Comparison of linear and nonlinear programming approaches for “worst case dose” and “minmax” robust optimization of intensity‐modulated proton therapy dose distributions

Abstract Robust optimization of intensity‐modulated proton therapy (IMPT) takes uncertainties into account during spot weight optimization and leads to dose distributions that are resilient to uncertainties. Previous studies demonstrated benefits of linear programming (LP) for IMPT in terms of delivery efficiency by considerably reducing the number of spots required for the same quality of plans. However, a reduction in the number of spots may lead to loss of robustness. The purpose of this study was to evaluate and compare the performance in terms of plan quality and robustness of two robust optimization approaches using LP and nonlinear programming (NLP) models. The so‐called “worst case dose” and “minmax” robust optimization approaches and conventional planning target volume (PTV)‐based optimization approach were applied to designing IMPT plans for five patients: two with prostate cancer, one with skull‐based cancer, and two with head and neck cancer. For each approach, both LP and NLP models were used. Thus, for each case, six sets of IMPT plans were generated and assessed: LP‐PTV‐based, NLP‐PTV‐based, LP‐worst case dose, NLP‐worst case dose, LP‐minmax, and NLP‐minmax. The four robust optimization methods behaved differently from patient to patient, and no method emerged as superior to the others in terms of nominal plan quality and robustness against uncertainties. The plans generated using LP‐based robust optimization were more robust regarding patient setup and range uncertainties than were those generated using NLP‐based robust optimization for the prostate cancer patients. However, the robustness of plans generated using NLP‐based methods was superior for the skull‐based and head and neck cancer patients. Overall, LP‐based methods were suitable for the less challenging cancer cases in which all uncertainty scenarios were able to satisfy tight dose constraints, while NLP performed better in more difficult cases in which most uncertainty scenarios were hard to meet tight dose limits. For robust optimization, the worst case dose approach was less sensitive to uncertainties than was the minmax approach for the prostate and skull‐based cancer patients, whereas the minmax approach was superior for the head and neck cancer patients. The robustness of the IMPT plans was remarkably better after robust optimization than after PTV‐based optimization, and the NLP‐PTV‐based optimization outperformed the LP‐PTV‐based optimization regarding robustness of clinical target volume coverage. In addition, plans generated using LP‐based methods had notably fewer scanning spots than did those generated using NLP‐based methods.

after PTV-based optimization, and the NLP-PTV-based optimization outperformed the LP-PTV-based optimization regarding robustness of clinical target volume coverage. In addition, plans generated using LP-based methods had notably fewer scanning spots than did those generated using NLP-based methods.

| INTRODUCTION
Intensity-modulated proton therapy (IMPT) is potentially one of the most effective ways to treat cancer because it can deliver highly conformal and homogenous dose distributions to a target with a complex shape while maximally sparing adjacent healthy tissues. 1 It is delivered using thin scanning beams (beamlets) of protons with a sequence of discrete energies. For a given energy, the dose from a proton beam or beamlet increases as a function of depth of penetration in the patient until it reaches a peak (the Bragg peak) and then falls sharply to near zero. The high potential of IMPT owes to the fact that protons have a finite range and a sharp dose falloff at the end of the range and that IMPT can control the range (energy) and intensity of individual beamlets. For IMPT to be effective, a high degree of precision and accuracy in delivery is required so that the dose distribution that is actually delivered is a good approximation of the dose distribution in the treatment plan. [2][3][4][5][6][7] Unfortunately, the characteristics of protons that make them suitable for radiotherapy also make them sensitive to various types of uncertainty. The two most important sources of uncertainty in IMPT are the beam range and patient setup uncertainties. These uncertainties can result in deviation of the delivered IMPT dose distribution from the planned distribution, which may lead to suboptimal treatment decisions and unforeseen outcomes. Therefore, these uncertainties must be considered during IMPT plan optimization.
In photon therapy, the conventional approach to handling patient setup uncertainties and organ motion is to expand the clinical target volume (CTV) by an empirically determined margin to form a planning target volume (PTV). The underlying assumption in the determination of the CTV-to-PTV margin is that the CTV will be sufficiently covered with high probability (e.g., 95%) in the face of uncertainties.
This approach works well for photon therapy because the variations in photon dose distributions when patient anatomy changes are relatively small. 7,8 However, for IMPT, uncertainties can cause substantial perturbations in the dose distributions not only in the CTV-to-PTV margins but also within the CTV as well as in regions distal and proximal to the target al.ong the beam paths. Dose distributions in normal tissues lateral to the CTV also may be substantially perturbed, especially when protons pass through complex heterogeneities. Thus, simply applying the concept of PTV to proton therapy cannot efficiently mitigate the impact of uncertainties, so alternative approaches to PTV-based optimization are required. 9 One such approach is robust optimization, which aims to produce optimal, resilient IMPT plans in the face of uncertainties.
Researchers have conducted several probabilistic and scenario-based studies to incorporate uncertainties into IMPT plan optimization. [2][3][4][5][6][7][8][9] In a probabilistic approach, the expectation value of the random objective function is optimized. 5,6 Three scenario-based approaches have been proposed, that is, the "worst case dose", 10 "minmax," 9 and "composite objective" 11 robust optimization. In reality, they are all worst case approaches. The first is based on the worst case dose in each voxel. The second considers the worst case value of the objective function for the dose distribution as a whole. The third takes the worst case value of the objective function for the dose distribution in each structure. All of these scenario-based approaches can work with a linear programming 12 or nonlinear programming (NLP) 7 model. Some groups have proposed a worst case dose robust optimization approach using an LP model to consider range uncertainties, 5 This nonlinear constrained model does not assume a probability distribution for uncertainties. 9 Chen et al. also reported on a multicriteria minmax optimization approach using a piecewise-linear convex constrained model similar to that proposed by Fredriksson and colleagues. 8 To date, only a handful of studies have compared different IMPT robust optimization approaches. Fredriksson generalized a class of robust optimization methods, including expected value and minmax optimization. 14 He studied and compared special cases and found that the minmax approach had advantages over other methods in controlling hot spots within the target and sparing the organs at risk (OARs). More recently, Fredriksson and Bokrantz compared three approaches to NLP-based worst case dose optimization. 11 However, they did not identify a dominant broadly applicable approach. They observed identical behavior of plan quality and robustness in all three approaches without any conflicting planning criteria but clear differences in the presence of conflicting criteria.
In all of these studies, researchers investigated robust optimization techniques for IMPT using either an LP or NLP model. However, a comparison of the performance of LP and NLP models with one or more robust optimization approaches has yet to be reported. Therefore, we performed this study to identify and outline differences in the behavior of various LP-and NLP-based approaches and models for IMPT robust optimization. To that end, we developed and evaluated worst case dose (voxel-wise) and minmax methods for both LP and NLP formalism models. To better understand the influence of LP-and NLP-based models on robust optimization results, we also performed PTV-based conventional optimization with both LP and NLP objective functions. Thus, we compared six optimization methods: LP-PTV-based, NLP-PTV-based, LP-worst case dose, NLP-worst case dose, LP-minmax, and NLP-minmax. Specifically, we compared the plan optimality and robustness and number of scanning spots (surrogate of plan efficiency) for the plans created with each of the six methods. Although Fredriksson and Bokrantz previously compared quadratic worst case dose and minmax approaches to account for setup uncertainties for two patients with prostate cancer, the performance of different methods may depend on the treatment site. 11 Therefore, in our study, we compared the six methods for two patients with prostate cancer, a patient with a skull base tumor, and two patients with head and neck cancer. In addition, we considered both beam range and patient setup uncertainties.

2.A | Patient data, beam configurations, and uncertainty scenarios
The relative performance of various robust optimization approaches was evaluated by regenerating treatment plans for two patients with prostate cancer, one with skull base cancer, and two with head and neck cancer, all of whom had undergone proton therapy at our institution. Two lateral fields were used for prostate cancer cases, whereas three fields were used for the other three cases. For each patient, eight uncertainty scenarios were assumed: two setup uncertainty scenarios (AE5 mm for prostate cancer and AE3 mm for the other cancers) in the x, y, and z directions and two range uncertainty scenarios (AE3.5% of the nominal range of the beams). For each patient, the six optimization methods described above and below were used to account for range and setup uncertainties. The PTV was chosen as the target for the conventional optimization approach and, appropriately, the CTV was used as the target for the robust optimization approaches. 7,9 The prescribed doses, target volumes, beam angles, dose grid resolutions, and margins used for the robust optimization approaches are listed in Table 1. All beams were coplanar (couch angle, 0°).

2.B.2 | Worst case dose robust optimization
For a voxel inside the target, the minimum dose of the voxel of all dose distributions corresponding to different uncertainty scenarios was selected. For any voxel outside the target, the maximum dose of the voxel was selected. This formed the worst case dose distribution as follows: 2,3 Where d r ij is the influence matrix, which denotes the dose contributed by the J th beamlet per unit weight, and is received by voxel i under scenario r.
T A B L E 1 Dose and beam configurations and uncertainty scenarios for the robust optimization approaches used to generate treatment plans for each patient in our study. RBE denotes relative biological effectiveness. Robust optimization was performed by substituting the worst case dose distribution for the nominal dose distribution in each iteration. The decision variable x j was the intensity of beamlet j. The optimization models applied to both CTV and OARs were as follows:

Case
LP : min

2.B.3 | Minmax robust optimization
An alternative approach to worst case dose robust optimization is the minmax method described by Fredriksson et al. 9 This method is designed to minimize the penalty of the worst case dose distribution scenario. Specifically, the objective function is evaluated under a number of treatment scenarios, and the worst calculated objective function is selected. In contrast with the worst case dose distribution, only physically realizable scenarios are considered. For this optimization approach, the dose distribution was calculated as Both CTV and OARs are incorporated in the optimization model.
Relative strengths of the minmax and voxel-wise robust optimization methods are described in the literature. 9,11

2.C | Plan generation and comparison
Six IMPT plans were generated using the six optimization methods and compared in terms of plan quality and robustness and delivery efficiency. Only dose constraints (hard constraints for LP-based models and soft constraints for NLP-based models) were used in plan optimization. Both dose and dose-volume constraints of nominal and worst case dose distributions were reviewed after optimization. If an optimized plan failed to meet such constraints, the plan could be reoptimized by adjusting objective weights or dose constraints until all constraints were satisfied. Some of the key criteria for plan quality we used are listed below. Those comparisons are discussed in the Results section.
To compare the robustness of the IMPT plans generated using the different methods, families of DVHs corresponding to different uncertainty scenarios were plotted along with the nominal DVHs.
The resulting envelopes were used to assess the sensitivity of the plans under the uncertainty scenarios. 15 The DVH-family bandwidth method was also used to evaluate and compare the robustness of the different methods. 16  based optimization methods (Fig. 1, first two rows) was notably less robust than that with plans generated using other methods. Specifically, the DVH bands for the CTV were wider for PTV-based optimization methods than for robust optimization methods, indicating that the plans generated using robust optimization were more robust regarding setup and range uncertainties than were those generated using conventional PTV-based methods. Among the PTV-based optimization methods, the NLP-PTV-based method outperformed the LP-PTV-based method in terms of robustness of CTV coverage. Furthermore, comparison of the robustness of the plans created using robust optimization (Fig. 1, bottom four rows) demonstrated that the worst case dose methods outperformed the corresponding minmax methods in covering the CTV. However, the robustness of normal tissue sparing was similar for all of the methods.
To further evaluate plan robustness, we compared the DVHfamily band widths at key dose-volume indices for the six methods for patient 1 (Fig. 2). For robust optimization, the LP-based methods covered the CTV D95 slightly more robustly than did the corresponding NLP-based methods. Using four robust optimization methods (LP-worst case dose, NLP-worst case dose, LP-minmax, and NLP-minmax), the NLP-minmax plan was less robust for CTV D95 than were the other methods. However, the robustness for the rectum V70 and bladder V65 did not differ markedly among the four methods. In addition, the bladder V65 in nominal plans for the PTVbased optimization methods was consistently improved by robust optimization.
DVH indices for nominal doses and DVH-family band widths for patient 2 (the other prostate cancer patient) are shown in Fig. 3. The results of plan quality and robustness evaluation for this patient were consistent with those for the other prostate cancer patient.
The robustness of CTV coverage for the PTV-based optimization methods was inferior to that for the robust optimization methods.
Among the PTV-based optimization methods, NLP-PTV-based optimization was superior to the LP-PTV-based method in robustness of CTV coverage. Using four robust optimization methods, robustness of CTV D95 for the NLP-minmax optimized plan was outperformed by other three robust optimization methods. We found no marked variations in the robustness of OAR (rectum and bladder) sparing among the different robust optimization methods. However, the rectum and bladder sparing with the NLP-based methods was inferior to that with the LP-based methods. Also, the robustness of CTV coverage for the worst case dose robust optimization methods was superior to that for the minmax methods. When comparing LP-and NLP-based robust optimization methods, we observed that the CTV D95 robustness for the NLP-based methods was inferior to that for similar LP-based methods.
The robustness of CTV coverage for the PTV-based methods was inferior to that for the robust optimization methods for patient 3 (the skull base cancer patient) (Fig. 4). In this case, CTV coverage robustness provided by the NLP-based optimization methods was superior to that provided by the corresponding LP-based methods.
Moreover, worst case dose robust optimization methods generated plans with better CTV coverage robustness than did minmax methods. However, the robustness for the brainstem and temporal lobes in terms of variations in V50 and V60 was comparable for all opti-  In addition, we compared the PTV coverage for the six optimization methods.  shows the number of spots for the optimal solution for the six methods. For each patient, the bar on the far right in Fig. 7 indicates the total number of spots in the initial spot arrangement. This number was reduced to a smaller number after optimization. In other words, some of the spots were turned off during optimization. We observed many more spots with NLP-based methods than with LP-based methods ( Table 3).

| DISCUSSION
The  T A B L E 3 Percentages of the total number of spots selected using LP-and NLP-based models averaged for the PTV-based, worst case dose, and minmax approaches.

Model
Percentage of spots selected the remaining "easy" scenarios are discarded in the optimization model. 11 Our results were consistent with those reported by Fredriksson and Bokrantz 11 for the two typical prostate cancer cases with the same beam arrangement (two lateral opposed beams) and the skull base cancer case, in which the worst case dose methods provided more robust target coverage than did the minmax methods.
We observed a clear difference between the LP-and NLP-based models in terms of the number of spots included in the optimized plans ( Table 3). One of the most important features of linear optimization is that the optimal solutions are sparser than those with nonlinear optimization. 18  resulting from truncation of monitor units to meet the minimum monitor unit constraint can be greatly avoided using LP-based models. 21 We also observed that LP-PTV-based optimization was outperformed by NLP-PTV-based optimization in terms of robustness of CTV coverage in all five cases. The robustness of sparing critical structures for NLP-PTV-based optimization was either superior to or comparable with that for LP-PTV-based optimization. This may have resulted from the fact that the plans generated using the LPbased methods were sparser than the plans created using the NLP- A limitation of the robust optimization approaches evaluated in this study was the limited set of predefined uncertainty scenarios.
These scenarios accounted only for setup uncertainties along the x, y, and z axes and range uncertainties. We did not consider movements in other directions, nonrigid patient movements, changes in patient anatomy, or other sources of uncertainties. However, incorporating more error scenarios requires more time to solve problems, which may not be practical. Some studies have used a larger number of scenarios. 22,23 However, evaluation of robustness on the basis of a limited number of setup uncertainties is a common practice and considered to be predictive of both robust optimization and robustness evaluation in IMPT planning. 22

| CONCLUSION
The findings of this study reiterate the importance of implementing robust optimization for IMPT. Our results also demonstrate that the robust optimization method ideally should be chosen on a site-bysite basis. None of the robust optimization methods (consisting of worst case dose and minmax methods with both LP-and NLP-based models) consistently outperformed the others in terms of either nominal plan quality or plan robustness against uncertainties. The LP-based methods provided more robust target coverage than did the NLP-based methods where all uncertainty scenarios were able to meet tight dose constraints, as shown for prostate cancer cases.
However, the robustness of the plans created using the LP-based methods was inferior in more challenging cases, such as head and neck cancer cases, in which some scenarios prevented LP with relatively tighter constraints from finding a feasible solution. In addition, the LP-based methods resulted in significantly fewer scanning spots than did the NLP-based methods for the same quality and robustness. With conventional PTV-based optimization, NLP-based method outperformed LP-based method regarding robustness of CTV coverage.

ACKNOWLED GMENTS
We are very grateful to Donald Norwood for editorial assistance in improving the manuscript. The research described in this paper was supported by grant number U19 CA021239 from the National Cancer Institute.