A model of fractional-order diffusion in the glial scar

Implantation of neuroprosthetic electrodes induces a stereotypical state of neuroinflammation, which is thought to be detrimental for the neurons surrounding the electrode. Mechanisms of this type of neuroinflammation are still not understood well. Recent experimental and theoretical results point out possible role of the diffusion species in this process. The paper considers a model of anomalous diffusion occurring in the glial scar around a chronic implant in two simple geometries – a separable rectilinear electrode and a cylindrical electrode, which are solvable exactly. We describe a hypothetical extended source of diffusing species and study its concentration profile in steady-state conditions. Diffusion transport is assumed to obey a fractional-order Fick law, which is derived from physically realistic assumptions using a fractional calculus approach. The derived fractional-order distribution morphs into regular order diffusion in the case of integer fractional exponents. The model presented here demonstrates that accumulation of diffusing species can occur and the scar properties (i.e. tortuosity, fractional order, scar thickness) can influence such accumulation. The observed shape of the concentration profile corresponds qualitatively with GFAP profiles reported in the literature. The main difference with respect to the previous studies is the explicit incorporation of the apparatus of fractional calculus without assumption of an ad hoc tortuosity parameter. Intended application of the approach is the study of diffusing substances in the glial scar after implantation of neural prostheses, although the approach can be adapted to other studies of diffusion in biological tissues, for example of biomolecules or small drug molecules.