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Analysis Of Magneto Hemodynamics Of Blood In A Stenosed Artery

Analysis Of Magneto Hemodynamics Of Blood In A Stenosed Artery

ABSTRACT

Human body responds to physical activities, external temperature, external magnetic field, and other factors by homeostatic-ally adjusting its blood flow in order to deliver nutrients such as oxygen and glucose to every organ in the body and allow them to function. Complications in this hemodynamic may arise due to acute coronary syndromes such as stenosis or arteriosclerosis leading to obstruction of coronary arteries. These complications do cause a change in the blood flow to the body organ as well as the amount of glucose and oxygen that is supplied to them, which may have serious effects on the functioning of some or all bodily systems. Moreover, high grade stenosis increases flow resistance in arteries which means the body has to raise the blood pressure to maintain the necessary blood flow velocity. Both the high pressure and the constricted vessels lead to high flow velocity, high shear stress and low shear stress.Thrombus formation, growth of atherosclerosis and plaque cap rupture which leads directly to stroke and heart attack are related to these irregular flows in blood. How this works is still not well known and under investigation. A Study of how this physiological process works leads to early diagnosis, prevention and treatment stenosis related diseases. In this thesis, a theoretical study of blood flow through a stenosed artery under the combined action of thermal radiation, viscous dissipation, buoyancy force, Joule heating and an externally applied magnetic field is provided. The stenosed artery is modelled as a symmetrical rigid wall channel containing a viscous incompressible, Newtonian, electrically conducting and optically dense bio magnetic fluid representing blood. The variable viscosity of blood depending on hematocrit (percentage by volume of erythrocytes) is taken into account in order to improve resemblance to the real situation. The governing equations of momentum and energy balance are obtained and solved both numerically using a shooting technique coupled with Runge-Kutta-Fehlberg integration method and analytically using a well-known perturbation technique. The effects of various determining parameters on the dimensionless velocity, temperature, pressure gradient, skin friction and Nusselt number are presented graphically and pertinent results are discussed.

TABLE OF CONTENTS

1 INTRODUCTION 1
1.1 HEMODYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 CARDIOVASCULAR SYSTEM . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 BLOOD VESSELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 STENOSED ARTERY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.5 BLOOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6 HEMATOCRIT AND BLOOD VISCOSITY . . . . . . . . . . . . . . . . 5
1.7 MAGNETO-HEMODYNAMICS . . . . . . . . . . . . . . . . . . . . . . 5
1.8 THERMAL RADIATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.9 PROBLEM STATEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.10 STUDY OBJECTIVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.11 SIGNIFICANCE OF STUDY . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.12 STRUCTURE OF STUDY . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.13 RESEARCH METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . 9
1.13.1 Regular Perturbation Method . . . . . . . . . . . . . . . . . . . . 9
1.14 SHOOTING METHOD WITH NEWTON RAPHSON ITERATIONS . . 9
1.14.1 RUNGE-KUTTA-FEHLBERG METHOD: . . . . . . . . . . . . . 11
2 LITERATURE REVIEW 12
2.1 Effects of stenosis on arteries . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Addressing stenosis through hemodynamics . . . . . . . . . . . . . . . 12
2.3 Some Models used for blood flow . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Applications of heat transfer in blood flow . . . . . . . . . . . . . . . . 14
2.5 More complex models for blood flow . . . . . . . . . . . . . . . . . . . . 14
2.6 Some Applications of MHD in medicine . . . . . . . . . . . . . . . . . . 15
2.7 Studies on carotid artery bifurcation . . . . . . . . . . . . . . . . . . . . 16
2.8 Studies on coronary artery stenosis . . . . . . . . . . . . . . . . . . . . . 16
i
ii
3 DERIVATION OF THE BASIC MHD FLUID EQUATIONS 18
3.1 THE CONTINUITY EQUATION . . . . . . . . . . . . . . . . . . . . . . 18
3.2 NAVIER STOKE EQUATIONS[17][27][35] . . . . . . . . . . . . . . . . 19
3.3 ENERGY EQUATION[17][27][35] . . . . . . . . . . . . . . . . . . . . . . 21
3.4 LORENTZ FORCE[52] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.5 MHD MODEL EQUATIONS[52] . . . . . . . . . . . . . . . . . . . . . . 24
4 MAGNETO-HEMODYNAMICS MIXED CONVECTION OF BLOOD FLOW
AND HEAT TRANSFER IN A STENOSED ARTERY 26
4.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 MODEL FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 PERTURBATION APPROACH . . . . . . . . . . . . . . . . . . . . . . . 31
4.4 NUMERICAL APPROACH . . . . . . . . . . . . . . . . . . . . . . . . . 33
5 RESULTS AND DISCUSSION 34
5.1 Streamline Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Velocity Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3 Temperature Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.4 Pressure Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.5 Skin friction and Nusselt Number . . . . . . . . . . . . . . . . . . . . . 45
5.6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.7 STUDY LIMITATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

CHAPTER ONE

INTRODUCTION

1.1 HEMODYNAMICS

This is a cardiovascular physiological study that deals with the forces the heart generates to circulate blood round the cardiovascular system. Proper blood flow ensures proper blood supply of oxygen and nutrients to all tissues which is a necessary condition for cardiovascular health which helps survival of patients during surgery, prolongs lifespan and improves on quality of life. In medical care, blood pressure and blood flow paired values at the nodes of the cardiovascular system are indicators of hemodynamic forces. The interest in hemodynamics is obvious:

A significant majority of all cardiovascular diseases such as stenosis and disorders result from hemodynamic dysfunction. Hypertension and congestive heart failure are good examples systemic hemodynamic disorders.

1.2 CARDIOVASCULAR SYSTEM

Blood and lymph circulation transport nutrients, oxygen, carbon dioxide, hormones, blood cells, etc. through the circulatory system, this helps in fighting diseases, stabilizing body temperature and pH, and in homeostasis. The circulatory system can be considered to be composed of cardiovascular system, which circulates blood and the lymphatic system which returns excess blood plasma from interstitials fluid as lymph. Humans have a closed cardiovascular system; more primitive diploblastic animal phyla lack the circulatory systems. On the other hand, the lymphatic system is an open system that provides an accessory route for excess interstitial fluid to get
returned to blood.

Figure 1.1: Hemodynamic in human circulatory system

1.3 BLOOD VESSELS

These are paths through which blood flows rapidly and effectively to and from the heart to every part of the body.The body regulates the volume of blood supplied to a particular tissue or organ by adjusting the diameter of the vessel. Blood flows through a visco-elastic lumen in the blood vessel. The wall of vessel that surrounds the lumen may be thick in the case of arteries or thin in the case of capillaries. Capillaries receive lower blood pressures than arteries. The squamous endothelium keeps the blood cells inside the blood vessels and prevents clots from forming. The endothelium runs through the entire cardiovascular system, all the way into the interior of the heart where it is now called the endocardium. There are three major types of blood vessels:
arteries, capillaries and veins.

Examples of blood vessels include: the brachiocephalic artery that carries blood into the brachial (arm) and cephalic (head) regions. One of its branches, runs under the clavicle; hence it’s named subclavian. At the axillary region, the subclavian artery is now known as the axillary artery.

1.4 STENOSED ARTERY

An artery is said to be stenosed if any of its sections narrows or constricts. This narrowing results from the deposition of fatty tissue called plaques inside the artery.

The process is known as atherosclerosis. For example carotid artery stenosis occurs when the carotid artery which is located on each side of the neck becomes narrowed

Figure 1.2: Illustrations showing the process of atherosclerosis and flow in the head, face and brain become disturbed and can likely cause brain stroke, dizziness, fainting and blurred vision which may be signs of the brain not receiving enough blood.

Risk factors for carotid artery stenosis include age, cigarette smoking, high blood pressure (hypertension), diabetes, obesity and an inactive lifestyle. Transient Ischemic Attack (TIA) or stroke is usually the first symptom of stenosis. Here, a small blood clot in the artery, resulting from blockage, becomes dislodged and travels to the brain and can plug up a smaller artery which supplies blood to a particular part of the brain. Symptoms of TIA and stroke are similar and include paralysis, blurred vision, headache, trouble speaking and difficulty responding to others. A stroke is often associated with a long lasting damage of part of the brain, disability or death.

Meanwhile a TIA is less dangerous since it is due to a temporary occlusion of a small artery but it is a warning sign.

1.5 BLOOD

Averagely, the human body contains about 4 to 5 liters of blood. Blood connects tissue, and it is the transport medium for many substances and like nutrients and oxygen round the body, and it also helps to maintain homeostasis of nutrients, wastes, and gases. The primary components of Blood are red blood cells, white blood cells, platelets, and liquid plasma. The red blood cells make up about 45% of blood volume, they are also known as erythrocytes, and are by far the most common type of blood cell.

Manufactured inside the red bone marrow from stem cells at the astonishing rate of about 2 million cells every second, erythrocytes are biconcave-disks in shape with a concave curve on both sides of the disk in a way that it is thinnest at its centre. The advantage of this unique shape of erythrocytes a high surface area to volume ratio that allows them to fold to fit into thin capillaries.

While immature, erythrocytes have a nucleus, at maturity, the nucleus is ejected from the cell to provide it with its unique shape and increases its flexibility. The lack of a nucleus implies red blood cells contain no DNA and cannot repair themselves once damaged. The principal function of Erythrocytes is transport of oxygen in the blood with the help of the red pigment hemoglobin. Hemoglobin molecule contains joined iron and proteins,together with their high surface area to volume ratio, these greatly enhance the oxygen carrying ability of erythrocytes.

The white blood cells, also known as leukocytes, though with a relatively very small fraction of the total number of cells in blood, have the important function to maintain the body’s immune system. We have either granular leukocytes or a granular leukocytes.

Blood also contains small cell fragments called Platelets (thrombocytes) which are responsible for the clotting of blood and the formation of scabs. Platelets are produced in the red bone marrow from large mega karyocyte cells that rupture and release thousands of pieces of membrane that become the platelets, periodically.

Platelets lack a nucleus and only survive in the body for up to a week before they are captured and digested by macrophages.

Making up about 55% of the bloods volume, plasma is the part of blood that contains no cell. Plasma is a mixture of water, proteins, and dissolved substances. Around 90% of plasma is made of water, this percentage varies depending upon the hydration levels of the individual. The proteins within plasma form antibodies and albumins. By binding to antigens on the surface of pathogens that infect the body, antibodies help the body’s immune system. Albumins provide an isotonic solution for cells of the body, this helps to maintain the body’s osmotic balance. The plasma functions as transportation medium for dissolved substances like glucose, oxygen, carbon dioxide, electrolytes, cell nutrients, and waste products from cells.

1.6 HEMATOCRIT AND BLOOD VISCOSITY

Hematocrit is a test that indicates the relative volume of red blood cells to the volume of the whole blood. Hematocrit ratio increases as number and size of RBCs in blood increases. It is also known as packed cell volume (PCV) or erythrocyte volume fraction (EVF). Hematocrit ratio is normally 45% for men and 40% for women. Like hemoglobin concentration, white blood cell, platelet counts and, hematocrit form an integral part of a person’s complete blood count result. Blood viscosity which is a measure of the resistance of blood to flow,is primarily determined by hematocrit and plasma viscosity.

The water-content and macromolecular components of blood determine plasma viscosity. Nevertheless, hematocrit has the strongest and most important impact on whole blood viscosity. Due to high shear rate along the wall, the red blood cells aggregate along the arterial centreline region, meanwhile the blood plasma is found near the arterial wall lumen. Consequently, the blood viscosity is low near the arterial lumen wall and high along the centreline due to hematocrit. Moreover, the normal blood viscosity may be within the following range:

1.7 MAGNETO-HEMODYNAMICS

This studies the fluid dynamics of blood in the presence of static high magnetic fields. The ferro-magnetic component of blood is iron contained in hemoglobin that resides inside the red blood cells. Hemoglobin molecules in red blood cells carry oxygen.

Deoxygenated hemoglobin (dHb) is more magnetic (paramagnetic)than oxygenated hemoglobin (Hb), which is virtually resistant to magnetism (diamagnetic). Moreover, with the innovative development of ferro fluid injection in nanomedicine for magnetic drug targeting, an excellent platform for interaction between an externally imposed magnetic field and blood flow is established.

1.8 THERMAL RADIATION

At temperatures above absolute zero, all particles of matter are in thermal motion and since they possess charge, they generate electromagnetic radiation due to charge acceleration and/ or dipole oscillation. This is electromagnetic radiation generated by the thermal motion of charged particles in matter. The wide spectrum of kinetic energies and accelerations of particles results in a wide spectrum of radiation wavelengths that occur even at a single temperature. Radiation is an importantly and often used as a form of anti-cancer therapy since the risk of recurrence after surgery is greatly reduced with radiation. Heat therapy or hyperthermia is most commonly used for rehabilitation purposes. The therapy also helps in effects of heat that include increasing the extensibility of collagen tissues; making joints less stiff; reducing or killing pain; relieving muscle spasms; reducing inflammation, edema, and facilitates the post acute phase of healing; and also increases blood flow rate. The increased blood flow to the affected area of the body provides nutrients, and even oxygen for better healing.

Figure 1.3: Illustration showing the thermal therapy process

1.9 PROBLEM STATEMENT

In our daily life we often use several kinds of electromagnetic instruments, such as cellular phones, transistors, computers, television etc., which emit radiation and have some magnetic field effect. This radiation and magnetic field affects the blood velocity and heat transfer characteristics leading to various kinds of health hazards such as vomiting tendency, headache, partial loss of vision, etc. Moreover, in pathological situations such arterial stenosis or arteriosclerosis, the combined effects of magnetic field and thermal radiation absorption on blood flow may produce a very complex hemodynamic response which may positively or negatively affect the body system.

In order to understand this complex scenario of magnetic-hemodynamic with thermal radiation absorption under pathological conditions, a mathematical model will be proposed and analyzed. The results obtained from the study may produce a new avenue for flow and heat transfer control at these stenotic regions without the need of in-vivo testing of the patient.

1.10 STUDY OBJECTIVES

The general objective of this study is to develop a mathematical model for analyzing the combined effects of magnetic field, thermal radiation, buoyancy force on blood flow and heat transfer in a stenosed artery. The specific objectives are:

I. To develop a mathematical model for blood flow and heat transfer in a stenosed artery under the combined actions of magnetic field, thermal radiation, buoyancy force, viscous and Joule heating.

II. To analyze the model problem using perturbation method and numerically using shooting technique coupled with Runge-Kutta Iteration method.

III. To obtain the effects of various biophysical parameters on the flow streamlines, blood velocity, temperature, pressure gradient, skin friction and Nusselt number at the stenotic region of the artery.

IV. To quantitatively and graphically discuss the hemodynamic response of blood flow in the stenotic region of artery to complex interaction with magnetic field and thermal radiation absorption.

1.11 SIGNIFICANCE OF STUDY

This study is significant in the following ways:

i. The result will be useful to health providers /practitioners for understanding how magnetic field and thermal radiation can be used to reduce or avoid any physical injury associated with cardiovascular disease.

ii. The obtained result could be used to predict pathological conditions such as the stenotic coronary arteries and give the opportunity for new avenues of flow and heat transfer control without the need of in-vivo testing of the patient.

iii. The results will be useful in improving the design of flow meters in bio-medical instrumentation for detecting cardiovascular pathological conditions.

iv. The model provides an excellent avenue for understanding the application of nano-medicine with respect to magnetic drug targeting under pathological condition.

v. The model provides an excellent avenue for understanding the application of deep heat therapy for rehabilitation purposes under pathological conditions.

1.12 STRUCTURE OF STUDY

Chapter one is devoted to background information with respect to physiological applications of magneto-hemodynamic of arterial blood flow, arterial stenosis and its pathological implication, external magnetic field effect, buoyancy force, bio-thermal radiation and heat transfer.

Chapter two deals with a detail review of relevant literature on hemodynamic of blood in stenosed arteries in the presence of magnetic field. The basic magnetohydrodynamic (MHD) equations with respect to conservation of mass, momentum and energy balance including Maxwell equations for electromagnetism were outlined in Chapter three.

In Chapter four, the model problem for stenosed artery, magneto-hemodynamic with heat transfer is formulated and analysed.

The numerical solutions and graphical results are presented and quantitatively discussed in chapter five. This is also followed by a concluding remark.

It is hoped that, our findings could be useful in improving the design of flow meters in bio-medical instrumentation for detecting cardiovascular pathological conditions.

1.13 RESEARCH METHODOLOGY

In this thesis, two major mathematical techniques are employed in order to tackle the derived model problem. They are regular perturbation method and numerical method base on shooting techniques.

1.13.1 Regular Perturbation Method

This provides an iterative way of obtaining a close form solution of parameter dependent ordinary differential equations. Suppose a differential equation of the form

yjj+lyyj = 0 with y(0) = 1; y(1) = 0 (1.1)

The regular perturbation solution to the problem can be taken as a power series in l that is:
y(x) = y0(x)+ly1(x)+l2y2(x)+l3y3(x)+ (1.2)

The corresponding equations for yis will be
yjj
0 =0 with y0(0)=1; y0(1)=0 yjj
n =
n􀀀1
å
i=0
yiyj
n􀀀1 with yn(0)=0; yn(1)=0; f or n>1 (1.3)

and the values of yis can be obtained iteratively for small parameter values l.

1.14 SHOOTING METHOD WITH NEWTON RAPHSON ITERATIONS

In this section we discuss ”pure shooting, where the integration proceeds from x1 to x2, and we try to match boundary conditions at the end of the integration. We describe shooting to the boundary point, where the solution to the equations and boundary conditions is found by launching shots from the first boundary of the interval and trying to match the conditions at the other boundary. Our implementation of the shooting method exactly uses multidimensional, globally convergent Newton-Raphson. It seeks to zero n2 functions of n2 variables. The functions are obtained by integrating N differential equations from x1 to x2. Let us see how this works:

The method At the starting point x1 there are N starting values yi to be specified, but only n1 conditions are given here. Therefore there are n2 = N 􀀀n1 freely specifiable starting values. Let us imagine that these freely specifiable values are the components of a vector ~V in a vector space of dimension n2. Then, for the boundaries, we
can write;

yi(x1) = yi(x1;V1; :::;Vn2 ); i = 1; :::;N
yi(x2) = yi(x2;V1; :::;Vn2 ); i = 1; :::;N

Given a particular, ~V , a particular ~y(x1) is thus generated. It can then be turned into a~y(x2) by integrating the ODEs from x1tox2 as an initial value problem (Using Runge Kutta method). Now, at x2, let us define a discrepancy vector, ~F, also of dimension n2, whose components measure how far we are from satisfying the n2 boundary conditions
at x2 .

Suppose the boundary conditions at the end are given as,
yk(x2) = ak f or k = 1 to n2; (1.4)
then we de f ine Fk as :
Fk = yk(x2;~V)􀀀ak (1.5)
so that ~F =~y(x2;V1;V2;V3; Vn2 )􀀀~a (1.6)

~F is a vector of n2 functions, each with n2 variables. The required components of ~V are those such that the discrepancy ~F is zero (the roots of ~F). Now we use Newton-Raphsons method to have an approximate value for the root of ~F. By Taylor expansion ~F about its root ~V0
Fi(~V) = Fi(~V0)+
n2å
j=1
¶Fi
¶Vj
DVj + (1.7)
where DVj =Vj 􀀀V0j with Fi(~V0) = 0
for i = 1 n2,

This is a matix system of n2 equations of the form
~F(~V) = b J:(~V 􀀀~V0) (1.8)
~V0 =~V +[ b J]􀀀1~F (1.9)
with b Ji j =
¶Fi
¶Vj
called the Jacobian o f ~F
Hence ~Vnew =~Vold +[ b J]􀀀1~F(~Vold) (1.10)

As the appropriate value of ~V is known after some iterations, the problem is now transformed to a complete Initial Value problem which is integrated using the Runge Kutta Fehlberg method

1.14.1 RUNGE-KUTTA-FEHLBERG METHOD:

One way to guarantee accuracy in solution of IVPs is to solve the problem twice using step size h and h=2 and compare answers at mesh points corresponding to large step size. But this requires a significant amount of computation for the smaller step size and must be repeated if it is determined that the arrangement is not good enough. Runge-Kutta-Fehlberg method is one way to try to resolve this problem. It has a procedure to determine if the proper step size h is being used. At each step, two different approximations are accepted. If the answers agree to more significant digits than the required, then the step size is increased.The RKF method has an error estimator of order h5.
Each step requires the following six set of values:

k1 = h f (tn;yn);
k2 = h f (tn; 1
4h; yn+ 1
4 k1);
k3 = h f (tn; 38
h; yn+ 3
32 k1+ 9
32 k2);
k4 = h f (tn; 12
13h; yn+ 1932
2197 k1􀀀 7200
2197 k2+ 7296
2197 k3);
k5 = h f (tn; h; yn+
439
216
k1􀀀8k2+ 3680
513 k3􀀀 845
4104 k4);
k6 = h f (tn; 12
h; yn􀀀 8
27 k1􀀀2k2+ 3544
2565 k3􀀀 1859
4104 k4􀀀 11
45 k5);
Then the approximation to the solution of the IVP is made using a fourth order Runge-Kutta method. That is:
yn+1 = yn+
25
216
k1+
1408
256
k3+
2197
4101
k4􀀀
1
5
k5 (1.11)


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