Method of Computer Modeling of Heart Rhythm based on the Vector of Stationary and Stationary-related Random Sequences

Abstract

The article discusses a method of computer modeling of the rhythm of an electrocardiosignal based on a mathematical model in the form of a vector of stationary-correlated random sequences. This computer modeling method allows for the formation of implementations of the vector rhythm of the cardio signal (components of the vector of stationary-correlated random sequences) for different types of electrocardiosignals, both in the norm and with various types of rhythm pathologies. Based on the obtained statistical information in the form of estimates of correlation functions of vector components, modeling of the rhythm of electrocardiosignals was carried out. The accuracy of the computer modeling by the proposed method was studied. Computer modeling, vector of stationary random sequences, electrocardiosignal, heart rhythm CITI'2023: 1st International Workshop on Computer Information Technologies in Industry 4.0, June 14-16, 2023, Ternopil, Ukraine ORCID: 0000-0002-9717-4538 (P. Oniskiv); 0000-0001-7311-4103 (I. Lytvynenko); 0000-0001-9137-7580 (O. Volianyk); 0000-0003-23627386 (G. Shymchuk); 0000-0003-2143-6818 (V. Hotovych)

1. Introduction

Automated diagnostic systems allow studying the state of the human cardiovascular system, and the effectiveness of such systems largely depends on their components, including hardware and software tools. It is known that the process of automated diagnosis based on such systems involves processing the electrocardiosignal (ECG) signal in two stages. At the first stage, diagnostic information is obtained based on the analysis of the morphological features of the patient's ECG. This involves analyzing the shape and amplitude of diagnostic zones of the ECG. At the second stage, information is obtained based on the analysis of rhythm characteristics, i.e., the temporal relationships between the durations of diagnostic zones (segments) of the ECG. The development of technical tools based on models and methods that allow processing and modeling the ECG rhythm is of great interest because this information allows assessing the adaptive-regulatory capabilities of the cardiovascular system, as well as the psycho-emotional state of the patient. To develop new models and methods in medicine, in addition to well-known databases of biological signals such as https://physionet.org/, more and more tools are being used to create new databases of modeled signal realizations [1]. Additionally, knowledge bases are being formed, for example, based on the application of ontologies, knowledge bases in medicine, including for traditional medicine. In the vast majority of cases, databases are used to test created methods and verify new mathematical models, so creating them is an important task.

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2023 Copyright for this paper by its authors.

2. Analysis of recent research

There are many developed automated diagnostic systems for processing electrocardiosignals (ECGs) that analyze the rhythm of the signal. Known systems use artificial intelligence and machine learning algorithms [2,3]. In such systems, the ECG is processed and a diagnostic conclusion is formed based on the training of neural network algorithms. Stochastic mathematical models have found wide application in some systems [4,5]. Mathematical models that allow for variability and rhythmic variability are presented in works [6,7], and these approaches to building mathematical models are possible through the use of a vector of cyclic rhythmically related random processes. New diagnostic features are proposed based on these new mathematical models and methods of rhythm processing [14]. Mathematical models in the form of cyclostationary signals for processing ECGs are considered in works [8-13]. ECG modeling is presented in works [14-16].

The existence of a significant number of mathematical models [18-26] and methods for processing ECGs [37-43] emphasizes the importance of evaluating diagnostic characteristics of morphological nature. However, rhythm analysis methods have been less widely applied, since not all mathematical models take into account the stochasticity and variability of the rhythm. This is due to the insufficient level of development of mathematical models and methods of rhythm analysis, which would allow for the consideration of both the stochastic nature of the signal, as manifested in the morphological and rhythmic features of the ECG [31-36]. Part of computer realisation of models is presented in works [25, 26, 36, 37, 43-45] 3. Mathematical model of the rhythm of the electrocardio signal In [7], a rhythmocardio signal with increased resolution is substantiated and described using a vector ____  of stationary and stationary-related random sequences ΞL ( , m)  Tl ( , m),∈ Ω, l  1, L, m∈ Z  . At   the same time, its elements can be elements from a vector VL (′, m) for the case when it is necessary to study the time distances between the same type phases of the electrocardio signal in two of its adjacent cycles. Dimensionality (number of components) of the L vector ΞL (′, m) determines the resolution of the rhythmocardio signal and is equal to the number of investigated time intervals between pre-allocated phases in the electrocardio signal. By clarifying and specifying the probabilistic characteristics of the vector ΞL (′, m) , applied a mathematical model of the rhythmocardio signal with increased resolution in the form of a vector of stationary and stationary-related random sequences [7], which is characterized by the invariance of its family of distribution functions to time shifts by an arbitrary integer k ∈ Z , namely, for the distribution function F x1,..., xp , m1,..., mp  of the order of p pTl1. .Tlp ( p ∈ N ) from the family of vector distribution functions ΞL ( , m) stationary and stationary-connected random sequences have the following equality:

F pTl1 . .Tlp x1,..., x p , m1,..., mp   FpTl1 . .Tlp x1,..., x p , m1  k,..., mp  k ,

____ x1,..., x p ∈ R, m1,..., mp ∈ Z, l1,...,l p ∈ 1, L, k ∈ Z . (1)  

The structure of probabilistic characteristics of a high-resolution electrocardiosignal, which follow from the invariance properties of the corresponding probability characteristics of stationary and stationary-related random sequences, has been studied in works [7, 14]. These characteristics complement the known probabilistic characteristics of the electrocardiosignal based on known models [37-45].

To implement the task of modeling the ECG rhythm based on a justified mathematical model in the form of a vector of stationary-related random sequences, we consider a five-component vector  ____  Ξ5 ( , m)  Tl (m), l  1,5, m  1,90  durations of the diagnostic zones of the electrocardiosignal, the 

 ____  model of which is a vector Ξ5 ( , m)  Tl ( , m),  ∈ Ω, l  1,5, m  1,90  stationary-associated random   sequences obtained based on ECG processing, which is shown in Figure 1a. The method described in [14] is used to generate vector component realizations. Figure 1b shows a fragment of the discrete rhythm function obtained using the methods described in [14]. The dashed line represents the continuous rhythm function, which is an estimate of the discrete rhythm function and characterizes the rhythm of the investigated ECG. Only a few vector components, are presented in Figures 2 and 3, including the first component of the T1( , m) is a random stationary sequence that describes a tooth P (diagnostic zone) in electrocardiosignal for all its 90 registered cycles. Implementation schedule T1 (m) of this component is presented in Figure 2, a. Figure 2b shows the third component T3 ( , m) of this vector, which is a random stationary sequence describing the duration of the diagnostic zones QRS of the complex in the electrocardiosignal. Figure 3 shows the fourth component, which in turn describes T4 ( , m) - tooth T in the electrocardiosignal. 0 10 20 30 40 50 60 70 80 90 Figure 3: Implementation fragment T4 (m) of the fourth component of the vector T4 ( ,m) , which describes the duration T - teeth in the electrocardiosignal

For data components of the vector of stationary-related random sequences, estimates of correlation functions were determined according to the formula: ˆ 1 M M1Tl1 (k)  сˆT (l1) Tl2  (k  u)  сˆT (l2 ) ryT (u)  rˆyT (m1  m2 ) M  M1  1 k 0 , (2) u  0, M1 1, m1, m2  1, M1,l1,l2  1,5 , y  1,3,4

____  ____  where Сˆ1L  cˆT (l), l  1,5  - estimates of mathematical expectations for each of the five components  of the vector ( L  5 ):

1 M ____ сˆT (l)  Tl (m) , l  1,5 , m  1,90 ; (3)

M m1

Where the M  90 - is the number of counts corresponding to the registered cycles of the investigated EKS implementation, M1 - is the element number in the sequence (correlation depth), y - the number of the correlation function for the corresponding component of the vector y  1,5 (for this example y  1,3,4).

The results of the obtained statistical estimates of the correlation functions are shown in Figures 4, 5.

r1T (u) 106 ˆ r3T (u) 105 ˆ electrocardiosignal; b) realization of correlation function estimation T3 (m) of the third component of the vector T3 ( , m) , which describes durations QRS - of the complex in the electrocardiosignal; 0 -0.8 r4T (u) 105 ˆ 0 10 20 30 40 50 60 u

Based on the computer modeling method presented in [17], modeling experiments were conducted, the results of which are the simulated realizations of vector components presented in Figures 6 and 7. During computer modeling, the obtained estimates of correlation functions presented in Figures 4 and 5 were taken into account.

rˆˆ1T (u) 106 third component of the vector electrocardiosignal; 0 10 20 30 40 50 60 0 10 20 30 40 50 60 which describes durations of P - teeth in the electrocardiosignal; b) implementation of T3 (m) of the T3 ( , m) , which describes durations of QRS-complex in the m u Tˆ1 (m), s the vector T3 ( , m) , which describes durations of QRS - of the complex in the electrocardiosignal 0 -0.8

rˆˆ4T (u) 105

We will investigate the accuracy of the developed method of computer simulation modeling [17], and estimate the errors of computer modeling. To do this, we will determine the absolute and relative errors of the obtained statistical estimates of correlation functions for the modeled components of the vector of stationary-correlated random sequences.

The absolute and relative errors of modeling were determined as follows: ____ , k, j  1, N , y  1,3,4.

(4) Cy (k)  1 N

 rˆyT ( j)  rˆˆyT ( j)

N j1 Cy (k) 

Cy (k ) 1 N rˆˆ

yT ( j) 9.6 7.9 6.2 4.5 2.8 1.1 -00.6 -2.3 estimates of the third component of vector T1( , m) ; c) absolute error of the third component of vector T3 ( , m) in computer modeling; d) relative error of computer modeling for estimates of the third component of vector T3 (, m) ; e) absolute error of the fourth component of vector T4 (, m) in computer modeling; f) relative error of computer modeling for estimates of the fourth component of vect or T4.( , m) 4. Discussion of obtained results

The obtained results suggest that for modeling the rhythm of an electrocardiosignal based on a model in the form of a vector of stationary-related random sequences, the maximum relative error of modeling statistical estimates of vector components does not exceed 17% for the studied realizations, indicating sufficient accuracy of computer modeling. In [6, 14], a structured diagram of a diagnostic complex is presented (Figure 11). We will show that this diagnostic complex includes an additional block for computer modeling of a vector cardiac signal (stationary-related random sequences) based on the obtained statistical estimates.

 ____  Dˆz  tˆi , i  1,С, j  1, Z   j 

Tˆ(t,1),t  W  (tk ), tk  W segmEevnatalulasttirounctoufre of rhythmocardiosignal signal

Evaluation of rhythmic

structure of rhythmocardiosignal signal ΞˆL ( , m)

Statistical processing of vector rhythmocardiosignal signal

____ С1L  сTl , l  1, L

  ____ D1L  dTl , l  1, L

 ˆ ___  RT  ryT (u), y  1, L

  Block of imitation modeling

of vector rhythmocardiosignal signal (A vector of stationary random sequences) (A vector of stationaryrelated random sequences) Decision-making based on morphological

features Decision-making based on rhythmic features

mˆ (t), t  W1 сl,l 0,1 dˆ (t ), t  W1

Normalization of statistical estimates

and their distributions in the

Chebyshev's basis SˆT  Sˆ 2Tl1Tl2  , l1,l2 1, L

Formation of vector rhythmocardiosignal

signal ΞL ( , m)

ΞˆL ( , m) Statistical processing of rhythmocardiosignal

signal Block of spectral analysis of statistical estimates of vector rhythmocardiosignal signal Diagnostic conclusion on the patient's condition based on morphological and rhythmic features

5. Conclusions

Based on a mathematical model of vector cardiac rhythm signals, a method for computer modeling of the heart rhythm electrocardiosignal (ECG) was developed in the form of a vector of stationary-linked random sequences. Statistical processing methods were applied to the components of the vector cardiac rhythm signal based on the mathematical model in the form of a vector of stationary-linked random sequences. During the computer modeling of realizations of the vector cardiac rhythm signal components, obtained statistical estimates of the real ECG were used. An assessment of the accuracy of computer modeling of vector cardiac rhythm signal components was carried out, and it was established that the relative error of computer modeling does not exceed 17%.

In further studies, it is planned to process ECGs with various types of rhythm pathologies such as tachycardia, bradycardia, arrhythmia, and others, while establishing those marker elements of correlation function estimates of the vector components that are sensitive to heart rhythm disorders.

6. References

[1]. Lupenko S. The Ontology as the Core of Integrated Information Environment of Chinese Image Medicine / Lupenko S., Orobchuk O., Mingtang Xu. // International Conference on Computer Science,

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