Characterization of anomalous diffusion in porous biological tissues using fractional order derivatives and entropy

In this high-resolution magnetic resonance imaging (MRI) study at 17.6 Tesla of a fixed rat brain, we used the continuous time random walk theory (CTRW) for Brownian motion to characterize anomalous diffusion. The complex mesoporus structure of biological tissues (membranes, organelles, and cells) perturbs the motion of the random walker (water molecules in proton MRI) introducing halts between steps (waiting times) and restrictions on step sizes (jump lengths). When such waiting times and jump lengths are scaled with probability distributions that follow simple inverse power laws (t ^-(1+α), |x|^-(1+β)) non-Gaussian motion gives rise to sub- and super- diffusion. In the CTRW approach, the Fourier transform yields a solution to the generalized diffusion equation that can be expressed by the Mittag-Leffler function (MLF), E_α(-D _α,β|q|^βΔ^α. We interrogated both white and gray matter regions in a 1 mm slice of a fixed rat brain (190 l m in plane resolution) with diffusion weighted MRI experiments using b-values up to 25,000 s/mm², by independently varying q and Δ. When fitting these data to our model, the fractional order parameters, α and β, and the entropy measure, H(q,Δ), were found to provide excellent contrast between white and gray matter and to give results that were sensitive to the type of diffusion experiment performed.