We analyze the effect of tumor repopulation on optimal dose delivery in radiation therapy. We are primarily motivated by accelerated tumor repopulation towards the end of radiation treatment, which is believed to play a role in treatment failure for some tumor sites. A dynamic programming framework is developed to determine an optimal fractionation scheme based on a model of cell kill due to radiation and tumor growth in between treatment days. We find that faster tumor growth suggests shorter overall treatment duration. In addition, the presence of accelerated repopulation suggests larger dose fractions later in the treatment to compensate for the increased tumor proliferation. We prove that the optimal dose fractions are increasing over time. Numerical simulations indicate potential for improvement in treatment effectiveness.
Perhaps the closest related work is @cite_36 , which considers both faster tumor proliferation and re-oxygenation during the course of treatment. While a dose intensification strategy is also suggested in @cite_36 , the primary rationale for increasing dose fractions is different: it is concluded that due to the increase in tumor sensitivity from re-oxygenation, larger fraction sizes are more effective at the end of treatment. Our work, on the other hand, suggests dose intensification (i.e., larger doses over time) as a direct consequence of a model of accelerated tumor repopulation during the course of treatment.
@cite_36
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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. © 2000 Elsevier Science Inc. Dynamic optimization, Linear‐ quadratic, Reoxygenation, Repair, Repopulation. Radiotherapy is fractionized to increase the therapeutic ratio. Fractionation in conventional treatment is determined as part of the prescription, and a fixed fraction size is used for the whole course of treatment. Due to patients' day-to-day variations on the relative distance between the tumor and the organs at risk (OAR), a better therapeutic ratio may be attained by using an adaptive fraction size. Intuitively, we want to use a larger fraction size when OAR and the tumor are far apart and a smaller fraction size when OAR and the tumor are close to each other. The concept and strategies of adaptive fractionation therapy (AFT) are introduced in this paper. AFT is an on-line adaptive technique that utilizes the variations of internal structures to get optimal OAR sparing. Changes of internal structures are classified as different configurations according to their feasibility to the radiation delivery. A priori knowledge is used to describe the probability distribution of these configurations. On-line processes include identifying the configuration via daily image guidance and optimizing the current fraction size. The optimization is modeled as a dynamic linear programming problem so that at the end of the treatment course, the tumor receives the same planned dose while OAR receives less dose than the regular fractionation delivery. Extensive simulations, which include thousands of treatment courses with each course consisting of 40 fractions, are used to test the efficiency and robustness of the presented technique. The gains of OAR sparing depend on the variations on configurations and the bounds of the fraction size. The larger the variations and the looser the bounds are, the larger the gains will be. Compared to the conventional fractionation technique with 2 Gy fraction in 40 fractions, for a 20 variation on tumor–OAR configurations and [1 Gy, 3 Gy] fraction size bounds, the cumulative OAR dose with adaptive fractionation is 3–8 Gy, or 7–20 less than that of the regular fractionation, while maintaining the same cumulative tumor dose as prescribed. Radiation therapy is fractionized to differentiate the cell killing between the tumor and organ at risk (OAR). Conventionally, fractionation is done by dividing the total dose into equal fraction sizes. However, as the relative positions (configurations) between OAR and the tumor vary from fractions to fractions, intuitively, we want to use a larger fraction size when OAR and the tumor are far apart and a smaller fraction size when OAR and the tumor are close to each other. Adaptive fractionation accounts for variations of configurations between OAR and the tumor. In part I of this series, the adaptation minimizes the OAR (physical) dose and maintains the total tumor (physical) dose. In this work, instead, the adaptation is based on the biological effective dose (BED). Unlike the linear programming approach in part I, we build a fraction size lookup table using mathematical induction. The lookup table essentially describes the fraction size as a function of the remaining tumor BED, the OAR tumor dose ratio and the remaining number of fractions. The lookup table is calculated by maximizing the expected survival of OAR and preserving the tumor cell kill. Immediately before the treatment of each fraction, the OAR-tumor configuration and thus the dose ratio can be obtained from the daily setup image, and then the fraction size can be determined by the lookup table. Extensive simulations demonstrate the effectiveness of our method compared with the conventional fractionation method. We conduct a theoretical study of various solution methods for the adaptive fractionation problem. The two messages of this paper are as follows: (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.