Heat flow in female breast. Temperature distribution in semi spherical shaped human breast with and without abnormalities

Doctoral Thesis / Dissertation , 2013 , 200 Pages , Grade: 9

Mathematics - Applied Mathematics

This thesis is divided in the following order. The first, two chapters respectively deal with the physiological and mathematical background necessary for the study of heat flow in female breast. Chapter-3 is devoted to analytical and numerical solutions to study temperature distribution in spherical organs of a human body for a one dimensional steady state case. Chapter-4 and 5 deals with two dimensional finite element models to study temperature distribution in semi spherical shaped human breast with and without abnormalities like tumor.

The development of quantitative methods and computer technology has facilitated the formation of new models in medical and biological sciences. Further, the application of mathematical methods in solving many real world problems of medicine and biology has yielded fruitful results. Despite advancements in instrumentations technology and biomedical equipments, it is not always possible to perform experiments in medicine and biology due to various reasons. Thus mathematical modeling and simulation is viewed as an viable alternative in such situations. The problems of biology and medicine pose new challenges for modeling and simulation.

Attempts are in progress world wide for mathematical modeling of control systems of human body as these systems are highly sophisticated systems and poorly understood. The temperature regulation system of a human body is one of the important control systems involved is maintaining structure and function of human body organs. The study of temperature regulation system of human body organs is quite interesting both from application and mathematical point of view. The modeling of these problems poses new challenges for mathematics. In view of above the theoretical study entitled “Computational Models To Study Temperature Distribution In Extended Spherical And Ellipsoidal Shaped Human Organs” has been taken up.

CHAPTER 1

PHYSIOLOGICAL BACKGROUND

1.1 INTRODUCTION

A large number of physical and physiological processes are taking place at each level of the hierarchy of the system and subsystems in a human or animal body in-order to maintain the structure and function of the each component of the body. A number of control systems are present in a human body to regulate these processes like; (1) Fluid control system (2) Temperature control system (3) Communication control system etc. The temperature control (regulation) system is one of the very important control systems of a human body. It regulates the body core temperature at a constant temperature of 37°C by maintaining balance between heat generation within the body and heat loss from the body to the environment [35,39], Any abnormality in these processes or subsystems can affect the control system, thus causing the diseases or any disorder in the human body organs due to some loss of function.

Various physical and physiological processes like blood flow, metabolic heat generation, thermal conduction, radiation, convection and evaporation are taking place in order to maintain this thermal balance of the body with the environment. Any abnormality in the structure, physiological parameters or environmental conditions can disturb this thermal balance of the body with the environment. A notable example is of cancer which is an abnormality of growth leading to abnormal rates of metabolic activity in the tissue 23, The study of thermal problems of a human body under normal and abnormal conditions can give us a better understanding of relationships among various parameters which can be useful to biomedical scientists for development of protocols for detection and treatment of diseases or disorders caused by these abnormalities in the system. In view of the above an attempt has been made here to study thermoregulation in extended spherical and ellipsoidal organs of a human body especially breast with special relevance to cancerous tumors [1,8].

1.2 THERMOREGULATION

Thermoregulation is the ability of an organism to keep its body temperature within certain boundaries, even when temperature surrounding is very different. This process is one aspect of homeostatic; a dynamic state of stability between living being’s internal environment and its external environment. If the body is unable to maintain a normal temperature and it increases significantly above normal, a condition known as heat stroke occurs [39,78], The opposite condition, when body temperature decreases below normal levels, is known as hypothermia. An organism that thermo regulates is one that keeps its core body temperature within certain limits; a thermo conformer, is subject to changes in body temperature according to changes in the temperature outside of its body. Heat production and heat loss vary considerably in different parts of the body, although the circulation of the blood tends to bring about a mean temperature of the internal parts 49, Humans have been able to adapt to a great diversity of climates, including hot humid and hot arid. High temperatures pose serious stresses to the human body, placing it in great danger of injury or even death. Disturbances say, an increase in metabolic heat production due to some abnormality upsets the thermal balance. Heat is stored in the body and core temperature rises. When the disturbance is too large the whole body is no longer able to function. In order to deal with these climatic conditions, humans have developed physiological and cultural modes of adaptation. The skin assists in keeping different the body temperature constant by reacting differently to the hot and cold conditions so that the inner body temperature remains more or less constant. The temperature regulation mechanism of the human body depends on the following mechanisms 35,

(i) Thermo analysis ( Heat loss or gain to / from the environment)
(ii) Thermo genesis (Heat generation)
(iii) Heat transport from body core to the surface through skin

1.2.1 THERMOANALYSIS

Humans and animals in general, are usually in a thermal steady state with respect to their surroundings. Heat generated by metabolic processes is lost to the environment from the skin surface by the following mechanisms 56 and are shown in Fig 1.1 .

(i) Convection
(ii) Conduction
(iii) Radiation
(iv) Evaporation

Unless the organism has more heat than which can be eliminated by radiation and convection, the evaporation (through perspiration) is not required and conduction is negligible 35,

The process of transfer of heat from the warmer to the cooler of two solid bodies that are in contact is called conduction. In the human body, the rate of conduction depends on the temperature gradient between the skin and the material with which the skin is in contact, and on the thermal properties of the material 40

Convection is the transfer of heat from particle to particle. It is the transfer of heat by the movement of fluids such as air or water. For example, the heated air around a stove will tend to rise to the ceiling. We all know hot air rises and cool air sinks. That is natural convection. Forced convection refers to the use of fans or pumps to move a fluid/air and the heat contained in it. We generally have forced-air furnaces in our homes. 41

Convection transfers heat (in order of importance)

1. Between fluids and solids
2. Within fluids.

If the body surface temperature is more than environmental air temperature, the heat flows from the body to the surrounding air. When the surface is at a temperature above that of the air, heat is transferred from the surface to the adjacent air by conduction, thereby changing the density of the heated air [56, 57, 55], When the ambient temperature is above body temperature, then radiation, conduction and convection all transfer heat into the body rather than out. Since there must be a net outward heat transfer, the only mechanism left under this condition is the evaporation or perspiration from the skin and the evaporative cooling from exhaled moisture. Even when one is unaware of perspiration, physiology texts quote an amount of about 600 grams per day of "insensate loss" of moisture from the skin. [59, 47]

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Figure 1.1 Thermoregulation 83

1.2.2 THERMOGENESIS

The phenomenon of heat generation within the body or tissue is known as thermo genesis Heat is generated within the body due to breathing, digestion and physical efforts. The most important factor responsible for heat generation in the human body is metabolism [83, 1], The overall process of metabolism consists of two generalized classes of chemical reactions.

(i) Anabolic reaction
(ii) Catabolic reaction

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Figure 1.2 Structure of ATP 98

The anabolism or constructive metabolism is the process of synthesis required for the growth of new cells and maintenances of all tissues.

Catabolism or destructive metabolism is a continuous process concerned with the production of the energy required for all external and internal physical activity.

Metabolism consists of anabolism (the constructive phase) and catabolism (the destructive phase, in which complex materials are broken down). The transformation of the macronutrients carbohydrates, fats, and proteins in food to energy, and other physiological processes are part of the metabolic process. ATP (adenosine Tri phosphate) is the major form of energy used for cellular metabolism 24.

Adenosine Tri Phosphate is arguably the most important molecule located within the body of human beings. It carries the energy necessary to facilitate all of the processes of human metabolism. The molecule itself is reasonably simple (especially in comparison to the structure of some of the other molecules commonly found in the human body) and consists of an adenine molecule attached to three phosphate molecules (hence the three phosphate part of the molecule name) [10, 11], The molecular weight of the molecule is around 507 grams of ATP per one mole (amole is 6.02 x 1023 molecules). (See fig 1.2)

1.2.3 HEAT TRANSPORT FROM CORE TO THE BODY SURFACE THROUGH SKIN

The body can be divided into two compartments: the thermal core and shell 34, Most of the heat produced within the core is dissipated onto the environment via the body surface including the lungs. The heat produced by metabolism must be transported from the core to the skin surface where it can be exchanged with the environment. Body temperature distribution thus strongly depends on convective heat transfer by the blood. Blood flow is driven by metabolic rate. Higher tissue temperature comprises an increase in metabolic rate, and consequently an increase in blood flow. By vasomotion the amount of blood flow is further regulated. Under thermo neutral conditions skin temperature is lower than the core and varies with ambient temperature 45, During exercise and heat exposure, cutaneous vasodilatation and sweating are triggered. On exposure to cold environments, skin blood flow decreases via cutaneous vasoconstriction. This results in reduced heat transfer from the core to the skin surface followed by decreased heat loss from the skin.

1.3 SKIN AND SUBDERMAL TISSUES

Skin is the largest organ in the body both by weight and surface area. In adults, the weight of the skin accounts for about 16% of your total body weight. The skin forms a protective covering around the living being partially responsible for retaining the body shape [75, 69], The skin and sub dermal tissues (SST) region of the human body undergoes temperature variation due to changes in environmental conditions and physiological parameters, like rates of metabolism, and blood mass flow, perspiration and structure of the SST region in order to maintain uniform body core temperature (37.4°C) 8, The heat transport from the body core to the surface takes place through this SST region 49,

There are two main layers of skin

- Epidermis
- Dermis

Epidermis and dermis make a skin layer and below this are subcutaneous (subdermal) tissues. The structure of skin is shown in Fig 1.3.

1.3.1 EPIDERMIS

This layer is seen on the surface of the skin. It is made up of cells called kératinocytes, which are stacked on top of each other, forming different sub-layers. The kératinocytes develop at the bottom and rise to the top, where they are shed from the surface as dead cells. So this layer is constantly renewing itself, the living cells are changing into dead, hard, flattened cells. Melanocytes and Langerhans cells are other important cells found in the epidermis which have special functions 19

Epidermis is divided into 5 layers

(a) Stratum Corneum

The stratum comeum is most superficially placed. The cells are keratinized here It is also called the homy layer.

(b) Stratum Lucidum

This is a thin and more or less transparent layer 3 to 5 cells deep placed below the stratum comeum. The cells out lines are indistinct and the nuclei are absent.

(c) Stratum Granulosum

It is situated below the stratum lucidem and consists of 3 to 5 layers of fattened cells filled with keratohyalin granules which take a deep satin with haemotorylin 19

(d) Stratum Spinosum

This is a broad layer of variable thickness and is made up to polyhedral cells. This layer is covered with minute spines.

(e) Stratum Germinativum

This growing layer is composed of a single layer of columnar epithelium which has got transverse thin short cytoplasm process on its vessel lamina by means of which they anchor the epithelium to the underlying dermis 39,

1.3.2 DERMIS:

The dermis consists mostly of connective tissue and is much thicker than the epidermis. It is responsible for the skin pliable and mechanical resistance and is also involved in the regulation of the body temperature. The dermis supplies the vascular epidermis with nutrients by means of its vascular network. It contains sense organs for touch, pressure, pain and temperature, as well as blood vessels, nerve fibers, sebaceous and sweat glands and hair follicles. [2,3,46]

- Blood Vessels

These are tiny pipes through which blood circulates. The blood vessels supply the skin with fresh blood, which contains nutrients and oxygen, and carry away waste products.

- Meissner's Corpuscle

These touch receptors are especially effective in detecting light touch and soft, fleeting movements.

- Pacinian Corpuscles

Pacinian corpuscles function as receptors for deep pressure and vibration.

- Free Nerve Endings

Free nerve endings are sensitive to pain, temperature changes and itchiness.

- Nerve Fibers

Nerve fibers forward information.

- Sebaceous Glands

Sebaceous or oil glands are small, sacculated organs that secrete sebum. This oily substance is a natural moisturizer which conditions the hair and skin. The sebaceous glands are found all over the body, but they are more numerous in the scalp area and around the forehead, chin, cheeks and nose. 4

- Sweat Glands

These are sweat-producing structures consisting of a single tube, a coiled body and a superficial duct. They are involved in thermoregulation, as they cool the skin by sweating.

- Hair Follicles

Hair follicles are downward growths into the dermis of epidermal tissue and produce hair. They are found all over the body except on the palms of the hands and soles of the feet as well as on the lips. When the body gets cold, the hair stands upright with the help of the arrector pili muscle, closing up the skin's pores and keeping the warmth in.

- Arrector Pili Muscle

This small muscle is attached to the base of the follicle. When it is stimulated by cold or fright, it pulls up the hair follicle, causing it to stand upright. 34

1.3.3 SUBCUTANEOUS TISSUES:

The subcutaneous layer under the dermis consists of loose connective tissue and much fat. It acts as a protective cushion and helps to insulate the body by monitoring heat gain and heat loss. Not all authors consider this layer a part of the skin, but it definitely has a strong impact on the way the skin looks. 46

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Figure 1.3 Structure of Skin [99, 94]

1.4 BLOOD CIRCULATORY SYSTEM

The Circulatory system is responsible for transporting materials throughout the entire body. The circulatory system is made up of the vessels and the muscles that help and control the flow of the blood around the body. This process is called circulation. The main parts of the system are the heart, arteries, capillaries and veins. As blood begins to circulate, it leaves the heart from the left ventricle and goes into the aorta. The aorta is the largest artery in the body. The blood leaving the aorta is full of oxygen. This is important for the cells in the brain and the body to do their work. The oxygen rich blood travels throughout the body in its system of arteries into the smallest arterioles. On its way back to the heart, the blood travels through a system of veins. As it reaches the lungs, the carbon dioxide (a waste product) is removed from the blood and replaces with fresh oxygen that we have inhaled through the lungs. 35

The blood circulatory system is shown in Fig 1.4. Arteries are tough, elastic tubes that carry blood away from the heart. As the arteries move away from the heart, they divide into smaller vessels. The larger arteries are about as thick as a thumb. The smallest arteries are thinner than hair. These thinner arteries are called arterioles. Arteries carry bright red blood! The color comes from the oxygen that it carries. Veins carry the blood to the heart. The smallest veins, also called phenols, are very thin. They join larger veins that open into the heart. The veins carry dark red blood that doesn't have much oxygen. Veins have thin walls. They don't need to be as strong as the arteries because the blood in it returns to the heart and it is under less pressure. The blood passes through the right side of the heart and goes to the lungs, Where it gets rid of carbon dioxide and picks up more oxygen and then it passes through the left side of the heart and is pumped back around the body 1. The blood always circulates through the body in the same direction. Apart from oxygen and carbon dioxide many other substances are carried in the blood. The blood circulating through the digestive system picks up digested food products and carries them to the liver to be used or stored 46,

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Figure 1.4 A Diagrammatic Representation Blood Circulatory System 100

1.5 BLOOD FLOW AND METABOLIC ACTIVITY IN SST REGION

The blood circulatory system of the skin can be divided into two major types of vessels (i) the nutritive arteries, capillaries and veins, and (ii) vascular structures concerned with the heating of the skin. It consists principally of (a) an extensive subcutaneous venous plexus which hold large quantities of blood, that can heat the surface of the skin, and (b) in some skin area arteriovenous anastomoses, which are large vascular communication directly between the arteries and veins [1, 46], The number of blood vessels in the dermis is very thin near the interface of epidermis and dermis, but it increases gradually and becomes large and almost uniformly distributed in the sub dermal part. The blood flow in the epidermis is negligible. The rate of blood flow in SST is among the most varied of any part of the body 83, as the same is controlled by nervous system which regulates the supply of blood depending on atmospheric temperature and metabolic activity of the tissues. Skin, like a kidney (but in contrast, muscle, brain and myocardium) normally have a blood flow which far exceeds the nutritional needs. Under ordinary cool conditions the blood flow to the skin is about twenty five c.c. per square centimeters by body surface area. On the other hand when skin is heated until maximum vasodilations has resulted, the blood flow can be much as seven times this value. When the skin is exposed to cold, the blood vessels constrict more and more and at a temperature of about 13°C they reach their maximum degree of constriction. Below 10°C , the vessels again start dilating to heat the living cells active in the peripheral region . At the minimum blood flow, the insulation property of SST is maximized. However, the degree of insulation differs from one person to another depending to a great extent on the quality of adipose tissue 46,

Metabolic circulatory control adjusts the blood flow to the metabolic rate of the organ. It exists because tissue metabolically required substances and /or metabolites are intrinsically linked to the smooth muscle of vessels in the tissue. Metabolic control increase blood flow when metabolic requirements increase and may or may not account for auto regulation. The postulated mechanism of metabolic control is that a metabolically required substance such as oxygen or a metabolite such as carbon dioxide acts directly through some chemical intermediate on vascular smooth muscle 9, In the case of carbon dioxide in the extracellular fluids which surrounds the vascular wall. This would cause the vascular smooth muscle to relax, increasing the vessel radius and blood flow. The increased flow would carry away the excess carbon dioxide, lowering the tissue level and the stimulus to vasodilation. In the same way the inadequate supply of required metabolic material like oxygen causes vasodilation. In most organs, 49 if the blood flow is interrupted by occluding the artery for a brief period, marked vasodilation will be observed. This phenomenon is called hyperthermia and is a manifestation of metabolic control. During the period of arterial occlusion, either the metabolism substrate is depleted or metabolite concentration is built up. Thus, the metabolic control is a coupling between the metabolic rate of a tissue and the blood flow through it , independent of nerves and reflexes 1.

1.6 BREAST

The breast or the mammary gland is the most important structure in the pectoral region. Both men and women have breasts; but they are well developed only in women. It is rudimentary in men. It is well developed in the female after puberty. The breast is a modified sweat gland. It forms an important accessory organ of the female reproductive system, and it provides nutrition to the newborn in the form of milk 77, The structure of the breast is shown in ( Fig 1.5).

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Figure 1.5 Structure of Breast 83

1.6.1 LOCATION:

It lies in the superficial fascia of the pectoral region. Auxiliary’s tail (of Spence) pierces the deep fascia and lies in the axilla. Some women discover this- especially when it may enlarge during a menstrual cycle- and become concerned that it may be a ‘lump’ or enlarged lymph nodes 93,

1.6.2 THE SKIN OF BREAST:

It covers the gland and has the following features.

1. A conical projection called the nipple is present just below the center of the breast at the level of the fourth intercostal space. However the position varies in multiparous women and therefore is not a reliable guide to fourth intercostal space in adult females. The nipple is pierced by 15-20 lactiferous ducts. It contains circular and longitudinal smooth muscle fibers which can make the nipple stiff or flatten it. It has few modified sweat and sebaceous glands. It is rich in its nerve supply and has many sensory end organs at the terminations of nerve fibers 77,
2. The skin surrounding the base of the nipple is pigmented and forms a circular area called the areola. This region is rich in modified sebaceous glands, particularly in its outer margin. These become enlarged during pregnancy and lactation to form raised tubercles (of Montgomery). An oily secretion of these glands lubricates the nipple and prevents them from cracking during lactation. Apart from sebaceous glands the areola also contains some sweat glands and accessory mammary glands. The skin of the areola and nipple is devoid of hair, and there is no fat subjacent to it 84,

1.6.3 STAGES OF BREAST DEVELOPMENT

A newborn baby has nipples, areolas, and the beginnings of breast tissue, but most of breast development occurs in two different periods of time in a woman's life: first in puberty, then during pregnancy. The first external signs of breast development appear at the age of 10or 11 - though it can be as late as 14 years. The ovaries start to secrete estrogen leading to an accumulation of fat in the connective tissue that causes the breast to enlarge. The duct system also begins to develop, but only to the point of forming cellular knobs at the end of the ducts. As far as we know the mechanism that secretes milk doesn’t develop until pregnancy. Although the breast may appear fully grown within a few years of puberty, strictly speaking, their development is not complete until they have fulfilled their biological function - that is, until the woman carries a pregnancy to term and breastfeeds her baby, when they will undergo further changes 81, Human breast tissue begins to develop during the sixth week of fetal life. Breast tissue initially develops along the lines of the armpits and extends to the groin (this is called the milk ridge). By the ninth week of fetal life, it regresses (goes back) to the chest area, leaving two breast buds on the upper half of the chest. In females, columns of cells grow inward from each breast bud, becoming separate sweat glands with ducts leading to the nipple. Both male and female infants have very small breasts and actually experience some nipple discharge during the first few days after birth 96. Female breasts do not begin growing until puberty—the period in life when the body undergoes a variety of changes to prepare for reproduction. Puberty usually begins for women around age 10 or 11. After pubic hair begins to grow, the breasts will begin responding to hormonal changes in the body. Specifically, the production of two hormones, estrogen and progesterone, signal the development of the glandular breast tissue. During this time, fat and fibrous breast tissue becomes more elastic. The breast ducts begin to grow and this growth continues until menstruation begins (typically one to two years after breast development has begun). Menstruation prepares the breasts and ovaries for potential pregnancy 77,

Breast development is a vital part of puberty in the human female. The different stages of development are shown in Fig 1.6 and are given below:

Stage 1

(Preadolescent) only the tip of the nipple is raised

Stage 2

Buds appear, breast and nipple raised, and the areola (dark area of skin that surrounds the nipple) enlarges

Stage 3

Breasts are slightly larger with glandular breast tissue present

Stage 4

The areola and nipple become raised and form a second mound above the rest of the breast

Stage 5

Mature adult breast; the breast becomes rounded and only the nipple is raised

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Figure 1.6 Stages of Breast 80

1.6.4 MATURITY OF THE BREASTS

Once a young woman reaches puberty, and ovulation and the menstrual cycle begin, the breasts start to mature, forming real secretary glands at the ends of the milk ducts. Initially these glands are very primitive and may consist of only one or two layers of cells surrounded by a base membrane 80,

Between this membrane and the glandular cells are cells of another type, called a Myo­epithelial cells, these cells are the ones that contract and squeeze milk from the gland if pregnancy occurs and milk production takes place .With further growth, the lobes of the glands become separated from one another by dense connective tissue and fat deposits. This tissue is easily stretched. This is where the natural enlargement formula comes in and allows the enlargement that normally occurs during pregnancy when the glandular elements swell and grow. The duct system grows considerably after conception and many more glands and lobules are formed. This causes the breast to increase in size as it matures to fulfill its role of providing food for the baby 85,

1.6.5 AGING OF THE BREAST

As we get older, our breasts tend to sag and flatten; the larger the breasts, the more they sag. With the menopause there is a reduction in stimulation by the hormone estrogen to all tissues of the body, including breast tissue; this results in a reduction in the glandular tissue of the breasts. So they lose their earlier fullness. Regular exercise would have however prevented or slowed down the ageing process. Much of the connective tissue in the breast is composed of a fibrous protein called collagen, which needs estrogen to keep it healthy 77, Without estrogen, it becomes dehydrated and inelastic. Once the collagen has lost its shape and stretch ability it "was" believed that it could not return to its former state or condition.

1.7 CANCER

Our bodies are made up of billions of cells that grow, divide, and then die in a predictable manner. Cancer occurs when something goes wrong with this system, causing uncontrolled cell division and growth. Cancer cells lump together and form a mass of extra tissue, also known as a tumor, which continues to grow 90, The exact causes of most types of cancer are still not known. However, it is now known that the risk of developing many types of cancer can be reduced by adopting certain lifestyle changes, such as quitting smoking and eating a better diet. 89 To understand briefly about cancer it is useful to define Neoplasia which means new growth. In medical terms Neoplasia is restricted to tumor growth, a process that serves no useful purpose and which is not controlled by the laws of normal growth. As the tumor grows, it may damage and invade nearby tissue. If a cancerous tumor outgrows its birthplace (called the primary cancer site) and moves on to another place (called the secondary cancer site), it's referred to as metastasizing. Thus in other words, a tumor is a mass of tissue that

Serves no useful purpose and generally exists at the expense of healthy tissue. The process of growth of the tumor is shown in Fig 1.8 87.

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Figure 1.7 Structure of Cancer 87

1.7.1 CAUSES OF CANCER

The exact cause of cancer is not known. Most cancers result from permanent damage to genes or from mutations, which occur either due to internal factors, such as hormones, immune conditions, metabolism, and the digestion of nutrients within cells, or by exposure to environmental or external factors. A chemical or other environmental agent that produces cancer is called a carcinogen. Overall, environmental factors, defined broadly to include tobacco. 87 . Use, diet, infectious diseases, chemicals, and radiation, are believed to cause between 75 and 80 percent of all cancer cases in the world. Tobacco use, including cigarettes, cigars, chewing tobacco, and snuff, can cause cancers of the lung, mouth, throat, larynx, bladder, kidney, esophagus, and pancreas. Heavy consumption of alcohol has also been shown to increase the risk of developing cancer of the mouth, pharynx, larynx, esophagus, liver, and breast. Overweight and obesity are associated with increased risk of cancers of the breast, colon, kidney, and gallbladder. The following chemicals have been found to cause cancer: coal tars and their derivatives, such as benzene; some hydrocarbons; aniline, a substance used to make dyes; and asbestos 90,

1.7.2 ORIGINS OF CANCER

All cancers begin in cells, the body's basic unit of life. To understand cancer, it's helpful to know what happens when normal cells become cancer cells. The body is made up of many types of cells. These cells grow and divide in a controlled way to produce more cells as they are needed to keep the body healthy. When cells become old or damaged, they die and are replaced with new cells 87, However, sometimes this orderly process goes wrong. The genetic material (DNA) of a cell can become damaged or changed, producing mutations that affect normal cell growth and division. When this happens, cells do not die when they should and new cells form when the body does not need them. The extra cells may form a mass of tissue called a tumor 96 .

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Figure 1.8 Growth of Tumor 89

1.7.3 TYPES OF TUMORS

The tumors are classified into following three types.

1. Benign or Innocent Tumor
2. Malignant Tumor
3. Metastasis

These are discussed in the subsequent subsections below:

1.7.3.1 BENIGN TUMOR

A benign tumor is a tumor that does not grow in an unlimited, aggressive manner, does not invade surrounding tissues, and does not metastasize. It is basically a tumor that doesn't come back and doesn't spread to other parts of the body. Common examples of benign tumors include moles and uterine fibroids. The term "benign" implies a mild and no progressive disease, and indeed, many kinds of benign tumor are harmless to the health 87, Benign tumors tend to grow more slowly than malignant tumors and are less likely to cause health problems. However, some 'benign tumors' which lack the invasive properties of a cancer, may still produce negative health effects. Benign tumors inhibit their ability to behave in a malignant manner. Nonetheless, many types of benign tumors have the potential to become malignant.

1.7.3.2 MALIGNANT TUMOR

Malignant tumors are ambitious. Unlike benign tumors that generally stay put, malignant tumors have two goals in life: to survive and to conquer new territory. In general, malignant tumors grow faster than benign tumors and are more likely to cause health problems. In deciding the malignancy of a tumor it is necessary to consider the arrangement of tumor cells and their relations to surrounding normal tissues, and the character of tumor cells and especially the character of their nucleus and nucleolus. The malignancy has two main functional qualities: Invasion i.e. intrusion on and destruction of adjacent tissues and formation of metastasis i.e. the spread of a disease from one organ or part to another non-adjacent organ or part 88,

1.7.3.2 METASTASIS

In many malignant tumors, as the cells spread, they come across blood vessels. If they actually spread into the blood vessel, they get carried around the body and eventually get stuck in a smaller blood vessel in another part of the body. Here they begin to divide and grow again eventually forming a new tumor. These are called secondary tumors or metastases. This process of cancers spreading around the body is called metastasis [87, 82],

1.7.4 BREAST CANCER:

A cancer that forms in tissues of the breast, usually the ducts (tubes that carry milk to the nipple) and lobules (glands that make milk). It occurs in both men and women, although male breast cancer is rare [63, 64], Breast cancer refers to cancers originating from breast tissue, most commonly from the inner lining of the milk ducts or the labels that supply the ducts with milk (See Fig 1.9). Cancers originating from ducts are known as ductal carcinomas; those originating from labels are known as lobular carcinomas. There are many different types of breast cancer, with different stages (spread), aggressiveness, and genetic makeup; survival varies greatly depending on those factors 19,

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Figure 1.9: Breast Cancer [85, 89]

1.7.5 VASCULAR BED IN TUMOR

The vascular network of tumor is different from that of normal tissue. The relative proportion of the relative components in each tumor is characterized by that of the tumor type as well as the stage of tumor growth. In early stages the tumor exists as a small nodule of tissue. However the establishment and progressive growth of malignant tumors is possible when the supply of essential nutrients is adequately maintained through vascular network. During the initial stages of tumor growth, the tumor is supported by the nutrients supplied by the adjacent vessels of the host tissue and become dilated and tortuous. Subsequently the endothelial cells of the altered host start to proliferate perhaps by the influence of angiogenesis factor from the tumor cells. Numerous sprouts grow out of the hypertrophic vessels and grow towards the tumor [48,70], The sprouts or their branches eventually give rise to loops. The growing capillaries slowly begin to perfuse through the new vessel. As the tumor mass increases more tumor capillaries are formed. The newly formed tumor capillaries consist of a single layered endothelial wall with no external basement membrane [64, 38], In many types of tumors, between the gaps of endothelial cell, neoplastic cells often protrude into the tumor of capillaries and obstruct the perfusion of blood. One of the striking features of the tumor vascular network is the presence of tortuous giant capillaries. The wall of such vessels is composed only of endothelial cells with same fibrous supporting tissues. These vessels are usually located at the periphery of tumors and contain vascular blood. Hence not all tumor vessels are newly formed vessels. Another interesting feature is that the host vessels are resistant to neoplastic growth and is rarely invaded by the tumor cells but often incorporated into the tumor mass [8, 1], The incorporated host vessels do not increase in number. Although the length and caliber may increase Proteins of such host vessels are incorporated into the tumor vessels, particularly in the necrotic area of the tumors.

1.7.6 METABOLIC ACTIVITY IN TUMOR

Cancer is characterized by uncontrolled growth and the energy which is required for this growth is derived from the host’s growth. Hence the metabolic activity of the tumor is also uncontrolled. In fact cancer is in essence a change in cell metabolism brought about by some failure in the intracellular enzyme systems. In cancer cells there is a tendency toward a common enzymatic pattern of activity whereas the cells of normal organs and tissue process specific chemical chrematistics [10, 11], Cancers are much more effective that normal adult tissue in synthesizing their protein at the expense of the energy made available to them by respiration and glysoysis. The tumors are well known for their metabolic superiority over the normal tissue to enhanced amino acid concentrating power of the tumors. The normal and neoplastic cells share a common volume of extracellular fluid and compete with each other for the available metabolites 8,

1.8 CONCLUDING REMARKS

The necessary biophysical background required for modeling thermal problems in extended spherical and ellipsoidal shaped human organs was presented in this chapter. This includes the basic concepts, description of thermoregulatory processes, structure of SST region and breast, cancer and biophysical processes taking place in SST region, breast and tumors [37,38], Based on this physiological background, the formulation of the mathematical model is presented in the next chapter.

CHAPTER -2

MATHEMATICAL BACKGROUND

2.1 INTRODUCTION

The advances in technology during the last few decades have propelled the growth of experimental research in biology and medicine. In spite of these advancements in instrumentation technology, there are many situations in which biological experiments are extremely difficult or very expensive. Apart from the above there are restrictions by government or ethical agencies for experiments on the animals. Thus the mathematical modeling is a viable alternative in such situations [35,69,70], The domains of biology and medicine provide many opportunities for fascinating research at all levels of mathematics. The most biological and medical material is highly variable and poses new challenges for mathematics. In order to achieve the real insight into the control of phenomena and processes involved, much more precise logical, logical instrument is required, which can be provided with the right kind of mathematics [13,31,27], The development of computer technology and computational sciences has increased the scope for the application of mathematical methods in solving real world problems of biology and medicine. In the present chapter the mathematical model of heat flow in human body organs is presented along with literature survey and mathematical methods for the solution in the subsequent sections 12,

2.2 MATHEMATICAL MODEL

The mathematical formulation of heat diffusion in solids is governed by the partial differential equation which was derived from W. Perl 74 and is given below:

If Qdenotes quantity of heat in tissue element at time t then is defined as the sum of equation which was derived from W. Perl [74] and is given below: rates of change in heat quantity due to diffusion d, perfusion P and metabolic heat generation m.

Abbildung in dieser Leseprobe nicht enthalten

The first term on the right hand side of equation (2.1) can be defined with the help of Fick; s perfusion principle which states that the rate of change of a quantity of a substance in an organ due to perfusion equals the rate of outflow of substance via the venous blood. The rate of inflow of substance is the product of concentration of substance in the arterial blood and the rate of flow of arterial blood.

The second term on the right hand side of equation (2.1) can be defined with the help of Fourier’s law and the third term can be denoted by S the rate of metabolic heat generation. Fick’s principle of perfusion can be written as [72, 69, and 30]

Abbildung in dieser Leseprobe nicht enthalten (2.2)

Quantity of substance per fused Rate of blood flow in arteries Rate of venous blood flow

Here CA, Cy is the concentration of the substances respectively in arteries and veins. For small volume dV the equation (2.2) is written as

Abbildung in dieser Leseprobe nicht enthalten

3Q Amount of substance in volume element dr

Dividing both sides of (2.3) by volume element dr we get

Abbildung in dieser Leseprobe nicht enthalten (2.4)

Where C = Tissue concentration of substance, which is a function of position and time

Abbildung in dieser Leseprobe nicht enthalten (2.5)

inflow of substance via arterial blood minus the rate of Where C = Tissue concentration of substance, which is a function of position and time

If the substance of interest is thermal energy, then equation (2.4) takes this following form

Local tissue temperature. Arteriolar blood temperature. Venular blood temperature. Density of tissues

Abbildung in dieser Leseprobe nicht enthalten

The conduction may be elaborated by using Fourier law of heat conduction as given below12:

Abbildung in dieser Leseprobe nicht enthalten(2.6)

Abbildung in dieser Leseprobe nicht enthalten(2.7)

K is thermal conductivity of various human body tissues 72, The third term on the right hand side of equation (2.1) is due to effect of metabolism [72, 12] and expressed as

Abbildung in dieser Leseprobe nicht enthalten (2.6)

Where mb is the blood mass flow, Cb is the specific heat of blood; S is rate of metabolic heat generation. The parameters K, 1Hb and S may depend on position, time and temperature.

The boundary conditions can be imposed based on the physical conditions. When outer surface of the skin is exposed to the environment, the heat loss takes place from outer surface to the environment by conduction, convection, radiation and evaporation. Further the inner boundary is the body core which is maintained at a uniform temperature of 37°C under normal conditions [56, 57], In view of the above the boundary conditions can be mathematically expressed as follows:

Abbildung in dieser Leseprobe nicht enthalten(2.11)

On boundary T (outer surface)

Abbildung in dieser Leseprobe nicht enthalten= Directional derivative along outward normal to 1 .

Where, h= heat transfer coefficient, Ta = atmospheric temperature, L = latent heat of evaporation, E= rate of evaporation.

(ii) At the inner boundary surface 7 = Tb

Tb is the body core temperature, Ta is the arterial blood temperature.

At lower atmospheric temperature the body core temperature is variable in extreme parts of human body like human limbs and breast. In such conditions the expression (2.12) can be replaced by appropriate function of position based on the physical conditions of the problem [31,55].

2.3 DEVELOPMENT OF THE SUBJECT

Earlier experimental investigations were made by Patterson 39 to obtain temperature profiles in the SST region. Some theoretical investigations have been carried out during the last few decades by Cooper, Trezek, and Chao et.al. [10,11] and Saxena et.al. [54, 55], to study the temperature distribution in the SST region under normal environmental and physiological conditions. Saxena et.al. 53 and Gurung et.al. [15 ,16], also studied similar transformation and finite element method to unsteady state heat migration problem in human SST. Also attempts have been made by Saxena and Pardasani [47, 48 &62] & Mittal et.al 32 to study problems involving abnormalities like tumor in the SST region of the human body. Saxena , Pardasani and Agrawal 61 developed an Unsteady state model heat flow in the epidermis and dermis of the human body . Saxena and Arya [56,57 &58] solved at one and two dimensional steady state problem using variational finite element approach. With all this background, Bindra 9 developed a mathematical study of heat transfer problems in cutaneous in vivo tissues. Saxena and Bindra 59 carried out the investigation and developed Steady state temperature distribution in dermal regions of human body with variable blood flow perspiration and self controlled metabolic heat generation using this approach. Further they improved the assumptions regarding physiological parameters like rate of metabolic heat generation. It was taken as a function of position, tissue temperature and atmospheric temperature. Also they assumed the mass blood flow rate and thermal conductivity as position dependent. Saxena and Bindra 60 used quadratic shape functions in variational finite element approach to study heat distribution in cutaneous and subcutaneous tissues for steady state cases. Some attempts have also been made to study the temperature distribution in skin and subcutaneous tissues involving abnormalities like tumors. Pardasani and Adlakha [44,45 &46 ], developed a model to study temperature variation in human limbs for one and two dimensional steady state cases under normal physiological and environmental conditions. Pardasani and Adlakha 43 obtained exact solutions to heat flow problem in dermal regions of the human body with a solid tumor. Pardasani 46, developed model mathematical investigations as human physiological heat flow problems with special relevance to cancerous tumors. Adlakha 1 has been developed a finite element approach to thermal study of tumors in cylindrical and spherical shaped organs. Gupta and Pardasani 14 developed mathematical approach for the formal study of malignant tumors in flat shaped organs for a one, two and three dimensional case. Some models have been developed by Pardasani and Jas 19 for temperature variation in human limbs for one and two dimensional steady state cases by using several techniques like the finite element method, Ritz method and pseudo analytical approaches etc. Under normal physiological and environmental conditions. Pardasani and Shakya [40, 41, and 42] have studied the temperature distribution in plane regions of the human body for two dimensional steady state cases involving finite and infinite domains. Agrawal, Adlakha and Pardasani [2, 3, 5 and 6] developed two and three dimensional thermal distribution models in the dermal layers of elliptical shaped human limbs involving a uniformly perfused tumor. Agrawal, Adlakha and Pardasani [4 and 7] developed cubic splines and Fourier series approach to study temperature variation in dermal layers of elliptical shaped human limbs and tapered shaped human organs. From above literature survey it is evident that very little attention has been paid to study the heat flow in spherical shaped human organs. The thermoregulation in human head has been investigated by Khanday et. Al. [21,22] under cold environment. The thermal modeling of women breast under normal environmental conditions has been carried out by researchers [33, 23and20] to study relationships among various biophysical parameters. Theoretical investigations have 36 also been carried out to study the effect of tumors in the deep tissues of women’s breast on the surface temperature of the breast.

Some attempts have been made by Osman et.al. [37&38] for thermal modeling of the normal and malignant woman’s breast under different conditions. Sudharsan 63 developed a model of surface temperature distribution of a breast with and without spherical shaped tumor. From the literature survey it is observed that no attempt is reported for the study of temperature distribution in the peripheral part of the human breast. Also no attempt has been made in the past by research workers in the study of effect of malignant tumors in peripheral regions of the human breast. Apart from this the investigators have made attempts to study the temperature distribution in the deep tissues of human breast of spherical shape. But the shape and size of human breast vary with the age, ethnicity, physical make up etc. No attempt is reported in the literature for study of temperature distribution in different ellipsoidal shaped breasts depending on their stages of development. In view of above an attempt has been made in this thesis to address some of the issues mentioned above.

2.4 MATHEMATICAL AND COMPUTATIONAL METHODS

The formulation of the model for temperature distribution in the human body organs in section

2.3 resulted in initial and boundary value problem involving partial differential equations. The next step is to explore mathematical and computational methods for their solution. The analytical solution is always preferable, but analytical methods have limited scope as they can be employed for simplified models. But when we incorporate more and more details of physiology and biological parameters the models become very complicated and therefore it becomes difficult to use analytical methods for their solution 73, Thus one has to explore advanced numerical and computational techniques like finite differences, finite volume and finite element methods etc. Some of the analytical and numerical techniques which have been used in present thesis are discussed in the subsequent subsections 72,

2.5 NUMERICAL TECHNIQUES

The numerical techniques like Finite Difference Method, finite volume method and finite element method can be employed to obtain the solution. Some of these methods are presented as given below in the subsequent series 66,

2.5.1. FINITE DIFFERENCES METHOD

The Finite Different Method (FDM) consists of transforming the partial derivatives in difference equations over a small interval. It is important to note that the finite difference method is capable of evaluating any type of linear and non-linear partial differential equation as well as ordinary differential equations [66, 28], Assuming that u is function of the independent variables x and y, we can divide the x-y plane in mesh points equal to dx = h & dy = It, we can evaluate u at point P by:

Abbildung in dieser Leseprobe nicht enthalten

The value of the second derivative of w at point P can be evaluated by 28:

Abbildung in dieser Leseprobe nicht enthalten (2.14)(2.15)

The value of the first derivative of w at point P can also be evaluated by the following three approximations [62 &66] :

1) Central Difference:

Abbildung in dieser Leseprobe nicht enthalten

2) Forward Difference

Abbildung in dieser Leseprobe nicht enthalten

3) Backward Difference:

Abbildung in dieser Leseprobe nicht enthalten

After substituting the derivative approximations in the PDE, PDE is converted into a set of finite difference equations that can be solved either by explicit or implicit method. Hereafter, we will think of the j index as time to expiration (so that j + 1 means more time to expiration, an earlier time instant), and i index as spatial point [50,72],

2.5.2 FINITE ELEMENT METHOD

Finite element method is an advanced mathematical cum numerical technique for solving the boundary value problem. The method involves dividing the solution domain into a finite number of simple sub domains known as elements and to work out an appropriate method of constituting solution in each sub region or group of sub regions 26, The solution may be exact or approximate depending on the type of the equations constituting a boundary value problem. The final solution is then obtained by assembling the entire domain. Since these elements (sub regions) can be put together in a number of ways, these can be used to represent exceedingly complex shapes [68,73] .In a continuum problem of any dimension the field variable is a function of each generic point in the body or solution region, so it possesses infinitely many values. Consequently, the problem is one with an infinite number of unknowns. The finite element discretization procedure reduces the problem to one of a finite number of unknowns by dividing the solution region into elements and by expressing the unknown field variable in terms of assumed approximating functions within each element [51, 52],

The mathematical approach in finite element method is based on any one of the classical methods 18,

I. Ritz Approach
II. Weighted Residual Approach
III. Collocation Method
IV. Galerkin’s Approach
V. Least Squares Approach

These approaches are discussed in the subsequent subsections below:

2.5.2.1 THE RITZ METHOD

Consider the functional (2.19)

This functional has a minimum point where 18: (2.20) and (2.20) is the Euler Lagrange equation for variational principle (2.19)

The Ritz approximation is given by:

Abbildung in dieser Leseprobe nicht enthalten(2-21)

For the solution of the minimization problem we take

Abbildung in dieser Leseprobe nicht enthalten(2.22)

It defines a space of dimension n, which leads to

Abbildung in dieser Leseprobe nicht enthalten (2.23)

for i = 1,2,3, « ,and 0, are trial functions. This (2.23) gives a system of algebraic equations in ut, which are than solved to obtain approximate solution [28,26, 18],

For the functional involving only one space variable x e [a, b ] is expressed as:

Abbildung in dieser Leseprobe nicht enthalten (2.24)

The Euler Lagrange equation takes the form

Here I (v) has the minimum at w = v. The functional involving two independent variables x and y is expressed as:

Abbildung in dieser Leseprobe nicht enthalten(2.26)

The Euler Lagrange equation takes the form [68] :

EL L o (2.27)

This formulation is called a weak formulation because it weakens the continuity requirements.

2.5.2.2 VARIATIONAL (RAYLEIGH - RITZ) FINITE ELEMENT METHOD

This method is an extension of Ritz method and is used for more complex boundary value problems. In the variational finite element method (v.f.e.m) the domain r is discretized into a finite number of sub domains (elements) and variational functional is obtained. The approximate solution for each element is expressed in terms of undetermined nodal values of the field variable, as appropriate shape functions (trial functions) or interpolating functions. The algebraic equations for the elements are assembled over the entire region and boundary conditions are suitably incorporated. The equations are solved for the nodal values and the approximate solutions are obtained as piecewise interpolation functions [46, 19], The variational finite element procedure can be divided into the following steps:

(i) Discretizing the domain (and the boundary)
(ii) Transformation to variational form
(iii) Selection of nodal points and interpolation functions
(iv) Formulation for each element
(v) Application of Ritz method in each element
(vi) Assembling equation for entire domain
(vii) Application of boundary conditions
(viii) Solution of equations for nodal values by additional computation if necessary
(ix) Solution of algebraic equations
(x) Assembling element wise solutions
(xi) Numerical interpretation.
(xii) Error estimates and other requirements.

2.5.2.2.1 TYPES OF ELEMENTS

We divide the body into an equivalent system of finite elements with associated nodes and choose the most appropriate element type to model most closely the actual physical behavior. The total number of elements to be used and their variation in size and type within a given body depends on the shape and size of the body. The primary line elements consist of bar and beam elements [51, 18], They have a cross-sectional area but are usually represented by line segments. In general, the cross-sectional area within the element can vary, but throughout this text it will be considered to be constant. The simplest line element has two nodes, one at each end, although higher-order elements having three or more nodes (called quadratic, cubic, etc. elements) also exist. The basic two-dimensional (or plane) elements are loaded by forces in their own plane (plane stress or plane strain conditions). They are triangular or quadrilateral elements 43, The simplest two-dimensional elements have corner nodes only (linear elements) with straight sides or boundaries although there are also higher-order elements, typically with midsize nodes (called quadratic elements) and curved sides .The elements can have variable thicknesses throughout or be constant. The most common three-dimensional elements are tetrahedral and hexahedral (or brick) elements; they are used when it becomes necessary to perform a three-dimensional stress analysis .Whereas higher-order elements with mid edge nodes (and possible mid face nodes) have curved surfaces for their sides 51, The axisymmetric element is developed by rotating a triangle or quadrilateral about a fixed axis located in the plane of the element through 360. Some of the element types are shown in Fig 2.1

Abbildung in dieser Leseprobe nicht enthalten

Figure 2.1 Types of elements [18]

There are three types of coordinate systems in finite element method 52,

The coordinate system used to define the points in the entire structure is called global coordinate system (see Fig 2.2).

Abbildung in dieser Leseprobe nicht enthalten

Figure 2.2 Global coordinates System 51

Abbildung in dieser Leseprobe nicht enthalten

For the convenience of deriving element properties in FEM many times for each element a separate coordinate system is used which is called local coordinate system. But the final equations are to be formed in the global coordinate system only. A natural coordinate system is a coordinate system which permits the specification of a point within the element of a set of dimensionless numbers whose magnitude never exceeds unity [26, 65],

2.5.2.3 WEIGHTED RESIDUAL APPROACH

The weighted residual methods also involve an integral. In these methods an approximate solution is substituted into the differential equation 52, Since the approximate solution does not satisfy the equation a residual or error term results. Let y = h (x) be the solution of differential equation then we have:

Abbildung in dieser Leseprobe nicht enthalten (2.28)(2.29)

The y = h ( x) does not satisfy the equation. The weighted residual methods require that

Abbildung in dieser Leseprobe nicht enthalten

[...]

• Grade: 9
• Author: Akshara Makrariya (Author)
• Publication Year: 2013
• Pages: 200
• Catalog Number: V1152372
• ISBN (PDF): 9783346581792
• Language: English
• Tags: heat, temperature