MCM: The Mathematical Contest in Modeling 2003 MCM Problems PROBLEM A: The stunt Person 问题A:特技人 An exciting action scene in a movie is going to be filmed, and you are the stunt coordinator! 要拍摄一部电影中的令人兴奋的表演场景,而你是特级表演的合作者 A stunt person on a motorcyde will jump over an elephant and land in a pile of cardboard boxes to cushion their fall 骑摩托车的特技人将要跳跃一头大象后落到能使他软着陆的一堆纸箱中, You need to protect the stunt person, and also use relatively few cardboard boxes(lower cost, not seen by camera, etc. 你需要保护特技人,但是还使用相对少的纸箱(降低成本,不被摄象机摄到等。) Your job is to determine what size boxes 确定要使用的箱子的尺寸 determine how many boxes to use 确定要使用多少箱子 determine how the boxes will be stacked 确定要如何堆放箱子 determine if any modifications to the boxes would help 确定如果对箱子做修改的话是否能有帮助, generalize to different combined weights(stunt person motorcycle) and different jump heights 推广到不同的组合重量(特技人与摩托车)和不同的跳跃高度 Note that, in"Tomorrow Never Dies", the James bond character on a motorcycle jumps over a helicopter. 注意,在“明日帝国”, the james bond character骑着摩托车跳过一架直升飞机 PROBLEM B: Gamma Knife Treatment Planning
MCM: The Mathematical Contest in Modeling 2003 MCM Problems PROBLEM A: The Stunt Person 问题 A： 特技人 An exciting action scene in a movie is going to be filmed, and you are the stunt coordinator! 要拍摄一部电影中的令人兴奋的表演场景，而你是特级表演的合作者。 A stunt person on a motorcycle will jump over an elephant and land in a pile of cardboard boxes to cushion their fall. 骑摩托车的特技人将要跳跃一头大象后落到能使他软着陆的一堆纸箱中。 You need to protect the stunt person, and also use relatively few cardboard boxes (lower cost, not seen by camera, etc.). 你需要保护特技人，但是还使用相对少的纸箱（降低成本，不被摄象机摄到等。） Your job is to: • determine what size boxes to use 确定要使用的箱子的尺寸 • determine how many boxes to use 确定要使用多少箱子 • determine how the boxes will be stacked 确定要如何堆放箱子 • determine if any modifications to the boxes would help 确定如果对箱子做修改的话是否能有帮助， • generalize to different combined weights (stunt person & motorcycle) and different jump heights 推广到不同的组合重量（特技人与摩托车）和不同的跳跃高度 Note that, in "Tomorrow Never Dies", the James Bond character on a motorcycle jumps over a helicopter. 注意，在“明日帝国”，the James Bond character 骑着摩托车跳过一架直升飞机。 PROBLEM B: Gamma Knife Treatment Planning
问题B:咖玛刀治疗方案 Stereotactic radiosurgery delivers a single high dose of ionizing radiation to a radiographically vell-defined, small intracranial 3D brain tumor without delivering any significant fraction of the prescribed dose to the surrounding brain tissue. 立体放射外科技术对一个由放射照相技术精确定位的颅内小型三维脑瘤发出高剂量的致电离辐 射 Three modalities are commonly used in this area; they are the gamma knife unit, heavy charged particle beams, and external high-energy photon beams from linear accelerators 该领域常使用三种方式,即从线性加速器发出的咖玛刀单元,充电重粒子束和外部高能光子束。 The gamma knife unit delivers a single high dose of ionizing radiation emanating from 201 cobalt-60 unit sources through a heavy helmet. 咖玛刀单元通过一个重型头盔从201钴60单元源发出一个高剂量致电离辐射 All 20 1 beams simul taneously intersect at the isocenter, resulting in a spherical (approximately) dose distribution at the effective dose levels 全部201致电离辐射束同时相交于脑瘤中心,导致一个近似球形的、剂量水平有效的剂量分布。 Irradiating the isocenter to deliver dose is termed a shot. Shots can be represented as different spheres. 照射脑瘤中心而发出剂量叫做一次“照射”。可用不同的球形表示若干次照射。 Four interchangeable outer collimator helmets with beam channel diameters of 4, 8, 14, and 18 mm are available for irradiating different size volumes. 对照射不同体积的脑瘤可使用致电离辐射束直径为4、8、14、和18mm的四个可互换的瞄准仪 头盔。 For a target volume larger than one shot, mul tiple shots can be used to cover the entire target. 对于比一次照射大的目标脑瘤体积,可使用多次照射以覆盖整个目标。 In practice, most target volumes are treated with 1 to 15 shots 实践中,大部分的脑瘤目标体积是用1-15次照射进行治疗的。 The target volume is a bounded, three-dimensional digital image that usually consists of millions of points. 目标脑瘤体积是有界限的、三维的数字图象,由数百万个点组成 The goal of radiosurgery is to deplete tumor cells while preserving normal structures
问题 B： 咖玛刀治疗方案 Stereotactic radiosurgery delivers a single high dose of ionizing radiation to a radiographically well-defined, small intracranial 3D brain tumor without delivering any significant fraction of the prescribed dose to the surrounding brain tissue. 立体放射外科技术对一个由放射照相技术精确定位的颅内小型三维脑瘤发出高剂量的致电离辐 射。 Three modalities are commonly used in this area; they are the gamma knife unit, heavy charged particle beams, and external high-energy photon beams from linear accelerators. 该领域常使用三种方式，即从线性加速器发出的咖玛刀单元，充电重粒子束和外部高能光子束。 The gamma knife unit delivers a single high dose of ionizing radiation emanating from 201 cobalt-60 unit sources through a heavy helmet. 咖玛刀单元通过一个重型头盔从 201 钴 60 单元源发出一个高剂量致电离辐射。 All 201 beams simultaneously intersect at the isocenter, resulting in a spherical (approximately) dose distribution at the effective dose levels. 全部 201 致电离辐射束同时相交于脑瘤中心，导致一个近似球形的、剂量水平有效的剂量分布。 Irradiating the isocenter to deliver dose is termed a “shot.” Shots can be represented as different spheres. 照射脑瘤中心而发出剂量叫做一次“照射”。可用不同的球形表示若干次照射。 Four interchangeable outer collimator helmets with beam channel diameters of 4, 8, 14, and 18 mm are available for irradiating different size volumes. 对照射不同体积的脑瘤可使用致电离辐射束直径为 4、8、14、和 18mm 的四个可互换的瞄准仪 头盔。 For a target volume larger than one shot, multiple shots can be used to cover the entire target. 对于比一次照射大的目标脑瘤体积，可使用多次照射 以覆盖整个目标。 In practice, most target volumes are treated with 1 to 15 shots. 实践中，大部分的脑瘤目标体积是用 1-15 次照射进行治疗的。 The target volume is a bounded, three-dimensional digital image that usually consists of millions of points. 目标脑瘤体积是有界限的、三维的数字图象，由数百万个点组成。 The goal of radiosurgery is to deplete tumor cells while preserving normal structures
放射外科技术的目的是要耗尽肿瘤细胞,同时保护正常的组织结构 Since there are physical limitations and biolo gical uncertainties involved in this therapy process, a treatment plan needs to account for all those limitations and uncertainties. 因为该治疗过程中涉及到一些人体限制和生物不确定性,所以治疗方案需要考虑所有这些限制和 不确定性 In general, an optimal treatment plan is designed to meet the following requirements 般说来,一个最佳的治疗方案的设计要满足如下要求: 1. Minimize the dose gradient across the target volume. 使通过目标脑瘤体积的剂量梯度为最小 2. Match specified isodose contours to the target volumes. 对目标脑瘤体积要匹配特定的同一剂量的外形轮廓 3. Match specified dose-volume constraints of the target and critical organ. 匹配特定的目标和关键器官的剂量体积限制。 4. Minimize the integral dose to the entire volume of normal tissues or organs 使给正常组织和器官的整体剂量为最小。 Constrain dose to specified normal tissue points below tolerance dose 把给特定正常组织的点位的剂量限制在可耐受剂量之下。 6. Minimize the maximum dose to critical volumes 使给关键脑瘤体积的最大剂量为最小 In gamma unit treatment planning, we have the following constraints 咖玛刀单元治疗方案的限制条件如下 1. Prohibit shots from protruding outside the targe 禁止突出目标脑瘤外部的照射。 2. Prohibit shots from overlapping( to avoid hot spots). 禁止重叠照射(以避免热点)
放射外科技术的目的是要耗尽肿瘤细胞，同时保护正常的组织结构。 Since there are physical limitations and biological uncertainties involved in this therapy process, a treatment plan needs to account for all those limitations and uncertainties. 因为该治疗过程中涉及到一些人体限制和生物不确定性，所以治疗方案需要考虑所有这些限制和 不确定性。 In general, an optimal treatment plan is designed to meet the following requirements. 一般说来，一个最佳的治疗方案的设计要满足如下要求： 1. Minimize the dose gradient across the target volume. 使通过目标脑瘤体积的剂量梯度为最小 2. Match specified isodose contours to the target volumes. 对目标脑瘤体积要匹配特定的同一剂量的外形轮廓。 3. Match specified dose-volume constraints of the target and critical organ. 匹配特定的目标和关键器官的剂量体积限制。 4. Minimize the integral dose to the entire volume of normal tissues or organs. 使给正常组织和器官的整体剂量为最小。 5. Constrain dose to specified normal tissue points below tolerance doses. 把给特定正常组织的点位的剂量限制在可耐受剂量之下。 6. Minimize the maximum dose to critical volumes. 使给关键脑瘤体积的最大剂量为最小。 In gamma unit treatment planning, we have the following constraints: 咖玛刀单元治疗方案的限制条件如下： 1. Prohibit shots from protruding outside the target. 禁止突出目标脑瘤外部的照射。 2. Prohibit shots from overlapping (to avoid hot spots). 禁止重叠照射（以避免热点）
3. Cover the target volume with effective dosage as much as possible. But at least 90% of the target volume must be covered by shots 用有效剂量尽可能多地覆盖目标脑瘤体积。但是照射必须覆盖至少90%的目标脑瘤体 积。 4. Use as few shots as possible 使用的照射次数尽可能少 Your tasks are to formulate the optimal treatment planning for a gamma knife unit as a sphere-packing problem, and propose an algorithm to find a solution 你的任务给作为球形包问题的咖玛刀单元制订最佳治疗方案,并提出能够找到解决方案的算法 Thile designing your algorithm, you must keep in mind that your algorithm must be reasonably efficient. 在设计你的算法的同时,你必须记住你的算法必须有效合理
3. Cover the target volume with effective dosage as much as possible. But at least 90% of the target volume must be covered by shots. 用有效剂量尽可能多地覆盖目标脑瘤体积。但是照射必须覆盖至少 90%的目标脑瘤体 积。 4. Use as few shots as possible. 使用的照射次数尽可能少。 Your tasks are to formulate the optimal treatment planning for a gamma knife unit as a sphere-packing problem, and propose an algorithm to find a solution. 你的任务给作为球形包问题的咖玛刀单元制订最佳治疗方案，并提出能够找到解决方案的算法。 While designing your algorithm, you must keep in mind that your algorithm must be reasonably efficient. 在设计你的算法的同时，你必须记住你的算法必须有效合理