The Degree of Mutual Anisotropy of Biological Liquid Crystals Net during the Diagnostics of Human Tissues Birefringence

To characterize the degree of consistency of parameters of the optically uniaxial birefringent liquid crystals (protein fibrils) nets of biological tissues a new parameter-complex degree of mutual anisotropy is suggested. The technique of polarization measuring the coordinate distributions of the complex degree of mutual anisotropy of biological tissues is developed. It is shown that statistic approach to the analysis of complex degree of mutual anisotropy distributions of biological tissues of various morphological and physiological states and optical thicknesses appears to be more sensitive and efficient in differentiation of physiological state in comparison with investigations of complex degree of mutual polarization of the corresponding laser images.


Introduction
In [1,2] for characterizing the consistency between the polarization states the laser object field in the points (r 1 , r 2 ) with the intensities I(r 1 ), I(r 2 ) a new parameter-complex degree of mutual polarization (CDMP) in the next form This "two-point" theoretical approach was extended to the analysis of polarization-inhomogeneous laser images of human BT with the aim of experimental diagnostics of optical anisotropic structure [3].In [4] a method of direct polarization measurement of the real part of CDMP for different points (r 1 , r 2 ) of optically thin image (attenuation coefficient τ ≤ 0, 1) of BT layers is proposed Re{V } ≡ V (r 1 , r 2 ) = U x (r 1 )U x (r 2 ) − U y (r 1 )U y (r 2 ) cos ϕ(r 1 ) − ϕ(r 2 ) 2 I(r 1 )I(r 2 ) , ( where ϕ(r 1 ) and ϕ(r 2 )are the phase shifts between the orthogonal components U x , U y of laser amplitude in object field.
The ranges of changes of the 1st-4th distribution order statistic moments V (x, y) of the corresponding laser images, important for diagnostics of the human connective tissue oncologic state were determined in [5,6].On the other hand, such analysis techniques of V (x, y) lead to disregarding the BT extracellular matrix birefringence, which is a principal physical mechanism of their polarizationheterogeneous images formation [7][8][9][10][11].That is why it appears to be important to search for new diagnostic parameters directly characterizing the degree of consistency of optical axes and birefringence orientations of various points of protein fibrils network forming the BT extracellular matrix [3].
This paper is aimed on investigation of diagnostic possibilities of optical-anisotropic structure of biological tissues extracellular matrix of different morphology and physiological state by means of statistic analysis of a new parameter-CDMA coordinate distribution.

Theoretical Analysis of Laser Radiation Parameters Transformation by the Network of Optically Uniaxial Birefringent Liquid Crystals
The processes of laser beam interaction with BT are discussed in the following model approximation [12][13][14][15][16]: (1) Morphologically all of variety of human tissues consists of four main types (connective, muscle, epithelial, and nervous), each of it is formed by a twocomponent amorphous-crystalline structure.
(2) The crystalline component of the BT (extracellular matrix) is formed by a network of coaxial cylindrical liquid crystals (collagen, elastin, myosin, etc, fibrils).
Here ρ(r)-the angle (orientation) of the optical axis, which is determined by the direction of packing of birefringent (Δn) protein fibrils with a transverse size d(r); δ(r) = (2π/λ)Δnd(r)-phase shift of laser beam with the wavelength λ at coordinate r.
It should be mentioned that further consideration of the possibilities of two-point Jones matrix description of complex anisotropy will be performed in a 2D approximation for (x, y) plane of biological crystals anisotropic layer-r ≡ r(x, y).
With accordance to the approach outlined in [2], use (1)-( 3) we obtain the following expression for the parameter W(r 1 , r 2 ) (complex degree of mutual anisotropy-CDMA) that characterizes the degree of coordination between the orientation (ρ 1 (r 1 ), ρ 2 (r 2 )) and phase (δ 1 (r 1 ); δ 2 (r 2 )) parameters of the anisotropy of different points (r 1 , r 2 ) in the plane of birefringent protein crystals network of the extracellular matrix where ( Here ϕ 0 -(constant and crystals network anisotropic parameters independent) the phase shift between the orthogonal components U 0x and U 0y of the illuminating laser beam amplitude, tgΩ 0 = U 0y /U 0x .Multipliers exp(−iϕ 0 ) and tgΩ 0 are defined by illuminated laser beam polarization state and birefringent biological crystals optical-geometric structure independent.
Analysis of ( 4) and (5) shows the dependence of W(r 1 , r 2 ) on rotation of the polarization plane of the illuminating laser wave or rotation of the BT layer to the irradiation direction.The only exception is a case of irradiation of BT by the circular polarized (tgΩ 0 = 1; ϕ 0 = 0, 5π) laser wave U 0 In accordance with (6) the equations ( 4) and (5) become the following form: Taking the real part (Re{d ik (r)} ≡ d ik (r)) of the Jones matrix elements (3), we obtain an expression for the real part of the CDMA (Re{W(r 1 , r 2 )} ≡ W(r 1 , r 2 )) of extracellular matrix of BT layer, which can be obtained by direct polarization measurements where ( Let us analyze the relationship of CDMA W(r 1 , r 2 ) of BT layer extracellular matrix with variations orientations ρ 1 (r 1 ), ρ 2 (r 2 ) of optical axes and the phase shifts δ 1 (r 1 ), δ 2 (r 2 ) that introduced by the protein crystals of the extracellular matrix in the points (r 1 , r 2 ).
Table 1 shows the main characteristic values of CDMA W(r 1 , r 2 ) in two arbitrary points (r 1 , r 2 ) of the extracellular matrix of BT layer.
From the obtained data one can see that CDMA has a wide range of value changes (0, 0 ≤ W(r 1 , r 2 ) ≤ 1, 0) against to changes of orientation (ρ(r)) and phase (δ(r)) anisotropic parameters in different points of biological crystals network.Therefore, the coordinate distribution W(x, y) can be used in diagnostics of optical-geometric structure of biological tissues layers.

Optical Scheme and Experimental Measurements of Coordinate CDMA Distributions of Biological Tissue
Experimental investigations were carried out in the classical polarimeter the main parts and elements of which are presented in Figure 1 [4,5].
It was illuminated by collimated (∅ = 10 4 μm) He-Ne laser beam (λ = 0.6328 μm) with the power of 50 μW.Polarization illuminator (quarter-wavelength plates 3, 5 and polarizer 4) formed the beam with arbitrary polarization azimuth and ellipticity.Polarization images of BT by means of microobjective 7 (focal distance: 1.5 cm, aperture: 0.2, magnification: 4x) were projected into the plane of lightsensitive area of CCD camera (overall amount of pixels: 800 × 600, light sensitive area size: 4000 × 3000 μm, deviation of photosensitive characteristics from linear no more then 15%), which provided the range of measuring the structural elements of BT with the resolution 2-2000 μm.Maximal resolution verification (2 μm) where performed using the stage micrometer (linear scale), which image was projected into the light sensitive area of CCD camera with the help of microobjective 7.
Minimal resolution (2000 μm) corresponds to the situation when the light sensitive area of CCD camera is entirely filled by two equal sized structural elements (light and dark) of stage micrometer.The conditions of the experiment were chosen in such a way that it enabled to reduce the spaceangular aperture filtering while forming the BT images.This was ensured by conformance of angular characteristics of indicatrices of light scattering by the BT samples (Δγ ≈ 16 0 ) and angular aperture of microobjective (Δω = 20 0 ).Here Δγ is the solid angle within which 98% of all the energy of lightscattered radiation is concentrated.
Initially, according to the classical technique given in [11], it was measured two-dimensional distribution of the real parts of Jones matrix elements d ik (m × n) of BT histological section.
Further rotation of the analyzer 9 transmission plane (Θ) in the limits Θ = 0 • ÷ 180 • it is possible to determine minimum (I min ) and maximum (I max ) levels of intensity for each pixel of CCD-camera 10.Thus, the arrays of the intensity extreme values I min (m × n), I max (m × n) of laser images of BT histological Section 6 and the corresponding angles of rotation Θ min (m × n) ⇔ I min (m × n) of the analyzer 9 can be obtained.
On the basis of the obtained data it is possible to calculate the coordinate distribution of azimuth α(m × n) and ellipticity β(m × n) of polarization and determine the two-dimensional distribution of phase shifts by using the following algorithms The value of CDMA W(r 1 , r 2 = r 1 + Δr) of the two points (r 1 , r 1 + Δr) shifted by the interval Δr of the network of protein crystals is calculated using the algorithm ( 9), (10).Coordinate distribution W(x, y) of the BT layer extracellular matrix is determined by scanning with the Δr = 1 pix step in two mutually transverse directions (x = 1 ÷ m, y = 1 ÷ n).

Experimental Researches of Coordinate Distribution of CDMA of BT with Different Morphological and Physiological States
It should be noted that the structure of the anisotropic component of the different types of BT is rather complicated and diverse [3].Therefore the analysis of coordinate distribution of optical parameters that are characterized the BT extracellular matrix of various types objectively requires complex statistical, correlation, fractal and topological [11] and other approaches.In our paper we will restrict by study of the relationship of statistical moments of the 1st-4th order [12], which characterize the coordinate distribution W(x, y) of the BT extracellular matrix, with pathological changes of its orientation-phase structure.
As objects of study (Figure 2) were chosen the histological sections of myocardium muscle tissue (MT) (Figure 2(a)) and skin derma layer (SD) (Figure 2(b)).This choice objects of study due to the fact that birefringence indexes (Δn) of myosin (MT-Δn = 1, 35 • 10 −2 [13]) and collagen (SD: Δn = 1, 5 • 10 −2 [14]) fibrils of the extracellular matrix of these tissues are similar.Therefore we can assume that the ranges of phase shifts values (δ(x, y)) are determined Table 2: Statistic moments of the 1st-4th orders of CDMA W(x, y) and CDMP V (x, y) distributions of optically thin (τ = 0, 09) layers of physiologically normal and pathological changed samples of MT and SD.
Therefore, comparative studies of CDMA coordinate distributions of such BT extracellular matrix will help to determine the range of statistical parameters variation of W(x, y) and on this basis, to implement the diagnostic of their physiological state.
The analysis of the obtained experimental data discovered that the coordinate distributions of CDMA (Figures 3(a The experimental data satisfactorily correlate with the proposed model analysis of the extracellular matrix structure with the help of CDMA.Thus, there will always be points (r, r + Δr) in the network of protein fibrils that define such relationships between the parameters of anisotropy (ρ, δ) (Table 1), which correspond to extreme values of CDMA On the other hand, the specificity of the morphological construction of BT extracellular matrix of various types is manifested in different probabilities of CDMA W(x, y) values.It is believed that for directional ordered (ρ(r = x, y)) and the geometric dimensions (d(r = x, y)) of MT myosin fibrils network (Figure 3(a)) the dispersion of the orientations of optical axes and the phase shifts are substantially less than for the disordered network of SD collagen fibrils (Figure 4(a)).Therefore, the distribution of CDMA random values of MT layer predominantly localized in a small range (0, 5 ≤ W ≤ 1, 0) relatively to the main Advances in Optical Technologies In order to obtain objective criteria of diagnostic efficiency, the comparative investigation of CDMP ( V (x, y)) [4,5] and CDMA W(x, y) techniques was performed in the conditions of single and multiple scattering of laser radiation by the layers of MT and SD.

Comparative Statistical Analysis of Coordinate Distributions of CDMA of BT Extracellular Matrix and CDMP of Its Laser Images
With the aim of determination the ranges of statistical moments changes that characterize the coordinate distribution of W(x, y) and V (x, y) four groups of histological sections of MT (sound: 21 samples; dystrophic changed: 19 samples) and SD layer samples were formed (sound: 20 samples; oncological changed: 19 samples).Statistical moments of the 1st-4th-order M k=1,2,3,4 of distributions were calculated with the help of Matlab software product on the basis of the following algorithms [12] where N = m × n-quantity of pixels of CCD-camera photosensitive area.In Tables 2 and 3 the comparative results of calculations of the average (M 1 ), dispersion (M 2 ), the skewness (M 3 ) and the kurtosis (M 4 ) coefficients of CDMA W(x, y) distributions of four groups of MT and SD of different optical thickness (attenuation coefficient τ = 0, 09 and τ = 0, 75) and of CDMP V (x, y) of their laser images are presented.
From the obtained data about the coordinate distributions of CDMA of optically thin (τ = 0, 09) layers of MT and SD tissue one can see the following.

(i) The average and dispersion of distributions W(x, y)
of physiologically normal and pathological changed MT and SD differ insufficiently within 1.3 (M 1 )-1.5 (M 2 ) times.For two-dimensional distributions V (x, y) of laser images there is practically no difference between M 1 and M 2 .
(ii) The skewness values M 3 of distributions W(x, y) of the investigated samples differ by 2.1 times; the kurtosis values-by 3.2 times.For CDMP distributions V (x, y) the values of the 3rd and 4th statistic moments vary for M 3 -by 1.3 times; for M 4 -by 1.8 times.
On the other hand, with increasing the light scattering the distribution of azimuths and ellipticities of polarization of the BT object field are changed [8-10, 18, 19].As a result, the interconnection between the parameters W(r 1 , r 2 ) and V (r 1 , r 2 ) is destroyed (Table 3): (i) As diagnostically actual parameters of the physiological state of optically thick BT the statistical moments of higher order can be used.Thus, skewness values M 3 of coordinate distributions W(x, y) for the sound and oncological changed SD tissue differ by 1.4 times; the kurtosis values M 4 -by 1.45 times.
(ii) The values of the 3rd-4th order statistic moments of distributions V (x, y) of CDMP of object field of optically thick BT layers do not depend much on their physiological state and are the an order less than the values of CDMP W(x, y) skewness and the kurtosis.

Conclusions
To characterize the degree of consistency of parameters of the optically uniaxial birefringent (liquid crystals) protein fibrils nets of biological tissues a new parameter-complex degree of mutual anisotropy is suggested.The technique of polarization measuring the coordinate distributions of the complex degree of mutual anisotropy of biological tissues is developed.It is shown that statistic approach to the analysis of complex degree of mutual anisotropy distributions of biological tissues of various morphological and physiological states and optical thicknesses appears to be more sensitive and efficient in differentiation of physiological state in comparison with investigations of complex degree of mutual polarization of the corresponding laser images.

Figure 2 :
Figure 2: Architectonics of the muscular tissue (a) and the dermal layer (b) in crossed polarizer and analyzer.

Table 3 :
Statistic moments of the 1st-4th orders of CDMA W(x, y) and CDMP V (x, y) distributions of optically thick (τ = 0, 75) layers of physiologically normal and pathological changed samples of SD tissue.