A method of evaluation of the shape of the human cerebellum: MRI study

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Introduction
According to previous studies, the functions of the cerebellum are to regulate balance, posture, and muscle coordination; however, it was later found to also include cognitive functions such as memory [15], speech [8], and emotion [1].The cerebellum was also associated with swallowing function in adults [13].At the same time, each of the functions of the cerebellum is localized in specific areas of the cerebellum and functions within the limits of the neural network [5,16,19].At present, studies of the morphology of the cerebellum are being conducted to create models of its development and track the periods of modification of individual functions, in particular in newborns and children [24,26].
Development of the cerebellum begins approximately from the fourth week of pregnancy and continues throughout the first postnatal year [23].The basic micromorphology of the cerebellum, formed as a result of the proliferation of neurons, processes of migration and differentiation, is finally established around the 20th week of pregnancy [6].In the subsequent period up to the 40th week of pregnancy, the cerebellum undergoes a faster increase in volume and surface complexity than other brain structures [3].As a result, the features of the morphology of the cerebellum at various stages of intrauterine development, such as its linear dimensions, the degree of development of individual lobes, as well as the volume and shape, serve as guidelines for determining the age of the fetus [2,28,29].Recent studies have also confirmed the significant role of the cerebellum in brain development [4,25].
Functional and morphological changes of the cerebellum are associated with various neurological and psychiatric disorders [7,12,14,22], such as autism, multiple sclerosis, Arnold-Chiari anomaly, cerebellar cognitive-affective syndrome, Parkinson's disease, Alzheimer's disease, and others.Research is being conducted to create criteria for the norm of diagnostic methods of neuroimaging of morphological changes in the cerebellum (linear dimensions, reduction in mass and volume) [26].However, information about the anatomical norm of the cerebellum, on which these criteria are based, does not take into account the peculiarities of its individual anatomical variability.The variability of the shape of the organ is one of the manifestations of its individual anatomical variability.One of the ways to determine the shape of an organ is to evaluate the ratio of its linear dimensions [17,21,30].Magnetic resonance imaging and other modern neuroimaging research methods make it possible to establish the morphological features of organs during life, while preserving their natural position.Studies of the shape of the cerebellum and its variability are few; they were performed on anatomical preparations of the cerebellum.A comprehensive assessment of the shape of the cerebellum on tomograms has not been carried out before.
The purpose of the work is to develop a method for evaluating the shape of the human cerebellum based on the results of morphometry of MR tomograms.

Materials
In this work, T2-weighted MR images of the cerebellum were examined.MRI was performed on 30 people (15 men and 15 women aged 20 to 40), and no visible brain pathology was detected.The study was performed on a 1.5 T MRI machine (Siemens Magnetom Symphony, Munich, Germany).Imaging parameters: TE (echo time) 122 ms, TR (repetition time) 4520 ms, and slice thickness 5 mm.The Commission on Bioethics of the Kharkiv National Medical University, Kharkiv (protocol No. 4 dated 15.09.2020) established that the research does not contradict the basic bioethical standards of the Helsinki Declaration, the Council of Europe Convention on Human Rights and Biomedicine (1977), relevant WHO regulations and laws of Ukraine.
The method of determining the size of the cerebellum When analyzing the MRI images, the dimensions of the cerebellum in the axial, coronal, and sagittal projections were determined as the dimensions of a rectangle that can be constructed to cover the contour of the cerebellum, or its right and left hemispheres, on the MRI image in the corresponding projection (the so-called "bounding rectangle", Fig. 1).The rectangle is oriented in such a way that two of its sides are parallel to the line that appears at the point of intersection of the median plane with the plane of the MRI image in the axial and coronal projections (further -the median line).In the sagittal projection, two sides of the indicated rectangle are parallel to the intercommissural line by Talairach [10].Thus, two dimensions of the cerebellum were determined in each projection.
In the axial projection (Fig. 1A): • L ax -the length of the cerebellum is equal to the length of the side of the rectangle parallel to the median line; • W ax -the width of the cerebellum is equal to the length of the side of the rectangle perpendicular to the median line; In the coronal projection (see Fig. 1B): • H cor -the height of the cerebellum is equal to the length of the side of the rectangle parallel to the median line; • W cor -the width of the cerebellum is equal to the length of the side of the rectangle perpendicular to the median line; In the sagittal projection, the height and length of the right and left hemispheres were measured (see Fig. 1C): • L sag -the length of the cerebellum is equal to the length of the side of the rectangle parallel to the intercommissural line; • H sag -the height of the cerebellum is equal to the length of the side of the rectangle perpendicular to the intercommissural line.
For further analysis, the maximum values of these dimensions were taken into account.

Methods of determining the shape of the cerebellum
To assess the shape of the cerebellum on MR images, shape factors were calculated -the ratio of cerebellum sizes determined in each projection: in axial projection W ax /L ax (1) in the coronal projection W cor /H cor (2) in the sagittal projection L sag /H sag (3) To assess the shape of the cerebellum as a threedimensional structure, MR images offer parameters that take into account the ratio of one linear size of the cerebellum to the other two, namely: the relative width of the cerebellum (W r ), the relative length of the cerebellum (L r ), and the relative height of the cerebellum (H r ).They are calculated according to the formulas: W r =(W ax x W cor ) / (L ax x H cor ) (4) L r =(L ax x L sag ) / (W ax x H sag ) (5) H r =(H cor x H sag ) / (W cor x L sag ) (6) Statistical data processing was carried out in Microsoft Excel 2016.The distribution of values was analyzed according to the Kolmogorov-Smirnov test criterion.It was determined that the distribution of values of all morphometric parameters, both linear and their ratios, does not differ from normal.Further statistical analysis included calculation of the sample mean (M), standard deviation (S), coefficient of variation (Cv), determination of the minimum and maximum value and standard error of the mean (m).Correlation analysis was carried out with the calculation of the Pearson correlation coefficient (r) to establish patterns of individual variability.Parametric methods of testing the significance of differences were used.The values of cerebellum sizes and their ratios were divided by the mean value and standard deviation into three groups: small (from minimum to (M-S)), medium (M±S) and large (from (M+S) to maximum).

Results
Variability of the linear dimensions of the cerebellum.Table 1 (A) shows the values of the height, width, and length of the cerebellum on tomographic images in the corresponding projections.
As can be seen from the data in Table 1 (A), there is variability in the values of the linear dimensions of the cerebellum of the studied sample, but it is insignificant; the greatest variability is observed in Lax (Cv=7.27%),smallest -in Lsag (Cv=4.92%).
Statistical indicators of the distribution of values of the same linear size of the cerebellum, determined in mutually perpendicular tomographic projections, differ; but only the difference in height values is statistically significant.The distribution of these values is shown in Figures 2-4.
As can be seen from the data of Figures 2-4, the width of the cerebellum is determined almost identically in two different projections, the discrepancy is greater when determining the length, the maximum difference in values is found when measuring the height.This fact is fully explained by the complex three-dimensional organization of the cerebellum (see Fig. 1).In addition, the value of the same size in different projections can belong to different  groups by the value of the sign.Thus, these discrepancies were observed in 3 cerebellums when comparing Wax and Wcor, in 10 cerebellums when comparing Lax and Lsag, and in another 10 cerebellums when comparing Hcor and Hsag.
Figures 5-7 show the distribution of values of mutually perpendicular dimensions in the corresponding projections.
Correlation analysis (Fig. 5-7) showed a statistically significant linear relationship between the values of W ax and L ax (r=0.48;p<0.01); average strength and statistically significant linear relationship between W cor and H cor (r=0.39;p<0.05), as well as average strength, but statistically insignificant linear relationship between L sag and H sag (r=0.33;p>0.05).
Variability of form factors.According to Table 1 (B), the shape of the cerebellum in individual projections can be characterized by the relative value of the shape factor [21].If the value of the shape factor of the cerebellum under     • relatively wide and short, with a large ratio value W ax / L ax (1.92÷2.05),or vice versa -relatively narrow and long, with a small value of this ratio (1.61÷1.68); • relatively wide and low, with a large ratio value W cor / H cor (1.94÷2.08),or vice versa -relatively narrow and tall, with a small value of this ratio (1.62÷1.71); • relatively long and low, with a large ratio value L sag / H sag (1.42÷1.57),or vice versa -relatively short and tall, with a small value of this ratio (1.2÷1.24).According to the ratio of width and length (in the axial projection), 19 cerebellums can be classified as proportional, 4 are relatively wide and short, and 7 are relatively narrow and long.According to the ratio of width and height (in the coronal projection), 21 cerebellums can be classified as proportional, 5 are relatively wide and low, and 4 are relatively narrow and high.According to the ratio of length and height (in the sagittal projection), 21 cerebellums can be classified as proportional, 4 -relatively long and low, 5 -relatively short and high.The distribution of the values of all three form factors by the size of the feature is compared together in Table 2.
As can be seen from the data in Table 2, there is a variety of combinations of variants of the shape of the cerebellum in different projections.8 cerebellums are proportional in all three shape factors (group 8), 16 have average values of two of the three parameters (groups 3, 5, 7, 9-11).In another 5 cerebellums, only one parameter is in the range of average values (groups 1,4,6,12).Only 1 cerebellum (group 2) belongs to the disproportionate size of all three shape factors.
Relative parameters of the cerebellum.These relative parameters of the cerebellum are shown in Table 1 (C).According to these data, the shape of the cerebellum as a whole can be characterized by the value of the relative parameter.Just as in the analysis of form factors, if the values of the parameters of the cerebellum under study are in the range of average values, such a cerebellum is characterized as proportional, while extreme values indicate disproportionality: • relatively wide, with large value W r (3.66÷4.08)or vice versa -relatively narrow with little value W r (2.76÷2.92).
• relatively long, with large value L r (0.81÷0.86) or vice versa -relatively short with little value L r (0.59÷0.68).
• relatively high, with large value H r (0.46 ÷ 0.48) or vice   versa -relatively small with little value H r (0.33÷0.37).
According to the relative width, 19 cerebellums can be classified as proportional, 5 are relatively wide, and 6 are relatively narrow.According to the relative length, 19 cerebellums can be classified as proportional, 6 are relatively long, and 5 are relatively short.According to the relative height, 20 cerebellums can be classified as proportional, 5 are relatively high, and 5 are relatively low.The distribution of the values of all three relative dimensions by the size of the feature is compared together in Table 3.
As can be seen from the data in Table 3, there is a variety of combinations of parameters of the shape of the cerebellum.Eleven cerebellums have average values of each of the three parameters (group 6), seven have average values of two of the three parameters (groups 4, 7, 8, 9, 11).In another 11 cerebellums, only one parameter is in the range of average values (groups 2, 3, 5, 10, 12), and 1 disproportionate cerebellum was also observed (group 1).

Discussion
The method of determining the size of the cerebellum used in this work is similar to the "bounding box" method of L. Xiang for analyzing a three-dimensional model of the brain [27].The method of orientation of the rectangle in different projections was chosen among the most common methods in stereotaxic neurosurgery, which were the most informative and based on the most stable and visible on MRI images of the brain structure [11]."Limiting rectangle" can be used as an additional examination of the cerebellum during morphometry of MR tomograms [9].
Measurement of the cerebellum on tomograms makes it possible to assess the size and shape in vivo, in its natural position in the skull cavity, but only in separate tomographic projections.To assess the shape of the cerebellum on MR images, shape factors were calculated -the ratio of cerebellum sizes determined in each projection.In our work, this method of assessing the shape of the cerebellum was adapted for morphometric studies on MR tomograms [21].
As shown in Figure 1, each linear dimension -length, width, height -is defined in two mutually perpendicular projections.Due to the peculiarities of conducting MR tomography (the presence of a "step" when conducting MR "slices" and their orientation in space), on the one hand, as well as the complex three-dimensional shape of the cerebellum, on the other hand, the extreme, most distant points on the surface of the cerebellum, on which the bounding rectangle rests, on mutually perpendicular MR images often do not coincide.As a result, the value of the same size defined in one projection differs from that defined in another projection.Also, the same linear dimension in two different projections is related to two different other linear dimensions.Therefore, to evaluate the shape of the cerebellum as a three-dimensional structure, parameters were proposed on MR images that take into account the ratio of one linear size of the cerebellum to the other two, namely: the relative width of the cerebellum (W r ), the relative length of the cerebellum (L r ) and the relative height of the cerebellum (H r ).
The obtained data on the variability of the linear dimensions of the cerebellum differ from those in [20], where the height was determined to be more variable than the width or length.This is explained by the peculiarities of the measurement technique and sample sizes.
The existence of a statistically significant linear relationship between the values of length and width was previously established when measuring anatomical preparations of the cerebellum [20].However, the significant linear relationship between W cor and Hcor found in this study may be due to the peculiarities of measuring cerebellar height on tomograms.The variability of the values of paired linear dimensions measured in one projection and the lack of a functional connection between them lead to the variability of the ratios of linear dimensions, hence the variability of the shape factors (Table 1 B), which characterize the shape of the cerebellum in tomographic projections.
As shown in the study of A. Yu.Stepanenko [21], the size of the form factor determined on an anatomical preparation of the cerebellum affects its external structure in individual projections.In our opinion, the size of the shape factors determined on MR tomograms affects the shape of intracerebellar structures, namely the nuclei of the cerebellum, first of all, the dentate nucleus, which has a complex three-dimensional organization [18].
The analysis of relative indicators allows, in our opinion, to determine which linear size has the greatest influence on the shape of the cerebellum as a whole and, thereby, on the shape of the lobes, the course of the furrows, the threedimensional organization of its nuclei and other anatomical features.Taking into account the shape of the cerebellum,

Reports of Morphology
A method of evaluation of the shape of the human cerebellum: MRI study in turn, will contribute to the improvement of the diagnosis of its diseases using MRI, will be useful when conducting neuromorphological studies.

Conclusion
1.The proposed complex method of assessing the shape of the cerebellum in the morphometry of tomograms.The method consists in measuring linear dimensions (width, length and height) on tomograms in three different projections, calculating their ratios (form factors: width / length, width / height and length / height ratios) and relative dimensions (relative width, length and height cerebellum) according to formulas.
2. Analysis of the shape of the cerebellum contributes to the intravital determination of the features of its structure, namely: the shape of the lobes, the course of the furrows, the three-dimensional organization of its nuclei, etc.

Fig. 1 .
Fig. 1.Determining the linear dimensions of the cerebellum on MRI images of the brain (A -axial, B -coronal, C -sagittal projection): a -a straight line arising at the intersection of the median plane with the plane of the MRI image; b -intercommissural line by Talairach.

Fig. 2 .
Fig. 2. Distribution of cerebellar width values in axial and coronal projections.Note: dashed lines correspond to M-S and M+S values (here and in Fig. 3-7).

Fig. 3 .
Fig. 3. Distribution of cerebellar length values in axial and sagittal projections.

Fig. 4 .
Fig. 4. Distribution of cerebellar height values in coronal and sagittal projections.

Fig. 5 .
Fig. 5. Distribution of values of the width and length of the cerebellum in the axial projection.

14 ISSN
1818-1295 eISSN 2616-6194 Reports of Morphology A method of evaluation of the shape of the human cerebellum: MRI study study is in the range of average values, such a cerebellum is characterized as proportional, while extreme values indicate disproportionality:

Fig. 6 .
Fig. 6.Distribution of values of the width and height of the cerebellum in the coronal projection.

Fig. 7 .
Fig. 7. Distribution of values of the height and length of the cerebellum in the sagittal projection.

Table 1 .
Statistical evaluation of the distribution of cerebellar size values, their ratios and relative parameters.
Note: * -the difference is statistically significant for p<0.05.

Table 2 .
Variants of observed cerebellar forms (by form factors).

Table 3 .
Variants of observed cerebellar forms (by relative parameters).