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

Greta Chiaravalli; Guglielmo Lanzani; Riccardo Sacco; Sandro Salsa; Greta Chiaravalli; Guglielmo Lanzani; Riccardo Sacco; Sandro Salsa

doi:10.3934/mine.2023075

Mathematics in Engineering

2023, Volume 5, Issue 4: 1-44. doi: 10.3934/mine.2023075

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Research article

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

1.
Center for Nanoscience and Technology, Istituto Italiano di Tecnologia, via Pascoli 70, 20133 Milano, Italy
2.
Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
3.
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

Received: 12 December 2022 Revised: 21 January 2023 Accepted: 23 January 2023 Published: 01 February 2023

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.
- multiscale and multiphysics modeling,
- nonlinear interface coupling,
- ion electrodiffusion,
- drift-diffusion charge transport,
- neuron depolarization,
- functional iterations
Citation: Greta Chiaravalli, Guglielmo Lanzani, Riccardo Sacco, Sandro Salsa. Nanoparticle-based organic polymer retinal prostheses: modeling, solution map and simulation[J]. Mathematics in Engineering, 2023, 5(4): 1-44. doi: 10.3934/mine.2023075

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Abstract

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.

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