Stability and reconstruction of a special type of anisotropic conductivity in magneto-acoustic tomography with magnetic induction

Abstract

We consider the issues of stability and reconstruction of the electrical anisotropic conductivity of biological tissues in a domain $\Omega \subset {\mathbb{R}}^3$ by means of the hybrid inverse problem of magneto-acoustic tomography with magnetic induction (MAT-MI). The class of anisotropic conductivities considered here is of type $\sigma (\cdot )=A(\cdot ,\gamma (\cdot ))$ in $\Omega$, where $[{\lambda }^{-1},\lambda ]\ni t\mapsto A(\cdot ,t)$ is a one-parameter family of matrix-valued functions which are \textit{a-priori} known to be $C^{1,\beta }$, allowing us to stably reconstruct $\gamma$ in $\Omega$ in terms of an internal functional $F(\sigma )$. Our results also extend previous results in MAT-MI where $\sigma (\cdot )=\gamma (\cdot )D(\cdot )$, with $D$ an \textit{a-priori} known matrix-valued function on $\Omega$ to a more general anisotropic structure which depends non-linearly on the scalar function $\gamma$ to be reconstructed.