Using weighted power mean for equivalent square estimation

Abstract Purpose Equivalent Square (ES) enables the calculation of many radiation quantities for rectangular treatment fields, based only on measurements from square fields. While it is widely applied in radiotherapy, its accuracy, especially for extremely elongated fields, still leaves room for improvement. In this study, we introduce a novel explicit ES formula based on Weighted Power Mean (WPM) function and compare its performance with the Sterling formula and Vadash/Bjärngard's formula. Methods The proposed WPM formula is ESWPMa,b=waα+1−wbα1/α for a rectangular photon field with sides a and b. The formula performance was evaluated by three methods: standard deviation of model fitting residual error, maximum relative model prediction error, and model's Akaike Information Criterion (AIC). Testing datasets included the ES table from British Journal of Radiology (BJR), photon output factors (S cp) from the Varian TrueBeam Representative Beam Data (Med Phys. 2012;39:6981–7018), and published S cp data for Varian TrueBeam Edge (J Appl Clin Med Phys. 2015;16:125‐148). Results For the BJR dataset, the best‐fit parameter value α = −1.25 achieved a 20% reduction in standard deviation in ES estimation residual error compared with the two established formulae. For the two Varian datasets, employing WPM reduced the maximum relative error from 3.5% (Sterling) or 2% (Vadash/Bjärngard) to 0.7% for open field sizes ranging from 3 cm to 40 cm, and the reduction was even more prominent for 1 cm field sizes on Edge (J Appl Clin Med Phys. 2015;16:125–148). The AIC value of the WPM formula was consistently lower than its counterparts from the traditional formulae on photon output factors, most prominent on very elongated small fields. Conclusion The WPM formula outperformed the traditional formulae on three testing datasets. With increasing utilization of very elongated, small rectangular fields in modern radiotherapy, improved photon output factor estimation is expected by adopting the WPM formula in treatment planning and secondary MU check.


| INTRODUCTION
Equivalent Square (ES) is a widely used and important concept in photon external beam radiation dose calculation. ES postulates that, for an arbitrary rectangular field, there exists an equivalent square field sharing certain dosimetric characteristics. That concept provides us with a pathway for estimating a rectangular field's properties (e.g., central axis percentage depth dose, scatter factor) from measurements performed on square fields.
The crucial step in the success of this approach is to identify the optimal formula that will predict the correct equivalent square. For a rectangular field with width a and length b, Sterling's formula 1 was historically the first widely used, explicit ES formula for such a purpose. It remains the primary choice in current medical physics practice.
Originally proposed in 1964 for studying the rectangular radiation field's central axis percentage depth dose that is generated by X-ray units and 60 Co machines, Sterling's formula has enjoyed success from that point to this, and now is almost a synonym for ES because of its simple mathematical structure and good prediction power for conventionally shaped and sized fields in many applications.
with an adjustable parameter A > 0. Therefore, the aim of this study, using two well-known datasets, was to propose a revised formula and demonstrate that it offers an improvement over the two most popular, conventional formulae.

| METHODS
Herein, a Weighted Power Mean (WPM) 5,6 based ES formula was introduced, with the aim of achieving better accuracy than that obtained by the Sterling's and VB's formulae: with two adjustable parameters: power index a and weighting factor It is worth mentioning that our formula [see Eq. w; a f g¼arg min where k is the number of free parameters in the formula, and L is the maximum value of the likelihood function for the formula.
Under the assumption that the residuals are distributed according to independent identical normal distributions (with zero mean), we have AIC ¼ 2k þ n ln RSS n þ constant: Here, n is the number of data points, RSS ¼ P n i¼1 y i À f x i jh ð Þ ð Þ 2 is the residual sum of squares for formula f with the optimal parameter set h, and the constant term is model-independent for a given dataset. We applied AIC as an objective comparison between our formula and the two established formulae. Please note that for the same dataset, only the relative value of AIC is meaningful; therefore, we will set the constant term to zero for the rest of this paper. For model comparison, the lower the AIC value, the better the model's performance.

| RESULTS
When we applied the WPM-based formula to the BJR ES table, due to the intrinsic symmetry between the two field sides of every rectangular field in the dataset, b must be 1 2 . Next, we performed least squares fitting to obtain the optimal a value: where ES a; b ð Þ is the known ES for a rectangular field with sizes a and b.
For the BJR dataset, the fitting procedure above led to a ¼ À1:25, and the standard deviation of residual error was reduced by 20% compared with using a = À1 (i.e., the Sterling-type ES formulae). The 99% confidence interval of a was CI 99% ¼ À1:34; À1:16 ½ .
The optimal value for a was unlikely to be À1 for this dataset. Therefore, using the BJR dataset to support the use of Sterling's formula was based more upon clinical practicality rather than statistical analysis.
The best fitting values of a and w for open field S cp in the Varian dataset for Eclipse are listed in Table 1 for some representative photon energies. Again, none of them selected a = À1.
All open-field values of S cp from the Varian dataset were plotted against the predicted ES values from the three explicit ES formulae for four different photon energies (See Fig. 1). We can see that our WPM formula did a better job minimizing the spread of data points around the measured square-field curve, indicating better modeling performance.
The largest magnitude relative errors for open fields at all photon energies from the Varian dataset were graphed in Fig. 2 Fig. 3). The lowest AIC values among the three formulae indicated that the newly proposed WPM-based ES formula outperformed both the Sterling and VB formulae, even after we took into account the number of adjustable model parameters.
The worst relative data fitting errors for all three formulae occurred at data entries where photon fields were very elongated, narrow fields. The relative model fitting error for the fields with a shorter side at 1cm is tabulated in Table 2. We interpolated Varian 10X FFF Edge S cp values of nonsquare shaped fields based only on the square field measurements and ES predicted by the three formulae. The improvements of the WPM formula over the other two traditional formulae were obvious for these hard-to-fit field cases of large aspect ratios and very small short side lengths. From Table 2, we can see that the VB's formula has lower maximum relative error than the Sterling's formula while the WPM consistently outperformed the other two traditional ES formulae when it was applied to elongated small fields.
When other factors are fixed, the relative error in dose calculation for a photon field equates to the relative error in the employed output factor. Therefore, we expect our new ES formula will improve the accuracy of dose calculation, particularly when the photon field has an elongated small rectangular shape.