In this paper, we study an interesting geometric partition problem, called optimal field splitting, which arises in Intensity-Modulated Radiation Therapy (IMRT). In current clinical practice, a multi-leaf collimator (MLC) with a maximum leaf spread constraint is used to deliver the prescribed intensity maps (IMs). However, the maximum leaf spread of a MLC may require to split a large intensity map into several overlapping sub-IMs with each being delivered separately. We develop a close-to-linear time algorithm for solving the field splitting problem while minimizing the total complexity of the resulting sub-IMs, thus improving the treatment delivery efficiency. Meanwhile, our algorithm strives to minimize the maximum beam-on time of those sub-IMs. Our basic idea is to formulate the field splitting problem as computing a shortest path in a directed acyclic graph, which expresses a special “layered” structure. The edge weights of the graph satisfy the Monge property, which enables us to solve this shortest path problem by examining only a small portion of the graph, yielding a close-to-linear time algorithm. To minimize the maximum beam-on time of the resulting sub-IMs, we consider an interesting min–max slope path problem in a monotone polygon which is solvable in linear time. The min–max slope path problem may be of interest in its own right. Experimental results based on real medical data and computer generated IMs showed that our new algorithm runs fast and produces high quality field splitting results.

在本文中，我们研究了一个有趣的几何分割问题，称为最佳场分裂，它出现在调强放射治疗(IMRT) 中。在目前的临床实践中，具有最大叶展的多叶准直器 (MLC)约束用于提供规定的强度图 (IM)。然而，MLC 的最大叶扩展可能需要将大强度图拆分为几个重叠的子 IM，每个子 IM 分别传送。我们开发了一种接近线性的时间算法来解决场分裂问题，同时最大限度地减少产生的子 IM 的总复杂度，从而提高治疗传递效率。同时，我们的算法力求最小化这些子 IM 的最大波束开启时间。我们的基本思想是将场分裂问题表述为计算有向无环图中的最短路径，它表达了一种特殊的“分层”结构。图的边权重满足 Monge 性质，这使我们能够通过检查图的一小部分来解决这个最短路径问题，产生接近线性的时间算法。为了最小化结果子 IM 的最大波束开启时间，我们考虑了一个有趣的最小-最大斜率路径问题，该问题可以在线性时间内解决。最小-最大斜率路径问题本身可能很有趣。基于真实医疗数据和计算机生成的 IM 的实验结果表明，我们的新算法运行速度快，并产生高质量的场分裂结果。