Effects of microwave heating on the thermal states of biological tissues

A mathematical analysis of microwave heating equations in one-dimensional multi-layer model has been discussed. Maxwell's equations and transient bioheat transfer equation were numerically calculated by using finite difference method to predict the effects of thermal physical properties on the transient temperature of biological tissues. This prediction of the temperature evolution in biological bodies can be used as an effective tool for thermal diagnostics in medical practices.


INTRODUCTION
The use of microwave radiation for heating is now common in many industrial situations such as smelting, sintering and drying, it also has many applications in joining mathematical and medical fields especially in clinical cancer therapy hyperthermia (Deuflhard and Seebass, 1998;Hill and Pincombe, 1992).Body surface temperature is controlled by the blood circulation under neath the skin, local metabolism and heat exchange between the skin and its environment.Changes in any of these parameters can induce variation of temperature and heat flux at the skin surface, reflecting the physiological state of human body.For example, highly vascurized skin tumor can lead to an increase of blood flow and thus to an increase in skin temperature.In burn injures low skin temperature can occur due to insufficient blood supply.Apparently the abnormal temperature or heat flux at the skin surface might indicate irregular peripheral circulation which can be used in clinical *Corresponding author.E-mail: sky_12@masrawy.com.
diagnosis (Liu and Xu, 2000).Effects of thermal properties and geometrical dimensions on the skin burn injures has been discussed by Jiang et al. (2002).Ng and Chua (2002) proposed a comparison of one and twodimensional programmes for predicting the state of skin burns.Lu et al. (1998) studied the simulation of the thermal wave propagation in biological tissues by dual reciprocity boundary element method.
Preliminary survey on the mechanism of the wave-like behaviors of heat transfer in living tissues has been discussed by Liu (2000) who introduced a new concept of multi-mode energy coupling, a phenomenological thermal wave model of bioheat transfer.Marchant and Liu (2001) considered the steady-state microwave heating of a finite one-dimensional slab.The temperature dependency of the electrical conductivity and thermal absorptivity were assumed to be governed by the Arrhenius law, while both the electrical permittivity and permeability were assumed constant.The microwave heating of three-dimensional blocks with a transverse magnetic waveguide mode in a long rectangular waveguide is considered by Lui and Marchant (2002).
The aim of our paper is to show the effect of microwave heating equations on the thermal states of biological tissues and to predict the effects of the thermal physical properties on the transient temperature of tissues and damage function.This research can be very beneficial in widening the idea of clinical thermal technology and thermal medical practices.

MATHEMATICAL MODELS
The diagram of one dimensional multi-layer tissue model is illustrated in Figure 1.The temperature, electric and magnetic fields for microwave heating are given respectively by The heat source arising from microwaves is proportional to the square of the modulus of the electric field intensity (Hill and Pincombe, 1992), hence the Maxwell's equations coupled with bioheat equation can be written as: where the analysis assumes that no interfacial resistance exists between heating source and skin.Therefore the surface temperature remains constant during exposure.The tissue assumed to be homogeneous within each layer.b ω is the mass flow rate of blood flow per unite volume (blood perfusion), thermal conductivity and heat capacity are assumed to be constants.The term in the left of eq. ( 4) is the energy storage in the tissue, the first term on the right caused by temperature gradient, the second term describes the heat transport between the tissue and microcirculatory blood perfusion and the term 2 E is the electric power density deposited in tissue.The following boundary and initial conditions were required for the solution of partial differential equations (2-4) Liu (2001) presents experimental evidence which indicates that the physical properties of material have power law dependence on temperature, so we can assume that the body heating coefficient has the form, ( ) ( ) The dimensionless Maxwell's and energy equations, after dropping stare mark can be written as ( ) Hill and Pincombe (1992) represented a similarity solution to solve the microwave heating equation ( 10) without studying the effect of convection due to the blood flow, he assumed that the electrical field decays exponentially with distance

Numerical technique
A one dimensional finite difference method is used to solve Maxwell's equations together with the bioheat transient equation which describe the skin burn process resulting from the application of a high temperature heat source to a skin surface.The space steps and time steps are small enough to ensure that the transient temperature were mesh independent.A finite difference scheme of Crank-Nicolson was applied to system of the partial differential equations ( ) ( ) 10 8 − to yield the following finite difference equations ( ) where the appropriate mesh sides

. x = ∆
and time steps

. t = ∆
are considered for calculations , i and j denote the grid functions which approximate the exact solutions of ( ),

The thermal dose calculation
The thermal dose at C 0 43 defined by Sapareto and Dewey (1984) can describe the extent of thermal damage or destruction of tissue, and its expression is used as following: < < numerical integration formula of eq. ( ) 14 can be written as (Chapra and Canale, 2002).

RESULTS AND DISCUSSION
A one dimensional multi-layer transient finite difference model for predicting temperatures in living tissues such as prostate tissue undergoing microwave heating is presented.The microwave energy can be transmitted into the body from applicator on the skin tissue.For example in the case of prostate tissue, microwave energy is emitted from the transurethral catheter.Pennes perfusion term is assumed to predict the effect of perfusion on heat transfer.The thickness of the tissue differs from different people and different location.
The transient temperature for different thickness is shown in Figure 2. The temperature distribution decreases with the increasing of tissue thickness L .In      tissue this thermal dose damping and then increases at inner tissue.Figure 8 illustrates the thermal dose distribution Ω with time t , it is found that the thermal dose decreases as time t increases.Finite difference method has been used to solve the system of partial differential equations which describe Maxwell's equations and bioheat equation in order to show the effect of microwave heating equations on the thermal states of biological tissues.The behaviors of temperature distributions for different thermal properties of tissue have been illustrated.The thermal dose distribution has been plotted for different thickness of tissue and time steps.The applications of this problem have a great benefit in various branches of science especially in cancer therapy using microwave devices such as the thermocouple.

E
and k .In our paper we will solve Maxwell's equations and transient bioheat transfer equation numerically and we will assume for simplicity 1 m = .

Figure 3
Figure 3 the temperature distribution of tissue decreases as time t increases.The effect of variation of blood perfusion b ω on the tissue is presented in Figure 4.It is clear the transient temperature increases with increasing of blood perfusion b ω .

Figure 5
Figure 5 illustrates the effect of three different values of thermal conductivity of tissue κ on the transient temperature.It is shown that the temperature distributiondecreases with increasing of thermal conductivity κ of epidermis and dermis and it is increases with the increasing of thermal conductivity of inner tissue.Figure6shows the relation between the electric field in the free space 0 Figure 6 shows the relation between the electric field in the free space 0 E and the temperature distribution.It is seen that the temperature distribution increases as 0 E increases.

Figure 7
Figure 7 shows the relation between the temperature distribution and time for different thermal conductivity κ of tissue.The temperature decreases with the increasing of κ .Figures 8 and 9 illustrates the relation between thickness of skin x and thermal dose Ω , which describes the extent of thermal damage or destruction of