Stochastic multi-scale models of competition within heterogeneous cellular populations: Simulation methods and mean-field analysis

Abstract

We propose a modelling framework to analyse the stochastic behaviour of heterogeneous, multi-scale cellular populations. We illustrate our methodology with a particular example in which we study a population with an oxygen-regulated proliferation rate. Our formulation is based on an age-dependent stochastic process. Cells within the population are characterised by their age (i.e. time elapsed since they were born). The age-dependent (oxygen-regulated) birth rate is given by a stochastic model of oxygen-dependent cell cycle progression. Once the birth rate is determined, we formulate an age-dependent birth-and-death process, which dictates the time evolution of the cell population. The population is under a feedback loop which controls its steady state size (carrying capacity): cells consume oxygen which in turn fuels cell proliferation. We show that our stochastic model of cell cycle progression allows for heterogeneity within the cell population induced by stochastic effects. Such heterogeneous behaviour is reflected in variations in the proliferation rate. Within this set-up, we have established three main results. First, we have shown that the age to the G1/S transition, which essentially determines the birth rate, exhibits a remarkably simple scaling behaviour. Besides the fact that this simple behaviour emerges from a rather complex model, this allows for a huge simplification of our numerical methodology. A further result is the observation that heterogeneous populations undergo an internal process of quasi-neutral competition. Finally, we investigated the effects of cell-cycle-phase dependent therapies (such as radiation therapy) on heterogeneous populations. In particular, we have studied the case in which the population contains a quiescent sub-population. Our mean-field analysis and numerical simulations confirm that, if the survival fraction of the therapy is too high, rescue of the quiescent population occurs. This gives rise to emergence of resistance to therapy since the rescued population is less sensitive to therapy.

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

“…Before proceeding further, we present a general overview of the stochastic multi-scale model of tumour growth as well as a summarised discussion of the different elements involved in the formulation of the stochastic multi-scale model. The model presented here is closely related to that presented in [17] , the main difference being the introduction of spatial heterogeneities, which were neglected in our previous work. The model we present in this paper accounts for processes with widely different characteristic time scales, as depicted in the scheme shown in Fig.…”

“…is explicitly described by models of cell behaviour of varying levels of complexity [3] , [40] , [61] , [50] , [19] , [55] , [58] , [70] , [15] , [39] . Further to the individual-based approach to multi-scale modelling of biological cell populations, we have recently introduced new stochastic models that allow to analyse the effects of fluctuations, both at the intracellular level (intrinsic noise in signalling pathways and gene regulatory networks) and at the level of the birth-and-death dynamics of cells [32] , [17] .…”

“…In this paper, we extend and further develop the hybrid method formulated by Spill et al [67] for stochastic reaction–diffusion systems to stochastic multi-scale models of tumour growth. Such models [32] , [17] consider fluctuations regarding both number of cells ( population noise) and the intracellular (cell-cycle) dynamics ( structure noise), and consequently any attempt to formulate a hybrid method for such systems must find a way to accommodate both types of noise. Structure noise is associated with noise at the intracellular level and it manifests itself in fluctuations of the birth rate.…”

“…In Section 2 , we present a summary of the stochastic multi-scale model. For an in-depth presentation, the reader is referred to de la Cruz et al [17] . Section 3 contains a multiple scale asymptotic analysis which concludes with the derivation of versions of both the full stochastic model and its mean-field limit where the intracellular dynamics (i.e.…”

“…In Section 5 , we proceed with an assessment of the accuracy of the coarse-grained and hybrid models for travelling wave solutions. We take as a benchmark the full stochastic multi-scale model solved by means of the age-structured Gillespie algorithm [17] . During this analysis we conclude that failure of the coarse-grained growth rate to describe the population dynamics at the leading edge of the front is responsible for the discrepancies between the travelling wave speeds.…”