## 2 Materials and Methods
### Base Models' Training and Their Hyperparameter Optimization  
As discussed in Section 1, ResNet, DenseNet, and EfficientNet are extremely successful in addressing many different problems related to medical imaging with CT data. However, their potential for addressing spine-related issues has been little explored. Motivated by the ensemble learning theory, which suggests using heterogeneous models to maximize diversity and improve generalization in stacking [27], and also given the recognized advantages in extracting different features, the models WideResNet-40 (WRN-40) [28], DenseNet-121 [29], and EfficientNet-B7 [30] are chosen as base models for the proposed methodology to detect C-spine vertebral fractures. These models have different architectures, depth, and feature extraction strategies, ensuring complementary learning. Additionally, they are computationally efficient and effectively alleviate the vanishing gradient problem using various strategies. ResNet-152 [31] and ResNet-200 [32] are also utilized to train on our dataset to validate the choice of base models. Both variants of ResNet are trained using the Adam optimizer, with a batch size of 32 and for 200 epochs. To avoid overfitting, L2 regularization is employed in both models during training. A similar hyperparameter setup, which is used for the chosen base models, is introduced in these two models. The training and validation accuracy patterns are shown in Figure 4a indicating significant fluctuations in ResNet-152, suggesting that the model is prone to underfitting and capturing noise rather than learning the underlying patterns in the data. In the case of ResNet-200, a significant gap is observed between the training and validation accuracy shown in Figure 4b, which demonstrates an overfitting issue. Additionally, the accuracy, specificity, sensitivity, and F1-score achieved by ResNet-152 are 62.9%, 58.37%, 61.09% and 59.72%, respectively. And, ResNet-200 achieved 64.17% accuracy, 56.02% specificity, 60.93% sensitivity, and 61.4% F1-score. These results are also illustrated in Section 3, which are not very encouraging, indicating that neither variant is a suitable candidate for a base learner. A brief overview of the base models along with their core components, and the training, as well as the fine-tuning of the hyperparameters, are explained below. A multi-layered network is required to detect tiny features, including subtle textures, shapes, and structures, such as hairline fracture. And deeper ResNet, which did not show promising results, might encounter vanishing gradient problems. To address this, WideResNet, an improved version of ResNet was proposed by [PERSON] et al., in which the number of filters per layer is increased. This reduced the need of the network to be deeper. The skip connection, shown in Figure 5a, is the core component of WRN-40 that improves feature flow and feature reuse across all layers by eliminating the vanishing gradient issue. The input \(X\) in Figure 5a is sent to \(1\times 1\) convolution, which reduces the number of channels in the feature maps. Then, the output feature maps are fed into a \(5\times 5\) convolution block, a larger filter, that extracts the features well, followed by a \(1\times 1\) convolution, which restores the original channel dimension of the feature maps. The number of kernels in the \(5\times 5\) convolution block is \(16k\), where \(k\) represents the widening factor, and the base number of channels is 16. Choosing the value \(k=4\), the number of kernels becomes 64 after widening. Dense connection is the core component of DenseNet, it improves feature propagation and allows the reuse of features across the DenseNet network. The architecture of the first dense block of DenseNet-121 is shown in Figure 5b, which demonstrates that each convolution layer in the dense block receives feature maps from all preceding layers, helping the network to capture and utilize information from different levels of the model.