## 2 Materials and Methods
### Validation
#### 2.3.2 Quantitative Validation  
Residuals of tumor extracted by our algorithm were quantitatively compared with manual annotations considered as ground truth. Manual segmentation in the iUS data is a complex task due to the unclear representation of tumorous structure borders. Therefore, the method validation was done in two steps, namely the comparison of (1) the localization of areas containing the tumor residuals and (2) voxel classification. First, the tumorous structures detected by the algorithm and the manual annotations were enclosed in 3D bounding boxes. The overlap coefficient (\(Overlap\)) of these boxes was used as a similarity measure to assess the spatial localization of tumor residuals as proposed by [PERSON] et al. [38]. Figure 4: Image fusion approach for residual brain tumor identification. The border of the resection cavity and highlighted structures are respectively extracted from B-mode and CEUS. Afterwards, they are combined on the feature-level fusion step. Finally, the expected result is obtained by selecting only specific structures based on the rules defined in the decision-level fusion step. Indeed, an \(Overlap\) value of 1 is reached when one box is completely enclosed in the other one. Moreover, a value of 0 occurs when there is no intersection between both boxes. Several boxes were used when different disconnected regions were detected. The final \(Overlap\) index was the average of indices calculated for each box. According to the application, this coefficient allows one to evaluate detection methods through a binary output based on a threshold value (i.e., detected or no detected). For instance, threshold values of 0.3 and 0.5 were set for target detection in [39] and [40], respectively. Thus, in our application, a threshold value of 0.5 has been selected for evaluating the proposed approach. The task of tumor residuals' detection was considered as succeed when \(Overlap\geq\) 0.5 and failed otherwise. This evaluation methodology, as illustrated in the 3D iUS images in Figure 5 for 3 patients (1, 6 and 16), provides information about the intersection rate between the two volume boxes. The green and red bounding boxes encompass respectively the brain areas identified by the algorithm and the ground truths. This similarity measure is described as follows:  
\[Overlap=\frac{BB_{al}\cap BB_{gt}}{min(BB_{al},BB_{gt})} \tag{2}\]  
where \(BB_{al}\) and \(BB_{gt}\) are the bounding boxes enclosing the brain areas detected by the algorithm and those manually annotated (ground truth), respectively. Second, the additional metrics, including accuracy (\(Acc\)), area under the ROC curve (\(AUC\)) [41] and error rate (\(Err\)) or percentage of wrong classifications [42], were calculated to evaluate the voxels classification as the tumor residual or healthy tissue by the method. This evaluation was carried out by interactively defining a region of interest enclosing the resection cavity where the remnant tumors can be found. Furthermore, these metrics were computed only for the cases where the method succeeds to identify tumor residuals based on the first quantitative metric (i.e., \(Overlap\geq\) 0.5). These similarity measures take values in the interval [0,1]. \(Acc\) and \(AUC\) values of 1, and \(Err\) of 0 value represent the best performance of the algorithm. They are calculated as:  
\[Acc=\frac{TP+TN}{TP+TN+FP+FN} \tag{3}\]  
\[AUC=\frac{1}{2}(\frac{TN}{TN+FP}+\frac{TP}{TP+FN}) \tag{4}\]  
\[Err=\frac{FP+FN}{TP+TN+FP+FN} \tag{5}\]  
where \(TP\), \(TN\), \(FP\) and \(FN\) are:  
True positive (voxels correctly classified as tumorous tissue), true negative (voxels correctly classified as healthy tissue), false positive (healthy tissue misclassified as tumor region) and false  
Figure 5: 3D representation of the quantitative evaluation approach on Patients 1, 6 and 16. \(BB_{d}\) is the algorithm result’s bounding box, and \(BB_{gt}\) is the ground truth’s bounding box. negative (undetected tumorous tissue), respectively.