Bulletin of Mathematical Biology

, Volume 63, Issue 5, pp 801–863

Mathematical modeling of capillary formation and development in tumor angiogenesis: Penetration into the stroma

  • Howard A. Levine
  • Serdal Pamuk
  • Brian D. Sleeman
  • Marit Nilsen-Hamilton
Article

Abstract

The purpose of this paper is to present a mathematical model for the tumor vascularization theory of tumor growth proposed by Judah Folkman in the early 1970s and subsequently established experimentally by him and his coworkers [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 53–65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323–328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58–64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: an experimental model using the rabbit cornea, J. Nat. Cancer Inst., 52, 413–419]. In the simplest version of this model, an avascular tumor secretes a tumor growth factor (TGF) which is transported across an extracellular matrix (ECM) to a neighboring vasculature where it stimulates endothelial cells to produce a protease that acts as a catalyst to degrade the fibronectin of the capillary wall and the ECM. The endothelial cells then move up the TGF gradient back to the tumor, proliferating and forming a new capillary network. In the model presented here, we include two mechanisms for the action of angiostatin. In the first mechanism, substantiated experimentally, the angiostatin acts as a protease inhibitor. A second mechanism for the production of protease inhibitor from angiostatin by endothelial cells is proposed to be of Michaelis-Menten type. Mathematically, this mechanism includes the former as a subcase.

Our model is different from other attempts to model the process of tumor angiogenesis in that it focuses (1) on the biochemistry of the process at the level of the cell; (2) the movement of the cells is based on the theory of reinforced random walks; (3) standard transport equations for the diffusion of molecular species in porous media.

One consequence of our numerical simulations is that we obtain very good computational agreement with the time of the onset of vascularization and the rate of capillary tip growth observed in rabbit cornea experiments [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 73–65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323–328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58–64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: An experimental model using the rabbit cornea, J. Nat. Cancer Inst., 52, 413–419]. Furthermore, our numerical experiments agree with the observation that the tip of a growing capillary accelerates as it approaches the tumor [Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58–64].

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References

  1. Alberts, B., D. Bray, J. Lewis, M. Raff, K. Roberts and J. D. Watson (1994). Molecular Biology of the Cell, 3rd edn, NY and London: Garland Pub. Inc.Google Scholar
  2. Anderson, R. (1985). Mammary gland, in Lactation, Bruce Larson (Ed.), Ames: Iowa State University Press, pp. 1–38.Google Scholar
  3. Ankoma-Sey, V., M. Matli, K. B. Chang, A. Lalazar, D. B. Donner, L. Wong, R. S. Warren and S. L. Friedman (1998). Coordinated induction of VEGF receptors in mesenchymal cell types during rat hepatic wound healing. Oncogene 17, 115–121.CrossRefGoogle Scholar
  4. Araki, S., Y. Shimada, K. Kaji and H. Hayashi (1990). Apoptosis of vascular endothelial cells by fibroblast growth factor deprivation. Biochem. Biophys. Res. Commun. 168, 1194–1200.CrossRefGoogle Scholar
  5. Ausprunk, D. H. and J. Folkman (1977). Migration and Proliferation of endothelial cells in preformed and newly formed blood vessels during tumor angiogenesis. Microvasc. Res. 14, 53–65.CrossRefGoogle Scholar
  6. Balding, D. and D. L. McElwain (1985). A mathematical model of tumour-induced capillary growth. J. Theor. Biol. 114, 53–73.CrossRefGoogle Scholar
  7. Baramova, E. N., K. Bajou, A. Remacle, C. L’Hoir, H. W. Krell, U. H. Weidle, A. Noel and J. M. Foidart (1997). Involvement of PA/plasmin system in the processing of pro-MMP-9 and in the second step of pro-MMP-2 activation. FEBS Lett. 405, 157–162.CrossRefGoogle Scholar
  8. Blasi, F. (1993). Urokinase and urokinase receptor: a paracrine/autocrine system regulating cell migration and invasiveness. Bioessays 15, 105–111.CrossRefGoogle Scholar
  9. Boffa, M. B., W. Wang, L. Bajzar and M. E. Nesheim (1998). Plasma and recombinant thrombin-activable fibrinolysis inhibitor (TAFI) and activated TAFI compared with respect to glycosylation, thrombin/thrombomodulin-dependent activation, thermal stability, and enzymatic properties. J. Biol. Chem. 273, 2127–2135.CrossRefGoogle Scholar
  10. Bourdoulous, S., G. Orend, D. A. MacKenna, R. Pasqualini and E. Ruoslahti (1998). Fibronectin matrix regulates activation of RHO and CDC42 GTPases and cell cycle progression. J. Cell Biol. 143, 267–276.CrossRefGoogle Scholar
  11. Brem, H. and J. Folkman (1975). Inhibition of tumor angiogenesis mediated by cartilage. J. Exp. Med. 141, 427–439.CrossRefGoogle Scholar
  12. Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997). Inhibition of neovascularization by an extract derived from vitreous. Am. J. Opthalmol. 84, 323–328.Google Scholar
  13. Cahill, A., T. C. Jenkins and I. N. H. White (1993). Metabolism of 3-amino-1,2,4-benzotriazine-1,4-dioxide (SR 4233) by purified DT-diaporase unde aerobic and anaerobic conditions. Biochem. Pharmacol. 45, 321–329.CrossRefGoogle Scholar
  14. Carmeliet, P. and R. K. Jain (2000). Angiogenesis in cancer and other diseases. Nature 407, 249–257.CrossRefGoogle Scholar
  15. Carney, D. H. and D. D. Cunningham (1977). Initiation of check cell division by trypsin action at the cell surface. Nature 268, 602–606.CrossRefGoogle Scholar
  16. Chaplain, M. A. J. and A. R. A. Anderson (1999). Modelling the growth and form of capillary networks, in On Growth and Form: Spatio-Temporal Pattern Formation in Biology, New York: Wiley, pp. 225–249.Google Scholar
  17. Chapman, H. A. (1997). Plasminogen activators, integrins, and the coordinated regulation of cell adhesion and migration. Curr. Opin. Cell Biol. 9, 714–724.CrossRefGoogle Scholar
  18. Cho, A., L. Mitchell, D. Koopmans and B. L. Langille (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
  19. Cliff, W. J. (1963). Observations on healing tissue: a combined light and electron microscopic investigation. Philos. Trans. R. Soc. London B 246, 305-ff.Google Scholar
  20. Crocker, D. J., T. M. Murad and J. C. Geer (1970). The role of the pericyte in wound healing: an ultrastructural study. Exp. Mol. Pathol 13, 51–65.CrossRefGoogle Scholar
  21. Curran, S. and G. I. Murray (1999). Matrix metalloproteinases in tumour invasion and metastasis. J. Pathol. 189, 300–308.CrossRefGoogle Scholar
  22. Davis, B. (1990). Reinforced random walks. Probability Theory Related Fields 84, 203–229.MATHCrossRefGoogle Scholar
  23. Dekker, A., A. A. Poot, J. A. van Mourik, M. P. Workel, T. Beugeling, A. Bantjes, J. Feijen and W. G. van Aken (1991). Improved adhesion and proliferation of human endothelial cells on polyethylene precoated with monoclonal antibodies directed against cell membrane antigens and extracellular matrix proteins. Thromb. Haemost. 66, 715–724.Google Scholar
  24. Edelstein-Keshet, L. (1988). Mathematical Models in Biology, Boston: McGraw-Hill.MATHGoogle Scholar
  25. Fields, G., S. J. Netzewl-Arnett, L. J. Windsor, J. A. Engler, H. Berkedal-Hansen and H. E. van Wart (1990). Proteolytic activities of human fibroblast collagenase; hydrolysis of a broad range of substrates at a single active site. Biochemistry 29, 6600–6677.CrossRefGoogle Scholar
  26. Folkman, J. (1976). The vascularization of tumors. Sci. Am. 234, 58–64.CrossRefGoogle Scholar
  27. Folkman, J. (1992). Angiogenesis-retrospect and outlook, in Angiogenesis: Key Principles-Science-Technology-Medicine, R. Steiner, P. B. Weisz and R. Langer (Eds), Basel: Birkhäuser.Google Scholar
  28. Frenzen, C. L. and P. K. Maini (1988). Enzyme kinetics for a two-step enzymic reaction with comparable initial enzyme-substrate ratios. J. Math. Biol. 26, 689–703.MATHMathSciNetGoogle Scholar
  29. Gamble, J. R., L. J. Matthias, G. Meyer, P. Kaur, G. Russ, R. Faull, M. C. Berndt and M. A. Vadas (1993). Regulation of in vitro capillary tube formation by anti-integrin antibodies. J. Cell Biol. 121, 931–943.CrossRefGoogle Scholar
  30. Gengrinovitch, S., B. Berman, G. David, L. Witte, G. Neufeld and D. Ron (1999). Glypican-1 is a VEGF165 binding proteoglycan that acts as an extracellular chaperone for VEGF165. J. Biol. Chem. 274, 10816–10822.Google Scholar
  31. Gimbrone, M. A. Jr., R. S. Cotran, S. B. Leapman and J. Folkman (1974). Tumor growth and neovascularization: an experimental model using the rabbit cornea. J. Natl. Cancer Inst. 52, 413–419.Google Scholar
  32. Gordon, S. R. and J. DeMoss (1999). Exposure to lysosomotropic amines and protease inhibitors retard corneal endothelial cell migration along the natural basement membrane during wound repair. Exp. Cell Res. 246, 233–242.CrossRefGoogle Scholar
  33. Gospodarowicz, D., G. Greenburg, H. Bialecki and B. R. Zetter (1978). Factors involved in the modulation of cell proliferation in vivo and in vitro: the role of fibroblast and epidermal growth factors in the proliferative response of mammalian cells. In Vitro 14, 85–118.Google Scholar
  34. Haas, T. L. and B. R. Duling (1997). Morphology favors an endothelial cell pathway for longitudinal conduction within arterioles. Microvasc Res. 53, 113–120.CrossRefGoogle Scholar
  35. Han, Z. C. and Y. Liu (1999). Angiogenesis: state of the art. Int. J. Hematol. 70, 68–82.Google Scholar
  36. Hanahan, D. and J. Folkman (1996). Patterns and emerging mechanisms of the angiogenic switch during tumorigenesis. Cell 86, 353–364.CrossRefGoogle Scholar
  37. Hicks, K. O., Y. Fleming, B. G. Siim, C. J. Koch and W. R. Wilson (1998). Extravascular diffusion of tirapazamine: effect of metabolic consumption assessed using the multicellular layer model. Int. J. Radiat. Oncol. Biol. Phy. 42, 641–649.CrossRefGoogle Scholar
  38. Hiraoka, N., E. Allen, I. J. Apel, M. R. Gyetko and S. J. Weiss (1998). Matrix metalloproteinases regulate neovascularization by acting as pericellular fibrinolysins. Cell 95, 365–377.CrossRefGoogle Scholar
  39. Holash, J., P. C. Maisonpierre, D. Compton, P. Boland, C. R. Alexander, D. Zagzag, G. D. Yancopoulos and S. J. Wiegand (1998). Vessel cooption, regression, and growth in tumors mediated by angiopoietins and VEGF. Science 284, 1994–1998.CrossRefGoogle Scholar
  40. Jaffee, E. A. and D. F. Mosher (1978). Synthesis of fibronectin by cultured human endothelial cells. J. Exp. Med. 147, 1779–1791.CrossRefGoogle Scholar
  41. Kabelic, T., S. Ganbisa, B. Glaser and L. A. Liotta (1983). Basement membrane collagen: degradation by migrating endothelial cells. Science 221, 281–283.Google Scholar
  42. Kendall, R. L., R. Z. Rutledge, X. Mao, A. J. Tebben, R. W. Hungate and K. A. Thomas (1999). Vascular endothelial growth factor receptor KDR tyrosine kinase activity is increased by autophosphorylation of two activation loop tyrosine residues. J. Biol. Chem. 274, 6453–6460.CrossRefGoogle Scholar
  43. Kuwano, M., S. Ushiro, M. Ryuto, K. Samoto, H. Izumi, K. Ito, T. Abe, T. Nakamura, M. Ono and K. Kohno (1994). Regulation of angiogenesis by growth factors. GANN Monograph on Cancer Research 42, 113–125.Google Scholar
  44. Lagelund, T. D. and P. A. Low (1987). A mathematical simulation of oxygen delivery in rat peripheral nerve. Microvascular Research 34, 211–222.CrossRefGoogle Scholar
  45. Landau, L. D. and E. M. Lifschitz (1982). Course of Theoretical Physics, Vol. 6, Fluid Mechanics, Oxford, UK: Pergamon Press.Google Scholar
  46. Levine, H. A. and B. D. Sleeman (1997). A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM J. Appl. Math. 57, 683–730.MATHMathSciNetCrossRefGoogle Scholar
  47. Levine, H. A., B. D. Sleeman and M. Nilsen-Hamilton. Mathematical modeling of the onset of capillary formation initiating angiogenesis. J. Math. Biol. (in press).Google Scholar
  48. Levine, H. A., B. D. Sleeman and M. Nilsen-Hamilton. A mathematical model for the roles of plasminogen activators, collagenases and heparanase on tumor angiogenesis, (in preparation).Google Scholar
  49. Levine, H. A., B. D. Sleeman and M. Nilsen-Hamilton (2000). A Mathematical model for the roles of pericytes and macrophages in the onset of angiogenesis: I. The role of protease inhibitors in preventing angiogenesis. Math. Biosci. 168, 77–115.MATHMathSciNetCrossRefGoogle Scholar
  50. Librach, C. L., Z. Werb, M. L. Fitzgerald, K. Chiu, N. M. Corwin, R. A. Esteves, D. Grobelny, R. Galardy, C. H. Damsky and S. J. Fisher (1999). 92-kD type IV collagenase mediates invasion of human cytotrophoblasts. J. Cell Biol. 189, 300–308.Google Scholar
  51. Mandriota, S. J., G. Seghezzi, J. D. Vassalli, N. Ferrara, S. Wasi, R. Mazzieri, P. Mignatti and M. S. Pepper (1995). Vascular endothelial growth factor increases urokinase receptor expression in vascular endothelial cells. J. Biol. Chem. 270, 9709–9716.CrossRefGoogle Scholar
  52. Moerman, D. G. (1999). A metalloprotease prepares the way. Curr. Biol. 9, R701–R703.CrossRefGoogle Scholar
  53. Morimoto, K., H. Mishima, T. Nishida and T. Otori (1993). Role of urokinase type plasminogen activator (u-PA) in corneal epithelial migration. Thromb. Haemost. 69, 387–391.Google Scholar
  54. Murphy, G. and J. Gavrilovic (1999). Proteolysis and cell migration: creating a path? Curr. Opin. Cell Biol. 11, 614–621.CrossRefGoogle Scholar
  55. Murray, J. D. (1989). Mathematical Biology, Biomathematics Texts, Springer-Verlag.Google Scholar
  56. Nelsen, N. J. (1998). Inhibitors of angiogenesis enter phase III testing. J. Natl. Cancer Inst. 90, 960–962.CrossRefGoogle Scholar
  57. Nerem, R. M., M. J. Levesque and J. F. Cornhill (1981). Vasuclar endothelial cell morphology as an indicator of the pattern of blood flow. J. Biomech. Eng. 103, 172–176.CrossRefGoogle Scholar
  58. Nicosia, R. F., E. Bonanno and M. Smith (1993). Fibronectin promotes the elongation of microvessels during angiogenesis in vitro. J. Cell Physiol. 154, 654–661.CrossRefGoogle Scholar
  59. Olofsson, B. et al. (1998). Vascular endothelial growth factor B (VEGF-B) binds to VEGF receptor-1 and regulates plasminogen activator activity in endothelial cells. Proc. Natl. Acad. Sci. U. S. A. 95, 11709–11714.Google Scholar
  60. Orme, M. E. and M. A. J. Chaplain (1996). A mathematical model of the first steps of tumour related angiogenesis: capillary sprout formation and secondary branching. I.M.A. J. Math. Appl. Med. Biol. 13, 73–98.MATHGoogle Scholar
  61. Orme, M. E. and M. A. J. Chaplain (1997). Two-dimensional models of tumour angiogenesis and anti-angiogenesis strategies. I.M.A. J. Math. Appl. Med. Biol. 14, 189–205.MATHGoogle Scholar
  62. Othmer, H. G. and A. Stevens (1997). Aggregation, blow up and collapse: the ABC’s of taxis and reinforced random walks. SIAM J. Appl. Math. 51.Google Scholar
  63. Pamuk, S. (May, 2000). Two dimensional models of tumor angiogenesis, PhD Thesis, Iowa State University.Google Scholar
  64. Paweletz, N. and M. Knierim (1989). Tumor related angiogenesis. Crit. Rev. Oncol. Hematol. 9, 197–242.Google Scholar
  65. Rakusan, K. (1995). Coronary Angiogenesis. From morphology to molecular biology and back. Ann. Ny Acad. Sci. 752, 257–266.Google Scholar
  66. Roberts, J. M. and J. V. Forrester (1990). Factors affecting the migration and growth of endothelial cells from microvessels of bovine retina. Exp. Eye Res. 50, 165–172.CrossRefGoogle Scholar
  67. Rochefort, H., M. Garcia, M. Glondu, V. Laurent, E. Liaudet, J. M. Rey and P. Roger (2000). Cathepsin D in breast cancer: mechanisms and clinical applications, a 1999 overview. Clin. Chim. Acta. 291, 85–118.CrossRefGoogle Scholar
  68. Saksela, O. (1985). Plasminogen activation and regulation of pericellular proteolysis. Biochim. Biophys. Acta 823, 35–65.Google Scholar
  69. Sato, H., T. Takino, Y. Okada, J. Cao, A. Shinagawa, E. Yamamoto and M. Seiki (1994). A matrix metalloproteinase expressed on the surface of invasive tumour cells. Nature 370, 61–65.CrossRefGoogle Scholar
  70. Schleef, R. R. and C. R. Birdwell (1982). The effect of proteases on endothelial cell migration in vitro. Exp. Cell Res. 141, 503–508.CrossRefGoogle Scholar
  71. Schoefl, G. I. (1963). Studies on inflammation. III. Growing capillaries: their structure and permeability. Virchows Arch. Pathol. Anat. 337, 97-ff.Google Scholar
  72. Schoefl, G. I. and G. Majno (1964). Regeneration of blood vessels in wound healing. Adv. Biol. Skin 5, 173-ff.Google Scholar
  73. Schor, A. M., A. E. Canfield, A. B. Sutton, T. D. Allen, P. Sloan and S. L. Schor (1992). The behavior of pericytes in vitro: relevance to angiogenesis and differentiation, in Angiogenesis: Key Principles-Science-Technology-Medicine, R. Steiner, P. B. Weisz and R. Langer (Eds), Basel: Birkhäuser.Google Scholar
  74. Segel, L. A. (1988). On the validity of the steady state assumption of enzyme kinetics. Bull. Math. Biol. 50, 579–593.MATHMathSciNetCrossRefGoogle Scholar
  75. Segel, L. A. and M. Slemrod (1989). The quasi steady state assumption: a case study in perturbation. SIAM Rev. 31, 446–477.MATHMathSciNetCrossRefGoogle Scholar
  76. Sethian, J. A. (1996). Level Set Methods and Fast Marching Methods, Cambridge, U.K.: Cambridge University Press.Google Scholar
  77. Sherratt, J. A. and J. D. Murray (1990). Models of epidermal wound healing. Proc. R. Soc. Lond. B. 241, 29–36.Google Scholar
  78. Sherratt, J. A., A. J. Perumpanani and M. R. Owen (1999). Pattern formation in cancer, in On Growth and Form: Spatio-Temporal Pattern Formation in Biology, New York: Wiley, pp. 47–73.Google Scholar
  79. Sholley, M. M., G. P. Ferguson, H. R. Seibel, J. L. Montour and J. D. Wilson (1984). Mechanisms of neovascularization. Vascular sprouting can occur without proliferation of endothelial cells. Lab. Invest. 51, 624–634.Google Scholar
  80. Sleeman, B. D. (1996). Solid tumor growth: a case study in mathematical biology. Nonlin. Math. Appl., Ed. P. J. Aston, C.U. P 237–256.Google Scholar
  81. Sleeman, B. D. and I. P. Wallis (2001). Tumor induced angiogenesis as a reinforced random walk: modelling capillary network formation without endothelial cell proliferation, Math. Copmp. Mod., (in press).Google Scholar
  82. Soldi, R., S. Mitola, M. Strasly, P. Defilippi, G. Tarone and F. Bussolino (1999). Role of alphavbeta3 integrin in the activation of vascular endothelial growth factor receptor-2. EMBO J. 18, 882–892.CrossRefGoogle Scholar
  83. Stack, M. S., S. Gately, L. M. Bafetti, J. Enghild, J. Soff and G. A. Soff (1999). Angiostatin inhibits endothelial and melanoma cellular invasion by blocking matrix-enhanced plasminogen activation. Biochem J. 340, 77–84.CrossRefGoogle Scholar
  84. Stokes, C. L. and D. A. Lauffenburger (1991). Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J. Theor. Biol. 152, 377–403.Google Scholar
  85. Takahashi, K., H. C. Kwaan, E. Koh and M. Tanabe (1992). Enzymatic properties of the phosphorylated urokinase-type plasminogen activator isolated from a human carcinomatous cell line. Biochem. Biophys. Res. Commun. 182, 1473–1481.CrossRefGoogle Scholar
  86. Terman, B. I., M. Dougher-Vermazen, M. E. Carrion, D. Dimitrov, D. C. Armellino, D. Gospodarowicz and P. Bohlen (1992). Identification of the KDR tyrosine kinase as a receptor for vascular endothelial cell growth factor. Biochem. Biophys. Res. Commun. 187, 1579–1586.CrossRefGoogle Scholar
  87. Terranova, V. P., R. DiFlorio, R. M. Lyall, S. Hic, R. Friesel and T. Maciag (1985). Endothelial cells are chemotactic to endothelial cell growth factor and heparin. J. Cell. Biol. 101, 2330–2334.CrossRefGoogle Scholar
  88. Thews, G. (1960). Dei sauerstoffdiffusion im gehirn: Ein beitrag tur frage der Saurstoffversorung der organe. Plugers Arch. 271, 197–226.CrossRefGoogle Scholar
  89. Unemori, E. N., N. Ferrara, E. A. Bauer and E. P. Amento (1992). Vascular endothelial growth factor induces interstitial collagenase expression in human endothelial cells. J. Cell Physiol. 153, 557–562.CrossRefGoogle Scholar
  90. Waltenberger, J., L. Claesson-Welsh, A. Siegbahn, M. Shibuya and C.H. Heldin (1994). Different signal transduction properties of KDR and Flt1, two receptors for vascular endothelial cell growth factor. J. Biol. Chem. 269, 26988–26995.Google Scholar
  91. Warren, B. A. (1970). The ultrastructure of the microcirculation at the advancing edge of Walker256 carcinoma. Microvasc. Res. 2, 443–453.CrossRefGoogle Scholar
  92. Yamada, K. M. and K. Olden (1978). Fibronectins—adhesive glycoproteins of cell surface and blood. Nature 275, 179–184.CrossRefGoogle Scholar
  93. Zhou, Z., S. S. Apte, R. Soininen, R. Cao, G. Y. Baaklini, R. W. Rauser, J. Wang, Y. Cao and K. Tryggvason (2000). Impaired endochondral ossification and angiogenesis in mice deficient in membrane-type matrix metalloproteinase I. Proc. Natl. Acad. Sci. U.S.A. 97, 4052–4057.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2001

Authors and Affiliations

  • Howard A. Levine
    • 1
  • Serdal Pamuk
    • 2
  • Brian D. Sleeman
    • 3
  • Marit Nilsen-Hamilton
    • 4
  1. 1.Department of MathematicsIowa State UniversityAmesUSA
  2. 2.Matematik BölümüKocaeli ÜniversitesiKocaeliTurkey
  3. 3.School of MathematicsUniversity of LeedsLeedsEngland, UK
  4. 4.Department of Biochemistry, Biophysics and Molecular BiologyIowa State UniversityAmesUSA

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