A three-phase-lag (TPL) model is proposed to describe heat transfer in a finite domain skin tissue with temperature dependent metabolic heat generation. The Laplace transform method is applied to solve the problem. Three special types of heat flux are applied to the boundary of skin tissue for thermal therapeutic applications. The depth of tissue is influenced by the different oscillation heat flux. The comparison between the TPL and dual-phase-lag (DPL) models is analyzed and the effects of phase lag parameters (τ_{q}, τ_{t} and τ_{v}) and material (k^{*}) on the tissue temperature distribution are presented graphically.
[1] Pennes H.H. (1948): Analysis of tissue and arterial blood temperature in the resting forearm. – J. Appl. Physiol. vol.1 pp.93-122.
[2] Cattaneo C. (1958): A form of heat conduction equation which eliminates the paradox of instantaneous propagation. – Comp. Rend. vol.247 pp.431-433.
[3] Vernotte P. (1958): Les paradoxes de la theorie continue de l’equation de la chaleur. – Comp. Rend. vol.246 pp.3154-3155.
[4] Tzou D.Y. (1995): Macro-to Microscale Heat transfer: The Lagging Behavior. – Washington: Taylor and Francis.
[5] Tzou D.Y. (1995): A unified field approach for heat conduction from macro-to-microscales. – ASME J. Heat Transf. vol.117 pp.8-16.
[6] Roy Choudhuri S.K. (2007): On a thermoelastic three-phase-lag model journal of thermal stresses. – J. Therm. Stress vol.30 pp.231-238.
[7] Sur Abhik and Kanoria M. (2015): Analysis of thermoelastic response in a functionally graded infinit space subjected to a Mode-I crack. – Int. J. Adv. Appl. Math. Mech vol.3 pp.33-44.
[8] Kumar R. and Gupta V. (2016): Plane wave propagation and domain of influence in fractional order thermoelastic materials with three-phase-lag heat transfer. – Mech. Adv. Mater. Struct. vol.23 pp.896-908.
[9] Shih T.C. Yuan P. Lin W.L. and Kou H.S. (2007): Analytical analysis of the Pennes bioheat transfer equation with sinusoidal heat flux condition on skin surface. – Med. Eng. Phys. vol.29 pp.946-953.
[10] Zhang Y. Chen J.K. and Zhou J. (2009): Dual-phase lag effects on thermal damage to biological tissues caused by laser irradiation. – Comput. Biol. Med. vol.39 pp.286-293.
[11] Zhou J. Zhang Y. and Chen J.K. (2009): An axisymmetric dual-phase-lag bioheat model for laser heating of living tissues. – Int. J. Therm. Sci. vol.48 pp.1477-1485.
[12] Zhang Y. (2009): Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues. – Int. J. Heat Mass Transf. vol.52 pp.4829-4834.
[13] Liu K.C. and Chen H.T. (2009): Analysis for dual phase lag bioheat transfer during magnetic hyperthermia treatment. – Int. J. Heat Mass Transf vol.52 pp.1185-1192.
[14] Majchrzak J.E. (2010): Numerical solution of dual phase lag model of bioheat transfer using the general boundary element method. – Comput. Modeling Eng. Sci. vol.69 pp.43-60.
[15] Poor H.Z. Moosavi H. and Moradi A. (2014): Investigation on the dual-phase-lag effects in biological tissues during laser irradiation. – Int. J. Mech. Syst. Eng. vol.4 pp.33-46.
[16] Majchrzak E. and Turchan L. (2015): The general boundary element method for 3D dual-phase lag model of bioheat transfer. – Eng. Anal. with Boundary Elements. vol.50 pp.76-82.
[17] Ahmadikia H. Fazlali R. and Moradi A. (2012): Analytical solution of the parabolic and hyperbolic heat transfer equations with constant and transient heat flux conditions on skin tissue. – Int. Commun Heat Mass Transf. vol.39 pp.121-130.
[18] Ahmadikia H. Moradi A. Fazlali R. and Parsa A.B. (2012): Analytical solution of non-Fourier and Fourier bioheat transfer analysis during laser irradiation of skin tissue. – J. Mech. Sci. Technol. vol.26 pp.1937-1947.
[19] Afrin N. Zhou J. Zhang Y Tzou D.Y. and Chen J.K. (2012): Numerical simulation of thermal damage to living tissues induced by laser irradiation based on a generalized dual phase lag model. – Numer. Heat Transf. vol.61 pp.483-501.
[20] Kengne E. Lakhssassi A. and Vaillancourt R. (2012): Temperature distribution in living biological tissue simultaneously subjected to oscillatory surface and spatial heating: analytical and numerical analysis. – Int. Mathematical Forum. vol.7 pp.2373-2392.
[21] Fazlali R. and Ahmadikia H. (2013): Analytical solution of thermal wave models on skin tissue under arbitrary periodic boundary conditions. – Int. J. Thermophys. vol.34 pp.139-159.
[22] Shahnazari M. Aghanajafi C. Azimifar M. and Jamali H. (2013): Investigation of bioheat transfer equation of Pennes via a new method based on wrm and homotopy perturbation. – IJRRAS vol.17 pp.306-314.
[23] Gupta P.K. Singh J. Rai K.N. and Rai S.K. (2013): Solution of the heat transfer problem in tissues during hyperthermia by finite difference-decomposition method. – Appl. Math. Comput. vol.219 pp.6882-6892.
[24] Askarizadeh H. and Ahmadikia H. (2014): Analytical analysis of the dual-phase-lag model of bioheat transfer equation during transient heating of skin tissue. – Heat Mass Transf. vol.50 pp.1673-1684.
[25] Kengne E. Mellal I. Hamouda M.B. and Lakhssassi A. (2014): A mathematical model to solve bio-heat transfer problems through a bio-heat transfer equation with quadratic temperature-dependent blood perfusion under a constant spatial heating on skin surface. – J. Biom. Sci. Eng. vol.7 pp.721-730.
[26] Kumar P. Kumar D. and Rai K.N. (2015): A numerical study on dual-phase-lag model of bio-heat transfer during hyperthermia treatment. – J. Therm. Biol. vol.49-50 pp.98-105.
[27] Majchrzak E. Turchan L. and Dziatkiewicz J. (2015): Modeling of skin tissue heating using the generalized dual phase-lag equation. – Arch. Mech. vol.67 pp.417-437.
[28] Jasinski M. Majchrazak E. and Turchan L. (2016): Numerical analysis of the interactions between laser and soft tissue using generalized dual-phase-lag equation. – Appl. Math. Model. vol.40 pp.750-762.
[29] Kumar D. and Rai K.N. (2016): A study on thermal damage during hyperthermia treatment based on DPL model for multilayer tissues using finite element Legendre wavelet Galerkin approach. – J. Therm. Biol. vol.62 pp.170-180.
[30] Agrawal M. and Pardasani K.R. (2016): Finite element model to study temperature distribution in skin and deep tissues of human limbs. – J. Therm. Biol. vol.62 pp.98-105.
[31] Green A.E. and Naghdi P.M. (1992): On undamped heat waves in an elastic solid. – J. Therm. Stress. vol.15 pp.253-264.
[32] Green A.E. and Naghdi P.M. (1993): Thermoelasticity without energy dissipation. – J. Elast. vol.31 pp.189-208.
[33] Mitchell J.W. Galvez T.L. Hangle J. Myers G.E. and Siebecker K.L. (1970): Thermal response of human legs during cooling. – J. Appl. Physiol. vol.29 pp.859-865.
[34] Arpaci V.C. (1996): Conduction heat transfer. – New York: Addisson Wesley.
[35] Yamada Y. Tien T. and Ohta M. (1995): Theoretical analysis of temperature variation of biological tissue irradiated by light. – ASME/JSME Thermal Eng. Con vol.4 pp.575-581.
[36] Torvi D.A. and Dale J.D. (1994): A finite element model of skin subjected to a flash fire. – ASME J. Biomech. Eng. vol.116 pp.250-255.