Evolution and Memory Effects in the Homogenization Limit for Electrical Conduction in Biological Tissues

Abstract

We study a problem set in a finely mixed periodic medium, modelling electrical conduction in biological tissues. The unknown electric potential solves standard elliptic equations set in different conductive regions (the intracellular and extracellular spaces), separated by a dielectric surface (the cell membranes), which exhibits both a capacitive and a nonlinear conductive behaviour. Accordingly, dynamical conditions prevail on the membranes, so that the dependence of the solution on the time variable t is not only of parametric character. As the spatial period of the medium goes to zero, the electric potential approaches in a suitable sense a homogenization limit u0, which keeps the prescribed boundary data, and solves the equation [Formula: see text]. This is an elliptic equation containing a term depending on the history of the gradient of u0; the matrices B0, A1 in it depend on the microstructure of the medium. More exactly, we have that, in the limit, the current is still divergence-free, but it depends on the history of the potential gradient, so that memory effects explicitly appear. The limiting equation also contains a term ℱ keeping trace of the initial data.

Stability and memory effects in a homogenized model governing the electrical conduction in biological tissues

“…Finally, integrating Equation (3-39) over time, changing the order in the double integral that results, and using (3-44), we obtain also the following formulation which shows that the homogenized equation has exactly the form of an equation with memory of the type derived in [Amar et al 2003;2004b] and studied in [Amar et al 2004a]. …”

“…Our model is designed to investigate the response of biological tissues to the injection of electrical currents in the radio frequency range, that is, the Maxwell-Wagner interfacial polarization effect [Foster and Schwan 1989;Bisegna et al 2001], at higher frequencies than those considered in [Amar et al 2003;2004b;2008]. This effect is relevant to clinical applications like electric impedance tomography and body composition [De Lorenzo et al 1997;Bronzino 1999].…”

“…Lemma 4.1 (Poincaré's inequality [Hummel 2000;Amar et al 2004b]). Let v belong to the space H 1 " .˝/ introduced in Equation (4-1).…”

Multiscale analysis in nonlinear thermal diffusion problems in composite structures

“…Using our assumptions on the data and the CauchySchwartz, Poincaré's, Young's and Gronwall's inequalities, we can obtain suitable energy estimates, independent of ε, for our solution (see [2], [13], [9], [15] and [20]). More precisely, if we multiply the first equation in (1) by ε , the second one by ε and we integrate formally by parts, we obtain, for 0 < < T ,…”

“…Our results constitute a generalization of those obtained in [3] and [12], by considering nonlinear sources, nonlinear dynamical transmission conditions and different techniques in the proofs. Similar problems have been considered, using different techniques, in [2] and [13], for studying electrical conduction in biological tissues.…”