## 2 Materials and Methods
### Classification  
In recent years, extensive research has been conducted on deep learning across various domains, notably in computer vision, image processing, authentication systems, and speech recognition. CNNs represent a kind of classification model [27]. In contrast to a traditional Neural Network, a CNN possesses the ability to learn intricate features, rendering it highly effective in tasks such as image classification, object recognition, and medical image analysis. The fundamental premise of a CNN lies in its capability to extract localized features from input data at higher levels and propagate them to lower layers to construct more complex feature representation. A CNN architecture typically comprises three layers, as illustrated in Figure 2[28]: (i) the convolution layer, responsible for feature extraction; (ii) the pooling layer, which reduces dimensionality; and (iii) the fully connected layer, which handles classification and transforms two-dimensional matrices into one-dimensional vectors [29]. In the convolutional layer, a learnable filter extracts features from an input image. Within this network, non-linearity is managed through the activation function, which induces a non-linear transformation of the neuron's inputs. In the binary classifier, the sigmoid function is employed in the output layer, yielding probabilities of a data point belonging to a specific class within the range of 0 to 1, as determined in Equation (5). A Rectified Linear Unit (ReLU) is utilized for all hidden layers due to the limitations of the sigmoid, producing zero results for negative input values. Consequently, the neurons remain inactive, thereby expediting computation and training, as expressed in Equation (6). \[f_{sigmoid}=\frac{1}{1+\exp(-x)} \tag{5}\]  
\[f_{Relu}=max(0,x) \tag{6}\]  
In the proposed CNN multi-classifier, the SoftMax function was utilized [30], which computes the probability of data values belonging to certain classes. Our research employed a Long Short-Term Memory (LSTM) network as the classification layer, which was fed by a Convolutional Neural Network (CNN). The LSTM architecture, denoted as LSTM [31; 32], tackles challenges faced by traditional RNNs like gradient vanishing and exploding. Unlike standard RNN units, LSTM employs memory units as a technique. Its crucial departure from RNNs lies in its incorporation of a cell memory, facilitating long-term memory encoding. LSTM efficiently retrieves and integrates data spanning previous time intervals up to the present moment, with three gateways: a \"forget\" gate as an input, and an output gateway. The current input is \(xt\), while the initial and modified cell memories are \(ct-1\) and \(ct\), respectively. The current and previous outputs are \(ht\) and \(ht-1\). The LSTM model architecture is shown in Figure 5.  
Figure 5: LSTM. The operational principle of the input gate in LSTM is outlined by the following equations:  
\[i_{t}=\sigma(W_{i}.[h_{t-1},x_{t}]+b_{i}), \tag{7}\]  
\[\hat{\mathsf{C}}_{t}=\tan(W_{i}.[h_{t-1},x_{t}]+b_{i}), \tag{8}\]  
\[\hat{\mathsf{C}}_{t}=f_{t}\hat{\mathsf{C}}_{t-1}+i_{t}\hat{\mathsf{C}}_{t} \tag{9}\]  
In the context described, Equation (7) quantifies the integration level of information from \(ht-1\) and \(xt\). Subsequently, the previous embedded state \(ht-1\) and the present data point \(xt\) are subject to a _tanh_ activation function, represented by a layer, which is mathematically expressed in Equation (8). The output obtained from this layer is denoted as \(\hat{\mathsf{C}}_{t}\). Equation (9) merges data from the current input, \(\hat{\mathsf{C}}_{t}\), and the previous long-term memory, \(\hat{\mathsf{C}}_{t}-1\), to adjust the current cell memory, \(\hat{\mathsf{C}}_{t}\). Symbols \(Wi\) and \(bi\) represent weight and the bias of the input gateway within the LSTM network. To elaborate on how LSTM contributes to improving the model's understanding of disease progression, let us break down each equation into its constituent parts and explain the significance of each term.