Division of focal plane polarimeter-based 3 × 4 Mueller matrix microscope: a potential tool for quick diagnosis of human carcinoma tissues

Abstract. A polarization microscope is a useful tool to reveal the optical anisotropic nature of a specimen and can provide abundant microstructural information about samples. We present a division of focal plane (DoFP) polarimeter-based polarization microscope capable of simultaneously measuring both the Stokes vector and the 3×4 Mueller matrix with an optimal polarization illumination scheme. The Mueller matrix images of unstained human carcinoma tissue slices show that the m24 and m34 elements can provide important information for pathological observations. The characteristic features of the m24 and m34 elements can be enhanced by polarization staining under illumination by a circularly polarized light. Hence, combined with a graphics processing unit acceleration algorithm, the DoFP polarization microscope is capable of real-time polarization imaging for potential quick clinical diagnoses of both standard and frozen slices of human carcinoma tissues.


Introduction
Polarization imaging is becoming an emerging technique for studies in material, 1,2 astronomy, 3 remote sensing, 4,5 and biomedicine. [6][7][8][9] When observing the biomedical and material specimens, polarization microscope is a useful tool to reveal their optical anisotropic nature and can provide abundant microstructural information about samples. The traditional polarization microscope adopts a polarizer to provide linear polarized illumination, and an analyzer is inserted for observation in cross-or parallel-polarized light mode, which is simple to implement but difficult for quantitative analysis. 10 Oldenbourg et al. 11 modified the traditional polarization microscope by adopting the liquid crystal retarders. The modified microscopy greatly improves the analytic power, providing quantitative birefringence distributions of thin specimens, such as the mitotic spindle isolated from the fertilized sea urchin egg. 11 Recently, Arteaga et al. 12 proposed a Mueller matrix microscope based on two continuous rotating wave plates to completely extract the polarization property of a specimen. In this scheme, the intensity at each pixel is analyzed by digital demodulation of thousands of continuously captured frames. 12 The polarization microscopes based on liquid crystal retarders and rotating wave plates are division of time technologies, which are suitable for stationary samples. For the polarization measurements of dynamic processes, however, division of the focal plane (DoFP) polarimeters can be adopted. For a microgrid polarizer DoFP polarimeter, an array consisting of numerous differently orientated pixel-size micropolarizers is fixed on the imaging sensor. [13][14][15] Several recent studies have incorporated the DoFP polarimeter into a microscope for simultaneous Stokes vector imaging. 13, 16 Liu et al. 13 developed a complementary fluorescence-polarization microscope using a DoFP polarimeter to enable real-time video rate polarization imaging without any moving parts. Hsu et al. 16 presented a polarization microscope using an infrared full-Stokes imaging polarimeter. The DoFP polarimeter has the same instrument size as the conventional image sensor and can capture the Stokes vector images in a single shot.
Recently, polarization imaging techniques have been used as potential tools for biomedical diagnosis, such as the detection of abnormal tissues in skin, 17,18 liver, 19,20 esophagus, 21 colon, 22,23 cervix, 24 bladder, 25 and so on. [26][27][28] Nowadays, the cancer incidence is growing fast worldwide, making pathological diagnoses time-consuming. Basically, during a diagnosis process, the pathologists need about half an hour to prepare the stained frozen slices of suspicious tissue samples from the patients. With careful observations using a transmission microscope, the pathologists need to provide quick assessments which are crucial for the surgeons. Compared to the diagnosis using the standard dewaxed slices cut from fixed tissues in paraffin specimens, the quick assessments always lead to a small portion of misdiagnose. In our experiments, we find that when using a DoFP polarimeter for polarization imaging of pathological tissue slices, the experimental results can provide additional useful information but will sometimes be confusing and misleading if the illumination state of polarization (SoP) and the sample are not well matched. For different illumination SoPs, there will be considerable differences in the specimen's Stokes vector images. In this situation, measuring the sample's Mueller matrix is conducive to our understanding of the sample's comprehensive polarization property. Moreover, it is also found that when applied to the thin dewaxed slices of tissues, the intrinsic anisotropic fibrous structures of the cancerous tissues are more prominently highlighted in the Mueller matrix elements related to the circularly polarized light, or the elements in the fourth row and column. Hence, a DoFP polarimeter-based microscope capable of measuring the 3 × 4 Mueller matrix may provide useful information of human carcinoma tissue slices.
In this article, we propose an optimal 4-SoP illumination scheme for the 3 × 4 Mueller matrix measurement using a DoFP-based polarization microscope. The experimental results show that the polarization features of the unstained human carcinoma tissue slices are more prominently reflected in the Mueller matrix elements m 24 and m 34 . It is also indicated that the pathological microstructures are less visible in the polarization images under illumination by the linear polarization light, but can be enhanced by "polarization staining" under illumination by a circularly polarized light. As a nonstained technique, the DoFP-based polarization microscope can provide additional microstructural information of both standard and frozen pathological slices, which may be used as a potential tool for the quick diagnosis of human carcinoma tissues.

Polarization Imaging Using the Division of Focal Plane Polarimeter
We use the DoFP polarimeter "PolarCam" (4D Technology Inc.) whose sensor size is 488 × 648 pixels and whose video frame rate can reach to 110 fps. 15 The diagram of the pixelated micropolarizer array is shown in Fig. 1(a). Four pixels with different polarization orientations comprise a super pixel, which are combined together to calculate the linear part of the Stokes vector where I 0 , I 45 , I 90 , and I 135 are the intensities of the linear polarization components at four different degrees. Since for a DoFP polarimeter, the image sensor response, the extinction ratio, and the transmission axis of each pixel inherently have significant nonuniformities, calculating the Stokes vector simply using Eq. (1) would introduce errors. 29 For accurate Stokes vector measurement, the nonuniformities for all pixels should be calibrated. We introduce light with different known SoPs and measure the instrument responses to obtain the instrument matrix of the polarimeter. 30 Using the obtained instrument matrix, we can calculate the Stokes vector using the polarimetric data reduction method. For each super pixel as shown in Fig. 1(a), the polarimetric measurement can be expressed as where I is the light intensity of the four pixels and A is a 4 × 3 instrument matrix for a super pixel. Then S can be calculated by S ¼ pinvðAÞ × I, where pinv (A) represents the pseudoinverse of A. It has been shown that if the condition number (CN: the ratio of the largest singular value to the smallest singular value of a matrix) of A is close to 1, the measurement errors in S produced by errors in I will be minimized. 31 In addition, we reduce the edge artifacts 32 and increase the spatial qualities of the Stokes vector images by a bilinear interpolation on the raw intensity images. For the DoFP polarimeter, the instrument matrices for every super pixel are different, and the Stokes vector computation in Eq. (2) is usually time-consuming. For example, for the construction of 488 × 648 pixels-sized Stokes vector images, the computation takes about 16 s when running in MATLAB ® R2014a with an Intel core i5-4460 CPU. However, using a GTX 780 graphics processing unit (GPU) card and algorithm we reduce the computation time of the same image to 0.047 s in double-precision calculation, which makes it feasible for online video rate visualization of Stokes vector images. The degree of linear polarization (DoLP) and angle of polarization (AoP) can be obtained from the Stokes vector E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 3 2 6 ; 2 6 3 DoLP E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 4 ; 3 2 6 ; 2 2 3 The three images for S 0 , DoLP, and AoP can be fused into a single "polarization staining" image using the hue-saturationintensity visualization scheme, which uses S 0 for the brightness, AoP for hue, and DoLP for saturation. 33 In real applications, polarization staining with only DoLP and AoP generates satisfactory effects. The mapping strategy is shown as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 5 ; 3 2 6 ; 1 3 0 AoP ⇒ Hue; 100% ⇒ Saturation; DoLP ⇒ Intensity: This strategy takes advantage of the colorimetric representation as shown in Fig. 1(b) and is easy to identify by the human vision Journal of Biomedical Optics 056002-2 May 2016 • Vol. 21 (5) system. It avoids the influence of the light intensity variation and focuses on the visualization of polarization properties.

Design of the Division of Focal Plane Polarimeter-Based 3 × 4 Mueller Matrix Microscope
The configuration of the DoFP polarimeter-based 3 × 4 Mueller matrix microscope is shown in Fig. 2. The original light source and the imaging sensor are replaced with the polarization states generator (PSG) and the DoFP polarimeter, respectively. In the PSG, the polarizer (Thorlabs) is fixed, thus no matter what SoP the light emitting diode (LED, Cree, 630 nm) irradiance is, the light passing through the polarizer will be linearly polarized with a constant light intensity. Measuring all columns of a Mueller matrix needs at least four illuminations with independent SoPs. The wave plate (Thorlabs) is rotated to four angles to produce the SoPs. For each illumination, the Stokes vector images of the specimen are calculated using the single-frame image of the DoFP polarimeter. The i'th (i ¼ 1;2; 3, and 4) measurement is described as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 6 ; 6 3 ; where M sample , M W , and M P are the Mueller matrices of the sample, the wave plate, and the polarizer, respectively. S LED is the Stokes vector of the LED irradiance and S out ðiÞ is the Stokes vector of the specimen for the i'th illumination. If the polarizer is fixed to 0 deg, Eq. (6) can be expressed as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 7 ; 3 2 6 ; 6 9 7 where δ is the retardance of the wave plate and θðiÞ is the i'th fast axis angle of the wave plate. The DoFP polarimeter records only the linear part of the Stokes parameters, hence only the first three rows of M sample are measurable. Taking all the four Stokes vector measurements into account, the following equation can be obtained: ; t e m p : i n t r a l i n k -; e 0 0 8 ; 6 3 ; 4 8 9 The Mueller matrix M sample can then be calculated using After the optimization of ½S in , the propagation of errors in Mueller matrix construction is minimized. However, for accurate measurement, the nonideal nature of the polarizing elements should be calibrated. The nonideal nature of the polarizing elements includes the rotation errors of the polarizer and the quarter-wave plate, and the retardance error of the quarter-wave plate. The calibration of the DoFP polarimeter-based Mueller matrix system includes the calibration of the DoFP polarimeter and calibration of the matrix ½S in . Because the DoFP polarimeter has already been calibrated, we only need to calibrate the matrix ½S in . When the sample is removed from the polarization microscope and the wave plate is rotated to four different angles (−45 deg, −19.6 deg, 19.6 deg, and 45 deg), the first three rows of the matrix ½S in will be directly measured by the DoFP polarimeter, and the three errors can be calculated. The calibration shows that the real retardance of the quarter-wave plate used in this study is 90.98 deg. After the optimization and calibration of the matrix ½S in , the mean errors for all the Mueller matrix elements reach to about 0.3% and 0.6% when measuring the Mueller matrices of the air and a linear polarizer, respectively.

Cancerous Tissue Samples
Cancer incidence is growing fast worldwide, making the pathological diagnoses of different cancer tissues crucial tasks. Using a conventional microscope, the features of carcinoma tissues are usually visible for stained slices but invisible for unstained ones. However, when observing the unstained slices using polarization imaging techniques, we can obtain abundant useful polarization information to distinguish the normal and abnormal tissues. To test the potential power of the DoFP polarimeterbased 3 × 4 Mueller matrix microscope, in this study, we apply it to human liver and cervical carcinoma tissue samples. Figure 4 shows the carcinoma tissue slices provided and prepared by Shenzhen Sixth People's Hospital. The slice shown in Fig. 4(a2) is a 4-μm thick hematoxylin-eosin (H-E) stained human liver carcinoma slice, and the slice shown in Fig. 4(a1) is the corresponding unstained, dewaxed 12-μm thick slice from the same biopsy sample archived in the hospital. The slice shown in Fig. 4(b2) is a 4-μm thick H-E stained human cervical carcinoma slice, and the slice shown in Fig. 4(b1) is the unstained, dewaxed 12-μm thick slice from the same biopsy sample. Because the two slices shown in Figs. 4(a1) and 4(a2) [and in Figs. 4(b1) and 4(b2)] are originally "twins" adjacent to each other, the cancerous areas should look similar. The microscopic images of the H-E stained slices are shown in Figs. 4(a3), 4(a4), 4(b3), and 4(b4). Many fibrous structures are visible across both slices, and the stained colors of the dysplastic regions are darker than that of healthy tissues. The microscopic images of the unstained, dewaxed 12-μm thick slices are shown in Figs. 6(a1) and 6(b1), respectively. From these images, we can see that the characteristic microstructures represented in the H-E stained slices are invisible in the unstained ones. This work was approved by the Ethics Committee of the Shenzhen Sixth People's (Nanshan) Hospital.  During the pathological process from hepatitis to cirrhosis and liver cancer, the proportion of fibrous structures rises. 20 Therefore, the fibrosis degree can serve as a quantitative indicator for the detection and scoring of liver carcinoma tissues. It can be observed from the Mueller matrix elements shown in Fig. 5 that the proportion of fibrous structures in the area represented by Fig. 5(b) is higher than that represented by Fig. 5  (a). On the other hand, the cervical carcinoma processes usually lead to breaking down of the well-aligned anisotropic structures in healthy cervix tissues, which can be used as diagnostic indicators from different cervical intraepithelial neoplasia stages to cervical cancer. 27 As indicated by the Mueller matrix elements shown in Fig. 5, the breakdown level of the well-aligned anisotropic structures in the area represented by Fig. 5(d) is higher than that represented by Fig. 5(c). It can be clearly observed from Fig. 5 that the intrinsic anisotropic fibrous structures of the cancerous liver and cervical tissues are mainly revealed in the m 24 and m 34 elements. Meanwhile, in other Mueller matrix elements, the microstructures are not significant or are even invisible, which are different from the characteristic features of Mueller matrices of bulk tissues measured in the reflection mode. 7 When measuring a bulk tissue, the strong scattering power usually leads to a Mueller matrix with prominent patterns in all the elements. Hence, for the biomedical polarimetry in vivo, the 3 × 3 Mueller matrix imaging is appropriate. However, for the standard thin dewaxed human carcinoma tissue slices with limited scattering, the 3 × 4 Mueller matrix images can provide more important pathological information than linear polarized 3 × 3 Mueller matrix images. The Mueller matrix can completely describe the polarization property of a specimen, but its physical meaning is not easy to understand. The diattenuation of a sample is a function of the first row of the Mueller matrix. For the thin carcinoma slice, the values of diattenuation are often very small. Since the diagonal m 11 , m 22 , and m 33 are close to one, we can safely assume that the samples are almost nondepolarizing. For the nondepolarizing samples, the linear retardance (δ) and its orientation angle (θ) can be derived as The value and orientation of the birefringence effect can be proven as potentially crucial information for pathologists. Since the tissue slices are usually very thin, their values of retardance Considering that the Mueller matrix elements m 14 , m 21 , and m 31 are close to zero as shown in Fig. 5, the following approximation can be obtained: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 1 ; 6 3 ; 5 9 9 m 11 ≈ S 0 ð4Þ; m 24 ≈ S 1 ð4Þ; m 34 ≈ S 2 ð4Þ: Thus, we can recast Eq. (9) as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 2 ; 6 3 ; 5 Equation (12) is similar to Eqs. (3) and (4). The information in δ and θ is essentially the same to those in DoLP and AoP.
Thus, under right-hand circularly polarized light illumination, the polarization staining images of the unstained dewaxed human liver and cervical carcinoma specimens calculated using Eqs. (2)-(5) are shown in Figs. 6(a2)-6(d2), in which the information of m 24 and m 34 are encoded. Compared with the intensity images shown in Figs. 6(a1)-6(d1), the polarization staining images reveal much more detailed information of the microstructures, especially the pathological-related fibrosis with strong birefringence. The values of DoLP, as well as the color saturation of the polarization image of the fibrosis region, will be higher than the background. In addition to the value of the DoLP, the alignment orientations of the microfibers can also be clearly revealed from polarization staining images. The color images of the H-E-stained slices and the polarization staining images of the unstained slices are based on different imaging contrast mechanisms. Therefore, the differences between the results shown in Fig. 4 and 6 are significant. Compared with Fig. 4, the anisotropic fibrosis structures in Fig. 6 are more prominent than other types of microstructures, meaning that the polarization imaging technique can be used to reveal the optical anisotropic nature of tissue samples in pathological assessments. The polarization staining imaging using circularly polarization light has several unique advantages: (a) many previous works have pointed out that the polarization imaging techniques are capable of providing additional structural information of the samples, especially information about structures smaller than the diffraction limit [26][27][28] and (b) the use of circularly polarized light illumination can avoid the orientation influence of the linearly or elliptically polarized light illuminations on the anisotropic microstructures widely existing in biological tissues.
Because both the intensity and the polarization staining images are obtained with single frames, using GPU acceleration algorithm, the DoFP polarimeter-based polarization microscope has the capacity for real-time polarization monitoring of dynamic processes such as living cell behaviors. Similarly, the use of the DoFP polarimeter and circularly polarized light illumination may make the implementation of the real-time polarization endoscope possible, too.

Conclusion
In this article, a DoFP polarimeter-based polarization microscope capable of simultaneously measuring both the Stokes vector and the 3 × 4 Mueller matrix with an optimal 4-SoP illumination is presented. We test the polarization microscope by measuring unstained human cancerous tissue slices. The experimental results show that the characteristic microstructures represented in the H-E-stained slices are invisible in the unstained ones. However, from the 3 × 4 Mueller matrix images, we can see that the fibrous structures are highlighted. The intrinsic anisotropic fibrous structures of the cancerous liver and cervical tissues are mainly revealed in the m 24 and m 34 elements. Meanwhile, in other Mueller matrix elements the microstructures are not significant or even invisible. The characteristic features of the m 24 and m 34 can be observed in the polarization staining images using the circularly polarized light as illumination. In this way, combined with GPU acceleration algorithm, the DoFP polarization microscope has the capacity for realtime polarization imaging of biomedical specimens to aid in clinical diagnosis.