Monte Carlo and analytic modeling of an Elekta Infinity linac with Agility MLC: Investigating the significance of accurate model parameters for small radiation fields

Abstract Purpose To explain the deviation observed between measured and Monaco calculated dose profiles for a small field (i.e., alternating open‐closed MLC pattern). A Monte Carlo (MC) model of an Elekta Infinity linac with Agility MLC was created and validated against measurements. In addition, an analytic model which predicts the fluence at the isocenter plane was used to study the impact of multiple beam parameters on the accuracy of dose calculations for small fields. Methods A detailed MC model of a 6 MV Elekta Infinity linac with Agility MLC was created in EGSnrc/BEAMnrc and validated against measurements. An analytic model using primary and secondary virtual photon sources was created and benchmarked against the MC simulations and the impact of multiple beam parameters on the accuracy of the model for a small field was investigated. Both models were used to explain discrepancies observed between measured/EGSnrc simulated and Monaco calculated dose profiles for alternating open‐closed MLC leaves. Results MC‐simulated dose profiles (PDDs, cross‐ and in‐line profiles, etc.) were found to be in very good agreements with measurements. The best fit for the leaf bank rotation was found to be 9 mrad to model the defocusing of Agility MLC. Moreover, a very good agreement was observed between results from the analytic model and MC simulations for a small field. Modifying the radial size of the incident electron beam in the BEAMnrc model improved the agreement between Monaco and EGSnrc calculated dose profiles by approximately 16% and 30% in the position of maxima and minima, respectively. Conclusion Accurate modeling of the full‐width‐half‐maximum (FWHM) of the primary photon source as well as the MLC leaf design (leaf bank rotation, etc.) is essential for accurate calculations of dose delivered by small radiation fields when using virtual source or MC models of the beam.


| INTRODUCTION
Monte Carlo (MC) techniques are accepted to be the most accurate method of dose calculation in radiotherapy and a reliable tool for modeling linear accelerators (linacs). 1 Creating a reliable dose calculation tool requires accurate and detailed knowledge of the geometry and material of the linac components as well as the characteristics of the incident electron beam through a precise benchmarking of MC model against measurements. 1,2 Many researchers have studied several linac designs using MC codes to model the geometry of the treatment head and to derive beam parameters for different beam energies. [3][4][5][6][7][8][9][10][11][12] The methodology adopted by these groups was to create a model of the linac head based on the vendor provided information and to match depth-dose and dose-profile curve simulations against measurements to determine the initial beam parameters. In some cases, as reported by Chibani and Ma,8 corrections to the information provided by the vendor might be required.
The sensitivity of the linac model to different parameters has been investigated by several groups. 5,6,9,11,[13][14][15] Sheikh-Bagheri and Rogers 5,6 studied beam parameters of nine megavoltage photon beams from different manufacturers (Varian, Elekta and Siemens) and concluded that MC simulations of photon beams are highly sensitive to the radial intensity and mean energy of the incident electron beam. They also reported that the accuracy of simulations is sensitive to the primary collimator opening and flattening filter material and density. Chibani and Ma 8,9 investigated the influence of different parameters of the incident electron beam on Varian photon beams of different energies. In addition to confirming results reported by Sheikh-Bagheri and Rogers, 5,6 they showed that large field sizes (e.g., 35 × 35 cm 2 ) are quite sensitive to the angular divergence of the electron beam. 9 Keall et al. 11 found that MC simulations are sensitive to changes in radial distribution and mean energy of the initial electron beam as well as the target density. Other groups [13][14][15] confirmed those results and showed that accurate tuning of the incident electron beam parameters is very important to achieve the best match between MC simulations and measurements.
Although Bush et al., 14 investigated the impact of deviating from Gaussian intensity distribution, the optimal shape of the electron radial intensity profile was confirmed to be Gaussian. This is the shape adopted in all studies that model beam parameters in MC simulations.
An alternative approach of modeling treatment beams is using virtual source models (VSMs). A VSM typically comprises of multiple virtual sources that simulate the contributions from different components of the treatment head. These typically consist of the photons from the target, primary collimator and flattening filter as well as electron contamination. [16][17][18][19][20][21] The data for particles (e.g., position and direction) generated by each source are derived from the phase space file calculated by MC simulations and scored in a specific plane. Tuning of parameters (e.g., virtual source size, energy fluence, weight of each source) of the VSMs can be achieved by comparison against MC simulations and/or measurements. Chabert et al. 20 created a virtual source model of the Elekta Synergy 6 MV photon beam using phase space data file calculated by the PENELOPE 22 MC code and scored below the flattening filter. Their VSM model included three virtual sources including a primary source (photons from the target) and two scattered sources (photons from the primary collimator and flattening filter). They implemented their VSM in PENELOPE and investigated the accuracy of dose calculations and portal image prediction with regard to different binning methods to process particle information. Sikora et al. 23 showed that for field sizes smaller than 2 × 2 cm 2 , precise modeling of the size and contribution of the primary photon source (i.e., photons from the target) is of high importance. They showed that to achieve a good agreement between calculated and measured cross-and in-line profiles for a 0.8 × 0.8 cm 2 field, the FWHM of the primary photon source needs to be reduced by at least 30% from its original value determined for larger field sizes. In any VSM, all calculations related to virtual sources and resultant photon fluence are according to analytic and mathematical functions describing the source properties.
Besides less complexity, another advantage of using VSMs is faster calculation time compared to full MC simulation. Recent years have seen rapid improvements in the techniques of radiation therapy delivery for cancer treatment. More advanced techniques like intensity modulated radiation therapy (IMRT), volumetric modulated radiation therapy (VMAT), and stereotactic body radiation therapy (SBRT) rely on small radiation fields for high precision conformal dose delivery to a target volume while sparing organs at risk (OAR). This introduces important challenges as small fields are associated with greater uncertainty in the accuracy of beam modeling and clinical dosimetry. [28][29][30][31] These challenges include charged particle disequilibrium, source occlusion and choice of small detectors (e.g., small ion chambers and diodes) to reduce the effect of volume averaging of large detectors. 28,30,31 A small radiation field is defined as one whose dimensions are comparable to or less than the lateral range of charged particles. 32 Based on this criterion, for a 6-MV photon beam, field sizes equal to or less than 3 × 3 cm 2 are considered to be small. 28 The focus of this work is to present a detailed MC model of an    calculations for the Elekta Infinity linac. Excellent agreement was observed between measured and simulated PDD curves as illustrated in Fig. 5(a). All dose points past the buildup region passed a 1%/1 mm gamma comparison. Also, over 90% of dose points from MC simulations were found to be within 0.5% of measurements as shown in Fig. 5(b).

The cross-line and in-line profiles [Figs. 6(a) and 6(b)] also
showed good agreement between measurements and MC calculations. For cross-line profiles, all points from MC calculations passed a 2%/1 mm gamma analysis when compared against measurements.
For the same criteria applied to in-line profiles, passing rates of 100% and over 95% were observed for field sizes smaller than or equal to 20 × 20 cm 2 and larger than 20 × 20 cm 2 , respectively.
Average DTA (left and right) values between MC calculated and measured data, in penumbra region (50% of the relative dose), for field sizes from 2 × 2 cm 2 to 40 × 40 cm 2 were found to be better than 0.1 mm.

Figure 7 shows a comparison of measured and MC calculated
ROFs for several field sizes. The agreement was found to be very good for all field sizes, with the largest discrepancy of less than 0.5% for the 40 × 40 cm 2 field size. values at the penumbra region (50% of the maximum dose) are shown in Table 1.
From the data presented in Fig. 8 and Table 1 Impact of interleaf air gap on leaf transmission for LBRTO value of 9 mrad is shown in Fig. 9. The nominal air gap was calculated to be 0.089 mm.
From this plot, we can see that as the interleaf air gap increases by 0.001 mm, the leaf transmission also increases by approximately 6.0%.
Parameters of the Elekta Infinity linac model that were derived based on the above analysis are shown in Table 2.
The leaf composition (i.e., tungsten alloy) was modified from the manufacturer provided values according to Table 3. Agreement of better than 1% was observed at the position of maximum fluence. Also, the average DTA was found to be 0.04 mm at the penumbra region (50% of the maximum dose).

3.C | Impact of analytic model parameters on the
fluence at the isocenter plane The impact of modifying the analytic model parameters on the fluence at the isocenter plane, as described in Section 2.D, is illustrated in Fig. 11. From Fig. 11(a), we can see that change in the maximum fluence due to increasing leaf bank rotation follows the same trend as in Fig. 8 and Table 1. Due to the source occlusion, the maximum fluence drops from approximately 40% higher to 10% lower than the fluence at the nominal LBROT (9 mrad) as leaf bank rotation increases from 0 to 12 mrad. Ignoring leaf attenuation (100% leaf transmission), as illustrated in Fig. 11(b), increases the maximum flu-  Table 4 shows percentage dose differences and DTA values as similarly reported in Table 1.
To understand the observed differences, results of the analytic

| DISCUSSION
Benchmarking of the MC model of a 6-MV Elekta Infinity linac using the method introduced by Almberg et al. 10 is presented in this study.
PDD curves for a 5 × 5 cm 2 field as well as cross-and in-line profile measurements of different field sizes were used to derive the mean energy and radial intensity (FWHM) of the incident electron beam, respectively. Almberg et al. 10 used film measurements of the penumbra and buildup regions to take advantage of the energy independent film response. In this work, similar measurements were performed using diodes combined with ion chamber to complement diode measurements and to account for the energy dependence of the diodes in large field sizes. Further adjustment of the FWHM of the radial intensity profile was performed using ROFs. The ROFs of small fields (e.g., 2 × 2 cm 2 ) were measured using small volume ion chamber and photon diodes. The angular distribution of the electron beam was determined from profile measurements of large field sizes.
A very good agreement was found between MC calculated and measured curves for all PDD, profiles and output factor measurements.
A passing rate of 100% was observed when comparing simulated PDD curves against measurements using a 1%/1 mm gamma criteria.
As for cross-and in-line profiles, all simulated dose points passed a 2%/1 mm gamma comparison against measurements for field sizes smaller than or equal to 20 × 20 cm 2 . The passing rate for larger field sizes was better than 95%. For ROFs, worst agreement was less than 0.5% for the 40 × 40 cm 2 field size. photon sources rather than a spatial energy distribution. These results are in agreement with findings from groups who studied sensitivity of the MC model parameters to the characteristics of the incident electron beam. 5,6,9,11,[13][14][15] Due to the fact that the secondary photon source only represents the scattered photons, the fluence showed to have negligible sensitivity to excluding this source or changing its parameters (e.g., mean energy). However, the contribution of the secondary photon source could become more important for larger field sizes compared to the ones investigated in this study.
Regarding the impact of the leaf bank rotation, the change in the calculated fluence follows the same trend as the dose differences at the maxima in the dose profile of the alternating field as presented in Table 1. Increasing the leaf bank rotation causes a decrease in the fluence due to increased occlusion of the source. Inappropriate modeling of the leaf transmission (e.g., leaf density, attenuation coefficient and leaf thickness) can also affect the fluence at the isocenter plane. However, the sensitivity of the fluence to this parameter was not found to be large since a 25% decrease in the leaf attenuation causes 4% error in the fluence. Thus, we can see that although it is calculate the photon fluence resultant from a treatment head. Moreover, with advancements in radiation delivery techniques, using small fields has become inevitable in radiotherapy. This introduces more complexity to fine tuning model parameters of MC or analytic source models (i.e., beam and collimation parameters) due to the challenges associated with small fields.
In this paper, we demonstrated the detailed MC modeling of a

ACKNOWLEDGMENTS
We thank Jason Smale of Elekta for helpful discussions. This project has been supported by funding from the Ontario Consortium for Adaptive Radiotherapy (OCAIRO).

CONF LICTS OF INTEREST
The authors have no relevant conflicts of interest to disclose.