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A Cell-based Model of Endothelial Cell Migration, Proliferation and Maturation During Corneal Angiogenesis

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Abstract

The motivation of this work stems from two critical experimental observations associated with corneal angiogenesis: (1) angiogenesis will not succeed without endothelial cell proliferation, and (2) proliferation mainly occurs at the leading edge of developing sprouts (Sholley et al., Lab. Invest. 51:624–634, 1984). To discover the underlying mechanisms of these phenomena, we develop a cell-based mathematical model that integrates a mechanical model of elongation with a biochemical model of cell phenotype variation regulated by angiopoietins within a developing sprout. This model allows for a detailed study of the relative roles of endothelial cell migration, proliferation, and maturation. The model is validated by quantitatively comparing its predictions with data derived from corneal angiogenesis experiments. We conclude that cell elasticity and cell-to-cell adhesion allow only limited sprout extension in the absence of proliferation, and the maturation process combined with bioavailability of VEGF can explain the localization of proliferation to the leading edge. We also use this model to investigate the effects of X-ray irradiation, Ang-2 inhibition, and extracellular matrix anisotropy on sprout morphology and extension.

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References

  • Addison-Smith, B., McElwain, D.L.S., Maini, P.K., 2008. A simple mechanistic model of sprout spacing in tumour-associated angiogenesis. J. Theor. Biol. 250, 1–15.

    Article  Google Scholar 

  • Amano, S., Rohan, R., Kuroki, M., Tolentino, M., Adamis, A.P., 1998. Requirement for vascular endothelial growth factor in wound- and inflammation-related corneal neovascularization. Invest. Ophthalmol. Vis. Sci. 39(1), 18–22.

    Google Scholar 

  • Anderson, A.R.A., Chaplain, M.A.J., 1998b. Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 60, 857–900.

    Article  MATH  Google Scholar 

  • Anderson, A.R.A., Chaplain, M.A.J., 1998a. A mathematical model for capillary network formation in the absence of endothelial cell proliferation. Appl. Math. Lett. 11, 109–114.

    Article  MATH  Google Scholar 

  • Arakelyan, L., Vainstein, V., Agur, Z., 2002. A computer algorithm describing the process of vessel formation and maturation, and its use for predicting the effects of anti-angiogenic and anti-maturation therapy on vascular tumor growth. Angiogenesis 5(3), 203–214.

    Article  Google Scholar 

  • Ashara, T., Chen, D., Takahashi, T., Fujikawa, K., Kearney, M., Magner, M., Yancopoulos, G.D., Isner, J.M., 1998. Tie-2 receptor ligands, angiopoietin-1 and angiopoietin-2, modulate VEGF-induced postnatal neovascularisation. Circ. Res. 83, 233–240.

    Google Scholar 

  • Augustin, H.G., Young Koh, G., Thurston, G., Alitalo, K., 2009. Control of vascular morphogenesis and homeostasis through the angiopoietin-tie system. Nat. Rev. Mol. Cell Biol. 10(3), 165–177.

    Article  Google Scholar 

  • Ausprunk, D.H., Folkman, J., 1977. Migration and proliferation of endothelial cells in preformed and newly-formed blood vessels during tumor angiogenesis. Microvasc. Res. 14, 53–65.

    Article  Google Scholar 

  • Balding, D., McElwain, D.L.S., 1985. A mathematical model of tumor-induced capillary growth. J. Theor. Biol. 114, 53–73.

    Article  Google Scholar 

  • Bauer, A., Jackson, T., Jiang, Y., 2007. A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis. Biophys. J. 92, 3105.

    Article  Google Scholar 

  • Beck, H., Acker, T., Wiessner, C., Allegrini, P.R., Plate, K.H., 2000. Expression of Angiopoietin-1, Angiopoietin-2, and Tie receptors after middle cerebral artery occlusion in the rat. Am. J. Pathol. 157, 1473–1483.

    Google Scholar 

  • Benjamin, L.E., Golijanin, D., Itin, A., Pode, D., Keshet, E., 1999. Selective ablation of immature blood vessels in established human tumors follows vascular endothelial growth factor withdrawal. J. Clin. Invest. 103(2), 159–165.

    Article  Google Scholar 

  • Byrne, H.M., Chaplain, M.A.J., 1995. Mathematical models for tumour angiogenesis: Numerical simulations and nonlinear wave solutions. Bull. Math. Biol. 57, 461–486.

    MATH  Google Scholar 

  • Byrne, H.M., Chaplain, M.A.J., 1996. Explicit solutions of a simplified model of capillary sprout growth during tumor angiogenesis. Appl. Math. Lett. 9, 69–74.

    Article  MATH  MathSciNet  Google Scholar 

  • Chan-Ling, T., 1997. Glial, vascular and neuronal cytogenesis in whole-mounted cat retina. Microsc. Res. Tech. 36, 1–16.

    Article  Google Scholar 

  • Chaplain, M.A.J., 1996. Avascular growth, angiogenesis and vascular growth in solid tumours: The mathematical modelling of the stages of tumour development. Math. Comput. Model. 23(6), 47–87.

    Article  MATH  Google Scholar 

  • Cho, A., Mitchell, L., Koopmans, D., Langille, B.L., 1997. Effects of changes in blood flow rate on cell death and cell proliferation in carotid arteries of immature rabbits. Circ. Res. 81, 328–337.

    Google Scholar 

  • Costa, K.D., Sim, A.J., Yin, F.C., 2006. Non-hertzian approach to analyzing mechanical properties of endothelial cells probed by atomic force microscopy. J. Biomech. Eng. 128(2), 176–184.

    Article  Google Scholar 

  • Cursiefen, C., Hofmann-Rummelt, C., Küchle, M., Schlötzer-Schrehardt, U., 2003. Pericyte recruitment in human corneal angiogenesis: an ultrastructural study with clinicopathological correlation. Br. J. Ophthalmol. 87, 101–106.

    Article  Google Scholar 

  • Davis, S., Aldrich, T.H., Jones, P.F., Acheson, A., Compton, D.L., Jain, V., Ryan, T.E., Bruno, J., Radziejewski, C., Maisonpierre, P.C., Yancopoulos, G.D., 1996. Isolation of angiopoietin-1, a ligand for the Tie2 receptor, by secretion-trap expression cloning. Cell 87(7), 1161–1169.

    Article  Google Scholar 

  • De Smet, F., Segura, I., De Bock, K., Hohensinner, P.J., Carmeliet, P., 2009. Mechanisms of vessel branching: Filopodia on endothelial tip cells lead the way. Arterioscler. Thromb. Vasc. Biol. 29(5), 639–649.

    Article  Google Scholar 

  • Fiedler, U., Scharpfenecker, M., Koidl, S., Hegen, A., Grunow, V., Schmidt, J.M., Kriz, W., Thurston, G., Augustin, H.G., 2004. The Tie-2 ligand Angiopoietin-2 is stored in and rapidly released upon stimulation from endothelial cell Weibel–Palade bodies. Blood 103(11), 4150–4156.

    Article  Google Scholar 

  • Fiedler, U., Reiss, Y., Scharpfenecker, M., Grunow, V., Koidl, S., Thurston, G., Gale, N.W., Augustin, H.G., 2006. Angiopoietin-2 sensitizes endothelial cells to tnf-α and has a crucial role in the induction of inflammation. Nat. Med. 12(2), 235–239.

    Article  Google Scholar 

  • Gerhardt, H., Golding, M., Fruttiger, M., Ruhrberg, C., Lundkvist, A., Abramsson, A., Jeltsch, M., Mitchell, C., Alitalo, K., Shima, D., Betsholtz, C., 2003. VEGF guides angiogenic sprouting utilizing endothelial tip cell filopodia. J. Cell Biol. 161, 1163–1177.

    Article  Google Scholar 

  • Gevertz, J.L., Torquato, S., 2006. Modeling the effects of vasculature evolution on early brain tumor growth. J. Theor. Biol. 243, 517–531.

    Article  MathSciNet  Google Scholar 

  • Gracheva, M.E., Othmer, H.G., 2004. A continuum model of motility in ameboid cells. Bull. Math. Biol. 66, 167–193.

    Article  MathSciNet  Google Scholar 

  • Griffioen, A.W., Molema, J., 2000. Angiogenesis: Potentials for pharmacologic intervention in the treatment of cancer, cardiovascular diseases, and chronic inflammation. Pharmacol. Rev. 52(2), 237–268.

    Google Scholar 

  • Guido, S., Tranquillo, R.T., 1993. A methodology for the systematic and quantitative study of cell contact guidance in oriented collagen gels. J. Cell Sci. 105, 317–331.

    Google Scholar 

  • Harrington, H.A., Maier, M., Naidoo, L., Whitaker, N., Kevrekidis, P.G., 2007. A hybrid model for tumor-induced angiogenesis in the cornea in the presence of inhibitors. Math. Comput. Model. 46, 513–524.

    Article  Google Scholar 

  • Holash, J., Wiegand, S.J., Yancopoulos, G.D., 1999. New model of tumor angiogenesis: dynamic balance between vessel regression and growth mediated by angiopoietins and VEGF. Oncogene 18, 5356–5362.

    Article  Google Scholar 

  • Horowitz, A., Simons, M., 2008. Branching morphogenesis. Circ. Res. 103(8), 784–795.

    Article  Google Scholar 

  • Jackson, T.L., Zheng, X., 2009. A continuous model of angiogenesis: initiation, extension and maturation modulated by vascular endothelial growth factor and angiopoietins (in preparation).

  • Jain, H.V., Nör, J.E., Jackson, T.L., 2008. Modeling the VEGF–Bcl-2–CXCL8 pathway in intratumoral angiogenesis. Bull. Math. Biol. 70, 89–117.

    Article  MATH  MathSciNet  Google Scholar 

  • Karl, E., Warner, K., Zeitlin, B., Kaneko, T., Wurtzel, L., Jin, T., Chang, J., Wang, S., Wang, C., Strieter, R.M., Nunez, G., Polverini, P.J., Nor, J.E., 2005. Bcl-2 Acts in a Proangiogenic Signaling Pathway through Nuclear Factor-kappaB and CXC Chemokines. Cancer Res. 65(12), 5063–5069.

    Article  Google Scholar 

  • Kearney, J.B., Kappas, N.C., Ellerstrom, C., DiPaola, F.W., Bautch, V.L., 2004. The VEGF receptor (VEGFR1) is a positive modulator of vascular sprout formation and branching morphogenesis. Blood 103, 4527–4535.

    Article  Google Scholar 

  • Lamalice, L., Boeuf, F.L., Huot, J., 2007. Endothelial cell migration during angiogenesis. Circ. Res. 100, 782–794.

    Article  Google 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, 113–123.

    Article  Google Scholar 

  • Levine, H.A., Nilsen-Hamilton, M., 2006. Angiogenesis-a biochemial/mathematical perspective. In: Friedman, A. (Ed.), Tutorials in Mathematical Biosciences III, Lecture Notes in Mathematics, vol. 1872, p. 65. Springer, Berlin, Chap. 2.

    Google Scholar 

  • Levine, H.A., Pamuk, S., Sleeman, B.D., Nilsen-Hamilton, M., 2001a. Mathematical modeling of capillary formation and development in tumor angiogenesis: Penetration into the stroma. Bull. Math. Biol. 63, 801–863.

    Article  Google Scholar 

  • Levine, H.A., Sleeman, B.D., Nilsen-Hamilton, M., 2001b. Mathematical modeling of the onset of capillary formation initiating angiogenesis. J. Math. Biol. 42, 195–238.

    Article  MATH  MathSciNet  Google Scholar 

  • Mac Gabhann, F., Popel, A.S., 2004. Model of competitive binding of vascular endothelial growth factor and placental growth factor to vegf receptors on endothelial cells. Am. J. Physiol. Heart Circ. Physiol. 286(1), H153–164.

    Article  Google Scholar 

  • Maisonpierre, P.C., Suri, C., Jones, P.F., Bartunkova, S., Wiegand, S.J., Radziejewski, C., Compton, D., McClain, J., Aldrich, T.H., Papadopoulos, N., Daly, T.J., Davis, S., Sato, T.N., Yancopoulos, G.D., 1997. Angiopoietin-2, a natural antagonist for Tie2 that disrupts in vivo angiogenesis. Science 277, 55–60.

    Article  Google Scholar 

  • Mantzaris, N., Webb, S., Othmer, H.G., 2004. Mathematical modeling of tumor-induced angiogenesis. J. Math. Biol. 49, 111–187.

    Article  MATH  MathSciNet  Google Scholar 

  • Milde, F., Bergdorf, M., Koumoutsakos, P., 2008. A hybrid model for three-dimensional simulations of sprouting angiogenesis. Biophys. J. 95, 3146–3160.

    Article  Google Scholar 

  • Munevar, S., Wang, Y.-L., Dembo, M., 2001. Traction force microscopy of migrating normal and H-ras transformed 3T3 fibroblasts. Biophys. J. 80, 1744–1757.

    Article  Google Scholar 

  • Oliner, J., et al., 2004. Suppression of angiogenesis and tumor growth by selective inhibition of angiopoietin-2. Cancer Cell 6(5), 507–516.

    Article  Google Scholar 

  • Othmer, H., Stevens, A., 1997. Aggregation, blowup, and collapse: The ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math. 57, 1044–1081.

    Article  MATH  MathSciNet  Google Scholar 

  • Paweletz, N., Knierim, M., 1989. Tumor related angiogenesis. Crit. Rev. Oncol. Hematol. 9, 197–242.

    Article  Google Scholar 

  • Peirce, S.M., Van Gieson, E.J., Skalak, T.C., 2004. Multicellular simulation predicts microvascular patterning and in silico tissue assembly. FASEB J., Feb. 2004.

  • Pettet, G.J., Byrne, H.M., McElwain, D.L.S., Norbury, J., 1996a. A model of wound-healing angiogenesis in soft tissue. Math. Biosci. 263, 1487–1493.

    Google Scholar 

  • Pettet, G., Chaplain, M.A.J., McElwain, D.L.S., Byrne, H.M., 1996b. On the role of angiogenesis in wound healing. Proc. R. Soc. Lond. Ser. B 263, 1487–1493.

    Article  Google Scholar 

  • Plank, M.J., Sleeman, B.D., 2003. A reinforced random walk model of tumor angiogenesis and anti-angiogenesis strategies. IMA J. Math. Appl. Med. Biol. 20, 135–181.

    Article  MATH  Google Scholar 

  • Plank, M.J., Sleeman, B.D., 2004. Lattice and non-lattice models of tumour angiogenesis. Bull. Math. Biol. 66, 1785–1819.

    Article  MathSciNet  Google Scholar 

  • Plank, M.J., Sleeman, B.D., Jones, P.F., 2004. A mathematical model of tumour angiogenesis, regulated by vascular endothelial growth factor and the angiopoietins. J. Theor. Biol. 229, 435–454.

    Article  MathSciNet  Google Scholar 

  • Pollard, T.D., Borisy, G., 2003. Cellular motility driven by assembly and disassembly of actin filaments. Cell 112, 453–465.

    Article  Google 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(6), 767–772.

    Article  Google Scholar 

  • Purhonen, S., Palm, J., Rossi, D., Kaskenpää, N., Rajantie, I., Ylä-Herttuala, S., Alitalo, K., Weissman, I.L., Salven, P., 2008. Bone marrow-derived circulating endothelial precursors do not contribute to vascular endothelium and are not needed for tumor growth. PNAS 105, 6620–6625.

    Article  Google Scholar 

  • Qutub, A., Popel, A., 2009. Elongation, proliferation & migration differentiate endothelial cell phenotypes and determine capillary sprouting. BMC Syst. Biol. 3(1), 13.

    Article  Google Scholar 

  • Scharpfenecker, M., Fiedler, U., Reiss, Y., Augustin, H.G., 2005. The Tie-2 ligand Angiopoietin-2 destabilizes quiescent endothelium through an internal autocrine loop mechanism. J. Cell Sci. 118, 771–780.

    Article  Google Scholar 

  • Schugart, R.C., Friedman, A., Zhao, R., Sen, C.K., 2008. Wound angiogenesis as a function of tissue oxygen tension: A mathematical model. PNAS 105, 2628–2633.

    Article  Google Scholar 

  • Semino, C.E., Kamm, R.D., Lauffenburger, D.A., 2006. Autocrine EGF receptor activation mediates endothelial cell migration and vascular morphogenesis induced by VEGF under interstitial flow. Exp. Cell Res. 312, 289–298.

    Google Scholar 

  • Serini, G., Ambrosi, D., Giraudo, E., Gamba, A., Preziosi, L., Bussolino, F., 2003. Modeling the early stages of vascular network assembly. EMBO 22, 1771–1779.

    Article  Google Scholar 

  • Sherratt, J.A., Murrat, J.D., 1990. Models of epidermal wound healing. Proc. R. Soc. Lond. B 241, 29–36.

    Article  Google Scholar 

  • Sholley, M.M., Wilson, J.D., 1987. Proliferation and migration of irradiated endothelial cells. In: Rifkin, D.B., Klagsbrun, M. (Eds.), Current Communications in Molecular Biology, pp. 139–144. Cold Spring Harbor Laboratory, Cold Spring Harbor.

    Google Scholar 

  • Sholley, M.M., Wilson, J.D., Montour, J.L., 1983a. Effect of X-irradiation on proliferation of microvascular endothelial cells. Radiat. Res. 94, 648–649.

    Google Scholar 

  • Sholley, M.M., Wilson, J.D., Montour, J.L., 1983b. Microvascular growth in X-irradiated rat corneas. Radiat. Res. 94, 649.

    Google Scholar 

  • Sholley, M.M., Ferguson, G.P., Seibel, H.R., Montour, J.L., Wilson, J.D., 1984. Mechanisms of neovascularization. Vascular sprouting can occur without proliferation of endothelial cells. Lab. Invest. 51, 624–634.

    Google Scholar 

  • Sleeman, B.D., Wallis, I.P., 2002. Tumour induced angiogenesis as a reinforced random walk: modeling capillary network formation without endothelial cell proliferation. J. Math. Comput. Model. 36, 339–358.

    Article  MATH  MathSciNet  Google Scholar 

  • Stokes, C.L., Lauffenburger, D.A., 1991. Analysis of the roles of microvessel endothelial cell random mobility and chemotaxis in angiogenesis. J. Theor. Biol. 152, 377–403.

    Article  Google Scholar 

  • Sun, S., Wheeler, M.F., Obeyesekere, M., Patrick, C., 2005. A deterministic model of growth factor-induced angiogenesis. Bull. Math. Biol. 67, 313–337.

    Article  MathSciNet  Google Scholar 

  • Sundberg, C., Kowanetz, M., Brown, L.F., Detmar, M., Dvorak, H.F., 2002. Stable expression of angiopoietin-1 and other markers by cultured pericytes: Phenotypic similarities to a subpopulation of cells in maturing vessels during later stages of angiogenesis in vivo. Lab. Invest. 82(4), 387–401.

    Google Scholar 

  • Suri, C., Jones, P.F., Patan, S., Bartunkova, S., Maisonpierre, P.C., Davis, S., Sato, T.N., Yancopoulos, G.D., 1996. Requisite role of angiopoietin-1, a ligand for the tie2 receptor, during embryonic angiogenesis. Cell 87(7), 1171–1180.

    Article  Google Scholar 

  • Szabo, A., Perryn, E.D., Czirok, A., 2007. Network formation of tissue cells via preferential attraction to elongated structures. Phys. Rev. Lett. 98, 038102.

    Article  Google Scholar 

  • Thompson, L.J., Wang, F., Proia, A.D., Peters, K.G., Jarrold, B., Greis, K.D., 2003. Proteome analysis of the rat cornea during angiogenesis. Proteomics 3, 2258–2266.

    Article  Google Scholar 

  • Thoumine, O., Ott, A., 1997. Time scale dependent viscoelastic and contractile regimes in fibroblasts probed by microplate manipulation. J. Cell Sci. 110, 2109–2116.

    Google Scholar 

  • Tong, S., Yuan, F., 2001. Numerical simulations of angiogenesis in the cornea. Microvasc. Res. 61, 14–27.

    Article  Google Scholar 

  • Tong, S., Yuan, F., 2008. Dose response of angiogenesis to basic fibroblast growth factor in rat corneal pocket assay: II. Numerical simulations. Microvasc. Res. 75, 16–24.

    Article  Google Scholar 

  • Wakui, S., Yokoo, K., Muto, T., Suzuki, Y., Takahashi, H., Furusato, M., Hano, H., Endou, H., Kanai, Y., 2006. Localization of Ang-1, -2, Tie-2, and VEGF expression at endothelial-pericyte interdigitation in rat angiogenesis. Lab. Invest. 86, 1172–1184.

    Google Scholar 

  • Wang, D., Lehman, R.E., Donner, D.B., Matli, M.R., Warren, R.S., Welton, M.L., 2002. Expression and endocytosis of VEGF and its receptors in human colonic vascular endothelial cells. Am. J. Physiol. Gastrointest. Liver Physiol. 282(6), G1088–1096.

    Google Scholar 

  • White, R.R., Shan, S., Rusconi, C.P., Shetty, G., Dewhirst, M.W., Kontos, C.D., Sullenger, B.A., 2003. Inhibition of rat corneal angiogenesis by a nuclease-resistant rna aptamer specific for angiopoietin-2. PNAS 100(9), 5028–5033.

    Article  Google Scholar 

  • Witzenbichler, B., Maisonpierre, P.C., Jones, P., Yancopoulos, G.D., Isner, J.M., 1998. Chemotactic properties of angiopoietin-1 and -2, ligands for the endothelial-specific receptor tyrosine kinase Tie-2. J. Biol. Chem. 273, 18514–18521.

    Article  Google Scholar 

  • Zeng, G., Taylor, S.M., McColm, J.R., Kappas, N.C., Kearney, J.B., Williams, L.H., Hartnett, M.E., Bautch, V.L., 2007. Orientation of endothelial cell division is regulated by VEGF signaling during blood vessel formation. Blood 109(4), 1345–1352.

    Article  Google Scholar 

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  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Trachette Jackson & Xiaoming Zheng

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  1. Trachette Jackson
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Jackson, T., Zheng, X. A Cell-based Model of Endothelial Cell Migration, Proliferation and Maturation During Corneal Angiogenesis. Bull. Math. Biol. 72, 830–868 (2010). https://doi.org/10.1007/s11538-009-9471-1

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