TY - JOUR A1 - Erb, Wolfgang A1 - Weinmann, Andreas A1 - Ahlborg, Mandy A1 - Brandt, Christina A1 - Bringout, Gael A1 - Buzug, Thorsten M. A1 - Frikel, Jürgen A1 - Kaethner, C. A1 - Knopp, Tobias A1 - März, T. A1 - Moddel, Martin A1 - Storath, Martin A1 - Weber, A. T1 - Mathematical analysis of the 1D model and reconstruction schemes for magnetic particle imaging JF - Inverse Problems N2 - Magnetic particle imaging (MPI) is a promising new in vivo medical imaging modality in which distributions of super-paramagnetic nanoparticles are tracked based on their response in an applied magnetic field. In this paper we provide a mathematical analysis of the modeled MPI operator in the univariate situation. We provide a Hilbert space setup, in which the MPI operator is decomposed into simple building blocks and in which these building blocks are analyzed with respect to their mathematical properties. In turn, we obtain an analysis of the MPI forward operator and, in particular, of its ill-posedness properties. We further get that the singular values of the MPI core operator decrease exponentially. We complement our analytic results by some numerical studies which, in particular, suggest a rapid decay of the singular values of the MPI operator. KW - FORMULATION KW - ill-posedness KW - INTERPOLATION KW - inverse problems KW - magnetic particle imaging KW - model-based reconstruction KW - MPI KW - NODES KW - reconstruction operator KW - RELAXATION KW - RESOLUTION Y1 - 2018 U6 - https://doi.org/10.1088/1361-6420/aab8d1 VL - 34 IS - 5 PB - IOP Publishing ER - TY - JOUR A1 - Bringout, Gaël A1 - Erb, Wolfgang A1 - Frikel, Jürgen T1 - A new 3D model for Magnetic Particle Imaging using realistic magnetic field topologies for algebraic reconstruction JF - Inverse Problems N2 - We derive a new 3D model for magnetic particle imaging (MPI) that is able to incorporate realistic magnetic fields in the reconstruction process. In real MPI scanners, the generated magnetic fields have distortions that lead to deformed magnetic low-field volumes with the shapes of ellipsoids or bananas instead of ideal field-free points (FFP) or lines (FFL), respectively. Most of the common model-based reconstruction schemes in MPI use however the idealized assumption of an ideal FFP or FFL topology and, thus, generate artifacts in the reconstruction. Our model-based approach is able to deal with these distortions and can generally be applied to dynamic magnetic fields that are approximately parallel to their velocity field. We show how this new 3D model can be discretized and inverted algebraically in order to recover the magnetic particle concentration. To model and describe the magnetic fields, we use decompositions of the fields in spherical harmonics. We complement the description of the new model with several simulations and experiments, exploring the effects of magnetic fields distortion and reconstruction parameters on the reconstruction. Y1 - 2020 U6 - https://doi.org/10.1088/1361-6420/abb446 VL - 36 IS - 12 PB - IOP Publishing ER -