Geodesic
Optimal control synthesis in therapy of solid tumor growth
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 6, pp. 946-966 
A. S. Bratus'; E. S. Chumerina. Optimal control synthesis in therapy of solid tumor growth. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 6, pp. 946-966. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_6_a1/
@article{ZVMMF_2008_48_6_a1,
     author = {A. S. Bratus' and E. S. Chumerina},
     title = {Optimal control synthesis in therapy of solid tumor growth},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {946--966},
     year = {2008},
     volume = {48},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_6_a1/}
}
TY  - JOUR
AU  - A. S. Bratus'
AU  - E. S. Chumerina
TI  - Optimal control synthesis in therapy of solid tumor growth
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2008
SP  - 946
EP  - 966
VL  - 48
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_6_a1/
LA  - ru
ID  - ZVMMF_2008_48_6_a1
ER  - 
%0 Journal Article
%A A. S. Bratus'
%A E. S. Chumerina
%T Optimal control synthesis in therapy of solid tumor growth
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2008
%P 946-966
%V 48
%N 6
%U http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_6_a1/
%G ru
%F ZVMMF_2008_48_6_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

A mathematical model of tumor growth therapy is considered. The total amount of a drug is bounded and fixed. The problem is to choose an optimal therapeutic strategy, i.e., to choose an amount of the drug permanently affecting the tumor that minimizes the number of tumor cells by a given time. The problem is solved by the dynamic programming method. Exact and approximate solutions to the corresponding Hamilton–Jacobi–Bellman equation are found. An error estimate is proved. Numerical results are presented.

[1] Aranjo R. P., Mcelwain D. L., “A history of the study of solid tumour growth: The contribution of mathematical modelling”, Bull. Math. Biol., 66 (2004), 1039–1091 | DOI | MR

[2] Byrne H. M., “Using mathematics study solid tumour growth”, Proc. 9the General Meetings of European Women in Math., 1999, 81–107 | MR

[3] Byrne H. M., “A weakly nonlinear analysis of a model of avascular solid tumour growth”, J. Math. Biol., 39 (1999), 151–181 | DOI | MR

[4] Matzavinos A., Chaplain M., Kuznetsov V., “Mathematical modelling of the spatio-temporal response of cytotoxic $T$-lymphocytes to a solid tumour”, Math. Medicine and Biology, 21 (2004), 1–34 | DOI | Zbl

[5] Guiot C., Degiorgis P. G., Delsanto P. P. et al., “Does tumour growth follow a “universal low””, J. Theor. Biol., 225 (2003), 147–151 | DOI | MR

[6] Murray J. D., Mathematical biology, v. II, Spatial models and biomedical applications, Springer, Berlin, 2003 | MR

[7] Kolmogorov A. H., Fomin C. B., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1968 | MR | Zbl

[8] Costa M. I., Boldini J. L., Bassanezi R. C., “Drug kinetics and drug resistance in optimal chemotherapy”, Math. Biosciences, 125 (1995), 191–209 | DOI | Zbl

[9] Costa M. I., Boldini J. L., Bassanezi R. C., “Chemotherapeutic treatments involving drug resistance and level of normal cells as a criterion of toxicity”, Math. Biosciences, 125 (1995), 211–228 | DOI | Zbl

[10] Hofbauer J., Sigmund K., The theory of evolution and dynamical systems, Cambridge Univ. Press, New York, 1988 | MR | Zbl

[11] Fleming W., Rishel R., Deterministic and stochastic optimal control, Springer, Berlin, 1975 | MR

[12] Chernousko F. A., “Avtomodelnye resheniya uravneniya Bellmana dlya zadach optimalnoi korrektsii sluchainykh vozmuschenii”, Prikl. matem. i mekhan., 35:2 (1971), 333–342 | MR

[13] Bratus A. C., Volosov K. A., “Tochnye resheniya uravneniya Gamiltona–Yakobi–Bellmana dlya zadach optimalnoi korrektsii s ogranichennym summarnym resursom upravleniya”, Prikl. matem. i mekhan., 68:4 (2004), 819–832 | MR | Zbl

[14] Kirschner D., Panetta J. C., “Modelling immunotherapy of the tumour-immune interaction”, J. Math. Biol., 37 (1998), 235–252 | DOI | Zbl

[15] Burden T. N., Ernstberger J., Fister K. R., “Optimal control applied to immunotherapy”, J. Discrete and Continuous Dynamical Systems, Ser. B, 4 (2004), 135–146 | MR | Zbl