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Computer Modeling of Ventricular Rhythm During Atrial Fibrillation and Ventricular Pacing

Jie Lian^*, Senior Member, IEEE, Dirk Müssig, Volker Lang

Abstract—We propose a unified AF-VP model to demonstrate the effects of ventricular pacing (VP) on the ventricular rhythm during atrial fibrillation (AF). In this model, the AV junction (AVJ) is treated as a lumped structure characterized by refractoriness and automaticity. Bombarded by random AF impulses, the AVJ can also be invaded by the VP-induced retrograde wave. The model includes bi-directional conduction delays in the AVJ and ventricle. Both refractory period and conduction delay of the AVJ are dependent upon its recovery time. The electrotonic modulation by blocked impulses is also considered in the model. Our simulations show that, with proper parameter settings, the present model can account for most principal statistical properties of the RR intervals during AF. We further demonstrate that the AV conduction property and the ventricular rate in AF depend on both AF rate and the degree of electrotonic modulation in the AVJ. Finally, we show that multi-level interactions between AF and VP can generate various patterns of ventricular rhythm that are consistent with previous experimental observations.

Index Terms—atrial fibrillation, ventricular pacing, RR interval, AV junction, electrotonic modulation, concealed conduction, refractory period, AV conduction

I. INTRODUCTION

trial fibrillation (AF) represents the most common sustained cardiac arrhythmia in clinical practice, and is associated with increased morbidity and mortality [1]. Converging evidence suggests that irregular ventricular rhythm during AF, independent of rapid heart rate, significantly contributes to patient symptoms and hemodynamic deterioration [2], [3].

The variation of RR interval during AF has been thought to result mainly from autonomic modulations of the electrophysiological properties of the atria and the atrioventricular (AV) node [4]. It was demonstrated that the ventricular response to AF was influenced by an increase in sympathetic tone and a decrease in parasympathetic tone, but was not necessarily influenced by an increase in parasympathetic dominance [5], despite the observation that the AF frequency was subject to parasympathetic modulations [6]. On the other hand, the irregular ventricular response in AF could further increase the sympathetic nerve activity [7], thus forming a vicious circle of modulations.

The characterization of ventricular rhythm in AF has been controversial, ranging from completely short-term randomness [8], to weak predictability [9], to short-term determinist [10]. Likewise, the mechanism of ventricular rhythm in AF has been a subject of debate for decades [11]-[23]. Things get more complicated when ventricular pacing (VP) is considered. Although it has been known that properly programmed VP could stabilize the irregular ventricular rhythm during AF [24]-[27], the underlying mechanisms remain unclear.

Conventionally, the irregular ventricular response during AF has been explained in terms of decremental conduction and repetitive concealment of the AF impulses within the AV node [11]. This concept was challenged by the postulate that the AV node functions as a pacemaker whose rate and rhythm are modulated by the AF impulses, based on the observation that the short RR intervals during AF could be eliminated by VP at relatively long cycle lengths [12]-[14]. However, that hypothesis could not explain the experimental findings obtained in a canine model, where the VP cycle length resulting in >95% of ventricular captures during AF was linearly related to the shortest spontaneous RR interval during AF [15]. An alternative hypothesis was that the electrotonic modulation of the AV nodal propagation by concealed AF impulses is responsible for the irregular RR intervals during AF [16]. This model predicted an inverse relationship between atrial rate and ventricular rate in AF, which was also indicated in an isolated rabbit heart model [17]. However, recent clinical experience failed to support this prediction [18].

Another quantitative model for the ventricular response during AF was developed by Cohen et al. [19]. In this model, random AF impulses continuously bombard the AV junction (AVJ), which was treated as a lumped structure with defined electrical properties including the refractory period and the automaticity. It was demonstrated that this model could adequately account for most principal statistical properties of the RR intervals during AF [19]. However, several important physiological properties of the heart were missing from this model, such as the conduction delays within the AVJ and the ventricle, and the recovery-dependency of the AVJ properties. Recently, Jorgensen et al. proposed an alternative method to model the AV node by comparing the observed ventricular response during atrial flutter and atrial fibrillation to the predictions of the model. In their model, the AV conduction time was related to the preceding recovery time through a finite difference equation, and the prolongation of AV nodal refractory period by blocked beat was also considered [20]. Several other theoretical models have also been developed by the same group to account for some nonlinear dynamics of the AV conductions or statistical properties of the RR intervals [21]-[23]. Nevertheless, all these models [19]-[23] omitted the effect of VP, thus could not be used to investigate the interactions between AF and VP.

In this paper, we describe a novel AF-VP model. This model can be viewed as an extension and enhancement of Cohen’s AF model, by taking into account ventricular pacing, physiological conduction delays, and electrotonic modulation in the AVJ. In this regard, our AF-VP model provides a unified platform wherein previous concepts can be incorporated. As we demonstrate later, with different model parameter settings, this model can account for most known experimental observations.

II. Methods

A. Model Components

Fig. 1 illustrates the schematic drawing of the present computer model, which consists of four inter-connected components: AF generator, AVJ, ventricle, and electrode.

The AF generator simulates the turbulent electrical activity in the atrium, generating a series of AF impulses bombarding the AVJ. The generation of the AF impulses is characterized as a truncated Poisson process that has a mean arrival rate l with an upper limit 20/s (i.e., mean AF cycle length or PP interval 1/l with a lower limit 0.05s [28]). (As in [19], we assume that the AF impulses arrive in AVJ randomly in time at a mean rate l. A truncation threshold 20/s is added to prevent generating non-physiologic AF frequency. Accordingly, the mean AF cycle length 1/l, which follows the exponential distribution, is truncated at a lower limit of 0.05s to simulate a certain degree of atrial refractoriness [28]. That is, following an atrial depolarization, a minimum of 50ms recovery period is needed for the atria to be depolarized again.) On the other hand, any retrograde penetration of the atrium by an activation wave escaping the AVJ after a retrograde AV delay (see below) collides with an incoming AF impulse, causing the AF generator to reset its timing cycle.

The AVJ is modeled as a lumped structure with defined electrical properties. The AVJ fires when its transmembrane potential, V_m, reaches the depolarization threshold, V_T, as a result of either antegrade activation or retrograde activation (see below). The firing of the AVJ generates an activation wave, which starts an antegrade or retrograde AV delay according to the direction of activation. If the AVJ is retrograde activated while an antegrade wave has not finished

Fig. 1. Schematic drawing of the present AF-VP model.

its AV delay (or vice versa), a collision within the AVJ occurs that annihilates the activation waves in both directions [13], [14], [29].

If an antegrade AV delay expires, an activation wave is generated in the ventricle and starts an antegrade conduction delay (VD_ant). The delivery of VP also generates an activation wave in the ventricle with a retrograde conduction delay (VD_ret), after which period the retrograde wave reaches the AVJ. When both antegrade and retrograde waves are detected in the ventricle, a ventricular fusion beat is predicted, causing the extinction of both waves [13], [14], [30].

The electrode is implanted in the ventricle, and is connected to a pacing device operating in demand mode VVI. If an activation wave propagates to the electrode after an antegrade conduction delay, a ventricular sense (VS) occurs that inhibits the scheduled VP, whereas the timeout of the pacing basic interval, BI, initiates the VP.

B. AVJ Properties

As in [19], the AVJ can be activated due to combined effect of spontaneous phase-IV depolarization and the AF bombardments. In addition, the AVJ can also be activated by the invasion of a VP-induced retrograde wave. As illustrated in Fig. 2(a), the activation of the AVJ starts a refractory period, t, during which time the AVJ is non-responsive to stimulation by both the AF impulses and the VP-induced retrograde waves. When the AVJ refractory period expires, V_m returns to the resting potential, V_R, and starts a linear increase at a rate dV/dt. Each time an AF impulse reaches the AVJ during phase-IV, V_m is increased by a discrete amount DV. On the other hand, if the AVJ is penetrated by a VP-induced retrograde wave during phase-IV, V_m is brought to V_T immediately. The AVJ recovery time, RT, is defined as the interval between the end of the last AVJ refractory period and the current AVJ activation time.

The AV delay (AVD) is known to depend on the AVJ recovery time [16], [20]-[23]. Their relationship is modeled by the following equation:

(1)

where AVD_min is the minimum AV delay when RT ® µ, a is the maximum extension of the AV delay when RT = 0, and t_c is the conduction-curve time constant [20]-[23]. The antegrade and retrograde AV delays may share the same set of parameters (as in the present study), or these parameters may be independently adjusted for bi-directional AV delays. Fig. 2(b) illustrates the recovery-dependency of the AV delay.

Furthermore, the refractory period (t) of the AVJ has been recognized as rate-dependent: the higher the heart rate, the shorter the t [19]. We further extend this model by assuming t depends on the AVJ recovery time, with their relationship given by:

(2)

where t_min is the shortest AVJ refractory period corresponding to RT = 0, b is the maximum extension of the refractory period when RT ® µ, and t_r is the refractory-curve time constant. The recovery-dependency of the AVJ refractory period is illustrated in Fig. 2(c). For simplicity, we allow only one activation wave in the AVJ. This is achieved by the additional constraint t ≥ AVD. That is, the AVJ is refractory if there is an antegrade or retrograde wave in the AVJ.

Finally, the electrotonic modulation of the AVJ refractory period by blocked impulses is also incorporated within our model [16], [20]. If an impulse (either antegrade or retrograde) is blocked by the refractory AVJ, the refractory period of the AVJ is prolonged by that concealed impulse, according to the following equation:

(3)

where t and t’ are respectively the original and prolonged refractory periods, and t is the time when the impulse is blocked, i.e., 0 < t < t. Note that the degree of the AVJ refractory period extension depends on both the timing and the strength of the blocked impulse, which are independently modulated by two positive parameters q and d, respectively. The longest possible extension of the AVJ refractory period is t_min, which is the shortest possible AVJ refractory period corresponding to zero recovery time. As illustrated in Fig. 2(d), this occurs when a supra-threshold AF impulse (i.e., DV/(V_T - V_R) ≥ 1) or a VP-induced retrograde wave (in which case we set DV/(V_T - V_R) = 1) is blocked at the end of the refractory period (i.e., t ® t). The extension will be shorter if the impulse is blocked at the early phase of the refractory period, or if the impulse is weaker in strength. Note that the degree of electrotonic modulation increases as the control parameters q and d decrease.

C. Simulation Protocols

We first examine the model’s behavior during AF in the absence of VP. The statistical properties of the RR intervals, namely, histograms and autocorrelation functions, are examined by varying model parameters. In addition, we study the impact of electrotonic modulation on the conduction property of AVJ, and the relationship between PP intervals and RR intervals.

TABLE I

PROGRAMMABLE MODEL PARAMETERS

Furthermore, we investigate the ventricular response when both AF and VP are present. We investigate the multi-level interactions between AF-induced antegrade waves and VP-induced retrograde waves. We further evaluate the effect of VP on RR intervals in AF, and compare the results with previous experimental observations.

Table I summarizes the basic set of programmable parameters that are used in the present model. The default values of these parameters are also listed unless otherwise specified in the text. As shown in [19], the first four model parameters listed in Table I actually have only two degrees of freedom, specified by the relative amplitude of the AF impulses, DV/(V_T - V_R), and the relative rate of the AVJ phase-IV depolarization, (dV/dt)/(V_T - V_R).

III. Results

A. RR Intervals During AF

In the absence of VP, the present model can be viewed as a more general version of the Cohen’s AF model [19], with additional considerations of the AVJ properties, including the recovery-dependent conduction delay, and the recovery-dependent refractory period which can be eletrotonically modulated by the concealed impulses. Therefore, with proper parameter settings, the present model can account for most principal statistical properties of the RR intervals during AF as shown in [19].

Fig. 3 shows four representative examples of the model generated RR interval sequences (500 beats each, top panels), together with their histograms (middle panels) and the autocorrelation functions (bottom panels). The BI is set sufficiently long (10s) to ensure that VP is inhibited. In the case shown in Fig. 3(a), the excitation of the AVJ is dominated by the arrival of AF impulses (l = 8/s, DV = 20mV), while the rate of spontaneous phase-IV depolarization is negligible (dV/dt = 0mV/s). Marked by a single broad peak, the resulting RR interval histogram resembles the unimodal Erlang function that has been observed experimentally and predicted theoretically. In the case shown in Fig. 3(b), both the spontaneous phase-IV depolarization (dV/dt = 50mV/s) and the AF bombardments (l = 4/s, DV = 10mV) significantly contribute to the AVJ excitation. The resulting RR interval histogram is characterized by multiple evenly spaced delta functions superimposed on the pieces of Erlang curves. In the case shown in Fig. 3(c), the AVJ has fast rate of spontaneous phase-IV depolarization (dV/dt = 150mV/s), while its transmembrane potential is also modulated by frequent but weak AF impulses (l = 15/s, DV = 6mV). The resulting RR interval histogram has a narrow peak superimposed on a smooth background. Fig. 3(d) shows another case where high degree of electrotonic modulation (q = 0.5, d = 1) is considered. The AVJ excitation is predominantly controlled by the AF bombardments (l = 6/s, DV = 45mV) without consideration of the spontaneous phase-IV depolarization (dV/dt = 0mV/s). The resulting RR intervals show multimodal distribution that could be attributed to the concealed AF impulses. Note that, in all examples shown above, the RR intervals separated by more than 1 beat are essentially uncorrelated. Although positive lag-1 autocorrelation is often observed in simulations (see Figs. 3(a), (b), and (d)), the correlation between two consecutive beats can be weak or negative (see Fig. 3(c)) [10], [23].

B. Effects of Electrotonic Modulation During AF

We further investigate the effects of electrotonic modulation on AV conduction property and RR intervals during AF.

Fig. 4(a) plots the mean RR intervals (each point is averaged over 500 beats) vs. the mean PP intervals during AF under three different parameter settings: (1) DV = 30mV, q = 10, d = 10, t_min = 0.03s; (2) DV = 40mV, q = 0.5, d = 0.5, t_min = 0.03s; and (3) DV = 40mV, q = 0.5, d = 0.5, t_min = 0.05s. The respective P/R ratios (the number of AF impulses divided by the number of conducted beats) are plotted in Fig. 4(b). Using the same sets of (q, d, t_min) while changing DV = 50mV, a similar PP-RR relationship is shown in Fig. 4(c), and the corresponding P/R ratios are plotted in Fig. 4(d). For all cases, the effect of spontaneous phase-IV depolarization is eliminated (dV/dt = 0mV/s). Therefore, the AVJ depolarization is solely determined by the AF rate l, which is varied from 5/s to 18/s (step size 1/s) to generate different mean PP intervals. Note that the PP-RR relationship is inherently linked to the P/R ratio, such that at any given AF frequency, the mean RR interval can be derived from the product of the mean PP interval and the P/R ratio.

We first test the model by considering random AF only. We show that the model can account for various patterns of RR interval distributions in AF that have been previously observed (see Fig. 3) [19]. We further demonstrate that both positive and negative correlations between atrial rate and ventricular rate are possible during AF [17], [18]. The AV conduction property and the ventricular rate in AF depend on both AF rate and the degree of electrotonic modulation in AVJ (see Fig. 4). Finally, we test the model by considering both AF and VP. We show that the VP-induced retrograde waves may interact with the AF-induced antegrade waves at different levels (see Fig. 5). Such multi-level AF-VP interactions can generate various RR interval patterns that are in agreement with experimental observations [12], [15]. For example, we demonstrate that VP can suppress short intrinsic RR cycles in AF, but the difference between the VP cycle length resulting in >95% ventricular pacing and the shortest RR interval is inversely related to the intrinsic ventricular rate in AF (see Fig. 6).

The present model provides a unified platform wherein previous concepts can be incorporated, such as AVJ automaticity and refractoriness, recovery-dependent AVJ properties, bi-directional conduction delays, electrotonic modulations, etc. Although Wittkampf’s model also considered AVJ automaticity [12]-[14], it did not include the AVJ refractoriness, and its hypothesis that VP suppressed AVJ automaticity was not supported by experimental findings [15]. The concealed conduction model states that the blocked impulses at AVJ affect the electrophysiological state of the AVJ [31]. This has been explained in terms of electrotonic modulation and demonstrated using multi-cell models [16]. A more simplified model prolongs the AVJ refractory period by a fixed interval after a blocked impulse [20]. We further extend this model by assuming the degree of electrotonic modulation depends on both the timing and strength of the blocked impulse. In accordance with Jorgensen et al. [20], the AV conduction delay in this model is recovery-dependent. We wish to emphasize that, within the same model framework, it is flexible to choose different formulas other than (1)-(3) to characterize the AV delay, refractory period, and the effect of electrotonic modulation. For example, the calculation of AV delay may be refined to incorporate other AVJ properties, such as facilitation and fatigue [21], [22]. Alternatively, the definition of recovery time may be decoupled for the AV delay and refractory period. That is, AVD may be a function of the conduction recovery time (defined as the interval from the end of AV conduction to the next AVJ activation), while t may be a function of the refractory recovery time (defined as the interval from the end of refractory period to the next AVJ activation).

The present AF-VP model may be useful in a variety of applications. For example, different antiarrhythmic drugs may have different effects on various electrical properties of the heart, such as the conduction time, the refractory period, the automaticity, etc. Therefore, the present model may provide a quantitative framework to investigate drug effects by fitting the model to experimental data involving pharmacological intervention. Although a simultaneous search over all model parameters is technically difficult, the dimension of the search space may be reduced by deriving some baseline parameters independently. For example, the basic parameters pertaining to AV conduction delay and refractory period (e.g., minimum value, maximum extension, time constant) may be derived from respective recovery curves (see Figs. 2(b) and (c)), which can be obtained during application of standard atrial stimulation protocols (associated with 1:1 AV conduction) [32]. The parameters pertaining to electrotonic modulation may be found through a scanning process as detailed in [20], or estimated through more elaborate pacing protocols involving concealed conduction [33]. Other model parameters (e.g., AF frequency, relative impulse strength and rate of phase-IV depolarization) may be deduced by fitting the model with recorded RR interval distribution as detailed in [34]. Also note that the AF generator can be easily modified to generate other random or determined processes, to simulate other types of supraventricular arrhythmia (e.g, atrial flutter and AV junctional tachycardia), or the sinus rhythm with certain heart rate variability. Although the ventricular pacing is simulated in VVI mode in the present study, it can be easily extended to incorporate other pacing control algorithms that were designed for ventricular rate stabilization during AF [24]-[27]. Therefore, the present model can be used to investigate possible interactions between AF and VP, and to evaluate the performance of different pacing algorithms in a simulation environment [35], [36].

While the present model can generate various patterns of ventricular rhythm that are consistent with previous experimental observations, a more direct validation of the model in real experiments is needed to confirm its concrete behavior. It has been recognized that the AV conduction not only depends on the recovery time, but also is affected by the autonomic modulation [37]-[39]. Other nonlinear dynamics of the AV conduction under specific conditions were also reported, such as alternans [40] and hysteresis [41], [42]. In addition, the ventricle is simplified as a conduction compartment in the present model, while its depolarization and repolarization properties are not addressed. Despite these limitations, the present AF-VP model provides a flexible framework to study the interactions between AF and VP. Further refinement of this model by adding more realistic features will be addressed in a future study.

Acknowledgment

The authors are grateful to Dr. S. E. Greenhut for helpful discussions on the AF model and to R. A. Schomburg for expert assistance in preparation of the manuscript.

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Jie Lian (S’98-M’02–SM’05) received the B.S. and M.S. degrees in biomedical engineering from Zhejiang University, China, in 1992 and 1995, respectively. He received the Ph.D. degree in bioengineering from the University of Illinois at Chicago, Chicago, Illinois, in 2002.

He was a Research Associate at Zhejiang University, China, from 1995 to 1997. Since 2002, he has been a Senior Clinical Research Biomedical Engineer at Micro Systems Engineering, Inc., Lake Oswego, Oregon. His research interests include feature design and algorithm development for implantable cardiac devices (pacemakers and defibrillators), cardiac and neural electrophysiology, high-resolution functional imaging of heart and brain electrical sources, physiological system modeling, biomedical signal processing and interpretation. He has published 2 book chapters and over 50 peer-reviewed journal and conference papers.

Dr. Lian is a member of Sigma Xi.

Dirk Müssig was born in Gera, Germany, in 1970. He received the M.S. degree in physics from the University of Jena, Germany and the Ph.D. degree in physics from the University of Erlangen, Germany, in 1996 and 2002, respectively.

He is currently a Senior Manager for Applied Research at Micro Systems Engineering, Inc., Oregon, USA. From 1996 to 2002, he worked as a Researcher at the Department of Biomedical Engineering of the University of Erlangen, Germany. His professional research interest is signal processing of intracardiac signals for implantable cardiac devices.

Volker Lang received the M.S. and Ph.D. degrees in physics from the Friedrich-Alexander-University in Erlangen, Germany, in 1994 and 1999, respectively. From 1994 to 1999, he was a Research Associate at the Institute for Biomedical Engineering at the University of Erlangen. Since 2000 he has been the Director for Applied Research at Micro Systems Engineering Inc., Lake Oswego, Oregon. His research interests include modeling of the heart on cellular levels for studying atrial arrhythmias, sensor development for cardiovascular applications, biomedical signal processing, and expert systems.

Manuscript received June 16, 2005; revised February 20, 2006. Asterisk indicates corresponding author.

*J. Lian is with the Micro Systems Engineering, Inc., Lake Oswego, OR 97035, USA (e-mail: jie.lian@ biotronik.com).

D. Müssig is with the Micro Systems Engineering, Inc., Lake Oswego, OR 97035, USA.

V. Lang is with the Micro Systems Engineering, Inc., Lake Oswego, OR 97035, USA.

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