Sunday, 18 December 2016

Mathematical Modeling Of Excess Fats On Blood Flow In Human Arteries

ABSTRACT  The study of blood flow has become more important and necessary coupling with the ever-increasing cardiac related problems which are usually as a result of one form of abnormality on blood flow or another. It is clear that under normal condition, the amount of fluid that enters a medium equals that which leaves the medium, but in a situation whereby there is a constriction at some point in the conduit through which the fluid flows due to some external substances such as sedimentation in pipes, accumulation of fats on the walls of an artery etc, the diameter of the conduit reduces hence affects the flow profile.
In this research work, we consider the case of excess fats as an external factor affecting blood flow. The continuity equation, the Navier Stokes equation for a cylindrical tube as well as Poissuile equation is used in formulating the model. The result shows that the accumulation of fats on the walls of artery is due to the drag force that exist between the blood and the walls of the artery as the velocity of the blood seems to be zero compare to the axial velocity. The geometry of an artery is used to describe the flow of blood in a healthy artery and that of an affected one. To correct this, there should be a constant head pump of blood from the heart which is best achieved through regular exercise as well as taking meals with less fats particularly people who are of age.                                 


Title page
1.0 Introduction 
1.1 Problem Statement   
1.2 Aim and Objectives  
1.3 Significance of Study
1.4 Scope and Limitations  
1.5.0Definition of physiological terms used  
1.5.1 Blood  
1.5.2 Fats  
1.5.3 Cardiovascular System
1.5.4 Blood Vessels 
1.5.5 Arteries  
1.5.6 Veins
1.5.7 Capillaries   
1.5.8 Fluid  
1.5.9 Definition of fluid terms
1.6.0 Review of Partial Differential Equations   
1.6.1 Introduction
1.6.1 Definitions  
1.6.2 Methods of forming PDE’S  
1.6.3 Classification of PDE’S 
2.0 Introduction            
2.1 Review of Related Literatures
2.2 Mathematical Models   
2.3 Model Assumptions
2.4 Momentum Equations (Navier-Stokes)  
3.0 Formation of Problem   
3.1 Application of  Model Assumption in Artery 

4.0 Regulation of Blood flow
1.0 Introduction:  One of the leading causes of deaths in the world today is due to heart diseases and the most common heard names are Ischemia, Atherosclerosis and Angina pectoris. Ischemia is the deficiency of oxygen in a part of the body, usually temporary. It is mostly due to a constriction (stenoses)or obstruction in the blood vessel supplying that part. Atherosclerosis on the other hand is a type of arteriosclerosis which comes from the Greek words Arthero (meaning gruel or paste) and sclerosis (hardness). It involves deposits of fatty substances, cholesterol, cellular waste products, calcium and fibrin (a clothing material in the blood) in the inner lining of an artery. (Pralhad and Schultz, 2014).
The build up in the walls of the artery forms what we call plaque. Plaque may partially or totally block the blood flow in an artery causing bleeding (hemorrhage) and secondly, forms blood clot (thrombus) on the plaque’s surface. If either of these occurs and blocks the arteries, a heart attack or stroke may occur. A symptom complex of Ischemia heart disease is characterized by paroxysmal attack of chest pain which is referred to as angina pectoris.

In cardiac-related problems, the affected arteries get hardened as a result of accumulation of fatty substances inside the lumen or because of formation of plaque as a result of hemorrhage. As the disease gets progress, the arteries get constricted and as such the flow behavior in the stenosed artery becomes quite different with that in the normal arteries. Also, stresses and resistance to flow parameters such as the velocity, flow rate, pressure drop will aid in formulating solutions to such problems. Hence, fluid mechanics aspect of the arterial stenoses need to be given a considerable attention now than before.

In this work we model blood flow in a stenose tube (artery) and assumed that the flow obeys Newtonian hypothesis, in which the flow variables are computed using basic Navier Stokes equations of motion, the continuity equation where by the solution is obtained using Poisseulle’s equation.

There is a lot of havoc during blood flow due to the deposition of fatty substances in the human blood vessels. The accumulation of these substances leads to what we called STENOSIS which causes resistance to blood flow leading to most of the heart  related diseases we encounter today such as    hypertension, stroke, ischemia, atherosclerosis, angina pectoris among others. The blood flow in human arteries and veins can be best represented using fluid flow equation of motions. To properly analyze these problems for better understanding, the stability of the factors governing blood flow such as velocity distribution, viscosity of blood, pressure drop,the resistance to flow, the diameter of the conduit(artery) through which the blood flows  as well as the length of the  conduit (artery) under study needs to be taken into consideration.

The aim of this research is to make use of mathematical equations to study the effects of excess fat as related to blood flow, what really happened when there is a constriction (a reduction in the diameter of the artery) and then compare the behavior of blood in a healthy artery with that of  a constricted artery. To achieve this, we make use of the following set of objectives below;

Ø   The continuity equation which describes fluid flow under mass conservation law
Ø   The Navier stokes equations of an incompressible cylindrical flow to explain the flow behaviour
Ø   Poisseulle equation to explain the basic terms associated with blood such as pressure drop, resistance to flow, the volume flow rate as well as the velocity of the blood.

With the ever increasing heart related problems today, which are usually as a result of abnormal blood flow, this study gives an insight on how to reduce these problems as well as create an  avenue for both students of physics and mathematical sciences with little previous knowledge on human physiology to refresh back. It will also help us to be conscious of these effects associated with blood flow and their possible way out as well as create a room for further research.         

1.4  SCOPE AND LIMITATIONS.                              
In  this study, we make use of  basically the continuity equation, the Navier- Stokes equation for an in compressible cylindrical flow and the Poisseuile equation in analyzing the flow and the study is limited to mammalian physiology particularly human beings.    

1.5.0    DEFINITION OF BASIC TERMS                                                                                                
  1.5.1 BLOOD: Just as an engine doesn’t function without fuels, lubricants and other forms of fluids, so also living things can’t function without fluids in their systems, specifically human beings. Generally, blood is essential for life. It constitutes the liquid in the human body called the plasma. It is 95% water and the remaining 5% is made up of dissolved substances including salts.  In adults, there are 10.6 pants of blood circulating around the system and an average adult has about 5 of the blood.

The blood transport substances such as hormones, white blood cells, red blood cells and platelets to the various part of the body where they are needed. Blood is a suspension of red blood cells(erythrocytes), white blood cells(leucocytes) and platelets in a complex solution called plasma. An increase in cells leads to an increase in the volume of blood i.e. the more the cells in the body, the more the function which implies a greater viscosity. The coefficient ofviscosity of man at room temperature i.e. 37 degree Celsius

1.5.2   FATS: This is a natural substance occurring in animal bodies especially when deposited as a layer under the skin or around certain organs. The food we eat contains nutrients that provides energy and other substances the body needs. Most of the nutrients fall into three major groups i.e. proteins, fats and carbohydrates.

Fats serve as a fuel source since they are the major storage form of energy in the body. There are two types of fats, saturated and unsaturated. Saturated fats donot have double bonds and are solid at room temperature e.g. butter, whereas, unsaturated fats have one or more double bonds and are liquid at room temperature e.g. fats and oils. It is advisable that people who are of age (old people) should reduce their intake of saturated fats, while fat intake should be more for people who engaged in heavy labor.                


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