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.