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Local Controllability of Models of Combined Anticancer Therapy with Delays in Control

Published online by Cambridge University Press:  20 June 2014

A. Świerniak*
Affiliation:
Department of Automatic Control, Silesian University of Technology, 44-100 Gliwice, Poland
J. Klamka
Affiliation:
Department of Automatic Control, Silesian University of Technology, 44-100 Gliwice, Poland
*
Corresponding author. E-mail: [email protected]
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Abstract

We present sufficient conditions of local controllability for a class of models of treatment response to combined anticancer therapies which include delays in control strategies. The combined therapy is understood as combination of direct anticancer strategy e.g. chemotherapy and indirect modality (in this case antiangiogenic therapy). Controllability of the models in the form of semilinear second order dynamic systems with delays in control enables to answer the questions of realizability of different objectives of multimodal therapy in the presence of PK/PD effects. We compare results for the models without delays and conditions for relative local controllability of models with delays.

Type
Research Article
Copyright
© EDP Sciences, 2014

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References

Bergers, G., Hanahan, D.. Modes of resistance to anti-angiogenic therapy. Nature Reviews Cancer, 8 (2008) 592603. CrossRefGoogle Scholar
Czornik, A., Świerniak, A.. On controllability with respect to the expectation of discrete time jump linear systems. Journal of the Franklin Institute, 338 (2001), no. 4, 443453. CrossRefGoogle Scholar
Czornik, A., Świerniak, A.. On direct controllability of discrete time jump linear system. Journal of the Franklin Institute, 341 (2004), no. 6, 491503. CrossRefGoogle Scholar
M. Dolbniak, A. Swierniak. Comparison of simple models of periodic protocols for combined anticancer therapy. Computational and Mathematical Methods in Medicine, (2013), Article ID 567213, doi: 11.1055/2013/567213.
D’Onofrio, A., Gandolfi, A.. A family of models of angiogenesis and antiangiogenesis anti-cancer therapy. Mathematical Medicine and Biology, 26 (2009), 6369. CrossRefGoogle Scholar
D’Onofrio, A., Gandolfi, A.. Chemotherapy of vascularised tumours: role of vessel density and the effect of vascular "pruning”. Journal of Theoretical Biology, 264 (2010), 253265. CrossRefGoogle ScholarPubMed
A. D’Onofrio, A. Gandolfi. Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al. (1999), Mathematical Biosciences, 191 (2004), 159–184.
Ebos, J.M.L., Kerbel, R.S.. Antiangiogenic therapy: impact on invasion, disease progression, and metastasis. Nature Reviews Clinical Oncology, 8 (2011), 210221. CrossRefGoogle Scholar
Ergun, A., Camphausen, K., Wein, L.M.. Optimal scheduling of radiotherapy and angiogenic inhibitors. Bulletin of Mathematical Biology, 65 (2003), 407. CrossRefGoogle Scholar
Folkman, J.. Tumor angiogenesis: therapeutic implications, N. Engl. J. Med., 295 (1971), 11821186. Google Scholar
Folkman, J.. Antiangiogenesis: new concept for therapy of solid tumors. Ann. Surg., 175 (1972), 409416. 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 Research, 59 (1999), 47704775. Google ScholarPubMed
Hanahan, D., Weinberg, R.A.. Hallmarks of Cancer: The Next Generation. Cell, 144 (2011), 647670. CrossRefGoogle ScholarPubMed
R.K. Jain. Normalization of tumor vasculature and microenvironment in antiangiogenic therapies. ASCO Annual Meeting, (2007), 412–417.
Kerbel, R.S.. A cancer therapy resistant to resistance. Nature, 390 (1997), 335340. CrossRefGoogle Scholar
J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht, Netherlands, 1991.
Klamka, J.. Constrained controllability of nonlinear systems. J. Math. Anal. Appl., 201 (1996), 365374. CrossRefGoogle Scholar
Klamka, J.. Constrained controllability of semilinear systems with multiple delays in control. Bull. PAS, Techn. Sci., 52 (2004), 25-30. Google Scholar
Klamka, J., Swierniak, A.. Controllability of a model of combined anticancer therapy. Control and Cybernetics, 42 (2013), 125138. Google Scholar
Li-Song, T., Ke-Tao, J., Kui-Feng, H., Hao-Hao, W., Jiang, C., De-Cao, Y.. Advances in combination of antiangiogenic agents targeting VEGF-binding and conventional chemotherapy and radiation for cancer treatment. Journal of the Chinese Medical Association, 73 (2010), no. 6, 281288. Google Scholar
Ma, J., Waxman, D.J.. Combination of anti-angiogenesis with chemotherapy for more effective cancer treatment. Molecular Cancer Therapeutics, 7 (2010), no. 12, 36703684. CrossRefGoogle Scholar
McDougall, S.R., Anderson, A.R., Chaplain, M.A., Sherratt, J.A.. Mathematical modeling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies. Bulletin of Mathematical Biology, 64 (2002), no. 4, 673702. CrossRefGoogle ScholarPubMed
Piotrowska, M.J., Forys, U.. Analysis of the Hopf bifurcation for the family of angiogenic models. Journal of Mathematical Analysis and Applications, 382 (2011), 180203. CrossRefGoogle Scholar
Swierniak, A.. Comparison of six models of antiangiogenic therapy. Applicationes Mathematicae, 36 (2009), no. 3, 333348. CrossRefGoogle Scholar
Swierniak, A.. Direct and indirect control of cancer populations. Bulletin of the Polish Academy of Sciences, Technical Sciences, 56 (2008), 367378. Google Scholar
A. Swierniak, J. Klamka. Control properties of models of antiangiogenic therapy. in: Advances in Automatics and Robotics (K. Malinowski and R. Dindorf R. Eds.), Monograph of Committee of Automatics and Robotics PAS, 16 (2011), no. 2, 300–312.