**6.1 Polarimetric approach**

Laser radiation, similarly to natural light, can be absorbed and scattered by BT. Each of these processes leads to enrichment of the field by information on micro- and macrostructure of the studied medium and its components. Spectrophotometric techniques are among the most widely used now for diagnostics of BTs. These techniques are based on the analysis of spatial and temporal changes of intensity of a field scattered by optically inhomogeneous samples. At the same time, other diagnostic techniques are also intensively developed, being based on fundamental concepts of *polarization* and *coherence*.

Spatial fluctuations of the parameters of optical fields are traditionally characterized in terms of the field's coherent properties [1]. The concept of *the measure of coherence* between two light disturbances is associated with ability of such disturbances to produce interference pattern and, consequently, with visibility of an interference pattern [17, 108]. It is just the measure of the sum of correlations between equal polarization projections of electrical fields at two specified ponts.

Another type of correlation characteristics of scattered laser fields is the degree of polarization defined as the maximal magnitude of correlations between the orthogonal polarization projections of a field at the fixed point [17]. Polarization properties of a field are experimentally investigated by measuring intensity of radiation passed various optically active elements of a medium. The techniques based on the use of the coherency matrix and the degree of polarization related to correlation of orthogonal in polarization components at one point of a field are often referred to as the polarization techniques [93].

Intense development of vector (polarization) approach in the study of morphological structure and physiological state of various BTs [96] formed the basis for elaborating model concepts on optically anisothropic and self-similar structure of BTs [98, 109 - 112]. So, Cowin [113] analyzes hierarchical self-similar structure of typical connective tissue, *viz.* tendon (tropocollagen, microfibril, subfibril, fascia etc.) Cowin emphasizes that threadlike structural elements are discrete, being characterized by scale recurrence over large interval of "optical sizes" (from 1 m to 103 m). For that, optical characteristics of such structure of BT correspond, generally, to "frozen" optically uniaxial liquid crystals.

The same approach for describing morphological structure of BT has been applied in Refs [111, 112], where a BT is considered as two-component amorphous-crystalline structure. Amorphous component of BT (fat, lipids, unstructured proteins) is isotropic in polarization, i.e. optically inactive.

Crystalline component of BT is formed by oriented birefringent protein fibrils (collagen proteins, myosin, elastin etc.). The properties of the each isolated fibril are modeled by optically uniaxial crystal, whose axis direction coincides with the direction of packing at the plane of BT, and the birefringence coefficient is determined by the fibril matter. Architectonic net formed by disordered birefringent fibrils constitutes higher level of BT organization.

This model provides explanation of polarization inhomogeneity of object fields produced by BTs of various types, such as osseous and muscular tissues, tissues of female reproductive organs (myometrium) [111, 112]. The interconnections between the azimuth of polarization and ellipticity of a field, on the one hand, and the directions of packing of fibrils, as well as

Laser radiation, similarly to natural light, can be absorbed and scattered by BT. Each of these processes leads to enrichment of the field by information on micro- and macrostructure of the studied medium and its components. Spectrophotometric techniques are among the most widely used now for diagnostics of BTs. These techniques are based on the analysis of spatial and temporal changes of intensity of a field scattered by optically inhomogeneous samples. At the same time, other diagnostic techniques are also intensively developed, being

Spatial fluctuations of the parameters of optical fields are traditionally characterized in terms of the field's coherent properties [1]. The concept of *the measure of coherence* between two light disturbances is associated with ability of such disturbances to produce interference pattern and, consequently, with visibility of an interference pattern [17, 108]. It is just the measure of the sum of correlations between equal polarization projections of electrical fields

Another type of correlation characteristics of scattered laser fields is the degree of polarization defined as the maximal magnitude of correlations between the orthogonal polarization projections of a field at the fixed point [17]. Polarization properties of a field are experimentally investigated by measuring intensity of radiation passed various optically active elements of a medium. The techniques based on the use of the coherency matrix and the degree of polarization related to correlation of orthogonal in polarization components at

Intense development of vector (polarization) approach in the study of morphological structure and physiological state of various BTs [96] formed the basis for elaborating model concepts on optically anisothropic and self-similar structure of BTs [98, 109 - 112]. So, Cowin [113] analyzes hierarchical self-similar structure of typical connective tissue, *viz.* tendon (tropocollagen, microfibril, subfibril, fascia etc.) Cowin emphasizes that threadlike structural elements are discrete, being characterized by scale recurrence over large interval of "optical sizes" (from 1 m to 103 m). For that, optical characteristics of such structure of BT

The same approach for describing morphological structure of BT has been applied in Refs [111, 112], where a BT is considered as two-component amorphous-crystalline structure. Amorphous component of BT (fat, lipids, unstructured proteins) is isotropic in polarization,

Crystalline component of BT is formed by oriented birefringent protein fibrils (collagen proteins, myosin, elastin etc.). The properties of the each isolated fibril are modeled by optically uniaxial crystal, whose axis direction coincides with the direction of packing at the plane of BT, and the birefringence coefficient is determined by the fibril matter. Architectonic net formed by disordered birefringent fibrils constitutes higher level of BT

This model provides explanation of polarization inhomogeneity of object fields produced by BTs of various types, such as osseous and muscular tissues, tissues of female reproductive organs (myometrium) [111, 112]. The interconnections between the azimuth of polarization and ellipticity of a field, on the one hand, and the directions of packing of fibrils, as well as

one point of a field are often referred to as the polarization techniques [93].

correspond, generally, to "frozen" optically uniaxial liquid crystals.

based on fundamental concepts of *polarization* and *coherence*.

**6.1 Polarimetric approach** 

at two specified ponts.

i.e. optically inactive.

organization.

the parameters characterizing birefringence, on the other hand, have been also determined for the single-scattering regime [113]. It allowed improving the technique of polarization visualizing the BT's architectonics by applying statistical analysis of 2D distributions of scattered fields [111, 112].

The papers [111, 114, 115] represent the results on determining the interconnections among the set of the 1st to the 4th order statistical moments characterizing microgeometry of a surface and orientation/phase structure of human BT's birefringent architectonics, on the one hand, and the set of the corresponding statistical moments of 2D distributions of the azimuth of polarization and ellipticity of the images of these objects, i.e. polarization maps, on the other hand. It has been stated that increasing asymmetry and excess characterizing the distributions of the azimuth of polarization and ellipticity at polarization maps result from increasing dispersion in orientation of birefringent fibril optical axes. Decreasing asymmetry and excess correspond to increasing dispersion of phase delays caused by biological crystals of architectonic nets.

Further development of laser polarimetry led to new techniques for measuring 2D arrays of polarization parameters that characterize nets of biological crystals inherent in various types of human's BTs. So, statistical analysis of the coordinate distributions of the Stokes parameters provide new information on microstructure (such as magnitudes and coordinate distributions of the parameters of optical anizotropy of architectonic nets formed by collagen or myosin) of physiologically normal and pathologically changed BTs [116]. Generally, intensive development of the techniques for diagnostic applications of laser radiation has been reflected into optical coherent tomography that became the most elaborated and convenient instrument for non-invasive study of BT structure.

The use of polarization of laser beam as a tool for contrasting of BT images resulted in a new branch of optical coherent tomography (OCT) [94, 104, 105], *viz.* polarization-sensitive optical coherent tomography (PSOCT) [95, 96, 101, 106, 107]. Note, special feature of *laser polarimetry of distributions of the azimuth of polarization and ellipticity* consists in point-by-point analysis of the object field's polarization parameters followed by searching for interconnections of these parameters with orientation and anisotropic parameters of BT's architectonics. For that, it leaves undetermined the data on peculiarities (statistical, fractal) of 2D distributions of polarization parameters of a field and orientation/phase characteristics of an object. So, further development of the techniques for non-invasive macro-diagnostics of geometrical optical structure of BT through improving conventional polarization-interference mapping and looking for new techniques for reconstruction of BT architectonics is among vital topics of modern optics.

## **6.2 Optical correlation approach**

It is known [5, 32, 98, 108] that polarization properties of light at specified point of space can be described by the coherency matrix. This formalism is comprehensive for of a light field as a whole, when the field in statistically homogeneous, i.e. when the field's characteristics are independent on spatial coordinates. However, for spatially inhomogeneous fields it is of importance to know not only coordinate distributions of polarization parameters, but also interconnections of the states of polarization and the degree of coherence at different points of a field.

The first attempt to describe spatially inhomogeneous in polarization optical fields consisted in direct generalization of the coherency matrix to the two-point coherency (polarization) matrix performed by Gori [98, 117]. (Note, this principle of representation of optical fields of general type, i.e. partially spatially coherent and inhomogeneously polarized fields, even without restrictions connected with the paraxial approximation, has been explicitly formulated yet at the morning of the era of lasers (*'litic age'*), in the early sixties, being summed up in seminal review by Wolf and Mandel [1].) In '*post-litic age'*, Gori shows that some magnitude of interference pattern's visibility corresponds to the each coherency matrix element; these patterns result from superposition of radiations from two point sources whose polarization characteristics are formed by the set of polarizers and phase plates. Matrix analysis of correlation properties of scattered coherent radiation has been generalized for vector (inhomogeneously polarized) fields [109, 110].

As much prospective is the development of the tools for direct measuring CDMP in problems of biomedical optics connected with processing of coherent, inhomogeneous in polarization images of BTs obtained by allying the OCT techniques. It has been shown [100] that the CDMP of BT's coherent image is the parameter sensitive to orientation/phase changes of BT's architectonics. Experimental study [118] of 2D distributions of the CDMP of BT's laser images for examples of muscular, skin and osseous tissues have corroborated existence of the interconnections between the coordinate structure of the CDMP at laser images and geometrical/optical structure of birefingent architectonic nets of physiologically normal and dystrophycally changed BTs. Taking into account diagnostic feasibilities of the CDMP, further searching for peculiarities of the coordinate distributions of this parameter for various BTs seems to be quite relevant and urgent problem. The following consideration is devoted to this problem.

#### **6.3 BT as birefringent extracellular matrix transforming laser light parameters**

As it has been mentioned above, BT consists of two components, *viz*. optically isotropic (amorphous) and anisotropic net (extracellular matrix) of birefringent optically uniaxial fibril [4, 94, 103, 111, 112], see Figure 20.

Fig. 20. *n* - d-diam birefringent fibril; *<sup>i</sup>* - the directions of the fibril packing at the plane of a BT sample.

The first attempt to describe spatially inhomogeneous in polarization optical fields consisted in direct generalization of the coherency matrix to the two-point coherency (polarization) matrix performed by Gori [98, 117]. (Note, this principle of representation of optical fields of general type, i.e. partially spatially coherent and inhomogeneously polarized fields, even without restrictions connected with the paraxial approximation, has been explicitly formulated yet at the morning of the era of lasers (*'litic age'*), in the early sixties, being summed up in seminal review by Wolf and Mandel [1].) In '*post-litic age'*, Gori shows that some magnitude of interference pattern's visibility corresponds to the each coherency matrix element; these patterns result from superposition of radiations from two point sources whose polarization characteristics are formed by the set of polarizers and phase plates. Matrix analysis of correlation properties of scattered coherent radiation has been

As much prospective is the development of the tools for direct measuring CDMP in problems of biomedical optics connected with processing of coherent, inhomogeneous in polarization images of BTs obtained by allying the OCT techniques. It has been shown [100] that the CDMP of BT's coherent image is the parameter sensitive to orientation/phase changes of BT's architectonics. Experimental study [118] of 2D distributions of the CDMP of BT's laser images for examples of muscular, skin and osseous tissues have corroborated existence of the interconnections between the coordinate structure of the CDMP at laser images and geometrical/optical structure of birefingent architectonic nets of physiologically normal and dystrophycally changed BTs. Taking into account diagnostic feasibilities of the CDMP, further searching for peculiarities of the coordinate distributions of this parameter for various BTs seems to be quite relevant and urgent problem. The following consideration

**6.3 BT as birefringent extracellular matrix transforming laser light parameters** 

*<sup>i</sup>* - the directions of the fibril packing at the plane

As it has been mentioned above, BT consists of two components, *viz*. optically isotropic (amorphous) and anisotropic net (extracellular matrix) of birefringent optically uniaxial

generalized for vector (inhomogeneously polarized) fields [109, 110].

is devoted to this problem.

fibril [4, 94, 103, 111, 112], see Figure 20.

Fig. 20. *n* - d-diam birefringent fibril;

of a BT sample.

The action of amorphous and architectonic (crystalline) components of BT, *A* and *C* , respectively, on coherent radiation is characterized by the following Jones matrix operators:

$$\mathbb{P}\{A\} = \begin{vmatrix} a\_{11} & a\_{12} \\ a\_{21} & a\_{22} \end{vmatrix} = \begin{vmatrix} \exp(-\tau l) & 0 \\ 0 & \exp(-\tau l) \end{vmatrix};\tag{24}$$

$$\begin{aligned} \{\mathbf{C}\} = \begin{vmatrix} \mathbf{c}\_{11} & \mathbf{c}\_{12} \\ \mathbf{c}\_{21} & \mathbf{c}\_{22} \end{vmatrix} = \begin{vmatrix} \cos^2 \rho + \sin^2 \rho \exp(-i\delta); & \cos \rho \sin \rho \begin{bmatrix} 1 - \exp(-i\delta) \end{bmatrix}; \\ \cos \rho \sin \rho \begin{bmatrix} 1 - \exp(-i\delta) \end{bmatrix}; & \sin^2 \rho + \cos^2 \rho \exp(-i\delta); \end{vmatrix} \end{aligned} \tag{25}$$

Here, is the absorption coefficient for BT of thickness *l* ; is the packing direction of anisotropic fibril (with birefringence coefficient *n* ) at the plane of a BT's sample introducing a phase shift 2 *nl* between the orthogonal polarization components, , *E E <sup>x</sup> <sup>y</sup>* , of the probing laser beam of wavelength .

#### **6.4 Mechanisms of forming inhomogeneous in polarization BTs laser images**

As it follows from analysis performed in Refs [93, 95, 97, 113] the mechanism of forming inhomogeneous in polarization boundary field of a BT at each point can be represented in the following form:

"decomposition" of an amplitude of laser wave *U* into orthogonal linearly polarized mutually coherent components:

$$
\begin{pmatrix} \mathcal{U}\_x(r) \\ 0 \end{pmatrix} \text{ and } \begin{pmatrix} 0 \\ \mathcal{U}\_y(r) \end{pmatrix};
$$

forming a phase shift (phase difference) between these components accounting birefringence, *r* ;

superposing the orthogonally polarized components that results, in general case, in elliptically polarized wave that is described by the following equation:

$$\frac{X^2}{\mathcal{U}\_x^2(r)} + \frac{Y^2}{\mathcal{U}\_y^2(r)} - \frac{2XY}{\mathcal{U}\_x(r)\mathcal{U}\_y(r)}\cos\delta(r) = \sin^2\delta(r) \tag{26}$$

#### **6.5 Statistical and fractal analysis of polarization images of BTs**

Two types of optically thin (extinction coefficient 0.1 ) of BT's histological tomes have been studied in Refs [4, 97]:


Coordinate distributions *r r* ; and histograms *W W* , of the magnitudes of the azimuth of polarization and ellipticity at images of histological tomes of physiologically normal osseous tissue (left part) and kidney tissue (right part) are shown in Figure 22 (fragments (a), (b) and (c), (d), respectively).

Fig. 21. Polarization images of osseous tissue (а, b) and tissue of kidney (c, d) for matched (а, c) and crossed (b, d) polarizer and analyzer, respectively.

Fig. 22. Polarization maps of osseous tissue (left part) and kidney tissue (right part). Fragments (a), (b) correspond to 2D distributions of the azimuth of polarization and ellipticity, respectively; fragments (c), (d) show histograms of the corresponding distributions.

(a) (b)

(c) (d) Fig. 21. Polarization images of osseous tissue (а, b) and tissue of kidney (c, d) for matched (а,

(c) (d) (c) (d)

Fig. 22. Polarization maps of osseous tissue (left part) and kidney tissue (right part). Fragments (a), (b) correspond to 2D distributions of the azimuth of polarization and ellipticity, respectively; fragments (c), (d) show histograms of the corresponding

(a) (b)

c) and crossed (b, d) polarizer and analyzer, respectively.

(a) (b)

distributions.

Distributions of the azimuth of polarization and ellipticity of the maps of BTs of two types characterize the set of statistical moments of the 1st to the 4th orders shown in Table 1.

The obtained data for statistical moments of the 1st to the 4th orders for distributions *W W* , of images of BTs of different morphological construction show that as birefringent architectonic nets are higher structured, as the magnitudes of the 3rd and the 4th statistical moments associated with the set of polarization parameters increase.


Table 1. Statistical momentums *Mi* of the coordinate distributions of the states of polarization at images of BTs of osseous and kidney BTs
