Approximating a spatially-heterogeneously mass-emitting object by multiple point sources in a diffusion model

TL;DR

This paper compares two models for simulating chemical secretion by cells, introducing a multi-Dirac point source approach that accurately approximates spatially heterogeneous fluxes and improves numerical stability.

Contribution

It proposes a multi-Dirac point source method with explicit expressions, enhancing approximation accuracy for heterogeneous fluxes in diffusion models.

Findings

Two to three Dirac points suffice for accurate approximation.

Multi-Dirac approach outperforms single-Dirac models.

Green's function method improves numerical stability.

Abstract

Various biological cells secrete diffusing chemical compounds into their environment for communication purposes. Secretion usually takes place over the cell membrane in a spatially heterogeneous manner. Mathematical models of these processes will be part of more elaborate models, e.g. of the movement of immune cells that react to cytokines in their environment. Here, we compare two approaches to modelling of the secretion-diffusion process of signalling compounds. The first is the so-called spatial exclusion model, in which the intracellular space is excluded from consideration and the computational space is the extracellular environment. The second consists of point source models, where the secreting cell is replaced by one or more non-spatial point sources or sinks, using -- mathematically -- Dirac delta distributions. We propose a multi-Dirac approach and provide explicit expressions for the intensities of the Dirac distributions. We show that two to three well-positioned Dirac points suffice to approximate well a temporally constant but spatially heterogeneous flux distribution of compound over the cell membrane, for a wide range of variation in flux density and diffusivity. The multi-Dirac approach is compared to a single-Dirac approach that was studied in previous work. Moreover, an explicit Green's function approach is introduced that has significant benefits in circumventing numerical instability that may occur when the Dirac sources have high intensities.