Robust domain of attraction estimation for a tumor growth model. (English) Zbl 1510.92099

Summary: This paper deals with the estimation of regions of attraction (RoAs) for a cancer dynamical model. The estimation of this type of sets is important in the field of control for cancer dynamics, since it provides the set of possible initial health indicators, for which a treatment protocol exists allowing to heal the patient. In this paper, a methodology is proposed to estimate the region of attraction of a nonlinear dynamical system describing the interaction between a tumor, the immune system and combined therapies of cancer. A method for characterizing the RoA for a given model parameter vector is provided and employed in order to derive an outer approximation of the robust RoA under parametric uncertainties.


92C50 Medical applications (general)
34C60 Qualitative investigation and simulation of ordinary differential equation models
34H05 Control problems involving ordinary differential equations
93D20 Asymptotic stability in control theory
Full Text: DOI HAL


[1] de Pillis, L.; Radunskaya, A., A mathematical tumor model with immune resistance and drug therapy: an optimal control approach, J. Theor. Med., 3, 2, 79-100 (2001) · Zbl 0985.92023
[2] Martin, R., Optimal control drug scheduling of cancer chemotherapy, Automatica, 28, 6, 1113-1123 (1992)
[3] Murray, J. M., Optimal control cancer growth, Math. Biosci., 98, 273-287 (1990) · Zbl 0693.92009
[4] Afenya, E., Acute leukemia and chemotherapy: a modeling viewpoint, Math. Biosci., 138, 2, 79-100 (1996) · Zbl 0885.92026
[5] Matveev, A. S.; Savkin, A. V., Application of optimal control theory to analysis of cancer chemotherapy regimens, Syst. Control Lett., 46, 5, 311-321 (2002) · Zbl 1002.92012
[6] Bratus, A.; Samokhin, I.; Yegorov, I.; Yurchenko, D., Maximization of viability time in a mathematical model of cancer therapy, Math. Biosci., 294, 110-119 (2017) · Zbl 1382.92157
[7] Feizabadi, M., Modeling multi-mutation and drug resistance: analysis of some case studies, Theor. Biol. Med. Modell., 14, 6 (2017)
[8] Eftimie, R.; Gillard, J. J.; Cantrell, D. A., Mathematical models for immunology: current state of the art and future research directions, Bull. Math. Biol., 78, 10, 2091-2134 (2016) · Zbl 1361.92033
[9] Stepanova, N., Course of the immune reaction during the development of a malignant tumour, Biophysics, 24, 917-923 (1980)
[10] d’Onofrio, A.; Ledzewicz, U.; Schättler, H., On the Dynamics of Tumor-Immune System Interactions and Combined Chemo- and Immunotherapy, 249-266 (2012), Springer Milan · Zbl 1316.92020
[11] Ledzewicz, U.; Faraji, M., On optimal protocols for combinations of chemo- and immunotherapy, Proceedings of the 51st IEEE Conference on Decision and Control, 7492-7497 (2015), Hawaii, USA
[12] Chesi, G., Estimating the domain of attraction for uncertain polynomial systems, Automatica, 40, 1981-1986 (2004) · Zbl 1067.93055
[13] Alamo, T.; Cepeda, A.; Limon, D., Improved computation of ellipsoidal invariant sets for saturated control systems, Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (2005), Sevilla, Spain
[14] Blanchini, F., Set invariance in control, Automatica, 35, 11, 1747-1767 (1999) · Zbl 0935.93005
[15] Topcu, U.; Packard, A. K.; Seiler, P.; Balas, G. J., Robust region-of-attraction estimation, IEEE Trans. Autom. Control, 55, 1 (2010) · Zbl 1368.93510
[16] Blanchini, F., Ultimate boundedness control for discrete-time uncertain systems via set-induced Lyapunov functions, IEEE Trans. Autom. Control, 39, 428-433 (1994) · Zbl 0800.93754
[17] Blanchini, F.; Miani, S., Set-Theoretic Methods in Control (2008), Springer · Zbl 1140.93001
[18] Zhou, B.; Duan, G.; Lin, Z., Approximation and monotonicity of the maximal invariant ellipsoid for discrete-time systems by bounded controls, IEEE Trans. Autom. Control, 55, 2 (2010) · Zbl 1368.93486
[19] Alamo, T.; Cepeda, A.; Fiacchini, M.; Camacho, E. F., Convex invariant sets for discrete-time lure systems, Automatica, 45, 1066-1071 (2009) · Zbl 1162.93371
[20] Fiacchini, M.; Alamo, T.; Camacho, E. F., On the computation of convex robust control invariant sets for nonlinear systems, Automatica, 46, 8, 1334-1338 (2010) · Zbl 1205.93037
[21] Fiacchini, M., Convex difference inclusions for systems analysis and design (2012), Universidad de Sevilla: Universidad de Sevilla Spain, Ph.D. thesis
[22] Fiacchini, M.; Tarbouriech, S.; Prieur, C., Polytopic control invariant sets for differential inclusion systems : a viability theory approach, Proceedings of the 2011 American Control Conference ACC, 1218-1223 (2011)
[23] Fiacchini, M.; Alamo, T.; Camacho, E. F., Invariant sets computation for convex difference inclusions systems, Syst. Control Lett., 61, 8, 819-826 (2012) · Zbl 1251.93078
[24] Riah, R., Théorie des ensembles pour le contrôle robuste des systèmes non linaires: application à la chimiothrapie et les thérapies anti-angiogniques (2016), Communauté Université Grenoble Alpes, Ph.D. thesis
[25] Korda, M.; Henrion, D.; Jones, C. N., Inner approximations of the region of attraction for polynomial dynamical systems, IFAC Proc. Vol. (IFAC-PapersOnline), 9, 1, 534-539 (2013)
[26] Henrion, D.; Korda, M., Convex computation of the region of attraction of polynomial control systems, IEEE Trans. Autom. Control, 59, 2, 297-312 (2014) · Zbl 1360.93601
[27] Moussa, K.; Fiacchini, M.; Alamir, M., Robust optimal control-based design of combined chemo- and immunotherapy delivery profiles, Proceedings of the 8th IFAC Conference on Foundations of Systems Biology in Engineering (2019), Valencia, Spain
[28] Moussa, K.; Fiacchini, M.; Alamir, M., Robust optimal scheduling of combined chemo- and immunotherapy: considerations on chemotherapy detrimental effects, Proceedings of the 2020 American Control Conference (2020), Denver, USA
[29] Doban, A. I.; Lazar, M., Domain of attraction computation for tumor dynamics, Proceedings of the 53rd IEEE Conference on Decision and Control, 6987-6992 (2014)
[30] Zarei, M.; Javadi, K.; Kalhor, A., Perturbed tumor immunotherapy domain of attraction estimation via the arc-length function, Proceedings of the 2018 25th National and 3rd International Iranian Conference on Biomedical Engineering (ICBME), 1-6 (2018)
[31] Merola, A.; Cosentino, C.; Amato, F., An insight into tumor dormancy equilibrium via the analysis of its domain of attraction, Biomed. Signal Process. Control, 3, 3, 212-219 (2008)
[32] Riah, R.; Fiacchini, M.; Alamir, M., Iterative method for estimating the robust domains of attraction of non-linear systems: application to cancer chemotherapy model with parametric uncertainties, Eur. J. Control, 47, 64-73 (2019) · Zbl 1412.93038
[33] Ledzewicz, U.; Schättler, H., On the role of the objective in the optimization of compartmental models for biomedical therapies, J. Optim. Theory Appl. (2020) · Zbl 1462.49078
[34] Sharifi, N.; Ozgoli, S.; Ramezani, A., Multiple model predictive control for optimal drug administration of mixed immunotherapy and chemotherapy of tumours, Comput. Methods Programs Biomed., 144, 13-19 (2017)
[35] Sharifi, N.; Zhou, Y.; Holmes, C. Y., Overcoming channel uncertainties in touchable molecular communication for direct drug targeting assisted immuno- chemotherapy, IEEE Trans. Nanobiosci., 19, 2, 249-258 (2020)
[36] Francomano, E.; Hilker, F. M.; Paliaga, M.; Venturinoc, E., An efficient method to reconstruct invariant manifolds of saddle points, Dolomities Research Notes on Apptoximation, 10, 25-30 (2017), Padova University Press · Zbl 1370.34078
[37] Farjami, S.; Kirk, V.; Osinga, H. M., Computing the stable manifold of a saddle slow manifold, SIAM J. Appl. Dyn. Syst., 17, 1, 350-379 (2018) · Zbl 1403.37036
[38] Netzer, T.; Plaumann, D.; Thom, A., Determinantal representations and the hermite matrix, Michigan Math. J., 62, 407-420 (2013) · Zbl 1273.15005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.