4.1 Polarization Correlometry of Microscopic Images of Biological Layers

4.1.1 Brief Theory of the Method

This section presents the results of the practical application of the theoretical correlation approach for combining the polarization [1,2,3,4,5,6,7,8,9,10,11,12,13,14] and correlation [15,16,17,18,19,20,21,22,23] approaches to the analysis of polarization-inhomogeneous fields by introducing a polarization-correlation parameter—a complex degree of mutual polarization [21].

In works [22, 23], an expression of the CDMP value was found for two points (\(r_{1} ,r_{2}\)) of a microscopic image of a histological section of biological tissue

$$V\left( {r_{1} ,r_{2} } \right) = \frac{{\left( {U_{x} \left( {r_{1} } \right)U_{x} \left( {r_{2} } \right) - U_{y} \left( {r_{1} } \right)U_{y} \left( {r_{2} } \right)} \right)^{2} + 4U_{x} \left( {r_{1} } \right)U_{x} \left( {r_{2} } \right)U_{y} \left( {r_{1} } \right)U_{y} \left( {r_{2} } \right)\exp \left( {i\left( {\delta_{y} \left( {r_{2} } \right) - \delta_{x} \left( {r_{1} } \right)} \right)} \right)}}{{\left( {U_{x}^{2} \left( {r_{1} } \right) + U_{y}^{2} \left( {r_{1} } \right)} \right)\left( {U_{x}^{2} \left( {r_{2} } \right) + U_{y}^{2} \left( {r_{2} } \right)} \right)}}.$$
(4.1)

Here, \(U_{x}\) and \(U_{y}\) are the orthogonal components of the amplitude of the laser wave in the plane of the microscopic image, \(\delta_{x}\) and \(\delta_{y}\) are the corresponding phase values of such components.

The main characteristic values of the parameter \(V\) for points \(\left( {r_{1} ,r_{2} } \right)\) with different types of polarization are given in Table 4.1.

Table 4.1 Value of the complex degree of mutual polarization \(V\left( {r_{1} ,r_{2} } \right)\) of two points \(\left( {r_{1} ,r_{2} } \right)\) with different types of light vibrations
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From the data in Table 4.1, it follows that the magnitude of the complex degree of mutual polarization \(V\left( {r_{1} ,r_{2} } \right)\) can be used as a diagnostic parameter for evaluating the coordinate polarization-inhomogeneous structure of microscopic images of biological objects.

For the purpose of the experimental application of this parameter, we determine the real (\(\text{Re} \left\{ V \right\}\)) part of the complex degree of mutual polarization \(V\):

$$\text{Re} \left\{ {V\left( {r_{1} ,r_{2} } \right)} \right\} = \frac{{\left( {U_{x} \left( {r_{1} } \right)U_{x} \left( {r_{2} } \right) - U_{y} \left( {r_{1} } \right)U_{y} \left( {r_{2} } \right)} \right)^{2} + 4U_{x} \left( {r_{1} } \right)U_{x} \left( {r_{2} } \right)U_{y} \left( {r_{1} } \right)U_{y} \left( {r_{2} } \right)\cos \delta_{12} }}{{\left( {U_{x}^{2} \left( {r_{1} } \right) + U_{y}^{2} \left( {r_{1} } \right)} \right)\left( {U_{x}^{2} \left( {r_{2} } \right) + U_{y}^{2} \left( {r_{2} } \right)} \right)}};$$
(4.2)

Chapter 2, paragraph 2.3.1 [relation (2.6) (2.7)], provides a methodology for experimental measurement of the coordinate distribution of the values of the CDMP modulus at the points of a microscopic image of a histological section of a biological tissue.

4.1.2 CDMP Mapping of Microscopic Images of Biological Layers with Ordered Architectonics

Figure 4.1 shows the results of an experimental study of the coordinate distributions of the values of the CDMP modulus (fragment 1) of a histological section of skeletal muscle tissue, as well as a histogram of the distribution of random values of the CDMP modulus (fragment 2).

Fig. 4.1
figure 1

CDMP map (left side) and a histogram (right side) of the distribution of CDMP modulus values in a microscopic image of a histological section of skeletal muscle

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From the obtained data, it is seen that the optical anisotropy of the histological section of the skeletal muscle with a spatially structured fibrillar network are manifested in the formation of a polarization-inhomogeneous microscopic image. From a physical point of view, the main factor in this is the presence of a certain spectrum of orientations of the optical axes and phase shifts introduced by birefringent fibrils with different geometric cross sections. On the other hand, the polycrystalline myosin network of the skeletal muscle is rather spatially ordered. Based on this, the structure of the histogram of the distribution of random values of the CDMP modulus becomes clear—asymmetric dependences with a fairly small dispersion and a clearly defined extremum.

Quantitatively, the results of a statistical analysis of the coordinate distributions of CDMP values of optically thin layers of the myocardium and cerebrospinal fluid film are shown in Table 4.2.

Table 4.2 Statistical moments \(Q_{i = 1;2;3;4}\) that characterize the coordinate distribution of the values of the CDMP modulus of the microscopic image of a histological section of the skeletal muscle
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Analysis of the full set of statistical moments that characterize the coordinate distributions of the values of the CDMP modulus of a microscopic image of a histological section of a biological tissue with a spatially ordered polycrystalline network revealed the most sensitive of them. These include higher-order statistical moments \(Q_{i = 3;4} \left( V \right)\) characterizing the skewness and sharpness of the peak of the histograms of the distributions of the values of the CDMP modulus of microscopic images of such objects.

4.1.3 CDMP Mapping of Microscopic Images of Biological Layers with Disordered Architectonics

Figure 4.2 illustrates the results of an experimental study of coordinate distributions of values of the CDMP modulus (left side) of a histological section of brain tissue, as well as a histogram of the distribution of random values of the CDMP modulus (right side).

Fig. 4.2
figure 2

CDMP map (left side) and a histogram (right side) of the distribution of values of the CDMP modulus in a microscopic image of a histological section of the brain

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A comparative analysis of the coordinate distributions of the values of the CDMP modulus of microscopic images of histological sections of the myocardium (see Fig. 4.1) and brain tissue (see Fig. 4.2) revealed significant differences. In particular, the structure of the polarization-inhomogeneous microscopic image is more complex in the sense of increasing the interval of random changes in the values of the CDMP modulus. The latter, in turn (see Table 4.1), are determined by the degree of similarity, or, conversely, by differences in polarization states at the points of a microscopic image of a histological section of brain tissue.

The discovered feature can be associated with a wide range of orientations of the optical axes of local fibrils of a given biological tissue. This morphological structure of the optically anisotropic fibrillar network makes it possible to form more “diverse” polarization states in different coordinates of the microscopic image. Based on this, a greater level of dispersion of the histogram of the distribution of random values of the CDMP modulus (see Fig. 4.2, right part) becomes clear, while reducing its skewness and peak sharpness - Table 4.3.

Table 4.3 Statistical moments \(Q_{i = 1;2;3;4}\) that characterize the coordinate distribution of values of the CDMP modulus of a microscopic image of a histological section of the brain
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A comparative analysis of the magnitude of the statistical moments of the first to fourth orders (see Tables 4.2 and 4.3) characterizing the coordinate distribution of the values of the CDMP modulus of polycrystalline networks of histological sections of biological tissues of both types was found:

  • Various types of polycrystalline networks of real biological tissues are characterized by an individual set of statistical moments \(Q_{i = 1;2;3;4} \left( V \right)\) characterizing the distribution of the degree of correlation consistency of azimuth and ellipticity of polarization at different points of the microscopic image of the object.

  • The most sensitive to changes in the polarization inhomogeneity of laser fields in the plane of microscopic images were all statistical moments \(Q_{i = 1;2;3;4} \left( V \right)\).

  • The following trends in the magnitude of statistical moments of the first to fourth orders that characterize the polarization-correlation structure of microscopic images with a birefringent fibrillar network disordered in the directions of the optical axes—\(Q_{1} \downarrow ;Q_{2} \uparrow ;Q_{3} \downarrow ;Q_{4} \downarrow\).

The results and certain patterns of formation of CDMP distributions of networks of spatially structured fibrillar networks of biological crystals were the basis for the development of a method for differentiating changes in optical anisotropy due to necrotic changes in the myocardium.

4.1.4 CDMP Mapping Diagnostic Features

This paragraph contains the results of studies on the possibility of polarization-correlation differentiation of microscopic images of histological sections of the myocardium that deceased due to mechanical asphyxiation and heart attack. Figure 4.3 shows a series of coordinate distributions of the values of the modulus of the polarization-correlation parameter CDMP (left parts) and histograms of their random values (right parts). A comparison of the statistical parameters \(Q_{i = 1;2;3;4} \left( V \right)\) characterizing the distribution of the values of the CDMP modulus of microscopic images of histological sections of the myocardium of both types revealed the following differences for mechanical asphyxiation compared with the case of a heart attack:

Fig. 4.3
figure 3

CDMP maps (left parts) and histograms (right parts) of the distribution of CDMP modulus values in microscopic images of histological sections of the myocardium that deceased from mechanical asphyxiation (top line) and heart attack (bottom line)

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  • Decrease in average \((Q_{1} \left( V \right) \downarrow \uparrow )\), skewness \((Q_{3} \left( V \right) \downarrow \uparrow )\) and kurtosis \((Q_{4} \left( V \right) \downarrow \uparrow )\);

  • Increase of dispersion \((Q_{2} \left( V \right) \uparrow )\), which characterizes the histograms of the distribution of the value of the CDMP modulus, for the image of the sample of the myocardium deceased due to mechanical asphyxiation.

Physically obtained results can be associated with the “destruction” of birefringence of the fibrillar network of the myocardium, which deceased due to a heart attack. As a result, the probability of formation of CDMP modulus values other than extreme decreases \(V = 1\). As a result, the dispersion decreases and, on the contrary, the average, skewness, and kurtosis increase, characterizing the distribution of this correlation parameter of the microscopic image of the histological section of the necrotic altered myocardium.

The extension of the range of variation of random values of the CDMP modulus in the plane of the microscopic image of the myocardium histological section deceased due to mechanical asphyxiation is associated with a high level of birefringence of the myosin network of this sample—\(Q_{1} \downarrow ;Q_{2} \uparrow ;Q_{3} \downarrow ;Q_{4} \downarrow\).

The results of the comparative analysis of the averaged (within the respective groups of samples) values of the statistical moments \(Q_{i = 1;2;3;4}\) characterizing the two-dimensional distribution of the values of the CDMP modulus are given in Table 4.4.

Table 4.4 Statistical moments \(Q_{i = 1;2;3;4}\) that characterize the distribution of the values of the CDMP modulus of the myocardium histological sections deceased due to mechanical asphyxiation (\(V\)) and heart attack (\(V^{ * }\))
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The data obtained revealed the most sensitive (\(\Delta Q = \hbox{max}\)) to changes in the polarization structure of microscopic images of polycrystalline myocardial networks—statistical moments \(Q_{1}\), \(Q_{2}\).

The following quantitative criteria for intergroup differences in the values of statistical moments of the first–second orders characterizing the distribution of the modulus of the polarization-correlation parameter CDMP of microscopic images of histological sections of the myocardium of both groups have been established—\(V\left( {m \times n} \right) \Leftrightarrow \Delta Q_{1} = 1,35;\Delta Q_{2} = 1,69.\)

At the same time, a high degree of balanced accuracy was achieved—Ac(V) = 79–85% which is higher than the accuracy of azimuthally invariant polarization and Mueller-matrix mapping (see Sect. 3, Tables 3.3, 3.4, 3.7 and 3.8).

4.2 Polarization Correlometry of Optically Anisotropic Networks of Biological Layers

4.2.1 CDMA Mapping of Biological Layers with Ordered Architectonics

Figure 4.4 shows the results of an experimental study of the coordinate distributions of CDMA (left side) of a histological section of skeletal muscle tissue, as well as a histogram of the distribution of random values of the CDMA modulus (right side).

Fig. 4.4
figure 4

CDMA map (left side) and a histogram (right side) of the distribution of values of the CDMA modulus of the histological section of the skeletal muscle

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It can be seen from the obtained data that the manifestations of optical anisotropy of the birefringent fibrillar network of the histological section of the myocardium spatially ordered along the directions of the optical axes differ significantly in the values of the statistical moments of the first to fourth orders characterizing the distribution of the values of the CDMA modulus from the similar results of polarization-correlation microscopy of this biological layer (see Figs. 4.1 and 4.2).

The main reason for this, in our opinion, is the other “information content” of this polarization-correlation method, which allows direct estimation of the coordinate structure of an optically anisotropic spatially ordered network of myosin fibrils. Based on this, the histograms of the distributions \(N\left( W \right)\) of random values of the CDMA modulus are even more asymmetric with a fairly small dispersion and a clearly defined extremum of the dependence (see Fig. 4.1, the right side and Fig. 4.4, the right side).

Quantitatively, the results of a statistical analysis of the coordinate distributions of the values of the CDMA modulus of optically thin histological sections of the skeletal muscle are shown in Table 4.5.

Table 4.5 Statistical moments \(Q_{i = 1;2;3;4}\) that characterize the coordinate distribution of the values of the CDMA modulus of the histological section of the skeletal muscle
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A comparative analysis of the magnitude of the set of statistical moments of the first to fourth orders that characterize the coordinate distributions of the values of the modulus CDMP and CDMA polycrystalline networks of the skeletal muscle layer was found (see Tables 4.2 and 4.5):

  • Various types of polarization-correlation distributions characterizing the structural heterogeneity of the azimuth and polarization ellipticity at the points of microscopic images and the corresponding orientations of the optical axes and birefringence are characterized by an individual set of statistical moments \(Q_{i = 1;2;3;4}\).

  • The following relationships between the values of statistical parameters:

    $$\begin{array}{*{20}c} {Q_{1} \left( W \right) \prec \prec Q_{1} \left( V \right);} \\ {Q_{2} \left( W \right) \approx Q_{2} \left( V \right);} \\ {Q_{3} \left( W \right) \succ Q_{3} \left( V \right);} \\ {Q_{4} \left( W \right) \succ Q_{4} \left( V \right).} \\ \end{array}$$
    (4.3)

  • Highest-order statistical moments \(Q_{i = 3;4} \left( W \right)\), which characterize the skewness and sharpness of the peak of the histograms of the distribution of the values of the CDMA modulus of such objects, turned out to be the most sensitive in magnitude to the manifestations of birefringence of spatially ordered polycrystalline networks of skeletal muscle.

4.2.2 CDMA Mapping of Biological Layers with Disordered Architectonics

Figure 4.5 shows the results of an experimental study of the coordinate distributions of CDMA (right side) of a histological brain section (tissue with a birefringent network disordered in the directions of the optical axes), as well as histograms of the distribution of random values of the CDMA modulus, characterizing the coordinate consistency of optical anisotropy parameters in the plane of the biological layer (left side).

Fig. 4.5
figure 5

CDMA map (left side) and a histogram (right side) of the distribution of CDMA modulus values in a microscopic image of a histological section of the brain

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Analysis of the obtained data shows that the manifestations of the optical anisotropy of the polycrystalline network of the histological section of brain tissue significantly differ from the similar results of polarization-correlation microscopy of the histological section of the skeletal muscle (see Figs. 4.4 and 4.5).

From a physical point of view, the main factor in such changes is the difference in the spatial-geometric structure of fibrillar networks, which determines the specific distribution of the directions of the optical axes. As a result, a larger spectrum of such directions is formed for a histological section of brain tissue. This, in turn, leads to a wider range of changes in the values of the CDMA modulus in the plane of the biological layer. Based on this, the histograms of the distributions \(N\left( W \right)\) of random values of the CDMA modulus are less asymmetric with a greater dispersion of the dependence (see Fig. 4.4, the right side and Fig. 4.5, the right side).

Quantitatively, the results of a statistical analysis of the coordinate distributions of the values of the CDMA modulus of the brain tissue layer are given in Table 4.6.

Table 4.6 Statistical moments \(Q_{i = 1;2;3;4}\) that characterize the coordinate distribution of the values of the CDMA modulus of the microscopic image of a histological section of the brain
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A comparative analysis of the aggregate of statistical parameters characterizing the coordinate distribution of the values of the CDMA modulus of polycrystalline networks of biological layers of both types (\(W\) is the skeletal muscle, \(W^{ * }\) is the brain tissue) revealed the following relationships:

$$\begin{array}{*{20}c} {Q_{1} (W) \prec Q_{1} \left( {W^{*} } \right)} \\ {Q_{2} (W) \prec Q_{2} \left( {W^{*} } \right)} \\ {Q_{3} (W) \succ Q_{3} \left( {W^{*} } \right)} \\ {Q_{4} (W) \succ Q_{4} \left( {W^{*} } \right).} \\ \end{array}$$
(4.4)

The obtained results and certain regularities in the formation of CDMA cataracts of networks of spatially ordered and disordered along the directions of the optical axes networks of biological crystals were the basis for the development of a polarization-correlation microscopy of biological layers in differentiating changes in optical anisotropy caused by necrotic changes in the myocardium.

4.2.3 Diagnostic Features of CDMA-Mapping

This section contains the results of studies on the possibility of polarization-correlation differentiation of changes in birefringence of myosin myocardial networks due to necrotic changes in the myocardium.

Figure 4.6 shows the polarization-correlation CDMA maps \(W(m \times n)\) (left parts) and histograms \(N(W)\) (right parts) of the distributions of CDMA modulus values in the plane of histological sections of the myocardium that deceased due to mechanical asphyxiation (upper line) and heart attack (lower line).

Fig. 4.6
figure 6

CDMA maps (left parts) and histograms (right parts) of the distribution of CDMA modulus values in microscopic images of histological sections of the myocardium that deceased from mechanical asphyxiation (top line) and heart attack (bottom line)

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Quantitative differences between polarization-correlation maps characterize the values of statistical moments of the first–fourth orders, which are given in Table 4.7.

Table 4.7 Statistical moments \(Q_{i = 1;2;3;4}\) that characterize the distribution of the values of the CDMA modulus of histological sections of the myocardium that deceased due to mechanical asphyxiation (\(W\)) and heart attack (\(W^{ * }\))
Full size table

From the analysis of statistical moments of the first–fourth orders \(Q_{i = 1;2;3;4}\), which characterize the coordinate distributions of the values of the modulus of the polarization-correlation parameter CDMA of histological sections of the myocardium of both types, the following main differences are revealed.

First, for both causes of death, there is a significant difference between the values of all four statistical moments \(Q_{i = 1;2;3;4} \left( W \right)\) and \(Q_{i = 1;2;3;4} \left( {W^{ * } } \right)\).

Second—for necrotic changes caused by a heart attack, there is a decrease in average (\(Q_{1} \left( W \right) \downarrow\)) and dispersion \((Q_{2} \left( W \right) \downarrow )\).

Third, for the case of mechanical asphyxiation, the values of the statistical moments increase \(Q_{3} \left( {W^{ * } } \right) \uparrow\) and \(Q_{4} \left( {W^{ * } } \right) \uparrow\).

The revealed scenarios can be associated with a different effect of necrotic changes on the birefringence of myosin fibrillar networks. For the samples deceased due to a heart attack, there is a morphological destruction of the myocardial fibrillar network. Optically, this is reflected in a decrease in birefringence. As a result, the probability of formation of CDMA modulus values other than extreme decreases \(W = 0\). As a result, the dispersion decreases and, conversely, the average, skewness, and kurtosis values increase, characterizing the distribution of this correlation parameter of the histological section of the necrotic altered myocardium.

The expansion of the range of variation of random values of the CDMA modulus in the plane of the birefringent network of the histological section of the myocardium that deceased due to mechanical asphyxiation is associated with a high level of birefringence of the myosin network of this sample—\(Q_{1} \left( W \right) \downarrow ;Q_{2} \left( W \right) \uparrow ;Q_{3} \left( W \right) \downarrow ;Q_{4} \left( W \right) \downarrow\).

The results of the comparative analysis of the averaged (within the respective groups of samples) values of the statistical moments characterizing the two-dimensional distribution of the values of the CDMA modulus are given in Table 4.7.

The data in Table 4.7 allowed us to establish criteria for intergroup differences in the values of statistical moments of the first–fourth orders, which characterize the polarization-correlation CDMA maps \(W\left( {m \times n} \right)\) of histological sections of the myocardium of both types

$$W\left( {m \times n} \right) \Leftrightarrow \left\{ {\Delta M_{1} = 1,22;\Delta M_{2} = 1,26;\Delta M_{3} = 1,34;\Delta M_{4} = 1,36.} \right.$$

The results revealed the sensitivity of all four statistical moments \(Q_{i = 1;2;3;4}\) to necrotic changes in the optical anisotropy of polycrystalline myosin networks of the myocardium. At the same time, the highest level of balanced accuracy was achieved—Ac(W) = 80–87% which is higher (10–20%) than the accuracy of azimuthally invariant polarization and Mueller-matrix mapping using wavelet analysis and spatial-frequency filtering.