Free Access
Math. Model. Nat. Phenom.
Volume 9, Number 4, 2014
Optimal control
Page(s) 88 - 104
Published online 20 June 2014
  1. B.M. Adams, H.T. Banks, H.-D. Kwon, H.T. Tran. Dynamic multidrug therapies for HIV: optimal and STI control approaches. Math. Biosci. Eng., 1 (2004), 223–241. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  2. A.V. Antipov, A.S. Bratus. Mathematical model of optimal chemotherapy strategy with allowance for cell population dynamics in a heterogeneous tumor. Zh. Vychisl. Mat. Mat. Fiz., 49 (2009), 1907–1919. [Google Scholar]
  3. F. Billy, J. Clairambault. Designing proliferating cell population models with functional targets for control by anti-cancer drugs. Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 865–889. [Google Scholar]
  4. T. Burden, J. Ernstberger, K.R. Fister. Optimal control applied to immunotherapy. Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 135–146. [MathSciNet] [Google Scholar]
  5. F. Castiglione, B. Piccoli. Optimal control in a model of dendritic cell transfection cancer immunotherapy. Bull. Math. Biol., 68 (2006), 255–274. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  6. A.J. Coldman, J.M. Murray. Optimal control for a stochastic model of cancer chemotherapy. Math. Biosci., 168 (2000), 187–200. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  7. M. Costa, J. Boldrini, R. Bassanezi. Drug kinetics and drug resistance in optimal chemotherapy. Math. Biosci., 125 (1995), 191–209. [CrossRef] [PubMed] [Google Scholar]
  8. M. Delitala, T. Lorenzi. A mathematical model for the dynamics of cancer hepatocytes under therapeutic actions. J. Theoret. Biol., 297 (2012), 88-102. [Google Scholar]
  9. L. Desvillettes, P.E. Jabin, S. Mischler, G. Raoul. On selection dynamics for continuous structured populations. Commun. Math. Sci., 6 (2008), 729–747. [CrossRef] [Google Scholar]
  10. O. Diekmann, P.E. Jabin, S. Mischler, B. Perthame. The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach. Theor. Pop. Biol., 67 (2005), 257–271. [Google Scholar]
  11. G. Dimitriu. Numerical approximation of the optimal inputs for an identification problem. Intern. J. Computer Math., 70 (1998), 197–209. [CrossRef] [Google Scholar]
  12. P. Dua, V. Dua, E. Pistikopoulos. Optimal delivery of chemotherapeutic agents in cancer. Comput. Chem. Eng., 32 (2008), 99–107. [CrossRef] [Google Scholar]
  13. G.P. Dunn, A.T. Bruce, H. Ikeda, L.J. Old, R.D. Schreiber. Cancer immunoediting: from immunosurveillance to tumor escape. Nature Immunol., 3 (2002), 991–998. [Google Scholar]
  14. M. DuPage, C. Mazumdar, L.M. Schmidt, A.F. Cheung, T. Jacks. Expression of tumour-specific antigens underlies cancer immunoediting. Nature, 482 (2012), 405–9. [CrossRef] [PubMed] [Google Scholar]
  15. M. Engelhart, D. Lebiedz, S. Sager. Optimal control for selected cancer chemotherapy ODE models: A view on the potential of optimal schedules and choice of objective function. Math. Biosci., 229 (2011), 123–134. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  16. K.R. Fister, J. Donnelly. Immunotherapy: an optimal control theory approach. Math. Biosci. Eng., 2 (2005), 499–510. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  17. K.R. Fister, J.C. Panetta. Optimal control applied to cell-cycle-specific cancer chemotherapy. SIAM J. Appl. Math., 60 (2000), 1059–1072. [CrossRef] [MathSciNet] [Google Scholar]
  18. K.R. Fister, J.C. Panetta. Optimal control applied to competing chemotherapeutic cell-kill strategies. SIAM J. Appl. Math., 63 (2003), 1954–1971. [CrossRef] [Google Scholar]
  19. W.H. Fleming, R.W. Rishel. Deterministic and Stochastic Optimal Control. Springer-Verlag, 1975. [Google Scholar]
  20. A. Ghaffari, N. Naserifar. Optimal therapeutic protocols in cancer immunotherapy. Comput. Biol. Med., 40 (2010), 261–270. [CrossRef] [PubMed] [Google Scholar]
  21. J. Goldie, A. Coldman. Drug resistance in cancer: mechanisms and models. Cambridge University Press, 1998. [Google Scholar]
  22. M. Gottesman. Mechanisms of cancer drug resistance. Annu. Rev. Med., 53 (2002), 615–627. [CrossRef] [PubMed] [Google Scholar]
  23. F.H. Igney, P.H. Krammer. Immune escape of tumors: apoptosis resistance and tumor counterattack. J. Leukoc. Biol., 71 (2002), 907–20. [PubMed] [Google Scholar]
  24. L.S. Jennings, M.E. Fisher, K.L. Teo, C.J. Goh. MISER3 Optimal Control Software: Theory and User Manual. Department of Mathematics, The University of Western Australia, Nedlands, WA 6907, Australia, 2004. [Google Scholar]
  25. M.I. Kamien, N.L. Schwartz. Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management, Advanced Textbooks in Economics. Second ed., vol. 31. North-Holland, 1991. [Google Scholar]
  26. N. Komarova, D. Wodarz. Drug resistance in cancer: Principles of emergence and prevention. Proc Natl Acad Sci USA, 102 (2005), 9714–9719. [CrossRef] [PubMed] [Google Scholar]
  27. U. Ledzewicz, A. d’Onofrio, H. Maurer, H. Schäettler. On optimal delivery of combination therapy for tumors. Math. Biosci., 222 (2009), 13–26. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  28. U. Ledzewicz, M. Naghnaeian, H. Schättler. An optimal control approach to cancer treatment under immunological activity. Appl. Math., 38 (2011), 17–31. [CrossRef] [MathSciNet] [Google Scholar]
  29. T. Lorenzi, A. Lorz, G. Restori. Asymptotic dynamics in populations structured by sensitivity to global warming and habitat shrinking. Acta Appl. Math., 2013, DOI 10.1007/s10440-013-9849-9. [Google Scholar]
  30. K. Liu. Role of apoptosis resistance in immune evasion and metastasis of colorectal cancer. World J. Gastrointest. Oncol., 15 (2010), 399–406. [CrossRef] [Google Scholar]
  31. A. Lorz, T. Lorenzi, M.E. Hochberg, J. Clairambault, B. Perthame. Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies. ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 377–399. [Google Scholar]
  32. A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil, B. Perthame. Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors. preprint, 2014. [Google Scholar]
  33. D.L. Lukes. Differential Equations: Classical to Controlled, vol. 162. Academic Press, 1982. [Google Scholar]
  34. R. Martin, K. Teo. Optimal Drug Administration in Cancer Chemotherapy. World Scientific, Singapore, 1994. [Google Scholar]
  35. A. Matveev, A. Savkin. Application of optimal control theory to analysis of cancer chemotherapy regimens. Syst. Control Lett., 46 (2002), 311–321. [CrossRef] [Google Scholar]
  36. L. Merlo, J. Pepper, B. Reid, C. Maley. Cancer as an evolutionary and ecological process. Nat. Rev Cancer, 6 (2006), 924–935. [Google Scholar]
  37. J. Murray. Optimal control for a cancer chemotherapy problem with general growth and loss functions. Math. Biosci., 98 (1990), 273–287. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  38. L.G. de Pillis, W. Gu, K.R. Fister, T. Head, K. Maples, A. Murugan, T. Neal, K. Yoshida. Chemotherapy for tumors: An analysis of the dynamics and a study of quadratic and linear optimal controls. Math. Biosci., 209 (2007), 292–315. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  39. L.G. de Pillis, W. Gu, A.E. Radunskaya. Mixed immunotherapy and chemotherapy of tumors: modelling, applications and biological interpretations. J. Theor. Biol., 238 (2006), 841–862. [CrossRef] [Google Scholar]
  40. L.G. de Pillis, A.E. Radunskaya, C.L. Wiseman. A validated mathematical model of cell-mediated immune response to tumor growth. Cancer Res., 65 (2005), 7950–7958. [PubMed] [Google Scholar]
  41. L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko. The Mathematical Theory of Optimal Processes. Gordon and Breach, 1962. [Google Scholar]
  42. V. Shankaran, H. Ikeda, A.T. Bruce, J.M. White, P.E. Swanson, L.J. Old, R.D. Schreiber. IFNgamma and lymphocytes prevent primary tumour development and shape tumour immunogenicity, Nature 410 (2001), 1107–1111. [CrossRef] [PubMed] [Google Scholar]
  43. G.W. Swan. Role of optimal control theory in cancer chemotherapy. Math. Biosci., 101 (1990), 237–284. [CrossRef] [PubMed] [Google Scholar]
  44. Z. Szymanska. Analysis of immunotherapy models in the context of cancer dynamics. Int. J. Appl. Math. Comput. Sci., 13 (2003), 407–418. [MathSciNet] [Google Scholar]
  45. C.L. Zindl, D.D. Chaplin. Tumor immune evasion. Science, 328 (2010), 697–698. [CrossRef] [PubMed] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.