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

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].

[1]  J. Folkman,et al.  Migration and proliferation of endothelial cells in preformed and newly formed blood vessels during tumor angiogenesis. , 1977, Microvascular research.

[2]  R. Hungate,et al.  Vascular Endothelial Growth Factor Receptor KDR Tyrosine Kinase Activity Is Increased by Autophosphorylation of Two Activation Loop Tyrosine Residues* , 1999, The Journal of Biological Chemistry.

[3]  D. Dimitrov,et al.  Identification of the KDR tyrosine kinase as a receptor for vascular endothelial cell growth factor. , 1992, Biochemical and biophysical research communications.

[4]  H. Birkedal‐Hansen,et al.  Proteolytic activities of human fibroblast collagenase: hydrolysis of a broad range of substrates at a single active site. , 1990, Biochemistry.

[5]  S. Mandriota,et al.  Vascular Endothelial Growth Factor Increases Urokinase Receptor Expression in Vascular Endothelial Cells (*) , 1995, The Journal of Biological Chemistry.

[6]  Erkki Ruoslahti,et al.  Fibronectin Matrix Regulates Activation of RHO and CDC42 GTPases and Cell Cycle Progression , 1998, The Journal of cell biology.

[7]  R. Soldi,et al.  Role of αvβ3 integrin in the activation of vascular endothelial growth factor receptor‐2 , 1999, The EMBO journal.

[8]  Y. Shimada,et al.  Apoptosis of vascular endothelial cells by fibroblast growth factor deprivation. , 1990, Biochemical and biophysical research communications.

[9]  O. Saksela,et al.  Plasminogen activation and regulation of pericellular proteolysis. , 1985, Biochimica et biophysica acta.

[10]  M. Boffa,et al.  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* , 1998, The Journal of Biological Chemistry.

[11]  C. Frenzen,et al.  Enzyme kinetics for a two-step enzymic reaction with comparable initial enzyme-substrate ratios , 1988, Journal of mathematical biology.

[12]  D. Carney,et al.  Initiation of chick cell division by trypsin action at the cell surface , 1977, Nature.

[13]  W. Cliff.  Observations on healing tissue: A combined light and electron microscopic investigation , 1963, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[14]  D. Balding,et al.  A mathematical model of tumour-induced capillary growth. , 1985, Journal of theoretical biology.

[15]  Kenneth M. Yamada,et al.  Fibronectins—adhesive glycoproteins of cell surface and blood , 1978, Nature.

[16]  G Murphy,et al.  Proteolysis and cell migration: creating a path? , 1999, Current opinion in cell biology.

[17]  H. Kwaan,et al.  Enzymatic properties of the phosphorylated urokinase-type plasminogen activator isolated from a human carcinomatous cell line. , 1992, Biochemical and biophysical research communications.

[18]  J. Gamble,et al.  Regulation of in vitro capillary tube formation by anti-integrin antibodies , 1993, The Journal of cell biology.

[19]  N. Nelson.  Inhibitors of angiogenesis enter phase III testing. , 1998, Journal of the National Cancer Institute.

[20]  R Langer,et al.  Inhibition of neovascularization by an extract derived from vitreous. , 1977, American journal of ophthalmology.

[21]  M. Stack,et al.  Angiostatin inhibits endothelial and melanoma cellular invasion by blocking matrix-enhanced plasminogen activation. , 1999, The Biochemical journal.

[22]  M. Shibuya,et al.  Vascular endothelial growth factor B (VEGF-B) binds to VEGF receptor-1 and regulates plasminogen activator activity in endothelial cells. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[23]  S. Gordon,et al.  Exposure to lysosomotropic amines and protease inhibitors retard corneal endothelial cell migration along the natural basement membrane during wound repair. , 1999, Experimental cell research.

[24]  H. Chapman,et al.  Plasminogen activators, integrins, and the coordinated regulation of cell adhesion and migration. , 1997, Current opinion in cell biology.

[25]  E. Jaffe,et al.  Synthesis of fibronectin by cultured human endothelial cells , 1978, Annals of the New York Academy of Sciences.

[26]  S. Pamuk.  Two dimensional models of tumor angiogenesis , 2000 .

[27]  J. Folkman,et al.  Inhibition of tumor angiogenesis mediated by cartilage , 1975, The Journal of experimental medicine.

[28]  G. Neufeld,et al.  Glypican-1 Is a VEGF165 Binding Proteoglycan That Acts as an Extracellular Chaperone for VEGF165 * , 1999, The Journal of Biological Chemistry.

[29]  B. Warren.  The ultrastructure of the microcirculation at the advancing edge of Walker 256 carcinoma. , 1970, Microvascular research.

[30]  L. Liotta,et al.  Basement membrane collagen: degradation by migrating endothelial cells. , 1983, Science.

[31]  M. Shibuya,et al.  Different signal transduction properties of KDR and Flt1, two receptors for vascular endothelial growth factor. , 1994, The Journal of biological chemistry.

[32]  H. Rochefort,et al.  Cathepsin D in breast cancer: mechanisms and clinical applications, a 1999 overview. , 2000, Clinica chimica acta; international journal of clinical chemistry.

[33]  E. Bauer,et al.  Vascular endothelial growth factor induces interstitial collagenase expression in human endothelial cells , 1992, Journal of cellular physiology.

[34]  B. Duling,et al.  Morphology favors an endothelial cell pathway for longitudinal conduction within arterioles. , 1997, Microvascular research.

[35]  J. Folkman.  Angiogenesis--retrospect and outlook. , 1992, EXS.

[36]  Motoharu Seiki,et al.  A matrix metalloproteinase expressed on the surface of invasive tumour cells , 1994, Nature.

[37]  H. Izumi,et al.  Regulation of angiogenesis by growth factors , 1994 .

[38]  R M Nerem,et al.  Vascular endothelial morphology as an indicator of the pattern of blood flow. , 1981, Journal of biomechanical engineering.

[39]  W. Wilson,et al.  Extravascular diffusion of tirapazamine: effect of metabolic consumption assessed using the multicellular layer model. , 1998, International journal of radiation oncology, biology, physics.

[40]  R. Schleef,et al.  The effect of proteases on endothelial cell migration in vitro. , 1982, Experimental cell research.

[41]  D A Lauffenburger,et al.  Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. , 1991, Journal of theoretical biology.

[42]  J. Forrester,et al.  Factors affecting the migration and growth of endothelial cells from microvessels of bovine retina. , 1990, Experimental eye research.

[43]  L. A. Segel,et al.  The Quasi-Steady-State Assumption: A Case Study in Perturbation , 1989, SIAM Rev..

[44]  A. Cho,et al.  Effects of changes in blood flow rate on cell death and cell proliferation in carotid arteries of immature rabbits. , 1997, Circulation research.

[45]  M. Chaplain,et al.  A mathematical model of the first steps of tumour-related angiogenesis: capillary sprout formation and secondary branching. , 1996, IMA journal of mathematics applied in medicine and biology.

[46]  B. Sleeman,et al.  A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. I. The role of protease inhibitors in preventing angiogenesis. , 2000, Mathematical biosciences.

[47]  J. Folkman.  The vascularization of tumors. , 1976, Scientific American.

[48]  V. Terranova,et al.  Human endothelial cells are chemotactic to endothelial cell growth factor and heparin , 1985, The Journal of cell biology.

[49]  P. Carmeliet,et al.  Angiogenesis in cancer and other diseases , 2000, Nature.

[50]  Z. Han,et al.  Angiogenesis: state of the art. , 1999, International journal of hematology.

[51]  M. Chaplain,et al.  Two-dimensional models of tumour angiogenesis and anti-angiogenesis strategies. , 1997, IMA journal of mathematics applied in medicine and biology.

[52]  Elena Bonanno,et al.  Fibronectin promotes the elongation of microvessels during angiogenesis in vitro , 1993, Journal of cellular physiology.

[53]  Lee A. Segel,et al.  On the validity of the steady state assumption of enzyme kinetics , 1988 .

[54]  J. Folkman,et al.  Tumor growth and neovascularization: an experimental model using the rabbit cornea. , 1974, Journal of the National Cancer Institute.

[55]  J. Feijen,et al.  Improved Adhesion and Proliferation of Human Endothelial Cells on Polyethylene Precoated with Monoclonal Antibodies Directed against Cell Membrane Antigens and Extracellular Matrix Proteins , 1991, Thrombosis and Haemostasis.