Efficient independent planar dose calculation for FFF IMRT QA with a bivariate Gaussian source model

Abstract The aim of this study is to perform a direct comparison of the source model for photon beams with and without flattening filter (FF) and to develop an efficient independent algorithm for planar dose calculation for FF‐free (FFF) intensity‐modulated radiotherapy (IMRT) quality assurance (QA). The source model consisted of a point source modeling the primary photons and extrafocal bivariate Gaussian functions modeling the head scatter, monitor chamber backscatter, and collimator exchange effect. The model parameters were obtained by minimizing the difference between the calculated and measured in‐air output factors (S c). The fluence of IMRT beams was calculated from the source model using a backprojection and integration method. The off‐axis ratio in FFF beams were modeled with a fourth degree polynomial. An analytical kernel consisting of the sum of three Gaussian functions was used to describe the dose deposition process. A convolution‐based method was used to account for the ionization chamber volume averaging effect when commissioning the algorithm. The algorithm was validated by comparing the calculated planar dose distributions of FFF head‐and‐neck IMRT plans with measurements performed with a 2D diode array. Good agreement between the measured and calculated S c was achieved for both FF beams (<0.25%) and FFF beams (<0.10%). The relative contribution of the head‐scattered photons reduced by 34.7% for 6 MV and 49.3% for 10 MV due to the removal of the FF. Superior agreement between the calculated and measured dose distribution was also achieved for FFF IMRT. In the gamma comparison with a 2%/2 mm criterion, the average passing rate was 96.2 ± 1.9% for 6 MV FFF and 95.5 ± 2.6% for 10 MV FFF. The efficient independent planar dose calculation algorithm is easy to implement and can be valuable in FFF IMRT QA.


| INTRODUCTION
Flattening filter-free (FFF) photon beams have recently gained popularity in intensity-modulated radiotherapy (IMRT) due to their high dose rate, reduced collimator scatter, reduced head leakage and, consequently, reduced out-of-field dose to the patient. 1,2 Flattening filter (FF) was once considered essential in the linear accelerator (linac) design to achieve uniform dose profiles at certain depths.
However, with the advent of advanced optimization techniques and beam shaping devices such as multi-leaf collimator (MLC), 3 FF is not necessary in delivering IMRT. The characteristics of FFF beams and their advantages over FF beams have been discussed extensively in the literature. 2,[4][5][6][7] With regard to IMRT, in addition to reduced beam-on time owing to the combination of increased dose rate and faster MLC moving speed of modern linacs, 2,8,9 dose reduction outside the treatment volume is mostly noteworthy. [10][11][12] IMRT has been linked with increased risks of inducing secondary cancers due to its low monitor unit efficiency. 13,14 Therefore, the dose reduction outside the treatment volume in IMRT with FFF beams is clinically significant, especially for pediatric patients. 15,16 In the study of pediatric IMRT of intracranial tumors using FFF beams, Cashmore et al. found an average reduction in peripheral doses of 23.7%, 29.9%, 64.9%, and 70.0% to the thyroid, lung, ovaries, and testes, respectively, compared to conventional IMRT with FF beams. 11 The complex three dimensional (3D) dose distributions of IMRT warrant rigorous pretreatment patient-specific quality assurance (QA) for safe delivery. 17,18 The conventional practice calls for IMRT QA to be performed with measurements using detectors or detector arrays inside phantoms, which is usually time consuming and labor intensive. Computer-based independent dose calculation also proves valuable in validating treatment planning system (TPS) though it cannot replace measurement-based QA. 19 Most independent dose calculation algorithms compute two-dimensional (2D) or 3D dose distributions with high spatial resolution (e.g., 1 mm) within a short amount of time. The computed dose distributions are then compared with the dose distributions computed by the TPS for validation. These independent dose calculation algorithms typically employ a source model which describes the source distribution inside the gantry head. 3,[20][21][22] For high accuracy, the dosimetric details of the beam shaping devices such as the collimator jaws and MLC need to be fully considered when calculating the fluence distribution inside the patient or phantom. 21,23 To speed up the calculation, the fluence distribution is usually convolved with a dose deposition kernel to obtain the dose distribution. Since independent dose calculation cannot be used as a substitute for measurementbased IMRT QA, 24 comprehensive geometric and dosimetric QA of the linac, especially the MLC, needs to be performed rigorously on a regular basis. 25 The removal of the flattening filter has a few effects on the independent dose calculation algorithms. In a conventional linac, the conical-shaped FF, which preferentially attenuates the forward-peaked photon beam at the center, acts as an extra-focal photon source. It contributes up to 11% of the fluence at the isocenter. 26 Other extra-focal sources include the primary collimator, the monitor ionization chamber and the collimator jaws, but their combined contribution to the fluence at the isocenter is only 3%-4%. 27 Thus, the removal of the flattening filter significantly reduces the head scatter which potentially simplifies the required source model for independent dose calculation. It also reduces the number of photons backscattered into the monitor ionization chamber from the collimator jaws which brings further simplification. Other major effects associated with the removal of FF include changes of the lateral beam profiles (from horn-shaped to cone-shaped) and the photon beam spectrum. The latter has profound effect on the depth dose distribution which needs special attention if the independent dose calculation algorithm calculates the full 3D dose distribution.
where A i , r x,i , and r y,i are the amplitude, the standard deviation along

2.A.2 | Off-axis ratio
Off-axis ratio refers to the ratio between the dose at a point away from the central axis and the dose at the central axis of the beam at the same depth. In the FFF mode, dose profiles at all depths exhibit a cone shape which can be modeled with rotationally symmetric off-axis ratio (OAR). We use a fourth degree polynomial R(x) to model the OAR, where a i are the coefficients. The fluence calculated using the backprojection and integration method is modified by multiplying with the OAR to bring it closer to a cone shape.

2.A.3 | Dose deposition kernel
Analytical dose calculation algorithms use dose deposition kernels to represent the energy transport and dose deposition of secondary particles originating from the initial interaction point in water. 29 Realistic dose deposition kernels could be calculated with Monte Carlo simulation. However, to keep the dose calculation algorithm and its commissioning simple yet accurate, we use the following analytical kernel which is the sum of three 2D Gaussian where A i and r i are the amplitude and the standard deviation for the ith Gaussian respectively.

2.A.4 | MLC modeling
The details of individual MLC leaves need to be modeled for accurate dose calculation for IMRT. 30,31 The characteristics of a MLC leaf affecting the dose calculation include the interleaf transmission, the tongue-and-groove effect, and the transmission through the rounded leaf end. The interleaf transmission factor was measured directly under closed MLC leaves. The Versa HD has a small tongue-andgroove gap with a projected width of 0.26 mm at the isocenter. 9 Its dosimetric influence was modeled with a tongue-and-groove transmission factor. The Versa HD uses a rounded leaf end design to ensure consistent beam profile penumbra (distance between 20% and 80% intensity) across the field in the leaf movement direction.
The transmission through the first centimeter within the leaf tip gradually declines and was modeled with an exponentially decaying function, where a and b are parameters determining the shape of the function.

2.B | Commissioning
The model commissioning process determined the parameters for The measurement for fields with field sizes larger than 6 9 6 cm 2 was performed at isocenter with a source-to-detector distance    Table 2. The best-fit parameters for the kernels are listed in Table 3.    | 129 dose threshold was used. When the 3%/3 mm criterion was used, the average passing rate was 99.2 AE 0.9% and 98.9 AE 1.2% for the 6 MV and 10 MV respectively. Figure 6 illustrates the close agreement between the measured and calculated profiles extracted from a typical 6 MV FFF IMRT plan. to the source occlusion effect and the VAE of the used dosimeter (IC-10), which is why they were excluded in the optimization.

| Discussions
For both energies, the range of S c reduces from 8% for FF beams to less than 3% for FFF beams between field sizes of 3 9 3 to 35 9 35 cm 2 which matches well with the results reported for the same type of linac. 5 Our results show that the relative contribution of the extra-focal source to S c at the isocenter was reduced by 34.7% from FF beams to FFF beams for 6 MV and by 49. The extra-focal source distribution for the FF beams of both energies has a wider core and taller tail than that for the FFF beams, which can be explained by the shape of the S c curve as well. Unlike   F I G . 6. Comparison between the measured (circle) and calculated (line) dose profiles extracted from a 6 MV FFF headand-neck step-and-shoot IMRT plan using the MapCHECK software.
6 MV FFF and 0.9% for 10 MV FFF. Our observation of FFF step-and-shoot IMRT delivery was that the actual dose rate stayed above 600 MU/min where the dose rate dependency was less than 0.4% for both energies. Thus, the dose rate dependence of the MapCHECK was not accounted for in the comparison, and excellent agreement between the calculated and measured dose distribution was still achieved. A very strict criterion (2%/2 mm with local dose-error criterion) was used in the evaluation which is critical for error detection. 18 The dose calculation model can be extended to calculate planar dose distributions at various depths.
However, the benefit of the extension in homogenous water phantom is limited. The algorithm uses back-projection through all beam-defining devices (jaws and MLC) to calculate fluence, therefore it can be easily extended for linacs with two sets of independent jaws. It is also worth pointing out that the authors have no intention to suggest acceptance criteria for FFF IMRT QA.
Instead, an efficient, easy-to-implement planar dose calculation algorithm with superior accuracy is provided to facilitate FFF IMRT QA.

| CONCLUSIONS
A direct comparison of the analytical source models for FF and FFF beams from an Elekta Versa HD treatment unit was performed in this study. A source model consisting of bivariate Gaussian functions was used and good agreement between the measured and calculated in-air output factors was achieved for both FF beams (<0.25%) and

CONF LICT OF I NTEREST
The authors declare no conflict of interest.