At first, let's consider analysis of the macro scale of bone tissue. We interpret macro scale data of the bone sample in the case when a whole bone slice is presented. In such data, usually cells (e.g., osteocyte) can't be seen properly, but information about blood vessels is complete. For such data it is important to calculate some traditional morphology information [16–19], such as bone surface (B.PM), bone area (B.Ar), porosity (Po), cortical thickness (Ct.Wi), canal number (N.Ca), canal area (Ca.Ar), and etc. Additionally, the degree of anisotropy can be measured. For these tasks it is important to split all data into, at least, three major groups: external environment, bone tissue, and inclusions.

Origin data is denoted as S.

At the first step, all data should be clustered into 2 clusters. Clusterization had been done by means of the Euclidean distance in color space K-means clustering algorithm for cluster classification was used. Clustered data let's denote as S_cl.

To separate inclusions from the external environment, data should be smoothed. This was achieved by means of the gradient filter as well as filling the transform and watershed components. The gradient filter provides data corresponding to the magnitude of the input's gradient and is computed using discrete derivatives of a Gaussian of pixel radius gfPixelRadius. For the gradient filter options, we should use about 1–5% of lower dimension. Filling transform fills all extended minima with depth fthMinima or less by lifting their values to the lowest value found among the surrounding pixels. Where an extended minimum is a connected to a set of pixels surrounded by pixels by a radius that all have a greater value than the pixels in the set. For filling transform maximum depth should be in range 0.002–0.02. Watershed components find basins at each regional minimum in the image.

Smoothed data is denoted as S_sm.

To get inclusions received data should be binary subtracted from clustered data, where inclusions data is denoted as S_in. Denote binary subtraction as an inversion of converse implication:

Proposed method depends only on values of input.

And this fact allows us to keep using the same option for sets of images, which improves the quality of automation [20–22]. On Figure 1 all mentioned sets are shown: set S (upper left), S_cl (upper right), S_sm (bottom left), and S_in (bottom right). In this case, histological parameters can be calculated easily:

Canal number can be calculated as number of related subsets in S_in. Thus, average canal area can be calculated:

To calculate Ct.Wi, a polar coordinate system is introduced in the center of mass of S_cl. Then, using radial vectors of the distance between interfaces (bone and void) can be calculated. Distribution of these values is a function of polar angle.

The algorithm of preparing the data can be described as:

To calculate the degree of anisotropy, a fabric tensor was used [23–26]. For this purpose, all data should meshed by 2-D Cartesian grid and fabric tensor should be calculated for every element. Inclusions data was used to improve the quality of the mesh. An adaptive algorithm was used to tailor the mesh. The algorithm based on the calculation of relative abundances of inclusions in an element. For every element B.Ar should be calculated, to check if there are some bone tissue compare B.Ar with preset infimum εB.AR. Then if the size of the element allows it can be remeshed. Then the presence of inclusions should be checked. To do this comparison of Po with preset infimum εPo needed. If true, then MIL can be calculated. According to the abundances, the element can be thrown away and then remeshed or taken into the calculation (see Figure 2). The algorithm of meshing can be described as:

Will consider fabric tensor as a quadratic approximation of mean intercept length (MIL) [23–26].

where L – MIL data, x – space vector, M – fabric tensor.

Degree of anisotropy can be calculated as an aspect ratio of the eigenvalues of fabric tensor.

where λ₁ – the 1st eigenvalue, λ₂ – the 2nd eigenvalue.

To expand analyze of inclusions distribution the eigenvectors field were investigated. For this purpose, additional data set S_rs should be formed. The polar coordinate system with a pole in the center of mass of clustered data is introduced below.

where φ, r – polar coordinate of i-th element's center, an investigated parameter can be a degree of anisotropy η in the i-th element, h – 1st or 2nd eigenvector (analogically in the i-th element). The algorithm calculation the fabric tensor distribution can be described as:

The description of the algorithm presented in detail in Semenova et al. [23] with an example for estimation the quality of collagen recovery using microscopy data. For the data, the correlation matrix was calculated and singular value decomposition was used. Not affecting the parameter coordinates can be determined by defining infinitesimal singular values. On the basis of this analysis, average parameter can be calculated. Described methods were used on μCT orthogonal slices of diaphysis femur. Origin data consists of about a 100 slices.

The methods described were used for analyzing bone tissue samples. For this purpose, μCT data of bone tissue slices (n = 100) were used [27]. Micro / nanofocal X-ray inspection system for CT and 2D inspections of Phoenix V|tome|X S240 was used for scanning. The system is equipped with two X-ray tubes: microfocus with a maximum accelerating voltage of 240 kV power of 320 W and nanofocus with a maximum accelerating voltage of 180 kV power of 15 W. For primary data processing and creating a volume (voxel) model of the sample based on x-ray images, the datos|x reconstruction software was used. The sample fixed in the holder was placed on the rotating table of the X-ray computed tomography chamber at the optimum distance from the X-ray source. The survey was carried out at an accelerating voltage of 90–100 kV, and current 140–150 mA. Described data can be useful in histology analysis [28–35]. The dimension of slices was 449 ×610 pixels, pixel's dimension −7.985 μm. The options used for the calculations depended on properties of inputs and it is an advantage of the method. It means that for all arrays of slices, the options selected will be constant. Nowadays options should be picked up manually. For the presented data, a gradient filter with 8 pixels value was used (gfPixelRadius), filling transform −0.03 (fthMinima). For meshing algorithm low Threshold was equal to 5% and upper Threshold was equal to 70%. On Figure 3 original data S (the first row), clustered data S_cl (the second row), smoothed data S_sm (the third row) and data of inclusions S_in (the fourth row) for the arbitrary slices are shown.

Marked above histological parameters were calculated and analyzed. On Figure 4 all results are shown. Po was linear in a longitudinal direction, correlation coefficient was −0.30 (r² = 0.921). B.Ar was linear in longitudinal direction, correlation coefficient was −0.26 (r² = 0.978). B.Pm. was quadratic in longitudinal direction (r² = 0.931). Ca.Ar was linear in longitudinal direction, correlation coefficient was −0.21 (r² = 0.962).

Results for fabric tensor were close to results from previous work [26], and the difference can be explained by dimensions of data (2D via 3D). The second eigenvector of the fabric tensor was almost radial directed (deviation about ± 10°), the first eigenvector of the fabric tensor was in the orthogonal direction. Eigenvectors do not depend on radial coordinates. Aspect ratio of eigenvalues was 0.58 ± 0.07 (13%).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.