British Journal of Radiology (2004) 77, 672-674
© 2004 British Institute of Radiology
doi: 10.1259/bjr/89967913
Implementation of a spreadsheet program for calculation of equivalent square field size
A W Seaby, BSc, MSc
M Thomas, BSc
and
S J S Ryde, MSc, PhD
Department of Medical Physics and Clinical Engineering, Singleton Hospital, Swansea NHS Trust, Swansea SA2 8QA, UK
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Abstract
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Equivalent square field size is a parameter widely used in calculations for external beam treatment planning. The algorithm and features of a spreadsheet program used to calculate the equivalent square, independently of the treatment planning system, are described. The program uses an approximation method based on the application of Lamé curves. Comparison of the output, for a variety of field shapes and sizes, with tabulated data and other calculation methods yielded differences typically 
1 mm.
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Introduction
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Independent checking of monitor unit calculations, for each radiotherapy treatment plan, is essential for quality assurance [1]. It is considered desirable if the beam data set and calculation algorithm are independent of those of the treatment planning system. The concept of equivalent square field size, proposed by Day [2], is frequently used in such calculations and must be determined for many regular and irregular field shapes. For regular fields, tabulated data (Day and Aird [3]) or empirical formulae, such as that based on the work of Sterling et al [4] may be used to find equivalent squares. The Clarkson method [5] is widely used for calculating doses in irregular fields. Locally, this has been implemented in several spreadsheet programs and requires the measurement of a number of radii at fixed angular intervals.
This note describes an algorithm, utilizing Lamé curves to define the field shape, that has been used in a program for the calculation of equivalent squares. Input is limited to the long and short axes of the field and modification of the field shape shown on the monitor until it is the best approximation to the shape of the planned field.
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Methods and materials
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In the 19th century, the French physicist Gabriel Lamé [6] studied the family of plane curves, described by the formula:
where,
2a=length of axis in the x direction
2b=length of axis in the y direction
The exponent n determines the shape of the curve. For n<1 the shape is an astroid, for n=1 a diamond and for n=2 an ellipse. As n
the shape becomes a rectangle (Figure 1).
The spreadsheet program was implemented as an Excel 97 Workbook (Microsoft Corporation). A treatment field, with the long and short axes defined by the user, is divided into quadrants by the principle axes. Each quadrant is a curve, given by the above equation and characterized by a and b components representing collimator settings, e.g. X2, Y1 and an exponent n. Initially, the default value of n is 20, giving a quarter of a square, or rectangle, with a rounded corner of small radius. A scroll-bar allows the operator to change the value of n, for each quadrant, until the curve is the best approximation to the shape of the field (Figure 2).
For each quadrant, the lengths of radii from the origin to the curve are found for 5° intervals. The corresponding values of the scatter–radius function [3] are calculated and summed. The mean scatter value is calculated and the corresponding radius found from an embedded look-up table. The side of the equivalent square is then determined using a conversion formula from circular to square fields [3].
The hardcopy for the program consists of a single page, showing all input data, the equivalent square field sizes at 100 cm and the treatment source–skin distance (SSD) and a graphical representation of three fields scaled at 0.7. This hardcopy can be superimposed on templates for each field, generated by the treatment planning system (Helax TMS; Nucletron B.V., Veenendaal, The Netherlands), to check the agreement of size and shape.
Quality control checks on the program included the preparation of 105 regularly shaped fields (circles, squares, rectangles and "square" diamonds). The aspect ratios for the rectangular fields ranged from 2:3 to 1:3. The equivalent squares for these fields were compared with tabulated values [3]. 15 irregularly shaped clinical fields were also checked by comparing them with a scatter integration method using 16 or 24 sectors. The irregular fields included the re-entrant shape shown in Figure 3. To input this shape, the axes indicated by the dashed lines were used and not the principle axes.
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Figure 3. A re-entrant field shape and the spreadsheet approximation. The rotated axes, shown as dashed lines, were used for the equivalent square program.
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Results
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For the 105 regular fields the agreement with tabulated data was generally very good. A maximum difference of 1 mm was noted in 15 instances. These differences were mainly for the circles and square diamonds, which required the operator to adjust the shape of the curves for all quadrants. For the 15 irregularly shaped fields, a maximum difference of 1 mm was noted for 6 fields. When another experienced treatment planner carried out these tests, the results were similar, although, a difference of 3 mm was noted for 1 field. This difference was ascribed to operator subjectivity since there were no features of the field that suggested that it was inherently more difficult to model than the others. For the size (10.0 x 7.8 cm), depth (12.2 cm) and modality (4 MV X-ray) of the clinical beam, a 3 mm difference in equivalent square corresponded to a difference of approximately 0.5% of local dose. For the re-entrant shape, the equivalent squares agreed to within 1 mm.
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Discussion and conclusion
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The program is suitable for a range of field shapes where the shielding is arranged around the periphery of the open field. An "island" block would not be acceptable and many irregular fields may be impossible to approximate. However, in the case of the re-entrant field, described above, selection of suitable axes allowed the shape to be successfully modelled.
This simple program has proved to be a useful tool for those engaged in the task of independent monitor unit calculations. It is straightforward to use and the numerical input required is very modest. Despite a degree of subjectivity in defining the irregular field approximation, the agreement with other methods is good.
Received for publication November 18, 2003.
Accepted for publication April 15, 2004.
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References
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- Institute of Physics and Engineering in Medicine. Physics aspects of quality control in radiotherapy. IPEM Report 81. York, UK: IPEM, 1999.
- Day MJ. A note on the calculation of dose in X-ray fields. Br J Radiol 1950;23:368–9.[Medline]
- Day MJ, Aird EGA. The equivalent field method for dose determination in rectangular fields. Br J Radiol 1996;(Suppl. 25):138–51.
- Sterling TD, Perry H, Katz L. Automation of radiation treatment planning. IV. Derivation of a mathematical expression for the per cent depth dose surface of cobalt-60 beams and visualisation of multiple field dose distribution. Br J Radiol 1964;37:544–50.
- Clarkson JR. A note on depth doses in fields of irregular shape. Br J Radiol 1941;14:265–8.[Abstract/Free Full Text]
- Gridgeman NT. Lamé ovals. Mathematical Gazette 1970;54:31–7.[CrossRef]
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Abstract
Introduction
