The application of the distributed-order time fractional Bloch model to magnetic resonance imaging

It is now well known that the magnetic resonance imaging (MRI) signal decay deviates from the classical mono-exponential relaxation. This deviation is referred to in the literature as anomalous relaxation. The modelling of this anomalous relaxation can provide a better understanding of MRI magnetization. The purpose of this work is to investigate the utility of the distributed-order time fractional Bloch equations to describe anomalous relaxation processes in human brain MRI data. Two choices of continuous distribution weight functions, which are parameterised by their mean μ and standard deviation σ, are studied to investigate their impact on the model solution behaviour. An implicit numerical method implemented on a graded mesh is proposed to solve the model and the stability and convergence analysis are presented. We also derive semi-analytical solutions of the fully coupled Bloch equations using the Laplace transform technique to assess the accuracy of the numerical scheme. Furthermore, three different voxel models of continuous distribution weight functions, namely a single continuous probability distribution (model 1), two distinct continuous probability distributions (model 2) and a mixture of two continuous probability distributions (model 3), are applied to the in vivo human brain MRI data, and a feasible and reliable parameter estimation method based on a modified hybrid Nelder-Mead simplex search and particle swarm optimization is presented to perform the voxel-level temporal fitting of the MRI data. The application of these distributed-order time fractional Bloch models highlights the validity of the proposed models, and based on the mean square error we conclude that models 2 and 3 might be more suitable than model 1 to characterize anomalous relaxation processes in human brain MRI data.