# Optimal Control of a Cancer Cell Model with Delay

C. Collins; K.R. Fister; M. Williams

Mathematical Modelling of Natural Phenomena (2010)

- Volume: 5, Issue: 3, page 63-75
- ISSN: 0973-5348

## Access Full Article

top## Abstract

top## How to cite

topCollins, C., Fister, K.R., and Williams, M.. "Optimal Control of a Cancer Cell Model with Delay." Mathematical Modelling of Natural Phenomena 5.3 (2010): 63-75. <http://eudml.org/doc/197660>.

@article{Collins2010,

abstract = {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.},

author = {Collins, C., Fister, K.R., Williams, M.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {cancer dynamics; optimal control},

language = {eng},

month = {4},

number = {3},

pages = {63-75},

publisher = {EDP Sciences},

title = {Optimal Control of a Cancer Cell Model with Delay},

url = {http://eudml.org/doc/197660},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Collins, C.

AU - Fister, K.R.

AU - Williams, M.

TI - Optimal Control of a Cancer Cell Model with Delay

JO - Mathematical Modelling of Natural Phenomena

DA - 2010/4//

PB - EDP Sciences

VL - 5

IS - 3

SP - 63

EP - 75

AB - 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.

LA - eng

KW - cancer dynamics; optimal control

UR - http://eudml.org/doc/197660

ER -

## References

top- I. Athanassios D. Barbolosi. Optimizing drug regimens in cancer chemotherapy by an efficacy-toxicity mathematical model. Comp. Biomedical Res., 33 (2000), 211–226.
- 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.
- P. C. Das, R. R. Sharma. On optimal controls for measure delay-differential equations. SIAM J. Control, 6 (1971) No. 1, 43–61.
- L. G. de Pillis, K. R. Fister, W. Gu, T. Head, K. Maples, A. Murugan, T. Neal, K. Kozai. Optimal control of mixed immunotherapy and chemotherapy of tumors. Journal of Biological Systems, 16 (2008), No. 1, 51–80.
- L.G. de Pillis, K. R. Fister, W. Gu, C. Collins, M. Daub, D. Gross, J. Moore B. Preskill. Mathematical Model Creation for Cancer Chemo-Immunotherapy. Computational and Mathematical Methods in Medicine, 10 (2009), No. 3, 165–184.
- L. G. de Pillis, K. R. Fister, W. Gu, C. Collins, M. Daub, D. Gross, J. Moore, B. Preskill. Seeking Bang-Bang Solutions of Mixed Immuno-chemotherapy of tumors. Electronic Journal of Differential Equations, (2007), No. 171, 1–24.
- R. D. Driver. Ordinary and Delay Differential Equations. Springer-Verlag, New York, 285–311, 1977.
- K. R. Fister, J. H. Donnelly. Immunotherapy: An Optimal Control Theory Approach. Mathematical Biosciences in Engineering, 2 (2005), No. 3, 499–510.
- 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.
- M. Kim, S. Perry, and K. B. Woo. Quantitative approach to the deisgn of antitumor drug dosage schedule via cell cycle kinetics and systems theory. Ann. Biomed. Eng., 5 (1977), 12–33.
- D. Kirschner J. C. Panetta. Modeling immunotherapy of the tumor-immune interaction. Journal of Mathematical Biology, 35 (1998), 235–252.
- W. Liu, T. Hillen, H. I. Freedman. A Mathematical model for M-phase specific chemotherapy including the Go-phase and immunoresponse. Mathematical Biosciences and Engineering, 4 (2007), No. 2, 239-259.
- D. McKenzie. Mathematical modeling and cancer. SIAM News, 31, Jan/Feb 2004.
- J. M. Murray. Some optimality control problems in cancer chemotherapy with a toxicity limit. Mathematical Biosciences, 100 (1990), 49–67.
- L. S. Pontryagin, V. G. Boltyanksii, R. V. Gamkrelidze, E. F. Mischchenko. The Mathematical theory of optimal processes. Wiley, New York, 1962.
- G. W. Swan, T. L. Vincent. Optimal control analysis in the chemotherapy of IgG multiple myeloma. Bulletin of Mathematical Biology, 39 (1977), 317–337.
- M. Villasana, A. Radunskaya. A delay differential equation model for tumor growth. Journal of Mathematical Biology, 47 (2003), 270–294.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.