School of Cancer Studies, University of Liverpool, Liverpool L69 3GA, United Kingdom.
Med Phys. 2011 Aug;38(8):4518-30. doi: 10.1118/1.3606457.
To derive limits on the numbers of beams needed to deliver near-optimal IMRT, and to assess the accuracy of the limits.
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.
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.
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.
推导出实现近似最优调强放疗（IMRT）所需光束数量的限制，并评估这些限制的准确性。
作者推导出了四种不同的限制。一种是 K(A)，通过将傅里叶技术与用于获得 Bortfeld 限制 K 的证明相结合而得出，如果一个多野计划的所有横截面轮廓都可以表示为半径为 [-R, + R] 的多项式，则可以使用 K 个场在半径为 R 的圆内创建相同的剂量分布。另外两个限制 K(H)和 K(N)是使用采样理论得出的，其中 K(N)限制描述了间隔为奈奎斯特频率的场。K(N)可以推广为 K(N,Fbeamlet)，该限制考虑了从调制场构建的有限大小的束斑。使用 Bortfeld 的理论框架，通过测试 8MV 双高斯铅笔束和 1cm 和 4cm 宽的场的横截面轮廓可以用阶数等于不同限制减去 1 的多项式来近似的程度，探索了限制的准确性。还研究了简单几何形状和头颈部（口咽）病例的 IMRT 计划的优化成本函数值随用于构建计划的光束数量的变化情况。
限制都是 R/W 的倍数（W 是宽场的 20%-80%半影宽度），对于 R = 10cm 和 W = 5.3mm，K = 27、K(A) = 43、K(H) = 34 和 K(N) = 68 个场。所有横截面轮廓都可以用 K(N)-1 阶多项式很好地近似，而 K-1 阶多项式则可以很好地近似铅笔束轮廓，这表明在推导限制 K 时使用的假设存在一些不准确。K(A)-1 阶多项式不能准确描述铅笔束轮廓，但可以很好地近似 1cm 和 4cm 轮廓，因为在这些较宽的场中，较高的空间频率会被衰减。所有的轮廓都可以用 K(N,Fbeamlet(-1))阶多项式很好地表示，随着束斑宽度的增加，它从 K(N)减小。在使用更多的场时，IMRT 计划研究中的成本函数会下降，然后在 K(N,Fbeamlet)场左右趋于平稳。
数值计算表明，实现近似最优调强放疗所需的最小光束数量约为广义奈奎斯特限制 K(N,Fbeamlet)。对于临床现实中 5.3mm 的 20%-80%半影宽度和 10cm 的感兴趣半径，当束斑宽度从 0 增加到 1cm 时，K(N,Fbeamlet)从 68 下降到 47 个场。
