Investigation of singularities inherent to Mueller matrix images of biological crystals: diagnostics of their birefringent structure
In this work, we have theoretically grounded conceptions of singularities observed in coordinate distributions of Mueller matrix elements for a network of human tissue biological crystals. Found is interrelation between polarization singularities of laser images inherent to these biological cryst...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2009
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| Цитувати: | Investigation of singularities inherent to Mueller matrix images of biological crystals: diagnostics of their birefringent structure / I.Z. Misevitch, Yu.O. Ushenko, O.G. Pridiy, A.V. Motrich, Yu.Ya. Tomka, O.V. Dubolazov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2009. — Т. 12, № 4. — С. 379-390. — Бібліогр.: 26 назв. — англ. |
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irk-123456789-1188412017-06-01T03:06:45Z Investigation of singularities inherent to Mueller matrix images of biological crystals: diagnostics of their birefringent structure Misevitch, I.Z. Ushenko, Yu.O. Pridiy, O.G. Motrich, A.V. Tomka, Yu.Ya. Dubolazov, O.V. In this work, we have theoretically grounded conceptions of singularities observed in coordinate distributions of Mueller matrix elements for a network of human tissue biological crystals. Found is interrelation between polarization singularities of laser images inherent to these biological crystals and characteristic values of above matrix elements. We have determined criteria for statistical diagnostics of pathological changes in the birefringent structure of biological crystal network by using myometrium tissue as an example. 2009 Article Investigation of singularities inherent to Mueller matrix images of biological crystals: diagnostics of their birefringent structure / I.Z. Misevitch, Yu.O. Ushenko, O.G. Pridiy, A.V. Motrich, Yu.Ya. Tomka, O.V. Dubolazov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2009. — Т. 12, № 4. — С. 379-390. — Бібліогр.: 26 назв. — англ. 1560-8034 PACS 87.64.-t http://dspace.nbuv.gov.ua/handle/123456789/118841 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
In this work, we have theoretically grounded conceptions of singularities
observed in coordinate distributions of Mueller matrix elements for a network of human
tissue biological crystals. Found is interrelation between polarization singularities of
laser images inherent to these biological crystals and characteristic values of above
matrix elements. We have determined criteria for statistical diagnostics of pathological
changes in the birefringent structure of biological crystal network by using myometrium
tissue as an example. |
| format |
Article |
| author |
Misevitch, I.Z. Ushenko, Yu.O. Pridiy, O.G. Motrich, A.V. Tomka, Yu.Ya. Dubolazov, O.V. |
| spellingShingle |
Misevitch, I.Z. Ushenko, Yu.O. Pridiy, O.G. Motrich, A.V. Tomka, Yu.Ya. Dubolazov, O.V. Investigation of singularities inherent to Mueller matrix images of biological crystals: diagnostics of their birefringent structure Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Misevitch, I.Z. Ushenko, Yu.O. Pridiy, O.G. Motrich, A.V. Tomka, Yu.Ya. Dubolazov, O.V. |
| author_sort |
Misevitch, I.Z. |
| title |
Investigation of singularities inherent to Mueller matrix images of biological crystals: diagnostics of their birefringent structure |
| title_short |
Investigation of singularities inherent to Mueller matrix images of biological crystals: diagnostics of their birefringent structure |
| title_full |
Investigation of singularities inherent to Mueller matrix images of biological crystals: diagnostics of their birefringent structure |
| title_fullStr |
Investigation of singularities inherent to Mueller matrix images of biological crystals: diagnostics of their birefringent structure |
| title_full_unstemmed |
Investigation of singularities inherent to Mueller matrix images of biological crystals: diagnostics of their birefringent structure |
| title_sort |
investigation of singularities inherent to mueller matrix images of biological crystals: diagnostics of their birefringent structure |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118841 |
| citation_txt |
Investigation of singularities inherent
to Mueller matrix images of biological crystals:
diagnostics of their birefringent structure / I.Z. Misevitch, Yu.O. Ushenko, O.G. Pridiy, A.V. Motrich, Yu.Ya. Tomka, O.V. Dubolazov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2009. — Т. 12, № 4. — С. 379-390. — Бібліогр.: 26 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
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2025-07-08T14:46:00Z |
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| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 379-390.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
379
PACS 87.64.-t
Investigation of singularities inherent
to Mueller matrix images of biological crystals:
diagnostics of their birefringent structure
I.Z. Misevitch, Yu.O. Ushenko, O.G. Pridiy, A.V. Motrich, Yu.Ya. Tomka, O.V. Dubolazov
Chernivtsi National University, Optics and Spectroscopy Dept., Chernivtsi 58000, Ukraine
Abstract. In this work, we have theoretically grounded conceptions of singularities
observed in coordinate distributions of Mueller matrix elements for a network of human
tissue biological crystals. Found is interrelation between polarization singularities of
laser images inherent to these biological crystals and characteristic values of above
matrix elements. We have determined criteria for statistical diagnostics of pathological
changes in the birefringent structure of biological crystal network by using myometrium
tissue as an example.
Keywords: polarization, Mueller matrix, singularity, biological crystal, birefringence,
statistical moment.
Manuscript received 28.05.09; accepted for publication 10.09.09; published online 30.10.09.
1. Introduction
In recent years, in laser diagnostics of biological tissue
(BT) structures they effectively use the model approach
[1] that allows considering this object as containing two
components: amorphous and optically anisotropic ones.
Topicality of this modeling is related with the possibility
to apply the Mueller matrix analysis of changes in
polarization properties caused by transformation of the
optic-and-geometric structure of anisotropic components
in these biological objects [2-7], optical properties of
which are often described using the Mueller matrix [8].
Being based on the approximation of a single light
scattering, they found interrelation between the set of
statistic distribution moments of the 1st to 4th orders
Z(j = 1, 2, 3, 4) that characterizes orientation and phase
structure of BT birefringent architectonics as well as the
set of respective moments [9] for two-dimensional
distributions of Mueller matrix elements or Mueller-
matrix images (MMI) [10-14]. In parallel with
traditional statistical investigations, formed in recent 10
to 15 years is the new optical approach to describe a
structure of polarizationally inhomogeneous fields in the
case of scattered coherent radiation. The main feature of
this approach is the analysis of definite polarization
states to determine the whole structure of coordinate
distributions for azimuths and ellipticities of
polarization. The so-called polarization singularities are
commonly used as these states [15-24]:
- states with linear polarization when the direction of
rotation for the electric field vector is indefinite, the
so-called S-points;
- circularly-polarized states when the azimuth of
polarization (α) for the electric field vector is
indefinite, the so-called C-points.
Investigations of polarizationally inhomogeneous
object fields for BT with different morphology [25 – 27]
allowed to ascertain that they possess a developed
network of S- and C-points. For example in [26], the
authors found interrelations between conditions providing
formation of polarization singular points and particularity
of the orientation-phase structure of biological crystals
present in territorial matrix of human tissue architectonic
network. These interrelations served as a base to make
statistical and fractal analyses of distribution densities for
the number of singular points in BT images. As a result,
the authors confirmed the efficiency of this method for
investigation of object fields to differentiate optical
properties of BT with a different morphological structure
and physiological state.
It should be noted that studied in the works [26, 27]
were only statistical sets of amounts of S- and C-points
for polarization inhomogeneous object fields. In these
cases, information about the structure of topological
distributions inherent to polarization states around these
points was outside researchers’ attention. Besides, the
mechanisms of direct formation of polarization
singularities in biological crystals as well as
interferential formation of these polarization states due
to a multiple scattering were distinguished not so clearly.
Starting from it, a further analysis of mechanisms
and scenarios providing formation of polarization
singularities in the field of laser radiation scattered by a
biological object seems actual.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 379-390.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
380
2. Interrelation between the Mueller matrix and
polarization images of biological tissues
As a mathematical apparatus to describe mechanisms of
polarization singularity formation by a biological object,
one can use the Mueller matrix that entirely describes its
optical properties
44434241
34333231
24232221
14131211
ffff
ffff
ffff
ffff
F . (1)
As known from [14], any “ik” element of the
Mueller matrix characterizes the degree of
transformation taking place with the Stokes vector
parameter kU 0 of the probing beam when transforming
it into the Stokes vector parameter iU of the object
beam
0
444
0
343
0
2424
0
434
0
333
0
2323
0
424
0
323
0
2222
UfUfUfU
UfUfUfU
UfUfUfU
(2)
Originating from (2), let us understand as
singularities of Mueller matrix elements for biological
objects the following “uncertainties”
.0ikf (3)
From the mathematical viewpoint, the relations (3)
mean that the mechanisms of transformation of the
Stokes vector parameter kU 0 into iU are indeterminate.
In other words, indeterminate is the scenario for
transformation of the laser radiation polarization state by
biological tissue.
Another singular “state” is described by the
following values of Mueller matrix elements
.1ikf (4)
From the physical viewpoint, the condition (4)
means that optical properties of this medium (isotropic,
anisotropic, etc.) and the respective mechanisms of
transformation of laser radiation polarization states are
indeterminate.
Some characteristic examples of optical property
singularities are demonstrated by biological tissues that
are combinations of isotropic A and anisotropic F
structures [15]. Each of these components is
characterized by intrinsic matrix operators
,
1000
0100
0010
0001
leA (5)
where is the extinction coefficient inherent to the
layer of biological tissue with the geometric thickness l
.
cossin2cos
sin2coscos2cos2sin
sin2sin)cos1(2sin2cos
00
sin2sin0
)cos1(2sin2cos0
cos2sin2cos0
01
22
22
F
(6)
Here, is the orientation of an protein fibril in the
architectonic network, the matter of which introduces the
phase shift between orthogonal components of the
laser wave amplitudes.
A comparative analysis of matrixes (5) and (6)
shows that the values of elements ika and ikf coincide
in definite conditions
.1
,0
ikik
ikik
fa
fa
(7)
It follows from (7) that it is impossible to separate
or differentiate properties of isotropic and anisotropic
optical components for biological objects in the points
where Mueller matrix elements acquire singular values.
Analysis of information about conditions providing
formation of singular values for Mueller matrix elements
describing biological tissues (Table 1) allows to predict
scenarios for formation of “field” polarization
singularities by them and to determine:
- a whole set of positive and negative C-points
( 2
) in the image of biological object, the
coordinate position of which is corresponded with
the following conditions 0332244 fff ;
- a whole set of S-points ( 0 ) for a polarization
image with arbitrary azimuths ( 0 ) which
are corresponded with the conditions 13322 ff .
Besides, it is possible to select different types of
polarization singularities in the image of biological
tissue:
- dextrogyrate ( 2
) – (+C) and laevogyrate
( 2
) – (–C) singular points
"."1
;""1
34,24
34,24
Cf
Cf
(8)
- “orthogonal” S-points, formation of which is
related with orthogonal orientations of the optical
axis in biological crystal
.
2
;0int,1
4
int,0
90,034
45,4534
forspoSf
forspoSf
(9)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 379-390.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
381
So, after determining the coordinate distributions of
the Mueller matrix for any biological tissue, one can
predict a further scenario of formation of polarization
singularities in the object field, which set its coordinate
topological structure in the form of an aggregate of S
contours [5 – 10]. Thereof, the next step was to study
interrelations between topological singular structures for
polarization images of BT and respective topological
structures of two-dimensional distributions for elements
of their Mueller matrixes.
Summarized in Table 1 are the data upon relations
of Mueller matrix element singular values for BT and
respective values of optical axis orientations as well as
phase shifts of the anisotropic components for biological
crystals.
Table 1. Conditions providing formation of singular values
for matrix elements fik
fik ρ δ
f22
0
all 2/
4/ all
1 all 0
f22 = f32
0
all 0
4/ all
1 all 0
f24 = f42
0
all 0
0 all
1 4/ 2/
-1 4/ 2/
f33
0
all 2/
2/ all
1 all 0
f34 = f43
0
all 0
4/ all
1 0 2/
–1 2/ 2/
f44
0 all 2/
1 all 0
–1 all
Our analysis of the data summarized in Table 1
shows that, among mechanisms providing formation of
singular values for Mueller matrix elements by a system
of optical single-axis birefringent fibrils in the
architectonic network of biological tissue, one can
separate two groups that correspond to formation of
circular polarization ( 2
):
“phase” ones, when singular values of the
element 044 f are formed only due to phase shifts
2
independently of directions inherent to optical
axes of biological crystals;
“orientational” ones, when singular values of
the elements 14224 ff are formed due to phase
shifts 2
in dependence on two directions
4
inherent to optical axes of biological
crystals or singular values of the elements
14334 ff are formed due to phase shifts
2
in dependence on two directions 2;0
inherent to optical axes of biological crystals.
3. The scheme of experimental studying and method
of measuring the singularities in Mueller matrix
images of biological tissues
Fig. 1 shows the traditional optical scheme of a po-
larimeter to measure sets of MMI for mounts of BT [14].
Illumination was performed with a parallel ( =
104 μm) beam of a He-Ne laser ( = 0.6328 μm,
W = 5.0 mW). The polarization illuminator consists of
the quarter-wave plates 3 and 5 as well as polarizer 4,
which provides formation of a laser beam with an
arbitrary azimuth 0
0
0 1800 or ellipticity
0
0
0 900 of polarization.
Polarization images of BT were projected using
the micro-objective 7 into the light-sensitive plane
(800x600 pixels) of CCD-camera 10 that provided
measurements of BT structural elements within the
range 2 to 2,000 μm.
Experimental conditions were chosen in such a
manner that spatial-anglular filtration was practically
eliminated when forming BT images. It was provided by
matching the angular characteristics of light scattering
indicatrixes by BT samples ( 016 ) and angular
aperture of the micro-objective ( 020 ). Here, is
the angular cone of an indicatrix where 98% of the total
scattered radiation energy is concentrated.
Fig. 1. Optical scheme of a polarimeter: 1 – He-Ne laser; 2 –
collimator; 3 – stationary quarter-wave plate; 5, 8 –
mechanically movable quarter-wave plates; 4, 9 – polarizer
and analyzer, respectively; 6 – studied object; 7 – micro-
objective; 10 – CCD camera; 11 – personal computer.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 379-390.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
382
Analysis of BT images was made using the
polarizer 9 and quarter-wave plate 8. As a result, we
determined the Stokes vectors for BT images 4,3,2,1jS
and calculated the ensemble of Mueller matrix elements
in one point illuminated with a laser beam (with the
cross-section 2rS ) in accord with the following
algorithm
.4,3,2,1,
;
;5.0
;5.0
1
4
4
1
3
3
21
2
21
1
iMSM
MSM
SSM
SSM
iii
iii
iii
iii
(10)
The indexes 1 to 4 corresponds to the following
polarization states of the beam illuminating BT: 1 - 00; 2
– 900; 3 - +450; 4 – (right circulation).
The method to measure MMI singularities for BT
samples was as follows:
BT mount was illuminated with a laser beam,
within the area of which in accord with the algorithm
(10) we determined the array ( 600800 nm ) of
values for each element of the Mueller matrix
30,...2,1
1
,
111
,..........
,........,
l
nm
ik
n
ik
m
ikik
ik
MM
MM
nmM . (11)
determined for each massif jik nmM were
coordinate distributions of its singular values.
As an object of the experimental study, we used
tissues of a woman matrix (myometrium).
Fig. 2. Topological structure characterizing the two-dimensional distributions of the matrix element f22 for myometrium (a) and
U2 parameter of the Stokes vector (b) describing its polarization image.
Fig. 3. Topological structure corresponding to the two-dimensional distributions of the matrix element f23 for myometrium
layer (a) and U3 parameter of the Stokes vector (b) describing its polarization image.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 379-390.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
383
4. Local topological structure of experimental
Mueller-matrix and polarization images
of biological tissues
In the series of Figs 2 to 5, we show the results of
experimental MMI measurements for ikf measurements
(fragments a and b) as well as Stokes-polarimetric study
of image topological structure jU (fragment b) inherent
to human conjunctive tissue.
It can be easily seen that the elements 22f and 23f
of the Mueller matrix as well as respective coordinate
distributions of 2U and 3U parameters of the Stokes
vector for myometrium polarization image are
characterized with closed matrix S-contours in the form
of a set of S- and ±С-lines (Fig. 2, fragments A and B;
Fig. 3, fragments A, B and C).
While for the elements 22f and 23f of the
Mueller matrix as well as respective coordinate
distributions of 2U and 3U parameters of the Stokes
vector for myometrium polarization image are
characterized with open matrix S-contours in the form of
a set of S- and ±С-lines (Figs 4 and 5, fragments A, B
and C).
It should be noted that for all the types of
topological distributions for MMI of myometrium we
Fig. 4. Topological structure characterizing the two-dimensional distributions of the matrix element f44 for myometrium (a)
and U4 parameter of the Stokes vector (b) describing its polarization image.
Fig. 5. Topological structure characterizing the two-dimensional distributions of the matrix element f33 for myometrium (a) and
U3 parameter of the Stokes vector (b) describing its polarization image.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 379-390.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
384
have revealed one universal regularity. Namely,
geometrical places of singular points belonging to one
type are necessarily separated with geometrical places
for singular points of another type. For instance,
;1),(
0),(1),(
44,33,22
10
10
44,33,22
10
44,33,22
yxf
yxfyxf
ik
ikik
f
ff
(12)
;1),(
0),(1),(
42,24,43,34
10
10
42,24,43,34
10
42,24,43,34
yxf
yxfyxf
ik
ikik
f
ff
(13)
.1),(
0),(1),(
32,23
10
10
32,23
10
32,23
yxf
yxfyxf
ik
ikik
f
ff
(14)
The expressions (12) to (14) are mathematical
equivalents of S-contours in coordinate distributions of
the whole set of matrix elements ikf that characterize
birefringence of this biological tissue.
Juxtaposition of topological distributions ),( yxfik
and ),( yxU j allows to find a coordinate correlation not
only for their singular values but for respective types of
“object” and “field” S-contours, too.
Thus, the topological structure of MMI for the
ensemble of elements ),( yxfik that characterizes optical
Fig. 6. Coordinate structure of the matrix element f22 for myometrium (a). A – coordinate distribution of singular values (b) –
S-points (f22 = 1) labeled as () and +C-points (f22 = 0) labeled as (Δ).
Fig. 7. Coordinate structure of the matrix element f44 for myometrium (a). A – coordinate distribution of singular values (b) –
S-points (f44 = 1) labeled as () and +C-points (f44 = 0) labeled as (Δ).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 379-390.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
385
anisotropy of BT properties is basic in formation of
singular structures (see distributions of S- and C-points)
of the object boundary field and can be used in a further
analysis of their interferential transformations.
5. Statistic singular structure of the Mueller matrixes
for biological tissues
Figs 6 to 11 show statistical coordinate distributions for all
the types of singularities inherent to the ensemble of
matrix elements represented in Table 1 that were
determined over the whole area of the sample under study.
As seen from these experimental data, the
coordinate distributions of all the Mueller matrix
elements for myometrium possess a developed network
of singular values. Being based on this fact, we have
offered singular differentiation of changes in the
distribution of optical axis orientations in biological
crystals that form the architectonic network, by using as
an example the woman matrix tissue.
Fig. 8. Structure of the matrix element f33 for myometrium (a). A – coordinate distribution of singular values (b) – S-points
(f33 = 1) labeled as () and +C-points (f33 = 0) labeled as (Δ).
Fig. 9. Coordinate structure of the matrix element f34 for myometrium (a). A – coordinate distribution of singular values (b) –
S-points (f34 = 1) labeled as (), +C-points (f34 = +1) - (Δ) and –C-points (f34 = –1) - ( ).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 379-390.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
386
6. Mueller-matrix diagnostics of orientation changes
of organic crystals in biological tissues
As objects for our experimental investigations, we used
mounts of myometrium tissue of two types:
biopsy of healthy tissue from a woman
matrix (type A);
biopsy of conditionally normal tissue from
the vicinity of a benign hysteromyoma (type B).
Fig. 12 shows MMI of the element 44f for
myometrium samples of A and B types.
From the optical viewpoint, the obtained two-
dimensional distributions ),(44 yxf characterize the
degree of anisotropy in the matter of studied samples.
Thereof, it can be easily seen that the birefringency
value of the samples A and B is practically identical. It is
confirmed by the close level of relative values for the
matrix element ),(44 yxf describing the tissues of A and
B types ( ),(),( 4444 yxfyxf АБ ). In parallel with it, one
can observe ordering the directions of optical axes
inherent to anisotropical structures of type B
myometrium.
Thus, the main parameter allowing differentiation
of optical properties for the samples of this type is the
orientation structure of their birefringent networks.
Fig. 10. Coordinate structure of the matrix element f24 for myometrium (a). A – coordinate distribution of singular values (b) –
S-points (f24 = 1) labeled as (), +C-points (f24 = +1) - (Δ) and –C-points (f24 = –1) - ( ).
Fig. 11. Coordinate structure of the matrix element f23 for myometrium (a). A – coordinate distribution of singular values (b) –
S-points (f23 = 1) labeled as (), +C-points (f23 = +1) - (Δ) and –C-points (f23 = –1) - ( ).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 379-390.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
387
Fig. 12. MMI of the element 44f for myometrium tissue of A (a) and B (b) types.
Fig. 13. Dependences of singularity numbers )0( N (a), )90( N (b), )45( N (c), )45( N (d) for
myometrium tissue of A type.
.
To obtain objective criteria for Mueller-matrix
singular differentiation of optical properties inherent to
the myometrium samples of A and B types, we have
used the following approach:
- measured in sequence were the MMI elements
42,24f and 43,34f , that are basic to determine singular
states (±1) formed in organic crystals with orthogonal
orientations of optical axes ( 190,0 43,34 f ;
145,45 42,24 f (Table 1);
- determined were graticubes for distribution of the
singular values 1),(43,34 yxf and 1),(42,24 yxf ;
- determined was the set of dependences for the
number of points corresponding to singular states MMI
)0( N , )90( N , )45( N , )45( N ;
- )( iN dependences were processed using the
following algorithms:
.
)45()45(
)45()45(
;
)90()0(
)90()0(
)(
)(
42,24
43,34
MM
MM
Z
MM
MM
Z
f
M
f
M
(15)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 379-390.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
388
.
)45()45(
)45()45(
;
)90()0(
)90()0(
)(
)(
42,24
43,34
f
f
Z
Z
(16)
Here, )( iM and )( i are the average and
dispersion of )( iN distributions.
Shown in Figs 13 and 14 are the distributions of
singularity numbers )()( xN j for the myometrium tissue
of A and B types.
In the case of myometrium tissue with pathological
changes, one can observe asymmetry between the ranges
of changes in values of the dependences 0)( N
(Fig. 14, a) and 90)( N (Fig. 14, b).
The above results can be explained as based on the
found relation between conditions providing formation of
MMI singular values and orientation-phase structure of
biological crystals in the myometrium tissue (Table 1).
Orientation structure of MMI for the element
44f describing the myometrium tissue of B type
(Fig. 12, b) contains singular points 1),(44 yxf
asymmetrically located in the direction 90 .
Thereof, one should expect a maximal number of
singular values for the element 1),(34 yxf as
compared to that of singular values 1),(43,34 yxf and
1),(42,24 yxf .
Statistically found asymmetry in distributions of
singular states for MMI describing the myometrium
tissue of both types was estimated using the coefficients
of asymmetry (15) and (16) introduced by us. Table 2
shows statistically averaged values of the coefficients
MZ and Z within two groups of myometrium samples
of A and B types.
Analysis of data represented in Table 2 allowed us
to conclude:
- first- and second-order statistical moments for
distributions of singular values ),(43,34 yxf and
),(42,24 yxf characterizing MMI of healthy myometrium
tissue do not practically differ from zero, which is
indicative of their azimuthal symmetry;
Fig. 14. Dependences of singularity numbers )0( N (a), )90( N (b), )45( N (c), )45( N
(d) for myometrium tissue of B type.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 379-390.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
389
Table 2. Asymmetry coefficients of Mueller-matrix
singularities
Z
Myometrium
(normal state)
(25 samples)
Myometrium
(pathological state)
(23 samples)
34,43f
MZ 0.03±0.005 0.45±0.063
34,43f
Z 0.02±0.004 0.12±0.019
24,42f
MZ 0.025±0.0036 0.28±0.037
24,42f
Z 0.03±0.0047 0.14±0.021
- values of the asymmetry coefficient for
distributions of MMI singular values ),(43,34 yxf and
),(42,24 yxf describing the pathologically changed
myometrium tissue of B type grow practically by one
order, which indicates formation of their azimuthal
asymmetry related with the direction of pathological
growth of birefringent protein fibrils.
Thus, the above analysis of statistical distributions
describing the number of points for MMI singular values
inherent to the set of elements ikf characterizing
biological tissues of different kinds seems to be efficient
in differentiation of phase and orientation changes in the
structure of their birefringent components, which are
related with changes in their physiological state.
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