Homogenization limit for electrical conduction in biological tissues in the radio-frequency range
Micòl Amar, Daniele Andreucci, Paolo Bisegna, Roberto Gianni · Comptes Rendus Mécanique · 2003
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 ∇ x u 0 + A 0 ∇ x u 0 + ∫ 0 t A 1 ( t - τ ) ∇ x u 0 ( x , τ ) d τ - 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.