Numbers of beam angles required for near‐optimal IMRT: Theoretical limits and numerical studies

Purpose: To derive limits on the numbers of beams needed to deliver near‐optimal IMRT, and to assess the accuracy of the limits. Methods: The authors four different limits have been derived. One, K A , has been obtained by coupling Fourier techniques with a proof used to obtain Bortfeld's limit, K, that if all the cross‐profiles of a many‐field plan can be represented as polynomials of order (K−1) over the range [−R, + R], then within the radius R circle an identical dose‐distribution can be created using just K fields. Two further limits, K H and K N, have been obtained using sampling theory, the K N limit describing fields spaced at the Nyquist frequency. K N can be generalized to K N,Fbeamlet , a limit that accounts for the finite size of the beamlets from which modulated fields are constructed. Using Bortfeld's theoretical framework, the accuracy of the limits has been explored by testing how well the cross‐profiles of an 8 MV double‐Gaussian pencil beam and of 1 and 4 cm wide fields can be approximated by polynomials of orders equal to the different limits minus one. The dependence of optimized cost function values of IMRT plans, generated for a simple geometry and for a head‐and‐neck (oropharynx) case, on the numbers of beams used to construct the plans has also been studied. Results: The limits are all multiples of R/W (W being the 20%–80% penumbra‐width of a broad field) and work out at K = 27, K A  = 43, K H  = 34, and K N  = 68 fields for R = 10 cm and W = 5.3 mm. All and none of the cross‐profiles are approximated well by polynomials of order K N −1 and K−1, respectively, suggesting some inaccuracy in the assumptions used to derive the limit K. Order K A −1 polynomials cannot accurately describe the pencil beam profile, but do approximate the 1‐ and 4‐cm profiles reasonably well because higher spatial frequencies are attenuated in these wider fields. All the profiles are represented well by polynomials of order K N,Fbeamlet −1, which decreases from K N as beamlet width increases. Cost functions generated in the IMRT planning study fall as greater numbers of fields are used, before plateauing out around K N,Fbeamlet fields. Conclusions: Numerical calculations suggest that the minimum number of fields required for near‐optimal IMRT lies around the generalized Nyquist limit K N,Fbeamlet . For a clinically realistic 20%–80% penumbra‐width of 5.3 mm and a radius of interest of 10 cm, K N,Fbeamlet falls from 68 to 47 fields as the beamlet width rises from 0 to 1 cm.