Abstract
We present a hybrid partial differential equation-agent-based model (PDE-ABM). In our framework, tumor cells secrete tumor angiogenic factor (TAF), while endothelial cells chemotactically migrate and branch in response. Reaction-diffusion PDEs for TAF, oxygen, and cytotoxic drug are coupled to discrete stochastic dynamics of tumor cells and endothelial tip cells, ensuring multiscale integration. Motivated by observed perfusion heterogeneity in tumors and its pharmacokinetic consequences, we conduct a linear stability analysis for a reduced endothelial-TAF reaction-diffusion subsystem and derive an explicit finite-domain threshold for Turing instability. We demonstrate that bidirectional coupling, where endothelial cells both chemotactically migrate along TAF gradients and secrete TAF, is necessary and sufficient to generate spatially periodic vascular clusters and inter-cluster hypoxic regions. These emergent patterns produce heterogeneous drug penetration and resistant niches. Our results identify TAF clearance, chemotactic sensitivity, and endothelial motility as effective levers to homogenize perfusion. The model is two-dimensional and employs simplified kinetics, and we outline necessary extensions to three dimensions and saturable kinetics required for quantitative calibration. The study links reaction-diffusion mechanisms with clinical principles and suggests actionable strategies to mitigate resistance by targeting endothelial-TAF feedback.