Polarization-correlation mapping of biological tissue coherent images

Oleg V. Angelsky; Alexander G. Ushenko; Yuriy A. Ushenko; Yevheniya G. Ushenko; Yuriy Y. Tomka; Vasyl P. Pishak

doi:10.1117/1.2148251

1 November 2005 Polarization-correlation mapping of biological tissue coherent images

Oleg V. Angelsky, Alexander G. Ushenko, Yuriy A. Ushenko, Yevheniya G. Ushenko, Yuriy Y. Tomka, Vasyl P. Pishak

Author Affiliations +

Journal of Biomedical Optics, Vol. 10, Issue 6, 064025 (November 2005). https://doi.org/10.1117/1.2148251

Abstract

We investigate the statistical polarization parameters of biological tissue histological section images with different morphological structure. First we outline the results of polarization coordinate mapping and analysis of the statistics of the first to fourth orders of biological tissue image polarization azimuth and ellipticities. Second, we study the statistics of the first to fourth orders of coordinate distributions of the complex degree of mutual polarization (CDMP) of biological tissue images. Finally, we consider the diagnostic possibilities of investigating 2-D distributions of CDMP of images that correspond to physiologically normal and degeneratively and/or dystrophycally changed biological tissues that are being analyzed.

1. Introduction

Historically optical methods for researching biological tissue (BT) structure can be divided into three main groups:

1. Spectrophotometric methods^{1, 2, 3} based on the analysis of spatial $(r)$ [temporal $(τ)$ ] changes of field intensity of radiation scattered by BT
2. Polarization methods using the coherency matrix of complex amplitude^{4, 5, 6} ${K (r, τ)}$ :
Eq. 1
$K (r, τ) = ∥ \begin{matrix} ⟨ E_{x} (r, τ) E_{x}^{*} (r, τ) ⟩ & ⟨ E_{x} (r, τ) E_{y}^{*} (r, τ) ⟩ \\ ⟨ E_{x}^{*} (r, τ) E_{y} (r, τ) ⟩ & ⟨ E_{y} (r, τ) E_{y}^{*} (r, τ) ⟩ \end{matrix} ∥,$
and based on the analysis of polarization degree $P (r)$ as correlation function between complex orthogonal components of electromagnetic oscillations $E_{x}$ and $E_{y}$ in that of point $(r)$ of the scattered radiation field:⁷
Eq. 2
$P (r) = {1 - \frac{4 ∣ [⟨ E_{x} (r, τ) E_{x}^{*} (r, τ) ⟩ ⟨ E_{y} (r, τ) E_{y}^{*} (r, τ) ⟩ - ⟨ E_{x} (r, τ) E_{y}^{*} (r, τ) ⟩ ⟨ E_{y} (r, τ) E_{x}^{*} (r, τ) ⟩] ∣}{{[⟨ E_{x} (r, τ) E_{x}^{*} (r, τ) ⟩ + ⟨ E_{y} (r, τ) E_{y}^{*} (r, τ) ⟩]}^{2}}} .$
3. Correlation methods based on the analysis of correlation degree $J$ between parallel polarization components $E_{x} (r_{1})$ , $E_{x} (r_{2})$ of light oscillations in different points of object field^{7, 8} $(r_{1}, r_{2})$ :

Eq. 3

J = \frac{⟨ E_{x} (r_{1}, τ) E_{x}^{*} (r_{2}, τ) ⟩ - ⟨ E_{x}^{*} (r_{1}, τ) E_{x} (r_{2}, τ) ⟩}{⟨ E_{x} (r_{1}, τ) E_{x}^{*} (r_{2}, τ) ⟩ + ⟨ E_{x}^{*} (r_{1}, τ) E_{x} (r_{2}, τ) ⟩} .

Real-object BT fields, including their images, have a change of both polarization and correlation characteristics.⁹ In the case of single scattering, there is an unequivocal interconnection between the morphological structure of BT histological section and polarization characteristics of an image. To analyze these fields a new methodological approach was proposed,^{10, 11, 12} which is based on the generalization of coherence matrix ${K (r, τ)}$ by two-point $(r_{1}, r_{2})$ polarization coherence matrix ${Φ (r_{1}, r_{2}, τ)}$ :

Eq. 4

K (r, τ) \Rightarrow Φ (r_{1}, r_{2}, τ),

Eq. 5

∥ \begin{matrix} ⟨ E_{x} (r, τ) E_{x}^{*} (r, τ) ⟩ & ⟨ E_{x} (r, τ) E_{y}^{*} (r, τ) ⟩ \\ ⟨ E_{x}^{*} (r, τ) E_{y} (r, τ) ⟩ & ⟨ E_{y} (r, τ) E_{y}^{*} (r, τ) ⟩ \end{matrix} ∥ \Rightarrow ∥ \begin{matrix} ⟨ E_{x} (r_{1}, τ) E_{x}^{*} (r_{2}, τ) ⟩ & ⟨ E_{x} (r_{1}, τ) E_{y}^{*} (r_{2}, τ) ⟩ \\ ⟨ E_{x}^{*} (r_{1}, τ) E_{y} (r_{2}, τ) ⟩ & ⟨ E_{y} (r_{1}, τ) E_{y}^{*} (r_{2}, τ) ⟩ \end{matrix} ∥ .

According to Eq. 5 the main parameter determining correlation between differently polarized points $(r_{1}, r_{2})$ of object field with intensities $I (r_{1}, τ)$ and $I (r_{2}, τ)$ is consideredto be complex degree of mutual polarization^{13, 14} (CDMP) $V^{2} (r_{1}, r_{2}, τ)$ . The latter is written in the following form

Eq. 6

V^{2} (r_{1}, r_{2}, τ) = 4 \frac{v_{1}^{2} + v_{2}^{2} + v_{3}^{2}}{I (r_{1}, τ) I (r_{2}, τ)},

where coefficients

v_{i}

are defined by relations

7.

v_{1} = \frac{⟨ E_{x} (r_{1}, τ) E_{x}^{*} (r_{2}, τ) ⟩ - ⟨ E_{y} (r_{1}, τ) E_{y}^{*} (r_{2}, τ) ⟩}{2},

v_{2} = \frac{⟨ E_{x} (r_{1}, τ) E_{y}^{*} (r_{2}, τ) ⟩ - ⟨ E_{y} (r_{1}, τ) E_{x}^{*} (r_{2}, τ) ⟩}{2},

v_{3} = i \frac{⟨ E_{x} (r_{1}, τ) E_{y}^{*} (r_{2}, τ) ⟩ - ⟨ E_{y} (r_{1}, τ) E_{x}^{*} (r_{2}, τ) ⟩}{2},

and have a meaning of value differences of interference pattern visibility from points

r_{1}

and

r_{2}

, which are measured through differently oriented polarization filter (

v_{1}

are axis rotation angles of the analyzer 0,

90 \deg

;

v_{2}

are 45,

135 \deg

; and

v_{3}

are right or left circulation).

Let us consider two differently polarized laser oscillations in points $r_{1}, r_{2}$ of a BT image:

Eq. 8

{\begin{cases} E_{x} (r_{1}) + E_{y} (r_{1}) \exp [- i δ_{1} (r_{1})], \\ E_{x} (r_{2}) + E_{y} (r_{2}) \exp [- i δ_{2} (r_{2})] . \end{cases}

Here

δ_{1} (r_{1})

and

δ_{2} (r_{2})

are phase shifts between orthogonal components of complex amplitudes

E_{x}

and

E_{y}

at points

r_{1}

and

r_{2}

.

For the situation of completely correlated laser BT image a generalized coherence matrix, Eq. 1 takes the following form:

Eq. 9

Φ (r_{1}, r_{2}) = ∥ \begin{matrix} E_{x} (r_{1}) E_{x}^{*} (r_{2}) & E_{x} (r_{1}) E_{y}^{*} (r_{2}) \\ E_{x}^{*} (r_{1}) E_{y} (r_{2}) & E_{y} (r_{1}) E_{y}^{*} (r_{2}) \end{matrix} ∥ .

Taking into account Eq. 8, the operator of Eq. 9 can be rewritten in the following way:

Eq. 10

Φ (r_{1}, r_{2}) = ∥ \begin{matrix} E_{x} (r_{1}) E_{x} (r_{2}) & E_{x} (r_{1}) E_{y} (r_{2}) \exp [i δ (r_{2})] \\ E_{y} (r_{1}) E_{x} (r_{2}) \exp [i δ (r_{1})] & E_{y} (r_{1}) E_{y} (r_{2}) \end{matrix} ∥ .

Based on Eq. 10 and taking into account Eqs. 5, 6, 7, one can show that CDMP for two image points considered is described by the relation

Eq. 11

V^{2} (r_{1}, r_{2}) = \frac{{[E_{x} (r_{1}) E_{x} (r_{2}) - E_{y} (r_{1}) E_{y} (r_{2})]}^{2} + 4 E_{x} (r_{1}) E_{x} (r_{2}) E_{y} (r_{1}) E_{y} (r_{2}) \exp {i [δ_{2} (r_{2}) - δ_{1} (r_{1})]}}{[E_{x}^{2} (r_{1}) + E_{y}^{2} (r_{1})] [E_{x}^{2} (r_{2}) + E_{y}^{2} (r_{2})]} .

Currently only several publications^{15, 16, 17, 18} dedicated to problems of CDMP direct measurement of statistical fields of scattered laser radiation are known. This approach is urgent in biomedical optics concerned with processing of coherent polarization-inhomogeneous biotissue images.¹⁹

This paper deals with the elaboration of polarization measuring of 2-D distributions of CDMP of BT images for searching their interrelationship with orientation morphological structure of birefringent architectonic nets during early, preclinical diagnostics of their physiological state.

2. Theoretical Modeling

2.1.

Polarization Maps (PMs) of Biotissue Images

The analysis of the formation of a BT image PM is based on an approach that enables the morphological presentation of these bio-objects by using two-component amorphous $A$ and crystalline $C$ matrices, and their optical properties can be described by the following Jones operators:²⁰

Eq. 12

A = ∥ \begin{matrix} \exp (- χ l) & 0 \\ 0 & \exp (- χ l) \end{matrix} ∥,

Eq. 13

C = ∥ \begin{matrix} \cos^{2} ρ + \sin^{2} ρ \exp (- i δ) & \cos ρ \sin ρ [1 - \exp (- i δ)] \\ \cos ρ \sin ρ [1 - \exp (- i δ)] & \sin^{2} ρ + \cos^{2} ρ \exp (- i δ) \end{matrix} ∥,

where

χ

is the coefficient of laser radiation absorption by the BT amorphous layer with geometrical thickness

l

; and

ρ

is package direction of birefringent fibrils in the plane of BT sample, the substance of which (fibrils) introduces phase-shift

δ

between orthogonal polarization components

E_{0 x}

and

E_{0 y}

of illuminating laser beam.

According to Eqs. 12, 13, the amplitude-phase structure of imaging amorphous and crystalline BT components can be described analytically by the following matrix equations

Eq. 14

[\begin{matrix} E_{x}^{A} (r_{A}) \\ E_{y}^{A} (r_{A}) \end{matrix}] = A (\begin{matrix} E_{0 x} \\ E_{0 y} \end{matrix}),

Eq. 15

[\begin{matrix} E_{x}^{C} (r_{C}) \\ E_{y}^{C} (r_{C}) \end{matrix}] = C (\begin{matrix} E_{0 x} \\ E_{0 y} \end{matrix}),

where

[E_{x}^{A} (r_{A}), E_{y}^{A} (r_{A})]

and

[E_{x}^{C} (r_{C}), E_{y}^{C} (r_{C})]

are complex amplitudes of orthogonally polarized laser oscillationsat points

r_{A}

of amorphous and

r_{C}

of the crystalline (architectonic²⁰) constituent parts of BT image.

For simplicity, let us consider the formation of BT image polarization structure under its probing by linearly polarized laser beam with azimuth $α = 0 \deg$ relatively to inclination plane:

Eq. 16

(\begin{matrix} E_{0 x} \\ E_{0 y} \end{matrix}) \Rightarrow (\begin{matrix} E_{0 x} \\ 0 \end{matrix}) .

Taking into account Eq. 16 relations 14, 15 are rewritten as follows:

17.

E_{x}^{A} (r) = E_{0 x} \exp (- χ l)

E_{y}^{A} (r) = 0,

18.

E_{x}^{C} (r) = E_{0 x} {\cos^{2} ρ (r) + \sin^{2} ρ (r) \exp [- i δ (r)]},

E_{y}^{C} (r) = E_{0 x} (\cos ρ (r) \sin ρ (r) {1 - \exp [- i δ (r)]}) .

To determine the local states of light oscillation polarization at the points of a BT image let us write their corresponding coherence matrices of amorphous $K^{A} (r)$ and crystalline $K^{C} (r)$ constituent parts:

Eq. 19

K^{A} (r) = E_{0 x}^{A} ∥ \begin{matrix} \exp (- χ l) & 0 \\ 0 & 0 \end{matrix} ∥,

Eq. 20

K^{C} (r) = ∥ \begin{matrix} E_{x}^{C} (r) E_{x}^{* C} (r) & E_{x}^{C} (r) E_{y}^{* C} (r) \\ E_{x}^{* C} (r) E_{y}^{C} (r) & E_{y}^{C} (r) E_{y}^{* C} (r) \end{matrix} ∥ .

According to Eqs. 19, 20, the coordinate distribution of azimuth $α (r)$ and ellipticity $β (r)$ polarization states of a BT image are represented by two types of PM—polarization homogeneous,

Eq. 21

{\begin{matrix} α^{A} (r) = 0 \deg \\ β^{A} (r) = 0 \deg, \end{matrix}

and polarization inhomogeneous ones,

Eq. 22

{\begin{cases} α^{C} (r) = 0.5 \arctan [\frac{E_{x}^{C} (r) E_{y}^{* C} (r) - E_{x}^{* C} (r) E_{y}^{C} (r)}{E_{x}^{C} (r) E_{x}^{* C} (r) - E_{y}^{C} (r) E_{y}^{* C} (r)}], \\ β^{C} (r) = 0.5 \arcsin {\frac{i [E_{x}^{C} (r) E_{y}^{* C} (r) - E_{x}^{* C} (r) E_{y}^{C} (r)]}{{(q_{1} + q_{2} + q_{3})}^{1 ∕ 2}}}, \end{cases}

where

23.

q_{1} = {[E_{x}^{C} (r) E_{x}^{* C} (r) - E_{y}^{C} (r) E_{y}^{* C} (r)]}^{2},

q_{2} = {[E_{x}^{C} (r) E_{x}^{* C} (r) - E_{y}^{* C} (r) E_{y}^{C} (r)]}^{2},

q_{3} = i {[E_{x}^{C} (r) E_{y}^{* C} (r) - E_{x}^{* C} (r) E_{y}^{C} (r)]}^{2} .

Here

r \equiv (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}})

is the totality of BT image coordinates defined by the pixel amount of registering area CCD camera.

2.2.

Complex Degree of Mutual Polarization of BT Images

The CDMP relation [Eq. 11] analysis shows that the values of local phase shifts $δ_{1} (r_{1})$ and $δ_{2} (r_{2})$ are concerned with parameters $α (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}})$ and $β (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}})$ of polarization map of BT image by the following relations:

24.

δ (r_{1}) = \arctan [\frac{\tan 2 β (r_{1})}{\tan α (r_{1})}],

δ (r_{2}) = \arctan [\frac{\tan 2 β (r_{2})}{\tan α (r_{2})}] .

The analysis of Eqs. 8, 9, 10 and Eq. 24 shows that the range of the CDMP $V^{2} (r_{1}, r_{2})$ change turns out to be wide enough for different polarization states of light oscillations $E (r_{1})$ and $E (r_{2})$ at points $(r_{1}, r_{2})$ , and it is situated within 0.0 (orthogonal states) and 1.0 (collinear states).

The significant cases of the correlation interconnections between the different polarization states (collinear and orthogonal, linear with different azimuths, linear and circular, etc.) at the points $(r_{1}, r_{2})$ are shown in the Table 1 .

Table 1

Complex degree of mutual polarization of different types light oscillations.

E(r1)	E(r2)	V2(r1,r2)
$E_{x} (r_{1})$ ; $E_{x} (r_{1}) + E_{y} (r_{1}) \exp (i δ)$	$E_{x} (r_{2})$ ; $E_{x} (r_{2}) + E_{y} (r_{2}) \exp (i δ)$	1
$E_{x} (r_{1})$	$E_{x} (r_{2}) + E_{y} (r_{2})$	$\frac{E_{x}^{2} (r_{2})}{E_{x}^{2} (r_{2}) + E_{y}^{2} (r_{2})}$
$E_{x} (r_{1})$	$E_{x} (r_{2}) + i E_{y} (r_{2})$	0.5
$E_{x} (r_{1})$ ; $E_{x} (r_{1}) + E_{y} (r_{1}) \exp (i δ)$	$E_{y} (r_{2})$ ; $E_{x} (r_{2}) - E_{y} (r_{2}) \exp (i δ)$	0

Thus, having the information on coordinate distribution of parameters $α (r)$ , $β (r)$ , $E_{x} (r)$ , and $E_{y} (r)$ one can define 2-D distributions of CDMP of BT images with a random discretization interval $Δ r_{i}$ :

25.

Δ r_{1} = r_{n + 1} - r_{n},

Δ r_{2} = r_{n + 2} - r_{n},

⋥

Δ r_{k} = r_{n + k} - r_{n} .

3. Experimental Measuring of BT Polarization-Correlation Maps (PCMs)

The optical scheme of measuring BT PCM is presented in Fig. 1 . Illumination was realized by collimated $(\emptyset = 10^{4} μ m)$ He–Ne laser beam ( $λ = 0.6328 μ m$ , $W = 5.0 mW$ ). Polarization illuminator consisted of quarter-wave plates 3, 5 and polarizer 4 forms illuminating beam with the range of azimuth $0 \deg ⩽ α_{0} ⩽ 180 \deg$ and with the range of ellipticity $0 \deg ⩽ β_{0} ⩽ 90 \deg$ of polarization.

Fig. 1

Optical scheme for measuring PCMs of BT images, where 1, He–Ne laser; 2, collimator; 3 and 5, quarter-wave plates; 4 and 8, polarizer and analyzer, respectively; 6, object of investigation; 7, microobjective; 9, CCD camera; and 10, personal computer.

Polarization BT images with the help of microscope objective 7 were projected into the plane of light-sensitive area $(800 \times 600 pixels)$ of CCD camera 9 provided with measuring range of structural BT elements for the following scales $2 to 2000 μ m$ .

The experimental conditions were designed to eliminate spatial-angular aperture filtration under BT image formation. This was provided by correspondence of indicatrix angular characteristics of light scattering by BT samples $(Ω_{BT} \approx 16 \deg)$ and microscope objective angular aperture $(Δ ω = 20 \deg)$ . Here $Ω_{BT}$ is the plane angle of cones where there is concentrated 98% of the total energy of scattered radiation. The analysis of BT images was realized by using polarizer 8.

The methodology of defining BT images polarization-correlation structure consists in the following sequence of actions:

1. By using CCD 9 [without analyzer 8] one can measure an array of intensities of BT image $I (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}})$ .
2. Then by setting up the analyzer 9, the optical axis of which is sequentially oriented at angles $Θ = 0 \deg$ and $Θ = 90 \deg$ , one can measure arrays of intensities $I^{(0)} (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}})$ and $I^{(90)} (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}})$ .
3. By rotating passage axis $Θ$ of analyzer within $Θ = 0 to 180 - \deg$ one can define arrays of minimum and maximum intensity levels $I_{\min} (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}})$ and $I_{\max} (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}})$ of laser image for each individual pixel $(m, n)$ of CCD camera and their corresponding rotation angles $Θ (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}}) [I (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}}) \equiv I_{\min}]$ .
4. Then calculate PM of BT image by using the following relationship:
26.
$α (\begin{matrix} r_{1}, \dots, r_{m} \\ ⋥ \\ r_{n}, \dots, r_{m} \end{matrix}) = Θ [I (r) \equiv I_{\min}] - \frac{π}{2},$
$β (\begin{matrix} r_{1}, \dots, r_{m} \\ ⋥ \\ r_{n}, \dots, r_{m} \end{matrix}) = \arctan \frac{I {(r)}_{\min}}{I {(r)}_{\max}} .$
5. The coordinate distribution of CDMP of BT images are calculated using the following algorithm:
Eq. 27
$V^{2} (r_{n + k}, r_{n}) = \frac{{{[I^{(0)} (r_{n + k}) I^{(0)} (r_{n})]}^{1 ∕ 2} - {[I^{(90)} (r_{n + k}) I^{(90)} (r_{n})]}^{1 ∕ 2}}^{2}}{I (r_{n + k}) I (r_{n})} + \frac{4 {[I^{(0)} (r_{n + k}) I^{(90)} (r_{n + k}) I^{(0)} (r_{n}) I^{(90)} (r_{n})]}^{1 ∕ 2} \cos [δ_{n + k} (r_{n + k}) - δ_{n} (r_{n})]}{I (r_{n + k}) I (r_{n})} .$

4. Characteristics of Research Objects

As research objects are “optically thin” $(τ ⩽ 0.1)$ histological BT cuts of different morphological structure:

1. Muscular tissue (myocardium)—MT [Figs. 2(a) and 2(b) ],
2. Kidney tissue—KT [Figs. 2(c) and 2(d)],
3. Spleen tissue—ST [Figs. 2(e) and 2(f)].

Fig. 2

Images of different types of BTs: (a) and (b) MT, (c) and (d) KT, and (e) and (f) ST in parallel (left column) and crossed (right column) polarizer (P) and analyzer (A). The arrows depict the orientation of the optical axes of both polarizer (A) and analyzer (B), respectively.

The research objects chosen have the similar optical anisotropic components with birefringence index $Δ n \approx 1.5 \times 10^{- 3}$ visualized in crossed polarizer and analyzer [Figs. 2(b), 2(d), 2(f)].

The morphological architectonic structure of such a BT is different. MT is formed by “quasiordered” bundles of birefringent myosin bundles [Fig. 2(b)]. KT architectonics is made by collagen fibers oriented chaotically [Fig. 2(d)]. ST includes “island” inclusions of anisotropic collagen [Fig. 2(f)].

A similar sample selection provides the possibility to do the analysis of the influence of orientation architectonics peculiarities of physiologically different BT in a PM speckle-image structure.

5. Analysis and Discussion of Experimental Data

The experimental part of research contains the following data:

1. PMs $α (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}})$ and $β (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}})$ of all types of BT images [Figs. 3(a), 3(b), 4(a), 4(b), 5(a), 5(b) ]
2. Probability distributions $W$ of azimuth $α (r)$ and ellipticity $β (r)$ polarization values and their corresponding statistical moments of the first to fourth orders (Figs. 3(c), 3(d), 4(c), 4(d), 5(c), 5(d)); Table 2 ;
3. PCMs $[2 D V^{2} (Δ r_{i})]$ of BT images with different intervals of discretization $Δ r_{i}$ [Figs. 6(a), 7(a), 8(a)].
4. Probability distributions $Q [V^{2} (Δ r_{i})]$ and corresponding statistical moments of first to fourth orders [Figs. 6(b), 7(b), 8(b)] and Table 3 .

Table 3

Statistics of the first to fourth orders of CDMP of BT images.

MT (37 samples)		KT (34 samples)		ST (35 samples)
$M_{V^{2}}$	$0.18 \pm 6 %$	$M_{V^{2}}$	$0.24 \pm 4 %$	$M_{V^{2}}$	$0.03 \pm 7 %$
$σ_{V^{2}}$	$0.11 \pm 8 %$	$σ_{V^{2}}$	$0.18 \pm 6 %$	$σ_{V^{2}}$	$0.02 \pm 9 %$
$A_{V^{2}}$	$67.4 \pm 10 %$	$A_{V^{2}}$	$259.3 \pm 9 %$	$A_{V^{2}}$	$17.4 \pm 12 %$
$E_{V^{2}}$	$324.2 \pm 14 %$	$E_{V^{2}}$	$874.9 \pm 11 %$	$E_{V^{2}}$	$26.9 \pm 15 %$

Fig. 6

(a) Coordinate and (b) statistical structure of CDMP of MT image.

Fig. 7

Polarization-correlation structure of kidney tissue speckle-image.

Fig. 8

Coordinate (a) and statistical (b) structure of CDMP of spleen tissue image.

Table 2

Statistics of the first to fourth orders of polarization states of BT images.

MT (37 samples)				KT (34 samples)				ST (35 samples)
α(ri)		β(ri)		α(ri)		β(ri)		α(ri)		β(ri)
$M_{α}$	$0.38 \pm$ 3%	$M_{β}$	$0.24 \pm$ 4%	$M_{α}$	$0.29 \pm$ 2%	$M_{β}$	$0.18 \pm$ 4%	$M_{α}$	$0.11 \pm$ 3%	$M_{β}$	$0.08 \pm$ 5%
$σ_{α}$	$0.25 \pm$ 4%	$α_{β}$	$0.21 \pm$ 5%	$σ_{α}$	$0.18 \pm$ 4%	$σ_{β}$	$0.16 \pm$ 6%	$σ_{α}$	$0.06 \pm$ 3%	$σ_{β}$	$0.05 \pm$ 7%
$A_{α}$	$93.8 \pm$ 9%	$A_{β}$	$89.7 \pm$ 11%	$A_{α}$	$79.4 \pm$ 7%	$A_{β}$	$68.3 \pm$ 12%	$A_{α}$	$20.4 \pm$ 14%	$A_{β}$	$16.1 \pm$ 17%
$E_{α}$	$981.6 \pm$ 16%	$E_{β}$	832.0 $\pm 19 %$	$E_{α}$	682.7 $\pm 14 %$	$E_{β}$	574.2 $\pm 18 %$	$E_{α}$	$11.4 \pm$ 9%	$E_{β}$	$8.25 \pm$ 10%

Fig. 3

PMs of (a) azimuth and (b) ellipticities of MT images and (c) and (d) their probability structure.

Fig. 4

Coordinate and probability distributions of (a) and (c) azimuths and (b) and (d) of the polarization states of points of kidney tissue image.

Fig. 5

PMs of (a) azimuth and (b) ellipticities of ST image and (c) and (d) their probability structures, respectively.

5.1.

PMs

The coordinate structure of PM of MT images [Figs. 3(a) and 3(b)] is considered to be ensembles of large-scale $(100 to 200 - μ m)$ monopolarized $[{α (r); β (r)} \approx const]$ regions (polarization domains). The areas of similar polarization for KT images [Figs. 4(a) and 4(b)] are more small-scale $(25 to 100 μ m)$ and have a rather chaotic form. PM of ST (Fig. 5(a) and 5(b)) are formed by polarized parts of small sizes $(5 to 20 μ m)$ and located equiprobable enough in the plane of laser image.

Histograms $W (α)$ and $W (β)$ [Figs. 3(c), 3(d), 4(c), 4(d), 5(c), 5(d)] characterize the peculiarities of probability distributions of polarization parameters ${α (r); β (r)}$ of BT images. It is peculiar to MT to have dependency extremes of $W (α)$ and $W (β)$ expressed distinctly [Figs. 3(c) and 3(d)]. They point out to statistical prevalence of a certain polarization state of its architectonic net in laser image. Histograms of PM of KT images [Figs. 4(c) and 4(d)] are formed by a series of local extremes in a wider range of value change of polarization parameters ${α (r); β (r)}$ . Probability polarization structure of ST image [Figs. 5(c) and 5(d)] is made by superposition of two components: polarization-homogeneous component, coinciding with polarization state of illuminating beam ( $α_{0} = 0 \deg$ , $β_{0} = 0 \deg$ ) and polarization-inhomogeneous ones. The first PM component is characterized by histogram extremes $W (α)$ and $W (β)$ expressed distinctly; the second one, by the totality of local extremes of a small value.

Actually dependencies $W (α)$ and $W (β)$ are characterized by the totality of statistical moments of first to fourth orders of azimuths and ellipticities values, and they are calculated using the following relationship:

28.

M_{α, β} = \frac{1}{N} \sum_{i = 1}^{N} ∣ z_{i} ∣,

σ_{α, β} = (\frac{1}{N} \sum_{i = 1}^{N} z_{i}^{2}),

A_{α, β} = \frac{1}{σ_{S}^{3}} \frac{1}{N} \sum_{i = 1}^{N} z_{i}^{3},

E_{α, β} = \frac{1}{σ_{S}^{2}} \frac{1}{N} \sum_{i = 1}^{N} z_{i}^{4},

where

N = m \times n

—the overall number of CCD camera pixels, and

z_{i}

corresponds to

α_{i}

or

β_{i}

values.

The quantitative values of parameters Eq. 28 averaged at statistically reliable amount of all types of BT samples are shown in Table 2.

The researched PM structure of different types of BT can concern the following peculiarities of morphological structure of their architectonics. The structures transforming laser radiation polarization for MT are large-scale domains of “mono-oriented” myosin fibers. According to Eqs. 22, 23 coordinate distribution of azimuth $α (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}})$ and ellipticity $β (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}})$ polarization turns to be homogeneous within such domains as $ρ, δ {(⋥_{r_{g}, \dots, r_{q}}^{r_{l}, \dots, r_{k}})}_{1 ⩽ l, k, g, q ⩽ m, n} \approx const$ [Figs. 3(a) and 3(b)] and corresponding histograms $W (α)$ , $W (β)$ have expressed extremes [Figs. 3(c) and 3(d)].

A wide spectrum of angles between polarization azimuth of illuminating beam $α_{0}$ and local angles $ρ$ in case of KT architectonic net is realized, which is formed by collagen fibers oriented chaotically. Hence, regional areas of identical polarization decrease [Figs. 4(a) and 4(b)], and their corresponding histograms can be represented as sets of a great number of local extremes [Figs. 4(c) and 4(d)].

The ST anisotropic component does not have a distinct fibrous, fibrillar structure. Therefore, the polarization-inhomogeneous component of its speckle-images is formed by values of polarization of azimuths and ellipticities distributed equiprobable enough [Figs. 5(a) and 5(b)], the probabilities of which are one order less than polarization parameters of illuminating laser beam [Figs. 5(c) and 5(d)].

The information on statistical moments of the first to fourth orders of the distribution of $W (α)$ and $W (β)$ of different tissues speckle-images help us to determine the following:

29.

M_{α, β} (MT) ⩾ M_{α, β} (KT) ⩾ M_{α, β} (ST),

σ_{α, β} (MT) ⩾ σ_{α, β} (KT) ⩾ σ_{α, β} (ST),

A_{α, β} (MT) ⩾ A_{α, β} (KT) ⩾ A_{α, β} (ST),

E_{α, β} (MT) ⩾ E_{α, β} (KT) ⩾ E_{α, β} (ST) .

It is clear that the more the probability distributions of optical-geometrical parameters of architectonic nets deviate from the statistical (equiprobable or normal law) character to a stochastic and self-similar one, the greater become values of corresponding statistical moments of polarization parameters of BT images.

5.2.

Polarization-Correlation Maps

Distributions of $2 D [V^{2} (Δ r_{i})]$ of MT images obtained for $Δ r_{i} = 1 pixel$ are mainly formed by large-scale $(25 to 100 μ m)$ regions [Fig. 6(a) ] with maximum correlated polarization states $(V^{2} (Δ r_{i}) \to 1)$ . Statistically, this points out to the basic histogram ${Q [V^{2} (Δ r_{i})]}$ extremum [Fig. 6(b)]. Along with this fact $2 D [V^{2} (Δ r_{i})]$ have regions where the CDMP value changes within a wide range $[0 ⩽ V^{2} (Δ r_{i}) ⩽ 1]$ , which has corresponding additional extremes on the histogram $Q [V^{2} (Δ r_{i})]$ .

Except for the areas of maximum polarization correlation, all-scale $(10 to 30 μ m)$ regions, where CDMP value changes within a wide range $[0 ⩽ V^{2} (Δ r_{i}) ⩽ 1]$ [Fig. 7(a) ], form a distribution of $2 D [V^{2} (Δ r_{i})]$ of the MT image. The corresponding histograms $Q [V^{2} (Δ r_{i})]$ also include the totality of considerably equiprobable extrema for what are actually whole range of value change of CDMP of KT images [Fig. 7(b)].

The correlation structure of polarization states of speckle-images points out to essential prevalence of their polarization homogeneous component. Distributions of $2 D [V^{2} (Δ r_{i})]$ [Fig. 8(a) ] are considered to be the totality of regions with the extreme CDMP value $V^{2} (Δ r_{i}) \to 1$ . The histogram [Fig. 8(b)] by its structure is close to a $δ$ functions of the form

Q [V^{2} (Δ r_{i})] = {\begin{matrix} 1 [V^{2} (Δ r_{i}) = 1] \\ \to 0 [V^{2} (Δ r_{i}) \neq 1] . \end{matrix}

The totality of statistical moments of the first to fourth value orders $Q [V^{2} (Δ r_{i})]$ of images of all types of BT are shown in Table 3.

The researched structure of $2 D [V^{2} (Δ r_{i})]$ BT images closely concerns the different morphological structures of their architectonics. The package order of MT myosin fibrils within considerably large geometrical domains according to Eqs. 11, 22, 23 conditions the high degree of correlation identity $[V^{2} (Δ r_{i}) \to 1]$ of polarization states of corresponding image points. The transition to other domains (parts with a different orientation architectonic structure) is accompanied with CDMP value decrease of the given image parts. However, within the domain the degree of mutual polarization of MT image points increases extremely again [Fig. 6(a)].

There is a more dynamic value change $V^{2} (Δ r_{i})$ for image points of different parts of architectonic net in case of a 2-D distribution of the CDMP of images of a collagen net that is oriented chaotically. Therefore, along with the main maximum statistical distributions $Q [V^{2} (Δ r_{i})]$ are formed by considerably equiprobable extremes [Fig. 7(b)], which are conditioned by a variety of KT architectonics orientation structure and by the decrease of its self-similar geometrical sizes as well [Fig. 7(a)].

The structure of $2 D [V^{2} (Δ r_{i})]$ ST images represents the high correlation interrelation between points’ polarization states of a completely polarization-homogeneous image [Figs. 8(a) and 8(b)].

A comparative analysis of statistical moments’ values of the first to fourth orders of parameter $V^{2} (Δ r_{i})$ shows

30.

M_{V^{2}} (MT) ⩾ M_{V^{2}} (KT) ⩾ M_{V^{2}} (ST),

σ_{V^{2}} (MT) ⩾ σ_{V^{2}} (KT) ⩾ σ_{V^{2}} (ST),

A_{V^{2}} (MT) ⩾ A_{V^{2}} (KT) ⩾ A_{V^{2}} (ST),

E_{V^{2}} (MT) ⩾ E_{V^{2}} (KT) ⩾ E_{V^{2}} (ST) .

A comparative analysis Eq. 30 and Eq. 29 determines the large sensitivity (within 2 to 5 times) to the orientation structure peculiarities of architectonic nets of statistical moments of third and fourth orders of parameter $V^{2} (Δ r_{i})$ in comparison with analogous parameters of polarization maps $α (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}})$ and $β (⋥_{r_{n}, \dots, r_{m}}^{r_{1}, \dots, r_{m}})$ of BT images. This is why the investigation of possibility of using the polarization-correlation analysis of BT images during the diagnostics of their physiological state, which is connected with architectonic nets structure changes, is topical.

6. Pathological Changes Diagnostics Using Complex Degree of Mutual Polarization of BT Images

This part of the paper deals with searching for an interconnection between the coordinate structure CDMP of BT images and the optical-geometrical parameters of their physiologically normal and pathologically changed birefringent architectonics nets.

The main idea consists in the fact that the values of $α (r)$ , $β (r)$ , $E_{x} (r)$ , and $E_{y} (r)$ at every point of the BT image appear to be connected with orientational $ρ (r)$ and phase $δ (r)$ parameters of its architectonics [Eqs. 11, 14, 22]:

31.

α (r) = 0.5 \arctan [\frac{\sin 4 ρ (r) \sin^{2} 0.5 δ (r)}{\cos^{2} 2 ρ (r) + \sin^{2} 2 ρ (r) \cos δ (r)}],

β (r) = 0.5 \arcsin [\frac{\tan 2 ρ (r)}{\sin δ (r)}],

32.

E_{x} (r) = {I_{0} [\sin^{2} α (r) + \cos^{2} α (r) \tan^{2} β (r)]}^{1 ∕ 2},

E_{y} (r) = {I_{0} [\cos^{2} α (r) + \sin^{2} α (r) \tan^{2} β (r)]}^{1 ∕ 2} .

It follows from Eqs. 31, 32 that the CDMP of BT image [Eqs. 23, 27] appears to be a parameter that is sensitive to orientation-phase architectonic changes.

“Optically thin” $(τ ⩽ 0.1)$ frozen histological sections of the BT of the following types were used as the objects of investigation:

1. Physiologically normal [Fig. 9(a) ] and dystrophically changed [Fig. 9(b)] myocardium muscle tissue (MT),
2. Physiologically normal [Fig. 9(c)] tissue of cannon bone and bone tissue (BnT) affected with osteoporosis [Fig. 9(d)].

Fig. 9

Images of architectonics of physiologically (a) normal MT, (c) BnT, and dystrophical changed (b) MT and (d) BnT (with osteoporosis) in crossed polarizer (P) and analyzer (A). The arrows depict the orientation of the optical axes of both polarizer (A) and analyzer (B), respectively.

The technique of freezing of BT thin layers to “nitrogen” temperatures provided practically complete identity with them morphologically in vivo and in vitro. Histological sections of physiologically changed MT and BnT were taken in areas that do not correspond to localization of pathological changes in architectonics (psoriasis, muscular dystrophy). From the medical point of view such samples are “conventionally” normal. Traditional histochemical investigations do not show any differences in their physiological state.

Morphological structure of such BTs architectonics is different. MT is formed by quasiordered myosin fibrils and fibers forming bundles [Fig. 9(b)] with the birefringence index of their substance $Δ n \approx 1.5 \times 10^{- 3}$ . The BnT architectonics is formed by oriented collagen fibers, which are mineralized by the hydroxylapathite crystals²¹ $(Δ n \approx 1.1 \times 10^{- 1})$ [Fig. 9(d)].

Early (preclinical) degenerative-dystrophic changes of such tissues are formed morphologically according to various “scenarios.”²² Myosin fibrils and fibers under the unchanged birefringence level disorder within mono-oriented MT bundles. The osteoporosis-affected MT presents a decrease of hydroxylapathite crystals concentration ( $Δ n \approx 10^{- 2}$ to $10^{- 3}$ ) under an unchanged orientation structure of the collagen net.

Coordinate distributions of the complex degree of mutual polarization of images of physiologically normal and degeneratively and/or dystrophically changed MT and BnT are illustrated in Figs. 10(a), 11(a), 10(b), 11(b) , correspondingly.

Fig. 11

(a) and (b) coordinate and (c) and (d) statistical structure of CDMP of images of healthy (a) and (c) and pathological changed (dystrophically changed) (b) and (d) MT.

Fig. 10

(a) and (b) coordinate and (c) and (d) statistical structure of CDMP of images of healthy (a) and (c) and pathological changed (osteoporosis) (b) and (d) BnT.

Measuring the intensity values, azimuths, and ellipticities of polarization necessary for calculation of $2 D [V^{2} (Δ r_{i})]$ of images of all types of BT was performed for discretization parameter $Δ r_{i} = 1 pixel$ .

Figures 10(c), 10(d), 11(c), 11(d) present histograms of probability distributions of CDMP ${2 D [V^{2} (Δ r_{i})]}$ values.

Distributions $2 D [V^{2} (Δ r_{i})]$ of images of physiologically normal BT of both types are rather close in their structure—mainly formed by the areas [Figs. 10(a) and 11(a)] with maximally correlated state of polarization $V^{2} (Δ r_{i}) \to 1$ . Statistically the principal extreme of corresponding histograms $Q [V^{2} (Δ r_{i})]$ points to this [Figs. 10(c) and 11(c)]. Alongside this, $2 D [V^{2} (Δ r_{i})]$ of such images contain a small number of areas of intermediate values of CDMP $[0 ⩽ V^{2} (Δ r_{i}) ⩽ 1]$ with additional extreme $Q [V^{2} (Δ r_{i})]$ corresponding to them.

Coordinate distributions CDMP of the degeneratively and/or dystrophically changed BT samples images is formed by the areas for which the value of CDMP changes within wide limits $[0 ⩽ V^{2} (Δ r_{i}) ⩽ 1]$ [Figs. 10(b) and 11(b)]. Corresponding histograms $Q [V^{2} (Δ r_{i})]$ contain the totality of equiprobable extrema, enough for the whole range of changing the CDMP value of images of both MTs [Fig. 10(d)] and BTs [Fig. 11(d)].

The dependences $Q [V^{2} (Δ r_{i})]$ objectively characterize the totality of statistic moments of values of CDMP of the first to fourth orders (see Table 4 ).

Table 4

Statistics of the first to fourth orders of CDMP of BT images.

MT (37 samples)			BnT (34 samples)
	Normal	DystrophicallyChanged		Normal	Affected byOsteoporosis
$M_{V^{2}}$	$0.18 \pm 4 %$	$0.96 \pm 11 %$	$M_{V^{2}}$	$0.24 \pm 6 %$	$0.85 \pm 13 %$
$σ_{V^{2}}$	$0.11 \pm 6 %$	$0.67 \pm 17 %$	$σ_{V^{2}}$	$0.18 \pm 8 %$	$0.45 \pm 16 %$
$A_{V^{2}}$	$67.4 \pm 11 %$	$589.7 \pm 19 %$	$A_{V^{2}}$	$87.3 \pm 13 %$	$967.7 \pm 23 %$
$E_{V^{2}}$	$32.2 \pm 15 %$	$451.3 \pm 23 %$	$E_{V^{2}}$	$74.9 \pm 17 %$	$798.3 \pm 25 %$

The investigated structure of $2 D [V^{2} (Δ r_{i})]$ of physiologically normal and degeneratively and/or dystrophically changed BT images is closely connected with various morphological structure of their architectonics.

The packing and well-ordering of myosin and collagen fibrils of physiologically normal MTs and BnTs within rather large geometric domains of their architectonic nets according to Eqs. 18, 19, 23 causes high correlation similarity $[V^{2} (Δ r_{i}) \to 1]$ of polarization states of corresponding points in their images. However, within the domain itself, the degree of mutual polarization of BT images points extremely increases again.

The inverse tendency is observed for 2-D distributions of CDMP images of physiologically changed samples of both types of BTs.

Morphological degenerative-dystrophic changes of MT are accompanied by fibril packing disorder in MT myosin bundles. From an optical point of view, such a process is similar to an increase of the angle interval between polarization azimuth of illuminating laser beam $α_{0}$ and the highest speed axis $ρ (r)$ direction of optically anisotropic structures. According to Eqs. 22, 23 various states of polarization $α (r)$ and $β (r)$ of laser oscillations will be formed in corresponding points of MT image. Thus, the fluctuations of CDMP value increase while corresponding statistic distributions $Q [V^{2} (Δ r_{i})]$ will be transformed into the totality of rather equiprobable extrema [Fig. 10(d)].

Optically, early manifestation of BnT osteoporosis can be represented by coordinate modulation of phase shifts $δ (r)$ , shown by hydroxylapathite crystals between orthogonal polarization components $E_{0 x}$ and $E_{0 y}$ of the illuminating laser radiation, forming a corresponding coherent image. Such modulation at the unchanged orientational structure $ρ (r)$ of collagen fibers is the main reason of forming polarizationally inhomogeneous $α (r)$ and $β (r)$ parts of the BnT image, and value of $V^{2} (Δ r_{i})$ connected with this becomes statistical.

A comparative analysis of statistic moments of first to fourth orders of the $V^{2} (Δ r_{i})$ value of coherent images ofthe groups (see Table 3) of physiologically normal $(M_{V^{2}}^{*}, σ_{V^{2}}^{*}, A_{V^{2}}^{*}, E_{V^{2}}^{*})$ and pathologically changed $(M_{V^{2}}, σ_{V^{2}}, A_{V^{2}}, E_{V^{2}})$ BT of both types demonstrates the following:

Eq. 33

M_{V^{2}} ⩾ M_{V^{2}}^{*}, σ_{V^{2}} ⩾ σ_{V^{2}}^{*}, A_{V^{2}} ⩾ A_{V^{2}}^{*}, E_{V^{2}} ⩾ E_{V^{2}}^{*} .

We can see that if the probability distributions of the following form

Q [V^{2} (Δ r_{i})] = {\begin{matrix} 1 [V^{2} (Δ r_{i}) = 1] \\ \to 0 [V^{2} (Δ r_{1}) \neq 1], \end{matrix}

[Figs. 10(c) and 11(c)] transform into a random

(0 ⩽ Q [V^{2} (Δ r_{i})] ⩽ 1)

. In this case, the corresponding statistical moments of CDMP values of degeneratively and/or dystrophycally BT images become larger. Such differences are the most pronounced (by one order of the value) for the third and fours statistical moments of CDMP.

7. Conclusions

On the basis of polarizational measurements technique of 2-D distributions of CDMP of BT images the interconnection between the statistical moments of CDMP values and the optical-geometrical structure of degeneratively and/or dystrophycally changed the MT and BnT architectonics nets. The obtained information concerning the polarization-correlation structure of images correspond to different BT morphological structures can be used for elaboration of new methods for their analysis of physiological state.

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Citation Download Citation

Oleg V. Angelsky, Alexander G. Ushenko, Yuriy A. Ushenko, Yevheniya G. Ushenko, Yuriy Y. Tomka, and Vasyl P. Pishak "Polarization-correlation mapping of biological tissue coherent images," Journal of Biomedical Optics 10(6), 064025 (1 November 2005). https://doi.org/10.1117/1.2148251

Published: 1 November 2005

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KEYWORDS

Polarization

Tissues

Biological research

Statistical analysis

Phase modulation

Collagen

Polarizers

1.

Introduction

Eq. 1

Eq. 2

Eq. 3

Eq. 4

Eq. 5

Eq. 6

7.

Eq. 8

Eq. 9

Eq. 10

Eq. 11

2.

Theoretical Modeling

2.1.

Polarization Maps (PMs) of Biotissue Images

Eq. 12

Eq. 13

Eq. 14

Eq. 15

Eq. 16

17.

18.

Eq. 19

Eq. 20

Eq. 21

Eq. 22

23.

2.2.

Complex Degree of Mutual Polarization of BT Images

24.

Table 1

25.

3.

Experimental Measuring of BT Polarization-Correlation Maps (PCMs)

Fig. 1

26.

Eq. 27

4.

Characteristics of Research Objects

Fig. 2

5.

Analysis and Discussion of Experimental Data

Table 3

Fig. 6

Fig. 7

Fig. 8

Table 2

Fig. 3

Fig. 4

Fig. 5

5.1.

PMs

28.

29.

5.2.

Polarization-Correlation Maps

30.

6.

Pathological Changes Diagnostics Using Complex Degree of Mutual Polarization of BT Images

31.

32.

Fig. 9

Fig. 11

Fig. 10

Table 4

Eq. 33

7.

Conclusions

References

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