In this study, the Pennes and Cattaneo–Vernotte bioheat transfer equations in the presence of fractal spatial dimensions are derived based on the product-like fractal geometry. This approach was introduced recently, by Li and Ostoja-Starzewski, in order to explore dynamical properties of anisotropic media. The theory is characterized by a modified gradient operator which depends on two parameters: R which represents the radius of the tumour and R₀ which represents the radius of the spherical living tissue. Both the steady and unsteady states for each fractal bioheat equation were obtained and their implications on living cells in the presence of growth of a large tumour were analysed. Assuming a specific heating/cooling by a constant heat flux equivalent to the metabolic heat generation in the tissue, it was observed that the solutions of the fractal bioheat equations are robustly affected by fractal dimensions, the radius of the tumour growth and the dimensions of the living cell tissue. The ranges of both the fractal dimensions and temperature were obtained, analysed and compared with recent studies. This study confirms the importance of fractals in medicine.

在本研究中，基于类产品分形几何导出了存在分形空间维数的 Pennes 和 Cattaneo-Vernotte 生物传热方程。Li 和 Ostoja-Starzewski 最近提出了这种方法，以探索各向异性介质的动力学特性。该理论的特征在于修改的梯度算子，其取决于两个参数：代表肿瘤半径的R和代表球形活组织的半径的R _{0 。}获得了每个分形生物热方程的稳态和非稳态，并分析了它们在大肿瘤生长的情况下对活细胞的影响。假设通过相当于组织中代谢热产生的恒定热通量进行特定加热/冷却，观察到分形生物热方程的解受到分形维数、肿瘤生长的半径和肿瘤生长的维数的强烈影响。活细胞组织。获得了分形维数和温度的范围，进行了分析并与最近的研究进行了比较。这项研究证实了分形在医学中的重要性。