Table of Contents
Declaration and Approval i
Dedication ii
Acknowledgments iii
Chapter 1: INTRODUCTION
1.1 The Bio Heat Transfer 1
1.1.1 Role of blood in heat transfer 1
1.1.2 Role of skin in bio heat transfer 1
1.2 Importance of heat transfer 2
1.3 Biological background of cancer and treatment 3
1.3.1 Cancer treatment 4
1.4 The Pennes Bio Heat Equation 4
1.4.1 The Background of The Pennes’ Bio Heat Equation 4
1.4.2 Pennes’ Bio Heat Equation 4
1.4.3 Definition of terms and derivation of Pennes’ Bioheat Equation 5
1.4.4 Assumptions in pennes’ bio heat equation 7
1.4.5 Advantages of pennes’ bio heat equation 7
1.4.6 Inadequacies in pennes equation 8
Chapter 2: Literature Review
2.1 The Pennes’ model 9
2.2 Gautherie’s model 10
2.3 The Chen-Holmes bio heat transfer model 10
2.4 The Weinbaum and Jiji Model 10
2.5 Baish et al 12
2.6 Preliminaries 12
2.6.1 Finite Element Method(FEM) 13
Chapter 3: Solution of Pennes’ Bio heat Equation
3.1 Finite Difference Method 14
3.1.1 Discretization 14
3.1.2 Deriving Finite Difference derivatives 15
3.1.3 solving a BVP using Finite difference method 17
3.1.4 How do we apply the finite difference formulas into solving an BVP? 18
3.1.5 1D Pennes’ Equation 19
3.1.6 Finite Difference Scheme for 1D Pennes’ Bio Heat Equation 20
3.2 Formulation of the problem 20
3.3 Solution to the problem 22
3.4 Mathematical simulation 25
CONCLUSION
3.5.1 Future work 27
Alternative RESULTS 28
Appendices 30
A MATLAB codes 31
Bibliography 35
Chapter 1
INTRODUCTION
The world is changing drastically and diseases are evolving over time. One kind of such is cancer. Modern clinical treatments have come up to treat cancer and thus under- standing of the temperature behaviour in human tissues is of essence. Thermal ablation is one kind of treatment to cancerous tissues by exposing them to very high temperature (hyperthermia) while protecting the temperatures of the surrounding tissue [1]. The main goal of thermal ablation is to raise the temperature of the cancerous tissue to a point where cancer cells are destroyed while maintaining normal temperature for tissues that surround. There are other modern clinical treatments that need the understanding of heat transfer in living tissues and these are cryosurgery, cryopreservation and thermal diagnostics [2]. Thus, study of bio heat transfer in living tissues has been an important topic to scientists over the last many years.
1.1 The Bio Heat Transfer
Bio heat transfer is the study of thermal energy transfer in living tissue. Biological processes are heat dependant hence heat transfer plays a critical role. In the living tissues Blood and skin plays a very important role in heat transfer.
1.1.1 Role of blood in heat transfer
In biological tissues blood plays an important role of transport of thermal energy. Some of the roles of blood in biological tissues are;
• Blood plays a role of body metabolism, this is by transporting oxygen to all the body parts and transporting of carbon-dioxide and waste from cells.
• Blood regulates blood pressure
• Blood transports heat which enhances thermo-regulation.
1.1.2 Role of skin in bio heat transfer
skin plays a very important role in thermal transfer. The skin has 3 layers: the dermis, epidermis and subcutaneous tissues shown in the figure below.
Figure 1. skin layers
In these layers are blood vessels which are essential in blood circulation. The human body has a circulatory system that is composed of blood vessels which are majorly arteries and veins, which carry blood from heart to the tissues and back to the heart. Blood leaves the heart through the largest artery called aorta. It then enters to the main supply arteries and veins, then to the primary arteries, to the secondary arteries, to the arterioles and finally to the capillaries which are in contact with the tissues and back to the heart through veins. This process of blood supply through the blood vessels is a major form of thermal transfer and thermal equilibrium.
Bio heat transfer includes evaporation, heat generation, heat absorption, heat transmission, evaporation and conduction. These biological processes takes place in the tissues which are solid and in the blood, which is the fluid. The table below shows significance of the thermal processes in a living tissue.
These processes are coupled up with physiological processes which includes blood circulation, metabolic heat generation and heat dissipation.[3] There are also various factors that include temperature distribution, tissues strain, tissue stress and damage of thermal tissues.
1.2 Importance of heat transfer
In living tissues, heat transfer plays an important role since biochemical processes are heat dependent and highly utilized in the medical field and physiological studies. For example, heat transfer in tissues is utilized in cryosurgery, frostbite, thermal ablation, hyperthermia/cancer treatment, skin burns and body thermal regulation. The temperature interact with tissues at different temperature ranges and are given different terminologies as shown in the table below;
1.3 Biological background of cancer and treatment
Bio heat transfer has been used in diagnostic and therapeutic applications [4] which rely on advanced computerized techniques. This has enhanced the development of mathematical models to study and analyze various bio heat transfer processes. Cancer is one of the diagnostic application of the bio heat transfer.
The study of cancer in medicine field is called oncology. Cancer has its existence since ancient times. Cancer is the abnormal growth of cells in human body, whether in the breast, in the throat, in the stomach, in the bones and any other tissue in the body. Cancer was first discovered by Greek physician called Hippocrates(460-370BC). Hippocrates believed that the body has four fluids, the blood, phlegm, yellow bile, and black bile. He suggested that the imbalance of these fluids, with an excess of black bile caused cancer. Since then many theories and research on cancer have been studied. The latest theory in 1920’s argued that trauma was the cause of cancer.
Tumors are detected in the body via temperature by evaluating temperature distribution around that particular tissue. Tissues with tumors are known to have a significant high temperatures than the body tissue [6].
1.3.1 Cancer treatment
In oncology, the term "hyperthermia" refers to the treatment of malignant tissues by use of very high temperatures [7]. The temperatures are usually in range of 40-42°C. This is also what we call thermal ablation, a process called coagulation necrosis. The use of hyper-thermia to treat cancerous cells dates back to 1898, by Swedish gynecologist Westermark, who treated cervical cancer using hot water. [8].
In treatment of cancer we ensure that the high temperature only applies to the affected cells and not the normal cells. In cancer treatment, heating facilitates radio and chemotherapy therapy. Cancer treatment through hyperthermia includes killing of the cancerous cells, while taking care of the normal cells surrounding the cancerous tissue. It also involves decrease in oxyhemoglobin saturation which decreases the tumor tissue PH hence killing it [7]. In killing the cancerous cells, laser is used in order to protect the surrounding tissues. To study the heat transfer in killing the tumors, many scientist have come up with models to describe heat transfer and fluid flow in biological processes when a tissue is heated and vascularized.
1.4 The Pennes Bio Heat Equation
1.4.1 The Background of The Pennes’ Bio Heat Equation
Several authors have come up to study the heat transfer within the tissues. Harry H. Pennes from the Department of Neurology, college of physicians and surgeons, Colombia University, and the Neurological Institute, New York is one of them who greatly contributed into this issue. In 1994, Harry H. Pennes published a study on temperature distribution in human body. He published “Analysis of tissue and arterial blood temperatures in the resting forearm” which appeared in volume 1, No. 2, published in August 1948. The purpose of Pennes’ study was to “to evaluate the applicability of heat flow theory to the forearm in basic terms of the local rate of tissue heat production and volume flow of blood" [9]. Many authors have adopted pennes’ work in developing mathematical models of heat transfer in human body for example analyzing digital cooling and developing a whole body human thermal model done by Eugene H. Wissler in 1958 and 1961 respectively.
1.4.2 Pennes’ Bio Heat Equation
Pennes (1948) came up with his famous heat transfer model called Pennes’ Bio heat equation. Pennes model considered the effect of blood flow in the region as a heat source or heat sink term added to the heat conduction equation.
Pennes formulated a model based on the temperatures at the forearm. He proposed a model to describe the effects of metabolism and blood perfusion on the energy balance within tissue. The Pennes Bio heat equation is given by;
where ;
1. T(x,t) is the sought temperature variable
2. Tb Temperature of arterial blood.
3. t is the time where t is greater than 0,
4. k is the thermal conductivity
5. is the mass density, c is the specific heat capacity
6. Qmet is the total internal heat generation per unit volume expressed as addition of heat generation and heat gain
Qmet = Qm + Qr,
and
Qr,
is the metabolic heat generation and
Qr,
is the heat deposition.
7. ρb, cb, ωb are the blood density, specific heat capacity of blood and blood perfusion rate respectively.
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