Studies of anomalous diffusion in the human brain using fractional order calculus

It is well known that diffusion-induced MR signal loss deviates from monoexponential decay, particularly at high b-values (e.g., >1500 sec/mm ² for human brain tissues). A number of models have been developed to describe this anomalous diffusion behavior and relate the diffusion measurements to tissue structures. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (Magin et al., J Magn Reson 2008;190:255-270; Zhou et al., Proc Int'l Soc Magn Reson Med 2008). Using a spatial Laplacian ∇^2β, this model yields a new set of parameters to describe anomalous diffusion: diffusion coefficient D, fractional order derivative in space β, and a spatial parameter μ (in units of mm). In this study, we demonstrate that the fractional calculus model can be successfully applied to analyzing diffusion images of healthy human brain tissues in vivo. Five human volunteers were scanned on a commercial 3-T scanner using a customized single-shot echo-planar imaging diffusion sequence with 15 b values ranging from 0 to 4700 sec/mm². The set of images was analyzed using the fractional calculus model, producing spatially resolved maps of D, β, and μ. The β and μ maps showed notable contrast between white and gray matter. The contrast has been attributed to the varying degree of complexity of the underlying tissue structures and microenvironment. Although the biophysical basis of β and μ remains elusive, the potential utility of these parameters to characterize the environment for molecular diffusion, as a complement to apparent diffusion coefficient, may lead to a new way to investigate tissue structural changes in disease progression, intervention, and regression.