Contour scanning for penumbra improvement in pencil beam scanned proton therapy

TEAM: Triangular-mEsh Adaptive and Multiscale proton spot generation method

BACKGROUND

Proton therapy is preferred for its dose conformality to spare normal tissues and organs-at-risk (OAR) via Bragg peaks with negligible exit dose. However, proton dose conformality can be further optimized: (1) the spot placement is based on the structured (e.g., Cartesian) grid, which may not offer conformal shaping to complex tumor targets; (2) the spot sampling pattern is uniform, which may be insufficient at the tumor boundary to provide the sharp dose falloff, and at the same time may be redundant at the tumor interior to provide the uniform dose coverage, for example, due to multiple Coulomb scattering (MCS); and (3) the lateral spot penumbra increases with respect to the depth due to MCS, which blurs the lateral dose falloff. On the other hand, while (1) the deliverable spots are subject to the minimum-monitor-unit (MMU) constraint, and (2) the dose rate is proportional to the MMU threshold, the current spot sampling method is sensitive to the MMU threshold and can fail to provide satisfactory plan quality for a large MMU threshold (i.e., high-dose-rate delivery).

PURPOSE

This work will develop a novel Triangular-mEsh-based Adaptive and Multiscale (TEAM) proton spot generation method to address these issues for optimizing proton dose conformality and plan delivery efficiency.

METHODS

Compared to the standard clinically-used spot placement method, three key elements of TEAM are as follows: (1) a triangular mesh instead of a structured grid: the triangular mesh is geometrically more conformal to complex target shapes and therefore more efficient and accurate for dose shaping inside and around the target; (2) adaptive sampling instead of uniform sampling: the adaptive sampling consists of relatively dense sampling at the tumor boundary to create the sharp dose falloff, which is more accurate, and coarse sampling at the tumor interior to uniformly cover the target, which is more efficient; and (3) depth-dependent sampling instead of depth-independent sampling: the depth-dependent sampling is used to compensate for MCS, that is, with increasingly dense sampling at the tumor boundary to improve dose shaping accuracy, and increasingly coarse sampling at the tumor interior to improve dose shaping efficiency, as the depth increases. In the TEAM method the spot locations are generated for each energy layer and layer-by-layer in the multiscale fashion; and then the spot weights are derived by solving the IMPT problem of dose-volume planning objectives, MMU constraints, and robustness optimization with respect to range and setup uncertainties.

RESULTS

Compared to the standard clinically-used spot placement method UNIFORM, TEAM achieved (1) better plan quality using <60% number of spots of UNIFORM; (2) better robustness to the number of spots; (3) better robustness to a large MMU threshold. Furthermore, TEAM provided better plan quality with fewer spots than other adaptive methods (Cartesian-grid or triangular-mesh).

CONCLUSIONS

A novel triangular-mesh-based proton spot placement method called TEAM is proposed, and it is demonstrated to improve plan quality, robustness to the number of spots, and robustness to the MMU threshold, compared to the clinically-used spot placement method and other adaptive methods.

Combined optimization of spot positions and weights for better FLASH proton therapy

Objective.In Intensity Modulated Proton Therapy (IMPT), the weights of individual pencil-beams or spots are optimized to fulfil dosimetric constraints. Theses spots are usually located on a regular lattice and their positions are fixed during optimization. In many cases, the range of spot weights may however be limited, leading sometimes to sub-optimal plan quality. An emblematic use case is the delivery of a plan at ultra-high dose rate (FLASH-RT), for which the spot weights are typically constrained to high values.Approach. To improve further the quality of IMPT FLASH plans, we propose here a novel algorithm to optimize both the spot weights and positions directly based on the objectives defined by the treatment planner.Main results. For all cases considered, optimizing the spot positions lead to an enhanced dosimetric score, while maintaining a high dose rate.Significance. Overall, this approach resulted in a substantial plan quality improvement compared to optimizing only the spot weights, and in a similar execution time.

Mixed-size spot scanning with a compact large momentum acceptance superconducting (LMA-SC) gantry beamline for proton therapy

Objective. Lowering treatment costs and improving treatment quality are two primary goals for next-generation proton therapy (PT) facilities. This work will design a compact large momentum acceptance superconducting (LMA-SC) gantry beamline to reduce the footprint and expense of the PT facilities, with a novel mixed-size spot scanning method to improve the sparing of organs at risk (OAR).Approach. For the LMA-SC gantry beamline, the movable energy slit is placed in the middle of the last achromatic bending section, and the beam momentum spread of delivered spots can be easily changed during the treatment. Simultaneously, changing the collimator size can provide spots with various lateral spot sizes. Based on the provided large-size and small-size spot models, the treatment planning with mixed spot scanning is optimized: the interior of the target is irradiated with large-size spots (to cover the uniform-dose interior efficiently), while the peripheral of the target is irradiated with small-size spots (to shape the sharp dose falloff at the peripheral accurately).Main results. The treatment plan with mixed-size spot scanning was evaluated and compared with small and large-size spot scanning for thirteen clinical prostate cases. The mixed-size spot plan had superior target dose homogeneities, better protection of OAR, and better plan robustness than the large-size spot plan. Compared to the small-size spot plan, the mixed-size spot plan had comparable plan quality, better plan robustness, and reduced plan delivery time from 65.9 to 40.0 s.Significance. The compact LMA-SC gantry beamline is proposed with mixed-size spot scanning, with demonstrated footprint reduction and improved plan quality compared to the conventional spot scanning method.

Optimization of proton pencil beam positioning in collimated fields

BACKGROUND

The addition of static or dynamic collimator systems to the pencil beam scanning delivery technique increases the number of options for lateral field shaping. The collimator shape needs to be optimized together with the intensity modulation of spots.

PURPOSE

To minimize the proton field's lateral penumbra by investigating the fundamental relations between spot and collimating aperture edge position.

METHODS

Analytical approaches describing the effect of spot position on the resulting spot profile are presented. The theoretical description is then compared with Monte Carlo simulations in TOPAS and in the RayStation treatment planning system, as well as with radiochromic film measurements at a clinical proton therapy facility. In the model, one single spot profile is analyzed for various spot positions in air. Further, irradiation setups in water with different energies, the combination with a range shifter, and two-dimensional proton fields were investigated in silico.

RESULTS

The further the single spot is placed beyond the collimating aperture edge ('overscanning'), the sharper the relative lateral dose fall-off and thus the lateral penumbra. Overscanning up to 5 mm $5\,\text{mm}$ reduced the lateral penumbra by about 20% on average after a propagation of 13 cm $13\,\text{cm}$ in air. This benefit from overscanning is first predicted by the analytical proofs and later verified by simulations and measurements. Corresponding analyses in water confirm the benefit in lateral penumbra with spot position optimization as observed theoretically and in air. The combination of spot overscanning with fluence modulation facilitated an additional improvement.

CONCLUSIONS

The lateral penumbra of single spots in collimated scanned proton fields can be improved by the method of spot overscanning. This suggests a better sparing of proximal organs at risk in smaller water depths at higher energies, especially in the plateau of the depth dose distribution. All in all, spot overscanning in collimated scanned proton fields offers particular potential in combination with techniques such as fluence modulation or dynamic collimation for optimizing the lateral penumbra to spare normal tissue.

Validation of pencil beam scanning proton therapy with multi-leaf collimator calculated by a commercial Monte Carlo dose engine

This study aimed to evaluate the clinical beam commissioning results and lateral penumbra characteristics of our new pencil beam scanning (PBS) proton therapy using a multi-leaf collimator (MLC) calculated by use of a commercial Monte Carlo dose engine. Eighteen collimated uniform dose plans for cubic targets were optimized by the RayStation 9A treatment planning system (TPS), varying scan area, modulation widths, measurement depths, and collimator angles. To test the patient-specific measurements, we also created and verified five clinically realistic PBS plans with the MLC, such as the liver, prostate, base-of-skull, C-shape, and head-and-neck. The verification measurements consist of the depth dose (DD), lateral profile (LP), and absolute dose (AD). We compared the LPs and ADs between the calculation and measurements. For the cubic plans, the gamma index pass rates (γ-passing) were on average 96.5% ± 4.0% at 3%/3 mm for the DD and 95.2% ± 7.6% at 2%/2 mm for the LP. In several LP measurements less than 75 mm depths, the γ-passing deteriorated (increased the measured doses) by less than 90% with the scattering such as the MLC edge and range shifter. The deteriorated γ-passing was satisfied by more than 90% at 2%/2 mm using uncollimated beams instead of collimated beams except for three planes. The AD differences and the lateral penumbra width (80%-20% distance) were within ±1.9% and ± 1.1 mm, respectively. For the clinical plan measurements, the γ-passing of LP at 2%/2 mm and the AD differences were 97.7% ± 4.2% on average and within ±1.8%, respectively. The measurements were in good agreement with the calculations of both the cubic and clinical plans inserted in the MLC except for LPs less than 75 mm regions of some cubic and clinical plans. The calculation errors in collimated beams can be mitigated by substituting uncollimated beams.

Dosimetric Comparison of Various Spot Placement Techniques in Proton Pencil Beam Scanning

Purpose

To present quantitative dosimetric evaluations of five proton pencil beam spot placement techniques.

Materials and Methods

The spot placement techniques that were investigated include two grid-based (rectilinear grid and hexagonal grid, both commonly available in commercial planning systems) and three boundary-contoured (concentric contours, hybrid, and optimized) techniques. Treatment plans were created for two different target volumes, one spherical and one conical. An optimal set of planning parameters was defined for all treatment plans and the impact of spot placement techniques on the plan quality was evaluated in terms of lateral/distal dose falloff, normal tissue sparing, conformity and homogeneity of dose distributions, as well as total number of spots used.

Results

The results of this work highlight that for grid-based spot placement techniques, the dose conformity is dependent on target cross-sectional shape perpendicular to beam direction, which changes for each energy layer. This variable conformity problem is mitigated by using boundary contoured spot placement techniques. However, in the case of concentric contours, the conformity is improved but at the cost of decreased homogeneity inside the target. Hybrid and optimized spot placement techniques, which use contoured spots at the boundary and gridlike interior spot patterns, provide more uniform dose distributions inside the target volume while maintaining the improved dose conformity. The optimized spot placement technique improved target coverage, homogeneity of dose, and minimal number of spots. The dependence of these results on spot size is also presented for both target shapes.

Conclusion

This work illustrates that boundary-contoured spot placement techniques offer marked improvement in dosimetry metrics when compared to commercially available grid-based techniques for a range of proton scanned beam spot sizes.

Simulation study using the spots deletion technique in spot scanning proton beam therapy for prostate cancers

The aim of the present study was to investigate the effects on the dose distribution and beam delivery time in spot scanning proton beam therapy (PBT) incorporating the spot deletion technique. A spot scanning plan was created for 30 patients with prostate cancer. The plan was then modified via two processes: Spots with lower weighting depositions were deleted (process A) and spots that were distant from the clinical target volume (CTV) were deleted (process B). The dose distribution to the organs at risk (OAR), the expanded CTV (exCTV), which was defined by a uniform expansion of the CTV by a radius of 5 mm, and the beam delivery time were compared among initial and modified plans. The V_{50Gy [relative biological effectiveness (RBE)]} to the rectum and bladder, and V_{60 Gy(RBE)} to the urethral bulb, inhomogeneity index (INH) of the exCTV showed a difference (P=1.1x10^-14, P=6.4x10^-14, P=2.7x10^-7, P=3.2x10^-17), although only changes by process B were significant. Modified plan by process B showed the V_{50 Gy(RBE)} to the rectum and bladder decreased by -2.4±1.6 and -2.3±1.4%, and the V_{60 Gy (RBE)} to the urethral bulb decreased by -15.9±19.4%. The INH of the exCTV increased by 0.05±0.03%. On the other hand, modification of the initial plan by process A did not affect the dose of the OAR, exCTV or beam delivery time. In spot scanning PBT, modification of the initial radiotherapy plan by systemic deletion of spots distant from the CTV could result in a dose reduction to the OAR.

An adaptive spot placement method on Cartesian grid for pencil beam scanning proton therapy

Pencil beam scanning proton radiotherapy (RT) offers flexible proton spot placement near treatment targets for delivering tumoricidal radiation dose to tumor targets while sparing organs-at-risk. Currently the spot placement is mostly based on a non-adaptive sampling (NS) strategy on a Cartesian grid. However, the spot density or spacing during NS is a constant for the Cartesian grid that is independent of the geometry of tumor targets, and thus can be suboptimal in terms of plan quality (e.g. target dose conformality) and delivery efficiency (e.g. number of spots). This work develops an adaptive sampling (AS) spot placement method on the Cartesian grid that fully accounts for the geometry of tumor targets. Compared with NS, AS places (1) a relatively fine grid of spots at the boundary of tumor targets to account for the geometry of tumor targets and treatment uncertainties (setup and range uncertainty) for improving dose conformality, and (2) a relatively coarse grid of spots in the interior of tumor targets to reduce the number of spots for improving delivery efficiency and robustness to the minimum-minitor-unit (MMU) constraint. The results demonstrate that (1) AS achieved comparable plan quality with NS for regular MMU and substantially improved plan quality from NS for large MMU, using merely about 10% of spots from NS, where AS was derived from the same Cartesian grid as NS; (2) on the other hand, with similar number of spots, AS had better plan quality than NS consistently for regular and large MMU.

Dose distribution effects of spot-scanning proton beam therapy equipped with a multi-leaf collimator for pediatric brain tumors

The present study simulated the effect of spot-scanning proton beam therapy (PBT) performed using a device equipped with a multi-leaf collimator (MLC) to calculate the dose distribution. Simulation studies using 18 pediatric patients with brain tumors in the posterior fossa were performed. Treatment plans were created for the MLC at different stages: Fully open (initial plan), fully closed to allow an irradiated area extending to 15 mm from the clinical target volume (CTV) (15-mm plan), or closing only the leaves where an organ at risk (OAR) overlapped with a border at 10 or 5 mm from the CTV (10- and 5-mm plans, respectively). The mean dose values for the brainstem, cervical cord, brain and cochlea in all MLC closure plans decreased as the MLC was closed (P=9.9×10⁻¹⁰, P=1.3×10⁻¹⁷, P=2.1×10⁻¹⁶ and P=2.0×10⁻⁵, respectively). The maximum dose (D_max) values of the cervical cord and cochlea in all MLC closure plans were also decreased as the MLC was closed (P=3.0×10⁻⁴ and P=1.1×10⁻⁵, respectively). The dose to the CTV was almost unchanged. In 10 patients, the D_max of the brain in all MLC-closure plans was higher than that of the initial plan, but the maximum increase was only 0.8 gray relative biological effectiveness [Gy(RBE)]. In conclusion, the existing MLC installed in the treatment device can be used to decrease the OAR dose significantly using spot-scanning PBT without a large capital investment. The dose from the scattered particles was small.

Proton therapy delivery: what is needed in the next ten years?

Proton radiation therapy has been used clinically since 1952, and major advancements in the last 10 years have helped establish protons as a major clinical modality in the cancer-fighting arsenal. Technologies will always evolve, but enough major breakthroughs have been accomplished over the past 10 years to allow for a major revolution in proton therapy. This paper summarizes the major technology advancements with respect to beam delivery that are now ready for mass implementation in the proton therapy space and encourages vendors to bring these to market to benefit the cancer population worldwide. We state why these technologies are essential and ready for implementation, and we discuss how future systems should be designed to accommodate their required features.

An optimized approach for robust spot placement in proton pencil beam scanning

Maintaining a sharp lateral dose falloff in pencil beam scanning (PBS) proton therapy is crucial for sparing organs at risk (OARs), especially when they are in close proximity to the target volume. The most common approach to improve lateral dose falloff is through the use of physical beam shaping devices, such as brass apertures or collimator based systems. A recently proposed approach focuses on proton beam spot placements, moving away from traditional grid-based placements to concentric-contours based schemes. This improves lateral dose falloff in two ways: (1) by better conforming all spots to the tumor boundary and (2) allowing for 'edge enhancement', where boundary spots deliver higher fluence than more central spots, thereby creating a steeper lateral dose falloff. However, these benefits come at the expense of maintaining uniformity of spot distribution inside the target volume. In this work we have developed a new optimized spot placement scheme that provides robust spot distributions inside the target. This approach achieves the boundary conformity of a concentric-contours based approach and uses a fast-iterative method to distribute the interior spots in a highly uniform fashion in an attempt to improve both the lateral dose falloff and uniformity. Furthermore, we quantified the impact of this new approach through direct comparison with grid, contour, and hybrid spot placements schemes, showing improvements for this new approach. The results were validated in homogeneous medium for two different target shapes having concave and convex geometry.

Proton therapy for non-squamous cell carcinoma of the head and neck: planning comparison and toxicity

To investigate optimal treatment planning using proton beams for non-squamous cell carcinoma of the head and neck (NSCHN), the dose distributions of plans involving pencil beam scanning (PBS) with or without a patient-specific aperture system (PSAS), passive-scattering proton therapy (PSPT) and X-ray intensity-modulated radiotherapy (IMRT) were compared. As clinical results, toxicities of PBS with PSAS, including changes in quality of life, were reported. Between April 2014 and August 2016, a total of 30 patients were treated using PBS with PSAS. In 20 patients selected at random, the dose distributions of PBS with or without the PSAS, PSPT and IMRT plans were compared. Neutron exposure by proton therapy was calculated using a Monte Carlo simulation. Toxicities were scored according to CTCAE ver. 4.0. Patients completed EORTC quality of life survey forms (QLQ-C30 and QLQ-HN35) before and 0-12 months after proton therapy. The 95% conformity number of PBS with the PSAS plan was the best, and significant differences were detected among the four plans (P < 0.05, Bonferroni tests). Neutron generation by PSAS was ~1.1-fold higher, but was within an acceptable level. No grade 3 or higher acute dermatitis was observed. Pain, appetite loss and increased weight loss were more likely at the end of treatment, but recovered by the 3 month follow-up and returned to the pretreatment level at the 12 month follow-up. PBS with PSAS reduced the penumbra and improved dose conformity in the planning target volume. PBS with PSAS was tolerated well for NSCHN.

Trimmer sequencing time minimization during dynamically collimated proton therapy using a colony of cooperating agents

The dynamic collimation system (DCS) can be combined with pencil beam scanning proton therapy to deliver highly conformal treatment plans with unique collimation at each energy layer. This energy layer-specific collimation is accomplished through the synchronized motion of four trimmer blades that intercept the proton beam near the target boundary in the beam's eye view. However, the corresponding treatment deliveries come at the cost of additional treatment time since the translational speed of the trimmer is slower than the scanning speed of the proton pencil beam. In an attempt to minimize the additional trimmer sequencing time of each field while still maintaining a high degree of conformity, a novel process utilizing ant colony optimization (ACO) methods was created to determine the most efficient route of trimmer sequencing and beamlet scanning patterns for a collective set of collimated proton beamlets. The ACO process was integrated within an in-house treatment planning system optimizer to determine the beam scanning and DCS trimmer sequencing patterns and compared against an analytical approximation of the trimmer sequencing time should a contour-like scanning approach be assumed instead. Due to the stochastic nature of ACO, parameters where determined so that they could ensure good convergence and an efficient optimization of trimmer sequencing that was faster than an analytical contour-like trimmer sequencing. The optimization process was tested using a set of three intracranial treatment plans which were planned using a custom research treatment planning system and were successfully optimized to reduce the additional trimmer sequencing time to approximately 60 s per treatment field while maintaining a high degree of target conformity. Thus, the novel use of ACO techniques within a treatment planning algorithm has been demonstrated to effectively determine collimation sequencing patterns for a DCS in order to minimize the additional treatment time required for trimmer movement during treatment.

Treatment planning for proton therapy: what is needed in the next 10 years?

Treatment planning is the process where the prescription of the radiation oncologist is translated into a deliverable treatment. With the complexity of contemporary radiotherapy, treatment planning cannot be performed without a computerized treatment planning system. Proton therapy (PT) enables highly conformal treatment plans with a minimum of dose to tissues outside the target volume, but to obtain the most optimal plan for the treatment, there are a multitude of parameters that need to be addressed. In this review areas of ongoing improvements and research in the field of PT treatment planning are identified and discussed. The main focus is on issues of immediate clinical and practical relevance to the PT community highlighting the needs for the near future but also in a longer perspective. We anticipate that the manual tasks performed by treatment planners in the future will involve a high degree of computational thinking, as many issues can be solved much better by e.g. scripting. More accurate and faster dose calculation algorithms are needed, automation for contouring and planning is required and practical tools to handle the variable biological efficiency in PT is urgently demanded just to mention a few of the expected improvements over the coming 10 years.

Contour scanning, multi-leaf collimation and the combination thereof for proton pencil beam scanning

In proton therapy, the lateral fall-off is often used to spare critical organs. It is therefore crucial to improve the penumbra for proton pencil beam scanning. However, previous work has shown that collimation may not be necessary for depths of  >15 cm in water. As such, in this work we investigate the effectiveness of a thin multi leaf collimator (just thick enough to completely stop protons with ranges of  <15 cm in water) for energy layer specific collimation in patient geometries, when applied in combination with both grid and contour scanned PBS proton therapy. For this, an analytical model of collimated beam shapes, based solely on data available in the treatment planning system, has been included in the optimization, with the resulting optimised plans then being recalculated using Monte Carlo in order to most accurately simulate the full physics effects of the collimator. For grid based scanning, energy specific collimation has been found to reduce the V30 outside the PTV by 19.8% for an example patient when compared to the same pencil beam placement without collimation. V30 could be even reduced by a further 5.6% when combining collimation and contour scanning. In addition, mixed plans, consisting of contour scanning for deep fields (max range  >15 cm WER) and collimated contour scanning for superficial fields (<15 cm), have been created for four patients, by which V30 could be reduced by 0.8% to 8.0% and the mean dose to the brain stem by 1.5% to 3.3%. Target dose homogeneity however is not substantially different when compared to the best un-collimated scenario. In conclusion, we demonstrate the potential advantages of a thin, multi leaf collimator in combination with contour scanning for energy layer specific collimation in PBS proton therapy.

A study of lateral fall-off (penumbra) optimisation for pencil beam scanning (PBS) proton therapy

The lateral fall-off is crucial for sparing organs at risk in proton therapy. It is therefore of high importance to minimize the penumbra for pencil beam scanning (PBS). Three optimisation approaches are investigated: edge-collimated uniformly weighted spots (collimation), pencil beam optimisation of uncollimated pencil beams (edge-enhancement) and the optimisation of edge collimated pencil beams (collimated edge-enhancement). To deliver energies below 70 MeV, these strategies are evaluated in combination with the following pre-absorber methods: field specific fixed thickness pre-absorption (fixed), range specific, fixed thickness pre-absorption (automatic) and range specific, variable thickness pre-absorption (variable). All techniques are evaluated by Monte Carlo simulated square fields in a water tank. For a typical air gap of 10 cm, without pre-absorber collimation reduces the penumbra only for water equivalent ranges between 4-11 cm by up to 2.2 mm. The sharpest lateral fall-off is achieved through collimated edge-enhancement, which lowers the penumbra down to 2.8 mm. When using a pre-absorber, the sharpest fall-offs are obtained when combining collimated edge-enhancement with a variable pre-absorber. For edge-enhancement and large air gaps, it is crucial to minimize the amount of material in the beam. For small air gaps however, the superior phase space of higher energetic beams can be employed when more material is used. In conclusion, collimated edge-enhancement combined with the variable pre-absorber is the recommended setting to minimize the lateral penumbra for PBS. Without collimator, it would be favourable to use a variable pre-absorber for large air gaps and an automatic pre-absorber for small air gaps.

Evaluation of dosimetric advantages of using patient-specific aperture system with intensity-modulated proton therapy for the shallow depth tumor

In this study, we evaluate dosimetric advantages of using patient-specific aperture system with intensity-modulated proton therapy (IMPT) for head and neck tumors at the shallow depth. We used four types of patient-specific aperture system (PSAS) to irradiate shallow regions less than 4 g/cm² with a sharp lateral penumbra. Ten head and neck IMPT plans with or without aperture were optimized separately with the same 95% prescription dose and same dose constraint for organs at risk (OARs). The plans were compared using dose volume histograms (DVHs), dose distributions, and some dose indexes such as volume receiving 50% of the prescribed dose (V₅₀ ), mean or maximum dose (D_mean and D_max ) to the OARs. All examples verified in this study had decreased V₅₀ and OAR doses. Average, maximum, and minimum relative reductions of V₅₀ were 15.4%, 38.9%, and 1.0%, respectively. D_max and D_mean of OARs were decreased by 0.3% to 25.7% and by 1.0% to 46.3%, respectively. The plans with the aperture over more than half of the field showed decreased V₅₀ or OAR dose by more than 10%. The dosimetric advantage of patient-specific apertures with IMPT was clarified in many cases. The PSAS has some dosimetric advantages for clinical use, and in some cases, it enables to fulfill dose constraints.