Table of Contents 
Journal of Nonlinear Dynamics
Volume 2013 (2013), Article ID 608598, 13 pages
http://dx.doi.org/10.1155/2013/608598
Research Article

Dynamical Behaviour of a Tumor-Immune System with Chemotherapy and Optimal Control

Swarnali Sharma and G. P. Samanta

Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah 711 103, India

Received 22 April 2013; Revised 7 June 2013; Accepted 23 June 2013

Academic Editor: Giovanni P. Galdi

Copyright © 2013 Swarnali Sharma and G. P. Samanta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. V. A. Kuznetsov and G. D. Knott, “Modeling tumor regrowth and immunotherapy,” Mathematical and Computer Modelling, vol. 33, no. 12-13, pp. 1275–1287, 2001. View at Publisher · View at Google Scholar · View at Scopus
  2. V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor, and A. S. Perelson, “Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis,” Bulletin of Mathematical Biology, vol. 56, no. 2, pp. 295–321, 1994. View at Google Scholar · View at Scopus
  3. D. Kirschner and J. C. Panetta, “Modeling immunotherapy of the tumor—immune interaction,” Journal of Mathematical Biology, vol. 37, no. 3, pp. 235–252, 1998. View at Google Scholar · View at Scopus
  4. M. Kolev, “Mathematical modelling of the competition between tumors and immune system considering the role of the antibodies,” Mathematical and Computer Modelling, vol. 37, no. 11, pp. 1143–1152, 2003. View at Publisher · View at Google Scholar · View at Scopus
  5. L. G. De Pillis, W. Gu, and A. E. Radunskaya, “Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations,” Journal of Theoretical Biology, vol. 238, no. 4, pp. 841–862, 2006. View at Publisher · View at Google Scholar · View at Scopus
  6. L. G. De Pillis and A. Radunskaya, “A mathematical tumor model with immune resistance and drug therapy: an optimal control approach,” Journal of Theoretical Medicine, vol. 3, no. 2, pp. 79–100, 2001. View at Google Scholar · View at Scopus
  7. J. C. Arciero, T. L. Jackson, and D. E. Kirschner, “A mathematical model of tumor-immune evasion and siRNA treatment,” Discrete and Continuous Dynamical Systems B, vol. 4, no. 1, pp. 39–58, 2004. View at Google Scholar · View at Scopus
  8. N. Bellomo, A. Bellouquid, and M. Delitala, “Mathematical topics on the modelling complex multicellular systems and tumor immune cells competition,” Mathematical Models and Methods in Applied Sciences, vol. 14, no. 11, pp. 1683–1733, 2004. View at Publisher · View at Google Scholar · View at Scopus
  9. N. Bellomo and L. Preziosi, “Modelling and mathematical problems related to tumor evolution and its interaction with the immune system,” Mathematical and Computer Modelling, vol. 32, no. 3-4, pp. 413–452, 2000. View at Publisher · View at Google Scholar · View at Scopus
  10. B. S. Chan and P. Yu, “Bifurcation analysis in a model of cytotoxic T-lymphocyte response to viral infections,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 64–77, 2012. View at Publisher · View at Google Scholar · View at Scopus
  11. L. G. De Pillis, A. E. Radunskaya, and C. L. Wiseman, “A validated mathematical model of cell-mediated immune response to tumor growth,” Cancer Research, vol. 65, no. 17, pp. 7950–7958, 2005. View at Publisher · View at Google Scholar · View at Scopus
  12. L. Derbel, “Analysis of a new model for tumor-immune system competition including long-time scale effects,” Mathematical Models and Methods in Applied Sciences, vol. 14, no. 11, pp. 1657–1681, 2004. View at Publisher · View at Google Scholar · View at Scopus
  13. A. D'Onofrio, “A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences,” Physica D, vol. 208, no. 3-4, pp. 220–235, 2005. View at Publisher · View at Google Scholar · View at Scopus
  14. F. Nani and H. I. Freedman, “A mathematical model of cancer treatment by immunotherapy,” Mathematical Biosciences, vol. 163, no. 2, pp. 159–199, 2000. View at Publisher · View at Google Scholar · View at Scopus
  15. S. T. R. Pinho, F. S. Bacelar, R. F. S. Andrade, and H. I. Freedman, “A mathematical model for the effect of anti-angiogenic therapy in the treatment of cancer tumors by chemotherapy,” Nonlinear Analysis: Real World Applications, vol. 14, pp. 815–828, 2013. View at Google Scholar
  16. H. Siu, E. S. Vivetta, R. D. May, and J. W. Uhr, “Tumor dormancy—I. Regression of BCL1 tumor and induction of a dormant tumor state in mice chimeric at the major histocompatibility complex,” Journal of Immunology, vol. 137, no. 4, pp. 1376–1382, 1986. View at Google Scholar · View at Scopus
  17. T. Takayanagi and A. Ohuchi, “A mathematical analysis of the interactions between immunogenic tumor cells and cytotoxic T lymphocytes,” Microbiology and Immunology, vol. 45, no. 10, pp. 709–715, 2001. View at Google Scholar · View at Scopus
  18. R. Yafia, “Dynamics analysis and limit cycle in a delayed model for tumor growth with quiescene,” Nonlinear Analysis. Modelling and Control, vol. 11, pp. 95–110, 2006. View at Google Scholar
  19. R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy, World Scientific, River Edge, NJ, USA, 1994.
  20. A. S. Matveev and A. V. Savkin, “Application of optimal control theory to analysis of cancer chemotherapy regimens,” Systems and Control Letters, vol. 46, no. 5, pp. 311–321, 2002. View at Publisher · View at Google Scholar · View at Scopus
  21. L. G. de Pillis, W. Gu, K. R. Fister et al., “Chemotherapy for tumors: an analysis of the dynamics and a study of quadratic and linear optimal controls,” Mathematical Biosciences, vol. 209, no. 1, pp. 292–315, 2007. View at Publisher · View at Google Scholar · View at Scopus
  22. L. G. De Pillis and A. Radunskaya, “The dynamics of an optimally controlled tumor model: a case study,” Mathematical and Computer Modelling, vol. 37, no. 11, pp. 1221–1244, 2003. View at Publisher · View at Google Scholar · View at Scopus
  23. M. Engelhart, D. Lebiedz, and S. Sager, “Optimal control for selected cancer chemotherapy ODE models: a view on the potential of optimal schedules and choice of objective function,” Mathematical Biosciences, vol. 229, no. 1, pp. 123–134, 2011. View at Publisher · View at Google Scholar · View at Scopus
  24. K. R. Fister and J. Donnelly, “Immunotherapy: an optimal control theory approach,” Mathematical Biosciences and Engineering, vol. 2, no. 3, p. 499, 2005. View at Google Scholar
  25. K. R. Fister and J. C. Panetta, “Optimal control applied to cell-cycle-specific cancer chemotherapy,” SIAM Journal on Applied Mathematics, vol. 60, no. 3, pp. 1059–1072, 2000. View at Google Scholar · View at Scopus
  26. K. R. Fister and J. C. Panetta, “Optimal control applied to competing chemotherapeutic cell-kill strategies,” SIAM Journal on Applied Mathematics, vol. 63, no. 6, pp. 1954–1971, 2003. View at Publisher · View at Google Scholar · View at Scopus
  27. U. Ledzewicz and H. Schättler, “Antiangiogenic therapy in cancer treatment as an optimal control problem,” SIAM Journal on Control and Optimization, vol. 46, no. 3, pp. 1052–1079, 2007. View at Publisher · View at Google Scholar · View at Scopus
  28. S. M. Mamat and A. Kartono, “Mathematical model of cancer treatment using immunotherapy, chemotherapy and biochemotherapy,” Applied Mathematical Sciences, vol. 7, no. 5, pp. 247–261, 2013. View at Google Scholar
  29. H. R. Joshi, “Optimal control of an HIV immunology model,” Optimal Control Applications and Methods, vol. 23, no. 4, pp. 199–213, 2002. View at Publisher · View at Google Scholar · View at Scopus
  30. G. Zaman, Y. Han Kang, and I. H. Jung, “Stability analysis and optimal vaccination of an SIR epidemic model,” BioSystems, vol. 93, no. 3, pp. 240–249, 2008. View at Publisher · View at Google Scholar · View at Scopus
  31. J. K. Hale, Theory of Functional Differential Equations, Springer, New York, Ny, USA, 1977.
  32. M. Kot, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, UK, 2001.
  33. L. Bannock, http://www.doctorbannock.com/nutrition.html.
  34. P. Calabresi and P. S. Schein, Eds., Medical Oncology: Basic Principles and Clinical Management of Cancer, McGraw-Hill, New York, NY, USA, 2nd edition, 1993.
  35. A. Diefenbach, E. R. Jensen, A. M. Jamieson, and D. H. Raulet, “Rae1 and H60 ligands of the NKG2D receptor stimulate tumour immunity,” Nature, vol. 413, no. 6852, pp. 165–171, 2001. View at Publisher · View at Google Scholar · View at Scopus
  36. M. C. Perry, Ed., The Chemotherapy Source Book, Lippincott Williams and Wilkins, 3rd edition, 2001.
  37. K. Blayneh, Y. Cao, and H.-D. Kwon, “Optimal control of vector-borne diseases: treatment and prevention,” Discrete and Continuous Dynamical Systems, vol. 11, no. 3, pp. 587–611, 2009. View at Publisher · View at Google Scholar · View at Scopus
  38. S. Lcnhart and J. T. Workman, Optimal Control Applied to Biological Mathods, Chapman and Hall/CRC, London, UK, 2007.
  39. G. W. Swan, Applications of Optimal Control Theory in Biomedicine, Marcel Dekker, New York, NY, USA, 1984.
  40. J. M. Tchuenche, S. A. Khamis, F. B. Agusto, and S. C. Mpeshe, “Optimal control and sensitivity analysis of an influenza model with treatment and vaccination,” Acta Biotheoretica, vol. 59, no. 1, pp. 1–28, 2011. View at Publisher · View at Google Scholar · View at Scopus
  41. W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, New York, NY, USA, 1975.
  42. D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1982.
  43. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Process, Gordon and Breach, 1962.