Numerical simulations of angiogenesis in the cornea.

Angiogenesis plays important roles in many physiologic and pathologic processes in the body. To understand mechanisms of angiogenesis, we developed a mathematical model for quantitative analysis of various biological events involved in angiogenesis. Our model was focused on two-dimensional angiogenesis in the cornea. The model considered diffusion of angiogenic factors, uptake of these factors by endothelial cells, and randomness in the rate of sprout formation and the direction of sprout growth. Our simulation results indicated that redistribution and uptake of angiogenic factors during angiogenesis had significant effects on the structure of vascular networks. A decrease in the uptake rate resulted in increases in vessel density, self-loop formation, and front migration speed of vascular networks. The randomness in the direction of sprout formation determined the curvature of vessels, whereas the probability of sprout formation from a vessel segment had a significant effect on the total number of vessels in vascular networks. The vascular networks generated in numerical simulations were similar to those observed experimentally. The mathematical model developed in this study can be used to evaluate effects of individual factors on angiogenesis, understand mechanisms of interactions among different factors during angiogenesis, and generate experimentally testable hypotheses.

[1]  D. Moscatelli,et al.  Increased capillary endothelial cell protease activity in response to angiogenic stimuli in vitro. , 1983, Proceedings of the National Academy of Sciences of the United States of America.

[2]  N. Bouck,et al.  Assay and purification of naturally occurring inhibitor of angiogenesis. , 1991, Methods in enzymology.

[3]  M. Markus,et al.  Simulation of vessel morphogenesis using cellular automata. , 1999, Mathematical biosciences.

[4]  R K Jain,et al.  Direct in vivo measurement of targeted binding in a human tumor xenograft. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[5]  R K Jain,et al.  Fluorescence photobleaching with spatial Fourier analysis: measurement of diffusion in light-scattering media. , 1993, Biophysical journal.

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

[7]  R. Auerbach,et al.  Assays for angiogenesis: a review. , 1991, Pharmacology & therapeutics.

[8]  M. Chaplain,et al.  Mathematical modelling, simulation and prediction of tumour-induced angiogenesis. , 1996, Invasion & metastasis.

[9]  W. Risau,et al.  Mechanisms of angiogenesis , 1997, Nature.

[10]  A. R. Gourlay,et al.  Hopscotch: a Fast Second-order Partial Differential Equation Solver , 1970 .

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

[12]  D. Gospodarowicz,et al.  Brain-derived fibroblast growth factor: a study of its inactivation. , 1983, Life sciences.

[13]  H. Dvorak.  Tumors: wounds that do not heal. Similarities between tumor stroma generation and wound healing. , 1986, The New England journal of medicine.

[14]  R. Jain,et al.  Role of tumor vascular architecture in nutrient and drug delivery: an invasion percolation-based network model. , 1996, Microvascular research.

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

[16]  Rakesh K. Jain,et al.  Quantitative angiogenesis assays: Progress and problems , 1997, Nature Medicine.

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

[18]  R K Jain,et al.  Quantitation and physiological characterization of angiogenic vessels in mice: effect of basic fibroblast growth factor, vascular endothelial growth factor/vascular permeability factor, and host microenvironment. , 1996, The American journal of pathology.

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

[20]  J. Folkman,et al.  Angiogenesis and angiogenesis inhibition: an overview. , 1997, EXS.

[21]  Takayuki Asahara,et al.  Isolation of Putative Progenitor Endothelial Cells for Angiogenesis , 1997, Science.

[22]  Napoleone Ferrara,et al.  Clinical applications of angiogenic growth factors and their inhibitors , 1999, Nature Medicine.

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

[24]  R. Montesano.  1992 MACK FORSTER AWARD LECTURE REVIEW , 1992, European journal of clinical investigation.

[25]  Berk,et al.  Scale-invariant behavior and vascular network formation in normal and tumor tissue. , 1995, Physical review letters.

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

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

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

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

[30]  H. Granger,et al.  B1 receptor involvement in the effect of bradykinin on venular endothelial cell proliferation and potentiation of FGF‐2 effects , 1998, British journal of pharmacology.

[31]  R. Jain,et al.  Microvascular architecture in a mammary carcinoma: branching patterns and vessel dimensions. , 1991, Cancer research.

[32]  J. Folkman,et al.  A model of angiogenesis in the mouse cornea. , 1996, Investigative ophthalmology & visual science.

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

[34]  S. Szabó,et al.  Duodenal ulcer : discovery of a new mechanism and development of angiogenic therapy that accelerates healing , 1991 .

[35]  R. Jain,et al.  Angiogenesis, microvascular architecture, microhemodynamics, and interstitial fluid pressure during early growth of human adenocarcinoma LS174T in SCID mice. , 1992, Cancer research.

[36]  A. Pries,et al.  Structural adaptation and stability of microvascular networks: theory and simulations. , 1998, American journal of physiology. Heart and circulatory physiology.

[37]  M. Nugent,et al.  Heparan sulfate proteoglycans control intracellular processing of bFGF in vascular smooth muscle cells. , 1998, Biochemistry.

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