Fractal analysis of images in medicine and morphology: basic principles and methodologies

Background. Fractal analysis is an informative and objective method of mathematical analysis that can complement existing methods of morphometry and provides a comprehensive quantitative assessment of the spatial configuration of irregular anatomical structures. Objective: a comparative analysis of fractal analysis methods used for morphometry in biomedical research. Methods. A comprehensive analysis of morphological studies, based on fractal analysis. Results. Different types of medical images with different preprocessing algorithms can be used for fractal analysis. The parameter determined by fractal analysis is the fractal dimension, which is a measure of the complexity of the spatial configuration and the degree of filling of space with a certain geometric object. The most known methods of fractal analysis are the following: box counting, caliper, pixel dilation, "mass-radius", cumulative intersection, grid intercept. The box counting method and its modifications is the most commonly used method due to the simplicity and versatility. Different methods of fractal analysis have a similar principle: fractal measures (different geometric figures) of a certain size completely cover the structure in the image, size of fractal measure is iteratively changed, and the minimum number of fractal measures covering the structure is calculated. Methods of fractal analysis differ in the type of fractal measure, which can be a linear segment, a square of a fractal grid, a cube, a circle, a sphere etc. Conclusion. The choice of the method of fractal analysis and image preprocessing method depends on the studied structure, features of its spatial configuration, the type of image used for the analysis, and the aim of the study.

Superconductivity and nucleation from fractal anisotropy and product-like fractal measure

Superconductivity is analysed based on the product-like fractal measure approach with fractal dimension α introduced by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal elastic media. Our study shows the emergence of a massless state at the boundary of the superconductor and the simultaneous occurrence of isothermal and adiabatic processes in the superconductor depending on the position of the electrons. Several physical quantities were found to be position-dependent comparable with those arising in heavy doping and p–n junction. At the boundary of the superconductor, a shrinkage of the magnetic field was observed, leading to a scenario equivalent to the Meissner–Ochsenfeld effect. An enhancement of the London penetration depth is revealed and such an improvement was observed in pnictides at the onset of commensurate spin-density-wave order inside the superconducting phase at zero temperature. The Bardeen–Cooper–Schrieffer theory was also analysed and the appearance of zero-energy states is detected. Nucleation of superconductivity in a bulk was also studied. The system acts as a quantum damped harmonic oscillator and our analysis showed that type-I superconductivity occurs when κ < 2 / ( 1 + α ) , whereas type II occurs for κ > 2 / ( 1 + α ) , where κ is the Ginzburg–Landau parameter. The transition at the passage from the ‘genuine’ to the ‘intermediate’ type-I estimates 0.767767 < α ≤ 1 .

Method Development to Determine the Fractal Structures Index into the Broiler Chickens’ Intestines

Microbial communities in the intestines of birds self-organize to carry out joint biochemical actions and extract nutrients from plant substrates. Depending on the feed entering the intestine, the configuration of the microbial biosystem is regularly reconstructed, evenly distributing the enzymatic load between the microbial components. To assess the qualitative and quantitative indicators of these processes, it is proposed to calculate the index of fractal structures of microbial communities existing in the intestines of birds. The technique is based on a fractal analysis of the frequency spectrum of operational taxonomic units (OTU) of microbial communities obtained by the molecular genetic method. Initially, the initial OTU spectrum is converted into an abbreviated spectrum of OTU groups, in which each group combines OTE that are similar in frequency of occurrence. Next, a two-dimensional fractal portrait of the microbial community is constructed, on which each OTU group is represented by a point with coordinates that depend on the frequencies of these groups. A fractal portrait is necessary for conducting a topological analysis of the location of points on it and isolating fractal structures in the microbial community. As a fractal measure, the analysis uses triangles with vertices at the points of the OTU groups. At the same time, the smallest heights of the triangles are selected with the scale of the fractal measure. The construction of triangles and their arrangement in increasing order of scale of the fractal measure allows us to select the first triangles in a row, since they are triangles degenerate into a segment by a straight line. OTU groups included in degenerate triangles are the sought-after determined OTU groups belonging to fractal structures in the microbial community. Their share in the microbial community determines the index of fractal structures (IFS), which also takes into account the uniform distribution of the enzymatic load between OTU. In the experiment, the molecular genetic method of NGS was used to study microbial communities in the intestines of broilers of NGS using two types of probiotics. The obtained frequency-taxonomic spectra were used to calculate the IFS. An analysis of the results showed that probiotics affect the self-organization of microbial communities.

METRICAL ESTIMATIONS OF DETERMINANT OF THE INTEGRAL PROBLEM FOR EQUATION OF VIBRATING OF STRING

Metrical theorems are inprocess set about estimations from below small denominators that arose up at investigational existence of periodic at times decision of task with integral conditions as moments after a spatial variable for equalization of small vibrating of string. For leading to of metrical estimations the concept of fractal measure and dimension of Hausdorff is from below applied.