zbMATH — the first resource for mathematics

Chemotherapeutic treatments involving drug resistance and level of normal cells as a criterion of toxicity. (English) Zbl 0821.92016
Summary: A system of differential equations for the control of tumor cells growth in a cycle nonspecific chemotherapy is presented. Drug resistance and toxicity conveyed through the level of normal cells are taken into account in a class of optimal control problems. Alternative treatments for the exponential tumor growth are set forth for cases where optimal treatments are not available.

92C50 Medical applications (general)
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
49N70 Differential games and control
49N75 Pursuit and evasion games
49K15 Optimality conditions for problems involving ordinary differential equations
Full Text: DOI
[1] Coldman, A.J.; Goldie, J.H., A model for the resistance of tumor cells to cancer chemotherapeutic agents, Math. biosci., 65, 291-307, (1983) · Zbl 0519.92008
[2] Coldman, A.J.; Goldie, J.H., A stochastic model for the origin and treatment of tumors containing drug-resistant cells, Bull. math. biol., 48, 279-292, (1986) · Zbl 0613.92006
[3] M. I. S. Costa, J. L. Boldrini, and R. C. Bassanezi, Drug kinetics and drug resistance in optimal chemotherapy, Math. Biosci., in press. · Zbl 0821.92015
[4] Costa, M.I.S.; Boldrini, J.L.; Bassanezi, R.C., Optimal chemical control of populations developing drug resistance, IMA J. math. appl. med. biol., 9, 215-226, (1992) · Zbl 0779.92011
[5] Costa, M.I.S.; Boldrini, J.L.; Bassanezi, R.C., Optimal chemotherapy: A case study with drug resistance, saturation effect and toxicity, IMA J. math. appl. med. biol., 11, 45-59, (1994) · Zbl 0816.92008
[6] Eisen, M., Mathematical models in cell biology and cancer chemotherapy, () · Zbl 0414.92005
[7] Goldie, J.H.; Coldman, A.J., A mathematical model for relating the drug sensitivity of tumors to their spontaneous mutation rate, Cancer treat. rep., 63, 11-12, 1727-1733, (1979)
[8] Harnevo, L.; Agur, Z., Drug resistance as a dynamic process in a model for multistep gene amplification under various levels of selection stringency, Cancer chemother. pharmacol., 30, 469-476, (1992)
[9] Kimmel, M.; Axelrod, D.E., Mathematical models for gene amplification with application to cellular drug resistance and tumorigenicity, Genetics, 125, 633-644, (1990)
[10] Kimmel, M.; Axelrod, D.E.; Wahl, G.M., A branching process model of gene amplification following chromosome breakage, Mut. res., 276, 225-239, (1992)
[11] Kirk, D., Optimal control theory, (1970), Prentice-Hall, Inc Englewood Cliffs, NJ
[12] Lee, E.B.; Markus, L., Foundations of optimal control theory, (1967), Wiley NY · Zbl 0159.13201
[13] Murray, J.M., Optimal control for a cancer chemotherapy problem with general growth and loss functions, Math. biosci., 98, 273-287, (1990) · Zbl 0693.92009
[14] Murray, J.M., Some optimal control problems in cancer chemotherapy with a toxicity limit, Math. biosci., 100, 49-67, (1990) · Zbl 0778.92012
[15] Sage, A.P., Optimum systems control, (1968), Prentice-Hall, Inc Englewood Cliffs, NJ · Zbl 0192.51502
[16] Skipper, H.E., The forty year old mutation theory of luria and delbruck and its pertinence to cancer chemotherapy, Adv. cancer res., 40, 331-363, (1983)
[17] Swan, G.W., General applications of optimal control theory in cancer chemotherapy, IMA J. math. appl. biol., 5, 303-316, (1988) · Zbl 0678.92003
[18] Swan, G.W., Optimal control analysis of a cancer chemotherapy problem, IMA J. math. appl. med. biol., 4, 171-184, (1987) · Zbl 0616.92002
[19] Swan, G.W., Optimal control in some cancer chemotherapy problems, Internat. J. sys. sci., 11, 223-237, (1980) · Zbl 0426.92007
[20] Swan, G.W., Role of optimal control theory in cancer chemotherapy, Math. biosci., 101, 237-284, (1990) · Zbl 0702.92007
[21] Swan, G.W.; Vincent, T.L., Optimal control analysis in the chemotherapy of Igg multiple myeloma, Bull. math. biol., 39, 317-337, (1977) · Zbl 0354.92041
[22] Vaidya, V.G.; Alexandro, F.J., Evaluation of some mathematical models for tumor growth, Internat. J. bio-med. comp., 13, 19-35, (1982)
[23] Vendite, L.L., Modelagem matemática para o crescimento tumoral e o problema de resistência celular aos Fármacos anti-blásticos, ()
[24] Zietz, S.; Nicolini, C., Mathematical approaches to optimization of cancer chemotherapy, Bull. math. biol., 41, 305-325, (1979) · Zbl 0404.92004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.