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Abstract
Mueller matrix microscopy is capable of polarization characterization of pathological samples and polarization imaging based digital pathology. In recent years, hospitals are replacing glass coverslips with plastic coverslips for automatic preparations of dry and clean pathological slides with less slide-sticking and air bubbles. However, plastic coverslips are usually birefringent and introduce polarization artifacts in Mueller matrix imaging. In this study, a spatial frequency based calibration method (SFCM) is used to remove such polarization artifacts. The polarization information of the plastic coverslips and the pathological tissues are separated by the spatial frequency analysis, then the Mueller matrix images of pathological tissues are restored by matrix inversions. By cutting two adjacent lung cancer tissue slides, we prepare paired samples of very similar pathological structures but one with a glass coverslip and the other with a plastic coverslip. Comparisons between Mueller matrix images of the paired samples show that SFCM can effectively remove the artifacts due to plastic coverslip.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Mueller matrix is a complete description of the sample’s polarization characteristics and has shown many applications, such as cancer diagnosis [1–17], thin film characterization [18–20] and image dehazing [21,22]. In particular, Mueller matrix microscopy has shown broad prospects in polarization digital pathology with distinctive advantages for the identification and diagnosis of pathological microstructures [11–17]. In a typical clinical histopathology practice based on microscopy imaging, a tissue slice is prepared with a coverslip that fixes and protects the sample from dust or accidental contact, or prevents the microscope’s objective lens from contacting the sample. Coverslip thickness and refractive index should be carefully controlled to avoid spherical aberration and resolution reduction. The most common materials for the coverslip are soda lime glass and borosilicate glass, or fused silica glass for fluorescence microscopy using ultraviolet light sources [23]. In recent years, plastic coverslips are becoming more and more popular. Compared to glass coverslips, plastic coverslips are easier to keep dry and clean, and avoid slide-sticking or air bubbles [24].
Many different mechanisms may introduce polarization artifacts in an optical system, such as residue birefringence in various optical components or dichroism due to surface reflection. In the design of a Mueller matrix imaging system, sophisticated calibration techniques have to be used to minimize the effects of such intrinsic polarization. Before measuring actual samples, an instrument matrix of the system has to be acquired by taking Mueller matrices of standard samples such as air, polarizers and wave plates. The instrument matrix is then used to calibrate experimental results [25–32]. However, such calibration methods do not apply if the errors are introduced along with the sample, for instance, the interference from the coverslips which are part of the pathological tissue slices, or from petri dishes when measuring living cells. If such sample-dependent polarization artifacts are spatially uniform and stay steady during measurements, they may be removed or reduced by measuring the Mueller matrix at the background area of the pathological tissue slices where there are no tissues behind the coverslip. However, such a calibration method is laborious and time-consuming, and often fails if the coverslip’s polarization property varies spatially. A more practical self-calibration method without much prior knowledge of the plastic component is desired.
The sample’s image contains spatial frequencies which can be utilized to extract spatial information related to periodic structures. Spatial frequency analysis has been used in the Mueller matrix image’s feature extraction [33]. The spatial frequency spectrum of the Mueller matrix image provides abundant information about the regularity and texture of the sample’s structure. Since the polarization artifacts due to a coverslip usually vary slowly across Mueller matrix images, they have distinct spatial frequency distributions compared to the polarization features of a complex pathological tissue. It is relatively easy to distinguish signals and interferences in the frequency domain. In this work, we applied this idea to the calibration process and proposed a spatial frequency based calibration method (SFCM) to remove the polarization artifacts in Mueller matrix microscopic images introduced by the plastic coverslip. The polarization information of the plastic coverslips and pathological tissues are separated by spatial frequency analysis of Mueller matrix images without the need for prior measurement of the coverslip, then the pathological tissue’s Mueller matrix can be restored from the interference of plastic coverslip by matrix inversion.
2. Materials and methods
2.1 Experimental setup and samples
Figure 1 shows the basic schematic diagram of the upright Mueller matrix microscope for pathological tissue slice measurements. The monochromatic 633 nm illumination light from the light-emitting diode (LED) is collimated by a collimating lens to generate a parallel beam. Its polarization state is modulated by the polarization states generator (PSG), then illuminated the pathological tissue slice. The scattered light from the sample is collected by the objective lens and tube lens. Its polarization states are analyzed by a polarization state analyzer (PSA) and detected by an imaging sensor. In this study, the PSG consists of a fixed-angle linear polarizer and a rotatable quarter-wave plate. The PSA and imaging sensor adopts the dual division-of-focal plane polarimeters (DoFPs) scheme [26]. The pathological tissue slice sample can be simplified into a three-layer structure. The bottom layer is the microscope glass slide of 1 mm thickness, whose polarization property is close to air and can be neglected. The middle layer is the pathological tissue of about 4 µm thickness. The top layer is the coverslip of 0.16 mm thickness. When the coverslip is made of plastic, its polarization property will be introduced into the measurement results of the sample’s Mueller matrix.
Fig. 1. Schematic diagram of the Mueller matrix microscope for pathological tissue slice measurement.
During the Mueller matrix measurement, PSG’s quarter-wave plate rotates to four angles to generate four independent input Stokes vectors $[{{{\boldsymbol S}_{in}}} ]$, and PSA detects the four output Stokes vectors $[{{{\boldsymbol S}_{\textrm{out}}}} ]$, then the Mueller matrix of the pathological tissue slice can be calculated by:
In this study, the adjacent 4-µm-thick hematoxylin-eosin (H&E) stained lung cancer pathological slices are prepared to provide a pair of samples with very similar microstructure but different types of coverslips. All the samples are provided by Peking University Shenzhen Hospital, as shown in Fig. 2. The pathological tissue slice with glass coverslip was prepared manually by a professional technician, and the pathological tissue slice with plastic coverslip was prepared by an automatic coverslip machine (Tissue-Tek Film Automated Coverslipper, Sakura Finetek, Inc., USA). This study was approved by the Ethics Committee of the Peking University Shenzhen Hospital.
Fig. 2. The H&E images of adjacent pathological tissue slices of human lung cancer with a glass coverslip (a), and a plastic coverslip (b), respectively.
2.2 Spatial frequency based calibration method (SFCM) for eliminating polarization artifacts
In polarization optics, when light passes through a sample consisting of several layers, the Mueller matrix of the sample can be represented by the cascade multiplication of the Mueller matrices of each layer:
For the pathological tissue slice with a coverslip, the equivalent Mueller matrix ${{\boldsymbol M}_{\textrm{measure}}}$ measured without calibration can be expressed as:
For the Mueller matrix of the plastic coverslip, the polarization properties are dominated by linear retardance caused by the birefringence of the plastic, ${{\boldsymbol M}_{\textrm{coverslip}}}$ can be expressed as the following equation with only ${m_{11}}$ and the lower-right 3${\times} $3 elements are non-zero:
The Mueller matrix of the pathological tissue mainly contains three polarization properties including diattenuation, retardance and depolarization [34–38]. Since polarization effects are usually very weak for the thin histopathology tissue, the above polarization parameters are close to zero, then the Mueller matrix ${{\boldsymbol M}_{\textrm{tissue}}}$ can be simplified as the following form [39]:
Equation (6) shows that the plastic coverslip does not affect the first row of ${{\boldsymbol M}_{\textrm{tissue}}}$, but seriously affect other elements, especially the lower-right 3${\times} $3 part of the Mueller matrix. The measured Mueller matrix of the sample ${{\boldsymbol M}_{\textrm{measure}}}$ before calibration can be approximated as the Mueller matrix of the plastic coverslip ${{\boldsymbol M}_{\textrm{coverslip}}}$. The serious interference on these elements of ${{\boldsymbol M}_{\textrm{tissue}}}$ from ${{\boldsymbol M}_{\textrm{coverslip}}}$ proves the necessity of plastic coverslip calibration.
If the Mueller matrix of the plastic coverslip is invertible and can be determined in advance, the polarization interference of the plastic coverslip can be removed by matrix left division:
To perform calibration, the most important problem is to determine the Mueller matrix of the plastic coverslip ${{\boldsymbol M}_{\textrm{coverslip}}}$. Since the polarization property of the plastic coverslip may vary between different manufacturers, measuring the coverslip alone without the tissue in advance is not only laborious but also unfeasible. One possible solution is to estimate ${{\boldsymbol M}_{\textrm{coverslip}}}$ by calculating the average value of the Mueller matrix of the coverslip by finding background region with the coverslip and without the tissue in the field of view (FOV). However, for samples without clear background regions or ${{\boldsymbol M}_{\textrm{coverslip}}}$ with spatial distributions, the above approach will fail.
The polarization properties of the plastic coverslips and the pathological tissues tend to be different in spatial distributions. The former usually varies slowly across the FOV and has lower spatial frequency components, while the latter often contains much more microstructural details and is dominated by higher spatial frequency components. Therefore, contributions to the Mueller matrix by the coverslip and pathological tissue can be differentiated through spatial frequency analysis. The process includes three steps: calculate the spatial frequency of the measured Mueller matrix ${{\boldsymbol M}_{\textrm{measure}}}$ by performing 2D Fourier transform (2D-FT), extract the frequency components of the coverslip by low pass filtering, estimate the Mueller matrix of the coverslip ${{\boldsymbol M}_{\textrm{coverslip}}}$ by performing a 2D inverse Fourier transform (2D-IFT).
Figure 3 shows the flowchart of estimating the Mueller matrix of the plastic coverslip ${{\boldsymbol M}_{\textrm{coverslip}}}$. We calculated the spatial spectrum image of the measured Mueller matrix ${{\boldsymbol M}_{\textrm{measure}}}$ by 2D-FT. The image shows the low spatial frequency component concentrates in the center of the spatial spectrum image of each Mueller matrix element, while high spatial frequency components are widely and unevenly distributed, which mainly comes from the details of the irregular structures of pathological tissues. By performing low pass filtering, high frequency components are eliminated, and only the polarization information with low spatial frequency from the plastic coverslip is retained. Since the distributions of Mueller matrices of the pathological tissue and the plastic coverslip overlap in the frequency domain, the Gaussian filter with flat distribution is better than the ideal filter. In addition, as shown in Fig. 4, since multiplying the ideal low pass filter in the frequency domain is equivalent to convolution with a sinc (sin(x)/x) function in spatial domain, and sinc function has prominent side lobes, ring effect would remain on the filtered results, while Gaussian low pass filter has similar response in both spatial and frequency domain, it can avoid the ring effect on the results, which is important for obtaining accurate polarization information for biomedical applications [40]. In order to ensure the polarization property of the plastic coverslip rather than the high-frequency pathological tissue is extracted, the standard deviation $\sigma$ of the Gaussian filter should be carefully seted, then the cut-off frequency of the Gaussian filter in x-axis or y-axis can be calculated by the following equation:
Fig. 3. Flowchart of estimating the Mueller matrix of the coverslip ${{\boldsymbol M}_{\textrm{coverslip}}}$. ${{\boldsymbol M}_{\textrm{coverslip}}}$ is estimated by performing low pass filtering on the spatial spectrum of the measured Mueller matrix ${{\boldsymbol M}_{\textrm{measure}}}$ using a Gaussian filter. For better visual display, the diagonal elements of the Mueller matrix are subtracted by 1, and the size of the spatial spectrum image and Gaussian filter is expanded.
Fig. 4. (a) Frequency distribution and (b) spatial distribution of Gaussian low pass filter. (c) Frequency distribution and (d) spatial distribution of ideal low pass filter.
3. Results and discussions
3.1 Simulation experiment of SFCM on polarization standard samples
Firstly, the calibration performance of SFCM was verified through a simulation experiment. As shown in Fig. 5(a), the simulated polarization standard sample using MATLAB includes the foreground region consisting of a ${0^\circ }$ linear polarizer (square area) and two quarter-wave plates with different angles (circular areas), as well as the background region of air. The relatively small size of the foreground region contains high spatial frequency polarization information of pathological tissue, while the uniform polarization properties inside the foreground area also contain low spatial frequency components. Figure 5(b) shows the theoretical Mueller matrix image of this polarization standard sample. By adding a ${20^\circ }$1/8 wave plate between the PSA and the polarization standard sample on each pixel to simulate the polarization interference from the plastic coverslip, the affected Mueller matrix image was obtained (Fig. 5(c)), and it can be seen that all but the first row of Mueller matrix elements have undergone significant distortions, with the most pronounced effect on the lower-right 3${\times} $3 Mueller matrix elements. The Mueller matrix of the polarization standard sample with plastic coverslip after SFCM calibration is shown in Fig. 5(d). The average root mean square error (RMSE) of the foreground region was calculated to evaluate polarization standard sample’s error level according to Eq. (8),
N is the number of pixels of foreground region in a Mueller matrix element image. ${x_i}$ and $tru{e_i}$ are the pixel value of Mueller matrix element with and without the plastic coverslip, respectively. After SFCM calibration, RMSE decreased from 0.0267 to 0.0035, which demonstrates that the error introduced by the plastic coverslip can be effectively eliminated and the original polarization properties of the foreground sample can be retained satisfactorily by SFCM.
Fig. 5. (a) The simulated polarization standard sample. (b) Mueller matrix image of the simulated polarization standard sample without plastic coverslip. (c) Mueller matrix image of the simulated standard sample with plastic coverslip before calibration. (d) Mueller matrix image of the simulated polarization standard sample with plastic coverslip after SFCM calibration.
3.2 Comparison of calibration performance between different methods
In this section, we compared the calibration performance between calibtation using SFCM, and calibration using the average value of background region’s Mueller matrices without the pathological tissue, which can be regarded as the plastic coverslip’s Mueller matrix. Figure 6(a), (d), and (g) show that after calibration using background region’s average value, lower-right 3${\times} $3 elements of the Mueller matrix remains considerable nonuniformity compared to the weak polarization information of the pathological tissue, while this phenomenon does not exist on the Muller matrix after calibration using SFCM.
Fig. 6. The Mueller matrix images of the pathological tissue slice (a) before calibration, (d) after calibration using the average value of the background region, and (g) after calibration using SFCM. For better visual display, the diagonal elements of the Mueller matrix are subtracted by 1. The linear retardance δ of the pathological tissue slice (b) before calibration, (e) after calibration using the average value of the background region and (h) after calibration using SFCM. The fast axis orientation θ of the pathological tissue slice (c) before calibration, (f) after calibration using the average value of the background region and (i) after calibration using SFCM.
In biological research, in addition to the Mueller matrix, we also concerned about the recovery performance of the sample’s Mueller matrix parameters. Here we calculate the Mueller matrix polar decomposition parameters (MMPD) [34–38], including the linear retardance δ and the fast axis orientation θ. As for the calibration performance of the linear retardance δ, the result of δ before calibration is dominated by the high birefringence of the plastic coverslip, while the result of δ calibrated by SFCM shown uniform background and the real polarization information of pathological tissue is well revealed, however, the result of δ calibrated by the average value of the background region shows obvious artifact at the edge of the FOV, mainly caused by the nonuniform distribution of plastic coverslip’s polarization property. As for the calibration performance of the fast axis orientation θ, the uncalibrated result cannot reveals much useful information about the pathological tissue, and the result calibrated by the average value of the background region is also seriously disturbed, while the result calibrated by SFCM can restore the polarization orientation information of the pathological tissue. Therefore, SFCM has better calibration performance to restore the pathological tissue’s true polarization properties.
3.3 Comparison of Mueller matrix imaging results for adjacent pathological slices
In this section, we compared the Mueller matrix images of adjacent pathological tissue slices with a glass coverslip and a plastic coverslip under a 20 ${\times} $ objective lens. The Mueller matrix images of tissue slices with a plastic coverslip before calibration, a plastic coverslip after SFCM calibration, a glass coverslip before calibration, and a glass coverslip after SFCM calibration were shown in Fig. 7(a), (b), (c), and (d), respectively. The Mueller matrix image in Fig. 7(a) clearly shows that the plastic coverslip affects the lower-right corner of Mueller matrix elements seriously, making it almost impossible to retrieve information on the retardance of the pathological tissue. The artifacts are mostly removed after SFCM calibration as shown in Fig. 7(b) and the restored Mueller matrix elements of the tissue slice under the plastic coverslip look similar to Fig. 7(c), which is regarded as the gold standard. Note, Fig. 7(b) shows that the elements at the lower-right 3${\times} $3 corner of the calibrated Mueller matrix are still different from Fig. 7(c). This is likely due to the retardance generated in the histological tissue slice preparation process. We also conduct SFCM calibration on Muller matrix image of the sample with glass coverslip (Fig. 7(d)), the result shows that SFCM slightly change the polarization information of the sample compared with Fig. 7(c), however, the high-frequency information belonging to the fibers in pathological tissue still preserved after SFCM calibration.
Fig. 7. Mueller matrix images of (a) the pathological tissue slice with a plastic coverslip before calibration, (b) a pathological tissue slice with a plastic coverslip after calibration using SFCM, (c) a pathological tissue slice with a glass coverslip, and (d) a pathological tissue slice with a glass coverslip after calibration using SFCM. For better visual display, the diagonal elements of the Mueller matrix are subtracted by 1.
Quantitative assessments of the SFCM show more clarity by comparing the frequency distribution histograms (FDHs) of Mueller matrix elements for pathological tissue slices with a glass coverslip, a plastic coverslip, and a plastic coverslip after SFCM calibration, as shown in Fig. 8. The results indicate Mueller matrix elements m24, m34, m42 and m43 that related to linear retardance were greatly affected by the plastic coverslip, as shown in red lines. After SFCM calibration, these retardance-related Mueller matrix elements were well restored as demonstrated by the approaching of the black lines towards the blue lines in Fig. 8. The results in Fig. 8 show that SFCM can effectively remove the plastic coverslip introduced artifacts and restore the retardance-related microstructural information of pathological samples. Figure 8 also demonstrates the plastic coverslip has little impact on diattenuation-related Mueller matrix elements including m12, m13, m21, and m31.
Fig. 8. Frequency distribution histogram (FDH) of the 16 Mueller matrix elements for pathological tissue slices with a glass coverslip (blue line), a plastic coverslip without calibration (red line), and a plastic coverslip after SFCM calibration. The graph of the Mueller matrix element m11 shows the FHD of the intensity image of the adjacent pathological slices. The other 15 Mueller matrix elements were normalized by m11.
3.4 Comparison of polarization parameters for adjacent pathological slides
The polarization parameters that derived with relatively clear physical meanings to characterize the microstructures of the lung cancerous tissues with a glass coverslip, a plastic coverslip, and the plastic coverslip with SFCM calibration were shown in Fig. 9. The Mueller matrix polar decomposition (MMPD) parameters D and δ represent the diattenuation and the linear retardance of the lung cancerous tissues [34–38]. The Mueller matrix transformation (MMT) parameters DL is calculated by Mueller matrix elements m12 and m13, and is related to the diattenuation. The MMT parameter rL is calculated by Mueller matrix elements m24 and m34, and is related to the linear retardance of lung cancerous tissues [41–42]. The calculation formulas of those polarization parameters are shown in Table 1. The results in Fig. 9 illustrate the plastic coverslip induces very small artifacts on diattenuation-related polarization parameters D and DL, which are usually more sensitive to cell nucleus microstructures of pathological tissue. Figure 9 also demonstrate the plastic coverslip induces large artifacts on linear retardance-related polarization parameters δ and rL, which generally are more sensitive to fibrous microstructures of pathological tissue. The comparison of polarization parameters for glass coverslip, plastic coverslip, and the plastic coverslip with SFCM calibration indicates the SFCM can effectively restore the linear retardance-related microstructures of pathological tissue. The removal of the artifacts introduced by plastic coverslip enables the acceleration of large-scale applications of polarization imaging in assisting the early detection, diagnosis, and prognosis of cancerous diseases.
Fig. 9. Polarization parameters of the lung cancer tissue slices with a glass coverslip, a plastic coverslip, and the plastic coverslip with SFCM calibration. MMPD parameter δ represents the linear retardance, MMPD parameter D represents diattenuation. Parameter ${D_L}$ is calculated from Mueller matrix elements m12 and m13, and is related to diattenuation. Parameter rL is calculated from Mueller matrix elements m24 and m34, and is related to linear retardance.
Table 1. Polarization basis parameters
Next, the quantitative analysis of Fig. 9 was performed. It is worth noting that since the pathological tissue slice with glass coverslip and the pathological tissue slice with plastic coverslip come from adjacent pathological tissue sample, therefore, it is difficult to evaluate the error of the polarization parameter image after calibration in pixel scale. In order to evaluate the spatial distribution of the error of polarization parameter after SFCM calibration, we divided the polarization parameter image into 20${\times} $20 patches. For each patch, based on the polarization parameter of the tissue slice with glass coverslip which regarded as the reference image, structural similarity index metric (SSIM) of the polarization parameter of the sample with plastic coverslip before and after calibration was calculated respectively [43]. SSIM is a referenced image quality evaluation index which evaluates the similarity between two images from three aspects: brightness, contrast, and structure. The value range of SSIM is between 0 and 1, and the closer it is to 1, the less error the polarization parameter image has.
Figure 10 shows the SSIM images of sample’s polarization parameters with plastic coverslip before and after SFCM calibration in patch scale. It can be seen that due to the introduction of birefringence interference by the plastic coverslip, SSIM of linear retardance-related parameters δ and rL decreased significantly before calibration, while the impact on the parameters related to diattenuation was not severe. After SFCM calibration, the average SSIM of all parameters was improved and greater than 0.8, which proves that SFCM calibration can effectively remove the interference of residual polarization properties of plastic coverslip. In addition, it can also be observed that due to the overlap of the Mueller matrix images of pathological tissue and plastic coverslip in frequency domain, the polarization information of the samples will be affected by the SFCM calibration, and the polarization parameters related to linear retardance are more difficult to be fully restored. A typical case is shown in Fig. 10(g) and (h), where the tissue in the central area of the image has high density and lower spatial frequency, SFCM cannot perfectly distinguish the polarization information between the tissue in this region and the plastic coverslip, resulting in lower SSIM and poorer calibration performance. For tissue slice with significant high-frequency information, SFCM is more suitable; whereas for samples with more low-frequency information, such as dense zones generated by the non-uniform thickness of the tissue sections [44], lower cutoff frequency of the Gaussian filter for SFCM calibration is prefered, or apply the calibration method using the average value of the background region.
Fig. 10. SSIM image of polarization parameters’ patches of sample with plastic coverslip before (1st row) and after (2nd row) SFCM calibration.
4. Conclusions
In conclusion, Mueller matrix imaging provides abundant histological microstructural information and has shown promising application potential for assisting pathologists in early detection, diagnosis, and prognosis of cancerous diseases through polarization imaging based digital pathology. Previous Mueller matrix imaging studies were performed on pathological slides with glass coverslips. The transformation from using glass coverslips to plastic coverslips by hospitals raises the necessity to remove the polarization artifacts introduced by plastic coverslips. This study proposed a spatial frequency based calibration method (SFCM) to separate the polarization information of the plastic coverslip and the pathological microstructures by the spatial frequency analysis of Mueller matrix images. Under a 4${\times} $ objective lens, a comparison of the calibration performance between SFCM and the calibration using the average value of the Mueller matrix of the background region was performed. The results indicate the calibration using the average value of the Mueller matrix of the background region show obvious artifacts at the edge of the FOV, which may be mainly caused by the nonuniform distribution of the plastic coverslip’s polarization property. However, the SFCM calibration can well restore the linear retardance of the pathological sample with a uniform background. Under a 20${\times} $ objective lens, comparisons of Mueller matrix images and polarization parameters of adjacent pathological tissue slides with a glass coverslip and a plastic coverslip before and after SFCM calibration were performed. The results indicate Mueller matrix elements and polarization parameters related diattenuation were less affected by the plastic coverslips. Those related to the linear retardance were greatly affected, but can be well restored by the SFCM calibration method. The removal of plastic polarization artifacts by the spatial frequency analysis is also applicable to remove the polarization artifacts of the plastic petri dish in polarization imaging of living cells.
Funding
National Natural Science Foundation of China (11974206, 61527826).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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