Validation of the geometric equivalent field concept in total scatter factor calculations, for half‐, quarter‐ and off‐isocenter asymmetric square fields

Abstract Objective Monitor unit (MU) verification for any symmetric or asymmetric field is performed using a total scatter factor (Scp), that is calculated based on the geometric equivalent square field (GESF) concept. In this study, we measured the Scp of various asymmetric square fields (ASFs) and their respective GESFs. Methods Square half‐fields (SHFs), square quarter‐fields (SQFs) and square off‐isocenter fields (SOFs), with sizes ranging from 3×3 cm2 to 20×20 cm2 were created, by varying the collimator jaws of two Varian iX Linacs (6/18 and 6/23 MV). A semi‐flex ion chamber was used to measure Scp at a depth of 10 cm within a water phantom, at the effective field center (EFC) of all ASFs, and at the isocenter (IC) of their respective GESFs. The later Scp values were corrected by the off‐axis ratio [OAR(r)] of the 40×40 cm2 field size, where r is the distance between EFC and IC. Results The results show that the Scp (EFC) is independent of the type of the ASF (SHF, SQF, or SOF) and no significant difference exists between the 18 and 23 MV beams. Compared with the Scp (IC), the Scp (EFC) increased with increasing r, by up to 2% and 4% for 18/23 and 6 MV, respectively. Conclusions The GESF concept provides acceptable accuracy (< 2%) for the calculation of Scp of the ASFs used in most clinical situations (except from SOF with EFC at large r), and thus can be used in MU verification calculations.


INTRODUCTION
The total scatter factor (S cp ), also referred to as output factor in water, has been always an important parameter for the calculation of monitor units (MUs) in radiotherapy, as a function of the treatment field (TF) size.Two different approaches have been used to calculate the S cp .In the first approach, which is also applicable in irregular TFs, S cp is derived by the product of two fac-tors, S c and S p . 1,2S c is the head or collimator scatter factor, which is related to the collimator opening dimensions.S c accounts for the increase in output observed with increased field size, attributed to the increased amount of radiation scattered by the collimator jaws that reaches the phantom (or the patient).S p is the phantom scatter factor, which is related to the amount of scatter produced within the irradiated volume of the phantom/patient.For the S p calculations it is taken into account that the presence of blocks or a multi-leaf collimator (MLC), modify the dimensions of the TF and therefore the amount of scattered radiation produced within the phantom/patient.In the second approach, the S cp is taken directly from tables, for different field sizes (collimator dimensions) and beam energies, 3 and incorporates both the head and phantom scatter.However, this approach is applicable in manual MU calculations for regular treatment fields only (when no blocks are used) or for a quick cross-check of the MU calculated by a treatment planning system (TPS).
TFs may be either symmetric or asymmetric, depending on the horizontal distances of the collimator jaws to the isocenter (IC).TFs are usually described using the notation (y 1 ,y 2 ,x 1 ,x 2 ), where y 1 , y 2 , x 1 and x 2 are the respective collimator jaw positions (in cm) determined at the level of the IC.Asymmetric fields are commonly used in radiotherapy, and regarding treatment planning, there are three special categories of interest (for singleisocenter techniques), according to the location of the IC with respect to the TF: (a) The IC is centered at an edge of the TF, as in the case of the half -beams used in the treatment of breast, 4 as shown in Figure 1a, head and neck, 5 and craniospinal tumors, 6,7 (b) The IC is located at a corner of the TF as in the case of a quarterbeams used for the treatment of the chest wall in breast cancer cases, 4,8 as shown in Figure 1b, c) The IC is outside TF, in special cases like the off -isocenter-beams are used for the treatment of the supraclavicular area and the boost field used for breast tumors, 9 as shown in Figure 1c and d, respectively.
In all special categories mentioned above, the IC is partially or completely blocked, so MUs are calculated at the effective field center (EFC) rather than the IC.The distance between EFC and IC is called off -axis distance (r).In MU calculations based on the geometric equivalent square field (GESF) concept, the use of dosimetric quantities like S cp , PDD, and TMR determined using symmetric fields will lead to significant errors, should the off -axis ratio (OAR) value of the maximum field size (i.e., 40 × 40 cm 2 ) at the location of the EFC is not taken into account.This problem has been studied for square half -fields (SHF s ) only, and the errors observed were up to 3% regarding the output factors (S c and S p ) calculation, 10 and 7% regarding dose calculations within the phantom/patient. 113][14][15][16][17] Thus, several mathematical formulas were developed to improve the accuracy of calculations for asymmetric fields based on the GESF concept, 12,[18][19][20][21] the simplest of which is the areaperimeter ratio. 22][25][26] In the referenced studies this conclusion has been validated for almost all Varian Linacs, such as Clinac 4/100 with 4 MV beam, 600C with 6 MV beam, 2100C with energies 6 and 10 MV, 2100CD with energies 6 and 15 MV, and the Linacs 2100C, 2300C/D, Clinac-DHX (with energies 6 and 18MV), but no relevant data exist for Varian Clinac IX accelerators, or the 23 MV beam energy.Most important, few studies have measured the S cp of half -fields, while fewer studies have measured the S cp of the off -isocenter and quarter-fields. 11,17oreover, there are no detailed data regarding the S cp measurements.
The aim of this study is to directly measure the S cp of Varian Clinac IX accelerators for the half -, quarter-and off -isocenter fields and compare it with the values measured at the respective GESFs.This study will be limited to asymmetric square fields (ASFs), for which the effective equivalent field is the same as that of symmetric square field (SSF) whose S cp are tabulated and therefore no approximation is needed.For example, using the notation (y 1 ,y 2 ,x 1 ,x 2 ) described above, a SHF with dimensions (0, 10, 5, 5), a square quarter-field (SQF) with dimensions (0, 10, 0, 10) and a square off -isocenter field (SOF) with dimensions (−10, 20, 5, 5) have all a respective GESF size of 10 × 10 cm 2 . 21,22 I G U R E 2 Examples of the geometry of the studied treatment fields are given.Red and green marks indicate the IC and EFC locations, respectively, and r is the distance between them.For SHF (b) and SQF (c) there are four different EFC locations (marked with 1, 2, 3 and 4), for the different field size sets.EFC, effective field center; IC, isocenter; SHF, square half -field.

Linear accelerators (LINAC)
This study was performed in two Varian IX Linacs (Varian Medical Systems, Palo Alto, California, USA).The first Linac (L 1 ) emits 6 and 23 MV energy photon beams, while the second Linac (L 2 ) emits 6 and 18 MV energy photon beams.For each Linac, the dimensions of the treatment field are determined by two pairs of secondary collimators jaws, the upper jaw (y 1 , y 2 ) and the lower jaw (x 1 , x 2 ).It should be noted that x 1 and y 1 can move beyond the IC, for a distance up to −2 and −10 cm, respectively.When y 1 , y 2 x 1 , x 2 , are all equal, the TF is SSF, as shown in Figure 2a.A half -field is obtained by setting one of these jaws to position 0, and therefore there are four possible SHF sets (y 1 = 0, or y 2 = 0, or x 2 = 0, or x 1 = 0), which have their EFC located either on the X-or Y-axis, as shown in Figure 2b.A quarter-field is obtained by setting one jaw of each pair equal to 0 and there are four possible SQF sets (y 1 = x 2 = 0, y 2 = x 2 = 0, y 1 = x 1 = 0 or y 2 = x 1 = 0), which have their EFC s located on the diagonal axes, as shown in Figure 2c.Finally, two SQF sets were studied by setting y 1 = −5 cm and y 1 = −10 cm, which have their EFCs located on the positive Y-axis, as shown in Figures 2d and e, respectively.The distance (r) between IC and EFC in quarter-fields was calculated using Equation (1).
where i = 1 and/or 2

Water phantom and ionization chamber description
A MP3 water phantom 600 × 500 × 408 mm 3 (PTW, Freiburg, Germany) was used, connected with TANDEM dual-channel electrometer, TBA control unit, and hand pendant.Its proper positioning was fine-tuned using three laser sources and an inclinometer (Tajima, SLANT 100).To measure the charge, a semi-flex chamber (PTW 31010, 0.125 cc) connected to a PTW UNIDOS electrometer was used.The chamber was located at a depth of 10 cm within the water phantom and the distance of the x-ray source focus to the water's surface was adjusted to 90 cm and the gantry angle was kept constant at 0 • .For each measurement, the ionization chamber was irradiated to 100 MU and the resulting charge (R) value was recorded by the electrometer.
The S cp (EFC) was calculated from Equation (3), which is similar to the head scatter (symbolized as H s or S c ) Equations, defined in previous studies. 12,13,14 where R EFC (y 1 , y 2 , x 1 , x 2 , r) and R IC (10 × 10 cm 2 ) are the charges measured at the location of EFC and IC, respectively, and OAR(r) is the OAR measured in a water phantom for a point at a perpendicular distance r cm from the central axis for the maximum field size (40 × 40 cm 2 ), [27][28][29][30] as defined in The off -axis points that correspond to EFC s points of the ASF s used in this study are shown.The red marks on the orthogonal and blue marks on diagonal axes denote the EFC s for the SHFs and SQF s , respectively.The green marks on the y 2 axis denote the EFC s for the SOF s .ASFs, asymmetric square fields; EFC, effective field center; SHFs, square half -fields; SOFs, square off -isocenter fields; SQFs, square quarter-fields.
where R r (40 × 40 cm 2 ) is the measured charge at a distance r from the IC and R IC (40×40 cm 2 ) is the measured charge at the IC for 40 × 40 cm 2 field size.In Figure 3 are shown the off -axis points that correspond to the EFCs of the ASFs used in this study.For the measurement of R r, the largest distance (r) from the IC was 15 cm, being within 80% of the largest field size and away from the field edges. 28,31,32o clarify the order of measurements and calculations made in this study, it must be stressed that before performing any measurements with ASFs, the values of the charge R IC (x, y) of all SSFs (which are the GESF of one or more of the ASFs studied, plus the 40 × 40 cm 2 ) were first recorded, with the ion chamber positioned at the IC and irradiation of the phantom with 100 MU.For S cp (EFC) measurements with ASFs, an example of the measurements performed will be given for the SQF 5 × 5 cm 2 .The collimator jaws were first adjusted at (5, 0, 5, 0) and the chamber was moved at the respective EFC, which is on the diagonal (at distance r = 3.54 cm), the phantom was irradiated using 100 MU, and the value of the charge R EFC (5, 0, 5, 0, r = 3.5 cm) was recorded (to get the numerator of Equation 3).Then, with the chamber kept the same place, the collimator jaws were adjusted to get the maximum SSF 40 × 40 cm 2 , and the reading R r (40 × 40 cm 2 ) was recorded for an irradiation with 100 MU (to get the numerator of Equation 4).The same procedure was followed for the measurements with all ASF s studied.
For the cases of SHF s and SQF s , the S cp (EFC) was calculated (using Equations 3 and 4), at four clinically possible locations having the same distance r from the IC, as shown in Figures 2b and c, respectively.The final S cp (EFC) was defined as the average of the four values as shown in Equation ( 5).This consideration in calculating S cp (EFC) is similar to the diagonal normalized flatness (F DN ) and OAR calculations defined in previous studies. 31,33,34 The difference (D f %) between S cp (IC) and S cp (EFC) was calculated using Equation (6).
For each Linac and ASF, at every EFC point located at a distance r to the IC, the measurements of both beam energies were performed, to reduce the uncertainty regarding chamber position.
The uncertainty of the S cp measurements was calculated as the standard deviation (STDEV.P) of the three measurements for the SSF s and SOF s including the respective OAR measurements, and the SD of the four averages of the measurements of the four alternative quadrants for the SHF s and SQF s using Equation (7).
where N is the number of repeated readings.N = 3 when calculating S cp for SSF s and SOF s , and N = 4 when calculating S cp for SHFs and SQF s .

RESULTS
More than 400 measurements were made in the two Linacs to calculate OAR(r) and then S cp at various points located on one horizontal plane parallel to the water phantom surface at a depth of 10 cm.These points represent the EFCs of the ASF s and the IC of the SSF.All measurements and calculations are listed in Tables 1-4.These tables display the comparison of the individual S cp values calculated for all ASFs (SHF s , SQF s , and SOF s ) with their respective GESF s .
Based on these results, there are no significant differences in S cp values calculated for two 6 MV energies for L 1 and L 2 , as illustrated in Figure 4.The same was true for 18 and 23 MV, except probably for the field 3 × 3 cm 2 off -isocenter fields, where the difference was about 1%, as shown in Figure 5.This confirms the accuracy of the calculation methodology followed and its compatibility for both accelerators, so that the average S cp values can be adopted for the two 6 MV beams and respectively for the 18/23 MV x-ray beam energies.
Tables 1-4 show that the uncertainty associated with the S cp calculation is small with maximum error being 0.3% and 0.2% for energies 6 and 18/23 MV, respectively.It was noted from Figures 4 and 5 that the maximum uncertainty when studying the difference in the S cp between the two accelerators (considering respective TF comparisons) reached 0.5% and 0.3% for 6 and 18/23 MV, respectively.On the other hand, it was found that the estimated uncertainty in calculating the difference in S cp between ASF s and SSF s is less than 1% and ranges from 0.1% to 0.3% and from 0.1% to 0.2%, as shown in Table 5 and Figure 6, for energies 6 and 18/23 MV respectively.
Table 5 shows the S cp differences between ASF s and their GESF s for the 6 and 18/23 MV photon energies, which are also represented graphically in Figure 6.From Table 5 and Figure 6 it becomes evident that the S cp differences between ASF and their respective GESF, increase gradually with the increasing field size (i.e., with increasing Table's 5 column number).Also, the more asymmetric the field is (SHF → SQF → SOF) the larger the more the increase is (i.e., with increasing Table's 5 row number), as the more asymmetric the field is the larger the distance r of its geometric center (EFC) to the IC is.However, these differences reduce with increasing beam energy.Consequently, there are two factors that may affect the increase of S cp (EFC), and it can be either both or one of them.The first factor is the method of the jaw setting (SHF, SQF or SOF) and the second factor is the off -axis distance (r).
The effect of the method of jaw setting can be investigated by comparing directly the ratio of the measured charge values at the center of two different ASF shapes (e.g., the SOF with dimensions 15 × 15 cm 2 and 5 × 5 cm 2 ) which have the same EFC (e.g., ) with the measured charge ratio of their GESF s in the IC (e.g., ).
Table 6 shows that the two ratios are almost equal at all studied energies and the difference between them is less than 0.05%.Since the OAR(r) used to correct the charge measurements is the same for all ASFs having the EFC at same distance r, the S cp (EFC) is independent of the asymmetric field size or the location of the field margin from the IC for these studied cases.Therefore, this implies that the main factor contributing TA B L E 1 Total scatter factors (S cp ) for symmetric (SSF), half (SHF), quarter (SQF) and off -isocenter (SOF) fields of 6 MV photon beam generated by Linac 1. r(cm): The distance between IC and EFC.OAR(r): Off -axis ratio.Ratio 1 = R(ASF)/R(SSF).Ratio 2 = S cp (ASF)/S cp (SSF).
The S cp , S cp , D f %, and  S cp were calculated from Equations (3), ( 5), (6), and (7) F I G U R E 4 S cp differences D f % between the L 1 and the L 2 for the 6 MV energy for all studied fields.S cp, scatter factor.
to the S cp increase is the off -axis distance (r) of the EFC.Indeed, in Figure 7 are shown the S cp differences between ASF s and GESF s (expressed as their ratio) for the 6 and 18/23 MV with relation to the EFC's r values.This figure shows that the average S cp ratios increase gradually with r and that the average values of 6 MV data, are in almost all cases larger than the respective 18/23 MV data.For 18/23 MV, the differences are within or close to 2% but for 6 MV, the differences exceed 4%.

DISCUSSION
The flattening filter (FF) has a Gaussian-like shape, and as a result the beam energy of the central ray passing through IC is relatively larger than the rays away from the IC, and thus more penetrating.As a result, OAR(r) profiles at small depths are not flat around the central axis and present their maximum values at a distance from the IC, depending on the field size (as shown in Figure 8). 16Daryoush et al. 35 confirmed that the 18 MV energy photons scatter preferentially in the forward direction, while the 6 MV energy photons scatter in the forward and sides direction with a relative predominance in the forward direction of 0.5%.On the other hand, the S cp increases with decreasing energy for larger field size due to the increasing phantom scatter contribution (S p ).
In asymmetric fields, the EFC shifts to the sides and thus the lateral scatter increases with increasing r due to decreasing energy, 16 as long as the measurement is performed away from the beam edges.This agrees with the results shown in Table 5, Figure 6 and Figure 7. Khan et al. 13 and John et al. 2 measurements confirmed that calculating the S cp based on the corrections TA B L E 2 Total scatter factors (S cp ) for symmetric (SSF), half (SHF), quarter (SQF) and off -isocenter (SOF) fields of 23 MV photon beam generated by Linac 1. r (cm): The distance between IC and EFC.OAR(r): Off -axis ratio.Ratio 1 = R(ASF)/R(SSF).Ratio 2 = S cp (ASF)/S cp (SSF).
The S cp , S cp , D f %, and  S cp were calculated from Equations (3), ( 5), (6), and (7) F I G U R E 5 S cp difference D f % between the beam energies of 18 MV (L 1 ) and 23 MV (L 2 ), for all studied fields.S cp, scatter factor.
using the OAR(r) 40 × 40 cm 2 field, leads to errors in the output factors greater than 5% when r > 10 cm.In this research, the errors were less, especially at the energy 18 MV, even though the distance r was up to 15 cm.Possibly, this is because in the previous studies a homogeneous field in all directions was considered (i.e., no difference between left and right, or up and down field halves), and OAR(r) values were derived from the crossbeam or diagonal profile only, regardless of the direction of the EFC.Das et al. 36 who studied the applicability of the GESF concept for rectangular fields ranging from 0.5 × 0.5 cm 2 to 5 × 5 cm 2 , exhibited that differences in field output factors exist between linacs from two different vendors, because of the different design of the collimators.Such differences were observed in our study when compar-ing S cp values between two Varian Linacs, especially for 3 × 3 cm 2 fields (see Figures 4 and 5).However, in the Das et al. 36 study, no asymmetric fields were included.
In Figure 6c there is a jump in D f % for the 5 × 5 cm 2 field size in the case of SOF (y 1 = −5 cm) for the 6 MV beams only, which was not observed in Figures 6a,b,d.Since this jump was observed in both Linacs, it may be assumed that is attributed to the design of the FF at the measurement point.
Jackson et al. 23 confirmed that the differences in MU between manual and TPS calculations for breast radiotherapy increase when these calculations require OAR(r) corrections, and compared 6 MV and Cobalt beams (the OAR correction is neglected for Co-60) under similar conditions while neglecting influencing factors such as tissue heterogeneities in both calculations.
The S cp , S cp , D f %, and  S cp were calculated from Equations (3), ( 5), (6), and (7)  Also, computerized investigations proved that these differences are small at the IC and large away from it under the same conditions. 3,24,37In addition, Ian et al. 25,26 proved, after correcting the missing tissue in breast, that these differences increase with the decreasing energy of the photon beam.Part of the remaining unresolved difference in MUs can be explained by the fact that the equivalent field in the TPS is defined as the symmetric field in which the PDD of its central axis has the same properties as the PDD in the effective axis of the asymmetric field,and this also applies to the S cp .This means that the effect of the OAR(r) is included in the TPS calculations, whereas in manual calculations the OAR(r) effect is only corrected for PDD, TPR or TMR and is not taken into account for correcting the output factor or total S cp which are computed directly from the GESF.
In breast radiotherapy with a single-isocenter technique, after correction of the missing tissue, the GESF for the half -or quarter-fields does not exceed TA B L E 4 Total scatter factors (S cp ) for symmetric (SSF), half (SHF), quarter (SQF) and off -isocenter (SOF) fields of 18MV photon beam generated by Linac 2. r (cm): The distance between IC and EFC.OAR(r): Off -axis ratio.Ratio 1 = R(ASF)/R(SSF).Ratio 2 = S cp (ASF)/S cp (SSF).
The S cp , S cp , D f %, and  S cp were calculated from Equations (3), ( 5), (6), and (7)  and the results can be used for manual MU check calculations of three-dimensional conformal radiotherapy (3D-CRT) plans to improve their accuracy.Based on these findings, it is preferable in treatment plans to have the EFC as close as possible to IC, to keep the difference D f % as low as possible, especially for the breast radiotherapy when chest wall is longer than 20 cm (single-isocenter technique without the use of half beam blocks), 9 as in the case previously shown in Figure 1c.

CONCLUSION
The results of this study suggest that in cases where the IC is partially or completely blocked, the S cp (EFC) is dependent on the distance of the measuring point (i.e., the EFC) from the IC but independent of the method of jaw setting (half -, quarter-and off -isocenter fields).For all measurements, the S cp (EFC) was larger than S cp (IC).This increase was up to 2% and 4% for energies 23/18 and 6 MV respectively, but generally it was within 2% for most clinical cases.In this study we have provided detailed information about S cp (EFC) measurements and calculations which are tabulated in Tables 1-4 and can be utilized as correction factors to improve the accuracy of quick, manual verifications of MU in the 3D-CRT treatment plans, which is still used in some developing countries.

AC K N OW L E D G M E N T S
This study was supported by Damascus University and Tishreen University Hospital (no funding).Open Access funding was provided by the Qatar National Library.

C O N F L I C T O F I N T E R E S T S TAT E M E N T
The authors have no conflict of interest to declare.

F I G U R E 1
Examples of clinical uses of asymmetric fields are given.(a) Supraclavicular field using a half -beam.(b) Tangential fields using quarter-beams.(c) Supraclavicular field using offisocenter beam.(d) Boost field using off -isocenter beams.Green marks indicate the EFC.EFC, effective field center.

F I G U R E 8
OAR(r) profiles at d max for 23 MV.OAR, off -axis ratio.
AU T H O R C O N T R I B U T I O N S Mohammad Samir Hmodi, Majeda Nahili and Ousamah Anjak conceived this project.Mohammad Samir Hmodi and Ali Hasan designed the experiments and performed the measurements.Mohammad Samir Hmodi and Karlos Shamout wrote the draft manuscript.Majeda Nahili, Ioannis A Tsalafoutas and Mohammad Hmodi analyzed the data, interpreted the results, and revised the manuscript.All authors have approved the manuscript's final version.
, respectively.Abbreviations: ASF, asymmetric square fields; EFC, effective field center; IC, isocenter; SHF, square half -field; SOF, square off -isocenter field; SQF, square quarter-field; SSF, symmetric square field.The S cp differences between ASFs (SHFs, SHFs and SOFs) and their equivalent fields SSF s or GESF s for the 6 and 18/23 MV photon energies.D f % : Average D f % values for every two equal energies from Tables1-4.The superscripts in parenthesis for the 10×10 fields(6 MV)indicate the distance r between the IC and the EFC.With bold are denoted differences from 1% to 2% and with bold italics differences above 2%.Average S cp difference between ASF s and GESF s for the 6 and 18/23 MV energies.ASFs, asymmetric square fields; GESF, geometric equivalent square field; S cp, scatter factor.