A fast jaw‐tracking model for VMAT and IMRT Monte Carlo simulations

Abstract Modern radiotherapy techniques involve routine use of volumetric arc therapy (VMAT) and intensity modulated radiotherapy (IMRT) with jaw‐tracking – dynamic motion of the secondary collimators (jaws) in tandem with multi‐leaf collimators (MLCs). These modalities require accurate dose calculations for the purposes of treatment planning and dose verification. Monte Carlo (MC) methods for radiotherapy dose calculation are widely accepted as capable of achieving high accuracy. This paper presents an efficiency‐enhancement method for secondary collimator modeling, presented in the context of a tool for MC‐based dose second checks. The model constitutes an accuracy trade‐off in the source model for the sake of efficiency enhancement, but maintains the advantages of MC transport in patient heterogeneities. The secondary collimator model is called Flat‐Absorbing‐Jaw‐Tracking (FAJT). Transmission through and scatter from the secondary collimators is neglected, and jaws are modeled as perfectly absorbing planes. To couple the motion of secondary collimators with MLCs for jaw‐tracking, the FAJT model was built into the VCU‐MLC model. Gamma‐index analysis of the dose distributions from FAJT against the full BEAMnrc MC simulations showed over 99% pass rate for a range of open fields, two clinical IMRT, and one VMAT treatment plan, for 2%/2 mm criteria above 10%. Using FAJT, the simulation speed of the secondary collimators for open fields increased by a factor of 237, 1489, and 1395 for 4 × 4, 10 × 10, and 30 × 30 cm2, respectively. In general, clinically oriented simulation times are reduced from “hours” to “minutes” on identical hardware. Results for nine representative clinical cases (seven with jaw‐tracking) are presented. The average 2%/2 mm γ‐test success rate above the 80% isodose was 96.8% when tested against the EPIDose electronic portal image‐based dose reconstruction method and 97.3% against the Eclipse analytical anisotropic algorithm.

bottleneck. The problem is partially resolved using phase-space sources, which store particle fluence after simulation of treatment plan-independent components. Additionally, some linac manufacturers provide phase-space models to users rather than geometrical specifications. As such, many strategies for fast source models involve the use of phase-space files. [9][10][11][12][13] When modeling the treatment head, many of the primary particles are absorbed in the secondary collimators and do not contribute to the dose in the volume of interest. This is the usual bottleneck on simulation time, particularly when combined with a fast MLC model 14,15 and a dose calculation code such as VMC++, 16 DPM 17 , and gDPM. 18 A number of fast MC codes have been summarized recently. 19 Analytical (or semi-analytical) source models can achieve high efficiency, particularly on GPU devices. 13,20 However, many users are dependent on phase-space sources and do not have secure access to GPU resourcesthis is the context for the present article.
A previous study 21 investigated the effects of simplified particle transport through the secondary collimators on calculation efficiency.
In particular, the authors presented a planar, completely absorbing collimator model positioned at the vertical midpoint of each jaw.
This model ignored collimator transmission and scatter, and achieved a 274-fold gain in overall efficiency and good agreement with a MC benchmark for a 6 MV 10 9 10 cm 2 field. However, clinical dose distributions had relatively low agreement with a benchmark, dropping from gamma-index test results (1%/1 mm) of 98%-97% for more rigorous models to 65%-68% for the flat-absorbing model.
These results were discouraging and no following investigations of similar models have been reported in the literature.
In this paper, we present and evaluate a similar secondary collimator model, but combined with a more sophisticated MLC model. collimator. This jaw model (called Flat-Absorbing-Jaw-Tracking, or FAJT) was integrated into the VCU-MLC model 14,15 to enable fast simulation of secondary collimator motions coupled with MLC motion in jaw-tracking. While full particle transport models with jawtracking have been previously reported, [22][23][24] the FAJT model   achieves higher efficiency while maintaining sufficient accuracy for a   range of clinical applications, particularly treatment planning dose verification.

| METHODS
The FAJT method is a fast alternative to performing MC simulation of photon and electron transport through secondary collimators. It presents the jaws as perfectly absorbing planes positioned at the top (closest to the target) surface of each of the secondary collimators.
An essential component of achieving high efficiency with this model is azimuthal particle redistribution 25,26 (APR). APR is a variance reduction technique usually used to suppress latent variance 17 from nonanalytic (phase-space) sources. In this work, we will refer to such an incident phase-space source as PhspA, as shown in Fig. 1. Each particle from the source is recycled a number of times and azimuthally redistributed. After performing APR, particles are ray-traced to the top of the secondary collimators to determine whether or not they pass through the collimator opening. Figure 2 illustrates an example of absorbed and allowed particles. Those particles that pass within the collimator opening are kept for further simulation (eventually written to an intermediate phase-space that will be used for dose calculation, called PhspB).   To avoid restarting the phase-space in the dose calculations, the phase-space PhspB scored below the secondary collimator is generated to contain exactly the number of particles that will be simulated (N requested ). The number of particles read from the input phase-space PhspA N read is determined on-the-fly based on the number of rejected particles, in order to achieve N requested . Recycling of the PhspA combined with APR is used to avoid very large input phasespace files, which allows for the data to be stored in RAM prior to the simulation for high-speed access. Additionally, particles outside the radius of the maximum collimator opening can be immediately discarded, while those within the field are recycled. The number of times to recycle each particle from PhspA is set to a fixed number, N recycle , which is chosen with the aim of being large enough to avoid reading PhspA more than once, and small enough to avoid latent variance. The first particle to be read from the PhspA is chosen at random to ensure the independence of parallelized calculations, and subsequent particles are read sequentially from the file.

2.B | The jaw-tracking model for VMAT and IMRT
The flat-absorbing jaws were integrated into the VCU-MLC model 14,15 that was designed for fast simulation of radiation transport through moving collimator leaves. The VCU-MLC software uses MLC control points from the treatment plan to specify the positions of each leaf during radiation delivery. Each control point is associated with fractional delivered monitor units (MUs). In the original implementation of VCU-MLC without an integrated jaw model, the particle source for MLC simulation was a PhspB from a BEAMnrc 27 simulation of the secondary collimators (scored just above the top of the MLCs). As each particle is read from the phase-space, positions of the MLCs for the particle to be transported through are determined by randomly sampling the fractional MU delivered. The particle is then transported through the MLCs using exact MLC geometry but an approximate transport model.
In jaw-tracking mode, every control point also contains positions of each jaw. To model jaw-tracking, the VCU-MLC code was modified to include modeling secondary collimators as flat-absorbing jaws.
In this case, particles originate from PhspA particle source stored in RAM, and APR is applied to each particle as described in the previ-

2.C | Absolute dose calculation
All MC simulations were performed using our in-house Monte Carlo software framework. 28,29 Within the framework, transport through a phantom is performed using either the DOSXYZnrc 30 or VMC++ 16 dose calculation software. It has been shown that the dose calculations from these codes are in excellent agreement. 31 However, implementation of the FAJT model is independent of the dose calculation code. We have chosen to highlight VMC++ in this text because its fast simulation speeds make it particularly well-suited to the task.
When the MC dose calculation is complete, the dose distribution where S b accounts for backscatter radiation from the secondary collimators into the monitor chamber. 32 The backscatter correction is necessary since MC models generally do not account for the experimental effect of backscatter into the monitor chamber, a mechanism that impacts the dose delivered and depends on collimator positions.
In full simulation of the treatment head S b can be obtained by recording the dose in the monitor chamber separately for the 2. An illustration of the FAJT collimation process. Only particles that strike the top surface of the collimator are absorbed (open arrows). Otherwise, the particles are projected to the next collimator without scattering (closed arrows).
forward (toward the phantom) or backward moving particles. However, without scatter from the secondary collimators modeled in FAJT, it was instead necessary to use a measurement-based S b lookup table, determined in previous work. 33 To account for dynamic collimator motion, a value of S b was determined separately for each control point in an IMRT plan, using a look-up table and the secondary collimator positions. Each S b factor was then assigned a weighting factor according to the fractional MU associated with the given control point relative to the total MU in the field. Finally, using a weighted average over the control points, S b was calculated for each field in IMRT jaw-tracking plans. VMAT plans were treated differentlygantry motion in our VMAT MC where the simulation is divided into a large number of independent parallel simulations to discretely model dynamic gantry rotation.

2.E | Validation and performance tests
The validation results presented in this paper include a range of clini-     regions above the 40% isodose consistently exceeded 95% (Table 3).
Lower pass rates were seen in the regions with doses in the 20%-   The presented benchmarks of the FAJT method demonstrated good agreement with BEAMnrc calculations as well as with the portal image-based dosimetry of EPIdose. 33 As with most dosimetry methods, the portal image-based method has its strengths and deficiencies. An advantage of this method is that it captures measured particle fluence that can be processed and used for the dose reconstruction. Therefore, it has valuable information on fluence modulation and MLC transmission, that is often less accurate in computational models. On the other hand, the dose reconstruction employs a relatively simple convolution-based algorithm that is very accurate in homogeneous media near the beam axis, but suffers reduced accuracy off-axis due to an invariable convolution kernel used in the process. This is where we see increased differences between EPIDose-based reconstructed dose and our FAJT MC as well as AAA calculations. We therefore attribute these differences more to the imperfection of EPID and convolution-based dose reconstruction method than to the inaccuracy of the FAJT model.
Previously, the impact of nine levels of simplification of particle transport through beam collimation systems (jaws and MLCs) was investigated. 21 The most rigorous of the nine methods was faithful MC simulation using EGSnrc, while the simplest was the "Flat-

| CONCLUSION
The FAJT model was shown to provide a substantial reduction in

CONFLI CT OF INTEREST
The authors have no relevant conflicts of interest to disclose.