Optical Anisotropy of Biological Polycrystalline Networks: Vector-Parametric Diagnostics 9819910862, 9789819910861

Table of contents :
Acknowledgments
Introduction
References
Contents
1 Mueller-Matrix Modeling and Diagnostics of Optically Anisotropic Biological Layers
1.1 Basic Principles of Polarimetry of Biological Layers
1.2 Theoretical Models for Describing the Polarization-Optical Properties of Biological Tissues with Anisotropic Components
1.3 Mueller Matrixes of Optically Anisotropic Biological Layers and Rotational Invariants
1.4 Mueller-Matrix Model of Linear Birefringence of Fibrillar Networks of Biological Layers
1.5 Generalized Mueller-Matrix Model of Phase Anisotropy of the Biological Layer
References
2 Methods and Systems of Polarization Mueller-Matrix Microscopy of Biological Samples
2.1 Physical Substantiation and Selection of Research Objects
2.2 Methods of Manufacturing Prototypes of Biological Tissues and Fluids
2.3 Characterization of Research Objects
2.4 Optical Scheme of Experimental Research and Their Characteristics
2.4.1 Optical Scheme of Two-Dimensional Spectrally Selective Stokes Polarimetry and Its Characteristics
2.4.2 Optical Scheme of Two-Dimensional Fourier–Stokes Polarimetry and Its Characteristics
2.5 Methods of Analytical Analysis and the Totality of Its Objective Parameters
2.5.1 Statistical Analysis
2.5.2 Fractal Analysis
2.5.3 Information Analysis and Method Strength
References
3 Mueller-Matrix Description of the Optically Anisotropy of Biological Layers
3.1 The Main Types of Optical Anisotropy and Partial Matrix Operators for Its Description
3.2 Mueller-Matrix Approach to the Description of Polycrystalline Layers with Phase and Amplitude Anisotropy
3.3 Partial Mueller-Matrix Operators Describing the Mechanisms of Phase and Amplitude Anisotropy
3.4 Generalized Mueller Matrix of a Biological Layer with Phase and Amplitude Anisotropy
3.5 Mueller-Matrix Reconstruction or Reproduction of Optical Anisotropy Parameters
References
4 Azimuthally Invariant Mueller-Matrix Mapping of Optically Anisotropic Networks of Biological Tissues and Fluids
4.1 Mueller-Matrix Images of Optically Anisotropic Networks of Biological Tissues
4.2 Mueller-Matrix Invariants Characterizing the Optical Anisotropy of Histological Sections of Biological Tissues
4.3 Mueller-Matrix Differentiation of Fibrillar Networks of Biological Tissues with Different Phase and Amplitude Anisotropy
4.3.1 Differentiation of Linear and Circular Birefringence Parameters
4.3.2 Differentiation of Linear and Circular Dichroism Parameters
4.4 Mueller-Matrix Mapping of Blood-Filled Biological Tissues
References
5 Azimuthally Invariant Mueller-Matrix Reconstruction of the Optical Anisotropy Parameters of the Polycrystalline Structure of Biological Tissues and Human Fluids
5.1 Mueller-Matrix Reconstruction of the Distribution of Parameter Values Characterizing Birefringence and Dichroism of Optically Anisotropic Networks of Biological Tissues in a Precancerous State
5.2 “Two-Wave” Mueller-Matrix Reconstruction of the Distribution of Optical Anisotropy Parameter Values of Polycrystalline Films of Biological Fluids (Methodological Justification)
5.3 “Two-Wave” Mueller-Matrix Reconstruction of the Distribution of Optical Anisotropy Parameter Values of Polycrystalline Films of Biological Fluids (Experimental Results)
5.3.1 Determination of a Statistically Significant Representative Sample of Patients with Known (Reference) Diagnosis
5.3.2 Checking the “Stability” of Polarization Reconstruction Algorithms
5.3.3 Polarization Reconstruction of the Optical Anisotropy Parameters of Plasma Films Blood
5.3.4 Polarization Reconstruction of the Optical Anisotropy Parameters of Bile Films of Man
References
6 Methods and Means of Fourier-Stokes Polarimetry and Spatial-Frequency Filtering of Phase Anisotropy Manifestations
6.1 Fourier-Stokes Polarimetry Manifestations of Linear Birefringence Mechanisms of Structured Fibrillar Networks of Histological Sections of Biological Tissues
6.1.1 Justification and Relevance of the Method
6.2 Theoretical Basis of the Method
6.3 Spatial-Frequency Fourier-Stokes Polarimetry of the Manifestations of Linear Birefringence of Histological Sections of Biological Tissues
6.4 Spatial-Frequency Fourier-Stokes Polarimetry of Birefringence Manifestations of Small-Scale Fibrillar Networks of Biological Tissues
6.5 Fourier-Mueller-Matrix Mapping of Parameter Distributions Characterizing the Phase Anisotropy of Histological Sections of Biological Tissues in a Precancerous State
6.5.1 Spatial-Frequency Fourier-Stokes Polarimetry of Linear Birefringence Endometrium
6.5.2 Spatial-Frequency Fourier-Stokes Polarimetry of Circular Birefringence of Endometrium
References
Conclusions

Citation preview

SpringerBriefs in Applied Sciences and Technology Lilia Trifonyuk · Iryna V. Soltys · Alexander G. Ushenko · Yuriy A. Ushenko · Alexander V. Dubolazov · Jun Zheng

Optical Anisotropy of Biological Polycrystalline Networks Vector-Parametric Diagnostics

SpringerBriefs in Applied Sciences and Technology

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Lilia Trifonyuk · Iryna V. Soltys · Alexander G. Ushenko · Yuriy A. Ushenko · Alexander V. Dubolazov · Jun Zheng

Optical Anisotropy of Biological Polycrystalline Networks Vector-Parametric Diagnostics

Lilia Trifonyuk Rivne Regional Hospital Rivne Oncological Center, Institute of Health University of Water Management and Environmental Engineering Rivne, Ukraine Alexander G. Ushenko Department of Optics and Publishing Chernivtsi National University Chernivtsi, Ukraine Alexander V. Dubolazov Department of Optics and Publishing Chernivtsi National University Chernivtsi, Ukraine

Iryna V. Soltys Department of Optics and Publishing Chernivtsi National University Chernivtsi, Ukraine Yuriy A. Ushenko Computer Science Department Chernivtsi National University Chernivtsi, Ukraine Jun Zheng Research Institute of Zhejiang University Taizhou, China

ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs in Applied Sciences and Technology ISBN 978-981-99-1086-1 ISBN 978-981-99-1087-8 (eBook) https://doi.org/10.1007/978-981-99-1087-8 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Acknowledgments

This work received funding from: National Research Foundation of Ukraine, Grant 2020.02/0061 and Scholarship of the Supreme Council for Young Scientists— Doctors of Sciences.

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Relevance of the Topic. Biological tissues are structurally heterogeneous media that can absorb optical radiation [1–10]. In order to describe the interaction of laser radiation with such complex systems, it is necessary to use an approach that uses the formalism of Mueller matrices and of information analysis [11, 14–16, 22–26, 30–42, 44, 49–51, 53, 54]. Currently, biological and medical research uses many practical methods based on the measurement and analysis of Mueller matrices of prototypes. In recent years, biomedical optics has formed an independent direction– laser polarimetry [12–14]. Within the framework of this research area, it was possible to establish the relationship between the coordinate distributions of the values of the matrix elements (Mueller-matrix images) and the corresponding distributions of the values of the birefringence of polycrystalline networks of optically thin layers of human biological tissues. On this basis, changes in the optically anisotropic structure of biological tissues (skin dermis, epithelial tissue, etc.) are differentiated, which are caused by oncological conditions of human organs [17–20, 27–29, 31, 36, 38, 39, 43–48, 50, 57]. At the same time, laser polarimetry methods require further development and deepening. First, not all elements of the Mueller matrix are convenient for characterizing biological samples. The reason for this is the azimuthal dependence of the majority of the matrix elements—in general, 12 out of 16 elements change as the sample rotates around the sounding axis [21, 22]. In addition, there are a number of azimuthally independent combinations of elements of the Mueller matrix or Mueller-matrix invariants (MMIs). Secondly, the mechanisms of optical anisotropy of biological layers are not limited to manifestations of birefringence of spatially structured fibrillar networks. Actual on the way to expanding the arsenal of diagnostic techniques is to take into account the influence of the mechanisms of amplitude anisotropy—linear and circular dichroism [6, 8, 10]. Thirdly, there is a wide range of optically anisotropic biological objects for which laser polarimetry methods are not yet widely used. Such objects include biological fluids (blood and its plasma, urine, bile, saliva, etc.), which are easily accessible and do not require for obtain a sample of a traumatic operation biopsy. Fourth, the manifestations of

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these mechanisms of optical anisotropy manifest themselves differently on different scales of the geometric dimensions of polycrystalline structures of biological layers [52–57]. Consequently, the relevance of the monograph is due to the need to develop new, information-complete and experimentally reproducible approaches to the analysis of optical anisotropy of biological tissues and fluids, to search for new azimuthally independent methods of Stokes polarimetry using algorithms of polarization reconstruction and of spatial-frequency filtering of object fields, which allows us to separate the manifestations of different mechanisms of phase and amplitude anisotropy of multiscale polycrystalline networks of biological layers in the development of objective criteria for assessing the degree of pathology and differentiation of the research samples. The purpose of the monograph is to develop new azimuthally independent methods of Stokes polarimetry and Mueller-matrix reconstruction of the distributions of optical anisotropy parameters using spatial-frequency filtering of phase (linear and circular birefringence) and amplitude (linear and circular dichroism) anisotropy to diagnose changes in the orientation-phase structure of fibrillar networks of histological sections of biological tissues and polycrystalline films of biological fluids. V. A. Ushenko Iryna V. Soltys

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Contents

1 Mueller-Matrix Modeling and Diagnostics of Optically Anisotropic Biological Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Basic Principles of Polarimetry of Biological Layers . . . . . . . . . . . . . . 1.2 Theoretical Models for Describing the Polarization-Optical Properties of Biological Tissues with Anisotropic Components . . . . . 1.3 Mueller Matrixes of Optically Anisotropic Biological Layers and Rotational Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Mueller-Matrix Model of Linear Birefringence of Fibrillar Networks of Biological Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Generalized Mueller-Matrix Model of Phase Anisotropy of the Biological Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Methods and Systems of Polarization Mueller-Matrix Microscopy of Biological Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Physical Substantiation and Selection of Research Objects . . . . . . . . 2.2 Methods of Manufacturing Prototypes of Biological Tissues and Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Characterization of Research Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Optical Scheme of Experimental Research and Their Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Optical Scheme of Two-Dimensional Spectrally Selective Stokes Polarimetry and Its Characteristics . . . . . . . . 2.4.2 Optical Scheme of Two-Dimensional Fourier–Stokes Polarimetry and Its Characteristics . . . . . . . . . . . . . . . . . . . . . . . 2.5 Methods of Analytical Analysis and the Totality of Its Objective Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Fractal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Information Analysis and Method Strength . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Mueller-Matrix Description of the Optically Anisotropy of Biological Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Main Types of Optical Anisotropy and Partial Matrix Operators for Its Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Mueller-Matrix Approach to the Description of Polycrystalline Layers with Phase and Amplitude Anisotropy . . . . . . . . . . . . . . . . . . . 3.3 Partial Mueller-Matrix Operators Describing the Mechanisms of Phase and Amplitude Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Generalized Mueller Matrix of a Biological Layer with Phase and Amplitude Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Mueller-Matrix Reconstruction or Reproduction of Optical Anisotropy Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Azimuthally Invariant Mueller-Matrix Mapping of Optically Anisotropic Networks of Biological Tissues and Fluids . . . . . . . . . . . . . . 4.1 Mueller-Matrix Images of Optically Anisotropic Networks of Biological Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Mueller-Matrix Invariants Characterizing the Optical Anisotropy of Histological Sections of Biological Tissues . . . . . . . . . 4.3 Mueller-Matrix Differentiation of Fibrillar Networks of Biological Tissues with Different Phase and Amplitude Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Differentiation of Linear and Circular Birefringence Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Differentiation of Linear and Circular Dichroism Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Mueller-Matrix Mapping of Blood-Filled Biological Tissues . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Azimuthally Invariant Mueller-Matrix Reconstruction of the Optical Anisotropy Parameters of the Polycrystalline Structure of Biological Tissues and Human Fluids . . . . . . . . . . . . . . . . . . 5.1 Mueller-Matrix Reconstruction of the Distribution of Parameter Values Characterizing Birefringence and Dichroism of Optically Anisotropic Networks of Biological Tissues in a Precancerous State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 “Two-Wave” Mueller-Matrix Reconstruction of the Distribution of Optical Anisotropy Parameter Values of Polycrystalline Films of Biological Fluids (Methodological Justification) . . . . . . . . . 5.3 “Two-Wave” Mueller-Matrix Reconstruction of the Distribution of Optical Anisotropy Parameter Values of Polycrystalline Films of Biological Fluids (Experimental Results) . . . . . . . . . . . . . . . . 5.3.1 Determination of a Statistically Significant Representative Sample of Patients with Known (Reference) Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.3.2 Checking the “Stability” of Polarization Reconstruction Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Polarization Reconstruction of the Optical Anisotropy Parameters of Plasma Films Blood . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Polarization Reconstruction of the Optical Anisotropy Parameters of Bile Films of Man . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Methods and Means of Fourier-Stokes Polarimetry and Spatial-Frequency Filtering of Phase Anisotropy Manifestations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Fourier-Stokes Polarimetry Manifestations of Linear Birefringence Mechanisms of Structured Fibrillar Networks of Histological Sections of Biological Tissues . . . . . . . . . . . . . . . . . . . 6.1.1 Justification and Relevance of the Method . . . . . . . . . . . . . . . . 6.2 Theoretical Basis of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Spatial-Frequency Fourier-Stokes Polarimetry of the Manifestations of Linear Birefringence of Histological Sections of Biological Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Spatial-Frequency Fourier-Stokes Polarimetry of Birefringence Manifestations of Small-Scale Fibrillar Networks of Biological Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Fourier-Mueller-Matrix Mapping of Parameter Distributions Characterizing the Phase Anisotropy of Histological Sections of Biological Tissues in a Precancerous State . . . . . . . . . . . . . . . . . . . . 6.5.1 Spatial-Frequency Fourier-Stokes Polarimetry of Linear Birefringence Endometrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Spatial-Frequency Fourier-Stokes Polarimetry of Circular Birefringence of Endometrium . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Chapter 1

Mueller-Matrix Modeling and Diagnostics of Optically Anisotropic Biological Layers

1.1 Basic Principles of Polarimetry of Biological Layers Polarized radiation plays an important role in our understanding of the nature of electromagnetic waves, elucidating the three-dimensional characteristics of chemical bonds, revealing the nature of asymmetric (choral) biological molecules, determining the concentration of sugar in industrial technologies and quantifying the properties of proteins in solutions, supporting various non-invasive and non-destructive diagnostic methods, developing fundamental concepts such as polarization entropy, creating sensitive sensors in meteorology and astronomy, differentiation of normal and precancerous cells in the layer of biological tissue, as well as in other biomedical applications [1–10]. Traditional polarimetry is well suited for use in optically homogeneous media, as well as for studying rough surfaces. However, the presence of light scattering in the thickness of biological tissues leads to a statistical change in the type and form of polarization of radiation. As a result, a polarized inhomogeneous field of scattered radiation is formed, which is statistically estimated by the degree of depolarization. In the future, the problem arises of measuring such polarization signals in both the statistical and local approximations. Multiple scattering also leads to local changes in polarization states. In particular, due to such processes, rotations of the plane of the linear polarization vector are formed [11–14]. This example combines well with other polarization effects, which is realized due to linear birefringence of muscle tissue fibers or other spatially structured protein fibrils and optical rotation of the plane of polarization by choral molecules and structures. So, solving the inverse problem (determining the optical properties of biological tissue) is an extremely complex, multiparameter (linear birefringence, optical activity, multiple interaction) problem [15]. A separate area in biomedical polarimetry was the use of optical images based on polarization filtering to separate the multiply scattered (depolarized) components of the light beam. This made it possible to significantly increase contrast and deepen © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 L. Trifonyuk et al., Optical Anisotropy of Biological Polycrystalline Networks, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-99-1087-8_1

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the resolution of images of biological tissues. This direction of fine polarimetry is generalized based on the use of various polarization states (linear, circular) for irradiation of biological layers [16–19]. As a result, a separate direction was created—fine Stokes polarimetry. Such optical systems use a description of biological tissue imaging processes within two approximations: • Jones matrix which uses the amplitude-phase approach to analyze scattering processes [1, 3, 16] • Mueller matrix which uses the Stokes vector formalism as an additive addition of intensities of differently polarized beams [2, 4, 8, 20, 21]. To measure the coordinate distributions of 16 elements of the Mueller matrix, various Stokes polarimeters have been developed, the optical schemes of which are described previously. Despite the large number of different experimental schemes, the measurement algorithm for most of them remains classical: • The sample is irradiated sequentially with four beams with different types of polarization—0°, 90°, 45° and right circulation. • For each of the beams, the Stokes vector transformed by the beam object is measured. • Based on the data obtained, 16 elements of the Mueller matrix are calculated. In our work, experimental studies mainly concern static biological layers; therefore, as an experimental basis, we chose the classical Mueller-matrix techniques. So, for the development of Mueller-matrix mapping in the sense of solving inverse problems, the necessary condition is the development of adequate models for describing the optical anisotropy of biological media. In this regard, we consider the main types of such models, information about which exist in the modern scientific literature.

1.2 Theoretical Models for Describing the Polarization-Optical Properties of Biological Tissues with Anisotropic Components Biological tissues are optically heterogeneous absorption media [7]. The propagation of light in such media depends on the scattering and absorbing properties of the components of the biological tissue [10]. In particular, such parameters as the particle size, their shape, their packing density, the properties surrounding the scattering particle of the basic substance, in one way or another affect the characteristics of the scattered radiation [3, 4, 6]. Therefore, optical methods for diagnosing biological tissues and visualizing their structure occupy one of the leading places due to their high information content. Equally important is the ability to carry out multifunctional monitoring of the studied environment, as well as their relative simplicity and low

1.2 Theoretical Models for Describing the Polarization-Optical Properties …

3

cost. Among them, polarization diagnostic methods have certain promising in biology and medicine. Interest in such methods is primarily due to the high sensitivity of the polarization characteristics of scattered optical fields to the optical properties and geometry of scattering media. An analysis of the polarization characteristics of scattered biological tissue radiation in some cases allows to obtain qualitatively new results in studies of the morphological and functional state of biological tissues, which is one of the most important areas of modern medical diagnostics [22–24]. The possibilities of polarizing diagnostics of biological structures were demonstrated in the works on the early diagnosis of the lens cataract and the assessment of glucose concentration in the tissues of patients with diabetes. Polarization measurements provide information on cell structures or type of biological tissue. In addition, when using probing linearly polarized radiation, the image quality of macroscopic inhomogeneities in a scattering medium is improved [16]. Optical methods combining spectral and polarization analysis of the interaction of light with biological tissues are constantly developing and finding wider application in biology and medicine. In particular, the possibility of visualizing structural heterogeneities of biological tissues at different depths, from tens of micrometers to several centimeters, was demonstrated previously [1, 2]. An important aspect in the development of methods for optical diagnostics and visualization of spatially inhomogeneous scattering media using polarized probe radiation is the analysis of the transformation of the state of polarization of light when it is scattered by the medium. The scattering of probe polarized radiation propagating in biological tissue as a randomly inhomogeneous medium leads to significant changes in its polarization state [20, 25]. Most biological tissues have optical anisotropy at both the microscopic and macroscopic levels [15–17]. For macroscopically homogeneous and isotropic biological tissues, fluctuations in local birefringence at the microscopic (cellular) level lead to small-scale polarization modulation of radiation, which propagates. Optical anisotropy at the macroscopic level is caused, in particular, by the orientationally ordered fibrillar structure of the tissue, which leads to a transformation of the type of polarization of the polarized radiation component. The ability of biological tissues to birefringent light is clearly visible when observing sections and tears of tissue in polarizing microscopy [1, 5]. For example, for skin tissue, birefringence can be observed in the stratum corneum of the epidermis and dermis. Birefringence is controlled by the main cellular components of the stratum corneum of the epidermis—keratinocytes. They have a sufficiently large difference in the main refractive indices Δn ≈ 0.0022. The birefringence of the dermis is mainly due to the optical anisotropy of collagen fibers and their orientational ordering at the macroscopic level. The bundles of collagen fibers in the dermis lie parallel to the surface of the skin and, being oriented at small angles relative to a certain selected direction, intersect each other, forming a multilayer network with rhombic cells [26]. The existence of a distinguished direction of the predominant orientation of collagen fibers determines, in particular, the anisotropy of skin stretch, which is well known in

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medicine and is described using the so-called Langer lines (skin stretch lines)—the direction of these lines is taken into account in surgical skin incisions. We recalled only two of the most accessible types of biological tissues for noninvasive optical diagnostics, which exhibit birefringence (muscle tissue, bone tissue, cornea tissue, tendons, cartilage, scleral tissue, dura mater, etc.) [17]. The presence of optical anisotropy in combination with structural heterogeneity makes analysis and the phenomenological description of the optical properties of biological tissue rather complicated. However, manifesting itself in the polarization characteristics of the biological object under study, optical anisotropy can provide valuable information about its structural features and physiological status. More general approximations, such as the Mueller–Stokes formalism, are needed to describe the interaction of polarized light with complex systems such as biological tissues [8–10].

1.3 Mueller Matrixes of Optically Anisotropic Biological Layers and Rotational Invariants The Mueller–Stokes formalism is based on the representation of the state of polarization of the light wave by the Stokes vector and the representation of the depolarizing optical system by the Mueller matrix 4 × 4 [1, 15]. Stokes vector components ⎞ S0 ⎜ S1 ⎟ ⎟ S=⎜ ⎝ S2 ⎠ S3 ⎛

(1.1)

are defined as: ) ( ) ( S0 = I0 = Ix + I y = I+π/4 + I−π/4 = Il + Ir S1 = Ix − I y S2 = I+π/4 − I−π/4 S3 = Ir − Il

(1.2)

where I0 —total light wave intensity, Ix , I y , I+π/4 , I−π/4 , Il and Ir —the intensity of a light wave passing through an ideal polarizer located in the path of this wave and transmitting linearly polarized light in the directions x, y, +π/4 and −π/4, as well as left and right-circularly polarized light, respectively. The result of the interaction of the light wave with the optical system can be calculated by multiplying the Stokes vector of the incident wave on the left by the

1.3 Mueller Matrixes of Optically Anisotropic Biological Layers …

5

Mueller matrix of the optical system considered. As a result of this, we get the Stokes vector of the original wave: ⎛

⎞ ⎛ S0 m 11 ⎜ S1 ⎟ ⎜ m 21 ⎜ ⎟ =⎜ ⎝ S2 ⎠ ⎝ m 31 S3 0 m 41

m 12 m 22 m 32 m 42

m 13 m 23 m 33 m 43

⎞⎛ ⎞ S0 m 14 ⎜ ⎟ m 24 ⎟⎜ S1 ⎟ ⎟ m 34 ⎠⎝ S2 ⎠ m 44 S3 I

(1.3)

This relation represents the basic law of the Stokes vector transformation of a partially polarized light wave propagating through an optical system. Now in biological and medical research, many practical methods are used, based on the measurement and analysis of the Mueller matrices of the samples. Several methods of polarizing mapping of biological tissues are known, based on an analysis of the spatial dependences of the elements of the Mueller matrices for transmitting and reflecting a sample [22–24, 27–31]. The elements of the Mueller matrix, which contain comprehensive information about the interaction of light and tissue, are nevertheless very convenient for characterizing the sample itself. The reason for this is the dependence of the elements of this matrix on the choice of the coordinate system: In general, 12 out of 16 elements change when the sample rotates around the sounding axis. In addition, the relationship between the optical properties of the object of study, represented by a set of elements of the Mueller matrix, and its structural characteristics is far from obvious: The necessary diagnostic information is “encrypted” in the Mueller matrix [21, 32]. One of the practically used methods of “decryption” is the method of polar decomposition of Mueller matrices. In this method, the object under study is formally represented as a set of three idealized elements (polarizer, phase plate and depolarizer), and the measured Mueller matrix is represented as the product of the Mueller matrices of these elements. In the geometry of detection of light passing when the sample rotates around the propagation direction of the probe beam, such elements and combinations of elements of the transmission Mueller matrix remain constant [32–37]: m 11 , m 14 , m 41 , m 44 , m 22 + m 33 , m 23 − m 32 ,

(1.4)

and also preserves the length of the vectors ( aH =

) ) ) ) ( ) ( ( ( m 22 − m 33 m 12 m 21 m 42 m 24 , aV = , bH = , bV = ,g = , m 13 m 31 m 43 m 34 m 23 + m 32 (1.5)

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1.4 Mueller-Matrix Model of Linear Birefringence of Fibrillar Networks of Biological Layers The main result of this approach was the formation of an original method of opticalphysical diagnostics—laser polarimetry of birefringence of fibrillar networks of histological sections of the main types of human biological tissues. Methodologically, this direction of diagnosis is based on the approximation according to which the biological layer consists of two components—an isotropic and linearly birefringent network created by protein filaments or fibrils [11–14, 18, 19, 33, 34, 38–42]. As a result of such a model approach to histological sections of various types of biological tissues, the relationship between the distributions of the polarization states of the points of laser images and the distributions of the directions of the optical axes and phase shifts was determined. An important methodological basis for the further development of the understanding of the optical anisotropy of biological layers is the cycle of work of the Savenkov group, which are systematically described previously. A new generalized approach to the analysis of the manifestations of mechanisms of not only phase, but also amplitude optical anisotropy is proposed. In accordance with this approach, any object can be represented as a sequence of layers with a certain type of anisotropy: • • • •

circular birefringence linear birefringence circular dichroism linear dichroism.

Proceeding from this, the spread of such an ideology to complex, structurally heterogeneous biological layers in different parts of the spectrum, where various mechanisms of optical anisotropy most clearly manifests, is relevant. For experimental studies of the coordinate distributions of the manifestations of various mechanisms of phase anisotropy in the red region of the optical radiation spectrum, the Stokes polarimetric method was developed previously, which is more informatively meaningful in the sense of mapping optically anisotropic structures. This approach is unified both for an optically thin layer of biological tissue and the situation of multiple scattering of laser radiation by multilayer networks of biological crystals. The analysis of laser radiation scattering by multilayer biological networks was systematically carried out previously. The optical properties of each partial layer of biological tissue (Fig. 1.1) were characterized by the Mueller matrix for an optically uniaxial birefringent crystal [36, 37].

1.4 Mueller-Matrix Model of Linear Birefringence of Fibrillar Networks …

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Fig. 1.1 On the analysis of the optical properties of a bilayer (A–B) anisotropic (birefringence indices Δn 1 , Δn 2 ) architectonic structure of virtual biological tissue. Here E ox , E oy ; E x(1) , E y(1) ; E x(2) , E y(2) —orthogonal components of the amplitude, respectively: probing, converted by layer A and converted by layer B of a laser wave with polarization states α0 , β0 ; α1 , β1 and α2 , β2

∥ ∥ ∥ 1; ∥ 0 0 0 ∥ ( ∥ ( ) 2 ρ ∥ 0; sin2 δ cos 2ρ + cos2 δ ); ∥ 0.5 sin 4ρ sin ; sin 2ρ sin δ); (− ∥ ∥ 2 )2 ( ) ( 2 {F} = ∥ , 2 ρ 2 δ 2 δ 0.5 sin 4ρ sin 2 ; −→ − sin 2 cos 2ρ + cos 2 ; (cos 2ρ sin δ); ∥ ∥ 0; ∥ ← ∥ ∥ ) ( ∥ 0; 2 cos2 2δ − 1 ; ∥ (sin 2ρ sin δ); (cos 2ρ sin δ); (1.6) where ρ—orientation angle of the optical axis of the biological crystal; δ—the magnitude of the phase shift. The analysis of the results of the study of the coordinate distributions of the Stokes vector parameters of polarization-inhomogeneous images of networks of crystal cylinders within the statistical, correlation and fractal approaches revealed interconnections between the distributions of the directions of the optical axes ρ(x, y) and phase shifts δ(x, y) that are introduced by the networks of virtual birefringent cylinders between the orthogonal components of the amplitude of the laser wave and the laws of variation of the statistical moments of the first fourth orders that characterize the distribution S i = 2,3,4 (x, y).

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1.5 Generalized Mueller-Matrix Model of Phase Anisotropy of the Biological Layer The basis for further diagnostic use of laser polarimetry techniques is a common unified model that describes the polarization properties of biological layers. The main postulate of this modeling is the consideration of any biological tissue in the form of a structure formed by two main components [43–58]. The first is optically isotropic, which does not change the polarization state of the laser wave, but only weakens it (Fig. 1.2 ({A})). The second component is optically anisotropic, the effect of which is manifested in the transformation of the polarization state of the probe laser beam due to phase modulation between the orthogonal components of the amplitude of the laser wave. Two main mechanisms of such modulation are distinguished—linear (Fig. 1.2 ({D})) and circular (Fig. 1.2 ({C})) birefringence. The application of this model approach provided the opportunity to interpret large amounts of experimental information on the image of various biological tissues from a single analytical position. Due to this, objective criteria have been formed for the diagnosis of pathological changes in biological tissues, fluid films and the classification of their optical properties:

Fig. 1.2 To the analysis of model representations. E 0 , α0 —amplitude and azimuth of polarization of the probe beam; E x , E y , α, β, φ—orthogonal components of the amplitude, azimuth, ellipticity and phase of the object beam E; {A}; {D}; {C}—Johns matrices of isotropic, linearly and circularly birefringent components of the biological layer

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• The relationship between the changes in the birefringence of fibrillar biological crystallites and the distribution of the values of the Mueller-matrix images of histological sections of biological tissues was revealed. On this basis, the effectiveness of a statistical analysis of the distribution of the values of the phase Mueller-matrix image of a birefringent protein network in differentiating optically thin histological sections of a benign and malignant uterine wall biopsy was demonstrated for the first time. • The Fourier–Stokes method of polarimetry of the spatial frequency spectra of laser images was developed, based on cross-correlation analysis of polarization maps with the determination of a set of statistical moments of the 1st–4th orders characterizing the distribution of orthogonal (in two mutually perpendicular scanning directions) autocorrelation functions and logarithmic dependences of power spectra of coordinate distributions of azimuth and polarization ellipticity. Based on this, the differentiation of pathological conditions of biological tissues (histological sections of a biopsy of a benign and malignant tumor of the rectum) in the Fourier plane. On the other hand, the analysis of polarization-inhomogeneous images within the framework of laser polarimetry representations requires further deepening and expansion of experimental capabilities in the direction of systemic and Muellermatrix differentiation of differentiation of optical anisotropy of biological layers. Thus, it can be stated that Mueller-matrix diagnostics requires the further development of new, information-complete and experimentally reproducible vectorparametric approaches to the analysis of optical anisotropy of biological tissues and fluids, the search for new azimuthally stable methods of Stokes polarimetry using polarization reconstruction algorithms and consistent spatial-frequency filtration of manifestations of various mechanisms of phase and amplitude anisotropy of biological layers for development objective criteria for assessing the degree of differentiation and severity of pathology.

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26. W. Bickel, W. Bailey, Stokes vectors, Mueller matrices, and polarized scattered light. Am. J. Phys. 53(5), 468–478 (1985) 27. A. Ushenko, A. Dubolazov, V. Ushenko, O. Novakovskaya, Statistical analysis of polarizationinhomogeneous Fourier spectra of laser radiation scattered by human skin in the tasks of differentiation of benign and malignant formations. J. Biomed. Opt. 21(7), 071110 (2016) 28. V.P. Ungurian, O.I. Ivashchuk, V.O. Ushenko, Statistical analysis of polarizing maps of blood plasma laser images for the diagnostics of malignant formations. Proc. SPIE 8338, 83381L (2011) 29. V.P. Prysyazhnyuk, Y.A. Ushenko, A.V. Dubolazov, A.G. Ushenko, V.A. Ushenko, Polarization-dependent laser autofluorescence of the polycrystalline networks of blood plasma films in the task of liver pathology differentiation. Appl. Opt. 55, B126–B132 (2016) 30. V.A. Ushenko, M.P. Gorsky, Complex degree of mutual anisotropy of linear birefringence and optical activity of biological tissues in diagnostics of prostate cancer. Opt. Spectrosc. 115, 290–297 (2013) 31. Y.A. Ushenko et al., Spatial-frequency Fourier polarimetry of the complex degree of mutual anisotropy of linear and circular birefringence in the diagnostics of oncological changes in morphological structure of biological tissues. Quantum. Electron. 42, 727–732 (2012) 32. V.A. Ushenko, N.D. Pavlyukovich, L. Trifonyuk, Spatial-frequency azimuthally stable cartography of biological polycrystalline networks. Int. J. Opt. 683174, 1–7 (2013) 33. V.A. Ushenko, M.S. Gavrylyak, Azimuthally invariant Mueller-matrix mapping of biological tissue in differential diagnosis of mechanisms protein molecules networks anisotropy. Proc. SPIE 8812, 88120Y (2013) 34. V.A. Ushenko, A.V. Dubolazov, Correlation and self similarity structure of polycrystalline network biological layers Mueller matrices images. Proc. SPIE 8856, 88562D (2013) 35. V.A. Ushenko, Complex degree of mutual coherence of biological liquids. Proc. SPIE 8882, 88820V (2013) 36. V.A. Ushenko, A.V. Dubolazov, L.Y. Pidkamin, M.Y. Sakchnovsky, A.B. Bodnar, Y.A. Ushenko, A.G. Ushenko, A. Bykov, I. Meglinski, Mapping of polycrystalline films of biological fluids utilizing the Jones-matrix formalism. Laser Phys. 28(2), 025602 (2019) 37. O. Ushenko, V. Zhytaryuk, V. Dvorjak, I.V. Martsenyak, O. Dubolazov, B.G. Bodnar, O.Y. Vanchulyak, S. Foglinskiy, Multifunctional polarization mapping system of networks of biological crystals in the diagnostics of pathological and necrotic changes of human organs. Proc. SPIE—The Int. Soc. Opt. Eng. 11087, 110870S (2019) 38. V. Devlaminck, Physical model of differential Mueller matrix for depolarizing uniform media. J. Opt. Soc. Am. A 30(11), 2196 (2013) 39. V. Ushenko, A. Sdobnov, A. Syvokorovskaya, A. Dubolazov, O. Vanchulyak, A. Ushenko, Y. Ushenko, M. Gorsky, M. Sidor, A. Bykov, I. Meglinski, 3D Mueller-matrix diffusive tomography of polycrystalline blood films for cancer diagnosis. Photonics 5(4), 54 (2018) 40. L. Trifonyuk, W. Baranowski, V. Ushenko, O. Olar, A. Dubolazov, Y. Ushenko, B. Bodnar, O. Vanchulyak, L. Kushnerik, M. Sakhnovskiy, 2D-Mueller-matrix tomography of optically anisotropic polycrystalline networks of biological tissues histological sections. Opto-Electron. Rev. 26(3), 252–259 (2018) 41. V.A. Ushenko, A.Y. Sdobnov, W.D. Mishalov, A.V. Dubolazov, O.V. Olar, V.T. Bachinskyi, A.G. Ushenko, Y.A. Ushenko, O.Y. Wanchuliak, I. Meglinski, Biomedical applications of Jones-matrix tomography to polycrystalline films of biological fluids. J. Inno. Opt. Health Sci. 12(6), 1950017 (2019) 42. M. Borovkova, L. Trifonyuk, V. Ushenko, O. Dubolazov, O. Vanchulyak, G. Bodnar, Y. Ushenko, O. Olar, O. Ushenko, M. Sakhnovskiy, A. Bykov, I. Meglinski, Mueller-matrixbased polarization imaging and quantitative assessment of optically anisotropic polycrystalline networks. PLoS ONE 14(5), e0214494 (2014) 43. V.O. Ushenko, Spatial-frequency polarization phasometry of biological polycrystalline networks. Opt. Mem. Neur. Netw. 22, 56–64 (2013)

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44. Dubolazov, A.V., Olar, O.V., Pidkamin, L.Y., Arkhelyuk, A.D., Motrich, A.V., Shaplavskiy, M.V., Bodnar, B.G., Sarkisova, Y., Penteleichuk, N. Polarization-phase reconstruction of polycrystalline structure of biological tissues. Proc. SPIE—The Int. Soc. Opt. Eng. 11087, 1108714 (2019) 45. Dubolazov, A.V., Olar, O.V., Pidkamin, L.Y., Arkhelyuk, A.D., Motrich, A.V., Petrochak, O., Bachynskiy, V.T., Litvinenko, O., Foglinskiy, S. Methods and systems of diffuse tomography of optical anisotropy of biological layers. Proc. SPIE—The Int. Soc. Opt. Eng. 11087, 110870 (2019) 46. Y.A. Ushenko et al., Jones-matrix mapping of complex degree of mutual anisotropy of birefringent protein networks during the differentiation of myocardium necrotic changes. Appl. Opt. 55, B113–B119 (2016) 47. L. Cassidy, Basic concepts of statistical analysis for surgical research. J. Surg. Res. 128(2), 199–206 (2005) 48. C.S. Davis, Statistical Methods of the Analysis of Repeated Measurements (Springer-Verlag, New York, 2002) 49. A. Petrie, C. Sabin, Medical Statistics at a Glance (Wiley-Blackwell, Chichester, UK, 2009) 50. A.G. Ushenko, A.V. Dubolazov, V.A. Ushenko, O.Y. Novakovskaya, Statistical analysis of polarization-inhomogeneous Fourier spectra of laser radiation scattered by human skin in the tasks of differentiation of benign and malignant formations. J. Biomed. Opt. 21(7), 071110 (2016) 51. A.G. Ushenko, O.V. Dubolazov, V.A. Ushenko, O.Yu. Novakovskaya, O.V. Olar, Fourier polarimetry of human skin in the tasks of differentiation of benign and malignant formations. Appl. Opt. 55(12), B56–B60 (2016) 52. Yu.A. Ushenko, V.T. Bachynsky, O.Ya. Vanchulyak, A.V. Dubolazov, M.S. Garazdyuk, V.A. Ushenko, Jones-matrix mapping of complex degree of mutual anisotropy of birefringent protein networks during the differentiation of myocardium necrotic changes. Appl. Opt. 55(12), B113– B119 (2016) 53. A.V. Dubolazov, N.V. Pashkovskaya, Yu.A. Ushenko, Yu.F. Marchuk, V.A. Ushenko, O.Yu. Novakovskaya, Birefringence images of polycrystalline films of human urine in early diagnostics of kidney pathology. Appl. Opt. 55(12), B85–B90 (2016) 54. M.S. Garazdyuk, V.T. Bachinskyi, O.Ya. Vanchulyak, A.G. Ushenko, O.V. Dubolazov, M.P. Gorsky, Polarization-phase images of liquor polycrystalline films in determining time of death. Appl. Opt. 55(12), B67–B71 (2016) 55. A. Ushenko, A. Sdobnov, A. Dubolazov, M. Grytsiuk, Y. Ushenko, A. Bykov, I. Meglinski, Stokes-correlometry analysis of biological tissues with polycrystalline structure. IEEE J. Sel. Top. Quantum Electron. 25(1), 8438957 (2019) 56. O. Vanchulyak, O. Ushenko, V. Zhytaryuk, V. Dvorjak, O. Pavlyukovich, O. Dubolazov, N. Pavlyukovich, N.P. Penteleichuk, Stokes-correlometry of polycrystalline films of biological fluids in the early diagnostics of system pathologies. Proc. SPIE—The Int. Soc. Opt. Eng. 11105, 1110519 (2019) 57. A.V. Dubolazov, O.V. Olar, L.Y. Pidkamin, A.D. Arkhelyuk, A.V. Motrich, V.T. Bachinskiy, O.V. Pavliukovich, N. Pavliukovich, Differential components of Muller matrix partially depolarizing biological tissues in the diagnosis of pathological and necrotic changes. Proc. SPIE—The Int. Soc. Opt. Eng. 11087, 1108713 (2019) 58. O. Pavlyukovich, N. Pavlyukovich, Y. Ushenko, O. Galochkin, M. Sakhnovskiy, M. Kovalchuk, A. Dovgun, S. Golub, O. Dubolazov, Fractal analysis of patterns for birefringence biological tissues in the diagnostics of pathological and necrotic states. Proc. SPIE—The Int. Soc. Opt. Eng. 11105, 1110518 (2019)

Chapter 2

Methods and Systems of Polarization Mueller-Matrix Microscopy of Biological Samples

2.1 Physical Substantiation and Selection of Research Objects The object of study is optically anisotropic layers of biological origin. These include histological sections of biological tissues and polycrystalline films of human fluids. This selection of objects has fundamental and applied aspects. Fundamental. From a physical point of view, such objects are complex optically inhomogeneous, light-scattering structures. These objects are characterized by the simultaneous presence of optically isotropic and anisotropic components. In accordance with this, a wide range of mechanisms for converting parameters of laser radiation is realized. The main mechanisms include optically isotropic reflection, refraction, and absorption. In addition, optically anisotropic interaction takes place—phase (optical activity and birefringence) and amplitude (linear and circular dichroism) anisotropy. Multiple interaction of laser radiation with such optically anisotropic structures occurs in the bulk of such a medium. The result is a complex, multiply scattered polarization-inhomogeneous optical field. In our work, we considered optically thin (attenuation coefficient) nondepolarizing biological layers. From a physical point of view, one-time acts of interaction with optically anisotropic structures are realized in the thickness of such objects. As a result, a laser field is formed with coordinate-modulated parameters characterizing the polarization of the field. We note that in the boundary zone of such a field (image of the biological layer), an unambiguous relationship is realized between the totality of the optically anisotropic parameters characterizing the object and the two-dimensional distributions of the azimuth and ellipticity of the polarization converted by the field. Under such conditions, it becomes possible to solve the inverse problem. In order to ensure the uniqueness of such solutions, we will rely on the analytically substantiated and experimentally tested Mueller-matrix formalism. Such an © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 L. Trifonyuk et al., Optical Anisotropy of Biological Polycrystalline Networks, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-99-1087-8_2

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approach from a unified methodological position makes it possible to fully describe the optical anisotropy of various biological layers. In the future, we will call the analytical solutions of such a problem “polarization reconstruction” of parameters characterizing the mechanisms of phase and amplitude anisotropy. A simultaneous (comparative) study of samples of biological tissues and films of human fluids will allow us to analyze the manifestations of the optical anisotropy mechanisms of various biological structures on the coordinate distributions of azimuthally independent elements of the Mueller matrix and their invariant combinations. Biological tissues are birefringent fibrillar networks formed by polypeptide chains of optically active protein molecules (collagen, elastin, myosin). Biological fluids that are studied in the form of smears, polycrystalline films of various optically active amino acids (albumin, globulin, etc.) and molecular complexes (bilirubin, bile acid crystals, etc.) Applied. Optically thin layers of histological sections of biological tissues and smears of human biological fluids are classical objects of light microscopic biomedical research. In particular, the histology of biopsies of various human organs, the results of which establish a reference diagnosis, or the “gold standard”, is one of example of traditional and most common biomedical research. Smears of biological fluids (blood, bile, mucous membrane of internal organs) form the basis of mass screening studies for the preliminary detection of various pathologies. Such objects are much more accessible and do not require a traumatic biopsy. This implies an urgent fundamental and applied task of a comparative study of the scenarios of the interaction of laser radiation with optically anisotropic structures of biological tissues and fluids in order to further develop a set of methods for early laser polarimetric diagnostics of such objects with various types of pathology.

2.2 Methods of Manufacturing Prototypes of Biological Tissues and Fluids Samples for research were made according to standard medical methods existing in medicine. Optically thin (geometric thickness l = 20 µm ÷ 40 µm) histological sections of a biopsy of human tissues and organs were made using a standard freezing microtome. Films of biological fluids were formed under identical conditions by applying a drop to optically uniform glass. The formed film was dried at room temperature (t = 22 °C). The sampling of the mucous membrane of the internal organs of the female reproductive sphere for cytology occurred after washing the surface with 0.9% Na–Cl solution. Using Volkman’s spoon, scrapings were made from the surface of the uterine wall and cervical canal. The resulting material was applied to a glass substrate.

2.3 Characterization of Research Objects

15

2.3 Characterization of Research Objects As objects of research in the work, we used two groups: • histological sections of biological tissues and human organs of various morphological structures (single and multi-layered) and physiological conditions (benign and malignant tumors, precancerous conditions) • polycrystalline planar films of biological fluids. In the series of Figs. 2.1, 2.2 and 2.3 show microscopic images of histological sections of single-layer (Fig. 2.1) and multilayer (Figs. 2.2 and 2.3) biological tissues recorded in coaxial (0°–0°)—fragments (1)—and crossed (0°–90°)—fragments (2)—polarizer—analyzer. The figures show that the microscopic images of histological sections of tissues of different morphological structures (Figs. 2.1, 2.2, 2.3, fragments (1)) are polarizedinhomogeneous. This is evidenced by a significant level of enlightenment in polarized-filtered (0°–90°) images.

Fig. 2.1 Microscopic images of a histological section of the myocardium. Explanation in the text

Fig. 2.2 Microscopic images of a histological section of the cervix endometrium. Explanation in the text

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Fig. 2.3 Microscopic images of a histological section of a rectal tumor. Explanation in the text

On the other hand, such an intensity distribution is heterogeneous, both in magnitude and in coordinate distribution (Figs. 2.1, 2.2, 2.3, fragments (2)). The fact revealed indicates the presence of a complex coordinate distribution of the manifestations of the mechanisms of optical anisotropy, which leads to the transformation of the state of polarization of laser radiation by various biological tissues. So, the common “denominator” of the optical properties of the examined histological sections is the transformation of the azimuth and ellipticity of the polarization of laser radiation and the formation of polarization-inhomogeneous microscopic images. A series of laser microscopic images of biological fluid films that are recorded in coaxial (0°–0°)—fragments (1)—and crossed (0°–90°)—fragments (2)—polarizer—analyzer is illustrated in Figs. 2.4, 2.5 and 2.6. A comparative analysis of the polarization structure of microscopic images of films of various fluids, as in the case of histological sections of biological tissues (Figs. 2.1, 2.2 and 2.3), revealed complex statistical and coordinate manifestations of optical anisotropy of such polycrystalline objects (Figs. 2.4, 2.5 and 2.6, fragments (2)). However, the phase anisotropy of the films of biological fluids is somewhat less

Fig. 2.4 Microscopic images of bile film. Explanation in the text

2.4 Optical Scheme of Experimental Research and Their Characteristics

17

Fig. 2.5 Microscopic images of the synovial fluid film of the joint. Explanation in the text

Fig. 2.6 Microscopic images of a blood plasma film. Explanation in the text

than for samples of biological tissues. This follows from a lower level of enlightenment, the image of optically anisotropic structures of polycrystalline films, which is recorded in crossed polarizer-analyzer.

2.4 Optical Scheme of Experimental Research and Their Characteristics In our work, we used two main optical schemes for experimental studies: • two-dimensional spectrally selective Stokes polarimetry [1–5] • two-dimensional Fourier–Stokes polarimetry [6–8].

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2.4.1 Optical Scheme of Two-Dimensional Spectrally Selective Stokes Polarimetry and Its Characteristics The measurement of coordinate distributions (two-dimensional arrays of values in the plane of the samples) of the values of the elements of the Mueller matrices was carried out in the arrangement (Fig. 2.7) of a standard Stokes polarimeter. Irradiation of samples 6 was sequentially carried out with a parallel (∅ = 2 × 103 µm) beam of “red” He–Ne (λ1 = 0.6328 µm) and a semiconductor “blue” (λ2 = 0.405 µm) laser 1. The polarizing irradiator consisted of a quarterwave plate 3 and a polarizer 4. Images of samples 6 using a polarizing microlens 7 (Nikon CFI Achromat P, focal length—30 mm, aperture—0.1, magnification— 4×) were projected onto the plane of the photosensitive area of CCD camera 10 (The Imaging Source DMK 41AU02.AS, monochrome 1/2'' CCD, Sony ICX205AL (progressive scan); resolution—1280 × 960, the size of the photosensitive area— 7600 × 6200 µm; sensitivity—0.05 lx; dynamic range—8 bit). Polarization analysis of images of samples 6 was carried out using a quarter-wave plate 8 and a polarizer 9. The calculation within each pixel of the digital camera 10 of the set of elements of the Mueller matrix Mik of sample 6 was carried out in accordance with the algorithm ) ( M11 = 0.5 V10 + V190 ; ) ( M12 = 0.5 V10 − V190 ; M13 = V145 − M11 ; M14 = V1⊗ − M11 ; ) ( M21 = 0.5 V20 + V290 ; ) ( M22 = 0.5 V20 − V290 ; M23 = V245 − M21 ; M24 = V2⊗ − M21 ; ) ( M31 = 0.5 V30 + V390 ; ) ( M32 = 0.5 V30 − V390 ;

Fig. 2.7 Optical scheme of the “two-wave” spectrally selective Stokes polarimeter. Explanation in the text

2.4 Optical Scheme of Experimental Research and Their Characteristics

19

M33 = V345 − M31 ;

M34 = V3⊗ − M31 ; ) ( M41 = 0.5 V40 + V490 ; ) ( M42 = 0.5 V40 − V490 ; M43 = V445 − M41 ;

M44 = V4⊗ − M41 .

(2.1)

0;45;90;⊗ Here Vi=2;3;4 are the parameters of the Stokes vector of the points of the digital image of sample 6, measured for a series of linearly (0°; 45°; 90°) and right-circular (⊗) polarized laser beams 0;45;90;⊗ 0;45;90;⊗ Vi=1 = I00;45;90;⊗ + I90 ; 0;45;90;⊗ 0;45;90;⊗ Vi=2 = I00;45;90;⊗ − I90 ; 0;45;90;⊗ 0;45;90;⊗ 0;45;90;⊗ = I45 − I135 ; Vi=3 0;45;90;⊗ Vi=4 = I⊗0;45;90;⊗ + I⊕0;45;90;⊗ .

(2.2)

Here I0;45;90;135;⊗;⊕ —the intensity of the light transmitted by an object passing through a linear polarizer 9 with an angle of rotation of the transmission plane Θ: 0°; 45°; 90°; 135°and also through the “quarter-wave plate 8—polarizer 9” system, which passes the right—(⊗) and left (⊕) circularly polarized components of the object laser radiation.

2.4.2 Optical Scheme of Two-Dimensional Fourier–Stokes Polarimetry and Its Characteristics In Fig. 2.8 presents the optical scheme of a laser Stokes polarimeter with spatialfrequency filtering, proposed by A. Karachevtsev [6]. Sample 6 was irradiated with a parallel (∅ = 104 µm) He–Ne laser light beam (λ = 0, 6328 µm, power W = 5 mW). The polarizing illuminator consists of quarter-wave plates 3, 5 and polarizer 4. Samples 6 were placed in the focal plane of the polarizing microlens 7 (focal length f = 30 mm, magnification 4×, digital aperture N.A. = 0.1). A spatial-frequency (low-frequency or high-frequency) filter 8 was located in the rear focal plane of the microlens 7. A polarizing microlens 9 (focal length f = 30 mm, 4× magnification, digital aperture N.A. = 0.1) was mounted at the focal length from the frequency plane of the lens 7, due to which the inverse Fourier transform was performed spatially and frequently filtered laser radiation field.

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Fig. 2.8 Optical scheme of a Stokes polarimeter using spatial-frequency filtering, where 1 is a He– Ne laser; 2—collimator; 3—stationary quarter-wave plate; 5, 10—mechanically movable quarterwave plates; 4, 11—polarizer and analyzer, respectively; 6—object of study; 7, 9—microlenses; 8—low-pass and high-pass filters; 12—CCD camera; 13—personal computer

The coordinate intensity distribution of such a field was recorded in the plane of the photosensitive CCD camera 12, which was also at the focal distance from the microlens 9. Calculations based on the totality of the data obtained were carried out in accordance with the algorithms (2.1) and (2.2). 45,90,⊗ Other two-dimensional distributions Vi=1;2;3;4 (m × n) are determined similarly, and the coordinate distributions of the values of the elements of the Mueller matrix Mik (m × n) of the biological layer under study are calculated (relation (2.2)). So, the main information array in our work is a combination of directly measured Mik (m × n) and spatially frequency filtered Mueller { matrix images F(Mik (m × n)) f j (Mik ) and solutions of the inverse problem ϕ(m × n) = —parameters of g j (F(Mik )) the optical anisotropy of the biological layer ⎞ (Mik )11 · · · (Mik )1n ⎟ ⎜ .. .. Mik (m × n) = ⎝ ⎠; . ··· . (Mik )m1 · · · (Mik )mn ⎧⎛ ⎞⎫ ⎪ ⎨ (Mik )11 · · · (Mik )1n ⎪ ⎬ ⎜ ⎟ .. .. F(Mik (m × n)) = F ⎝ ⎠ . ··· . ⎪ ⎪ ⎩ ⎭ (Mik )m1 · · · (Mik )mn ⎞ ⎛ f j (Mik )11 · · · f j (Mik )1n ⎟ ⎜ .. .. ϕik (m × n) = ⎝ ⎠. . ··· . ⎛

f j (Mik )m1 · · · f j (Mik )mn

(2.3)

(2.4)

(2.5)

2.5 Methods of Analytical Analysis and the Totality of Its Objective Parameters

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2.5 Methods of Analytical Analysis and the Totality of Its Objective Parameters 2.5.1 Statistical Analysis The statistical moments of the first (Z 1 ), second (Z 2 ), third (Z 3 ), and fourth (Z 4 ) orders, which were calculated by the following algorithms, were used as the main analytical tool for estimating the distributions of MMI (hereinafter q(m × n), relation (2.7)–(2.9)) 1 ∑ qj; N j=1 ┌ | N |1 ∑ ( ) q2 j ; Z2 = √ N j=1 N

Z1 =

N 1 1 ∑( 3 ) q j; Z3 = 3 Z 2 N j=1

Z4 =

N 1 1 ∑( 4 ) q j, Z 24 N j=1

(2.6)

where N —the number of sampling elements, which is determined by the number of pixels of the photosensitive area of the CCD camera.

2.5.2 Fractal Analysis A fractal analysis [9–14] of the distribution of the values of the MMI q(m × n) [10–13] consists in such a sequence of actions: • Power spectra (J (q)) of coordinate value distributions q(m × n) were calculated. • The log–log dependences of the power spectra were calculated log J (q) − log(ν), ν = l −1 —spatial frequency, l—the size of the structural element in the coordinate distribution of values q(m × n). • The dependences log J (q) − log(ν) were approximated by the least squares method in the curves V (η), η—the slope of the approximating curve. • With a linear nature of the dependence V (η = const) for changes within 2–3 decades of the size l of the structural elements of the crystalline network—the distribution of values q(m × n) is fractal • In the presence of several constant slope angles V (η)—the distribution of values q(m × n) is multifractal.

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• In the absence of stable tilt V (η) angles in the entire size l range, the distribution of values q(m × n) is random. As one of the parameters characterizing the distribution of dependence lg J (q) − lg(ν) values, we chose the second-order statistical moment ┌ | N |1 ∑ ( ) lg J (q) − lg(ν)2 j . D=√ N j=1

(2.7)

In the future, this parameter D will be called the dispersion of the logarithmic dependences lg J (q) − lg(ν) of the distribution of parameter q(m × n) values.

2.5.3 Information Analysis and Method Strength The term “method strength” determines the degree of success of a diagnostic test based on a comprehensive statistical and fractal⎛analysis of experimen⎞ (Mik )11 · · · (Mik )1n ⎟ ⎜ .. .. tally determined value distributions Mik (m × n) = ⎝ ⎠; . ··· .

F(Mik (m × n)) ⎛

=

(M ) · · · (Mik )mn ⎧⎛ ⎞⎫ ik m1 ⎪ ⎨ (Mik )11 · · · (Mik )1n ⎪ ⎬ ⎜ ⎟ .. .. F ⎝ = ⎠ ; ϕik (m × n) . · · · . ⎪ ⎪ ⎩ ⎭ (Mik )m1 · · · (Mik )mn ⎞

f j (Mik )11 · · · f j (Mik )1n ⎟ .. .. ⎠ in the task of using this method in determining . ··· . f j (Mik )m1 · · · f j (Mik )mn the degree of influence of the obtained results on the adoption of a medical decision. Important criteria that characterize the informativeness of the methods of the Mueller-matrix analysis of biological preparations are the reliability, reproducibility and coincidence of the research results [14–16]. The reliability or validity of the method shows the extent to which the resulting set of distributions of values q(m × n) characterizing the optical anisotropy of biological preparations corresponds to the specific condition of the patient, determined using the gold standard—the method with the highest diagnostic accuracy. When conducting our research on samples of biological layers taken from different groups of patients, we will use the terminology of medical opinion generally accepted in evidence-based medicine: ⎜ ⎝

1. The interpretation is “positive” for patients with the presence of the disease. This is a “truly positive case”—(TP). 2. The interpretation is “negative” for patients with no disease. This is a “truly negative case”—(TN).

References

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3. The interpretation is “positive” for patients with no disease. This is a “false positive case”—(FP). 4. The interpretation is “negative” for patients with the presence of the disease. This is a “wrong negative case”—(FN). To characterize the information content of any diagnostic method, objective parameters are used, which are called operational characteristics. There are the main and auxiliary characteristics. Key features include: • Sensitivity (Se)—this is the proportion of correct positive results (TP) of the diagnostic method among all sick patients (D+ ) Se =

TP 100%. D+

(2.8)

• Specificity (Sp)—this is the proportion of correct negative results (TN) of the method among a group of healthy patients (D− ) Sp =

TN 100%. D−

(2.9)

• Accuracy (Ac)—proportion of correct test results (TP + TN) among all examined patients (D+ + D− ) Ac =

TP + TN 100%. D+ + D−

(2.10)

Accuracy reflects the number of correct diagnoses obtained by the method of Mueller-matrix images of biological preparations.

References 1. M. Borovkova, M. Peyvasteh, O. Dubolazov, Y. Ushenko, V. Ushenko, A. Bykov, S. Deby, J. Rehbinder, T. Novikova, I. Meglinski, Complementary analysis of Mueller-matrix images of optically anisotropic highly scattering biological tissues. J. Eur. Opt. Soc. 14(1), 20 (2018) 2. V. Ushenko, A. Sdobnov, A. Syvokorovskaya, A. Dubolazov, O. Vanchulyak, A. Ushenko, Y. Ushenko, M. Gorsky, M. Sidor, A. Bykov, I. Meglinski, 3D Mueller-matrix diffusive tomography of polycrystalline blood films for cancer diagnosis. Photonics 5(4), 54 (2018) 3. L. Trifonyuk, W. Baranowski, V. Ushenko, O. Olar, A. Dubolazov, Y. Ushenko, B. Bodnar, O. Vanchulyak, L. Kushnerik, M. Sakhnovskiy, 2D-Mueller-matrix tomography of optically anisotropic polycrystalline networks of biological tissues histological sections. Opto-Electron. Rev. 26(3), 252–259 (2018) 4. V.A. Ushenko, A.V. Dubolazov, L.Y. Pidkamin, M.Y. Sakchnovsky, A.B. Bodnar, Y.A. Ushenko, A.G. Ushenko, A. Bykov, I. Meglinski, Mapping of polycrystalline films of biological fluids utilizing the Jones-matrix formalism. Laser Phys. 28(2), 025602 (2018)

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5. V.A. Ushenko, A.Y. Sdobnov, W.D. Mishalov, A.V. Dubolazov, O.V. Olar, V.T. Bachinskyi, A.G. Ushenko, Y.A. Ushenko, O.Y. Wanchuliak, I. Meglinski, Biomedical applications of Jones-matrix tomography to polycrystalline films of biological fluids. J. Inno. Opt. Health Sci. 12(6), 1950017 (2019) 6. M. Borovkova, L. Trifonyuk, V. Ushenko, O. Dubolazov, O. Vanchulyak, G. Bodnar, Y. Ushenko, O. Olar, O. Ushenko, M. Sakhnovskiy, A. Bykov, I. Meglinski, Mueller-matrixbased polarization imaging and quantitative assessment of optically anisotropic polycrystalline networks. PLoS ONE 14(5), e0214494 (2019) 7. A. Doronin, C. Macdonald, I. Meglinski, Propagation of coherent polarized light in turbid highly scattering medium. J. Biomed. Opt. 19(2), 025005 (2014) 8. A. Doronin, A. Radosevich, V. Backman, I. Meglinski, Two electric field Monte Carlo models of coherent backscattering of polarized light. J. Opt. Soc. Am. A 31(11), 2394 (2014) 9. A. Ushenko, V. Pishak, Laser polarimetry of biological tissue: principles and applications, in ed. by V. Tuchin, editors, Handbook of Coherent-Domain Optical Methods: Biomedical Diagnostics, Environmental and Material Science (2004), pp. 93–138 10. V.A. Ushenko, M.S. Gavrylyak, Azimuthally invariant Mueller-matrix mapping of biological tissue in differential diagnosis of mechanisms protein molecules networks anisotropy. Proc. SPIE 8812, 88120Y (2013) 11. V.O. Ushenko, Spatial-frequency polarization phasometry of biological polycrystalline networks. Opt. Mem. Neur. Netw. 22, 56–64 (2013) 12. V.A. Ushenko, N.D. Pavlyukovich, L. Trifonyuk, Spatial-frequency azimuthally stable cartography of biological polycrystalline networks. Int. J. Opt. 683174, 2013 (2013) 13. V.A. Ushenko, Complex degree of mutual coherence of biological liquids. Proc. SPIE 8882, 88820V (2013) 14. L. Cassidy, Basic concepts of statistical analysis for surgical research. J. Surg. Res. 128(2), 199–206 (2005) 15. C.S. Davis, Statistical Methods of the Analysis of Repeated Measurements (Springer-Verlag, New York, 2002) 16. A. Petrie, C. Sabin, Medical Statistics at a Glance (Wiley-Blackwell, Chichester, UK, 2009)

Chapter 3

Mueller-Matrix Description of the Optically Anisotropy of Biological Layers

3.1 The Main Types of Optical Anisotropy and Partial Matrix Operators for Its Description The main mechanisms of optical anisotropy of biological layers [1–10] include the following: • Phase anisotropy mechanisms—linear and circular birefringence. The mechanism of circular birefringence is associated with the optical activity of chiral molecules containing closed rings. As a result, the wave decomposes into orthogonal, rightand left-circularly polarized components propagating at different speeds. With a superposition of such components, a rotation (rotation) of the plane of polarization of the laser radiation occurs. The linear birefringence mechanism is formed due to the spatial structuring of molecular complexes. Optically active protein molecules form polypeptide chains. The latter are combined into different-sized filamentary fibrillar structures: “Microfibril—fibril—fiber—bundle”. As a result, a spatially oriented and ordered optically uniaxial birefringent structure is formed. When a laser wave interacts with such a biological crystal, the wave decomposes into orthogonal linearly polarized components propagating at different speeds and an elliptically polarized wave forms. • The mechanisms of amplitude anisotropy, linear and circular (circular) dichroism of biological structures, are due to polarization-dependent absorption of laser radiation. The mechanism of circular dichroism is associated with the spiral structure of molecules. In its presence, right- and left-circularly polarized components are absorbed differently by the substance. As a result, an elliptically polarized radiation is formed that has passed through such a layer. The mechanism of linear dichroism is interconnected with linear birefringence of fibrillar networks. When a laser wave passes through such spatially structured media, the wave decomposes

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 L. Trifonyuk et al., Optical Anisotropy of Biological Polycrystalline Networks, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-99-1087-8_3

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3 Mueller-Matrix Description of the Optically Anisotropy of Biological …

into orthogonal linearly polarized components for which the absorption coefficients are different. Due to this, an elliptically polarized wave arises, passing through such a polycrystalline biological medium.

3.2 Mueller-Matrix Approach to the Description of Polycrystalline Layers with Phase and Amplitude Anisotropy Let us consider the possibility of an analytical description of the parameters of the phase and amplitude anisotropy of the biological layer. Such a need, first of all, requires forecasting and objective assessment of the processes of interaction of laser radiation with various types of biological objects. This will allow avoiding research empiricism and expanding the functionality of optical diagnostic methods for the polycrystalline structure of biological layers of various biochemical and morphological structures. Secondly, the implementation of such an approach requires the use of unified, most informative analytical approaches. The most adequate in this sense is a well analytically substantiated and widely experimentally tested vector-parametric or Mueller-matrix approach. In further analytical consideration, we will use the basic principles of the theory of generalized anisotropy. In our case, we apply such a theory to the polycrystalline structure of biological objects. In other words, we consider a biological layer in which four types of optical anisotropy exist simultaneously. For the Mueller-matrix description of the optical properties of such an object, we use the methodological approach, the structural— logical scheme of which is shown in Fig. 3.1. According to this approach, we distinguish the following steps: • Consider a biological layer in which all types of phase (δ, θ ) and amplitude anisotropy (Δτ, Δg) are simultaneously realized • Each( of these types of anisotropy is associated with a partial matrix oper) ator {Ω}, {D}, {ϕ}, {Ψ} , which exhaustively describes the mechanisms of conversion of laser radiation • We determine the Mueller matrix ({M}) of such a biological layer in the form of a product of partial matrix operators • Analyze the azimuthal independence of the obtained Mueller matrix and determine the corresponding elements and invariants (Mik (Θ) = const ; ΔMik (Θ) = const) • We determine the relationships describing the interconnection of the anisotropy parameters and MMI, and we obtain algorithms for the polarization reproof( the phase))shifts (duction ( of∗ the coordinate ) ( (distributions))of the values θ = w (Mik , ΔM(ik∗ , δ = u ))Mik∗ , ΔMik∗ linear Δτ = v Mik∗ , ΔMik∗ and circular Δg = h Mik∗ , ΔMik∗ dichroism coefficients that characterize the polycrystalline structure of the biological layer.

3.3 Partial Mueller-Matrix Operators Describing the Mechanisms of Phase …

27

Fig. 3.1 Structural-logical diagram of the Mueller-matrix simulation of optical anisotropy of the biological layer

3.3 Partial Mueller-Matrix Operators Describing the Mechanisms of Phase and Amplitude Anisotropy In the framework of the model of complex anisotropy of the biological layer developed in numerous studies, we can write the following expressions for the partial Mueller matrices [11–13]: Round or circular birefringence The matrix operator {Ω} that describes the optical activity of the molecules of the medium has the following form ∥ ∥1 ∥ ∥0 {Ω} = ∥ ∥0 ∥ ∥0

0 ω22 ω32 0

0 ω23 ω33 0

∥ 0∥ ∥ 0∥ ∥, 0∥ ∥ 1∥

(3.1)

where { ωik =

ω22 = ω33 = cos 2θ, ω23 = −ω32 = sin 2θ.

(3.2)

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3 Mueller-Matrix Description of the Optically Anisotropy of Biological …

Here θ —angle of rotation of the plane of polarization. Linear birefringence The optical manifestations of this mechanism of spatially structured fibrillar networks are characterized by a partial Mueller matrix {D} ∥ ∥1 ∥ ∥0 {D} = ∥ ∥0 ∥ ∥0

0 d22 d32 d42

0 d23 d33 d43

∥ 0 ∥ ∥ d24 ∥ ∥, d34 ∥ ∥ d44 ∥

(3.3)

where ⎧ ⎪ d ⎪ ⎪ 22 ⎪ ⎪ ⎪ ⎪ d23 ⎨ d33 dik = ⎪ d42 ⎪ ⎪ ⎪ ⎪ d 34 ⎪ ⎪ ⎩ d44

= cos2 2ρ + sin2 2ρ cos δ, = d32 = cos 2ρ sin 2ρ(1 − cos δ), = sin2 2ρ + cos2 2ρ cos δ, = −d24 = sin 2ρ sin δ, = −d43 = cos 2ρ sin δ, = cos δ.

(3.4)

Here ρ is the direction of the optical axis, which is determined by the orientation of the position of the fibril in the plane of the biological layer or polypeptide chain Δnl—phase shift between linearly orthogonally polarized of amino acids. δ = 2π λ components of the amplitude of the laser beam; λ—wavelength; Δn—birefringence value; l—geometric layer thickness. Circular dichroism This mechanism of optically anisotropic absorption is described by the following matrix operator ∥ ∥ 1 0 ∥ ∥ 0 ϕ22 {ϕ} = ∥ ∥ 0 0 ∥ ∥ϕ 0 41

0 0 ϕ33 0

∥ ϕ14 ∥ ∥ 0 ∥ ∥, 0 ∥ ∥ 1 ∥

(3.5)

where { ϕik =

2

ϕ22 = ϕ33 = 1−Δg , 1+Δg 2 2Δg ϕ14 = ϕ41 = ± 1+Δg 2.

(3.6)

⊕ , —absorption ratios right—(⊗) and left—(⊕)circularly Here Δg = gg⊗⊗ −g +g⊕ g⊗ , g⊕ polarized components of the amplitude of the laser radiation.

3.4 Generalized Mueller Matrix of a Biological Layer with Phase …

29

Linear dichroism The partial matrix operator that describes such a mechanism of optically anisotropic absorption has the following analytical form ∥ ∥ 1 ∥ ∥ φ21 {Ψ} = ∥ ∥ φ31 ∥ ∥ 0

φ12 φ22 φ32 0

φ13 φ23 φ33 0

∥ 0 ∥ ∥ 0 ∥ ∥, 0 ∥ ∥ φ44 ∥

(3.7)

where ⎧ ⎪ ⎪ φ12 ⎪ ⎪ ⎪ φ13 ⎪ ⎪ ⎨ φ22 φik = ⎪ φ23 ⎪ ⎪ ⎪ ⎪ φ33 ⎪ ⎪ ⎩ φ44

= φ21 = (1 − Δτ ) cos 2ρ, = φ31 = (1 − Δτ ) sin 2ρ, √ = (1 + Δτ ) cos2 2ρ + 2 Δτ sin2 2ρ, = φ32 = (1 − Δτ ) sin 2ρ, √ = (1√+ Δτ ) sin2 2ρ + 2 Δτ cos2 2ρ, = 2 Δτ .

(3.8)

{

τx = τ cos ρ; , τx , τ y —absorption coefficients of linearly τ y = τ sin ρ polarized orthogonal components of the amplitude of the laser radiation.

Here Δτ =

τx , τy

3.4 Generalized Mueller Matrix of a Biological Layer with Phase and Amplitude Anisotropy In the presence of four mechanisms of the optical anisotropy of the biological layer (relations (3.1)–(3.8)), its generalized Mueller matrix can be represented by the product of partial matrix operators

4 {M} = πi=1

∥ ∥ 1 ∥ ∥ M21 −1 {M}i = M11 × ∥ ∥ M31 ∥ ∥M 41

M12 M22 M32 M42

M13 M23 M33 M43

∥ M14 ∥ ∥ M24 ∥ ∥. M34 ∥ ∥ M44 ∥

(3.9)

When analyzing the expression (3.9), the question arises. In what sequence are the partial matrix operators {Mi } written in the generally speaking non-commuting 4 {M}i . The answer to this question was considered previously. product πi=1 In our case, the sequence of writing partial matrix operators acquires fundamental content in the sense of solving the inverse problem, finding the Mueller-matrix reconstruction algorithms for the parameters of optical anisotropy

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) ( θ = w Mik∗ , ΔMik∗ , ( ) δ = u Mik∗ , ΔMik∗ , ( ) Δg = h Mik∗ , ΔMik∗ , ( ) Δτ = v Mik∗ , ΔMik∗

(3.10)

Expressions (3.10) should correspond to the main condition—independence from a change in the position of the partial operators in (3.9) of the analytical determination of the optical anisotropy parameters (δ, θ , Δτ, Δg) based on a combination of azimuthally independent MMI.

3.5 Mueller-Matrix Reconstruction or Reproduction of Optical Anisotropy Parameters In studies of the azimuthal symmetry of matrix operators, it is shown that the following matrix elements and their combinations are azimuthally independent {

M11 (Θ) = const; M14 (Θ) = const, M41 (Θ) = const; M44 (Θ) = const,

{

[M22 + M33 ](Θ) ≡ ∑ M22;33 (Θ) = const, [M23 − M32 ](Θ) ≡ ΔM23;32 (Θ) = const. (3.11)

Based on (3.11), it becomes clear the search for algorithms (3.10) taking into account precisely the azimuthally independent Mueller-matrix invariants with analytical stability of solutions (3.9). We consider two cases. First, when the mechanisms of optical anisotropic absorption can be neglected Δτ = 1, Δg = 0.

(3.12)

Such a situation is experimentally realized by appropriate selection of the wavelength of the laser radiation probing the biological layer. It is known that the spectral absorption maxima of most protein molecules are in the ultraviolet region. Therefore, experimentally analytical conditions (3.12) are realized in the “red—λ1 ” region of the spectral range. So, taking into account conditions (3.12), the matrix product (3.9) can be rewritten in the following form ∥ ∥1 0 0 ∥ ∥ 0 (d22 ω22 + d23 ω32 ) (d22 ω23 + d23 ω33 ) {F(λ1 )} = {D}{Ω} = ∥ ∥ 0 (d32 ω22 + d33 ω32 ) (d32 ω23 + d33 ω33 ) ∥ ∥ 0 (d ω + d ω ) (d ω + d ω ) 42 22 43 32 42 23 43 33

∥ 0 ∥ ∥ d24 ∥ ∥. d34 ∥ ∥ d ∥ 44

(3.13)

3.5 Mueller-Matrix Reconstruction or Reproduction of Optical Anisotropy …

31

With (3.13) we can obtain the ratio of the polarization reproduction of the parameters of the phase anisotropy of the polycrystalline layer through the MMI {

δ = arccos f 44 (λ1 ), Δ f 23;32 (λ1 ) . θ = 0.5 arctan ∑ f 22;33 (λ1 )

(3.14)

In order to verify the stability of solutions (3.14), we considered another sequence of partial matrix operators in the product (3.13)—{D}{Ω} → {Ω}{D} {

∥ ∥ ∥ ∥1 0 0 0 ∥ ∥ ∥ 0 (d22 ω22 + d32 ω23 ) (d23 ω22 + d33 ω23 ) (d24 ω22 + d34 ω23 ) ∥ ∥ ˜ 1 ) = {Ω}{D} = ∥ F(λ ∥ 0 (d22 ω32 + d32 ω33 ) (d23 ω32 + d33 ω33 ) (d24 ω32 + d34 ω33 ) ∥ ∥ ∥ ∥ ∥0 d42 d43 d44 (3.15) }

It is easy to see that the solution of the system of Eq. (3.15) with respect to the parameters characterizing the linear (relations (3.3), (3.4)) and circular (relations (3.1), (3.2)) birefringence gives a result similar to (3.14). Therefore, we can state the uniqueness of the Mueller-matrix azimuthally invariant reconstruction of the phase anisotropy parameters. From a physical point of view, this fact can be related to the fact that in a polycrystalline layer, phase shifts between orthogonal linearly (δ ↔ M44 ) and circularly (θ ↔ ΔM) polarized components are formed almost simultaneously. To analyze the manifestations of amplitude anisotropy, we used the shortwavelength region of the spectrum (λ2 ), where the maxima of the optically anisotropic absorption of protein molecules are localized. In this situation, it is relevant to take into account all types of optical anisotropy, characterized by the set described in clause 3.3, of the partial matrix operators (relations (3.1)–(3.8)). To this end, we consider the ( possibility ∑ of obtaining ) Mueller∗ M Δg = h , M22;33 , ΔM23;32 ,Δτ = matrix reconstruction 11;14;41;44 ( ) ∑ u ∗ M11;14;41;44 , M22;33 , ΔM23;32 algorithms taking into account expressions (3.9), (3.14). Let us analyze two analytical situations of the recording sequence of partial matrix operators that characterize the mechanisms of amplitude anisotropy (relations (3.5)– (3.8)) in the product of operators (3.9) {

{M(λ2 )} = {ϕ}{Ψ}{F}, . {M ∗ (λ2 )} = {Ψ}{ϕ}{F}.

(3.16)

A comparative analysis of the symmetry of the matrix operators of phase anisotropy (relations (3.1)–(3.4) and (3.13), (3.15)) and the Mueller matrix for generalized anisotropy (relations (3.9), (3.16)) found that the information on amplitude anisotropy is contained in values of three azimuthally independent matrix elements M14;41;44 (λ2 ). . Based on this, from (3.16), taking into account (3.1)–(3.8), we obtain

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3 Mueller-Matrix Description of the Optically Anisotropy of Biological …

⎧ ⎨ M14 (λ2 ) = φ12 f 24 + φ13 f 34 + ϕ14 φ44 f 44 = ϕ14 φ44 f 44 , M (λ ) = ϕ41 , ⎩ 41 2 M44 (λ2 ) = ϕ41 φ12 f 24 + ϕ41 φ13 f 34 + φ44 f 44 = φ44 f 44 .

(3.17)

Here f ik are the elements of the phase anisotropy matrix of the form (3.13) with allowance for the “spectral” correction of the phase shift δ(λ2 ) = λλ21 δ(λ1 ). Note that for the matrix operator of phase anisotropy of another symmetry (relation (3.15)), the right-hand sides of the system of Eq. (3.17) do not change. The solutions of the system of Eq. (3.17) are the following algorithms of the Mueller-matrix reconstruction of the parameters of linear and circular dichroism Δτ = 0.25

M44 (λ2 ) λ2 × , M44 (λ1 ) λ1

(3.18)

Δg = 1 + (1 − M41 (λ2 ))0.5 .

(3.19)

An additional condition for the adequacy proposed previously for the model of generalized anisotropy (relation (3.9)) is the following relation between the values of the Mueller-matrix invariants M41 (λ2 ) =

M14 (λ2 ) . M44 (λ2 )

(3.20)

To assess the stability of solutions (3.18) and (3.19), we considered another sequence of partial matrix operators {ϕ}{Ψ} → {Ψ}{ϕ} in the product (3.9). The result obtained is illustrated by the relations ⎧ ∗ ⎨ M14 (λ2 ) = ω22 φ12 f 24 + ω33 φ13 f 34 + ω14 f 44 = ω14 f 44 , M ∗ (λ2 ) = ω41 φ44 , ⎩ 41 ∗ M44 (λ2 ) = φ44 f 44 .

(3.21)

Solutions (3.21) are the following algorithms of Mueller-matrix reconstruction or polarization reproduction of the values of linear and circular dichroism ∗ M44 (λ2 ) λ2 × , ∗ M44 (λ1 ) λ1 ) ( M ∗ (λ2 ) 0.5 . Δg = 1 + 1 − 41 0.5 2τ

Δτ = 0.25

(3.22)

(3.23)

The adequacy condition for the solutions of (3.22) and (3.23) is the “two-wave” condition defined from (3.13), (3.15) and (3.21) {

( f 44 (λ2 ) =

∗ ∗ M14 (λ2 )M44 (λ2 ) ∗ M41 (λ2 )

)0.5 }

( → cos

λ1 arccos( f 44 (λ1 )) λ2

)

References

33

( =

∗ ∗ M14 (λ2 )M44 (λ2 ) ∗ M41 (λ2 )

)0.5 (3.24)

So, a comparative analysis of the Mueller-matrix reconstruction algorithms (3.18), (3.19) and (3.22), (3.23) revealed the coincidence of the expressions for calculating the linear dichroism parameter Δτ . The uncertainty in the calculated values of circular dichroism Δg can be removed by an additional analysis of relations (3.20) and (3.24) for the experimentally measured elements of the Mueller matrix. Correct is the determination algorithm Δg for which either relation (3.20) or relation (3.24) holds.

References 1. V. Tuchin, L. Wang, D. Zimnjakov, Optical Polarization in Biomedical Applications (Springer, New York, USA, 2006) 2. R. Chipman, Polarimetry, in M. Bass, editors, Handbook of Optics: Vol I—Geometrical and Physical Optics, Polarized Light, Components and Instruments (McGraw-Hill Professional, New York, 2010), p. 22.1–22.37 3. N. Ghosh, M. Wood, A. Vitkin, Polarized light assessment of complex turbid media such as biological tissues via Mueller matrix decomposition, in Handbook of Photonics for Biomedical Science. ed. by V. Tuchin (CRC Press, Taylor & Francis Group, London, 2010), pp.253–282 4. S. Jacques, Polarized light imaging of biological tissues, in Handbook of Biomedical Optics. ed. by D. Boas, C. Pitris, N. Ramanujam (CRC Press, Boca Raton, London, New York, 2011), pp.649–669 5. N. Ghosh, Tissue polarimetry: concepts, challenges, applications, and outlook. J. Biomed. Opt. 16(11), 110801 (2011) 6. M. Swami, H. Patel, P. Gupta, Conversion of 3 × 3 Mueller matrix to 4×4 Mueller matrix for non-depolarizing samples. Optics Commun. 286, 18–22 (2013) 7. D. Layden, N. Ghosh, A. Vitkin, Quantitative polarimetry for tissue characterization and diagnosis, in Advanced Biophotonics: Tissue Optical Sectioning. ed. by R. Wang, V. Tuchin (CRC Press, Taylor & Francis Group, Boca Raton, London, New York, 2013), pp.73–108 8. T. Vo-Dinh, Biomedical Photonics Handbook: 3 Volume Set, 2nd ed (CRC Press, Boca Raton, 2014) 9. A. Vitkin, N. Ghosh, A. Martino, Tissue polarimetry, in Photonics: Scientific Foundations, Technology and Applications, 4th edn., ed. by D. Andrews (John Wiley & Sons, Inc., Hoboken, New Jersey, 2015), pp.239–321 10. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis, 2nd edn. (SPIE Press, Bellingham, Washington, USA, 2007) 11. A. Ushenko, V. Pishak, Laser polarimetry of biological tissue: principles and applications, in V. Tuchin, editors, Handbook of Coherent-Domain Optical Methods: Biomedical Diagnostics, Environmental and Material Science (2004), pp. 93–138 12. O. Angelsky, A. Ushenko, Y. Ushenko, V. Pishak, A. Peresunko, Statistical, correlation and topological approaches in diagnostics of the structure and physiological state of birefringent biological tissues, in Handbook of Photonics for Biomedical Science (2010), pp. 283–322 13. Y. Ushenko, T. Boychuk, V. Bachynsky, O. Mincer, Diagnostics of structure and physiological state of birefringent biological tissues: statistical, correlation and topological approaches, in V. Tuchin, editors, Handbook of Coherent-Domain Optical Methods (Springer Science+Business Media, 2013)

Chapter 4

Azimuthally Invariant Mueller-Matrix Mapping of Optically Anisotropic Networks of Biological Tissues and Fluids

4.1 Mueller-Matrix Images of Optically Anisotropic Networks of Biological Tissues Experimental studies of the coordinate distributions of the values of the elements of the Mueller matrices of histological sections of biological tissues [1–10] were carried out at the location of the Stokes polarimeter, the optical scheme and characteristics of which are given in Chap. 2, Sect. 2.4.1 and Fig. 2.1. Experimental samples were irradiated with a blue wavelength semiconductor laser λ2 = 0.405 µm [11–14]. The calculation of the values of the matrix elements in each pixel of the digital camera was carried out according to the algorithms (2.1) and (2.2). [1–5, 8–10]. Statistical analysis of the distribution of the values of the matrix elements was carried out using the relation (2.10) [15–21]. Figure 4.1 shows a series of Mueller-matrix images of a histological section of a single-layer skeletal muscle tissue (a spatially ordered fibrillar network formed by optically active myosin molecules). The statistical and scale-like structure of coordinate distributions of the values of the three main types—“orientation”, “orientation-phase”, “phase”—matrix elements M33 ; M34 ; M44 —is illustrated in Fig. 4.2. Such a choice of matrix elements has a physical justification, which is described previously. It is shown here that the coordinate distribution of values M33 (m × n) is related to the distribution of the directions of the optical axes of the linearly birefringent network of the biological layer. The distribution of values M34 (m × n) carries information about the manifestations of the linear and circular birefringence mechanisms of the polycrystalline component of the object. Finally, the coordinate structure of the map M44 (m × n) is determined by the distribution of the values of the phase shifts introduced by the birefringent structures between the orthogonal components of the amplitude of the laser beam.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 L. Trifonyuk et al., Optical Anisotropy of Biological Polycrystalline Networks, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-99-1087-8_4

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Fig. 4.1 Mueller-matrix images of a histological section of the skeletal muscle. Explanation in the text

A comparative analysis of the structure of Mueller-matrix images characterizing the phase and amplitude anisotropy of the biological layer allows, at different levels of the morphological organization of the structure (molecules—crystallites—fibrils— fibers—beams) of the tissue, to expand physical ideas about the spectrum of the mechanisms of interaction of laser radiation with fibrillar networks. In addition, it is possible to establish objective statistical (statistical moments of the first to fourth orders of magnitude) and fractal (logarithmic dependences of power spectra) criteria by which the processes of changes in the anisotropy of biological layers associated with the pathology of human tissues or organs are evaluated. The structure of such objects was analytically “laid down” in computer modeling of various types of Mueller-matrix images of planar polycrystalline networks with controlled parameters (Chap. 3, Sect. 3.6). Therefore, on this basis, one can evaluate the adequacy of the proposed model concepts of the Mueller 4 matrix as a superposition of partial operators that characterize the mechanisms of linear (relations (3.3), (3.4)), circular (relations (3.1), (3.2)) birefringence and also linear (relations (3.7), (3.8)) and circular (relations (3.5), (3.6)) dichroism presented in Chap. 3 (relations (3.9), (3.13), (3.15), (3.16)). If we compare all 16 experimentally measured distributions of the values Mik (m × n) of both types of samples (Fig. 4.1), then we can state the correspondence of the obtained data to the considered model concepts of the phase and amplitude anisotropy of the biological layer. As you can see all the matrix elements Mik /= 0.

4.1 Mueller-Matrix Images of Optically Anisotropic Networks …

37

Fig. 4.2 Statistical and large-scale self-similar structure of the distribution of the values of the Mueller-matrix images of the histological section of the skeletal muscle

This fact indicates a deeper informational content of the measured Mueller matrix as compared to Mueller-matrix mapping of biological layers only in the “red” (λ1 = 0.6328 µm) spectral range. Here, the symmetry of the matrix operator and the set of Mueller-matrix images (relations (3.13), (3.15)) are mainly associated with manifestations of the mechanisms of phase anisotropy. When mapping occurs in the “blue” (λ2 = 0.405 µm) spectral range, the functional “load” of the partial Muellermatrix images is different. Let us consider it in more detail. From the analysis of the generalized matrix {M} (relations (3.9), (3.13), (3.15)) and its partial components {Ω} (relations (3.1), (3.2)), {D} (relations (3.3), (3.4)), {ϕ} (relations (3.5), (3.6)) and {Ψ} (relations (3.7), (3.8)) follows: • a set of element values Mi=1; k=1;2;3;4 characterizes the mechanisms of optically anisotropic absorption; • elements Mi=2;3; k=1;2;3;4 characterize phase modulation (δ, θ ) of laser radiation against the background of its optically anisotropic absorption (Δg, Δτ ); • the values of the elements Mi=4; k=1;2;3;4 carry complex information about the superposition of the manifestations of the mechanisms of linear birefringence and dichroism.

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4 Azimuthally Invariant Mueller-Matrix Mapping of Optically …

When conducting a model analysis of the formation of Mueller-matrix images of polycrystalline networks of cylinders with circular optical axes (Chap. 3, Figs. 3.1– 3.10), the formation of a cluster topographic structure was detected in the presence of modulation of the anisotropy parameters. An analysis of the experimental distributions Mik (m × n) also confirmed the presence of such structurality for the Mueller-matrix images characterizing the optical anisotropy of the ordered fibrillar network of skeletal muscle tissue. Examples of such topographic distributions are shown in enlarged fragments of the Mueller-matrix images of Fig. 4.1. Thus, using mutually complementary statistical (Chap. 2, Sect. 2.6.1) and fractal (Chap. 2, Sect. 2.6.2) approaches, objective criteria for the diagnosis and differentiation of the polycrystalline structure of biological layers can be determined. As the main information arrays, we will use the data of analytical modeling (relation (3.11))—a set of values of azimuthally independent Mueller-matrix invariants

4.2 Mueller-Matrix Invariants Characterizing the Optical Anisotropy of Histological Sections of Biological Tissues This section presents the results of the study of the laws and formation of the values of MMI, which characterize the phase and amplitude anisotropy of the most common multilayer tissues of human organs. For the purpose of possible application, a case of a pathological condition is considered—a formed benign tumor. The results of the study of the statistical (histogram of the distribution of random values) and scale self-similar (logarithmic dependences of power spectra) structure of the MMI (relation (3.11)) characterizing the linear M44 (Fig. 4.3) and circular ΔM (Fig. 4.4) birefringence, as well as linear M14 (Fig. 4.5) and circular M41 (Fig. 4.6) dichroism of the histological section of a biopsy of a benign prostate adenoma tumor have been obtained. An analysis of the data revealed an individual topographic structure of the set of all MMI (relation (3.11)) characterizing the optical anisotropy of the biological

Fig. 4.3 Two-dimensional m ×n (fragment (1)), statistical N (M44 ) (fragment (2)) and lg J (M44 )− lg ν (fragment (3)) distribution of the values of the Mueller-matrix invariant M44 of the histological section of the prostate adenoma

4.2 Mueller-Matrix Invariants Characterizing the Optical Anisotropy …

39

Fig. 4.4 Two-dimensional m ×n (fragment (1)), statistical N (ΔM) (fragment (2)) and lg J (ΔM)− lg ν (fragment (3)) distribution of values of the Mueller-matrix invariant ΔM of the histological section of the prostate adenoma

Fig. 4.5 Two-dimensional m ×n (fragment (1)), statistical N (M41 ) (fragment (2)) and lg J (M41 )− lg ν (fragment (3)) distribution of values of the Mueller-matrix invariant M41 of the histological section of the prostate adenoma

Fig. 4.6 Two-dimensional m ×n (fragment (1)), statistical N (M14 ) (fragment (2)) and lg J (M14 )− lg ν (fragment (3)) distribution of values of the Mueller-matrix invariant M14 of the histological section of the prostate adenoma

layer of prostate adenoma. The ranges of variation and eigenvalues of the invariants M44 , ΔM (Figs. 4.3 and 4.4, fragments (1)), characterizing the mechanisms of phase anisotropy, are slightly larger than similar quantities and their element variations M14 , M41 of amplitude anisotropy (Figs. 4.5 and 4.6, fragments (1)). This fact can be physically associated with the spectral selectivity of the manifestations of the mechanisms of linear and circular dichroism. It is known that the extrema of the absorption spectra of protein molecules that form optically anisotropic

40

4 Azimuthally Invariant Mueller-Matrix Mapping of Optically …

structures are in the ultraviolet region of the spectrum, while the mechanisms of phase modulation operate in a wide spectral range. Differences in the statistical distributions of values M(44;14;41 (m × n); ) ΔM(m × n) are quantified by the series of histograms N M44;14;41 , ΔM shown in fragments (2) of Figs. 4.3, 4.4, 4.5 and 4.6. A comparative analysis of such distributions shows that for the histograms of phase Mueller-matrix invariants M44 , ΔM, , a large half width is inherent. In addition, the main extrema are localized in the region of large values of the indicated( parameters ) (Figs. 4.3 and 4.4, fragments (2)). On the contrary, histograms N M14;41 are narrower, are asymmetric and have sharp peaks (Figs. 4.5 and 4.6, fragments (2)). (The experimentally ) determined histograms of the distributions of random values N M44;14;41 , ΔM correspond to the results of computer modeling (Chap. 3, Sect. 3.6, Figs. 3.1–3.10). However, the MMI histograms characterizing phase anisotropy N (M44 , (ΔM) are ) characterized by a lower half width and peak sharpness. The distributions N M14;41 characterizing the mechanisms of amplitude anisotropy are also narrower and asymmetric. The revealed differences can be explained by the aforementioned features of the spectral manifestations of various types of anisotropy. In addition, the quantitative parameters of optical anisotropy used in computer modeling are deterministic and understandable, not fully such that reflect the specifics of optical anisotropy of real polycrystalline biological layers. Coordinate value distributions M44;14;41 (m × n), ΔM(m × n) are multifractal. This statement follows from the presence of several (three) ( tilt angles of) approximating (broken) V (η) to the set of dependencies lg J M44;14;41 , ΔM − lg(ν) (Chap. 2, Sect. 2.6.2). This fact correlates well with the computer simulation of the formation of spatialfrequency spectra of the distribution of the MMI values of planar networks of curved crystals with harmonious modulation of the optical anisotropy parameters (Chap. 3, Figs. 3.6–3.10, fragments (3)). In other words, the coordinate modulation of the values of such physical parameters in the case of real fibrillar networks is manifested in the formation of a set of different-scale self-similar sets of values M44;14;41 (m × n), ΔM(m × n) . Table 4.1 contains data on the calculation of the magnitude of statisand the dispersion D of the logarithmic dependences tical( moments Z i=1;2;3;4 ) lg J M44;14;41 , ΔM − lg(ν) of the power spectra of the distribution of the values M44;14;41 (m × n), ΔM(m × n) of the histological section of the prostate adenoma with a fibrillar network formed by optically active protein molecules of myosin and collagen. The analysis of the magnitudes of the statistical moments of the first and fourth orders characterizing the distribution of values M44;14;41 (m × n), ΔM(m × n) revealed the maximum average Z 1 and dispersion Z 2 values for the Mueller-matrix images of invariants associated with the influence of linear birefringence (M44 ). For histograms of random values of matrix elements M14 ↔ Δτ and M41 ↔ Δg

4.3 Mueller-Matrix Differentiation of Fibrillar Networks of Biological …

41

Table 4.1 Statistical Z i=1;2;3;4 and fractal D parameters characterizing the value of MMI M44;14;41 (m × n), ΔM(m × n), histological section of prostate adenoma Parameters

M44

ΔM

M14

M41

Z1

0.46

0.12

0.73

0.16

Z2

0.29

0.13

0.19

0.11

Z3

0.48

0.23

0.57

1.14

Z4

0.47

0.61

0.41

0.93

D

0.23

0.29

0.26

0.22

maximum values, statistical moments of the 3rd and The disper( 4th orders acquire. ) sion value D characterizing the dependences lg J M44;14;41 , ΔM − lg(ν) of the power spectra of the distribution of the values M44;14;41 (m × n); ΔM(m × n) of the histological section of the prostate adenoma varies from 25 to 45% for the coordinate distributions of the values of various MMI. So, we have discovered the sensitivity of various objective parameters for estimating the coordinate structure of the MMI, which are determined by the optical manifestations of various physical mechanisms of optical anisotropy. The obtained results will form the basis for the development of methods for differentiating changes in optical anisotropy associated with benign and malignant conditions, based on azimuthally independent Mueller-matrix mapping of such biological layers.

4.3 Mueller-Matrix Differentiation of Fibrillar Networks of Biological Tissues with Different Phase and Amplitude Anisotropy This section contains the results of a comparative study in the framework of statistical and fractal approaches to differentiating the manifestations of various mechanisms of optical anisotropy of histological sections of a biopsy of benign (polyp) and malignant (carcinoma) rectal tumors. From a physical point of view, the biological layers chosen for the study differ in both phase and amplitude anisotropy parameters. Such differences are associated with biochemical and tissue changes characteristic of a particular pathology. Therefore, we solved the problem of optical differentiation of the severity of the pathology using the method of selection of such manifestations on the basis of azimuthally invariant Mueller-matrix mapping of histological sections of tumor biopsy with a known reference diagnosis: • polyp (36 samples)—group 1; • carcinoma (36 samples)—group 2.

42

4 Azimuthally Invariant Mueller-Matrix Mapping of Optically …

4.3.1 Differentiation of Linear and Circular Birefringence Parameters The series of Figs. 4.7 and 4.8 are two-dimensional (fragments (1), (2)), statistical N (M44 , ΔM) (fragments (3), (4)) and lg J (M44 , ΔM) − lg ν (fragments (5), (6)) distributions of the values of the Mueller-matrix invariants M44 , ΔM of phase anisotropy of histological sections of a polyp biopsy (fragments (1), (3), (5)) and carcinomas (fragments (2), (4), (6)). A comparative analysis of the data shown in Figs. 4.7 and 4.8, discovered: Linear birefringence. For a histological section of a carcinoma tumor, the value of the Mueller-matrix element M 44 and the range of its change are smaller compared with a benign polyp sample (Fig. 4.7, fragments (1), (2)). Therefore, to distribute the values of the MMI characterizing the linear birefringence of a carcinoma sample, a

Fig. 4.7 Two-dimensional m × n (fragments (1), (2)), statistical N (M44 ) (fragments (3), (4)) and lg J (M44 ) − lg ν (fragments (5), (6)) the distribution of MMI M44 values of histological sections of a polyp biopsy (fragments (1), (3) (5)) and carcinomas (fragments (2), (4), (6))

4.3 Mueller-Matrix Differentiation of Fibrillar Networks of Biological …

43

Fig. 4.8 Two-dimensional m × n (fragments (1), (2)), statistical N (ΔM) (fragments (3), (4)) and lg J (ΔM) − lg ν (fragments (5), (6)) the distribution of MMI ΔM values of histological sections of a polyp biopsy (fragments (1), (3) (5)) and carcinomas (fragments (2), (4), (6))

decrease in the average (Z 1 ↓) and dispersion (Z 2 ↓) that characterize the histograms N (M44 ) is found. Moreover, the magnitude of the statistical moments of higher orders (asymmetry (Z 3 ↑) and excess (Z 4 ↑)) increases. From the physical point of view, the results can be associated with an increase in the phase modulation of laser radiation (M44 → min), which passes through the optically anisotropic fibrillar network of the histological section of carcinoma. It is known that the oncological state is characterized by the formation of more structured and geometrically larger newly formed tumor sprouts. Due to this, the linear birefringence of such biological crystallites increases and, conversely, the level of the corresponding Mueller-matrix image decreases M44 (m × n). This relationship was also established on the basis of the generalized anisotropy model (Sect. 3, relations (3.3), (3.4), (3.13), (3.15)) tested by computer simulation (Sect. 3, Figs. 3.2, 3.3, 3.7 and 3.8). In the framework of the fractal approach, the multifractality of the distributions of the values M44 (m × n) of the samples of both groups was revealed. As can be seen,

44

4 Azimuthally Invariant Mueller-Matrix Mapping of Optically …

the approximating curves V (η) of the logarithmic dependences of the power spectra lg J (M44 ) − lg ν of the distributions of the values of the phase matrix element M44 of the histological section of the polyp (Fig. 4.13, fragment (3)) are broken lines. In addition, due to the increase in the geometric dimensions of the newly formed fibrillar network, the dispersion D increases, characterizing the distribution of values lg J (M44 ) − lg ν determined for histological sections of rectal carcinoma. Circular birefringence. Other, compared with previously determined (Fig. 4.7), trends in the statistical moments of the 1st-4th orders that characterize the histograms N(ΔM) were revealed. This fact can be associated with an increase in the concentration of proteins necessary for the growth of a malignant tumor, the optical activity (θ ) of its optically anisotropic component increases. Quantitatively, such a scenario illustrates the growth of dispersion and the sharpness of the peak in the distribution of values N(ΔM) (Fig. 4.8, fragments (3), (6)). That is, the oncological state is characterized by such a scenario of changing the values of the set of statistical moments of the 1st–4th orders— Z 1 (ΔM) ↑, Z 2 (ΔM) ↑, Z 3 (ΔM) ↓, Z 4 (ΔM) ↓ . Fractal analysis revealed the transformation of multifractal distributions of the values ΔM(m × n) of samples from group 1 to random for group 2. Broken V (η) dependences lg J (ΔM) − lg ν of the histological section of the polyp (Fig. 4.8, fragment (3)) “turn” into curves without a certain angle of inclination for the dependences of the MMI, which characterize the optical activity of carcinoma samples (Fig. 4.8, fragment (6)).The revealed fact can be physically explained by the increase in the optical activity of the histological section from group 2.

4.3.2 Differentiation of Linear and Circular Dichroism Parameters The m )× n (fragments (1), (2)), statistical ) of studies of two-dimensional ( results ( N M14;41 (fragments (3), (4)) and lg J M14;41 − lg ν (fragments (5), (6)) distributions of MMI M14 , M41 values characterizing the amplitude anisotropy of histological sections of a polyp biopsy (fragments (1), (3), (5)) and carcinomas (fragments (2), (4), (6)) are presented in the series of Figs. 4.9 and 4.10. Analysis of the results revealed: Linear dichroism. The growth of optically anisotropic absorption by samples of a malignant tumor compared with samples of a benign polyp (Fig. 4.9, fragments (1), (2)). In addition, additional extremes of the histograms N (M14 ) of samples from group 2 are formed (Fig. 4.10, fragment (2)). Therefore, according to the analysis of the manifestations of linear dichroism Z 1 ↑, Z 2 ↑, Z 3 ↑, Z 4 ↑ there is a quantitative indicator of changes in the amplitude anisotropy associated with the oncological state, which characterize the histograms N (M14 ). The physical reason for the growth of linear dichroism can be considered the formation of a more structured and more geometrically scaled newly formed fibrillar network of the carcinoma layer. This

4.3 Mueller-Matrix Differentiation of Fibrillar Networks of Biological …

45

Fig. 4.9 Two-dimensional m × n (fragments (1), (2)), statistical N (M14 ) (fragments (3), (4)) and lg J (M14 ) − lg ν (fragments (5), (6)) the distribution of MMI M14 values of histological sections of a polyp biopsy (fragments (1), (3) (5)) and carcinomas (fragments (2), (4), (6))

relationship was analytically analyzed by computer simulation (Chap. 3, Figs. 3.4 and 3.9). Fractal analysis revealed an increase in the dispersion characterizing the distribution of the values of the logarithmic dependences of the power spectra log J (M14 ) − log ν of the matrix element M14 determined for the histological section of carcinoma (Fig. 4.9, fragment (3)). The optical-physical reason for this scenario is also associated with an increase in the geometric dimensions of the optically anisotropic fibrillar network of the malignant tumor. Circular dichroism. We have already noted that the growth of a malignant tumor requires an increase in the concentration of proteins. The indicated biochemical tendency is optically manifested in the growth of circular dichroism (Δg ↑) of the indicated molecular complexes. Therefore (Sect. 3, relations (3.5), (3.6)), quantitatively, this process of changing the amplitude anisotropy illustrates a decrease in the range of values M41 in the asymmetric distribution with an acute peak

46

4 Azimuthally Invariant Mueller-Matrix Mapping of Optically …

Fig. 4.10 Two-dimensional m × n (fragments (1), (2)), statistical N (M41 ) (fragments (3), (4)) and lg J (M41 ) − lg ν (fragments (5), (6)) the distribution of MMI M41 values of histological sections of a polyp biopsy (fragments (1), (3) (5)) and carcinomas (fragments (2), (4), (6))

distribution N (M41 ) (Fig. 4.9, fragments (3), (6)).That is, for the oncological condition, there is such a change in the values of the set of statistical moments: Z 1 (M41 ) ↓, Z 2 (M41 ) ↓, Z 3 (M41 ) ↑, Z 4 (M41 ) ↑ . For the Mueller-matrix images M41 (m × n) characterizing the manifestations of circular dichroism of benign tumor samples, a multifractal (V (η1 , η2 )) structure of their distributions was found (Fig. 4.10, fragment (3)). Similar distributions of MMI values determined for histological sections of malignant carcinoma are random (η /= const) (Fig. 4.10, fragment (6)). For the possible clinical use of the method of azimuthally independent Muellermatrix mapping of histological sections of biopsy of rectal wall tumors, the average values (within group 1 and group 2 -n 1 = n 2 = 36) of statistical moments Z i=1;2;3;4 and their standard deviations ±σ were determined—Table 4.2, as well as balanced accuracy (Ac)—Table 4.3.

0.99 ± 0.11

0.52 ± 0.041

0.81 ± 0.074

0.41 ± 0.037

Z3

Z4

D

0.34 ± 0.028

0.15 ± 0.012

0.43 ± 0.039

0.19 ± 0.015 1.23 ± 0.13

1.14 ± 0.11

0.11 ± 0.013

0.11 ± 0.009

Group 1

Z1

ΔM

Group 2

M44

Group 1

Z2

Parameters

1.04 ± 0.097

0.92 ± 0.086

0.14 ± 0.012

0.14 ± 0.013

Group 2

0.39 ± 0.025

0.99 ± 0.088

0.12 ± 0.011

0.11 ± 0.011

Group 1

M14

0.47 ± 0.038

1.28 ± 0.11

0.16 ± 0.015

0.13 ± 0.012

Group 2

1.52 ± 0.12

0.41 ± 0.038

0.09 ± 0.0074

0.13 ± 0.009

Group 1

M41

1.25 ± 0.11

0.34 ± 0.028

0.11 ± 0.008

0.16 ± 0.014

Group 2

Table 4.2 Statistical parameters characterizing the distribution of MMI values M44;14;41 (m × n); ΔM(m × n) of histological sections of a biopsy of rectal wall tumors (n 1 = n 2 = 36)

4.3 Mueller-Matrix Differentiation of Fibrillar Networks of Biological … 47

48

4 Azimuthally Invariant Mueller-Matrix Mapping of Optically …

Table 4.3 Balanced accuracy of the method of Mueller-matrix mapping of histological sections of biopsy of rectal wall tumors Parameters Ac(Z i ), %

Zi

M44

ΔM

M14

M41

Z1

58.3

61.1

63.9

66.7

Z2

55.6

58.3

61.1

63.9

Z3

72.2

75

77.8

80.6

Z4

72.2

75

77.8

80.6

Differentiation of changes in the anisotropy of samples of both groups, which is described by each MMI, was carried out by comparing the distribution histograms Z i=1;2;3;4 (M44 , ΔM, M14 , M41 ). If the average value of one or another statistical moment Z i=1;2;3;4 in the group of samples of benign tumors is outside the standard deviation σ of the group of samples of malignant tumors, then the difference is considered statistically significant. At the same time, the analysis of histogram Z i=1;2;3;4 overlap turns out to be relevant, which determines the sensitivity Se, specificity Sp and balanced accuracy Ac (Chap. 2, Sect. 2.6, relations (2.14)—(2.16)). Table 4.3 shows the values of the balanced accuracy of the azimuthally independent method of Mueller-matrix mapping of polycrystalline networks of histological sections of biopsy of rectal tissue tumors. A comparative analysis of the balanced accuracy values of the Mueller-matrix mapping method for the manifestations of optical anisotropy of histological sections of (highlighted { (of the ) rectum }wall revealed { ( optimal ) } { ( gray) ) parameters } { tissue M44 { Ac( Z 3;4 ) = 72.5%}, ΔM Ac Z 3;4 = 75% , M14 Ac Z 3;4 = 77.8% , M41 Ac Z 3;4 = 80.6% . The results obtained suggest a fairly high level of balanced accuracy of azimuthally independent Mueller-matrix mapping in the differentiation of changes in the optical anisotropy of benign and malignant tumors of human organs. According to the criteria of evidence-based medicine, the parameter R(M44 , ΔM) 70% meets satisfactory quality and R(M14 , M41 ) 80% meets good quality.

4.4 Mueller-Matrix Mapping of Blood-Filled Biological Tissues As the objects of study, histological sections of operationally taken biological tissues with two types of inflammation were used: acute (group 1—36 samples) and serous { appendicitis (group 2—36 samples). The results of measuring the MMI ΔM23;32 q ≡ M44 (m × n), ∑ characterizing the optical anisotropy of histological M22;33 sections of appendicitis of both groups are shown in Figs. 4.11 and 4.12.

4.4 Mueller-Matrix Mapping of Blood-Filled Biological Tissues

(a)

(c)

49

(b)

(d)

Fig. 4.11 2D (fragments (a), (b)) and 3D (fragments (c), (d))—distribution of the matrix element M44 of the histological sections of acute (fragments (a), (c)) and serous (fragments (b), (g)) appendicitis

A {comparative analysis of the data revealed an individual distribution structure ΔM23;32 , both in size (fragments (a), (b)) and in the topographic q ≡ M44 (m × n), ∑ M22;33 structure (fragments (c), (d)). For a sample from group 1, the most probable (P) values of these parameters ) ( ΔM23;32 0.3 ÷ 0.35; for a sample from group lie within P(M44 ) 0.45 ÷ 0.5 and P ∑ ) M22;33 ( ΔM23;32 0.1 ÷ 0.15. 2—P(M44 ) 0.8 ÷ 0.85 and P ∑ M22;33 The results can be associated with the “destruction” of the optical anisotropy of histological sections of group 2 (serous appendicitis) due to the morphological destruction of fibrillar networks. This process is most clearly manifested for optically active protein molecules that form optically anisotropic networks. Analytically, this scenario is reflected in the following trends in the optical anisotropy parameters { δ → 0, . Qualitatively, this is detected by the corresponding changes in the values θ →0 of the MMI

50

4 Azimuthally Invariant Mueller-Matrix Mapping of Optically …

(a)

(b)

(c)

(d)

Fig. 4.12 2D (fragments (a), (b)) and 3D (fragments (c), (d))—distribution of the MuellerΔM of histological sections of acute (fragments (a), (c)) and serous matrix rotational invariant ∑ M23;32 22;33 (fragments (b), (g)) appendicitis

⎧ ⎨ M44 (m × n) → 1, ∑ M22;33 (m × n) → 1, ⎩ ΔM23;32 (m × n) → 0.

(4.1)

It is easy to see that in the limiting case, the Mueller matrix of the histological section from group 2 is transformed into a single diagonal matrix of the optically ∥ ∥ ∥1 0 0 0∥ ∥ ∥ ∥0 1 0 0∥ ∥. ∥ isotropic layer {M} → ∥ ∥ ∥0 0 1 0∥ ∥0 0 0 1∥ For the possible { clinical application of the Mueller-matrix mapping technique for ΔM23;32 , it was tested within two statistically significant invariants q ≡ M44 (m × n), ∑ M22;33 groups (confidence interval p < 0.001) of samples of both types (Table 4.4).

4.4 Mueller-Matrix Mapping of Blood-Filled Biological Tissues

51

Table 4.4 Statistical moments of the 1st–4th orders characterizing the distribution of the MMI values of histological sections of acute and serous appendicitis Zi

Z1

ΔM23;32 ∑ M22;33

M44 (n 1 = n 2 = 36)

(n 1 = n 2 = 36)

Group 1

Group 2

Group 1

Group 2

0.47 ± 0.038

0.86 ± 0.075

0.19 ± 0.014

0.11 ± 0.011

Z2

0.09 ± 0.008

0.26 ± 0.023

0.17 ± 0.015

0.11 ± 0.009

Z3

1.23 ± 0.12

0.56 ± 0.046

0.83 ± 0.092

1.69 ± 0.15

Z4

2.19 ± 0.21

0.87 ± 0.082

1.07 ± 0.12

2.15 ± 0.21

Table 4.5 Operational characteristics of the Mueller-matrix mapping method ΔM23;32 ∑ M22;33

Zi

M44 Se, %

Sp, %

Ac, %

Se, %

Sp, %

Ac, %

Z1

84

72

78

78

72

75

Z2

94

68

81

84

74

79

Z3

90

76

83

82

74

78

Z4

92

74

83

84

72

78

Almost all statistical moments of the first and fourth orders, characterizing the ΔM23;32 , turned out to be sensitive to changes in distribution of the MMI M44 and ∑ M22;33 anisotropy (Table 4.5). The results can be associated with the destruction of optically anisotropic structures of serous appendicitis. The decrease in phase shifts (δ ↓) is accompanied by an increase in the magnitude and range of the matrix element (M44 = cos δ) ↑. Therefore, for the histogram of the distribution of random values of such a parameter, the average (Z 1 ↑), dispersion (Z 2 ↑) increases, and, conversely, the asymmetry (Z 3 ↓) and excess (Z 4 ↓) decrease. Another picture is observed for changing the values of statistical parameters (Z i ), ΔM23;32 tg2θ . which characterize the distribution of the values of MMI ∑ M22;33 The degradation of optical activity (θ ↓) is manifested in the formation of asymmetric (Z 3 ↑) with a sharp peak (Z 4 ↑) distribution of small (Z 1 ↓, Z 2 ↓) values ΔM23;32 in the plane of the histological section of serous appendicitis. ∑ M22;33 Thus, Mueller-matrix mapping of blood-filled biological tissues has proven effective in the differential diagnosis of inflammatory processes of appendicitis ΔM23;32 (Ac = 75% − 79%). tissue—M44 (balanced accuracy Ac = 78% − 83%) and ∑ M22;33

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4 Azimuthally Invariant Mueller-Matrix Mapping of Optically …

References 1. A. Ushenko, V. Pishak, Laser polarimetry of biological tissue: principles and applications, in Handbook of Coherent-Domain Optical Methods: Biomedical Diagnostics, Environmental and Material Science, ed. by V. Tuchin (2004), pp. 93–138 2. O. Angelsky, A. Ushenko, Y. Ushenko, V. Pishak, A. Peresunko, Statistical, correlation and topological approaches in diagnostics of the structure and physiological state of birefringent biological tissues, in Handbook of Photonics for Biomedical Science (2010), pp. 283–322 3. Y. Ushenko, T. Boychuk, V. Bachynsky, O. Mincer, Diagnostics of structure and physiological state of birefringent biological tissues: statistical, correlation and topological approaches, in Handbook of Coherent-Domain Optical Methods, ed. by V. Tuchin (Springer Science+Business Media, 2013) 4. O. Angelsky, A. Ushenko, Y. Ushenko, Investigation of the correlation structure of biological tissue polarization images during the diagnostics of their oncological changes. Phys. Med. Biol. 50(20), 4811–4822 (2005) 5. V. Ushenko, O. Dubolazov, A. Karachevtsev, Two wavelength Mueller matrix reconstruction of blood plasma films polycrystalline structure in diagnostics of breast cancer. Appl. Opt. 53(10), B128 (2016) 6. Y. Ushenko, G. Koval, A. Ushenko, O. Dubolazov, V. Ushenko, O. Novakovskaia, Muellermatrix of laser-induced autofluorescence of polycrystalline films of dried peritoneal fluid in diagnostics of endometriosis. J. Biomed. Opt. 21(7), 071116 (2016) 7. A. Ushenko, A. Dubolazov, V. Ushenko, O. Novakovskaya, Statistical analysis of polarizationinhomogeneous Fourier spectra of laser radiation scattered by human skin in the tasks of differentiation of benign and malignant formations. J. Biomed. Opt. 21(7), 071110 (2016) 8. V. Ushenko, N. Pavlyukovich, L. Trifonyuk, Spatial-frequency azimuthally stable cartography of biological polycrystalline networks. Int. J. Optics 2013, 1–7 (2013) 9. S. Manhas, M.K. Swami, P. Buddhiwant, N. Ghosh, P.K. Gupta, K. Singh, Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry. Opt. Exp. 14, 190–202 (2006) 10. Y. Deng, S. Zeng, Q. Lu, Q. Luo, Characterization of backscattering Mueller matrix patterns of highly scattering media with triple scattering assumption. Opt. Exp. 15, 9672–9680 (2007) 11. S.Y. Lu, R.A. Chipman, Interpretation of Mueller matrices based on polar decomposition. J. Opt. Soc. Am. A 13, 1106–1113 (1996) 12. Y. Guo, N. Zeng, H. He, T. Yun, E. Du, R. Liao, H. Ma, A study on forward scattering Mueller matrix decomposition in anisotropic medium. Opt. Exp. 21, 18361–18370 (2013) 13. A. Pierangelo, S. Manhas, A. Benali, C. Fallet, J.L. Totobenazara, M.R. Antonelli, P. Validire, Multispectral Mueller polarimetric imaging detecting residual cancer and cancer regression after neoadjuvant treatment for colorectal carcinomas. J. Biomed. Opt. 18, 046014 (2013) 14. V.P. Ungurian, O.I. Ivashchuk, V.O. Ushenko, Statistical analysis of polarizing maps of blood plasma laser images for the diagnostics of malignant formations. Proc. SPIE 8338, 83381L (2011) 15. V.A. Ushenko, O.V. Dubolazov, A.O. Karachevtsev, Two wavelength Mueller matrix reconstruction of blood plasma films polycrystalline structure in diagnostics of breast cancer. Appl. Opt. 53, B128–B139 (2014) 16. V.P. Prysyazhnyuk, Y.A. Ushenko, A.V. Dubolazov, A.G. Ushenko, V.A. Ushenko, Polarization-dependent laser autofluorescence of the polycrystalline networks of blood plasma films in the task of liver pathology differentiation. Appl. Opt. 55, B126–B132 (2016) 17. V.A. Ushenko, M.S. Gavrylyak, Azimuthally invariant Mueller-matrix mapping of biological tissue in differential diagnosis of mechanisms protein molecules networks anisotropy. Proc. SPIE 8812, 88120Y (2013) 18. V.A. Ushenko, M.P. Gorsky, Complex degree of mutual anisotropy of linear birefringence and optical activity of biological tissues in diagnostics of prostate cancer. Opt. Spectrosc. 115, 290–297 (2013)

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Chapter 5

Azimuthally Invariant Mueller-Matrix Reconstruction of the Optical Anisotropy Parameters of the Polycrystalline Structure of Biological Tissues and Human Fluids

5.1 Mueller-Matrix Reconstruction of the Distribution of Parameter Values Characterizing Birefringence and Dichroism of Optically Anisotropic Networks of Biological Tissues in a Precancerous State As the objects of study, we used optically thin (geometric thickness d ≈ 25 µm ÷ 30 µm, attenuation coefficient τ ≺ 0.1) histological sections of a biopsy of cervical tissue—endometrium in a precancerous state of two types: • simple endometrial atrophy—group 1 (36 samples); • endometrial polyp—group 2 (36 samples). The series of Figs. 5.1, 5.2, 5.3 and 5.4 show the results of the Mueller-matrix reconstruction of the distributions of the values of the parameters of the phase (δ) and amplitude (Δτ ) anisotropy of the endometrium with simple atrophy (Figs. 5.1 and 5.2) and with a polyp (Figs. 5.3 and 5.4). Each figure consists of coordinate (fragments (1), (2)) and topographic (fragments (3)) distributions of phase shift values δ (Figs. 5.1 and 5.2) and the coefficient of linear dichroism Δτ (Figs. 5.3 and 5.4). Topographic distributions are a system of lines of equal values. [δ = 0.1 π ](m × n) and [Δτ = 0.5](m × n). Let us analyze the obtained distributions of values δ(m × n) and Δτ (m × n) polycrystalline networks of histological sections of the endometrium in the framework of the statistical approach. A comparative analysis of such polarized reproducible distributions revealed a commensurate range of phase shift values in the plane of histological sections of samples of both types (Figs. 5.1 and 5.2, fragments (1)). Quantitatively, this fact illustrates the proximity of the magnitude of statistical moments of the 1st–4th order, characterizing the distribution of the values of phase shifts introduced by histological sections of the endometrium: © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 L. Trifonyuk et al., Optical Anisotropy of Biological Polycrystalline Networks, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-99-1087-8_5

55

56

5 Azimuthally Invariant Mueller-Matrix Reconstruction of the Optical …

Fig. 5.1 Two-dimensional 2D − δ(m × n), three-dimensional 3D − δ(m × n) and topographic [δ = 0.1π ](m × n) distributions of phase shift values δ, which are formed by a histological section of endometrial biopsy with simple atrophy

Fig. 5.2 Two-dimensional 2D − δ(m × n), three-dimensional 3D − δ(m × n) and topographic [δ = 0.1 π ](m × n) distribution of the values of phase shifts δ, which are formed by a histological section of the endometrial polyp

• with atrophy—( Z 1 (δ) = 0.24, Z 2 (δ) = 0.12, Z 3 (δ) = 0.85, Z 4 (δ) = 1.29 ); • polyp—( Z 1 (δ) = 0.21, Z 2 (δ) = 0.09, Z 3 (δ) = 1.26, Z 4 (δ) = 1.91 ). As you can see, the maximum differences between the values of the statistical moments of the 1st and 2nd order Z i=1;2 (δ) do not exceed 30–35%. For the magnitude of statistical moments of higher orders Z i=3;4 (δ), the range of differences increases— up to 50%.

5.1 Mueller-Matrix Reconstruction of the Distribution of Parameter Values …

57

Fig. 5.3 Two-dimensional 2D − Δτ (m × n), three-dimensional 3D − Δτ (m × n) and topographic [Δτ = 0.1π ](m × n) distributions of the values of the linear dichroism Δτ parameter of a histological section of an endometrial biopsy with simple atrophy

Fig. 5.4 Two-dimensional 2D −Δτ (m × n), three-dimensional 3D −Δτ (m × n) and topographic [Δτ = 0.1 π ](m × n) distributions of the values of the linear dichroism Δτ parameter of the histological section of the biopsy of the endometrial polyp

A statistical analysis of coordinate distributions (Figs. 5.3 and 5.4, fragments (1), (2)) of the values of the linear dichroism Δτ (m × n) index of the endometrial layers in the precancerous state revealed a slightly larger range (up to 60–70%) of the differences between the values Z i=1;2;3;4 (Δτ ). The following data on the linear dichroism of histological endometrial sections were obtained:

58

5 Azimuthally Invariant Mueller-Matrix Reconstruction of the Optical …

Table 5.1 Statistical moments of the 1st–4th order characterizing the distribution of the values of the parameters of linear birefringence and dichroism of histological sections of the endometrium Zi

δ, n 1 = n 2 = 36

Δτ , n 1 = n 2 = 36

Atrophy

Polyp

Atrophy

Polyp

Z1

0.23 ± 0.019

0.22 ± 0.018

0.48 ± 0.037

0.53 ± 0.041

Z2

0.13 ± 0.011

0.16 ± 0.014

0.18 ± 0.013

0.23 ± 0.021

Z3

1.23 ± 0.13

0.88 ± 0.072

0.69 ± 0.058

0.95 ± 0.089

Z4

1.81 ± 0.17

1.27 ± 0.11

0.81 ± 0.077

1.04 ± 0.099

• with atrophy— Z 1 (Δτ ) = 0.49, Z 2 (Δτ ) = 0.24, Z 3 (Δτ ) = 0.63, Z 4 (Δτ ) = 0.84 ; • polyp— Z 1 (Δτ ) = 0.52, Z 2 (Δτ ) = 0.19, Z 3 (Δτ ) = 0.91, Z 4 (Δτ ) = 1.09 . As can be seen, for the polyp biopsy sample, there is a decrease (Z 2 ↓) in the statistical moment of the 2nd order (1.26 times) and an increase in the magnitude of the statistical moments of the 3rd (Z 3 ↑—1.44 times) and 4th (Z 4 ↑—1.3 times) orders. This fact indicates a higher sensitivity (compared with the reconstruction of the distributions of the linear birefringence) (the range of differences between the statistical moments characterizing the distribution of the values of the anisotropy parameters) of the polarization reproduction of the linear dichroism Δτ parameter. The values of statistical moments Z i=1;2;3;4 (δ) and Z i=1;2;3;4 (Δτ ) averaged within both groups of samples are shown in Table 5.1. The value of the balanced accuracy of the method of Mueller-matrix differentiation of optical anisotropy of biological layers that are in a precancerous state are shown in Table 5.2. This method is aimed at differentiating the manifestations of optical anisotropy of samples very close in morphological structure that are found in a precancerous state. Therefore, the achieved level of accuracy (Ac(δ) = 75%, Ac(Δτ ) = 77.8%) in terms of evidence-based medicine corresponds to a good level. Based on this, the next step in the study was the development of an azimuthally independent Mueller-matrix method for reproducing the polycrystalline structure of biological fluid films with the aim of early detection of objective criteria for systemic and oncological diseases. Table 5.2 Balanced accuracy of the method of Mueller-matrix reconstruction of the polycrystalline structure of histological sections of the endometrial biopsy

Zi

Ac(δ), %

Ac(Δτ ), %

Z1

61.1

61.1

Z2

61.1

63.9

Z3

75

77.8

Z4

75

77.8

5.2 “Two-Wave” Mueller-Matrix Reconstruction of the Distribution …

59

5.2 “Two-Wave” Mueller-Matrix Reconstruction of the Distribution of Optical Anisotropy Parameter Values of Polycrystalline Films of Biological Fluids (Methodological Justification) In this part of the work, a model of the mechanisms of optical anisotropy of polycrystalline films of bile and blood plasma is presented. The informational possibilities of Mueller-matrix reproduction in two spectrally selective (blue λ2 = 0.405 µm and red λ1 = 0.614 µm laser radiation) parameter ranges of phase (linear and circular birefringence) and amplitude (linear and circular dichroism) anisotropy of polycrystalline bile and blood plasma films are considered. In the framework of statistical analysis, a set of statistical moments of the 1st– 4th orders characterizing the distribution of optical anisotropy parameter values is determined. On this basis, the criteria for differentiation of samples were found: donors; patients with type II diabetes; and patients with breast cancer, and a balanced accuracy of diagnosis of such conditions has been established. This material is a development of research [1–12], which examined the fundamental, applied and diagnostic aspects of laser polarimetry of blood plasma films taken from healthy and cancer patients: Fundamental. The polycrystalline structure of optically thin, non-depolarizing films of blood plasma and other biological fluids was considered in the linear birefringence approximation of needle-shaped crystals of amino acids, bilirubin, bile acids, etc. Model concepts were expanded taking into account the optical activity of such molecules. The analysis of experimental data was carried out in the framework of the search for solutions to the direct problem—the conversion of the polarization of laser radiation by objects of study. At the same time, until recently, the question of polarization reconstruction of the distributions of the optical anisotropy parameters of such films remains open. The indicated inverse problem requires taking into account the influence of not only phase mechanisms, but also amplitude anisotropy—optically anisotropic absorption. Applied. We used the methods of polarization (distribution of azimuths and elliptic polarization) and Mueller-matrix (distribution of the values of matrix elements) mapping of laser images of biological fluid films. In order to determine the objective parameters of the change in optical anisotropy, a statistical analysis of the obtained polarization maps and Mueller-matrix images was used. To deepen the accuracy of laser polarimetry, we used the methods of singular (determining the distributions of the characteristic values of the Mueller-matrix images), scale-selective wavelet analysis of the obtained data. The main drawback of polarimetric techniques was the azimuthal dependence and, as a consequence of this, poor reproducibility of the obtained data. Diagnostic. The possibility of differentiating blood plasma samples of healthy and sick patients with high sensitivity (a large range of differences between the values of statistical moments characterizing the distribution of the values of the matrix

60

5 Azimuthally Invariant Mueller-Matrix Reconstruction of the Optical …

elements) was demonstrated. For clinical use, this indicator alone is not enough. A necessary factor is the requirement of not only high sensitivity, but also stable reproducibility of the results. In order to experimentally separate (differentiate) the manifestations of the mechanisms of phase and amplitude optical anisotropy, we used the spectrally selective approach using short- and long-wave radiation from two semiconductor lasers. In the visible spectrum of the optical radiation, the manifestations of phase and amplitude anisotropy are “spaced” into different spectral regions. In the red region of the spectrum, the mechanisms of circular and linear birefringence of amino acid molecules and their complexes substantially prevail. The maximum optically anisotropic absorption (Δτ → 0, C → 1) of protein molecules lies in the ultraviolet region of the spectrum. That is, in the red region (λ1 = 0.614) of the spectrum, linear and circular dichroism is practically absent and the condition Δτ → 1, C → 0 is satisfied (Sect. 3, relation (3.12)). In this approximate symmetry of the matrix of generalized anisotropy, (relation (3.9)) is simplified to the form 1 0 0 0 0 f 22 f 23 f 24 {F(λ1 )} = . An analytical solution to this approximation is unique 0 f 32 f 33 f 34 0 f 42 f 43 f 44 (independent of the switching of partial phase anisotropy matrices—relations (3.1)– (3.4), (3.13), (3.15)) polarization reproducing algorithms for linear (δ) and circular (θ ) birefringence (relation (3.14)). To analyze the manifestations of amplitude anisotropy, we chose the shortwavelength region of the spectrum, where the maxima of the optically anisotropic absorption of protein molecules of biological films are localized. A semiconductor laser with a wavelength λ2 = 0.45 µm was used as a probing source. For this situation, when analyzing the experimentally measured Mueller matrices (relation (3.9)), an actual account of all types of anisotropies characterizing biological polycrystalline films.

5.3 “Two-Wave” Mueller-Matrix Reconstruction of the Distribution …

61

5.3 “Two-Wave” Mueller-Matrix Reconstruction of the Distribution of Optical Anisotropy Parameter Values of Polycrystalline Films of Biological Fluids (Experimental Results) 5.3.1 Determination of a Statistically Significant Representative Sample of Patients with Known (Reference) Diagnosis The diagnosis of patients (breast cancer or type II diabetes) (group 2) was determined by the gold standard method—biopsy of an operationally removed tumor and histochemical analysis data. As a control group (group 1), previously examined donors were used. Using the StatMate software product, a statistically significant number of patients—n = 57 was established for a 95% confidence interval ( p ≺ 0.05).

5.3.2 Checking the “Stability” of Polarization Reconstruction Algorithms The analytically obtained “two-wave” values of the Mueller-matrix invariants (relations (3.18), (3.19) and (3.22), (3.23)) were checked for compliance with the model relations (3.20) and (3.24). It is found that conditions (3.20) are satisfied to within 10%, which determines the adequacy of the polarization reconstruction of the amplitude anisotropy parameters (relations (3.18), (3.19)). For another boundary condition (3.24), the deviations reach 30–40%. Therefore, in the future, we used phase reconstruction algorithms (relation (3.14)), as well as expressions (3.18), (3.19) for reproducing the amplitude anisotropy parameters.

5.3.3 Polarization Reconstruction of the Optical Anisotropy Parameters of Plasma Films Blood Series of Figs. 5.5, 5.6, 5.7 and 5.8 illustrates the results of the Mueller-matrix reconstruction of the parameters δ; θ ; Δτ ; C of polycrystalline blood plasma films for three randomly selected samples. A comparative analysis of the data revealed trends common for the samples studied: Linear birefringence. It was found that polycrystalline films of donor blood plasma (Fig. 5.5, fragments (1)) are characterized by a higher degree of birefringence of the networks of needle-shaped albumin crystals in comparison with samples (Fig. 5.5, fragments (3))) of blood from cancer patients. The main extremes of the

62

5 Azimuthally Invariant Mueller-Matrix Reconstruction of the Optical …

Fig. 5.5 Coordinate ((1), (3)) and probabilistic ((2), (4)) distributions of phase shifts δ formed by a polycrystalline film of blood plasma of a group of donors ((1), (2)) and cancer patients ((3), (4))

Fig. 5.6 Coordinate ((1), (3)) and probabilistic ((2), (4)) distributions of rotations of the plane of polarization θ formed by a polycrystalline film of blood plasma of a group of donors ((1), (2)) and cancer patients ((3), (4))

Fig. 5.7 Coordinate ((1), (3)) and probabilistic ((2), (4)) distributions of linear dichroism Δτ index values formed by a polycrystalline film of blood plasma of a group of donors ((1), (2)) and cancer patients ((3), (4))

5.3 “Two-Wave” Mueller-Matrix Reconstruction of the Distribution …

63

Fig. 5.8 Coordinate ((1), (3)) and probabilistic ((2), (4)) distributions of the values of the circular dichroism Δg index formed by a polycrystalline film of blood plasma of a group of donors ((1), (2)) and cancer patients ((3), (4))

histograms N (δ) of group 1 (Fig. 5.5, fragments (2)) are localized in the region δ = (0.25 ÷ 0.35) × 10−1 π . For group 2, the most probable (Fig. 5.5, fragments (4)) lower values of phase shifts—δ → (0.05 ÷ 0.1) × 10−1 π . Therefore, a statistical indicator of the oncological state was a decrease in the average (Z 1 ↓) and dispersion (Z 2 ↓), which characterize the histograms N (δ). At the same time, the magnitudes of the statistical moments of higher orders (asymmetry (Z 3 ↑) and excess (Z 4 ↑)) increase. From a physical point of view, the results can be associated with the data of biochemical analysis—for healthy patients, the concentration of albumin in the polycrystalline film of blood plasma is higher. So, the modulation (Chap. 3, relations (3.3), (3.4) of phase shifts between the orthogonal components of the amplitude by a network of needle-shaped birefringent albumin crystals is higher for donors than for sick patients. Circular birefringence. Inverse trends, compared with the phase shifts δ distributions, are observed in statistical changes in the parameter θ (Fig. 5.6) characterizing the optical activity of block-like crystals of globulin proteins (Chap. 3, relations (3.1), (3.2)) in polycrystalline blood plasma films. This fact can be associated with an increase in the concentration of this protein in the blood plasma of cancer patients. Therefore, the probability of large values of the rotation of the plane of polarization θ , which is formed by blood plasma films from group 2, increases (Fig. 5.10, fragments (3)). Quantitatively, this illustrates the increase in the probability of the formation of large values of the optical activity θ = (0.3 ÷ 0.45) × 10−1 π parameter in the distributions N (θ ) (Fig. 5.10, fragments (4)). In other words, for the oncological condition, the following changes in the magnitude of statistical moments take place Z 1 (θ ) ↑, Z 2 (θ ) ↑, Z 3 (θ ) ↓, Z 4 (θ ) ↓ . Linear dichroism. For polycrystalline films of donor blood plasma, a higher level of linear dichroism was revealed (Fig. 5.7, fragments (1)). The probability of the values of the indicator of this mechanism of amplitude anisotropy in the range Δτ = 0.85 ÷ 0.95 of 1.5–2 times more (Fig. 5.7, fragments (2)) is compared with the linear

64

5 Azimuthally Invariant Mueller-Matrix Reconstruction of the Optical …

dichroism of samples of plasma films of blood of patients with cancer (Fig. 5.7, fragments (4)). The indicated transformation of the histograms N (Δτ ) is illustrated by such a tendency for changes in the set of statistical moments of the 1st–4th orders Z 1 (Δτ ) ↓, Z 2 (Δτ ) ↓, Z 3 (Δτ ) ↑, Z 4 (Δτ ) ↑ . Circular dichroism. The polarization reconstruction of the indicator of circular dichroism (Fig. 5.8) revealed an increase in its value for blood plasma films of cancer patients (Fig. 5.8, fragments (3)). This fact can be associated with a higher concentration of globulin type proteins in blood plasma from group 2. The corresponding histograms N (Δg) (Fig. 5.8, fragments (4)) are characterized by a greater (1.5– 2.5 times) level of probability of the magnitude of circular dichroism in the area Δg = (0.65 ÷ 0.85) × 10−2 . The consequence of this is an increase in the average Z 1 (Δg) ↑ and dispersion Z 2 (Δg) ↑ compared to similar statistical moments characterizing the histograms N (Δg) (Fig. 5.8, fragments (2)). The opposite tendency is realized for the magnitude of the statistical moments of higher orders Z 3 (C) ↓ and Z 4 (C) ↓. Statistically averaged within both groups of samples values of statistical moments, Z i=1;2;3;4 (δ), Z i=1;2;3;4 (θ ), Z i=1;2;3;4 (Δg) and Z i=1;2;3;4 (Δτ ) are shown in Table 5.3. A comparative analysis of the data obtained (Table 5.3 and Fig. 5.9) found that the differences between the average Z i=1;2;3;4 (q) moments of all orders are statistically significant. At the same time, for all histograms of distributions of random values of statistical moments of the 1st–4th orders N (Z i ) there is an intergroup overlap. Moreover, the magnitude of the range of such overlap is inversely proportional to the magnitude of the difference between the averages Z i=1;2;3;4 (q). We have established the following differences between the average statistical moments Z i (q): • Linear birefringence—for the Mueller-matrix reconstructed distributions of the phase shift values δ(λ1 ) converted by polycrystalline plasma films of both groups, the differences between the average values of the statistical moments Z i=1;2;3;4 (δ) are {ΔZ 1 (δ) = 1.26, ΔZ 2 (δ) = 1.27, ΔZ 3 (δ) = 1.53, ΔZ 4 (δ) = 1.56. • Circular birefringence—for average values of statistical moments Z i=1;2;3;4 (θ ) characterizing the distribution of rotations of the plane of polarization of laser radiation θ (λ1 ) formed by optically active globulin structures in blood plasma films of both groups of patients, it was found:{ΔZ 1 (θ ) = 1.25, ΔZ 2 (θ ) = 1.17, ΔZ 3 (θ ) = 1.57, ΔZ 4 (θ ) = 1.47. • Linear dichroism—for the distributions Δτ (λ2 ) of the magnitude of the index of amplitude anisotropy of polycrystalline films of blood plasma, the following differences were determined Z i=1;2;3;4 (Δτ ): {ΔZ 1 (Δτ ) = 1.27, ΔZ 2 (Δτ ) = 1.31, ΔZ 3 (Δτ ) = 1.66, ΔZ 4 (Δτ ) = 1.69. • Circular dichroism—the distributions of the values Δg(λ2 ) index differ so of the amplitude anisotropy {ΔZ 1 (Δg) = 1.27, ΔZ 2 (Δg) = 1.29, ΔZ 3 (Δg) = 1.68; ΔZ 4 (Δg) = 1.71.

0.19 ± 0.012

0.15 ± 0.012

1.33 ± 0.12

0.92 ± 0.089

0.24 ± 0.019

0.19 ± 0.016

0.87 ± 0.071

0.59 ± 0.047

Z3

Z4

1.43 ± 0.13

1.69 ± 0.14

0.12 ± 0.011

0.12 ± 0.011

Norm

Z1

θ (λ1 ) (n = 57) Norm

Cancer

δ(λ1 ) (n = 57)

Z2

q

0.97 ± 0.088

1.08 ± 0.12

0.14 ± 0.012

0.15 ± 0.013

Cancer

0.36 ± 0.031

0.48 ± 0.031

0.21 ± 0.018

0.19 ± 0.015

Norm

Δτ (λ2 ) (n = 57)

0.61 ± 0.049

0.79 ± 0.064

0.16 ± 0.013

0.15 ± 0.012

Cancer

1.18 ± 0.14

1.38 ± 0.13

0.17 ± 0.013

0.11 ± 0.013

Norm

C(λ2 ) (n = 57)

0.69 ± 0.059

0.82 ± 0.077

0.22 ± 0.017

0.14 ± 0.012

Cancer

Table 5.3 Average (Z i=1;2;3;4 ) and standard deviations (±σ ) of the statistical moments characterizing the distribution of optical anisotropy parameters of blood plasma films

5.3 “Two-Wave” Mueller-Matrix Reconstruction of the Distribution … 65

66

5 Azimuthally Invariant Mueller-Matrix Reconstruction of the Optical …

( ) Fig. 5.9 Histograms of the distributions N Z i of the average values of statistical moments of the 1st–4th orders within both groups of blood plasma films

Fig. 5.10 Coordinate ((1), (3)) and probabilistic ((2), (4)) distributions of phase shifts δ formed by a polycrystalline bile film of a group of donors ((1), (2)) and patients with type II diabetes ((3), (4))

As can be seen from the data presented, the most sensitive (ΔZ i=1;2;3;4 (q) = max) were the values of statistical moments of the 3rd and 4th orders, which characterize the histograms N (q) of plasma films of blood groups of both groups of patients. On the other hand, the more ΔZ i=1;2;3;4 (q), the less “overlap” histograms. Therefore, for such parameters, the Mueller method of matrix reproduction of the distributions of optical anisotropy parameter values turned out to be more informative ( Se ↑, Sp ↑, Ac ↑ —Chap. 2, Sect. 2.6).

5.3 “Two-Wave” Mueller-Matrix Reconstruction of the Distribution …

67

Table 5.4 Operational characteristics of the method of Muller-matrix reconstruction of the polycrystalline structure of blood plasma films q

Zi

δ(λ1 ) (%)

θ (λ1 ) (%)

Δτ (λ2 ) (%)

C(λ2 ) (%)

Se(Z i )

Z1

66.6

71.9

77.2

77.2

Z2

70.2

70.2

78.9

82.4

Z3

82.4

85.9

91.2

92.9

Z4

80.7

89.4

94.7

94.7

Z1

63.1

66.6

73.6

73.6

Z2

66.6

66.6

75.4

78.9

Z3

80.7

82.4

87.8

89.4

Z4

77.2

85.9

91.2

92.9

Z1

64.85

69.25

75.4

75.4

Z2

68.4

68.4

77.15

80.65

Z3

81.55

84.15

89.5

91.15

Z4

78.95

87.65

92.95

93.8

Sp(Z i )

Ac(Z i )

Table 5.4 shows data on the information content of the azimuthally stable method of the Mueller-matrix reconstruction of the phase and amplitude anisotropy of polycrystalline blood plasma films. A comparative analysis of the operational characteristics of the two-wave Muellermatrix reconstruction method of the polycrystalline structure of blood plasma films revealed the following optimal parameters (highlighted in gray—Table 5.4) from a clinical point of view: ⎧ ) ( ) ( ) } { ( δ(λ1 ) → R(δ) ≡ Se Z 3;4 = 81% − 82%, Sp Z 3;4 = 77% − 81%; Ac Z 3;4 = 79% − 82% , ⎪ ⎪ ) ( ) ( ) } { ( ⎪ ⎨ θ (λ1 ) → R(θ ) ≡ Se Z 3;4 = 86% − 89%, Sp Z 3;4 = 83% − 86%; Ac Z 3;4 = 84% − 88% ) ( ) ( ) } { ( ⎪ Δτ (λ2 ) → R(Δτ ) ≡ Se Z 3;4 = 91% − 95%, Sp Z 3;4 = 88% − 92%; Ac Z 3;4 = 90% − 93% , ⎪ ⎪ ) ( ) ( ) } { ( ⎩ C(λ2 ) → R(C) ≡ Se Z 2;3;4 = 93% − 95%, Sp Z 3;4 = 90% − 93%; Ac Z 2;3;4 = 91% − 94% .

The results obtained allow us to speak of a high level of balanced accuracy (Chap. 2, Sect. 2.6.3, relations (2.14)) of the proposed “Two-wave” method of azimuthally stable Mueller-matrix mapping. According to the criteria of evidencebased medicine R(δ, θ ) 80%, the parameters correspond to good quality, and R(Δτ, C) 90% − 95%—to the high quality of the diagnostic test.

5.3.4 Polarization Reconstruction of the Optical Anisotropy Parameters of Bile Films of Man In this paragraph of the monograph, the results of applying the “Two-wave” method of azimuthally invariant Mueller-matrix mapping for another type of biological fluids—bile films—are given. Such an object has a different polycrystalline

68

5 Azimuthally Invariant Mueller-Matrix Reconstruction of the Optical …

structure compared to blood plasma films. From an optical point of view, bile is a multicomponent phase-inhomogeneous liquid, which consists of three main fractions: • optically isotropic—optically homogeneous micellar solution with a small number of cylindrical epithelial cells, white blood cells, big round cells, mucus; • optically anisotropic—liquid crystalline phase consisting of a combination of three types of liquid crystals: needle crystals of fatty acids, crystals of cholesterol monohydrate, crystals of calcium bilirubinate; • optically crystalline—solid crystalline phase formed due to dendritic and disclination crystallization mechanisms. The dendritic crystallization mechanism leads to the transformation of the liquid crystal optic-anisotropic fraction into a set of solid needle-shaped optically uniaxial birefringent crystals (Figs. 5.10, 5.11, 5.12, and 5.13). The values of the statistical moments Z i=1;2;3;4 (δ), Z i=1;2;3;4 (θ ), Z i=1;2;3;4 (Δg) and Z i=1;2;3;4 (Δτ ) characterizing the phase and amplitude anisotropy of such layers averaged within both groups of samples of bile films are shown in Table 5.5. A comparative analysis of the obtained data (Table 5.5) found that the statistical moments of the 1st and 2nd orders, which characterize the histograms N (q) of bile films of both groups of patients, are the most sensitive (ΔZ i=1;2;3;4 (q) = max) to changes in the distributions of phase and amplitude anisotropy parameters. Therefore, for such parameters, the Mueller-matrix method for reproducing the

Fig. 5.11 Coordinate ((1), (3)) and probabilistic ((2), (4)) distributions of phase shifts θ formed by a polycrystalline bile film of a group of donors ((1), (2)) and patients with type II diabetes ((3), (4))

Fig. 5.12 Coordinate ((1), (3)) and probabilistic ((2), (4)) distributions of linear dichroism Δτ parameter values formed by a polycrystalline bile film of a group of donors ((1), (2)) and patients with type II diabetes ((3), (4))

5.3 “Two-Wave” Mueller-Matrix Reconstruction of the Distribution …

69

Fig. 5.13 Coordinate ((1), (3)) and probabilistic ((2), (4)) distributions of circular dichroism Δg parameter values formed by a polycrystalline bile film of a group of donors ((1), (2)) and patients with type II diabetes ((3), (4))

optical anisotropy parameters of bile layers turned out to be more informative ( Se ↑; Sp ↑; Ac ↑ )—Table 5.6. The following optimal parameters were obtained (highlighted in gray—Table 5.6) from a clinical point of view ⎧ ⎪ ⎪ ⎨

δ(λ1 ) → R(δ) ≡{{Ac(Z ( 12)) = 78.95% − 81.5%},} θ (λ1 ) → R(θ ) ≡ Ac { Z (1;2 =) 84.15% − 87.65% ,} ⎪ → R(Δτ ≡ Δτ ) ) (λ 2 ⎪ { Ac( Z 1;2 ) = 89.5% − 92.95%}, ⎩ Δg(λ2 ) → R(Δg) ≡ Ac Z 1;2 = 91.15% − 93.8% . The results obtained suggest a high level of balanced accuracy of the proposed “Two-wave” method for reconstructing the parameters of polycrystalline bile films. According to the criteria of evidence-based medicine, the parameters R(δ, θ ) 80% correspond to good quality and R(Δτ, Δg) 90% − 95%—to the high quality of the diagnostic test.

0.49 ± 0.037

0.25 ± 0.019

0.44 ± 0.039

0.39 ± 0.027

0.34 ± 0.029

0.17 ± 0.012

0.57 ± 0.051

0.45 ± 0.033

Z3

Z4

0.54 ± 0.045

0.69 ± 0.044

0.16 ± 0.013

0.21 ± 0.016

Norm

Norm

Z1

θ (λ1 ) (n = 57)

Type II diabetes

δ(λ1 ) (n = 57)

Z2

q

0.47 ± 0.037

0.58 ± 0.051

0.21 ± 0.016

29 ± 0.18

Type II diabetes

0.31 ± 0.027

0.41 ± 0.039

0.21 ± 0.018

0.31 ± 0.025

Norm

Δτ (λ2 ) (n = 57)

0.26 ± 0.022

0.32 ± 0.024

0.29 ± 0.023

0.44 ± 0.038

Type II diabetes

0.61 ± 0.054

0.71 ± 0.066

0.15 ± 0.013

0.21 ± 0.017

Norm

C(λ2 ) (n = 57)

0.54 ± 0.049

0.65 ± 0.051

0.23 ± 0.017

0.29 ± 0.019

Type II diabetes

Table 5.5 The average (Z i=1;2;3;4 ) and standard deviations (±σ ) of the statistical moments Z i=1;2;3;4 characterizing the distribution of the optical anisotropy of the bile films

70 5 Azimuthally Invariant Mueller-Matrix Reconstruction of the Optical …

References

71

Table 5.6 Balanced accuracy of the method of Mueller-matrix reconstruction of the polycrystalline structure of bile films q Ac(Z i )

Zi

δ(λ1 ) (%)

θ (λ1 ) (%)

Δτ (λ2 ) (%)

Δg(λ2 ) (%)

Z1

81.55

84.15

89.5

91.15

Z2

78.95

87.65

92.95

93.8

Z3

64.85

69.25

75.4

69.25

Z4

68.4

68.4

77.15

75.4

References 1. Y.A. Ushenko, G.D. Koval, A.G. Ushenko, O.V. Dubolazov, V.A. Ushenko, O.Y. Novakovskaia, Mueller-matrix of laser-induced autofluorescence of polycrystalline films of dried peritoneal fluid in diagnostics of endometriosis. J. Biomed. Opt. 21(7), 071116 (2016) 2. A.G. Ushenko, O.V. Dubolazov, V.A. Ushenko, O.Yu. Novakovskaya, O.V. Olar, Fourier polarimetry of human skin in the tasks of differentiation of benign and malignant formations. Appl. Optics 55(12), B56–B60 (2016) 3. Yu.A. Ushenko, V.T. Bachynsky, O.Ya. Vanchulyak, A.V. Dubolazov, M.S. Garazdyuk, V.A. Ushenko, Jones-matrix mapping of complex degree of mutual anisotropy of birefringent protein networks during the differentiation of myocardium necrotic changes. Appl. Optics 55(12), B113–B119 (2016) 4. A.V. Dubolazov, N.V. Pashkovskaya, Yu.A. Ushenko, Yu.F. Marchuk, V.A. Ushenko, O.Yu. Novakovskaya, Birefringence images of polycrystalline films of human urine in early diagnostics of kidney pathology. Appl. Optics 55(12), B85–B90 (2016) 5. M.S. Garazdyuk, V.T. Bachinskyi, O.Ya. Vanchulyak, A.G. Ushenko, O.V. Dubolazov, M.P. Gorsky, Polarization-phase images of liquor polycrystalline films in determining time of death. Appl. Optics 55(12), B67–B71 (2016) 6. M. Borovkova, M. Peyvasteh, O. Dubolazov, Y. Ushenko, V. Ushenko, A. Bykov, S. Deby, J. Rehbinder, T. Novikova, I. Meglinski, Complementary analysis of Mueller-matrix images of optically anisotropic highly scattering biological tissues. J. Eur. Opt. Soc. 14(1), 20 (2018) 7. V. Ushenko, A. Sdobnov, A. Syvokorovskaya, A. Dubolazov, O. Vanchulyak, A. Ushenko, Y. Ushenko, M. Gorsky, M. Sidor, A. Bykov, I. Meglinski, 3D Mueller-matrix diffusive tomography of polycrystalline blood films for cancer diagnosis. Photonics 5(4), 54 (2018) 8. L. Trifonyuk, W. Baranowski, V. Ushenko, O. Olar, A. Dubolazov, Y. Ushenko, B. Bodnar, O. Vanchulyak, L. Kushnerik, M. Sakhnovskiy, 2D-Mueller-matrix tomography of optically anisotropic polycrystalline networks of biological tissues histological sections. Optoelectronics Rev. 26(3), 252–259 (2018) 9. V.A. Ushenko, A.V. Dubolazov, L.Y. Pidkamin, M.Y. Sakchnovsky, A.B. Bodnar, Y.A. Ushenko, A.G. Ushenko, A. Bykov, I. Meglinski, Mapping of polycrystalline films of biological fluids utilizing the Jones-matrix formalism. Laser Phys. 28(2), 025602 (2018) 10. V.A. Ushenko, A.Y. Sdobnov, W.D. Mishalov, A.V. Dubolazov, O.V. Olar, V.T. Bachinskyi, A.G. Ushenko, Y.A. Ushenko, O.Y. Wanchuliak, I. Meglinski, Biomedical applications of Jones-matrix tomography to polycrystalline films of biological fluids. J. Innov. Opt. Health Sci. 12(6), 1950017 (2019) 11. M. Borovkova, L. Trifonyuk, V. Ushenko, O. Dubolazov, O. Vanchulyak, G. Bodnar, Y. Ushenko, O. Olar, O. Ushenko, M. Sakhnovskiy, A. Bykov, I. Meglinski, Mueller-matrixbased polarization imaging and quantitative assessment of optically anisotropic polycrystalline networks. PLoS ONE 14(5), e0214494 (2019) 12. A. Ushenko, A. Sdobnov, A. Dubolazov, M. Grytsiuk, Y. Ushenko, A. Bykov, I. Meglinski, Stokes-Correlometry analysis of biological tissues with polycrystalline structure. IEEE J. Selected Topics Quantum Electron. 25(1), 8438957

Chapter 6

Methods and Means of Fourier-Stokes Polarimetry and Spatial-Frequency Filtering of Phase Anisotropy Manifestations

6.1 Fourier-Stokes Polarimetry Manifestations of Linear Birefringence Mechanisms of Structured Fibrillar Networks of Histological Sections of Biological Tissues 6.1.1 Justification and Relevance of the Method It is known that at the tissue level, traditionally pathological processes are detected using a light microscope. The manufacture of preparations for microscopic examination is a complex process that entails a deterioration in the image quality of histological preparations. Overcoming difficulties associated with image distortion requires high professionalism of the researcher and the implementation of complex routine work to adjust the results. The use of computer technology greatly simplifies the solution of such problems, facilitating the work with medical images. At the moment, methods and algorithms for isolating objects in optical microscopy images are predominantly developing. The development and addition of computational techniques for processing the structure of microscopic images has become a new method—laser polarimetry of histological sections of biological tissues. This method allows to obtain new information on the optical anisotropy of various biological objects, inaccessible to histological and mathematical methods of analysis. At the same time, optically anisotropic structures are not unified; they differ both in physical mechanisms of optical anisotropy and in geometric scales. We have already stated (Chap. 4, Sect. 4.2) that different optical anisotropies are inherent for different-scale structural elements. Phase anisotropy is associated with the spatial structuring of optically active molecular protein complexes into birefringent filamentous fibrillar networks. At

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 L. Trifonyuk et al., Optical Anisotropy of Biological Polycrystalline Networks, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-99-1087-8_6

73

74

6 Methods and Means of Fourier-Stokes Polarimetry …

the same time, an intermediate state remains outside the scope of this approach— small-scale, spatially unstructured protein structural elements of the morphological structure of biological tissue. The manifestations of their anisotropy against the background of large-scale fibrillar fibers and bundles are masked. On the other hand, pathological changes in the structure of biological tissue begin precisely with small-scale structures. In addition, the tissue structure of most human organs is multilayer. Moreover, individual layers have not only different optical properties, but also geometric scales. Therefore, the urgent task is to improve the method of azimuthally invariant Mueller-matrix mapping according to the method of spatial-frequency filtering of the measured coordinate distributions of the values of MMI, describing the optical anisotropy of biological structures.

6.2 Theoretical Basis of the Method In this part, we consider optical modeling of the manifestations of linear birefringence of a multilayer biological tissue [1–7] of the rectal wall. Such an analysis of the conversion of laser radiation parameters by polycrystalline networks of such a layer with linear and circular birefringence is proposed: • the wall of the rectum consists of two main optically anisotropic layers—fibrous connective tissue as the submucosa and muscle tissue of the rectum; • the optically anisotropic network of muscle tissue of the rectal membrane is formed by large-scale (transverse size range l–30 µm ÷ 100 µm) filamentous myosin fibrils with linear birefringence; • optical anisotropy of such structures is described by the coordinate distribution of MMI M44 (δ) values in the histological section plane—Chap. 3, relation (3.3), (3.4); • the optically anisotropic component of the fibrous connective tissue layer is formed by collagen molecular complexes (l–5 µm ÷ 10 µm, L ≈ l) without a specific direction of the optical axis ρ and with a correspondingly lower level of linear birefringence M44 (δ ∗ ). From a medical point of view, the separation of birefringence of optically anisotropic fibrillar networks of the layers of the rectum wall is an urgent task. The fact is that the early stages of oncological changes in human organs manifest themselves in an increase in the concentration of optically active proteins in different layers and then “generalize” at later stages in the changes in the morphological structure of birefringent fibrillar networks. In order to select the manifestations of various birefringence mechanisms of optically anisotropic formations with different scales, we used the method of spatial-frequency filtering of polarization-inhomogeneous fields in the Fourier plane [7–12].

6.2 Theoretical Basis of the Method

75

A detailed review of the method of spatial-frequency filtering of polarizationinhomogeneous fields in the Fourier plane is shown in (relations (6.1)–(6.16)). Here, we give brief calculations that are needed to facilitate the perception of experimental material. The main analytical tool in this problem is the use of the Jones matrix formalism. Proceeding from it, for the process of laser radiation conversion by a polycrystalline layer with phase anisotropy, one can write such a Jones matrix relation E = {G}E 0 ,

(6.1)

( ) ( ) where E 0 E 0x eiφ0x , E 0y ei φ0y , E E x eiφx , E y ei φ y —Jones vectors of the laser irradiating and transformed by the object; {G}—The Jones matrix of the phase (linear and circular birefringence) anisotropy of the biological crystal, which has the form {G} = {D}{C},

(6.2)

where {D}—linear birefringence Jones matrix; {C}—Jones matrix optical activity: [

] d11 d12 {D} = . d21 d22

(6.3)

Here, ⎧ 2 ⎨ d11 = sin κ + cos2 κ exp(−i φ), . d = cos2 κ + sin2 κ exp(−i φ), ⎩ 22 d12 = d21 = sin κ cos κ(1 − exp(−i φ)). ] [ c11 c12 {C} = , c21 c22

(6.4)

(6.5)

where ⎧ ⎨ c11 = c22 = cos ψ; . c = sin ψ; ⎩ 12 c21 = − sin ψ.

(6.6)

In expanded form, Eq. (6.1) can be rewritten as: (

E x eiφx E y eiφ y

)

)] ( φ cos2 κ ) amp; cos κ sin κ 1 − e−i φ sin2 κ + e−i ( = cos κ sin κ 1 − e−i φ amp; cos2 κ + e−i φ sin2 κ ) [ ]( cos ψ amp; sin ψ E 0x eiφ0x . × E 0x eiφ0y − sin ψ amp; cos ψ [

(6.7)

76

6 Methods and Means of Fourier-Stokes Polarimetry …

From (6.7), we obtain the expressions for the orthogonal linearly polarized components of the complex amplitude of the laser wave at an arbitrary point in the image of the crystal grid ) ( [( ) ] sin2 κ + e−i φ cos2 κ cos ψ + cos κ sin κ 1 − e−i γ sin ψ ) ( [( ) ] iφ + e 0y E 0y sin2 κ + e−i φ cos2 κ sin ψ + cos κ sin κ 1 − e−i γ sin ψ ,

E x eiφx = ei φ0x E 0x

)( ( ) ] E y ei φ y = eiφ0x E 0x cos κ sin κ 1 − e−i γ cos ψ + cos2 κ + e−iγ sin2 κ 1 − e−i γ sin ψ [ ( ) ( ) ] iφ + e 0y E 0y cos κ sin κ 1 − e−i γ sin ψ + cos2 κ + e−i γ sin2 κ sin ψ . [

(

(6.8)

)

(6.9)

Expressions (6.8) and (6.9) are the “input” parameters for diffraction integrals that describe the direct Fourier transform of the amplitude-phase distributions of the object field. Ux (η, μ) =

1 +∞ ∫ ∫ E x (x, y) exp[−i2π (xη + yμ)d xd y], i λ f −∞

(6.10)

U y (η, μ) =

1 +∞ ∫ ∫ E y (x, y) exp[−i2π (xη + yμ)d xd y], i λ f −∞

(6.11)

where f —lens focal length; λ—laser wavelength; x, y—coordinates of points in the image plane, and η = λx1f , μ = λy1f —spatial frequencies. If you place a low-pass R or high-pass R −1 filter in the central part of the Fourier plane {

1 → η∗ ∈ Δη; μ∗ ∈ Δμ, / Δη; μ∗ ∈ / Δμ, 0 → η∗ ∈ { / Δη; μ∗ ∈ / Δμ, 1 → η∗ ∈ R −1 (Δη, Δμ) = ∗ 0 → η ∈ Δη; μ∗ ∈ Δμ, R(Δν, Δμ) =

(6.12)

(6.13)

then it is possible to distinguish the corresponding large- or small-scale component of the Fourier spectra of the distributions of the amplitudes of the boundary field, formed either predominantly by the effects of linear Uˆ (κ, γ , η, μ) or circular U˙ (ψ, η, μ) birefringence {

Uˆ (κ, γ , η, μ) = R(Δη, Δμ)U (η, μ), U˙ (ψ, η, μ) = R −1 (Δη, Δμ)U (η, μ).

(6.14)

The corresponding distribution of complex amplitudes in the image plane of the ] [ Eˆ x (κ, γ , x, y) biological layer can be restored by the inverse Fourier transform ˙ E x (ψ, x, y) ] [ ˆ E (κ, γ , x, y) and ˙ y E y (ψ, x, y)

6.2 Theoretical Basis of the Method

⎧ ⎪ ⎪ ⎨ Eˆ x (κ, γ , x, y) = ⎪ ⎪ ⎩ E˙ x (ψ, x, y) =

+∞

∫ ∫ R(Δη, Δμ)Uˆ x (η, μ) exp[i2π (xη + yμ)]dηdμ,

−∞ +∞

∫ ∫ R −1 (Δη, Δμ)U˙ x (η, μ) exp[i2π (xη + yμ)]dηdμ,

1 iλ f

⎧ ⎪ ⎪ ⎨ Eˆ y (κ, γ , x, y) = ⎪ ⎪ ⎩ E˙ y (ψ, x, y) =

1 iλf

77

−∞

(6.15) 1 iλf

1 iλf

+∞

∫ ∫ R(Δη, Δμ) · Uˆ y (η, μ) exp[i2π (xη + yμ)]dηdμ,

−∞ +∞

∫ ∫ R −1 (Δη, Δμ) · U˙ y (η, μ) exp[i2π (xη + yμ)]dηdμ.

−∞

(6.16) Using the algorithm (6.1)–(6.16), one can obtain such expressions for calculating the value of the matrix element M44 for each pixel (m × n) )⊗ ( − Eˆ x2 + Eˆ y2 ⊗ ⊕ M44 (δ) = ( )⊗ ( )⊗ 2 2 2 2 ˆ ˆ ˆ ˆ Ex + E y + Ex + E y ⊗ ⊕ ⎛( )90 ( )90 ⎞ )0 ( )0 ( 2 2 2 ˆ ˆ ˆ − Eˆ x2 + Eˆ y2 Eˆ x2 + Eˆ y2 E x + E y − E x + Eˆ y2 ⎜ ⊗ ⊕ ⎟ ⊗ ⊕ − 0.5⎝ ( )0 ( )0 + ( )90 ( )90 ⎠ Eˆ x2 + Eˆ y2 + Eˆ x2 + Eˆ y2 Eˆ x2 + Eˆ y2 + Eˆ x2 + Eˆ y2 ⊗ ⊕ ⊗ ⊕ ) ( 0 I⊗90 − I⊕90 I⊗ − I⊕0 I⊗⊗ − I⊕⊗ . (6.17) = ⊗ − 0.5 0 + 90 I⊗ + I⊕0 I⊗ + I⊕90 I⊗ + I⊕⊗ )⊗ ( )⊗ ( 2 E˙ x + E˙ y2 ⊗ − E˙ x2 + E˙ y2 ⊕ M44 (θ ) = ( )⊗ ( )⊗ E˙ x2 + E˙ y2 ⊗ + E˙ x2 + E˙ y2 ⊕ ⎛( )0 ( )0 )90 ( )90 ⎞ ( 2 E˙ x2 + E˙ y2 ⊗ − E˙ x2 + E˙ y2 ⊕ E˙ x + E˙ y2 ⊗ − E˙ x2 + E˙ y2 ⊕ − 0.5⎝ ( )0 ( )0 + ( )90 ( )90 ⎠ E˙ x2 + E˙ y2 ⊗ + E˙ x2 + E˙ y2 ⊕ E˙ x2 + E˙ y2 ⊗ + E˙ x2 + E˙ y2 ⊕ ) ( 0 I⊗90 − I⊕90 I⊗ − I⊕0 I⊗⊗ − I⊕⊗ . (6.18) = ⊗ − 0.5 0 + 90 I⊗ + I⊕0 I⊗ + I⊕90 I⊗ + I⊕⊗ (

Eˆ x2 + Eˆ y2

)⊗

Here, I⊗⊗;0;90 ; I⊕⊗;0;90 are the intensities of the points of the spatial-frequency filtered image in the presence of polarization analysis, the right (⊗) and left (⊕) circularly polarized filters for each polarizing state of the irradiating beam ( ⊗; 0◦ ; 90◦ ).

78

6 Methods and Means of Fourier-Stokes Polarimetry …

6.3 Spatial-Frequency Fourier-Stokes Polarimetry of the Manifestations of Linear Birefringence of Histological Sections of Biological Tissues Experimental studies were carried out at the Fourier location of the Stokes polarimeter, the optical scheme of which is given in Chap. 2, Sect. 2.4.2 of Fig. 2.8. Two groups of optically thin (attenuation coefficient τ ≈ 0.087 ÷ 0.098) histological sections of adenoma biopsy (group 1) and carcinoma (group 2) of the rectal wall were studied as objects. Figure 6.1 shows classical microscopic images of histological sections of both groups. As you can see, the coordinate and scale structure of such classic microscopic images are similar. This fact significantly complicates the histological differentiation of benign and malignant conditions of the rectal wall tumor. Figure 6.2 shows the results of statistical and fractal analysis of the low-frequency (relation (6.12)) coordinate distribution of the values of the Mueller-matrix invariant M44 (δ) calculated by relation (6.16). A comparative analysis of the obtained data characterizing the statistical and largescale self-similar structure of maps M44 (δ) describing the linear birefringence of the myosin networks of the muscular membrane of the rectal wall revealed significant differences between a benign and malignant state. In particular, the main extrema of the histograms of the distribution of random values M44 (δ) of histological sections of samples of both types are in different areas. So, the most probable values for the histological section of a malignant tumor is the value M44 (δ) 0.2 (Fig. 6.2, fragment (5)) for a sample of a benign tumor M44 (δ) 0.8 (Fig. 6.2, fragment (2)). Such differences, in our opinion, are associated with the formation of a fibrillar network of the muscle membrane of the malignant tumor that is more structured along the orientations in comparison with the similar optical-anisotropic network of benign rectal adenoma. No significant differences were found in the framework of the fractal analysis—the logarithmic dependences lg J (M44 (δ)) − lg ν of the power spectra of the distribution of values M44 (δ) are characterized by a similar slope of the approximating curves V (η) (Fig. 6.2, fragments (3), (6)). Fig. 6.1 Microscopic images of histological sections of a biopsy of a benign (1) and malignant (2) rectal wall tumor

6.3 Spatial-Frequency Fourier-Stokes Polarimetry of the Manifestations …

79

Fig. 6.2 “Low-frequency” maps M44 (δ) (1), (4), histograms (2), (5), logarithmic dependences of power spectra (3), (6) histological sections of adenoma biopsy ((1)–(3)) and carcinoma ((4)–(6))

So, the main indicator of oncological changes in the structure of the optically anisotropic fibrillar structure of the muscular membrane of the rectal wall is the growth of linear birefringence. Quantitatively, the differences between the maps M44 (δ) characterizing the optical anisotropy of the subsurface muscle layer of both types of samples are illustrated by the data given in Table 6.1. In addition, the balanced accuracy Ac of this Fourier-Mueller-matrix mapping method is presented here. A comparative analysis of the data obtained using the method of spatial polarimetry with spatial-frequency filtering found that: • all statistical moments of the 1st–4th order turned out to be diagnostically sensitive, characterizing the distribution of the values of the Mueller-matrix image Table 6.1 Parameters of the statistical and large-scale self-similar structure of linear birefringence maps M44 (δ) of polycrystalline networks of the muscular membrane of the rectal wall

Parameters

M44 (δ) n 1 = n 2 = 36

Ac (%)

Adenoma

Carcinoma

Z1

0.39 ± 0.035

0.67 ± 0.054

94.4

Z2

0.28 ± 0.025

0.18 ± 0.015

91.7

Z3

1.16 ± 0.12

0.93 ± 0.086

77.8

Z4

0.55 ± 0.053

0.43 ± 0.036

77.8

D

0.33 ± 0.025

0.28 ± 0.021

52

80

6 Methods and Means of Fourier-Stokes Polarimetry …

M44 (δ), which is determined by the linear birefringence of the large-scale fibrillar network of the muscular membrane of benign and malignant tumors; • achieved such a level of balanced accuracy Z i=3;4 = 77.8% (good quality diagnostic test), Z i=1;2 = 91.7% − 94.4% (excellent quality diagnostic test); • the balanced accuracy of the Fourier method of Stokes polarimetry with spatialfrequency filtering is 20% higher than that of the azimuthally independent Mueller-matrix mapping of histological sections of biopsy of rectal wall tumors.

6.4 Spatial-Frequency Fourier-Stokes Polarimetry of Birefringence Manifestations of Small-Scale Fibrillar Networks of Biological Tissues The diagnostic possibilities for differentiating the manifestations of linear birefringence of collagen networks of the surface layer of fibrous connective tissue of pathologically altered samples of adenoma and carcinoma using high-pass filtering illustrate the data shown in Fig. 6.3. A comparative analysis of the structure of the high-frequency component of the linear birefringence maps (Fig. 6.3, fragments (1), (4)) of the surface layer of the fibrous collagen network revealed that: • histograms of the distribution M44 (δ ∗ ) of the carcinoma sample are characterized by a wide range of random values and a lower peak sharpness (Fig. 6.3, fragment

Fig. 6.3 “High-frequency” maps M44 (δ ∗ ) (1), (4), histograms (2), (5), logarithmic dependences of power spectra (3), (6) histological sections of adenoma biopsy ((1)–(3)) and carcinoma ((4)–(6)) the walls of the rectum

6.4 Spatial-Frequency Fourier-Stokes Polarimetry of Birefringence …

81

(5)) compared with similar dependences obtained for histological sections of benign adenoma (Fig. 6.3, fragment (2)). The revealed fact of an increase in the optical activity of a substance of a malignant tumor can be associated with an increase in the concentration of protein collagen molecules in small-scale birefringent collagen networks of the fibrous connective tissue. • the logarithmic dependences lg J (M44 (δ ∗ )) − lg ν of the power spectra of the distribution of values M44 (δ ∗ ) indicate their multifractal structure (broken approximating curves V (η1 , η2 )) (Fig. 6.3, fragments (3), (6)). The differences between the cards M44 (δ ∗ ) are quantified by the objective parameters shown in Table 6.2. A comparative analysis of the data obtained using the Fourier method of high-pass filtering Stokes polarimetry revealed the following: • all statistical moments of the 1st–4th order turned out to be diagnostically sensitive, characterizing the distribution of the values of the Mueller-matrix image M44 (δ ∗ ) of the surface layer of benign and malignant tumors of the rectum wall; • a satisfactory level of balanced accuracy Z i=1;2;3;4 ≤ 70% in the quality of the diagnostic test has been achieved; • the balanced accuracy of the Fourier-Stokes polarimetry method with highfrequency filtering corresponds to the azimuthally independent Mueller-matrix mapping of histological sections of the rectal wall tumor biopsy. The obtained high level of balanced accuracy of differentiation of samples of formed benign and malignant tumors using the Fourier-Stokes polarimetry method makes it urgent to study its capabilities in a more difficult situation—the diagnosis of the severity of the pathology, in other words, in determining the objective criteria for detecting changes in the optical anisotropy of biological tissues of human organs in the early, precancerous stages of their pathological changes. Table 6.2 Parameters of the statistical and scale-self-similar structure of circular birefringence maps M44 (δ ∗ ) of polycrystalline collagen networks of fibrous connective tissue

Parameters

M44 (δ ∗ ) n 1 = n 2 = 36 Adenoma

Carcinoma

Z1

0.13 ± 0.011

0.16 ± 0.014

66.7

Z2

0.11 ± 0.08

0.18 ± 0.015

72.2

Z3

0.74 ± 0.072

0.68 ± 0.059

66.7

Z4

0.89 ± 0.085

0.78 ± 0.074

63.9

D

0.31 ± 0.029

0.28 ± 0.027

59.4

Ac (%)

82

6 Methods and Means of Fourier-Stokes Polarimetry …

6.5 Fourier-Mueller-Matrix Mapping of Parameter Distributions Characterizing the Phase Anisotropy of Histological Sections of Biological Tissues in a Precancerous State As the objects of study, we used the same samples as in previous studies of the possibilities of the Mueller-matrix reconstruction method for the distribution of optical anisotropy parameters of biological tissue layers (Chap. 5, Sect. 5.2, Figs. 5.5–5.8). Namely, optically thin (geometric thickness d = 25 µm ÷ 30 µm, attenuation coefficient τ ≺ 0.1) histological sections of a biopsy of the cervical tissue—endometrium in a precancerous state of two types: simple endometrial atrophy—group 1 (36 samples) and endometrial polyp—group 2 (36 samples).

6.5.1 Spatial-Frequency Fourier-Stokes Polarimetry of Linear Birefringence Endometrium Figure 6.4 shows the results of statistical and fractal analysis of the low-frequency (relation (6.12)) coordinate distribution of the values of the Mueller-matrix invariant M44 (δ) of histological sections of the endometrium of both types, calculated by the relation (6.16). A comparative analysis of the distributions of random values of such a Muellermatrix invariant (Sect. 3.3, relation (3.14)) revealed the same range of phase shifts in the plane of histological sections of samples of both types (Fig. 6.4, fragments (1), (4)). Quantitatively, this fact illustrates the proximity of the magnitude of the statistical moments of the 1st—4th order, characterizing the distribution of the values of phase shifts introduced by histological sections of the endometrium: • with simple atrophy—(Z 1 = 0.29, Z 2 = 0.11, Z 3 = 1.02, Z 4 = 1.39); • polyp—(Z 1 = 0.34, Z 2 = 0.15, Z 3 = 0.91, Z 4 = 1.18). The maximum differences between the statistical moments of the 1st–4th order do not exceed 30%. In the framework of the fractal approach, no significant differences between coordinate distributions M44 (δ) were found. The values of statistical moments Z i=1;2;3;4 and variance of the logarithmic dependences of power spectra M44 (δ) averaged within both groups of histological sections samples are shown in Table 6.3. A comparison of such indicators with the data obtained by the Mueller-matrix reconstruction of the linear birefringence parameters of the formed tumors (Chap. 5, Sect. 5.2, Figs. 5.5, 5.7 and Tables 5.3, 5.4) revealed a similar level of balanced accuracy of the Fourier method of Stokes polarimetry. However, this method is aimed at differentiating the manifestations of optical anisotropy of samples very close in morphological structure that are found in a

6.5 Fourier-Mueller-Matrix Mapping of Parameter Distributions …

83

Fig. 6.4 “Low-frequency” maps M44 (δ) (1), (2), histograms (3), (4), logarithmic dependences of power spectra (5), (6) histological sections of an endometrial biopsy with simple atrophy ((1), (3), (5)) and of endometrial polyp ((2), (4), (6)) Table 6.3 Parameters of the statistical and scale-like structure of circular birefringence maps M44 (δ) of polycrystalline networks of the cervix endometrium

Parameters

M44 (δ) n 1 = n 2 = 36

Ac (%)

Atrophy

Polyp

Z1

0.27 ± 0.023

0.35 ± 0.028

75

Z2

0.12 ± 0.011

0.18 ± 0.014

77.8

Z3

1.05 ± 0.12

0.92 ± 0.086

63.9

Z4

1.28 ± 0.12

1.07 ± 0.096

63.9

D

0.31 ± 0.029

0.28 ± 0.024

58.2

84

6 Methods and Means of Fourier-Stokes Polarimetry …

precancerous state. Therefore, a balanced accuracy is achieved (Ac = 75%÷77.8%) in terms of evidence-based medicine corresponds to a good level of diagnostic test. Figure 6.5 shows the results of statistical and fractal analysis of the high-frequency (relation (6.12)) coordinate distribution calculated by relation (6.17), the values of the Mueller-matrix invariant M44 (δ ∗ ) of endometrial layers in a precancerous state. Inverse trends, compared with maps M44 (δ), are observed in statistical changes in the coordinate distribution of parameter M44 (δ ∗ ) values (Fig. 6.5), which characterizes the birefringence of small-scale endometrial fibrillar networks. This fact can be associated with the similarity of the morphological structure of collagen networks at various stages of precancerous changes. In addition, this confirms the identity of the logarithmic dependences of the power spectra of the coordinate distributions M44 (δ ∗ ).

Fig. 6.5 “High-frequency” maps M44 (δ ∗ ) (1), (2), histograms (3), (4), logarithmic dependences of power spectra (5), (6) histological sections of an endometrial biopsy with simple atrophy ((1), (3), (5)) and endometrial polyp ((2), (4), (6))

6.5 Fourier-Mueller-Matrix Mapping of Parameter Distributions … Table 6.4 Parameters of the statistical and scale-like structure of circular birefringence maps M44 (δ ∗ ) of polycrystalline networks of the cervix endometrium

Parameters

M44 (δ) n 1 = n 2 = 36

85 Ac (%)

Atrophy

Polyp

Z1

0.15 ± 0.013

0.19 ± 0.018

72.2

Z2

0.21 ± 0.017

0.23 ± 0.024

66.7

Z3

1.44 ± 0.13

1.29 ± 0.11

61.3

Z4

1.28 ± 0.12

1.07 ± 0.106

63.9

D

0.31 ± 0.027

0.28 ± 0.024

54.7

The averaged values of the statistical moments Z i=1;2;3;4 characterizing the coordinate distribution of the values of the Mueller-matrix invariants M44 (δ ∗ ) of histological sections of the endometrium are shown in Table 6.4. As can be seen from a comparative analysis of the obtained data, the balanced accuracy (Ac ≤ 70%) of high-frequency filtering of the manifestations of linear birefringence of collagen networks of the endometrium in a precancerous state is inferior to the possibilities of low-frequency filtering. From a physical point of view, this result corresponds to the idea of the formation of a linear birefringence mechanism, as a manifestation of the spatial orientation of protein optically active molecules. Based on this, the task of studying the manifestations of the optical activity of such objects using spatial-frequency filtering of the coordinate distributions of the values of another azimuthally independent Mueller-matrix invariant becomes urgent ΔM(θ ).

6.5.2 Spatial-Frequency Fourier-Stokes Polarimetry of Circular Birefringence of Endometrium Figure 6.6 shows the results of statistical and fractal analysis of the low-frequency (relation (6.12)) coordinate distribution of the values of the Mueller-matrix invariant ΔM(θ ) of histological endometrial sections of both types of pathology. Analysis of the obtained MMI ΔM(θ ) of histological endometrial sections of both types revealed a slightly higher level of optical activity of large-scale collagen fibrils of the polyp sample (Fig. 6.6, fragments (1), (4)). This fact quantitatively illustrates the shift of the histogram extremum N (θ ) to the region of higher values θ and the increase in the range of their scatter (Fig. 6.6, fragments (2), (5)). This transformation of histograms corresponds to such changes in the set of statistical moments of the 1st–4th order characterizing the distribution of parameter values θ determined by the concentration level of optically active collagen molecules in the endometrial layer: • with simple atrophy—( Z 1 = 0.17, Z 2 = 0.14, Z 3 = 1.32, Z 4 = 1.51 ); • polyp—( Z 1 = 0.24, Z 2 = 0.21, Z 3 = 1.19, Z 4 = 1.38 ).

86

6 Methods and Means of Fourier-Stokes Polarimetry …

Fig. 6.6 “Low-frequency” maps ΔM(θ ) (1), (2), histograms (3), (4), logarithmic dependences of power spectra (5), (6) histological sections of an endometrial biopsy with simple atrophy ((1), (3), (5)) and endometrial polyp ((2), (4), (6))

As you can see, the maximum differences between the values of statistical moments of the 1st–4th orders are {ΔZ 1 (θ ) = 1.41, ΔZ 2 (θ ) = 1.5, ΔZ 3 (θ ) = 1.11, ΔZ 4 (θ ) = 1.09. It has been established that statistical moments of the first and second orders, which characterize the coordinate distributions of the MMI values, are most sensitive in the differentiation of the manifestations of the optical activity of large-scale collagen fibrils ΔM(θ ). The logarithmic dependences of the power spectra of the distributions of the MMI ΔM(θ ) values are very close in structure, which indicates the identity of the scales of optical inhomogeneities with circular birefringence. The values of statistical moments Z i=1;2;3;4 and the value of balanced accuracy Ac averaged within both groups of histological sections were shown in Table 6.5. Comparison of balanced accuracy indicators Ac 85% with the data obtained by the Fourier-Stokes polarimetry method of the manifestations of linear birefringence

6.5 Fourier-Mueller-Matrix Mapping of Parameter Distributions … Table 6.5 Parameters of the statistical and scale-like structure of circular birefringence maps ΔM(θ ) of polycrystalline networks of the cervix endometrium

Parameters

ΔM(θ ) n 1 = n 2 = 36

87 Ac (%)

Atrophy

Polyp

Z1

0.09 ± 0.01

0.14 ± 0.014

83.3

Z2

0.08 ± 0.007

0.13 ± 0.011

86.1

Z3

1.32 ± 0.11

1.19 ± 0.088

63.9

Z4

1.62 ± 0.14

1.51 ± 0.13

61.3

D

0.19 ± 0.016

0.21 ± 0.017

58.3

of collagen fibrillar networks (Figs. 6.3, 6.4 and Table 6.3) revealed a good level of balanced accuracy of Mueller-matrix mapping of the manifestations of the optical activity of such endometrial networks in precancerous condition. Figure 6.7 shows the results of statistical and fractal analysis of the high-frequency (relation (6.12)) coordinate distribution of the values of the Mueller-matrix invariant ΔM(θ ∗ ) of the endometrial layers of both types of pathology. Inverse trends, compared with maps, ΔM(θ ) are observed in statistical changes in the coordinate distribution of map values ΔM(θ ∗ ) (Fig. 6.5), which characterizes the optical activity of small-scale collagenic endometrial fibrillar networks. As you can see, for such an optically anisotropic network of the histological section of the endometrial polyp, somewhat larger values ΔM(θ ∗ ) are inherent (Fig. 6.7, fragments (2), (5)). The revealed fact can be associated with a higher concentration of optically active molecules at the stage of transition from simple atrophy to the formation of an endometrial polyp. The values of the statistical moments Z i=1;2;3;4 averaged within both groups of samples characterizing the coordinate distributions of the values of the Muellermatrix invariants ΔM(θ ) and ΔM(θ ∗ ) histological sections of the endometrial biopsy of both groups of patients are shown in Table 6.6. As can be (seen )from a comparative analysis of the obtained data, the balanced accuracy (Ac Z 1;2 = 72.2% ÷ 75%) of the thod of high-frequency filtering of MMI ΔM(θ ∗ ) distributions describing the manifestations of circular birefringence of collagen networks of the endometrium in a precancerous state reaches a satisfactory level, although it is inferior to the capabilities of low-frequency filtering of distributions ΔM(θ ).

88

6 Methods and Means of Fourier-Stokes Polarimetry …

Fig. 6.7 “High-frequency” maps ΔM(θ ∗ ) (1), (2), histograms (3), (4), logarithmic dependences of power spectra (5), (6) histological sections of an endometrial biopsy with simple atrophy ((1), (3), (5)) and endometrial polyp ((2), (4), (6)) Table 6.6 Parameters of the statistical and large-scale self-similar structure of circular birefringence maps ΔM(θ ∗ ) of polycrystalline networks of the cervix endometrium

Parameters

ΔM(θ ∗ ) n 1 = n 2 = 36

Ac (%)

Atrophy

Polyp

Z1

0.18 ± 0.013

0.24 ± 0.018

75

Z2

0.14 ± 0.013

0.18 ± 0.015

72.2

Z3

1.64 ± 0.14

1.39 ± 0.12

63.9

Z4

1.08 ± 0.107

0.87 ± 0.076

63.9

D

0.31 ± 0.025

0.28 ± 0.024

54.6

References

89

References 1. V. Tuchin, L. Wang, D. Zimnjakov, Optical Polarization in Biomedical Applications (Springer, New York, USA, 2006) 2. C.R. Polarimetry, in Handbook of Optics: Vol I—Geometrical and Physical Optics, Polarized Light, Components and Instruments, ed. by M. Bass (McGraw-Hill Professional, New York, 2010), pp. 22.1–22.37 3. N. Ghosh, M. Wood, A. Vitkin, Polarized light assessment of complex turbid media such as biological tissues via Mueller matrix decomposition, in Handbook of Photonics for Biomedical Science. ed. by V. Tuchin (CRC Press, Taylor & Francis Group, London, 2010), pp.253–282 4. S. Jacques, Polarized light imaging of biological tissues, in Handbook of Biomedical Optics. ed. by D. Boas, C. Pitris, N. Ramanujam (CRC Press, Boca Raton, London, New York, 2011), pp.649–669 5. N. Ghosh, Tissue polarimetry: concepts, challenges, applications, and outlook. J. Biomed. Opt. 16(11), 110801 (2011) 6. M. Swami, H. Patel, P. Gupta, Conversion of 3 × 3 Mueller matrix to 4 × 4 Mueller matrix for non-depolarizing samples. Optics Commun. 286, 18–22 (2013) 7. D. Layden, N. Ghosh, A. Vitkin, Quantitative polarimetry for tissue characterization and diagnosis, in Advanced Biophotonics: Tissue Optical Sectioning. ed. by R. Wang, V. Tuchin (CRC Press, Taylor & Francis Group, Boca Raton, London, New York, 2013), pp.73–108 8. A. Ushenko, V. Pishak, Laser polarimetry of biological tissue: principles and applications, in Handbook of Coherent-Domain Optical Methods: Biomedical Diagnostics, Environmental and Material Science, ed. by V. Tuchin (2004), pp. 93–138 9. O. Angelsky, A. Ushenko, Y. Ushenko, V. Pishak, A. Peresunko, Statistical, correlation and topological approaches in diagnostics of the structure and physiological state of birefringent biological tissues, in Handbook of Photonics for Biomedical Science (2010), pp. 283–322 10. Y. Ushenko, T. Boychuk, V. Bachynsky, O. Mincer, Diagnostics of structure and physiological state of birefringent biological tissues: statistical, correlation and topological approaches, in Handbook of Coherent-Domain Optical Methods, ed. by V. Tuchin (Springer Science+Business Media, 2013) 11. O. Angelsky, A. Ushenko, Y. Ushenko, Investigation of the correlation structure of biological tissue polarization images during the diagnostics of their oncological changes. Phys. Med. Biol. 50(20), 4811–4822 (2005) 12. V. Ushenko, O. Dubolazov, A. Karachevtsev, Two wavelength Mueller matrix reconstruction of blood plasma films polycrystalline structure in diagnostics of breast cancer. Appl. Opt. 53(10), B128 (2016)

Conclusions

1. The model of phase and amplitude optical anisotropy was first proposed and analytically justified to describe polycrystalline networks of histological sections of biological tissues and films of biological fluids. On this basis, the Mueller matrix of the generalized anisotropy of an optically thin biological layer is calculated as a superposition of partial matrix operators of phase (linear and circular birefringence) and amplitude (linear and circular dichroism) anisotropy. 2. A set of azimuthally independent Mueller-matrix invariants (MMI) describing the of biological layers is determined ( manifestations of optical anisotropy ) / − M ) (M 23 32 M44 (δ); ΔM(θ ) = , ; M M (Δτ ); (Δg) 14 41 (M22 + M33 ) and the first time a set of solutions to the inverse problem is obtained—algorithms for polarized reproduction of coordinate distributions of the values of phase parameters (phase shift δ and angle of rotation of the plane of polarization θ ) and amplitude (linear Δτ and circular dichroism Δg) anisotropy indices for experimentally measured distributions of azimuthally independent matrix elements and combinations thereof. 3. The relationship between the set of statistical moments 1–4 orders of magnitude characterizing the distribution of the MMI values of histological sections of biological tissues and films of biological fluids and the distribution of optical anisotropy parameters (the direction of the optical axis, phase shift between the orthogonal linearly and circularly polarized components of the laser wave amplitude, the ratio of the coefficients absorption of such orthogonal components) polycrystalline networks. 4. For the first time, an azimuthally independent Mueller-matrix mapping and differentiation of the manifestations of phase and amplitude anisotropy of histological sections of a biopsy of benign and malignant tumors of a human organ. It is established that the physical cause of the increase in the values of asymmetry (Z 3 ↑) and excess (Z 4 ↑), which characterize the distribution of the values of the invariants of linear birefringence M44 and dichroism M14 , is a decrease in the dispersion of the distribution of orientations of the optical axes of the fibrils of biological © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 L. Trifonyuk et al., Optical Anisotropy of Biological Polycrystalline Networks, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-99-1087-8

91

92

Conclusions

tissues; inverse to the above processes, changes in the statistical moments of the 3rd (Z 3 ↓) and 4th (Z 4 ↓) orders characterizing the distribution of MMI values due to optical activity ΔM and circular dichroism M41 correspond to an increase in the concentration of optically active molecules of protein networks. On this basis, for the first time, differentiation of representative samples of histological sections of a biopsy of benign (adenoma) and malignant (carcinoma) tumors of the rectum wall with balanced accuracy R 80% corresponding to a good quality diagnostic test was made. 5. For the first time, the “two-wave” method of azimuthally independent Muellermatrix reconstruction of parameters that describe the mechanisms of phase and amplitude anisotropy of planar polycrystalline films of biological fluids has been developed. It has been established that the physical cause of the increase in the statistical moments of the 1st (Z 1 ↑) and 2nd (Z 2 ↑) orders characterizing the distribution of the values of the parameters of linear birefringence and dichroism, as well as circular birefringence and dichroism, is an increase in the concentration of optically active molecules (albumin, globulin, fibrin, as well as bilirubin, cholesterol and bile acids) and the formation of spatially structured polycrystalline networks. On the basis of this, differentiation of representative samples of bile films was realized, and for the first time, type II diabetes was diagnosed with good (R(δ, θ ) 80%) and excellent (R(Δτ, Δg) 90% − 95%) balanced accuracy. A similar level of diagnostic test was achieved in the differentiation of polycrystalline plasma films of blood plasma from donors and patients with breast cancer. 6. The method of Stokes polarimetry with spatial-frequency filtering of the distributions of MMI values is analytically substantiated to differentiate the manifestations of the phase anisotropy mechanisms of different-scale fibrillar networks of histological sections of biological tissues of different morphological structures and physiological states. It has been established that all statistical moments of the first and fourth orders characterizing the distribution of the values of the matrix element M44 of the linear birefringence of the large-scale fibrillar optically anisotropic component of benign and malignant tumors are diagnostically sensitive. The following levels of balanced accuracy were achieved: Ri=3;4 = 77.8% (good quality of the diagnostic test), Ri=1;2 = 91.7–94.4% (excellent quality of the diagnostic test), which is 20% higher than that of the azimuthally independent Mueller-matrix mapping of histological sections of the biopsy of rectal wall tumors. 7. For the first time, a polarization reconstruction method with spatial-frequency differentiation of the distributions of MMI ΔM values characterizing circular birefringence of histological sections of multilayer biological tissues was developed to differentiate phase anisotropy changes (θ ) of polycrystalline networks in the early stages of cancer pathology—precancerous conditions (simple atrophy, polyp) of the organs of the female reproductive system. The physical relationships between such changes and the transformation of the statistical structure of the distribution of the values of MMI (Z 1 (θ ) ↑; Z 2 (θ ) ↑; Z 3 (θ ) ↓; Z 4 (θ ) ↓) are established. On this basis, a high level of balanced accuracy R 85% of the

Conclusions

93

diagnostic test for differentiating the manifestations of the optical activity of endometrial networks in a precancerous state has been achieved. 8. The fractality of coordinate distributions of MMI values, characterizing the manifestations of partial mechanisms of optical anisotropy of polycrystalline films of biological fluids of healthy donors, has been established. The physical reason for the change in the scale-like structure (formation of multifractal distributions) is an increase in the concentration of optically active molecules that crystallize into birefringent networks of biological fluid films taken from sick patients. Due to the formation of optically anisotropic newly formed fibrillar networks of malignant tumors randomly oriented along the directions, the coordinate distributions of MMI values are transformed into random. A quantitative criterion for such a process is an increase in the dispersion of the distribution of the logarithmic dependences of the power spectra of the ensemble of values of MMI. On this basis, the criteria for differentiating the optical anisotropy of polycrystalline bile films (normal—type II diabetes) and blood plasma (normal—breast cancer) as well as precancerous conditions (simple atrophy and polyp) of the endometrium were determined.