Radiation Physics
A New Technique for the Calculation of Scattered Radiation from 60CO_ Teletherapy Beams 1 Fred H. Edwards, M.S., M.D.2, and Charles W. Coffey, II, Ph.D.
An alternative to the Clarkson technique for calculating total scattered radiation is presented. Resultant TAR values are compared with those obtained by the Clarkson technique for the same field. The new scatter technique is always more than 2.8 times faster than the Clarkson technique and more than 4 times faster on a large number of fields. The new approach appears to offer a simple and expedient alternative to the more traditional methods for irregular field dose calculations. INDEX TERMS:
Radiations, measurement • Therapeutic radiology, physics
Radiology 132: 193-196, July 1979
HE ABSORBED
dose delivered to a given point in a tel-
Tetherapy beam is often calculated by separately evaluating the primary and scattered components of radiation. This approach is termed the' 'Clarkson scatter technique" (1), the heart of which is the use of scatter-air ratio (SAR) tables to determine the contribution of scattered radiation to the total dose. The SAR at depth d for a circular field of radius r is given by SAR(d,r) = TAR(d,r) -
TAR(d,O)
(1)
where TAR(d,r) is the tissue-air ratio for depth d and radius r, and TAR(d,O) is the extrapolated "zero area tissue-air ratio" that represents the attenuated primary beam. Clarkson (1) and Cunningham (2) point out that the scattered radiation to a given point in a beam of any cross-sectional shape can always be represented by a summation of circular beam sectors (Fig. 1). The total scattered radiation is given by S
=L j
!1fJ .
SAR(d,rj) ~ 21r
D E Fig. 1.
The Clarkson scatter technique for irregular field
ABCOE. Coordinates (Xj,Yi) define vector Vi' To find the scattered radiation from segment AOB to point 0, a series of radii, rj, are projected from 0 to side AB at intervals of ~{Jj- Each radius is associated with an SAR value which is added to cumulative SAR
until the entire field has been covered.
(2)
where rj and !If}j are defined in Figure 1. The classic computer application of this technique is Cunningham's program IRREG (3), in which tables of SAR(d,r) and TAR(d,O) are permanently stored for look-up and interpolation procedures. The user enters coordinate points (Xj,Yi) of the corners of the field in a plane perpendicular to the central axis (Fig. 1), and with !lfJj set to approximately 10 0 , vector relationships between Vi and Vj+ 1 are used to generate rj- S is then found from equation (2)
and added to TAR(d,0)3 to give the TAR for the point in question. Once the TAR has been calculated, it is a relatively simple matter to determine the absorbed dose (2, 3). A significant drawback to this technique is the repetitious algorithm used to generate the scattered dose. Generally, it is necessary to calculate a minimum of 36 radii even for simple rectangular fields. For the more ir-
1 From the Department of Radiation Medicine, University of Kentucky, Albert B. Chandler Medical Center, Lexington, Ky. Received Oct. 3, 1978; accepted and revision requested Jan. 18, 1979; revision received Feb. 28. 2 Present address: Capt. Fred H. Edwards, M.S., M.D., Department of Surgery, Eisenhower Army Medical Center, Augusta, Ga. 30905. Supported in part by USPHS Research Grants P01-CA-17786 and R10-CA-20255 from the National Cancer Institute. 3 There is a modifying factor", f(x,y), that accounts for the variation of primary beam intensity across the field. Generally, this should be multiplied by TAR(d,O) and that product added to S to give the TAR. as
193
194
FRED
H. EDWARDS AND CHARLES W. COFFEY, II
B
B
July 1979
B
o
A
A
(c)
(b)
Fig. 2. The modified scatter te_chnique for field segmen~ AOa. . _ _ a. The coordinate points define Vi as in the Clarkson technique. (J IS the angl~ from Vi to Vi+1 and may have either a positive or negative value. The SAR of segment AOa IS S; . b. A circular sector of radius fa and center 0 is inscribed within AOa. The SAR of the circular sector grossly approximates S; but the SAR of the remaining shaded area must be accounted for in order to obtain acceptable accuracy. . c The shaded annular sector has area ~ A chosen so that SAR(d,feff) = S: If certain assurnptions are made, then ~ A may be considered equal to the shaded area in b.
regular fields in which negative contributions to scatter must be considered, the number of calculated radii often exceeds 60. The amount of computer time required for these calculations is quite significant and as a result, often leads to considerable inconvenience and expense for the user. METHOD
The total scattered radiation, S, can be calculated by a more expedient technique. In this approach, the triangular segment defined by Vj and Vi+ 1 (Fig. 2) is considered as a whole, with only a single "effective radius", reft, used to obtain a tabular SAR(d,reft) value. This value multiplied by H127f represents the scattered component of radiation, S; for the triangular segment in question. Each segment is evaluated in turn until the entire field has been covered. The value of S for the field is then given by L~=1 S, where N is the total number of segments. Consider a general segment as shown in Fig. 2, a. The scatter contribution from this segment can be represented by a single circular sector having a scatter-air ratio equal to S; The radius of that sector, by definition, is reft. In order to calculate reft, one inscribes a circular sector of radius r0 in such a way as to occupy as much of the area of the triangle as possible. As suggested by Fig. 2, b, the scatter-air ratio associated with this r o sector, SAR (d,ro ) • ()I 27f, represents a reasonable approximation of S; Since the whole area of the triangle has not been accounted for, however, it is certain that the value of SAR (d,ro ) is too small. If the area of the r o sector is augmented by an area ~ A, distributed as a concentric annulus as shown in Fig. 2, c, then with the proper value of ~ A it becomes possible to form a circular sector having a SAR = S; This sector
with the added area Do A will have the desired radius reft. Since the relationship between these parameters is given by f)
Do A = -27f . 7f(r 2 eft
-
r
2)
0'
(3)
then
reft =
[~. !l. A + r 2]"2. 0
(4)
It is possible to find r o from simple geometric relationships. By finding an appropriate value for D. A, one may determine the reftand the SAR(d,reft) for the field segment in question. To examine how the SAR varies as a function of radius and area, standard tables (4) were consulted. Graphs were constructed to plot the change in SAR per unit area versus the radius from the central axis out to the unit area. In order to do this, concentric circles with radii differing by 1 em were examined. The SAR of the inner circle was subtracted from that of the larger circle and that difference was divided by the area of the 1 em annulus. The result, D. SARI A, is assumed to provide a measure of the scattered radiation from a unit area at a radius r as shown in Figure
3. Radii from 1 to 25 em at depths from 0.5 to 30 em were investigated and the results plotted to yield curves similar to those in Figure 4. At radii greater than 10 em, the curves become approximately linear and have a very small slope. The small slope indicates that !:1SARI A in this region is relatively independent of the radius. It follows, then, that a segment of area contributes virtually the same amount of scattered radiation regardless of its location in this region. Therefore the area lying outside the ro circular sector
195
CALCULATION OF SCATTERED RADIATION
Vol. 132
Radiation Physics
(Fig, 2, b, shaded) may be redistributed in the annulus configuration of Figure 2, c without significantly changing the SAR contribution of that area. The resultant circular sector, then, will have a SAR S, and the resultant radius of that sector must be feft. Equations (3) and (4) demonstrate how this value of feftmay be determined. Essentially, the major assumption is that 6. A may be represented by the magn itude of the shaded area in Figure 2, b. This means that 6. A will be given by the difference between the area of the triangular segment (112 v.v, + 1
=
sinO) and the area of the circular sector
6. A = 1/2
VjVj+1
sine -
(:~ ,rro2):
~ 7ff02. 27f
Fig. 3.
(5)
shaded annulus will be ilsAR = SAR(dh) - SAR(d,f1)' The contribution per unit area is D. SARI A where A is the area of the shaded t-ern annulus. The area is assumed + 0.5 cm from the to lie at a radius r = center.
Substituting this into equation (4) gives 'elf
=
[v;v;r~ sinO]
112
=
V1*1
Derivation of the D.SARI A vs,
r curves. Here the added scatter from the
'1
(6)
v
where = Vj X Vj + 1. Equation (6) shows that, given the assumptions above, feft isnot a function of fo. The region in which f o ~ 10 em has been considered up to this point. In the region in which f o 10 em, there is a marked dependence of ~SAR/ A upon f o (Fig. 4). Here, a unit area at a smaller f o offers a significantly larger scatter contribution than a unit area at a more distant radius. Assume that the shaded area of Figure 2, c is equal to the shaded area of Figure 2, b. Note that the radii associated with an annular area like that of Figure 2, c will not be as large as some of the radii associated with the shaded area of Figure 2, b. It follows that the SAR of the area augmented by the annulus will be larger than S; i.e., the technique outlined above for ro > 10 cm will always overestimate the SAR if applied in the ro < 10 cm region. By trial and error, we found that it is possible to compensate for this overestimation by reducing feft by 5 %. With this correction factor, the value of feft for all ro then becomes
l~
A New Technique for the Calculation of Scattered Radiation from 60CO_ Teletherapy Beams 1 Fred H. Edwards, M.S., M.D.2, and Charles W. Coffey, II, Ph.D.
An alternative to the Clarkson technique for calculating total scattered radiation is presented. Resultant TAR values are compared with those obtained by the Clarkson technique for the same field. The new scatter technique is always more than 2.8 times faster than the Clarkson technique and more than 4 times faster on a large number of fields. The new approach appears to offer a simple and expedient alternative to the more traditional methods for irregular field dose calculations. INDEX TERMS:
Radiations, measurement • Therapeutic radiology, physics
Radiology 132: 193-196, July 1979
HE ABSORBED
dose delivered to a given point in a tel-
Tetherapy beam is often calculated by separately evaluating the primary and scattered components of radiation. This approach is termed the' 'Clarkson scatter technique" (1), the heart of which is the use of scatter-air ratio (SAR) tables to determine the contribution of scattered radiation to the total dose. The SAR at depth d for a circular field of radius r is given by SAR(d,r) = TAR(d,r) -
TAR(d,O)
(1)
where TAR(d,r) is the tissue-air ratio for depth d and radius r, and TAR(d,O) is the extrapolated "zero area tissue-air ratio" that represents the attenuated primary beam. Clarkson (1) and Cunningham (2) point out that the scattered radiation to a given point in a beam of any cross-sectional shape can always be represented by a summation of circular beam sectors (Fig. 1). The total scattered radiation is given by S
=L j
!1fJ .
SAR(d,rj) ~ 21r
D E Fig. 1.
The Clarkson scatter technique for irregular field
ABCOE. Coordinates (Xj,Yi) define vector Vi' To find the scattered radiation from segment AOB to point 0, a series of radii, rj, are projected from 0 to side AB at intervals of ~{Jj- Each radius is associated with an SAR value which is added to cumulative SAR
until the entire field has been covered.
(2)
where rj and !If}j are defined in Figure 1. The classic computer application of this technique is Cunningham's program IRREG (3), in which tables of SAR(d,r) and TAR(d,O) are permanently stored for look-up and interpolation procedures. The user enters coordinate points (Xj,Yi) of the corners of the field in a plane perpendicular to the central axis (Fig. 1), and with !lfJj set to approximately 10 0 , vector relationships between Vi and Vj+ 1 are used to generate rj- S is then found from equation (2)
and added to TAR(d,0)3 to give the TAR for the point in question. Once the TAR has been calculated, it is a relatively simple matter to determine the absorbed dose (2, 3). A significant drawback to this technique is the repetitious algorithm used to generate the scattered dose. Generally, it is necessary to calculate a minimum of 36 radii even for simple rectangular fields. For the more ir-
1 From the Department of Radiation Medicine, University of Kentucky, Albert B. Chandler Medical Center, Lexington, Ky. Received Oct. 3, 1978; accepted and revision requested Jan. 18, 1979; revision received Feb. 28. 2 Present address: Capt. Fred H. Edwards, M.S., M.D., Department of Surgery, Eisenhower Army Medical Center, Augusta, Ga. 30905. Supported in part by USPHS Research Grants P01-CA-17786 and R10-CA-20255 from the National Cancer Institute. 3 There is a modifying factor", f(x,y), that accounts for the variation of primary beam intensity across the field. Generally, this should be multiplied by TAR(d,O) and that product added to S to give the TAR. as
193
194
FRED
H. EDWARDS AND CHARLES W. COFFEY, II
B
B
July 1979
B
o
A
A
(c)
(b)
Fig. 2. The modified scatter te_chnique for field segmen~ AOa. . _ _ a. The coordinate points define Vi as in the Clarkson technique. (J IS the angl~ from Vi to Vi+1 and may have either a positive or negative value. The SAR of segment AOa IS S; . b. A circular sector of radius fa and center 0 is inscribed within AOa. The SAR of the circular sector grossly approximates S; but the SAR of the remaining shaded area must be accounted for in order to obtain acceptable accuracy. . c The shaded annular sector has area ~ A chosen so that SAR(d,feff) = S: If certain assurnptions are made, then ~ A may be considered equal to the shaded area in b.
regular fields in which negative contributions to scatter must be considered, the number of calculated radii often exceeds 60. The amount of computer time required for these calculations is quite significant and as a result, often leads to considerable inconvenience and expense for the user. METHOD
The total scattered radiation, S, can be calculated by a more expedient technique. In this approach, the triangular segment defined by Vj and Vi+ 1 (Fig. 2) is considered as a whole, with only a single "effective radius", reft, used to obtain a tabular SAR(d,reft) value. This value multiplied by H127f represents the scattered component of radiation, S; for the triangular segment in question. Each segment is evaluated in turn until the entire field has been covered. The value of S for the field is then given by L~=1 S, where N is the total number of segments. Consider a general segment as shown in Fig. 2, a. The scatter contribution from this segment can be represented by a single circular sector having a scatter-air ratio equal to S; The radius of that sector, by definition, is reft. In order to calculate reft, one inscribes a circular sector of radius r0 in such a way as to occupy as much of the area of the triangle as possible. As suggested by Fig. 2, b, the scatter-air ratio associated with this r o sector, SAR (d,ro ) • ()I 27f, represents a reasonable approximation of S; Since the whole area of the triangle has not been accounted for, however, it is certain that the value of SAR (d,ro ) is too small. If the area of the r o sector is augmented by an area ~ A, distributed as a concentric annulus as shown in Fig. 2, c, then with the proper value of ~ A it becomes possible to form a circular sector having a SAR = S; This sector
with the added area Do A will have the desired radius reft. Since the relationship between these parameters is given by f)
Do A = -27f . 7f(r 2 eft
-
r
2)
0'
(3)
then
reft =
[~. !l. A + r 2]"2. 0
(4)
It is possible to find r o from simple geometric relationships. By finding an appropriate value for D. A, one may determine the reftand the SAR(d,reft) for the field segment in question. To examine how the SAR varies as a function of radius and area, standard tables (4) were consulted. Graphs were constructed to plot the change in SAR per unit area versus the radius from the central axis out to the unit area. In order to do this, concentric circles with radii differing by 1 em were examined. The SAR of the inner circle was subtracted from that of the larger circle and that difference was divided by the area of the 1 em annulus. The result, D. SARI A, is assumed to provide a measure of the scattered radiation from a unit area at a radius r as shown in Figure
3. Radii from 1 to 25 em at depths from 0.5 to 30 em were investigated and the results plotted to yield curves similar to those in Figure 4. At radii greater than 10 em, the curves become approximately linear and have a very small slope. The small slope indicates that !:1SARI A in this region is relatively independent of the radius. It follows, then, that a segment of area contributes virtually the same amount of scattered radiation regardless of its location in this region. Therefore the area lying outside the ro circular sector
195
CALCULATION OF SCATTERED RADIATION
Vol. 132
Radiation Physics
(Fig, 2, b, shaded) may be redistributed in the annulus configuration of Figure 2, c without significantly changing the SAR contribution of that area. The resultant circular sector, then, will have a SAR S, and the resultant radius of that sector must be feft. Equations (3) and (4) demonstrate how this value of feftmay be determined. Essentially, the major assumption is that 6. A may be represented by the magn itude of the shaded area in Figure 2, b. This means that 6. A will be given by the difference between the area of the triangular segment (112 v.v, + 1
=
sinO) and the area of the circular sector
6. A = 1/2
VjVj+1
sine -
(:~ ,rro2):
~ 7ff02. 27f
Fig. 3.
(5)
shaded annulus will be ilsAR = SAR(dh) - SAR(d,f1)' The contribution per unit area is D. SARI A where A is the area of the shaded t-ern annulus. The area is assumed + 0.5 cm from the to lie at a radius r = center.
Substituting this into equation (4) gives 'elf
=
[v;v;r~ sinO]
112
=
V1*1
Derivation of the D.SARI A vs,
r curves. Here the added scatter from the
'1
(6)
v
where = Vj X Vj + 1. Equation (6) shows that, given the assumptions above, feft isnot a function of fo. The region in which f o ~ 10 em has been considered up to this point. In the region in which f o 10 em, there is a marked dependence of ~SAR/ A upon f o (Fig. 4). Here, a unit area at a smaller f o offers a significantly larger scatter contribution than a unit area at a more distant radius. Assume that the shaded area of Figure 2, c is equal to the shaded area of Figure 2, b. Note that the radii associated with an annular area like that of Figure 2, c will not be as large as some of the radii associated with the shaded area of Figure 2, b. It follows that the SAR of the area augmented by the annulus will be larger than S; i.e., the technique outlined above for ro > 10 cm will always overestimate the SAR if applied in the ro < 10 cm region. By trial and error, we found that it is possible to compensate for this overestimation by reducing feft by 5 %. With this correction factor, the value of feft for all ro then becomes
l~
