A 2 × 2-inch (128 × 128-pixels) fragments containing the midsagittal sections of the cerebella were copied from the digital MR images. Afterwards, the fragments of MR images were pre-processed with the help of Adobe Photoshop CS5 software (Adobe Inc., California, USA) for further skeletonizing procedure and fractal analysis. During this pre-processing, the structures surrounding the cerebella were initially removed from the images (Figure 1A), and the pixels in these areas were colored white (brightness value of 255). Next, the segmentation of the MR images of the cerebella into two components was performed - into the studied structures (cerebellar tissue as a whole) and the background. For that purpose, we used the “threshold” tool which converted the MR images to binary format which resulted in MR image segmentation according to the pixel brightness value. An empirical brightness threshold value of 100 was used for the localization of cerebellar tissue on MR images. As a result of the pre-processing and segmentation procedures of the MR images, the pixels corresponding to the cerebellar tissue were black (brightness value of 0) and the pixels corresponding to the background were white (brightness value of 255) (Figure 1B). Further processing and FD analysis of the binary images were performed with the Image J software.

Figure 1

The digital image skeletonizing procedure was performed using the “skeletonize” tool. This tool revealed the main branches of the cerebellar white matter and formed a linear digital skeleton of the white matter — the “arbor vitae” of the cerebellum (Figure 1C).

Furthermore, the skeletonized images were analyzed using the “analyze skeleton” tool (Figure 1D). The following parameters were estimated: the number of branches, the number of junctions, the amount of end-point voxels, junction voxels and slab voxels, the average and maximum branch lengths, the longest-shortest patch length, and the number of triple and quadruple points.

We also analyzed the individual branches of the digital skeletons of the cerebellar white matter. The following parameters were determined: branch length variability, Euclidean distance, and branch length/Euclidean distance ratio. In total, the morphologic parameters of 4130 branches of 30 cerebella were calculated.

The fractal analysis was carried out using the box counting method tool of the Image J software. The values of the two-dimensional FD were determined. The fractal analysis was performed twice: prior to and after the skeletonizing procedure. As a result, two FD were obtained: FD (100) — fractal dimension of non-skeletonized MR images (which characterized the cerebellar tissue as a whole), and FD (S) — fractal dimension of skeletonized MR images (which characterized the skeleton of the cerebellar white matter).

A statistical data processing was done using Excel 2010 software. The following values were calculated: the sample mean (M), the standard error of the mean (m), standard deviation (σ), coefficient of variation (CV, relative standard deviation), the minimum (min) and the maximum (max) values, the median (Me), and the mode (Mo). The distribution of the values by the percentiles was determined. The significance of the statistical differences between the values of FD(S) and FD(100) was assessed using Student's T test. The significance of statistical differences between the values of the quantitative parameters of different cerebellar clusters was assessed using the Mann-Whitney test. Pearson's correlation coefficient (R) was calculated to establish the relationships between the values. The significance of the correlation was assessed using Student's T test. The significance level for all results was accepted as p < 0.05.

Table 1

Parameter	M	mm	Me	σ	C ν	mcv	min	max
Branches	120.03	4.70	124.50	25.73	21.44	3.056	67	168
Junctions	60.60	2.55	63.50	13.94	23.00	3.197	30	86
End-point voxels	40.83	1.50	40.00	8.24	20.18	2.759	28	58
Junction voxels	239.10	10.32	239.50	56.52	23.64	3.408	134	353
Slab voxels	734.03	25.94	747.00	142.09	19.36	2.428	405	973
Average branch length, m-3	4.02	0.08	3.98	0.44	11.00	1.40	3.19	5.10
Branch length variability (CV, %)	59.31	1.21	57.14	6.62	11.16	0.662	48.89	77.54
Maximum branch length, m-3	12.97	0.43	12.95	2.39	18.42	2.34	7.93	16.59
Euclidean distance m-3	3.46	0.08	3.37	0.45	13.00	1.65	2.78	4.59
Branch length/Euclidean distance	1.13	0.003	1.080	0.17	14.88	0.164	1.00	4.16
Longest shortest patch m-3	73.70	1.13	73.13	6.31	8.56	1.09	64.07	86.04
Triple points	46.97	1.97	50.00	10.78	22.96	3.030	19	65
Quadruple points	10.50	0.78	10.00	4.25	40.46	5.700	3	24

The values of the quantitative parameters of the cerebellar white matter are presented in Table 1. Most of the parameters had mean values of the coefficient of variation; the value of quadruple points had a high coefficient of variation. The mean values of these parameters and the confidence interval of the values (calculated as M ± 2σ) may be used as norm criteria to interpret the results of the quantitative morphological assessment of the cerebellar MR skeletonized images.

Morphometric parameters of 4130 branches of the skeletonized MR images of the cerebellum were estimated. The mean value of the branch length was 3.85 ± 0.04 m^-3 (min — 0.40 m^-3, max — 16.59 m^-3) (Figure 2). As can be seen from the data in Figure 3, the statistical distribution of the branch length had a right-hand asymmetry; the values of the sample mean (3.85 m^-3), the mode (1.92 m^-3) and the median (3.43 m^-3) did not coincide. Thus, the distribution of the values by the percentiles was determined to describe the statistical distribution. The value of the percentile 2.5 was 0.56 m^-3, percentile 16 – 1.75 m^-3, percentile 25 – 2.25 m^-3, percentile 50 – 3.43 m^-3, percentile 75 – 4.98 m^-3, percentile 84 – 5.86 m^-3, and percentile 97.5 – 9.81 m^-3. The mean value of the Euclidean distance of the white matter branches was 3.4 ± 0.03 m^-3 (min — 0.40 m^-3, max — 15.33 m^-3). The mean value of the ratio of branch length and Euclidean distance was 1.13 ± 0.003 (min — 1, max — 4.16). The branch length value was very close to the value of the Euclidean distance and often coincided with it or exceeded it by 1.1 - 1.2 times, sometimes more than 4 times. There was a significantly strong positive linear relationship between the values of branch length and Euclidean distance (R = 0.97, p ≈ 0). The distributions of these parameters are presented in Figure 3.

Figure 5

Figure 6

Figure 7

Table 2

Parameter		1	2	3	4	5	6	7	8	9	10	11	12
1	Branches	-	0.993	0.723	0.983	0.713	-0.522	-0.593	-0.441	-0.32	0.492	0.943	0.773
2	Junctions	0.993	-	0.643	0.973	0.753	-0.482	-0.633	-0.441	-0.26	0.482	0.963	0.763
3	End-point voxels	0.723	0.643	-	0.643	0.341	-0.633	-0.25	-0.462	-0.532	0.381	0.653	0.431
4	Junction voxels	0.983	0.973	0.643	-	0.683	-0.502	-0.573	-0.401	-0.31	0.472	0.893	0.803
5	Slab voxels	0.713	0.753	0.341	0.683	-	0.18	-0.653	-0.06	0.361	0.703	0.723	0.593
6	Average branch length, m-3	-0.522	-0.482	-0.633	-0.502	0.18	-	0.01	0.552	0.933	0.18	-0.482	-0.31
7	Branch length variability (CV, %)	-0.593	-0.633	-0.25	-0.573	-0.653	0.01	-	0.431	-0.21	-0.33	-0.653	-0.421
8	Maximum branch length, m-3	-0.441	-0.441	-0.462	-0.401	-0.06	0.552	0.431	-	0.441	0.03	-0.502	-0.19
9	Euclidean distance m-3	-0.32	-0.26	-0.532	-0.31	0.361	0.933	-0.21	0.441	-	0.23	-0.25	-0.19
10	Longest shortest patch m-3	0.492	0.482	0.381	0.472	0.703	0.18	-0.33	0.03	0.23	-	0.441	0.401
11	Triple points	0.943	0.963	0.653	0.893	0.723	-0.482	-0.653	-0.502	-0.25	0.441	-	0.583
12	Quadruple points	0.773	0.763	0.431	0.803	0.593	-0.31	-0.421	-0.19	-0.19	0.401	0.583	-

The strength and direction of the correlation relationships between the quantitative characteristics of the skeletonized MR images of human cerebella were different. As can be seen from Table 2, there were significant positive linear correlational relationships between the number of branches, the number of junctions, the amount of end-point voxels, junction voxels and slab voxels, the average and maximum branch lengths, the longest-shortest patch length, and the number of triple and quadruple points. The numbers of branches, junctions, end-point voxels, junction voxels, triple points and quadruple points had significant negative linear correlation relationships with the average branch length, the branch length variability (the coefficient of variation), and the maximum branch length.

Therefore, the more branched the white matter of the cerebellum was, the more branches, junctions and end points it had. The cerebellum with a high degree of branching of the white matter had a lot of short branches and junctions. The cerebellum with less branched white matter did not have numerous branches and junctions; their length of branches was greater and more variable than in those cerebella with greater branching degree.

Taking into consideration features of the branching characteristics of the digital skeletons of the cerebellar white matter, we can assume that the key characteristics of its branching pattern are the number of branches and the number of junctions. These parameters (number of branches and junctions) were related by a correlation relationship close to the functional one (R = 0.99, p = 0.11 × 10^-35). It was easier to determine the number of branches than number of junctions (if manual count was used and image skeletonizing procedure was not applied). Thus, the number of branches was chosen as a key characteristic to divide cerebella into clusters and to determine features of other quantitative parameters in different cerebellar clusters. We divided cerebella into two clusters according to the median value of the number of branches (130). The number of branches in cerebella of the first cluster was 130 or less, and the second cluster was 131 or more. These clusters had some differences in the distribution of the values of quantitative parameters of the digital skeletons of MR images of cerebellar white matter (Figure 4, Figure 5).

The mean number of junctions in cerebella of the first cluster was 48.67 ± 2.16, and of the second cluster was 74.38 ± 2.31; the mean number of the end-point voxels was 36.40 ± 1.71 (1^st cluster) and 46.38 ± 2.17 (2^nd cluster); the mean number of the triple points was 38.27 ± 1.94 (1^st cluster) and 56.56 ± 1.50 (2^nd cluster); and the mean number of the quadruple points was 7.73 ± 0.62 (1^st cluster) and 14.00 ± 1.23 (2^nd cluster).

The mean value of the average branch length in cerebella of the first cluster was 4.24 ± 0.12 m^-3, of the second cluster — 3.86 ± 0.07 m^-3; the mean value of the maximum branch length in cerebella of the first cluster was 13.64 ± 0.54 m^-3, of the second cluster — 12.65 ± 0.60 m^-3.

The differences between the values of these parameters of the cerebella of the first and second clusters were statistically significant (p < 0.05).

The fractal analysis was performed to evaluate the degree of complexity of the spatial organization of the cerebellar tissue as a whole and its skeletonized MR images. The mean value of FD(100) (fractal dimension of the cerebellar tissue as a whole) was 1.738 ± 0.009 (min — 1.634, max — 1.823). The coefficient of variation of FD(100) was 3.06% which indicated a low degree of variation. The average value of FD(S) (fractal dimension of skeletonized MR images) was 1.469 ± 0.007 (min — 1.360, max — 1.551). The coefficient of variation of FD(S) was 2.76%, which also showed a low degree of variation. Both values of FD were statistically significantly different from each other (p = 0.215 × 10^-25). There was a moderate positive linear correlation relationship between FD(100) and FD(S) (R = 0.41, p = 0.016). These values of the FD had correlation relationships of different strength and direction with the quantitative parameters of the skeletonized MR images (Figure 6, Figure 7).

As can be seen from Figure 6, FD(S) had significant positive linear correlation relationships with the number of branches (p = 0.143 × 10^-7), the number of junctions (p = 0.644 × 10^-7), the number of end-point voxels (p = 0.959 × 10^-4), the number of junction voxels (p = 0.14 × 10^-7), the number of slab voxels (p = 0.43 × 10^-4), the longest-shortest patch length (p = 0.788 × 10^-3), the number of triple points (p = 0.786 × 10^-5), and the number of quadruple points (p = 0.413 × 10^-5). FD(S) had significant negative linear correlation relationships with the average branch length (p = 0.019), the branch length variability (the coefficient of variation of branch length) (p = 0.028), the maximum branch length (p = 0.125), and the Euclidean distance (p = 0.049).

The FD of the cerebellar tissue as a whole (FD(100)) had a similar tendency to increase or decrease depending on the quantitative parameters of the skeletonized MR images but it had weaker strength of correlation relationships with these parameters than FD(S) (Figure 7). As can be seen from Figure 7, FD(100) had significant positive linear correlation relationships with the number of branches (p = 0.012), the number of junctions (p = 0.007), the number of junction voxels (p = 0.018), the number of slab voxels (p = 0.103 × 10^-6), the longest-shortest patch length (p = 0.603 × 10^-3), the number of triple points (p = 0.01), and the number of quadruple points (p = 0.031). FD(100) had a significant negative linear correlation relationship with the coefficient of variation of branch length (p = 0.002). However, FD(100) had no significant correlation relationships with the number of end-point voxels (p = 0.952) or maximum branch length (p = 0.997), and had significant positive linear correlation relationships with the average branch length (p = 0.012) and Euclidean distance (p = 0.603 × 10^-3).

The mean value of FD(S) of cerebella of the first cluster (described according to the number of branches) was 1.438 ± 0.01, of the second cluster — 1.495 ± 0.01. The difference between FD(S) values of cerebella of the first and second clusters was significant (p < 0.05). The mean FD(100) value of cerebella of the first cluster was 1.720 ± 0.01, of the second cluster — 1.753 ± 0.01; however, the difference between these values was not significant (p > 0.05).

Hence, there were two groups of quantitative parameters of digital skeletons which had an impact on the values of the FD. The first group included parameters that characterized the branching of the digital skeletons (branches, junctions, junction voxels, end-point voxels, and triple and quadruple points). They were interconnected by positive correlation relationships (inside the group), had positive correlation relationships with the fractal dimensions FD(S) & FD(100), and showed negative correlation relationships with the parameters of the second group, including the average and maximum branch length, Euclidean distance, and the coefficient of variation of branch length (branch length variability, CV). The parameters of the second group exhibited negative correlation relationships with the fractal dimensions.

The cerebella, which had the maximum values of the FD, contained a larger number of small branches approximately equal in length which were connected by numerous junctions and which formed numerous endpoints. Those cerebella that had larger values of the average branch length and greater branch length variability demonstrated lower values of the FD.

It became obvious that the key characteristics of the digital skeletons that determine the values of the FD of the cerebellum and its skeletonized MR images were the number of branches and the number of junctions that showed the strongest correlation relationships with the FD of the skeletonized MR images FD(S) (Figure 7). The same parameters (the number of branches and their junctions) were used for calculation of the confidence intervals of the FD values of the skeletonized MR images of the human cerebellum.

The number of branches and their junctions remain constant throughout life, as they characterize the innate anatomical features of the white matter of the cerebellum. However, the FD of the cerebellum can change during some brain diseases10, 11, 12, 13 and may be used as an objective morphometric criterion for the diagnosis of these pathologies. Therefore, considering the fact that the FD strongly depends on the number of branches and their junctions, which have a distinct individual variability and do not change throughout life, we propose to consider these parameters for determination of the normal value of FD as a diagnostic criterion.