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 u(o), which keeps the prescribed boundary data, and solves the equation div [B(o)del(x)u(o) + integral(o)(t) A(1)(t - tau)del(x)u(o)(tau) dtau - F] = 0. This is an elliptic equation containing a term depending on the history of the gradient of uo; the matrices B-o, A(1) 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 T keeping trace of the initial data.

Amar, M., Andreucci, D., Gianni, R., Bisegna, P. (2004). Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues. MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES, 14(9), 1261-1295 [10.1142/S0218202504003623].

Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues

BISEGNA, PAOLO
2004-01-01

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 u(o), which keeps the prescribed boundary data, and solves the equation div [B(o)del(x)u(o) + integral(o)(t) A(1)(t - tau)del(x)u(o)(tau) dtau - F] = 0. This is an elliptic equation containing a term depending on the history of the gradient of uo; the matrices B-o, A(1) 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 T keeping trace of the initial data.
2004
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore ICAR/08 - SCIENZA DELLE COSTRUZIONI
Settore ING-IND/34 - BIOINGEGNERIA INDUSTRIALE
Settore MAT/05 - ANALISI MATEMATICA
English
Con Impact Factor ISI
Dynamical condition; electrical conduction in biological tissues; evolution equation with memory; homogenization
Amar, M., Andreucci, D., Gianni, R., Bisegna, P. (2004). Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues. MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES, 14(9), 1261-1295 [10.1142/S0218202504003623].
Amar, M; Andreucci, D; Gianni, R; Bisegna, P
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/23552
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