Volume 03 Issue 12-2023
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VOLUME
03
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12
Pages:
185-197
SJIF
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A
BSTRACT
This article is devoted to determining the Geometric modeling of self-similar fractal structures and fractal
size of organs. There is a detailed description of the various mathematical methods for determining the
size of fractal organs, and an analysis of errors in determining the fractal size of organs. In the article, the
fractal structure, fractal size, properties of human organs were determined using the Mandelbrot-
Richardson scale (or cell method). The fractal structure of the human lung was also studied by comparing
the fractal structure of tree branches. In particular, tree branches, vascular systems in the human retina,
and fractal measurements of the lungs were calculated. In determining the fractal scale, changes in human
div parts were not taken into account. Most articles have used fractal measurement only in relation to
geometric shapes. In this article, the fractal structure of human organisms is studied on the basis of
mathematical formulas and special methods are used to calculate fractal dimensions, as well as the results
of an appropriate number of experiments.
K
EYWORDS
Journal
Website:
http://sciencebring.co
m/index.php/ijasr
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Research Article
GEOMETRIC MODELING OF SELF-SIMILAR FRACTAL
STRUCTURES
Submission Date:
December 11,
Accepted Date:
December 16, 2023,
Published Date:
December 21, 2023
Crossref doi:
https://doi.org/10.37547/ijasr-03-12-33
Jabbarov J.S.
Samarkand State University named after Sharof Rashidov, 140104, University Boulevard, 15, Samarkand,
Uzbekistan
Mukhtorov D.N.
Branch of National University of Uzbekistan Jizzakh, Jizzakh, Uzbekistan
Mustaffaqulov M.A.
Samarkand State University Urgut branch, Samarkand, Uzbekistan
Baxromov A.B.
Samarkand State University Urgut branch, Samarkand, Uzbekistan
Volume 03 Issue 12-2023
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VOLUME
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Pages:
185-197
SJIF
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(2021:
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Fractal, fractal measurement, Mandelbrot-Richadson measurement, pulmonary vascular systems.
I
NTRODUCTION
Fractals are used in computer systems,
telecommunications, industrial production and
many other fields of science and technology [1-7].
One of the most interesting areas is the
application of these fractals in medicine. The
fractal organs of the human div are the entire
respiratory tract, vascular system, lymphatic
vessels, liver and bile ducts, as well as the nervous
system [8], the fractal dimensions of which are
determined in different ways.
The vascular systems in the human retina are
statistically similar and fractal [9]. It corresponds
to an irregular but limited growth process and
can affect the embryological development of the
vascular system. In addition, the human bronchial
and respiratory tracts also have a fractal
structure.
At present, it is important to determine the fractal
size of organs with a fractal structure in the
human div, on the basis of which to help to
predict and treat various diseases in humans [10].
Therefore, this article focuses on determining the
fractal dimensions of human organs. Methods for
determining the fractal size of tree branches were
used to find fractal measurements of human
organs.
The fractal size of the lungs was determined by
R.V. Genny using the vector method, [9] and in
this paper the fractal size of the lungs was
determined using the grid method, also known as
the Mandelbrot-Richardson gauge method or the
grid method. In determining the fractal
dimensions of the vascular systems in the human
lung and retina, the location of the vessels was
studied by comparing them to tree branches. It
was therefore first found in the fractal
dimensions of tree branches and then applied to
the human limb, which has a fractal structure.
2.
The main findings and results
Fractal model of world cognition, which is
inextricably linked with the processes of self-
organization. This concept is derived by us from
several existing problems of brain research and
the above postulates. Those areas of knowledge
that form the synergetics of fractal systems are
genetically revealed. These sources are reflected
as the history of individual sciences:
mathematics, physiology of the senses, and
psychology [12,13].
The fractal model is based on a mathematical
basis, and in philosophical research, the results of
mathematics and physics very often serve as
proof of theoretical conclusions. It is proved that
the basis of nonlinear structures of the external
world is a geometric self-similar fractal [14].
The study of natural fractal structures enables us
to gain a deeper understanding of the processes
of self-organization and development of
nonlinear systems. Most of the surfaces of objects
in the world around us are fractal and are jagged
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(2023:
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and rough. Although for a long time only polished,
geometrically smoothed surfaces of objects have
been studied in science. The mathematician B.
Mandelbrot [1] introduces the concept of a
fractal, which more adequately reflects the
morphology of formless bodies, taking into
account the peculiarities of the structure of the
world. He describes a wide variety of objects,
ranging from the coastline, metal alloys, cloud
silhouettes, and to many other natural and
artificial objects. In the study, a fractal is
understood as a certain formation that has the
property of self-similarity and self-affinity. It has
a regular geometric structure, where each
fragment of the fractal repeats the entire
structural structure as a whole.
Based on the similarity of self-organization
processes, a hierarchy of levels of interconnected
systems is built: the physical world, neural
networks of the human brain and the
psychological system. It is these levels that define
the geometrical order in the world. According to
scientists, the fractal picture of the world,
reflected in the human brain, is self-organizing in
neural networks at a different, higher level and
will also have a fractal structure, that is, a
structure that is self-similar and self-affine to the
inner world. Consequently, the newly formed
neural connections will be able to set the
structure of human interaction, expressed in the
development of the psychological system. Taking
into account the conditions that the formation of
a fractal occurs according to a single
mathematical formula in which the parameters
are strictly defined, and the forming structure is
harmonious, then all three levels: the structure of
the external world, the structure of neural
networks of the human brain and the structure of
the interaction of individuals - will be determined
by one self-similar fractal. It is the creation of such
a fractal geometry that corresponds to the self-
organization of the systems of the integral real
world. We extend the fractal property of systems
to the psychological world, interpreting them
from the broadest positions of development and
unity of the world, which differ from each other in
space-time scales. We synergistically combine the
selected levels of the fractal picture of the world
into a dynamic nonlinear system with many
different dimension values. We get a holistic
structure and characterize it as a multifractal. We
introduce a refinement, according to which a
similarity is observed between a fractal and an
attractor. At the same time, many phenomena in
inanimate and living nature are described in
terms of sinks, cycles, attractors and strange
attractors. Such phenomena should be revealed
on the basis of a single fractal structure. When the
dimension is more than two, but less than three,
the properties of the strange attractor of Lorentz
appear [15]. In translation from English, a strange
attractor means "attractor". It is understood as a
set of trajectories in the phase space. It also
attracts various trajectories that lie in the vicinity
of the attractor. Thus, the fractal structure
acquires a certain order, and the fractal itself
becomes a very convenient model for studying
self-organization and the development of
nonlinear systems [16, 17]. So, the dialectical
concept of development is confirmed by
synergetics and the theory of fractals. It relies on
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physical thermodynamics and mathematical set
theory, systemic and structural approaches that
interpret the processes of development of
inanimate and living nature using nonlinear
methods of understanding the world, due to the
universality of self-organizing processes of
various levels.
The human brain is also represented as a fractal
hologram. The brain reflects the world around it.
It is part of a broader system. Emphasis is placed
on the structural and functional mechanisms of
the brain. The human brain, like many other
systems, operates in a nonlinear, chaotic mode
[18]. The processes of self-organization of neural
structures are constantly going on in it.
Nonlinear dynamics is the basis of little-studied
phenomena. The functioning of the human brain
is presented as an integral non-linear system of
the external and internal world. To understand
the self-organization of such nonlinear systems, a
new synergetic approach is needed, because the
study of individual levels of its constituent
systems by various separate methods does not
give a complete, all-embracing picture of the
whole. Although rather complex measurements
of the electrical and magnetic fields of the brain
are used, which are created in the process of
interaction of many neurons. It is important to
note that, in contrast to the traditional approach,
the synergistic approach operates not with
individual cells, but with a neural network.
One of the directions of modeling neurosystems is
a neurocomputer, which functions on the basis of
a self-organization model and, accordingly, makes
it possible to study the union of neurons into a
system with certain behavior properties. It
should be noted, however, that there are
fundamental differences between brain activity
and the machine that simulates it. The parallel
processes of a functioning brain do not
correspond to the sequential processes of a PC. If
in the computer algorithm the program is rigidly
specified, then in the new interpretation the
neural network is self-organizing.
Fractal dimension of organs and organisms.
Fractals not only surround us, they are also inside
us and many animals and plants, since many
organs of the human and animal div, as well as
plants, have fractal properties. Using the
capabilities of fractal structures, nature designed
the human div extremely effectively.
The most thoroughly studied is the fractal
structure of the airways, through which air enters
the lungs.
The lungs are vital organs responsible for the
exchange of oxygen and carbon dioxide in the
human div and perform the respiratory
function. The diagram of the lungs includes three
important
structural
elements:
bronchi,
bronchioles and pulmonary alveoli.
The framework of the lungs is a branched
bronchial system. Each lung is made up of many
structural units. Each slice has a pyramidal shape
with an average size of 15x25 mm. The bronchus
enters the apex of the lung lobule, the branches of
which are called small bronchioles. In total, each
bronchus is divided into 5-20 bronchioles. At the
ends of the bronchioles there are special
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formations - acini, consisting of several dozen
alveolar branches covered with many alveoli. The
most important structural elements of the lungs
are the alveoli, on which the normal exchange of
oxygen and carbon dioxide in the div depends.
Pulmonary alveoli are small vesicles with very
thin walls, braided by a dense network of
capillaries. Thanks to microscopic alveoli, the
average diameter of which does not exceed 0.3
mm, the area of the respiratory surface of the
lungs increases to 80 square meters. They
provide a large area for gas exchange and
continuously supply oxygen to the blood vessels.
In the course of gas exchange, oxygen and carbon
dioxide penetrate through the thin walls of the
alveoli into the blood, where they “meet” with
erythrocytes [19]. Thus, the lungs are an example
of how a large area is “squeezed in” into a rather
small space.
The bronchi and bronchioles of the lung form a
“tree” with numerous branches. A quantitative
analysis of the branching of the airways showed
that it has fractal geometry.
Methods for calculating fractal dimension. In
practical problems, the calculation of the fractal
dimension is most often carried out on the basis
of the cubes method, the coating method, the local
dispersion method, the prism method, and a
number of others. However, even when
processing the same image using different
methods, the results are often different from each
other.
Figure 1. The structure of the respiratory tract
When calculating the fractal dimension in
practice, one should choose an appropriate
algorithm
based
on
considerations
of
computational accuracy, speed, and system
resources [18].
3.
Mandelbrot-Richardson measurement or
grid method.
Complex geometric objects with fractal structures
can be described and studied by mathematical
methods. The analysis of the placement density of
tree branches can be viewed as a quantitative
determination of the filling of this gap. Thus, the
closer the value of the fractal dimension of two-
dimensional branched horns to two, the more
effectively the tree fills the space, because the
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VOLUME
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Pages:
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(2023:
6.741
)
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upper limit of the fractal dimension corresponds
to the topological dimension. The fact is that the
images of fractal geometric objects are usually
always considered to be in the plane [8,9]. We can
see how much area fractal images occupy in the
plane. To do this, divide the plane into cells, the
size of which is denoted by a, and calculate how
many cells intersect the fractal images.
Determination of the fractal scale of tree branches
using the Mandelbrot-Richardson scale:
Figure 2.
Three different-sized squares were drawn on the horns of Darat. From this it was determined
that: a is the size of the cell, conditionally a = 48 mm, the number of cells in the drawing is corresponding,
the number of cells in black is N
1
= 7, the number of cells in yellow is N
2
= 22, the number of cells in blue is
N
3
= 73.
The above
𝑁
and
𝑎
are related to the Mandelbrot-Richardson formula: That is,
𝑁 = 𝐶 × 𝑎
−𝐷
(1)
where,
𝐷 −
is the fractal dimension,
𝐶 −
is the size characteristic of fractal geometry. Fractal measurement
indicates the degree to which a flat surface is filled with a fractal drawing.
Table 1.
The results of measuring the number of cells in which the lines.
The size of the
cell
а
9
16
48
Number of cells
N
73
22
7
𝒚 = 𝒍𝒏𝑵
4,2904 3,0910 1,9459
𝒙 = 𝒍𝒏𝒂
2,1972 2,7726 3,8712
Analysis of the placement density of tree branches is done by calculating their fractal dimensions. We will
look at this in more detail in the example in Figure 2. The results of measuring the number of cells in which
the lines of the drawing are located depend on the cell size are given in
Table 1.
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(2023:
6.741
)
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1368736135
Figure 3.
Creating a straight line
Based on these, logarithmic values were also calculated. As can be seen from the graph in Figure 3, a straight
line is formed between the two points. That is,
𝑦 = −𝐷 ∙ 𝑥 + 𝑐 (2)
This is the fractal dimension sought in formula
𝐷 −
Now we add all
𝑥
and
𝑦
in Table 1:
∑ 𝑦
𝑖
= 𝑛 ∙ 𝑐 − 𝐷 ∙ ∑ 𝑥
𝑖
𝑛
𝑖=1
𝑛
𝑖=1
(3)
(3) is multiplied by both sides of the formula:
∑
i
x
𝑦
𝑖
= 𝑐 ∙
=
n
i
i
x
1
− 𝐷 ∙ ∑ 𝑥
𝑖
2
𝑛
𝑖=1
𝑛
𝑖=1
(4)
The formula for finding the fractal scale using the Mandelbrot-Richardson scale.
𝐷 = (∑ 𝑥
𝑖
∑ 𝑦
𝑖
− 𝑛 ∑ 𝑥
𝑖
𝑦
𝑖
𝑛
𝑖=1
𝑛
𝑖=1
𝑛
𝑖=1
)
(𝑛 ∑ 𝑥
𝑖
2
𝑛
𝑖=1
− (∑ 𝑥
𝑖
𝑛
𝑖=1
)
2
) (5)
⁄
If we put the data given in Table 1 above in (5), the fractal size of the tree branches is determined. That is,
𝐷 = (∑ 𝑥
𝑖
∑ 𝑦
𝑖
− 𝑛 ∑ 𝑥
𝑖
𝑦
𝑖
𝑛
𝑖=1
𝑛
𝑖=1
𝑛
𝑖=1
)
(𝑛 ∑ 𝑥
𝑖
2
𝑛
𝑖=1
− (∑ 𝑥
𝑖
𝑛
𝑖=1
)
2
) = 1,3531 (6)
⁄
Determination of the fractal scale of the complex structure of tree branches using the Mandelbrot-
Richardson scale:
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(2021:
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(2022:
5.636
)
(2023:
6.741
)
OCLC
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1368736135
Figure 4.
On the horns of the tree are drawn squares of three different sizes. From this it was determined
that: a is the size of the cell, conditionally a = 48 mm, the number of cells in the drawing is corresponding,
the number of cells in black is N
1
= 8, the number of cells in yellow is N
2
= 27, the number of cells in blue is
N
3
= 89
In Fig. 4, the fractal dimension in the complex case of tree horns was determined (it has a more
complex structure than in Fig. 2).
Table 2.
The number of cells in which the tree branches are located
The size of the
cell а
9
16
48
Number of cells
N
89
27
8
𝒚 = 𝒍𝒏𝑵
4,4886 3,2958
2,0794
𝒙 = 𝒍𝒏𝒂
2,1972 2,7726
3,8712
Based on the above data, the fractal dimension in the complex case of tree branches is given by Equation 5
as follows.
𝐷 = (∑ 𝑥
𝑖
∑ 𝑦
𝑖
− 𝑛 ∑ 𝑥
𝑖
𝑦
𝑖
𝑛
𝑖=1
𝑛
𝑖=1
𝑛
𝑖=1
)
(𝑛 ∑ 𝑥
𝑖
2
𝑛
𝑖=1
− (∑ 𝑥
𝑖
𝑛
𝑖=1
)
2
) = 1,3952 (7)
⁄
Hence, instead of concluding from the results of 6 and 7, it can be said that the higher the surface coverage
of a given shape, the more accurate its fractal dimension.
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(2023:
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Figure 5.
This picture shows that the human lung has a tree-like structure
.
Evaluation of the fractal scale can also be used to characterize the human retina, various tumor formations
[17], as well as to analyze the three-dimensional arterial tree of the human lung obtained using computed
tomography data. It appears that, at the very least, the arterial system of the lung consists of a combination
of two components: a capillary network that uniformly fills the cavity, and a scattered fractal structure of
large vessels.
Using the above method, we determine the fractal size of the human lung:
Figure 6.
Three different sized cells are drawn over the human lung. From this it was determined that: a is
the size of the cell, conditionally a = 48 mm, the number of cells in the drawing is corresponding, the
number of cells in black is N
1
= 6, the number of cells in yellow is N
2
= 26, the number of cells in blue is N
3
= 87.
Based on the data in Figure 6, the human lung has a fractal structure and its fractal size is defined as follows:
Table 3.
The results of the location of the vascular systems of the human lung in several cells
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The size of the cell а
9
16
48
Number of cells N
89
27
8
𝒚 = 𝒍𝒏𝑵
4,4659 3,2580 2,7917
𝒙 = 𝒍𝒏𝒂
2,1972 2,7726 3,8712
From Table 3, it can be seen from the values of N that the human lung has a complex structure. Because as
the cell size decreases, the number of cells increases. This indicates an infinite distribution of blood vessels
in the lungs. Based on the above data, when the fractal size of the human lung is calculated using the cell
method.
According to formula 5 is equal to the following.
𝐷 = (∑ 𝑥
𝑖
∑ 𝑦
𝑖
− 𝑛 ∑ 𝑥
𝑖
𝑦
𝑖
𝑛
𝑖=1
𝑛
𝑖=1
𝑛
𝑖=1
)
(𝑛 ∑ 𝑥
𝑖
2
𝑛
𝑖=1
− (∑ 𝑥
𝑖
𝑛
𝑖=1
)
2
) = 1,5626 (8)
⁄
In addition, fractal measurements of the pulmonary vascular system of patients have been evaluated in
several studies and are widely used to describe the vascular systems in various diseases. For example,
computed tomography angiography can assess the deterioration in survival due to a decrease in the fractal
size of the pulmonary arterial tree and an increased risk of insulin in people with pulmonary hypertension.
The vascular systems in the retina, which is another part of the human div with a fractal structure, provide
oxygen to every tissue in the div and improve blood circulation, nourishment, and prevent tissue damage
and functional impairment. Human blood vessels have the same fractal structure as above, and its fractal
size can also be determined.
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SJIF
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(2021:
5.478
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(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
Figure 7.
Three different sizes of cells are drawn over the vascular systems in the human retina. From this
it was determined: a is the size of the cell, conditionally a = 48 mm, the number of cells in the drawing is
corresponding, the number of cells in black is N
1
= 6, the number of cells in yellow is N
2
= 24, the number
of cells in blue is N
3
= 116.
Based on the data in Figure 7, we determine the fractal size of the vascular systems in the human retina as
follows:
Table 4.
Results of the location of human vascular systems in multiple cells
The size of the cell а
9
16
48
Number of cells N
116
24
6
𝒚 = 𝒍𝒏𝑵
4,7536 3,1780 1,7917
𝒙 = 𝒍𝒏𝒂
2,1972 2,7726 3,8712
The values of N in Table 4 show that the vascular systems in the human retina have a complex structure.
Because the smaller the cell size, the higher the number of cells. This indicates that the vascular systems in
the retina are infinitely distributed. Based on the above data, the fractal size of the vascular systems in the
human retina is calculated using the grid method, i.e., according to formula 5, is as follows.
𝐷 = (∑ 𝑥
𝑖
∑ 𝑦
𝑖
− 𝑛 ∑ 𝑥
𝑖
𝑦
𝑖
𝑛
𝑖=1
𝑛
𝑖=1
𝑛
𝑖=1
)
(𝑛 ∑ 𝑥
𝑖
2
𝑛
𝑖=1
− (∑ 𝑥
𝑖
𝑛
𝑖=1
)
2
) = 1,7021 (9)
⁄
The vascular systems in the human retina have their own characteristics, and as a person grows older, the
blood vessels in his retina grow like trees, like a king. This means that the blood vessels in the human retina
change fractal size over time. However, the value of the fractal dimension does not change much relative to
the original dimension. That is, the results of the study show that the fractal size of the blood vessels in the
human retina varies by ± 0.073.
C
ONCLUSION
1. It was found that the human airways have a fractal structure, and on this basis the fractal size of the human
lungs was found.
2. It was found that the fractal size of the human lung does not depend on div size, but varies from 1.57 to
1.58.
3. It was found that the vascular system in the human retina has a fractal structure.
Volume 03 Issue 12-2023
196
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
12
Pages:
185-197
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
4. The fractal size of the vascular systems in the human retina was found to be 1.7. Through this
measurement, it is possible to predict whether Nisa has diabetes mellitus.
5. It was found that the fractal size of the vascular system in the human retina may change with age, i.e., its
value may increase.
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197
International Journal of Advance Scientific Research
(ISSN
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2750-1396)
VOLUME
03
ISSUE
12
Pages:
185-197
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
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1368736135
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