A parabolic-hyperbolic free boundary problem modelling tumor treatment with virus

Mathematical Models and Methods in Applied Sciences

Volumn 17, Issue 1, 2007, Pages 63-80

  Tao, Youshan ^a   Zhang, Hui ^a  

^a DONGHUA UNIVERSITY   (China)

Author keywords

Chemotaxis; Free boundary problems; Immune response; Tumors; Virus

Indexed keywords

EID: 33846256703     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202507001838     Document Type: Article

References (27)

1. J. Adam and N. Bellomo, A Survey of Models for Tumor-Immune System Dynamics (Birkhäuser, 1997).
2. N. Bellomo, E. De Angelis and L. Preziosi, Multiscale modelling and mathematical, problems related to tumor evolution and medical therapy, J. Theor. Med. 5 (2003) 111-136.
3. N. Bellomo, A. Bellouquid and M. Delitala, Mathematical topics on the modelling complex multicellular systems and tumor immune cells competition, Math. Mod. Meth. Appl. Sci. 14 (2004) 1683-1733.
4. N. Bellomo and G. Forni, Dynamics of tumor interaction with the host immune system, Math. Comput. Model. 20 (1994) 107-122.
5. A. Bellouquid and M. Delitala, Mathematical methods and tools of kinetic theory towards modelling complex biological system, Math. Mod. Meth. Appl. Sci. 15 (2005) 1639-1666.
6. A. Bertuzzi, A. Fasano and A. Gandolfi, A free boundary problem with unilateral constraints describing the evolution of a tumor cord under the influence of cell killing agents, SIAM J. Math. Anal. 36 (2004) 882-915.
7. A. Bertuzzi, A. Fasano and A. Gandolfi, A mathematical model for tumor cords incorporating the flow of interstitial fluids, Math. Mod. Meth. Appl. Sci. 15 (2005) 1735-1778.
8. H. M. Byrne, J. R. King, D. L. McElwain and L. Preziosi, A. two-phase model of solid tumour growth, Appl. Math. Lett. 16 (2003) 567-573.
9. M. A. J. Chaplain, Mathematical modelling of angiogenesis, J. Neuro-Oncology 50 (2000) 37-51.
10. M. A. J. Chaplain, V. A. Kuznetsov, Z. H. James and L. A. Stepanova, Spatio-temporal dynamics of the immune system reponse to cancer, Proc. of the Mathematical Models in Medical and Health Sciences Conference, eds. M. A. Horn, G. Simonett and G. Webb (Vanderbilt Univ. Press, 1998).
11. A. Friedman, Partial Differential Equations of Parabolic Type (Prentice-Hall, 1964).
12. A. Friedman and G. Lolas, Analysis of a mathematical model of tumor lymphangiogenesis, Math. Mod. Meth. Appl. Sci. 15 (2005) 95-107.
13. A. Friedman and Y. Tao, Analysis of a model of a virus that replicates selectively in tumor cells, J. Math. Biol. 47 (2003) 391-423.
14. T. L. Jackson and H. M. Byrne, A mathematical, model, to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy, Math. Biosci. 164 (2000) 17-38.
15. R. Jain, Barriers to drug delivery in solid tumors, Sci. Am. 271 (1994) 58-65.
16. O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Transl., Vol. 23 (Amer. Math. Soc., 1968).
17. J. Nemuuaitis, C. Cuuningham, A. Buchanan, A. Blackburn, G. Edelman, P. Maples, G. Netto, A. Tong, B. Randley, S. Olson and D. Kirn, Intravenous infusion of a replication-selective adenovirus (ONYX-015) in cancer patients: Safety, feasibility and biological activity, Gene Therapy 8 (2001) 746-759.
18. J. Nemunaitis, I. Ganly, F. Khuri, J. Arseneau, J. Kuhn, T. McCarty, S. Landers, P. Maples, L. Romel, B. Randley, T. Reid, S. Kaye and D. Kirn, Selective replication and oncolysis in p53 mutant tumors with ONYX-015, an E1B-55kD gene-deleted adenovirus, in patients with advanced head and neck cancer: a phase II trial, Cancer Res. 60 (2000) 6359-6366.
19. M. R. Owen, H. M. Byrne and C. E. Lewis, Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites, J. Theor. Biol. 226 (2004) 377-391.
20. M. Owen and J. A. Sherratt, Pattern formation, and spatio-temporal irregularity in a model for macrophage-tumour interactions, J. Theor. Biol. 189 (1997) 63-80.
21. Y. Tao and M. Chen, An elliptic-hyperbolic free boundary problem modelling cancer therapy, Nonlinearity 19 (2006) 419-440.
22. Y. Tao and Q. Guo, The competitive dynamics between tumor cells, a replication-competent virus and an immune response, J. Math. Biol. 51 (2005) 37-74.
23. Y. Tao, N. Yoshida and Q. Guo, Nonlinear analysis of a model of vascular tumour growth and treatment, Nonlinearity 17 (2004) 867-895.
24. J. P. Ward and J. R. King, Mathematical modelling of drug transport in tumour multicell spheroids and monolayer cultures, Math. Biosci. 181 (2003) 177-207.
25. L. M. Wein, J. T. Wu and D. H. Kirn, Validation and analysis of a mathematical model of a replication-competent oncolytic virus for cancer treatment: Implications for virus design and delivery, Cancer Res. 63 (2003) 1317-1324.
26. J. T. Wu, H. M. Byrne, D. H. Kirn and L. M. Wein, Modelling and analysis of a virus that replicates selectively in tumor cells, Bull. Math. Biol. 63 (2001) 731 768.
27. J. T. Wu, D. H. Kirn and L. M. Wein, Analysis of a three-way race between tumor growth, a replication-competent virus and an immune response, Bull. Math. Biol. 66 (2004) 605-625.