The Application of Power Series Expansion to Optimal Control of an Immune-Oncology Nonlinear Dynamic Problem

Document Type : Research Article


1 Faculty of Electrical, Biomedical and Mechatronic, Qazvin Branch, Islamic Azad University, Qazvin, Iran

2 Department of Electrical, Biomedical and Mechatronics Engineering, Qazvin Branch, Islamic Azad University

3 Department of Electrical Engineering, Amirkabir University of Technology, Hafez Ave.,15875-4413, Tehran, Iran

4 Qazvin Branch, Islamic Azad University


This paper is concerned with the eradication of tumor cells in the human body by defining an optimal protocol using a polynomial approximation technique for the injection of chemotherapy drugs. The dynamics of the system are described based on immune-oncology. Variation of host, tumor, and immune cells’ populations are studied in the model during the injection of the chemotherapeutic drugs. The objective is the minimization of cancerous cells' average population by minimum drug injection to avoid the destructive side-effects of these chemotherapeutic substances. It should be done by stabilizing the population of host and immune cells around a free-tumor desirable health condition. This optimization problem by considering the nonlinear model of the system makes applying nonlinear optimal control inevitable. Solving Hamilton-Jacobi-Bellman (HJB) nonlinear partial differential equation (PDE) for the system is put into our perspective to cope with this problem. Since the dynamics of the system are not polynomial, it comprises fractional terms, this PDE cannot be solved straightforwardly. We take advantage of the power series expansion technique to approximate the solution of the PDE with satisfactory accuracy. Finally, a series of simulations are carried out to prove the capability of the controller in terms of robustness and sensitivity, increasing convergence rate for the elimination of cancerous cells, and enlargement of the domain of attraction.


Main Subjects

[2] J.A. Adam, N. Bellomo, A survey of models for tumor-immune system dynamics, Springer Science & Business Media, 2012.
[3] A. Swierniak, M. Kimmel, J. Smieja, Mathematical modeling as a tool for planning anticancer therapy, European journal of pharmacology, 625(1-3) (2009) 108-121.
[4] M. Marušić, Ž. Bajzer, J. Freyer, S. Vuk‐Pavlović, Analysis of growth of multicellular tumour spheroids by mathematical models, Cell proliferation, 27(2) (1994) 73-94.
[5] V.A. Kuznetsov, I.A. Makalkin, M.A. Taylor, A.S. Perelson, Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis, Bulletin of mathematical biology, 56(2) (1994) 295-321.
[6] L.G. De Pillis, A. Radunskaya, A mathematical tumor model with immune resistance and drug therapy: an optimal control approach, Computational and Mathematical Methods in Medicine, 3(2) (2001) 79-100.
[7] L.G. De Pillis, A. Radunskaya, The dynamics of an optimally controlled tumor model: A case study, Mathematical and computer modelling, 37(11) (2003) 1221-1244.
[8] A. GHAFARI, N. Naserifar, Mathematical modeling and lyapunov-based drug administration in cancer chemotherapy,  (2009).
[9] D. Kirschner, J.C. Panetta, Modeling immunotherapy of the tumor–immune interaction, Journal of mathematical biology, 37(3) (1998) 235-252.
[10] L.G. de Pillis, W. Gu, A.E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations, Journal of theoretical biology, 238(4) (2006) 841-862.
[11] A. Ashyani, H. Mohammadinejad, O. RabieiMotlagh, Hopf bifurcation analysis in a delayed system for cancer virotherapy, Indagationes Mathematicae, 27(1) (2016) 318-339.
[12] M. Itik, M.U. Salamci, S.P. Banks, SDRE optimal control of drug administration in cancer treatment, Turkish Journal of Electrical Engineering & Computer Sciences, 18(5) (2010) 715-730.
[13] D.A. Drexler, L. Kovács, J. Sápi, I. Harmati, Z. Benyó, Model-based analysis and synthesis of tumor growth under angiogenic inhibition: a case study, IFAC Proceedings Volumes, 44(1) (2011) 3753-3758.
[14] G.W. Swan, Role of optimal control theory in cancer chemotherapy, Mathematical biosciences, 101(2) (1990) 237-284.
[15] H.K. Khalil, J.W. Grizzle, Nonlinear systems, Prentice hall Upper Saddle River, NJ, 2002.
[16] T.-L. Chien, C.-C. Chen, C.-J. Huang, Feedback linearization control and its application to MIMO cancer immunotherapy, IEEE Transactions on Control Systems Technology, 18(4) (2009) 953-961.
[17] H. Moradi, M. Sharifi, G. Vossoughi, Adaptive robust control of cancer chemotherapy in the presence of parametric uncertainties: A comparison between three hypotheses, Computers in biology and medicine, 56 (2015) 145-157.
[18] N. Babaei, M.U. Salamci, Personalized drug administration for cancer treatment using model reference adaptive control, Journal of theoretical biology, 371 (2015) 24-44.
[19] H. Nasiri, A.A. Kalat, Adaptive fuzzy back-stepping control of drug dosage regimen in cancer treatment, Biomedical Signal Processing and Control, 42 (2018) 267-276.
[20] A. Aghaeeyan, M.J. Yazdanpanah, J. Hadjati, A New Tumor-Immunotherapy Regimen based on Impulsive Control Strategy, Biomedical Signal Processing and Control, 57 (2020) 101763.
[21] U. Ledzewicz, H. Schättler, Optimal bang-bang controls for a two-compartment model in cancer chemotherapy, Journal of optimization theory and applications, 114(3) (2002) 609-637.
[22] T.N. Burden, J. Ernstberger, K.R. Fister, Optimal control applied to immunotherapy, Discrete and Continuous Dynamical Systems Series B, 4(1) (2004) 135-146.
[23] L.G. De Pillis, K.R. Fister, W. Gu, C. Collins, M. Daub, Seeking bang-bang solutions of mixed immuno-chemotherapy of tumors,  (2007).
[24] Y. Batmani, H. Khaloozadeh, Optimal chemotherapy in cancer treatment: state dependent Riccati equation control and extended Kalman filter, Optimal Control Applications and Methods, 34(5) (2013) 562-577.
[25] D.E. Kirk, Optimal control theory: an introduction, Courier Corporation, 2004.
[26] A. Fakharian, M. Hamidi Beheshti, A. Davari, Solving the Hamilton–Jacobi–Bellman equation using Adomian decomposition method, International Journal of Computer Mathematics, 87(12) (2010) 2769-2785.
[27] C.L. Navasca, A.J. Krener, Solution of hamilton jacobi bellman equations, in:  Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No. 00CH37187), IEEE, 2000, pp. 570-574.
[28] T. Hunt, A.J. Krener, Improved patchy solution to the Hamilton-Jacobi-Bellman equations, in:  49th IEEE Conference on Decision and Control (CDC), IEEE, 2010, pp. 5835-5839.
[29] R.W. Beard, G.N. Saridis, J.T. Wen, Galerkin approximations of the generalized Hamilton-Jacobi-Bellman equation, Automatica, 33(12) (1997) 2159-2177.
[30] M. Sassano, A. Astolfi, Dynamic solution of the HJB equation and the optimal control of nonlinear systems, in:  49th IEEE Conference on Decision and Control (CDC), IEEE, 2010, pp. 3271-3276.
[31] M. Sassano, A. Astolfi, Dynamic approximate solutions of the HJ inequality and of the HJB equation for input-affine nonlinear systems, IEEE Transactions on Automatic Control, 57(10) (2012) 2490-2503.
[32] M. Nazari Monfared, M. Yazdanpanah, Friction Compensation for Dynamic and Static Models Using Nonlinear Adaptive Optimal Technique, AUT Journal of Modeling and Simulation, 46(1) (2014) 1-10.
[33] M.N. Monfared, M.H. Dolatabadi, A. Fakharian, Nonlinear optimal control of magnetic levitation system based on HJB equation approximate solution, in:  2014 22nd Iranian Conference on Electrical Engineering (ICEE), IEEE, 2014, pp. 1360-1365.
[34] R.M. Milasi, M.J. Yazdanpanah, C. Lucas, Nonlinear optimal control of washing machine based on approximate solution of HJB equation, Optimal Control Applications and Methods, 29(1) (2008) 1-18.
[35] Z. Bajzer, M. Marušić, S. Vuk-Pavlović, Conceptual frameworks for mathematical modeling of tumor growth dynamics, Mathematical and computer modelling, 23(6) (1996) 31-46.
[36] A. Tsoularis, J. Wallace, Analysis of logistic growth models, Mathematical biosciences, 179(1) (2002) 21-55.
[37] E.A. Sarapata, L. de Pillis, A comparison and catalog of intrinsic tumor growth models, Bulletin of mathematical biology, 76(8) (2014) 2010-2024.
[38] A. Merola, C. Cosentino, F. Amato, An insight into tumor dormancy equilibrium via the analysis of its domain of attraction, Biomedical Signal Processing and Control, 3(3) (2008) 212-219.
[39] A. Dini, M. Yazdanpanah, Estimation of the domain of attraction of free tumor equilibrium point of perturbed tumor immunotherapy model, in:  2016 4th International Conference on Control, Instrumentation, and Automation (ICCIA), IEEE, 2016, pp. 69-74.
[40] F.L. Lewis, D. Vrabie, V.L. Syrmos, Optimal control, John Wiley & Sons, 2012.
[41] Y. Batmani, M. Davoodi, N. Meskin, Nonlinear suboptimal tracking controller design using state-dependent Riccati equation technique, IEEE Transactions on Control Systems Technology, 25(5) (2016) 1833-1839.