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

Entropy

Carson Ingo¹  Richard L. Magin² 

[1] C.J. Gorter Center for High Field MRI, Department of Radiology, Leiden University Medical Center, Albinusdreef 2, 2333ZA Leiden, The Netherlands;Department of Bioengineering, University of Illinois at Chicago, 851 S. Morgan St, Chicago, 60607, IL, USA; E-Mail:

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

DOI  :  10.3390/e16115838

来源: mdpi

【 摘 要 】

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.

【 授权许可】

CC BY   
© 2014 by the authors; licensee MDPI, Basel, Switzerland