Homogenization limit for electrical conduction in biological tissues in the radio-frequency range

Abstract

We study an evolutive model for electrical conduction in biological tissues, where the conductive intra-cellular and extracellular spaces are separated by insulating cell membranes. The mathematical scheme is an elliptic problem, with dynamical boundary conditions on the cell membranes. The problem is set in a finely mixed periodic medium. We show that the homogenization limit u 0 of the electric potential, obtained as the period of the microscopic structure approaches zero, solves the equation − div(σ 0∇ xu 0+A 0∇ xu 0+∫ 0 tA 1(t−τ)∇ xu 0(x,τ) dτ− F(x,t))=0 where σ 0>0 and the matrices A 0, A 1 depend on geometric and material properties, while the vector function F keeps trace of the initial data of the original problem. Memory effects explicitly appear here, making this elliptic equation of non standard type. To cite this article: M. Amar et al., C. R. Mecanique 331 (2003).