Mathematical model of blood flow through a tapered artery with multiple stenosis of different heights

Project Abstract

<p> </p><p>We studied and modeled mathematically the blood flow in a tapered cylindrical tube (artery)<br>with multiple stenosis of different heights using Mathematical Modeling. The Models were<br>developed and analyzed/solved analytically. From the analytical solutions, it was observed that<br>the change in the height of the stenosis presents different velocities both radially and axially at a<br>particular point of the stenosis. The graphical representations have been made to validate the<br>analytical findings with a view of its applicability to stenotic diseases and blood flow<br>complication analysis and provision of remedy</p><p>&nbsp;</p><p><strong>&nbsp;</strong></p> <br><p></p>

Project Overview

<p> INTRODUCTION<br>Death rate associated with cardiovascular disease is on the increase due to changes in life<br>style. Cardiovascular diseases such as stroke, heart attack, and heart failure are associated with<br>some form of abnormal flow of blood in stenotic arteries, Mahrabi and Setayeshi, (2012).<br>Arterial stenosis is a disease in the diameter of an artery or narrowing leading to<br>restriction of blood flow. The blood flow obstruction leads to lack of enough oxygenated blood<br>causing some symptoms such as chest pain, shortness of breath and damage to the tissues.<br>Severe stenosis may cause critical flow conditions related to artery collapse/blockage<br>which lead directly to heart attack, stroke, heart failure or even sudden death. It also causes<br>pressure changes at the throat of stenosis and shear stress changes in the distal region Carrocio et<br>al, (2002), Frank et al, (2002).<br>The exact mechanism of the formation of arterial stenosis is not well known, but<br>deposition of various substances such as cholesterol and other fatty materials called plaque on<br>the endothelium of the arterial wall and proliferation of connective tissues are believed to be the<br>factors that accelerate the formation of stenosis, Prakash and Makinde, (2011).<br>How Blood Flows Through the Body<br>As the heart pumps, blood is pushed through the body through the entire circulatory<br>system. Oxygenated blood is pumped away from the heart to the rest of the body, while<br>deoxygenated blood is pumped to the lungs where it is reoxygenated before returning to the<br>heart.<br>Blood Flow Away from the Heart<br>With each rhythmic pump of the heart, blood is pushed under high pressure and velocity<br>away from the heart, initially along the main artery, the aorta. In the aorta, the blood travels at 30<br>cm/sec. From the aorta, blood flows into the arteries and arterioles and, ultimately, to the<br>capillary beds. As it reaches the capillary beds, the rate of flow is dramatically (one-thousand<br>times) slower than the rate of flow in the aorta. While the diameter of each individual arteriole<br>2<br>and capillary is far narrower than the diameter of the aorta, the rate is actually slower due to the<br>overall diameter of all the combined capillaries being far greater than the diameter of the<br>individual aorta.<br>Fig. 1.1 The Heart<br>The slow rate of travel through the capillary beds, which reach almost every cell in the<br>body, assists with gas (especially oxygen and carbon dioxide) and nutrient exchange. Blood flow<br>through the capillary beds is regulated depending on the body’s needs and is directed by nerve<br>and hormone signals. For example, after a large meal, most of the blood is diverted to the<br>stomach by vasodilation (widening) of vessels of the digestive system and vasoconstriction<br>(narrowing) of other vessels. During exercise, blood is diverted to the skeletal muscles through<br>vasodilation, while blood to the digestive system would be lessened through vasoconstriction.<br>The blood entering some capillary beds is controlled by small muscles called precapillary<br>sphincters . A sphincter is a ringlike band of muscle that surrounds a bodily opening,<br>constricting and relaxing as required for normal physiological functioning. If the precapillary<br>sphincters are open, the blood will flow into the associated branches of the capillary bed. If all of<br>the sphincters are closed, then the blood will flow directly from the arteriole to the venule<br>through the thoroughfare channel. These muscles allow the body to precisely control when<br>3<br>Figure 1.2: Figure 1.2:<br>capillary beds receive blood flow. At any given moment, only about 5-10 percent of our<br>capillary beds actually have blood flowing through them.<br>Precapillary sphincters<br>(a) Precapillary sphincters are rings of smooth muscle that regulate the flow of blood through<br>capillaries; they help control the blood flow to where it is needed. (b) Valves in the veins<br>prevent blood from moving backward.<br>Blood Flow to the Heart<br>After the blood has passed through the capillary beds, it enters the venules, veins, and<br>finally the two main venae cavae (singular, vena cava) that take blood back to the heart. The<br>flow rate increases again, but is still much slower than the initial rate in the aorta. Blood<br>primarily moves in the veins by the rhythmic movement of smooth muscle in the vessel wall and<br>by the action of the skeletal muscle as the body moves. Because most veins must move blood<br>against the pull of gravity, blood is prevented from flowing backward in the veins by one-way<br>valves. Thus, because skeletal muscle contraction aids in venous blood flow, it is important to<br>get up and move frequently after long periods of sitting so that blood will not pool in the<br>extremities.<br>4<br>The following figures illustrate how arterial stenosis forms due to deposition of plaque.<br>Figure (1.3) shows that plaque narrows the internal diameter of artery which can cause<br>the abnormal flow of blood.<br>Figure (1.4) shows the obstruction of the flow of blood to heart muscle. As a<br>consequence, the tissues and cells supplied by the artery become short of oxygen, Fallon and<br>Mary, (2001).<br>Figure (1.5) shows normal and diseased artery.<br>Figure 1.3, Stenosed Artery shows the formation of plaque.<br>Figure 1.4, Artery with cholesterol build up.<br>5<br>Figure 1.5, Normal and diseased artery.<br>Blood is made up of a suspension of particles in a solution of protein and electrolytes<br>called plasma.<br>Erythrocytes, leukocytes and platelets are the main constituents of blood. The<br>erythrocytes or red blood cells (RBCS) are more than a thousand times more than the leukocytes<br>or white blood cells (WBCS) and much longer than platelets.<br>For this reason, the flow properties of blood mainly involve the red blood cells (RBCS).<br>The Hematocrit (percentage of the blood volume that is made up of red blood cells) is the major<br>determinant of blood viscosity. When the flow of blood to a part of the body is reduced, the<br>oxygen supply to that part of the body is cut off and cells begin to die, resulting in heart attack<br>(Bonn, 1999).<br>6<br>The major risk factors that create such situations in the lumen of arteries are the<br>following:<br>Ø High blood pressure<br>Ø High cholesterol<br>Ø Smoking<br>Ø Some diseases, such as diabetes, obesity etc.<br>Ø A family history of early heart disease.<br>Among all these condition, high blood pressure is more dangerous as the excess strain on<br>the arteries causes them to become weak and calcium and fatty deposits tend to form in these<br>weakened areas causing the blood pressure to become even higher. Figure (1.5), shows the<br>reduced blood flow and clotting of blood due to plaque in an artery.<br>Stenosis symptoms depend on which arteries are affected. For example:<br>Ø If stenosis is present in the heart arteries, the person may have symptoms similar to those<br>of a heart attack, such as chest pain (angina).<br>Ø If stenosis is present in the arteries which are leading to the brain, the person may have<br>symptoms such as sudden numbness or weakness in the arms or legs, difficulty in<br>speaking or drooping muscles in the face.<br>Ø If stenosis is present in the arteries which are related to arms and legs, the person may<br>have symptoms of peripheral arterial disease, such as leg pain when walking etc.<br>Sometimes stenosis causes erectile dysfunction in men.<br>1.2 AIMS AND OBJECTIVES OF THE STUDY<br>The general objective of this study is to develop a mathematical model for<br>analyzing blood flow in a tapered artery with multiple stenosis of different heights.<br>The specific objectives are;<br>Ø To develop a mathematical model for blood flow in a tapered stenosed artery.<br>Ø To determine the solutions of the model equation.<br>7<br>Ø To explain how tapering affect flow in the artery.<br>Ø To improve understanding management of medical implications of human health.<br>Ø To have the knowledge about the effects of plaque build-up in human blood<br>arteries.<br>1.3 SCOPE OF THE STUDY<br>This study will be concerned with formulation of model that will describe the<br>flow pattern of blood in tapered cylindrical tubes, (arteries) with multiple stenosis of different<br>heights. That is, the study interest will be on the blood flow and not the mechanism of<br>treatment.<br>1.4 LIMITATION OF THE STUDY<br>This study was purely set for effective actualization of its aims and objectives.<br>The geometry of blood flow in a tapered cylindrical artery with multiple stenosis had<br>made the work so involving that the model could not talk about permeability of the artery<br>and measure the blood flow pattern in the branches of the artery.<br>8 <br></p>