Journal of Radiology and Imaging
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Journal of Radiology and Imaging
Volume 7, Issue 1, August 2023, Pages 1–4
Original researchOpen Access
Further development of the preceding Gaussianpencilbeammodel used for calculation of the inwater dose 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; and Shingo Terashima, Graduate School of Health Sciences, Hirosaki University, 661 Honcho, Hirosaki, Aomori 0368564, Japan. Tel: +81172395525, Email: stera@hirosakiu.ac.jp
Received 14 March 2023 Revised 1 June 2023 Accepted 16 June 2023 Published 28 June 2023
DOI: http://dx.doi.org/10.14312/23998172.20231
Copyright: © 2023 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: We perform further development for our previous Gaussianpencilbeammodel used for calculating the electron dose in water under clinical electronbeam irradiation. The main purpose is to evaluate accurately the parallel beam depthdoses at deep depths beyond about the extrapolated range (R_{p}) under an infinite field. Methods: Sets of parallel beam depthdoses under an infinite field were reconstructed for beams of E=6, 12, and 18 MeV in light of the electron Monte Carlo (eMC) datasets reported by Wieslander and Knöös (2006), separating the datasets into the direct electron beam and directplusindirect electron beam groups. The datasets at the deep depths were then reconstructed using each factor of . Results and conclusions: The following results were obtained by comparing the calculated datasets of depth dose (DD) and offaxis dose (OAD) with the eMC datasets: (i) The further revised Gaussian pencil beam model is of practical use without using complicated correction factors; and (ii) The DD and OAD datasets are yielded effectively over wide ranges of depths and offaxis distances.
Keywords: Gaussian pencil beam model; dose calculation; electron MC; electron beams; linear accelerator
Research highlightsTop
The dose in water caused by the clinical electronbeam irradiation is mainly composed of the doses due to the direct electrons, the indirect electrons, and the contaminant Xrays. In light of the electron Monte Carlo (eMC) datasets reported by Wieslander and Knöös (2006), this paper describes further development of the preceding Gaussianpencilbeammodel for calculating doses in water. The main subject is how to evaluate accurately the parallel beam depthdoses at deep depths beyond about the extrapolated range (R_{p}) under an infinite field. Characteristic shapes were yielded effectively at shallow and deep depths for both of the depth dose curves and the offaxis dose profiles.
IntroductionTop
In 2022, we reported [1] a revised Gaussianpencilbeam model, which was constructed on the basis of electron Monte Carlo (eMC) datasets performed by Wieslander and Knöös [2, 3] for 6, 12, and 18MeV electron beams taking 10 × 10 cm^{2} and 10 × 10/ 14 × 14 cm^{2} applications. However, we realize that there remain two problems: (a) Parallelbeam depthdose datasets (D_{∞}) at deep depths in an infinitely broad field; and (b) Characteristic shape differences of the offaxis dose (OAD) profiles at shallow and deep depths.
For resolving these problems, we use the almost same dose calculation procedures as in the previous paper, only excepting the usage purpose of the factor [1] (this factor is originally introduced for recalculating reasonable OAD datasets at deep depths from ones illustrated in paper diagrams). Although this paper also utilizes a parallelbeam depthdose dataset (D_{∞}) of infinite field for a given irradiation, the parallelbeam depthdose dataset (D_{∞}) is partially recalculated using the factor for a point that is situated beyond about the extrapolated range (R_{p}). Actually, reporting this technique is the main subject of the paper.
Materials and methodsTop
Based on the depth dose (DD) and offaxis dose (OAD) datasets of the WK eMC work [3], we similarly develop this study. Here, we would like to emphasize that each of the DD or OAD datasets is normalized with a dose of 1.0 Gy per 100 MU at the maximum dose depth (d_{max}) caused under the use of an open electron applicator of A_{appl} = 20 × 20 cm^{2}. This paper is also use the same dose unit of Gy/100 MU for each of the DD and OAD datasets.
We use KK numbers of KK=1 to 24 for the DD and OAD datasets, originally introduced in the previous paper on compiling and handling of the beam energy (E), the electron applicator (A_{appl}), and the utilized electron Monte Carlo (eMC). It should be noted that the eMCbased dose is separated into the dose due to (i) the direct electrons getting no interactions with the electron applicator, the dose due to (ii) the indirect electrons getting interactions with the electron applicator, and the dose due to (iii) the contaminant Xrays from within the treatment head. This paper is described only taking the direct electron beams and the directplusindirect electron beams (Tables 1 and 2). The Supplementary Figure (Supp. Fig.) numbers below are the corresponding ones described in the paper of the WK eMC work using standard and commercial eMC techniques (it should be noted that the OAD datasets for a given irradiation are yielded on two horizontal planes at shallow and deep Z_{c} depths).
For the direct electron beams (Table 1), the standard eMC datasets are classified into:
Under Supp. Fig. 3(a)DD, we set KK=1 for Supp. Fig. 3(b)OAD (Z_{c}=1 cm) and KK=2 for Supp. Fig. 3(c)OAD (Z_{c} =5 cm);
Under Supp. Fig. 5(a)DD, we set KK=3 for Supp. Fig. 5(b)OAD (Z_{c} =2 cm) and KK=4 for Supp. Fig. 5(c)OAD (Z_{c} =10 cm);
Under Supp. Fig. 3(d)DD, we set KK=5 for Supp. Fig. 3(e)OAD (Z_{c} =3 cm) and KK=6 for Supp. Fig. 3(f)OAD (Z_{c} =15 cm).
Similarly, the commercial eMC datasets are classified into:
Under Supp. Fig. 3(a)DD, we set KK=7 for Supp. Fig. 3(b)OAD (Z_{c} =1 cm) and KK=8 for Supp. Fig. 3(c)OAD (Z_{c} =5 cm);
Under Supp. Fig. 5(a)DD, we set KK=9 for Supp. Fig. 5(b)OAD (Z_{c} =2 cm) and KK=10 for Supp. Fig. 5(c)OAD (Z_{c} =10 cm);
Under Supp. Fig. 3(d)DD,we set KK=11 for Supp. Fig. 3(e)OAD (Z_{c} =3 cm) and KK=12 for Supp. Fig. 3(f)OAD (Z_{c} =15 cm).
For direct electron beams 
D_{a} (Gy/100 MU) 
Z_{a} (cm) 
D_{b} (Gy/100 MU) 
Z_{b} (cm) 
(Z_{a}) (no unit) 

(i) Standard eMC  
KK=1 & 2 (E=6 MeV)  1.45E03 
5 
8.36E02 
3.35 
0.619 

KK=3 & 4 (E=12 MeV)  3.38E03 
10 
1.08E02 
6.99 
39.063 

KK=5 & 6 (E=18 MeV)  5.38E03 
15 
1.30E02 
9.72 
36.115 

(ii) Commercial eMC  
KK=7 & 8 (E=6 MeV)  1.49E03 
5 
8.88E02 
3.33 
1.548 

KK=9 & 10 (E=12 MeV)  3.00E03 
10 
1.08E02 
7.00 
34.278 

KK=11 & 12 (E=18 MeV)  5.63E03 
15 
1.41E02 
9.89 
50.819 
For the directplusindirect electron beams (Table 2), the standard eMC datasets are classified into:
Under Supp. Fig. 3(a)DD, we set KK=13 for Supp. Fig. 3(b)OAD (Z_{c}=1 cm) and KK=14 for Supp. Fig. 3(c)OAD (Z_{c} =5 cm);
Under Supp. Fig. 5(a)DD, we set KK=15 for Supp. Fig. 5(b)OAD (Z_{c} =2 cm) and KK=16 for Supp. Fig. 5(c)OAD (Z_{c} =10 cm);
Under Supp. Fig. 3(d)DD, we set KK=17 for Supp. Fig. 3(e)OAD (Z_{c} =3 cm) and KK=18 for Supp. Fig. 3(f)OAD (Z_{c} =15 cm).
Similarly, the commercial eMC datasets are classified into:
Under Supp. Fig. 3(a)DD, we set KK=19 for Supp. Fig. 3(b)OAD (Z_{c} =1 cm) and KK=20 for Supp. Fig. 3(c)OAD (Z_{c} =5 cm);
Under Supp. Fig. 5(a)DD, we set KK=21 for Supp. Fig. 5(b)OAD (Z_{c} =2 cm) and KK=22 for Supp. Fig. 5(c)OAD (Z_{c} =10 cm);
Under Supp. Fig. 3(d)DD,we set KK=23 for Supp. Fig. 3(e)OAD (Z_{c} =3 cm) and KK=24 for Supp. Fig. 3(f)OAD (Z_{c} =15 cm).
For direct & indirect electron beams 
D_{a} (Gy/100 MU) 
Z_{a} (cm) 
D_{b} (Gy/100 MU) 
Z_{b} (cm) 
(Z_{a}) (no unit) 

(i) Standard eMC  
KK=13 & 14 (E = 6 MeV)  1.33E03 
5 
8.96E02 
3.36 
0.159 

KK=3 & 4 (E=12 MeV)  3.38E03 
10 
1.08E02 
6.99 
39.063 

KK=15 & 16 (E=12 MeV)  3.65E03 
10 
3.01E02 
6.94 
0.214 

(ii) Commercial eMC  
KK=19 & 20(E=6 MeV)  1.33E03 
5 
8.96E02 
3.36 
0.156 

KK=21 & 22 (E=12 MeV)  3.25E03 
10 
3.77E02 
6.36 
0.180 

KK=23 & 24 (E=18 MeV)  5.66E03 
15 
1.23E02 
9.88 
1.72E+02 
We propose another usage of the factor, which is used for obtaining datasets of parallel beam depthdose (D_{∞}) in an infinitely broad field using a mathematical expression as a function of depth Z in a region beyond about the extrapolated range (R_{p}).
For constructing such datasets of parallel beam depthdose (D_{∞}), we use again the same DD datasets plotted in graphs of the paper reported by Wieslander and Knöös [3] (the construction method is the same as reported in the previous paper [1]). It has been found that, as shown later in Supp. Fig. 1, the reconstructed D_{∞} datasets reported in the previous paper usually contain relatively large errors in the lowdose region at deep depths beyond about the extrapolated range (R_{p}). In order to avoid this situation, the previous paper performed the dose calculation introducing the factor as a dose correction factor only at each specific depth.
This paper proposes a method to reconstruct a more reasonable D_{∞} dataset at deep depths for each beam irradiation. Supp. Fig. 1 shows its procedure (utilizing the DD dataset of KK=15 & 16, as illustrated below in Supp. Fig. 5b). The orange line shows a raw dataset as used in the previous paper, and the blue line shows its modified dataset proposed in the present dose calculation, setting two points along the beam axis. Let the point at Z=Z_{a} indicate the position of the dose (D_{a}) that is equal to the dose obtained using the factor of as proposed in the previous paper, and let the point at Z=Z_{b} (<Z_{a}) indicate the common position of another reasonable dose of D_{b} estimated by eye on both dose lines. Then along the blue line for Z ≥ Z_{b}, we set
where α is a constant, determined by the doses of D_{a} and D_{b}, as pointed up by using yellow dots in Supp. Fig. 1, at the corresponding two positions of Z_{a} and Z_{b}.
Table 1 lists sets of (D_{a}, Z_{a}), (D_{b}, Z_{b}), and (Z_{a}) values for KK=112 based on the standard or commercial eMC for the direct electron beams of E=6, 12, and 18 MeV. Similarly, Table 2 lists the corresponding datasets for KK=1324 taking the direct and indirect electron beams. It should be noted that the Z=Z_{a} point is situated beyond the extrapolated range (R_{p}) for each irradiation [1], and that the factor at each Z=Z_{a} depth all takes much smaller or larger values than unity (we need no correction for D_{a} in case of =1).
We use the same dose calculation procedure as the previous one [1], only except the purpose of the usage of the factor as described above.
Supp. Fig. 2 shows how the effective square field of at each dose calculation Z_{c} depth (as shaped using a square electron applicator (A_{appl})) is divided equally into small sections of 40 × 40. In the previous paper [1], we has reported that the function spreads exponentially with Z_{c} depth in water. Almost the same phenomenon is illustrated in a MonteCarlo drawing picture on the cover of Klevenhagen’s textbook [4]. However, the perfect squareshaped fields would not be held with increasing depth in phantom like the case of the primary Xray beam intensity distribution.
Results and discussionTop
First, we describe how the parallelbeam depthdose datasets (D_{∞}) of infinite field are varied in (i) the whole Z region and (ii) the deep Z region on large scale, as follows:
Supp. Fig. 3ac shows the direct electron beam cases on the standard eMC for (a) KK=1 & 2 (E=6 MeV), (b) KK=3 & 4 (E=12 MeV), and (c) KK=5 & 6 (E=18 MeV). Similarly, Supp. Fig. 4ac shows the cases on the commercial eMC for (a) KK=7 & 8 (E=6 MeV), (b) KK=9 & 10 (E=12 MeV), and (c) KK=11 & 12 (E=18 MeV).
Supp. Fig. 5ac shows the direct & indirect electron beam cases on the standard eMC for (a) KK=13 & 14 (E=6 MeV), (b) KK=15 & 16 (E=12 MeV), and (c) KK=17 & 18 (E=18 MeV). Similarly, Supp. Fig. 6ac shows the cases on the commercial eMC for (a) KK=19 & 20 (E=6 MeV), (b) KK=21 & 22 (E=12 MeV), and (c) KK=23 & 24 (E=18 MeV).
It could be understood more clearly from the diagrams on large scale that each blue line at depths of Z ≥ Z_{b} is connected smoothly with the corresponding orange line at depths of Z < Z_{b}.
Lastly, we describe how the OAD curve pattern varies with Z_{c} depth by setting regions of relatively (i) shallow and (ii) deep Z_{c} depths, as follows:
Supp. Fig. 7ac shows the direct electron beam cases on the standard eMC for (a) KK=1 & 2 (E=6 MeV), (b) KK=3 & 4 (E=12 MeV), and (c) KK=5 & 6 (E=18 MeV). Similarly, Supp. Fig. 8ac shows the cases on the commercial eMC for (a) KK=7 & 8 (E=6 MeV), (b) KK=9 & 10 (E=12 MeV), and (c) KK=11 & 12 (E=18 MeV).
Supp. Fig. 9ac shows the direct & indirect electron beam cases on the standard eMC for (a) KK=13 & 14 (E=6 MeV), (b) KK=15 & 16 (E=12 MeV), and (c) KK=17 & 18 (E=18 MeV). Similarly, Supp. Fig. 10ac shows the cases on the commercial eMC for (a) KK=19 & 20 (E=6 MeV), (b) KK=21 & 22 (E=12 MeV), and (b) KK=23 & 24 (E=18 MeV).
It can be seen from the OAD curves that the shapes in the very shallow Z_{c} depths (less than ∼1E04 cm) form triangles, the ones in the middle Z_{c} depths form squarish trapezoids, and the ones in the deep Zc depths form round trapezoids.
The revised Gaussianpencilbeammodel uses a mathematical expression, being reconstructed based on datasets of for E = 6, 10, 14, and 20 MeV as reported by Bruinvis et al. [5]. On the other hand, judging from the OAD curves at very shallow depths for each of the 6, 12, and 18 MeV beam energies, we should see steep DD descents toward each zero depth. Actually, we have obtained a fact that each calculated DD curve forms zero dose values at very shallow Z_{c} depths (less than ∼1E15 cm), where it should be noted that we do not use any set of parallel beam depthdoses of infinite field that descends very sharply near the zero depth, as seen from Supp. Fig. 36. Examining Khan’s text book [6], such dose descents are not described (details will be reported in the next article). We would like to let it be a great subject when taking inhomogeneous phantoms for dose calculations with this Gaussianpencilbeammodel.
ConclusionTop
We conducted two supplementary studies for the previous paper: one is for the parallelbeam depthdose dose (D_{∞}) at deep depths in an infinitely broad field, and the other is for the characteristic shape differences of offaxis dose (OAD) profiles at shallow and deep depths, including in the very shallow depths (less than ∼1E04 cm). We believe that these two studies must be important for dose calculations especially when using heterogeneous phantoms or when considering the dose calculation on a cellbycell basis.
Acknowledgments
The authors thank Dr. Tabata for his close attention to useful suggestions for revising this paper.
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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