Planar dose calculation of electron therapy

Purpose. The aim of this study is to determine the planar dose distribution of irregularly-shaped electron beams at their maximum dose depth (z max) using the modied lateral build-up ratio (LBR) and curve-fitting methods. Methods. Circular and irregular cutouts were created using Cerrobend alloy for a 14 × 14 cm2 applicator. Percentage depth dose (PDD) at the standard source-surface-distance (SSD = 100 cm) and point dose at different SSD were measured for each cutout. Orthogonal profiles of the cutouts were measured at z max. Data were collected for 6, 9, 12, and 15 MeV electron beam energies on a VERSA HDTM LINAC using the IBA Blue Phantom2 3D water phantom system. The planar dose distributions of the cutouts were also measured at z max in solid water using EDR2 films. Results. The measured PDD curves were normalized to a normalization depth (d 0) of 1 mm. The lateral-buildup-ratio (LBR), lateral spread parameter (σ R (z)), and effective SSD (SSD eff ) for each cutout were calculated using the PDD of the open applicator as the reference field. The modified LBR method was then employed to calculate the planar dose distribution of the irregular cutouts within the field at least 5 mm from the edge. A simple curve-fitting model was developed based on the profile shapes of the circular cutouts around the field edge. This model was used to calculate the planar dose distribution of the irregular cutouts in the region from 3 mm outside to 5 mm inside the field edge. Finally, the calculated planar dose distribution was compared with the film measurement. Conclusions. The planar dose distribution of electron therapy for irregular cutouts at z max was calculated using the improved LBR method and a simple curve-fitting model. The calculated profiles were within 3% of the measured values. The gamma passing rate with a 3%/3 mm and 10% dose threshold was more than 96%.


Introduction
Electron beam therapy is an integral part of treatment plans for a wide variety of tumor sites.The rapid dose deposition of electron beams followed by a characteristically sharp dose drop-off as a function of depth makes electron therapy an important treatment option in radiation oncology [1][2][3][4].Accurate and fast calculation of the dose distribution from irregularly-shaped cutouts is critical for the clinical implementation of electron therapy.This calculation has a complex dependence on the shape and energy of the electron beam as it travels from the accelerator head to the patient.A complete electron dosimetry calculation requires calculating the depth dose along the central axis and isodose line distribution.The isodose line distribution of the electron beams at a given depth strongly depends on the cutout shape, beam energy, and calculation depth.
As an electron beam penetrates a medium, it expands rapidly below the surface owing to scattering.However, the individual spread of the isodose curves varies, depending on the isodose line, beam energy, field size, and collimation.All the isodose curves of the low-energy beams show some expansion.However, only low-value isodose lines of higher energy beams bulge.Higher isodose lines tend to show lateral constriction, which worsens with decreasing field size.The shape of the isodose lines inside the electron field also depends on the cutout shape.For an electron beam created by an irregular cutout, the wider side of the cutout tends to have higher isodose lines than the narrower side.This is mainly owing to the increase in the output factor with the cutout size.Profiles/isodose curves of simple shapes (circles, squares, etc.) are often measured during the commissioning of LINAC machines [5,6].However, most clinical electron cutouts are irregularly shaped.This requires determining the shape of the isodose curves at least at selected depths [6].Currently, several treatment planning systems calculate the 3D dose distribution of electron beams: ADAC Pinnacle 3 [7] using the pencil beam algorithm [8] and Eclipse using the Monte Carlo algorithm [9,10].The accuracy of these treatment planning systems has been verified by several authors [1,9,10].Depending on the complication, creating a 3D electron plan using these treatment planning systems can take several hours to days.
In certain situations, the electron treatment field parameters (energy, shape, applicator size, gantry angle, collimator angle, couch angle, and SSD) are determined during patient setup in the treatment room and the patient must be treated on the same day.Radiation therapy clinics maintain a library of simpleshaped cutouts, along with their corresponding measured isodose lines and output factors, and use the cutouts if their shape matches the desired electron beam shape.Otherwise, clinics make new cutouts and measure their output factor and isodose distributions, commonly at d max , in solid water using ion chambers (or diodes) and films [6].In addition to being timeconsuming, the result is susceptible to human error and the way the films are processed.
Khan et al [11] suggested a semi-experimental model based on the lateral spread of pencil beams using the lateral build-up ratio (LBR), which is the ratio of the arbitrary cutout and a broad field dose as a function of depth, to calculate the percentage depth dose and dose per monitor unit (MU) of irregularlyshaped electron beams.They determined the LBR of arbitrary circular cutout using its radius (R) and a lateral spread parameter σ R (z), which they assumed is independent of cutout size for a given energy.
In our previous publications [12,13], we demonstrated that σ R (z) is dependent on the size of the cutout.By considering this size dependence of σ R (z), we illustrated a significant improvement in the accuracy of PDD and output factor calculations.In this study, we expand upon our previous work to compute the relative dose distribution of irregular cutouts on a plane perpendicular to the central axis at z max using an Elekta VERSA HD TM linear accelerator.

Material and methods
This research was conducted using an Elekta VERSA HD TM linear accelerator equipped with a 14 × 14 cm 2 applicator and 6, 9, 12 and, 15 MeV electron energies.Circular cutouts of various diameters (2.0, 3.0, 4.0, 5.0, 6.0, 7.0, and 9.0 cm) as well as several irregular cutouts were prepared.The cutouts were 1.5 cm thick and made with Lipowitz metal, which is known by its brand name Cerrobend.Depth dose curves were acquired at standard SSD using an IBA blue-phantom 2 [14] water tank and PTW Dosimetry Diode E detectors.The experimental setup for depth dose measurements has been previously detailed in our publications [12,13].The depth of the maximum dose (d max ) for each cutout and electron beam energy was determined from the measured depth dose curves.The effective point of measurement of the PTW Semiflex ion chamber was positioned at d max and the ionization charge for 100 MU was recorded for SSD values of 97.5, 100.0, 105.0, 110.0, 115.0, and 120.0 cm.Subsequently, the ionization charge readings were utilized to determine the effective SSD (SSD eff ) values of the cutouts.Additionally, for each cutout and electron beam energy, EDR2 film was placed at d max in solid water and irradiated with 200 MU.To mitigate the impact of film-to-film variations, films for each energy were taken from the same pack.[15,16] The irradiated films were developed using a Konica SRX-101A film processor and scanned using an EPSON 10 000 scanner.Employing a calibration curve prepared for each electron energy, the scanned images were converted into doses and the analysis was conducted using RIT V.8.64 software.
It is discussed by Khan et al [11] and Tyner et al [17] that the incident fluence of electrons is uniform in intensity and energy distribution inside the cutout regions at the surface of the phantom.Normalizing the depth dose curves at a depth between the surface and a depth (d 0 ) of 2-3 mm removes the electron fluence dependence.Figure 1 shows the PDD curves of the Versa HD TM LINAC measured using 9 MeV for circular cutouts with diameters of 2.0, 2.5, 3.0, and 4.0 cm and an open 14 × 14 cm 2 applicator.The measured depth dose (PDD) curves were normalized at depth d 0 = 1.0 mm.The figure demonstrates that, at a given depth, the dose increases with the cutout size

=
) of the electron beams.Further increase in the cutout size beyond R eq has little effect on the shape of PDD and is commonly equated to the PDD of the open field.
In our prior publication [13], we have shown that output factor and PDD of an irregular cutout at standard SSD and depth z can be determined using the equation   is the gap factor of a circular cutout with radius R i,c at distance g 0 ; LBR 0 (R i,c , z) is the lateral build-up ratio of the circular cutout of radius R i,c at depth z. becomes equal to one.The value of g 0 was determined using both film and diodes with good agreement between the two methods.The measured value of g 0 for VERSA HD TM was close to -2.4 cm.Note that g 0 is negative since it is located between the standard SSD  and the bottom of the applicator.Since g 0 is negative, the gap factor is greater than one.The lateral build-up ratio, LBR 0 (R i,c , z), was defined by Khan et al. [11] utilizing the dose at depth z and the incident fluence of the electron field at the phantom surface of circular cutout of radius R i,c and open field applicator.They showed that normalizing the depth curves at a depth between the surface and a depth of 2-3 mm removes the dependence of the depth dose curves on the electron fluence and LBR 0 (R i,c , z) can be calculated from the relation where σ R (z) is a lateral mean square spread parameter.In this research, we used d 0 = 1 mm as the normalization depth for the depth dose curves.For a monoenergetic electron beam incident on the surface of a homogeneous scattering medium perpendicularly, σ R (z) can be approximated by a Gaussian distribution.In a clinical setup, however, the electrons interact with various parts of the LINAC components, including the edge of the cutout block, as they travel from the scattering foil to the surface of the medium.This causes σ R (z) to lose its Gaussian distribution.Gebreamlak et al [12] demonstrated that at a given depth, the σ R (z) value increases linearly with cutout size until the cutout size reaches the equilibrium range of the electron beam (R eq ).It is important to note that equation (1) can be used to calculate the dose distribution at any point inside the irregular cutout field that is at least a distance l away from the field edge.The magnitude of l is primarily dictated by the smallest measured cutout size utilized in the model.In this article, we will use l = 5 mm.
represents the dose distribution of a cutout field at r on the calculation plane, the relative  Since the goal of this research is to calculate the relative planar dose distribution of irregular electron field up to 3 mm outside the field edge, we will divide the entire calculation region into two: region 1 and region 2. Region 1 represents the region inside the cutout field that is at least 5 mm away from the field edge and region 2 represents the region from 3 mm outside to 5 mm inside the cutout field edge.

Data analysis
Within 'region 1', the relative planar dose distribution at z max can be calculated by substituting equation (1) into equation (3).Simplifying the resulting equation yields , where R i c , 0 and R i c , ¢ are the sector radii of the cutout with respect to the reference and calculation points, respectively.
To compute the relative planar dose distribution inside 'region 2', a simple curve fitting model was employed.The profile shapes of various circular cutouts around their field edges were compared for each energy.Within the region of interest ('region 2'), their profiles exhibited similar shapes.As a result, a single curve fitting equation was generated for each energy.The fitting equation has the form where a, b, c, d, and e represent fitting parameters dependent on the electron energy, while x denotes the distance from the field edge.A negative/positive x value indicates that the calculation point is outside/ inside the cutout field.The calculated values of 'region 2' were then scaled such that they are equivalent to those of 'region 1' at their shared points, which is along the intersection curve of the two regions.Finally, the overall relative planar dose distribution of the cutout field was determined by combining the relative planar distributions of the two regions.Relevant relative isodose lines of the cutout field could then be displayed.).For a central axis passing through the origin, the calculated z max value of the cutout for 6 MeV was 1.2 cm. Figure 4 compares the calculated versus measured isodose lines of the cutout, while figure 5 compares the calculated versus measured cross-profiles.The difference between the crossprofiles within the cutout field was less than 3%.Employing the EDR2 film measurement as a reference, the gamma pass rate with a 3%/3 mm and 10% threshold was 96%.z max of the cutout for 9 MeV was 1.6 cm.Figures 6 and 7 compare the isodose lines and cross-profiles of the cutout for 9 MeV, respectively.The gamma pass rate with 3%/3 mm and 10% threshold was 98.5%. Figure 3(b) showcases an image of another irregularly shaped cutout created using a VERSA HD TM 14 × 14cm 2 cone, along with the two planar dose calculation regions ('region 1' and 'region 2' ).For a central axis passing through the origin, the calculated z max value of the cutout for 12 MeV was 2.1 cm. Figure 8 compares the calculated versus measured isodose lines of the cutout, while figure 9 compares the calculated versus measured cross-profiles.The discrepancy between the cross-profiles was less than 3%.Utilizing the EDR2 film measurement as a reference, the gamma pass rate with a 3%/3 mm and 10% threshold was 100.0%.The z max of the cutout for 15 MeV was 2.0 cm.Figures 10 and 11   the cutout for 15 MeV, respectively.The gamma pass rate with a 3%/3 mm and 10% threshold was 100.0%.

Results and discussion
In our previous publications [12,13] we calculated the percentage depth dose curves and output factors of irregular cutouts.In this article, we extended our work to calculate the relative dose distribution of irregular cutouts at d max .When an electron plan already exists, our work can serve as a secondary verification tool.In cases where the electron cutout shape is determined during clinical setup without an existing electron plan, our work can calculate key parameters such as PDD, output factor, and relative dose distribution at d max for the cutout.

Conclusion
The generation of planar dose distributions for irregularly shaped electron cutouts typically involves the use of EDR2 films placed at various depths in solid water.However, this method is labor-intensive and vulnerable to both human error and film processing inaccuracies.In this study, we devised a rapid technique for computing the planar dose distribution of any irregular cutout at its maximum dose depth, employing the improved LBR method and simple curve fitting.Comparison between the calculated and measured isodose lines demonstrated favorable agreement, with the discrepancy between the calculated and measured relative cross profiles falling within 3%.Moreover, the gamma pass rate, utilizing a 3%/3 mm and 10% relative dose threshold criterion, exceeded 96%.

Figure 1 .
Figure 1.Normalized depth dose curves of Versa HD TM 9 MeV for cutout diameters of 2.0, 2.5, 3.0, and 4.0 cm and the open field using 14 × 14 cm 2 cone at SSD = 100 cm.The normalization depth (d 0 ) is 1.0 mm.

Figure 2 .
Figure 2. Comparison between measured profiles of different cutouts and models for region within ± 15 mm of field edge for 6 MeV and 15 MeV electron beam energies, respectively.The equation utilized to generate the model curve is shown at the top of each figure.
until the cutout radius reaches the lateral equilibrium range ( where D 0 (u, z) is the dose per monitor unit (MU) of cutout u; K 0 is the dose/MU for a broad field reference applicator of size R 0 at the depth z max of the dose maximum for the broad field; PDD 0 (R 0 , z) is the relative

Figure 3 .
Figure 3.The schematic diagrams of two irregular cutouts (cutout1 and cutout2) prepared using VERSA HD TM 14 × 14cm 2 cone.The diagrams show the cutouts' edge and the two planar dose calculation regions.The solid line (black) represent cutouts' edge.The empty circles (red) and filled squares (blue) represent the calculation grid points of the 'region 1' and 'region 2' regions, respectively.The grid size is 2 mm in both vertical and horizontal directions.

Figure 4 .
Figure 4. Comparison of the measured (Ref.) and the calculated (Tar.)isodose lines at depth = 1.2 cm of cutout1 for an electron beam energy of 6 MeV.
The gap distance, g 0 , represents the distance from the standard SSD to the position long the central axis at which the electron fluence becomes the same for different cutout sizes.At this distance,

Figure 5 .
Figure 5.Comparison of the measured (Ref.) and the calculated (Tar.)cross-profiles at depth = 1.2 cm of cutout1 for an electron beam energy of 6 MeV.

Figure 6 .
Figure 6.Comparison of the measured (Ref.) and the calculated (Tar.)isodose lines at depth = 1.6 cm of cutout1 for an electron beam energy of 9 MeV.

Figure 7 .
Figure 7.Comparison of the measured (Ref.) and the calculated (Tar.)cross-profiles at depth = 1.6 cm of cutout1 for an electron beam energy of 9 MeV.

Figure 8 .
Figure 8.Comparison of the measured (Ref.) and the calculated (Tar.)isodose lines at depth = 2.1 cm of cutout2 for an electron beam energy of 12 MeV.

Figure 9 .
Figure 9.Comparison of the measured (Ref.) and the calculated (Tar.)cross-profiles at depth = 2.1 cm of cutout2 for an electron beam energy of 12 MeV.

Figure 10 .
Figure 10.Comparison of the measured (Ref.) and the calculated (Tar.)isodose lines at depth = 2.0 cm of cutout2 for an electron beam energy of 15 MeV.

Figures 2 (
Figures 2(a) and (b) compare measured electron profiles of various circular cutouts with the model-generated curves at z max for 6 MeV and 15 MeV, respectively.The curve fitting equation is shown at the top of each figure.Both figures show good agreement between the model and the measured profiles of different cutout sizes.Figure3(a) illustares a schematic diagram of an irregularly shaped cutout (cutout1) prepared using a VERSA HD TM 14 × 14 cm 2 cone alongside the two planar dose calculation regions ('region 1' and 'region 2' ).For a central axis passing through the origin, the calculated z max value of the cutout for 6 MeV was 1.2 cm.Figure4compares the calculated versus measured isodose lines of the cutout, while figure 5 compares the calculated versus measured cross-profiles.The difference between the crossprofiles within the cutout field was less than 3%.Employing the EDR2 film measurement as a reference, the gamma pass rate with a 3%/3 mm and 10% threshold was 96%.z max of the cutout for 9 MeV was 1.6 cm.Figures6 and 7compare the isodose lines and cross-profiles of the cutout for 9 MeV, respectively.The gamma pass rate with 3%/3 mm and 10% threshold was 98.5%.Figure3(b)showcases an image of another irregularly shaped cutout created using a VERSA HD TM 14 × 14cm 2 cone, along with the two planar dose calculation regions ('region 1' and 'region 2' ).For a central axis passing through the origin, the calculated z max value of the cutout for 12 MeV was 2.1 cm.Figure8compares the calculated versus measured isodose lines of the cutout, while figure9compares the calculated versus measured cross-profiles.The discrepancy between the cross-profiles was less than 3%.Utilizing the EDR2 film measurement as a reference, the gamma pass rate with a 3%/3 mm and 10% threshold was 100.0%.The z max of the cutout for 15 MeV was 2.0 cm.Figures10 and 11represent the comparison of the isodose lines and cross-profiles of

Figure 3 (
a) illustares a schematic diagram of an irregularly shaped cutout (cutout1) prepared using a VERSA HD TM 14 × 14 cm 2 cone alongside the two planar dose calculation regions ('region 1' and 'region 2' represent the comparison of the isodose lines and cross-profiles of

Figure 11 .
Figure 11.Comparison of the measured (Ref.) and the calculated (Tar.)cross-profiles at depth = 2.0 cm of cutout2 for an electron beam energy of 15 MeV.