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

TL;DR

This work addresses how to approximate a spatially heterogeneously mass-emitting cell, modeled by a spatial exclusion PDE, with a diffusion-based point-source approach. It develops a multi-Dirac framework that places off-center Dirac sources (and a central source) inside the cell to reproduce the boundary flux pattern, deriving explicit intensities for $n=1$ and $n=2$ Fourier modes of the boundary flux. To tackle numerical instability from large Dirac intensities, it introduces an explicit Green's function strategy that separates a semi-analytic unbounded-domain contribution from a regular correction on the domain, enabling stable simulations and efficient computation. The authors quantify when multi-Dirac representations outperform single-Dirac ones, using $L^2$ and $H^1$ metrics and Monte Carlo analyses across diffusion and heterogeneity parameters, and show that diffusion strength and flux frequency govern the method’s applicability. The findings suggest practical benefits for simulating cell-to-cell signaling in systems with moving or many cells, offering a scalable alternative to dense meshing in spatial exclusion models.

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.

Figures (17)

• Figure 1.1: Geometric configuration of the spatial domain in the models. Panel (a): A schematic presentation of one circular cell $\Omega_C$ embedded in the domain $\Omega$. The boundaries of the cell and the entire domain are denoted by $\partial\Omega_C$ and $\partial\Omega$, respectively. We define the normal vector $\boldsymbol{n}$ of $\partial\Omega_C$ pointing towards the center of the cell. This figure is taken from Peng2023. Panel (b): A biological cell has a more irregular shape. The heterogeneous flux density over its boundary can be converted into a circular cell with (different) heterogeneous flux density. Then, using the method proposed in this article, the circular cell can be converted to mass-emitting points, i.e. sources (blue) and sinks (red), with the cell boundary now virtual.
• Figure 3.1: The locations of the Dirac points for approximating the inhomogenoues flux density $\phi_n(\boldsymbol{x}_\theta)=\phi_0 + A\sin(n\theta)$ over the circular cell boundary of radius $R$ are shown for the cases (a) of a polarized flux distribution($n=1$), and (b) of an axially oriented flux distribution ($n=2$). The off-centre Dirac points (blue) are each on a line segment connecting the cell's centre point (red) to a point on the cell boundary where $\phi_n(\boldsymbol{x})$ attains its maximum value $\phi_{max}$ (green). The distance between the cell centre and any off-centre Dirac point(s) is $r$ ($0<r<R$).
• Figure 3.2: For $n=1$, $\phi_0 = A = 1.0$ and cell radius $R=1.0$ the desired flux density $\phi(\boldsymbol{x}_\theta) = 1+\sin(\theta)$ is compared as function of the angle $\theta$ with (a) $\widetilde{\phi}(\boldsymbol{x}_\theta,t)$, computed from Equation \ref{['eq:expresion tilde Phi']} for indicated times, and (b) the limit of $\widetilde{\phi}(\boldsymbol{x}_\theta,t)$ as $t\to\infty$, given by Equation \ref{['Eq_flux_stst_1']} for various values of $r$. The solid curve is the prescribed flux density $\phi_1$ in the spatial exclusion approach and the dashed curves show the approximants $\widetilde{\phi}(\cdot,t)$ or $\widetilde{\phi}_\infty$; different colours of the curves represent different values of time $t$ or the distance between the off-centre Dirac point and the cell centre $r$.
• Figure 3.3: Selecting $\rho = 1.0$ and cell radius $R=1.0$ for $\phi = 1+\sin(2\theta)$, the flux over the cell boundary is shown versus the angle $\theta$. The solid curve is the predefined flux density in the spatial exclusion approach and the dashed curves are the flux density computed by Equation \ref{['Eq_2_Dirac_bnd_flux_t']} with $n=2$; different colours of the curves represent different values of the distance between the off-centre Dirac point and the cell centre $r$.
• Figure 5.1: The local norm difference between the solution to $(\rm BVP_S)$ using the homogeneous flux density $\phi(\boldsymbol{x}) = 1$ and the inhomogeneous flux density $\phi(\boldsymbol{x}) = 1+\rho\sin(n\theta)$, where $n\in\{1,2,3\}$ and $\rho = 0.001, 1$. Here, we show the relative $L^2-$ and $H^1-$norm difference in Panel (a) and (b), respectively. In simulations, standard parameter values have been used from Table \ref{['tab:para_all']}.

Theorems & Definitions (9)

• Proposition 2.1
• Lemma A.1
• proof
• Corollary A.1
• Corollary A.2
• Proposition B.1
• proof
• Proposition B.2
• proof