## 3 Results
### Correlation Analysis  
The correlation between the TRE and the magnitude of the deformation in the reference DVF was investigated on a voxel basis within the body contour, and a significant correlation (\(p\)-values < 0.01 across all patients) was found, with a Pearson correlation coefficient of 0.79 \(\pm\) 0.14 (range 0.48-0.97), meaning that the DIR spatial errors are a function of the deformation magnitude of each registration.  
As reported in the previously published studies [13; 17; 29], the effect of the DVF on the dose map is complex. Theoretically, it is expected to depend on the local dose gradient and on the spatial registration error [17]. The correlation between the dose map error and TRE was tested on an OAR-by-OAR basis, and the results are shown in Figure 5. The upper panel shows, in a correlation graph, the median value of the TRE versus the median value of the dose error for each OAR. The correlation values (indicated by the diameter of each marker size) varied significantly across the OARs.  
Figure 5: Upper panel: Correlation graph of the median TRE and dose errors values for each OAR. The diameter of each marker is proportional of the Pearson’s correlation coefficient value (see lower panels for individual numeric values). A linear correlation fit line is also reported. Lower panels: Correlation graph of the TRE and dose errors, across 20 registrations divided for each OAR. A linear correlation fit line is also reported. In the lower right corner of all panels, the [PERSON]’s coefficient was reported between TRE and dose error. A “\(\ast\)” indicates that Pearson test confirmed statistically significant correlation.  
The Figure 5 lower panels show, in correlation graphs, the TRE values versus the dose errors for each of the 20 registrations for each OAR. The Pearson coefficients, ranging from 0.21 (larynx) to 0.84 (right parotid), were reported for each panel. Significant correlations for the body, right parotid, mandibula and brainstem were found.  
The correlation between the dose error and TRE was tested also with a voxel-based approach inside the body. A statistically significant correlation was found (\(p\)-values \(<\) 0.01), albeit with a low mean Pearson coefficient of 0.31 \(\pm\) 0.10 (range 0.15-0.53) across all registrations [30]. This confirmed the complex correlation between the dose difference and TRE found on an OAR basis and reported above.  
The dose error showed a moderate but statistically significant correlation with the dose gradient, with a mean Pearson coefficient of 0.51 \(\pm\) 0.06 (range 0.37-0.61) and \(p\)-values \(<\) 0.01 across all patients on a voxel basis, confirming the hypothesis that the local dose gradient influences dose errors related to the DIR uncertainties, as reported also in the previous studies [17; 31].  
As found in the previous studies [13; 17; 29], the dose errors were also investigated as a function of the combination of the dose gradient and the TRE (here the scalar product between the two was considered). A highest mean Pearson coefficient of 0.52 \(\pm\) 0.07 (range 0.41-0.66) and \(p\)-values \(<\) 0.001 were found, with respect to the single factors analysis (TRE and dose gradient).