2010_a procedure for generating finite element models (fem) of abdominal aortic aneurysms with fluid...

8/6/2019 2010_A Procedure for Generating Finite Element Models (FEM) of Abdominal Aortic Aneurysms With Fluid Boundary

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A Procedure for Generating Finite Element Models (FEM) of Abdominal Aortic Aneurysms withFluid Boundary Conditions Derived from Magnetic Resonance Imaging (MRI) Velocimetry

M.S. Thesis

Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the

Graduate School of The Ohio State University

By

Mark Allen McElroy, B.S.

Graduate Program in Mechanical Engineering

The Ohio State University

2010

Thesis Committee

Samir N. Ghadiali, Advisor

Orlando Simonetti


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Copyright by

Mark Allen McElroy

2010


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Abstract

Abdominal Aortic Aneurysms (AAAs) are localized bulges in the lower aortic artery

tissue. AAAs are prone to rupture, an extremely dangerous event, in which the aorta rips open

and blood is allowed to flow freely into the bodys internal cavity. The fatality rate for ruptured

AAAs is over 50% so preventative surgery is the preferred method of treatment. However many

AAAs never rupture and the risks involved with preventative surgery are not negligible.

Clinicians therefore must decide when the risks of AAA rupture outweigh those of preventative

surgery.

The current clinical metric for determining the risk of AAA rupture is the

transverse diameter. A 5.5 cm diameter is the suggested max allowable size. As many as

20-30% of AAAs below this threshold rupture and in practice, the operating surgeon

must account for other risk factors too such as blood pressure and aneurysm shape. As a

result, the decision is no more than an educated guess based on a series of known risk

factors. There is a clinical desire for a more reliable and comprehensive AAA rupture risk

metric

Studies have shown that maximum arterial wall Von-Mises stress, calculated

using patient-specific finite element (FE) models outperforms diameter with regards to

predicting AAA rupture. Modern AAA FE models employ fully coupled dynamic fluid-


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structure interaction (FSI) techniques in an effort to accurately measure max wall stress

in-vivo and non-invasively.

Published boundary conditions (BCs) for dynamic AAA model fluid domains

typically involve standard flow rate and pressure conditions being applied at the inlet

and outlet of the model respectively. Our lab proposes using in-vivo blood velocity

measurements from phase-encoded velocimetry MRI scans to generate patient-specific

fluid BCs. A patient-specific flow rate condition is applied at the inlet matching the

velocimetry data read in at the inlet. A patient-specific downstream pressure is applied

at the outlet. This pressure BC is derived from an optimization routine which seeks to

match the modeled and measured outlet flow rates by altering the impedance at the

outlet.

To date, only one model has been run to convergence, due to a computation run

time of over 1 month. While changes were made to the pressure condition at the outlet

(a 14% increase in dynamic range) during optimization, these changes had almost no

effect of the max arterial wall stress.


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Dedication

I would like to dedicate this work to my loving and supportive parents, Paul and Laura McElroy

and to my brother, Matt McElroy


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Acknowledgments

I would like to acknowledge all those who have worked so hard to help bring this project

to fruition. I would like to thank Dr. Yu Ding for aiding with the medical imaging aspects of this

project, and Dr. Georgeta Mihai for working so hard on recruiting and scanning patients for this

project. Thank you Dr. Sanjay Rajagopalan for providing a hands-on, clinical perspective to the

project. Finally, thank you Dr. Orlando Simonetti and Dr. Samir Ghadiali for orchestrating and

funding this project.


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Vita

June 2004 ............................................................. Stow-Munroe Falls High School

Fall 2006, Summer 2006, 2007, 2008 .................. Lexmark International

March 2009 .......................................................... B.S. Mechanical Engineering, The Ohio State

University

June 2009 to present ........................................... Graduate Research Associate, Department of

Biomedical Engineering, The Ohio State

University

Fields of Study

Major Field: Mechanical Engineering


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Table of Contents

Abstract ............................................................................................................................... ii

Dedication .......................................................................................................................... iv

Acknowledgments............................................................................................................... v

Vita ..................................................................................................................................... vi

Table of Contents .............................................................................................................. vii

List of Tables ...................................................................................................................... ix

List of Figures ...................................................................................................................... x

Chapter 1: Abdominal Aortic Aneurysm Anatomy and Physiology .................................... 1

Chapter 2: Dealing with Abdominal Aortic Aneurysms Clinically ....................................... 5

Chapter 3: Abdominal Aortic Aneurysms from an Engineering Perspective ...................... 8

Chapter 4: In Depth Description of Models ...................................................................... 16

Chapter 5: Results ............................................................................................................. 39

Chapter 6: Future Work .................................................................................................... 52

References ........................................................................................................................ 56

Appendix A: How to Generate CAD data from MRI images ............................................. 59


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Appendix B: Generating a CAD model in Rhinoceros ....................................................... 71

Appendix C: Extracting Flow Rate Data from Velocimetry MRI Images ........................... 82

Appendix D: Putting it All Together in The Finite Element Model .................................... 86

Appendix E: The Optimization Routine and Running A Model ......................................... 94

Appendix F: Matlab Script Descriptions .......................................................................... 103

Appendix G: Visual Basic Script Descriptions .................................................................. 112

Appendix H: Utilizing the Ohio Supercomputing Center ................................................ 114

Appendix I: Helpful Contacts .......................................................................................... 115

Appendix J: Matlab Code ................................................................................................ 117

Appendix K: Visual Basic Code ........................................................................................ 176


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List of Tables

Table 1: Change in lumen volume with respect to sampling ......................................................... 21

Table 2: Important Dicom properties and descriptions ................................................................. 64

Table 3: Suggested time stepping scheme .................................................................................... 91

Table 4: Required fields for settings structure .............................................................................. 95

Table 5: Required fields for ADINAwithMATLAB2 ......................................................................... 98

Table 6: Matlab Script Summary .................................................................................................. 103

Table 7: Visual Basic Script Summary .......................................................................................... 113


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List of Figures

Figure 1: Comparison of healthy and aneurismal aorta .................................................................. 1

Figure 2: The composition of a healthy aortic wall with no atherosclerosis ................................... 3

Figure 3: Simplified arterial cross section ........................................................................................ 3

Figure 4: Aortic Dissection ............................................................................................................... 5

Figure 5: Examples of the 2 preventative surgeries ........................................................................ 6

Figure 6: Force diagram for a pipe under pressure ......................................................................... 9

Figure 7: Example of axisymmetric geometry ............................................................................... 10

Figure 8: Modeled blood flow in AAA ............................................................................................ 12

Figure 9: Relationship between material stiffness, pressure and flow rate .................................. 15

Figure 10: Example MRI images ..................................................................................................... 17

Figure 11: The 2D image analysis process ..................................................................................... 20

Figure 12: Sampling artifacts in spline curves ................................................................................ 21

Figure 13: Sampling artifact in lofting ............................................................................................ 21

Figure 14: Initial data in Rhino ....................................................................................................... 23

Figure 15: Rhino processing ........................................................................................................... 23

Figure 16: The development of the blood domain ........................................................................ 23

Figure 17: The solid domain ........................................................................................................... 24

Figure 18: The ILT domain wall inner boundary ............................................................................ 24
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Figure 19: Selecting a velocimetry region of interest .................................................................... 25

Figure 20: Comparison of flow rate data before and after processing ......................................... 26

Figure 21: Example of rough wall, ILT, and fluid mesh .................................................................. 30

Figure 22: Fluidic circuit diagram of the downstream impedance condition ................................ 31

Figure 23: Frequency response of 4-parameter Windkessel ......................................................... 32

Figure 24: The process for developing a pressure curve ............................................................... 33

Figure 25: Types of Nelder-Mead iterations shown in 2D ............................................................. 35

Figure 26: Convergence Chart........................................................................................................ 36

Figure 27: Iterative cycle for generating a patient specific impedance condition ........................ 37

Figure 28: Refined mesh used to generate mesh independent solutions ..................................... 38

Figure 29: Comparison of initial and final flow rate conditions for patient TK ............................. 40

Figure 30: Comparison of initial and final pressure conditions for Patient TK .............................. 40

Figure 31: Comparison of initial and final max stress values for Patient TK .................................. 41

Figure 32: Comparison of initial and experimental outlet flow rates for Patient MM .................. 42

Figure 33: Comparison of initial and experimental outlet flow rates for Patient AK .................... 42

Figure 34: Peak stress of initial and converged arterial wall ......................................................... 43

Figure 35: Artery wall stiffness variational study .......................................................................... 45

Figure 36: ILT stiffness variational study ........................................................................................ 46

Figure 37: Impedance parameter R1 variational study ................................................................. 47

Figure 38: Impedance parameter L variational study .................................................................... 47

Figure 39: Impedance parameter R2 variational study ................................................................. 48
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Figure 40: Impedance parameter C variational study ................................................................... 48

Figure 41: Comparison of input function with impedance condition ............................................ 50

Figure 42: Stress Maps for unconverged Patient MM and Patient AK models ............................. 51

Figure 43: The measured flow rate ................................................................................................ 52

Figure 44: Typical daw data set ..................................................................................................... 59

Figure 45: How to view data sets using a viewer program ............................................................ 61

Figure 46: A data set after being organized into folders ............................................................... 63

Figure 47: How to use the Crop GUI .............................................................................................. 66

Figure 48: How to use the Edging GUI ........................................................................................... 69

Figure 49: Fixing spline curves in Rhino ......................................................................................... 72

Figure 50: Generating surface A and the blood domain ................................................................ 73

Figure 51: Generating surface B and surface C .............................................................................. 74

Figure 52: Rhino geometries .......................................................................................................... 75

Figure 53: How to create a cutting surface .................................................................................... 76

Figure 54: The cutting plane does not intersect across the whole surface. .................................. 76

Figure 55: The cutting planes. ........................................................................................................ 77

Figure 56: Rhino geometries .......................................................................................................... 78

Figure 57: Joining outwall surfaces ................................................................................................ 79

Figure 58: Dealing with problems during outerwall surface joining .............................................. 80

Figure 59: A typical velocimetry image and an image with a well defined lumen ........................ 83

Figure 60: Example region of interest ............................................................................................ 83
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Figure 61: Inlet and outlet flow rate data both before and after processing. ............................... 85


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Chapter 1: Abdominal Aortic Aneurysm Anatomy and Physiology

An aneurysm is the localized bulging of

an artery. This bulge can include an enlarged

blood cavity, a thickening of the artery wall, or

both. Aneurysms can develop anywhere, but

have strong tendencies towards specific

locations, such as the aorta and brain [9].

The aorta is the largest artery in the

human body. It emerges directly from the top of

the heart, curves downward in what is known as the aortic arch, and then proceeds through the

thorax (or chest) to the abdomen. It terminates at the aortic bifurcation, located at the fourth

lumbar vertebra, where it bifurcates into the right and left common iliac arteries [10]. Directly

above the aortic bifurcation, at the end of the aorta is a common location for aneurysm

development. Aneurysms in this abdominal section of the aorta are simply referred to as

Abdominal Aortic Aneurysms (AAA).

While a clear understanding of why AAAs develop is not well understood, the

mechanism through which it occurs is. The bulge in the artery occurs due to the force of the

blood flowing through it acting on damaged or diseased tissue. Known risk factors include aging,

smoking, high blood pressure, atherosclerosis, and diseases that inflame blood vessels, such as

vasculitis [9]. Males and elderly patients are at higher risk for developing AAAs.

Figure 1: Comparison of healthy and aneurismal

aorta. Reprinted from [2].


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For simplification and modeling purposes

these layers are often grouped into 3 separate

materials that from here on will be referred to as

wall tissue, ILT tissue, and calcified tissue. Wall tissue

represents the adventia, media, and intima layers

and is the primary load bearing surface. ILT is the

blood clot like material buildup in the lumen and is

relatively soft. The calcified tissue is the very stiff

Figure 2: The composition of a healthy aortic wall with no atherosclerosis. It is divided into

3 sections: the intima (I), media (M) and adventitia (A). Reprinted from [7].

Figure 3: Simplified arterial cross section.

Reprinted from [6].


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calcium deposits that form in ILT or artery wall as shown in Figure 3. This modeling ignores any

heterogenatiy in these three materials.


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Chapter 2: Dealing with Abdominal Aortic Aneurysms Clinically

The reason aneurysms are a clinical

concern, is that they are prone to dissection

and rupture. Dissection occurs when a layer or

a few layers of the artery wall tear, and blood

begins flowing between the layers of the wall.

Rupture is similar to dissection but in this case,

the artery wall completely tears, and blood

flows freely out of the artery and into the

internal cavities of the body. Both are very

serious, often proving fatal. We will discuss the implications of rupture only. Since AAAs occur in

such a large artery, internal bleeding is extremely dangerous. The naturally occurring pressure

and flow rate in the aorta are much higher than seen in smaller arteries and as a result victims

can bleed out very quickly. Less than 50% of AAA rupture victims make it to the hospital before

dying from this internal bleeding [12]. The speed at which AAA rupture can kill makes surgical

repair an unreliable solution. Instead, a much safer course of action is to try to prevent AAA

rupture before it occurs. Current medical practice is to perform preventative surgery when

concern of AAA rupture arises. The goal of these preventative surgeries are to greatly reduce the

chance of AAA rupture occurring, in hope of preventing rupture and thehigh mortality rate

associated with it.

Figure 4: Aortic Dissection. Reprinted from [3].


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There are 2 preventative surgeries

for AAA rupture open repair and

endovascular stent grafting. Open repair

involves the abdomen being opened and the

diseased AAA vessel being replaced with a

synthetic artery. Endovascular stent grafting

involves small incisions being made in the

femoral artery so that the stent and surgical

catheters may be inserted into the arterial

tree downstream of the aneurysm. The stent is

installed inside the lumen of the AAA to relieve

pressure on the diseased wall tissue from the blood. Open repair is much more invasive,

requiring more recovery time and is more prone to complications such as infection.

Endovascular surgery is a quick and easy fix comparatively, but often requires recurring

surgeries to make adjustments to the stent. With either method, the risk of complication is not

negligible, and while AAA rupture is devastating, not all AAAs rupture. Performing either of

these surgeries unnecessarily exposes the patient to unwarranted risk. Therefore, the question

of when to perform preventative surgery must somehow be answered.

In determining when surgery should be performed, the goal should be to minimize the

total risk to the patient, in which case the moment the risk of AAA rupture outweighs the risks

involved with surgery is the correct moment. Due to the somewhat unpredictable nature of AAA

Figure 5: Examples of the 2 preventative surgeries. Open

repair (left) and endovascular stent (right). Reprinted

rom [5].


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rupture, there is no definitive way to answer this question. Experience and empirical evidence

have produced AAA diameter as the clinically suggested method for gauging rupture risk. As the

diameter of an AAA increases, evidence has shown that the chance of rupture does as well.

Common practice is to hold off on preventative surgery until the diameter of the AAA exceeds

5.5cm, or the aneurysm is expanding faster than 0.5 1.0 cm/year [1, 6, 13-15]. If either of

these values are exceeded, the risk of rupture is deemed to be greater than the risks involved

with preventative surgery; if they are not exceeded, then surgery is deemed too risky and the

usual recommendation is to wait and continue monitoring the aneurysm.

Once an AAA is detected, patients are typically monitored yearly, or in some cases every

6 months, to ensure the AAA is relatively healthy and surgery is not necessary. The AAA is

imaged, usually through ultrasound, and the diameter is approximated. While the 5.5cm and

rapid expansion rules are considered a good rule of thumb, the final decision is up to the

surgeon, who takes into account not only the diameter and growth rate, but other known risk

factors such as blood pressure and age. The result is often more of an educated guess as to

when it is appropriate to perform surgery, than a carefully evaluated risk assment. This

subjectivity is necessary due to the weak correlation between aneurysm diameter and rupture

risk. Studies by Simao da Silva, Darling, and Sterpetti all suggest that as many as 20 30% of

AAAs with a diameter less than 5.5cm rupture before exceeding 5.5cm [16-18]. Furthermore,

AAAs have been recorded to get as large as 17cm before rupturing [16].The clinical community

is aware of the weakness of the diameter risk metric, and has expressed a desire for a better

metric for determining AAA rupture risk [13, 19].


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Chapter 3: Abdominal Aortic Aneurysms from an Engineering Perspective

Rupture is a well documented and studied form of material failure in solid mechanics.

There is general agreement in the engineering community that stress is the best way to

determine if a material will fail. Stress is a measure of force per area in a material (such as

lbs/ft2). While traditional stress values are directional, when analyzing material failure, often a

scalar quantity known as von Mises stress, or effective stress is used. This is simply a way of

incorporating the stress in all directions into one scalar quantity, which can be used to asses the

total amount of stress a point is under independent of direction. Many materials have been

noted to have critical amounts of von Mises stress they can withstand before they begin to fail.

There are various definitions of material failure, but the first form, and the form that will be

referred to in this paper involves a phenomenon known as yield. Yield occurs when a solid

begins to permanently change shape. Up to a certain stress, solids will bounce back to their

original shape when unloaded, but if loaded past their yield stress the solids will not return to

their original shape naturally and instead returns to some new natural shape. While a material

yielding does not necessarily mean it will break or rupture, no material can break without first

reaching its yielding point. The statement if the stress in an arterial wall is less than the walls

yield stress, it will not rupture, but if the stress is greater than the yield stress, the artery has

already begun to fail is true by definition. This suggests that stress or perhaps the ratio of stress

to yield stress would be an ideal metric for determining if an aneurysm might rupture [20].


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While a correlation between diameter and AAA rupture does exist, it is relatively weak.

20 30% of aneurysms with a diameter under 5.5cm still rupture. Stress has the potential to be

a much better indicator than diameter. In fact, diameter being correlated with rupture is firmly

predicted within solid mechanics theory. The equation for stress in the wall of a thin walled pipe

under constant pressure, often referred to in medicine as the Law of Laplace, is shown in

Equation (1).

(1)

P is the pressure difference across the pipe wall, tis

the thickness of the pipe wall, dis the internal

diameter of the pipe, and is the stress in the wall. In

this equation, the stress and diameter are

proportional. This equation can be derived from the

force diagram shown in Figure 6. The blue arrows

represent pressure pushing the pipe apart, and the

red arrows represent the stress in the pipe wall

holding everything together. Unless the pipe breaks

open, these forces must be in balance. As the

diameter of the pipe increases, the interior cross section of the pipe increases. Since the same

internal pressure is acting on a larger area, the force exerted by the fluid, and thus the amount

Figure 6: Force diagram for a pipe under

pressure. As the diameter increases, so does

the force from pressure, p. Modified from [4].


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of force the wall must exert to maintain balance increases. Since a larger force is required of the

same amount of material in the wall, the stress (force/area) increases. While modeling an

aneurysm as a thin walled pipe is an oversimplification, the logic relating diameter and stress is

still intact. In both cases, when the amount of force required to contain the pressure becomes

greater than what the material can provide (in other words when the stress exceeds the yield

stress) it will rupture.

While diameter and stress are related, diameter does not take into account as much

information as stress does. Quantities such as blood pressure and variations in wall thickness,

which are both clinically relevant risk factors, are ignored when using diameter to measure

patient risk. However, stress can be used to capture all of these risk factors in one metric. It is

generally agreed that stress would be a better metric for AAA rupture, however accurately

measuring stresses in-vivo is a tricky problem that has not yet been perfected. While there are

many solutions for simple geometry, such as the Law of Laplace, for more complex problems

such as stress in an aneurysm, the solutions can be much more mathematically intense to

derive.

In order to better determine in-vivo

stresses, researchers began constructing

computer models of the artery mechanics to

help calculate these stresses [8]. Using a

mathematical modeling technique known asFigure 7: Example of axisymmetric geometry.

Reprinted from [8].


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finite elements (FE), often used in engineering to determine stress, researches have begun

modeling aneurysm mechanics. Early models were static with simplified axisymmetric geometry

(see Figure 7), due largely to computational constraints [8]. With the concept proven,

researches began to look for ways to improve upon these models.

Many of the first improvements involved the material models used to describe the

mechanical properties of arterial tissue. Mechanical material modeling involves creating a

relationship between force and material deformation. Due to the highly heterogeneous, multi-

scale and largely varying properties of living tissue, modeling biological materials properly can

be difficult. Clinical efforts are ongoing to properly define the properties and variations of

arterial tissue [13, 21, 22]. Many computational models have been proposed [19, 23, 24]. The

latest material models are anisotropic, meaning the artery tissue reacts differently when

stressed circumferentially as opposed to longitudinally. This makes sense due to the

organizational nature of tissue layers in the artery wall. However the most commonly used

material model is the isotropic hyperelastic model first proposed by Raghavan in 2000 mostly

due to the easy of application [24].

In 1999 David Vorps lab at the University of Pittsburgh made the jump to creating 3

dimensional patient-specific models [25-27]. This was a vast improvement over the previous,

idealized, and often axisymmetric models. These models were generated using image analysis

techniques to extract data from computed tomography (CT) scans.


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The inclusion of the intraluminal thrombus

(ILT) by Wang in 2002 proved to be a significant

improvement [11]. As mentioned earlier, the

intraluminal thrombus is a fatty buildup within the

inner layers of the wall with vastly different material

properties than the arterial wall tissue. While this

tissue is usually considered unhealthy and related to

atherosclerosis, FE models have shown it to help

distribute forces uniformly through the wall and

thus reduce peak stresses. Efforts soon followed to

do the same with respect to calcium deposits in the tissue [6, 28]. There is currently no

consensus on the effect of calcium deposits on wall stress.

More recently a push has developed to improve the accuracy of the loading conditions.

Past models have applied a uniform systolic pressure force normal to the surface of the lumen

boundary. In reality, the abnormal geometry of aneurysms is known to often cause atypical

blood flow patterns as shown in Figure 8 and these complex flow patterns can push harder on

the artery wall in some places than in other. To capture these complex forces, the blood flow

must be modeled in the lumen and allowed to exert force on the wall tissue, instead of applying

loads directly to the inside of the wall. Computer models in which solid and fluid domains are

allowed to interact with each other are known as fluid-structure interaction (FSI) models. Some

of these models are static and solve for stresses assuming constant unchanging blood flow in

Figure 8: Modeled blood flow in AAA. Reprinted

rom [1].


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the artery [1, 29], while others model the flow dynamically, and as a result also capture changes

in stress due to the dynamic fluctuations in the loads [14, 15, 30]. One of the problems with FSI

modeling in-vivo is how much more information is required to model both the fluid and in some

cases the dynamics. In dynamic models this additional information comes in the form of 2 fluid

boundary conditions (BC). A standard cardiac flow rate curve for the aorta is applied at the inlet

and a standard pressure curve is applied at the outlet. These are used to model the blood flow,

which then exerts time varying and complex forces on the artery wall.

These improvements have been vital in creating a more accurate model for predicting

patient in-vivo stresses. In 2003 Fillinger showed that stress values in his models were 12% more

accurate than diameter at predicting rupture [31]. While beating aneurysm diameter as a

predictor is promising, the goal is to predict rupture and no model has yet proven itself able to

distinguish between aneurysms near rupture and those that are not with any statistical

significance. Therefore, the push to create a more accurate model continues. There are

concerns that current FSI models dont have enough measured data to validate their results and

that until the stresses produced from FSI models can somehow be validated, they cannot be

trusted to reflect the actual system. If the modeled blood flow is incorrect for some reason, the

stresses the model predicts will be incorrect as well. In hopes of validating their fluid models,

some researchers have begun coupling their FSI models with experiments to reproduce the flow

in a laboratory environment, allowing the computational and experimental results to be

compared [29, 32].


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Our lab is looking for ways to further validate these FSI models computationally. By

developing patient-specific fluid boundary conditions, assumptions regarding flow rate and

pressure can be relaxed. MRI velocimetry data is utilized to measure the inlet and outlet flow

rate of the modeled section of artery. A patient-specific impedance condition is then

determined by selecting the impedance which maximizes the agreement between the modeled

downstream flow rate and the flow rate measurement extracted from MRI velocimetry images

at the downstream boundary. This impedance condition is then used to define what pressure

the blood is under through the cardiac cycle at the outlet. One weakness of this approach is that

changes to the material properties of the artery wall will directly affects the amount of pressure

necessary to create a specific downstream flow rate. This requires the assumption that the

patient-specific impedance condition is only patient-specific as long as the wall material

properties are correct. As shown in Figure 9 varying the material properties will alter the

downstream flow in the same way changing the pressure can. Therefore the amount of pressure

required to produce a specific flow rate is dependant on the assumed material properties.


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Figure 9: Relationship between material stiffness (A) and downstream pressure (B) with respect to downstream

flow rate


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Chapter 4: In Depth Description of Models

The modeling procedure described below generates a computational finite element (FE)

model that simulates the mechanics of an abdominal aortic aneurysm (AAA). It is a dynamic fully

coupled fluid-structure interaction (FSI) model developed using patient-specific geometry and

in-vivo blood velocity measurements from phase encoded velocimetry magnetic resonance

imaging (MRI) scans. The arterial wall is subdivided into 2 different materials arterial wall

tissue and intraluminal thrombus (ILT), each with their own material models. The measured in-

vivo velocity data is utilized along with the patients brachial diastolic blood pressure to

generate a patient-specific downstream impedance condition relating the blood flow to

pressure that can then be utilized in modeling the blood flow.

The procedure begins with a series of MRI scans, lasting approximately 1 hours. The

MRI machine is a 1.5T strength MRI capable of achieving 1x1x1 mm spatial resolution and

performing phase contrast acquisition. A T1 weighted 3D anatomical scan is performed over the

diseased section of the artery. Also, using MRI phase contrast data, the in-vivo velocity is

measured at the upstream and downstream boundaries of the anatomical data. The location of

a given velocity measurement within the cardiac cycle is measured by recording the amount of

time after the patients R-wave the image was taken. The R-wave is the electrocardiogram signal

which signifies the beginning of ventricular contraction in the heart.


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It is important that the velocimetry

data be taken as close to the boundaries of

the model as possible, as this is where they

will be applied. Also, the phase shift

between the 2 velocity data sets, resulting

from the time it takes for pressure waves to

propagate down the modeled section of

aorta is important in the optimization of the

impedance condition. Typically, the velocity

is measured in the 2 in-plane directions of

the image as well as in the out of the plane

direction. Due to noise, only the out-of-

plane data is utilized, so the velocimetry

images should also be taken as orthogonal

to the direction of flow as possible. Since in-

plane motion is ignored, efforts should be

made to measure the velocity in a location

where there are no flow artifacts that might

produce excessive in-plane motion, such as

circular flows. The patients brachial diastolic Figure 10: MRI images. (A) is a slice of the anatomicalscan. (B) is the out-of-plane velocity map. (C) is an in-

plane velocity map in the anterior-posterior direction


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pressure is also taken during this time. Efforts must be made to record an appropriate brachial

pressure, as patients might suffer from stress induced high blood pressure during testing. Once

all the required data is collected, construction of the model can begin.

The anatomical MRI data is run through an in-house image analysis program created

using Matlab, which walks the user through the image analysis procedure. It begins with

capturing the luminal opening using edge detection. The program first performs some

preliminary 3D median pre-filtering (Figure 11.2). During this process each voxel is assigned a

value equal to the median value of all the voxels within its 3D neighborhood. In this case, the

neighborhood is defined to be a 3x3x3 pixel volume.

Then, a series of 2D image analysis procedures for detecting a 2D edge are implemented

as shown in Figure 11. The 2D process begins by enlarging the images using bicubic interpolation

(Figure 11.3) and applying a 2D median filter to the image (Figure 11.4). The filter neighborhood

in this case is a 2D square of pixels. The size is defined by the user. Edge detection is performed

using the Canny algorithm (Figure 11.5). This algorithm utilizes 2 different thresholds. The main

or primary threshold defines how much of a change in pixel intensity is required before an edge

exists. The lower the primary threshold, the more edges will be detected in an image. The

secondary threshold also detects edges, however any edge detected using the secondary

threshold is only kept as an edge if it touches an existing primary edge. In this way, the

secondary threshold can control how long a detected edge is or how many times it branches off.

The lower the secondary threshold, the longer and more complex edges appear. After edge


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detection is performed, a morphological closing operation is implemented on the 2D image

(Figure 11.6). This procedure, also referred to as a flood-fill procedure, increases the thickness of

all the detected lines a user specified amount, and then erodes the added pixels away again.

Anywhere where 2 edges were joined during the flood, they will remain joined after the pixels

are eroded away. This helps ensure that the edges detected are closed and define a body of

points. The parameters in each of these process steps (magnification, filter neighborhood size,

edging thresholds, closure radius) can be altered in real time by the user and an image of the

detected edges overlaid onto the pre-filtered image is available to check the accuracy of the

detected lumen (Figure 11 .7). Once a series of 2D bodies have been defined, the user is asked

to select the body in each image which represents the artery lumen (Figure 11.8).


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Figure 11: The 2D image analysis process. Raw (1). Pre-filtered (2). Enlarged (3). Filtered (4). Edge detected (5).

Closed (6). Overlay (7). Final lumen body (8)

Once the lumen has been completely defined, the next step is to define the outer

boundary of the artery wall. The anatomy of an artery is such that the outer edge of an artery is

poorly defined and cannot be automatically extracted using any type of edge detection.

Therefore, the user is prompted to hand draw the outer boundary for a subset of the slices.

Both the luminal and outer boundaries are recorded as a series of points. These points are then

written to a Rhinoceros command file in such a way that only every 4th slice is considered and


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the points in each slice are connected using a closed 6th

order spline curve. The reduced

sampling and high order spline curves were determined empirically in order to prevent high

frequency oscillations in the surface that are obvious artifacts of the sampling. Figure 12 shows

this sampling artifact in the spline curves, and Figure 13 shows it when the curves are lofted

together. Table 1 has data regarding the change in volume of the models with regards to slice

sampling. The change to the model volume as a result of this resampling is only about 1%.

Figure 12: Sampling artifacts in spline curves

Figure 13: Sampling artifact in lofting. (A) 1 for 1

sampling. (B) 1 for 2 sampling. (C) 1 for 4 sampling

Table 1: Change in lumen volume with respect to

sampling

Lumen Volume

(cm3)

% Change in

Volume

1 for 1 14.88 0.00

1 for 2 14.94 0.39

1 for 4 15.03 1.02


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Rhinoceros is a computer aided design (CAD) package based on non-

uniform rational B-splines (NURBS) instead of line segments like traditional CAD

packages. As a result, Rhino is excellent for smoothing out digitized geometry

extracted from the MRI scans into smooth and continuous shapes. In

Rhinoceros, the command file exported from Matlab is read in as a stack of 2

dimensional spline curves a shown in Figure 14. The lumen boundary curves are

lofted together and then caped at the ends to form the fluid domain. Due to the

resolution of the MRI, many defining features of the wall cannot be resolved

such as calcium deposits or the boundary between the artery wall and

intraluminal thrombus (ILT). For that reason, the thickness of the arterial wall

Figure 14: Initial

data in Rhino

Figure 16: The development

of the blood domainFigure 15: Offsetting the outer surface 1.8mm (center) and

subtracting the blood domain (right)


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tissue is assumed to be 1.8mm. The arterial wall spline curves

are lofted together and the resulting surface is offset inwards

1.8mm. The inset wall surface and lumen surface then act as

the outer and inner boundaries of the intraluminal thrombus

(ILT). To ensure the wall has uniform thickness, the outermost

boundary of the combined fluid and ILT domains is then offset

1.8mm out. The arterial wall is defined as the body between

these 2 surfaces. Since the outer boundary of the ILT was

defined to be 1.8mm in from the defined arterial wall curves,

the majority of the arterial wall is exactly the

same as the surface generated from the

hand drawn outer boundary. Wherever the

fluid domain extends through the ILT layer, however, the artery wall is

expanded slightly past this surface to ensure a 1.8mm thick boundary around

these protrusions. Once the 3 domains (blood, ILT, and wall) are defined,

the solid and fluid domains of the model are exported from Rhinoceros

separately as parasolid files (.x_t).

Figure 18: Both the ILT domain (left)

and wall inner boundary (right) are

derived from combinations of the

lumen boundary and inset outer

boundary

Figure 17: The solid

domain. The thick

spots are places

where the surface

was deemed too thin


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In order to obtain flow rate data

from the MRI velocimetry images, the

images are loaded into another in-house

image analysis program written in Matlab.

This program prompts the user to select a

region of interest (ROI) at each end. It then

records the flow rate through that region

for each image in the stack by averaging

the velocity over the region and

multiplying it by the area. The trigger time

property, recorded in the Dicom images, is

used to place each data point in time. This

property records how long after the trigger event, in this case the patients R-wave, the image

was taken. Doing this for the inlet and outlet of the model results in flow rate vs. time curve

defined at both ends of the model. To enforce a steady state flow, the period and flow volume

per cycle of the outflow is scaled to match the inlet flow. This prevents the volume of model

from slowly changing across many cardiac cycles. Also, a running average is applied to the data

to help dampen out noise. Figure 20 compares these raw and processed flow curves.

Figure 19: Selecting a region of the velocimetry image to

calculate the flow rate through


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transitory (meaning it is sometimes laminar and sometimes turbulent). This affect the fluid flow

because turbulent flow dissipates kinetic energy through the use of eddies while laminar flow

does not. While the flow could be modeled as turbulent, this is ill advised due to a lack of

information regarding the amount of energy being dissipated. To combat this flaw flow-

condition-based interpolation (FCBI) elements are used instead of traditional finite fluid

elements. FCBI elements are a hybrid between the finite element and finite volume

mathematical techniques for solving fluids models [33]. They utilize the Petrov-Galerkin finite

element formulation Equations (2)-(3) to solve the incompressible Navier-Stokes Equations.

(2)

(3)

The Petrov-Galerkin method is a virtual work method for finding the minimum energy solution

to a given energy equation. In this case vandp represent a descretized and interpolated

solution for the real velocities and pressures of the solution while wand q represent small

changes to the velocity and pressure solution. The energy will be minimized when infinitely

small changes in either of the imaginary variables have no effect on the energy. In variational

calculus, this is known as a stationary point and it signifies a maximum or minimum. When

dealing with energy functions, there is typically no maximum value, and so this is equivalent to

solving for a minimum. Typically the interpolation equation used to represent vis also used to

represent w. This is known as the Bubnov-Galerkin method. In the case of FCBI elements,


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however, the vfunction contains exponential terms dependent upon the Reynolds number and

the wfunction is defined as a step function. During the solution process, the exponential terms

allow the solution to automatically account for upwinding effects, while the discontinuous

virtual velocity function ensures that interpolation between nodes perfectly captures the

function shape. This results in conservation of both mass and momentum at the element scale.

By enforcing conservation laws and accounting for upwinding effects, FCBI elements can model

highly turbulent flows well without the use of a turbulence model [34].

At the inlet to the fluid domain a velocity boundary condition (BC) is chosen to produce

the measured inlet flow rate for the given area of the inlet. This is done simply because ADINA

does not support a flow rate BC. Applying a uniform velocity across the opening assumes the

flow can be modeled as plug flow. A pressure BC is applied to the outlet. The source of this

pressure BC comes from the downstream impedance condition and the downstream flow rate

velocimetry data. The remaining surfaces are constrained by the solid domain through a fully

coupled FSI BC. This BC actually incorporates kinematic, velocity and stress requirements shown

in Equations (4)-(6) all into one BC.

(4)

(5)

(6)


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The kinematic constraint requires the fluid and solid surfaces remain fixed to one

another and displace together. The velocity BC is the no-slip condition, specifying the velocity of

the fluid at the wall to be 0, and the stress BC requires the stress to be continuous across the

boundary.

When the solid model is loaded into ADINA, there will be 2 separate bodies the wall

and the ILT. At the shared surface between these 2 bodies, they must be linked together,

physically attaching them to one another. If this were not true, the wall and ILT would be able to

slide against one another. The arterial wall tissue is modeled using the Mooney-Rivlin

hyperelastic material model proposed by Raghavan [24].

(7)

Where Wis the strain energy and I1 is the first invariant of the left Cauchy-Green tensor.

and

are empirical values derived from uniaxial tensile loading of AAA wall tissue excised during

surgical repair equal to 174 kPa and 372 kPa respectively. The ILT tissue is modeled using the

linear elastic model developed by Di Martino [30, 35], where the Youngs Modulus is 110 kPa

and the material is incompressible (Poissons Ratio = 0.49). Other than the FSI BC on the inner

surface of the combined solids the only other BCs are at the upper and lower ends of the model.

At the top surfaces of the model, both bodies are fully constrained. While this is admittedly non-

realistic, it is required to enforce the fluid flow rate BC at the inlet. Since the flow rate BC is

defined as a velocity BC (because ADINA does not support flow rate BCs) it requires the area of

the inlet to remain constant. Otherwise, the velocity BC will no longer represent the desired


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flow rate. At the outlet, the bodies are only constrained in the out-of-plane direction. This is

commonly referred to as being longitudinally tethered in the literature and is a better reflection

of the true boundary. No BC is applied to the outer surface of the model. This assumes any

stabilizing effects from the loose connective tissue surrounding the aorta are negligible.

The fluid and solid models are solved using a direct FSI algorithm. Each body is meshed

using linear elements at a density of about 2 mm. This mesh is optimized for speed, not

accuracy, and does not claim to be mesh independent. Once a model converges to an

impedance condition, the mesh can be refined to eliminate mesh dependence, and then rerun

through the optimization routine. This speeds up the procedure by minimizing the number of

times a dense mesh required to generate a mesh independent solution is run. The mesh initially

used is shown in Figure 21. It consists of 34k 1st

order fluid elements, 25k 1st

order arterial wall

elements and 33k 1st order ILT elements.

Once the FE models are constructed, the

downstream impedance condition can be

considered. The downstream impedance

condition is a time varying relationship between

the amount of blood leaving the model (outlet

flow rate) and the amount of resistance to that

blood flow (downstream pressure). This complexFigure 21: Example of rough wall, ILT, and fluid

mesh (left to right) used during optimization


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relationship results from the network of

arteries downstream of the model. Each artery,

like a pipe, resists fluid flow. This resistance is

related to the length and diameter of each

artery in the network. Because the arteries are

compliant the artery diameter is related to

pressure and the artery material properties too

and so the impedance condition depends not

only on length and diameter of every

downstream artery, but also pressure and

compliance. The actual impedance condition

for an arterial tree is therefore much too

complex to derive directly. Instead simplifications are made. The Windkessel 4 parameter model

is a common way to approximate the impedance condition in large arteries [36]. Mathematically

it is represented by the transfer function shown in Equation (8).

(8)

Where R1, R2, L, and C are 4 constants, j is the imagina ry number, and is the frequency of the

input. Using the electrical-fluid flow analogy, this impedance condition can also be depicted

using as shown in Figure 22, where the parameters of each component correspond to the

Figure 22: Fluidic circuit diagram of the downstream

impedance condition


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constants in Equation (8). R1 and

L represent a low-pass filter in

which the gain is determined by

R1 and the cut-off frequency is

determined by L. In a similar

way, R2 and C represent a high-

pass filter in which the gain is

determined by R2 and the cut-off

frequency is determined by C. The

result of combining these 2 effects is a

notch filter, which attenuates signals within the 2 cut-off frequencies while amplifying high

frequency signals by R1 and low frequencies by R2 as shown by the bode plot in Figure 23.

One of the downsides to this method of creating a pressure curve is that it can only

react to changes in flow rate and cannot detect any type of baseline pressure. The pressure

curve resulting from the Windkessel model assumes the pressure is zero when the flow rate is

zero. In order to sidestep this pitfall, the patients brachial diastolic pressure is added to the

pressure curve, making the pressure equal to the brachial diastolic pressure when the flow rate

is zero and preserving the dynamic pressure trends. This of course assumes that the artery is at

diastole when the flow rate is zero, and that diastole in the aorta is the same pressure as

diastole in the brachial artery. Since the outlet flow rate is known from MRI velocimetry images,

Figure 23: Frequency response of 4-parameter Windkessel

0

50

100

150

200

Magnitude(dB)

10-8

10-6

10-4

10-2

100

102

90

135

180

225

270

Phase(deg)

Bode Diagram

Frequency (rad/sec)


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outlet pressure curves can be

derived through transforming

the flow rate to a pressure

using the Windkessel model

and then adding the measured

diastolic pressure to the

resulting pressure curve as

shown in Figure 24. Note that

changing the 4 parameters of

the Windkessel model will

alter the final pressure curve

independent of the measured

flow rate data. This process is

totally automated using in-house

Matlab code.

In order to calibrate the impedance condition to the patient, an outlet pressure curve is

derived from an initial guess of the correct impedance condition. The pressure BC is imported

into the ADINA FE model and the model is run to completion. Once the model has completed

running, the downstream flow rate vs. time curve from the model is exported to Matlab where

it is compared with the MRI velocimetry measured flow rate vs. time curve. The error in

agreement between the curves is measured as the total area between the curves squared. This

0 0.5 1 1.5-50

0

50

100

150

200Experimental Flow Rate

Flow

Rate(cm

3/s)

0 0.5 1 1.5-1

-0.5

0

0.5

1Impedance Transformed Pressure Curve

Pressure

(kPa)

0 0.5 1 1.510

10.5

11

11.5

Final Offset Pressure Curve

Pressure(kPa)

Time (s)

Figure 24: The process for developing a pressure curve.

Transform the experimental flow rate (middle) and then add the

diastolic pressure (bottom)


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error in flow rate agreement is then minimized by varying the 4 impedance parameters. The

optimization routine systematically selects values for the 4 parameters of the impedance model,

derives pressure curves based on those selected impedance parameters, and then runs a FE

model with that pressure curve to completion. Once completed, the error in flow rate

agreements is calculated for that specific model. This process is repeated until the functional

value has converges to some minimum amount of error. Once the model is converged, the

impedance condition is said to be calibrated to the patient. This process is automated using a

series of custom made Matlab and Visual Basic programs.

The optimization routine used is the Nelder-Mead simplex method which is a non-

gradient method. This routine makes decisions regarding which impedance parameters to

evaluate next by considering a simplex of points (in this case a simplex is 5 points). At the

beginning of each iteration, the largest functional valued point in the simplex is reflected across

the centroid of the remaining simplex points. Based on the evaluation of this point, a decision is

made regarding which point to try next. The method chooses to either end the iteration,

expand, contract inside, contract outside or shrink [37].


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If the reflected point is not as good as the best point, but

better than the 2nd worst point, the reflected point is chosen to

replace the worst point in the active simplex and the iteration ends.

If the new point is less than the lowest value in the simplex,

expansion is attempted. A point is calculated which lies further in the

same direction than the reflected point did. If the expansion point is

less than the reflected point it replaces the largest point in the

simplex, otherwise, the reflected point replaces the largest point.

If the reflected point is better than the worst point in the

simplex but not better than any others, an outside contraction is

performed. Another point is evaluated closer to the simplex than the

initial reflected point. The best point between these 2 replaces the

worst point in the simplex.

If the reflected point is worse than any point currently in the

simplex, a contract inside operation is performed. During this step, a

point close to the centroid is evaluated. If the new point is better than

the worst point in the simplex, replace the worst point with it. If the

new point is still not better than current worst point, a shrink

operation is performed. This means that all of the current points in the

Figure 25: Types of Nelder-

Mead iterations shown in 2D.

From top to bottom:

reflection, expansion,

contract outside, contract

inside, shirnk.


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simplex (except the best

one) are brought towards

the best simplex point.

A proper

convergence criterion for

the optimization routine has

not yet been decided.

Convergence is currently

declared when the lowest 5

function values (the active simplex) are all within 10-15 of each other. This was chosen based on

an observed lack of improvement in the functional values at this point as shown in Figure 26,

and an r2

value greater than 0.995. The impedance parameters were still steadily changing value

at this point which suggest that while the function appears to be close to convergence, it is

not, by definition converged. Due to the large r2 value, an argument can be made that the

functional value is close to convergence, however, the impedance parameters cannot be

referred to as patient-specific until they also converge. Since the functional value is of primary

importance, this is acceptable, however in order to develop a patient-specific impedance

condition, more iterations would be necessary.

Initial attempts at optimization utilized the Broyden-Fletcher-Goldfarb-Shanno (BFGS)

gradient based method. Gradient based methods such as BFGS determine the direction of

0 10 20 30 40 50 60 70 80 90 1007.5

8

8.5

9

9.5

10x 10-11

Iteration

Error(m3/entiremodel)

Convergence Chart for Unconstrained Optimization of Patient TK

Figure 26: Convergence Chart


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steepest decent by calculating the gradient at the operating point, usually through a finite

difference scheme. They then move in that direction towards a minimum value before

recalculating the gradient and changing the direction of search. Gradient methods traditionally

outperform non-gradient methods in most cases. They can however become unstable when

dealing with signal noise and discontinuous functions due to the impact these have on

calculating a reliable gradient value. Reliably calculating a gradient value was the reason the

Nelder-Mead method was eventual chosen over the BFGS method despite BFGSs better rate of

convergence. Since each function evaluation requires running a complex FE simulation, the

whole optimization routine has proven very computationally expensive and can require literally

hundreds of FE models to run before convergence is reached.

Figure 27: Iterative cycle for generating a patient specific impedance condition. Red denotes steps taken in Matlab,

while blue represents steps in ADINA

Optimization RoutineSelects an Impedance

Condition

A Pressure Curve thatmatches that ImpedanceCondition is Generated

The FE Model isModified to Include the

New Pressure Curve

The FE Model is Run toCompletion

The Outlet Flow Rate isExtracted From The

Model Results

The Modeled Flow Rateis Used to Calculate the

Error


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Once the model has converged using a

minimalistic mesh, the mesh must be refined to assure

a mesh independent solution. Figure 28 shows the

refined mesh used after the impedance was

optimized. This improved mesh contains 111k 1st

order fluid elements, 58k 1st order wall elements, and

117k 1st order ILT elements.

Figure 28: Refined mesh used to generate

mesh independent solutions


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Chapter 5: Results

In application, deriving a patient-specific impedance condition has proven time

consuming. Once the images are received, generating a model ready to be used in the

optimization routine takes anywhere from 4 hours to a couple days depending on what type of

problems are encountered, usually with regards to the validity of the geometry in Rhino, or the

linking of the two solid bodies in ADINA. With more practice and experience, this will likely drop

considerably Each Finite Element (FE) model then takes approximately 5-7 hours to run when

given two 3 GHz, 64-bit processing nodes, and requires about 2 Gb of RAM. Since this large

computation time results more from the FSI iterations and time steps than from the number of

degrees of freedom in the model, a large diminishing return is seen when parallelizing these

models across many cores. The Patient TK model required 161 models to be evaluated before

converging (functional simplex range less than 10-15). The model therefore required

approximately 1000 hours (41 days) of run time to converge starting from a standard

Windkessel impedance value for the abdominal aorta [36]. Figure 29 compares the initial and

final modeled flow rates with the experimentally measured one, and Figure 30 shows the initial

and final outlet pressure boundary condition applied to the model. Figure 31 then compares the

calculated max stress values of the converged and unconverged models.


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Figure 29: Comparison of initial and final flow rate conditions for patient TK

Figure 30: Comparison of initial and final pressure conditions for Patient TK

0 0.5 1 1.5-50

0

50

100

150

200

OutletFlow

Rate(cm

3/s)

Time (s)

Comparision of Outlet Flow Rates

Experimental

Standard

Converged

0 0.5 1 1.510

10.2

10.4

10.6

10.8

11

11.2

11.4

11.6

11.8

12

DownstreamP

ressure(kPa)

Time (s)

Downstream Pressures Derived from Impedance Conditions

Standard

Converged


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Figure 31: Comparison of initial and final max stress values for Patient TK

While solid conclusions cannot be drawn from a single data set, the data comparing the

converged and initial conditions for Patient TK suggests that generating a patient-specific

downstream impedance condition has almost no impact on the calculated maximum wall stress.

While the range of dynamic pressures increased by about 200 Pa (14%) the max wall stress

changed by less than 0.1%. While a much larger change in the ILT stress (15%) occurred, the

stress experienced by the ILT is not typically considered when predicting AAA rupture. Figures

32 and 33 compares experimental outlet flow rate with the standard outlet flow rate generated

by using a typical impedance condition for 2 other patients. When compared with Patient TKs

0

20

40

60

80

100

120

140

160

Wall ILT

MaxVonMisesStress(kPa)

Max Stress Values Before and After Convergence

Initial

Converged


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initial data, both of these data sets show more potential for improvement through the

derivation of a patient-specific boundary than Patient TK did.

Figure 32: Comparison of initial and experimental outlet flow rates for Patient MM

Figure 33: Comparison of initial and experimental outlet flow rates for Patient AK

0 0.5 1 1.5-20

0

20

40

60

80

100

120Comparision of Outlet Flow Rate for Patient MM

OutletFlow

Rate(cm

3/s)

Time (s)

Experimental

Standard

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-20

0

20

40

60

80

100

120

140

160

180

OutletFlow

Rate(cm

3/s)

Time (s)

Comparision of Outlet Flow Rates for Patient AK

Experimental

Standard


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Figure 34: Peak stress of initial (left) and converged (right) arterial wall

The mapped stress values for the converged and unconverged Patient TK models during

their peak stress values are shown in Figure 34. While the peak values themselves did not

increase much, the amount of total strain the wall is experiencing is slightly larger. This suggests

that the increased load is being distributed across a larger area of wall by the ILT. This behavior

allows the total load to increase, while not increasing the maximum wall stress values. It is of

note that Patient TK has a very large amount of ILT tissue when compared to the other patients,

and having less ILT would likely increase the sensitivity of the maximum stress to changes in the


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Figure 35: Artery wall stiffness variational study

The astricks markers represent the operating point, in this case the assumed material

properties. The ILT shows to be very sensitive to changes in the wall stress. The wall itself shows

significant change in max stress as well, though not at the operating point since the astricks is

nearly a minimum. If the operating point were moved up to about 6*103 kPa, the sensativity

would be about 30 kPA/decade. Figure 36 shows similar results in which the stiffness of the ILT

was varied over about 1 decade. Since the ILT is a linear elastic model, the stiffness is constant

and equivalent to the Youngs Modulus.

102

103

104

105

0

50

100

150Effects of Artery Wall Initial Stiffness on Max Stress


ILT Young's Modulus (kPa)


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Figure 36: ILT stiffness variational study

Changes to the stiffness of the ILT affect wall stress significantly more (approximately 110

kPa/decade at the operating point). This helps validate the theory that the ILT is helping

distribute forces over the artery wall. Figures 37 40 show the impact of independently varying

the 4 impedance parameters around the initial assumed impedance condition.

101

102

103

104

0

20

40

60

80

100

120

140Effects of ILT Stiffness on Max Stress


ILT Young's Modulus (kPa)


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Figure 37: Impedance parameter R1 variational study

Figure 38: Impedance parameter L variational study

105

106

107

108

20

40

60

80

100

120

140

Fluid Resistance (Ns/m5)

M

axVonMisesStress(kPa)

Effects of the R1 Impedance Parameter on Max Stress

Wall

ILT

104

105

106

107

20

30

40

50

60

70

80

90

100

110

120

Fluid Inertance (m8/kgs)


Effects of the L Impedance Parameter on Max Stress

Wall

ILT


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Figure 39: Impedance parameter R2 variational study

Figure 40: Impedance parameter C variational study

10-5

100

105

1010

1015

1020

20

30

40

50

60

70

80

90

100

110

120

Fluid Resistance (Ns/m5)

M

axVonMisesStress(kPa)

Effects of the R2 Impedance Parameter on Max Stress

Wall

ILT

10-5

100

105

1010

20

40

60

80

100

120

140

Compliance (m5/N)


Effects of the C Impedance Parameter on Max Stress

WallILT


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The max stresses do not appear to be as sensitive to changes in the impedance model as they

are to changes in the material models. This suggests that the optimization routine might be

better implemented, if a downstream boundary condition was assumed and patient specific

material properties were generated using the optimization routine. Due to the nature of the

routine, this should be a generally easy change to make.

The lack of change in functional value with respect to the R2 and C parameters is a result

of the assumed impedance condition used. As mentioned, the R2 and C parameters act as a low-

pass filter, allowing only low frequency components of the outlet flow rate data through, while

attenuating higher frequencies. The C value defines what a low frequency is, by affecting the

location of the cutoff frequency (the point at which the filter begins attenuating signals). Under

the assumed impedance condition, the experimental flow rate curve, which acts as the input to

the impedance condition has no low-frequency" components, as shown in Figure 41, where the

impedance condition and input flow rate data are compared in the frequency domain. As shown

in Figure 40, if the capacitance value were changed enough to include components of the input

signal on the low-frequency side of the notch filter, then both C and R2 would have an effect on

the stresses.


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resistance values, which make no physical sense. The optimization routines erratic behavior with

respect to the R2 parameter is a result of the functions complete lack of dependence on R2.

While the bulk of data collected so far has been with the Patient TK data set, other

models have been generated and proven to work using these techniques. While these models

work, they have not yet been run to convergence, and so they do not represent patient-specific

impedance data sets. Figure show examples of other geometries and mapped stress that have

been successfully calculated using the techniques presented.

Figure 42: Stress Maps for unconverged Patient MM (left) and Patient AK (right) models


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Chapter 6: Future Work

While the optimization tool developed appears to work well, more data is necessary

before any firm conclusions can be made with regards to its significants. Also, there is need for

some further validation such as proving the solution is independent of the selected time

stepping scheme and creating a more physically significant definition of convergence.

With respect to the tool itself,

experience suggests it would be a more

accurate strategy to define an average

velocity from the MRI velocimetry images

instead of a flow rate. This would allow the

region of interest defined during velocimetry

image processing to be placed well within the

visible lumen barrier instead of on it. The

lumen cross sectional area varies by as much

as 20% through the cardiac cycle and

attempting to define a stationary boundary of

the artery over which to calculate flow rate is

an exercise in futility. If too large a region of i