Scheduler automaton

The above model consists of two main parts a scheduler and the plant to schedule. Since the scheduler defines exactly when to start and finish the tasks, the behavior of the entire system is deterministic. Running both parts, the scheduler and the plant, corresponds to a simulation in which one single behavior of the composed system is obtained. Obviously, this situation requires the presence of a scheduler which knows the (optimal) schedule. If such a scheduler is absent, then the resource automata in the plant receive no signals. Scheduling of a plant can be understood as the task of finding a scheduler automaton which provides the optimal schedule with respect to an optimization criterion. In the sequel it is assumed that a scheduler does not exist and the automata model is designed as shown in Figure 10.4. 

The effort to find the best solution can be reduced by defining an appropriate guiding Junction. A guiding function (third line of the algorithm) attempts to select those nodes from W, which are assessed to be best or cheapest. If a perfect guiding function was given (similar to a scheduler automaton which knows the optimal solution), it would always make the optimal decisions and step-by-step select the nodes in Table 10.1 from W. 

In the context of reachability analysis, this graph is called symbolic reachability graph of the automaton A and can be searched using shortest path search techniques as widely used in computer science. Hence, the task of finding the cost-optimal schedule is to find the shortest (or cheapest) path in a (priced) symbolic reachability graph. 

To clarify the reason why different circadian schedules of 5-FU delivery have distinct cytotoxic effects, we used the cell cycle automaton model to determine the time evolution of the fraction of cells in S phase in response to different patterns of circadian drug administration, for a cell cycle variability of 15%. The results, shown in Fig. 10.5, correspond to the case considered in Fig. 10.4, namely, entrainment of a 22-h cell cycle by the circadian clock. The data for Fig. 10.5a clearly indicate why the circadian schedule with a peak at 4 a.m. is the least toxic. The reason is that the fraction of cells in S phase is then precisely in antiphase with the circadian profile of 5-FU. Since 5-FU only affects cells in the S phase, the circadian delivery of the anticancer drug in this case kills but a negligible amount of cells. 

The cases of peak delivery at 10 a.m. (Fig. 10.5b) or 10 p.m. (Fig. 10.5d) are intermediate between the two preceding cases. Overlap between the peak of 5-FU and the peak of cells in S phase is only partial, but it is still greater in the case of the peak at 10 a.m., so that this pattern is the second most toxic, followed by the circadian delivery centered around 10 p.m. The comparison of the four panels Fig. 10.5a-d explains the results of Fig. 10.4a on the marked differences in cytotoxic effects of the four 5-FU circadian delivery schedules. The use of the cell cycle automaton helps clarify the dynamic bases that underlie the distinctive effects of the peak time in the circadian pattern of anticancer drug delivery. 

The case of the continuous infusion of 5-FU is considered in Fig. 10.5e. Because the total amount of 5-FU administered over 24 h is the same as for the circadian semi-sinusoidal patterns, the level of 5-FU - and hence the cytotoxic effect of the drug - is sometimes below and sometimes above that reached with the circadian schedule. The numerical simulations of the automaton model indicate that the cytoxicity is comparable to that observed for the most toxic circadian pattern, with peak delivery of 5-FU at 4 p.m. 

The results presented here point to the interest of measuring, both in normal and tumor cell populations, parameters such as the duration of the cell cycle phases and their variability, as well as the presence or absence of entrainment by the circadian clock. As shown by the results obtained with the cell cycle automaton model, these data are crucial for using the model to predict the differential outcome of various anticancer drug delivery schedules on normal and tumor cell populations. In a sub-... 

The present modeling approach to circadian cancer chronotherapy is based on an automaton model for the cell cycle. Continuous approaches to cell cycle progression have also been used to study the link between cell proliferation and circadian rhythms [44] and to determine, in conjunction with optimal control theory, the most efficient circadian schedules of anticancer drug administration [45]. Including more molecular details of the cell cycle in continuous models for cell populations represents a promising line for future research. Hybrid models incorporating molecular details into the automaton approach presented here will also likely be developed. 

Here, as in a previous publication [33], we used the cell cycle automaton model to probe the cytotoxic effect of various patterns of circadian or continuous 5-FU delivery. The results provide a framework to account for experimental and clinical observations, and to help us predict optimal modes of drug delivery in cancer chronotherapy. By explaining the differential cytotoxicity of various circadian schedules of 5-FU delivery, the model clarifies the foundations of cancer chronothera-peutics. In view of its versatility and reduced number of parameters, the automaton model could readily be applied to probe the administration schedules of other types of anticancer medications active on other phases of the cell cycle. 

A behavior FSM has incompletely specified time behavior— in contrast to an RTFSM, we do not know the exact clock cycle on which each iiq>ut event or output event will occur. The I/O behavior of an automaton can be defined in terms of events, where each event is a pin, value, timestamp) triples. An RTFSM s inputs and outputs are totally ordered the time of every I/O event is completely determined. A behavior FSM s I/O behavior, in contrast, is partially ordered. The formulas that describe the times of events need not have a single solution. Each unique solution is a schedule of events for the automaton. 

Our joint work with Miriam Leeser on automaton models for scheduling started at the Fourth International Workshop on High-Level Synthesis our work with Miriam on BFSMs is central to PUBSS. Thanks to Mike McFarland and Raul Camposano for valuable feedback on a variety of problems. 

---