Abstract
This paper uses optimal control theory in conjunction with a Gompertzian type model for cellular growth to determine the optimal method of administering cycle non-specific chemotherapy or more generally the optimal durations of treatment and rest periods during chemotherapy. The performance critera employed to determine the relative merits of the therapy include not only the destruction of malignant cells, but also the sparing of a critical normal tissue. Since these criteria are at odds with one another, the solutions are found which satisfy the Pareto optimality conditions.
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Literature
Almquist, K. J. and H. T. Banks. 1976. “A Theoretical and Computational Method for Determining Optimal Treatment Schedules in Fractionated Radiation Therapy”Math. Biosci.,29, 159–179.
Aroesty, J., T. Lincoln, N. Shapiro and G. Boccia. 1973. “Tumor Growth and Chemotherapy: Mathematical Methods, Computer Simulation, and Experimental Foundations.”Math. Biosci.,17, 243–300.
Berenbaum, M. C. 1969. “Dose-response Curves for Agents that Impair Cell Reproductive Integrity.”Br. J. Cancer,23, 434–445.
Bremermann, H. 1970. “A Global Method for Unconstrained Optimization.”Math Biosci.,9, 1–15.
Goodman, L. and A. Gilman. 1970.The Pharmacological Basis of Therapeutics, 4th edn. pp. 17–19. New York: Macmillan.
Laird, A. K. 1969. “Dynamics of Growth in Tumors and Normal Organisms.”Human Tumor Cell Kinetics, NCI Monograph V 30, pp. 15–28.
Leitmann, G. 1974.Cooperative and Non-cooperative Many Player Differential Games, International Centre for Mechanical Sciences Course Lectures. No. 190. Springer Verlag Udine.
Leitmann, G. 1966.An Introduction to Optimal Control. New York: McGraw-Hill.
Nicolini, C. 1976. “The Principles and Methods of Cell Synchrony in Cancer Chemotherapy.”BBA 458 (1976), pp. 243–282.
Nicolini, C., E. Milgram, F. Kendall and W. Giaretti. 1977. “Mathematical Models for Drug Actionin vitro”Growth Kinetics and Biochemical Regulation of Normal and Malignant Cells. (Eds. B. Drewinko and R. M. Humphrey. Baltimore: Williams & Williams.
Nicolini, N. and F. Kendall. 1975. “Monte Carlo Simulation of Drug Action in Intact Animals.”Mathematical Modeling in Cell Kinetics (Ed. A. J. Valleron) pp. 81–83. Ghent: European Press Medikon.
Swan, G. and T. Vincent. 1977. “Optimal Control Analysis in the Chemotherapy of IgC Multiple Myeloma.”Bull. Math. Biol.,39, 317–337.
Yu, P. L. and G. Leitmann. 1974. “Non-dominated decisions and Cone Convexity in Dynamic Multicriteria Decision Problems.”JOTA,14, (5) 573–584.
Zeleny, M. 1974. Linear Multiobjective Programming Lecture Notes in Economics and Mathematical Systems.” Vol. 95. New York: Springer-Verlag.
Zietz, S. 1977. “Mathematical Modeling of Cellular Kinetics and Optimal Control Theory in the Service of Cancer Chemotherapy.” Ph.D. Thesis, Dept. of Math., University of California, Berkeley.
Zietz, S., C. Desaire, M. Grattarola, and C. Nicolini. 1978. “Deterministic Mathematical Models to Obtain Optimized Drug Metabolism and Cell Kinetic Parameters as a Function of Dosage from Both Autoradiographic and Flow Microfluorimetric Analysis.” In:Biomathematics and Cell Kinetics (Ed. A.-J. Valleron), pp. 431–438. Amsterdam: Elsevier.
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Zietz, S., Nicolini, C. Mathematical approaches to optimization of cancer chemotherapy. Bltn Mathcal Biology 41, 305–324 (1979). https://doi.org/10.1007/BF02460814
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DOI: https://doi.org/10.1007/BF02460814