Hostname: page-component-7d684dbfc8-w65q4 Total loading time: 0 Render date: 2023-09-24T07:43:05.991Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false

Optimal Control of a Cancer Cell Model with Delay

Published online by Cambridge University Press:  28 April 2010

C. Collins
Department of Mathematics, University of Tennessee, Knoxville, TN 37996 USA
K.R. Fister*
Department of Mathematics and Statistics, Murray State University, Murray, KY 42071 USA
M. Williams
Department of Mathematics, University of Nebraska, Lincoln, NE 68588 USA
*Corresponding author. E-mail:
Get access


In this paper, we look at a model depicting the relationship of cancer cells in different development stages with immune cells and a cell cycle specific chemotherapy drug. The model includes a constant delay in the mitotic phase. By applying optimal control theory, we seek to minimize the cost associated with the chemotherapy drug and to minimize the number of tumor cells. Global existence of a solution has been shown for this model and existence of an optimal control has also been proven. Optimality conditions and characterization of the control are discussed.

Research Article
© EDP Sciences, 2010

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.)


Athanassios, I. Barbolosi, D.. Optimizing drug regimens in cancer chemotherapy by an efficacy-toxicity mathematical model . Comp. Biomedical Res., 33 (2000), 211226.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, 131–183, Springer-Verlag, Berlin, 2006.
Das, P. C., Sharma, R. R.. On optimal controls for measure delay-differential equations . SIAM J. Control, 6 (1971) No. 1, 4361. CrossRefGoogle Scholar
de Pillis, L. G., Fister, K. R., Gu, W., Head, T., Maples, K., Murugan, A., Neal, T., Kozai, K.. Optimal control of mixed immunotherapy and chemotherapy of tumors . Journal of Biological Systems, 16 (2008), No. 1, 5180. CrossRefGoogle Scholar
de Pillis, L.G., Fister, K. R., Gu, W., Collins, C., Daub, M., Gross, D., Moore, J. Preskill, B.. Mathematical Model Creation for Cancer Chemo-Immunotherapy . Computational and Mathematical Methods in Medicine, 10 (2009), No. 3, 165184.CrossRefGoogle Scholar
de Pillis, L. G., Fister, K. R., Gu, W., Collins, C., Daub, M., Gross, D., Moore, J., Preskill, B.. Seeking Bang-Bang Solutions of Mixed Immuno-chemotherapy of tumors . Electronic Journal of Differential Equations, (2007), No. 171, 124. Google Scholar
R. D. Driver. Ordinary and Delay Differential Equations. Springer-Verlag, New York, 285–311, 1977.
Fister, K. R., Donnelly, J. H.. Immunotherapy: An Optimal Control Theory Approach. Mathematical Biosciences in Engineering, 2 (2005), No. 3, 499510. Google ScholarPubMed
R. Fletcher. Practical methods of optimization. Wiley and Sons, New York, 1987.
M. I. Kamien, N. L. Schwartz. Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management, Vol. 31 of Advanced Textbooks in Economics. North-Holland, 2nd edition, 1991.
Kim, M., Perry, S., and Woo, K. B.. Quantitative approach to the deisgn of antitumor drug dosage schedule via cell cycle kinetics and systems theory . Ann. Biomed. Eng., 5 (1977), 1233.CrossRefGoogle Scholar
Kirschner, D. Panetta, J. C.. Modeling immunotherapy of the tumor-immune interaction . Journal of Mathematical Biology, 35 (1998), 235252.CrossRefGoogle Scholar
Liu, W., Hillen, T., Freedman, H. I.. A Mathematical model for M-phase specific chemotherapy including the Go-phase and immunoresponse . Mathematical Biosciences and Engineering, 4 (2007), No. 2, 239-259. Google Scholar
D. McKenzie. Mathematical modeling and cancer. SIAM News, 31, Jan/Feb 2004.
Murray, J. M.. Some optimality control problems in cancer chemotherapy with a toxicity limit . Mathematical Biosciences, 100 (1990), 4967.CrossRefGoogle Scholar
L. S. Pontryagin, V. G. Boltyanksii, R. V. Gamkrelidze, E. F. Mischchenko. The Mathematical theory of optimal processes. Wiley, New York, 1962.
Swan, G. W., Vincent, T. L.. Optimal control analysis in the chemotherapy of IgG multiple myeloma . Bulletin of Mathematical Biology, 39 (1977), 317337. CrossRefGoogle Scholar
Villasana, M., Radunskaya, A.. A delay differential equation model for tumor growth . Journal of Mathematical Biology, 47 (2003), 270294. CrossRefGoogle ScholarPubMed