Fractional order measures of anomalous diffusion in healthy aging of neural tissue

In this paper we use fractional calculus to characterize diffusion in brain tissue by generalizing Fick´s 2^nd Law. This approach, rooted in the physics of the Continuous Time Random Walk (CTRW) Theory expresses separate measures of tissue complexity through the fractional order of the time derivative, α, and the space derivative, ß. We also calculate the entropy of the characteristic function of the probability distribution function (pdf) as a measure of the heterogeneity of the tissue. We applied this theory to the analysis of high-field (17.6 Tesla) diffusion-weighted MRI data to characterize the structural complexity of neural tissue. When interpreted in the context of anomalous diffusion, healthy aging in the normal rat brain is observed in both white matter (WM) and gray matter (GM).