Probabilistic definition of the clinical target volume—implications for tumor control probability modeling and optimization

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.