Delay differential equations and the dose‐time dependence of early radiotherapy reactions

The dose‐time dependence of early radiotherapy reactions impacts on the design of accelerated fractionation schedules—oral mucositis, for example, can be dose limiting for short treatments designed to avoid tumor repopulation. In this paper a framework for modeling early reaction dose‐time dependence is developed. Variation of stem cell number with time after the start of a radiation schedule is modeled using a first‐order delay differential equation (DDE), motivated by experimental observations linking the speed of compensatory proliferation in early reacting tissues to the degree of tissue damage. The modeling suggests that two types of early reaction radiation response are possible, stem cell numbers either monotonically approaching equilibrium plateau levels or overshooting before returning to equilibrium. Several formulas have been derived from the delay differential equation, predicting changes in isoeffective total radiation dose with schedule duration for different types of fractionation scheme. The formulas have been fitted to a wide range of published animal early reaction data, the fits all implying a degree of overshoot. Results are presented illustrating the scope of the delay differential model: most of the data are fitted well, although the model struggles with a few datasets measured for schedules with distinctive dose‐time patterns. Ways of extending the current model to cope with these particular dose‐time patterns are briefly discussed. The DDE approach is conceptually more complex than earlier descriptive dose‐time models but potentially more powerful. It can be used to study issues not addressed by simpler models, such as the likely effects of increasing or decreasing the dose‐per‐day over time, or of splitting radiation courses into intense segments separated by gaps. It may also prove useful for modeling the effects of chemoirradiation.