## 2 Materials and Methods
### Develop Machine Learning-Based Models to Predict Tumor Response to Chemotherapy  
Pearson correlation coefficient (PCC) and LASSO (Least Absolute Shrinkage and Selection Operator) approaches were used to reduce the dimensionality of our feature space. First, the PCC was calculated, and a heatmap was generated to investigate the feature dependency. After the computation, feature pairs with a Pearson correlation coefficient (PCC) exceeding the pre-set threshold were identified, and one of the correlated features was removed [29] to reduce the redundant information. Next, LASSO was applied on the rest of the feature pool to create an optimal feature cluster [30], as it has the capability to handle high-dimensional data with a limited number of observations [31]. Essentially, this method employs a variation of the least squares regression method that employs L1 regularization to produce sparse variable coefficients. Accordingly, LASSO constrains the sum of the absolute values of the regression weights to be smaller than a fixed threshold. Therefore, it improves model interpretability by removing variables irrelevant to the response variable  
Figure 1: Overview of the feature extraction process [11]. and reduces the possibility of overfitting [32]. The LASSO objective function is defined in Equation (1). \[\sum_{i=1}^{n}\left(y_{i}-\sum_{j}x_{ij}\beta_{j}\right)^{2}+\lambda\ \sum_{i=1}^{p}\big{|}\beta_{j}\big{|},\qquad\textit{ subject to }\sum_{j=1}^{p}\big{|}\beta_{j}\big{|}<t \tag{1}\]  
In the above error function, \(t\) is the upper-bound threshold for the sum of the absolute values of feature weights \(\beta\), and \(\lambda\) is a non-negative tuning parameter that regulates the degree of the penalty. We employed the LASSO-cross validation (LASSO-CV) approach to tune the LASSO parameters [33]. After applying LASSO, features with non-zero weights were chosen to be included in the model. Finally, we adopted the synthetic minority oversampling technique (SMOTE) to add more non-responder samples and balance the dataset, which generates samples by interpolating between one selected minority sample with its nearest neighbors, which was performed using scikit-learn library [34]. Next, a support vector machine (SVM) was used to predict the tumor responses (i.e., 6-month PFS). SVM has been widely validated as an effective classifier in various medical imaging tasks [35; 36; 37; 38]. The method constructs one or more hyperplanes to classify the feature vectors into two classes with minimal generalization error [39]. Mathematically, the SVM model is established by solving the following optimization problem: \(\min\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
Figure 4a-c represents the heatmap of all the features extracted from 2D, incomplete 3D, and 3D tumor, respectively. The maps were produced by a total of 188 observations. Each entry in the map indicates the colorized Pearson correlation coefficient ranging from 0 (Blue) to 1 (red). The maps illustrate that most of the features extracted from 2D tumor masks have relatively low correlation. The similar pattern can be observed in the features obtained from incomplete 3D tumor masks. However, the correlation structure of the features extracted from whole 3D tumor masks was notably different, particularly in logarithmically, exponential, square, and square root-based features, exhibiting higher correlation among different classes. The high correlation may be attributed to the enhancement of certain 3D structures when applying these filters, which leads to consistent value changes on some categories of the features. Furthermore, Figure 4d-f demonstrates the histograms of all feature categories for 2D, incomplete 3D, and 3D features. According to the histograms, more than 80% of correlation coefficient values for 2D and incomplete 3D features are less than 0.5, whereas a higher percentage of 3D features have correlation coefficients greater than 0.5. These results imply that 2D and incomplete-3D tumor features cover comprehensive tumor characteristics with low information redundancy.