Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy

Mathematical Biosciences and Engineering

Volumn 8, Issue 2, 2011, Pages 307-323

  Ledzewicz, Urszula ^a   Maurer, Helmut ^b   Schättler, Heinz ^c  

^a SOUTHERN ILLINOIS UNIVERSITY EDWARDSVILLE   (United States)

^b UNIVERSITY OF MÜNSTER   (Germany)

^c WASHINGTON UNIVERSITY IN ST LOUIS   (United States)

Author keywords

Combination therapy; Optimal and suboptimal controls; Tumor anti angiogenesis

Indexed keywords

ANGIOGENESIS INHIBITOR; ANTINEOPLASTIC AGENT;

ANIMAL; BIOLOGICAL MODEL; CELL PROLIFERATION; COMPUTER ASSISTED DRUG THERAPY; COMPUTER SIMULATION; CONFERENCE PAPER; DOSE RESPONSE; DRUG EFFECT; HUMAN; METHODOLOGY; NEOPLASM; NEOVASCULARIZATION (PATHOLOGY); PATHOPHYSIOLOGY; TREATMENT OUTCOME;

ANGIOGENESIS INHIBITORS; ANIMALS; ANTINEOPLASTIC AGENTS; ANTINEOPLASTIC COMBINED CHEMOTHERAPY PROTOCOLS; CELL PROLIFERATION; COMPUTER SIMULATION; DOSE-RESPONSE RELATIONSHIP, DRUG; DRUG THERAPY, COMPUTER-ASSISTED; HUMANS; MODELS, BIOLOGICAL; NEOPLASMS; NEOVASCULARIZATION, PATHOLOGIC; TREATMENT OUTCOME;

EID: 79955036414     PISSN: 15471063     EISSN: 15510018     Source Type: Journal    
DOI: 10.3934/mbe.2011.8.307     Document Type: Conference Paper

References (34)

1. T. Boehm, J. Folkman, T. Browder and M. S. O'Reilly, Antiangiogenic therapy of experimental cancer does not induce acquired drug resistance, Nature, 390(1997), 404-407. (Pubitemid 28027321)
2. B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory", Springer Verlag, Paris, 2003.
3. A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control", American Institute of Mathematical Sciences, 2007.
4. C. Büskens, "Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer-und Zustands- Beschränkungen", Dissertation, Institut für Numerische Mathematik, Universität Münster, Germany, 1998.
5. A. d'Onofrio, Rapidly acting antitumoral anti-angiogenic therapies, Physical Review E, 76(2007), Art. No. 031920.
6. A. d'Onofrio and A. Gandolfi, Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al. (1999), Mathematical Biosciences, 191(2004), 159-184. (Pubitemid 39195231)
7. A. D'Onofrio and A. Gandolfi, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy, Mathematical Medicine and Biology, 26(2009), 63-95.
8. A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Mathematical Biosciences, 222(2009), 13-26.
9. A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bulletin of Mathematical Biology, 65(2003), 407-424. (Pubitemid 36521826)
10. J. Folkman, Antiangiogenesis: new concept for therapy of solid tumors, Annals of Surgery, 175(1972), 409-416.
11. J. Folkman, Angiogenesis inhibitors generated by tumors, Molecular Medicine, 1(1995), 120-122.
12. 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 Research, 59(1999), 4770-4775. (Pubitemid 29472873)
13. R. K. Jain, Normalizing tumor vasculature with anti-angiogenic therapy: A new paradigm for combination therapy, Nature Medicine, 7(2001), 987-989. (Pubitemid 32937367)
14. R. K. Jain and L. L. Munn, Vascular normalization as a rationale for combining chemotherapy with antiangiogenic agents, Principles of Practical Oncology, 21(2007), 1-7.
15. M. Klagsburn and S. Soker, VEGF/VPF: The angiogenesis factor found?, Current Biology, 3(1993), 699-702.
16. R. S. Kerbel, A cancer therapy resistant to resistance, Nature, 390(1997), 335-336. (Pubitemid 28027292)
17. R. S. Kerbel, Tumor angiogenesis: Past, present and near future, Carcinogensis, 21(2000), 505-515. (Pubitemid 30137936)
18. I. A. K. Kupka, The ubiquity of Fuller's phenomenon, in "Nonlinear Controllability and Optimal Control" (H. Sussmann, Ed.), Marcel Dekker, (1990), 313-350.
19. U. Ledzewicz, H. Maurer and H. Schättler, Bang-bang and singular controls in a mathematical model for combined anti-angiogenic and chemotherapy treatments, Proc. 48th IEEE Conference on Decision and Control, Shanghai, China, (2009), 2280-2285.
20. U. Ledzewicz, J. Munden and H. Schättler, Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models, Discrete and Continuous Dynamical Systems, Series B, 12(2009), 415-438.
21. U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy, Journal of Optimization Theory and Applications - JOTA, 114(2002), 609-637. (Pubitemid 39583314)
22. U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy, J. of Biological Systems, 10(2002), 183-206. (Pubitemid 135711981)
23. U. Ledzewicz and H. Schättler, The influence of PK/PD on the structure of optimal control in cancer chemotherapy models, Mathematical Biosciences and Engineering (MBE), 2(2005), 561-578.
24. U. Ledzewicz and H. Schättler, A synthesis of optimal controls for a model of tumor growth under angiogenic inhibitors, Proc. 44th IEEE Conference on Decision and Control, Sevilla, Spain, (2005), 945-950.
25. U. Ledzewicz and H. Schättler, Anti-angiogenic therapy in cancer treatment as an optimal control problem, SIAM J. on Control and Optimization, 46(2007), 1052-1079.
26. U. Ledzewicz and H. Schättler, Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy, Mathematical Biosciences, 206(2007), 320-342. (Pubitemid 46483526)
27. U. Ledzewicz and H. Schättler, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis, J. of Theoretical Biology, 252(2008), 295-312.
28. U. Ledzewicz and H. Schaettler, Singular controls and chattering arcs in optimal control problems arising in biomedicine, Control and Cybernetics, 38(2009), 1501-1523.
29. H. Maurer, C. Büskens, J. H. R. Kim and C. Y. Kaya, Optimization methods for the verification of second order sufficient conditions for bang-bang controls, Optimal Control: Appliations and Methods, 26(2005), 129-156. (Pubitemid 40892084)
30. H. Schaettler, U. Ledzewicz and B. Cardwell, Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis, Mathematical Biosciences and Engineering (MBE), this volume, 355-369.
31. A. Swierniak, Modelling combined angiogenic and chemo-therapy, Proc. of the Fourteenth National Conference on Applications of Mathematics in Biology and Medicine, Leszno, Poland, (2008), 127-133.
32. A. Swierniak, Direct and indirect control of cancer populations, Bulletin of the Polish Academy of Sciences, Technical Sciences, 56(2008), 367-378.
33. A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. of Applied Mathematics and Computer Science, 13(2003), 357-368.
34. M. I. Zelikin and V. F. Borisov, "Theory of Chattering Control with Applications to Astronautics, Robotics, Economics and Engineering", Birkhäuser, Boston, 1994.