Nanoparticle-based organic polymer retinal prostheses: modeling, solution map and simulation

In this article we investigate a mathematical model for a retinal prosthesis made of organic polymer nanoparticles (NP) in the stationary regime. The model consists of a Drift-Diffusion system to describe free charge transport in the NP bulk; a Poisson-Nernst-Planck system to describe ion electrodiffusion in the solution surrounding the NP; and nonlinear transmission conditions at the NP-solution interface. To solve the model we use an iteration procedure for which we prove the existence and briefly comment the uniqueness of a fixed point under suitable smallness assumptions on model parameters. For system discretization we use a stabilized finite element method to prevent unphysical oscillations in the electric potential, carrier number densities and ion molar densities. Model predictions describe the amount of active chemical molecule accumulating at the neuron surface and highlight electrostatic effects induced by the sole presence of the nanoparticle. These results support the use of mathematical modeling as a virtual laboratory for the optimal design of bio-hybrid systems, whose investigation may be impervious due to experimental limits.