  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.
