Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-16T09:20:57.822Z Has data issue: false hasContentIssue false
  • English
  • Français

Delay Dynamics of Cancer and Immune Cell Model

Published online by Cambridge University Press:  25 January 2012

D. Adongo  and
K.R. Fister
Get access

Abstract

We investigate optimal control of a cancer-immune cell interactive model with delay inthe interphase compartment. By applying the optimal control theory, we seek to minimizethe cost associated with the chemotherapy drug, minimize the accumulation of cancer cells,and increase the immune cell presence. Optimality conditions and characterization of thecontrol are provided. Numerical analyses are given to enhance the understanding of thedifficulties that occur in the control of cancer.

Keywords

optimal controlchemotherapy drugcancer cells
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 1: Cancer modeling , 2012 , pp. 261 - 278
Copyright
© EDP Sciences, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Athanassios, I., Barbolosi, D.. Optimizing drug regimens in cancer chemotherapy by an efficacy-toxicity mathematical model. Compu. Biomedical Res., 33 (2000), 211–226. Google Scholar
M. Chaplain, A. Matzavinos. Mathematical modelling of spatio-temporal phenomena in tumour immunology tutorials in mathematical biosciences III. Cell Cycle, Proliferation, and Cancer, (2006), 131–183.
W. Cheney, D. Kincaid. Numerical mathematics and computing. Thomson Brooks/Cole, Belmont, 2008.
Collins, C., Fister, K. R., Williams, M.. Optimal control of a cancer cell model with delay. Math. Model. Nat. Phen., 5 (2010), No. 3, 63–71. CrossRefGoogle Scholar
Das, P. C., Sharma, R. R.. On optimal controls for measure delay-differential equations. SIAM J. Control, 6 (1971), No. 1, 43–61. CrossRefGoogle Scholar
dePillis, L. G., Radunskaya, A. E.. A mathematical tumor model with immune resistance and drug therapy : an optimal control approach. J. Theoretical Medicine, 3 (2001), 79–100. CrossRefGoogle Scholar
dePillis, L. G., Radunskaya, A. E.. The dynamics of an optimally controlled tumor model : a case study. Math. Comp. Model., 37 (2003), No. 11, 1221–1244. CrossRefGoogle Scholar
dePillis, L. G., Radunskaya, A. E., Wiseman, C. L.. A validated mathematical model of cell-mediated immune response to tumor growth. Cancer Research, 61 (2005), No. 17, 7950–7958. CrossRefGoogle Scholar
dePillis, L. G., Fister, K. R., Gu, W., Collins, C., Daub, M., Moore, J., Preskill, B.. Mathematical model creation for cancer chemo-immunotherapy. Computational and Math. Methods in Medicine, 10 (2009), No. 3, 165–184. CrossRefGoogle Scholar
R. D. Driver. Ordinary and delay differential equations. Springer-Verlag, New York, 1977.
R. Fletcher. Practical methods of optimization. Wiley and Sons, New York, 1987.
Lu, W., Hillen, T., Freedman, H. I.. A mathematical model for M-phase specific chemotherapy including the Go-phase and immunoresponse. Math. Biosci. and Engng., 4 (2007), No. 2, 239–259. Google Scholar
M.I. Kamien, N. L. Schwartz. Dynamic optimization : the calculus of variations and optimal control in economics and management, Advanced Textbooks in Economics. North-Holland, 1991.
Kim, M., Perry, S., Woo, K. B.. Quantitative approach to the design of antitumor drug dosage schedule via cell cycle kinetics and systems theory. Ann. Biomed. Engng, 5 (1977), 12–33. CrossRefGoogle ScholarPubMed
Kirschner, D., Panetta, J. C.. Modeling immunotherapy of the tumor-immune interaction. J. Math. Bio., 35 (1998), 235–252. CrossRefGoogle Scholar
Ledzewicz, U., Brown, T., Schattler, H.. Comparison of optimal controls for a model in cancer chemotherapy with l1 and l2 type objectives. Optimization Methods and Software, 19 (2004), No. 3-4, 339–350. CrossRefGoogle Scholar
Ledzewicz, U., Schattler, H.. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete and Continuous Dynamical Systems - Series B, 6 (2006), No. 1, 129–150. Google Scholar
McKenzie, D.. Mathematical modeling and cancer. SIAM News, 31 (2004), 1–2. Google Scholar
Murray, J. M.. Some optimality control problems in cancer chemotherapy with a toxicity limit. Math. Biosci., 100 (1990), 49–67. CrossRefGoogle Scholar
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko. The mathematical theory of optimal processes. Gordon and Breach, 1962.
Swan, G. W., Vincent, T. L.. Optimal control analysis in the chemotherapy of IgG multiple myeloma. Bull. of Math. Bio., 39 (1977), 317–337. CrossRefGoogle ScholarPubMed
Swierniak, A., Ledzewicz, U., Schattler, H.. Optimal control for a class of compartmental models in cancer chemotherapy. Int. J. Appl. Math. Comput. Sci., 13 (2003), No. 3, 357–368. Google Scholar
Villasana, M. Radunskaya, A.. A delay differential equation model for tumor growth. J. Math. Bio., 47 (2003), 270–294. CrossRefGoogle ScholarPubMed