Numerical Study of Heat Propagation in Living Tissue Subjected to Instantaneous Heating

Transcription

1 Indian Journal of Biomechanics: Special Issue (NCBM 7-8 March 9) Numerical Study of Heat Propagation in Living Tissue Sujected to Instantaneous Heating P. R. Sharma 1, Sazid Ali, V. K. Katiyar 1 Department of Mathematics, University of Rajasthan, Jaipur, India Department of Mathematics, University of Rajasthan, Jaipur, India Department of Mathematics, IIT Roorkee, India Astract Aim of this paper is to investigate the thermal ehavior of heat propagation in living tissue sujected to instantaneous heating with traditional Pennes s equation and the newly developed thermal wave model of ioheat transfer (TWMBT). The finite difference scheme is used to solve the one dimensional thermal wave model of ioheat transfer in finite medium. The deviations etween the thermal wave model of ioheat transfer and the Pennes model are discussed. It is oserved that the lood perfusion and the relaxation time significantly affect the thermal propagation ehavior and the thermal wave model of ioheat transfer provide the realistic prediction of temperature distriution in living tissue. 1. Introduction Knowledge of heat transfer in living tissue has een widely used in therapeutic applications. To predict the ehavior of ioheat transfer in living tissue, the Pennes s model is the most commonly used one among many ioheat transfer models (Arkin and Holmes, 1994) for simplicity and validity. The Pennes s ioheat transfer equation descries the thermal ehavior ased on the classical Fourier s law. As is well known, Fourier s law depicts an infinitely fast propagation of thermal signal, oviously incompatile with physical reality. Thus a modified flux model for transfer processes with finite speed wave is suggested (Weymann, 1967 and Ozisik, 1994). The literature (Luikov, 1968; Kaminski, 199; Braznikov et al. 1975) reported the relaxation time in iological odies to e -s. Recently Mitra et al.(1995) performed experimental measurement in processed meat andτ 15 was otained. The experimental investigate made y Roetzel et al. () showed the value of relaxation time to e aout s for processed meat. The preceding literature further support the phenomena of finite thermal propagation velocity in the process of ioheat transfer. As a result, the thermal wave model of ioheat transfer (Yang,199; Liu et al, 1999) is introduced to investigate the physical mechanics and ehaviors of heat transfer in living tissues. The relationship etween the heat flux vector and the thermal disturance in the thermal wave model is descried as (Weymann, 1967 and Ozisik, 1994). q q τ + = K T (1) whereτ is the relaxation time in homogenous sustance, which ranges from s (Kaminski, 199). The time derivative term of heat flux in equation (1) mathematically descried the effect of relaxation time. Some literatures have studied and analyzed the prolems of ioheat transfer in skin. Most of their computational analysis is performed 5

2 using Pennes ioheat transfer equation. In accordance with the contents of literature (Weymann, 1967 and Ozisik, 1994), the thermal ehaviour of heat transfer is ovious in the shorter heating processes. This paper attends to investigate the temperature distriution in skin with instantaneous surface heating for more understanding to the thermal wave propagation ehavior of ioheat transfer. One dimensional solution of the TWMBT equation in a finite medium was otained using finite difference method for a given constant surface temperature condition (Ozen et al., 8). The prediction from TWMBT equation was then compared with those y the Pennes s equation to reveal the distinct difference in heat transfer process.. Materials and Methods Based on the equation (1) for heat flux including the relaxation time τ as well as the Pennes s equation (Pennes, 1948), a general form of the thermal wave model of ioheat transfer (TWMBT) in living tissues was initially introduced y Liu et al. (1995). T Qm Qr T T.( K T ) + WC ( T T ) + Qm + Qr + τ WC + + = ρc τ + () Where ρ,c, T denote the density, specific heat and temperature of tissue, respectively. K is thermal conductivity, C is specific heat of lood, W is the perfusion rate of lood, T is Q m is the metaolic heat generation, and Q r is the heat source of spatial heating, arterial temperature and regarded constant. Equation () is hyperolic equation and is more mathematically complex than the traditional Pennes s ioheat transfer equation. Asτ =, equation () ecome a Pennes s ioheat equation. This paper focuses on studying the 1-D ioheat transfer prolem in skin with the instantaneous surface heating. Heat assumed to e incident on the skin, and so the spatial heating is equal to zero. The 1-D form of equation () with constant thermal parameter for Q = constant and Q = is written as K T + W ( ) C T T T T T + Qm τwc = ρc τ + () x Then the initial steady state temperature T ( x,) in skin can e derived from equation () as T K + W ( C T T ) + Qm = (4) x Sutracting equation (4) from equation () leads to θ θ θ + ( ρc + τwc ) + WCθ K = (5) x where the elevation temperature θ is defined as θ = T T. For the purpose of numerical solution, assuming heat flux approaches zero deep in tissue( x L) which is also realistic for iological ody, then the oundary conditions can e listed as follows: m r 6

3 θ θ = θ x=, =, x= L x L, (6) and initial conditions are θ θ, t = = =, t= A one dimensional finite difference method is used to solve the equation (5) (7) θ j+ 1 i ( ) ( ) ( ρ τ )( ) WC t K t j = θ i + A t C + A t x ( ) K t A t + θ θ + θ + + A t ( )( ) ( ) A t x j j j 1 i i i (8) The staility condition of the difference solution related to time step difference x in temperature calculation is t ( x) ( ) x W C + K where A = τwc + ρc t and spatial (9). Results and Discussion Typical values of thermal properties for skin tissue and other parameters have een chosen as ρ = 1 kg / m, C = C = 4 J / kg C (Liu et al., 1997). The distance etween the skin and ody core is L =.18m, the surface temperature θ = 1 C andq r =. Fig.1 shows that the tissue temperature elevation from these two equations can deviate sustantially under the constant surface temperature heating. Results from TWMBT equation showed that the tissue temperature inside the ody was undistured at the initial stage of heating and then took an instantaneous jump, which can e viewed at a wave front resulting from a step change in temperature at the skin. This clearly reflects the ehavior caused y thermal wave propagating at a finite speed on the other hand the temperature from the Pennes s equation immediately responded to the step heating every where in tissue. The maximum difference etween the TWMBT and Pennes s equation appears near the skin surface. The magnitude of temperature rise from the Pennes s equation is larger than that from the TWMBT equation. Fig. shows that the effect of the perfusion rate of lood on the temperature distriution in living tissue. The perfusion rate of lood plays an important role in ioheat transfer. The lood perfusion develops a cooling function since the skin temperature is higher than the arterial temperature. The heat energy taken away y the lood is proportional to the perfusion rate. Thus it is clear from the fig. that the skin temperature for W =. kg / m. s is lower than that for W = 1.5 kg / m. s andw =. kg / m. s. 7

4 Tem perature ( C) x=.8 m K=. W/m. C W =.5 kg/m.s TWMBT (=s) Pennes's equation (=s) Time t (sec) Fig.1. Temperature elevation in the skin due to constant surface heating at x= t =16s =s K =. W/m. C Temperature ( C) W =. kg/m.s W =1.5 kg/m.s W =. kg/m.s Distance X (m) Fig.. Temperature distriutions versus skin depth with K=.W/m C and = s at t= 16s for different lood perfusion rate. 8

5 4. Conclusion The thermal wave model of ioheat transfer and Pennes model are applied to investigate the thermal response in living tissue with instantaneous surface heating. A one dimensional TWMBT in finite medium was solved using finite difference method. It is oserved that the TWMBT which accounts for finite thermal wave propagation may provide realistic prediction of temperature distriution in living tissue. The lood perfusion develops the cooling function to prevent the tissue rising, ut does not affect the thermal propagation velocity. References 1. Arkin H, Xu LX, Holmes K.R. Recent developments in modeling heat transfer in lood perfused tissues, IEEE Trans. Biomed. Engrg. 1994; 41, Braznikov AM, Karpychev VA, Luikova AV, One Engineering Method of Calculating Heat Conduction Process, Inzhenerno Fizicheskij Zhurnal, 1975, 8, No. 4, Kaminski W, Hyperolic heat conduction equation for material with a nonhomogenous inner structure, ASME J. Heat Transfer 199; 11, Liu J, Chen X, Xu LX, New thermal wave aspects on urn evaluation of skin sujected to instantaneous heating, IEEE Trans. Biomed. Engrg. 1999; 46, Liu J, Ren Z and Wang C. Interpretation of living tissue s temperature oscillations y thermal wave theory, Chinese Sci. Bull., 1995; 4, Liu J, Zhang X, and Wang C, Generalized time delay ioheat equation and preliminary analysis on its wave nature, Chinese Sci. Bull., 1997; 4, Luikov V, Analytical Heat Diffusion Theory, Academic Press, NewYork, 1968; Mitra K, Kumar S,Vedavarz A, Moallemi MK, Experimental evidence of hyperolic heat conduction in processed meat, ASME J. Heat Transfer, 1995; 117, Ozen S, Helhel S, Cerezci O, Heat Analysis of Biological Tissue Exposed to Microwave y Using Thermal Wave Model of LIU AND CHENG 781 Bio-Heat Transfer, Burns, Including Thermal Injury, 8; 4, No.1, Ozisik MN, Tzou DY, On the Wave Theory in Heat Conduction, Journal of Heat Transfer, 1994; 116, No., Pennes HH, Analysis of tissue and arterial temperature in the resting human forearm, J. Appl. Physiol., 1948; 1, Roetzel W, Putra N, Das SK, Experiment and analysis for non-fourier conduction in materials with non-homogeneous inner structure, Int. J.Thermal Sci. ; 4, Weymann, HD, Finite Speed of Propagation in Heat Conduction, Diffusion, and Viscous Shear Motion, American Journal of Physics, 1967; 5, No. 6, Yang WH. Thermal (Heat) Shock Biothermomechanical Viewpoint, Journal of Biomed.Engineering, 199; 115, No. 4B, Inzhenerno Fizicheskij Zhurnal,1975; 8, No.4,