MODELING OF BLOOD FLOW IN STENOSED ARTERY WITH A CLOT

JALILAH BINTI RUBAI

UNIVERSITI TEKNOLOGI MALAYSIA

Modeling of Blood Flow in Stenosed Artery with a Clot

JALILAH BINTI RUBAI

A dissertation submitted in partial fulfilment of the

requirement for the award of degree of

Master of Science (Engineering Mathematics)

Faculty of Science

Universiti Teknologi Malaysia

June 2014

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To my beloved mother.

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ACKNOWLEDGMENT

Foremost, I would like to express my sincere gratitude to my supervisor Prof. Dr.

Norsarahaida Binti Saidina Amin for her patience, motivation, enthusiasm, and

immense knowledge in guiding me to complete this dissertation.

I would like to thank my family for supporting me throughout my studies. Last but

not least to all my colleagues for their encouragement and understanding.

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ABSTRACT

The presence of blood clot in the blood stream can lead to partial or even

complete blockage of the blood vessel. The condition can get worse in a stenosed

artery. The clot will alter the flow field and modify the pressure distribution. To

understand the changes in the blood flow due to presence of the clot, the fluid

mechanics of blood flow in a stenosed artery with a clot is studied through a

mathematical analysis. Blood is modelled as an incompressible Newtonian fluid and

the flow is treated to be axi-symmetric. The artery is modelled as a rigid tube with a

stenosis and an elongated shape representing a clot inside it. Both the stenosis and

the clot are axi-symmetric over the length of the artery. The governing equations of

motion are solved by using regular perturbation analysis. The results show that in

the presence of the clot, the important physiologically flow characteristics such as

pressure gradient and wall shear stress vary markedly across the stenotic lesion. The

height of the peak of the clot increases the wall shear stress. Results are sensitive to

not only the geometry of the stenotic wall but also to the position of the clot.

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ABSTRAK

Kehadiran darah beku di dalam aliran darah boleh menyebabkan salur darah

tersumbat separa atau tersumbat sepenuhnya. Keadaan boleh menjadi lebih buruk di

dalam arteri dengan stenosis. Gumpalan akan mengubah medan aliran dan

mengubah taburan tekanan. Untuk memahami perubahan dalam aliran darah

disebabkan kehadiran gumpalan, mekanik bendalir aliran darah dalam arteri yang

mengandungi gumpalan dikaji melalui analisis matematik. Darah dimodelkan

sebagai cecair Newtonian yang tidak boleh mampat dan aliran dianggap simetri

terhadap paksi. Arteri dimodelkan sebagai tiub yang rigid dengan stenosis dan

sebuah bentuk memanjang yang mewakili gumpalan di dalamnya. Stenosis dan

gumpalan adalah simetri terhadap paksi di sepanjang arteri. Persamaan-persamaan

kawalan gerakan diselesaikan dengan menggunakan analisa perturbasi biasa. Hasil

kajian menunjukkan bahawa dengan kehadiran gumpalan, ciri-ciri fisiologi aliran

yang penting seperti kecerunan tekanan dan tegasan ricih dinding berubah agak

ketara di seluruh kawasan stenosis. Ketinggian puncak gumpalan meningkatkan

tegasan ricih dinding. Keputusaan tidak hanya sensitif terhadap geometri dinding

stenosis, tetapi juga terhadap posisi gumpalan.

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TABLE OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

ABSTRAK vi

LIST OF FIGURES ix

LIST OF SYMBOLS/NOTATIONS xi

LIST OF APPENDICES xiii

1 INTRODUCTION 1

1.1 Research Background 1

1.2 Problem Statement 2

1.3 Objectives of Study 3

1.4 Scope of Study 3

1.5 Significance of Study 4

2 LITERATURE REVIEW 5

2.1 Introduction 5

2.2 Anatomy of Blood Vessel 5

2.3 Physiology of Blood Flow 7

2.4 Stenosis and Blood Clot 10

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2.5 Studies on the Blood Flow in Artery 13

3 MATHEMATICAL FORMULATION 18

3.1 Introduction 18

3.2 Formulation of the Problem 19

3.3 Derivation of Continuity Equation 20

3.4 Cylindrical Continuity Equation 22

3.5 Derivation of Momentum Equation 23

3.6 Non-dimensionalisation of the Governing Equations 29

3.7 Boundary Conditions 33

4 SOLUTION PROCEDURE 34

4.1 Introduction 34

4.2 Series Solutions 35

4.3 Solution for )1(O at 0 37

4.4 Steady Streaming of )(O 43

5 RESULTS AND DISCUSSIONS 46

5.1 Introduction 46

5.2 The Equations of the Clot and Stenosis 47

5.3 Streamline Pattern 49

5.4 Pressure Gradient 52

5.5 Wall Shear Stress 53

5.6 Discussion 54

6 CONCLUSIONS AND SUGGESTION 56

6.1 Summary 56

6.2 Conclusions 57

6.3 Suggestion for Future Work 57

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REFERENCES 58

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LIST OF FIGURES

FIGURE NO. TITLE PAGE

2.1 Blood vessels: Artery, capillary and vein 6

2.2: Blood vessels: The wall of an artery and vein 6

2.3 Velocity of blood flow changes throughout the systemic

circuit.Velocity changes according to the total crosssectional 9

2.4 Blood pressure changes throughout the systemic circuit

Blood pressure decreases with distance from the left ventricle 9

2.5a Stenosis in blood vessel 10

2.5b The forming of stenosis 10

2.6 The process of blood clotting 11

2.7 Schematic diagram of blood vessel and catheter 14

2.8 Schematic representation of a catheterized artery 14

2.9 The schematic diagram of the catheterized stenosed artery 15

2.10 The schematic diagram of the flow geometry corresponding

to a catheterized curved artery with stenosis 16

3.1 Diagram of the stenosed artery with a clot inside 19

3.2 Finite volume control fixed in space 20

3.3 Infinitesimally small, moving fluid element 24

4.1 Flowchart of solution porcedure 34

5.1 The Matlab coding to plot the clot and the stenosis model 47

5.2 The model of clot and stenosis with different value of zd.

(a) zd = 0 and (b) zd = 0.125 48

5.3 The steady streamline patterns for 2 = 0.2. = 5 and zd = 0 51

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5.4 The steady streamline patterns for 2 = 0.2, = 5 and zd = 0.125 51

5.5 Variation of induced pressure gradient '

sp along axial

direction for different values of 2 52

5.6 Variation of s along axial direction for different values of 2 53

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LIST OF SYMBOLS/NOTATIONS

Roman Letters

d - Diameter

f(z) - function describing wall variation

k - Catheter radius (ratio of catheter size to vessel size, 𝑘 𝑅𝑜 𝑅𝑜

l - Length

𝑝 - Dimensional pressure

p - Non-dimensional pressure

Q - Dimensional flow rate

q - Non-dimensional flow rate

𝑟 - Dimensional radial distance

r - Non-dimensional radial distance

𝑅𝑜 - Undisturbed/ mean radius of the stenosed artery

𝑅𝑠𝑡 - Steady streaming Reynolds number (𝑈𝑜2 𝜔𝜇 )

𝑅𝑒𝑛 - Reynolds number (𝑈𝑜𝑅𝑜 𝜇 )

𝑅𝑠 - Dimensional radius of stenosed artery

𝑟𝑠 - Non-dimensional radius of stenosed artery

𝑅𝑐 - Dimensional radius of clot artery

𝑟𝑐 - Non-dimensional radius of clot artery

𝑡 - Dimensional time

𝑈𝑜 - Characteristic axial velocity

(𝑈, 𝑊) - Dimensional velocity in (𝑟 , 𝑧 ) direction

(u,w) - Non-dimensional velocity in (𝑟, 𝑧) direction

𝑧 - Dimensional axial distance

z - Non-dimensional axial distance

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𝐼𝑜 , 𝐾𝑜 - Modified Bessel‟s function of order 0

𝐼1, 𝐾1 - Modified Bessel‟s function of order 1

Greek Letters

α - Womersley number (𝑅𝑜 𝜔 𝜇 )

ρ - Density of the fluid

λ - Length of stenosis

δ - Geometric parameter 𝑅𝑜 𝜆

δ2 - Radius of peak of clot

θ - Dimensional azimuthal distance

ω - Angular frequency

μ - Kinematic viscosity

τ - Non-dimensional time

τo - Phase angle

ψ - Stream function

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LIST OF APPENDICES

APPENDIX TITLE PAGE

A Bessel‟s Function and its Solutions 60

B Matlab Coding to Obtain the Modified Bessel‟s

Function of Order 0 and 1 66

CHAPTER 1

INTRODUCTION

1.1 Research Background

The medical term for a blood clot is a thrombus (pl. thrombi). When the

thrombus becomes larger, the shear forces that act on it also increase due to the

acceleration of blood around the thrombus. If the shear forces exceed the strength of

the inter-cellular/inter-molecular bonds, a portion of the thrombus can break off and

enter the circulation. A portion of the thrombus that breaks off of the main thrombus

is called a thromboembolus or embolus (pl. emboli). Emboli that enter the

circulation can have many devastating effects within the body. As they pass through

the circulatory system, it is possible that emboli with different sizes become lodged

in different sections of the cardiovascular system. This will reduce or completely cut

off the oxygen supply to the cells in that area. As a result, the part of the heart

muscle that is deprived of oxygen dies, and a heart attack can occur. If the emboli

become stuck in the cerebral circulation, a stroke can occur.

There are many situations in which a blood clot is more likely to form

including being on long-term bed rest, crossing legs for long periods of time when

sitting, or sitting for long periods of time, such as in a plane or car, during and after

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pregnancy, not having enough water in body (dehydration), taking birth control pills

or estrogen hormones (especially in women who smoke) and using an intravenous

catheter long-term. Blood clots are also more likely in people with cancer, recent

surgery or injury, obesity, and liver or kidney disease. People who have high

cholesterol level are also at risk for blood clots. A build up of cholesterol that

narrows an artery may change or slow the flow of blood, making it easier for a blood

clot or thrombus to form. Conditions that are passed down through families

(inherited) may make you more likely to form abnormal blood clots. Inherited

conditions such as Factor V Leiden thrombophilia, Prothrombin G20210A mutation

and other rare conditions such as protein C, protein S, and antithrombin III

deficiencies which affect clotting can also make a person more likely to form

abnormal blood clots.

Blood clot can also enter the blood stream during the insertion of catheter into

the blood vessel. The catheter may cause a thrombus on the wall of the blood vessel

to break free and enter the blood circulation. There is also a risk of bringing

pathological constituents into the blood stream during the catheterization process.

The constituents that go inside the blood vessel will be attached to the vessel wall but

later some constituents can get detached from the wall, then join the blood stream

again, and form a clot that can lead to partial or even complete blockage of the blood

vessel. One example of the constituents is bacteria such as Coagulase negative

staphylococci. Coagulase negative staphylococci can produce an enzyme called

coagulase which can cause blood clot formation.

1.2 Problem Statement

In the previous research, most of the studies on blood flow in the arteries

involved the insertion of catheter and the existence of stenosis. Only a few of them

included clot formation in the blood vessel. The existence of blood clot in the blood

vessel especially in the stenosed artery will affect the blood flow. An annular region

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will be formed in the blood vessel between the blood clot and the wall of the

stenosed blood vessel. Thus this research will investigate whether there are any

significant changes on the flow field due to the existence of clot inside the stenosed

artery.

1.3 Objectives of Study

The purpose of this study is to determine the blood flow characteristics such as

the pressure gradient and the wall shear stress in a stenosed artery when there is a

presence of a clot inside it. The objectives of the study are

a) to formulate a mathematical model to represent the flow of blood in the

annular region between a clot and a stenosed artery.

b) to solve the governing equations of the blood flow by using perturbation

method.

c) to investigate the effect of the clot on the blood flow in terms of pressure

gradient, wall shear stress and streamline pattern.

1.4 Scope of Study

This work is based on the journal paper by Sarkar and Jayaraman (2005)

titled „Nonlinear analysis of arterial blood flow: steady streaming effect‟. The

stenosis and the clot are assumed to be axi-symmetic over the length of the artery.

The flow is considered axi-symmetric and oscillatory in nature. The blood is

considered Newtonian and incompressible.

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1.5 Significance of Study

The blood clot in the blood circulation can lead to partial or even complete

blockage of the blood vessels. The seriousness can be very acute in the case of a

stenotic artery. This study is significant to be carried out in order to understand the

seriousness of the situation by investigating the blood flow characteristic in the

artery. In reality, clot formation in blood vessel after the catheter insertion procedure

can lead to serious problems such as heart attack and stroke due to the blockage in

the blood vessel. Usually, after an insertion of catheter procedure into the blood

vessel, the doctor will likely recommend the patient to take aspirin and another

anticlotting medicine to prevent blood clot formation. So the investigation of blood

flow in stenosed artery together with blood clot is significance to be carried out.

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