Nonlinear analysis of a model of vascular tumour growth and treatment

Nonlinearity

Volumn 17, Issue 3, 2004, Pages 867-895

  Tao, Youshan ^a   Yoshida, Norio ^b   Guo, Qian ^a  

^a DONGHUA UNIVERSITY   (China)

^b UNIVERSITY OF TOYAMA   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 2442670526     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/17/3/008     Document Type: Article

References (34)

1. Adam J 1986 A simplified mathematical model of tumor growth Math. Biosci. 81 224-9
2. Anderson A R A and Chaplain M A J 1998 Continuous and discrete mathematical models of tumour-induced angiogenesis Bull. Math. Biol. 60 857-99
3. Bazaliy B V and Friedman A 2003 A free boundary problem for an elliptic-parabolic system: application to a model of tumor growth Commun. PDE 28 517-60
4. Bazaliy B V and Friedman A 2003 Global existence and stability for an elliptic-parabolic free boundary problem: an application to a model of tumor growth Indiana Univ. Math. J. 52
5. Bicknell R, Lewis C E and Ferrara N 1997 Tumor Angiogenesis (Oxford: Oxford University Press)
6. Breward C J W, Byrne H M and Lewis C E 2003 A multiphase model describing vascular tumor growth Bull. Math. Biol. 65 609-40
7. Britton N and Chaplain M 1993 A qualitative analysis of some models of tissue growth Math. Biosci. 113 77-89
8. Byrne H M and Chaplain M A J 1995 Growth of nonnecrotic tumours in the presence and absence of inhibitors Math. Biosci. 130 151-81
9. Byrne H M and Chaplain M A J 1996 Growth of necrotic tumors in the presence and absence of inhibitors Math. Biosci. 135 187-216
10. Carmeliet P and Jain R K 2000 Angiogenesis in cancer and other diseases Nature 407 249-57
11. Franks S J, Byrne H M, King J R, Underwood J C E and Lewis C E 2003 Modelling the early growth of ductal carcinoma in situ of the breast J. Math. Biol. 47 424-52
12. Friedman A and Reitich F 1999 Analysis of a mathematical model for the growth of tumors J. Math. Biol. 38 262-84
13. Friedman A and Reitich F 2000 Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth Trans. Am. Math. Soc. 353 1587-634
14. Friedman A and Reitich F 2001 On the existence of spatially patterned dormant malignancies in a model for the growth of non-necrotic vascular tumor Math. Models Methods Appl. Sci. 77 1-25
15. Friedman A and Tao Y 2003 Analysis of a model of a virus that replicates selectively in tumor cells J. Math. Biol. 47 391-423
16. Gilbarg D and Trudinger N S 1983 Elliptic Partial Differential Equations 2nd edn (Berlin: Springer)
17. Greenspan H 1972 Models for the growth of solid tumor by diffusion Stud. Appl. Math. 51 317-40
18. Greenspan H 1976 On the growth and stability of cell cultures and solid tumors J. Theor. Biol. 56 229-42
19. Hahnfield P, Panigraphy D, Folkman J and Hlatky L 1999 Tumor development under angiogenics signalling: a dynamic theory of tumor growth, treatment response and postvascular dormancy Cancer Res. 59 4770-5
20. Jackson T L and Byrne H M 2000 A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy Math. Biosci. 164 17-38
21. Jackson T L 2002 Vascular tumor growth and treatment: consequence of polyclonality, competition and dynamic vascular support J. Math. Biol. 44 201-26
22. Ladyzhenskaya O A and Ural'tseva N N 1968 Linear and Quasilinear Elliptic Equations (New York: Academic)
23. Li Ta-tsien 1994 Global Classical Solutions for Quasilinear Hyperbolic System (New York: Wiley)
24. Marmé D 2003 The impact of anti-angiogenic agents on cancer therapy J. Cancer Res. Clin. Oncol. 129 607-20
25. McDougall S R, Anderson A R A, Chaplain M A J and Sherratt J A 2002 Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies Bull. Math. Biol. 64 673-702
26. Orme M E and Chaplain M A J 1996 A mathematical model of vascular tumor growth and invasion Math. Comput. Modelling 23 43-60
27. Sherrat J and Chaplain M 2001 A new mathematical model for avascular tumor growth J. Math. Biol. 43 291-312
28. Sum B G, Lee A E, Shalal-Zwain S, Pruijn F B, Mckeage M J and Wilson W R 2003 Marked potentiation of the antitumour activity of chemotherapeutic drugs by the antivascular agent 5,6-dimethylxanthenone-4-acetic acid (DMXAA) Cancer Chemother. Pharmacol. 51 43-52
29. Ward J P and King J R 1997 Mathematical modelling of avascular tumor growth IMA J. Math. Appl. Med. Biol. 14 39-69
30. Ward J P and King J R 1998 Mathematical modelling of avascular-tumor growth II: modelling growth saturation IMA J. Math. Appl. Med. 15 1-42
31. Ward J P and King J R 2003 Mathematical modelling of drug transport in tumour multicell spheroids and monolayer cultures Math. Biosci. 18 1177-207
32. Wu J T, Byrne H M, Kirn D H and Wein L M 2001 Modeling and analysis of a virus that replicates selectively in tumor cells Bull. Math. Biol. 63 731-68
33. Yancopoulos G D, Davis S, Gale N W, Rudge J S, Wiegand S J and Holash J 2000 Vascular-speific growth factors and blood vessel formation Nature 407 242-8
34. Ychou M et al 2002 High-dose, single-agent irinotecan as first-line therapy in treatment of metastatic colorectal cancer Cancer Chemother. Pharmacol. 50 383-91