# Simulation-Driven Machine Learning for Predicting Stent Expansion in Calcified Coronary Artery
## 2 Materials and Methods
The FE simulation was validated against the OCT measurements of lumen area, lumen shape, and malapposition of stent struts. Linear regression is one of the fundamental supervised machine learning algorithms. For a given dataset (\(y_{i}\), \(x_{i1}\), \(x_{i2}\),... \(x_{i8}\)), i represents 90 sets of training data, \(x_{i1}\) to \(x_{i8}\) represent eight features, and \(y_{i}\) is the stent-induced lumen area. The linear model is:  
\[\mathbf{y}=\mathbf{X}\mathbf{w}+\mathbf{\varepsilon} \tag{4}\]  
where \(\mathbf{w}\) is the regression coefficient and \(\mathbf{\varepsilon}\) is the correction term. Both are obtained using the least squares method to minimize the error between the predicted value and the measured value:  
\[\min\sum_{i=0}^{n}(y_{i}-\mathbf{x}_{i}\mathbf{w})^{2} \tag{5}\]  
On the other hand, the SVR method regulates the error to a certain degree as illustrated in Figure 2. The objective function of the SVR method is to minimize the margin of the hyperplane, especially the l2-norm of the normal vector \(\mathbf{w}\):  
\[min\frac{1}{2}\|\mathbf{w}\| \tag{6}\]  
under the constraints:  
\[\left\{ \begin{array}{l}y_{i}-\mathbf{w}\mathbf{x}_{i}-b\leq\varepsilon\\ \mathbf{w}\mathbf{x}_{i}+b-y_{i}\geq\varepsilon\end{array} \right. \tag{7}\]  
Besides the linear kernel, the SVR models could transfer the feature space into a higher dimensional feature space with nonlinear kernel functions, such as polynomial and RBF kernels. All machine learning models were developed within Python 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Tissue** & **C\({}_{10}\) (MPa)** & **C\({}_{01}\) (MPa)** & **C\({}_{11}\) (MPa)** & **C\({}_{20}\) (MPa)** & **C\({}_{02}\) (MPa)** & **C\({}_{30}\) (MPa)** & **C\({}_{03}\) (MPa)** \\ \hline Artery & 0.10881 & \(-\)0.101 & \(-\)0.1790674 & 0.0885618 & 0.062686 & & \\ Fibrotic tissue & 0.04 & & & & 0.003 & & 0.02976 \\ Calcification & \(-\)0.49596 & 0.50661 & 1.19353 & 3.6378 & & 4.73725 & \\ \hline \end{tabular}
\end{table} 
Table 1: Material coefficients. Figure 2: Schematic of support vector regression method. The accuracy of these four ML models was evaluated in terms of the percentage error between the predicted value and the measured value from the FE model. The influence of the feature selection and neighboring cross sections on prediction accuracy were further studied using SVR_P. Two subgroups, stretch features (Area_L, ArcLength_C, and ArcLength_F) and calcification features (Area_C, Angle_C, and Thickness_C), were selected for inspecting the influence of feature selection on the prediction accuracy. In addition, the influence of neighboring cross sections on the prediction accuracy was also investigated by using the average value of each feature over eight neighboring cross sections (i.e., four cross sections, if any, immediately before and after the current cross section) for training and testing the regression model. Statistical analysis was performed with _p_-value < 0.05 considered statistically significant. The ANOVA test was used to compare the means among more than two groups of the prediction errors, and a simple t-test was used to compare the means between two groups.