[A Margaritis] Spoudastiki Ergasia (Report)

Θεσσαλονίκη, 2016

Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης

Πολυτεχνική Σχολή

Τμήμα Μηχανολόγων Μηχανικών Εργαστήριο Μηχανικής Ρευστών και Στροβιλομηχανών

Αριθμητική Ανάλυση της Ροής Αίματος

στην Ανθρώπινη Καρωτιδική Διακλάδωση

Σπουδαστική Εργασία

Αθανάσιος ΜΑΡΓΑΡΙΤΗΣ

[email protected]

AEM: 5516

Επιβλέπων: Ανέστης Ι. ΚΑΛΦΑΣ, Αναπληρωτής Καθηγητής

Θεσσαλονίκη, 2016

Aristotle University of Thessaloniki

Faculty of Engineering

School of Mechanical Engineering Laboratory of Fluid Mechanics and Turbomachinery

Numerical Analysis of Blood Flow through

the Human Carotid Artery Bifurcation

Engineering Diploma First Cycle Thesis

By Athanasios MARGARITIS

[email protected]

AEM: 5516

Supervisor: Anestis I. KALFAS, Associate Professor

ΣΥΝΟΨΗ Στην παρούσα Σπουδαστική Εργασία μελετάται η ροή του αίματος διαμέσου ανθρώπινων

καρωτιδικών διακλαδώσεων, χρησιμοποιώντας αριθμητικές προσομοιώσεις και εμπορικό

λογισμικό υπολογιστικής ρευστομηχανικής. Το πεδίο της αιμοδυναμικής, της δυναμικής

συμπεριφοράς ροών αίματος διαμέσου του αρτηριακού δέντρου, έχει αποτελέσει ακμάζον

αντικείμενο μελέτης τις τελευταίες δεκαετίες, υποσχόμενο επαναστατικές μεθόδους πρόγνωσης,

διάγνωσης και θεραπείας παθολογικών καταστάσεων του καρδιαγγειακού συστήματος. Σε αυτή

την μελέτη, παρουσιάζεται μια ανασκόπηση σημαντικών προηγούμενων εργασιών στον τομέα,

επισημαίνοντας κενά και αντιφάσεις μεταξύ των δημοσιευμένων αποτελεσμάτων, ενώ

πραγματοποιείται η προσπάθεια να συμπληρωθούν προηγούμενες μελέτες, επικυρώνοντας τα

αποτελέσματα και τα μοντέλα τους και προτείνοντας κατευθύνσεις για μελλοντική έρευνα.

Οι μέθοδοι που χρησιμοποιήθηκαν στην παρούσα μελέτη περιλαμβάνουν τις πλέον σύγχρονες

τεχνικές στον τομέα. Χρησιμοποιήθηκε απεικόνιση μέσω μαγνητικής τομογραφίας για την λήψη

τομών των αρτηριακών γεωμετριών από 3 υγιείς εθελοντές, οι οποίες έπειτα

ανακατασκευάστηκαν ψηφιακά ώστε να παραχθούν υπολογιστικά πλέγματα στα οποία

εφαρμόστηκαν αριθμητικές μέθοδοι υπολογιστικής ρευστομηχανικής. Η χρονική και η χωρική

διακριτοποίηση της λύσης επιλέχθηκαν με βάση προηγούμενες έρευνες και τρέχοντα

αποτελέσματα δοκιμών. Εξετάστηκαν διαφορετικά μοντέλα ιξώδους του αίματος, ώστε να

εκτιμηθεί η επίδρασή τους στην υπολογιστική ακρίβεια, ενώ μελετήθηκαν διαφορετικές

αρτηριακές γεωμετρίες και συγκρίθηκαν τα αποτελέσματα.

Τα αποτελέσματα που παρουσιάζονται σε αυτή την έρευνα εστιάζουν στις επιπτώσεις του

μοντέλου ιξώδους του αίματος, επιβεβαιώνοντας ότι το Νευτώνειο μοντέλο είναι μια αποδεκτή

υπόθεση για την ροή αίματος στην καρωτιδική διακλάδωση, ενώ δίνεται επίσης έμφαση και στην

ποιοτική αξιολόγηση συσχετίσεων μεταξύ περιοχών χαμηλών διατμητικών τάσεων τοιχώματος

και περιοχών έντονα ελικοειδών, δευτερογενών ροών ή ανακυκλοφορίας. Οι παραπάνω

αποτελούν παράγοντες ευνοϊκούς για την ανάπτυξη αθηρωματικής πλάκας, ως αποτέλεσμα της

παρατεταμένης παραμονής Λιποπρωτεϊνών σε αυτές τις περιοχές. Τα αποτελέσματα των

προσομοιώσεων βρίσκονται σε απόλυτη συμφωνία με την πλειονότητα των σχετικών μελετών,

επιβεβαιώνοντας φαινόμενα που παρατηρήθηκαν από προηγούμενους ερευνητές και

υπολογίζοντας τιμές σε φυσιολογικό εύρος για όλες τις εξεταζόμενες μεταβλητές.

Η βελτίωση της παρούσας εργασίας είναι εφικτή αίροντας τους βασικούς περιορισμούς της, οι

οποίοι έγκεινται στην εξαίρεση της ελαστικότητας και της ενδοτικότητας των αρτηριακών

τοιχωμάτων και της πολυφασικής φύσης του αίματος από το μοντέλο αριθμητικής

προσομοίωσης. Μία μακράν πιο ρεαλιστική προσέγγιση των συνθηκών στο εσωτερικό του

οργανισμού είναι δυνατή με την συμπερίληψη αυτών των παραμέτρων στο αριθμητικό μοντέλο.

Λέξεις-Κλειδιά: αιμοδυναμική, καρωτιδική διακλάδωση, υπολογιστική ρευστομηχανική, CFD

ABSTRACT In this Semester’s thesis, the flow of blood through human Carotid Artery Bifurcations has been

studied, using numerical simulations and commercial codes for Computational Fluid Dynamics. The

field of hemodynamics, the dynamics of blood flows through the arterial tree, has been a flourishing

subject for research over the last few decades, promising to provide revolutionary methods for

successful prognosis, diagnosis and treatment of pathological conditions affecting the cardiovascular

system. In this study, a review of important previous work done in the field has been presented,

noting gaps and contradictions in their results, while an attempt is made to complement previous

studies, validate their results and models and propose directions for future work.

Methodologies used in this study include the current state-of-the-art techniques in the field.

Magnetic Resonance Imaging has been used to create 2D cross-sectional images of the arterial

geometries from 3 healthy subjects, which were then digitally reconstructed to produce

computational meshes on which the numerical simulations (Computational Fluid Dynamics – CFD)

were applied. Time and spatial discretization of the solution have been carefully selected, based on

previous research and current test-case results. Different models for blood’s viscosity have been

examined to evaluate their effect on simulation accuracy, while various arterial geometries have

studied and the results have been compared.

Results presented in this study focus on the effects of blood’s viscosity model, confirming that the

Newtonian model is a reasonable assumption for blood flow through the carotid artery, while

emphasis is also placed upon qualitatively assessing correlations of low wall shear stress regions with

highly helical, secondary or recirculating flows. These are factors that favour the development of

atheromatous plaque, as a result of prolonged residence of Lipoproteins in these regions. Simulation

results are in perfect agreement with the majority of previous studies, confirming phenomena

reported by previous researchers and calculating physiological values for the variables examined.

Improvement of the present work is possible by deducting its main limitations, which are the

exclusion of the elasticity and the compliance of the arterial walls as well as the multiphase nature of

blood from the numerical simulation model. A far more realistic approach of the in-vivo conditions

is possible by including these effects into the simulation model.

Keywords: hemodynamics, carotid artery, bifurcation, blood flow, computational fluid dynamics, CFD

ACKNOWLEDGMENTS The completion of this thesis would not have been possible without the help, guidance and advice

of a few specific people, whose support has been invaluable during the progress of this research.

First of all, I have to acknowledge the contribution of my professor Anestis Kalfas, whose support

and teaching have been invaluable through all the stages of this work, by continually motivating me

and suggesting additions and ways to improve it, without hesitating to even sacrifice his own

personal time in order to instruct me.

Furthermore, I have to express my deepest gratitude to Panagiotis Kalozoumis who has been

exceedingly helpful, sharing his expertise and insight with me, and whose Diploma thesis provided

the basis for this paper to build upon. I am also thankful for the assistance provided by Chrysa

Kottarakou, whose Diploma thesis and previous experience has been a great source of help for the

present thesis.

Additionally, I have to extend my thanks to all my university professors, whose assistance and

guidance have provided me with all the resources needed in order to complete this study.

Finally, this accomplishment would not have been feasible without the encouragement and support

of my family and friends, whose advice and occasional suggestions always provided a different,

refreshing scope to this research, and for that I owe them all my utmost gratitude.

Contents Numerzical Analysis of Blood Flow through the Human Carotid Artery Bifurcation

1

CONTENTS INTRODUCTION ..................................................................................................................................................................... 3

THEORETICAL BACKGROUND ......................................................................................................................................... 5

A.1 THEORY OF FLUID MECHANICS ............................................................................................................................................................................... 5

A.1.1 Theoretical Background of Fluid Mechanics ..................................................................................................................................... 5

A.1.2 Computational Fluid Dynamics (CFD) and Applications .............................................................................................................. 7

A.2 HEMODYNAMICS AND BLOOD VESSELS .................................................................................................................................................................. 8

A.2.1 Physiology and Properties of Blood ..................................................................................................................................................... 8

A.2.2 Biochemical Composition and Properties of Blood Vessels ....................................................................................................... 10

A.3 ARTERIAL DISEASES ................................................................................................................................................................................................ 12

A.4 MEASUREMENT AND IMAGING METHODS ............................................................................................................................................................ 15

A.4.1 Invasive Methods ...................................................................................................................................................................................... 15

A.4.2 Non-Invasive Measurement Methods ............................................................................................................................................... 15

A.4.3 Non-Invasive Geometry Reconstruction Methods ......................................................................................................................... 15

A.5 CAROTID ARTERY BIFURCATION (CAB) ................................................................................................................................................................ 16

A.6 CARDIAC CYCLE AND HEARTBEAT ......................................................................................................................................................................... 16

A.7 WAVE PROPAGATION............................................................................................................................................................................................. 17

A.7.1 Wave Speed ............................................................................................................................................................................................... 17

A.7.2 Wave Travel and Reflections ................................................................................................................................................................ 17

A.7.3 Arterial Input Impedance ...................................................................................................................................................................... 17

A.7.4 Windkessel models .................................................................................................................................................................................. 17

LITERATURE REVIEW ...................................................................................................................................................... 19

B.1 PREVIOUS CFD APPLICATIONS TO BIOLOGICAL FLOWS ....................................................................................................................................... 19

B.1.1 CFD Applications to Carotid Artery Bifurcations (CAB) ............................................................................................................... 19

B.1.2 Other Research on Blood Flows through the Arterial Tree ......................................................................................................... 26

B.2 GENERAL OBSERVATIONS ....................................................................................................................................................................................... 27

METHODS ........................................................................................................................................................................ 29

C.1 GEOMETRICAL RECONSTRUCTION OF CAB MODELS ........................................................................................................................................... 29

C.2 COMPUTATIONAL MESH CONSTRUCTION ............................................................................................................................................................ 29

C.2.1 Carotid Artery Bifurcation Models – Geometry and Shape ........................................................................................................ 30

C.3 SOLUTION ............................................................................................................................................................................................................... 37

C.3.1 Solution Methods and Equations ........................................................................................................................................................ 37

C.3.2 Boundary Conditions .............................................................................................................................................................................. 37

C.4 ACCURACY .............................................................................................................................................................................................................. 39

C.4.1 Mesh Independence................................................................................................................................................................................. 39

C.4.2 Time-Step Independence ....................................................................................................................................................................... 43

C.4.3 Periodicity ................................................................................................................................................................................................... 43

RESULTS ........................................................................................................................................................................... 45

D.1 IMPORTANCE OF THE VISCOSITY MODEL .............................................................................................................................................................. 45

D.2 FLOW FIELD AND WALL SHEAR STRESS DISTRIBUTION ........................................................................................................................................ 46

DISCUSSION ..................................................................................................................................................................... 51

E.1 IMPORTANCE OF THE VISCOSITY MODEL ............................................................................................................................................................... 51

E.2 WALL SHEAR STRESS DISTRIBUTION ...................................................................................................................................................................... 51

E.3 HELICAL SECONDARY FLOWS AND FLOW VELOCITY PROFILES ............................................................................................................................. 52

CONCLUSIONS ................................................................................................................................................................. 53

F.1 LIMITATIONS ............................................................................................................................................................................................................ 54

F.2 SUGGESTIONS FOR FURTHER RESEARCH ................................................................................................................................................................ 54

REFERENCES ......................................................................................................................................................................... 55

Athanasios Margaritis LFMT Aristotle University of Thessaloniki

2

Introduction Numerzical Analysis of Blood Flow through the Human Carotid Artery Bifurcation

3

INTRODUCTION The topic of this paper lies where the subjects of Fluid Mechanics, especially Computational Fluid

Dynamics (CFD), Bioengineering and Hemodynamics overlap each other. To date, atherosclerotic

diseases remain one of the main causes of mortality and morbidity in the developed world, while

one of the most suitable ways to study such flows is by applying CFD methodologies, since

measurement data are difficult to obtain. The author’s interest in this topic, i.e. blood flow through

the carotid artery bifurcation, was raised after studying similar research, regarding simulation of

blood flow through the arterial tree in order to predict and evaluate the risk for cardiovascular

diseases, a growing field in Biomechanical Engineering. The supervisor for this paper is Anestis Kalfas,

Associate Professor at the School of Mechanical Engineering of the Faculty of Engineering, at the

Aristotle University of Thessaloniki.

The purpose of the present study is to explore applications of CFD simulations in hemodynamics,

review previous work done in the field and assess the current state of research. Additionally, this

research aims to compare and evaluate different blood viscosity models, validate findings with

previously reported results in relevant literature and ultimately examine and assess the implications

of this research to clinical applications for cardiovascular disease prognosis, diagnosis and treatment.

In order to complete this research, several stages were required. Firstly, digitally reconstructed

Carotid Artery Bifurcations were taken from previous work done by Kalozoumis (2009) and were then

imported to ANSA software by BETA CAE Systems SA in order to create the computational grid

required for the following simulations. Afterwards, time- and space-varying functions for flow

velocity and pressure at the inlet and outlet boundaries, respectively, were created in MATLAB and

then coded in C in order to simulate the cardiac cycle, based on values from previous research. The

problem was then solved in ANSYS FLUENT, calculating the flow field for different blood models and

comparing the results.

This thesis consists of six chapters. Chapter A provides the fundamental theoretical background

required in order to proceed to Chapter B, which includes a literature review of numerous previous

important research work in the field of CFD applications for the study of blood flows. Chapter C then

describes the methods used in the present study in order to get the results described in Chapter D.

Finally, Chapter E includes a discussion of these results, concerning their accuracy and restrictions,

while final conclusions are drawn in Chapter F, together with a review of this study’s limitations and

suggestions for further research and improvement of the aforementioned methods.


4

Theoretical Background Numerzical Analysis of Blood Flow through the Human Carotid Artery Bifurcation

5

THEORETICAL BACKGROUND In the first chapter of this paper, a concise background of the fundamental theory required to

implement and interpret the following computational methods is provided. First of all, a theoretical

background for Fluid Mechanics is provided, emphasized on application of CFD for numerical

solutions to complex problems. Then, a few topics on Blood and Blood Vessels physiology are

discussed, followed by some of the most common relevant diseases and their respective methods of

treatment. Finally, a more specific discussion of the shape and importance of the Carotid Artery

Bifurcation is included, followed by a short discussion of measurement methods for blood flows.

A.1 THEORY OF FLUID MECHANICS In the first part of Chapter A, a short introduction to the area of fluid mechanics is provided, only

up to the level required to apply this fundamental knowledge to the complex problem of studying

the flow of blood through human carotid arteries. After this theoretical background, a presentation

of numerical methods and CFD is provided.

A.1.1 Theoretical Background of Fluid Mechanics Fluid Mechanics is the physics of fluid substances (liquids, gases or steam) in comparison to Solid

Mechanics, which is the physics of solid bodies. The fundamental difference between the two is that

when a shear stress is applied on a solid body, it undergoes a finite shear strain, while when a shear

stress is applied on a fluid substance, it undergoes a continuously growing shear strain. In the

following paragraphs, a few elementary topics of Fluid Mechanics are presented. In this paper, fluids

are considered as continuous mass and no reference is made to discrete particles modelling.

A.1.1.1 Viscosity

Viscosity or Absolute Viscosity or Dynamic Viscosity (𝜇) of a fluid is the ratio of shear stress applied

on the fluid over the shear rate this shear stress causes. Shear rate (�̇�) is the incremental difference

in velocity between two adjacent layers of fluid, that is, the Velocity Gradient, and, supposing that

these are moving in parallel lines with an axial velocity (𝑢) and 𝑦 is the normal direction then shear

rate may be expressed as: �̇� =𝜕𝑢

𝜕𝑦.

Shear stress and shear rate are connected through Newton’s law for Newtonian fluids,

𝜏 = 𝜇 ∙ �̇� = 𝜇 ∙𝜕𝑢

𝜕𝑦, and Kinematic Viscosity (𝜈) is defined as Viscosity (𝜇) divided by Density (𝜌), thus

𝜈 =𝜇

𝜌. Substances that satisfy Newton’s law with a constant viscosity, independent of shear stress or

shear rate, where shear stress and shear rate are linearly related are called Newtonian. However, there

are other fluids whose viscosity is a function of shear rate and/or time, called Non-Newtonian fluids.

There are different types of Non-Newtonian fluids:

o Shear-thinning fluids, for which the apparent viscosity decreases with increasing shear rate.

o Shear-thickening fluids¸ for which the apparent viscosity increases with increasing shear rate.

o Bingham plastics, which behave as solids up to a certain level of shear stress (yield shear stress)

and flow as fluids for shear stresses of greater value.

o Thixotropic and Rheopectic liquids¸ for which the apparent viscosity decreases and increases

respectively with time while undergoing shear stress.

Viscosity is a function of temperature for all fluids, and also a function of pressure for gases.


6

A.1.1.2 Conservation Equations

This paragraph includes a small summary of Fluid Mechanics equations used in the following

calculations, providing the background for the following discussions. First of all, the flow is modeled

as three-dimensional, since there are complex phenomena which cannot be described using 1D or

2D axisymmetric modelling as can be done for flows through a plain tube.

In general, flows may be modeled as steady or unsteady. The difference is that for steady flows, all

time derivatives are considered equal to zero, i.e. 𝜕

𝜕𝑡= 0, which means that flow properties, such as

velocity and pressure, do not vary with time at any point in the flow field. Blood flow through large

arteries close to the heart is unsteady, since there is periodicity due to the cardiac cycle. This is the

case for Carotid Arteries and all modelling in this work is using the equations for unsteady flows.

Real flows are compressible, which means that density is not constant through the flow field, but

is a function of flow velocity. However, most low speed flows, for velocities up to one third of speed

of sound, may be modeled as incompressible flows, with constant density through the flow field with

sufficient accuracy. This simplifies the equations which need to be solved. Such is the case for almost

all biological flows, hence for blood flows through Carotid Bifurcation Arteries.

The most important equations in Fluid Mechanics are the Continuity Equation, which is an

expression of the conservation of mass, and the Momentum Equation, also called Navier-Stokes

Equations, which is a result of Newton’s second law (Munson, et al., 2013).

o Continuity Equation: The Continuity Equation is an expression of conservation of mass for

fluids. In its general form, it can be written as 𝜕𝜌

𝜕𝑡+ ∇ ∙ (𝜌�⃗� ) = 0

while for incompressible flows it is simplified to

∇ ∙ �⃗� = 0

where

𝜌, is the density of the fluid,

�⃗� = 𝑢 ∙ 𝑒𝑥⃗⃗⃗⃗ + 𝑣 ∙ 𝑒𝑦⃗⃗⃗⃗ + 𝑤 ∙ 𝑒𝑧⃗⃗ ⃗, is the vector of velocity of the fluid,

∇= 𝑒𝑥⃗⃗⃗⃗ 𝜕

𝜕𝑥+ 𝑒𝑦⃗⃗⃗⃗

𝜕

𝜕𝑦+ 𝑒𝑧⃗⃗ ⃗

𝜕

𝜕𝑧, in Cartesian coordinates, and

𝑒𝑖⃗⃗⃗ , 𝑓𝑜𝑟 𝑖 = {𝑥, 𝑦, 𝑧}, is the unit vector in the direction 𝑖.

o Momentum (Navier-Stokes) Equations: The Momentum Equations are the main equations

governing fluid flows, as an expression of Newton’s second law, relating velocity and pressure

gradient. For a 3D flow, with no external forces exerted onto the fluid, they are written as

𝜌 (𝜕

𝜕𝑡+ �⃗� ∙ ∇) 𝑢 = −

𝜕𝑝

𝜕𝑥+ 𝜌𝑔𝑥 + 𝜇 (

𝜕2𝑢

𝜕𝑥2+

𝜕2𝑢

𝜕𝑦2+

𝜕2𝑢

𝜕𝑧2)

𝜌 (𝜕

𝜕𝑡+ �⃗� ∙ ∇) 𝑣 = −

𝜕𝑝

𝜕𝑦+ 𝜌𝑔𝑦 + 𝜇 (

𝜕2𝑣

𝜕𝑥2+

𝜕2𝑣

𝜕𝑦2+

𝜕2𝑣

𝜕𝑧2)

𝜌 (𝜕

𝜕𝑡+ �⃗� ∙ ∇)𝑤 = −

𝜕𝑝

𝜕𝑧+ 𝜌𝑔𝑧 + 𝜇 (

𝜕2𝑤

𝜕𝑥2+

𝜕2𝑤

𝜕𝑦2+

𝜕2𝑤

𝜕𝑧2)

where

𝜌 is the density of the fluid.

𝑝 is the pressure of the fluid.

�⃗� = 𝑢 ∙ 𝑒𝑥⃗⃗⃗⃗ + 𝑣 ∙ 𝑒𝑦⃗⃗⃗⃗ + 𝑤 ∙ 𝑒𝑧⃗⃗ ⃗ is the vector of velocity of the fluid.

∇= 𝑒𝑥⃗⃗⃗⃗ 𝜕

𝜕𝑥+ 𝑒𝑦⃗⃗⃗⃗

𝜕

𝜕𝑦+ 𝑒𝑧⃗⃗ ⃗

𝜕

𝜕𝑧 in Cartesian coordinates.

𝑔𝑖 𝑓𝑜𝑟 𝑖 = {𝑥, 𝑦, 𝑧} is the gravitational acceleration in the direction 𝑖.

𝑒𝑖⃗⃗⃗ 𝑓𝑜𝑟 𝑖 = {𝑥, 𝑦, 𝑧} is the unit vector in the direction 𝑖.


7

A.1.1.3 Womersley Parameter for Oscillatory Flow

Assuming a stiff tube and a laminar, sinusoidal flow, the flow velocity profile depends on the

Womersley parameter (𝑎𝑊) which is defined as

𝑎𝑊2 =

(2𝜋𝑓)𝜌𝑟2

𝜇

where 𝑓 is the frequency of the pulse, 𝜌 is the fluid density, 𝜇 is the absolute viscosity and 𝑟 is the

radius of the tube. Womersley parameter is a kind of Reynold’s number, since it indicates the relative

importance of inertia effects over viscous effects. For low values of Womersley parameter (𝑎𝑊 < 3)

the viscous effects dominate and flow velocity profiles are mainly parabolic. For very low values

(𝑎𝑊 < 1) a quasi-steady flow may be assumed. This is the case for smaller blood vessels. For medium

values of Womersley parameter (3 ≤ 𝑎𝑊 ≤ 10) profiles are flatter and maximum velocity does not

occur at the center of the vessel but at some intermediate distance between the center and the wall.

For high values of Womersley parameter (𝑎𝑊 > 10) the inertia effects dominate, which is the case

for either high frequency flows or larger vessels. The flow velocity profiles in this case are mostly

blunt due to high inertia effects and change of flow direction (Ku, 1997).

Womersley parameter is valid only for sinusoidal flows, hence Fourier analysis is required for

application on hemodynamics, where complex blood flow must be analyzed to a superposition of

sinusoidal flows.

A.1.1.4 Wall Shear Stress (WSS)

Wall Shear Stress is defined as 𝑊𝑆𝑆 = 𝜏𝑊 = 𝜇 ∙ |𝑑𝑢

𝑑𝑟|𝑟=𝑟𝑤𝑎𝑙𝑙

where 𝜇 is the viscosity and |𝑑𝑢

𝑑𝑟|𝑟=𝑟𝑤𝑎𝑙𝑙

is

the local velocity gradient at the wall (Ku, 1997). WSS is important in hemodynamics, since the

endothelium has receptors which respond according to WSS values by adjusting their shape.

A.1.2 Computational Fluid Dynamics (CFD) and Applications Analytical solutions to the partial differential equations governing fluid flows are rarely available,

only for special, highly simplified cases. In all other cases when further simplifying assumptions

cannot be made, approximate numerical solutions are obtained using computers. Computational

Fluid Dynamics (CFD) involve replacing partial differential equations with discretized algebraic ones

that are approximations of the original PDEs. These algebraic equations can then be numerically

solved in trivial ways to obtain values for flow field variables at the discrete points, grid or mesh

nodes, in space and/or time. At every other point, flow variables may be calculated using various

interpolation schemes. The most common discretization techniques are the finite difference method,

the finite element/volume method and the boundary element method (Munson, et al., 2013).

o For the finite difference method, Taylor series are typically used in order to convert partial

differential equations to algebraic ones and approximate the partial derivatives.

o For the boundary element method, only the boundary of the flow field is discretized, and then

appropriate singularities, such as sources/sinks, doublets and vortices, are distributed along the

boundary and the flow field variables are calculated by calculating each singularity’s

contribution at every point. This method is commonly used for airfoils, called panel method.

o For the finite element/volume method, the flow field is discretized into small volume elements

of appropriate size, shape and number, for which the conservation equations are written in

approximate, algebraic form and then solved. This is the method used for discretization of the

flow field of the carotid artery bifurcation in the present paper.

Application of CFD gives the opportunity to obtain solutions for cases which cannot be easily

studied experimentally, such as flows in arteries. It should always be noted that approximating the


8

partial differential equations with algebraic ones induces errors which must always be accounted for

and minimized, through choosing appropriate number, shape and size for the grid’s elements.

Grid size and shape is an important problem for CFD applications, without a trivial solution. It must

necessarily represent accurately the case’s geometry and capture the complexity of the specific flow

field. Coarse meshes, with a small number of large elements, may lead to loss of accuracy and

absolutely irrelevant results, while fine meshes, with a large number of tiny elements, rapidly increase

computational time, with the additional danger of over-enhancing weak, small scale phenomena,

which are not being studied. Therefore, it is evident that the computational mesh must be as coarse

as possible in order to minimize computational cost, while being fine enough to provide accurate

solutions and a realistic approximation of the flow field.

Boundary Conditions are an important factor in CFD, since they are the way the solution of the

Navier-Stokes and Continuity equations is adapted to the specific geometry and problem.

CFD Algorithms can be either developed for each specific case, or commercial CFD Programs with

embedded code may be used. The first case is quite common for more complex, sophisticated

problems, such as the flow field through carotid artery bifurcation, while commercial CFD programs

are appropriate for most usual problems, mainly external flows. In the present paper, commercial

program ANSYS Fluent™ is used for CFD Application, since it is considered sufficient for the accuracy

required and CFD code development is beyond the scope of this research.

A.2 HEMODYNAMICS AND BLOOD VESSELS In this section of Chapter A, properties of blood are discussed first, including its biochemical

composition and its main properties from the standpoint of fluid mechanics, such as density and

viscosity. The biochemical composition and the resulting properties of Blood Vessels are then

presented, before a short discussion of the most important Cardiovascular Diseases of the present

days. Finally, a more specific discussion about the Carotid Artery Bifurcation (CAB) is presented, after

a short description of the most common measurement methods for hemodynamic parameters.

A.2.1 Physiology and Properties of Blood This part includes a short description of the most important properties of Blood, which is a

multiphase liquid which consists of Plasma and Particles.

A.2.1.1 Viscosity

Chemical composition of blood is what determines its viscosity. Blood is a multiphase liquid

composed by around 50-55% plasma, a Newtonian liquid containing mostly water, ions and proteins.

The remaining volume contains particles, the vast majority of which, around 99% of particle volume,

are the Red Blood Cells (RBC) or erythrocytes, which transport oxygen, while the rest are lymphocytes,

which are one of the main human defence mechanisms, and platelets, which take part in hemostasis

(Kottarakou, 2015; Kalozoumis, 2009). This means that RBCs are the main cause of the difference

between plasma and blood viscosity. Therefore, it may be argued that the viscosity of blood depends

on three factors; the viscosity of the plasma, the hematocrit (Ht) which is the volume percentage of

RBCs and their deformability (Westerhof, et al., 2010; Ku, 1997). As the hematocrit rises and

deformability decreases, the blood viscosity becomes higher. There are a lot of formulas that

correlate blood viscosity and hematocrit, but no such formulas are used in the present paper. The

viscosity of plasma is about 0.015 𝑃𝑜𝑖𝑠𝑒 and the viscosity of blood at a physiological hematocrit of

40-45% is about 0.032 𝑃𝑜𝑖𝑠𝑒 (Westerhof, et al., 2010). The parameter of RBC deformability is more

complex and cannot be easily quantified.


9

Blood demonstrates a Non-Newtonian behaviour, which means that its viscosity depends on shear

rate. At higher shear rates, the viscosity of blood decreases, thus blood is a shear-thinning liquid. The

physiological explanation of this phenomenon is that, at higher shear rates RBCs orient themselves

in the direction of flow, hence viscosity is lower, while at lower shear rates RBC aggregates may occur,

hence increasing the viscosity. In larger arteries, such as Carotid Arteries, these effects may be smaller

and can be neglected, since shear rates are relatively high (higher than 100 𝑠−1) (Westerhof, et al.,

2010; Ku, 1997). However, in a lot of studies these phenomena, such as shear-thinning viscosity,

viscoelasticity and thixotropy, which become apparent at low shear rates, are included in the

calculations using various blood viscosity models. A mathematical model that takes into account the

shear-thinning effects of blood viscosity, widely used for CFD studies on blood flows, is the Carreau-

Yassuda model, described in the next section. Physiological values of wall shear stress, which can be

related to shear rate, are about 1 − 2 𝑃𝑎.

Other Non-Newtonian phenomena are apparent in microcirculation, where this behaviour of blood

is enhanced. For Carotid Arteries, these non-linear effects may be neglected.

Blood viscosity also depends on temperature, however this correlation is of little importance since

body temperature may be assumed constant at 36.6 °𝐶 for human bodies.

A.2.1.1.1 Carreau-Yassuda Model

The Carreau-Yassuda viscosity model quantifies the shear-thinning behaviour of blood, using two

asymptotic apparent viscosities and other parameters (Kim, et al., 2006). The mathematical model is

𝜇(�̇�) = 𝜇∞ + (𝜇0 − 𝜇∞) ∙ [1 + (𝜆 ∙ �̇�)𝛼]𝑛−1𝛼

where

𝜇(�̇�) is the apparent viscosity at any shear rate �̇�.

𝜇∞ = 0.0035 [𝑘𝑔

𝑚∙𝑠] is the apparent viscosity at infinite shear rate �̇� → ∞.

𝜇0 = 0.0560 [𝑘𝑔

𝑚∙𝑠] is the apparent viscosity at zero shear rate �̇� = 0.

𝜆 = 3.313 [𝑠] is a constant defining the transition from Newtonian to Power-Law region.

𝛼 = 2 is a constant defining the transition from Newtonian to Power-Law region.

𝑛 = 0.3568 is the Power-Law slope.

These values for the above constants used in this paper were selected, taking into account previous

research on CFD for blood flows (Janela, et al., 2010).

A.2.1.2 Density

In this paper, blood density is assumed constant, 𝜌 = 1060 [𝑘𝑔

𝑚], which is a valid assumption, based

on previous research and considering the constant temperature of the human body, around 36.6 °𝐶.

A.2.1.3 Turbulence

The vast majority of physiological, healthy blood flows through the arterial tree is laminar, where

turbulence intensity is negligible. Physiological turbulent blood flow occurs in arteries near the heart,

such as the aorta and the aortic arch, while it may also occur in jet flows, caused by arterial stenosis.

In the present paper, no turbulent models where examined, although such pathological cases in the

Carotid Artery Bifurcation are of great significance in clinical applications. It has been proved in

various papers that the same process may be expanded and applied using turbulent models, which

in healthy, physiological cases only marginally increase simulation accuracy.


10

A.2.2 Biochemical Composition and Properties of Blood Vessels The role of vessels is to transfer blood from the heart to every organ in the body, through a variety

vessels, from the large conduit arteries to the small peripheral vessels (Kottarakou, 2015). Arteries

have a complex geometry which may change under varying stress. Their properties are highly non-

linear, due to viscoelasticity and residual stresses, as a result of their composition (Kalozoumis, 2009).

Arteries consist of three different layers or tunicae: tunica intima, tunica media and tunica

adventitia. Tunica Adventitia is the outer layer of human arteries, which consists mainly of collagen

fibers, connective tissue cells, macrophages and small blood vessels, and mainly connects the artery

with surrounding tissues. Tunica Media is intermediate layer and consists of mostly smooth muscle

cells, collagen and elastin. Tunica Intima is the inner layer of arterial walls and, together with Tunica

Adventitia, they are responsible for artery’s mechanical properties. Tunica Intima consists of the

endothelium and the basement membrane, and is the most important part of arterial walls, since the

endothelium contains micro-receptors that are responsible for controlling the permeability of the

arterial wall. Additionally, endothelium provides resistance to thrombosis and other pathological

conditions by reacting to blood flow characteristics.

While studying blood flows, some properties of the arterial walls are of great importance in wave

propagation studies and pressure measurements; therefore, some of them are presented below.

A.2.2.1 Resistance

Resistance (𝑅) is defined as the pressure drop through a vessel (𝛥𝑝) over the volumetric flow (𝑄),

thus

𝑹 =𝜟𝒑

𝑸

where 𝛥𝑝 is the pressure drop through the blood vessel and 𝑄 is the blood volumetric flow which

corresponds to this pressure drop. From Poiseuille ‘s law it can be deduced that

𝑅 =8𝜇𝑙

𝜋𝑟4

where 𝜇 is the absolute viscosity, 𝑙 is the length of the vessel and 𝑟 is the vessel radius. This is not an

appropriate way for calculations of Resistance, since Poiseuille’s law assumes a uniform vessel and a

Newtonian fluid. Additionally, it would require extremely good measurement of the vessel’s radius,

due to the 4th power. Therefore, Resistance is usually calculated indirectly, by measuring the

volumetric flow and the pressure drop through a blood vessel (Westerhof, et al., 2010).

In the human arterial tree, most of the total Resistance is located in arterioles, since it can be seen

from the above formula that for extremely small radius the Resistance becomes very large. The

Resistance of all arterioles is added in parallel, thus the total Resistance is lower. As a result, only

around 1% of the total pressure drop in the arterial tree occurs in the aorta. The total Resistance of

small arteries and arterioles is usually referred to as Peripheral Resistance (𝑅𝑃) (Westerhof, et al.,

2010). Addition rules for Resistance are similar to those of electronics, thus for serial or parallel

addition of Resistances 𝑅𝑖 the equivalent total Resistance is

𝑅𝑒𝑞𝑠𝑒𝑟𝑖𝑎𝑙= ∑𝑅𝑖

𝑖

𝑎𝑛𝑑 1

𝑅𝑒𝑞𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙

= ∑1

𝑅𝑖𝑖


11

A.2.2.2 Inertance

Inertance (𝐿) is defined as the pressure drop through a vessel (𝛥𝑝) as the flow is accelerated over

the volumetric acceleration (𝑑𝑄

𝑑𝑡), thus

𝛥𝑝 = 𝐿 ∙𝑑𝑄

𝑑𝑡𝑜𝑟 𝑳 =

𝜟𝒑

𝒅𝑸 𝒅𝒕⁄

where 𝛥𝑝 is the pressure drop through the vessel and 𝑑𝑄

𝑑𝑡 is the volumetric flow acceleration. From

Newton’s 2nd law it can be deduced that

𝐿 =𝜌𝑙

𝐴

where ρ is the fluid density, 𝑙 is the vessel length and 𝐴 is the vessel area. Inertance is important in

large vessels, where the viscous resistance is small and the flow is highly pulsatile, since the heartbeat

causes periodic accelerations and decelerations of the flow. It can be seen that Inertance is inversely

proportional to the vessel area, or to 𝑟2, while Resistance is inversely proportional to 𝑟4, as discussed

in A.2.2.1. Therefore, in small arteries, Resistance is the dominating parameter, while in larger arteries

Inertance plays the most important role (Westerhof, et al., 2010). Addition rules for Inertance are

similar to those for Resistance, that is

𝐿𝑒𝑞𝑠𝑒𝑟𝑖𝑎𝑙= ∑𝐿𝑖

𝑖

𝑎𝑛𝑑 1

𝐿𝑒𝑞𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙

= ∑1

𝐿𝑖𝑖

A.2.2.3 Compliance

Compliance (𝐶) is used mostly for blood vessels and is

defined as

𝑪 =𝒅𝑽

𝒅𝒑> 0

where 𝑉 is the vessel volume and 𝑝 is the transmural

pressure. For blood vessels, Compliance decreases at

higher pressure and volume, thus the volume over pressure

diagram is concave. Usually, compliance is calculated and

referred to at a specific working point, that is specific

pressure and volume. Compliance is often defined as

𝐶 =𝑑𝐴

𝑑𝑝 𝑜𝑟 𝐶 =

𝑑𝐷

𝑑𝑝

where 𝐴 is the vessel area and 𝐷 is the vessel diameter.

Serial and Parallel addition of Compliance follows the same rules as Resistance, that is

𝐶𝑒𝑞𝑠𝑒𝑟𝑖𝑎𝑙= ∑𝐶𝑖

𝑖

𝑎𝑛𝑑 1

𝐶𝑒𝑞𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙

= ∑1

𝐶𝑖𝑖

In general, arterial Compliance decreases with age, which is the main reason for the increase in

transmural pressure which has been observed with an increase in age (Westerhof, et al., 2010).

A.2.2.4 Residual Stresses

It has been proven that Residual Stresses exist in arteries, since when a ring of artery is excised

from the body and cut longitudinally, it opens and forms and open arc. This is referred to as the Zero

Stress State. When an artery forms a closed cylindrical surface at zero transmural pressure, the inner

wall is under compression while the outer wall is under tension. However, under normal arterial

pressure inside the artery, a uniform tension distribution occurs (Westerhof, et al., 2010).

Figure A.2.1. Compliance definition

(Westerhof, et al., 2010).


12

A.2.2.5 Elasticity and Viscoelasticity

Εlastic materials are characterized by a linear stress-strain

relation, where the Elastic or Young’s modulus is defined as

𝐸 =𝑑𝜎

𝑑𝜀=

𝜎

𝜀

where 𝜎 is the stress and 𝜀 is the corresponding strain, and 𝐸

is constant and independent of stress and strain values, and

time. Most biological tissues demonstrate a convex stress-

strain relation, thus the incremental Elastic modulus 𝐸𝑖𝑛𝑐 =𝑑𝜎

𝑑𝜀

is increased at higher values of stress or strain. This means

that biological tissues become stiffer with increasing stress or

strain, and it further indicates a limitation in strain, hence for strains larger than a specific value the

slope becomes infinite and the corresponding stress exorbitantly high (Bergel, 1961).

This behaviour of arterial walls is explained by their structure. Arterial walls consist of vascular

smooth muscle, Elastin and Collagen (Alexopoulos, 2005). Elastin fibers are highly extensible with a

low value of Elastic modulus, which remains almost constant. On the other hand, Collagen fibers are

hard and stiff, with a Young’s modulus around 1000 times higher. Collagen fibers in arterial walls are

found in coils. At low stresses, collagen fibers are still wavy and bear no load, hence Elastin and

smooth muscle determine the elastic properties of arterial wall, which subsequently demonstrates

low Elastic modulus and stress-strain slope at low stresses. However, at larger stresses, Collagen coils

unravel and start to bear some load, which results in a rapid increase in stiffness, quantified by the

Elastic modulus or the stress-strain slope (Westerhof, et al., 2010; Zalger, 2010). Therefore, orientation

of these Collagen fibers is essential and vascular tissue is highly anisotropic.

Biological tissues also demonstrate viscous properties, thus being characterized as viscoelastic

materials (Avolio, 1980). Viscoelasticity describes phenomena such as hysteresis due to creep and

stress relaxation, which must be taken into consideration while studying mechanical behaviour of

arteries; these phenomena require complex mathematical modelling (Bergel, 1961).

A.3 ARTERIAL DISEASES This part presents some of the most common pathological conditions of the cardiovascular system.

A.3.1.1 Stenosis and Development of Atherosclerosis

The term Arterial Stenosis is used to describe a localized narrowing in the arterial lumen, typically

as a result of Atherosclerosis. Atherosclerosis involves the development of atheromatous plaques in

the subintimal layer of the arterial walls, usually at regions of lower WSS, which then extend into the

arterial lumen, thus constricting the area available for blood to flow.

Atheromatous plaque consists of lipids and foam cells, cholesterol and other products of

metabolism and result in a narrowing of the arterial lumen. Development of atheromatous plaque

begins with high concentration of lipoproteins which intrude the endothelium, as a result of high

cholesterol concentration, high arterial pressure and smoking. These result in damage of the

endothelium and subsequently stenosis. More specifically, high concentration of Low Density

Lipoprotein (LDL) in Tunica Intima leads to high concentration of monocytes in the arterial wall.

Lymphocytes then intrude the vessel wall and multiply, absorb proteins and convert to foam cells.

Then smooth muscle cells cover the Tunica Intima and multiply to cover the area. This leads to an

increase in the thickness of Tunica Intima and plaque (Kalozoumis, 2009).

Figure A.2.2. Elasticity definition

(Westerhof, et al., 2010).


13

As described elsewhere (Westerhof, et al., 2010), the endothelial layer cells fully align with the

direction of blood flow at high 𝑊𝑆𝑆 > 1 𝑃𝑎, while at lower values, especially at 𝑊𝑆𝑆 < 0.4 𝑃𝑎 they

do not. In the latter case, NO-synthase inhibits the area and apoptosis increases the endothelial

dysfunction, which may lead to enhanced adhesion of monocytes, increased platelet activation and

vasoconstriction and increased smooth muscle proliferation and oxidant activity (Alexopoulos, 2005).

The above create a favourable environment for atherogenesis.

Therefore, atheromatous plaque usually develops in regions of low, oscillating WSS which result in

a prolonged time of residence for particles, thus leading to accumulations of atherogenetic

lipoproteins and plaque development (Kalozoumis, 2009). This is the reason that Atherosclerosis is

usually localized around bifurcations, where low and oscillatory 𝑊𝑆𝑆 < 0.4 𝑃𝑎 occur, especially in the

Carotid Sinus, where flow separation occurs during the decelerating phase of the cardiac cycle,

resulting in low and oscillatory shear stresses (Westerhof, et al., 2010).

The composition of atheromatous plaque is important for its stability. High hardness leads to fragile

plaque, which is prone to rupture. This increases the risk for a stroke, since the ruptured plaque is

carried away by the blood flow and may damage or obstruct the artery downstream (Kottarakou,

2015). Vortices tend to lead to stable plaque, while low, oscillating WSS lead to vulnerable plaque.

On the other hand, high WSS induce an atheroprotective endothelial phenotype, decreasing the

expression of adhesion molecules and oxidants (Westerhof, et al., 2010).

An artery with stenosis consists of a converging part, followed by the throat, i.e. the minimum

lumen area and then a diverging part. Maximum flow velocities appear at the throat, while after the

diverging part of the arterial stenosis, flow usually separates and high turbulence occurs.

Quantification of the severity of the arterial stenosis is possible by calculating the percentage of

lumen diameter or area that is obstructed by atheromatous plaque, that is

(1 −𝐷𝑠

𝐷𝑜) 100% ή (1 −

𝐴𝑠

𝐴𝑜) 100%

where 𝐷𝑠 𝑎𝑛𝑑 𝐴𝑠 denote the stenotic lumen diameter and area, respectively, and 𝐷𝑜 𝑎𝑛𝑑 𝐴𝑜 denote

the unstenosed lumen diameter and area, respectively. Arterial stenosis leads to severe pressure

losses, high jet velocities and relatively high shear stresses, often accompanied by transition to

turbulent flow. Characterization of a stenosis is possible by calculating the relationship between

pressure losses and blood mass flow (Westerhof, et al., 2010). Pressure drop in low severity stenosis

is mostly due to boundary layers interaction and flow separation, while in high severity stenosis

((1 − 𝐷𝑠 𝐷0⁄ )100% > 75%) it is mostly caused by turbulence (Kottarakou, 2015; Ku, 1997).

Post-stenotic dilatation refers to a phenomenon where the arterial diameter distal of a stenosis is

increased. While its mechanism remains unclear, it might be due to increased turbulence and shear

stresses, or vessel wall vibrations (Westerhof, et al., 2010). Another interesting phenomenon

occurring at stenotic arteries is the relatively low pressure at the throat, as a result of high velocity.

Therefore, in arteries with compliant walls higher external pressure may lead to extra narrowing or

even collapse. Afterwards, flow obstruction reduces flow velocity hence pressure is increased and the

artery reopens, and the process, referred to as an unstable wall oscillation (Ku, 1997), repeats itself.

After the occurrence of stenosis in an artery, higher wall shear stresses appear, in the stenotic

section, which may now lead to rupture of atheromatous plaque. When this happens in arteries near

the brain, such as the Carotid Artery (specifically the Internal Carotid Artery – ICA), the plaque carried

away by the blood flow, if it is large enough, may obstruct the lumen area at a subsequent section

of the artery which is narrower, thus leading to blockage of blood flow to the brain, hence to a stroke.


14

A.3.1.1.1 Treatment Methods

Treatment of atherosclerosis is possible by various methods, most of them interventional. The most

common of these are described shortly in this chapter.

A.3.1.1.1.1 Atherectomy

Atherectomy is a highly interventional treatment in which the atheromatous plaque and part of the

endothelium are removed using some medical probe. This probe may either be like a razor blade or

a rotary probe. Atherectomy requires a longitudinal cut on the arterial wall, so stitching or a patch is

required after the procedure (Kalozoumis, 2009).

A.3.1.1.1.2 Angioplasty and Stenting

Angioplasty using stent is a less interventional treatment method than Atherectomy. A stent is used

for increasing the lumen area and supporting the stenotic artery. To implant the stent, a probe with

a balloon is used, which is brought in the stenotic artery and then inflated, thus extending the stent

and the artery. The stent has sufficient strength to support the artery and keep the lumen area

unobstructed. This method is mainly used in large arteries with concentrated stenosis with only a few

occurrences of atheromatous plaque and is not appropriate for dispersed plaque development

A common post-treatment problem is restenosis of the artery due to Intima hyperplasia in regions

of low flow velocity and lower than normal WSS (Westerhof, et al., 2010). It must be noted that some

endothelium development is desirable in order to achieve normal blood flow and minimize the flow

disturbance caused by the stent, but further increase in thickness leading to hyperplasia may lead to

worse stenosis than before the treatment (Westerhof, et al., 2010).

A.3.1.1.1.3 Bypass

Surgical Bypass is a method used in cases of stenosis when endovascular techniques are not easily

applied. In general, it requires a more extensive and difficult neck dissection and has higher morbidity

rates. In general, the connection of two arteries requires careful planning and various considerations

before the surgery, otherwise blood flow after the surgery is completed may not be optimal.

A.3.1.2 Thrombosis

Thrombosis is a common disease where a blood clot or thrombus forms inside a blood vessel,

obstructing the flow of blood through the vessel. Thrombosis is a result of Hemostasis, which is the

normal physiological response of the body that slows and subsequently stops blood loss after a

vascular injury. To treat the injury, various proteins are activated enabling fibrin and platelets to form

a blood clot in order to prevent blood loss and patch the injured region.

Although Hemostasis is an important mechanism for the body, pathological conditions, namely

Thrombosis, may arise when Hemostasis is activated incorrectly, thus forming a blood clot which may

obstruct the flow of blood inside an uninjured vessel. Thrombi may be arterial or venous, depending

on where they are formed. Arterial Thrombosis is often a result of rupture of atheroma

(Atherothrombosis), since Thrombi tend to form around atherosclerotic plaques, and may lead to

arterial embolism or stroke. Thrombi obstructing blood flow to tissues may lead to hypoxia or anoxia

which are the reduction or complete deprivation of oxygen, respectively.

The main causes of Thrombosis are summarized in Virchow’s Triad and are Hypercoagulability,

which is genetic predisposition to Thrombus formation, Endothelial Cell Injury, which may be a result

of trauma, infection or turbulent flow at bifurcations, and Disturbed Blood flow, which mainly refers

to stagnation of blood at specific regions. Treatment methods for thrombosis usually do not require

surgery and rely on appropriate medication.


15

A.4 MEASUREMENT AND IMAGING METHODS In this chapter, measurement methods for hemodynamic parameters and arterial imaging

techniques are described. These are divided into two categories: invasive and non-invasive methods.

A.4.1 Invasive Methods Invasive methods are akin to those used in Fluid Mechanics, but rarely applied in Hemodynamics.

A.4.2 Non-Invasive Measurement Methods Non-invasive measurement methods for hemodynamic parameters are used widely, since it is

desirable to be able to measure hemodynamic parameters without any kind of surgery. They are

important for applications of CFD in hemodynamics in order to use realistic boundary conditions and

validate computational results. Two techniques are able to measure the waveform of blood velocity:

Pulse-Doppler Ultrasound (US) and Phase Contrast Magnetic Resonance Imaging (PC-MRI).

The Pulse-Doppler Ultrasound allows sufficiently small time-step, less than 15 𝑚𝑠, thus is the most

appropriate technique to measure blood flow velocities (Kottarakou, 2015). It is extremely sensitive

to velocity changes and provides ample accuracy for superficial vessels (Atkinson & Wells, 1977).

PC-MRI is an MRI technique, which provides greater accuracy and has no limitations concerning

the position of the vessel in the body. However, its application has some limitations regarding the

patients on whom it may be used, due to the presence of magnets, and needs consistent cardiac

timing, so it cannot be used in cases of arrhythmia. It is able to measure volumetric flow and velocities

in blood vessels with diameters larger than a few millimeters (Gatehouse, et al., 2005).

A.4.3 Non-Invasive Geometry Reconstruction Methods

A.4.3.1 Black Blood Magnetic Resonance Imaging (BB-MRI)

MRI is currently the most widely used imaging method for in-vivo arterial geometry acquisition. It

provides high accuracy 2D images, which can be segmented and semi-automatically reconstructed,

manually corrected and then aligned and smoothed (Glor, et al., 2003).

Using BB-MRI, it is possible to obtain arterial geometry without limitations by bones, such as the

bottom jaw in the case of Carotid Artery, and it can be applied for both superficial and deeper vessels.

Additionally, during the MRI procedure there is limited freedom of movement, which leads to better

posture of the body and higher quality of images (Kalozoumis, 2009; Kottarakou, 2015). However,

MRI is an expensive and time-consuming technique and its application is complex, with limitations,

regarding suitability for patients, due to the use of magnets and the limited freedom of movement.

A.4.3.2 3D Ultrasound (3D-US)

3D Ultrasound is a cheaper alternative to BB-MRI, with limited applications to superficial vessels,

such as Carotid or Femoral arteries. In principle, 3D-US relies on the Doppler effect, thus detecting

the different regions of blood and walls. As well as BB-MRI, it can provide information about the

shape and thickness of blood vessels, after segmentation and manual reconstruction of 2D images.

3D-US is cheaper than BB-MRI, while it is able to provide images of equal accuracy (Glor, et al.,

2003). 3D-US requires a much quicker procedure, which is easier for both the patient and the

operator, since 3D-US devices are very widespread. During 3D-US it is possible for the patient to

reposition their body, thus allowing a greater resolution and quality of images but resulting in a

distortion of the shape of the arteries. Additionally, application of 3D-US is limited only to superficial

vessels due to obstruction of visibility by bones and tissues (Kalozoumis, 2009; Kottarakou, 2015).


16

A.5 CAROTID ARTERY BIFURCATION (CAB) Carotid Arteries (CA) play an important role in the circulatory system. Each human has two, almost

symmetrical Common Carotid Arteries (CCA): right and left. They both ascend vertically through the

neck to the head, but start at different positions; the right CA branches from the brachiocephalic

artery at the neck, while the left CA starts at the aortic arch in the thoracic area (Kalozoumis, 2009).

Both Carotid Arteries split at the Carotid Artery Bifurcation (CAB), located around the 4th vertebra,

into two branches, Internal Carotid Artery (ICA) and External Carotid Artery (ECA), which both keep

ascending vertically. ECA is responsible for supplying blood to the neck and face muscles, while ICA

proceeds internally to supply blood to the brain. It is, therefore, evident that partial or total

obstruction of blood flow in the CCA or the ICA may lead to hypoxia or anoxia, and ultimately to an

ischemic episode or stroke, which makes the study of blood flow through CA clinically significant.

Carotid Sinus or Bulb, located at the root of the ICA, is a section of increased lumen area, compared

with the ICA downstream (Kottarakou, 2015). Atherosclerotic plaque development often occurs on

the outer walls at the CAB, usually at the bulb. Flow disturbances appear in this region, increasing

the risk for Cerebrovascular Accident; therefore, study of blood flow through the bulb is essential.

A.6 CARDIAC CYCLE AND HEARTBEAT Heart is the central organ of the circulatory or cardiovascular system, circulating blood as a pump.

It consists of four sections, right and left atrium and right and left ventricle. Deoxygenated blood

coming from the head and body moves through veins to the right atrium. It is then transported to

the right ventricle from where it is pumped to the lungs through the pulmonary artery in order to be

oxygenated. Oxygenated blood is then transported to the left atrium and then to the left ventricle,

from where it is pumped through the aorta to the body (Kottarakou, 2015). The coronary circulation

system supplies blood to the heart itself, through the left and right coronary arteries. Blood is allowed

to flow through the heart only in one direction, which is achieved by two atrioventricular (AV) valves,

the mitral and the tricuspid valve, and two semilunar (SL) valves, the aortic and the pulmonary valve.

Heartrate is defined as the number of contractions of the heart per minute and regulated only by

the sinoatrial node (SA node), and by sympathetic and parasympathetic input to the SA node

(Kottarakou, 2015). Pulse rate, measured by compressing an artery against a bone, may differ from

the heartrate, although they are usually equivalent. Cardiac cycle refers to a complete heartbeat, from

its generation to the beginning of the next beat, and includes three phases, the systole, the diastole

and the intervening pause, and five stages (Kottarakou, 2015). Its frequency is described as the

heartrate in beats per minute [𝑏𝑝𝑚]. The stages of a cardiac cycle are the following:

o Diastole (1st stage), when the SL valves are closed and the AV valves are open, and the whole

heart is relaxed, hence pressure is almost constant while ventricular volume increases.

o Atrial Systole (2nd stage), when the atria contract and blood flows to the ventricles through

the AV valves, with both increasing ventricular pressure and ventricular volume.

o Isovolumic Contraction (3rd stage), when all the AV and SL valves are closed and the ventricles

begin to contract and there is no change in blood volume inside the heart, hence ventricular

pressure increases, while ventricular volume remains constant.

o Ventricular Ejection (4th stage), when the SL valves are open and the ventricles are contracting

to pump blood outside of the heart, hence ventricular volume decreases while pressure first

increases and then decreases.

o Isovolumic Relaxation (5th stage), when no blood enters the ventricles, since the AV and SL

valves are closed, hence pressure decreases, while volume remains constant.


17

Aortic pressure and arterial pressure in general during the cardiac cycle are different to the left

ventricle pressure, due to elasticity and compliance of the large arteries, which induce various

pressure wave propagation phenomena (Westerhof, et al., 2010).

A.7 WAVE PROPAGATION Wave propagation through the arterial and venous tree is a complex procedure and specific

phenomena must be taken into account in order to achieve successful modelling. Some elementary

principles are discussed in brief, since they are not analyzed in depth in the present study, but may

improve further research.

A.7.1 Wave Speed The cardiac pressure pulse generated by the heart propagates through the arterial tree at a certain

speed, which is called wave speed. Wave speed or phase velocity (𝑐) depends on the shape and

properties of the vessel. It has been deduced that wave speed increases with increasing elastic

modulus (𝐸𝑖𝑛𝑐) and increasing wall thickness (ℎ), while it decreases with increasing vessel radius (𝑟)

and increasing density (𝜌) and the Moens-Korteweg formula describes the relation between these

parameters for a non-viscous fluid as

𝑐 = √ℎ ∙ 𝐸𝑖𝑛𝑐

2𝑟 ∙ 𝜌

The above formula does not include reflection effects. When included, using Fourier analysis of the

pressure waveform, the apparent wave velocity may be calculated. For high frequencies, these

reflection effects become negligible (Westerhof, et al., 2010).

A.7.2 Wave Travel and Reflections Wave reflections take place at every bifurcation or discontinuity of the vasculature, most

importantly in arterioles where reflections are extremely frequent, since the number of arterioles

increases exponentially with decreasing size. As a result, at any point in the arterial tree, measured

quantities such as pressure (𝑝) , velocity (�⃗� ) or volumetric flow rate (𝑄) are a superposition of

forward and backward running waves. Therefore, a Fourier waveform analysis is required in order to

distinguish the magnitude and phase of each wave. Transfer functions may be formed and used for

a mathematical model which enables calculation of the pressure pulse at any other point by

measuring the pulse at a specific point of the arterial tree (Westerhof, et al., 2010).

A.7.3 Arterial Input Impedance Input impedance (𝑍𝑖𝑛) characterizes a part of the arterial tree, providing a relation between the

pressure difference (𝛥𝑝) through this part and the volumetric flow rate (𝑄) in a linear system for

harmonic flows. Input impedance is a complex number and is the result of waveform analysis,

described by a magnitude and a phase, so its value changes at different frequencies. At higher

frequencies, its phase becomes almost equal to zero and it takes a constant, real value, namely the

characteristic impedance (𝑍𝐶) of the proximal aorta (Westerhof, et al. , 2010).

A.7.4 Windkessel models Windkessel models are mathematical modelling tools similar to elementary electrical circuits, used

to substitute the arterial tree with its calculated properties, namely the peripheral resistance (𝑅𝑃),

the compliance (𝐶) , the characteristic impedance (𝑍𝐶) and the Inertance (𝐿) . Different types of

Windkessel models have been developed, with different complexity (Westerhof, et al., 2010).


18

The two element Windkessel model includes only the peripheral resistance (𝑅𝑃), which is mainly

located at the resistances of the smaller arteries, the arterioles and the capillaries, and the compliance

(𝐶), which is mainly located at the conduit arteries, near the heart. Higher compliance indicates an

ability to store larger quantities of blood in the arterial tree, while higher peripheral resistance

indicates a lower blood flow rate and velocity.

The three element Windkessel model is an expansion of the two element Windkessel model, with

the addition of the characteristic impedance (𝑍𝐶).

Furthermore, the four element Windkessel model has been developed as an expansion of the three

element Windkessel model, by including the inertance (𝐿).

The above models are widely used to provide outlet boundary conditions for CFD, modelling the

downstream branches of the arterial tree as Windkessel models (Ku, 1997).

Apart from Windkessel models, other types of models, such as tube models, have also been

developed. Their use is the same, but the principles on which they rely are different (Westerhof, et

al., 2010). All these various mathematical models serve the same purpose; that is, calculations of

hemodynamic parameters at different points in the arterial tree and at any time during the cardiac

cycle. Other types of models are not described in this paper, since they are not relevant to the present

study. A great example of an elastic tube model of the arterial tree is given in (Avolio, 1980).

Literature Review Numerzical Analysis of Blood Flow through the Human Carotid Artery Bifurcation

19

LITERATURE REVIEW This chapter contains a short literature review of previous research conducted in the specific topic

of CFD applications for studying biological flows and more specifically, blood flow through Carotid

Artery Bifurcation. This review must not be perceived as complete, since the field of study is rather

new, with most research conducted in the past two decades. Therefore, knowledge of the subject

has not yet reached a mature state, hence only elementary principles have been established and

more sophisticated conclusions are not yet sturdy.

B.1 PREVIOUS CFD APPLICATIONS TO BIOLOGICAL FLOWS Applications of CFD simulations, which are extremely common in industrial practice, are steadily

increasing in medical science and research. They are of course not limited to blood flow, but cover

every aspect of biological fluid motion and dynamics in the body, such as the respiratory system. In

this review, only previous research in blood flows is presented, in mainly chronological order.

Firstly, papers studying blood flow through the Carotid Artery Bifurcation where the vessel walls

are assumed rigid are presented, which were the most common method until a decade ago.

Afterwards, studies of blood flow through the Carotid Artery using simulations including Fluid-

Structure Interactions are described, which are the most popular practice during the last few years,

due to the increasing capabilities of computers and increased accuracy requirements. Finally, some

general papers presenting studies of blood flow through the arterial tree are mentioned, before

drawing some general conclusions, which provide a guideline for this paper, aiming to confirm or

reject these statements and evaluate research methods and their applications.

B.1.1 CFD Applications to Carotid Artery Bifurcations (CAB)

B.1.1.1 Rigid Body CFD Applications

One of the earliest applications of CFD simulations studying blood flow through CAB is the one by

Wells et al. (1996), which used transient, 2-dimensional simulations to study blood flow through 3

different geometries, which represented 3 possible Carotid Bifurcation reconstructions after medical

procedure, for both Newtonian and Non-Newtonian viscosity models for blood. This comparison

concluded that abrupt geometric changes at the CAB induce flow disturbances and increase WSS

gradients and WSS peak values. The authors proved that smoothly tapered CAB geometry, without

bulb and wit a small bifurcation angle leads to favourable blood flow characteristics, regarding both

atherosclerosis and hyperplasia, hence proposed this geometry as a possible optimum

reconstruction. They also stated that bulb reconstruction using a patch transfers flow disturbances

downstream. In that study, high WSS gradients where assumed as indicators of proneness to arterial

disease, instead of low, oscillatory WSS regions, which was a widely accepted concept at that time.

One of the most important and complete reviews of early research on the subject is the one by Ku

(1997). That paper provides a summary of the early conclusions of the studies of blood flow in the

arterial tree. The author discusses elementary principles of fluid mechanics as applied to blood flows,

concluding that normal blood flow is laminar, forming secondary flows at branches and curves,

identifying low, oscillatory WSS as the main cause of atherosclerotic diseases and stenosis. Ku also

summarizes the main effects of stenosis, which are turbulence and losses, reduced mass flow and

even choking. On the contrary, high WSS may induce thrombosis due to platelets activation.

Physiological Reynolds number regime is found to be between 1 and 4000, with highest values near

the aorta. The author also discusses the validity of the rigidity hypothesis for the vessel walls, which


20

is stated to be reasonable, since large arteries are quite stiff with negligible elasticity effects. In

addition, Ku provides a comparison and evaluation of different measurement techniques available,

namely MRI and Pulse-Doppler velocimetry, while also refers to research stating that endothelial cells

sense WSS and respond accordingly, maintaining physiological values of around 1.5 Pa. More

specifically on Carotid Artery, the author discusses flow characteristics, which include a Reynolds

number around 300, and supports a correlation of low WSS and atheromatous plaque development.

Ku provides references stating that Non-Newtonian blood viscosity and wall elasticity have small

effects on the flow compared to geometry. Finally, Ku notes that stenosis induces turbulent flow

downstream, flow separation in the diverging part and higher velocities and WSS values, while

extremely high stenosis levels may induce to turbulence during the whole cardiac cycle even at very

low Reynolds numbers.

In their research, Milner et al. (1998), attempted to investigate the concept of a “geometric risk

factor” for atherosclerosis. They used MRI to create the geometry models (BB-MRI) and measure flow

rate waveforms at the boundaries, while the images were later segmented producing 3D geometries

using ICEM CFD/CAE. The authors’ hypothesis was that helical flow patterns induced by asymmetric

geometry play a crucial role in WSS distribution. A Newtonian viscosity model was selected for the

simulation, assuming rigid vessel walls and fully developed velocity profiles at the boundaries.

Comparison of 2 in-vivo geometries with a symmetric idealized one concluded that curvature and

geometry play a significant role in WSS distribution, inducing secondary flows which were absent

from the symmetric idealized Y-tube model. Finally, a remark by the authors is that helicity at the

inlet may alter WSS distribution, thus questioning the validity of simulations using ideal inlet profiles.

Later on, Tortoli et al. (2003) studied velocity profiles of blood flow through Carotid Arteries,

concluding that real velocity profiles are neither parabolic nor flat and cannot be predicted by

Womersley theory. Assuming rigid vessel walls and Newtonian fluid model for blood, the authors

found that during the deceleration phase (late systole) higher velocities occur near the walls, instead

of the centreline, thus leading to a W (or M) profile. Additionally, the authors reported that when

Common Carotid Artery is curved, non-axisymmetric velocity profiles occur, while the velocity profiles

are axisymmetric if CCA is straightened. This may be explained by the momentum equation in the

radial direction, which leads to increased velocities, causing asymmetric W (or M) velocity profiles.

Further studies (Marshall, et al., 2004) attempted to identify reversed flow regions for both healthy

and stenosed Carotid Artery Bifurcations, calculating WSS vectors from in-vivo velocity

measurements. Reversed flow was found greater and more extensive during deceleration, as

expected, while the main reversed flow zone was on the outer ICA wall for healthy the healthy CAB,

and two zones occurred on the outer ICA wall right downstream of the stenosis and on the inner ICA

wall, further downstream. One of the limitations encountered by the authors was the vast amount of

time needed for the acquisition of WSS vectors using MRI.

Moyle et al. (2006) studied the effect of the velocity profile used as inlet boundary condition on

the CFD simulation results. The authors reported that inlet secondary flows break down after a few

diameters length, thus having a much smaller effect than geometry variation and uncertainty, hence

identifying geometry as the primary influence factor for simulations. Curvature of the artery, which

changes according to body and head posture, creates centrifugal forces and therefore change in

angular momentum, inducing pressure gradients. These results note that patient posture must be

controlled during measurements, since it may significantly affect velocity profiles through the CAB.

Research by Bressloff (2007) attempted to identify critical parameters of blood flow through Carotid

Artery by simulating only a fraction of the cardiac cycle, thus radically limiting the computational


21

time required. To achieve this, the author examined correlations among geometry and flow

parameters, and the most significant geometry parameter was found to be the sinus width. Bressloff

concluded that only half of a pulse had to be simulated in order for the solution to converge and to

achieve ample accuracy, though stating that further research is required to generalize these results.

Further studies (Younis, et al., 2007) examine the occurrence of turbulence due to stenosis, focusing

on the flow of blood through a diseased (stenosed) human Carotid Artery. The authors found

unstable jets in both the ICA and the ECA, caused by stenosis, which lead to increased turbulence

downstream. Similar results are reported by Lorenzini and Casalena (2008), which used CFD

simulations for an idealized cylindrical vessel with rigid walls, approximating blood viscosity as a

Cassonian fluid. The authors investigated the effects of plaque shape, dimensions and location on

blood flow through the vessel. Their results supported that higher stenosis levels lead to higher

strains, while concentration of plaques leads to greater local disturbance of the flow and turbulence.

Important recent studies of blood flow through the right coronary artery (Jung & Hassanein, 2008)

also include the different phases of blood in CFD simulations, namely plasma, red blood cells and

leukocytes, which lead to a highly Non-Newtonian viscous behaviour. The authors proved the known

fact that blood viscosity is mainly determined by RBC concentration, which is a result applicable to

blood flow through the Carotid Artery. A significant limitation noted by the authors is the amount of

computational time required to complete these multi-phase CFD simulations.

Research by Lee et al. (2009) aimed to determine which of the indices and factors used to

characterize disturbed blood flow are relevant and which are redundant. Using a sample of 50 normal

CAB, the authors reported a strong correlation among Time-Averaged WSS (TAWSS), OSI and

Relative Residence Time (RRT). The model used assumed Newtonian behaviour for blood and rigid

vessel walls, and included transient, pulsatile flow. The authors recommended RRT as an indicator of

regions of low, oscillating WSS.

Kalozoumis (2009) in his Engineering Diploma Thesis, which was used as a basis for the present

study, examined applications of CFD simulations to blood flow through CAB, using both laminar and

turbulent models for both steady and transient flow. This research used 6 geometry models taken

using MRI from 3 healthy volunteers. Kalozoumis’ results in principle agree with previous research,

calculating physiological velocities and predicting a separated flow region in ICA sinus, although very

high WSS were calculated. The author presumes this was caused by unsuitable boundary conditions,

since measured data were not available for inlet mass flow.

An interesting study of the effect of blood rheology into CFD simulations was presented by

Morbiducci (2011), evaluating the significance of taking into account the shear-thinning behaviour.

The author compared simulation results using the Newtonian, the Carreau-Yassuda and the Ballyk

viscosity model, concluding that the simulations are less sensitive to blood rheology, in comparison

with the effect of geometry reconstruction or boundary conditions uncertainty.

Research by Gallo et al. (2012) focused on helicity of blood flow through the Carotid Artery,

suggesting that helicity develops as a means of suppressing athero-prone hemodynamics, that is to

limit flow separation and oscillatory shear stress. More specifically, the authors proposed that

secondary flows at CAB are induced in order to mitigate flow recirculation, therefore helical flow

patterns could indicate disturbed shear stress at the bifurcation.

A recent study (Campbell, et al., 2012) attempted to identify the most accurate idealized inlet

velocity profile among blunt, parabolic and Womersley, to be used in simulations in case real

measured data are not available. The authors reported the idealized parabolic velocity profile proved

to be the most accurate at approximating results using measured velocity data, though noting that


22

the inlet boundary conditions selected are of little importance when compared with geometry

variation. This is emphasized by the results of (Zhang, et al., 2012), which indicated a significant

correlation of planarity and large curvature of Internal CA with low, oscillating WSS values, signalizing

the importance of geometry for these simulations.

Recent studies by Dong et al. (2013) attempted to model peripheral vascular impedance of the

arterial tree using a porous medium with transient permeability for the wall. In these simulations the

viscosity of blood was modelled by the Carreau-Yassuda mode, while the authors succeeded in

modelling the effects of peripheral vascular impedance, proposing that the same way of modelling

could also be implement in simulations including Fluid-Structure interaction, in order to additionally

include the vessel compliance effect. In another study, during the same year (Dong, et al., 2013), the

same authors evaluated a transitional SST turbulence model for blood flow in their simulations. Their

results indicated that plaque development increases mass flow and velocities, hence increasing WSS

values, which may cause plaque rupture.

Further research (Sharifi & Niazmand, 2015) attempted to investigate the significance of the Non-

Newtonian behaviour of blood by using multi-phase models in simulations. The authors examined

LDL concentration in blood, concluding that LDL accumulates in some specific shapes of vessel

geometry and noting the importance of studying LDL mass transport, since LDL concentration plays

a crucial role in the mechanism of atheromatous plaque development. Additionally, Non-Newtonian

effects were found to become much more important in sharp angles.

Kottarakou (2015) in her Engineering Diploma Thesis, based on Kalozoumis (2009), provides a

thorough literature review on CFD simulations of blood flow through CAB, including methods for

imaging, segmentation. reconstruction, measurements and simulations. Kottarakou also studies the

importance of acceleration in the flow field through CAB, focusing on the G-LOC phenomenon.

Mainly experienced by pilots. In her research, she investigates both laminar and turbulent flow

models for steady and transient flow, under variable inertial acceleration in the vertical direction.

Results agree with previous studies, showing helical flows in the CAB, mostly during the systolic

deceleration phase, accompanied by low WSS and high OSI values. Kottarakou also concludes that

increasing inertial acceleration leads to an increased number of smaller, scattered separation regions.

Finally, most recent studies (Sousa, et al., 2016) mainly focus on evaluating CFD simulation methods

for different models. These authors used 3D-US to capture the geometry and the flow parameters of

a patient. CFD simulation results, using Newtonian, incompressible, homogeneous fluid models,

assuming rigid vessel walls, were compared with measured values, confirming high accuracy of the

simulation methods. The outlet boundary condition used was a fixed ratio of mass flow through ICA

over CCA, rather common in recent research, also used in the current study.

As it may be seen by this short review, due to the relative immaturity of this field of study, most

research is focused on establishing some standards for the CFD simulations, evaluating models and

assumptions. In this context, there is still a lot of research to be conducted in order to fully understand

and predict hemodynamics of the CAB and their relation with plaque development, including the use

of more complex and complete models and determining the most suitable boundary conditions. A

step in that direction is including Fluid-Structure interactions in simulation models, thus accounting

for wall elasticity, compliance and impedance, which affect the cardiac pulse wave propagation and

the hemodynamics.


23

B.1.1.2 Fluid-Structure Interaction (FSI) CFD Applications

Elastic properties of vessel walls are an important factor in the flow field created during the cardiac

pulse and wave propagation through the arterial tree, therefore it is essential to understand the

underlying mechanisms and include these parameters into further studies of blood flow through

simulations. Early research by Bergel for both the static elastic (Bergel, 1961) and the dynamic elastic

properties of arterial walls (Bergel, 1961) provided a solid basis on which further research and analysis

built upon. However, due to increased complexity and computational time requirements of

simulations including Fluid-Structure Interactions (FSI) it is reasonable that such simulations became

readily applicable only during the past few years.

Early research using CFD simulations including FSI was conducted by Tang et al. using two different

specifications discribed in two different papers. In the first paper (Tang, et al., 1999) the authors

simulated a 3D tube with thin walls with symmetric or asymmetric stenosis, assuming the material to

be homogeneous, isotropic, incompressible and hyperelastic. The reported results include high

velocity and maximum WSS values at the throat and low velocity and WSS values, flow separation

and wall compression distal to the stenosed region. This situation may lead to thrombosis, due to

high WSS which activate the blood platelets, or plaque cap rupture, due to high shear strain.

Additionally, wall compression, or collapse in extreme cases, was reported distal to the stenosis, while

WSS at the throat increased with increased stenosis severity. Examining the asymmetric stenosis case,

the authors reported increased resistance and wall compression, higher values of WSS, while flow

velocities are reduces. In the second paper (Tang, et al., 1999), the authors the case of a 3D tube with

thick walls, assuming the material to be homogeneous, isotropic, incompressible and hyperelastic.

The velocity and pressure fields were reported to be similar to the thin wall case, although wall

deformation differs due to increased stiffness, while a large flow separation downstream of the

stenosed area also occured. Maximum WSS values still appear at the throat, while the wall was

reported to be stiffer for asymmetic stenosis than the symmetric case, which resulted to lower WSS

values and reduced flow. Wall compression was also noticed for extreme stenosis severity.

Researchers continued to model the hyperelasticity of vessel walls, using a strain energy function

(Daly, et al., 2000). In another paper (Cebral, et al., 2001) FSI simulations were used to account for the

vessel wall compliance, obtaining the CAB geometry using BB-MRI and the velocity field upstream

and downstream of the CAB using PC-MRA. The authors reported significant differences in the

velocity and WSS distributions between the cases of simulations with and without the FSI.

Later research by Tang et al. (2002), confirmed simulation methods and experimentally determined

the elastic modulus, bending stiffness and other properties of arterial walls. In their experiments, the

authors reported observing periodic wall compression and collapse, validating simulation results.

In their research, Zhao et al. (2002) suggested that regions of low WSS and high tensile mechanical

stresses are prone to atheromatous plaque development. For their studies, they used MRI and 3D-

US to acquire the geometry and velocity measurements, using these data, after segmentation of the

2D cross-sectional images and creation and smoothing of the 3D surface, for both CFD and FSI

simulations. The wall material was assumed to be linearly elastic, the fluid was modelled as

Newtonian, and the simulations were computed using CFX and ABAQUS software. WSS peak values

were reported to occur at the inner ICA wall, close to the CAB, while the lowest WSS were observed

at the outer wall of the ICA sinus, near the roots, where also the maximum mechanical stress value

was observed. Large variations were reported by the authors between the two models.

Cebral et al. (2002) used BB-MRI for geometry reconstruction and PC-MRI to measure physiological

boundary conditions for their studies, while blood was assumed to be a Newtonian fluid. Their


24

method was reported to be very robust, fast and efficient, and similar ones are used commonly in

literature. Their findings shown that including arterial wall compliance using FSI simulations it is of

great importance for hemodynamics simulations, since it resulted in significant differences on WSS.

Later studies by Tang et al. (2003) focused on the case of asymmetric stenosis, modelling the arterial

wall material as non-linearly elastic, assuming hyperelastic material with a pre-axial stretch, while

blood was assumed as Newtonian and the flow as laminar and incompressible. The authors reported

that stenosis asymmetry resulted in higher wall compression distal to the stenosis, increased WSS

values and larger flow separation region. Stenosis asymmetry also caused, according to the authors,

asymmetric distribution of WSS, with higher values at one side.

In further research, Gerbeau et al. attempted to establish an FSI simulation strategy from medical

imaging, using a robust and stable algorithm for the coupled Fluid-Structure problem. However,

most researchers emphasized on evaluating the importance of including FSI to the simulations

(Younis, et al., 2004). These authors used both CFD and FSI simulations to compute the flow field in

the CAB, assuming laminar flow, Newtonian and incompressible fluid and thick elastic arterial walls.

Their results from simple CFD simulations confirm previous findings reported in literature, with low,

oscillating WSS at the ICA bulb and high WSS values near the bifurcation apex. Although the authors

expected the FSI to have a negligible influence in the simulations’ results, taking into account the

arterial wall compliance greatly affected low OSI regions, while also having a secondary effect on

WSS calculated values. Maximum values of WSS Time-Gradient (WSSTG) was reported at the

bifurcation apex and it was suggested by the authors that this may lead to atherogenesis.

Comparison of CFD and FSI simulation results was also presented by Mauritis et al. (2007), who

emphasize on the importance of outflow conditions used for the simulations. According to these

authors’ findings, accounting for wall compliance resulted in a reduction in WSS values and lower

velocities during the systolic phase of the cardiac cycle, while velocities during the diastolic phase of

the cardiac cycle remained almost unchanged, suggesting that they are mainly affected by peripheral

resistance, rather than compliance of the larger arteries.

In studies reported by Kim et al. (2006) FSI CFD simulations were carried out, using the Carreau-

Yassuda viscosity model and Poiseuille’s formula, which results from the electric circuit analogy, as

boundary conditions at the outlets. The authors reported a significant effect of altered gravity, similar

to that reported by Kottarakou (2015) for inertial acceleration variation.

Further studies using FSI simulations confirmed or expanded previous results from simple CFD

simulations. Tan et al. (2008) reported their findings, suggesting that turbulent models may be

significantly more accurate than laminar ones. Kock et al. (2008) attempted to assess plaque

vulnerability and estimate the mechanical stresses occurring in the fibrous cap, using FSI simulations.

Their findings suggest that increased blood pressure and hypertension may result in higher stresses

in the wall and the plaque, with a risk for plaque rupture.

An interesting study by Nguyen et al. (2008) proposed a risk scale in order to quantify the influence

of carotid geometry on the danger for development of vascular disease, using an idealized,

parametric CAB geometry model. Reported findings by the authors suggested that larger ICA angles

result in lower WSS values and increased recirculation frequency. Additionally, larger off-plane angles

(non-planarity of the CAB) were found to increase the risk of atherogenesis, especially at high ICA

bifurcation angles. The authors reported that symmetric bifurcations were the most favourable,

regarding the risk for atherogenesis.

In later studies, Gao et al. (2009) intended to make an atherogenesis risk assessment in terms of

mechanical factors, using a reconstructed plaque morphology by multi-spectral MRI, a coupled FSI


25

simulation for laminar flow and a non-linear wall material model. In their findings, the authors

reported a relation between WSS and Wall Tensile Stress (WTS) in the arterial wall and plaque at the

stenosis, with higher WTS values in the luminal wall.

A more sophisticated way of modelling the arterial wall was proposed by Valencia & Baeza (2009),

using a two-layer anisotropic material to account for compliance and solving using the ADINA code.

The authors investigated the case of symmetric stenosis, using the Carreau-Yassuda viscosity model

for blood, and reported exponential increase in maximum values of WSS and velocity at the stenosis

throat and the size of the recirculation bubble distal to the stenosis, with increasing stenosis severity.

Later studies by Yang et al. (2010) investigated the progression of plaque development, which

initiates as a result of low, oscillating WSS. The authors carried out 3D FSI simulations, reporting that

plaque development led to an increase in both the fluid shear stress and plaque WSS, indicating

significant positive correlations between plaque thickness and fluid shear stress or plaque WSS. They

also noted that vessels are stressed and pressurized internally in-vivo.

Further studies (Vavourakis, et al., 2011) compared arterial geometries at the original in-vivo state

and the zero-pressure state. The authors reported significant variations, stating that not taking this

effect into account, simulations may underestimate the WSS and predict a larger area of low WSS

values, indicating the importance of correcting the arterial geometry before applying FSI simulations.

Findings by Lee et al. (2012) from FSI simulations using elastic wall material model which were

compared with previous results from CFD simulations, suggested that including wall elasticity to the

simulations leads to lower axial and secondary flow velocities.

In recent research, Toloui et al. (2012) investigated the influence of blood viscosity model and the

wall deformability, during different phases of the cardiac cycle. The authors’ findings suggested that

WSS distribution during steady-state flow, or almost steady-state, such as the final phase of the

cardiac cycle, is mostly affected by blood’s viscosity model, hence the shear-thinning behaviour of

blood is important during the end-diastolic phase, while wall compliance and deformability is

important during the first 2 phases of the cardiac cycle, when blood pressure is highest. For these

studies, the authors compared simple CFD and FSI simulations, using either the Newtonian or the

Carreau-Yassuda model for blood, parabolic velocity profiles at the boundaries and an idealized CAB

geometry. In summary, the Newtonian model predicted lower WSS values and larger recirculation

region, especially during the final phases of the cardiac pulse, and lower WSS peak values.

Recent studies by Deyranlou et al. (2015) focused on LDL mass transfer for simulations using a

Non-Newtonian blood model and permeable, elastic walls, in order to include the effect of

endothelium pores expansion on LDL accumulation. According to the authors’ findings, this effect is

increased for Non-Newtonian blood viscosity model and elastic walls, which may be explained since

Non-Newtonian behaviour of blood is enhanced close to the arterial walls. Increased wall elasticity

at high blood pressure was found to increase LDL accumulation, which may be an indicator that

hypertension could lead to or intensify cardiovascular diseases.

It is evident that the constant increase of computational power available is a favourable factor for

the progress of computational research concerning blood flow through the arterial tree. This

continuous development offers new possibilities, one of which is the ability to include wall

deformability into simulations. This led to intensive research during the past few years, which resulted

in many significant observations regarding the initiation and development of cardiovascular diseases.

The ultimate purpose of relevant research is to develop robust, stable and efficient methodologies

for practical, clinical applications in the future, which may vastly reduce diagnosis and treatment time

and improve current medical methods.


26

B.1.2 Other Research on Blood Flows through the Arterial Tree This section includes significant research on blood flow through the arterial tree in general, that is

not based on simulations of flow through the CAB. This research mainly focuses on wave propagation

and flow characteristics across the whole arterial tree, as modelled for computational simulations.

One of the first arterial tree models developed was the one by Avolio (1980), who improved the

electric circuit model developed by Taylor in 1966. It was a digital, multi-branched elastic tube model

used to study wave propagation through the arterial tree. Later research (Liu, et al., 1989) focused on

the dynamic characteristics of wave propagation and the effects of altering compliance, characteristic

impedance and the viscoelastic properties of vessel walls on the input impedance.

Further contribution by Holdsworth et al. (1999) focused on creating a universal flow waveform for

cardiac pulse, at the CAB, that could be used in the absence of patient-specific boundary conditions.

The results were produced by characterizing blood flows of 17 volunteers, using Pulse-Doppler US.

Wang & Parker (2004) developed a model including the 55 largest human arteries, terminated by

the arterial tree’s peripheral resistance, including the effect of complicated first and secondary wave

reflections at bifurcations. Their model allowed, by measuring the pulse wave at some point and

knowing the required transfer functions, to calculate pulse wave at any given point across the model.

Reymond et al. (2009) used a non-linear, viscoelastic wall material, 1D model to approximate the

systemic arterial tree and study wave propagation. This model was terminated by 3D Windkessel

models to account for distal vasculature, included the effect of wall compliance and assumed blood

to be a Newtonian fluid. The authors improved a previous model, by also including a model of the

heart, additional arteries and accounting for arterial wall viscoelasticity.

In their research, Janela et al. (2010) developed a stable, well-structured mathematical model for

the coupled problem including Fluid-Structure interaction between blood and hyperelastic arterial

walls. Their model could be adapted for both Newtonian and Non-Newtonian viscosity, such as the

Carreau-Yassuda. Their computational method for coupling the two problems was found to be stable

and robust for both the Newtonian and the Non-Newtonian viscosity models, allowing their

comparison. In further research (Janela, et al., 2010), the same authors applied linear absorbing

boundary conditions at the outflows, which used a relation between pressure and volumetric flow,

for a Carreau-Yassuda viscosity model and a hyperelastic structural model. This study evaluated a

way of incorporating local 3D FSI simulations into larger 1D models.

Other research (Harloff & Markl, 2012) investigated by in-vivo measurements (4D-MRI) and

confirmed significant correlations between geometrical parameters, such as the bifurcation angle,

tortuosity and ratio of ICA diameter over CCA diameter, and OSI or WSS distribution.

An interesting study of different methods for simulating blood flow in the aorta was presented by

Brown et al. (2012), who compared CFD simulations with rigid arterial walls and FSI simulations using

a compressible fluid to reproduce the effects of wall compliance. Their findings suggested that

compressible flow may be able to sufficiently approximate wall compliance for specific applications,

thus not needing the complexity of coupled FSI simulations to achieve the desired accuracy levels.

In a recent study, Kanaris et al. (2012) used the Cason viscosity model for blood, investigating the

viscosity and Hematocrit (Ht) effects on heart pumping power and pressure drop through the

arteries. Another comparison by Reymond et al. (Reymond, et al., 2013) focused on the differences

of a 3D model with rigid vessel walls, a 3D model including wall elasticity and FSI and a 1D model,

reporting that they all produced similar results for the pressure and flow waveforms, but varied

significantly in the WSS distributions.


27

Finally, studies by students of Aristotle University of Thessaloniki provided a thorough illustration

of current research methods and techniques. Such studies include the work of Vardoulis (2006), who

investigated 3D vessel reconstruction techniques from raw MRI data and studies of Vlachodimos

(2013) and Savvopoulos (2015), which focused on wave propagation through the arterial tree.

B.2 GENERAL OBSERVATIONS Chapter B included an overall literature review regarding previous research on CFD applications for

studies of blood flows, specifically through Carotid Artery Bifurcations. The main findings observed

while reviewing the aforementioned references, and numerous more sources, provide a guideline for

the present study, which is set to validate previously observed results.

It is expected that this study will confirm the common observation of low and oscillating WSS on

the outer walls of the ICA, especially for carotid arteries which have a large sinus. Another commonly

accepted result is the peak value of WSS at the bifurcation apex, while secondary flows are expected

to occur downstream of the bifurcation. Physiological ranges for the values of velocity, pressure and

shear stress have also been established and results of this study are expected to validate them.

This literature research also provides guidance and suggestions for the appropriate methods to

apply in order to study the problem of blood flow through CAB, using commercially available

software. Viscosity model selection for blood has been reported to play minor role in the simulation

results, which is expected to be the case for this study. Furthermore, the most suitable boundary

conditions and solver settings, widely tested in other references, were carefully selected in order to

achieve sufficient accuracy and approximate physiological and realistic results.


28

Methods Numerzical Analysis of Blood Flow through the Human Carotid Artery Bifurcation

29

METHODS This chapter includes a report of the different stages in the progress of completing the present

work. First, the method for geometrical reconstruction of carotid artery bifurcation models is

described, followed by a description of the construction of the computational meshes. Afterwards,

geometry parameters and details for the six CAB models studied are described. Solution methods

and boundary conditions are then presented, before an accuracy analysis for the above methods.

C.1 GEOMETRICAL RECONSTRUCTION OF CAB MODELS Geometrical reconstruction of the Carotid Artery models, based on MRI images, begins with 2D

cross-sectional images of the arteries, taken using MRI, and finishes with a digital stereolithography

(STL) geometry file. The complete process is described elsewhere (Vardoulis, 2006; Kottarakou, 2015;

Kalozoumis, 2009); however, a short summary is presented in this section.

As described in the aforementioned papers, the CAB geometries were taken from 3 health

volunteers, aged 24-26, non-smokers, with no history of cardiovascular diseases. An intravenous

contrast serum injection helped to enhance contrast between blood and other substances (bones,

tissues, etc.). A typical procedure of MRI followed, producing 208 cross-sectional images for each

volunteer. Image segmentation and 3D geometry reconstruction was completed using the ITK-SNAP

software, specialized in management of medical images, through a combination of automatic and

manual techniques. Finally, irrelevant vessels connected to the Carotid Artery were removed, imaging

errors and anomalies were corrected and surface smoothing was applied, resulting in the final

geometry, exported to a stereolithography file.

Stereolithography (STL) files for the present paper were taken from previous work done by

Kalozoumis (2009) and were ready to be imported into a Pre-Processing software for final geometry

corrections and mesh construction.

C.2 COMPUTATIONAL MESH CONSTRUCTION In order to create the final computational meshes which were used for the CFD simulations from

STL files, ANSA, a Pre-Processing software package by BETA CAE Systems, was used. The STL files

were imported to ANSA, then were translated and rotated in order to achieve an almost parallel inlet

flow into the CCA with the +Z axis. A combination of automatic and manual smoothing techniques

was then applied, especially in the vicinity of the bifurcation apex, where MRI resolution provided

poor imaging quality. This smoothing may be the main source of error, since it might significantly

alter the bifurcation geometry, depending on the software operator’s caution.

After surface grid smoothing and refinement, internal volume mesh was generated. Inflatable layers

were created first, in order to provide sufficient accuracy for calculations in the boundary layers near

the wall, while layer sizing was selected following a mesh independence study (see C.4.1). These

volume layers were 8 in number, with a first layer height of 0.01 𝑚𝑚 and a growth factor of 1.2.

Internal volume mesh, after constructing these layers, could be completed using either hexahedral

or tetrahedral elements. Comparison of both types was carried out using a steady-state CFD analysis

(see C.4.1) and hexahedral elements were selected, since the differences in results between

hexahedral and tetrahedral meshes were negligible, while the computational time required for

solving the problem with a tetrahedral computational grid was significantly increased. This may be

explained by the parallel orientation of hexahedral elements to the main direction of flow, which can

lead to an easier computational process in simple flow cases.


30

Mesh elements had to be fine enough to provide sufficient resolution to study flow characteristics.

Their size was selected after completing a mesh independence study (see C.4.1). Firstly, meshes with

different cell sizes were examined and it was found that a mesh with cell size selected such that the

number of elements was around 400K-500K was sufficient to provide the accuracy needed, in a

reasonable amount of computational time, and further refinement did not improve the accuracy

significantly. In this scope, minimum element length was fixed at 0.3 mm, while maximum element

length was fixed at 0.5 mm. Close to the bifurcation apex, further refinement was needed, so

maximum element length was selected to be 0.15-0.20 mm. The specified cell sizes resulted in

meshes with 400K-500K elements in all the cases studied.

Quality of the mesh was ensured using automatic quality improvements ANSA provides, after

manually setting the criteria according to which these modifications were implemented.

C.2.1 Carotid Artery Bifurcation Models – Geometry and Shape A summary of the most important geometric characteristics for the 6 computational meshes

created, is given in Table C.2.1.

Carotid Shell

Elements Number

Volume Elements Number

CCA Inlet Equiv.

Diameter [mm]

ICA/CCA Diameter

Ratio

ECA/CCA Diameter

Ratio

ICA/ECA Diameter

Ratio

L1 25392 464748 5.7902 58.5% 35.1% 166%

R1 24673 429366 5.7388 63.9% 31.2% 205%

L2 25197 453835 6.4318 68.7% 39.0% 176%

R2 24868 414471 6.3413 79.4% 45.5% 175%

L3 27602 495230 6.6125 74.6% 52.1% 143%

R3 24945 443211 6.6236 71.5% 54.0% 132%

Table C.2.1. Important geometric characteristics of the meshes for the 6 Carotid Artery Bifurcations studied.

As it can be deduced from the data presented in Table C.2.1, the size of the 6 meshes ranges

between 400K and 500K volume elements, while all 6 were generated from surface meshes of around

25K shell elements.

Additionally, it can be seen that there is a symmetry between left and the right Common Carotid

Artery of the same Subject, since the inlet diameters are almost equal, and the same can be noticed

for the outlet diameter ratio ICA/ECA, except for the first Subject, although these should not be

considered as some fixed geometric parameters, since there are not physiological flow data taken

from each volunteer at specific outlet points across the ICA and ECA, therefore the outlet boundaries

are not going to be applied arbitrarily at locations selected by the author who constructed the mesh.

These outlet areas were chosen as perpendicular as possible to the main flow direction, and at a

reasonable distance from the bifurcation, in order to let the blood flow develop physiologically.

In the following sections, each Carotid Artery Bifurcation’s main geometric parameters are

presented. The angles mentioned are qualitatively described, since their definition is ambiguous,

because there are no clear centerlines. In case such centerlines are calculated with certainty, which

requires much more accurate scanning and reconstruction, the bifurcation and the off-plane angles

may be defined with respect to the CCA, ICA and ECA centerlines.


31

C.2.1.1 Subject 1

C.2.1.1.1 Left Carotid Artery Bifurcation L1

The computational mesh generated for the Left CAB of Subject 1 (L1) can be seen in Figures C.2.1

and C.2.2. The exact number and distribution of elements can be seen on the left of the figures, while

other qualitative characteristics may be deduced from the figures.

There is a small, almost non-existent ICA bulb, with a maximum diameter of about 7.4 mm which

fades smoothly into the ICA. The bifurcation angle is small and the flow through the ICA and ECA

quickly becomes again parallel to the flow in the CCA. The off-plane angle is also small, since the two

branches deviate only slightly from the main plane, defined as the plane through the centerline on

which the largest area can be projected.

Figure C.2.1. Top view (-Z direction) of the Left Carotid Artery Bifurcation of Subject 1 (L1).

Figure C.2.2. Front view (+X direction) of the Left Carotid Artery Bifurcation of Subject 1 (L1).


32

C.2.1.1.2 Right Carotid Artery Bifurcation R1

The computational mesh generated for the Right CAB of Subject 1 (R1) can be seen in Figures C.2.3



There is a small ICA bulb in this case, too, with a maximum diameter of about 7.5 mm which fades

smoothly into the ICA. The bifurcation angle is small and the flow through the ICA and ECA quickly

becomes again parallel to the flow in the CCA, while the off-plane angle is also small. In general, this

case is very similar to the Left CAB of the same Subject, which has been described above.

Figure C.2.3. Top view (-Z direction) of the Right Carotid Artery Bifurcation of Subject 1 (R1).

Figure C.2.4. Front view (+X direction) of the Right Carotid Artery Bifurcation of Subject 1 (R1).


33

C.2.1.2 Subject 2


The computational mesh generated for the Left CAB of Subject 2 (L2) is presented in Figures C.2.5



This geometry is different from Subject 1, with a distinctive ICA bulb, narrowing rapidly from a

diameter of about 7.8 mm to about 5mm, before changing direction abruptly. The bifurcation angle

is larger, around 80° and then there is another angle of around 90° in the ICA, while the ECA continues

from the CCA almost in a straight line. Finally, the off-plane angle is very small in this case as well.




34


The computational mesh generated for the Right CAB of Subject 2 (R2) can be seen in Figures C.2.7



This geometry is very different from the Left CAB of Subject 2, while it is more similar to those of

Subject 1, since there is no visible ICA bulb. The bifurcation angle is small, the direction of the ICA is

almost parallel to the CCA, while at the ECA there is a significant bend. There is a larger off-plane

angle in this case, since the ECA and the ICA are moving out of the main plane, in opposite directions.




35

C.2.1.3 Subject 3


The computational mesh generated for the Left CAB of Subject 3 (L3) can be seen in Figures C.2.9

and C.2.10. The exact number and distribution of elements can be seen on the left of the figures,

while other qualitative characteristics may be deduced from the figures.

In this geometry there is a small ICA bulb fading from a diameter of about 7.5 mm to about 6.2

mm very smoothly, while the bifurcation angle is in a physiological range and the off-plane angle

almost zero.




36


The computational mesh generated for the Right CAB of Subject 3 (R3) can be seen in Figures

C.2.11 and C.2.12. The exact number and distribution of elements can be seen on the left of the

figures, while other qualitative characteristics may be deduced from the figures.

This last geometry also has no significant ICA bulb, since the change in ICA diameter is very smooth,

the bifurcation angle is very small, with the ICA and the ECA developing almost parallel to each other,

and the off plane angle is almost zero.




37

C.3 SOLUTION In this section, the complete methodology used for solving the problem is described. ANSYS Fluent

solver was used for the CFD simulations, due to its variety of possibilities and simplicity.

C.3.1 Solution Methods and Equations The Pressure-Based solver was selected, since there are no significant compressibility effects in the

problem investigated. A transient solver was used, in order to study the periodic, pulsatile flow of

blood. Gravity was neglected, since they are not relevant to this study and the direction of gravity

varies with body posture during normal movement. However, gravitational and inertial acceleration

must be taken into account in special cases, such as aircraft pilots (Kottarakou, 2015). Reference

values were set for each CAB model according to its geometry, using inlet diameter, area and velocity.

A coupled equation scheme was used, with second order pressure and momentum spatial

discretization, and second order implicit transient formulation. For other parameters, ANSYS Fluent

default values were used. Solution convergence was assumed when residuals were less than 10−4 for

the continuity equation and less than 10−5 for the 3 components of the momentum equation.

C.3.1.1 Laminar Model

Laminar model was used for blood flow, since no turbulence occurs at such low Reynolds number.

Previous researchers have reported that laminar modelling may sufficiently simulate blood flow

through arteries, except for the large conduit arteries. Blood density was assumed constant at

1060 𝑘𝑔 𝑚3⁄ . Two different viscosity models were studied, the Newtonian and the Carreau-Yassuda

model.

C.3.1.1.1 Newtonian Model

For the Newtonian model, a constant viscosity of 0.0035𝑘𝑔

𝑚∙𝑠 was assumed for blood, a value

commonly used in research (Reymond, et al., 2013).

C.3.1.1.2 Non-Newtonian (Carreau-Yassuda) Model

The Carreau-Yassuda model, a Non-Newtonian viscosity model, was selected since it approximates

the shear-thinning behaviour of blood under a wide range of strain rates, which is why it is used

widely in literature (Kim, et al., 2006; Morbiducci, et al., 2011; Dong, et al., 2013; Valencia & Baeza,

2009; Janela, et al., 2010; Janela, et al., 2010). The values selected were previously used in numerous

studies, such as the aforementioned papers by Janela et al. (2010).

C.3.1.2 Turbulent Model

Solution using turbulent models was not attempted in the present study, since for physiological

blood flow through non-stenotic CAB, maximum Reynolds number is around 200-300, which is far

below the laminar flow limit. However, studying stenotic arteries requires turbulence modelling, since

it has been reported repeatedly that turbulence occurs downstream of a stenotic area.

C.3.2 Boundary Conditions The boundary conditions used in the present study is its main limitation and source of error, since

it was not possible to obtain real physiological measurements for the 3 Subjects, hence physiological

boundary conditions according to relevant literature were used. This methodology certainly

introduces a significant error, which however should not prevent the extraction of valuable

conclusions through this study. Different combinations of inlet and outlet boundary conditions were


38

evaluated before selecting the most appropriate one, taking into account both qualitative criteria for

the solution and the computational time required, in cases of similar results.

C.3.2.1 Inlet

For the Common Carotid Artery Inlet, the volumetric flow pulse presented in Figure C.3.1 was used,

which is composed of a series of exponential terms, manually selected to approximate a realistic

pulse, in agreement with literature (Zhao, et al., 2002; Cebral, et al., 2002), based on previous work

done by Kottarakou (2015). Volumetric flow is equivalent to mass flow for blood, whose density is

constant.

Figure C.3.1. Volumetric flow pulse, used as a transient boundary condition for the inlet.

The pulse presented in Figure C.3.1 was adapted to create a parabolic profile in a UDF file coded

in C, using this value to estimate a mean mass flux, since density and inlet area are known. From that

mean mass flux value, the maximum value at the centreline was calculated and imposed in the

geometric centre of the CCA inlet and a parabolic profile was generated. Therefore, no extension was

needed at the inlet to ensure fully developed flow, since the parabolic profile was directly imposed.

In addition to the mass flux pulse boundary condition, a pressure pulse initial condition as depicted

in Figure C.3.2 was used, in order to enhance convergence.

C.3.2.2 Outlets

For the outlets, pressure pulses were imposed as boundary conditions. The pressure pulses were

selected in a physiological range for the CAB, according to literature (Cebral, et al., 2002), while ICA

outlet pressure was selected 3 mmHg lower than CCA inlet pressure and ECA outlet pressure was

selected 2 mmHg lower than ICA pressure. Apart from these pressure boundary conditions, another

condition for a fixed mass flow distribution of 65% 𝐼𝐶𝐴 − 35%𝐸𝐶𝐴 was selected and imposed as a

target mass flow, using the relevant ANSYS Fluent option, after coding a UDF file, in a similar way to

the one for the inlet mass flow boundary condition, with no need for a parabolic profile in this case.


39

Figure C.3.2. Pressure pulse, used as a transient boundary condition for the CCA inlet and the ICA and ECA outlets.

C.4 ACCURACY This section covers a description of the methodologies followed for selecting appropriate

parameters and characteristics for the simulation, such that the solution achieved may be trusted,

independently of the time and spatial discretization of the model.

C.4.1 Mesh Independence Two different parameters of mesh size were investigated. Firstly, the size and shape of the inflation

layers had to be selected, which was examined while keeping the shell mesh unchanged. After

selecting suitable layer height and inflation factor, a mesh size independence study was carried out

in order to select appropriate shell and volume element size.

C.4.1.1 Layer Height Independence

Concerning layer height, 4 different cases for layers’ size were investigated, by comparing different

selections for the first layer height and for the total number of layers, for two different shell and

volume element size settings (1R_2_ and 1R_4_). The results are presented in Figures C.4.1, C.4.2 and

C.4.3, where some characteristic variables are plotted for the different cases of layers.

As it can be seen in Figure C.4.1, the pressure distribution along a straight line through the CCA

and the ICA was almost the same for all the cases studied, with main differences located at the CCA

and a perfect convergence downstream of the bifurcation. The error is considered negligible;

therefore, this graph did not indicate a significant criterion to select appropriate layer size. In

addition, examining Figure C.4.2, WSS distribution on the outer walls of the CCA, ICA and ECA is also

relatively independent of layer size.


40

Figure C.4.1. Pressure distribution along a straight line through the Common Carotid Artery (CCA) and the Internal

Carotid Artery (ICA) of the Carotid Artery Bifurcation model R1.

Figure C.4.2. Wall Shear Stress (WSS) distribution along two curves on the outer walls of the Internal Carotid Artery

(ICA), External Carotid Artery (ECA) and the Common Carotid Artery (CCA) of the Carotid Artery Bifurcation model

R1, in the +Z direction. These curves resulted as the intersection of the main plain and the outer walls of the CAB.


41

Figure C.4.3. Wall Shear Stress (WSS) distribution along a closed curve around the bifurcation apex area of the

Carotid Artery Bifurcation model R1.

As depicted in Figure C.4.3, the WSS distribution varies significantly with different layer size, and

relative convergence is achieved for layers with a first layer height of 0.01 mm and a total layer height

of about 0.0358 mm, distributed in 8 layers (1R_4_hexa). Further refinement of the layers provides no

significant improvement in accuracy.

C.4.1.2 Element Size Independence

A similar study was performed for the volume mesh, in order to choose an appropriate maximum

element size. According to relevant literature, for similar artery sizes, around 500K-1M elements are

enough to achieve mesh independence. Therefore, meshes of 650K elements (1R_1_hexa), 250K

elements (1R_2_hexa), 350K elements (1R_4_hexa), 1M elements (1R_3_hexa) and 1.3M elements

(1R_5_hexa) were constructed, in order to estimate the element size required for mesh independence

of the CAB models. All geometries are similar in dimensions and shape; therefore, similar mesh sizes

were created for all 6 cases.

As it can be seen in Figure C.4.4, pressure along a straight line through the CCA and the ICA is

relatively independent of volume element size.

Additionally, as seen in Figure C.4.5, WSS distribution of WSS on the outer walls of the Carotid

Arteries is also independent of volume element size.

However, examining figure C.4.6, it can be seen that WSS distribution around the bifurcation apex

area, a region critical for this study, converges for mesh sizes larger than 500-600K, which was the

reason for selecting the appropriate settings to construct computational meshes of around 500K.

Another parameter taken into account was the computational time needed to reach a final solution,

since the clinical application of this process shall not be disregarded; in order for widespread use of

CFD simulations for blood flows in the future, the whole procedure has to be fast and flexible.


42

Figure C.4.4. Pressure distribution along a straight line through the Common Carotid Artery (CCA) and the Internal

Carotid Artery (ICA) of the Carotid Artery Bifurcation model R1.

Figure C.4.5. Wall Shear Stress (WSS) distribution along a closed curve around the bifurcation apex area of the

Carotid Artery Bifurcation model R1.


43

Figure C.4.6. Wall Shear Stress (WSS) distribution along two curves on the outer walls of the Internal Carotid Artery

(ICA), External Carotid Artery (ECA) and the Common Carotid Artery (CCA) of the Carotid Artery Bifurcation model

R1, in the +Z direction. These curves resulted as the intersection of the main plain and the outer walls of the CAB.

C.4.2 Time-Step Independence A time-step independence study was not performed, since it is reported in literature that time-step

independence is easily achieved for time-step of the order of 0.005 s or lower. For the present study,

a time-step of 0.005 s was selected, using a smaller time-step of 0.001 s for the first 10% of total

solution time in order to favour early convergence.

C.4.3 Periodicity A period of 𝑇 = 0.8 𝑠 was selected for the cardiac cycle (75 𝑏𝑝𝑚), which is in the physiological

range for healthy subjects. The pulsatile boundary conditions at the inlet and the outlets were

imposed according to this period. The solution in the whole flow field has to be periodic and in order

to achieve such periodicity, a total of 10 cardiac cycles were simulated for each case. According to

literature, periodicity was expected to occur after completing 2-5 cardiac cycles, hence it is expected

that the rest of the cardiac cycles are going to be identical.

The findings firmly confirm this hypothesis. According to the present study, periodic solution was

established after just one complete cardiac cycle, as confirmed by Figures C.4.8 and C.4.9. Although

these figures present results for the case L1, similar behavior was noticed in all 6 cases. In Figure

C.4.7, velocity at a specific point in the ICA bulb segment near the inner wall demonstrates relatively

fast convergence, since the error during the systolic phase of the first cardiac cycle is less than 0.1%

when compared with the rest cardiac cycles.


44

Figure C.4.7. Time progress of velocity magnitude at a point inside the Internal Carotid Artery bulb, close to the

inner wall for the case L1. It is evident that periodic solution is achieved very early, during the first cardiac cycle.

In Figure C.4.8, WSS is plotted versus time for the duration of 10 cardiac cycles. Significant deviation

is noticed in the systolic phase of the first cardiac cycle, compared with the rest. However, after just

one cardiac cycle, solution convergence is essentially achieved.

Figure C.4.8. Time progress of velocity magnitude at a point inside the Internal Carotid Artery bulb, close to the

inner wall for the case L1. It is evident that periodic solution is achieved very early, during the first cardiac cycle.

Therefore, considering the aforementioned results, computational results suggest that periodic

solution convergence for pulsatile flow is established after simulating just 2 cardiac cycle periods and

further continuation of the simulations does not improve the solution accuracy.

In the rest of this report, results are going to be presented based on the tenth and final cardiac

cycle simulated, since the solution is periodic after the first cycle and the results are identical.

Results Numerzical Analysis of Blood Flow through the Human Carotid Artery Bifurcation

45

RESULTS This section provides a presentation of the results of the simulations, carried out following the

methodology described in the previous chapter. For the presentation of the results, time-steps were

saved only every 0.05 seconds (16 time-steps per cardiac cycle) due to extremely large file size.

D.1 IMPORTANCE OF THE VISCOSITY MODEL One of this study’s goals is to evaluate the importance of blood’s viscosity model concerning

simulation accuracy. Previous researchers (Morbiducci, et al., 2011) have reported that the blood

viscosity model plays a minor role in the accuracy, while geometry is the most significant parameter.

Examining Figure D.1.1, it is evident that WSS at a point on the outer wall of the ICA bulb for two

of the cases studied, is relatively indifferent to blood’s viscosity model used. In this figure, the use of

a Non-Newtonian viscosity model, the Carreau-Yassuda model, to account for blood’s shear-thinning

behaviour, provides a smoother, more realistic time distribution of WSS, reducing, or even

eliminating sharp peaks. However, using a more sophisticated viscosity model for blood provides

only minor improvement in accuracy, which indicates that blood viscosity through the Carotid Artery

is relatively constant, around its infinite shear-rate value, thus shear-rates occurring are high.

Since the results are indifferent to the viscosity model used, the rest of this and the following

chapters present the results for the Newtonian model, unless differently stated.

Figure D.1.1. Comparison of Wall Shear Stress on the outer wall of the ICA bulb for the cases L1 and L2, for different

blood viscosity models (Newtonian and Carreau-Yassuda viscosity models).


46

D.2 FLOW FIELD AND WALL SHEAR STRESS DISTRIBUTION In this section, results regarding blood flow near the bifurcation area are presented. These results

are compared with previous findings reported in literature, in order to validate their credibility.

Firstly, in Figures D.2.1, D.2.2, D.2.3 the Wall Shear Stress and Velocity distributions are presented

for Subjects 1, 2, 3, respectively. In each of the figures, the top, middle and bottom rows correspond

to the systolic acceleration (velocity peak) phase (𝑡 = 𝑇 8⁄ ), systolic deceleration phase (𝑡 = 3𝑇 16⁄ )

and diastole phase (𝑡 = 13𝑇 16⁄ ), respectively, while the left column depicts the Left CAB and the

right column depicts the Right CAB for each subject.

Figure D.2.1. Blood flow through the Left L1 (left column) and the Right R1 (right column) Carotid Artery

Bifurcations during the systolic acceleration (top), systolic deceleration (middle) and diastole (bottom) for Subject

1. In each picture, the left colourbar corresponds to the Wall Shear Stress magnitude [in Pa] and the right colourbar

corresponds to the Velocity magnitude [in m/s].

In Figure D.2.1 the blood flow through the Left and Right CAB of Subject 1 is presented. As it may

be seen in the top row, during the systolic acceleration phase of the cardiac cycle, no recirculation is

possible, due to the rapid flow increase. Blood flow passes mainly close to the inner walls of the

bifurcation, with velocity decreasing smoothly towards the outer walls. The flow field drastically

changes after the peak velocity, during the systolic deceleration phase, as seen in the middle row.

There is a further increase in velocity magnitude near the inner walls of the CAB, while velocity at the

centers of the ICA and the ECA and close to the outer walls decreases, creating secondary flows, i.e.

two counter-rotating vortices in each artery. In the ICA, the vortex near the outer wall is vastly


47

larger than the one near the inner wall, while in the ECA the two vortices are of the same size. During

diastole, velocity remains very low near the outer walls, although the two vortices disappear.

Concerning WSS distribution, which is further discussed later, there is a peak value at the bifurcation

apex, which is of the order of 70 ÷ 110 𝑃𝑎 at the end of the systolic acceleration phase, while the

values near the apex on the inner walls, typically of the order of 10 ÷ 15 𝑃𝑎, are higher than those

on the outer walls. Minimum WSS, even lower than 0.5 𝑃𝑎 which is considered as the lower limit for

physiological conditions, occurs mainly at the roots of the ICA and ECA branches, on the outer walls.

WSS values are significantly lower during the rest of the cardiac cycle, when velocity values are lower.





A similar picture for Subject 2 is presented in Figure D.2.2. As discussed in Chapter C, Subject 2 has

geometrically different Left and Right CAB. The Right CAB is relatively straight, with almost parallel

outflows through the ICA and the ECA, while the Left CAB has a relatively large bifurcation angle,

with the ICA almost perpendicular to the ECA after the bifurcation. During the systolic acceleration

phase, there is, as expected, no indication of recirculating flow, due to the rapid increase in flow

velocity. However, during the systolic deceleration phase secondary flow structures appear, in the

same form as for Subject 1, with two counter-rotating vortices in each Carotid branch. This happens

in both the Left CAB, with the abrupt change in geometry, and the Right CAB, which is seemingly

smoother. In the Right CAB, the largest part of the ICA bulb is occupied by two vortices, one larger


48

than the other, while in the Left CAB, due to the abrupt change in flow direction, a recirculation vortex

appears lower in the ICA bulb, at its root, so it is not visible in the figure. In the ECA, two small vortices

are evident for both the Left and the Right CAB. During the diastole, again flow velocity is extremely

low, especially in the ICA of the Right CAΒ, while there are no significant secondary flows.

Similarly to the previous subject, WSS peak values of the order of 70 ÷ 110 𝑃𝑎 occur at the

bifurcation apex at the end of the systolic acceleration phase, while significantly lower values occur

during the other phases of the cardiac cycle. Minimum values of WSS also appear on the outer walls

of both the ICA and the ECA of Subject 2.





Finally, the corresponding results for Subject 3 are presented in Figure D.2.3. The findings are very

similar to the other two subjects, with no secondary flows during the systolic acceleration phase, due

to the rapid increase in velocity, and the diastolic phase, due to extremely low velocity. As expected,

secondary flow patterns, with two counter-rotating vortices in each branch, appear during the

systolic deceleration phase. There is a larger ICA sinus at the L3, hence this phenomenon is more

intense in the Left ICA than the Right ICA of Subject 3.

Regarding WSS distribution, similar results to the previous two subjects are observed. Peak WSS

values occur at the bifurcation apex at the end of the systolic acceleration, with lower values during

the rest of the cardiac cycle and minimum values on the outer walls of the ICA and the ECA.


49

Figure D.2.4. Blood flow streamlines through the Left (left column) and the Right (right column) Carotid Artery

Bifurcations for Subject 1 (top – L1 and R1), Subject 2 (middle – L2 and R2) and Subject 3 (bottom – L3 and R3)

during the systolic deceleration phase of the cardiac cycle. In each picture, the colourbar corresponds to the Velocity

magnitude [in m/s].

The streamlines for Subjects 1, 2, 3 during the systolic deceleration phase, when the secondary flow

patterns are more intense, are presented in Figure D.2.4. Helical flows are evident near the outer walls

for all 6 cases, with varying severity.

For Subject 1, at L1 the main core of the blood flow passes through narrow segments of the ICA

and the ECA near the inner walls, while extremely helical flow with low velocity is evident near the

outer walls. Due to the smaller diameter of the ECA, the secondary flow is limited at the root of the

ECA branch, while helical flow through the ICA continues to flow further downstream of the

bifurcation. At R1, the flow core is even narrower, with larger secondary flow regions near the outer

walls of the ICA and ECA.

For Subject 2, at R2 the flow patterns are similar to the case of Subject 1, with secondary vortices

near the outer walls at the roots of the ICA and the ECA, with helicity through the ICA retained further

downstream. However, at L2, due to the larger bifurcation angle the flow field is slightly different. A

very large helical flow region appears very early, upstream of the bifurcation, near the outer wall of

the ICA, but through the ICA bulb the helicity decreases rapidly.

For Subject 3, the flow field is similar to Subject 1, with large helical flow regions near the outer

walls of the ICA bulbs at both L3 and R3, larger for the ICA at L3, and smaller, shorter secondary flow

regions near the outer walls of the ECA for both the L3 and the R3 cases.


50

Figure D.2.5. Low Wall Shear Stress regions (𝑾𝑺𝑺 < 𝟎. 𝟓 𝑷𝒂) for the Left (left column) and the Right (right column)

Carotid Artery Bifurcations for Subject 1 (top – L1 and R1), Subject 2 (middle – L2 and R2) and Subject 3 (bottom –

L3 and R3) at the end of the systolic acceleration phase of the cardiac cycle. In each picture, the colourbar

corresponds to the Wall Shear Stress magnitude [in Pa].

Figure D.2.5 presents the critical regions for atherogenesis in the vicinity of the Carotid Artery

Bifurcation for Subjects 1, 2, 3. As critical for atherogenesis are considered regions of extremely low

and oscillating WSS, i.e. 𝑊𝑆𝑆 < 0.5 𝑃𝑎 at the end of the systolic acceleration phase of the cardiac

cycle, while oscillation may be quantified using the OSI, which has been found to correlate strongly

with the WSS (Lee, et al., 2009), hence it has not been examined in the present study. Blue-coloured

areas in Figure D.2.5 represent regions of 𝑊𝑆𝑆 < 0.5 𝑃𝑎 , which are prone to development of

atheromatous plaque. It is evident that these regions occur on the outer walls of the ICA and ECA

arterial branches, with typically larger regions on the ICA walls than the ECA walls, especially in the

case of Subject 1. For Subject 2, as discussed earlier, a low WSS region occurs relatively early

upstream of the bifurcation, on the outer wall of the ICA sinus.

It must be noted that for calculations using the Non-Newtonian, Carreau-Yassuda viscosity model

the above areas were slightly smaller, thus confirming the statement that the shear-thinning

behaviour of blood contributes to limiting extreme values of WSS, either low or high. Moreoever,

areas of high helicity and vorticity values were also slightly decreased. No further changes occurred

for the simulations using the Carreau-Yassuda viscosity model, apart from this negligible reduction

in the size of low WSS and high helicity regions.

Discussion Numerzical Analysis of Blood Flow through the Human Carotid Artery Bifurcation

51

DISCUSSION This chapter includes a quick review of the results presented in Chapter D, concerning their validity

and their agreement with results previously reported in relevant literature.

E.1 IMPORTANCE OF THE VISCOSITY MODEL Summarizing the results of the present study, numerical evidence suggests that the viscosity model

does not play a crucial role in the hemodynamics of the Carotid Artery Bifurcation, as previously

reported by Morbiducci et al. (2011). More specifically, a comparison between the Newtonian and

the Non-Newtonian Carreau-Yassuda models was carried out, indicating only minor differences

concerning the flow field through the CAB for all the 6 cases studied.

As explained in the previous chapter, using a Non-Newtonian viscosity model for blood, the

Carreau-Yassuda model, to account for blood’s shear-thinning properties results in only small

changes in WSS, velocity and other flow variables. These alterations are negligible and are mostly

located at peak values, both maximum and minimum, while average values and trends are perfectly

predicted using the Newtonian viscosity model, as seen in Figure D.1.1; hence the Newtonian model

is considered sufficient for blood flows in arteries with high shear-rates, especially when variables

and values close to the walls, such as WSS, are of great importance. The Carreau-Yassuda viscosity

model is found to smooth excessively high or low values, providing more realistic results, although

this improvement is negligible. Additionally, using the Carreau-Yassuda model, smaller low-WSS and

high-helicity regions were calculated; therefore, using the Newtonian viscosity model, simulations

remain on the safe side, according to the results of this study.

E.2 WALL SHEAR STRESS DISTRIBUTION It has been evident and thoroughly described in literature that the most significant factor regarding

arterial hemodynamics and plaque development at the Carotid Artery Bifurcation is the Wall Shear

Stress. WSS distributions calculated in the present study, as depicted in Figures D.2.1-D.2.3 and D.2.5,

are in perfect agreement with previously reported results. Maximum values of WSS occur at the

bifurcation apex in all 6 cases examined, with values ranging between 70 ÷ 110 𝑃𝑎 at the end of the

systolic acceleration phase, and lower values during the rest of the cardiac cycle. WSS values decrease

rapidly while moving away from the bifurcation apex, with physiological values of the order of 10 𝑃𝑎

at the peak of the systolic acceleration phase and 1.5 𝑃𝑎 during the rest of the cardiac cycle, similar

to those previously reported in literature. Lowest WSS values occur on the outer walls of the ECA and

the ICA bulb, near the roots of the two branches, with maximum values at the end of the systolic

acceleration phase of the order of 0.5 ÷ 1 𝑃𝑎.

Bifurcation angle is confirmed to be a significant geometric parameter, greatly affecting WSS

distribution on the CAB walls, determining the location of secondary flow and low WSS regions.

Numerical results indicate that higher bifurcation angle leads to significant helical flow region at the

root of the ICA bulb, though the flow downstream remains physiological with low helicity, while lower

bifurcation angle and seemingly more favourable straight ICA and ECA branches lead to increased

helicity in the ICA bulb which is retained further downstream due to the absence of curvature to

damp disturbances of the flow. Similar results are reported in previous research (Milner, et al., 1998;

Gallo, et al., 2012), with helical flows occurring at disturbed flow regions, correlating with low WSS

values. As suggested by Gallo et al., helical flow may be a response of blood flow to mitigate flow

separation and recirculation, limiting flow disturbances, a hypothesis supported by present findings.


52

E.3 HELICAL SECONDARY FLOWS AND FLOW VELOCITY PROFILES Velocity profiles at the inlet of the CCA were assumed parabolic in the present study, a common

assumption in relevant literature, inducing minor errors since there is sufficient length in the CCA for

the flow profile to become fully developed. However, velocity profiles immediately upstream and

downstream of the bifurcation are far from parabolic; present findings suggest that flow is limited in

the central part of the CCA near the bifurcation, with small flow separation and recirculation regions

near the outer wall, while extreme flow disturbances occur in the bifurcation area. Secondary flows

are induced due to the branching of the CCA into the ECA and the ICA near the outer walls, with

strongly altered velocity profiles; excessively high velocity values appear near the inner walls and

lower velocity values appear close to the outer walls, where there are helical flow regions and maybe

recirculating, separated flows areas, during the deceleration phase of the cardiac cycle. During

diastole, flow inversion may occur in these areas, since average flow velocity is significantly lower

and secondary flows cannot be retained, in order to suppress flow separation (Gallo, et al., 2012).

Helicity and secondary flows, which appear at the bifurcation due to curvature of the flow, are

retained further downstream, depending on the subsequent geometry of the artery. Considering the

rest of the arterial tree, with narrower vessels and more frequent branching downstream of the CAB,

the flow may not have the time to develop parabolic or general Womersley profiles after the Carotid

Artery. According to numerical results of this study, helical flows are retained further downstream

and obstruct a bigger part of the ICA in case of low bifurcation angles and relatively straight Carotid

Artery branches, while in case of higher bifurcation angle, flow separation and recirculation occurs in

a limited area near the outer wall of the ICA sinus, at the root of the ICA branch, with no significant

secondary flows downstream of the sinus.

Conclusions Numerzical Analysis of Blood Flow through the Human Carotid Artery Bifurcation

53

CONCLUSIONS Studying blood flow through the arterial tree is a broad, complex field which requires elements

from Medical Science, Biomechanical Engineering and Computational Fluid Dynamics. The intention

driving the present study was to explore these growing fields of research, assess the current state of

knowledge and evaluate commonly used models.

Having concluded this work, a thorough review of relevant literature has been carried out, to

summarize progress over the last years in this field. Assessing the current state and needs in the field,

this research set out to validate the significance of the choice of blood viscosity model and evaluate

the different stages required to fully implement a numerical analysis of the blood flow through the

Carotid Artery Bifurcation, from its inception to the presentation of the results, in order to examine

the possibilities for clinical application.

Concerning the viscosity models tested for blood, the Non-Newtonian Carreau-Yassuda model did

not result in significant variation from the Newtonian model, thus indicating that the shear-thinning

behaviour of blood, accounted for by the Carreau-Yassuda model, is unimportant in such simulations.

Insignificant differences between the two models were noticed mostly at peak values and regions of

minimum Wall Shear Stress and high helicity. Numerical simulations of blood flow through Carotid

Artery Bifurcations are relatively insensitive to shear-thinning effects, since the shear-rates that occur

in the CAB, especially near the wall, are sufficiently high; therefore, the apparent dynamic viscosity

value of blood is relatively constant. However, the importance of the viscosity model selected has to

be further examined in the cases of accounting for blood’s multiple phases and the compliant and

elastic arterial walls. Since the Carreau-Yasuda model does not increase the complexity of the model

significantly and was not found to require more computational time, it may be used even in case it

does not greatly improve accuracy, provided that the required parameters are carefully selected.

Calculated Wall Shear Stress time and spatial distributions are in perfect agreement with previous

reports in relevant literature, validating the accuracy of the models used in the present study, using

the commercial software package ANSYS Fluent, which is easy to use and widely available, hence it

has numerous possibilities for clinical application. Existence of regions of secondary flows and

recirculation, correlated with low, oscillating WSS was confirmed, located at the roots of the ICA bulb

and the ECA branch. These regions cover the largest part of the vessel and are often characterized

by excessively helical flows, which, according to previous studies (Gallo, et al., 2012), are induced in

order to limit flow recirculation. Flow velocity in these secondary flow regions is extremely low,

leading to flow inversion during the diastolic phase of the cardiac cycle, low and oscillating WSS and

resulting in long particle residence time, providing favourable conditions for the development of

atheromatous plaque, since LDL molecules are able to intrude the openings among endothelial cells.

The present study complements previous research work, clarifying and confirming results regarding

the significance of blood’s viscosity model and WSS distribution in such simulations. Past, current

and future research in this field, i.e. Numerical analysis of blood flow through arteries, shall aim to

allow for fast, robust and accurate clinical applications of CFD codes, in order to improve prognosis

and diagnosis techniques for cardiovascular diseases.


54

F.1 LIMITATIONS The most important limitation of this study is the use of generic boundary conditions, instead of

patient-specific data obtained from in-vivo measurements. The use of a parabolic velocity profile at

the inlet introduces a minor error, since the CCA length is sufficient for damping the inlet boundary

condition. However, the 65% − 35% split of the flow between ICA and ECA, considering the arbitrary

determination of the outlets where the pressure boundary conditions are imposed, introduces errors.

In general, physiological, average pressure outlet conditions minimize such errors, although highest

accuracy is achieved by imposing patient-specific boundary conditions from measurements.

Another important source of error lies in the imaging methods used, which can be improved

following the technological progress. Cross-sectional 2D MRI images used in this study were taken

by Kalozoumis (2009) and recent advances in MRI and US technologies may provide better accuracy

and further limit imaging errors. Moreover, MRI and US imaging procedures are not standardized

and variations of patient posture, machine and operator controls may affect their outcome.

Furthermore, 2D images segmentation and 3D geometry reconstruction require a combination of

manual and automatic methods, hence subjective parameters may alter the resulting arterial

geometry. Development and implementation of fully automatic and robust algorithms for the whole

process from 2D image segmentation to 3D geometry reconstruction and computational mesh

generation, is essential for successful and efficient clinical application of CFD for hemodynamics.

F.2 SUGGESTIONS FOR FURTHER RESEARCH The broad field of CFD simulations in Hemodynamics cannot be fully investigated in a single paper;

this study aimed to evaluate the possibilities of CFD applications and to validate widely used models,

comparing its findings with results reported in literature. However, this work is not exhaustive and

there is progress to be made in order to fully explore the field of hemodynamics and to allow for

clinical utilization of CFD codes for the prognosis, diagnosis and treatment of cardiovascular diseases.

The first step in that direction is to establish suitable CFD models for hemodynamics research. There

has been significant research in this direction, though the diverse results cannot be combined and

often contradict each other. The use of turbulent instead of laminar models and the use of Non-

Newtonian models must be further investigated, while considering other non-linear effects apart

from blood’s shear-thinning behaviour. The effects of multiple phases in blood flows have to be

further examined regarding the Carotid Artery, emphasizing on LDL accumulation.

The most important expansion possible for the present study is including the interactions between

blood and arterial walls (FSI) in the simulations. Fully coupled FSI simulations provide much more

realistic results and capture the complete dynamic response of the phenomenon in question,

allowing for thorough examination of flow dynamics in the region of the CAB. However, FSI

simulations must not be carried out without solid mathematical modelling of the non-linear

properties of arterial walls. Accurate FSI modelling is the way to distinguish over-simplifications from

reasonable assumptions, and create complete CFD models and tools for clinical implementation.

Finally, a quantitative relation between parameters of blood flow and the development of

atheromatous plaque or the progress of cardiovascular diseases has to be developed. A

mathematical model between the arterial geometry parameters, measured data and simulation

results, estimating risk factors for patients with varying stenosis severity, will be a powerful tool

providing guidance for appropriate treatment methods and evaluating the need for invasive, surgical

procedures by medical scientists. This shall be the ultimate goal for all relevant research in the field.

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55

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