Abstract
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 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. In this paper 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 e.g. Swierniak etal., 1996b; Polariski etal., 1997; Swierniak etaL, 1998c), into the integro‐differential form. It enables application of the necessary conditions of optimality given by the appropriate version of Pontryagin′s maximum principle, e.g. (Gabasov and Kirilowa, 1971). 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 (Duda 1994; 1997).
Highlights
Despite a long history of mathematical modeling of cancer chemotherapy its practical application to development of chemotherapy protocols has beenJAROSLAW SMIEJA et nlIn this paper, we would like to move the practical importance of mathematical modeling a step forward
Mathematical modeling of gene amplification has provided good fits to experimental data (Axelrod et al 1993, Harnevo and Agur 1991: 1992; 1993, Kimme1 and Axelrod 1990, Kimmel et al 1992, Kimmel and Stivers 1994). These results suggest that drug resistance and other processes altering the behavior of cancer cells may be better described by multistage mechanisms, including a gradual increase in number of discrete units, rather than by classical irreversible mutation models described by Coldman and Goldie (1979; 1983)
Mathematical modeling of cancer populations taking into account both stochastic changes in number of gene copies in cells from one generation to another and the stochastic variability of cell lifetime can be based on branching random walk (Kimmel and Axelrod, 1990; Kimmel and Stivers, 1994). This approach leads to an infinite system of differential equations which may be used to model controling a cell population with evolving drug resistance caused by gene amplification or other mechanisms
Summary
We would like to move the practical importance of mathematical modeling a step forward. Mathematical modeling of gene amplification has provided good fits to experimental data (Axelrod et al 1993, Harnevo and Agur 1991: 1992; 1993, Kimme and Axelrod 1990, Kimmel et al 1992, Kimmel and Stivers 1994) These results suggest that drug resistance and other processes altering the behavior of cancer cells may be better described by multistage mechanisms, including a gradual increase in number of discrete units, rather than by classical irreversible mutation models described by Coldman and Goldie (1979; 1983). The multistage stepwise model of gene amplification or, more generally, of transformations of cancer cells, leads to new mathematical problems and resultr in novel dynamic properties of the systems involved These problems were first studied mathematically in (Kimmel and Axelrod, 1990) for the discrete-time models and in (Kimmel and Stivers, 1994) for the continuous-time models.
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