Polarization-Singular Approach to Imaging Mueller-Matrix Polarimetry in the Differential Diagnosis of Histological Sections of Biopsy of Tumors of the Uterus and Prostate

The possibilities of the diagnostic use of the singular approach of the distributions of the number of characteristic values of the MMI is effective for differentiating the polarization properties of histological biopsy sections of benign and malignant tumours of the uterus and prostate. Within the framework of evidence-based medicine, the sensitivity, specificity and accuracy of the azimuthal-invariant express (∼15 min) method of Mueller-matrix mapping of polarization-singular states in the differential diagnosis of uterine myoma and adenocarcinoma, as well as adenocarcinoma of the prostate with varying degrees of differentiation have been determined.

One of the main tasks of MMP is to detect and differentiate the type of oncological changes that occur in the tissues of human organs. Using MMM methods [14][15][16][23][24][25][26], the possibility of differential diagnosis of histological sections of biopsy of benign and malignant tumours of the prostate, endometrium, cervix, breast, etc. has been demonstrated. However, the further successful development of MMP restrains a number of theoretical and experimental problems that have not been resolved to the end: ❖ Azimuthal dependence of the value of 12 out of 16 elements of the Mueller matrix, which worsens the accuracy of MMM methods in the process of serial measurements of histological preparations; ❖ Significant "computational" time required for processing large arrays of experimental data obtained by polar decomposition of Mueller matrices and twodimensional MMM; ❖ Distorting effect of the depolarized background on the Mueller-matrix detection of optical anisotropy of tissue samples of human organs [34].
One of the options for a comprehensive solution to these MMP problems can be the synthesis of MMM methods of optically thin (non-depolarizing) biological preparations and the principles of polarization-singular analysis of their object fields [1-5, 7, 9]. The main idea of this approach is that for such layers there are direct relationships between the points of linear ("L") and circular ("C") polarization states of the microscopic image and the characteristic values of azimuthally invariant matrix elements that characterize the optical anisotropy of fibrillar networks.
At this point in time in biomedical optics, singular approaches were applied in so far, few publications [35][36][37]. Here, for the first time, the analytical conditions for the formation of singly (linear) and doubly degenerate (circular) polarization singularities of images of linearly birefringent biological tissues are determined. The distributions of the number of polarization singular states in the images of such biological tissues are experimentally investigated. It has been demonstrated that the statistical moments, which characterize the distributions of the number of singular points in the object field, are sensitive to changes in morphological structure of biological tissues of various physiological state.
However, this direction of biomedical diagnostics remains poorly researched and requires further extension of the ideology of the singularity of optical fields to methods and means of one of the most effective optical technologies-azimuthally invariant MMP of the polycrystalline component of pathologically altered tissues of various human organs.
Our work is aimed at: ❖ Identification of analytical relationships between polarization-singular states of the object field of optically thin anisotropic layers of tissues of the uterus and prostate and characteristic values of their Mueller-matrix images; ❖ Development of a new azimuthally invariant Muellermatrix polarization-singular technique for serial and express (∼15 min) measurements for differential diagnosis of changes in optical anisotropy caused by tumours of uterus (malignant myoma and malignant adenocarcinoma) and prostate (adenocarcinoma with different malignant grades) tissues.
The relevance of such studies is associated with the widespread prevalence and high mortality caused by these cancers.
Prostate cancer is the second most common cancer globally in men, and in some countries is now the most diagnosed form of cancer [38,39]. Early diagnosis, intervention, and management can give significantly improved patient outcomes [40]. It is necessary to differentiate between malignant (carcinoma) grades of tumour tissues [41]. A similar situation is realized for uterine cancer, which ranks fourth among women oncological diseases [42].

BRIEF THEORY AND BASIC RELATIONS
In a polarization-inhomogeneous field, the existence of lines or surfaces is possible, at each point of which an indefinite (singular) one of its parameters [1-5, 7, 9]: 1) points of circular polarization of the field ("C"-point), in which the polarization ellipse degenerates into a circle and, accordingly, the direction of the main axis (azimuth) of the polarization ellipse is uncertain; 2) points with linear polarization ("L"-point), which are degenerate in the direction of rotation of the electric vector.
The characteristic values of the fourth parameter VS 4 of the Stokes vector VS are used as the main "detector" of the presence and coordinate position of polarization-singular points in the object field of laser radiation [6,8,10].
VS 4 (r) 05L ϕ kπ , k 0; 1; 2; . . . . ; (1) Here r-spatial coordinate in a polarization-inhomogeneous object field; ϕ-phase shift between orthogonal amplitude components at a point r. At the same time, the analytical representation of the grid of polarization singularities of the object field of laser radiation does not carry direct information about the polycrystalline structure of the biological layer-the distributions of the directions of the optical axes χ(r) of biological crystals and the phase shifts φ(r) that they form.
The fact is that the process of formation of a polarizationinhomogeneous field can be represented by a superposition of two main mechanisms: ❖ "object"-phase modulation [φ(r)] of probing laser radiation with optically anisotropic birefringent biological crystals (protein fibrillar networks); ❖ "diffractive"-secondary phase modulation [φ(r) φ(r) + δ(r)] as a result of cross-interference of partial laser waves formed by an object of a polarizationinhomogeneous object field during propagation in free space. Therefore, in each zone of laser radiation diffraction, the maps of polarization singularities [L(r) and C(r)] change. As a result, the problem of polarimetric diagnostics of changes in the structure of the biological layer turns out to be ambiguous.
Overcoming this problem can be the use of the Mueller-matrix formalism [10] in the description of the processes of the formation of the polarization-singular structure of the object field of an optically anisotropic biological layer. This approach provides unambiguous information on the relationships between the maps of polarization singularities and the parameters of the polycrystalline birefringent structure of fibrillar networks.
Within the framework of the model of birefringence of spatially structured fibrillar networks of the biological layer, developed in numerous studies [12-16, 18, 27-32], one can write the following expression for the Mueller matrix where Here χ-direction of the optical axis, determined by the orientation of the position of the fibril in the plane of the biological layer; φ 2π/λΔnz-object phase shift between linearly orthogonally polarized components of the laser beam amplitude; λ-wavelength; Δn-birefringence; z-geometric layer thickness.
The Mueller-matrix formalism for describing the birefringent properties of biological tissues makes it possible to exhaustively describe the formation of polarization at the points (r) of the object field in terms of the parameters of the Stokes' vector VS [10].
HereVS 0 (r) j VS p (r)-Stokes' vectors of probing and transformed by a layer of biological tissue of laser beams.
Hence (expressions Eqs 10, 11) there are significant limitations in the use of the Stokes-parametric approach in diagnostic serial measurements of maps of polarization singularities of a large number of biological preparations. Even in the situation of a constant polarization state ((VS 0 2 ; VS 0 3 ) const) of the probing laser beam, the coordinate position of the L(r; χ) and C(r; χ) -points turns out to be dependent on the angle of rotation (Ψ) of the biological preparation -L(r 1 ; χ ± Ψ) and C(r 1 ; χ ± Ψ). The indicated azimuthal irreproducibility of polarization-singular data is multiplied within the limits of representative samplings of biological samples.
To overcome this problem of Stokes polarimetry, we considered the diagnostic capabilities of the polarizationsingular approach using the Mueller-matrix formalism.
Based on relations Eqs 3-5, it is possible to determine diagnostically important relationships between the characteristic values of the elements of the Mueller matrix of the conditions of formation L(r; χ) and C(r; χ) polarization-singular states that are formed by a birefringent fibrillar network- Table 1.
The analysis of the ratios given in Table 1 revealed an azimuthal dependence of the formation of characteristic values f ik (r, χ)5q(L); f ik (r, χ)5g(±C) of practically all elements of the Mueller matrix of the birefringent biological layer. The exception (highlighted in green) is the matrix element f 44 , which in what follows we will call the Mueller-matrix invariant-MMI.

Measurement and Analysis of Experimental Data
In Figure 1 shows the optical scheme of Stokes polarimetry [14,15] of the coordinate distributions of the Mueller-matrix invariant f 44 , which characterizes the birefringence of histological sections of uterine tumours. The optical and metrological parameters of the Stokespolarimeter are presented in detail in a series of publications [14,15,[27][28][29][30][31][32]. Here we give a brief description of them, which is necessary for a better understanding of the further presentation of the experimental material.
Illumination of samples was performed by parallel (Ø 2 × 10 3 μm) weakly intensive (p 5 mW) beam 1 of He- Here I ⊗;⊕ -the intensity of transmitted by the object light that passed through a system of "quarter-wave plate-polarizer" of polarization analysis unit which transmits right-(⊗) and left-(⊕) circularly polarized components of the object laser radiation. 6) The accuracy of determining the value f 44 (m × n) is 2% and was determined through a series of measurements using model phase-shifting plates }0.25λ} and }0.5λ}.

Characteristics of Research Objects
For the purpose of express (∼15 min) differential diagnosis of benign (myoma) and malignant (adenocarcinoma of varying degrees of differentiation), optically thin single-scattering histological sections of uterine tumours were made on a microtome with rapid freezing during the operation. Three representative groups of histological biopsy sections of tumours were formed: • Group 1 consisted of n 40 myoma samples; • Group 2 consisted of n 36 high differentiated adenocarcinoma samples; • Group 3 consisted of n 36 poorly differentiated adenocarcinoma samples.
The type of uterus tumour was determined by an independent assessment of stained histological samples ( Figure 2).  Table 2 presents the optical and geometric parameters of the samples of native histological sections of prostate tumour biopsies from each of the groups.
The geometric thickness (z, μm) of histological sections of prostate tissue was determined by the standard values of the freezing microtome scale.
The extinction coefficient (τ, cm −1 ) of uterus tissue samples was measured according to the standard method of photometric measurement of the attenuation by the sample of the intensity of the illuminating beam [44] using an integral light scattering sphere [45].
The measurement of the integral degree of depolarization (Λ, %) of samples of histological sections of uterus tissue was carried out in the scheme of a standard Mueller-matrix polarimeter, the optical scheme of which is presented in [14,15,[27][28][29][30][31][32].
The experimental data presented in Table 2 indicate the adequacy of our model analysis of the phase modulation of laser radiation by birefringent fibrillar networks (relations Eqs 4-11, Table 1)-within the statistically reliable samplings of samples of all groups, the conditions of single scattering are realized (τ ≤ 0.1; Λ → 0 [16]) in the volume of histological sections of biopsy of uterine tumours.
To determine the statistical significance of a representative sampling of the number of samples by the cross-validation method [46], the standard deviation θ 2 of each of the calculated values of the central statistical moments Z i 1;2 (n), which characterize the distribution of the values of parameters of N(f 44 0) and M(f 44 ±1), was determined. The specified number (40 for each group) of samples provided the level θ 2 ≤ 0.025. This standard deviation corresponds to a confidence interval p30.05, which demonstrates the statistical reliability of the polarization-interference mapping method.  Table 1), which characterize the mechanisms of formation of L-states. On the contrary, the probability of the formation of characteristic values (f 44 0) (relations (5), Table 1), which characterize the mechanisms of formation of ±C-states, decreases.
These processes quantitatively illustrate the distributions N(f 44 ±1) and N(f 44 0), which are shown in Figures 6, 7.
The results of the statistical analysis of the distributions N(f 44 0) [fragment (1)] and M(f 44 ±1) within the representative samplings of histological sections of uterine tumours of all types are presented in Tables 3, 4.
Here are the mean ( Z i 1;2 ) and standard deviations (θ i 1;2 ) within all groups of samples ( Z i 1;2 ± 2θ i 1;2 2) values of the statistical moments of the first and second orders, which characterize the distributions N(f 44 0) ( Table 3) and M(f 44 ±1) ( Table 4).
From the data given in Table 3     From the data given in Table 4 it follows:

TUMOURS OF THE UTERUS
The results of information analysis using statistical processing of the set of distributions of characteristic values N(f 44 0) and M(f 44 ±1) within representative samples of histological sections of biopsy of myoma and adenocarcinoma with different degrees of differentiation are presented in Tables 5, 6.
The results shown in Table 8 indicate a high (90-93.75%) efficiency of differential diagnosis of prostate tumours by Mueller matrix mapping of histological sections of biopsy of adenocarcinoma with varying degrees of differentiation by means of a statistical analysis of distributions N(f 44 0) that determine the mechanisms of formation ±C-states.
The efficiency of differentiation of the degree of statistical analysis of the experimentally obtained distributions of characteristic values M(f 44 ±1), which determine the mechanisms of formation of-states, turned out to be rather low and did not exceed 60-82.5%, - Table 9.