Qualitative analysis of a mathematical model for capillary formation in tumor angiogenesis

Mathematical Models and Methods in Applied Sciences

Volumn 13, Issue 1, 2003, Pages 19-33

  Pamuk, Serdal ^a  

^a KOCAELI UNIVERSITY   (Turkey)

Author keywords

Angiogenesis; Capillary; Probability density function; Quasi steady state; Stability

Indexed keywords

EID: 0037289527     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202503002362     Document Type: Review

References (28)

1. D. Ambrosi, A. Gamba, E. Giraudo, G. Serini, L. Preziosi and F. Bussolino, Burgers-like dynamics governs the self-assembly of endothelial cell networks, Rapporto Dipartimento di Matematica, Politecnico di Torino, No. 24, giugno (2001).
2. D. Ambrosi and L. Preziosi, On the closure of mass balance models for tumour growth, Math. Models Methods Appl. Sci. 12 (2002) 731-749.
3. A. R. A. Anderson and M. A. J. Chaplin, A mathematical model for capillary network formation in the absence of endothelial cell proliferation, Appl. Math. Lett. 11 (1998) 109-114.
4. E. D. Angelis and L. Preziosi, Advection-diffusion models for solid turnout evolution in vivo and related free boundary problem, Math. Models Methods Appl. Sci. 10 (2000) 379-407.
5. D. H. Ausprunk and J. Folkman, Migration and proliferation of endothelial cells in preformed and newly formed blood vessels during tumor angiogenesis, Microvascular Res. 14 (1977) 53-65.
6. J. C. Bowersox and N. Sorgento, Chemotaxis of aortic endothelial cells in response to fibronectin, Cancer Res. 42 (1982) 2547-2551.
7. M. A. J. Chaplin, S. M. Giles, B. D. Sleeman and R. J. Jarvis, A mathematical analysis of a model for turnout angiogenesis, J. Math. Biol. 33 (1995) 774-770.
8. M. A. J. Chaplin and L. Preziosi, Macroscopic modelling of the growth and development of tumor masses, Rapporto Dipartimento di Matematica, Politecnico di Torino, No. 27 (November 2000).
9. B. Davis, Reinforced random walks, Probal. Theory Related Fields 84 (1990) 203-229.
10. J. Folkman and Y. Shing, Angiogenesis, J. Biol. Chem. 267 (1992) 10931-10934.
11. A. Friedman and F. Reitich, On the existence of spatially patterned dormant malignancies in a model for the growth of non-necrotic vascular tumors, Math. Models Methods Appl. Sci. 11 (2001) 601-625.
12. A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol. 38 (1999) 262-284.
13. H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of the onset of capillary formation initiating angiogenesis, J. Math. Biol. 42 (2001) 195-238.
14. H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, A mathematical model for the roles pericytes and macrophages in the onset of angiogenesis, Math. Biosci. 168 (2000) 77-115.
15. 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.
16. H. A. Levine et al., Mathematical modeling of tumor angiogenesis and the action of angiostatin as a protease inhibitor, Isaac Newton Institute for Mathematical Sciences, Preprint Series, NI01054-ICB (10 December 2001).
17. K. Morimoto, H. Mishima, T. Nishida and T. Otori, Role of urokinase type plasminogen activator (u-PA) in corneal epithelial migration, Thromb. Haemost. 69 (1993) 387-391.
18. R. F. Nicosia, E. Bonanno and M. Smith, Fibronectin promotes the elongation of microvessels during angiogenesis in vitro, J. Cell. Physiol. 154 (1993) 654-661.
19. M. E. Orme and M. A. Chaplin, Two-dimensional models of tumor angiogenesis and anti-angiogenesis strategies, IMA J. Math. Appl. Med. Biol. 14 (1997) 189-205.
20. H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math. 57 (1997) 1044-1081.
21. S. Pamuk, Two-dimensional models of tumor angiogenesis, Ph.D. Thesis, Iowa State University, May 2000.
22. S. Pamuk, Computational analysis of linear/nonlinear models of capillary formation, in Proc. of the First Int. Conf. on Nonlinear Analysis and Nonlinear Modeling, July 17-23, 2001 Fethiye, Turkey, pp. 297-307.
23. N. Paweletz and M. Knierim, Tumor-related angiogenesis, Crit. Rev. Oncology/Hematology 9 (1989) 197-243.
24. R. R. Schleef and C. R. Birdwell, The effect of proteases on endothelial cell migration in vitro, Exp. Cell Res. 141 (1982) 503-508.
25. A. M. Schor and S. L. Schor, Tumor angiogenesis, J. Pathol. 141 (1983) 385-413.
26. B. D. Sleeman, A. R. A. Anderson and M. A. J. Chaplin, A mathematical analysis of a model for capillary network formation in the absence of endothelial cell proliferation, Appl. Math. Lett. 12 (1999) 121-127.
27. K. M. Yamada and K. Olden, Fibronectins - Adhesive glycoproteins of cell surface and blood, Nature 275 (1978) 179-184.
28. E. Zauderer, Partial Differential Equations of Applied Mathematics (Wiley-Interscience, 1989), 2nd edition.