Analytical solution of the bioheat equation for thermal response induced by any electrode array in anisotropic tissues with arbitrary shapes containing multiple-tumor nodules

Abstract

The Pennes bioheat transfer equation is the most used model to calculate the temperature induced in a tumor when physical therapies like electrochemical treatment, electrochemotherapy and/or radiofrequency are applied. In this work, a modification of the Pennes bioheat equation to study the temperature distribution induced by any electrode array in an anisotropic tissue containing several nodules (primary or metastatic) with arbitrary shape is proposed. For this, the Green functions approach is generalized to include boundaries among two or more media. The analytical solution we obtain in a very compact way, under quite general suppositions, allows calculating the temperature distributions in the tumor volumes and their surfaces, in terms of heat sources, initial temperature and calorific sources at the boundary of tumors.

Simulations of surface charge density changes during the untreated solid tumour growth

“…Loss dielectric is a dielectric that has finite electrical conductivity and its induced electrical charges can move but not as freely as they would in a perfect conductor. If σ 12 = 0 at Σ (η 1 ε 2 −η 2 ε 1 = 0), η 1 = η 2 and ε 1 = ε 2 , in contrast with the experiment [10,13,14,[28][29][30]. Therefore, σ 12 at Σ must be considered in cancer and its TGK.…”

“…The first assumption may be approached from the physical point of view. As solid tumour and surrounding healthy tissue are anisotropic and heterogeneous media (formed by cells, water, ions, molecules, macromolecules, among others) [10,29,30], we consider that ηfalse↔ and εfalse↔ are real symmetric second-order tensors (3 × 3 matrix symmetric) of electrical conductivity and electrical permittivity, respectively. When the electrical conductivity and electrical permittivity are referred to the principal axes and both the electric field and the current density are related to the same coordinate system, then all nondiagonal elements are equal to zero and this 3 × 3 symmetric matrix becomes diagonal.…”

“…9: 220552 conductivities according to these main axes. If these diagonal elements are replaced by their mean value, named η (η = (η 1 + η 2 + η 3 )/3) in this approximation, the tensor h $ corresponds to the scalar matrix ηI, where I is the identity matrix of order 3, as in [30]. The tensor 1 $ is treated in the same manner and its mean value is ε.…”

“…where h $ is the symmetric second-order tensor of the electrical conductivity of any linear, anisotropic and non-homogeneous medium (for example, a biological tissue). This tensor is used in previous studies [29,30]. Equation (2.1) is obtained by combining the continuity equation (r †J þ @r=@t ¼ 0) for the static case (@r=@t ¼ 0) and law of Ohm…”

“…Secondly, the cancer and its surrounding healthy tissue are in contact and heterogeneous [28]. Thirdly, both tissue types differ significantly in their electrical properties and thermal [10,13,14,29,30] and physiological parameters [8,14]. Fourthly, σ 12 is due to synergism between an external volumetric current density (the source of electricity) and the Maxwell-Wagner-Sillars interfacial polarization.…”