[Paper Review] Individual-based and continuum models of phenotypically heterogeneous growing cell populations

Fiona R. Macfarlane, Xinran Ruan+1 more|arXiv (Cornell University)|Feb 14, 2022

Mathematical Biology Tumor Growth4 citations

TL;DR

This paper develops an individual-based model for phenotypically heterogeneous growing cell populations, where cells vary in proliferation and migration rates via a phenotypic structuring variable. It formally derives a non-local continuum PDE as the deterministic limit, showing excellent quantitative agreement between IB and PDE simulations in large-population regimes, while highlighting discrepancies due to demographic stochasticity in low-cell-number scenarios.

ABSTRACT

Existing studies comparing individual-based models of growing cell populations and their continuum counterparts have mainly focused on homogeneous populations, in which all cells have the same phenotypic characteristics. However, significant intercellular phenotypic variability is commonly observed in cellular systems. Therefore, we develop here an individual-based model for the growth of phenotypically heterogeneous cell populations. In this model, the phenotypic state of each cell is described by a structuring variable that captures intercellular variability in cell proliferation and migration rates. The model tracks the spatial evolutionary dynamics of single cells, which undergo pressure-dependent proliferation, heritable phenotypic changes and directional movement in response to pressure differentials. We formally show that the continuum limit of this model comprises a non-local partial differential equation for the cell population density, which generalises earlier models of growing cell populations. Results of the individual-based model illustrate how proliferation-migration tradeoffs shaping the evolution of single cells can lead to the formation of travelling waves at the population level where highly-mobile cells locally dominate at the invasive front, while more-proliferative cells are found at the rear. We demonstrate that there is an excellent quantitative agreement between these results and the results of numerical simulations and formal travelling-wave analysis of the continuum model, when sufficiently large cell numbers are considered. We provide numerical evidence of scenarios in which the predictions of the two models may differ due to demographic stochasticity, which cannot be captured by the continuum model. This indicates the importance of integrating individual-based and continuum approaches when modelling the growth of phenotypically heterogeneous cell populations.

Motivation & Objective

• To model phenotypically heterogeneous cell populations with variable proliferation and migration rates using an individual-based (IB) approach.
• To formally derive the deterministic continuum limit of the IB model as a non-local partial differential equation (PDE).
• To compare the dynamics of the IB model with those of the derived PDE model through numerical simulations and travelling-wave analysis.
• To identify conditions under which the IB and PDE models agree or diverge, particularly due to demographic stochasticity.
• To demonstrate the utility of integrating IB and continuum models for studying complex spatiotemporal dynamics in growing cell populations.

Proposed method

• Models each cell as an individual agent with a phenotypic state variable capturing variation in proliferation and migration rates.
• Implements a discrete-time branching random walk on physical space and phenotypic state space, incorporating pressure-dependent proliferation and directional movement.
• Uses heritable phenotypic changes to model evolutionary dynamics at the single-cell level.
• Applies asymptotic and limiting procedures to derive the deterministic continuum limit, resulting in a non-local PDE for cell population density.
• Performs numerical simulations of the IB model under varying population sizes and compares results with numerical solutions and formal travelling-wave analysis of the PDE.
• Integrates hybrid modelling techniques to explore potential extensions involving chemical species such as nutrients or chemoattractants.

Experimental results

Research questions

• RQ1 How does phenotypic heterogeneity in proliferation and migration rates influence the emergent spatial dynamics of growing cell populations?
• RQ2 To what extent does the deterministic continuum PDE derived from the IB model accurately reproduce the population-level dynamics observed in the IB model?
• RQ3 In what parameter regimes do the IB and PDE models diverge, and what role does demographic stochasticity play in these discrepancies?
• RQ4 Can the derived non-local PDE model capture the formation of travelling waves with spatially structured phenotypes, such as highly mobile cells at the front and proliferative cells at the rear?
• RQ5 How can individual-based and continuum modelling approaches be integrated to improve the robustness and biological fidelity of models of phenotypically heterogeneous cell populations?

Key findings

• The IB model successfully reproduces the formation of travelling waves in which highly mobile cells dominate the invasive front and more proliferative cells accumulate at the rear, consistent with observations in gliomas.
• There is excellent quantitative agreement between numerical simulations of the IB model and both numerical solutions and formal travelling-wave analysis of the derived non-local PDE model when sufficiently large cell numbers are used.
• Discrepancies between the IB and PDE models arise in low-cell-number regimes due to demographic stochasticity, which the deterministic PDE cannot capture.
• The formal derivation confirms that the continuum limit of the IB model is a non-local PDE, generalizing earlier models to the case of phenotypic heterogeneity.
• The IB model and its continuum counterpart are robustly consistent in large-population regimes, validating the use of the PDE as a mean-field approximation.
• The framework is extendable to higher spatial dimensions, irregular lattices, off-lattice spatial representations, and hybrid models incorporating chemical species.