Recommended dose voxel size and statistical uncertainty parameters for precision of Monte Carlo dose calculation in stereotactic radiotherapy

Abstract Monte Carlo (MC)‐based treatment planning requires a choice of dose voxel size (DVS) and statistical uncertainty (SU). These parameters effect both the precision of displayed dose distribution and time taken to complete a calculation. For efficient, accurate, and precise treatment planning in a clinical setting, optimal values should be selected. In this investigation, 30 volumetric modulated arc therapy (VMAT) stereotactic radiotherapy (SRT) treatment plans, 10 brain, 10 lung, and 10 spine were calculated in the Monaco 5.11.02 treatment planning system (TPS). Each plan was calculated with a DVS of 0.1 and 0.2 cm using SU values of 0.50%, 0.75%, 1.00%, 1.50%, and 2.00%, along with a ground truth calculation using a DVS of 0.1 cm and SU of 0.15%. The variance at each relative dose level was calculated for all SU settings to assess their relationship. The variation from the ground truth calculation for each DVS and SU combination was determined for a range of DVH metrics and plan quality indices along with the time taken to complete the calculations. Finally, the effect of defining the maximum dose using a volume of 0.035 cc was compared to 0.100 cc when considering DVS and SU settings. Changes in the DVS produced greater variations from the ground truth calculation than changes in SU across the values tested. Plan quality metrics and mean dose values showed less sensitivity to changes in SU than DVH metrics. From this study, it was concluded that while maintaining an average calculation time of <10 min, 75% of plans could be calculated with variations of <2.0% from their ground truth values when using an SU setting of 1.50% and a DVS of 0.1 cm in the case of brain or spine plans, and a 0.2 cm DVS in the case of lung plans.

treatment planning. 1,4 These optimizations have seen the implementation of virtual source models, transmission filters, and photon cutoff energies along with variance reduction techniques such as interaction forcing, electron history repetition, and Russian Roulette, all of which are used within Monaco (Elekta, Stockholm, Sweden) to achieve acceptable calculation times. 1,[5][6][7][8] The choice of different user settings, however, such as the dose voxel size (DVS) and statistical uncertainty (SU), still play a large role in the time required to complete a calculation.
When completing an MC-based calculation, the dose distribution region is divided into voxels of equal volume, defining the spatial resolution. The dose deposited within a given voxel is represented at a single location before an interpolation between voxels is completed to display a complete distribution. 6 The SU setting is unique to MC-based algorithms and controls the level of statistical noise remaining within the final calculation. As the SU value is decreased, the number of simulated histories is increased, resulting in a lower level of statistical noise present in the final display. 6 For a single beam, the dose displayed at a given point, i, in a dose distribution calculated with a given SU, D SU i , can be assumed to be a summation of a theoretical noise-free, or 0% SU, dose, D 0 i , and an error sampled at random from a Gaussian distribution with a mean of 0, G(σ), and a variance, σ, proportional to the square root of the dose in that voxel 1,[9][10][11] where σ Dmax is the variance in the voxel with the maximum dose, D max . A specification of σ Dmax typically constitutes the user-definition of SU.
A number of studies have investigated the effects of varying the SU on the ability to assess the suitability of a treatment plan in a theoretical fashion using MC codes not customized for treatment planning in clinical practice. 3,10,11 Others have looked at the possibility of removing the noise completely. [12][13][14][15] Although these studies can suggest SU settings, which may be appropriate, it can be hard to understand or quantify the direct effects they have on a clinical distribution due to their limited examples or purely theoretical nature.
In addition, the suggested SU settings are made at the discretion of the authors, they may not be applicable to all treatment techniques or clinical requirements and they may not directly translate to commercial systems using variance reduction techniques.
Treatments requiring the highest levels of accuracy, such as stereotactic radiotherapy (SRT), which uses small fields and steep dose gradients to allow dose escalation to small targets, stand to benefit strongly from using more accurate dose calculation algorithms. 1,4,16 In particular, areas of rapid change in density, such as the lung and spine (bone), are expected to show significant improvements in accuracy when calculated using MC algorithms. 11,16 The complexity of SRT treatment plans have been shown to require a DVS of not >2 mm in any direction to ensure the dosimetry is accurately represented. [16][17][18] To the knowledge of the authors, no previous studies have investigated directly, or quantified, the effects of the SU settings used in commercial TPS upon SRT dose distributions and their assessment of fitness for clinical use.
Reductions in the SU require additional particle histories to be considered such that for a fixed DVS setting where N is the number of histories used 1,6,11 . It should be noted that Eq. (3) may not always be directly applicable for treatment plans which use multiple beams or control points (beamlets), which may not all contribute to calculation at the D max voxel; however, it is a reasonable approximation even in these cases. 6,10 Combining Eqs. (2) and (3), one can therefore write

| MATERIALS AND METHODS
A total of 30 previously treated VMAT SRT treatment plans developed and delivered at GenesisCare using the Monaco 5.11.02 TPS were considered in this investigation. The plan cohort consisted of 10 brain, 10 lung, and 10 spine (bone) treatment plans. Cases were planned with one of two beam models with an energy of 6 MV or flattening filter free 6 MV (6FFF).

2.A | Monaco
Upon setting a value of SU for a given plan, the Monaco 5.11.02 TPS considered in this investigation determines the number of particle histories required for the calculation using an empirical formula assuming all control points to have an equal weight. Once the simulation of these particles is complete, the achieved SU is calculated and additional histories are simulated if required, or the simulation is ceased. 6 The SU can be set per plan or per control point, corresponding to the SU of the total dose distribution, or the dose distribution from a given control point, respectively. Within this investigation, SU per plan will be referred to unless explicitly stated otherwise.
For a given patient and plan combination, Monaco initiates a MC calculation from a fixed initial seed point. As such, the calculation of a plan completed with a low SU setting contains the same initial histories as the calculation of the plan with a high SU setting, combined with further additional histories. This implementation of a fixed initial seed point ensures that multiple repeat calculations using the same SU and DVS settings yield the exact same result. This is ideal for TPS quality assurance but prevents an assessment of the SU setting using the same plan and patient combination by limiting the sample size at each combination of settings to one.
The DVS is defined in Monaco by the "Grid Spacing" setting. This is a single value which results in the generation of equal volume, cubic, isotropic voxels. The minimum allowable setting is 0.10 cm and the resolution for incremental increases from this value is 0.01 cm.

2.B | Ground truth calculation
Each plan was calculated with DVS and SU settings of 0.1 cm and 0.15%, respectively. The plan was then normalized to ensure that 98% of the PTV were covered by the prescription dose for brain or lung plans, and 90% of the PTV for spine plans as per clinical protocols. Throughout this investigation, for a given plan, the dose distribution calculated using these settings was considered the ground truth distribution of the TPS. These distributions represent the most precise calculations completed in this study; however, it is important to note that inaccuracies in the calculation may still be present. This assumption that a low SU calculation can be treated as noise-free has been made in other publications. 10

2.D | Assessment of the SU setting
To determine the applicability of Eq. The variance of the combined values of ΔD SU i within a given relative dose interval, across all plans, was calculated to give a single value of σ int , as an estimate of the variance of the random errors for the given dose interval.

2.E | Assessing DVH metrics
The value for a given DVH metric was recorded from each calculation of the treatment plan and compared to the ground truth for the given plan. The following DVH metrics were assessed during this investigation: • Plan maximum dose The PTV within this investigation was created via an isotropic growth from the GTV. For brain plans, the growth margin was 1 mm, for spine 2 mm, and for lung 5 mm. For spine plans, the planning risk volume (PRV) was a 2 mm isotropic growth of the spinal cord contour. When analyzing the variations in cord maximum dose, variations in the spinal cord and spinal cord PRV were considered as two separate points taken from the same calculation. This resulted in a total of 20 data points per SU and DVS combination, two from each plan.
The ability to precisely determine these metrics allows comparison of clinical outcomes across practices and informs clinicians on the likely clinical outcomes. ICRU report 91 recommends the near maximum, near minimum dose, and median dose of the target volume are recorded for level 2 reporting. 4 The near minimum and median dose provide descriptions of the dose that the target will receive during treatment, allowing the clinician to advise on the probability of local control, based on similar patient outcomes. High maximum doses are often desirable in SRT due to the ablative intent of the treatment; however, precise calculation of the maximum dose is required to accurately report the prescription dose if defined as an isodose line relative to the maximum dose. 4 When producing SRT plans for spinal treatments, the maximum spinal cord dose is often the limiting factor. If an excessive dose is delivered to the spinal cord, there are serious risks to the patient including myelopathy and a compromise of the PTV coverage may therefore be required. [20][21][22] 2.F | Assessing plan quality indices Two plan quality metrics commonly considered in SRT treatment planning were also investigated for every plan. The first was the Paddick conformity index (PCI) calculated as: where TV PIV is the target volume covered by the prescription isodose line, TV is the target volume, and V RI is the volume of the prescription isodose. This index was originally developed to evaluate conformity for brain radiosurgery treatment plans but has value in all treatment planning. 4,23 The second was the gradient index (GI) where V R50 is the volume of the isodose line equal to 50% of the prescription dose. For dose-escalated SRT treatments, the gradient of the dose fall off outside of the treatment volume is important and can be assessed using this metric. 4,16 Although this value was also primarily introduced in the context of stereotactic brain treatment planning, it can provide guidance on plan quality, particularly if compared to local center baselines. It is included here for all plans for these reasons in addition to its ability to allow consideration of the variation of the 50% isodose line compared to the prescription (100%) isodose line when changing the SU.

2.G | Treatment plan calculation time
The Monaco optimization console was used to determine the time between calculation initiation and the final control point completion.
All calculations were completed using the GenesisCare clinical TPS servers with Intel ® Xeon ® CPU E5-2690 v4 processors @ 2.60 GHz with 14 Cores and 128 GB of random access memory (RAM).
Assignment to a specific server was via Citrix to the sever with lowest use at the time of opening the Monaco application. Each server was limited to a maximum of two users at any time; however, no record was kept of the number of users on the server while the calculations were completed. The only exception to this is the calculation of the ground truth dose distributions which were calculated overnight and the time for these calculations is therefore not reported.

3.A | Assessment of the SU setting
The variance of the dose within a given interval relative to D max for all plans considered in this investigation is shown in Fig. 1   The dose to a single voxel in a MC calculation should never be used during a plan assessment as some voxels will display doses multiple times the SU from their noise-free value. 1,4 When defining the maximum dose, one must therefore compromise between using a number of voxels to reduce the influence of statistical noise and the associated volume of those voxels. If too few voxels are used, the reported maximum dose will be artificially high; however, if too

3.B.2 | PTV coverage
The PTV coverage was defined as the dose received by 98% of the PTV for either brain or lung plans, and 90% of the PTV for spine plans. In the ground truth plans, the PTV coverage was achieved by the prescription dose. Figure 3 shows the relative variation in PTV coverage from the ground truth calculation.
As the SU value is increased, the reported target coverage is When using the highest tested SU setting of 2.00%, the PTV coverage was recorded above −0.7% of the ground truth plan, for 75% of plans when using a DVS of 0.1 cm, and −1.5% when using a 0.2 cm DVS.  Fig. 1. As the number of simulated histories is increased, the fraction of those which deposit dose in the spinal cord or PRV is significantly lower than those that contribute to the target region.

3.B.3 | Spinal cord and PRV maximum dose
As a consequence, lower SU settings are required to reduce the IQR associated with the OAR to those observed for the plan maximum doses. 2 As discussed with regard to PTV coverage, one should consider the purpose of the spinal cord or PRV maximum dose criteria. Plans are designed with the maximum dose to not exceed a given value to avoid associated treatment complications. As shown in Fig. 4, the statistical noise adds an additional safety net with a tendency to overestimate the dose. The use of a higher SU or DVS setting will therefore not generally lead to planning errors in which the calculated dose is falsely determined as planning constraint compliant.
Conversely, meeting the constraint with an added safety net could lead to a reduction in target coverage as cord dose is prioritized over target coverage. [20][21][22] Lower values of SU should therefore be considered to ensure precise reporting of the maximum dose in conjunction with the ability to achieve an optimal coverage of the target volume. For calculations using a 0.1 cm DVS with a SU setting of up to 2.00%, the maximum reported dose did not vary by more than 2.0% from the ground truth value for 75% of plans. This was reduced to 0.9% by reducing the SU to 1.50%. When using a 0.2 cm DVS, a lower SU setting of 0.75% was required to maintain the reported maximum dose for 75% of plans within 2.0% of the ground truth value.

3.B.4 | GTV mean dose
Due to the steep dose gradients across SRT targets, the mean or median dose can be a more consistent way to report the delivered dose. 4 The reported GTV mean dose did not vary systematically with

3.C | Assessing plan quality indices
When assessing the effect of changing SU on the calculated PCI as No overall trends were observed between the GI and SU settings as shown in Fig. 7. A larger offset was observed for calculations with a 0.2 cm DVS compared to the ground truth. It is important to note that these changes in plan quality indices are very small relative to the value of the index, typically in the range of 0.7 to 1.0 for the PCI, or 3 to 7 for the GI.
Due to the sensitivity of near minimum and maximum doses to targets and OAR to changing SU, it is desirable to also report on less sensitive metrics and indices such as the PCI and GI. From these results, it could be suggested that when planning with MC-based systems, the roles of plan quality indices, such as the PCI or GI, can provide reliable information about a dose distribution for all settings of SU. To generate treatment plans which can be calculated, on average, in under than 10 min, and for which the deviation of 75% of plans remains within 2.0% of the ground truth for the metrics considered here, the following SU and DVS settings are recommended.

3.D | Clinical implications
• Brain plans with DVS of 0.1 cm and SU of 1.50% • Lung plans with DVS of 0.2 cm and SU of 1.50% • Spine plans with DVS of 0.1 cm and SU of 1.50% In clinical practice, an additional safety margin on the choice of a 2.0% deviation and 75% of plans may be desirable. A reduction of the SU settings above to 1.00-1.25% could provide added confidence for a relatively small cost in time.

ETHICAL STATEMENT
This study was approved by Cabrini Institute of Cabrini Health Limited, Victoria, Australia.