Derivation of equations to define inflection point and its analysis in flattening filter free photon beams based on the principle of polynomial function

Purpose: The objective of this work is to (1) present a mechanism for calculating inflection points on profiles at various depths and field sizes, and (2) study the doses at the inflection points for various field sizes at depth of maximum dose (Dmax) for flattening filter free (FFF) photon beam profiles. Methods: Graphical representation was done on percentage of dose versus inflection points. Also, using the polynomial function, the author formulated equations for calculating spot-on inflection point on the profiles for both the 6MV and 10 MV energies for different field sizes at various depths. Results: In a 10 MV FFF radiation beam, the dose at inflection point of the profile decreases as the field size increases. However, in 6MV FFF radiation beam, the dose at the inflection point initially increases with an increase in the field size up to 10 ×10 cm2 and decreases after 10 ×10 cm2. The polynomial function was fitted for both the 6 MV and 10 MV FFF beams for all field sizes and depths. Conclusion: Polynomial function is one of the easiest ways of identifying the inflection point in FFF beam for various field sizes and depths. Graphical representation of dose versus inflection point for both FFF energies was derived.


Introduction
In conventional linear accelerators, a flat photon beam is produced with a flattening filter to simplify treatment planning in radiotherapy by delivering homogeneous dose distribution. In latest radiotherapy technologies such as intensity-modulated radiosurgery (IMRS), stereotactic body radiotherapy (SBRT) and gated treatments, treatment time is prolonged. By removing the flattening filter, treatment time can be reduced due to increment in dose rate. 1 A number of studies have analysed and reported the flattening filter free (FFF) beam characteristics.
Standard parameters to define symmetry, flatness, and penumbra for a flattened beam (FB) are different from a FFF beam since FB is a forward peaked beam with reduced scattering. Many cancer centers use the parameters from the American Association of Physicists in Medicine (AAPM) Task Group 142 for the quality assurance of flat photon beams. 2 Parameters such as beam flatness, symmetry, and penumbra may not be directly useful for the new FFF beams. Therefore, there is a need to find new parameters, which can be used for both the standard FF and FFF radiation beams. This can be explained with the concept of an "inflection point" (IP).
An inflection point is defined as the mid-point on either side of the high gradient region (sharply descending part) of the FFF beam profile. From the inflection point, it is possible to evaluate penumbra and the field size. A challenge in FFF beam profile is the assessment of location of inflection point. It is complicated by general beam characteristics 3-7 , FFF mode, forward peaked beam, the large variations of flatness and central axis positions, beam energy and depth doses [8][9][10][11] , back scatter 12 , and electron contamination. 13 The ultimate goal of inflection point evaluation is location of inflection point for FFF beam with various field sizes at different depths for 6 MV (hereafter referred as 6X) and 10 MV (hereafter referred as 10X). To date, an accurate and computationally efficient solution is far from reach, and simplified evaluation is frequently performed in a clinical environment.
Hence, it is very important and necessary to find the accurate value of inflection point, because  It is located in the region of highest gradient.  In this region, the variation of dose is in the order of 10% / mm.  To date, there is no fine measurement stepping and detector size for the accurate measurement of this point.
 Dose can be affected by 10%. It can also influence the central axis dose by up to 40% in FFF beam due to normalization at the inflection point. 14 Literature for a mathematical model or formula to find the inflection point is limited.
The main purpose of this study is to develop a formula to identify an inflection point on any FFF beam profile curve for 6X and 10X photon beams for field sizes ranging from 1 × 1 cm 2 to 40 × 40 cm 2 at any depth, and this will take into account percentage of doses at inflection point versus field size. At the heart of the framework is a mechanism for calculating inflection point on FFF beam profile in both in-line and cross-line. Furthermore, detailed analysis of doses at inflection points versus field sizes for 6X and 10X FFF photon beams is presented in this paper.

Inflection point calculation on 6X FFF and 10X FFF profiles in both inline and cross-line
The field size in FFF is defined using inflection points, which is defined as the mid-point on either side of the high gradient region (sharply descending part) of the beam profile. In the present study, using the dose at the inflection point (IP) verses field size at Dmax, we formulated equations for computing location of the inflection point on any field size and depth in 6X and 10X FFF beam profiles. First, the graphs comparing doses at inflection point versus field size for both energies at various depths were plotted. These inflection points were derived for each profile from the method which was explained in the above paragraph. Based on the principle of polynomial equation, the inflection point on FFF curve is computed using the trend line option for curves. All fields were normalized at Dmax to 100%. Second, readings were then analysed at various depths. where, y =percentage depth dose at inflection point, x = field size, R² = R-Squared value, and DDF = depth dependent factor [see Table 1]. 5. Combination of (multiplication) these two (3 and 4) gives us the factor "y", which is percentage depth dose at inflection point.

Relation between percentage of dose versus field size in FFF beam
The Figures 1-4 shows that the dose at inflection point for 6X FFF beam increases as the field size increase from 1 × 1 cm 2 to 15 × 15 cm 2 in in-line and up to 12 × 12 cm 2 in cross-line. Beginning from the field sizes (15 × 15 cm 2 in in-line and 12 × 12 cm 2 in cross-line) to the maximum field size (40 × 40 cm 2 ), continuous decrement in dose at inflection point was observed. However, in 10X, the dose at inflection point decreases continuously as the field size increases.  For both the energies, the shapes of the curves are different. In fact, the shape of the curves in 6 MV is different between in-line and cross-line as well. The reason behind the differences of these curves are energy spectrum, scatter dose, lateral photon fluence fall off. As each energy having its own energy spectrum and all other said characters which influenced the shape of the curves. Reduction in head scatter and electron contamination is also contributing to the difference in the shape of these curves.
The in-line direction is more sensitive to the presence or absence of the scattered photons originating from y-jaws, because the y-jaws are closest to the filter location. In 6X, the presence of scatter is more in-line compared with 10X due to its properties and energy spectrum.
Irrespective of field size, in 6XFFF, the maximum values of % depth dose at inflection point were 48.05% and 59.12% in-inline and cross-line, respectively. In 10X FFF, these values were 51.37% and 57.17% in-inline and cross-line, respectively. This is very interesting new relationship, which is presented in Figures 1 to 4

Inflection point calculation on 6X FFF and 10X FFF profiles in both in-line and cross-line
From Figure 1, the reference dose of inflection point at a given location inside profile is y = [-0.0141x 2  In this study, we revealed a connection between the percentage of the dose at the inflection point to the field size and depth. We extended the applications of Y to calculate the inflection point for any field size and depth for both energies. In statistics, the coefficient of determination is denoted by R 2 and the pronounced R squared indicates how well data points fit a line or curve. It is a statistic used in the context of statistical models whose main purpose is either the prediction of future outcomes or the testing of hypotheses, on the basis of other related information. It provides a measure of how well observed outcomes are replicated by the model, as the proportion of total variation of outcomes explained by the model. 15 The shape of the curve was observed different between in-line and cross-line. This may be due to in-plane direction is more sensitive to the presence or absence of the scattered photons originating from the Y jaws, which are closest to the filter location.