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Mathematics 2019, 7(4), 348;

Fractional Order Complexity Model of the Diffusion Signal Decay in MRI

Department of Bioengineering at University of Illinois at Chicago, Chicago, IL 60607, USA
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA
Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 211100, China
Author to whom correspondence should be addressed.
Received: 11 March 2019 / Revised: 1 April 2019 / Accepted: 8 April 2019 / Published: 12 April 2019
(This article belongs to the Special Issue Fractional Order Systems)
PDF [2875 KB, uploaded 12 April 2019]


Fractional calculus models are steadily being incorporated into descriptions of diffusion in complex, heterogeneous materials. Biological tissues, when viewed using diffusion-weighted, magnetic resonance imaging (MRI), hinder and restrict the diffusion of water at the molecular, sub-cellular, and cellular scales. Thus, tissue features can be encoded in the attenuation of the observed MRI signal through the fractional order of the time- and space-derivatives. Specifically, in solving the Bloch-Torrey equation, fractional order imaging biomarkers are identified that connect the continuous time random walk model of Brownian motion to the structure and composition of cells, cell membranes, proteins, and lipids. In this way, the decay of the induced magnetization is influenced by the micro- and meso-structure of tissues, such as the white and gray matter of the brain or the cortex and medulla of the kidney. Fractional calculus provides new functions (Mittag-Leffler and Kilbas-Saigo) that characterize tissue in a concise way. In this paper, we describe the exponential, stretched exponential, and fractional order models that have been proposed and applied in MRI, examine the connection between the model parameters and the underlying tissue structure, and explore the potential for using diffusion-weighted MRI to extract biomarkers associated with normal growth, aging, and the onset of disease. View Full-Text
Keywords: anomalous diffusion; complexity; magnetic resonance imaging; fractional calculus anomalous diffusion; complexity; magnetic resonance imaging; fractional calculus

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Magin, R.L.; Karani, H.; Wang, S.; Liang, Y. Fractional Order Complexity Model of the Diffusion Signal Decay in MRI. Mathematics 2019, 7, 348.

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