Debye Relaxation in Model-Based Multi-Dimensional Magnetic Particle Imaging

Abstract: Model-based reconstruction approaches for the medical imaging modality Magnetic Particle Imaging (MPI) are typically based on the Langevin model, which assumes instantaneous alignment of the particles magnetic momenta with the applied field. Regarding the application to real data, Langevin model-based reconstruction methods require model transfer functions (MTF) obtained from calibrations to preprocess the data. There are also model-based reconstruction approaches that include relaxation effects and other particle-level dynamics. However, they are limited either to 1D or 1D-like scanning scenarios when considering real data, or are limited to simulated data in the case of multi-dimensional field-free point (FFP) MPI. Thus, fully model-based reconstructions from multi-dimensional FFP scanning data that incorporate relaxation effects without using an MTF have not yet been demonstrated. In this work, we incorporate relaxation effects by considering a multi-dimensional Debye model and provide reconstruction formulae. In particular, we show that the Debye model-based signal is the response of a linear time-invariant system with exponential memory applied to a Langevin model-based signal. We provide a reconstruction algorithm for the introduced multi-dimensional Debye model. To this end, we devise a relaxation adaption step. For the resulting relaxation-adapted Debye signal, we show that it can be expressed by the well-studied MPI core operator derived from the Langevin theory. This results in a three-stage algorithm with low additional cost over the Langevin model, as the relaxation adaption scales linearly in the input data. We provide numerical results for the proposed algorithmic approach. In particular, we obtain fully model-based reconstructions from real 2D MPI data without involving any specific MTF analogous to the Langevin model case.