  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.
