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Angiogenesis and Neovascularization

The formation of new blood vessels from pre-existing ones. This process is critically regulated by growth factors like Vascular Endothelial Growth Factor (VEGF), ensuring oxygen and nutrient supply.
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The statement of the theorem

Consider the concentration of Vascular Endothelial Growth Factor (VEGF), CVEGFC_{VEGF}, and the vessel density, ρv\rho_v. The dynamics are governed by a reaction-diffusion system:\n∂CVEGF∂t=DVEGF∇2CVEGF−kbindCVEGF[Endothelial]+SVEGF(ρv)\frac{\partial C_{VEGF}}{\partial t} = D_{VEGF} \nabla^2 C_{VEGF} - k_{bind} C_{VEGF} [\text{Endothelial}] + S_{VEGF}(\rho_v) \n∂ρv∂t=kangiCVEGFρv(1−ρvρmax)−kpruneρv\frac{\partial \rho_v}{\partial t} = k_{angi} C_{VEGF} \rho_v (1 - \frac{\rho_v}{\rho_{max}}) - k_{prune} \rho_v \nwhere kangik_{angi} models the growth rate stimulated by VEGF, and kprunek_{prune} accounts for vessel regression.