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2011, 8(2): 355-369. doi: 10.3934/mbe.2011.8.355

Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis

Heinz Schättler 1, , Urszula Ledzewicz 2, and  Benjamin Cardwell 3,

1. 

Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri, 63130-4899
2. 

Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653
3. 

Dept. of Mathematics and Statistic, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653, United States

Received  February 2010 Revised  September 2010 Published  April 2011

We describe optimal protocols for a class of mathematical models for tumor anti-angiogenesis for the problem of minimizing the tumor volume with an a priori given amount of vessel disruptive agents. The family of models is based on a biologically validated model by Hahnfeldt et al. [9] and includes a modification by Ergun et al. [6], but also provides two new variations that interpolate the dynamics for the vascular support between these existing models. The biological reasoning for the modifications of the models will be presented and we will show that despite quite different modeling assumptions, the qualitative structure of optimal controls is robust. For all the systems in the class of models considered here, an optimal singular arc is the defining element and all the syntheses of optimal controlled trajectories are qualitatively equivalent with quantitative differences easily computed.
Keywords: singular arcs, optimal control, Tumor anti-angiogenesis, synthesis..
Mathematics Subject Classification: Primary: 49N35; Secondary: 49K15, 92C5.
Citation: Heinz Schättler, Urszula Ledzewicz, Benjamin Cardwell. Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis. Mathematical Biosciences & Engineering, 2011, 8 (2) : 355-369. doi: 10.3934/mbe.2011.8.355
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]

S. Davis and G. D. Yancopoulos, The angiopoietins: Yin and Yang in angiogenesis, Current Topics in Microbiology and Immunology, 237 (1999), 173-185. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

U. Ledzewicz, J. Marriott, H. Maurer and H. Schättler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Mathematical Medicine and Biology, 27 (2010), 157-179. doi: 10.1093/imammb/dqp012.  Google Scholar

[15]

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

[16]

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), 934-939. doi: 10.1109/CDC.2005.1582277.  Google Scholar

[17]

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

[18]

U. Ledzewicz and H. Schättler, Analysis of a mathematical model for tumor anti-angiogenesis, Optimal Control Applications and Methods, 29 (2008), 41-57. doi: 10.1002/oca.814.  Google Scholar

[19]

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

[20]

L. Norton and R. Simon, Growth curve of an experimental solid tumor following radiotherapy, J.of the National Cancer Institute, 58 (1977), 1735-1741. Google Scholar

[21]

L. Norton, A Gompertzian model of human breast cancer growth, Cancer Research, 48 (1988), 7067-7071. Google Scholar

[22]

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

[23]

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

[24]

A. Swierniak, Comparison of six models of antiangiogenic therapy, Applied Mathematics, 36 (2009), 333-348. 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]

S. Davis and G. D. Yancopoulos, The angiopoietins: Yin and Yang in angiogenesis, Current Topics in Microbiology and Immunology, 237 (1999), 173-185. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

U. Ledzewicz, J. Marriott, H. Maurer and H. Schättler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Mathematical Medicine and Biology, 27 (2010), 157-179. doi: 10.1093/imammb/dqp012.  Google Scholar

[15]

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

[16]

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), 934-939. doi: 10.1109/CDC.2005.1582277.  Google Scholar

[17]

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

[18]

U. Ledzewicz and H. Schättler, Analysis of a mathematical model for tumor anti-angiogenesis, Optimal Control Applications and Methods, 29 (2008), 41-57. doi: 10.1002/oca.814.  Google Scholar

[19]

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

[20]

L. Norton and R. Simon, Growth curve of an experimental solid tumor following radiotherapy, J.of the National Cancer Institute, 58 (1977), 1735-1741. Google Scholar

[21]

L. Norton, A Gompertzian model of human breast cancer growth, Cancer Research, 48 (1988), 7067-7071. Google Scholar

[22]

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

[23]

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

[24]

A. Swierniak, Comparison of six models of antiangiogenic therapy, Applied Mathematics, 36 (2009), 333-348. Google Scholar

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