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Numerical simulation of dual-phase-lag bioheat transfer model during thermal therapy. (English) Zbl 1348.92084

Summary: This paper theoretically investigates the thermal behavior in a living biological tissue under various coordinate systems and different non-Fourier boundary conditions with the dual-phase-lag bioheat transfer model during thermal therapy. The properties of Legendre wavelets together with the finite difference scheme are used to find an approximate analytical solution of the present problem. It has been observed that surrounding healthy tissues are less affected in second and third kind of boundary condition when applied along with spherical symmetric coordinate system. Also greater temperature rise and fast achievement of peak hyperthermia temperature is achieved when second and third kind of boundary conditions are used in combination with Cartesian coordinate system. It is observed that due to the presence of blood perfusion and temperature dependent metabolic heat generation term, the dual-phase-lag bioheat transfer model reduces to Pennes bioheat transfer model only when \(\tau_q = \tau_T = 0 s,\) not for arbitrary \(\tau_q = \tau_T\). Further, in case of dual-phase-lag bioheat transfer model wave-like or diffusion-like behavior will dominate depends whether the ratio \(\tau_q/\tau_{T} > 1\) or \(\tau_{q}/\tau_{T} < 1\). Effect of temperature dependent metabolic heat generation rate, thermal conductivity and blood perfusion rate on dimensionless temperature are discussed in details. The whole analysis is presented in dimensionless form.

MSC:

92C50 Medical applications (general)
65T60 Numerical methods for wavelets
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[1] Habash, R. W.Y.; Bansal, R.; Krewski, D.; Alhafid, H. T., Thermal therapy, part 1: an introduction to thermal therapy, Crit. Rev. Biomedic. Eng., 34, 6, 459-489 (2006)
[2] Gupta, P. K.; Singh, J.; Rai, K. N., Numerical simulation for heat transfer in tissues during thermal therapy, J. Therm. Biol., 35, 6, 295-301 (2010)
[3] Dimichele, F.; Pizzichellia, G.; Mazzolai, B.; Sinibaldi, E., On the preliminary design of hyperthermia treatments based on infusion and heating of magnetic nanofluids, Math. Biosci., 262, 105-116 (2015) · Zbl 1315.92045
[4] Padra, C.; Salva, N. N., Locating multiple tumors by moving shape analysis, Math. Biosci., 245, 103-110 (2013) · Zbl 1308.92052
[5] Das, K.; Singh, R.; Mishra, S. C., Numerical analysis for determination of the presence of a tumor and estimation of its size and location in a tissue, J Ther. Biol., 38, 32-40 (2013)
[6] Dombrovsky, L. A.; Timchenko, V.; Jackson, M., Indirect heating strategy for laser induced hyperthermia: an advanced thermal model, Int. J. Heat Mass Transf., 55, 4688-4700 (2012)
[7] Pennes, H. H., Analysis of tissue and arterial blood temperature in the resting forearm, J. Appl. Physiol., 1, 93-122 (1948)
[8] Chen, M. M.; Holmes, K. R., Micro-vascular contributions in tissue heat transfer, Ann. NY Acad. Sci., 335, 137-150 (1980)
[9] Weinbaum, S.; Jiji, L. M., A new simplified bioheat equation for the effect of blood flow on local average tissue temperature, ASME J. Biomech. Eng., 107, 131-139 (1985)
[10] Nakayama, A.; Kuwahara, F., A general bioheat transfer model based on the theory of porous media, Int. J. Heat Mass Transf., 51, 3190-3199 (2008) · Zbl 1143.80323
[11] Bhowmik, A.; Singh, R.; Repaka, R.; Mishra, S. C., Conventional and newly developed bioheat transport models in vascularized tissues: a review, J. Therm. Biol., 38, 107-125 (2013)
[12] Mitra, K.; Kumar, S.; Vedavarz, A.; Moallemi, M. K., Experimental evidence of hyperbolic heat conduction in processed meat, ASME J. Heat Transf., 117, 568-573 (1995)
[13] Cattaneo, C., Sur une forme de i’equation de la chaleur elinant le paradox d’une propagation instantance, C.R. Acad. Sci., 247, 431-433 (1958)
[14] Vernotee, M. P., Les paradoxes de la theorie continue de i equation de la chleur, C.R. Acad. Sci., 246, 3154-3155 (1958) · Zbl 1341.35086
[15] Tzou, D. Y., A unified field approach for heat conduction from micro to macroscale, J. Heat Transf., 117, 8-16 (1995)
[16] Tzou, D. Y., Macro- to Micro-Scale Heat Transfer: the Lagging Behavior (1996), Taylor & Francis: Taylor & Francis Washington
[17] Zhang, Y., Generalized dual-phase lag bioheat equations based on non-equilibrium heat transfer in living biological tissues, Int. J. Heat Mass Transf., 52, 4829-4834 (2009) · Zbl 1176.80054
[18] Vadasz, P., Lack of oscillations in dual-phase-lagging heat conduction for a porous slab subject to imposed heat flux and temperature, Int. J. Heat Mass Transf., 48, 2822-2828 (2005) · Zbl 1189.76613
[19] Okajima, J.; Maruyama, S.; Takeda, H.; Komiya, A., Dimensionless solutions and general characteristics of bioheat transfer during thermal therapy, J. Therm. Biol., 34, 377-384 (2009)
[20] Haugk, M.; Sterz, F.; Grassberg, M.; Uray, T.; Kliegel, A.; Janata, A.; Richling, N.; Herkner, H.; Laggner, A. N., Feasibility and efficacy of a new non-invasive surface cooling device in post-resuscitation intensive care medicine, Resuscitation, 75, 76-81 (2007)
[21] Cheng, P.-J.; Liu, K.-C., Numerical analysis of bio-heat transfer in spherical tissue, J. Appl. Sci., 9, 5, 962-967 (2009)
[22] Kengne, E.; Lakhssassi, A., Bioheat transfer problem for one-dimensional spherical biological tissues, Math. Biosci., 269, 1-9 (2015) · Zbl 1351.92029
[23] Akbarzadeh, A. H.; Chen, Z. T., Heat conduction in one-dimensional functionally graded media based on the duel-phase-lag theory, J. Mech. Eng. Sci., 227, 4, 744-759 (2012)
[24] Rodrigues, D. B.; Pereira, P. J.S.; Limao-Vieira, P.; Stauffer, P. R.; Maccarini, P. F., Study of one-dimensional and transient bioheat transfer equation: multi-layer solution development and applications, Int. J. Heat Mass Transf., 62, 153-162 (2013)
[25] Xu, F.; Lu, T. J.; Seffen, K. A.; Ng, E. Y.K., Mathematical modeling of skin bioheat transfer, Appl. Mech. Rev., 62, 050801-050835 (2009)
[26] Liu, K. C.; Chen, H. T., Analysis for the dual-phase-lag bio-heat transfer during magnetic hyperthermia treatment, Int. J. Heat Mass Transf., 52, 1185-1192 (2009) · Zbl 1157.80350
[27] Liu, K. C.; Chen, H. T., Investigation for the dual phase lag behavior of bio-heat transfer, Int. J. Therm. Sci., 49, 1138-1146 (2010)
[28] Andra, W.; d’Ambly, C. G.; Hergt, R.; Hilger, I.; Kaiser, W. A., Temperature distribution as function of time around a small spherical heat source of local magnetic hyperthermia, J. Magn. Magn. Mater., 194, 197-203 (1999)
[29] Kumar, D.; Singh, S.; Rai, K. N., Analysis of classical fourier, SPL and DPL heat transfer model in biological tissues in presence of metabolic and external heat source, Heat Mass Transf., 52, 6, 1089-1107 (2016)
[30] Stolwijk, J. A.J.; Hardy, J. D., Temperature regulation in man-a theoretical study, Pflug Arch, 291, 129-162 (1966)
[31] Wang, L. Q.; Zhou, X.; Wei, X., Heat Conduction: Mathematical Models and Analytical solutions (2008), Springer Verlag: Springer Verlag Berlin
[32] Zhao, J. J.; Zhang, J.; Kang, N.; Yang, F., A two level finite difference scheme for one dimensional Pennes’ bioheat equation, Appl. Math. Comput., 171, 320-331 (2005) · Zbl 1086.65090
[33] Gupta, P. K.; Singh, J.; Rai, K. N., Solution of the heat transfer problem in tissues during hyperthermia by finite difference-decomposition method, Appl. Math. Comput., 219, 6882-6892 (2013) · Zbl 1286.65098
[34] Saghatchi, R.; Ghazanfarian, J., A novel SPH method for the solution of dual-phase-lag model with temperature-jump boundary condition in nanoscale, Appl. Math. Model., 39, 1063-1073 (2015) · Zbl 1432.80005
[35] Ahmadikia, H.; Fazlali, R.; Moradi, A., 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., 39, 121-130 (2012)
[36] Askarizadeh, H.; Ahmadikia, H., Analytical study on the transient heating of a two-dimensional skin tissue using parabolic and hyperbolic bioheat transfer equations, Appl. Math. Model., 39, 3704-3720 (2015) · Zbl 1443.80012
[37] Kumar, P.; Kumar, D.; Rai, K. N., A mathematical model for hyperbolic space-fractional bioheat transfer during thermal therapy, Procedia Eng., 127, 56-62 (2015)
[38] Kumar, D.; Kumar, P.; Rai, K. N., A study on DPL model of heat transfer in bi-layer tissues during MFH treatment, Comput. Biol. Med., 75, 160-172 (2016)
[39] Hariharan, G.; Kannan, K., Review of wavelet methods for the solution of reaction-diffusion problems in science and engineering, Appl. Math. Model., 38, 799-813 (2014) · Zbl 1427.65429
[40] Chen, Y.; Ke, X.; Wei, Y., Numerical algorithm to solve system of non-linear fractional differential equations based on wavelets method and the error analysis, Appl. Math. Comput., 251, 475-488 (2015) · Zbl 1328.65171
[41] Kumar, P.; Kumar, D.; N. Rai, K., A numerical study on dual-phase-lag model of bio-heat transfer during hyperthermia treatment, J. Therm. Biol., 49-50, 98-105 (2015)
[42] Mitchell, J. W.; Galvez, T. L.; Hangle, J.; Myers, G. E.; Siebecker, K. L., Thermal response of human legs during cooling, J. Appl. Physiol., 29, 859-865 (1970)
[43] Kumar, P.; Kumar, D.; Rai, K. N., Non-linear dual-phase-lag model for analyzing heat transfer phenomena in living tissues during thermal ablation, J. Therm. Biol., 60, 204-212 (2016)
[44] Karaa, S.; Zhang, J.; Yang, F., A numerical study of a 3d bioheat transfer problem with different spatial heating, Math. Comput. Simul., 68, 375-388 (2005) · Zbl 1062.92018
[45] Scheid, F., Schaum’s outline of theory and problems: numerical analysis (1989), McGraW-Hill · Zbl 0318.65001
[46] Gu, J. S.; Jiang, W. S., The Haar wavelets operational matrix of integration, Int. J. Syst. Sci., 27, 623-628 (1996) · Zbl 0875.93116
[47] Razzaghi, M.; Yosefi, S., Legendre wavelets direct method for variational problems, Math. Comput. Simul., 53, 3, 185-192 (2000)
[48] Razzaghi, M.; Yosefi, S., The legendre wavelets operational matrix of integration, Int. J. Syst. Sci., 32, 4, 495-502 (2001) · Zbl 1006.65151
[49] Henriques, F., Studies of thermal injury, Arch. Pathol., 43, 489-502 (1947)
[50] Dombrovsky, L. A.; Timchenko, V., Laser induced hyperthermia of superficial tumors: computational models for radiative transfer, combined heat transfer, and degradation of biological tissues, Therm. Process. Eng., 7, 1, 24-36 (2015)
[51] Kumar, D.; Rai, K. N., 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. (2016)
[52] Zhou, J.; Chen, J. K.; Zhang, Y., Dual-phase lag effects on thermal damage to biological tissues caused by laser irradiation, Comput. Biol. Med., 39, 286-293 (2009)
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