Albin Fredriksson

Minimax optimization for handling range and setup uncertainties in proton therapy.

PURPOSE Intensity modulated proton therapy (IMPT) is sensitive to errors, mainly due to high stopping power dependency and steep beam dose gradients. Conventional margins are often insufficient to ensure robustness of treatment plans. In this article, a method is developed that takes the uncertainties into account during the plan optimization. METHODS Dose contributions for a number of range and setup errors are calculated and a minimax optimization is performed. The minimax optimization aims at minimizing the penalty of the worst case scenario. Any optimization function from conventional treatment planning can be utilized by the method. By considering only scenarios that are physically realizable, the unnecessary conservativeness of other robust optimization methods is avoided. Minimax optimization is related to stochastic programming by the more general minimax stochastic programming formulation, which enables accounting for uncertainties in the probability distributions of the errors. RESULTS The minimax optimization method is applied to a lung case, a paraspinal case with titanium implants, and a prostate case. It is compared to conventional methods that use margins, single field uniform dose (SFUD), and material override (MO) to handle the uncertainties. For the lung case, the minimax method and the SFUD with MO method yield robust target coverage. The minimax method yields better sparing of the lung than the other methods. For the paraspinal case, the minimax method yields more robust target coverage and better sparing of the spinal cord than the other methods. For the prostate case, the minimax method and the SFUD method yield robust target coverage and the minimax method yields better sparing of the rectum than the other methods. CONCLUSIONS Minimax optimization provides robust target coverage without sacrificing the sparing of healthy tissues, even in the presence of low density lung tissue and high density titanium implants. Conventional methods using margins, SFUD, and MO do not utilize the full potential of IMPT and deliver unnecessarily high doses to healthy tissues.

A characterization of robust radiation therapy treatment planning methods—from expected value to worst case optimization

PURPOSE To characterize a class of optimization formulations used to handle systematic and random errors in radiation therapy, and to study the differences between the methods within this class. METHODS The class of robust methods that can be formulated as minimax stochastic programs is studied. This class generalizes many previously used methods, ranging between optimization of the expected and the worst case objective value. The robust methods are used to plan intensity-modulated proton therapy (IMPT) treatments for a case subject to systematic setup and range errors, random setup errors with and without uncertain probability distribution, and combinations thereof. As reference, plans resulting from a conventional method that uses a margin to account for errors are shown. RESULTS For all types of errors, target coverage robustness increased with the conservativeness of the method. For systematic errors, best case organ at risk (OAR) doses increased and worst case doses decreased with the conservativeness. Accounting for random errors of fixed probability distribution resulted in heterogeneous dose. The heterogeneities were reduced when uncertainty in the probability distribution was accounted for. Doing so, the OAR doses decreased with the conservativeness. All robust methods studied resulted in more robust target coverage and lower OAR doses than the conventional method. CONCLUSIONS Accounting for uncertainties is essential to ensure plan quality in complex radiation therapy such as IMPT. The utilization of more information than conventional in the optimization can lead to robust target coverage and low OAR doses. Increased target coverage robustness can be achieved by more conservative methods.

A critical evaluation of worst case optimization methods for robust intensity-modulated proton therapy planning

PURPOSE To critically evaluate and compare three worst case optimization methods that have been previously employed to generate intensity-modulated proton therapy treatment plans that are robust against systematic errors. The goal of the evaluation is to identify circumstances when the methods behave differently and to describe the mechanism behind the differences when they occur. METHODS The worst case methods optimize plans to perform as well as possible under the worst case scenario that can physically occur (composite worst case), the combination of the worst case scenarios for each objective constituent considered independently (objectivewise worst case), and the combination of the worst case scenarios for each voxel considered independently (voxelwise worst case). These three methods were assessed with respect to treatment planning for prostate under systematic setup uncertainty. An equivalence with probabilistic optimization was used to identify the scenarios that determine the outcome of the optimization. RESULTS If the conflict between target coverage and normal tissue sparing is small and no dose-volume histogram (DVH) constraints are present, then all three methods yield robust plans. Otherwise, they all have their shortcomings: Composite worst case led to unnecessarily low plan quality in boundary scenarios that were less difficult than the worst case ones. Objectivewise worst case generally led to nonrobust plans. Voxelwise worst case led to overly conservative plans with respect to DVH constraints, which resulted in excessive dose to normal tissue, and less sharp dose fall-off than the other two methods. CONCLUSIONS The three worst case methods have clearly different behaviors. These behaviors can be understood from which scenarios that are active in the optimization. No particular method is superior to the others under all circumstances: composite worst case is suitable if the conflicts are not very severe or there are DVH constraints whereas voxelwise worst case is advantageous if there are severe conflicts but no DVH constraints. The advantages of composite and voxelwise worst case outweigh those of objectivewise worst case.

Automated improvement of radiation therapy treatment plans by optimization under reference dose constraints

A method is presented that automatically improves upon previous treatment plans by optimization under reference dose constraints. In such an optimization, a previous plan is taken as reference and a new optimization is performed toward some goal, such as minimization of the doses to healthy structures under the constraint that no structure can become worse off than in the reference plan. Two types of constraints that enforce this are discussed: either each voxel or each dose-volume histogram of the improved plan must be at least as good as in the reference plan. These constraints ensure that the quality of the dose distribution cannot deteriorate, something that constraints on conventional physical penalty functions do not. To avoid discontinuous gradients, which may restrain gradient-based optimization algorithms, the positive part operators that constitute the optimization functions are regularized. The method was applied to a previously optimized plan for a C-shaped phantom and the effects of the choice of regularization parameter were studied. The method resulted in reduced integral dose and reduced doses to the organ at risk while maintaining target homogeneity. It could be used to improve upon treatment plans directly or as a means of quality control of plans.

Necessary and sufficient conditions for Pareto efficiency in robust multiobjective optimization

We provide necessary and sufficient conditions for robust efficiency (in the sense of Ehrgott et al., 2014) to multiobjective optimization problems that depend on uncertain parameters. These conditions state that a solution is robust efficient (under minimization) if it is optimal to a strongly increasing scalarizing function, and only if it is optimal to a strictly increasing scalarizing function. By counterexample, we show that the necessary condition cannot be strengthened to convex scalarizing functions, even for convex problems. We therefore define and characterize a subset of the robust efficient solutions for which an analogous necessary condition holds with respect to convex scalarizing functions. This result parallels the deterministic case where optimality to a convex and strictly increasing scalarizing function constitutes a necessary condition for efficiency. By a numerical example from the field of radiation therapy treatment plan optimization, we illustrate that the curvature of the scalarizing function influences the conservatism of an optimized solution in the uncertain case.

Deliverable navigation for multicriteria IMRT treatment planning by combining shared and individual apertures

We consider the problem of deliverable Pareto surface navigation for step-and-shoot intensity-modulated radiation therapy. This problem amounts to calculation of a collection of treatment plans with the property that convex combinations of plans are directly deliverable. Previous methods for deliverable navigation impose restrictions on the number of apertures of the individual plans, or require that all treatment plans have identical apertures. We introduce simultaneous direct step-and-shoot optimization of multiple plans subject to constraints that some of the apertures must be identical across all plans. This method generalizes previous methods for deliverable navigation to allow for treatment plans with some apertures from a collective pool and some apertures that are individual. The method can also be used as a post-processing step to previous methods for deliverable navigation in order to improve upon their plans. By applying the method to subsets of plans in the collection representing the Pareto set, we show how it can enable convergence toward the unrestricted (non-navigable) Pareto set where all apertures are individual.

Maximizing the probability of satisfying the clinical goals in radiation therapy treatment planning under setup uncertainty.

PURPOSE This paper introduces a method that maximizes the probability of satisfying the clinical goals in intensity-modulated radiation therapy treatments subject to setup uncertainty. METHODS The authors perform robust optimization in which the clinical goals are constrained to be satisfied whenever the setup error falls within an uncertainty set. The shape of the uncertainty set is included as a variable in the optimization. The goal of the optimization is to modify the shape of the uncertainty set in order to maximize the probability that the setup error will fall within the modified set. Because the constraints enforce the clinical goals to be satisfied under all setup errors within the uncertainty set, this is equivalent to maximizing the probability of satisfying the clinical goals. This type of robust optimization is studied with respect to photon and proton therapy applied to a prostate case and compared to robust optimization using an a priori defined uncertainty set. RESULTS Slight reductions of the uncertainty sets resulted in plans that satisfied a larger number of clinical goals than optimization with respect to a priori defined uncertainty sets, both within the reduced uncertainty sets and within the a priori, nonreduced, uncertainty sets. For the prostate case, the plans taking reduced uncertainty sets into account satisfied 1.4 (photons) and 1.5 (protons) times as many clinical goals over the scenarios as the method taking a priori uncertainty sets into account. CONCLUSIONS Reducing the uncertainty sets enabled the optimization to find better solutions with respect to the errors within the reduced as well as the nonreduced uncertainty sets and thereby achieve higher probability of satisfying the clinical goals. This shows that asking for a little less in the optimization sometimes leads to better overall plan quality.

Scenario-based radiation therapy margins for patient setup, organ motion, and particle range uncertainty

This work extends and validates the scenario-based generalization of margins presented in Fredriksson and Bokrantz (2016 Phys. Med. Biol. 61 2067-82). Scenario-based margins are, in their original form, a method for robust planning under setup uncertainty where the sum of a plan evaluation criterion over a set of scenarios is optimized. The voxelwise penalties in the summands are weighted by a distribution of coefficients defined such that the method is mathematically equivalent to the use of conventional geometric margins if the scenario doses are calculated using the static dose cloud approximation. The purpose of this work is to extend scenario-based margins to general types of geometric uncertainty and to validate their use on clinical cases. Specifically, we outline how to incorporate density heterogeneity in the calculation of coefficients and demonstrate the extended methods ability to safeguard against setup errors, organ motion, and range shifts (and combinations thereof). For a water phantom with a high-density slab partly covering the target, the extended form of scenario-based margins method led to improved target coverage robustness compared to the original method. At most minor differences in robustness were, however, observed between the extended and original method for a prostate and two lung patients, all treated with intensity-modulated proton therapy, yielding evidence that the calculation of weighting coefficients is generally insensitive to tissue heterogeneities. The scenario-based margins were, furthermore, verified to provide a comparable level of robustness to expected value and worst case optimization while circumventing some known shortcomings of these methods.

Robust radiotherapy planning

Motion and uncertainty in radiotherapy is traditionally handled via margins. The clinical target volume (CTV) is expanded to a larger planning target volume (PTV), which is irradiated to the prescribed dose. However, the PTV concept has several limitations, especially in proton therapy. Therefore, robust and probabilistic optimization methods have been developed that directly incorporate motion and uncertainty into treatment plan optimization for intensity modulated radiotherapy (IMRT) and intensity modulated proton therapy (IMPT). Thereby, the explicit definition of a PTV becomes obsolete and treatment plan optimization is directly based on the CTV. Initial work focused on random and systematic setup errors in IMRT. Later, inter-fraction prostate motion and intra-fraction lung motion became a research focus. Over the past ten years, IMPT has emerged as a new application for robust planning methods. In proton therapy, range or setup errors may lead to dose degradation and misalignment of dose contributions from different beams - a problem that cannot generally be addressed by margins. Therefore, IMPT has led to the first implementations of robust planning methods in commercial planning systems, making these methods available for clinical use. This paper first summarizes the limitations of the PTV concept. Subsequently, robust optimization methods are introduced and their applications in IMRT and IMPT planning are reviewed.

4D robust optimization including uncertainties in time structures can reduce the interplay effect in proton pencil beam scanning radiation therapy

PURPOSE Interplay effects in proton radiotherapy can create large distortions in the dose distribution and severely degrade the plan quality. Standard methods to mitigate these effects include abdominal compression, gating, and rescanning. We propose a new method to include the time structures of the delivery and organ motion in the framework of four-dimensional (4D) robust optimization to generate plans that are robust against interplay effects. METHODS The method considers multiple scenarios reflecting the uncertainties in the delivery and in the organ motion. In each scenario, the pencil beam scanning spots are distributed to different phases of the breathing cycle according to each individual spot time stamp, and a partial beam dose is calculated for each phase. The partial beam doses are accumulated on a reference phase through deformable image registrations. Minimax optimization is performed to take all scenarios into account simultaneously. For simplicity, the uncertainties in this proof of concept study are limited to variations in the breathing pattern. The method is evaluated for three different nonsmall cell lung cancer patients and compared to plans using conventional 4D robust optimization both with and without rescanning. We assess the ability of the method to mitigate distortions from the interplay effect over multiple evaluation scenarios using 4D dose calculations. This interplay evaluation is performed in an experimentally validated framework, which is independent of the optimization in the plan generation step. RESULTS For the three studied patients, 4D optimization including time structures is efficient, especially for large tumor motions, where rescanning of conventional 4D robustly optimized plans is not sufficient to mitigate the interplay effect. The most efficient approach of the new method is achieved when it is combined with rescanning. For the patient with the largest motion, the mean V95% is 99.2% and mean V107% is 3.65% for the best rescanned 4D plan optimized with time structure. This can be compared to conventional 4D optimized plans with mean V95% of 92.7% and mean V107% of 13.1%. CONCLUSIONS The current study shows the potential of reducing interplay effects in proton pencil beam scanning radiotherapy by incorporating organ motion and delivery characteristics in a 4D robust optimization.