Abstract: The purpose of this paper is to derive the Trefftz functions (T-functions) for the Pennes’ equation and for the single-phase-lag (SPL) model (hyperbolic equation) with perfusion and then comparing field of temperature in a flat slab made of skin in the case when perfusion is taken into account, with the situation when a Fourier model is considered. When considering the process of heat conduction in the skin, one needs to take into account the average values of its thermal properties. When in biological bodies relaxation time is of the order of 20 s, the thermal wave propagation appears. The initial-boundary problems for Pennes’ model and SPL with perfusion model are considered to investigate the effect of the finite velocity of heat in the skin, perfusion and thickness of the slab on the rate of the thermal wave attenuation. As a reference model, the solution of the classic Fourier heat transfer equation for the considered problems is calculated. A heat flux has direction perpendicular to the surface of skin, considered as a flat slab. Therefore, the equations depend only on time and one spatial variable.
B I B L I O G R A F I A[1] Peshkov V, Second sound in Helium H. J. Physics USSR 1944
8: 381-386.
[2] Tzou DY, A Unified Field Approach for Heat Conduction From Macro- to Micro-Scales, ASME J. Heat Transfer 1995
117(1) : 8–16.
[3] Cattaneo C, Sur une forme de l'equation de la chaleur elinant le paradoxe d'une propagation instantance, C.R.Acad. Sci. 1958
247 : 431-432.
[4] Vernotte P, Les paradoxes de la theorie continue de l'equation de la chaleur, C.R. Acad. Sci. 1958
246
3154-3155.
[5] Tzou DY, Experimental support for the lagging behavior in heat propagation. J.of Thermophysics and Heat Transfer 1995
9(4)
686-693.
[6] Fan QM, Lu WQ, A New Numerical Method to Simulate the Non-Fourier Heat Conduction in a Single-Phase Medium, Int. J. Heat Mass Transfer 2002
45(13) : 2815–2821.
[7] Pennes HH, Analysis of tissue and arterial blood temperatures in the resting human forearm, J. Appl. Physiol.
1948 : 1 : 93–122.
[8] Field SB, Bleehen NM, Hyperthermia in the treatment of cancer, Cancer Treat. Rev. 1979: 6 : 63–69
[9] Liu J, Xu LX, Boundary information based diagonestics on the thermal states of biological bodies, Int. J. Heat Mass Transf. 2000
43 : 2827–2839.
[10] Zhang J, Sandison GA, Murthy JY, Xu LX, Numerical simulation for heat transfer in prostate cancer cryosurgery, J. Biomed. Eng. 2005
127 : 279–294.
[11] Chen MM, Holmes KR, Rupinskas V, Pulse-decay method for measuring the thermal conductivity of living tissues, J. Biomed. Eng. 1981
103 : 253–260.
[12] Yilbus Z, Sami M, Patiroglu T, Study into penetration speed during laser cutting on brain tissues, J. Med. Eng. Technol. 1998
22 : 274–279.
[13] Seip R, Ebbini ES, Noninvasive estimation of tissue temperature response to heating fields using diagnostic ultrasound, IEEE Trans. Biomed. Eng. 1995
42 : 828–839.
[14] Liu J, Zhu L, Xu LX, Studies on the three-dimensional temperature transients in the canine prostate during transurethral micro wave thermal therapy, J. Biomech. Eng. 2000
122 : 372–379.
[15] Lu WQ, Liu J, Zeng Y, Simulation of the thermal wave propagation in biological tissues by the dual reciprocity boundary element method, Eng. Anal. Bound. Elem. 1998
22 : 167–174.
[16] Zolfaghari A, Maerefat M , A new simplified thermoregulatory bioheat model for evaluating thermal response of the human body to transient environments, Build. Environ. 2010
45 : 2068–2076.
[17] 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. Modelling 2015
39 : 3704–3720.
[18] Kuznetsov AV, Optimization problems for bioheat equation, Int. Comm. in Heat and Mass Transfer 2006
33 : 537–543.
[19] Xu F, Lu TJ, Seffen KA, Ng EYK, Mathematical Modeling of Skin Bioheat Transfer, Applied Mechanics Reviews, 2009
62 : 050801-1-35.
[20] Sharma PR, Ali S, Katiyar VK, Numerical Study of Heat Propagation in Living Tissue Subjected to Instantaneous Heating, Indian Journal of Biomechanics: Special Issue NCBM-2009, ISSN 0974-0783 : 205-209
[21] Vishwakarma V, Das AK, Das PK, Analysis of non-Fourier heat conduction using smoothed particle hydrodynamics, Applied Thermal Engineering 2011
31 : 2963-2970.
[22] Ghazanfarian J, Saghatchi R, Patil DV, Implementation of smoothed-particle hydrodynamics for non-linear Pennes’ bioheat transfer equation, Appl. Math. Comput. 2015
259 : 21–31.
[23] Kałuża G, Ładyga E, Application of BEM for numerical solution of thermal wave model of bioheat transfer, Scientific Research of the Institute of Mathematics and Computer Science, 2011
10(1) : 83-91.
[24] Ng EYK, Tan HM, Ooi EH, Prediction and parametric analysis of thermal profiles within heated human skin using the boundary element method, Phil. Trans. Royal Society A 2010
368 : 655–678.
[25] Zhang ZW, Wang H, Qin QH, Method of Fundamental Solutions for Nonlinear Skin Bioheat Model, Journal of Mechanics in Medicine and Biology 2014
14(4) : 1450060-1-20.
[26] Loureiro FS,Mansur WJ, Wrobel LC, Silva JEA, The Explicit Green’s Approach with stability enhancement for solving the bioheat transfer equation, Int. J. of Heat and Mass Transfer 2014
76 : 393–404.
[27] Kengne E, Lakhssassi A, Bioheat transfer problem for one-dimensional spherical biological tissues, Mathematical Biosciences 2015
269 : 1-9.
[28] Deng ZS, Liu J, Analytical study on bioheat transfer problems with spatial or transient heating on skin surface or inside biological bodies, J. Biomech. Eng. 2002
124 : 638–649.
[29] Deng ZS, Liu J, Blood perfusion-based model for characterizing the temperature fluctuation in living tissues, Physica A 2001
300 : 521-530.
[30] Brixa G, Seebassb M, Hellwig G, Griebel J, Estimation of heat transfer and temperature rise in partial-body regions during MR procedures: An analytical approach with respect to safety considerations, Magnetic Resonance Imaging 2002
20 : 65-76.
[31] Grysa K, Maciag A, Adamczyk-Krasa J, Trefftz Functions Applied to Direct and Inverse Non-Fourier Heat Conduction Problems, Journal of Heat Transfer, 2014 : 136(9) : 091302-1-9, doi: 10.1115/1.4027770.
[32] Hożejowska S., Kaniowski R., Poniewski M.E., Application of adjustment calculus to the Trefftz method for calculating temperature field of the boiling liquid flowing in a minichannel, Int. J. Num. Meth. Heat & Fluid Flow 2014
24(4)
811–824.
[33] Maciejewska B., Strąk K, Piasecka M., The solution of a two-dimensional inverse heat transfer problem using two methods: The Trefftz method and the Beck method, Int. J. Num. Meth. Heat & Fluid Flow 2018
28(1)
206–219.
[34] Ciałkowski MJ, New Type of Basic Functions of FEM in Application to Solution of Inverse Heat Conduction Problems. J. of Thermal Science 2002
11(2) : 163-171.
[35] Qin QH, The Trefftz Finite and Boundary ElementMethod, WIT Press, Southampton, Boston, MA, 2000.
[36] Ciałkowski MJ, Frąckowiak A, Heat Functions and Their Application for Solving Heat Transfer and Mechanical Problems, Poznań University of Technology Publishers, Poznań, Poland (in Polish), 2000.
[37] Li ZC, Lu, TT, Hu HY, Cheng AHD, The Trefftz and Collocation Methods, WIT Press, Southampton, UK, 2008.
[38] Kołodziej J, Zieliński, AP, Boundary Collocation Techniques and Their Application in Engineering, WIT Press, Southampton, UK, 2009.
[39] Maciag A, Trefftz Functions for Some Direct and Inverse Problems of Mechanics, Kielce University of Technology Publishers, Kielce, Poland (in Polish), 2009
[40] Maciag A., Trefftz function for a plate vibration problem, Journal of Theoretical and Applied Mechanics 2011, Vol. 49, No. 1, pp. 97–116.
[41] Grysa K, Trefftz Functions and Their Applications in Solving the Inverse Problems, Kielce University of Technology Publishers, Kielce, Poland (in Polish), 2010.
[42] Ciałkowski MJ, Frąckowiak A, Thermal and Related Functions Used in Solving Certain Problems of Mechanics, Part I. Solving Some Differential Equations With the Use of Inverse Operator, Modern Problems of Technics, Studies and Materials -Technics 3, J. Mielniczuk and B. Pietrulewicz eds., Univ. of Zielona Góra Publishers, 2003 : 7–70.
[43] Rosenbloom PC, Widder DV, Expansion in terms of heat polynomials and associated functions. Trans. Am. Math. Soc. 1956
92 : 220-266.
[44] Hożejowski L, Thermal polynomials and their application in the direct and inverse heat conduction problems. PhD Dissertation, Politechnika Świętokrzyska, Kielce 1999 (in Polish).
[45] de Dear RJ, Arens E, Hui Z, Oguro M, Convective and radiative heat transfer coefficientsfor individual human body segments, Int J Biometeorol (1997) 40:141–156.
[46] Properties of solids, http://webserver.dmt.upm.es/~isidoro/dat1/eSol.pdf, available on Dec 28, 2017
1995-2017 by Isidoro Martinez.
[47] Maciag A., The usage of wave polynomials in solving direct and inverse problems for two-dimensional wave equation, Int. J. Numer. Meth. Biomed. Engng. 2011
27 (7):1107–1125. 
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