  Magnetic particle imaging (MPI) is a relatively new imaging modality. The
nonlinear magnetization behavior of nanoparticles in an applied magnetic field
is employed to reconstruct an image of the concentration of nanoparticles.
Finding a sufficiently accurate model for the particle behavior is still an
open problem. For this reason the reconstruction is still computed using a
measured forward operator which is obtained in a time-consuming calibration
process. The state of the art model used for the imaging methodology and first
model-based reconstructions relies on strong model simplifications which turned
out to cause too large modeling errors. Neglecting particle-particle
interactions, the forward operator can be expressed by a Fredholm integral
operator of the first kind describing the inverse problem. In this article we
give an overview of relevant mathematical models which have not been
investigated theoretically in the context of inverse problems yet. We consider
deterministic models which are based on the physical behavior including
relaxation mechanisms affecting the particle magnetization. The behavior of the
models is illustrated with numerical simulations for monodisperse as well as
polydisperse tracer. We further motivate linear and nonlinear problems beyond
the solely concentration reconstruction related to applications. This model
survey complements a recent topical review on MPI [30] and builds the basis for
upcoming theoretical as well as empirical investigations.
