ORIGINAL ARTICLE

Differential Mueller matrix imaging of partially depolarizing opticallyanisotropic biological tissues

L. Trifonyuk1 & A. Sdobnov2 & W. Baranowski3 & V. Ushenko4& O. Olar4 & A. Dubolazov4 & L. Pidkamin4

& M. Sidor4 &

O. Vanchuliak5 & A. Motrich4& M. Gorsky4 & I. Meglinski2,6,7,8

Received: 15 May 2019 /Accepted: 6 September 2019# The Author(s) 2019

AbstractSince recently, a number of innovative polarization-based optical imaging modalities have been introduced and extensively usedin various biomedical applications, with an ultimate aim to attain the practical tool for the optical biopsy and functionalcharacterization of biological tissues. The techniques utilize polarization properties of light and Mueller matrix mapping ofmicroscopic images of histological sections of biological tissues or polycrystalline films of biological fluids. The main drawbackof currently developed laser polarimetry approaches and Mueller matrix mapping techniques is poor reproducibility of experi-mental data. This is due to azimuthal dependence of polarization and ellipticity values of most matrix elements to sampleorientation in respect to incidence light polarization. Current study aims to generalize the methods of laser polarimetry fordiagnosis of partially depolarizing optically anisotropic biological tissues. A method of differential Mueller matrix mappingfor reconstruction of linear and circular birefringence and dichroism parameter distributions of partially depolarizing layers ofbiological tissues of different morphological structure is introduced and practically implemented. The coordinate distributions ofthe value of the first-order differential matrix elements of histological sections of brain tissue with spatially structured, opticallyanisotropic fibrillar network, as well as of parenchymatous tissue of the rectum wall with an “islet” polycrystalline structure aredetermined. Within the statistical analysis of polarization reproduced distributions of the averaged parameters of phase andamplitude anisotropy, the significant sensitivity of the statistical moments of the third and fourth orders to changes in thepolycrystalline structure of partially depolarizing layers of biological tissue is observed. The differentiation of female reproduc-tive sphere connective tissue is realized with excellent accuracy. The differential Mueller matrix mapping method for recon-struction of distributions of linear and circular birefringence and dichroism parameters of partially depolarizing layers of bio-logical tissues of different morphological structures is proposed and substantiated. Differential diagnostics of changes in thephase (good balanced accuracy) and amplitude (excellent balanced accuracy) of the anisotropy of the partially depolarizing layersof the vagina wall tissue with prolapse of the genitals is realized. Themaximum diagnostic efficiency of the first-order differentialmatrix method was demonstrated in comparison with the traditional methods of polarization and Mueller matrix mapping ofhistological sections of light-scattering biological tissues.

Keywords Polarized light . Mueller matrix . Optical anisotropy . Birefringence . Partial depolarization . Biomedical imaging

* I. Meglinskiigor.meglinski@oulu.fi

1 Rivne State Medical Center, 78 Kyivska Str, Rivne 33007, Ukraine2 Faculty of Information Technology and Electrical Engineering,

University of Oulu, 90570 Oulu, Finland3 Warsaw Military Institute of Medicine, 04141 Warsaw, Poland4 Chernivtsi National University, 2 Kotsiubynskyi Str,

Chernivtsi 58012, Ukraine

5 Bukovinian State Medical University, 3 Theatral Sq,Chernivtsi 58000, Ukraine

6 Institute of Engineering Physics for Biomedicine (PhysBio), NationalResearch Nuclear University MEPhI, Moscow 115409, Russia

7 Interdisciplinary Laboratory of Biophotonics, National ResearchTomsk State University, Tomsk 634050, Russia

8 School of Engineering & Applied Science, Aston University,Birmingham, UK and School of Life & Health Sciences, AstonUniversity, Aston University, Birmingham, UK

https://doi.org/10.1007/s10103-019-02878-2Lasers in Medical Science (2020) 35:877–891

/Published online: 20 2019November

Introduction

The study of biological tissue structure in various pathologicaland physiological states is one of the most important tasks formode rn m i c r o s cop i c b i omed i c a l imag i ng [ 1 ] .Morphologically, most biological tissues have anisotropicstructure (fibrillar protein networks of eye tissues, skinderma,muscle, bone tissue, etc.) [2–4], which leads to thelinear birefringence and dichroism. Additionally, many com-ponents (chiral molecules) of biological tissues have opticalactivity and circular dichroism [5–7]. Traditional light micros-copy does not provide information on optical anisotropy,which limits its diagnostic capabilities. The main method forobtaining this information is the use of polarized radiation [8,9]. Such techniques can be called a “polarization introscopy”(visualization of optical anisotropy parameters) of the poly-crystalline structure of biological tissues. Particularly, Muellermatrix polarimetry (MMP) became one of the most effectivepolarization methods for tissue diagnostics [9]. This methodprovides the most complete information about the polarizationmanifestations of a set of mechanisms of optical anisotropy ofbiological objects [10, 11]. At the moment, MMPmethods aredeveloping in two main directions. The first one is the inves-tigation of the structure and symmetry of the light scatteringmatrices (LSM)-angular dependences (indicatrices) of matrixelements [12, 13]. The second one is the investigation ofMueller matrix images (MMI)—coordinate distributions ofmatrix elements [14, 15]. Methods based on LSM allowobtaining statistically averaged information about the size,shape, and optical properties of ensembles of scattering struc-tural elements of biological tissues [16–21]. The experimentaldetermination of biological object MMI is an innovative de-velopment of LSM techniques [22]. The obtained MMI, as asuperposition of depolarization, dichroism, and birefringencematrices, allow to investigate the integral structure of the op-tically anisotropic component of biological tissues [23, 24]using the polar decomposition method [25, 26]. This algo-rithm proved to be the most effective for analysis of the poly-crystalline structure in the case of single scattering in biolog-ical layers. In this approximation, we first determined theMueller matrix reconstruction algorithms for the coordinatedistributions of the linear and circular birefringence and di-chroism of biological tissues [27, 28]. Statistical analysis ofthe obtained maps of optical anisotropy revealed objectivecriteria (statistical moments of the first to fourth order), whichwere used as the basis for differentiation of oncological chang-es in human organs (benign and malignant tumors of the pros-tate, myometrium, and endometrium) [29–33]. The main lim-itation of this technique is the hardly achievable requirementof single scattering or the absence of depolarization. The mostcommon type of laboratory samples of biological tissues ispartially depolarizing optically anisotropic layers. Modern mi-croscopic diagnosis of such changes does not provide a

sufficient level of accuracy. For example, the accuracy of theanalogous diagnosis of the first stage of endometrial cancer is75%, the second stage is 66%, and only in the third stage it isincreased to 88% [34]. At the preclinical stage of no lesssevere pathology of prolapse of the genitals, quantitative esti-mation of changes in the morphological structure of the con-nective tissue component of the uterine ligament is not preciseenough [35, 36]. Therefore, the further development of theMMP method in the diagnostics of the polycrystalline struc-ture of light-scattering biological tissues of various morpho-logical structures and the physiological state is relevant. Oneof the ways to solve this problem is described in references[37–43], which are based on their presentation of the Muellermatrix as a superposition of a completely polarized (first-orderdifferential matrix) and depolarized (second-order differentialmatrix) components. Differential, in contrast to traditional[14–22] Mueller matrix mapping and polar decompositionmethods [25, 26], provides separate information about thedistribution of average values (polarized component of theMueller matrix) and magnitudes (depolarized component ofthe Mueller matrix) of fluctuations of the phase and amplitudeanisotropy parameters.

However, this theoretical differential Mueller matrix meth-od has not yet found practical application in biomedicaldiagnostics.

Current paper is aimed to the development and experimen-tal approbation of the method of differential Mueller matrixmapping and reconstruction of the parameters of optical an-isotropy of the partially depolarizing layers of biological tis-sues of various morphological structures (brain, rectal wall)and the pathological state “norm-prolapse of genitals” in orderto obtain quantitative criteria for evaluating and differentiatingthe polycrystalline structure of such objects.

Theoretical background

In this section, we give a brief outline of the theory [37–43]which describes algorithms for experimental measurement ofa first-order differential matrix of a partially depolarizing op-tically anisotropic biological layer. In the case of multiplescattering, the Mueller matrix of a depolarizing layer variesalong the propagation direction of light z. Analytically, thisdependence is illustrated by the equation

d Mf g zð Þdz

− Mf g zf g mf g zð Þ ð1Þ

where {M}(z) is the Mueller matrix object in the plane z and{m}(z) is the differential Mueller matrix.

For optically thin layers, the non-depolarizing differentialmatrix {m}(z) consists of six elementary polarization proper-ties which fully characterize the optical anisotropy of a

Lasers Med Sci (2020) 35:877–891878

biological layer. These parameters fully characterize the am-plitude (linear LD0; 90, LD45; 135, and circular CD⊗; ⊕ dichro-ism) and phase (linear LB0; 90, LB45; 135, and circular CB⊗; ⊕

birefringence) anisotropy of the biological layer

mf g ¼0 m12 m13 m14

m21 0 m23 −m24

m31

m41

−m32

m42

0 m34

−m43 0

��������������

¼0 LD0;90 LD45;135 CD⊗;⊕

LD0;90 0 CB⊗;⊕ −LB45;135

LD45;135

CD⊗;⊕

−CB⊗;⊕LB45;135

0 LB0;90

−LB0;90 0

��������������

ð2Þ

Here, the indices “0; 90”, “45; 135” denote the unit vectorsof the linearly polarized components;⊗ right,⊕ left circularlypolarized components of the amplitude of the laser wave.

The partially depolarizing medium for expression (2) canbe represented as average ⟨{m}⟩ (polarization part {m}(z)) andfluctuating⟨{Δm2}⟩ (depolarizing part {m}(z)) components

mf g zð Þ ¼ mf gh i þ Δm2� �� � ð3Þ

It should be noted that there is always a feedback betweenthe differential matrix and the Mueller matrix

M zð Þ ¼ exp mf g zð Þð Þ ð4Þ

A combined analysis of ratios (1)-(4) is performed and alogarithmic expression of matrix algorithm is derived as

L zð Þ ¼ ln M zð Þf g ¼ Lp þ Ld ð5Þ

which is determined as superposition of antisymmetric Lp(polarization) and symmetric Ld (depolarization) componentsof L(z)

Lp ¼ mf gh iz;Ld ¼ 0; 5 Δm2

� �� �z2;

�ð6Þ

where

Lp ¼ 0:5 L−GLTG�

;Ld ¼ 0:5 Lþ GLTG

� ;

G ¼ diag 1;−1;−1;−1ð Þ:

8<: ð7Þ

Here, T is the transpose operation and G is the Minkowskimetric matrix.

Taking into account relations (2)–(7), polarization compo-nent of a logarithmic matrix algorithm L(z) takes the form of

Lp ¼0 j12 þ j21ð Þ j13 þ l31ð Þ j14 þ j41ð Þ

j21 þ l12ð Þ 0 j23− j32ð Þ j24− j42ð Þj31 þ l13ð Þj41 þ l14ð Þ

j32−l23ð Þj42−l24ð Þ

0 j34− j43ð Þj43− j34ð Þ 0

�������������� ð8Þ

where

jik ¼ lnMik ;jik þ jki ¼ ln Mik �Mkið Þ;jik− jki ¼ ln

Mik

Mki

�8>><>>: ð9Þ

On the basis of relations (8) and (9), we found a relation-ship between the elements of the first-order differential matrix⟨{mik}⟩ (equations (2) and (3)) and combinations of the aver-aged by depth l elements Mikof the Mueller matrix (equation(3)) of the partially depolarizing layer of the biological fabrics

mf gh i ¼ z−1 � Lp ¼ z−1

�

0 ln M12M21ð Þ ln M 13M31ð Þ ln M 14M 41ð Þln M12M21ð Þ 0 ln

M23

M32

�ln

M 24

M 42

�

ln M13M31ð Þln M14M41ð Þ

lnM 32

M 23

�

lnM 42

M 24

� 0

lnM43

M34

� lnM 34

M 43

�0

��������������

��������������ð10Þ

The combined analysis of relations (2) and (10) allowed usto obtain algorithms of polarization reconstruction of averagevalues (over the entire depth l of the biological layer) of pa-rameters of phase and amplitude anisotropy of polycrystallinestructure of optically thick biological layer

LB0;90 ¼ l−1lnM 34

M 43

�ð11Þ

LB45;135 ¼ l−1lnM24

M42

�ð12Þ

CB⊗;⊕ ¼ l−1lnM 23

M 32

�ð13Þ

LD0;90 ¼ l−1ln M 12M 21ð Þ ð14ÞLD45;135 ¼ l−1ln M 13M 31ð Þ ð15ÞCD⊗;⊕ ¼ l−1ln M 14M 41ð Þ ð16Þ

Thus, the use of differential analysis of Mueller matrixmapping data allowed us to obtain a set of algorithms (ratios(11)–(16)) of polarization reconstruction of average values ofphase and amplitude anisotropy parameters of polycrystallinecomponent of partially depolarized biological layer.

Experimental approach

In this part of the paper, the theory for experimental mappingof the distributions of elements of a first-order differentialmatrix ⟨{mik}⟩using the MMP technique is presented. As alighting probe, linearly polarized (with azimuths 0°, 45°,

Lasers Med Sci (2020) 35:877–891 879

90°) and right circularly polarized helium-neon laser beams(wavelength 632.8 μm, power 10 mW) were used in series.

For each of the illuminating beams, a polarization analysisof the microscopic image of the biological layer was carriedout. For this purpose, multichannel polarization filtering wasused. More detailed description can be found in our previouspapers [27–29]. Therefore, we do not give a description of thecomputation of the elementsMik, but the main attention is paidto the methods for determination of the distributions⟨{mik}⟩.

Taking into account relations (8)–(10), we obtain the ex-pressions for calculation of elements of differential matrix ofthe first order, which characterize the linear and circular:

& birefringence (LB0;90; LB45;135; CB⊗;⊕ ):

m34h i ¼ lnM34

M43

�¼ ln

S⊗3 −0; 5 S03 þ S903�

S454 −0; 5 S04 þ S904�

!

m43h i ¼ lnM43

M34

�¼ ln

S454 −0; 5 S04 þ S904�

S⊗3 −0; 5 S03 þ S903�

!9>>>>=>>>>;⇒LB0;90; ð17Þ

m42h i ¼ lnM42

M24

�¼ ln

0; 5 S04−S904

� S⊗2 −0; 5 S02 þ S902

� !

m24h i ¼ lnM24

M42

�¼ ln

S⊗2 −0; 5 S02 þ S902�

0; 5 S04−S904

� !

9>>>>=>>>>;⇒LB45;135; ð18Þ

m23h i ¼ lnM23

M32

�¼ ln

S452 −0; 5 S02 þ S902�

0; 5 S03−S903

� !

m32h i ¼ lnM32

M23

�¼ ln

0; 5 S03−S903

� S452 −0; 5 S02 þ S902

� !

9>>>>=>>>>;⇒CB⊗;⊕; ð19Þ

& dichroism (LD0;90; LD45;135; CD⊗;⊕ ):

m12h i ¼ m21h i ¼ ln M 12M 21ð Þ ¼¼ ln 0; 25 S01−S

901

� S02 þ S902� �

⇒LD0;90; ð20Þ

m13h i ¼ m31h i ¼ ln M 13M 31ð Þ ¼¼ ln 0; 25 S451 −0; 5 S01 þ S901

� � S03 þ S903� �

⇒LD45;135;ð21Þ

m14h i ¼ m41h i ¼ ln M 14M 41ð Þ ¼¼ ln 0; 25 S⊗1 −0; 5 S01 þ S901

� � S04 þ S904� �

⇒CD⊗;⊕: ð22Þ

S0;45;90;⊗i¼1 ¼ I0;45;90;⊗0 þ I0;45;90;⊗90 ;S0;45;90;⊗i¼2 ¼ I0;45;90;⊗0 −I0;45;90;⊗90 ;S0;45;90;⊗i¼3 ¼ I0;45;90;⊗45 −I0;45;90;⊗135 ;S0;45;90;⊗i¼4 ¼ I0;45;90;⊗⊗ þ I0;45;90;⊗⊕ :

ð23Þ

Here, S0;45;90;⊗i¼1;2;3;4 is the parameters of the Stokes vector of an

image of a partially depolarizing histological section of bio-logical tissue; I0; 45; 90; 135; ⊗ ; ⊕ is the intensity of light trans-mitted through the object that passed through the linearpolarizer with the rotation angle of the transmission plane

Θ: 00; 450; 900; 1350 , and through a system of “quarterwave plate polarizer” of polarization analysis unit that trans-mits right (⊗) and left (⊕) circularly polarized components ofthe object laser radiation [24]. Within the ensemble (m × n) ofCCD camera pixels (The Imaging Source DMK41AU02.AS,monochrome 1/2″ CCD, Sony ICX205AL (progressive scan);resolution—1280 × 960; size of light-sensitive plate—5952m× 4464 m; sensitivity—0.05 lx; dynamic range—8 bit;SNR—9 bit, nonlinearity does not exceed 3–5%), we obtainthe coordinate distributions of the elements of the first-orderdifferential matrix.

The next step is the statistical analysis of the optical anisot-ropy maps obtained with the help of algorithms (17)–(23)LB0;90;LB45;135;CB⊗;⊕;LD0;90; LD45;135;CD⊗;⊕

�m� nð Þ the calculation of the

statistical moments of the first to fourth orders Zi = 1; 2; 3;

4which characterize the average (Z1), dispersion (Z2), skew-ness (Z3), and kurtosis (Z4) of the coordinate distributions ofthe values of the phase and amplitude anisotropy of the poly-crystalline structure of a sample of biological tissue [29].

Brief description of the facilities

Our choice of the objects of study was based on the followingfundamental and applied considerations.

Fundamental aspect—there are two limiting cases in thestructure of optically anisotropic biological layers [1, 9, 14,21]. The first case—spatially structured fibrillary networks.For such networks, due to availability of the “far” order, themechanism of linear birefringence prevails on the backgroundof circular birefringence of chiral moleculesLB0;90; LB45;135 > CB⊗;⊕;LD0;90; LD45;135 > CD⊗;⊕

. As an example of such a tissue,

we chose a histological section of the brain tissue of dead pa-tients. The polycrystalline structure of this object is mainlyformed by large-scale (up to 10 cm) fibrillary nerve fiber net-works (LB0; 90; LB45; 135; LD0; 90; LD45; 135). In addition, theneuron networks of the brain tissue contain thin filamentary neu-rofibrils, which are formed by neuroalbumins (LB0; 90; LB45; 135;LD0; 90; LD45; 135) and neuroglobulins (CB⊗; ⊕; CD⊗; ⊕).

The second one—parenchymatous or “islet” optically an-isotropic structures. For such formations, there is no “far” dueto a threadlike fibrous network. The predominant type of mor-phological structure here is the presence of spatially non-oriented polypeptide chains that are generated by opticallyactive molecules with circular birefringence. In other words,LB0;90; LB45;135≤CB⊗;⊕;LD0;90; LD45;135≤CD⊗;⊕

. As an example of such tissue, we

chose histological section of the rectum wall. The morphologicalstructure of this tissue is formed by three layers. The inner layer ismucous on the basis of epithelial tissue (CB⊗, ⊕; CD⊗, ⊕).Mu s c u l a r l a y e r f o rmed by smoo t h myocy t e s

Lasers Med Sci (2020) 35:877–891880

(LB0;90; LB45;135≈CB⊗;⊕;LD0;90; LD45;135≈CD⊗;⊕

) and fibers of loose connective tissue

(LB0;90; LB45;135;LD0;90; LD45;135

).

The study of differential matrices of the first order of selectedsamples provides information on themagnitude and ranges of thevariation of the statistical parameters (Zi = 1; 2; 3; 4), which char-acterize the distribution of phase (LB0, 90; LB45; 135; CB⊗, ⊕) andamplitude (LD0, 90; LD45; 135; CD⊗, ⊕) anisotropy of biologicaltissues with different polycrystalline structure.

Applied aspect—the actual task of differentialMuellermatrixmapping is the determination of the criteria for differentiationbetween healthy and pathologically altered samples of the humanbody’s biological tissue. As an example, we chose the histolog-ical sections of the operatively extracted vaginal wall during theprolapse of genitals of two types—healthy (group 1) and patho-logically altered (group 2) patients. We already mentioned that atthe preclinical stage of no less severe pathology, quantitativeestimation of changes in the morphological structure of the con-nective tissue component of the uterine ligament is ineffective[35, 36]. The morphological structure of the vaginal wall isformed by the mucosa with the epithelial base (CB⊗, ⊕; CD⊗,

⊕) and the fibers of the loose connective tissue (LB0;90; LB45;135;LD0;90; LD45;135

), as well as themuscular fibrillar spiral and longitudinal layers of

the myosin fibrils (LB0;90; LB45;135;LD0;90; LD45;135

). Prolapse of the geni-

tals is accompanied by destructuring and thinning of thefibrillar nets of the muscular layer. This pathological pro-cess leads to a decrease in structural optical anisotropy.Therefore, the obtained optical anisotropy mapsLB0;90; LB45;135;CB⊗;⊕;LD0;90; LD45;135;CD⊗;⊕

�m� nð Þ can be relevant for

the development of criteria for early (preclinical) diagno-sis of this pathology at a stage when there are no obvi-ous morphological changes in the vaginal wall.

Histological sections of all types of biological tissues weremade according to the standard method on a microtome withfreezing. The obtained samples are characterized by the follow-ing optical geometric parameters:

& brain tissue-geometric thickness l = 60μm; coefficient ofattenuation (extinction) τ = 0.21; degree of depolarizationof laser radiation Λ = 43%;

& rectal wall tissue-l = 60μm; τ = 0.32; Λ = 58%;& tissue samples of the vaginal wall-l = 60μm; τ = 0.26 ÷

0.29; Λ = 47 % − 52%.

Figure 1 shows microscopic images (×4) of the polycrystal-line structure of samples of biological layers, polarizationally

Fig. 1 Polarizationally visualized images of an optically anisotropicstructure of histological sections of brain (fragment (a), l = 60μm; τ =0.21; Λ = 43%), of rectal wall (fragment (b), l = 60μm; τ = 0.32; Λ =

58%) tissues, healthy (fragment (c), l = 60μm; τ = 0.27; Λ = 49%), andpathologically altered (fragment (d), l = 60μm; τ = 0.29;Λ = 52%) vaginawall. See explanation in the text

Lasers Med Sci (2020) 35:877–891 881

visualized in crossed polarizer-analyzer, namely, partiallydepolarizing histological sections of brain (fragment (1)), rectalwall (fragment (2)), healthy (fragment (3)), and pathologicallyaltered (fragment (4)) vagina wall tissues.

The polarization images of biological tissue histological sec-tions presented in Fig. 1 illustrate the diversity of topographicaland geometric structure (bright areas) of optically anisotropicstructures:

& birefringent spatially structured fibrillar nerve fiber networksof brain tissue (fragment (a));

& “islands” of optically active birefringent fibers of looseconnective tissue of the rectum wall (fragment (b));

& optically anisotropic fibers of loose connective tissue and& ibrillar myosin nets of healthy (fragment (c)) and patho-

logically altered (fragment (d)) vaginal wall.

As can be seen (fragments (a) and (b)), the topographic struc-ture of the polarization images of optically anisotropic tissue ofdifferent morphological structures is significantly different bothin size and shape. In contrast, a comparative qualitative analysisof images of histological sections of the vaginal wall samplesrevealed a similarity not only of morphological, but also poly-crystalline structure.

Results and discussion

This part of the article presents the results of experimentalapprobation of the method of differential Mueller matrix

mapping of histological sections of partially depolarizing bi-ological tissues of different morphological structure and phys-iological state with the aim of:

& studying the structure and symmetry of first-order dif-ferential matrices of all types of samples (Figs. 2, 3, 4,and 5);

& determination of the magnitude and ranges of the changesin the statistical moments of the first to fourth orders Zi = 1;

2 ; 3 ; 4 , which charac te r i ze the d i s t r ibu t ionsLB0;90; LB45;135;CB⊗;⊕;LD0;90; LD45;135;CD⊗;⊕

(Tables 1 and 2);

& finding objective criteria (Zi = 1; 2; 3; 4) for differentiatingthe polycrystalline structure of healthy and pathologicallyaltered tissues of the vaginal wall during genital prolapse(Figs. 6 and 7, Tables 3 and 4);

& comparative analysis of the diagnostic efficiency of ourmethod and the traditional methods of direct polarizationand Mueller matrix mapping (Table 5).

Differential matrices of the first order of spatiallystructured fibrillar networks

Figures 2 and 3 present the maps and histograms of thedistributions of the polarization-reconstructed distribu-tions of the phase parameters (see Fig. 2) and the ampli-tude (see Fig. 3) anisotropy of the partially depolarizinglayer of brain tissue.

Fig. 2 Maps (1–3) and histograms (4–6) of the partially depolarizing layer of brain tissue phase anisotropy parameters (l = 60μm; τ = 0.21; Λ = 43%)

Lasers Med Sci (2020) 35:877–891882

The analysis of the obtained results revealed:

& A good correlation between the symmetry of the experi-mentally determined differential matrix of the first orderand the theoretical data [37–43]:

m11f gh i ¼ m11f gh i ¼ m11f gh i ¼ m11f gh i;m12f gh i ¼ m21f gh ið Þ; m13f gh i ¼ m31f gh ið Þ; m14f gh i ¼ m41f gh ið Þ;m23f gh i ¼ − m32f gh ið Þ; m24f gh i ¼ − m42f gh ið Þ; m34f gh i ¼ − m43f gh ið Þ:

:

& The asymmetric structure of the histograms of distribu-tions of the linear and circular dichroism (LD0; 90; LD45;

135; CD⊗; ⊕) parameters (fragments (4)–(6) and, converse-ly, sufficiently symmetrical bell-shaped structure of histo-grams of distributions of the linear and circular birefrin-gence (LB0; 90; LB45; 135; CB⊗; ⊕) parameters (fragments(1)–(3)). From the physical point of view, these facts canbe related to the different multiplicities of the “non-ab-sorbing” (phase anisotropy—“q-acts”) and “absorbing”(amplitude anisotropy—“k-acts”) interaction of laser radi-ation with optically anisotropic structures of biologicaltissue of brain—q ≻ ≻ k. This difference is also determinedby the fact that in the red «section of the spectrum, theprobability of absorption is significantly lower than that ofFresnel transformations of laser waves by birefringent re-fractive collagen fibrils formed by optically active proteinmolecules. Due to the influence of these two factors in

accordance with the central boundary theorem [44], theaverage values of the linear (LB0; 90; LB45; 135) and circular(CB⊗; ⊕) birefringence appear to be almost normally dis-tributed. On the contrary, the values of the linear (LD0; 90;LD45; 135) and circular (CD⊗; ⊕) dichroism parameters ofcollagen networks are distributed asymmetrically.

& The predominance of the mechanisms of linear birefrin-gence and dichroism (average values and range of histo-gram variation presented in fragments (4), (5)) over opticalactivity and circular dichroism (fragment (6))—LB0;90; LB45;135 > CB⊗;⊕;LD0;90; LD45;135 > CD⊗;⊕

is due to the “developed” fi-

brillar structure of brain tissue.

The results of the statistical analysis of the distribu-tion of the parameters of the optical anisotropy of thepolycrystalline structure of the histological section ofbrain tissue are presented in Table 1.

The analysis of the data provided by the statisticalanalysis showed that due to different probability of ab-sorption acts and phase shift formation between the or-thogonal components of the laser radiation amplitude inthe volume of brain tissue, significant differences (with-in one order of magnitude—highlighted in gray) areformed between the statistical moments of the 3st; 4thorders that characterize the distribution of the values ofparameters LB0, 90; CB⊗, ⊕andLD0, 90; CD⊗, ⊕.

The predominance of linear birefringence and dichro-ism is quantitatively “detected” by large (1.5–2.5 times)values of average (Z1) and dispersion (Z2) of the

Fig. 3 Maps (1–3) and histograms (4–6) of the partially depolarizing layer of brain tissue amplitude anisotropy parameters (l = 60μm; τ = 0.21;Λ = 43%)

Lasers Med Sci (2020) 35:877–891 883

dis t r ibu t ionsLB0;90; LB45;135;LD0;90; LD45;135

in compar i son wi th

distributionsCB⊗;⊕;CD⊗;⊕

.

Differential matrices of the first order of “islet”polycrystalline structures of parenchymatous tissue

Figures 4 and 5 present maps and histograms of the partiallyd e p o l a r i z i n g l a y e r o f t h e r e c t u m w a l l

distributionLB0;90; LB45;135;CB⊗;⊕;LD0;90; LD45;135;CD⊗;⊕

.

The comparative analysis of the obtained data on the struc-ture of the first-order differential matrices of partiallydepolarizing layers of rectal wall (see Figs. 4 and 5) and brain(see Figs. 2 and 3) tissues revealed statistically similar mani-festations of the set of optical anisotropy mechanisms of thefibrillar network and of the ensemble of spatially non-structured “islet” protein chains. In particular, distributionsof the linear and circular dichroism parameters are also asym-metric (see Fig. 5, fragments (4)–(6)), while those of the pa-rameters of linear and circular birefringence are “bell-shaped”(see Fig. 4, fragments (4)–(6)). The revealed fact can be relat-ed to the large multiplicity of light scattering by phase-shiftingbiological nets and “islets”. Due to this (repeated averaging),the spatial-geometric structure of tissue samples of both typesis largely graded.

An important distinctive feature of the polycrystallinestructure of the rectum wall tissue is the commensurabilityof the mechanisms of the linear (fragments (4), (5)) and

circular (fragments (6)) birefringence. The revealed fact isquantitatively “detected” by the close by magnitude valuesof the mean (Z1) and variance (Z2), which characterize the

distributionsLB0;90; LB45;135;LD0;90; LD45;135

andCB⊗;⊕;CD⊗;⊕

, Table 2.

Comparative analysis of the data presented in Tables 1 and2 revealed the most sensitive diagnostic parameters—

skewness Z3LB0;90; LB45;135;CB⊗;⊕;LD0;90; LD45;135;CD⊗;⊕

�and kurtosis

Z4LB0;90; LB45;135;CB⊗;⊕;LD0;90; LD45;135;CD⊗;⊕

�. The differences between the

statistical moments of higher orders for distributions of differ-ential matrix elements for the samples of spatially structuredand parenchymatous tissues reach two to three times.

The obtained results were put in the basis of clinicalapplication—differential diagnosis of pathological changes inthe polycrystalline structure of partially depolarizing layers of bi-ological tissues of human organs on the example of prolapse ofgenitals.

Differential diagnosis of pathological changesin the polycrystalline structure of partiallydepolarizing layers of biological tissues

This part of the paper presents the results of a possible clinicalapplication of the differential Mueller matrix mapping methodfor partially depolarizing layers of the vaginal wall of differentphysiological states (group 1 and group 2). The topicality ofsuch studies is also due to the fact that this tissue is

Fig. 4 Maps (1–3) and histograms (4–6) of the partially depolarizing layer of rectum wall tissue phase anisotropy parameters (l = 60μm; τ = 0.32; Λ =58%)

Lasers Med Sci (2020) 35:877–891884

characterized by a high level of blood filling. Therefore, itshistological sections contain a significant amount of light-scattering uniform elements of blood, as well as coagulatedprotein fibrillar structures. As a result, it is practically impos-sible to obtain optically thin (τ < 0.1), non-depolarizing sec-tions of such tissue.

Maps and histograms of the distribution of the parametersof phaseLB0, 90; LB45; 135; CB⊗, ⊕ (see Fig. 6) and theamplitudeLD0, 90; LD45; 135; CD⊗, ⊕ (see Fig. 7) of anisotropyof the vaginal wall samples obtained for group 1 and group 2.

The comparative analysis of the histograms of parameterdistributions of the mechanisms of phase (see Fig. 6) andamplitude (see Fig. 7) anisotropy of layers of the tissue ofthe vaginal wall of both groups confirmed the conclusionconcerning the decrease of linear birefringence (LB0; 90;LB45; 135) and dichroism (LD0; 90; LD45; 135) levels in the caseof prolapse of genitals. This is indicated by the decrease in therange of changes and value of the main extremes of the

corresponding histograms (see Figs. 6 and 7, fragments (2b)and (4b)).

This can be explained by degenerative-dystrophic changesin the network of myosin fibrils of the spiral and longitudinallayers of the vaginal wall of patients from group 2. As a resultof such changes (disorientation of the packing directions, thin-ning [32, 33]), the linear birefringence and dichroismdecrease.

The level of optical activity (CB⊗; ⊕) and circular dichro-ism (CD⊗; ⊕) of samples of both types varies significantly.This is indicated by the similar structure of the extreme and theranges of changes in the structure of the histogramsH(CB⊗; ⊕)and H(CD⊗; ⊕) (see Figs. 6 and 7, fragments (6a) and (6b)).This can be explained by the fact that the concentration ofcircularly birefringent structures of the epithelial base of sam-ples of a healthy and pathologically altered vaginal wall variesinsignificantly. In order to determine the criteria for differen-tial diagnosis of tissue samples of the vaginal wall, we used a

Table 1 Statistical moments of the first to fourth orders characterizingdistributions of value of optical anisotropy parameters of a histologicalsection of brain tissue (z = 60μm; τ = 0, 21; Λ = 43%)

Zi LB0; 90 LB45; 135 CB⊗; ⊕ LD0; 90 LD45; 135 CD⊗; ⊕

Z1 0.015 0.025 0.01 0.074 0.085 0.065

Z2 0.03 0.04 0.02 0.087 0.07 0.06

Z3 0.11* 0.21 0.17* 0.82* 0.57 1.37*

Z4 0.18* 0.27 0.23* 1.12* 0.33 1.76*

Table 2 Statistical moments of the first to fourth orders, whichcharacterize the distribution of the of optical anisotropy parameters ofrectum wall (l = 60μm; τ = 0.32; Λ = 58%) histological section

Zi LB0; 90 LB45; 135 CB⊗; ⊕ LD0; 90 LD45; 135 CD⊗; ⊕

Z1 0.011 0.017 0.015 0.055 0.05 0.025

Z2 0.02 0.025 0.02 0.048 0.06 0.04

Z3 0.38* 0.24* 0.41 0.52* 0.18* 0.97

Z4 0.44* 0.32* 0.53 0.88* 0.34* 0.58

Fig. 5 Maps (1–3) and histograms (4–6) of the partially depolarizing layer of rectum wall tissue amplitude anisotropy parameters (l = 60μm; τ = 0.32;Λ = 58%)

Lasers Med Sci (2020) 35:877–891 885

Fig. 6 Maps (1a, 1b, 3a, 3b) andhistograms (2a, 2b, 4a, 4b) of thedistributions of the vaginal wallsamples from group 1 (fragments1a–6a) and group 2 (fragments1b–6b) phase (LB0; 90; LB45; 135;CB⊗; ⊕) anisotropy

Lasers Med Sci (2020) 35:877–891886

Fig. 7 Maps (1a, 1b, 3a, 3b) andhistograms (2a, 2b, 4a, 4b) of thedistributions of the vaginal wallsamples from group 1 (fragments1a–6a) and group 2 (fragments1b–6b) amplitude (LB0; 90; LB45;135; CB⊗; ⊕) anisotropy

Lasers Med Sci (2020) 35:877–891 887

well-tested method of statistical and information analysis,which is described in details in references [45, 46]. Here, wegive a brief description of its main stages:

& in the operatively extracted (prolapse of the genitalia) wallof the vagina it were visually determined areas of healthyand pathologically altered tissue;

& two groups of samples were formed: histological sectionsof tissue from such areas;

& by means of histological method (“gold standard”) foreach sample, its state was determined—“norm” (group1,32 samples) or “pathology” (group 2, 32 samples);

& within the limits of each group of samples with the veri-

fied diagnosis, the average values (Zi¼1;2;3;4 ) and standarddeviations (±2σ) of the magnitude of the statistical mo-me n t s t h a t c h a r a c t e r i z e t h e d i s t r i b u t i o n sLB0;90; LB45;135;CB⊗;⊕;LD0;90;LD45;135;CD⊗;⊕

�were calculated;

& the parameter Zi was considered statistically reliable if its

average value Z1ð Þi in group 1 does not coincide with the

value Z2ð Þi � 2σ in group 2 and vice versa;

& on this basis, the parameters Z*i¼1;2;3;4 of objective differ-

entiation of the samples of both groups were determined.

For possible clinical application of the method of differen-tial Mueller matrix mapping for each group of samples tradi-tional for evidence-based medicine [42, 43], operating char-acteristics that define the diagnostic potentiality of the methodwere determined. Namely—sensitivity (Se ¼ a

aþb 100% );

specificity (Sp ¼ ccþd 100% ); balanced accuracy (Ac ¼ SeþSp

2

), where a and b are the numbers of correct and wrong diag-noses within group 2; c and d are the same within group 1.

Tables 3 and 4 show the values (Zi¼1;2;3;4 � 2σ ) of thes t a t i s t i c a l p a r am e t e r s t h a t c h a r a c t e r i z e t h eLB0;90;LB45;135;CB⊗;⊕;LD0;90; LD45;135;CD⊗;⊕

�distributions of vagina wall his-

tological sections of the patients from group 1 and group 2.The comparative analysis of the set of statistical moments

of the first to fourth orders revealed the following ones, themost sensitive to changes in phase and amplitude anisotropyof vagina wall layers (highlighted in gray). As in the case ofanalysis of the polycrystalline structure of fibrillar (seeTable 1) and parenchymal (see Table 2) tissues, the most di-agnostically efficient parameters were the statistical momentsof the third and fourth orders:

& Z3; 4(LB0, 90; LB45; 135; CB⊗, ⊕)—good level Ac > 80%(see Table 3);

& Z3; 4(LD0, 90; LD45; 135; CD⊗, ⊕)—excellent quality Ac >90% (see Table 4).

This is arguably related to a high probability of scatteringof laser radiation by optically anisotropic structures of thevagina wall tissue. Due to this, the distribution histograms ofphase anisotropyH(LB0, 90; LB45; 135; CB⊗, ⊕) are close to the“bell-like” ones (see Fig. 6). Therefore, the difference betweenasymmetry and excess (Z3; 4(LB0, 90; LB45; 135; CB⊗, ⊕)) ofsuch distributions are less pronounced than in the case ofsimilar statistical parameters Z3; 4(LD0, 90; LD45; 135; CD⊗,

⊕), which characterize the distribution of the amplitudeanisotropy.

Table 3 Statistical moments of the first to fourth orders characterizing distributions (LB0, 90; LB45; 135;CB⊗, ⊕) of vagina wall histological sections ofthe patients from group 1 and group 2, and balanced accuracy of the method

Zi LB0, 90 Ac, % LB45; 135 Ac, % CB⊗, ⊕ Ac, %

Z1 0.04 ± 0.003 0.03 ± 0.002 74 0.045 ± 0.003 0.033 ± 0.002 78 0.025 ± 0.002 0.028 ± 0.002 60

Z2 0.06 ± 0.004 0.045 ± 0.003 76 0.055 ± 0.004 0.04 ± 0.003 80 0.021 ± 0.001 0.031 ± 0.002 66

Z3 0.31 ± 0.017* 0.48 ± 0.029* 82* 0.43 ± 0.027* 0.87 ± 0.052* 88* 0.32 ± 0.018* 0.51 ± 0.029* 84*

Z4 0.44 ± 0.027* 0.69 ± 0.041* 86* 0.62 ± 0.039* 0.96 ± 0.062* 86* 0.38 ± 0.022* 0.62 ± 0.036* 88*

Table 4 Statistical moments of the first to fourth orders characterizing distributions (LD0, 90; LD45; 135;CD⊗, ⊕) of vagina wall histological sections ofthe patients from group 1 and group 2 and balanced accuracy of the method

Zi LD0, 90 Ac, % LD45; 135 Ac, % CD⊗, ⊕ Ac, %

Z1 0.08 ± 0.005 0.065 ± 0.004 72 0.07 ± 0.004 0.05 ± 0.003 82 0.065 ± 0.004 0.046 ± 0.003 78

Z2 0.09 ± 0.006 0.07 ± 0.004 78 0.09 ± 0.006 0.06 ± 0.004 86 0.075 ± 0.005 0.057 ± 0.004 76

Z3 0.48 ± 0.028* 0.92 ± 0.0049* 90* 0.41 ± 0.025* 0.24 ± 0.014* 94* 0.29 ± 0.017* 0.12 ± 0.007* 92*

Z4 0.34 ± 0.019* 0.76 ± 0.042* 94* 0.32 ± 0.019* 0.18 ± 0.011* 92* 0.36 ± 0.021* 0.15 ± 0.008* 90*

Lasers Med Sci (2020) 35:877–891888

Comparative studies of the diagnostic efficiencyof direct polarization, Mueller matrix, and differentialmatrix mapping

In this part of the paper, we present comparative informationon the balanced accuracy of our method and other most com-monly spread polarimetric techniques—direct polarization(coordinate distributions of the azimuth α and polarizationellipticity β) [27] and Mueller matrix (MMI{Mik}) [28, 29]mapping of partially depolarizing biological tissues (seeTable 5).

Comparative analysis of balanced accuracy of diagnosticmethods of the change in phase and amplitude anisotropy ofpartially depolarizing layers of vagina wall tissue showed

& Polarization mapping {α, β} is not applicable (Ac < 70%)for differential diagnosis of the vagina wall with genitalprolapse.

& Accuracy of differential diagnosis by direct Mueller ma-trixmapping {Mik}method reaches satisfactory (Ac = 71–72%) level.

& Efficiency of differential matrix mapping lies within good(Ac > 80%) and excellent quality (Ac > 90%) (Table 5)(highlighted in gray).

Summary and conclusion

1. The differential Mueller matrix mapping method for re-construction of distributions of linear and circular birefrin-gence and dichroism parameters of partially depolarizinglayers of biological tissues of different morphologicalstructures is introduced and practically implemented.

2. Coordinate distributions of values of the parameters ofphase and amplitude anisotropy of histological sectionsof brain tissue with spatially structured optically aniso-tropic fibrillar network, as well as of parenchymatous tis-sue of rectum wall with the “islet” polycrystalline struc-ture, were determined.

3. Within the statistical analysis of polarizationally recon-structed distributions of the averaged parameters of phaseand amplitude anisotropy, significant sensitivity of the

statistical moments of the third and fourth orders to thechanges of the polycrystalline structure of partiallydepolarizing layers of biological tissues was determined.

4. Differential diagnostics of changes in the phase (goodbalanced accuracy) and amplitude (excellent balanced ac-curacy) of the anisotropy of the partially depolarizinglayers of the vagina wall tissue with prolapse of the gen-itals is realized.

5. The maximum diagnostic efficiency of the first-order dif-ferential matrix method was demonstrated in comparisonwith the traditional methods of polarization and Muellermatrix mapping of histological sections of light-scatteringbiological tissues.

Funding information Grant sponsor: Academy of Finland; Grant num-ber: 325097; Grant sponsor: INFOTECH; Grant sponsor: MEPhIAcademic Excellence Project; Grant number: 02.a03.21.0005; Grantsponsor: National Research Tomsk State University Academic D.I.Mendeleev Fund Program. This project has received funding from theATTRACT funded by the EC under Grant Agreement 777222.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict ofinterest.

Ethical approval This study was conducted in accordance with the prin-ciples of the Declaration of Helsinki and in compliance with theInternational Conference on Harmonization-Good Clinical Practice andlocal regulatory requirements. The study was reviewed and approved bythe appropriate Independent Ethics Committees, and written informedconsent was obtained from all subjects prior to study initiation

Open Access This article is distributed under the terms of the CreativeCommons At t r ibut ion 4 .0 In te rna t ional License (h t tp : / /creativecommons.org/licenses/by/4.0/), which permits unrestricted use,distribution, and reproduction in any medium, provided you give appro-priate credit to the original author(s) and the source, provide a link to theCreative Commons license, and indicate if changes were made.

References

1. Tuchin VV (2007) Tissue Optics: Light Scattering Methods andInstruments for Medical Diagnosis, 2nd edn. SPM 166, SPIEPress, Bellingham

2. Waigh TA (2007) Applied Biophysics: Molecular Approach forPhysical Scientists. Wiley, Chichester

Table 5 Balanced accuracy ofpolarization mapping methods forthe polycrystalline structure ofpartially depolarizing tissue layersof a healthy and pathologicallyaltered vagina wall

Zi {α; β} {Mik} (LB0, 90; LB45; 135;CB⊗, ⊕) (LD0, 90; LD45; 135;CD⊗, ⊕)

Z1 54–60% 58–62% 60–78% 72–82%

Z2 55–62% 60–64% 66–80% 76–86%

Z3 63–65% 65–69% 82–88%* 90–94%*

Z4 61–67% 64–68% 86–88%* 90–94%*

Lasers Med Sci (2020) 35:877–891 889

3. Wang XJ, Milner TE, Chang MC, Nelson JS (1996) Group refrac-tive index measurement of dry and hydrated type I collagen filmsusing optical low-coherence reflectometry. J Biomed Opt 1:212–216

4. Hemenger RP (1996) Refractive index changes in the ocular lensresult from increased light scatter. J Biomed Opt 1:268–272

5. Wang LV, Cote GL, Jacques SL (2002) Special section guest edi-torial: tissue polarimetry. J Biomed Opt 7:278

6. Hadley KC, Vitkin IA (2002) Optical rotation and linear and circu-lar depolarization rates in diffusively scattered light fromchiral, ra-cemic, and achiral turbid media. J Biomed Opt 7:291–299

7. Tuchin VV,Wang L, Zimnyakov DA (2006) Optical Polarization inBiomedical Applications. Springer, New York

8. Wang LV, Wu HI (2007) Biomedical Optics: Principles andImaging. Wiley-Interscience, Hoboken

9. Tuchin VV (2015) Tissue Optics: Light Scattering Methods andInstruments for Medical Diagnostics, vol PM 254, 3rd edn. SPIEPress, Bellingham

10. Ambiraijan A, Look DC (1997) A backward Monte Carlo studyofthe multiple scattering of a polarized laser beam. J QuantSpectrosc Radiat Transf 58:171–192

11. Zimnyakov DA, Sinichkin YP, Zakharov PV, Agafonov DN (2001)Residual polarization of non-coherently back- scattered linearly po-larized light: the influence of the anisotropy parameter of the scat-tering medium. Waves Random Media 11:395–412

12. Martino DE (2009) A polarization-based optical techniques appliedto biology and medicine. in Proc. European Workshop,EcolePolytechnique, Massy

13. Cote GL, Cameron BD (2009) A noninvasive glucose sensor basedon polarimetric measurements through the aqueous humor oftheeye. In: Tuchin VV (ed) Handbook of Optical Sensing ofGlucose in BiologicalFluids and Tissues. CRC Press, Taylor&Francis Group, London, pp 183–211

14. Gosh N, Vitkin IA (2011) Tissue polarimetry: concepts, challenges,applications and outlook. J Biomed Opt 16:110801

15. Jacques SL (2011) Polarized light imaging of biological tissues. In:Boas D, Pitris C, Ramanujam N (eds) Handbook of BiomedicalOptics. CRC Press, Boca Raton, pp 649–669

16. Maksimova IL, Tuchin VV, Shubochkin LP (1986) Polarizationfeatures of eyes cornea. Opt Spectrosc 60:801–807

17. Greenfield DS, Knighton RW, Huang XR (2000) Effect of cornealpolarization axis on assessment of retinal nerve fiber layer thicknessby scanning laser polarimetry. Am J Ophthalmol 129:715–722

18. Maksimova IL, Tuchin VV, Shubochkin LP (1988) Light scatteringmatrix of crystalline lens. Opt Spectrosc 65:615–619

19. Tuchin VV (1997) Light scattering study of tissues. J Phys Uspekhi40:495–515

20. Vitkin A, Ghosh N, de Martino A (2015) Tissue polarimetry.In: Andrews DL (ed) Photonics: Scientific Foundations,Technology and Applications, vol 4. Wiley, Hoboken, pp239–321

21. Sun M, He H, Zeng N, Du E, Guo Y, Peng C, He Y, Ma H (2014)Probing microstructural information of anisotropic scatteringmediausing rotation-independent polarization parameters. Appl Opt53(14):2949–2955

22. Spandana KU, Mahato KK, Mazumder N (2019) Polarization-resolved Stokes-Mueller imaging: a review of technology and ap-plications. Lasers Med Sci. https://doi.org/10.1007/s10103-019-02752-1

23. Ushenko YA (2013) Diagnostics of structure and physiolog-ical state of birefringent biological tissues: statistical, corre-lation and topological approaches. in Coherent-DomainOptical Methods: Biomedical Diagnostics. In: Tuchin VV(ed) Environmental Monitoring andMaterial Science, 2ndedn. Springer Reference, Science + Business Media, NewYork, pp 107–148

24. DebooB,SasianJ,ChipmanRA(2004)TDegreeofpolarizationsurfacesandmaps for analysis of depolarization. Opt Express 12:4941–4958

25. Ghosh N,Wood FG, Vitkin IA (2010) Polarized light assessment ofcomplex turbid media such as biological tissues viaMueller matrixdecomposition. In: Tuchin VV (ed) in Handbook of Photonics forBiomedical Science. CRC Press, Taylor & Francis Group, London,pp 253–282

26. Ghosh N, Wood MF, Vitkin IA (2008) Mueller matrix de-composition for extraction of individual polarization param-eters from complexтturbid media exhibiting multiple scatter-ing, optical activity,and linear birefringence. J Biomed Opt13(4):044036

27. Ushenko AG, Pishak VP (2004) Laser Polarimetry of BiologicalTissue: Principles and Applications. In: Tuchin VV (ed) Handbookof Coherent-Domain Optical Methods: Biomedical Diagnostics,Environmental and Material Science, vol 1. Kluwer AcademicPublishers, Boston, pp 93–138

28. Ushenko V, Sdobnov A, Syvokorovskaya A, Dubolazov A,Vanchulyak O, Ushenko A, Meglinski (2018) 3D Mueller-MatrixDiffusive Tomography of Polycrystalline Blood Films for CancerDiagnosis. Photonics 5:54

29. Ushenko A, Sdobnov A, Dubolazov A, Grytsiuk M, Ushenko Y,Bykov A, Meglinski I (2019) Stokes-Correlometry Analysis ofBiological Tissues With Polycrystalline Structure. IEEE J Sel TopQuant Electron 25:1–12

30. Angelsky OV, Ushenko AG, Ushenko YG (2005)Investigation of thecorrelation structure of biological tissuepolarization images during the diagnostics of their oncolog-ical changes. Phys Med Biol 50:4811

31. Angelski OV, Ushenko AG, Arkhelyuk AD, Ermolenko SB,Burkovets DN (2000) Scattering of laser radiation by multifractalbiological structures. Opt Spectrosc 88:444–447

32. Ushenko AG (1999) Laser diagnostics of biofractals. QuantumElectron 29:1078

33. Angelsky OV, Maksimyak AP, Maksimyak PP, Hanson SG (2005)Interference diagnostics of white-light vortices. Opt Express 13:8179–8183

34. Goksedef BP, Akbayr O, Corbacolu A, Guraslan H, Sencan F, ErolO, Cetin A (2012) Comparison of Preoperative EndometrialBiopsyGrade and Final Pathologic Diagnosis in Patients withEndometrioid Endometrial Cancer. J Turk Germ Gynecol Assoc13:106–110

35. Salter SA, Batra RS, Rohrer TE, Kohli N, Kimball AB (2006)Striaeand pelvic relaxation: Two disorders of connective tissue withstrong association. J Invest Dermatol 126:1745–1748

36. Labopuesse M (2012) Role of the extracellular matrix in epithelialmorphogenesis: A view from C. elegans. Organogenesis 8N:65–70

37. Azzam RMA (1978) Propagation of partially polarized lightthroughanisotropic media with or without depolarization. J OptSoc Am 68:1786–1767

38. Ortega-Quijano N, Arce-Diego JL (2011) Mueller matrixdifferentialdecomposition. Opt Lett 36:1942–1944

39. Ortega-Quijano N, Arce-Diego JL (2011) Depolarizing differentialMueller matrices. Opt Lett 36:2429–2431

40. Ossikovski R, Devlaminck V (2014) General criterion for the phys-ical realizability of the differential Mueller matrix. Opt Lett 39:1216–1219

41. Devlaminck V, Ossikovski R (2014) Uniqueness of the differentialMueller matrix of uniform homogeneous media. Opt Lett 39:3149–31452

42. Devlaminck V (2013) Physical model of differential Mueller matrixfor depolarizing uniform media. J Opt Soc Am 30:2196–2204

43. Ossikovski R, Arteaga O (2005) Statistical meaning of the differ-ential Mueller matrix of depolarizing homogeneous media. Opt Lett39:4470–4473

Lasers Med Sci (2020) 35:877–891890

44. Goodman JW (1975) Statistical properties of laser speckle patterns.In: Dainty JC (ed) Laser Speckle and Related Phenomena.Springer-Verlag, Berlin, pp 9–75

45. Cassidy LD (2005) Basic concepts of statistical analysis for surgicalresearch. J Surg Res 128:199–206

46. Davis CS (2002) Statistical methods of the analysis of repeatedmeasurements. Springer-Verlag, New York

Publisher’s note Springer Nature remains neutral with regard to jurisdic-tional claims in published maps and institutional affiliations.

Lasers Med Sci (2020) 35:877–891 891