Chemotherapy for tumors: An analysis of the dynamics and a study of quadratic and linear optimal controls. (English) Zbl 1122.49020

Summary: We investigate a mathematical model of tumor-immune interactions with chemotherapy, and strategies for optimally administering treatment. In this paper we analyze the dynamics of this model, characterize the optimal controls related to drug therapy, and discuss numerical results of the optimal strategies. The form of the model allows us to test and compare various optimal control strategies, including a quadratic control, a linear control, and a state-constraint. We establish the existence of the optimal control, and solve for the control in both the quadratic and linear case. In the linear control case, we show that we cannot rule out the possibility of a singular control. An interesting aspect of this paper is that we provide a graphical representation of regions on which the singular control is optimal.


49K15 Optimality conditions for problems involving ordinary differential equations
92C50 Medical applications (general)
93A30 Mathematical modelling of systems (MSC2010)
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49N90 Applications of optimal control and differential games
Full Text: DOI


[1] L. Bannock, Nutrition. Available from: <http://www.doctorbannock.com/nutrition.html>.
[2] Bellomo, N.; Bellouquid, A.; Delitala, M., Mathematical topics on the modelling of multicellular systems in competition between tumor and immune cells, Math. mod. meth. appl. sci., 14, 1683, (2004) · Zbl 1060.92029
[3] Bellomo, N.; Preziosi, L., Modelling and mathematical problems related to tumor evolution and its interaction with the immune system, Math. comput. model., 32, 413, (2000) · Zbl 0997.92020
[4] Blattman, J.N.; Greenberg, P.D., Cancer immunotherapy: a treatment for the masses, Science, 305, July, 200, (2004)
[5] Burden, Thalya; Ernstberger, Jon; Renee Fister, K., Optimal control applied to immunotherapy, Discrete contin. dyn. syst. ser. B, 4, 1, (2004) · Zbl 1042.92013
[6] ()
[7] Chandawarkar, R.Y.; Guyton, D.P., Oncologic mathematics – evolution of a new specialty, Arch surg., 137, 1428, (2002)
[8] Chester, K.A.; Mayer, A.; Bhatia, J.; Robson, L.; Spencer, D.I.R.; Cooke, S.P.; Flynn, A.A.; Sharma, S.K.; Boxer, G.; Pedley, R.B.; Begent, R.H.J., Recombinant anti-carcinoembryonic antigen antibodies for targeting cancer, Cancer chemother. pharmacol., 46, S8, (2000)
[9] Coldman, Andrew J.; Murray, J.M., Optimal control for a stochastic model of cancer chemotherapy, Math. biosci., 168, 2, 187, (2000) · Zbl 0961.92019
[10] Couzin, J., Select T cells, given space, shrink tumors, Science, 297, September, 1973, (2002)
[11] Cui, S.B., Analysis of a mathematical model for the growth of tumors under the action of external inhibitors, J. math. biol., 44, 395, (2002) · Zbl 1019.92017
[12] Dalgleish, A., The relevance of non-linear mathematics (chaos theory) to the treatment of cancer, the role of the immune response and the potential for vaccines, Qjm, 92, 347, (1999)
[13] De Angelis, E.; Jabin, P.E., Qualitative analysis of a Mean field model of tumor – immune system competition, Math. mod. meth. appl. sci., 13, 187, (2003) · Zbl 1043.92012
[14] De Angelis, E.; Mesin, L., Modelling of the immune response: conceptual frameworks and applications, Math. mod. meth. appl. sci., 11, 1609, (2001) · Zbl 1013.92016
[15] de Pillis, L.G.; Gu, W.; Radunskaya, A.E., Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations, J. theor. biol., 238, 4, 841, (2006)
[16] de Pillis, L.G.; Radunskaya, A.E., A mathematical tumor model with immune resistance and drug therapy: an optimal control approach, J. theor. med., 3, 79, (2001) · Zbl 0985.92023
[17] de Pillis, L.G.; Radunskaya, A.E., The dynamics of an optimally controlled tumor model: a case study, Math. comput. model., 37, 11, 1221, (2003) · Zbl 1043.92018
[18] de Pillis, L.G.; Radunskaya, A.E.; Wiseman, C.L., A validated mathematical model of cell-mediated immune response to tumor growth, Cancer res., 61, 17, 7950, (2005)
[19] Delitala, M., Critical analysis and perspectives on kinetic (cellular) theory of immune competition, Math. comput. model., 35, 63, (2002) · Zbl 1003.92019
[20] Derbel, L., Analysis of a new model for tumor – immune system competition including long time scale effects, Math. mod. meth. appl. sci., 14, 1657, (2004) · Zbl 1057.92036
[21] Diefenbach, A.; Jensen, E.R.; Jamieson, A.M.; Raulet, D., Rae1 and H60 ligands of the NKG2D receptor stimulate tumor immunity, Nature, 413, September, 165, (2001)
[22] Donnelly, J., Cancer vaccine targets leukemia, Nat. med., 9, 11, 1354, (2003)
[23] d’Onofrio, A., A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical references, Physica D, 208, 220, (2005) · Zbl 1087.34028
[24] Ermentrout, Bard, Simulating, analyzing, and animating dynamical systems: A guide to XPPAUT for researchers and students, (2002), SIAM · Zbl 1003.68738
[25] Ferreira, S.C.; Martins, M.L.; Vilela, M.J., Reaction-diffusion model for the growth of avascular tumor, Phys. rev. E, 65, (2002)
[26] Fister, K.R.; Donnelly, J., Immunotherapy: an optimal control theory approach, Math. biosci. eng., 2, 3, 499, (2005) · Zbl 1180.92038
[27] Renee Fister, K.; Panetta, John Carl, Optimal control applied to cell-cycle-specific cancer chemotherapy, SIAM J. appl. math., 60, 3, 1059, (2000) · Zbl 0991.92014
[28] Renee Fister, K.; Panetta, John Carl, Optimal control applied to competing chemotherapeutic cell-kill strategies, SIAM J. appl. math., 63, 6, 1954, (2003) · Zbl 1058.92025
[29] Fleming, Wendell H.; Rishel, Raymond W., Deterministic and stochastic optimal control, (1975), Springer-Verlag · Zbl 0323.49001
[30] Friedrich, S.W.; Lin, S.C.; Stoll, B.R.; Baxter, L.T.; Munn, L.L.; Jain, R.K., Antibody-directed effector cell therapy of tumors: analysis and optimization using a physiologically based pharmacokinetic model, Neoplasia, 4, 449, (2002)
[31] Garcia-Penarrubia, P.; Lorenzo, N.; Galvez, J.; Campos, A.; Ferez, X.; Rubio, G., Study of the physical meaning of the binding parameters involved in effector-target conjugation using monoclonal antibodies against adhesion molecules and cholera toxin, Cell immunol., 215, 141, (2002)
[32] Gatenby, R.A.; Gawlinski, E.T., The glycolytic phenotype in carcinogenesis and tumor invasion: insights through mathematical models, Cancer res., 63, 3847, (2003)
[33] Gatenby, R.A.; Maini, P.K., Modelling a new angle on understanding cancer, Nature, 410, December, 462, (2002)
[34] Gurney, H., How to calculate the dose of chemotherapy, Brit. J. cancer, 86, 1297, (2002)
[35] Hardy, K.; Stark, J., Mathematical models of the balance between apoptosis and proliferation, Apoptosis, 7, 373, (2002)
[36] B. Hauser, Blood tests. Technical report, International Waldenstrom’s Macroglobulinemia Foundation. Available from: <http://www.iwmf.com/Blood_Tests.pdf>, January 2001 (accessed May 2005).
[37] Janeway, C.A.; Travers, P.; Walport, M.; Shlomchik, M.J., Immunobiology: the immune system in health and disease, (2005), Garland Science Publishing
[38] L.S. Jennings, M.E. Fisher, K.L. Teo, and C.J. Goh, MISER3 Optimal Control Software: Theory and User Maual. Department of Mathematics, The University of Western Australia, Nedlands, WA 6907, Australia, 2004. Version 3. Available from: <http://www.cado.uwa.edu.au/miser/>.
[39] Kamien, M.I.; Schwartz, N.L., Dynamic optimization: the calculus of variations and optimal control in economics and management, Advanced textbooks in economics, vol. 31, (1991), North-Holland · Zbl 0709.90001
[40] Keil, D.; Luebke, R.W.; Pruett, S.B., Quantifying the relationship between multiple immunological parameters and host resistance: probing the limits of reductionism, J. immunol., 167, 4543, (2001)
[41] Kirschner, Denise; Panetta, John Carl, Modeling immunotherapy of the tumor – immune interaction, J. math. biol., 37, 235, (1998) · Zbl 0902.92012
[42] Kolev, M., Mathematical modelling of the competition between tumors and immune system considering the role of the antibodies, Math. comput. model., 37, 1143, (2003) · Zbl 1043.92013
[43] Krener, Arthur J., The high order maximal principle and its application to singular extremals, SIAM J. control optim., 15, 2, 256, (1977) · Zbl 0354.49008
[44] Kuznetsov, V.; Knott, G.D., Modeling tumor regrowth and immunotherapy, Math. comp. model., 33, 1275, (2001) · Zbl 1004.92021
[45] Kuznetsov, V.; Makalkin, I.; Taylor, M.; Perelson, A., Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis, Bull. math. bio., 56, 2, 295, (1994) · Zbl 0789.92019
[46] Kuznetsov, V.A.; Knott, G.D., Modeling tumor regrowth and immunotherapy, Math. comput. model., 33, 1275, (2001) · Zbl 1004.92021
[47] Ledzewicz, Urszula; Brown, Tim; Schattler, Heinz, Comparison of optimal controls for a model in cancer chemotherapy with l_{1} and l2 type objectives, Optim. meth. software, 19, 3-4, 339, (2004) · Zbl 1056.92033
[48] Ledzewicz, Urszula; Schattler, Heinz, Drug resistance in cancer chemotherapy as an optimal control problem, Discrete contin. dynam. syst. ser. B, 6, 1, 129, (2006) · Zbl 1088.92040
[49] Lucia, U.; Maino, G., Thermodynamical analysis of the dynamics of tumor interaction with the host immune system, Physica A, 313, 569, (2002) · Zbl 0998.92501
[50] Lukes, D.L., Differential equations: classical to controlled, vol. 162, (1982), Academic Press · Zbl 0509.34003
[51] Maplesoft. Maple. Version 9. Available from: <http://www.maplesoft.com/>.
[52] Marincola, F.M.; Wang, E.; Herlyn, M.; Seliger, B.; Ferrone, S., Tumors as elusive targets of T-cell-based active immunotherapy, Trends immunol., 24, 335, (2003)
[53] The Mathworks. MATLAB. Version 7. Available from: <http://www.mathworks.com/>.
[54] Matveev, Alexey; Savkin, Andrey, Application of optimal control theory to analysis of cancer chemotherapy regimens, Syst. control lett., 46, 311, (2002) · Zbl 1002.92012
[55] Murray, J.M., Some optimality control problems in cancer chemotherapy with a toxicity limit, Math. biosci., 100, 49, (1990) · Zbl 0778.92012
[56] Nani, F.; Freedman, H.I., A mathematical model of cancer treatment by immunotherapy, Math. biosci., 163, 159, (2000) · Zbl 0997.92024
[57] Owen, M.R.; Sherratt, J.A., Mathematical modelling of macrophage dynamics in tumours, Math. mod. meth. appl. sci., 9, 513, (1999) · Zbl 0932.92019
[58] Pardoll, D.M., Cancer vaccines, Nat. med., 4, 5, 525, (1998), (Vaccine Supplement)
[59] ()
[60] Pontryagin, L.S.; Boltyanskii, V.G.; Gamkrelidze, R.V.; Mishchenko, E.F., The mathematical theory of optimal processes, (1962), Gordon and Breach · Zbl 0102.32001
[61] Rosenberg, S.A.; Yang, J.C.; Restifo, N.P., Cancer immunotherapy: moving beyond current vaccines, Nat. med., 10, 9, 909, (2004)
[62] A. Schwartz, E. Polak, Y. Chen, RIOTS: Recursive Integration Optimal Trajectory Solver. A Matlab Toolbox for Solving Optimal Control Problems, 1997. Version 1 for Windows. Available from: <http://www.schwartz-home.com/adam/RIOTS/>.
[63] Sharma, R.A.; Dalgleish, A.G.; Steward, W.P.; O’Byrne, K.J., Angiogenesis and the immune response as targets for the prevention and treatment of colorectal cancer (review), Oncol. rep., 10, 1625, (2003)
[64] Sotolongo-Costa, O.; Molina, L.M.; Perez, D.R.; Antoranz, J.C.; Reyes, M.C., Behavior of tumors under nonstationary therapy, Physica D, 178, 242, (2003) · Zbl 1011.92028
[65] Stengel, R.F.; Ghigliazza, R.; Kulkarni, N.; Laplace, O., Optimal control of innate immune response, Optim. control appl. meth., 23, 91, (2002) · Zbl 1072.92510
[66] Swierniak, Andrzej; Ledzewicz, Urszula; Schattler, Heinz, Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. appl. math. comput. sci., 13, 3, 357, (2003) · Zbl 1052.92032
[67] Szymanska, Zuzanna, Analysis of immunotherapy models in the context of cancer dynamics, Int. J. appl. math. comput. sci., 13, 3, 407, (2003) · Zbl 1035.92023
[68] Takayanagi, T.; Ohuchi, A., A mathematical analysis of the interactions between immunogenic tumor cells and cytotoxic T lymphocytes, Microbiol. immunol., 45, 709, (2001)
[69] O. von Stryk, Numerical solution of constrained optimal control problems by DIRect COLlocation. Lehrstuhl M2 Numerische Mathematik, Technische Universitaet Muenchen, 1999. Version 2.1. Available from: <http://www.sim.informatik.tu-darmstadt.de/sw/dircol/dircol.html>.
[70] Wallace, R.; Wallace, D.; Wallace, R.G., Toward cultural oncology: the evolutionary information dynamics of cancer, Open syst. inf. dyn., 10, 159, (2003) · Zbl 1030.92018
[71] Webb, S.D.; Sherratt, J.A.; Fish, R.G., Cells behaving badly: a theoretical model for the fas/fasl system in tumour immunology, Math. biosci., 179, 113, (2002) · Zbl 1014.92022
[72] Wein, L.M.; Wu, J.T.; Kirn, D.H., Validation and analysis of a mathematical model of a replication-competent oncolytic virus for cancer treatment: implications for virus design and delivery, Cancer res., 63, 1317, (2003)
[73] Wheeler, C.J.; Asha, D.; Gentao, L.; Yu, J.S.; Black, K.L., Clinical responsiveness of glioblastoma multiforme to chemotherapy after vaccination, Clin. cancer res., 10, 5316, (2004)
[74] Wodarz, D., Viruses as antitumor weapons: defining conditions for tumor remission, Cancer res., 61, 3501, (2001)
[75] Wodarz, D.; Jansen, V.A.A., A dynamical perspective of ctl cross-priming and regulation: implications for cancer immunology, Immunol. lett., 86, 213, (2003)
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.