Russian version English version
Volume 9   Issue 1   Year 2014
Yegorov I. Ye.

Optimal Feedback Control in a Mathematical Model of Malignant Tumour Treatment with the Immune Reaction Taken Into Account

Mathematical Biology & Bioinformatics. 2014;9(1):257-272.

doi: 10.17537/2014.9.257.


  1. Aranjo RP, Mcelwain DG. A history of the study of solid tumour growth: The contribution of mathematical modelling. Bulletin of Mathematical Biology. 2004;66:1039-1091. doi: 10.1016/j.bulm.2003.11.002
  2. Handbook of Cancer Models with Applications. Eds. Tan W.-Y., Hanin L. World Scientific Publishing; 2008.
  3. Stepanova NV. Immune response dynamics during malignant tumor development. Biofizika (Biophysics). 1979;24(5):897-902 (in Russ.).
  4. Ledzewicz U, Naghnaeian M, Schattler H. An optimal control approach to cancer treatment under immunological activity. Applicationes Mathematicae. 2011;38(1):17-31. doi: 10.4064/am38-1-2
  5. Ledzewicz U, Schattler H. The influence of PK/PD on the structure of optimal controls in cancer chemotherapy models. Mathematical Biosciences and Engineering. 2005;2(3):561-578. doi: 10.3934/mbe.2005.2.561
  6. Bratus AS, Chumerina ES. Optimal control synthesis in therapy of solid tumor growth. Computational Mathematics and Mathematical Physics. 2008;48(6):892-911.  doi: 10.1134/S096554250806002X
  7. Chumerina ES. Choice of optimal strategy of tumor chemotherapy in Gompertz model. J. Comput. Systems Sci. Internat. 2009;48(2):325-331. doi: 10.1134/S1064230709020154
  8. Antipov AV, Bratus' AS. Mathematical model of optimal chemotherapy strategy with allowance for cell population dynamics in a heterogeneous tumour. Computational Mathematics and Mathematical Physics. 2009;49(11):1825-1836. doi: 10.1134/S0965542509110013
  9. Bratus' AS, Zaichik SYu. Smooth solutions of the Hamilton–Jacobi–Bellman. equation in a mathematical model of optimal treatment of viral infections. Differential Equations. 2010;46(11):1571-1583. doi: 10.1134/S0012266110110054
  10. Bratus AS, Fimmel E, Todorov Y, Semenov YS, Nuernberg F. On strategies on a mathematical model for leukaemia therapy. Nonlinear Analysis: Real World Applications. 2012;13:1044-1059. doi: 10.1016/j.nonrwa.2011.02.027
  11. Bratus A, Todorov Y, Yegorov I, Yurchenko D. Solution of the feedback control problem in the mathematical model of leukaemia therapy. Journal of Optimization Theory and Applications. 2013;159(3):590-605. doi: 10.1007/s10957-013-0324-6
  12. Clarke FH, Ledyaev YuS, Stern RJ, Wolenski PR. Nonsmooth Analysis and Control Theory. New York: Springer-Verlag; 1998. 278 p.
  13. Subbotina NN. The method of characteristics for Hamilton—Jacobi equations and applications to dynamical optimization. J. Mathematical Sciences. 2006;13(3):2955-3091. doi: 10.1007/s10958-006-0146-2
  14. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishechenko EF. The Mathematical Theory of Optimal Processes. New York/London, John Wiley & Sons; 1962. DOI: 10.1002/zamm.19630431023 doi: 10.1002/zamm.19630431023
  15. De Pillis LG, Radunskaya A. A mathematical tumor model with immune resistance and drug therapy: An optimal control approach. Journal of Theoretical Medicine. 2001;3:79-100. doi: 10.1080/10273660108833067
Table of Contents Original Article
Math. Biol. Bioinf.
doi: 10.17537/2014.9.257
published in Russian

Abstract (rus.)
Abstract (eng.)
Full text (rus., pdf)


  Copyright IMPB RAS © 2005-2024