zbMATH — the first resource for mathematics

Chemotherapeutic treatments: A study of the interplay among drug resistance, toxicity and recuperation from side effects. (English) Zbl 0899.92024
Summary: A system of differential equations for the control of tumor growth cells in a cycle nonspecific chemotherapy is analyzed. Spontaneously acquired drug resistance is taken into account, and a criterion for the selection of chemotherapeutic treatment is used. This criterion purports to describe the possibility of improvement of the patient’s health when treatment is discontinued. Contrary to our early results [see the authors and R. C. Bassanezi, Math. Biosci. 125, No. 2, 191-209 (1995; Zbl 0821.92015); ibid. 211-228 (1995; Zbl 0821.92016)] which also take drug resistance into account, in this context strategies of continuous chemotherapy in which rest periods take part may be better than maximum drug concentration throughout the treatment (which appears to be in accordance with clinical practice). This bears out our previous conjecture that when drug resistance is accounted for, the imperfections in the usual modelling of treatment criteria, which in general do not allow for patient recuperation, ruled out the possibility of rest periods in optimal continuous chemotherapy.

92C50 Medical applications (general)
49N70 Differential games and control
49N75 Pursuit and evasion games
93C15 Control/observation systems governed by ordinary differential equations
PDF BibTeX Cite
Full Text: DOI
[1] Bellomo, N. and G. Forni. 1994. Dynamics of tumor interaction with the host immune system.Math. Comput. Modelling 20, 107–122. · Zbl 0811.92014
[2] Bernardo Strada, M. R., G. Bernardo and G. Robustelli Della Cuna. 1983. Principi di immunoterapia antitumorale. InManuale de Oncologia Medica, G. Bonadonna (Ed.) Paris: Masson.
[3] Clarke, F. H. 1983.Optimization and Nonsmooth Analysis, New York: Wiley. · Zbl 0582.49001
[4] Coldman, A. J. and J. H. Goldie. 1983. A model for the resistance of tumor cells to cancer chemotherapeutic agents.Math. Biosci. 65, 291–307. · Zbl 0519.92008
[5] Coldman, A. J. and J. H. Goldie. 1986. A stochastic model for the origin and treatment of tumors containing drug-resistant cells.Bull. Math. Biol. 48, 279–292. · Zbl 0613.92006
[6] Costa, M. I. S., J. L. Boldrini and R. C. Bassanezi. 1992. Optimal chemical control of populations developing drug resistance.IMA J. Math. Appl. Med. Biol. 9, 215–226. · Zbl 0779.92011
[7] Costa, M. I. S., J. L. Boldrini and R. C. Bassanezi. 1994. Optimal chemotherapy: a case study with drug resistance, saturation effect and toxicity.IMA J. Math. Appl. Med. Biol. 11, 45–59. · Zbl 0816.92008
[8] Costa, M. I. S., J. L. Boldrini and R. C. Bassanezi. 1995a. Drug kinetics and drug resistance in optimal chemotherapy.Math. Biosci. 125, 191–209. · Zbl 0821.92015
[9] Costa, M. I. S., J. L. Boldrini and R. C. Bassanezi. 1995b. Chemotherapeutic treatments involving drug resistance and level of normal cells as a criterion of toxicity.Math. Biosci. 125, 211–228 · Zbl 0821.92016
[10] Eisen, M. 1978.Mathematical Models in Cell Biology and Cancer Chemotherapy. Lecture Notes in Biomathematics, Vol. 30. New York: Springer-Verlag. · Zbl 0414.92005
[11] Goldie, J. H. and A. J. Coldman. 1979. A mathematical model for relating the drug sensitivity of tumors to their spontaneous mutation rate.Cancer Treat. Rep. 63, 1727–1733.
[12] Harnevo, L. and Z. Agur. 1992. 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.
[13] Herson, J. 1980. Evaluation of toxicity: statistical considerations.Cancer Treat. Rep. 64, 463–468.
[14] Kimmel, M. and D. E. Axelrod. 1990. Mathematical models for gene amplification with application to cellular drug resistance and tumorigenicity.Genetics 125, 633–644.
[15] Kimmel, M., D. E. Axelrod and G. M. Wahl. 1992. A branching process model of gene amplification following chromosome breakage.Mut. Res. 276, 225–239.
[16] Kuznetsov, V. A., I. A. Makalin, M. A. Taylor and A. S. Perelson. 1994. Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis.Bull. Math. Biol. 56, 295–321. · Zbl 0789.92019
[17] Marusic, M., Z. Bajzer, S. Vuk-Pavlovic and J. P. Fryer. 1994. Tumor growthin vivo and as multicellular spheroids compared by mathematical models.Bull. Math. Biol. 56, 617–631. · Zbl 0800.92117
[18] Mohler, R. R., K. S. Lee, A. L. Asachenkov and G. I. Marchuk. 1994. A system approach to immunology and cancer.IEEE Trans. Syst. Cybernetics 24, 632–641.
[19] Murray, J. M. 1990a. Optimal control for a cancer chemotherapy problem with general growth and loss functions.Math. Biosci. 98, 273–287. · Zbl 0693.92009
[20] Murray, J. M. 1990b. Some optimal control problems in cancer chemotherapy with a toxicity limit.Math. Biosci. 100, 49–67. · Zbl 0778.92012
[21] Murray, J. M. 1995. An example of the effects of drug resistance on the optimal schedule for a single drug in cancer chemotherapy.IMA J. Math. Appl. Med. Biol. 12, 55–71. · Zbl 0832.92009
[22] Schandl, F. R. 1989. Optimal treatment strategies in cancer chemotherapy. Forschungsbericht Nr. 116, Institut für Ökonometrie, OR und Systemtheorie, Technische Universität Wien.
[23] Skipper, H. E. 1983. The forty year old mutation theory of Luria and Delbruck and its pertinence to cancer chemotherapy.Adv. Cancer Res. 40, 331.
[24] Swan, G. W. 1987. Tumor growth models and cancer chemotherapy. InCancer Modeling, J. R. Thompson and B. W. Brown (Eds). New York: Dekker.
[25] Swan, G. W. 1990. Role of optimal control theory in cancer chemotherapy.Math. Biosci. 101, 237–284. · Zbl 0702.92007
[26] Swan, G. W. and T. L. Vincent. 1977. Optimal control analysis in the chemotherapy of IgG multiple myeloma.Bull. Math. Biol. 39, 317. · Zbl 0354.92041
[27] Thornton, B. S. 1988. Prescheduling graphic displays for optimal cancer therapies to reveal possible tumor regression or stabilization.J. Med. Syst. 12, 31–41.
[28] Vaidya, V. G. and F. J. Alexandro, Jr. 1982. Evaluation of some mathematical models for tumor growth.Int. J. Bio-Med. Comp. 13, 19–35.
[29] Vendite, L. L. 1988. Modelagem matemática para o crescimento tumoral e o problema de resistência celular aos fármacos anti-bláticos. Ph.D. thesis, Faculdade de Engenharia Elétrica, Universidade Estadual de Campinas, SP, Brazil.
[30] Vietti, T. J. 1980. Evaluation of toxicity: clinical issues.Cancer Treat. Rep. 64, 457–461.
[31] Zietz, S. and C. Nicolini. 1979. Mathematical approaches to optimization of cancer chemotherapy.Bull. Math. Biol. 41, 305–324. · 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.