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Dynamics of Pure Parasystole

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Dynamics of Pure Parasystole Dynamics of pure parasystole LEON GLASS, ARY L. GOLDBERGER, AND JACQUES BELAIR Department of Physiology, McGill University, Montreal, Quebec H3G 1 Y6, and Centre de Recherches Mathimatiques, Universitk de Montrial, Montrkal, Qutibec H3C 3J7, Canada; and Cardiovascular Division, Beth Israel Hospital, Boston, Massachusetts 02215 GLASS, LEON, ARY L. GOLDBERGER, AND JACQUES B~LAIR. not affected by the normal sinus rhythm. The ectopic Dynamics of pure parasystole. Am. J. Physiol. 251 (Heart Circ. focus generates a propagated action potential if the ec- Physiol. 20): H841-H847, 1986.-A mathematical model is topic depolarization falls outside of the normal refractory proposed for ventricular parasystole. In this model, there are two separate rhythms, a sinus rhythm and a ventricular ectopic period of the myocardium. rhythm. An ectopic beat will occur if the ectopic depolarization The concept of ventricular parasystole was formulated falls during a time interval when the ventricles are not refrac- over 70 yr ago (5, 13). Subsequent research has shown tory. Following an ectopic beat there is a compensatory pause. that there can be complex interactions between the par- Analysis of this model, utilizing numerical simulation and asystolic focus and the sinus rhythm. For example, Levy techniques in number theory, demonstrates several new rules et al. (17) demonstrated in experiments in dogs that for parasystole.7 Specifically, for any set of fixed values for the ectopic beats lead to alterations in blood pressure which sinus and ectopic frequencies and the ventricular refractory in turn affect the sinus rhythm via baroreceptor and time, there are at most three different values for the number of sinus beats between ectopic beats. One and only one of these other reflexes. More recently, Jalife, Moe, and co-work- values is odd, and the sum of the two smaller values is one less ers (g-11,20) have argued that a parasystolic focus might than the larger value. The variation in the allowed values of be influenced by the underlying normal sinus rhythm via the number of sinus beats between ectopic beats, as a function electrotonic coupling. The sinus rhythm impulse would of the parameters of the model, is classified. Clinical cases lead to a phase-resetting of the ectopic rhythm. The found in the literature display certain aspects of the predictions resulting rhythm, termed “modulated” parasystole would of the theoretical model. Theoretical analysis of this kind show deviation from the dynamics expected in “pure” provides new approaches to assessing the mechanism of com- parasystole. A number of previously reported clinical plex ventricular arrhythmias. studies were shown to be consistent with modulated parasystole (9). Recent studies have shown that phase ventricular cardiac arrhythmia; number theory; cardiac oscil- lator resetting curves can be derived from electrocardiogram (ECG) records in favorable situations (3,9,21). Further- more, arrhythmias usually ascribed to reentry (such as ventricular bigeminy or trigeminy with fixed coupling VENTRICULAR ARRHYTHMIAS are of major clinical impor- tance because of their role in the pathogenesis of the between sinus and ectopic beats) may also result from a sudden cardiac death syndrome. Ventricular ectopic modulated parasystolic mechanism (9, 10). beats may be due to at least one of the two following Although several theoretical analyses of the dynamics major mechanisms (4): abnormalities in cardiac conduc- of modulated parasystole have appeared (8, 20, 26), a tion (e.g., reentry) and abnormalities in impulse forma- careful theoretical analysis of pure parasystole has not tion (e.g., enhanced automaticity). Perhaps theobest char- yet been made. At first inspection, pure parasystole acterized example of abnormal automaticity is ventricu- appears to present a straightforward situation devoid of lar parasystole. dynamical subtleties. Detailed analysis, however, reveals Ventricular parasystole is a cardiac arrhythmia char- that pure parasystole generates a surprising richness of acterized by three distinctive features on the electrocar- dynamical behavior, including striking regularities and diogram (19, 22) 1) variable coupling intervals between structure that have not been previously described. In the the sinus and the ventricular ectopic rhythm; 2) inter- main body of this paper, directed toward physiologists ectopic intervals that are multiples of a common divisor; and cardiologists, we present the model for ventricular and 3) the presence of fusion beats. The postulated parasystole, describe the dynamics of this model, and mechanism for ventricular parasystole is the presence of discuss the results in the context of clinical cardiology. an autonomous ectopic focus generating beats at a fixed In the APPENDIX, we give derivations of many of the rate. Because of entrance block, this automatic focus is theoretical results. This work shows how techniques in 0363-6135/86 $1.50 Copyright 0 1986 the American Physiological Society H841 H842 DYNAMICS OF PURE PARASYSTOLE number theory (25) can be used in the analysis of cardiac and iterate for 1,000 sinus cycles. By choosing ratios for arrhythmias. which @E equals qts only for large values of p and 4, we minimize the effects of the initial condition, and conse- Theoretical Model of Pure Parasystole quently simulations are carried out only from one initial condition. It would also be possible to choose values of We assume the mechanism for pure parasystole pro- tE and ts so that ptE equals #s for small values for p and posed by Fleming (5) and Kaufman and Rothberger (13), 4, but doing so would require averaging the results over illustrated in Fig. 1. Assume that there is a normal sinus several initial conditions. The rules given in the next rhythm of period ts and an ectopic rhythm with period section are alSO valid for t&s rational (Smt? APPENDIX). tE with tE 2 ts. After each sinus beat there is a refractory In summary, in the mathematical model we system- period, 8. If the ectopic rhythm generates an impulse atically vary the refractory period and the ratio t&s and during the refractory period, then it is blocked. If the determine the resulting dynamics. ectopic rhythm generates an impulse outside of the re- fractory period it leads to an ectopic beat. After each Dynamics ectopic beat, the next sinus beat is assumed to be blocked, resulting in a “compensatory pause”. One way to characterize cardiac rhythms such as par- To simulate this on a digital computer it is essential asystole in which there are frequent extrasystoles is to to choose ts, tE, 8, and the initial phase of the ectopic count the number of normal sinus beats intervening beat in the sinus cycle. For each different set of four between ectopic beats. Studies have shown that occa- parameters, a time sequence giving the number of sinus sionally the number of normal beats between ectopic beats between ectopic beats is generated. Analysis of the beats follows some simple arithmetic laws as in the dynamics as a function of the four parameters is ob- situations in which all intervals are composed of odd viously unwieldy. However, consideration of the mathe- numbers (“concealed bigeminy”), or all intervals assume matical structure of the model for parasystole leads to a a value (3N-1) (“concealed trigeminy”; 22-24). Variants simplification that eliminates two parameters. The fol- of these rhythms have also been described (16). Although lowing considerations simplify the numerical simula- the sequences giving the number of sinus beats between tions: 1) Independent variation of both ts and tE is not ectopic events and histograms of these values can be needed, since we can use ts to set the scale of time and readily determined from ECG data, and readily computed measure tE relative to ts. This reduces the number of in mathematical models, a systematic analysis of such parameters by 1. 2) Two numbers, x1 and x2, are said to distributions has not yet appeared. be rationally related if it is possible to find two integers, We computed the histograms showing the number of p and 4, such that pxl equals qx2. In this case, the ratio, sinus beats between ectopic events over a range of ratios xJ;xz equals 4/p, which is a rational number. In the of ts and tE and for several values of 8. The results of the situation that t, and tE are rationally related, the dynam- simulations can be presented graphically. Let p(a), p(b), ics may be sensitive to the initial condition. This is p(c) be the probability that there are a, b, or c, respec- illustrated in Fig. 2, in which we show the dynamics for tively, sinus beats between two given ectopic events. Note two initial conditions with t&s = 1.5 and O/ts = 0.4. In that p(a) + p(b) + p(c) equals 1. We now plot p(a), p(b), one case there is a periodic pattern in which there are and p(c) as a function of t&s for e/ts equals 0.4 in Fig. two sinus beats between each ectopic beat (Fig. ZA), and 3A, and for O/t, = 0.6 in Fig. 3B. The values of a, b, and in the second case there is a periodic pattern in which c are given on the curves. It is also possible to show the there are alternately no sinus beats and one sinus beat allowed values of a, b, and c in the (O/ts, t&s) plane. between ectopic beats (Fig. 2B). For the situation in This is done in Fig. 4, which is based on numerical which ts and tE are not rationally related (i.e., t&s is calculations and a theoretical construction described in irrational), the asymptotic properties of the dynamics the APPENDIX. The labeled regions show the values for are insensitive to the initial condition. In practice, it is the number of sinus beats between ectopic events.
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