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Dynamic Inverse Problems in Magnetic Particle Imaging (D-MPI)

Subject Area Mathematics
Term from 2019 to 2023
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 426078691
 
Final Report Year 2024

Final Report Abstract

Magnetic particle imaging (MPI) is an imaging modality with promising potential medical applications. In MPI, the spatial distribution of magnetic nanoparticle tracers is determined from voltage signals induced by their nonlinear magnetization behavior in a dynamic applied magnetic field. This technique offers alternative and complementary applications to established imaging systems such as CT, PET or MRI. In particular, its high temporal resolution, high sensitivity and high spatial resolution make MPI suitable for several in-vivo applications such as imaging blood flow, long-term monitoring by utilizing a circulating tracer, flow estimation, tracking and guiding medical instruments, cancer detection, and cancer treatment by hyperthermia. MPI is still mainly in the preclinical research phase, i.e., its potential is established on scanners built for the size of small animals. To simplify modeling, data acquisition and reconstruction, several dynamic aspects, for instance particle dynamics, have so far been left out of consideration for the purpose of image reconstruction. However, these aspects are crucial to derive reliable mathematical models and to provide reconstructions of sufficiently good quality. This project intended to explore dynamic aspects in MPI associated to modeling the system function (Part A), reconstructing time-dependent particle concentrations (Part B) and enlarging the FOV (Part C). Part A has been devoted to particle magnetization models, their numerical realization and parameter identification therein. From a general perspective, it is split into two main topics, namely mathematical modeling/simulation and parameter identification. By using more sophisticated magnetization models such as Néel dynamics with particle anisotropy also in combination with a fluid approximation we made an important step towards a solution to the model-based reconstruction problem in MPI. We obtained further improvements by addressing suitable parameter identification problems within these models. Here, we also analyzed the resulting parameter identification problems in the underlying Fokker-Planck equation theoretically. Despite the fast data acquisition, the time-dependency of the concentration during the acquisition of the voltage measurements has to be properly accounted for in order to avoid motion artifacts. In Part B, we respectively adapted the MPI forward model while accounting for the inherently fast measurement times of MPI, which allows to assume that the searched-for concentration is piecewise constant in time, as well as exploiting correlations between individual states of the concentration. We then solved the dynamic image reconstruction task by interpreting the unknown object motion as inexactnesses in the forward model. In particular, this approach does not require extensive or unrealistic a priori information, only rough estimates on the overall modelling error which, in the context of MPI, can be extracted directly from the measured voltage data. An efficient but computationally expensive alternative to compensate for the dynamic behavior is to incorporate explicit deformation models as motion priors. To obtain such motion information, we made use of image registration techniques as well as the optical flow equation as constraint. Such strong priors are particularly essential in the context of long measurement periods arising when enlarging the field of view which was studied in Part C of the project. Using an explicit motion model allows to include information on regions which would otherwise be neglected by the current measurement and, hence, can improve the reconstruction quality. Another challenge in this context represents the long simulation time for model-based system functions. Overcoming this drawback was achieved by a deep learning approach through the use of Fourier Neural Operators (FNOs). First results show that we can simulate system functions for given parameters with this strategy in the time of less than five seconds for a 2D matrix, where a conventional simulation would take upwards of 20 minutes. This demonstrates that model-based methods are indeed a viable option for MPI image reconstruction. Overall, we succeeded within this project to develop solution methods which account for the aforementioned dynamic aspects, enabling a faster set-up of the system function and significantly improving the reconstruction quality for time-dependent concentrations. This represents an important step towards real in-vivo applications of MPI and, in the long run, to the next phase of this promising imaging modality.

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