Introduction

Following the advances in radiotherapy techniques, intensity modulated radiation therapy (IMRT) and volumetric modulated arc therapy (VMAT) have been widely adopted for various cancers, by delivering an optimal dose distribution to the target volumes while sparing the surrounding normal tissues. To ensure the required treatment accuracy of IMRT and VMAT, patient-specific quality assurance (QA) has become an important task in clinics. Generally, the calculated dose distributions are compared with the measured using several dosimeters for patient-specific QA. This process is labor intensive and time consuming. Hence, several institutions have tried to assure the deliverability of IMRT and VMAT plans using independent computer calculations instead of measurements [1,2,3,4]. In an independent verification program, the in-air output ratio (conventionally denoted as Sc) for each beam segment must be accurately calculated to verify an IMRT or VMAT plan, which often comprises a significant number of beam segments or control points [2, 3].

There were three important factors to determine Sc: (1) Scattered radiation from the linear accelerator head to the phantom (2) Backscattered radiation from the jaws to the monitor chamber, and (3) Source obscuring effect for small field sizes [5,6,7,8,9]. To calculate the Sc values, a source plane was introduced [10,11,12,13] and implemented through modeling methods such as single [14], dual [7], and three-source models [15]. Yan et al. compared the three models and demonstrated that the measured and calculated Sc values were in best agreement when using the dual-source model (DSM) [16]. In the DSM, a primary source at the position of the X-ray target and an extra-focal source at the position of the flattening filter were used. For an arbitrary field shape defined by both jaws and multi-leaf collimators (MLCs), the DSM was used to calculate Sc based on the detector’s eye view (DEV) and Gaussian integration methods [11].

The American Association of Physicists in Medicine (AAPM) Task Group (TG) 74 report clearly updated the definition of Sc as the change in the scattered radiation of the output beam due to the primary collimator, flattening filter, jaws, and MLCs [9]. Moreover, it stated that the scattered radiation from MLCs is no longer negligible for Sc calculation, particularly when small and irregular fields are used in IMRT and VMAT plans [9]. Nevertheless, the scattered radiation from MLCs has not been considered in previous source models, wherein MLCs only played a role of blocking the area to define the DEV. The DEV-based method cannot account for the effect of scattered radiation from MLCs when the leaves are not in the DEV field. Therefore, the calculated Sc of an MLC-defined field often differs from the measurement that originates from the MLC scatter effect. Several studies have been conducted to correct this discrepancy. Kim et al. quantified the amount of scattered radiation from the MLC of a Varian machine and attempted to include it in the Sc calculation using the equivalent field concept at the source plane and a field mapping method [11,12,13]. Alaei and Higgins introduced fitting curves for Sc to parameterize the aperture effect for any given jaw and MLC setting [17]. Zhu et al. developed an integrated extra-focal source involving MLC scatter and other components [18, 19]. Although these studies improved the calculation accuracy of Sc by accounting for the MLC scatter effect, residual discrepancies between the measured and calculated Sc values remained. The reason for these discrepancies was that scattered radiation from MLCs was considered as an independent source. To the best of our knowledge, no attempt has been made to explicitly model an individual extra-focal source accounting for only scattered radiation from MLCs.

This study aims to model an extra-focal source for the scattered radiation from MLCs, namely an MLC scatter source, and to correct Sc calculated using the conventional DSM to increase the accuracy of point dose calculation. The MLC scatter source was designed to have line-based and area-based sources for scattered radiation from the rounded-edge of MLCs and the exposed MLC areas, respectively, and then the parameters were iteratively optimized based on the measured Sc. To validate the effectiveness of the proposed method, measurements were taken for various MLC-defined square and irregular fields. The Sc values calculated using the proposed MLC scatter source and conventional DSM were compared with the measured data.

Methods

Measurements for extra-focal source modeling

To model the extra-focal source, several sets of measured Sc values were obtained. A 6 MV photon beam from a Varian Clinac® iX linear accelerator equipped with 60 pairs of millennium MLCs (Varian Medical Systems, Palo Alto, CA, USA) was used. The measurements were taken using a 0.125 cm3 cylindrical ionization chamber (Model 31010, PTW-Freiburg, Germany) in a water-equivalent miniphantom (Model 670, CIRS Inc., Norfolk, VA, USA) at a depth of 10 cm. The source-to-chamber distance (SCD) was 100 cm.

For the conventional dual-source modeling, we measured Sc for various square and rectangular fields defined by the jaws with the MLCs fully retracted. On the other hand, to model the MLC scatter source, MLC-defined square fields were used with fixed jaw sizes ranging from 10 × 10 to 30 × 30 cm2. The MLC-defined square field sizes ranged from 4 × 4 cm2 to the fixed jaw-defined field size in increments of 1 cm. The reference field size was 10 × 10 cm2. In addition to the basic measurement data for modeling, the Sc values for various MLC-defined irregular fields were measured for evaluation. To reduce uncertainty in the Sc measurements, at least five readings were taken for every experiment and the standard deviations were within 0.2%. Table 1 summarizes the measurement cases.

Table 1 Measurement cases for modeling and evaluating the dual-source and multi-leaf collimator (MLC) scatter source
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Dual-source model (DSM)

The DSM proposed by Jiang et al. was chosen as the basic source model to which the MLC scatter source was added [7]. The DSM simultaneously accounts for the scattered radiation from the flattening filter and the backscattered radiation into the monitor chamber. Sc,DSM of a certain field size fs can be expressed as.

$$S_{c,DSM} (fs) = \frac{{\left( {1 + F_{efs} \left( {fs} \right)} \right) \cdot \left( {1 - F_{mbs} \left( {fs} \right)} \right)}}{{\left( {1 + F_{efs} \left( {fs_{ref} } \right)} \right) \cdot \left( {1 - F_{mbs} \left( {fs_{ref} } \right)} \right)}},$$
(1)

where Fefs is the ratio of the scattered radiation from the extra-focal source for flattening filter ESsf to the contribution from the primary point source, Fmbs is the decreased ratio of the output beam due to the backscattered radiation into the monitor chamber (i.e., beam output ratio with the backscattered radiation to that without it), and fsref is the reference jaw setting (10 × 10 cm2 at the isocenter). The location of ESsf is at the bottom of the flattening filter, and Fefs can be calculated by integrating over the back-projected area on ESsf through the DEV [7, 10].

The DSM parameters in this study were iteratively optimized using the trust-region-reflective algorithm for nonlinear least squares (MathWorks, Inc., Natick, MA, USA) to match the calculated Sc with the measured one for the jaw-defined square and rectangular fields (see Table 1). The objective function was the chi-square difference between the measured and calculated Sc values for the jaw-defined fields with the MLCs fully retracted. The optimization was done within the maximum number of iterations (400) until the objective function was below a termination tolerance (10−6).

MLC scatter source model

Basic concept of the MLC scatter source model: Fig. 1 shows a comparison of the measured Sc data with those calculated using the DSM for MLC-defined square field sizes ranging from 4 × 4 to 20 × 20 cm2 with a jaw setting of 20 × 20 cm2. As shown in Fig. 1, noticeable discrepancies are observed, implying that the scattered radiation from the MLCs influences Sc. A correction factor was introduced to reduce the discrepancies as follows:

Fig. 1
figure 1

In-air output ratio (Sc) for a 6 MV photon beam as a function of the multi-leaf collimator (MLC)-defined square field size ranging from 4 × 4 to 20 × 20 cm2 at a fixed jaw setting of 20 × 20 cm2. The calculated Sc values are derived from the dual-source model (DSM)

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$$S_{c} = S_{c,DSM} \cdot S_{c,MLC} .$$
(2)

In Eq. (2), Sc,DSM is the DSM-calculated Sc value, and Sc,MLC is the correction factor for the scattered radiation from the MLCs. To determine Sc,MLC of an arbitrary field shaped by both jaws and MLCs, an MLC scatter source (ESmlc) was developed. It is assumed that the position for the ESmlc is located at the center of the MLC (51.0 cm downward from the source target in this study). ESmlc comprises line-based and area-based source models, namely ESline and ESarea, respectively, to consider the characteristics of the scattered radiation from the MLCs empirically.

$$S_{c,MLC} = S_{c,line} \cdot S_{c,area} .$$
(3)

Sc,MLC can be calculated by multiplying Sc,line and Sc,area using ESline and ESarea, respectively. Sc,line is a portion of the scattered radiation from the perimeter of the MLC-defined field (i.e., from the rounded-edge of MLCs). It increases when the MLC-defined field size increases or when it becomes irregularly shaped. Sc,area is a portion of the scattered radiation from the radiation-exposed area of the MLCs. It decreases when the MLC-defined field size increases and when the exposed MLC area decreases.

Classification of the MLC scatter source model based on beam’s eye view (BEV) and DEV

Based on the pattern of the discrepancy between the measured and calculated Sc values, as shown in Fig. 1, we designed ESmlc to account for both BEV and DEV of a certain MLC-defined field. Three categories are established as shown in Fig. 2: (1) The MLC is in a retracted position out of the jaw-defined BEV (2) The MLC is in the jaw-defined BEV but does not affect the change in the DEV, and (3) The MLC is in the jaw-defined BEV and simultaneously affects the change in the DEV.

Fig. 2
figure 2

Schematics of the geometrical relationship between the jaws and multi-leaf collimators (MLCs) in terms of the beam’s eye view (BEV), detector’s eye view (DEV), and scatter interface for (a) category 1, (b) category 2, and (c) category 3

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To distinguish the categories 2 and 3, a rectangular plane, i.e., a scatter interface, was introduced. As shown in Fig. 2, the scatter interface is defined as an area projected by the jaw-based DEV at the mid MLC plane. Details regarding the scatter interface are given in the next section.

In category 1, the effect of ESmlc was negligible or non-existent (i.e., Sc,MLC = 1). On the other hand, two individual ESmlc values were required to explain the categories 2 and 3, wherein the MLC scatter contributions were assumed to be different. Therefore, using the scatter interface as a border, ESmlc was derived.

Scatter interface

Figure 3 shows a schematic of the geometry of the scatter interface in a linear accelerator. As mentioned above, the scatter interface size varies with respect to the jaw-defined field size. According to the geometrical relationship between the jaws and the scatter interface, a constant coefficient α was determined to calculate the scatter interface size as follows:

Fig. 3
figure 3

Schematic of the scatter interface geometry. With the constant α, the geometry of the scatter interface can be determined based on the jaw opening

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$$FX_{{m,{{int}} }} = \alpha \times FX_{m,jaw} ,$$
(4)
$$FY_{{m,{{int}} }} = \alpha \times FY_{m,jaw} ,$$
(5)

where FXm,int and FYm,int are the x and y sizes of the scatter interface at the mid-MLC plane, respectively, and FXm,jaw and FYm,jaw are the x and y sizes of the jaw-defined field at the same plane, respectively. From Fig. 3, α can be calculated as follows:

$$\alpha = \frac{{SCD_{x} \cdot \left( {SAD - SMD} \right)}}{{SMD \cdot \left( {SAD - SCD_{x} } \right)}}\left( {0.577\;{\text{in}}\;{\text{this}}\;{\text{study}}} \right),$$
(6)

where SCDx is the distance between the source and the upper edge of the X jaw, SMD is the distance from the source to the mid MLC plane, and SAD is the source-to-axis distance.

Empirical analysis of the MLC scatter source model

To account for the scatter contribution from the MLCs, ESline and ESarea were assumed. ESmlc can be described as follows:

$$ES_{mlc,out} \supset ES_{line,out}\, \& \,ES_{area,out} ,$$
(7)
$$ES_{mlc,in} \supset ES_{line,in}\, \& \,ES_{area,in} ,$$
(8)

where ESline,out and ESline,in denote the scattered radiations from the perimeter of the MLC-defined fields (reaching to the point of interest) in categories 2 and 3, respectively, as shown in Fig. 2. Sc,line can be directly calculated using the source models with the perimeters of the MLC-defined fields.

We chose a fitting function to express ESline as follows:

$$ES_{line} = a\left( {FP^{b} - RP^{b} } \right) + 1,$$
(9)

where FP is the perimeter of the MLC-defined field. RP is the perimeter of the scatter interface. For categories 2 (i.e., outside the scatter interface) and 3 (i.e., inside the scatter interface), two parameters a and b need to be determined for the line-based source.

ESarea,out and ESarea,in denote the scattered radiations from the radiation-exposed area of the MLCs in categories 2 and 3, respectively. ESarea can be calculated as follows.

$$ES_{area} = e^{{ - MA/2\sigma^{2} }} .$$
(10)

In Eq. (10), σ is a parameter of the Gaussian distribution. MA is the exposed MLC area at the mid-MLC plane and was calculated by subtracting the MLC-defined field size from the jaw-defined field size. From empirical evidence, it was found that the optimal parameters which were a, b, and σ for the MLC scatter sources vary with different field sizes defined by the jaws and type of category. These parameters were empirically– determined as a quadratic function of the area to perimeter (AP) ratio of the jaw-defined field size as follows:

$$\begin{gathered} \left\{ {\begin{array}{*{20}l} {a_{out} = 3.067e^{ - 5} \cdot A\Pr atio^{2} + 5.333e^{ - 5} \cdot A\Pr atio + 0.015} \\ {b_{out} = 7.810e^{ - 4} \cdot A\Pr atio^{2} - 6.042e^{ - 3} \cdot A\Pr atio + 0.840} \\ {\sigma_{out} = - 1.762e^{ - 4} \cdot A\Pr atio^{2} + 26.360e^{ - 5} \cdot A\Pr atio - 37.103} \\ \end{array} } \right. \hfill \\ \left\{ {\begin{array}{*{20}l} {a_{in} = 3.724e^{ - 4} \cdot A\Pr atio^{2} + 5.120e^{ - 5} \cdot A\Pr atio + 0.025} \\ {b_{in} = 3.303e^{ - 4} \cdot A\Pr atio^{2} - 4.135e^{ - 3} \cdot A\Pr atio + 1.648} \\ {\sigma_{in} = - 14.072e^{ - 4} \cdot A\Pr atio^{2} + 224.207e^{ - 3} \cdot A\Pr atio - 440.218} \\ \end{array} } \right. \hfill \\ \end{gathered}$$
(11)

Similarly, the parameters were optimized using the same optimization methods as in DSM to attain the best fit between the measured and calculated Sc,MLC values.

General formalism of Sc,MLC for an arbitrary field shape

For an arbitrary irregular field defined by the MLCs, as shown in Fig. 4, Sc,MLC can be calculated by multiplying the contributions from the two regions divided by the scatter interface. A general formula for Sc,MLC is given as follows:

Fig. 4
figure 4

Schematic of Sc,MLC calculation for an arbitrary irregular multi-leaf collimator (MLC)-defined field using the MLC scatter source and the scatter interface. Here, MAin and MAout are the exposed MLC areas inside and outside the scatter interface, respectively. FPin and FPout are the perimeters of the MLC-defined field inside and outside the scatter interface, respectively

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$$S_{c,MLC} = \left\{ {ES_{line,out} \left( {FP_{out} } \right) \cdot ES_{line,in} \left( {FP_{in} } \right)} \right\} \cdot \left\{ {ES_{area,out} \left( {MA_{out} - RA_{out} } \right) \cdot ES_{area,in} \left( {MA_{in} - RA_{in} } \right)} \right\},$$
(12)

where MAin and MAout are the radiation-exposed areas of the MLCs inside and outside the scatter interface, respectively. FPin and FPout are the perimeters of the MLC-defined field inside and outside the scatter interface, respectively. RA is a reference for the radiation-exposed areas of the MLCs (scatter interface area and 3 × 3 cm2 at the isocenter for categories 2 and 3, respectively).

Evaluation of the MLC scatter source model

The calculated Sc values for various MLC-defined square and irregular fields were compared with the corresponding measurement values to evaluate the efficacy of the developed MLC scatter source. The size of the MLC-defined squares ranged from 4 × 4 cm2 up to the fixed position of the jaw-defined field sizes (ranging from 10 × 10 to 30 × 30 cm2). For the evaluation, the data sets with jaw-defined field sizes of 15 × 15 and 25 × 25 cm2 were added to the data sets previously used for modeling ESmlc. The Sc measurements were taken at least five times using a cylindrical ionization chamber at a depth of 10 cm (SSD: 90 cm) in a water-equivalent miniphantom and then averaged.

In addition, three irregular field shapes (cross, mirrored E, and maze) were considered, as shown in Fig. 5. Each irregular shape included three different sizes in accordance with jaw-defined field sizes of 15 × 15, 20 × 20, and 25 × 25 cm2 [20]. For the MLC-defined irregular fields, the dose and Sc values were calculated and measured at the central point. When measuring Sc values for irregular shaped fields, a brass build-up cap with a diameter of 1 cm was used as a miniphantom to provide electron equilibrium conditions. Clarkson’s method was used for the dose calculation, which integrates the scattered component of each angular sector for irregularly shaped fields [21]. With the Clarkson integration, values of phantom scatter factor (Sp) and tissue maximum ratio (TMR) for three irregular field shapes were obtained by summing scatter contributions from each angular section of 1° (total angular section is 360°). Dose calculations with these parameters were performed by using an in-house program that was written in MATLAB (R2016a, Mathworks Inc., Natick, MA, USA). For the dose measurements, at least five readings were taken using a cylindrical ionization chamber at a depth of 10 cm (SSD: 90 cm) in a solid water phantom with a size of 30 × 30 × 30 cm3 and then multiplied by dose conversion factor (cGy/nC) to obtain the measured dose values. The dose conversion factors could be obtained by using measured readings for a given dose value.

Fig. 5
figure 5

Three multi-leaf collimator (MLC)-defined irregular fields designed to verify the MLC scatter source. The irregular fields are used in three different sizes in accordance with jaw-defined field sizes of 15 × 15, 20 × 20, and 25 × 25 cm2

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Results

Dual-source model evaluation

The DSM in this study was evaluated using jaw-defined square fields ranging from 4 × 4 to 40 × 40 cm2 and rectangular fields with one pair of jaws fixed at 4, 10, or 40 cm while the other pair is varied from 4 to 40 cm. Figure 6 shows a comparison between the calculated and measured Sc values for seven sets of fields. The calculated and measured Sc values were in good agreement (the difference was lower than 0.47%) for square fields and rectangular fields with one pair of jaws fixed at 10 cm. The maximum discrepancies were 0.61% and 0.39% for the rectangular fields with one pair of jaws fixed at 4 and 40 cm, respectively. It was demonstrated that the DSM could accurately predict Sc for the jaw-defined fields with the MLCs fully retracted.

Fig. 6
figure 6

Comparison between the calculated and measured Sc values of a 6 MV photon beam from a Varian machine for (a) square fields ranging from 4 × 4 to 40 × 40 cm2 and rectangular fields with one pair of jaws fixed at 10 cm, (b) rectangular fields with one pair of jaws fixed at 4 cm while the other pair is varied from 4 to 40 cm, and (c) rectangular fields with one pair of jaws fixed at 40 cm while the other pair is varied from 4 to 40 cm. The conventional dual-source model is used for the calculation

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MLC scatter source model evaluation

The optimum values of the parameters of ESmlc,out and ESmlc,in were determined through a nonlinear least squares method using the trust-region-reflective algorithm. Table 2 lists the results. The values of the line-based and area-based source models did not show a noticeable trend.

Table 2 Optimal parameter values of the multi-leaf collimator (MLC) scatter source for a 6 MV beam
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The sources were empirically assumed based on the characteristics of the scattered radiation from the MLCs. Figure 7 shows the tendencies of the scattered radiation from the line-based and area-based sources. With the increase in the MLC-defined field size under the fixed jaw condition, the perimeter of the MLC-defined field increases, thereby increasing the scattered radiation from the perimeter. On the other hand, the scattered radiation from the radiation-exposed area of the MLCs decreases.

Fig. 7
figure 7

Sc,line and Sc,area calculated from multi-leaf collimator (MLC) scatter source comprising line-based and area-based source models as a function of the MLC-defined field at the mid-MLC plane. The calculated Sc,MLC values are determined by multiplying Sc,line and Sc,area

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Evaluation of MLC-defined square fields

Figure 8 shows a comparison between three data sets: measured Sc, Sc calculated using the DSM (i.e., DSM only), and Sc calculated using the DSM in conjunction with the MLC scatter source (denoted as DSM + MLC). The mean differences between the measured and calculated Sc values in the DSM case were 0.03%, 0.07%, 0.19%, 0.16%, and 0.25% for jaw-defined field sizes of 10 × 10, 15 × 15, 20 × 20, 25 × 25, and 30 × 30 cm2, respectively, whereas these values were − 0.06%, − 0.06%, − 0.03%, − 0.04%, and − 0.00% in the DSM + MLC case. The DSM + MLC results showed improved Sc calculation accuracy compared with the DSM results with statistical significance (p value < 0.03). When the MLC-defined fields were smaller than 8 × 8, 13 × 13, and 15 × 15 cm2 for jaw-defined field sizes of 20 × 20, 25 × 25, and 30 × 30 cm2, the difference between the DSM calculated and measured Sc values was less initially but increases thereafter. This is also due to only modeling the scattered radiation from linac components above the jaw in addition to interacting on the complicated contributions of the scattered radiation from the MLCs and jaws. Table 3 summarizes the numerical results of the differences between the measured and calculated Sc values for the MLC-defined square fields.

Fig. 8
figure 8

Comparison between the measured and calculated Sc values. Sc is calculated based on the dual-source model (DSM) and DSM in conjunction with the multi-leaf collimator (MLC) scatter source (DSM + MLC). The dashed line in each graph indicates the scatter interface for each jaw-defined square field

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Table 3 Differences between the measured and calculated Sc values for multi-leaf collimator (MLC)-defined square field sizes ranging from 4 × 4 cm2 to the fixed jaw opening size
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Evaluation of MLC-defined irregular fields

Table 4 lists the evaluation results of the MLC-defined irregular fields. The measured and calculated values of both Sc and dose were compared at the central point (shown in Fig. 5). For the cross field shape with three different sizes of 15 × 15, 20 × 20, and 25 × 25 cm2, the deviations between the measured and calculated Sc values in the DSM case were 0.64%, 0.52%, and 0.21%, respectively, whereas the values were 0.27%, − 0.15%, and − 0.11% in the case of DSM + MLC. For the mirrored E field shape with three different sizes of 15 × 15, 20 × 20, and 25 × 25 cm2, the deviations in the dose values in the DSM case were 1.45%, 0.79%, and 0.86%, respectively, whereas the values were 0.37%, 0.45%, and 0.25% in the DSM + MLC case. The larger the irregular field, the better the agreement between the measured and calculated values. Similar to the square field results, the irregular field results showed that the MLC scatter source could improve the Sc calculation accuracy and therefore the dose calculation accuracy with statistical significance (p value < 0.05).

Table 4 Comparison between the measured and calculated Sc values and dose values for irregular fields
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Discussion

This study attempted to explicitly model the scattered radiation from MLCs as an additional extra-focal source, which was not considered in conventional models including the DSM developed by Jiang et al. Jiang et al. reported that Sc calculated using the DSM through the DEV was lower than the measured value for certain MLC-defined fields [7]. This was because the model was intended only for jaw-defined fields, and the energy fluence variation was irregular depending on where the MLCs were located in terms of blocking the DEV, as shown in Fig. 1. To account for the irregularity, a simple Gaussian approximation as the extra-focal source may not be sufficient. Previous studies used a curve fitting method and an equivalent field method to calculate Sc for MLC-defined small and irregular fields [7, 11,12,13, 17]. On the other hand, Zhu et al. developed an algorithm based on an empirical model wherein the MLC scatter contribution was implicitly included [19]. The effective head scatter sources for the flattening filter and MLCs could be simultaneously determined using a scaling factor. Compared with the source model proposed by Zhu et al., our MLC scatter source was independently developed to explicitly account for the MLC scattered radiation, and the limitation in defining the DEV in the conventional DSM was overcome by adding the MLC scatter source [19].

Both the DSM developed by Jiang et al. and MLC scatter source model developed in this study are fundamentally based on the measured Sc values, and it was demonstrated that it should be ensured that these data have small uncertainty. During measurements, there were several uncertainties related to field apertures defined by jaw or MLCs, cylindrical ionization chamber position in miniphantom, and Linac output variation. In this study, to reduce these uncertainties, several measurements were taken to achieve a small standard deviation of 0.2%. All data were qualified to model the sources and verify them.

The scatter interface is the boundary of the jaw-based DEV located at the mid-MLC plane and can be easily calculated using the parameter α for a given field size. It is used to distinguish between categories 2 and 3 geometrically and to define the boundary of approximately 50% of the beam intensity due to primary and scattered radiations for a certain jaw-defined field shape at the mid-MLC plane physically. The spatial beam intensities on the mid-MLC plane were calculated using the DSM and then normalized using the central intensity. The beam intensity inside the scatter interface has higher photon energy than outside the scatter interface because it might be based on several Gaussian-distribution sources from the target and flattening filter, and peripheral scattered radiation generally consists of low energy components. However, there are limitations in the empirical understanding of scatter components inside and outside the scatter interface in this study, and Monte Carlo (MC) simulation should be performed for more accurate analysis. We assumed that the irregular patterns of the scattered radiation from the MLCs could be explained using this scatter interface. Using the MLC scatter source and scatter interface, we could appropriately account for the MLC scatter contributions at all MLC positions relative to the jaws. As shown in Fig. 8, the scatter interface shows the irregular patterns of the scattered radiation from the MLCs for jaw-defined field sizes of 10 × 10, 15 × 15, and 20 × 20 cm2. However, for jaw-defined field sizes of 25 × 25 and 30 × 30 cm2, the physical scatter interfaces were smaller than the geometric scatter interfaces. The geometric scatter interfaces for large field sizes underestimated the normalized intensity value because of an increase in the low energy component of the peripheral scattered radiation. Slightly greater maximum discrepancies between the measured and calculated Sc values were observed compared with that under smaller jaw-defined field sizes (i.e., 10 × 10, 15 × 15, and 20 × 20 cm2) (Fig. 8).

In this approach, line-based and area-based source functions were used to account for the complicated patterns of the scattered radiation from the MLCs. In principle, the line-based and area-based sources represent scattered radiation reaching the detector from the rounded-edge and radiation-exposed areas on the MLCs, respectively. When the MLC-defined field becomes larger at a fixed jaw setting, the scattered radiation from the MLC edge increases, whereas those from the irradiated areas of the MLCs decreases. This requires two functions, one that increases with the MLC field size and one that decreases with it, as shown in Fig. 7. The same condition is applied to the areas inside and outside the scatter interface to account for each component independently. The line-based and area-based sources change depending on the jaw-defined field size considering optimal parameters because the contribution of the scattered radiation from the jaw aperture changes with different apertures. The area-based source, expressed in Eq. (10), was based on a Gaussian function, which is widely used in obtaining good approximations of scattered radiation from radiation-exposed materials. However, for a line-based source, no such general function exists. Thus, a function was empirically chosen focusing on meeting the trend in the data rather than satisfying the physical meaning.

The parameters of the MLC scatter source model were well optimized with the objective function value below a termination tolerance of 10−6 using the nonlinear least squares method. Two parameters (a and b) in the line-based source and one parameter (σ) in the area-based source were used. Each parameter was classified based on categories 2 and 3. For the line-based source model expressed in Eq. (9), the parameter a helped determine the intensity of the scattered radiation from the radiation-exposed rounded-edges of the MLCs, relative to the intensity of the scattered radiation from each jaw aperture. The jaw-defined field size increased with the increase in aout whereas ain decreased simultaneously. It was assumed that most of the scattered radiation from the rounded-edges of the MLCs outside the scatter interface could reach the detector, while the radiation inside the scatter interface could reach the detector and the monitor chamber via backscattering. The other parameter b was found to decide the intensity gradient as a function of the perimeter of the MLC-defined field shape. The initial intensity gradient with a low value of b is high for a large jaw-defined field because the contribution of the scattered radiation from the jaw aperture increased with the increase in the jaw aperture size. The parameter σ in Eq. (10) for the area-based source increased exponentially with the increase in the jaw-defined field size (see Table 2). Sc,area was calculated using the contribution of the scattered radiation from an exposed MLC area. The high value of σ decreased the gradient of the cumulative distribution function for Sc,area. The portion of the scattered radiation from an exposed MLC area, normalized with that from the reference field, decreases with the increase in the jaw-defined field size.

Figure 8 shows the variation in Sc with respect to the MLC-defined field sizes for five fixed jaw-defined fields. With the increase in the jaw aperture, a certain trend was more obvious: the measured Sc values for MLC-defined field sizes greater than a certain value were greater than the corresponding DSM-calculated Sc values (e.g., over 8 × 8 cm2 for a jaw aperture of 20 × 20 cm2). This is because the DSM was modeled with only jaw-defined Sc data, and therefore, the DEV could not consider the MLC positions in category 2 where scattered radiation from MLCs certainly exists. The scattered radiation from the MLCs was not considered in the DSM for Sc calculation. In this study, the MLC scatter source was developed to consider this contribution.

The effect of MLC scatter has generally been considered negligible and is not explicitly included in standard Sc calculations used for radiation therapy treatment planning. It is true that the MLC scatter contribution is insignificant in conventional treatments. However, in complicated delivery techniques, such as IMRT and VMAT, a dose to a point is a combined effect of many irregularly shaped fields including ones having leaves blocking the point in the BEV. In such cases, the relative importance of the MLC scatter contribution is significant. For example, in our clinic, a typical prostate VMAT plan having a total of 90 fields contains approximately 30 to 40 fields with the isocenter blocked. If we assume equal weighting from each field and an MLC scatter contribution of approximately 0.3%, the total MLC effect becomes approximately 0.5%, which is a justifiable level of correction considering that many correction factors (e.g., ion recombination correction factor, Pion) for output calibration have similar magnitudes. In fact, a higher MLC scatter effect is expected in more complicated plans. Five head-and-neck VMAT plans were used to verify the effectiveness of the MLC scatter source model. The VMAT plans had 178 control points for one arc. For simplicity, the measurement and calculation were performed with gantry angles fixed at 0°. The mean deviations in the central dose in the DSM and DSM + MLC cases were 3.25% and 1.02%, respectively, for the five plans. This shows that the monitor unit (MU) calculation accuracy for IMRT or VMAT plans was improved when the proposed method was incorporated in the MU calculation program. As future work, comprehensive analysis including various tumor sites and types of plans will be conducted to demonstrate the advantage of this method.

In this work, we developed an MLC scatter source and a scatter interface to account for the scattered radiation from various MLC-defined fields. It was difficult to simplify the model because the behavior of radiation scattered from the complicated structure of MLC leaves is not simple. MC simulations that can track the life of each simulated photon with all interaction events in surrounding materials will be performed as a future work. With this, we can analyze the accurate energy fluence within the scatter interfaces to define all the MLC scatter components more clearly and further improve the model.

Conclusion

An extra-focal source model, namely the MLC scatter source, was developed to accurately calculate the scatter components from the head of a linear accelerator. The MLC scatter source comprises line-based and area-based sources. We demonstrated that in conjunction with the conventional DSM, the developed MLC scatter source could improve the accuracy of Sc calculation for both square and irregular fields.