Developing the analytical solution for the nonlinear bioheat transfer equation through homotopy analysis method along with an optimal convergence-control parameter

Yıl 2024, Cilt: 10 Sayı: 3, 613 - 621, 21.05.2024

Rouhollah Ostadhossein

Öz

The Homotopy Analysis Method (HAM) is an effective technique to achieve the analytical solution of a broad range of problems, mainly with nonlinear governing equations. The solution of Pennes’ bioheat equation in nonlinear form arising from the linear temperature-dependent nature of specific heat capacity of a biological tissue using the Homotopy Analysis Method has been obtained analytically and validated with the numerical results obtained from the Finite Difference Method (FDM) the first time in this study. The analysis demonstrated that considering the various values of the convergence parameter and computing the Mean Squared Error (MSR) to achieve the optimum values ensures accurate results even at the low-order approximations of the solution. Investigating the effect of the nonlinear term’s magnitude on the solution indicated a direct relationship; However, the effect was not remarkable even at the major values, thus it is possible to consider the specific heat capacity of a living tissue, a constant value through thermal simulations. According to this research, the Homotopy Analysis Method can be a proper method to derive the analytical solution of either the linear or nonlinear form of Pennes’ bioheat equation.

Anahtar Kelimeler

Homotopy Analysis Method (HAM), Mean Squared Error, Optimum Convergence-Control Parameter, Pennes

Kaynakça

• [1] Pennes HH. Analysis of tissue and arterial blood temperatures in the resting human forearm. J Appl Physiol 1948;1:93–122. [CrossRef]
• [2] Hristov J. Bio-heat models revisited: Concepts, derivations, nondimensalization and fractionalization approaches. Front Phys 2019;7:00189. [CrossRef]
• [3] Shih TC, Yuan P, Lin WL, Sen Kou H. Analytical analysis of the Pennes bioheat transfer equation with sinusoidal heat flux condition on skin surface. Med Eng Phys 2007;29:946–953. [CrossRef]
• [4] Shahnazari, Aghanajafi, Azimifar. Investigation of bioheat transfer equation of pennes via a new method based on wrm & homotopy perturbation. 2014. Available at: https://api.semanticscholar.org/CorpusID:212596037. Accessed May 2, 2024.
• [5] Qin Y, Wu K. Numerical solution of fractional bioheat equation by quadratic spline collocation method. 2016. J Nonlinear Sci Appl 2016;9:5061–5072. [CrossRef]
• [6] Al-Humedi HO, Al-Saadawi FA. The numerical solution of bioheat equation based on shifted Legendre polynomial. Int J Nonlinear Anal Appl 2021;12:1061–1070. [CrossRef]
• [7] Liu KC, Tu FJ. Numerical solution of bioheat transfer problems with transient blood temperature. Int J Comput Methods 2019;16:1843001. [CrossRef]
• [8] Singh J, Gupta PK, Rai KN. Solution of fractional bioheat equations by finite difference method and HPM. Math Comput Model 2011;54:2316–2325. [CrossRef]
• [9] Kumar P, Kumar D, Rai KN. A mathematical model for hyperbolic space-fractional bioheat transfer during thermal therapy. Procedia Engineer 2015;56:56–62. [CrossRef]
• [10] Wang X, Qi H, Yang X, Xu H. Analysis of the time-space fractional bioheat transfer equation for biological tissues during laser irradiation. Int J Heat Mass Transf 2021;177:121555. [CrossRef]
• [11] Kabiri A, Talaee MR. Analysis of hyperbolic Pennes bioheat equation in perfused homogeneous biological tissue subject to the instantaneous moving heat source. SN Appl Sci 2021;3:398. [CrossRef]
• [12] Irfan M, Shah FA, Nisar KS. Fibonacci wavelet method for solving Pennes bioheat transfer equation. Int J Wavelets Multiresolut Inf Process 2021;19:2150023. [CrossRef]
• [13] Irfan M, Shah FA. Fibonacci wavelet method for solving the time-fractional bioheat transfer model. Optik (Stuttg) 2021;241:167084. [CrossRef]
• [14] Lakhssassi A, Kengne E, Semmaoui H. Modified Pennes’ equation modelling bio-heat transfer in living tissues: Analytical and numerical analysis. Nat Sci (Irvine) 2010;2:1375–1385. [CrossRef]
• [15] Gupta PK, Singh J, Rai KN. Numerical simulation for heat transfer in tissues during thermal therapy. J Therm Biol 2010;35:295–301. [CrossRef]
• [16] Majchrzak E. Numerical Solution of dual phase lag model of bioheat transfer using the general boundary element method. CMES Comput Model Engineer Sci 2010;69:43–60.
• [17] Majchrzak E, Turchan L. The general boundary element method for 3D dual-phase lag model of bioheat transfer. Engineer Anal Bound Elem 2015;50:76–82. [CrossRef]
• [18] Ostadhossein R, Hoseinzadeh S. The solution of Pennes’ bio-heat equation with a convection term and nonlinear specific heat capacity using Adomian decomposition. J Therm Anal Calorim 2022;147:12739–12747. [CrossRef]
• [19] Ostadhossein R, Hoseinzadeh S. Developing computational methods of heat flow using bioheat equation enhancing skin thermal modeling efficiency. Int J Numer Methods Heat Fluid Flow 2023;34:1380–1398. [CrossRef]
• [20] Forghani P, Ahmadikia H, Karimipour A. Non-fourier boundary conditions effects on the skin tissue temperature response. Heat Transfer - Asian Research 2017;46:29–48. [CrossRef]
• [21] Akbari M, Hemmat Esfe M, Rostamian SH, Karimipour A. Non fourier heat conduction in a semi infinite body exposed to the pulsatile and continuous heat sources. J Curr Res Sci 2013;1:144–150. [CrossRef]
• [22] Karimipour A, Hossein Nezhad A, D’Orazio A, Shirani E. Investigation of the gravity effects on the mixed convection heat transfer in a microchannel using lattice Boltzmann method. Int J Therm Sci 2012;54:142–152. [CrossRef]
• [23] Karimipour A, Taghipour A, Malvandi A. Developing the laminar MHD forced convection flow of water/FMWNT carbon nanotubes in a microchannel imposed the uniform heat flux. J Magn Magn Mater 2016;419:420–428. [CrossRef]
• [24] Karimipour A. New correlation for Nusselt number of nanofluid with Ag / Al2O3 / Cu nanoparticles in a microchannel considering slip velocity and temperature jump by using lattice Boltzmann method. Int J Therm Sci 2015;91:146–156. [CrossRef]
• [25] Karimipour A, Alipour H, Akbari OA, Semiromi DT, Esfe MH. Studying the effect of indentation on flow parameters and slow heat transfer of water-silver nano-fluid with varying volume fraction in a rectangular two-dimensional micro channel. Indian J Sci Technol 2015;8:51707. [CrossRef]
• [26] Menni Y, Azzi A, Chamkha A. Turbulent heat transfer and fluid flow over complex geometry fins. Defect Diffus Forum 2018;388:378–393. [CrossRef]
• [27] Menni Y, Azzi A, Chamkha A. Modeling and analysis of solar air channels with attachments of different shapes. Int J Numer Methods Heat Fluid Flow 2019;29:1815–1845. [CrossRef]
• [28] Menni Y, Azzi A, Chamkha A. Enhancement of convective heat transfer in smooth air channels with wall-mounted obstacles in the flow path: A review. J Therm Anal Calorim 2019;135:1951–1976. [CrossRef]
• [29] Menni Y, Saim R. Numerical simulation of turbulent forced convection in a channel fitted with baffles of various shapes. Université Abou Bekr Belkaid de Tlemcen Faculté de Technologie Le 3ème Séminaire sur les Technologies Mécaniques Avancées, 8-9 November 2014.
• [30] Liao S. Homotopy analysis method: A new analytical technique for nonlinear problems. Commun Nonlinear Sci Numer Simul 1997;2:95–100. [CrossRef]
• [31] Chakraverty S, Mahato N, Karunakar P, Rao TD. Advanced Numerical and Semi‐Analytical Methods for Differential Equations. Homotopy Analysis Method. Hoboken: Wiley; 2019. pp. 149–156. [CrossRef]
• [32] Abbasbandy S. Homotopy analysis method for heat radiation equations. ICHMT 2007;34:380–387. [CrossRef]
• [33] Hariharan G. A homotopy analysis method for the nonlinear partial differential equations arising in engineering. JCMSE 2017;18:191–200. [CrossRef]
• [34] Gupta VG, Gupta S. Application of homotopy analysis method for solving nonlinear cauchy problem. Surv Math Appl 2012;7:105–116. [CrossRef]
• [35] Lu D, Liu J. Application of the homotopy analysis method for solving the variable coefficient kdv-burgers equation. Abstr Appl Anal 2014:309420. [CrossRef]
• [36] Hesameddini E, Latifizadeh H. Homotopy analysis method to obtain numerical solutions of the Painlevé equations. Math Methods Appl Sci 2012;35:1423–1433. [CrossRef]
• [37] Hammouch Z, Mekkaoui T, Sadki H. Homotopy analysis method for solving MHD free convection flow from a cooling sheet. Math Natural Sci 2019;3:39–47. [CrossRef]
• [38] Odibat Z. An improved optimal homotopy analysis algorithm for nonlinear differential equations. J Math Anal Appl 2020;488. [CrossRef]
• [39] Biswal U, Chakraverty S. Investigation of Jeffery-Hamel flow for nanofluid in the presence of magnetic field by a new approach in the optimal homotopy analysis method. J Appl Comput Mech 2022;8:48–59.
• [40] Obalalu AM, Ajala AO, Akindele AO, Oladapo OA, Adepoju O, Jimoh MO. Unsteady squeezed flow and heat transfer of dissipative casson fluid using optimal homotopy analysis method: An application of solar radiation. Part Differ Equat Appl Math 2021;4:100146. [CrossRef]
• [41] Rabbani M. Modified homotopy method to solve non-linear integral equations. Int J Nonlinear Anal Appl 2015;6:133–136.
• [42] Jbr RK, Al-Rammahi A. Q-homotopy analysis method for solving nonlinear fredholm integral equation of the second kind. Int J Nonlinear Anal Appl 2021;12:2145–2152.
• [43] Ahmad AA. Solving partial differential equations via a hybrid method between homotopy analytical method and Harris hawks optimization algorithm. Int J Nonlinear Anal Appl 2022;13:663–671.
• [44] Huang HW, Horng TL. Bioheat transfer and thermal heating for tumor treatment. In: Becker SM, Kuznetsov AV, eds. Heat Transfer and Fluid Flow in Biological Processes. Amsterdam: Elsevier; 2015. pp. 1–42. [CrossRef]
• [45] Ghanmi A, Abbas IA. An analytical study on the fractional transient heating within the skin tissue during the thermal therapy. J Therm Biol 2019;82:229–233. [CrossRef]
• [46] Liao S. Beyond Perturbation. New York: Chapman and Hall/CRC; 2003. [CrossRef]
• [47] Fürnkranz J. Encyclopedia of Machine Learning. Boston, MA: Springer US; 2010. [CrossRef]