For the sake of computational efficiency and for theoretical purposes, in mathematical modelling, the Dirac Delta distributions are often utilized as a replacement for cells or vesicles, since the size of cells or vesicles is much smaller than the size of the surrounding tissues. Here, we consider the scenario that the cell or the vesicle releases the diffusive compounds to the immediate environment, which is modelled by the diffusion equation. Typically, one separates the intracellular and extracellular environment and uses homogeneous Neumann boundary condition for the cell boundary (so-called spatial exclusion approach), while the point source approach neglects the intracellular environment. We show that extra conditions are needed such that the solutions to the two approaches are consistent. We prove a necessary and sufficient condition for the consistency. Suggested by the numerical results, we conclude that an initial condition in the form of Gaussian kernel in the point source approach compensates for a time-delay discrepancy between the solutions to the two approaches in the numerical solutions. Various approaches determining optimal amplitude and variance of the Gaussian kernel have been discussed.

翻译：为提升计算效率及满足理论需求，在数学建模中常以狄拉克δ分布替代细胞或囊泡，因其尺寸远小于周围组织。本研究考虑细胞或囊泡向周围环境释放扩散性化合物的情景，该过程由扩散方程建模。通常采用空间排斥方法，将胞内与胞外环境分离，并在细胞边界施加齐次诺伊曼边界条件；而点源方法则忽略胞内环境。研究表明，两种方法解的一致性需满足附加条件，并证明了其充要条件。数值结果表明，点源方法中采用高斯核形式的初始条件可补偿两种方法数值解之间的时间延迟偏差。本文还讨论了确定高斯核最优振幅与方差的多类方法。