In this thesis, a new mathematical model for the description of the magnetization dynamics during MRI measurements was developed. This model, labelled Phase Distribution Graphs (PDG), is based closely on the established Extended Phase Graphs (EPG). It is a full description of the spatial and off-resonance frequency distribution of isochromats. With PDG, fully spatially resolved simulations of imaging sequences under the consideration of relaxation, diffusion, magnetic field inhomogeneities and 𝑇′2 dephasing are possible. In contrast to EPG, this is possible for sequences with arbitrary timing, encoding, and spoiling patterns without adjustments to the simulation. No prior assumptions to the mechanisms of the simulated sequence are made and possible artifacts are accurately depicted. These properties were previously only fulfilled by isochromat simulations, which PDG aims to fully replace. Through its origins in phase graphs, PDG inherits their strengths: As an analytical simulation, PDG is typically multiple orders of magnitude faster than the numerical approximation done by isochromat simulations. In addition, the computed magnetizations and signals do not suffer from stochastic noise, as they are exact solutions to the Bloch equations under the few assumptions made by PDG. Furthermore, PDG provides valuable insights into the formation of spin and gradient echoes of MRI measurements. This is what phase graphs were originally developed for and what still remains in PDG, augmented to more dimensions of dephasing by an extended mathematical model compared to EPG. The mathematical model of PDG is only a small portion of this work. It is the implementation which makes the theory applicable to arbitrary MRI sequences. New metrics were developed to efficiently rank the importance of different magnetization pathways, to merge and exclude them from calculations in order to minimize the computational cost. The efficiency and accuracy of the resulting simulation was proven by comparisons with signal equations, measurements, and other proven MRI simulations. PDG analysis of different sequences showcase its usefulness in sequence development. The simulation is fully differentiable by backpropagation, enabling the use of efficient gradient-descent optimizations of sequences or the physical properties of the simulated tissue. To facilitate the use of PDG even further, a likewise differentiable integration with the widespread Pulseq library was written. This makes it possible to use a single source for simulating and optimizing MRI sequence, as well as exporting and measuring the result on a real scanner. During this thesis, the work was made available as part of the open-source MR-zero project. Its fast adoption did not only prove the practicability and advantages of the PDG simulation but also adds to the list of comparisons to prove its speed and accuracy.
Jonathan Endres (Thu,) studied this question.