MATHEMATICAL MODELING OF PHASE CHANGE HEAT TRANSFER PROCESS IN BIOLOGICAL TISSUES

Abstract

Heat transfer involving phase change phenomena in biological tissues is a complex process. It involves several mechanisms such as thermal conduction, convection, radiation, metabolic heat generation, blood perfusion and phase change. The analysis of many biological heat transfer applications by the physiologists, physicians and engineers in the field of bio-heat transfer have resulted in improvement of treatment, preservation, destroying tumors and the protection of humans from extreme environmental conditions. Phase change heat transfer problems are also known as moving boundary problems which are encountered in many practical applications like metal casting, environmental engineering, thermal energy storage system, freezing and thawing of foodstuff, cryopreservation and cryosurgery etc. Cryosurgery is a therapeutic technique that uses extreme freezing to treat the diseased tissues. The basic feature of this technique is that it is low invasive and offer the advantages of less expensive, shorter hospitalization and recovery period. The objective of cryosurgery is to treat the affected tissues and minimize the damage of healthy tissues in the vicinity of the tumor tissues. A number of investigations have been carried out to study the applications of cryosurgery. Various heat transfer models are used to analyze phase change phenomena in heat transfer problems. The purpose of most of the heat transfer models is to find the temperature field and heat flux in a biological tissue under the set of constraints: general heat equation, initial and boundary conditions and distribution of sources or sinks, etc. During phase change, interface between the frozen and unfrozen regions is moving with time and the boundary conditions at this interface require specific treatment. Except initial and boundary conditions, two more conditions are needed on the moving boundary, one to determine the boundary itself and another to complete the solution of the heat equation in each region. The phase change problems are non-linear in nature due to the unknown position of the freezing front and the direction of ice growth. In advance, it is difficult to predict the position and velocity of moving interface. The required mathematical analysis is much more complicated, when the physical properties of the system are temperature dependent. ii Phase change heat transfer problems have a limited number of analytical solutions, which are confined up to one-dimensional problems along with some simple boundary conditions. Therefore, for solving such type of problems it is essential to employ the numerical methods because they appear to offer a more practical perspective. Based on front tracking, non-front tracing and fixed domain approaches, various numerical methods have been proposed for the solution of phase change problems. Numerical methods based on enthalpy and effective heat capacity formulation are well known methods to solve phase change heat transfer problems. The present thesis deals with some Mathematical models to study phase change heat transfer problems in biological tissues during cryosurgery. The study of the thermal gradient inside the tissue is an important issue for the optimization of cryosurgery. The transient temperature profiles in tumor and normal tissue are useful to diagnose whether the tumor is damage or not and also try to minimize injury to healthy tissues during cryosurgery. Numerical solutions are obtained using finite difference method based on temperature dependent enthalpy. A computer code has been developed using MATLAB software on “Intel core i5 processor @ 3.30 GHz with 6GB RAM”. Results obtained are interpreted in the graphical form. The present thesis is compiled in six chapters and the chapter wise description is given below. Chapter 1 is an introductory and contains some basic concepts of heat transfer. Different heat transfer models are also discussed in this chapter. It gives a brief description of freezing process of biological tissue during cryosurgery, mechanism and mathematical formulation of cryosurgery and the solution methodology. In Chapter 2, a two-dimensional hyperbolic bio-heat model is developed by modifying the classical Pennes bio-heat model. Non-ideal property of tissue, metabolic heat generation and blood perfusion are also taken into consideration to study the phase change heat transfer during cryosurgery process of lung cancer. An enthalpy based finite difference scheme is adopted to solve the present model. We have examined the effect of different values of relaxation time on transient temperature, lower and upper interfaces during freezing process. Information obtained is useful to predict that the tumor tissue has been damaged or not and minimization of the damage of surrounding normal tissues by over-freezing, which could be helpful to improve the treatment planning. iii In Chapter 3, a two-dimensional dual phase lag model is proposed to study the phase change heat transfer process during cryosurgery of lung tumor tissue. This model is based on dual phase lag constitutive relation and also includes the discontinuity of temperature at the frozen-unfrozen interface. The temperature dependent enthalpy formulation and finite difference method is used to solve the mathematical model. The effects of phase lag of heat flux and temperature gradient on temperature profiles and position of phase change interfaces have been studied numerically. The results of this study are significant for successful cryosurgical treatment. In Chapter 4, we have studied the freezing behavior of triple layer skin tissue using a three-dimensional hyperbolic bio-heat model. The complexities of the problem are due to moving interface, discontinuity in the temperature at interface and triple layer skin tissue which has different thermal properties in different layers. The finite difference method is adopted to analyze the effect of relaxation time on freezing interfaces and temperature distribution in skin tissue. It is noted that relaxation time has important effect on phase change interfaces and temperature distribution. In Chapter 5, to study the effects of two phase lags in triple layer skin tissue freezing, a three-dimensional dual phase lag model is proposed. The difficulties of the problem are temperature discontinuity and movement of freezing interfaces and different thermal properties of layers of skin tissue. The finite difference approximation based on temperature dependent enthalpy has been used to solve the dual phase lag model. Temperature profiles and motion of freezing interface are plotted to see the effects of both the phase lags in freezing procedure. It is found that the freezing is fast for small value of phase lag of heat flux. Finally, Chapter 6 presents the conclusion drawn from the thesis and possible directions of the future scope.