The Inverse Problem of Magnetorelaxometry Imaging

TL;DR

This paper provides a rigorous mathematical analysis of the inverse problem in Magnetorelaxometry Imaging, addressing ill-posedness, uniqueness, and proposing a regularization method with numerical validation for biomedical imaging applications.

Contribution

It introduces a detailed mathematical framework for the inverse problem in MRXI, including an idealized model, and proposes a variational regularization approach with numerical results.

Findings

Regularization yields stable solutions for synthetic data

Analysis of uniqueness issues in idealized models

Numerical improvements over previous methods

Abstract

The aim of this paper is to provide a solid mathematical discussion of the inverse problem in Magnetorelaxometry Imaging (MRXI), a currently developed technique for quantitative biomedical imaging using magnetic nanoparticles. We provide a detailed discussion of the mathematical modeling of the forward problems including possible ways to activate and measure, leading to a severely ill-posed linear inverse problem. Moreover, we formulate an idealized version of the inverse problem for infinitesimal small activation coils, which allows for a more detailed analysis of uniqueness issues. We propose a variational regularization approach to compute stable approximations of the solution and discuss its discretization and numerical solution. Results on synthetic are presented and improvements to methods used previously in practice are demonstrated. Finally we give an outlook to further questions and in particular experimental design.