## 3 Results and Discussions
### Separation of Tendon Tissues in Different States Using NN Classifiers  
The above section shows that, there are several indicators of the MMT parameters \(\alpha_{1}\) and \(t_{1}\) that could distinguish among the four stretching states of tendon tissues. However, to reduce the complexity of the NN and improve the classification speed of the model, here we selected two of them for input into the model. The reason for this is that the first-order moment of the parameter \(t_{1}\) can discriminate the samples in the TDO and MDO states, and the first-order moment of the parameter \(\alpha_{1}\) can discriminate the samples in the ODO state. Thus, in this section, the NN has two input nodes, one is \(I_{1}\) of parameter \(t_{1}\) indicated as \(x_{1}\) in Equation (8), and the second input node is \(I_{1}\) of parameter \(\alpha_{1}\) indicated as \(x_{2}\) in Equation (8). For testing, the parameters \(x_{1}\) and \(x_{2}\) have been normalized by subtracting the minimum value from the current value and dividing by the difference between the maximum and minimum values. The NN also has four output nodes corresponding to four stretching states. Training tests showed that when there are two hidden layers with six nodes in each layer, as illustrated in Figure 3, a good classification can be achieved by the NN model. We use vectors to describe the entire network. The two nodes of the input layer are represented as Equation (8):  
\[X=[x_{1}\ x_{2}]^{\mathrm{T}} \tag{8}\]  
The weights \(R\) on the connection line between the input layer and the first hidden layer are shown in Equation (9), where \(r_{11}\) represents the weights on the line from the first neuron of the input layer to the first neuron of the first hidden layer, and so on for the rest of the parameters:  
\[R= \begin{bmatrix}r_{11}&r_{12}&r_{13}&r_{14}&r_{15}&r_{16}\\ r_{21}&r_{22}&r_{23}&r_{24}&r_{25}&r_{26}\end{bmatrix} ^{\mathrm{T}} \tag{9}\]  
Then, the data received by each node of the first hidden layer are \(R\times X\), and on this basis, the bias term \(B_{1}\) of each node is added as Equation (10):  
\[B_{1}=[b_{11}\quad b_{12}\quad b_{13}\quad b_{14}\quad b_{15}\quad b_{16}\ ]^{\mathrm{T}} \tag{10}\]  
After the activation of function \(Leaky\_Relu\), the output \(F_{1}\) of the first hidden layer is obtained as Equation (11): \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \multirow{2}{*}{\(t_{1}\)} & \multicolumn{3}{c}{**Origin**} & \multicolumn{3}{c}{**TDO**} & \multicolumn{3}{c}{**MDO**} & \multicolumn{3}{c}{**ODO**} \\ \cline{2-13}  & \(S_{1}\) & \(S_{2}\) & \(S_{3}\) & \(S_{1}\) & \(S_{2}\) & \(S_{3}\) & \(S_{1}\) & \(S_{2}\) & \(S_{3}\) & \(S_{1}\) & \(S_{2}\) & \(S_{3}\) \\ \hline \(I_{1}\) & 0.10 & 0.20 & 0.17 & 0.03 & 0.09 & 0.12 & 0.08 & 0.24 & 0.21 & 0.05 & 0.21 & 0.15 \\ \(I_{2}\) & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.01 & 0.01 & 0.00 & 0.01 & 0.00 \\ \(I_{3}\) & 0.13 & \(-0.23\) & \(-0.31\) & 1.43 & 0.84 & 0.33 & 0.19 & \(-0.17\) & 0.42 & 0.70 & \(-0.15\) & 0.03 \\ \(I_{4}\) & 2.93 & 3.23 & 3.28 & 7.95 & 4.41 & 3.20 & 3.31 & 2.78 & 3.54 & 3.82 & 3.07 & 3.08 \\ \hline \end{tabular}
\end{table} 
Table 2: Central moments \(I_{1}\) to \(I_{4}\) of MMT parameters \(t_{1}\) for bovine tendon tissue samples. \[F_{1}=leaky\_relu(R\times X+B_{1})=leaky\_relu\left( \begin{bmatrix}r_{11}\times x _{1}+r_{21}\times x_{2}+b_{11}\\ r_{12}\times x_{1}+r_{22}\times x_{2}+b_{12}\\ r_{13}\times x_{1}+r_{23}\times x_{2}+b_{13}\\ r_{14}\times x_{1}+r_{24}\times x_{2}+b_{14}\\ r_{15}\times x_{1}+r_{25}\times x_{2}+b_{15}\\ r_{16}\times x_{1}+r_{26}\times x_{2}+b_{16}\end{bmatrix} \right) \tag{11}\]  
Similarly, if the weight on the connection line between the first hidden layer and the second hidden layer is \(H\) (six rows and six columns), the data received by the nodes of the second hidden layer are \(H\times F_{1}\), and the bias term \(B_{2}\) (six rows and one column) of each node will be added on this basis.