Abstract
We consider a procedure for cancer therapy which consists of injecting replication-competent viruses into the tumor. The viruses infect tumor cells, replicate inside them, and eventually cause their death. As infected cells die, the viruses inside them are released and then proceed to infect adjacent tumor cells. However, a major factor influencing the efficacy of virus agents is the immune response that may limit the replication and spread of the replication-competent virus. The immune response is cytokine-mediated. The expression of viruses in tumor cells sensitize cells to lysis by the TNF (tumor necrosis factor) cytokine. The competition between tumor cells, a replication-competent virus and an immune response is modelled as a free boundary problem for a nonlinear system of partial differential equations, where the free boundary is the surface of the tumor. In this model, the immune response equation is a non-standard parabolic equation due to the chemotaxis (spatial gradients of diffusible chemicals) of the immune response. The purpose of this paper is to give the numerical methods for solving this kind of free boundary problems. Several simulation results are also given.
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References
H.M. Byrne and M.A.J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Mathematical Biosciences, 181 (1995), 130–151.
H. Byrne and M. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci. 135 (1996), 187–216.
M.A.J. Chaplain, V.A. Kuznetsov, Z.H. James and L.A. Stepanova, Spatio-temporal dynamics of the immune system response to cancer. Proceedings of the mathematical Models in Medical and Health Sciences Conference (eds. M.A. Horn, G. Simonett, G. Webb), Vanderbilt University Press, 1998, ISBN 0-8265-1310-7.
A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol. 38 (1999), 262–284.
A. Friedman and Y. Tao, Analysis of a model of a virus that selectively in tumor cells, J. Math. Biol. 47 (2003), 391–423.
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.
R. Jain, Barriers to drug delivery in solid tumors, Sci. Am. 271 (1994), 58–65.
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.
J. Nemunaitis, C. Cunningham, 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.
M. Owen and J.A. Sherratt, Pattern formation and spatio-temporal irregularity in a model for macrophage-tumor interactions, J. Theor. Biol. 189 (1997), 63–80.
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.
Y. Tao, N. Yoshida and Q. Guo, Nonlinear analysis of a model of vascular tumor growth and treatment, Nonlinearity 17 (2004), 867–895.
J. P. Ward and J. R. King, Mathematical modelling of avascular-tumor growth, IMA J. Math. Appl. Med. Biol. 14 (1997), 39–69.
J.P. Ward and J.R. King, Mathematical modelling of drug transport in tumor multi-cell spheroids and monolayer cultures, Math. Biosci. 181 (2003), 177–207.
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.
J.T. Wu, H.M. Byrne, D.H. Kirn and L.M. Wein, Modeling and analysis of a virus that replicates selectively in tumor cells, Bull. Math. Biol., 63 (2001), 731–768.
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.
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© 2006 Birkhäuser Verlag Basel/Switzerland
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Tao, Y., Guo, Q. (2006). Simulation of a Model of Tumors with Virus-therapy. In: Figueiredo, I.N., Rodrigues, J.F., Santos, L. (eds) Free Boundary Problems. International Series of Numerical Mathematics, vol 154. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7719-9_42
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DOI: https://doi.org/10.1007/978-3-7643-7719-9_42
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