The Use of the State Observer in the Simulation of the Process of the Anti-Angiogenesis Therapy
# 12, December 2016
1 Bauman Moscow State Technical University, Moscow, Russia
Anti-angiogenesis therapy is one of the modern and progressive methods for treatment of cancerous disease in which the growth of new vessels is suppressed.
In the paper a model of cancerous tumour evolution in the framework of anti-angiogenesis therapy is analyzed. Earlier a treatment for this model was developed which was determined by a distribution of medicament dose along the treatment. To reach the required result of the tumour volume reduction upto the given level the treatment requires 120 days. This result is refined using the differential-geometric methods and the new scheme requires only 60 days.
The proposed treatment schemes are based on the full state vector of the system but this can be hardly provided by measurements. Hence, the problem of observer design arises which obtains a state vector estimate from the measured tumour volume.
In this note the normal form of the system is refined and an observer with high gain is designed. An estimate of the full state vector of the system obtained by the observer is used in the output feedback which stabilizes the system. An identification algorithm for parameters of the nonlinear system is also presented which is based on lower sample of the output measurement. The identification algorithm is based on using the observer. The theoretical results obtained in this work are verified by numerical simulation.
- 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, 1999, vol. 59, iss. 19, pp. 4770–4775.
- Kerbel R., Folkman J. Clinical translation of angiogenesis inhibitors. Nature Reviews Cancer, 2002, vol. 2, no. 10, pp. 727–739. DOI: 10.1038/nrc905
- Stepanova E.V. Anti-angiogenesis Therapy: new ability of treatment of malignancies. Prakticheskaja onkologija = Practical oncology, 2002, vol. 3, no. 4, pp. 246–252. (in Russian).
- Drexler D., Kovacs L., Sapi J., Harmati I., Benyo Z. Model-based analysis and synthesis of tumor growth under angiogenic inhibition: a case study. IFAC Proceedings Volumes, 2011, vol. 44, iss. 1, pp. 3753–3758. DOI: 10.3182/20110828-6-IT-1002.02107
- Gauthier J.P., Kupka I. Deterministic observation theory and applications. Cambridge University Press, 2001. 226 p.
- Astolfi D., Marconi L. A High-Gain Nonlinear Observer with Limited Gain Power. arXiv.org, 2015, arXiv:1501.04330 [cs.SY]. DOI: 10.1109/TAC.2015.2408554
- Khalil H.K., Praly L. High-gain observers in nonlinear feedback control. Int. J. of Robust and Nonlinear Control, 2014, vol. 24, iss. 6, pp. 993–1015. DOI: 10.1002/rnc.3051
- Mukhomorova O.Yu., Krishchenko A.P. Cancerous Tumour Model Analysis and Constructing schemes of Anti-angiogenesis Therapy at an Early Stage. Matematika i matematicheskoe modelirovanie. MGTU im. N.E. Baumana = Mathematics and Mathematical Modelling of the Bauman MSTU, 2015, no. 3, pp. 39–58. DOI: 10.7463/mathm.0315.0790877 (in Russian).
- Ergun A., Camphausen K., Wein L.M. Optimal scheduling of radiotheraphy and angiogenic inhibitor. Bulletin of Mathematical Biology, 2003, vol. 65, iss. 3, pp. 407–424. DOI: 10.1016/S0092-8240(03)00006-5
- Krasnoshchechenko V.I., Krishchenko A.P. Nelineinye sistemy: geometricheskie metody analiza i sinteza [Nonlinear systems: geometric methods of analysis and synthesis]. Moscow, Bauman MSTU Publ., 2005. 520 p. (in Russian).
- Isidori A. Nonlinear Control Systems. 3rd ed. London, Springer, 1995. 549 p. DOI: 10.1007/978-1-84628-615-5
- Zhevnin A.A., Krishchenko A.P., Glushko Yu.V. Controllability, observability of non-linear control-systems, and synthesis of terminal control. Doklady Akademii nauk SSSR, 1982, vol. 266, no. 4, pp. 807–811. (in Russian).
- Vinogradova M.S. Parametrical identification of a model of cooperating cellular populations on the basis of Bayesian approach. Nauka i obrazovanie MGTU im. N.E. Baumana} = Science and Education of the Bauman MSTU, 2012, no. 11, pp. 155–182. DOI: 10.7463/1112.0490900 (in Russian).
- Vinogradova M.S. Point Estimates and Probability Distribution of Mathematical Model Parameters of Evolving Cell Population System Taking into Account Contact Inhibition. Nauka i obrazovanie MGTU im. N.E. Baumana = Science and Education of the Bauman MSTU, 2015, no 11, pp. 406–425. DOI: 10.7463/1115.0826730 (in Russian).