## 2 Materials and Methods
### The Proposed Soft-Voting Strategy  
The proposed SV strategy is a post-processing step following a 2D-input image classification model. As shown in Figure 2, after slice-wise malignancy prediction, the proposed voxel weighting strategy takes both tumor sizes and predicted probabilities (benign and malignant) into consideration for tumor classification.  
At first, a convolutional neural network (CNN) is employed for slice-wise prediction. After the CNN model is trained, to an unseen input image slice (\(s_{i}\)), its output is two prediction probabilities (benign, \(p_{i}^{b}\); malignant, \(p_{i}^{m}\); and \(p_{i}^{b}+p_{i}^{m}=1\)).  
Then, the proposed SV strategy is implemented with voxel-level weighting. To a voxel in slice \(s_{i}\) of tumor region, its predicted probability is \(p_{i}^{b}\) and \(p_{i}^{m}\). Since a tumor is a volume with many slices and voxels, we assume that each voxel contributes to the prediction of tumor malignancy, and thus, voxel weighting becomes reasonable.  
Specifically, assume that a volumetric tumor \(v\) contains \(n\) slices \(\{s_{i}\}_{i=1}^{n}\), and the voxel numbers in slices is \(\{r_{i}\}_{i=1}^{n}\) correspondingly. And thereby, the benign probability \(p_{v}^{b}\) and the malignant probability \(p_{v}^{m}\) of the tumor can be defined in Equation (1). The numerators stand for the contribution of voxel-weighted benign or malignant probabilities, and the denominators denote the number of voxels or the tumor size. Since the denominators are the same \(\sum_{i=1}^{n}r_{i}\), the values of the numerators are directly related to the final tumor classification.  
\[\left\{\begin{aligned} p_{v}^{b}&=\frac{\sum_{i=1}^{n }p_{i}^{b}r_{i}}{\sum_{i=1}^{n}r_{i}}\propto\sum_{i=1}^{n}p_{i}^{b}r_{i}\\ p_{v}^{m}&=\frac{\sum_{i=1}^{n}p_{i}^{m}r_{i}}{\sum_ {i=1}^{n}r_{i}}\propto\sum_{i=1}^{n}p_{i}^{m}r_{i}\end{aligned}\right. \tag{1}\]  
Figure 1: The distribution of voxel numbers of tumor regions.  
Figure 2: The proposed soft-voting strategy. After slice-wise prediction, the strategy accounts both tumor sizes and voxel probabilities for tumor classification.  
In the end, the classification \(p_{v}^{sv}\) of the volumetric tumor is determined by the larger value of \(p_{v}^{b}\) and \(p_{v}^{m}\) as shown in Equation (2). The core idea of voxel weighting or the SV strategy is that the voxel numbers of a tumor and the voxel probabilities of malignancy derived from slice-wise prediction are utilized.  
\[p_{v}^{sv}=\max\{p_{v}^{b},p_{v}^{m}\} \tag{2}\]  
Compared to the baseline CNN model used in the workflow, as a post-processing strategy, voxel weighting slightly increases the computing time in tumor classification, while it enhances the prediction robustness and decision-making confidence.  
In contrast to voxel weighting, a more straightforward strategy is HV, or slice-level weighting. It mainly depends on the number of slices predicted as benign or as malignant. In other words, among the tumor with \(n\) slices of lesion regions, if \(k\) slices are predicted as malignant (\(p^{m}>p^{b}\)) and \(k>(n-k)\), the volumetric tumor is voted as malignant, and vice versa. Equation (3) shows the comparison of the number of benign (\(k\)) and malignant (\(n-k\)) slices in the volumetric tumor.  
\[p_{v}^{lv}=\max\{n-k,k\} \tag{3}\]