Numerical simulation of anomalous diffusion with application to medical imaging

Anomalous diffusion is applicable in environments that are not locally homogeneous, such as human brain tissue. In such environments the model of restricted diffusion commonly employed in the analysis of diffusion magnetic resonance data is not valid. Anomalous diffusion displays a nonlinear time dependence for the mean-squared displacement, and provides a prediction of a stretched exponential form for the signal decay. Anomalous diffusion probes tissue complexity in a way that is not possible using standard diffusion tensor imaging. Fractional order dynamics, particularly when applied to diffusion, leads to an extension of the concept of Brownian motion through a generalisation of the Gaussian probability function. Water molecule diffusion in the brain can be measured using a magnetic resonance imaging method, and the anisotropy of the diffusion tensor is of particular interest in brain images. In physics and chemistry, specifically in nuclear magnetic resonance (NMR) or magnetic resonance imaging (MRI), the Bloch equations are used to calculate the nuclear magnetization as a function of time. NMR usually assumes an averaging process over a large number of nuclear spins, that while suitable for mm-scale resolution, may not be suitable if more localized information on the structure, or substructure, of water diffusing in the human brain is required. A fractional Bloch-Torrey model could be more useful to study anomalous diffusion in the human brain.