  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.
