Journal of Radiology and Imaging
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Journal of Radiology and Imaging
Volume 4, Issue 2, February 2020, Pages 7–16
Original researchOpen Access
An analytical method for 3dimensional calculation of the contaminant Xray dose in water caused by clinical electronbeam irradiation

Akira Iwasaki^{1,*}, Shingo Terashima^{2}, Shigenobu Kimura^{3}, Kohji Sutoh^{3}, Kazuo Kamimura^{3}, Yoichiro Hosokawa^{2} and
Masanori Miyazawa^{4}
 ^{1 }2324 Shimizu, Hirosaki, Aomori 0368254, Japan
 ^{2 }Graduate School of Health Sciences, Hirosaki University, 661 Honcho, Hirosaki, Aomori 0368564, Japan
 ^{3 }Department of Radiology, Aomori City Hospital, 11420 Katta, Aomori 0300821, Japan
 ^{4 }Technology of Radiotherapy Corporation, 212 Koishikawa, Bunkyoku, Tokyo 1750092, Japan
*Corresponding authors: Akira Iwasaki, 2324 Shimizu, Hirosaki, Aomori 0368254, Japan. Tel.: +172332480; Email: fmcch384@ybb.ne.jp or fmcch384@gmail.com; and Shingo Terashima, Graduate School of Health Sciences, Hirosaki University, 661 Honcho, Hirosaki, Aomori 0368564, Japan. Tel.: +81172395525; Email: stera@hirosakiu.ac.jp
Received 20 November 2019 Revised 14 January 2020 Accepted 24 January 2020 Published 31 January 2020
DOI: http://dx.doi.org/10.14312/23998172.20202
Copyright: © 2020 Iwasaki A, et al. Published by NobleResearch Publishers. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.
AbstractTop
Purposes: In this paper, an analytical method for 3dimensional (3D) calculation of the contaminant Xray dose in water caused by clinical electronbeam irradiation is proposed in light of the two groups of Monte Carlo (MC) datasets reported by Wieslander and Knöös (2006). Methods: The dose calculation was performed based on Clarkson’s sector method. We used a plane called the isocenter plane, which is set perpendicular to the beam axis, containing the isocenter on it. On the isocenter plane, we defined the applicator field formed by an electron applicator and the cerrobend area field formed by a cerrobend insert if any, as well as other physical terms that are important for the dose calculations. The original sector method was modified to consider the following terms: (a) the vague beamfield margins formed by the dualfoil system; (b) the inair dose distribution of the contaminant Xray beam; (c) the Xray spectrum change between the contaminant Xray PDD datasets and the published radiotherapy Xray PDD datasets; and (d) the contaminant Xray attenuation for the cerrobent insert, if any. Results and conclusions: By comparing the calculated datasets of depth dose (DD) and offaxis dose (OAD) with the MC results for electron beams of E=6, 12, and 18 MeV, it can be concluded that the analytical calculation method is of practical use for various irradiation conditions. In particular, it should be noted that the analytical method can give almost the same calculation results as the MCbased dose calculation algorithm used in a commercial treatment planning system (TPS).
Keywords: clinical electronbeams; contaminant Xray dose; electron applicator; linear accelerator; scattering foil; Clarkson’s sector method
Research highlightsTop
Based on Clarkson’s sector method, we developed an analytical method for calculation of the contaminant Xray dose in water caused by clinical electronbeam irradiation. The analytical method was constructed by considering the following terms: (a) the vague beamfield margins formed by the dualfoil system; (b) the inair dose distribution of the contaminant Xray beam; (c) the Xray spectrum change between the contaminant Xray PDD datasets and the published radiotherapy Xray PDD datasets; and (d) the contaminant Xray attenuation for the cerrobent insert, if any. The dose calculation was performed in light of the two groups of Monte Carlo (MC) datasets reported by Wieslander and Knöös (2006). We conclude that the analytical method can achieve accurate dose calculations, even for beams with cerrobent inserts.
IntroductionTop
Khan [1] describes the physical outline of highenergy electrons used in radiation therapy as follows: The most useful energy for electrons is 6 to 20 MeV. At these energies, the electron beams can be used to treat superficial tumors (less than 5 cm deep) with a characteristically sharp dropoff in dose beyond the tumor. The principal applications are (a) the treatment of skin and lip cancers, (b) chest wall irradiation for breast cancer, (c) administering boost dose to nodes, and (d) the treatment of head and neck cancers. Although many of these sites can be treated with superficial Xrays, brachytherapy, or tangential photon beams, the electronbeam irradiation offers distinct advantages in terms of dose uniformity in the target volume and in minimizing the dose to deeper tissues.
Acceptable field flatness and symmetry are obtained [1] with a proper design of beam scatterers and beam defining collimators. Accelerators with magnetically scanned beam do not require scattering foils. Others use one or more scattering foils, usually made up of lead, to widen the beam as well as give a uniform dose distribution across the treatment field. In recent years, linear accelerators with scattering foil, having photon and multienergy electronbeam capabilities, have become increasingly available for clinical use. Regarding the unnecessary contaminant Xray dose caused when electronbeam irradiation occurs using linear accelerators, Mahdavi et al. [2] summarize as follows: (a) the contaminant Xray dose provides a small dose to the patient; (b) the major part of the contaminant Xrays is produced by the accelerator exit window tube, monitor chamber, scattering foil, upper and lower pairs of collimator jaws, and electron applicator; (c) the scattering foil is the main source [3] and causes Xray contamination, especially at high energies [4]; (d) the atomic number and thickness of the scattering foil have a mass effect on the Xray contamination [5]; and (e) recently, a dualfoil system has been used, which is composed of the first and second foils (Figure 1), and this foil system can generate broader electron beams with less Xray contamination, especially at high energies.
It should be emphasized that the contaminant Xray dose is relatively small by comparison with the maximum electronbeam dose on the isocenter axis; however, the contaminant Xray dose should not generally be ignored for accurate dose evaluation. It seems that nobody has reported how to calculate the 3D contaminant Xray dose analytically. The analytical dealing is important for understanding the process of producing the contaminant Xray dose. Monte Carlo (MC) methods are today widely used in many electronbeam therapy applications because of the complexity of photon and electron transport. Wieslander and Knöös [6, 7] proposed implementation of a virtual linear accelerator (based on MC simulations) into a commercial treatment planning system (TPS) to verify the TPS. The characterization set for the TPS includes depth doses, profiles, and output factors. The authors also emphasize that problems associated with conventional measurements can be avoided and properties that are considered unmeasurable can be studied because the MC method can divide the dose into the separate doses yielded by direct electrons, indirect electrons, and contaminant Xrays. They summarized two groups of MC dose datasets [7] for electronbeams of energies of 6, 12, and 18 MeV under a series of electron applicators using an Elekta Precise linear accelerator. Here, we refer to the datasets of interest as “the WK MC dose datasets” or “the WK MC dose work.” These datasets were yielded using a common virtual accelerator for two MC simulation techniques: one is the standard simulation, and the other is the simulation used in a commercial TPS.
The present paper will propose an analytical method for calculation of the contaminant Xray dose in water in light of the two groups of WK MC dose datasets. The analytical method is based on Clarkson’s sector method [1], but considers the vague beamfield margins caused by using the dualfoil system.
Materials and methodsTop
Symbols and units
This paper uses the following units: the lengths are expressed in cm; the areas are expressed in cm^{2}; the angles are expressed in radian; and the doses are expressed in Gy. It should also be noted that some other physical quantities are dimensionless.
Theoretical background
Figure 1 is redrawn with reference to the textbook by Khan [1], showing a typical arrangement for electronbeam irradiation using the dualfoil system. The first foil (S_{1}) widens the narrow electron beam by multiple scattering, and the second foil (S_{2}) is designed to make the widened electron beam uniform in crosssection (these two foils are installed in the treatment head of the accelerator). The thickness of the S_{2} foil is differentially varied across the beam to produce the desired degree of beam widening and flattening. The beamdefining collimators are designed to provide a variety of field sizes and to maintain or improve the crosssectional flatness of the beam. Basically, the beamdefining collimators provide a primary collimation close to the S_{1} foil that defines the maximum field size and a secondary collimation close to the patient (or the isocenter plane) to define the treatment field. The secondary collimation is performed using the Xray collimator jaws and a series of electron applicators. It should be noted that the Xray collimator jaws are usually opened to a size larger than the electron applicator opening (in rectangles). Because the Xray collimator jaws give rise to extensive electron scattering, they are interlocked with the individual electron applicators to open automatically to a fixed predetermined size.
It should be stressed that the contaminant Xrays are primarily generated in the S_{1} foil, and the amount of the contaminant Xrays varies in general with the field size (or electron applicator opening). We use a simple assumption here that the contaminant Xrays are all produced at a point () on the isocenter axis within the S_{1} foil, as shown in Figure 1 (this paper does not directly refer to the dose caused by the bremsstrahlung produced by the electrons running through the phantom). This diagram also shows a geometrical arrangement for the contaminant Xray dose calculation using a semiinfinite water phantom whose surface coincides with the isocenter plane on which the isocenter (O) is situated. Let SSD_{X} be the distance between the point and the isocenter (O), and let SSD_{a} be the distance between the point and the beam exit side of the electron applicator. Furthermore, we set orthogonal coordinates of X_{beam}, Y_{beam}, and Z_{beam} whose origin is placed at the isocenter (O), where the X_{beam} and Y_{beam} axes are set on the isocenter plane, and the Z_{beam} axis is drawn down from the isocenter (O), coinciding with the isocenter axis (or the beam axis). Let the dose calculation be performed at an arbitrary point P(X_{cal}, Y_{cal}, Z_{cal}) within the water phantom, definined as
(Eq. 1)
Let the intersection of the line P and the isocenter plane be denoted Q(X_{iso}, Y_{iso}) (P is one of the fanlines radiating from point ), defined as
(Eq. 2)
Let Z_{0} be the water length of point P, measured from point Q(X_{iso}, Y_{iso}) along the line P; then, we have
(Eq. 3)
The present analytical method is constructed for calculation of the contaminant Xray dose at point Q(X_{iso}, Y_{iso}), which is based on Clarkson’s sector method [1], described as follows:
(a) Let E be the electronbeam energy (MeV). Here, it is assumed that the contaminant Xrays are all produced in the S_{1} foil by the EMeV electrons coming out from the accelerator with an acceleration voltage of E (MV).
(b) We utilize published radiotherapy Xray percentage depth dose (PDD) datasets for a sourcesurface distance (SSD) of 100 cm. Here, we use Z_{max}(E) as the depth of maximum dose, letting it be simply determined only by E(MV) around a field of 10×10 cm^{2}.
(c) Let the contaminant Xray dose calculation be performed in a semiinfinite water phantom placed at an SSD of 100 cm, assuming that the dose is under lateral electron equilibrium and that the contaminant Xray beam intensity in air at SSD=100 cm is the inair dose measured in a small mass of water under forward and lateral electron equilibrium.
(d) We use electron applicators forming rectangular beam fields (the dose calculations are performed by neglecting the fine structures of the applicators). Let be denoted as the beam field measured at the beam exit side of the electron applicator. Let be the beam field measured on the isocenter plane. As the field is shaped by the fanlines emanating from the point, the field can be described as
(Eq. 4)
Conversely, we let be the field size that the Xray collimator jaws form on the isocenter plane. As described earlier, we have >. Therefore, it can be seen that the more accurate intensity of the contaminant Xray beam should be evaluated [1] based on . This paper uses cerrobend inserts only within the electron applicator field (this is because the WK MC dose datasets are all collected under such irradiation conditions). It should be noted that the WK MC dose datasets are produced with SSD_{X}=100 cm and SSD_{a}=95 cm using =10×10,14×14, and 20×20 cm^{2} (these dimensions are defined at SSD=SSD_{a} [7]) for E = 6, 12, and 18 MeV.
(e) Let (, E ) be the relative inair dose intensity (refer to (c)) of the contaminant Xray beam of E at the isocenter (O) when using an electron applicator of with no cerrobend insert.
(f) Let () be the attenuation factor for the contaminant Xray beam of E for a cerrobend insert with a thickness of (≤1, setting (=0, E )=1).
(g) For calculation of the relative inair dose for the contaminant Xray beam of E with no beam shielding insert for a point of Q(X_{iso}, Y_{iso}) on the isocenter plane, we utilize the following function:
(Eq. 5)
The relative inair dose calculation for any combination of and E is then simply performed symmetrically with respect to the isocenter (O) on the isocenter plane, taking . Because the jaw field () determined by the electron applicator field () forms a perfect or approximate square field, the above F_{0} function is practically reasonable for use.
(h) The WK MC dose datasets show that, for the contaminant Xray beams, the offaxis dose (OAD) curves (or the dose profiles along lines perpendicular to the isocenter axis) at any depth do not sharply change around the field border of the electron applicator and around the field border of the cerrobend insert (as illustrated in Figure C1(b) in Appendix C). Consequently, it has been found that, on the isocenter plane, there is a need to introduce special factors for each sector with respect to the field borders of the electron applicator and the cerrobend insert as follows:
For one of the lines (k=1,2,3,…) extending radically on the isocenter plane from point Q (refer to Figures 26, given later), if the corresponding line intersects the field border of the electron applicator or the cerrobend insert at distances of R_{k_1}, R_{k_2}, R_{k_3}, etc., measured from point Q, it is necessary to set the special factors of interest for R_{k_i} (i=1,2,3,…) as
(Eq. 6)
Moreover, it has been found (Figures C1 (a) and (b)) that α(E)=1 is a good and simple model for any contaminant Xray beam energy (E) and for any electron applicator field (). It should be noted that the present paper does not directly use the h_{0}(E) function.
(i) When using Clarkson’s sector method, we assume that, on the isocenter plane, a square field with a side of S is equivalent to a circular field with a radius of . The present paper utilizes this assumption for both the PDD function and the SF function taking the field size measured on the isocenter plane.
(j) For each energy E, the WK MC dose datasets are expressed using the normalized valuation obtained when a common virtual accelerator is set up to deliver 1.0 Gy per 100 MU at a depth of d_{max} on the isocenter axis in water, where d_{max} is the depth at which the maximum dose caused by the electronbeam irradiation with an open electron applicator of =20×20 cm^{2} is yielded (this means that the WK MC dose datasets are all expressed in Gy/100 MU for each electron beam). Conversely, the present analytical dose calculation is performed for contaminant Xray beams, based on published radiotherapy Xray PDD datasets. Therefore, when comparing the analytical and MC datasets for each combination of and E, we need to take into account the Xray spectrum difference between the contaminant Xray beam and the published radiotherapy PDD Xray beam, and we should introduce a conversion factor of CF_{MC/PDD}(Gy/100 MU/%) for setting both datasets at the same dose valuation level. However, the present study does not directly use the CF_{MC/PDD} factor for the dose calculation.
(k) Under the assumption that the contaminant Xrays are all emitted from the point at a distance of SSD_{X}=100 cm from the isocenter (O) along the isocenter axis (Figure 1), we may suppose that there is no change in PPD with SSD between the contaminant Xray PDD function of SSD_{X}=100 cm and the published radiotherapy Xray PDD function [8, 9] with SSD_{0}=100 cm (the following PDD functions are described under SSD_{0}=SSD_{X}=100 cm). For a given dose evaluation depth Z_{0}, taking as a beam field on the isocenter plane, we let the published radiotherapy Xray PDD function be expressed as PDD_{0}=PDD_{0} (Z_{0},, E), and let the contaminant Xray PDD function be expressed as PDD_{X}=PDD_{X} (Z_{0},, E).
(l) It has been found that the contaminant Xray PDD_{X} can be approximated as follows:
For Z_{0}<Z_{max} (E)
(Eq. 7)
letting be the equivalent square field side of , where
(Eq. 8)
(Eq. 9)
Next for Z_{0}<Z_{max} (E)
(Eq. 10)
Here, the pair of Q_{a}(E) and V_{0}(E) and the pair of Q_{b}(E) and β(E) are introduced to consider the Xray spectrum change between the PDD_{X} and PDD_{0} Xray beams of energy E.
(m) Figure 2 shows two arrangements for point Q(X_{iso}, Y_{iso}) on the isocenter plane. One is set in the field, and the other is outside the field. For the dose calculation relating to each Q point using Clarkson’s sector method, we take the line with an inclination angle θ_{k} radiating from point Q on the isocenter plane, setting θ_{k}=(k1) Δθ_{0}+ Δθ_{0}/2 (k = 1 −360) with Δθ_{0} =2π/360 (radian), taken as anticlockwise rotation angles measured from the X_{beam} axis direction.
(n) Based on the above preconditions, we describe how to calculate the dose for point P(X_{cal}, Y_{cal}, Z_{cal} ) by summing up each dose element (ΔD) obtained from the corresponding sector of and Δθ_{0}. Figure 2 shows the case containing no cerrobend insert (=0). Let (j=14) be the line vectors for the sides of the field, taking the rectangular field corners anticlockwise as ①,②,…,⑤.
First, we set point Q(X_{iso}, Y_{iso}) inside the field, letting the line intersect with the field side of as an example, and letting the distance between the point Q and the intersection point be R_{k_1}. Then, we can calculate the dose of ΔD as
ΔD(X_{cal}, Y_{cal}, Z_{cal})
(Eq. 11)
where and (T_{cerrow}=0, E)=1 (refer to (f)); SF(R_{k_1}) is the scatter factor (SF), evaluated using R_{k_1} as the field radius (the SF can be set not as a function of E for MV photon beams [8]); and PDD_{X} (Z_{0}, R_{k_1}, E) is expressed using the field radius of R_{k_1}. It should be emphasized that the term h_{0}(E)R_{k_1}^{αE} is introduced to take into account the vague beamfield margin formed by the dualfoil system. We can then rewrite equation 11 as
ΔD(X_{cal}, Y_{cal}, Z_{cal})=
(Eq. 12)
with
(Eq. 13)
As described later, we will attempt to evaluate the FAC function in one lump for a given irradiation condition.
(O) Second, we set point Q(X_{iso}, Y_{iso}) outside the field (Figure 2). Then, the line concerns the sector dose calculation at two distances R_{k_1} and R_{k_2} from point Q. Let the line intersect with the electron applicator field sides as an example. We can then calculate the dose of ∆D as
ΔD(X_{cal}, Y_{cal}, Z_{cal})=
(Eq. 14)
where and . It should be emphasized that, when the line does not intersect with the field sides, we need to calculate the relational sector dose as ∆D(X_{cal}, Y_{cal}, Z_{cal} )=0. This fact shows one of the defects for the present sector method. However, it has been found that the α(E) function can effectively deal with such dose calculation defects, as shown in the offaxis dose (OAD) curves calculated in the next section.
(p) Figure 3 shows another irradiation case, in which a cerrobend insert with a hollow region in itself is set within an field. Then, we take points at distances of R_{k_1}, R_{k_2}, etc., measured from point Q(X_{iso}, Y_{iso}) along the line, depending on the position. Figure 4 shows a case in which point Q is placed in the hollow region of the cerrobend insert, where (j=1, 2, …, 10) are straight continuous lines forming the cerrobend insert shape anticlockwise (the numbers ①,②,…,⑪ start from a point on the outside border of the cerrobend insert). Then, the dose ∆D from one sector of and Δθ_{0} within the cerrobend area can be similarly calculated as
ΔD(X_{cal}, Y_{cal}, Z_{cal})=
(Eq. 15)
with
(Eq. 16)
Subsequently, we attempt to calculate the dose from the regions outside the cerrobend area.
Figure 5 shows how to take points at distances of R_{k_1}, R_{k_2}, etc., measured from point Q(X_{iso}, Y_{iso}) along the line, depending on the position. Referring to Figure 6, in which point Q is set in the hollow region of the cerrobend insert, the dose ∆D from one sector of and Δθ_{0} can be calculated as
ΔD(X_{cal}, Y_{cal}, Z_{cal})=
(Eq. 17)
where
(q) Finally, we reassess the factor X_{att} using equations 13 and 16. This is given by
(Eq. 18)
Correspondence with the WK MC dose datasets
The WK MC dose datasets are produced using a common virtual accelerator for two MC simulation techniques: one is performed with BEAMnrc [10, 11] as the dose calculation simulation using a Cartesian voxel grid with the DOSXYZnrc code [1214] as the phantom simulation (let the combination of these simulations be called the standard simulation technique); the other is performed using the MCbased dose calculation simulation in a commercial TPS.
The WK MC dose datasets are performed in water phantoms for E=6, 12, and 18 MeV using =10×10 cm^{2},=10×10/14×14 cm^{2} (where 10×10 is the field produced using a cerrobend insert placed just inside the 14×14 applicator), and =20×20 cm^{2}. The dose datasets are separated into depth dose (DD) curves and offaxis dose (OAD or profiledose) curves, which are normalized with the dose obtained when the virtual accelerator is set up to deliver 1.0 Gy per 100 MU at the maximum dose depth (d_{max}) in water using =20×20 cm^{2} for each electronbeam energy (E). The dose datasets acquired using the standard simulation technique are composed of stepped curves of DD and OAD; and the dose datasets acquired using the commercial TPS are composed of dotted curves of DD and OAD. It should be noted that both the stepped and dotted datasets of OAD are classified as OAD profiles in the X and Y directions; however, the present paper does not refer to the OAD differences in the X and Y directions.
Results and discussionTop
The functions and constants used for each of the DD and OAD calculations were determined by trial and error. Table 1 summarizes values for the functions and constants, excluding α(E)=1, U_{0} ( ) as given by equation 8, and V_{0}(E) as given by equation 9. Table 1 is classified into four groups: (a) the stepped curves of DD (Case1 to 8), (b) the stepped curves of OAD (Case9 to 16), (c) the dotted curves of DD (Case17 to 24), and (d) the dotted curves of OAD (Case25 to 32). For each case number, the corresponding reference datasets are given using figure numbers of the WK MC dose work as follows:
Regarding the group (a),
Case1 (=10×10 cm^{2}, E=6 MeV) is for DD in Figure 3(a) under OAD in Figure 2(b);
Case2 (=10×10 cm^{2}, E=6 MeV) is for DD in Figure 3(a) under OAD in Figure 3(b);
Case3 (=10×10 cm^{2}, E=6 MeV) is for DD in Figure 3(a) under OAD in Figure 3(c);
Case4 (=10×10/14×14 cm^{2}, E=12 MeV) is for DD in Figure 5(a) under OAD in Figure 5(b);
Case5 (=10×10/14×14 cm^{2}, E=12 MeV) is for DD in Figure 5(a) under OAD in Figure 5(c);
Case6 (=10×10 cm^{2}, E=18 MeV) is for DD in Figure 3(d) under OAD in Figure 2(b));
Case7 (=10×10 cm^{2}, E=18 MeV) is for DD in Figure 3(d) under OAD in Figure 3(e);
Case8 (=10×10 cm^{2}, E=18 MeV) is for DD in Figure 3(d) under OAD in Figure 3(f).
Regarding the group (b),
Case9 (=20×20 cm^{2}, E=6 MeV) is for OAD (Z_{cal}=5 cm) in Figure 2(b) under DD in Figure 3(a);
Case10 (=10×10 cm^{2}, E=6 MeV) is for OAD (Z_{cal}=1 cm) in Figure 3(b) under DD in Figure 3(a);
Case11 (=10×10 cm^{2}, E=6 MeV) is for OAD (Z_{cal}=5 cm) in Figure 3(c) under DD in Figure 3(a);
Case12 (=10×10/14×14 cm^{2}, E=12 MeV) is for OAD (Z_{cal}=2 cm) in Figure 5(b) under DD in Figure 5(a);
Case13 (=10×10/14×14 cm^{2}, E=12 MeV) is for OAD (Z_{cal}=10 cm) in Figure 5(c) under DD in Figure 5(a);
Case14 (=20×20 cm^{2}, E=18 MeV) is for OAD (Z_{cal}=15 cm) in Figure 2(b) under DD in Figure 3(d);
Case15 (=10×10 cm^{2}, E=18 MeV) is for OAD (Z_{cal}=3 cm) in Figure 3(e) under DD in Figure 3(d));
Case16 (=10×10 cm^{2}, E=18 MeV) is for OAD (Z_{cal}=15 cm) in Figure 3(f) under DD in Figure 3(d).
Regarding the group (c),
Case17 (=10×10 cm^{2}, E=6 MeV) is for DD in Figure 3(a) under OAD in Figure 2(b);
Case18 (=10×10 cm^{2}, E=6 MeV) is for DD in Figure 3(a) under OAD in Figure 3(b);
Case19 (=10×10 cm^{2}, E=6 MeV) is for DD in Figure 3(a) under OAD in Figure 3(c);
Case20 (=10×10/14×14 cm^{2}, E=12 MeV) is for DD in Figure 5(a) under OAD in Figure 5(b);
Case21 (=10×10/14×14 cm^{2}, E=12 MeV) is for DD in Figure 5(a) under OAD in Figure 5(c);
Case22 (=10×10 cm^{2}, E=18 MeV) is for DD in Figure 3(d) under OAD in Figure 2(b);
Case23 (=10×10 cm^{2}, E=18 MeV) is for DD in Figure 3(d) under OAD in Figure 3(e);
Case24 (=10×10 cm^{2}, E=18 MeV) is for DD in Figure 3(d) under OAD in Figure 3(f).
Regarding the group (d),
Case25 (=20×20 cm^{2}, E=6 MeV) is for OAD (Z_{cal}=5 cm) in Figure 2(b) under DD in Figure 3(a);
Case26 (=10×10 cm^{2}, E=6 MeV) is for OAD (Z_{cal}=1 cm) in Figure 3(b) under DD in Figure 3(a);
Case27 (=10×10 cm^{2}, E=6 MeV) is for OAD (Z_{cal}=5 cm) in Figure 3(c) under DD in Figure 3(a);
Case28 (=10×10/14×14 cm^{2}, E=12 MeV) is for OAD (Z_{cal}=2 cm) in Figure 5(b) under DD in Figure 5(a);
Case29 (=10×10/14×14 cm^{2}, E=12 MeV) is for OAD (Z_{cal}=10 cm) in Figure 5(c) under DD in Figure 5(a);
Case30 (=20×20 cm^{2}, E=18 MeV) is for OAD (Z_{cal}=15 cm) in Figure 2(b) under DD in Figure 3(d);
Case31 (=10×10 cm^{2}, E=18 MeV) is for OAD (Z_{cal}=3 cm) in Figure 3(e) under DD in Figure 3(d);
Case32 (=10×10 cm^{2}, E=18 MeV) is for OAD (Z_{cal}=15 cm) in Figure 3(f) under DD in Figure 3(d).
Table 1 also lists values of Z_{max}(E), Applicator ( & =equivalent square field side of ), Q_{a}(E), Q_{b}(E), β(E), f_{0}E), FAC(, =0, E), FAC(, >0,E), and (≥0,E).
By analyzing the datasets of the stepped curves of DD and OAD (Case1 to 16) regarding the functions of Q_{a}(E), Q_{b}(E), β(E), f_{0}E, we constructed the following regression equations:
(Eq. 19)
(Eq. 20)
(Eq. 21)
(Eq. 22)
Conversely, for the FAC function, we used and E as variables (letting be defined as the equivalent square field side of ). By analyzing the FAC datasets of Case1 to 16, we constructed a FAC regression function of
(Eq. 23)
Similarly, for the dotted curves of DD and OAD under Case17 to 32, we built the following regression equations:
(Eq. 24)
(Eq. 25)
(Eq. 26)
(Eq. 27)
(Eq. 28)
These regression functions may be useful for estimating reasonable values for the corresponding functions for given irradiation conditions. Details are described in Appendix A. Further in Appendix B, we refer to detailed results for calculated and MCbased DD and OAD datasets; and in Appendix C, we refer to the working of the function (E).
Caseno. E(MeV) 
Z_{max} (E) cm  Applicator / cm^{2} 
Q_{a}(E)  Q_{b}(E)  β(E)  f_{0}(E) (cm^{1})  FAC(, )(for =0)  FAC (, )(for >0)  (, )(for ≥0) 
Case1 E=6 MeV 
1.5  10 × 10 =10.5 cm) 
1.000  9.428E04  1.946  2.638E03  1.287E05  no existing  1 
Case2 E=6 MeV 
1.5  (10×10 =10.5 cm) 
1.000  9.428E04  1.946  4.411E03  1.287E05  no existing  1 
Case3 E=6 MeV 
1.5  (10×10 =10.5 cm) 
1.000  9.428E04  1.946  3.819E02  1.287E05  no existing  1 
Case4 E=12 MeV 
2.6  (10×10 =14.7 cm) 
1.043  1.097E02  1.097  4.401E02  3.123E05  1.875E05  0.600 
Case5 E=12 MeV 
2.6  (10×10 =14.7 cm) 
1.043  1.097E02  1.097  3.264E02  3.205E05  1.722E05  0.537 
Case6 E=18 MeV 
3.2  (10×10 =10.5 cm) 
1.100  3.000E03  1.501  4.898E02  5.993E05  no existing  1 
Case7 E=18 MeV 
3.2  (10×10 =10.5 cm) 
1.100  3.000E03  1.501  3.853E02  5.993E05  no existing  1 
Case8 E=18 MeV 
3.2  (10×10 =10.5 cm) 
1.100  3.000E03  1.501  5.305E02  5.993E05  no existing  1 
Caseno. E(MeV) 
Z_{max} (E) cm  Applicator / cm^{2} 
Q_{a}(E)  Q_{b}(E)  β(E)  f_{0}(E) (cm^{1})  FAC(, )(for =0)  FAC (, )(for >0)  (, )(for ≥0) 
Case9 E=6 MeV 
1.5  20×20 =21.1 cm) 
1.000  9.428E04  1.946  2.638E03  6.495E06  no existing  1 
Case10 E=6 MeV 
1.5  (10×10 =10.5 cm) 
1.000  9.428E04  1.946  4.411E03  1.335E05  no existing  1 
Case11 E=6 MeV 
1.5  (10×10 =10.5 cm) 
1.000  9.428E04  1.946  3.819E02  1.433E05  no existing  1 
Case12 E=12 MeV 
2.6  (10×10/ 14×14 =14.7 cm) 
1.043  1.097E02  1.097  2.739E02  3.084E05  1.852E05  0.600 
Case13 E=12 MeV 
2.6  (10×10/ 14×14 =14.7 cm) 
1.043  1.097E02  1.097  4.287E02  3.126E05  1.679E05  0.537 
Case14 E=18 MeV 
3.2  (20×20 =21.1 cm) 
1.100  3.000E03  1.501  4.898E02  3.282E05  no existing  1 
Case15 E=18 MeV 
3.2  (10×10 =10.5 cm) 
1.100  3.000E03  1.501  3.853E02  6.424E05  no existing  1 
Case16 E=18 MeV 
3.2  (10×10 =10.5 cm) 
1.100  3.000E03  1.501  5.305E02  6.427E05  no existing  1 
Caseno. E(MeV) 
Z_{max} (E) cm  Applicator / cm^{2} 
Q_{a}(E)  Q_{b}(E)  β(E)  f_{0}(E) (cm^{1})  FAC(, )(for =0)  FAC (, )(for >0)  (, )(for ≥0) 
Case17 E=6 MeV 
1.5  10×10 =10.5 cm) 
1.400  3.428E04  2.396  2.763E02  1.291E05  no existing  1 
Case18 E=6 MeV 
1.5  (10×10 =10.5 cm) 
1.400  3.428E04  2.396  3.017E02  1.291E05  no existing  1 
Case19 E=6 MeV 
1.5  (10×10 =10.5 cm) 
1.400  3.428E04  2.396  2.107E02  1.291E05  no existing  1 
Case20 E=12 MeV 
2.6  (10×10/ 14×14 =14.7 cm) 
1.200  1.780E02  1.110  2.975E02  3.042E05  1.319E05  0.434 
Case21 E=12 MeV 
2.6  (10×10/ 14×14 =14.7 cm) 
1.200  1.780E02  1.110  2.477E02  3.182E05  1.010E05  0.318 
Case22 E=18 MeV 
3.2  (10×10 =10.5 cm) 
0.957  7.903E02  0.677  5.659E02  7.730E05  no existing  1 
Case23 E=18 MeV 
3.2  (10×10 =10.5 cm) 
0.957  7.903E02  0.677  1.963E02  7.730E05  no existing  1 
Case24 E=18 MeV 
3.2  (10×10 =10.5 cm) 
0.957  7.903E02  0.677  3.751E02  7.730E05  no existing  1 
Caseno. E(MeV) 
Z_{max} (E) cm  Applicator / cm^{2} 
Q_{a}(E)  Q_{b}(E)  β(E)  f_{0}(E) (cm^{1})  FAC(, )(for =0)  FAC (, )(for >0)  (, )(for ≥0) 
Case25 E=6 MeV 
1.5  20×20 =21.1 cm) 
1.400  3.428E04  2.396  2.763E02  7.429E06  no existing  1 
Case26 E=6 MeV 
1.5  (10×10 =10.5 cm) 
1.400  3.428E04  3.017E02  4.411E03  1.463E05  no existing  1 
Case27 E=6 MeV 
1.5  (10×10 =10.5 cm) 
1.400  3.428E04  2.396  2.107E02  1.347E05  no existing  1 
Case28 E=12 MeV 
2.6  (10×10/ 14×14 =14.7 cm) 
1.200  1.780E02  1.110  2.868E02  2.963E05  1.285E05  0.434 
Case29 E=12 MeV 
2.6  (10×10/ 14×14 =14.7 cm) 
1.200  1.780E02  1.110  2.752E02  3.199E05  1.016E05  0.318 
Case30 E=18 MeV 
3.2  (20×20 =21.1 cm) 
0.957  7.903E02  0.677  5.659E02  4.469E05  no existing  1 
Case31 E=18 MeV 
3.2  (10×10 =10.5 cm) 
0.957  7.903E02  0.677  1.963E02  7.577E05  no existing  1 
Case32 E=18 MeV 
3.2  (10×10 =10.5 cm) 
0.957  7.903E02  0.677  3.751E02  8.867E05  no existing  1 
ConclusionTop
We attempted to develop an analytical method for 3dimensional (3D) calculation of the contaminant Xray dose in water caused by clinical electronbeam irradiation in light of the two groups of Monte Carlo (MC) datasets reported by Wieslander and Knöös (2006). The analytical method is based on Clarkson’s sector method. However, the original sector method was modified to take into account the following terms: (a) the vague beamfield margins formed by the dualfoil system; (b) the inair dose distribution of the contaminant Xray beam; (c) the difference between the Xray spectrum used for constructing the contaminant Xray PDD datasets and that used for constructing the published radiotherapy Xray PDD datasets; and (d) the contaminant Xray attenuation for the cerrobent insert, if any. We can conclude that the analytical method can achieve accurate dose calculations, even for beams with cerrobent inserts. In particular, it should be emphasized that the analytical method can give almost the same calculation results as the MCbased dose calculation algorithm in a commercial TPS.
Acknowledgements
We would like to thank Editage (www.editage.com) for English language editing.
Conflicts of interest
This study was carried out in collaboration with Technology of Radiotherapy Corporation, Tokyo, Japan. This sponsor had no control over the interpretation, writing, or publication of this work.
Supplementary dataTop
ReferencesTop
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