Modeling the magnetization dynamics for large ensembles of immobilized magnetic nanoparticles in multi-dimensional magnetic particle imaging

Magnetic nanoparticles (MNPs) play an important role in biomedical applications including imaging modalities such as magnetic resonance imaging (MRI) and magnetic particle imaging (MPI). The latter one exploits the non-linear magnetization response of a large ensemble of magnetic nanoparticles to magnetic fields which allows determining the spatial distribution of the MNP concentration from measured voltage signals. The image-to-voltage mapping is linear and described by a system matrix. Currently, modeling the voltage signals of large ensembles of MNPs in an MPI environment is not yet accurately possible, especially for liquid tracers in multi-dimensional magnetic excitation fields. As an immediate consequence, the system matrix is still obtained in a time consuming calibration procedure. While the ferrofluidic case can be seen as the typical setting, more recently immobilized and potentially oriented MNPs have received considerable attention. By aligning the particles magnetic easy axis during immobilization one can encode the angle of the particle's magnetic easy axis into the magnetization response providing a relevant benchmark system for model-based approaches. In this work we address the modeling problem for immobilized and oriented MNPs in the context of MPI. We investigate a model-based approach where the magnetization response is simulated by a Neel rotation model for the particle's magnetic moments and the ensemble magnetization is obtained by solving a Fokker-Planck equation approach. Since the parameters of the model are a-priori unknown, we investigate different methods for performing a parameter identification and discuss two different models: One where a single function vector is used from the space spanned by the model parameters and another where a superposition of function vectors is considered. We show that our model can much more accurately reproduce the orientation dependent signal response when compared to the equilibrium model, which marks the current state-of-the-art for model-based system matrix simulations in MPI.