Temperature Distributions for Regional Hypothermia Based on Nonlinear Bioheat Equation of Pennes Type: Dermis and Subcutaneous Tissues

Affiliation(s)

Laboratoire d’Ingénierie des Microsystèmes Avancés, Département d’Informatique et d’Ingénierie, Université du Québec en Outaouais, Succursale Hull, Gatineau, Canada.

Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada.

Laboratoire d’Ingénierie des Microsystèmes Avancés, Département d’Informatique et d’Ingénierie, Université du Québec en Outaouais, Succursale Hull, Gatineau, Canada.

Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada.

ABSTRACT

We have used a nonlinear one-dimensional heat transfer model based on temperature-dependent blood perfusion to predict temperature distribution in dermis and subcutaneous tissues subjected to point heating sources. By using Jacobi elliptic functions, we have first found the analytic solution corresponding to the steady-state temperature distribution in the tissue. With the obtained analytic steady-state temperature, the effects of the thermal conductivity, the blood perfusion, the metabolic heat generation, and the coefficient of heat transfer on the temperature distribution in living tissues are numerically analyzed. Our results show that the derived analytic steady-state temperature is useful to easily and accurately study the thermal behavior of the biological system, and can be extended to such applications as parameter measurement, temperature field reconstruction and clinical treatment.

We have used a nonlinear one-dimensional heat transfer model based on temperature-dependent blood perfusion to predict temperature distribution in dermis and subcutaneous tissues subjected to point heating sources. By using Jacobi elliptic functions, we have first found the analytic solution corresponding to the steady-state temperature distribution in the tissue. With the obtained analytic steady-state temperature, the effects of the thermal conductivity, the blood perfusion, the metabolic heat generation, and the coefficient of heat transfer on the temperature distribution in living tissues are numerically analyzed. Our results show that the derived analytic steady-state temperature is useful to easily and accurately study the thermal behavior of the biological system, and can be extended to such applications as parameter measurement, temperature field reconstruction and clinical treatment.

Cite this paper

E. Kengne, A. Lakhssassi and R. Vaillancourt, "Temperature Distributions for Regional Hypothermia Based on Nonlinear Bioheat Equation of Pennes Type: Dermis and Subcutaneous Tissues,"*Applied Mathematics*, Vol. 3 No. 3, 2012, pp. 217-224. doi: 10.4236/am.2012.33035.

E. Kengne, A. Lakhssassi and R. Vaillancourt, "Temperature Distributions for Regional Hypothermia Based on Nonlinear Bioheat Equation of Pennes Type: Dermis and Subcutaneous Tissues,"

References

[1] H. H. Pennes, “Analysis of Tissue and Arterial Blood Temperatures in the Resting Forearm,” Journal of Applied Physiology, Vol. 1, No. 2, 1948, pp. 93-122.

[2] A. Szasz and G. Vincze, “Dose Concept of Oncological Hyperthermia: Heat-Equation Considering the Cell Destruction,” Journal of Cancer Research and Therapeutics, Vol. 2, No. 4, 2006, pp. 171-181. doi:10.4103/0973-1482.29827

[3] C. K. Charney, “Mathematical Models of Bioheat Transfer,” Advanced Heat Transfer, Vol. 22, 1992, pp. 19-155. doi:10.1016/S0065-2717(08)70344-7

[4] P. Deuflhard and R. Hochmuth, “Multiscale Analysis of Thermoregulation in the Human Microvascular System,” Mathematical Methods in the Applied Sciences, Vol. 27, No. 8, 2004, pp. 971-989. doi:10.1002/mma.499

[5] T. R. Gowrishankar, D. A. Stewart, G. T. Martin and J. C. Weaver, “Transport Lattice Models of Heat Transport in Skin with Spatially Heterogeneous, Temperature-Dependent Perfusion,” BioMedical Engineering on Line, Vol. 3, No. 4, 2004, pp. 1-17.

[6] A. Lakhssassi, E. Kengne and H. Semmaoui, “Modified Pennes’ Equation Modelling Bio-Heat Transfer in Living Tissues: Analytical and Numerical Analysis,” Natural Science, Vol. 2, No. 12, 2010, pp. 1375-1385. doi:10.4236/ns.2010.212168

[7] C. R. Davies, G. M. Saidel and H. Harasaki, “Sensitivity Analysis of One-Dimensional Heat Transfer in Tissue with Temperature-Dependent Perfusion,” Journal of Biomechanical Engineering, Vol. 119, No. 1, 1997, pp. 77-80. doi:10.1115/1.2796068

[8] J. Lang, B. Erdmann and M. Seebass, “Impact of Nonlinear Heat Transfer on Temperature Control in Regional Hyperthermia,” IEEE Transactions on Biomedical Engineering, Vol. 46, No. 9, 1999, pp. 1129-1138. doi:10.1109/10.784145

[9] J. Liu and L. X. Xu, “Estimation of Blood Perfusion Using Phase Shift in Temperature Response to Sinusoidal Heating at the Skin Surface,” IEEE Transactions on Biomedical Engineering, Vol. 46, No. 9, 1999, pp. 10371043. doi:10.1109/10.784134

[10] 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,” Journal of Biomechanical Engineering, Vol. 106, No. 4, 1984, pp. 321330. doi:10.1115/1.3138501

[11] P. F. Byrd and M. D. Friedman, “Handbook of Elliptic Integrals for Engineers and Scientists,” 2nd Edition, Springer-Verlag, Berlin, 1971.

[12] Z. S. Deng and J. Liu, “Analytical Study on Bioheat Transfer Problems with Spatial or Transient Heating on Skin Surface or Inside Biological Bodies,” Journal of Biomechanical Engineering, Vol. 124, No. 6, 2002, pp. 638-650. doi:10.1115/1.1516810

[13] S. Karaa, J. Zhang and F. Yang, “A Numerical Study of a 3D Bioheat Transfer Problem with Different Spatial Heating,” Mathematics and Computers in Simulation, Vol. 68, No. 4, 2005, pp. 375-388. doi:10.1016/j.matcom.2005.02.032

[14] W. Shen and J. Zhang, “Modeling and Numerical Simulation of Bioheat Transfer and Biomechanics in Soft Tissue,” Mathematical and Computer Modelling, Vol. 41, No. 11-12, 2005, pp. 1251-1265. doi:10.1016/j.mcm.2004.09.006

[15] A. T. Patera, B. B. Mikic, G. Eden and H. F. Bowman, “Prediction of Tissue Perfusion from Measurement of the Phase Shift between Heat Flux and Temperature,” Proceedings of ASME Winter Annual Meeting, Advances in Bioengineering, 1979, pp. 187-191.

[1] H. H. Pennes, “Analysis of Tissue and Arterial Blood Temperatures in the Resting Forearm,” Journal of Applied Physiology, Vol. 1, No. 2, 1948, pp. 93-122.

[2] A. Szasz and G. Vincze, “Dose Concept of Oncological Hyperthermia: Heat-Equation Considering the Cell Destruction,” Journal of Cancer Research and Therapeutics, Vol. 2, No. 4, 2006, pp. 171-181. doi:10.4103/0973-1482.29827

[3] C. K. Charney, “Mathematical Models of Bioheat Transfer,” Advanced Heat Transfer, Vol. 22, 1992, pp. 19-155. doi:10.1016/S0065-2717(08)70344-7

[4] P. Deuflhard and R. Hochmuth, “Multiscale Analysis of Thermoregulation in the Human Microvascular System,” Mathematical Methods in the Applied Sciences, Vol. 27, No. 8, 2004, pp. 971-989. doi:10.1002/mma.499

[5] T. R. Gowrishankar, D. A. Stewart, G. T. Martin and J. C. Weaver, “Transport Lattice Models of Heat Transport in Skin with Spatially Heterogeneous, Temperature-Dependent Perfusion,” BioMedical Engineering on Line, Vol. 3, No. 4, 2004, pp. 1-17.

[6] A. Lakhssassi, E. Kengne and H. Semmaoui, “Modified Pennes’ Equation Modelling Bio-Heat Transfer in Living Tissues: Analytical and Numerical Analysis,” Natural Science, Vol. 2, No. 12, 2010, pp. 1375-1385. doi:10.4236/ns.2010.212168

[7] C. R. Davies, G. M. Saidel and H. Harasaki, “Sensitivity Analysis of One-Dimensional Heat Transfer in Tissue with Temperature-Dependent Perfusion,” Journal of Biomechanical Engineering, Vol. 119, No. 1, 1997, pp. 77-80. doi:10.1115/1.2796068

[8] J. Lang, B. Erdmann and M. Seebass, “Impact of Nonlinear Heat Transfer on Temperature Control in Regional Hyperthermia,” IEEE Transactions on Biomedical Engineering, Vol. 46, No. 9, 1999, pp. 1129-1138. doi:10.1109/10.784145

[9] J. Liu and L. X. Xu, “Estimation of Blood Perfusion Using Phase Shift in Temperature Response to Sinusoidal Heating at the Skin Surface,” IEEE Transactions on Biomedical Engineering, Vol. 46, No. 9, 1999, pp. 10371043. doi:10.1109/10.784134

[10] 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,” Journal of Biomechanical Engineering, Vol. 106, No. 4, 1984, pp. 321330. doi:10.1115/1.3138501

[11] P. F. Byrd and M. D. Friedman, “Handbook of Elliptic Integrals for Engineers and Scientists,” 2nd Edition, Springer-Verlag, Berlin, 1971.

[12] Z. S. Deng and J. Liu, “Analytical Study on Bioheat Transfer Problems with Spatial or Transient Heating on Skin Surface or Inside Biological Bodies,” Journal of Biomechanical Engineering, Vol. 124, No. 6, 2002, pp. 638-650. doi:10.1115/1.1516810

[13] S. Karaa, J. Zhang and F. Yang, “A Numerical Study of a 3D Bioheat Transfer Problem with Different Spatial Heating,” Mathematics and Computers in Simulation, Vol. 68, No. 4, 2005, pp. 375-388. doi:10.1016/j.matcom.2005.02.032

[14] W. Shen and J. Zhang, “Modeling and Numerical Simulation of Bioheat Transfer and Biomechanics in Soft Tissue,” Mathematical and Computer Modelling, Vol. 41, No. 11-12, 2005, pp. 1251-1265. doi:10.1016/j.mcm.2004.09.006

[15] A. T. Patera, B. B. Mikic, G. Eden and H. F. Bowman, “Prediction of Tissue Perfusion from Measurement of the Phase Shift between Heat Flux and Temperature,” Proceedings of ASME Winter Annual Meeting, Advances in Bioengineering, 1979, pp. 187-191.