HOMOGENIZATION LIMIT AND ASYMPTOTIC DECAY FOR ELECTRICAL CONDUCTION IN BIOLOGICAL TISSUES IN THE HIGH

We derive a macroscopic model of electrical conduction in biological 
tissues in the high radio-frequency range, which is relevant in applications 
like electric impedance tomography. 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 solution for large times. Our results imply that 
periodic boundary data lead to an asymptotically periodic solution.


Introduction
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]. For example, diagnosis of pulmonary emboli, monitoring of heart function and blood flow, and breast cancer detection can benefit from a measurement of the dielectric properties of the living tissue. Indeed, Electric Impedance Tomography (EIT) is the inverse problem of determining the impedance in the interior of a body, given simultaneous measurements of direct or alternating electric currents and voltages at the boundary [11]. Clearly, an effective numerical reconstruction must be based on a reliable mathematical model of electric conduction.
In practice quite different frequency ranges of alternating currents are employed, calling for different modelling set-ups. 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. For higher frequencies, also 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 . 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)).
The geometrical and functional complexity of the problem at the microscopic cellular scale, as opposed to the macroscopical scale of clinical measurements, suggests to perform a homogenization limit, by letting the characteristic cellular length ε go to zero.
The homogenization of the Maxwell equations has been treated extensively in [24], where however no interfaces, and therefore no jumps, are allowed, and only homogeneous initial data are considered. This motivates a rigorous mathematical investigation of this problem in the framework of the homogenization with active interfaces ( [21,22,2,4,5,13]).
This study is based on the method due to Tartar [27] of the "oscillating test functions", which in this case must be determined in a peculiar way, due to the presence of both the time derivative of the gradient in  and the dynamical interface condition (2)(3)(4).
We prove that the homogenization of (2-2)-(2-7) leads to an equation with memory, similar to the one derived in [4], with the relevant difference that now the time derivative of the gradient (see ) appears under the divergence operator.
In view of the applications, it is also of interest to study the evolution in time of the homogenized potential. From a mathematical point of view, the asymptotic behavior of evolutive equations with memory is a classical problem [15,26,14,20], currently drawing much interest in the literature, e.g. [16,19,17,23,8].
In [7] the exponential decay of the homogenized potential with homogeneus Dirichlet boundary data is proved, however the most interesting case in applications involves periodic boundary data. Indeed, 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.
In this paper we prove that this assumption is essentially correct, since time-periodic, not necessarily time-harmonic, boundary data elicit a time-periodic solution for large times, also in the high radio-frequency range. The time derivative of the gradient appearing in the present homogenized equation requires new estimates in order to extend to the present case the argument in [6], where the low radio-frequency range was investigated.
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 finally in Section 5 we establish Theorem 2.3, i.e. the asymptotic behaviour of the solution.

Position of the problem and main results
Let Ω be an open bounded subset of R N . Following [4], [6], [7], we introduce a periodic open subset E of R N , so that E + z = E for all z ∈ Z N . We assume that Ω, E have regular boundary, say of class C ∞ for the sake of simplicity. We also employ the notation Y = (0, 1) N , and We stipulate that E 1 is a connected smooth subset of Y such that dist(E 1 , ∂Y ) > 0. Some generalizations may be possible, but we do not dwell on this point here. We introduce the set: Here E 1 is the light gray region and Γ is its boundary. The remaining part of Here Ω ε 1 is the light gray region and Γ ε is its boundary. The remaining part of Ω (the white region) is Ω ε 2 .
For all ε > 0 we define Clearly, dist(Γ ε , ∂Ω) > γε for some constant γ > 0 independent of ε since, by the choice of Z N ε , we dropped all the inclusions contained in the cells ε(Y + z), z ∈ Z N which intersect ∂Ω. The typical geometry we have in mind is depicted in Figure 1.
Since u ε is not in general continuous across Γ ε we have set A similar convention is employed for the current flux density across the membrane (κ∇u εt + σ∇u ε ) · ν. We assume that the initial data s ε and g ε satisfy: is continuous in x, uniformly over y ∈ Y , and periodic in y, for each x ∈ Ω; (H4) s 1 L ∞ (Ω×Γ ) < ∞, s 1 (x, y) is continuous in x, uniformly over y ∈ Γ , and periodic in y, for each x ∈ Ω; where g 0 : Ω × E i → R, i = 1, 2, and s 1 : Ω × Γ → R are the leading order terms in the two scale expansion of g ε and s ε (see  and (4)(5)). Our main result concerning the homogenization of problem (2-2)-(2-7) is stated in the following theorem.
The limit function u 0 introduced above satisfies the following exponential time-decay [7].
Remark 4.2. We note that in the definition of the function u 1 (see ), the cell function χ 0 : Y → R N is standard. In addition to this function, a new cell function χ 1 : Y → R N is required, owing to the dynamical terms in equations (4-14)- (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16). The definition of such a function involves a transform T , which plays an essential role. From the point of view of physics, the transform T associates to the initial data the evolution of the potential itself, in the process determining the discharge of the membrane in the unit cell Y under periodic boundary conditions. Memory effects appear in the homogenized equation (see  and (4-52)) just as a consequence of the transform T .
4B. The structure of the limit equation.
Proof. Existence and regularity of χ 0 , χ 1 (·, 0) follow by standard application of Lax-Milgram Lemma, whereas existence and regularity of v can be obtained reasoning as in the proof of Theorem 3.2.
Indeed the Gronwall argument of Lemma 7.2 of [4] carries over to the present case, which however deals with a second order equation, and hence needs the L 2 -estimate on ∇u 0 (·, 0) implied by  and . Note that (4-92) and (4-93) are independent of the sought after regularity for t positive. Thus, a.e. (x, t). Clearly div ξ 0 = 0 in the sense of distributions (see e.g., (4-86) above). This shows that (4-47) is in force.
Finally, from an inspection of the differential equation in , one might infer that from the mathematical point of view the most natural initial data to be assigned is div(κ∇u ε ) in the sense of distributions. However, by the remarks above, this amounts again to prescribing z ε .
The case k = 0 is dealt with in §5A, where the subscript k is dropped throughout for the sake of simplicity, and an alternative expression for A ω k is given (equation (5-41)). The case k = 0 is dealt with in §5B.
In particular, as a consequence, it follows that the problem − div(A ω ∇v) = 0 , in Ω; is uniformly elliptic with respect to k and admits a unique solution v ∈ H 1 (Ω). Moreover, the function v 0 = lim ε→0 v ε , which was proved to satisfy the problem above, coincides with v. Hence, v 0 ∈ H 1 (Ω) and the following estimate holds: for a constant γ independent of k. We note that the uniqueness of v 0 also implies that actually the whole sequence {v ε } converges to v 0 .
5B. Homogenization limit of time-harmonic solutions: the case k = 0. Here we prove Theorem 5.3 in the case k = 0. Let us distinguish the cases: β = 0 and β = 0.
5C. Time-periodic solutions. In this Section we prove Theorem 5.4.