Investigation of nonlinear temperature distribution in biological tissues by using bioheat transfer equation of Pennes’ type

ABSTRACT

In this paper, a two level finite difference scheme of Crank-Nicholson type is constructed and used to numerically investigate nonlinear temperature distribution in biological tissues described by bioheat transfer equation of Pennes’ type. For the equation under consideration, the thermal conductivity is either depth-dependent or tem-perature-dependent, while blood perfusion is temperature-dependent. In both cases of depth- dependent and temperature-dependent thermal conductivity, it is shown that blood perfusion decreases the temperature of the living tissue. Our numerical simulations show that neither the localization nor the magnitude of peak tempera-ture is affected by surface temperature; however, the width of peak temperature increases with surface temperature.

In this paper, a two level finite difference scheme of Crank-Nicholson type is constructed and used to numerically investigate nonlinear temperature distribution in biological tissues described by bioheat transfer equation of Pennes’ type. For the equation under consideration, the thermal conductivity is either depth-dependent or tem-perature-dependent, while blood perfusion is temperature-dependent. In both cases of depth- dependent and temperature-dependent thermal conductivity, it is shown that blood perfusion decreases the temperature of the living tissue. Our numerical simulations show that neither the localization nor the magnitude of peak tempera-ture is affected by surface temperature; however, the width of peak temperature increases with surface temperature.

Cite this paper

Lakhssassi, A. , Kengne, E. and Semmaoui, H. (2010) Investigation of nonlinear temperature distribution in biological tissues by using bioheat transfer equation of Pennes’ type.*Natural Science*, **2**, 131-138. doi: 10.4236/ns.2010.23022.

Lakhssassi, A. , Kengne, E. and Semmaoui, H. (2010) Investigation of nonlinear temperature distribution in biological tissues by using bioheat transfer equation of Pennes’ type.

References

[1] Pennes, H.H. (1948) (1998) Analysis of tissue and arterial blood temperatures in the resting human forearm. Journal of Applied Physics, 1, 93-122; 85, 5-34.

[2] Meyer, C., Philip, P. and Troltzsch, F. (2004) Optimal control of semilinear PDE with nonlocal radiation interface conditions. IMA Preprint Series, 2002.

[3] Yamamoto, M. and Zou, J. (2001) Simultaneous rec- onstruction of the initial temperature and heat radiative coefficient. Investment Problems, 17, 1181-1202.

[4] Ganzler, T., Volkwein, S. and Weiser, M. (2006) SQP methods for parameter identification problem arising in hyperthermia. Optimization Methods and Software, 21, 869-887.

[5] Arkin, H., et al. (1986) Thermal pulse decay method for simultaneous measurement of local thermal conductivity and blood perfusion: A theoritical snalysis. Journal of Biomechanical Engineering, 108, 208-214.

[6] Deuflhard, P. and Seebass, M. (1998) Adaptive multilevel FEM as decisive tools in clinical cancer therapy hyper- thermia. Konrad-Zuse-Zentrum für Information stechn, Berlin Takustr, Berlin, 7.

[7] Hill, J. and Pincombe, A. (1992) Some similarity temp- erature profiles for the microwave heating of a half-space. Journal of the Australian Mathematical Society, Series B, 33, 290-320.

[8] Chato, J.C. (1980) Heat transfer to blood vessels. Journal of Biomechanical Engineering, 102, 110-118.

[9] Weinbaum, S. and Jiji, L.M. (1985) A two simplified bioheat equation for the effect of blood flow on average tissue temperature. Journal of Biomechanical Engineer- ing, 107, 131-139.

[10] Chen, M.M. and Holmes, K.R. (1980) Microvascular contributions in tissue heat transfer. Annals of the New York Academy of Sciences, 335, 137-150.

[11] Charney, C.K. (1992) Mathematical models of bioheat transfer. Advanced Heat Transfer, 22, 19-155.

[12] Davies, C.R., Saidel, G.M. and Harasaki, H. (1997) Sensitivity analysis of 1-D heat transfer in tissue with temperature-dependent perfusion. Journal of Biomechan- ical Engineering, 119, 77.

[13] Lang, J., Erdmann, B. and Seebass, M. (1999) Impact of nonlinear heat transfer on temperature control in regional hyperthermia. IEEE Transactions on Biomedical Engin- eering, 46, 1129-1138.

[14] Weierstrass, K. (1915) Mathematische Werke V, New York, Johnson, 4-16; Whittaker, E.T. and Watson, G.N. (1927) A Course of Modern Analysis. Cambridge University Press, Cambridge, 454.

[15] Zhao, J.J., Zhang, J., Kang, N. and Yang, F. (2005) A two level finite difference scheme for one dimensional Pennes’ bioheat equation. Applied Mathematics and Computation, 171, 320-331.

[16] Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vallerling, W.T. (1992) Numerical Recipes in Fortran. Cambridge University Press, Cambridge.

[17] Heath, M.T. (2002) Scientific Computing, an Introductory Survey, second edtion, McGraw-Hill, New York.

[18] Liu, J., Chen, X. and Xu, L.X. (1999) New thermal wave aspects on burn evaluation of skin subjected to ins- tantaneous heating. IEEE Transactions on Biomedical Engineering, 46, 420-428.

[19] Tun?, M., ?amdali, ü., Parmaksizoglu, C. and Cikrik ?i, S. (2006) The bioheat transfer equation and its applica-tions in hyperthermia treatments. Engineering Computa-tions, 23, 451-463.

[1] Pennes, H.H. (1948) (1998) Analysis of tissue and arterial blood temperatures in the resting human forearm. Journal of Applied Physics, 1, 93-122; 85, 5-34.

[2] Meyer, C., Philip, P. and Troltzsch, F. (2004) Optimal control of semilinear PDE with nonlocal radiation interface conditions. IMA Preprint Series, 2002.

[3] Yamamoto, M. and Zou, J. (2001) Simultaneous rec- onstruction of the initial temperature and heat radiative coefficient. Investment Problems, 17, 1181-1202.

[4] Ganzler, T., Volkwein, S. and Weiser, M. (2006) SQP methods for parameter identification problem arising in hyperthermia. Optimization Methods and Software, 21, 869-887.

[5] Arkin, H., et al. (1986) Thermal pulse decay method for simultaneous measurement of local thermal conductivity and blood perfusion: A theoritical snalysis. Journal of Biomechanical Engineering, 108, 208-214.

[6] Deuflhard, P. and Seebass, M. (1998) Adaptive multilevel FEM as decisive tools in clinical cancer therapy hyper- thermia. Konrad-Zuse-Zentrum für Information stechn, Berlin Takustr, Berlin, 7.

[7] Hill, J. and Pincombe, A. (1992) Some similarity temp- erature profiles for the microwave heating of a half-space. Journal of the Australian Mathematical Society, Series B, 33, 290-320.

[8] Chato, J.C. (1980) Heat transfer to blood vessels. Journal of Biomechanical Engineering, 102, 110-118.

[9] Weinbaum, S. and Jiji, L.M. (1985) A two simplified bioheat equation for the effect of blood flow on average tissue temperature. Journal of Biomechanical Engineer- ing, 107, 131-139.

[10] Chen, M.M. and Holmes, K.R. (1980) Microvascular contributions in tissue heat transfer. Annals of the New York Academy of Sciences, 335, 137-150.

[11] Charney, C.K. (1992) Mathematical models of bioheat transfer. Advanced Heat Transfer, 22, 19-155.

[12] Davies, C.R., Saidel, G.M. and Harasaki, H. (1997) Sensitivity analysis of 1-D heat transfer in tissue with temperature-dependent perfusion. Journal of Biomechan- ical Engineering, 119, 77.

[13] Lang, J., Erdmann, B. and Seebass, M. (1999) Impact of nonlinear heat transfer on temperature control in regional hyperthermia. IEEE Transactions on Biomedical Engin- eering, 46, 1129-1138.

[14] Weierstrass, K. (1915) Mathematische Werke V, New York, Johnson, 4-16; Whittaker, E.T. and Watson, G.N. (1927) A Course of Modern Analysis. Cambridge University Press, Cambridge, 454.

[15] Zhao, J.J., Zhang, J., Kang, N. and Yang, F. (2005) A two level finite difference scheme for one dimensional Pennes’ bioheat equation. Applied Mathematics and Computation, 171, 320-331.

[16] Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vallerling, W.T. (1992) Numerical Recipes in Fortran. Cambridge University Press, Cambridge.

[17] Heath, M.T. (2002) Scientific Computing, an Introductory Survey, second edtion, McGraw-Hill, New York.

[18] Liu, J., Chen, X. and Xu, L.X. (1999) New thermal wave aspects on burn evaluation of skin subjected to ins- tantaneous heating. IEEE Transactions on Biomedical Engineering, 46, 420-428.

[19] Tun?, M., ?amdali, ü., Parmaksizoglu, C. and Cikrik ?i, S. (2006) The bioheat transfer equation and its applica-tions in hyperthermia treatments. Engineering Computa-tions, 23, 451-463.