Analysis and Mean-Field Limit of a Hybrid PDE-ABM Modeling Angiogenesis-Regulated Resistance Evolution

Mathematical modeling is indispensable in oncology for unraveling the interplay between tumor growth, vascular remodeling, and therapeutic resistance. We present a hybrid modeling framework (continuum-discrete) and present its hybrid mathematical formulation as a coupled partial differential equation–agent-based (PDE-ABM) system. It couples reaction–diffusion fields for oxygen, drug, and tumor angiogenic factor (TAF) with discrete vessel agents and stochastic phenotype transitions in tumor cells. Stochastic phenotype switching is handled with an exact Gillespie algorithm (a Monte Carlo method that simulates random phenotype flips and their timing), while moment-closure methods (techniques that approximate higher-order statistical moments to obtain a closed, tractable PDE description) are used to derive mean-field PDE limits that connect microscale randomness to macroscopic dynamics. We provide existence/uniqueness results for the coupled PDE-ABM system, perform numerical analysis of discretization schemes, and derive analytically tractable continuum limits. By linking stochastic microdynamics and deterministic macrodynamics, this hybrid mathematical formulation—i.e., the coupled PDE-ABM system—captures bidirectional feedback between hypoxia-driven angiogenesis and resistance evolution and provides a rigorous foundation for predictive, multiscale oncology models.