statistical structure of skin derma mueller matrix images in the process of cancer changes

ISSN 1060�992X, Optical Memory and Neural Networks (Information Optics), 2011, Vol. 20, No. 2 pp. 145–154. © Allerton Press, Inc., 2011.

145

1. INTRODUCTION

Among many directions of optical diagnostics of organic phase�inhomogeneous objects an new tech�nique—laser polarimetry [1–29]—has been formed within recent 10 years. It enables to obtain informa�tion about optical anisotropy [2–7, 9–24] of BT in the form of coordinate distributions of the biological tis�sues (BT) Mueller matrix elements, azimuths and ellipticities of their object fields' polarization [8, 10–13].To analyze this polarimetric information the following model approach was elaborated [2, 5, 12]:

—all the variety of human BT can be represented as four main types—connective, muscle, epithelialand nerve tissues;

—morphological structure of any BT type is regarded as a 2�component amorphous�crystalline struc�ture;

—the crystalline component or extracellular matrix is an architectonic net consisting of coaxial cylin�drical protein (collagen, myosin, elastin, etc.) fibrils;

—optically, the protein fibrils possess the properties of uniaxial birefringent crystals;

—interaction of laser radiation with the BT layer is considered in the single scattering approximation,when the attenuation factor satisfies corresponds to ( —attenuation coefficient).

Specifically, the above mentioned model was used for finding and substantiating the interconnectionsbetween the ensemble of statistic moments of the 1st–4th orders that characterize the orientation�phasestructure (distribution of optical axes and phase shifts of protein fibrils networks directions) of birefringentBT architectonics and that of 2D distributions of the elements of the corresponding Mueller matrix[21, 23]. It was determined [22] that the 3rd and the 4th statistic moments of coordinate distributions of'phase' matrix elements ( ) are the most sensitive to the change (dystrophic and oncological pro�cesses) of optical anisotropy of protein crystals. These statistic moments characterize the BT extracellularmatrix birefringence. On this basis the criteria of early diagnostics of muscle dystrophy, pre�cancer statesof connective tissue, collagenosis, etc. were determined.

However, such techniques do not take into account the coordinate heterogeneity of orientation�phasestructure of protein crystals nets of the BT layer, as well as the order of scattering in its depth. Thus it isimportant to investigate the distribution of statistic moments of the 1st–4th orders that characterize the2D elements of Mueller matrix not only in the section of the probing laser beam, but also within the wholeBT layer of various optical thickness and physiological state.

For this we shall consider the potentiality of matrix modeling of polarization properties of opticallythick BT layer.

0.1τ ≤ τ

24,z 34,z 44z

Statistical Structure of Skin Derma Mueller Matrix Images in the Process of Cancer Changes

Yu. A. Ushenkoa, O. V. Dubolazovb, and A. O. Karachevtsevb

aCorrelation Optics Department, Chernivtsi National University, 2 Kotsyubinsky Str., 58012, Chernivtsi, UkrainebOptics and Spectroscopy Department, Chernivtsi National University, 2 Kotsyubinsky Str., 58012, Chernivtsi, Ukraine

e�mail: [email protected] February 24, 2011; in final form, April 12, 2011

Abstract—This research is aimed to investigate the reliability of Mueller�matrix differentiation ofbirefringence change of optically thick layers of biological tissues at the early stages of the change intheir physiological state. This is performed by measuring the set of the skewness and the kurtosis valuesof Mueller matrix image of the phase element M44 in various points of the object under investigation.

Keywords: polarization, correlation, biological tissue, statistics, Mueller matrix.

DOI: 10.3103/S1060992X1102010X

146

OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS) Vol. 20 No. 2 2011

USHENKO et al.

2. MUELLER MATRIX MODELING OF THE PROPERTIES OF OPTICALLY THICK BIOLOGICAL TISSUE

Let us represent the layer of such a biological object as the set of successively located optically thin par�tial layers (Fig. 1).

Polarization properties of birefringent nets of every other BT layer ( ) are described by the Muellermatrix representing a superposition of matrix operators of separate optically coaxial protein fibrils [2]

(1)

where

(2)

Here —the direction of optical axis determined by the direction of packing of the birefringent fibril,—phase shift introduced between the orthogonal components of the amplitude of laser wave

with the length λ passing through the fibril with linear size of its geometrical section d and birefringenceindex

Mueller matrix elements of the net of protein fibrils of partial BT layer (n) are determined by thefollowing algorithms:

• for finite number (u = 1–N) of fibrils

(3)

• for “infinite” ( ) number of fibrils

(4)

Here and —distribution function of orientation (ρ) and phase (δ) parameters of biological crys�tals network.

n

{ } jz

{ } 22 23 24

32 33 34

42 43 44

1 0 0 0

0,

0

0

j

z z zz

z z z

z z z

=

( )

( )

2 222

23,32

2 233

34,43

24,42

44

cos 2 sin 2 cos ,

cos 2 sin 2 1 cos ,

sin 2 cos 2 cos , .cos 2 sin ,

sin 2 sin ,

cos .

ik j

j

z

z

zzz

z

z

⎛⎧ ⎞= ρ + ρ δ⎜ ⎟⎪ = ρ ρ − δ⎜ ⎟⎪⎜ ⎟⎪⎪ = ρ + ρ δ= ⎜ ⎟⎨

= ± ρ δ⎜ ⎟⎪⎜ ⎟⎪ = ± ρ δ⎜ ⎟⎪⎜ ⎟⎪ = δ⎝⎩ ⎠

ρ

2 ndδ = π λΔ

.nΔ

( )ik nZ

( ) ( )1 1

,N N

ik ikn u

u u

Z z= =

=∑∑N → ∞

( ) ( )

2

0 0

, .ik iknZ Q W z d d

π π

ρ δ= ρ δ ρ δ∫ ∫

Qρ Wδ

{z}1 {z}2 {z}3 {z}n

Fig. 1. On the analysis of modeling polarization properties of anisotropic component of optically thick biological tissue.


STATISTICAL STRUCTURE OF SKIN DERMA MUELLER MATRIX IMAGES 147

Mueller matrix of optically thick BT is determined by multiplication of partial matrix operators

(5)

To make it simpler (without decreasing the analysis depth) further we shall consider only a 2�layered BT

(6)

In the expanded form the matrix operator elements (6) are written as follows:

(7)

The analysis of relations (7) shows that optical properties of anisotropic component of 2�layered BT( ) are described by complex superposition of orientation and phase parameters of its partial layers' bio�logical crystals ( , ). The concrete form of such dependences for “phase” matrix elements [2, 18] are illustrated by the following relations

(8)

The analysis of relations (8) shows that a strict solution of the inverse problem —

revealing the changes in the structure of biological crystals network of one of the layers on the basis of dataabout matrix elements and , is both mathematically incorrect and physically ambiguous.

Thus, it is important to elaborate approximated statistical methods of experimental solution of suchdiagnostic task.

3. SCHEME OF EXPERIMENTAL CHANGES OF COORDINATE DISTRIBUTIONS OF BIOLOGICAL TISSUE MUELLER MATRIX ELEMENTS

Conventional optical scheme of polarimeter for measuring 2D distributions of the BT Mueller matrixelements is presented in Fig. 2 [2].

Histological sections of BT were illuminated by a parallel beam of He–Ne laser (λ = 0.6328 μm, W =5.0 μW) with the radius r = 1 mm. Polarization illuminator consists of quarter�wave plates 3, 5 and polar�izer 4, providing the formation of laser beam with random azimuth 0° ≤ α0 ≤ 180° or ellipticity 0° ≤ β0 ≤ 90°of polarization. Polarization images of BT by means of microobjective 7 were projected into the plane ofsensitized plate (m × n = 800 × 600 pixels) if CCD�camera 10. The analysis of BT images was carried outby means of polarizer 9 and quarter�wave plate 8. As a result, the Stokes vector parameters for every pixel

of the BT image were determined and the set of elements of Mueller matrix was calculatedaccording to algorithm [16, 18, 21]:

(9)

Indices 1–4 correspond to the following polarization states of the beam probing the BT layer: 1—0°;2—90°; 3—+45°; 4—⊗ (right�hand circulation).

{ } { } { } { } { }1 2 1... ,

n nZ Z Z Z Z

−

=

{ } { } { } { }{ }2 1.Z Z Z Y X= ≡

( )

22 22 22 23 32 24 42

23 22 23 23 33 24 43

32 32 22 33 32 34 42

33 32 23 33 33 34 43

34 32 24 33 34 34 44

43 42 23 43 33 44 43

24 22 24 23 34 24 44

42 42 22

,

,

,

,

, , , ,

,

,

ik x y x y

Z y x y x y x

Z y x y x y x

Z y x y x y x

Z y x y x y x

Z Z y x y x y x

Z y x y x y x

Z y x y x y x

Z y x

= + += + += + += + +

ρ ρ δ δ = = + += + += + += 43 32 44 42

44 42 24 43 34 44 44

,

.

y x y x

Z y x y x y x

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪ + +⎪

= + +⎩

ikZ( ),ik x xx ρ δ ( ),ik y yy ρ δ 42,43,44z

( ) ( )[ ]

( ) ( )[ ]24

34

44

sin cos 2 sin 2 cos sin 2 cos 2 sin cos sin 2 ;

sin sin 2 sin 2 cos cos 2 cos 2 cos sin cos 2 ;

cos(2 2 ) sin sin cos cos .

y x x y x x x y x y x

x y x y y y x y x y y

x y x y x y

z

z

z

= δ ρ ρ − ρ + δ ρ ρ + ρ + δ δ ρ⎧⎪

= δ ρ ρ − ρ + δ ρ ρ + ρ + δ δ ρ⎨⎪ = ρ − ρ δ δ + δ δ⎩

( )

( )

, , ;

, ,

x ik y y

x ik y y

q z

g z

ρ = ρ δ⎧ ⎫⎨ ⎬δ = ρ δ⎩ ⎭

ikz iky

{ }=1,2,3,4iS

( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 3 41 2 3 1 4 10.5 , 0.5 , , , 1,2,3,4.i i i l i i l ii i i iZ S S Z S S Z S Z Z S Z i⎡ ⎤ ⎡ ⎤= + = − = − = − =

⎣ ⎦ ⎣ ⎦

148


USHENKO et al.

Fig. 2. Optical scheme of polarimeter, where 1—He–Ne laser; 2—collimator; 3—stationary quarter�wave plate; 5, 8—mechanically movable quarter�wave plates; 4, 9—polarizer and analyzer correspondingly; 6—object of investigation;7—microobjective; 10—CCD camera; 11—PC.

The technique of measuring the ensemble ( —the number of probing areas) of 2D matrix elements

of BT consisted in the following sequence of actions:

—the plane of BT histological sections ( ) was scanned line by line by the laser beamwith the radius with the step of linear displacement ;

—within each of 50 areas ( ) of illumination of the BT layer plane according to the algo�

rithm (9) the local array ( )of values of Mueller matrix elements was deter�mined

(10)

—for every local array statistic moments of the 3rd–4th orders were calculated;

—the histograms of the values of statistical moments of higher orders within the wholeplane of the investigated BT histological sections were determined.

4. INVESTIGATION OF THE SKEWNESS AND THE KURTOSIS DISTRIBUTIONS OF 2D PHASE ELEMENT OF MUELLER MATRIX OF THE SAMPLES

OF SKELETAL MUSCLE TISSUE

Distributions of the values of the skewness and the kurtosis ) of 2D phase

( ) element of Mueller matrix of skin derma (SD) tissue were investigated. Histological sections of var�ious optical thickness ( and ) and physiological state were the object of investigation.

The choice of matrix element as an analytical parameter is explained by the fact that it is the mostsensitive to the changes of birefringence of protein fibrils nets, connected with their pathological changes[16, 18].

The choice of a SD tissue of a rat as the object of investigation is connected with the possibility of directexperimental formation of oncological changes and monitoring the control of their optical manifestationsunder the condition of scattering of laser beam of various multiplication factor.

The series of Fig. 3 and Fig. 4 presents the 2D�distributions of the element of optically thinhistological sections of healthy (Fig. 3) and oncological changed (Fig. 4) SD tissue.

It can be seen from the data obtained that:

—2D distributions of matrix elements of both types of tissues are coordinately heteroge�neous;

—for the sample of healthy muscle tissue (Fig. 3) the lesser range of values change (from –0.7 to 0.7)of the matrix element is typical in comparison with the changes (from –0.85 to 0.85) of the anal�ogous element of Mueller matrix of the oncological changed tissue (Fig. 4).

The structure of 2D distributions of elements can be connected with the following peculiar�ities of birefringent architectonics of the SD tissue samples. Firstly, while a laser wave (with the wave�

r( )( )

rikZ m n×

mm mm10 20≈ ×

mm1r = mm2rΔ =

1, 2, ..., 50r =

800 600m n× = ×( )( )

rikZ m n×

( )

11 1

( )

11,2,...50.

...

... ... ...

...

mik ik

rik

n nmik ik j

Z Z

Z m n

Z Z=

⎛ ⎞⎜ ⎟

× = ⎜ ⎟⎜ ⎟⎝ ⎠

( )( )

rikZ m n×

( )= 3; 4sM

( )( )=3;4=1,2,...50s

jN M

( )( )=3;4sN M ( )=3sM ( )= 4sM

44Z

1 0.08τ = 2 1.47τ =

44Z

( )44 ,Z m n

( )44 ,Z m n

( )44 ,Z m n

( )44 ,Z m n

1 2 3 4 5 6 8 9 10 11

P A

7



length ) propagates through the network of optically anisotropic (birefringence index [13, 18]) collagen fibrils a wide range of values of phase shifts , proportional to their geometrical sizes

is formed

(11)

Thus, phase elements distributions are coordinately heterogeneous and dependent on thepeculiarities of morphological structure of extracellular matrix of SD tissue samples.

Secondly, the cancer process is accompanied by the increase of birefringence due to formation ofmyosin fibrils edema [19, 21]. Optically, it causes the increase of the range of phase shifts changes andthe growth of fluctuations of the element connected with it (Fig. 4).

The histograms of distribution of statistic moments of higher orders of coordinate

distributions of investigated samples of optically thin histological sections is presented inFig. 5.

λ31.5 10n −

Δ = ×

kδ

kd

( )1

~ 2 : .k

k

d

n

d

⎧⎪

δ πΔ λ × ⎨⎪⎩

( )44 ,Z m n

nΔ

kδ

( )44 ,Z m n( )( )=3;4sN M ( )= 3; 4sM

( )( )

=1 5044 ,jZ m n−

Z440.8

0.4

0

–0.4

–0.8600

400

2000

5001000

x, µm

Z44(xy)

200 µm

0.750–0.75

Fig. 3. Coordinate and 3D distribution of the values of phase matrix element of histological section of optically

thin ( ) layer of the healthy SD tissue.

( )44 ,Z m n

1 0.08τ =

Z440.90

0.45

0

–0.45

–0.90600

400

2000

5001000

x, µm

Z44(xy)

200 µm

0.850–0.85

y, µm


thin ( ) layer of the oncological changed SD tissue.

( )44 ,Z m n

1 0.08τ =

y, µm

150


USHENKO et al.

The analysis of experimentally measured histograms proved that:

—the change ranges of the values of the skewness and the kurtosis of coordinate distri�

butions , measured for the healthy (a, c) and oncological changed (b, d) SD tissues do notactually coincide;

—extreme values of the skewness in the distribution for phase element of the healthy

tissue is by 4–6 times higher than analogous values of the given statistic moment for the oncologicalchanged SD tissue;

—extreme values of the kurtosis in the distribution for the oncological changed tissue is by

2–3 times higher than the values of the statistic moment for the healthy SD tissue;

Thus, it can be stated that for single differentiation of physiological state of optically thin layers of SD

tissue it is enough to measure the 2D phase matrix element in one domain ( ) of irradiationby the laser beam and to calculate the skewness and the kurtosis of its values distribution.

The series of Figs. 6 and 7 the results of experimental measurements of 2D phase elements ofMueller matrix of optically thick ( ) of the SD tissue layers.

It can be seen from the data obtained that:

—for the samples of the healthy and oncological changed SD tissue the same range of values change(from –0.55 to 0.55) is typical in the distribution of matrix elements ;

( )( )=3;4sN M =

( )=3sM ( )= 4sM( )

( )=1 50

44 ,jZ m n−

( )( )=3sN M 44Z( )=3sM

( )( )= 4sN M( )= 4sM

( )44 ,Z m n2rπ

( )44 ,Z m n

2 1.47τ =

( )44 ,Z m n

(a) (b)

8

10

12

6

4

2

0 906040

N

M (s = 3)

8020

8

10

12

6

4

2

0 906040

N

M(s = 3)

8020

8

10

14

6

4

2

0 1406040

N

M (s = 4)

8020

12

120100

8

10

6

4

2

0 1406040

N

M (s = 4)

8020

12

120100

(c) (d)

Fig. 5. Histograms of the set of the skewness values (a), (b) and the kurtosis (c), (d) of coordinate distri�

bution of the element of histological section of optically thin ( ) layer of the healthy (a), (c) andoncological changed (b), (d) SD tissue.

( )= 3sM

( )= 4sM

( )( )

=1 5044 ,

jZ m n

−

1 0.08τ =



—comparative visual analysis of coordinate distributions of matrix elements of histologicalsections of oncological changed SD tissue of both types didn’t show any sufficient difference betweenthem.

Similarity of the distribution structure of both types of SD tissue samples can be explained bymultiple light�scattering. As a result of every local ( �th) act of interaction between laser radiation and sep�

arate fibril random value of phase shift is formed, which is multiplied in the process of prop�

agation in the BT depth reaching equiprobable random values 0 to ,

Figure 8 shows histograms of the set of values of statistic moments of the skewness

and the kurtosis of distributions of the matrix element of optically thick samples ofSD tissue.

( )44 ,Z m n

( )44 ,Z m ni

( )iδ

=1

*R

i

i

⎛ ⎞⎜ ⎟δ = δ⎜ ⎟⎝ ⎠

∑2kπ 1,2,3...k =

( )( )=3;4sN M ( )=3sM

( )= 4sM ( )( )

=1 5044 ,jZ m n−

Z440.6

0.3

0

–0.3

–0.6600

400

2000

5001000

x, µm

Z44(xy)

200 µm

0.50–0.5

y, µm0.25–0.25


thick ( ) layer of the healthy SD tissue.

( )44 ,Z m n

1 1.47τ =

Z440.6

0.3

0

–0.3

–0.6600

400

2000

5001000

x, µm

Z44(xy)

200 µm

0.50–0.5

y, µm0.25–0.25


thick ( ) layer of the oncological changed SD tissue.

( )44 ,Z m n

1 1.47τ =

152


USHENKO et al.

The analysis of experimental data showed that:

—the skewness values distribution of 2D phase changes within the whole planeof samples of the layer of the healthy (a) and inflamed (b) SD tissue practically coincide and cannot beused as an objective criterion for their optical properties differentiation;

—histograms of the values of the statistic moment of elements dis�tribution possess individual structure that depends on physiological state of SD tissue;

—extreme values of the kurtosis in the histogram of the healthy tissue (c) are by4–5 times less than analogous values of the given statistic moment of the oncological changed SD tissue (d).

Thus, at early stages of changing the physiological state of optically thick BT layers the measurement

of the kurtosis distributions of phase elements ensemble becomes sensitive forMueller�matrix differentiation.

CONCLUSIONS

The interrelations between the statistic moments of the first to the fourth orders of Mueller�matrix ele�

ments and the optical�geometric structure of optically thick biological tissues have beenfound.

It has been shown that the kurtosis of statistic distributions of phase matrix elements isthe most sensitive to changing of physiological structure of birefringent optically thick biological tissues.

Data obtained can be useful for creation of systems for early diagnostics of degenerative�dystrophicand oncological changes of biological tissues.

( )=3sM ( )( )

=1 5044 , ,jZ m n−

( )( )= 4sN M ( =4)sM ( )( )

=1 5044 , ,jZ m n−

( =4)sM ( )( )= 4sN M

( )( )= 4sN M ( )( )

=1 5044 , ,jZ m n−

( )( )

=1 5044 ,jZ m n−

( )( )

=1 5044 ,jZ m n−

(a) (b)

810

1412

642

0 15010050

N

M (s = 4)

81012

642

0 503020

N

M (s = 3)

4010

(c) (d)

810

1412

642

0 15010050

N

M (s = 4)

81012

642

0 503020

N

M (s = 3)

4010

Fig. 8. Histograms of the set of the skewness values (a), (b) and the kurtosis (c), (d) of coordinate distri�

bution of the element of histological section of optically thick ( ) layer of the healthy (a), (c) andoncological changed (b), (d) SD tissue.

( )3sM = ( )= 4sM

( )( )

=1 5044 ,

jZ m n

−

2 1.17τ =



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statistical structure of skin derma mueller matrix images in the process of cancer changes

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