Table of Contents Author Guidelines Submit a Manuscript
Computational and Mathematical Methods in Medicine
Volume 2014, Article ID 206287, 13 pages
http://dx.doi.org/10.1155/2014/206287
Research Article

Optimal Treatment Strategy for a Tumor Model under Immune Suppression

Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea

Received 24 January 2014; Revised 2 May 2014; Accepted 18 May 2014; Published 23 July 2014

Academic Editor: Shenyong Chen

Copyright © 2014 Kwang Su Kim et al. 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. A. K. Abbas and A. H. Lichtman, Cellular and Molecular Immunology, Elsevier Saunders, St. Louis, Miss, USA, 7th edition, 2011.
  2. L. A. Emens, “Chemotherapy and tumor immunity: an unexpected collaboration,” Frontiers in Bioscience, vol. 13, no. 1, pp. 249–257, 2008. View at Publisher · View at Google Scholar · View at Scopus
  3. 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 Publisher · View at Google Scholar · View at Scopus
  4. 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 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 MathSciNet · View at Scopus
  6. L. G. de Pillis and A. E. Radunskaya, “A mathematical model of immune response to tumor invasion,” Computational Fluid and Solid Mechanics, vol. 2, pp. 1661–1668, 2003. View at Google Scholar
  7. L. de Pillis, K. R. Fister, W. Gu et al., “Mathematical model creation for cancer chemo-immunotherapy,” Computational and Mathematical Methods in Medicine, vol. 10, no. 3, pp. 165–184, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. 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 Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. 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 MathSciNet · View at Scopus
  10. T. L. Whiteside, “Immune suppression in cancer: effects on immune cells, mechanisms and future therapeutic intervention,” Seminars in Cancer Biology, vol. 16, no. 1, pp. 3–15, 2006. View at Publisher · View at Google Scholar · View at Scopus
  11. J. Vaage, “Concomitant immunity and specific depression of immunity by residual or reinjected syngeneic tumor tissue.,” Cancer Research, vol. 31, no. 11, pp. 1655–1662, 1971. View at Google Scholar · View at Scopus
  12. S. Chouaib, C. Asselin-Paturel, F. Mami-Chouaib, A. Caignard, and J. Y. Blay, “The host-tumor immune conflict: from immunosuppression to resistance and destruction,” Immunology Today, vol. 18, no. 10, pp. 493–497, 1997. View at Publisher · View at Google Scholar · View at Scopus
  13. M. Robertson-Tessi, A. El-Kareh, and A. Goriely, “A mathematical model of tumor-immune interactions,” Journal of Theoretical Biology, vol. 294, pp. 56–73, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering, Academic Press, New York, NY. USA, 1982. View at MathSciNet
  15. M. I. Kamien and N. L. Schwartz, Dynamic Optimization, vol. 31 of Advanced Textbooks in Economics, North-Holl, Amsterdam, The Netherlands, 2nd edition, 1991. View at MathSciNet
  16. J. Hasskamp, J. Zapas, and G. Elias, “Dendritic cells in patients with melanoma,” Annals of Surgical Oncology, vol. 15, no. 6, p. 1807, 2008. View at Publisher · View at Google Scholar · View at Scopus
  17. M. D. McCarter, J. Baumgartner, G. A. Escobar et al., “Immunosuppressive dendritic and regulatory T cells are upregulated in melanoma patients,” Annals of Surgical Oncology, vol. 14, no. 10, pp. 2854–2860, 2007. View at Publisher · View at Google Scholar · View at Scopus
  18. S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall, CRC, London, UK, 2007.
  19. G. Zaman, Y. H. Kang, and I. H. Jung, “Optimal treatment of an SIR epidemic model with time delay,” BioSystems, vol. 98, no. 1, pp. 43–50, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. R. Ullah, G. Zaman, and S. Islam, “Multiple control strategies for prevention of avian influenza pandemic,” The Scientific World Journal, vol. 2014, Article ID 949718, p. 9, 2014. View at Publisher · View at Google Scholar
  21. R. Ullah, G. Zaman, and S. Islam, “Prevention of influenza pandemic by multiple control strategies,” Journal of Applied Mathematics, vol. 2012, Article ID 294275, 14 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet