@2024 Afarand., IRAN

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ISSN: 2251-8215    Sarem Journal of Reproductive Medicine 2017;1(1):3-8

Background
Scientists have long been thinking of defining a mathematical model for natural objects; in 1979, a ‎German scientist, Benoit Mandelbrot, solved this problem by defining a non- Euclidean geometry‏ ‏called fractal geometry. Indeed, using this kind of non-Euclidean geometry, it is possible to ‎mathematically define structures such as trees, snow crystals, and galactic clusters that are ‎morphologically non-compliant with Euclidean geometry [1].‎
Previous Studies
‎… [2]. Fractals are objects or phenomena that dispersion of each of its component is equal to total ‎dispersion. The dispersion of components in a building or fractal phenomenon follows the formula ‎N=KrDf, in which N is the number of the components, R is the radius of the sphere or circle that limits ‎those components. K is the constant number and Df is the fractional coefficient of that object. Df ‎represents any fractal object, which means that with the help of Df, the dispersion of the components of ‎the object can be calculated and predict its geometric shape. In recent years, in medical sciences like ‎other sciences such as physics and chemistry, fractal geometry has been used to define the physical ‎structure of body components. Examples of these studies can be found in the study of blood vessel ‎structures [3] or the diffusion of brain neurons [4]. Also, these studies have been used to determine the ‎pathological pattern of various lesions. For example, to interpret the dispersion of pigmented skin ‎lesions [5], abnormal chest image in interstitial lung tissue diseases (ILD) [6] and the interpretation of ‎Pap smear [7], fractal geometry has been used and numerous other examples have been recorded in ‎various journals. ‎ The structure of bone marrow protects the three-dimensional structure and contrasts the various ‎components of bone marrow. These conflicts are controlled by complex mechanisms, which result in ‎the creation of multiplicative patterns and averages that we call them the Normal Range [8]. …various ‎growth factors such as lymphokines, cytokines, growth factors and their receptors along with complex ‎molecules, such as integrins, play an essential role in the evolution and development of bone marrow ‎cells [9] and produce a number of specific structures that in morphology they appear to be formed of ‎‎“repetitive units” [10]. These units have placed between fatty spaces, and when geometric dimensions ‎and geometric structure relationship are protected, it controls the release of adult cells. The important ‎thing about the fractal factor is that the fractal factor of the numerical biological structure is between ‎‎1.6-1.7 (because the fractal factor in growth phenomena is limited to propagation of 1.6-1.7).‎
Aim(s)
The purpose of this study was to investigate the physical structure of bone marrow based on ‎mathematical model of fractals and growth-limited phenomena of propagation. ‎
Research type

Research society, place and time
‎31 slides of bone marrow patients were selected from the archives of pathological samples of ‎Pathology Department of Rasoul-e-Akram Hospital from October 1991 to October 1996. The samples ‎were all from the lilac Crest area of the patients, and they were related to patients who had been ‎referred to the center for the study of anemia of thrombocytopenia, leukemia, multiple myeloma, or ‎metastatic tumors (before or after treatment).‎
Sampling method and number
Out of 31 selected samples, 21 were from males and 10 were from females. The average age was 50 ‎years (from 18 to 75 years old) and the length of each sample was about 1 to 2 cm. and they had been ‎dehydrated and paraffin embedded with routine fixation and decalcified processes. The samples were ‎cut to 3-5 micrometer in diameter and stained with haematoxylin and eosin. ‎
Used devices&materials
The samples were examined under a light microscope as well as bone marrow diseases. All of the ‎slides entered the study were morphologically normal and no abnormal building or disease process ‎was observed. To capture the morphology of the samples with a magnification of 20, a computer shot ‎was captured by a Cas 200 device. A random image was taken from each slide in four different regions, ‎and in all images, a cell group (not fat) was placed in the center of the image. A total of 31 bone marrow ‎samples and 124 microscopic images were analyzed by Image Cytometry using the Image Pro Plus ‎software. In each image, eight circular concentric circles were analyzed. Each circle had twice the ‎surface of tis inner circle. At each circular surface, the overall cell structure was measured (its size was ‎obtained by subtracting the entire surface of the circle from the fat level) and the total surface of cell ‎nuclei was also analyzed. In addition, 9 samples of 36 microscopic images, which were selected ‎randomly from 31 samples, were printed and they were investigated with four concentric circular ‎circuits in the number of cores. Therefore, three different, but at the same time, related parameters ‎were analyzed in terms of the shape and dimension of the fractal: 1) total surface cell (total surface ‎area of the bone marrow from the fat level); 2) the surface of nucleus, and 3) the number of bone ‎marrow cells (number of nucleus).‎ If x represents the Sn pattern of the bone marrow from the fractal dimensions with the fractal factor Df, ‎then for each radius of the circle r, the formula X ~ rDf will be applied. Therefore, Df is obtained from ‎the slope of the log regression line Log r-Log X. The inner radius of the circle was 7.57 micrometers ‎‎(equivalent to 377 pixels). To calculate the regression line angle coefficient, Excel 5 software was used. ‎All statistical calculations and comparisons (T test) were performed using SPSS 7 software. ‎
Findings by Text
Based on the three analyzed parameters, the special structure of fractal bone marrow cells are shown ‎‎(Fig. 1; a to c). The mean cellularity in 124 microscopic image was 55.51±22.00 %. In order to reject any errors in the ‎calculation of the mean fractal coefficient, the fractal coefficient for the five concentric circle surface ‎was also recalculated. The results of this study did not show significant difference with the results of ‎the main study (p<0.05). The finding showed that if the larger circular surface was used in the study, ‎the results did not show a significant difference with the current result. The results of this study were ‎compared with the results of the study by Naeem et al. [11] by the t-test, which did not show a ‎significant difference (p<0.05; Table 3).‎‎
Main comparison to the similar studies
‎… [12-14].In this study, the bone marrow biopsy under light microscope forms a two dimensional ‎system. In this system, the total fractional coefficient (for cellularity 40-60%) is calculated to be ‎DfMean=1.68±0.13 which is coordinated with the fractal coefficient derived from the growth ‎phenomena. … [15].‎ In this study, the bone marrow adheres to a fractal pattern and the fact that its fractal coefficient is ‎unmatched by similar amounts in limited growth phenomena indicates that the spatial structure of the ‎hematopoietic cells within the bone marrow is dependent on the release of regulatory cytokines in the ‎environment. If the mechanism for creating a three-dimensional bone marrow structure in healthy ‎people follows a pattern of growth factors limited to spread, the 3D-strucutre of the bone marrow will ‎have a fractional coefficient of 2.45. This 3D structure is very complex and branchy [16]. The existence ‎of similar structures of growth factors limited to the spread in the bone marrow may reveal some of its ‎pathological processes. ‎ In 1998, Sahimi et al., during publication of article titled “Bone marrow fractal dimension in metastatic ‎complications”, proved that when bone marrow cancer metastasizes, the fractal factor increases from ‎normal value 1.72±0.10% to 1.98±0.20. This finding clearly shows that the lack of fractal structure in ‎metastatic lesions can lead to a loss of control of structural adjustment mechanisms of the bone ‎marrow and the Milieu-interior internal environment of the bone marrow in such a way that it creates ‎peripheral blood changes such as roblastosis Leuko-eryth. This finding also may provide a way for ‎interpretation of the physiopathology of bone marrow diseases. S. Organ et al. from the Izmir ‎University of Medical Sciences in Turkey published an article in 2009 in which the calculation of the ‎bone fractal coefficient in calcium absorption images using the X-Ray method was applied to detect the ‎effect of parathyroid adenoma hyperparathyroidism in order to distinguish it from other osteoporosis ‎factors. ‎
Suggestions
To determine the non-fractal nature of bone marrow pathological processes, there is a need for further ‎research in each of the areas of disease, and our advice to respected researchers is to study and ‎research in the field of fractal numbers in other blood diseases.‎
Limitations

Conclusions
Under normal conditions, the fractal factor is about 1.7 in the two dimensional and three-dimensional ‎levels. This study, which was conducted for the first time in Iran, proved that the structure of the bone ‎marrow is fractal. ‎
Acknowledgments

Conflict of interest

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TABLES and CHARTS

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