The Interconnection between the Coordinate Distribution of Mueller-Matrixes Images Characteristic Values of Biological Liquid Crystals Net and the Pathological Changes of Human Tissues

Abstract

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

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 optic-and-geometric 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 first to fourth orders that characterizes orientation and phase structure of BT birefringent architectonics as well as the set of respective moments [9] for two-dimensional distributions of Mueller matrix elements or Mueller-matrix images (MMIs) [10–14], that is, as it was done during the investigation of random phase objects [15]. In parallel with traditional statistical investigations, formed in the recent 10 to 15 years is the new optical approach to describe a structure of polarization 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 so-called polarization singularities are commonly used as the following states [15–32].(i)States with linear polarization when the direction of rotation for the electric field vector is indefinite, the so-called 𝐿-points.(ii)Circularly-polarized states when the azimuth of polarization for the electric field vector is indefinite, the so-called 𝐶-points.

Investigations of polarization inhomogeneous object fields for BT with different morphology [33–35] allowed us to ascertain that they possess a developed network of 𝐿- and 𝐶-points. For example in [34], the authors found interrelations between conditions providing formation of polarization singular points and particularity of the orientation-phase 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.

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 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 structure its optical properties can be described by combination of Mueller matrixes of isotropic {𝐴} and anisotropic {𝐹} structures. Each of these components is characterized by intrinsic matrix operators{𝐴}=1000010000100001⋅𝑒^−𝜏𝑙, (1) where 𝜏 is the extinction coefficient inherent to the layer of biological tissue with the geometric thickness 𝑙{𝐹}=10000𝑓₂₂𝑓₂₃𝑓₂₄0𝑓₃₂𝑓₃₃𝑓₃₄0𝑓₄₂𝑓₄₃𝑓₄₄=10000cos²2𝜌+sin²2𝜌⋅cos𝛿cos2𝜌sin2𝜌(1−cos𝛿)sin2𝜌sin𝛿0cos2𝜌sin2𝜌(1−cos𝛿)sin²2𝜌+cos²2𝜌cos𝛿cos2𝜌sin𝛿0−sin2𝜌sin𝛿−cos2𝜌sin𝛿cos𝛿. (2)

Here, 𝜌 is the orientation of a protein fibril in the architectonic network, the matter of which introduces the phase shift 𝛿 between orthogonal components of the laser wave amplitudes.

The analysis of (1) and (2) shows that the major role in laser beam polarization state transformation belongs to birefringent fibrils network. Among the different values of 𝜌 and 𝛿 one can separate the particular (characteristic) values of orientation 𝜌^∗ and phase 𝛿^∗ protein fibrils of BT extracellular matrix 𝜌^∗=0^∘,±45^∘,90^∘;𝛿^∗=0^∘,±90^∘,180^∘. (3)

As it can be seen from relations (3) the necessary terms for forming polarization singular states of the optically 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 the following:𝑆₄(𝜌^∗,𝛿^∗𝑆)=0⇔𝐿-point,₄(𝜌^∗,𝛿^∗𝑆)=+1,0⇔+𝐶-point,₄(𝜌^∗,𝛿^∗)=−1,0⇔−𝐶-point. (4)

On the other hand, the values (3) for biological crystals network parameters are connected with particular (characteristic) values of Mueller matrix elements 𝑓^∗_𝑖𝑘(𝜌^∗,𝛿^∗).

Considering expressions (2)–(4) the characteristic values 𝑓^∗_𝑖𝑘 were defined, corresponding to the 𝐿- and 𝐶-points in laser image of the extracellular matrix of the BT layer, as follows:(i)the values 𝑓₄₄=0 correspond to the complete set of ±C-points;(ii)the complete set of 𝐿-points of the laser image is caused by the terms 𝑓₂₂=𝑓₃₃=𝑓₄₄=1.

Mueller-matrix analysis enables us to perform the sampling of polarization singularities of the laser image, formed by biological crystals with orthogonally oriented (𝜌=0^∘−90^∘ and 𝜌=±45^∘≡45^∘−135^∘) optical axes to(i)“orthogonal” ±𝐶-points 𝑓₃₃=0,𝑓_34,43=±1−±𝐶_0;90−(𝜌=0^∘-90^∘𝑓),₂₂=0,𝑓_24,42=±1−±𝐶_45;135−(𝜌=45^∘-135^∘); (5)(ii)“orthogonal” 𝐿_0;90- and 𝐿_45;135-points 𝑓_24,42=0−𝐿_0;90−(𝜌=0^∘-90^∘𝑓),_34,43=0−𝐿_45;135−(𝜌=45^∘-135^∘). (6)

Thus, measuring the coordinate distributions of the characteristic values (𝑓^∗_𝑖𝑘=0,±1) of the BT Mueller matrix elements enables us not only to foresee the scenario of forming the ensemble of polarization singularities of its image, but also to additionally realize their differentiation, conditioned by the specificity of orientation structure of biological crystals.

3. The Scheme of Experimental Measuring the Coordinate Distributions of Characteristic Points in Mueller Matrix Images of Biological Tissues

Figure 1 shows the traditional optical scheme of a polarimeter to measure the sets of MMI of BT [14].

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.

Illumination was performed with a parallel (∅=10⁴ 𝜇m) beam of a He-Ne laser (𝜆=0.6328 𝜇m, 𝑊=5.0 mW). The polarization illuminator consists of the quarter-wave plates 3 and 5 as well as polarizer 4, which provides formation of a laser beam with an arbitrary azimuth 0^∘≤𝛼₀≤180^∘ or ellipticity 0^∘≤𝛽₀≤90^∘ of polarization.

Polarization images of BT were projected using the micro-objective 7 into the light-sensitive plane (800 × 600 pixels) of CCD-camera 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 spatial-angular filtration was practically eliminated when forming BT images. It was provided by matching the angular characteristics of light scattering indicatrices by BT samples (Ω≈16^∘) and angular aperture of the micro-objective (Δ𝜔=20^∘). Here, Ω 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 quarter-wave plate 8. As a result, we determined the Stokes vectors for BT images {𝑆_𝑗=1,2,3,4} and calculated the ensemble of Mueller matrix elements in one point illuminated with a laser beam in accord with the following algorithm: 𝑓_𝑖1=0.5[𝑆_𝑖⁽¹⁾+𝑆_𝑖⁽²⁾𝑓],_𝑖2=0.5[𝑆_𝑖⁽¹⁾−𝑆_𝑖⁽²⁾𝑓],_𝑖3=𝑆_𝑖⁽³⁾−𝑓_𝑖1,𝑓_𝑖4=𝑆_𝑖⁽⁴⁾−𝑓_𝑖1,𝑖=1,2,3,4. (7)

The indexes 1 to 4 correspond to the following polarization states of the beam illuminating BT: 1: 0^∘; 2: 90^∘; 3: +45^∘; 4: ⨂ (right circulation).

The method used to measure MMI characteristic values for BT samples was as follows.(i)BT mount was illuminated with a laser beam, within the area of which, in accord with the algorithm (7) we determined the array (𝑚×𝑛=800×600) of values for each element of the Mueller matrix 𝑓_𝑖𝑘𝑓(𝑚×𝑛)=^11𝑖𝑘;…;𝑓^1𝑚_𝑖𝑘;𝑓^𝑛1_𝑖𝑘;…;𝑓^𝑛𝑚_𝑖𝑘.. (8)(ii)Determined for each massif 𝑓_𝑖𝑘(𝑚×𝑛) were coordinate distributions of its characteristic values 𝑓^∗_𝑖𝑘𝑓(𝑚×𝑛)=^11𝑖𝑘=0;±1,0;…;𝑓^1𝑚_𝑖𝑘𝑓=0;±1,0;_{𝑛1𝑖𝑘,}=0;±1,0;…;𝑓^𝑛𝑚_𝑖𝑘=0;±1,0.. (9)

As an object of the experimental study, we used tissues of a woman matrix (myometrium).

4. Coordinate Distributions of the Mueller Matrixes Images Characteristic Values of Biological Tissues

Figures 2, 3, 4, 5, 6, and 7 show coordinate distributions for all the types of characteristic values 𝑓^∗_𝑖𝑘 inherent to the ensemble of protein birefringence liquid crystal net Mueller-matrixes images.

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(b)

Figure 2 

Coordinate structure of the matrix element 𝑓22 for myometrium (a). A: coordinate distribution of characteristic values 𝑓22 = 1,0 labeled as (□) and 𝑓22 = 0 labeled as (▵).

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(b)

Figure 3 

Coordinate structure of the matrix element 𝑓44 for myometrium (a). A: coordinate distribution of characteristic values 𝑓44 = 1,0 labeled as (□) and 𝑓44 = 0 labeled as (▵).

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(b)

Figure 4 

Coordinate structure of the matrix element 𝑓33 for myometrium (a). A: coordinate distribution of characteristic values 𝑓33 = 1,0 labeled as (□) and 𝑓33 = 0 labeled as (▵).

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(b)

Figure 5 

Coordinate structure of the matrix element 𝑓34 for myometrium (a). A: coordinate distribution of characteristic values 𝑓34 = 0 labeled as (□): 𝑓34 = +1,0 (▵) and 𝑓34 = –1,0 (▿).

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(b)

Figure 6 

Coordinate structure of the matrix element 𝑓24 for myometrium (a). A: coordinate distribution of characteristic values 𝑓24 = 0 labeled as (□), 𝑓24 = +1,0 (▵) and 𝑓24 = −1,0 (▿).

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Figure 7 

Coordinate structure of the matrix element 𝑓23 for myometrium (a). A: coordinate distribution of characteristic values 𝑓23 = 0 labeled as (□), 𝑓23 = +1,0 (▵) and 𝑓23 = –1,0 (▿).

As seen from these experimental data, the coordinate distributions of all the Mueller matrix elements for myometrium possess a developed network of characteristic values. Being based on this fact, we have offered Mueller-matrixes 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.

5. Mueller-Matrix Diagnostics of Orientation Changes of Liquid Crystals Nets in Biological Tissues

As objects for our experimental investigations, we used mounts of myometrium tissue of two types:(i)biopsy of healthy tissue from a woman matrix (type A),(ii)biopsy of conditionally normal tissue from the vicinity of a benign hysteromyoma (type B).

Figure 8 shows MMI of the element 𝑓₄₄ for myometrium samples of A and B types.

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(b)

Figure 8 

Mueller-matrixes image of the element 𝑓44 for myometrium tissue of A (a) and B (b) types.

From the optical viewpoint, the obtained two-dimensional distributions 𝑓₄₄(𝑥,𝑦) characterize the degree of anisotropy in the matter of studied samples. Thereof, it can be easily seen that the birefringence value of the samples A and B is practically identical. It is confirmed by the close level of relative values for the matrix element 𝑓₄₄(𝑥,𝑦) describing the tissues of A and B types (𝑓^B₄₄(𝑥,𝑦)≈𝑓^A₄₄(𝑥,𝑦)). In parallel with it, one can observe the ordering of the directions of optical axes inherent to anisotropic 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 Mueller-matrix differentiation of optical properties inherent to the myometrium samples of A and B types, we have used the following approach:(i)measured in sequence were the MMI elements 𝑓_24,42 and 𝑓_34,43, that are basic to determine characteristic states (±1) formed in biological crystals with orthogonal orientations of optical axes (𝜌=0^∘, 90^∘→𝑓_34,43=±1,0; 𝜌=45^∘,135^∘→𝑓_24,42=±1,0 ((5) and (6));(a)two-dimension array 𝑥=1÷𝑚; 𝑦=1÷𝑛 of CCD-camera pixels can be represented by the set of columns shifted along the 𝑥-direction by Δ𝑥=1 pix 𝑟𝑚×𝑛=¹¹⋅⋅𝑟_1𝑛≡𝑋₁,𝑟₁₁+Δ𝑥≡𝑟₂₁⋅⋅𝑟_1𝑛+Δ𝑥≡𝑟_2𝑛≡𝑋₂𝑟;…,₁₁+(𝑚−1)Δ𝑥≡𝑟_(𝑚−1)1⋅⋅𝑟₁₁+(𝑚−1)Δ𝑥≡𝑟_(𝑚−1)1≡𝑋_𝑚−1,𝑟₁₁+𝑚Δ𝑥≡𝑟_𝑚1⋅⋅𝑟₁₁+𝑚Δ𝑥≡𝑟_𝑚1≡𝑋_𝑚, (10)(ii)the amount 𝑁_𝑗=1÷𝑚 of characteristic points 𝑓_34,43(𝑥,𝑦) = ±1,0 within each column 𝑋_𝑗=1÷𝑚 was calculated, and 𝑓_24,42(𝑥,𝑦)=±1,0 within Mueller matrix images obtaining the distributions of 𝑁_𝜌(𝑋) for different optical axes orientations 𝜌 = 0^∘; 90^∘,+45^∘;135^∘ of biological crystals birefringent network. 𝑁_𝜌=0(𝑋)=𝑁₁(𝑓_34;43=+1,0);𝑁₂(𝑓_34;43=+1,0);…;𝑁_𝑚−1(𝑓_34;43=+1,0);𝑁_𝑚(𝑓_34;43𝑁=+1,0),_𝜌=90(𝑋)=𝑁₁(𝑓_34;43=−1,0);𝑁₂(𝑓_34;43=−1,0);…;𝑁_𝑚−1(𝑓_34;43=−1,0);𝑁_𝑚(𝑓_34;43𝑁=−1,0),_𝜌=45(𝑋)=𝑁₁(𝑓_24;42=+1,0);𝑁₂(𝑓_24;42=+1,0);…;𝑁_𝑚−1(𝑓_24;42=+1,0);𝑁_𝑚(𝑓_24;42𝑁=+1,0),_𝜌=135(𝑋)=𝑁₁(𝑓_24;42=−1,0);𝑁₂(𝑓_24;42=−1,0);…;𝑁_𝑚−1(𝑓_24;42=−1,0);𝑁_𝑚(𝑓_24;42=−1,0). (11)(iii)𝑁_𝜌(𝑋) dependences were processed using the following algorithms 𝑄^(𝑓34,43)_𝑀=𝑀(𝑁_𝜌=0^∘)−𝑀(𝑁_𝜌=90^∘)𝑀(𝑁_𝜌=0^∘)+𝑀(𝑁_𝜌=90^∘),𝑄^(𝑓24,42)_𝑀=𝑀(𝑁_𝜌=45^∘)−𝑀(𝑁_𝜌=135^∘)𝑀(𝑁_𝜌=45^∘)+𝑀(𝑁_𝜌=135^∘).𝑊^(𝑓34,43)_𝜎=𝜎(𝑁_𝜌=0^∘)−𝜎(𝑁_𝜌=90^∘)𝜎(𝑁_𝜌=0^∘)+𝜎(𝑁_𝜌=90^∘),𝑊^(𝑓24,42)_𝜎=𝜎(𝑁_𝜌=45^∘)−𝜎(𝑁_𝜌=135^∘)𝜎(𝑁_𝜌=45^∘)+𝜎(𝑁_𝜌=135^∘). (12)

Here, 𝑀(𝑁_𝜌) and 𝜎(𝑁_𝜌) are the average and dispersion of 𝑁_𝜌(𝑋) distributions.

Shown in Figures 9 and 10 are the distributions 𝑁_𝜌(𝑋) of Mueller-matrixes images for the myometrium tissue of A and B types.

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(b)

(c)

(d)

Figure 9 

𝑁𝜌(𝑋)-dependences for myometrium tissue of A type.

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(b)

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(d)

Figure 10 

𝑁𝜌(𝑋)-dependences 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 𝑁_𝜌=0(𝑋) (Figure 10(a)) and 𝑁_𝜌=90(𝑋) (Figure 10(b)).

The above results can be explained as based on the relation found between conditions, providing formation of MMI characteristic values and orientation-phase structure of biological crystals in the myometrium tissue ((5) and (6)).

Orientation structure of MMI for the element 𝑓₄₄ describing the myometrium tissue of B type (Figure 8(b)) contains characteristic points 𝑓₄₄(𝑥,𝑦)=1,0 asymmetrically located in the direction 𝜌=90^∘. Thereof, one should expect a maximal number of characteristic values for the element 𝑓₃₄(𝑥,𝑦)=−1,0 as compared to that of characteristic values 𝑓_34,43(𝑥,𝑦)=1,0 and 𝑓_24,42(𝑥,𝑦)=±1,0.

Statistically found asymmetry in distributions of characteristic points for MMI describing the myometrium tissue of both types was estimated using the asymmetry coefficients (12) introduced by us. Table 1 shows statistically averaged values of the coefficients 𝑄_𝑀 and 𝑊_𝜎 within two groups of myometrium samples of A and B types.

Table 1 

The skewness coefficients of characteristic values distributions in Mueller-matrix images.

Analysis of data represented in Table 1 allowed us to conclude the following:(i)first- and second-order statistical moments for distributions of characteristic values 𝑓_34,43(𝑥,𝑦) and 𝑓_24,42(𝑥,𝑦) of healthy myometrium tissue do not practically differ from zero, which is indicative of their azimuthal symmetry;(ii)values of the skewness coefficient for distributions of MMI characteristic values 𝑓_34,43(𝑥,𝑦) and 𝑓_24,42(𝑥,𝑦) describing the pathologically changed myometrium tissue of B type grow practically by one order, which indicates the formation of their azimuthal asymmetry related with the direction of pathological growth of birefringent protein fibrils.

6. Conclusion

Thus, the above analysis of statistical distributions describing the number of points for MMI characteristic values inherent to the set of elements 𝑓_𝑖𝑘 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.