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Open AccessArticle

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

1
C.J. Gorter Center for High Field MRI, Department of Radiology, Leiden University Medical Center, Albinusdreef 2, 2333ZA Leiden, The Netherlands
2
Department of Bioengineering, University of Illinois at Chicago, 851 S. Morgan St, Chicago, 60607, IL, USA
3
Department of Radiology, Northwestern University, 737 Michigan Ave 16th Floor, Chicago, 60611 IL, USA
*
Author to whom correspondence should be addressed.
Entropy 2014, 16(11), 5838-5852; https://doi.org/10.3390/e16115838
Received: 17 October 2014 / Accepted: 31 October 2014 / Published: 6 November 2014
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
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. View Full-Text
Keywords: entropy; kurtosis; fractional derivative; continuous time random walk;anomalous diffusion; magnetic resonance imaging; Mittag–Leffler function entropy; kurtosis; fractional derivative; continuous time random walk;anomalous diffusion; magnetic resonance imaging; Mittag–Leffler function
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.

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