Magnetic particle imaging (MPI) is a tracer-based imaging modality developed to detect the concentration of superparamagnetic iron oxide nanoparticles. It is highly sensitive to the nanoparticle's nonlinear response to an applied magnetic field and allows a rapid data acquisition. The imaging problem is a linear inverse problem given by a Fredholm integral equation of the first kind. The problem of modeling MPI, with respect to finding the correct integral kernel/system function in the forward operator, remains an unsolved problem. Existing model-based reconstruction techniques incorporate particle behavior based on the theory of paramagnetism which have not yet reached the necessary quality of measured system functions. As a result the state of the art reconstruction method still relies on a full calibration of the system function by moving a delta sample through the field-of-view, which is time-consuming, limits the spatial resolution, and does not generalize with respect 3 to particle and scanner parameters. This defines a second important problem in MPI, the calibration problem. One particular challenge of modeling the signal acquisition chain and thus the development of model-based
calibration methods is the magnetization dynamic of large ensembles of magnetic nanoparticles. The application MPI motivates the considerations of general and specific problems in various directions such
as (i) physical modeling and analysis of the imaging operator, (ii) sophisticated reconstruction methods
simultaneously determining parameters in the forward operator, (iii) learning-based methods for image
reconstruction, (iv) model-based calibration, and (v) time-dependent concentration reconstruction. After an introduction to the imaging modality, we mainly focus on selected problems in the directions (i), (ii), and (iii). More precisely, (i) includes a discussion of the degree of ill-posedness in the equilibrium model and the consideration of nanoparticles' magnetization dynamics to improve model-based reconstruction. In (ii) we investigate a variational approach for joint image reconstruction with improved resolution and parameter identification in the imaging operator. Finally, in (iii) we investigate general deep image prior concepts for inverse problems and their application to the imaging problem. The talk is concluded with an outlook on the directions (iv) and (v).