A proof that uniform dose gives the greatest TCP for fixed integral dose in the planning target volume

Abstract

In this note it is shown that for a fixed integral dose to the planning target volume, the highest tumour control probability (TCP) arises when the dose is spatially uniform. This 'uniform dose theorem' is proved both for (i) a specific TCP model based on Poisson/independent voxel statistics, and (ii) any model for voxel control probability having a specific shape with respect to increasing dose.