Local controllability and optimal control for\newline a model of combined anticancer therapy with control delays

Jerzy Klamka; Helmut Maurer; Andrzej Swierniak; Jerzy Klamka; Helmut Maurer; Andrzej Swierniak

doi:10.3934/mbe.2017013

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 1: 195-216. doi: 10.3934/mbe.2017013

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Local controllability and optimal control for\newline a model of combined anticancer therapy with control delays

1.
Silesian University of Technology, Department of Automatic Control, Akademicka 16, 44101 Gliwice, Poland
2.
University of Münster, Institute of Computational and Applied Mathematics, Einsteinstr. 62,48149 Münster, Germany
3.
Silesian University of Technology, Department of Automatic Control, Akademicka 16, 44101 Gliwice, Poland

Received: 06 October 2015 Accepted: 15 July 2016 Published: 01 February 2017
MSC : Primary: 49J15; Secondary: 93B05, 37N25, 92C50, 93C15, 65K10

We study some control properties of a class of two-compartmental models of response to anticancer treatment which combines anti-angiogenic and cytotoxic drugs and take into account multiple control delays. We formulate sufficient local controllability conditions for semilinear systems resulting from these models. The control delays are related to PK/PD effects and some clinical recommendations, e.g., normalization of the vascular network. The optimized protocols of the combined therapy for the model, considered as solutions to an optimal control problem with delays in control, are found using necessary conditions of optimality and numerical computations. Our numerical approach uses dicretization and nonlinear programming methods as well as the direct optimization of switching times. The structural sensitivity of the considered control properties and optimal solutions is also discussed.
- Combined cancer therapy,
- local controllability,
- optimal control,
- control delays
Citation: Jerzy Klamka, Helmut Maurer, Andrzej Swierniak. Local controllability and optimal control for\newline a model of combined anticancer therapy with control delays[J]. Mathematical Biosciences and Engineering, 2017, 14(1): 195-216. doi: 10.3934/mbe.2017013

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Abstract

We study some control properties of a class of two-compartmental models of response to anticancer treatment which combines anti-angiogenic and cytotoxic drugs and take into account multiple control delays. We formulate sufficient local controllability conditions for semilinear systems resulting from these models. The control delays are related to PK/PD effects and some clinical recommendations, e.g., normalization of the vascular network. The optimized protocols of the combined therapy for the model, considered as solutions to an optimal control problem with delays in control, are found using necessary conditions of optimality and numerical computations. Our numerical approach uses dicretization and nonlinear programming methods as well as the direct optimization of switching times. The structural sensitivity of the considered control properties and optimal solutions is also discussed.

References

[1]	[ G. Bergers,D. Hanahan, Modes of resistance to anti-angiogenic therapy, Nature Reviews Cancer, 8 (2008): 592-603.
[2]	[ A. C. Billioux, U. Modlich and R, Bicknell, The Cancer Handbook: Angiogenesis, 2nd Edition, John Wiley & Sons, 2007.
[3]	[ R. F. Brammer, Controllability in linear autonomous systems with positive controllers, SIAM J. Control, 10 (1972): 339-353.
[4]	[ C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse Für Optimale Steuerprozesse mit Steuer-und Zustands-Beschränkungen, PhD thesis, Institut für Numerische Mathematik, Universität Münster, Germany, 1998.
[5]	[ C. Büskens,H. Maurer, SQP methods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and real time control, J. Comput. Appl. Math., 120 (2000): 85-108.
[6]	[ J. M. Collins,D. S. Zaharko,R. L. Dedrick,B. A. Chabner, Potential roles for preclinical pharmacology in phase Ⅰ clinical trials, Cancer Treat. Rep., 70 (1986): 73-80.
[7]	[ V. T. Devita and J. Folkman, Cancer: Principles and Practice of Oncology, 6th edition, Lippincott Williams & Wilkins Publishers, 2001.
[8]	[ M. Dolbniak and A. Swierniak, Comparison of simple models of periodic protocols for combined anticancer therapy, Computational and Mathematical Methods in Medicine, 2013 (2013), Article ID 567213, 11pp.
[9]	[ A. D'Onofrio,A. Gandolfi, Tumor eradication by anti-angiogenic therapy: Analysis and extensions of the model by Hahnfeldt et al. (1999), Mathematical Biosciences, 191 (2004): 159-184.
[10]	[ A. D'Onofrio,A. Gandolfi, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy, Mathematical Medicine and Biology, 26 (2009): 63-95.
[11]	[ A. D'Onofrio,A. Gandolfi, Chemotherapy of vascularised tumors: Role of vessel density and the effect of vascular "pruning", Journal of Theoretical Biology, 264 (2010): 253-265.
[12]	[ A. D'Onofrio,U. Ledzewicz,H. Maurer,H. Schaettler, On optimal delivery of combination therapy for tumors, Math. Biosciences, 222 (2009): 13-26.
[13]	[ A. Ergun,K. Camphausen,L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bulletin of Mathematical Biology, 65 (2003): 407-424.
[14]	[ J. Folkman, Anti-angiogenesis: New concept for therapy of solid tumors, Annals of Surgery, 175 (1972): 409-416.
[15]	[ J. Folkman, Tumor angiogenesis therapeutic implications, New England Journal of Medicine, 285 (1971): 1182-1186.
[16]	[ R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks-Cole Publishing Company, 1993.
[17]	[ G. Gasparini,R. Longo,M. Fanelli,B. A. Teicher, Combination of anti-angiogenic therapy with other anticancer therapies: Results, challenges, and open questions, Journal of Clinical Oncology, 23 (2005): 1295-1311.
[18]	[ L. Göllmann,H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Special Issue on Computational Methods for Optimization and Control, J. of Industrial and Management Optimization, 10 (2014): 413-441.
[19]	[ P. Hahnfeldt,D. Panigrahy,J. Folkman,L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 59 (1999): 4770-4775.
[20]	[ D. Hanahan,R. A. Weinberg, Hallmarks of cancer: The next generation, Cell, 144 (2011): 646-674.
[21]	[ R. S. Kerbel, Inhibition of tumor angiogenesis as a strategy to circumvent acquired resistance to anti-cancer therapeutic agents, BioEssays, 13 (1991): 31-36.
[22]	[ M. Kimmel and A. Swierniak, Control theory approach to cancer chemotherapy: Benefiting from phase dependence and overcoming drug resistance, Tutorials in Mathematical Biosciences Ⅲ: Cell Cycle, Proliferation, and Cancer (A. Friedman-Ed. ), Lecture Notes in Mathematics, Mathematical Biosciences Subseries, Springer, Heidelberg, 1872 (2006), 185-221.
[23]	[ J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1991.
[24]	[ J. Klamka, Constrained controllability of nonlinear systems, J. Math. Anal. Appl., 201 (1996): 365-374.
[25]	[ U. Ledzewicz,H. Schaettler, Anti-angiogenic therapy in cancer treatment as an optimal control problem, SIAM Journal on Control and Optimization, 46 (2007): 1052-1079.
[26]	[ U. Ledzewicz,H. Schaettler, Analysis of optimal controls for a mathematical model of tumor anti-angiogenesis, Optimal Control Applications and Methods, 29 (2008): 41-57.
[27]	[ U. Ledzewicz,H. Schaettler, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis, J. Theor. Biol., 252 (2008): 295-312.
[28]	[ U. Ledzewicz,H. Schaettler, On the optimality of singular controls for a class of mathematical models for tumor antiangiogenesis, Discrete and Continuous Dynamical Systems, Series B, 11 (2009): 691-715.
[29]	[ U. Ledzewicz,J. Marriott,H. Maurer,H. Schaettler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Mathematical Medicine and Biology, 27 (2010): 157-179.
[30]	[ U. Ledzewicz, H. Maurer and H. Schaettler, Minimizing tumor volume for a mathematical model of anti-angiogenesis with linear pharmacokinetics, in Recent Advances in Optimization and its Applications in Engineering, 267-276, Springer, 2010.
[31]	[ U. Ledzewicz,H. Maurer,H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor antiangiogenesis in combination with chemotherapy, Mathematical Biosciences and Engineering, 8 (2011): 307-323.
[32]	[ J. Ma,D. J. Waxman, Combination of antiangiogenesis with chemotherapy for more effective cancer treatment, Molecular Cancer Therapeutics, 7 (2008): 3670-3684.
[33]	[ H. Maurer,C. Büskens,J. H. R. Kim,C. Y. Kaya, Optimization methods for the verification of second order sufficient conditions for bang-bang control, Optimal Control Appl. Meth., 26 (2005): 129-156.
[34]	[ N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control, SIAM Advances in Design and Control, Vol. DC 24, SIAM Publications, Philadelphia, 2012.
[35]	[ M. J. Piotrowska,U. Forys, Analysis of the Hopf bifurcation for the family of angiogenesis models, Journal of Mathematical Analysis and Applications, 382 (2011): 180-203.
[36]	[ R. K. Sachcs,L. R. Hlatky,P. Hahnfeldt, Simple ODE models of tumor growth and anti-angiogenic or radiation treatment, Math. Comput. Mod, 33 (2001): 1297-1305.
[37]	[ H. Schättler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples, Springer, New York, 2012.
[38]	[ A. Swierniak, Direct and indirect control of cancer populations, Bulletin of the Polish Academy of Sciences: Technical Sciences, 56 (2008): 367-378.
[39]	[ A. Swierniak, Modelling combined anti-angiogenic and chemo-therapies, in: Proc. 14th National Conf. Appl. Math. Biol Medicine, Leszno, 2008,127-133.
[40]	[ A. Swierniak, Comparison of six models of anti-angiogenic therapy, Applicationes Mathematicae, 36 (2009): 333-348.
[41]	[ A. Swierniak, A. d'Onofrio and A. Gandolfi, Control problems related to tumor angiogenesis, Proc. of the 32nd Annual Conference on IEEE Industrial Electronics (IECON '06), Paris, 677-681, November 2006.
[42]	[ A. Swierniak,J. Klamka, Local controllability of models of combined anticancer therapy with delays in control, Math. Model. Nat. Phenom., 9 (2014): 216-226.
[43]	[ L. S. Teng,K. T. Jin,K. F. He,H. H. Wang,J. Cao,D. C. Yu, Advances in combination of anti-angiogenic agents targeting VEGF-binding and conventional chemotherapy and radiation for cancer treatment, Journal of the Chinese Medical Association, 73 (2010): 281-288.
[44]	[ The Internet Drug Index, (2015), http://www.rxlist.com/avastin-drug/clinical-pharmacology.html
[45]	[ US National Institutes of Health, Clinical Trials, (last updated June 2015), http://www.clinicaltrials.gov/ct2/show/NCT00520013
[46]	[ A. Wächter,L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006): 25-57.

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