Effective wedge angles with a universal wedge

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1985 Phys. Med. Biol. 30 985

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Phys. Med. Biol., 1985, Vol. 30, No. 9, 985-991. Printed in Great Britain

Effective wedge angles with a universal wedge

Paula L Petti and Robert L Siddon Joint Center for Radiation Therapy and Department of Radiation Therapy, Harvard Medical School, Boston, MA02115, USA

Received 1 March 1985, in final form 7 May 1985

Abstract. Some recently designed x-ray-producing accelerators are equipped with a single built-in wedge, and different 'effective' wedge angles are obtained by combining an open (unwedged) and a wedged field in the appropriate proportions. This paper describes a technique for determining these proportions from measured isodose distributions of the two component fields. Our data for the Philips SL/75 6 MV accelerator are compared with two existing theoretical models. One model, in which the beams are weighted by the ratio of the tangents of the effective and nominal wedge angles, agrees with the data to within 3" over the range of effective wedge angles and square field sizes examined. The second and simpler model, i n which the beams are weighted by the ratio of the wedge angles directly, results in errors of as much as 11". It is shown that both of these models are approximations to an exact theoretical solution which may be formulated in terms of one free parameter. This parameter may be interpreted physically as the ratio of the slopes of the central-axis depth-dose curves for the open and wedged fields.

1. Introduction

Wedge filters are used routinely in radiation therapy to optimise dose distributions with x-ray beams. Most accelerators are provided with a selection of wedges that are mounted externally on the head of the machine. A single, steep wedge, mounted inside the head and positioned by remote control, may also be used. This is similar to the principle of the 'universal wedge' (Bentel et al 1982). Recently, our centre acquired a Philips SL/75 linear accelerator which has a single wedge of this kind with a nominal wedge angle of 60". Beams with wedge angles less than 60" are obtained by delivering an appropriate portion of the dose with the 60" wedged field and the remaining portion with the open (unwedged) field.

The purpose of this study was to evaluate two existing theoretical models for determining the fraction of dose required from the 60" wedged field for a given desired effective wedge angle over a range of field sizes from 5 cm x 5 cm to 20 cm x20 cm. One of these methods was originally proposed by Tatcher (1970), and the other was suggested by Philips Medical Systems Division (1983). A refinement of the latter model is also proposed and examined.

2. Methods

Isodose distributions for the Philips SL/75 6 MV x-ray beam were measured in a water phantom using a sealed 0.1 cm3 ionisation chamber and a computerised scanning system. The plane of measurement contained the central axis and was along the wedged direction of the field. Field sizes of 5 cm X5 cm, 10 cm X 10 cm, 15 cm X 15 cm and

0031-9155/85/090985 +07$02.25 @ 1985 The Institute of Physics 985

986 P L Petri and R L Siddon

20 cm X20 cm were considered for both the open and 60" wedged fields. Measurements were made at increments of 5 mm in depth for all field sizes. Perpendicularly to the central axis, measurements were taken at intervals of 2, 3, 3 and 4mm intervals for field sizes of 5 cm x 5 cm, 10 cm x 10 cm, 15 cm X 15 cm and 20 cm x20 cm, respectively.

Combining the measured open, Do, and wedged, Dw, isodose scans in the propor- tions A and B, where A + B = 1, yields the effective isodose distribution D, where

D = ADo+ BDw. (1)

Figure 1 illustrates this combination procedure for 10 cm x 10 cm open and wedged fields. The effective wedge angle, OE, in this illustration is 20". As indicated by a cross in figure 1, Do and Dw are both normalised at a depth of 10 cm on the central axis. As A + B = 1, the effective dose D is also normalised at 10 cm depth. In the following, we adopt the ICRU (1976) recommendation that the wedge angle is defined as the complement of the angle between the central axis and a line tangent to the isodose curve at the depth of 10 cm.

The calculations indicated in equation (1) were carried out in FORTRAN 77 on a Vax 11/780 computer and subsequently visualised with a Lexidata 3400 graphic image processor.

Open Wedge Ef fect ive wedge

A

Figure 1. The effective wedge distribution D is given by the combination of the open D, and wedged Dw fields in the proportions A and B respectively, where A + B = 1. The example shown is for a 10 cm X 10 cm field.

3. Mathematical models

A simple approximate model for combining open and wedged fields was proposed by Tatcher (1970), who suggested that the effective wedge angle &, resulting by the addition of the two beams, is equal to the nominal wedge angle Ow for the wedged beam, weighted by the fraction of wedged field, B:

OE = BBw. (2)

The angles Ow and OE are shown in figure 1. This model has been reported to be accurate for energies ranging from 6 o C ~ to

25 MV, for nominal wedge angles ew less than 60" and for field sizes less than 20 cm x20 cm (Tatcher 1970, Abrath and h r d y 1980, Mansfield et a1 1974, Zwicker et a1 1985).

Efective wedge angles with a universal wedge 987

An exact relation between the two angles eE and Ow may be derived as follows. Since, by definition, an isodose curve satisfies the condition D = constant, the gradient of D is orthogonal to the isodose curve. Consequently, the ratio of the components of the gradient V D is equal to the tangent of the effective wedge angle, i.e.

tan OE = axD/dZD

where X is the transverse direction, Z is the depth, and the symbols 8, and a , represent the X and Z components of the gradient respectively.

Substituting D (equation (1)) into equation (3) yields,

As the isodose curve for the open field Do is symmetric about the central axis of the field, axDo is equal to zero. Furthermore, for the wedged field alone

tan Ow = axDw/azDw.

Combining equations (4) and (5) yields the result

tan eE = B tan Ow/(Af+ B )

where the quantitiy f is given by the ratio

The factor J ; the ratio of the slopes of the central-axis depth-dose curves for the open and wedged fields, differs from unity due to the presence of scattered radiation and as a result of beam hardening introduced by the wedge filter. In the absence of these effects, the ratio of the slopes of the depth-dose curves are equal (f= l ) , and equation (6) reduces to the simpler result

B =tan &/tan Ow. (7)

Equation (7) was proposed by Philips Medical Systems Division (1983) and has also been discussed by Zwicker et a1 (1985). In the limit of small angles, equation (7) reduces to Tatcher’s result (equation (2)). Predictions based on these two models are compared to measured data for the Philips SL/75 in the following sections.

Equation (6) depends on depth, energy and field size implicitly through the parameters Ow and f: The values of these parameters must be determined experi- mentally. Following ICRU (1976), we have restricted this study to a single depth of 10 cm. Since we were interested specifically in the Philips SL/75 6 MV accelerator, the energy dependence was not studied. The dependence of Ow and f on field size was, however, considered (figure 2) and is discussed in 0 5.

4. Results

Using the more general result, equation (6), the fraction B may be expressed in terms of the desired effective wedge angle eE and the ratio f :

B =f/[(tan Owltan 6,) +f- l]. Both the Philips and Tatcher models are approximations to this general equation. Predictions based on these two models were compared to measured data. The results are shown in figure 3 for field sizes from 5 cm x 5 cm to 20 cm x20 cm. Each figure shows (i) exact calculations based on measured data according to equation (1) (full

988 P L Petti and R L Siddon

65

i

0 5 10 15 20

S ( c m 1

Figure 2. The nominal wedge angle 8, as a function of field size S.

circles), (ii) a line corresponding to predictions based on Tatcher's model, (iii) a curve describing the Philips model, and (iv) a curve corresponding to a least-squares fit of the data to the extended model, equation (8).

5. Discussion

For the Philips SL/75, the nominal wedge angle Ow is a nearly linear function of field size, as shown in figure 2. The nominal wedge angle increases from 56" for the 5 cm X 5 cm field to 63" for the 20 cm x20 cm field, an increase of approximately 0.5" cm". Such a linear dependence of wedge angle on field size was observed by Wu et a1 (1984), who measured a slope of 0.5" cm" for the 15, 30 and 45" wedges for the Clinac 6/ 100.

Figure 3(a)-(d) demonstrate that at any field size, the Philips model (equation (7)) sufficiently describes the data, whereas the Tatcher model does not. The maximum deviation of the Tatcher model from the data is approximately 10" which is much larger than the &l" estimated error in the measurements. The Philips model agrees with the data to within 3" over all field sizes between 5 cm X 5 cm and 20 cm x 20 cm.

Other authors have reported satisfactory agreement between the Tatcher model and measured data for a wide range of effective and nominal wedge angles, generally less than 60". The agreement for small values of 8, may be understood if one considers the difference between the Philips and Tatcher models:

6eE = tan"( B tan e,) - Be,. ( 9 )

Maximising this difference with respect to B yields the results shown in table 1. The maximum difference between the Tatcher and Philips models is only 1" for Ow = 30". However, for angles Ow greater than 45" the discrepancy between the two models increases markedly with increasing Ow, reaching approximately 11" for Ow = 60". Thus,

B

B

Efective wedge angles with a universal wedge

l , l

1 . O l la) 5 c m X 5 c m 4

0.4 t l

0. i

Philips , ~l ~

0.2 - 0,:58.6'~

0 10 20 30 40 50 60 70

l l

0 . 4 1

1 8

l ' 0,=63.2Ol

70 0 10 20 30 40 50 60 70

989

eE l d e g l

Figure 3. Comparison of results calculated from measured data for each component field (O), Tatcher's model, Philips' model and the extended model described by equation (8) for field sizes ( a ) 5 cm X 5 cm, ( 6 ) 10 cm x 10 cm, (c ) 15 cm X 15 cm and ( d ) 20 cm x20 cm. The nominal wedge angle Ow for each field size is also indicated on each figure.

Table 1. Maximum value of the quantity 80E (described in text) as a function of

the nominal wedge angle angle Ow.

15 20 25 30 35 40 45 50 55 60

0.1 0.3 0.6 1 .o 1.8 2.8 4.1 5.8 8.0

10.9

990 P L Petti and R L Siddon

while the Tatcher model is approximately correct for small values of Ow, it is inadequate if Ow exceeds 45", in particular for Ow = 60" as is the case for the Philips SL/75.

The curves in figures 3 ( a ) - ( d ) describing the exact model (equation (8)) were obtained by performing a least-squares fit to the measured data to determine the parameter f: Values o f f derived in this manner are given in table 2 as a function of field size. We also determined f directly by fitting central-axis depth-dose curves for open and wedged fields to exponential functions and calculating the ratio of the derivatives at a depth of 10 cm. These values are also listed in table 2. The two sets of results for f agree to within S % , which is roughly the level of uncertainty in each estimate. Thus, our data support the interpretation o f f as the ratio of the slopes of the depth-dose curves and confirm the extended model described by equation (8).

Table 2. The parameter 1:

Field size (cm xcm) f? f t

5 x5 1 .os 1.03 10 x 10 1.02 1.03 15 X 15 1.06 1 .oo 20 x 20 1.17 1.19

t Determined by fitting measured data to equation (S) . $ Calculated directly from the slopes of central- axis depth-dose curves.

6. Conclusions

Effective wedge angles for the Philips SL/75 may be obtained by combining open and 60" wedged fields in the appropriate proportions. The results of this investigation are consistent with the exact theoretical model

B =f/[(tan Owltan e,)+f- 13

where the parameter f is the ratio of the slopes of the central-axis depth-dose curves for the open and wedged beams.

In the limit where f = 1 and 0, and Ow are small, the exact result reduces to Tatcher's model:

B = e,/ ew. This approximation is reasonable only for values of Ow less than 45". For accelerators such as the Philips SL/75, where the nominal wedge angle Ow is 60", and in most other situations of practical interest, the Tatcher equation is inadequate.

Since for all of the field sizes considered in this study, the parameter f is close to 1, the exact model may be approximated by

B =tan @,/tan Ow.

This equation was proposed by Philips Medical Systems Division and relates with sufficient accuracy the fraction B of the dose required from the wedged field to the desired effective wedge angle 6,. Predictions based on this model agree with our data for the Philips SL/75 6 MV accelerator to within 3" for all field sizes considered.

Eflective wedge angles with a universal wedge 99 1

Acknowledgments

The authors are grateful to Dr Bengt E Bjarngard for stimulating discussions and a thoughtful critique of this work. This work was supported in part by Public Health Service Grants Numbers CA-09234 and CA-34964, awarded by the National Cancer Institute, DHHS.

RBsumB

Angles de filtres en coin effectifs obtenus avec un filtre en coin universel.

Certains acctltrateurs rdcents produisant des FS sont Bquipts d'un filtre en coin unique, difftrents 'angles effectifs' ttant obtenus en combinant un champ sans coin et un champ avec coin avec une pondtration convenable. Cet article prtsente une technique permettant de dtterminer les pondtrations B partir des distributions de dose exptrimentales correspondant aux deux champs utilises. Nous y avons compart les donntes que nous avons obtenues pour l'acctltrateur Philips SL/75 6 MV avec les rtsultats de deux modbles thtoriques. L'un des modeles, dans lequel les faisceaux sont pondtrts en fonction du rapport des tangentes des angles des coins nominal et effectif, conduit B un accord B mieux que 3" sur la gamme des angles de coins effectifs et des dimensions de champs carrds ttudits. Le second modble, plus simple, dans lequel les faisceaux sont directement pondtrts en fonction du rapport des angles de coin, conduit B des erreurs allant jusqu'i 1 I". On montre que ces deux modbles sont des approximations d'une thtorie exacte qui peut ttre exprimte en fonction d'un seul parambtre libre. Ce parambtre peut &re interprttt physiquement comme le rapport des pentes des courbes de rendement en profondeur pour les champs respectivement avec et sans coins.

Zusammenfassung

Effektiver Keilwinkel bei einem Universalkeilfilter.

Einige kurzlich entwickelte Rontgenstrahlen erzeugende Beschleuniger sind ausgestattet mit einem einzigen Einbau-Keilfilter und verschiedene 'effektive' Keilwinkel konnen erhalten werden durch Kombination von Feldem entsprechender GroBenverhaltnisse mit und ohne Keilfilter. In dieser Arbeit wird ein Verfahren zur Bestimmung dieser GroBenverhaltnisse aus gemessenen Isodosenverteilungen fur zwei-Komponenten- Felder beschrieben. Die Werte fur einen Philips SL/75 6 MV-Beschleuniger werden verglichen mit zwei theoretischen Modellen. Ein Modell, bei dem die Strahlen gewichtet werden mit dem Verhaltnis der Tangenten der effektiven und der nominalen Keilwinkel stimmt mit den Werten iiberein innerhalb 3" uber den untersuchten Bereich von effektiven Keilwinkeln und quadratischen FeldgroBen. Das zweite und einfachere Modell, bei dem die Strahlen gewichtet werden durch das Verhaltnis der Keilwinkel, ergibt einen Fehler von mehr als 11". Es wird gezeigt, daO beide Modelle Naherungen fur eine exakte theoretische Losung sind, die in Form eines freien Parameters formuliert werden kann. Dieser Parameter kann phy- sikalisch interpretiert werden als das Verhaltnis der Richtungskoeffizienten der Tiefendosiskurven auf dem Zentralstrahl fur Fleder mit und ohne Keilfilter.

References

Abrath F G and Purdy J A 1980 Radiology 136 757-62 Bentel G C, Nelson C E and Noell K T 1982 Treatment Planning and Dose Calculation in Radiation Oncology

ICRU 1976 Determination of Absorbed Dose in a Patient Irradiated by Beams of X or Gamma Rays in

Mansfield C M, Suntharalingam N and Chow M 1974 Am. J. Roenrgenol. 120 699-702 Philips Medical Systems Division 1983 Product Data 764 (Eindoven, The Netherlands: Philips) Tatcher M 1970 Radiology 97 132 Wu A, Zwicker R D, Krasin F and Sternick E S 1984 Med. Phys. 11 186-8 Zwicker R D, Shahabi S, Wu A and Stemick E S 1985 Med. Phys. 12 347-9

(New York: Pergamon)

Radiotherapy Procedures, Report No 24 (Washington, DC: ICRU)