Stability and memory effects in a homogenized model governing the electrical conduction in biological tissues
Micòl Amar, Daniele Andreucci, Paolo Bisegna, Roberto Gianni · Journal of mechanics of materials and structures · 2009
We present a macroscopic model of electrical conduction in biological tissues.This model is derived via a homogenization limit by a microscopic formulation based on Maxwell's equations, taking into account the periodic geometry of the microstructure.We also study the asymptotic behavior of the model for large times.Our results imply that periodic boundary data lead to an asymptotically periodic solution.The model is relevant to applications like electric impedance tomography.Á 1 which models the cell cytosol, coated by a shell Á which models the cell membrane, included in a phase E Á 2 which models the extracellular fluid [Foster and Schwan 1989].In particular, the permittivity Ä in E Á 1 and E Á 2 is lower, and the conductivity is higher, than in Á .The diameter of the cell is of the order of tens of micrometers, while the width of the membrane is of the order of ten nanometers.This suggests that the thin shell Á could be preferably modeled as a two dimensional interface , in order to get a simpler model and, possibly, a better understanding of the effect of the geometric features of the microscopic structure.This simpler model can be obtained from Equation (1-1) via a concentration-ofcapacity procedure [Amar et al. 2006], leading to Problem (2-1)-(2-6), below.In particular, Equation (2-3) takes into account the conductive/capacitive behavior of the concentrated membrane.As shown in (2-3), the electric potential jumps across the interface , and its jump satisfies a dynamical condition (roughly speaking, in the form of a hyperbolic differential equation on the interface itself).Our model is designed to investigate the response of biological tissues to the injection of electrical currents in the radio frequency range, that is, the Maxwell-Wagner interfacial polarization effect [Foster and Schwan 1989;Bisegna et al. 2001], at higher frequencies than those considered in [Amar et al. 2003; 2004b; 2005; 2006;2008].This effect is relevant to clinical applications like electric impedance tomography and body composition [De Lorenzo et al. 1997; Bronzino 1999].