Modeling tumor progression in heterogeneous microenvironments: A cellular automata approach

TL;DR

This study develops a stochastic 2D hexagonal cellular automaton to model tumor progression within a spatially heterogeneous microenvironment, distinguishing normal and tumor stem and non-stem cells and incorporating dynamic tumor–microenvironment feedback. Proliferation, death, differentiation, and mutation rates are modulated by a local microenvironment index $m$, with $m$ itself evolving under tumor- and normal-cell influences; mutations in normal cells drive malignant transitions. Key findings show that lowering mutation rates mitigates tumor growth, a healthy microenvironment can substantially delay progression, and microenvironmental deterioration kinetics critically shape outcomes, with therapeutic simulations targeting $k_{11}$ or stabilizing $m$ yielding strong, timing-dependent suppression. The work demonstrates the utility of CA for capturing spatial heterogeneity and stochasticity in tumor evolution and suggests microenvironment-targeted strategies as part of combination therapies; code is publicly available for reproducibility.

Abstract

Understanding how microenvironmental heterogeneity influences tumor progression is essential for advancing both cancer biology and therapeutic strategies. In this study, we develop a cellular automata (CA) model to simulate tumor growth under varying microenvironmental conditions and genetic mutation rates, addressing a gap in existing studies that rarely integrate these two factors to explain tumor dynamics. The model explicitly incorporates the cellular heterogeneity of stem and non-stem cells, dynamic cell-cell interactions, and tumor-microenvironment crosstalk. Using computational simulations, we examine the synergistic effects of gene mutation rate, initial tumor burden, and microenvironmental state on tumor progression. Our results demonstrate that lowering the mutation rate significantly mitigates tumor expansion and preserves microenvironmental integrity. Interestingly, the initial tumor burden has a limited impact, whereas the initial condition of the microenvironment critically shapes tumor dynamics. A supportive microenvironment promotes proliferation and spatial invasion, while inhibitory conditions suppress tumor growth. These findings highlight the key role of microenvironmental modulation in tumor evolution and provide computational insights that may inform more effective cancer therapies.

Figures (11)

• Figure 1: Spatial structure of the cellular automaton and cell-type transition diagram.(a) Schematic of the two-dimensional cellular automaton, where each hexagonal grid cell can be in one of five states: empty, normal stem cells ($N_1$), normal cell ($N_2$), tumor stem cell ($T_1$), or tumor cell ($T_2$). (b) Diagram of cell-type transitions. All cell types are capable of self-renewal, while stem cells ($N_{1}$, $T_1$) can also differentiate into non-stem cells ($N_{2}$, $T_2$). Non-stem cells can undergo death at a rate $\mu$. Normal cells may acquire mutations and convert to tumor cells, with $\alpha_{1}$ and $\alpha_{2}$ denoting mutation rates for normal stem cells and normal non-stem cells, respectively. The heterogeneous microenvironment is represented by a continuous variable $m$ ($0 < m< 1$) assigned to each grid point. The microenvironment is dynamically influenced by the local densities of tumor and normal cells, and all cell-state transition rates are modulated by the local microenvironmental index.
• Figure 2: Overview of the stochastic simulation algorithm. The left panel depicts the main flowchart of the simulation loop. The right panels detail the two core sub-modules: the calculation of update rates for cell states and the microenvironment, and the execution of state updates for current cells.
• Figure 3: Calibration of model parameters against experimental data. Data on tumor volume derived from lung cancer Benzekry2014ClassicalMM are shown as blue dots, and the corresponding model simulation results are shown as a black solid line.
• Figure 4: Evolution dynamics of cells and the microenvironment in the absence of gene mutations. Top row (a-d): Simulation initialized with $20$ normal stem cells and a randomly distributed microenvironment $m \in (0, 0.3)$. Bottom row (e-h): Simulation initialized with $20$ normal stem cells and $20$ tumor stem cells, with the same initial microenvironment range. (a, e) Temporal evolution of cell populations. (b, f) Temporal evolution of the average microenvironment value. (c, g) Spatial distribution of all cell types at day $250$. (d, h) Spatial distribution of the microenvironment at day $250$. Results represent a single stochastic realization.
• Figure 5: Evolution dynamics of cells and microenvironment under varying gene mutation rates. Mutation rates for normal stem cells ($\alpha_1$) and normal non-stem cells ($\alpha_2$) were set as follows: (a)$(\alpha_1, \alpha_2) = (1.1\times 10^{-4}, 2.0\times 10^{-4})$. (b)$(\alpha_1, \alpha_2) = (2.2\times 10^{-4}, 4.0\times 10^{-4})$. (c)$(\alpha_1, \alpha_2) = (4.4\times 10^{-4}, 8.0\times 10^{-4})$. Data are shown from a representative single simulation for each mutation rate combination.