An approach to processing and analyzing ECG data based on highprecision determination of R peaks in QRS complexes and a statistical analysis of the QRS complex duration and ECG signal dispersion online is studied. A method is proposed for effectively finding R peaks in the ECG. Conducted statistical sequential analysis to study the behavior of R peaks, the significant difference of which is that the number of observations necessary to make a decision on the hypothesis depends on the test results and is a random variable. The method of successive testing of a hypothesis involves at each stage of monitoring the state of the heart rhythm making a decision on the presence or absence of violations. It turned out that a consistent assessment of the variance of the cardiogram in a healthy person has a pronounced linear character. With a poor heart rate, the areas where the heart works ambiguously are clearly expressed.
In the online monitoring problems of the state of the cardiovascular system, a huge amount of data is obtained, which needs automated processing for making medical decisions and recommendations. The quality of these decisions and the effectiveness of subsequent recommendations depends on the processing efficiency of the resulting data array.
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By heart rhythm disturbances, or arrhythmias, is meant any heart rhythm that is not a regular sinus rhythm of a normal frequency. Arrhythmias can be caused by a change in one or more functions of the heart (see surveys [1],[2]).
Various pathologies can lead to heart rhythm disturbances and behavioral changes. Depending on the heart rhythm disturbances, several types of arrhythmia can be distinguished: tachycardia (an increase in the heart rate of more than 90 beats per minute at state of rest), bradycardia (a slowing of the pulse of less than 60 beats per minute), extrasystole (a violation of cardiac activity, which consists in premature reduction myocardium or its individual parts), atrial fibrillation (chaotic atrial contraction).
The occurrence of arrhythmia is primarily associated with a violation of the heart rhythm. It’s mean that the formation of a sequence of data that reflects the rhythmic structure of the heart is so important. As a rule, the QRS complex acts as data elements [3]. Algorithms based on the differentiated ECG are computationally efficient for real-time analysis of datasets. A frequently used mathematical tool in various algorithms for QRS complex selection are derivatives.
When processing the data of the QRS complex, it is necessary to take into account possible omissions of the QRS complex and formation of a false sequence element, called false alarm. Also an important characteristic is the accuracy of the binding of data sequence elements to the time position of the QRS complex. 2
An important problem of ECG analysis is the construction of methods and computer algorithms for detecting and correctly detecting the QRS complex. For these purposes, an effective method for detecting the QRS complex is widely used — the PanTompkins algorithm [4]. As characteristic features for highlighting QRS complexes, the slope, amplitude and width of the ECG signal are used. It is important to distinguish other characteristic features from the ECG signal. So in the paper [5], as characteristic features for the successful detection of anomalies, it is proposed to use amplitude, duration, pre-gradient, post-gradient and so on. In the paper [6] dynamic features would mean extracting RR interval features, heart rate, HRV and the R/P ratio. Note that an ECG signal is the combination of various peaks, waves, valleys, segments, intervals, complexes and points. It should be noted that the data obtained from the ECG signal are non-stationary functions [7], and its derivatives are calculated by differentiating the interpolation functions. In many studies, piecewise polynomial curves are chosen as such functions. An analysis of various interpolation methods is contained in papers [8] and [9]. Various interpolation methods — linear interpolation, Lagrange interpolation, Hermits interpolation, and cubic spline interpolation — have been studied to find the optimal method for choosing the RR interval when analyzing heart rate variability. As a result of experiments, it was shown that the third-order Lagrange interpolation polynomial is the most suitable algorithm for selecting the RR interval for estimating the autoregressive spectrum, since it requires a short processing time and shows the lowest error rates in calculating the heart rate. Cubic splines are widely used to interpolate heart rate, although despite a large number of works on this topic, new where follows:
where ones are constantly being created and existing mathematical algorithms are developed for processing and analyzing ECG data. In paper [10], a combination of wavelet transform with cubic interpolation by splines is used to increase the accuracy of detection of QRS complexes. Additional requirements for mathematical algorithms are presented in the case of online ECG processing [11]. The identification of the Rpeak is crucial in the analysis of the signal of the ECG. In the study [12] proposed an adaptive and time-efficient algorithm for detecting R-peaks for ECG processing.
Thus, the problem of approximating ECG signals is important and relevant. In this study, a parametric piecewise polynomial continuous curve that has a continuous first derivative will be used to approximate ECG signals [13]. The advantages of this spline are the possibilities, using controlled parameters, of fitting its shape to the unsteady behavior of real ECG signals. Also, the proposed curve allows you to approximate the heart signal and calculate the first derivatives at arbitrary points. Of particular importance will be attached to the correct finding of R-peaks [14]. The article proposes methods of statistical analysis that allow recursively evaluating the characteristics of the ECG signal, which is especially important for real-time processing and verification and confirmation of some hypotheses about the distribution of R- peaks. The data for the study were taken from open access [15], as well as from [16], [17]. 3
Let’s introduce mathematical problem statement: will be to supose that the sequence of points be given on time interval [a, b]: ∆τ : a = τ 1 < τ 2 < ... < τ N = b , and at points τ i , the values of the cardiac signal Fi are given.
Along with the grid ∆τ , a grid ∆ x : τ1 = x1 < x2 < ... < xN +1 = τN is introduced,
xi =ηiτ i−1 + (1 −ηi )τ i , 0 <ηi < 1 , i = 3, N − 1 , 0 <η2 ≤ 1 ,
xi+1 = γ iτ i−1 + (1 − γ i )τ i , 0 < γ i < 1 , ηi+1 = 1 − γ i , i = 2, N − 1 , 0 < γ N −1 ≤ 1 .
For segments [xi , xi+1] , i = 2, N − 1 the interpolation polynomial is written as
x − xi + fi x − xi+1 + ( x − xi )( x − xi+1)( Ax + B) ,
si ( x) = fi+1 xi+1 − xi xi − xi+1
(
fi′+ fi′+1 − 2( fi+1 − fi ) , (hi + hi+1)2 (hi + hi+1)3 B = 2(hi + hi+1) fi′+1 − fi′ − ( fi′+1 + fi′)( xi + xi+1) + ( fi+1 − fi )( xi + xi+1) ,
2(hi + hi+1)2 (hi + hi+1)3 fi =ηi Fi−1 + (1 −ηi )Fi , fi+1 = γ i Fi+1 + (1 −γ i )Fi , fi′ = Fi − Fi−1 , fi′+1 = Fi+1 − Fi , τ i −τ i−1 τ i+1 −τ i hi =ηi (τ i −τ i−1) , hi+1 = γ i (τ i+1 −τ i ) .
Consider the examples of using polynomials (
Note that it is the squares of the first derivatives that play a key role in the algorithm for detecting R peaks. After separating the intervals with the values of the square of the derivative exceeding the specified threshold, the maximum value on the interval and the corresponding abscissa value are extracted from the signal. To automate the peak extraction process, a certain threshold value is set. 4
It is believed that the distribution of the lengths of the R-R intervals has a normal distribution for the cardiogram of a healthy person.
For various cardiac arrhythmias, it becomes necessary to test the hypothesis of equality of means for the normal distribution, as well as the hypothesis of equality of variances with an equal average.
In this paper, use the method of sequential analysis to solve these problems is proposed. A significant difference of this method is that the number n of observations z1 ,..., zn required to make a decision on the hypothesis depends on the outcome of the tests and is a random variable. The method of consecutively testing a hypothesis assumes at each stage of observation the adoption of one of the possible solutions: accept the hypothesis, reject it, or continue observing.
Before planning a sequential analysis procedure, acceptable values of the probabilities of permissible errors are assigned: α - the probability of accepting the hypothesis H1 , when the hypothesis is true H 0 (error of the first kind) and β - the probability of accepting the hypothesis H 0 , when the hypothesis is true H1 (error of the second kind).
Sequential analysis suggests that α << β , or β << α , when α ,β are quantities of different order of smallness.
To test the null hypothesis H 0 : µ = µ 0 against the alternative H1 : µ = µ1 , (µ 0 < µ1 ) when the variances are equal σ 12 = σ 22 = σ 2 , they are calculated: C = To further study the distribution of R-R intervals, we consider the ECG of a healthy person [15], i.e. cardiogram where there is no heart rhythm disturbance. We studied a ECG lasting one minute, the data were taken in step ∆t = 0,003sec and contain N = 21600 points. The graph of the initial ECG is shown in Fig. 4. Using the method proposed in this study (see Section 3), we find the R peaks in the original cardiogram and calculate the lengths of the R-R intervals for this cardiogram (see Fig. 5). Distribution of lengths of R-R intervals is demonstrated on the Fig. 6.
Note, that investigated ECG contains M = 74 R-R intervals. The graph of the R-R intervals lengths distribution is given in Fig. 6 demonstrates that their distribution is close to the normal distribution. The average value is equal σ µ 0 = 292.42 , varianceσ 2 = 182.30 , coefficient of variation kv = = 0.046. At a µ 0 confidential level ε = 0.95 we can conclude that the hypothesis that the distribution of the lengths of the R-R intervals has a normal distribution is confirmed.
Dispersion distribution of the ECG for healthy person
Next, we turn to the study of the dispersion of the cardiogram for a healthy person
(see Fig. 3.). Consider the variance σ 2 (m, n) of the cardiogram as an observed
parameter, the observation of variance of length m . The variance was estimated by the
next formula
σ 2 (n, m) = min(n −11, m − 1) k =max∑(n0,znk−2m) − k =mmianx∑(n(0m,zn,k−nm)) 2 , n = 0,1...N −1 . (
The length of the observation segment may change. we investigate in the given paper two segments: segment of the length m = 400 (one cycle of the heart rate) and m = 1200 (three cycles of the heart rate) are considered. On the Fig. 7, and Fig.8. the changes in the variance of the cardiogram are shown.
Let’s consider a ECG for person with changes in heart rate [15] with the aim of investigate distribution of lengths of R-R intervals and dispersion changes base on approach given Section 4. ECG lasting 4.5 minutes, the data was taken in steps ∆t = 0,0025sec and contains N = 108000 points. A graph of a given cardiogram is presented on the Fig. 9.
In fig. 10 and fig. 11 is the variance estimate for m = 400 and m = 1200 , respectively.
The variance changes shown in the graphs show at least three different sections of the heart rhythm. The ECG elements of these areas are shown in Fig. 12, Fig. 13., Fig.14 respectively.
As can be seen from the graphs, the hypothesis of inequality of the average variance in these areas is confirmed.
Consider, as an obsession, the lengths of R-R intervals. For the entire ECG, using spline analysis, R-R intervals were isolated and their lengths were calculated. An estimate of the density of the distribution of the length of R-R intervals is constructed.
On the Fig.15. a graph of the lengths of R-R intervals is presented, and a graph of histogram of the distribution of lengths of R-R intervals is presented in the Fig.16.
It can be seen that the lengths of the R-R intervals have pronounced three sections of different behaviour.
The average value of the R-R intervals is 338.9, the variance is 10239.8, and the standard deviation is 101.2.
From the graphs of the R-R intervals it can be seen that in the first segment (points 1 ... 90): • the average value of R-R intervals is 323.8, • the variance is 1142.8, • coefficient of variation 0.10; on the second segment (points 91 ... 203): • the average value of R-R intervals is 329.2, • the variance is 25957.2, • the coefficient of variation is 0.49; on the third segment (points 204 ... 318): • the average value of R-R intervals is 358.9, • the variance is 1363.5, • coefficient of variation 0.10.
Thus, the hypothesis was confirmed that the average is equal in these areas and that the variance is different.
The heart rate variability in the first segment corresponded to the norm, in the second and third heart rate variability has features.
What caused these features is unknown, a person could run, and then stop, or perform some kind of action.
The use of these algorithms allows you to select areas of changes in heart rate online. Depending on the state of the person for whom these measurements are carried out, the results can be used to make decisions about the state of the person. 6
The article proposes an approach to the processing and analysis of data in cardiology, based on high-precision determination of R peaks in QRS complexes and a statistical analysis of the QRS complex duration and ECG signal dispersion online.
In this investigation proposes the use of statistical sequential analysis for these problems, the significant difference of which is that the number of observations necessary to make a decision on the hypothesis depends on the outcome of the tests and is a random variable. The method of successive testing of a hypothesis involves at each stage of monitoring the state of the heart rhythm making a decision on the presence or absence of violations.
It was found that a consistent assessment of the variance of the cardiogram of a healthy person has a pronounced linear character. With a bad heart rate, areas where the heart works ambiguously are clearly expressed.
The use of these algorithms allows you to select areas of changes in heart rate online. The results can be used to make decisions about the condition of a person whose heart rate is being monitored. It is unknown what caused these features, a person could run away, and then stop, or perform some kind of action, or he has heart problems.