## 3 Materials and Methods
### Model Description and Primary End-Point Measurement  
The neural network employed for the experiments was built from scratch, following the description provided in the paper from [PERSON] et al. [51]. This 3D segmentation method is based on the U-Net 3D fully convolutional neural network [42]. As classical CNNs, U-Net 3D contains a contraction and an expansion path. The former, also called the encoder, is used to capture the context in the image. The latter, the decoder, is a symmetric expanding path whose goal is to enable precise localization using transposed convolutions. In the encoder path, we employed three down sampling achieved by means of two convolution set, ReLU, and batch normalization, followed by a maxpooling layer. On the other hand, the decoder path is composed by three up sampling phases, each one composed by a convolution, ReLU, and batch normalization, followed by a transposed convolution. Skip connecting are employed among latent space with the same spatial resolution as in the standard UNet. All the layers are detailed in Appendix A. In this specific context, the decoder produces a probability map assigning to each voxel of the input volume a score between 0 and 1. This output score has been later assessed by the threshold to distinguish voxels inside and outside the volume. The PyTorch-based [34] implementation of this model can be found on [[https://ditto.ing.unimore.it/maxillo/](https://ditto.ing.unimore.it/maxillo/)]([https://ditto.ing.unimore.it/maxillo/](https://ditto.ing.unimore.it/maxillo/)) (accessed on 28 February 2023). The primary endpoint has been the measurement of the accuracy of the annotations created by the CNN, compared to the ground-truth volume generated by the medical team. Sorensen-Dice similarity (_Dice_) score and Intersection over union (_IoU_) measurements have been performed for every test. These allow us to calculate the similarity and the amount of overlapping between two objects. They both range between 0 and 1; the closer the value is to 1, the greatest is the accuracy of the model. _IoU_ is the Intersection over Union, and in this study, serves to measure the quality _of_ model prediction. More specifically, we name \(P\) and \(G\) the set of foreground voxels in the predicted segmentation and ground-truth volume, respectively. The _IoU_ is then calculated as:  
\[IoU=\frac{\#\left(P\ \cap\ \ G\right)}{\#\left(P\ \cup\ \ G\right)}\]  
where # is the cardinality of the set. If the prediction is equal to the ground-truth, the score is 1. If the predicted voxels have no overlap with the ground-truth, the score is 0. The _Dice_ score, also known as Sorensen-Dice index, is given by the following formula:  
\[Dice=\frac{2\;*\;\;\#\;(P\;\cap\;\;G)}{\#\;P+\#\;G}\]  
In order to ensure comparison with other papers, both Intersection over Union and Sorensen-Dice similarity (_Dice_ score) are presented in the experimental Results section, even though their values are very strictly correlated. Their values have been calculated on the test set. Three experiments with specific aims have been conducted. They are schematized in Figure 4 and listed below:  
1. In the first experiment, the CNN was trained twice on the primary dataset. First, only using the sparse annotations extended with the circular expansion technique (experiment 1A). Second, the CNN was trained on the same number of volumes, but this time using dense annotations (experiment 1B). The results in terms of IoU and Dice score were then compared. 2. In the second experiment, the results of the CNN obtained with two different techniques were compared (experiment 2). In one case, the CNN was trained using only the secondary dataset with circular expansion. In the second case, the CNN was trained with a dataset composed of the whole secondary dataset and of the sparse annotations on the primary dataset, both undergoing a circular expansion. 3. In the third experiment, two attempts were made.