Use of the Green Function in the Study of Morpho-structural Changes
in Tissues
SERGEY KOSTAREV1, TATYANA SEREDA2
1Department of the Informatics
Perm Institute of National Guard Forces of the Russian Federation
1, Gremyachy Log, 614030, Perm, RUSSIA
2Department of Infectious Diseases
Perm State Agrarian-Technological University named after academician D N Pryanishnikov
23, Petropavlovskaja Str, 614990, Perm
RUSSIA
Abstract: - Currently, automation and robotization of research is being introduced into all areas of medicine and
veterinary medicine, including histological analysis. At the same time, it is necessary to provide an automated
decision-making system for quality control of histological images preparation. The technological process of
histologic analysis running in spatial and temporal basis has been studied. The process of histologic analysis is
a complex dynamic system including the stages of biomaterial preparation and study of morphostructural
changes in tissues. The problem of process flow description is based on the law of mass conservation during
biosphere transfer, which takes into account the equation of flow continuity in Euler and Lagrange variables. In
controlling the histologic process, the errors associated with process failure were taken into account. The
solution was obtained using an impulsive transient Green's function. In the period 2000-2023, according to
statistical data, an increase in the number of cancer cases was observed, which makes the development of an
automated histological analyzer relevant. The aim of the study was to build a control model for the automated
process of histological analysis. Research Methods. The approaches to the device design were based on the
theory of histologic analysis, application of continuum mechanics methods, methods of solving differential
equations using Green's function. Results. The technique of histologic analysis was studied. The analytical
solution describing the control of the automated technological process of histological analysis in conditions of
possible disturbances caused by perturbations, such as "marriage" and time delay in the preparation of
histological specimens has been obtained. Preparation of high-quality histological images will accelerate the
diagnosis in the study of morphostructural changes in tissues, which will help to reduce the risks of developing
not only cancer, but also other diseases. Conclusion. Express analyzer of histological analysis will reduce the
time of preparation of histological images and the burden on highly qualified medical personnel.
Key-Words: - histologic analysis, automated system, Green's function.
Received: March 22, 2022. Revised: November 2, 2023. Accepted: November 23, 2023. Published: December 31, 2023.
1 Introduction
Currently, much attention is paid to the automation
of histologic analysis [1, 2]. Histologic examination
is widely used in the study of morpho-structural
changes in tissues [3, 4]. In the preparation of
histological preparation, the technological process is
automated in a fragmentary way, the known
equipment is mainly foreign. Within the framework
of import-substitution strategy, the issue of
automation of histologic analysis management
becomes especially urgent. The technological
process of histologic analysis includes many
operations [5] (Figure 1). Histologic analysis is a
deterministic, multi-parameter system, which
creates a number of difficulties for the development
of an automatic control system.
Fig. 1: Structural diagram of the robot histologist
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DOI: 10.37394/232020.2023.3.13
Sergey Kostarev, Tatyana Sereda
E-ISSN: 2732-9941
90
Volume 3, 2023
The processes of automation and modeling of
technological flows and resources in the preparation
of histological images and diagnosis in medicine
and veterinary medicine are currently an urgent task.
The classification of models used in histologic
analysis can be divided into flow-continuous,
discrete-logic, network, neural network, wavelet
technology, macro and microanalysis, and others.
Flow-continuous models describe a spatially
distributed operational flow, the realization of which
guarantees the fulfillment of the specified planned
indicators of product preparation (histological
images) within the stipulated period [6].
Discrete-logic models are characterized by a
combinatorial approach [7]. Models describing the
step-by-step process of histological image
preparation and pathology analysis based on tree-
structured indicators are presented in [8].
Network models are represented as graphs, the
vertices of which are technological operations, and
the arcs represent biomaterial movements.
Neural network methods have now found
application in recognizing pathologies in cancer and
diseases caused by chlamydia infection [9].
The application of macro and micro analysis
models in histological express diagnostics is
considered on the example of diagnosing oncologic
diseases realized in the ATLANT diagnostic
complex. The results of laboratory experiments
allowed us to identify the parameters of histological
analysis processes and gave us an opportunity to
develop and build a model of automated control
when designing a histological analyzer.
2 Materials and Methods
Methods of histological analysis, characteristics of
systems with distributed parameters, methods of
continuum mechanics, theory of automatic control,
methods of mathematical and simulation modelling
were used to substantiate methods and algorithms of
control of histological process.
3 Results of the study
3.1 Description of the technological process
To describe the technological process of moving the
biosphere during histological analysis, let us define
the spatial and temporal basis (l, t). Let us denote
conjugate variables characterizing the local place (l,
) on the technological route and instantaneous time
(t,) of histological analysis processing.
In the process of biomaterial passage on the
technological route, disturbances may occur due to
processing failure at technological operations
causing non-critical A(l, ) (Figure 2) and gross
errors leading to biomaterial sample loss C(, t)
(Figure 3).
Fig. 2: Uneven coloring
Fig. 3: Faulty staining leading to loss of histologic
section
Interference correction on the process route
will be by controlling the biomaterial flow q(l,t)
and the processing rate v(l,t) (Figure 4).
Fig. 4: Management system
We replace the Lagrangian coordinates of
biomaterial passage by Euler variables, in this case
we can use differential-integral calculus, in
particular the Green's function [10], to describe the
technological process.
As criteria for controlling the technological
process of histologic analysis, we can use the
indices of relative deviation of processing speed and
biomaterial flow:
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DOI: 10.37394/232020.2023.3.13
Sergey Kostarev, Tatyana Sereda
E-ISSN: 2732-9941
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Volume 3, 2023
,mindd,
;mindd,
0 0
0 0
k k
k k
t
t
l
l
t
t
l
l
tltlq
tltlv
(1)
Let us write down the biomaterial flow control
by the integral expression
,,,
0
1dttlqztlv k
t
t
(2)
The regulation of biomaterial movement control
on the technological route with respect to
biomaterial distribution density (l,t), let us write
down by the integral expression
2
1
,d,ρ
2
l
l
ltlztq
(3)
The system of equations characterizing the state
of biomaterial processing is described by the mass
conservation equation (Euler equation) taking into
account biomaterial rejects (losses) on the
technological route and the control equation [6]:
.,,,
,,,,
: conditions initialunder
,),(,
,
,,
00
0000
21
ltltqtlq
tqtlqttlll
dttlqlztztlAtlv
tlC
l
tlq
t
tl
kk
k
t
(4)
3.2 Synthesis of analytical solution
At known initial and boundary conditions system
(4) has an analytical solution based on the use of
Green's function (G):
t
t
l
l
tlGtlq
0 0
,dd,,,,,
(5)
where G(l, , t, ) is the transition function of the
system (Green's function);
-
,
standardizing function of external
influences.
The expression for the Green's function will take
the following form
.1exp
1
,,
lltl
d
tlG
The standardizing function has the form
.,1where
,,,ω
21 zz
txfxtqtldtl
(6)
In the one-dimensional (scalar) case the delta
function is described by the expression
.
)(
)(
kk
k
lf
ll
lf
When interference is given as a polynomial form
of the Dirac function
,
1
,2
l
t
tlС
(7)
.
1
,2
t
l
tlA
the solution of the problem of controlling the flow
and distributed density of biomaterial from the
failure causing rejection will be written by
expressions
,, 2
2
2
tl
z
tlq
(8)
,
1
,2
2
1
tl
z
tl
where
.,1 21 zz
The solution of the control problem taking into
account the technological failure causing time
delays on the operating line is described by the
equations
,arctg
2
),( 2
2
2
2
tl
t
tl
t
tlq
(9)
,arctg
2
,2
2
2
2
tl
t
tl
tl
tl
where
.,1 21 zz
The analytical expressions (8) and (9) express
the control of flow and distributed density of
biomaterial advancement taking into account the
interference caused by equipment failure on the
process route, which will contribute to the
development of control systems for automated
histological analysis processes.
3.3 Numerical modelling
In order to analyze the solution (9) of the set
problem and the formation of the control action,
numerical simulation was carried out at coefficients
z1 = 2, z2 = 1 (Table 1).
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DOI: 10.37394/232020.2023.3.13
Sergey Kostarev, Tatyana Sereda
E-ISSN: 2732-9941
92
Volume 3, 2023
Table 1. Numerical modelling
t
Stage of technological operation, l
1
2
3
4
5
6
7
8
9
1
-0.38
-0.71
-0.91
-1.04
-1.13
-1.19
-1.24
-1.28
-1.31
2
-0.19
-0.44
-0.63
-0.76
-0.87
-0.95
-1.02
-1.07
-1.11
3
-0.12
-0.29
-0.45
-0.58
-0.69
-0.77
-0.85
-0.92
-0.96
4
-0.08
-0.21
-0.34
-0.46
-0.56
-0.64
-0.72
-0.78
-0.83
5
-0.05
-0.16
-0.26
-0.37
-0.46
-0.54
-0.61
-0.68
-0.73
6
-0.04
-0.12
-0.21
-0.30
-0.38
-0.46
-0.53
-0.59
-0.65
7
-0.03
-0.09
-0.17
-0.25
-0.33
-0.39
-0.46
-0.52
-0.57
8
-0.02
-0.08
-0.14
-0.21
-0.28
-0.35
-0.42
-0.46
-0.51
9
-0.02
-0.06
-0.12
-0.18
-0.24
-0.30
-0.36
-0.41
-0.46
10
-0.02
-0.05
-0.10
-0.16
-0.21
-0.27
-0.32
-0.37
-0.42
Figure 5 shows the graph of biological media
flow control based on the calculations presented in
the Table1.
4 Conclusion
The use of mathematical physics methods allowed
to formalize the control system of biomaterial
movement on the technological route of histological
analyzer taking into account perturbations at
operations. The analytical solution of the system of
differential equations in deterministic formulation
formalizing the control model of histological
analysis is given. At introduction of the automated
line at histological analysis it is possible to realize
control functions for minimization of disturbances
caused by loss (rejection) of biomaterial and failure
of technological equipment causing time delays that
will improve the quality of morpho-structural
diagnostics of tissues at diagnosis.
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DOI: 10.37394/232020.2023.3.13
Sergey Kostarev, Tatyana Sereda
E-ISSN: 2732-9941
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Volume 3, 2023
quantities, Fractals, Vol. 5, No. 3, 1997, pp.
473-491.
Author Contributions:
Conceptualization and research, methodology and
formal analysis Sergey Kostarev; writing-reviewing
and editing Tatyana Sereda.
Sources of funding for research presented in a
scientific article:
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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DOI: 10.37394/232020.2023.3.13
Sergey Kostarev, Tatyana Sereda
E-ISSN: 2732-9941
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