  We conduct a theoretical study of various solution methods for the adaptive
fractionation problem. The two messages of this paper are: (i) dynamic
programming (DP) is a useful framework for adaptive radiation therapy,
particularly adaptive fractionation, because it allows us to assess how close
to optimal different methods are, and (ii) heuristic methods proposed in this
paper are near-optimal, and therefore, can be used to evaluate the best
possible benefit of using an adaptive fraction size.
  The essence of adaptive fractionation is to increase the fraction size when
the tumor and organ-at-risk (OAR) are far apart (a "favorable" anatomy) and to
decrease the fraction size when they are close together. Given that a fixed
prescribed dose must be delivered to the tumor over the course of the
treatment, such an approach results in a lower cumulative dose to the OAR when
compared to that resulting from standard fractionation. We first establish a
benchmark by using the DP algorithm to solve the problem exactly. In this case,
we characterize the structure of an optimal policy, which provides guidance for
our choice of heuristics. We develop two intuitive, numerically near-optimal
heuristic policies, which could be used for more complex, high-dimensional
problems. Furthermore, one of the heuristics requires only a statistic of the
motion probability distribution, making it a reasonable method for use in a
realistic setting. Numerically, we find that the amount of decrease in dose to
the OAR can vary significantly (5 - 85%) depending on the amount of motion in
the anatomy, the number of fractions, and the range of fraction sizes allowed.
In general, the decrease in dose to the OAR is more pronounced when: (i) we
have a high probability of large tumor-OAR distances, (ii) we use many
fractions (as in a hyper-fractionated setting), and (iii) we allow large daily
fraction size deviations.
