Multi-axis goniometric 3D-visualization of vector diagrams of optical characteristics of dispersed and biological structures on a chip using different laser scanning regimes and trajectories
The angular dependence of the optical characteristics of biological structures in terms of biophysics may be considered as a correlate of their ultramorphology, whereby physical descriptors of cells and subcellular structures correlated can be used as the key ones simultaneously correlated with ultrastructural (which is identical to nanostructured) characteristics of cells, and colloid or in general, dispersed physical characteristics of biological media.
The angular dependence of the intensity of light scattering (indicatrix) represents, in terms of polarization and nephelometric analysis and cytometric determination of cell types, the most complex descriptor of their identification, based on the individual properties of radiation-scattering particles . In the experimental conditions, the natural cells have deviations (essential for the geometrical (and hence morphological) identification) in the scattering indicatrices from approximation by the idealized- shape particles with scattering indicatrices, computable based on Mie theory, T-matrix method or discrete dipoles. These deviations (or approximation disparities) are also the descriptors of the cell characteristics, in addition to the cell morphology. Proceeding from the indicatrix analysis, it is possible to characterize complementary descriptors, which include, in particular, the refractive index and the absorption coefficient, size, as well as the polarization/depolarization of the radiation beam [2,3] in the cellular microenvironment, based on the complex of which a relatively complete classification of particles/cells can be made [4, 5].
Naturally, the indicatrix can be plotted not only for beam scattering on the cellular structure, but also for reflection , as well as the scattering indicatrix can be fixed or calculated using reflection factor references . The concept of relative scattering indicatrix  allows us to extend the concept of indicatrix to any formats of measuring radiation, reflection, transmission and scattering. Currently, not only abroad, but also in Russia, the term "fluorescence indicatrix" is used in a standard way .
In a certain approximation, it is possible to compare the indicatrices of optical characteristics to radiation patterns in radioelectronics (or the corresponding projection of directograms in the physics of radio waves propagation), determining the indicatrix as a function, determining the spatial (ideally, in volume, not in planar projection) radiation intensity distribution at the output of the studied sample, represented in the form of a vector diagram with a proportionality of the radius-vectors of parameter intensity in the mapping direction with given angular coordinates (e. g., in quaternion form or Euler angles). In this case, the fundamental value for the analysis and facet classification or identification (fingerprinting) of cells in these coordinates can be expressed not only by the angular distribution of the radiation scattered by the cell, but also by the angular and polarization features of the intrinsic radiation of a stained cell or containing native chromophores cell. The limiting case of this problem will be the practically coherent propagation of radiation of proteins with fluorescent properties, such as GFP (green fluorescent protein) [10, 11], in a cell under nanosecond pumping; the opposite case is the normal distribution of bioluminescence beyond the conditions of pulsed pumping, which is inherent in the results of conventional fluorescence microscopy .
In this connection, the adequacy of the scanning geometry ("goniometric optical path") to radiation pattern at the sample exit is essential for all methods using the angular distribution of optical parameters as a metrological signal. In cell counters, this usually involves either reducing the number of data collection positions up to two positions, which is fraught with loss of the main array of relevant information, or detectors on a rotating optomechanical/mechatronic lock, which gives the problem of non-synchronous scanning (sometimes leading to the impossibility of comparison and colocalization of measurement data in a multi-angle rendering of a non-stationary particle/cell), or discrete arrays of detectors that give a discrete display of descriptors, both in angular and spatial coordinates. Despite the importance of this problem (since the angular position of the minima gives the possibility of calculating the properties of particles/cells along with the ratiometric indices of intensities at the angles of the maximum and minimum), it has not been satisfactorily solved to the present time, since, usually, the proposed constructive solutions have a palliative nature, worsening certain target technical characteristics while improving others. There are no solutions for scanning trajectories of complex shapes, such as surfaces of the third order, as well as for displacements in non-standard coordinate systems, as shown by deep automated bibliographic analysis; moreover, even in the researches where a consistent scanning (according to the geometry of the sample) should be performed, this problem is not usually posed. As a result, a qualitatively new layer of data is missing, representing descriptors unique in the complex that are one-to-one comparable with the geometry of a given sample (morphisms, in the terminology of the category theory) projected onto its geometry, thereby characterizing it with positional and angular sensitivity.
It should be mentioned that three-dimensional photometric indicatrices have been used since the 1970s  for the characterization of remote objects and are taken into account in the methods of optical triangulation . Radiation, including convective-radiant, heat transfer in the volume of a microheterogeneous medium is calculated only using indicatrices having a corresponding metric , for which the technique of three-dimensional analysis on indicatrices can be used . Since the 1960s, the solutions of the radiation transfer equation for the scattering indicatrix, which is very different from the spherical one, are known . However, most authors try to reduce the geometry of model systems to a spherical form, based on the computational features of the solution of the inverse problem, at least in the first approximation [18–23]. The scattering indicatrices of ellipsoidal and spheroidal (ellipsoid of revolution) particles, which degenerate into a sphere when all three semi-axes are equal are considered less frequently [24, 25]; even more rarely, cylindrical particles in the Wentzel-Kramers-Brillouin  or Rayleigh-Hans-Debye [27, 28] approximations; the studies on particles of the toroidal (and more complex) form are individual and exotic . In this case, as a rule, the geometry of the particle is taken into account, but the possible scanning geometry of the given experiment is not included in the calculation, which quite significantly limits the nomenclature of possible particle geometries. A paradox of the optimality of hematological analyzers is known which consists in the fact that in using a standard system analyzing the scattering of radiation in two solid angles (2–6°, 8–15° of the polar angle), the hematological analyzer is not optimal for measuring normal red blood cells, but is suitable for measurements of the parameters of the spherical erythrocytes , i. e. the cells different from the native form that arise either in the process of sample preparation (spherizing with preservation of volume by the Kim-Ornstein method) or as a consequence of anomaly (spherocytosis associated with defects in the glycolytic pathway and ATP deficiency; hereditary spherocytosis associated with the loss of a portion of the membrane lipids; reactive spherocytosis with drug oxidation of sulfhydryl groups; paroxysmal nocturnal hemoglobinuria; membrane degradation and loss of Heinz’s bodies when red blood cells pass through the spleen, as well as erythrocyte injury in collisions with intravascular fibrin). As a result, the heuristic/diagnostic value of the results of these cytometric measurements is critically reduced. Therefore, a fundamentally new technique for measuring and rendering biological structures and patterns of interacting radiation with different trajectories and angular scanning modes is needed, allowing for compensation of defects and possible artifacts, knowing the change in signal intensity at all scanning angles of a biological sample.
MATERIALS AND METHODS
The aim of this publication is to illustrate the results of the analysis and the volume rendering of the visual patterns of radiation patterns or optical indicatrix in the multi-axis geometries of the experiments at different scanning trajectories and speed regimes. Such methods may be implemented on the "Quaternion"-type setups for lens-free imaging of biological structures with angular resolution. This setup family has been assembled in 2016–2017 by our group and the work is supported by RFBR. Measurements in spherical geometry were carried out on a five-axis system similar to that described in  for rendering in the coordinates adequate to the sphere, even though the cells in the sample could be of arbitrary shape . If in the standard methods of analyzing the particle size distributions according to scattering indicatrices the measurements were made in the presence of speckle/speckle noise , then for this type of instrument, the speckle was also an informative signal. The rendering algorithm was based on obtaining a gradient map of the laser beam signal before and after passing through the biological sample. Indicatrices/radiation patterns were obtained by extrusion of the isophotes from the surface of the coordinate projection grid to a radius-vector proportional to the intensity of the signal (according to the definition given in the introduction). As a source of coherent radiation, DPSS (diode-pumped solid-state laser) was used to scan the yttrium orthovanadate crystal with an NIR diode (λ = 808 nm); in this case, the doubling of the orthovanadate frequency (λ = 1064 nm) occurred on KTP (KTiOPO4), resulting in the scanner output beam with λ = 532 nm. Scanning at different speeds and physical trajectories was performed, observing the condition of complete passage of the sample geometry and the projection surface during the frame acquisition time on the CMOS chip, except for obtaining impulse responses at a fixed projection point (as shown in Fig. 1). Subsequently, the rendering was performed in cylindrical coordinates and, for topological comparison, on the torus.
Fig. 1 shows a reference speckle-subtracted (contrasted) signal that characterizes the position of the laser beam on the sphere from the angular projection at some arbitrary time point without the preparation. The sweep of this beam during the scanning gives the projection of the sample. An example of an analysis of a calibration dispersion medium on a chip with a projection to the hemisphere is shown in Fig. 2, which demonstrates the noise of diffuse scattering on the sample particles. The more the system is coarsely dispersed, the more pronounced its component (blue and violet) is different from the background (i. e. red) shade, as can be seen in Fig. 3.
By introducing a rather clearly localized cellular structure on the projected chip (here – a fragmented biopsy sample), one can observe a localized response in the regions correlated with this structure (violet part of the projection on the RGB map in the pseudo colors in Fig. 4). A similar pattern is observed for other scanning trajectories and in other coordinates (cylindrical as a contradictory example, since the alternative use of spherical and cylindrical coordinates, as well as an alternative description in the Euler or Lagrange variables in the theory of rotating optical sources, e. g. stars, analyzed by their speckles, is a classic for comparing similar techniques). Thus, Fig. 5 and 6 show reconstructions for gauge heterogeneous-disperse systems, the spherical projections of which are given in Fig. 2 and 3, respectively. The corresponding rendering format for a fragmented biopsy specimen is shown in Fig. 7. As you can see, the spherical and cylindrical projections provide valuable information about the structure of the sample.
The measurement targets are easily rendered and can be machine-recognized (using software tools for machine learning of stereometric identification of structures) after filtering diffuse/dispersion noise on spherical and cylindrical projections. The problem of identification on the torus is significantly less optimistic, from the standpoint of identification, due to the intersection of the rays and the cross-noise of the channels. Often, three-dimensional ray renderings, similar to that shown in Fig. 2–7, are heuristically meaningless and do not say anything about the nature of the sample, for topological reasons, when a part of the projection from the sample is inside the torus. This also leads to errors in the calibration process, when calibration is carried out on a false-negative background, and the presence of non-rendered structures is not taken into account. This effect can be shown with sufficient accuracy on models. Fig. 8-a shows a pattern for a heterodisperse medium with uniform sedimentation in a toroidal projection. Fig. 8-b shows a pattern of a localized sample projected on a torus (the same notation for pseudocolors as in Figure 4, Figure 7), and Fig. 8-c, which could be taken as the absolute zero of calibration in this projection, shows the same sample of a fragmented biopsy, but smaller and biased (in the space of the recording chamber).
Since 3D-rendering takes significant computer time, it is not optimal for scaling and rotating internal projections on the torus when calibrated in a fast experiment on live cells, colloids deposited, etc. Therefore, in order to avoid operator errors, we cannot recognize toroidal projections as heuristically valuable as spherical or cylindrical projections. For this reason, the final software (for biocolloid chemistry and laser biophysics) does not have rendering on the torus integrated.
Thus, a multi-angle system for scanning mapping of biological samples in different geometries is created, qualitatively different both from the diffraction structure meters of the past  registering the structures using scattering indicatrices, and from the modern devices for measuring particles in the nanoscale , primarily by the geometry of the projection and the dimension of the rendering. It should be noted that the stereometric "Quaternion"-type setups are not identical to the standard laser radiation pattern 3D-scanners [36, 37], since they analyze the parameters of the samples (cells, colloid particles) rather than the actual beam used only in calibration with subsequent subtraction of the profile or in speckle-mediated methods of analyzing biological structures on a chip, which, however, does not exclude the need for this type of setup in lasers with a stably scanning directivity pattern  or using specialized acoustic and optical deflectors for scanning. The available know-how makes it possible to organize such measurements on a simpler hardware level, however this is the subject of registration of intellectual property and, in part, the subject of the following articles.
The work was supported by the RFBR grant 16–32–00914.