

ORIGINAL ARTICLE 



Year : 2013  Volume
: 38
 Issue : 2  Page : 8286 

Determination of square equivalent field for rectangular field in electron therapy
Mohammad J Tahmasebi Birgani^{1}, Mohammad A Behrouz^{2}, Saeedeh Aliakbari^{2}, Seyed M Hosseini^{3}, Davood Khezerloo^{2}
^{1} Department of Medical Physics and Radiation Therapy, University of Jundi Shapoor, School of Medicine, Ahwaz, Iran ^{2} Department of Medical Physics, University of Jundi Shapoor, School of Medicine, Ahwaz, Iran ^{3} Department of Radiation Therapy, University of Jundi Shapoor, Golestan Hospital, Ahwaz, Iran
Date of Submission  16Jul2012 
Date of Decision  07Jan2013 
Date of Acceptance  08Jan2013 
Date of Web Publication  3May2013 
Correspondence Address: Saeedeh Aliakbari Department of Medical Physics, University of Jundi Shapoor, School of Medicine, Ahwaz Iran
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/09716203.111317
Abstract   
Equivalent field for electron beams is considered by using pencil beam theory. According to the FermiEyges model the dose distribution of an electron pencil beam has a Gaussian profile. For this function determination of mean square radial displacement scattering of electrons is important. In this study the contribution of back scatter electron has been taken into account by using the multiple scattering theories for calculating mean square radial displacement scattering. The dimension of standard equivalent field depends on depth and shape of treatment field. Here the depth under study is the depth that mean square radial displacement scattering is extremum and the shape of treatment field is rectangular. In this study four energies were used 6, 9,12 and 15 MeV electron beams of 2100C/D Varian Linac. Findings of this study are based on analytical calculations, which are in good agreement with other experimental data. The findings of this study that were resulted from formula, shows, for all circular fields of radius ≥LSE (lateral scattering equilibrium) were considered broad field and equivalent. For validating the findings, Percentage Depth Dose (PDD) and Output factors were measured in 15 MeV electron beams for 7 × 3cm, 6 × 4cm and 4 × 2cm and their equivalent squares and equivalent circular fields and compared.
Keywords: Dosimetry, electron therapy, field equivalence, mass angular scattering power, pencil beam theory
How to cite this article: Tahmasebi Birgani MJ, Behrouz MA, Aliakbari S, Hosseini SM, Khezerloo D. Determination of square equivalent field for rectangular field in electron therapy. J Med Phys 2013;38:826 
How to cite this URL: Tahmasebi Birgani MJ, Behrouz MA, Aliakbari S, Hosseini SM, Khezerloo D. Determination of square equivalent field for rectangular field in electron therapy. J Med Phys [serial online] 2013 [cited 2022 Sep 28];38:826. Available from: https://www.jmp.org.in/text.asp?2013/38/2/82/111317 
Introduction   
The pencil beam theory has been established in considering of the behavior of electron beams. ^{[1],[2]} By using FermiEyges model, one can show that the distribution of an electron pencil beam has a Gaussion profile. ^{[3]} It is well known that a fundamental parameter of the pencil beam method of electron beam treatment planning is mean square radial displacement scattering . This parameter has been determined by several methods and tabulated. ^{[4],[5]} In different studies they have been stated that the definition of equivalent field for electron beams is not generally possible. ^{[1],[6]} On the other hand in several researches determination of equivalent field for electron beams are investigated by using FermiEyges multiple scattering theory. ^{[6],[9]} In this research the standard equivalent field of rectangular fields have been calculated by minimizing of in different energy electron beams of 2100C/D Varian Linac and compared with other studies.
Materials and Methods   
The dose distribution of electron pencil beam in a phantom looks like an onion. The lateral spread followed the Gaussian function in a depth. Gaussian function is characterized by σ_{r} (z) which shows the raduis extending gaussian function, as well as scatter radius of electron pencil beam in water phantom. The scatter spread parameter σ_{r} (z) was theoretically predicted by Eyges. ^{[10]}
Is the mass angular scattering power in the water, ρ is density of the slab phantom and z is depth. But there are limitations to the Eyges equation. As pointed out by Werner, Khan and Deible ^{[3],[5]} σ that was given by Eyges equation, increases with depth infinitely, which is contrary to what is observed experimentally, also Eyges equation is based on smallangle multiple coulomb scattering and under estimate large angle scattering. One can solve this problem by changing the limitations of integral to 0 to practical range Rp.
The energy dependency of is obvious from Eq.4 by R _{p} .
The depth dose at central axis for irregular shaped field is calculated as the following form.
Where the field is divided into n sectors at angular intervals of Δθ and is the radius of the i ^{th} sectors and D_{∞}(0,0,z) is the central axis depth dose per unit incident flounce for an infinitely wide parallel beam(2a,2b≥σ(z)). σ_{r} (z) is the root mean square radial spread of the Gaussian pencil beam as a function of depth.
For a beam of circular cross section of radius R, the central axis depth dose distribution Eq.5 is given by.
With equalizing Eqs.5 and 6 and solving the integrals for square and rectangular fields, the equivalent radius for rectangular fields (2a × 2b) of [Figure 1] becomes:
Lateral scattering equilibrium was calculated by khan formula to determine of limits to define the equivalent fields. ^{[8]}
Jaws were opened 14 × 14 cm for 10 × 10 applicator and Nine cerrobend cutouts fields (7 cm × 3 cm, 6 cm × 4 cm and 4 cm × 2 cm Rectangular fields, their equivalent circular and square fields) were used to perform dosimetry.
Dosimetric measurements were performed for a 15 MeV beam from Varian 2100 C/D linac, at 100 cm sourcetosurface distance (SSD), using CC13 ionization chamber (0.13 cm ^{3} volume, total active length 5.8 mm, cylinder length 2.8 mm, inner diameter of cylinder 6.0 mm, wall thickness 0.4 mm, diameter of inner electrode 1.0 mm and length of inner electrode 3.3 mm) in a 50 × 50 × 50 cm ^{3} Scanditronix water phantom. The isodose curves, PDD and Output factor for each cutout were drawn and tabulated by omniproaccept and Excel software.
Results   
The values of σ have been calculated from Eq.4 by using from ICRU report 21 and practical range R _{p} for 6, 9, 12 and 15 MeV energies of electron beam for Varian 2100C/D Linac were determined. The data are tabulated in [Table 1].
From Eq.7 the equivalent radiuses for rectangular field at 6, 9, 12 and 15 MeV energies are calculated and tabulated in [Table 2],[Table 3],[Table 4] and [Table 5].  Table 2: Req for rectangle fi eld sizes, E=6 MeV and σ^{2} =0.38 (radian^{2} cm^{2}). 2a and 2b (cm) are the sides of rectangle
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 Table 3: R_{eq} for rectangle field sizes, E=9 MeV and σ^{2} =0.67 (radian^{2} cm^{2}) 2a and 2b (cm) are the sides of rectangle
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 Table 4: R_{eq} for rectangle field sizes, E=12 MeV and σ^{2} =1.03 (radian^{2} cm^{2}) 2a and 2b (cm) are the sides of rectangle
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 Table 5: R_{eq} for rectangle field sizes, E=15 MeV and σ^{2} =1.3 (radian^{2} cm^{2}). 2a and 2b (cm) are the sides of rectangle
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An equation was fitted to main diagonals of data [Table 2],[Table 3],[Table 4] and [Table 5] by Matlab software with license No.161052. This equation shows the simple relation between R _{eq} and a _{eq} as the following form:
Equation 8 would be true if 2a is less than Lateral Scattering Equilibrium (2a ≥ LSE) because when the sides of a rectangular field were larger than LSE, all fields were considered broad fields and are equivalent. ^{[8]}
For example, according to [Table 5] for a 6 cm × 4 cm rectangular field size, equivalent radius derived R _{eq} = 2.4 and from Eq.8 a _{eq}= 4.4 cm.
For validating the tabulated data, dosimetric measurements (PDD and Output factors) were performed in 15 MeV electron beam for 7 cm × 3 cm, 6 cm × 4 cm and 4 cm × 2 cm Rectangular field and their equivalent circular and equivalent square fields, for comparing measured data were tabulated in [Table 6],[Table 7],[Table 8] and [Table 9]. Also this comparison is achieved for percentage depth dose by the PDD curve [Figure 2],[Figure 3] and [Figure 4].  Figure 2: Comparison of depth dose distribution of rectangular field 7 × 3 cm and equivalent circular fi eld R = 1.9 cm and equivalent square field 3.5 × 3.5 cm at 15 MeV
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 Figure 3: Comparison of depth dose distribution of rectangular fi eld 6 × 4 cm and equivalent circular field R = 2.4 cm and equivalent square field 4.4 × 4.4 cm at 15 MeV
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 Figure 4: Comparison of depth dose distribution of rectangular field 4 × 2 cm and equivalent circular field R = 1.4 cm and equivalent square field 2.5 × 2.5 cm at 15 MeV
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 Table 6: Measured depth dose distribution of rectangular fi eld 7×3 cm and equivalent circular fi eld r=2 cm and equivalent square field 3.5×3.5 cm for 15 MeV
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 Table 7: Measured depth dose distribution of rectangular fi eld 4×2 cm and equivalent circular fi eld r=1.4 cm and equivalent square field 2.5×2.5 cm for15 MeV
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 Table 8: Measured depth dose distribution of rectangular fi eld 6×4 cm and equivalent circular field r=2.4 cm and equivalent square field 4.4×4.4 cm for 15 MeV
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 Table 9: Measured output factors of rectangular, equivalent square and equivalent circular field for 15 MeV electron beam at 3.7 cm
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Discussion   
The calculated σ ^{2} for 6, 9, 12 and 15 MeV at are in good agreement by practical data that was resulted by Khan and Brunivis ^{[1],[11]} By applying the pencil beam theory and Equations (5), (6) for irregular and circular field one can derive R _{eq} and aeq for any arbitrary shaped electron field.
In this study for deriving equivalent field, the sector integration method for rectangular field is applied. Khan and Higgins have applied Gaussian pencil beam theory to this problem and derived an equation that can be used to find approximation equivalence circular or square fields of any shaped.
According to the definition of equivalent field, each two fields have the same percentage depth dose on central axis are equivalent. From the [Table 2],[Table 3],[Table 4] and [Table 5] one can obtain the equivalent radius and finally by applying Eq.8, equivalent square field was derived. The dosimetric measurement (PDD and Output factor) validated the tabulated data with good agreement.
Conclusions   
Findings of this study are based on analytical which is in good agreement with other semi empirical. ^{[8]} This method was suggested to use in treatment planning system (TPS) for calculating the equivalent square field for rectangular treatment field in electron therapy.
Acknowledgment   
The author would like to thank radiotherapy and oncology center of Golestan Hospital of Jundi Shapour medical university for any cooperation in duration of this study and Vice Chancellor of Jundi Shapour medical university for financial support.
References   
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[Figure 1], [Figure 2], [Figure 3], [Figure 4]
[Table 1], [Table 2], [Table 3], [Table 4], [Table 5], [Table 6], [Table 7], [Table 8], [Table 9]
