On expedient properties of common biological score functions for multi-modality, adaptive and 4D dose optimization

Frequently, radiotherapy treatments are comprised of several dose distributions computed or optimized in different patient geometries. Therefore, the need arises to compute the comprehensive biological effect or physical figure of merit of the combined dose of a number of distinct geometry instances. For that purpose the dose is typically accumulated in a reference geometry through deformation fields obtained from deformable image registration. However, it is difficult to establish precise voxel-by-voxel relationships between different anatomical images in many cases. In this work, the mathematical properties of commonly used score functions are exploited to derive an upper boundary for the maximum effect for normal tissue and a lower boundary for the minimum effect for the target of accumulated doses on multiple geometry instances.