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Vein graft remodeling

University of Florida Institutional Repository
Permanent Link: http://ufdc.ufl.edu/UFE0002823/00001

Material Information

Title: Vein graft remodeling
Physical Description: 90 p.
Language: English
Creator: Fernandez, Chessy 1978- ( Dissertant )
Tran-Son-Tay, Roger ( Thesis advisor )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2003
Copyright Date: 2003

Subjects

Subjects / Keywords: Biomedical Engineering thesis, M.E
Dissertations, Academic -- UF -- Biomedical Engineering

Notes

Abstract: To improve the long-term patency of peripheral vein grafting, a mechanistic understanding of early vein graft adaptation is necessary. Critical to these events are the imposed hemodynamic forces, which regulate the balance between intimal thickening and expansive remodeling ultimately defining the morphologic characteristics of the vein graft. The objective of this project is to characterize the dynamic shear and wall tensile stresses during early vein graft remodeling by developing a mathematical model for shear and tensile stresses, based on the work of Womersley and Patel using in vivo flow rate data collected from an experimental rabbit vein graft model. Previously ex vivo models were used to determine the hemodynamic forces in the steady state only. Our mathematical model uses an in vivo bilateral carotid vein graft with distal branch ligation model to collect the experimental flow rate data in a pulsatile hemodynamic environment. The experimental animal model created two flow environments with reduced flow/shear through the ligated vein graft and elevated flow/ shear in the contralateral vein graft. Using thirty-four New Zealand white male rabbits, vein grafts were implanted and then harvested at 1, 3, 7, 14, and 28 days after initial surgical procedure. Hemodynamic and video measurements were collected before and after ligation for calculation of the shear and wall tensile forces. A computational program was developed to determine the velocity and shear stress profiles based on the collected hemodynamic data. Due to the size and length of the rabbit vein graft model, the pressure gradient is difficult to ascertain; therefore the flow measurement is used for the basis of our calculations. This approach is slightly different from previous models that relied upon the derived pressure gradient. Vein grafts were exposed to distinct flow environments characterized with a 6-fold difference in mean flow rate. Accelerated intimal hyperplasia and reduced outward remodeling were observed in the low flow grafts. At day 7, there was a peak in maximum and minimum shear stress with a delayed increase in lumen diameter leading to normalization of wall shear by day 28. At day 3, the intramural wall tension was at its maximum and there was an increase in wall thickness leading to a significant reduction of these stresses by day 14. There was no difference in incremental modulus of elasticity despite the significant difference in remodeling between the high and low flow grafts. Our mathematical model provides a simple way to determine dynamic wall shear and tension in a pulsatile hemodynamic environment, using readily available technology. Our mathematical model reveals a correlation among shear stress, flow, and intimal thickening, which coincides with ex vivo studies modeling steady flow. Future research may entail a more realistic development of computational modeling of pulsatile blood flow in this model as well as other more complicated configurations such as bifurcated blood vessels, which in the past have dealt with steady flow in ex vivo models.
Subject: hyperplasia, intimal, shear, stress, tension, wall
General Note: Title from title page of source document.
General Note: Includes vita.
Thesis: Thesis (M.E.)--University of Florida, 2003.
Bibliography: Includes bibliographical references.
Original Version: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0002823:00001

Permanent Link: http://ufdc.ufl.edu/UFE0002823/00001

Material Information

Title: Vein graft remodeling
Physical Description: 90 p.
Language: English
Creator: Fernandez, Chessy 1978- ( Dissertant )
Tran-Son-Tay, Roger ( Thesis advisor )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2003
Copyright Date: 2003

Subjects

Subjects / Keywords: Biomedical Engineering thesis, M.E
Dissertations, Academic -- UF -- Biomedical Engineering

Notes

Abstract: To improve the long-term patency of peripheral vein grafting, a mechanistic understanding of early vein graft adaptation is necessary. Critical to these events are the imposed hemodynamic forces, which regulate the balance between intimal thickening and expansive remodeling ultimately defining the morphologic characteristics of the vein graft. The objective of this project is to characterize the dynamic shear and wall tensile stresses during early vein graft remodeling by developing a mathematical model for shear and tensile stresses, based on the work of Womersley and Patel using in vivo flow rate data collected from an experimental rabbit vein graft model. Previously ex vivo models were used to determine the hemodynamic forces in the steady state only. Our mathematical model uses an in vivo bilateral carotid vein graft with distal branch ligation model to collect the experimental flow rate data in a pulsatile hemodynamic environment. The experimental animal model created two flow environments with reduced flow/shear through the ligated vein graft and elevated flow/ shear in the contralateral vein graft. Using thirty-four New Zealand white male rabbits, vein grafts were implanted and then harvested at 1, 3, 7, 14, and 28 days after initial surgical procedure. Hemodynamic and video measurements were collected before and after ligation for calculation of the shear and wall tensile forces. A computational program was developed to determine the velocity and shear stress profiles based on the collected hemodynamic data. Due to the size and length of the rabbit vein graft model, the pressure gradient is difficult to ascertain; therefore the flow measurement is used for the basis of our calculations. This approach is slightly different from previous models that relied upon the derived pressure gradient. Vein grafts were exposed to distinct flow environments characterized with a 6-fold difference in mean flow rate. Accelerated intimal hyperplasia and reduced outward remodeling were observed in the low flow grafts. At day 7, there was a peak in maximum and minimum shear stress with a delayed increase in lumen diameter leading to normalization of wall shear by day 28. At day 3, the intramural wall tension was at its maximum and there was an increase in wall thickness leading to a significant reduction of these stresses by day 14. There was no difference in incremental modulus of elasticity despite the significant difference in remodeling between the high and low flow grafts. Our mathematical model provides a simple way to determine dynamic wall shear and tension in a pulsatile hemodynamic environment, using readily available technology. Our mathematical model reveals a correlation among shear stress, flow, and intimal thickening, which coincides with ex vivo studies modeling steady flow. Future research may entail a more realistic development of computational modeling of pulsatile blood flow in this model as well as other more complicated configurations such as bifurcated blood vessels, which in the past have dealt with steady flow in ex vivo models.
Subject: hyperplasia, intimal, shear, stress, tension, wall
General Note: Title from title page of source document.
General Note: Includes vita.
Thesis: Thesis (M.E.)--University of Florida, 2003.
Bibliography: Includes bibliographical references.
Original Version: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0002823:00001


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VEIN GRAFT REMODELING


By


CHESSY FERNANDEZ


























A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REOUIREMENTS FOR THE DEGREE OF
MASTER OF ENGINEERING


UNIVERSITY OF FLORIDA


2003





































Copyright 2003

by

Chessy Fernandez





































I dedicate this thesis to my father, R.D. Fernandez.

















ACKNOWLEDGMENTS

I would like to extend my warm thanks and gratitude to the many people who have

helped me through this journey. I would like to take this opportunity to thank my

advisor, Dr. Roger Tran-Son-Tay, for providing me with insight and guidance during this

process. I would also like to thank Dr. Scott Berceli for giving me the chance to work on

this project as well as provide me with the opportunity to see first hand how engineering

is applied to clinical applications. My greatest appreciation extends to everyone in Dr.

Berceli and Dr. Ozaki's lab at the VA hospital. Without them, this project would not

have been accomplished. This project is the hard work and effort of all in the lab and to

them Itruly am indebted.

I would also like to thank my friends, family, and professors who have all helped

me become the person I am today. I would like to thank my parents for providing the

encouragement and loving support as I achieve my life's goals. I would like to extend

my appreciation to my friends for their moral support, friendship, and advice. Finally, I

would like to thank all my professors for providing me with the knowledge and

background needed for becoming an engineer in today's world.


















TABLE OF CONTENTS



ACKNOWLEDGMENTS ........................................ iv

LIST OF TABLES ................... ................... ................... .........

LIST OF FIGURES ........................................ ix

ABSTRACT................................ xi

CHAPTER

1 INTRODUCTION ........._____ ........._____ ........._____ ..........

Arterial Occlusive Disease ........................................
Objective ................... ................. .............. ...........
Specific Aims....................................

2 BACKGROUND AND SIGNIFICANCE............................


Anatomy and PhysioloW of Blood Vessels .................. .................. .................. ...........3
Current Understanding of Blood Vessel Remodeling .................. .................. ..............4
Morphologic Changes and Physical Forces ........................................
Biochemical Events ................... .............. ............. ..........
Vein Graft Adaptation .............. ............. ............... .........
In-Vive Model .............. ............. ............... ..........
Wall Stress Models .............. ............. ............... .........
Hemodvnamics ................... ................... ................... ...........
Poiseuille model for steady flow ................... ................... ................... .......
Womersley analysis for pulsatile flow ................... ................... ................... ..8
Modified Womersley approach ................... ................... ................... .........
Other mathematical models for estimation of velocity and wall shear stress 9
Intramural Wall Stresses. ................... ................... ................... .........
Lame's equation for wall tension-static circumferential wall stress............l 1
Isotropic model for wall tension. ................... ................... ................... .......
Other mathematical models that study intramural wall stress. ................... ..13
Significance ................... ................... ................... ...........

3 MATERIALS AND METHODS .........____ .........____ .........____ ........


Experimental Vein Graft Model ................... ................... ................... ........











Surgical Methods ................... ................... ................... ..........
Hemodynamic Measurements ................... ................... ................... ..........
Video ................... ................... ................... ...........
Flow rate ................... ................... ................... ..........
Pressure ................... ................... ................... ...........
Morphometric Analysis ................... ................... ................... ..........
Modeling of Wall Stresses ................... ................... ................... ........
Modified Womersley Approach ................... ................... ................... .........
Numerical Computations ................... ................... ................... ..........
Intramural Wall Stresses................................
Lame's equation for static circumferential wall stress ................... ..............30
Isotropic model for wall tension. ................... ................... ................... .......

4 RESULTS .........____ .........____ .........____ ..........


Hemodynamics ........................................
Flow, Velocity, and Shear Stress..................................
Wall Stress and Elastic Modulus ................... ................... ................... .......
Vein Graft Remodeling..............................

5 DISCUSSION..............................


Mathematical Model: Modified Womersley Approach. ................... ................... .......45
Experimental Model ........................................
Vein Graft Remodeling..............................
Future Research ................... ................... ................... ..........
Conclusion ................... ................... ................... ...........


APPENDIX


A DERIVATION OF WOMERSLEY EOUATIONS ........................................

B TABLES USED TO DETERMINE CONSTANTS BASED ON WOMERSLEY
NUMBER .........____ .........____ .........____ ........._

C FLOW CHART OF MATLAB PROGRAM.................................

D MATLAB CODE FOR DETERMINING VELOCITY AND SMEAR STRESS
PROFILES .........____ .........____ .........____ ........._

Code for Discrete Fourier Transform ................... ................... ................... .......
Code for Determining Velocity Profile ........................................
Code for Determining Wall Shear Stress Profile ................... ................... .................. 65
Code for Main Driver of Program .................. .................. .................. .................. ......67













LIST OF REFERENCES .........___ .........___ .........___ .........


BIOGRAPHICAL SKETCH ........................................

















LIST OF TABLES


Table

4-1. Flow environment in the ligated/ low flow vein graft and the contralateral/ high flow
vein graft. ........................................

4-2. Maximum centerline velocity at each time point in the ligated/ low flow vein graft
and the contralateral/ high flow vein graft. ........................................

4-3. Minimum centerline velocity at each time point in the ligated/ low flow vein graft
and the contralateral/ high flow vein graft. ........................................

4-4. Mean centerline velocity at each time point in the ligated/ low flow vein graft and
the contralateral/ high flow vein graft. ........................................

4-5. Dynamic wall shear stress values- Maximum shear stress ................... ................... ..40

4-6. Dynamic wall shear stress values- Minimum shear stress ........................................

4-7. Dynamic wall shear stress values- Poiseuille shear stress..................................

4-8. Static circumferential wall shear stress comparing ligated vein graft and
contralateral vein graft. ........................................

4-9. Dynamic wall shear stress and elastic modulus at day 28. ................... ................... ..42

B-l. Table of solutions to Bessel functions used to calculate Mo and Bo. .................. ......55

B-2. Table of solutions to Bessel Functions to determine M1 and 81 ..............................56

B-3. Table to determined((:, and E:, based on the Womersley number. ................... ......57

















LIST OF FIGURES


Firure

2-1. Anatomy of the blood vessels walls with the layers of the artery wall on the left and
the layers of the vein wall on the right. ........................................

2-2. Bilateral interposition vein graft with distal ligation. ............... .............. ............7

2-3. Mechanical models of viscoelastic behavior where L is the elongation and S is the
stress. ........................................ 10

2-4. Wall stress acting on arterial wall. T,, and Tee are the longitudinal and
circumferential wall stresses. S, and S, are the normal and fluid shear stresses. ...12

3- 1. Schematic of anastomotic cuff technique used in vein grafting procedure. ..............17

3-2. Bilateral carotid vein grafting with distal ligation model. ................... ................... ...17

3-3. Pre-measured marker next to left carotid distal artery unligated. ................... ...........19

3-4. Measurement of in vive flow rate and pressure with waveform output from Chart
Recorder Program. ................... ................... ................... ..........

3-5. Histologic cross-section illustrating the longitudinal section of anastomotic segment
using Masson's staining. ........................................

3-6. Cross-sectional area of 28-day ligated/low flow rabbit vein graft at harvest 20x and
40x illustrating different wall layers. ........................................

3-7. Original flow waveform (solid line) compared to approximated DFT flow waveform
(dashed line) and 0-9 DFT harmonics (dotted lines). ................... ................... .........25

3-8. Centerline velocity and wall shear stress derived from the DFT approximation of the
original flow waveform. ........................................

3-9. Velocity profile at a specified time points encompassing the diameter of the vein
graft. ........................................

3-10. Wall shear stress at a specified point in time over one cardiac cycle......................31

4-1. Three-dimensional velocity profile within bilateral vein grafts one day after
implantation. ........................................










4-2. Three-dimensional velocity profile within bilateral vein grafts three days after
implantation. ........................................

4-3. Three-dimensional velocity profile within bilateral vein grafts seven days after
implantation. ........................................

4-4. Three-dimensional velocity profile within bilateral vein grafts fourteen days after
implantation. ........................................

4-5. Three-Dimensional Velocity profile within bilateral vein grafts 28 days after
implantation. ........................................

4-6. Dynamic wall shear stress vs. time within bilateral vein grafts one day after
implantation. ................... ................... ................... ...........

4-7. Dynamic wall shear stress vs. time within bilateral vein grafts three days after
implantation. ................... ................... ................... ...........

4-8. Dynamic wall shear stress vs. time within bilateral vein grafts seven days after
implantation. ................... ................... ................... ...........

4-9. Dynamic wall shear stress vs. time within bilateral vein grafts 14-days after
implantation. ................... ................... ................... ...........

4-10. Dynamic wall shear stress vs. time within bilateral vein grafts 28 days after
implantation. ................... ................... ................... ...........

4-11. Vein graft remodeling. ................... ................... ................... .....

A-i. Schematic of flow through a blood vessel. ................ ............... ................ ..

C-l. Flowchart of MATLAB code....................................

















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Engineering

VEIN GRAFT REMODELING

By

Chessy Fernandez

December 2003

Chair: Roger Tran-Son-Tay
Major Department: Biomedical Engineering

To improve the long-term patency of peripheral vein grafting, a mechanistic

understanding of early vein graft adaptation is necessary. Critical to these events are the

imposed hemodynamic forces, which regulate the balance between intimal thickening

and expansive remodeling ultimately defining the morphologic characteristics of the vein

graft. The objective of this project is to characterize the dynamic shear and wall tensile

stresses during early vein graft remodeling by developing a mathematical model for shear

and tensile stresses, based on the work of Womersley and Patel using in vive flow rate

data collected from an experimental rabbit vein graft model.

Previously ex vive models were used to determine the hemodynamic forces in the

steady state only. Our mathematical model uses an in vive bilateral carotid vein graft

with distal branch ligation modelto collect the experimental flow rate data in a pulsatile

hemodynamic environment. The experimental animal model created two flow

environments with reduced flow/shear through the ligated vein graft and elevated flow/

shear in the contralateral vein graft. Using thirty-four New Zealand white male rabbits,










vein grafts were implanted and then harvested at i, 3, 7, 14, and 28 days after initial

surgical procedure. Hemodynamic and video measurements were collected before and

after ligation for calculation of the shear and wall tensile forces. A computational

program was developed to determine the velocity and shear stress profiles based on the

collected hemodynamic data. Due to the size and length of the rabbit vein graft model,

the pressure gradient is difficult to ascertain; therefore the flow measurement is used for

the basis of our calculations. This approach is slightly different from previous models

that relied upon the derived pressure gradient.

Vein grafts were exposed to distinct flow environments characterized with a 6-fold

difference in mean flow rate. Accelerated intimal hyperplasia and reduced outward

remodeling were observed in the low flow grafts. At day 7, there was a peak in

maximum and minimum shear stress with a delayed increase in lumen diameter leading

to normalization of wall shear by day 28. At day 3, the intramural wall tension was at its

maximum and there was an increase in wall thickness leading to a significant reduction of

these stresses by day 14. There was no difference in incremental modulus of elasticity

despite the significant difference in remodeling between the high and low flow grafts.

Our mathematical model provides a simple way to determine dynamic wall shear

and tension in a pulsatile hemodynamic environment, using readily available technology

Our mathematical model reveals a correlation among shear stress, flow, and intimal

thickening, which coincides with ex vive studies modeling steady flow. Future research

may entail a more realistic development of computational modeling of pulsatile blood

flow in this model as well as other more complicated configurations such as bifurcated

blood vessels, which in the past have dealt with steady flow in ex vive models.
















CHAPTER 1
INTRODUCTION

Arterial Occlusive Disease

Arterial occlusive disease, causing myocardial infarctions and strokes, affects

millions of people and is one of the leading causes of death in the United States. In many

cases, invasive surgicaltechniques, such as bypass vein grafting and angioplasties, have

been used to alleviate vascular occlusion. Bypass vein grafting uses a vein taken from

another part of the body to replace the obstructed blood vessel and inserts it into the

arterial system to provide better blood flow [1]. However, restenosis or occlusion can

still occur in the time frame of months to years. The current standard of care for

peripheral vein grafting is 80% i-year and 60% 5- year patency rates [2, 3]. Patency is

defined as a blood vessel that remains open to flow without thrombosis [1]. Because of

these reasons, many researchers have been attempting to improve these long-term

patency results. An understanding of early vein graft adaptation and progression must be

established in order to improve the long-term results. Currently, it is known that many

factors, including physical forces, morphologic changes, and biochemical events, are

involved in this adaptation process. Playing a key role in the remodeling process are the

biomechanical forces[4, 5]. The changes in these biomechanical forces regulate the

balance between intimal thickening and expansive remodeling, which govern the

morphologic changes in the vein graft [6, 7].










Objective

The arterial hemodynamic environment is comprised of intricate branching

geometries. Characterizing this environment is complex and consists of many non-

linearities [8]. Dynamic non-linearities on the biologic response of the arterial

vasculature has been well established; however the understanding of the interface

between these imposed physical forces and the secondary remodeling processes has

traditionally been based on time-invariant, linear estimations of wall shear and tensile

Eorces[9]. The objective of this study is to characterize early vein graft remodeling by

finding the correlation between the physical forces and morphologic changes by

characterizing the dynamic shear and wall tensile forces with the intimal thickening and

expansive remodeling that occurs in vein graft arterialization.

Specific Aims

The specific aims for this thesis are to:

i. Develop a theoretical model to calculate wall shear and velocity profiles in the
arterial circulation.

2. Perform experiments to gather flow and pressure measurements using a rabbit vein
graft model in order to estimate shearing and tensile forces to which the graft is
subjected.

3. Examine the correlation between the physical forces and morphologic changes,
during the vein graft remodeling process.
















CHAPTER 2
BACKGROUND AND SIGNIFICANCE

Anatomy and Physiology of Blood Vessels

The cardiovascular system consists of the heart, blood, and blood vessels. The

heart pumps the blood throughout the body via the blood vessels. The blood transports

oxygen and nutrients from the lungs to the rest of the tissues in the body while carrying

carbon dioxide back to the lungs. The blood vessels consist of the arteries and veins.

The arteries carry blood away from the heart while the veins carry the blood back to the

heart. The blood vessel walls consist of three layers: the intima, the media, and the

externa (Figure 2-1) [10]. The intima is the innermost layer consisting of endothelial

lining and an underlying layer of connective tissue. There is a thick layer of elastic fiber

called the internal elastic lamina (membrane) in arteries. The media is made up of

smooth muscle tissue and connective tissue. Another layer of elastic fiber called the

external elastic lamina separates the media from the externa. The externa, or adventitia,

is the outermost layer. It is the muscular and elastic components of these layers that

change the diameter to compensate for blood pressure and blood flow changes [10].

It can be seen from Figure 2-1 that the arterial walls are generally thicker than the

vein walls. This is because the artery has more smooth muscle and elastic fibers. The

arteries are elastic and contractile. It is this elasticity that allows passive changes in the

vessel diameter in response to changes in blood pressure. When there is no opposition to

pressure, the elastic fibers recoil in the artery, causing the lumen to constrict. Veins

however have thinner walls because the blood pressure is lower than in the arteries.










Because of tlus low blood pressure, vems have valves m order to oppose the force of

gravity and to prevent back fow [10]



Tunica extern
Smooth r
muscle .Sot
Tunica mu mscl
interna


Lumen



.: Il~ 1t m
I *otefrm / Endothe llum.


ARTERY
Luman of arte VEIN

Figure 2-1 Anatomy of the blood vessels walls wilth the layers of the artery wall on the
left and the layers of the vem wall on the nght


Current Understanding of Blood Vessel Remodeling

Morphologic Changes and Physical Forces

Vem graft remodehng after bypass graft or angioplasty procedures is compnsed of

two different processes Constnctive remodehng mvolves a tluckemung of the vessel wall

wluch leads to a narrowmg of the inside lumen Positive remodehng mvolves the

enlargement of the blood vessel and an merease m the lumen diameter It is the balance

between these two processes that is mfluenced by the local hemodynanucs of pressure

and flow [11, 12] Tlus balance also determmes the patency of the treated artenes or

venous bypass grafts [13-15] Little is 1mnown about how exactly the mechamusms of

shear and tensile forces mfluence the remodehng process










Biochemical Events

Extracellular proteins form a scaffold with the vein graft wall, and breakdown of

these proteins is involved with vascular reorganization [16, 17]. This reorganization

leads to permanent changes in the size and composition of the blood vessel. More

specifically, matrix metalloproteinases (MMP) are specialized enzymes important in the

reorganization of the extracellular proteins. The balance between the MMP enzymes and

its tissue inhibitors control the blood vessel reorganization in response to outside stimuli.

Two enzymes, MMP-2 and MMP-9 appear to play important roles in this reorganization

of the blood vessel wall under the outside stimuli.

In previous studies, it has been found that active MMPs are controlled by:

transcriptional regulation, translation into protein, release into extracellular space, and

proteolytic activation of latent enzyme [17, 18]. The MMP activity is regulated to

maintain the architecture of the blood vessel wall [19]. The MMPs involved in the

reorganization process deal with the breakdown of elastic lamellae to permit smooth

muscle cell migration and proliferation in the intima. They also deal with the

incorporation of the new cellular matrix components, which accompanies an increase in

luminal area [16].

Vein Graft Adaptation

Significant structural changes in the vein graft wall are caused by the changes in the

pressure and flow environment from the venous to arterial circulation. Characteristics of

this change from the low pressure/low flow venous system to the high pressure/high flow

arterial system include an increase in the intimal and medial thickness in the wall, as well

as a burst of smooth cell proliferation [7, 20]. Vein grafts are exposed to four forces:

circumferential (hoop), radial, longitudinal tensile forces, and surface shearing forces,










which are directed along the axis of flow [21, 22]. Researchers have been unable to

separate these variables but have found evidence suggesting that there is a correlation

between the medial thickening and circumferential tensile forces and another correlation

between the intimal thickening and fluid shear forces [6].

In-Vive Model

A rabbit vein graft will serve as the experimental system for these experiments. In

this model, a reversed segment of extemaljugular vein is used for the interposition

grafting into the common carotid artery. Several investigators have used this particular

model to study the mechanisms of vein graft failure and intimal hyperplasia[6, 7, 23-26].

Various vascular constructions are used to create distinct regions of altered shear stress

and wall tension. The bilateral carotid interposition graft model was adapted to examine

the effects of shear stress (Figure 2-2). The model is modified to include a distal ligation

of the internal carotid and ligation of three of the four primary branches of the external

carotid artery [23]. By doing this, there is an 8-10 fold difference in the flow rates

between the ligated/low flow and contralateral/high flow vein grafts. Creating this

change in hemodynamic environment, the remodeling process caused by the difference in

flow can be studied.

Wall Stress Models

The local hemodynamics of pressure and flow influence the remodeling process. It

is the forces caused by pressure and flow that appear to have an effect on biochemical

events and the morphologic changes. To characterize these shear stress and wall tensile

forces, an understanding of the hemodynamics is needed in order to develop a well-suited

mathematical model for vein graft remodeling.






















Ligated Graft


Contralateral Graft


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Womersley analysis for pulsatile flow

Pulsatility adds a whole new set of components to the blood flow. For example, the

pulsatility adds more inertial forces and the diameter of the vessel varies with time

throughout the cardiac cycle [29]. To account for the unsteady pulsatility in blood, a

more complex model must be used. The Womersley analysis relates flow and the

pressure gradient for pulsatile blood flow[30-32]. The Womersley analysis, derived from

the Navier-Stokes equations, makes the following assumptions:

Laminar flow
Newtonian flow
Uniform cylindrical tube with a rigid wall
Infinite length

The Womersley analysis drops the nonlinear terms in the Navier-Stokes equations

because these terms are relatively small when applied to blood flow. This is done to

linearize the equations and create a general solution for blood flow velocity (Appendix

A). The Navier-Stokes equations are further reduced by assuming that velocity is only in

one axial direction and thus removing the radial velocity. Since the blood vessel is

assumed to be a non-moving rigid wall, the no-slip condition applies to the wall meaning

there is no velocity occurring at the wall [29, 33, 34].

WP
From the Womersley analysis, the Womersley number, u = R ~ where a is the



Womersley number, w is the frequency, p is the density, and ~u is the fluid viscosity.

This non-dimensional number is introduced and relates the oscillatory flow to the

viscosity. The Womersley number is an indicator of the stability of laminar flow [35].










Modified Womersley approach

The physical nature of the rabbit model makes it difficult to accurately assess the

pressure gradient directly because the vessel is small in length and in diameter. In the

past, the experimental pressure gradient in the arterial system was determined by

measuring the pressure at two points along the arterial tree [30]; however, this approach

is more suitable when making hemodynamic measurements in large diameter arteries.

Applying this approach to smaller vessels such as vein grafts, the estimation of dynamic

intraluminal pressure is prone to significant artifact. Because of this and recent advances

in instrumentation, flow rate is used for estimating the velocity and shear stress profiles

rather than the pressure gradient [33, 35].

Other mathematical models for estimation of velocity and wall shear stress

Realistically, blood vessels exhibit more of a viscoelastic behavior. The blood

vessel is not completely elastic but has both viscous and elastic properties. Viscoelastic

behavior can be shown as three different types of models as shown in Figure 2-3 [29].

The mechanical models are all combinations of linear springs, which instantaneously

produce a deformation proportional to a load, and dashpots, which produce a velocity

proportional to the load at any instant. The force in the Maxwell model is transmitted

~om the spring to the dashpot. The Voigt model deals with the idea that the spring and

dashpot have the same displacement. The Kelvin model is considered to be the standard

linear solid model. Viscoelasticity can be experimentally determined when the material

undergoes periodic oscillations [36].

With a viscoelastic behavior a phase lag between the stress and strain is produced.

Womersley also modified his original analysis to incorporate this behavior by

incorporating the effects of wall viscosity [29]. The blood vessel is suddenly strained and








the strain is constant afterwards. Viscoelasticity complicates our model and our modified

approach still provides substantial information in a simplified form. Due to the

significant complexities and inability to obtain an analytical solution with the addition of

viscoelasticity, these terms were omitted in our current model.


A B C


Dashpot II E,

Spring






V~ L/- :"


Ir= I~
Maxwell Voigt St. Venant
Figure 2-3. Mechanical models of viscoelastic behavior where L is the elongation and S
is the stress. (A) Maxwell model. (B) Voigt Model. (C) St. Venant model.

Intramural Wall Stresses

Blood flow causes viscous drag between the outermost laminae of the fluid and the

vessel wall. This viscous drag is a possible factor in causing arterial disease. It is this

shear stress imposed on the wall that can affect the functional and structural integrity of

the endothelial cells. In atherosclerosis, there are intracellular deposits of cholesterol in

the intimal layers of the vessel walls. The movement of these proteins into and out of the

intima can be affected by wall shear and tensile stresses. This causes plaques to bulge










into the vessel lumen with differences in local stress potentially playing a role in the

localization [29].

Stress is the force distributed over an area across a surface, and can be divided into

two components: normal stress, which is stress directed perpendicular to the surface and

shear stress, which is the stress along the transverse cross section. Shear stress occurs in

the tangential plane along the wall. Strain is the deformation of the stressed object. It is

the ratio of change in a given dimension to its original unstressed state [29, 37]. To study

these intramural stresses, the Lame and isotropic models are used in our analysis.

Lame's equation for wall tension-static circumferential wall stress

Static circumferential wall stress can be determined using a variation of Laplace's

PR
Law called Lame's equation, where P is the pressure, R is the lumen radius, and h is


the wall thickness [33]. While easy to use, the wall stress using Lame's equation neglects

pulsatility and fails to take into account the differences in stress in the successive vessel

wall layers. It also assumed that the wall thickness does not change when stretched thus

forcing Poisson's ratio to be zero[29]. Poisson's ratio is defined as the ratio of lateral or

perpendicular strain to the longitudinal or axial strain [37].

Isotropic model for wall tension

A more realistic model may be used to account for the pulsatility in blood flow

providing a more realistic model of the stresses within a vein graft. The dynamic wall

stress is based on the models of Patel and Fry[38] and the Kuchar and Ostrach[39] and

Bergel [40], which use the elastic modulus to determine three wall stresses: radial (S,),

axial(T,), and circumferential (Tss) stresses (Figure 2-4) [41].











The Patel and Fry model assumes that there is a non-linearity in the arterial wall

stress-strain relationship and assumes the blood vessel as a cylindrically orthotropic tube.

The Bergel model is based on this model and is used to determine the elastic modulus[3g,

40, 41].




STRESSES ACTING ON SMALL ELEMENT OF
ARTERIAL WALL



_~

,,,





Tse

Figure 2-4. Wall stress acting on arterial wall. T,, and Tee are the longitudinal and
circumferential wall stresses. S, and S, are the normal and fluid shear
stresses.



The incremental elastic modulus relates stress and strain. It is a measure of a material's


resistance to distortion by a tensile or compressive load and in its simplest form can be

considered the inverse of compliance [42]. In a purely elastic body, the elastic modulus

is linearly proportional to stress. This is in accordance with Hooke's law. However, the

blood vessel walls produce a curvilinear stress-strain relationship, which is seen in most

nonhomogeneous materials [29]. The Kuchar and Ostrach model assumes an elastic,

isotropic model for arterial wall stress. Assuming a linear stress-strain relationship and

the above assumptions from Patel and Fry and Kuchar and Ostrach, a model can be

developed using a general isotropic model over the range of blood vessel deformations.

With this generalized model, the incremental elastic modulus is derived by Bergel, and










from this modulus, the radial, axial, and circumferential stresses can be calculated

through algebraic manipulation of the general isotropic model [29].

Other mathematical models that study intramuralwall stress

Other modified models have been used to estimate wall stresses. The model used


in Sokolis, et al. [43] studies the stress-strain relationship in uniaxial tension. This model

removes the viscoelastic phenomena through preconditioning prior to tensile-testing. The

FT 1
stress is calculated based on the Kirchhoff expression, where S = and /Z =
A/Z /Z 1


where S is the Kirchoff stress, F is the force, A, is the area, T is the Lagrangian stress


is the change in length. The model uses the nonlinear Green-St. Venant strain



(E), E_1~2 _1) where h is the longitudinal stretch ratio and the elastic modulus (M) is


dS
h(l = where S is stress and E is the Green-St. Venant strain [43].
dE


Rachev, et al. [44] developed a mathematical model for stress-induced thickening

of the arterial wall close to the implanted stent. They assumed the host artery to be a

cylindrical shell with a constant thickness, and the stent to be non-deformable in the

circumferential direction. Similar to the Sokolis model, the Rachev model used Green-

St. Venant strain. These models would not work for our study due to the fact that

Kirchoff stresses and the definition of the Green-St. Venant strain deals with large


deformations such as those seen in the pulmonary artery, which is not the case in the

rabbit vein graft [36].












Significance

It is unclear how all these factors, physical forces, morphologic changes, and

biochemical factors, together play a role in vein graft remodeling and adaptation. It is

the biomechanical forces that influence the remodeling process that are central to

understanding vein graft adaptation [4, 5]. Previous studies have studied the

hemodynamic forces based on ex vive models [38, 41, 45, 46]. What makes this project

different is the use of an in vive experimental model to study the hemodynamic forces

that affect the remodeling process. While many in vive models have studied

hemodynamic forces in the mean or steady state, they have not studied the forces in a

pulsatile environment due to the complexity of calculations [22, 47]. However, cell

culture experiments studying endothelial and smooth muscle cells [48-50], and anatomic

correlation studies [9] have shown the impact of oscillatory shear stress on blood vessel

morphology and cellular function. In recent years complex numerical analyses for

studying these hemodynamic forces in a pulsatile environment have been easier to

achieve with advancements in computational numerical methods. Several experimental

studies have investigated the effects of these forces in a pulsatile environment; however

they have used model casts of blood vessels, examined in bench-top perfusion systems

[41, 51]. There is a limited amount of in vive studies available to provide an

understanding of the interaction between the dynamic components of the biomechanical

forces and the intact vein graft or blood vessel. By utilizing the advancements in

computer capabilities and accurate instrumentation, hemodynamic forces can be

determined in an in vive pulsatile environment. By obtaining a mechanical understanding







15


of early vein graft adaptation, we can gain further insight into the early remodeling

process to provide strategies for the patency of vein grafts.
















CHAPTER 3
MATERIALS AND METHODS

Experimental Vein Graft Model

Surgical Methods

Bilateral carotid vein grafting with selective distal branch ligation was performed to

create defined regions of differential blood flow [52]. Using thirty-four New Zealand

white male rabbits (3.0-3.5 kg), an anastomotic cuff technique was performed to implant

the bilateral externaljugular vein segments into the common carotid arteries. The rabbits

were anesthetized through intramuscular injection with ketamine hydrochloride (30.0

mg/kg), and anesthesia was maintained with endotracheal intubation and inhaled

isoflurane (2. 5-3.0%). Heparin was also given intravenously, 1000 units, at the start of

the procedure [52].

The technique harvested external jugular veins, 3 cm in length, for the creation of

an interposition graft into the common carotid artery (Figure 3-1) [52]. The external

jugular vein ends were passed through polymer cuffs, which were fashioned from a 4-

French introduced sheath. An arteriotomy was performed and the reversed cuffed vein

ends were inserted and fixed to the artery. Ligatures were placed distal to the vein graft

in order to unilaterally reduce the graft blood flow. This completely occludes the internal

carotid artery and three of the four primary branches of the external carotid artery, which

can be seen in Figure 3-2 [52]. At 1, 3, 7, 14, and 28 days after the initial implantation,

the vein grafts were exposed via a mid-line neck incision. Rabbits received BrdU








mlections 24 hours before the harvest day Rabbis were euthanized by overdose of 5ml
pentobarbital (50mg/ml) mtravenously pnor to vein graft tissue harvest [52]


(4)


Figure 3-1 Schematic of arrastomotic cuff techm~que used m vem graftmg procedure
The external~ugular vem is excised and polymer cuffs are placed at the vem
ends (steps 1-3)and then the vein is inserted (steps 4-6) and fixed to the
carotid artegy (steps 7-8)


Lgated
Vein Graft


Vein Graft


Figure 3-2 Bilateral carotid vem graftmg with distal ligation model


(2)


(3)


(1)


(7) (8)


~" ~9"













Hemodjnamic Measurements

Hemodynamic measurements (flow rate and pressure) and video analysis were

acquired at implantation and harvest of the vein grafts. In vive hemodynamic

measurements, both flow rate and pressure, and video were recorded in real time. All

measurements were obtained with the aid of an IBM compatible PC computer through a

MacLab/Xe AD Instruments analog to digital board at 1000 Hz.

Video

At implantation and harvest, real-time video was obtained using a Sony Analog
09


camera and a Diagnostic Instrument Wild 5A Dissecting Microscope connected to a

computer. The video camera setup uses the microscope to obtain a clearer more focused

image. Once set up, a small pre-measured marker is placed next to the blood vessel for

later analysis (Figure 3-3). Video is saved as .bmp files and images are captured at 30

frames per second. The video images were analyzed using Zeiss Imaging software

(AxioVision v. 3.1)to determine the external vein graft radius and dynamic wall motion.

Flow rate

The flow rate in the implanted vein graft is measured using a Transonic Systems

small animal ultrasonic blood flow meter, modelT106. The Transonic Systems 2SB flow

probe, with 100 cm cable length was used for a digital measurement of flow rate in 2mm

vessel diameters. The flow meter ranges from 0-1 V, which corresponds to 0.0-

100ml/min and a relative accuracy of f 2 mi/min [53]. The flow rate waveform is

collected through the A/D board and computer using the AD Instruments Chart recorder

program (Figure 3-4). The Chart recorder program interpreted the flow rate waveform








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Mrsurwe asducr fi-vre ressre Wae ca
'i



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grafmodel ( B)Tnocnwea()Transonicorl~owmetr()Cateorehoga w
pressureandilowrate



Themt.alutmnalpeeinalprss ured l sma sullarmo iedwta naia~hlr ir-ictee

transducer(modelSR61,c~hhsbe lSPR-67),which a lous bee t pla 2g ae d t helovseva 2-u e










catheter. The output signal ranges from 0-1V, which corresponds to 0-100mmHg. The

sensor is side-mounted at the tip and the catheter size is 1.1 French [54]. The transducer

unit is connected to the computer and data is recorded with the Chart recorder program

(Figure 3-4). The pressure waveform data is represented in (mmHg) in graphical form in

the Chart program. The pressure is taken at various locations along the vein graft and

artery. However because of the invasive method of retrieving the measurement, the

intraluminal pressure is only recorded when the vein graft is harvested.

Morphometric Analysis

The vein graft segments were fixed in 10% buffered formalin and embedded in

paraffin. Histologic cross sections were stained with Masson's trichrome and van

Gieson's elastic stains (Figure 3-5) [52]. Masson's trichrome stain is used to differentiate

between collagen and smooth muscle cells [55]. The van Gieson's stain is used to show

the elastic lamina[56]. Both these stains show the different layers in the vein graft wall.

Using Zeiss imaging software, digitized images were collected and analyzed to measure

the intimal, medial, and adventitial areas (Figure 3-6). In vive lumen diameter and graft

(extemal elastic lamina (EEL)) diameter were calculated using the following morphology

data: intimal, medial and lumen areas. The lumen diameter was later calculated based on

the morphology data and the external diameter measured from the video analysis.

I_. 4(A
Lumen diameter =,ID'vldeo ~?-;~i (3.1)
Y Ir

where Dvrdeo is the video diameter and Aadv and A~ are the cross- sectional areas within the

adventitia and lumen respectively.

The graft diameter is determined to be





Lumen ----~---,

~_e~r4~4~44~~
,Intima ---~
~
~~ I;
~---
rWedia
-


Graft diameter = D2,ao 4(A,, Assu


where AEEz is the cross-sectional area within the EEL.


~
~.
P~




~v ; ~,-;


20b


Figure 3-5


Histologic cross-section illustrating the longitudinal section of anastomotic
segment using Masson's staining.


~I
.~~ =Y


Figure 3-6. Cross-sectional area of 28-day ligated/low flow rabbit vein graft at harvest at
(A) 20x and (B) 40x illustrating different wall layers.


To estimate lumen diameter at the time of graft implantation, normal external jugular

vein segments were harvested, embedded, and sectioned for analysis. Image analysis of


A drentitia *










these sections were used to determine a "standard" vein wall thickness with the lumen

diameter at graft implantation being determined as

Lumen Diameter = D,,,,, 2h (3.3)


where h is the "standard" graft wall thickness.

Modeling of Wall Stresses

Hemodynamic forces have been shown to influence vein graft arterialization during

the remodeling process. How exactly these forces influence the remodeling process is

unknown. To look at shear stress more closely, a mathematical model isolating this force

is developed. From the experimental hemodynamic data and the calculated lumen

diameter, the velocity and shear stress profiles was calculated based on our mathematical

model, a modified Womersley approach. To study the intramural wall stresses, i.e. the

circumeferential, axial, and radial stresses, the Lame and isotropic models were used.

Modified Womersley Approach

Assuming that the pressure gradient is a function oftime, the pressure waveform

can be approximated as a series of sinusoidal functions in the form of

ap ap, aF ,,,,
=--+C-~ (3.4)
L L L

aP aP
where P- is the mean pressure gradient, and is the oscillating pressure gradientw
L L


is the frequency, and t is the time.

Incorporating the equation for the lumen diameter (equation 3.2) in the steady state

and oscillatory equations from the Womersley analysis, we can derive the velocity and

shear stress profiles. The derivation can be seen in Appendix A.

Velocity profile vjrtl:










u-
\'(r, t)= RZ~Po (1 r RZ~P ~lp "Rr I,
4~uL (RZ~Po RZ.=, ~I 1---~- (3.5)
4~uL I~uLuZ Jo(1"Y



where vjrtl is the velocity as a function of vessel radius and time, r is the radial

coordinate along the vessel diameter, R is the vessel radius at the wall, ~u is the fluid

viscosity, ~ are the Womersley numbers for harmonics n=l through N, Jo is a zero order

Bessel function, J1 is a first order Bessel function, w is the frequency, and t is the time.

The shear stress profile, zjrtl, is derived from the velocity profile using equation (3.5)

dv
and knowing that z = ~u


"Rr
u
(3.6)
.=, RapL I:"~J,('",) 1,,,



where zjrtl is the shear stress as a function of radius and time [35].

The volumetric flow rate can now be obtained knowing the pressure gradient equation

(equation 3.4) and from the direct integration of the velocity profile equation (3.5) across

Idl
the vessel lumen, e =
Idl


ld14~ ld14~P
C ~I ZJi'" ) 1~
e(t)=~ 1- (3.7)
8~uL I~uLu


where ejt/ is the blood flow with respect to time, ~ are the Womersley numbers for

harmonics n=l through N, Jo is a zero order Bessel function, J1 is a first order Bessel










function, w is frequency, and t is time. The dimensionless Womersley number is

calculated at each harmonic as [35].

WP
a=R Jw"P (3.8)


where p is the density of the fluid assumed to be 1.05 g/cm'[29].

The experimental flow rate waveform will be simplified into its sinusoidal

wave components using the Discrete Fourier Transform (DFT) in the simple form of


e(r)- e, +i~~Eos(lrdi.i+C.. ) 0.9)


where ejtl is the time-dependent blood flow rate, e, is mean flow rate, and A,, B,, C,

are the magnitude, frequency, and phase shift, respectively, of the harmonic waveforms

n= 1 through N and are determined via the solution to the DFT [28, 35].

To handle the complex nature of the hemodynamic data, Fourier series

approximation is used. The Fourier series approximation deals with oscillating systems

and utilizes trigonometric functions for modeling these systems. The Fourier analysis

uses both the time and frequency domains, and is used to simplify the complex waveform

into sinusoidal functions. Here, the Discrete Fourier Transform (DFT) represents the

waveform as a finite set of discrete values [28, 57]. With enough individual harmonic

waveform components the original waveform can sufficiently be approximated as seen in

Figure 3-7. Here ten harmonics are used to adequately approximate the original

waveform since higher frequency harmonics have little effect [35, 58]. The DFT data is

converted into sinusoidal functions and is represented in the form of magnitude,

~equency, and phase shift.










0-10 Harmonics Summed to Approximate Onglnal Flow Waveform


01 02 03 04 05 06 07 08 09 1


One Cardiac Cycle

Figure 3-7. Original flow waveform (solid line) compared to approximated DFT flow
waveform (dashed line) and 0-9 DFT harmonics (dotted lines).


Converting the flow rate equation (3.9) into a polar coordinate system yields


ld14aP, i~4~P, Mbcos(mri
e(t)=~ +
8~uL .=, ~uLuZ 2 'O


where ~ is the phase shift, h~~ and E:, are constants that are dependent on u and can be

determined using tables found in Appendix B [29]. Comparing the two forms of the flow

rate equations (3.9) and (3.10) lead to expressions for the mean flow rate, magnitude,

~equency and phase shift

Idl"h(c:,aq
A= (3.11)
~U",ZL








B =


c, =-~--+Eto
2


Substituting equations (3.11-3.13) into the polar coordinate form of equation (3.6) gives
the final expression for the velocity profile:


zQ, (,
"('f)= rd(' (


+CN Ah~~ cos(~lrBt+C,-E:,+
~=1 xRZh~:,


where


r,

,,, I


"R
2 cos a,
"(" ~) [( .jR


ai,,,~,


~i Rr
~sm
M,(a, )


s,(a~)


~(~ ~1
M,(a, )


where h~o and so are tabulated functions [59] dependent on the Womersley number and

a~. dl.~.Rli~.l... radius. (~ ;YIY~I~ Flllbr tOLnld Lu 1PPCII~IX B
Similarly, the shear stress profile, equation (3.6), can also be converted into polar
coordinate form and with substitution of equations (3.1 1-3.13) can be finalized as


Z(T, t)= 4eOlUT C a~ 'M~M,(a,) ,,,(
~a~4 4










where h~ and 81 are tabulated functions dependent on the Womersley number and the

dimensionless radius, which can be found in Appendix B [59].

Therefore, equations (3.14) and (3.17) quantify the velocity and shear stress profiles

along the diameter of the vein graft using morphology data and experimental flow rate

data. The advantages to using this particular model is that it accounts for pulsatile blood

flow, and does not depend on a knowledge of the pressure gradient, which is a difficult

and inaccurate measurement in small vessels. To compensate for this, all calculations are

based on the flow rate data rather than pressure gradient.

Numerical Computations

All major computations were performed using MATLAB version 6.5 Release 13.

A computer program was written to calculate the velocity and shear stress profiles using

the experimental flow rate data and the calculated lumen diameter. A flowchart diagram

of the program layout is seen in Appendix C while the MATLAB code is shown in

Appendix D. The experimental flow rate data used in the computational analysis will

consist of one waveform cycle for simplicity. Using more than one cycle will only

produce repetitive data during each cycle and would be computationally inefficient.

Once the experimental flow rate data and calculated lumen diameter are entered, the

program will perform the DFT and separate the experimental flow rate data into ten

separate harmonic components in the form of magnitude, frequency, and phase shift. The

program will then determine the velocity and shear stress profile at 100 time points and

100 spatial points along the diameter of the vein graft. The program allows the user to

select several choices. A more detailed description of each part of the program is

described in Appendix C.










Using the FFT function in MATLAB, the DFT was determined with the

magnitude, frequency, and phase shift obtained accordingly. Once the DFT waveform

was generated, the velocity and wall shear stress profiles were determined from the

experimental flow rate waveform using our modified Womersley approach as seen in

Figure 3-8. Figure 3-8B illustrates the centerline velocity at several harmonics and the

tinal approximated centerline velocity summing all ten harmonics. Figure 3-8C similarly

illustrates the wall shear stress starting from the first harmonic to the summation of all ten

harmonics.

Once the DFT was determined, the data was used to first determine the centerline

velocity in the vein graft over one cardiac cycle. At the centerline of the vein graft, it can

be assumed that at this point the velocity will reach the maximum value. Therefore, to

check the validity of our modified approach, Poiseuille's law was used to estimate the

~PRZ
maximum velocity at the vein graft's center, ~max [33]. This value was
4~uL

compared to our centerline velocity calculated from our modified Womersley model,

which were found to be very close proving that the calculated data from the numerical

analysis is valid. From our modified Womersley approach, the velocity was then

calculated at each grid point (100 points) along the diameter of the vein graft over the

cardiac cycle (Figure 3-8B). The velocity profile can be determined at any specified

time point encompassing the diameter of the vein graft (Figure 3-9). A similar approach

is used for the calculation of the shear stress profile, at any time during the cardiac cycle

(Figure 3-10).













019 na Wau~iorm wth DiT npprox mat on
Ongnl F~W~RD
-O I~hllmonlc
x- I~ ~nd 2nd *lmMle
r I2r3 hnm~nr~
C:: O(Ohprmonrp


ImD (j)


Cenfeiiine VeiaaN Approximated wih DFT Harmanicr

-*- 0+1 harmonics
---- 0-3 harmonics
0-10 harmonics











C

M 08 12 15 Zii
Tlme(j)


Wail Sheai Strerr Approximated w~w~m DFi Harmon os

*- 0+1 harmonics
---- 0-3 harmonics
0-10 harmonics









:::. *



b I os 16 20
~me(5)


Centerline velocity and wall shear stress derived from the DFT
approximation of the ongmal flow waveform (A) Ongmal flow waveform
(solld ime) compared with the DFT approximation at first harmonic, 0-1
harmonics, 0-3 harmonics summed, and 0-10 harmonics summed (B)
Centerline velocity approximated from (A) (C) Wall shear stress
approximated from (A)


Figure 3-8







30



Velocity Profile at SpeclRed Time Points


0 5 10 15 20 25


Velocity (cm/s)

Figure 3-9. Velocity profile at a specified time points encompassing the diameter of the
vein graft.



Intramural Wall Stresses


Lame's equation for static circumferentialwall stress


A variation of Laplace's law, Lame's equation, is used to calculate the static

circumferential wall stress


PR
Z,,,,,, (3.18)



where zsa~t~h~ is the static circumferential wall stress, P is the pressure, R is the radius, h


is the wallthickness.


This static circumferential wall stress is used in comparison to the dynamic


circumferential wall stress.











Wall Shear Stress over One Cycle


01 02 03 04 05 06 07 08 09 1


One Cardiac Cycle

Figure 3-10. Wall shear stress at a specified point in time over one cardiac cycle.



Isotropic model for wall tension

Assuming a linear stress-strain relationship over the range of blood vessel deformations

[40], the incremental elastic modulus is used to calculate the axial and circumferential

stresses and is calculated as


P,-P, 2(1-o)R,:R,,
E
'"' R,, R,, R~- R:


where E,,, is the incremental elastic modulus, P is the pressure, ois Poisson's ratio


assumed to be 0.5, R is the radius, I, o, 1,2, and 3 represent the inside, outside, minimum,


mean, and maximum values [33].









The isotropic model by Kuchar and Ostrach [39]is used to calculate the radial,

axial, and circumferential components of the wall stress based on the following

assumptions:

Graft is elastic, isotropic, axisymmetric and semi-infinite in length
Radial displacement of the wall is small compared to the radius
Graft undergoes minimal longitudinal displacement [39, 41]


Using the incremental modulus of elasticity equation (3.19) the dynamic wall stress

components can be derived from a general isotropic model by Love [29] based on the

above assumptions as


rwa~.l slr..._ v~ =[p_2~ dv(r)dr

oE~~ 17
Axial Stress, T~~ =~-
R~1

E,, 17 EhZ17
Circumferential Stress, T,, = (1-OZ)' R,+12R: (l-oZ)


where vjrl is the velocity of the fluid in the radial direction, 17 is the radial displacement,

h is the wall thickness, E,,, is the incremental elastic modulus, and R is the lumen radius

[38, 41, 60].

The incremental elastic modulus was determined by using the dynamic in vive pressure

and the video morphometry data [40, 60]. Required experimental input for the model

equation (equation 3.19) are obtained from intraluminal pressure waveforms and video

inputs used to calculate radial displacement.
















CHAPTER 4
RESULTS

Hemodynamics

Flow, Velocity, and Shear Stress

The bilateral vein graft with distal ligation model created two flow environments as

well as isolated the hemodynamic forces involved in vein graft remodeling. The first

flow environment created a reduced flow/reduced shear stress through the ligated vein

graft and the second, an elevated flow/elevated shear stress through the contralateral vein

graft. These flow environments can be seen in Table 4-1. There was an approximate 6-

fold difference in mean flow rate maintained throughout the 28-day perfusion period

(P<0.001, two-way ANOVA, ligated vs. contralateral grafts). Data are presented as mean

f standard error of the mean (SEM).

The modified Womersley mathematical model was used to determine the velocity

profile along the diameter of the vein graft over one cardiac cycle. Based on the flow rate

waveform and calculated in vive lumen diameter, the velocity profile determined from

our mathematical model is the mean steady component plus its oscillatory component.

The velocity profile from equation 3-14 can be illustrated in three dimensions (time,

location, and velocity) and representative profiles for vein grafts at 1, 3, 7, 14, and 28

days can be seen in Figures 4-1 4-5. Five grafts were studied for each time point and

the group averaged data for the maximum, minimum, and centerline velocities for

bilateral vein grafts at i, 3, 7, 14, and 28 days are shown (Table 4-2 4-4).












Table 4-1. Flow environment in the ligated/ low flow vein graft and the contralateral/
flow vein

Ligated Vein Graft Flow Contralateral Vein Graft
DAYS
Flow (mVsec
1 0.12 ~ 0 02 0 78 011
3 0 09 0 02 0 71 0 21
7 013 0 02 0 78 0 09
14 0 07 ~ 0 02 0 82 ~ 0 08
28 014~ 0 06 0 73 ~ 013


Table 4-2. Maximum centerline velocity at each time point in the ligated/ low flow vein
graft and the contralateral/ high flow vein graft.

Ligated Vein Graft ContralateralVein Graft
DAYS
Velocity (cm/sec) Velocity (cm/sec)
1 9 48 ~1 44 36 21 ~ 4 92
3 5 40 ~1 04 33 96 ~ 9 65
7 7 80 ~1 88 46 97 ~11 15
14 4 90 ~1 57 45 46 ~ 714
28 10 37 ~ 2 68 18 86 ~ 4 70


Table 4-3. Minimum centerline velocity at each time point in the ligated/ low flow vein
i and the contralateral/ high flow vein

Ligated Vein Graft ContralateralVein Graft
DAYS
Velocity (cm/sec~ I Velocity (cm/sec
1 1 39 ~ 0 67 17 73 ~ 3 60
3 1 35 ~ 0 35 15 67 ~ 8 26
7 1 97 ~ 0 86 15 76 ~ 3 92
14 1 26 ~ 0 40 16 09 ~ 3 73
28 356~131 769~2 29


Table 4-4. Mean centerline velocity at each time point in the ligated/ low flow vein graft
and the contralateral/ high flow vein graft.

Ligated Vein Graft ContralateralVein Graft
DAYS
Velocity (cm/sec) Velocity (cm/sec)
1 3 61 ~ 0 81 26 24~ 3 99
3 2 80 ~ 0 47 25 58 ~ 9 15
7 4 74~1 27 33 61 ~ 7 20
14 328~110 31 70~6 41
28 6 08 ~1 72 12 00 ~ 3 07


1P< O 001, Two-Way ANOVA, ilgated vs contralateral grafts











































21-

,,





J;_
D


1Day Contra atera Ve n Glaff


1Day L gated Ve n Glaft


CPnlPr np 15
12
08
gn
Wa Imel')


on
we rme(r)


Contol no


Figure 4-1. Three-dimensional velocity profile within bilateral vein grafts one day after
implantation. (A) Ligated vein graft. (B) Contralateral vein graft.


JDay L gated Ve n Glaft


JDay Contra atera Ve n Glaff


Contol no
12
08
gn
Wa
Tme (r)


Figure 4-2. Three-dimensional velocity profile within
after implantation. (A) Ligated vein graft.




A 7 Day L gated Ve n Glaff B


bilateral vein grafts three days
(B) Contralateral vein graft.



r oav contra atera ve n Glaff


b;-e-e
~

:,a ~:'i;?~


12
08
gn
Wa Tmel')


,,
,,


Figure 4-3. Three-dimensional velocity profile within bilateral vein grafts seven days
after implantation. (A) Ligated vein graft. (B) Contralateral vein graft.


~


gn 08
Wa
me (r)









14 Day L gated Ve n Glaff 14Day Contra atera Ve n Glah
A B



~ICr






,, w, a -- -rr

Figure 4-4. Three-dimensional velocity profile within bilateral vein grafts fourteen days
after implantation. (A) Ligated vein graft. (B) Contralateral vein graft.


A B
Llga~ed Van Gian Contraiateral Vein GlaR




~~



Time(r) Time(r)

Figure 4-5. Three-Dimensional Velocity profile within bilateral vein grafts 28 days after
implantation. (A) Ligated Vein Graft. ~) Contralateral vein graft.


Representative dynamic wall shear stress profiles, using equation 3.17, for

individual rabbits at each time point are provided in Figures 4-6 4-10 and reveal the

hemodynamic changes have taken place over the 28-day time course. Group averaged

data for the maximum, minimum, and Poiseuille shear stresses can be seen in Tables

32~u~
4-5 4-7. Poiseuille shear stress [22] was calculated for comparison as z =
i~~,,,

The Poiseuille shear stress is the mean steady shear stress and is provided for

comparison. At day 7, the dynamic shear stresses peaks and drop off by day 28 in the

contralateral/ high flow vein graft. Tables 4-5 4-7 also show a considerable difference










m wall shear stress between the hgated/ low flow and contmalateral/ Ingh flow vein grafts

(P <0 001, two-way ANOVA, 11gated vs contralateral grafts) It can be seen that the

maximum dynanne wall shear stress withm the contralateral/ Ingh flow vem gmafts was

elevated at day 7 and day 14 By day 28, there was a considerable reduction (P=0 003, 7

vs 28 days, P=0 03, 14 vs 28 days) Corresponding to thns, the nummu dynamic wall

shear stress exhlubted the same increase at day 7 and a reduction in day 28 (P=0 01, 7 vs

28 days)



1-ayu Wal Shear Stress vs Time
-Llgated Ve n Graft
8 -Contralatoera Veln Graft






7







O 00 008 O 12 O 16 02
Time (s)

Figure 4-6 Dynamic wall shear stress vs tune wi~thmn blatemal vem gmafts one day after
Implantation











3-Day Wal Shear Stress vs Time


0004 008 O 12 O 16 02
TIme (s)

Figure 4-7 Dynamic wall shear stress vs tune wi~thmn blatemal vem gmafts three days
after plantation


I~~ay Llgte Vein GraftvsTm


0004 008 O 12 O 16 02
TIme (s)

Figure 4-8 Dynamic wall shear stress vs tune withm bilatemal vem gmafts seven days
after plantation











14-Day Wall Shear Stresrus lime
Lgated Vein Grft
20 Contralateral Ve n Graft
















D 00 0 08 0 12 0 16 02
Time (s)

Figure 4-9 Dynamic wall shear stress vs time wi~thmn blatemal vem gmafts 14-days after
Implantation



28-Day Wall Shear Stress vs T me

-Llgated Ve n Graff
--Contralateral Ve n Graft













2I /


0 004 008 012 016 02
Time (s)

Figure 4-10 Dynanne wall shear stress vs time wi~thin bilateral vem gmitls 28 days after
Implantation











Table 4-5. Dynamic wall shear stress values- Maximum shear stress'
Ligated Vein Graft Contralateral Vein Graft
DAYS
Shear Stress cm2 Shear Stress
1 6 69 0 44 23 39 1 47
3 3 73 0 28 23 95 2 70
7 6 05 ~ 0 77 36 97 ~ 4 62
14 414~ 0 55 32 80 ~ 2 84
28 443~1 41 11 30~2 43


Table 4-6. Dynamic wall shear stress values- Minimum shear stress'
Ligated Vein Graft Contralateral Vein Graft
DAYS
Shear Stress (d cm Shear Stress(
1 0 48 016 10 71 0 82
3 0 79 0 06 11 54 214
7 1 59 0 22 15 65 1 55
14 1 40 0 21 13 68 1 61
28 1 19 ~ 0 45 4 32 ~ 0 92


Table 4-7. Dynamic wall shear stress values- Poiseuille shear stress

Ligated Vein Graft Contralateral Vein Graft
oaus emz) Shear Stress(
Shear Stress (d
1 2 06 ~ 0 52 15 23 ~ 2 64
3 1 65 ~ 0 29 15 92 ~ 6 29
7 2 99 ~ 0 96 23 29 ~ 6 89
14 2 28 ~ 0 84 20 28 ~ 5 38
28 219 ~ 0 71 6 92 ~1 47


Wall Stress and Elastic Modulus

PR
The static circumferential wall stress was determined using Lame's Law,



equation 3.18. The static wall stress used a mean pressure determined from the


experimental in vive pressure. Table 4-8 provided the group averaged static


circumferential wall stress (equation 3.18)in vein graft segments ipsilateral, and


contralateral to the distal ligation. Static circumferential wall stress in the contralateral


vein appeared to decrease over the entire time course within both ipsilateral and


Z P=O 003, 7 vs 28 days, P=O 03, 14 vs 28 days

'P=O 01, 7 vs 28 days











contralateral vein grafts. This corresponded to the increase in wall thickness and showed

that the circumferential wall stress after 28 days reduced to 30% at Day 1 (P<0.001, Day

7,14, and 28 vs. Day i). For comparison, the estimated static wall stress for a vein in the

venous circulation was 0.053 x10~6 dynes/cm2. There was no consistent difference in the

static circumferential stress found when comparing the ligated and contralateral vein

grafts.

The dynamic wall shear stress components for the ligated and contralateral vein

grafts were calculated for the 28-day grafts using equations 3.20-3.22 and can be seen in

Table 4-9. There were no significant differences in the dynamic wall stresses between

the ligated and contralateral vein grafts. Observed was an order of magnitude difference

between the static and dynamic circumferential wall stresses at day 28. The elastic

moduli, calculated from equation 3.19, were similar in both vein grafts showing that the

elastic properties for both cases were similar. This showed that the hemodynamic

simulations and the secondary impact on intimal thickening had little impact on the

mechanical properties of the vein graft wall.



Table 4-8. Static circumferential wall shear stress comparing ligated vein graft and
contralateral vein graft. 4
Contralateral Vein
Ligated Vein Graft
Graft
Circumferential
DAYS Circumferential Wall
Wall Stress
Stress
(g106 dynes/cm') xlU~b
1 2 62 ~ 0 29 219 ~ 0 24
3 2 91 ~ 0 90 2 02 ~ 0 34
7 1 12 ~ 0 22 1 28 ~ 0 25
14 0 22 ~ 0 04 0 63 ~ 0 09
28 0 35 0 053 0 86 016



4 P









Table 4-9. Dynamic wall shear stress and elastic modulus at day 28.

Ligated Vein Graft ContralateralVein Graft
Wall Stress


Radial, S,
(X 10" dynes/cm2) 5 68 ~ 3 50 4 8 ~ 3 51
Axial, T~
(X 10" dynes/cm2) 417 ~ 3 37 5 92~1 03
Circumferential, Tss
x 10" dynes/cm? 3 37 ~ 3 74 11 9~ 2 05

Elastic Modnlns, Ei~,
(x 106 dynes/cm) 12 7 ~ 3 48 12 2 ~ 2 27

Vein Graft Remodeling

Figure 4-11 showed the intimal area, medial area, and the graft diameter based on

the morphologic data. Figure 4-11A showed significant variation overall starting at day

7. The most significant change in the intimal area occurred in the ligated vein graft

segment. The intimal area dramatically increased over the time course for the ligated

vein segment while the intimal area in the contralateral vein segment increased slightly.

Intimal hyperplasia can be inferred from Figure 4-11A at day 7 showing an enhanced

intimal thickening in the low flow/ligated vein graft (P<0.001, two-way ANOVA). A 9-

fold difference in the intimal area at day 14 was found when comparing the hyperplasia

rates in the ligated/low flow and contralateral/high flow vein grafts (P <0.001, 1.3 if

0.25~Lm vs. 0.15 f 0.02 ~un for ligated vs. contralateral grafts). At day 28, there was a 6-

fold difference in the intimal area(P<0.001, 1.21 f 0.17mmZ vs. 0.22 f 0.17 mm2 for

ligated vs. contralateral grafts). Figure 4-11B showed the medial area increasing in both

the ligated and contralateral vein grafts starting at day 7. This accompanied the intimal

hyperplasia seen in Figure 4-11A. This increase in medial area was time-dependent with

the maximum areas occurring at days 14 and 28 (P< 0.001, Day 14 and day 28 vs. Day










1). The graft diameter in Figure 4-11C remained stable for both the ligated and

contralateral vein graft segments until day 28 when there was a slight increase in

diameter when the grafts were exposed to high flow conditions (P=0.002, 3.25 f 0.15 mm

vs. 3.72 f 0.24 mm for ligated vs. contralateral grafts).

As early as day 7, it can be inferred that some form of vein graft remodeling has

taken place. In these findings, it can be seen that there was a correlation between intimal

thickening and the hemodynamic forces by day 7. It can be seen that under low shear

conditions the intimal area was markedly increased by day 7. It can also be seen that

under low flow conditions the intimal area also increased, signifying intimal hyperplasia

has occurred.











EZZI 1galai
I -Cntratera


*@


1 2.


mm Panmle-.m


EzzLgaid
Im Coiin~rb~


Figure 4-11


Veingrah remodelmg


(A) Intimal area comparmy ligated/low now vem


graft and contralateral/Ig Bwvn ghat Ho ven ra "(B Mdrlarea companng
ligated vein graft and contralateral vemn graht d (C) EEL graft diameter
throughout time course


'P< 0 001, 11gated vs contralateral Day l4, P < 001, 11gated vs contralateral Day 28

SP< 0 001, Day 1 vs Day 14, P <0 001, Day 1 vs Day 28


1 P= 0 002,11gaeu otaatedvs cont um ralaterlDy2 engat
















CHAPTER 5
DISCUSSION

Mathematical Model: Modified Womersley Approach

Our mathematical model adapted from Womersley uses the DFT of the

experimental flow rate waveform as the basis for the velocity and shear stress

calculations. Many investigators have validated the Womersley analysis using MRI

velocity measurements. In their studies, the experimental root-mean-square differences

between the analytical and measured velocities approached 5% and correlation

coefficients are greater than 0.99 [51, 61]. This approach is limited to model systems

with uniform, cylindrical conduits, distant to branch regions or vessel curvature.

Our first-order modified mathematical model is best suited for this study since it

accounts for pulsatile blood flow and uses experimental flow rate data, which is more

accurately measured than the pressure gradient. Measurement of the intraluminal

pressure within the small vessels of the rabbit carotid system is difficult and can cause

significant artifact. To verify the accuracy of our numerical computation, the

experimental flow rate is compared to the approximated flow rate data calculated from

the DFT (Figure 3-7). A known mean velocity and shear stress was compared to the

computational results obtained from our mathematical model as well. All of these

verifications show that our mathematical model is fairly accurate and valid for

approximating the velocity and wall shear stress.










Experimental Model

Our experimental model [52] used isolates the hemodynamic forces, i.e. shear

stress and wall tension, involved in vein graft remodeling. This follows the work initially

performed by Zarins [62], which defined the impact of shear stress on lesion development

by studying the localization of arterial plaques to low shear regions of the carotid

bifurcation. Since then, it has been inferred that the structure of the blood vessel wall is

closely linked to its surrounding hemodynamic environment [11, 12, 62]. Many

experimental studies demonstrate a balance between the local shear stress and tensile

forces predicting vessel morphometry during physiologic and pathologic remodeling [63-

65]. From these findings, more recent studies reaffirm the concept of lumen preservation

and positive remodeling by observing an inverse relationship between wallthickness and

shear stress in the human coronary circulation [15, 66, 67].

Our experimental model is a modified version of the standard techniques used for

studying vein graft adaptation [6, 23, 52]. There is variability in the extent of intimal

thickening under hand-sewn anastomotic technique using the rabbit carotid interposition

vein graft. Our experimental model uses a cuff anastomotic technique, which was

adapted from a rodent lung transplantation model [68]. Adapting this technique to our

experimental rabbit vein graft model offers a more reproducible method for vein graft

placement [52]. Our experimental model provides a more robust biologic response by

using extensive distal branch ligation, which was suggested by Meyerson [69]. High

flow/ high shear conditions were created in the contralateral vein graft while the low

flow/low shear conditions were created in the ligated vein graft. Our model created a 6-

Eold difference in mean flow rate and a 5-fold difference in intimal cross-sectional area.










Vein Graft Remodeling

Studies in understanding vein graft remodeling and secondary pathologic occlusion

began with Dobrin [21], who realized that intimal thickening was closely correlated with

low shear while medial thickening was correlated with circumferential strain. In more

detail, Zwolak [7] and Schwartz [6] applied these observations to the rabbit carotid

system and found that an increased wall tension correlated with an increase in myointmal

thickening.

Using our mathematical model, we were able to calculate velocity and wall shear

stress profiles. By day 28, there is an increase in intimal thickening under low flow

conditions. This was consistent with the findings of Galt[23], who focused on the effect

of wall shear stress in the rabbit vein graft model. Different from our model, he left the

internal carotid artery widely patent, ligating only the external carotid branch. The Gait

study found a 50% reduction in mean wall shear stress, which led to a 70% increase in

intimal area. Our model shows a 30% wall stress reduction from day 1 to the end of the

time course, day 28.

In addition, our findings show that the intimal thickness is inversely proportional to

flow. The wall thickness, comprised of the media and the intima, is also inversely

proportional to the flow [7]. These findings suggest a correlation between intimal

thickening and wall shear stress. Looking at the mean Poiseuille wall shear stress, we see

under low shear conditions, intimal thickening is markedly increased by day 28.

At high static wall circumferential stress the intimal area is very low while the

opposite could be said for low static wall shear stress. This data agrees with Zwolak's

hypothesis that early in the vein graft remodeling process, there is an elevated wall stress

with less thickness. As the remodeling process continues, by day 28, the wall stress of










the vein graft is normalized toward the normal arterial values [7]. The derived isotropic

model [29]isolates wall tension showing that deformation does occur. Circumferential

stress is elevated and is considered to be the largest stress showing an increase in

diameter [22]. This increase in diameter shows the vein graft has adapted to the arterial

system.

We also find that determining the circumferential wall stress within vein grafts

using Lame's equation is limited. Comparing the results from this static circumferential

wall stress to the dynamic circumferential wall stress determined from the dynamic wall

motion and incremental modulus of elasticity implies the stress is approximately l0-fold

less than the Lame's equation estimation. This result is consistent with other studies that

found Lame's equation inaccurate when the wall thickness to radius ratio is greater than

0.10. This ratio during vein graft remodeling ranged from 0.11 to 0.15, which is the

reason for the inaccuracy [7].

Future Research

Future research can be taken into different aspects. Computationally, a more

realistic development of the model can be performed in other blood vessels such as

studying blood flow in a bifurcation or a curved blood vessel under similar hemodynamic

conditions using a more sophisticated computational technique, such as finite element

analysis. Altering the hemodynamic conditions such as assuming the blood vessel to be

rigid to demonstrate blood flow under stenosis could also be studied. A more detailed


and sophisticated computational technique based on this in vive experimental data could

be developed to eliminate any future and unnecessary in vive experiments.

Mathematically, improvements in our modified model could be to apply the model

to other blood vessels in the arterial system. For example, we can make more realistic










assumptions about the blood vessel by taking into account the viscoelasticity, wave

propagation, or reflected waves. These were neglected in our assumptions. By altering

the model to include non-linear terms in the Navier-Stokes equations as well as define a

different model for the intramural stresses, we can better simulate non-linear in vive

pulsatile blood vessel systems. Another improvement would to assume a thick vessel

wall. Studies have applied this assumption and found that there was a 12% higher value

in phase velocity than the Womersley analysis [29]. Many studies have taken reflected

waves and wave propagation into account, these studies differ in the assumptions, i.e.

constrained or unconstrained tubes. The differences disappear when the vessel becomes

large such as in arteries. For our study with small vessels, the wave propagation and

viscoelasticity may have some effects depending on the geometry and length of the vessel

[29, 70] and may have to be accounted for.

The use of our model can be used in comparison with other aspects that influence

vein graft remodeling and adaptation such as biochemical factors. Since hemodynamic

forces appear to have an influence on the biochemical factors, our mathematical model

could be used in studying the effects of dynamic shear stress on biochemical factors such

as matrix metalloproteinases (MMP). Other future research could look at how the wall

shear stress calculated in this study affects the individual layers of the blood vessel wall.

Our mathematical model only looks at the shear stress that affects the innermost and

outermost layers. Other research could entail studying the changes caused by intramural

pressure rather than the present study's look at changes caused by differential flow rate

environments.










Conclusion

The understanding of how hemodynamic forces impact vein graft remodeling has

been limited to the use of the mean or steady state models for estimating shear stress and

wall tension. Our modified Womersley analytical approach provides a simple technique

for estimation of dynamic shear stress and wall tension in an in vive environment. Our

model provides a way to study the biomechanical changes that occur during the early

stages of vein graft remodeling. This technique provides a basis for understanding how

the physical hemodynamic forces along with the biological changes control the balance

between physiological remodeling and pathologic stenosis, eventually leading to

strategies for the enhancement of vein graft patency.
















APPENDIX A
DERIVATION OF WOMERSLEY EQUATIONS

The following derives the equations were used to determine the velocity and shear

stress profiles. Starting with the Navier-Stokes equations in cylindrical coordinate form

we can derive the Womersley equations. The Womersley equations were derived as a

fully developed pipe flow with an oscillating pressure gradient.

From the Navier-Stokes momentum equation, we can derive the oscillatory portion

of the Womersley solution (Figure A- 1) [33].


V Vaial


------,
Tube wall
Velocity --t


Figure A- i. Schematic of flow through a blood vessel.


z-momentum Navier-Stokes equation:


dv, ];~ la dv, ) 1 dZv, dZv,
,, +
dz r dr dr rZ dBZ dzZ


dv, +~~ dv, +V, dv, +
dt dr r dB


pg (A.1)


where p is density, v is the velocity in the r, 8, and z directions, r is the radius, ~u is

ap
viscosity, is the pressure gradient, and g is gravity [27]









The following assumptions can be made:

1. Laminar Flow
2. Incompressible Flow
3. Axisymmetric Flow
4. Parallel flow
5. Ignoring body forces.

The z-momentum Navier-Stokes equation can be reduced to:


r-
dvdt p dzIdpt~( dvdr (A.2

dp.
Since Is an oscillating pressure gradient, we can assume that it takes the form
dz

Ael". To simplify things, we assume that the velocity profile is v(rt)=ReCf( r ) elw').


and the boundary conditions, v,(r ~R, t)=O and v(r=O, t)


finite number. By adding the


r ,,




(A.3)




(A.4)


above assumptions into equation A.2, and introducing the parameter, y =


can transform the equation to

u' f"+uf'+u' f=o

Equation A.3 is the Bessel equation and has the general solution of

f(u)= RI~ (y)+BY, (y)


where J, and Y, are the zero order Bessel functions of the first and second kind [71].

The term Y,(y) ~co as y~ O because of the assumption that the velocity reaches a finite

value at r=O. Because ofthis, B is zero and substituting the no-slip conditions back into

equation A.4 the velocity profile for an oscillating pressure gradient becomes


A
\'(r, t)= Iwp I


(-"(~)1









[33, 72]


equation (3.4), we can replace A


where cc is the Womersley numberu=R andr

aPaP ~P
Since the pressure gradient is = + C -~'""' ,
L L L


in equation A.5 and reach the final equation for the velocity profile as equation (3.5)


RapI~LI~LUZ


1--~-

I(''I )


,rjr.T)= R' AP, (
4~uL (
















APPENDIX B
TABLES USED TO DETERMINE CONSTANTS BASED ON WOMERSLEY
NUMBER

The following tables were used to determine the constants based on the Womersley

numbercc, which was calculated using equation (3.6). Table B-l [59] is referenced to


ascertain the values of Mo and Bo, which are used in order to determine E:, and h~~


(equations 3.13 and 3.14). Table B-2 [59] is referenced to ascertain the M1 and 81, which

are used in order to determine the shear stress profile (equation 3.15). Table B-3 [29] is

referenced to ascertain h~:, and E:,,which are both used in determining both the


velocity and shear stress profiles (equations 3.12 and 3.15) .







55


Table B-1 Table of solutions to Bessel functions used to calculate Mo and 60
TAB~LES

TABLE 27

Jo(zil) = M,(z)e' => = ber z+i bel


o+1s IUOOD I FAmM


J.11(r) o ( 3I(}


a


(1,216, represents 1,216x 10'.)

Whecn a > 4.5, Mo(r) and 00(z) canl be foulnd to 4 dercimall plaes and to thle
nearest 0-001', respectively, from thea ormurllrae
0 0384
logslo ne(s) = 030)7093;-| z --03009)- 1 logno =.
5-Dor
Oa(s) 2 40 51423:n~- 225 (degrr ees)

The function logno(?lzlfo(s)} is tabulated in case it is desairedl to intcrplolate.
Proportional parts may be used~ for in~terpolation withlout introdueslng an
error > 1 in thle last figure, exceplt fo~r A,,(l) when1) z~ > 2- andl fr logno{ 1'e3/o(i)}
whecn a < 075. Wheon a > 27, joRao(Vzrlle(;)) sholdl~ be found from Llho liabin
and MoI(z) can he dedulced theurerom. Scot also thle formnulso in problems 32, 33
at the endt of Chap. W11.













Table B-2 Table of solutions to Bessel Functions to determne M1 and 81







J1(i(i) = M (z)elBI=) i; beri z+i beit z

J M(z) logs41.(4z,2)) fz(s) a 3 1.,(1) lolhe(4.11s(z)) 0421)
0.05 0I 025 3-4 1:IT5e 02 r3 1 11? 0 a 713 I172 0.
0*1 0075 a43 145n 10 le2 Ill 5-uli 11 rxum 17tu I

0 25 1250 11 2 -7850 135 li 1 2-50 1 1474 04308 178 30-
o*su 0-30 u2 0147 133I 04 U1 5 I-1 0 j332 180 013
O SE. I U1750 i011 l135 88 I ? 0l 1-40 039 1 TOJil dli
8:: 40 0-2000 r1a 10? j3-3 B I'r 034i 184 39
u1r45 Ir2250 7-78 13045 E 1 n0 u-4004 185tol
0L 50 (I-20 billife 13 0 2 0 I-(il5 U 1:7 188 37
O~s f& o*275k 14006l 11.5717 2-9 1-705 I 442i I I8 1.8 111
9*60 ~ ~ ~ "~j 000 1300 178 a 0 u*4UJr 119571
0s 00WI: 0 8232 3-18, 13*0 s t L II 5267 100-371
0-70 860 T:5C 1649i' I(Lt* 0 II UY: U-5j:"r 2.5j
0-75 3753n 1-500 1'.ith:11 3 :11 1 191 1) .586.5 200 83*
o-as u-40mr T-ta lat 34 2.14 0-0171 21002 sl
0.85 U*4258 7*5037 140 17:a 3, Gol 2-7 6 l, r
000 0-50 T-Hti 140l-80 3-6 I51 88 I)li.Ol-JUI
D00 O*600U I-700k 14-1= -80 liUI 07UL05 228 07*

14 55-'1 1.62 ]:1-110 4) 23~ 3 Ot 088l 24.5 77
I1 5 5776 T-7tijo 144-4 4 SO 4 27 0-9;170 233 87
I*20 0*03 U8to 545St 5uo 1 81 LI13 17JBSS
I1.2 O 6 200 1 8471 140 17* 5- 7 ?Dlj 1 2tl02 2L93 48"
140 0-6518 i 87:11 147-07Ul1-5 5 25 334
1-35 .18018 1 8018" 148-OS 6+5 14-90j Iln I70 33 4
1-4 0 070 04 148OO 7 04.1-0 1 7343 353 6
146 0-7833 T-40 150-00 7- 8-"7 L-8889 II7350
1*60 0-7508 i-9088 151*04* 8- 39O 07a s0-184 303-69'
1*55 0 718 0 81 OlK)9 1518 I 0I 741 51 352 433-0
U0 -BlasU 0 01Y5 1531-?3 10-0 144 7 I 6604 474 28
] as 0-8408 0 0335L 154-38 11-0 280-4 2-90185 514-03I
190U 0 U8084 O 0530U 155'SSO 1i-0 543 U 3 2705 655 OS
1-76 O'8002~~~~ 003 57* o ,8 8000 63584*
1.80 (1 n244 09338 158?n 00 I I3 -5(l72 71.?i!
I85 U*900 U-1127 159217 l80 3,2., -1222 79-6
IV95 1 UH & 0 13 19 14t1-~ 1k ) U :4,753, 7.3738 1980 0
2V 00 UI 1-4 6 103-7 ?U 0h 1,78 8si 8000 I 123 5
2*1 110 0*2015 IMOSal 400si 1,04 :lu I I!s1118 817 8 168-4
8-05 i184 040 187+53 450~iU Jj 3,td J13417 189008
PIU 1.1(11 11 (1,1 l 1 rpeut I178 x 510'j lLB Ii


Whn z.n > 50, 31I,(z) an~d 0,(s) can be; found to 4I decimnal p~luces andc to theo
neresot. 0-001", respectivetyly, fon the formulae:
01152
log~nor,(s) e 0307093L-~ -039!00-109-logioz
1510,
0,(r) a 4051423;+--+--4-87- (dogrees).

The0 function logy,(VzMd,(z)) is tabu~llated ini clas it, ir kesiredc to interpolate.
Proportionlal parts may be used for inter~polation, without inltroducing an
error > 1 in the lasit Iigure, except for n1,(a) wheln a > 3*3 and for logno(Vzbl(z)}
when s < 1-3. When; > 3*3, log,.(. II,(z)} should be foundl from the table and
AlI(,a) ded~cuced thelrefron. See alho the~ formlaofn inl problems 34, 35 at thle end
of Chap. VII.








57




Table B-3 Table to determmeM;,' and e'o based on the Womersley number


TABLE. C I Wol~me-ley's (Wactins "I1'<. .md d-'t for ir 0-40


16~ ~3 0
16 56 3 OS
16j (9910
16 3 1
1607 3 20

1591 i 25
I57 ib130
15~ 61 j 5
15i 46 -o
15j 32 3 4

15 i8 3 50
15 4 l55
14 90 8 0
I(477 8 65
14 63 1 "O

14 5 3
t'4 38 3 O
1435 3.85
I14 13 89
14O 0 .95

13.59 900
13 77 9 05
13 66 9 to
(3 54 9 15
13.43 9 -0

13 32 9.15
iL3 21 9
1311i935
13 00 9 40
129 945

L2.30 9 50
12.70 955
(1260 9 60
12.50 9.65
1Z.41 970

1^.31 9-75
12.22 9.30
12.13 9.35
i104 9 9
11 95 9.95

11 37 10
IiS 75 1
1 1.70 14i
1161 16
11.j3 18

11.15 20
Il 37 Z5
i i 19 30
1121 35
II.14 U0


II W23
-U 1X50


n i~l 2 **


5402 39 96 5 50
5882 39 05 5 55
5959 13 16 5 0
6032 37 32 5 65
0102 365 570 :

6169 35710 5 75
6233 34 93 580
o24 34 18 5 35
6353 33 -46 590
6409 32.77 5 95

64613 32 09 6 00
6514 31 45 6 05
6563 30 82 6. to
6611 302' 61 i
6656 29 6 6 LO

6700 29 08 6 25
67412 28 53 6.30
6783 iS.OL635
6822 1.7 51 6 40
6860 i7 02 6 45

6896 26 55 650
6931 1 6.10 6 55
o965 25.66 6 60
6999 25 24 6.65
7031 14 83 6 70

7062 2443 6 75
7092 24 05 6 80
7122 23.68 65 d
7151 23 32 6.90
7179 "2.98 695

7236 22 64 7 CO
7233 22.32 705
7259 22.00 7 10
7285 21.70 7 15
i721 2140 70 1'

7334 21 11 725
7358 20.34 7 30
'382 -)0 56 7 35
u)45 20 30 7.40
7428 20 05 7 45

7450) 19 80 7 50
:472 19.55 7 55
.7493 19 32 7 60
7515 19 09 7 65
7536 1886 770

7556 18.63 75
7576 13 43 7 80
7196 (8 '3 7 85
7616 18 0: 790
7635 1733 95

~64 17 63
7673 17 44
~691 17 26
770 1 0
777 1690


19 98X 2,80
(9 901 255
19) 79 2 90
(19 2.95

190 300 1
39 14 305
JJ J3 3.10
18_.7 3.15
ds 07 1 0

17 61 3 25
J7 11 13 30
1s 57 J 35
XJ 97 340
55 33 3 45

J4 65 3.50
63,91 3.55
lj 14 .00
J3 32 3 65
J1.45i 3 70

SO.55 375
19 60 3 SO
78 61 3 5
7" 59 3 90
76 53 3 95


1 5 1886 75 44 4 00
S30 20Z9 7i 31 4 85
135i i37 316 4 10
1 0 322 ~1 8 4 15
tas 242 70-77 420

[ 0 624 69 14 4 25
15 1776 08.30 4.30
. 0 1930 67 03 4 35
bi 65 083 6j 76 40
:.7 3"37 6447 4.45

1 S 3389 63.18 .0
1 0 3540 61 39 4 55
1 5 3690 60 59 60
i 9 3837 59 30 4 65
jgs jf38 8.02 4i 70

2.CO 4125 56.74 475
2j 05b 426 55 47 SO
:.10 -1400 54 22 4 35




:o 20 905 lu9 39 5 0

n 5133 4711 5.15
-li 5 24 6.01 5.20

ii j34 -L 93 15 S
i" 55 5 4 -3 58 5 30

55031 i1 36 5.40
OD 5713 4090o 15.5
F -om Woumersley (1957a)
valuesr of e a re in desres~.
















APPENDIX C
FLOW CHART OF MATLAB PROGRAM

A MATLAB code was developed to determine the velocity and shear stress profiles

within the vein graft. Figure C.1 shows a flowchart describing the code. A pop-up menu

appears providing the user with several selections when analyzing the experimental

waveform data collected in vive. The menu has the following selections:

1. Load waveform
2. Perform Velocity and Shear Stress calculations
3. Analyze Pressure calculation
4. Display flow rate, velocity, and shear stress vs. time
5. Create and display a 2-D velocity profile movie
6. Create and display a 3-D velocity profile movie
7. Save all calculated data in program
8. Exit program

Once the user selects a menu choice, the program performs the operation and then

once complete, the program returns to the menu until the user chooses to exit the program

completely. Choice 1 allows the user to input the flow waveform to be analyzed as well

as the lumen diameter determined from the morphology data. Choice 2 processes all the

input data given from choice 1 and determines the centerline velocity and shear stress. It

also provides a display of the maximum, minimum, and mean values for both the velocity

and shear stress calculations. Choice 3 allows the user to input the in vive pressure

waveform and tabulates the maximum, minimum, and mean pressure. Choice 4 provides

a graphical display of the in vive flow rate waveform, the derived centerline velocity, and

the derived wall shear stress versus time. Choice 5 and 6 create .avi movies of the


calculated velocity profile in 2-D or 3-D. Choice 7 allows the user to save all calculated










data into a text file. The text file saves the flow waveform, the velocity profile, the shear

stress profile, and the minimum/maximum/mean values for the profiles. Choice 8 exits

the program completely.






































(2)
Centerline
Velocity/
Shear Stress
Calculation


Figure C-l. Flowchart of MATLAB code.

















APPENDIX D
MATLAB CODE FOR DETERMINING VELOCITY AND SMEAR STRESS
PROFILES

Code for Discrete Fourier Transform


function d=dft(data)
%=======================================
%This function determines the Discrete Fourier Transform
%assuming frequency rate of 1000 samples/sec and displays
%data in the form of Frequency, Magnitude, and Phase Shift
%Input data: Original waveform data
%Output data: Frequency, Magnitude, Phase Shift of DFT
%=======================================


chessy=data/60;

y=fft(chessy) ;
N=length(y);

y=v/N;
m=abs(y);
Es=1000;
~es=(o: (N-l))*l;
p=-(angle(y));
fourier=[freq' mp

~freq(l:ll);
phase=p(l:ll);


%converting data from mi/min to ml/s

%DFT of data
%calculates number of data points

%shift zero harmonic to mean flow
%magnitude ofy

%frequency for one cycle
%phase shift offft
I ; %displays data

%Retrieves fmag, phase of first 11 points


%0-10 harmonics


mag=m(l:ll);
fourier=[f mag phase];

d=fourier;











Code for Determining Velocity Profile

function velocity=createmov(dataD)
%=======================================
%Program to analyze velocity profile from flow data
%==Inputs: DFT Data and Diameter tin cm) of blood vessel
%==Outputs: Velocity profile data over one cardiac cycle
%== Velocity (cm/s)
%=======================================
%=======================================
f=data;
a=xlsread('Table C1'); %Table C.1 for Womersley values, M10,e10
b=xlsread('Table 27b');
%=======================================
%=======================================
p=1.05; %--------- ---------------- --g/mL density
R=D/2; %---------------------------radius of vessel
u=.04; %------------------------g/(cm sc) viscosity
%=======================================
%--------------Loop to calculate womersley #, M'o, M'10
g=0:1/(50-1):1;%--------------------Calc 100 points along the radius
for z=l:length(g)
'=R*g(z);
y=r/R;
number=l0;
for n=l:number


alpha(n)=R* sqrt((2*pi*f(n+l, l)*p)/u);%alpha(n)=Womersley #~n harmonic

zl=alpha(n)*y; %zl and 22= z for M'o, M'10
z2=alpha(n);

%Interpolate using function interplq (xmxi) x- alpha values from
%table; m- corresponding mprime 10 values; xi- alpha(n) calculated form
%above


mlO(n)= interplq(a(:, l),a(:,2),z2); % mprimel0 from Table C.1

% values taken from Table 27
thetal(n)=interp iq(b(:, l),b(:,5),zl); % theta at alpha* (r/R)
theta2(n)=interp iq(b(:, l),b(:,5),z2); % theta at alpha

Mo l(n)= interplq(b(:,l),b( :,2),21); %M() from Table 27
Mo2(n)= interplq(b(:, l),b(:,2),z2);

elO(n)=interp 1 q(a(:,l),a(:,5), alpha(n));
k=Mol(n)/Mo2(n);












Mprimeo(n)=sqrt(l-2 *k* cos(theta 1 (n)-theta2(n))+k^2);
% As defined by eqn C.7 from Appendix C
eo(n)=atan(k* sin(theta l(n)-theta2(n)))/(l -k* cos(theta l(n)-theta2(n)));

end


i-i,
%for each time point, calculating the corresponding flow at each harmonic
for t=0:.01:1
%initialize array to zero for summation of harmonics
sum=zeros(n, 1);
for n=l:number
x=cos(2*pi*f(n+ll )*t-f(n+l,3 )+eo(n)+e 1 O(n));
%calculation of pressure gradient at harmonics 1-10
Q(")=f(n+l,2)/(pi* R^2*mlO(n));
sum(n)=sum(n)+(Q(n)*Mprimeo(n)*x); %summation ofharmonics


v(ni)=sum(n);


%velocity profile at each time point


n=n+l;


end
i=i+l;


for n=l:number


%calculate max v
maxv(n)=max(v(n,:));

end


tspan=[0:.01:l];
for t=1:101 %calculate zeroth harmonic of velocity
k(t)=((2*f( 1,2)/(pi* R^2))*(1 -(r^2/R^2)));
end


tsum=zeros(li-l);


for ts=l:(i-l)
test=v(:,ts);


%summation of harmonics 0-10


total=cumsum(test)+k( i),
tsum(ts)=total(n);

end


if z==lsl=tsum;







64


elseifz==2,s2=tsum;
elseifz==3,s3=tsum;
elseifz==4,s4=tsum;
elseifz==5,s5=tsum;
elseifz==6,s6=tsum;
elseifz==7,s7=tsum;
elseifz==gsg=tsum;
elseifz==9,s9=tsum;
elseif z==10, slO=tsum;
elseif z==1 i, sl l=tsum;end

end


sll=zeros(ll0l); %-----sets Velocity at wall equal to zero
%=======================================
%=====Final Output: Velocity profile
velocity=[s il;sl O;s9;s8;s7;s6;s5 ;s4;s3;s2;sl; s2; s3; s4; s5; s6; s7; s8; s9; s10; sil];











Code for Determining Wall Shear Stress Profile

function shear=stress(dataD)
%=======================================
%===Program to analyze Wall Shear Stress
%===INPUTS: DFT Data and Diameter of Blood Vessel
%===OUTPUTS: Wall Shear Stress (dynes/cm^2)
%=======================================
~data;
a=xlsread('Table C1'); %Table C.1 for Womersley values, M10,e10
b=xlsread('Table 27b');
c=xlsread('table28');
%=======================================
p=1.05; %-----------------------------g/mL density
R=D/2;%------------------------cm radius ofvessel
u=.04; %-----------------------g/(cm sc) viscosity
%=======================================
%---------Loop to calculate womersley #, M'o, M'10
g=0:1/(50-1):1;
for z=l:length(g)
'=R*g(z);
y=r/R;
numb er=lO;
for n=l:number
alpha(n)=R*sqrt((2*pi*f(n+ll)*p)/u);%al #~n harmonic
zl=alpha(n)*y; %zl and 22= z for M'o, M'10
z2=alpha(n);

%Interpolate using function interplq (xmxi) x- alpha values from
%table; m- corresponding mprime 10 values; xi- alpha(n) calculated form
%above


mlO(n)= interplq(a(:,l),a(: ,2),22); % mprime 10 from Table C.1
% values taken from Table 27
thetal(n)=interp iq(c(:, 1),c(:,5),zl); % theta at alpha* (r/R)from Table 28
theta2(n)=interp iq(b(:, 1),b(:,5),z2); % theta at alpha

Mol(n)= interplq(c(:,l ),c(:,2),zl); %M() from Table 28
Mo2(n)= interplq(b(:,l),b( :,2),22); %M() from Table 27
elO(n)=interp 1 q(a(:,l),a(:,5 ),alpha(n));
end


i-i,
%for each time point, calculating the corresponding flow at each harmonic
for t=0:.01:1
%initialize array to zero for summation of harmonics
sum=zeros(n, 1);











for n=l:number
x=cos(-(pi/4)+2*pi *f(n+ll)*t+thetal (n)-theta2(n)-e 1 O(n)+f(n+l,3));
%calculation of pressure gradient at harmonics 1-10
Q(n)=(f(n+lz)*u* alpha(n)*Mo l(n))/(pi* R^3*m 10(n)* Mo2(n));
sum(n)=sum(n)+(Q(n)*x); %summation of harmonics
tau(ni)=sum(n); %velocity profile at each time point
n=n+l;
end
i=i+l;
end


tspan=[0:.01:l];
for t=1:101 %------------calculate zeroth harmonic of shear
k(t)=((4*f(1,2)* u*r)/(pi*R^4));
end
n=number;

tsum=zeros(li-l);

for ts=l:(i-l) %-----------summation of harmonics 0-10
test=tau(:,ts);
total=cumsum(test)+k( 1);
tsum(ts)=total(n);
end


for k=l:length(g)
sh(z, :)=tsum;
end
end
sll=sh(length(g),: );
j=length(s 11);
%=======================================
%=========Final output for wall shear stress===============
for i=l:length(sll)
shear(i)=s il~);i=j-l;











Code for Main Driver of Program

%=======================================
%==================MAIN MENU=============================
%==Following menu will provide options for analyzing
%== the flow or pressure waveform and determining the
%== velocity and wall shear stress profiles.
%== Also provides user with the chance to save the file
%== as well as create a 2-D avi file or 3-D movie of the
%== velocity profile.
%=======================================
clear all
warning off
N=1000;


forj=l :N
k=menu('Select option to calculate:','load waveform to Analyze','Centerline Velocity and
Wall Shear Stress Profile', 'Pressure Waveform', 'Create Graph of Flow, Centerline
Velocity and Wall Shear Stress','2-D Velocity Movie','3-D Velocity Movie','Save
Data','Exit');

%=======================================
ifk==l %------------------LOAD WAVEFORM
clear
rabbit=input('Rabbit Number:');
x=O;
%------------------------------------Inp Flow Data
rootname=input('Naam offlow waveform file:','s');
extension='.txt';
filename=[rootnameextension];
[file]= eval('load(filename)');
%---------------------------------------
loc=input('Enter number of diameter locations(l or 3).'),
for w=1:3
ifloc ==1
diameter(l)= input(' Medial Diameter in cm:');
break
elseif loc==3
diameter(l)=input(' Proximal Diameter in cm:');
diameter(2)=input(' Medial Diameter in cm:');
diameter(3)=input(' Distal Diameter in cm:');
break
else
disp('Error: Please enter number of diameter locations.')
end
end
%--------------------------------------- DFT










z=dft(file);

%=======================================
elseif k==2%---------------------Velocity and Wall Stress
u=0.04;
for d=l:loc
vel=createmov(zdiameter(d));
velocity(d,: )=vel(ll,:);
shear(d,: )=stress(zdiameter(d));
maxv(d)=max(velocity(d, :));
minv(d)=min(velocity(d, :));
meanv(d)=mean(velocity(d, :));

maxstress(d)=max(shear(d, :));
minstress(d)=min(shear(d, :));
meanstress(d)=mean(shear(d, :));
meansteadyshear(d)=(4* u*z(l,2))/(pi* (diameter(d)/2)^3);
end


MVG=[maxv maxstress;minv minstress; meany meanstress]
Poiseuille shear=meansteadyshear

%=======================================
elseif k==3 %-------------------------Evaluate Pressure Waveform
P=input('Name of Pressure Waveform file:');
press=[Pextension];
[Pressure]- eval('load(press)');
x=l;
%--------------------------------Pressur in mmHg

maxp=max(Pressure);
minp=min(Pressure);
meanp=mean(Pressure);
PRESSURE=[maxp minp meanpl

%=======================================


elseif k==4 %--------------------Create Graphs

tspan=0:.01:l;
%Plot of Original Flow Waveform
time=0: l/(length(file)-l): i,
subplot(3,1,1)
flow=file'/60;
plot(timeflow)
title('Original Flow Waveform')











ylabel('Flow (ml/s)')

%Velocity
subplot(3, 1,2)
plot(tspan, velocity)
ylabel('Velocity (cm/s)')
if loc==3
legend('PVG','MVG','DVG')
end

%Plot of Shear Stress
subplot(3, 1,3)
plot(tspanshear)
xlabel('One Cycle')
ylabel('Shear Stress(dynes/cmY)')
if loc==3
legend('PVG','MVG','DVG')
end

%======================================


elseifk==5 %-------------------------2-D Velocity Movie
disp('Note: The avi file is saved as vprofile.avi')
tigure(2)

mov=avifile('vprofile avi')

for k=1:101
h=plot(vel(:,k),r);
set(h,'EraseMode','xor')
axis([-l 15-111)

F=getframe(gca);
mov=addframe(movF);
end
disp('Note: The avi file is saved as vprofile.avi')


%======================================


elseif k==6%------------------------3-D Velocity movie
aviobj =avifile('vprofile3 .avi')
for m=1:101
k=vel(:,m);


%--------------------------performs coordinate transform from 2-d to 3-d










forj=l:length(r)
for theta=1:361
x(thetaj )=r~)*cos(theta*pi/ 180);
y(thetaj )=-r~)*sin(theta* pi/i 80);
z(thetaj)=k~);
theta=theta+l;
end
j=j+i;
end


h=figure('visible','off);
none=[l;
%-----3-D profile
surf(zyx,'FaceAlpha','flat', 'AlphaDataMapping', 'scaled','AlphaData', gradient(z), 'FaceCol
or','blue','MeshStyle','column','Facelig
set(gc~'Xlim',[-l 151)
set(gca,'YTickLabel',none,'ZTickLabel',n [172. 56-12.87
4.191)
aviobj=addframe(aviobjh);
end
aviobj=close(aviobj);


%=======================================

elseif k==7 %----------------------------SAVE DATA
flow=file'/60;
tspan=0:.01:l;
disp('This will save the velocity and shear stress profiles in the graph.')
root=input('Save velocity and shear stress profiles as:');
name= [rootextension];
%The following will save centerline velocity and shear stress to one file.
eval(['save ', name ,' tspan flow velocity shear -ascii'l)
disp('The following prompt will ask you to save the maximum/minimum/mean data for
velocity and shear stress')
r=input('Save min/max data as:');
a=[rextension];

%The following will save min/max velocity/shear stress values and mean
%steady flow value into separate file.

if x==l
eval(['save ',a,' rabbit mary minv meany maxstress minstress meanstress
meansteadyshear maxp minp meanp -ascii'l)
else x==O












eval(['save ',a,' rabbit mary minv meany maxstress minstress meanstress
meansteadyshear -ascii'l)


elseif k=
break


%----------------EXITS MENU

















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BIOGRAPHICAL SKETCH

Chessy Fernandez was born on May 4, 1978, in Norfolk, VA. Her father's military

transfer brought the family to Pensacola, FL, where she later graduated from high school

in 1996. After growing up in the sunny Florida weather, she decided to go to college

further north in the nation's capital. It was in Washington, DC, that she was able to

explore different aspects of engineering through volunteering and internships while

pursuing a varied education, which included studying photography and web design. She

attended The Catholic University of America in Washington, DC, where she graduated

with a bachelor's degree in biomedical engineering in May 2000. After a break from

school, she decided to return to Florida to be closer to home and continue her education.

In the spring of 2001 she began her graduate school career in biomedical engineering at

the University of Florida.





PAGE 1

VEIN GRAFT REMODELING By CHESSY FERNANDEZ A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERING UNIVERSITY OF FLORIDA 2003

PAGE 2

Copyright 2003 by Chessy Fernandez

PAGE 3

I dedicate this thesis to my father, R.D. Fernandez.

PAGE 4

ACKNOWLEDGMENTS I would like to extend my warm thanks and gratitude to the many people who have helped me through this journey. I would like to take this opportunity to thank my advisor, Dr. Roger Tran-Son-Tay, for providing me with insight and guidance during this process. I would also like to thank Dr. Scott Berceli for giving me the chance to work on this project as well as provide me with the opportunity to see first hand how engineering is applied to clinical applications. My greatest appreciation extends to everyone in Dr. Berceli and Dr. Ozakis lab at the VA hospital. Without them, this project would not have been accomplished. This project is the hard work and effort of all in the lab and to them I truly am indebted. I would also like to thank my friends, family, and professors who have all helped me become the person I am today. I would like to thank my parents for providing the encouragement and loving support as I achieve my lifes goals. I would like to extend my appreciation to my friends for their moral support, friendship, and advice. Finally, I would like to thank all my professors for providing me with the knowledge and background needed for becoming an engineer in todays world. iv

PAGE 5

TABLE OF CONTENTS Page ACKNOWLEDGMENTS ................................................................................................. iv LIST OF TABLES........................................................................................................... viii LIST OF FIGURES ........................................................................................................... ix ABSTRACT....................................................................................................................... xi CHAPTER 1 INTRODUCTION........................................................................................................1 Arterial Occlusive Disease ...........................................................................................1 Objective.......................................................................................................................2 Specific Aims................................................................................................................2 2 BACKGROUND AND SIGNIFICANCE....................................................................3 Anatomy and Physiology of Blood Vessels .................................................................3 Current Understanding of Blood Vessel Remodeling ..................................................4 Morphologic Changes and Physical Forces ..........................................................4 Biochemical Events...............................................................................................5 Vein Graft Adaptation...........................................................................................5 In-Vivo Model.......................................................................................................6 Wall Stress Models.......................................................................................................6 Hemodynamics......................................................................................................7 Poiseuille model for steady flow....................................................................7 Womersley analysis for pulsatile flow...........................................................8 Modified Womersley approach......................................................................9 Other mathematical models for estimation of velocity and wall shear stress 9 Intramural Wall Stresses......................................................................................10 Lames equation for wall tension-static circumferential wall stress............11 Isotropic model for wall tension...................................................................11 Other mathematical models that study intramural wall stress......................13 Significance ................................................................................................................14 3 MATERIALS AND METHODS ...............................................................................16 Experimental Vein Graft Model.................................................................................16 v

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Surgical Methods.................................................................................................16 Hemodynamic Measurements .............................................................................18 Video ............................................................................................................18 Flow rate.......................................................................................................18 Pressure ........................................................................................................19 Morphometric Analysis.......................................................................................20 Modeling of Wall Stresses..........................................................................................22 Modified Womersley Approach..........................................................................22 Numerical Computations.....................................................................................27 Intramural Wall Stresses......................................................................................30 Lames equation for static circumferential wall stress.................................30 Isotropic model for wall tension...................................................................31 4 RESULTS...................................................................................................................33 Hemodynamics ...........................................................................................................33 Flow, Velocity, and Shear Stress.........................................................................33 Wall Stress and Elastic Modulus.........................................................................40 Vein Graft Remodeling...............................................................................................42 5 DISCUSSION.............................................................................................................45 Mathematical Model: Modified Womersley Approach..............................................45 Experimental Model ...................................................................................................46 Vein Graft Remodeling...............................................................................................47 Future Research ..........................................................................................................48 Conclusion..................................................................................................................50 APPENDIX A DERIVATION OF WOMERSLEY EQUATIONS ...................................................51 B TABLES USED TO DETERMINE CONSTANTS BASED ON WOMERSLEY NUMBER ...................................................................................................................54 C FLOW CHART OF MATLAB PROGRAM..............................................................58 D MATLAB CODE FOR DETERMINING VELOCITY AND SHEAR STRESS PROFILES..................................................................................................................61 Code for Discrete Fourier Transform .........................................................................61 Code for Determining Velocity Profile ......................................................................62 Code for Determining Wall Shear Stress Profile........................................................65 Code for Main Driver of Program ..............................................................................67 vi

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LIST OF REFERENCES...................................................................................................72 BIOGRAPHICAL SKETCH .............................................................................................78 vii

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LIST OF TABLES Table page 4-1. Flow environment in the ligated/ low flow vein graft and the contralateral/ high flow vein graft..................................................................................................................34 4-2. Maximum centerline velocity at each time point in the ligated/ low flow vein graft and the contralateral/ high flow vein graft...............................................................34 4-3. Minimum centerline velocity at each time point in the ligated/ low flow vein graft and the contralateral/ high flow vein graft...............................................................34 4-4. Mean centerline velocity at each time point in the ligated/ low flow vein graft and the contralateral/ high flow vein graft......................................................................34 4-5. Dynamic wall shear stress valuesMaximum shear stress........................................40 4-6. Dynamic wall shear stress valuesMinimum shear stress........................................40 4-7. Dynamic wall shear stress valuesPoiseuille shear stress.........................................40 4-8. Static circumferential wall shear stress comparing ligated vein graft and contralateral vein graft. ...........................................................................................41 4-9. Dynamic wall shear stress and elastic modulus at day 28.........................................42 B-1. Table of solutions to Bessel functions used to calculate M 0 and 0 .........................55 B-2. Table of solutions to Bessel Functions to determine M 1 and 1. ..............................56 B-3. Table to determine and 10M 10 based on the Womersley number..........................57 viii

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LIST OF FIGURES Figure page 2-1. Anatomy of the blood vessels walls with the layers of the artery wall on the left and the layers of the vein wall on the right.......................................................................4 2-2. Bilateral interposition vein graft with distal ligation...................................................7 2-3. Mechanical models of viscoelastic behavior where L is the elongation and S is the stress. .......................................................................................................................10 2-4. Wall stress acting on arterial wall. T xx and T are the longitudinal and circumferential wall stresses. S rr and S rx are the normal and fluid shear stresses....12 3-1. Schematic of anastomotic cuff technique used in vein grafting procedure...............17 3-2. Bilateral carotid vein grafting with distal ligation model..........................................17 3-3. Pre-measured marker next to left carotid distal artery unligated...............................19 3-4. Measurement of in vivo flow rate and pressure with waveform output from Chart Recorder Program....................................................................................................19 3-5. Histologic cross-section illustrating the longitudinal section of anastomotic segment using Massons staining...........................................................................................21 3-6. Cross-sectional area of 28-day ligated/low flow rabbit vein graft at harvest 20x and 40x illustrating different wall layers........................................................................21 3-7. Original flow waveform (solid line) compared to approximated DFT flow waveform (dashed line) and 0-9 DFT harmonics (dotted lines)................................................25 3-8. Centerline velocity and wall shear stress derived from the DFT approximation of the original flow waveform............................................................................................29 3-9. Velocity profile at a specified time points encompassing the diameter of the vein graft..........................................................................................................................30 3-10. Wall shear stress at a specified point in time over one cardiac cycle......................31 4-1. Three-dimensional velocity profile within bilateral vein grafts one day after implantation. ............................................................................................................35 ix

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4-2. Three-dimensional velocity profile within bilateral vein grafts three days after implantation. ............................................................................................................35 4-3. Three-dimensional velocity profile within bilateral vein grafts seven days after implantation. ............................................................................................................35 4-4. Three-dimensional velocity profile within bilateral vein grafts fourteen days after implantation. ............................................................................................................36 4-5. Three-Dimensional Velocity profile within bilateral vein grafts 28 days after implantation. ............................................................................................................36 4-6. Dynamic wall shear stress vs. time within bilateral vein grafts one day after implantation..............................................................................................................37 4-7. Dynamic wall shear stress vs. time within bilateral vein grafts three days after implantation..............................................................................................................38 4-8. Dynamic wall shear stress vs. time within bilateral vein grafts seven days after implantation..............................................................................................................38 4-9. Dynamic wall shear stress vs. time within bilateral vein grafts 14-days after implantation..............................................................................................................39 4-10. Dynamic wall shear stress vs. time within bilateral vein grafts 28 days after implantation..............................................................................................................39 4-11. Vein graft remodeling..............................................................................................44 A-1. Schematic of flow through a blood vessel................................................................51 C-1. Flowchart of MATLAB code....................................................................................60 x

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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Engineering VEIN GRAFT REMODELING By Chessy Fernandez December 2003 Chair: Roger Tran-Son-Tay Major Department: Biomedical Engineering To improve the long-term patency of peripheral vein grafting, a mechanistic understanding of early vein graft adaptation is necessary. Critical to these events are the imposed hemodynamic forces, which regulate the balance between intimal thickening and expansive remodeling ultimately defining the morphologic characteristics of the vein graft. The objective of this project is to characterize the dynamic shear and wall tensile stresses during early vein graft remodeling by developing a mathematical model for shear and tensile stresses, based on the work of Womersley and Patel using in vivo flow rate data collected from an experimental rabbit vein graft model. Previously ex vivo models were used to determine the hemodynamic forces in the steady state only. Our mathematical model uses an in vivo bilateral carotid vein graft with distal branch ligation model to collect the experimental flow rate data in a pulsatile hemodynamic environment. The experimental animal model created two flow environments with reduced flow/shear through the ligated vein graft and elevated flow/ shear in the contralateral vein graft. Using thirty-four New Zealand white male rabbits, xi

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vein grafts were implanted and then harvested at 1, 3, 7, 14, and 28 days after initial surgical procedure. Hemodynamic and video measurements were collected before and after ligation for calculation of the shear and wall tensile forces. A computational program was developed to determine the velocity and shear stress profiles based on the collected hemodynamic data. Due to the size and length of the rabbit vein graft model, the pressure gradient is difficult to ascertain; therefore the flow measurement is used for the basis of our calculations. This approach is slightly different from previous models that relied upon the derived pressure gradient. Vein grafts were exposed to distinct flow environments characterized with a 6-fold difference in mean flow rate. Accelerated intimal hyperplasia and reduced outward remodeling were observed in the low flow grafts. At day 7, there was a peak in maximum and minimum shear stress with a delayed increase in lumen diameter leading to normalization of wall shear by day 28. At day 3, the intramural wall tension was at its maximum and there was an increase in wall thickness leading to a significant reduction of these stresses by day 14. There was no difference in incremental modulus of elasticity despite the significant difference in remodeling between the high and low flow grafts. Our mathematical model provides a simple way to determine dynamic wall shear and tension in a pulsatile hemodynamic environment, using readily available technology. Our mathematical model reveals a correlation among shear stress, flow, and intimal thickening, which coincides with ex vivo studies modeling steady flow. Future research may entail a more realistic development of computational modeling of pulsatile blood flow in this model as well as other more complicated configurations such as bifurcated blood vessels, which in the past have dealt with steady flow in ex vivo models. xii

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CHAPTER 1 INTRODUCTION Arterial Occlusive Disease Arterial occlusive disease, causing myocardial infarctions and strokes, affects millions of people and is one of the leading causes of death in the United States. In many cases, invasive surgical techniques, such as bypass vein grafting and angioplasties, have been used to alleviate vascular occlusion. Bypass vein grafting uses a vein taken from another part of the body to replace the obstructed blood vessel and inserts it into the arterial system to provide better blood flow [1]. However, restenosis or occlusion can still occur in the time frame of months to years. The current standard of care for peripheral vein grafting is 80% 1-year and 60% 5year patency rates [2, 3]. Patency is defined as a blood vessel that remains open to flow without thrombosis [1]. Because of these reasons, many researchers have been attempting to improve these long-term patency results. An understanding of early vein graft adaptation and progression must be established in order to improve the long-term results. Currently, it is known that many factors, including physical forces, morphologic changes, and biochemical events, are involved in this adaptation process. Playing a key role in the remodeling process are the biomechanical forces[4, 5]. The changes in these biomechanical forces regulate the balance between intimal thickening and expansive remodeling, which govern the morphologic changes in the vein graft [6, 7]. 1

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2 Objective The arterial hemodynamic environment is comprised of intricate branching geometries. Characterizing this environment is complex and consists of many non-linearities [8]. Dynamic non-linearities on the biologic response of the arterial vasculature has been well established; however the understanding of the interface between these imposed physical forces and the secondary remodeling processes has traditionally been based on time-invariant, linear estimations of wall shear and tensile forces[9]. The objective of this study is to characterize early vein graft remodeling by finding the correlation between the physical forces and morphologic changes by characterizing the dynamic shear and wall tensile forces with the intimal thickening and expansive remodeling that occurs in vein graft arterialization. Specific Aims The specific aims for this thesis are to: 1. Develop a theoretical model to calculate wall shear and velocity profiles in the arterial circulation. 2. Perform experiments to gather flow and pressure measurements using a rabbit vein graft model in order to estimate shearing and tensile forces to which the graft is subjected. 3. Examine the correlation between the physical forces and morphologic changes, during the vein graft remodeling process.

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CHAPTER 2 BACKGROUND AND SIGNIFICANCE Anatomy and Physiology of Blood Vessels The cardiovascular system consists of the heart, blood, and blood vessels. The heart pumps the blood throughout the body via the blood vessels. The blood transports oxygen and nutrients from the lungs to the rest of the tissues in the body while carrying carbon dioxide back to the lungs. The blood vessels consist of the arteries and veins. The arteries carry blood away from the heart while the veins carry the blood back to the heart. The blood vessel walls consist of three layers: the intima, the media, and the externa (Figure 2-1) [10]. The intima is the innermost layer consisting of endothelial lining and an underlying layer of connective tissue. There is a thick layer of elastic fiber called the internal elastic lamina (membrane) in arteries. The media is made up of smooth muscle tissue and connective tissue. Another layer of elastic fiber called the external elastic lamina separates the media from the externa. The externa, or adventitia, is the outermost layer. It is the muscular and elastic components of these layers that change the diameter to compensate for blood pressure and blood flow changes [10]. It can be seen from Figure 2-1 that the arterial walls are generally thicker than the vein walls. This is because the artery has more smooth muscle and elastic fibers. The arteries are elastic and contractile. It is this elasticity that allows passive changes in the vessel diameter in response to changes in blood pressure. When there is no opposition to pressure, the elastic fibers recoil in the artery, causing the lumen to constrict. Veins however have thinner walls because the blood pressure is lower than in the arteries. 3

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4 Because of this low blood pressure, veins have valves in order to oppose the force of gravity and to prevent backflow [10]. Figure 2-1. Anatomy of the blood vessels walls with the layers of the artery wall on the left and the layers of the vein wall on the right. Current Understanding of Blood Vessel Remodeling Morphologic Changes and Physical Forces Vein graft remodeling after bypass graft or angioplasty procedures is comprised of two different processes. Constrictive remodeling involves a thickening of the vessel wall which leads to a narrowing of the inside lumen. Positive remodeling involves the enlargement of the blood vessel and an increase in the lumen diameter. It is the balance between these two processes that is influenced by the local hemodynamics of pressure and flow [11, 12]. This balance also determines the patency of the treated arteries or venous bypass grafts [13-15]. Little is known about how exactly the mechanisms of shear and tensile forces influence the remodeling process.

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5 Biochemical Events Extracellular proteins form a scaffold with the vein graft wall, and breakdown of these proteins is involved with vascular reorganization [16, 17]. This reorganization leads to permanent changes in the size and composition of the blood vessel. More specifically, matrix metalloproteinases (MMP) are specialized enzymes important in the reorganization of the extracellular proteins. The balance between the MMP enzymes and its tissue inhibitors control the blood vessel reorganization in response to outside stimuli. Two enzymes, MMP-2 and MMP-9 appear to play important roles in this reorganization of the blood vessel wall under the outside stimuli. In previous studies, it has been found that active MMPs are controlled by: transcriptional regulation, translation into protein, release into extracellular space, and proteolytic activation of latent enzyme [17, 18]. The MMP activity is regulated to maintain the architecture of the blood vessel wall [19]. The MMPs involved in the reorganization process deal with the breakdown of elastic lamellae to permit smooth muscle cell migration and proliferation in the intima. They also deal with the incorporation of the new cellular matrix components, which accompanies an increase in luminal area [16]. Vein Graft Adaptation Significant structural changes in the vein graft wall are caused by the changes in the pressure and flow environment from the venous to arterial circulation. Characteristics of this change from the low pressure/low flow venous system to the high pressure/high flow arterial system include an increase in the intimal and medial thickness in the wall, as well as a burst of smooth cell proliferation [7, 20]. Vein grafts are exposed to four forces: circumferential (hoop), radial, longitudinal tensile forces, and surface shearing forces,

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6 which are directed along the axis of flow [21, 22]. Researchers have been unable to separate these variables but have found evidence suggesting that there is a correlation between the medial thickening and circumferential tensile forces and another correlation between the intimal thickening and fluid shear forces [6]. In-Vivo Model A rabbit vein graft will serve as the experimental system for these experiments. In this model, a reversed segment of external jugular vein is used for the interposition grafting into the common carotid artery. Several investigators have used this particular model to study the mechanisms of vein graft failure and intimal hyperplasia[6, 7, 23-26]. Various vascular constructions are used to create distinct regions of altered shear stress and wall tension. The bilateral carotid interposition graft model was adapted to examine the effects of shear stress (Figure 2-2). The model is modified to include a distal ligation of the internal carotid and ligation of three of the four primary branches of the external carotid artery [23]. By doing this, there is an 8-10 fold difference in the flow rates between the ligated/low flow and contralateral/high flow vein grafts. Creating this change in hemodynamic environment, the remodeling process caused by the difference in flow can be studied. Wall Stress Models The local hemodynamics of pressure and flow influence the remodeling process. It is the forces caused by pressure and flow that appear to have an effect on biochemical events and the morphologic changes. To characterize these shear stress and wall tensile forces, an understanding of the hemodynamics is needed in order to develop a well-suited mathematical model for vein graft remodeling.

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7 Figure 2-2. Bilateral interposition vein graft with distal ligation. Contralateral/ high flow graft shows constrictive remodeling and positive remodeling while the ligated/ low flow graft shows constrictive remodeling without positive remodeling. Hemodynamics Poiseuille model for steady flow Analyzing blood flow in arteries or vein is analogous to analyzing a Newtonian fluid in circular pipe flow. Poiseuilles law relates pressure, flow and radius in rigid tubes under steady flow [27]. Poiseuilles Law for steady flow is determined as LPRQ84 (2.1) where Q is the blood flow, R is the radius, and LP is the pressure gradient, and is the fluid viscosity (0.04 poise(or g/cms)) for blood). However, this is limited in its applications when applied to the pulsatile flow of blood[28].

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8 Womersley analysis for pulsatile flow Pulsatility adds a whole new set of components to the blood flow. For example, the pulsatility adds more inertial forces and the diameter of the vessel varies with time throughout the cardiac cycle [29]. To account for the unsteady pulsatility in blood, a more complex model must be used. The Womersley analysis relates flow and the pressure gradient for pulsatile blood flow[30-32]. The Womersley analysis, derived from the Navier-Stokes equations, makes the following assumptions: Laminar flow Newtonian flow Uniform cylindrical tube with a rigid wall Infinite length The Womersley analysis drops the nonlinear terms in the Navier-Stokes equations because these terms are relatively small when applied to blood flow. This is done to linearize the equations and create a general solution for blood flow velocity (Appendix A). The Navier-Stokes equations are further reduced by assuming that velocity is only in one axial direction and thus removing the radial velocity. Since the blood vessel is assumed to be a non-moving rigid wall, the no-slip condition applies to the wall meaning there is no velocity occurring at the wall [29, 33, 34]. From the Womersley analysis, the Womersley number, nR where is the Womersley number, is the frequency, is the density, and is the fluid viscosity. This non-dimensional number is introduced and relates the oscillatory flow to the viscosity. The Womersley number is an indicator of the stability of laminar flow [35].

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9 Modified Womersley approach The physical nature of the rabbit model makes it difficult to accurately assess the pressure gradient directly because the vessel is small in length and in diameter. In the past, the experimental pressure gradient in the arterial system was determined by measuring the pressure at two points along the arterial tree [30]; however, this approach is more suitable when making hemodynamic measurements in large diameter arteries. Applying this approach to smaller vessels such as vein grafts, the estimation of dynamic intraluminal pressure is prone to significant artifact. Because of this and recent advances in instrumentation, flow rate is used for estimating the velocity and shear stress profiles rather than the pressure gradient [33, 35]. Other mathematical models for estimation of velocity and wall shear stress Realistically, blood vessels exhibit more of a viscoelastic behavior. The blood vessel is not completely elastic but has both viscous and elastic properties. Viscoelastic behavior can be shown as three different types of models as shown in Figure 2-3 [29]. The mechanical models are all combinations of linear springs, which instantaneously produce a deformation proportional to a load, and dashpots, which produce a velocity proportional to the load at any instant. The force in the Maxwell model is transmitted from the spring to the dashpot. The Voigt model deals with the idea that the spring and dashpot have the same displacement. The Kelvin model is considered to be the standard linear solid model. Viscoelasticity can be experimentally determined when the material undergoes periodic oscillations [36]. With a viscoelastic behavior a phase lag between the stress and strain is produced. Womersley also modified his original analysis to incorporate this behavior by incorporating the effects of wall viscosity [29]. The blood vessel is suddenly strained and

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10 the strain is constant afterwards. Viscoelasticity complicates our model and our modified approach still provides substantial information in a simplified form. Due to the significant complexities and inability to obtain an analytical solution with the addition of viscoelasticity, these terms were omitted in our current model. Figure 2-3. Mechanical models of viscoelastic behavior where L is the elongation and S is the stress. (A) Maxwell model. (B) Voigt Model. (C) St. Venant model. Intramural Wall Stresses Blood flow causes viscous drag between the outermost laminae of the fluid and the vessel wall. This viscous drag is a possible factor in causing arterial disease. It is this shear stress imposed on the wall that can affect the functional and structural integrity of the endothelial cells. In atherosclerosis, there are intracellular deposits of cholesterol in the intimal layers of the vessel walls. The movement of these proteins into and out of the intima can be affected by wall shear and tensile stresses. This causes plaques to bulge

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11 into the vessel lumen with differences in local stress potentially playing a role in the localization [29]. Stress is the force distributed over an area across a surface, and can be divided into two components: normal stress, which is stress directed perpendicular to the surface and shear stress, which is the stress along the transverse cross section. Shear stress occurs in the tangential plane along the wall. Strain is the deformation of the stressed object. It is the ratio of change in a given dimension to its original unstressed state [29, 37]. To study these intramural stresses, the Lame and isotropic models are used in our analysis. Lames equation for wall tension-static circumferential wall stress Static circumferential wall stress can be determined using a variation of Laplaces Law called Lames equation, hPR where P is the pressure, R is the lumen radius, and h is the wall thickness [33]. While easy to use, the wall stress using Lames equation neglects pulsatility and fails to take into account the differences in stress in the successive vessel wall layers. It also assumed that the wall thickness does not change when stretched thus forcing Poissons ratio to be zero[29]. Poissons ratio is defined as the ratio of lateral or perpendicular strain to the longitudinal or axial strain [37]. Isotropic model for wall tension A more realistic model may be used to account for the pulsatility in blood flow providing a more realistic model of the stresses within a vein graft. The dynamic wall stress is based on the models of Patel and Fry[38] and the Kuchar and Ostrach[39] and Bergel [40], which use the elastic modulus to determine three wall stresses: radial (S rr ), axial (T xx ), and circumferential (T ) stresses (Figure 2-4) [41].

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12 The Patel and Fry model assumes that there is a non-linearity in the arterial wall stress-strain relationship and assumes the blood vessel as a cylindrically orthotropic tube. The Bergel model is based on this model and is used to determine the elastic modulus[38, 40, 41]. Figure 2-4. Wall stress acting on arterial wall. T xx and T are the longitudinal and circumferential wall stresses. S rr and S rx are the normal and fluid shear stresses. The incremental elastic modulus relates stress and strain. It is a measure of a materials resistance to distortion by a tensile or compressive load and in its simplest form can be considered the inverse of compliance [42]. In a purely elastic body, the elastic modulus is linearly proportional to stress. This is in accordance with Hookes law. However, the blood vessel walls produce a curvilinear stress-strain relationship, which is seen in most nonhomogeneous materials [29]. The Kuchar and Ostrach model assumes an elastic, isotropic model for arterial wall stress. Assuming a linear stress-strain relationship and the above assumptions from Patel and Fry and Kuchar and Ostrach, a model can be developed using a general isotropic model over the range of blood vessel deformations. With this generalized model, the incremental elastic modulus is derived by Bergel, and

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13 from this modulus, the radial, axial, and circumferential stresses can be calculated through algebraic manipulation of the general isotropic model [29]. Other mathematical models that study intramural wall stress Other modified models have been used to estimate wall stresses. The model used in Sokolis, et al. [43] studies the stress-strain relationship in uniaxial tension. This model removes the viscoelastic phenomena through preconditioning prior to tensile-testing. The stress is calculated based on the Kirchhoff expression, where TAFSo and oll where S is the Kirchoff stress, F is the force, A o is the area, T is the Lagrangian stress oAF oll is the change in length. The model uses the nonlinear Green-St. Venant strain (E), E= 12 21 where is the longitudinal stretch ratio and the elastic modulus (M) is dEdS M where S is stress and E is the Green-St. Venant strain [43]. Rachev, et al. [44] developed a mathematical model for stress-induced thickening of the arterial wall close to the implanted stent. They assumed the host artery to be a cylindrical shell with a constant thickness, and the stent to be non-deformable in the circumferential direction. Similar to the Sokolis model, the Rachev model used Green-St. Venant strain. These models would not work for our study due to the fact that Kirchoff stresses and the definition of the Green-St. Venant strain deals with large deformations such as those seen in the pulmonary artery, which is not the case in the rabbit vein graft [36].

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14 Significance It is unclear how all these factors, physical forces, morphologic changes, and biochemical factors, together play a role in vein graft remodeling and adaptation. It is the biomechanical forces that influence the remodeling process that are central to understanding vein graft adaptation [4, 5]. Previous studies have studied the hemodynamic forces based on ex vivo models [38, 41, 45, 46]. What makes this project different is the use of an in vivo experimental model to study the hemodynamic forces that affect the remodeling process. While many in vivo models have studied hemodynamic forces in the mean or steady state, they have not studied the forces in a pulsatile environment due to the complexity of calculations [22, 47]. However, cell culture experiments studying endothelial and smooth muscle cells [48-50], and anatomic correlation studies [9] have shown the impact of oscillatory shear stress on blood vessel morphology and cellular function. In recent years complex numerical analyses for studying these hemodynamic forces in a pulsatile environment have been easier to achieve with advancements in computational numerical methods. Several experimental studies have investigated the effects of these forces in a pulsatile environment; however they have used model casts of blood vessels, examined in bench-top perfusion systems [41, 51]. There is a limited amount of in vivo studies available to provide an understanding of the interaction between the dynamic components of the biomechanical forces and the intact vein graft or blood vessel. By utilizing the advancements in computer capabilities and accurate instrumentation, hemodynamic forces can be determined in an in vivo pulsatile environment. By obtaining a mechanical understanding

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15 of early vein graft adaptation, we can gain further insight into the early remodeling process to provide strategies for the patency of vein grafts.

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CHAPTER 3 MATERIALS AND METHODS Experimental Vein Graft Model Surgical Methods Bilateral carotid vein grafting with selective distal branch ligation was performed to create defined regions of differential blood flow [52]. Using thirty-four New Zealand white male rabbits (3.0-3.5 kg), an anastomotic cuff technique was performed to implant the bilateral external jugular vein segments into the common carotid arteries. The rabbits were anesthetized through intramuscular injection with ketamine hydrochloride (30.0 mg/kg), and anesthesia was maintained with endotracheal intubation and inhaled isoflurane (2.5-3.0%). Heparin was also given intravenously, 1000 units, at the start of the procedure [52]. The technique harvested external jugular veins, 3 cm in length, for the creation of an interposition graft into the common carotid artery (Figure 3-1) [52]. The external jugular vein ends were passed through polymer cuffs, which were fashioned from a 4-French introducer sheath. An arteriotomy was performed and the reversed cuffed vein ends were inserted and fixed to the artery. Ligatures were placed distal to the vein graft in order to unilaterally reduce the graft blood flow. This completely occludes the internal carotid artery and three of the four primary branches of the external carotid artery, which can be seen in Figure 3-2 [52]. At 1, 3, 7, 14, and 28 days after the initial implantation, the vein grafts were exposed via a mid-line neck incision. Rabbits received BrdU 16

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17 injections 24 hours before the harvest day. Rabbits were euthanized by overdose of 5ml pentobarbital (50mg/ml) intravenously prior to vein graft tissue harvest [52]. Figure 3-1. Schematic of anastomotic cuff technique used in vein grafting procedure. The external jugular vein is excised and polymer cuffs are placed at the vein ends (steps 1-3)and then the vein is inserted (steps 4-6) and fixed to the carotid artery (steps 7-8) Figure 3-2. Bilateral carotid vein grafting with distal ligation model.

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18 Hemodynamic Measurements Hemodynamic measurements (flow rate and pressure) and video analysis were acquired at implantation and harvest of the vein grafts. In vivo hemodynamic measurements, both flow rate and pressure, and video were recorded in real time. All measurements were obtained with the aid of an IBM compatible PC computer through a MacLab/8e AD Instruments analog to digital board at 1000 Hz. Video At implantation and harvest, real-time video was obtained using a Sony Analog camera and a Diagnostic Instrument Wild 5A Dissecting Microscope connected to a computer. The video camera setup uses the microscope to obtain a clearer more focused image. Once set up, a small pre-measured marker is placed next to the blood vessel for later analysis (Figure 3-3). Video is saved as .bmp files and images are captured at 30 frames per second. The video images were analyzed using Zeiss Imaging software (AxioVision v. 3.1) to determine the external vein graft radius and dynamic wall motion. Flow rate The flow rate in the implanted vein graft is measured using a Transonic Systems small animal ultrasonic blood flow meter, modelT106. The Transonic Systems 2SB flow probe, with 100 cm cable length was used for a digital measurement of flow rate in 2mm vessel diameters. The flow meter ranges from 0-1 V, which corresponds to 0.0-100ml/min and a relative accuracy of 2 ml/min [53]. The flow rate waveform is collected through the A/D board and computer using the AD Instruments Chart recorder program (Figure 3-4). The Chart recorder program interpreted the flow rate waveform

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19 data and displayed the information in graphical form in (ml/min). The flow rate measurements are taken in vivo after implantation and at harvest. Figure 3-3. Pre-measured marker next to left carotid distal artery unligated. A BC Figure 3-4. Measurement of in vivo flow rate and pressure with waveform output from Chart Recorder Program. (A) Flow meter and Pressure Transducer in vein graft model. (B) Transonic Flow meter. (C) Chart Recorder Program with pressure and flow rate. Pressure The intraluminal pressure is measured with an animal Millar micro-tip catheter transducer (model SPR-671), which has been placed into the blood vessel via a 22-guage

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20 catheter. The output signal ranges from 0-1V, which corresponds to 0-100mmHg. The sensor is side-mounted at the tip and the catheter size is 1.1 French [54]. The transducer unit is connected to the computer and data is recorded with the Chart recorder program (Figure 3-4). The pressure waveform data is represented in (mmHg) in graphical form in the Chart program. The pressure is taken at various locations along the vein graft and artery. However because of the invasive method of retrieving the measurement, the intraluminal pressure is only recorded when the vein graft is harvested. Morphometric Analysis The vein graft segments were fixed in 10% buffered formalin and embedded in paraffin. Histologic cross sections were stained with Massons trichrome and van Giesons elastic stains (Figure 3-5) [52]. Massons trichrome stain is used to differentiate between collagen and smooth muscle cells [55]. The van Giesons stain is used to show the elastic lamina[56]. Both these stains show the different layers in the vein graft wall. Using Zeiss imaging software, digitized images were collected and analyzed to measure the intimal, medial, and adventitial areas (Figure 3-6). In vivo lumen diameter and graft (external elastic lamina (EEL)) diameter were calculated using the following morphology data: intimal, medial and lumen areas. The lumen diameter was later calculated based on the morphology data and the external diameter measured from the video analysis. Ladvvideo2AA4D diameter Lumen (3.1) where D video is the video diameter and A adv and A L are the crosssectional areas within the adventitia and lumen respectively. The graft diameter is determined to be

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21 EELadv2AA4D diameter Graft video (3.2) where A EEL is the cross-sectional area within the EEL. Figure 3-5. Histologic cross-section illustrating the longitudinal section of anastomotic segment using Massons staining. Figure 3-6. Cross-sectional area of 28-day ligated/low flow rabbit vein graft at harvest at (A) 20x and (B) 40x illustrating different wall layers. To estimate lumen diameter at the time of graft implantation, normal external jugular vein segments were harvested, embedded, and sectioned for analysis. Image analysis of

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22 these sections were used to determine a standard vein wall thickness with the lumen diameter at graft implantation being determined as h2DDiameterLumen video (3.3) where h is the standard graft wall thickness. Modeling of Wall Stresses Hemodynamic forces have been shown to influence vein graft arterialization during the remodeling process. How exactly these forces influence the remodeling process is unknown. To look at shear stress more closely, a mathematical model isolating this force is developed. From the experimental hemodynamic data and the calculated lumen diameter, the velocity and shear stress profiles was calculated based on our mathematical model, a modified Womersley approach. To study the intramural wall stresses, i.e. the circumeferential, axial, and radial stresses, the Lame and isotropic models were used. Modified Womersley Approach Assuming that the pressure gradient is a function of time, the pressure waveform can be approximated as a series of sinusoidal functions in the form of inwtnoeLPLPLP (3.4) where LPo is the mean pressure gradient, and LPn is the oscillating pressure gradient, is the frequency, and t is the time. Incorporating the equation for the lumen diameter (equation 3.2) in the steady state and oscillatory equations from the Womersley analysis, we can derive the velocity and shear stress profiles. The derivation can be seen in Appendix A. Velocity profile v(r,t):

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23 tinNnnnnneiJRriJLiPRRrLPRtrv1230230222202114, (3.5) where v(r,t) is the velocity as a function of vessel radius and time, r is the radial coordinate along the vessel diameter, R is the vessel radius at the wall, is the fluid viscosity, n are the Womersley numbers for harmonics n=1 through N, J 0 is a zero order Bessel function, J 1 is a first order Bessel function, is the frequency, and t is the time. The shear stress profile, (r,t), is derived from the velocity profile using equation (3.5) and knowing that rv : tinNnnnnneiJiRriJLPRLrPtr12302323102, (3.6) where (r,t) is the shear stress as a function of radius and time [35]. The volumetric flow rate can now be obtained knowing the pressure gradient equation (equation 3.4) and from the direct integration of the velocity profile equation (3.5) across the vessel lumen, dAvdAQ : tinnonnnoeiJiiJLiPRLPRtQ2323231244218 (3.7) where Q(t) is the blood flow with respect to time, n are the Womersley numbers for harmonics n=1 through N, J 0 is a zero order Bessel function, J 1 is a first order Bessel

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24 function, is frequency, and t is time. The dimensionless Womersley number is calculated at each harmonic as [35]: nR (3.8) where is the density of the fluid assumed to be 1.05 g/cm 3 [29]. The experimental flow rate waveform will be simplified into its sinusoidal wave components using the Discrete Fourier Transform (DFT) in the simple form of NnnnnCtBAQtQ102cos)( (3.9) where Q(t) is the time-dependent blood flow rate, Q o is mean flow rate, and A n B n C n are the magnitude, frequency, and phase shift, respectively, of the harmonic waveforms n= 1 through N and are determined via the solution to the DFT [28, 35]. To handle the complex nature of the hemodynamic data, Fourier series approximation is used. The Fourier series approximation deals with oscillating systems and utilizes trigonometric functions for modeling these systems. The Fourier analysis uses both the time and frequency domains, and is used to simplify the complex waveform into sinusoidal functions. Here, the Discrete Fourier Transform (DFT) represents the waveform as a finite set of discrete values [28, 57]. With enough individual harmonic waveform components the original waveform can sufficiently be approximated as seen in Figure 3-7. Here ten harmonics are used to adequately approximate the original waveform since higher frequency harmonics have little effect [35, 58]. The DFT data is converted into sinusoidal functions and is represented in the form of magnitude, frequency, and phase shift.

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25 Figure 3-7. Original flow waveform (solid line) compared to approximated DFT flow waveform (dashed line) and 0-9 DFT harmonics (dotted lines). Converting the flow rate equation (3.9) into a polar coordinate system yields NnnnotMLPRLPRtQ110102442cos8 (3.10) where is the phase shift, 10M and 10 are constants that are dependent on and can be determined using tables found in Appendix B [29]. Comparing the two forms of the flow rate equations (3.9) and (3.10) lead to expressions for the mean flow rate, magnitude, frequency and phase shift LPMRAnnn2104 (3.11)

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26 2nnB (3.12) 102 nC (3.13) Substituting equations (3.11-3.13) into the polar coordinate form of equation (3.6) gives the final expression for the velocity profile: NnnnnCtBMRMARrRQtrv1010102022202cos12, (3.14) where 2120000000cos21nnnnnnMRrMRrMRrMM (3.15) nnnnnnnnRrMRrMRrMRrM0000000010cos1sintan (3.16) where M 0 and 0 are tabulated functions [59] dependent on the Womersley number and the dimensionless radius, Rr which can be found in Appendix B. Similarly, the shear stress profile, equation (3.6), can also be converted into polar coordinate form and with substitution of equations (3.11-3.13) can be finalized as tBCRrMMRRrMARrQtrnnnnnNnnnnn24cos4,100110103140 (3.17)

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27 where M 1 and 1 are tabulated functions dependent on the Womersley number and the dimensionless radius, which can be found in Appendix B [59]. Therefore, equations (3.14) and (3.17) quantify the velocity and shear stress profiles along the diameter of the vein graft using morphology data and experimental flow rate data. The advantages to using this particular model is that it accounts for pulsatile blood flow, and does not depend on a knowledge of the pressure gradient, which is a difficult and inaccurate measurement in small vessels. To compensate for this, all calculations are based on the flow rate data rather than pressure gradient. Numerical Computations All major computations were performed using MATLAB version 6.5 Release 13. A computer program was written to calculate the velocity and shear stress profiles using the experimental flow rate data and the calculated lumen diameter. A flowchart diagram of the program layout is seen in Appendix C while the MATLAB code is shown in Appendix D. The experimental flow rate data used in the computational analysis will consist of one waveform cycle for simplicity. Using more than one cycle will only produce repetitive data during each cycle and would be computationally inefficient. Once the experimental flow rate data and calculated lumen diameter are entered, the program will perform the DFT and separate the experimental flow rate data into ten separate harmonic components in the form of magnitude, frequency, and phase shift. The program will then determine the velocity and shear stress profile at 100 time points and 100 spatial points along the diameter of the vein graft. The program allows the user to select several choices. A more detailed description of each part of the program is described in Appendix C.

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28 Using the FFT function in MATLAB, the DFT was determined with the magnitude, frequency, and phase shift obtained accordingly. Once the DFT waveform was generated, the velocity and wall shear stress profiles were determined from the experimental flow rate waveform using our modified Womersley approach as seen in Figure 3-8. Figure 3-8B illustrates the centerline velocity at several harmonics and the final approximated centerline velocity summing all ten harmonics. Figure 3-8C similarly illustrates the wall shear stress starting from the first harmonic to the summation of all ten harmonics. Once the DFT was determined, the data was used to first determine the centerline velocity in the vein graft over one cardiac cycle. At the centerline of the vein graft, it can be assumed that at this point the velocity will reach the maximum value. Therefore, to check the validity of our modified approach, Poiseuilles law was used to estimate the maximum velocity at the vein grafts center, LPRv42max [33]. This value was compared to our centerline velocity calculated from our modified Womersley model, which were found to be very close proving that the calculated data from the numerical analysis is valid. From our modified Womersley approach, the velocity was then calculated at each grid point (100 points) along the diameter of the vein graft over the cardiac cycle (Figure 3-8B). The velocity profile can be determined at any specified time point encompassing the diameter of the vein graft (Figure 3-9). A similar approach is used for the calculation of the shear stress profile, at any time during the cardiac cycle (Figure 3-10).

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29 A B C Figure 3-8. Centerline velocity and wall shear stress derived from the DFT approximation of the original flow waveform. (A) Original flow waveform (solid line) compared with the DFT approximation at first harmonic, 0-1 harmonics, 0-3 harmonics summed, and 0-10 harmonics summed. (B) Centerline velocity approximated from (A). (C) Wall shear stress approximated from (A).

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30 Figure 3-9. Velocity profile at a specified time points encompassing the diameter of the vein graft. Intramural Wall Stresses Lames equation for static circumferential wall stress A variation of Laplaces law, Lames equation, is used to calculate the static circumferential wall stress hPRstatic (3.18) where -static is the static circumferential wall stress, P is the pressure, R is the radius, h is the wall thickness. This static circumferential wall stress is used in comparison to the dynamic circumferential wall stress.

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31 Figure 3-10. Wall shear stress at a specified point in time over one cardiac cycle. Isotropic model for wall tension Assuming a linear stress-strain relationship over the range of blood vessel deformations [40], the incremental elastic modulus is used to calculate the axial and circumferential stresses and is calculated as 2122221313inc12ERRRRRRPPooioo (3.19) where E inc is the incremental elastic modulus, P is the pressure, is Poissons ratio assumed to be 0.5, R is the radius, i, o, 1,2, and 3 represent the inside, outside, minimum, mean, and maximum values [33].

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32 The isotropic model by Kuchar and Ostrach [39]is used to calculate the radial, axial, and circumferential components of the wall stress based on the following assumptions: Graft is elastic, isotropic, axisymmetric and semi-infinite in length Radial displacement of the wall is small compared to the radius Graft undergoes minimal longitudinal displacement [39, 41] Using the incremental modulus of elasticity equation (3.19) the dynamic wall stress components can be derived from a general isotropic model by Love [29] based on the above assumptions as Radial Stress, 1)(2iRrrrrrvP S (3.20) Axial Stress, 121iincxxRE T (3.21) Circumferential Stress, 2332121121iinciincRhERE T (3.22) where v(r) is the velocity of the fluid in the radial direction, is the radial displacement, h is the wall thickness, E inc is the incremental elastic modulus, and R is the lumen radius [38, 41, 60]. The incremental elastic modulus was determined by using the dynamic in vivo pressure and the video morphometry data [40, 60]. Required experimental input for the model equation (equation 3.19) are obtained from intraluminal pressure waveforms and video inputs used to calculate radial displacement.

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CHAPTER 4 RESULTS Hemodynamics Flow, Velocity, and Shear Stress The bilateral vein graft with distal ligation model created two flow environments as well as isolated the hemodynamic forces involved in vein graft remodeling. The first flow environment created a reduced flow/reduced shear stress through the ligated vein graft and the second, an elevated flow/elevated shear stress through the contralateral vein graft. These flow environments can be seen in Table 4-1. There was an approximate 6-fold difference in mean flow rate maintained throughout the 28-day perfusion period (P<0.001, two-way ANOVA, ligated vs. contralateral grafts). Data are presented as mean standard error of the mean (SEM). The modified Womersley mathematical model was used to determine the velocity profile along the diameter of the vein graft over one cardiac cycle. Based on the flow rate waveform and calculated in vivo lumen diameter, the velocity profile determined from our mathematical model is the mean steady component plus its oscillatory component. The velocity profile from equation 3-14 can be illustrated in three dimensions (time, location, and velocity) and representative profiles for vein grafts at 1, 3, 7, 14, and 28 days can be seen in Figures 4-1 4-5. Five grafts were studied for each time point and the group averaged data for the maximum, minimum, and centerline velocities for bilateral vein grafts at 1, 3, 7, 14, and 28 days are shown (Table 4-2 4-4). 33

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34 Table 4-1. Flow environment in the ligated/ low flow vein graft and the contralateral/ high flow vein graft. 1 DAYS Ligated Vein Graft Flow (ml/sec) Contralateral Vein Graft Flow (ml/sec) 1 0.12 0.02 0.78 0.11 3 0.09 0.02 0.71 0.21 7 0.13 0.02 0.78 0.09 14 0.07 0.02 0.82 0.08 28 0.14 0.06 0.73 0.13 Table 4-2. Maximum centerline velocity at each time point in the ligated/ low flow vein graft and the contralateral/ high flow vein graft. DAYS Ligated Vein Graft Velocity (cm/sec) Contralateral Vein Graft Velocity (cm/sec) 1 9.48 1.44 36.21 4.92 3 5.40 1.04 33.96 9.65 7 7.80 1.88 46.97 11.15 14 4.90 1.57 45.46 7.14 28 10.37 2.68 18.86 4.70 Table 4-3. Minimum centerline velocity at each time point in the ligated/ low flow vein graft and the contralateral/ high flow vein graft. DAYS Ligated Vein Graft Velocity (cm/sec) Contralateral Vein Graft Velocity (cm/sec) 1 1.39 0.67 17.73 3.60 3 1.35 0.35 15.67 8.26 7 1.97 0.86 15.76 3.92 14 1.26 0.40 16.09 3.73 28 3.56 1.31 7.69 2.29 Table 4-4. Mean centerline velocity at each time point in the ligated/ low flow vein graft and the contralateral/ high flow vein graft. DAYS Ligated Vein Graft Velocity (cm/sec) Contralateral Vein Graft Velocity (cm/sec) 1 3.61 0.81 26.24 3.99 3 2.80 0.47 25.58 9.15 7 4.74 1.27 33.61 7.20 14 3.28 1.10 31.70 6.41 28 6.08 1.72 12.00 3.07 1 P< 0.001, Two-Way ANOVA, ligated vs. contralateral grafts.

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35 A A B Figure 4-1. Three-dimensional velocity profile within bilateral vein grafts one day after implantation. (A) Ligated vein graft. (B) Contralateral vein graft. A B Figure 4-2. Three-dimensional velocity profile within bilateral vein grafts three days after implantation. (A) Ligated vein graft. (B) Contralateral vein graft. A B Figure 4-3. Three-dimensional velocity profile within bilateral vein grafts seven days after implantation. (A) Ligated vein graft. (B) Contralateral vein graft.

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36 A B Figure 4-4. Three-dimensional velocity profile within bilateral vein grafts fourteen days after implantation. (A) Ligated vein graft. (B) Contralateral vein graft. Figure 4-5. Three-Dimensional Velocity profile within bilateral vein grafts 28 days after implantation. (A) Ligated Vein Graft. (B) Contralateral vein graft. Representative dynamic wall shear stress profiles, using equation 3.17, for individual rabbits at each time point are provided in Figures 4-6 4-10 and reveal the hemodynamic changes have taken place over the 28-day time course. Group averaged data for the maximum, minimum, and Poiseuille shear stresses can be seen in Tables 4-5 4-7. Poiseuille shear stress [22] was calculated for comparison as 332LumenDQ The Poiseuille shear stress is the mean steady shear stress and is provided for comparison. At day 7, the dynamic shear stresses peaks and drop off by day 28 in the contralateral/ high flow vein graft. Tables 4-5 4-7 also show a considerable difference

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37 in wall shear stress between the ligated/ low flow and contralateral/ high flow vein grafts (P <0.001, two-way ANOVA, ligated vs. contralateral grafts). It can be seen that the maximum dynamic wall shear stress within the contralateral/ high flow vein grafts was elevated at day 7 and day 14. By day 28, there was a considerable reduction (P=0.003, 7 vs. 28 days; P=0.03, 14 vs. 28 days). Corresponding to this, the minimum dynamic wall shear stress exhibited the same increase at day 7 and a reduction in day 28 (P=0.01, 7 vs. 28 days). Figure 4-6. Dynamic wall shear stress vs. time within bilateral vein grafts one day after implantation.

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38 Figure 4-7. Dynamic wall shear stress vs. time within bilateral vein grafts three days after implantation. Figure 4-8. Dynamic wall shear stress vs. time within bilateral vein grafts seven days after implantation.

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39 Figure 4-9. Dynamic wall shear stress vs. time within bilateral vein grafts 14-days after implantation. Figure 4-10. Dynamic wall shear stress vs. time within bilateral vein grafts 28 days after implantation.

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40 Table 4-5. Dynamic wall shear stress valuesMaximum shear stress 2 DAYS Ligated Vein Graft Shear Stress (dynes/cm 2 ) Contralateral Vein Graft Shear Stress (dynes/cm 2 ) 1 6.69 0.44 23.39 1.47 3 3.73 0.28 23.95 2.70 7 6.05 0.77 36.97 4.62 14 4.14 0.55 32.80 2.84 28 4.43 1.41 11.30 2.43 Table 4-6. Dynamic wall shear stress valuesMinimum shear stress 3 DAYS Ligated Vein Graft Shear Stress (dynes/cm 2 ) Contralateral Vein Graft Shear Stress (dynes/cm 2 ) 1 0.48 0.16 10.71 0.82 3 0.79 0.06 11.54 2.14 7 1.59 0.22 15.65 1.55 14 1.40 0.21 13.68 1.61 28 1.19 0.45 4.32 0.92 Table 4-7. Dynamic wall shear stress valuesPoiseuille shear stress DAYS Ligated Vein Graft Shear Stress (dynes/cm 2 ) Contralateral Vein Graft Shear Stress (dynes/cm 2 ) 1 2.06 0.52 15.23 2.64 3 1.65 0.29 15.92 6.29 7 2.99 0.96 23.29 6.89 14 2.28 0.84 20.28 5.38 28 2.19 0.71 6.92 1.47 Wall Stress and Elastic Modulus The static circumferential wall stress was determined using Lames Law, hPR equation 3.18. The static wall stress used a mean pressure determined from the experimental in vivo pressure. Table 4-8 provided the group averaged static circumferential wall stress (equation 3.18) in vein graft segments ipsilateral, and contralateral to the distal ligation. Static circumferential wall stress in the contralateral vein appeared to decrease over the entire time course within both ipsilateral and 2 P=0.003, 7 vs. 28 days; P=0.03, 14 vs. 28 days 3 P=0.01, 7 vs. 28 days

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41 contralateral vein grafts. This corresponded to the increase in wall thickness and showed that the circumferential wall stress after 28 days reduced to 30% at Day 1 (P<0.001, Day 7,14, and 28 vs. Day 1). For comparison, the estimated static wall stress for a vein in the venous circulation was 0.053 x10 -6 dynes/cm 2 There was no consistent difference in the static circumferential stress found when comparing the ligated and contralateral vein grafts. The dynamic wall shear stress components for the ligated and contralateral vein grafts were calculated for the 28-day grafts using equations 3.20-3.22 and can be seen in Table 4-9. There were no significant differences in the dynamic wall stresses between the ligated and contralateral vein grafts. Observed was an order of magnitude difference between the static and dynamic circumferential wall stresses at day 28. The elastic moduli, calculated from equation 3.19, were similar in both vein grafts showing that the elastic properties for both cases were similar. This showed that the hemodynamic simulations and the secondary impact on intimal thickening had little impact on the mechanical properties of the vein graft wall. Table 4-8. Static circumferential wall shear stress comparing ligated vein graft and contralateral vein graft. 4 DAYS Ligated Vein Graft Circumferential Wall Stress (x10 -6 dynes/cm 2 ) Contralateral Vein Graft Circumferential Wall Stress (x10 -6 dynes/cm 2 ) 1 2.62 0.29 2.19 0.24 3 2.91 0.90 2.02 0.34 7 1.12 0.22 1.28 0.25 14 0.22 0.04 0.63 0.09 28 0.35 0.053 0.86 0.16 4 P <0.001, Day 1 vs. Day 7; P <0.001, Day 1 vs. Day 14; P <0.001, Day 1 vs. Day 28

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42 Table 4-9. Dynamic wall shear stress and elastic modulus at day 28. Wall Stress Ligated Vein Graft Contralateral Vein Graft Radial, S rr (x 10 4 dynes/cm 2 ) 5.68 0.50 4.8 0.51 Axial, T xx (x 10 4 dynes/cm 2 ) 4.17 0.37 5.92 1.03 Circumferential, T (x 10 4 dynes/cm 2 ) 8.37 0.74 11.9 2.05 Elastic Modulus, E inc (x 10 6 dynes/cm) 12.7 3.48 12.2 2.27 Vein Graft Remodeling Figure 4-11 showed the intimal area, medial area, and the graft diameter based on the morphologic data. Figure 4-11A showed significant variation overall starting at day 7. The most significant change in the intimal area occurred in the ligated vein graft segment. The intimal area dramatically increased over the time course for the ligated vein segment while the intimal area in the contralateral vein segment increased slightly. Intimal hyperplasia can be inferred from Figure 4-11A at day 7 showing an enhanced intimal thickening in the low flow/ligated vein graft (P<0.001, two-way ANOVA). A 9-fold difference in the intimal area at day 14 was found when comparing the hyperplasia rates in the ligated/low flow and contralateral/high flow vein grafts (P <0.001, 1.31 0.25m vs. 0.15 0.02 m for ligated vs. contralateral grafts). At day 28, there was a 6-fold difference in the intimal area (P<0.001, 1.21 0.17mm 2 vs. 0.22 0.17 mm 2 for ligated vs. contralateral grafts). Figure 4-11B showed the medial area increasing in both the ligated and contralateral vein grafts starting at day 7. This accompanied the intimal hyperplasia seen in Figure 4-11A. This increase in medial area was time-dependent with the maximum areas occurring at days 14 and 28 (P< 0.001, Day 14 and day 28 vs. Day

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43 1). The graft diameter in Figure 4-11C remained stable for both the ligated and contralateral vein graft segments until day 28 when there was a slight increase in diameter when the grafts were exposed to high flow conditions (P=0.002, 3.25 0.15 mm vs. 3.72 0.24 mm for ligated vs. contralateral grafts). As early as day 7, it can be inferred that some form of vein graft remodeling has taken place. In these findings, it can be seen that there was a correlation between intimal thickening and the hemodynamic forces by day 7. It can be seen that under low shear conditions the intimal area was markedly increased by day 7. It can also be seen that under low flow conditions the intimal area also increased, signifying intimal hyperplasia has occurred.

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44 Figure 4-11. Vein graft remodeling. (A) Intimal area comparing ligated/low flow vein graft and contralateral/high flow vein graft. 5 (B) Medial area comparing ligated vein graft and contralateral vein graft. 6 (C) EEL graft diameter throughout time course. 7 5 P< 0.001, ligated vs. contralateral Day 14; @ P < 0.001, ligated vs. contralateral Day 28 6 # P< 0.001, Day 1 vs. Day 14; % P <0.001, Day 1 vs. Day 28 7 $ P= 0.002, ligated vs. contralateral Day 28 vein grafts

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CHAPTER 5 DISCUSSION Mathematical Model: Modified Womersley Approach Our mathematical model adapted from Womersley uses the DFT of the experimental flow rate waveform as the basis for the velocity and shear stress calculations. Many investigators have validated the Womersley analysis using MRI velocity measurements. In their studies, the experimental root-mean-square differences between the analytical and measured velocities approached 5% and correlation coefficients are greater than 0.99 [51, 61]. This approach is limited to model systems with uniform, cylindrical conduits, distant to branch regions or vessel curvature. Our first-order modified mathematical model is best suited for this study since it accounts for pulsatile blood flow and uses experimental flow rate data, which is more accurately measured than the pressure gradient. Measurement of the intraluminal pressure within the small vessels of the rabbit carotid system is difficult and can cause significant artifact. To verify the accuracy of our numerical computation, the experimental flow rate is compared to the approximated flow rate data calculated from the DFT (Figure 3-7). A known mean velocity and shear stress was compared to the computational results obtained from our mathematical model as well. All of these verifications show that our mathematical model is fairly accurate and valid for approximating the velocity and wall shear stress. 45

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46 Experimental Model Our experimental model [52] used isolates the hemodynamic forces, i.e. shear stress and wall tension, involved in vein graft remodeling. This follows the work initially performed by Zarins [62], which defined the impact of shear stress on lesion development by studying the localization of arterial plaques to low shear regions of the carotid bifurcation. Since then, it has been inferred that the structure of the blood vessel wall is closely linked to its surrounding hemodynamic environment [11, 12, 62]. Many experimental studies demonstrate a balance between the local shear stress and tensile forces predicting vessel morphometry during physiologic and pathologic remodeling [63-65]. From these findings, more recent studies reaffirm the concept of lumen preservation and positive remodeling by observing an inverse relationship between wall thickness and shear stress in the human coronary circulation [15, 66, 67]. Our experimental model is a modified version of the standard techniques used for studying vein graft adaptation [6, 23, 52]. There is variability in the extent of intimal thickening under hand-sewn anastomotic technique using the rabbit carotid interposition vein graft. Our experimental model uses a cuff anastomotic technique, which was adapted from a rodent lung transplantation model [68]. Adapting this technique to our experimental rabbit vein graft model offers a more reproducible method for vein graft placement [52]. Our experimental model provides a more robust biologic response by using extensive distal branch ligation, which was suggested by Meyerson [69]. High flow/ high shear conditions were created in the contralateral vein graft while the low flow/low shear conditions were created in the ligated vein graft. Our model created a 6-fold difference in mean flow rate and a 5-fold difference in intimal cross-sectional area.

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47 Vein Graft Remodeling Studies in understanding vein graft remodeling and secondary pathologic occlusion began with Dobrin [21], who realized that intimal thickening was closely correlated with low shear while medial thickening was correlated with circumferential strain. In more detail, Zwolak [7] and Schwartz [6] applied these observations to the rabbit carotid system and found that an increased wall tension correlated with an increase in myointmal thickening. Using our mathematical model, we were able to calculate velocity and wall shear stress profiles. By day 28, there is an increase in intimal thickening under low flow conditions. This was consistent with the findings of Galt[23], who focused on the effect of wall shear stress in the rabbit vein graft model. Different from our model, he left the internal carotid artery widely patent, ligating only the external carotid branch. The Galt study found a 50% reduction in mean wall shear stress, which led to a 70% increase in intimal area. Our model shows a 30% wall stress reduction from day 1 to the end of the time course, day 28. In addition, our findings show that the intimal thickness is inversely proportional to flow. The wall thickness, comprised of the media and the intima, is also inversely proportional to the flow [7]. These findings suggest a correlation between intimal thickening and wall shear stress. Looking at the mean Poiseuille wall shear stress, we see under low shear conditions, intimal thickening is markedly increased by day 28. At high static wall circumferential stress the intimal area is very low while the opposite could be said for low static wall shear stress. This data agrees with Zwolaks hypothesis that early in the vein graft remodeling process, there is an elevated wall stress with less thickness. As the remodeling process continues, by day 28, the wall stress of

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48 the vein graft is normalized toward the normal arterial values [7]. The derived isotropic model [29]isolates wall tension showing that deformation does occur. Circumferential stress is elevated and is considered to be the largest stress showing an increase in diameter [22]. This increase in diameter shows the vein graft has adapted to the arterial system. We also find that determining the circumferential wall stress within vein grafts using Lames equation is limited. Comparing the results from this static circumferential wall stress to the dynamic circumferential wall stress determined from the dynamic wall motion and incremental modulus of elasticity implies the stress is approximately 10-fold less than the Lames equation estimation. This result is consistent with other studies that found Lames equation inaccurate when the wall thickness to radius ratio is greater than 0.10. This ratio during vein graft remodeling ranged from 0.11 to 0.15, which is the reason for the inaccuracy [7]. Future Research Future research can be taken into different aspects. Computationally, a more realistic development of the model can be performed in other blood vessels such as studying blood flow in a bifurcation or a curved blood vessel under similar hemodynamic conditions using a more sophisticated computational technique, such as finite element analysis. Altering the hemodynamic conditions such as assuming the blood vessel to be rigid to demonstrate blood flow under stenosis could also be studied. A more detailed and sophisticated computational technique based on this in vivo experimental data could be developed to eliminate any future and unnecessary in vivo experiments. Mathematically, improvements in our modified model could be to apply the model to other blood vessels in the arterial system. For example, we can make more realistic

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49 assumptions about the blood vessel by taking into account the viscoelasticity, wave propagation, or reflected waves. These were neglected in our assumptions. By altering the model to include non-linear terms in the Navier-Stokes equations as well as define a different model for the intramural stresses, we can better simulate non-linear in vivo pulsatile blood vessel systems. Another improvement would to assume a thick vessel wall. Studies have applied this assumption and found that there was a 12% higher value in phase velocity than the Womersley analysis [29]. Many studies have taken reflected waves and wave propagation into account, these studies differ in the assumptions, i.e. constrained or unconstrained tubes. The differences disappear when the vessel becomes large such as in arteries. For our study with small vessels, the wave propagation and viscoelasticity may have some effects depending on the geometry and length of the vessel [29, 70] and may have to be accounted for. The use of our model can be used in comparison with other aspects that influence vein graft remodeling and adaptation such as biochemical factors. Since hemodynamic forces appear to have an influence on the biochemical factors, our mathematical model could be used in studying the effects of dynamic shear stress on biochemical factors such as matrix metalloproteinases (MMP). Other future research could look at how the wall shear stress calculated in this study affects the individual layers of the blood vessel wall. Our mathematical model only looks at the shear stress that affects the innermost and outermost layers. Other research could entail studying the changes caused by intramural pressure rather than the present studys look at changes caused by differential flow rate environments.

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50 Conclusion The understanding of how hemodynamic forces impact vein graft remodeling has been limited to the use of the mean or steady state models for estimating shear stress and wall tension. Our modified Womersley analytical approach provides a simple technique for estimation of dynamic shear stress and wall tension in an in vivo environment. Our model provides a way to study the biomechanical changes that occur during the early stages of vein graft remodeling. This technique provides a basis for understanding how the physical hemodynamic forces along with the biological changes control the balance between physiological remodeling and pathologic stenosis, eventually leading to strategies for the enhancement of vein graft patency.

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APPENDIX A DERIVATION OF WOMERSLEY EQUATIONS The following derives the equations were used to determine the velocity and shear stress profiles. Starting with the Navier-Stokes equations in cylindrical coordinate form we can derive the Womersley equations. The Womersley equations were derived as a fully developed pipe flow with an oscillating pressure gradient. From the Navier-Stokes momentum equation, we can derive the oscillatory portion of the Womersley solution (Figure A-1) [33]. Figure A-1. Schematic of flow through a blood vessel. z-momentum Navier-Stokes equation: gzpzvvrrvrrrzvvvrvrvvtvzzzzzzzrz2222211 (A.1) where is density, v is the velocity in the r, and z directions, r is the radius, is viscosity, zp is the pressure gradient, and g is gravity [27]. 51

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52 The following assumptions can be made: 1. Laminar Flow 2. Incompressible Flow 3. Axisymmetric Flow 4. Parallel flow 5. Ignoring body forces. The z-momentum Navier-Stokes equation can be reduced to: rvrrrzptvzz1 (A.2) Since zp is an oscillating pressure gradient, we can assume that it takes the form Ae it To simplify things, we assume that the velocity profile is v(r,t)=Re{f ( r ) e it }. and the boundary conditions, v z (r =R, t)=0 and v(r=0, t)= finite number. By adding the above assumptions into equation A.2, and introducing the parameter, iry we can transform the equation to 0'"22fyyffy (A.3) Equation A.3 is the Bessel equation and has the general solution of yBYyAJyfo0 (A.4) where J o and Y o are the zero order Bessel functions of the first and second kind [71]. The term Y o (y) as y 0 because of the assumption that the velocity reaches a finite value at r=0. Because of this, B is zero and substituting the no-slip conditions back into equation A.4 the velocity profile for an oscillating pressure gradient becomes tiooeRriJRriJiAtrv23231, (A.5)

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53 where is the Womersley number, R and [33, 72]. Since the pressure gradient is inwtnoeLPLPLP equation (3.4), we can replace A in equation A.5 and reach the final equation for the velocity profile as equation (3.5) tinNnnnnneiJRriJLiPRRrLPRtrv1230230222202114, (3.5)

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APPENDIX B TABLES USED TO DETERMINE CONSTANTS BASED ON WOMERSLEY NUMBER The following tables were used to determine the constants based on the Womersley number,, which was calculated using equation (3.6). Table B-1 [59] is referenced to ascertain the values of M 0 and 0 which are used in order to determine and 10 0M (equations 3.13 and 3.14). Table B-2 [59] is referenced to ascertain the M 1 and 1 which are used in order to determine the shear stress profile (equation 3.15). Table B-3 [29] is referenced to ascertain and 10M 10 which are both used in determining both the velocity and shear stress profiles (equations 3.12 and 3.15) 54

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55 Table B-1. Table of solutions to Bessel functions used to calculate M 0 and 0

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56 Table B-2. Table of solutions to Bessel Functions to determine M 1 and 1.

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57 Table B-3. Table to determine 10M and 10 based on the Womersley number.

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APPENDIX C FLOW CHART OF MATLAB PROGRAM A MATLAB code was developed to determine the velocity and shear stress profiles within the vein graft. Figure C.1 shows a flowchart describing the code. A pop-up menu appears providing the user with several selections when analyzing the experimental waveform data collected in vivo. The menu has the following selections: 1. Load waveform 2. Perform Velocity and Shear Stress calculations 3. Analyze Pressure calculation 4. Display flow rate, velocity, and shear stress vs. time 5. Create and display a 2-D velocity profile movie 6. Create and display a 3-D velocity profile movie 7. Save all calculated data in program 8. Exit program Once the user selects a menu choice, the program performs the operation and then once complete, the program returns to the menu until the user chooses to exit the program completely. Choice 1 allows the user to input the flow waveform to be analyzed as well as the lumen diameter determined from the morphology data. Choice 2 processes all the input data given from choice 1 and determines the centerline velocity and shear stress. It also provides a display of the maximum, minimum, and mean values for both the velocity and shear stress calculations. Choice 3 allows the user to input the in vivo pressure waveform and tabulates the maximum, minimum, and mean pressure. Choice 4 provides a graphical display of the in vivo flow rate waveform, the derived centerline velocity, and the derived wall shear stress versus time. Choice 5 and 6 create .avi movies of the calculated velocity profile in 2-D or 3-D. Choice 7 allows the user to save all calculated 58

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59 data into a text file. The text file saves the flow waveform, the velocity profile, the shear stress profile, and the minimum/maximum/mean values for the profiles. Choice 8 exits the program completely.

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60 Menu Choice (5) 2-D Velocity Movie (4) Graphs Input Menu Choice (1) Load Wavefor m (3) Pressure Calculation (2) Centerline Velocity/ Shear Stress Calculation (6) 3-D Velocity Movie (7) Save Data (8) Exit Program START END Figure C-1. Flowchart of MATLAB code.

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APPENDIX D MATLAB CODE FOR DETERMINING VELOCITY AND SHEAR STRESS PROFILES Code for Discrete Fourier Transform function d=dft(data) %================================================ %This function determines the Discrete Fourier Transform %assuming frequency rate of 1000 samples/sec and displays %data in the form of Frequency, Magnitude, and Phase Shift %Input data: Original waveform data %Output data: Frequency, Magnitude, Phase Shift of DFT %================================================= chessy=data/60; %converting data from ml/min to ml/s y=fft(chessy) ; %DFT of data N=length(y); %calculates number of data points y=y/N; %shift zero harmonic to mean flow m=abs(y); %magnitude of y fs=1000; freq=(0:(N-1))*1; %frequency for one cycle p=-(angle(y)); %phase shift of fft fourier=[freq' m p] ; %displays data f=freq(1:11); %Retrieves f,mag, phase of first 11 points phase=p(1:11); %0-10 harmonics mag=m(1:11); fourier=[f' mag phase]; d=fourier; 61

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62 Code for Determining Velocity Profile function velocity=createmov(data,D) %======================================================= %Program to analyze velocity profile from flow data %==Inputs: DFT Data and Diameter (in cm) of blood vessel %==Outputs: Velocity profile data over one cardiac cycle %== Velocity (cm/s) %======================================================= %======================================================= f=data; a=xlsread('Table C_1'); %Table C.1 for Womersley values, M10,e10 b=xlsread('Table 27b'); %======================================================= %==============================================constants p=1.05; %-------------------------------g/mL density R=D/2; %---------------------------radius of vessel u=.04; %------------------------g/(cm sc) viscosity %====================================================== %--------------Loop to calculate womersley #, M'o, M'10 g=0:1/(50-1):1;%--------------------Calculates 100 points along the radius for z=1:length(g) r=R*g(z); y=r/R; number=10; for n=1:number alpha(n)=R*sqrt((2*pi*f(n+1,1)*p)/u);%alpha(n)=Womersley # @ n harmonic z1=alpha(n)*y; %z1 and z2= z for M'o, M'10 z2=alpha(n); %Interpolate using function interp1q (x,m,xi) xalpha values from %table; mcorresponding mprime 10 values; xialpha(n) calculated form %above m10(n) = interp1q(a(:,1),a(:,2),z2); % mprime10 from Table C.1 % values taken from Table 27 theta1(n)=interp1q(b(:,1),b(:,5),z1); % theta at alpha* (r/R) theta2(n)=interp1q(b(:,1),b(:,5),z2); % theta at alpha Mo1(n)= interp1q(b(:,1),b(:,2),z1); %M() from Table 27 Mo2(n)= interp1q(b(:,1),b(:,2),z2); e10(n)=interp1q(a(:,1),a(:,5),alpha(n)); k=Mo1(n)/Mo2(n);

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63 Mprimeo(n)=sqrt(1-2*k*cos(theta1(n)-theta2(n))+k^2); % As defined by eqn C.7 from Appendix C eo(n)=atan(k*sin(theta1(n)-theta2(n)))/(1-k*cos(theta1(n)-theta2(n))); end i=1; %for each time point, calculating the corresponding flow at each harmonic for t=0:.01:1 %initialize array to zero for summation of harmonics sum=zeros(n,1); for n=1:number x=cos(2*pi*f(n+1,1)*t-f(n+1,3)+eo(n)+e10(n)); %calculation of pressure gradient at harmonics 1-10 Q(n)=f(n+1,2)/(pi*R^2*m10(n)); sum(n)=sum(n)+(Q(n)*Mprimeo(n)*x); %summation of harmonics v(n,i)=sum(n); %velocity profile at each time point n=n+1; end i=i+1; end for n=1:number %calculate max v maxv(n)=max(v(n,:)); end tspan=[0:.01:1]; for t=1:101 %calculate zeroth harmonic of velocity k(t)=((2*f(1,2)/(pi*R^2))*(1-(r^2/R^2))); end tsum=zeros(1,i-1); for ts=1:(i-1) %summation of harmonics 0-10 test=v(:,ts); total=cumsum(test)+k(1); tsum(ts)=total(n); end if z==1,s1=tsum;

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64 elseif z==2,s2=tsum; elseif z==3,s3=tsum; elseif z==4,s4=tsum; elseif z==5,s5=tsum; elseif z==6,s6=tsum; elseif z==7,s7=tsum; elseif z==8,s8=tsum; elseif z==9,s9=tsum; elseif z==10,s10=tsum; elseif z==11, s11=tsum;end end s11=zeros(1,101); %-----sets Velocity at wall equal to zero %============================================================== %=====Final Output: Velocity profile velocity=[s11;s10;s9;s8;s7;s6;s5;s4;s3;s2;s1; s2; s3; s4; s5; s6; s7; s8; s9; s10; s11];

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65 Code for Determining Wall Shear Stress Profile function shear=stress(data,D) %================================================ %===Program to analyze Wall Shear Stress %===INPUTS: DFT Data and Diameter of Blood Vessel %===OUTPUTS: Wall Shear Stress (dynes/cm^2) %================================================ f=data; a=xlsread('Table C_1'); %Table C.1 for Womersley values, M10,e10 b=xlsread('Table 27b'); c=xlsread('table28'); %========================================constants p=1.05; %-----------------------------g/mL density R=D/2;%------------------------cm radius of vessel u=.04; %-----------------------g/(cm sc) viscosity %================================================= %---------Loop to calculate womersley #, M'o, M'10 g=0:1/(50-1):1; for z=1:length(g) r=R*g(z); y=r/R; number=10; for n=1:number alpha(n)=R*sqrt((2*pi*f(n+1,1)*p)/u);%alpha(n)=Womersley # @ n harmonic z1=alpha(n)*y; %z1 and z2= z for M'o, M'10 z2=alpha(n); %Interpolate using function interp1q (x,m,xi) xalpha values from %table; mcorresponding mprime 10 values; xialpha(n) calculated form %above m10(n) = interp1q(a(:,1),a(:,2),z2); % mprime10 from Table C.1 % values taken from Table 27 theta1(n)=interp1q(c(:,1),c(:,5),z1); % theta at alpha* (r/R)from Table 28 theta2(n)=interp1q(b(:,1),b(:,5),z2); % theta at alpha Mo1(n)= interp1q(c(:,1),c(:,2),z1); %M() from Table 28 Mo2(n)= interp1q(b(:,1),b(:,2),z2); %M() from Table 27 e10(n)=interp1q(a(:,1),a(:,5),alpha(n)); end i=1; %for each time point, calculating the corresponding flow at each harmonic for t=0:.01:1 %initialize array to zero for summation of harmonics sum=zeros(n,1);

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66 for n=1:number x=cos(-(pi/4)+2*pi*f(n+1,1)*t+theta1(n)-theta2(n)-e10(n)+f(n+1,3)); %calculation of pressure gradient at harmonics 1-10 Q(n)=(f(n+1,2)*u*alpha(n)*Mo1(n))/(pi*R^3*m10(n)*Mo2(n)); sum(n)=sum(n)+(Q(n)*x); %summation of harmonics tau(n,i)=sum(n); %velocity profile at each time point n=n+1; end i=i+1; end tspan=[0:.01:1]; for t=1:101 %------------calculate zeroth harmonic of shear k(t)=((4*f(1,2)*u*r)/(pi*R^4)); end n=number; tsum=zeros(1,i-1); for ts=1:(i-1) %-----------summation of harmonics 0-10 test=tau(:,ts); total=cumsum(test)+k(1); tsum(ts)=total(n); end for k=1:length(g) sh(z,:)=tsum; end end s11=sh(length(g),:); j=length(s11); %========================================================== %=========Final output for wall shear stress=============== for i=1:length(s11) shear(i)=s11(j);j=j-1; end

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67 Code for Main Driver of Program %======================================================== %==================MAIN MENU============================= %==Following menu will provide options for analyzing %== the flow or pressure waveform and determining the %== velocity and wall shear stress profiles. %== Also provides user with the chance to save the file %== as well as create a 2-D avi file or 3-D movie of the %== velocity profile. %======================================================= clear all warning off N=1000; for j=1:N k=menu('Select option to calculate:','Load waveform to Analyze','Centerline Velocity and Wall Shear Stress Profile', 'Pressure Waveform', 'Create Graph of Flow, Centerline Velocity and Wall Shear Stress','2-D Velocity Movie', '3-D Velocity Movie', 'Save Data','Exit'); %======================================================= if k==1 %------------------LOAD WAVEFORM clear rabbit=input('Rabbit Number:'); x=0; %------------------------------------Input Flow Data rootname=input('Name of flow waveform file:','s'); extension='.txt'; filename=[rootname,extension]; [file]= eval('load(filename)'); %-----------------------------------------------DIAMETER loc=input('Enter number of diameter locations(1 or 3):'); for w=1:3 if loc ==1 diameter(1)= input(' Medial Diameter in cm:'); break elseif loc==3 diameter(1)=input(' Proximal Diameter in cm:'); diameter(2)=input(' Medial Diameter in cm:'); diameter(3)=input(' Distal Diameter in cm:'); break else disp('Error: Please enter number of diameter locations.') end end %------------------------------------------Performs DFT

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68 z=dft(file); %======================================================== elseif k==2%---------------------Velocity and Wall Stress u=0.04; for d=1:loc vel=createmov(z,diameter(d)); velocity(d,:)=vel(11,:); shear(d,:)=stress(z,diameter(d)); maxv(d)=max(velocity(d,:)); minv(d)=min(velocity(d,:)); meanv(d)=mean(velocity(d,:)); maxstress(d)=max(shear(d,:)); minstress(d)=min(shear(d,:)); meanstress(d)=mean(shear(d,:)); meansteadyshear(d)=(4*u*z(1,2))/(pi*(diameter(d)/2)^3); end MVG=[maxv maxstress;minv minstress; meanv meanstress] Poiseuille_shear=meansteadyshear %========================================================= elseif k==3 %-------------------------Evaluate Pressure Waveform P=input('Name of Pressure Waveform file:'); press=[P,extension]; [Pressure]= eval('load(press)'); x=1; %--------------------------------Pressure in mmHg maxp=max(Pressure); minp=min(Pressure); meanp=mean(Pressure); PRESSURE=[maxp minp meanp] %========================================================== elseif k==4 %--------------------Create Graphs tspan=0:.01:1; %Plot of Original Flow Waveform time=0:1/(length(file)-1):1; subplot(3,1,1) flow=file'/60; plot(time,flow) title('Original Flow Waveform')

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69 ylabel('Flow (ml/s)') %Velocity subplot(3,1,2) plot(tspan, velocity) ylabel('Velocity (cm/s)') if loc==3 legend('PVG','MVG','DVG') end %Plot of Shear Stress subplot(3,1,3) plot(tspan,shear) xlabel('One Cycle') ylabel('Shear Stress(dynes/cm^2)') if loc==3 legend('PVG','MVG','DVG') end %========================================================== elseif k==5 %-------------------------2-D Velocity Movie disp('Note: The avi file is saved as vprofile.avi') figure(2) mov=avifile('vprofile.avi') r=-1:0.1:1; for k=1:101 h=plot(vel(:,k),r); set(h,'EraseMode','xor') axis([-1 15 -1 1]) F=getframe(gca); mov=addframe(mov,F); end disp('Note: The avi file is saved as vprofile.avi') %========================================================== elseif k==6%------------------------3-D Velocity movie aviobj=avifile('vprofile3.avi') for m=1:101 k=vel(:,m); %--------------------------performs coordinate transform from 2-d to 3-d

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70 for j=1:length(r) for theta=1:361 x(theta,j)=r(j)*cos(theta*pi/180); y(theta,j)=-r(j)*sin(theta*pi/180); z(theta,j)=k(j); theta=theta+1; end j=j+1; end h=figure('visible','off'); none=[]; %-----3-D profile surf(z,y,x,'FaceAlpha','flat','AlphaDataMapping','scaled','AlphaData',gradient(z),'FaceColor','blue','MeshStyle','column','FaceLighting','flat','BackFaceLighting','reverselit'); set(gca,'Xlim',[-1 15]) set(gca,'YTickLabel',none,'ZTickLabel',none,'CameraPosition',[172.56 -12.87 4.19]) aviobj=addframe(aviobj,h); end aviobj=close(aviobj); %========================================================== elseif k==7 %----------------------------SAVE DATA flow=file'/60; tspan=0:.01:1; disp('This will save the velocity and shear stress profiles in the graph.') root=input('Save velocity and shear stress profiles as:'); name=[root,extension]; %The following will save centerline velocity and shear stress to one file. eval(['save ', name tspan flow velocity shear -ascii']) disp('The following prompt will ask you to save the maximum/minimum/mean data for velocity and shear stress') r=input('Save min/max data as:'); a=[r,extension]; %The following will save min/max velocity/shear stress values and mean %steady flow value into separate file. if x==1 eval(['save ',a rabbit maxv minv meanv maxstress minstress meanstress meansteadyshear maxp minp meanp -ascii']) else x==0

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71 eval(['save ',a rabbit maxv minv meanv maxstress minstress meanstress meansteadyshear -ascii']) end %=========================================================== elseif k==8 %----------------EXITS MENU break %=========================================================== end end

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BIOGRAPHICAL SKETCH Chessy Fernandez was born on May 4, 1978, in Norfolk, VA. Her fathers military transfer brought the family to Pensacola, FL, where she later graduated from high school in 1996. After growing up in the sunny Florida weather, she decided to go to college further north in the nations capital. It was in Washington, DC, that she was able to explore different aspects of engineering through volunteering and internships while pursuing a varied education, which included studying photography and web design. She attended The Catholic University of America in Washington, DC, where she graduated with a bachelors degree in biomedical engineering in May 2000. After a break from school, she decided to return to Florida to be closer to home and continue her education. In the spring of 2001 she began her graduate school career in biomedical engineering at the University of Florida. 78