半渗透球形细胞从外部平面扩散场中摄取纳米颗粒的研究 II. 基于有限元法验证的时空发展数值研究
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
AI总结：
本文研究球形细胞在平面扩散场中的纳米颗粒摄取问题，利用拉普拉斯变换、分离变量法和渐近展开推导大时间渐近解，并通过有限元数值验证其精度。
摘要：
本文对暴露于一侧传播平面扩散前沿的球形生物细胞内外粒子扩散进行了数学研究。球形细胞内外介质通过各自的扩散常数加以区分。通过拉普拉斯变换、分离变量法和渐近级数展开的综合手段，推导出大时间尺度的闭式渐近解。求解过程借助有效的远场边界条件，该条件对于解决平面几何与球形几何之间的冲突具有关键作用。本文的重点是通过数值比较来确定渐近解相对于使用有限元法获得的完全数值解的精度。结果表明，在一系列扩散条件下，渐近解在捕捉细胞内部和外部系统动态行为方面均具有很高的有效性。
英文摘要：
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.
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