D Mueller-Matrix Diffusive Tomography of Polycrystalline Blood Films for Cancer Diagnosis

The decomposition of the Mueller matrix of blood films has been carried out using differential matrices with polarized and depolarized parts. The use of a coherent reference wave is applied and the algorithm of digital holographic reconstruction of the field of complex amplitudes is used. On this basis, the 3D Mueller-matrix diffuse tomography method—the reconstruction of distributions of fluctuations of linear and circular birefringence of depolarizing polycrystalline films of human blood is analytically justified and experimentally tested. The dynamics of the change in the magnitude of the statistical moments of the first-fourth order, which characterize layer-by-layer distributions of fluctuations in the phase anisotropy of the blood film, is examined and analyzed. The most sensitive parameters for prostate cancer are the statistical moments of the third and fourth orders, which characterize the asymmetry and kurtosis of fluctuations in the linear and circular birefringence of blood films. The excellent accuracy of differentiation obtained polycrystalline films of blood from healthy donors and patients with cancer patients was achieved.


Introduction
Nowadays, Mueller-matrix polarimetry (MMP) [1][2][3][4][5] approaches are extensively used for the visualization of malformation of biological tissue structure and determination of its functional physiological variations.The MMP now includes a number of popular research directions, such as studies of scattering matrices [6][7][8][9][10], polar decomposition of Mueller matrices [11,12], two-dimensional Mueller-matrix mapping [13][14][15][16], and others.The main disadvantage of MMP is associated with invasive procedure of preparation of bio-tissue samples that significantly limits its application for biomedicine.In fact, the easily accessible biological liquids, including those obtained from the particular organs, could be used as the main subjects of tissue samples in MMP.Thus, one of the most promising applications of MMP is screening of the so-called 'integral' fluid of the human body, such as blood and plasma.The thin film of such biological fluids represents a complex spatially inhomogeneous optically anisotropic structure, which is formed by various types of biochemical and molecular crystalline complexes.Monitoring of the dynamics of crystallization within such thin films provides an opportunity to characterize the internal processes at the macro-level of molecular interaction and to carry out the early diagnosis of various diseases.
Here, we examine the polarimetric diagnosis towards the possibility of quantitative characterization of the optical anisotropy of thin polycrystalline films of whole blood.This object of investigation can be considered both as a heterogeneous complex-structured liquid and/or heterogeneous colloidal polymer solution.Traditionally, the molecular components (including various proteins-albumin monomers (~60%), globulin (~40%), and fibrinogen) and the blood cells (erythrocytes, platelets, and leukocytes) are investigated by biochemical methods.At normal conditions, the protein fraction is typically presented in the form of monomers, whereas at the oncological pathology, the tertiary and quaternary structure of proteins are changed with the invariance of biochemical parameters.A few quite effective methods are available and used for studying supramolecular (dimers, trimers, and monomers) protein structures (100-2000 nm) of blood [17,18].In fact, a development of new and more sensitive and informative screening technologies for diagnosis of structural protein changes in tissue samples are required.The polarimetry of polycrystalline films of biological fluids is one of extremely promising technique for this role.
The feasibility studies of polarimetric examinations of optically thin non-depolarizing polycrystalline plasma films have been presented [19][20][21][22][23].In the frame of these studies, a development of algorithms for reconstructing the distributions of phase and amplitude anisotropy parameters of the protein polycrystalline networks of sampling layers have been performed.Later, the potential of this diagnostic approach has been demonstrated [24][25][26][27][28].
In practical use, the MMP of blood and plasma samples is associated with two main limitations.The first one is that most of polycrystalline plasma films are partially depolarized incident light, which leads to a decrease of accuracy of differential diagnosis.The second one is that the biochemical composition of plasma is not so informative, as compared to blood itself.At the same time, due to spatially inhomogeneous structure and the presence of uniform elements, the blood samples provide a higher level of depolarization [29,30].
These factors complicate the direct reconstruction of the optical anisotropy distributions in the samples with subsequent differentiation of its morphological changes.Therefore, to develop a new generalized MMP approach for examination of multi-layered polycrystalline structures, such as diffuse biological tissue samples, are urgently required.This goal can be achieved by the combination (synthesis) of the two following techniques:

•
Isolation of the depolarized component of Mueller matrix of thin blood films by its decomposition on the basis of differential matrices of first order (i.e., the polarized part, which is the distribution of the mean values of the optical anisotropy parameters of polycrystalline structures of proteins and formed elements) and the second one (i.e., the depolarized part, defined by distribution of fluctuations of linear and circular birefringence and dichroism) [31][32][33][34][35][36]; The use of a coherent reference wave and the algorithm for digital holographic reconstruction of the complex amplitudes field in different sections of the blood film [37,38].
This work is aimed to developing and experimental testing of the Mueller-matrix diffuse tomography approach for the layer-by-layer reconstruction of fluctuations of optical anisotropy within the thin tissue samples, with the final target to differentiate polycrystalline blood films of healthy donors and patients with prostate cancer.
The theoretical background of the Mueller matrix approach to describe the interaction of optical radiation with depolarizing layers is presented in detail elsewhere [31][32][33][34].Analytically, this dependence is illustrated by the equation: where M(r) is the Mueller matrix of the object in the plane r and Ψ(r) -differential matrix operator.
For optically thin layers that do not depolarize but transform the polarization of probing light, the matrix operator Ψ(r) consists of six parameters (ξ j=1−6 ) that completely describe phase and amplitude anisotropy of the biological layer: where the parameters are defined as: • ξ 1 = BL 0,90 ; ξ 2 = BL 45,135 are the linear birefringence for orthogonal components 0 The matrix operator Ψ(r) in the expression (1) can be represented as average components ("differential matrix of the 1st order"-the polarized part Ψ(r) ) and fluctuating ("differential matrix of the second order"-the depolarized part Ψ(r) ), as: It should be pointed out here that there is always a valid relation: The simultaneous analysis of relations (1)-( 4), described in References [31][32][33][34], allows obtaining the expression of the logarithmic matrix algorithm: that is determined as a superposition of antisymmetric P(r) and symmetric D(r) components Λ(r) according to: where: Here, K is the metric Minkovsky matrix [33].Devlaminck et al. developed and generalized this theory to the case of a physical medium with weak absorption and fluctuations in the parameters of optical anisotropy [35,36].For such a layer, the value of each of the six elementary polarization properties ξ j=1−6 ≡ (Ψ ik ) j=1−6 (see Equation ( 2)) is considered as the average θ j = Ψ ik j=1÷6 and the fluctuating σ j = Ψ ik j=1÷6 components: Taking into account Equation ( 8), the differential matrix operator Ψ(r) (see Equation ( 3)) it can be determined by the superposition of: On this basis, the following expressions are obtained for differential matrix operators of the first ( Ψ(r) ) and second ( Ψ(r) ) orders Here, The practical use of Equations ( 9) and (10) with the application to depolarizing biological layers can be found using the results of Ghosh et al. [6].Here, it was experimentally shown that the Muller matrix (1) of a depolarizing biological phantom is diagonalized M(r) = diag(M 11 ; M 22 ; M 33 ; M 44 ).Whereas, in the frame of Devlaminck's theoretical approach [35,36], this result indicates that in a multiply dispersing depolarizing medium, fluctuations of various phase (σ i=4−6 ), and amplitude (σ j=1−3 ) anisotropy mechanisms turn out to be uncorrelated ( σ i σ j = 0), and mean values θ i=1−6 = 0.
In such conditions, the differential operator Ψ(r) turns out to be informative and diagnostically relevant, which also transforms into a diagonal matrix diag Ψ 11 ; Ψ 22 ; Ψ 33 ; Ψ 44 .
Therefore, in the current work, the main attention was focused on the analysis of coordinate distributions the diagonal elements Ψ 11;22;33;44 (x, y), which will be called further diffuse tomograms DT Ψ ii .
Thus, on the basis of theoretical conclusions, we propose the following principle for obtaining diffuse tomograms DT Ψ ii .
In the current work, the Equation ( 12) was considering in red region of spectrum, where the absorption of optically anisotropic protein structures of the polycrystalline blood film is about two orders smaller compared to the magnitudes in short-wavelengths of spectrum [29,30].Therefore, we are able to re-write Equation ( 12) in terms of fluctuations of linear and circular birefringence, as: Let us consider the algorithms for determining the depolarization component Ψ(r) .It was shown in References [37,38] that these components are determined (Equations ( 6) and ( 7)) by the matrix operator of the next symmetry:  (14) Here J ik = lnM ik .Thus, the set of diffuse tomograms DT red of the depolarizing biological layer can be experimentally determined using the following algorithm

Optical Scheme and 3D Mueller-Matrix Polarimetry Technique
Figure 1 shows the optical setup for 3D Mueller-matrix polarimetry of biological layers [35][36][37][38].The parallel (O = 2 × 10 3 µm) laser beam (He-Ne laser 1, λ = 0.6328 µm) has been formed using collimator two.Further, a laser beam has been separated on two beams ("irradiating" and "reference") using beam splitter three.The "irradiating" beam has been passed through the polarization filters six-eight using rotary mirror four.Further, this beam illuminated the sample of the biological layer nine.

Optical Scheme and 3D Mueller-Matrix Polarimetry Technique
Figure 1 shows the optical setup for 3D Mueller-matrix polarimetry of biological layers [35][36][37][38].using collimator two.Further, a laser beam has been separated on two beams ("irradiating" and "reference") using beam splitter three.The "irradiating" beam has been passed through the polarization filters six-eight using rotary mirror four.Further, this beam illuminated the sample of the biological layer nine.
The image of the object nine has been obtained using lens 10 (Nikon CFI Achromat P, focal length 30 mm, numerical aperture 0.1, and magnification 4×) and photosensitive pad ( The "reference" beam has been passed through the polarization filters 11-13 using mirror five.Further, this beam illuminated image plane of the object 9. The resulting interference pattern passed through the polarizer-analyzer 14 has been recorded using digital camera 15 (The Imaging Source DMK 41AU02.AS, monochrome 1/2 "CCD, Sony ICX205AL (progressive scan), resolution-1280 × 960, the size of the light-sensitive pad-7600 × 6200 μm, sensitivity-0, 05 lx, dynamic range-8 bit, and SNR-9 bit).
The polarization states of the "irradiating" and "reference" beams have been formed by means of polarization filters six-eight and 11-13.
Each filter consists of: • "input" polarizers six and 11 forming plane-polarized beams with azimuth 0°; The image of the object nine has been obtained using lens 10 (Nikon CFI Achromat P, focal length 30 mm, numerical aperture 0.1, and magnification 4×) and photosensitive pad (m × n = 1280 × 960 pixels) of the digital camera 15.
The "reference" beam has been passed through the polarization filters 11-13 using mirror five.Further, this beam illuminated image plane of the object 9.
The polarization states of the "irradiating" and "reference" beams have been formed by means of polarization filters six-eight and 11-13.

•
The procedure of determination of the layered distributions of diffuse tomograms DT red ϕ j ; x; y includes the following steps: • Formation in the irradiating and reference laser beams the next polarization states-0 • ; 90 • ; 45 • ; ⊗. • Registration of each interference pattern through the polarizer-analyzer 14 with a consistent orientation of the plane of transmission at angles Ω = 0 Application of Fourier transform procedure to interference pattern and using the inverse Fourier transform to obtaining the distributions of complex amplitudes U(x, y) = |U|(x, y)expiϕ j in different phase planes ( ϕ j = 2π λ r; 0 ≤ r ≤ z ) of the object field with an arbitrary step ∆ϕ j=0...q is realized [39,40].

•
In each phase plane ϕ j (x; y), for a series of planar (with azimuths 0 • ; 90 • ; 45 • ) and the right of circularly (⊗) polarized irradiating beams, the distributions of the four sets of parameters of the Stokes vector S i (0 • On the basis of Equation ( 16) layered Muller-matrix images M ik ϕ j , x, y are determined as: Based on the set of distributions ( 17), a series of layer wise distributions of phase anisotropy x; y is determined.

Samples
Blood from 18 healthy donors (group 1) and 18 patients with prostate cancer (group 2) were used in this study.Each sample film has been obtained by applying a drop of biological fluid to a substrate of optically homogeneous glass, followed by drying at room temperature.The optical thickness (τ) of the sample films varied within the range 0.68 ≤ τ ≤ 0.75, the degree of depolarization ∆-41% ≤ ∆ ≤ 52%.

Layered Maps of Fluctuations in the Parameters of Phase Anisotropy of a Partially Depolarizing Polycrystalline Film of Blood
fourth orders (marked by * in Table 1).The range of their variations reaches:   Analysis of the diffuse tomograms DT red ϕ j , x, y of an optically anisotropic blood film, presented at Figure 2, revealed a good correlation between the experimental data (Equations ( 16) and ( 17)) and theoretical data (Equations ( 10)-( 15)):

•
In each phase cross-section, the coordinate distributions DT red ϕ j ; x; y were estimated by calculating the aggregate of statistical moments of the first-fourth order X i=1;2;3;4 [3][4][5].By means of MATLAB software, we calculated the histograms (operator "hist") and statistical moments of the first-fourth order (operator mean, STD, skewness, excess), which characterize the distributions DT red ϕ j ; x; y (DT red ϕ j ; x; y − X 1 ) j 2 ; (DT red ϕ j ; x; y − X 1 ) Here, K is the number of pixels on the CCD-camera.These parameters characterize the mean value (X 1 ), dispersion (X 2 ), skewness (X 3 ), and kurtosis or "peak sharpness" (X 4 ) of the distributions DT red ϕ j ; x; y .
Table 1 presents a series of "phase" dependences of the value X i=1;2;3;4 that characterize the distributions DT red B L 0;90 ; DT red B L 45;135 ; (x; y) of the polycrystalline blood film from healthy donors.Analysis of the data presented in Table 1 revealed the following statistical scenario for the variation of the distribution of fluctuations in the parameters of optical anisotropy: Such changes in magnitude X i=1;2;3;4 can be attributed to the change in the multiplicity of light scattering in the volume of a polycrystalline film of blood.In the region of small ϕ j distributions DT red B L 0;90 ; DT red B L 45;135 ; DT red B C ⊗;⊕ , they are asymmetric (X 3;4 > X 1;2 ) [3][4][5].Due to the increase in multiplicity ( ϕ j ↑ ) according to the central boundary theorem [41], the structure of such distributions tends to be normal ( X 1;2 ↑; X 3;4 → 0 ).
The most sensitive to changes in the polarization manifestations of fluctuations in the phase anisotropy parameters of the polycrystalline blood film were statistical moments of the third and fourth orders (marked by * in Table 1).The range of their variations reaches:

3D Mueller-Matrix Differentiation of Diffuse Polycrystalline Films of Blood
Optical technology for differential diagnosis of depolarizing polycrystalline whole blood films from healthy donors (group 1) and patients with prostate cancer (group 2) includes: 1.

Conclusions
The method of 3D Mueller-matrix diffuse tomography of distributions DT red B L 0;90 ; B L 45;135 ; B C ⊗;⊕ (ϕ 1 = 0.3rad; 2ϕ 1 ; , . . ., 6ϕ 1 ) of fluctuations in the parameters of linear and circular birefringence of partially depolarizing polycrystalline films of biological fluids has been theoretically substantiated and experimentally tested.The dynamics of the change in the magnitude of the statistical moments of the first-fourth orders characterizing layer-by-layer distributions of the fluctuations of the linear and circular birefringence of a polycrystalline film of blood of a healthy donor in different "phase" sections of its volume has been investigated and analyzed.The most sensitive to prostate cancer parameters are statistical moments of the third and fourth orders ( ∆X * i=;3;4 ≡ ∆X i=3;4 (ϕ * = 0.85rad) → max ), which characterize the distribution of fluctuations B L 0;90 ; B L 45;135 ; B C ⊗;⊕ in the parameters of phase anisotropy of polycrystalline blood films of practically healthy donors and patients with prostate cancer.

Figure 3 .
Figure 3.An example of maps and histograms of the

Figure 3 .
Figure 3.An example of maps and histograms of the DT red spatial distribution fluctuations of the linear birefringence B L 0;90 of polycrystalline blood films taken from healthy donor ((1) and (2)) and cancer patient ((3) and (4)).

Figure 3 .
Figure 3.An example of maps and histograms of the

Figure 4 .
Figure 4.An example of maps and histograms of the

Figure 5 .
Figure 5.An example of maps and histograms of the

Figure 5 .
Figure 5.An example of maps and histograms of the DT red spatial distribution fluctuations of the circular birefringence B L ⊗;⊕ of polycrystalline blood films taken from healthy donor ((1) and (2)) and cancer patient ((3) and (4)).

Table 1 .
The statistical structure parameters of layered diffuse tomograms of the fluctuation phase anisotropy of a polycrystalline blood film from healthy donors.
* statistical moments the most sensitive to the polarization changes in the phase anisotropy parameters of the polycrystalline blood film.

Table 2 .
Efficiency of statistical analysis of diffuse tomograms of polycrystalline blood films for the differentiation between healthy donors and patients with prostate cancer patients.'good' and 'excellent' level of balanced accuracy, obtained based on the statistical examination of the spatial distributions of the magnitude of fluctuations observed in the maps of linear and circular birefringence of polycrystalline blood films (see further details in the text). *