Tutorials and Reviews International Journal of Bifurcation and Chaos, Vol. 11, No. 10 (2001) 2531–2548 c World Scientific Publishing Company
TEMPORAL EVOLUTION OF NONLINEAR DYNAMICS IN VENTRICULAR ARRHYTHMIA MICHAEL SMALL∗ , DEJIN YU and ROBERT G. HARRISON Department of Physics, Heriot-Watt University, Edinburgh, UK RICHARD CLAYTON School of Biomedical Sciences, University of Leeds, Leeds, UK TRYGVE EFTESTØL Signal Processing Group, Høgskolen i Stavanger, Stavanger, Norway KJETIL SUNDE and PETTER ANDREAS STEEN Department of Anesthesiology, Ulleval University Hospital, Oslo, Norway Received June 23, 2000; Revised November 14, 2000 Ventricular fibrillation (VF) is a rapidly lethal cardiac arrhythmia and one of the leading causes of sudden death in many industrialized nations. VF appears at random, but is produced by a spatially extended excitable system. We generated VF-like “pseudo-ECG” signals from a numerical caricature of cardiac tissue of 100 × 100 × 50 elements. The VF-like “pseudo-ECG” signals represent the propagation and break-up of an excitation scroll wave under FitzHugh– Nagumo dynamics. We use surrogate data and correlation dimension techniques to show that the dynamics observed in these computational simulations is consistent with the evolution of spontaneous VF in humans. Furthermore, we apply a novel adaptation of the traditional first return map technique to show that scroll wave break-up may be represented by a characteristic structural transition in the first return plot. The patterns and features identified by the first return mapping technique are found to be independent of the observation function and location. These methods offer insight into the evolution of VF and hint at potential new methods for diagnosis and analysis of this rapidly lethal condition.
1. Introduction Ventricular fibrillation (VF) is characterized by disorganized “chaotic”1 electrocardiogram (ECG) and no net transport of blood [Wiggers, 1940]. The heart muscles twitch and writhe, but the usual rhythmic beating is absent. Untreated, VF is usually fatal within minutes. Treatment for VF is typically via administration of a massive electrical impulse (defibrillation) and (occasionally) anti-
arrhythmic drugs. The mechanism underlying the onset of human ventricular fibrillation and the subsequent efficacy of defibrillation is not well known. One model of the onset of VF is a re-entrant scroll wave of electrical activity interacting with itself and fragmenting into “daughter” wavelets [Jalife & Gray, 1996; Panfilov, 1998]. We apply a computational model based on this premise to produce time series of VF-like “pseudo-ECG” [Biktashev & Holden, 1998; Clayton et al., 1999].
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Address for correspondence: Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong. E-mail:
[email protected] 1 Not necessarily in the mathematical sense. 2531
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In this paper we analyze this data and compare the results to the analysis of recordings of spontaneous VF in humans. We apply two main analysis techniques: (i) correlation dimension and surrogate techniques and (ii) temporal first return maps.
1.1. Introducing: Dimension and surrogate analysis There has been conflicting evidence that VF may be best described as linear noise [Goldberger et al., 1986; Kaplan & Cohen, 1990], or deterministic chaos [Ravelli & Antolini, 1990; Govindan et al., 1998]. Recently, our own analysis has shown that VF in pig hearts is not consistent with linearly filtered noise, but is consistent with a noise driven nonlinear model [Small et al., 2000c]. The correlation dimension of fibrillating pig hearts is estimated to be approximately 6 [Small et al., 2000c; Yu et al., 2000c; Small et al., 1999; Yu et al., 1999]. It is likely that VF in humans may also be described as a nonlinear process. In this paper we test this hypothesis by applying the same correlation dimension and linear surrogate techniques to episodes of human VF and computational simulations of VF-like dynamics. Our results demonstrate that spontaneous VF in humans is not consistent with linearly filtered noise.
1.2. Introducing: Temporal first return maps Furthermore, we calculate first return maps for simulated data and compare the results to the analysis of human VF recorded in a coronary care unit (CCU) [Small et al., 2000a] and during paramedical emergency resuscitation. First return maps have commonly been used as a method of analysis for ECG time series, i.e. analysis of the RR intervals (or HRV) [Kaplan & Cohen, 1986; Woo et al., 1992; Garfinkel et al., 1995; Hastings et al., 1996; Addio et al., 1998; Witkowski et al., 1998; Vallverd´ u et al., 1998]. However, in this paper we do not assume the presence of an easily definable QRS complex, and we work with a more general class of possible first return maps. This additional flexibility allows us to extract equivalent features from linearly independent time series in a manner that would not be possible with standard linear statistics. Using our computational simulations we confirm that it is possible to reconstruct the underlying
dynamics of a spatiotemporal system from measurements of only one point. Furthermore, the choice of that measurement point is not critical. The implication for clinical ECG signals is that the exact ECG pad position is not important for this type of analysis. We demonstrate the application of this technique in a clinical setting by applying these methods to time series recorded by Laerdal defibrillators on-board ambulance in the Oslo Ambulance Service. From a dynamical systems standpoint, these time series are extremely short, noisy, poorly digitized, and under-sampled. Applying these methods we demonstrate that, in many recordings, the same qualitative features that are present in the computer simulations may also be observed in clinical recordings. However, these ambulance data sets are of fairly low quality and provide less convincing results. We have recently established a new data collection facility capable of recording large amounts of high resolution ECG data during VF [Small et al., 2000a]. We also apply this analysis to data collected from this facility. The results for this data are consistent with the other findings but are somewhat more compelling.
1.3. Overview In Sec. 2 we describe the surrogate and correlation dimension techniques to be used in this paper. Section 2.3 describes the results. In Sec. 3 we describe the time dependent first return maps we utilize — Secs. 4.1 and 4.2 demonstrate the application of this technique to “pseudo-ECG” and human ECG data during VF (recorded by ambulance defibrillators). Section 4.3 presents similar results for data collected from patients in a CCU. In Appendices A–C we describe the computational simulations and clinical VF data acquisition.
2. Correlation Dimension and Surrogate Analysis Correlation dimension is a measure of the structural complexity of an attractor. For scalar times series the method of time delay embedding (see Sec. 3.1) is used to reconstruct the attractor. In Sec. 2.1 we provide a brief review of the dimension estimation routine we employ in this paper. Section 2.2 describes the surrogate data techniques and Sec. 2.3 shows the results of applying these methods to human and simulated data of ventricular arrhythmias.
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2.1. Judd’s algorithm Several algorithms have been proposed to estimate the correlation dimensions of a nonlinear system from a scalar time series. The original Grassberger–Procaccia algorithm [Grassberger & Procaccia, 1983a, 1983b] has several well known problems when applied to short or noisy time series. Other algorithms have been proposed that allow for substantial system noise in the underlying model of the attractor — Diks’ algorithm [Diks, 1996; Yu et al., 2000b] estimates the correlation dimension, entropy, and noise level simultaneously. The algorithm we employ here models the underlying attractor as the Cartesian cross-product of a multidimensional “Cantor-type” set and an Euclidean space [Judd, 1992]. Judd’s algorithm has been shown to be robust enough to reliably estimate correlation dimension from short noisy data
sets [Judd, 1992, 1994; Galka et al., 1998]. Whereas the Grassberger–Procaccia model estimates correlation dimension as log(P ()) →0 N →∞ log
dc = lim lim
where P () is an estimate of the probability that two points on the attractor are less than a distance apart (the distribution of inter-point distances), N is the number of sample points and dc is the correlation dimension. Judd showed that assuming the attractor is the cross-product of a “Cantor-type” set and an Euclidean space then for all less than some cut-off, 0 , the correlation dimension dc (0 ) may be estimated as
P () log p() dc (0 ) = lim lim →0 N →∞ log
Fig. 1. Correlation dimension estimates for simulated VF data. Correlation dimension estimates for the three computational simulations of evolution of VF (Appendix A). The scale on each plot is identical, except for the center plot. Each row corresponds to the Ex , Ey and Ez components of a given simulation. Each row is a different simulation. Curves are shown for a range of embedding dimensions de ∈ {4, 5, 6, 7, 8, 10}. There is some variation between estimates for different components of the same system and between simulations. However, in general the correlation dimension at the smallest scales is around 4. Some components exhibit only a one- to two-dimensional system — the additional dynamics is presumably orthogonal to those directions for those simulations. The time series corresponding to the top three plots is shown in Fig. 5.
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where p(·) is a polynomial of order less than the topological dimension of the attractor and dc (0 ) is the correlation dimension, expressed as a function of 0 , the viewing scale [Judd, 1992]. For a fixed value of 0 both the correlation dimension, and the form of the polynomial p(·) are estimated from the distribution of interpoint distances P () using a nonlinear optimization routine.
2.2. Surrogates Estimating correlation dimension alone does not provide a statistical guide to the significance of the results obtained. For this reason we apply the method of surrogate data. Surrogate data are generated by manipulating a given data set so that it appears “like” the original time series, but is consistent with some class of systems, the null hypothesis. The three most common null hypotheses are [Theiler et al., 1992]: (0) i.i.d. noise; (1) linearly filtered noise; and (2) a monotonic nonlinear trans-
formation of linearly filtered noise. These three algorithms are commonly known as algorithm 0, algorithm 1 and algorithm 2. Data consistent with these three hypotheses are generated by [Theiler et al., 1992]: (0) shuffling the original time series; (1) shuffling the phases of the Fourier transform of the original time series; and (2) reordering the original so that it has the same rank distribution as an algorithm 1 surrogate. Several generalizations and extensions of this principle have been proposed by many authors. In particular Small and Judd [1998b] suggested that simulations from nonlinear models fitted to the data may act as a kind of nonlinear surrogate. In this paper we verify our correlation dimension estimation with the application of linear surrogates.
2.3. Results Figures 1 and 2 show the estimates of correlation dimension for simulated and patient VF data
Fig. 2. Correlation dimension estimates for human ambulance VF data. Correlation dimension estimates for nine different recordings of human VF from Laerdal defibrillators (Appendix B). The horizontal and vertical scales on each plot are identical to those in Fig. 1. Curves are shown for a range of embedding dimensions de ∈ {4, 5, 6, 7, 8, 10}. The correlation dimension at the smallest scales varies from around 5.5 to 7. Time series representative of those used in our calculations are shown in Fig. 7.
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described in Appendices A and B. The correlation dimension estimates and surrogate analysis for the data described in Appendix C produced similar results [Yu et al., 2000a]. Correlation dimension estimates from the simulated data show some clear variation between orthogonal components of the same simulations although these different channels are measuring the same underlying dynamics. As these simulations are short and they are constructed so that the initial traveling wave of excitation is virtually orthogonal to one of the coordinate directions we conclude that this orthogonality means that measurements in that direction only represent an orthogonal subsystem. There is a similar variation, or though not as marked, between estimates of correlation dimension from recordings of human VF. This is to be expected. All these recordings are fairly short
and noisy, in each case the clinical arrhythmia has been in progress for a different length of time (before the arrival of the paramedics) and each patient may have different cardiac disorders that are symptomatic of their VF. Furthermore, the estimates of correlation dimension for human VF are substantially higher than the simulations — typically around 6 versus about 4 for simulations. The computation simulations we have used are a gross simplification of the human cardiac system and this is to be expected. Furthermore, this result is in good agreement with our previous estimates for induced swine VF [Small et al., 2000c; Yu et al., 2000c]. These calculations show closest agreement between human VF and the computational simulation initiated with a single bent scroll was (file14). To distinguish the correlation dimension estimates we have observed from typical results for
Fig. 3. Surrogate analysis for simulated VF data. We computed correlation dimension estimates for the three computational simulations of evolution of VF (Appendix A), and for 30 algorithm two surrogates of each data set. For each set of 30 surrogates we calculated the mean µ and standard deviation σ of the distribution. These plots show the number of standard deviations from the mean value and the true value dc for the original data set, i.e. significance = (µ − dc )/σ. The scale on each plot is identical. Each row of plots corresponds to the Ex , Ey and Ez component of a given simulation. Curves are shown for a range of embedding dimensions de ∈ {4, 5, 6, 7, 8, 10}. In each case there is a clear distinction between the data and the surrogate: these computational simulations of VF may not be modeled as a monotonic nonlinear transformation of linearly filtered noise. The estimates of correlation dimension for the time series used here are given in Fig. 1.
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Fig. 4. Surrogate analysis for human ambulance VF data. We computed the number of standard deviations between the correlation dimension estimates for nine different recordings of human VF from Laerdal defibrillators (Appendix B) and 30 surrogates, i.e. significance = (µ − dc )/σ. The horizontal and vertical scales on each plot are identical to Fig. 3. Curves are shown for a range of embedding dimensions de ∈ {4, 5, 6, 7, 8, 10}. In each case there is a clear distinction between the data and the surrogate: spontaneous VF in humans is not a monotonic nonlinear transformation of linearly filtered noise. The estimates of correlation dimension for the original time series used here are given in Fig. 2.
linear noise we apply linear surrogate techniques. Figures 3 and 4 compare the estimates of correlation dimension for the original data (in Figs. 1 and 2) to algorithm 2 surrogates. In each case the significance is defined as the number of standard deviations (of the distribution of surrogate values) separating the mean value for the surrogates from the value estimated for the data. That is, significance =
µ − dc , σ
where µ and σ are the mean and standard deviation estimated from the surrogates, and dc is the corresponding value for the original data. All these quantities are functions of the viewing scale 0 . The null hypothesis represented by algorithm 2 surrogates is that the dynamics can be described as a monotonic nonlinear transformation of linearly filtered noise. These surrogates test the hypothesis that there is no dynamic nonlinearity in the system. For the
computational simulations we know this not to be the case. Furthermore, it is natural to suppose that human VF is a nonlinear dynamical system. The computations depicted in Figs. 3 and 4 clearly confirm this. In each case the underlying dynamical system cannot be described as a monotonic nonlinear transformation of linear filtered noise. In particular, this demonstrates that although the human VF recordings from ambulance defibrillators are short and extremely noisy, they do contain nontrivial information concerning the underlying dynamics, and the estimates of correlation dimension in Fig. 2 are not due to the experimental noise in the system.
3. Time Dependent First Return Maps In Sec. 2 we demonstrated that human VF and computational VF cannot be described as a linear
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noise system, and that both systems exhibited similar dynamic features. To examine the temporal evolution of the dynamic features of these systems we construct first return maps. First return maps are a commonly used and widely accepted technique [Kaplan & Glass, 1996] and have often been used for the analysis of “semi-periodic” experimental time series. In particular, study of RR intervals (the time between successive heart beats) has been the subject of considerable interest [Addio et al., 1998; Censi et al., 1998; Woo et al., 1992; Huikuri et al., 1996; Wessel et al., 2000; M¨ akikallio et al., 1999b; Valkama et al., 1993; Huikuri et al., 1993; Mani et al., 1999]. We apply this powerful technique to describe the dynamics during VF-like simulations, and for recordings of VF in a clinical setting. However, the observed ECG routinely varies from patient to patient — the position of ECG leads is standardized but the leads selected for monitoring will be influenced by clinical concerns. For this reason we require a technique which is independent of the observation function. Such invariance is guaranteed (theoretically) by Takens’ embedding theorem [Takens, 1981] and its corollaries. We show that the technique outlined below ensures that this is true for practical experimental data. Further, we employ first return mapping techniques to examine the temporal evolution of VF. To do this we color successive points of the standard first return mapping according to their temporal ordering. In Sec. 3.1 we describe the preprocessing techniques to project the observed time series onto its dominant direction in embedded space. To perform this operation it is necessary to embed the scalar time series. Section 3.2 briefly describes our selection of delay coordinates. Section 3.3 describes the peak detection algorithm to extract time series of successive returns, and Sec. 4 presents our experimental results.
3.1. Preprocessing Time delay embedding can be applied to ensure that (in theory) the underlying dynamics are reconstructed uniquely. The vector time series {vt }nt=1 is reconstructed from the scalar time series {yt }N t=1 according to vt = (yt , yt−τ , yt−2τ , . . . , yt−(de −1)τ )T .
(1)
Selection of the embedding parameters de and τ is crucial but techniques vary [Abarbanel, 1996].
In Sec. 3.2 we describe the method we employ to select suitable values. The embedding (1) produces a cloud of points, the temporal evolution of which is topologically equivalent to the underlying dynamical system. After normalizing the data so that they have zero mean, we apply the singular value decomposition (SVD) [Press et al., 1988] to the cloud X = [v1 : v2 : · · · vn ] (n = N − (de − 1)τ ) to estimate the dominant spatial orientation of X. Let ui denote the eigenvector corresponding to the ith largest singular value. We then compute the derived time series z (zi ∈ R) from X by projecting X onto u1 , z = (z1 , z2 , . . . , zn ) = u1 · X .
(2)
Some time series have two (or possibly more) singular values of comparable size. In these cases it is difficult to determine which eigenvector to use, and the analysis is repeated with each. From the time series z we estimate the time and location of successive peaks and troughs according to the method described in Sec. 3.3. Either the sequence of peak values, trough values, the difference between peak and trough values or the inter-peak (or inter-trough) time series may be used to produce a time dependent first return map. We choose to examine the time series of successive amplitudes (the difference between adjacent peak and trough values). A similar scalar time series could be obtained directly from the raw data — without need for delay embedding, SVD and projection. However, the experimental time series that we are working with are poorly digitized (only 8 or 10 bit sampling precision). By first embedding the scalar time series and then projecting it to obtain a derived scalar time series we are utilizing additional information evident from successive measurements but not available from the original scalar measurement. Despite the poor digitization and observational noise in the raw data, the embedded vector time series should be (approximately) diffeomorphic to the underlying dynamical system. By projecting onto a direction not orthogonal to a coordinate axis (hence the SVD) we are able to view the data in such a way that we extract as much information as possible from the underlying system.
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3.2. Embedding parameters The choice of embedding parameters for time delay reconstruction is of considerable importance. However, there is currently no absolute statement concerning which method one should employ. To some extent, the selection of embedding parameters will depend on the purpose of reconstruction. For example, embedding may be considered as an intrinsic part of the modeling process [Judd & Mees, 1998], and doing so seems to improve modeling results [Small & Judd, 1998a; Judd & Small, 2000]. Furthermore by embedding time it is possible to consider nonstationary issues as a modeling problem [Small et al., 2000]. For our purposes we choose fairly straightforward and widely accepted techniques. We estimate embedding lag τ using the first zero of the autocorrelation function and embedding dimension de using the plateau onset of the proportion of false nearest neighbors. These methods, and others, are discussed in more detail in, for example [Abarbanel, 1996; Kantz & Schreiber, 1997].
3.3. Peak detection Peak detection is a substantial problem. Though it seems “obvious” what constitutes a peak of a time series it is difficult to get an automated algorithm to detect all the appropriate peaks and no others. While a review of these techniques is somewhat tangential to the current discussion we will briefly
describe our own approach that provides a parameter free method to select peaks from a time series. We wish to identify the peaks at a rate approximately equal to the fundamental frequency of the time series. That is, we find one “peak” for each “quasi-period”. Hence, our peak detection algorithm proceeds as follows. 1. Calculate the approximate period p of a time series. This can be achieved with either a Fourier transform [Press et al., 1988] or autocorrelation [Priestly, 1989] based technique. 2. Examine a sliding window, the length of which is several times the period p (say 5p). 3. For each sliding window, identify the underlying period pw by the same algorithm as used in step 1. Find the maximum value among the first pw points of the window, record this as a peak. 4. Move the sliding window along so that the last peak is excluded, set p = pw , and proceed from step 2. We find that this method is capable of reliably extracting the peaks of successive periods and adapting to changes in the underlying period, and is independent of the general morphology of the waveform. The algorithm we have described here is reliant on the data having a clearly identifiable (and relatively sharp) peak for every period. This is precisely the type of behavior commonly observed in ECG traces.
Fig. 5. One of the three computational simulations. The Ex , Ey and Ez components are shown here (see Appendix A). This simulation was initiated with a single bent scroll wave. The horizontal axis is datum number, the vertical axis is arbitrary but equal in each case. Note that each of the three components appear visually quite distinct. As the scroll wave becomes orthogonal to a given direction the amplitude in that component approaches 0 while the amplitude in the other components increases. Correlation dimension and surrogate calculations for this data are shown in Figs. 1 and 3 respectively.
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(a)
(b)
(c) Fig. 6. Estimated first return maps for the data illustrated in Fig. 5. From left to right, we have illustrated the reconstructed first return maps for Ex , Ey and Ez . The successive returns are ordered and colored according to that ordering. The ordering is the same from figure to figure, but as the data used in each case is different the exact number of returns (and therefore the exact coloration) is not the same. In each set of panels, the top panel is the two-dimensional embedding (first return map) of successive scalar returns, shown (as a time series) in the lower panel. The choice of embedding parameters (de and τ ) is the same for each case (de = 5, τ = 8).
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4. Results In this section we present the application of the first return map techniques developed in the Sec. 3. Section 4.1 describes the results for computational simulations of VF. Sections 4.2 and 4.3 give the corresponding results for human VF recorded by ambulance defibrillators and in a CCU.
4.1. Simulation results Figure 5 shows the x-, y-, and z-components (Ex , Ey , and Ez ) of a computational simulation of “VF-like” behavior in a numerical caricature of cardiac tissue. A detailed description of the computation of these time series is given in Appendix A. Figure 6 shows the three first return maps computed from each of the three scalar time series depicted in Fig. 5. The qualitative features in each reconstruction are identical — the orientation and proportion are different for each reconstruction, but the temporal evolution of the shape is essentially the same. That is, these three reconstructions share the same topology. The dynamics in each case are clearly equivalent. Furthermore the dynamics revealed here exhibit apparent time dependence. The initial first return map is basically “L” shaped, this corresponds to a stage when substantial organization is still present in the simulation (colored in blues). The first return map dynamics then change to the regions colored in light blues through to light
oranges. Finally, a third distinct region is evident in the part of the first return map colored red. These same three distinct regions may be observed in each of these three reconstructions from three linearly independent time series (Ex , Ey and Ez ). These calculations have also been conducted for the other two simulations described in Appendix A Similar results were obtained for these data. In each case the three components (Ex , Ey and Ez ) produced equivalent results. Furthermore, the same broad features were observed in each simulation. The results we have presented in Fig. 6 are representative. Whereas the correlation dimension estimation varied between orthogonal components the first return map extracted the same features from each orthogonal view.
4.2. Ambulance VF time series In this section we present the results of the estimation of some first return maps for clinically recorded data. The main point we wish to emphasize is that some of these data sets exhibit features that are qualitatively similar to the results in Sec. 4.1 (Fig. 6). We do not claim to observe the same features in all the clinical data set, nor do we claim to provide a clinically significant discriminant. We only wish to demonstrate that the computational simulations, and real VF share some common, nontrivial, features. The data and their classification are described in Appendix B.
Fig. 7. Ambulance defibrillator VF data. Three clinical recordings from a Laerdal defibrillator on board a Norwegian ambulance: (a) h104cmbw1s1, (b) h383cmbw2s2, and (c) h282cmbw5s4. The horizontal axis is datum number, the data was recorded at 100 Hz. Vertical axis is arbitrary — it is the digitized value. Note that panels (a) and (c) exhibit no obvious time dependent features in their dynamics, panel (b) does. Correlation dimension and surrogate calculations for data similar to this is shown in Figs. 2 and 4.
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(a)
(b)
(c) Fig. 8. Estimated first return maps for the three time series illustrated in Fig. 7. Panels (a)–(c) of this figure correspond, respectively, to panels (a)–(c) of Fig. 7. The successive returns are ordered and colored according to that ordering. The ordering is the same from figure to figure, but as the time series are recordings of distinct events, the number of returns (and therefore the coloration) is not the same. In each set of panels, the top panel is the two-dimensional embedding (first return map) of successive scalar returns, shown (as a time series) in the lower panel. The choice of embedding parameters (de and τ ) are: (a) (de = 9, τ = 2), (b) (de = 5, τ = 9) and (c) (de = 5, τ = 8). Panels (a) and (b) exhibit obvious similarities to the first returns maps represented in Fig. 6. Panel (c) exhibits no obvious time dependence. Panels (a)–(c) correspond respectively to recordings exhibiting pulse, no pulse, and continuous VF (according to the classification described in Appendix B).
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Figure 7 depicts three of the clinically recorded data sets. Figure 8 shows three dynamical reconstructions from these three recordings in three different subjects. Two of these data sets exhibit obvious similarities to the results obtained for the computational simulations, the third [Fig. 8, panel (c)] exhibits no time dependent dynamics in the first return maps. Panels (a) and (b) exhibit not only the characteristic “L” shape of the first return maps in Fig. 6, but also three separate phases in the dynamics. In Fig. 8 panel (a), these are colored dark blue, light blue to green, and orange to red, respectively. In panel (b) the dark blue and red phases seem to coincide, marking a return to the initial dynamics late in the time series. The intermediate phase however is distinct (colored light blue) and similar to the second and third phases in the computational simulations. Figure 8, panel (c) has no apparent time dependent structure. We applied this technique to 47 recordings from 14 subjects, each with embedding parameters (de , τ ) selected according to the criteria described in Sec. 3.2. Of these, 28 recordings from 13 subjects exhibited some time dependent dynamics in the first return map. Only one subject did not exhibit time dependent dynamics in any recording (only one recording was available from that subject). Some of these dynamical changes were fairly obvious and may have been observed directly from the time series, some of them were fairly subtle. Many of the time series were particularly short
making discrimination a difficult and possibly subjective task. However, the results presented in Fig. 8 are typical. Figure 8 panels (a) and (b) are definite positive results, panel (c) shows no temporal pattern.
4.3. CCU VF data The results of the previous section are encouraging, but the ambulance data are extremely short and noisy. In this section we present some preliminary results we have obtained, measuring higher resolution ECG from patients in the CCU of the Royal Infirmary of Edinburgh. Figure 9 shows a recording of spontaneous VF from a patient in the CCU. Analysis of this data set is provided in Fig. 10. These results clearly show that there are three distinct dynamical phases — sinus rhythm, ventricular tachycardia (VT) and VF. The first return plots in Fig. 10 clearly distinguish among these phases. It is significant that this method provides such a clear distinction between VT and VF (this is not always a trivial clinical task [Zhang et al., 1999]). However, there appears to be no obvious change in the underlying dynamics during VF. The dynamics observed during VF and VT are consistent with the distinct phases observed in the first return plots of the simulations (Fig. 6). The lack of obvious change in the dynamics during VF is consistent with the observations in which 19 of 47 recordings of VF recorded by ambulance defibrillators also lacked temporal evolution in the dynamics.
Fig. 9. A recording of spontaneous VF from a patient in the CCU. The horizontal axes are datum number (relative to the start of arrhythmia). The vertical axis is surface ECG voltage (in mV). The data was recorded at 500 Hz, so the data shown covers 48 seconds. Defibrillation shock was successfully applied 51 sec after onset of arrhythmia. The first panel shows mostly VT, while the bottom two show VF. The first peak in the top plot corresponds to the last “normal” heart beat (QRS complex).
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(a) 2 minutes before onset to defibrillation
(b) onset to defibrillation
(c) 16 seconds after onset to defibrillation Fig. 10. Estimated first return maps for the time series depicted in Fig. 9. Panels (a)–(c) of this figure are different depictions of the same calculation. Panel (a) is the analysis for the full recording (including two minutes of sinus rhythm preceding arrhythmia. Panel (b) shows only the data depicted in Fig. 9, that is from the onset of the arrhythmia. Panel (c) depicts the section of the data corresponding to well-developed VF. The successive returns are ordered and colored according to that ordering. The ordering is the same from figure to figure, but the coloration is not the same in each plot. In each set of panels, the top panel is the two-dimensional embedding (first return map) of successive scalar returns, shown (as a time series) in the lower panel. The choice of embedding parameters (de and τ ) are (de = 5, τ = 25). Panel (a) clearly shows the transition from sinus rhythm (colored in blue) to arrhythmia (red). Panel (b) shows the transition from VT (dark blue) to VF (light blue to red — from datum 50 onwards), and panel (c) shows no obvious trend during VF.
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5. Conclusion Correlation dimension estimates and surrogate data techniques demonstrate that human ECG recordings during spontaneous VF and computational simulations of VF are clearly distinct from a monotonic nonlinear transformation of linearly filtered noise. The dimension estimates themselves demonstrate that the computational simulations are approximately four dimensional, while human VF recorded by surface ECG is approximately six dimensional. There is some variation in the results, both between recordings (in humans) and between channels (for the simulations). Variation in results between channels would suggest that a single surface measurement may be insufficient to accurately estimate the correlation dimension of the underlying dynamics. However, this is not the case in general. Estimates from multiple recordings from many subjects showed a smaller amount of variation. Perhaps the underlying dynamics in real VF is somehow more easily observable than the somewhat artificial orthogonal components of the pseudoECG. Furthermore, the simulations are short and show fairly obvious changes in the dynamics (that the human VF data sets lack). The transitions are more or less evident in the various channels of the simulated VF recordings. Within the variation observed for human VF we are able to make a reasonable estimate of the dimension of VF dynamics. The estimate of 6 is consistent with our own results from induced VF in pigs [Yu et al., 2000; Small et al., 2000]. Even if it were not possible to estimate the instantaneous dimension of VF from a single lead ECG the values of dimension estimated in this way may still provide a clinically significant indicator [Skinner et al., 1991; Skinner et al., 1992; M¨ akikallio et al., 1999a; M¨ akikallio et al., 1999b; Ivanov et al., 1999], as may other measures from nonlinear dynamics [Wessel et al., 2000; Zhang et al., 1999; Huikuri et al., 1996]. The observation that the noisy time series of VF in humans is distinct from a linear noise process indicates that the noise in these recordings is not excessive. The substantially lower correlation dimension for “pseudo-ECG” time series indicates that this model does not adequately describe the complete mechanism underlying surface ECG measured during VF. A re-entrant scroll wave mechanism may lead to VF, but the measured surface ECG also contains additional high dimensional dynamics and
substantial noise. We observed the closest agreement between real VF and the computational simulations initiated with a single bent scroll wave. This indicates that propagation of a single bent scroll wave is perhaps the most realistic model (of these three) of initiation of VF. Dimension and surrogate analysis provide evidence of nonlinearity in these time series and demonstrate that the results for simulations and human VF and previous results from animal experiments [Small et al., 2000; Yu et al., 2000] are consistent. Therefore, we choose to apply an adaption of the well established technique of first return maps to extract time dependent dynamics from these recordings. The technique of first return maps, as described in this paper, appears to provide a useful tool for tracking the evolution and structure of VF. Three orthogonal measurements of a spatially extended excitable dynamical system exhibited identical structural characteristics. Furthermore, from three separate simulations the first return maps all tracked the evolution and break-up of scroll waves in the original system. Computational biology suggests that scroll wave break-up may be the mechanism underlying spontaneous VF. However, this hypothesis has proved difficult to test. First return maps appear to offer a method to track change in the underlying dynamics with time, so we applied these techniques to clinical data in an effort to extract similar structures. For some recordings (28 of 47) we found significant similar transition in the first return maps. Only one subject from 14 showed no significant transition in any of the recordings — for this subject only one recording was available. Furthermore a scroll wave break-up type mechanism was clearly detected in data recorded from the CCU. This transition corresponded precisely to the transition from VT to VF. This does not provide conclusive proof that a scroll wave break-up underlies spontaneous VF. However, our results are certainly consistent with this mechanism.
Acknowledgments The research is currently funded through a Research Development Grant by the Scottish Higher Education Funding Council (SHEFC), No. RDG/078. We wish to thank K. Morallee for the Laerdal defibrillator data. We are also grateful for the assistance of N. Grubb of the Royal Infirmary of Edinburgh,
Temporal Evolution of Ventricular Arrhythmia 2545
and wish to thank J. Simonotto for assistance in collecting the CCU data.
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Appendix A Computational Simulation of VF We used a cuboid of 100 × 100 × 50 units with FitzHugh–Nagumo excitability (diffusion coefficient 1, β = 0.75, γ = 0.50, and ε = 0.30) [Winfree, 1991] to produce a numerical caricature of re-entrant waves in the myocardium. We solved the three-dimensional cable equations for the excitation variable u and the recovery variable v (time step of 0.03 time units, space step of 0.50 space units). The computational details of this algorithm are discussed in [Biktashev & Holden 1998; Clayton et al., 1999].
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Fig. 11. Scroll wave break-up. A snapshot of the excitation variable u during one of our computational simulations of multiple re-entry waves in cuboid of “heart-like” tissue.
We initiated the medium with three distinct configurations, each intended to generate multiple re-entrant waves; (i) initiation with a single bent wave, (ii) initiation with a ring shaped wave, and (iii) initiation with three separate waves. A typical snapshot of the excitation variable u is shown in Fig. 11. In each case we computed the pseudo ECG E along each of the coordinate axes as a weighted average of the gradient of u; Ex =
100 X 1 i=2
Ey =
[u(i−1,0,0) − u(i,0,0) ]
(A.1)
[u(0,j−1,0) − u(0,j,0) ]
(A.2)
[u(0,0,k−1) − u(0,0,k) ].
(A.3)
100 X 1 j=2
Ez =
i
j
50 X 1 k=2
k
Each simulation consists of a vector time series (Ex , Ey , Ez ) of 5000 points sampled at 0.03 time units.
Appendix B Clinical VF Recordings Episodes of VF and response to defibrillation shocks
were recorded by a Laerdal defibrillator on board ambulances in the Oslo ambulance service. Data were recorded at a frequency of 100 Hz and a resolution of 8 bits. Each data set was then classified according to the dominant rhythm; w1: w2: w3: w4: w5:
Pulse (sinus rhythm), No pulse, Isoelectric (asystole), Noncontinuous ventricular fibrillation, and Continuous ventricular fibrillation.
The length of recording in each case varies, but in general it corresponds to the period from the end of cardio-pulmonary resuscitation until the next defibrillation shock. The period preceding the first shock or subsequent to the last may also be recorded. In some recordings interference from CPR is noticeable at the start of the recording (these time series have been excluded from this study). From 836 recordings of 108 subjects we selected 47 recordings from 14 subjects on the basis of the length of recording, digital resolution and lack of noise and artifacts for further analysis. Results for these 47 data sets are described in this paper. Correlation dimension and surrogate calculations were conducted on the nine longest stationary recordings.
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Appendix C Coronary Care Unit Recordings High resolution recordings of spontaneous VF and its precursors in humans are extremely rare. Episodes of VF occur without warning and monitoring of potential victims is expensive, and somewhat cumbersome. In an effort to obtain high quality recordings of VF and its precursors we have developed and installed a unique data collection facility [Small et al., 2000] in the Coronary Care Unit (CCU) of the Royal Infirmary of Edinburgh.
This facility constantly monitors patients in the CCU and automatically records potential VF waveforms and their precursors. Data are recorded at 10 bits and 500 Hz. Although the digitization is still fairly low, the spatial resolution tends to be adequate (bedside monitors automatically amplify the ECG waveform to provide optimal digitization). Recordings are then manually checked against hospital records to identify true episodes of arrhythmia.
