Fractional Order Complexity Model of the Diffusion Signal Decay in MRI

Richard L. Magin, Hamid Karani, Shuhong Wang and Yingjie Liang
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Richard L. Magin: Department of Bioengineering at University of Illinois at Chicago, Chicago, IL 60607, USA
Hamid Karani: Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA
Shuhong Wang: Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 211100, China
Yingjie Liang: Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 211100, China

Mathematics, 2019, vol. 7, issue 4, 1-16

Abstract: 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.

Keywords: anomalous diffusion; complexity; magnetic resonance imaging; fractional calculus (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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