IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i3p389-d735373.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/3/389/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/3/389/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:10:y:2022:i:3:p:389-:d:735373. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.