Abstract
We derive a macroscopic model of electrical conduction in biologicaltissues in the high radio-frequency range, which is relevant in applicationslike electric impedance tomography. This model is derived via a homogenizationlimit by a microscopic formulation, based on Maxwell’s equations, takinginto account the periodic geometry of the microstructure. We also study theasymptotic behavior of the solution for large times. Our results imply thatperiodic boundary data lead to an asymptotically periodic solution.
Highlights
Recent developments in diagnostic techniques are drawing attention to the problem of modeling the response of biological tissues to the injection of electrical current [9]
Most of the models available in the literature rely on a quasi-static assumption, implying that the variation in time of the magnetic field may be neglected [12], so that the electric field is given by the gradient of an electric potential
Even under this general assumption, different equations for the potential are derived in different frequency ranges: for frequencies up to 1 MHz the behavior of the intra and extra cellular phases is of Ohmic type, i.e., the current density is proportional to the gradient of the electric potential
Summary
Recent developments in diagnostic techniques are drawing attention to the problem of modeling the response of biological tissues to the injection of electrical current [9]. The electric displacement current, which is proportional to the time derivative of the gradient, must be taken into account This is the case we deal with here; namely we consider the equation for the electric potential given by (2-2). A peculiar feature of biological tissues is that the intra and extra-cellular phases are separated by an interface, that is the cell membrane, displaying a capacitive behavior This leads to a dynamical jump condition for the electric potential across the interface [2, 4] (see equations (2-3)–(2-4)). Experimental measurements are currently performed by assigning time-harmonic boundary data and assuming that the resulting electric potential is time-harmonic, too This assumption, which is often referred to as the limiting amplitude principle, leads to the commonly accepted mathematical model based on the complex elliptic problem (5-22)–(5-23) for the electric potential. The paper is organized as follows: in Section 2 we present the problem and state our main results (Theorems 2.1 and 2.3); in Section 3 we prove some preliminary results of existence and compactness; in Section 4 we prove Theorem 2.1, i.e. the homogenization result, and in Section 5 we establish Theorem 2.3, i.e. the asymptotic behaviour of the solution
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