Evaluation approach for whole dose distribution in clinical cases using spherical projection and spherical harmonics expansion: spherical coefficient tensor and score method

Whole dose distribution results from well-conceived treatment plans including patient-specific (location, size and shape of tumor, etc.) and facility-specific (clinical policy and goal, equipment, etc.) information. To evaluate the whole dose distribution efficiently and effectively, we propose a method to apply spherical projection and real spherical harmonics (SH) expansion, thus leading to the expanded coefficients as a rank-2 tensor, SH coefficient tensor, for every patient-specific dose distribution. To verify the feature of this tensor, we introduce Isomap from the manifold learning method and multi-dimensional scaling (MDS). Subsequently, we obtained the MDS distance representing similarity, η, and the SH score, ζ, which is a Frobenius norm of the SH coefficient tensor. These were then validated in the intensity-modulated radiation therapy (IMRT) data sets of: (i) 375 mixing treated regions, (ii) 135 head and neck (HN), and (iii) 132 prostate cases, respectively. The MDS map indicated that the SH coefficient tensor enabled a quantitative feature extraction of whole dose distributions. In particular, the SH score systematically detected irregular cases as the deviation higher than +1.5 standard deviations (SD) from the average case, which matched up with clinically irregular case that required very complicated dose distributions. In summary, the proposed SH coefficient tensor is a useful representation of the whole dose distribution. The SH score from the SH coefficient tensor is a convenient and simple criterion used to characterize the entire dose distributions, which is not dependent on the data set.