The Completion of the ABM: A Unified Continuum Operator for Tumour Growth Under Mechanical and Evolutionary Dynamics

This repository provides a companion mathematical framework for the agent-based model (ABM) developed by Macnamara, Ramis-Conde, Lorenzi, and Caiazzo (2024) for tumour growth incorporating mechanical and evolutionary cell dynamics. The ABM models cancer cells as viscoelastic spheres with Hertzian contact forces, stochastic HIF-linked phenotype fluctuations, and oxygen-dependent fitness. While the ABM captures rich single-cell behaviour, it lacks a single governing equation that enables analytical tractability. This work extracts the full mathematical structure of the ABM and assembles it into a unified phenotype-structured partial differential equation (PDE). The derivation follows standard mean-field and diffusion limits, producing three coupled components: (i) nonlinear spatial diffusion and pressure-driven drift from overdamped Langevin mechanics, (ii) exact phenotype diffusion from the discrete random walk, and (iii) a selection term from the mitosis and death rules. The resulting operator reproduces all qualitative behaviours observed in the ABM, including phenotype concentration under homogeneous oxygen, spatial segregation under heterogeneous oxygen, interface mixing, and crowding-dependent growth suppression. The repository contains: A complete mathematical derivation of the continuum operator from the ABM rules (extracted, not reinterpreted). Analytical characterisation of steady states, stability, and non-oscillatory dynamics. A calibration protocol for the mechanical diffusion term using ABM simulation data. A numerical validation framework for comparing PDE and ABM outputs. Identified extensions to multi-phenotype systems, additional environmental fields, evolutionary games, and therapy-driven selection. This is not a replacement for the authors' ABM but a mathematical completion: their biological and computational insights form the core, and this framework wraps those insights in the continuum-level structure that allows them to be analysed, compared, and extended with analytical clarity. Keywords: agent-based model, continuum limit, phenotype-structured PDE, tumour growth, hypoxia-inducible factors, mechanical interactions, evolutionary dynamics, mean-field limit References: Macnamara, C.K., Ramis-Conde, I., Lorenzi, T., Caiazzo, A. (2024). An agent-based modelling framework for tumour growth incorporating mechanical and evolutionary aspects of cell dynamics. bioRxiv preprint, doi:10.1101/2024.05.30.596685. % --- INTRODUCTION OF THE OPERATOR --- \[The continuum evolution of the phenotype-structured density ρ(o,x,t) is governed by the unified operator extracted directly from the ABM:\] % --- MAIN PDE --- \[∂ ρ(o,x,t)/∂ t=∇_x · ( D(x,t)\,∇_x ρ(o,x,t) + v(x,t)\,ρ(o,x,t))+ D_o\,∂^2 ρ/∂ o^2+ R\!(o,\,c(x),\,n(x,t))\,ρ(o,x,t)\] \[n(x,t) = ∫_0^1 ρ(o,x,t)\,do\] \[∂ ρ/∂ o = 0 o=0,\,1\] % --- TRANSPORT COEFFICIENTS --- \[D_o = β\,(Δ o)^2/2\] \[D_0 = a_f^2/2γ^2\] \[D(x,t) = D_0\,g\!(n(x,t)),(x,t) = -1/γ\,∇_x P\!(n(x,t))\] \[g(n) ≥ 0,(0)=1,\ monotone\] \[P(n) ≥ 0,(0)=0,\ monotone increasing\] % --- FITNESS AND SELECTION --- \[R(o,c,n) = p(o,c) - κ\,p(n)\] \[p(o,c)= 4_h(1-o^2)+ p_e\,c/c+c_T\,[1-(1-o)^2]\] \[p(n) = 5n\] % --- STEADY STATES --- \[õ(c)= p_e\,c/4_h(c+c_T) + p_e\,c\] \[n^*= {p\!(õ(c),\,c)}{5κ}\] % --- ABM MECHANICAL FORCE --- \[Fᵢⱼ(t)=[ 4/3EeffReff1/2hᵢⱼ3/2 - α(R_i - {hᵢⱼ}{4})hᵢⱼ]{dᵢⱼ}{|dᵢⱼ|}\] \[hᵢⱼ = R_i + R_j - |dᵢⱼ|,ᵢⱼ = X_i - X_j\] Contact: For enquiries or research questions related to this work, email matthewcarlo.research@gmail.com