Introduction

According to the WHO and the Medical Association of Cardiologists, it has been established that the development trend of human heart valve lesions occupies a dominant place among all cardiovascular diseases in Europe [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The primary causes of such lesions are congenital (arising during fetal development) and acquired (complications of rheumatism, infectious endocarditis or after surgical treatment of mitral stenosis) defects of the human heart.

In medical practice, such basic non-invasive instrumental research methods as cardiography [1,2], auscultation [17,18], echocardiography [4,6,13,14,15] and phonocardiography [17,19] are used to diagnose mitral insufficiency. Phonocardiography along with other methods makes it possible to objectify unclear sound phenomena of human heart valves (degree of weakening and presence of heart sounds, intensity, duration and form of systolic murmur) (Sacks, Roberts and Evans) according to registered phonocardiographic signals (PCG signals) (Figure 1). Of particular value is the possibility of long-term (over a year or more) objective dynamic monitoring of the PCG signal during the formation of a valve defect. To implement the phonocardiography method in medical practice, hardware devices in the form of computer phonocardiographs are used, such as Cardio+ (Ukraine, Metekol LLC), PCG-02 (India, Mumbai), Techbook Scientech 2356 (India, Scientech Technologies Pvt . Ltd.) and Audio-Technica ATR288W (Japan). Well-known phonocardiographs based on the principle of obtaining diagnostic information about the condition of human heart valves are built according to the scheme "bioobject (human heart valves)-mathematical support (mathematical model-processing method)-algorithmic support-softwarediagnosis result" [21,24,26,28,29,38,39,40]. In phonocardiographs, software and algorithmic support, which form the result of diagnostics, are implemented on the correlation, spectral and synphase methods of FCS processing, the core of which is a mathematical model in the form of a deterministic harmonic function (Dodge and Cabot), periodic functions of different frequencies (Manheimer), almost periodic functions (Kasyrskyi G.I.), a stationary stochastic process (Metin Akay (Houston)), a mixture of a stationary stochastic process and a deterministic function (Metin Akay (Houston)), periodically correlated stochastic process (Osukhivska H.M., Dragan Ya.P. [22,23,35,36], Palianytsia Y.B. [31,32]), relaxation multipulsator (Osukhivska H.M.) [37]. Such mathematical support (models and methods developed on their basis) of processing by its structure and properties does not provide tracking the temporal dynamics of changes in the amplitude and phase indicators of the functioning of the valves of the human heart, which is important in detecting the manifestations of mitral insufficiency over time.

Therefore, the development of mathematical support (a processing method based on its adequate mathematical model) and algorithmic-software processing of PCG signal for detecting mitral valve insufficiency of the human heart is an urgent scientific task.

Mathematical support for phonocardiographic signal processing

The experimentally recorded implementation of the PCG signal in the case of mitral insufficiency is shown in Figure 2. The PCG signal in mitral insufficiency as a stochastic and cyclical process by the nature of the correlation function satisfies the conditions of cyclicity of the average characteristics and their variations:

    t m T t m   , (1)    s t r T s T t r , ,    . (2)

In this case, T corresponds to the duration of one PCG signal cycle in mitral insufficiency. Expressions (1-2) indicate the cyclic nature of the characteristics, namely that the values of the average characteristics and their variations due to the cardiac cycle are repeated, but with changes caused by chance.

PCG signal in the case of mitral class insufficiency provides the study of its type of harmonicity as a process of stochastic and cyclic nature in the form of PСSP from the standpoint of the energy approach (developer Ya.P. Dragan [20,22,23,30,31,32,33,34]).

Since the PCG signal in mitral insufficiency is presented in a stochastic process with cyclic characters of a finite nature, it can be represented through the expression of translational components [20]:

               Z p k Z k t q t pT t H t   ,(3)

where is a    H single function that indicates the place of generation of the PCG signal tone in mitral insufficiency; T -is the cycle period of the PCG signal in case of mitral insufficiency;   t k  -PCG signal tones in case of mitral insufficiency within the cardiac cycle; k -cardiac cycle number.

But based on the equivalence of representations through translational components (3) and stationary components, the expression of the PCG signal model in mitral insufficiency has the form [20][21][22][23]28,31,32]:

       Z k t T ik k e t t    2 , R  t (4)

where are   t k  the steady-state components of the PCG signal in mitral insufficiency.

Mitral insufficiency by nature has the appearance of a noise similar in shape to white noise. Such similarity should be reflected in model ( 4) additively in the form of an expression:

          Z k t T ik k t n e t t    2 , R  t (5)

where   t n is white noise as a sign of mitral insufficiency in the realization of the PCG signal (according to the nature of its formation).

Expression (5) ensures the use of the component method for the processing of the PCG signal in the case of mitral insufficiency as a stochastic process with cyclic characters of a finite nature.

The component method of PCG signal processing based on model (5) ensures the calculation of correlation components according to the expression [20,24,27,28]:

             T t T ik k dt e kT t kT u t T u B 0 2 0 0 1    , R  t (6)

where is

      t m t t      0 0 a centered PCG signal with mitral valve insufficiency;   t m 

-the average PCG signal in the case of mitral valve insufficiency.

Algorithmic support for phonocardiographic signal processing

Having considered the expression ( 6), the main operations of its implementation are highlighted: 1. Finding the T period of the FСG signal in the case of mitral valve insufficiency [25].

2. Calculation of the estimate of the mathematical expectation  m of the PCG signal in the case of mitral valve insufficiency. of the PCG signal in case of mitral valve insufficiency.

In Figure 3, we show all the listed operations in the form of an algorithm, which displays all the operations of implementing the component method of processing.

Software and results of phonocardiographic signal processing

Software with a graphical user interface was developed in the MATLAB environment to automate the process of processing the PCG signal when mitral insufficiency is detected.

The According to the results of the analysis of the correlation components, a slight change in the power intensity of the components is visible, namely, with mitral insufficiency, the power is more scattered than in the normal state. The components allow only visual detection of changes in the operation of human heart valves without quantitative evaluation.

For the quantitative evaluation of correlation (spectral) components, the evaluation of their averaging by time shifts was used (Figure 9 In Figure 9-10 it is visually and quantitatively visible that the maximum values of the averaged components for a patient in a physiologically normal state (patient A) and a patient with mitral valve insufficiency (patient B) are concentrated at the same frequencies (identical in structure), but the amplitude values of the estimates slightly differ from each other, namely, for patients A and B, the maximum amplitudes are concentrated on the 1st and 7th components. However, an increased power level of the 2nd, 3rd, 4th, and 5th components, as well as an increase in the maximums of the 1st and 7th components, were noticed in patient B. Such changes quantitatively reflect the manifestations of mitral valve insufficiency of the human heart.

Therefore, the calculation of correlation (spectral) components is a quantitative parameter for detecting early changes in the functioning of the valves of the human heart, in particular valve insufficiency.

Conclusions

The work substantiates the mathematical model of the PCG signal as a mixture of a periodically correlated stochastic process and noise for solving the problem of detecting the insufficiency of the mitral valves of the human heart. A method and algorithm for processing the PCG signal based on the component method was developed, which provided a procedure for detecting the insufficiency of the mitral valves of the human heart based on the averaged estimates of the correlation components. Software was developed using the MATLAB tool for processing the empirical data of the PCG signal, which provided the process of calculating correlation components as diagnostic signs that quantitatively reflect indicators of mitral valve insufficiency of the human heart.