Determine the actual time-dependent Dirac source intensities in the point-source diffusion model

Determine the actual time-dependent source intensities Φ_i(t) for the clustered Dirac delta points used in the point-source diffusion model, as defined by the right-hand side of ∂u/∂t − DΔu = Σ Φ_i(t) δ(x − x_i), that generate the target inhomogeneous sinusoidal boundary flux density φ(θ) over the boundary of a single circular cell without resorting to the mean-intensity approximation Φ_i(s) ≈ Φ_i(t) and the neglect of the deviation term δ_t(s) in the convolution-based boundary-flux expression.

Background

The paper studies two formulations for diffusion of secreted compounds from a single cell: a spatial exclusion model with an inhomogeneous sinusoidal flux density prescribed over the circular cell boundary, and a point-source model where the cell is represented by a cluster of Dirac delta sources/sinks inside the cell domain. Exact agreement of the two models requires matching the boundary flux generated by diffusion from the point sources to the prescribed boundary flux.

Because the boundary flux induced by point sources involves a challenging time-convolution integral, the authors approximate the time-dependent source intensities Φ_i(s) by a time-dependent mean \u007FΦ_i(t) and ignore the deviation term δ_t(s). They then solve for these approximate intensities by enforcing equality of the flux at the extreme points of the sinusoidal boundary flux. In the numerical section, they explicitly state that determining the actual Φ_i(t) was unfeasible at the moment, highlighting an unresolved task to compute the true, non-approximated intensities consistent with the full convolution expression.