New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis.

Todd Parrish,Carson Ingo,Richard Magin

doi:10.3390/e16115838

Abstract

Fractional order derivative operators offer a concise description to model multi-scale, heterogeneous and non-local systems. Specifically, in magnetic resonance imaging, there has been recent work to apply fractional order derivatives to model the non-Gaussian diffusion signal, which is ubiquitous in the movement of water protons within biological tissue. To provide a new perspective for establishing the utility of fractional order models, we apply entropy for the case of anomalous diffusion governed by a fractional order diffusion equation generalized in space and in time. This fractional order representation, in the form of the Mittag–Leffler function, gives an entropy minimum for the integer case of Gaussian diffusion and greater values of spectral entropy for non-integer values of the space and time derivatives. Furthermore, we consider kurtosis, defined as the normalized fourth moment, as another probabilistic description of the fractional time derivative. Finally, we demonstrate the implementation of anomalous diffusion, entropy and kurtosis measurements in diffusion weighted magnetic resonance imaging in the brain of a chronic ischemic stroke patient.

Highlights

The ischemic tissue (IT), in the right hemisphere of the patient’s brain, has a diffusion coefficient, D, value (∼ 3 × 10−3 mm2 /s), which is similar to the typical value found for the cerebral spinal fluid (CSF) of the ventricles
From a continuous time random walk (CTRW) physical model perspective, as the diffusion time increases in the spatial domain, we argue that the distribution widens and the entropy increases as a dynamic measure by which the uncertainty in predicting the location of the diffusing particle increases
Both the space (β) and the time (α) fractional order dependence are expressed separately as special cases governed by the Mittag–Leffler function

Summary

Introduction

The fundamental concept in continuous time random walk (CTRW) theory is to extend the diffusion equation, such that the fractional order partial derivatives can be used as the governing mathematical operators to describe the diffusion propagator, P (x, t): ∂ α P (x, t) ∂ β P (x, t) = D , α,β ∂tα ∂|x|β (1). The justification for making use of the fractional derivative operators is to provide a mathematical means to interpolate from homogeneous and relatively simple systems that exhibit local, Gaussian behavior to heterogeneous and relatively complex systems that exhibit non-local, power-law behavior [2,3,4,5,6,7,8]. In the CTRW context, the order of the fractional operators, α and β, provides a description of a random walker’s likelihood to have broader distributions of waiting times and jump lengths, respectively, in comparison to classical Brownian motion. In the most basic form, the one-dimensional MSD is expressed by a composite power law as: hx (t)i ∼ t2α/β

Methods

Results

Conclusion