Numerical determination of anomalies in multifrequency electrical impedance tomography

The multifrequency electrical impedance tomography consists in retrieving the conductivity distribution of a sample by injecting a finite number of currents with multiple frequencies. In this paper we consider the case where the conductivity distribution is piecewise constant, takes a constant value outside a single smooth anomaly, and a frequency dependent function inside the anomaly itself. Using an original spectral decomposition of the solution of the forward conductivity problem in terms of Poincaré variational eigenelements, we retrieve the Cauchy data corresponding to the extreme case of a perfect conductor, and the conductivity profile. We then reconstruct the anomaly from the Cauchy data. The numerical experiments are conducted using gradient descent optimization algorithms. 1. The mfEIT Mathematical Model Experimental research has found that the conductivity of many biological tissues varies strongly with respect to the frequency of the applied electric current within certain frequency ranges [GPG]. In [AGGJS], using homogenization techniques, the authors analytically exhibit the fundamental mechanisms underlying the fact that effective biological tissue electrical properties and their frequency dependence reflect the tissue composition and physiology. The multifrequency electrical impedance tomography (mfEIT) is a diffusive imaging modality that recovers the conductivity distribution of the tissue by using electrodes to measure the resulting voltage on its boundary, induced by two known injected currents and for many frequency values. The principal idea behind the (mfEIT) is that the dependance of the effective conductivity of the tissue with respect to the frequency of the electric current is extremely related to its state. In fact, its frequency dependence changes with its composition, membrane characteristics, intra-and extra-cellular fluids and other factors [AGGJS]. Therefore, the frequency dependence of the conductivity of the tissue can provide some information about the tissue microscopic structure and its physiological and pathological conditions. In other words, the frequency dependence of the conductivity of the tissue can help to determine if it is healthy or cancerous. The advantages of the (mfEIT) is canceling out errors due to boundary shape, the electrode positions, and other systematic errors that appear in -the more conventional imaging modalityelectric impedance tomography (EIT) [Bor]. In the following we introduce the mathematical model of the (mfEIT). Let Ω be the open bounded smooth domain in R2, occupied by the sample under investigation and denote by ∂Ω its boundary. The mfEIT forward problem is to determine the potential u(·, ω) ∈ H1(Ω) := {v ∈ L2(Ω) : ∇v ∈ Date: April 17, 2017. 1991 Mathematics Subject Classification. Primary: 35R30.