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

Ervin K. Lenzi, Haroldo V. Ribeiro, Marcelo K. Lenzi, Luiz R. Evangelista and Richard L. Magin
Additional contact information
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

Mathematics, 2022, vol. 10, issue 3, 1-11

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.

Keywords: comb model; fractional diffusion equation; memory effects; anomalous diffusion; magnetic resonance imaging (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
https://www.mdpi.com/2227-7390/10/3/389/pdf (application/pdf)
https://www.mdpi.com/2227-7390/10/3/389/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:10:y:2022:i:3:p:389-:d:735373

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().