Aust. Radiol. (1975), 19, 145
Isoexposure Curves for @CoOphthalmic Applicators A. H. BEDDOE Department of Radiotherapy, Christchurch Hospital, Christchurch, New Zealand. INTRODUCTION Various methods exist for the radiotherapeutic treatment of retinoblastoma and other ocular neoplasms. Of these, perhaps the most commonly utilized is some form of radioactive applicator containing a radioisotope such as 2P2Rn, 2-bRa, ..>. 9oSror “OCo. A set of such applitors, manufactured by the Radiochemical Centre, Amersham, to a design developed at St. Bartholomew’s Hospital, London, has been described by Stallard (1962) and Innes ( 1962). These applicators consist of platinum discs containing O0Coin circular, “D’and crescentic shapes, and have two fixing lugs by which they can be sutured to the eye. They are described in the Radiochemical Centre Catalogue (Radiopharmaceuticals and Clinical Radiation Sources, 1971). In the initial work by Innes (1962), values for “design depth” were calculated for each applicator, the “design depth” being the depth of the neoplasm below the sclera. The values appear in Table 1. Further work by Magnus (1967) and Magnus, Gobbeler and Strotges ( 1968) indicated methods of determining exposure at depths along the central axis other than the “design depth.” Recently, Casebow ( 1971 ) has presented a method of calculating exposure at points in two planes, assuming inverse square law attenuation with no tissue absorption. He also presents values for “design depth” based on his calculations, these values *Since this paper was submitted for publication in October, 1971, The Radiochemical Centre, Amersham, has recalculated “design activities.” The “design activities” now supplied will of course produce the same relative exposure distributions as shown in Figure 3, but the treatment time necessary to achieve a given set of actual isoexposure curves will of course no longer be 144 hours. Treatment times in any case need adjusting because of source decay, SO that an exposure time, t, for each applicator is give3 by : “design activity” T t = actual activity where the “design activity” is taken from Table 1. Australasia7 Radiology, Vol. XIX, No. 2, June, 1975
also appearing in Table 1. This paper indicates another method of computation of exposure contributions fom the various distributions of G°Coin each applicator, including the effects of tissue absorption, and presents sets of isoexposure curves constructed from computer calculated data. METHOD OF CALCULATION For computation purposes, a set of Cartesian co-ordinate axes (X, Y, Z) was defined for each applicator, and these are shown in Figure 1. Co-ordinates were chosen such that the origin (0,0, 0) was the centre of curvature of the concentric rings and semi-rings for each case except CKAlO. For the latter, in order to avoid the inconvenience of an origin not located within the applicator area, it was assumed that the centre of curvature of the semi-rings is as shown, and calculations were based on a mean radius for each semi-ring. The exposure was calculated at points in the X-Y plane, this k i n g the plane of highest exposure, and therefore of prime importance to treatment planning. The exposure at any point in this plane is the sum of contributions from the rings, semi-rings, arcs and point sources that form the distribution of ‘ T o in each applicator. An activity for each source was calculated from data given in the Radiochemical Centre Catalogue (Radiopharmaceuticals and Clinical Radiation Sources, 1971) on the basis of its fraction of the total D°Coarea of the applicator. The total activity of each applicator was considered to be the “design activity”, this being the activity necessary to produce 4000 R in 6 days at “design depth”. In this paper the “design activity” is considered as fixed for each applicator, since this is the activity supplied initially by the Radiochemical Centre*. PRESENTADDRESS: Bone Dosimetry Research, Dept. of Medical Physics, Cookridge Hospital, Leeds, LS 16 6QB.
145
A. H. BEDDOE
TABLEI COMPARISON OF CALCULATED
Applicator
- -_-
DESIGN DEPTHS
Innes (1962)
CKAll
THOSE GIVENIN
Design Depth (mm) Casebow
Design Activity (equivalent mCi
5.7 7.1 8.1 10.5 4.7 5.8 9.3 7.8 8.5 10.6 11.3
0.87 1.38 1.84 2.90 0.68 1.oo 2.36 1.80 2.05 2.50 3.15
5.8 7.4 8.65 11.3 5.0 6.15 10.0 8.4 9.1 11.3 12.0
4.x 6.5 8. I 11.5 4.0 5.25 9.35 7.9 8.75 11.5 12.4
THE LITERATURE
Preser.t Work
(1971)
- - - -
CKAI CK A 2 CK A 3 CKA4 CK A 5 CKA6 CK A 7 CKAR CKAY CK A I O
WITH
e'%o)
IZ
CKA.2
CKA.1
CKA.3
CKA.4
GlIZ
x~
CKA 6
CKA 5
v
CKA.7
CKA.8
CKA.9
CKA.10
CKA.11
FIGURE1-The eleven T o ophthalmic applicators showing the position of the axes with respect to each applicator. The dimensions of the applicators may be obtained from tha Radiochemical Centre catalogue, "Radiopharmaceuticals and Clinical Radiation Sources ( 1971)".
146
Australasian Radiology, Vol. XIX, N o . 2. lurre, 1975
ISOEXPOSURE CURVES FOR
6oc0 OPHTHALMIC APPLICATORS
I\ a
cc
in the text. (A) Projection on X-Z plane. (B) Projection on Y-z plane.
(a)
The following summarizes the method of calculation of the total exposure at any point (x, y) from the four possible types of contributing sources that form the 6oCodistribution in each applicator. The inverse square contribution was calculated first, since this is the major factor in determining exposure. r is the specific y-ray constant (taken to be 13.2 R/mCi/hr, as given in the Radiochemical Manual, 1962), T is the “design exposure” time (144 hours) and Qi is the activity of a contributory source, such that: Q = B Qi = design activity of the applicator.
C . Semi-Ring Source Contribution The symmetry of a semi-ring source about the X-axis allows expression of the exposure at (x, y) by a simple integral of the form:
Di (x, y) = d6 [(x
+ a cos8)2 + (y-pl2 + a* sin’ 61
which when integrated reduces to:
i
Other symbols are indicated in Figure 2. The total exposure at any point (x, y) from a given applicator is given by: D (x,Y) = BDi (x,y) i
A . Point Source Contribution
r T Q, Di (x, y) =
+
+
(x-d)’ (y-p)’ P B . Ring Source Contribution r T Q, Di (x,Y) = [(xz [y--pl2 a2)2-4 a2x2]i which is the well-known relationship (for example, Loevinger, Japha and Brownell, 1956).
+
Australasian Radiology, Vol.
+
XIX,No. 2, June, 1975
I$[
4FTQt
Di (x, Y) =
tan-’ R* (y-p)’ a2 -2 a xli
?r R,
where R, = [xz andR, = [x2
+
+ (y-p)*+
+
a2
+ 2axfl
D. Arc Source Contribution The “stroke” of the “D’source is an arc of a circle radius r mm, where r is the radius of the eye (in this work taken as 11 mm, being the radius of the infant eye) plus 0.5 mm platinum sheath. This arc is in a plane parallel to the Y-Z plane, and its symmetry about the X-axis again allows the expression of exposure at a point (x, y) by a simple integral of the form: 147
A. H. BEDDOE
which when integrated reduces to:
2rTQi
D,f x , y ) = R,R,tan whereR,{ = [ ( x - d ) ' + and R, = [(x-df'
y2
+ y']i
[
C
( ,
r' - c 2 )
+ 4a'-
&
]
4ayli
The above expressions were then modified for tissue absorption and the absorption OF the platinum sheath. For the latter, 5.5 % was subtracted, representing the reduction in transmission for 0.5 mm platinum. In this work n o correction was made €or oblique radiation paths through the platinum, nor €or edge effects due to various platinum thicknesses. Thus the resultant isoexposure curves are subject to a greater errur at the field edges than along the Y-axis. However, extreme oblique radiation paths, including triple-thickness paths, were not considered in the calculations, since knowledge of exposure in such areas is in general not of importance to the treatment of ocular neoplasms. In the calculations the source distributions were considered to be line distributions, an approximation which is reasonable over the area of interest, but which decreases in accuracy as one approaches the applicator.
ISOEXPOSURE CURVES A PDP-8/1 Computer was used to calculate exposure at points (x, y) on a 1 x 1 mm matrix. Programs were written in FOCAL. Graphs were then constructed enabling isoexposure curves to be plotted as shown in Figure 3. The designations, CKAl to CKAl1, aro the ones used by the Radiochemical Centre, Amersham, in their catalogue (mentioned above). The isoexposure curves are based on a 6-day treatment time using the design activity for each applicator. 148
tan
DISCGSSION It is of interest to compare the design depths as calculated and measured from Figure 3 wit! those given in the literature. In the former cast design depths were measured by subtracting 0.2 mm from the Y-axis value €or the 4000 R isoexposure curve, thus giving the depth belom the sclera of the 4000 R curve along the defined Y-axis. The values for design depth are given alongside those of Innes (1962) and Casebow (1971). In general there is much closer agreement between the values of Casebow and those of the author. However, it is apparent that the discrepancy between Casebow's values and those presented in this work increases as a function of increasing design depth. This is because the second order effect of tissue absorption is included in this work. However, it is unlikely that such differences would have clinical significance. ABSTRACT A method of calculation of exposure in the plane of interest is presented for a commonly used set of O0Co ophthalmic applicators. From computer calculated data, sets of isoexposure curves are presented and discussed. The design depths obtained from these curves are compared with corresponding design depths available in the literature. ACKNOWLEDGEMENTS I wish to thank Dr. A. J. Campbell, Director of Radiotherapy, for permission to publish this paper, the Radiotherapy and Photographic Department staffs for valuable assistance given, and Mr. J. J. Tait, Senior Physicist, for his valuable help and encouragement. Australasian Radiology, Vol. XIX, No. 2, June, 1975
ISOEXPOSURE CURVES FOR ""CoOPHTHALMIC APPLICATORS
2
2
4
4
6
6
P
8
10
10
12
12
14
14
16
16
18
18
20
20
CKA3
CKAl FIGURE3 4 t s of isoexposure curves for CKAl to CKAl I . Curves are numbered in kilo-roentgens and represent the design exposure. The scale in each case 1s in mm.
2
2
4
4
6
6
8
8
10
10
12
12
14
14
16
16
16
16
20
20
CKA2 Australasian Radiology, Yol. XIX,No. 2, June, 1975
CKA4 149
A. H. BEDDOE
2
2
4
4
6
6
8
8
10
10
12 14
12
16
16
14
18
18
20
20
CKA 7
CKAS
2 4
6 8 10 12
14 16 18
20
CKA6 150
CKA8 Australasian Radiology, Vol. X I X , No. 2, June, 1975
ISOEXPOSURECURVES FOR ‘j0CoOPHTHALMIC APPLICATORS
2
2
4
4
6
6
8
8
10
10
12
12
14
14
16
16
18
18
20
20
CKAll
CKA9
REFERENCES Casebow, M. P. (1971): Brit. 1. Radiol.. 44, 618-624. Innes, G. (1962): “Proceedings of Symposium on Ocular and Adnexal Tumours.” (C. I.: Mosby Conipany, Missouri, U.S.A.). Loevinger, R., Japha, E. M., and Brownell, G. L. 2 (1956): Radiation Dosimetry, Eds. Hine G. I., and 4 Brownell, G. L., Chapter 16 (Academic Press, New York). 6 Magnus, L. (1967): Strahlentherapie, 132, 379-386. 8 10 Magnus, L., Gobbeler, T., and Strotges, W. (1968): Strahlentherapie. 136, 170-177. 12 Stallard, H. B. (1962): Trans. Ophthal. SOC.U.K.. 82, 14 473. 16 The Radiochemical Centre Catalogue (1971): Radiopharmaceuticals and Radiation Sources. 18 The Radiochemical Centre Handbook (1962): The 20 Radiochemical Manual, Part 1, Physical Data.
Australasian Radiology, Vol. XIX,No. 2, June, 1975
151
Isoexposure Curves for @CoOphthalmic Applicators A. H. BEDDOE Department of Radiotherapy, Christchurch Hospital, Christchurch, New Zealand. INTRODUCTION Various methods exist for the radiotherapeutic treatment of retinoblastoma and other ocular neoplasms. Of these, perhaps the most commonly utilized is some form of radioactive applicator containing a radioisotope such as 2P2Rn, 2-bRa, ..>. 9oSror “OCo. A set of such applitors, manufactured by the Radiochemical Centre, Amersham, to a design developed at St. Bartholomew’s Hospital, London, has been described by Stallard (1962) and Innes ( 1962). These applicators consist of platinum discs containing O0Coin circular, “D’and crescentic shapes, and have two fixing lugs by which they can be sutured to the eye. They are described in the Radiochemical Centre Catalogue (Radiopharmaceuticals and Clinical Radiation Sources, 1971). In the initial work by Innes (1962), values for “design depth” were calculated for each applicator, the “design depth” being the depth of the neoplasm below the sclera. The values appear in Table 1. Further work by Magnus (1967) and Magnus, Gobbeler and Strotges ( 1968) indicated methods of determining exposure at depths along the central axis other than the “design depth.” Recently, Casebow ( 1971 ) has presented a method of calculating exposure at points in two planes, assuming inverse square law attenuation with no tissue absorption. He also presents values for “design depth” based on his calculations, these values *Since this paper was submitted for publication in October, 1971, The Radiochemical Centre, Amersham, has recalculated “design activities.” The “design activities” now supplied will of course produce the same relative exposure distributions as shown in Figure 3, but the treatment time necessary to achieve a given set of actual isoexposure curves will of course no longer be 144 hours. Treatment times in any case need adjusting because of source decay, SO that an exposure time, t, for each applicator is give3 by : “design activity” T t = actual activity where the “design activity” is taken from Table 1. Australasia7 Radiology, Vol. XIX, No. 2, June, 1975
also appearing in Table 1. This paper indicates another method of computation of exposure contributions fom the various distributions of G°Coin each applicator, including the effects of tissue absorption, and presents sets of isoexposure curves constructed from computer calculated data. METHOD OF CALCULATION For computation purposes, a set of Cartesian co-ordinate axes (X, Y, Z) was defined for each applicator, and these are shown in Figure 1. Co-ordinates were chosen such that the origin (0,0, 0) was the centre of curvature of the concentric rings and semi-rings for each case except CKAlO. For the latter, in order to avoid the inconvenience of an origin not located within the applicator area, it was assumed that the centre of curvature of the semi-rings is as shown, and calculations were based on a mean radius for each semi-ring. The exposure was calculated at points in the X-Y plane, this k i n g the plane of highest exposure, and therefore of prime importance to treatment planning. The exposure at any point in this plane is the sum of contributions from the rings, semi-rings, arcs and point sources that form the distribution of ‘ T o in each applicator. An activity for each source was calculated from data given in the Radiochemical Centre Catalogue (Radiopharmaceuticals and Clinical Radiation Sources, 1971) on the basis of its fraction of the total D°Coarea of the applicator. The total activity of each applicator was considered to be the “design activity”, this being the activity necessary to produce 4000 R in 6 days at “design depth”. In this paper the “design activity” is considered as fixed for each applicator, since this is the activity supplied initially by the Radiochemical Centre*. PRESENTADDRESS: Bone Dosimetry Research, Dept. of Medical Physics, Cookridge Hospital, Leeds, LS 16 6QB.
145
A. H. BEDDOE
TABLEI COMPARISON OF CALCULATED
Applicator
- -_-
DESIGN DEPTHS
Innes (1962)
CKAll
THOSE GIVENIN
Design Depth (mm) Casebow
Design Activity (equivalent mCi
5.7 7.1 8.1 10.5 4.7 5.8 9.3 7.8 8.5 10.6 11.3
0.87 1.38 1.84 2.90 0.68 1.oo 2.36 1.80 2.05 2.50 3.15
5.8 7.4 8.65 11.3 5.0 6.15 10.0 8.4 9.1 11.3 12.0
4.x 6.5 8. I 11.5 4.0 5.25 9.35 7.9 8.75 11.5 12.4
THE LITERATURE
Preser.t Work
(1971)
- - - -
CKAI CK A 2 CK A 3 CKA4 CK A 5 CKA6 CK A 7 CKAR CKAY CK A I O
WITH
e'%o)
IZ
CKA.2
CKA.1
CKA.3
CKA.4
GlIZ
x~
CKA 6
CKA 5
v
CKA.7
CKA.8
CKA.9
CKA.10
CKA.11
FIGURE1-The eleven T o ophthalmic applicators showing the position of the axes with respect to each applicator. The dimensions of the applicators may be obtained from tha Radiochemical Centre catalogue, "Radiopharmaceuticals and Clinical Radiation Sources ( 1971)".
146
Australasian Radiology, Vol. XIX, N o . 2. lurre, 1975
ISOEXPOSURE CURVES FOR
6oc0 OPHTHALMIC APPLICATORS
I\ a
cc
in the text. (A) Projection on X-Z plane. (B) Projection on Y-z plane.
(a)
The following summarizes the method of calculation of the total exposure at any point (x, y) from the four possible types of contributing sources that form the 6oCodistribution in each applicator. The inverse square contribution was calculated first, since this is the major factor in determining exposure. r is the specific y-ray constant (taken to be 13.2 R/mCi/hr, as given in the Radiochemical Manual, 1962), T is the “design exposure” time (144 hours) and Qi is the activity of a contributory source, such that: Q = B Qi = design activity of the applicator.
C . Semi-Ring Source Contribution The symmetry of a semi-ring source about the X-axis allows expression of the exposure at (x, y) by a simple integral of the form:
Di (x, y) = d6 [(x
+ a cos8)2 + (y-pl2 + a* sin’ 61
which when integrated reduces to:
i
Other symbols are indicated in Figure 2. The total exposure at any point (x, y) from a given applicator is given by: D (x,Y) = BDi (x,y) i
A . Point Source Contribution
r T Q, Di (x, y) =
+
+
(x-d)’ (y-p)’ P B . Ring Source Contribution r T Q, Di (x,Y) = [(xz [y--pl2 a2)2-4 a2x2]i which is the well-known relationship (for example, Loevinger, Japha and Brownell, 1956).
+
Australasian Radiology, Vol.
+
XIX,No. 2, June, 1975
I$[
4FTQt
Di (x, Y) =
tan-’ R* (y-p)’ a2 -2 a xli
?r R,
where R, = [xz andR, = [x2
+
+ (y-p)*+
+
a2
+ 2axfl
D. Arc Source Contribution The “stroke” of the “D’source is an arc of a circle radius r mm, where r is the radius of the eye (in this work taken as 11 mm, being the radius of the infant eye) plus 0.5 mm platinum sheath. This arc is in a plane parallel to the Y-Z plane, and its symmetry about the X-axis again allows the expression of exposure at a point (x, y) by a simple integral of the form: 147
A. H. BEDDOE
which when integrated reduces to:
2rTQi
D,f x , y ) = R,R,tan whereR,{ = [ ( x - d ) ' + and R, = [(x-df'
y2
+ y']i
[
C
( ,
r' - c 2 )
+ 4a'-
&
]
4ayli
The above expressions were then modified for tissue absorption and the absorption OF the platinum sheath. For the latter, 5.5 % was subtracted, representing the reduction in transmission for 0.5 mm platinum. In this work n o correction was made €or oblique radiation paths through the platinum, nor €or edge effects due to various platinum thicknesses. Thus the resultant isoexposure curves are subject to a greater errur at the field edges than along the Y-axis. However, extreme oblique radiation paths, including triple-thickness paths, were not considered in the calculations, since knowledge of exposure in such areas is in general not of importance to the treatment of ocular neoplasms. In the calculations the source distributions were considered to be line distributions, an approximation which is reasonable over the area of interest, but which decreases in accuracy as one approaches the applicator.
ISOEXPOSURE CURVES A PDP-8/1 Computer was used to calculate exposure at points (x, y) on a 1 x 1 mm matrix. Programs were written in FOCAL. Graphs were then constructed enabling isoexposure curves to be plotted as shown in Figure 3. The designations, CKAl to CKAl1, aro the ones used by the Radiochemical Centre, Amersham, in their catalogue (mentioned above). The isoexposure curves are based on a 6-day treatment time using the design activity for each applicator. 148
tan
DISCGSSION It is of interest to compare the design depths as calculated and measured from Figure 3 wit! those given in the literature. In the former cast design depths were measured by subtracting 0.2 mm from the Y-axis value €or the 4000 R isoexposure curve, thus giving the depth belom the sclera of the 4000 R curve along the defined Y-axis. The values for design depth are given alongside those of Innes (1962) and Casebow (1971). In general there is much closer agreement between the values of Casebow and those of the author. However, it is apparent that the discrepancy between Casebow's values and those presented in this work increases as a function of increasing design depth. This is because the second order effect of tissue absorption is included in this work. However, it is unlikely that such differences would have clinical significance. ABSTRACT A method of calculation of exposure in the plane of interest is presented for a commonly used set of O0Co ophthalmic applicators. From computer calculated data, sets of isoexposure curves are presented and discussed. The design depths obtained from these curves are compared with corresponding design depths available in the literature. ACKNOWLEDGEMENTS I wish to thank Dr. A. J. Campbell, Director of Radiotherapy, for permission to publish this paper, the Radiotherapy and Photographic Department staffs for valuable assistance given, and Mr. J. J. Tait, Senior Physicist, for his valuable help and encouragement. Australasian Radiology, Vol. XIX, No. 2, June, 1975
ISOEXPOSURE CURVES FOR ""CoOPHTHALMIC APPLICATORS
2
2
4
4
6
6
P
8
10
10
12
12
14
14
16
16
18
18
20
20
CKA3
CKAl FIGURE3 4 t s of isoexposure curves for CKAl to CKAl I . Curves are numbered in kilo-roentgens and represent the design exposure. The scale in each case 1s in mm.
2
2
4
4
6
6
8
8
10
10
12
12
14
14
16
16
16
16
20
20
CKA2 Australasian Radiology, Yol. XIX,No. 2, June, 1975
CKA4 149
A. H. BEDDOE
2
2
4
4
6
6
8
8
10
10
12 14
12
16
16
14
18
18
20
20
CKA 7
CKAS
2 4
6 8 10 12
14 16 18
20
CKA6 150
CKA8 Australasian Radiology, Vol. X I X , No. 2, June, 1975
ISOEXPOSURECURVES FOR ‘j0CoOPHTHALMIC APPLICATORS
2
2
4
4
6
6
8
8
10
10
12
12
14
14
16
16
18
18
20
20
CKAll
CKA9
REFERENCES Casebow, M. P. (1971): Brit. 1. Radiol.. 44, 618-624. Innes, G. (1962): “Proceedings of Symposium on Ocular and Adnexal Tumours.” (C. I.: Mosby Conipany, Missouri, U.S.A.). Loevinger, R., Japha, E. M., and Brownell, G. L. 2 (1956): Radiation Dosimetry, Eds. Hine G. I., and 4 Brownell, G. L., Chapter 16 (Academic Press, New York). 6 Magnus, L. (1967): Strahlentherapie, 132, 379-386. 8 10 Magnus, L., Gobbeler, T., and Strotges, W. (1968): Strahlentherapie. 136, 170-177. 12 Stallard, H. B. (1962): Trans. Ophthal. SOC.U.K.. 82, 14 473. 16 The Radiochemical Centre Catalogue (1971): Radiopharmaceuticals and Radiation Sources. 18 The Radiochemical Centre Handbook (1962): The 20 Radiochemical Manual, Part 1, Physical Data.
Australasian Radiology, Vol. XIX,No. 2, June, 1975
151
