A spatially second-order accurate implicit numerical method for the space and time fractional Bloch-Torrey equation

In recent years, it has been found that many phenomena in engineering, physics, chemistry and other sciences can be described very successfully by models using mathematical tools from Fractional Calculus. Recently, a new space and time fractional Bloch-Torrey equation (ST-FBTE) has been proposed (Magin et al., J. Magn. Reson. 190(2), 255-270, 2008), and successfully applied to analyse diffusion images of human brain tissues to provide new insights for further investigations of tissue structures. In this paper, we consider the ST-FBTE with a nonlinear source term on a finite domain in three-dimensions. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we propose a spatially second-order accurate implicit numerical method (INM) for the ST-FBTE whereby we discretize the Riesz fractional derivative using a fractional centered difference. Secondly, we prove that the implicit numerical method for the ST-FBTE is uniquely solvable, unconditionally stable and convergent, and the order of convergence of the implicit numerical method is O (τ+τ+h +h}+h ). Finally, some numerical results are presented to support our theoretical analysis

Can anomalous diffusion models in magnetic resonance imaging be used to characterise white matter tissue microstructure?

During the time window of diffusion weighted magnetic resonance imaging experiments (DW-MRI), water diffusion in tissue appears to be anomalous as a transient effect, with a mean squared displacement that is not a linear function of time. A number of statistical models have been proposed to describe water diffusion in tissue, and parameters describing anomalous as well as Gaussian diffusion have previously been related to measures of tissue microstructure such as mean axon radius. We analysed the relationship between white matter tissue characteristics and parameters of existing statistical diffusion models.A white matter tissue model (ActiveAx) was used to generate multiple b-value diffusion-weighted magnetic resonance imaging signals. The following models were evaluated to fit the diffusion signal: 1) Gaussian models - 1a) mono-exponential decay and 1b) bi-exponential decay; 2) Anomalous diffusion models - 2a) stretched exponential, 2b) continuous time random walk and 2c) space fractional Bloch-Torrey equation. We identified the best candidate model based on the relationship between the diffusion-derived parameters and mean axon radius and axial diffusivity, and applied it to the in vivo DW-MRI data acquired at 7.0 T from five healthy participants to estimate the same selected tissue characteristics. Differences between simulation parameters and fitted parameters were used to assess accuracy and in vivo findings were compared to previously reported observations.The space fractional Bloch-Torrey model was found to be the best candidate in characterising white matter on the base of the ActiveAx simulated DW-MRI data. Moreover, parameters of the space fractional Bloch-Torrey model were sensitive to mean axon radius and axial diffusivity and exhibited low noise sensitivity based on simulations. We also found spatial variations in the model parameter β to reflect changes in mean axon radius across the mid-sagittal plane of the corpus callosum.Simulations have been used to define how the parameters of the most common statistical magnetic resonance imaging diffusion models relate to axon radius and diffusivity. The space fractional Bloch-Torrey equation was identified as the best model for the characterisation of axon radius and diffusivity. This model allows changes in mean axon radius and diffusivity to be inferred from spatially resolved maps of model parameters