3D Mueller-matrix-based azimuthal invariant tomography of polycrystalline structure within benign and malignant soft-tissue tumours

We introduce a method of azimuthally invariant 3D Mueller-matrix (MM) layer-by-layer mapping of the phase and amplitude parameters of anisotropy of the partially depolarizing layers of benign (adenoma) and malignant (carcinoma) prostate tumours. The technique is based on the analysis of spatial variations of Mueller matrix invariant (MMI) of histological sections of benign (adenoma) and malignant (carcinoma) tissue samples. The phase dependence of magnitudes of the first-to-fourth order statistical moments is applied to characterize 3D spatial distributions of MMI of linear and circular birefringence and dichroism of prostate tumours. The high order statistical moments and phase sections of the optimal differentiation of the polycrystalline structure of tissue samples are revealed. The obtained results are compared with the results obtained by conventional methods utilizing polarized light, including 2D and 3D Mueller matrix imaging.

Therefore, it is actually to develop 3D azimuthally invariant MM polarimetry of the most common type of objectspartially depolarizing biological tissues. The basis for this can be the joint use of probing and reference coherent beams. This will ensure the possibility of obtaining layered MM images based on the synthesis of methods: • 2D mapping of Mueller-matrix invariants (MMIs) of biological layers-azimuthally independent distributions of matrix elements or their analytical combinations [24,36,37]; • digital holographic restoration of distributions of complex amplitudes of the object field of such layers [32,33].
Our article is aimed at the development and experimental validation of the diagnostic effectiveness of the method of azimuthally invariant MM tomography of the polycrystalline structure of histological sections of biopsy of benign and malignant tumors of the prostate.

{F} ==
(1) where Θ is the angle of rotation of the sample.
and dichroism (LD, CD) Figure 1 presents the optical arrangement for 3D MM polarimetry of biological layers. Parallel (Ø = 2 × 10 3 µm) beam of He-Ne (λ = 0.6328 µm) laser 1, formed by optical collimator 2 by means of 50/50 beamsplitter 3, divided on illuminating and reference ones. Illuminating laser beam is directed through the polarization filter 5-7 on the sample of biological layer 8. Polarizationinhomogeneous image of the sample 8 by means of strainfree objective 9 (Nikon CFI Achromat P, focal distance-30 mm, numerical aperture-0,1, magnification-4x) is projected in the plane of digital camera 14 (The Imaging Source DMK 41AU02.AS, monochrome 1/2 ′′ CCD, Sony ICX205AL (progressive scan); resolution-1280 × 960; sizes of light-sensitive plate-7600 × 6200 µm; sensitivity-0,05 lx; dynamic range-8 bit, SNR-9 bit). Reference laser beam by means of reflected mirror 4 is directed through the polarization filter 10-12 in the plane of polarizationinhomogeneous image of sample 8. As the result the interference pattern registered by digital camera 14 is formed. Formation of different polarization states of illuminating and reference beams is created by both polarization filters 5-7 and 10-12, each of them consists of two linear polarizers (B + W Kaesemann XS-Pro Polarizer MRC Nano) and quarterwave plate (Achromatic True Zero-Order Waveplate).
The technique of polarization-interference measurements of MM elements consists in the following sequence of operations: • Formation of six polarization states ( The basis for the determination of a series of layered distributions of q (x, y, z) is the use of a reference laser radiation wave, which is superimposed on the polarizationinhomogeneous image of the biological layer [31,32].
The resulting interference pattern is recorded with a digital camera.
Further, apply Fourier transform to interference pattern and using the inverse Fourier transform obtained the distributions of complex amplitudes |A of the object field with an arbitrary step ∆ϕ j=0.p .
The optical scheme and the technique of experimental measurements of the MMI aggregate of biological layers (relations (6)-(9)) are described in detail and presented in [19-21, 24, 36, 37].
The optical technology of differential diagnostics of such samples includes the following steps: .
where k 1 and k 2 is the number of correct and incorrect diagnoses within group 2; g 1 and g 2 -the same within group.

Analysis and discussion of the experimental results
In Let us analyze the results obtained from the physical point of view.
For the distributions of linear LB (figure 1) and circular CB (figure 2) birefringence of samples of both types due to different pathology of the prostate tissue, the following transformations of the polycrystalline structure take place.
A benign tumor contains a developed newly formed linear birefringent network, which is formed due to an increase in the concentration of optically active protein molecules [2-4, 19-21, 34, 35].
Oncological processes are accompanied by destruction of the fibrillar network. Optically, this is manifested in a decrease in the structural anisotropy-linear birefringence (LB ↓) of the prostate tissue. Due to this, the mean and range of variation of the random variables of the MMI distribution F 44 (ϕ, m × n) ∼ cos LB (relation (2)) increases (figure 2, fragments (1), (2)). In parallel with this, the concentration of optically active protein molecules decreases due to necrosis. As a result, for the distribution ∆M (ϕ, m × n) ∼ tan CB (relation (2)), there is a reverse scenario-a decrease in the mean and range of variation of the random variables of this MMI (figure 2, fragments (1), (2)).
The most pronounced such scenario (relations (11), (12)) is realized in the scattering region of insignificant multiplicity, which corresponds to small values (0. The circular (CD ↑, (equation (4)) dichroism of oncological changed prostate layers-increases Z group1 The greatest difference between statistical moments Z i=1;2;3;4 (F 41 ;F 14 ) is achieved in the range of somewhat larger values of the 'phase' cross sections (0.6 rad ≺ ϕ ≺ 0.9 rad ⇒ ϕ * = 0, 75 rad).   (1)) and carcinomas (fragment (2)) of the prostate. The most sensitive to the differences between the polycrystalline structure of the adenoma samples and the prostate carcinoma parameters have been identified-the statistical moments of the third and fourth orders, which characterize the asymmetry and kurtosis of the distributions [F 44 ; ∆M;F 41 ;F 14 ] (ϕ * , m × n).
The result stimulated an additional study-the determination of the comparative efficacy of traditional [1-4, 12-15, 19-21, 24, 34, 35, 38-43] and the methods of polarimetric diagnostics proposed in this work, depending on the depolarizing ability of biological tissue samples.

Comparative characteristics of the diagnostic efficiency of the methods of polarization, 2D and 3D MM mapping
The comparative results of the diagnostic efficiency of differentiation of benign and malignant prostate tumors with different depolarizing ability by azimuthally invariant polarization [19,38,[42][43][44]46], 2D [20,21,24,35,38,39,41,45] and 3D MM mapping. For this purpose, groups (16 samples each) of histological sections of adenoma biopsies and carcinomas with various geometric (l, µm), optical (τ ) and depolarizing (Λ, %) characteristics were studied. Table 2 shows the maximum values of balanced accuracy, which are achieved by the methods considered.
From the comparative analysis of the accuracy of the polarimetric methods for diagnosing the changes in the phase and amplitude anisotropy of the depolarizing layers of benign and malignant prostate tumors:  decreases from good (Ac (τ = 0.25; Λ = 28%) = 83% − 84%) to satisfactory (Ac (τ = 0.62; Λ = 43%) = 75%)). (c) Azimuthal invariant 3D MM tomography proved to be the most diagnostically effective (70% ≺ Ac ≺ 96%) in a wide range of optical thickness (0.25 ≺ τ ≺ 1.37 ⇔ 28% ≺ Λ ≺ 68%) histological sections of biopsy of benign and malignant tumors of the prostate. Comparative studies of the diagnostic efficiency of azimuthally-invariant methods of polarization, 2D and 3D MM mapping of prostate tumor samples with different optical thickness (τ ) and depolarizing ability (Λ) were carried out. On this basis, the levels of balanced accuracy (Ac) and limits (0.25 ≺ τ ≺ 1.37 ⇔ 28% ≺ Λ ≺ 68%) of using the methods are determined. [