A space-fractional reaction-diffusion system with cylindrical symmetry

TL;DR

This paper models diffusion in porous biological tissues using a space-fractional reaction-diffusion system with cylindrical symmetry, providing analytical solutions involving Fox H-functions and numerical methods.

Contribution

It introduces a space-fractional model for reaction-diffusion in cylindrical tissues, offering analytical solutions and a flexible approach for parameter estimation from experimental data.

Findings

Analytical steady-state solutions using Fox H-functions.

Reduction to Bessel functions in the integer-order case.

Numerical solutions via quadrature of Bessel integrals.

Abstract

Diffusion within porous media, such as biological tissues, exhibits departures from conventional Fick's laws, which could result in space-fractional diffusion. The paper considers a reaction-diffusion system with two spatial compartments -- a proximal one of finite radius having a source, and an outer one extending to infinity where the source is not present but first-order decay of the diffusing species takes place. The system models the foreign body reaction around an implanted electrode. Microscopic heterogeneity inside the tissue was modeled by a space-fractional Riesz Laplacian acting on the concentration. This allows for a flexible approach when estimating transport parameters from experimental data. The steady-state of the system is solved in terms of Hankel and Mellin transforms, resulting in a Fox H-function. In the integer-order case, the analytical solution reduces to a superposition of modified Bessel functions of the first and second kinds. Solutions are exhibited by numerical quadrature of the involved Bessel function integrals.