Free Access
Math. Model. Nat. Phenom.
Volume 9, Number 4, 2014
Optimal control
Page(s) 204 - 215
Published online 20 June 2014
  1. B. Bodnar, U. Foryś. Influence of time delays on the Hahnfeldt et al. angiogenesis model dynamics. Appl. Math. (Warsaw), 36 no. 3 (2009), 251–262. [CrossRef] [MathSciNet] [Google Scholar]
  2. J.M. Brown, A.J. Giaccia. The unique physiology of solid tumors: opportunities (and problems) for cancer therapy. Cancer Res., 58 (1998), 1408–1416. [PubMed] [Google Scholar]
  3. L. Cesari. Optimization-theory and applications: problems with ordinary differential equations, volume 17. Springer-verlag New York, 1983. [Google Scholar]
  4. R. Cooke. Dr. Folkman’s War: Angiogenesis and the struggle to defeat cancer. Random House, New York, 2001. [Google Scholar]
  5. M. Dolbniak, A. Świerniak. Comparison of Simple Models of Periodic Protocols for Combined Anticancer Therapy. Computational and Mathematical Methods in Medicine, 1 (2013), 1–11. [CrossRef] [Google Scholar]
  6. A. d’Onofrio, A. Gandolfi. Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al.(1999). Math. Biosci., 191 (2004), 159–184. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  7. A. Ergun, K. Camphausen, L.M. Wein. Optimal scheduling of radiotherapy and angiogenic inhibitors. Bull. Math. Biol., 65 (2003), 407–424. [Google Scholar]
  8. J. Folkman. Tumor angiogenesis: therapeutic implications. N. Engl. J. Med., 18 (1971), 1182–1184. [Google Scholar]
  9. B. Gompertz. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Phil. Trans. R. Soc. B, 115 (1825), 513–583. [CrossRef] [Google Scholar]
  10. P. Hahnfeldt, D. Panigrahy, J. Folkman, L. Hlatky. Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Res., 59 (1999), 4770–4775. [PubMed] [Google Scholar]
  11. Rakesh K Jain. Normalization of tumor vasculature: an emerging concept in antiangiogenic therapy. Science, 307 (2005), 58–62. [CrossRef] [PubMed] [Google Scholar]
  12. Rakesh K Jain. Taming vessels to treat cancer. Sci. Am., 298 (2008), 56–63. [CrossRef] [PubMed] [Google Scholar]
  13. J. Klamka, A. Świerniak. Controllability of a model of combined anticancer therapy. Control and Cybernetics, 42 (2013), 125–138. [Google Scholar]
  14. U. Ledzewicz, H. Schättler. Analysis of optimal controls for a mathematical model of tumour anti-angiogenesis. Optim. Contr. Appl. Met., 29 (2008), 41–58. [CrossRef] [Google Scholar]
  15. U. Ledzewicz, H. Schättler. Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis. J. Theor. Biol., 252 (2008), 295–312. [CrossRef] [PubMed] [Google Scholar]
  16. L.A. Loeb. A mutator phenotype in cancer. Cancer Res., 61 (2001), 3230–3239. [PubMed] [Google Scholar]
  17. I. H. Mufti. Computational Methods in Optimal Control Problems. Springer-Verlag, 1970. [Google Scholar]
  18. M.J. Piotrowska, U. Foryś. Analysis of the Hopf bifurcation for the Family of Angiogenesis Models. J. Math. Anal. Appl., 382 (2011), 180–203. [CrossRef] [Google Scholar]
  19. J. Poleszczuk. Mathematical modelling of tumour angiogenesis. Mathematica Applicanda, 41 (2013), 1–12. [CrossRef] [MathSciNet] [Google Scholar]
  20. J. Poleszczuk, M. Bodnar, U. Foryś. New approach to modeling of antiangiogenic treatment on the basis of Hahnfeldt et al. model. Math. Biosci. Eng., 8 (2011), 591–603. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  21. J. Poleszczuk, U. Foryś. Derivation of the Hahnfeldt em et al. model (1999) revisited. Proceedings of the XVI National Conference Applications of Mathematics to Biology and Medicine, (2010), 87–92. [Google Scholar]
  22. J. Poleszczuk, U. Foryś„ M.J. Piotrowska. New approach to anti-angiogenic treatment modelling and control. In Proceedings of the XVII National Conference Applications of Mathematics to Biology and Medicine, (2011), 73–78. [Google Scholar]
  23. Jan Poleszczuk, Iwona Skrzypczak. Tumour angiogenesis model with variable vessels effectiveness. Applicationes Mathematicae, 38 1 (2011), 33–49. [CrossRef] [MathSciNet] [Google Scholar]
  24. L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko. The Mathematical Theory of Optimal Processes. MacMillan, New York, 1964. [Google Scholar]
  25. A. Świerniak. Comparison of six models of antiangiogenic therapy. Applicationes Mathematicae, 36 (2009), 333–348. [CrossRef] [MathSciNet] [Google Scholar]
  26. A. Świerniak. Combined anticancer therapy as a control problem. In Advances in Control Theory and Automation. Monograph of Committee of Automatics and Robotics PAS, 2012. [Google Scholar]
  27. A. Świerniak. Control problems related to three compartmental model of combined anticancer therapy. In 20 IEEE Mediterenian Conference on Automation and Control MED 12, Barcelona, (2012), 1428–1433. [Google Scholar]
  28. A. Świerniak, Z. Duda. Singularity of optimal control in some problems related to optimal chemotherapy. Mathematical and Computer Modelling, 19 (1994), 255–262. [CrossRef] [Google Scholar]
  29. A. Świerniak, G. Gala, A. d’Onofrio„ A. Gandolfi. Optimization of anti-angiogenic therapy as optimal control problem. in Proc 4th IASTED Conf. on Biomechanics, ACTA Press (ed. M. Doblaré), (2006), 56–60. [Google Scholar]
  30. O. von Stryk, R. Bulirsch. Direct and indirect methods for trajectory optimization. Ann. Oper. Res., 37 (1992), 357–373. [CrossRef] [MathSciNet] [Google Scholar]

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