Two-point Stokes vector diagnostic approach for characterization of optically anisotropic biological tissues

The purpose of the study is to demonstrate a new method of Stokes-correlometric evaluation of polarization-inhomogeneous images of optically thin (optical thickness smaller than 0.01) histological sections from optically anisotropic biological tissues of different morphological structure. This method is based on a correlation (‘two-point’) generalization of traditional optical methods for analyzing ‘one-point’ distributions of polarization states of microscopic images of biological tissues. Analytical algorithms are obtained for describing the ‘two-point’ complex parameters of the Stokes vector image of a birefringent biological tissue. An experimental technique has been developed for measuring polarization-correlation maps, i.e. the coordinate distributions of the magnitude and phase of the ‘two-point’ Stokes vector parameters. Within the framework of the statistical and correlation analysis of the obtained data, new quantitative criteria for the differentiation of the optical properties of biological tissues of various morphological structures are found. A comparative analysis of the distribution of the ‘single-point’ and ‘two-point’ parameters of the Stokes vector of polarizationally inhomogeneous images was performed. It revealed a higher sensitivity (2–5 times) of the Stokes-correlometry method to variations in orientation-phase structure of biological tissues compared to the single-point approach.


Introduction
Polarization methods occupy an important place in the diagnosis and visualization of the optically anisotropic component of biological tissues [1][2][3][4][5]. For the most complete description of the interaction of polarized light with such complex objects, the Stokes vector-parametric formalism is used [6][7][8][9][10][11]. Stokes polarimetry is based on the principles of 'single-point' mapping, i.e. obtaining coordinate distributions (maps) of the azimuth and polarization ellipticity at points of microscopic images of histological sections of biological tissues [12][13][14][15][16][17]. In the framework of statistical, correlation and fractal analysis of polarization maps, objective criteria for differential diagnosis of various stages of pre-cancer and cancer of human tissues were found [4,6,7,[12][13][14][15][16][17]. However, development of practical applications of the methods of 'single-point' Stokes polarimetry limits the azimuthal dependence of the magnitude of the polarization parameters on the rotation of the plane of the sample of biological tissue relative to the incident laser beam. One of the directions in solving this problem may be the use of the azimuthally invariant correlation approach in polarimetric diagnostics. In [18], a new 'two-point' analytic parameter (complex degree of mutual polarization, CDMP) was proposed to describe the degree of matching of polarization states at various points of optical fields.
This polarization-correlation approach was developed and successfully used for a group of differentiating pathological changes in optical anisotropy of representative samples of histological sections of biological tissues [19,20]. At the same time, the further development and diagnostic application of new 'two-point' methods of polarization correlometry of the biological tissue structure of various morphological structures and physiological conditions requires overcoming a number of unresolved problems: (a) improvement of 'CDMP theory' using more general 'twopoint' formalism of the Stokes vector theoretically introduced in [21,22]; (b) development of universal methods for measuring coordinate distributions of 'two-point' parameters of the Stokes vector (polarization-correlation maps) of polarization-inhomogeneous object fields of biological tissues; (c) obtaining and substantiation of new diagnostic relationships between optically anisotropic structures of biological tissues and their polarization-correlation maps ('Stokes-correlometry parameters', SCP). Our work is aimed at theoretical substantiation and experimental testing of the diagnostic capabilities of the 'two-point' method of Stokes correlometry with the aim of differentiating polarization-correlation SCP maps of optically thin (optical thickness smaller than 0.01) histological sections of biological tissues with different spatial-angular structures of the optically anisotropic component.

Theoretical background
Let us consider the object field of an optically anisotropic biological layer. The complex amplitudes E (r) of each point r of such a field are described by the Jones vector [4,12,13] in the following form: Here, ρ is orientation of optical axis; tgρ (r) = |Ey|(r) |Ex|(r) and δ (r) = (δ y − δ x ) (r) are phase shifts between the orthogonal components (|E x | (r) , |E y | (r)) of laser wave amplitude.
To describe the correlation structure of the stationary distributions of the fields of complex amplitudes of laser radiation converted by optically anisotropic layers, a biological matrix of mutual spectral density can be used in the following form [21,22]: Herer 1 and r 2 are coordinates of the neighboring points in the laser radiation field.
Using this matrix operator one can introduce the following relations for the 'two-point' Stokes vector parameters (3) Here It is known that the first Stokes vector parameter characterizes the full intensity at the point r; the second S 2 (r) and third S 3 (r) ones characterize changes in polarization azimuth and ellipticity, and the fourth S 4 (r) one characterize the value of polarization ellipticity. On this basis we will carry out further detailed analytical and experimental analysis of the potentiality of polarimetry of 'two-point' Stokes vector parameters using S 3 (r 1 , r 2 ) and S 4 (r 1 , r 2 ) as examples.
In the future, algorithms (8) and (9) will be used in the analysis and forecast of changes in the orientation-phase structure of experimental polarization-correlation SCP maps of the optically anisotropic component of biological tissue samples of different morphological structures.

Setup
Measurement of the coordinate distribution values is carried using a Stokes-polarimeter, which optical scheme is shown in figure 1 [4].
A low-intensity (W = 5.0 mW) He-Ne laser 1 radiation with a wavelength of 633 nm (Lasos HeNe Laser, Edmund Optics, USA) is used as an optical probe. The collimator 2 consists of two micro lenses, the foci of which coincide. As a result, a parallel illuminating beam is formed-a probe with a diameter of 2 mm. To realize the conditions of azimuthally invariant SCP mapping, a circular polarization of the laser beam is formed.
To this end, we use a multifunctional polarizing filter, which consists of sequentially placed quarter-wave plates 3; 5 (Achromatic True Zero-Order Waveplate (APAW 15 mm, Astropribor, Ukraine) and polarizer 4 (B + W XS-Pro Polarizer MRC Nano, Kaesemann, Germany). Histological section 6 converts the circular polarization of the optical probe according to the topographic structure of the optical anisotropic components of biological tissue.
As a result, a polarization-inhomogeneous image of the biological sample under study is formed. A polarizing microobjective 7 (CFI Achromat P, focal length: 30 mm, numerical aperture: 0.1, magnification: 4x, Nikon, Japan) projects an image of a histological section of biological tissue 6 into matrix plane (m × n = 1280 × 960 pixels) of the photosensitive area of the digital CCD-camera 10 (CFI Achromat P, focal length: 30 mm, numerical aperture: 0.1, magnification: 4x, Nikon, Japan).
Achromatic True Zero-Order Waveplate (APAW 15 mm, Astropribor, Ukraine) and a polarizer 9 (B + W XS-Pro Polarizer MRC Nano, Kaesemann, Germany) are placed in front of the pixel matrix. The polarization filter passes various linear and circular polarization states of the image of the biological tissue sample 6. As a result, a set of digital (discretized by total number of pixels) polarization-filtered images of the histological section 6 is formed. Then, using the computer 11, algorithmic calculation of the coordinate distributions of the SCP value is performed.
The method of measuring the absolute value |(S i=3;4 (∆x, ∆y))| and the phase Arg (S i=3;4 (∆x, ∆y)) of the SCP consists of the following sequence of steps: Arg (S i=4 (∆x; ∆y)) were calculated by the following ratios: ) .
Here I 0 and vI 90 are the intensities at the orientation of transmission plane of polarizer 0 0 and 90 0 ; δ i are phase shifts between the orthogonal components of the amplitude of the laser radiation in the points with coordinates r 1 and r 2 [4].

Biological samples
In the region of 'single-point' polarimetry, relationships were found between the maps of the polarization parameters and the optical anisotropy of the fibrillary and parenchymal structures of biological tissues [1-4, 6, 11, 23]. The information obtained was effectively used in oncology for the differential diagnosis of various stages of cancer [8,[12][13][14][15][16][17]24]. Therefore, in our work, we tested the technique of 'two-point' polarizationcorrelation mapping specifically for these types of biological tissues. This approach makes it possible to conduct a comparative analysis of the sensitivity of various polarimetric techniques and determine the diagnostic potential of Stokes correlometry of pathological changes in the orientation-phase structure of biological tissues.
Histological sections of biological tissues were obtained by a microtome from frozen samples.
The selected samples are ultimate (extreme) types of morphological structure of most human biological tissues, both in orientation and of amorphous-anisotropic structure. This selection of samples will provide comparative information about the patterns and scenarios of changes in the polarizationcorrelation structure of microscopic images of such objects. This will provide an information basis in the search for relationships between changes in the orientation-phase anisotropy of biological tissue and their polarization-correlation manifestations. The retrieved information can be used for development of new, more sensitive objective criteria for the Stokescorrelometric diagnosis of oncological changes in human organs (formation of spatially oriented fibrillary networks and the growth of birefringence [4,7,10,12,17,20]), necrotic changes in the myocardium (degradation fibrillary networks and a decrease in birefringence [12,13]), inflammatory septic processes (degradation of parenchymal organs-lungs, liver, spleen, kidneys, etc [1,14,16,19]).
Comparative analysis of these microscopic images (figure 2) revealed their individual polarizationally inhomogeneous topographical structure-the coordinate distributions of different polarization states visualized as spots of varying intensity-fragments (4)- (6). Correlation treatment of such polarizationally inhomogeneous images is the basis for a 'twopoint' Stokes-polarimetry technique. The aim of such studies is to identify objective statistical, correlation, and fractal criteria that characterize the correlation coherence of optical anisotropy parameters and can be the basis for the differential diagnosis of biological tissues birefringence changes.   4), (5)) and colon wall parenchyma (6).

Correlation analysis.
The correlation approach in the analysis of polarization-correlation SCP maps of microscopic images of histological sections of biological tissues is based on: • calculation of the set of autocorrelation functions of SCPparameters by means of line scanning with a scanning step ∆x = 1pix of according to the algorithm given in [12]; • calculation of the resulting autocorrelation function of the coordinate distribution of SCP-parameters by averaging the partial correlation dependencies over all (n) lines (1, . . . , m) of photosensitive matrix of the CCD camera 10 (figure 1).

Tissue with ordered (rectilinear) birefringent fibrillar networks-myocardium atrium
Analysis of the obtained polarization-correlation maps of the SCP module ( figure 3(a)) revealed the following: • coordinate heterogeneity of the distributions of the magnitude of the third and fourth 'two-point' parameters |S i=3;4 (∆x, ∆y)| of the Stokes vector of the microscopic image of a spatially ordered network of myosin fibrils atrium myocardium ( figure 3(a), fragments (1), (5)); • histograms of distributions of random variables |S i=3;4 (∆x, ∆y)| are individual for the third and fourth 'twopoint' parameters (figure 3(a), fragments (2), (6)); • probabilistic distributions N are characterized by the presence of major extrema localized in the vicinity |S i=3;4 (∆x, ∆y)| = 0, as well as a significant scatter in SCP, asymmetry and peak acuity (figure 3(a), fragments (2), (6)); For polarization-correlation maps of the SCP phase of the third and fourth 'two-point' parameters of the Stokes vector ( figure 3(b)) the following is found: • individual and coordinate-inhomogeneous topographic structure of the distribution of quantities Arg (S i=3;4 (∆x, ∆y)) (figure 3(b), fragments (1), (5)); • high sharpness of the peak, asymmetry, and the range of variation in the phase magnitude of the third and fourth 'twopoint' Arg (S i=3;4 (∆x, ∆y)) parameters relative to the main extrema ( figure 3(b), fragments (2), (6)).
From the physical point of view, the results of the experimental structure of the polarization-correlation maps of the module |S i=3;4 (∆x, ∆y)| can be explained as follows.
In the 'one-point' approximation, the third parameter of the Stokes vector characterizes the magnitude of the polarization azimuth [4,6]. The magnitude of the polarization azimuth is proportional to the direction of the optical axis ρ (relation (1)) of the biological crystal, which is determined by the spatial orientation of the birefringent fibrils [13,15,17]. Therefore, the degree of correlation 'point-to-point' matching (|S i=3 (∆x, ∆y)|) of polarization azimuths at the points of the microscopic image of spatially ordered atrium myocardium myosin fibrils practically does not change.
As a result, the polarization-correlation map of this SCS parameter is quite homogeneous (figure 3(a), fragment (1)), and the distribution histogram is characterized by a pronounced extremum (figure 3(a), fragment (2)).
The fourth 'one-point' parameter of the Stokes vector characterizes the magnitude of the ellipticity of polarization [4,11,12]. This parameter is more pronounced in comparison with the azimuth associated with the direction of the optical axis ρ of the birefringent fibrils (relation (1)). Therefore, the degree of correlation 'point-to-point' matching (|S i=4 (∆x, ∆y)|) of the ellipticity of polarization at the points of the microscopic image of myosin fibrils of myocardium atrium is more sensitive to variations ρ.
As a result, the degree of correlation matching of the ellipticity of polarization at different points decreases. As a result, the coordinate heterogeneity of the polarizationcorrelation maps of this SCP parameter (figure 3(a), fragment (5)) increases, and additional 'decorrelation' extremes (figure 3(a), fragment (6)) are formed in the distribution histogram.
The experimentally obtained structure of polarizationcorrelation maps of the modulus |S i=3;4 (∆x, ∆y)| and phase Arg (S i=3;4 (∆x, ∆y)) is in good agreement with the developed model representations of the processes of formation of the magnitude of the third and fourth 'two-point' parameters of the Stokes vector of the microscopic image of a birefringent spatially ordered network of atrium myocardium myosin fibrils.
The SCP phase value Arg (S 3;4 (∆x, ∆y)) ( figure 3(b), fragments (1), (5)) is determined by the degree of coordinate coherence of both the directions of the geometry axes ρ (r) of myocardium myosin fibrils and the value of the phase shift between the orthogonal components of the amplitude of laser radiation (equations (8), (9)). It comes from the analysis of the above relations that the extreme values Arg (S 3 (∆x, ∆y)) → 0 and Arg (S 4 (∆x, ∆y)) → 0.5π are determined by these orientation-phase conditions ) fulfilment.
Such conditions are most probable particularly for the ensemble of myosin fibrils of the ordered birefringent network. Quantitatively, this is confirmed by the formation of major extremes of histograms N (Arg (S 3 (∆x, ∆y)) = 0) and N (Arg (S 4 (∆x, ∆y)) = 0.5π) (figure 3(b), fragments (2), (6)). However, this mechanism is not the only one. The value of the phase shift is also changing due to the difference in the size of birefringent fibrils and their curvature. As a result of these two factors a wide range of values (0 ⩽ Arg (S i=3;4 (∆x, ∆y)) ⩽ 0.5π) change of SCP phase is formed in polarizationally inhomogeneous image of myocardium atrium histological sections ( figure 3(b), fragments (2), (6)).

Tissue with disordered birefringent fibrillar networks-myocardium ventricle
In the series of dependencies in figure 4 the statistical, correlation and fractal characteristics of values |S i=3;4 (∆x; ∆y)| distributions (figures 4(a) and Arg (S i=3;4 (∆x, ∆y)) (b), calculated for histological sections of myocardium ventricle with the network of myosin fibers disordered by the packaging directions are presented.
The revealed scenario of transformation of the topographic and statistical structure of the distributions of the magnitude of the 3rd and 4th 'two-point' parameters of the Stokes vector of the microscopic image of the histological section of myocardium ventricle can be associated with a wider range of spatial orientations of the optical axes of myosin fibrils. As a result, at the points of a polarization-inhomogeneous image of a myocardial sample of this type, the degree of correlation between the azimuth and ellipticity of polarization decreases.
For polarization-correlation maps of the SCP phase of parameters ( figure 4(b)), similar processes of transformation of coordinate ( figure 4(b), fragments (1), (5)) and probability ( figure 4(b), fragments (2), (6)) distributions of the value Arg (S i=3;4 (∆x; ∆y)) are revealed. This scenario is caused by an increase in the dispersion of orientations ρ of the optical axes and phase shifts δ introduced by birefringent myosin fibrils of various geometric thicknesses (relation (1)).
From the physical point of view the determined features of the statistical structure of SCP-maps can be associated with having a significant impact of optically isotropic component in the substance of histological sections of rectal wall (figure 5(a), fragments (2), (6)). Within these areas the field in the plane of the microscopic image is polarizationally homogeneous. Thus the consistency of the Stokes vector parameters is maximal: . Due to this the probability of extreme values of |S i=3 (∆x; ∆y)| = 0; |S i=4 (∆x; ∆y)| = 1 and Arg (S i=3 (∆x; ∆y)) = 0; Arg (S i=4 (∆x; ∆y)) = 0.5π in the total distribution of the SCP modulus and phase value of the corresponding microscopic image of the rectal wall histological sections significantly increases.

Correlation analysis of SCP-maps
As it was mentioned above, in each point (r) of polarizationally non-uniform microscopic images of histological sections of myocardium and amorphous anisotropic rectum wall, an individual value of Stokes vector parameters S i (r) is formed. In other words, the SCP-maps (S i=3;4 (∆x; ∆y)) and Arg (S i=3;4 (∆x, ∆y)) are also coordinately inhomogeneous. This fact is substantiated by the rapid decrease of autocorrelation functionsK ((|S i=3;4 (∆x; ∆y)|))   andK (Arg (S i=3;4 (∆x; ∆y))) (figures 3(b)-5, fragments (3), (7)). Comparative analysis of such dependencies showed the most rapid 'decrease' for the sample of myocardium with the structure disordered by the directions of birefringent myosin fibrils (figures 4(a) and (b), fragments (3), (7)). The autocorrelation functions of the distributions |S i=3;4 (∆x; ∆y)|defined for the polarizationally inhomogeneous image of rectal wall (figures 5(a) and (b), fragments (3), (7)) are the slowest in decrease. This may be related to the lowest period of the coordinate modulation of the values of modulus and phase of the SCP image of myocardium ventricle (figure 5(a), fragments (1), (5)) due to the largest variance of birefringence parameters. On the contrary, for amorphous anisotropic tissue of rectum wall (figure 5(b), fragments (3), (7)) fluctuations of such parameters are significantly lower or minimal. Due to this fact, the sharpness of the peak of autocorrelation function K (|S i=3;4 (∆x; ∆y)|) andK (Arg (S i=3;4 (∆x; ∆y))) decreases, while the half-width increases. In other words, the statistical moments of the 2nd and 4th order can be selected as the objective criteria of changes in the topographic structure of distributions |S i=3;4 (∆x; ∆y)| and Arg (S i=3;4 (∆x, ∆y))that will be further referred to as correlation moments Z k 2 and Z k 4 .

Fractal analysis of SCP-map
It is known that the geometry of fibrillary and polycrystalline structures of most biological tissues is hierarchical or self-similar in scale. In particular, for filamentous protein networks this is implemented in the following hierarchical sequence-'polypeptide chain-microfibril-fibril-fiberbundle' [4,12]. On this basis we should expect the formation of fractal or multifractal distributions of values of not only polarization parameters, but also of the polarizationcorrelation ones of microscopic images of biological layers. Within Stokes-correlometry mapping it was determined that experimentally measured SCP-maps (|S i=3;4 (∆x; ∆y)| and Arg (S i=3;4 (∆x; ∆y))) have fractal or multifractal structure. The approximating lines to the logarithmic dependences of power spectra log J (|S i=3;4 (∆x, ∆y)|) − log(d −1 ) and log J (Arg (S i=3;4 (∆x, ∆y))) − log(d −1 ) are either straight or broken with two inclination angles (figures 3-5, fragments (6), (8) (6), (8)). The smallest variation is observed for polarizationally inhomogeneous image of rectum wall tissue (figures 5(a) and (b), fragments (6), (8)). From a physical point of view it can be associated with the geometric features dimensions of the SCP-maps structural elements. The largest range of changes due to the specific morphological structure of biological tissue is observed for myocardium ventricle; while the smallest one is observed for the amorphous anisotropic tissue of the rectum wall.

Intergroup statistical, correlation and fractal analysis of the modulus and phase distributions of SCP-maps
The results of statistical (Z i=1;2;3;4 ), correlation (Z k i=2;4 ) and fractal (D f ) analysis of coordinate distributions of the values of |S i=3;4 (∆x; ∆y)| and Arg (S i=3;4 (∆x; ∆y)) of polarizationally inhomogeneous images of histological sections of all types of biological tissues are presented in tables 1 and 2, respectively. For this purpose, three statistically reliable sample groups of biological tissues-39 samples in each group-were formed. Further, within each group the average values ofq and average errors ±χ were calculated.
The obtained results of the Stokes-correlometry mapping were compared with the similar results of the technique of the Stokes-polarimetry mapping of microscopic images of histological sections of three types for biological tissues samples (table 3).

Conclusions
A new method of Stokes-correlometry is suggested and analytically substantiated. It reveals coordinate distributions of modulus and phase of 'two-point' Stokes vector parameters of polarizationally inhomogeneous images for optically thin (optical thickness smaller than 0.01) histological sections of biological tissues with different morphological structure.
Within statistic, correlation and fractal analysis the objective criteria characterizing the SCP-maps of polarizationally non-uniform microscopic images for three groups of samples (with the ordered, disordered birefringent fibrillary networks and the 'islet' isotropic, anisotropic structure) are revealed.
Comparative analysis of the results based on statistical, correlation and fractal analysis of distributions of 'single-point' and 'two-point' Stokes vector parameters of polarizationally inhomogeneous sample images revealed a greater (2-5 times) sensitivity of the Stokes-correlometry method compared to the single-point approach.
The results of the study show than the direct Stokes polarimetry mapping can be a basis for the differential diagnosis of changes in optical anisotropy of the human biological tissues of different morphological structure and physiological state.