Steady-state analysis of a mathematical model for capillary network formation in the absence of tumor source

Mathematical Biosciences

Volumn 189, Issue 1, 2004, Pages 21-38

  Pamuk, Serdal ^a  

^a KOCAELI UNIVERSITY   (Turkey)

Author keywords

Angiogenesis; Extracellular matrix; Steady state; Threshold; Tumor angiogenesis factor

Indexed keywords

MATHEMATICAL MODELS; TUMORS;

CAPILLARY NETWORKS; CELL MIGRATION;

NEURAL NETWORKS;

ANGIOGENIC FACTOR;

MATHEMATICAL ANALYSIS; TUMOR;

ANGIOGENESIS; ARTICLE; BASEMENT MEMBRANE; CAPILLARY; CAPILLARY WALL; CARCINOGENESIS; CELL MIGRATION; ENDOTHELIUM CELL; EXTRACELLULAR MATRIX; MATHEMATICAL MODEL; MAXIMUM ALLOWABLE CONCENTRATION; STEADY STATE; TRANSPORT KINETICS; TUMOR VASCULARIZATION;

ALGORITHMS; ANGIOGENESIS INDUCING AGENTS; ANGIOSTATINS; ANIMALS; BASEMENT MEMBRANE; CELL DIVISION; CELL MOVEMENT; COMPUTER SIMULATION; ENDOPEPTIDASES; ENDOTHELIAL CELLS; EXTRACELLULAR MATRIX; FIBRONECTINS; HUMANS; KINETICS; MODELS, THEORETICAL; NEOVASCULARIZATION, PATHOLOGIC; PROTEASE INHIBITORS;

EID: 1642348159     PISSN: 00255564     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mbs.2004.01.006     Document Type: Article

References (23)

1. D. Ambrosi, A. Gamba, E. Giraudo, G. Serini, L. Preziosi, 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, L. Preziosi, On the closure of mass balance models for tumour growth, Rapporto Dipartimento di Matematica, Politecnico di Torino, No. 27, giugno, 2001
3. Anderson A.R.A., Chaplin M.A.J. A mathematical model for capillary network formation in the absence of endothelial cell proliferation. Appl. Math. Lett. 11:1998;109.
4. Angelis E.D., Preziosi L. Advection-diffusion models for solid tumour evolution in vivo and related free boundary problem. Math. Models Meth. Appl. Sci. 10(3):2000;379.
5. Ausprunk D.H., Folkman J. Migration and proliferation of endothelial cells in preformed and newly formed blood vessels during tumor angiogenesis. Microvasc. Res. 14:1977;53.
6. Berg P.W., McGregor J.L. Elementary Partial Differential Equations. Holden-Day Series in Mathematics. 1966;Holden-Day, Oakland, CA, USA.
7. Carter S.B. Principles of cell motility: the direction of cell movement and cancer invasion. Nature. 208:1965;1183.
8. Chaplin M.A.J., Giles S.M., Sleeman B.D., Jarvis R.J. A mathematical analysis of a model for tumour angiogenesis. J. Math. Biol. 33:1995;770.
9. M.A.J. Chaplin, L. Preziosi, Macroscopic modeling of the growth and development of tumor masses, Rapporto Dipartimento di Matematica, Politecnico di Torino, No. 27, November, 2000
10. Davis B. Reinforced random walks. Probal. Theory Relat. Fields. 84:1990;203.
11. Folkman J., Shing Y. Angiogenesis. J. Biol. Chem. 267(16):1992;10931.
12. Levine H.A., Sleeman B.D., Nilsen-Hamilton M. Mathematical modeling of the onset of capillary formation initiating angiogenesis. J. Math. Biol. 42(3):2001;195.
13. Levine H.A., Sleeman B.D., Nilsen-Hamilton M. A mathematical model for the roles pericytes and macrophages in the onset of angiogenesis. Math. Biosci. 168:2000;77.
14. Levine H.A., Pamuk S., Sleeman B.D., Nilsen-Hamilton M. Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma. Bull. Math. Biol. 63(5):2001;801.
15. Levine H.A., Pamuk S., Sleeman B.D.et al. Mathematical modeling of tumor angiogenesis and the action of angiostatin as a protease inhibitor. J. Theor. Med. 4(2):2002;133.
16. Morimoto K., Mishima H., Nishida T., Otori T. Role of urokinase type plasminogen activator (u-PA) in corneal epithelial migration. Thromb. Haemost. 69:1993;387.
17. Orme M.E., Chaplin M.A. Two-dimensional models of tumor angiogenesis and anti-angiogenesis strategies. IMA J. Math. Appl. Med. Biol. 14:1997;189.
18. Pamuk S. Qualitative analysis of a mathematical model for capillary formation in tumor angiogenesis. Math. Models Meth. Appl. Sci. 13(1):2003;19.
19. S. Pamuk, Two dimensional models of tumor angiogenesis, Ph.D. Thesis, Iowa State University, 2000
20. S. Pamuk, Computational analysis of linear/nonlinear models of capillary formation, in: Proceedings of the First International Conference on Nonlinear Analysis and Nonlinear Modeling, July 17-23, 2001, Fethiye, Turkey, pp. 297-307
21. Paweletz N., Knierim M. Tumor-related angiogenesis. Crit. Rev. Oncol./Hematol. (9):1989;197.
22. Schleef R.R., Birdwell C.R. The effect of proteases on endothelial cell migration in vitro. Exp. Cell Res. 141(2):1982;503.
23. Schor A.M., Schor S.L. Tumor angiogenesis. J. Pathol. 141:1983;385.