Dynamic optimization of a linear-quadratic model with incomplete repair and volume-dependent sensitivity and repopulation.

PURPOSE: The linear-quadratic model typically assumes that tumor sensitivity and repopulation are constant over the time course of radiotherapy. However, evidence suggests that the growth fraction increases and the cell-loss factor decreases as the tumor shrinks. We investigate whether this evolution in tumor geometry, as well as the irregular time intervals between fractions in conventional hyperfractionation schemes, can be exploited by fractionation schedules that employ time-varying fraction sizes.
METHODS: We construct a mathematical model of a spherical tumor with a hypoxic core and a viable rim, which is most appropriate for a prevascular tumor, and is only a caricature of a vascularized tumor. This model is embedded into the traditional linear-quadratic model by assuming instantaneous reoxygenation. Dynamic programming is used to numerically compute the fractionation regimen that maximizes the tumor-control probability (TCP) subject to constraints on the biologically effective dose of the early and late tissues.
RESULTS: In several numerical examples that employ five or 10 fractions per week on a 1-cm or 5-cm diameter tumor, optimally varying the fraction sizes increases the TCP significantly. The optimal regimen incorporates large Friday (afternoon, if 10 fractions per week) fractions that are escalated throughout the course of treatment, and larger afternoon fractions than morning fractions.
CONCLUSION: Numerical results suggest that a significant increase in tumor cure can be achieved by allowing the fraction sizes to vary throughout the course of treatment. Several strategies deserve further investigation: using larger fractions before overnight and weekend breaks, and escalating the dose (particularly on Friday afternoons) throughout the course of treatment.