# Simulation-Driven Machine Learning for Predicting Stent Expansion in Calcified Coronary Artery
## 2 Materials and Methods  
The workflow integrating the FE method and the ML method is depicted in Figure 1. A patient-specific coronary artery model was reconstructed from OCT images and used for simulating the stent deployment, as described in our previous work [16]. Briefly, the calcified artery model was reconstructed based on 131 OCT images with a pixel size of 12.5 um and spacing of 200 um. Both calcification and fibrotic tissue were considered as components of plaque, corresponding to the late stage of a calcified artery. Stent deployment was implemented in an OCT-based patient-specific coronary artery model through the FE model as described in the next paragraph. There were 120 cross sections with spacing of 0.1 mm extracted from the FE model, within the stented region (i.e., 7 mm away from both ends of the artery) at pre- and post-stenting stages, respectively. From pre-stenting images, eight features were extracted: area of the artery (Area_A), fibrotic tissue (Area_F), calcification (Area_C) and lumen (Area_L), calcification angle (Angle_C) and maximum thickness of the calcification (Thickness_C), inner circumferential length of the calcification and fibrotic tissue (ArcLength_C and ArcLength_F, respectively). All these features were measured using an in-house code developed in Python 3. Correlational studies were performed to investigate the dependent relationships between different features. The centroid of the lumen was identified for measuring the calcification angle. From post-stenting images, the lumen area was extracted at each cross section, which was also the output parameter from ML models. In total, 120 pairs of datasets, eight pre-stenting features as the input variables, and one lumen area value as the output variable, were obtained for the regression learning and prediction. Specifically, the first 90 pairs of datasets were selected for training, and the rest of the 30 pairs were used for testing. A support vector regression (SVR) model with three different kernels, linear, polynomial, and radial basic function (RBF) (SVR_L, SVR_P, and SVR_RBF, respectively), were adopted and compared with a linear regression (LR) model. The artery had a length of 26 mm, which was 5 mm longer than the stent at each end. An Express stent with a length of 16 mm was deployed in the artery model. The hyperelastic material behavior was adopted to describe the mechanical behavior of the tissue. A reduced third-order polynomial was defined for the strain density function of the material:  
\[U=\sum_{i,j=1}^{3}C_{ij}(I_{1}-3)^{i}(I_{2}-3)^{j} \tag{1}\]  
\[I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2} \tag{2}\]  
\[I_{2}=1/\lambda_{1}^{2}+1/\lambda_{2}^{2}+1/\lambda_{3}^{2} \tag{3}\]  
where \(I_{1}\) and \(I_{2}\) are the first and second invariants of the Cauchy-Green deformation tensor, and \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) are the principal stretches. The coefficients \(C_{ij}\) are adopted in the literature [23; 24], as summarized in Table 1. In addition, a perfect plastic behavior was adopted for the fibrotic plaque with the yield stress of 0.07 MPa and the corresponding yield strain of 34% [25]. The stent material was 316L stainless steel, with a Young's modulus of 190 GPa, Poisson's ratio of 0.3, and yield strength of 207 MPa [24]. Symmetric constraints were enforced at both ends of the artery. The stent was first crimped from its nominal outer diameter of 3 mm into 1 mm to mimic its delivery state. At the stenotic location, the stent was radially expanded to a diameter of 3.3 mm, which compressed and pushed the stenotic lesion outwards [26]. After unloading, the stent recoiled to its final shape. A frictionless contact was  
Figure 1: Workflow of simulation-driven machine learning (ML) methods to predict post-stenting lumen area in a calcified coronary artery. enforced between the stent and the lesion [27]. The model was solved using the commercial software Abaqus (Dassault Systemes Simulia Corp., Providence, RI, USA).