A viscoelastic model of blood capillary extension and regression: derivation, analysis, and simulation
 Xiaoming Zheng,
 Chunjing Xie
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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 onedimensional 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 timeindependent 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 blowups 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.
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Within this Article
 Introduction
 Onedimensional viscoelastic model of blood capillary extension and regression
 Inviscid problem
 Numerical simulations of blood capillary extension/regression
 Conclusion
 References
 References
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 Title
 A viscoelastic model of blood capillary extension and regression: derivation, analysis, and simulation
 Journal

Journal of Mathematical Biology
Volume 68, Issue 12 , pp 5780
 Cover Date
 20140101
 DOI
 10.1007/s0028501206248
 Print ISSN
 03036812
 Online ISSN
 14321416
 Publisher
 Springer Berlin Heidelberg
 Additional Links
 Topics
 Keywords

 Angiogenesis
 Viscoelastic
 Growth pressure
 Extension
 Regression
 92C10
 35K61
 Industry Sectors
 Authors

 Xiaoming Zheng ^{(1)}
 Chunjing Xie ^{(2)}
 Author Affiliations

 1. Department of Mathematics, Central Michigan University, Mount Pleasant, MI, 48859, USA
 2. Department of Mathematics, Institute of Natural Sciences, Ministry of Education Key Laboratory of Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai, 200240, China