Comptes Rendus
no. 7
Homogenization limit for electrical conduction in biological tissues in the radio-frequency range
[Limite d'homogénéisation pour la conduction électrique dans les tissus biologiques dans le domaine des radiofréquences]

Présenté par : Évariste Sanchez-Palencia

Micol Amar1 ; Daniele Andreucci1 ; Paolo Bisegna2 ; Roberto Gianni1
1 Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, via A. Scarpa 16, 00161 Roma, Italy
2 Dipartimento di Ingegneria Civile, Università di Roma “Tor Vergata”, via del Politecnico 1, 00133 Roma, Italy
Comptes Rendus. Mécanique, Volume 331 (2003) no. 7, pp. 503-508.

On étudie un modèle d'évolution pour la conduction électrique dans les tissus biologiques, où les espaces conducteurs intracellulaires et extracellulaires sont séparés par des membranes cellulaires isolantes. Le schéma mathématique est celui d'un problème elliptique avec des conditions aux limites dynamiques sur les membranes des cellules. Le problème est formulé dans un milieu périodique finement mixte. On démontre que la limite d'homogénéisation u0 du potentiel électrique, obtenue pour une période de la structure microscopique approchant de zéro, est solution de l'équation -div(σ0∇xu0+A0∇xu0+∫0tA1(t-τ)∇xu0(x,τ)dτ-ℱ(x,t))=0 où σ0>0 et les matrices A0, A1 dépendent de la géométrie et des propriétés du matériau, tandis que la fonction vectorielle ℱ gardes une trace des données initiales du problème originaire. Des effets de mémoire se montrent ici explicitement, rendant cette équation elliptique de type non classique.

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 u0 of the electric potential, obtained as the period of the microscopic structure approaches zero, solves the equation -div(σ0∇xu0+A0∇xu0+∫0tA1(t-τ)∇xu0(x,τ)dτ-ℱ(x,t))=0 where σ0>0 and the matrices A0, A1 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-0721(03)00107-4
Keywords: Continuum mechanics, Electrical conduction, Homogenization, Biomathematics
Mots-clés : Mécanique des milieux continus, Conduction électrique, Homogénéisation, Biomathématique

Micol Amar 1 ; Daniele Andreucci 1 ; Paolo Bisegna 2 ; Roberto Gianni 1

1 Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, via A. Scarpa 16, 00161 Roma, Italy
2 Dipartimento di Ingegneria Civile, Università di Roma “Tor Vergata”, via del Politecnico 1, 00133 Roma, Italy 
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Micol Amar; Daniele Andreucci; Paolo Bisegna; Roberto Gianni. Homogenization limit for electrical conduction in biological tissues in the radio-frequency range. Comptes Rendus. Mécanique, Volume 331 (2003) no. 7, pp. 503-508. doi : 10.1016/S1631-0721(03)00107-4. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(03)00107-4/

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