# American Institute of Mathematical Sciences

June  2013, 18(4): 1031-1051. doi: 10.3934/dcdsb.2013.18.1031

## Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost

 1 Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653, United States 2 Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri, 63130-4899

Received  December 2011 Revised  April 2012 Published  February 2013

An optimal control problem for combination of cancer chemotherapy with immunotherapy in form of a boost to the immune system is considered as a multi-input optimal control problem. The objective to be minimized is chosen as a weighted average of (i) the number of cancer cells at the terminal time, (ii) a measure for the immunocompetent cell densities at the terminal point (included as a negative term), the overall amounts of (iii) cytotoxic agents and (iv) immune boost given as a measure for the side effects of treatment and (v) a small penalty on the free terminal time that limits the overall therapy horizon. This last term is essential in obtaining a mathematically well-posed problem formulation. Both analytical and numerical results about the structures of optimal controls will be presented that give some insights into the structure of optimal protocols, i.e., the dose rates and sequencing of drugs in these combination treatments.
Citation: Urszula Ledzewicz, Mozhdeh Sadat Faraji Mosalman, Heinz Schättler. Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 1031-1051. doi: 10.3934/dcdsb.2013.18.1031
##### References:
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Schättler, Bang-bang and singular controls in a mathematical model for combined anti-angiogenic and chemotherapy treatments, Proc. 48th IEEE Conf. on Dec. and Contr., Shanghai, China, (2009), 2280-2285. Google Scholar [17] U. Ledzewicz, M. Naghnaeian and H. Schättler, An optimal control approach to cancer treatment under immunological activity, Applicationes Mathematicae, 38 (2011), 17-31. doi: 10.4064/am38-1-2.  Google Scholar [18] U. Ledzewicz, M. Naghnaeian and H. Schättler, Bifurcation of singular arcs in an optimal control problem for cancer immune system interactions under treatment, Proceedings of the 49th IEEE Conf. on Decision and Control, Atlanta, USA, (2010), 7039-7044. Google Scholar [19] U. Ledzewicz, M. Naghnaeian and H. Schättler, Dynamics of tumor-immune interactions under treatment as an optimal control problem, Proc. of the 8th AIMS Conf., Dresden, Germany, (2010), 971-980. Google Scholar [20] U. Ledzewicz, M. Naghnaeian and H. Schättler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics, J. of Mathematical Biology, 64 (2012), 557-577. doi: 10.1007/s00285-011-0424-6.  Google Scholar [21] U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy, J. of Biological Systems, 10 (2002), 183-206. Google Scholar [22] U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Control and Optimization, 46 (2007), 1052-1079. doi: 10.1137/060665294.  Google Scholar [23] U. Ledzewicz and H. Schättler, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis, J. of Theoretical Biology, 252 (2008), 295-312. Google Scholar [24] L. Norton and R. Simon, Growth curve of an experimental solid tumor following radiotherapy, J. of the National Cancer Institute, 58 (1977), 1735-1741. Google Scholar [25] L. Norton, A Gompertzian model of human breast cancer growth, Cancer Research, 48 (1988), 7067-7071. Google Scholar [26] A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences, Physica D, 208 (2005), 220-235 doi: 10.1016/j.physd.2005.06.032.  Google Scholar [27] A. d'Onofrio, Tumor-immune system interaction: modeling the tumor-stimulated proliferation of effectors and immunotherapy, Math. Models and Methods in Applied Sciences, 16 (2006), 1375-1401. doi: 10.1142/S0218202506001571.  Google Scholar [28] A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On Optimal Delivery of Combination Therapy for Tumors, Mathematical Biosciences, 222 (2009), 13-26. doi: 10.1016/j.mbs.2009.08.004.  Google Scholar [29] L. G. de Pillis and A. Radunskaya, A mathematical tumor model with immune resistance and drug therapy: an optimal control approach, J. of Theoretical Medicine, 3 (2001), 79-100. Google Scholar [30] A. V. Rao, D. A. Benson, G. T. Huntington, C. Francolin, C. L. Darby and M. A. Patterson, "User's Manual for GPOPS: A MATLAB Package for Dynamic Optimization Using the Gauss Pseudospectral Method," University of Florida Report, 2008. Google Scholar [31] H. Schättler and U. Ledzewicz, "Geometric Optimal Control: Theory, Methods and Examples," Springer Verlag, 2012. doi: 10.1007/978-1-4614-3834-2.  Google Scholar [32] N.V. Stepanova, Course of the immune reaction during the development of a malignant tumour, Biophysics, 24 (1980), 917-923. Google Scholar [33] G. W. Swan, Role of optimal control in cancer chemotherapy, Mathematical Biosciences, 101 (1990), 237-284. Google Scholar [34] A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, Proceedings of the 12th IMACS World Congress, Paris, 4 (1988), 170-172.  Google Scholar [35] A. Swierniak, Cell cycle as an object of control, J. of Biological Systems, 3 (1995), 41-54. Google Scholar [36] A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. Applied Mathematics and Computer Science, 13 (2003), 357-368.  Google Scholar [37] H. P. de Vladar and J. A. González, Dynamic response of cancer under the influence of immunological activity and therapy, J. of Theoretical Biology, 227 (2004), 335-348. doi: 10.1016/j.jtbi.2003.11.012.  Google Scholar

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##### References:
 [1] D. J. Bell and D. H. Jacobson, "Singular Optimal Control Problems," Academic Press, 1975.  Google Scholar [2] D. A. Benson, "A Gauss Pseudospectral Transcription for Optimal Control," Ph.D. thesis, MIT, 2004. Google Scholar [3] D. A. Benson, G. T. Huntington, T. P. Thorvaldsen and A. V. Rao, Direct trajectory optimization and costate estimation via an orthogonal collocation method, J. of Guidance, Control, and Dynamics, 29 (2006), 1435-1440. Google Scholar [4] B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory," Springer Verlag, Series: Mathematics and Applications, Vol. 40, 2003.  Google Scholar [5] A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control," American Institute of Mathematical Sciences, 2007.  Google Scholar [6] T. Burden, J. Ernstberger and K. R. Fister, Optimal control applied to immunotherapy Discrete and Continuous Dynamical Systems - Series B, 4 (2004), 135-146.  Google Scholar [7] F. Castiglione and B. Piccoli, Optimal control in a model of dendritic cell transfection cancer immunotherapy, Bulletin of Mathematical Biology, 68 (2006), 255-274. doi: 10.1007/s11538-005-9014-3.  Google Scholar [8] M. Eisen, "Mathematical Models in Cell Biology and Cancer Chemotherapy," Lecture Notes in Biomathematics, Vol. 30, Springer Verlag, 1979.  Google Scholar [9] A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bulletin of Mathematical Biology, 65 (2003), 407-424. Google Scholar [10] K. R. Fister and J. Hughes Donnelly, Immunotherapy: an optimal control approach, Mathematical Biosciences and Engineering (MBE), 2 2005, 499-510. doi: 10.3934/mbe.2005.2.499.  Google Scholar [11] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer Verlag, New York, 1983.  Google Scholar [12] G. T. Huntington, "Advancement and Analysis of a Gauss Pseudospectral Transcription for Optimal Control," Ph.D. thesis, MIT, 2007. Google Scholar [13] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. of Mathematical Biology, 37 (1998), 235-252. Google Scholar [14] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis, Bull. Mathematical Biology, 56 (1994), 295-321. Google Scholar [15] U. Ledzewicz, J. Marriott, H. Maurer and H. Schättler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Math. Medicine and Biology, 27 (2010), 157-179. doi: 10.1093/imammb/dqp012.  Google Scholar [16] U. Ledzewicz, H. Maurer and H. Schättler, Bang-bang and singular controls in a mathematical model for combined anti-angiogenic and chemotherapy treatments, Proc. 48th IEEE Conf. on Dec. and Contr., Shanghai, China, (2009), 2280-2285. Google Scholar [17] U. Ledzewicz, M. Naghnaeian and H. Schättler, An optimal control approach to cancer treatment under immunological activity, Applicationes Mathematicae, 38 (2011), 17-31. doi: 10.4064/am38-1-2.  Google Scholar [18] U. Ledzewicz, M. Naghnaeian and H. Schättler, Bifurcation of singular arcs in an optimal control problem for cancer immune system interactions under treatment, Proceedings of the 49th IEEE Conf. on Decision and Control, Atlanta, USA, (2010), 7039-7044. Google Scholar [19] U. Ledzewicz, M. Naghnaeian and H. Schättler, Dynamics of tumor-immune interactions under treatment as an optimal control problem, Proc. of the 8th AIMS Conf., Dresden, Germany, (2010), 971-980. Google Scholar [20] U. Ledzewicz, M. Naghnaeian and H. Schättler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics, J. of Mathematical Biology, 64 (2012), 557-577. doi: 10.1007/s00285-011-0424-6.  Google Scholar [21] U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy, J. of Biological Systems, 10 (2002), 183-206. Google Scholar [22] U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Control and Optimization, 46 (2007), 1052-1079. doi: 10.1137/060665294.  Google Scholar [23] U. Ledzewicz and H. Schättler, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis, J. of Theoretical Biology, 252 (2008), 295-312. Google Scholar [24] L. Norton and R. Simon, Growth curve of an experimental solid tumor following radiotherapy, J. of the National Cancer Institute, 58 (1977), 1735-1741. Google Scholar [25] L. Norton, A Gompertzian model of human breast cancer growth, Cancer Research, 48 (1988), 7067-7071. Google Scholar [26] A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences, Physica D, 208 (2005), 220-235 doi: 10.1016/j.physd.2005.06.032.  Google Scholar [27] A. d'Onofrio, Tumor-immune system interaction: modeling the tumor-stimulated proliferation of effectors and immunotherapy, Math. Models and Methods in Applied Sciences, 16 (2006), 1375-1401. doi: 10.1142/S0218202506001571.  Google Scholar [28] A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On Optimal Delivery of Combination Therapy for Tumors, Mathematical Biosciences, 222 (2009), 13-26. doi: 10.1016/j.mbs.2009.08.004.  Google Scholar [29] L. G. de Pillis and A. Radunskaya, A mathematical tumor model with immune resistance and drug therapy: an optimal control approach, J. of Theoretical Medicine, 3 (2001), 79-100. Google Scholar [30] A. V. Rao, D. A. Benson, G. T. Huntington, C. Francolin, C. L. Darby and M. A. Patterson, "User's Manual for GPOPS: A MATLAB Package for Dynamic Optimization Using the Gauss Pseudospectral Method," University of Florida Report, 2008. Google Scholar [31] H. Schättler and U. Ledzewicz, "Geometric Optimal Control: Theory, Methods and Examples," Springer Verlag, 2012. doi: 10.1007/978-1-4614-3834-2.  Google Scholar [32] N.V. Stepanova, Course of the immune reaction during the development of a malignant tumour, Biophysics, 24 (1980), 917-923. Google Scholar [33] G. W. Swan, Role of optimal control in cancer chemotherapy, Mathematical Biosciences, 101 (1990), 237-284. Google Scholar [34] A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, Proceedings of the 12th IMACS World Congress, Paris, 4 (1988), 170-172.  Google Scholar [35] A. Swierniak, Cell cycle as an object of control, J. of Biological Systems, 3 (1995), 41-54. Google Scholar [36] A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. Applied Mathematics and Computer Science, 13 (2003), 357-368.  Google Scholar [37] H. P. de Vladar and J. A. González, Dynamic response of cancer under the influence of immunological activity and therapy, J. of Theoretical Biology, 227 (2004), 335-348. doi: 10.1016/j.jtbi.2003.11.012.  Google Scholar
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