## 2 Materials and Methods
### Tumor Grading  
The tumors were graded using the original 3D CNN model designed by us. In this model, grading was performed on the volume of interest (VOI) that was cropped according to the size of the tumor region. The VOIs were cubical, and the size to be extracted was taken as twice the maximum side of the circumscribed rectangle of the tumor; this was done to consider the information regarding the surrounding brain parenchyma. Data augmentation was performed to prevent overfitting when training the 3D CNN for grading. All the VOIs were laterally flipped, and lateral and cephalocaudal rotations were performed between \(-10^{\circ}\) and \(10^{\circ}\). Because a gap existed in the number of cases between grade II/III and grade IV tumors, the rotation interval was adjusted to \(5^{\circ}\) for grades II and III and \(10^{\circ}\) for grade IV to balance the training data. Consequently, the number of training images for grades II and III were 3200 VOIs and 3402 for grade IV. Additionally, the images were distorted by randomly expanding the coordinates of the eight apexes of each VOI along the lateral, cephalocaudal, and anteroposterior directions; the maximum expansion and contraction distances were 10% the matrix size on each side of the VOIs. The VOIs were then resized to 64 \(\times\) 64 \(\times\) 64 voxels, and after normalization, they were fed to the 3D CNN for grading. The voxel values were normalized by dividing each voxel value of VOI by the maximum voxel value of all VOIs in the dataset. Here, the maximum value was calculated using denoised VOI after applying the median filter to the original images.  
Figure 2: 3D CNN architecture used for extracting the brain tumor region. This model was provided by the Clara project developed by NVIDIA. It comprises ResNet-like blocks and skip connections.  
The architecture of the 3D CNN model used for grading is shown in Figure 3. This model is the 3D extension of ResNet and comprises residual blocks [25]. Each ResNet-like block comprises three sets of batch-normalization layers, ReLU function, and convolutional layer. Additionally, the skip connections in the residual blocks reduce the vanishing gradient. Convolution was performed in the convolutional layer using filters with a kernel size of 1 \(\times\) 1 \(\times\) 1 or 3 \(\times\) 3 \(\times\) 3 and a stride of 1, following which image features were extracted. A filter with a kernel size of 1 \(\times\) 1 \(\times\) 1 was used to reduce the dimension, and subsequent filters with a kernel size of 3 \(\times\) 3 \(\times\) 3 and 1 \(\times\) 1 \(\times\) 1 was used to restore the dimension. This bottleneck architecture prevents the computational efficiency degradation due to multi-layered structures [26]. The first residual block has max pooling just before the first convolution and the image size was halved. Furthermore, when three or four residual blocks were passed, the stride was changed to 2, and the image size was halved. Finally, in the fully connected layer, these image features were integrated, and the grading result of the tumor was obtained using the softmax function. The batch size was 16, the number of epochs 30, and SGD (learning rate = 1 \(\times\) 10\({}^{-4}\), momentum = 0.9) was adopted as the optimization algorithm [27]. During the training period, 20% of the training dataset was randomly selected and validated. The NVIDIA GeForce RTX 2080 Ti 11 GB GPU was used to train this model, which was implemented using the Keras and TensorFlow AI platforms.