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 L2(0,T;H1). A locally frozen one-dimensional reduction admits monotone fronts precisely at or above the threshold B+2√(DR) for B≥−√(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.
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BibTeX
@misc{yu2026nonlocal,
title = {{A Nonlocal Perfusion-Gated Angiogenesis System: Global Weak Solutions, Fast-Signal Limits, and Invasion Fronts}},
author = {Jiguang Yu and Louis Shuo Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13753},
eprint = {2609.13753},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13753},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Nonlocal Perfusion-Gated Angiogenesis System: Global Weak Solutions, Fast-Signal Limits, and Invasion Fronts
AU - Jiguang Yu
AU - Louis Shuo Wang
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13753
UR - https://arxiv.org/abs/2609.13753
AB - 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√(DR) for B≥−√(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.
ER -