MATHEMATICAL MODELING OF BLOOD FLOW THROUGH...

MATHEMATICAL MODELING OF BLOOD FLOW THROUGH AN

ECCENTRIC CATHETERIZED ARTERY

SIMA SARV AHRABI

A dissertation submitted in partial fulfilment of the

requirements for the award of the degree of

Master of Science (Engineering Mathematics)

Faculty of Science

Universiti Teknologi Malaysia

DECEMBER 2013

iii

To my beloved Father and 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.

Last but not least; I would like to thank my family: my parents DAVOOD

and HAMIDE, for giving birth to me and supporting me spiritually throughout my

life; As well as my lovely sisters Maryam and Mahshad for all their kindness towards

me. Furthermore, my aunts Saadaat, Minoo, Nikoo and Mahboobe who did not leave

me alone; in addition, all of my friends who really love and help me to finish this

dissertation.

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ABSTRACT

The mathematical model of blood flow through a catheterized stenosed artery

is considered. A catheter is a tube, which is used in medicine for patients who are

bedridden and whose blood pressure needs to be measured and monitored

continuously. An example is the use of catheter during X-ray angiography or

coronary balloon angioplasty in cardiac patients. Inserting a catheter in an artery will

alter some characteristics of blood flow. This project investigates the effect on blood

flow characteristics such as the velocity, the wall shear stress, the resistance

impedance and the streamlines when a catheter is inserted into a stenosed artery.

The catheter and the artery are assumed to be in a co-axial and eccentric position

while blood is assumed to be Newtonian. The governing Navier-Stokes equations are

solved analytically using perturbation method. The results show that a catheter

placed in an eccentric position does alter the blood flow characteristics such that the

axial velocity and the wall shear stress distribution are higher while the resistance

impedance values are lower compared to their values in an artery where the catheter

is concentrically placed. It is also found that under the same situation, the position of

trapping moves closer to the wall of the stenosis while the size of the trapped bolus

increases.

vi

ABSTRAK

Pemodelan matematik bagi aliran darah apabila kateter dimasukkan ke dalam

arteri berstenosis dipertimbangkan. Kateter adalah suatu tiub yang digunakan dalam

perubatan bagi pesakit yang memerlukan tekanan darah mereka diukur dan dipantau

secara berterusan. Contohnya, kateter digunakan semasa prosidur X-ray angiografi

atau belon angioplasti untuk pesakit jantung. Apabila kateter dimasukkan ke dalam

arteri, ciri-ciri aliran darah akan berubah. Projek ini mengkaji kesan kateter terhadap

ciri-ciri aliran darah seperti halaju, tekanan ricih diding, rintangan impedan dan garis

arus apabila kateter dimasukkan pada kedudukan sepaksi eksentrik , manakala darah

diandaikan sebagai bendalir Newtonan. Persamaan menakluk Navier-Stokes

diselesaikan seecara analisis menggunakan kaedah usikan. Hasil kajian menunjukkan

bahawa halaju sepaksi dan tekanan ricih dinidng adalah lebih tinggi sementara nilai

rintangan impedan adalah lebih rendah berbanding dengan ciri-ciri aliran pada arteri

di mana kateter diletakkan secara konsentrik. Kajian juga mendapati pada situasi

serupa, kedudukan trapping makin hampir dengan dinding stenosis sementara saiz

bolus trapping bertambah.

vii

TABLE OF CONTENTS

CHAPTER TITLES PAGE

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENTS iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES x

LIST OF FIGURES xi

LIST OF SYMBOLS xiv

1 INTRODUCTION 1

1.1 Research Background 1

1.2 Problem Statement 4

1.3 Objectives of the Study 5

1.4 Scope of the Study 5

1.5 Significance of the Study 5

1.6 Outline of Dissertation

6

viii

2 LITERATURE REVIEW 7

2.1 Introduction 7

2.2 Blood Flow through Catheterized Artery 7

3 PROBLEM FORMULATION 14

3.1 Introduction 14

3.2 Geometery of Arterial Wall 14

3.3 Governing Equations 16

3.4 Non-Dimensionalization of Equations 18

4 SOLUTION PROCEDURE 25

4.1 Introduction 25

4.2 Solution by Perturbation Method 25

4.3 Solution of the Axial Velocity 26

4.4 Solution of the Stream Function 32

4.5 Solution of the Wall Shear Stress 35

4.6 Solution of Resisance Impedance 35

5 RESULTS AND DISCUSSIONS 38

5.1 Introduction 38

5.2 Axial Velocity 38

5.3 Wall Shear Stress 42

5.4 Resistance Impedance 47

5.5 Streamlines 49

ix

6 CONCLUSION 56

6.1 Summary of Work 56

6.2 Conclusion 56

6.3 Recommendation 57

REFERENCES 59

x

LIST OF TABLES

TABLE NO. TITLE

PAGE

2.1 Summary of Literature Review 11

xi

LIST OF FIGURES

FIGURE NO. TITLE PAGE

1.1 Arterial Catheter

1

1.2a A Catheter through an artery

2

1.2b Balloon Angioplasty

2

1.3 X-ray Angiography

3

1.4 Intravascular Ultrasound and Using Catheter

3

2.1 Catheterized curved artery

8

2.2 Catheterized artery with a clot

9

2.3 Catheterized artery with a balloon

10

3.1 Schematic diagram of an overlapping stenosed artery

15

3.2 Schematic diagram of a tapered overlapping stenosed

artery

15

3.3 Schematic diagram of eccentric catheter through stenosed

artery

16

5.1 Variation of axial velocity zv with radial distance r for

different values of catheter radius

39


catheter radius 0.1 and different values of eccentricity

parameter

40


different values of velocity of catheter

41

xii


different values of angle of circumferential direction

41

5.5 Variation of wall shear stress distributionw in the stenotic

region for difference values of maximum height of stenosis

42


region for difference values of taper angle

43


region for difference values of catheter radius

44


region for difference values of eccentricity parameter

45


region for difference values of velocity of catheter

46


region for difference values of angle of circumferential

direction

46

5.11 Variation of the resistance impedance with the maximum

height of stenosis * for different values of catheter radius

47


height of stenosis * for different values of eccentricity

parameter

48


height of stenosis * for different values of velocity of

catheter

48


height of stenosis * for different values of taper angle

49

5.15 Plot showing streamlines for different values of the radius

50

5.16 Plot showing streamlines for different values of the

eccentricity parameters

51

xiii

5.17 Plot showing streamlines for different values of the angle

of circumferential direction

52

5.18 Plot showing streamlines for different values of the

velocity of catheter

53

5.19 Plot showing streamlines for different values of maximum

height of stenosis

54

5.20 Plot showing streamlines for different values of taper

angle.

55

xiv

LIST OF SYMBOLS

d - Location of the stenosis

eR - Reynolds number

( , )R z t - Radius of the tapered arterial region

0R - Radius of the normal artery without stenosis

p - Fluid pressure

t - Time

- Density of fluid

- Angular frequency

- Dynamic viscosity

- Angle of tapering

- Shear stress

- Resistance impedance

- Radius of catheter

- Eccentricity parameter

- Stream function

CHAPTER 1

INTRODUCTION

1.1 Research Background

A catheter is a thin, flexible, hollow and lengthy plastic tube that is generally

inserted into an artery (large blood vessel) placed in the wrist; but can also be

inserted into the elbow, groin, foot or the inside of the arm. However, in theory,

every artery could be used for injection of catheter but in practical situation, arteries

of the mentioned organs are commonly placed for the injection. (Figure 1.1)

Figure 1.1 Arterial Catheter, [wikipedia.org/wiki/File:Arterial_kateter,(n.d.)]

http://en.wikipedia.org/wiki/File:Arterial_kateter_(Seldinger).jpg

2

Catheters have an important role in medicine for heart problems. For

instance, they are usually used to diagnose and treat heart conditions. During

catheterization, small tubes (catheters) are inserted into the circulatory system under

x-ray guidance in order to obtain information about blood flow and pressures within

the heart and to determine if there are obstructions within the blood vessels feeding

the heart muscle (coronary arteries). Usage of the catheter is well known in medicine

of various arterial diseases as ―coronary balloon angioplasty‖ (Figure 1.2a,b), ―X-ray

angiography‖(Figure 1.3),―and ―intravascular ultrasound‖ (Figure 1.4).

Figure 1.2a A Catheter through an artery, [ arterydisease.org/(n.d.)]

Figure 1.2b Balloon Angioplasty, [ arterydisease.org/(n.d.)]

http://virchicago.com/peripheral-artery-disease/http://virchicago.com/peripheral-artery-disease/

3

Figure 1.3 X-ray Angiography, [ arterydisease.org/(n.d.)]

Figure 1.4 Intravascular Ultrasound and Using Catheter, [Tekes et al.,(2009)]

http://virchicago.com/peripheral-artery-disease/

4

In addition, a catheter is used for the measurement of various physiological

flow characteristics such as arterial blood pressure or pressure gradient and flow

velocity or flow rate. Examples include patients in the intensive care unit (ICU)

requiring isotropic support or patients with severe cardiovascular disease undergoing

surgery.

Furthermore, catheter can be used for periodic arterial blood gas analysis in

patients with respiratory failure, or severe acid/base disturbance. When a patient has

a lung problem that is so severe it requires checking the levels of oxygen or carbon

dioxide of the blood more than 3 to 4 times a day regularly, the arterial catheter is

used to draw blood without having to repeatedly stick a needle into the patient body.

1.2 Problem Statement

When a catheter is inserted into a blood vessel, blood clots can be formed on

the tips of arterial catheters, the clots can block blood flow. Furthermore,

bleeding can occur at the time of inserting the catheter. Patient becomes

uncomfortable resulting from the injection. However, more important, the injection

of a catheter alters the flow field and disturbs the hemodynamic conditions in the

artery. In practical situation, the injection of a catheter cannot be exactly concentric

with the artery. It is in fact placed in an eccentric position. Hence, this study

investigates what happen to the blood flow characteristics when catheter is inserted

eccentrically through an artery.

5

1.3 Objectives of the Study

The main objective of this research is to determine the effects of eccentric

catheterization on blood flow characteristics in a tapered and stenosed artery.

Specific objectives are to calculate the axial velocity, stream function and trapping,

resistance impedance and wall shear stress.

1.4 Scope of the Study

This study is based on the work of Mekheimer and Kot (2012). The blood is

considered Newtonian fluid, incompressible, steady and laminar; In addition,

catheter and artery supposed to be two tubes, which are co-axial together, and inner

tube is catheter that assumed to be in eccentric position with the outer tube (tapered

artery).

1.5 Significance of the Study

A motivation of the present analysis is the hope that such a problem will be

applicable in many clinical applications. It is hoped that with this investigation, the

injection can be carried out more proficiently. In addition, the results are used to

obtain the estimates of increased or decreased of blood flow characteristics across an

artery stenosis during catheterization.

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1.6 Outline of Dissertation

In Chapter 1, the nature and use of catheter the problem statement and

objectives are presented. In Chapter 2, overview of previous studies on catheterized

artery and blood flow changes are discussed. In Chapter 3, problem is formulated

mathematically. In Chapter 4, method to solve the problem is expressed and the

equations of some characteristics of blood flow are obtained. In Chapter 5, the results

are displayed graphically. In Chapter 6, conclusions and comments about the results

and recommendation for future studies are interpreted.

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University Journal of Science and Technology, 4(2), 7-16.

Daripa, P., & Dash, R. K. (2002). A numerical study of pulsatile blood flow in an

eccentric catheterized artery using a fast algorithm. Journal of engineering

mathematics, 42(1), 1-22.

Dash, R., Jayaraman, G., & Mehta, K. (1996). Estimation of increased flow

resistance in a narrow catheterized artery—a theoretical model. Journal of

Biomechanics, 29(7), 917-930.

Dash, R., Jayaraman, G., & Mehta, K. (1999). Flow in a catheterized curved artery

with stenosis. Journal of Biomechanics, 32(1), 49-61.

Doffin, J., & Chagneau, F. (1981). Oscillating flow between a clot model and a

stenosis. Journal of Biomechanics, 14(3), 143-148.

Hoffmann, K. A., & Chiang, S. T. (2000). {Computational fluid dynamics, Vol. 1}.

Wichita, KS: Engineering Education System.

Ismail, Z., Abdullah, I., Mustapha, N., & Amin, N. (2008). A power-law model of

blood flow through a tapered overlapping stenosed artery. Applied

Mathematics and Computation, 195(2), 669-680.

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steady streaming effect. Nonlinear Analysis: Theory, Methods &

Applications, 63(5), 880-890.

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Biological Engineering and Computing, 33(5), 720-724.

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