The solution of Pennes' bio-heat equation with a convection term and nonlinear specific heat capacity using Adomian decomposition

Pennes' bio-heat equation is the most widely used equation to analyze the heat transfer phenomenon associated with hyperthermia and cryoablation treatments of cancer. In this study, the semi-analytical and numerical solutions of Pennes' equation in a highly nonlinear form derived from renal cell carcinoma tissue's nonlinear specific heat capacity along with a freezing convection term were obtained and analyzed for the first time. Here, the governing equation was reduced to a lumped capacity form for simplification and exerted on a solid spherical renal tumor. In the following, two semi-analytical techniques, the adomian decomposition method (ADM) and the Akbari–Ganji's method (AGM) were evaluated in solving the governing ODE. The comparison revealed full conformity between ADM and AGM, in addition to an excellent agreement between the semi-analytical and the numerical results before the phase transition. The analysis highlighted a deviation between the semi-analytical and numerical results for the limited convergence of power-series-based semi-analytical methods throughout the phase change and beyond. For the investigated case with Dt=0.025,Biot=0.075\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{t} = 0.025,\quad {\text{Biot}} = 0.075$$\end{document}, the convergence occurred while τ∈[0,0.7]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \in [0,0.7]$$\end{document} for both methods. Consequently, both semi-analytical techniques ADM and AGM, are applicable to find the solution of Pennes' bio-heat equation before the phase change, and there is no superiority in favor of one in accuracy. In contrast, numerical methods are reliable during the phase transition and after that. The analysis of the numerical solution showed that with the growth of the tumor, achieving the necrosis of the malignant tissue takes longer, and large tumors' temperature may not decrease to necrosis temperature. The interpretation of the numerical results indicated that cryoablation could be considered an effective thermal treatment for renal tumors with a diameter lower than 2.0 cm.