Quantitatively differentiating microstructures of tissues by frequency distributions of Mueller matrix images

Abstract. We present a new way to extract characteristic features of the Mueller matrix images based on their frequency distributions and the central moments. We take the backscattering Mueller matrices of tissues with distinctive microstructures, and then analyze the frequency distribution histograms (FDHs) of all the matrix elements. For anisotropic skeletal muscle and isotropic liver tissues, we find that the shapes of the FDHs and their central moment parameters, i.e., variance, skewness, and kurtosis, are not sensitive to the sample orientation. Comparisons among different tissues further indicate that the frequency distributions of Mueller matrix elements and their corresponding central moments can be used as indicators for the characteristic microstructural features of tissues. A preliminary application to human cervical cancerous tissues shows that the distribution curves and central moment parameters may have the potential to give quantitative criteria for cancerous tissues detections.

Quantitatively differentiating microstructures of tissues by frequency distributions of Mueller matrix images Chao He, a,b Honghui He, a Xianpeng Li, a,c Jintao Chang, a,c Ye Wang, a,c Shaoxiong Liu, d Nan Zeng, a Yonghong He, a and Hui Ma a,c, *

Introduction
Polarization imaging can provide rich microstructural and optical information of tissues for diagnostic purposes. [1][2][3][4][5] Since a Mueller matrix provides the most comprehensive characterization of the polarization features, 6 it has been applied to differentiate various abnormal tissues, such as skin cancer, 7 cervical cancer, [8][9][10] colon cancer, [11][12][13] liver fibrosis, 14 and so on. [15][16][17][18] For anisotropic tissues, previous studies have shown that the Mueller matrix elements may change significantly with the orientation of the sample, making quantitative characterization of the microstructural features very difficult. [19][20] It was also pointed out that the structural information encoded in a Mueller matrix can be presented by other transformed parameters with more explicit physics meanings. 10 In fact, the abundant information carried by the Mueller matrix may allow us to identify the characteristic features of abnormal tissues without using high-resolution images. The two-dimensional (2-D) images of Mueller matrix elements can be reduced into a group of quantitative or semiquantitative, orientation insensitive parameters that reveal clearly the key structural features of the samples.
In this paper, we present a new way based on the statistical method to transform the 2-D images of the Mueller matrix elements into frequency distribution histogram (FDH), and central moment parameters. We study the quantitative influence of sample orientation on the frequency distributions of Mueller matrix elements, and then analyze the relations between the microstructures of tissues and the shapes of the FDHs. The experimental results show that the central moment analysis can provide us a group of orientation insensitive parameters representing the dominant features of tissues. The preliminary results of human cervical cancerous tissues show that the analysis method presented in this paper may serve as quantitative or semiquantitative criteria for cancerous tissue detection. For the experimental setup shown in Fig. 1, the polarizers P1 and P2 are fixed in the horizontal direction, while the wave plates R1 and R2 are rotated with a fixed rate θ 1 ¼ 5θ 2 , where θ 1 and θ 2 are the rotation angles of the wave plates R1 and R2, respectively. The Fourier series intensities can be given by Eq. (1), E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 6 3 ; 3 8 ðα n cos 2nθ 1 þ β n sin 2nθ 1 Þ; (1) where α n and β n are the Fourier coefficients. Using α n and β n , we can calculate the Mueller matrix elements according to Ref. 21. In order to record robust images, we measure the Mueller matrices of standard samples such as air and wave plate in the transmission mode and then calibrate the imaging system using the method proposed by Chenuault et al. 24 The Mueller matrix elemental accuracy is tested by measuring the high extinction polarizers. After the calibration in transmission direction, the PSA arm of the system is rotated to the backscattered direction as shown in Fig. 1. We also measure the backscattering Mueller matrices of some samples with known polarization properties such as the microsphere solutions. In this work, the mean measurement errors of the diagonal and nondiagonal elements are less than 0.5% and 0.3%, respectively. It is shown in our analysis of experimental results that the elemental uncertainty of the Mueller matrix measurement does not change the main characteristic features of the FDH, as well as the central moment parameters.

Biological Tissue Samples
In previous studies, we have taken the 2-D backscattering Mueller matrix images of different tissues, and analyzed the relations between their characteristic microstructural features and the Mueller matrix elements. 10 The tissue samples are shown in Fig. 2: (a) bovine skeletal muscle, (b) chicken heart muscle, (c) porcine liver, and (d) porcine fat. Both the bovine skeletal muscle and the chicken heart muscle are anisotropic, but the fibers are mostly aligned in the same direction for the bovine muscle sample while aligned concentrically around the ventricle for the chicken heart sample. 25 Porcine liver and fat tissues are close to isotropic, but the liver sample contains many hexagonal boundaries of hepatic lobules, which are connective tissues of birefringence. 10 The use of the animal tissues in this study was approved by the Administrative Committee on Animal Research of the Graduate School at Shenzhen, Tsinghua University. From the backscattering Mueller matrix images shown in Fig. 3, we can obtain abundant structural information of these samples. For instance, the anisotropy of tissues may originate from both the optical birefringence and cylindrical scatterers, which can be distinguished by the features in different Mueller matrix elements. 10 The contributions due to the fibrous scatterers are encoded in the m12, m21, m13, and m31 elements (m12 ¼ m21, m13 ¼ m31), while those due to the birefringence are encoded in the m24, m42, m34, and m43 elements (m24 ¼ −m42, m34 ¼ −m43). The orientation of fibrous structures and the depolarization power can also be extracted from the Mueller matrix elements. 10 Since the Mueller matrix contains abundant information on the tissue samples, we may not have to rely on the 2-D images to identify their characteristic features. It can be helpful to find a method to transform the 2-D images of Mueller matrix elements into a group of quantitative or semiquantitative, orientation insensitive parameters, which are crucial for the extraction of the dominant microstructural information of samples.

Central Moment Analysis
To quantitatively evaluate the Mueller matrix elements, we adopt the central moment method for statistical analysis of frequency distributions. 26   μ ¼ P1 ¼ EðXÞ; Suppose we have a random variable X, whose central moments: expected value, variance, skewness, and kurtosis are exactly represented by Eq. (2). 27 Here, the expected value P1 is the mean value of an FDH. The second central moment P2 is called the variance usually denoted by σ 2 , where σ represents the standard deviation of the FDH. A small P2 indicates that the measured data tend to be distributed close to the expected value, while a large P2 indicates that the data points are spread out around the expected value and from each other. The third and fourth central moments P3 and P4 represent the skewness and kurtosis of the FDH, respectively. P3 (skewness) shows the asymmetry of the FDH. The skewness value can be positive or negative. A negative (or positive) skewness value indicates that the tail on the left side (or the right side) of the FDH is longer or fatter than the right side (or the left side). P4 (kurtosis) indicates the "peakedness" of the FDH. It is a descriptor of the shape of a probability distribution. The detailed physical interpretations of kurtosis, skewness, and the central moments can be found in Refs. 26 and 28. In this paper, we first record the 2-D backscattering Mueller matrix images of samples, and then transfer the pixel images to FDHs by statistical analysis. Finally, the central moment method is applied to the FDHs of Mueller matrix elements.

Characteristic Features in Mueller Matrix
Elements of Tissues at Different Orientations Figure 4 shows the FDH of the backscattering Mueller matrix elements of the bovine skeletal muscle tissue. Previous studies have shown that the anisotropic muscle fibers can seriously affect the polarization measurement, resulting in difficulties of the structural information extraction. 19 In this study, we rotate the skeletal muscle sample, measure the Mueller matrix images in different orientations, and then choose a square area of 700 × 700 pixels at the same location of the sample. In Fig. 4, the horizontal axis of each FDH represents the value of the pixel from the corresponding Mueller matrix element, while the vertical axis represents the distributing probability. There are four experimental curves of the bovine skeletal muscle sample along 30 deg (black lines), 60 deg (red lines), 120 deg (green lines), and 150 deg (blue lines) directions [the angle is indicated as γ in Fig. 1(c)]. To make the evaluation quantitative, we apply the central moment method to the FDHs, and calculate the parameters P1, P2, P3, and P4, which are listed in Table 1. Figure 4 and Table 1 show that the FDHs are transformed into the quantitative central moment parameters: the expected value P1, variance P2, skewness P3, and kurtosis P4 which all together characterize the position and shape of the FDH curves. It can be observed from Fig. 4 and Table 1 that, as the orientation direction of the fibrous sample changed, most FDHs of the elements move except the m11, m14, m41, and m44, showing the influence of sample orientation on Mueller matrix measurements. Compared with the 2-D images, the FDHs and central moment parameters still reveal clearly and quantitatively the same main structural features of tissues as summarized in our previous studies: 10,18 (1) the Mueller matrix shown in Fig. 4 and Table 1 is nondiagonal, the m22 and m33 elements are not equal (for instance, for the muscle sample along 30 deg the P1 of m22 and m33 are 0.114 and 0.241, respectively), and testifying that the bovine skeletal muscle is anisotropic. (2) We also notice that the FDHs of the m24, m34, m42, and m43 elements represent slight variations (the absolute value of P1 varies from 0.001 to 0.022 in different angles), indicating the existence of birefringence in this sample. Further analysis of the positive or negative values of these elements can provide the orientation axis information. 18 (3) The values of the diagonal elements m22, m33, and m44 are relatively small (P1 mostly distributed in 0.02 to 0.3), indicating a large depolarization property. 10 The influences of sample orientation on the Mueller matrix elements are serious. Mueller matrix images of an anisotropic tissue sample along different directions may look like from different samples. However, Fig. 4 shows that the FDH curves can help us to distinguish that it is a different sample or just the same sample along a different orientation; as the sample rotated, the positions of the curves move but their shapes almost remain the same. Table 1 quantitatively confirms that when we rotate the muscle sample, the values of P1 change periodically, which indicates the variations of the positions for the FDHs. Meanwhile, the values of P2 (variance), P3 (skewness), and P4 (kurtosis) display very small changes, indicating the similar shapes (distributing width, asymmetry, and peakedness) of the FDHs. It should be pointed out that, compared to P2 and P4, the variation of P3 seems to be prominent in Table 1. It is because the length of the confidence interval of P3 is related to the absolute value of the skewness. When the skewness is very small, the confidence     interval becomes wider. 27 Therefore, according to the small values of P3 in this work, its confidence interval is about −0.3 to 0.3. The values of P3 shown in Table 1 mean that the asymmetry of the FDH curves shown in Fig. 4 can be treated as almost the same. For comparisons, we also take the backscattering Mueller matrix images of porcine liver tissue along different sample orientations. The liver tissues are primarily isotropic, but contain many thin hexagonal structures around the isotropic liver tissues, which are identified as birefringent connective tissues. 10 Figure 5 shows the FDHs of the liver sample at 30 deg (black lines), 60 deg (red lines), 120 deg (green lines), and 150 deg (blue lines) directions, respectively. The central moment parameters P1, P2, P3, and P4 are listed in Table 2.
It can be observed from Fig. 5 and Table 2 that, the FDHs of porcine liver tissues have isotropic structural features, which are different from those of the fibrous skeletal muscle shown in Fig. 4 and Table 1: (1) the Mueller matrix shown in Fig. 5 and Table 2 is nearly diagonal (the P1 values of the m12, m21, m13, and m31 are close to 0), and the m22 and m33 elements are equal, representing isotropic dominant properties.
(2) The m24, m34, m42, and m43 elements show slight differences (almost m24 ¼ −m42, m34 ¼ −m43), indicating the existence of birefringent connective tissues. (3) Compared to the skeletal muscle sample, the values of the diagonal elements of the liver tissue are larger (P1 mostly distributed in 0.25 to 0.5), reminding a smaller depolarization property. We also notice that when the sample rotates, the FDHs of the Mueller matrix elements all keep the same except the m24, m34, m42, and m43, confirming that for isotropic samples, the influence of orientation on polarization measurements is limited. Again, the FDHs and central moment parameters reveal more clearly and quantitatively the characteristic features of Mueller matrix elements for isotropic samples. 10 From the data shown in Tables 1 and 2, we can conclude that: for the same sample placed along different orientations, the values of P2, P3, and P4 for the Mueller matrix elements almost do not change, while the value of P1 can be varied (for the anisotropic sample) or constant (for the isotropic sample). The central moment analysis of the FDHs provides us a tool to transform the complicated 2-D Mueller matrix images to a group of quantitative indicators of dominant structural properties of tissues. More importantly, using the P2, P3, and P4, we can obtain the main intrinsic properties of samples without the influence from orientation variations.

Comparisons of Tissue Samples with Distinctive Microstructures
In Sec. 3.1, we have found that the shapes of the FDHs of Mueller matrix elements are orientation insensitive, therefore may be used as indicators for intrinsic microstructural features of different tissues. To study the relationship between the FDHs and the structural properties, we take the Mueller matrices of the tissue samples as shown in Fig. 2. Figure 6 represents the experimental results of bovine skeletal muscle tissue (black lines), porcine liver tissue (red lines), chicken heart tissue (green lines), and porcine fat tissue (blue lines), the imaging area is a square of 700 × 700 pixels. During the measurements, the orientations of the samples are kept the same, for the bovine skeletal muscle sample its fibers are along the 30 deg direction (γ ¼ 30 deg).
We can see from Fig. 6 that the FDHs of different tissues have very different distributions. The corresponding central    m12  m13  m14  m21  m22  m23  m24  m31  m32  m33  m34  m41  m42  m43  m44   30 Table 3. In the discussions above, we have summarized the characteristic features of anisotropic and isotropic tissues using Figs. 4, 5 and Tables 1, 2. Here, we can confirm the relations between the microstructures and the distributions of Mueller matrix elements from Fig. 6 and Table 3 more clearly. First, the anisotropic and isotropic tissues can be distinguished by using the diagonal elements. The porcine liver (red lines) and fat (blue lines) tissues are predominantly isotropic; therefore, their m22 and m33 curves are almost the same (for example, for the fat tissue, P1 of the m22 and m33 are 0.041 and 0.041, P2 are 0.013 and 0.012). The anisotropic bovine skeletal muscle (black lines) and chicken heart (green lines) tissues, however, display differences between the m22 and m33, which become more prominent as the anisotropy increases. Table 3 shows that the differences in P1 of m22 and m33 elements for anisotropic skeletal muscle and heart samples are 0.127 and 0.014, respectively. This is because that the fibers in skeletal muscle sample are well aligned in almost the same direction, while in heart sample the fibers are distributed in different orientations. For isotropic fat and liver tissues, the differences in P1 of m22 and m33 elements are 0 and 0.002. This is because the fat tissue is totally isotropic, while the liver sample has a small portion of birefringent connective tissues. Second, we also notice that the distribution widths of the FDHs (the values of P2) for bovine skeletal muscle, chicken heart, and porcine liver samples are larger than the fat sample, indicating more complicated microstructures for these metabolic exuberant tissues. The FDHs of the m24, m42, m34, and m43 elements for skeletal muscle, heart, and liver tissues show small positive or negative values, which are related to the birefringent structures in these tissues. The signs of the elements can be used to determine the aligned fibers directions. 18 At last, the different depolarization power of tissues can also be observed from Fig. 6 and Table 3: the liver tissue sample has the largest P1 values of the diagonal elements, showing the smallest depolarization power, while the smallest P1 values of the diagonal elements indicate the most prominent depolarization property of the fat tissue. 10 Although more studies are still needed to reveal the relationships between the derived parameters and tissue morphology, it has been shown that the parameter P2 should be sensitive to the complexity of a sample: a large value of P2 means that the measured polarization data are distributed in a wider range, indicating a complex structural feature of the tissue. The parameter P3 should be sensitive to the heterogeneity of a sample: a large value of P3 means that the measured polarization data are unequally distributed around the expected value. The parameter P4 can also be used to reflect the complexity of a sample: a large P4 shows that most measured polarization data are distributed very close to the mean value, meaning that the microstructural features are similar.
In summary, from the results discussed above, we can conclude that: (1) the shapes of FDHs (values of P2, P3, and P4) are orientation insensitive, therefore can reflect some intrinsic structural properties of the samples.

Application to Human Cervical Cancerous Tissues
To testify the potential applications of the pixel FDHs and central moment parameters on diagnosis, we take the Mueller matrices of an unstained 28-μm-thick slice of human cervical cancerous tissue prepared and provided by the Shenzhen Sixth People's (Nanshan) Hospital (ID: 120900924), the detailed information of the tissue can be found in Ref. 10. This work was approved by the Ethics Committee of the Shenzhen Sixth People's (Nanshan) Hospital. We also choose 700 × 700 squares from both the normal and abnormal regions of the 2-D images, and then calculate the FDHs and central moments of Mueller matrix elements shown as shown in Fig. 7 and Table 4. It can be observed from Fig. 7 and Table 4 that the normal and abnormal cervical tissues represent different structural features. The normal region has larger anisotropy (more prominent difference between the m22 and m33 elements) and depolarization properties (smaller values of the m22, m33, and m44 elements) than the abnormal region. The positive and negative values of the m34 and m43 elements also indicate the existence of birefringence in normal tissue (P1 of the m34 and m43 are −0.014 and 0.020), while for abnormal tissue, the birefringent    effect becomes limited. 9-10 Besides, we also notice that the values of P3 and P4 for normal and abnormal tissues represent large difference in some elements, such as the m22, m33, and m44. These preliminary studies show that the FDHs and central moment parameters may have the potential to give quantitative or semiquantitative criteria for cancerous tissues detections.

Conclusion
In this work, we take the backscattering Mueller matrices of tissues of distinctively different microstructures: bovine skeletal muscle tissue, porcine liver tissue, chicken heart tissue, and porcine fat tissue, then use the pixel FDH and central moment analysis to transform the 2-D Mueller matrix images to a group of quantitative indicators for characterizing the dominant structural properties of tissues. By rotating anisotropic skeletal muscle sample and isotropic liver tissue, we find that the central moment parameters P2, P3, and P4 are insensitive to sample orientation directions. Comparisons among different tissues testify that the distribution behavior and corresponding central moment parameters of Mueller matrix elements are good indicators of the microstructures of tissues. A preliminary application to human cervical cancerous tissues shows that the distribution curves and central moment parameters may have the potential to give quantitative or semiquantitative criteria for cancerous tissues detections.