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.
There is a significant amount of literature, especially from the mathematical biology community, on the use of control theory and dynamic programming (DP) for optimal cancer therapy. Several of these works ( @cite_57 @cite_26 @cite_54 @cite_4 ) have looked into optimization of chemotherapy. For radiation therapy fractionation, some studies ( @cite_46 @cite_34 @cite_36 ) have used the DP approach based on deterministic biological models, as done in this paper. However, these works have not carried out a detailed mathematical analysis of the implications of optimal dose delivery in the presence of accelerated repopulation. Using imaging information obtained between treatment days, dynamic optimization models have been developed to adaptively compensate for past accumulated errors in dose to the tumor ( @cite_53 @cite_14 @cite_42 @cite_47 ). There also has been work on online approaches which adapt the dose and treatment plan based on images obtained immediately prior to treatment ( @cite_39 @cite_55 @cite_2 @cite_48 @cite_27 @cite_5 @cite_13 ).
@cite_47 @cite_26 @cite_4 @cite_14 @cite_36 @cite_48 @cite_54 @cite_53 @cite_42 @cite_55 @cite_39 @cite_57 @cite_27 @cite_2 @cite_5 @cite_46 @cite_34 @cite_13
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State-of-the-art methods for optimizing cancer treatment over several weeks of external beam radiotherapy take a static–deterministic view of the treatment planning process, mainly focusing on spatial distribution of dose. Recent progress in quantitative functional imaging as well as mathematical models of tumor response to radiotherapy is increasingly enabling treatment planners to monitor predict a patient’s biological response over weeks of treatment. In this paper we introduce dynamic biologically conformal radiation therapy (DBCRT), a mathematical framework intended to exploit these emerging technological and biological modeling advances to design patient-specific radiation treatment strategies that dynamically adapt to the spatiotemporal evolution of a patient’s biological response over several treatment sessions in order to achieve the best possible health outcome. More specifically, we propose a discrete-time stochastic control formalism where we use the patient’s biological condition to model the system state and the beam intensities as controls. Three approximate control schemes are then applied and compared for efficiency. Numerical simulations on test cases show that DBCRT results in a 64–98 improvement in treatment efficacy as compared to the more conventional static–deterministic approach. 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. While ART has been studied for years, the specific quantitative implementation details have not. In order for this new scheme of radiation therapy (RT) to reach its potential, an effective ART treatment planning strategy capable of taking into account the dose delivery history and the patient's on-treatment geometric model must be in place. This paper performs a theoretical study of dynamic closed-loop control algorithms for ART and compares their utility with data from phantom and clinical cases. We developed two classes of algorithms: those Adapting to Changing Geometry and those Adapting to Geometry and Delivered Dose. The former class takes into account organ deformations found just before treatment. The latter class optimizes the dose distribution accumulated over the entire course of treatment by adapting at each fraction, not only to the information just before treatment about organ deformations but also to the dose delivery history. We showcase two algorithms in the class of those Adapting to Geometry and Delivered Dose. A comparison of the approaches indicates that certain closed-loop ART algorithms may significantly improve the current practice. We anticipate that improvements in imaging, dose verification and reporting will further increase the importance of adaptive algorithms. 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.