## 3 Results and Discussions
### MMT Parameters for Tendon Tissues in Different States  
We first plot the FDH curves of the MMT parameters \(\alpha_{1}\) and \(t_{1}\) for the tendon tissues in four different states in the same subplots. The \(\alpha_{1}\) and \(t_{1}\) parameters of these four different states have been normalized by dividing the current value by the corresponding parameter value at the maximum absolute value of the four states. As shown in Figure 5, the blue solid lines represent the MDO state, the orange solid lines with circle markers indicate the TDO state, the yellow solid lines with triangle markers indicate the ODO state and the purple dashed lines represent the origin state. From the FDHs of parameter \(\alpha_{1}\) shown in Figure 5a, we can see that: (1) among the four states, the FDH curve of the MDO sample has the steepest shape, which is similar to the FDH curve of origin sample; (2) the FDH curve of the ODO sample is far away from those of the other three states; and (3) the FDH curve of the TDO sample is lower than those of the other three states and has the flattest shape. From the FDHs of parameter \(t_{1}\) shown in Figure 5b, it can be seen that the mean value of the FDH curves of the four states increase sequentially as TDO, ODO, origin, MDO. To statistically verify the above conclusions, we performed tensile experiments on three randomly selected bovine tendon tissue samples and calculated the central moments (\(l_{1}\) to \(l_{4}\) moments) of their MMT parameters \(\alpha_{1}\) and \(t_{1}\)  
Figure 4: Frequency distribution histograms (FDHs) of 3 \(\times\) 3 Mueller matrix elements for bovine tendon tissues in four states: origin, TDO, MDO and ODO. The horizontal axis is divided into 400 intervals, and the vertical axis coordinates represent the probability that the data points will fall into one of the intervals of the horizontal axis. Here, the coordinate ranges are determined according to Ref. [24]. As shown in Table 1, for parameter \(\alpha_{1}\) in the four states of origin, TDO, MDO and ODO, the average values of the first-order moment of the three samples are \(0.023\), \(-0.003\), \(-0.030\) and \(-0.443\), respectively. It can be seen that the first-order moment of the ODO sample is very different from those of the other three states. Because the oblique stretching makes the fiber structure very chaotic, the longitudinal or transverse stretch of its fiber structure changes are ordered, so that the general anisotropy of ODO is very different from MDO and TDO. Moreover, the average values of the second-order moment of the three samples in the origin, TDO, MDO and ODO states are \(0.016\), \(0.173\), \(0.026\) and \(0.103\), respectively. We can notice that the second-order moments of the origin and MDO samples are much smaller compared with the other two states, and the second-order moment of the TDO sample is the largest among all four states. This is because the lateral stretching makes the arrangement of the fiber structure become loose, and the closer to the center, the greater the degree of looseness, so that the range of its total anisotropy values becomes wider and the variance in the lateral stretch state becomes larger. It is also found that the mean value of the third-order moments of the ODO state is \(2.123\), which is larger than the mean values of third-order moments of the origin, TDO and MDO states, and the mean value of the fourth-order moments of the TDO state is \(3.680\), which is smaller than the mean values of fourth-order moments of the origin, ODO and MDO states. As shown in Table 2, for the MMT parameter \(t_{1}\) of the three tendon tissue samples in four states of origin, TDO, MDO and ODO, the average values of the first-order moment are \(0.156\), \(0.080\), \(0.176\) and \(0.136\), respectively. It shows that the first-order moment, or the expected value of the MMT parameter \(t_{1}\), can be used to distinguish the stretching states of the samples.