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Fractional Diffusion with Geometric Constraints: Application to Signal Decay in Magnetic Resonance Imaging (MRI)

Author

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  • Ervin K. Lenzi

    (Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa 84040-900, Paraná, Brazil)

  • Haroldo V. Ribeiro

    (Departamento de Física, Universidade Estadual de Maringá, Maringá 87020-900, Paraná, Brazil)

  • Marcelo K. Lenzi

    (Departamento de Engenharia Química, Universidade Federal do Paraná, Av. Cel. Francisco H. dos Santos 210, Curitiba 81531-980, Paraná, Brazil)

  • Luiz R. Evangelista

    (Departamento de Física, Universidade Estadual de Maringá, Maringá 87020-900, Paraná, Brazil)

  • Richard L. Magin

    (Department of Biomedical Engineering, University of Illinois at Chicago, Chicago, IL 60607, USA)

Abstract

We investigate diffusion in three dimensions on a comb-like structure in which the particles move freely in a plane, but, out of this plane, are constrained to move only in the perpendicular direction. This model is an extension of the two-dimensional version of the comb model, which allows diffusion along the backbone when the particles are not in the branches. We also consider memory effects, which may be handled with different fractional derivative operators involving singular and non-singular kernels. We find exact solutions for the particle distributions in this model that display normal and anomalous diffusion regimes when the mean-squared displacement is determined. As an application, we use this model to fit the anisotropic diffusion of water along and across the axons in the optic nerve using magnetic resonance imaging. The results for the observed diffusion times (8 to 30 milliseconds) show an anomalous diffusion both along and across the fibers.

Suggested Citation

  • Ervin K. Lenzi & Haroldo V. Ribeiro & Marcelo K. Lenzi & Luiz R. Evangelista & Richard L. Magin, 2022. "Fractional Diffusion with Geometric Constraints: Application to Signal Decay in Magnetic Resonance Imaging (MRI)," Mathematics, MDPI, vol. 10(3), pages 1-11, January.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:3:p:389-:d:735373
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    References listed on IDEAS

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    1. Ilya Yurchenko & Joao Marcos Vensi Basso & Vladyslav Serhiiovych Syrotenko & Cristian Staii, 2019. "Anomalous diffusion for neuronal growth on surfaces with controlled geometries," PLOS ONE, Public Library of Science, vol. 14(5), pages 1-21, May.
    2. Richard L. Magin & Ervin K. Lenzi, 2021. "Slices of the Anomalous Phase Cube Depict Regions of Sub- and Super-Diffusion in the Fractional Diffusion Equation," Mathematics, MDPI, vol. 9(13), pages 1-29, June.
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    Cited by:

    1. Richard L. Magin & Ervin K. Lenzi, 2022. "Fractional Calculus Extension of the Kinetic Theory of Fluids: Molecular Models of Transport within and between Phases," Mathematics, MDPI, vol. 10(24), pages 1-20, December.
    2. 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.

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    1. Richard L. Magin & Ervin K. Lenzi, 2022. "Fractional Calculus Extension of the Kinetic Theory of Fluids: Molecular Models of Transport within and between Phases," Mathematics, MDPI, vol. 10(24), pages 1-20, December.

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