A continuous model of Angiogenesis: Initiation, extension, and maturation of new blood vessels modulated by vascular endothelial growth factor, Angiopoietins, platelet-derived growth factor-B, and pericytes

This work presents a continuous model for three early stage events in angiogenesis: initiation, sprout extension, and vessel maturation. We carefully examine the regulating mechanisms of vascular endothelial growth factor (VEGF) and angiopoietins (Ang1 and Ang2) on the proliferation, migration and maturation of endothelial cells through their endothelium-specific receptor tyrosine kinase VEGFR2 and Tie2, respectively. We also consider the effect of platelet-derived growth factor-B (PDGF-B) on the proliferation and migration of pericytes. For growth factors, we present a mathematical model integrating molecular reactions on blood vessels with tissue-level diffusion. For capillary extension, we develop a visco-elastic model to couple tip cell protrusion, endothelium elasticity, and stalk cell proliferation. Our model reproduces corneal angiogenesis experiments and several anti-angiogenesis therapy results. This model also demonstrates that (1) the competition between Ang1 and Ang2 is the angiogenic switch; (2) the maturation process modulated by pericytes and angiopoietins is crucial to vessel normalization and can explain the resistance to anti-VEGF therapy; (3) combined anti-pericyte and anti-VEGF therapy enhances blood vessel regression over anti-VEGF therapy alone.