A viscoelastic model of capillary growth: derivation, analysis, and simulation

This work studies a fundamental problem in blood capillary growth: how the cell proliferation or death induces the stress response and the capillary extension or regression. We develop a one-dimensional viscoelastic model of blood capillary extension/regression under nonlinear friction with surroundings, analyze its solution properties, and simulate various growth patterns in angiogenesis. The mathematical model treats the cell density as the growth pressure eliciting a viscoelastic response from the cells, which again induces extension or regression of the capillary. Nonlinear analysis provides some conditions to guarantee the global existence of biologically meaningful solutions, while linear analysis and numerical simulations predict global biological solutions exist provided the change in cell density is sufficiently slow in time. Examples with blow-ups are captured by numerical approximations and the global solutions are recovered by slow growth processes. Numerical simulations demonstrate this model can reproduce angiogenesis experiments under several biological conditions including blood vessel extension without proliferation and blood vessel regression.