Open Stent Design (PDF Available)

Open Stent Design (PDF Available)
Open Stent Design
Craig Bonsignore
NDC
47533 Westinghouse Drive
Fremont, CA, 94566
[email protected]
December 7, 2010
2
©2010 Nitinol Devices & Components, Inc. Some rights reserved.
This work is licensed under the Creative Commons Attribution-Share Alike 3.0 United
States License. To view a copy of this license, visit http://creativecommons.org/licenses/bysa/3.0/us/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.
This document is distributed in the hope that it will be useful, but WITHOUT ANY
WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE.
Contents
Introduction
5
1 Basic Elements of Stent Design
1.1 Introduction to Nitinol . . . . . . . . . . . . . . . . .
1.2 Stent Anatomy . . . . . . . . . . . . . . . . . . . . .
1.3 Transformations . . . . . . . . . . . . . . . . . . . .
1.3.1 Diameter Transformation . . . . . . . . . . .
1.3.2 Material Removal . . . . . . . . . . . . . . . .
1.3.3 Dimensionality and Coordinate Systems . . .
1.3.4 Simultaneous Configurations and Constraints
1.4 Computer Aided Design of a nitinol Stent . . . . . .
2 Parametric Solid Model
2.1 Input Parameters and Equations . .
2.2 Master Strut Sketch . . . . . . . . .
2.3 Creating a Planar Unit Cell . . . . .
2.4 Creating a Full Planar Stent Model .
2.5 Creating a Wrapped Unit Cell Model
2.6 Creating a Full Wrapped Model . . .
2.7 Transforming the State of the Model
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3 Stent Calculator Formulas
3.1 Stent Design Inputs . . . . . . . . . . . . . .
3.2 Stent Process Inputs . . . . . . . . . . . . . .
3.3 Material Property Inputs . . . . . . . . . . .
3.4 Service Parameters . . . . . . . . . . . . . . .
3.5 Stent Dimension Calculations . . . . . . . . .
3.6 Strut Angle and Deflection Calculations . . .
3.7 Stent Length Calculations . . . . . . . . . . .
3.8 Surface Areas, Volume, and Mass Estimation
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4
CONTENTS
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
Moment of Inertia Calculations . . . . . .
Force and Strain Calculations . . . . . . .
Pressure and Stiffness Calculations . . . .
Calculating the Stiffness of the Vessel . .
Balanced Diameters of the Stented Vessel
Strut Deflections at Balanced Diameters .
Strain Values . . . . . . . . . . . . . . . .
Safety Factor Calculations . . . . . . . . .
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5 Finite Element Analysis Confirmation
5.1 Abaqus model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 FEA results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Stent Calculator Applications
4.1 Stent Calculator Spreadsheet . . . . . .
4.1.1 Trend Analysis: Vessel Diameter
4.1.2 Trend Analysis: Wall thickness .
4.1.3 Trend Analysis: Strut Length . .
4.2 Stent Calculator Python . . . . . . . . .
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Introduction
Why Open Stent Design?
This project is intended to bring the collaborative principles of open source to the typically
closed and proprietary world of medical device development. NDC has a long history
pioneering development of Nitinol stents and similar components, and has also been a
leading publisher and educator in the field. Contributing to the open source and creative
commons movements is a natural evolution of our commitment to Nitinol education. It is
our hope that providing these tools and resources in an unlimited way to the community of
medical device developers, as well as academic researchers, reviewers, and others, we will
inspire a new generation of designers with ideas that will advance the state of the art, and
the practice of medicine.
Thoughts on Intellectual Property
It is nearly impossible to separate commercial medical device development from intellectual
property. Bringing a medical component to market, especially in the case of an implant, is
an exceptionally expensive affair. The level of upfront investment is high, the road is long,
the outcome is uncertain. If the constellations align, the rewards are great. Consequently,
any investor (whether a venture capitalist, or the R&D department of a large corporation)
considering funding a medical device development endeavour is very concerned about intellectual property (IP) ownership and rights. IP is a broad term that includes creative works
that may be protected by trade secrets, copyrights, trade secrets, or patents. In the case
of medical component design, the focus is typically on patents and the issue for an investor
is simple: after I invest my capital in developing this design, will someone else be able to
simply copy my work and unfairly reap the benefit of my investment? Without patent
protection, the risks to the potential investor may be unacceptably high, and consequently
the investment may not be made, and the invention may never come to fruition.
5
6
CONTENTS
So in the medical device business, just about everything we do is covered by patents,
patent applications, trade secrets, and/or confidentiality agreements. The objective of any
design is to create something novel and differentiated that can be protected by patents
and distinguished in the marketplace. Companies work hard to preserve these benefits
by enforcing strict practices of secrecy. So the medical device industry is proprietary by
nature; the natural incentives in the indusrtry promote secrecy and discourage sharing. In
this way, the theory goes, innovation is enabled by providing the economic rationale for
investments in expensive projects with long development cycles.
The proprietary nature of the medical device industry creates some difficulties when those
of us inside the industry are motivated to share design guidance, principles, and techniques:
Every realistic example we know about is proprietary! This manuscript seeks to circumvent
this problem by creating a realistic medical component, the “Open Stent” that is completely
generic, and using it as an example to discuss useful techniques and procedures relating to
design and analysis of similar components.
In the microelectronics, software, and entertainment industries, by contrast, the development cycle is much shorter, the regulatory barriers are much lower, and intellectual
property flows much more freely. The speed of innovation in these industries is ferocious.
This accelerated culture of collaboration is enabled by the principles of open source development. In this model, creative individuals contribute their effort to a project, in exchange
for a promise: I will donate my efforts to the commons, and in exchange the community
can build upon my work, and society will enjoy the benefits of our collective efforts. This
approach works quite well in the case of software, where IP is neatly embodied in lines of
code that can be easily exchanged.
In the medical device industry, there is no direct equivalent of “lines of code.” Instead,
there is a constellation of resources, including sketches, drawings, specifications, protocols,
procedures, processes, and so on. In practice, though, much of this gets reduced to “lines of
code,” in a figurative and often literal sense. Design specifications are often created using
computer aided (CAD) systems, detailed in spreadsheets, and analyzed using sophisticated
computer simulations. All of these things share the character of software code: they neatly
capture creative effort, they are readily portable, and are easily shared and extended.
So this brings us to the purpose of this manuscript. Stents have been around for quite
some time, thousands of stent related patents have been granted, and many more have
been applied for. In IP terms, this means that there is a significant amount of prior art
in the field. Because of all the published patents and other works in the public domain,
it is now exceedingly difficult to develop novel designs in this field. The stent design used
throughout this manuscript, instead, takes the opposite approach: it is intended to be
completely generic, and intentionally not novel.
While the stent design itself is quite general, the techniques and resources that are described
CONTENTS
7
here are creative works that have not been previously published, and (we hope) are useful,
practical, and can be extended, expanded upon, and applied to new, different, and novel
designs. Our motivation for this is simple: we want the medical design community to have
the best tools and resources available for designing Nitinol medical devices. In doing so,
the community benefits, society benefits, and NDC benefits along the way.
Thoughts on Licensing
In the past few years, a variety of standardized licensing strategies have been developed to
aid and encourage efforts such as these. Typically, the copyright for creative works such as
this manuscript is typically held by the author. In the era of the printing press, an author
assigned his or her copyrights to a publisher, because only publishers had the means to
duplicate and distribute content to reach a large audience. In the modern internet era, any
content can be made available instantaneously throughout the world, with minimal cost.
The intent of this publication is to reach as wide an audience as possible, and to make it
as easy as possible to apply and adapt the content for new purposes.
To serve this purpose, this document is offered with a Creative Commons AttributionShare Alike 3.0 United States License. A simple explanation of the license can be found
at http://creativecommons.org/licenses/by-sa/3.0/us/, but it basically means that you are
free to share, copy, distribute, and adapt this work under two simple conditions: 1) any
copies or adaptations must provide attribution to the original author, and 2) any derivitive
works must carry the same freedoms as those afforded by this license.
Publication, Attribution and Feedback
The version of this manuscript that you are currently reading is an unfinished working
draft. We intend to continue to add and edit content, and hope to incorporate thoughts
and feedback from the community. We plan to publish this through formal channels at
some point in the near future, simply because a bound volume is often more convenient
than an electronic version, and futher, it is helpful to have a more formal means to cite the
work in other publications. The terms of the license require any copies or derivative works
to include a reference to the title and author. Though not strictly required, the author is
quite interested in your thoughts on this work, and any improvements or adaptations you
may make. The online home for this work can be found at http://nitinoluniversity.com,
and we encourage you to visit us there to provide your feedback, and check for updates
or new revisions as they become available. There you will also find additional resources
relating to this work, including links to the design files, spreadsheets, finite element analysis
input files, and related items.
8
Now go forth, remix, reuse, recycle. Everybody wins.
CONTENTS
Chapter 1
Basic Elements of Stent Design
The word stent is used to describe any artificial structure that is used to provide support
or scaffolding to a lumen or cavity within the body. The term is often credited to Charles
Stent, a nineteenth century English dentist [3], and came into common use the medical
field with the introduction of the Wallstent, co-invented by Hans I. Wallstén in the 1980’s
[6] [5]. Modern stents are used throughout the human body, most commonly in arteries
of the heart, neck, and lower limbs. Many other stent applications exist throughout the
cardiovascular, pulmonary, and gastrointestinal systems of the body. Stents are typically
fabricated from metals like stainless steel, cobalt alloys, or nitinol, and some polymer based
designs are also being investigated.
1.1
Introduction to Nitinol
Nitinol, a nearly equiatomic alloy of nickel and titanium, is one of many materials that
is commonly used to fabricate cardiovascular implants such as stents. Unlike traditional
engineering materials like stainless steel and cobalt alloys, Nitinol exhibits the unusual
properties of shape memory and superelasticity. These properties are manifestations of a
phase change that occurs in the material as it transitions between a higher temperature
austenite phase and a lower temperature martensite phase. The mechanical properties
of these phases are quite different, and the transition between the phases creates unusual
properties that are useful for many medical applications.
The temperature at which the phase change occurs, the transition temperature is critically
important to the mechanical properties of the finished component. More specifically, it
is the difference between the transition temperature and the environmental temperature
that dictates performance. This is one of the reasons that Nitinol works so well in medical
9
10
CHAPTER 1. BASIC ELEMENTS OF STENT DESIGN
applications: the environmental temperature of the human body is very stable, therefore
the mechanical properties of a Nitinol implant are also stable.
When the material is substantially below its transition temperature, it is fully martensitic, and has material characteristics much like soft lead. It is easily deformed, and will
remain deformed, just like any ordinary material. The unusual properties of Nitinol are
demonstrated when the material is heated above its transition temperature to return to
its austenitic phase. Now, this same material will spontaneously recover to its original
shape, as if it had never been deformed. This demonstrates the shape memory property of
nitinol.
When the material is substantially above its transition temperature, it is fully austenitic,
and has material properties more like steel than lead. It is very elastic, with much higher
stiffness than it had in the marensitic phase, though lower stiffness than typical stainless steels or other conventional engineering materials. Unlike typical materials, though,
austenitic Nitinol can be deformed to a very substantial degree, and still recover to its
original shape. This is a demonstration of superelasticity, and it is enabled by the stress
induced transformation from austenite to martensite in local regions of high stress.
For all of these reasons, the transition temperature of a Nitinol component is of critical
importance. It is commonly measured using a bend free recovery technique, wherein the
component is cooled until it is fully martensitic, deformed to a specific shae, then slowly
warmed to higher temeperatures while measuring the recovery to its original shape. Results
from a typical transition temperature test are shown in Figure 1.1. The test begins at position 1. When heated, the shape begins to recover at position 2, and completes its recover
fully by position 3. Tangent lines are drawn as indicated to establish As , the austenite start
temperature, and the more commonly used Af , the austenite finish temperature.
Figure 1.2 below illustrates typical stress vs. strain response for superelastic Nitinol in
a uniaxial tensile test, for material having a transition temperature of approximately 25
degrees C. From position 1 to 2, the material is in its austenite phase. From position 2 to 3,
the material is undergoing a transition from austenite to martnesite; this region is typically
described as the upper plateau. From position 3 to 4, the material is fully martensitic; note
that the slope in the 3-4 region is less steep than that in the 1-2 region, demonstrating
the relatively lower modulus of martensite compared with austenite. When material is
unloaded, it follows a different stress-strain path from position 4 to 5, then transitions
along the lower plateau to position 6, before full recovering to its original shape at position
1.
The Stent Calculator formulations described later can theoretically apply to any type of
metallic stent, but are especially well suited to Nitinol designs. This has nothing to do with
the unusual shape memory or superelastic properties of the material, but rather the unusually high strains that can be achieved in the linear elastic region of the austenite phase.
1.2. STENT ANATOMY
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5.0
Af
4.5
➍
➌
4.0
deflection (arbitrary units)
3.5
3.0
2.5
2.0
➋
1.5
➊
1.0
As
0.5
0.0
-10
-5
0
5
10
15
20
25
30
35
Temperature (°C)
Figure 1.1: Typical transition temperature measurement results, using a bend free recover technique
The stress-strain curve between position 1 and 2 is substantialy linear for strains of 1% to
2% in typical superelastic nitinol, which is an order of magnitude higher than the comparable level of fully recovereable strain in conventional engineering materials like stainless
steel. Because of this, the stess vs. strain relationship for nitinol stents is approximately
constant for relatively large, and often clinically relevant, range of deformations.
1.2
Stent Anatomy
Stents typically are comprised of an array of repeating structural elements commonly described as struts. These struts are generally oriented with their long dimension aligned
with the axis of the cylindrical form of the stent. Struts are typically disposed around the
circumference of a stent, and joined at alternating ends to form a series of “V” or “W”
shapes. The union of adjacent struts is commonly described as a tip, elbow, or apex. A series of struts and apices that traverses one complete circumference is commonly described
as a ring or column. Adjacent columns of struts are typically joined by bridges which
connect some or all apices according to some regular pattern.
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CHAPTER 1. BASIC ELEMENTS OF STENT DESIGN
600
➍
500
➌
True Stress (MPa)
400
➋
300
200
➎
➏
100
➊
0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Log Strain (%)
Figure 1.2: Typical uniaxial tensile test data for superelastic Nitinol
1.3
Transformations
Stents experience some important transformations during fabrication and service. Several
of these are described in the following sections.
1.3.1
Diameter Transformation
A stent must be able to transform from a small diameter during insertion and delivery to
a larger diameter at the implantation site, and in some cases repeat this cycle one or more
times. If the stent is designed at or near its intended minimum diameter, the designer
must craft features that can be fabricated at this small diameter, and expand to the
intended maximum diameter, while providing the intended strength, scaffolding, flexibility
and durability at a range of service diameters. Alternatively, if a stent is designed and
fabricated at or near its maximum diameter, the designer must assure that the features
1.3. TRANSFORMATIONS
13
can crimp, fold, or otherwise pack efficiently allowing the structure to be constrained to
the intended minimum diameter. In both cases, it is very easy to design structures that
appear compelling in their fabricated state, but fail to transform to the opposite end of
the expansion range. To avoid this unsatisfying end, the stent must be designed with both
the crimped and expanded configuration in mind.
1.3.2
Material Removal
Another important transformation that occurs during manufacturing is material removal.
Stents are commonly fabricated using a laser machining process that leaves a heat affected
zone (HAZ) of some thickness adjacent to cut surfaces. Furthermore, the tubing from
which stents are fabricated commonly have draw lines or other undesired features on their
inner or outer surfaces. For these and other reasons, material is typically removed from
the raw, or as-cut component by some combination of mechanical or chemical processes.
Consequently, the stent must be designed to be fabricated according to one set of feature
dimensions, then processed to remove a specified amount of material from each surface,
such that the features achieve some desired finished dimensional targets. Here again, the
stent must be designed with both the raw and finished configuration in mind.
1.3.3
Dimensionality and Coordinate Systems
One additional transformation is simply an engineering abstraction, albeit an important
one. Stents are typically cylindrical structures that naturally exist in a cylindrical coordinate system. However, they are typically designed in planar form, using a cartesian
coordinate system. Both are essential. The laser cut pattern for a stent must be developed in a two dimensional planar form, wherein the vertical height of the “unwrapped”
stent is equivalent to the circumference of the tube on which it will be cut. The motion
controller that reads the machine code will transform the vertical coordinates to theta coordinates, or rotational motions to fabricate the stent. While the two dimensional planar
representation is essential for fabrication, a three dimensional cylindrical, or wrapped, representation is helpful for visualizing the actual component, and is essential for simulation
and analysis.
1.3.4
Simultaneous Configurations and Constraints
Within a single instance of a single iteration of a single stent design, lie a multitude
of embodiments, all of which must be considered simultaneously to achieve a successful
design. The designer must consider:
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CHAPTER 1. BASIC ELEMENTS OF STENT DESIGN
• Crimped and expanded diameter configurations
• Raw and finished feature dimensions
• Planar and cylindrical representations
• And more. . .
Beyond these items, a design family may be comprised of a matrix of different stent lengths
and expansion diameters, and may have multiple design features within a single stent.
And it is likely that these features will be iterated many times during the design and
development process to achieve optimal performance, reliability, and manufacturability.
The combination of all of these simultaneous configurations and constraints creates an
important opportunity to apply the tools of computer aided design, or CAD.
1.4
Computer Aided Design of a nitinol Stent
While Computer Aided Design, or CAD, has come to be associated with computerized
drafting or solid modeling, in a more general sense the term applies to any computer based
techniques that can be applied to the design or development process. This manuscript
considers three interrelated elements of computer aided stent design. Each is driven by
some essential design inputs, and provides some prediction of relevant performance outputs.
1. Parametric Solid Modeling: A three dimensional solid model is developed to
programmatically create a stent design in any combination of crimped, expanded,
raw, finished, planar or cylindrical forms.
2. Stent Calculator: A series of formulas is developed to predict the strength, strain,
durability, and other performance features of a stent design on the basis of a finite
set of input parameters.
3. Finite Element Analysis: The stent design is simulated using the techniques of
finite element analysis to confirm the Stent Calculator results.
Each of these are described in the following chapters. The order of the first two is somewhat
arbitrary, and does not imply dependency. Stent Calculator and the Parametric Solid
Model both require the same input data, and neither require the results of the other.
The tools of Stent Calculator are very easy to apply to many “what-if” design scenarios
very quickly, making it a very useful screening tool. Consequently, it is best to explore
many potential designs using Stent Calculator before investing energy into any Parametric
Solid Modeling. In any event, Finite Element Analysis (FEA) is the most complex of the
three, and is typically used sparingly. In this manuscript, the Parametric Solid Modeling
1.4. COMPUTER AIDED DESIGN OF A NITINOL STENT
15
chapter is presented first with the intention of providing a visual and geometric basis for
understanding the math and discussion in the later Stent Calculator chapter.
16
CHAPTER 1. BASIC ELEMENTS OF STENT DESIGN
Chapter 2
Parametric Solid Model
It would be challenging to describe the design process for a stent without referring to an
actual stent design. Since virtually every every stent design in existence is proprietary, a
new generic design was created for this exercise. The Open Source Stent (OSS) is designed
for no particular purpose other than provide a realistic example for utilizing the tools of
computer aided stent design.
The Open Source Stent described here was designed with SolidWorks Professional 2010
(Dassault Systèmes SolidWorks Corporation, Concord, MA). SolidWorks is a widely used
commercial CAD software package that is commonly used in the medical device industry.
The source part file that is provided under the same terms as this manuscript can be edited
and manipulated with SolidWorks 2010 or higher, and can be viewed using the eDrawings
Viewer application available from http://edrawingsviewer.com.
As noted in the previous chapter, the stent must be designed with a number of simultaneous
constraints in mind. This SolidWorks part is parameter driven, allowing the user to easily
select the raw or finished state, crimped or expanded state, and planar or wrapped configuration. The following sections provide step by step details describing exactly how the
stent geometry is built, and how the model can be transformed between states. With the
detail provided, a moderately experienced user of Solidworks, or other solid CAD packages
(Pro/Engineer, Autodesk Inventor, or others) should be able to recreate the design, and
more importantly extend or customize the design to be suitable for various applications. In
the spirit of the Creative Commons, the community is encouraged to create “translated”
versions of this design, and share them with the same licensing terms.
17
18
CHAPTER 2. PARAMETRIC SOLID MODEL
Table 2.1: Global variables defined using SolidWorks equations
title name of the table
Global Variable
"finishing"=1
"expanded"=1
"N_col"=10
"N_struts"=42
"D_tube"=1.915
"D_set"=8
"t_raw"=0.17
"L_strut_inner"=1.2
"w_apex_raw"=0.13
"X_bridge"=.15
"Y_bridge"=0
"w_bridge_raw"=0.125
"N_bridges"=7
"w_kerf"=0.025
"m_width"=0.036
"m_thickness"=0.059
2.1
Comment
A logic flag to define whether the design is in a raw/as-cut state (0) or
finished state (1).
A logic flag to define whether the design is in a crimped state (0) or expanded state (1).
Numer of columns of struts along
the length of the stent.
Number of struts around the circumference of the stent.
Outer diameter of the tubing from
which the stent is cut.
Expanded diameter of the stent.
Wall thickness of the tubing from
which the stent is cut.
Length of the strut.
Width of an apex in the raw state.
Axial gap between adjacent columns
of struts
Circumferential offset for each
bridge.
Width of a bridge in the raw state.
Number of bridges around the circumference
Minimum circumferential gap between struts in the crimped state.
Amount of material removal from
feature widths.
Amount of material removal from
wall thickness.
Input Parameters and Equations
The variables and equations described by Tables 2.1 and 2.2 can all be defined before
beginning any geometry creation. In SolidWorks, the equations are entered by accessing
2.1. INPUT PARAMETERS AND EQUATIONS
19
Table 2.2: Equations to define global variables linked to feature dimensions
Equation
"D_model"=
"D_tube"+"expanded"*("D_set"-"D_tube")
Comment
"Y_strut"=
"D_model"*pi/"N_struts"
Circumferential distance occupied
by a single strut at the analysis diameter.
"w_strut"=
(("D_tube"*pi)/"N_struts")
-"w_kerf"-"m_width"*"finishing"
"w_bridge"=
"w_bridge_raw"-"m_width"*"finishing"
"w_apex"=
"w_apex_raw"-"m_width"*"finishing"
"t"=
"t_raw"-"m_thickness"*"finishing"
Diameter of the model, considering
the state of the expanded logic flag.
Width of a strut, considering the
state of the finishing logic flag.
Width of a bridge, considering the
state of the finishing logic flag.
Width of an apex, considering the
state of the finishing logic flag.
Wall thickness, considering the state
of the finishing logic flag.
"inner_radius"=
("w_kerf"+"m_width"*"finishing")/2
Inner radius of an apex
"outer_radius"=
"inner_radius"+"w_strut"
Outer radius of an apex
"L_strut_rectangle"=
"L_strut_inner"-"inner_radius"*2
"D_inner"=
"D_model"-("t"*2)
Length of the perfectly rectangular
section of a strut between apieces
Inner diameter of the stent
Tools > Equations... from the menu. When defining dimensions in sketches and features, the driving value can be linked to these global variables as shown in Figures 2.1 and
2.2. The completed global variables and equations appear as shown in Figure 2.3.
20
CHAPTER 2. PARAMETRIC SOLID MODEL
Figure 2.1: Define dimension by selecting “Link values. . . ”
Figure 2.2: Select desired global variables to link to feature dimension.
2.2
Master Strut Sketch
The first sketch in the stent part is the Master Strut Sketch. This sketch is carefully
constructed such that all of its feature dimensions are driven by global variables, and it
can reliably transform from the crimped to expanded state, and raw to finished state,
without “breaking”. The sketch is fully constrained without being overconstrained, which
2.2. MASTER STRUT SKETCH
21
Figure 2.3: SolidWorks global variables and equations driving design features
is a balance that can be challenging to achieve with stent models in SolidWorks.
The sketch is created on the front plane, and the first geometry features placed in the
sketch form a rectangle comprised of construction lines. The bottom horizontal line is
anchored at one end to the origin, and the top horizontal line is placed at a distance of
Y_strut above the bottom line. These two lines define the bounds of the strut in the
vertical or circumferential direction, and the spacing between them will vary depending
upon the crimped or expanded diameter of the stent. Vertical lines are then placed at
the left and right, forming a construction rectangle that will bound the strut at all times.
Figure 2.4 shows the master strut sketch in the crimped state, and Figure 2.5 shows the
same master strut sketch in the expanded state. Note that the horizontal length of the
construction rectangle is not defined, but rather it is dependent upon the defined length of
22
CHAPTER 2. PARAMETRIC SOLID MODEL
the strut, and the angle at which the strut is expanded. It should also be noted that the
expanded configuration assumes that the strut will be perfectly straight, while in reality
an expanded strut bend with a curvature that is too complex to represent in this simple
model. Consequently, the expanded configuration constructed in this CAD model should
be considered a visual approximation only.
Figure 2.4: Master strut sketch, with the model in the crimped state
2.3. CREATING A PLANAR UNIT CELL
23
Figure 2.5: Master strut sketch, with the model in the expanded state
2.3
Creating a Planar Unit Cell
The first sketch is preserved as a “master”, leaving it available for future use if necessary.
To achieve this, a new sketch is created for the first extrude feature, and the “convert
entities” function in the sketch module is used to create new entities that are linked to
those in the master sketch. With this technique, no new dimensions are defined in the
extrude feature, other than the thickness of the extrusion itself (which is linked to the
global variable t). The first strut is shown in Figure 2.6. After creating this first strut,
it is mirrored to form a strut pair. In the next step, the strut pair is copied twice. The
number of “copies” to define in this step depends on the pattern of bridge connections in
the desired design. In this 42-strut case, there are seven bridges around the circumference,
or one for every three strut pairs; therefore, two copies are required for this design. The
mirrored, copied, and combined strut are shown in Figure 2.7.
Once the first set of struts has been created, the bridge features are added to each end,
as shown in Figure 2.8. The combined structure now represents half of the unit cell, or
smallest repeating geometry unit in the stent. This structure is next copied in the axial
direction, forming the first elements of the second column of the design, as seen in Figure
2.9. The struts are now in an inconvenient arrangement, so a split and move features are
created to form the final combined planar unit cell geometry as shown in Figure 2.10
24
CHAPTER 2. PARAMETRIC SOLID MODEL
Figure 2.6: The first strut is extruded into solid form.
Figure 2.7: The strut is mirrored, copied, and combined.
2.3. CREATING A PLANAR UNIT CELL
Figure 2.8: The bridge geometry is sketched and extruded.
Figure 2.9: The first partial column of struts is copied to form the second partial column.
25
26
CHAPTER 2. PARAMETRIC SOLID MODEL
Figure 2.10: The struts are realigned to form the final unit cell.
2.4. CREATING A FULL PLANAR STENT MODEL
2.4
27
Creating a Full Planar Stent Model
The unit cell geometry developed in the previous section is the repeating unit that forms
the basis for the full stent geometry. However, the ends of a stent typically have some
unique geometry features that require special treatment. At very least, the bridges should
not be present at the ends of the stent, so they need to be removed. The approach, then,
is to copy the unit cell in the axial direction as many times as necessary, then modify
the end features before copying the full axial unit around the circumference. These steps
are created in the same SolidWorks part file, but using a new configuration. With this
approach, the full model can build from the unit cell model, and when both are completed,
the user can easily switch between them.
Figure 2.11 shows the first axial copying step,1 in this case creating four copies of the first
pair of columns, for a total of ten columns in the finished design. To correct the geometry
at each end, the strut pairs are split in such as way that the apex with the bridge can be
deleted with a Delete Body, and the apex without the bridge can be copied in its place
with a Body Move/Copy feature. These steps are shown in Figures 2.12 and 2.13. Finally,
the geometry is patterned around the circumference using copy and combine features, with
the final result shown in Figure 2.14.
1
Using a Linear Pattern feature may be more obvious and intuitive, but this feature does not allow
the offset distance to be derived directly from existing geometry. With the Body Move/Copy feature,
corresponding points at the right and left bridges are used to define the translation for each copy. Now,
as the geometry is modified, these points move automatically, and the copy feature always uses the correct
translation.
28
CHAPTER 2. PARAMETRIC SOLID MODEL
Figure 2.11: Master strut sketch, with the model in the crimped state.
Figure 2.12: Defining the split feature. Prior to this step, a reference plane was created by
selecting the midpoints of the struts shown. Now, the split feature is always positioned at the
correct location on the strut.
2.4. CREATING A FULL PLANAR STENT MODEL
29
Figure 2.13: After deleting the apex with the unwanted bridge, the clean apex is copied into
place.
Figure 2.14: The full planar stent geometry, ready to create a two dimensional CAD file for laser
coding.
30
2.5
CHAPTER 2. PARAMETRIC SOLID MODEL
Creating a Wrapped Unit Cell Model
As described in the previous section, the wrapped unit cell model is created as a new
configuration that builds upon the planar unit cell shown above in Figure 2.10. The first
step in building this configuration is to actually delete the planar unit cell using the Delete
Body feature, as seen in Figure 2.15. Next, in Figure 2.16, a cylinder is created, onto which
the unit cell geometry will be projected and embossed. A Wrap feature is created, along
with a new sketch on the front plane. While in this sketch, the deleted final planar unit
cell geometry is selected from the model tree. One face is selected, the Convert Entities
function in the sketch module traces the unit cell, and creates new cloned geometry for
the wrap feature that is linked back to the original planar unit cell, shown in Figure 2.17.
When the wrap feature is completed, the unit cell geometry becomes embossed on the inner
surface of the cylinder, as shown in Figure 2.18. Next, in Figure 2.19, the cylinder is cut
away, leaving behind the finished wrapped unit cell, as shown in Figure 2.20.
Figure 2.15: The planar unit cell is deleted from the model, but can still be used later to define
the wrapping geometry.
2.5. CREATING A WRAPPED UNIT CELL MODEL
31
Figure 2.16: A cylinder with its inner diameter equal to the desired outer diameter of the wrapped
stent. The thickness of this cylinder is arbitrary.
Figure 2.17: A clone of the unit cell is created by linking features in a new sketch back to the
planar unit cell created above.
32
CHAPTER 2. PARAMETRIC SOLID MODEL
Figure 2.18: The unit cell is projected onto the inner surface of the cylinder and embossed to
create a wrapped unit cell.
Figure 2.19: The cylinder is stripped away from the unit cell using a cut feature.
2.5. CREATING A WRAPPED UNIT CELL MODEL
Figure 2.20: The final wrapped unit cell.
33
34
CHAPTER 2. PARAMETRIC SOLID MODEL
2.6
Creating a Full Wrapped Model
The fourth and final configuration to create is a full wrapped stent model. The process for
creating this model is similar to that used to extend the planar unit cell model to a full
planar model above. First, in Figure 2.21, the wrapped unit cell is patterned in the axial
direction using Body Move/Copy as before. Next, in Figure 2.22, the struts are split, and
the unwanted apex is deleted. For the next step, the new apex can not be simply translated
to the empty place as before; rather, it must be translated and rotated to align properly.
To accomplish this, the apex is first copied to an area away from the stent, as shown in
Figure 2.23. This detached apex is next moved into place using mating functionality of
Body Move/Copy 2 . As shown in Figure 2.24 two pairs of coincident points are selected on
the detached apex and the stent itself. This provides adequate constraints to position the
new apex properly. This is repeated on both ends of the stent. In Figure 2.25, an axis is
created at the intersection of the top and front planes to define the central axis of the stent.
Finally, in Figure 2.26, the Circular Pattern feature is used to complete the wrapped stent
geometry. After a final combine feature, the full wrapped stent geometry can be seen in
Figure 2.27.
Figure 2.21: The wrapped unit cell is copied in the axial direction.
2
Unfortunately, SolidWorks does not allow mating alignments when copying bodies – this only works
when moving bodies. It is for this reason that the intermediate step of creating a detached apex is required.
2.6. CREATING A FULL WRAPPED MODEL
Figure 2.22: After splitting the strtus, the unwanted apex is deleted.
Figure 2.23: The new apex is copied away from the stent temporarily.
35
36
CHAPTER 2. PARAMETRIC SOLID MODEL
Figure 2.24: The detached apex is moved into place using to coincident point pairs.
Figure 2.25: An axis is placed at the center of the stent.
2.6. CREATING A FULL WRAPPED MODEL
Figure 2.26: A circular pattern feature completes the circumferential patterning.
Figure 2.27: The final fully wrapped stent.
37
38
2.7
CHAPTER 2. PARAMETRIC SOLID MODEL
Transforming the State of the Model
The completed solid part can now be easily transformed into alternate configurations. To
change from the raw to the finished state, change the finishing global variable from 0
to 1, as shown in Figure 2.28. After making the change, the user must manually trigger a
rebuild of the model by clicking the appropriate icon, or pressing Ctrl+B. The transformed
finished part is shown in Figure 2.29. To change from the crimped to expanded state,
change the expanded global variable from 0 to 1, as shown in Figure 2.30. After rebuilding
the model, the result is shown in Figure 2.31.
The wrapped and planar state, as well as the unit cell and full states for each, are controlled
by SolidWorks Configurations.3 . Figure 2.32 shows the configuration selection for the
wrapped unit cell case, and Figure 2.33 shows the configuration selection for the planar
unit cell case. Figure 2.34 depicts four possible unit cell configurations in the crimped
state. The bottom right case, a wrapped unit cell with finished dimensions, is one that
might be used for finite element analysis simulation. Figure 2.35 is a planar full stent,
in the crimped configuration, with raw dimensions – this is a configuration that might be
used to generate a laser cutting program. Finally, Figure 2.36 depicts the full stent in its
finished state – this configuration might be used to support a finished specification for the
component.
3
Ideally, the crimped or expanded state, and raw or finished state would also be controlled by Configurations Unfortunately, this is not possible because of a limitation that prevents global variables from being
controlled in a configuration design table. Reference SolidWorks Knowledge Base issue S-04428: “Can a
design table establish global variables which are used in SolidWorks equations?”
2.7. TRANSFORMING THE STATE OF THE MODEL
39
Figure 2.28: Change from the raw to finished state by editing the “finishing” global variable.
Figure 2.29: Transformation to the finished state.
40
CHAPTER 2. PARAMETRIC SOLID MODEL
Figure 2.30: Change from the crimped to expanded state by editing the “expanded” global
variable.
Figure 2.31: Transformation to the expanded state.
2.7. TRANSFORMING THE STATE OF THE MODEL
41
Figure 2.32: Change to the wrapped unit cell state using the SolidWorks configuration manager.
Figure 2.33: Change to the planar unit cell state using the SolidWorks configuration manager.
42
CHAPTER 2. PARAMETRIC SOLID MODEL
Figure 2.34: Crimped unit cell configurations. Top: Planar. Bottom: Cylindrical. Left: Raw.
Right: Finished.
Figure 2.35: Planar full stent, in the raw state. This geometry is suitable for laser cutting.
2.7. TRANSFORMING THE STATE OF THE MODEL
43
Figure 2.36: Cylindrical full stent, in the finished state. This geometry is suitable for supporting
a final component specification.
44
CHAPTER 2. PARAMETRIC SOLID MODEL
Chapter 3
Stent Calculator Formulas
This chapter details the variables and formulas used in the Stent Calculator application.
Each section focuses on a specific aspect of design or performance, and generally the results
from each section are used for further calculations in later sections. Throughout this text,
example values are provided based on the Open Source Stent design described in the
previous section. Where applicable, the example values are provided with corresponding
SI units of measure.
3.1
Stent Design Inputs
The variables below define the key aspects of stent geometry. These inputs are typically
drawn from an engineering drawing or related specification.
Ncol is the number of columns of struts along the length of the stent.
Ncol = 10
(3.1)
Nstruts is the number of columns of struts around the circumference of the stent.
Nstruts = 42
(3.2)
Dtube is the outer diameter of the tube from which the stent is fabricated, in millimeters.
45
46
CHAPTER 3. STENT CALCULATOR FORMULAS
Dtube = 1.915 mm
(3.3)
traw is the wall thickness of the tube from which the stent is fabricated, in millimeters.
traw = 0.17 mm
(3.4)
Lstrut inner is the length of a strut, as measured between the quadrants of the inner arcs of
opposite apices, in millimeters.
Lstrut inner = 1.200 mm
(3.5)
wapex raw is the width of an apex in the raw, or as-cut, state. This width may be equal
to the strut width, but it does not necessarily need to be. It is often designed to be some
multiple of a strut width (i.e. 1.0x, 1.1x, 1.2x, etc.).
wapex raw = 0.130 mm
(3.6)
Xbridge is the axial gap between adjacent columns of struts, as measured by the axial
distance between the closest points on the outer arc of adjacent apices.
Xbridge = 0.125 mm
(3.7)
Ybridge is the circumferential distance traversed by a single bridge, or the offset in the
circumferential direction between like points of corresponding adjacent apices.
Ybridge = 0.000 mm
(3.8)
wbridge raw is the width of a bridge element in the raw, or as-cut, state.
wbridge raw = 0.125 mm
(3.9)
3.2. STENT PROCESS INPUTS
47
Nbridges is the number of bridges around the circumference of the stent. Typically, this
Nstruts
value must be a factor of
. In the case of this example, with 42 struts, Nbridges =21
2
would imply that every internal apex is connected to an adjacent apex. Nbridges =7 would
imply that every third internal apex is connected to a corresponding adjacent apex. The
only other option, Nbridges =3 suggests that every seventh internal apex is connected.
Nbridges = 7
ZBDSH[BUDZ
ZBVWUXWBUDZ
WBUDZ
/BVWUXWBLQQHU
ZBEULGJHBUDZ
(3.10)
ZBNHUI
;BEULGJH
Figure 3.1: Unit cell stent geometry in the raw, or as-cut, state
3.2
Stent Process Inputs
The variables below relate to various assumptions regarding the manufacturing processes
used to fabricate the stent.
wkerf is the effective kerf width between struts when fabricated (or in the crimped state).
In the typical case of laser micromachining of stent from tubing, this is the effective width
of the laser beam.
wkerf = 0.025 mm
(3.11)
mwidth is the total amount of material removal from feature widths during finishing operations, i.e. after laser cutting is complete. Typically, the raw feature widths are planned
48
CHAPTER 3. STENT CALCULATOR FORMULAS
W
ZBDSH[
X
VWU
ZB
ZBEULGJH
W
;BEULGJH
;BFHOO
Figure 3.2: Unit cell stent geometry in the expanded and finished state
3.3. MATERIAL PROPERTY INPUTS
49
to be larger than finished feature widths to allow for effective removal of the heat affected
zone, or HAZ, of material that may be embrittled by the cutting operation. mwidth is
selected to allow for HAZ removal, as well as provide for sufficient surface smoothing and
edge rounding as required by the design.
mwidth = 0.036 mm
(3.12)
mthickness is the total amount of material removal from the wall thickness during finishing
operations, i.e. after laser cutting is complete. This is commonly greater than mwidth
because is it often desirable to remove additional material from the inner surface of the
stent to eliminate tubing draw lines or other unwanted surface features from the inner
and/or outer surfaces of the stent.
mthickness = 0.059 mm
(3.13)
Af is the austenite finish temperature of the finished component. This is the temperature
at which the transformation from martensite to austenite is complete, as measure by bend
free recovery techniques.
Af = 27◦ C
3.3
(3.14)
Material Property Inputs
The Stent Calculator application is particularly well suited to analyze nitinol designs because of the unique linear elastic behavior of the material with strains of 1-2%. In this
regime, the material is dominated by the properties of the austenite phase. The elastic
modulus of this material in this phase varies with the transformation temperature. With
Af temperatures progressively lower than body temperature, the stiffness of the material
at body temperature increases (See Figure 3.3). Understanding this relationship, Stent
Calculator can adjust the elastic modulus of the material as a function of specified Af
temperatures using the curves fit to data as shown in Figure 3.3.
EAf,low is the elastic modulus of the austenite phase having an Af temperature of Aflow .
EAf,low = 94, 000 MPa
(3.15)
Aflow is the first Af temperature at which the austenite elastic modulus is defined.
Aflow = −5◦ C
(3.16)
50
CHAPTER 3. STENT CALCULATOR FORMULAS
100!103
75!103
50!103
25!103
-8
-4
0
4
8
12
16
20
24
28
32
36
40
Figure 3.3: Relationship between initial (austenite) modulus and Af temperature for superelastic
nitinol at an environmental temperature of 37◦ C. The points shown were obtained experimentally
from nitinol tubing heat treated to achieve desired Af temperatures. The green and orange curves
were developed manually to fit the data, and reasonably reflect the expected performance of the
material. The green curve applies for Af temperatures less than 19◦ C, and the orange curve applies
for Af temperatures greater than 19◦ C.
3.3. MATERIAL PROPERTY INPUTS
51
EAf,high is the elastic modulus of the austenite phase having an Af temperature of Afhigh .
EAf,high = 34, 000 MPa
(3.17)
Afhigh is the second Af temperature at which the austenite elastic modulus is defined.
Afhigh = 37◦ C
(3.18)
Afinf lection is the temperature at which the Af vs E relationship transitions from the low
temperature curve to the high temperature curve, as shown in Figure 3.3.
Afinf lection = 19◦ C
(3.19)
The calculated value of E, the austenite elastic modulus for a material having the specified
Af , depends on the Af temperature. For Af less than Aflow :
Ecase1 = EAf,low
(3.20)
For Af between Aflow and Afinf lection , the green curve of Figure 3.3 applies:
Ecase2 = EAf,low − 1.9 · (Af − Aflow )3
(3.21)
For Af between Afinf lection and Afhigh , the orange curve of Figure 3.3 applies:
Ecase3 = EAf,high + 5.9 · (Afhigh − Af )3
(3.22)
For Af equal to or above Afhigh :
Ecase4 = EAf,high
(3.23)
In this example, with Af = 27, Ecase3 applies.
E = Ecase3 for Af = 27◦ C
E = 34059 MPa
(3.24)
52
CHAPTER 3. STENT CALCULATOR FORMULAS
ρ is the mass density of nitinol, used later to estimate the mass of the stent on the basis
of its estimated volume.
ρ = 6.7g/cm3
ρ = 6.7mg/mm3
(3.25)
f sl is the fatigue strain limit of the material. For mean strains less than 4%, Pelton et.
al. report a strain amplitude fatiuge strain limit of 0.4% for nitinol test samples fabricated
and processed using techniques representative of those used for stents.[2]
f sl = 0.4%
3.4
(3.26)
Service Parameters
This section defines the diameter to which the stent is expanded, and the diameter of the
vessel into which the stent is placed. The Analysis Diameter is also defined here, typically
equal to the vessel diameter. Various properties of the stent, including strength and strain,
are calculated at this diameter.
This section also defines the mechanical properties of the vessel into which the stent is
placed. Commonly, the compliance of a vessel is reported in terms of a percentage change
in diameter related to a specific applied pressure range. This compliance is typically defined
based on arterial, venous, or other data derived experimentally or drawn from literature.
The systolic and diastolic pressures considered in the fatigue analysis are also defined in
this section.
Dset is the expanded, or thermal shape set, diameter of the stent. This is the maximum
diameter to which the stent is expanded for any given usage case.
Dset = 8.0 mm
(3.27)
Dves is the diameter of the vessel into which the stent is placed. This is typically smaller
than Dset , the fully expanded diameter of the vessel. In this example, the stent is oversized
by 1.5mm.
Dves = 6.5 mm
(3.28)
3.4. SERVICE PARAMETERS
53
D is the diameter at which strains are calculated.
D = Dves = 6.5 mm
(3.29)
Cpercent is part of the definition for vessel compliance. This is the percent change in effective
diameter for a defined change in pressure, Cpressure . In the literature, this is often reported
as ∆D/D. The compliance in this example is arbitrary, but similar to values commonly
used in the arterial system.
Cpercent = 6%
(3.30)
Cpressure is part of the definition for vessel compliance. This is the change in pressure1
that is related to a ∆D/D = Cpercent .
Cpressure = 100 mmHg
(3.31)
Psystolic is the systolic pressure experienced at the site of stent implantation.
Psystolic = 150 mmHg
(3.32)
Pdiastolic is the diastolic pressure experienced at the site of stent implantation.
Pdiastolic = 50 mmHg
(3.33)
Pmean is the mean pressure experienced at the site of stent implantation, assuming a simple
sinusoidal pressure wave.
Psystolic + Pdisatolic
2
Pmean = 100 mmHg
Pmean =
1
(3.34)
This is an arbitrary value, not necessarily related to a physiologic pressure. Rather, it is simply the
pressure half of the definition for compliance, as reported in literature or by experiment
54
3.5
CHAPTER 3. STENT CALCULATOR FORMULAS
Stent Dimension Calculations
This section explains the calculations of a number of derived stent characteristics and
dimensions.
Ncells is the number of “crowns,” “tips,” or “cells” around the circumference of the stent.
This is equal to half the number of struts around the circumference.
Nstruts
2
= 21
Ncells =
Ncells
(3.35)
Dcrimp is the fully constrained outer diameter of the stent within its delivery sheath.
Dcrimp = Dtube
Dcrimp = 1.915 mm
(3.36)
Lstrut is the effective length of the strut, as measured between the centerlines of opposite
apices. This measurement is not easily measured or defined, so it is derived here based on
the inner strut length and the width of the apices in the raw state.
Lstrut = Lstrut inner + 2 ·
wapex raw
2
(3.37)
Lstrut = 1.330 mm
wstrutraw is the width of each strut in the as-cut, or raw, state. This value is derived
based on the tubing diameter, number of struts around the circumference, and the kerf
width.
Dtube · π
− wkerf
Nstruts
= 0.118 mm
wstrut raw =
wstrut raw
wstrut is the width of each strut in the finished state.
(3.38)
3.6. STRUT ANGLE AND DEFLECTION CALCULATIONS
wstrut = wstrut raw − mwidth
wstrut = 0.082 mm
55
(3.39)
wbridge is the width of each bridge in the finished state.
wbridge = wbridge raw − mwidth
wbridge = 0.089 mm
(3.40)
wapex is the width of each apex in the finished state.
wapex = wapex raw − mwidth
wapex = 0.094 mm
(3.41)
t is the wall thickness of the stent in the finished state.
t = traw − mthickness
t = 0.111 mm
3.6
(3.42)
Strut Angle and Deflection Calculations
This section calculates a variety of derived strut angle and deflection calculations. Angles
and deflections are calculated here for the vessel diameter, and for a diameter 1mm less
than the fully expanded diameter.
θset is the angle a single strut is deflected between the crimped state and the fully expanded
(thermal shape set) state.

D
·
π
−
D
·
π
set
crimp


180
Nstruts

=
sin 


π
Lstrut

θset
θset = 20.0 degrees
(3.43)
56
CHAPTER 3. STENT CALCULATOR FORMULAS
θd is the angle a single strut is deflected between the crimped state and the analysis
diameter.

D · π − Dcrimp · π


Nstruts

sin 


Lstrut

θd =
180
π
(3.44)
θd = 14.9 degrees
∆θd is the change in angle of a single strut between the fully expanded diameter and the
analysis diameter.
∆θd = θset − θd
(3.45)
∆θd = 5.1 degrees
2θ is the maximum included angle, or the angle between a pair of circumferentially adjacent
struts in the expanded state.
2θ = 2 · θset
(3.46)
2θ = 40.0 degrees
δd is the deflection of a single strut between the expanded state and the analysis diameter.
δd = 2 · Lstrut · sin
∆θd
2
(3.47)
δd = 0.118 mm
θ1mm is the deflection of a single strut between the expanded state and one millimeter less
than the expanded diameter.

(D
−
1)
·
π
−
D
·
π
set
crimp


180
Nstruts

=
sin 


π
Lstrut

θ1mm
θ1mm = 16.6 deg
(3.48)
3.7. STENT LENGTH CALCULATIONS
57
∆θ1mm is the change in angle of a single strut between the fully expanded diameter and
one millimeter less than the expanded diameter.
∆θ1mm = θset − θ1mm
(3.49)
∆θ1mm = 3.4 degrees
δ1mm is the deflection of a single strut between the expanded state and one millimeter less
than the expanded diameter.
δ1mm = 2 · Lstrut · sin
∆θ1mm
2
(3.50)
δ1mm = 0.079 mm
3.7
Stent Length Calculations
Xcell crimp is the axial length of a repeating unit cell (a full strut plus half a bridge on each
end) in the constrained state.
Xcell crimp
Xcell crimp
Xbridge
= Lstrut inner + 2 · wapex raw +
2
= 1.610 mm
(3.51)
Xtotal crimp is the axial length of the full stent in the constrained state.
Xtotal crimp = Xcell crimp · Ncol −
Xbridge
2
·2
(3.52)
Xtotal crimp = 15.950 mm
Xcell is the axial length of a repeating unit cell (a full strut plus half a bridge on each
end) at the analysis diameter. This differs from Xcell crimp by estimating the change in cell
length, or foreshortening, that occurs as the cell is expanded in diameter.
Xcell
Xcell
Xbridge
= Lstrut inner · cos (θd ) + 2 · wapex raw +
2
= 1.569 mm
(3.53)
58
CHAPTER 3. STENT CALCULATOR FORMULAS
Xtotal is the axial length of the full stent at the expanded diameter. Here again, this
formulation accounts for estimated foreshortening.
Xtotal = Xcell · Ncol −
Xbridge
2
·2
(3.54)
Xtotal = 15.544 mm
F S is the foreshortening of the stent, or percentage reduction in length as the stent expands
from the constrained state to the analysis diameter. This tend to underestimate the actual
amount of foreshortening experienced in a real stent, because this model assumes that the
struts act as perfectly straight beams with perfect hinges. This formulation is useful for
comparing relative foreshortening between different designs.
F S = Xcell · Ncol −
Xbridge
2
·2
(3.55)
F S = 2.54 %
3.8
Surface Areas, Volume, and Mass Estimation
Astrut is the outer surface area of a single strut, as measured in the rectangular area between
apices.
Astrut = (Lstrut inner − wkerf ) · wstrut
Astrut = 0.092 mm2
(3.56)
Rapex is the outer radius of an apex.
wkerf
mwidth
+
2
2
= 0.113 mm
Rapex = wstrut +
Rapex
Aapex is the outer surface area of a single apex.
(3.57)
3.8. SURFACE AREAS, VOLUME, AND MASS ESTIMATION
Aapex
hw
1
mwidth i2
kerf
2
= · π [Rapex ] ) − π
+ 2 · Rapex · (wapex − wstrut )
+
2
2
2
Aapex = 0.021 mm
59
(3.58)
2
Abridge is the outer surface area of a single bridge.
Abridge =
q
(Xbridge ) + (Ybridge )2 · wbridge
Abridge = 0.013 mm2
(3.59)
Acontact is an estimate of the the total outer surface area of the stent, which is also the
total area in contact with the vessel. Note that for each strut in the stent, there is half an
apex at one end of the strut, and half an apex at the opposite end of the strut; therefore,
the total number of struts is equal to the total number of apices in the model.
Acontact = (Astrut + Aapex ) · Nstruts · Ncol
+ Abridge · Nbridges · (Ncol − 1)
Acontact = 50.3 mm
(3.60)
2
Acylinder is the cylindrical area of the vessel occupied by the stent, at a length corresponding
to the analysis diameter.
Acylinder = π · D · Xtotal
Acylinder = 317.4 mm2
(3.61)
P CA is the percent coverage area, also known as percent metal area. This is the proportion
of the cylindrical vessel area occupied by the stent that is actually in contact with the stent.
This is reported at the analysis diameter.
P CA =
Acontact
Acylinder
P CA = 15.9 %
(3.62)
60
CHAPTER 3. STENT CALCULATOR FORMULAS
P OA is the percent open area, or the proportion of the cylindrical vessel area that is not
in contact with the stent. This is reported at the analysis diameter.
P OA = 1 − P M A
P OA = 84.1 %
(3.63)
A typical strut has a wedge shaped cross-section, wherein the width at the outer surface
is larger than the width at the inner surface. wstrut id is the width of a strut at the inner
surface.
wstrut id
wstrut id
π · (Dtube − 2t)
=
− wkerf − mwidth
Nstruts
= 0.066 mm
(3.64)
In the next series of formulas, Astrut id , Aapex id , and Abridge id estimate the surface area of
the inner surface of each of these features. The inner surface area is estimated by multiplying the outer surface areas, calculated above, with the ratio of wstrut id with wstrut od .
wstrut id
wstrut od
= 0.077 mm2
(3.65)
wstrut id
wstrut od
= 0.017 mm2
(3.66)
wstrut id
wstrut od
= 0.011 mm2
(3.67)
Astrut id = Astrut ·
Astrut id
Aapex id = Aapex ·
Aapex id
Abridge id = Abridge ·
Abridge id
Now, knowing the surface area at the outer surface and inner surfaces of each feature, the
volume of each feature can be estimated by multiplying the average of these by the wall
thickness.
3.9. MOMENT OF INERTIA CALCULATIONS
Vstrut = t ·
Vapex = t ·
(3.68)
3
Aapex + Astrut id
2
Vapex = 0.002 mm
Vbridge = t ·
Astrut + Astrut id
2
Vstrut = 0.010 mm
61
(3.69)
3
Aapex + Astrut id
2
Vbridge = 0.001 mm
(3.70)
3
With the volumes of each feature know, the total volume of the stent can be calculated
using a formulation similar to that for Atotal above.
Vtotal = (Vstrut + Vapex ) · Nstruts · Ncol
+ Vbridge · Nbridges · (Ncol − 1)
(3.71)
Vtotal = 5.021 mm3
mass is the estimated mass of the stent based on Vtotal and density ρ.
mass = ρ · Vtotal
mass = 33.640 mg
3.9
(3.72)
Moment of Inertia Calculations
The cross section of a typical stent strut can be approximated as rectangular, but can be
more accurately modeled as a sector of a hollow circle. The formulation for the moment of
inertia for such a section is detailed below in Figure 3.4, and the following formulas.
R, as defined in Figure 3.4 above, is the outer radius of the tubing from which the stent is
cut.
62
CHAPTER 3. STENT CALCULATOR FORMULAS
Figure 3.4: Moment of Inertia for a typical strut cross section. [4]
Dtube
2
R = 0.958 mm
R=
(3.73)
t, as defined in Figure 3.4 above, is the finished wall thickness of the stent.
t = 0.111 mm
(3.74)
wstrut is the finished width of each strut, which is required to derive the α parameter.
wstrut = 0.082 mm
(3.75)
α is the angle occupied by half the strut cross section, as defined in Figure 3.4 above.
1
wstrut
α= ·
· π · 2π
2
Dtube
α = 0.043 radians
(3.76)
I is the moment of inertia for a strut having a cross section described by a sector of a
hollow circle. I is calculated for bending about the y axis depicted in Figure 3.4.
t2
t3
3t
+
−
· (α − sin (α) cos (α))
I =R t· 1−
2R R2 6R3
3
I = 4.32 · 10
−6
mm
4
(3.77)
3.10. FORCE AND STRAIN CALCULATIONS
3.10
63
Force and Strain Calculations
The relationships between stress, load, deflection, and strain have been thoroughly documented for a variety of beam loading conditions. Force and strain related to a specified
strut deflection are based on the formulation for a beam fixed at one end, and free but
guided at the other as documented in Machinery’s Handbook [1].
Strain = ! =
Force = F =
3w
"#
L2
FL
2
F
12EI
!"
L3
F
L
FL
2
E = modulus of elasticity
I = moment of inertia, beam cross section
w = strut width
L = strut length
Figure 3.5: Beam fixed at one end, and free but guided at the other.
Fhoop is the hoop component of the force exerted by a single strut when the stent is
constrained from the fully expanded state to the analysis diameter. This is equal to F
in Figure 3.5 by the definition of the “free but guided” beam as described in Machinery’s
Handbook [1].
12 · E · I
· δd
(Lstrut )3
= 1.03 · 10−1 N
Fhoop =
Fhoop
(3.78)
Fhoop 1mm is the hoop component of the force exerted by a single strut when the stent
is constrained from the fully expanded state to a diameter one millimeter less than the
analysis diameter. This allows for later calculation of stent forces normalized per millimeter
diameter constraint.
12 · E · I
· δ1mm
(Lstrut )3
= 6.92 · 10−2 N
Fhoop 1mm =
Fhoop 1mm
(3.79)
64
CHAPTER 3. STENT CALCULATOR FORMULAS
d is the maximum strain experienced within the strut when the stent is constrained from
the fully expanded state to the analysis diameter. This is equal to in Figure 3.5 by the
definition of the “free but guided” beam as described in Machinery’s Handbook [1].
3wstrut
· δd
(Lstrut )2
d = 1.64 %
d =
(3.80)
1mm is the maximum strain experienced within the strut when the stent is constrained
from the fully expanded state to one millimeter less than the analysis diameter.
3wstrut
· δ1mm
(Lstrut )2
= 1.10 %
1mm =
1mm
3.11
(3.81)
Pressure and Stiffness Calculations
In this section, the forces and other calculations derived above are used to estimate radial
resistive force in terms that are common for bench testing.
RFhoop is the hoop component of the force exerted when the stent is constrained from the
fully expanded state to 1mm less than the expansion diameter, normalized by length in
centimeters. This value is consistent with radial resistive force type measurement (RRF)
generated from a collar type fixture. By convention, it is expressed in terms of Newtons
per centimeter length, and is thus normalized by length.
Fhoop 1mm h
mm i
· 10 ·
Xcell
cm
= 0.44 N/cm
RFhoop =
RFhoop
(3.82)
RFtrf is the true radial component of the force exerted when the stent is constrained from
the fully expanded state to 1mm less than the expanded diameter, normalized by length in
centimeters. This value is consistent with radial resistive force type measurement (RRF)
generated from a Blockwise or MSI type testing fixture. This is also expressed in terms of
newtons per centimeter length, and is thus also normalized by length, and evaluated for a
1mm diameter constraint.
3.11. PRESSURE AND STIFFNESS CALCULATIONS
RFtrf = 2π · RFhoop
RFtrf = 2.77 N/cm
65
(3.83)
Peq estimates the amount of outward pressure that could replace the effect of the stent,
when the stent is constrained from its maximum diameter to the analysis diameter. This
equivalent pressure is derived from the formulation for hoop stress in a thin walled cylinder:
σhoop = P · r/t, in combination with the formulation relating hoop force with hoop stress:
Fhoop = σhoop · t · L. Peq is derived by combining and rearranging these formulas to solve
for pressure P in terms of a known force Fhoop , radius r = (D/2), and length L = Xcell . It
is expressed in clinically familiar pressure units of millimeters of mercury, or mmHg, also
known as torr.
Peq
Peq
Fhoop
mmHg
· 75, 600.6
=
D
MPa
Xcell ·
2
= 151.9 mmHg
(3.84)
Pcontact estimates the contact pressure at the interface between the outer surface of the stent
and the surrounding vessel. This value is derived by dividing the total radial outward force
of the stent by the outer surface area of the stent. This estimates the pressure experienced
by individual endothelial cells in contact with the struts of the stent, and is expressed in
units of kilopascals.
2π · Fhoop · Ncol
Acontact
= 129.0 kPa
Pcontact =
Pcontact
(3.85)
kstent is another normalized expression of the stiffness of the stent, in terms of a ”spring
constant” describing the hoop force exerted per millimeter diameter constraint.
kstent
kstent
Fhoop
mmHg
· 75, 600.6
=
D
MPa
Xcell ·
2
= 0.069 N/mm
(3.86)
66
3.12
CHAPTER 3. STENT CALCULATOR FORMULAS
Calculating the Stiffness of the Vessel
This section considers the cyclic change in diameter expected within the vessel as a result of
pulsatile nature of blood flow, where the maximum pressure occurs at systole and minimum
pressure occurs at diastole. The actual compliance of the unstented vessel depends upon
the defined pressure differential between systolic and diastolic pressures, combined with the
compliance definition provided in the definition of Cpercent (Equation 3.30) and Cpressure
(Equation 3.31).
CVpresure is the vessel compliance pressure defined above, here converted to megapascal
units.
1
MPa
7500.6 mmHg
= 0.013 MPa
CVpressure = Cpressure ·
CVpressure
(3.87)
Vessel compliance was defined by stating a percent change in vessel diameter associated
with a change in pressure. DVlow is the diameter related to the low (or zero) pressure state.
This is assumed to be equal to the nominal vessel diameter.
DVlow = Dves
DVlow = 6.50 mm
(3.88)
Vessel compliance was defined by stating a percent change in vessel diameter associated
with a change in pressure. DVhigh is the diameter related to the high pressure state.
DVhigh = Dves · (1 + Cpercent )
DVhigh = 6.89 mm
(3.89)
Next, the hoop force in the vessel wall, F Vhoop is calculated using the thin walled cylinder
equation as in Equation 3.84. The hoop force is calculated for a length of vessel that is
equal to the length of a single cell so it can be directly compared with stent hoop forces
for a single cell.
F Vhoop = CVpressure ·
F Vhoop = 0.72 mm
DVhigh
· Xcell
2
(3.90)
3.13. BALANCED DIAMETERS OF THE STENTED VESSEL
67
Now, the change in hoop force related to a change in diameter can be expressed in terms of
a “spring constant” kvessel that is comparable to kstent calculated above. In this example,
the vessel is more than twice as stiff as the stent.
kvessel =
F Vhoop
DVhigh − DVlow
(3.91)
kvessel = 0.185 N/mm
3.13
Balanced Diameters of the Stented Vessel
The nominal vessel diameter is specified above as Dves , and the diastolic and systolic
pressures are specified above as well. The analysis assumes that the nominal vessel diameter
relates to the mean pressure, and is defined below as Dv,mean
Dv,mean = Dves
Dv,mean = 6.5 mm
(3.92)
Dv,diastolic , the diameter of the native vessel at diastolic pressure, is calculated based on
the compliance definitions given above.
Dv,diastolic
Pmean − Pdiastolic
= Dv,mean − Dv,mean Cpercent ·
Cpressure
(3.93)
Dv,diastolic = 6.31 mm
Dv,systolic , the diameter of the native vessel at systolic pressure, is calculated similarly.
Dv,systolic
Psystolic − Pmean
= Dv,mean + Dv,mean Cpercent ·
Cpressure
(3.94)
Dv,systolic = 6.70 mm
Now, these diameters can be recalculated considering the effects of an implanted stent.
Db,mean is the balanced diameter of the stented vessel at mean pressure by relating the
stiffness of the stent and vessel.
68
CHAPTER 3. STENT CALCULATOR FORMULAS
(kstent · Dset ) + (kvessel · Dv,mean )
kstent + kvessel
= 6.91 mm
Db,mean =
Db,mean
(3.95)
Db,diastolic repeats this calculation to derive the balanced diameter of the stented vessel at
diastolic pressure.
(kstent · Dset ) + (kvessel · Dv,diastolic )
kstent + kvessel
= 6.77 mm
Db,diastolic =
Db,diastolic
(3.96)
And Db,systolic repeats this calculation to derive the balanced diameter of the stented vessel
at systolic pressure.
(kstent · Dset ) + (kvessel · Dv,systolic )
kstent + kvessel
= 7.05 mm
Db,systolic =
Db,systolic
3.14
(3.97)
Strut Deflections at Balanced Diameters
Next, having calculated the balanced diameter for diastolic, mean, and systolic pressures,
the change in strut angle and strut deflection are calculated for each case as they were in
Equation 3.44 for θd , Equation 3.45 for ∆θd , and Equation 3.47 for δd .
First, at the mean diameter, strut angle is calculated, followed by the change in strut angle
between the set diameter and mean diameter, and finally the strut deflection.

Db,mean · π − Dcrimp · π


Nstruts

sin 


Lstrut

θmean =
180
π
(3.98)
θmean = 16.306 degrees
∆θmean = θset − θmean
∆θmean = 3.706 degrees
(3.99)
3.14. STRUT DEFLECTIONS AT BALANCED DIAMETERS
δmean = 2 · Lstrut · sin
∆θmean
2
69
(3.100)
δmean = 0.086 mm
Next, these calculations are repeated for the balanced diameter of the stented vessel at
diastolic pressure.

Db,distolic · π − Dcrimp · π


Nstruts

sin 


Lstrut

θdiastolic =
180
π
(3.101)
θdiastolic = 15.830 degrees
∆θdiastolic = θset − θdiastolic
(3.102)
∆θdiastolic = 4.183 degrees
δdiastolic = 2 · Lstrut · sin
∆θdiastolic
2
(3.103)
δdiastolic = 0.097 mm
Finally, these calculations are repeated for the balanced diameter of the stented vessel at
systolic pressure.

Db,systolic · π − Dcrimp · π


Nstruts

sin 


Lstrut

θsystolic =
180
π
(3.104)
θsystolic = 16.784 degrees
∆θsystolic = θset − θsystolic
(3.105)
∆θsystolic = 3.229 degrees
δsystolic = 2 · Lstrut · sin
δsystolic = 0.075 mm
∆θsystolic
2
(3.106)
70
3.15
CHAPTER 3. STENT CALCULATOR FORMULAS
Strain Values
Knowing the strut deflections relating to the balanced mean, diastolic, and systolic pressure
cases, the maximum strain can be calculated for each of these cases according to the
formulation described in Figure 3.5.
First, the strain is calculated at the nominal diameter of the vessel. This is somewhat
arbitrary, because the stent will cause the vessel to increase in diameter, so it will not be
expected experience this strain during service.
3 · wstrut
· δd
L2strut
= 1.64 %
vessel =
vessel
(3.107)
Next, the strain is calculated at the diameter of the stented vessel at mean pressure.
3 · wstrut
· δmean
L2strut
= 1.20 %
P,mean =
P,mean
(3.108)
Next, the strain is calculated at the diameter of the stented vessel at diastolic pressure. This
is the maximum strain experienced during the pulsatile cycle; at the minimum pressure,
the vessel is at its minimum diameter, and the stent is therefore smallest relative to its
fully expanded diameter.
3 · wstrut
· δdiastolic
L2strut
= 1.35 %
P,diastolic =
P,diastolic
(3.109)
Finally, the strain is calculated at the diameter of the stented vessel at systolic pressure.
This represents the minimum strain experienced by the stent during the pulsatile cycle;
at maximum pressure, the vessel is at its maximum diameter, and the stent is therefore
closest to its fully expanded diameter.
3 · wstrut
· δsystolic
L2strut
= 1.05 %
P,systolic =
P,systolic
(3.110)
3.16. SAFETY FACTOR CALCULATIONS
3.16
71
Safety Factor Calculations
As described above, the pulsatile cycling of pressure within the vessel creates a cyclic change
in vessel diameter. The stent contributes some outward force to the vessel, thus increasing
its mean diameter from Dves to Db,mean . The stent also contributes some damping to the
pulsatile cycle, because the stented vessel has a stiffness that is greater than the native vessel
alone. Consequently, the pulsatile range of the native vessel, (Dv,systolic − Dv,diastolic ), will
be reduced to a smaller range in the balanced stented vessel, (Db,systolic − Db,diastolic ).
The durability performance of a nitinol component is determined as a function of the mean
strain and strain amplitude related to the cycling of the structure between Db,systolic and
Db,diastolic , as illustrated in Figure 3.6 below.
4
Systolic
Pressure / Diameter / Strain
Strain Amplitude
3
2
Mean Strain
1
-3
-2
-1
0
1
2
-1
3
4
5
6
7
8
Diastolic
Pressure / Diameter / Strain
Figure 3.6: Mean strain and strain amplitude, as related to cyclic pressure and diameter
-2
The mean strain, mean , is calculated by averaging the strain at the systolic and diastolic
balanced diameters.
P,diastolic + P,systolic
2
= 1.20 %
mean =
mean
(3.111)
Now, the strain amplitude amplitude can be calculated as half the difference between the
strain at systolic and diastolic pressures.
P,diastolic − P,systolic
2
= 0.15 %
amplitude =
amplitude
(3.112)
Finally, a fatigue safety factor Nsf can be estimated by comparing the strain amplitude
with the fatigue strain limit defined above in Equation 3.26.
72
CHAPTER 3. STENT CALCULATOR FORMULAS
Nsf =
f sl
amplitude
Nsf = 2.59
(3.113)
Chapter 4
Stent Calculator Applications
The formulas described in the previous chapter have been implemented in the form of a
spreadsheet model, and Python code. Each will be explained in the next section.
4.1
Stent Calculator Spreadsheet
Each of the formulas detailed above has been transcribed into a Stent Calculator Spreadsheet application. In the spreadsheet format, each calculation is contained within a row,
and therefore each column can represent a unique combination of design input parameters
and corresponding performance predictions. The Stent Calculator Spreadsheet is therefore
a useful tool for conducting design explorations, “what if” analysis, and understanding
design tradeoffs. It can also be a useful tool for documenting design history and rationale.
The format of the Spreadsheet can be seen in Figure 4.1.
The Stent Calculator Spreadsheet is also useful for understanding design sensitivity and
trends, and the cause/effect relationship between input parameters and performance measures of interest. For example, one might explore the impact of changing a single design
input variable while holding all others constant, as illustrated for Strut Length in Figure
4.2. The following sections demonstrate this capability by exploring the performance trends
associated with changing the target vessel diameter, wall thickness, and strut length.
73
74
CHAPTER 4. STENT CALCULATOR APPLICATIONS
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++T;
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C"=3
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#33
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Figure 4.1: The first several rows of the Stent Calculator Spreadsheet define the input parameters
for the design.
4.1. STENT CALCULATOR SPREADSHEET
75
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Figure 4.2: In this example, a separate tab has been created to study the impact of varying strut
length while holding all other input parameters constant.
76
CHAPTER 4. STENT CALCULATOR APPLICATIONS
4.1.1
Trend Analysis: Vessel Diameter
The Open Stent design described above assumes placement in a vessel having a nominal
diameter of 6.5mm. This section applies the Stent Calculator Spreadsheet to consider
the impact of placing the the stent in a vessel ranging from 6.0mm to 7.0mm. This is a
typical scenario for nitinol stents, which are commonly indicated for use in vessels having
a specified range of nominal diameters.
7.6
7.4
Diameter (mm)
7.2
7.0
6.8
6.6
6.4
6.2
5.9
6.1
6.3
6.5
6.7
6.9
7.1
Vessel Diameter (mm)
Balanced Diameter, Diastolic Pressure
Balanced Diameter, Systolic Pressure
Figure 4.3: Balanced Diameters sensitivity to Vessel Diameter. The relationship here is substantially linear, as expected.
4.1. STENT CALCULATOR SPREADSHEET
77
1.8%
1.6%
1.4%
Strain
1.2%
1.0%
0.8%
0.6%
0.4%
0.2%
0.0%
5.9
6.1
6.3
6.5
6.7
6.9
7.1
Vessel Diameter (mm)
mean strain
strain amplitude
Figure 4.4: Strain sensitivity to Vessel Diameter. Mean strain decreases with increasing vessel
diameter, as this reduces the amount of “oversizing” experienced by the stent. Strain amplitude
is substantially constant, with a slight trend toward increasing with increasing vessel diameter, as
the larger stented vessel is slightly less stiff.
78
CHAPTER 4. STENT CALCULATOR APPLICATIONS
3.0
2.5
Nsf
2.0
1.5
1.0
0.5
0.0
5.9
6.1
6.3
6.5
6.7
6.9
7.1
Vessel Diameter (mm)
Fatigue Safety Factor
Figure 4.5: Fatigue Safety Factor sensitivity to Vessel Diameter. The slightly increasing value
of strain amplitude noted above results in a slightly decreasing safety factor with increasing vessel
diameter. All else being equal, the maximum vessel diameter (minimum oversizing) is therefore
a worse case than minimum vessel diaemter (maximum oversizing) when fatigue performance is
driven by strain amplitude.
4.1. STENT CALCULATOR SPREADSHEET
79
0.6
0.5
RRF (N/cm)
0.4
0.3
0.2
0.1
0.0
5.9
6.1
6.3
6.5
6.7
6.9
7.1
Vessel Diameter (mm)
Radial Resistive Force
Figure 4.6: Radial Resistive Force sensitivity to Vessel Diameter. RRF is virtually constant as
vessel diameter varies.
80
CHAPTER 4. STENT CALCULATOR APPLICATIONS
4.1.2
Trend Analysis: Wall thickness
The baseline Open Stent Design assumed a nominal starting wall thickness of 0.170mm.
This section applies the Stent Calculator Spreadsheet to consider the impact of using a
starting wall thickness ranging from 0.120mm to 0.220mm.
7.2
7.1
Diameter (mm)
7.0
6.9
6.8
6.7
6.6
6.5
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
Starting Wall Thickness (mm)
Balanced Diameter, Diastolic Pressure
Balanced Diameter, Systolic Pressure
Figure 4.7: Balanced Diameters sensitivity to Wall Thickness. The nominal diameter of the vessel
is 6.5mm in this example. Nearly doubling the wall thickness has minimal impact on the balanced
diameter of the stented vessel.
4.1. STENT CALCULATOR SPREADSHEET
81
1.6%
1.4%
1.2%
Strain
1.0%
0.8%
0.6%
0.4%
0.2%
0.0%
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
Starting Wall Thickness (mm)
mean strain
strain amplitude
Figure 4.8: Strain sensitivity to Wall Thickness. Mean strain and strain amplitude decrease
slightly with increasing wall thickness.
82
CHAPTER 4. STENT CALCULATOR APPLICATIONS
3.0
2.5
Nsf
2.0
1.5
1.0
0.5
0.0
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
Starting Wall Thickness (mm)
Fatigue Safety Factor
Figure 4.9: Fatigue Safety Factor as a function of Wall Thickness. The slight decrease in strain
amplitude leads to a slight increase in fatigue safety factor with increasing wall thickness, as this
increases the overall stiffness of the stented vessel. Consequently, the minimum wall thickness
condition will tend to be more critical for fatigue than the maximum wall thickness condition.
4.1. STENT CALCULATOR SPREADSHEET
83
0.6
0.5
RRF (N/cm)
0.4
0.3
0.2
0.1
0.0
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
Starting Wall Thickness (mm)
Radial Resistive Force
Figure 4.10: Radial Resistive Force as a function of Wall Thickness. RRF has a nearly 1:1 linear
relationship with wall thickness. As wall thickness doubles, the predicted RRF also doubles.
84
CHAPTER 4. STENT CALCULATOR APPLICATIONS
4.1.3
Trend Analysis: Strut Length
The Open Stent described above assumes a strut length of 1.2mm. This section applies
the Stent Calculator Spreadsheet to consider the impact of changing the strut length from
0.7mm to 1.7mm. These lengths are used for illustraion only; in reality, strut lengths less
than 1.0mm may not be feasible for this design.
7.8
7.6
Diameter (mm)
7.4
7.2
7.0
6.8
6.6
6.4
6.2
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
Strut Length (mm)
Balanced Diameter, Diastolic Pressure
Balanced Diameter, Systolic Pressure
Figure 4.11: Balanced Diameters sensitivity to Strut Length. As the strut length is decreased,
the stiffness of the stent increases dramatically. Consequently, with short struts, the balanced
diameter rises steeply to approach 8.0mm, the set diameter of the stent. The trend of decreased
pulse variability with increasing stiffness is also very appearant in this figure.
4.1. STENT CALCULATOR SPREADSHEET
85
1.8%
1.6%
1.4%
Strain
1.2%
1.0%
0.8%
0.6%
0.4%
0.2%
0.0%
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
Strut Length (mm)
Mean Strain
Strain Amplitude
Figure 4.12: Strain sensitivity to Strut Length. Strut length has a powerful influence on the
expected strain levels in the stent. Shorter struts generally increase the expected magnitude of
mean strain and strain amplitude. The local maximum in mean strain at a strut length of 0.8mm
represents the point at which the stent and vessel stiffnesses are equivalent, as seen in the next
Figure.
86
CHAPTER 4. STENT CALCULATOR APPLICATIONS
0.30
0.25
k (N/mm)
0.20
0.15
0.10
0.05
0.00
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
Strut Length (mm)
stent stiffness
vessel stiffness
Figure 4.13: k sensitivity to Strut Length. At approximately 0.8mm, the “k” (effective “spring
stiffness”) of the stent and vessel cross each other, creating the characteristic curve observed in
Figure 4.12. The k value for the vessel varies with strut length here because this value is normalized
by diameter, not normalized by length. Rather, kvessel is calculated for a length of vessel equal to
the axial length of a stent unit cell. Because the stent length is a variable in this study, so too is
kvessel .
4.1. STENT CALCULATOR SPREADSHEET
87
4.5
4.0
3.5
Nsf
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
Strut Length (mm)
Fatigue Safety Factor
Figure 4.14: Fatigue Safety Factor as a function of Strut Length. The predicted fatigue safety
factor generally increases with increasing strut length.
88
CHAPTER 4. STENT CALCULATOR APPLICATIONS
3.0
2.5
RRF (N/cm)
2.0
1.5
1.0
0.5
0.0
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
Strut Length (mm)
Radial Resistive Force
Figure 4.15: Radial Resistive Force as a function of Strut Length. RRF is very strongly influenced
by strut length, with sharply increasing strength as strut length decreases.
4.2. STENT CALCULATOR PYTHON
4.2
89
Stent Calculator Python
A future draft will describe a Python adaptation of the Stent Calculator, along with some
more advanced design exploration topics.
90
CHAPTER 4. STENT CALCULATOR APPLICATIONS
Chapter 5
Finite Element Analysis
Confirmation
A future draft will describe Finite Element Analysis (FEA) techniques used to simulate
pulsatile fatigue conditions using more sophisticated techniques that account for the nonlinear nature of the material and loading conditions.
5.1
Abaqus model
5.2
FEA results
91
92
CHAPTER 5. FINITE ELEMENT ANALYSIS CONFIRMATION
Bibliography
[1] Erik Oberg, Machinery’s handbook, Industrial Press, New York, NY, 2001.
[2] A. R. Pelton, V. Schroeder, M. R. Mitchell, X. Y. Gong, M. Barney, and S. W. Robertson, Fatigue and durability of nitinol stents, J Mech Behav Biomed Mater 1 (2008),
no. 2, 153–64, Journal Article.
[3] M E Ring, How a dentist’s name became a synonym for a life-saving device: the story
of dr. charles stent, J Hist Dent 49 (2001), no. 2, 77–80.
[4] Raymond J. Roark, Warren C. Young, and Richard G. Budynas, Roark’s formulas for
stress and strain, 7th ed., McGraw-Hill, New York, 2002, 2001278787 Warren C. Young,
Richard G. Budynas. ill. ; 24 cm. Originally published under title: Formulas for stress
and strain. Includes bibliographies and indexes.
[5] H Rousseau, J Puel, F Joffre, U Sigwart, C Duboucher, C Imbert, C Knight, L Kropf,
and H Wallsten, Self-expanding endovascular prosthesis: an experimental study, Radiology 164 (1987), no. 3, 709–14.
[6] Hans I. Wallsten and Christian Imbert, Self-expanding prosthesis, United States Patent
(1991), no. 5,061,275.
93
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