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Investigation of singularities inherent to Mueller matrix images of biological crystals: diagnostics of their birefringent structure Yu.O. Ushenko, O.G. Pridiy, A.V.Motrich, O.V. Dubolazov, I.Z.Misevitch Chernivtsi National University, Correlation Optics Dept., Chernivtsi, Ukraine, 58000 Abstract. In this work, we have theoretically grounded conceptions of singularities observed in coordinate distributions of Mueller matrix elements for a network of human tissue biological crystals. Found is interrelation between polarization singularities of laser images inherent to these biological crystals and characteristic values of above matrix elements. We have determined criteria for statistical diagnostics of pathological changes in the birefringent structure of biological crystal network by using myometrium tissue as an example. Ключові слова: поляризація, матриця Мюллера, сингулярність, біологічний кристал, двопроменезаломлення, статистичні моменти Keywords: polarization, Mueller matrix, singularity, biological crystal, birefringence, statistical moment. Manuscript received; accepted for publication; published online. 1. Introduction In recent years, in laser diagnostics of biological tissue (BT) structures they effectively use the model approach [1] that allows considering this object as containing two components: amorphous and optically anisotropic ones. Topicality of this modeling is related with the possibility to apply the Mueller matrix analysis of changes in polarization properties caused by transformation of the opticandgeometric structure of anisotropic components in these biological objects [2 – 7], optical properties of which are often described using the Mueller matrix [8]. Being based on the approximation of a single light scattering, they found interrelation between the set of statistic distribution moments of the 1^{st} to 4^{th} orders Z^{(}^{j}^{ = 1, 2, 3, 4}^{)} that characterizes orientation and phase structure of BT birefringent architectonics as well as the set of respective moments [9] for twodimensional distributions of Mueller matrix elements or Muellermatrix images (MMI) [10 – 14]. In parallel with traditional statistical investigations, formed in recent 10 to 15 years is the new optical approach to describe a structure of polarizationally inhomogeneous fields in the case of scattered coherent radiation. The main feature of this approach is the analysis of definite polarization states to determine the whole structure of coordinate distributions for azimuths and ellipticities of polarization. The socalled polarization singularities are commonly used as these states [15 – 24]:
Investigations of polarizationally inhomogeneous object fields for BT with different morphology [25 – 27] allowed to ascertain that they possess a developed network of S and C points. For example in [26], the authors found interrelations between conditions providing formation of polarization singular points and particularity of the orientationphase structure of biological crystals present in territorial matrix of human tissue architectonic network. These interrelations served as a base to make statistical and fractal analyses of distribution densities for the number of singular points in BT images. As a result, the authors confirmed the efficiency of this method for investigation of object fields to differentiate optical properties of BT with a different morphological structure and physiological state. It should be noted that studied in the works [26, 27] were only statistical sets of amounts of S and C points for polarization inhomogeneous object fields. In these cases, information about the structure of topological distributions inherent to polarization states around these points was outside researchers’ attention. Besides, the mechanisms of direct formation of polarization singularities in biological crystals as well as interferential formation of these polarization states due to a multiple scattering were distinguished not so clearly. Starting from it, a further analysis of mechanisms and scenarios providing formation of polarization singularities in the field of laser radiation scattered by a biological object seems actual. ^ As a mathematical apparatus to describe mechanisms of polarization singularity formation by a biological object, one can use the Mueller matrix that entirely describes its optical properties . (1) As known from [14], any “ik” element of the Mueller matrix characterizes the degree of transformation taking place with the Stokes vector parameter of the probing beam when transforming it into the Stokes vector parameter of the object beam (2) Originating from (2), let us understand as singularities of Mueller matrix elements for biological objects the following “uncertainties” (3) From the mathematical viewpoint, the relations (3) mean that the mechanisms of transformation of the Stokes vector parameter into are indeterminate. In other words, indeterminate is the scenario for transformation of the laser radiation polarization state by biological tissue. Another singular “state” is described by the following values of Mueller matrix elements (4) From the physical viewpoint, the condition (4) means that optical properties of this medium (isotropic, anisotropic, etc.) and the respective mechanisms of transformation of laser radiation polarization states are indeterminate. Some characteristic examples of optical property singularities are demonstrated by biological tissues that are combinations of isotropic and anisotropic structures [15]. Each of these components is characterized by intrinsic matrix operators (5) where is the extinction coefficient inherent to the layer of biological tissue with the geometric thickness l (6) Here, is the orientation of an organic fibril in the architectonic network, the matter of which introduces the phase shift between orthogonal components of the laser wave amplitudes. A comparative analysis of matrixes (5) and (6) shows that the values of elements and coincide in definite conditions (7) It follows from (7) that it is impossible to separate or differentiate properties of isotropic and anisotropic optical components for biological objects in the points where Mueller matrix elements acquire singular values. Analysis of information about conditions providing formation of singular values for Mueller matrix elements describing biological tissues (Table 1) allows to predict scenarios for formation of “field” polarization singularities by them and to determine:
Besides, it is possible to select different types of polarization singularities in the image of biological tissue:  dextrogyrate () – (+C) and laevogyrate () – (–C) singular points (8)  “orthogonal” Spoints, formation of which is related with orthogonal orientations of the optical axis in organic crystal (9) So, after determining the coordinate distributions of the Mueller matrix for any biological tissue, one can predict a further scenario of formation of polarization singularities in the object field, which set its coordinate topological structure in the form of an aggregate of S contours [5 – 10]. Thereof, the next step was to study interrelations between topological singular structures for polarization images of BT and respective topological structures of twodimensional distributions for elements of their Mueller matrixes. Summarized in Table 1 are the data upon relations of Mueller matrix element singular values for BT and respective values of optical axis orientations as well as phase shifts of the anisotropic components for biological crystals. Table 1. Conditions providing formation of singular values for matrix elements
Our analysis of the data summarized in Table 1 shows that, among mechanisms providing formation of singular values for Mueller matrix elements by a system of optical singleaxis birefringent fibrils in the architectonic network of biological tissue, one can separate two groups that correspond to formation of circular polarization ():
^ На рис. 1 показано традиційну оптичну схему поляриметра для вимірювання сукупності ММЗ гістологічних зрізів БТ [14]. Fig. 1 shows the traditional optical scheme of a polarimeter to measure sets of MMI for mounts of BT [14]. Fig. 1. Optical scheme of a polarimeter: 1 – HeNe laser; 2 – collimator; 3 – stationary quarterwave plate; 5, 8 – mechanically movable quarterwave plates; 4, 9 – polarizer and analyzer, respectively; 6 – studied object; 7 – microobjective; 10 – CCD camera; 11 – personal computer. Illumination was performed with a parallel ( = 10^{4} μm) beam of a HeNe laser ( = 0.6328 μm, W = 5.0 mW). The polarization illuminator consists of the quarterwave plates 3 and 5 as well as polarizer 4, which provides formation of a laser beam with an arbitrary azimuth or ellipticity of polarization. Polarization images of BT were projected using the microobjective 7 into the lightsensitive plane (800x600 pixels) of CCDcamera 10 that provided measurements of BT structural elements within the range 2 to 2,000 μm. Experimental conditions were chosen in such a manner that spatialanglular filtration was practically eliminated when forming BT images. It was provided by matching the angular characteristics of light scattering indicatrixes by BT samples () and angular aperture of the microobjective (). Here, is the angular cone of an indicatrix where 98% of the total scattered radiation energy is concentrated. Analysis of BT images was made using the polarizer 9 and quarterwave plate 8. As a result, we determined the Stokes vectors for BT images and calculated the ensemble of Mueller matrix elements in one point illuminated with a laser beam (with the crosssection ) in accord with the following algorithm (10) The indexes 1 to 4 corresponds to the following polarization states of the beam illuminating BT: 1  0^{0}; 2 – 90^{0}; 3  +45^{0}; 4 – (right circulation). The method to measure MMI singularities for BT samples was as follows:
. (11)
As an object of the experimental study, we used tissues of a woman matrix (myometrium). ^ In the series of Figs 2 to 5, we show the results of experimental MMI measurements for measurements (fragments a and b) as well as Stokespolarimetric study of image topological structure (fragment b) inherent to human conjunctive tissue. Fig. 2. Topological structure characterizing the twodimensional distributions of the matrix element for myometrium (a) and parameter of the Stokes vector describing its polarization image. Fig. 3. Topological structure corresponding to the twodimensional distributions of the matrix element for myometrium layer (a) and parameter describing its polarization image. It can be easily seen that the elements and of the Mueller matrix as well as respective coordinate distributions of and parameters of the Stokes vector for myometrium polarization image are characterized with closed matrix Scontours in the form of a set of S and ±Сlines (Fig. 2, fragments A and B; Fig. 3, fragments A, B and C). Fig. 4. Topological structure characterizing the twodimensional distributions of the matrix element for myometrium (a) and parameter of the Stokes vector describing its polarization image. Fig. 5. Topological structure characterizing the twodimensional distributions of the matrix element for myometrium (a) and parameter of the Stokes vector describing its polarization image. While for the elements and of the Mueller matrix as well as respective coordinate distributions of and parameters of the Stokes vector for myometrium polarization image are characterized with open matrix Scontours in the form of a set of S and ±Сlines (Figs 4 and 5, fragments A, B and C). It should be noted that for all the types of topological distributions for MMI of myometrium we have revealed one universal regularity. Namely, geometrical places of singular points belonging to one type are necessarily separated with geometrical places for singular points of another type. For instance, , (12) , (13) . (14) The expressions (12) to (14) are mathematical equivalents of Scontours in coordinate distributions of the whole set of matrix elements that characterize birefringence of this biological tissue. Juxtaposition of topological distributions and allows to find a coordinate correlation not only for their singular values but for respective types of “object” and “field” Scontours, too. Отже, топологічна структура ММЗ сукупності елементів , які характеризують оптичну анізотропію властивостей біологічної тканини, є базисною у формуванні сингулярних структур (розподіли S і С  точок) її об’єктного граничного поля і може бути використана для подальшого аналізу їх інтерференційної трансформації. Thus, the topological structure of MMI for the ensemble of elements that characterizes optical anisotropy of BT properties is basic in formation of singular structures (see distributions of S and Cpoints) of the object boundary field and can be used in a further analysis of their interferential transformations. ^ Figs 6 to 11 show statistical coordinate distributions for all the types of singularities inherent to the ensemble of matrix elements represented in Table 1 that were determined over the whole area of the sample under study. Fig. 6. Coordinate structure of the matrix element for myometrium (a). A – coordinate distribution of singular values – Spoints ( = 1) labeled as () and +Cpoints ( = 0) labeled as (Δ). Fig. 7. Coordinate structure of the matrix element for myometrium (a). A – coordinate distribution of singular values – Spoints ( = 1) labeled as () and +Cpoints ( = 0) labeled as (Δ). Fig. 8. Structure of the matrix element for myometrium (a). A – coordinate distribution of singular values – Spoints ( = 1) labeled as () and +Cpoints ( = 0) labeled as (Δ). Fig. 9. Coordinate structure of the matrix element for myometrium (a). A – coordinate distribution of singular values – Spoints ( = 1) labeled as (), +Cpoints ( = +1) (Δ) and –Cpoints ( = –1) (). Fig. 10. Coordinate structure of the matrix element for myometrium (a). A – coordinate distribution of singular values – Spoints ( = 1) labeled as (), +Cpoints ( = +1) (Δ) and –Cpoints ( = –1) (). Fig. 11. Coordinate structure of the matrix element for myometrium (a). A – coordinate distribution of singular values – Spoints ( = 1) labeled as (), +Cpoints ( = +1) (Δ) and –Cpoints ( = –1) (). As seen from these experimental data, the coordinate distributions of all the Mueller matrix elements for myometrium possess a developed network of singular values. Being based on this fact, we have offered singular differentiation of changes in the distribution of optical axis orientations in biological crystals that form the architectonic network, by using as an example the woman matrix tissue. ^ As objects for our experimental investigations, we used mounts of myometrium tissue of two types:
Fig. 12 shows MMI of the element for myometrium samples of A and B types. Fig. 12. MMI of the element for myometrium tissue of A (a) and B (b) types. From the optical viewpoint, the obtained twodimensional distributions characterize the degree of anisotropy in the matter of studied samples. Thereof, it can be easily seen that the birefringency value of the samples A and B is practically identical. It is confirmed by the close level of relative values for the matrix element describing the tissues of A and B types (). In parallel with it, one can observe ordering the directions of optical axes inherent to anisotropical structures of type B myometrium. Thus, the main parameter allowing differentiation of optical properties for the samples of this type is the orientation structure of their birefringent networks. To obtain objective criteria for Muellermatrix singular differentiation of optical properties inherent to the myometrium samples of A and B types, we have used the following approach:  measured in sequence were the MMI elements and , that are basic to determine singular states (±1) formed in organic crystals with orthogonal orientations of optical axes (; (Table 1);  determined were graticubes for distribution of the singular values and ;  determined was the set of dependences for the number of points corresponding to singular states MMI , , , ;  dependences were processed using the following algorithms: (15) (16) Here, and are the average and dispersion of distributions. Shown in Figs 13 and 14 are the distributions of singularity numbers for the myometrium tissue of A and B types. Fig. 13. Dependences of singularity numbers for myometrium tissue of A type. Fig. 14. Dependences of singularity numbers for myometrium tissue of B type. In the case of myometrium tissue with pathological changes, one can observe asymmetry between the ranges of changes in values of the dependences (Fig. 14, a) and (Fig. 14, b). The above results can be explained as based on the found relation between conditions providing formation of MMI singular values and orientationphase structure of biological crystals in the myometrium tissue (Table 1). Orientation structure of MMI for the element describing the myometrium tissue of B type (Fig. 12, b) contains singular points asymmetrically located in the direction . Thereof, one should expect a maximal number of singular values for the element as compared to that of singular values and . Statistically found asymmetry in distributions of singular states for MMI describing the myometrium tissue of both types was estimated using the coefficients of asymmetry (15) and (16) introduced by us. Table 2 shows statistically averaged values of the coefficients and within two groups of myometrium samples of A and B types. Table 2. Asymmetry coefficients of Muellermatrix singularities
З вказаних в таблиці 2 даних про значення коефіцієнтів асиметрії зразків тканин міометрію «А» і «Б» типів випливає: Analysis of data represented in Table 2 allowed us to conclude:  first and secondorder statistical moments for distributions of singular values and characterizing MMI of healthy myometrium tissue do not practically differ from zero, which is indicative of their azimuthal symmetry;  values of the asymmetry coefficient for distributions of MMI singular values and describing the pathologically changed myometrium tissue of B type grow practically by one order, which indicates formation of their azimuthal asymmetry related with the direction of pathological growth of birefringent protein fibrils. Thus, the above analysis of statistical distributions describing the number of points for MMI singular values inherent to the set of elements characterizing biological tissues of different kinds seems to be efficient in differentiation of phase and orientation changes in the structure of their birefringent components, which are related with changes in their physiological state. References
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