# A viscoelastic model of blood capillary extension and regression: derivation, analysis, and simulation

## Abstract

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 captures two cases when the biologically meaningful solution exists: (1) the cell density decreases from root to tip, which may occur in vessel regression; (2) the cell density is time-independent and is of small variation along the capillary, which may occur in capillary extension without proliferation. The linear analysis with perturbation in cell density due to proliferation or death predicts the global biological solution exists 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, which validate the linear analysis theory. Numerical simulations demonstrate this model can reproduce angiogenesis experiments under several biological conditions including blood vessel extension without proliferation and blood vessel regression.

## Keywords

Angiogenesis Viscoelastic Growth pressure Extension Regression## Mathematics Subject Classification

92C10 35K61## Notes

### Acknowledgments

The authors thank Dapeng Du in Northeast Normal University (China) and Jeffrey Rauch in University of Michigan for helpful discussions. Xie thanks the support from University of Michigan where part of the work was done. Xie was supported in part by an NSFC Grant 11241001 and a startup grant from Shanghai Jiao Tong University. Zheng thanks Central Michigan University ORSP Early Career Investigator Grant #C61373.

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