УДК 577.3 Biophysics, Vol. 46, No. 2, 2001, pp. 313–323. ## Quantitative Analysis of Electrocardiographic Variability Typical of Polymorphic Arrhythmias
A. V. Moskalenko 1
## INTRODUCTIONAt present, electrocardiography is the most common method for differential diagnosis of cardiac arrhythmias. Moreover, many arrhytmias can be detected only by electrocardiographic analysis. There are sufficient grounds to expect that the role of this method in differential diagnosis of cardiac arrhythmias will increase in the future. With this expected increase in its usage comes the need for continually improving the methods for processing the electrocardiographic data. Developing more precise diagnostic criteria and new processing algorithms may help attain higher sensitivity and better specificity in differential diagnosis of arrhythmias. When interpreting electrocardiographic data, cardiologists usually analyze the QRS complex (amplitude, duration, and shape of individual waves and the entire complex), determine various intrabeat and interbeat segments (for example, RR intervals), examine temporal variations of these parameters, and take into account whether the recordings contain missing or extra waves. Knowing this information, the cardiologist can judge how electrical excitation propagates over the myocardium. Polymorphic ventricular tachycardias (PVTs), or high-frequency arrhythmias, pose a special problem in analysis of patient electrocardiograms. They are highly variable and, moreover, individual QRS complexes cannot be reliably delimited in them. No difference is often made between polymorphic ventricular tachycardias and ventricular fibrillation, a harbinger of sudden death. The classic definition of fibrillation (as given in the Unabridged Encyclopedia of Medicine) is as follows: cardiac ventricular fibrillation is pathological (chaotic, uncoordinated, and asynchronous) contraction of individual muscle fibers of the cardiac ventricles that cannot sustain their proper functioning. (…) In the electrocardiogram, the waveforms vary with the fibrillation stage, depending on the severity of myocardial hypoxia and the extent of metabolic alterations. Fibrillation is graded according to the oscillation types and the relationship between them (rhythmic isomorphic versus arrhythmic polymorphic) [10]. No doubt, analysis of variability in polymorphic ventricular tachycardia is a problem of considerable interest, which cannot be solved without developing new methods for quantitatively processing electrocardiographic data. In this study, we sought ways to solve this problem. Earlier [11], following Coumel’s recommendations [12], we set down quantitative criteria for identifying the electrocardiogram type (polymorphic, monomorphic, or quasi-monomorphic), which relied on the amplitudes and frequencies of the signals recorded. No criterion was then proposed for quantitatively assessing the amount of polymorphism present in electrocardiograms. In this study, we focused on the parameters suitable for estimating variations in electrocardiograms of patients with polymorphic ventricular tachycardia. To this end, we developed a new approach to analysis of arrhythmias. This approach is free of the need to identify individual QRS complexes; rather, we assessed the variability by numerically comparing neighboring fragments of the electrocardiogram. The entire electrocardiogram or any its fragment is described with two parameters, of which one is an index of variability and the other specifies how this index changes with time. These two parameters are useful in assessing the polymorphism within one electrocardiogram and in comparing different electrocardiograms. The approach proposed is promising in diagnosing polymorphic arrhythmias and in studying their mechanisms. ## NORMALIZED-VALUE ANALYSIS OF ELECTROCARDIOGRAPHIC VARIABILITY IN POLYMORPHIC VENTRICULAR TACHYCARDIA
A family of methods that met these criteria will be referred to hereafter as normalized-value analysis of electrocardiographic variability (NVAEV) in polymorphic ventricular tachycardia. The method that we describe in this study belongs to this family.
To assess the amount of electrocardiographic variability in simple cases with identifiable individual
QRS complexes, let us define the variability index i+ 1)th complex as
where A - are the peak-to-peak amplitudes of two consecutive complexes;
and_{i+1}T and _{i}T - are their widths.
Note that_{i+1}w= 1 if _{A} A= _{i } A and _{i+1} w= 1 if _{T} T = _{i}T._{i+1}
Hence, the variability index w= 0 if the _{i} ith and the (i + 1)th complexes have equal amplitudes and widths: A= _{i } A and _{i+1} T = _{i}T . _{i+1} The overall estimate of electrocardiographic variability of a fragment will be a normalized sum of the variability index values for all transitions in this fragment:
where When validating the normalized-value method proposed for assessing the electrocardiographic variability, we simulated simple cases of polymorphic
ventricular tachycardia by using what is called pseudoelectrocardiograms. This approach facilitates the validation procedure. Pseudoelectrocardiograms are the
curves with known
T= 182 ms.
Hence, we had _{b}w = 2.56 and_{T}w = 1.56._{A}
w = 3.
As follows from (2), the overall variability index _{5}V = 0.16 in this case.
The pseudoelectrocardiograms shown in Figs. 2-I
and 3-I are used to demonstrate how our method of normalized-value analysis of electrocardiographic variability works.
For each electrocardiogram, we determine the
i corresponding to the current position of the sampling window with segment j, the first segment most closely resembling segment i (reference sample) in the subsequent recording.
Formally, the procedure of their comparison is as follows. For any segment of the electrocardiogram, vector F is defined:
f),_{k}, f_{k+h}, ... ,f_{k+(p-1)h}
where n; n varies from k to k+(p-1)h, where p is the dimension of the so-called embedding space, and h is the embedding step (see, for example, [4, 5]).
The distance between the ith and jth vectors is determined by the norm:
Note that, for periodic signals, i-j| = mT,
where m = 0,1,2,3, etc. For aperiodic signals, the smaller the difference
between the ith and the jth segments, the smaller the r.
_{ij}
Segments = _{i}j-i, j>i
Seeking the jth segment and determining the t
value are based on the assumption that _{i}j is a unique function of i.
If t p-dimensional embedding space.
Therefore, we postulate:
where i.
To automatically seek a segment most closely corresponding to the current sample and separated from it by the
shortest interval, we systematically scan an electrocardiogram segment of length is formally
set into correspondence with the position of the norm (3) minimum.
In other words, we assume that the _{i}r minimum
determines the segment _{ij}j position in the scanning window. In this case, expression (4) reads
This approach considerably saves the time for computing L is chosen to be as wide as the greatest QRS width in the biological species under study.
Note that a search for segments analogous to the current sample is, in general, a procedure independent of the comparison procedure. In fact, whereas the polymorphic properties of patient electrocardiograms make the basis for assessing their variability, there is no need to have search criteria also related to these properties.
Fig. 2) with a single transition for which w is nonzero and for relatively
intricate pseudoelectrocardiograms (as those shown in _{i}Fig. 3).
The elements used to construct these pseudoelectrocardiograms have the same width as the elements
shown in Fig. 1-I; only the amplitude of element b varies.
The same pseudoelectrocardiograms ( Fig. 3-II, the I values averaged over time tend to covary with _{i}V
(table).
Contrary to expectation, the w values are equal.
_{i}Figure 2 shows simple pseudoelectrocardiograms (panel I)
along with the I functions calculated for them (_{i}panel II).
The oscillation period changes at the point of transition between nonidentical elements
in pseudoelectrocardiograms a and b, but not c or d.
Pseudoelectrocardiograms b and d
are obtained from a and c, respectively, by transposition of their fragments; therefore,
the electrocardiograms in each pair (a and b; c and d) do not differ
in w (1) at the point of junction of nonidentical elements.
However, their _{i}I functions are different. In our opinion, some uncompensated error in recognizing
and locating the segments analogous to the segment in the current sampling window is responsible for this difference.
_{i}
The presence of such recognition errors may be suspected from the shape of t
Fig. 2-III.
In electrocardiogram d, the interval between QRS complexes
does not vary with time. Therefore, t
is expected to be constant.
Contrary to expectation, t
_{i} changes dramatically at the transition from smaller to larger amplitudes,
suggesting that there is an error in locating the segment-analog.
Such an error is likely to arise when the neighboring complexes are very different,
unlike those in electrocardiogram _{i}b (Fig. 2), which have comparable amplitudes.
The characteristics may vary from complex to complex, but so great a difference between
two consecutive complexes is rare to occur in experimental or clinical electrocardiograms.
The algorithm that we propose for seeking the segments analogous to that in the current sampling
window may generate errors of two types.
First, if the segment-analog is outside the sampling window, a segment for calculating the norm is necessarily
at the end of the sampling window, which results in overestimating the I value turns out underestimated. In our experience, when errors of
both types are present, their effects on the _{i}I value are often offset, at least in part.
_{i}A long fragment of the electrocardiogram is usually analyzed to assess its polymorphism. Hence, it is desirable that the estimate of electrocardiographic variability obtained with the normalized-value method would be largely independent of the local characteristics of the signal and the errors in their determination. To overcome the shortcomings of the algorithm, we use the characteristics of electrocardiographic variability averaged over time.
the electrocardiographic variability index V, and
(ii) the coefficient of variation of the
_{1}I function, _{i}V [13].
Obviously (table), _{2}V tends to increase as we go over to curves
(pseudoelectrocardiograms) with higher values of the variability index _{1}V (2).
Hence, the method proposed is sensitive to the variability of the signal.
Recall that, unlike V, V can be calculated without identifying the QRS complexes.
The different procedures for their calculation may account for the fact that, from curve _{1}f to curve i,
the changes in V and V do not follow the same pattern (table).
_{1}
V for the segment in some fixed-width window, that is, the averaging window (AvW).
Shifting it along the time axis, we obtain a trajectory in the (_{2i}V, _{1i}V) parameter plane,
which reflects the dynamics of changes in the electrocardiogram. Note that the properties of the end segment
of the electrocardiogram whose length is equal to the averaging window width are uncertain; therefore,
the averaging window should not be too wide. Basing on the definition of ventricular tachycardia [9],
we chose the width of the averaging window to be six QRS widths (about 400 ms in the electrocardiograms under study).
Below, we assume that, unlike _{2i}V and _{1i}V,
_{2i}V and _{1}V (see _{2}Section 5 for their definitions) characterize
segments that are considerably longer than the averaging window width.
Figure 4
shows an intricate pseudoelectrocardiogram and the temporal evolution
of the corresponding V and _{1i}V functions.
This pseudoelectrocardiogram (_{2i}Fig. 4-I) was obtained by connecting in series
2000-ms-long fragments of curves i, h, g, and f (Fig. 3-I)
in the order of decreasing Vvalues (table).
As can be seen, the I, _{i}V, and _{1i}V functions
(_{2i}Figs. 4-II, 4-IV, and 4-V, respectively) reflect the changes in the pseudoelectrocardiogram.
Shifting the averaging window along the time axis of this electrocardiogram ( V) parameter plane
(Fig. 5),
which allow us to judge the dynamics of changes in the _{2i}i, h, g, and f segments.
Note that their projections onto the (V, _{1i}V) parameter plane are narrow and
are relatively far from one another.
Importantly, these trajectories differ significantly even when the corresponding _{2i}V
and _{1}V are such (cf., for example, segments _{2}i and g in Fig. 5) that it is difficult
to conclude whether the variability changes from segment to segment.
In other words, constructing trajectories in the index space, we obtain an additional source
of information concerning the electrocardiographic variability.
We emphasize that, with the normal-value method for assessing the dynamics of changes in pseudoelectrocardiograms ( V will be useful in assessing real polymorphic electrocardiograms, especially when individual
QRS complexes are difficult or impossible to identify.
As a first step to this goal, the normalized-value method proposed in this study is used
to analyze the variability in pseudoelectrocardiograms constructed of the fragments recorded experimentally.
_{2i}
Figure 6
shows three such pseudoelectrocardiograms and their trajectories in the
( V) parameter plane.
One can see that, with an apparent increase in the variability (from _{2i}Fig. 6a to Fig. 6b), the trajectory
is shifted to larger Vand _{1i}V values.
The corresponding integral parameters (variability index _{2i}V and coefficient _{1}V; not shown),
which characterize the entire electrocardiogram, change similarly.
It is easy to see (_{2}Fig. 6c) that the movement from more polymorphic segment 1
of the electrocardiogram to segment 2 where oscillations are relatively regular corresponds
to the passage from one area in the (V, _{1i}V) parameter plane to another.
On average, both parameters are higher for the polymorphic segment than for the regular segment.
_{2i}
Figure 7
shows two pseudoelectrocardiograms in which no difference in variability can be detected by visual inspection.
However, the types and locations of their trajectories in the ( V)
parameter plane clearly indicate that the two electrocardiograms differ in both
_{2i}V and _{1}V values and that the electrocardiogram in
_{2}Fig. 7a corresponds to a more severe case of arrhythmia.
## DISCUSSIONThe method proposed in this study makes it possible to numerically analyze the electrocardiographic variability.
An essential result of this study is that the electrocardiographic variability is characterized by two parameters:
the variability index I function, and the coefficient
of variation of the _{i}I function _{i}V.
The latter parameter describes how the variability index varies with time, affording
a possibility of distinguishing among arrhythmias that are similar in the variability index but differ
in the pattern of its variation with time. In other words, it comes to be possible to score polymorphism
as "more stable" or "less stable."
Note, however, that changes in _{2}V are usually correlated fairly
well with changes in _{1}V.
Research into the nature of this fact will shed new light on the mechanisms of polymorphic arrhythmias.
It is also important to address the issue of whether changes in the myocardial excitation patterns are correlated with
the changes in the trajectories in the (_{2}V, _{1i}V) parameter plane.
_{2i}
Algorithm (5), which is a version of universal algorithm (4) for analysis of electrocardiographic variability
in polymorphic arrhythmia, has some limitations.
For example, the algorithm proposed sometimes select a segment-analog that does not coincide with the
segment visually determined by an expert observer.
In such cases (recognition errors), the I almost to zero in a segment preceding first fibrillations.
The other is the offset phenomenon: an abrupt rise in the _{i}I and _{i}V
values against a background of a relatively small change in the _{1i}V.
The trajectory in the (_{2i}V, _{1i}V) parameter plane is a kind of "smile" in this case, which does
not reflect the actual changes in the electrocardiographic variability
(Fig. 8).
Both artifacts arise during the normalization for the peak-to-peak amplitude used in algorithm (5).
Developing better algorithms for analysis of electrocardiographic polymorphism may become a focus of our future studies.
_{2i}The normalized-value method or its modifications may be promising in classifying arrhythmias: the vague qualitative definition of polymorphism is replaced with its quantitative measure, the electrocardiographic variability index. The index allows the electrocardiographic variability to be assessed both for any moment of time and for arbitrarily long intervals. ## ACKNOWLEDGMENTThis work was supported in part by the Medical University of South Carolina, grant no. RB0-676. ## REFERENCES1.The European Society of Cardiology and the North American Society of Pacing and Electrophysiology. Heart Rate Variability, Circulation. 1996. V. 93. P. 1043-1065. 2.Kneppo, P., in Teoreticheskie osnovy elektrokardiografii (Theory of Electrocardiography), Moscow: Meditsina, 1979, pp. 410-432. 3.Babloyantz A., Destexhe A., Biol. Cybern. 1988. V. 58. P.203-211. 4.Kaplan D., Glass L., Understanding Nonlinear Dynamics. New-York: Springer, 1995. 5.Kantz H., Schreiber T., Nonlinear tine series analysis. Cambridge: Cambridge University, 1997. 6.Skinner J.E., Carpeggiani C., Landisman C.E., Fulton K.W., Circ. Res. 1991. V. 68. P. 966-976. 7.Absil P.-A., Sepulchre R., Bilge A., Gerard P., Physica A. 1999. V. 272. P.235-244. 8.Robles de Medina E.O., Fisch Ch., Bernard R., Coumel Ph., Damato A.N., Krikler D., Mazur N.A., Meijler F.L., Mogensen L., Moret P., Pisa Z., Rosenbaum M.B., Wellens H.J., Kardiologiya, 1982, vol. 22, pp. 86-89. 9.Waldo A.L., Akhtar M., Brugada P., Henthorn R.W., Scheinman M.M., Ward D.E., Wellens H.J.J., JACC. 1985. V. 6. P. 1174-1177. 10.Semenov, V.N., Bol'shaya Meditsinskaya Entsiklopediya (Unabridged Encyclopedia of Medicine), Moscow: Sov. Entsiklopediya, 1985, vol. 26, p. 275. 11.Kukushkin, N.I., Sidorov, V.Yu., Medvinsky, A.B., Romashko, D.N., Burashnikov, A.Yu., Starmer, C.F., Sarancha, D.Yu., and Baum, O.V., Biofizika, vol. 43, no. 6, pp. 1043-1059. 12.Coumel Ph., Antiarrhythmic Drugs. Berlin, Heidelberg: Springer. 1989. P. 86. 13.Voitinskii, E.Ya., Livshits, M.E., Romm, B.I., and Ryzhikov, V.S., Analiz biopotentsialov na tsifrovoi adaptivnoi sisteme (Analysis of Biopotentials Using a Digital Adaptive System), Leningrad: Nauka, 1972. ## FIGURES
Numerical characteristics of electrocardiographic variability in pseudoelectrocardiograms shown in
Note that, because electrocardiographic signals in polymorphic ventricular tachycardia are of irregular shape, the very entity of polymorphic ventricular tachycardia is not defined unambiguously in the known classifications of arrhythmias; its definition frequently overlaps with definitions of other types of cardiac rhythm disorders. For example, according to the 1981 classification of cardiac rhythm and conduction disorders, it is recommended, in diagnosing polymorphic ventricular tachycardia, to discriminate between its (i) bipolar (alternating), (ii) fusion-complex, (iii) torsado de pointes, and (iiii) mixed types [8]. In the 1985 classification by the North American Society of Pacing and Electrophysiology, ventricular tachycardia is defined as polymorphic if the QRS configuration varies in the electrocardiogram, whatever the recording lead [9]. The electrocardiographic term fibrillation in the same 1985 classification is defined as ventricular tachycardia in which the QRS complexes are difficult to identify in electrocardiograms measured from the body surface. Leaving aside the qualitative nature of such definitions, we only emphasize that they are almost of no value in constructing a continuous quantitative scale for scoring polymorphic ventricular tachycardias. |