## 2 Materials and Methods
### Models and Methods  
#### 2.3.1 U-Net Architecture  
By executing the FEM model on a range of arterial wall geometries, one can acquire a collection of input and output image data. This data can then be utilized to train a deep learning model in a supervised learning manner. An encoder-decoder network maps the input image to the output stress and strain map [29], depicted in Figure 3a. The binary maps provided as input depict the positions of calcium, lumen, and fibrous tissue. The encoder-decoder network converts these input images into a latent space with lower dimensions, subsequently mapped back to the stress and strain field. This approach assumes that the input and target spaces share a common latent space. The U-Net architecture [24], based on the FEM mapping data of images to stress and strain field, has been utilized for this purpose. The weights of architecture are trained through learning. Initially introduced in 2015 for medical image segmentation, the U-Net architecture is named after its U-shape, consisting of an encoder and a decoder. It has proven effective in capturing latent representations for various types of images. The encoder extracts feature from the input image using convolutional and max pooling layers. In contrast, the decoder generates the output segmentation map using up-convolutional and concatenation layers. One of the significant advantages of the U-Net architecture is its ability to produce high-quality segmentation results even when trained on a limited amount of data. U-Net has been widely used in various medical image segmentation tasks, such as segmenting brain tumors, retinal blood vessels, and cell membranes. The architecture has also been extended and modified for other applications, such as semantic segmentation, instance segmentation, and object detection. The standard architecture in this study incorporates contracting and expanding layers along with skip connections to propagate context information and improve output resolution. The U-Net architecture utilized here is a slightly modified version of the original design. The encoder section consists of six repeating blocks, each containing a 2 \(\times\) 2 max pooling operation, two consecutive 3 \(\times\) 3 2D convolutions (except for the first and last blocks), batch normalization, and ReLU \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(\mathbf{C_{10}}\) (MPa) & \(\mathbf{C_{01}}\) (MPa) & \(\mathbf{C_{11}}\) (MPa) & \(\mathbf{C_{20}}\) (MPa) & \(\mathbf{C_{02}}\) (MPa) & \(\mathbf{C_{30}}\) (MPa) & \(\mathbf{C_{03}}\) (MPa) \\ \hline Artery & 0.108 & \(-\)0.101 & \(-\)0.179 & 0.088 & 0.062 & & \\ Fibrous & 0.040 & & & & 0.003 & & 0.0297 \\ Calcium & \(-\)0.495 & 0.506 & 1.193 & 3.637 & & 4.737 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties of arterial wall features [27]. Figure 2: 2D arterial wall cross-section geometry meshed with four-node constant strain element and corresponding von Mises stress and strain distribution. The maximum stress and strain are considered 300 kPa and 0.45, respectively. activation. The first decoder block employs a transpose convolution layer, while the last block consists of two consecutive convolution-batch norm-ReLU layers without a transpose convolution. The encoder and decoder blocks are followed by a final \(1\times 1\) convolutional layer, which reduces the 64-channel decoder output to a single channel. During weight training, the loss function is the weighted mean squared error (MSE) between the predicted and true von Mises stress and strain maps derived from the training data. The U-Net architecture was implemented using the TensorFlow and Keras libraries [30]. Figure 3b illustrates a diagram of the modified U-Net architecture. The training loss function is defined using the \(MSE\), which is denoted as:  
\[MSE=L[Y,\ f(X)]=\frac{1}{n}\sum_{i=1}^{n}[Y-f(X)]^{2} \tag{1}\]  
Figure 3: (**a**) An encoder–decoder-based network that can serve as an efficient surrogate for the FEM mapping and (**b**) the U-Net architecture.