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The “Ring” method: A semi-empirical method to calculate dose distributions of irregularly shaped photon beams
Radiotherapyand Oncology, 7 (1986) 363-369 Elsevier
363
RTO 00288
Review Article
The "Ring" method: A semi-empirical method to calculate dose distributions of irregularly shaped photon beams Frits W. Jacobs and H e r m a n J. van Kleffens* Rotterdam Radiotherapeutic Institute~Dr. Dan&l den Hoed Cancer Centre, Groene Hilledijk 301, 3075 EA Rotterdam, The Netherlands
(Received 13 March 1986, revisionreceived4 July 1986, accepted26 August 1986)
Key words:
Irregularfield;Dose calculation;Scatteredradiation; Radiotherapytreatmentplanning
Summary The "Ring" method provides a fast dose calculation and isodose presentation for photon beams with blocks. The method takes into account the change in scatter due to the blocks at each calculation point. Firstly, the dose in a point is calculated assuming that no blocks are present. Secondly, the scatter reduction caused by the blocks is calculated and subtracted. To determine the scatter reduction the irradiated surface is divided in concentric rings around a point at the surface at the intersection with a ray line between focus and calculation point. The scatter reduction caused by blocks for each ring is calculated. The effect of scatter for rings with an outer radius > 15 cm where the scatter contribution is less than 1.0% is neglected. Results of the method for 4 MV photons using eight rings are presented. Comparison of dose measurements with calculations in an arrow-shaped photon field showed maximum deviations of 4.0%, using the IRREG program of Cunningham [5], 6.5% using the BLKINP program of Schlegel [7], which is based on Clarkson's method [3], 5.0% using the method of Wrede [18] and 2.2% using the "Ring" method. Contrary to the first two calculation programs, the programs using the last two calculation methods provide isodose lines dose values at points.
Introduction The use of computed tomography has been an important step forward in assessing anatomical data of the patient. One of the results of the improved tomography is a better description of target volFrits W. Jacobs, Gestelseweg5, 5296 KP Esch, The Netherlands.
Present address:
* To whom reprint requests shouM be addressed.
umes for radiotherapy treatment planning purposes. At the same time, the lack of algorithms which calculate dose distributions covering these more accurately described target volumes satisfactory, is experienced daily. An example is the need for accurate and fast calculations of dose distributions for irregularly shaped photon beams. Important steps have already been set by Cunningham et al. [5] by means of the I R R E G program, calculating doses in specified points for flat surfaces, by Quast
0167-8140/86/$03.50 9 1986ElsevierSciencePublishers B.V. (BiomedicalDivision)
364 et al. [13] with their weighted beam zone method, providing a possibility to estimate the dose with " h a n d " calculation, and by others [14,6-8,18]. These steps, however, have not yet led to a general available method:
n
Dscat (x,y,d)
The "Ring" method which has been implemented on an array processor, is intended to meet these criteria.
Dscat.i (x,y,a).
E i
=
(2)
1
The total dose at point Q is
D(x,y,d) - providing a complete isodose calculation instead of calculations in a number of points - t a k i n g into account the real anatomy of the patient - taking into account the effect of the blocks on scattered radiation - giving a proper calculation in the shadow of the blocks as has been demonstrated in dosimetric studies [7,10-16] - having an acceptable calculation time.
----
Oprim(x,y,d)
=
+ D,c,t
(x,y,d).
(3)
If no shielding is applied:
D(x,y,d)
=
Dopen(x,y,d).
(4)
In situations with shielding, the scatter contribution to the dose in point Q for ring i is proportional to the irradiated area of the ring, assuming that the annular width is such that the scatter contribution from each point in the ring to point Q is the same. Similarly, the shielded part Pi (x,y) of the i-th ring is responsible for a proportional loss o f scatter at point Q. The dose in point Q becomes:
D(x,y,d)
=
Dopen (x,y,d)
(5)
n
Theory
-
~, Pi (x,y) Dscat,i (x,y,d).
-
i
A point Q(x,y,d) is situated at an arbitrary depth d in a water phantom which is irradiated with an irregularly shaped photon beam. Let us consider the projection of the shielding blocks on a plane passing through point Q' (the intersection of the rayline through Q with the surface of the phantom) and perpendicular to the central axis of the irradiating beam. Consider in this plane n concentric rings around Q' (see Fig. 1). In our situation where the method has been tested for 4 MV photons, eight rings are used with a width of 2 cm at the surface. The relation between the outer radius, ri, of ring i and the depth d is given by the formula:
ri =
SSD + d SSD
=
1
/I t- -
_
. . . . . . . .
/
SSD
DQ = Dopen- i=l PiDscat,i
- •
9 (2i -
1) cm
1 ~< i ~< n.
(1)
The total scattered radiation at point Q is assumed to be the sum of the scattered radiation arriving from the volume determined by the projection of each ring
876543
Fig. 1, The "Ring" method.
3456
Y
36: The "ring scatter" Dscat,i is energy and machine-dependent and can be determined empirically (see next paragraph). If point Q is situated in the shadow of the blocks also the primary radiation has to be subtracted: n
D(x,y,d) = Dope. (x,y,d) -
~. i
=
P, (x,y) D~r 1
(x,y,d) - Dprim (x,y,d) 9 (1 - T).
(6)
Oprim(x,y,d)
has to be corrected for the block transmission T. Scattered radiation arriving from transmitted primary radiation (through blocks) is neglected. The " R i n g " method starts with the dose in the open field situation and applies a correction factor f to correct for the shielded situation: D(x,y,d) f(x,y,d)
-
Dopen ( x , y , d )
(7)
"
Determination of the ring scatter
Dscat,i
In order to determine the effect of shielding blocks on the absorbed dose, an experiment has been performed using a 40 x 40 x 40 cm 3 water phantom, a Wellhofer 0.15 cm 3 ionization chamber, a Keithley electrometer (type 602) and a Varian 4 MV linear accelerator. The experiment was started with the ionization chamber in the center of the water phantom. The ionization chamber remained fixed at the position of the isocenter of the linear accelerator. During the experiment, the water phantom was shifted in steps of 0.2 cm out of the beam (field size 30 x 30 cm 2 at the isocenter) until the ionization chamber reaches a position 1 cm from the edge of the phantom. This was repeated for several depths: (1 cm, i.e. normalization depth, 10, 20 and 30 cm). F r o m the results of this experiment, it was concluded that the influence on the dose from the shielding blocks outside a radius of 15 cm (projected at surface level) is less than 1.0% of the maximum dose on the central axis for 4 MV X-rays. The circular area around the point of calculation is divided in eight rings. The smallest "ring" is a
circle with 1 cm radius, the other rings have an annular width of 2 cm at the surface. F r o m the experiment described above it was derived that the scatter contributions of each unshielded ring is les,~ than 2.0% of the maximum dose on the central axis The values of the ring scatter Dscat,i are determinec from this experiment by moving the water phantorr stepwise out of the beam. In this sequence the measured dose value decreases because the scatter con tribution decreases. At the start position the scatter contribution from all rings is present. First the scatter contribution from the largest ring (no. 8) will decrease. The percentage P of the "shielded" area of ring 8 relative to its total area is calculated frorr the intersection of the ring and the edge o f the water phantom. From the measurements and Eqn. (6), a value of Dscat,8 at the measured depth can be obtained. If the water phantom is moved further out, the scatter contribution o f the seventh ring will alsc diminish together with a greater amount o f the 8th ring and, accordingly, a value for D~,t,7 is derived using P7 (x,y), P8 (x,y) and the value o f D~at,8. The step size of 0.2 cm provides 10 measurements to derive a value for each D~ata. The measurements have to be made for several depths as the scattered radiation changes with depth. At depths'where nc measurements have been made, inter- and extrapolation using data at the measured depths is performed. A more directmethod to determine D~r is to measure the dose when the whole field i: blocked except the ring for which the scatter contribution has to be determined. This implies, however, some practical shielding problems. A comparison between Dprim, Dscat value. and Tissue Air Ratios (TAR) will be the subject ot another paper. The values of Dprim are found by considering a point, which is totally shielded by blocks. Subtraction of the total scatter contribution from the dose in the open field Dopen gives Oprim (see Eqns. 3 and
4): Dprim (x,y,d) = Dopen (x,y,d) 8
-
~ i
=
D~r 1
(x,y,d).
(8)
366 It should be noted that the dose under a block can be expressed, using Eqns. (6) and (8) as:
A
8
D(x,y,d)
:
2 i = i
(1
-
Pi (x,y))
Dscat,i
(x,y,d)
(9) + T . Dprim
(x,y,d).
Methods and results
The " R i n g " method has been implemented in the Siemens therapy planning system SIDOS using an array processor. The time required to calculate a complete isodose pattern is about 5 s. The dose corrections for irregular fields have also been performed according to Wrede's method [I 8], which is normally applied in the SIDOS system. The " R i n g " method has been tested in two situations, namely an arrow-shaped field simulating a mantle field (see Fig. 2) and a half-shielded field (obtained by shielding h a l f a 20 x 2 0 c m 2 field to a 10 x 2 0 c m 2 field, asymmetric with respect to the central axis). Furthermore, the I R R E G program (Cunningham [5]) using a TP 11 system o f A E C L and the B L K I N P program again on the SIDOS system (Clarkson's method [6]) have been applied. Measurements are made with photon radiation beams of a 4 MV therapy unit (Varian, Clinac), using a 50 x 50 x 50 cm 3 water phantom and 0.15 cm 3 ionization chambers (Wellhofer).
20 cm
32 cm
Fig. 2. Arrow-shaped photon field.
Fig. 3. Isodose presentation in the sagittal plane AB o f the arrow-shaped photon field calculated by means o f the "Ring" method. - - - , 32 x 20 cm 2 rectangular field; - - - , arrowshaped field.
The arrow-shapedphoton field A photon field (4 MV) o f 3 2 • 2 0 c m 2 at 8 0 c m SSD is transformed into an arrow-shaped field by the use of four blocks (see Fig. 2). Figure 3 shows the change in the isodose pattern in the sagittal plane AB as a result of this transformation calculated by means of the " R i n g " method. Both isodose patterns are normalized to the maximum dose of the non-blocked field. The maximum dose of the mantle field reaches a value of 97.1% of the maximum dose of the non-blocked field. Because the left side of the field is more shielded, the dose values at that side are smaller. A comparison between dose measurements and calculations for 6 points in this arrow field has been presented elsewhere by van Kleffens et al. [17]. In that work the calculations were done with the I R R E G program (using 4 MV T A R - S A R tables and profile data for our 4 MV linear accelerator as required in the TP 11 manual), the SIDOS program (using Wrede's method for irregular fields) and the B L K I N P program. For the same 6 points, calculations were performed using the "Ring" method and compared with the previous data (see Table I). The " R i n g " method performs better than Wrede's method and at least as good as the I R R E G or B L K I N P program. The rather large deviations of the B L K I N P program in points 5 and 6 are caused by the fact that at the time of the comparison the beam profile could not yet be taken into account properly. The deviations
367 TABLE I Differences between total dose calculation and measurement (in percent relative to the maximum dose on the central axis). 4 MV (IRREG) Depth (cm)
2 5 10 15 20
Point no. 1
2
3
4
5
6
+0.3 +0.7 +3.4 +2.4 +2.3
--0.3 0 +1.0 +0.8 +0.8
+0.6 +0.5 +0.8 +0.5 +0.7
-0.6 --0.2 +0.2 +1.0 +0.7
+0.1 +2.7 +2.6 +2.0 +2.2
+4.0 +2.3 +1.7 +1.6 +0.9
4 MV (BLKINP) Depth (cm)
2 5 I0 15 20
Point no. I
2
3
4
5
6
-1.4 +0.6 +2.0 +2.2 +2.1
+0.3 0 +2.0 +0.6 +0.7
+0.8 +0.5 +0.6 +0.9 +0.8
+0.2 +0.3 +0.6 +I.1 +1.3
--3.0 --0.6 +0.6 +1.1 +1.0
+6.5 +4.2 +3.3 +2.9 +2.8
1
2
3
4
5
6
-1.5 +1.5 0 0 0
+I.0 +1.0 0 0 0
+1.5 +1.5 0 --I.0 --0.5
-0.5 -1.0 -3.0 -2.0 --2.0
-1.5 +2.5 -0.5 -1.0 --1.0
+4.0 +5.0 +3.0 +3.0 +1.0
1
2
3
4
5
6
-0.7 0 -0.7 --0.4 --0.6
+0.5 +0.6 --0.1 +0.3 --0.5
+0.2 +0.8 +0.9 +0.3 -0.1
-0.2 -0.2 +0.2 -0.5 -0.7
-2.2 +1.5 +0.3 --0.2 -0.5
+2.1 +1.8 +0~5 -0.2 -0.7
4 MV (Wrede) Depth (cm)
2 5 10 15 20
Point no.
4 MV (Ring) Depth (cm)
2 5 10 15 20
Point no.
Fig. 4. Isodose lines of a half blocked square (20 x 20 c m 2) photon field calculated by means of the "Ring" method ........ Measured (half blocked); - - , calculated (half blocked); - -, calculated (unblocked); +, *, the maximum dose of the unblocked and blocked field, respectively. o f the " R i n g " m e t h o d at a d e p t h o f 2 c m in p o i n t s 5 a n d 6 ( a p p r o x i m a t e l y 2 . 0 % ) are p r o b a b l y caused by inaccuracies in the c a l c u l a t i o n o f the fractions o f the smallest two rings which are blocked.
The half-shielded field T h e water p h a n t o m is exposed to a p h o t o n field (4 MV) o f 20 x 20 cm 2 at 80 cm SSD, o f which one half is blocked. T h e b r o k e n lines in Fig. 4 show the isodose lines, o f the n o n - b l o c k e d field, n o r m a l i z e d to the m a x i m u m dose. The solid lines show the isodose lines of the h a l f - b l o c k e d fields n o r m a l i z e d to the m a x i m u m dose o f the n o n - b l o c k e d field. T h e little asterisk designates the m a x i m u m dose o f the half-field, which is 99.2% o f the m a x i m u m dose o f the n o n - b l o c k e d field (large asterisk). T h e d o t t e d lines represent the m e a s u r e d isodose lines. It c a n be seen, that the isodose lines o f the half-field app r o a c h those o f the n o n - b l o c k e d field in regions at a large distance f r o m the block, because scatter changes, as a result o f the presence of the block, are negligible there. I n the vicinity o f the b l o c k e d p a r t o f the field, the differences b e t w e e n the isodose lines o f the half-field a n d those o f the n o n - b l o c k e d field increase, because the absence o f scatter f r o m the right half o f the field becomes m o r e a n d m o r e imp o r t a n t . T h e difference b e t w e e n b l o c k e d a n d u n blocked field reaches a value o f a b o u t - 7 . 0 % o f the m a x i m u m dose o n the central axis of the n o n -
368 blocked field, if the p e n u m b r a is excluded. Measured and calculated isodose lines in the blocked situation agree rather well except in the region within 3 cm from the edge of the block. This might be caused either by the inaccuracy in the calculation of the scatter contribution of the smallest two rings or by an inaccurate algorithm which describes the p e n u m b r a region (the standard algorithm in the SIDOS system is used). This will be a subject of a further investigation. The difference however in the vicinity of the blocks between the isodose lines in the blocked and non-blocked situation is quite remarkable.
Discussion and conclusion It can be concluded that a convenient method providing isodose lines has been derived which brings into account scatter changes due to the application of shielding blocks. The calculation time for an isodose pattern with the " R i n g " method using an array processor is about 5 s. It should be noted that an array processor speeds up calculations whatever method is used. The accuracy seems to be at least as good as achieved with the above mentioned programs (see Table I). The power of the method lies in the fact that the unshielded dose distribution (taking into account the CT Hounsfield values on the line from the focus to the calculation point) is used to obtain the shielded situation by bringing into account scatter changes due to the shielding. This approach distinguishes the method from the method of Clarkson where all scatter contributions in each point have to be summed to obtain the total dose. The method does not yet take into account the effect of inhomogeneities on the scattered radiation. A problem with the " R i n g " method as well as with the T A R method, is that the scatter data are determined for a homogeneous water phantom. Perhaps either additional scatter data for lung tissue and bone structures or the application of a scaling method [12] will provide acceptable accuracy. The step to real three-dimensional calculation seems to be less complicated: in principle the calculation of
the scatter contribution has already been three-dimensional. A considerable extension is to bring into account the different patient a n a t o m y outside the plane of calculation. A combination of the method of Clarkson and the " R i n g " method might be a good approach. The method can be easily applied to other energies. The number of rings and the annular width can be adapted to obtain the required accuracy.
Acknowledgements This study was supported by a grant f r o m Siemens A.G., Erlangen, West Germany. The authors wish to thank R. Thieme and A. Lutz of Siemens A.G. for their support concerning SIDOS-software. They also wish to thank B. van der Leije, P. Storchi, H. Huizenga, C. Blom-van Otterloo, B. Mijnheer and A. Visser for m a n y helpful discussions and A. van Eijk-van Santen for typing the manuscript.
References 1 Agarwal, S. K., Wakley, J., Scheele, R. V. and Normansell, A. A method of dosimetry for irregularly shaped fields. Int. J. Radiat. Oncol. Biol. Phys. 2:199 203, 1977. 2 Bukovitz, A. G. Computer calculation of dose for irregularly shaped fields for Cobalt-60 and 6 MV photons. Radiology 113: 181-185, 1974. 3 Clarkson, J. R. A note on depth doses in fields of irregular shape. Br. J. Radiat. 14: 265-268, 1941. 4 Cundiff, J. H., Cunningham, J. R., Golden, R., Lanzl, L. H., Meurk, M. L., Ovadia, J., Page Last, V., Pope, R. A., Sampiere, V. A., Saylor, W. L., Shalek, R. J. and Sutharalingham, N. A method for the calculation of dose in the radiation treatment of Hodgkin's disease. Am. J. Roentgenol. 117: 30-44, 1973. 5 Cunningham, J. R., Shrivastava, P. N. and Wilkinson, J. M. Program IRREG-calculation of dose from irregularly shaped radiation beams. Comp. Prog. Biomed. 2: 192-199, 1972. 6 Edwards, F. H. and Coffey, C. W. A new technique for the calculation of scattered radiation from 60 Co teletherapie beams. Radiology 132: 193-196, 1979. 7 Hartmann, G. H., Scharfenberg, H., Schlegel, W. and Thieme, R. Methoden der Bestrahlungsplanungen bei der Therapie rnit irregulaeren Feldern: zwei Verfahren der Dosisberechnung im Vergleich. Elektromedica 3:107-110, 1983.
369 8 Khan, F. M., Levitt, S. H., Moore, V. C. and Jones, T. K. Computer and approximation methods of calculating depth dose in irregularly shaped fields. Radiology 106: 433-436, 1973. 9 Landberg, T., Svahn-Tapper, G. and Wintzell, K. Mantle treatment of Hodgkin's disease. Acta Radiol. Ther. Phys. Biol. 10: 174-186, 1971. 10 Meurk, M. L., Green, J. P,, Nussbaum, H. and Vaeth, J. M. Phantom dosimetry study of shaped Cobalt-60 fields in the treatment of Hodgkin's disease. Radiology 91: 554-558, 1968. I 1 Page, V , Gardner, A. and Karzmark, C. J. Physical and dosimetric aspects of the radiotherapy of malignant lymphomas. I. The mantle technique. Radiology 96: 609-618, 1970. 12 Pruitt, J. S. and Loevinger, R. The photon-fluence scaling theorem for Compton-scattered radiation. Med. Phys. 9: 176-179, 1982. 13 Quast, U. and Glaeser, L, Irregular field dose determination with the weighted beam-zone method. Int. J. Radiat. Oncol. Biol. Phys. 8: 1637-1645, 1982.
14 Simpson, L. D., Holt, J. G., Edelstein, G. R. and D'Angio, G. Radiation dosimetry for large irregularly shaped fields in the treatment of acute lymphocytic leukemia. Radiology 109: 205-207, 1973. 15 Svahn-Tapper, G. Dosimetric studies of mantle fields in Cobalt-60 therapy of malignant lymphomas. Acta Radiol. Ther. Phys. Biol. 9: 190-204, 1970. 16 Svahn-Tapper, G. Mantle treatment. Absorbed dose measurements in patients compared with dose planning. Acta Radiol. Ther. Phys. Biol. 15: 340-356, 1976. 17 Van Kleffens, H. J., Jacobs, F. W., Linden, P. M., van der Venselaar, J. and Thieme, R. Investigation of the accuracy of calculations for irregularly shaped photon beams. Proceedings of the 8th International Conference on the Use of Computers in Radiation Therapy, Toronto, Canada, 9-12 July 1984: 41-44. 18 Wrede, D. E. Central axis tissue-air ratios as a function of area/perimeter at depth and their applicability to irregularly shaped fields. Phys. Med. Biol. 17: 548-554, 1972.
363
RTO 00288
Review Article
The "Ring" method: A semi-empirical method to calculate dose distributions of irregularly shaped photon beams Frits W. Jacobs and H e r m a n J. van Kleffens* Rotterdam Radiotherapeutic Institute~Dr. Dan&l den Hoed Cancer Centre, Groene Hilledijk 301, 3075 EA Rotterdam, The Netherlands
(Received 13 March 1986, revisionreceived4 July 1986, accepted26 August 1986)
Key words:
Irregularfield;Dose calculation;Scatteredradiation; Radiotherapytreatmentplanning
Summary The "Ring" method provides a fast dose calculation and isodose presentation for photon beams with blocks. The method takes into account the change in scatter due to the blocks at each calculation point. Firstly, the dose in a point is calculated assuming that no blocks are present. Secondly, the scatter reduction caused by the blocks is calculated and subtracted. To determine the scatter reduction the irradiated surface is divided in concentric rings around a point at the surface at the intersection with a ray line between focus and calculation point. The scatter reduction caused by blocks for each ring is calculated. The effect of scatter for rings with an outer radius > 15 cm where the scatter contribution is less than 1.0% is neglected. Results of the method for 4 MV photons using eight rings are presented. Comparison of dose measurements with calculations in an arrow-shaped photon field showed maximum deviations of 4.0%, using the IRREG program of Cunningham [5], 6.5% using the BLKINP program of Schlegel [7], which is based on Clarkson's method [3], 5.0% using the method of Wrede [18] and 2.2% using the "Ring" method. Contrary to the first two calculation programs, the programs using the last two calculation methods provide isodose lines dose values at points.
Introduction The use of computed tomography has been an important step forward in assessing anatomical data of the patient. One of the results of the improved tomography is a better description of target volFrits W. Jacobs, Gestelseweg5, 5296 KP Esch, The Netherlands.
Present address:
* To whom reprint requests shouM be addressed.
umes for radiotherapy treatment planning purposes. At the same time, the lack of algorithms which calculate dose distributions covering these more accurately described target volumes satisfactory, is experienced daily. An example is the need for accurate and fast calculations of dose distributions for irregularly shaped photon beams. Important steps have already been set by Cunningham et al. [5] by means of the I R R E G program, calculating doses in specified points for flat surfaces, by Quast
0167-8140/86/$03.50 9 1986ElsevierSciencePublishers B.V. (BiomedicalDivision)
364 et al. [13] with their weighted beam zone method, providing a possibility to estimate the dose with " h a n d " calculation, and by others [14,6-8,18]. These steps, however, have not yet led to a general available method:
n
Dscat (x,y,d)
The "Ring" method which has been implemented on an array processor, is intended to meet these criteria.
Dscat.i (x,y,a).
E i
=
(2)
1
The total dose at point Q is
D(x,y,d) - providing a complete isodose calculation instead of calculations in a number of points - t a k i n g into account the real anatomy of the patient - taking into account the effect of the blocks on scattered radiation - giving a proper calculation in the shadow of the blocks as has been demonstrated in dosimetric studies [7,10-16] - having an acceptable calculation time.
----
Oprim(x,y,d)
=
+ D,c,t
(x,y,d).
(3)
If no shielding is applied:
D(x,y,d)
=
Dopen(x,y,d).
(4)
In situations with shielding, the scatter contribution to the dose in point Q for ring i is proportional to the irradiated area of the ring, assuming that the annular width is such that the scatter contribution from each point in the ring to point Q is the same. Similarly, the shielded part Pi (x,y) of the i-th ring is responsible for a proportional loss o f scatter at point Q. The dose in point Q becomes:
D(x,y,d)
=
Dopen (x,y,d)
(5)
n
Theory
-
~, Pi (x,y) Dscat,i (x,y,d).
-
i
A point Q(x,y,d) is situated at an arbitrary depth d in a water phantom which is irradiated with an irregularly shaped photon beam. Let us consider the projection of the shielding blocks on a plane passing through point Q' (the intersection of the rayline through Q with the surface of the phantom) and perpendicular to the central axis of the irradiating beam. Consider in this plane n concentric rings around Q' (see Fig. 1). In our situation where the method has been tested for 4 MV photons, eight rings are used with a width of 2 cm at the surface. The relation between the outer radius, ri, of ring i and the depth d is given by the formula:
ri =
SSD + d SSD
=
1
/I t- -
_
. . . . . . . .
/
SSD
DQ = Dopen- i=l PiDscat,i
- •
9 (2i -
1) cm
1 ~< i ~< n.
(1)
The total scattered radiation at point Q is assumed to be the sum of the scattered radiation arriving from the volume determined by the projection of each ring
876543
Fig. 1, The "Ring" method.
3456
Y
36: The "ring scatter" Dscat,i is energy and machine-dependent and can be determined empirically (see next paragraph). If point Q is situated in the shadow of the blocks also the primary radiation has to be subtracted: n
D(x,y,d) = Dope. (x,y,d) -
~. i
=
P, (x,y) D~r 1
(x,y,d) - Dprim (x,y,d) 9 (1 - T).
(6)
Oprim(x,y,d)
has to be corrected for the block transmission T. Scattered radiation arriving from transmitted primary radiation (through blocks) is neglected. The " R i n g " method starts with the dose in the open field situation and applies a correction factor f to correct for the shielded situation: D(x,y,d) f(x,y,d)
-
Dopen ( x , y , d )
(7)
"
Determination of the ring scatter
Dscat,i
In order to determine the effect of shielding blocks on the absorbed dose, an experiment has been performed using a 40 x 40 x 40 cm 3 water phantom, a Wellhofer 0.15 cm 3 ionization chamber, a Keithley electrometer (type 602) and a Varian 4 MV linear accelerator. The experiment was started with the ionization chamber in the center of the water phantom. The ionization chamber remained fixed at the position of the isocenter of the linear accelerator. During the experiment, the water phantom was shifted in steps of 0.2 cm out of the beam (field size 30 x 30 cm 2 at the isocenter) until the ionization chamber reaches a position 1 cm from the edge of the phantom. This was repeated for several depths: (1 cm, i.e. normalization depth, 10, 20 and 30 cm). F r o m the results of this experiment, it was concluded that the influence on the dose from the shielding blocks outside a radius of 15 cm (projected at surface level) is less than 1.0% of the maximum dose on the central axis for 4 MV X-rays. The circular area around the point of calculation is divided in eight rings. The smallest "ring" is a
circle with 1 cm radius, the other rings have an annular width of 2 cm at the surface. F r o m the experiment described above it was derived that the scatter contributions of each unshielded ring is les,~ than 2.0% of the maximum dose on the central axis The values of the ring scatter Dscat,i are determinec from this experiment by moving the water phantorr stepwise out of the beam. In this sequence the measured dose value decreases because the scatter con tribution decreases. At the start position the scatter contribution from all rings is present. First the scatter contribution from the largest ring (no. 8) will decrease. The percentage P of the "shielded" area of ring 8 relative to its total area is calculated frorr the intersection of the ring and the edge o f the water phantom. From the measurements and Eqn. (6), a value of Dscat,8 at the measured depth can be obtained. If the water phantom is moved further out, the scatter contribution o f the seventh ring will alsc diminish together with a greater amount o f the 8th ring and, accordingly, a value for D~,t,7 is derived using P7 (x,y), P8 (x,y) and the value o f D~at,8. The step size of 0.2 cm provides 10 measurements to derive a value for each D~ata. The measurements have to be made for several depths as the scattered radiation changes with depth. At depths'where nc measurements have been made, inter- and extrapolation using data at the measured depths is performed. A more directmethod to determine D~r is to measure the dose when the whole field i: blocked except the ring for which the scatter contribution has to be determined. This implies, however, some practical shielding problems. A comparison between Dprim, Dscat value. and Tissue Air Ratios (TAR) will be the subject ot another paper. The values of Dprim are found by considering a point, which is totally shielded by blocks. Subtraction of the total scatter contribution from the dose in the open field Dopen gives Oprim (see Eqns. 3 and
4): Dprim (x,y,d) = Dopen (x,y,d) 8
-
~ i
=
D~r 1
(x,y,d).
(8)
366 It should be noted that the dose under a block can be expressed, using Eqns. (6) and (8) as:
A
8
D(x,y,d)
:
2 i = i
(1
-
Pi (x,y))
Dscat,i
(x,y,d)
(9) + T . Dprim
(x,y,d).
Methods and results
The " R i n g " method has been implemented in the Siemens therapy planning system SIDOS using an array processor. The time required to calculate a complete isodose pattern is about 5 s. The dose corrections for irregular fields have also been performed according to Wrede's method [I 8], which is normally applied in the SIDOS system. The " R i n g " method has been tested in two situations, namely an arrow-shaped field simulating a mantle field (see Fig. 2) and a half-shielded field (obtained by shielding h a l f a 20 x 2 0 c m 2 field to a 10 x 2 0 c m 2 field, asymmetric with respect to the central axis). Furthermore, the I R R E G program (Cunningham [5]) using a TP 11 system o f A E C L and the B L K I N P program again on the SIDOS system (Clarkson's method [6]) have been applied. Measurements are made with photon radiation beams of a 4 MV therapy unit (Varian, Clinac), using a 50 x 50 x 50 cm 3 water phantom and 0.15 cm 3 ionization chambers (Wellhofer).
20 cm
32 cm
Fig. 2. Arrow-shaped photon field.
Fig. 3. Isodose presentation in the sagittal plane AB o f the arrow-shaped photon field calculated by means o f the "Ring" method. - - - , 32 x 20 cm 2 rectangular field; - - - , arrowshaped field.
The arrow-shapedphoton field A photon field (4 MV) o f 3 2 • 2 0 c m 2 at 8 0 c m SSD is transformed into an arrow-shaped field by the use of four blocks (see Fig. 2). Figure 3 shows the change in the isodose pattern in the sagittal plane AB as a result of this transformation calculated by means of the " R i n g " method. Both isodose patterns are normalized to the maximum dose of the non-blocked field. The maximum dose of the mantle field reaches a value of 97.1% of the maximum dose of the non-blocked field. Because the left side of the field is more shielded, the dose values at that side are smaller. A comparison between dose measurements and calculations for 6 points in this arrow field has been presented elsewhere by van Kleffens et al. [17]. In that work the calculations were done with the I R R E G program (using 4 MV T A R - S A R tables and profile data for our 4 MV linear accelerator as required in the TP 11 manual), the SIDOS program (using Wrede's method for irregular fields) and the B L K I N P program. For the same 6 points, calculations were performed using the "Ring" method and compared with the previous data (see Table I). The " R i n g " method performs better than Wrede's method and at least as good as the I R R E G or B L K I N P program. The rather large deviations of the B L K I N P program in points 5 and 6 are caused by the fact that at the time of the comparison the beam profile could not yet be taken into account properly. The deviations
367 TABLE I Differences between total dose calculation and measurement (in percent relative to the maximum dose on the central axis). 4 MV (IRREG) Depth (cm)
2 5 10 15 20
Point no. 1
2
3
4
5
6
+0.3 +0.7 +3.4 +2.4 +2.3
--0.3 0 +1.0 +0.8 +0.8
+0.6 +0.5 +0.8 +0.5 +0.7
-0.6 --0.2 +0.2 +1.0 +0.7
+0.1 +2.7 +2.6 +2.0 +2.2
+4.0 +2.3 +1.7 +1.6 +0.9
4 MV (BLKINP) Depth (cm)
2 5 I0 15 20
Point no. I
2
3
4
5
6
-1.4 +0.6 +2.0 +2.2 +2.1
+0.3 0 +2.0 +0.6 +0.7
+0.8 +0.5 +0.6 +0.9 +0.8
+0.2 +0.3 +0.6 +I.1 +1.3
--3.0 --0.6 +0.6 +1.1 +1.0
+6.5 +4.2 +3.3 +2.9 +2.8
1
2
3
4
5
6
-1.5 +1.5 0 0 0
+I.0 +1.0 0 0 0
+1.5 +1.5 0 --I.0 --0.5
-0.5 -1.0 -3.0 -2.0 --2.0
-1.5 +2.5 -0.5 -1.0 --1.0
+4.0 +5.0 +3.0 +3.0 +1.0
1
2
3
4
5
6
-0.7 0 -0.7 --0.4 --0.6
+0.5 +0.6 --0.1 +0.3 --0.5
+0.2 +0.8 +0.9 +0.3 -0.1
-0.2 -0.2 +0.2 -0.5 -0.7
-2.2 +1.5 +0.3 --0.2 -0.5
+2.1 +1.8 +0~5 -0.2 -0.7
4 MV (Wrede) Depth (cm)
2 5 10 15 20
Point no.
4 MV (Ring) Depth (cm)
2 5 10 15 20
Point no.
Fig. 4. Isodose lines of a half blocked square (20 x 20 c m 2) photon field calculated by means of the "Ring" method ........ Measured (half blocked); - - , calculated (half blocked); - -, calculated (unblocked); +, *, the maximum dose of the unblocked and blocked field, respectively. o f the " R i n g " m e t h o d at a d e p t h o f 2 c m in p o i n t s 5 a n d 6 ( a p p r o x i m a t e l y 2 . 0 % ) are p r o b a b l y caused by inaccuracies in the c a l c u l a t i o n o f the fractions o f the smallest two rings which are blocked.
The half-shielded field T h e water p h a n t o m is exposed to a p h o t o n field (4 MV) o f 20 x 20 cm 2 at 80 cm SSD, o f which one half is blocked. T h e b r o k e n lines in Fig. 4 show the isodose lines, o f the n o n - b l o c k e d field, n o r m a l i z e d to the m a x i m u m dose. The solid lines show the isodose lines of the h a l f - b l o c k e d fields n o r m a l i z e d to the m a x i m u m dose o f the n o n - b l o c k e d field. T h e little asterisk designates the m a x i m u m dose o f the half-field, which is 99.2% o f the m a x i m u m dose o f the n o n - b l o c k e d field (large asterisk). T h e d o t t e d lines represent the m e a s u r e d isodose lines. It c a n be seen, that the isodose lines o f the half-field app r o a c h those o f the n o n - b l o c k e d field in regions at a large distance f r o m the block, because scatter changes, as a result o f the presence of the block, are negligible there. I n the vicinity o f the b l o c k e d p a r t o f the field, the differences b e t w e e n the isodose lines o f the half-field a n d those o f the n o n - b l o c k e d field increase, because the absence o f scatter f r o m the right half o f the field becomes m o r e a n d m o r e imp o r t a n t . T h e difference b e t w e e n b l o c k e d a n d u n blocked field reaches a value o f a b o u t - 7 . 0 % o f the m a x i m u m dose o n the central axis of the n o n -
368 blocked field, if the p e n u m b r a is excluded. Measured and calculated isodose lines in the blocked situation agree rather well except in the region within 3 cm from the edge of the block. This might be caused either by the inaccuracy in the calculation of the scatter contribution of the smallest two rings or by an inaccurate algorithm which describes the p e n u m b r a region (the standard algorithm in the SIDOS system is used). This will be a subject of a further investigation. The difference however in the vicinity of the blocks between the isodose lines in the blocked and non-blocked situation is quite remarkable.
Discussion and conclusion It can be concluded that a convenient method providing isodose lines has been derived which brings into account scatter changes due to the application of shielding blocks. The calculation time for an isodose pattern with the " R i n g " method using an array processor is about 5 s. It should be noted that an array processor speeds up calculations whatever method is used. The accuracy seems to be at least as good as achieved with the above mentioned programs (see Table I). The power of the method lies in the fact that the unshielded dose distribution (taking into account the CT Hounsfield values on the line from the focus to the calculation point) is used to obtain the shielded situation by bringing into account scatter changes due to the shielding. This approach distinguishes the method from the method of Clarkson where all scatter contributions in each point have to be summed to obtain the total dose. The method does not yet take into account the effect of inhomogeneities on the scattered radiation. A problem with the " R i n g " method as well as with the T A R method, is that the scatter data are determined for a homogeneous water phantom. Perhaps either additional scatter data for lung tissue and bone structures or the application of a scaling method [12] will provide acceptable accuracy. The step to real three-dimensional calculation seems to be less complicated: in principle the calculation of
the scatter contribution has already been three-dimensional. A considerable extension is to bring into account the different patient a n a t o m y outside the plane of calculation. A combination of the method of Clarkson and the " R i n g " method might be a good approach. The method can be easily applied to other energies. The number of rings and the annular width can be adapted to obtain the required accuracy.
Acknowledgements This study was supported by a grant f r o m Siemens A.G., Erlangen, West Germany. The authors wish to thank R. Thieme and A. Lutz of Siemens A.G. for their support concerning SIDOS-software. They also wish to thank B. van der Leije, P. Storchi, H. Huizenga, C. Blom-van Otterloo, B. Mijnheer and A. Visser for m a n y helpful discussions and A. van Eijk-van Santen for typing the manuscript.
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