Polarization and depolarization metrics as optical markers in support to histopathology of ex vivo colon tissue

: Tissue polarimetry holds great promise to improve the effectiveness of conventional cancer diagnostics and staging, being a fast, minimally invasive, and low-cost optical technique. We introduce an enhanced diagnostic method for ex vivo colon specimens assessment by utilizing Stokes and Mueller matrix polarimetry. The proposed method makes use of experimental Mueller matrices, measured from healthy and tumor zones of a colon specimen, as input data for post-processing algorithms that include physical realisability filtering, symmetric decomposition and estimation of various polarization and depolarization metrics for colon specimen diagnostics. We validated our results with the gold standard histological diagnostics provided by pathologists. It was found that the Stokes-Mueller matrix polarimetry, combined with the appropriate filtering, decomposition algorithms and polarization/depolarization metrics calculations provides relevant optical markers of the colon tissue pathological conditions (healthy


Introduction
Cancer is one of the most socially significant health problems nowadays and its early theranostics is vital in order to increase life expectancy [1,2]. According to the data from The Global Cancer Observatory [3], in 2020 19.3 M of total tumor cases were reported worldwide; out of them 9.96 M ended with mortality. In 2040 the total number of tumor cases is estimated to grow up to 30.2 M with an expected mortality of 16.3 M [3]. Colorectal cancers are in the leading position in terms of incidence and mortality. Contrary to the cancer of skin, the cancer of colon is often detected at the advanced stages due to its anatomical location out of direct sight of view. Therefore, multiple endoscopic tests and biopsy examinations are required for early diagnosis of cancer of the digestive system. For these particular reasons, biophotonics can provide an adequate assistance [4][5][6].
Tissue polarimetry is a non-invasive, optical diagnostic technique that exploits the extreme sensitivity of optical polarization to early changes in tissue morphology, induced by the malignancy, without using any contrast agents [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Any polarization properties of biological tissues affecting both the polarization state of the incident light beam and its degree of polarization (DOP) can be characterised by the transfer function of the optical medium, known as 4 × 4 Mueller matrix (MM) [24]. However, not every real-valued 4 × 4 matrix can be considered as a Mueller matrix, since the input Stokes vector of the incident light that is modified by the transfer function should be transformed into an output Stokes vector with a degree of polarization less or equal to 1 [25]. Furthermore, physically realizable, depolarizing MMs must be representable as weighted averages of non-depolarizing MMs, i.e. MMs preserving the DOP of totally polarized input light. Due to measurement errors and noise, this condition may not be met. Such matrices need to undergo a physical realizability filtering, according to the method proposed by Cloude [25,26]. A direct quantitative representation of the optical properties of the sample, derived from its Mueller matrix elements, is possible for homogeneous media only. This is typically not the case with biological samples that are heterogeneous and anisotropic, thus exhibiting a complex dependence of their Mueller matrix elements on their optical properties [27]. For this particular reason, a decomposition algorithm should be implemented -usually the polar decomposition of Lu and Chipman [28] or the symmetric decomposition (SD) [24,29]. Both algorithms of non-linear data compression allow for extracting information on depolarization, diattenuation and retardance of the sample under study. However, the SD of a Mueller matrix is better suited angular-resolved measurements that are commonly used in materials science [30,31]. For instance, SD may provide better representation of the polarization and depolarization phenomena for reflection geometry, by placing a MM of pure/canonical depolarizer between MMs of retardars and diattenuators. To the best of our knowledge, few reports are available on using the SD for the analysis of the polarimetric properties of biological tissues [32,33] or in biomedical applications [23]. In addition, more detailed information on the tissue depolarizating properties can be extracted by calculating the various depolarization parameters and constructing the corresponding depolarization spaces [34][35][36]. The above mentioned data processing algorithms hold promise in providing quantitative criteria for the discrimination between healthy and pathological tissue zones that are linked to the micro-structure of tissue.
We start by presenting an indirect method for measuring the full Mueller matrix of a sample with a Stokes polarimeter and then apply this method to measure polarimetrically both healthy and malignant zones of thick formalin-fixed colon specimen at oblique incidence. A combined data processing algorithm including physical realizability filtering, symmetric decomposition and calculation of various depolarization metrics from the Mueller matrix was implemented for processing of the experimental data of the colon specimen. Both healthy and cancerous zones of the colon specimen were analyzed in terms of their polarization and depolarization properties calculated with the algorithm. Next, we constructed the 3D canonical, natural, and indices of polarimetric purity (IPP) depolarization spaces and plotted all polarization and depolarization parameters for both healthy and cancerous zones of colon measured at different points. The depolarization parameters for healthy and cancer colon tissue form distinct clusters. The results provide better insight into the structural modifications caused by cancer and suggest using this approach for the optical diagnostics of colon specimens.

Theory
First, we define four input polarization states of a light beam characterized by their normalized (S 0 = 1) Stokes vectors : S Hi = (1, 1, 0, 0) T , S Vi = (1, −1, 0, 0) T , S Pi = (1, 0, 1, 0) T and S RCi = (1, 0, 0, 1) T . The respective output Stokes vectors (also normalized), resulting from probing an arbitrary sample with the above input polarized light beams, can be measured with a Stokes polarimeter: S Ho , S Vo , S Po and S RCo . The subscripts denote horizontal linear (H), vertical linear (V), +45 linear (P), and right-circular (RC) polarization states; i -stands for the input states, o -for the output states and the superscript T is for transposition. The input and output Stokes vectors are connected by the relation S o =M · S i [24,29,37] where M is the Mueller matrix of the sample under measurement. An important relation between Stokes vector elements accounting for the potential presence of depolarization is given by [24,29,37]: By knowing the total degree of polarization ρ and the detected signal power ℘, the normalized output Stokes vectors reads: The four output Stokes vectors S dep can be represented as algebraic sums of the columns C l (l = 1, . . . , 4) of the sample Mueller matrix as follows: where all superscripts in Eq. (3) denote the respective input polarization states. By measuring the output Stokes vector for each one of the four different input polarization states, all columns of the Mueller matrix C l and therefore, the complete Mueller matrix, can be derived [38]: Biological tissues are typically depolarizing media [39] and can be generally characterized by their ability to change both the initial polarization state of the probing beam and its degree of polarization. The former is affected by both (surface) Fresnel reflection and (volume) sample anisotropy whereas the latter is due to the coherence loss resulting from the light-tissue interaction taking the form of multiple scattering events, originating from macroscopic spatial fluctuations of the refractive index. In order to extract both polarization and depolarization properties of a given biological specimen, the symmetric decomposition of the Mueller matrix of the sample can be applied [24,29]: where the canonical forms of M ∆ , M R and M D is defined as: In Eq. (6), M ∆ describes a diagonal depolarizer of type I [24] with depolarization coefficients , where subscript 1 denotes depolarization of H and V linear polarization, subscript 2 -of ±45 • linear polarization and subscript 3 -of right and left circular polarization states, respectively. M R and M D denote respectively the Mueller matrices of a (general elliptical) retarder and a diattenuator. The 3 × 3 submatrix m ϕ is 3D rotation matrix through the angle specified by the phase shift or retardance φ, whereas the 3 × 3 submatrix m D is constructed from the diattinuation ⃗ D. Finally, the net depolarization ∆, diattenuation D, polarizance P and retardance φ can be expressed as [24]: Next, each Mueller matrix can be cast into the form of the so called covariance matrix H, a Hermitian matrix defined as [34][35][36]: where σ k are the four Pauli spin matrices and the symbol ⊗ denotes the Kronecker product. [25] The eigenvalues of the matrix H, arranged in decreasing order (λ 1 ≥ λ 2 ≥ λ 3 ≥ λ 4 ), are used to derive the indices of polarimetric purity (IPP) [34][35][36]40,41]: where tr denotes the trace of a matrix. It is known that high concentrations of scattering particles and/or high structural sample anisotropy enhance the scattering of light and, consequently, the depolarization effect. [26] According to the values of the indices of polarimetric purity P i (i = 1, . . . , 3), there are two limiting cases: ideal depolarizing medium when P i = 0 and a non-depolarizing one when P i = 1. From the IPP values one can calculate the polarimetric purity index PI and the Gil-Bernabeu's depolarization index P ∆ likewise [42]: At this point, it is practical to draw a parallel between the extrinsic and intrinsic depolarization parameters of a given M. The former are described by the eigenvalues of the covariance matrix H associated with M whereas the latter are defined by the eigenvalues of the covariance matrix H ∆ of the canonical depolarizer M ∆ . In the absence of diattenuation the matrices of both diattenuators in the symmetric decomposition are equal to the identity matrix and the extrinsic and intrinsic depolarization parameters are equal. Otherwise, the diattenuators have the capability to alter the degree of polarization of partially polarized light by introducing repolarizing effect and thus, making the difference between the two sets of depolarization parameters.
Eventually, by introducing the Cloude's entropy parameter S in addition to PI and P ∆ , one may enrich the depolarization parameterization. S can be calculated from the eigenvalues λ i of the covariance matrix H from the following expression: The value of S = 1 corresponds to an ideal depolarizer (i.e. λ 1 = λ 2 = λ 3 = λ 4 ); the respective turbid medium exhibits polarimetric properties that fluctuate completely at random. Conversely, a turbid medium with S = 0 is a non-depolarizing one, i.e. it completely preserves the degree of polarization of totally polarized light [29].

Sample preparation
A formalin-fixed human colon specimen was used for ex vivo polarimetric measurements (see Fig. 1). The tissue specimen was provided in the framework of a cooperation agreement for optical investigations of cancerous tissues between the Institute of Electronics -Bulgarian Academy of Sciences and the Surgical Department of University Hospital "Tsaritsa Yoanna -ISUL", Sofia under approval #286/2012 of the local Ethical Committee. After a surgical excision procedure, the biological specimen was cut into two parts for histological and optical examination, where the tumor staging was performed by the pathologist. The histology analysis confirmed the presence of a tumor zone containing G2-adenocarcinoma (T2, N0, M0) [43] with moderately differentiated tumor cells. No spread to regional lymph nodes was found. However, the tumor had grown into the muscularis propria layer of colon wall. Thus, the results of the histological examination were used as a "gold standard" and reference to polarimetric diagnostic measurements. Both healthy and tumor zone thicknesses were about 1 mm; therefore, the polarimetric measurements were done in reflection configuration.

Experimental setup and data processing algorithms
In order to measure the full Mueller matrix, at least sixteen independent measurements have to be conducted with different polarization states generated by the polarization state generator (PSG) and analyzed by the polarization state detector (PSD) channels of the polarimetric equipment. Instead of using discrete polarization states, one may use continuous polarization modulation, typically achieved either with liquid crystal retarders or with rotating wave-plates. In our work, the PSG polarization states were set discretely, while those of the PSD were modulated continuously, thus reducing the number of discrete measurements to only four. In Fig. 2 the PSG and PSD channels are presented in a side view for better clarity. All optical elements were inserted in tube systems, in order to block stray light that could interfere with the probing or detected beam. The field of view (FoV) of the system was found to be 100×100 µm. In the reflection geometry used, the angles of incidence and detection were respectively set to 55 • and 30 • . Such configuration was found earlier [23] to enrich the polarimetric response of the turbid medium and is good to be combined with the SD.  [44].
For our experiments, a supercontinuum fiber laser -SC (Leukos Ltd., France), connected to an acousto-optic tunable filter -AOTF (Leukos Ltd., France), generated the selected measurement wavelength of 635 nm with a spectral width of 8 nm and output power of 2 mW. Two irises were used to obtain a collimated beam. A half-wave plate was inserted to vary the azimuth of the linearly polarized laser beam, while an electrically-driven liquid crystal variable quarter-wave plate was used to generate circular polarization. By means of the lens L 1 the light beam was focused to a particular depth of the sample volume. Objective lenses with 10× magnification collected the diffusely scattered light, whereas the lens L 2 , the 100 µm pinhole and the lens L 3 were employed to eliminate any out-of-focus photons. By using another lens, L 4 , the light was focused onto a 90-10 beam splitter; the reflected beam was detected by a CMOS camera for accurate focus adjustment while the transmitted beam was analyzed by the polarimetric device (Thorlabs Ltd., USA). A rotating quarter-wave plate and a fixed linear polarizer inside the polarimetric device continuously modulate the light beam before it reaches the photo-detector Si photodiode. Prior to conducting the experiments with the biological specimen, the optical set-up was tested by measuring a mirror MM (M exp ), which was compared to a theoretical MM of reflection (M th ) without any depolarization properties -i.e. diag (1,1,1,1). Afterwards, a calculation of the RMSE for each matrix element, except for m 11 was performed.
By using a motorized translation stage, both healthy and cancer colon zones were scanned independently over the region of interest (ROI) of 1 mm 2 and a step size of 0.2 mm in x-y directions. For each one of the four input polarization state measurements, the degree of polarization ρ of the measured output Stokes vector was calculated and a normalization procedure was applied (see Eq. (2)). A total of 36 Mueller matrices were obtained for all scanning points of each tissue zone (healthy and cancerous) via Eq. (4). All Mueller matrices were tested and filtered in accordance with the physical realizability criterion; as a result, three matrices per tissue zone were discarded from the data set due to non-compliance. The remaining matrices were decomposed using the symmetric decomposition and the depolarization metric calculations were performed. In Eq.    (14) In the figures of the following section either scatter or surface plots were generated. For the latter, a linear interpolation between the experimental data points was implemented.

Polarization metrics and symmetric decomposition
First, we investigated the ability of the net diattenuation D, the polarizance P, and of the two pairs of polarization parameters (D i and R i , i = 1, 2) from the symmetric decomposition to discriminate the tumor zone of the colon specimen from the healthy one. Figure 3(a,b) show the scatter plots of D versus P and D 2 versus D 1 , respectively. The polarization parameters were calculated from the experimental Mueller matrices for all measurement points. Both plots hold great potential to be considered as diagnostically relevant and to support colon histopathology analysis, because the points that represent two different zones (healthy and cancer) clearly form two distinctive clusters. The diattenuation values are significant for both tissue sections, which justifies the subsequent use of the extrinsic depolarization space. The D 2 parameter demonstrates a larger spread of values for the tumor zone compared to the healthy one, suggesting the use of the D 2 parameter as a decision variable for colon tissue diagnostics. The diagnostic value of the diattenuation was also demonstrated in [12], but the values of the diattenuation were calculated from the Lu-Chipman decomposition. Next, the retardance values demonstrate large spatial fluctuations within both healthy and cancerous zones and cannot provide accurate diagnostic information (see Fig. 3(c)). It is worth to mention that polarimetric experiments on a fresh colon specimen reported low retardance values [18]. Note that the retardance matrices can also be used to calculate the optical activity from the relation: tan ψ = [(m ϕ (1, 2) − m ϕ (2, 1)) · (m ϕ (1, 1) + m ϕ (2, 2)) −1 ]. In our studies, no optical activity was found within both cancerous and healthy zones of the colon specimen. Next, we present the 3D spatial plots of both polarizance (P) and output diattenuation (D 2 ) within the cancerous and the healthy tissue zones since these two polarization parameters exhibit the most distinct polarimetric responses for the two zones.
As can be seen in Fig. 4, both P and D 2 values are very different for the cancerous and the healthy zones of the colon specimen. As the polarimetric measurements were performed at angles of incidence and detection different from the normal to the sample surface, the impact of surface topography is enhanced. The latter may explain the increased diagnostic value of both P and D 2 parameters which are likely affected by surface and/or volume scattering. Indeed, the colon specimen used in our studies was found to have higher surface roughness within the tumor zone. However, the detailed analysis of the contributions of the surface and volume scattering to the values of P and D 2 polarimetric parameters is beyond the scope of this study and is a subject of future work. Finally, the intrinsic depolarization can be characterized with the spatial distribution of ∆ (see Fig. 5). Low values of ∆ indicate low depolarization of the probing light by the sample. Tumor tissues are known to be less depolarizing, as the morphological alterations induced by the cancer break the fine complex structure of the healthy tissue and, consequently, decrease light scattering and favor Rayleigh-Mie regime transition [45]. Hence, the low values of ∆ can be used as an optical indicator of a tumor spread throughout the scanned tissue zone.

Depolarization metrics
We likewise analyzed our experimental data by using various depolarization metrics and the relevant depolarization spaces.
Upon structural modifications in tissues due to malignancy, an alteration of the ECM is followed by destruction of the collagen cross-links. Also, the effective size of the comprising scattering particles is changed too [46,47]. Thus, both the polarization and depolarization properties of tissues are inevitably affected when tumors are developing. For this particular reason, an indirect measurement of the polarization entropy S could be regarded as a representative quantity of the tissue spatial heterogeneity. Fig. 6 reveals a lower entropy for the cancer zone in comparison to the healthy tissue zone. Specifically, the same spatial location within the cancerous zone is characterized by both lower depolarization ∆ from Fig. 5 and lower entropy S. Hence, the polarization entropy S can be a good diagnostic indicator of the sample inner structure arrangement and organization. Note that, since S is derived from the eigenvalues of the covariance matrix H, it is directly correlated to P ∆ and PI. In fact, all three depolarization parameters belong to the extrinsic depolarization space. Conversely, the relation between S and ∆ is a complex indirect one because ∆ is an intrinsic depolarization space parameter. Even more depolarization information can be extracted by adopting the concept of polarimetric purity [34][35][36]40,41,48] and by relating the values of the parameters from the extrinsic depolarization space to the health conditions of the tissue samples under study. Thus, Fig. 7 reveals two important features for both histological conditions, namely a higher polarimetric purity and an increased value of P ∆ depolarization index for the tumor tissue zone compared to the healthy one. As a result, the cancerous tissue zone acts as weaker depolarizer compared to the healthy colon tissue, as previously discussed. At this point, a parallel should be drawn between the depolarization ∆ and the depolarization index P ∆ . Their interchangeable use should be avoided since these parameters originate from different depolarization spaces. Moreover, as can be seen from Fig. 5 and 7 the same scanned area is characterized by opposite magnitude values of ∆ and P ∆ .
It is worth to visualize all 3D depolarization spaces (see Fig. 8), as such representation potentially reinforces the polarimetric support to histopathology. In Fig. 8 there are two sets of points, marked by green and red color, that represent the respective values for cancerous and healthy tissue zones of the parameters belonging to different depolarization spaces (canonical, natural and IPP). One notices that some of the tumor measurements are very closely grouped to the healthy cluster, making these values diagnostically irrelevant. Fortunately, the rest of the tumor data points are grouped separately from the healthy zone ones; they are to be related to the higher polarimetric purity and Gil-Bernabeu depolarization index values in Fig. 7, as well as to the lower depolarization and entropy in Fig. 5 and 6. Hence, these plots enable us to differentiate the two health conditions, provided there is a significantly decreased depolarization in the corresponding ROI.

Conclusions
In our previous work [22], healthy versus tumor differentiation was successfully achieved by tracing changes in the polarization states represented on the surface of the Poincaré sphere. As a logical continuation, in this manuscript the same ROIs were analysed in terms of polarization and depolarization parameters. We believe that both methods could be potentially useful for optical diagnostics and may assist pathologists in colon cancer detection. In this study, we combined both Stokes and Mueller matrix polarimetry to measure the complete MMs of the scanned ROIs of an ex vivo colon specimen. In-house algorithm was developed and applied for data post-processing, matrix filtering and decomposition, as well as for constructing three depolarization spaces for healthy versus tumor differentiation. The physical realizability criterion allowed us to filter our data for noise and experimental errors, before decomposing the experimental MMs from the scanned ROIs. To sum up, the colon specimen exhibits non-negligible polarizance and diattenuation values, determining the separation between external and internal depolarization spaces. From our results, both polarizance and diattenuation can be classified as highly discriminating polarimetric parameters. Moreover, the diattenuation can be employed as a diagnostic quantity, obtained either from its net value or from the symmetric decomposition. Further, no optical activity was found for both tissue sections. All depolarization metrics indicate larger depolarization and higher spatial heterogeneity for the healthy section, while higher polarimetric purity with less depolarization and heterogeneity was revealed for the malignant zone of the colon specimen. It was shown earlier [22,49], that the circular polarization is very sensitive to morphological alterations and provides good opportunities to detect cancer within particular ROIs. However, by measuring and decomposing the full MMs of both health conditions, one is able to add more polarimetric parameters in the analysis, potentially enriching the sample characterization and consequently, enhancing the diagnostic support to colon histopathology. Finally, measuring more colon tissue specimens with different cancer stages, will potentially contribute to building a polarimetric model, performing tissue classification with machine-learning algorithms and providing clinicians with more accurate and advanced polarimetric diagnostics using optical markers in support to colon histopathology.