard of care is to biopsy suspicious pigmented lesions due to the low survival rates
for oral melanoma. Since the number of amalgam pigmented lesions is comparable to the
number of melanin pigmented lesions, a method to non-invasively discriminate between
these categories of pigmented lesions could greatly reduce the number of biopsies required,
saving patients from additional pain and cost of the biopsy.
Techniques based on diffuse reflectance spectroscopy have shown promise to detect
tissue morphological changes and distinguish between cancerous and normal tissues [10–
13, 141–144]. In skin, diffuse reflectance spectroscopy has been used to measure chromophore concentrations using absorption profiles [145]. Diffuse reflectance spectroscopy
can be performed non-invasively and requires minimal equipment. Other methods could
be adapted to differentiate amalgam and melanin pigmentation in the oral cavity such as
confocal microscopy [146] and energy dispersive X-ray microanalysis [147]. However, such
equipment is considerably more costly than diffuse reflectance spectroscopy.
Dental amalgams vary widely in their exact mixture but they are composed of primarily silver (40–70% by weight), tin (15–30%), copper (12–30%) that is then mixed with
roughly an equal weight of mercury. Modern conventional amalgams are considered high
copper amalgams while some of the older amalgams used a low copper content amalgam
of less than 6%. Other metals have been observed in trace quantities including indium,
cadmium, lead, and antimony in various amalgams [148] while zinc may be present in
quantity less than 1%. Corrosion of amalgam in the oral environment leads to a large
host of possible products. Silver and mercury are least likely to electrochemically dissolve
but may form nearly insoluble sulfide layers [149]. Tin forms mostly soluble ionic compounds while zinc is thermodynamically the most active component of amalgam followed
by copper in dissolution [149]. However, amalgam tattoos react differently embedded in
tissue than amalgam in the oral environment. Large particles are encapsulated and little
breakdown occurs to the amalgam [150], while fine particles are digested by macrophages
150
and giant cells resulting in the progressive loss of tin and mercury leaving behind fine particles containing silver and sulphur [150, 151]. The age of the amalgam tattoo affects the
materials in a tattoo; younger tattoos will contain degradation products and the oldest
tattoos contains only particles of fine silver and sulphur and possibly large particles of
almost unaltered amalgam.
The goal of this study was to evaluate the functionality of a dual-fiber diffuse reflectance
probe to differentiate oral pigmented lesions caused by amalgam particles or by melanin.
The results of a pilot study are reported using reflectance spectroscopy to differentiate
measurements obtained from pigmented oral lesions and nearby non-pigmented sites from
30 patients. An example of an amalgam tattoo is shown in figure 7.1 along with the corresponding radiographic evidence for the presence of amalgam for one subject enrolled in
the study. In contrast, figure 7.2 shows a typical subject with natural racial pigmentation.
For all subjects, clearly defined borders between pigmented areas and the non-pigmented
areas were observed. Melanin pigmentation exhibited a wide range of coloring in the oral
cavity similar to the range observed in skin pigmentation.
151
Amalgam
tattoo
Amalgam
Figure 7.1: Above: An amalgam tattoo on the alveolar ridge in one subject is found
where a tooth has been extracted leaving a bluish-black lesion as indicated by the arrow.
Below: The radiograph of the above subject clearly showing a large cluster of amalgam
particles emedded in the soft tissue. Large amalgam particles may be seen in radiographs
whereas fine particles are not always observable, therefore the dimensions of the tattoo in
the picture do not match the radiograph image.
152
Melanin site
Unpigmented
site
Figure 7.2: Measurements of melanin sites were performed on subjects with obvious racial
pigmentation. This typical subject had melanin along the gum line and highly delineated
non-pigmented area nearby which is measured as a control site. The darkness of the
melanin pigmentation varied widely among subjects as well as the intra-subject variation
shown here. The melanin is visibly more concentrated at the point of the arrow for the
melanin site than to the left of the arrow.
153
7.2
7.2.1
Materials and methods
Study population
Thirty volunteers participated in the study that were recruited from the patients receiving dental care through the Oregon Health & Science University (OHSU) Dental School.
The study was approved by the Institutional Review Board at OHSU. Potential subjects
were approached if they exhibited natural melanin pigmentation of the gums (which is
sometimes present in individuals with darker skin color) or had radiographic evidence of
an amalgam tattoo that corresponded with a pigmented oral lesion. All volunteers were
informed of the details of the experiment and their consent was obtained prior to the experiments. Half of the subjects had an amalgam tattoo, while sixteen subjects had obvious
melanin pigmentation. One subject had both racial pigmentation and an amalgam tattoo,
and was included in both groups of measurements. Volunteers were then taken to a dental
examination chair in the clinic with the spectroscopy cart. An oral pathologist placed
the fiber-optic probe onto the surface of the oral lesion and on nearby non-pigmented site.
Three measurements were recorded at each site. After recording data from the subject, an
additional measurement of a teflon block was made for use as a normalization reference.
7.2.2
Device
A spectroscopic system (fig. 7.3) was used to collect diffuse reflectance spectra from
the oral mucosa in vivo. The system included a xenon arc lamp (6251, Oriel Instruments,
Stratford, CT), with a water filter to reduce longer IR wavelengths while providing a
broadband light source, a fiber optic probe that directs light to the tissue and collects
diffusely reflected light at two positions, and a silicon-diode array grating spectrometer
(1451 photodetector and 1471 controller, EG &G Corp., Fremont, CA) for light detection.
The detector was purged with dry nitrogen gas to prevent water condensation on the
detector. A fiber optic probe similar to the differential-path-length fiber probe of Amelink
et al. [41] consisted of two optical fibers at the probe-face. One fiber both emits and collects
light while the second fiber collects light emitted by the first fiber. All fibers (CeramOptec
industries, Inc., East Longmeadow, MA) were 400 microns in diameter with a numerical
154
aperture of 0.22. The emission fiber was bifurcated fiber by end-coupling two fibers to a
single fiber to allow for simultaneous emission and collection of light as described in section
6.4. Light was intentionally lost while coupling between fibers at the bifurcation by using
fibers of the same diameter. The loss of light kept the magnitude of the collected light by
the emitting fiber comparable to the collection-only fiber for measurements on the block
of Teflon used as a reflectance standard. This allowed both the emission and collection
fibers to use a comparable dynamic range while being switched into a single detector. The
fibers were encased in Teflon tubing for robustness and the distal end of the fibers were
fixed into a clear square-cylinder of acrylic then polished together to a 65 degree angle to
the fiber-axis to prevent collection of specular reflected light. The probe tip was placed in
contact with the tissue after the surface was patted dry with sterile gauze.
The diffuse reflectance of oral pigmented lesions and adjacent non-pigmented (normal)
sites were collected with this system from 415–800 nm every 0.5 nm. The spectral resolution was 7 nm FWHM. The detector integration time was 0.6 s for all measurements. For
most subjects, three measurements are recorded for both fibers on each site. The probe
was placed by an oral pathologist on a pigmented oral lesion six times (for three consecutive measurements with each fiber) and then on a nearby non-pigmented site (6x). After
each patient was measured, measurements on standards were recorded. These include a
measurement on a white Teflon block 2.4 cm thick and 2.5 cm in diameter (Mstandard ) and
a dark measurement by placing the probe tip in a black box (Mdark ). The reflectance,
R(λ) was normalized by
R(λ)[a.u.] =
Mtissue (λ) − Mdark (λ)
Mstandard (λ) − Mdark (λ)
where Mtissue is the measured reflectance of the tissue. The probe was disinfected between
patients by soaking in Cidex OPA(Johnson & Johnson) disinfectant followed by a spray
rinsing of ethanol.
7.2.3
Amalgam characterization
The dominant chromophores in skin in the visible spectrum are hemoglobin (both oxygenated and deoxygenated) and melanin [152]. The absorption properties of hemoglobin
155
Figure 7.3: The device consists of two 400 micron fibers that are held together in an acrylic
housing. One of the the fibers is bifurcated by end-coupling two 400 micron diameter fibers
to a single 400 micron diameter for illumination and collection (the emission fiber) while
the second fiber (collection fiber) collects light emitted by the emission. The fibers are
switched into a spectrometer to record diffuse reflection for both fibers sequentially with
a 0.6 s integration time. All optical fibers are encased in teflon tubing for robustness.
156
[153, 154] and melanin [155, 156] are well-documented. Dental amalgam is the exogenous
chromophore causing the tattoos of interest in this study, but whose absorption properties
are undocumented. Absorption by amalgam particles was characterized for three common alloys of amalgam: New True Dentalloy, Tytin and Dispersalloy. Each ot the three
amalgam have differing ratios of metals. The three alloys were mixed and pressed into
pellets. After 1 week, filings of each alloy were taken from the pellets allowing sufficient
time for all chemical interactions in the amalgam. The fillings were placed between glass
microscope slides with 0.15 mm cover-slips used as spacers. A reference blank slide was
prepared without particles for signal normalization. All slides were cleaned with ethanol
prior to use to remove any extraneous dust. Total diffuse reflection and transmission measurements were made with a double integrating sphere experiment since the particles both
scatter and absorb light.
A pair of 203.2 mm diameter integrating spheres (IS-080-SF, Labsphere, Inc., North
Sutton, NH) with a spectraflect coating were used for all measurements. The reflectance
sphere had three open ports: a 6.3 mm port for the entrance, 19.1 mm sample port and
a 12.7 mm detector port. A 1 mm diameter fiber (PWF1000T, CeramOptec Industries,
East Longmeadow, MA) guided light from the detector port to a scanning grating photomultiplier tube detector in a spectrofluorometer (Fluorolog-3, Instruments S. A.,Inc., Edison, NJ); this fiber was used for collection from both spheres in sequential measurements
of reflection and transmission. Spectra were recorded for visible wavelengths 400–800 nm
in 1 nm increments with a 5 nm band-pass. A 600 micron optical fiber delivered white light
from a tungsten lamp (LS-1, Ocean Optics Inc., Dunedin, FL) with a 5 mm diameter spot
size. The fiber was held by a 4 mm steel tube painted with flat white spray paint inserted
through the entrance port. The transmission sphere used identical port dimensions as
the reflection sphere for the sample and detector ports, but no open entrance port. The
relative absorption with respect to wavelength was determined using the combined inverse
adding-doubling and Monte Carlo model. Results were considered to be relative since the
concentration of amalgam particles was not precisely determined in the optical path of
the beam nor was the particle distribution uniform across the sample port.
157
4
Absorption coefficient [cm-1]
10
3
10
melanin
2
10
deoxy-Hb
1
10
0
10
oxy-Hb
!1
10
!2
10
water
!3
10
!4
10
400
500
600
700
800
900
1000
Wavelength [nm]
Figure 7.4: The absorption coefficients for the chromophores in tissue are used to fit to
the spectral shape of the fiber measurements made with the collection fiber to determine
if other optical characteristics of the measured spectra beyond melanin absorption can
discriminate among sites.
7.2.4
Data Analysis
Spectral analysis was applied to the collection fiber data to determine the fraction of
blood and melanin, the oxygen saturation using the absorption spectra of these components and evaluate the scattering properties from the shape of the measured reflectance.
These parameters would then be examined to see which may be used as indicators to
discriminate between sites. The absorption spectra for each component is shown in figure
7.4. A wavelength independent instrument scalar and a baseline offset were included into
the fit to allow for variations due to contact variation of the fiber to the tissue but do not
158
alter the spectral shape of the fitting in of themselves. The reflectance measurements of
the oral sites varied widely even within a single subject and compensation is necessary for
a better fitting of the measurements. Kollias and Baqer [155, 157] showed that melanin
concentration in skin may be estimated by the spectral region of 620–720 nm. The concentration of melanin was shown to be proportional to the slope (m) of a linear fit of the
negative logarithm of the reflectance spectra in between 650-750 nm given by
− log10 R(λ) = mλ + b.
This linear relationship is then applied to obtain the melanin fraction as was shown by
Jacques [158]. The melanin concentration M was then
M = (m − mave,
where mave,
nonmelanin
nonmelanin )
∗ 100
= 0.00033 OD/nm was the average slope for all the sites without
melanin which indicates the slope in the absence of melanin pigmentation. Typical values
for m in melanin pigmented sites were 0.0015 OD/nm. The proportionality constant of
100 was not established experimentally but rather a value was chosen arbitrarily which set
the melanin fraction to values that are published values of the melanin fraction in darker
skinned individuals [158]. A 20% change to the proportionality constant did not affect the
other fitting parameters other than the scattering.
A minimization routine was used to reduced the absolute error between the measurement and a predicted measurement based on the diffusion approximation in the wavelength
range 450–725 nm. The predicted measurement was evaluated using the diffusion equation to get a reflectance between points spaced 200 microns apart with a melanin term
separated out given by
predicted Measurement = Ke−M µa,melanin D Rdiffusion (µa,net , aµ0s ) + b
where fit values were K the instrument scaling factor, a a scattering coefficient scaling
factor, and b a baseline offset. The melanin concentration M was determined as described
above to prevent cross-talk between scattering scaling factor and the melanin concentration. The absorption spectrum of melanin was given by
µa,melanin = 6.6 ∗ 1011 λ−3.33
159
and a constant path-length factor D = 0.0060 cm was used. Rdiffusion is the diffusion
model equation with an extrapolated boundary given by Farrell [17] using an absorption
coefficient given by
µa,net = B(Sµa,oxy−Hb + (1 − S)µa,deoxy−Hb ) + 0.7µa,water
with fit parameters B the fraction of blood and S the oxygen saturation used in conjunction
with the absorption coefficient for oxygenated hemoglobin (µa,oxy−Hb )and de-oxygenated
hemoglobin (µa,deoxy−Hb ). The absorption spectrum of water assumed a constant concentration at 70% in tissue. The scattering coefficient µ0s was a linear combination of Rayleigh
and Mie scattering terms given by
µ0s = 1.4 ∗ 1012 λ−4 + 4.59 ∗ 103 λ−0.913
where λ in the wavelength in units of nm. The measured reflectance was scaled as a
function of wavelength by the reflectance that would be collected by the collection fiber
on the teflon standard Rteflon . This value was determined by calculating the absorption
and reduced scattering properties of the teflon by integrating sphere measurements and
then calculating the expected reflection by the collection fiber using the Monte carlo model
using a refractive index of 1.35 for the teflon at all wavelengths.
Two approaches were employed to analyze the reflectance spectra, each to answer
separate questions regarding the pigmentation rely upon melanin absorption as a means
to distinguish sites.. The first approach focused on the quantity of melanin as observed in
the reflectance spectrum. As previously stated, Kollias and Baqer [155, 157] showed that
melanin concentration in skin may be estimated by the spectral region of 620–720 nm.
The concentration of melanin was shown to be proportional to the slope (m) of a linear
fit of the negative logarithm of the reflectance spectra in that wavelength range given by
− log10 R(λ) = mλ + b.
I found that absorption by oxy-hemoglobin affected the reflectance spectra in our probe
measurements below 640 nm. Therefore, a linear fit was used to establish the value of m
for each measurement (both emission and collection fibers) between 640 and 720 nm.
160
The values of m were then separated into 4 groups: melanin sites, non-pigmented
sites on melanin subjects (melanin controls), amalgam sites and non-pigmented sites on
amalgam subjects (amalgam controls) as seen in table 7.1. Sites that contain melanin
pigmentation were determined by using a threshold value of the slope. A Receiver Operator Characteristic (ROC) [159] was determined for measurements with the emission
and collection fibers independently for different threshold values. Measurements from all
4 groups were included giving an N= 48 for melanin sites and N= 132 for non-melanin
sites. The sensitivity at each threshold was calculated by
Sensitivity =
ΣTrue positive
ΣTrue positive + ΣFalse negative
and the specificity by
Specificity =
ΣTrue negative
.
ΣTrue negative + ΣFalse positive
True positives are defined as melanin sites identified as containing melanin, false negatives
are melanin sites identified as containing no melanin pigmentation, true negatives are
amalgam and unpigmented sites identified as containing no melanin pigmentation, and
false positives are amalgam and unpigmented sites identified as containing melanin.
Discriminant analysis was used as a second alternative analysis to classify measurements into one of three groups: melanin pigmentation, amalgam pigmentation, or nonpigmented. For the discriminant analysis, the reflectance spectra were modified to simulate
the reflectance measurements that would be produced using several spectral filters, white
light, and a broadband absorbing detector (e. g. silicon). The spectral filters were centered
every 25 nm starting at 450 nm and extending to 700 nm for 11 simulated measurements
(Fig. 7.5). Each filter was modeled as a normalized gaussian function (area under the
gaussian equal to one) with full width at half max (FWHM) of 12 nm given by
−
T (λ) = Ce
(λ0 −λ)2
2σ 2
where C is a normalization constant, λ0 is the centroid wavelength, and σ = 5 nm. The filter functions were multiplied by each of the reflectance spectra recorded from the subjects.
This effectively generated a set of 22 measurements at each site, 11 different wavelength
bands and collection by both fibers.
161
0.05
Transmission [au]
0.04
0.03
0.02
0.01
0
400
450
500
550
600
650
700
750
Wavelength [nm]
Figure 7.5: Each fiber measurements was multiplied by the 11 gaussians shown here to
simulate the collection of light using spectral filters. Each filter is a gaussian with a 12 nm
at FWHM and normalized to have an area under the curve of one.
162
The simulated spectral filter data was split into a classification group (roughly 2/3 of
all data) and a test group (the remaining 1/3). All measurements for a single subject
were included together in a single group. To prevent bias in selection, the test group
consisted of every third subject for the cases with melanin pigmentation and every third
with amalgam pigmentation. The number of melanin subjects was not divisible by three
and so the one extra subject was included in the test group. In total, the classification
group consisted of 10 subjects with an amalgam tattoo, 10 subjects with natural melanin
pigmentation and 20 non-pigmented sites on those subjects. The test group consisted
of 5 subjects with an amalgam tattoo, 6 subjects with melanin pigmentation and the
corresponding non-pigmented sites for those subjects. The classification group was used
to determine the discriminant functions and which variables (spectral filters) to include
or exclude.
It is expected that some of the wavelength measurements will be dependently related to
each other and therefore will not improve the discriminant model. The reduction of redundant data was performed using only data in the classification group. All statistical tests
were performed using SPSS software. The reflectance populations at each wavelength band
were tested for normality, an assumption of discriminant analysis. Discriminant functions
were calculated with the software giving weights to each input variable (wavelength bands)
and showing the classification table for those functions (eg. table 7.1). An organized trial
and error approach was employed to determine the best discriminant functions. The first
discriminant functions used only the wavelength bands where the reflectance was normally
distributed. A second set of discriminant functions were calculated using a subset of inputs
where the lowest weighted (≤ 5%) inputs were excluded. The process was repeated to find
the minimum number of wavelength bands that provided the best classification of only the
melanin sites (amalgam and non-pigmented sites were not yet considered). Separation of
the melanin sites from the other sites required just two reflectance measurements by the
collection fiber.
Next, the process was repeated to evaluate the wavelength bands that best differentiate
the amalgam sites from the non-pigmented sites using a two-group discriminant analysis
omitting melanin site data. The first discriminant function used all normally distributed
163
wavelength bands. New discriminant functions were determined eliminating the least
contributing inputs (≤ 5% weight). This step was repeated while correct classification
of the two groups did not drop. This led to the identification of six measurements that
provided the best discrimination between amalgam and non-pigmented sites.
Of a possible 22 inputs measurements, eight measurements were identified as key to discriminating the three groups. Classification using those eight measurements was compared
to classification with discriminant functions that included a ninth input measurement.
Each of the omitted measurements was individually included with the eight key measurements and classified with discriminant analysis. Finally, classification was performed using
all combinations with 1 of the 8 measurements removed. The correct classification rate of
the three groups was reduced whenever a measurement was removed. Classification was
unimproved with a ninth measurement.
Classification in one of the three groups is decided by using two discriminant functions
to score each site. The first discriminant function (F1 ) separates the melanin sites from
the non-pigmented sites and amalgam sites,
F1 = −1.72 − 35.61Rem,
+ 19.02Rcoll,
500
475
+ 65.34Rem,
− 24.57Rcoll,
525
500
− 27.35Rem,
+ 80.75Rcoll,
650
550
+ 6.539Rcoll,
− 76.86Rcoll,
475
675
where Rem is an emission fiber reflectance, Rcoll is a collection fiber reflectance and the
number in the subscript denotes the centroid wavelength.
The second discriminant function then divides the non-pigmented sites from the amalgam sites and is given by
F2 = −2.16 − 3.605Rem,
− 3.906Rcoll,
500
475
+ 29.03Rem,
− 6.306Rcoll,
525
500
− 23.52Rem,
− 35.55Rcoll,
650
550
+ 0.1358Rcoll,
+ 45.32Rcoll,
475
675
A scatter plot of all measurements in the classification group is shown in figure 7.6.
The measurements in the test group are evaluated by the discriminant function scores.
Measurements are then classified by the nearest group centroid from the classification
group set. The centroids of the melanin sites, amalgam sites and non-pigmented sites are
respectively: (F1 , F2 ) = (−2.667, 0.236), (0.315, −1.049), and (1.222, 0.402).
164
7.3
Results
The optical properties of dental amalgam must be different from those of melanin in
order to differentiate lesions. The absorption properties of three formulations of dental
amalgam are shown in figure 7.7. Absolute absorption coefficients could not be established
from our measurements but all three amalgams exhibited a spectrally uniform absorption
from 400–800 nm.
All the measurements for both fibers are shown in figure 7.8 separated by the site
pigment type. The mean reflectance spectra obtained from all subjects is shown for the
emission fiber in figure 7.9 and the collection fiber in figure 7.10. The reflectance for any
individual measurement is similar to the mean reflectance for its site type. Though the
standard deviation for each site type is large relative to the difference between the means,
some observations were consistent within an individual subject. The non-pigmented sites
produce the highest measured reflectance in measurements with both fibers. The absorption of light by hemoglobin (producing valleys at 550 and 570 nm) is obvious in the
reflectance by the non-pigmented sites and by the amalgam sites though to a lesser degree.
In contrast, the absorption by hemoglobin is almost completely obscured in measurements
at melanin sites with the emission fiber. The most distinguishing feature between reflectance collected on amalgam and melanin sites is the change in the reflectance slope
with respect to wavelength.
Component spectral fitting was performed on the data for the collection fiber to ascertain which known variables might be useful for discrimination. Typical fit spectra are
shown with the measurement being fit in figure 7.11. Five parameters were allowed to
vary in the fitting, the fraction of blood, the blood oxygen saturation, a total reflectance
scalar, a scattering scalar, and the baseline offset. A handful of measurements did not fit
well; these measurements exhibited much less blood absorption or reflected blue light and
red light nearly equally well. Histograms of the the spectral components parameters are
shown in figures 7.12, 7.14, 7.15, and 7.16. The oxygen saturation tended to have lower
values with drop in the fraction of blood and was likely due to inability to resolve with a
small hemoglobin signature in the reflectance measurement. The method for establishing
165
Actual
count
Actual
percentage
Group
non-pigmented
melanin
amalgam
non-pigmented
melanin
amalgam
Predicted
non-pigmented
27
0
7
90
0
46.7
Predicted
melanin
0
17
0
0
94.4
0
Predicted
amalgam
3
1
8
10
5.6
53.3
Total
30
18
15
100
100
100
Table 7.1: Classification table of measurements from the test group as predicted using
discriminant function analysis on an independent classification group of measurements.
Overall, 82.5% of measurements are correctly classified.
the fraction of melanin scaled the melanin sites within expectations but allowed the nonmelanin sites to have a small negative melanin content if the slope was less than the mean
for non-melanin pigmented sites. The melanin concentrations are shown in figure 7.13.
Histograms of the scaler to the scattering coefficient is shown in figure 7.14. Melanin sites
that show an elevated scattering properties were due to outliers in cause by poor fitting.
The values for blood concentration, oxygen saturation, scattering, instrument scaling and
baseline offset all show little is if any difference between site types. Therefore, melanin is
the only factor that consistently differentiates the spectral shape of the measurements for
the collection fiber as is shown in figure 7.17 in comparison to the blood fraction..
Figure 7.18 shows box plot distributions of the melanin slope m for the emission and
collection fibers. Using a simple threshold value to determine sites that are predominantly
influenced by melanin absorption, the sensitivity and specificity are shown in figures 7.19
and 7.20 as ROC curves of the threshold value.
Figure 7.21 shows a scatter-plot of the two discriminant function scores of the measurements in the test group. The discriminant functions are described by equations 7.1
and 7.1. Measurements are classified into the group of the nearest group centroid of the
classification data set. A summary of the classification results are shown in table 7.1.
166
8
6
F2
4
2
unpigmented
melanin
0
-2
amalgam
-6
-4
-2
0
2
4
F
1
Figure 7.6: A scatterplot of the measurements in the classification group in the discriminant function space. Points indicated by diamonds represent measurements from melanin
sites, circles indicate amalgam sites, and triangles indicate the non-pigmented sites. The
centroids of each group is shown as a filled square. Future cases are determined by the
nearest group centroid.
167
30
Absorption [a.u.]
25
NTD
20
Tytin
15
Dispersalloy
10
5
0
400
450
500
550
600
650
700
750
800
Wavelength [nm]
Figure 7.7: The fraction of light absorbed of three commercial alloys of dental amalgams
determined from single integrating sphere measurements. In the visible spectrum amalgam
absorbs light uniformly acting like a neutral density filter. The absorption coefficient was
not determinable due to the unknown concentration of particles in the light-path for each
alloy. In this regard, it is not determinable whether one ally absorbs more strongly than
any other.
1.5
1
0.5
0
400
MR
probe 1 amalgam
600
800
1000
! [nm]
probe 1 melanotic
MR
1.5
1
0.5
0
400
MR
600
800
1000
! [nm]
probe 1 normal near amalgam
1.5
1
0.5
0
400
600
800
1000
! [nm]
probe 1 normal near melanotic
1.5
1
0.5
0
400
600
800
1000
! [nm]
MR
MR
MR
MR
MR
168
1.5
1
0.5
0
400
1.5
1
0.5
0
400
probe 2 amalgam
600
800
1000
! [nm]
probe 2 melanotic
600
800
1000
! [nm]
probe 2 normal near amalgam
1.5
1
0.5
0
400
600
800
1000
! [nm]
probe 2 normal near melanotic
1.5
1
0.5
0
400
600
800
1000
! [nm]
Figure 7.8: The normalized measured reflectance spectra for all sites is shown with the
emission fiber indicated as probe 1 and the collection fiber as probe 2. Measurement
variation necessitates the use of a baseline adjustment and instrument scaling factor in
order to fit the measurements using the spectral shape of the known components.
169
1
unpigmented
0.6
amalgam
0.4
R
emission
[a.u.]
0.8
melanin
0.2
0
400
450
500
550
600
650
700
750
Wavelength [nm]
Figure 7.9: The mean reflectance collected by the emission fiber for all amalgam, melanin
and non-pigmented sites is shown. The error bars represent the standard deviation.
170
0.5
unpigmented
0.3
amalgam
0.2
R
collection
[a.u.]
0.4
0.1
melanin
0
400
450
500
550
600
650
700
750
Wavelength [nm]
Figure 7.10: The mean reflectance collected by the collection fiber for all amalgam, melanin
and non-pigmented sites is shown. The error bars represent the standard deviation.
171
26
mR [a.u.]
1.5
1
B = 0.006
S = 0.699
M = 0.011
a = 0.629
K = 5.931e!005
base = 2.880e!005
CNT = 961
12
!4
x 10
2.5
Amalgam sites
2
mR [a.u.]
!4
x 10
1.5
B = 0.000
S = 0.645
M = 0.058
a = 3.204
K = 1.432e!004
base = 2.213e!005
CNT = 832
Melanin sites
1
0.5
0.5
500
mR [a.u.]
1
700
0
400
800
B = 0.008
S = 0.658
M = 0.004
a = 0.710
K = 7.650e!005
base = 2.788e!005
CNT = 729
500
x 10
Normal sites
near amalgam
1
600
! [nm]
700
800
12
!4
26
!4
x 10
1.5
600
! [nm]
mR [a.u.]
0
400
B = 0.008
S = 0.637
M = 0.017
a = 0.668
K = 5.432e!005
base = 2.268e!005
CNT = 1201
Normal sites
near melanin
0.5
0.5
0
400
500
600
! [nm]
700
800
0
400
500
600
! [nm]
700
800
Figure 7.11: The measured reflectance spectra with the collection fiber are fit using the
known absorption spectra for blood, water and melanin, to evaluate concentrations and
scattering properties. Typical fits are shown over-layed on the measured reflectance for an
amalgam site (upper left) and a nearby non-pigmented site on the same patient (lower left)
and a melanin site (upper right) and its nearby non-pigment site (lower right). Five parameters were varied in the fitting: the blood fraction, the oxygen saturation, a reflectance
scaling factor, a scattering scaling factor, and a wavelength invariant baseline.
172
15
15
amalgam sites
10
Count
Count
amalgam sites
5
0
10
5
0
0
0.01
0.02
0.03
0.04
Blood fraction
0
0.2
15
15
Count
Count
10
5
0.01
0.02
0.03
Count
Count
5
0.02
0.03
Count
Count
5
0.02
0.03
Blood fraction
1
0.4
0.8
O2 saturation
10
5
0
0.01
0.8
Normal sites
near melanin
10
0
0
15
Normal sites
near melanin
0
0.6
5
0.04
Blood fraction
15
0.4
O2 saturation
10
0
0.01
0.2
Normal sites
near amalgam
10
0
0
15
Normal sites
near amalgam
0
1
5
0.04
Blood fraction
15
0.8
10
0
0
0.6
Melanin sites
Melanin sites
0
0.4
O2 saturation
0.04
0
0.2
0.4
0.6
0.8
1
O saturation
2
Figure 7.12: Histograms of the fraction of blood in the tissue and oxygen saturation for
all measurements with the collection fiber are shown separated by type of site. The blood
content is similar for all measurements but the oxygen saturation is given a low value
when the blood content is relatively low.
173
15
15
Melanin sites
10
Count
Count
amalgam sites
5
0
-0.04
-0.02
0
0.02
10
5
0
0.04
0
0.1
Melanin fraction
0.4
15
Normal sites
near amalgam
Normal sites
near melanin
10
Count
Count
0.3
Melanin fraction
15
5
0
0.2
-0.04
-0.02
0
0.02
Melanin fraction
0.04
10
5
0
-0.04
-0.02
0
0.02
0.04
Melanin fraction
Figure 7.13: The melanin concentration was proportionally fixed to the negative slope to
the logarithm of the reflectance between 650-750nm for the collection fiber measurements.
This prevented cross-talk between melanin concentration and the scattering scalar. An
increase in melanin has a similar overall effect to the reflectance measurement as does
a decrease in scattering in the fitting model. In establishing the melanin concentration
linearly to the slope, an offset slope equal to the mean of sites without melanin is used to
account for the slope with the fiber without melanin absorption. However, this approach
forces the non -pigmented sites to register a small negative melanin content.
174
25
25
amalgam sites
15
10
5
0
Melanin sites
20
Count
Count
20
15
10
5
0
2
4
6
8
10
0
0
Scattering scalar, a
6
8
10
35
Normal sites
near amalgam
30
Normal sites
near melanin
30
25
Count
25
Count
4
Scattering scalar, a
35
20
15
10
5
0
2
20
15
10
5
0
2
4
6
8
Scattering scalar, a
10
0
0
2
4
6
8
10
Scattering scalar, a
Figure 7.14: Histograms of the scaler to the scattering coefficient that has a wavelength
dependence the follows a linear combination of Rayleigh and Mie scattering terms. Melanin
sites show a wider variability in scattering but the values above 4 were poorly fits with
the model.
175
10
10
amalgam sites
6
4
2
0
Melanin sites
8
Count
Count
8
6
4
2
0
5x10-5
0.0001
0.00015
0.0002
0
0
Instrument scalar, K
0.00015
0.0002
10
Normal sites
near amalgam
6
4
Normal sites
near melanin
8
Count
8
Count
0.0001
Instrument scalar, K
10
2
0
5x10-5
6
4
2
0
5x10-5
0.0001
0.00015
Instrument scalar, K
0.0002
0
0
5x10-5
0.0001
0.00015
0.0002
Instrument scalar, K
Figure 7.15: The overall predicted measurement is scaled by a wavelength independent
constant to correct for differences in fiber contact to the tissue among measurements.
176
15
15
10
5
0
Melanin sites
Count
Count
amalgam sites
0
2x10-5
4x10-5
6x10-5
8x10-5
10
5
0
0
Baseline offset [au]
6x10-5
8x10-5
15
Normal sites
near amalgam
10
Normal sites
near melanin
Count
Count
4x10-5
Baseline offset [au]
15
5
0
2x10-5
0
2x10-5
4x10-5
6x10-5
Baseline offset [au]
8x10-5
10
5
0
0
2x10-5
4x10-5
6x10-5
8x10-5
Baseline offset [au]
Figure 7.16: The baseline for each measured site was allowed to differ to account for contact
variability. It is expected that the offset would be similar for all measurements with the
exception of the amalgam sites due to absorption by the amalgam which is spectrally flat.
however, no discernible difference is seen.
177
0.4
Melanin fraction
0.3
0.2
0.1
0
-0.1
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Blood fraction
Figure 7.17: TThe fraction of blood versus the fraction of melanin is shown for all measurements with the collection fiber. Melanin sites are shown as squares, amalgam sites
by circles, and non-pigmented sites nearby amalgam and melanin sites are respectively
diamonds and triangles. The blood fraction does not distinguish among site types as
expected, while the melanin sites are clearly differentiated.
178
0
m [a.u.]
-0.001
-0.002
-0.003
-0.004
-0.005
melanin sites melanin control sites amalgam site amalgam control sites
N = 48,
N = 44,
N = 45,
N = 43
0
m [a.u.]
-0.002
-0.004
-0.006
-0.008
-0.01
melanin sites melanin control sites amalgam sites amalgam control sites
N = 48,
N = 44,
N = 45,
N = 43
Figure 7.18: This is a box plot of the melanin slope m with the emission fiber (top) and
collection fiber (bottom). The control sites refer to the non-pigmented sites adjacent to
a type of pigmented site. The control sites and amalgam sites show similar distributions
while the melanin sites generally have a greater magnitude of slope which has a wide
distribution that mirrors the skin color range of the subjects.
179
1
-0.0014
-0.0016
Sensitivity
0.8
-0.0015
-0.0017
-0.0018
-0.0019
0.6
-0.0020
0.4
-0.0025
0.2
0
0
0.2
0.4
0.6
0.8
1
1 − Specificity
Figure 7.19: ROC curve for sites containing melanin as determined by a threshold value
for the melanin slope m by the emission fiber. As the threshold value increases (values
labeled), the sensitivity increases at the sacrifice of a decrease in specificity (shown as
1 − specificity).
180
1
-0.0016
-0.0017
-0.0018
0.95
Sensitivity
-0.0019
-0.0020
0.9
-0.0021
-0.0022
0.85
0.8
0
0.05
0.1
0.15
0.2
1 − Specificity
Figure 7.20: ROC curve for sites containing melanin as determined by a threshold value
for the melanin slope m by the collection fiber. For clarity, the domain and range are
truncated to show detail. As the threshold value increases (values labeled), the sensitivity
increases at the sacrifice of a decrease in specificity (shown as 1 − specificity).
181
8
6
F2
4
unpigmented
melanin
2
0
amalgam
-2
-6
-4
-2
0
2
4
F1
Figure 7.21: The discriminant functions scores of measurements in the test case group are
shown in a scatterplot. Measurements are classified according to the shortest distance to
one of the group centroids (shown as filled squares) established from the the classification
group measurements. Points indicated by diamonds represent measurements from melanin
sites, circles indicate amalgam sites, and triangles indicate the non-pigmented sites.
182
7.4
Discussion
Diffuse reflectance spectroscopy provides an information rich signal regarding the
chemical content of tissues [6]. The primary endogenous absorbers in the visible and
near-infrared wavelengths are oxy- and deoxy-hemoglobin, melanin, and water. In this
study, dental amalgam is the lone exogenous chromophore. The problem addressed in this
study is whether sites that contain melanin can be differentiated from the sites that contain amalgam. Spectral fitting of the shape of the measured sites to the known absorbers
confirmed that melanin absorption is the only consistently distinguishing characteristic.
Yet, a secondary goal of this study is to provide a basis for the design of a second generation probe that would effectively test pigmented oral lesions but require less expensive
instrumentation. It is this goal that informed the remainder of the method of analysis.
Melanin could be distinguished from normal and amalgam sites using diffuse reflectance
over the wavelength range 640–720 nm (figs. 7.19 and 7.20). Though the reflectance spectrum is processed in the same manner, both the sensitivity and specificity of identifying
melanin pigmented lesions is approximately 10% greater with the collection fiber than for
reflectance collected with the emission fiber. The difference may be attributed to volume of tissue being sampled and the location of melanin within the oral mucosa. Like in
skin, oral melanin pigmentation arises from melanocytes at the epitheleal-conective tissue
junction [132] and is found in the epithelium as shown in figure 7.22. The light reflected
into the collection fiber travels a longer distance than light collected by the emission fiber
(as shown in chapter 6), and thus a greater attenuation of light due to melanin occurs
with the collection fiber. A probe that discriminates based on a threshold due to melanin
absorption could be reduced to a pair of side-by-side fibers, one for illumination and the
other for light collection.
Though a threshold for melanin absorption discrimination approach still requires a
spectrometer, the main limitation is that the discrimination between pigmented and nonpigmented sites without melanin is needed to ensure accurate placement of the probe.
The diffuse reflectance spectra was used to simulate narrow spectral bandwidth reflectance
measurements that would be obtained using an array of lamp filters or LEDs as a light
183
source and the same probe geometry. Discriminant function analysis was used to evaluate
these pseudo-measurements for classification of sites as pigmented by melanin, amalgam
or non-pigmented. Discriminant analysis is comparable to multiple logistic regression,
but that the predictor variables are continuous. Separation of sites containing melanin
pigmentation was determined by reflectance at the wavelength bands at 650 and 675 nm.
The wavelength bands at 550 and 575 nm also correlated with separation of melanin sites,
but were relatively weak predictors in comparison to the 650 and 675 nm wavelength bands
as given by the weights when used together in the discriminant analysis. The 550 and
575 nm wavelength bands correspond with absorption by hemoglobin, which had a less
pronounced dip in reflectance in that spectral band for sites with melanin pigmentation
than either non-pigmented or amalgam sites.
The amalgam and non-pigmented sites had a smaller separation between group centroids than with melanin sites as shown in table 7.1 and therefore accuracy suffered in discriminating between these two groups. Discrimination of the amalgam and non-pigmented
sites was best established by the reflectance at 475, 500 and 550 nm with the emission fiber
and 475, 500, and 525 nm with the collection fiber as predictors. The overall reflectance
was lower for amalgam sites than non-pigmented sites with intra-subject measurements;
but, the reflectance distribution of the amalgam sites and non-pigmented sites overlap
considerably leading to a large overlapping of the two groups. It is for this reason that
only 53% of amalgam sites in the test group were classified correctly while all incorrectly
classified amalgam tattoos were attributed as non-pigmented sites and all mis-classified
cases of non-pigmented sites were attributed as amalgam sites.
A possible pitfall to the grouping overlap between amalgam and non-pigmented sites
is the misplacement of the probe in cases of amalgam tattoo. Misclassification of amalgam
sites in this manner could lead to misplaced measurements where the probe is positioned
proximal to the pigmented site but misses the actual pigmented site. To avoid this problem, site identification may be achieved by a real-time indicator of the reflectance for nearly
any visible wavelength, though wavelengths longer than 640 nm are preferable since they
are less affected by hemoglobin absorption artifacts. Moving the probe over the pigmented
site would then drop the reflectance relative to the surrounding non-pigmented mucosa
184
localizing the pigmented site for assessment.
Other investigators have explored solutions of similar problems regarding the discrimination of melanin pigmented lesions. Marchesini et al. [142] used discriminant function
analysis to differentiate nevi from melanoma of the skin but differed in their analysis.
Rather than using absolute reflectance, spectral features such as the second derivative
with respect to wavelength in specific wavelength bands were shown to be capable classifiers in the discriminant functions. Another difference in their analysis was the use of
leave-one-out classification. I chose to use separate sets of measurements for classification and testing of future cases since a more conservative and realistic representation of
classification is determined.
7.5
Conclusions
The design of the fiber probe was constrained by practical limitations. The dimensions
of surface contact of the probe with the tissue is restricted by two conflicting factors.
Small areas of contact lead to large variations in the measured reflectance due to changes
in blood perfusion caused by contact pressure instability (as demonstrated in chapter 5).
On the other hand, larger areas of contact implicitly increase the physical size of the probe
thereby restricting the placement of the probe, such as in close proximity to a tooth on the
alveolar ridge. One potential subject was rejected for this study by the dental pathologist
performing the measurements, since it was felt that the probe could not be placed on the
amalgam tattoo due to its proximity to a molar.
In addition, poor probe design can lead to discomfort during measurements. Natural
racial oral pigmentation in subjects generally occurs along the gum-line as shown in fig.
7.2. Non-pigmented sites typically occur further away from the teeth on the gums. On two
occasions, a subject felt a short pinching sensation with placement of the probe on this
non-pigmented location of the gums. Pressure imparted by the probe placement onto one
of the shallow nerves in this area was considered to be the cause. In no other instances did
any subject feel discomfort during measurements. It is possible that the beveled fiber tip of
the probe aided in placement over a probe with a perpendicular viewing angle. The beveled
185
tip reduced the necessary bend in the optical fibers to place the probe on the surface of the
inner gums. A side-looking fiber probe design is a promising potential solution for future
probe designs to provide ease of placement in wide variety of tight compartments found
in the oral cavity while minimizing the pressure of the probe pressing into the mucosa.
Amalgam tattoo and melanin oral pigmented lesions can be distinguished using diffuse reflectance from a dual optical-fiber probe. Two methods of discrimination were
investigated. The first method used the spectral features of melanin in the 640–720 nm
wavelength band. The pigmented lesions containing melanin exhibited a higher change in
reflectance with respect to wavelength over this band in comparison to amalgam tattoos
or non-pigmented sites. The sensitivity and specificity for identifying melanotic lesions
from amalgam tattoo was 90% and 99% respectively when the threshold was −0.0021
using the collection fiber reflectance measurement. A better threshold was −0.0017 giving
a sensitivity of 98% and a specificity of 92% due to preference that melanin pigmented
sites be correctly identified for further screening by biopsy.
A second method of discriminating amalgam tattoo, melanin pigment and non-pigmented
sites uses discriminant function analysis on uniformly spaced wavelength bands of reflectance to simulate spectrally filtered reflected light. The discriminant functions are
created using two-thirds of the measurements to determine the classification functions
and the remaining one-third of the measurements to be used for testing. The sensitivity
and specificity was 94% and 100% respectively for classifying melanin pigmented sites. A
single melanin pigmented site (6%) was falsely classified as being amalgam.
The discriminant function performed poorly in classifying amalgam tattoos. The sensitivity was 53% with a specificity of 92%. All cases of misclassified amalgam tattoo were
classified as non-pigmented sites. The poor classification of amalgam is attributed to
the large inter-subject variability for reflectance measurements on amalgam tattoos and
non-pigmented sites. However, intra-subject measurements produce higher reflectance on
non-pigmented sites relative to the amalgam tattoo. It is suggested that a device show
real-time reflectance to assist in localization of the probe on the pigmented lesion.
186
epithelium
melanin
100µm
100µm
epithelium
Amalgam
Figure 7.22: The top image shows normal oral buccal mucosa with melanin pigmentation.
The melanin is formed along the epithelium border to the underlying connective tissue.
The bottom image is from a biopsy of an amalgam tattoo showing both large and fine
amalgam particles. The fine particles are digested by macrophages and giant cells resulting
in the progressive loss of tin and mercury leaving behind fine particles containing silver
and sulphur [150, 151]. Amalgam is not found in the epithelium as it is typically shed
every 5-6 days.
Chapter 8
General Discussion and Conclusions
This thesis has presented methods of reflectance spectroscopy with fiber-optics using
the shortest source-detector separation possible with the source fiber also acting as the
collection fiber. Two probe configurations were presented that incorporate the use of a
single fiber to emit and collect its own back-scattered light. A Monte Carlo was developed
and tested against measurements on phantoms. The model was used to relate two separate
diffuse reflectance measurements to the absorption and reduced scattering coefficients of
the medium for both probe designs. The model was used to establish that the device
behavior for both probes was not independent of the scattering anisotropy. The optical
sampling path-length for 200, 400, 600, and 1000 micron diameter fiber measurements
was evaluated with the Monte Carlo model in cases of perpendicularly polished fibers
and for bevel-tipped fibers. In addition, analysis was presented for the ability to resolve
the absorption and reduced scattering optical properties in the range commonly found in
tissue. Finally, a clinical study was presented using a fiber-optic probe to distinguish two
kinds of pigmented oral lesions, naturally occurring melanin and amalgam tattoos.
Complimentary work to the optical fiber probe characterization was presented in chapters 2 and 3. Experimental measurements on phantoms with known optical properties was
used to validate the Monte Carlo model. A method for fabricating optically-stable phantoms was developed for use at multiple wavelengths. The stability of phantoms make them
useful as a diffuse reflectance standard. In order to independently assess the optical properties of the phantom, an experimental method using integrating spheres was presented
with a model to interpret both single and double sphere experiments. A term for the
sphere gain was introduced showing the influence of the sphere on total diffuse reflectance
187
188
and transmittance in terms of the sphere’s dimensions (eg. wall area, port areas, and wall
reflectance). A combined inverse adding-double and Monte Carlo model was developed
to characterize the optical properties of phantoms with a slab geometry using integrating
sphere reflectance and transmittance measurements.
8.1
Fabrication of optically stable phantoms
A method to fabricate tissue phantoms that simulate the optical properties of living
tissue at the 690 and 830 nm wavelengths was presented. The optical properties of the
phantoms has been consistent over a period of 14 months from the date of casting. The
two molecular dyes provided independent absorption between our chosen wavelengths for
absorption coefficients up to 5 and 3 cm−1 at 690 and 830 nm respectively. These dyes
exhibited stable absorption through the curing process of polyurethane. In addition, the
high molecular extinction coefficient of each dye allowed us to keep the addition of dye
stock to less than 0.1 percent of the volume of the polyurethane. The polyurethane itself
was the only component that changed optical properties after casting. The absorption due
to the polyurethane decreased approximately by a factor of 2 over the visible spectrum.
This change is assumed to occur gradually though evidence was not obtained to support
this belief. Though polyurethane slowly changes absorption properties, the effect is to
make the material more colorless that helps to make any added dyes the more dominant
absorber. Four linear relationships (eqs. 1–4) that describe the absorption coefficient as
a function of dye concentration and the reduced scattering coefficient as a function of
the titanium dioxide concentration were determined. These relationships predicted the
measured absorbing and scattering properties with less than 4% error. Degassing of the
polyurethane was a time-consuming yet necessary step to obtain uniformity across a batch
of phantoms. The optical properties of the phantoms have shown to be stable over a period
of 14 months making the phantoms suitable for use as reference standards.
189
8.2
Combined Monte Carlo and inverse adding-doubling model
for use with integrating sphere measurements.
The combined IAD/MC model was demonstrated to accurately determine optical properties of homogenous optically turbid samples with a reasonable precision using multiple
sample thicknesses and sample port sizes for both single and double sphere experiments.
The nomenclature for integrating sphere measurements was simplified and rationalized
using the concept of sphere gain to express the results. Explicit directions for determining sphere parameters were shown. Formulas were given that work for diffuse incidence
or collimated incidence or any combination thereof. Inversion of optical properties from
integrating sphere measurements requires a theoretically compatible inversion model and
experimental methods otherwise gross error occurs. With substitution method sphere
measurements, the inversion model must correct for the sphere gain which increases the
measured reflectance and decreases transmittance. A 3D model such as Monte Carlo is
essential to correct for light lost through sample edges which improves the resolution and
accuracy of samples with small absorption coefficients. With these two corrections, experimental error becomes comparable in magnitude to the error in the predicted optical
properties for the presented IAD/MC model.
8.3
Sized-fiber reflectometry
Sized-fiber reflectance spectroscopy was the first method for measuring absorption and
reduced scattering of tissue using 200 and 600 µm diameter optical fibers. Backscattered
light measurements for solutions with absorption coefficients of 0.1–2.0 cm−1 and reduced
scattering coefficients of 5–50 cm−1 demonstrate that the device is most sensitive for the
highest scattering materials. Monte Carlo simulations suggest the device is insensitive to
the fiber illumination distribution and that the light returning to the fiber is nearly uniform
over all directions. Finally, experiments and Monte Carlo simulations of the sized-fiber
device indicate that 50% of the signal arises from roughly 1.2 and 1.9 reduced mean free
paths for the 200 and 600 µm fibers respectively and that in general larger fibers sample
deeper optically.
190
8.4
Specular reflectance noise with single fiber measurements
Specularly reflected light behaves as a noise source since it does not contain any information about the absorption or scattering properties of the tissue sample being measured.
Additionally, the amount of specular light was shown to be highly correlated to fiber contact. It was shown 5.5 that the specular reflection signal from a perpendicular polished
fiber changes by 6.5% with only a 4 degree rotation of the fiber relative to the normal
of the surface being measured. Elimination of the specular light from the fiber face was
tested for a reduction in signal variation using a beveled-tip fiber. The measurement
variability decreases 6 fold on in vivo skin and nearly 8 fold on a resin tissue phantom
with the 200 micron fiber with the beveled-tip fiber. On a solid resin tissue phantom, the
difference between the highest and lowest of 12 measurements is 6.5 and 2.4 percent for
the 200 and 1000 micron fiber respectively. On in vivo skin where the surface is compliant,
the difference between the highest and lowest of 13 measurements drops to 4.5 percent
difference with a 200 micron fiber while the 1000 micron fiber is 2.2 percent difference.
8.5
Two dual fiber reflectance probe designs
Two dual-fiber devices with beveled-tips were presented with corresponding methods to invert optical properties from diffuse reflection measurements, a sized-fiber probe
that uses a 200 and 1000 micron diameter fibers both emitting and collecting light independently and a dual 400 micron diameter fiber probe with one emitting fiber and both
fibers collecting reflected light. Only minor differences exists in performance between the
probes. The single emission fiber probe has more compact physical dimensions and a
slightly longer optical sampling path-length (a 25-30% increase) relative to the sized-fiber
probe. In the visible spectrum for typically expected range of tissue optical properties, the
greatest mean optical path-lengths measured are 2.5 mm or less, restricting the resolution
of absorption where 100% error is possible. Absolute determination of absorption with
either probe will be poor for such short path-lengths which appears as cross-talk between
the measured absorption and scattering properties. Yet the relative absorption spectral
191
shape was demonstrated on measurements of an optical phantom. The dual 400 micron
fiber probe is a better design than the sized-fiber probe for two reasons. First, it is a
less complex design with a single source fiber that is bifurcated for simultaneous emission
and collection of light. Second, dual 400 micron fiber probe has a slightly larger optical
sampling path and a more narrow overall diameter. However for the inversion of optical
properties, the linear interpolation of optical properties from a Monte Carlo generated
grid demonstrated better performance than the empirically derived relations for the single
emitting fiber probe, despite its greater mean sampling optical path-length.
8.6
Clinical discrimination of amalgam tattoos from melanin
pigmented oral lesions
A clinical study was presented using the dual 400 micron fiber probe to discriminate
between pigmented oral lesions due to melanin or amalgam tattoo. Amalgam tattoos,
though benign, are often biopsied to rule out melanoma which have a similar appearance.
Two methods of analysis were performed with the clinical data. The first method analysis
relied on the correlation of slope of the reflected spectrum to the concentration of melanin
from 640–720 nm. A sensitivity of 98% was obtained with a specificity of 92% with a slope
threshold of −0.0017 for the collection fiber to identify melanin sites.
The second method of analysis used the reflctance spectra obtained in the clinic to simulate data collected using either spectral filters or LED sources centered at 11 wavelengths
from 450-700 nm. Discriminant function analysis was used to create two discriminant functions with two-thirds of the measurements. The classification of the discriminant functions were evaluated using the other remaining third as an independent data set. Melanin
sites were identified with a sensitivity and specificity of 94% and 100% respectively. A
single melanin pigmented site (6%) was falsely classified as being amalgam. The discriminant function performed poorly in classifying amalgam tattoos. The sensitivity was 53%
with a specificity of 92%. All cases of misclassified amalgam tattoo were classified as
non-pigmented sites. The poor classification of amalgam is attributed to the large intersubject variability for reflectance measurements on amalgam tattoos and non-pigmented
192
sites. However, intra-subject measurements produce higher reflectance on non-pigmented
sites relative to the amalgam tattoo. We suggest that a device show real-time reflectance
to assist in localization of the probe on the pigmented lesion.
Appendix A
Angled-Fiber Monte Carlo Program
This program will calculate the diffuse reflectance emitted by a bevel-tipped fiber into
a semi-infinite medium that is collected by the emitting fiber and a second collection
fiber in contact to the side (perpendicular to the direction of the bevel). The angle of
the fiber tip is specified for a bevel in the x-z plane. For each fiber, the mean path of
travel for collected photons is calculated. The results are calculated for absorption and
scattering coefficients which are input as arrays as well as the fiber radius, so that multiple
combinations are run sequentially.
Important assumptions in this program are that the face of the beveled-tip is parallel
to the medium boundary. Both the emission and collection fibers have identical properties
(eg. diameter, numerical aperture, bevel angle etc.). The fibers are assumed to be stripped
to the cladding and in contact with each other in the y-axis with a center to center spacing
of two fiber radius plus twice the cladding thickness.
A sample output for a 400 micron diameter fiber is given for 1,000,000 photons with
the fibers at the surface of the medium.
193
g
--12.00
12.00
12.00
12.00
12.00
mu_s
1/cm
0.9000
0.9000
0.9000
0.9000
0.9000
Em_core
Em_Path
Col_core
--cm
--cm
0.001148 0.0373378 0.0001471 0.1000460
0.0012247 0.0362642 0.0001593 0.0991180
0.0011750 0.0372791 0.0001425 0.0972004
0.0011279 0.0379130 0.0001618 0.0984582
0.0011574 0.0371960 0.0001542 0.0985450
(microns)
Col_Path
sec
106.38
106.27
106.42
106.25
106.38
@Y
time
@Y
Figure A.1: Sample output of five runs with the same optical parameters using a 65 degree beveled fiber 400 micron in diameter.
84.62
mu_a
1/cm
10.0000 200.0000
10.0000 200.0000
10.0000 200.0000
10.0000 200.0000
10.0000 200.0000
Scattering Anisotropy
Medium Refractive Index
Fiber Refractive Index
Fiber Depth
Number of photons
total elapsed time =
diameter
microns
400
400
400
400
400
0.9000
1.3500
1.4570
0
1000000
Fiber MC Reflectance by Scott Prahl (http://omlc.ogi.edu)
1 W Uniform Fiber Illumination of Semi-Infinite Medium
modified by Ted Moffitt for beveled-tip fibers
194
195
char t1[80] = "Fiber MC Reflectance by Scott Prahl (http://omlc.ogi.edu)";
char t2[80] = "1 W Uniform Fiber Illumination of Semi-Infinite Medium";
char t3[80] = "modified by Ted Moffit for beveled-tip fibers";
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <time.h>
#define RND ((rand()+1.0)/(RAND MAX+1.0))
#define PI 3.141592653589
10
long iphoton, photon kind, photons = 100000;
double mu a = 1.0;
/* background excitation absorption in 1/cm */
double mu s = 200;
/* Scattering Coefficient in 1/cm */
double g = 0.9;
/* Scattering Anisotropy -1<=g<=1 */
double n medium = 1.40;
/* Index of refraction of medium */
double n fiber = 1.457;
/* Index of refraction of fiber */
double fiber depth = 0.0;
/* Depth fiber is immersed in cm */
double numerical aperture = 0.22; /* acceptance in air */
double fiber radius = 0.03;
/* Number of elements run from radius array */
double clad thickness = 0.0015; /* the thickness of the fiber cladding */
double clad rd, fiber1 rd, fiber2 rd, fiber accept angle, fiber crit angle,fiber radius;
double mu a,mu s,x,y,z,u,v,w,weight, rs, air rd, bit, air crit angle,mu t, heat, pathsum;
fiber1 path, fiber2 path, clock t first time, start time, finish time;
20
double bevel angle = 25;
/* angle of bevel on fiber from perpendicular in degrees */
#define COS bevel angle cos(bevel angle*PI/180.0)
#define TAN bevel angle tan(bevel angle*PI/180.0)
#define SIN bevel angle TAN bevel angle*COS bevel angle
30
static void launch(void) /* Start the photon */
{
double phi,theta,nx,ny,nz,nu,mu,xx,yy,zz;
/* launch uniformly over fiber face - modified for an elliptical face*/
do {x=2.0*RND − 1.0; y=2.0*RND − 1.0;} while (x*x+y*y>1);
x *= fiber radius;
y *=fiber radius;
z = fiber depth;
/* This section gives translates light with a gaussian launch distribution from the last */
/* whole plane perpendicular to the fiber axis and propagates out remaining fiber */
/* to the angled fiber face then refracts the light as it enters the medium */
phi = rand pi pi();
40
196
/* random gaussian launch angles in spherical coordinates */
do {theta = gaussian rand(numerical aperture/n fiber);} while (theta<0);
u = sin(theta)*cos(phi);
/* convert spherical to cartesian coordinates*/
v = sin(theta)*sin(phi);
w = cos(theta);
50
/* find intercept of photon with the plane of the fiber face, (xx,yy,zz) */
xx = (fiber radius*TAN bevel angle + x*w/u − z)/(w/u +TAN bevel angle);
yy = v*(xx−x)/u + y;
zz = w*(xx−x)/u + z;
/*Condition for reflecting photons off the interior wall of the fiber*/
while (xx*xx+yy*yy>fiber radius*fiber radius)
{
/*determine fiber wall intercept coordinates (nx,ny,nz)*/
if (u==0) {
if (v>0)
ny = sqrt(fiber radius*fiber radius − x*x);
else
ny = −sqrt(fiber radius*fiber radius − x*x);
nx = x;
nz = w*(ny−y)/v +z;
}
else if (u>0) {
nx = (v*v*x − u*v*y + sqrt(u*u*(fiber radius*fiber radius*(u*u + v*v)
−(v*x − u*y)*(v*x − u*y))))/(u*u + v*v);
ny = (nx−x)*v/u + y;
nz = w*(nx−x)/u +z;
}
else if (u<0) {
nx = (v*v*x − u*v*y − sqrt(u*u*(fiber radius*fiber radius*(u*u + v*v)
−(v*x − u*y)*(v*x − u*y))))/(u*u + v*v);
ny = (nx−x)*v/u + y;
nz = w*(nx−x)/u +z;
}
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x = nx; /* photon reflection point position (x,y,z) */
y = ny;
z = nz;
/* reflect photon off fiber wall */
nx /= fiber radius;
ny /= fiber radius;
nz = 0;
nu = nx*u + ny*v + nz*w; /* cosine of incoming angle with normal to fiber wall */
mu = −2*nu;
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u = u + mu*nx; /* reflection photon direction cosine vector (u,v,w) after reflection*/
v = v + mu*ny;
w = w + mu*nz;
/* find new intercept of photon with the plane of the fiber face, (xx,yy,zz) */
xx = (fiber radius*TAN bevel angle + x*w/u − z)/(w/u +TAN bevel angle);
yy = v*(xx−x)/u + y;
zz = w*(xx−x)/u + z;
}
100
/* map coordinaes so that (xx,yy,zz) is rotated about the y-axis */
/* such that the fiber face is in the z = 0 plane */
x = fiber radius/COS bevel angle + (xx−fiber radius)/(2*COS bevel angle)
− zz/(2*SIN bevel angle);
y = yy;
z = (xx−fiber radius)*SIN bevel angle + zz*COS bevel angle;
/* rotate the u and w direction coines accordingly */
u = −u*COS bevel angle − w*SIN bevel angle;
w = u*SIN bevel angle + w*COS bevel angle;
110
/* refract the cosines due to index mismatch */
u = n fiber*u/n medium;
v = n fiber*v/n medium;
w = sqrt(1 − u*u − v*v);
/* end of angled fiber launching code */
120
weight = 1.0 − rs;
mu t = mu s + mu a;
pathsum = 0;
/* fiber surface reflection */
}
static double transmitted weight(double n index) /* weight out of medium*/
{
double t
= sqrt(1.0−n index*n index*(1.0−w*w)); /* cos of exit angle*/
double r par = (w − n index*t)/(w + n index*t);
130
double r per = (t − n index*w)/(t + n index*w);
/* refl = 0.5*(temp1**2+temp**2) */
return (1.0 − 0.5*(r par*r par+r per*r per)) * weight;
}
static void bounce (void) /* Interact with top surface (at z=0) */
{
198
double exit weight;
w = −w; z = −z;
if (w <= air crit angle) return;
/* total internal reflection */
exit weight = transmitted weight(n medium);
air rd += exit weight;
weight −= exit weight;
140
}
static void fiber bounce (void) /* Interact with fiber surface at z=surface */
{
double exit weight, dot product;
double delta = (z−fiber depth) / w;
double xx = x − delta * u;
double yy = y − delta * v;
/* x-direction cosine for refracted angle of fiber axis */
double exit vector u = n fiber*sin(bevel angle)/n medium;
Tab8/* z-direction cosine for refracted angle of fiber axis */
double exit vector w = sqrt(1 − exit vector u*exit vector u);]
if (xx*xx*COS bevel angle*COS bevel angle + yy*yy > fiber radius*fiber radius
&& xx*xx*COS bevel angle*COS bevel angle + (yy−2*fiber radius−2*clad thickness)
*(yy−2*fiber radius−2*clad thickness)
> fiber radius*fiber radius) return;
/* missed fiber faces */
w = −w; z = 2*fiber depth−z;
150
160
/* product of direction cosine and refracted axis of fiber */
dot product = u*exit vector u + w*exit vector w;
fiber accept angle = numerical aperture/n medium;
fiber accept angle = sqrt(1−fiber accept angle*fiber accept angle); /* cos in medium */
if (dot product>fiber accept angle) /* condition for transmission through fiber */
{
exit weight = transmitted weight(n medium/n fiber);
170
pathsum += delta/mu t;
if (xx*xx*COS bevel angle*COS bevel angle + yy*yy <= fiber radius*fiber radius)
{fiber1 rd += exit weight;
fiber1 path += pathsum*exit weight;
}
else {
fiber2 rd += exit weight;
fiber2 path += pathsum*exit weight;
}
}
else
180
clad rd += exit weight;
199
weight −= exit weight;
}
static void move() /* move to next scattering or absorption event */
{
double z old = z;
double d = −log(RND);
x += d * u / mu t;
y += d * v / mu t;
z += d * w / mu t;
if ( z old>fiber depth && z<fiber depth) fiber bounce();
if ( z<0 ) bounce();
pathsum += d/mu t;
/* tracks the pathlength travled for a packet of photons */
}
static void roulette() /* play roulette to terminate photon */
{
if (weight > 0.001) return;
bit −= weight;
if (RND > 0.1) weight = 0; else weight /= 0.1;
bit += weight;
}
190
200
static void absorb () /* absorb or convert to a fluorescent photon */
{
heat
+= weight * mu a/mu t;
weight *= mu s / mu t;
roulette();
/* bkgrnd absorption */
/* new weight for photon */
/* try to kill photon */
210
}
static void scatter() /* Scatter photon and establish new direction */
{
double x1, x2, x3, t, mu;
do {
x1=2.0*RND − 1.0;
x2=2.0*RND − 1.0;
} while ((x3=x1*x1+x2*x2)>1);
if (g==0) { /* isotropic */
u = 2.0 * x3 −1.0;
v = x1 * sqrt((1−u*u)/x3);
w = x2 * sqrt((1−u*u)/x3);
return;
220
200
}
mu = (1−g*g)/(1−g+2.0*g*RND);
mu = (1 + g*g−mu*mu)/2.0/g;
if ( fabs(w) < 0.9 ) {
t = mu * u + sqrt((1−mu*mu)/(1−w*w)/x3) * (x1*u*w−x2*v);
v = mu * v + sqrt((1−mu*mu)/(1−w*w)/x3) * (x1*v*w+x2*u);
w = mu * w − sqrt((1−mu*mu)*(1−w*w)/x3) * x1;
} else {
t = mu * u + sqrt((1−mu*mu)/(1−v*v)/x3) * (x1*u*v + x2*w);
w = mu * w + sqrt((1−mu*mu)/(1−v*v)/x3) * (x1*v*w − x2*u);
v = mu * v − sqrt((1−mu*mu)*(1−v*v)/x3) * x1;
}
u = t;
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}
static void print header(void)
{
printf("%s\n%s\n\n",t1,t2);
printf("%9.4f \tScattering Anisotropy\n",g);
printf("%9.4f \tMedium Refractive Index\n",n medium);
printf("%9.4f \tFiber Refractive Index\n",n fiber);
printf("%9.0f \tFiber Depth
\t(microns)\n",fiber depth*10000);
printf( "%9ld \tNumber of photons\n\n",photons);
250
printf("%9s\t %9s \t %9s\t %9s\t %9s\t %10s\t %10s\t %10s\t %10s\t %9s\n",
"diameter","g","mu_a","mu_s","Em_core","Em_Path","Col_core","Col_Path","time");
printf("%9s\t %9s \t %9s\t %9s\t %9s\t %10s\t %10s\t %10s\t %10s\t %9s\n",
"microns","---","1/cm","1/cm","---","cm","---","cm","sec");
fflush(NULL);
}
static void print results(void)
260
{
printf("%9.0f\t%9.2f\t%9.4f\t%9.4f\t%10.7f\t%10.7f\t%10.7f\t%10.7f\t%10.2f\n",
fiber radius*2*10000, g, mu a, mu s,
fiber1 rd/(photons+bit), fiber1 path/fiber1 rd, fiber2 rd/(photons+bit), fiber2 path/fiber2 rd,
(double)(finish time−start time)/CLOCKS PER SEC);
fflush(NULL);
}
int main ()
{
int iradius, iabs, isca, mua cnt, mus cnt, radius cnt;
double mu a array[20] = {.1,.2,.5,1,2,5};
double mu s array[20] = {100,100,100,200,200,200,500,500,500,1000,1000,1000};
double radius array[3] = {0.02,0.015,0.03};
270
201
mua cnt = 2;
mus cnt = 1;
radius cnt = 1;
srand (15);
rs = (n medium−n fiber)/(n medium+n fiber); /* specular reflection */
rs = rs * rs;
air crit angle = sqrt(1.0−1.0/n medium/n medium);
/* cos of critical angle */
fiber crit angle = (n medium<n fiber) ? 0.0 : sqrt(1−n fiber*n fiber/n medium/n medium);
280
first time = clock();
print header();
for (iradius = 0; iradius < radius cnt; iradius++) {
for (iabs = 0; iabs < mua cnt; iabs++) {
for (isca = 0; isca < mus cnt; isca++) {
290
mu a = mu a array[iabs];
mu s = mu s array[isca];
fiber radius = radius array[iradius];
air rd = 0; clad rd = 0; fiber1 rd = 0; fiber2 rd = 0; heat = 0; bit = 0;
fiber1 path = 0; fiber2 path = 0;
start time = clock();
for (iphoton = 1; iphoton <= photons; iphoton++){
launch ();
while (weight > 0) {
move ();
absorb ();
scatter();
}
300
}
finish time = clock();
print results();
fflush(NULL);
}
310
}
}
printf("total elapsed time = %10.2f\n",(double)(finish time−first time)/CLOCKS PER SEC);
return 0;
}
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