Non-Linear Mathematical Model of the Interaction between Tumor and Oncolytic Viruses

Affiliation(s)

Department of Mathematics, The Madura College, Madurai, India.

Department of Mathematics, V V College of Engineering, Tisaiyanvilai, India.

Department of Mathematics, The Madura College, Madurai, India.

Department of Mathematics, V V College of Engineering, Tisaiyanvilai, India.

ABSTRACT

A mathematical modeling of tumor therapy with oncolytic viruses is discussed. The model consists of two coupled, deterministic differential equations allowing for cell reproduction and death, and cell infection. The model is one of the conceptual mathematical models of tumor growth that treat a tumor as a dynamic society of interacting cells. In this paper, we obtain an approximate analytical expression of uninfected and infected cell population by solving the non-linear equations using Homotopy analysis method (HAM). Furthermore, the results are compared with the numerical simulation of the problem using Matlab program. The obtained results are valid for the whole solution domain.

A mathematical modeling of tumor therapy with oncolytic viruses is discussed. The model consists of two coupled, deterministic differential equations allowing for cell reproduction and death, and cell infection. The model is one of the conceptual mathematical models of tumor growth that treat a tumor as a dynamic society of interacting cells. In this paper, we obtain an approximate analytical expression of uninfected and infected cell population by solving the non-linear equations using Homotopy analysis method (HAM). Furthermore, the results are compared with the numerical simulation of the problem using Matlab program. The obtained results are valid for the whole solution domain.

Cite this paper

S. Usha, V. Abinaya, S. Loghambal and L. Rajendran, "Non-Linear Mathematical Model of the Interaction between Tumor and Oncolytic Viruses,"*Applied Mathematics*, Vol. 3 No. 9, 2012, pp. 1089-1096. doi: 10.4236/am.2012.39160.

S. Usha, V. Abinaya, S. Loghambal and L. Rajendran, "Non-Linear Mathematical Model of the Interaction between Tumor and Oncolytic Viruses,"

References

[1] D. H. Kirn and F. McCormick, “Replicating Viruses as Selective Cancer Therapeutics,” Molecular Medicine Today, Vol. 2, No. 12, 1996, pp. 519-527. doi:10.1016/S1357-4310(97)81456-6

[2] K. A. Parato, D. Senger, P. A. Forsyth and J. C. Bell, “Recent Progress in the Battle between Oncolytic Viruses and Tumours,” Nature Reviews Cancer, Vol. 5, No. 12, 2005, pp. 965-976. doi:10.1038/nrc1750

[3] F. McCormick, “Cancer-Specific Viruses and the Development of ONYX-015,” Cancer Biology and Therapy, Vol. 2, No. 4, 2003, pp. S157-S160.

[4] H. Kasuya, S. Takeda, S. Nomoto and A. Nakao, “The Potential of Oncolytic Virus Therapy for Pancreatic Cancer,” Cancer Gene Therapy, Vol. 12, No. 9, 2005, pp. 725-736. doi:10.1038/sj.cgt.7700830

[5] D. Kirn, T. Hermiston and F. McCormick, “ONYX-015: Clinical Data Are Encouraging,” Nature Medicine, Vol. 4, No. 12, 1998, pp. 1341-1342. doi:10.1038/3902

[6] F. R. Khuri, J. Nemunaitis, I. Ganly, J. Arseneau, I. F. Tannock, L. Romel, M. Gore, J. Ironside, R. H. MacDougall, C. Heise, B. Randlev, A. M. Gillenwater, P. Bruso, S. B. Kaye, W. K. Hong and D. H. Kirn, “A Controlled Trial of Intratumoral ONYX-015, a SelectivelyReplicating Adenovirus, in Combination with Cisplatin and 5-Fluorouracil in Patients with Recurrent Head and Neck Cancer,” Nature Medicine, Vol. 6, No. 8, 2000, pp. 879-885. doi:10.1038/78638

[7] J. Nemunaitis, F. Khuri, I. Ganly, J. Arseneau, M. Posner, E. Vokes, J. Kuhn, T. McCarty, S. Landers, A. Blackburn, L. Romel, B. Randlev, S. Kaye and D. Kirn, “Phase II Trial of Intratumoral Administration of ONYX-015, a Replication-Selective Adenovirus, in Patients with Refractory Head and Neck Cancer,” Journal of Clinical Oncology, Vol. 19, No. 2, 2001, pp. 289-298.

[8] A. C. Shah, D. Benos, G. Y. Gillespie and J. M. Markert, “Oncolytic Viruses: Clinical Applications as Vectors for the Treatment of Malignant Gliomas,” Journal of Neurooncology, Vol. 65, No. 3, 2003, pp. 203-226. doi:10.1023/B:NEON.0000003651.97832.6c

[9] H. L. Kaufman, G. Deraffele, J. Mitcham, D. Moroziewicz, S. M. Cohen, K. S. Hurst-Wicker, K. Cheung, D. S. Lee, J. Divito, M. Voulo, J. Donovan, K. Dolan, K. Manson, D. Panicali, E. Wang, H. Horig and F. M. Mar-incola, “Targeting the Local Tumor Microenvironment with Vaccinia Virus Expressing B7.1 for the Treatment of Melanoma,” Journal of Clinical Investigation, Vol. 115, No. 7, 2005, pp. 1903-1912. doi:10.1172/JCI24624

[10] T. Reid, R. Warren and D. Kirn, “Intravascular Adenoviral Agents in Cancer Patients: Lessons from Clinical Trials,” Cancer Gene Therapy, Vol. 9, No. 12, 2002, pp. 979-986. doi:10.1038/sj.cgt.7700539

[11] D. L. Lichtenstein, K. Toth, K. Doronin, et al., “Functions and Mechanisms of Action of the Adenovirus E3 Proteins,” International Review of Immunology, Vol. 23, 2004, pp. 75-111. doi:10.1080/08830180490265556

[12] A. Zou, I. Atencio, W. M. Huang WM, et al., “Over Expression of Adenovirus E3-11.6K Protein Induces Cell Killing by Both Caspase-Dependent and Caspase-Independent Mechanisms,” Virology, Vol. 326, 2004, pp. 240-249. doi:10.1016/j.virol.2004.06.007

[13] H. Takakuwa, F. Goshima, N. Nozawa, et al., “Oncolytic Viral Therapy Using a Spontaneously Generated Herpes Simplex Virus Type 1 Variant for Disseminated Peritoneal Tumor in Immunocompetent Mice,” Archives of Virology, Vol. 148, 2003, pp. 813-825. doi:10.1007/s00705-002-0944-x

[14] H. Wakimoto, P. R. Johnson, D. M. Knipe, et al., “Effects of Innate Immunityon Herpes Simplex Virus and Its Abilityto Kill Tumor Cells,” Gene Therapy, Vol. 10, 2003, pp. 983-990. doi:10.1038/sj.gt.3302038

[15] K. Ikeda, T. Ichikawa, H. Wakimoto, et al., “Oncolytic Virus Therapy of Multiple Tumors in the Brain Requires Suppression of Innate and Elicited Antiviral Responses,” Nature Medicine, Vol. 5, No. 8, 1999, pp. 881-887. doi:10.1038/11320

[16] D. Wodarz and N. Komarova, “Computational Biology of Cancer,” World Scientific Publishing Company, 2005. http://www.worldscibooks.com/lifesci/5642.html

[17] G. P. Karev, A. S. Novozhilov and E. V. Koonin, “Mathematical Modeling of Tumor Therapy with Oncolytic Viruses: Effects of Parametric Heterogeneity on Cell Dynamics,” Biology Direct, Vol. 1, No. 30, 2006. doi:10.1186/1745-6150-1-30

[18] S. J. Liao, “The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems,” Shanghai Jiao Tong University, Shanghai, 1992.

[19] S. J. Liao, “Beyond Perturbation: Introduction to the Homotopy Analysis Method,” Chapman & Hall/CRC Press, Boca Raton, 2003. doi:10.1201/9780203491164

[20] S. J. Liao, “A Kind of Approximate Solution Technique Which Does Not Depend upon Small Parameters (II): An Application in Fluid Mechanics,” International Journal of Non-Linear Mechanics, Vol. 32, 1997, pp. 815-822. doi:10.1016/S0020-7462(96)00101-1

[21] S. J. Liao, “On the Homotopy Analysis Method for Nonlinear Problems,” Applied Mathematics and Computation, Vol. 147, 2004, pp. 499-513. doi:10.1016/S0096-3003(02)00790-7

[22] S. J. Liao and Y. Tan, “A General Approach to Obtain Series Solutions of Nonlinear Differential Equations.” Studies in Applied Mathematics, Vol. 119, 2007, pp. 297-355. doi:10.1111/j.1467-9590.2007.00387.x

[23] S. J. Liao, “Beyond Perturbation: A Review on the Basic Ideas of the Homotopy Analysis Method and Its Applications,” Advanced Mechanics, Vol. 38, No. 1, 2008, pp. 1-34.

[1] D. H. Kirn and F. McCormick, “Replicating Viruses as Selective Cancer Therapeutics,” Molecular Medicine Today, Vol. 2, No. 12, 1996, pp. 519-527. doi:10.1016/S1357-4310(97)81456-6

[2] K. A. Parato, D. Senger, P. A. Forsyth and J. C. Bell, “Recent Progress in the Battle between Oncolytic Viruses and Tumours,” Nature Reviews Cancer, Vol. 5, No. 12, 2005, pp. 965-976. doi:10.1038/nrc1750

[3] F. McCormick, “Cancer-Specific Viruses and the Development of ONYX-015,” Cancer Biology and Therapy, Vol. 2, No. 4, 2003, pp. S157-S160.

[4] H. Kasuya, S. Takeda, S. Nomoto and A. Nakao, “The Potential of Oncolytic Virus Therapy for Pancreatic Cancer,” Cancer Gene Therapy, Vol. 12, No. 9, 2005, pp. 725-736. doi:10.1038/sj.cgt.7700830

[5] D. Kirn, T. Hermiston and F. McCormick, “ONYX-015: Clinical Data Are Encouraging,” Nature Medicine, Vol. 4, No. 12, 1998, pp. 1341-1342. doi:10.1038/3902

[6] F. R. Khuri, J. Nemunaitis, I. Ganly, J. Arseneau, I. F. Tannock, L. Romel, M. Gore, J. Ironside, R. H. MacDougall, C. Heise, B. Randlev, A. M. Gillenwater, P. Bruso, S. B. Kaye, W. K. Hong and D. H. Kirn, “A Controlled Trial of Intratumoral ONYX-015, a SelectivelyReplicating Adenovirus, in Combination with Cisplatin and 5-Fluorouracil in Patients with Recurrent Head and Neck Cancer,” Nature Medicine, Vol. 6, No. 8, 2000, pp. 879-885. doi:10.1038/78638

[7] J. Nemunaitis, F. Khuri, I. Ganly, J. Arseneau, M. Posner, E. Vokes, J. Kuhn, T. McCarty, S. Landers, A. Blackburn, L. Romel, B. Randlev, S. Kaye and D. Kirn, “Phase II Trial of Intratumoral Administration of ONYX-015, a Replication-Selective Adenovirus, in Patients with Refractory Head and Neck Cancer,” Journal of Clinical Oncology, Vol. 19, No. 2, 2001, pp. 289-298.

[8] A. C. Shah, D. Benos, G. Y. Gillespie and J. M. Markert, “Oncolytic Viruses: Clinical Applications as Vectors for the Treatment of Malignant Gliomas,” Journal of Neurooncology, Vol. 65, No. 3, 2003, pp. 203-226. doi:10.1023/B:NEON.0000003651.97832.6c

[9] H. L. Kaufman, G. Deraffele, J. Mitcham, D. Moroziewicz, S. M. Cohen, K. S. Hurst-Wicker, K. Cheung, D. S. Lee, J. Divito, M. Voulo, J. Donovan, K. Dolan, K. Manson, D. Panicali, E. Wang, H. Horig and F. M. Mar-incola, “Targeting the Local Tumor Microenvironment with Vaccinia Virus Expressing B7.1 for the Treatment of Melanoma,” Journal of Clinical Investigation, Vol. 115, No. 7, 2005, pp. 1903-1912. doi:10.1172/JCI24624

[10] T. Reid, R. Warren and D. Kirn, “Intravascular Adenoviral Agents in Cancer Patients: Lessons from Clinical Trials,” Cancer Gene Therapy, Vol. 9, No. 12, 2002, pp. 979-986. doi:10.1038/sj.cgt.7700539

[11] D. L. Lichtenstein, K. Toth, K. Doronin, et al., “Functions and Mechanisms of Action of the Adenovirus E3 Proteins,” International Review of Immunology, Vol. 23, 2004, pp. 75-111. doi:10.1080/08830180490265556

[12] A. Zou, I. Atencio, W. M. Huang WM, et al., “Over Expression of Adenovirus E3-11.6K Protein Induces Cell Killing by Both Caspase-Dependent and Caspase-Independent Mechanisms,” Virology, Vol. 326, 2004, pp. 240-249. doi:10.1016/j.virol.2004.06.007

[13] H. Takakuwa, F. Goshima, N. Nozawa, et al., “Oncolytic Viral Therapy Using a Spontaneously Generated Herpes Simplex Virus Type 1 Variant for Disseminated Peritoneal Tumor in Immunocompetent Mice,” Archives of Virology, Vol. 148, 2003, pp. 813-825. doi:10.1007/s00705-002-0944-x

[14] H. Wakimoto, P. R. Johnson, D. M. Knipe, et al., “Effects of Innate Immunityon Herpes Simplex Virus and Its Abilityto Kill Tumor Cells,” Gene Therapy, Vol. 10, 2003, pp. 983-990. doi:10.1038/sj.gt.3302038

[15] K. Ikeda, T. Ichikawa, H. Wakimoto, et al., “Oncolytic Virus Therapy of Multiple Tumors in the Brain Requires Suppression of Innate and Elicited Antiviral Responses,” Nature Medicine, Vol. 5, No. 8, 1999, pp. 881-887. doi:10.1038/11320

[16] D. Wodarz and N. Komarova, “Computational Biology of Cancer,” World Scientific Publishing Company, 2005. http://www.worldscibooks.com/lifesci/5642.html

[17] G. P. Karev, A. S. Novozhilov and E. V. Koonin, “Mathematical Modeling of Tumor Therapy with Oncolytic Viruses: Effects of Parametric Heterogeneity on Cell Dynamics,” Biology Direct, Vol. 1, No. 30, 2006. doi:10.1186/1745-6150-1-30

[18] S. J. Liao, “The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems,” Shanghai Jiao Tong University, Shanghai, 1992.

[19] S. J. Liao, “Beyond Perturbation: Introduction to the Homotopy Analysis Method,” Chapman & Hall/CRC Press, Boca Raton, 2003. doi:10.1201/9780203491164

[20] S. J. Liao, “A Kind of Approximate Solution Technique Which Does Not Depend upon Small Parameters (II): An Application in Fluid Mechanics,” International Journal of Non-Linear Mechanics, Vol. 32, 1997, pp. 815-822. doi:10.1016/S0020-7462(96)00101-1

[21] S. J. Liao, “On the Homotopy Analysis Method for Nonlinear Problems,” Applied Mathematics and Computation, Vol. 147, 2004, pp. 499-513. doi:10.1016/S0096-3003(02)00790-7

[22] S. J. Liao and Y. Tan, “A General Approach to Obtain Series Solutions of Nonlinear Differential Equations.” Studies in Applied Mathematics, Vol. 119, 2007, pp. 297-355. doi:10.1111/j.1467-9590.2007.00387.x

[23] S. J. Liao, “Beyond Perturbation: A Review on the Basic Ideas of the Homotopy Analysis Method and Its Applications,” Advanced Mechanics, Vol. 38, No. 1, 2008, pp. 1-34.