Insights into polycrystalline microstructure of blood films with 3D Mueller matrix imaging approach

This study introduces a novel approach in the realm of liquid biopsies, employing a 3D Mueller-matrix (MM) image reconstruction technique to analyze dehydrated blood smear polycrystalline structures. Our research centers on exploiting the unique optical anisotropy properties of blood proteins, which undergo structural alterations at the quaternary and tertiary levels in the early stages of diseases such as cancer. These alterations manifest as distinct patterns in the polycrystalline microstructure of dried blood droplets, offering a minimally invasive yet highly effective method for early disease detection. We utilized a groundbreaking 3D MM mapping technique, integrated with digital holographic reconstruction, to perform a detailed layer-by-layer analysis of partially depolarizing dry blood smears. This method allows us to extract critical optical anisotropy parameters, enabling the differentiation of blood films from healthy individuals and prostate cancer patients. Our technique uniquely combines polarization-holographic and differential MM methodologies to spatially characterize the 3D polycrystalline structures within blood films. A key advancement in our study is the quantitative evaluation of optical anisotropy maps using statistical moments (first to fourth orders) of linear and circular birefringence and dichroism distributions. This analysis provides a comprehensive characterization of the mean, variance, skewness, and kurtosis of these distributions, crucial for identifying significant differences between healthy and cancerous samples. Our findings demonstrate an exceptional accuracy rate of over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90\%$$\end{document}90% for the early diagnosis and staging of cancer, surpassing existing screening methods. This high level of precision and the non-invasive nature of our technique mark a significant advancement in the field of liquid biopsies. It holds immense potential for revolutionizing cancer diagnosis, early detection, patient stratification, and monitoring, thereby greatly enhancing patient care and treatment outcomes. In conclusion, our study contributes a pioneering technique to the liquid biopsy domain, aligning with the ongoing quest for non-invasive, reliable, and efficient diagnostic methods. It opens new avenues for cancer diagnosis and monitoring, representing a substantial leap forward in personalized medicine and oncology.


I. INTRODUCTION
Over the past decades, there have been extensive studies on the formation of complex patterns arising during the evaporation of liquid droplets [1].Distinct patterns, including coffee rings [2], cracking [3], and gelation [4], have been observed in biofluid droplets during drying.
These patterns hold potential as straightforward diagnostic tools for assessing the health of both humans and livestock [5,6].
The dried blood droplet displays a discernible structure comprising three zones with varying thickness [7].Peripheral zone, characterized by a polycrystalline layer of albumin exhibiting linear birefringence and dichroism.Transitional zone, consisting of external and internal layers of optically isotropic cubic crystals of Na-Cl salt, with an intermediate layer of globulin demonstrating circular birefringence and dichroism.Central zone, featuring an external layer of cubic crystals of Na-Cl salt.All zones contain multiple-scattering optical radiation elements-erythrocytes, platelets, and leukocytes-with manifestations of circular birefringence and dichroism [8].Alterations in the cellular and macromolecular constituents of blood, induced by diseases, are believed to contribute to variations in the dried drop patterns of both plasma and whole blood [5,6].
Spectroscopic techniques, including Raman, surfaceenhanced Raman spectroscopy (SERS), infrared (IR), Fourier Transform IR (FTIR), and vibrational spectroscopy, have demonstrated the ability to characterize biomolecular presence and generate a biochemical fingerprint, offering implications for indicating disease states through the detection of protein imbalances within the drop during liquid evaporation [6].Alternatively to the spectroscopic techniques, the study explores spatially non-uniform, optically anisotropic biological structures with multiple-scattering layers using the Mueller matrix (MM)-based polarimetry approach [9][10][11][12].This method extracts mediated information, represented by 16 matrix elements, and integrates it to provide a comprehensive understanding of the polycrystalline structure within the biological layer, covering all scattering (depolarizing) inhomogeneities throughout the volume.
In contradistinction to conventional tissue specimens, anisotropic structures, exemplified by dry blood smears, offer a readily accessible alternative that eliminates the necessity for invasive biopsy procedures.The optical scrutiny of blood smears emerges as a promising and expeditious screening modality, particularly in the context of conditions such as prostate cancer, which instigates discernible alterations in the optical anisotropy characteristics [9,13].This MM polarimetry approach presents a non-traumatic and straightforward methodology for screening purposes.
For practical clinical applications, it is crucial to extend MM polarimetry diagnostic techniques to assess the 3D polycrystalline structures of biological layers characterized by varying light scattering multiplicities and distinctive depolarizing capabilities.The layer-by-layer mapping of individual elements within the MM, specifically characterizing the parameters of phase and amplitude anisotropy, holds the potential to furnish critical diagnostic information.Achieving this goal involves the integration of previously reported differential MM techniques [14][15][16][17][18][19] with holographic mapping of phaseinhomogeneous layers [9,13].
The layer-by-layer distributions of depolarization degree, when aggregated, offer a thorough threedimensional representation of depolarization and individual anisotropy parameters at a local scale.Preliminary investigations utilizing this methodology have unveiled a correlation between tissue features and three-dimensional Mueller matrix (3D MM) imaging, as evidenced in prior research [11].This correlation establishes the groundwork for an effective and highly accurate differential diagnosis of prostate tumor tissues [10].
In current study, we introduce the 3D MM mapping technique employing digital holographic reconstruction for the layer-by-layer profiling of partially depolarizing dry blood smears -thin films.This technique facilitates the extraction of optical anisotropy parameters.Our results establish criteria for distinguishing polycrystalline blood films from those of healthy donors and patients with prostate cancer.Notably, through the integration of polarization-holographic and differential MM methodologies, we introduce, to the best of our knowledge, a novel approach for the spatial 3D characterization of polycrystalline structures within blood films.
Figure 1 shows a schematic of the 3D MM imaging experimental setup used in the studies of blood films (see also Supplementary video 1).
FIG. 1: The optical scheme used for MM imaging approach: LS -He-Ne laser; BC -collimator; BS -50-50 beam splitter; M1,M2 -rotating mirrors; P1,P2,P3,P4 -linear polarisers; QWP1, QWP2quarter wave plate ; S -the polycrystalline blood film sample under investigation; O -objective; P5 -linear polariser (analyser); CCD -digital camera; PCpersonal computer.See also Supplementary video 1 A parallel (2000 µm in diameter) beam of He-Ne (633 nm) laser is collimated before being split into equal "irradiating" and "reference" beams.Each beam passed through an equivalent polarising filter set to control the polarisation.The irradiating beam passed through the sample, and the image is projected by an objective, through a polariser, into the imaging plane of the digital camera.The reference beam is also guided into the imaging plane of the camera, and an interference pattern is formed from the superposition of the two beams.The camera records the intensity distribution of the interference pattern, which is then computationally analysed.
The showcased setup functions as a laboratory prototype of a 3D MM tomography tailored for layer-by-layer imaging of the polycrystalline structure in biological tissues and fluids from human organs.Future enhancements are targeted at automating the optical and polarization elements, fine-tuning reconstruction algorithms, and obtaining 3D distributions of anisotropy parameters with the ultimate goal to amalgamate the principles of 3D MM reconstruction with fiber-optic systems, thereby extending the methodology for measuring optical anisotropy parameters in vivo.
The presented experimental setup serves as a laboratory prototype for a 3D MM tomography designed specifically for the layer-by-layer imaging of the polycrystalline structure in biological tissues and fluids from human organs [10,11].Future developments are aimed at automating both the optical and polarization components, refining the reconstruction algorithms, and achieving 3D distributions of anisotropy parameters with the ultimate objective to extend the methodology for robotic automatic standalone optical biopsy and definitive histopathology diagnostics.

B. Samples of blood films
In the current study, blood smears are considered as a primary example of evaporated biological liquids.These tiny (2−5 µm) blood films exhibit a heterogeneous, complex 3D polycrystalline structure (Fig. 2) characterized by varying light scattering multiplicities and distinctive depolarizing capabilities.In essence, a biological fluid film represents a spatially inhomogeneous and optically anisotropic structure, comprised of various biochemical and molecular crystalline complexes.This film contains elements characterized by multiple optical scattering, including erythrocytes, platelets, and leukocytes, which exhibit circular birefringence and dichroism [20].
For the experiment, polycrystalline blood film samples were collected from both healthy and diseased volunteers.The blood film samples were prepared by applying a blood drop to an optically homogeneous cover glass heated up to 36  The extinction coefficient (τ, cm −1 ) of polycrystalline blood film samples is determined following the established photometric method, measuring the attenuation of illuminating beam intensity by the sample [21].This process utilized an integral light-scattering sphere [22].Additionally, the integral degree of depolarization (Λ, %) for polycrystalline blood film samples is assessed utilizing standard MM polarimetry [23][24][25].
To determine the statistical significance of a representative sampling of the number of samples by the crossvalidation method [26], the standard deviation σ 2 of each of the calculated values of the statistical moments Z i=1;2;3;4 (k) is determined.The specified number (36 for each group) of samples provided the level σ 2 ≤ 0.025.This standard deviation corresponds to a confidence interval p < 0.05, demonstrating the statistical reliability of the 3D MM mapping method.
The sample preparation procedure adhered to the principles of the Declaration of Helsinki and complied with the International Conference on Harmonization-Good Clinical Practice and local regulatory requirements.The study received review and approval from the appropriate Independent Ethics Committees, and written informed consent is obtained from all subjects prior to study initiation.Traditionally, samples containing spatially inhomogeneous optically anisotropic diffuse layers are studied using MM polarimetry approaches [27][28][29][30][31][32][33][34].In this approach, only indirect and averaged data (in the form of 16 matrix elements) can be obtained, representing the entire volume of scattering (depolarizing) inhomogeneities.To develop a new, more sensitive, and unambiguous method for tissue diagnosis, it is necessary to comprehensively address several theoretical and experimental challenges and synthesize the obtained results.The principles and steps of the proposed research are outlined in Table II.

Differential Mueller Matrix Mapping
When photons traverse a depolarizing medium and experience multiple scattering events, alterations occur in the MM of the depolarizing layer along the direction of light propagation [14][15][16][17][18][19].Analytically, this relationship is described as: where {R}(z) is the MM of an object in a plane at z in the direction of propagation, and {W }(z) is the corresponding differential MM.For optically thin, non-depolarizing layers, the differential matrix W (z) incorporates six elementary polarization properties, collectively providing a complete characterization of the optical anisotropy of the biological layer Here, LD and LB are the linear dichroism and birefringence for a direction of the optical axis y = 0 • ; LD ′ and LB ′ are the linear dichroism and birefringence for a direction of the optical axis y = 45 • ; and CD and CB are the circular dichroism and birefringence.For a diffuse medium, the matrix 2 can be represented as separate average <{W }> (polarisation part {W }(z)) and fluctuating While the connections between the fluctuating component W (z) of the differential matrix ( 3) and the depolarization parameters of the scattered radiation have been employed in prior diagnostic studies [17,35], our focus here is on exploring the potential for reproducing the polarization component <{ W }>(z) of the diffuse biological layer.ased on the preceding theoretical analysis [32-34, 36, 37], we derive a conclusive expression for the matrix operator By collectively examining equations ( 2) and ( 4), algorithms can be deduced for reproducing the average values of the phase and amplitude anisotropy parameters: < LD>(∆τ (0

Polarization-holographic recording and restoration of the object field
To determine the layer-by-layer distributions of matrix elements r ik six distinct polarization states are formed in the illuminating (Ir) and reference (Re) beams ({Ir − Re} ⇒ 0 • ; 90 • ; 45 • ; 135 • ; ⊗; ⊕).For each polarization state (p i r), two partial interference patterns are registered through a polarizer-analyzer oriented at Ω = 0 • ; Ω = 90 • .For each partial interference distribution, two-dimensional discrete Fourier transform F (u, v) is further performed.The F (u, v) of a two-dimensional array I Ω=0 • ;90 • ) (m, n) (the obtained image) is a function of two discrete variables coordinates (m, n) camera pixels defined by [11]: where are the coordinate distributions of the intensity of the interference pattern filtered by the analyser with the orientation of its transmission axis at Ω = 0 • ; Ω = 90 • ; * denotes the complex conjugation operation; (u, v) are the spatial frequencies in the x and y directions respectively; and (M, N ) are the number of pixels of the CCD camera in the m and n directions respectively, such that 0 ≤ m,u ≤ M and 0 ≤ n,v ≤ N .The subsequent application of the two-dimensional inverse discrete Fourier transform on the obtained spectrum can be expressed as Here, (ΦT * ) Ultimately, the complex amplitude distribution for each polarization state can be derived in various phase planes θ k = (δ y − δ x ) of the object field, separated by an arbitrary step of ∆θ: 3. Phase scanning of the amplitude-phase structure of the object field The algorithm, described by ( 15) and ( 16), for scanning the phase of the complex amplitude field ( 13) and ( 14) directly corresponds to the physical depth h i of an optically anisotropic biological layer in the case of single scattering: In the case of multiple scattering, the physical depth is multiplied (effective optical depth h * i ) and becomes a multiple of the geometric thickness value of the biological layer z.
In each phase plane θ k the corresponding sets of parameters of the Stokes vector and polarization parameters of the object field of the biological layer can be calculated as: Based on relations 19, the set of elements of the MM {R} is calculated using the following Stokes-polarimetric relations: .
(20) Using the set of distributions 4-10 a series of layer-by-layer distributions of the mean values of linear and circular birefringence and dichroism (G(<LB>, <LB ′ >, <CB>, <LD <LD ′ >, <CD>)) can be obtained : According to [14][15][16][17][18][19], we will further operate with the generalized quantities of linear birefringence and dichroism: In this manner, the synthesis of the 1st order differential matrix ( 1)-( 10) and the algorithms for layer-bylayer polarization-holographic determination of matrix elements ( 11)-( 20) allows the acquisition of layer-by-layer maps of linear and circular birefringence and dichroism of the polycrystalline structure of the dehydrated blood films ( 21)-( 28).

Quantitative Evaluation of Polarization Maps
To assess the layer-by-layer maps of optical anisotropy (G), statistical moments of the first (Z 1 ), second (Z 2 ), third (Z 3 ), and fourth (Z 4 ) orders are utilized and calculated as follows [23][24][25]: where P = m × n is the x-y resolution of the camera.These measures (Z 1 -Z 4 ) most fully characterize the mean, variance, skewness, and kurtosis of the layer-bylayer (θ k ) distributions G(θ k , m, n) of linear and circular birefringence (LB, ⟨CB⟩) and dichroism (LD, ⟨CD⟩).
The methodology for implementing this statistical approach involves several steps.Initially, groups of blood polycrystalline film samples are formed from both healthy and diseased patients.For each sample within each group, anisotropy maps, denoted as G(θ k , m, n), are obtained for a series (H) of phase sections θ (k=1...H) .Subsequently, a set of statistical moments of 1st to 4th orders, denoted as Z i=1−4 , is calculated using Eq.29.

Quantitative Analysis of Anisotropy Maps
For each statistically significant parameter Z * i=1−4 (LB, <CB>, LD, <CD>) the criteria of evidence-based medicine has been used [38][39][40]: 1) Sensitivity (Se) -proportion of true positive results (T P ) among the group of diseased (D + ) patients 2) Specificity (Sp) -proportion of true negative results (T N ) among the control group of healthy patients (D − ) 3) Accuracy (Ac) -proportion of true results (T P + T N ) among all the patients (D + + D − ) In our study, accuracy refers to the quantity of accurate diagnoses achieved through the utilization of 3D layerby-layer MM reconstruction for anisotropy mapping of the polycrystalline structure in blood films.Figure 3 illustrates an example of polarizationinterference measurement using digital holographic reconstruction of layered 3D elements of the MM of a dehydrated blood film.Each 3D matrix element comprises a set of 200 phase-resolved 2D layers (ranging from 0 to 2π with a scanning step of 0.01π).For each phase-resolved layer, the complex amplitude field (16) was reconstructed using algorithm (13).Subsequently, in this phase plane, the coordinate distributions of the MM elements are computed utilizing (20) and (21).Thus, through phase scanning of 3D matrix elements, optical anisotropy maps (corresponding to Eqs. ( 5)-( 10)) are reconstructed in each phase plane.
In current study, three phase slices at 0.2, 0.6, and 1.0 radians were utilized.The provided example illustrates the presence of all types of optical anisotropy mechanisms in the dehydrated blood film, as evidenced by the asymmetry in experimentally measured matrix elements.This observation can be theoretically explained by considering the presence of a complex mechanism involving linear birefringence and linear dichroism.The optical manifestations of linear birefringence in this mechanism are characterized by the partial MM: where Here, ρ is the optical axis direction, determined by the orientation of the polypeptide chain of amino acids; δ = 2π λ ∆nl is the phase shift between linearly orthogonal polarized components of the laser beam amplitude; λ is the wavelength, ∆n is the magnitude of birefringence; l represents the geometric thickness of the layer.

Linear dichroism
The analytical expression for the partial matrix operator that characterizes linear dichroism in optically anisotropic absorption is as follows: where Here, ∆τ = τx τy , τ x = τ cos ρ; τ y = τ sin ρ, and τ x , τ y is the coefficients of absorption for linearly polarized orthogonal components of laser radiation amplitude.
The resulting operator of two optical anisotropy mechanisms:

D. Diagnostic Algorithmic Framework
An analytical protocol for distinguishing between normal (healthy) and abnormal (e.g., cancerous prostate) tissues is outlined as follows.Initially, the identification of the phase plane, denoted as θ * , which demonstrates heightened sensitivity to pathological alterations in the optical anisotropy parameters of the polycrystalline blood structure, is undertaken as: 1) An initial "macro" phase scanning step θ max k = 0.05, rad is selected.
2) The layer-by-layer coordinate distributions G(θ k , m, n) are reconstructed for each θ max k .
3) The statistical moments Z i=1;2;3;4 are calculated.4) The differences between the values of each of the statistical moments are calculated 5) The phase interval ∆θ * = θ max j+1 − θ max j within which the monotonic increase in the value of (∆Z ) is calculated with a discrete "micro" phase scanning step θ min q = 0.01 rad.7) For each optical anisotropy parameter in G, the optimal phase plane θ * , in which ∆Z i (θ * ) = max, is determined.8) In these planes (θ * ), the mean Z * i=1;2;3;4 and standard deviations σ(∆Z * i ) are determined for comparing between groups 1 and 2, as well as for comparing between group 2 and 3.The sensitivity (Se), specificity (Sp) and balanced accuracy (Ac) are also calculated [26,[38][39][40].

III. RESULTS AND DISCUSSIONS
A. Layer-by-layer Phase and Amplitude Anisotropy Mapping in Polycrystalline Blood Films: Insights from Healthy Donors By employing phase scanning techniques ( 15) and ( 16) on the reconstructed object field of complex amplitudes, we extract layer-by-layer coordinate distributions of optical anisotropy parameters ( 21)-( 28) in blood film samples.The selection of phase planes θ i in the object field of complex amplitudes, along with their corresponding physical depths h i (17) and effective depths h * i (18) in our samples, is guided by optical-geometric approximations.Specifically, we consider ∆n ∼ 10 −3 [24,41,42] and a wavelength of λ = 0.63µm.
Utilizing ( 9) and ( 17), we estimate the phase intervals for scattering of various multiplicities in the object plane.A single pass of laser radiation through the polycrystalline blood film corresponds to the value θ 1 ≈ 0.7rad ⇔ z 1 ≈ 70µm.Similarly, double and triple passes correspond to θ 2 ≈ 1.4rad ⇔ z * 2 ≈ 140µm and θ 3 ≈ 2.1rad ⇔ z * 2 ≈ 210µm, respectively.In other words, phase shifts θ ≤ 0.7rad predominantly represent single scattering, while shifts in the range 0.7rad ≤ θ ≤ 1.4rad indicate low multiplicity scattering.For θ ≥ 1.4rad, multiple scattering prevails.Performing scanning in the range of phase shifts 0.15rad ≤ θ ≤ 0.7rad enables a significant reduction in the influence of depolarized background and enhances the signal for 1µm ≤ h ≤ 70µm.It's important to note that this evaluation doesn't account for the scattering multiplicity of optically active shaped elements at all depths in the polycrystalline blood film.Thus, we choose three phase planes corresponding to three regimes of laser light interaction with inhomogeneities in the blood film: 1) θ = 0.2 rad -Characterized by single scattering at both fibrillar networks of proteins (albumin, elastin, fibrin) and optically active shaped elements (erythrocytes, monocytes, leukocytes) in blood.
2) θ = 0.6 rad -Predominantly single scattering at fibrillar networks of proteins with an increased scattering multiplicity at optically active shaped elements in blood.
3) θ = 1.0 rad -Mainly associated with multiple scattering at optically active shaped elements in blood.
Figure 4 and Figure 5 depict maps illustrating the phase and amplitude anisotropies of the blood film for a series of phase planes θ k = 0.2rad; 0.6rad; 1.0rad.The analysis of the layer-by-layer maps of the phase (Figure 4) and amplitude (Figure 5) anisotropies of the polycrystalline blood film reveals several key findings.Firstly, all types of optical anisotropy, denoted as G and including linear birefringence (LB), circular birefringence (⟨CB⟩), linear dichroism (LD), and circular dichroism (⟨CD⟩), are present in the polycrystalline structure of the blood film.This observation indicates the existence of supramolecular structural anisotropy, specifically LB and LD, formed by the polycrystalline networks of protein molecules.It also suggests that optically active shaped elements of blood contribute to the formation of circular birefringence (⟨CB⟩) and dichroism (⟨CD⟩).Furthermore, the individual topological structure (m, n) of optical anisotropy maps (G) can be discerned at each phase section (θ) of the blood film object field.The coordinate heterogeneity of G (m, n) distributions can be explained by the specificity of processes involving supramolecular spatial-angular crystallization of protein molecules and blood film dehydration.Lastly, the average level and range of optical anisotropy parameters (G ≡ LB, ⟨CB⟩ , LD, ⟨CD⟩) exhibit an increase with the increment of physical h i and effective h * i depths in the polycrystalline blood film.This is attributed to the fact that the increase in physical depth (θ ≤ 0.7 rad) corresponds to an enhancement in the degree of spatial-angular ordering of supramolecular protein networks (LB ↑, LD ↑) and the number of formed elements (⟨CB⟩ ↑, ⟨CD⟩ ↑) in the polycrystalline blood film.Within the range of multiple scattering (0.7 rad ≤ θ ≤ 1.4 rad), this process is intensified for h * i .To quantitatively assess the transformation dynamics of algorithmically reconstructed optical anisotropy maps (G ≡ LB, ⟨CB⟩ , LD, ⟨CD⟩) at each phase plane θ k , a statistical analysis is conducted according to (29). Figure 6 illustrates a series of "phase" dependencies for the magnitudes of the 1st to 4th orders statistical moments (Z i=1;2;3;4 (θ)).The analysis of the obtained data revealed two contrasting scenarios for the behavior of Z i=1;2;3;4 (θ).The first scenario involves a monotonic "phase" increase in the 1st and 2nd statistical moments, which characterize the mean and variance of the G (θ, m, n) distribution.The second scenario entails a decrement in the 3rd and 4th statistical moments, which characterize the skewness and kurtosis of optical anisotropy parameters.This behavior is attributed to the increased scattering multiplicity at h * i (θ ≥ 0.7 rad).A multitude of optically anisotropic domains collectively contribute to the establishment of the average level of phase and amplitude anisotropy.Simultaneously, diverse geometric and concentration parameters within the fibrillar networks of proteins and optically active shaped elements of blood lead to an augmentation in the dispersion of linear and circular birefringence and dichroism in the polycrystalline blood film.The quantitative impacts of these processes are reflected in the values of the statistical moments.In the limit case, in accordance with the central limit theorem, the distribution G (θ ↑) ≡ LB, ⟨CB⟩ , LD, ⟨CD⟩ tends toward the normal distribution, and Z 3;4 → 0. Comparing the changes in the 1st to 4th orders statis-tical moments, it was observed that skewness (Z 3 ) and FIG.6: Phase-dependent magnitudes of the 1st (black, squares -×10 −3 ), 2nd (red, circles -×10 −3 ), 3rd (blue, upwards triangles), and 4th (green, downwards triangles) statistical moments characterizing the distributions of (A) linear birefringence, (B) circular birefringence, (C) linear dichroism, and (D) circular dichroism in a polycrystalline blood film from a healthy donor.
kurtosis (Z 4 ) exhibit greater sensitivity to phase changes in the distributions of linear and circular birefringence and dichroism G (θ).This heightened sensitivity may be attributed to the fact that small variations in (Z 2 ) lead to larger changes in higher-order statistical moments.In the range 0.2 rad ≤ θ ≤ 0.7 rad, the dynamic range of ∆Z 3;4 changes corresponding to linear birefringence and dichroism is 2.5-3 times, while for circular birefringence and dichroism, it is 3-4 times.Therefore, the layer-bylayer assessment of the polycrystalline structure in the phase shift range 0.2 rad ≤ θ ≤ 0.7 rad holds the potential for early-stage detection of oncological changes in the optical anisotropy of fibrillar networks of proteins and optically active shaped elements.

B. Polycrystalline Blood Film Diagnosis
The optimal phase planes for diagnostic purpose have been unidentified: θ * (LB, ⟨CB⟩) = 0.65 rad and θ * (LB, ⟨CB⟩) = 0.45 rad.The optical anisotropy parameters obtained in these planes are illustrated in Figure 6 and Figure 7 for samples from group 1 (healthy) and group 2 (moderately differentiated prostate adenocarcinoma with a Gleason's pattern scale of 3 + 3).
In samples from group 2, a decrease in both the average level and fluctuations of linear birefringence and dichroism was observed.Conversely, an increase in both the average level and fluctuations of circular birefringence and dichroism was noted in the same group.From a physical perspective, these results can be linked to changes in the ratio between the concentrations of albumin and blood globulin proteins.It is well-documented [36,37,[43][44][45] that early malignant processes are accompanied by an The reduction in the concentration of albumin molecules, in turn, leads to a decrease in the level of linear birefringence and dichroism of supramolecular protein networks.These biological changes are reflected in the intergroup differences ∆Z i of the statistical moments Z i characterizing the optical anisotropy maps of polycrystalline blood films from groups 1 and 2 (Table III).
The 4th-order statistical moment, representing the kurtosis of the distributions of phase (LB, ⟨CB⟩) and amplitude (LD, ⟨CD⟩) anisotropy parameters in polycrystalline blood films, demonstrates remarkable sensitivity to early signs of an oncological state.
Table IV presents the sensitivity, specificity (Sp), and balanced accuracy (Ac) values for the early diagnosis of prostate cancer using the 3D layer-by-layer MM mapping method.These values are calculated based on the intergroup difference in the fourth-order statistical moment for each of the four optical anisotropy parameters.The results reveal an excellent level of balanced accuracy, indicating high levels of selectivity and specificity in the diagnostic approach.A comparative analysis of diagnostic efficacy was conducted with three existing polarimetric methods, as outlined in Table V.The considered methods are: i. Azimuth-invariant polarization mapping of the distributions of polarization azimuth α(m, n) in the object field of the biological layer [36,37,43,44,[46][47][48]; ii. Azimuth-invariant polarization mapping of the distributions of polarization ellipticity β(m, n) in the object field of the biological layer [23,24,30,31,41,42,44]; iii.MM (R ik (m, n)) mapping of biological layers [44,45,49]; iv.This work: 3D MM reconstruction (3D − LB, ⟨CB⟩ , LD, ⟨CD⟩) of the parameters of phase and amplitude anisotropy in biological layers in 3D.An assessment of the diagnostic effectiveness of 2D and 3D polarization mapping methods for prostate tumor layers with varying optical thickness revealed that, for partially depolarizing polycrystalline blood films (Λ = 40 − 45%), the balanced accuracy of coordinate polarization methods (α, β(m, n)) and MM mapping mostly falls below a satisfactory level.However, the accuracy of early differential diagnosis achieved through the 3D MM reconstruction method described in this work represents a significant improvement.The presented analysis includes sensitivity (Se), specificity (Sp), and balanced accuracy (Ac) for the comparison of group 2 (moderately differentiated prostate adenocarcinoma, 3+3 on Gleason's pattern scale) and group 3 (poorly differentiated prostate adenocarcinoma, 4 + 4 on Gleason's Pattern scale) (see Table VI).
The results in Table VI indicate a high level of efficiency (ranging from 90.3% to 95.8%) in diagnosing prostate tumors through MM mapping of polycrystalline blood films from patients with prostate adenocarcinoma at varying degrees of differentiation.In conclusion, our study employed a 3D MM reconstruction approach for multiparameter polarimetry studies on the polycrystalline structure of dehydrated blood smears.The investigation revealed method's sensitivity to subtle changes in optical anisotropy properties resulting from alterations in the quaternary and tertiary structures of blood proteins, leading to disturbances in crystallization structures at the macro level at the very early stage of a disease.More specifically, the developed 3D MM diagnostic approach demonstrated discernible early cancer-related alterations in optical anisotropy proper-ties.This included an examination of spatial distributions of linear and circular birefringence and dichroism in partially depolarizing polycrystalline blood films sourced from healthy tissues and cancerous prostate tissues across various stages of adenocarcinoma.Observable and quantifiable changes in the 1st to 4th order statistical moments, characterizing the distributions of optical anisotropy parameters, were identified in different "phase" sections of the blood smear volumes.
Emphasizing the advantages of the presented diagnostic approach over traditional methods, we highlighted its cost-effectiveness and simplicity, requiring only a basic polarization-based optical setup without the need for reagents.Additionally, the analysis of dehydrated blood samples is prompt, providing express results compared to the time-consuming nature of biochemical analysis.Notably, during measurements, all parameters of the polycrystalline structure can be assessed simultaneously.An excellent accuracy (> 90%) for early cancer diagnosis and differentiation of its stages is achieved, demonstrating the technique's significant potential for rapid and accurate definitive cancer diagnosis compared to existing screening approaches.This pioneering work marks an initial step toward the development of an advanced, practical, and cost-effective toolkit for expedited, minimally invasive cancer diagnosis, integrated with conventional blood tests.

FIG. 2 :
FIG. 2: Microscopic images of a structure of representative blood film samples normal (left) and abnormal (right) used in the study; 300 × 200 µm each.

FIG. 4 :
FIG. 4: Maps of the linear LB (θ k , m, n) (A,C,E) and circular CB (θ k , m, n) birefringence (B,D,F) of a polycrystalline film of the blood of a healthy donor at "phase" sections of 0.2rad (A-B), 0.6 rad (C-D) and 1.0 rad (E-F).

C
. From Bench to Bedside: Envisioning the Clinical Role of MM Mapping

Table I .
.6 • in advance.The blood drops fully dehydrated within 40-45 minutes.Optical properties of polycrystalline blood film samples for the groups.

TABLE I :
Optical parameters of polycrystalline blood film samples

TABLE III :
Optical parameters of polycrystalline blood film samples

TABLE IV :
Operational characteristics of the diagnostic power of the 3D MM method

TABLE VI :
Operational Characteristics of the Diagnostic Power of the 3D MM Method for Prostate Adenocarcinoma Stage Differentiation.