Coupling volume-excluding compartment-based models of diffusion at different scales: Voronoi and pseudo-compartment approaches

Abstract

Numerous processes across both the physical and biological sciences are driven by diffusion. Partial differential equations are a popular tool for modelling such phenomena deterministically, but it is often necessary to use stochastic models to accurately capture the behaviour of a system, especially when the number of diffusing particles is low. The stochastic models we consider in this paper are 'compartment-based': the domain is discretized into compartments, and particles can jump between these compartments. Volume-excluding effects (crowding) can be incorporated by blocking movement with some probability. Recent work has established the connection between fine-and coarse-grained models incorporating volume exclusion, but only for uniform lattices. In this paper, we consider non-uniform, hybrid lattices that incorporate both fine-and coarse-grained regions, and present two different approaches to describe the interface of the regions. We test both techniques in a range of scenarios to establish their accuracy, benchmarking against fine-grained models, and show that the hybrid models developed in this paper can be significantly faster to simulate than the fine-grained models in certain situations and are at least as fast otherwise.

Accurate and efficient discretizations for stochastic models providing near agent-based spatial resolution at low computational cost

“…The CBM also encodes additional biologically inspired features, such as crowding effects, whereby potential movement and proliferation events cannot occur if the target compartment is fully occupied with agents. Inclusion of crowding distinguishes the CBM from other agent-based models that describe reaction-diffusion processes [27][28][29][30][31][32][33][34] and provides additional biological realism, since many experimental observations confirm that crowding effects are very important in proliferation assays and scratch assays [1,3,5,6,11].…”

Accurate and efficient discretisations for stochastic models providing near agent-based spatial resolution at low computational cost

“…We consider simulations of scratch assays where clustering is absent (see cell line 1 in Figure 1, Figure 6(a)(e)) and simulations of scratch assays where clustering is present (cell line 2 in Figure 1, Figure 6(f )(j)). To apply the CBM, we must rst consider how the diusion term in Equation (2), D∇ 2 C, changes when varying m. Previous examination of the continuum limit of diusion-only compartmentbased models [28,29] reveals that the jump rates between adjacent compartments scales with 1/m 2 , for m > 1. However, the models proposed in [28,29] assume that an isolated agent will always leave its compartment, rather than having non-zero probability to remain within its compartment.…”

“…To apply the CBM, we must rst consider how the diusion term in Equation (2), D∇ 2 C, changes when varying m. Previous examination of the continuum limit of diusion-only compartmentbased models [28,29] reveals that the jump rates between adjacent compartments scales with 1/m 2 , for m > 1. However, the models proposed in [28,29] assume that an isolated agent will always leave its compartment, rather than having non-zero probability to remain within its compartment. Since agents in the CBM attempt to move out of a particular compartment with probability 1/m, we divide the scaled diusivity proposed in [28,29], m 2 D, by m. Therefore, the diusivity D of the CBM continuum limit with m = 1 becomes mD for the CBM continuum limit with m > 1, and the continuum limit description of CBM simulations of scratch assays can be written as…”

“…However, the models proposed in [28,29] assume that an isolated agent will always leave its compartment, rather than having non-zero probability to remain within its compartment. Since agents in the CBM attempt to move out of a particular compartment with probability 1/m, we divide the scaled diusivity proposed in [28,29], m 2 D, by m. Therefore, the diusivity D of the CBM continuum limit with m = 1 becomes mD for the CBM continuum limit with m > 1, and the continuum limit description of CBM simulations of scratch assays can be written as…”

Coarse-graining and hybrid methods for efficient simulation of stochastic multi-scale models of tumour growth

“…According to this methodology, the mean-field limit of the system is used in low-noise regions which are then coupled to the full stochastic dynamics describing the high-noise regions. The coupling between both descriptions is achieved by means of appropriately defined boundary conditions at the interface(s) between mean-field and stochastic regions [49] , [22] , [34] , [23] , [62] , [67] , [75] , [71] .…”