British Journal of Radiology (2004) 77, 941-943
© 2004 British Institute of Radiology
doi: 10.1259/bjr/65139264
Equivalent diameters of elliptical fields
G S J Tudor, BA, MSci
and
S J Thomas, MA, MSc, PhD
Medical Physics, Addenbrookes NHS Trust, Cambridge CB2 2QQ, UK
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Abstract
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The equivalent field method is well established as a means of performing dose calculations in rectangular and irregular fields. There is, however, no consensus on the equivalent diameter (D) of elliptical fields, despite their common applications in kilovoltage radiotherapy. Measurements have been performed on 15 elliptical fields and 6 circular fields, comprising all possible combinations of 3 cm, 4 cm, 5 cm, 6 cm, 9 cm and 12 cm diameters, using 150 kV X-rays, with a half-value thickness of 8 mmAl. Equivalent diameters were calculated by a number of methods, including the equal area, ratio of perimeter to area, 2AB/(A+B) and sector integration. The best agreement with measurement was obtained using sector integration, which agreed with measurements within the limits of experimental error. The formula D=2AB/(A+B) was the best of the analytic formulae; at shallow depths it gave predictions of dose within better than 0.5%, whilst at 5 cm deep its greatest error was 1.6%. The equal area formula (
) gave the worst predictions, with errors up to 5% at shallow depths, and 9% at a depth of 5 cm.
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Introduction
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The equivalent field method is well established as a means of performing dose calculations in rectangular and irregular fields [1, 2]. The equivalent square or equivalent diameter field can be used for selection of depth doses, and for selection of other parameters that depend on scatter, such as backscatter factors (BSF). For rectangular fields, the use of the tables in British Journal of Radiology Supplement 25 [1], and the use of the 2XY/(X+Y) formula [3] are well established. Recent correspondence on the medical physics list server [4] has shown that there is a lack of similar consistency on methods of determining the equivalent diameter of elliptical fields.
Elliptical fields are commonly used in superficial radiotherapy, for beams with a half value layer (HVL) of the order of 8 mmAl. Equivalent diameters are required so that BSF and depth doses can be calculated for radiotherapy treatments.
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Methods of calculating equivalent diameter
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A number of methods of calculating the equivalent diameter of an elliptical field have been proposed. If the two diameters of the ellipse are A and B, then the simplest formula is based on the circle with the same area: 
This will overstate the diameter, especially for very elongated ellipses.
Another option is to use the formula that an equivalent square is equal to 4 times the area divided by the perimeter [5], combined with the formula that equivalent diameter=equivalent square/0.9 [1]. However, this has an obvious limitation for circular fields, since 4 times area/perimeter yields the diameter of a circle, so dividing by 0.9 would give an 11% overestimation. Therefore this formula can only be used if one takes 4 times area/perimeter as directly yielding the equivalent diameter [6]. An approximate formula for the perimeter of an ellipse is given by:
Where a=A/2, b=B/2. The area of an ellipse is
, hence the equivalent diameter becomes:
Another method is to apply the Worthley formula [3] directly to ellipses:
The Clarkson method [7] is based on an integration of a scatter radius function. The radius (r) of an ellipse is given by:
A scatter function S(r) is numerically summed over 360° in 1° steps, then divided by 360 to give an average value of S. The equivalent radius is one that gives the same value of S for a circular field.
Three possible scatter-radius functions were investigated. These were:
- The function given by Day and Aird [1]
where 
is a parameter of dimension length–1 and µ is a dimensionless parameter. S(
), the scatter for a field of infinite radius, cancels in all calculations, since we are only interested in relative values of the scatter function.
- Tabulated backscatter factors S(r)=BSF(r,E). Backscatter factors for 8 mmAl X-rays were used [8].
- Measured dose/monitor unit for circular fields. S(r)=Dose/Monitor unit.
Equations (1)
, (2) and (3), and scatter integration using functions (a), (b) and (c) together give six possible equivalent diameters to compare against experimental measurements.
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Experimental methods
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Measurements were performed on a Pantak DXT300 X-ray unit (Pantak Medical Systems, Reading, Berkshire) set at 150 kV. The HVL of the beam was 8.0 mmAl. The unit has a transmission monitor chamber.
Circular and elliptical fields were created using lead cut-outs. Six circular cut-outs, of diameters 3 cm, 4 cm, 5 cm, 6 cm, 9 cm, and 12 cm; and 15 elliptical cut-outs, consisting of all the possible combinations of the above dimensions (i.e. 3 cm x 4 cm, 3 cm x 5 cm, ..., 9 cm x 12 cm) were created using 1.5 mm lead sheet. From inspection, the random deviation of the edge of the collimator from the correct position was estimated to be <3 mm, and the systematic error for a whole collimator was estimated to be <1 mm.
Measurements were made in a phantom of water-equivalent WT1, using an NE2572 0.2 cm3 ionization chamber and an NE2570 electrometer [NE Technology (Thermo Electron Corporation), Beenham, UK].
Measurements were made at two depths; 1 cm and 5 cm. In each case, there was 10 cm of material beneath the measuring point to provide effectively full backscatter. The 15 cm x 15 cm square, 50 cm focus–skin distance (FSD) applicator was used, with a stand-off of around 5 mm in order to allow ease of collimator change. The lead cut-out collimators themselves measured approximately 17 cm x 17 cm square, to ensure full attenuation of the edges of the beam.
Measurements were made of electrometer reading per 100 MU at 1 cm deep and per 150 MU at 5 cm, and were in each case normalized to the readings at the same depth for the largest circular field. The effective diameter of each ellipse was found by constructing a plot of relative dose against circle diameter, and reading off that diameter of a circle that would produce the same measured dose as the ellipse.
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Results and discussion
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Figure 1 shows the relative dose per monitor unit for the circular fields, at depth 1 cm and 5 cm. The doses are normalized to the dose resulting from the use of the 12 cm diameter circle collimator.
Table 1 shows the predicted and measured equivalent diameters. For Equations (1), (2), (3) and method (b), the predictions were independent of depth. For method (a) the values of and µ were chosen to optimize the agreement, giving =0.344 cm–1 1 cm deep, and 0.343 cm–1 5 cm deep, and µ=0 at 1 cm deep, 0.4 at 5 cm deep. For method (c) the measured data from Figure 1 were used as the input.
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Table 1. Predicted and measured equivalent diameters. Columns (1), (2) and (3) refer to |<|(|>|, D=4 x Area/Perimeter and D=2AB/(A+B), respectively. (a), (b) and (c) refer to sector integrations using the function in BJR25 [1], backscatter function and measured dose/MU, respectively. For cases a and c separate calculations were done for 1 cm and 5 cm deep. The final two columns are the measured equivalent diameters
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Figures 2–5 show the ratio of predicted to measured doses for the Equations (1) and (2) at depths 1 cm and 5 cm. The error bars in each case show the estimated uncertainty in the experimental measurement. Sources of error considered include misalignment of the collimator with the treatment unit; drift in the output of the treatment unit and imperfect collimator manufacture.
The best agreement, as expected, was obtained for a sector integration done using measured dose/MU for circular fields at the depth of interest (method c). As can be seen from Table 1, this method agreed with measurement for all sizes, within the limits of experimental method. The dose predicted by this method at each depth and for each elliptical field was within 0.3% of the measured dose. Similar results were obtained using the formula from Day and Aird [1], using depth dependent parameters. The sector integration using BSF (formula b) also agreed within experimental error at a depth of 1 cm, but gave marginally worse agreement 5 cm deep; for all ellipses with an aspect ratio below 3 the agreement with measurement was within 1%, whilst the largest error (for a 3 cm x 12 cm ellipse) was 1.6%.
For daily use, it is advantageous to have a simple formula that does not require sector numerical integration. Of the three such formulae considered, overwhelmingly the best agreement with measurement was found for Equation (3), 2AB/(A+B). Its values of equivalent diameter were in all cases within 0.1 mm of the results of the BSF based integration. Hence at a depth of 1 cm deep, it predicted the dose for all ellipses within experimental error, as shown in Figure 3, whilst at 5 cm deep gave results within 1.6%, as shown in Figure 5.
The simplest formula, |<|(|>|, predicts doses that are in error by up to 5% at 1 cm deep, and 9% at 5 cm deep. These are for the extreme case of a 3 cm x 12 cm ellipse. For most ellipses encountered clinically the errors are unlikely to exceed 2%; however the formula is no easier to use than Equation (3), and so has no benefit.
Equation 2 (4 x area/perimeter) gives results between these two extremes, with maximum errors of 1.1% at 1 cm deep, 5.6% at 5 cm deep. Again, there is no reason to use this formula in preference to Equation (3).
The prediction of depth doses is slightly more complex. Since the two best sector integrations have depth dependent parameters, it is impossible to obtain an exact fit (within experimental errors) for any method. The best agreement is found with the two sector integrations optimized 5 cm deep, which agree with measurements within 0.7% in all cases. The BSF based integration, and Equation (3), give depth doses within 1% of measurements for all ellipses with an aspect ratio below 3:1, and within 1.7% in all cases.
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Conclusions
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The best agreement between predicted and measured equivalent diameters is obtained using sector integration with input data specific to the beam quality and depth of measurement. However nearly as good agreement can be obtained with the simple equation D=2AB/(A+B). For most aspect ratios encountered clinically, this will give predictions of dose that agree with measurement within 0.5%.
Received for publication January 14, 2004.
Revision received May 19, 2004.
Accepted for publication June 22, 2004.
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References
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- Day MJ, Aird EGA. The equivalent field method for dose determinations in rectangular fields. Br J Radiol 1996;Suppl. 25:138–47.
- Day MJ. The equivalent field method for axial dose determinations in rectangular fields. Br J Radiol 1961;Suppl. 10:77–82.
- Worthley B. Equivalent squares of rectangular fields. Br J Radiol 1966;39:559.[Abstract/Free Full Text]
- Medical-Physics-Engineering list server, 2003, http://www.jiscmail.ac.uk/cgi-bin/wa.exe?A0=medical-physics-engineering
- Sterling TD, Perry H, Katz L. Automation of treatment planning. IV. Derivation of a mathematical expression for the per cent depth dose surface of cobalt-60 beams and visualisations of multiple field dose distributions. Br J Radiol 1964;37:544–50.
- Rosenberg I. 2003, http://www.jiscmail.ac.uk/cgi-bin/wa.exe?A2=ind0306&L=medical-physics-engineering&P=R22252
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- Grosswendt B. Dependence of the photon backscatter factor for water on source-to-phantom distance and irradiation field size. Phys Med Biol 1990;35:1233–45.[CrossRef]
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Abstract
Introduction
) and 5 cm (
). The doses are normalized to the dose resulting from the use of the 12 cm diameter circular collimator.
