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

TL;DR

This paper investigates the stability and reconstruction of a specific class of anisotropic electrical conductivities in biological tissues using magneto-acoustic tomography with magnetic induction, extending previous models to more general nonlinear anisotropic structures.

Contribution

It introduces a new framework for stable reconstruction of anisotropic conductivities that depend non-linearly on a scalar function, broadening the applicability of MAT-MI techniques.

Findings

Stable reconstruction of anisotropic conductivity in 3D tissues.

Extension of previous models to nonlinear anisotropic structures.

Use of internal functionals for reconstruction.

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.