2.1.

Imaging Mueller Polarimetry System

We studied fresh and fixed animal brain thick sections using the custom-built multi-spectral wide-field imaging Mueller polarimetric system operating in reflection configuration in the visible wavelength range (450 to 700 nm). The IMP systems that acquire the complete Mueller matrix (MM) of a sample have been successfully used for the characterization and medical diagnosis of different pathologies.^{14,16,18–21} In this study, all presented measurements were performed at 550 nm. The details on the design, calibration, and optimization of our IMP instrument have been already reported elsewhere.^{20,28,59–61} For the sake of completeness, we remind the basic principles of our IMP system operation. The incident light beam produced by a Xenon (SOPRO Comeg 230) source passes through the polarization state generator (PSG) before interacting with a sample. The PSG consists in an assembly of fixed polarizer and two bistable V-shaped ferroelectric liquid crystals (FLCs) in smectic C phase⁶² operating as waveplates with fixed linear retardance and variable azimuth that can be electrically controlled. A further waveplate is placed between the two FLCs.⁵⁹ After reflection/scattering by a sample, the light beam passes through the polarization state analyzer (PSA), which is made from the same optical components as PSG but assembled in a reverse order, before reaching the camera (Stingray F080B ASG). The modulation of polarization of probing light beam is performed sequentially. To reconstruct the MM of a sample, 16 intensity measurements have to be performed (4 linearly independent polarization states generated by the PSG that are analysed with the 4 linearly independent polarization states of PSA). To increase the signal to noise ratio, the series of 16 intensity measurements has been performed and averaged 8 times. The measurement accuracy was guaranteed by implementing the eigenvalue calibration procedure.⁶³

The post-processing of the recorded MM images was done with the Lu-Chipman polar decomposition algorithm⁶⁴ that was applied pixel-wise. The calculated maps of the depolarization, linear retardance, and azimuth of the optical axis of the brain specimens were obtained afterwards, as described in Rodriguez-Nunez et al.²⁸ The azimuth of the optical axis was used to assess the effects on the orientation of the fiber tracts before and after the FF of the same specimen.²⁴ The depolarization and linear retardance have been proved to be informative for ex vivo brain tissue differentiation,^24,27 and hence, we decided to focus our quantitative work on the examination of the images of these three parameters.

2.2.

Brain Samples and Measurement Protocol

Fresh cadaveric pig brain hemispheres were obtained from a local butcher shop 6 hrs post-mortem. A total of 10 brain hemispheres were used for this study to ensure the study of images from a diverse set of brains. The brains were dissected at the level of corpus callosum and brainstem in the midline. Thirty coronal sections, each $\sim 3\mathrm{cm}$ thick, were obtained using a scalpel [Fig. 1(a)] and placed flat in a glass Petri dish. For the measurements, sections were selected based on whether the quality of the tissue was sufficient. The quality requirements included (i) absence of tissue deterioration and cutting artifacts and (ii) presence of both gray and white matter in the imaged zone of a specimen. The coronal sections were obtained from multiple brain areas (some sections were occipital while others were frontal) to achieve a global representation of brain tissue in the samples.

Fig. 1

The pipeline of the measurement and annotation process. (a) Color photo of a 3-cm-thick pig coronal section slice (top) and gray-scale reflected intensity image of the central zone shown by a red rectangle (bottom), (b) measurements were performed on the same section after FF (see text), and (c) gray and white matter masks were created manually from gray-scale reflected intensity images.

We used the wide field IMP system and Lu-Chipman decomposition of measured MM images to produce the maps of the depolarization, of the linear retardance, and of the azimuth of the optical axis of the thick coronal sections of fresh brain tissue. This first measurement was performed on fresh brain tissue right before the fixation of the sections and represents the first time point (0 h). The sections were fixed by being placed in small plastic recipients containing formalin (10% formaldehyde). For the measurements performed after FF, the samples were removed from formalin, washed with distilled water, and then measured. For accurately assessing the time evolution of the polarimetric parameters after FF, four sections were measured at five different time points following FF: +12 hrs, +24 hrs, +36 hrs, +48 hrs, +7 days, making up a total of 6 time points for which the parameters were obtained. A visual qualitative comparison of the depolarization, of the linear retardance, and of the azimuth of the optical axis maps was performed for all measurements of the same sample at different time points. Thirty fresh sections and 30 fixed sections measured with the polarimeter 10 days post-fixation [Fig. 1(b)] were selected for the assessment of the polarimetric values in the border between gray and white matter (described in Sec. 2.4). Finally, both gray and white matter masks were annotated manually for all measurements [Fig. 1(c)].

2.3.

Quantitative Analysis

To quantitatively analyze the evolution of polarimetric properties before and after fixation, we generated automatically and randomly, to avoid any bias linked to a manual selection, 25 squared regions of interest (ROIs) (by selecting with an uniform random variable a point in the image that we used as top left corner of the ROI), each $20\times 20\text{pixels}$, completely within gray matter, and 25 ROIs, each $20\times 20\text{pixel}$, within white matter, using the corresponding fresh measurement tissue gray-scale reflected intensity image [Fig. 2(a)]. The use of small ROIs to analyze the local effect of FF is necessary because of the spatial inhomogeneity of the polarimetric parameters in the tissue. The size of the square was chosen to be $20\times 20\text{pixels}$ to analyze the effects of FF for local brain tissue regions while keeping the number of pixels (400) large enough for robust statistical analyses. The gray-scale reflected intensity images at the different fixation time points were co-registered with the fresh one, using a custom pipeline based on Elastix^65,66 to ensure the analysis of the same areas in the images acquired after FF. The gray-scale reflected intensity image of fresh tissue was registered to the images of FF tissue [Fig. 2(a)], allowing to propagate the selected ROIs on the images of FF tissue. The ROIs registered as gray/white matter in the images of fresh tissue and as white/gray matter or background [defined as neither white nor gray matter, Fig. 1(c)] in the images of FF tissue were removed for further analysis. The histograms of the depolarization, linear retardance, and azimuth of the optical axis were then extracted for each ROI [Fig. 2(b)]. For the first two parameters, the medians $m_{\mathrm{ROI}}$ were computed for each ROI [Fig. 2(c)] and used as a descriptive statistics, based on the presence of skewed distributions within single ROIs (Fig. S1 in the Supplementary Material). The value of $m_{\mathrm{ROI}}$ for the fixed tissue was then divided by the corresponding value for the fresh tissue, to obtain the fold change $f{c}_{\mathrm{ROI}}$ in the medians. The mean and standard deviation of the $f{c}_{\mathrm{ROI}}$ values across the images were then obtained for both depolarization and linear retardance and compared. The angular nature of the azimuth of the optical axis prevented using a similar comparison. We computed the difference ${\theta }_{\mathrm{ROI}}$ in the mean angle observed before and after fixation in each ROI as a metrics. The mean and standard deviation of the ${\theta }_{\mathrm{ROI}}$ values across the images were then obtained and compared. Statistical differences between the means of the fold changes and the angle differences at the different fixation time points were assessed using SciPy’s Mann-Whitney U test (v1.9.3).⁶⁷ For depolarization and linear retardance, the medians $m_{\mathrm{ROI}}$ were also averaged across the four images obtained at each time point to check for the preservation of the general contrast between gray and white matter after FF. All the error bars represented in the result plots correspond to the 95% confidence intervals.

Fig. 2

The sequence of main steps in the quantitative analysis of the fixation effects. (a) ROIs (white square box is shown as a typical example) were selected automatically (see text) in the gray-scale reflected intensity image of fresh tissue using the masks generated in Fig. 1. These masks were then propagated to the images of FF tissue using a co-registration pipeline on both gray-scale reflected intensity images. (b) The polarimetric parameters of depolarization, linear retardance, and azimuth of the optical axis were then extracted using the ROIs generated in the previous steps. (c) Statistical data describing the distributions of polarimetric parameters for both fresh and FF tissues were then obtained and compared.

2.4.

Segmentation Analysis

For analyzing the behavior of the tissue at the gray/white matter border region, we reused the previously created masks of gray and white matter [Figs. 3(a) and 3(b)] and generated a line of the pixels located at the border by checking pixels assigned to the white matter located next to pixels assigned to gray matter [Fig. 3(c)]. We subsequently dilated this line using OpenCV (v 4.6.0.66)⁶⁸ with a number of six iterations and the kernel, being a $3\times 3$ identity matrix, for a total of nine times [Fig. 3(d), here only three regions were represented for the sake of simplicity]. Sequential regions, consisting of the pixels incorporated during the current dilation step and absent from the previous one, were generated through this process. These regions were therefore located further and further away from the border region. The pixels included in the newly dilated regions were considered for the analysis in case they belonged to either white or gray matter labeled regions but were discarded in case they belonged to the background. The depolarization values were then extracted for these regions and combined across different images. Descriptive statistical values, such as the mean $\mu$ and the standard deviation $\sigma$, were computed and compared between the regions defined previously [Fig. 3(e)]. The position of each region was then correlated to a physical distance related to the border region using the following equivalence relations

Fig. 3

The illustration of the main steps used to investigate the behaviour of the depolarization values at the border region. (a) Gray-scale reflected intensity image of a pig coronal section for which (b) masks of gray and white matter were manually drawn (see Sec. 2.2). (c) The pixels located at the border region between gray and white matter were selected. (d) These pixels were then dilated to create regions of fixed size at a different distance of the border region (here only three regions are represented for the sake of simplicity). (e) The polarimetric parameters were then extracted for the different regions, and descriptive statistical values were further computed and compared.

Thus, the size of a pixel in the $x$ and $y$ axis corresponds to

We then aimed to determine the so-called uncertainty region, which would correspond to the region in which the parameters are not similar to the white matter ones nor the gray matter ones (see Sec. 3.3). For that purpose, the means $\mu$ in the regions aforementioned were fitted with a sigmoid using SciPy’s curve fit function (v1.9.3).⁶⁷ We then used the kneed package (v0.8.1)⁶⁹ to define the location of the two elbows on the curve corresponding to the transition point between white matter and uncertainty region and between uncertainty region and gray matter, on the left and right sides of the plots [see Figs. 7(c) and 7(d)], respectively. The estimated width of the uncertainty region, defined as the difference between the two elbow points, was then computed.

3.1.

General Effects of Formalin Fixation on Polarimetric Properties of Brain Tissue

The polarimetric maps of depolarization, linear retardance, and azimuth of the optical axis obtained during the measurements of coronal sections of a pig brain at different times post-fixation are shown in Fig. 4. The gray-scale reflected intensity images [Figs. 4(a)–4(f)] were used to generate the masks of gray and white matter (see Sec. 2.2), as from the gray-scale reflected intensity image and with basic knowledge of neuroanatomy it is possible to differentiate between two types of brain tissue. One can also appreciate the apparition of physiological tissue shrinkage and tissue deformation, especially in the white matter region following FF. It should also be noted that specular reflections appear on the right-hand side of the image following FF, possibly as a consequence of the tissue deformation. In the depolarization maps [Figs. 4(g)–4(l)], we observed an important difference in the value of this parameter between gray and white matter. The former presents a lower depolarization than the latter, allowing contrast visualization of both tissue types. The same pattern of contrast remained visible after FF for all different time points at which the measurements were performed. Moreover, it was possible to observe the contrast between both tissue types using the linear retardance maps [Figs. 4(m)–4(r)], with the white matter displaying higher values than the gray matter. However, the contrast between gray and white matter in the retardance maps was substantially reduced by the fixation process compared to the depolarization maps. Finally, the azimuth of the optical axis maps was used to track nerve fibers in white matter of brain tissue, as described in previous studies.²⁴ Slight changes in the maps of the azimuth were detected after FF. Most of the observed changed is believed be due to tissue shrinkage and deformation (described in Sec. 3.3), especially after 7 days. The FF did not disturb significantly the azimuth values in the examined images of brain sections [Figs. 4(s)–4(x)] preserving the orientation of the fiber tracts. Consequently, it seems that even if minor changes are induced by FF, especially in linear retardance values, they do not greatly impair the contrast between the two tissue types and do not perturb fiber orientation identification.

Fig. 4

Polarimetric maps calculated from the experimental MM images of coronal sections of pig brain at different fixation time points (0 hrs, +12 hrs, +24 hrs, +36 hrs, +48 hrs, and +7 days). Each row represents the polarimetric maps acquired for a time point. The gray-scale reflect intensity (a)–(f) images and the polarimetric maps of depolarization (g-l), linear retardance (m)–(r), and azimuth of the optical axis (s)–(x) are demonstrated. The colored lines depicting the orientation of the optical axis were plotted in the regions, where all three conditions were fulfilled: the region was assigned as gray or white matter (see Sec. 2.2 for more details), the values of linear retardance $>4\mathrm{deg}$, depolarization $>90\%$ and gray-scale reflected intensity > mean intensity.

3.2.

Quantitative Analysis of Polarimetric Parameters After Formalin Fixation

The next step consisted of quantifying the differences in the polarimetric parameters induced by FF. To do so, we selected 25 areas ($20\times 20\text{pixels}$ each) within the gray and white matter zones in the gray-scale reflected intensity images of four coronal sections of pig brain. The fold changes in the median of the parameters were then obtained (see Sec. 2.3). Their evolution with time is represented in Fig. 5(a) for the linear retardance and Fig. 5(b) for the depolarization. There was an important evolution in the linear retardance fold change, when comparing fresh and 12 hrs-fixed tissue, for both gray (U = 1600, $n_1=85$, $n_2=100$, $p_{\text{value}}=2\times {10}^{-15}$) and white matter (U = 1400, $n_1=94$, $n_2=100$, $p_{\text{value}}=1\times {10}^{-19}$). The magnitude of the change was a reduction of about $\sim 27\%$ in gray matter and $\sim 28\%$ in white matter. Interestingly, little to no change of values was observed in linear retardance after the first two measurements (Table 1). For the depolarization parameter, there was a statistically significant change of fold change, when comparing fresh and fixed tissue, for gray (U = 7800, $n_1=85$, $n_2=100$, $p_{\text{value}}=1\times {10}^{-26}$) but not for white matter (U = 4900, $n_1=94$, $n_2=100$, $p_{\text{value}}=0.78$). The magnitude of the change was rather small, an increase in about $\sim 5\%$ for the gray matter and virtually no change for the white matter. Furthermore, there were small to no changes for the subsequent measurements after the initial one (Table 2). Although two value changes were found to be statistically significant for the depolarization in the white matter zone (+48 hrs, U = 5417, $n_1=94$, $n_2=96$, $p_{\text{value}}=6\times {10}^{-3}$ and (+ 7 days, U = 5332, $n_1=94$, $n_2=96$, $p_{\text{value}}=0.035$), the magnitude of the observed change was minimal with a maximum of 1% of depolarization value. The averages of the medians in each ROI were also obtained to verify the stability of the contrast between gray and white matter (see Sec. 2.3). Their evolutions are represented in Fig. 5(c) for the linear retardance and Fig. 5(d) for the depolarization. Despite a change in the polarimetric parameters, the contrast between gray and white matter was still present after fixation. Similar conclusions to the ones obtained with the fold change analysis could be drawn when comparing the averages of the medians (Table S1 and S2 in the Supplementary Material). Thus, both the depolarization and the linear retardance seem to be useful polarimetric markers to differentiate gray from white matter in both fresh and fixed brain tissues. The average of the differences in the mean azimuth of the optical axis angle observed before and after fixation for each ROI is represented in Fig. 5(e). There was an important change in the observed azimuth angle, when comparing fresh and 12 hrs-fixed tissue, for both gray (U = 8500, $n_1=85$, $n_2=100$, $p_{\text{value}}=3\times {10}^{-37}$) and white matter (U = 9400, $n_1=94$, $n_2=100$, $p_{\text{value}}=3\times {10}^{-38}$). However, there was no statistically significant variation in the mean azimuth angle difference observed for later time points (Table 3). The observed variation was expected and could be explained by the fact that the sample was placed and aligned manually at each measurement time point with the position mismatch of a few degrees. This is also indicated by the important values of standard deviations reported. The tissue morphological changes induced by FF also explain the observed changes. The mean azimuth angle difference was larger for gray matter, which is expected, because the absence of highly organized brain fibers in gray matter results in low retardance values, thus, impacting the orientation calculations. Globally, these results indicate that the time of fixation did not amplify the observed changes when comparing fresh and fixed tissue and that the contrast between gray and white matter observed in the images of fresh brain tissue is still preserved in the images of brain tissue after FF.

Fig. 5

Evolution of the values of polarimetric parameters with time after FF. The mean value and 95% confidence intervals of the fold changes computed for areas of $20\times 20\text{pixels}$ are plotted with respect to post-fixation time for (a) linear retardance and (b) depolarization. The mean value and 95% confidence intervals of the local medians for areas of $20\times 20\text{pixels}$ are plotted with respect to post-fixation time for (c) linear retardance and (d) depolarization. (e) Plot of the evolution of the average of the difference in the value of the azimuth of the optical axis in $20\times 20\text{pixels}$ areas before and after fixation with respect to fixation time.

Table 1

Summary of the statistics for the comparison between the linear retardance fold change mean values at different time points post-fixation. All the reported U and p values correspond to the test of the null hypothesis, stating that the mean variable is the same as the one 12 hrs post-fixation.

Table 2

Summary of the statistics for the comparison between the depolarization fold change mean values at different time points post-fixation. All the reported U and p values correspond to the test of the null hypothesis, stating that the mean variable is the same as the one 12 hrs post-fixation.

Table 3

Summary of the statistics for the comparison between the azimuth angle differences at different time points post-fixation. All the reported U and p values correspond to the test of the null hypothesis, stating that the mean variable is the same as the one 12 hrs post-fixation.

3.3.

Manual Segmentation of Gray and White Matter

The gray-scale reflected intensity images [Figs. 4(a)–4(f)] suggested the presence of tissue shrinkage and deformation induced by the FF process. This is mostly visible in Figs. 6(a)–6(f), where brain tissue considerably shrinks following FF. This was indicated by the presence of pixels that were attributed manually to white matter zone before FF and to gray matter zone after FF [see, for example, the pixels rendered in white, Fig. 6(g)]. They represent 13.9% of the total amount of pixels in the image.

Fig. 6

Illustration of the of tissue shrinkage following FF. (a) Gray-scale reflected intensity image of the fresh coronal brain section and (b) zoom on a specific sub-region ($160\times 126\text{pixels}$) of the gray-scale reflected intensity image and (c) corresponding labels. (d) Gray-scale reflected intensity image of the same region after FF ($t=+48\mathrm{hrs}$) and (e) zoom on a specific sub-region ($160\times 126\text{pixels}$) of the gray-scale reflected intensity image and (f) corresponding labels; (g) two overlapped binary masks for gray and white matter that were created from the binary masks (c) and (f).

To investigate the effect of such a shrinkage on polarimetric parameters, the depolarization values averaged over the dilated border zones (see details in Sec. 2.4) are plotted against the distance to the center of the dilated region from the initial border line for the fresh and fixed brain tissue measurements in Figs. 7(a) and 7(b), respectively. There appears to be an uncertainty region where it is not clear whether we are detecting gray or white matter, making the segmentation ground truth somewhat unclear in this area. As the data points obtained were able to be represented by a “S”-shaped curve, with the value of depolarization in gray and white matter being the two plateau regions, we fitted a sigmoid function the depolarization data for both fresh and fixed brain tissue [Figs. 7(c) and 7(d), respectively]. The best-fit sigmoid functions had the following coefficients:

Fig. 7

Determination of the width uncertainty region for both fresh and fixed brain tissue using data of polarimetric measurements. Mean and standard deviation of the depolarization values for the regions described in Sec. 2.4, plotted against distance to the physical gray and white matter border for fresh (a) and fixed (b) tissue (n = 30 for both fresh and fixed brain tissue measurements). Estimation of the uncertainty region was done using a fitted sigmoid curve for the depolarization data for fresh (c) and fixed (d) tissue. Over-imposition of the gray matter, uncertainty region and white matter on sample gray-scale reflected images for fresh (e) and fixed (f) tissue.

We then aimed to determine the width of the uncertainty region, which was estimated to be 1.53 mm for fresh tissue and 1.59 mm for fixed tissue. This finding suggested that despite the tissue’s physiological shrinkage after FF, the size of the so-called uncertainty region remains almost constant. The uncertainty region is represented over-imposed on the gray-scale reflected images for both fresh tissue [Fig. 7(e)] and fixed tissue [Fig. 7(f)]. In summary, we report the existence of an uncertainty region of a width close to 1.55 mm, where the depolarization values are changing continuously when moving from gray to white matter. We also prove that the size of this uncertainty region does not differ between fresh and fixed brain tissues.