Fractal Theory And Its Applications In Bioheat Transfer

This paper focuses on the study of the fractal theory and its applica-tions in biological tissue heat transfer.It is composed of three chapters which are correlative and independent to each other. In the first chapter, it gives a brief introduction to the development history and the applications of the frac-tal theory,the definition and properties of some special functions and integral transformation which are applied in this paper.In the second chapter,bioheat transfer equation in fractal tissue is established.Then we get the analytical so-lutions of the problem,and analyze some special cases.In the three chapter.we give the thermal wave model of bioheat transfer in fractal tissue,and get the analytical solutions of the model.Also,some special cases are analyzed and we get the numerical figures at last.Chapter1is the elementary knowledge and gives a brief introduction of the mathematical theory,tools and methods we used in this paper,so it is the basis of the following chapters of the thesis.In section§1.1,the basic defini-tion,the history and the development of fractal theory are given.In section§1.2,we introduce its applications in physics,chemical,biology,medical science and geography. In section§1.3,we give an introduction to the special function-s.Bessel function is familiar to everyone,so we just refer to its definition and its two frequently-used recurrence relations.However,Mittag-Leffler function is widely used in mathematics.we give its definitions and important formulae which are used in the solutions.In section§1.4,we give integral transforms which are applied in the paper,such as.fractional finite Hankel transform and Laplace transform.Fractional finite Hankel transform plays an important role in solving the following problems. In chapter2,we apply the fractal theory to the classical Pennes bioheat transfer equation,and get the fractal model of heat transport. In section§2.1,the development of the bioheat transfer and the academic work of classi-cal bioheat transfer are given.In section§2.2,the bioheat transfer equation in fractal tissue is given by: Then,by applying the fractional finite Hankel transform,Laplace transform and their inverse transforms to the equation,we can get the solution of the problem in the form of exponential function,as: In section§2.3,we discuss the situations of tâ†'∞and df=2. When tâ†'oo,the expression of temperature is: When df=2,the expression is: At last,we discuss the influences of fractal dimension and blood perfusion rate on temperature. In section§2.4,we give conclusions.In chapter3,based on the second chapter,we get the thermal wave model of bioheat transfer in fractal tissue.In section§3.1,we give a brief introduction to the development of oscillation theory of bioheat transfer.In section§3.2,the model of the above problem is established as: The solution we get by using the method of integral transforms is: In section§3.3,we study the solution of classic thermal wave equation,it is: And we give the graphics and relative analysis of the solution in different blood perfusion rates and fractal dimensions. In section§3.3,we give conclusions of this chapter.