Cancer modelling
Open Access
Math. Model. Nat. Phenom.
Volume 15, 2020
Cancer modelling
Article Number 69
Number of page(s) 23
Published online 03 December 2020
  1. B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, in In Vol. 40 of Mathématiques & Applications. Springer, Paris (2003). [Google Scholar]
  2. A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control. American Institute of Mathematical Sciences, California (2007). [Google Scholar]
  3. A.E. Bryson, Jr. and Y.C. Ho, Applied Optimal Control, Revised Printing. Hemisphere Publishing Company, New York (1975). [Google Scholar]
  4. C.S. Chou and A. Friedman, Introduction to Mathematical Biology - Modeling, Analysis and Simulation. Springer Verlag (2016). [CrossRef] [Google Scholar]
  5. M. Eisen, Mathematical Models in Cell Biology and Cancer Chemotherapy. Vol. 30 of Lecture Notes in Biomathematics, Springer, Berlin (1979). [CrossRef] [Google Scholar]
  6. L.A. Fernández and C. Pola, Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy. Discr. Cont. Dyn. Syst. Ser. B 24 (2019) 2577–2612. [Google Scholar]
  7. P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Res. 59 (1999) 4770–4775. [Google Scholar]
  8. A. Källén, Computational Pharmacokinetics. Chapman and Hall, CRC, London (2007). [CrossRef] [Google Scholar]
  9. H.K. Khalil, Nonlinear Systems, 3rded. Prentice Hall, Upper Saddle River, NJ (2002). [Google Scholar]
  10. M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy. Sci. Bull. Silesian Tech. Univ. 65 (1983) 120–130. [Google Scholar]
  11. U. Ledzewicz, H. Maurer and H. Schättler, Minimizing tumor volume for a mathematical model of anti-angiogenesis with linear pharmacokinetics, in Recent Advances in Optimization and its Applications in Engineering, edited by M. Diehl, F. Glineur, E. Jarlebring and W. Michiels. Springer, Heidelberg (2010) 267–276. [Google Scholar]
  12. U. Ledzewicz, H. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy. Math. Biosci. Eng. 8 (2011) 3-7–323. [Google Scholar]
  13. U. Ledzewicz, H. Maurer and H. Schättler, Optimal combined radio- and anti-angiogenic cancer therapy. J. Optim. Theory Appl. 180 (2019) 321–340. [CrossRef] [MathSciNet] [Google Scholar]
  14. U. Ledzewicz and H. Moore, Dynamical systems properties of a mathematical model for the treatment of CML. Appl. Sci. 6 (2016) 291. [CrossRef] [Google Scholar]
  15. U. Ledzewicz and H. Moore, Optimal control applied to a generalized Michaelis-Menten model of CML therapy. Dicr. Cont. Dyn. Syst. Ser. B 23 (2018) 331–346. [Google Scholar]
  16. U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy. J. Optim. Theory Appl. 114 (2002) 609–637. [CrossRef] [MathSciNet] [Google Scholar]
  17. U. Ledzewicz and H. Schättler, The influence of PK/PD on the structure of optimal control in cancer chemotherapy models. Math. Biosci. Eng. 2 (2005) 561–578. [CrossRef] [MathSciNet] [Google Scholar]
  18. U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem. SIAM J. Control Optim. 46 (2007) 1052–1079. [CrossRef] [MathSciNet] [Google Scholar]
  19. U. Ledzewicz and H. Schättler, Controlling a model for bone marrow dynamics in cancer chemotherapy. Math. Biosci. Eng. 1 (2004) 95–110. [CrossRef] [MathSciNet] [Google Scholar]
  20. U. Ledzewicz and H. Schättler, Singular controls and chattering arcs in optimal control problems arising in biomedicine. Control Cybern. 38 (2009) 1501–1523. [Google Scholar]
  21. U. Ledzewicz and H. Schättler, Multi-input optimal control problems for combined tumor anti-angiogenic and radiotherapy treatments. J. Optim. Theory Appl. 153 (2012) 195–224. [Google Scholar]
  22. M. Leszczyński, The Role of Pharmacometrics in Optimal Controls Problems for Mathematical Models of Cancer Therapies. Ph.D. thesis, Lodz University of Technology, Lodz, Poland (2019). [Google Scholar]
  23. M. Leszczyński, E. Ratajczyk, U. Ledzewicz and H. Schättler, Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics. Opuscula Math. 37 (2017) 403–419. [CrossRef] [MathSciNet] [Google Scholar]
  24. M. Leszczyński, U. Ledzewicz and H. Schättler, Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics. Discr. Cont. Dyn. Syst. Ser. B 24 (2019) 2315–2334. [Google Scholar]
  25. P. Macheras and A. Iliadin, Modeling in Biopharmaceutics, Pharmacokinetics and Pharmacodynamics, in Vol. 30 of Interdisciplinary Applied Mathematics, 2nd ed. Springer, New York (2016). [CrossRef] [Google Scholar]
  26. R. Martin and K.L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy. World Scientific Press, Singapore (1994). [Google Scholar]
  27. A. d’Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors. Math. Biosci. 222 (2009) 13–26. [CrossRef] [MathSciNet] [Google Scholar]
  28. L.G. de Pillis and A. Radunskaya, A mathematical tumor model with immune resistance and drug therapy: an optimal control approach. J. Theor. Med. 3 (2001) 79–100. [CrossRef] [Google Scholar]
  29. L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The Mathematical Theory of Optimal Processes. Macmillan, New York (1964). [Google Scholar]
  30. M. Rowland and T.N. Tozer, Clinical Pharmacokinetics and Pharmacodynamics, Wolters Kluwer Lippicott, Philadelphia (1995). [Google Scholar]
  31. H. Schättler and U. Ledzewicz, Geometric Optimal Control, Interdisciplinary Applied Mathematics, Vol. 38, Springer, New York (2012). [CrossRef] [Google Scholar]
  32. H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies, Interdisciplinary Applied Mathematics, Vol. 42, Springer, New York (2015). [CrossRef] [Google Scholar]
  33. H. Schättler, U. Ledzewicz and H. Maurer, Sufficient conditions for strong locak optimality in optimal control problems with L2 -type objectives and control constraints, Dicr. Cont. Dyn. Syst. Ser. B 19 (2014) 2657–2679. [Google Scholar]
  34. S. Shimoda, K. Nishida, M. Sakakida, Y. Konno, K. Ichinose, M. Uehara, T. Nowak and M. Shichiri, Closed-loop subcutaneous insulin infusion algorithm with a short-acting insulin analog for long-term clinical application of a wearable artificial endocrine pancreas, Front. Med. Biol. Eng. 8 (1997) 197–211. [PubMed] [Google Scholar]
  35. H.E. Skipper, On mathematical modeling of critical variables in cancer treatment (goals: better understanding of the past and better planning in the future), Bull. Math. Biol. 48 (1986) 253–278. [CrossRef] [MathSciNet] [Google Scholar]
  36. G.W. Swan, Applications of Optimal Control Theory in Medicine, Marcel Dekker, New York (1984). [Google Scholar]
  37. G.W. Swan, General applications of optimal control theory in cancer chemotherapy, IMA J. Math. Appl. Med. Biol. 5 (1988) 303–316. [Google Scholar]
  38. G.W. Swan, Role of optimal control in cancer chemotherapy, Math. Biosci. 101 (1990) 237–284. [CrossRef] [Google Scholar]
  39. A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, Proc. of the 12th IMACS World Congress, Paris 4 (1988), pp. 170–172. [Google Scholar]
  40. A. Swierniak, Cell cycle as an object of control, J. Biol. Syst. 3 (1995) 41–54. [CrossRef] [Google Scholar]

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