Continuous and discrete mathematical models of tumor-induced angiogenesis

Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then grow and develop, driven initially by endothelial-cell migration, and organize themselves into a dendritic structure. Subsequent cell proliferation near the sprout tip permits further extension of the capillary and ultimately completes the process. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumors. In this paper we present both continuous and discrete mathematical models which describe the formation of the capillary sprout network in response to chemical stimuli (tumor angiogenic factors, TAF) supplied by a solid tumor. The models also take into account essential endothelial cell-extracellular matrix interactions via the inclusion of the matrix macromolecule fibronectin. The continuous model consists of a system of nonlinear partial differential equations describing the initial migratory response of endothelial cells to the TAF and the fibronectin. Numerical simulations of the system, using parameter values based on experimental data, are presented and compared qualitatively with in vivo experiments. We then use a discretized form of the partial differential equations to develop a biased random-walk model which enables us to track individual endothelial cells at the sprout tips and incorporate anastomosis, mitosis and branching explicitly into the model. The theoretical capillary networks generated by computer simulations of the discrete model are compared with the morphology of capillary networks observed in in vivo experiments.

[1]  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.

[2]  G. Schoefl,et al.  STUDIES ON INFLAMMATION. III. GROWING CAPILLARIES: THEIR STRUCTURE AND PERMEABILITY. , 1963, Virchows Archiv fur pathologische Anatomie und Physiologie und fur klinische Medizin.

[3]  S. Carter,et al.  Principles of Cell Motility: The Direction of Cell Movement and Cancer Invasion , 1965, Nature.

[4]  P. Shubik,et al.  The growth of the blood supply to melanoma transplants in the hamster cheek pouch. , 1966, Laboratory investigation; a journal of technical methods and pathology.

[5]  S. Carter,et al.  Haptotaxis and the Mechanism of Cell Motility , 1967, Nature.

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

[7]  Hans Meinhardt,et al.  Models and Hypotheses , 1976 .

[8]  I. Lapidus,et al.  Model for the chemotactic response of a bacterial population. , 1976, Biophysical journal.

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

[10]  L Wolpert,et al.  Thresholds in development. , 1977, Journal of theoretical biology.

[11]  G M Saidel,et al.  Diffusion model of tumor vascularization and growth , 1977, Bulletin of mathematical biology.

[12]  E. Unanue,et al.  Activated macrophages induce vascular proliferation , 1977, Nature.

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

[14]  G. Nicolson,et al.  Identification, localization, and role of fibronectin in cultured bovine endothelial cells. , 1978, Proceedings of the National Academy of Sciences of the United States of America.

[15]  E. Macarak,et al.  Synthesis of cold-insoluble globulin by cultured calf endothelial cells. , 1978, Proceedings of the National Academy of Sciences of the United States of America.

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

[17]  A. Bell,et al.  Branching patterns: the simulation of plant architecture. , 1979, Journal of theoretical biology.

[18]  D. Gospodarowicz,et al.  Vascular endothelial cells maintained in the absence of fibroblast growth factor undergo structural and functional alterations that are incompatible with their in vivo differentiated properties , 1979, The Journal of cell biology.

[19]  W. Alt Biased random walk models for chemotaxis and related diffusion approximations , 1980, Journal of mathematical biology.

[20]  A. R. Mitchell,et al.  The Finite Difference Method in Partial Differential Equations , 1980 .

[21]  Judah Folkman,et al.  Angiogenesis in vitro , 1980, Nature.

[22]  A. Brasier,et al.  Two-dimensional peptide mapping of fibronectins from bovine aortic endothelial cells and bovine plasma. , 1980, Biochemical and biophysical research communications.

[23]  A. Schor,et al.  The effects of fibronectin on the migration of human foreskin fibroblasts and Syrian hamster melanoma cells into three-dimensional gels of native collagen fibres. , 1981, Journal of cell science.

[24]  R. Clark,et al.  Fibronectin in delayed-type hypersensitivity skin reactions: associations with vessel permeability and endothelial cell activation. , 1981, Journal of immunology.

[25]  L. Wolpert Positional information and pattern formation. , 1981, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[26]  R K Jain,et al.  Dynamics of neovascularization in normal tissue. , 1981, Microvascular research.

[27]  T. K. Hunt,et al.  Regulation of wound-healing angiogenesis-effect of oxygen gradients and inspired oxygen concentration. , 1981, Surgery.

[28]  H. Seppä,et al.  Role of collagen and fibronectin in neural crest cell adhesion and migration. , 1981, Developmental biology.

[29]  E Ruoslahti,et al.  Deposition of plasma fibronectin in tissues. , 1981, Proceedings of the National Academy of Sciences of the United States of America.

[30]  H. Meinhardt Models of biological pattern formation , 1982 .

[31]  P. Janmey,et al.  Conformational states of fibronectin. Effects of pH, ionic strength, and collagen binding. , 1982, The Journal of biological chemistry.

[32]  R. Auerbach,et al.  Tumor-induced neovascularization in the mouse eye. , 1982, Journal of the National Cancer Institute.

[33]  R. Clark,et al.  Blood vessel fibronectin increases in conjunction with endothelial cell proliferation and capillary ingrowth during wound healing. , 1982, The Journal of investigative dermatology.

[34]  J. Bowersox,et al.  Chemotaxis of aortic endothelial cells in response to fibronectin. , 1982, Cancer research.

[35]  L. Liotta,et al.  Tumor invasion and the extracellular matrix. , 1983, Laboratory investigation; a journal of technical methods and pathology.

[36]  R. Clark,et al.  Fibronectin beneath reepithelializing epidermis in vivo: sources and significance. , 1983, The Journal of investigative dermatology.

[37]  T. K. Hunt,et al.  Oxygen tension regulates the expression of angiogenesis factor by macrophages. , 1983, Science.

[38]  R. Clark,et al.  Fibronectin beneath reepithelializing epidermis in vivo: sources and significance. , 1983, The Journal of investigative dermatology.

[39]  P. Rudland,et al.  Topographical arrangement of basement membrane proteins in lactating rat mammary gland: comparison of the distribution of type IV collagen, laminin, fibronectin, and Thy-1 at the ultrastructural level. , 1983, Proceedings of the National Academy of Sciences of the United States of America.

[40]  N. Sorgente,et al.  Fibronectin and laminin distribution in bovine eye. , 1983, Japanese journal of ophthalmology.

[41]  D. Deno,et al.  Kinetics of endogenously labeled plasma fibronectin: incorporation into tissues. , 1983, The American journal of physiology.

[42]  J. Quigley,et al.  Fibronectin enhancement of directed migration of B16 melanoma cells. , 1984, Cancer research.

[43]  Ferguson Gp,et al.  Mechanisms of neovascularization. Vascular sprouting can occur without proliferation of endothelial cells. , 1984 .

[44]  M. Klagsbrun,et al.  Endothelial cell mitogens derived from retina and hypothalamus: biochemical and biological similarities , 1984, The Journal of cell biology.

[45]  A. Schreiber,et al.  Heparin binds endothelial cell growth factor, the principal endothelial cell mitogen in bovine brain. , 1984, Science.

[46]  D. Gospodarowicz,et al.  Isolation of brain fibroblast growth factor by heparin-Sepharose affinity chromatography: identity with pituitary fibroblast growth factor. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[47]  R. Lobb,et al.  Purification of two distinct growth factors from bovine neural tissue by heparin affinity chromatography. , 1984, Biochemistry.

[48]  D. Lauffenburger,et al.  Traveling bands of chemotactic bacteria in the context of population growth , 1984 .

[49]  J. McCarthy,et al.  Laminin and fibronectin promote the haptotactic migration of B16 mouse melanoma cells in vitro , 1984, The Journal of cell biology.

[50]  D. Newgreen,et al.  Do cells show an inverse locomotory response to fibronectin and laminin substrates? , 1985, The EMBO journal.

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

[52]  M. Klagsbrun,et al.  Purification of cartilage-derived growth factor by heparin affinity chromatography. , 1985, The Journal of biological chemistry.

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

[54]  J. Folkman Tumor angiogenesis. , 1985, Advances in cancer research.

[55]  Volcanic deposits: pyroclastic rocks. , 1985, Science.

[56]  J. Krachmer,et al.  Immunohistochemical characterization of extracellular matrix in the developing human cornea. , 1986, Current eye research.

[57]  A. Bell The simulation of branching patterns in modular organisms , 1986 .

[58]  H. Arnstein The molecular biology of the cell : B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts and J.D. Watson Garland Publishing; New York, London, 1983 xxxix + 1181 pages. $33.95 (hardback); $27.00, £14.95 (paperback, only in Europe) , 1986 .

[59]  J. Madri,et al.  Endothelial cell-matrix interactions: in vitro models of angiogenesis. , 1986, The journal of histochemistry and cytochemistry : official journal of the Histochemistry Society.

[60]  P. Gullino,et al.  Interaction of gangliosides with fibronectin in the mobilization of capillary endothelium. Possible influence on the growth of metastasis. , 1986, Invasion & metastasis.

[61]  S. Johansson,et al.  Identification of a fibronectin receptor specific for rat liver endothelial cells. , 1987, Experimental cell research.

[62]  Stuart K. Williams 8 Isolation and Culture of Microvessel and Large-Vessel Endothelial Cells: Their Use in Transport and Clinical Studies , 1987 .

[63]  T. Ishibashi,et al.  Immunofluorescent studies of fibronectin and laminin in the human eye. , 1987, Investigative ophthalmology & visual science.

[64]  S Parodi,et al.  Chemotaxis of 3T3 and SV3T3 cells to fibronectin is mediated through the cell-attachment site in fibronectin and a fibronectin cell surface receptor , 1987, The Journal of cell biology.

[65]  J. Folkman,et al.  Angiogenic factors. , 1987, Science.

[66]  H. Konomi,et al.  Immunoelectronmicroscopic localization of extracellular matrix components produced by bovine corneal endothelial cells in vitro. , 1987, Experimental cell research.

[67]  I. Wallow,et al.  Fibronectin distribution in the rat eye. An immunohistochemical study. , 1987, Investigative ophthalmology & visual science.

[68]  H. Dienes,et al.  Sinusoidal endothelial cells from guinea pig liver synthesize and secrete cellular fibronectin in vitro , 1987, Hepatology.

[69]  M. Rocco,et al.  Models of fibronectin. , 1987, The EMBO journal.

[70]  D. Woodley,et al.  Laminin inhibits human keratinocyte migration , 1988, Journal of cellular physiology.

[71]  M A Rupnick,et al.  Quantitative analysis of random motility of human microvessel endothelial cells using a linear under-agarose assay. , 1988, Laboratory investigation; a journal of technical methods and pathology.

[72]  G. Bard Ermentrout,et al.  Models for branching networks in two dimensions , 1989 .

[73]  W. Düchting Computer simulation in cancer research , 1989 .

[74]  N Paweletz,et al.  Tumor-related angiogenesis. , 1989, Critical reviews in oncology/hematology.

[75]  M. E. Gottlieb Modelling Blood Vessels: A Deterministic Method With Fractal Structure Based On Physiological Rules , 1990, [1990] Proceedings of the Twelfth Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[76]  Przemyslaw Prusinkiewicz,et al.  The Algorithmic Beauty of Plants , 1990, The Virtual Laboratory.

[77]  Jonathan A. Sherratt,et al.  Models of epidermal wound healing , 1990, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[78]  Werner Düchting Tumor growth simulation , 1990, Comput. Graph..

[79]  M A Rupnick,et al.  Chemotaxis of human microvessel endothelial cells in response to acidic fibroblast growth factor. , 1990, Laboratory investigation; a journal of technical methods and pathology.

[80]  B. Davis,et al.  Reinforced random walk , 1990 .

[81]  Stuart K. Williams,et al.  Migration of individual microvessel endothelial cells: stochastic model and parameter measurement. , 1991, Journal of cell science.

[82]  E Marc VASCULAR NETWORKS: FRACTAL ANATOMIES FROM NON-LINEAR PHYSIOLOGIES , 1991 .

[83]  F. Arnold,et al.  Angiogenesis in wound healing. , 1991, Pharmacology & therapeutics.

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

[85]  M. E. Gottlieb The VT model: a deterministic model of angiogenesis and biofractals based on physiological rules , 1991, Proceedings of the 1991 IEEE Seventeenth Annual Northeast Bioengineering Conference.

[86]  A. Hudetz,et al.  Computer simulation of growth of anastomosing microvascular networks. , 1991, Journal of theoretical biology.

[87]  S. Paku,et al.  First steps of tumor-related angiogenesis. , 1991, Laboratory investigation; a journal of technical methods and pathology.

[88]  C. Graham,et al.  Mechanisms of placental invasion of the uterus and their control. , 1992, Biochemistry and cell biology = Biochimie et biologie cellulaire.

[89]  L. Naldini,et al.  Hepatocyte growth factor is a potent angiogenic factor which stimulates endothelial cell motility and growth , 1992, The Journal of cell biology.

[90]  W. Düchting,et al.  Simulation of Malignant Cell Growth , 1992 .

[91]  L. Watson,et al.  Diffusion and wave propagation in cellular automaton models of excitable media , 1992 .

[92]  G Landini,et al.  Simulation of corneal neovascularization by inverted diffusion limited aggregation. , 1993, Investigative ophthalmology & visual science.

[93]  A. Ullrich,et al.  High affinity VEGF binding and developmental expression suggest Flk-1 as a major regulator of vasculogenesis and angiogenesis , 1993, Cell.

[94]  M. Chaplain,et al.  A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor. , 1993, IMA journal of mathematics applied in medicine and biology.

[95]  G B Ermentrout,et al.  Cellular automata approaches to biological modeling. , 1993, Journal of theoretical biology.

[96]  J A Sherratt,et al.  Chemical control of eukaryotic cell movement: a new model. , 1993, Journal of theoretical biology.

[97]  P. Allavena,et al.  Cytokine regulation of tumour-associated macrophages. , 1993, Research in immunology.

[98]  D. Cheresh,et al.  Integrin α v β 3 antagonists promote tumor regression by inducing apoptosis of angiogenic blood vessels , 1994, Cell.

[99]  Lars Holmgren,et al.  Angiostatin: A novel angiogenesis inhibitor that mediates the suppression of metastases by a lewis lung carcinoma , 1994, Cell.

[100]  J. Gross,et al.  Inhibition of angiogenesis as a strategy for tumor growth control , 1994, Molecular and chemical neuropathology.

[101]  M. Gertsenstein,et al.  Dominant-negative and targeted null mutations in the endothelial receptor tyrosine kinase, tek, reveal a critical role in vasculogenesis of the embryo. , 1994, Genes & development.

[102]  J. Sherratt Chemotaxis and chemokinesis in eukaryotic cells: the Keller-Segel equations as an approximation to a detailed model. , 1994, Bulletin of Mathematical Biology.

[103]  Th. M. Liebling,et al.  Culture analysis and external interaction models of mycelial growth , 1994, Bulletin of mathematical biology.

[104]  M. A. J. Chaplain,et al.  The mathematical modelling of tumour angiogenesis and invasion , 1995, Acta biotheoretica.

[105]  J. Folkman Angiogenesis in cancer, vascular, rheumatoid and other disease , 1995, Nature Medicine.

[106]  4217 Constitutive expression of the VEGF receptor KDR/FLK-1 in corneal endothelial cell mediates their proliferation , 1995, Vision Research.

[107]  Thomas N. Sato,et al.  Distinct roles of the receptor tyrosine kinases Tie-1 and Tie-2 in blood vessel formation , 1995, Nature.

[108]  H. M. Byrne,et al.  Mathematical models for tumour angiogenesis: Numerical simulations and nonlinear wave solutions , 1995 .

[109]  A. Harris,et al.  Clinical importance of the determination of tumor angiogenesis in breast carcinoma: much more than a new prognostic tool. , 1995, Journal of clinical oncology : official journal of the American Society of Clinical Oncology.

[110]  L. Ellis,et al.  Angiogenesis and breast cancer metastasis , 1995, The Lancet.

[111]  A. Paetau,et al.  Expression of endothelial cell-specific receptor tyrosine kinases and growth factors in human brain tumors. , 1995, The American journal of pathology.

[112]  A. Bikfalvi Significance of angiogenesis in tumour progression and metastasis. , 1995, European journal of cancer.

[113]  Philip K. Maini,et al.  Cellular pattern formation during Dictyostelium aggregation , 1995 .

[114]  Re: Tumor angiogenesis as a prognostic assay for invasive ductal breast carcinoma. , 1995, Journal of the National Cancer Institute.

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

[116]  J. A. Norton Tumor angiogenesis. The future is now. , 1995, Annals of surgery.

[117]  H. Berg,et al.  Spatio-temporal patterns generated by Salmonella typhimurium. , 1995, Biophysical journal.

[118]  J. Rossant,et al.  Role of the Flt-1 receptor tyrosine kinase in regulating the assembly of vascular endothelium , 1995, Nature.

[119]  E. Haber,et al.  Downregulation of vascular endothelial growth factor receptors by tumor necrosis factor-alpha in cultured human vascular endothelial cells. , 1996, The Journal of clinical investigation.

[120]  H M Byrne,et al.  On the rôle of angiogenesis in wound healing , 1996, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[121]  W Düchting,et al.  Cancer: a challenge for control theory and computer modelling. , 1996, European journal of cancer.

[122]  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.

[123]  P. Hewett,et al.  Coexpression of flt-1, flt-4 and KDR in freshly isolated and cultured human endothelial cells. , 1996, Biochemical and biophysical research communications.

[124]  M. Chaplain Avascular growth, angiogenesis and vascular growth in solid tumours: The mathematical modelling of the stages of tumour development , 1996 .

[125]  P. Hewett,et al.  Coexpression offlt-1,flt-4 andKDRin Freshly Isolated and Cultured Human Endothelial Cells , 1996 .

[126]  F Nekka,et al.  A model of growing vascular structures. , 1996, Bulletin of mathematical biology.

[127]  H. Othmer,et al.  A discrete cell model with adaptive signalling for aggregation of Dictyostelium discoideum. , 1997, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[128]  Hans G. Othmer,et al.  Aggregation, Blowup, and Collapse: The ABC's of Taxis in Reinforced Random Walks , 1997, SIAM J. Appl. Math..

[129]  P. Maini,et al.  A mathematical model for the capillary endothelial cell-extracellular matrix interactions in wound-healing angiogenesis. , 1997, IMA journal of mathematics applied in medicine and biology.

[130]  Identification of a VEGF receptor (KDR/FLK) promoter element which binds an endothelial cell-specific protein conferring endothelial selective expression , 1997 .

[131]  A. Harris Antiangiogenesis for cancer therapy , 1997, The Lancet.

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

[133]  S. Fox,et al.  Expression of the angiogenic factors vascular endothelial cell growth factor, acidic and basic fibroblast growth factor, tumor growth factor beta-1, platelet-derived endothelial cell growth factor, placenta growth factor, and pleiotrophin in human primary breast cancer and its relation to angiogenes , 1997, Cancer research.

[134]  Bryan S. Griffiths,et al.  Nematode movement along a chemical gradient in a structurally heterogeneous environment. 1. Experiment , 1997 .

[135]  Douglas Hanahan,et al.  Signaling Vascular Morphogenesis and Maintenance , 1997, Science.

[136]  Alexander R. A. Anderson,et al.  A Mathematical Model for Capillary Network Formation in the Absence of Endothelial Cell Proliferation , 1998 .

[137]  S. Abdulla Angiogenesis and inflammation , 1999 .