Preparation of data
A preconfigured reference beam dataset (reference data) generated by the vendor has been stored in the treatment planning system when a new Halcyon is installed. The reference data included the lateral dose profiles for field size of 2 × 2, 4 × 4, 6 × 6, 8 × 8, 10 × 10, 20 × 20 and 28 × 28 cm2 as function of depth at 1.3, 5, 10, and 20 cm into the water phantom. In order to compare the field size from the reference data with that from the measured data, the measurement was performed under the same conditions in the reference data. A source to surface distance (SSD) was set at 90 cm. The CC13 ionization chamber and the Blue Phantom water tank (IBA Dosimetry, Schwarzenbruck, Germany) were used to measure the relative dose profiles for field size > 4 × 4 cm2. For field sizes ≤ 4 × 4 cm2, an edge diode detector (Sun Nuclear, Melbourne, FL, USA) was used. The scanning step for the acquisition of the profile on the measurement line along the off axis position was 0.1 cm with increment per step. All measurement values were processed by using OminiPro Accept7 (version 7.4.24.0) software (IBA dosimetry, Schwarzenbruck, Germany).
Definition of fitting using sigmoidal curve
The sigmoidal curve is originated from the sigmoid function which has been used at the field of the signal process. The shape of the sigmoidal curve is given by Eq, (1),
The coefficients α, β, γ, and δ are used to determine shape of the curve f(x). The coefficient α controls the gradient of the sigmoidal curve. The higher value of α makes the curve gradient steeper. The coefficient β is related with the horizontal movement of the whole sigmoidal curve. The higher value of β let the sigmoidal curve move further to the right hand side of the curve. The coefficient γ determines the location of the only upper end of the sigmoidal curve. The higher value of γ let the upper end of the sigmoidal curve locate at a higher position side. The coefficient δ determines the vertical movement of the whole sigmoidal curve. The higher value of δ let the sigmoidal curve move to more upward direction. Thus, the coefficients α and γ contribute to transform the shape of the curve. The coefficients β and δ change the location of the sigmoidal curve.
After the upload of the profile to MATLAB (2019 version, MathWorks Inc, Sherborn, MA, USA), the SCF has been performed by the change of each coefficient until the sigmoidal curve overlapped on the profile. However, because the physical range of the sigmoidal curve cannot cover the whole range of the profile due to the original shape of the curve, the physical range of the sigmoidal curve was limited until the profile can cover the maximum range using the only sigmoidal curve. Through this methodology, all SCFs have been done to all profiles.
Verification of agreement for fitting curves
To verify the accuracy of the fitting curve based on the sigmoidal curve with the profiles, the average agreement ratio (AAR) between the values in the fitting curve (fi) and the values in profiles (xi) at the same step position were calculated using Eq. (2), which shows the agreement between values from profile and fitting.
In this study, if the AAR is higher than 97%, the optimization for the fitting terminates because the sufficient accuracy for the fitting has been obtained and the four coefficients (α, β, γ, and δ) are used to define shape of the final fitting curve. Moreover, an additional verification was performed through the evaluation of goodness of fit R2 (Eq. (3))
where is the mean of all fi values on fitting curve, and the yi is a value on profile. The same validation procedures were applied to the measured data and to the reference data.
Identification of specific regions & points
In this study, in order to describe the sigmoidal curve, three regions and two points have been assigned for the definition to the half-side of SCF (Fig. 1(a)). The three regions included the introductory region (IR), the growing region (GR), and plateau region (PR). The IR is the region starting to increase on the sigmoidal curve. The GR is continuously increasing region on the sigmoidal curve. The PR is the region slowing down to increase. These regions can be identified through the second derivative of the sigmoidal curve as shown Fig. 1(a). The range between the rightmost point and the maximum point on the second derivative curve was defined as the IR. The range between the maximum point and the minimum point on the second derivative curve was defined as the GR. The region between the minimum point and the leftmost point on the second derivative curve was defined as the PR. Because of the specification of the sigmoidal shape, there are two specific points as singular point (SP) and inflection point (IP), both of which these couple of points could be identified from the third derivative curve of the sigmoidal curve. The SP is the minimum point between the range of IR and the GR (Eq. (4)). The IP is another minimum point at the range between the GR and the PR as below (Eq. (5)). When there is no point either the IP or the SP, the re-fitting process was performed from 2. 2 part in this session.
Determination of the Field Size
After the SP and the IP have been obtained, the determined field size (DFS) can be identified as the maximum point on the third derivative curve between the SP and the IP as below Eq. (6). Figure 1(b) shows the conceptual DFS on the third derivative curve and the actual example of the DFS has been demonstrated with fitting to the profile at Fig. 1(b)
The factor 2 in the equation (Eq. (6)) was due to that fact which only right hand half side of the symmetric open beam profile was used for the curve fitting.