## M. Amar - D. Andreucci - P. Bisegna - R. Gianni

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

created on 30 Apr 2003

modified on 25 Jul 2003

[

BibTeX]

*Published Paper*

**Inserted:** 30 apr 2003

**Last Updated:** 25 jul 2003

**Journal:** C.R. Mecanique

**Volume:** 331

**Pages:** 503-508

**Year:** 2003

**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
\begin{equation**}
-{\rm Div}\Big(\sigma**_{0\nabla}_{{x}} u_{{0}} + A^{{0}} \nabla_{{x}} u_{{0}} +
\int_{{0}}^{{t}} A^{{1}}(t-\tau) \nabla_{{x}} u_{{0}}(x,\tau) d\tau
-{\mathcal F}(x,t)\Big)=0 \,,
\end{equation}
where $\sigma_0>0$ and the matrices $A^{0}$, $A^{1}$ depend on geometric
and material properties, while the vector function ${\mathcal F}$ keeps trace
of the initial data of the original problem. Memory effects
explicitly appear here, making this elliptic equation of non
standard type.

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