Optimal schedules of fractionated radiation therapy by way of the greedy principle: biologically-based adaptive boosting

We revisit a long-standing problem of optimization of fractionated radiotherapy and solve it in considerable generality under the following three assumptions only: (1) repopulation of clonogenic cancer cells between radiation exposures follows linear birth-and-death Markov process; (2) clonogenic cancer cells do not interact with each other; and (3) the dose response function s(D) is decreasing and logarithmically concave. Optimal schedules of fractionated radiation identified in this work can be described by the following 'greedy' principle: give the maximum possible dose as soon as possible. This means that upper bounds on the total dose and the dose per fraction reflecting limitations on the damage to normal tissue, along with a lower bound on the time between successive fractions of radiation, determine the optimal radiation schedules completely. Results of this work lead to a new paradigm of dose delivery which we term optimal biologically-based adaptive boosting (OBBAB). It amounts to (a) subdividing the target into regions that are homogeneous with respect to the maximum total dose and maximum dose per fraction allowed by the anatomy and biological properties of the normal tissue within (or adjacent to) the region in question and (b) treating each region with an individual optimal schedule determined by these constraints. The fact that different regions may be treated to different total dose and dose per fraction mean that the number of fractions may also vary between regions. Numerical evidence suggests that OBBAB produces significantly larger tumor control probability than the corresponding conventional treatments. (paper)