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A semiempirical method for the description of relative crossbeam dose profiles at depth from linear accelerators
Copyright
Medical Dosimetry, Vol. 22, No. I, pp. 63-67, 1997 0 1997 American Association of Medical Dosimetrists Printed in the USA. All rights reserved
0958.3947197
$17.00 + .OO
PI1 SO958-3947( 96) 00153-7
ELSEVIER
A SEMIEMPIRICAL METHOD FOR THE DESCRIPTION OF RELATIVE CROSSBEAM DOSE PROFILES AT DEPTH FROM LINEAR ACCELERATORS IOANNIS TSALAFOUTAS, M.Sc ., ’ STELIOS XENOFOS, PH.D., ’ and ION E. STAMATELATOS, PH.D.’ ’ Departmentof Medical Physics,Agios SavvasHospital, Athens, Greece; ’ Institute of NuclearTechnology and RadiationProtection, National Centreof Scientific ResearchDemokritos, Athens, Greece Abstract-A semiempirical method for the calculation of the relative crossbeam dose profiles at depth is described. The parameters required to set up the formulae and their dependence with field size and depth are investigated. Using the above method, measured crossbeam dose profiles at depth from two linear accelerators, Philips (SL-18) and AEC (Therac-6) are reproduced. The results indicate that this method Association of Medical is applicable within a wide range of depths and field sizes. 0 1997 American Dosimetrists Key Words:
Radiotherapy,
Linear
accelerator,
Megavoltage
beam,
Crossbeam
profile.
component can be farther analysed into collimator and phantom scatter components. Hence any theoretical formula that describes the crossbeamprofiles has the practical difficulty of determining the contribution of each component to the measuredcrossbeam profile. Based on the simple geometrical model delineated in Fig. 1, a semiempiricalformula hasbeen obtained and is used to describe the relative crossbeamdose profiles at any specified depth. In Fig. 1(a) a geometrical model of the head of a radiotherapy unit is shown. The dashed line is the line acrosswhich a smalldosimeterwith buildup cap is moving, in order to acquire the primary crossbeam profile. In Fig. 1(b) the geometric primary crossbeam profile is indicated by the dashedline, whereasthe actual primary crossbeamprofile is indicated by the solid line. The main aspectsthat differentiate the two profiles are that the source is not a sharply defined disk, there will be scatteringwithin the sourceand from the collimators, and transmissionthrough collimators and inverse squarelaw effects. The actual primary crossbeamprofile from a linear accelerator will be more complicated, becauseof the presenceof the flattening filter. Furthermore, the crossbeamdose profiles at depth will present increasedpenumbra compared to the primary crossbeam profile becauseof phantom scatter and loss of lateral electronic equilibrium. ’ The function that has been used to approximate the geometric primary crossbeamprofile is:
INTRODUCTION There are several semiempirical methods found in the literature which describe the primary crossbeamprofiles or the crossbeamdose profiles at depth for radiotherapy machines.‘,’ The accuracy of such methods is very important when these are used for treatment planning where the required accuracy in dose calculation should be greater than 3%. In this study we present a semiempirical method for the description of the relative crossbeamdose profiles at depth. It is basically a modification of a known relationship that describesprimary crossbeamprofiles.’ This modification was necessary in order to describe the crossbeamprofiles from linear accelerators which present certain characteristics that could not be accurately reproduced. These characteristics are a “shoulder” and a “tail” of different curvature, an abrupt fall-off of the dose in the penumbra region and, in some cases, a concavity in the central axis region due to the effect of the flattening filter. The method proposed produces profiles with the above mentioned characteristics by using two components, geometric primary and secondary components, along with a correction function to improve the fitting near the central axis.
METHOD The total dose to a point in a medium is the sum of primary and scattered components. The scattered
b(x)
= (1 + exp(SF,(lxl
- x0)))-’
(1)
where, x is the distance of the point of interest from the central ray of the beam, x0 the geometric edge of the field at the specified depth and SF, is proportional to the derivative of FP(x) at x = x0. FP(x ) is symmetric
Reprint requests: Dr. Ion E. Stamatelatos, Institute of Nuclear Technology & Radiation Protection, National Centre of Scientific Research, “Demokritus,” Agia Paraskevi, Athens 153.10, Greece. 63
64
Medical Dosimetry
Volume 22, Number 1, 1997
-,+sd
Hence, the formula that would describe symmetric relative crossbeam dose profiles specified depth would be written as:
ssd
simple, at any
F(x) = P Fp(x) + S F,(x)
-‘lo
/
.
Xl
-b
;6
where P and S = 100 - P are integer numbers that would give the percent contribution of each component to F(x). The above formula, however, predicts that if F( 0) = 100 then F(2xo) = 0. This is not the case for actual profiles, especially for large fields, as the dose persists well beyond 2x0. It must be noted that if the value of SFs is made adequately small for the given x0, then F,( 0) < 1 and F,( 2x,) > 0. Thus, if the S factor is increased by a certain amount (So) then we can produce a symmetric F(x) curve with F(0) = 100 (instead of 100 + S,) and F(2xo) = So (instead of 0). Thus:
.. x2
-‘4
-‘2
10
I2
4
6D 8 1 distance
Off-ax15
F(x) = P Fp(x) + (S + S,)F,(x)
Fig. 1. (a) The geometrical factors that lead to beam penumbra. (b) Geometric and actual primary crossbeam profiles in air along x1-x2.
= (1 + exp(%(lxl
- %J))Y1
(4)
Equation (4) still needs to be corrected as it cannot account for the concavity which sometimes appears near the central axis in the measured crossbeam profiles. The function used for this purpose is:
about the point (x0, FP( x0)), and converges to 1 at x = 0 and to 0 at x = 2x0. The convergence rate, for a given x0, depends on the SF, value. For large values of SF,, FP( x) approximates closely the shape of the geometric primary crossbeam profile. It was observed, however, that smaller values of SFp pkduce a slowly varying profile which presents a shoulder and a tail. Based on this observation it was assumed that a profile with all the desired characteristics could be produced by adding a slowly varying term named secondary component to the geometric primary term. This assumption was empirically proven to provide satisfactory results for the purposes of this study. Thus, in order to describe the secondary component the following function has been used: F,(x)
(3)
F,(x)
= 1 - H exp(-SF,/xj)
(5)
where H, SFc are parameters calculated from best fitted and measured profiles (Appendix A). The correction function ( Fc (x)) is applied to the second term of equation (4) and its effect is twofold. It can produce the concavity around the central axis and it also can modify the curvature of the shoulder. Hence, the final formula that is used to describe the relative crossbeam dose profile at any specified depth is: F-r(x) = P FAX) + F,(x)
(S + S,)Fs(x)
(6)
The effects that differentiate the geometric primary crossbeam profile from the actual primary crossbeam profile, flattening filter effect and phantom scatter, have been implicitly included in the second term of equation (6). In order to find the best fitting parameters S, So, SF,, SFs and x0, equation (4) was used to fit data points from crossbeam profiles obtained from two lin-
(2)
Equation (2) is of the same form as equation ( 1 ), SF, is proportional to the derivative of Fs (x) at x = x0 but SFs values are very small compared to SFp values.
Table 1. Parameters obtained by least squares for SL- 18 data Field
6 x 6 cm’
Depth (cm)
1.5
S SO SF, (cm-‘) Sf, (cm-‘)
9 2
x0 (cm)
H SF, (cm-‘)
15.3 1.3 3.03 0.30
2.18
6 14 2 13.9 0.9 3.14
0.11 1.81
15
10 X 10 cm* 30
1.5
6
15
15 X 15 cm’ 30
1.5
6
15
18
21
11
21
26
30
13
22
32
4 12.7 0.7 3.45
6 11.5 0.6 3.88 0.16
2 16.1 0.8 5.09 0.32
2 14.2 0.8 5.28 0.15 0.98
4 12.8 0.5 5.73 0.13
5 10.3 0.4 6.48
4 14.7 0.5 7.58 0.37 0.28
3 13.6 0.5 7.89 0.17 0.45
4 11.7 0.4 8.56
0.14 1.14
0.64
0.95
1.25
0.11 0.93
20 X 20 30
1.5
38
15
5
5
10.1 0.4 9.74
0.10
0.08
0.37
0.48
16.2 0.7 10.12 0.40 0.15
6
cm* 15
30
21
3:
4;
5 14.2 0.4 10.52 0.26 0.22
10.6 0.3 11.45 0.10 0.31
9.5 0.3 12.97 0.06 0.41
A semiempiricalmethod for relative cross beam dose 0
I. TSALAFOUTAS
65
et al.
Table 2. Parameters obtained by least squares for AEC Therac-6 data Field
6 x 6
Depth (cm)
1.5
S so
9 6 7.8 0.5 3.07 0.44 1.05
SFp (cm-‘) Sf, (cm-‘) x0 (cm)
H SF,- (cm-’ )
cm’
15 X 15 cm’
6
12
20
13 I 7.4 0.4 3.22 0.17 0.73
19 7 7.0 0.4 3.4 0.08
22 9 6.7 0.4 3.66 0.13 0.68
1.11
ear accelerators. Data points beyond 100% (toward the field edge) only are taken into account. The 100% corresponds to the maximum value of the measured crossbeam profile. The best fitted profiles are then corrected by equation (5 ) to yield the final profile FT (x) . The values for H and SF, are calculated as shown in Appendix A. The profiles were obtained from measurements at Agios Savvas Hospital using a profile scanning system and a water phantom. The measurements were carried out for 6 MV x-ray beams, on an SL-18 (Philips) and a Therac-6 (AEC) linear accelerator.
RESULTS
AND DISCUSSION
The results obtained using a simple least squares fitting procedure are shown in Table 1 (SL-18) and Table 2 (Therac-6). In Fig. 2 the relative crossbeam profile FT( x) and
1.5
6
18
12
26
31 5 7.1 0.3 8.43 0.07 0.81
6
5 8.2 0.3 7.69 0.14 0.37
7.3 0.3 8.02
0.11 0.33
20 34 6 5.9 0.3 9.05 0.09 0.44
its components-P Fp(x), (S + S,) F,(x) and F,(x) (S + S,) F,(x) -are separately plotted for a field size 10 x 10 cm* at d = 15 cm (SL-18). The profiles produced using equation (6) and the best fitting parameters tabulated for field size 15 X 15 cm2 and depths 6, 15 and 30 cm are shown in Fig. 3 for SL-18. The corresponding profiles for Therac-6 at depths 6, 12 and 20 cm for the same field size are shown in Fig. 4. From Figs. 3 and 4 agreement between experimental data and the reproduced
profiles can be seen within
all dose regions. The correction function F,(x) works satisfactorily, and errors are less than 2.5%. From Tables 1 and 2, it is shown that the parameters S, SF, and SFs have different values for the two accelerators. They change, however, with field size and depth in a similar way. The values of S increase with increasing depth and field size much as scatter radiation does. The variation of the S factor with field size for different depths is shown for SL-18 in Fig. 5. It is observed that S increases with field size more rapidly
d=1.5
cm
80 2ic
P
0
m
80
0.00
4.00 off-axis
8.00 distance
12.00
(cm)
Fig. 2. Fitted relative crossbeam profiles for a 10 X 10 cm’ field at depth 15 cm. F(x), FT(x) and the terms formingup equation (6) are separately plotted: (a) data points, (b) P b(x), (~1 (S + &)Fs(x), Cd) F(x), (e) Fdx)(S + S,)Fs(x), (f) FT.
off-axis
distance
(cm)
Fig. 3. Fitted profiles for a 15 X 15 cm’ field (SL-18). The height of the profiles has been adjusted to avoid overlapping.
Medical Dosimetry
Volume 22, Number 1, 1997
d=6
6.0 00.00i
5.00 II
11
1 ’ 1
10.00 II 11
1 ’ 11
11
15.00 I 11
i-m--m--n 0
5
10 square
off-axis
distance
15 field
side
20
cm
d=15
cm
d=30
cm
25
(cm)
(cm)
Fig. 4. Fitted profiles for a 15 x 15 cm’ field (Therac6). The height of the profiles has been adjusted to avoid overlapping.
at larger depths and seems to level-off at large field sizes. SFr values decrease with increasing depth. They also decrease with increasing field size but this variation is prominent only for large depths. This behaviour is shown for SL-18 in Fig. 6. Finally, SF, decreases with increasing depth and field size. The variation of SFs with field size for different depths is shown for SL-18 in Fig. 7. The possibility of obtaining lacking crossbeam pro-
Fig. 6. SFr factors as a function of field size for different depths (SL-18).
files was investigated for a supposedly missing profile of a field size 12 X 12 cm2 at depth 30 cm. Using Figs. 5, 6, and 7 and linear interpolation the parameters S, So, SF,, SFs : H and SF, were estimated. The lacking profile produced using equation (6) is compared with the experimental data and the best fitted profile in Fig. 8. It can be seen that the differences between the two profiles are not significant. Both profiles present a maximum error less than 2% when compared with data points. It may be concluded therefore, that equation (6) may also be used to obtain lacking crossbeam profiles.
0
d=30
cm
d=15
cm
d=6
d=1.5
d=1.5
cm
cm
cm
0
I,,.,,,,,,,,,,,~~~,~,,,,,,,,~,,~,,,,,~, 0
5
10
square
Fig. 5. S factors as a function of field size for different depths (SL-18).
15
field
side
20
25
(cm)
Fig. 7. SF, factors as a function of field size for different depths (SL-18).
A semiempirical
method
00000 ~------~
for relative
Data points Interpolated Best Fitted
cross beam dose 0 I. TSALAFOUTAS
et nl.
67
and depths, and which may be useful in treatment planning. Lacking crossbeam profiles can also be obtained from a limited set of data. It is possible that the method could also be used in conjunction with negative field method3 to produce crossbeam profiles in blocked fields. This is currently under investigation, and will be presented in the future. The advantages of this method are that only crossbeam profiles are required, and a small number of equations and fitting parameters are used. REFERENCES
off-axis
Fig.
8. Interpolated
distance
and best fitted cm2 field at depth
1. Johns, H.E.; Cunningham, J.R. The physics of radiology. 4th ed. Springfield, IL: CC. Thomas, 1983:369-372, 376-377. 2. International Commission on Radiation Units and Measurements. Report 42. Use of computers in external beam radiotherapy procedures with high-energy photons and electrons. Bethesda, Maryland: ICRU; 1987. 3. Khan, F.M. The physics of radiation therapy. Baltimore, MD: Wilkins & Wilkins; 1984.
(cm)
profiles 30 cm.
for
a 12 X 12
From the results tabulated we made the following conclusions concerning the parameters involved. The S, So parameters were chosen to be integer and SS, SFs real numbers with one decimal digit, as further accuracy has no meaning. We must note that small changes in S, So, SF, and SFs do not significantly alter the curve. In contrast, x0 is denoted by two decimal digits as it strongly affects the best fitted relative crossbeam profile. The SF, factors for Therac-6 are much lower than the corresponding for SL-18. This has the meaning that the FP( x) curves fall slower for Therac6, presenting a larger penumbra region.
CONCLUSIONS A semiempirical method has been presented which can describe relative crossbeam dose profiles at depth for two linear accelerators for different fieldsizes
APPENDIX
A
The H and SFc values are calculated rather than fitted in order to avoid the introduction of many fitting parameters. The calculation method that has been used is illustrated in the following: For the point on the central axis, x = 0, equation (6) becomes:
FT(0)=PFp(O)+(l-H)(S+So)Fs(0) Setting
F,(O)
= F,(O)
and solving
H = (F(O) Solving have: SFc
equation
= ( I/x~~~)
(6)
(A.])
for H we have:
- FE(O))/((S for SFc and setting
+ So) F,(O))
(24.2)
x = xioO ( xloo # 0) we
ln(H(S + So)Fs(xm)~(F(x,ce)
-
FE(XIOO)))
(A.3)
where xlDo is the off-axis distance corresponding to 100% of relative depth dose in the measured crossbeam profile, FE(O), FE(~,,XI) are the measured values and F(O), F(xloo) are the best fitted values at x = 0 and x = x,~, respectively. Thus, the correction function Fc( x) equates the contribution of the second term of equation (6) to S for x = XpJo. In case that more than one point corresponds to lOO%, the point closer to the central axis is used in equation (A.3). If xIM) = 0, then SF, can be calculated by any other point close to the central axis.
Medical Dosimetry, Vol. 22, No. I, pp. 63-67, 1997 0 1997 American Association of Medical Dosimetrists Printed in the USA. All rights reserved
0958.3947197
$17.00 + .OO
PI1 SO958-3947( 96) 00153-7
ELSEVIER
A SEMIEMPIRICAL METHOD FOR THE DESCRIPTION OF RELATIVE CROSSBEAM DOSE PROFILES AT DEPTH FROM LINEAR ACCELERATORS IOANNIS TSALAFOUTAS, M.Sc ., ’ STELIOS XENOFOS, PH.D., ’ and ION E. STAMATELATOS, PH.D.’ ’ Departmentof Medical Physics,Agios SavvasHospital, Athens, Greece; ’ Institute of NuclearTechnology and RadiationProtection, National Centreof Scientific ResearchDemokritos, Athens, Greece Abstract-A semiempirical method for the calculation of the relative crossbeam dose profiles at depth is described. The parameters required to set up the formulae and their dependence with field size and depth are investigated. Using the above method, measured crossbeam dose profiles at depth from two linear accelerators, Philips (SL-18) and AEC (Therac-6) are reproduced. The results indicate that this method Association of Medical is applicable within a wide range of depths and field sizes. 0 1997 American Dosimetrists Key Words:
Radiotherapy,
Linear
accelerator,
Megavoltage
beam,
Crossbeam
profile.
component can be farther analysed into collimator and phantom scatter components. Hence any theoretical formula that describes the crossbeamprofiles has the practical difficulty of determining the contribution of each component to the measuredcrossbeam profile. Based on the simple geometrical model delineated in Fig. 1, a semiempiricalformula hasbeen obtained and is used to describe the relative crossbeamdose profiles at any specified depth. In Fig. 1(a) a geometrical model of the head of a radiotherapy unit is shown. The dashed line is the line acrosswhich a smalldosimeterwith buildup cap is moving, in order to acquire the primary crossbeam profile. In Fig. 1(b) the geometric primary crossbeam profile is indicated by the dashedline, whereasthe actual primary crossbeamprofile is indicated by the solid line. The main aspectsthat differentiate the two profiles are that the source is not a sharply defined disk, there will be scatteringwithin the sourceand from the collimators, and transmissionthrough collimators and inverse squarelaw effects. The actual primary crossbeamprofile from a linear accelerator will be more complicated, becauseof the presenceof the flattening filter. Furthermore, the crossbeamdose profiles at depth will present increasedpenumbra compared to the primary crossbeam profile becauseof phantom scatter and loss of lateral electronic equilibrium. ’ The function that has been used to approximate the geometric primary crossbeamprofile is:
INTRODUCTION There are several semiempirical methods found in the literature which describe the primary crossbeamprofiles or the crossbeamdose profiles at depth for radiotherapy machines.‘,’ The accuracy of such methods is very important when these are used for treatment planning where the required accuracy in dose calculation should be greater than 3%. In this study we present a semiempirical method for the description of the relative crossbeamdose profiles at depth. It is basically a modification of a known relationship that describesprimary crossbeamprofiles.’ This modification was necessary in order to describe the crossbeamprofiles from linear accelerators which present certain characteristics that could not be accurately reproduced. These characteristics are a “shoulder” and a “tail” of different curvature, an abrupt fall-off of the dose in the penumbra region and, in some cases, a concavity in the central axis region due to the effect of the flattening filter. The method proposed produces profiles with the above mentioned characteristics by using two components, geometric primary and secondary components, along with a correction function to improve the fitting near the central axis.
METHOD The total dose to a point in a medium is the sum of primary and scattered components. The scattered
b(x)
= (1 + exp(SF,(lxl
- x0)))-’
(1)
where, x is the distance of the point of interest from the central ray of the beam, x0 the geometric edge of the field at the specified depth and SF, is proportional to the derivative of FP(x) at x = x0. FP(x ) is symmetric
Reprint requests: Dr. Ion E. Stamatelatos, Institute of Nuclear Technology & Radiation Protection, National Centre of Scientific Research, “Demokritus,” Agia Paraskevi, Athens 153.10, Greece. 63
64
Medical Dosimetry
Volume 22, Number 1, 1997
-,+sd
Hence, the formula that would describe symmetric relative crossbeam dose profiles specified depth would be written as:
ssd
simple, at any
F(x) = P Fp(x) + S F,(x)
-‘lo
/
.
Xl
-b
;6
where P and S = 100 - P are integer numbers that would give the percent contribution of each component to F(x). The above formula, however, predicts that if F( 0) = 100 then F(2xo) = 0. This is not the case for actual profiles, especially for large fields, as the dose persists well beyond 2x0. It must be noted that if the value of SFs is made adequately small for the given x0, then F,( 0) < 1 and F,( 2x,) > 0. Thus, if the S factor is increased by a certain amount (So) then we can produce a symmetric F(x) curve with F(0) = 100 (instead of 100 + S,) and F(2xo) = So (instead of 0). Thus:
.. x2
-‘4
-‘2
10
I2
4
6D 8 1 distance
Off-ax15
F(x) = P Fp(x) + (S + S,)F,(x)
Fig. 1. (a) The geometrical factors that lead to beam penumbra. (b) Geometric and actual primary crossbeam profiles in air along x1-x2.
= (1 + exp(%(lxl
- %J))Y1
(4)
Equation (4) still needs to be corrected as it cannot account for the concavity which sometimes appears near the central axis in the measured crossbeam profiles. The function used for this purpose is:
about the point (x0, FP( x0)), and converges to 1 at x = 0 and to 0 at x = 2x0. The convergence rate, for a given x0, depends on the SF, value. For large values of SF,, FP( x) approximates closely the shape of the geometric primary crossbeam profile. It was observed, however, that smaller values of SFp pkduce a slowly varying profile which presents a shoulder and a tail. Based on this observation it was assumed that a profile with all the desired characteristics could be produced by adding a slowly varying term named secondary component to the geometric primary term. This assumption was empirically proven to provide satisfactory results for the purposes of this study. Thus, in order to describe the secondary component the following function has been used: F,(x)
(3)
F,(x)
= 1 - H exp(-SF,/xj)
(5)
where H, SFc are parameters calculated from best fitted and measured profiles (Appendix A). The correction function ( Fc (x)) is applied to the second term of equation (4) and its effect is twofold. It can produce the concavity around the central axis and it also can modify the curvature of the shoulder. Hence, the final formula that is used to describe the relative crossbeam dose profile at any specified depth is: F-r(x) = P FAX) + F,(x)
(S + S,)Fs(x)
(6)
The effects that differentiate the geometric primary crossbeam profile from the actual primary crossbeam profile, flattening filter effect and phantom scatter, have been implicitly included in the second term of equation (6). In order to find the best fitting parameters S, So, SF,, SFs and x0, equation (4) was used to fit data points from crossbeam profiles obtained from two lin-
(2)
Equation (2) is of the same form as equation ( 1 ), SF, is proportional to the derivative of Fs (x) at x = x0 but SFs values are very small compared to SFp values.
Table 1. Parameters obtained by least squares for SL- 18 data Field
6 x 6 cm’
Depth (cm)
1.5
S SO SF, (cm-‘) Sf, (cm-‘)
9 2
x0 (cm)
H SF, (cm-‘)
15.3 1.3 3.03 0.30
2.18
6 14 2 13.9 0.9 3.14
0.11 1.81
15
10 X 10 cm* 30
1.5
6
15
15 X 15 cm’ 30
1.5
6
15
18
21
11
21
26
30
13
22
32
4 12.7 0.7 3.45
6 11.5 0.6 3.88 0.16
2 16.1 0.8 5.09 0.32
2 14.2 0.8 5.28 0.15 0.98
4 12.8 0.5 5.73 0.13
5 10.3 0.4 6.48
4 14.7 0.5 7.58 0.37 0.28
3 13.6 0.5 7.89 0.17 0.45
4 11.7 0.4 8.56
0.14 1.14
0.64
0.95
1.25
0.11 0.93
20 X 20 30
1.5
38
15
5
5
10.1 0.4 9.74
0.10
0.08
0.37
0.48
16.2 0.7 10.12 0.40 0.15
6
cm* 15
30
21
3:
4;
5 14.2 0.4 10.52 0.26 0.22
10.6 0.3 11.45 0.10 0.31
9.5 0.3 12.97 0.06 0.41
A semiempiricalmethod for relative cross beam dose 0
I. TSALAFOUTAS
65
et al.
Table 2. Parameters obtained by least squares for AEC Therac-6 data Field
6 x 6
Depth (cm)
1.5
S so
9 6 7.8 0.5 3.07 0.44 1.05
SFp (cm-‘) Sf, (cm-‘) x0 (cm)
H SF,- (cm-’ )
cm’
15 X 15 cm’
6
12
20
13 I 7.4 0.4 3.22 0.17 0.73
19 7 7.0 0.4 3.4 0.08
22 9 6.7 0.4 3.66 0.13 0.68
1.11
ear accelerators. Data points beyond 100% (toward the field edge) only are taken into account. The 100% corresponds to the maximum value of the measured crossbeam profile. The best fitted profiles are then corrected by equation (5 ) to yield the final profile FT (x) . The values for H and SF, are calculated as shown in Appendix A. The profiles were obtained from measurements at Agios Savvas Hospital using a profile scanning system and a water phantom. The measurements were carried out for 6 MV x-ray beams, on an SL-18 (Philips) and a Therac-6 (AEC) linear accelerator.
RESULTS
AND DISCUSSION
The results obtained using a simple least squares fitting procedure are shown in Table 1 (SL-18) and Table 2 (Therac-6). In Fig. 2 the relative crossbeam profile FT( x) and
1.5
6
18
12
26
31 5 7.1 0.3 8.43 0.07 0.81
6
5 8.2 0.3 7.69 0.14 0.37
7.3 0.3 8.02
0.11 0.33
20 34 6 5.9 0.3 9.05 0.09 0.44
its components-P Fp(x), (S + S,) F,(x) and F,(x) (S + S,) F,(x) -are separately plotted for a field size 10 x 10 cm* at d = 15 cm (SL-18). The profiles produced using equation (6) and the best fitting parameters tabulated for field size 15 X 15 cm2 and depths 6, 15 and 30 cm are shown in Fig. 3 for SL-18. The corresponding profiles for Therac-6 at depths 6, 12 and 20 cm for the same field size are shown in Fig. 4. From Figs. 3 and 4 agreement between experimental data and the reproduced
profiles can be seen within
all dose regions. The correction function F,(x) works satisfactorily, and errors are less than 2.5%. From Tables 1 and 2, it is shown that the parameters S, SF, and SFs have different values for the two accelerators. They change, however, with field size and depth in a similar way. The values of S increase with increasing depth and field size much as scatter radiation does. The variation of the S factor with field size for different depths is shown for SL-18 in Fig. 5. It is observed that S increases with field size more rapidly
d=1.5
cm
80 2ic
P
0
m
80
0.00
4.00 off-axis
8.00 distance
12.00
(cm)
Fig. 2. Fitted relative crossbeam profiles for a 10 X 10 cm’ field at depth 15 cm. F(x), FT(x) and the terms formingup equation (6) are separately plotted: (a) data points, (b) P b(x), (~1 (S + &)Fs(x), Cd) F(x), (e) Fdx)(S + S,)Fs(x), (f) FT.
off-axis
distance
(cm)
Fig. 3. Fitted profiles for a 15 X 15 cm’ field (SL-18). The height of the profiles has been adjusted to avoid overlapping.
Medical Dosimetry
Volume 22, Number 1, 1997
d=6
6.0 00.00i
5.00 II
11
1 ’ 1
10.00 II 11
1 ’ 11
11
15.00 I 11
i-m--m--n 0
5
10 square
off-axis
distance
15 field
side
20
cm
d=15
cm
d=30
cm
25
(cm)
(cm)
Fig. 4. Fitted profiles for a 15 x 15 cm’ field (Therac6). The height of the profiles has been adjusted to avoid overlapping.
at larger depths and seems to level-off at large field sizes. SFr values decrease with increasing depth. They also decrease with increasing field size but this variation is prominent only for large depths. This behaviour is shown for SL-18 in Fig. 6. Finally, SF, decreases with increasing depth and field size. The variation of SFs with field size for different depths is shown for SL-18 in Fig. 7. The possibility of obtaining lacking crossbeam pro-
Fig. 6. SFr factors as a function of field size for different depths (SL-18).
files was investigated for a supposedly missing profile of a field size 12 X 12 cm2 at depth 30 cm. Using Figs. 5, 6, and 7 and linear interpolation the parameters S, So, SF,, SFs : H and SF, were estimated. The lacking profile produced using equation (6) is compared with the experimental data and the best fitted profile in Fig. 8. It can be seen that the differences between the two profiles are not significant. Both profiles present a maximum error less than 2% when compared with data points. It may be concluded therefore, that equation (6) may also be used to obtain lacking crossbeam profiles.
0
d=30
cm
d=15
cm
d=6
d=1.5
d=1.5
cm
cm
cm
0
I,,.,,,,,,,,,,,~~~,~,,,,,,,,~,,~,,,,,~, 0
5
10
square
Fig. 5. S factors as a function of field size for different depths (SL-18).
15
field
side
20
25
(cm)
Fig. 7. SF, factors as a function of field size for different depths (SL-18).
A semiempirical
method
00000 ~------~
for relative
Data points Interpolated Best Fitted
cross beam dose 0 I. TSALAFOUTAS
et nl.
67
and depths, and which may be useful in treatment planning. Lacking crossbeam profiles can also be obtained from a limited set of data. It is possible that the method could also be used in conjunction with negative field method3 to produce crossbeam profiles in blocked fields. This is currently under investigation, and will be presented in the future. The advantages of this method are that only crossbeam profiles are required, and a small number of equations and fitting parameters are used. REFERENCES
off-axis
Fig.
8. Interpolated
distance
and best fitted cm2 field at depth
1. Johns, H.E.; Cunningham, J.R. The physics of radiology. 4th ed. Springfield, IL: CC. Thomas, 1983:369-372, 376-377. 2. International Commission on Radiation Units and Measurements. Report 42. Use of computers in external beam radiotherapy procedures with high-energy photons and electrons. Bethesda, Maryland: ICRU; 1987. 3. Khan, F.M. The physics of radiation therapy. Baltimore, MD: Wilkins & Wilkins; 1984.
(cm)
profiles 30 cm.
for
a 12 X 12
From the results tabulated we made the following conclusions concerning the parameters involved. The S, So parameters were chosen to be integer and SS, SFs real numbers with one decimal digit, as further accuracy has no meaning. We must note that small changes in S, So, SF, and SFs do not significantly alter the curve. In contrast, x0 is denoted by two decimal digits as it strongly affects the best fitted relative crossbeam profile. The SF, factors for Therac-6 are much lower than the corresponding for SL-18. This has the meaning that the FP( x) curves fall slower for Therac6, presenting a larger penumbra region.
CONCLUSIONS A semiempirical method has been presented which can describe relative crossbeam dose profiles at depth for two linear accelerators for different fieldsizes
APPENDIX
A
The H and SFc values are calculated rather than fitted in order to avoid the introduction of many fitting parameters. The calculation method that has been used is illustrated in the following: For the point on the central axis, x = 0, equation (6) becomes:
FT(0)=PFp(O)+(l-H)(S+So)Fs(0) Setting
F,(O)
= F,(O)
and solving
H = (F(O) Solving have: SFc
equation
= ( I/x~~~)
(6)
(A.])
for H we have:
- FE(O))/((S for SFc and setting
+ So) F,(O))
(24.2)
x = xioO ( xloo # 0) we
ln(H(S + So)Fs(xm)~(F(x,ce)
-
FE(XIOO)))
(A.3)
where xlDo is the off-axis distance corresponding to 100% of relative depth dose in the measured crossbeam profile, FE(O), FE(~,,XI) are the measured values and F(O), F(xloo) are the best fitted values at x = 0 and x = x,~, respectively. Thus, the correction function Fc( x) equates the contribution of the second term of equation (6) to S for x = XpJo. In case that more than one point corresponds to lOO%, the point closer to the central axis is used in equation (A.3). If xIM) = 0, then SF, can be calculated by any other point close to the central axis.