The replicability of oncolytic virus: Defining conditions in tumor virotherapy

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

The replicability of oncolytic virus: Defining conditions in tumor virotherapy

Jianjun Paul Tian ^1,

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

Received June 2010 Revised October 2010 Published June 2011

Abstract
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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.

Keywords: Hopf bifurcation., Virotherapy, burst size.

Mathematics Subject Classification: Primary: 34C23, 34C10; Secondary: 92B9.

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:

[1]	M. Aghi and R. L. Martuza, Oncolytic viral therapy-the clinical experience, Oncogene, 24 (2005), 7802-7816. 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), 109-122. doi: 10.1016/j.jtbi.2008.01.016.
[3]	J. Carr, "Applications of Centre Manifold Theory," Applied Mathematics Sciences, 35, Springer-Verlag, New York-Berlin, 1981.
[4]	E. A. Chiocca, Oncolytic viruses, Nature Reviews, Cancer, 2 (2002), 938-950. 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), 55-78. 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), 2314-2319. doi: 10.1158/0008-5472.CAN-05-2661.
[7]	B. A. Fuchs and V. I. Levin, "Functions of A Complex Variable," Pergamon Press, London, 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), 12873-12878.
[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), 2832-2839. 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), 1-18. 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), 37-74. 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), 1587-1597. 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), 1317-1324.
[15]	D. Wodarz, Viruses as antitumor weapons: Defining conditions for tumor remission, Cancer Res., 61 (2001), 3501-3507.
[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), e4271. 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), 153-159. 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), 731-768. doi: 10.1006/bulm.2001.0245.

show all references

References:

[1]	M. Aghi and R. L. Martuza, Oncolytic viral therapy-the clinical experience, Oncogene, 24 (2005), 7802-7816. 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), 109-122. doi: 10.1016/j.jtbi.2008.01.016.
[3]	J. Carr, "Applications of Centre Manifold Theory," Applied Mathematics Sciences, 35, Springer-Verlag, New York-Berlin, 1981.
[4]	E. A. Chiocca, Oncolytic viruses, Nature Reviews, Cancer, 2 (2002), 938-950. 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), 55-78. 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), 2314-2319. doi: 10.1158/0008-5472.CAN-05-2661.
[7]	B. A. Fuchs and V. I. Levin, "Functions of A Complex Variable," Pergamon Press, London, 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), 12873-12878.
[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), 2832-2839. 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), 1-18. 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), 37-74. 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), 1587-1597. 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), 1317-1324.
[15]	D. Wodarz, Viruses as antitumor weapons: Defining conditions for tumor remission, Cancer Res., 61 (2001), 3501-3507.
[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), e4271. 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), 153-159. 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), 731-768. doi: 10.1006/bulm.2001.0245.

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