Evidence has been presented that moving beyond the binary definition of clinical target volume (CTV) towards a probabilistic CTV can result in better treatment plans. The probabilistic CTV takes the likelihood of disease spread outside of the gross tumor into account. An open question is: how to optimize tumor control probability (TCP) based on the probabilistic CTV. We derive expressions for TCP under the assumptions of voxel independence and dependence. For the dependent case, we make the assumption that tumors grow outward from the gross tumor volume. We maximize the (non-convex) TCP under convex dose constraints for all models. For small numbers of voxels, and when a dose-influence matrix is not used, we use exhaustive search or Lagrange multiplier theory to compute optimal dose distributions. For larger cases we present (1) a multi-start strategy using linear programming with a random cost vector to provide random feasible starting solutions, followed by a local search, and (2) a heuristic strategy that greedily selects which subvolumes to dose, and then for each subvolume assignment runs a convex approximation of the optimization problem. The optimal dose distributions are in general different for the independent and dependent models even though the probabilities of each voxel being tumorous are set to the same in both cases. We observe phase transitions, where a subvolume is either dosed to a high level, or it gets ‘sacrificed’ by not dosing it at all. The greedy strategy often yields solutions indistinguishable from the multi-start solutions, but for the 2D case involving organs-at-risk and the dependent TCP model, discrepancies of around 5% (absolute) for TCP are observed. For realistic geometries, although correlated voxels is a more reasonable assumption, the correlation function is in general unknown. We demonstrate a tractable heuristic that works very well for the independent models and reasonably well for the dependent models. All data are provided.

已有证据表明，将临床目标体积 (CTV) 的二元定义转向概率 CTV 可以产生更好的治疗计划。概率 CTV 将疾病传播到大体肿瘤之外的可能性考虑在内。一个悬而未决的问题是：如何基于概率 CTV 优化肿瘤控​​制概率 (TCP)。我们在体素独立和依赖的假设下推导出 TCP 的表达式。对于依赖病例，我们假设肿瘤从总肿瘤体积向外生长。我们在所有模型的凸剂量约束下最大化（非凸）TCP。对于少量体素，并且不使用剂量影响矩阵时，我们使用穷举搜索或拉格朗日乘数理论来计算最佳剂量分布。对于较大的情况，我们提出（1）使用线性规划的多开始策略和随机成本向量来提供随机可行的起始解决方案，然后进行局部搜索，以及（2）贪婪地选择要给药的子体积的启发式策略，以及然后对于每个子体积分配运行优化问题的凸近似。对于独立模型和依赖模型，最佳剂量分布通常是不同的，即使在两种情况下每个体素肿瘤的概率都设置为相同。我们观察到相变，其中一个子体积要么被投加到高水平，要么被“牺牲”而根本不投加。贪婪策略通常会产生与多开始解决方案无法区分的解决方案，但对于涉及器官风险和依赖 TCP 模型的 2D 案例，观察到 TCP 的差异约为 5%（绝对值）。对于现实几何，虽然相关体素是一个更合理的假设，但相关函数通常是未知的。我们展示了一种易于处理的启发式方法，它对独立模型非常有效，对依赖模型也相当好。提供所有数据。