TIME-HARMONIC MAXWELL EQUATIONS IN BIOLOGICAL CELLS — THE DIFFERENTIAL FORM FORMALISM TO TREAT THE THIN LAYER

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.

[1]  Ralf Hiptmair,et al.  Discrete Compactness for the p-Version of Discrete Differential Forms , 2009, SIAM J. Numer. Anal..

[2]  Tomaz Slivnik,et al.  Sequential finite element model of tissue electropermeabilization , 2005, IEEE Transactions on Biomedical Engineering.

[3]  Erwan Faou,et al.  Elasticity on a thin shell: Formal series solution , 2002 .

[4]  Serge Nicaise,et al.  Corner Singularities of Maxwell Interface and Eddy Current Problems , 2004 .

[5]  C. Poignard.  About the transmembrane voltage potential of a biological cell in time-harmonic regime. , 2009 .

[6]  Karl F. Warnick,et al.  Teaching Electromagnetic Field Theory Using Differential Forms , 1997, Teaching Electromagnetics.

[7]  M. Vogelius,et al.  Gradient Estimates for Solutions to Divergence Form Elliptic Equations with Discontinuous Coefficients , 2000 .

[8]  C. Poignard.  Rigorous Asymptotics For The Elecric Field In TM Mode At Mid-Frequency In A Bidimensional Medium With Thin Layer , 2006, math/0607403.

[9]  Serge Nicaise,et al.  Singularities of eddy current problems , 2003 .

[10]  H. Flanders.  Differential Forms with Applications to the Physical Sciences , 1964 .

[11]  Matti Lassas,et al.  Maxwell's equations with a polarization independent wave velocity: direct and inverse problems , 2006 .

[12]  G. Pucihar,et al.  Numerical Determination of Transmembrane Voltage Induced on Irregularly Shaped Cells , 2006, Annals of Biomedical Engineering.

[13]  S. Agmon,et al.  Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I , 1959 .

[14]  Sergei Petrovich Novikov,et al.  The geometry of surfaces, transformation groups, and fields , 1984 .

[15]  K. Foster,et al.  Dielectric properties of tissues and biological materials: a critical review. , 1989, Critical reviews in biomedical engineering.

[16]  V. Rovenski,et al.  Differential Geometry of Curves and Surfaces , 1952, Nature.

[17]  C. Balanis.  Advanced Engineering Electromagnetics , 1989 .

[18]  Karl F. Warnick,et al.  Electromagnetic Green functions using differential forms , 1996 .

[19]  Victor Péron,et al.  Modélisation mathématique de phénomènes électromagnétiques dans des matériaux à fort contraste , 2009 .

[20]  C. Poignard.  Asymptotics for steady‐state voltage potentials in a bidimensional highly contrasted medium with thin layer , 2008 .

[21]  J. Lions,et al.  Problèmes aux limites non homogènes (VI) , 1963 .

[22]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[23]  G. Schwarz.  Hodge Decomposition - A Method for Solving Boundary Value Problems , 1995 .

[24]  Luc Paquet,et al.  Problèmes mixtes pour le système de Maxwell , 1982 .

[25]  Victor Péron,et al.  On the influence of the geometry on skin effect in electromagnetism , 2010, ArXiv.

[26]  E.C. Fear,et al.  Modelling assemblies of biological cells exposed to electric fields , 1997, CCECE '97. Canadian Conference on Electrical and Computer Engineering. Engineering Innovation: Voyage of Discovery. Conference Proceedings.

[27]  Clair Poignard.  APPROXIMATE TRANSMISSION CONDITIONS THROUGH A WEAKLY OSCILLATING THIN LAYER , 2009 .

[28]  T. Tsong,et al.  Electroporation of cell membranes. , 1991, Biophysical journal.

[29]  P. Dular,et al.  Approximate Conditions Replacing Thin Layers , 2008, IEEE Transactions on Magnetics.

[30]  C. Poignard,et al.  Approximate transmission conditions through a rough thin layer: The case of periodic roughness , 2009, European Journal of Applied Mathematics.

[31]  W. Marsden.  I and J , 2012 .

[32]  J. Lions,et al.  Problèmes aux limites non homogènes et applications , 1968 .

[33]  Elise C. Fear,et al.  Modeling assemblies of biological cells exposed to electric fields , 1998 .

[34]  Monique Dauge,et al.  EIGENMODE ASYMPTOTICS IN THIN ELASTIC PLATES , 1999 .

[35]  J L Sebastián,et al.  Analysis of the influence of the cell geometry, orientation and cell proximity effects on the electric field distribution from direct RF exposure. , 2001, Physics in medicine and biology.

[36]  Monique Dauge,et al.  Comportement asymptotique à haute conductivité de l'épaisseur de peau en électromagnétisme , 2010 .

[37]  Erwan Faou,et al.  Multiscale Expansions for Linear Clamped Elliptic Shells , 2005 .

[38]  D Miklavcic,et al.  A validated model of in vivo electric field distribution in tissues for electrochemotherapy and for DNA electrotransfer for gene therapy. , 2000, Biochimica et biophysica acta.