A POROUS MODEL OF TUMOR IN HYPERTHERMIA THERAPY WITH CONVECTION OF BLOOD FLOW

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1 ISTP-16, 005, PRAGUE 16 TH INTERNATIONAL SYMPOSIUM ON TRANSPORT PHENOMENA A POROUS MODEL OF TUMOR IN HYPERTHERMIA THERAPY WITH CONVECTION OF BLOOD FLOW Ping Yuan Department of Mechanical Engineering Lee-Ming Insitute of Technology Taipei, Taiwan, ROC pyuan@mail.lit.edu.tw Keywords: Tumor, Hyperthermia, Non-Local Equilibration, Temperature, Thermal Dose Abstract This study investigates the temperature and thermal dose of a tumor in hyperthermia therapy with non-local equilibrium. The thermal model assumes the tissue with blood vessel distribution as a porous media and uses the convection terms instead of perfusion terms in energy conservation equations for both tissue and blood. By using numerical method, this study calculates the temperature and thermal dose response of tissue for different vessel diameters and blood velocities with keeping constant porosity. Through the accuracy and model comparison, the results show that the temperature responses of tissue for different cases by this numerical method are reliable, and the model is suitable for various vessel diameters and porosity. 1 Introduction Hyperthermia therapy is a method to destroy or control the tumor. Since Pennes [1] proposes a simple bioheat equation model with a perfusion term, many literatures discuss the validation of perfusion term, and the simplicity of one equation in Pennes model. Arkin, et al. [] reviews many new bioheat transfer models by researchers from 1948 to The comment of Arkin, et al. is that the new models still lack experimental grounding, and Pennes model may still be the best practical approach for modeling bioheat transfer because of its simplicity. In fact, the one-equation bioheat transfer model is basing on the assumption of local thermal equilibration, which is the temperature of tissue is equal to the temperature of blood. Because local thermal equilibration occurs when heat exchange ability between tissue and blood is good, it is valid in the capillary bed having large density of vessels and area of heat transfer. According to the literature of Weinbaum [3][4], the largest local temperature difference between the blood and tissue is the order of 0. C when the tissue has only metabolic heat generation, and this occurs in large vessels of μm diameters. Therefore, most models of one equation are suitable to the case of d<300μm on the Table 1 of literature by Arkin, et al. []. On the other hand, due to the simple concept of porous media, many authors investigate the heat transfer in biological tissue by using theory of porous media. Khaled and Vafai [5] make a review about the role of porous media in modeling flow and heat transfer in biological tissues. The literature concludes that the developing advanced heat transfer models based on thermal non-equilibrium states between the blood and tissue is an important task. Xuan and Roetzel [6] use the porous media concept to analyze the steady temperature fields of tissue, artery and vein by a system of three energy equations. Although the governing equations are three dimensional, authors 1

2 simplify them into a two-dimensional equation for tissue and two one-dimensional equations for bloods based on analyzing an arm. The literature calculates the steady temperatures of tissue and bloods by numerical method. Later, Roetzel and Xuan [7] extend to investigate the transient heat transfer problem between the artery, vein, and tissue with cylinder physical model using porous media concept. Meanwhile, the equations of arterial and venous blood are one dimensional along the blood flow direction. The equation of tissue has two dimensions, which are the blood flow direction and radius. It is worth to note that the authors use the local volumetric heat transfer coefficient instead of the perfusion rate to govern the heat transfer between the blood and tissue. As mentioned by Roetzel and Xuan, the porosity and specific surface area depend upon many factors, so these parameters should be experimentally determined. Unfortunately, this paper has no information about the value of these parameters. Recently, Kou, et al. [8] investigate the effect of the directional blood flow on thermal dose distribution during thermal therapy by using Green s function method. This paper uses porous media concept and assumption of local thermal equilibration to combine the energy conservation equations of tissue and blood into single energy equation. The results show that the domain of thermal lesion may extend to the down stream normal tissue if the porosity is high and the averaged blood velocity has a larger value. Based on the survey of previous literature, this study plans to use two energy equations to govern the physical model of tissue with blood flow in hyperthermia therapy by porous media concept. The calculation domain is three dimensional, and the transfer mechanism between tissue and blood is the convection. In addition, this study simultaneously shows the temperature response and thermal dose of tissue by Pennes model and Kou s model in the figures. The results indicate that the Kou s model is suitable for the case of low porosity and blood velocity, and the Pennes model is suitable for the case of tissue with larger vessel distribution. The contour plot of tissue temperature shows that the temperature field predicted by Pennes model is always symmetrical to the heating center due to the lack of velocity term in the model. Therefore, the Pennes model is not only unsuitable for the capillary bed, but also unsuitable for the tissue with large blood velocity. Through the accuracy comparison and model comparison, this three dimensional and two-equation model with convection term is reasonable to predict the effect of hyperthermia therapy over wider conditions. Analysis This study investigates the transient temperature of a tumor in hyperthermia therapy. The calculation domain is a cubic with 0mm in each side, and the cubic heating zone with 10mm in each side is at the center of calculation domain, which is shown in Fig. 1. This study considers the blood flows along the x direction, and the initial temperature and boundary temperature are 37. Because the distribution of blood vessels in tissue is too complicated to simulate, this study considers the calculation domain to be a porous media of a tissue with uniform blood distribution. In this porous media, all blood vessels are assumed straight and parallel to each other in x direction. In Fig. 1, the d represents the diameter of blood, and represents the distance between vessels. Based on the arrangement of vessels, this study defines the porosity and transfer area per unit volume as following: V a d blood 0.5 (1) Vtotal A d V () Before formulating the governing equations, this study assumes the thermal properties of tissue and blood, heat transfer coefficient, and the blood flowing velocity are constant. By using the energy conservation for tissue and blood, the simultaneous differential equations can be formulated as following: T t 1 ( c p ) t 1 k t T t ha T b T t 1 q t (3) t

3 PAPER TITLE T b ( cp ) b c t b p b u T b k b T b ha T t T b q b (4) Meanwhile, a is volumetric transfer area between tissue and blood, and q is the power density in the heating zone. 3. Simulation This study selects the thermal properties of tissue and blood according to the paper of Kou et al. [8]. The thermal conductivity of tissue and blood is 0.5 W/m, the density of tissue and blood is 1050 kg/m 3, and the specific heat capacity of tissue and blood is 3770 J/kg. This study considers two kinds of heating conditions, which are 10 sec ( W/m 3 ) and 50 sec ( 10 6 W/m 3 ) for the tissue. Because the heat absorbed by blood is less than that by tissue in ultrasonic heating, this study estimates the power density in blood is tenth of that in tissue. Based on the result of Payne et al. [9], this study sets the heat transfer coefficient to be 173 W/m. In addition, the diameter of capillary is between 5 m and 15 m according to the paper of Arkin et al. [], as well as the distance between capillaries is larger than 140 m in brain based on the paper of Kocher et al. [10]. Therefore, this study selects the vessel diameter and distance of vessels is 15 m and 140 m to estimate the porosity and volumetric transfer area from Eqs. (1) and (). Based on the estimating porosity of 0.01, this study selects the range of porosity to be 0.005< <0.05 to investigate the temperature response of tissue in hyperthermia therapy. In this study, author keeps the porosity as a constant and calculates different volumetric transfer area for various diameters of blood vessels. Table 1 lists the relationship of volumetric heat transfer area and diameters of blood vessel. Meanwhile, the velocity of blood is 0.1, 3, 6, and 8cm/s for the diameter of blood vessel of 10, 100, 500, and 1000 μ m, respectively. The velocity data cite from the literature [11-13], and they are relating to the blood vessel of capillaries, terminal braches, secondary branches, and main branches of a 13- kg dog [11]. This study uses both finite difference method and a software package, FlexPDE, to simulate the transient temperature field of tissue and blood. This study discretizes the differential equations of (3) and (4) into finite difference equations by a uniform grid generation, as well as uses the implicit method to treat the unknown variables. In addition, the Fortran code uses the ADI method to solve the tridigonal matrix. On the other hand, because FlexPDE is a flexible solver for partial differential equations, this study utilizes FlexPDE software directly to solve the governing equations of (3) and (4). Table is a relative error of tissue temperature at center point for different grid dimension to that with grid dimension in this numerical method when the u=mm/s, q= , τ=10s, and a =167m /m 3. According to the results of Table, this study selects the dimension of grid to be because of the relative error lower than 0.5%. Figure is the accuracy comparison between this numerical method and previous literature [8] for one-equation model when the heating duration is 10sec, heating power density is 10 7 W/m 3, porosity is 0.05, and blood velocity is mm/s. In the comparison case, this study combines the equations of (3) and (4) into one-equation by assuming local equilibration, which is identical to the governing equation of literature [8]. In this figure, it is obvious that the results calculated by Fortran program coincide with the results of analytical solution by Kou, et al. Therefore, this numerical method is reliable. In order to make sure the accuracy of this numerical method for solving two-equation model, this study compares the results calculated by this numerical program with the results calculated by FlexPDE software in Fig. 3. Figure 3 compares the results of this study and numerical package for two-equation model with different volumetric transfer area when the heating duration is 10sec, heating power density is 10 7 W/m 3, porosity is 0.05, and blood velocity is mm/s. It shows that the symbols are similar to the line in this figure, so the numerical method for solving two-equation model is also accurate. 3

4 4. Results and Discussion Figure 4(a) depicts the tissue temperature at center point of heating zone versus time for different volumetric transfer area and blood velocity at ε =0.005 and h=173w/m. Meanwhile, the continuous, dashed, dash-dotted, and dash-doubledotted line represents the volumetric transfer area of 000, 00, 40, and 0 m /m 3 as well as the blood velocity of 0.1, 3, 6, and 8cm/s. The symbol of square is the result of Pennes model when the perfusion rate is 6.56kg/m 3 s. Based on the research of Payne, et al. [9], the heat transfer mechanism with w b =6.56 is same to that with h=173w/m. In addition, the symbol of delta is the result of Kou s one-equation model [8], which is in porous media with local thermal equilibrium. In this figure, the symbol of delta coincide with the continuous line, which is u=0.1cm/s and a =000 m /m 3 (d=10μm, =15μm). This is reasonable that the local thermal equilibrium is valid in porous media when both the diameter of blood vessel and the distance between blood vessels are small. In other words, the Kou s model is valid for the immense distribution of microvasculars, i.e. the capillary bed. In this figure, the symbol of square sounds like matching the dashed line, which is the u=3cm/s and a =00 m /m 3 (d=100μm, =153μm). This result indicate that the Pennes model is not valid for capillary, but for the tissue with larger vessels distribution. This is agreeable with the statement by Charny and Levin [14], The perfusion term describes heat loss due to breedoff from the largest branching vessels and not capillary heat exchange at the terminal end of the microvascular network. In Fig. 4(a), the temperature response of tissue does not regularly decrease with the increase of blood velocity when the porosity is constant. Because the effect factors for tissue temperature include the blood velocity and volumetric transfer area when we keep other parameters to be unchanged, both ability of heat exchange between blood and tissue and the capacity of carrying away by blood will influence the temperature response of tissue. Therefore, the temperature response with time looks like not related to blood velocity in this figure. According to the temperature profile from high to low, the rank is the cases of u=0.1cm/s ( a =000 m /m 3 ), u=8cm/s ( a =0 m /m 3 ), u=6cm/s ( a =40 m /m 3 ), and u=3cm/s ( a =00 m /m 3 ). Figure 4(b) depicts the temperature of blood versus time in the same conditions of Fig. 4(a). In this figure, it is obvious that the case of u=0.1cm/s ( a =000 m /m 3 ) is under thermal equilibrium because the blood temperature profile is similar to the tissue temperature in Fig. 4(a). Furthermore, the other three cases are not under thermal equilibrium because of the obvious temperature difference between the blood and tissue in Figs. 4(a) and 4(b). In Fig. 4(b), the temperature profile form high to low is regular drop when the velocity rises. The reason is that the heat transfers from tissue to blood and then be carried away by the blood flow, the more volumetric transfer area and the less blood velocity has the more heat transferring into the blood but the less heat carried away by blood flow. Therefore, the order of blood temperature profile form high to low is the case of u=0.1cm/s( a =000 m /m 3 ), u=3cm/s( a =00 m /m 3 ), u=6cm/s( a =40 m /m 3 ), u=8cm/s( a =0 m /m 3 ). Figure 4(c) depict the thermal dose at center point when the conditions are same to Figs. 4(a) and 4(b). The thermal dose is the quantity of relationship between time and temperature. The number of cumulative equivalent minutes at 43 is the most popular clinical means to characterize treatment capacity, which is defined as [15] CEM 43 = R ( 43 T ) t (5) where R=0.5 for T>43, R=0.5 for 39 <T<43, and R=0 for T<39. It is obvious that the thermal dose at center point predicted by Pennes model is agreeable to the dashed line when the porosity is 0.005, blood velocity is 3cm/s, and the volumetric transfer area is 00 m /m 3. In addition, the thermal dose predicted by Kou s model is same to the continuous line. In this figure, the order of thermal dose from high to low is the case of u=0.1cm/s( a =000 m /m 3 ), u=8cm/s( a =0 m /m 3 ), u=6cm/s( a =40 m /m 3 ), and u=3cm/s( a =00 m /m 3 ), which is similar to 4

5 PAPER TITLE those in Fig. 4(a). It is reasonable that the thermal dose is calculated from the tissue temperature based on the definition equation (5). Figures 5 and 6 depicts the tissue temperature, blood temperature, and thermal dose at the center point of heating zone for different blood velocity and volumetric transfer area when the heat transfer coefficient is 173 and the porosity is 0.01 and 0.05, respectively. In Figs. 5(a) and 6(a), it is clear that the Kou s model is still agreeable with the case of u=0.1cm/s ( a =4000 m /m 3 ) and u=0.1 cm/s ( a =0000 m /m 3 ), which likes the capillary bed and close to the situation of local thermal equilibrium. Comparing the temperature response predicted by Pennes model in Figs. 4(a), 5(a), and 6(a) shows that all of them are similar, because the Pennes model is not function of porosity except the heating term. In Fig. 5(a), the symbol of square is between dashed line and dash-dotted line, and the symbol move to dash-doubledotted line in Fig. 6(a). This means that Pennes model is suitable for tissue with larger vessel distribution when the porosity increase. Comparing Figs. 4(a), 5(a), and 6(a) indicates that the temperature profile of each case drops when the porosity rise from ε=0.005 to ε=0.05. It is reasonable that the more porosity means more blood in tissue and causes more heat carried away by blood. In addition, although the rank of tissue temperature is different to the rank of blood temperature, the order of tissue temperature and blood temperature from high to low is same in Figs. 4 to 6. For the thermal dose at center point of heating zone, Figs. 5(c) and 6(c) shows that the results by Kou, et al. are also coincide with the case of u=0.1cm/s ( a =4000 m /m 3 ) and u=0.1 cm/s ( a =0000 m /m 3 ), like showing in Fig. 4(c). Therefore, the Kou s one-equation model is suitable for the capillary bed in hyperthermia therapy. Figure 7 is the contour plots of tissue temperature on the x-y plane of z=1cm at 5, 50, 100, and 150 seconds when the porosity is 0.005, the blood velocity is 0.1cm/s, and the volumetric transfer area is 000 m /m 3. Meanwhile, the continuous, dashed, and dashdotted line represents the isotherm of this study model, Pennes model, and Kou s model. In this figure, the continuous line coincides with dashdotted line over whole plane. Because the case is suitable for local thermal equilibrium, the temperature predicted by two-equation model of this study is identical to that by one-equation model of Kou, et al. when the calculation domain is capillary bed. Moreover, the dashdotted line is obviously different to other lines, because Pennes model is suitable for the tissue with large vessel distribution, which has discussed in Figs. 4 and 5. Note that the isotherm of Pennes model is symmetrical to the center point for all time. This means the blood velocity does not affect the temperature distribution, because blood velocity does not exist in Pennes model. On the contrary, the isotherm of this study s model and Kou s model slightly move toward the down stream of blood flow, because the blood velocity is only 0.1cm/s. In addition, the isotherm of continuous line includes larger area than the isotherm of dashed line does when the time are 5 and 50 seconds, but the isotherm of continuous line includes smaller area than the isotherm of dashed line does when the time are 100 and 150 seconds. This means that the temperature rise predicted by this study is quicker than that by Pennes model in the heating period, but the temperature drop predicted by this study is slower than that predicted by Pennes model after heating. The result indicates that the perfusion term in Pennes model is overcooling the tissue when the tissue is capillary bed. Figure 8 is the contour plot of tissue temperature on the x-y plane of z=1cm at 5, 50, 100, and 150 seconds when the porosity is 0.005, the blood velocity is 3cm/s, and the volumetric transfer area is 00 m /m 3. In this figure, it is obvious to observe the effect of blood velocity on the temperature distribution. Because Pennes model has not the velocity term, the dashed line is still symmetrical to the center of x-y plane. If we focus on the isotherm of continuous line and dash-dotted line, we find that the isotherm of Kou s model move along the blood flow direction more than that of this study. Because the case is not suitable for the 5

6 local thermal equilibrium due to the obvious temperature difference between tissue and blood in Figs. 4(a) and 4(b), the Kou s model is not valid here. If we check Kou s model, its velocity term sounds like the heat carried away by the tissue flow. When the local thermal equilibrium is valid, the tissue temperature is equal to the blood temperature. Therefore, the result calculated by Kou s model is same to the result of this study, and both of them show the effect of blood. However, the heat carried away by tissue flow must be larger than that carried away by blood flow when the blood temperature is obviously lower than tissue temperature under non-local thermal equilibrium. In addition, although the tissue temperature at center point by Pennes model is same to that by this study in the case of u=3cm/s and a =00 m /m 3 in Fig. 4(a), the temperature distribution at other region is obviously different between Pennes model and this study. Therefore, Pennes model cannot accurately predict the temperature domain with considering blood flowing in hyperthermia therapy. 5 Conclusions This study investigates the temperature and thermal dose of a tissue with blood vessel distribution in hyperthermia therapy. Assuming the vessels be straight and uniform distribution, this study formulates the energy equations of tissue and blood with porous media concept. Furthermore, this study uses the convection term instead of perfusion term used in most models before. By using the numerical method, this study calculates the temperature field and thermal dose field versus time for different conditions. Through the accuracy comparison with previous literature and commercial package, this numerical method is reliable. In addition, this study simultaneously shows the results calculated by this study model, Pennes model, and Kou s model. According to the results, this study concludes that Kou s model is suitable for small blood vessel and low porosity, which satisfies the local equilibration assumption. Pennes model is suitable for tissues with large blood distribution and low blood velocity, because the isotherm of temperature is symmetrical to the heating center due to the lack of velocity in the model. Comparing the three models indicates that the two-equation model of this study can predict the temperature and thermal dose in hyperthermia therapy for wider conditions of tissue and blood. References [1] H.H. Pennes, Analysis of Tissue and Arterial Blood Temperature in Resting Forearm, Journal of Applied Physiology, vol. 1, pp. 93-1, [] H. Arkin, L.X. Xu, and K.R. Holmes, Recent Developments in Modeling Heat Transfer in Blood Perfused Tissues, IEEE Transactions on Biomedical Engineering, vol. 41, no., pp , [3] S. Weinbaum, L.M. Jiji, and D.E. Lemons, Theory and Experiment for the Effect of Vascular Microstructure on Surface Tissue Heat Transfer- Part I: Anatomical Foundation and Model Conceptualization, ASME Journal of Biomechanical Engineering, vol. 106, pp , [4] L.M. Jiji, S. Weinbaum, and D.E. Lemons, Theory and Experiment for the Effect of Vascular Microstructure on Surface Tissue Heat Transfer- Part II: Model Formulation and Solution, ASME Journal of Biomechanical Engineering, vol. 106, pp , [5] A.R.A. Khaled and K. Vafai, The Role of Porous Media in Modeling Flow and Heat Transfer in Biological Tissues, International Journal of Heat and Mass Transfer, vol. 46, pp , 003 [6] Y. Xuan and W. Roetzel, Bioheat Equation of the Human Thermal System, Chemical Engineering and Technology, vol. 0, pp , [7] W. Roetzel and Y. Xuan, Transient Response of the Juman Limb to an External Stimulust, International Journal of Heat and Mass Transfer, vol. 41, no. 1, pp. 9-39, [8] H.S. Kou, T.C. Shih, and W.L. Lin, Effect of the directional blood flow on thermal dose distribution during thermal therapy: an application of a Green s function based on the porous model, Physics in Medicine and Biology, vol. 48, pp , 003. [9] A. Payne, M. Mattingly, J. Shelkey, E. Scott, and R. Roemer, A dynamic two-dimensional phantom for ultrasound hyperthermia controller testing, International Journal of Hyperthermia, vol. 17, no., pp , 001. [10] M. Kocher, H. Treuer, J. Voges, M. Hoevels, V. Sturm, and R.P. Muller, Computer simulation of cytotoxic and vascular effects of radiosurgery in solid and necrotic brain metastases, Radiotherapy 6

7 PAPER TITLE & Oncology, vol. 54, pp , 000. [11] J. Crezee and J.J.W. Lagendijk, Temperature uniformity during hyperthermia: the impact of large vessels, Phys. Med. Biol., vol. 37, no. 6, pp , 199. [1] J.C. Chato, Heat Transfer to Blood Vessels, Journal of Biomechanical Engineering, vol. 10, pp , [13] S. Weinbaum and L.M. Jiji, A New Simplified Bioheat Equation for the Effect of Blood Flow on Local Average Tissue Temperature, Journal of Biomechanical Engineering, vol. 107, pp , [14] C.K. Charny and R.L. Levin, An Evaluation of the Weinbaum-Jiji Bioheat Equation for Normal and Hyperthermic Conditions, Journal of Biomechanical Engineering, vol. 11, pp , 1990 [15] D. Arora, M. Skliar, and R.B. Roemer, Model- Predictive Control of Hyperthermia Treatments, IEEE Transactions on Biomedical Engineering, vol. 49, no. 7, pp , 00. Table 1 Volumetric transfer area at different porosities and vessel diameters Transfer Area =0.005 =0.01 =0.05 (m /m 3) d =10 m d =100 m d =500 m d =1000 m Table The relative error of tissue temperature with different grid dimension to that with grid dimension when the u=mm/s, q=10 7, τ =10s, and a =167m /m 3. Relative Error(%) Time(s) Dim=1 Dim=41 Dim=61 Dim= y x z Heating Zone 10mm 0mm Calculation Domain Fig. 1 Schematic diagram of a porous media with tissue and blood in hyperthermia therapy Kou, et al[003] Fortran u= mm/s Fig. Accuracy comparison between this study and previous literature for one-equation model whenτ= 10s, q=10 7 W/m 3, ε= 0.05, and u= mm/s a=167, this study a=333, this study a=1334, this study a=13336, this study 55 a=167, FlexPDE a=333, FlexPDE 55 a=1334, FlexPDE a=13336, FlexPDE 50 stagger arrangement 50 Heating Duration=10sec 45 u=mm/s Time(sec) Fig. 3 Accuracy comparison between this study and numerical package for two-equation model whenτ= 10s, q=10 7 W/m 3, ε= 0.05, and u= mm/s. 7

8 h=173 u=0.1cm/s, a=000 u=3cm/s, a =00 u=6cm/s, a =40 u=8cm/s, a =0 55 Kou(u=0.1cm/s) 55 h=173 u=0.1cm/s, a=4000 u=3cm/s, a =400 u=6cm/s, a =80 u=8cm/s, a =40 55 Kou(u=0.1m/s) (a) tissue tempeature (a) tissue tempeature h=173 u=0.1cm/s, a=000 u=3cm/s, a =00 u=6cm/s, a =40 u=8cm/s, a =0 55 Kou(u=0.1cm/s) 55 h=173 u=0.1cm/s, a=4000 u=3cm/s, a =400 u=6cm/s, a =80 u=8cm/s, a =40 55 Kou(u=0.1m/s) 55 Tb Tb (b) blood temperature (b) blood temperature h= h= Thermal Dose u=0.1cm/s, a= u=3cm/s, a=00 u=6cm/s, a =40 u=8cm/s, a = Kou(u=0.1cm/s) Thermal Dose u=0.1cm/s, a=4000 u=3cm/s, a =400 u=6cm/s, a = u=8cm/s, a = Kou(u=0.1m/s) (c) thermal dose Fig. 4 Temperature and thermal dose versus time for different blood velocity and volumetric transfer area at ε=0.005 and h=173w/m when heating duration is 50sec and power density is 10 7 W/m (c) thermal dose Fig. 5 Temperature and thermal dose versus time for different blood velocity and volumetric transfer area at ε=0.010 and h=173w/m when heating duration is 50sec and power density is 10 7 W/m 3 8

9 PAPER TITLE h=173 u=0.1cm/s, a=0000 u=3cm/s, a =000 u=6cm/s, a =400 u=8cm/s, a =00 55 Kou(u=0.1cm/s) 55 This Study Kou(0.001m/s) 5sec sec (a) tissue temperature h=173 u=0.1cm/s, a=0000 u=3cm/s, a =000 u=6cm/s, a =400 u=8cm/s, a =00 55 Kou(u=0.1cm/s) sec 150sec Tb (b) blood temperature h= Fig. 7 Isotherm at different time on the x-y plane of z=1cm at ε=0.005, u=0.1cm/s, a =000 m /m 3, and h=173w/m when heating duration is 50sec and power density is 10 7 W/m 3 This Study Thermal Dose u=0.1cm/s, a=0000 u=3cm/s, a= u=6cm/s, a = u=8cm/s, a = Kou(u=0.1cm/s) 10 0 Kou(0.030m/s) 5sec 50sec (c) thermal dose Fig. 6 Temperature and thermal dose versus time for different blood velocity and volumetric transfer area at ε=0.05 and h=173w/m when heating duration is 50sec and power density is 10 7 W/m 3 100sec 150sec Fig. 8 Isotherm at different time on the x-y plane of z=1cm at ε=0.005, u=3cm/s, a =00 m /m 3, and h=173w/m when heating duration is 50sec and power density is 10 7 W/m 3 9