Nanoparticle uptake by a semi-permeable, spherical cell from an external planar diffusive field. II. Numerical study of temporal and spatial development validated using FEM

TL;DR

This paper develops an asymptotic mathematical model for nanoparticle diffusion around a spherical cell exposed to a planar diffusive front, validated against finite element numerical solutions, demonstrating high accuracy in dynamic conditions.

Contribution

It introduces a novel asymptotic solution for diffusion around a spherical cell and validates it with FEM, improving understanding of nanoparticle transport in biological systems.

Findings

Asymptotic solution accurately predicts diffusion dynamics

Model effectively captures internal and external nanoparticle behaviour

Validation shows strong agreement with FEM results

Abstract

In this paper we present a mathematical study of particle diffusion inside and outside a spherical biological cell that has been exposed on one side to a propagating planar diffusive front. The media inside and outside the spherical cell are differentiated by their respective diffusion constants. A closed form, large-time, asymptotic solution is derived by the combined means of Laplace transform, separation of variables and asymptotic series development. The solution process is assisted by means of an effective far-field boundary condition, which is instrumental in resolving the conflict of planar and spherical geometries. The focus of the paper is on a numerical comparison to determine the accuracy of the asymptotic solution relative to a fully numerical solution obtained using the finite element method. The asymptotic solution is shown to be highly effective in capturing the dynamic behaviour of the system, both internal and external to the cell, under a range of diffusive conditions.