# American Institute of Mathematical Sciences

2011, 8(3): 841-860. doi: 10.3934/mbe.2011.8.841

## The replicability of oncolytic virus: Defining conditions in tumor virotherapy

 1 Mathematics Department, College of William and Mary, Williamsburg, VA 23187, United States

Received  June 2010 Revised  October 2010 Published  June 2011

The replicability of an oncolytic virus is measured by its burst size. The burst size is the number of new viruses coming out from a lysis of an infected tumor cell. Some clinical evidences show that the burst size of an oncolytic virus is a defining parameter for the success of virotherapy. This article analyzes a basic mathematical model that includes burst size for oncolytic virotherapy. The analysis of the model shows that there are two threshold values of the burst size: below the first threshold, the tumor always grows to its maximum (carrying capacity) size; while passing this threshold, there is a locally stable positive equilibrium solution appearing through transcritical bifurcation; while at or above the second threshold, there exits one or three families of periodic solutions arising from Hopf bifurcations. The study suggests that the tumor load can drop to a undetectable level either during the oscillation or when the burst size is large enough.
Citation: Jianjun Paul Tian. The replicability of oncolytic virus: Defining conditions in tumor virotherapy. Mathematical Biosciences & Engineering, 2011, 8 (3) : 841-860. doi: 10.3934/mbe.2011.8.841
##### References:
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##### References:
 [1] M. Aghi and R. L. Martuza, Oncolytic viral therapy-the clinical experience,, Oncogene, 24 (2005), 7802. doi: 10.1038/sj.onc.1209037. [2] Z. Bajzer, T. Carr, K. Josic, S. J. Russel and D. Dingli, Modeling of cancer virotherapy with recombinant measles viruses,, J. Theoretical Biology, 252 (2008), 109. doi: 10.1016/j.jtbi.2008.01.016. [3] J. Carr, "Applications of Centre Manifold Theory,", Applied Mathematics Sciences, 35 (1981). [4] E. A. Chiocca, Oncolytic viruses,, Nature Reviews, 2 (2002), 938. doi: 10.1038/nrc948. [5] D. Dingli, M. D. Cascino, K. Josić, S. J. Russell and Z. Bajzer, Mathematical modeling of cancer radiovirotherapy,, Math. Biosci., 199 (2006), 55. doi: 10.1016/j.mbs.2005.11.001. [6] A. Friedman, J. P. Tian, G. Fulci, E. A. Chiocca and J. Wang, Glioma virotherapy: Effects of innate immune suppression and increased viral replication capacity,, Cancer Research, 66 (2006), 2314. doi: 10.1158/0008-5472.CAN-05-2661. [7] B. A. Fuchs and V. I. Levin, "Functions of A Complex Variable,", Pergamon Press, (1961). [8] G. Fulci, et al, Cyclophosphamide enhances glioma virotherapy by inhibiting innate immune responses,, PNAS Proceedings of the National Academy of Sciences of the United States of America, 103 (2006), 12873. [9] B. D. Hassard, N. D. Hazzarinoff and Y.-H. Wan, "Theory and Applications of Hopf Bifurcation,", Cambridge, (1981). [10] H. Kambara, H. Okano, E. A. Chiocca and Y. Saeki, An oncolytic HSV-1 mutant expressing ICP34.5 under control of a nestin promoter increases survival of animals even when symptomatic from a brain tumor,, Cancer Res., 65 (2005), 2832. doi: 10.1158/0008-5472.CAN-04-3227. [11] A. S. Novozhilov, F. S. Berezovskaya, E. V. Koonin and G. P. Karev, Mathematical modeling of tumor therapy with oncolytic viruses: Regimes with complete tumor elimination within the framework of deterministic models,, Biology Direct, 1 (2006), 1. doi: 10.1186/1745-6150-1-6. [12] 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. doi: 10.1007/s00285-004-0310-6. [13] D. Vasiliu and J. P. Tian, Periodic solutions of a model for tumor virotherapy,, Discrete and Continuous Dynamical Systems Ser. S, 4 (2011), 1587. doi: 10.3934/dcdss.2011.4.1587. [14] 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. [15] D. Wodarz, Viruses as antitumor weapons: Defining conditions for tumor remission,, Cancer Res., 61 (2001), 3501. [16] D. Wodarz and N. Komarova, Towards predictive computational models of oncolytic virus therapy: Basis for experimental validation and model selection,, PloS ONE, 4 (2009). doi: 10.1371/journal.pone.0004271. [17] D. Wodarz, Gene therapy for killing p53-negative cancer cells: Use of replicating versus nonreplicating agents,, Hum. Gene Ther., 14 (2003), 153. doi: 10.1089/104303403321070847. [18] 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. doi: 10.1006/bulm.2001.0245.
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