Optimal Protocols for the Anti-VEGF Tumor Treatment

J. Poleszczuk¹ ; M. J. Piotrowska² ; U. Foryś²

¹ College of Inter-Faculty Individual Studies in Mathematics and Natural Sciences University of Warsaw, 02-089 Warsaw, Poland
² Faculty of Mathematics, Informatics and Mechanics University of Warsaw, 02-097 Warsaw, Poland

Mathematical modelling of natural phenomena, Tome 9 (2014) no. 4, pp. 204-215.

Résumé

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.

Détail

DOI : 10.1051/mmnp/20149412

Bibliographie

[1] B. Bodnar, U. Foryś Appl. Math. (Warsaw) 2009 251 262

[2] J.M. Brown, A.J. Giaccia Cancer Res. 1998 1408 1416

[3] L. Cesari. Optimization-theory and applications: problems with ordinary differential equations, volume 17. Springer-verlag New York, 1983.

[4] R. Cooke. Dr. Folkman’s War: Angiogenesis and the struggle to defeat cancer. Random House, New York, 2001.

[5] M. Dolbniak, A. Świerniak Computational and Mathematical Methods in Medicine 2013 1 11

[6] A. D’Onofrio, A. Gandolfi Math. Biosci. 2004 159 184

[7] A. Ergun, K. Camphausen, L.M. Wein Bull. Math. Biol. 2003 407 424

[8] J. Folkman N. Engl. J. Med. 1971 1182 1184

[9] B. Gompertz Phil. Trans. R. Soc. B 1825 513 583

[10] P. Hahnfeldt, D. Panigrahy, J. Folkman, L. Hlatky Cancer Res. 1999 4770 4775

[11] Rakesh K Jain Science 2005 58 62

[12] Rakesh K Jain Sci. Am. 2008 56 63

[13] J. Klamka, A. Świerniak Control and Cybernetics 2013 125 138

[14] U. Ledzewicz, H. Schättler Optim. Contr. Appl. Met. 2008 41 58

[15] U. Ledzewicz, H. Schättler J. Theor. Biol. 2008 295 312

[16] L.A. Loeb Cancer Res. 2001 3230 3239

[17] I. H. Mufti. Computational Methods in Optimal Control Problems. Springer-Verlag, 1970.

[18] M.J. Piotrowska, U. Foryś J. Math. Anal. Appl. 2011 180 203

[19] J. Poleszczuk Mathematica Applicanda 2013 1 12

[20] J. Poleszczuk, M. Bodnar, U. Foryś Math. Biosci. Eng. 2011 591 603

[21] 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.

[22] 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.

[23] Jan Poleszczuk, Iwona Skrzypczak Applicationes Mathematicae 2011 33 49

[24] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko. The Mathematical Theory of Optimal Processes. MacMillan, New York, 1964.

[25] A. Świerniak Applicationes Mathematicae 2009 333 348

[26] 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.

[27] 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.

[28] A. Świerniak, Z. Duda Mathematical and Computer Modelling 1994 255 262

[29] 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.

[30] O. Von Stryk, R. Bulirsch Ann. Oper. Res. 1992 357 373