Time-Harmonic Maxwell Equations in Biological Cells — The Differential Form Formalism to Treat the Thin Layer

Abstract

We study the behavior of the electromagnetic field in a biological cell modelled by a medium surrounded by a thin layer and embedded in an ambient medium. We derive approximate transmission conditions in order to replace the membrane by these conditions on the boundary of the interior domain. Our approach is essentially geometric and based on a suitable change of variables in the thin layer. Few notions of differential calculus are given in order to obtain the first order conditions in a simple way, and numerical simulations validate the theoretical results. Asymptotic transmission conditions at any order are given in the last section of the paper. This paper extends to the time-harmonic Maxwell equations the previous works presented in [30, 33, 31, 6].

Thin Layer Models for Electromagnetism

“…(2) and (5)) : this is the model proposed in [7], where E 0 is computed, then E 1 is computed with a right-hand-side depending on E 0 . Once E 0 and E 1 computed, the approximated solution is given as E 0 + εE 1…”

“…Then, complex phenomenon known as electropermeabilization (or electroporation) may occur [21]: the cell membrane In order to avoid the meshing of the thin membrane, it is convenient to approximate the solution to problem (1), by replacing the thin layer by appropriate conditions across the surface Γ. The idea, as presented in [7], consists in rewriting the operator curl curl in the domain O ε m in local coordinates (x T , x 3 ) [1,7]. The variable x 3 ∈ (0, ε) is the Euclidean distance to Γ and x T denotes tangential coordinates on Γ.…”

“…Then performing the change of variable x 3 = εη, rewriting the operator in (x T , η)-coordinates and assuming that E ε can be developed as a formal expansion in power series of ε, we obtain the approximation of E ε at the desired order of accuracy. The rigorous derivation of the expansion is not in the scope of the present paper and we refer the reader to [7] for a detailed description of the calculation. Only the two first terms of the multiscale expansion are presented in this paper, namely:…”

“…In the mid-frequency range, the cell is a soft contrast material. In this section, we recall the first terms of the expansion given in [7]. 5 Two first orders of the asymptotic expansion.…”

“…We perform an asymptotic expansion as done in [7] by assuming that κ m is independent of ε, instead of assuming that κ m is independent of ε in [7]. Thus the derivation of the expansion becomes [6]:…”