A pencil beam kernel model for flattening filter-free X-ray beams

Fast and accurate dose computation is an important requirement for algorithms that are often used in optimization schemes. Decreasing the number of variables and parameters and the amount of tabulated data can reduce computation time. Flattening Filter-Free (FFF) beams provide reduced profile shape variations with depth relative to flattened beams. Therefore, the pencil beam kernel of a FFF beam must exhibit the reduced variation with depth when compared to the kernel of flattened beams. In this paper, a kernel with a minimal number of parameters is derived for the FFF beams. Moreover, some of the parameters are defined as depth independent. A finite-size pencil beam dose calculation model was used for kernel generation. The grid size for the dose calculation was set to 2.5 mm. During the kernel generation, the parameters (pre-exponential constants and exponential constants) of the kernel were determined in such a way that the difference between the computed and measured profiles is minimized by the global gamma analysis technique. The criteria for this technique were 1 % dose difference at distance of 1 mm with a 10 % threshold. Profiles for each field (5 x 5 cm², 10 x 10 cm², and 20 x 20 cm²) at five standard depths (dmax, 5 cm, 10 cm, 20 cm, and 30 cm), a total of 15 profiles, were used to generate the kernels. The multi-objective, non-derivative, unconstrained, non-linear optimization method in the programming package MATLAB (Mathworks, Natick, MA) optimization toolbox was used to generate kernel parameters. Commissioning of the model was performed for the static fields and the intensity-modulated radiation therapy (IMRT) fields. In static fields and dynamic IMRT fields, more than 95 % of data points satisfied the criteria defined in the global gamma analysis with 3 % and 3 mm. There was a good agreement between modelled and measured data in both cases. It is demonstrated that the pencil beam model developed in this study could be used for FFF x-ray beams. Pencil beam kernel parameters do not need to be defined at each depth.