Massachusetts General Hospital and Harvard Medical School, Department of Radiation Oncology, 30 Fruit St, Boston, MA 02114, USA.
Phys Med Biol. 2010 Jan 7;55(1):83-97. doi: 10.1088/0031-9155/55/1/006.
The first purpose of this paper is to shed some new light on the old question of selecting the number of beams in intensity-modulated radiation therapy (IMRT). The second purpose is to illuminate the related issue of discrete static beam angles versus rotational techniques, which has recently re-surfaced due to the advancement of volumetric modulated arc therapy (VMAT). A specific objective is to find analytical expressions that allow one to address the points raised above. To make the problem mathematically tractable, it is assumed that the depth dose is flat and that the lateral dose profile can be approximated by polynomials, specifically Chebyshev polynomials of the first kind, of finite degree. The application of methods known from image reconstruction then allows one to answer the first question above as follows: the required number of beams is determined by the maximum degree of the polynomials used in the approximation of the beam profiles, which is a measure of the dose variability. There is nothing to be gained by using more beams. In realistic cases, in which the variability of the lateral dose profile is restricted in several ways, the required number of beams is of the order of 10-20. The consequence of delivering the beams with a 'leaf sweep' technique during continuous rotation of the gantry, as in VMAT, is also derived in an analytical form. The main effect is that the beams fan out, but the effect near the axis of rotation is small. This result can serve as a theoretical justification of VMAT. Overall the analytical derivations in this paper, albeit based on strong simplifications, provide new insights into, and a deeper understanding of, the beam angle problem in IMRT. The decomposition of the beam profiles into well-behaved and easily deliverable smooth functions, such as Chebyshev polynomials, could be of general interest in IMRT treatment planning.
本文的首要目的是为强度调制放射治疗（IMRT）中光束数量选择这一古老问题提供新的见解。第二个目的是阐明与离散静态射束角度与旋转技术相关的问题，由于容积调强弧形治疗（VMAT）的进步，该问题最近再次出现。一个具体目标是找到解析表达式，以解决上述问题。为使问题在数学上易于处理，假设深度剂量是平坦的，并且侧向剂量分布可以用多项式近似，特别是第一类切比雪夫多项式，其阶数有限。然后，应用来自图像重建的方法可以使我们回答如下上述第一个问题：所需的光束数量由用于光束轮廓逼近的多项式的最大阶数决定，这是剂量变化的度量。使用更多的光束没有任何好处。在实际情况下，侧向剂量分布的变化受到多种方式的限制，所需的光束数量约为 10-20。在连续旋转机架的情况下，采用“叶片扫描”技术输送光束的结果（如 VMAT 中那样）也以解析形式得出。主要影响是光束散开，但在旋转轴附近的影响很小。这一结果可以作为 VMAT 的理论依据。总的来说，本文中的分析推导尽管基于很强的简化，但为 IMRT 中的射束角度问题提供了新的见解和更深入的理解。将光束轮廓分解为易于处理的规则和光滑的函数，如切比雪夫多项式，可能对 IMRT 治疗计划具有普遍意义。
