October  2017, 10(5): 1079-1093. doi: 10.3934/dcdss.2017058

Dynamical behavior of a new oncolytic virotherapy model based on gene variation

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

* Corresponding author: Zhiming Guo

Received  August 2016 Revised  January 2017 Published  June 2017

Fund Project: The authors are supported by NNSF of China grant 11371107 and Program for Changjiang Scholars and Innovative Research Team in University grant IRT1226

Oncolytic virotherapy is an experimental treatment of cancer patients. This method is based on the administration of replication-competent viruses that selectively destroy tumor cells but remain healthy tissue unaffected. In order to obtain optimal dosage for complete tumor eradication, we derive and analyze a new oncolytic virotherapy model with a fixed time period $τ $ and non-local infection which is caused by the diffusion of the target cells in a continuous bounded domain, where $τ $ is assumed to be the duration that oncolytic viruses spend to destroy the target cells and to release new viruses since they enter into the target cells. This model is a delayed reaction diffusion system with nonlocal reaction term. By analyzing the global stability of tumor cell eradication equilibrium, we give different treatment strategies for cancer therapy according to the different genes mutations (oncogene and antioncogene).

Citation: Zizi Wang, Zhiming Guo, Huaqin Peng. Dynamical behavior of a new oncolytic virotherapy model based on gene variation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1079-1093. doi: 10.3934/dcdss.2017058
References:
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Amar, M. Ben, C. Chatelain and P. Ciarletta, Contour instabilities in early tumor growth models, Physical review letters, 106 (2011), 970-978.Google Scholar

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Ž. BajzerT. Carr and K. Josić, Modeling of cancer virotherapy with recombinant measles viruses, Journal of Theoretical Biology, 252 (2008), 109-122. doi: 10.1016/j.jtbi.2008.01.016. Google Scholar

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R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley and Sons, 2003. doi: 10.1002/0470871296. Google Scholar

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D. DingliK. W. Peng and M. E. Harvey, Image-guided radiovirotherapy for multiple myeloma using a recombinant measles virus expressing the thyroidal sodium iodide symporter, Blood, 103 (2004), 1641-1646. doi: 10.1182/blood-2003-07-2233. Google Scholar

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S. Fedotov and A. Iomin, Migration and proliferation dichotomy in tumor-cell invasion, Physical Review Letters, 98 (2007), 1-5. Google Scholar

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R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Research, 56 (1996), 5745-5753. Google Scholar

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N. L. Komarova and D. Wodarz, ODE models for oncolytic virus dynamics, Journal of Theoretical Biology, 263 (2010), 530-543. doi: 10.1016/j.jtbi.2010.01.009. Google Scholar

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Y. LinH. Zhang and J. Liang, Identification and characterization of alphavirus M1 as a selective oncolytic virus targeting ZAP-defective human cancers, Proceedings of the National Academy of Sciences, 111 (2014), 4504-4512. doi: 10.1073/pnas.1408759111. Google Scholar

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R. L. Martuza and A. Malick, Experimental therapy of human glioma by means of a genetically engineered virus mutant, Science, 252 (1991), 854-856. doi: 10.1126/science.1851332. Google Scholar

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S. A. Menchón and C. A. Condat, Cancer growth: Predictions of a realistic model, Physical Review E, Statistical Nonlinear & Soft Matter Physics, (78) (2008), 2pp.Google Scholar

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J. D. Murray, Mathematical Biology Ⅱ Spatial Models and Biomedical Applications, Springer, 2003. Google Scholar

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J. W. H. SoJ. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. Ⅰ Travelling wavefronts on unbounded domains, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 457 (2001), 1841-1853. doi: 10.1098/rspa.2001.0789. Google Scholar

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Y. Tao and Q. Guo, The competitive dynamics between tumor cells, a replication-competent virus and an immune response, Journal of Mathematical Biology, 51 (2005), 37-74. doi: 10.1007/s00285-004-0310-6. Google Scholar

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H. R. Thieme and X. Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Analysis: Real World Applications, 2 (2001), 145-160. doi: 10.1016/S0362-546X(00)00112-7. Google Scholar

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H. R. Thieme, Convergence results and a Poincareć-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of mathematical biology, 30 (1992), 755-763. doi: 10.1007/BF00173267. Google Scholar

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M. X. WangY. J. Li and P. Y. Lai, Model on cell movement, growth, Differentiation and de-differentiation: Reaction-diffusion equation and wave propagation, European Physical Journal E, 36 (2013), 1-18. doi: 10.1140/epje/i2013-13065-4. Google Scholar

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S. WangS. Wang and X. Song, Hopf bifurcation analysis in a delayed oncolytic virus dynamics with continuous control, Nonlinear Dynamics, 67 (2012), 629-640. doi: 10.1007/s11071-011-0015-5. Google Scholar

[23]

Z. WangZ. Guo and H. Peng, A mathematical model verifying potent oncolytic efficacy of M1 virus, Mathematical Biosciences, 276 (2016), 19-27. doi: 10.1016/j.mbs.2016.03.001. Google Scholar

[24]

D. Wodarz, Gene therapy for killing p53-negative cancer cells: Use of replicating versus nonreplicating agents, Human Gene Therapy, 14 (2003), 153-159. doi: 10.1089/104303403321070847. Google Scholar

[25]

D. Wodarz, Viruses as antitumor weapons defining conditions for tumor remission, Cancer Research, 61 (2001), 3501-3507. Google Scholar

[26]

X. Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with timedelay, Canadian Applied Mathematics Quarterly, 17 (2009), 271-281. Google Scholar

show all references

References:
[1]

Amar, M. Ben, C. Chatelain and P. Ciarletta, Contour instabilities in early tumor growth models, Physical review letters, 106 (2011), 970-978.Google Scholar

[2]

Ž. BajzerT. Carr and K. Josić, Modeling of cancer virotherapy with recombinant measles viruses, Journal of Theoretical Biology, 252 (2008), 109-122. doi: 10.1016/j.jtbi.2008.01.016. Google Scholar

[3]

Becker and M. Wayne, et al, The World of the Cell, Vol. 6. San Francisco: Benjamin Cummings, 2003.Google Scholar

[4]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley and Sons, 2003. doi: 10.1002/0470871296. Google Scholar

[5]

D. DingliK. W. Peng and M. E. Harvey, Image-guided radiovirotherapy for multiple myeloma using a recombinant measles virus expressing the thyroidal sodium iodide symporter, Blood, 103 (2004), 1641-1646. doi: 10.1182/blood-2003-07-2233. Google Scholar

[6]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 1998. Google Scholar

[7]

S. Fedotov and A. Iomin, Migration and proliferation dichotomy in tumor-cell invasion, Physical Review Letters, 98 (2007), 1-5. Google Scholar

[8]

R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Research, 56 (1996), 5745-5753. Google Scholar

[9]

N. L. Komarova and D. Wodarz, ODE models for oncolytic virus dynamics, Journal of Theoretical Biology, 263 (2010), 530-543. doi: 10.1016/j.jtbi.2010.01.009. Google Scholar

[10]

Y. LinH. Zhang and J. Liang, Identification and characterization of alphavirus M1 as a selective oncolytic virus targeting ZAP-defective human cancers, Proceedings of the National Academy of Sciences, 111 (2014), 4504-4512. doi: 10.1073/pnas.1408759111. Google Scholar

[11]

R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Transactions of the American Mathematical Society, 321 (1990), 1-44. doi: 10.2307/2001590. Google Scholar

[12]

R. L. Martuza and A. Malick, Experimental therapy of human glioma by means of a genetically engineered virus mutant, Science, 252 (1991), 854-856. doi: 10.1126/science.1851332. Google Scholar

[13]

S. A. Menchón and C. A. Condat, Cancer growth: Predictions of a realistic model, Physical Review E, Statistical Nonlinear & Soft Matter Physics, (78) (2008), 2pp.Google Scholar

[14]

J. D. Murray, Mathematical Biology Ⅱ Spatial Models and Biomedical Applications, Springer, 2003. Google Scholar

[15]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[16]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Soc. , Providence, RI, 1995. Google Scholar

[17]

J. W. H. SoJ. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. Ⅰ Travelling wavefronts on unbounded domains, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 457 (2001), 1841-1853. doi: 10.1098/rspa.2001.0789. Google Scholar

[18]

Y. Tao and Q. Guo, The competitive dynamics between tumor cells, a replication-competent virus and an immune response, Journal of Mathematical Biology, 51 (2005), 37-74. doi: 10.1007/s00285-004-0310-6. Google Scholar

[19]

H. R. Thieme and X. Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Analysis: Real World Applications, 2 (2001), 145-160. doi: 10.1016/S0362-546X(00)00112-7. Google Scholar

[20]

H. R. Thieme, Convergence results and a Poincareć-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of mathematical biology, 30 (1992), 755-763. doi: 10.1007/BF00173267. Google Scholar

[21]

M. X. WangY. J. Li and P. Y. Lai, Model on cell movement, growth, Differentiation and de-differentiation: Reaction-diffusion equation and wave propagation, European Physical Journal E, 36 (2013), 1-18. doi: 10.1140/epje/i2013-13065-4. Google Scholar

[22]

S. WangS. Wang and X. Song, Hopf bifurcation analysis in a delayed oncolytic virus dynamics with continuous control, Nonlinear Dynamics, 67 (2012), 629-640. doi: 10.1007/s11071-011-0015-5. Google Scholar

[23]

Z. WangZ. Guo and H. Peng, A mathematical model verifying potent oncolytic efficacy of M1 virus, Mathematical Biosciences, 276 (2016), 19-27. doi: 10.1016/j.mbs.2016.03.001. Google Scholar

[24]

D. Wodarz, Gene therapy for killing p53-negative cancer cells: Use of replicating versus nonreplicating agents, Human Gene Therapy, 14 (2003), 153-159. doi: 10.1089/104303403321070847. Google Scholar

[25]

D. Wodarz, Viruses as antitumor weapons defining conditions for tumor remission, Cancer Research, 61 (2001), 3501-3507. Google Scholar

[26]

X. Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with timedelay, Canadian Applied Mathematics Quarterly, 17 (2009), 271-281. Google Scholar

Figure 1.  By Theorem 3.1, it is easy to see that the stability of tumor eradication equilibrium $E$ is independent of diffusion coefficients $d_i, (i=1, 2, 3)$ and $\tau$. Here, we set $d_1=1, d_2=1, d_3=1, \tau=0$, $d=1, \mu=1, a_1=1, b_1=1, c_1=2$, $a_2=1, b_2=1, c_2=1, B=0.2$, $\Gamma(x,y,\tau)=1$, as $x= y$, and $\Gamma(x,y,\tau)=0$, as $x\neq y$. Thus, by direct calculations, we get $\frac{d}{\mu}(a_2-\frac{a_1b_2}{b_1})=0$, $\frac{d}{\mu}(a_2-\frac{a_1c_2}{c_1})=0.5$. Then $B>\frac{d}{\mu}(a_2-\frac{a_1b_2}{b_1})$ in Theorem 3.1 holds. But the component tumor cells $u_2$ doesn't tend to 0 as $t\rightarrow\infty$
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