=Paper=
{{Paper
|id=Vol-2864/paper4
|storemode=property
|title=Intelligent Big Data System Based on Scientific Machine Learning of Cyber-physical Systems of Medical and Biological Processes
|pdfUrl=https://ceur-ws.org/Vol-2864/paper4.pdf
|volume=Vol-2864
|authors=Vasyl Martsenyuk,Aleksandra Klos-Witkowska,Andriy Sverstiuk,Oksana Bahrii-Zaiats,Marcin Bernas,Krzysztof Witos
|dblpUrl=https://dblp.org/rec/conf/cmis/MartsenyukKSBBW21
}}
==Intelligent Big Data System Based on Scientific Machine Learning of Cyber-physical Systems of Medical and Biological Processes==
https://ceur-ws.org/Vol-2864/paper4.pdf
Intelligent Big Data system based on scientific machine
learning of cyber‐physical systems of medical and biological
processes
Vasyl Martsenyuka, Aleksandra Klos-Witkowskaa, Andriy Sverstiukb, Oksana Bahrii-Zaiatsb,
Marcin Bernasa, Krzysztof Witosa
a
University of Bielsko-Biala, Willowa St., 2, Bielsko-Biala, 43-300, Poland
b
I. Horbachevsky Ternopil National Medical University, Maidan Voli St., 1, Ternopil, 46002, Ukraine
Abstract
The work focuses on developing a Big Data system allowing us to process medical data
intelligently, which are ingested from different sources and of various types. It aims to design
biosensor devices with desired qualitative characteristics, namely their operational stability.
The article suggests the mathematical representation of the discrete population dynamics and
the dynamic logic of the studied models of the most advanced biosensors as biopixels arrays
which are used at the analytics stage of scientific machine learning. Application software for
the intelligent Big Data system for investigating the stability of cyber-physical systems for
medical purposes using the R package has been developed. The software package for the
intelligent Big Data system for investigating the stability of cyber-physical systems for
medical applications consists of the following main software modules: unit of ingesting and
processing Big Data used to identify input parameters of the model of cyber-physical
systems; the software module of research of dynamic behavior of cyber-physical systems; the
software module of research of dynamic logic of cyber-physical systems; the block of
decision-making on the stability of cyber-physical systems; the block of visualization, lattice
images of macrophage/monoclonal antibody data source, a database that receives lattice
images of macrophage binding to monoclonal antibodies and a database that receives images
of fluorescent biopixels. Using the software package, the results of computer simulation of
mathematical models of cyber-physical systems of medical and biological processes in the
form of images of macrophage, monoclonal antibodies, connections of macrophages with
monoclonal antibodies, fluorescent pixels, and an electrical signal from the converter are
obtained.
Keywords 1
cyber-physical system, Big Data, stability of the model, differential equations, difference
equations, rectangular lattice, hexagonal lattice
1. Introduction
The most popular areas of use of Big Data in medicine are diagnostics using medical images in
ophthalmology [1-3], oncology [4, 5], dermatology [6], cardiology [7]. These are the branches of
modern medicine where most images are used for diagnosis.
Over the last decade, we have seen a rapid growth of digital developments, including a variety of
sensors, microcontrollers, communication systems to meet societal needs. Thanks to the Internet of
Things (IoT) and the use of biosensors, digital electronic health services are increasingly used every
year. Biosensors play a critical role in the Internet of Things when it comes to eHealth. There are a
CMIS-2021: The Fourth International Workshop on Computer Modeling and Intelligent Systems, April 27, 2021, Zaporizhzhia, Ukraine
EMAIL: vmartsenyuk@ath.bielsko.pl (V. Martsenyuk); awitkowska@ath.bielsko.pl (A. Klos-Witkowska); sverstyuk@tdmu.edu.ua (A.
Sverstiuk); bagrijzayats@tdmu.edu.ua (O. Bahrii-Zaiats); mbernas@ath.bielsko.pl (M. Bernas); kwitos@ath.bielsko.pl (K. Witos)
ORCID: 0000-0001-5622-1038 (V. Martsenyuk); 0000-0003-2319-5974 (A. Klos-Witkowska); 0000-0001-8644-0776 (A. Sverstiuk); 0000-
0002-5533-3561 (O. Bahrii-Zaiats); 0000-0002-0099-1647 (M. Bernas); 0000-0001-7959-7329 (K. Witos)
© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
number of sensors available on the market that help people monitor their daily fitness, blood glucose
levels, and many sensible applications of primary diagnostics at home. In [8] the problems of safe and
confidential use of the most modern biosensors for the Internet of Medical Things are considered.
The architecture of intelligent monitoring of electronic health of patients with chronic diseases is
presented in [9]. Elements of the architecture include portable devices, biosensors, and smartphones
for collecting medical and biological indicators [10]. The intelligent system uses scientific machine
learning (SciML) as a tool for analytics in the context of complex applications across science,
engineering, and medicine [11, 12]. It focuses on Big Data, obtained from various hospitals, and data
obtained from the patient, to diagnose and generate warnings of critical and crisis situations. The
system of health monitoring using prognostic calculations using Big Data analysis is presented in
[13]. The use of Big Data in endoprosthesis is considered in [14]. Google works with US clinics to
track searches for the COVID-19 pandemic and to predict in which region the outbreak will occur
[15].
In 2013, thanks to an electronic medical card, which stored all the information about the client,
scientists established a relationship between brain degradation and diabetes. Scientists from the
University of Cape Town, thanks to a correct analysis using Big Data, analyzed the most common
types of cancer and found that malignant tumors of the intestine, lung and ovary have clear genetic
markers. Medical Big Data helps prevent the development of the disease at an early stage through the
analysis of cardiovascular pressure, pulse, respiration and blood sugar [16, 17].
A few years ago, Apple and IBM joined forces to give IBM Watson even more structured
information from the iPhone and Apple Watch. One of the most promising vectors of Big Data
development in medicine is surgery [18, 19], where the relevant systems analyze all available
protocols of operations in order to assess the possible risks of surgical interventions in the form of
visualization in different colors. At home, Americans simply connect a stethoscope to the iPhone and
send the data to their doctor. At the same time, a large cluster collects information on all patients. It is
sorted by criteria and then looks for dependencies, confirm or refute myths.
Nowadays, the integration between physical resources and computing leads to the creation of
complicated computing systems with distributed parameters. Such systems are controlled or managed
by integrated into the Internet computing resources[20, 21]. In recent years, we have seen an interest
in cyber-physical systems (CPS), which are tools for controlling and monitoring the studied indicators
using computer technology, in which software is closely related to physical objects. CPS for medical
applications are used to measure and analyze biochemical parameters in biological fluids, detection of
cancer, markers of heart failure, pathogenic bacteria, to determine the level of contamination of food
and the environment. In recent years, CPSs have been used effectively, in which the function of
selective elements is performed by biosensors. Such systems are called cyber-physical biosensor
systems (CPBSS), which have a number of advantages: high selectivity, stability, performance,
affordable cost, the ability to use a wide range of users.
As a result of the use of CPBSS, Big Data is obtained, which requires processing and interrelation
with the measured medical and biological parameters. The analytical analysis of the obtained data in
the form of useful knowledge for use by software services that control the studied systems is carried
out.
For Big Data analysis, SciML is used as a data-driven method for studying computer algorithms
that are automatically improved through experience [22, 23].
In CPS, “experience” is usually presented in the form of data, and SciML allows you to create
mathematical models based on sample data for forecasting and decision making. SciML has made
significant strides in a variety of fields, including computer vision, language recognition, and control
systems, as well as related scientific fields in physics, chemistry, and biology [24].
Processing data ingested from CPS of medical and biological processes is one of the important
solutions which fits the “5 V’s of Big Data” characterization of Big Data [25]. The volume of such
type of data grows drastically. So, the amount of healthcare data gathered from 2013 to 2020 has
grown 15 times to 2314 exabytes [26]. The functioning of most CPS in medical applications is in real-
type mode, corresponding to the requirements of velocity for BigData. Data sources used for medical
CPS varies from numerical and literal signals (ECG, EEG, EMG, etc.) to medical images (CT, MRT,
USG, etc.), which fits a variety of data sources. The veracity of medical CPS data is assured by the
quality of standardized measurements used. Finally, the Value of Big Data solutions within medical
data-driven-CPS is supported by new results on diagnostics, treatment, and prophylaxis, which can be
exceptionally obtained using SciML.
Algorithms and methods for processing measurement parameters in the CPS of medical and
biological processes using Big Data are created on the basis of their mathematical models.
Mathematical models of biosensors for CPBSS on rectangular lattice using lattice differential
equations with delay have been developed [27, 28]. The mathematical models of biosensors for
CPBSS on rectangular lattice using lattice difference equations with time delay are developed in the
article [29]. A detailed description of the considered model is presented in [30, 31].
When developing the modern CPS with the desired qualitative characteristics we need to process
the medical information of the different kinds, which corresponds to the notion of Big Data. Namely,
the medical data fit the main features of Big Data such as variety, value, velocity, and veracity.
Medical data in CPBSS come from different sources and types, so in their operation it is necessary
to ensure independence and operational stability. Therefore, there is a need to develop a software
package for the intelligent system BigData in the study of the sustainability of medical CPS. These
studies are particularly relevant for autonomy, which is defined as the improvement of enzyme
activity, protein when maintained under certain conditions, as well as operational stability to ensure
the performance of CPS when measuring biomedical parameters. The usefulness of the studied
systems is determined by the stability of the sensitive element, which is located in the receptor, and
the functioning of the corresponding components of the matrix of the system during its use.
The purpose of the study: to develop a software package for the intelligent system BigData in the
study of the stability of medical CPS.
1. Mathematical models of biosensors of the intelligent Big Data system
for medical aplications
1.1. Mathematical model of biosensor on rectangular lattice using
differential equations with time delay
In models of lattice differential equations, the spatial structure has a discrete character and lattice
dynamics is widely used in the examples considered in [30, 31] and the mathematical model of the
immunosensor, which characterizes the change in the concentration of macrophages M i , j (t ) and
monoclonal antibodies Ai , j (t ) was substantiated:
dM i , j (t )
( - Ai , j (t - ) - M i , j (t - )) M i , j (t ) S M i , j ,
∧
dt (1)
dAi , j (t )
(- f M i , j (t - ) - f Ai , j (t )) Ai , j (t ), t 0,
dt
where - the speed of detection of macrophages;
- birth rate constant of macrophages;
- time delays in the formation of the macrophage-monoclonal antibody complex;
M - the average rate of decrease in the concentration of macrophages;
A - the average rate of decrease in the concentration of monoclonal antibodies;
- probabilistic rate of formation of the macrophage-monoclonal antibody complex;
- distance between immunopixels;
D - macrophage diffusion coefficient;
∧
{ }
S M i , j - operator of diffusion processes.
The model of the immunosensor (1) using lattice differential equations most fully takes into
account all parameters of immunosensory systems, namely: to take into account when measuring the
concentration of macrophages M i , j (t ) and monoclonal antibodies Ai , j (t ) and their binding properties,
constant fertility of macrophages, time delay formation of a macrophage-monoclonal antibody
complex; the average rate of decrease in the concentration of macrophages and monoclonal
antibodies; probabilistic rate of formation of the macrophage-monoclonal antibody complex; the
distance between the immunopixels; macrophage diffusion coefficient; diffusion process operator.
These parameters are used by immunosensory systems.
Model (1) is given by initial functions (2)
M i , j (t ) M i0, j (t ) ≥ 0, Ai , j (t ) Ai0, j (t ) ≥ 0,
(2)
t ∈[- ,0), M i , j (0), Ai , j (0) 0.
For a square array N × N , the discrete diffusion is used for the spatial operator, taking into account
the imbalance constant ndsbn
D-2 M 1, 2 M 2,1 M i , j -1 - 2ndsbn M 1,1 , i, j 1
-2
D M 2, j M 1, j -1 M 1, j 1 M i , j 1 - 3ndsbn M i , j , i 1, j ∈2, N - 1
-2
D M 1, N -1 M 2, N - 2ndsbn M 1, N , i, j ∈2, N - 1
-2 (3)
D M i -1, N M i 1, N M i , N -1 - 3ndsbn M i , N , i ∈2, N - 1, j N
S M i , j D- 2 M N -1, N M N , N -1 - 2ndsbn M N , N , i N , j N
∧
-2
D M 00 N -1, j M N , j -1 M N , j 1 M i , j 1 - 3ndsbn M N , j , i N , j ∈2, N - 1
D M N -1,1 M N , 2 - 2ndsbn M N ,1 , i N , j 1
-2
D- 2 M M
i 1,1 M i , 2 - 3ndsbn M i ,1 , i ∈2, N - 1, j 1
i -1,1
D- 2 M i -1, j M i 1, j M i , j -1 M i , j 1 - 4ndsbn M i , j , i, j ∈2, N - 1.
Each pixel is exposed to macrophages from four adjacent pixels that are separated by equal
distances .
The boundary condition M i , j 0 for the edges of the array i, j 0, N 1 is used.
Figure 1: A rectangular grid that connects four adjacent pixels in the biosensor model using a
Cartesian coordinate system
1.2. Mathematical model of biosensor on hexagonal lattice using lattice
difference equations with time delay
For the intelligent Big Data system of medical and biological processes dynamics, we use the
mathematical description with the help of nonlinear difference equations with delay.
The model of the biosensor on the basis of a hexagonal lattice is considered. In this case, for the
numbering of biopixels (i, j , k ) , i, j, k N , N , i j k 0 the cubic coordinate system is used [32,
33].
Let M i , j , k (t ) is the concentration of macrophages, Ai , j , k (t ) is the concentration of monoclonal
antibodies in the biopixel (i, j , k ) ; i, j, k N , N , i j k 0 .
In the model it is used next parameters:
0 - the speed of detection of macrophages;
- birth rate constant of macrophages;
- time delays in the formation of the macrophage-monoclonal antibody complex;
M - the average rate of decrease in the concentration of macrophages;
A - the average rate of decrease in the concentration of monoclonal antibodies;
- probabilistic rate of formation of the macrophage-monoclonal antibody complex;
0 - distance between immunopixels;
D - macrophage diffusion coefficient;
∧
{ }
S M i , j - operator of diffusion processes.
When colonies of monoclonal antibodies are absent, colonies of macrophages are regulated by a
logistic equation with time delay:
M i , j ,k ( n 1) (1 v M i , j ,k (n r ))M i , j ,k (n) , (4)
where and – positive numbers, and r 0 mean latency of the negative responce of the
macrophages’ colonies.
The diffusion of macrophages from six adjacent pixels is considered (i 1, j , k 1) , (i 1, j 1, k ) ,
(i, j 1, k 1) , (i 1, j, k 1) , (i 1, j 1, k ) і (i, j 1, k 1) (Fig. 2) with diffusion speed D 2 .
Fig. 2. Hexagonal lattice, which binds six neighboring pixels in the model of the biopixels using the
cubic coordinates:
1, 3, 5, 8, 9, 11 - D M (t ) ; 2 - 2 M i 1, j , k 1 (t ) ;
D
i , j ,k
2
4 - D M (t ) ; 6 - 2 M i , j 1, k 1 (t ) ; 7 - 2 M i 1, j , k 1 (t ) ; 10 - 2 M i 1, j 1, k (t ) ; 12 - 2 M i , j 1, k 1 (t ) .
D D D D
i 1, j 1, k
2
The mathematical model of late- macrophage - monoclonal antibody interaction for a hexagonal
array of biopixels using based on the well-known Marchuk model [34-36] and uses the spatial
operator S proposed in [37].
M i , j , k ( n 1) M i , j , k ( n) exp - Ai , j , k n - r - M i , j ,k n - r Sˆ M i , j ,k ( n),
(5)
Ai , j ,k ( n 1) Ai , j ,k ( n) exp- f M i , j ,k ( n - r ) - f Ai , j , k ( n), n 0
where S M i , j , k is a discrete diffusion for a spatial operator S .
∧
M i , j ,k (n) M i0, j ,k (n) ≥ 0, Ai , j ,k (n) Ai0, j ,k (n) ≥ 0,
(6)
n ∈[-r ,0), M i , j ,k (0), Ai , j ,k (0) 0.
2. Dynamic logical simulation of intelligent Big Data system for
investigation the stability of CPS of medical and biological processes.
In order to simulate the dynamic logic of an intelligent Big Data system for investigation the
stability of CPS of medical and biological processes, the syntax which is proposed by A. Platzer for
the general CPS [38, 39]it is used. The CPS uses the Hybrid Programming Language (HP), which has
more features than differential equations. Consider the dynamic logical simulation of intelligent Big
Data system for investigation the stability of CPS for medical applications on the example of the
mathematical model of biosensor on hexagonal lattice using lattice difference equations with time
delay. Dynamic program is the first level of HP and it is defined by the following grammar:
a :: M i , j ,k (n 1) M i , j ,k (n) exp Ai , j ,k (n r ) x Ai , j ,k (n r ) Sˆ M i , j , k (n) ,
(7)
Ai , j ,k (n 1) Ai , j ,k (n) exp f M i , j ,k (n r ) y Ai , j ,k (n) & t .
where t is an evolutionary domain constraint in the form of a formula for the logic of the first order
of real arithmetic
def
t M min Vi , j ,k (n) M max
(8)
Amin Ai , j ,k (n) Amax i, j, k N , N n 0, i j k 0
The functioning of the biopixel (i, j , k ) is determined by two states, with respect to
fluorescence. Namely, s f l is a state of fluorescence and snon f l is one of the non-fluorescence states.
The use of the first order of semantics of logic and the satisfaction ratio s L for the first-order
formula L of real arithmetic and state s can be determined for some pixels
i, j, k ; i, j, k N , N , i j k 0 states s f l and snon f l as
s ft k fl M i , j ,k (n) Ai , j ,k (n) ≥ fl ,
(9)
snonfl k fl M i , j ,k (n) Ai , j ,k (n) fl
Discrete changes occur in computer programs when they accept new values for variables. This
situation occurs when a fluorescence phenomenon occurs in a pixel i, j , k ; i, j , k N , N ,
i j k 0 . The state s f l ,i , j ,k : 1 is assigned a value of 1 to the variable s f l ,i , j ,k . This leads to a
discrete, jump-like change, as the value s f l ,i , j ,k does not change smoothly, but rapidly when it
suddenly changes from 1 to s f l ,i , j ,k , causing a discrete jump of values s f l ,i , j ,k . In this way, we obtain
a discrete model of change s f l ,i , j ,k : 1 , except for the model of change (9).
3. Software complex of the intelligent Big Data system for investigation the
stability of CPS of medical and biological processes
The software package for the intelligent Big Data system for investigation the stability of CPS of
medical and biological processes consists of the following main software modules: unit of ingesting
and processing Big Data used to identify input parameters of the model of CPS, the software module
of research of dynamic behavior of CPS, the software module of research of dynamic logic of CPS,
the block of decision-making on stability of CPS, the block of visualization, lattice images of
macrophage/ monoclonal antibody data source, a data source that receives lattice images of
macrophage binding to monoclonal antibodies and a database that receives images of fluorescent
biopixels.
A software package for the study of CPS phase diagrams using the R package (http://www.r-
project.org/) has been developed [40]. The block diagram of the software package of the intelligent
Big Data system for investigation the stability of CPS of medical and biological processes is shown in
Fig. 4.
Unit for simulation Unit for obtaining
Unit of ingesting and of lattice images of lattice images of
processing Big Data antigens / antigen-antibody
used to identify
antibodies bonds
input parameters of
the model of CPS
Software module
for studying the Software module for
dynamic behavior studying the discrete
of CPBSS dynamics of CPBSS
Unit for obtaining Unit for obtaining Unit for receiving
bifurcation images of an electrical signal
diagrams fluorescent from the converter
biopixels
Phase diagrams The decision-
study unit Visualization unit making unit on the
stability of CPBSS
Data source of Data source of
Data source of
lattice images of fluorescent
lattice images of
antigens / biopixel images
antigen-antibody
antibodies
binding
Figure 4: Block diagram of the software package of the intelligent Big Data system for investigation
the stability of CPS of medical and biological processes
The software module for studying the dynamic behavior of CPS consists of blocks for obtaining
bifurcation and phase diagrams. Using software to study the stability of intelligent Big Data system
for investigation the stability of CPS of medical and biological processes, obtained the results of
computer simulation in the form of bifurcation and phase diagrams of macrophage populations
relative to monoclonal antibodies, lattice images of macrophages, monoclonal antibodies,
probabilities of macrophage binding to monoclonal antibodies and monoclonal antibodies, the signal
from the converter. The software module for studying the dynamic logic of CPS consists of blocks for
modeling lattice images of macrophages/ monoclonal antibodies, block for obtaining lattice images of
macrophage binding to monoclonal antibodies, obtaining images of fluorescent biopixels and the
electrical signal from the converter. The study of CPS stability of medical and biological processes
based on the Big Data intelligent system is an important data source: lattice images of macrophage/
monoclonal antibody data source, a data source that receives lattice images of macrophage binding to
monoclonal antibodies, and a database that receives images of fluorescent biopixels.
4. Computer simulation of mathematical models of biosensors using an
intelligent Big Data system for investigation the stability of CPS for medical
applications
4.1. Parameters of mathematical models of biosensors for investigation the
stability of CPS for medical applications
To study the occurrence of bifurcation and deterministic chaos in compartmental mathematical
models of lattice type on a rectangular lattice using differential equations, models (1), (4) at N 4
ml
and values of parameters 2 min 1 , 2 , f 1 min 1 , 0 .8 / ,
minꞏmkg
2
0 .5
ml , f 0.5 ml , D 0.2 nm , 0.3nm are considered. The concentrations
min mkg min mkg min
of macrophages populations Vi, j (t ) , Vi , j , k (t ) and monoclonal antibodies populations Fi, j (t ) ,
Fi, j , k (t ) are measured in mkg .
ml
4.2. Results of computer simulation of mathematical model of biosensor on
rectangular lattice using lattice differential equations with delay
The computer simulation was implemented for different values . The long-term behavior of the
model (1) – (3) for 0.05 , = 0.22 , = 0.2865 with a set of parameter values, which are presented
above is analyzed. Qualitative changes in the behavior of biopixels and the intelligent Big Data
system model for the investigation of the stability of the CPS of medical and biological processes are
observed.
Figures 5 and 6 show the result of computer simulation of the discrete dynamics of the intelligent
Big Data system for the investigation of the stability of CPS of medical and biological processes in
the form of lattice images of macrophages and monoclonal antibodies in pixels of the studied system.
Figures 5 (a) and 6 (a) show the results of computer simulation of lattice images of macrophages
and monoclonal antibodies in the pixels of the system (1) at = 0.05 , which corresponds to a stable
focus. For = 0.22 there is a less pronounced (Fig. 5 (b) and Fig. 6 (b)), and for = 0.2865 more
pronounced traveling wave of monoclonal antibodies, which is presented in Figures 5 (c) and 6 (c).
а) b) c)
Figure 5: Lattice images of macrophages in pixels of system (1) at = 0.05 (a), = 0.22 (b),
= 0.2865 (c)
а) b) c)
Figure 6: Lattice images of monoclonal antibodies in pixels of system (1) at 0.05 (a), = 0.22 (b),
= 0.2865 (c)
In the second stage of computer modeling of intelligent Big Data system for investigation of the
stability of CPS of medical and biological processes lattice graphs are used on which for each pixel
the probability of contact of macrophages with monoclonal antibodies, as Vi , j × Fi , j at = 0.05 ,
= 0.22 , = 0.287 is presented. They are shown in Figures 7 (a - c).
а) b) c)
Figure 7: Lattice images of connections of macrophages with monoclonal antibodies in pixels of
system (1) at = 0.05 (а), = 0.22 (b), = 0.2865 (c)
Graphs of fluorescent pixels are shown in Figures 8 (a - c).
а) b) c)
Figure 8: Fluorescence image of system (1) as a result of numerical simulation at = 0.05 (а),
= 0.22 (b), = 0.2865 (c)
Figure 8 (a) shows the result of numerical simulation of system (1) at = 0.05 , which
corresponds to a stable focus. For = 0.22 there is a less pronounced (Fig. 8 (b)), and for
= 0.2865 it is more pronounced traveling wave of fluorescent pixels, which is shown in Figure 8
(c). Figure 8 (c) shows the result of numerical simulation of system (1), which corresponds to the
approach to the limit cycle (there is a traveling wave of fluorescent pixels). In the case = 0.2865,
chaotic behavior is observed, which begins with wave-like changes in fluorescent pixels (Fig. 8 (c))
and quickly progresses to chaotic changes. Figure 8 (c) shows the result of numerical simulation of
system (1), in which a chaotic wave of fluorescent pixels is observed.
To control the measurement process in the intelligent Big Data system for investigation of the
stability of the CPS for medical applications, the result of computer simulation of the electrical signal
from the converter (Fig. 9 (a, b)), which characterizes the number of fluorescent pixels at values of
= 0.05 , = 0.2865 plays an extremely important role. Analyzing the type of electrical signal in
Figure 9, we see that when the value changes qualitatively changes the behavior of the pixels and
the whole intelligent Big Data system to investigate the stability of the CPS for medical and biological
applications. In Figure 9 (a) there is a steady state at = 0.05 . In Figure 9 (b) ( = 0.2865) a
traveling wave of non-fluorescent pixels is clearly visible.
а) b)
Figure 9: Electrical signal from the converter, which characterizes the number of fluorescent pixels at
= 0.05 (a), = 0.2865 (b)
As shown by numerical analysis, fluorescent states in biopixels change according to the laws of
discrete dynamics. The threshold value for fluorescence is fl = 1,5 . Taking into account the
continuous dynamics of the immunological response, each biopixel is considered as a CPS.
4.3. Results of computer simulation of mathematical model of biosensor on
hexagonal lattice using lattice difference equations with delay
The long - term behavior of model (4) at r = 5 , r = 17 , r = 22 with a set of parameter values,
which are presented above (Fig. 10 - 14), is analyzed. We observe qualitative changes in the behavior
of biopixels and the biosensor model on hexagonal lattice using lattice difference equations with delay
in general. Figures 10 and 11 show the first stage of computer modeling of discrete dynamics of an
intelligent Big Data system for investigation of the stability of CPS of medical and biological
processes in the form of lattice images of macrophages and monoclonal antibodies in pixels of the
studied system. Figure 10 (a) shows the result of numerical simulation of system (4) at r = 5 , which
corresponds to a stable focus. For r = 17 there is less pronounced (Fig. 10 (b)), and for r = 22 more
pronounced wavy changes in the lattice images of macrophages and monoclonal antibodies in the
pixels of the system (4), as shown in Figures 10 (c).
а) b) c)
Figure 10: Lattice images of macrophages in pixels of system (4) at r = 5 (a), r = 17 (b), r = 22 (c)
а) b) c)
Figure 11: Lattice images of monoclonal antibodies in pixels of the system (4) at r = 5 (a), r = 17 (b),
r = 22 (c)
In the second stage of computer modeling of the intelligent Big Data system, lattice graphs were
used to investigate the stability of the CPS of medical and biological processes. Firstly, the
corresponding graphs are constructed, on which for each pixel the image of probability of contact of
macrophages with monoclonal antibodies, as Vi , j , k × Fi , j , k at r = 5 , r = 17 , r = 22 is presented in
Figure 12 (a - c).
а) b) c)
Figure 12: Lattice images of macrophage‐ monoclonal antibody binding in pixels of system (4) at
r = 5 (a), r = 17 (b), r = 22 (c)
Figure 12 (a) shows the result of numerical simulation of system (4) at r = 5 , which corresponds
to a stable focus. For r = 17 there is less pronounced (Fig. 12 (b)), and for r = 22 more pronounced
wavy changes in the images of the probability of contact of macrophages with monoclonal antibodies
of system (4), as shown in Figure 12 (c).
In the third stage of computer modeling of the intelligent Big Data system for investigation of the
stability of CPS of medical and biological processes, lattice graphs of fluorescent pixels were
obtained based on the fulfillment of condition (6), which are shown in Figures 13 (a - c).
Figure 13 (a) shows the result of numerical simulation of system (4) at r = 5 , which corresponds
to a stable focus. For r = 17 there is a less pronounced (Fig. 13 (b)), and for r = 22 - it is more
pronounced traveling wave of fluorescent pixels of the system (4), which is shown in Figure 13 (c).
а) b) c)
Figure 13: Fluorescence image of the system (4) as a result of numerical simulation at r 5 (a),
r 17 (b), r 22 (c)
Hopf bifurcation is observed with increasing time delay [41]. Figure 13 (c) shows the result of
numerical simulation of the system (4) at r = 17 , which corresponds to the limit cycle (there is a
traveling wave of fluorescent pixels). In the case r = 22 , chaotic behavior is observed, which begins
with wave-like changes in fluorescent pixels (Fig. 13 (c)) and quickly turns into chaotic changes.
Figure 14 shows the electric signal from the converter (Fig. 14), which characterizes the number of
fluorescent pixels at r = 17 , r = 22 .
а) b)
Figure 14: Electrical signal from the converter, which characterizes the number of fluorescent pixels
at r = 17 (a), r = 22 (b)
To control the measurement process using an electrical signal to an intelligent big data system for
investigation of the stability of the CPS of medical and biological processes is extremely important
the result of computer simulation.
Analyzing the type of electrical signal in Figure 14, we can conclude that changing the value
qualitatively changes the behavior of the pixels and the entire biosensor model based on lattice
difference equations. In Figure 14 (a) ( r = 17 ) occurs, and in Figure 14 (b) ( r = 22 ) there is a
traveling wave of non-fluorescent pixels. The threshold value for fluorescence is fl = 1,5 .
A two-dimensional array of biopixels was used to develop the dynamic logic of an intelligent Big
Data system to investigate the stability of the CPS of medical and biological processes on a hexagonal
lattice using delay equations. As shown by the results of numerical analysis, fluorescent states in
biopixels change according to the laws of discrete dynamics.
5. Conclusion
In the work, the general scheme of the cyber-physical sensor system proposed in [42] was
extended to the usage of Big Data. Peculiarities of immunosensors functioning are taken into account
in CPS mathematical models. According to the laws of discrete dynamics, lattice images in
immunopixels are modified. The proposed mathematical models take into account the interaction of
immunopixels through the diffusion of macrophages.
The intelligent Big Data system uses the discrete dynamics of macrophages and antibodies, which
is based on dynamic logic, to study the stability of CPS in medicine. The classes of rectangular lattice
using differential equations or hexagonal lattice using difference equations are used in the work as
modeling of the interaction of macrophages and monoclonal antibodies in immunopixels. This also
takes into account the presence of macrophage and monoclonal antibodies colonies, that are localized
in pixels, as well as the diffusion of macrophage colonies between pixels using the R package.
According to a series of experiments of intelligent Big Data system for investigation of the
stability of CPS of medical and biological processes using lattice differential equations with delay it is
established that in the case of using a rectangular lattice at constant time delay ∈[0, 0.22] the
solution of the studied system is a stable focus. When 0.23 (in the case of a rectangular lattice)
there is a Hopf bifurcation and all subsequent trajectories correspond to stable boundary cycles for all
imunopixels. With a further increase in the delay constant , chaotic behavior of the CPS of medical
processes occurs on the basis of lattice differential equations with time delay.
According to the results of a series of experiments of intelligent Big Data system for investigation
of the stability of CPS of medical processes using lattice difference equations with delay it is
established that in case of using hexagonal lattice at r ∈[0, 16] the solution of the investigated system
tend to endemic states, which are a stable focus. For r 17 (in the case of a hexagonal lattice) there is
a Hopf bifurcation and all subsequent trajectories correspond to stable boundary cycles for all
imunopixels. With a further increase in the delay constant r , chaotic behavior of the CPS occurs on
the basis of lattice difference equations.
The developed computer programs for the study stability of intelligent Big Data system for
investigation the stability of CPS of medical processes in the form of phase diagrams, lattice images
of macrophages / monoclonal antibodies, lattice images of macrophage binding to monoclonal
antibodies, images of fluorescent imunopixels and the electrical signal from the converter should be
used in research, design organizations, medical and laboratory centers in the development and testing
of cyber-physical systems of medical processes.
In the following researches is necessary to offer mathematical models of intelligent systems of
Big Data on the basis of other geometrical structures and to carry out research of their stability.
6. Aknowledgements
The work was co-funded by the European Union's Erasmus + Programme for Education under
KA2 grant (project no. 2020-1-PL01-KA203-082197 “Innovations for Big Data in a Real World”)
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