This paper presents a review of the ways in which optimal control theory interacts with cancer chemotherapy. There are three broad areas of investigation. One involves miscellaneous growth kinetic models, the second involves cell cycle models, and the third is a classification of "other models." Both normal and tumor cell populations are included in a number of the models. The concepts of deterministic optimal control theory are applied to each model in such a way as to present a cohesive picture. There are applications to both experimental and clinical tumors. Suggestions for designing better chemotherapy strategies are presented.

Role of optimal control theory in cancer chemotherapy.

A method for DNA histogram analysis is described that depends only on the simple assumption that the data are normally distributed and a requirement that a G1 peak is present. A probability density function was derived from the assumption that extracted the S-phase component from the whole histogram. The model was tested with simulated data, and good agreement between predicted and known proportions in G1, S, and G1 + M was found. Good agreement was also found between duplicates of experimentally derived data. Some systematic errors are present in the analysis of certain types of histograms. However, these result in small errors when compared with biological and experimental variation and are less than the average of variation and are less than the average of algorithms in current use. The program required only two queued requests, those of the start and the end channels over which the analysis is to be performed. The algorithms perform rapidly on a microcomputer with only 28K addressable memory. Only two failures occurred in over 350 analyses and the method can be used for drug- and radiation-perturbed populations as well as with unperturbed.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/cyto.990080101

A pragmatic approach to the analysis of DNA histograms with a definable G1 peak

A class of mathematical models for cancer chemotherapy which have been described in the literature take the form of an optimal control problem over a finite horizon with control constraints and dynamics given by a bilinear system In this paper, we analyze a two-dimensional model in which the cell cycle is broken into two compartments The cytostatic agent used as control to kill the cancer cells is active only in the second compartment where cell division occurs and the cumulative effect of the drug is used to model the negative effect of the treatment on healthy cells It is shown that singular controls are not optimal for this model and the optimality properties of bang-bang controls are established Specifically, transversality conditions at the switching surfaces are derived In a nondegenerate setting, these conditions guarantee the local optimality of the flow if satisfied, while trajectories will be no longer optimal if they are violated

/pdf/optimal-bang-bang-controls-for-a-two-compartment-model-in-33nte9skhm.pdf

Optimal bang-bang controls for a two-compartment model in cancer chemotherapy

. Twelve methods for analysing FCM-histograms were compared using the same set of data. Some of the histograms that were analysed were simulated by computer and some were taken from experiments. Simulated data were generated assuming asynchronously growing cell populations and (i) measurement coefficients of variation (CV) from 2 to 16%; (ii) constant measurement CV or CV's increasing from G1 to G2 phase, and (iii) varying fractions of cells in each phase. Simulated data were also generated assuming synchronous cell populations in which a block in early S phase was applied and released. DNA histograms were measured for L-929 cells at various times after mitotic selection. Labelling indices were also measured for these cells at the same time.



The fractions of cells in the G1, S, and (G2+ M) phases were calculated by each analytical method and compared with the actual fractions used for simulation, or in case of experimental data, with autoradiographic results. Generally, all methods yielded reasonably accurate fractions of cells in each phase with relative errors in the range of 10–20%. However, most methods tended to overestimate G1 fractions and underestimate S fractions. In addition, variations in the shape of the S phase distribution often caused considerable errors. Phase fractions were also calculated for histograms of kinetically perturbed populations, simulated as well as experimental The errors were only slightly larger than for histograms from asynchronously growing cell populations.

A comparison of mathematical methods for the analysis of DNA histograms obtained by flow cytometry

A new model for obtaining the distributions of cells in the cell cycle by computer analysis of flow cytometry data is proposed. The model is able to analyze distributions from both asynchronous and synchronous cell populations. A culture of HeLa cells is pulse-labeled with tritiated thymidine at various times after synchronization by the mitotic selection method, then analyzed by flow cytometry and autoradiography. The results show good agreement between the percentage S-phase cells determined by the model and by autoradiography.

A model for the computer analysis of synchronous DNA distributions obtained by flow cytometry

Mathematical methods for determining cell DNA synthesis rate and age distribution utilizing flow microfluorometry

A discrete-time kinetic model for chemotherapy was developed to deal with the effects of antitumor drugs on the cell cycle and proliferation kinetics of experimental tumor cell populations in which cell kinetic responses of chemotherapy are represented in terms of perturbation of cell kinetic parameters—cell age, cell size and DNA content distributions. The time-course behavior of these cell kinetic parameters was predicted by solving the discrete-time state equations which characterize the dynamics of tumor-drug interactions. The amount of antitumor drug administered was expressed to be the control function of the state equations and the transition matrix representing two modes of drug action, namely, cell kill and progression delay or accumulation of cells due to drug, was derived. The performance of the model, assessed by examining the effects of cell cycle stage-specific agents such as cytosine arabinoside on spontaneous AKR leukemia, compared favorably with experimental data. Utilizing an optimization scheme in engineering systems studies, an analytical method is described for optimizing the regimen of drug administration so as to maximize the effectiveness of drug dosage schedules and minimize the use of toxic amounts of the drug. The superiority of the schedule designed by an optimization scheme was evident at the termination of therapy, although the schedule designed by experimental trials reduced the number of surviving tumor cells more effectively than the one designed by an optimization scheme during the earlier therapy period. In the model, the proposed schedule will function more effectively for the entire therapy period when additional parameters of drug characteristics, such as the toxicity to the host and drug resistance, are encompassed.

M. Kim

Papers

Mathematical methods for determining cell DNA synthesis rate and age distribution utilizing flow microfluorometry

A quantitative approach to the design of antitumor drug dosage schedule via cell cycle kinetics and systems theory.

Mathematical description and analysis of cell cycle kinetics and the application to Ehrlich ascites tumor.