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Gradient method for finding optimal scheduling in infinite dimensional models of chemotherapy. (English) Zbl 0986.92018

Summary: One of the major obstacles against succesful chemotherapy of cancer is the emergence of resistance of cancer cells to cytotoxic agents. Applying optimal control theory to mathematical models of cell cycle dynamics can be a very efficient method to understand and, eventually, overcome this problem. Results that have been hitherto obtained have already helped to explain some observed phenomena, concerning dynamical properties of cancer populations. Because of recent progress in understanding the way in which chemotherapy affects cancer cells, new insights and more precise mathematical formulation of the control problem, in the meaning of finding optimal chemotherapy, became possible. This, together with a progress in mathematical tools, has renewed hopes for improving chemotherapy protocols.
We consider a population of neoplastic cells stratified into subpopulations of cells of different types. Due to the mutational event a sensitive cell can acquire a copy of the gene that makes it resistant to the agent. Likewise, the division of resistant cells can result in the change of the number of gene copies. We convert the model in the form of an infinite dimensional system of ordinary differential state equations discussed in our previous publications [see, i.e., A. Swiernak et al., Control Cybern. 28, No. 1, 61-73 (1999; Zbl 0949.93070); J. Math. Syst. Estim. Control 8, No. 1, 1-16 (1998; Zbl 0897.92015)] into the integro-differential form. It enables application of the necessary conditions of optimality given by the appropriate version of Pontryagin’s maximum principle. The performance index which should be minimized combines the negative cumulated cytotoxic effect of the drug and the terminal population of both sensitive and resistant neoplastic cells. The linear form of the cost function and the bilinear form of the state equation result in a bang-bang optimal control law. To find the switching times we propose to use a special gradient algorithm developed similarly to the one applied in our previous papers to finite dimensional problems.

MSC:

92C50 Medical applications (general)
49N90 Applications of optimal control and differential games
45J05 Integro-ordinary differential equations
49K22 Optimal control problems with integral equations (nec./ suff.) (MSC2000)
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