Phase distribution graphs for fast, differentiable, and spatially encoded Bloch simulations of arbitrary MRI sequences

Abstract

Purpose: An analytical approach to Bloch simulations for MRI sequences is introduced that enables time efficient calculations of signals free of Monte-Carlo noise, while providing full flexibility and differentiability in RF flip angles, RF phases, magnetic field gradients and time, as well as insights into image formation. Theory and Methods: We present an implementation of the extended phase graph (EPG) concept implemented in PyTorch, including an efficient state selection algorithm. This simulation is compared with an isochromat-based Bloch simulation with random isochromat distribution as well as with in vivo measurements using the Pulseq standard. Additionally, different sequences are tested and analyzed using this novel simulation approach. Results: Our simulation outperforms isochromat-based simulations in terms of computation time as well as signal quality, without exhibiting any kind of Monte-Carlo noise. The novel approach allows extracting useful information about the magnetization evolution not available otherwise. Our approach extends the common state-tensor-based EPG simulation approach for the contribution of dephased states including spatial encoding and (Formula presented.) effects, and arbitrary timing. This allows calculation of echo shapes in addition to echo amplitudes only. Our implementation provides full differentiability in all input parameters allowing gradient descent optimization. Simulation of non-instantaneous pulses via hard-pulse approximation is left for future work, as the performance and accuracy characteristics are not yet analyzed. Conclusions: Phase distribution graphs provide fast, differentiable, and spatially encoded Bloch simulations for most MRI sequences. It allows efficient simulation and optimization of arbitrary MRI sequences, which was previously only possible via high isochromat counts.