Development, analysis and application of mathematical methods for Magnetic Particle Imaging (MathMPI)

Magnetic Particle Imaging (MPI) is an emerging imaging modality that determines the spatial distribution of magnetic nanoparticles by measuring the non-linear magnetization response of the particles to an applied magnetic field. MPI offers a high dynamic spatial and temporal resolution and, in contrast to other tomographic methods, it does not employ any ionizing radiation. This makes MPI a very promising imaging modality for biomedical diagnostics. This project addresses for the first time the systematic study of mathematical questions related to MPI. The central goal of this network is the development, analysis and application of mathematical methods to improve the reconstruction quality in MPI. In particular, taylored to the specific needs of MPI we develop elaborate reconstruction algorithms, analyze and refine the underlying MPI models and test the new methods numerically on real biomedical data. To achieve these goals this network provides an interdisciplinary platform for researchers from various scientific fields including applied mathematics, modeling, image processing, medical physics as well as electrical engineering. Two of the meetings organized in this network are planned in form of study groups. This format is particularly suitable for interdisciplinary teamwork, generating synergies and strengthen the communication within the network.

The central goal of the interdisciplinary DFG-funded young researcher network ”Mathematical methods for Magnetic Particle Imaging (MathMPI)” is the development, analysis and application of mathematical methods to improve the reconstruction quality of the novel imaging modality Magnetic Particle Imaging (MPI). To achieve these goals the scientific network MathMPI attracted researchers from various scientific backgrounds including applied mathematics, medical physics, image processing, modeling, as well as medical engineering. The network organized five interdisciplinary research meetings in the period between 1.8.2014 and 31.07.2016. In particular, a kick-off meeting in Lübeck, two study groups in Munich and Ettlingen, a workshop combined with a Minisymposium in Hamburg, and a final conference in Osnabrück were organized. During the period of this project and based on the meetings of the network a considerable progress in research has been achieved. Tailored to the specific needs of MPI a fast edge preserving and noise reducing reconstruction algorithm for MPI was developed in which the nonnegative fused lasso model is used for regularization. Concerning the analysis and the modeling of the MPI system kernel a structured decomposition of the MPI imaging equation was developed which led to new reconstruction formulae in 2D and 3D. Regarding the discretization of the MPI system function, sampling on the node points of Lissajous curves was studied which led to new reduced system matrices and reconstruction algorithms adapted to the sampling paths of MPI scanners. Although a unified software toolbox for MPI reconstruction could not be finalized, several new reconstruction algorithms were implemented in single subprojects of the network. The performance of these novel algorithms was tested numerically in experimental setups and for some of them an improvement in reconstruction quality over the state of the art methods could be verified. More tests of these methods, in particular on real biomedical data, have still to be conducted in order to improve the quality of the reconstructions even further.