Mathematical Model of Blood Flow Through a Tapered Artery with Multiple Stenosis of Different Heights
We studied and modeled mathematically the blood flow in a tapered cylindrical tube (artery) with multiple stenosis of different heights using Mathematical Modeling. The Models were developed and analyzed/solved analytically. From the analytical solutions, it was observed that the change in the height of the stenosis presents different velocities both radially and axially at a particular point of the stenosis. The graphical representations have been made to validate the analytical findings with a view of its applicability to stenotic diseases and blood flow complication analysis and provision of remedy.
TABLE OF CONTENTS
Certification: – – – – – – – – – – i
Dedication: – – – – – – – – – – ii
Acknowledgement: – – – – – – – – – iii
Abstract: – – – – – – – – – – v
Table of contents: – – – – – – – – – vi
1.1 Introduction – – – – – – – – – 1
1.2 Aims and objectives of the study – – – – – – 6
1.3 Scope of the study – – – – – – – – 7
1.4 Limitation of the study – – – – – – – – 7
2.1 Literature Review – – – – – – – – – 8
3.1 The circulatory system – – – – – – – – 12
3.2 Blood and its composition – – – – – – – 12
3.2.1 Functions of blood – – – – – – – – 12
3.3 Blood vessels and its characteristics – – – – – – 13
3.4 Structure of artery – – – – – – – – – 14
3.5 The heart as a pump – – – – – – – – 19
3.6 Blood flow dynamics – – – – – – – – 19
3.7 Flow rate of blood – – – – – – – – 20
3.8 Viscosity/ Viscosity variation – – – – – – – 20
3.9 Bypass in artery / Bypass surgery – – – – – – 21
3.10 Blood flow through the artery with changes in diameter – – – 21
3.11 Blood flow through the artery with stenosis – – – – – 23
3.12 Assumptions of the model – – – – – – – 26
4.1 Formulation of the problem – – – – – – – 27
4.2 Methods of solution – – – – – – – – 30
4.3 Analysis of model equations – – – – – – – 43
5.1 Discussion of results – – – – – – – – 51
5.2 Summary — – – – – – – – – – 53
5.3 Conclusion and Recommendation – – – – – – – 54
REFERENCES – – – – – – – – – 55
Death rate associated with cardiovascular disease is on the increase due to changes in life style. Cardiovascular diseases such as stroke, heart attack, and heart failure are associated with some form of abnormal flow of blood in stenotic arteries, Mahrabi and Setayeshi, (2012).
Arterial stenosis is a disease in the diameter of an artery or narrowing leading to restriction of blood flow. The blood flow obstruction leads to lack of enough oxygenated blood causing some symptoms such as chest pain, shortness of breath and damage to the tissues.
Severe stenosis may cause critical flow conditions related to artery collapse/blockage which lead directly to heart attack, stroke, heart failure or even sudden death. It also causes pressure changes at the throat of stenosis and shear stress changes in the distal region Carrocio et al, (2002), Frank et al, (2002).
The exact mechanism of the formation of arterial stenosis is not well known, but deposition of various substances such as cholesterol and other fatty materials called plaque on the endothelium of the arterial wall and proliferation of connective tissues are believed to be the factors that accelerate the formation of stenosis, Prakash and Makinde, (2011).
How Blood Flows Through the Body
As the heart pumps, blood is pushed through the body through the entire circulatory system. Oxygenated blood is pumped away from the heart to the rest of the body, while deoxygenated blood is pumped to the lungs where it is deoxygenated before returning to the heart.
Blood Flow Away from the Heart
With each rhythmic pump of the heart, blood is pushed under high pressure and velocity away from the heart, initially along the main artery, the aorta. In the aorta, the blood travels at 30 cm/sec. From the aorta, blood flows into the arteries and arterioles and, ultimately, to the capillary beds. As it reaches the capillary beds, the rate of flow is dramatically (one-thousand times) slower than the rate of flow in the aorta. While the diameter of each individual arteriole and capillary is far narrower than the diameter of the aorta, the rate is actually slower due to the overall diameter of all the combined capillaries being far greater than the diameter of the individual aorta.
Fig. 1.1 The Heart
The slow rate of travel through the capillary beds, which reach almost every cell in the body, assists with gas (especially oxygen and carbon dioxide) and nutrient exchange. Blood flow through the capillary beds is regulated depending on the body’s needs and is directed by nerve and hormone signals. For example, after a large meal, most of the blood is diverted to the stomach by vasodilation (widening) of vessels of the digestive system and vasoconstriction (narrowing) of other vessels. During exercise, blood is diverted to the skeletal muscles through vasodilation, while blood to the digestive system would be lessened through vasoconstriction.
The blood entering some capillary beds is controlled by small muscles called pre-capillary sphincters . A sphincter is a ringlike band of muscle that surrounds a bodily opening, constricting and relaxing as required for normal physiological functioning. If the pre-capillary sphincters are open, the blood will flow into the associated branches of the capillary bed. If all of the sphincters are closed, then the blood will flow directly from the arteriole to the venule through the thoroughfare channel. These muscles allow the body to precisely control when
Figure 1.2: Figure 1.2:
capillary beds receive blood flow. At any given moment, only about 5-10 percent of our
capillary beds actually have blood flowing through them.
(a) Pre-capillary sphincters are rings of smooth muscle that regulate the flow of blood through capillaries; they help control the blood flow to where it is needed. (b) Valves in the veins prevent blood from moving backward.
Blood Flow to the Heart
After the blood has passed through the capillary beds, it enters the venules, veins, and finally the two main venae cavae (singular, vena cava) that take blood back to the heart. The flow rate increases again, but is still much slower than the initial rate in the aorta. Blood primarily moves in the veins by the rhythmic movement of smooth muscle in the vessel wall and by the action of the skeletal muscle as the body moves. Because most veins must move blood against the pull of gravity, blood is prevented from flowing backward in the veins by one-way valves. Thus, because skeletal muscle contraction aids in venous blood flow, it is important to get up and move frequently after long periods of sitting so that blood will not pool in the extremities.
The following figures illustrate how arterial stenosis forms due to deposition of plaque.
Figure (1.3) shows that plaque narrows the internal diameter of artery which can cause the abnormal flow of blood.
Figure (1.4) shows the obstruction of the flow of blood to heart muscle. As a consequence, the tissues and cells supplied by the artery become short of oxygen, Fallon and Mary, (2001).
Figure (1.5) shows normal and diseased artery.
Figure 1.3, Stenosed Artery shows the formation of plaque.
Figure 1.4, Artery with cholesterol build up.
Figure 1.5, Normal and diseased artery.
Blood is made up of a suspension of particles in a solution of protein and electrolytes called plasma.
Erythrocytes, leukocytes and platelets are the main constituents of blood. The erythrocytes or red blood cells (RBCS) are more than a thousand times more than the leukocytes or white blood cells (WBCS) and much longer than platelets.
For this reason, the flow properties of blood mainly involve the red blood cells (RBCS).
The Hematocrit (percentage of the blood volume that is made up of red blood cells) is the major determinant of blood viscosity. When the flow of blood to a part of the body is reduced, the oxygen supply to that part of the body is cut off and cells begin to die, resulting in heart attack (Bonn, 1999).
The major risk factors that create such situations in the lumen of arteries are the following:
Ø High blood pressure
Ø High cholesterol
Ø Some diseases, such as diabetes, obesity etc.
Ø A family history of early heart disease.
Among all these condition, high blood pressure is more dangerous as the excess strain on the arteries causes them to become weak and calcium and fatty deposits tend to form in these weakened areas causing the blood pressure to become even higher. Figure (1.5), shows the reduced blood flow and clotting of blood due to plaque in an artery.
Stenosis symptoms depend on which arteries are affected. For example:
Ø If stenosis is present in the heart arteries, the person may have symptoms similar to those of a heart attack, such as chest pain (angina).
Ø If stenosis is present in the arteries which are leading to the brain, the person may have symptoms such as sudden numbness or weakness in the arms or legs, difficulty in speaking or drooping muscles in the face.
Ø If stenosis is present in the arteries which are related to arms and legs, the person may have symptoms of peripheral arterial disease, such as leg pain when walking etc.
Sometimes stenosis causes erectile dysfunction in men.
1.2 AIMS AND OBJECTIVES OF THE STUDY
The general objective of this study is to develop a mathematical model for analyzing blood flow in a tapered artery with multiple stenosis of different heights.
The specific objectives are;
Ø To develop a mathematical model for blood flow in a tapered stenosed artery.
Ø To determine the solutions of the model equation.
Ø To explain how tapering affect flow in the artery.
Ø To improve understanding management of medical implications of human health.
Ø To have the knowledge about the effects of plaque build-up in human blood arteries.
1.3 SCOPE OF THE STUDY
This study will be concerned with formulation of model that will describe the flow pattern of blood in tapered cylindrical tubes, (arteries) with multiple stenosis of different heights. That is, the study interest will be on the blood flow and not the mechanism of treatment.
1.4 LIMITATION OF THE STUDY
This study was purely set for effective actualization of its aims and objectives.
The geometry of blood flow in a tapered cylindrical artery with multiple stenosis had made the work so involving that the model could not talk about permeability of the artery and measure the blood flow pattern in the branches of the artery.