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 sensitivit y and repopulation are constant over the time course of radiotherapy. Howev er, evidence suggests that the growth fraction increases and the cell-loss factor decreases as the tumor shrinks. We investigate whether this evolutio n in tumor geometry, as well as the irregular time intervals between fracti ons in conventional hyperfractionation schemes, can be exploited by fractio nation schedules that employ time-varying fraction sizes. Methods: We construct a mathematical model of a spherical tumor with a hypo xic core and a viable rim, which is most appropriate for a prevascular tumo r, and is only a caricature of a vascularized tumor. This model is embedded into the traditional linear-quadratic model by assuming instantaneous reox ygenation, Dynamic programming is used to numerically compute the fractiona tion regimen that maximizes the tumor-control probability (TCP) subject to constraints on the biologically effective dose of the early and late tissue s. 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 size s increases the TCP significantly. The optimal regimen incorporates large F riday (afternoon, if 10 fractions per week) fractions that are escalated th roughout the course of treatment, and larger afternoon fractions than morni ng 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: usin g larger fractions before overnight and weekend breaks, and escalating the dose (particularly on Friday afternoons) throughout the course of treatment . (C) 2000 Elsevier Science Inc.