Cattaneo-Vernotte bio-heat transfer equation. Identificaton of external heat flux and relaxation time in domain of heated skin tissue.

Marek Paruch; Bohdan Mochnacki

doi:10.24423/cames.243

Marek Paruch Silesian University of Technology
Bohdan Mochnacki University of Occupational Safety Management in Katowice

DOI: http://dx.doi.org/10.24423/cames.243

Abstract

A cylindrical skin tissue domain subjected to an external heat flux is considered. Thermal processes in the domain considered are described by the Cattaneo-Vernotte equation supplemented by the appropriate boundary and initial conditions. The aim of considerations is the identification of external heat flux and relaxation time on the basis of ‘measured’ heating/cooling curves at the set of selected points located on the surface of the skin. The direct problem is solved using the implicit scheme of the Finite Difference Method (FDM), while at the stage of the inverse problem solution, the evolutionary algorithm is applied. In the final part of the paper the examples of computations are presented.

Keywords

bio-heat transfer, inverse problems, Cattaneo-Vernotte equation, evolutionary algorithms, finite difference method,

References

[1] H.H. Pennes. Analysis of tissue and arterial blood temperatures in the resting human forearm. Journal of Applied Physiology, 1: 93–122, 1948, https://doi.org/10.1152/jappl.1948.1.2.93.
[2] S. Kumar, A. Srivastava. Finite integral transform-based analytical solutions of dual phase lag bio-heat transfer equation. Applied Mathematical Modelling, 52: 378–403, 2017, https://doi.org/10.1016/j.apm.2017.05.041.
[3] K.C. Liu, J.Ch. Wang. Analysis of thermal damage to laser irradiated tissue based on the dual-phaselag model. International Journal of Heat and Mass Transfer, 70: 621–628, 2014, https://doi.org/10.1016/j.ijheatmasstransfer.2013.11.044.
[4] B. Mochnacki, E. Majchrzak. Numerical model of thermal interactions between cylindrical cryoprobe and biological tissue using the dual phase lag equation. International Journal of Heat and Mass Transfer, 108: 1–10, 2017, https://doi.org/10.1016/j.ijheatmasstransfer.2016.11.103.
[5] A.N. Smith, P.M. Norris. Microscale Heat Transfer. John Willey & Sons, New York, 2003.
[6] M.C. Cattaneo. A form of heat conduction equation which eliminates the paradox of instantaneous propagation. Compte Rendus, 247: 431–433, 1958.
[7] W. Kaminski. Hyperbolic heat conduction equation for materials with a nonhomogeneous inner structure. Journal of Heat Transfer, 112: 555–560, 1990, https://doi.org/10.1115/1.2910422.
[8] E. Majchrzak, L. Turchan, J. Dziatkiewicz. Modeling of skin tissue heating using the generalized dual-phase lag equation. Archives of Mechanics, 67(6): 417–437, 2015.
[9] M. Ciesielski, B. Mochnacki. Hyperbolic model of thermal interactions in a system biological tissue-protective clothing subjected to an external heat source. Numerical Heat Transfer Part A-Applications, 74(11): 1685–1700, 2018, https://doi.org/10.1080/10407782.2018.1541292.
[10] K. Mitra, S. Kumar, A. Vedavarz, M.K. Moallemi. Experimental evidence of hyperbolic heat conduction in processed meat. Journal of Heat Transfer, 117: 568–573, 1995, https://doi.org/10.1115/1.2822615.
[11] F. Xu, K.A. Seffen, T.J. Lu. Non-Fourier analysis of skin biothermomechanics. International Journal of Heat and Mass Transfer, 51: 2237–2259, 2008, https://doi.org/10.1016/j.ijheatmasstransfer.2007.10.024.
[12] M.I.A. Othman, M.G.S. Ali, M.R. Farouk. The effect of relaxation time on the heat transfer and temperature distribution in tissues. World Journal of Mechanics, 1: 283–287, 2011, https://doi.org/10.4236/wjm.2011.16035.
[13] Y. Zhang. Generalized dual-phase lag bioheat equations based on non equilibrium heat transfer in living biological tissues. International Journal of Heat and Mass Transfer, 52: 4829–4834, 2009, https://doi.org/10.1016/j.ijheatmasstransfer.2009.06.007.
[14] E. Majchrzak. Numerical solution of dual phase lag model of bioheat transfer using the general boundary element method. Computer Modeling in Engineering & Sciences, 69(1): 43–60, 2010, https://doi.org/10.3970/cmes.2010.069.043.
[15] E. Majchrzak, B. Mochnacki. Application of numerical methods for solving the non-Fourier equations. A review of our own and collaborators’ works. Journal of Applied Mathematics and Computational Mechanics, 17(2): 43–50, 2018, https://doi.org/10.17512/jamcm.2018.2.04.
[16] E. Majchrzak, B. Mochnacki. Implicit scheme of the finite difference method for a second-order dual phase lag equation. Journal of Theoretical and Applied Mechanics, 56: 393–402, 2018, http://dx.doi.org/10.15632/jtampl.56.2.393.
[17] D. Deng, Y. Jiang, D. Liang. High-order finite difference method for a second-order dual-phase-lagging models of microscale heat transfer. Applied Mathematics and Computation, 309: 31–48, 2017, https://doi.org/10.1016/j.amc.2017.03.035.
[18] E. Majchrzak, B. Mochnacki. First and second order dual phase lag equation. Numerical solution using the explicit and implicit schemes of the finite difference method. MATEC Web of Conferences, 240: Article Number 05018, 2018, https://doi.org/10.1051/matecconf/201824005018.
[19] B. Mochnacki, M. Paruch. Estimation of relaxation and thermalization times in microscale heat transfer. Journal of Theoretical and Applied Mechanics, 51(4): 837–845, 2013.
[20] B. Mochnacki, E. Majchrzak, M. Paruch. Soft tissue freezing process. Identification of the dual phase lag model parameters using the evolutionary algorithms. AIP Conference Proceedings, 1922: Article Number 060001, 2018, https://doi.org/10.1063/1.5019062.
[21] E. Majchrzak, M. Paruch. Identification of electromagnetic field parameters assuring the cancer destruction during hyperthermia treatment. Inverse Problems in Science and Engineering, 19(1): 45–58, 2011, https://doi.org/10.1080/17415977.2010.531473.
[22] M. Paruch. Identification of the degree of tumor destruction on the basis of the Arrhenius integral using the evolutionary algorithm. International Journal of Thermal Sciences, 130: 507–517, 2018, https://doi.org/10.1016/j.ijthermalsci.2018.05.015.
[23] Z. Michalewicz. Genetic Algorithms + Data Structures = Evolution Programs. Springer-Verlag, Berlin, 1996.