Skip to main content Accessibility help
×
Home

Optimal Protocols for the Anti-VEGF Tumor Treatment

  • J. Poleszczuk (a1), M. J. Piotrowska (a2) and U. Foryś (a2)

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.

Copyright

Corresponding author

Corresponding author. E-mail: j.poleszczuk@mimuw.edu.pl

References

Hide All
[1] Bodnar, B., Foryś, U.. Influence of time delays on the Hahnfeldt et al. angiogenesis model dynamics. Appl. Math. (Warsaw), 36 no. 3 (2009), 251262.
[2] Brown, J.M., Giaccia, A.J.. The unique physiology of solid tumors: opportunities (and problems) for cancer therapy. Cancer Res., 58 (1998), 14081416.
[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] Dolbniak, M., Świerniak, A.. Comparison of Simple Models of Periodic Protocols for Combined Anticancer Therapy. Computational and Mathematical Methods in Medicine, 1 (2013), 111.
[6] 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), 159184.
[7] Ergun, A., Camphausen, K., Wein, L.M.. Optimal scheduling of radiotherapy and angiogenic inhibitors. Bull. Math. Biol., 65 (2003), 407424.
[8] Folkman, J.. Tumor angiogenesis: therapeutic implications. N. Engl. J. Med., 18 (1971), 11821184.
[9] 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), 513583.
[10] 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), 47704775.
[11] K Jain, Rakesh. Normalization of tumor vasculature: an emerging concept in antiangiogenic therapy. Science, 307 (2005), 5862.
[12] K Jain, Rakesh. Taming vessels to treat cancer. Sci. Am., 298 (2008), 5663.
[13] Klamka, J., Świerniak, A.. Controllability of a model of combined anticancer therapy. Control and Cybernetics, 42 (2013), 125138.
[14] Ledzewicz, U., Schättler, H.. Analysis of optimal controls for a mathematical model of tumour anti-angiogenesis. Optim. Contr. Appl. Met., 29 (2008), 4158.
[15] Ledzewicz, U., Schättler, H.. Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis. J. Theor. Biol., 252 (2008), 295312.
[16] Loeb, L.A.. A mutator phenotype in cancer. Cancer Res., 61 (2001), 32303239.
[17] I. H. Mufti. Computational Methods in Optimal Control Problems. Springer-Verlag, 1970.
[18] Piotrowska, M.J., Foryś, U.. Analysis of the Hopf bifurcation for the Family of Angiogenesis Models. J. Math. Anal. Appl., 382 (2011), 180203.
[19] Poleszczuk, J.. Mathematical modelling of tumour angiogenesis. Mathematica Applicanda, 41 (2013), 112.
[20] 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), 591603.
[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] Poleszczuk, Jan, Skrzypczak, Iwona. Tumour angiogenesis model with variable vessels effectiveness. Applicationes Mathematicae, 38 1 (2011), 3349.
[24] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko. The Mathematical Theory of Optimal Processes. MacMillan, New York, 1964.
[25] Świerniak, A.. Comparison of six models of antiangiogenic therapy. Applicationes Mathematicae, 36 (2009), 333348.
[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] Świerniak, A., Duda, Z.. Singularity of optimal control in some problems related to optimal chemotherapy. Mathematical and Computer Modelling, 19 (1994), 255262.
[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] von Stryk, O., Bulirsch, R.. Direct and indirect methods for trajectory optimization. Ann. Oper. Res., 37 (1992), 357373.

Keywords

Optimal Protocols for the Anti-VEGF Tumor Treatment

  • J. Poleszczuk (a1), M. J. Piotrowska (a2) and U. Foryś (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed