Convolution‐based modified Clarkson integration (CMCI) for electron cutout factor calculation

Abstract Electron therapy is widely used to treat shallow tumors because of its characteristic sharp dose fall‐off beyond a certain range. A customized cutout is typically applied to block radiation to normal tissues. Determining the final monitor unit (MU) for electron treatment requires an output factor for the cutout, which is usually generated by measurement, especially for highly irregular cutouts. However, manual measurement requires a lengthy quality assurance process with possible errors. This work presents an accurate and efficient cutout output factor prediction model, convolution‐based modified Clarkson integration (CMCI), to replace patient‐specific output factor measurement. Like the Clarkson method, we decompose the field into basic sectors. Unlike the Clarkson integration method, we use annular sectors for output factor estimation. This decomposition method allows calculation via convolution. A 2D distribution of fluence is generated, and the output factor at any given point can be obtained. We applied our method to 10 irregularly shaped cutouts for breast patients for 6E, 9E, and 15E beams and compared the results with measurements and the electron Monte Carlo (eMC) calculation using the Eclipse planning system. While both the CMCI and eMC methods showed good agreement with chamber measurements and film measurements in relative distributions at the nominal source to surface distance (SSD) of 100 cm, eMC generated larger errors than the CMCI method at extended SSDs, with up to −9.28% deviations from the measurement for 6E beam. At extended SSD, the mean absolute errors of our method relative to measurements were 0.92 and 1.14, while the errors of eMC were 1.42 and 1.79 for SSD 105 cm and 110 cm, respectively. These results indicate that our method is more accurate than eMC, especially for low‐energy beams, and can be used for MU calculation and as a QA tool for electron therapy.


| INTRODUCTION
Electron beams have been widely used in radiotherapy to escalate or boost dose for superficial lesions, where the fast dose fall-off feature of electron beams is desirable for reducing radiation damage to distal tissues. However, variations in electron scattering with respect to beam energy, collimation, and treatment distances make it difficult to predict or create a standard dosimetry model for calculating dose per monitor unit (MU) or, equivalently, output factor.
In most clinics, the output factors of irregularly shaped cutouts have to be measured individually at nominal or extended source to surface distance (SSD), while the output factors of simply shaped cutouts are referred to the institutional linear accelerator data book. However, measuring every cutout factor is not practical for busy clinics, and not every clinic can use an electron treatment planning system (TPS) because of capital and workload costs. Moreover, a simple calculation that includes percentage depth dose, SSD, and output factor is needed for secondary independent MU verification. Therefore, a fast and accurate in-house tool that predicts the electron output factor for both regular and irregularly shaped cutouts is desirable. This work presents an accurate and efficient cutout output factor prediction model to replace time-consuming patient-specific output factor measurement.
The dose calculation algorithms adapted by commercial electron treatment planning systems are the pencil beam method 1,2 and the Monte Carlo simulation method. [3][4][5] The CadPlan TPS (Varian Medical Systems Inc., Palo Alto, CA, USA) uses a generalized Gaussian pencil beam model to calculate electron dose. It could predict relative cutout output factors for extended SSDs within a clinically acceptable uncertainty (1-2%) for 18E beams. However, lower energy beams, such as 6E and 12E, produce large errors (>10%) in calculated relative output factors for small or extremely elongated rectangular fields. 6 The Xio eMC TPS (Elekta CMS Software GmbH, Freiburg, Germany) uses a voxel-based Monte Carlo planning system developed by Kawrakow et al. 7 Edimo et al. validated XiO eMC for electron output factors but concluded that it should be used with caution at lower electron energies, as the calculation showed a maximum error of 4.22% compared to measurement. 8  tions and measurements was within 3% for cutouts greater than 5 9 5 cm and 5% for cutouts smaller than 5 9 5 cm. However, the agreement was significantly poorer for cutouts of 3 9 3 cm, reaching up to 8% error. Therefore, commercial electron treatment planning systems have limited accuracy for calculating cutout output factors at low-energy and small-field dimensions (3 cm or less). 9 In the literature, the methods for predicting cutout output factors for electron beams include the equivalent square, 10,11 square root, 12 one-dimensional, 13 and sector-integration methods. 14 These algorithms predict the output factor of an irregular electron field using a database of parameters based on measurements.
Following the modified Clarkson Integration (MCI) presented by Kung et al., 15 we propose a new model that predicts the cutout output factor via convolution: the convolution-based modified Clarkson integration (CMCI). The MCI method uses annular fields to estimate scatter components instead of the pie sectors used in the original Clarkson integration method. Our CMCI method further assumes the cutout factors' invariance under shifts in the cutout shape. Two-dimensional convolution can integrate all output contributions in the opening area of an electron cutout and make the CMCI method very efficient. We verified CMCI's accuracy by comparing it with the results from eMC, film dosimetry, and ion chamber measurements at nominal and extended SSD using 10 highly irregularly shaped cutouts.  Assuming radially isotropic contributions, the CMCI method also uses annular fields to calculate an electron output contribution.
CMCI further assumes that the cutout output factor is shift invariant for a given cutout shape. This assumption neglects scattering's angular dependence on off-axis beams and central beams. However, such negligence is minor, as the opening area of the cutout is most likely around the center beam. These assumptions allow simple modeling of 2D fluence maps for irregular cutouts and efficient calculation via convolution.
The detailed CMCI method is summarized in Algorithm 2. In the first step, the annular output factor (AOF) for an annular field 2) 2D output kernel, OK n 1 ; n 2 ½ , can be calculated using AOF: where AOF ¼ annular output factor, and NP = number of pixels : 3) Final cutout output factor is calculated by convolution between the cutout shape f½x; y and the output kernel We conducted measurements in a Varian EX linear accelerator

| Validation and data analysis
We   we evaluated a correlation between the gamma passing rate and cutout shape complexity using a shape complexity measure: perimeter 2 / area ratio (P2A).
We tested our CMCI method using 10 irregular cutouts from our institute's clinical database (Fig. 1). These cases were for breast patients, especially breast boost irradiation cases. Generally, the breast boost was treated with a 1-1.5 cm margin, but we intentionally reduced the margin to simulate extreme irregularity in two cases [ Figure 1. (9) and (10)].

| Using limited cutout measurements to generate the kernels
In an attempt to reduce the cutout measurements, we used the lim- To compare two output factors from convolution kernels with a full and limited set of circular fields with an inverse square law, we calculated a percentage differences as %difference ¼ jCOFf ÀCOF l j 0:5ðCOFf ÇCOF l Þ Á 100 where COF f and COF l are the cutout output factors from kernels with a full set and limited set of circular fields.

| RESULTS
The output factors of circular fields for 6E, 9E, and 15E at nominal and extended SSDs were stabilized around 4 cm and converged to 1, 0.9, and 0.8 for 6E, 9E, and 15E, respectively (Fig. 2). The over-
While both results showed good agreement with the chamber measurements at all energies, we found slightly larger errors at 15E.
The relative output distributions of the CMCI method with EDR2 film results are presented in Fig. 4. The comparison view of the sample patient (case number 7) presented in Fig. 4 highly irregular case (case number 9) presented in Fig. 4 0.94% (À3.19% to 2.44%), and 1.09% (À2.48% to 1.63%) for 6E, 9E, and 15E, respectively ( Table 3). The larger errors were found in the highly irregular field in both methods. However, eMC generated more significant errors than the CMCI method, showing more than 5% (up to À9.28%) error in three test cases.
We found larger errors at 6E in eMC results because of an innate problem at low-energy beam (≤6E), which we will discuss later. Because of its measurement basis, the CMCI method generated accurate results even at a low-energy beam, except in two highly irregular fields.

| DISCUSSION
We presented a new CMCI method for predicting an electron cutout factor, especially for irregularly shaped cutouts. In contrast to the original Clarkson integration method, our method used annular sectors to estimate the output factor, and its outcome is a 2D distribution, which offers the relative output distribution as well as the output factor at the specific point. However, these errors are insignificant in clinical cutout shapes and ranges. Except for test cases 9 and 10, in which we intentionally reduced the margin to simulate extreme irregularity in our study, the differences between CMCI and measurements were within AE 2.5%.
Our CMCI method proved effective in terms of accuracy, calculation time, and area covered for a single calculation.
Known limitations for eMC calculation of electron beams with energies ≤6E include 16 differences in up to 5% between measured and calculated outputs for 6E electron beams and differences in up to 14% for circular inserts with a diameter of 3 cm at an extended SSD of 115 cm. 17 Approximations of the electron path in the direction distribution and dose deposition determine whether large sphere sizes for the electron transport are used in eMC. 18 However, several studies demonstrate that eMC can predict dose distributions for high-energy electron beams with high accuracy. 16,19,20 This explains why we observed large errors at 6E electrons at extended SSDs. However, eMC's overall calculation accuracy showed good agreement with the measured values of the other electron beam energies without showing a specific trend related to energy and cutout shape.
One of our convolution method's strengths is its ability to detect a maximum cutout output value as it calculates the 2D output distribution. The convolution maximum point is useful when the ion chamber measurement is required. In the clinic, using irregular cutouts with an extended SSD setting requires taking chamber measurements. Ideally, a measurement should be performed at the expected maximum dose point, but such measurements are prone to point variability, particularly for narrow and irregular cutout fields. Our convolution method offers a stable position for determining an ion chamber measurement point by offering a convolution maximum point.
In this study, we tested our method for the 15 9 15 cone size as a proof of concept. One drawback of the CMCI method is that it requires multiple circular cutout factor measurements for different energies, cone sizes, and SSD settings. In this study, our model was built based on four cutout measurements (ranging from 3 to 6 cm) at three beam energies and SSDs. To solve this issue, we evaluated the convolution kernels from a limited set of circular fields with an inverse square law using an effective SSD and verified them by comparing percentage differences between results from convolution kernels with a full and limited set of circular fields. The results were promising, and we are continuing to investigate the minimum measurements required to obtain reliable results and comparable kernels at large sizes (15 9 15, 20 9 20, 25 9 25) and high electron beam energies.

| CONCLUSION
We have developed an efficient and accurate model for predicting electron cutout outputs for arbitrary-shaped cutouts. Our CMCI method efficiently and accurately calculates entire 2D distributions of cutout factors. Our method can generate comparable results to the eMC method at clinically used cutout settings and can be used for the second MU verification calculation.

ACKNOWLEDG MENTS
The authors would like to thank Dr. Jonathan Feinberg for his extensive editing work on this manuscript.

CONFLI CT OF INTEREST
No conflict of interest.
F I G . 5. The percentage differences between results from the kernels with a full and limited set of circular fields at 6E, 9E, and 15E at extended SSD of (a) 105 cm and (b) 110 cm.