## 2 Materials and Methods
### Mueller Matrix Transformation (MMT) Parameters  
The Mueller matrix transformation method was proposed to derive groups of quantitative parameters for describing the microstructural features of biomedical tissue samples by fitting the Mueller matrix elements to the trigonometric functions [14]. Several MMT parameters such as \(t_{2}\), \(t_{3}\), \(x_{3}\) and \(1-b\) were introduced in our previous studies [14; 29], where \(t_{2}\) (also denoted as \(t_{1213}\) shown in Equation (9)) reveals the linear diattenuation property, \(t_{3}\) (also denoted as \(t_{2434}\) and \(t_{4243}\) shown in Equations (10) and (11)) is related to the linear retardance of the tissue samples, while \(x_{3}\) (also denoted as \(\phi_{2434}\) and \(\phi_{4243}\) shown in Equations (12) and (13)) can characterize the orientation of the birefringence structure and \(1-b\) (shown in Equation (14)) represents the linear depolarization of the tissue samples.  
\[t_{1213}=\sqrt{m_{12}^{2}+m_{13}^{2}} \tag{9}\]  
\[t_{2434}=\sqrt{m_{24}^{2}+m_{34}^{2}} \tag{10}\]  
\[t_{4243}=\sqrt{m_{42}^{2}+m_{43}^{2}} \tag{11}\]  
\[\phi_{2434}=\frac{1}{2}\tan^{-1}(\frac{-m_{24}}{m_{34}}) \tag{12}\]  
\[\phi_{4243}=\frac{1}{2}\tan^{-1}(\frac{m_{42}}{-m_{43}}) \tag{13}\]  
\[1-b=1-\frac{m_{22}+m_{33}}{2} \tag{14}\]  
However, as these parameters are derived based on specific models [14], when we apply these parameters on complex tissue samples, their applicability needs further discussion. Our previous studies showed that there is a deviation between the values of \(t_{2434}\) and \(t_{4243}\) when multiple crossing linear birefringence effects coexist, as the information of linear retardance are distributed among the \(m_{24}\), \(m_{34}\), \(m_{42}\) and \(m_{43}\) elements [14; 26; 30]. Such deviation also occurs in the birefringence orientation parameters \(\phi_{2434}\) and \(\phi_{4243}\)Here, according to tissue structures, we propose two modified MMT parameters \(t_{qr}\) and \(\phi_{qr}\) (shown in Equations (15) and (16)), which take m\({}_{24}\), m\({}_{34}\), m\({}_{42}\) and m\({}_{43}\) elements into account, to calculate the linear retardance and birefringence orientation of the layered tissue sample more accurately. Besides, we also propose another two MMT parameters \(t_{121314}\) and \(1-b_{m}\) (Equations (17) and (18)) to describe the overall diattenuation and depolarization properties of the tissue samples by taking the elements \(m_{14}\) and \(m_{44}\) into account, which contain the information of circular diattenuation and circular depolarization.  
\[t_{qr}=\sqrt{\frac{t_{2434}^{2}+t_{2433}^{2}}{2}}=\sqrt{\frac{m_{24}^{2}+m_{4 2}^{2}+m_{34}^{2}+m_{43}^{2}}{2}} \tag{15}\]  
\[\phi_{qr}=\frac{1}{2}\tan^{-1}(\frac{m_{42}-m_{24}}{m_{34}-m_{43}}) \tag{16}\]  
\[t_{121314}=\sqrt{m_{12}^{2}+m_{13}^{2}+m_{14}^{2}} \tag{17}\]  
\[1-b_{m}=1-\frac{m_{22}+m_{33}{+}|m_{44}|}{3} \tag{18}\]  
Among the available Mueller matrix analyzing methods, the calculation speed of MMT is relatively fast, which is advantageous for clinical applications [5; 24]. However, the principle of MMT calculations is based on the fitting parameters obtained by experiments and simulations, and its accuracy needs to be further studied when applied to quantitative characterization of specific tissues and cells. On the other hand, the MMPD parameters are widely used in the field of biomedical and pre-clinical detection. The applicability of the MMPD method on tissue samples has been proven in various ex vivo and in vivo measurements [5; 6].