Development of a fast and feasible spectrum modeling technique for flattening filter free beams

To develop a fast and robust technique for the determination of optimized photon spectra for flattening filter free (FFF) beams to be applied in convolution/superposition dose calculations.

A two-step optimization method was developed to derive optimal photon spectra for FFF beams. In the first step, a simple functional form of the photon spectra proposed by Ali ["Functional forms for photon spectra of clinical linacs," Phys. Med. Biol. 57, 31-50 (2011)] is used to determine generalized shapes of the photon spectra. In this method, the photon spectra were defined for the ranges of field sizes to consider the variations of the contributions of scattered photons with field size. Percent depth doses (PDDs) for each field size were measured and calculated to define a cost function, and a collapsed cone convolution (CCC) algorithm was used to calculate the PDDs. In the second step, the generalized functional form of the photon spectra was fine-tuned in a process whereby the weights of photon fluence became the optimizing free parameters. A line search method was used for the optimization and first order derivatives with respect to the optimizing parameters were derived from the CCC algorithm to enhance the speed of the optimization. The derived photon spectra were evaluated, and the dose distributions using the optimized spectra were validated.

The optimal spectra demonstrate small variations with field size for the 6 MV FFF beam and relatively large variations for the 10 MV FFF beam. The mean energies of the optimized 6 MV FFF spectra were decreased from 1.31 MeV for a 3 × 3 cm(2) field to 1.21 MeV for a 40 × 40 cm(2) field, and from 2.33 MeV at 3 × 3 cm(2) to 2.18 MeV at 40 × 40 cm(2) for the 10 MV FFF beam. The developed method could significantly improve the agreement between the calculated and measured PDDs. Root mean square differences on the optimized PDDs were observed to be 0.41% (3 × 3 cm(2)) down to 0.21% (40 × 40 cm(2)) for the 6 MV FFF beam, and 0.35% (3 × 3 cm(2)) down to 0.29% (40 × 40 cm(2)) for the 10 MV FFF beam. The first order derivatives from the functional form were found to improve the speed of computational time up to 20 times compared to the other techniques.

The derived photon spectra resulted in good agreements with measured PDDs over the range of field sizes investigated. The suggested method is easily applicable to commercial radiation treatment planning systems since it only requires measured PDDs as input.