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.
In intensity-modulated radiotherapy (IMRT), a treatment is designed to deliver high radiation doses to tumors, while avoiding the healthy tissue. Optimization-based treatment planning often produces sharp dose gradients between tumors and healthy tissue. Random shifts during treatment can cause significant differences between the dose in the “optimized” plan and the actual dose delivered to a patient. An IMRT treatment plan is delivered as a series of small daily dosages, or fractions, over a period of time (typically 35 days). It has recently become technically possible to measure variations in patient setup and the delivered doses after each fraction. We develop an optimization framework, which exploits the dynamic nature of radiotherapy and information gathering by adapting the treatment plan in response to temporal variations measured during the treatment course of a individual patient. The resulting (suboptimal) control policies, which re-optimize before each fraction, include two approximate dynamic programming schemes: certainty equivalent control (CEC) and open-loop feedback control (OLFC). Computational experiments show that resulting individualized adaptive radiotherapy plans promise to provide a considerable improvement compared to non-adaptive treatment plans, while remaining computationally feasible to implement. Copyright Springer Science+Business Media, LLC 2012 In this paper we consider the problems of modeling the tumor growth and optimize the chemotherapy treatment. A biologically based model is used with the goal of solving an optimization problem involving discrete delivery of antineoplastic drugs. Our model is formulated via compartmental analysis in order to take into account the cell cycle. The cost functional measures not only the final size of the tumor but also the total amount of drug delivered. We propose an algorithm based on the discrete maximum principle to solve the optimal drug schedule problem. Our numerical results show nice interpretations from the medical point of view.  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. 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.  Phase specific models for cancer chemotherapy are described as optimal control problems. We review earlier results on scheduling optimal therapies when the controls represent the effectiveness of chemotherapeutic agents, or, equivalently, when the simplifying assumption is made that drugs act instantaneously. In this paper we discuss how to incorporate more realistic medical aspects which hitherto have been neglected in the models. They include pharmacokinetic equations (PK) which model the drug's plasma concentration and various pharmacodynamic models (PD) which describe the effect the concentrations have on cells. We also briefly discuss the important medical issue of drug resistance. resistance Radiotherapy treatment is often delivered in a fractionated manner over a period of time. Emerging delivery devices are able to determine the actual dose that has been delivered at each stage facilitating the use of adaptive treatment plans that compensate for errors in delivery. We formulate a model of the day-to-day planning problem as a stochastic program and exhibit the gains that can be achieved by incorporating uncertainty about errors during treatment into the planning process. Due to size and time restrictions, the model becomes intractable for realistic instances. We show how heuristics and neuro-dynamic programming can be used to approximate the stochastic solution, and derive results from our models for realistic time periods. These results allow us to generate practical rules of thumb that can be immediately implemented in current planning technologies. We investigate an on-line planning strategy for the fractionated radiotherapy planning problem, which incorporates the effects of day-to-day patient motion. On-line planning demonstrates significant improvement over off-line strategies in terms of reducing registration error, but it requires extra work in the replanning procedures, such as in the CT scans and the re-computation of a deliverable dose profile. We formulate the problem in a dynamic programming framework and solve it based on the approximate policy iteration techniques of neuro-dynamic programming. In initial limited testing, the solutions we obtain outperform existing solutions and offer an improved dose profile for each fraction of the treatment. 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. 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. This paper uses optimal control theory in conjunction with a Gompertzian type model for cellular growth to determine the optimal method of administering cycle non-specific chemotherapy or more generally the optimal durations of treatment and rest periods during chemotherapy. The performance critera employed to determine the relative merits of the therapy include not only the destruction of malignant cells, but also the sparing of a critical normal tissue. Since these criteria are at odds with one another, the solutions are found which satisfy the Pareto optimality conditions. The goal in external beam radiotherapy for cancer is to maximize tumor-damage while limiting toxic eects of radiation on nearby healthy anatomies. This is achieved through spatial localization and temporal dispersion of radiation dose. Once a radi- ation intensity prole that achieves the maximum possible spatial localization is designed at the beginning of a multi-week treatment-course, the total planned dose is split into a series of predetermined equal-dosage fractions delivered daily so that healthy cells can recover between sessions. Thus, existing mathematical methods for treatment planning employ static-deterministic optimization techniques, and hence, cannot adapt to a tumor's uncertain biological response over time. In this tutorial, we review a recently proposed stochastic control framework, where the ultimate objective is to design individualized treatment strategies that dynamically adapt to tumor- response, to deliver the right dose to the right location at the right time. The current state of the art in cancer treatment by radiation optimizes beam intensity spatially such that tumors receive high dose radiation whereas damage to nearby healthy tissues is minimized. It is common practice to deliver the radiation over several weeks, where the daily dose is a small constant fraction of the total planned. Such a 'fractionation schedule' is based on traditional models of radiobiological response where normal tissue cells possess the ability to repair sublethal damage done by radiation. This capability is significantly less prominent in tumors. Recent advances in quantitative functional imaging and biological markers are providing new opportunities to measure patient response to radiation over the treatment course. This opens the door for designing fractionation schedules that take into account the patient's cumulative response to radiation up to a particular treatment day in determining the fraction on that day. We propose a novel approach that, for the first time, mathematically explores the benefits of such fractionation schemes. This is achieved by building a stylistic Markov decision process (MDP) model, which incorporates some key features of the problem through intuitive choices of state and action spaces, as well as transition probability and reward functions. The structure of optimal policies for this MDP model is explored through several simple numerical examples. 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. The reactions of a tumor cell population and a normal tissue cell population to irradiation are described by cell population kinetic models which consider factors such as repair, reoxygenation, and... Abstract Using a mathematical model based on existing models in the literature, the response of tumor and tumor-bed cell populations to fractionated radiation therapy is investigated. Problems of determining the optimal dose schedule for a given treatment calendar (specified number of doses and time intervals between doses) are formulated. A theoretical and computational method for solving such problems is proposed. Representative results, which support the efficacy of the method, are presented and discussed. 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.