Comparing methods for modelling spreading cell fronts

Abstract

Spreading cell fronts play an essential role in many physiological processes. Classically, models of this process are based on the Fisher-Kolmogorov equation; however, such continuum representations are not always suitable as they do not explicitly represent behaviour at the level of individual cells. Additionally, many models examine only the large time asymptotic behaviour, where a travelling wave front with a constant speed has been established. Many experiments, such as a scratch assay, never display this asymptotic behaviour, and in these cases the transient behaviour must be taken into account. We examine the transient and the asymptotic behaviour of moving cell fronts using techniques that go beyond the continuum approximation via a volume-excluding birth-migration process on a regular one-dimensional lattice. We approximate the averaged discrete results using three methods: (i) mean-field, (ii) pair-wise, and (iii) one-hole approximations. We discuss the performance of these methods, in comparison to the averaged discrete results, for a range of parameter space, examining both the transient and asymptotic behaviours. The one-hole approximation, based on techniques from statistical physics, is not capable of predicting transient behaviour but provides excellent agreement with the asymptotic behaviour of the averaged discrete results, provided that cells are proliferating fast enough relative to their rate of migration. The mean-field and pair-wise approximations give indistinguishable asymptotic results, which agree with the averaged discrete results when cells are migrating much more rapidly than they are proliferating. The pair-wise approximation performs better in the transient region than does the mean-field, despite having the same asymptotic behaviour. Our results show that each approximation only works in specific situations, thus we must be careful to use a suitable approximation for a given system, otherwise inaccurate predictions could be made.

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

“…where T = λt and C(0) is the initial agent density. Previous examination of stochastic models with m = 1 and r p /r m 1 [5,7,13,16] reveal that the averaged model data agrees well with the solution of the continuum limit, Equation (5). Under these conditions, pairwise correlations between the occupancy status of lattice sites are negligible.…”

“…Cell-level processes, including proliferation, death, and migration, drive tissue-level processes during regeneration, development and repair [13]. Traditionally, mathematical models of development and repair account for such tissue-level processes by modelling the cell population density with ordinary dierential equations (ODEs) [47] and partial dierential equations (PDEs) [810].…”

“…Traditionally, mathematical models of development and repair account for such tissue-level processes by modelling the cell population density with ordinary dierential equations (ODEs) [47] and partial dierential equations (PDEs) [810]. These continuum descriptions can be parametrised to predict the cell population density growth prole in commonlyused cell biology assays, such as proliferation assays [57,11,12] and scratch assays [810,13,14]. In a proliferation assay (Figure 1), cells are seeded uniformly on a two-dimensional substrate.…”

Assessing the role of spatial correlations during collective cell spreading

“…Unlike mean–field models, individual–based models explicitly incorporate spatial correlation effects202122 and allow us to visualise the cell spreading process in a way that is directly comparable with experimental images10112324. However, individual–based models are computationally expensive and many realisations are required to obtain reliable statistics, meaning that it is often difficult to simulate realistic biological systems22.…”

“…However, individual–based models are computationally expensive and many realisations are required to obtain reliable statistics, meaning that it is often difficult to simulate realistic biological systems22. Mean–field models are more amenable to analytical exploration and hence can be advantageous over individual–based models provided that the mean–field assumption is an accurate representation of the relevant system1722.…”