A Nonlocal Perfusion-Gated Angiogenesis System: Global Weak Solutions, Fast-Signal Limits, and Invasion Fronts

Abstract

Continuum angiogenesis systems often use local endothelial or vessel density as an oxygen-delivery proxy without distinguishing structurally formed vessels from pressure-supported vascular function. We formulate a two-dimensional model in which lumenized density determines a normalized nonlocal conductivity, a globally solved pressure field, and a flow-functional density that gates oxygen delivery and vessel regression. For fixed positive regularization parameters and signal relaxation times, we prove Lipschitz stability of the pressure--perfusion map and global bounded weak solvability; strong vessel compactness follows from a nonlocal ordinary differential equation stability estimate rather than spatial smoothing. As the oxygen and vascular endothelial growth factor timescales vanish simultaneously, weak solutions converge along a subsequence to a parabolic--elliptic--ordinary differential equation system, with both fast fields converging strongly in \(L^2(0,T;H^1)\). A locally frozen one-dimensional reduction admits monotone fronts precisely at or above the threshold \(B+2\sqrt{DR}\) for \(B\geq-\sqrt{DR}\) and \(-DR/B\) otherwise. Finite-volume experiments verify the analytical mechanisms and show that equal-mass vessel fields can generate distinct functional masses and oxygenation levels.