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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

Urszula Ledzewicz 1, , Helmut Maurer 2, and  Heinz Schättler 3,

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.
Keywords: Combination therapy, tumor anti-angiogenesis, optimal and suboptimal controls..
Mathematics Subject Classification: Primary: 92C50; Secondary: 49K15, 49M9.
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.  Google Scholar

[2]

B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory," Springer Verlag, Paris, 2003.  Google Scholar

[3]

A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control," American Institute of Mathematical Sciences, 2007.  Google Scholar

[4]

C. Büskens, "Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer-und Zustands-Beschränkungen," Dissertation, Institut für Numerische Mathematik, Universität Münster, Germany, 1998. Google Scholar

[5]

A. d'Onofrio, Rapidly acting antitumoral anti-angiogenic therapies, Physical Review E, 76 (2007), Art. No. 031920. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. Folkman, Antiangiogenesis: new concept for therapy of solid tumors, Annals of Surgery, 175 (1972), 409-416. doi: 10.1097/00000658-197203000-00014.  Google Scholar

[11]

J. Folkman, Angiogenesis inhibitors generated by tumors, Molecular Medicine, 1 (1995), 120-122. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[16]

R. S. Kerbel, A cancer therapy resistant to resistance, Nature, 390 (1997), 335-336. doi: 10.1038/36978.  Google Scholar

[17]

R. S. Kerbel, Tumor angiogenesis: Past, present and near future, Carcinogensis, 21 (2000), 505-515. doi: 10.1093/carcin/21.3.505.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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), (): 355.   Google Scholar

[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. Google Scholar

[32]

A. Swierniak, Direct and indirect control of cancer populations, Bulletin of the Polish Academy of Sciences, Technical Sciences, 56 (2008), 367-378. Google Scholar

[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. Google Scholar

[34]

M. I. Zelikin and V. F. Borisov, "Theory of Chattering Control with Applications to Astronautics, Robotics, Economics and Engineering," Birkhäuser, Boston, 1994. Google Scholar

show all references

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.  Google Scholar

[2]

B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory," Springer Verlag, Paris, 2003.  Google Scholar

[3]

A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control," American Institute of Mathematical Sciences, 2007.  Google Scholar

[4]

C. Büskens, "Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer-und Zustands-Beschränkungen," Dissertation, Institut für Numerische Mathematik, Universität Münster, Germany, 1998. Google Scholar

[5]

A. d'Onofrio, Rapidly acting antitumoral anti-angiogenic therapies, Physical Review E, 76 (2007), Art. No. 031920. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. Folkman, Antiangiogenesis: new concept for therapy of solid tumors, Annals of Surgery, 175 (1972), 409-416. doi: 10.1097/00000658-197203000-00014.  Google Scholar

[11]

J. Folkman, Angiogenesis inhibitors generated by tumors, Molecular Medicine, 1 (1995), 120-122. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[16]

R. S. Kerbel, A cancer therapy resistant to resistance, Nature, 390 (1997), 335-336. doi: 10.1038/36978.  Google Scholar

[17]

R. S. Kerbel, Tumor angiogenesis: Past, present and near future, Carcinogensis, 21 (2000), 505-515. doi: 10.1093/carcin/21.3.505.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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), (): 355.   Google Scholar

[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. Google Scholar

[32]

A. Swierniak, Direct and indirect control of cancer populations, Bulletin of the Polish Academy of Sciences, Technical Sciences, 56 (2008), 367-378. Google Scholar

[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. Google Scholar

[34]

M. I. Zelikin and V. F. Borisov, "Theory of Chattering Control with Applications to Astronautics, Robotics, Economics and Engineering," Birkhäuser, Boston, 1994. Google Scholar

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