A continuous model of Angiogenesis: Initiation, extension, and maturation of new blood vessels modulated by vascular endothelial growth factor, Angiopoietins, platelet-derived growth factor-B, and pericytes

Discrete and Continuous Dynamical Systems - Series B

Volumn 18, Issue 4, 2013, Pages 1109-1154

  Zheng, Xiaoming ^a   Koh, Gou Young ^b   Jackson, Trachette ^c  

^a CENTRAL MICHIGAN UNIVERSITY   (United States)

^b Korea Advanced Institute of Science and Technology (KAIST)   (South Korea)

^c UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Angiogenesis; Angiopoietin 1; Angiopoietin 2; Antiangiogenic therapy; Corneal angiogenesis; Endothelial cells; Initiation; Maturation; Migration; Pericytes; Platelet derived growth factor B; Tie2; Vascular endothelial growth factor; Viscoelastic model

Indexed keywords

EID: 84874736686     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2013.18.1109     Document Type: Conference Paper

References (162)

1. A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. Math. Biol., 60 (1998), 857-900.
2. A. R. A. Anderson and M. A.J. Chaplain, A mathematical model for capillary network formation in the absence of endothelial cell proliferation, Appl. Math. Lett., 11 (1998), 109-114.
3. L. Arakelyan, V. Vainstein and Z. Agur, 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 (2002), 203-214.
4. A. Armulik, A. Abramsson and C. Betsholtz, Endothelial/pericyte interactions, Circ. Res., 97 (2005), 512-523.
5. G. Ateshian, On the theory of reactive mixtures for modeling biological growth, Biomech. Model. Mechanobiol., 6 (2007), 423-445.
6. H. G. Augustin, G. Y. Koh, G. Thurston and K. Alitalo, Control of vascular morphogenesis and homeostasis through the angiopoietin-Tie system, Nat. Rev. Mol. Cell Biol., 10 (2009), 165-177.
7. D. Balding and D. L. S. McElwain, A mathematical model of tumor-induced capillary growth, J. Theor. Biol., 114 (1985), 53-73.
8. K. Bartha and H. Rieger, Vascular network remodeling via vessel cooption, regression and growth in tumors, J. Theor. Biol., 21 (2006), 903-918.
9. A. Bauer, T. Jackson and Y. Jiang, A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis, Biophys. J., 92 (2007), 3105-3121.
10. A. Bauer, T. Jackson, Y. Jiang and T. Rohlf, Receptor cross-talk in angiogenesis: mapping environmental cues to cell phenotype using a stochastic, boolean signaling network model, J. Theor. Biol., 264 (2010), 838-846.
11. A. R. Bausch, F. Ziemann, A. A. Boulbitch, K. Jacobson and E. Sackmann, Local measurements of viscoelastic parameters of adherent cell surfaces by magnetic bead microrheometry, Biophys. J., 75 (1998), 2038-2049.
12. K. Bentley, H. Gerhardt and P. A. Bates, Agent-based simulation of notch-mediated tip cell selection in angiogenic sprout initialisation, J. Theor. Biol., 250 (2008), 25-36.
13. K. Bentley, G. Mariggi, H. Gerhardt and P. A. Bates, Tipping the balance: Robustness of tip cell selection, migration and fusion in angiogenesis, PLoS Comput. Biol., 5 (2009), e1000549.
14. G. Bergers and D. Hanahan, Modes of resistance to anti-angiogenic therapy, Nat. Rev. Cancer, 8 (2008), 592-603.
15. G. Bergers, S. Song, N. Meyer-Morse, et al, Benefits of targeting both pericytes and en-dothelial cells in the tumor vasculature with kinase inhibitors, J. Clin. Invest., 111 (2003), 1287-1295.
16. F. Billy, B. Ribba, O. Saut, et al, A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy, J. Theor. Biol., 260 (2009), 545-562.
17. M. Bjarnegard, M. Enge, J. Norlin, et al, Endothelium-specific ablation of PDGF-B leads to pericyte loss and glomerular, cardiac and placental abnormalities, Development, 131 (2004), 1847-1857.
18. E. Bogdanovic, V. P. Nguyen and D. J. Dumont, Activation of Tie2 by angiopoietin-1 and angiopoietin-2 results in their release and receptor internalization, J. Cell Sci., 119 (2006), 3551-3560.
19. R. M. Bowen, "Introduction to Continuum Mechanics for Engineers," Springer, 2007.
20. H. M. Byrne and M. A. J. Chaplain, Mathematical models for tumour angiogenesis: Numerical simulations and nonlinear wave solutions, Bull. Math. Biol., 57 (1995), 461-486.
21. H. M. Byrne and M. A. J. Chaplain, Explicit solutions of a simplified model of capillary sprout growth during tumor angiogenesis, Appl. Math. Lett., 9 (1996), 69-74.
22. J. Cai, O. Kehoe, G. M. Smith, et al, The angiopoietin/Tie-2 system regulates pericyte survival and recruitment in diabetic retinopathy, Invest. Ophthalmol. Vis. Sci., 49 (2008), 2163.
23. V. Capasso and D. Morale, Stochastic modelling of tumour-induced angiogenesis, J. Math. Biol., 58 (2009), 219-233.
24. P. Carmeliet, Angiogenesis in life, disease and medicine, Nature, 438 (2005), 932-936.
25. P. Carmeliet and R. K. Jain, Molecular mechanisms and clinical applications of angiogenesis, Nature, 473 (2011), 298-307.
26. R. Carmeliet and R. K. Jain, Angiogenesis in cancer and other diseases, Nature, 407 (2000), 249-257.
27. S. Ćebe-Suarez, A. Zehnder-Fjällman and K. Ballmer-Hofer, The role of VEGF receptors in angiogenesis; complex partnerships, Cell. Mol. Life Sci., 63 (2006), 601-615.
28. B. Cohen, D. Barkan, Y. Levy, et al, Leptin induces angiopoietin-2 expression in adipose tissues, J. Biol. Chem., 276 (2001), 7697-7700.
29. K. D. Costa, A. J. Sim and F. C.-P. Yin, Non-Hertzian approach to analyzing mechanical properties of endothelial cells probed by atomic force microscopy, J. Biomech. Eng., 128 2006, 176-184.
30. S. C. Cowin, Tissue growth and remodeling, Annu. Rev. Biomed. Eng., 6 (2004), 77-107.
31. S. Davis, T. H. Aldrich, P. F. Jones, et al, Isolation of angiopoietin-1, a ligand for the Tie2 receptor, by secretion-trap expression cloning, Cell, 87 (1996), 1161-1169.
32. F. De Smet, I. Segura, K. De Bock, et al, Mechanisms of vessel branching, Arterioscler. Thromb. Vasc. Biol., 29 (2009), 639-649.
33. N. Desprat, A. Richert, J. Simeon and A. Asnacios, Creep function of a single living cell, Biophys. J., 88 (2005), 2224-2233.
34. H. F. Dvorak, Vascular permeability factor/vascular endothelial growth factor: A critical cytokine in tumor angiogenesis and a potential target for diagnosis and therapy, J. Clin. Oncol., 20 (2002), 4368-4380.
35. L. M. Ellis and D. J. Hicklin, Vegf-targeted therapy: Mechanisms of anti-tumour activity, Nat. Rev. Cancer, 8 (2008), 579-591.
36. R. Erber, A. Thurnher, A. D. Katsen, et al, Combined inhibition of vegf and pdgf signaling enforces tumor vessel regression by interfering with pericyte-mediated endothelial cell survival mechanisms, FASEB J., 18 (2004), 338-340.
37. Y. Feng, F. Vom Hagen, F. Pfister, et al, Impaired pericyte recruitment and abnormal retinal angiogenesis as a result of angiopoietin-2 overexpression, Thromb. Haemostasis, 97 (2007), 99-108.
38. P. Fernandez, L. Heymann, A. Ott, N. Aksel and P. A Pullarkat, Shear rheology of a cell monolayer, New J. Phys., 9 (2007), 419.
39. P. Fernandez and A. Ott, Single cell mechanics: Stress stiffening and kinematic hardening, Phys. Rev. Lett., 100 (2008), 238102.
40. N. Ferrara, The role of VEGF in the regulation of physiological and pathological angiogenesis, in "Mechanisms of Angiogenesis, Experientia Supplementum" (eds. Matthias Clauss and Georg Breier), Birkḧauser Basel, (2005), 209-231.
41. N. Ferrara, G. Hans-Peter and L. Jennifer, The biology of VEGF and its receptors, Nat. Med., 9 (2003), 669-676.
42. U. Fiedler, T. Krissl, S. Koidl, et al, Angiopoietin-1 and angiopoietin-2 share the same binding domains in the Tie-2 receptor involving the first Ig-like loop and the epidermal growth factor-like repeats, J.Biol. Chem., 278 (2003), 1721-1727.
43. U. Fiedler, M. Scharpfenecker, S. Koidl, et al, The Tie-2 ligand Angiopoietin-2 is stored in and rapidly released upon stimulation from endothelial cell Weibel-Palade bodies, Blood, 103 (2004), 4150-4156.
44. J. Folkman, Tumor angiogenesis: Therapeutic implications, New Engl. J. Med., 285 (1971), 1182-1186.
45. J. Folkman and R. Kalluri, Tumor angiogenesis, in "Cancer Medicine" (eds. D.W. Kufe, R.E. Pollock, R.R. Weichselbaum, et al.), BC Decker Inc., (2003), chapter 11.
46. K. Forsten-Williams, C. C. Chua and M. A. Nugent, The kinetics of fgf-2 binding to heparan sulfate proteoglycans and map kinase signaling, J. Theor. Biol., 233 (2005), 483-499.
47. J. A. Fozard, H. M. Byrne, O. E. Jensen and J. R. King, Continuum approximations of individual-based models for epithelial monolayers, Mathematical Medicine and Biology, 27 (2010), 39-74.
48. M. Franco, P. Roswall, E. Cortez, D. Hanahan and K. Pietras, Pericytes promote endothelial cell survival through induction of autocrine vegf-a signaling and bcl-w expression, Blood, 118 (2011), 2906-2917.
49. S. Fukuhara, K. Sako, K. Noda, et al, Tie2 is tied at the cell-cell contacts and to extracellular matrix by angiopoietin-1, Exp. Mol. Med., 41 (2009), 133.
50. S. Fukuhara, K. Sako, K Noda, et al, Angiopoietin-1/Tie2 receptor signaling in vascular quiescence and angiogenesis, Histol. Histopathol., 25 (2010), 387-96.
51. K. Gaengel, G. Genove, A. Armulik and C. Betsholtz, Endothelial-mural cell signaling in vascular development and angiogenesis, Arterioscler. Thromb. Vasc. Biol., 29 (2009), 630-638.
52. A. Gamba, D. Ambrosi, A. Coniglio, et al, Percolation, morphogenesis, and burgers dynamics in blood vessels formation, Phys. Rev. Lett., 90 (2003), 118101.
53. J. R. Gamble, J. Drew, L. Trezise, et al, Angiopoietin-1 is an antipermeability and anti-inflammatory agent in vitro and targets cell junctions, Circ. Res., 87 (2000), 603-607.
54. K. Garikipati, The kinematics of biological growth, Appl. Mech. Rev., 62 (2009), 030801.
55. H. Gerber, A. McMurtrey, J. Kowalski, et al, Vascular endothelial growth factor regulates endothelial cell survival through the phosphatidylinositol 3'-kinase/Akt signal transduction pathway. Requirement for Flk-1/KDR activation, J. Biol. Chem., 273 (1998), 30336-30343.
56. H. Gerhardt, M. Golding, M. Fruttiger, et al, VEGF guides angiogenic sprouting utilizing endothelial tip cell filopodia, J. Cell Biol., 161 (2003), 1163-1177.
57. J. L. Gevertz and S. Torquato, Modeling the effects of vasculature evolution on early brain tumor growth, J. Theor. Biol., 243 (2006), 517-531.
58. V. Goede, T. Schmidt, S. Kimmina, D. Kozian and HG Augustin, Analysis of blood vessel maturation processes during cyclic ovarian angiogenesis, Lab. Invest., 78 (1998), 1385.
59. H. P. Hammes, J. Lin, P. Wagner, et al, Angiopoietin-2 causes pericyte dropout in the normal retina: Evidence for involvement in diabetic retinopathy, Diabetes, 53 (2004), 1104-1110.
60. D. Hanahan and J. Folkman, Patterns and emerging mechanisms of the angiogenic switch during tumorigenesis, Cell, 86 (1996), 353-364.
61. R. Harfouche and S. N. A. Hussain, Signaling and regulation of endothelial cell survival by angiopoietin-2, Am. J. Physiol. Heart Circ. Physiol., 291 (2006), 1635-1645.
62. A. Hegen, S. Koidl, K. Weindel, et al, Expression of angiopoietin-2 in endothelial cells is controlled by positive and negative regulatory promoter elements, Arterioscler. Thromb. Vasc. Biol., 24 (2004), 1803-1809.
63. M. Hellstr̈om, M. Kaèn, P. Lindahl, A. Abramsson and C. Betsholtz, Role of PDGF-B and PDGFR-beta in recruitment of vascular smooth muscle cells and pericytes during embryonic blood vessel formation in the mouse, Development, 126 (1999), 3047-3055.
64. K. K. Hirschi, S. A. Rohovsky, L. H. Beck, S. R. Smith and P. A. D'Amore, Endothelial cells modulate the proliferation of mural cell precursors via platelet-derived growth factor-BB and heterotypic cell contact, Circ. Res., 84 (1999), 298-305.
65. J. Holash, P. C. Maisonpierre, D. Compton, et al, Vessel cooption, regression, and growth in tumors mediated by angiopoietins and VEGF, Science, 284 (1999), 1994-1998.
66. M. J. Holmes and B. D. Sleeman, A mathematical model of tumour angiogenesis incorporating cellular traction and viscoelastic effects, J. Theor. Biol., 202 (2000), 95-112.
67. J. Huang, J. O. Bae, J. P. Tsai, et al, Angiopoietin-1/Tie-2 activation contributes to vascular survival and tumor growth during VEGF blockade, Int J Oncol, 34 (2009), 79-87.
68. T. L. Jackson and X. Zheng, A cell-based model of endothelial cell elongation, proliferation and maturation during corneal angiogenesis, Bull. Math. Biol., 72 (2010), 830-868.
69. R. K. Jain, Normalization of tumor vasculature: an emerging concept in antiangiogenic therapy, Science, 307 (2005), 58-62.
70. C. Jang, Y. J. Koh, N. K. Lim, et al, Angiopoietin-2 exocytosis is stimulated by sphingosine-1-phosphate in human blood and lymphatic endothelial cells, Arterioscler. Thromb. Vasc. Biol., 29 (2009), 401-407.
71. N. Jo, C. Mailhos, M. Ju, et al, Inhibition of platelet-derived growth factor B signaling enhances the efficacy of anti-vascular endothelial growth factor therapy in multiple models of ocular neovascularization, Am. J. Pathol., 168 (2006), 2036-2053.
72. E. Karl, K. Warner, B. Zeitlin, et al, Bcl-2 acts in a proangiogenic signaling pathway through nuclear factor-κB and CXC chemokines, Cancer Res., 65 (2005), 5063-5069.
73. I. Kim, H. G. Kim, J. So, et al, Angiopoietin-1 regulates endothelial cell survival through the phosphatidylinositol 3'-Kinase/Akt signal transduction pathway, Circ. Res., 86 (2000), 24-29.
74. I. Kim, J. H. Kim, Y. S. Ryu, M. Liu and G. Y. Koh, Tumor necrosis factor-α upregulates angiopoietin-2 in human umbilical vein endothelial cells, Biochem. Biophys. Res. Commun., 269 (2000), 361-365.
75. K. Kim et al, Oligomerization and multimerization are critical for angiopoietin-1 to bind and phosphorylate tie2, J. Biol. Chem., 280 (2005), 20126-20131.
76. S. Koch, S. Tugues, X. Li, L. Gualandi and L. Claesson-Welsh, Signal transduction by vascular endothelial growth factor receptors, Biochem. J., 437 (2011), 169-183.
77. Y. J. Koh, H.-Z. Kim, S.-I. Hwang, et al, Double antiangiogenic protein, DAAP, targeting VEGF-A and angiopoietins in tumor angiogenesis, metastasis, and vascular leakage, Cancer Cell, 18 (2010), 171-184.
78. R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588.
79. K. Larripa and A. Mogilner, Transport of a 1D viscoelastic actin-myosin strip of gel as a model of a crawling cell, Physica A, 372 (2006), 113-123.
80. H. A. Levine and M. Nilsen-Hamilton, Angiogenesis-A biochemial/ mathematical perspective, in "Tutorials in Mathematical Biosciences III" (editor Aver Friedman), Springer, (2006), chapter 2.
81. H. A. Levine, S. Pamuk, B. D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of capillary formation and development in tumor angiogenesis: Penetration into the stroma, Bull. Math. Biol., 63 (2001), 801-863.
82. H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. I. the role of protease inhibitors in preventing angiogenesis, Math. Biosci., 168 (2000), 77-115.
83. F. Li and X. Zheng, Singularity analysis of a reaction-diffusion equation with a solution-dependent dirac delta source, Appl. Math. Lett., 25 (2012), 2179-2183.
84. G. Liu, A. Qutub, P. Vempati, F. Mac Gabhann and A. Popel, Module-based multiscale simulation of angiogenesis in skeletal muscle, Theor. Biol. Med. Model., 8 (2011), 6.
85. I. B. Lobov, P. C. Brooks and R. A. Lang, Angiopoietin-2 displays VEGF-dependent modulation of capillary structure and endothelial cell survival in vivo, PNAS, 99 (2002), 11205-11210.
86. N. R. London, K. J. Whitehead and D. Y. Li, Endogenous endothelial cell signaling systems maintain vascular stability, Angiogenesis, 12 (2009), 149-158.
87. B. Loret and F. M. F. Simes, A framework for deformation, generalized diffusion, mass transfer and growth in multi-species multi-phase biological tissues, Eur. J. Mech. A-Solid, 24 (2005), 757-781.
88. C. Lu, A. A. Kamat, Y. G. Lin, et al, Dual targeting of endothelial cells and pericytes in antivascular therapy for ovarian carcinoma, Clin. Cancer Res., 13 (2007), 4209-4217.
89. C. Lu, P. H. Thaker, Y. G. Lin, et al, Impact of vessel maturation on anti-angiogenic therapy in ovarian cancer, Am. J. Obstet. Gynecol., 198 (2008), 477.
90. R. Mabry, D. G Gilbertson, A. Frank, et al, A dual-targeting pdgfrbeta/vegf-a molecule assembled from stable antibody fragments demonstrates anti-angiogenic activity in vitro and in vivo, mAbs, 2 (2010), 20-34.
91. F. Mac Gabhann and A. S. Popel, 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 (2004), 153-164.
92. F. Mac Gabhann and A. S. Popel, Targeting neuropilin-1 to inhibit vegf signaling in cancer: Comparison of therapeutic approaches, PLoS Comput. Biol., 2 (2006), e180.
93. F. Mac Gabhann and A. S. Popel, Dimerization of VEGF receptors and implications for signal transduction: A computational study, Biophys. Chem., 128 (2007), 125-139.
94. P. C. Maisonpierre, C. Suri, P. F. Jones, et al, Angiopoietin-2, a natural antagonist for Tie2 that disrupts in vivo angiogenesis, Science, 277 (1997), 55-60.
95. S. J. Mandriota and M. S. Pepper, Regulation of angiopoietin-2 mRNA levels in bovine microvascular endothelial cells by cytokines and hypoxia, Circ. Res., 83 (1998), 852-859.
96. D. Manoussaki, A mechanochemical model of angiogenesis and vasculogenesis, ESAIM: Mathematical Modelling and Numerical Analysis, 37 (2003), 581-599.
97. N. Mantzaris, S. Webb and H. G. Othmer, Mathematical modeling of tumor-induced angio-genesis, J. Math Biol., 49 (2004), 111-187.
98. K. Matsushita, M. Yamakuchi, C. N. Morrell, et al, Vascular endothelial growth factor regulation of Weibel-Palade-body exocytosis, Blood, 105 (2005), 207.
99. S. R. McDougall, A. R. A. Anderson and M. A. J. Chaplain, Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: Clinical implications and therapeutic targeting strategies, J. Theor. Biol., 241 (2006), 564-589.
100. S. R. McDougall, M. A. J. Chaplain, A. St́ephanou and A. R. A. Anderson, Modelling the impact of pericyte migration and coverage of vessels on the efficacy of vascular disrupting agents, Math. Model. Nat. Phenom., 5 (2010), 163-202.
101. Q. Mi, D. Swigon, et al, One-dimensional elastic continuum model of enterocyte layer migration, Biophys. J., 93 (2007), 3745-3752.
102. F. Milde, M. Bergdorf and P. Koumoutsakos, A hybrid model for three-dimensional simulations of sprouting angiogenesis, Biophys. J., 95 (2008), 3146-3160. ́
103. R. Muñoz Ch́apuli, A. R. Quesada and M. Angel Medina, Angiogenesis and signal trans-duction in endothelial cells, Cell. Mol. Life Sci., 61 (2004), 2224-2243.
104. G. N. Naumov, E. Bender, Zurakowski, et al, A model of human tumor dormancy: An angiogenic switch from the nonangiogenic phenotype, J. Natl. Cancer Inst., 98 (2006), 316-325.
105. J. Nor and P. Polverini, Role of endothelial cell survival and death signals in angiogenesis, Angiogenesis, 3 (1999), 101-116.
106. H. Oh, H. Takagi, K. Suzuma, et al, Hypoxia and vascular endothelial growth factor selectively up-regulate angiopoietin-2 in bovine microvascular endothelial cells, J. Biol. Chem., 274 (1999), 15732-15739.
107. J. Oliner et al, Suppression of angiogenesis and tumor growth by selective inhibition of angiopoietin-2, Cancer Cell, 6 (2004), 507-516.
108. M. R. Owen, T. Alarcon, P. K. Maini and H. M. Byrne, Angiogenesis and vascular remodelling in normal and cancerous tissues, J. Theor. Biol., 58 (2009), 689-721.
109. S. M. Parikh, T. Mammoto, A. Schultz, et al, Excess circulating angiopoietin-2 may contribute to pulmonary vascular leak in sepsis in humans, PLoS Med., 3 (2006), 356.
110. S. M. Peirce, Computational and mathematical modeling of angiogenesis, Microcirculation, 15 (2008), 739-751.
111. S. M. Peirce, E. J. Van Gieson and T. C. Skalak, Multicellular simulation predicts microvas-cular patterning and in silico tissue assembly, FASEB J., 18 (2004), 731-733
112. K. Pietras and D. Hanahan, A multitargeted, metronomic, and maximum-tolerated dose chemo-switch regimen is antiangiogenic, producing objective responses and survival benefit in a mouse model of cancer, J. Clin. Oncol., 23 (2005), 939-952.
113. M. J. Plank and B. D. Sleeman, A reinforced random walk model of tumor angiogenesis and anti-angiogenesis strategies, IMA J. Math. Med. Biol., 20 (2003), 135-181.
114. M. J. Plank and B. D. Sleeman, Lattice and non-lattice models of tumour angiogenesis, Bull. Math. Biol., 66 (2004), 1785-1819.
115. M. J. Plank, B. D. Sleeman and P. F. Jones, A mathematical model of tumour angiogenesis, regulated by vascular endothelial growth factor and the angiopoietins, J. Theor. Biol., 229 (2004), 435-454.
116. M. Prass, K. Jacobson, A. Mogilner and M. Radmacher, Direct measurement of the lamel-lipodial protrusive force in a migrating cell, J. Cell Biol., 174 (2006), 767-772.
117. A. A. Qutub, F. Mac Gabhann, E. D. Karagiannis, P. Vempati and A. S. Popel, Multiscale models of angiogenesis, IEEE Eng. Med. Biol. Mag., 28 (2009), 14-31.
118. A. A. Qutub and A. Popel, Elongation, proliferation & migration differentiate endothelial cell phenotypes and determine capillary sprouting, BMC Syst. Biol., 3 (2009), 13.
119. A. Ramasubramanian and L. Taber, Computational modeling of morphogenesis regulated by mechanical feedback, Biomech. Model. Mechanobiol., 7 (2008), 77-91.
120. A. Raza, M. J. Franklin and A. Z. Dudek, Pericytes and vessel maturation during tumor angiogenesis and metastasis, Am. J. Hematol., 85 (2010), 593-598.
121. Y. Reiss, J. Droste, M. Heil, et al, Angiopoietin-2 impairs revascularization after limb ischemia, Circ. Res., 101 (2007), 88-96.
122. E. K. Rodriguez, A. Hoger and A. D. McCulloch, Stress-dependent finite growth in soft elastic tissues, J. Biomech., 27 (1994), 455-467.
123. P. Saharinen, L. Eklund, J. Miettinen, et al, Angiopoietins assemble distinct Tie2 signalling complexes in endothelial cell-cell and cell-matrix contacts, Nat. Cell Biol., 10 (2008), 527-537.
124. R.C. Schugart, A. Friedman, R. Zhao and C. K. Sen, Wound angiogenesis as a function of tissue oxygen tension: A mathematical model, PNAS, 105 (2008), 2628-2633.
125. C. E. Semino, R. D. Kamm and D. A. Lauffenburger, Autocrine EGF receptor activation mediates endothelial cell migration and vascular morphogenesis induced by VEGF under interstitial flow, Exp. Cell Res., 312 (2006), 289-298.
126. G. Serini, D. Ambrosi, E. Giraudo, et al, Modeling the early stages of vascular network assembly, EMBO J., 22 (2003), 1771-1779.
127. C. Sfiligoi, A. de Luca, I. Cascone, et al, Angiopoietin-2 expression in breast cancer correlates with lymph node invasion and short survival, Int. J. Cancer, 103 (2003), 466-474.
128. J. Shen et al, An antibody directed against pdgf receptor enhances the antitumor and the anti-angiogenic activities of an anti-vegf receptor 2 antibody, Biochem. Biophys. Res. Commun., 357 (2007), 1142-1147.
129. J. A. Sherratt and J. D. Murrat, Models of epidermal wound healing, Proc. R. Soc. Lond. B., 241 (1990), 29-36.
130. M. M. Sholley, G. P. Ferguson, H. R. Seibel, et al, Mechanisms of neovascularization. Vascular sprouting can occur without proliferation of endothelial cells, Lab. Invest., 51 (1984), 624-634.
131. B. D. Sleeman and I. P. Wallis, Tumour induced angiogenesis as a reinforced random walk: modeling capillary network formation without endothelial cell proliferation, Math. Comput. Model., 36 (2002), 339-358.
132. A. Stephanou, S. R. McDougall, A. R. A. Anderson and M. A. J. Chaplain, Mathematical modelling of the influence of blood rheological properties upon adaptative tumour-induced angiogenesis, J. Theor. Biol., 44 (2006), 96-123.
133. C. L. Stokes and D. A. Lauffenburger, Analysis of the roles of microvessel endothelial cell random mobility and chemotaxis in angiogenesis, J. Ther. Biol., 152 (1991), 377-403.
134. S. Sun, M. F. Wheeler, M. Obeyesekere and C. Patrick, A deterministic model of growth factor-induced angiogenesis, Bull. Math. Biol., 67 (2005), 313-337.
135. C. Sundberg, M. Kowanetz, L.F. Brown, M. Detmar and H. F. Dvorak, 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 (2002), 387-401.
136. C. Suri, P. F. Jones, S. Patan, et al, Requisite role of angiopoietin-1, a ligand for the Tie2 receptor, during embryonic angiogenesis, Cell, 87 (1996), 1171-1180.
137. A. Szabo, E. D. Perryn and A. Czirok, Network formation of tissue cells via preferential attraction to elongated structures, Phys. Rev. Lett., 98 (2007), 038102.
138. D. Szczerba, H. Kurz and G. Szekely, A computational model of intussusceptive microvas-cular growth and remodeling, J. Theor. Biol., 261 (2009), 570-583.
139. C. R. Tait and P. F. Jones, Angiopoietins in tumours: the angiogenic switch, J. Pathol., 204 (2004), 1-10.
140. K. Teichert-Kuliszewska, P. C. Maisonpierre, N. Jones, et al, Biological action of angiopoietin-2 in a fibrin matrix model of angiogenesis is associated with activation of Tie2, Cardiovasc. Res., 49 (2001), 659-670.
141. L. J. Thompson, F. Wang, A. D. Proia, et al, Proteome analysis of the rat cornea during angiogenesis, Proteomics, 3 (2003), 2258-2266.
142. O. Thoumine and A. Ott, Time scale dependent viscoelastic and contractile regimes in fibroblasts probed by microplate manipulation, J. Cell Sci., 110 (1997), 2109-2116.
143. G. Thurston, J. S. Rudge, E. Ioffe, et al, Angiopoietin-1 protects the adult vasculature against plasma leakage, Nat. Med., 6 (2000), 460-463.
144. G. Thurston, C. Suri, K. Smith, et al, Leakage-resistant blood vessels in mice transgenically overexpressing angiopoietin 2, Science, 286 (1999), 2511-2514.
145. S. Tong and F. Yuan, Numerical simulations of angiogenesis in the cornea, Microvasc. Res., 61 (2001), 14-27.
146. R. D. M. Travasso, E. Corvera Poir, M. Castro, et al, Tumor angiogenesis and vascular patterning: A mathematical model, PLoS ONE, 6 (2011), e19989.
147. R. Tyson, L. G. Stern and R. J. LeVeque, Fractional step methods applied to a chemotaxis model, J. Math. Biol., 41 (2000), 455-475.
148. K. Y. Volokh, Stresses in growing soft tissues, Acta Biomater., 2 (2006), 493-504.
149. S. Wakui, K. Yokoo, T. Muto, et al, Localization of Ang-1,-2, Tie-2, and VEGF expression at endothelial-pericyte interdigitation in rat angiogenesis, Lab. Invest., 86 (2006), 1172-1184.
150. R. Wcislo, W. Dzwinel, D. Yuen and A. Dudek, A 3-D model of tumor progression based on complex automata driven by particle dynamics, J. Mol. Model., 15 (2009), 1517-1539.
151. M. Welter, K. Bartha and H. Rieger, Vascular remodelling of an arterio-venous blood vessel network during solid tumour growth, J. Theor. Biol., 259 (2009), 405-422.
152. R. R. White, S. Shan, C. P. Rusconi, et al, Inhibition of rat corneal angiogenesis by a nuclease-resistant RNA aptamer specific for angiopoietin-2, PNAS, 100 (2003), 5028-5033.
153. J. L. Wilkinson-Berka, S. Babic, T. de Gooyer, et al, Inhibition of platelet-derived growth factor promotes pericyte loss and angiogenesis in ischemic retinopathy, Am. J. Pathol., 164 (2004), 1263-1273.
154. J. Wu, Q. Long, Xu S. and A. R. Padhani, Study of tumor blood perfusion and its variation due to vascular normalization by anti-angiogenic therapy based on 3d angiogenic microvas-culature, J. Biomech., 42 (2009), 712-721.
155. C. Xue, A. Friedman and C. K. Sen, A mathematical model of ischemic cutaneous wounds, PNAS, 106 (2009), 16782-16787.
156. S. Yang and T. Saif, Reversible and repeatable linear local cell force response under large stretches, Exp. Cell Res., 305 (2005), 42-50.
157. H. T. Yuan, E. V. Khankin, S. A. Karumanchi and S. M. Parikh, Angiopoietin 2 is a partial agonist/antagonist of Tie2 signaling in the endothelium, Mol. Cell. Biol., 29 (2009), 2011-2022.
158. H. T. Yuan, P. G. Tipping, X. Z. Li, D. A. Long and A. S. Woolf, Angiopoietin correlates with glomerular capillary loss in anti-glomerular basement membrane glomerulonephritis, Kidney Int., 61 (2002), 2078-2089.
159. L. Zhang, N. Yang, J. Park, et al, Tumor-derived vascular endothelial growth factor up-regulates angiopoietin-2 in host endothelium and destabilizes host vasculature, supporting angiogenesis in ovarian cancer, Cancer Res., 63 (2003), 3403-3412.
160. X. Zheng, Y. Kim, L. Rakesh and E.-B. Lin, A conservative multiresolution finite volume method for reaction and diffusion in angiogenesis, Submitted.
161. X. Zheng, S. Wise and V. Cristini, Nonlinear simulation of tumor necrosis, neovasculariza-tion and tissue invasion via an adaptive finite-element/level-set method, Bull. Math. Biol., 67 (2005), 211-259.
162. X. Zheng and C. Xie, A viscoelastic model of blood capillary extension and regression: Derivation, analysis, and simulation, J. Math. Biol., (2012).