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Hindawi Publishing Corporation Advances in Optical Technologies Volume 2010, Article ID 130659, 10 pages doi:10.1155/2010/130659 Research Article The Interconnection between the Coordinate Distribution of Mueller-Matrixes Images Characteristic Values of Biological Liquid Crystals Net and the Pathological Changes of Human Tissues Oleg V. Angelsky,Yuriy A. Ushenko, AlexanderV.Dubolazov,and Olha Yu.Telenha Correlation Optics Department, Chernivtsi National University, 2 Kotsyubynsky Street, 58012 Chernivtsi, Ukraine Correspondence should be addressed to Oleg V. Angelsky, angelsky@itf.cv.ua Received 1 December 2009; Revised 18 March 2010; Accepted 6 April 2010 Academic Editor: Igor I. Mokhun Copyright © 2010 Oleg V. Angelsky et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We have theoretically grounded conceptions of characteristic points observed in coordinate distributions of Mueller matrix elements for a network of human tissue biological crystals. The interrelation between polarization singularities of laser images inherent to these biological crystals and characteristic values of above matrix elements is found. We have determined the criteria for statistical diagnostics of pathological changes in the birefringent structure of biological crystal network by using myometrium tissue as an example. 1. Introduction inhomogeneous fields in the case of scattered coherent radiation. The main feature of this approach is the analysis of In recent years, in laser diagnostics of biological tissue (BT) definite polarization states to determine the whole structure structures they effectively use the model approach [1] that of coordinate distributions for azimuths and ellipticities allows considering this object as containing two components: of polarization. The so-called polarization singularities are amorphous and optically anisotropic ones. Topicality of this commonly used as the following states [15–32]. modeling is related with the possibility to apply the Mueller (i) States with linear polarization when the direction of matrix analysis of changes in polarization properties caused rotation for the electric field vector is indefinite, the by transformation of the optic-and-geometric structure of so-called L-points. anisotropic components in these biological objects [2–7], (ii) Circularly-polarized states when the azimuth of optical properties of which are often described using the polarization for the electric field vector is indefinite, Mueller matrix [8]. the so-called C-points. Being based on the approximation of a single light scattering, they found interrelation between the set of Investigations of polarization inhomogeneous object statistic distribution moments of the first to fourth orders fields for BT with different morphology [33–35]allowed that characterizes orientation and phase structure of BT us to ascertain that they possess a developed network birefringent architectonics as well as the set of respective of L-and C-points. For example in [34], the authors moments [9] for two-dimensional distributions of Mueller found interrelations between conditions providing forma- matrix elements or Mueller-matrix images (MMIs) [10–14], tion of polarization singular points and particularity of the that is, as it was done during the investigation of random orientation-phase structure of biological crystals present in phase objects [15]. In parallel with traditional statistical territorial matrix of human tissue architectonic network. investigations, formed in the recent 10 to 15 years is the These interrelations served as a base to make statistical and new optical approach to describe a structure of polarization fractal analyses of distribution densities for the number 2 Advances in Optical Technologies of singular points in BT images. As a result, the authors of Mueller matrixes of isotropic {A} and anisotropic {F} confirmed the efficiency of this method for investigation of structures. Each of these components is characterized by object fields to differentiate optical properties of BT with a intrinsic matrix operators different morphological structure and physiological state. The present work is devoted to investigation of new possibilities and differentiation of such objects on the basis of statistic analysis of Mueller matrix characteristic values coordinate distributions, which correspond to polarization 0100 −τl {A}= · e,(1) singular states in laser image of BT layer. 2. Characteristic Values of the Mueller-Matrix Images of Biological Tissues In accordance with a two-component biological tissue struc- where τ is the extinction coefficient inherent to the layer of ture its optical properties can be described by combination biological tissue with the geometric thickness l 1 000 10 0 0 0 f f f 0cos 2ρ +sin 2ρ · cos δ cos 2ρ sin 2ρ(1 − cos δ) sin 2ρ sin δ 22 23 24 {F}= = . (2) 0 f f f 32 33 34 0cos2ρ sin 2ρ(1 − cos δ) sin 2ρ +cos 2ρ cos δ cos 2ρ sin δ 0 f f f 0 − sin 2ρ sin δ − cos 2ρ sin δ cos δ 42 43 44 Here, ρ is the orientation of a protein fibril in the (i) the values f = 0 correspond to the complete set of architectonic network, the matter of which introduces the ±C-points; phase shift δ between orthogonal components of the laser wave amplitudes. (ii) the complete set of L-points of the laser image is The analysis of (1)and (2) shows that the major role caused by the terms f = f = f = 1. 22 33 44 in laser beam polarization state transformation belongs to birefringent fibrils network. Among the different values of Mueller-matrix analysis enables us to perform the sam- ρ and δ one can separate the particular (characteristic) pling of polarization singularities of the laser image, formed ∗ ∗ values of orientation ρ and phase δ protein fibrils of BT by biological crystals with orthogonally oriented (ρ = 0 − extracellular matrix ◦ ◦ ◦ ◦ 90 and ρ =±45 ≡ 45 − 135 ) optical axes to ∗ ◦ ◦ ◦ ρ = 0 ,±45 ,90 ; (3) (i) “orthogonal” ±C-points ∗ ◦ ◦ ◦ δ = 0 ,±90 , 180 . ◦ ◦ As it can be seen from relations (3) the necessary terms f = 0, f =±1−±C − ρ = 0 –90 , 33 34,43 0;90 for forming polarization singular states of the optically (5) ◦ ◦ ∗ ◦ ◦ f = 0, f =±1−±C − ρ = 45 –135 ; 22 24,42 45;135 birefringent crystal laser images are (L:(δ = 0 , 180 )and ±C:(δ =±90 ) points). The values of the fourth Stokes vector parameter, which correspond to above mentioned polarization singular states of the points in laser image are (ii) “orthogonal” L -and L -points 0;90 45;135 the following: ∗ ∗ ◦ ◦ S ρ , δ = 0 ⇐⇒ L-point, f = 0 − L − ρ = 0 –90 , 4 24,42 0;90 (6) ∗ ∗ ◦ ◦ S ρ , δ = +1, 0 ⇐⇒ +C-point, (4) f = 0 − L − ρ = 45 –135 . 4 34,43 45;135 ∗ ∗ S ρ , δ =−1, 0 ⇐⇒ −C-point. On the other hand, the values (3) for biological crystals Thus, measuring the coordinate distributions of the network parameters are connected with particular (charac- characteristic values ( f = 0,±1) of the BT Mueller matrix ik ∗ ∗ ∗ teristic) values of Mueller matrix elements f (ρ , δ ). elements enables us not only to foresee the scenario of ik Considering expressions (2)–(4) the characteristic values forming the ensemble of polarization singularities of its f were defined, corresponding to the L-and C-points in image, but also to additionally realize their differentiation, ik laser image of the extracellular matrix of the BT layer, as conditioned by the specificity of orientation structure of follows: biological crystals. Advances in Optical Technologies 3 1 2 34 56 78 9 10 11 Figure 1: Optical scheme of a polarimeter: 1: He-Ne laser; 2: collimator; 3: stationary quarter-wave plate; 5, 8: mechanically movable quarter-wave plates; 4, 9: polarizer and analyzer, respectively; 6: studied object; 7: micro-objective; 10: CCD camera; 11: personal computer. f (x, y) f (x, y) 22 22 0.5 0.5 (a) (b) Figure 2: Coordinate structure of the matrix element f for myometrium (a). A: coordinate distribution of characteristic values f = 1,0 22 22 labeled as ()and f = 0 labeled as (). 3. The Scheme of Experimental Measuring an indicatrix where 98% of the total scattered radiation energy is concentrated. the Coordinate Distributions of Analysis of BT images was made using the polarizer 9 and Characteristic Points in Mueller Matrix quarter-wave plate 8. As a result, we determined the Stokes Images of Biological Tissues vectors for BT images {S } and calculated the ensemble j=1,2,3,4 of Mueller matrix elements in one point illuminated with a Figure 1 shows the traditional optical scheme of a polarime- laser beam in accord with the following algorithm: ter to measure the sets of MMI of BT [14]. Illumination was performed with a parallel (∅= (1) (2) f = 0.5 S + S , i1 10 μm) beam of a He-Ne laser (λ = 0.6328 μm, W = i i 5.0 mW). The polarization illuminator consists of the (1) (2) f = 0.5 S − S , quarter-wave plates 3 and 5 as well as polarizer 4, which i2 i i (7) provides formation of a laser beam with an arbitrary azimuth (3) ◦ ◦ ◦ ◦ 0 ≤ α ≤ 180 or ellipticity 0 ≤ β ≤ 90 of polarization. f = S − f , 0 0 i3 i1 Polarization images of BT were projected using the (4) f = S − f , i = 1, 2, 3, 4. i4 i1 micro-objective 7 into the light-sensitive plane (800 × 600 pixels) of CCD-camera 10 that provided measurements of BT The indexes 1 to 4 correspond to the following polariza- structural elements within the range 2 to 2,000 μm. ◦ ◦ ◦ tion states of the beam illuminating BT: 1: 0 ;2:90 ;3:+45 ; Experimental conditions were chosen in such a manner 4: (right circulation). that spatial-angular filtration was practically eliminated The method used to measure MMI characteristic values when forming BT images. It was provided by matching the forBTsamples wasasfollows. angular characteristics of light scattering indicatrices by BT samples (Ω ≈ 16 ) and angular aperture of the micro- (i) BT mount was illuminated with a laser beam, within objective (Δω = 20 ). Here, Ω is the angular cone of the area of which, in accord with the algorithm (7)we CCD 4 Advances in Optical Technologies f (x, y) f (x, y) 44 44 1 1 0.5 0.5 0 0 (a) (b) Figure 3: Coordinate structure of the matrix element f for myometrium (a). A: coordinate distribution of characteristic values f = 1,0 44 44 labeled as ()and f = 0 labeled as (). f (x, y) f (x, y) 33 33 0.5 0.5 0 0 (a) (b) Figure 4: Coordinate structure of the matrix element f for myometrium (a). A: coordinate distribution of characteristic values f = 1,0 33 33 labeled as ()and f = 0 labeled as (). f (x, y) f (x, y) 0.5 0.5 −0.5 −0.5 −1 −1 (a) (b) Figure 5: Coordinate structure of the matrix element f for myometrium (a). A: coordinate distribution of characteristic values f = 0 34 34 labeled as (): f = +1,0 ()and f = –1,0 (). 34 34 Advances in Optical Technologies 5 f (x, y) 24 f (x, y) 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 (a) (b) Figure 6: Coordinate structure of the matrix element f for myometrium (a). A: coordinate distribution of characteristic values f = 0 24 24 labeled as (), f = +1,0 ()and f = −1,0 (). 24 24 f (x, y) f (x, y) 23 23 0.5 0.5 −0.5 −0.5 −1 −1 (a) (b) Figure 7: Coordinate structure of the matrix element f for myometrium (a). A: coordinate distribution of characteristic values f = 0 23 23 labeled as (), f = +1,0 ()and f = –1,0 (). 23 23 f (x, y)“A” f (x, y)“B” 44 44 ◦ 1 ◦ 1 90 90 0.5 0.5 0 0 −0.5 −0.5 100 μm 100 μm −1 −1 ◦ ◦ ρ 0 ρ 0 (a) (b) Figure 8: Mueller-matrixes image of the element f for myometrium tissue of A (a) and B (b) types. 44 6 Advances in Optical Technologies 50 40 25 20 0 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 (x, μm) (x, μm) (a) (b) 45 45 10 10 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 (x, μm) (x, μm) (c) (d) Figure 9: N (X )-dependences for myometrium tissue of A type. determined the array (m × n = 800 × 600) of values ensemble of protein birefringence liquid crystal net Mueller- for each element of the Mueller matrix matrixes images. As seen from these experimental data, the coordi- ⎛ ⎞ 1m f ;... ; f ; nate distributions of all the Mueller matrix elements for ik ik ⎝ ⎠ f (m × n) = . (8) ik n1 nm myometrium possess a developed network of characteristic f ;... ; f . ik ik values. Being based on this fact, we have offered Mueller- matrixes differentiation of changes in the distribution of (ii) Determined for each massif f (m × n) were coordi- ik optical axis orientations in biological crystals that form the nate distributions of its characteristic values architectonic network, by using as an example the woman ⎛ ⎞ matrix tissue. 11 1m f = 0;±1, 0;... ; f = 0;±1, 0; ik ik ⎝ ⎠ f (m × n) = . (9) ik n1 nm f = 0;±1, 0;... ; f = 0;±1, 0. ik, ik 5. Mueller-Matrix Diagnostics of Orientation Changes of Liquid Crystals As an object of the experimental study, we used tissues of Nets in Biological Tissues a woman matrix (myometrium). As objects for our experimental investigations, we used mounts of myometrium tissue of two types: 4. Coordinate Distributions of the Mueller Matrixes Images Characteristic Values of (i) biopsy of healthy tissue from a woman matrix (type Biological Tissues A), Figures 2, 3, 4, 5, 6,and 7 show coordinate distributions (ii) biopsy of conditionally normal tissue from the vicin- for all the types of characteristic values f inherent to the ity of a benign hysteromyoma (type B). ik (ρ=−45 ) (ρ=0 ) N N (ρ=−45 ) (ρ=90 ) N N Advances in Optical Technologies 7 45 35 0 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 (x, μm) (x, μm) (a) (b) 40 40 35 35 30 30 25 25 15 15 10 10 5 5 0 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 (x, μm) (x, μm) (c) (d) Figure 10: N (X )-dependences for myometrium tissue of B type. Figure 8 shows MMI of the element f for myometrium orthogonal orientations of optical axes (ρ = 0 , ◦ ◦ ◦ samples of A and B types. 90 → f =±1, 0; ρ = 45 , 135 → f =±1, 0 34,43 24,42 From the optical viewpoint, the obtained two-dimen- ((5)and (6)); sional distributions f (x, y) characterize the degree of (a) two-dimension array x = 1 ÷ m; y = 1 ÷ n of anisotropy in the matter of studied samples. Thereof, it can CCD-camera pixels can be represented by the be easily seen that the birefringence value of the samples A set of columns shifted along the x-direction by and B is practically identical. It is confirmed by the close level Δx = 1 pix of relative values for the matrix element f (x, y) describing B A ⎛ ⎞ ⎛ ⎞ the tissues of A and B types ( f (x, y) ≈ f (x, y)). In parallel 44 44 r r + Δx ≡ r 11 11 21 with it, one can observe the ordering of the directions of ⎜ ⎟ ⎜ ⎟ · · ⎜ ⎟ ⎜ ⎟ optical axes inherent to anisotropic structures of type B m × n = ⎜ ⎟ ≡ X , ⎜ ⎟ ≡ X ;... , 1 2 ⎝ · ⎠ ⎝ · ⎠ myometrium. r r + Δx ≡ r 1n 1n 2n Thus, the main parameter allowing differentiation of ⎛ ⎞ optical properties for the samples of this type is the r + (m − 1)Δx ≡ r 11 (m−1)1 ⎜ ⎟ orientation structure of their birefringent networks. ⎜ ⎟ ⎜ ⎟ ≡ X , m−1 To obtain objective criteria for Mueller-matrix differen- ⎝ ⎠ tiation of optical properties inherent to the myometrium r + (m − 1)Δx ≡ r 11 (m−1)1 samples of A and B types, we have used the following ⎛ ⎞ approach: r + mΔx ≡ r 11 m1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ≡ X , ⎝ · ⎠ (i) measured in sequence were the MMI elements f 24,42 r + mΔx ≡ r 11 m1 and f , that are basic to determine characteris- 34,43 (10) tic states (±1) formed in biological crystals with ◦ ◦ (ρ=45 ) (ρ=0 ) N N ◦ ◦ (ρ=45 ) (ρ=90 ) N N 8 Advances in Optical Technologies (ii) the amount N of characteristic points N (X)for different optical axes orientations ρ = 0 ; j=1÷m ρ ◦ ◦ ◦ f (x, y) = ±1, 0 within each column X 90 ,+45 ; 135 of biological crystals birefringent 34,43 j=1÷m was calculated, and f (x, y) =±1, 0 within network. 24,42 Mueller matrix images obtaining the distributions of N (X ) = N f = +1, 0 ; N f = +1, 0 ;... ; N f = +1, 0 ; N f = +1, 0 , ρ=0 1 34;43 2 34;43 m−1 34;43 m 34;43 N (X ) = N f =−1, 0 ; N f =−1, 0 ;... ; N f =−1, 0 ; N f =−1, 0 , ρ=90 1 34;43 2 34;43 m−1 34;43 m 34;43 (11) N (X ) = N f = +1, 0 ; N f = +1, 0 ;... ; N f = +1, 0 ; N f = +1, 0 , ρ=45 1 24;42 2 24;42 m−1 24;42 m 24;42 N (X ) = N f =−1, 0 ; N f =−1, 0 ;... ; N f =−1, 0 ; N f =−1, 0 . ρ=135 1 24;42 2 24;42 m−1 24;42 m 24;42 Table 1: The skewness coefficients of characteristic values distribu- Orientation structure of MMI for the element f tions in Mueller-matrix images. describing the myometrium tissue of B type (Figure 8(b)) contains characteristic points f (x, y) = 1, 0 asymmetrically Myometrium Myometrium located in the direction ρ = 90 . Thereof, one should expect (normal state) (pathological state) a maximal number of characteristic values for the element (25 samples) (23 samples) f f (x, y) =−1, 0 as compared to that of characteristic values 43,34 34 0.03 ± 0.005 0.45 ± 0.063 f (x, y) = 1, 0 and f (x, y) =±1, 0. 34,43 24,42 43,34 σ 0.02 ± 0.004 0.12 ± 0.019 Statistically found asymmetry in distributions of charac- 42,24 0.025 ± 0.0036 0.28 ± 0.037 teristic points for MMI describing the myometrium tissue of 42,24 σ 0.03 ± 0.0047 0.14 ± 0.021 both types was estimated using the asymmetry coefficients (12) introduced by us. Table 1 shows statistically averaged values of the coefficients Q and W within two groups of M σ (iii) N (X ) dependences were processed using the follow- myometrium samples of A and B types. ing algorithms Analysis of data represented in Table 1 allowed us to conclude the following: ◦ ◦ M N − M N ρ=0 ρ=90 ( f ) 34,43 Q = , ◦ ◦ M N + M N ρ=0 ρ=90 (i) first- and second-order statistical moments for dis- ◦ ◦ tributions of characteristic values f (x, y)and M N − M N 34,43 ρ=45 ρ=135 ( f ) 24,42 Q = . M f (x, y) of healthy myometrium tissue do not 24,42 ◦ ◦ M N + M N ρ=45 ρ=135 practically differ from zero, which is indicative of (12) their azimuthal symmetry; σ N ◦ − σ N ◦ ρ=0 ρ=90 ( f34,43 ) W = , ◦ ◦ σ N + σ N ρ=0 ρ=90 (ii) values of the skewness coefficient for distributions of MMI characteristic values f (x, y)and f (x, y) 34,43 24,42 σ N ◦ − σ N ◦ ρ=45 ρ=135 ( f ) 24,42 W = . describing the pathologically changed myometrium ◦ ◦ σ N + σ N ρ=45 ρ=135 tissue of B type grow practically by one order, which indicates the formation of their azimuthal asymmetry Here, M(N )and σ (N ) are the average and dispersion of related with the direction of pathological growth of ρ ρ N (X ) distributions. birefringent protein fibrils. ShowninFigures 9 and 10 are the distributions N (X)of Mueller-matrixes images for the myometrium tissue of A and Btypes. 6. Conclusion In the case of myometrium tissue with pathological changes, one can observe asymmetry between the ranges of Thus, the above analysis of statistical distributions describing changes in values of the dependences N (X)(Figure 10(a)) the number of points for MMI characteristic values inherent ρ=0 and N (X)(Figure 10(b)). to the set of elements f characterizing biological tissues ρ=90 ik The above results can be explained as based on the of different kinds seems to be efficient in differentiation relation found between conditions, providing formation of of phase and orientation changes in the structure of their MMI characteristic values and orientation-phase structure of birefringent components, which are related with changes in biological crystals in the myometrium tissue ((5)and (6)). their physiological state. Advances in Optical Technologies 9 References Optical Methods. Biomedical Diagnostics, Environmental and Material Science, V. 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Published: Jun 8, 2010
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