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Fractional Order Complexity Model of the Diffusion Signal Decay in MRI

Author

Listed:
  • 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)

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.

Suggested Citation

  • Richard L. Magin & Hamid Karani & Shuhong Wang & Yingjie Liang, 2019. "Fractional Order Complexity Model of the Diffusion Signal Decay in MRI," Mathematics, MDPI, vol. 7(4), pages 1-16, April.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:4:p:348-:d:222288
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    References listed on IDEAS

    as
    1. Li, Yuanlu & Liu, Fawang & Turner, Ian W. & Li, Tao, 2018. "Time-fractional diffusion equation for signal smoothing," Applied Mathematics and Computation, Elsevier, vol. 326(C), pages 108-116.
    2. Guoxing Lin, 2018. "Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches," Mathematics, MDPI, vol. 6(2), pages 1-16, January.
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    Cited by:

    1. Ervin Kaminski Lenzi & Luiz Roberto Evangelista & Luciano Rodrigues da Silva, 2023. "Aspects of Quantum Statistical Mechanics: Fractional and Tsallis Approaches," Mathematics, MDPI, vol. 11(12), pages 1-15, June.
    2. Pereira-Alves, Felipe & Soares-Pinto, Diogo O. & Paiva, Fernando F., 2024. "NMR diffusion in restricted environment approached by a fractional Langevin model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 641(C).

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