## Abstract

Atrial fibrillation (AF) is characterized by short and irregular ventricular cycle lengths (VCL). While the beneficial effects of heart rate slowing (i.e., the prolongation of VCL) in AF are well recognized, little is known about the impact of irregularity. In 10 anesthetized dogs, R-R intervals, left ventricular (LV) pressure, and aortic flow were collected for >500 beats during fast AF and when the average VCL was prolonged to 75%, 100%, and 125% of the intrinsic sinus cycle length by selective atrioventricular (AV) nodal vagal stimulation. We used the ratio of the preceding and prepreceding R-R intervals (RR_{p}/RR_{pp}) as an index of cycle length irregularity and assessed its effects on the maximum LV power, the minimum of the first derivative of LV pressure, and the time constant of relaxation by using nonlinear fitting with monoexponential functions. During prolongation of VCL, there was a pronounced decrease in curvature with the formation of a plateau, indicating a lesser dependence on RR_{p}/RR_{pp}. We conclude that prolongation of the VCL during AF reduces the sensitivity of the LV performance parameters to irregularity.

- vagus
- relaxation
- hemodynamics
- atrial fibrillation

the worsening of left ventricular (LV) performance during atrial fibrillation (AF) is due to an increase of both the ventricular rate and the ventricular rate irregularity (5, 6, 32). Our previous work (29,32) has shown that ventricular rate slowing during AF improves ventricular performance, although it is still associated with substantial irregularity.

The randomly timed beats during AF result in a complex hemodynamic sequence of events. Contractility during AF changes on a beat-by-beat basis and its exact evaluation remains debatable (3, 4,28). Previous computational models, as well as experimental data, have shown that beat-to-beat contractility in AF may be reasonably estimated from the ratio of preceding and prepreceding R-R intervals (RR_{p}/RR_{pp}) (26) that determine mechanical restitution and potentiation (26, 27, 31). However, these studies used a simple linear model, whereas the relationship between contractility and RR_{p}/RR_{pp} (26) is intrinsically nonlinear. Moreover, the effects of the underlying average ventricular rate on the shape of the contractility-to-RR_{p}/RR_{pp} relationship have never been assessed. Finally, whereas mechanical restitution also affects LV relaxation (18), the relaxation to RR_{p}/RR_{pp} relationship in AF was not investigated.

The major aim of this study was to assess the role of the cycle length irregularity on ventricular performance during AF by studying the relationships between several hemodynamic parameters [maximal LV (LV_{max}) power, the minimum of the first derivative of LV pressure (dP/d*t*
_{min}), and the time constant of isovolumic pressure decay (τ)] and the ratio RR_{p}/RR_{pp} during various underlying average ventricular rates. We applied nonlinear estimation methods to determine the LV performance-interval relationships.

## METHODS

The study was approved by the Institutional Animal Research Committee and is in compliance with the National Institutes of Health*Guide for the Care and Use of Laboratory Animals*.

#### Terminology.

The ventricular cycle length (VCL) is the time interval between two consecutive ventricular depolarizations. It can be measured between the QRS complexes of the surface ECG or by the use of electrical signals recorded directly from the ventricles. In this study, an individual VCL was determined as the R-R interval between two consecutive right ventricular electrograms.

During regular rhythm there is a simple relationship between the constant VCL and the heart rate (HR), namely, HR (beats/min) = 60,000/VCL (ms). However, because the VCL changes from beat to beat (especially during AF), only an average HR can be determined based on the average VCL of a large number of beats. According to the above equation, a prolongation of the average VCL is equivalent to a slowing of the average HR. In the following text, we will quantify the changes in the HR by using the parameter (average) VCL. Accordingly, its irregularity will be measured as a ratio of two consecutive R-R cycle lengths (see subsequent text).

#### Surgical preparation.

The study procedures have been previously described (29,32). The study was performed using 10 healthy open-chest dogs. Briefly, anesthesia was maintained with 1–2% isoflurane, with monitoring of volume status, arterial blood gases, and body temperature. Subcutaneous needle electrodes obtained a standard ECG. Custom-made quadripolar Ag-AgCl plate electrodes were sutured to the high right atrium and right ventricular apex for recording local electrical activity. Similar bipolar plate electrodes were sutured to two epicardial fat pads that contain parasympathetic (vagal) neural pathways selectively innervating the sinus node and the AV node (23). A Millar catheter was inserted through the left carotid artery and advanced into the left ventricle for pressure measurements. A flowmeter probe (model 16A/20A, HT 207, Transonic Systems; Ithaca, NY) was placed around the ascending aorta. All signals were amplified, displayed, and stored on a dedicated recording system (CardioLab, GE Marquette Medical Systems).

#### Study protocol.

After the recording of the hemodynamics at normal sinus rhythm, AF was induced by rapid right atrial pacing (20 Hz, 1 ms) and was further maintained by subthreshold stimulation (20 Hz, 50 μs) of the sinus node fat pad located near the right upper pulmonary vein-atrial junction (23). In several cases, AF was maintained simply by continuing the rapid right atrial pacing.

The following four steps were then performed in each experiment. In*step 1*, after the hemodynamics stabilized for a minimum of 15 min, data were recorded during “fast” AF. The term fast is used to stress that this was the step with the fastest ventricular rate (or the shortest VCL) in each animal, which was observed before any modifications of the rate were attempted.

In *steps 2*–*4*, while the atria continued to fibrillate, three longer average VCLs were achieved by graded AV nodal vagal stimulation. For quantitative purposes, these VCLs were expressed as a percentage from the spontaneous sinus cycle length (SCL) in each animal. Because the average VCL observed during fast AF was only 58 ± 5% of SCL, the three longer VCL were chosen as 75%, 100%, and 125% of the SCL (29).

The computer-controlled vagal stimulation consisted of brief bursts of current impulses synchronized with the ECG (Fig.1) and was applied onto the fat pad located at the left atrium-inferior vena cava junction (23). This epicardial structure contains vagal pathways to the AV node and its activation produces selective dromotropic effects and a slowing of the ventricular rate (22, 23). The role of the computer was the following: *1*) to determine the duration of each VCL on a beat-to-beat basis, *2*) to compare the measured value with the predetermined target level (75%, 100%, or 125% of SCL), and *3*) to adjust instantaneously the amplitude of the vagal stimulation so that the error between the measured VCL and the target is minimized (32).

We randomized the order of the protocol steps, and a rest period of 5–10 min was allowed after each step, when the fibrillation of the atria was maintained uninterrupted. Thus a total of 40 data sets (fast AF plus 3 levels of vagal stimulation in each of the 10 animals) was obtained.

#### Data analysis.

Electrograms and hemodynamic data were digitized at 1 kHz per channel for up to 1,000 ventricular beats in each AF period. The beats numbered 200–700 were analyzed by customized software.

We used three parameters to assess LV performance. First, mechanical performance was characterized by LV_{max} power calculated as a product of the peak aortic flow and peak LV pressure and expressed in watts (2). Second, peak relaxation was characterized by dP/d*t*
_{min}. Finally, the time course of relaxation was characterized by τ. It was calculated by fitting the pressure-time data (from the point of dP/d*t*
_{min}to LV pressure 5 mmHg higher than next LV end-diastolic pressure) to the equation (24)
where P_{b} is the pressure decay asymptote, P_{o} is the pressure at dP/d*t*
_{min}, and*t* is a time at P(*t*) referenced to time of dP/d*t*
_{min} occurrence. An interval of >25 ms was considered necessary to calculate τ.

We measured the duration of each VCL electronically by using the electrograms recorded at the right ventricular apex, as the time interval R-R (in ms) between the upstrokes (maximum first derivative) of two consecutive responses. The VCL irregularity was expressed as a ratio of two consecutive R-R intervals. As shown in Fig. 1, for each ventricular beat (e.g., current LV response in Fig. 1) two characteristic cycle lengths were determined. RR_{p} is the cycle length immediately preceding the analyzed beat. RR_{pp} is the prepreceding one. The ratio RR_{p}/RR_{pp} was used in subsequent calculations related to the analyzed beat (see below).

The parameter RR_{p}/RR_{pp} has been previously used to estimate the average values of hemodynamic parameters during AF. However, only linear regressions have been considered (26, 27). In the present study, the relationship between LV_{max} power or (dP/d*t*
_{min}) and RR_{p}/RR_{pp} was modeled by the following nonlinear equation
Equation 1
where R_{max} is a maximum response (plateau of the relationship), RR_{p}/RR_{pp}
_{min} is minimum RR_{p}/RR_{pp} at which aortic flow could still be observed, and *C* is the curvature of the relationship.

Similarly, the relationship between τ and RR_{p}/RR_{pp} was fitted to the equation
Equation 2where τ_{max} is maximum τ, τ_{min} is minimum τ (plateau of the relationship), and RR_{p}/RR_{pp}
_{min} is the minimum RR_{p}/RR_{pp} at which τ could be determined.

#### Comparison of LV_{max} power and contractility index E_{max}.

Because LV_{max} power is preload dependent (11) and therefore is not an optimal contractility index, we compared it to the theoretical normalized maximal elastance (*E*
_{max}; the slope of LV end-systolic pressure volume relationships) to evaluate how close the two indexes were correlated in the framework of the present experimental model. We used the equation derived by Yue et al. (31) that has been previously validated in a numerical simulation of contractility in AF (26) and calculated the normalized*E*
_{max} from actual RR_{p} and RR_{pp} intervals in six randomly picked data sets. The normalized *E*
_{max} was then compared by linear regression to the LV_{max} power measured in the same data sets.

We also determined the relationship between the normalized*E*
_{max} and RR_{p}/RR_{pp} during fast AF and when the VCL was prolonged to 125% SCL.

#### Statistical analysis.

Data are represented as means ± SD. The presence of nonlinear components in LV performance − RR_{p}/RR_{pp} relationships was confirmed in six randomly chosen data sets. For this purpose, after initial simple linear regression analysis, a fourth-order polynomial equation was fitted to the residuals.

The impact of VCL prolongation on the parameters in *Eqs.1
* and *
2
* (i.e., *C*, plateau, and RR_{p}/RR_{pp}
_{min}) was tested by one-way repeated-measures ANOVA, followed by Huynh-Feldt correction or with Friedman ANOVA if a strong violation of sphericity or variance homogeneity assumptions was observed.

The *C* of the three hemodynamic parameters (LV_{max}power, dP/d*t*
_{min}, and τ) was compared by two-way repeated-measures ANOVA, followed by contrast analysis. A*P* value of <0.05 was considered significant.

## RESULTS

#### Comparison of theoretical normalized E_{max} and LV_{max} power.

The coefficients of correlation between the normalized*E*
_{max} and the LV_{max} power in six evaluated data sets ranged from 0.86 to 0.92 (*P* < 0.0001 for all). However, as shown in Fig.2, the LV_{max} power underestimated the normalized *E*
_{max} when*E*
_{max} values were <0.75.

As shown in Fig. 3, the relationship between the theoretical normalized *E*
_{max} and RR_{p}/RR_{pp} could be approximated with a linear function during fast AF but showed a marked nonlinear component when the VCL was prolonged to 125% SCL, indicating a reduced dependence of contractility on the RR_{p}/RR_{pp}.

While these observations cannot be generalized to indicate that LV_{max} power is a surrogate measure of contractility, the close correlation with *E*
_{max} suggests that a similar nonlinear dependence on RR_{p}/RR_{pp} should be expected for LV_{max} power. The following results confirmed this expectation.

#### Global characterization of LV performance parameters during AF at various average VCL.

The parameters characterizing the ventricular rate and its variance in the studied animals are shown in Table 1. The average VCL during fast AF (285 ± 54 ms) was 58% of the intrinsic SCL (489 ± 81 ms). During vagally induced slowing, average VCL was prolonged to 371 ± 62 ms (75% SCL), 486 ± 81 ms (100% SCL), and 601 ± 91 ms (125% SCL), respectively. In an average animal, the standard deviation of both the VCL and the ratio RR_{p}/RR_{pp} showed a tendency to increase during vagal nerve stimulation.

To illustrate the impact of VCL prolongation on LV performance, in Fig.4 the measured values for each of the hemodynamic parameters in a typical animal are presented in the scatterogram format of Lorenz (or Poincaré) plots (1, 10,16). Briefly, each point in the Lorenz plot has as an*x*-coordinate the “current beat” value of a parameter and as a *y*-coordinate the “preceding beat” value of same parameter. The Lorenz plots permit an easy and fast evaluation of tendencies, as well as some quantification of the properties of distribution. Thus it can be seen that prolongation of the average VCL (from fast AF to 125% SCL) produced uniformed changes in all studied parameters. First, the original “clouds” present during fast AF (Fig. 4, *left*) were transformed into progressively more tightly packed groups, indicating less variable LV_{max}power, dP/d*t*
_{min}, and τ. Quantitatively, this resulted in a smaller scattering index *S* for longer VCL (8). Second, the scatterogram mean (red dots) indicated that with longer VCL there was an increase in LV_{max} power and dP/d*t*
_{min}, whereas τ decreased. For example, the proportion of the beats with LV_{max} power <1 dropped from 50.4% (during fast AF) to 5.3% (at VCL = 125% SCL).

#### Nonlinear relationship between LV performance parameters and RR_{p}/RR_{pp} at various average VCL.

The above scatterograms do not permit quantification of the role of the cycle length irregularity. For that purpose, we determined the nonlinear relationships between LV performance parameters (LV_{max} power, dP/d*t*
_{min}, and τ) and RR_{p}/RR_{pp}. Figure5 illustrates the results obtained in one representative experiment. The regression lines for each level of VCL are shown along with the raw data points. Figure 5, *E*,*J*, and *O*, contain the four superimposed regression lines for each parameter, respectively.

As shown in Fig. 5, *A*–*E*, for the LV_{max} power, during VCL prolongation the plateau, the RR_{p}/RR_{pp}
_{min}, and*C* progressively decreased (*P* < 0.0001 for all three parameters). The combined result of these changes was an increasing value of LV_{max} power at RR_{p}/RR_{pp} = 1, shown with the prolongation of VCL (Table 2,*P* < 0.0001).

Similarly, as shown in Fig. 5, *F*–*J*, for the parameter dP/d*t*
_{min}, during VCL prolongation the plateau, the RR_{p}/RR_{pp}
_{min}, and *C* progressively decreased (*P* < 0.04 for all three parameters). Again, this resulted in a larger absolute value of dP/d*t*
_{min} at RR_{p}/RR_{pp} = 1 with longer VCL (Table3, *P* < 0.0001).

Finally, as shown in Fig. 5, *K*–*O*, for τ, during VCL prolongation the plateau and the *C* did not change (Table 4, *P* = 0.08 and 0.1, respectively). However, τ at RR_{p}/RR_{pp} = 1 still decreased, i.e., the relaxation improved (Table 4, *P* < 0.0001). This was due to a leftward shift of τ − RR_{p}/RR_{pp} relations at longer VCL (Fig. 5
*O*).

There was a highly significant difference in *C* between the three LV performance parameters (*P* < 0.0001). In particular, the *C* for LV_{max} power was larger than for dP/d*t*
_{min} and τ (*P* = 0.003 for both vs. LV maximal power).

## DISCUSSION

#### Major findings.

These data demonstrate that during AF the relationship between VCL irregularity (expressed as the ratio of two subsequent coupling intervals, RR_{p}/RR_{pp}) and LV performance parameters is curvilinear and that this curvilinearity increases as the average VCL is prolonged. For this reason, during fast AF, both LV_{max} power and relaxation are more dependent on RR_{p}/RR_{pp}, whereas this dependence is attenuated at slower ventricular rates (longer average VCL).

Previous studies have established that cardiac contractility is not constant during AF (3) and that mechanistically its beat-to-beat irregularity is governed in part by the effects of restitution and potentiation (26). While both of these effects depend nonlinearly on the RR_{p} and RR_{pp} coupling intervals (31), a unique linear relationship has been described between contractility and the RR_{p}/RR_{pp} (26). In fact, we have proposed a similar linear relationship as a tool for evaluation not only of contractility but also for assessment of a broader selection of hemodynamic parameters during AF (27).

The present study extends previous observations by elucidating the role of the prevailing average ventricular rate (or average VCL) during AF and by defining the relaxation-interval relationships during AF. In particular, we established that nonlinear dependence on the RR_{p}/RR_{pp} better describes ventricular performance, especially during longer average VCL, and thus provided a more comprehensive explanation for the benefits of the slowed ventricular rate during AF.

#### Assessment of LV mechanical performance and relaxation during AF.

Evaluating cardiac contractility during AF is still problematic in view of the absence of a standard approach. The maximum rate of change of the LV pressure (+dP/d*t*
_{max}) has been frequently used to evaluate cardiac contractility during AF (28), although its usefulness is limited by beat-to-beat variations in preload. The LV maximal elastance (*E*
_{max}, the slope of end-systolic pressure-volume relationships) is frequently referred to as the “gold standard” measure of cardiac contractility (12). This parameter is preload independent, but its determination requires the generation of a family of stable pressure-volume loops while varying the diastolic filling. The determination of an accurate *E*
_{max} during AF is therefore associated with significant difficulties (30) and a model-derived theoretical normalized *E*
_{max}(26, 31), has been proposed as an approximation.

In this study, we found a good correlation between LV_{max}power and the normalized *E*
_{max} calculated as a function of the RR_{p} and RR_{pp}intervals, using the equation derived by Suzuki et al. (26) and Yue et al. (31). As shown in Fig. 2, these two variables were closely and positively correlated. Furthermore, we also demonstrated that both the experimentally measured LV_{max} power (Fig. 5) and the calculated normalized*E*
_{max} (Fig. 3) exhibit a nonlinear dependence on the ratio RR_{p}/RR_{pp}. Despite these similarities, however, the present data establish only the LV_{max} power as a practically convenient index of LV mechanical performance during AF, rather than as a surrogate of*E*
_{max}.

In accordance with previous studies by Prabhu and Freeman (18-20), we used both τ and dP/d*t*
_{min} as relaxation parameters. While both parameters showed nonlinear behavior with a “plateau” (Fig. 5), the latter was more pronounced for τ. A possible explanation for this observation is that values for this parameter had a distribution that was highly skewed to the left.

#### Irregularity ratio RR_{p}/RR_{pp} as predictor of cardiac performance during AF.

It has been well established that the varying contractile function during AF depends on the complex interaction of mechanisms that are triggered by changes in both volume and interval. The former refers to the Frank-Starling relationship (14), which predicts that increased venous return (and thus end-diastolic volume) would produce greater stroke volume during the next heart cycle. However, the precise molecular mechanisms governing the Frank-Starling relationship (17) and its involvement during AF remain unclear (7, 9).

On the other hand, cardiac performance in a given beat during AF depends also on the particular time sequence of several preceding beats (30). However, a good approximation is achieved by taking into account just the RR_{p} and the RR_{pp} intervals underlying the mechanisms of mechanical restitution and potentiation, respectively (31). Both canine (30) and human (3) studies found that RR_{p} and RR_{pp} were the predominant predictors of contractility in AF.

Recently, several investigators (26, 30) noted that, in addition to RR_{p} and RR_{pp}, the ratio RR_{p}/RR_{pp} is a strong predictor of LV performance during AF. In particular, a linear relationship between LV systolic parameters and the ratio RR_{p}/RR_{pp} has been reported. Moreover, it was demonstrated that values at RR_{p}/RR_{pp} = 1 in the linear regression lines can estimate the average values of various parameters (Doppler stroke volume, ejection fraction, peak aortic flow rate, and +dP/d*t*
_{max}) during AF (27).

#### Nonlinear relationship between ventricular performance and RR_{p}/RR_{pp} ratio during AF.

In view of the nonlinear dependence of both the restitution and the potentiation on the RR_{p} and RR_{pp}intervals, respectively, the previously reported existence of a linear relationship between a number of systolic left ventricular performance parameters during AF and the ratio RR_{p}/RR_{pp} appears somewhat surprising (26, 27, 30). However, our mathematical modeling (Fig. 3) and careful inspection of the data by Yue et al. (31) suggest that this linearity holds true only for a certain range of average VCL during AF. The simple linear equation used for description of the force-interval relations during fast AF may be inadequate if the average R-R interval during AF is prolonged. Such prolongation, of course, is not just an experimental utility. It is a major therapeutic goal during treatment of patients with AF (15).

Our experimental data showed that the normalized theoretical*E*
_{max} (31), as well as all measured LV performance parameters deviated from linear dependency on RR_{p}/RR_{pp} with prolongation of the VCL (Fig. 5). The curvilinear effects were less noticeable during fast AF (Fig. 5, *A*, *F*, and *K*), and were progressively accentuated as the cycle length was prolonged to 75% (Fig. 5, *B*, *G*, and *L*), 100% (Fig. 5,*C*, *H*, and *M*), and 125% (Fig. 5,*D*, *I*, and *N*) of the spontaneous SCL. Our data also showed that all hemodynamic parameters estimated at RR_{p}/RR_{pp} = 1 exhibited an improvement when the VCL was prolonged (Fig. 5, *E*, *J*, and*O*). This resulted from the combined effect of the VCL prolongation on the plateau, *C*, and position of the studied relationships (Tables 2-4).

Finally, the relaxation-interval relationships were more curvilinear than the LV_{max} power-interval relationship (Tables 1 and 4, Fig. 5). This confirms previous observations that extrasystolic relaxation restitution is faster (more curvilinear) than the mechanical one (18).

#### Clinical implications and limitations.

Our findings suggest that in patients with AF and controlled slower average ventricular rates (e.g., by appropriate pharmacological agents, such as β-blockers, Ca^{2+} channel blockers, or adenosine), the clinical benefit of AF conversion with subsequent regularization of the rate may be relatively small. Such a speculation is further supported by recent clinical studies that found that rate control is more important than rhythm regularization for improving quality of life and exercise capacity in patients with permanent AF (15). We should stress, however, that our experiments were performed on anesthetized open-chest dogs, which precludes direct clinical extrapolation.

To describe the characteristics of the observed nonlinear components for multiple parameters, we used simple exponential equations, which are widely used in the assessment of biological processes. However, no specific physiological correlates of our monoexponential fitting parameters were given, and possibly, another model may better reflect intrinsic organ behavior, especially in humans.

In conclusion, our data imply that ventricular rate has a major impact on curvilinearity of LV_{max} power and relaxation-interval relationships. A computer simulation indicating similar behavior of contractility index *E*
_{max} strengthens this observation. These findings may have a major implication in assessing novel AF treatment strategies, based on controlled slowing of the ventricular rate (29, 32).

## Acknowledgments

We thank Dr. Stan Dannemiller for guidance in the care and well- being of the animals and William J. Kowalewski for expert help during the surgical preparation and assistance in the experiments. We also acknowledge technical support from St. Jude Medical (St. Paul, MN) and Medtronic (Minneapolis, MN) for leads and pacing equipment.

## Footnotes

This study was supported in part by National Heart, Lung, and Blood Institute Grant RO1-HL-60833.

Address for reprint requests and other correspondence: T. N. Mazgalev, Research Institute FF1-02, The Cleveland Clinic Foundation, 9500 Euclid Ave., Cleveland, OH 44195 (E-mail:mazgalt{at}ccf.org).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

*advertisement*” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.August 22, 2002;10.1152/ajpheart.00571.2002

- Copyright © 2002 the American Physiological Society