Differential Equation Techniques for Modeling a Cycle-Specific Oncolytic Virotherapeutic

• Joanna R. Wares
• Joseph J. Crivelli
• Peter S. Kim
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 107)

Abstract

The development of a mathematical model of oncolytic virotherapeutic vesticular stomatitis virus (VSV) is presented in stages. Standard mathematical tools are discussed along with the development and analysis of the model. A defining property of VSV is that it only affects tumor cells when they are in the active phases of the cell cycle. To model this characteristic, we first model tumor growth and separate cells into active and resting, which takes the form of a linear system of differential equations. We then take into account the minimum time needed for cells to travel through the active phases of the cell cycle, first using delay-differential equations and then later age-structured partial differential equations. Our basic tumor growth model allows us to investigate linear systems analysis (eigenvalue analysis). We then study similar techniques for delay differential equations, after adding the minimum time necessary to travel through the active phases of the cell cycle to the model. After tumor growth alone has been modeled, we include viral dynamics, which takes the form of a nonlinear system of ordinary differential equations. We investigate how linearization helps us understand how to properly develop the model. Finally we add the minimum biological time to the viral model. With the model fully developed, we arrive at a system of differential equations, one of which is an age-structured partial differential equation, which provides a nice example for discussing the method of characteristics. Finally, we show how our model can be used to investigate the dynamics of the tumor-virus system. As we travel through the development of our model, we discuss various techniques to analyze ordinary, delay, and partial differential equations.

References

1. 1.
Au, G.G., Lindberg, A.M., Barry, R.D., Shafren, D.R.: Oncolysis of vascular malignant human melanoma tumors by Coxsackievirus A21. Int. J. Oncol. 26(6), 1471–1476 (2005)Google Scholar
2. 2.
Bajzer, Z., Carr, T., Dingli, D., Josic, K.: Optimization of tumor virotherapy with recombinant measles virus. In: Lim, G.J., Lee, E.K. (eds.) Optimization in Medicine and Biology. Auerbach Publications, New York (2008)Google Scholar
3. 3.
Bajzer, Z., Carr, T., Josic, K., Russell, S.J., Dingli, D.: Modeling of cancer virotherapy with recombinant measles viruses. J. Theor. Biol. 252(1), 109–122 (2008)
4. 4.
Balachandran, S., Barber, G.N.: Vesicular stomatitis virus (VSV) therapy of tumors. IUBMB Life 50(2), 135–138 (2000)
5. 5.
Bell, J., Parato, K., Atkins, H.: Vesicular stomatitis virus. In: Harrington, K.J., Vile, R.G., Pandha, H.S. (eds.) Viral Therapy of Cancer. Wiley, Hoboken (2008)Google Scholar
6. 6.
Biesecker, M., Kimn, J.H., Lu, H., Dingli, D., Bajzer, Z.: Optimization of virotherapy for cancer. Bull. Math. Biol. 72(2), 469–489 (2010)
7. 7.
Cain, J.W., Reynolds, A.M.: Ordinary and Partial Differential Equations. An Introduction to Dynamical Systems. Creative Commons. Virginia Commonwealth University, Richmond (2010)Google Scholar
8. 8.
Comins, C., Spicer, J., Protheroe, A., Roulstone, V., Twigger, K., White, C.M., Vile, R., Melcher, A., Coffey, M.C., Mettinger, K.L., Nuovo, G., Cohn, D.E., Phelps, M., Harrington, K.J., Pandha, H.S.: REO-10: a phase I study of intravenous reovirus and docetaxel in patients with advanced cancer. Clin. Cancer Res. 16(22), 5564–5572 (2010)
9. 9.
Crivelli, J.J., Foldes, J., Kim, P.S., Wares, J.R.: A mathematical model for cell cycle-specific cancer virotherapy. J. Biol. Dyn. 6(Suppl 1), 104–120 (2012)
10. 10.
Dingli, D., Peng, K.W., Harvey, M.E., Greipp, P.R., O’Connor, M.K., Cattaneo, R., Morris, J.C., Russell, S.J.: Image-guided radiovirotherapy for multiple myeloma using a recombinant measles virus expressing the thyroidal sodium iodide symporter. Blood 103(5), 1641–1646 (2004)
11. 11.
Friedman, A., Tian, J.P., Fulci, G., Chiocca, E.A., Wang, J.: Glioma virotherapy: effects of innate immune suppression and increased viral replication capacity. Cancer Res. 66(4), 2314–2319 (2006)
12. 12.
Heise, C., Sampson-Johannes, A., Williams, A., McCormick, F., Von Hoff, D.D., Kirn, D.H.: ONYX-015, an E1B gene-attenuated adenovirus, causes tumor-specific cytolysis and antitumoral efficacy that can be augmented by standard chemotherapeutic agents. Nat. Med. 3(6), 639–645 (1997)
13. 13.
Komarova, N.L., Wodarz, D.: ODE models for oncolytic virus dynamics. J. Theor. Biol. 263(4), 530–543 (2010)
14. 14.
Liu, W., Hillen, T., Freedman, H.I.: A mathematical model for M-phase specific chemotherapy including the G0-phase and immunoresponse. Math. Biosci. Eng. 4(2), 239–259 (2007)
15. 15.
Novozhilov, A.S., Berezovskaya, F.S., Koonin, E.V., Karev, G.P.: Mathematical modeling of tumor therapy with oncolytic viruses: regimes with complete tumor elimination within the framework of deterministic models. Biol. Direct 1, 6 (2006)
16. 16.
Oliere, S., Arguello, M., Mesplede, T., Tumilasci, V., Nakhaei, P., Stojdl, D., Sonenberg, N., Bell, J., Hiscott, J.: Vesicular stomatitis virus oncolysis of T lymphocytes requires cell cycle entry and translation initiation. J. Virol. 82(12), 5735–5749 (2008)
17. 17.
Park, B.H., Hwang, T., Liu, T.C., Sze, D.Y., Kim, J.S., Kwon, H.C., Oh, S.Y., Han, S.Y., Yoon, J.H., Hong, S.H., Moon, A., Speth, K., Park, C., Ahn, Y.J., Daneshmand, M., Rhee, B.G., Pinedo, H.M., Bell, J.C., Kirn, D.H.: Use of a targeted oncolytic poxvirus, JX-594, in patients with refractory primary or metastatic liver cancer: a phase I trial. Lancet Oncol. 9(6), 533–542 (2008)
18. 18.
Pecora, A.L., Rizvi, N., Cohen, G.I., Meropol, N.J., Sterman, D., Marshall, J.L., Goldberg, S., Gross, P., O’Neil, J.D., Groene, W.S., Roberts, M.S., Rabin, H., Bamat, M.K., Lorence, R.M.: Phase I trial of intravenous administration of PV701, an oncolytic virus, in patients with advanced solid cancers. J. Clin. Oncol. 20(9), 2251–2266 (2002)
19. 19.
Perko, L.: Differential Equations and Dynamical Systems, 3rd edn. Springer, New York (2001)
20. 20.
Reddy, P.S., Burroughs, K.D., Hales, L.M., Ganesh, S., Jones, B.H., Idamakanti, N., Hay, C., Li, S.S., Skele, K.L., Vasko, A.J., Yang, J., Watkins, D.N., Rudin, C.M., Hallenbeck, P.L.: Seneca Valley virus, a systemically deliverable oncolytic picornavirus, and the treatment of neuroendocrine cancers. J. Natl. Cancer Inst. 99(21), 1623–1633 (2007)
21. 21.
Todo, T., Martuza, R.L., Rabkin, S.D., Johnson, P.A.: Oncolytic herpes simplex virus vector with enhanced MHC class I presentation and tumor cell killing. Proc. Natl. Acad. Sci. USA 98(11), 6396–6401 (2001)
22. 22.
Villasana, M., Radunskaya, A.: A delay differential equation model for tumor growth. J. Math. Biol. 47(3), 270–294 (2003)
23. 23.
Wodarz, D.: Computational approaches to study oncolytic virus therapy: insights and challenges. Gene Ther. Mol. Biol. 8, 137–146 (2004)Google Scholar
24. 24.
Wodarz, D., Komarova, N.: Towards predictive computational models of oncolytic virus therapy: basis for experimental validation and model selection. PLoS ONE 4(1), e4271 (2009)
25. 25.
Wu, J.T., Byrne, H.M., Kirn, D.H., Wein, L.M.: Modeling and analysis of a virus that replicates selectively in tumor cells. Bull. Math. Biol. 63(4), 731–768 (2001)

© Springer Science+Business Media New York 2014

Authors and Affiliations

• Joanna R. Wares
• 1
• Joseph J. Crivelli
• 2
• Peter S. Kim
• 3
1. 1.University of RichmondRichmondUSA
2. 2.Weill Cornell Medical CollegeNew YorkUSA
3. 3.University of SydneysydneyAustralia