Phd-diss-Pulsatile Laminar and Turbulent Blood Flow Simulation in Large Stenosed Arteries and...

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FARZAN GHALICHI

Pulsati le Laminar and Turbulent Blood Flow

Simulation in Large Stenosed Arteries and

Stenosed Carotid Artery Bifurcation

These

presentee

a la Faculte des etudes superieures

de lUniversite Laval

pour lobtentiondu grade de Philosophiae Doctor (Ph.D.)

Departement de genie mecanique

FACULTE DES SCIENCES ET DE GENIE

UNIVERSITE LAVAL

QUEBEC

Septembre 1998

Farzan Ghalichi, 1998



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UNIVERSITY

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ont assists kfa soutenance de cette thdse.


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Abstract

In this dissertation, the effect of a minor and a severe stenosis was studied on various

aspec ts of flow downstream of stenosis in a femoral artery and a human carotid artery

bifurcation. The major p arameters of in terest in this research were the tim e-averaged

velocities, time-dependent shear stress, separation zone and reattachment length.

Comparing our results of the rea ttach m ent length to the laminar flow simulation with

experimental results for the Reynolds number higher than critical Reynolds number,

we did believe that the numerical results of laminar flow simulation beyond the critical

Reynolds number were not reliable. Therefore, a new methodology had to be used to

provide new numerical information. We found th at low-Re k u; turbulence model

was a very appropriate model with accurate data to simulate blood flow in the entire

flow domain. The predicted results by the low-Re model were in very good agreement

with the experimental measurements.

In the second pa rt of this thesis, th e evolution of atherosclerotic disease was stud

ied under the presence of various degrees of stenosis. The role of carotid artery

bifurca tion geometry was also taken into account. The finite elem ent calculations

of the stenosed carotid artery bifurcation were performed under laminar flow condi

tions at a mean Reynolds number of 200 and a flow division ratio of about 70/30,simulating an entire systolic and diastolic pulse wave. Two different geometries with

various degrees of stenoses were considered. The presence of a stenosis greater than

25% created two distinct flow zones in the internal carotid artery, a high wall shear

stress area at the stenosis which may cause mechanical damage to the endothelial

lining, and an elongated flow recirculation zone with low wall shear stress leading to

an increased duration of flow reversal in a pulse cycle which retards mass transport

through the arterial wall and may in turn accelerate the development of atheroscle

rosis downstream of the stenosis. Furthermore, the results obtained regarding the

streamlined contour, velocity profiles and duration of reversed flow in a pulse cycle

showed that the atherosclerotic lesions may develop very rapidly up to a stenosis of

between 25% and 40%. Beyond a 40% stenosis and up to 75% stenosis, hemodynam-

ically, the development of lesions occurred but not at the same rate as before, that



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means its progress ra te decreased. In contrast, the 75% stenosis showed a very sig

nificant variations in flow behavior leading to a fast progression of the atherosclerotic

lesions. These interesting findings have prom pted us to pursue our study for a more

severe stenosis. The presence of a severe stenosis (> 70%) does change the lam inar

flow regime to turbulent flow regime.

Low-i?e turbulence modeling was therefore used and successfully applied in the

pulsat ile flow simulation to de tect both laminar and turbulen t flow regimes. The

results showed that even in a healthy artery, the weak instabilities could be found at

least for a portion of the pulse cycle and in different areas. The presence of a 40%

and 55% stenoses in both test models did not alter significantly the flow properties

with regard to turbulence characteristics. On the other hands the presence of a 75%

stenosis altered the flow properties from laminar to turbulent, significantly. By using

more realistic conditions in the computations and applying the methodology used in

this research program, we believe that a better understanding of the progression of

atherosclerotic plaques and the measurement of stenosis in carotid artery bifurcation

would be possible.

Dr. Alain DeChamplain

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R e s u m e

Les desordres cardiovasculaires tels que latherosclerose, sont une des causes prin-

cipaies de mortalite dans les societes modernes. Cest une maladie degenerat ive qui

afFecte les grandes arteres causant pa r la progression de lepaississement e t du durcisse-

ment de certains vaisseaux par l accumulation de riche materiel lipidique. Les etudes

cliniques et postmortem indiquent que, chez I'homme les lesions atherosclerotiques ne

se developpent pas aleatoirement, et pas partout dans la circulation sanguine, mais

elles se localisent a certains endroits choisis dans le systeme arteriel ou un ecoulement

complexe se produit (cest a dire, les arteres coronaires, carotides, abdominaies, et

femorales) [147].

Des perturbations hemodynamiques ont e te fortement correlees avec la localisation des

lesions atherosclerotiques sur les murs des vaisseaux. Dans ces regions, lecoulement

deja perturbe peut etre a tous les regimes decoulement, sont laminaire, transitionnel

ou fortement turbulent.

Afin de comprendre Ie comportement normal et pathologique du systeme vas-

culaire humain, la connaissance detaillee de lecoulement du sang et la reponse des

vaisseaux sanguins sont exigees. La comparaison des parametres qu an tita tifs dans

les modeles stenoses et non-stenoses de geometrie differente des arteres a pu rendre

possible la detection de la stenose en debut de maladie.

Par consequent, dans cette these, nous avons etudie lecoulement sanguin dans les

grosses arteres avec une stenose dans un modele decoulement permanent et pulse.

Au Chapitre 1, la motivation, la problematique et les objectifs de la these sont

presentes. Une revue complete de la litteratu re est aussi donnee.

Au Chapitre 2, une description complete de la methode des elements finis et des

equations de Navier-Stokes utilisees dans ce tte etude est presentee. Les simplifica

tions de letude numerique de simulation decoulement dans les grosses arteres sont

egalement expliquees. La discretisation de Galerkin des equations de N-S, les con

ditions limites, les conditions initiales et les procedures de solution sont egalement

apportees dans ce chapitre.

Au Chapitre 3, la simulation laminaire en regime permanent est effectuee avec

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differents degres de stenose et les resulta ts sont compares avec les resu ltats experiment-

aux. Trois regimes decoulement sont p rodu its dans une artere obstruee comm engant

avec lecoulement laminaire. Dans la stenose faible pour des nombres de Re tres bas

nous navons observe aucune separation e t lecoulement eta it laminaire parto ut. Pou r

un nombre de Re plus grand, lecoulement demeure laminaire, mais la separation

se pro du it et une zone de recircula tion se developpe. Dans le troisieme regime,

lecoulement en aval du point de separation devient instable, eventuellement tur

bulent et persiste bien au-dela du point de ra ttachem ent . En comparant les re su ltats

de la longueur de rattachem ent obtenus de ce tte etude par la simulation decoulement

laminaire pour le nombre de Reynolds plus grand que le nombre de Reynolds critique

avec les resultats experimentaux [6, 140], nous croyons que les resultats numeriques

de simulation decoulement laminaire au-dela du nombre de Reynolds critique ne

sont pas fiables. Pa r consequent, la simulation decoulement turb ulen t utilisant le

modele de turbulence k uj a faible nombre de Reynolds (bas-Re) est effectuee a

differents degres de stenose. La comparaison prouve que les resulta ts prevus pax le

modele k u>sont en tres bonne concordance avec les mesures experimentales. En

utilisant ce modele, nous reproduisons exactement le nombre de Reynolds critique

auquel lecoulement de sang devient transitoire ou turbulent en aval de la stenose.

Un au tre point interessant, nous avons trouve que dans un ecoulement laminaire,la longueur de recirculation prevue par ce modele est en accord avec la longueur de

vortex prevue pax une simulation decoulement completement laminaire, proposant

que le modele de k ui est non seulement approprie pour modeliser des regimes

transitoires e t turbulents, mais egalement po ur lecoulement laminaire en amont de la

stenose. Pour verifier plus a fond, nous avons compare la prediction pou r la pression

statique de la paxoi dans une stenose a 50% avec un nombre de Reynolds de 500

aux resultats dune simulation en regime laminaire. La comparaison montre que le

modele actuel peut etre utilise pour faire la simulation dun ecoulement laminaire.En terme de pression statique sur la paxoi et de lintensite de turbulence, le modele

de turbulence de bas-Re donne des resultats beaucoup plus precis que le modele k e

stan dard. Lintensite prevue de la pression et de la turbulence pax le modele de bas-

Resont semblables aux mesures experimentales, alors que le modele k edonne une

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reprise excessive de pression dans la zone de deceleration et une prediction tres faible

de lintensite de turbulence. Nous concluons que le modele de turbulence de bas- .f te

peut fournir des informations satisfaisantes de 1ecoulement sanguin dams les arteres

stenosees, compte tenu que cest tres difficile dobtenir experimentalement (in vitroet

in vivo) avec exac titude ou en simulant numeriquement par u n ecoulement laminaire.

Cliniquement, les donnees sont particulierement interessantes pour la detection des

plaques atherosclerotiques localisees.

Au Chapitre 5, la simulation decoulement pulse est effectuee sur des modeles

normaux et stenoses de la bifurcation carotidienne. Linfluence de la variation de la

stenose et de la variation de la geometrie sur le profil de vitesse ainsi que le point de

rattachement et la contrainte de cisaillement sur les parois sont etudies pour evaluer

la possibility de detection de la stenose dans des conditions variables decoulement.

Lexistence possible de turbulence est egalement evaluee en utilisant un modele de

turbulen ce a bas-fte . Pour comprendre le role de la geometrie dun vaisseau sangu in,

deux modeles de bifurcation carotides avec un sinus different sont etudies. Les deux

modeles ont differentes formes de sinus. Le modele M-l a un sinus droit tand is que

le modele M-2 a un sinus profile. La severite de la stenose est mise en compte pour

juger si 1ecoulement reste laminaire pe ndan t le cycle de pulsation.

Pour evaluer linfluence de la forme de la stenose sur les comportements decoulement , deux stenoses differentes sont comparees. Dans M -l la stenose brusque est

utilisee, tandis q uune coupe profilee est utilisee en M-2. Bien que la stenose 25%

dans une artere droite naffecte pais les comportements decoulement en aval de la

stenose (Saad et Giddens [3]), la presente etude indique que la presence dune stenose

faible (25%) dans le sinus de la caxotide a une influence relativement petite sur la

comportement decoulement en aval de la stenose. Cependant, quand la stenose

grimpe entre 40% et 55%, le changement de comportement de lecoulement et de la

distribu tion de la contrainte de cisaillement sur la paxoi est prononce. La presencedune stenose tres severe cree deux zones distinctes d ecoulement dams 1axtere caxotide

interne. Le premier est a linter ieur du sinus de la caxotide et affiche des con train tes

de cisaillement elevees sur la paxoi. Le deuxieme est place imm ediatem ent en aval de

la stenose ou une inversion decoulement avec de faibles contraintes de cisaiillement

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sur la paroi existent. Dans certaines periodes dun cycle, cet ecoulement renverse

setend au-dela de la section de sinus, atteignant la paxoi diviseur de lecoulement.

Linfluence la plus importante de la stenose se manifeste dans les modifications de la

distribution des contraintes de cisaillement sur la paroi. Une stenose tres severe mene

a une variation remarquable de la contrainte de cisaillement sur la paxoi. En labsence

dune stenose, la valeur absolue de contra inte de cisaillement sur la paxoi dans le milieu

du sinus change entre 2.6 et 7 dy n/cm2 pour le modele caxotide M -l, tandis quelle

change entre 0 et 6 dy n/cm2 dans le modele M-2. En presence dune stenose 25%,

la contrain te de cisaillement se modifie entre 5 et 187 dyn /cm 2. Quand la stenose

devient 40% et 55%, la variation devient 4-370 dyn/cm2 et 30-470, respectivement

pour le modele M -l. Ces variations significatives de la contrainte de cisaillement de la

paxoi en presence d une stenose sont non physiologiques, favorisant lepaississement

intimal et sont probablement responsables de rupture de plaques daxteriosclerose et

de la des embolies de fragments [20]. Pendant un cycle d impulsion, les contraintes

de cisaillement dans le sinus de caxotide peuvent etre aussi hautes que 370 dyn/cm 2

pour la stenose de 40% et 470 dyn/cm2 pour la stenose de 55% dans le modele M -l.

Les contraintes de cisaillement elevees resultant de lexistence de la stenose peuvent

mener a des dommages mecaniques de la couche endotheliale dans le sinus [35].

Dans la deuxieme paxtie du Chapitre 5, la simulation decoulement turbulent est

effectuee dans une bifurcation caxotidienne stenosee. Cette section es t divisee en

deux parties: premierement, la validation du modele et deuxiement lapplication de

ce modele a differents degres de stenose avec des geometries diverses . En labsence

des resultats experimentaux et numeriques, la validation du modele de ku> dans

lecoulement pulse est realisee en compaxant les resultats de la simulation de l ecoulem-

ent laminaire a la meme forme de pulsation et de geometrie. La validation du modele

est effectuee sur le modele M-l sans stenose. Dans ce modele, les calculs sont effect ues

en ecoulement laminaire et en utilisan t egalement le modele de turbulence de k uj. De

lechelle du nombre de Reynolds utilises dans cette etude, lecoulement est laminaire

dans laxtere caxotide normale. Les calculs avec le modele de turbulence k u>confirme

cette similitude. Les profils de vitesse et les resultats de contraintes de cisaillement

sur la paxoi sont egalement compares. II y a bonne concordance en tre les resulta ts



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de Fecoulement laminaire et le modele de turbulence. Les calculs sont etendus aux

stenoses de 40%, 55% et 75% situees dans la sinus de carotide in terne. Lobjec tif

important est de documenter si ces degres de stenoses ont une influence sur lintensite

de la turbulence ou non. Les resultats de nos calculs precedents pour Fecoulement

laminaire sont egalement utilises pour comparer les deux methodes pour les stenoses.

Les profils de vitesse et les resultats des contraintes de cisaillement de la paroi pour

la stenose 40% dans les deux modeles M-l et M-2 sont en tres bonne accord avec

les resultats de simulation de Fecoulement laminaire. Dailleurs, Fenergie cinetique

de turbulence demontre que Fecoulement laminaire tend a devenir transitoire dans

une certaine partie du cycle et dans differentes sections de Fartere carotide interne.

Lintensite de lenergie cinetique de turbulence nest pas suffisamment forte pour

affecter la con trainte de cisaillement de fagon importante. De ceci, nous concluons

que la stenose 40% ne rend pas Fecoulement turbulent en aval de la stenose. Po ur la

stenose de 55% le profil de vitesse et les resulta ts des contraintes de cisaillement dans

le modele turbulence de kuisont conformes pour la majeure pa rtie de la periode du

cycle dimpulsion avec la simulation decoulement laminaire jusqua la region de la

stenose. Cependant, les valeurs de contraintes de cisaillement du mu r ne changent pas

de maniere significative, particulierement dans le modele M-l. Dans cette stenose les

valeurs moyennes et maximales de lintensite de turbulence dans le cycle dimpulsionne varient pas beaucoup. Le maximum est situe a lemplacement de la stenose, mais

pas en aval. Pax consequent, la stenose 55% ne change pas Fecoulement de lam inaire

a tu rbu lent . Des changements significatifs sobservent avec la stenose de 75% dans

le modele M-l. La vitesse maximale atte int 100 cm/sec au centre de la stenose et la

longueur de la zone de recirculation augmente en aval de la stenose. Les resul tats de

distribution de contraintes de cisaillement le long du cote exterieur de Fartere nous

demontre une augmentation considerable de la valeur des contraintes de cisaillement.

La valeur de ces contraintes de cisaillement atteint 40 dyn/cm2, compare a 7 et4 dyn/cm 2 sur le meme modele pour les stenoses de 40% et 55%. II est possible

de conclure que cette augmentation de la contrainte de cisaillement est due a la

turbulence intense en aval de la stenose. La valeur de Fenergie cinetique turbu lente

est en accord avec cette conclusion. A la difference des stenoses de 40% et de 55%



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qui affectent peu lecoulement dans la region centrale de la stenose du a 1energie

cinetique plus petite et a sa dissipation immediate, la stenose de 75% provoque une

augm entation de la valeur de lenergie cinetique turbulen te en aval de la stenose. Ces

resultats concordent avec des observations cliniques quune obstruction de plus de

70% produit des sons qui sont discernables avec un stethoscope.

Dr. Alain DeChatnplain



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Aknowledgment

I consider myself very fortunate to have Professor Alain DeChamplain as my

adviser. I would like to take this op portunity to especially thank him for patience,

kind guidance, and advice he has given me through my Ph.D. study. Together we

came up with a q uite interesting and challenging research topic.

I wish my sincere appreciation to my co-supervisor, Prof. Robert Guidoin, for his

suggestion of the dissertation topic, and for his constant encouragement and suppo rt

for this project. I am indebted to him specially for his efforts to get my workstation

being an essential equipment in this work.

I would like to thank to Dr. Xiaoyan Deng for his generous help. His guidance in

interpreting the results was both stimulating and enlightening.

I am very obliged to Dr. Yvan Douvilie for helping me in the im plem entation of

some of the ideas presented in this thesis.

I also wish to thank Prof. Robert Guenette and Prof. Allan Marble for their

careful reading of the manuscript and for their useful suggestions.

I would like to whole-heartedly express my gratitude to my parents. Thank you,

my mother, and my father, for providing me with the life I have and for being the

wonderful parents you have been. I wish my mother lives happily always, and a to tal

improvement to my lovely father who is suffering from a cerebral attack. Thank you

to my sisters for always being there and wishing me the best.

Finally, I thank my wife Fariba and my two beautiful daughters Fatemeh and

Faezeh for providing the emotioned support necessary to navigate the stresses of

pro ject and for making me a very happy man. W ithou t their endless supp ort and

love for me, I would have never achieved my current position.

I would like to offer my many thanks to the government of the Islamic Republic

of Iran to give me a chance to continue my studies towards my Ph.D. degree.

This research has been supported by Saint-Frangois dAssise Hospital of Quebec

city and Quebec Biomaterials Institute.



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Nomenclature

Cti , C( 2 ...... Turbulence model empirical constants

h............................ Height of the asymmetric stenosis

d ...... Stenosis diam eter

D ...... Unobstructed artery diameter and reference length

r ...... Radial distance from the symm etry line

R ...... Radial direction of flow

p ...... Fluid density

v ...... Fluid kinematic viscosity

p ...... Fluid dynamic viscosity

Um.................... ...... Mean flow velocity

Re ...... Reynolds num ber Re=

R e ' ...... Critical Reynolds number

/ ...... Pulse frequency

a ...... Womersley number a =

u> ...... Essential angular velocity

t ...... Non-dimensional wall shear stress

u , v ,w ...... Velocity components

p ...... Pressure

t ...... Time

Reference time

it* ...... Non-dimensional velocity vector

u ...... Velocity vector

p' ...... Non-dimensional pressure

pU l ...... Reference pressure

Uij ...... Components of velocity vector u

ur ...... Radial velocity in the axisymmetric flow

Ug ...... Axial velocity in the axisymmetric flow

Z ...... Axial direction of flow



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L ...... Stenosis length

m ...... Mass flow ra te

La ...... Reattachment length

~Z ...... Non-dimensional distance from stenosis

k...................... ...... Turbulent kinetic energy

e ...... Turbulent dissipation energy

/lj ...... Turbulent viscosity

y + ...... Y-plus param eter


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Contents

1 Introd uction 1

1.1 Motivation ........................................................................................................ 1

1.2 Problematics .................................................................................................. 3

1.3 Objectives ........................................................................................................ 9

1.4 Review of L ite ra tu re ..................................................................................... 11

1.4.1 Long-segment a r t e r y ........................................................................ 11

1.4.2 End-to-side and side-to-end anas to m osis ..................................... 16

1.4.3 Carotid Artery Bifurcation (CAB) .............................................. 20

1.4.4 Grid size and computational ti m e .................................................. 24

2 Nu m erical H emodynam ics 272.1 Simplifications.................................................................................................. 27

2.2 Numerical ba ck g ro un d .................................................................................. 31

2.2.1 Galerkin d isc re tiz at io n .................................................................... 31

2.2.2 Governing e q u a tio n s ........................................................................ 32

2.2.3 Formulation of the Discrete P ro b le m ........................................... 33

2.2.4 Penalty a p p ro a ch .............................................................................. 37

2.2.5 Nondimensionalization ..................................................................... 38

2.2.6 Boundary and initial co nd itio ns ..................................................... 40

2.2.7 Solution pro cedu res ........................................................................... 42

2.3 Computational co ns ide ratio ns ..................................................................... 43

2.3.1 Th e choice of ele m en t........................................................................ 43

2.3.2 Convergence c r it e r ia ........................................................................ 48

xii



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CONTENTS xiii

2.4 Time in teg ra tio n .......................................................................................... 48

2.4.1 Second Order Scheme (Trapezoidr u l e ) ......................................... 50

2.5 The Finite Element Package (F ID A P) ....................................................... 52

3 Lam inar Flow Sim ulations 54

3.1 Introduction .................................................................................................... 54

3.2 Numerical calculation of steady laminar fl o w .......................................... 56

3.2.1 Assumptions........................................................................................ 56

3.2.2 Modeling of the axisymmetric f l o w .............................................. 56

3.2.3 Governing e q u a ti o n s ........................................................................ 57

3.2.4 Geometry and boundary co nd itio ns .............................................. 58

3.3 Results ............................................................................................................. 61

3.3.1 Separation and reattachment p o i n t .............................................. 62

3.3.2 Pressure distribution along the vessel w a l l ................................. 66

3.3.3 Wall shear s t r e s s ............................................................................... 67

3.4 Discussion ....................................................................................................... 70

4 Tu rbulence M odeling 76

4.1 Introduction .................................................................................................... 76

4.2 Axisymmetric geometrical models and flowconditions ........................... 794.2.1 Boundary c o n d iti o n s ........................................................................ 80

4.3 Turbulence m o de lin g .................................................................................... 81

4.3.1 k eturbulence m od elin g ............................................................... 82

4.3.2 Low Reynolds k u> turbulence m o d e l ........................................ 86

4.4 F.E.M. analyses for turbulence m o d e ls .................................................... 88

4.5 Results ............................................................................................................. 93

4.5.1 Validation of the numerical results .............................................. 93

4.5.2 Recirculation and reattachment l e n g t h ........................................ 934.5.3 Velocity pro fi le s .................................................................................. 96

4.5.4 Centerline v e lo c it y ........................................................................... 104

4.5.5 Velocity d istu rb an ce s ........................................................................ 105

4.5.6 Pressure d ist rib u tio n ........................................................................ 108



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CONTENTS xiv

4.5.7 Wall sheax s t r e s s .............................................................................. 109

4.6 Hemodynamics of protein-impregnated arterial pros these s.................... I l l

4.6.1 Results .................................................................................................. 115

4.7 Pulsatile flow in an asymm etric stenos is..................................................... 115

4.7.1 Introduction ........................................................................................ 115

4.7.2 G eo m etry ........................................................................................... 116

4.7.3 Boundary C on ditio ns ........................................................................ 116

4.7.4 Results.................................................................................................. 119

4.8 Discussion ........................................................................................................ 122

4.9 Conclusion........................................................................................................ 127

5 C arotid A rtery Bifurcation 129

5.1 Introduction ..................................................................................................... 129

5.2 CAB simplifications........................................................................................ 132

5.3 CAB G eo m etry .............................................................................................. 133

5.4 Geometrical model and flow co n d ition s ...................................................... 134

5.5 Laminar flow si m u la tio n .............................................................................. 136

5.5.1 Mathematical e q u a tio n s .................................................................. 136

5.5.2 Boundary and initial co nd itio ns ..................................................... 141

5.5.3 Results.................................................................................................. 143

5.5.4 Discussion ........................................................................................... 159

5.5.5 Conclusion........................................................................................... 165

5.6 Turbulent flow si m u la tio n ........................................................................... 166

5.6.1 Results .................................................................................................. 168

5.6.2 Discussion ........................................................................................... 183

5.6.3 Conclusion........................................................................................... 190

6 C onclusions andrecommendations 191

A 216

A.l User subroutines ............................................................................................... 216

A.1.1 Applied flux boundarycondi t ions ........................................................216



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CONTENTS xv

A.1.2 Applied nodal boundary co nd iti on s ............................................. 217

A.2Carotid pulse wave d a t a ................................................................................ 218

A.3 Streamfunction for Axisymmetric Flow: Cylindrical coordinates . . . 223

B 225

B.lFIDAP Input File for Laminar Flow Sim u lat io n ............................................225

B.2FIDAP Input File for Turbulent Flow S im u la tio n .................................... 226

B.3FIDAP Input File for a Pulsatile Laminar F lo w ....................................... 227



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List of Figures

1.1 Development of artherosclerotic lesions. Presen tation of norm al, mod

erate and severe narrowing of the lumen due to atherosclerotic lesions

axe given from left to right respectively. ..................................................... 21.2 Diagram of the human body showing major blood vessels which may

be affected by arterial stenosis. (Courtesy of Human Anatom y and

Physiology, the Benjamin/Cum mings Publishing Company, Inc .,1989.) 4

1.3 Examples of geom etric configurations of an asym metric stenosis, a)

h/D = 0.66; b) h /D = 0.5; c) h/D =0.29 .............................................. 8

1.4 Geometric configuration of a carotid artery bifurcation .......................... 9

1.5 Geometric configurations for bypass graft................................................... 17

2.1 The quadrila teral Crouzeix-Raviart element in its reference (1 turbulence modeling and

laminar flow simulation. The solid lines represent the lam inar flow

where the dotted lines represnt the turbulent flow results......................



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Chapter 1

Introduction

In this chapter, the motivations fo r the thesis are discussed. A summary o f previous

research works on related topics is presented. The specific objectives of the thesis are

listed.

1.1 M otivation

Deposition of cholesterol, and other fatty tissues along human arteria l walls can form a

constriction and therefore restric t the blood flow. The disease which causes the buildup of plaque in the artery is part a the broader circulatory disease called atheroscle

rosis, and it is a form of arteriosclerosis 1. As the plaques form, the walls become

thick, fibrotic and later calcified, and the lumen narrows, reducing the flow of blood

to the tissues the artery supplies. This build-up can lead to stroke or heart attac k,

or the disfunction of other organs. Arteriosclerosis is the leading cause of death and

disability among North Americans and Europeans. Stroke is a portmanteau term

covering any disease or neurological disorder tha t results in th e m arked res triction or

cessation of flow affecting the brain. There are basically two different kinds of stroke:

hemorrhagicand ischemic2. Hemorrhagic strokes, which account for 20 percent of ail

1A group of diseases characterized by thickening and loss of elasticity of the arterial walls occurringin three forms, atherosclerosis, Monckebergs arteriosclerosis, and arteriolosderosis.

2Ischemia refers to decreased blood supply to a tissue, a potentially reversible condition; uncorrected, it leads to infarction, or tissue death due to anoxia.

1



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CHA PTER 1. INTROD UCTION 2

strokes, take place when vascular lesions ruptu re, releasing blood into the surrounding

brain tissue. The remaining 80 percent of strokes are ischemic in charac ter, caused

by the obstruction or clogging of the m ajo r arteries in th e cerebral circulation.

The earliest microscopic change in the formation of atherosclerosis is the accumu

lation of lipids in th e intima 3. These deposits called fa tty streaks are covered by an

intact layer of endothelium 4 which leaves the vessel wall undeformed and narrow the

cross-sectional area of am artery, as shown in Figure 1.1. The narrowed cross-section

is often called a stenosis. At subsequent stages, ri sing plaque levels become visible,

the endothelium may breaik down, amd pairticles are directly deposited by the blood

stream a t c ritical locations. The exact mechanisms by which these fatty substamces

initiadly are found in th e arte ry wadis to form a locally-growing plaque aire no t cleanly

understood. However, in general, a pati en ts inhe rited physiology, hemostatic factors,

hypertension, homocysteine, biochemical processes, smoking habits, daily diet, and

stress levels may all affect atherogenesis. 5

Figure 1.1: Development of artherosclerotic lesions. Pre sen tation of normal, moderate

and severe narrowing of the lumen due to atherosclerotic lesions aire given from left

to right respectively.

3The innermost coat of a blood vessel.4The layer of epithelial cells that lines the cavities of the heart as well as the blood and lymph

vessels.5Formation of abnorm al fatty deposits in an arterial wall.



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CHA PTER 1. INTROD UCTION 3

The failure of arterial prostheses has been found to be caused by intimal hyper

plasia, a feature of the normal healing response of arteries at graft anastomoses 6

[43], or the progression of atherosclerosis. Proximal or distal to the prostheses, inti

mal hyperplasia, is commonly found at the distal end of any vascular synthetic grafts

[26, 54, 29] and following many endovascular procedures (e.i., dilatation, stenting of

an artery, ...). While no clear cause of graft intimal hyperplasia has been identified,

numerous biological and biomechanical factors have been proposed.

While, it is unlikely that investigating a few risk factors may lead to a successful

description of the multicomponent disease, a better understanding of key physical

factors causing atherosclerotic lesions is vital to developing a comprehensive theory,

and may be helpful in the early detection and fight against atherosclerosis.

1.2 Problematics

Clinical and postmortem studies indicate that atherosclerotic lesions on human blood

vessel walls do not develop randomly and do not occur throughout the circulation but

rather locate at certain selected sites in the arterial tree, such as the branching sites

and curved segments of large arteries [121, 127, 112, 15, 133]. Examples of arteries

most often affected by atherosclerosis include the carotid arteries in the neck region,

the coronary arteries in the heart, the iliac arteries in the abdominal region, and the

femoral and t ibia l arteries in the legs (see Figure 1.2). From a fluid mechanics point

of view, these sites are where flow phenomena exhibit unique characteristics . In these

sites, blood flow is disturbed, and the separation of streamlines from the vessel wall

and the formation of eddies are likely to occur [15, 113, 77]. This may suggest that

the physiological processes themselves are not the sole factor.

For the past 30 years, it has been accepted that the physics of blood flow and hemo

dynamic factors are of importance in the initiation and development of atherosclerotic

lesions and intimal hyperplasia [114, 125, 143, 23, 15, 35, 145]. Among the hypothe

ses proposed to account for the localization of atherosclerosis, th e causative effects of

6 An abnormal increase in the number of normal cells in normal arrangem ent in an organ or tissuewhich reduces the internal diameter and increases its total volume.



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CH APT ER 1. INTRODUCTION 4

Internal carotid artery

External carotid artery -

Verteoral artery

IAnterior bOai artery

Posterior ttxaJ artery

Dorsal arcn

- Common carotid artenes

Bracrtoceonahc artery

Coronary artery

Cekac trunfc

Left gastnc arteryCommon hepatic artery Seierxc artery

Penal artery

PadM artery

Ulnar artery

AOdommai aorta

Suoenor mesentenc artery-

Gonadal artery

Inferior mesentenc artery-

Common *ae artery------

External artery internal aacartery

Oeeo palmar arcn

Superficial palmar arcn

i

Figure 1.2: Diagram of the human body showing major blood vessels which may

be affected by arterial stenosis. (Courtesy of Hum an Anatomy and Physiology, the

Benjamin/Cummings Publishing Company, Inc.,1989.)



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CHAPTER1. INTRODUCTION 5

high shear stress, originally claimed by Fry [35], and of low shear stress, as claimed by

Caro et al. [15], have received much attent ion . Fry believed th a t endothelial injury

caused by high shear stress was responsible for atherogenesis. However, observations

by Caro confirmed tha t atherosclerotic lesions developed more frequently in areas

with low shear stress and with flow recirculation than in areas with high shear stress

and unidirectional flow conditions [145, 64].

Several hypotheses have been put forward [5] to explain the mechanism by which

low vessel wall shear stress might promote the development of atherosclerotic lesions.

The endothelial cells undergo morphological alterations in response to changes in the

degree and orientation of shear stress; elongated endothelial cells located in regions

of high shear stress see their long axes aligned parallel to the direction of flow, and

polygonal endothelial cells in low shear stress regions become aligned in haphazard di

rections. It has been postu lated by Asakura and Karino [5] tha t these a lterations may

be responsible for changes in endothelial cell permeability to atherogenic lipoprotein

particles. Low shear stress stimulates the expression of endothelin mRNA as well as

the release of endothelin into the culture medium from cultured porcine endothelial

cells [144]. An increased synthesis of endothelin may in turn prom ote local, smooth

muscle cell and fibroblast proliferation. Caro [15] also suggested t hat low wall shear

rates retard the transpo rt of circulating particles away from the wall, resulting in theincreased intimal accumulation of lipids. Moreover, as blood flow through healthy

vessels may influence the formation of deposits, so may the appearance of atheroscle

rotic plaques on hem odynamics in the vicinity of the lesion [10, 20] . From a fluid

mechanics perspective, any obstruction has a pronounced effect on flow. The down

stream flow from a stenosis becomes irregular and causes changes to local parameters

such as velocity field, pressure drops, and wall shear stress distribu tion . Therefore,

the additional changes in flow and shear further contribute to build-up and helps the

progression of the disease.

Assessment of the actual risks to a patient with arterial disease must consider all

of these factors. Therefore, detailed insight regarding the flow phenomena occurring

in the bends and bifurcations contributes to a better understanding of the role of

hemodynamics in the initiation and progression process of atherosclerosis. The ability



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CHAP TER 1. INTROD UC TION 6

to completely describe the flow through stenosed vessels would therefore provide the

added possibility of early diagnosis of the disease, and hence preventive treatments

would become clinically possible.

In vivodata particularly for human subjects are usually difficult to obtain under

well-controlled conditions with accurate instrumentation. In vitro experiments that

simulate important characteristics of an in vivo situation axe useful, however, the

measurement of i m p o r t a n t pa rameters such as wall shear stress , is difficult. On the

other hand data collection using hot-film or hot-wire instrumentation suffers from

inaccuracies in areas of high turbulence intensities, laxge flow angularities, and regions

of flow separation and reversal flow.

Numerical simulation of blood flow could be used to stud y various aspects of car

diovascular disease and, consequently, help explore possible diagnostic techniques. It

offers a non-invasive means of obtaining detailed flow patte rns associated with disease

by supplying information beyond th at which is available in an experimental study.

The particular role of wall geometry together with the type and character of the flow

can be defined widely in a numerical study. While different authors have provided

useful information pertaining to flow patterns, in many cases important restrictions

and limitations were applied (e.g. the use of only low Reynolds numbers [69], the

application of a steady [75, 23], or simple pulsatile flow instead of a pulsatile physiological flow [86], the assumption of a square occlusion [102], the ignorance of the

possible turbulence flow in certa in geometries [17], th e violation of boundary condi

tions due to a too short computational region, or the termination of the simulation

before a full cycle because of com putational difficulties. One of the important fea tures

neglected in the majority of numerical works is the assumption of laminar flow where

flow disturbances can be found distal to the stenosis by experimentation.

Therefore, it is appropriate to numerically study the blood flow through a stenosis

in large arteries under m ore realistic conditions. Finer mesh, appropriate boundary

conditions with large computational domain, physiological pulse wave form and flow

disturbances will be of great consideration. The hu man carotid a rtery bifurcation is

a typical area where the relationship between local hemodynamics and atherogenesis



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CHAP TER 1. INTRO DU CTIO N 7

ran be studied [145]. Many studies on th e flow pat tern s in carotid artery bifurca

tions have been carried out, both theoretically under laminar flow [105, 106, 98] and

experimentally [145, 10, 11, 66, 64], yet very little information is available regard

ing blood flow in a caro tid bifurcation with a stenosis. Moreover, a comparison of

flow characteristics between diseased arteries and healthy ones may help in identi

fying hemodynamic properties that could be employed in a non-invasive diagnostic

procedure.

Therefore, to arrive at a b ette r understanding of the role of hemodynamics in the

genesis of atherosclerosis in the normal and stenosed arteries and stenosed carotid

artery bifurcation, numerical two and three-dimensional simulations of pulsatile, dis

cretized Navier-Stokes equations were carried out over the entire flow domain, thus

providing more detailed physical information with regard to space and time.

The following four major projects were formulated for this Ph.D. program:

Steady laminar flow simulation through a severe axisymmetric stenosis

Pulsatile laminar flow simulation through a severe asym metric stenosis (Figure

1.3)

Steady turbulen t flow simulation through various degrees of axisymmetric steno

sis

Pulsatile laminar and turb ulent flow through a stenosed carotid bifurcation( Figure

1.4).



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CH APTE R 1. INTROD UCTION 9

Stenosis,

Common carotid

Internal carotid

External carotid

Figure 1.4: Geometric configuration of a carotid artery bifurcation.

1.3 Objectives

It goes without saying that studying the evolution of the disease in relation with flow

dynamics is of the utmost importance. The analysis of the velocities measured at

and downstream of stenoses and the comparison with healthy arteries can be used

to estimate the severity of vessel constriction, and may be helpful in early detection

of atherosclerosis by a non-invasive diagnostic procedure. Early detection would also

create possibilities for a large-scale investigation of the population of carotid artery

bifurcation diseases, hence promoting a more adequate approach to the disease. Since

it is believed that the hemodynamical aspects of blood flow play an important role

in both the genesis and diagnosis of atherosclerotic disease, this dissertation exposes

the effect of a minor and a severe stenosis on various aspects of flow downstream

of stenosis under the following forms: axisym metr ic and asym metric large segment

arteries, and human carotid artery bifurcation. The m ajor parameters of interest in



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CHAP TER 1. INTRODU CTION 10

this research are the time-averaged velocities, time-dependent shear stress, separation

zone and reattachment length, and also the detection of any flow turbulence when

a stenosis is presen t. Comparing the results of the flow field in non-stenosed and

stenosed carotid artery bifurcation models may help define the parameters of distur

bance which may possibly be used for early de tect ion of atherosclerotic disease and

a better understanding of the progression of the disease.

In the laminar flow simulations, the computer simulations were based on the

steady/time-dependent (pulsatile), two/three-dimensional Navier-Stokes equations

for an incompressible Newtonian fluid:

A

p[~^+ (u *V)u] = pB Vp + fid iv{W u) (1.1)

du dv dwdivu= 0 i.e. - ( - r - + y " = 0 (1-2)

dx Qy oz

In the case of any turbulent flow simulation, the averaged-Reynolds equations were

applied.

The objectives of the theoretical studies were as follows:

1. Sim ulate the blood flow at various stages of atherogenesis

2. Simulate lam inar and turbu lent flow in an axisymmetric and asymmetric steno

sis

3. Simulate the blood flow in a stenosed caro tid arte ry bifurcation.



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CHAPTER 1. INTRO DU CTION 11

1.4 R eview o f Literature

1.4.1 Lo ng-seg m ent artery

The steady and unsteady flow downstream of a stenosis can be laminar, transitioned

or turbulent. The flow Reynolds number (UmD fu ) and the stenosis percentage [1

(d/D )2)] axe the main parameters th at determine which flow regime is present. In

this study, Um was the cross-sectional mean velocity in the unobstructed portion of

the artery, D the unobstructed diameter of the artery, and v the kinematic viscosity

of the fluid. T he param eter dwas the diameter of the stenosis at its narrowest point.

If the flow was pulsa tile, the Womersley number a = where u>is the essential

angular frequency of the velocity waveform, was also important, as it measures the

relative importance of unsteady to viscous influences. The effects of pulsatility had

to be considered in every method prior to applying any steady flow results to an in

vivosituation.

Steady laminar flow has been examined by many authors, both theoretically and

experimentally, but not many have investigated pulsatile flow. Numerical solutions

using turbulence models may be used where the flow becomes turbule nt. These models

involve two equations, one describing the turbulent kinetic energy k and the other

describing the dissipation rate e or turbulent frequency u;. These equations are used

to model the turbulen t shear stress term s in the time-averaged momentum equation.

The standard turbulent models have been designed for high Reynolds numbers and

can not be used in low Reynolds num ber flow simulations such as in the case of arterial

blood flow [138].

Experimental analysis

Back and Roschke [6] studied flow patterns through an 86% axisymmetric stenosis.

They considered three distinct regimes of flow reat tachment. In the first regime, at

low Reynolds numbers, the reattachment length was governed by the growth of the

laminar shear layer, and the reattachment point moved downstream with increasing

flow rate . In the second regime, while simultaneously developing instabilities in the



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CHAPTER I. IN TRODUCTIO N 12

shear layer that corresponded with a critical Reynolds number of 90, the reattach

ment point moved back towards the stenosis. In the third regime, which occurred

beyond a flow Reynolds number of 325, the shear layer was highly disturb ed and

the reattachment point was near the stenosis, moving very slowly downstream with

increasing flow rate.

Young and Tsai [142] studied some flow characteristics in arterial stenoses models

under steady flow conditions. In the case of steady flow these experiments yielded, a

description of the extent of separated flow regions and a measure of pressure losses

across the constriction. The nature of flow distal to the partial occlusions (lami

nar , transit ional or turbulen t) was also discussed. Moreover, the observations that

streamlines departed from laminar behavior for relatively low Reynolds numbers were

of interest in their work.

Flow disturbance distal to modeled stenoses under steady and pulsatile flow was

studied by Cassanova and Giddens [16] who used Reynolds numbers of 318 to 2540

and a pulsatile flow frequency param eter of 15. Their results indicated th at the

more abrupt and sharp-edged the stenosis the grea ter the flow disturbance at a given

Reynolds number when compared to the smoothly contoured configuration. The

greater the degree of blockage, the greater the disorder created in the distal field.

The effect of the distal wall interactions they obtained at low Reynolds numberswas that the wadi retarded the development of vortices, whereas at high Reynolds

numbers, it reduced the energy transferred to the vortex ring struc ture and increased

the rate at which the energy was transferred into a random distribution of eddy sizes.

Finally, the pulsatility destabilized the flow, which was clearly evident in the energy

spectra results.

Yongchareon and Young [140] investigated the initiation of turbulence in models

of arterial stenosis. Three severely constricted models (89% area reduction) were used

with Reynolds numbers ranging from 200 to 1000. From this work it can be said thatthe critical Reynolds number for the development of turbulence under pulsatile flow

through a stenotic obstruction depended on numerous factors including the shape and

size of the stenosis and the natu re of the base flow waveform. Turbulence developed at

Reynolds numbers well below the critical value for an obstructed tube. Also, for the



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CHAPTER 1. INTRODU CTION 13

severe stenoses the critical Reynolds number decreased as th e stenosis shape became

more abrupt. The critical Reynolds number varied with the frequency param eter with

the flow first becoming less stable and then more stable as the frequency parameter

was increased. The critical Reynolds number decreased with the axea ratio (as a first

approximation) decreasing in direct proportion to this ratio.

Clark [19] studied the propagation of turbulence produced by a stenosis during a

pulse cycle. The range of Reynolds num ber corresponding to th e biological conditions

and stenosis percentage were 1140 4170 and 89%, respectively. He observed tha t

with the onset of flow, a lam inar sheax layer was formed, and obtained that turbulence

produced by a given flow pulse was always associated with particles th at had been

upstream of the stenosis prior to the pulse. Under a post-stenotic flow turbulence

production occurred only in the shear layer; th e process represen ted th e ex trac tion

of energy from the mean flow by the action of Reynolds stresses. Vortex stretch ing

due to the non-uniformity of the flow resulted in a cascade of energy from the larger

energy-carrying eddies through progressively smaller eddies. They eventually reached

the size where turbulence dissipation to heat occu rred th rough the action of viscosity.

Poststenotic turbulence was not isotropic, but rather moving down the energy cas

cade. He explained that beyond the reattachm ent position (end of the shear layer)

turbulence production ceased, followed by progressive decay as dissipation continued.The result was th at in the region of flow beyond the turbulence produced by a single

pulse, there was not sufficient tim e during the pu lse for boundary layer disturbances

to amplify and propagate across the tube section. Eddies with a scale of the aortic

diameter would probably require more time to be damped than was available in dias

tole at normal hea rt ra tes. The viscous diffusion distance (z /r)1 2 was approxim ately

1.3 mm at a ra te of 70 beat/m in. Thus, during the next pulse, these residual dis

turbances m ay amplify, particularly during the deceleration phase when there was an

unfavorable pressure gradient.

The flow patterns under steady flow through axisymmetric stenoses at moderate

Reynolds numbers (500 < Re


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CHAPTERl . INTRODUCTION 14

Stenoses of 25, 50 and 75% area reduction were studied . Their results showed tha t

flow disturbances of discrete oscillation frequency may be more valuable than tur

bulence as an indicator of early stenosis development. In addition, despite the fact

that post-stenotic turbulence existed for the greater degrees of stenosis and Reynolds

numbers, the resulting wall shear stresses were only three to four times greater than

the Poiseuille value and were considerably less than the wall shear stress within the

stenosis itself.

Ahmed and Giddens [1] reported flow disturbance measurements through a con

stricted tube at moderate Reynolds numbers und er steady flow. The upstream

Reynolds numbers ranged between 500 and 2000. Depending on the degree of stenosis

and the Reynolds number, the flow field contained discrete oscillation disturbances

of a frequency, of a turbu len t nature, or both. For mild stenoses (50% area reduc

tion), the intensity of flow disturbances was relatively low until the Reynolds number

exceeded 1000. The authors verified the following factors. Flow separation and asso

ciated intense turbulence were expected to occur in the immediate poststenotic field of

locally constricted arteries prior to the stenoses becoming flow-restricting or hemo-

dynamically significant. An area of relatively constant centerline velocity occurred

in the poststenotic field. This velocity rapidly decreases when trans ition to turbu lence

occurred. In the area immediately downstream of the constriction, the mean velocityprofiles exhibited a jet-like response with large velocity gradients. Flow disturbances

originated in this shear layer for steady upstream flow conditions.

Ahmed and Giddens [2] studied the pulsatile flow field distal to axisymmetric

constrictions in a straight tube using laser Doppler anemometry. The upstream cen

terline velocity waveform was sinusoidal, w ith a Womersley number of 7.5 and a mean

Reynolds num ber of 600. Stenosis models of 25, 50 and 75% area reduction were used.

The authors found that a permanent area of poststenotic flow separation did not ex

ist, even for the severest constriction, in contrast to results for steady flow. Values of

wall shear stress were greatest near the throat of the constriction and were relatively

low in the poststenotic region, including that of the most intense flow disturbance.

In addition, turbulence was found only in the 75% stenosis model and was created

only during one segment of the cycle. According to their results, in the Reynolds



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CHA PTER 1. INTRODU CTION 15

number typical of that found in the human carotid artery, turbulence did not occur

until the stenosis exceeded 50% in area reduction. For the 25% area reduction th e

flow was stable throughout with no notable flow disturbances. For the 50% stenosis,

an organized disturbance was associated with the systolic acceleration phase ; how

ever, no turbulence was detected, however. For most of the cycle in th e 75% stenosis

areas of intense turbulence were observed, however, the authors mentioned that more

moderate constrictions may not, in fact, create a turbu len t flow. They added that

the effect of pulsatility was to disturb the distal flow somewhat more than steady

flow. The permanen t recirculation area of steady flow did not exist under pulsatile

conditions.

An in vivo demonstration of flow recirculation and turbulence downstream of

graded stenoses was performed by Hutchison et al. [53], who found that the devel

opment of post-stenotic turbulence was shown to follow the development of vorticity

in the shear layer between the je t and the recirculation zone. Also, they showed (in

lower Reynolds numbers and degree of stenosis) that tru e turbulence did not develop,

but ra th er a coherent disturbance (vorticity) was manifested by discrete frequency

velocity oscillations.

Siouffi et al. [118] discovered a major difference between the pulsatile flow and

steady flow recirculation zones. Under pulsatile flow, th e recirculation was not isola tedfrom other parts of the flow. The fluid in this zone was swept downstream with

each cycle. Under steady flow, fluid elements remained in the recirculation zone for

significantly longer periods of time.

Numerical analysis

Lee and Fung [69] were the first to use numerical approach to the problem, simulated

blood flow through an axisymmetric constriction for Reynolds numbers up to 25.

Their calculations was done for a low Reynolds number and there was no significantresults, physiologically.

Deshpande et al. [23] obtained results for much higher Reynolds numbers. For a

constriction with a 56% area reduction, the Reynolds number results were as high as

2000. These numerical results concurred reasonably well with experimental results



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CHA PTER 1. INTRODUCTION 16

(Young and Tsai [142]) in terms of pressure drop, and separation and reattachment

points. Disagreements were a ttributed to difficulties in measuring the separation and

reattachment points, and to the increasingly three-dimensional nature of the flow as

the Reynolds number increased.

Tu et al. [134] studied pulsatile flow through arter ial stenoses using the finite

element simulation method. According to their results, pulsatile blood flow through

a stenosis demonstrated that the unsteadiness effect played a very important role on

measured parameters such as wall shear stress and recirculation length. They found

that the flow pattern changed remarkably with time; the pressure and wall shear

stress also showed time dependence. Also, the pressure drop a t the stenosis increased

with an increase in the Womersley param eter, and the sheax stress on the wall showedthat 1), the maximum value coincided with the maximum flow rate, and 2), the peals

value was slightly larger for a smaller Womersley number.

Rosenfeld [110]numerically studied pulsatile flow distal to a constriction. He im

posed a pulsating incoming flow with a non-vanishing mean at the entrance, and

investigated the flow field for a wide range of Reynolds and Strouhal 8 numbers

(45 < Re


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CH APTE R 1. INTRODUCTION 17

occlusive lesion a t the distal end-to-side anastomosis remains a major cause of pros

the tic bypass graft failure. Thus for, accura te characterization of anastomotic intim al

thickening is lacking, however it has been widely accepted that local hemodynamic

factors, particularly low and oscillatory wall shear, have been shown to correlate with

regions of intimal thickening.

Figure 1.5: Geometric configurations for bypass graft.

Experimental analysis

Detailed experimental studies of flow in physiological geometries are relatively rare.

It is generally very difficult to obtain in vivod at a on the instantaneous flow field, and

most in vitro studies have used injected dye or particle tracking techniques, which

prec luded th e ex trac tion of useful quanti ta tive information.

Ojha et al. [87] reported the results of a 3-D experimental study of flow in an

end-to-side anastomosis model. Evidence was presented that suggested a correlation

existed between low and fluctuating wall shear stress and intimal hyperplasia.

They also studied the flow pat tern s in a side-to-end anastomosis [89]. T heir repor ts

revealed that the intimal hyperplasia at the distal side-to-end anastomosis may be

promoted by low wall shear stress at th e toe and heel, and probably by high shea r



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C H A P T E R !. INTRODUCTION 18

stress or shear stress gradients on the bed. In addition, they found that th e shear

stress in the vicinity of the toe for the distal anastomosis wasmuch smaller than that

observed at the proximal junction where the peak instantaneous value was found to

be roughly four tim es the normal value.

Anastomotic intimal hyperplasia was studied by Bassiouny et al. [8]. They con

cluded th at there axe two different types of anastomotic intimal thickening : su ture

line thickening related to compliance mismatch and focal geometric deformations that

result in complex secondary flow patte rns . Arterial floor intimal thickening far from

the suture line developed in regions of flow reattachment and low oscillating shear.

White et al. [137] stud ied the flow pat tern s in two models of end-to-side vascular

graft anastomoses. Effects of pulsatility, flow division, Reynolds number, and hoodlength were considered. They found that strong, three-dimensional helical flow pa t

terns which formed in the anastomotic junction were prominent features of the flow

fields. Regions of low wall shear, oscillatory wall shear, and long part icle residence

tim e were identified from the flow visualization experiments. Comparing with the

limited qualitative da ta available on in timal thickening in vascular graft anastomoses

suggests a connection between localization of vascular intimal thickening and those

surfaces experiencing low shear aswell as long particle residence time.

Num erical analysis

Lu et al. [73] simulated steady flow in 2-D end-to-side anastomosis for a variety of

graft angles, and assumed some outflow through the proximal host artery, based on

velocity profiles measured via Laser Doppler Anemometry (LDA). High shear stress

was recorded at the heel and at the bed, and low wall shear stress near the toe in all

cases.

Sottiurai et al. [119] reported tha t for the end-to-side configuration anastomotic

intima l hyperplasia occurred preferentially at the heel and toe of the graft and on the

bed of th e host vessel. It was postulated that hemodynamic factors such as unphys-

iological flow structures and/or wall shear stresses promoted intimal hyperplasia in

the end-to-side anastomosis geometry. A thorough understanding of flow patterns in

anastomotic geometries was therefore necessary to determine which flow features (if



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CH APTE R 1. INTRODU CTION 19

any) influenced the development of intimal hyperplasia.

Pietrabissa et al. [101] also considered steady flow through a bypass implanted

around a stenosis in one branch of a symmetric bifurcation, and found that recircu

lation zones were minimized with a 45 rather than 90 graft angle.

Henry and Collins [50] investigated unsteady sinusoidal flow in a 3-D end-to-side

anastomosis model at various graft angles and flow ratios, noting reduced recirculation

for geometries with shallow graft angles. No wall shear stress data were provided,

and th e authors conceded tha t their meshes were most likely under-resolved. No

definitive association was noted between wall shear stress behavior and sites of intimal

hyperplasia, as both high and low wall shear stress regions were observed at these

sites throughout the flow cycle.Using a numerical simulation of flow in a two-dimensional end-to-side anastomosis

model Steinman et al. [122] found that intim al hyperplasia may be correlated to wall

shear stress over the cardiac-cycle, high wall shear stress gradients, or a combination

of the two.

Fei et al. [30] investigated flow pa tte rns in distal end-to-side anastomosis using a

steady-flow, three-dimensional numerical model. The results, presented for the center-

line plane only, indicated that areas associated with intimal hyperplasia experienced

flow separation and stagnation, and therefore subsequent low shear stress.Th e effect of wall distensibility for unsteady flow in a two-dimensional end-to-side

anastomosis was studied numerically by Steinman et al. [124]. Flow was simulated in

a 2-D end-to-side anastomosis model with a 45 degree graft angle, equal graft/host

artery channel heights, and full occlusion at the proximal end of the host artery.

Using a finite element method in th eir simulation th e total execution times for rigid

simulations were 3 to 4 hours on a Sim SPARC2 workstation, and 7 to 8 hours

for distensible-walled simulations. They concluded that it was more im po rtant to

accurately m odel the geometry and flow waveform than it was to include distensibility,

such as exte nt of secondary flows, par ticle residence times, etc.

Zhang et al. [148] used a solver for the 3-D unsteady incompressible Navier-S tokes

equations to sim ulate blood flow in an end-to-side anastomosis. Fully-developed ve

locity profiles were specified at the graft inlet, a traction-free condition was imposed



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on the artery outlet, and the proximal portion of the artery was assumed to be fully

occluded. They observed maximum shear stress on the bed of the host artery down

stream of the stagnation point. Low shear stress was seen on the bed of the host

artery upstream of the site of maximum shear (in sinus of the hood) distal to the

graft toe. They concluded th at the development of intimal hyperplasia correlated

with low mean shear stress in conjunction with a highly oscillatory stress pattern.

Numerical analysis of steady flow in an aorto-coronary bypass 3-D model was

studied by Inzoli et al. [55]. Their results confirmed the sensitiv ity of flow behavior

to the models geometrical parameters, and stressed the importance of reproducing

the anastomosis junction as closely as possible in order to evaluate the effective shear

stress distribution.

Hofer et al. [51] studied wall mechanics and fluid dynamics in end-to-side anasto

moses and their correlation to intimal hyperplasia. The numerical results on the flow

pattern s revealed strongly skewed axial velocity profiles downstream of the junction,

a large secondary motion, as well as flow separation and recirculation on the artery

floor opposite the junction and at the in ner wall downstream of the toe. In these

regions, a correlation was observed between the time-averaged fluid wall shear stress

and intimal thickening found in th e animal experimental model.

1.4 .3 Carotid A rtery B ifurcat ion (CA B)

A human artery that has received much attention is the carotid artery bifurcation

(CAB), in which atherosclerotic lesions are frequently observed. Atherosclerosis in

this region is the major cause of transient ischemic attacks which may result from a

reduction in blood flow due to narrowing of the arterial lumen, but are most often

caused by thrombus 9 or emboli 10. By encouraging the aggregation of plate lets,

atherosclerotic plaques promote the development of emboli or thrombi. Ischemic

attacks are more common in men, and the prevalence increases with age, meaning

9A thrombus is a blood clot, an aggregation of platelets and fibrin formed in response either toan atherosclerotic lesion or to vessel injury.

10An emboli is any traveling obstruction, commonly a platelet aggregate, dislodged from a p laquebut also potentially a bubble of gas transported through the vasculature until it lodges in and blocksa vessel.



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CHAPTER L INTRODU CTION 21

that around age 60 the gender difference begin to disappear. In one study of 909

men and women aged 40 to 79 years, atherosclerotic lesions were detected in 47.8%

of the men and 36.3% of the women [61]. Another study of 1350 men and women

aged 18 to 99 years reported the global prevalence rate of carotid atherosclerosis to

be approximately 25%. However, the rates were much lower for those aged 39 or less

than for those older than 39 [103]. Hemodynamic phenomena are considered to be

among the possible initiating factors in atherogenesis. A brief review of investigations

this phenomena are given as follows.

Experimen tal analysis

Bharadvaj et al. [10] carried out an experimental study on steady flow in the CAB

and found complex axial and secondary flow patterns. Comparing the flow field in the

model bifurcation with the sites of atherosclerotic involvement their results indicated

that areas of predilection for disease coincided with regions of flow or reversed axial

flow and low or oscillatory shear stress.

In his study, Olson [91] concluded that the flow phenomena occurring in the

daughter branches of a symmetrical bifurcation originated mainly from curvature

effects.

Ku and Giddens [66] investigated flow behavior under physiological flow conditions. Hydrogen-bubble visualizations showed tha t during systolic accele ration, a low

shear region was formed at the non-divider side of the carotid sinus 11. This region

extended during the systolic deceleration. At the onset of diastole, a small vortex was

observed at the edge of th e low shear region near the divider wall. The same authors

performed LDA experiments in a three-dimensional model of th e carotid bifurcation.

The flow behavior in mildly stenosed carotid artery bifurcations (less than 25%

area reduction) has been less intensively investigated. Van de Vosse et al. [135] com

puted the velocity field in a two-dimensional model of the carotid artery bifurcation

under pulsatile flow. They compared two-dimensional axial velocity profiles in the

bifurcation. The effect of the geometry variation on the ax ial velocity profiles and

11A dilatation of the proximal portion of the internal carotid or distal portion of the commoncarotid artery, containing in its wall, prereceptors that are stimulated by changes in blood pressure.



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the wall shear stress appeared to be relatively small.

Palmen [92] et al. studied hydrogen-bubble visualization experiments in Plexi

glass models of a non-stenosed and a 25% stenosed carotid artery bifurcation. The

experiments showed that vortex formation occurred symmetrically at the onset of

diastole. This vortex formation was found in a shear layer located a t the interface of

an axea with low shear rates at the flow divider wall. Comparisons of the hydrogen

bubble profiles in th e 0% and 25% stenosed models showed th at th e stenosis only

slightly changed th e global flow phenomena. However, striking differences were found

in the stability of the shear layer. In the same study, the effect of the shape of the

flow pulse was also investigated. The shape of the flow pulse had a significant impact

on the velocity field.Palm en [93] applied LDA to show th at significant differences existed between th e

flow field in a non-stenosed carotid artery bifurcation and a mildly stenosed one. The

stability of the shear layer and the area of flow reversal were affected by the stenosis

in the sinus. Analysis of the velocity signals in the temporal and frequency domain

provided promising parameters to characterize the presence of the stenosis.

Num erical analysis

Numerical studies of steady and pulsatile models were carried out in recent yearsin hea lthy carotid arteries. Siouffl et al. [118] studied the effect of unstead iness on

the flow through bifurcations with rectangular cross sections. Com paring between

steady and unsteady velocity profiles by using flow visualization, the specific effect

of unsteadiness was brought to

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