Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Immunotherapy: An Optimal Control Theory Approach

1. Department of Mathematics and Statistics, Murray State University, 6C Faculty Hall, Murray, KY 42071

We investigate mathematical models for the dynamics between tumor cells, immune-effector cells, and cytokine interleukin-2 (IL-2). To better determine under what circumstances the tumor can be eliminated, we implement optimal control theory. We design two control functionals, the first functional having one control and the second having two controls, to maximize the effector cells and interleukin-2 concentration and to minimize the tumor cells. Next, we show that bang-bang optimal controls exist for each problem. Then, we characterize our optimal controls in terms of the solutions to the optimality system, which is the state system coupled with the adjoint system. Finally, we analyze the various optimal controls and optimality systems using numerical techniques.
  Article Metrics

Keywords optimal control; immunotherapy; ordinary differential system.; cancer

Citation: K. Renee Fister, Jennifer Hughes Donnelly. Immunotherapy: An Optimal Control Theory Approach. Mathematical Biosciences and Engineering, 2005, 2(3): 499-510. doi: 10.3934/mbe.2005.2.499


This article has been cited by

  • 1. 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 and Continuous Dynamical Systems - Series B, 2013, 18, 4, 1031, 10.3934/dcdsb.2013.18.1031
  • 2. G. Dimitriu, T. Lorenzi, R. Ştefănescu, S. Anita, N. Hritonenko, G. Marinoschi, A. Swierniak, Evolutionary Dynamics of Cancer Cell Populations under Immune Selection Pressure and Optimal Control of Chemotherapy, Mathematical Modelling of Natural Phenomena, 2014, 9, 4, 88, 10.1051/mmnp/20149406
  • 3. Zhiming Li, Zhidong Teng, Hui Miao, Modeling and Control for HIV/AIDS Transmission in China Based on Data from 2004 to 2016, Computational and Mathematical Methods in Medicine, 2017, 2017, 1, 10.1155/2017/8935314
  • 4. Svetlana Bunimovich-Mendrazitsky, Benzion Shklyar, Optimization of Combined Leukemia Therapy by Finite-Dimensional Optimal Control Modeling, Journal of Optimization Theory and Applications, 2017, 175, 1, 218, 10.1007/s10957-017-1161-9
  • 5. Subhas Khajanchi, Dibakar Ghosh, The combined effects of optimal control in cancer remission, Applied Mathematics and Computation, 2015, 271, 375, 10.1016/j.amc.2015.09.012
  • 6. F. Castiglione, B. Piccoli, Cancer immunotherapy, mathematical modeling and optimal control, Journal of Theoretical Biology, 2007, 247, 4, 723, 10.1016/j.jtbi.2007.04.003
  • 7. K. Kassara, A. Moustafid, Angiogenesis inhibition and tumor-immune interactions with chemotherapy by a control set-valued method, Mathematical Biosciences, 2011, 231, 2, 135, 10.1016/j.mbs.2011.02.010
  • 8. Tunde T. Yusuf, Francis Benyah, Optimal strategy for controlling the spread of HIV/AIDS disease: a case study of South Africa, Journal of Biological Dynamics, 2012, 6, 2, 475, 10.1080/17513758.2011.628700
  • 9. Swarnali Sharma, G. P. Samanta, Dynamical Behaviour of a Tumor-Immune System with Chemotherapy and Optimal Control, Journal of Nonlinear Dynamics, 2013, 2013, 1, 10.1155/2013/608598
  • 10. Sanaz Nazari, Hadi Basirzadeh, Natural killer or T-lymphocyte cells: Which is the best immune therapeutic agent for cancer? An optimal control approach, International Journal of Control, Automation and Systems, 2014, 12, 1, 84, 10.1007/s12555-013-0030-z
  • 11. Urszula Ledzewicz, Mohammad Naghnaeian, Heinz Schättler, Optimal response to chemotherapy for a mathematical model of tumor–immune dynamics, Journal of Mathematical Biology, 2012, 64, 3, 557, 10.1007/s00285-011-0424-6
  • 12. Khalid Kassara, A Unified Set-Valued Approach to Control Immunotherapy, SIAM Journal on Control and Optimization, 2009, 48, 2, 909, 10.1137/07070591X
  • 13. Mehdi Afshar, Mohammad Reza Razvan, Optimal Control of Differential Infectivity Models, International Journal of Applied and Computational Mathematics, 2017, 3, 1, 65, 10.1007/s40819-015-0089-8
  • 14. Francesco Pappalardo, Marzio Pennisi, Filippo Castiglione, Santo Motta, Vaccine protocols optimization: In silico experiences, Biotechnology Advances, 2010, 28, 1, 82, 10.1016/j.biotechadv.2009.10.001
  • 15. Swarnali Sharma, G. P. Samanta, Analysis of the Dynamics of a Tumor–Immune System with Chemotherapy and Immunotherapy and Quadratic Optimal Control, Differential Equations and Dynamical Systems, 2016, 24, 2, 149, 10.1007/s12591-015-0250-1
  • 16. G. P. Samanta, Ricardo Gómez Aíza, Swarnali Sharma, Analysis of a mathematical model of periodically pulsed chemotherapy treatment, International Journal of Dynamics and Control, 2017, 5, 3, 842, 10.1007/s40435-015-0204-z
  • 17. L.G. de Pillis, W. Gu, K.R. Fister, T. Head, K. Maples, A. Murugan, T. Neal, K. Yoshida, Chemotherapy for tumors: An analysis of the dynamics and a study of quadratic and linear optimal controls, Mathematical Biosciences, 2007, 209, 1, 292, 10.1016/j.mbs.2006.05.003
  • 19. C. Collins, K.R. Fister, M. Williams, Optimal Control of a Cancer Cell Model with Delay, Mathematical Modelling of Natural Phenomena, 2010, 5, 3, 63, 10.1051/mmnp/20105305
  • 20. Elham Ahmadi, Jafar Zarei, Roozbeh Razavi-Far, Mehrdad Saif, A dual approach for positive T–S fuzzy controller design and its application to cancer treatment under immunotherapy and chemotherapy, Biomedical Signal Processing and Control, 2020, 58, 101822, 10.1016/j.bspc.2019.101822
  • 21. M. Younus Baba, M. Saleem, M. Noman, Abdur Raheem, A Mixed Therapy Minimal Model: Some Strategies for Eradication or Minimization of Cancer, Computer Methods and Programs in Biomedicine, 2020, 105433, 10.1016/j.cmpb.2020.105433

Reader Comments

your name: *   your email: *  

Copyright Info: 2005, K. Renee Fister, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved