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2011, 8(2): 307-323. doi: 10.3934/mbe.2011.8.307

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

 1 Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653 2 Institut für Numerische und Angewandte Mathematik, Universität Münster, D-48149 Münster, Germany 3 Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri, 63130-4899

Received  March 2010 Revised  September 2010 Published  April 2011

We consider the problem of minimizing the tumor volume with a priori given amounts of anti-angiogenic and cytotoxic agents. For one underlying mathematical model, optimal and suboptimal solutions are given for four versions of this problem: the case when only anti-angiogenic agents are administered, combination treatment with a cytotoxic agent, and when a standard linear pharmacokinetic equation for the anti-angiogenic agent is added to each of these models. It is shown that the solutions to the more complex models naturally build upon the simplified versions. This gives credence to a modeling approach that starts with the analysis of simplified models and then adds increasingly more complex and medically relevant features. Furthermore, for each of the problem formulations considered here, there exist excellent simple piecewise constant controls with a small number of switchings that virtually replicate the optimal values for the objective.
Citation: Urszula Ledzewicz, Helmut Maurer, Heinz Schättler. Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy. Mathematical Biosciences & Engineering, 2011, 8 (2) : 307-323. doi: 10.3934/mbe.2011.8.307
##### References:
 [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. doi: 10.1038/37126. [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. doi: 10.1016/j.mbs.2004.06.003. [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. doi: 10.1093/imammb/dqn024. [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. doi: 10.1016/j.mbs.2009.08.004. [9] A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bulletin of Mathematical Biology, 65 (2003), 407-424. doi: 10.1016/S0092-8240(03)00006-5. [10] J. Folkman, Antiangiogenesis: new concept for therapy of solid tumors, Annals of Surgery, 175 (1972), 409-416. doi: 10.1097/00000658-197203000-00014. [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. [13] R. K. Jain, Normalizing tumor vasculature with anti-angiogenic therapy: A new paradigm for combination therapy, Nature Medicine, 7 (2001), 987-989. doi: 10.1038/nm0901-987. [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. doi: 10.1016/0960-9822(93)90073-W. [16] R. S. Kerbel, A cancer therapy resistant to resistance, Nature, 390 (1997), 335-336. doi: 10.1038/36978. [17] R. S. Kerbel, Tumor angiogenesis: Past, present and near future, Carcinogensis, 21 (2000), 505-515. doi: 10.1093/carcin/21.3.505. [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. doi: 10.3934/dcdsb.2009.12.415. [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. [22] U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy, J. of Biological Systems, 10 (2002), 183-206. doi: 10.1142/S0218339002000597. [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. doi: 10.1109/CDC.2005.1582277. [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. doi: 10.1137/060665294. [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. doi: 10.1016/j.mbs.2005.03.013. [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. doi: 10.1016/j.jtbi.2008.02.014. [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. doi: 10.1002/oca.756. [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.

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##### References:
 [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. doi: 10.1038/37126. [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. doi: 10.1016/j.mbs.2004.06.003. [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. doi: 10.1093/imammb/dqn024. [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. doi: 10.1016/j.mbs.2009.08.004. [9] A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bulletin of Mathematical Biology, 65 (2003), 407-424. doi: 10.1016/S0092-8240(03)00006-5. [10] J. Folkman, Antiangiogenesis: new concept for therapy of solid tumors, Annals of Surgery, 175 (1972), 409-416. doi: 10.1097/00000658-197203000-00014. [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. [13] R. K. Jain, Normalizing tumor vasculature with anti-angiogenic therapy: A new paradigm for combination therapy, Nature Medicine, 7 (2001), 987-989. doi: 10.1038/nm0901-987. [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. doi: 10.1016/0960-9822(93)90073-W. [16] R. S. Kerbel, A cancer therapy resistant to resistance, Nature, 390 (1997), 335-336. doi: 10.1038/36978. [17] R. S. Kerbel, Tumor angiogenesis: Past, present and near future, Carcinogensis, 21 (2000), 505-515. doi: 10.1093/carcin/21.3.505. [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. doi: 10.3934/dcdsb.2009.12.415. [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. [22] U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy, J. of Biological Systems, 10 (2002), 183-206. doi: 10.1142/S0218339002000597. [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. doi: 10.1109/CDC.2005.1582277. [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. doi: 10.1137/060665294. [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. doi: 10.1016/j.mbs.2005.03.013. [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. doi: 10.1016/j.jtbi.2008.02.014. [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. doi: 10.1002/oca.756. [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.
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