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

Abstract

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), Eα (− Dα, β|q|βΔα). We interrogated both white and gray matter regions in a 1 mm slice of a fixed rat brain (190 μm 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,Δfalse‒), 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.

Particular Solutions of the Confluent Hypergeometric Differential Equation by Using the Nabla Fractional Calculus Operator

“…Magin et al [21] characterized anomalous diffusion in porous biological tissues using fractional order derivatives and entropy. Ingo et al [22] applied entropy for the case of anomalous diffusion governed by the time and space fractional order diffusion equation.…”

A mechanical picture of fractional-order Darcy equation

“…Indeed the time evolution of the concentration and velocity profile predicted by Fick or Darcy is described by exponential-type solution and, several deviations from experimental results have been found in scientific literature regarding fluid flows in biological tissues [4,5] usually referred to long-tails of the diffusion processes as well as through biological membranes [6,7]. The difference among Fick prediction and experimental results have been captured, recently, considering particle transport at nanometric scale by means of molecular dynamics simulations [8][9][10][11].…”

Numerical Approximation of the Fractional Laplacian via $$hp$$ h p -finite Elements, with an Application to Image Denoising

“…Among others, fractional calculus has proved a viable tool in material science [2], finance [7], image processing [3], porous media flow [22], and bioengineering applications, such as anomalous diffusion in biological tissues [16].…”