Fractional Order Generalization of Anomalous Diffusion as a Multidimensional Extension of the Transmission Line Equation

In this paper, a new fractional order generalization of the diffusion equation is developed to describe the anisotropy of anomalous diffusion that is often observed in brain tissues using magnetic resonance imaging (MRI). The new model embeds three different fractional order exponents-corresponding to the principal directions of water diffusion-into the governing Bloch-Torrey equation. The model was used to analyze diffusion weighted MRI data acquired from a normal human brain using a 3T clinical MRI scanner. Analysis of the data revealed normal Gaussian diffusion in the cerebral spinal fluid (isotropic fractional order exponent of, and anomalous diffusion in both the white and the gray matter. In addition, we observed anisotropy in the fractional exponent values for white mater (along the fibers versus across the fibers), but not for gray matter. This model introduces new parameters to describe the complexity of the tissue microenvironment that may be sensitive biomarkers of the structural changes arising in neural tissues with the onset of disease.