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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{α}/∂t

^{α}represents the Caputo fractional derivative in time for 0 < α ≤ 1, ∂

^{β}/∂|x|

^{β}represents the order of the Riesz fractional derivative in space for 1 < β ≤ 2 and D

_{α,β}is the generalized diffusion constant (distance

^{β}/time

^{α}) [1–3].

^{−(1+}

^{α}

^{)}and |x|

^{−(1+}

^{β}

^{)}[5]. As a result, bulk non-Gaussian motions emerge, and the mean squared displacement (MSD) no longer increases with a linear dependence on time. In the most basic form, the one-dimensional MSD is expressed by a composite power law as:

## 2. Experimental Section

#### 2.1. Theory

_{α}represents the single-parameter Mittag–Leffler function (MLF), which describes a generalized power law decay for the range, 0 < α < 1, and a damped, but oscillating function for 1 < α < 2 [9,10]. Specific functions encapsulated by the MLF are a monoexponential decay when α = 1 and the cosine function when α = 2. Thus, for a fixed wavenumber and diffusion coefficient, Equation (4) corresponds to the time decay (with or without oscillations) of one spectral component. As the order of the fractional time derivative α decreases from one, there is a shift away from the single exponential decay toward an apparent multi-exponential relaxation in time. In the case of α = 1 and β = 2, we have the characteristic form for a Gaussian distribution,

#### 2.2. Kurtosis

_{MLF}= 0. For 0 < α < 1, K

_{MLF}> 0 with the maximum excess kurtosis value limited to max(K

_{MLF}) = 3 when α → 0. Equation (19) is plotted in Figure 3, which shows a nearly inverse linear relationship between K

_{MLF}and α.

#### 2.3. Entropy

_{i}) are the individual samples in the the discrete pdf and H

_{x}is the Shannon information entropy [12]. With the consideration of information formulated in the context of statistical uncertainty, we have a tool to compare systems governed by differing stochastic processes. For example, when comparing two distributions, both normalized with the same full-width, half maximum values, the heavier-tailed distribution can be shown to have greater information entropy, pertaining to greater statistical uncertainty. Another approach to measure the uncertainty in a system is to analyze the characteristic function in terms of the Fourier transform in space, P (x) → p(k), with spectral entropy,

_{k}, and the term, ln(N) (i.e., discrete uniform distribution of N samples), is a normalization factor applied so that the spectral entropy, H

_{k}, is between zero and one [13,14]. Furthermore, as Equation (21) is generally defined to measure the uncertainty of a characteristic function, we can adapt this formalism to compare the CFs for anomalous diffusion,

^{−1}) and $\overline{\mathrm{\Delta}}$ is the effective diffusion time (e.g., units of ms) to be rewritten as,

#### 2.4. Methods for Diffusion MRI Experiments

^{2}, 3 orthogonal diffusion weighted directions, number of averages NA = 6, in-plane voxel resolution = 2 × 2 mm, voxel thickness = 4 mm, 20 axial slices, scan time ∼ 6 minutes. The raw diffusion weighted data were Rician noise corrected by estimating the variance (σ

^{2}) in the signal intensity of the ventricle at each b-value, such that ${S}_{\mathit{rn}}=\sqrt{{S}^{2}-{\sigma}^{2}}$. The Rician noise-corrected diffusion weighted images were skull-stripped utilizing the Brain Extraction Tool [15]. All skull-stripped and Rician noise-corrected diffusion weighted images were co-registered to the b = 0 image space using statistical parametric mapping software (SPM8). Using the Levenberg–Marquardt minimization algorithm in MATLAB (Natick, MA, USA), the average of the three diffusion weighted direction data were fit on a voxel-wise basis to Equation (9) with the MLF algorithm in [16,17]. Following estimations of D and α, the excess kurtosis, K

_{MLF}, was computed using the conversion provided in Equation (19). Following estimations of D and α, the CF in Equation (9) for p(k, t) was constructed using N = 100 increments arrayed over variable b-values between 0 and 10 000 s/mm

^{2}. Then, the entropy (defined in Equation (22)) in the diffusion process, as modeled by the MLF, was computed as H

_{MLF}. The isotropic parameter maps of D, α, K

_{MLF}and H

_{MLF}for the same axial slice through the stroke patient’s brain are shown in Figure 4.

#### 2.5. Methods to Evaluate Entropy in the Mittag–Leffler Function

_{i}= 0 – 5 for N = 500 points. The overall results for entropy in the MLF are presented in Figure 5 as a three-dimensional entropy surface drawn above a plane defined by the positive values of α and β. The floor of the plot is essentially the phase diagram shown in Figure 2. Figures 6 and 7 plot the entropy in the diffusion process as a function of the order of the fractional derivative in space and time, respectively. Figures 8 and 9 plot the individual contributions to the total entropy of the diffusion process as a function of the wavenumber, k

_{i}, for selected cases of α and β.

## 3. Results and Discussion

#### 3.1. Diffusion MRI Experiments

^{−3}mm

^{2}/s), which is similar to the typical value found for the cerebral spinal fluid (CSF) of the ventricles. As can be seen in the contralateral hemisphere, prior to the onset of the stroke, the brain slice would have appeared symmetrical with white matter (WM) and gray matter (GM) voxels. However, as these data were acquired ∼ 2 years following onset, the IT microstructure has degenerated (necrosis), such that the bulk diffusion coefficient has increased to an unhindered value. Furthermore, the diffusion in the IT is close to Gaussian as α ∼ 1, indicating a monoexponential behavior, which is also the case for the CSF. The trace values for D in the healthy WM and GM are ∼ 1/3 of the values in the IT and CSF, with the WM possessing an overall slower diffusion than measured in the GM.

^{−3}mm

^{2}/s, the contrast between WM and GM is difficult to discern; however, in the α map, the WM/GM contrast is clearly visible with the WM demonstrating more subdiffusive behavior compared to the GM. The K

_{MLF}map also has clearly visible GM/WM contrast and appears as a negative image in the α map, due to the nearly inverse relationship between K

_{MLF}and α in Equation (19). The entropy, H

_{MLF}, map provides a stable image in which there is visible and smooth GM/WM contrast, with the IT and CSF exhibiting low values of entropy due to the unhindered Gaussian diffusion dynamics. Interestingly, voxels exhibiting low values of α, representing highly subdiffusive dynamics, also have relatively high entropy estimations (particularly in the WM) to indicate a diffusion propagator with a heavy-tailed pdf. In correspondence with entropy, there is high kurtosis estimated for the diffusion propagator pdf in regions of low values of α. However, kurtosis and entropy are not interchangeable measures of the diffusion propagator pdf, evident not only in the images shown in Figure 4, but also mathematically distinct in the forms of Equations (19) and (22). Specifically, in consideration of the CF, α and K

_{MLF}are measures of the deviation from a monoexponential form with a rate of the diffusion coefficient, D, whereas entropy considers the entire CF, which includes both the monoexponential component, D, and the non-Gaussian component, α, of the diffusion profile. As kurtosis is defined as the normalized fourth moment in Equation (16), it can be readily seen that the variance, or D, is canceled out when dividing Equation (18) by Equation (13).

#### 3.2. Measuring the Mittag–Leffler Function with Spectral Entropy

_{1}

_{,β}= 1, t = 1, and starting at β = 2, we observe that the entropy increases as β gets smaller, with an approximately 20% increase in the normalized spectral entropy when β = 1 (the Cauchy distribution), whereas travel in the direction of increasing β is mostly flat by this measure of entropy. From the Gaussian location, β = 2, the entropy appears to converge to a value near 0.5 for increasing β, while for decreasing β, the entropy increases in a monotonic manner at short times. The effect of increasing diffusion time (or larger values of the diffusion coefficient) results in a decrease in the magnitude for the normalized entropy values, as demonstrated going from t = 0.5 to t = 2. In the phase diagram (Figure 2: α = 1 and β < 2), Figure 6 evaluates the case of super-diffusion, and it is encouraging that this perspective portrays this regime, which includes the CF of the Cauchy distribution, as one of higher entropy (in comparison with the Gaussian diffusion case).

_{α,}

_{2}= 1, t = 1, and starting at α = 1, we observe the entropy increasing in both directions, overall. Again, the depth of the minimum grows for longer times, but in this cross-sectional view, the location is in the direction of higher values of α. As is shown in the phase diagram (Figure 2), when β = 2, values of α > 1 are in a region of super-diffusion and values of α < 1 are in a region of subdiffusion. Furthermore, in Figure 7, we observe that for a specific value of time (and diffusion coefficient constant), the entropy generally increases (from the Gaussian diffusion case of α = 1) as the value of α increases, as well as as it decreases. Thus, both higher and lower values of the order of the fractional derivative α (relative to α = 1) give higher entropy values.

## 4. Conclusions

_{MLF}and entropy H

_{MLF}measures to provide additional information about biological tissue microstructure beyond the classical diffusion coefficient, D.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**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, K

_{MLF}, computed in the Mittag–Leffler representation of subdiffusion versus the time-fractional derivative, α.

**Figure 4.**Trace parameter maps of α, D, K

_{MLF}and H

_{MLF}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 (D

_{α,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 D

_{1}

_{,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 D

_{α,}

_{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, D

_{1}

_{,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, D

_{α,}

_{2}= 1 and t = 1.

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**MDPI and ACS Style**

Ingo, C.; Magin, R.L.; Parrish, T.B.
New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis. *Entropy* **2014**, *16*, 5838-5852.
https://doi.org/10.3390/e16115838

**AMA Style**

Ingo C, Magin RL, Parrish TB.
New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis. *Entropy*. 2014; 16(11):5838-5852.
https://doi.org/10.3390/e16115838

**Chicago/Turabian Style**

Ingo, Carson, Richard L. Magin, and Todd B. Parrish.
2014. "New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis" *Entropy* 16, no. 11: 5838-5852.
https://doi.org/10.3390/e16115838