Journal of Radiology and Imaging
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Journal of Radiology and Imaging
Volume 6, Issue 1, December 2022, Pages 1–9
Original researchOpen Access
A revised Gaussian pencil beam model 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} , Masanori Miyazawa^{4} and Tatsuo Tabata^{5}
 ^{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
 ^{5 }Institute for Data Evaluation and Analysis, Sakai, Osaka 5938311, Japan
*Corresponding authors: Akira Iwasaki, 2324 Shimizu, Hirosaki, Aomori 0368254, Japan. Tel: +172332480; Email: fmcch384@ybb.ne.jp; Shingo Terashima, Graduate School of Health Sciences, Hirosaki University, 661 Honcho, Hirosaki, Aomori 0368564, Japan. Tel: +81172395525; Email: stera@hirosakiu.ac.jp.
Received 22 August 2022 Revised 31 October 2022 Accepted 12 November 2022 Published 23 November 2022
DOI: http://dx.doi.org/10.14312/23998172.20221
Copyright: © 2022 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: The dose in water caused by clinical electronbeam irradiation is mainly composed of the doses due to the direct electrons, the indirect electrons, and the contaminant Xrays. In this paper, an analytical method for 3dimensional (3D) dose calculation in water for the direct electrons and for the directplusindirect electrons is proposed in light of the electron Monte Carlo (eMC) datasets. Methods: The dose calculation was performed for square fields, based on a revised Gaussian pencil beam model, where the parallel beam depthdose dataset under an infinitely large field was reconstructed from the depthdose dataset of a fan beam with a finite field. We used a semiinfinite water phantom, setting the beam incident surface on the isocenter plane that is perpendicular to the beam axis. The dose calculation model sets an effective field at each dose calculation depth, where the effective field is larger than the geometrical field that the electron applicator forms on the basis of its divergent spread under the effective sourcesurface distance (SSD_{eff}). Results and conclusions: By comparing the calculated datasets of depth dose (DD) and offaxis dose (OAD) in water with the eMC datasets for electron beams of E=6, 12, and 18 MeV by Wieslander and Knöös (2006), it has been found that the revised Gaussian pencil beam model is of practical use and has a bright prospect to give almost the same calculation results as the eMC datasets.
Keywords: electronbeam calculation; Gaussian pencil beam model; electron beams; electron applicator; 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. This paper introduces a revised Gaussian pencil beam model for 3dimensional (3D) calculation of the direct electron dose or directplusindirect dose in water in light of the electron Monte Carlo (eMC) datasets for square fields as reported by Wieslander and Knöös (2006). The revised Gaussian pencil beam model uses a parallel beam depthdose dataset under an infinitely large field. This paper proposes a method for reconstructing the parallel beam depthdose dataset under an infinitely large field using the depthdose dataset of a fan beam with a finite field, and also proposes a generalized expression that is yielded based on the pencil beam parameter for offaxis to onaxis dose ratio calculations as produced by Bruinvis et al. (1983). By comparison with the eMC datasets, it may be concluded that the revised Gaussian pencil beam model can achieve accurate dose calculations in water.
IntroductionTop
First of all, the present authors would like to emphasize the importance of this paper dealing with the Gaussian pencil beam model for electron beam dose calculations. This model is one of the older analytical dose calculation methods. However, the authors believe that the analytical methods are generally useful in understanding how electron beam radiation penetrates media and are generally fast to quantitatively calculate the transferred energy as dose. All the better if there is an analytical method that can accurately calculate the dose. The next section elaborates on the subject of this paper.
With respect to the electron beam therapy performed using highenergy electron beams from linear accelerators, Khan’s textbook [1] summarizes the following two characteristics, one as to the linear accelerators and the other as to the clinical points:
As to the linear accelerators: The field flatness and symmetry are obtained with a proper design of beam scatterers and beam defining collimators using one or more scattering foils (Figure 1), usually made up of lead, to widen the beam as well as to give a uniform dose distribution across the treatment field; and linear accelerators with scattering foil, having photon and multienergy electron beam capabilities, have become increasingly available for clinical use. Recently, a dualfoil system has been used, which is composed of the first and second foils, and this foil system can generate broader electron beams with less Xray contamination, especially at high energies.
As to the clinical points: It should be noted that 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 Xray beams, the electronbeam irradiation offers distinct advantages in terms of dose uniformity in the target volume and in minimizing the dose to deeper tissues.
The textbook also refers to the electron beam dose computation. The outline is as follows: (a) Major limitations of the empirical methods based on broad beams and slab geometries are their inability to predict effects on dose distribution of small fields, sudden changes in surface contour, small inhomogeneities, and oblique beam incidence. (b) An improvement over the empirical methods came about with the development of algorithms based on the agediffusion equation first by Kawachi [2] in 1975. Although these algorithms are able to use semiempirically derived pencil beams that can be placed along the surface contour to predict effects of small fields and surface irregularity, their accuracy to calculate inhomogeneity correction is limited. (c) Major advancement in electron beam treatment planning occurred in the early 1980s. Methods were developed that were based on Gaussian pencil beam distributions calculated with the application of the FermiEyges multiple scattering theory [3]. The methods are based on smallangle multiple scattering approximation. As a result, the elementary pencil beam penetrating a scattering medium is very nearly Gaussian in its lateral spread at all depths. Largeangle scattering events could cause deviations from a pure Gaussian distribution, but their overall effect on dose distributions is considered to be small.
On the other hand, electron Monte Carlo (eMC) methods [47] are today widely used in many electronbeam therapy applications because of the complexity of photon and electron transport (the eMC systems were first commercially built around 2000). Wieslander and Knöös [8, 9] have proposed implementation of a virtual linear accelerator (based on eMC simulations) into a commercial treatment planning system (TPS) to verify the TPS with respect to depth doses (DDs), offaxis doses (OADs or dose profiles), and output factors. They emphasize [9] that problems associated with conventional measurements can be avoided and properties that are considered unmeasurable can be studied. The eMCbased dose is separated into the dose from (a) direct electrons that have not interacted in the electron applicator, the dose from (b) indirect electrons that have interacted in the electron applicator, and the dose from (c) contaminant Xrays from the treatment head. They have summarized two groups of eMCbased dose datasets in water for electron beams of energies of 6, 12, and 18 MeV under a series of two electron applicators (shaping squarefields of 10×10 cm^{2} and 10×10/14×14 cm^{2}). These groups of eMCdatasets are produced under a common virtual accelerator using two eMC simulations; one is installed in a given commercial TPS and the other is used as a standard simulation. The present paper refers to the datasets as “the WK eMC dose datasets” or “the WK eMC dose work”.
Iwasaki et al. [10] have developed an analytical method of calculating the 3dimensional (3D) dose in water for the contaminant Xrays in light of the WK eMC dose work. On the other hand, this present paper will propose another type of 3D dose calculation method, as a revised type of Gaussian pencil beam model, for both the direct electron beams and the directplusindirect electron beams, similarly in light of the WK eMC dose work. This dose calculation method utilizes a parallel beam depthdose dataset in an infinitely broad field with the same incident electronbeam strength on the isocenter plane as the pencil beam composed of direct electrons or directplusindirect electrons. We then propose a method for yielding the depthdose dataset in an infinitely broad field from a depthdose dataset yielded using a given electron applicator forming a finite field with the same incident electronbeam strength (Supplementary Figure 1). It should be emphasized that, for each electron beam, this method utilizes an effective beam field at each dose calculation depth level (Figure 2), letting the effective beam field be larger than the fanbeam field constructed simply using a given electron applicator. Finally, for the σ function used in the Gaussian pencil beam method, this paper uses a expression (as expressed later in equation 13) as functions of the pencilbeam axis depth and the electron beam energy, as regressed on the basis of the factor datasets proposed by Bruinvis et al. [11].
Materials and methodsTop
Symbols and units
This paper uses the following units: the length or depth is expressed in cm; the beam field is expressed in cm^{2}; the electronbeam energy (E) is expressed in MeV; the dose is expressed in Gy; the extrapolated range or is expressed in cm; the Gaussian σ ( or ) function is expressed in cm; and the angle in the sine formula is expressed in radian. It should also be noted that some other quantities are dimensionless (describing as “dimensionless”).
Equational format
Strictly speaking, the experimental functions used in this paper usually have a lot of variables. For such functions, we take simplified formats as: If an expression of F(x,y) is established under a certain major premise of U_{0}=1.0 and V_{0}=2.0, we express this function as F(x,y), F (x, y: U_{0}, V_{0}), F(x,y: U_{0}), F(x,y: V_{0}), F(x,y: U_{0} = 1.0, V_{0} = 2.0) and so on case by case.
Dose calculation background
We use a clinical linear accelerator with dualfoil system as illustrated in Figure 1. On referring to Khan’s textbook [1] and Klevenhagen’s textbooks [12, 13], the present dose calculation method is described only for square beam fields (this is because the present dose calculation method is described in light of the WK eMC dose work published only for square beam fields). On the other hand, it should also be emphasized that although the actual electronbeam irradiation is performed using an electron applicator forming a fan beam as shown in Figure 2a, the dose calculation is performed using its equivalent parallel beam irradiation as shown in Figure 2b. Let the surface perpendicular to the beam axis at the isocenter (O) be called the isocenter plane, and let the beam axis coincide with the Z_{beam} axis. We make an attempt to calculate the doses at arbitrary points in a semiinfinite water phantom whose surface coincides with the isocenter plane.
First, we enter general details of the dose calculation:
(a) The sourcetosurface distance (SSD) used for electron beam irradiations is characterized [1] as: The virtual SSD (=SSD_{X} in Figure 1) cannot give accurate inverse square law correction for dose at extended SSDs under all clinical conditions. This deviation from the inverse square law is caused by an additional decrease in dose because of a loss of sidescatter equilibrium in air and in phantom that is significant for small field sizes and low electron energies. An alternative method of correcting dose for the air gap between the electron applicator end and the isocenter plane (or the patient) is to determine the effective SSD (SSD_{eff}), which gives the correct inverse square law relationship for the change in dose with distance. For the (SSD_{eff}) evaluation for a given electron beam irradiation, this paper uses (SSD_{eff}) datasets taken from a catalogue of varian, as listed in Table 1.
E(MeV)  A_{appl}=6×6 cm^{2}  A_{appl}=10×6 cm^{2}  A_{appl}=10×10 cm^{2}  A_{appl}=15×15 cm^{2}  A_{appl}=20×20 cm^{2}  A_{appl}=25×25 cm^{2} 
6  64.2 cm  69.7 cm  82.9 cm  88.1 cm  91.9 cm  93.3 cm 
9  78.6  80.8  89.1  92.4  94.8  96.1 
12  82.0  84.3  88.0  91.5  94.0  95.0 
15  83.8  86.2  88.3  91.1  93.6  94.8 
16  86.3  88.8  92.8  95.1  96.2  96.2 
18  83.6  86.0  86.9  91.2  93.7  95.1 
20  82.2  83.6  86.5  91.9  94.5  96.2 
22  81.7  83.4  86.6  92.3  95.0  96.6 
(b) For a given (SSD_{eff})fan beam irradiation as shown in Figure 2a, we take rectangular coordinate axes of X_{beam}, Y_{beam}, and Z_{beam} by letting the Z_{beam} axis coincide with the beam axis and by letting the origins of the three axes be set at the isocenter (O). We set a dose calculation point of (X_{c}, Y_{c}, Z_{c}) on a Z_{c}depth plane with an origin mark (O^{'}) on the Z_{beam} axis. As described above, we take square beam fields, letting each beam field side (S_{c}) on the Z_{c}plane be parallel to the X_{beam} or Y_{beam} axis. A_{appl} expresses the beam field measured at the electron applicator end; A_{0} expresses the beam field measured on the isocenter plane; and A_{c} expresses the beam field on the Z_{c} plane, letting A_{c}=S_{c} × S_{c}.
(c) Figure 2b shows a parallel pencilbeam irradiation equivalent to the above fan beam irradiation, with respect to the dose at a point (X_{c}, Y_{c}, Z_{c}) on the Z_{c} plane, taking the same A_{c} field at Z_{beam} = Z_{c} with the same rectangular coordinate axes as illustrated in Figure 2a.
(d) This paper doses not perform the dose calculation directly using this A_{c} field, but performs it by setting a different field of on the Z_{c} plane as illustrated in Figure 2b. Let this field be called the effective beam field. As described later in the discussion paragraph, the dose calculation is performed under by taking into account of the side scatter of electrons beyond the A_{c} field.
Next, we describe the dose calculation procedure using numerical expressions:
(e) Setting the parallel beam field on the Z_{c} plane, we perform the dose calculation based on the Gaussian pencilbeam algorithm. The dose at point (X_{c}, Y_{c}, Z_{c}) in Figure 2a is then set to be equal to the sum of the doses at point (X_{c}, Y_{c}, Z_{c}) in Figure 2b due to each small parallel pencil beam of within the parallel beam field of .
(f) We use the σ function as being evaluated by the beam energy (E) and also by the fan pencilbeam axis depth as illustrated in Figures 2, where point () is set at the center point of each on the Z_{c} plane. It should be noted that the present study evaluates the σ function without taking into account of the air layer above the phantom.
(g) The Gaussian pencilbeam algorithm also contains a special dose function of D_{para} as described later in equation 4, expressing the parallelbeam depthdose in an infinitely broad field, caused by the same incident beam intensity at the phantom surface position as shown in Figure 2a. This function is usually determined [1] from the measured depthdose data of a broad field (e.g., 20 × 20 cm^{2}). However, the clinical linear accelerators cannot generally set such large fields using electron applicators while keeping the same SSD_{eff} index for a given electronbeam energy (Table 1). For this situation, we take the following procedures.
(h) First, by taking a fan beam depthdose function D_{fan} with a finite A_{0} field, we reconstruct [1] a parallel beam depthdose dataset of as
In Supplementary Figure 1, “DD line of Eq. 1” shows the dataset of equation 1. Using the extrapolated range (R_{p}) of the dataset, we obtain the depth (E) of the point where the tangent to the descending linear portion of the eMC dotted line intersects the depth (Z or Z^{'}) line. Next, we reconstruct the corresponding parallel beam dose dataset of infinite field (A_{0} = ∞) as
with , where is another extrapolated range evaluated without a loss of sidescatter equilibrium in water, being recommended by the ICRU [14] as
“DD line of Eq. 2” shows the dataset of the function in the lefthand side of equation 2, forming the extrapolated range that is equal to the value. However, it has been found that this function cannot always express accurate doses at shallow depths of the phantom (“DD line of Eq. 4” is rebuilt under this adjustment, as described in the next item). It should also be noted [1, 8] that the field size of A0 performed by the given electron applicator should be relatively large. This is because, when the field size is relatively too small, the depth dose line shape is much different at the shallow depths from the shapes yielded using greater fields.
(i) It should be emphasized that each DD or OAD dataset of the WK eMC dose work for a given electron beam energy (E) is normalized with a dose of 1.0 Gy per 100 MU at the maximum dose depth (d_{max}) caused by the mixed irradiation of direct electrons, indirect electrons, and contaminant photons under the use of an open electron applicator of A_{appl} = 20 × 20 cm^{2}. Accordingly, the dose normalization for each beam is somewhat complicated. Here we introduce three factors of FAC_{adjust}, FAC_{1}, and FAC_{2} to the (0,0,Z:A_{0}=∞) function of equation 2 by setting (Figure 2), describing as
(j) For the three factors, we let have the following meanings as:
• FAC_{adjust} (E) is used as the adjusting factor for the factors of FAC_{1} and FAC_{2} by letting it be expressed simply as a function of the beam energy E (Supplementary Tables 1 and 2), showing FAC_{adjust} (E) 0.85. This factor is determined after evaluation of the FAC_{1} and FAC_{2} factors for a given irradiation condition.
WK eMC DD & OAD datasets for the direct electron beams (KK=112)  E(MeV)  A_{appl}(cm^{2})  SSDeff cm)  (cm)  (cm)  FAC_{adjust}(E)(no unit)  Z_{1} (E)(cm)  Z_{2} (E)(cm)  Θ_{0} (E)(rad)  W_{α} (E)(no unit) 
(i) Using the standard eMC (in stepped curves)  
KK=1: fig. 3(b)OAD (Z_{c}=1 cm) under fig. 3(a)DD  6  10×10  82.9  2.834  3.291  0.884  1.5  2.5  1.776  0.104 
KK=2: fig. 3(c)OAD (Z_{c} =5 cm) under fig. 3(a)DD  6  10×10  82.9  2.834  3.291  0.884  1.5  2.5  1.776  0.104 
KK=3: fig. 5(b)OAD (Z_{c} =2 cm) under fig. 5(a)DD  12  10×10/14×14  91.5  5.915  6.002  0.914  1.0  4.3  1.56  0.0653 
KK=4: fig. 5(c)OAD (Z_{c} =10 cm) under fig. 5(a)DD  12  10×10/14×14  91.5  5.915  6.002  0.914  1.0  4.3  1.56  0.0653 
KK=5: fig. 3(e)OAD (Z_{c} =3 cm) under fig. 3(d)DD  18  10×10  86.9  8.996  8.453  0.806  0.95  8.0  1.68  0.150 
KK=6: fig. 3(f)OAD (Z_{c} =15 cm) under fig. 3(d)DD  18  10×10  86.9  8.996  8.453  0.806  0.95  8.0  1.68  0.150 
(ii) Using the commercial eMC (in dotted curves)  
KK=7: fig. 3(b)OAD (Z_{c} =1 cm) under fig. 3(a)DD  6  10×10  82.9  2.834  3.291  0.908  1.5  2.5  1.754  0.0738 
KK=8: fig. 3(c)OAD (Z_{c} =5 cm) under fig. 3(a)DD  6  10×10  82.9  2.834  3.291  0.908  1.5  2.5  1.754  0.0738 
KK=9: fig. 5(b)OAD (Z_{c} =2 cm) under fig. 5(a)DD  12  10×10/14×14  91.5  5.915  6.101  0.910  1.0  4.5  1.560  0.0630 
KK=10: fig. 5(c)OAD (Z_{c} =10 cm under fig. 5(a)DD  12  10×10/14×14  91.5  5.915  6.101  0.910  1.0  4.5  1.560  0.0630 
KK=11: fig. 3(e)OAD (Z_{c} =3 cm) under fig. 3(d)DD  18  10×10  86.9  8.996  8.575  0.793  0.9  9.0  1.90  0.170 
KK=12: fig. 3(f)OAD (Z_{c} =15 cm) under fig. 3(d)DD  18  10×10  86.9  8.996  8.575  0.793  0.9  9.0  1.90  0.170 
WK eMC DD & OAD datasets for the direct electron beams (KK=112)  Z_{c}, (E) (cm)  α_{1} (E) (cm)  b_{1} (E) (no unit)  c_{1} (E)(no unit)  T_{0} (Zc, E) (cm)  α_{2} (E) (cm)  b_{2} (E)(no unit)  c_{2} (E) (no unit)  (Z_{c})(no unit)  
(i) Using the standard eMC (in stepped curves)  
KK=1: fig. 3(b)OAD (Z_{c} =1 cm) under fig. 3(a)DD  12.90  3.52E08  19.72  9.80E03  2.50  1.01E37  86.10  6.60E03  0.990  
KK=2: fig. 3(c)OAD (Z_{c} =5 cm) under fig. 3(a)DD  17.65  3.52E08  19.72  9.80E03  6.27  1.01E37  86.10  6.60E03  0.619  
KK=3: fig. 5(b)OAD (Z_{c} =2 cm) under fig. 5(a)DD  12.41  3.45E19  44.77  8.45E03  2.17  2.46E36  82.08  1.20E02  0.994  
KK=4: fig. 5(c)OAD (Z_{c} =10 cm) under fig. 5(a)DD  22.99  3.45E19  44.77  8.45E03  10.90  2.46E36  82.08  1.20E02  39.063  
KK=5: fig. 3(e)OAD (Z_{c} =3 cm) under fig. 3(d)DD  13.20  3.71E23  53.73  8.50E03  2.53  5.10E37  83.30  1.30E02  1.039  
KK=6: fig. 3(f)OAD (Z_{c} =15 cm) under fig. 3(d)DD  27.86  3.71E23  53.73  8.50E03  15.10  5.10E37  83.30  1.30E02  36.115  
(ii) Using the commercial eMC (in dotted curves)  
KK=7: fig. 3(b)OAD (Z_{c} =1 cm) under fig. 3(a)DD  13.36  2.28E07  17.89  9.83E03  2.80  3.26E38  87.35  6.28E03  0.993  
KK=8: fig. 3(c)OAD (Z_{c} =5 cm) under fig. 3(a)DD  17.77  2.28E07  17.89  9.83E03  6.80  3.26E38  87.35  6.28E03  1.548  
KK=9: fig. 5(b)OAD (Z_{c} =2 cm) under fig. 5(a)DD  12.36  1.95E15  36.15  9.46E03  2.10  2.36E35  79.86  1.10E02  1.000  
KK=10: fig. 5(c)OAD (Z_{c} =10 cm under fig. 5(a)DD  21.60  1.95E15  36.15  9.46E03  8.84  2.36E35  79.86  1.10E02  34.278  
KK=11: fig. 3(e)OAD (Z_{c} =3 cm) under fig. 3(d)DD  13.20  1.70E23  54.50  8.50E03  2.47  1.60E37  84.42  1.30E02  1.028  
KK=12: fig. 3(f)OAD (Z_{c} =15 cm) under fig. 3(d)DD  28.16  1.70E23  54.50  8.50E03  15.10  1.60E37  84.42  1.30E02  50.819 
• is used as the relative beamstrength factor, as shown later in Supplementary Figure 3, for point () at the center of a small field of placed wholly inside the effective beam field of on a Z_{c} plane (Figure 2 & Supplementary Figure 2). Note that we set FAC1 (T_{min}:Z_{c},=0 for points () outside the field, where T_{min} is the minimum among the values of (T_{1}, T_{2}, T_{3}, T_{4}), expressing the distances to each of the four sides of from point () on the Z_{c} plane as shown in Supplementary Figure 2. We let T_{0} (Z_{c}:) be a constant determined for the given square field on the Z_{c} plane, then in Supplementary Figure 3, we set
• is introduced to estimate more accurate doses at relatively shallow depths of the phantom. This is because the dose of equation 2 in this region is generally smaller than the reasonable dose (as may be understood from the dose differences between the DD lines of Eqs. 2 and 4 in Supplementary Figure 1). We divide the depth along the beam ray line () into three regions by setting two points at depths of Z_{1} and Z_{2} (Z_{1} < Z_{2}) on the beam ray line as illustrated in Supplementary Figure 4. We then express the FAC_{2} factor as: For ≤ Z_{1}, we set
where θ_{0} is a constant (radian) given by the irradiation condition. It can be seen that the FAC_{2} factor ranges from at (where θ_{0}>0 because the dose in the region of ≤Z_{1} is smaller than the reasonable dose). For Z_{1} ≤ Z_{2}, we set
i.e., ranging from at =Z_{2}. For >Z_{2}, we set
(k) The present work depends largely on the D_{para} (0,0,:A_{0}=∞) function of equation 4 for accurately reproducing the WK eMC datasets printed in the published paper. However, it has been found that it is very difficult to read off the printed data accurately in cases when they are at large depths especially beyond (Supplementary Figure 1). This is because these data are almost zero in the diagrams on a uniform scale. Particularly, serious badness happens for the OAD datasets on deep Z_{c} () planes. In this case, by introducing a scalereading factor of for each Z_{c} plane, we reevaluate the D_{para} function as follows:
It has been found that, as listed in Table 2 and Supplementary Table 1, we take (Z_{c})1 on shallow Z_{c} planes (as described later, we should have taken (Z_{c})=1; however, this paper takes values of (Z_{c})<1 for simply making wellbalanced OAD patterns with the corresponding eMC datasets). On deep Z_{c} planes, we take values much different from 1 (the general treatment is described in (r)).
(l) The integration of the Gaussian distribution function within finite limits cannot be performed analytically. To evaluate this function necessitates the use of error function (erf). Thus convolution calculus shows that, for an electron beam of a rectangular cross section of on the Z_{c} plane (Supplementary Figure 2), the spatial dose for point (X_{c}, Y_{c}, Z_{c}) can be given [1] by
where is the mean square radial displacement of electrons at depth as a result of multiple Coulomb scattering.
(m) The error function (erf) is defined thus:
This function is normalized so that erf (∞) = 1. It should be noted that, for a negative value of ε (this case happens when the dose calculation point is placed outside the field), we can deal with equation 11 as
(n) Bruinvis et al. [7] have reported sets of values for E=6,10,14 and 20 MeV, determined from electron beam OAD profiles measured at Zdepths using a 10×10 cm^{2} field. We found that the function can take the place of the function in equation 10 (see the Appendix). Thus we reconstructed a expression based on the sets of values, as follows:
with
In this paper, we express the function also as (Z) or () in a brief form only taking a phantom depth of Z or , respectively, for a given Evalue. Supplementary Table 2 lists values (dimensionless) of a(E), b(E), c(E), d(E), and e(E) for each beam energy (E), [when a^{'} (E) is used instead of a(E)∙ (E), the goodnessoffit of equation 13 has been worse]. Supplementary Figure 5 compares two datasets of (Z,E) as a function of phantom depth Z for each beam energy (E): One is derived from equation 13 with the respective set of α(E) to e(E) values in Supplementary Table 2, and the other is the dataset of Bruinvis et al. [7]. Excellent agreement is seen between the two.
(o) To incorporate the dependence on beam energy (E) into equation 13, we analyzed each dataset of α(E) to e(E) in Supplementary Table 2 and obtained the following regression functions:
Supplementary Figure 6 compares two datasets of α(E)e(E) on a semilogarithmic scale: The straight lines show equations 1519 in the energy range of E=025 MeV; and the dots, the values listed in Supplementary Table 2 for E=6, 10, 14, and 20 MeV. Supplementary Figure 7 compares two datasets of (Z) for E=6, 10, 14, and 20 MeV: The solid curves show equation 13 with equations 1519; and the broken curves, the datasets reported by Bruinvis et al. [7] (Supplementary Figure 5). Looking at Supplementary Figures 5 and 7, we see that the agreement of each calculated dataset of (Z) in the latter with the reported dataset is less satisfying. This is considered mainly due to the deviation of e(E) values given by equation 19 from the corresponding values in Supplementary Table 2 (Supplementary Figure 6). However, the noticeable differences from the datasets of Bruinvis et al. [7] due to this effect happen mainly at large Zdepths beyond the maximum of (Z) where the dose is relatively small. Accordingly, this effect is considered to cause no serious problems in ordinary circumstances. Supplementary Figure 8 shows the curves of (Z) for E=230 MeV at intervals of 2 MeV generated by the generalized expression, i.e. equation 13 with equations 1519.
(p) The WK eMC dose datasets are arranged for each of the electronbeam energies of E=6, 12, and 18 MeV using applicators of A_{appl}=10×10 cm^{2}, A_{appl}=10×10/14×14 cm^{2} (10×10 is the opening field produced with a cerrobend insert in the original 14×14 shape), and A_{appl}=20×20 cm^{2}. The dose datasets for each electronbeam energy (E) are separated into DD curves and OAD curves, where each dataset is normalized as described in (i). In the dose datasets acquired through the standard eMC simulation are composed of stepped curves of DD and OAD; and the dose datasets acquired through 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 in the X and Y directional OAD profiles; however, the present paper does not refer to the OAD differences caused by the X and Y directions. It should also be noted that the field size by each of the three applicators expresses the one measured at the applicator end position. As the present study uses the phantom whose surface coincides with the isocenter plane; therefor, these applicators are installed with an air gap of 5 cm above the phantom (Figure 1).
(q) We describe how to determine both values of [Figure 2b & Supplementary Figure 2] and T_{0} (in equations 5a and 5b and Supplementary Figure 3) on an arbitrary Z_{c} plane. Each DD and OAD data combination by WK eMC dose work represents a DD dataset and a pair of OAD datasets at specific shallow and deep Z_{c} depths, dividing into groups of stepped and dotted line datasets for each beam, as being divided into separate groups of the direct electron beams (Table 2) and the directplusindirect electron beams (Supplementary Table 1). By analyzing the DD and OAD datasets through trial and error for each beam, we construct expressions to yield each pair of reasonable and T_{0} values over a wide range of Z_{c}. For the and T_{0} functions, this paper proposes the following expressions:
and
letting both sets of a_{1}, b_{1}, c_{1} values and of a_{2}, b_{2}, c_{2} values be constants simply determined by the beam energy (E) (Table 2 and Supplementary Table 1).
(r) Lastly, we describe how to generally perform the dose calculation at a point (X_{c}, Y_{c}) on a Z_{c} plane introducing a factor of R_{scale} (X_{c}, Y_{c}, Z_{c}) (this procedure happened because we cannot correctly read off the figures of the WK eMC datasets of DD in case the dose points are at relatively large depths and/or large offaxis distances). Accordingly, using equation 4, we rewrite equation 10 as
This paper simply uses R_{scale} (0,0, Z_{c}) = 1 for the DD calculation. For the OAD calculation, we use values of R_{scale} (X_{c}, Y_{c}, Z_{c})＝ (Z_{c}) as given in Table 2 and Supplementary Table 1 (each of the values was obtained by taking a wellbalanced OAD pattern between the WK eMC dataset and the calculated one).
ResultsTop
The DD or OAD datasets reported in the WK eMC dose work express the doses caused by (i) direct electrons and by (ii) directplusindirect electrons. Moreover, each of the DD or OAD datasets is illustrated using stepped and dotted curves, where the stepped curves are yielded using the standard eMC simulation and the dotted curves are yielded using the commercial TPS. Here, we classify them into two for avoiding the confusion. One is listed in Table 2 taking the direct electron beams (KK=112), and the other is listed in Supplementary Table 1 taking the directplusindirect electron beams (KK=1324). It should be noted that the DD datasets were transcribed from Supplementary Figures 3(a), 5(a), and 3(d) in the WK eMC dose work, and that the OAD datasets were similarly from Supplementary Figures 3(b), 3(c), 5(b), 5(c), 3(e), and 3(f).
Supplementary Figures 9 to 12 illustrate DD lines for (a) E=6 MeV, (b) E=12 MeV, and (c) E=18 MeV using the direct electron beams or the directplusindirect electron beams through the standard eMC or the commercial eMC simulation approach.
• Supplementary Figure 9 illustrates sets of DD curves yielded using (i) the standard eMC for “the direct electron beams” of (a) E=6 MeV with A_{appl}=10×10 cm^{2} (KK=1 and 2), (b) E=12 MeV with A_{appl}=10×10/14×14 cm^{2} (KK=3 and 4), and (c) E=18 MeV with A_{appl}=10×10 cm^{2} (KK=5 and 6), where the KK numbers are listed in Table 2. Each set starts from the orange line showing the dataset of D_{fan} (0,0, Z^{'}:A_{0} ) letting Z=Z^{'} (FanBeamDD of FiniteField in Eq. 1), then reaches the gray line showing the dataset of (0,0,Z^{'}:A_{0}) letting Z=Z^{'} (ParallelBeamDD of FiniteField in Eq. 1), then reaches the yellow line showing the dataset of (0,0, Z:A_{0}=∞) (ParallelBeamDD of InfiniteField in Eq. 2), and lastly reaches the blue line showing the dataset of D_{para} (0,0, :A_{0}=∞) letting Z= (ParallelBeamDD of InfiniteField of Eq. 4). On each Z line in the diagrams of (a)(c), the points of Z_{1} and Z_{2} are set (refer to equations 6 and 7); the point of is set on the basis of the DD datasets on the orange and gray lines (refer to equation 2); and the point of is set on the basis of the DD datasets on the blue and yellow lines (refer to equation 2). It should be noted that diagram (a) shows <, diagram (b) shows , and diagram (c) shows < (we have no reasons why diagram (a) shows <).
• Similarly, Supplementary Figure 10 illustrates sets of DD curves yielded using the commercial eMC for “the direct electron beams” of (a) E=6 MeV with A_{appl}=10×10 cm^{2} (KK=7 and 8), (b) E=12 MeV with A_{appl}=10×10/14×14 cm^{2} (KK=9 and 10), and (c) E=18 MeV with A_{appl}=10×10 cm^{2} (KK=11 and 12). It should also be noted that diagram (a) shows <, diagram (b) shows , and diagram (c) shows < (we also have no reasons why diagram (a) shows <).
• Similarly, Supplementary Figure 11 illustrates sets of depthdose (DD) curves yielded using the standard eMC for “the directplusindirect electron beams” of (a) E=6 MeV with A_{appl}=10×10 cm^{2} (KK=13 and 14), (b) E=12 MeV with A_{appl}=10×10/14×14 cm^{2} (KK=15 and 16), and (c) E=18 MeV with A_{appl}=10×10 cm^{2} (KK=17 and 18). It should also be noted that diagram (a) shows <, diagram (b) shows , and diagram (c) shows < (we also have no reasons why diagram (a) shows <).
• Similarly, Supplementary Figure 12 illustrates sets of DD curves yielded using the commercial eMC for “the directplusindirect electron beams” of (a) E=6 MeV with A_{appl}=10×10 cm^{2} (KK=19 and 20), (b) E=12 MeV with A_{appl}=10×10/14×14 cm^{2} (KK=21 and 22), and (c) E=18 MeV with A_{appl}=10×10 cm^{2} (KK=23 and 24). It should also be noted that diagram (a) shows <, diagram (b) shows , and diagram (c) shows < (we also have no reasons why diagram (a) illustrates <).
Under the and situations described above, the following features were obtained:
(a) Regarding the extrapolated range (E), of the WK eMC depthdose dataset for each electron beam energy (E), both the standard eMC simulation and the commercial eMC TPS yield the almost the same extrapolated range even if the beam is composed of direct electrons or directplusindirect electrons.
(b) Both the standard eMC simulation and the commercial eMC TPS perform (E)< (E) only for the 18 MeV electron beams; however, both perform (E)> (E) for the other energy beams.
(c) It should be noted that, even in the case of (E)> (E), we can build the expression of (0,0, Z: A_{0}=∞) of equation 2 holding Z
We finally describe the accuracy of the present dose calculation method, by comparing with the DD and OAD datasets reported in the WK eMC dose work. Here, it should be noted for the factors listed from (E) to (Z_{c}) in Table 2 and Supplementary Table 1 that each dataset of the factors of (E) to c_{2}(E) have been derived from the corresponding DD dataset and the value for the (Z_{c}) factor is derived from the corresponding OAD dataset. This paper has introduced a factor of R_{scale} in equation 22. However, this factor is adopted only as for each pair of OAD profiles on shallow and deep Z_{c} planes. For the shallow Z_{c} planes, the values of 1 are used for making a wellbalanced OAD pattern between the WK eMC dataset and the calculated dataset. On the other hand, for the deep Z_{c} planes, the values of >>1 are used for making accurate D_{para} (0,0, : A_{0}=∞) values (equation 9). The reason is that we cannot help using such large values because we use relatively much inaccurate data of D_{para} (0,0, : A_{0}=∞) or (0,0, : A_{0}=∞) at deep depths of (equations 4, 10, and 22).
• Each of Supplementary Figures 1318 illustrates a DD dataset and a pair of OAD datasets for the 6, 12, or 18MeV direct electron beam using the standard or commercial eMC, indicating the corresponding KKnumbers in Table 2, where the sets of dots express the corresponding eMC dose results and the lines express the corresponding calculated dose results. It should be noted that diagram (a) is for DD, diagram (b) is for OAD on the shallow Z_{c} plane, and diagram (c) is for OAD on the deep Z_{c} plane. The eMCOAD datasets on the deep Z_{c} planes illustrate large amounts of electron sidescatter.
• Each of Supplementary Figures 1924 similarly illustrates a DD dataset as diagram (a) and a pair of OAD datasets as diagrams (b) and (c) for the 6, 12, or 18MeV directplusindirect electron beam using the standard or commercial eMC, taking the corresponding KKnumbers in Supplementary Table 1. It can be understood that almost the same dose calculation tendency as in Supplementary Figures 1318 is obtained.
DiscussionTop
Based on the concept of this revised Gaussian pencil beam model, we would like to discuss our future studies as follows:
(a) Khan’s textbook describes that the Gaussian pencil beam model calculates the dose only within the fanbeam field of A_{c} (Figure 2) formed by a given electron applicator, namely by letting the incident beam strength be zero outside the field. On the other hand, the present paper proposes to use the effective field of (Figure 2) that is greater than the Ac field. It should then be emphasized that the dose outside the A_{c} field can also be calculated with accuracy. In this case, the dose is produced by electrons mainly scattered from inside the fanbeam field (A_{c}). Accordingly, it may be emphasized that, even for such electrons, the parallel beamaxis DD function of infinite field (equation 2) can be used reasonably.
(b) In Figure 2a, if a phantom layer exists just above the isocenter plane or if an air gap layer exists just below the isocenter plane, these layers become kinds of heterogeneities for the standard irradiation. For the future study, we would take account of these heterogeneities also including heterogeneities within the phantom.
(c) This paper does not mention other types of beam fields and beam energies. These research themes are important for clinical use.
ConclusionTop
We have constructed a revised Gaussian pencil beam model on referring to the Khan’s textbook. We then utilized electron Monte Carlo (eMC) datasets of depth dose (DD) and offaxis dose (OAD) in water obtained using electron beams of E=6 MeV (A_{appl}=10×10 cm^{2}), E=12 MeV (A_{appl}=10×10/14×14 cm^{2}), and E=18 MeV (A_{appl}=10×10 cm^{2}) as reported by Wieslander and Knöös (2006). The revised model has the following three features: (a) The parallel beam depthdose dataset under an infinitely large field for each beam is reconstructed using the corresponding fanbeam depthdose dataset. (b) The expression (equation 13 and Appendix A) is constructed based on the dataset produced by Bruinvis et al. (1983), as a function of depth (Z) and beam energy (E). (c) The function (equation 20) is useful for estimating how the effective field varies with depth (Z).
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.
ReferencesTop
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