Shadowless amyloid imaging with quantitative birefringence contrast

Abstract

In the diagnosis of amyloidosis using Congo red staining, the apple-green birefringence observed under crossed Nicols has been considered the gold standard for over fifty years. However, a variety of colors, including orange and blue, not just apple-green, are observed in clinical fields. Moreover, the issue of polarization shadows under the crossed Nicols also impedes detailed observations of amyloid deposits, such as senile plaques. The introduction of advanced birefringence microscopes into clinical settings is also limited due to equipment-related challenges. In this study, we propose a simple but robust method that equips a standard optical microscope with polarizers to determine the birefringence and dichroism of Congo red-stained amyloid deposits. By employing an innovative image processing technique that averages multiple images, we achieved a correlation of over 95% with the birefringence distribution. This approach not only minimizes commonly encountered polarization shadows but also enhances the clarity and accuracy of amyloidosis diagnoses. Our strategy simplifies the diagnostic process and facilitates the incorporation of quantitative imaging technologies into routine pathological practices. The findings suggest that characteristic birefringence, independent of azimuthal angle, serves as a reliable marker for amyloidosis, potentially revolutionizing diagnostic protocols in the field.

Introduction

Amyloid is a substance consisting of misfolded proteins that form fibrous masses insoluble in water and blood^1,2,3. Atomic force microscopy has revealed that amyloids are elongated fibrils with a diameter of approximately 10–15 nm and a length of several micrometers^4,5,6. These insoluble amyloid fibrils deposit in the spaces between cells and tissues, causing tissue dysfunction through repeated aggregation and accumulation^7,8. This condition is known as amyloidosis. Understanding the mechanisms of amyloidosis is crucial as it aids in enhancing diagnostic and therapeutic strategies for amyloid-related diseases.

For a long time, Congo red staining has been the gold standard for the definitive diagnosis of amyloidosis⁹. The dye molecules that infiltrate the gaps in the β-sheet structures of amyloid fibrils bind to them. This binding creates molecular orientation order and anisotropy in the dielectric properties, which are the origins of birefringence. When observed under an optical microscope with a pair of polarizers in a crossed Nicols arrangement¹⁰, an apple-green color appears. Traditionally, the observation of this color has been considered a definitive diagnosis of amyloidosis. However, the color observed by optical microscopy under crossed Nicols is not confined to apple green. Observed color also includes yellow, orange, and even blue, reflecting a combination of amyloid deposits and Congo red staining. This diversity has led to questioning the traditional use of the apple-green color as an indicator for diagnosing amyloidosis^11,12,13,14.

The discovery of amyloid dates back to 1854, as referenced in¹⁵. Before this, numerous medical scientists had identified misfolded protein states, referred to by various names without a fixed disease designation. The history of amyloid and Congo red staining is well described in Refs^15,16,17. In their investigations, the expression of birefringent color in amyloid deposits with Congo red was first discussed in 1927, spurred by advancements in optical and staining techniques^18,19. In the previous survey¹⁶, reports from²⁰ indicate that there was confusion regarding the birefringence of amyloid deposits in 1953. A proposal to associate the apple-green color with the birefringence of amyloid deposits is also believed to have originated in the 1973²¹. At the same time, researchers understood that this birefringence, when induced by Congo red staining, depended on the staining conditions, leading to questions about the reliability of the apple green color, as noted in Refs^11,12,13,14. Congo red staining of amyloid deposits goes beyond merely observing the presence or absence of birefringence; it also requires consideration of the magnitude of birefringence and azimuthal angle, along with the linear polarization dichroism and the wavelength-dependent absorption of the dye. This presents an optically complex problem, which had not been satisfactorily explained until 2008¹². At the 2016 International Amyloid Society meeting, a definitive diagnosis of amyloidosis was shown to involve observing green, yellow, or orange birefringence under a polarized light microscope in samples stained with Congo red²². However, in 2018, the biennial meeting returned to using the term “apple green birefringence.” Discussions about color resurfaced in 2020, and in 2022, researchers mentioned that the staining level of Congo red varies significantly between microscopes^23,24,25. In 2024, experts recommended adopting the neutral term “Characteristic birefringence” in place of “Apple green birefringence”²⁶. Nearly a century after the observation of birefringence in amyloid deposits by Congo red staining in 1927, the definitive diagnosis of amyloidosis remains an unresolved issue.

Birefringence is characterized by “retardance” and an “azimuthal angle,” which impart color tints depending on the relative angle with the polarizer²⁷. The crossed Nicols setting changes the observed color depending on this angle. Therefore, different colors are observed for the same sample depending on the angle of the polarizer. Due to these advancements in optical technology, several new methods have been developed that enable the detection of amyloidosis^{28,29,30,31,32,33}. Among these, birefringence quantification has proven effective in understanding amyloidosis; several studies have demonstrated the effectiveness of separating dichroism from birefringence in detecting amyloid deposits^29,31,32. In addition to these methods, label-free imaging techniques that do not require staining have been proposed, including proposals for autofluorescence, Raman microscopy, and deep learning^{34,35,36,37,38}. Unfortunately, there are no reports of the adoption of advanced microscopy in clinical practice because most microscopes installed in clinical settings are biological microscopes, not even polarizing microscopes. At best, they are equipped with a pair of polarizers, and thus are based on crossed Nicols manipulation, with no space for other optical elements to be incorporated. It is challenging for clinical investigators to operate complex polarizing optical elements such as Berek compensators²⁷. Furthermore, previous studies have not established a clear correlation between traditional crossed Nicols observations and birefringence distribution. Therefore, introducing new microscopy technologies requires careful consideration, as it involves understanding complex polarization analysis and detailed birefringence patterns.

Considering these clinical problems, we focus on the optical microscopes used in clinical practice and demonstrate a simple technique to separate birefringence and dichroism by introducing a pair of polarizers. Here, the molecular orientation of Congo red stained amyloid fibrils indicates the anisotropy of the dielectric constant, essentially defining birefringence. The quantity of Congo red molecules aligned with this orientation influences the degree of linear polarization dichroism. We effectively measure birefringence, dichroism, and their azimuthal angle. The findings affirm the effectiveness of previous birefringence imaging techniques. Furthermore, we evaluated the level of birefringence in amyloid deposits stained with Congo red. We also established a correlation between the birefringence distribution and crossed Nicols images—a topic not previously discussed—thereby deepening the understanding of the birefringence distribution in Congo red stained amyloid deposits. While the birefringence distribution has proven highly effective for amyloidosis detection, its complex programming presents a significant challenge in clinical settings. On the other hand, traditional polarization observation under crossed Nicols, using just a pair of polarizers, offers a straightforward method for determining amyloid deposits. However, a major obstacle arises from the ‘Polarization shadow’^39,40. This is a black shadow seen when the azimuthal angle derived from birefringence aligns with the transmitting axis of polarizers. Nevertheless, there is an issue of unclear image of amyloid deposits due to the polarization shadow. Therefore, we propose a shadowless amyloid imaging technique with quantitative birefringence contrast, which provides precise amyloidosis diagnosis in clinical fields. Specifically, we here demonstrate precise evaluation of birefringence in amyloid deposits stained with Congo red by simply adding a pair of polarizers to a standard optical microscope used in pathology. Notably, this strategy produces images free from shadows, enabling rapid and efficient diagnosis of amyloidosis.

Results

Shadowless amyloid imaging

Our strategy is outlined in Fig. 1. The microscope used in the experiments is a standard optical microscope (Fig. 1a). We positioned a polarizer in front of the condenser lens and an analyzer in front of the two-dimensional detector, each equipped with a rotation mechanism. Figure 1b shows an optical microscopic image of stained amyloid deposits without the analyzer, while Fig. 1c displays the image obtained when the analyzer is inserted. In Fig. 1(c1) and 1(c2), the relative angles between the polarizer and the analyzer differ by 30°. However, both maintain the crossed Nicols condition. Due to the variation in relative angles, the positions of the polarization shadows derived from the crossed Nicols differ in each case. The intensity distribution, shifting from green to yellow, is influenced by the azimuthal angle of birefringence from Congo red staining of the amyloid deposits. This distribution changes with the relative angle of the polarizers. While maintaining the crossed Nicols condition, shifting the relative angles of the polarizer and analyzer by 10° altered the observed image, as shown in Fig. 1d. By analyzing the polarization of the obtained image, we determined the birefringence distribution, as presented in Fig. 1e. The sample, taken from a renal artery, exhibited changes in retardance due to amyloid deposition. Plotting this as a histogram (Fig. 1f), five peaks are found. The small to large level of the retardances is thought to be derived from Congo red staining of amyloid deposits. Averaging the 18 images shown in Fig. 1d generated the intensity distribution shown in Fig. 1g. Unlike Fig. 1c, this image is free of polarization shadows, providing an image that aligns well with Fig. 1e. The agreement of these images was evaluated as shown in Fig. 1h. Sections in Method detail the optics, the method for separating birefringence and dichroism, the technique for producing birefringence images without shadows, and the measurement sample.

Fig. 1

Concept for shadowless amyloid imaging. (a) Optical microscope equipped with a pair of polarizers that have a rotation function. (b) Image of a Congo red stained renal artery without the analyzer. (c) Crossed Nicols view of the Congo red stained renal artery with polarization shadow changes due to polarizer angle θ = 0° (c1) and θ = 30° (c2). The vascular structure is not clearly visible. (d) Sequence of images showing changes in the polarizer angle relative to the crossed Nicols, from θ = 0° to 170°. This sequence highlights pronounced changes in polarization shadow and color distribution. (e) 2D map of retardance created through polarization analysis, utilizing the intensity distribution observed in (d). (f) Histograms derived from the 2D map of retardance in (e) distinguish between staining levels of Congo red. (g) Shadowless image obtained by averaging the intensity distributions shown in (d). (h) Left panel: Comparison of the agreement between images at polarizer angles θ = 0° and 30°. Right panel: Comparison of the agreement among the intensities at θ = 0°, 30°, and the averaged values from (d), against the retardance in (e). Scale bars indicate 10 μm.

Experimental results

Experiments were conducted to separate birefringence and dichroism in amyloid deposits specimens stained with Congo red from the American flamingo. (See the measurement sample in Method) Figure 2a shows an image of the Congo red stained specimen alongside HE-stained adjacent sections, with the detector-side polarizer removed. This image reveals thickening of the vessel walls and band-like structures. Figure 2b displays an image of the Congo red stained specimen. This was entirely stained red, suggesting amyloid deposition. To perform crossed Nicol observations, which are used for definitive diagnosis, a polarizer was inserted. The resulting image was shown in Fig. 2c. While maintaining crossed Nicol conditions, 18 images were captured by adjusting the angle of the polarizer by each 10°. The distribution of the retardance was determined using Eq. (19) (Fig. 2d, See Method). The retardance indicates the magnitude of molecular orientation order due to the binding of Congo red molecules that have infiltrated the β-sheet structure of the amyloid fibrils. The direction of molecular orientation order and the dichroism induced by the Congo red can be calculated using Eqs. (15.1) to (15.3). The optical intensity distribution of the specimen was analyzed as described in Method. The images were captured while rotating the polarizer on the light source side every 10° with the analyzer removed. The resulting azimuthal angle and dichroism were shown in Fig. 2e, f, respectively. Here, with the angle θ of the crossed Nicol condition, the enlarged images in the white box illustrated in Fig. 2c were shown in Fig. 2g. When the images are displayed every 30°, the color distribution shifts from green to yellow as the polarization shadow changes. Here, characteristic birefringence shifting from green to yellow was observed changes in the polarization shadow, due to the varying relative angles between the azimuthal angle of the sample and a pair of polarizers. The two-dimensional distributions of retardance (Fig. 2h), azimuthal angle (Fig. 2i), and dichroism (Fig. 2j) were displayed. The strong molecular orientation associated with the binding of Congo red molecules to the β-sheet structure of the amyloid fibrils was indicated by the retardance, and the results of the dichroism clearly show that the staining was well performed. These results support previous studies.

Fig. 2

Birefringence and dichroism imaging of renal arteries. (a) HE-stained image of adjacent pathology sections. (b) Congo red stained image (without analyzer). (c) Crossed Nicols observation of Congo red staining at θ = 0°. 2D distribution of (d) retardance, (e) azimuthal angle, and (f) dichroism, calculated using the intensity distribution of the green band. (g) Crossed Nicols images for polarizer angles θ = 0°, 30°, 60°, 90°, 120°, and 150°, showing characteristics birefringence shifting from yellow to green as the polarization shadow changes. 2D distribution of (h) retardance, (i) azimuthal angle, and (j) dichroism within the white frame shown in (c). Scale bars of (a–f) and (g–j) indicate 50 μm and 10 μm, respectively.

To investigate the agreement between the retardances and the crossed Nicols images, we analyzed the intensity distributions across the blue, green, and red channels of the CMOS camera. The agreement was employed for the blue (Table 1), green (Table 2), and red (Table 3) channels of the retardance. The mean values within five 500 × 500 pixel areas are displayed in the upper triangular matrix of each table, while the standard deviations are shown in the lower triangular matrix. In these intensity distributions, the images varied significantly when the crossed Nicols alignments were crossed at different angles. However, regardless of the angle of the crossed Nicols, the retardance consistently achieved an agreement score of C ≥ 0.56. This is attributed to the strong reflection of the retardance in the crossed Nicols images. Meanwhile, the polarization shadow results from the effect of the azimuthal angle of birefringence due to the Congo red staining of the amyloid deposits, causing the agreement to vary with the polarizer angle θ.

Table 1 Comparison of agreement between images observed under crossed Nicols (Six angles in blue channel) and retardance in three colored channels.

Table 2 Comparison of agreement between images observed under crossed Nicols (Six angles in green channel) and retardance in three colored channels.

Table 3 Comparison of agreement between images observed under crossed Nicols (Six angles in red channel) and retardance in three colored channels.

Furthermore, as described in Section of Method of evaluating the agreement, we evaluated the agreement between a shadowless amyloid image and the retardance. Figure 3a shows the retardance map. We herein compared crossed Nicols images at θ = 0° and 45° shown in Fig. 3b, c. Figure 3d–g show the intensity distribution for each averaged number of images. The relationship between the agreement and the number of averaged images is summarized in Fig. 3h. The results indicate that the agreement between the shadowless amyloid image and the retardance improves as the number of averaged images increases. This method allows us to easily obtain detailed pathological images without polarization shadows, avoiding the use of complicated analysis algorithms. Shadowless amyloid images can directly demonstrate the characteristic birefringence, significantly contributing to the quantitative evaluation of amyloidosis.

Fig. 3

Comparison of retardance and agreement. (a) Retardance image of amyloid deposits stained with Congo red. (b) Image observed under crossed Nicols with the polarizers at θ = 0° and (c) θ = 45°, showing Congo red staining. (d–g) Shadowless amyloid images averaged at N = 2, 4, 8, and 18, respectively, for different angles in a crossed Nicols arrangement. (h) Results evaluating the agreement between the intensity and retardance distributions above. Scale bars indicate 10 μm.

We generated histograms from the retardance measurements and conducted Gaussian fitting for analysis, with the results shown in Fig. 4. Figure 4a–c show the distributions of retardance in the blue, green and red bands, respectively. The histogram analysis identified approximately five peaks, with the superposition of adjacent peaks resulting in about three distinct peaks. Peaks at retardances of 40° and 20° represent the strong and moderate molecular orientation of amyloid deposits, respectively. Peaks occurring at less than 10° correspond to weak molecular orientation of amyloid deposits, highlighting the specific staining of amyloid by Congo red in this pathological specimen. This specificity allows for the quantitative analysis of staining level and characteristic birefringence.

Fig. 4

Histograms of 2D distribution of retardance (blue, green, red). Results of the histogram analysis show five peaks related to the level of retardance, derived from 2D distribution images calculated using (a) blue, (b) green, and (c) red color filters. These can generally be categorized into three distinct groups. Scale bars indicate 10 μm. Note that we have not performed precise identification of the fibril types, although we have compared the results with those of Picro-Sirius red staining by Fig. 5.

To investigate the issue of co-staining of collagen fibers with Congo red, we examined the distribution of collagen and muscle fibers using Picro-Sirius red staining on sections adjacent to those stained with Congo red. The Picro-Sirius red image in Fig. 5a shows collagen fibers in red and muscle fibers in yellow, with collagen being the predominant component. Figure 5b presents the results from the crossed Nicols examination performed with a polarized light microscope. Given the issues with polarization shadow in both Congo red and Picro-Sirius red staining, we created shadowless amyloid images as part of our approach. The enlarged images of both staining techniques within the white box of Fig. 5b are displayed in Fig. 5c, d. These images highlight the stark differences between the two stains. To quantify these differences, we compared their agreement, similar to the analysis done for Congo red staining, with the findings detailed in Table 4. Like in Tables 1, 2 and 3, the upper right triangle of Table 4 shows the mean agreement values, while the lower left triangle shows the standard deviations. The analysis revealed high agreement among the three-color channels of same staining. However, such agreement was not observed between the three-color channels of different staining, confirming the absence of collagen co-staining in this experiment.

Fig. 5

Adjacent sections of renal arteries stained with Picro-Sirius red. (a) Picro-Sirius red stained renal artery. (b) Shadowless image of Picro-Sirius red stained image. (c) Magnified image of the area within the white frame in (b). (d) Congo red stained image of adjacent sections. Note: These images were analyzed without polarization shadow, utilizing the a₀ component in polarization analysis. Scale bars in (a) and (c) are 50 μm, and in (b) and (d) are 10 μm, respectively.

Table 4 Comparison of congol red staining and Picro-Sirius red staining (3 bands: blue, green, red).

Discussion

Our imaging strategy for separating birefringence and dichroism enables us to quantify the molecular orientation order induced by Congo red molecules entering the β-sheet structures of amyloid fibrils, as shown in Fig. 2d. Additionally, we can assess the amount of Congo red staining within that orientation through the dichroism of the stained molecules, as shown in Fig. 2f. Essentially, the method allows for the quantitative evaluation of staining quality in pathological specimens. Traditionally known as apple green birefringence, the coloration of Congo red stained amyloid deposits actually varies. The findings indicate that this variation depends on the angle of the extinction ratio method and the azimuthal angle of birefringence. This supports the statement made by the International Amyloid Society in 2022²⁵.

When images captured at every 10° using the rotational extinction ratio method were analyzed, the agreement in intensity distribution varied depending on the angle of crossed Nicols, as demonstrated in Tables 1, 2 and 3. This setting for the crossed Nicols complicates the diagnosis of amyloidosis across hospitals. Moreover, polarization shadows, a significant issue with crossed Nicols observations, have impaired the clarity of detailed images^39,40. However, the retardance map developed in this study, alongside the newly proposed shadowless amyloid imaging, addresses these challenges. A comparison of the shadowless amyloid imaging shown in Fig. 3 revealed that an agreement exceeding 95% could be achieved with the retardance results by averaging just eight images. However, acquiring eight images may pose practical limitations in pathological workflows where rapid diagnosis is required. As shown in Eq. (22), the birefringence distribution associated with amyloid can theoretically be extracted using only two polarization images. Experimentally, the approach still achieved approximately 80% agreement. The primary cause of residual discrepancies is image shift and blurring induced by tilt in the simplified polarizer utilized as polarization analyzer. Addressing this issue could enable high-accuracy detection with fewer acquisitions.

In addition, the staining level of Congo red is known to be sensitive to pH and solution preparation conditions, often resulting in uneven coloration or variation in staining (as shown by the histogram analysis in Fig. 4). While the current histogram shown in Fig. 4 does not yet establish a direct causal relationship with staining level, future work should involve stricter control of pH and ionic strength, along with correlation studies between staining and optical quantification methods.

Building on the points mentioned above, the histogram of retardance in Fig. 4 shows that retardance can be differentiated by the level of Congo red staining. While this comparison primarily focused on retardance, the shadowless amyloid image, which aligns closely with retardance as shown in Fig. 3, also indicates that staining level can be distinguished. This signifies our success in separating characteristic birefringence by staining level, affirming the 2024 statement of the International Amyloid Society²⁶.

The 2D distribution of retardance offers clearer and more functional evaluations of birefringence than crossed Nicol images, traditionally considered the gold standard^31,32. Using the strategy, any pathology lab can now easily obtain high-quality images free from shadows. This is achieved by employing a simple averaging process, eliminating the need for complex birefringence analysis algorithms. This method provides a new approach to evaluate retardance, which reflects molecular orientation order independent of the azimuthal angle. Since the rotational extinction ratio method can be more easily implemented with standard optical microscope and polarized light microscope than advanced polarization microscope, it could significantly contribute to advancing the diagnosis of amyloidosis. Furthermore, we anticipate that the shadowless amyloid images are useful as training data for advanced deep learning. We expect that the strategy will open new doors for artificial intelligence in diagnostics^{34,35,36,37,38}.

Finally, a limitation of this study is that the available pathology specimens were restricted to renal amyloidosis in the American flamingo. Detecting amyloidosis that does not bind to Congo red staining molecules is likely challenging. We have not yet evaluated amyloidosis in other organs or from human sources^41,42,43. This innovative approach not only enhances the accuracy of amyloidosis diagnosis but also holds promise for detecting small senile plaques commonly associated with cerebral amyloidosis, significantly contributing to the field^44,45,46. Cerebral amyloidosis represents another potential application of the proposed technique. However, due to the high risk of complications such as bleeding and neurological damage, brain biopsy is generally not performed during life, and non-invasive PET imaging has become the mainstream diagnostic method. In contrast, postmortem neuropathological diagnosis relies on polarized observation of Congo red-stained brain tissue, which remains the gold standard⁴⁷. Cerebral amyloidosis is triggered by amyloid deposition along the inner walls of the microvasculature of the brain, where blood flow is relatively slow, similar to other organs. Conventional polarized light microscopy has limitations in detecting such small vascular deposits due to polarization shadows. The proposed shadowless amyloid imaging method has the potential to visualize amyloid in cerebral microvessels with a high signal-to-noise ratio (SNR), comparable to that of advanced birefringence microscopy. Given the effectiveness of our method, its application to other organs—such as the heart and liver, where biopsy-based diagnosis is more common—should be straightforward. Therefore, the technique has a significant impact on the translation of clinical practice. Further analysis of these studies is expected to enhance the ability to quantitatively assess amyloidosis, a challenge that has persisted for over a century.

The strategy to reduce polarization shadow began with a quantitative analysis of birefringence, which reflects the molecular orientation order induced by Congo red staining. Through meticulous analysis, we found that the histogram intensities of birefringence are distinguished based on staining level. We developed a mathematical approach that led to the creation of a shadowless amyloid imaging technique, which remains unaffected by the azimuthal angle of birefringence. The findings demonstrate that averaging just eight images can achieve a correlation exceeding 95% with the birefringence distribution, significantly enhancing image clarity and accuracy. Therefore, we have shown the potential of these advanced imaging techniques to significantly enhance the accuracy of amyloidosis diagnoses, offering valuable insights for both pathological research and clinical applications.

Methods

Optical microscope and polarizing optics

The optical microscope (Nikon, Eclipse V100) was used to determine Congo red stained amyloid deposits. A schematic diagram of the optical microscope and optical system is shown in Fig. 6a. In this experiment, the observation under a dark-field microscope and a 20x objective lens (Nikon, CFI LU Plan Fluor) was adopted. The objective lens with NA = 0.45 was used. The camera (IDS Imaging, UI-3590CP-C-HQ Rev.2) had a 1/2.3″ sensor (6.140 mm × 4.605 mm) with a pixel size of 1.25 μm. The spatial resolution of the microscope was estimated to be 1.24 μm using the formula d = 0.61l / (NA_ill + NA_obj) to be 1.24 μm. An automatic rotation stage (Thorlabs, K10CR1A2/M) was employed on the light source side to manipulate a pair of polarizers (Thorlabs, LPUV050), which were controlled by a PC. The polarizer was introduced in front of the condenser lens. A simple polarizer mounted on a microscope was used as the analyzer. Although the polarizers featured a rotation mechanism, they lacked a scale. Consequently, a scale was created, and the analyzer was then employed with a resolution of 1°. The maximum power of the light source was 50 W, and the exposure time of the camera was set to 999 ms. While a polarized light microscope is the preferred choice, a pair of polarizers installed in an optical microscope is sufficient for clinical settings. Even if the microscope does not have a pair of polarizers, the addition of a rotatable polarizer provides sufficient functionality. Introducing a rotating holder that can house polarizers is desirable to ensure the effectiveness of this algorithm.

Fig. 6

Optical layout of optical microscope and analysis procedure. (a) Optical microscope with an additional pair of rotational polarizers. (b) our analysis procedure.

Birefringence and dichroism separation techniques

A flowchart of the separation technique for birefringence and dichroism is shown in Fig. 6b. First, the intensities of the light source and the light transmitted through the lens system are corrected. Birefringence is then determined by measuring the dichroism and its azimuthal angle, which are influenced by the molecular orientation in the Congo red stained amyloid deposits. The polarization calculation can be easily introduced using a 4 × 4 Muller matrix and the Stokes vector algorism⁴⁸. The incident polarization from the light source is assumed to be an unpolarized halogen lamp or LED. The Stokes vector of the light source is given by S_in = [α(λ), 0, 0, 0]^T, where ‘a’ represents the optical intensity of the light source, and ‘T’ denotes transposition. The Muller matrices for the polarizer D, the retarder LB, and the rotation matrices R(φ) and R(− φ) are detailed below,

$$\:\varvec{D}=\left[\begin{array}{cc}\begin{array}{cc}1&\:0\\\:0&\:p\end{array}&\:\begin{array}{cc}0&\:0\\\:0&\:0\end{array}\\\:\begin{array}{cc}0&\:0\\\:0&\:0\end{array}&\:\begin{array}{cc}s&\:0\\\:0&\:1\end{array}\end{array}\right]$$

(1)

$$\:\varvec{L}\varvec{B}=\left[\begin{array}{cc}\begin{array}{cc}1&\:0\\\:0&\:1\end{array}&\:\begin{array}{cc}0\:\:\:\:\:\:\:\:\:&\:0\:\:\\\:0\:\:\:\:\:\:\:\:\:&\:0\:\:\end{array}\\\:\begin{array}{cc}0&\:0\\\:0&\:0\end{array}&\:\begin{array}{cc}\text{cos}{\Delta\:}&\:-\text{sin}{\Delta\:}\\\:\text{sin}{\Delta\:}&\:\text{cos}{\Delta\:}\end{array}\end{array}\right]$$

(2)

$$\:\varvec{R}\left(\varphi\:\right)=\left[\begin{array}{cc}\begin{array}{cc}1&\:0\\\:0&\:\text{cos}2\varphi\:\end{array}&\:\begin{array}{cc}0&\:0\:\\\:\text{sin}2\varphi\:&\:0\end{array}\\\:\begin{array}{cc}\:\:\:0&\:-\text{sin}2\varphi\:\\\:\:\:\:0&\:0\end{array}&\:\begin{array}{cc}\text{cos}2\varphi\:&\:0\\\:0&\:1\end{array}\end{array}\right]$$

(3)

$$\:\varvec{R}\left(-\varphi\:\right)=\left[\begin{array}{cc}\begin{array}{cc}1&\:0\\\:0&\:\text{cos}2\varphi\:\end{array}&\:\begin{array}{cc}0&\:0\\\:-\text{sin}2\varphi\:&\:0\end{array}\\\:\begin{array}{cc}0&\:\text{sin}2\varphi\:\\\:0&\:0\end{array}&\:\begin{array}{cc}\text{cos}2\varphi\:&\:\:\:\:0\\\:0&\:\:\:\:1\end{array}\end{array}\right]$$

(4)

.

Using the azimuthal angle and the transmission axis denoted by f, the combined Mueller matrices for the polarizer and retarder, $\:{\varvec{M}}_{\varvec{D}}\left(p,s,\varphi\:\right)$ and $\:{\varvec{M}}_{\varvec{R}}\left(\varDelta\:,\varphi\:\right)$, are calculated as

$$\:{\varvec{M}}_{\varvec{D}}\left(p,s,\varphi\:\right)=\varvec{R}\left(-\varphi\:\right)\bullet\:\varvec{D}\bullet\:\varvec{R}\left(\varphi\:\right)$$

$$\:=\left[\begin{array}{cc}\begin{array}{cc}\:q+r&\:\left(q-r\right)\text{c}\text{o}\text{s}2\varphi\:\\\:\left(q-r\right)\text{c}\text{o}\text{s}2\varphi\:&\:\left(q+r\right){\text{c}\text{o}\text{s}}^{2}2\varphi\:+2\sqrt{qr}{\text{s}\text{i}\text{n}}^{2}2\varphi\:\end{array}&\:\begin{array}{cc}\left(q-r\right)\text{s}\text{i}\text{n}2\varphi\:&\:\:\:\:\:\:\:\:\:\:0\\\:\frac{1}{2}\left(q+r+\left(q+r\right)\right)\text{c}\text{o}\text{s}4\varphi\:&\:\:\:\:\:\:\:\:\:\:0\end{array}\\\:\begin{array}{cc}\left(q-r\right)\text{s}\text{i}\text{n}2\varphi\:&\:\:\:\:\:\:\:\:\:\:\frac{1}{2}\left(q+r+\left(q+r\right)\right)\text{c}\text{o}\text{s}4\varphi\:\\\:\:\:0&\:-\text{sin}\varDelta\:\text{s}\text{i}\text{n}2\varphi\:\end{array}&\:\begin{array}{cc}\left(q+r\right){\text{s}\text{i}\text{n}}^{2}2\varphi\:+2\sqrt{qr}{\text{c}\text{o}\text{s}}^{2}2\varphi\:&\:0\\\:\text{sin}\varDelta\:\text{c}\text{o}\text{s}2\varphi\:&\:2\sqrt{qr}\end{array}\end{array}\right]$$

(5)

$$\:{\varvec{M}}_{\varvec{R}}\left(\varDelta\:,\varphi\:\right)=\varvec{R}\left(-\varphi\:\right)\bullet\:\varvec{L}\varvec{B}\bullet\:\varvec{R}\left(\varphi\:\right)$$

$$\:=\left[\begin{array}{cc}\begin{array}{cc}1&\:0\\\:0&\:{\text{cos}}^{2}2\varphi\:+\text{cos}\varDelta\:{\text{s}\text{i}\text{n}}^{2}2\varphi\:\end{array}&\:\begin{array}{cc}0&\:0\\\:\left(1-\text{c}\text{o}\text{s}\varDelta\:\right)\text{c}\text{o}\text{s}2\varphi\:\text{s}\text{i}\text{n}2\varphi\:&\:\text{sin}\varDelta\:\text{s}\text{i}\text{n}2\varphi\:\end{array}\\\:\begin{array}{cc}0&\:\left(1-\text{c}\text{o}\text{s}\varDelta\:\right)\text{c}\text{o}\text{s}2\varphi\:\text{s}\text{i}\text{n}2\varphi\:\\\:0&\:-\text{sin}\varDelta\:\text{s}\text{i}\text{n}2\varphi\:\end{array}&\:\begin{array}{cc}{\text{sin}}^{2}2\varphi\:+\text{cos}\varDelta\:{\text{c}\text{o}\text{s}}^{2}2\varphi\:&\:-\text{sin}\varDelta\:\text{c}\text{o}\text{s}2\varphi\:\\\:\text{sin}\varDelta\:\text{c}\text{o}\text{s}2\varphi\:&\:\text{cos}\varDelta\:\end{array}\end{array}\right]\:$$

(6)

The Muller matrix $\:{\varvec{M}}_{\varvec{D}\varvec{R}}\left(p,s,\varDelta\:,\varphi\:\right)$ that combines both dichroism and birefringence is derived from the product of M_D and M_R.

$$\:{\varvec{M}}_{\varvec{D}\varvec{R}}\left(p,s,\varDelta\:,\varphi\:\right)={{\varvec{M}}_{\varvec{D}}\left(p,s,\varphi\:\right)\bullet\:\varvec{M}}_{\varvec{R}}\left(\varDelta\:,\varphi\:\right)$$

$$\:=\left[\begin{array}{cc}\begin{array}{cc}{m}_{00}&\:{m}_{01}\\\:{m}_{10}&\:{m}_{11}\end{array}&\:\begin{array}{cc}{m}_{02}&\:{m}_{03}\\\:{m}_{12}&\:{m}_{13}\end{array}\\\:\begin{array}{cc}{m}_{20}&\:{m}_{21}\\\:{m}_{30}&\:{m}_{31}\end{array}&\:\begin{array}{cc}{m}_{22}&\:{m}_{23}\\\:{m}_{32}&\:{m}_{33}\end{array}\end{array}\right]$$

(7)

where m₀₀ − m₃₃ denote the elements of the Muller matrix, respectively. Using terms of dichroism p, s and retardance D, and azimuthal angle f, elements of Mueller matrices can be expressed as

$$\:{m}_{00}=q+r$$

(8.1)

$$\:{m}_{01}={m}_{10}=\left(q-r\right)\text{c}\text{o}\text{s}2\varphi\:$$

(8.2)

$$\:{m}_{02}={m}_{20}=\left(q-r\right)\text{s}\text{i}\text{n}2\varphi\:$$

(8.3)

$$\:{m}_{03}={m}_{30}=0$$

(8.4)

$$\:{m}_{11}=\frac{1}{2}\left(q+r+\left(q+r\right)\right)\text{c}\text{o}\text{s}4\varphi\:+4\sqrt{qr}\text{cos}\varDelta\:{\text{s}\text{i}\text{n}}^{2}2\varphi\:$$

(8.5)

$$\:{m}_{12}=\left(q+r+2\sqrt{qr}\text{c}\text{o}\text{s}\varDelta\:\right)\text{c}\text{o}\text{s}2\varphi\:\text{s}\text{i}\text{n}2\varphi\:$$

(8.6)

$$\:{m}_{13}={-m}_{31}=-2\sqrt{qr}\text{sin}\varDelta\:\text{s}\text{i}\text{n}2\varphi\:$$

(8.7)

$$\:{m}_{21}=\left(q+r-2\sqrt{qr}\text{c}\text{o}\text{s}\varDelta\:\right)\text{c}\text{o}\text{s}2\varphi\:\text{s}\text{i}\text{n}2\varphi\:$$

(8.8)

$$\:{m}_{22}=\sqrt{qr}\text{c}\text{o}\text{s}\varDelta\:\left(1+\text{c}\text{o}\text{s}4\varphi\:\right)+\left(q+r\right){\text{s}\text{i}\text{n}}^{2}2\varphi\:$$

(8.9)

$$\:{m}_{23}={-m}_{32}=2\sqrt{qr}\text{sin}\varDelta\:\text{c}\text{o}\text{s}2\varphi\:$$

(8.10)

$$\:{m}_{33}=2\sqrt{qr}\text{c}\text{o}\text{s}\varDelta\:$$

(8.11)

From the calculation of the Muller matrix and Stokes vector, the unknown parameters, optical intensity a, dichroism p, s and retardance D, and azimuthal angle φ are determined, respectively.

Let us consider the scenario where the optical intensity is measured with only the incident polarizer in place. The Stokes vector s₁, representing the optical intensity, is measured using a two-dimensional detector and is given by:

$$\:{\varvec{S}}_{1}={\varvec{S}}_{\varvec{i}\varvec{n}}=\left[\begin{array}{c}\begin{array}{c}a\\\:0\end{array}\\\:\begin{array}{c}0\\\:0\end{array}\end{array}\right]$$

(9)

This process determines the optical intensity a of the light source. A sample exhibiting both dichroism and birefringence is then introduced. The optical intensity is then measured while the polarizer on the incident side is rotated. Based on the Muller matrix and Stokes vector calculations, this measurement yields:

$$\:{\varvec{S}}_{2}={\varvec{M}}_{\varvec{D}\varvec{R}}\left(p,s,\:\varDelta\:,\varphi\:\right)\bullet\:\varvec{P}\left(\theta\:\right)\bullet\:{\varvec{S}}_{\varvec{i}\varvec{n}}$$

(10)

$$\:=\frac{{I}_{0}}{\text{２}}\bullet\:\left[\begin{array}{c}\begin{array}{c}q＋r＋\left(q-r\right)\text{c}\text{o}\text{s}2\left(\varphi\:-\theta\:\right)\\\:\left\{q-r＋\left(q+r\right)\text{c}\text{o}\text{s}2\left(\varphi\:-\theta\:\right)\right\}\text{c}\text{o}\text{s}2\varphi\:+2\sqrt{qr}\text{sin}\varDelta\:\text{s}\text{i}\text{n}2\left(\varphi\:-\theta\:\right)\text{s}\text{i}\text{n}2\varphi\:\end{array}\\\:\begin{array}{c}\left\{q-r＋\left(q+r\right)\text{c}\text{o}\text{s}2\left(\varphi\:-\theta\:\right)\right\}\text{s}\text{i}\text{n}2\varphi\:-2\sqrt{qr}\text{cos}\varDelta\:\text{s}\text{i}\text{n}2\left(\varphi\:-\theta\:\right)\text{c}\text{o}\text{s}2\varphi\:\\\:2\sqrt{qr}\text{sin}\varDelta\:\text{s}\text{i}\text{n}2\left(\varphi\:-\theta\:\right)\end{array}\end{array}\right]\:$$

(11)

Assume us that $\:\varvec{P}\left(\theta\:\right)={\varvec{M}}_{\varvec{D}}\left(\text{1,0},\theta\:\right)$. The optical intensity I(θ), derived from the Stokes vector of a sample exhibiting both dichroism and birefringence, is given by

$$\:I\left(\theta\:\right)=q＋r＋\left(q-r\right)\text{c}\text{o}\text{s}2\left(\varphi\:-\theta\:\right)$$

$$=q＋r＋\left(q-r\right)\text{c}\text{o}\text{s}2\varphi\:\text{c}\text{o}\text{s}2\theta\:+\left(q-r\right)\text{s}\text{i}\text{n}2\varphi\:\text{s}\text{i}\text{n}2\theta\:$$

(12)

Considering the Fourier series expansion, we obtain the following equation:

$$\:I\left(\theta\:\right)=\frac{{a}_{0}}{2}+\sum\:_{n=1}^{\infty\:}\left({a}_{n}\text{cos}n\theta\:+{b}_{n}\text{sin}n\theta\:\right)$$

(13)

a_n and b_n are the Fourier cosine and sine coefficient, respectively, and n is a natural number. From this equation, the bias and amplitude components are extracted as follows,

$$\:{a}_{0}=q+r$$

(14.1)

$$\:{a}_{2}=\left(q-r\right)\text{c}\text{o}\text{s}2\varphi\:$$

(14.2)

$$\:{b}_{2}=\left(q-r\right)\text{s}\text{i}\text{n}2\varphi\:$$

(14.3)

Solving the three simultaneous equations Eqs. (14.1)–(14.3), we obtain:

$$\:q=\frac{{a}_{0}+\sqrt{{{a}_{2}}^{2}+{{b}_{2}}^{2}}}{2}$$

(15.1)

$$\:r=\frac{{a}_{0}-\sqrt{{{a}_{2}}^{2}+{{b}_{2}}^{2}}}{2}$$

(15.2)

$$\:\varphi\:=\frac{1}{2}{\text{t}\text{a}\text{n}}^{-1}\left(\frac{{b}_{2}}{{a}_{2}}\right)$$

(15.3)

For these equations, the dichroism of the sample and its azimuthal angle can be determined. An observation polarizer is inserted and placed at an angle of 90° with the incident polarizer. The analysis is performed using the rotational extinction method, which involves simultaneously rotating both the incident and observation polarizers. In this case, the calculation between Muller matrix and the Stokes vector is obtained as,

$$\:{\varvec{S}}_{3}=\varvec{P}\left(\theta\:+\frac{\pi\:}{2}\right)\bullet\:{\varvec{M}}_{\varvec{D}\varvec{R}}\left(p,s,\:\varDelta\:,\varphi\:\right)\bullet\:\varvec{P}\left(\theta\:\right)\bullet\:{\varvec{S}}_{\varvec{i}\varvec{n}}$$

$$\:=\frac{{I}_{0}}{4}\bullet\:\left[\begin{array}{c}\begin{array}{c}\left(q＋r-2\sqrt{qr}\text{cos}\varDelta\:\right){\text{s}\text{i}\text{n}}^{2}\left[2\left(\varphi\:-\theta\:\right)\right]\\\:\left[-\left(q＋r\right)+2\sqrt{qr}\text{cos}\varDelta\:\right]{\text{s}\text{i}\text{n}}^{2}\left[2\left(\varphi\:-\theta\:\right)\right]\text{c}\text{o}\text{s}2\varphi\:\end{array}\\\:\begin{array}{c}\left[-\left(q＋r\right)+2\sqrt{qr}\text{cos}\varDelta\:\right]{\text{s}\text{i}\text{n}}^{2}\left[2\left(\varphi\:-\theta\:\right)\right]\text{s}\text{i}\text{n}2\varphi\:\\\:0\end{array}\end{array}\right]$$

(16)

From Eq. (16), the intensity distribution captured by the two-dimensional detector is expressed as

$$\:I\left(\theta\:\right)=\frac{{I}_{0}}{4}\bullet\:\left[\left(q＋r\right)-2\sqrt{qr}\text{cos}\varDelta\:\right]{\text{s}\text{i}\text{n}}^{2}\left[2\left(\varphi\:-\theta\:\right)\right]$$

$$\:=\frac{{I}_{0}}{8}\bullet\:\left[\left(q＋r\right)-2\sqrt{qr}\text{cos}\varDelta\:\right]-\frac{{I}_{0}}{8}\bullet\:\left[\left(q＋r\right)-2\sqrt{qr}\text{cos}\varDelta\:\right]\text{c}\text{o}\text{s}\left[4\left(\varphi\:-\theta\:\right)\right]\:$$

(17)

Considering the Fourier series expansion and following the calculations for the previously mentioned dichroism⁴⁹, we obtain a bias component a₀:

$$\:{a}_{0}=\frac{{I}_{0}}{8}\bullet\:\left[\left(q＋r\right)-2\sqrt{qr}\text{cos}\varDelta\:\right]$$

(18)

Since we have already determined the optical intensity of the source, a₀​, as well as the dichroism parameters q and r, the retardance D can be expressed as:

$$\:\varDelta\:={\text{c}\text{o}\text{s}}^{-1}\left[\frac{\left(q＋r\right)-\frac{8{a}_{4}}{{I}_{0}}}{2\sqrt{qr}}\right]$$

(19)

From our theory, the retardance can be determined.

Measurement sample

The Congo red-stained amyloid deposits sample was a kidney specimen from an American Flamingo (Phoenicopterus ruber). This specimen was sectioned to a thickness of 5 μm using a microtome. The specimen consisted of three sections. The sections adjacent to the Congo red-stained specimen were stained with hematoxylin and eosin (HE) for the initial diagnosis of suspected amyloidosis, and Picro-Sirius red (SR) for investigating co-staining with collagen fibers. Small-diameter renal arteries with amyloid deposits were identified by their hue shifting from green to orange under a crossed Nicols microscope after Congo red staining. Five 500 × 500 pixel squares of the renal arteries on the pathology specimen slides were analyzed for comparison.

Congo red staining

The Congo red staining was performed using the Puchtler-Sweet method. Paraffin-embedded tissue Sect. (5 μm thick) were air-dried overnight at 38 °C, then deparaffinized through three changes of xylene. The slides were rehydrated through a descending ethanol series (100%, 95%, 80%, and 70%), rinsed in running tap water, and finally washed in distilled water. Sections were stained with Mayer’s hematoxylin (Muto Pure Chemicals) for 5 min and checked under a light microscope to confirm nuclear staining. After thorough rinsing in tap water, the slides were “blued” in distilled water to develop the hematoxylin and then rinsed again in distilled water. Mordanting was carried out for 20 min in a freshly prepared solution of 2 g NaCl dissolved in 100 mL of 80% ethanol, to which 1 mL of 0.1% NaOH had been added immediately before use. For Congo red staining, 0.12 g Congo red (Nakarai Tesque, Cat. 09402-12) was dissolved in 100 mL of 80% ethanol with gentle heating, filtered, and then 1.2 g NaCl was added and mixed. Just before use, 1 mL of 0.1% NaOH was added to 100 mL of the filtered solution, which was then re-filtered and applied to the sections for 20 min. Following a brief tap-water rinse to remove excess dye, characteristic birefringence was confirmed under polarized light microscopy. Finally, the sections were dehydrated through graded ethanol, cleared in xylene, and coverslipped with Malinol mounting medium.

Shadowless amyloid imaging technique

Equation (18) shows that the a₀ component of the Fourier series does not incorporate the azimuthal angle, unlike the a₄ component as demonstrated in Eq. (17). In other words, isolating the a₀ component alone allows for the generation of an image free from ‘polarization shadows’. Moreover, the a₄ components can neutralize each other due to the relative angles of the polarizers. For example, with the transmission angles of polarizer set at 0° and 45° in a cross Nicol configuration, the equations of optical intensity can be expressed as follows:

$$\:I\left(0^\circ\:\right)=\frac{{I}_{0}}{8}\bullet\:\left[\left(q＋r\right)-2\sqrt{qr}\text{cos}\varDelta\:\right]-\frac{{I}_{0}}{8}\bullet\:\left[\left(q＋r\right)-2\sqrt{qr}\text{cos}\varDelta\:\right]\text{c}\text{o}\text{s}\left[4\varphi\:\right]\:\:$$

(20)

$$\:I\left(45^\circ\:\right)=\frac{{I}_{0}}{8}\bullet\:\left[\left(q＋r\right)-2\sqrt{qr}\text{cos}\varDelta\:\right]+\frac{{I}_{0}}{8}\bullet\:\left[\left(q＋r\right)-2\sqrt{qr}\text{cos}\varDelta\:\right]\text{c}\text{o}\text{s}\left[4\varphi\:\right]$$

(21)

The average of Eqs. (20) and (21) is calculated as follows:

$$\:I\left(0^\circ\:\right)+I\left(45^\circ\:\right)=\frac{{I}_{0}}{8}\bullet\:\left[\left(q＋r\right)-2\sqrt{qr}\text{cos}\varDelta\:\right]$$

(22)

This equation includes dichroism parameters q and r but lacks the azimuthal angle φ associated with birefringence. Consequently, the resulting image is unaffected by polarization shadows. For instance, this effect can be achieved by setting a relative angle of 45° between the polarizer and the analyzer under crossed Nicol conditions.

Method of evaluating the agreement

In this study, the rotational extinction method was employed, enabling the comparative evaluation of image agreement at multiple extinction angles relative to the azimuthal angle of the sample. The results of the retardance were evaluated for agreement with the images observed every 10°, respectively. The degree of agreement was quantified using Pearson’s correlation coefficient⁵⁰. The correlation coefficient C is expressed as

$$\:C=\frac{{\sum\:}_{x=1}^{m}{\sum\:}_{y=1}^{n}({I}_{A}\left(x,y\right)-\overline{{I}_{A}})({I}_{A}\left(x,y\right)-\overline{{I}_{B}})}{\sqrt{{{\sum\:}_{x=1}^{m}{\sum\:}_{y=1}^{n}\left[{I}_{A}\left(x,y\right)-\overline{{I}_{A}}\right]}^{2}}\sqrt{{{\sum\:}_{x=1}^{m}{\sum\:}_{y=1}^{n}\left[{I}_{B}\left(x,y\right)-\overline{{I}_{B}}\right]}^{2}}}$$

(23)

This equation represents that the numerator is the covariance of the two images I_A and I_B, and the denominator is the product of their standard deviations. The optical intensities of each pixel (x, y) in the two images are represented as I_A (x, y) and I_B (x, y). The means of the respective optical intensities are denoted as $\:\overline{{I}_{A}}$ and $\:\overline{{I}_{B}}$, and m and n represent the number of pixels in the x and y directions, respectively.