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

- 343 Downloads
- 4 Citations

## 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.

## References

- Anderson ARA, Chaplain MAJ (1998) Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull Math Biol 60:857–900CrossRefMATHGoogle Scholar
- Ausprunk DH, Folkman J (1977) Migration and proliferation of endothelial cells in preformed and newly formed blood vessels during tumor angiogenesis. Microvasc Res 14:53–65CrossRefGoogle Scholar
- Balding D, McElwain DLS (1985) A mathematical model of tumor-induced capillary growth. J Theor Biol 114:53–73CrossRefGoogle Scholar
- Bauer AL, Jackson TL, Jiang Y (2007) A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis. Biophys J 92:3105CrossRefGoogle Scholar
- Bauer AL, Jackson TL, Jiang Y (2009) Topography of extracellular matrix mediates vascular morphogenesis and migration speeds in angiogenesis. PLoS Comput Biol 5:e1000445CrossRefMathSciNetGoogle Scholar
- Bausch AR, Ziemann F, Boulbitch AA, Jacobson K, Sackmann E (1998) Local measurements of viscoelastic parameters of adherent cell surfaces by magnetic bead microrheometry. Biophys J 75:2038–2049CrossRefGoogle Scholar
- Bentley K, Gerhardt H, Bates PA (2008) Agent-based simulation of notch mediated tip cell selection in angiogenic sprout initialisation. J Theor Biol 250:25–36CrossRefGoogle Scholar
- Bentley K, Mariggi G, Gerhardt H, Bates PA (2009) Tipping the balance: robustness of tip cell selection, migration and fusion in angiogenesis. PLoS Comput Biol 5(10):e1000549CrossRefGoogle Scholar
- Byrne HM, Chaplain MAJ (1995) Mathematical models for tumour angiogenesis: numerical simulations and nonlinear wave solutions. Bull Math Biol 57:461–486MATHGoogle Scholar
- Capasso V, Morale D (2009) Stochastic modelling of tumour-induced angiogenesis. J Math Biol 58:219–233CrossRefMATHMathSciNetGoogle Scholar
- Carmeliet P, Jain RK (2011) Molecular mechanisms and clinical applications of angiogenesis. Nature 473:298–307CrossRefGoogle Scholar
- Costa KD, Sim AJ, Yin FCP (2006) Non-hertzian approach to analyzing mechanical properties of endothelial cells probed by atomic force microscopy. J Biomech Eng 128:176–184CrossRefGoogle Scholar
- CRC (1992–1993) Handbook of chemistry and physics, 73rd edn. Chemical Rubber Publishing Company, Boca RatonGoogle Scholar
- Dastjerdi MH, Al-Arfaj KM, Nallasamy N, Hamrah P, Jurkunas UV, Pavan-Langston D, Dana R (2009) Topical bevacizumab in the treatment of corneal neovascularization: results of a prospective, open-label, noncomparative study. Arch Ophthalmol 127:381–389CrossRefGoogle Scholar
- De Smet F, Segura I, De Bock K, Hohensinner PJ, Carmeliet P (2009) Mechanisms of vessel branching: filopodia on endothelial tip cells lead the way. Arterioscler Thromb Vasc Biol 29:639–649CrossRefGoogle Scholar
- Fernandez P, Ott A (2008) Single cell mechanics: stress stiffening and kinematic hardening. Phys Rev Lett 100:238102CrossRefGoogle Scholar
- Fozard JA, Byrne HM, Jensen OE, King JR (2010) Continuum approximations of individual-based models for epithelial monolayers. Math Med Biol 27:39–74CrossRefMATHMathSciNetGoogle Scholar
- Fung YC, Tong P (2001) Classical and computational solid mechanics. World Scientific, SingaporeGoogle Scholar
- Garikipati K (2009) The kinematics of biological growth. Appl Mech Rev 62:030801CrossRefGoogle Scholar
- Gerhardt H et al (2003) VEGF guides angiogenic sprouting utilizing endothelial tip cell filopodia. J Cell Biol 161:1163–1177CrossRefGoogle Scholar
- Gerhardt H (2008) VEGF and endothelial guidance in angiogenic sprouting. Organogenesis 4:241–246CrossRefGoogle Scholar
- Gerhardt H, Betsholtz C (2005) How do endothelial cells orientate? EXS 94:3–15Google Scholar
- Gracheva ME, Othmer HG (2004) A continuum model of motility in ameboid cells. Bull Math Biol 66:167–193CrossRefMathSciNetGoogle Scholar
- Hamilton NB, Attwell D, Hall CN (2010) Pericyte-mediated regulation of capillary diameter: a component of neurovascular coupling in health and disease. Front Neuroenerg 2:1CrossRefGoogle Scholar
- Holmes MJ, Sleeman BD (2000) A mathematical model of tumour angiogenesis incorporating cellular traction and viscoelastic effects. J Theor Biol 202:95–112CrossRefGoogle Scholar
- Jackson T, Zheng X (2010) A cell-based model of endothelial cell migration, proliferation and maturation during corneal angiogenesis. Bull Math Biol 72:830–868CrossRefMATHMathSciNetGoogle Scholar
- Jo N, Mailhos C, Ju M et al (2006) Inhibition of platelet-derived growth factor B signaling enhances the efficacy of anti-vascular endothelial growth factor therapy in multiple models of ocular neovascularization. Am J Pathol 168(6):2036–2053CrossRefGoogle Scholar
- Lamalice L, Le Boeuf F, Huot J (2007) Endothelial cell migration during angiogenesis. Circ Res 100:782–794CrossRefGoogle Scholar
- Larripa K, Mogilner A (2006) Transport of a 1d viscoelastic actin-myosin strip of gel as a model of a crawling cell. Physica A 372:113–123CrossRefGoogle Scholar
- Levine HA, Nilsen-Hamilton M (2006) Angiogenesis-a biochemial/mathematical perspective. In: Friedman A (ed) Tutorials in mathematical biosciences III. Springer, Berlin, p 65Google Scholar
- Levine HA, Pamuk S, Sleeman BD, Nilsen-Hamilton M (2001) Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma. Bull Math Biol 63:801–863CrossRefGoogle Scholar
- Lieberman GM (1996) Second order parabolic differential equations. World Scientific, SingaporeCrossRefMATHGoogle Scholar
- Liu G, Qutub AA, Vempati P, Popel AS (2011) Module-based multiscale simulation of angiogenesis in skeletal muscle. Theor Biol Med Model 8:6CrossRefGoogle Scholar
- Manoussaki D (2003) A mechanochemical model of angiogenesis and vasculogenesis. ESAIM Math Model Numer Anal 37:581–599CrossRefMATHMathSciNetGoogle Scholar
- McDougall SR, Anderson AR, Chaplain MA, Sherratt JA (2002) Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies. Bull Math Biol 64(4):673–702CrossRefGoogle Scholar
- McDougall SR, Anderson ARA, Chaplain MAJ (2006) Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: clinical implications and therapeutic targeting strategies. J Theor Biol 241:564–589CrossRefMathSciNetGoogle Scholar
- Mantzaris N, Webb S, Othmer HG (2004) Mathematical modeling of tumor-induced angiogenesis. J Math Biol 49:111–187CrossRefMATHMathSciNetGoogle Scholar
- Mi Q, Swigon D, Rivière R, Selma C, Vodovotz Y, Hackam DJ (2007) One-dimensional elastic continuum model of enterocyte layer migration. Biophys J 93:3745–3752CrossRefGoogle Scholar
- Milde F, Bergdorf M, Koumoutsakos P (2008) A hybrid model for three-dimensional simulations of sprouting angiogenesis. Biophys J 95:3146–3160CrossRefGoogle Scholar
- Peirce SM (2008) Computational and mathematical modeling of angiogenesis. Microcirculation 15:739–751CrossRefGoogle Scholar
- Peirce SM, Van Gieson EJ, Skalak TC (2004) Multicellular simulation predicts microvascular patterning and in silico tissue assembly. FASEB J 18:731–733Google Scholar
- Pettet GJ, Byrne HM, McElwain DLS, Norbury J (1996) A model of wound-healing angiogenesis in soft tissue. Math Biosci 263:1487–1493Google Scholar
- Plank MJ, Sleeman BD (2003) A reinforced random walk model of tumor angiogenesis and anti-angiogenesis strategies. Mathe Med Biol 20:135–181CrossRefMATHGoogle Scholar
- Plank MJ, Sleeman BD (2004) Lattice and non-lattice models of tumour angiogenesis. Bull Math Biol 66:1785–1819CrossRefMathSciNetGoogle Scholar
- Plank MJ, Sleeman BD, Jones PF (2004) A mathematical model of tumour angiogenesis, regulated by vascular endothelial growth factor and the angiopoietins. J Theor Biol 229:435–454CrossRefMathSciNetGoogle Scholar
- Prass M, Jacobson K, Mogilner A, Radmacher M (2006) Direct measurement of the lamellipodial protrusive force in a migrating cell. J Cell Biol 174:767–772CrossRefGoogle Scholar
- Pries AR, Secomb TW, Gaehtgens P (1998) Structural adaptation and stability of microvascular networks: theory and simulations. Am J Physiol 275:349–360Google Scholar
- Pries AR, Reglin B, Secomb TW (2001) Structural adaptation of microvascular networks: functional roles of adaptive responses. Am J Physiol Heart Circ Physiol 281:H1015–H1025Google Scholar
- Qutub A, Mac Gabhann A (2009) Multiscale models of angiogenesis: integration of molecular mechanisms with cell- and organ-level models. IEEE Eng Med Biol 28:14–31CrossRefGoogle Scholar
- Qutub A, Popel A (2009) Elongation, proliferation and migration differentiate endothelial cell phenotypes and determine capillary sprouting. BMC Syst Biol 3:13CrossRefGoogle Scholar
- Schmidt M et al (2007) EGFL7 regulates the collective migration of endothelial cells by restricting their spatial distribution. Development 134:2913–2923CrossRefGoogle Scholar
- Schugart RC, Friedman A, Zhao R, Sen CK (2008) Wound angiogenesis as a function of tissue oxygen tension: a mathematical model. PNAS 105:2628–2633CrossRefGoogle Scholar
- Semino CE, Kamm RD, Lauffenburger DA (2006) Autocrine EGF receptor activation mediates endothelial cell migration and vascular morphogenesis induced by VEGF under interstitial flow. Exp Cell Res 312:289–298Google Scholar
- Shirinifard A, Gens JS, Zaitlen BL, Popawski NJ, Swat M et al (2009) 3D multi-cell simulation of tumor growth and angiogenesis. PLoS One 4:e7190CrossRefGoogle Scholar
- Sholley MM, Ferguson GP, Seibel HR, Montour JL, Wilson JD (1984) Mechanisms of neovascularization. Vascular sprouting can occur without proliferation of endothelial cells. Lab Investig 51:624–634Google Scholar
- Sleeman BD, Wallis IP (2002) Tumour induced angiogenesis as a reinforced random walk: modeling capillary network formation without endothelial cell proliferation. J Math Comput Model 36:339–358CrossRefMATHMathSciNetGoogle Scholar
- Stokes CL, Lauffenburger DA (1991) Analysis of the roles of microvessel endothelial cell random mobility and chemotaxis in angiogenesis. J Theor Biol 152:377–403CrossRefGoogle Scholar
- Sun S, Wheeler MF, Obeyesekere M, Patrick C (2005) A deterministic model of growth factor induced angiogenesis. Bull Math Biol 67:313–337CrossRefMathSciNetGoogle Scholar
- Swartz MA, Fleury ME (2007) Interstitial flow and Its effects in soft tissues. Annu Rev Biomed Eng 9:229–256CrossRefGoogle Scholar
- Thoumine O, Ott A (1997) Time scale dependent viscoelastic and contractile regimes in fibroblasts probed by microplate manipulation. J Cell Sci 110:2109–2116Google Scholar
- Tong S, Yuan F (2001) Numerical simulations of angiogenesis in the cornea. Microvasc Res 61:14–27CrossRefGoogle Scholar
- Travasso RDM, Corvera Poir E (2011) Tumor angiogenesis and vascular patterning: a mathematical model. PLoS One 6:e19989CrossRefGoogle Scholar
- Xue C, Friedman A, Sen CK (2009) A mathematical model of ischemic cutaneous wounds. PNAS 106:16782–16787CrossRefGoogle Scholar
- Vakoc BJ et al (2009) Three-dimensional microscopy of the tumor microenvironment in vivo using optical frequency domain imaging. Nat Med 15:1219–1223CrossRefGoogle Scholar
- Volokh KY (2006) Stresses in growing soft tissues. Acta Biomater 2:493–504CrossRefGoogle Scholar
- Wcislo R, Dzwinel W, Yuen D, Dudek A (2009) A 3-D model of tumor progression based on complex automata driven by particle dynamics. J Mol Model 15:1517–1539CrossRefGoogle Scholar
- Zeng G, Taylor SM, McColm JR, Kappas NC, Kearney JB, Williams LH, Hartnett ME, Bautch VL (2007) Orientation of endothelial cell division is regulated by VEGF signaling during blood vessel formation. Blood 109:1345–1352CrossRefGoogle Scholar