Multi-term time-fractional Bloch equations and application in magnetic resonance imaging

Magnetic resonance imaging can reveal exquisite details about the complex structure and function of human tissue. However, magnetic resonance imaging signal behaviour at high or ultra-high field has shown increased deviation from the classically expected mono-exponential relaxation. The underlying mechanism of anomalous relaxation can contribute to a better understanding of the interaction of spins with their surroundings. The purpose of this work is to explore the utility of the multi-term time-fractional Bloch equations in relation to the anomalous relaxation processes. We proposed an effective predictor–corrector method to solve the multi-term equations. Voxel-level temporal fitting of the magnetic resonance imaging signal was performed based on the model developed from the multi-term time-fractional Bloch equations. A feasible parameter estimation method based on hybrid Nelder–Mead simplex search and particle swarm optimisation was introduced to perform the curve fitting. The extra time-fractional terms provide flexibility in the relaxation process.