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Optimal Protocols for the Anti-VEGF Tumor Treatment

Published online by Cambridge University Press:  20 June 2014

J. Poleszczuk ,
M. J. Piotrowska  and
U. Foryś
J. Poleszczuk*
Affiliation:
College of Inter-Faculty Individual Studies in Mathematics and Natural Sciences University of Warsaw, 02-089 Warsaw, Poland
M. J. Piotrowska
Affiliation:
Faculty of Mathematics, Informatics and Mechanics University of Warsaw, 02-097 Warsaw, Poland
U. Foryś
Affiliation:
Faculty of Mathematics, Informatics and Mechanics University of Warsaw, 02-097 Warsaw, Poland
*
⋆Corresponding author. E-mail: j.poleszczuk@mimuw.edu.pl
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Abstract

Cancer treatment using the antiangiogenic agents targets the evolution of the tumor vasculature. The aim is to significantly reduce supplies of oxygen and nutrients, and thus starve the tumor and induce its regression. In the paper we consider well established family of tumor angiogenesis models together with their recently proposed modification, that increases accuracy in the case of treatment using VEGF antibodies. We consider the optimal control problem of minimizing the tumor volume when the maximal admissible drug dose (the total amount of used drug) and the final level of vascularization are also taken into account. We investigate the solution of that problem for a fixed therapy duration. We show that the optimal strategy consists of the drug-free, full-dose and singular (with intermediate values of the control variable) intervals. Moreover, no bang-bang switch of the control is possible, that is the change from the no-dose to full-dose protocol (or in opposite direction) occurs on the interval with the singular control. For two particular models, proposed by Hahnfeldt et al. and Ergun et al., we provide additional theorems about the optimal control structure. We investigate the optimal controls numerically using the customized software written in MATLAB®, which we make freely available for download. Utilized numerical scheme is based on the composition of the well known gradient and shooting methods.

Keywords

angiogenesisbevacizumaboptimal controlcancer
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 4: Optimal control , 2014 , pp. 204 - 215
Copyright
© EDP Sciences, 2014

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References

Bodnar, B., Foryś, U.. Influence of time delays on the Hahnfeldt et al. angiogenesis model dynamics. Appl. Math. (Warsaw), 36 no. 3 (2009), 251–262. CrossRefGoogle Scholar
Brown, J.M., Giaccia, A.J.. The unique physiology of solid tumors: opportunities (and problems) for cancer therapy. Cancer Res., 58 (1998), 1408–1416. Google ScholarPubMed
L. Cesari. Optimization-theory and applications: problems with ordinary differential equations, volume 17. Springer-verlag New York, 1983.
R. Cooke. Dr. Folkman’s War: Angiogenesis and the struggle to defeat cancer. Random House, New York, 2001.
Dolbniak, M., Świerniak, A.. Comparison of Simple Models of Periodic Protocols for Combined Anticancer Therapy. Computational and Mathematical Methods in Medicine, 1 (2013), 1–11. CrossRefGoogle Scholar
d’Onofrio, A., Gandolfi, A.. Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al.(1999). Math. Biosci., 191 (2004), 159–184. CrossRefGoogle Scholar
Ergun, A., Camphausen, K., Wein, L.M.. Optimal scheduling of radiotherapy and angiogenic inhibitors. Bull. Math. Biol., 65 (2003), 407–424. CrossRefGoogle Scholar
Folkman, J.. Tumor angiogenesis: therapeutic implications. N. Engl. J. Med., 18 (1971), 1182–1184. Google Scholar
Gompertz, B.. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Phil. Trans. R. Soc. B, 115 (1825), 513–583. CrossRefGoogle Scholar
Hahnfeldt, P., Panigrahy, D., Folkman, J., Hlatky, L.. Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Res., 59 (1999), 4770–4775. Google ScholarPubMed
K Jain, Rakesh. Normalization of tumor vasculature: an emerging concept in antiangiogenic therapy. Science, 307 (2005), 58–62. CrossRefGoogle Scholar
K Jain, Rakesh. Taming vessels to treat cancer. Sci. Am., 298 (2008), 56–63. CrossRefGoogle Scholar
Klamka, J., Świerniak, A.. Controllability of a model of combined anticancer therapy. Control and Cybernetics, 42 (2013), 125–138. Google Scholar
Ledzewicz, U., Schättler, H.. Analysis of optimal controls for a mathematical model of tumour anti-angiogenesis. Optim. Contr. Appl. Met., 29 (2008), 41–58. CrossRefGoogle Scholar
Ledzewicz, U., Schättler, H.. Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis. J. Theor. Biol., 252 (2008), 295–312. CrossRefGoogle Scholar
Loeb, L.A.. A mutator phenotype in cancer. Cancer Res., 61 (2001), 3230–3239. Google Scholar
I. H. Mufti. Computational Methods in Optimal Control Problems. Springer-Verlag, 1970.
Piotrowska, M.J., Foryś, U.. Analysis of the Hopf bifurcation for the Family of Angiogenesis Models. J. Math. Anal. Appl., 382 (2011), 180–203. CrossRefGoogle Scholar
Poleszczuk, J.. Mathematical modelling of tumour angiogenesis. Mathematica Applicanda, 41 (2013), 1–12. CrossRefGoogle Scholar
Poleszczuk, J., Bodnar, M., Foryś, U.. New approach to modeling of antiangiogenic treatment on the basis of Hahnfeldt et al. model. Math. Biosci. Eng., 8 (2011), 591–603. CrossRefGoogle ScholarPubMed
J. Poleszczuk, U. Foryś. Derivation of the Hahnfeldt em et al. model (1999) revisited. Proceedings of the XVI National Conference Applications of Mathematics to Biology and Medicine, (2010), 87–92.
J. Poleszczuk, U. Foryś„ M.J. Piotrowska. New approach to anti-angiogenic treatment modelling and control. In Proceedings of the XVII National Conference Applications of Mathematics to Biology and Medicine, (2011), 73–78.
Poleszczuk, Jan, Skrzypczak, Iwona. Tumour angiogenesis model with variable vessels effectiveness. Applicationes Mathematicae, 38 1 (2011), 33–49. CrossRefGoogle Scholar
L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko. The Mathematical Theory of Optimal Processes. MacMillan, New York, 1964.
Świerniak, A.. Comparison of six models of antiangiogenic therapy. Applicationes Mathematicae, 36 (2009), 333–348. CrossRefGoogle Scholar
A. Świerniak. Combined anticancer therapy as a control problem. In Advances in Control Theory and Automation. Monograph of Committee of Automatics and Robotics PAS, 2012.
A. Świerniak. Control problems related to three compartmental model of combined anticancer therapy. In 20 IEEE Mediterenian Conference on Automation and Control MED 12, Barcelona, (2012), 1428–1433.
Świerniak, A., Duda, Z.. Singularity of optimal control in some problems related to optimal chemotherapy. Mathematical and Computer Modelling, 19 (1994), 255–262. CrossRefGoogle Scholar
A. Świerniak, G. Gala, A. d’Onofrio„ A. Gandolfi. Optimization of anti-angiogenic therapy as optimal control problem. in Proc 4th IASTED Conf. on Biomechanics, ACTA Press (ed. M. Doblaré), (2006), 56–60.
von Stryk, O., Bulirsch, R.. Direct and indirect methods for trajectory optimization. Ann. Oper. Res., 37 (1992), 357–373. CrossRefGoogle Scholar