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

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.

Keywords: Mittag–Leffler function; anomalous diffusion; continuous time random walk; entropy; fractional derivative; kurtosis; magnetic resonance imaging.

Conflict of interest statement

Conflicts of Interest The authors declare no conflict of interest.

Figures

Figure 1

Sketches of the mean squared displacement for the cases of Gaussian diffusion (2α/β = 1), subdiffusion (2α/β < 1) and superdiffusion (2α/β > 1).

Figure 2

Anomalous diffusion phase diagram with respect to the order of the fractional derivative in space, β, and the order of the fractional derivative in time, α.

Figure 3

Plot of Equation (19) for the kurtosis, KMLF, computed in the Mittag–Leffler representation of subdiffusion versus the time-fractional derivative, α.

Figure 4

Trace parameter maps of α, D, KMLF and HMLF for an axial slice through a brain of a chronic stroke patient.

Figure 5

Spectral entropy surface plot for the Mittag–Leffler function (MLF) in Equation (4) with respect to the order of the fractional space derivative, β, and the order of the fractional time derivative, α (Dα,β = 1, t = 1). The floor of the plot corresponds to the anomalous diffusion phase diagram.

Figure 6

Spectral entropy for Equation (15) with respect to the order of the fractional space derivative, β, with diffusion time cases where t = 0.5, 1, 1.5, 2 for α = 1 and D1,β = 1.

Figure 7

Spectral entropy for Equation (9) with respect to the order of the fractional time derivative, α, for four diffusion time cases where t = 0.5, 1, 1.5, 2 for β = 2 and Dα,2 = 1.

Figure 8

Plot of the individual wavenumber contributions to the spectral entropy of Equation (15) when the order of the fractional space derivative β = 0.5, 0.75, 1, 2, 4 for α = 1, D1,β = 1 and t = 1.

Figure 9

Plot of the individual wavenumber contributions to the spectral entropy of Equation (9) when the order of the fractional time derivative α = 0.5, 1, 1.5, 2 for β = 2, Dα,2 = 1 and t = 1.