=Paper=
{{Paper
|id=Vol-2623/paper4
|storemode=property
|title=Modification of the Software System for the Automated Determination of Morphological and Rhythmic Diagnostic Signs by Electrocardio Signals
|pdfUrl=https://ceur-ws.org/Vol-2623/paper4.pdf
|volume=Vol-2623
|authors=Serhii Lupenko,Iaroslav Lytvynenko,Nataliia Stadnyk,Halyna Osukhivska,Natalia Kryvinska
|dblpUrl=https://dblp.org/rec/conf/intelitsis/LupenkoLSOK20
}}
==Modification of the Software System for the Automated Determination of Morphological and Rhythmic Diagnostic Signs by Electrocardio Signals==
https://ceur-ws.org/Vol-2623/paper4.pdf
Modification of the Software System for the Automated
Determination of Morphological and Rhythmic
Diagnostic Signs by Electrocardio Signals
Serhii Lupenko1[0000-0002-6559-0721], Iaroslav Lytvynenko1[0000-0001-7311-4103],
Nataliia Stadnyk1[0000-0002-7781-7663], Halyna Osukhivska 1[0000-0003-0132-1378],
Natalia Kryvinska2[0000-0003-3678-9229]
1 Ternopil Ivan Puluj National Technical University, Ternopil, Ukraine
2 University of Vienna, Vienna, Austria; Comenius University, Bratislava, Slovakia
lupenko.san@gmail.com
iaroslav.lytvynenko@gmail.com
natalya.stadnik15@gmail.com
osukhivska@tstu.edu.ua Ukraine
natalia.kryvinska@univie.ac.at
Abstract. In this paper we consider a modernized of the software system for the
automated determination of morphological and rhythmic diagnostic signs of
electrocardio signal. The modification of the system is to create a new method
and appropriate software that involves processing electrocardio signals by re-
ducing the discrete cyclic random process, as a model of electrocardio signal to
isomorphic random periodic sequence. The use of a new mathematical model of
electrocardio signals in the form of conditional discrete cyclic random process
allowed to take into account and carry out automatic determination of both
morphological and rhythmic diagnostic sings of electrocardio signal within the
same mathematical model. The use of a new method of statistical processing
based on the new model, allowed to obtain statistical characteristics that are in-
fomative diagnostic signs (morphological and signs of rhythm) of the electro-
cardio signal. The application of the method of reducing a discrete cyclic ran-
dom process to isomorphic random periodic sequence before the procedures of
statistical processing of the electrocardio signal, in particular before obtaining
morphological and rhythmic sings allowed to increase the speed of automated
processing of the electrocardio signal in comparison with the previously devel-
oped methods which were based on it model in the form of a cyclic random
process and did not account for the double stochastic model.In the structure of
the modified software system, after the evaluation of the rhythmic structure of
the electrocardio signal and the procedure of reduction to a random periodic se-
quence of a discrete cyclic random process, its processing is branched into two
parallel stages. The first stage carries out the morphological analysis, which in-
volves the statistical processing of the electrocardio signal, normalization of sta-
tistical estimates and their distribution in the Chebyshev base and decision mak-
ing on the obtained morphological sings.
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0). IntelITSIS-2020
Keywords: Software System, Electrocardio Signal, Analysis Of Heart Rhythm,
Morphological Analysis Of Electrocardio Signals.
1 Introduction
Known processes and phenomena of reality are those that reflect over a repeating
structure in time. Electrocardio signals are known and well-studied among such pro-
cesses, signals. The current state of information technology development allows to
approach the processing of such signals in a new way and solve the advanced tasks of
modern medicine efficiently in the construction of diagnostic apparatus by creating
new effective mathematical means which make it possible to increase the accuracy
and informativeness of the processing of cyclic signals, in particular electrocardio
signals [1-12]. Therefore, the development of diagnostic software systems for auto-
mated diagnosing the human heart condition by registered electrocardio signals is a
relevant scientific and technical task, the solution of which will enable to improve the
quality and efficiency of diagnosing of the functional state of the heart and cardiovas-
cular system of the human body as a whole.
2 Related works
New mathematical models and methods of processing cyclic electrocardio signals
have been developed and substantiated in papers [13, 14]. The use of new mathemati-
cal models has permitted to improve the accuracy and reliability of diagnosing the
functional state of the heart by increasing the informativeness of morphological anal-
ysis and heart rhythm analysis. The software system for modeling and conducting
morphological analysis, electrocardio signal analysis of heart rhythm on the basis of
the developed mathematical models is presented in paper [15]. This system has been
modernized and embodied newly developed methods for the electrocardio signals
processing.
3 Overview of the Research
Increasing the speed of methods of statistical processing of electrocardio signals,
along with increasing the informativeness of their automated analysis is achieved
through the use of a new mathematical model that reflects the double stochasticity of
the investigated electrocardio signals (stochastic morphological and rhythmic struc-
tures) and applying a new method of reducing a discrete cyclic random process to an
isomorphic random periodic sequence.
This paper is devoted to the improvement of the software system, where in contrast
to the previous development [13, 14], the system includes the new method that pro-
vides the processing of electrocardio signals by reducing to an isomorphic random
periodic sequence of a discrete cyclic random process, as a model of electrocardio
signal, which allowed to increase the speed of its processing in comparison with the
previously developed methods by reducing the computational complexity of known
statistical methods of estimating the probabilistic characteristics of cyclic random
processes of a discrete argument.
4 Proposed model
The purpose of the research is to modernize the software system for morphological
analysis and heart rhythm analysis with increased informativeness, on the account of
using a new mathematical model of heart signals, in the form of a conditional discrete
circular random process, and a new statistical processing method, namely the method
of reducing a discrete cyclic random process to an isomorphic random periodic se-
quence, that by reducing the computational complexity of the electrocardiographic
signal processing method, made it possible to speed up their processing in computer
cardiac diagnostic systems compared to previously developed methods.
5 Results & Discussion
Here are presented the basic mathematical relations that underlie mathematical sup-
port in the modified software system. Still, we will focus on the developed part of the
complex, which relates to the reduction to an isomorphic random periodic sequence of
a discrete cyclic random process, as a model of the electrocardio signal. The software
system enables the automated analysis of the electrocardio signal, particularly its
morphological and rhythmic signs.
The mathematical model of electrocardio signal in the form of a cyclic random
process and conditional cyclic random process.
It is known from [13] that a discrete random process: {ξ (ω , tml ), ω ∈ Ω, tml ∈ D} is
called a cyclic discrete random process if there is such a discrete function T (t ml , n )
(rhythm function) that satisfies the conditions: 1) T (t ml , n ) > 0 , if n > 0 ; 2)
T (t ml , n ) = 0 , if n = 0 ; 3) T (t ml , n ) < 0 , if n < 0 ; 4) for any t m1l1 ∈ D and t m2l2 ∈ D ,
for which t m2l2 > t m1l1 , for function T (t ml , n ) inequality
( ) ( )
t m1l1 + T t m1l1 , n < t m2l2 + T t m2l2 , n , ∀n ∈ Z is applied; that finite-dimensional vectors (
ξ (ω , t m1l1 ) , ξ (ω , t m2l2 ) ,..., ξ (ω , t mk lk ) ) and ( )
( ξ (ω , t m1l1 + T t m1l1 , n ) ,
( )
ξ (ω , t m2l2 + T (t m2l2 , n )) ,..., ξ (ω , t mk lk + T t mk lk , n ), n ∈ Z , at all integer k ≥ 1 is sto-
chastically equivalent in a broad sense.
The method of reducing the statistical processing (estimation, analysis, prediction)
of a cyclic random process of a discrete argument to the corresponding statistical
processing of an isomorphic periodic random sequence consists in gradual execution
of the following steps:
1) transformation of a ω -realization ξ1ω (t ml ), ω ∈ Ω, t ml ∈ D of a cyclic random
process ξ1 (ω , t ml ), ω ∈ Ω, t ml ∈ D in a ω -realization ξ 2ω (i ), i ∈ Z of an isomorphic
(for the process) in relation to order and values of a L -periodic sequence
ξ 2 (ω , i ), ω ∈ Ω, i ∈ Z , by the action of a scale transformation operator G y (t ml ) {}
⋅ with
a scale transformation function y (t ml ) = L ⋅ (m − 1) + l ;
2) application of known methods of processing periodic random sequences and ob-
taining their results (statistical point and interval estimates of certain probabilistic
characteristics);
3) obtaining statistical estimates of the probabilistic characteristics of a cyclic ran-
dom process ξ1 (ω , t ml ), ω ∈ Ω, t ml ∈ D , by the application inversion operator of the
scale convension to previously obtained appropriate statistical estimations for L -
periodic random sequence.
We will assume, that is recorded M cycles by L counts in each cycle of the in-
vestigated cyclic signal whose mathematical model is a cyclic random process
____ ____
ξ1 (ω , t ml ), ω ∈ Ω, t ml ∈ R, m = 1, M , l = 1, L (for the simplified further marking
ξ1 (ω , t ml ) will be written). Relatively, mathematical model of a cyclic signal registro-
____ ____
gram will be a ω -realization ξ1ω (t ml ), t ml ∈ R, m = 1, M , l = 1, L (for the simplifying
of further marking ξ1ω (t ml ) will be written) of this cyclic random process of a discrete
argument. The isomorphic for the investigated discrete process in relation to order and
________
values of a L -periodic sequence ξ 2 (ω , i ), ω ∈ Ω, i = 1, M ⋅ L (for the simplifying of
further marking ξ 2 (ω , i ) will be written) is obtained, by the action of a scale trans-
formation operator with a scale transformation function y (t ml ) = L ⋅ (m − 1) + l to initial
random process ξ1 (ω , t ml ) , namely:
ξ 2 (ω , i ) = G y (t ml ) {ξ1 (ω , t ml )} (1)
which is equivalent to a such system of equations:
_____ ____
i = y (t ml ) = L ⋅ (m − 1) + l , m = 1, M , l = 1, L,
________ (2)
ξ 2 (ω , i ) = ξ1 (ω , t ml ), i = 1, M ⋅ L, t ml ∈ R.
The same scale transformation operator is related M -cyclic ω -realization cyclic
________
random process ξ1 (ω , t ml ) and M -cyclic ω -realization ξ 2ω (i ), i = 1, M ⋅ L (for the
simplifying of further marking ξ 2ω (i ) will be written).
The analytical formula for calculating the value of statistical estimation of the ini-
tial moment function of the first order (mathematical expectation) L -periodic se-
quence ξ 2 (ω , i ) , which is isomorphic in relation to the order and values of the cyclic
process ξ1 (ω , t ml ) , has the form of
1 M −1 ___
mˆ ξ 2 (l ) = ∑ξ 2ω (l + L ⋅ n ), l = 1, L (3)
M n=0
The analytical formula for the calculating the value of statistical estimation of cen-
tral moment function of the second order (dispersion) L -periodic sequence ξ 2 (ω , i ) ,
which is isomorphic in relation to the order and values of the cyclic process ξ1 (ω , t ml )
, has the form of
∑( )
1 M −1 ___
dˆξ 2 (l ) = ξ 2ω (l + L ⋅ n ) − mˆ ξ 2 (l ) 2 , l = 1, L (4)
M − 1 n=0
Due to the statistical procedure for the electrocardio signal processing normaliza-
tion received statistical estimation and their reduction in Chebyshev basis, that is in-
vestigated in paper [13].
A mathematical model of electrocardio signal will be considered below. A model
takes into account their double stochasticity, namely, morphological structure stochas-
ticity and stochasticity of the rhythmic structures of electrocardio signal. The condi-
tional cyclic random process is called a process {ξ (ω , ω ′, t ), ω ∈ Ω, ω ′ ∈ Ω′, t ∈ R} ,
that is set on the Cartesian product of two stochasticly independent probabilistic spac-
es with the sample sets Ω and Ω′ , and on the set of real numbers R , and for which
such conditions are satisfied:
1. a such random function exists T (ω ′, t , n ), ω ′ ∈ Ω′, t ∈ R, n ∈ Z , that for each ω ′ ,
relatively ω ′ -realization Tω ′ (t , n ) of this function, satisfies the conditions of
rhythm function;
2. for each ω ′ from Ω′ finite-dimensional vectors ( ξ ω ′ (ω , t1 ) , ξ ω ′ (ω , t 2 ) ,…,
ξ ω ′ (ω , t k ) ) and ( ξ ω ′ (ω , t1 + Tω ′ (t1 , n )) , ξ ω ′ (ω , t 2 + Tω ′ (t 2 , n ))) ,..., ξ ω ′ (ω , t k + Tω ′ (t k , n )
), n ∈ Z , where {t1 , t 2 ,..., t k } - separable set of the process
ξ ω ′ (ω , t ), ω ′ ∈ Ω′, ω ∈ Ω, t ∈ R , for all integer k ∈ N is a stochastic equivalent in a
broad sense;
3. for any different ω1′ ∈ Ω′ and ω 2′ ∈ Ω′ random processes ξ ω1′ (ω , t ) and ξ ω 2′ (ω , t )
are isomorphic in relation to the order and values of the cyclic random process.
A mathematical model of a rhythm cardio signal with increased resolution, accord-
ing to the paper [14], is a discrete random process
____
T (ω ′, t ml , n), ω ′ ∈ Ω′, t ml ∈ R, m ∈ Z, l = 1, L, L ≥ 2, n ∈ Z , which is embedded in a
random rhythm function T (ω ′, t , n), ω ′ ∈ Ω′, t ∈ R, n ∈ Z of a conditional cyclic ran-
dom process {ξ (ω , ω ′, t ), ω ∈ Ω, ω ′ ∈ Ω′, t ∈ R} . The first stage of a heart rhythm
analysis on the basis of a rhythm cardio signal with increased resolution is a for-
mation of a vector of random stationary and stationary connected sequences
____ ______
Ξ L (ω ′, m) = Tl (ω ′, m), ω ′ ∈ Ω′, l = 1, L, m = 1, M . Then the statistical processing
of a vector component is conducted. At the same time a mathematical expectation,
dispersion, the type of distribution (checking it for normality) is evaluated, by the
building a histogram and the use of Pearson's agreement criterion χ 2 . We present the
basic mathematical relations for estimating the probabilistic characteristics of the
components of this vector of random sequences.
An expression for calculating a realization of a statistical estimation of сˆ1T of the
l
relative vector component of the first-order initial moment с1T (mathematical expec-
l
tation) of a stationary random sequence Tl (ω ′, m) , namely:
1 M ____
сˆ1T = ∑ Tlω′ (k ), l ∈ 1, L (5)
l
M k =1
where M - the number of cycles of registered realization of an electrocardio signal,
Tlω′ (k ) - l vector component rhythmic cardio signal.
Statistic estimation of autocorrelation function will have the form of:
rˆ2T T (u ) = rˆ2T T (m1 − m2 ) =
l1 l2 l1 l2
∑( )( )
M − M1
Tl (k ) − сˆ1T ⋅ Tl 2 ′ (k + u ) − сˆ1T ,
1
M − M 1 + 1 k = 0 1ω′ l1 ω l2
__________
_____ ___
u = 0, M 1 − 1, m1 , m2 ∈ 1, M 1 , l1 , l 2 ∈ 1, L (6)
where M 1 - number of counts of correlation function, depth of correlation.
The decreasing of the number of diagnostic signs in an information system of
heart rhythm analysis on the basis of the main vector of a rhythm cardio signal with
increased informativeness is achieved by the use of a spectral decompositions of tri-
ˆ = rˆ
___ ____
angular matrix elements R T Tl1Tl2
2 (u ), l1 = 1, L , l 2 = l1 , L , in particular, by the us-
ing of a discrete Fourier transform for the estimation of the autocorrelation and inter-
correlation functions of this matrix. Namely, instead of a triangular matrix
ˆ = rˆ
___ ____
R T 2Tl1Tl2 (u ), l1 = 1, L, l 2 = l1 , L of correlation functions, a triangular matrix
___ ____
Sˆ T = Sˆ 2T T (ν ), l1 = 1, L, l 2 = l1 , L could be used, elements of which are the Fourier
l1 l2
transform of the image of relative estimations of a correlation function from the ma-
trix R̂ T . Namely, the Fourier transform of the image from the matrix Ŝ T are calcu-
lated as:
M 1 −1 − j 2πuν
Sˆ 2T T (ν ) = ∑ rˆ2T T (u ) ⋅ e
__________
M1
, ν = 0, M 1 − 1,
l1 l 2 l1 l 2
u =0 (7)
___ ____
l1 = 1, L, l 2 = l1 , L, j = − 1
The structural and functional scheme of the modified software system is presented
on the fig. 1. The system of programs is implemented in a programming language
Object Pascal.
EVALUATION OF EVALUATION OF
THE SEGMENT THE RHYTHMIC
REDUCTION OF A DISCRETE CYCLIC RANDOM
STRUCTURE OF STRUCTURE OF
PROCESS (ELECTROCARDIOGRAM) TO AN
THE THE ISOMORPHIC PERIODIC RANDOM SEQUENCE
ELECTROCARDIO- ELECTROCARDIO-
SIGNAL SIGNAL
MAKING NORMALIZATION OF STATISTICAL
DECISIONS BASED STATISTICAL
PROCESSING OF
ON ESTIMATES AND
THEIR SCHEDULES IN ELECTROCARDIO-
MORPHOLOGICAL
THE CHEBYSHEV BASIS SIGNAL
FEATURES
DIAGNOSIS RESULT
THE BLOCK OF
MAKING SPECTRAL ANALYSIS STATIC PROCESSING THE FORMATION
OF STATISTICAL
DECISIONS BASED OF A VECTOR OF VECTOR`S
ESTIMATES OF THE
ON RHYTHMIC VECTOR `S RHYTHMOCARDIO- RHYTMOCARDIO-
FEATURES RHYTHMOCARDIO- SIGNAL SIGNAL
SIGNAL
Fig. 1. Structural-functional scheme of the software system for the analysis of morphological
and rhythmic diagnostic of the electrocardio signal signs
The processing procedure of the investigated electrocardio signal involves the es-
timation of the segmental by means of the segmentation methods. The estimation of
rhythm function is conducted by interpolation of rhythmic structure (discrete rhythm
function).
After the evaluation of a rhythmic structure and the procedure of reduction to a
random sequence of a discrete cyclic random process of processing branches into two
parallel stages. The first stage conducts a morphological analysis which according to
the given structure provides a statistical processing of the electro cardio signal, nor-
malization of statistical estimations and their reduction in Chebyshev basis and mak-
ing decision due to the obtained morphological sings. The second stage conducts the
rhythm analysis and consists in the formation of the vector rhythm cardio signal, the
statistical vector processing and spectral analysis of the obtained statistical estima-
tions [13].
As an example, the Fig. 2 shows the general view of the program interface is given
for the statistical processing of the electrocardio signal, which provides the use of the
method of reducing the cyclic random process of a discrete argument to isomorphic
random sequence.
120 ξ ω (t i ), y.o. 340 Tω ′ (t i ,1)
100 320
80 300
60 280
40 260
20 240
0 220
-20 200
-40 ti ti
180
0 500 1000 1500 2000 2500 0 400 800 1200 1600 2000 2400
a) b)
Fig. 2. Results of the processing: а) a few cycles of the examined electro cardio signal; b)
rhythmic structure of the electro cardio signal, the red color defines the samples corresponding
to R-R - intervals
Figures 3-6 show the graphs for the explanation of the stages of the processing of
electrocardio signals by the software system.
120 ξ 2ω (i ), y.o.
100
80
60
40 T (t ml ,1) = T
2500
20
0 2000
-20
i ⋅101
1500
-40 t ml
1000
0 400 800 1200 1600 2000 0 500 1000 1500 2000 2500
a) b)
Fig. 3. Graphs:
a) a few cycles of the realization of the L -periodic random sequence, which is obtained from
the electro cardiogram, the action of a scale transformation operator;
b) the estimations of rhythm function T (t ml , n ) (at n = 1 ) of a L -periodic random sequence
120 ˆ 2 (i )
m
100
80 dˆ2 (i )
60 80
40 60
20 40
0
20
i ⋅101
-20
-40 i ⋅101 0
0 400 800 1200 1600 2000 0 400 800 1200 1600 2000
a) b)
Fig. 4. Graphs:
а) a few estimation cycles m ()
ˆ ξ 2 i of the initial moment function of the first order of the L -
periodic random sequence ξ 2 (ω , i ) ;
b) a few estimation cycles dˆξ 2 (i ) of the central moment function of the second order of the L
- periodic random sequence ξ 2 (ω , i )
A separate vector component of the rhythm cardio signal and the results of its sta-
tistical processing is depicted on the figures 4-6.
86
84
T3ω ′ ( m)
82
80
78
76
74
72
70
68
66
64
62
60 m
0 20 40 60 80 100 120 140 160 180 200
a) b)
Fig. 5. Graphs of:
а) a realization T3ω ′ (m) of the third vector component of rhythmic cardio signal T3 (ω ′, m) ,
which describes the duration of the T -intervals in the electro cardio signal;
b) histogram for the third component T3 (ω ′, m) , which describes the duration of the T -
intervals in the electro cardio signal
In this paper, the mathematical support of the software system for increasing the
speed of electrocardio signal processing in comparison with previously known meth-
ods is substantiated;the structural and functional scheme of the modernized software
system is developed; developed a program that implements a new method of reducing
a discrete random process to an isomorphic random periodic sequence, which will
achieve faster processing of electrocardio signals in computer cardiodiagnostic sys-
tems; the modernization of the modernized software system on real electrocardio
signals was carried out.
rˆ3T T (u ) 20
15 3 3
Sˆ3T T (u )
3 3
10
15
5
10
-0
-5 5
-10 u u
0
0 10 20 30 40 50 60 70 0 5 10 15 20 25 30 35
a) b)
Fig. 6. A realization graph of:
а) the rˆ3T T (u ) statistical estimation of the autocorrelation function r3T T (u ) ( l1 = 3, l 2 = 3 )
3 3 3 3
of the third vector component Ξ3 (ω ′, m) ;
b) Sˆ3T T (ν ) statistical estimations of a spectral power densities S 3T T (ν ) ( l1 = 3, l 2 = 3 ) of
3 3 3 3
the third vector component Ξ3 (ω ′, m)
6 Conclusion
Modernized software system due to the extension of its mathematical software, which
is based on a new approach to the processing of electrocardiograms based on a math-
ematical model in the form of a conditional cyclic random process and a meth-od of
reducing their mathematical model in the form of a discrete cyclic random process to
an isomorphic random sequence morphological analysis and analysis of the rhythm of
the cardio signals with increased informativeness, which made it possible to increase
the speed of their processing and increase the accuracy and reliability of diagnosis of
the cardiovascular system of the human body.
The created program system can be used as a component of specialized software in
automated diagnostic systems for system morphoanalysis and heart rhythm analysis
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