Introduction

We have already established that left ventricular (LV) O2 consumption (VO2) is reasonably predictable by VO2 = aPVA + bEmax + c on a per-beat basis in regular beats [1, 2]. Here, PVA is the LV pressure-volume (P–V) area as a measure of the LV total mechanical energy [1], Emax is the LV end-systolic elastance or P–V ratio as an index of contractility [3], a is a constant O2 cost of PVA, b is a constant O2 cost of Emax, and c is the basal metabolic VO2, all per beat [1]. The above formula proved to hold in steady-state beats under stable LV contractile and loading conditions where VO2 per beat is simply equal to VO2 per min divided by the regular heart rate [1]. We previously extrapolated the above formula to individual arrhythmic beats and reported its feasibility to predict the respective VO2 on a per-beat basis even under atrial fibrillation (AF) [4]. The purpose of the present study is to determine whether the EW efficiency of individual beat under AF is reasonably predictable from the measured EW and the predicted VO2/beat.

In the present study, we applied the above-established formula for the first time to estimate the external work (EW) efficiency of individual arrhythmic beats in six normal canine in situ LVs under electrically induced AF. We measured LV Emax and PVA as well as EW of individual arrhythmic beats under AF. We then substituted these Emax and PVA data into our VO2 formula shown above with the physiologically normal representative a and b values and the c value as the basal metabolic VO2. We then calculated the EW efficiency of each individual arrhythmic beat by dividing the measured EW with the estimated VO2 of the beat.

As the result, we found that the EW efficiency under AF varied around a mean of 13%, ranging between 5 and 18%, on average. The distribution of the EW efficiency tended to skew to the higher side with a mode of 15% on average. Although the mean efficiency under AF was significantly smaller than the efficiency under regular tachycardiac pacing, the mode efficiency was comparable to the efficiency of the regular beat and the maximum efficiency exceeded it in the respective hearts. We reasonably accounted for these statistical results by the ratio of the normally distributing EW of each beat under AF to the respective VO2 linearly correlating with EW, but always containing a considerable PVA-independent VO2 consisting of both basal metabolic VO2 and excitation–contraction coupling VO2.

Methods

Surgical preparation

We performed the canine experiments in conformity with the guiding principles for the care and use of animals in the field of physiological sciences of the Physiological Society of Japan and the American Physiological Society. We anesthetized six adult mongrel dogs (mean ± SD: 8.6 ± 1.0 kg) with pentobarbital sodium (25 mg/kg, i.v.) after premedication with ketamine hydrochloride (50 mg/kg, i.m.) and intubated for artificial respiration in each experiment. We maintained the anesthesia by fentanyl (100 μg/h per dog, i.v.) as usual in our laboratory [4].

After midsternal thoracotomy, we inserted a 3F catheter-tip micromanometer into the LV from the apex to measure the LV pressure (P). We introduced a 7F eight-electrode conductance catheter (Webster Laboratories, Baldwin Park, CA, USA) into the LV through an apical stab and placed it along the ventricular long axis to measure instantaneous LV volume (V) continuously. We described the method for measuring LVV with this catheter elsewhere [4].

Briefly, the catheter measured continuously the time-varying electrical conductance G(t) of the blood within the LV cavity of the beating heart [5, 6]. Our custom-made signal conditioner–processor (SI Medicotech Co., Ltd., Osaka, Japan) converted the G(t) to LV conductance volume after calibrating blood conductivity in the sampling cuvette. We obtained the parallel conductance Gp attributable to the conductance of the LV wall and the surrounding tissues and fluid by the standard hypertonic saline dilution method. From this Gp, we calculated a constant offset volume (Vc) and obtained the absolute LVV by subtracting Vc from the LV conductance volume [5, 6].

We attached a pair of stimulation electrodes to the left atrial appendage. Suprathreshold electrical stimulation at 20 Hz via these electrodes induced and maintained AF. AF started and stopped the stimulation [5]. We maintained AF for 2 min and recorded LVP and LVV during the latter 1 min at sampling intervals of 3 ms in a computer (Fig. 1a). We also recorded LVP and LVV while pacing the left atrium at a constant rate for 2 min after we raised the regular pacing rate to approach the mean arrhythmic heart rate during AF until pulsus alternans barely started. As the result, the regularly paced tachycardiac heart rate (mean ± SD: 170 ± 17 beats/min) reached 78% on average of the mean arrhythmic heart rate (219 ± 51 beats/min) in the six hearts.

Fig. 1
figure 1

a shows the left ventricular (LV) pressure (P)–volume (V) loops of 14 arbitrarily chosen continuous arrhythmic contractions under electrically maintained atrial fibrillation in a canine heart. b shows one of the P–V loops to explain abbreviations. In a, three P–V loops show their Emax values as the slopes of the respective end-systolic P–V lines drawn from the common dead volume on the volume axis. In b, the area within the quasi-rectangular P–V loop, mostly shaded, is the external work (EW), and the quasi-triangular area, also mostly shaded, under the Emax line on the origin side of the EW area is the mechanical potential energy (PE). The systolic pressure–volume area (PVA) is the sum of these two areas, namely, PVA = PE + EW. The stroke volume (SV) is the width of the P–V loop; the end-systolic pressure (ESP) is the pressure at the left upper corner of the P–V loop; the end-diastolic volume (EDV) is the volume at the right lower corner of the P–V loop. V0 is the dead volume at which end-systolic pressure is zero

Full size image

The LV, including the septum, weighed 46.5 ± 15.1 g after each experiment. Using these LV weights, we normalized LV mechanoenergetic variables relative to 100 g in each LV.

Mechanoenergetic variables

We obtained such key mechanoenergetic variables as EW, Emax, PVA, dead or unstressed volume (V 0), mechanical potential energy (PE), stroke volume (SV), end-diastolic volume (EDV), and end-systolic pressure (ESP) of individual arrhythmic beats under electrically induced AF as well as regular stable tachycardiac beats in each LV. Figure 1 illustrates these variables under AF in the P–V diagram. We have described in detail the methods to calculate EW, Emax, and PVA elsewhere [7, 8]. Briefly, we first obtained V 0 (Fig. 1) as the V-axis intercept of the straight line drawn through the end-systolic P–V points of the shrinking P–V loops while clamping the inferior vena cava during 10–20 regular beats, as described elsewhere [1, 3]. We then obtained LV Emax as the slope of the line connecting V 0 and the left-upper, end-systolic corner of each P–V loop (Fig. 1).

We also calculated PVA as usual on a computer as the P–V area scanned by the instantaneous P–V line connecting the instantaneous P–V point to V 0 in each arrhythmic beat [7, 8]. This calculation also yielded EW and PE (shaded rectangular and triangular areas, respectively, in Fig. 1) as the two components of PVA. We normalized Emax, PVA, EW, and PE relative to 100 g LV.

VO2 estimation

We estimated VO2 of each individual arrhythmic beat from the normalized Emax and PVA by calculating it with the following equation:

$$ V{\text{O}}_{2} = {a}\,{\text{PVA}} + b\,{\rm E}_{\rm max} + c $$
(1)

where a = O2 cost of PVA, b = O2 cost of Emax, and c = basal metabolic VO2 of each beat, all normalized for 100 g LV. We already established the reasonable VO2 predictability of this equation for a stable regular beat [1] and proposed the VO2 per beat prediction method for each individual arrhythmic beat [4]. Although the VO2 predictability for a stable regular beat is solid empirically [1], that of an arrhythmic beat is not yet so because no direct method is yet available to confirm it on a per-arrhythmic-beat basis.

As the representative a and b values of canine LVs, we used a = 1.8 × 10−5 ml O2/(mmHg ml) and b = 1.0 × 10−3 ml O2/(mmHg/ml) [1, 9]. We could reasonably assume these a and b values to be independent of the LV loading and contractile conditions (1). As for the c value, we used c = 1.0 × RR/60,000 ml O2/beat, all per 100 g LV. Here, 1.0 is in ml O2/min, and both RR (ECG R-R interval) and 60,000 are in ms. Since these representative values have some variations even among normal canine hearts (1), the above calculated VO2 values could serve as representatively estimated values in normal canine LVs.

External work (EW) efficiency

The EW efficiency is defined as EW/VO2. We obtained EW as the area within each arrhythmic P–V loop (Fig. 1b). We divided it by the above-estimated VO2. Namely,

$$ {\text{EW}}/V{\text{O}}_{2} = {\text{EW}}/\left( {{a}\,{\text{PVA}} + {b}\,{{\text{E}}_{\max}} + {c}} \right) $$
(2)

EW in mmHg ml was converted to EW in joule (J), using 1 mmHg ml = 1.33 × 10−4 J. VO2 in ml O2 was also converted to VO2 in J by 1 ml O2 = 20 J (1). Therefore, EW/VO2 became dimensionless between 0 and 1 or percent between 0 and 100%.

Statistical analyses

We analyzed the basic statistics [10, 11] of the obtained mechanoenergetic variables of individual arrhythmic beats sampled for 1 min in each of the six hearts. We had shown that 1 min was long enough to obtain a large enough number of arrhythmic beats for statistical analyses since we had found both Emax and PVA to distribute normally even within 1 min [8]. We used Microsoft Office 2004 Excel and SAS StatView 5.0 for the statistical analyses.

Results

Figure 1 shows a representative set of continuous LV P–V loops of arrhythmic beats under AF in one canine heart (Panel a) and a representative one of these loops (Panel b). The loops clearly show that their Emax fluctuated among the arrhythmic beats, and their SV, EW, PE, PVA (=PE + EW), EDV, and ESP also fluctuated simultaneously. As for these fluctuations under AF, we have already analyzed them in detail [4, 5, 8, 12–15].

Figure 2 plots LV EW (open circles) and its efficiency (solid circles) against the predicted LV VO2 of arrhythmic beats for 30 s in one canine heart. The EW − VO2 data fell fairly linearly with an extrapolated VO2 intercept of ~0.3 J/100 g per beat. Since the mean arrhythmic heart rate was ~200 beats/min, this extrapolated VO2 intercept per beat corresponds to ~60 J/100 g per min. Although this VO2 intercept was about three times a representative basal metabolic rate of 1 ml O2/min or 20 J/100 g/min under KCl arrest, we consider it physiologically reasonable because it was comparable to a representative PVA-independent VO2 of 0.3 J/100 g per regular beat in a relatively low Emax (1), consisting of both basal metabolic VO2 and excitation–contraction coupling VO2. Similar EW − VO2 relations were observed in all the other hearts.

Fig. 2
figure 2

Scattergram of left ventricular (LV) external work (EW) (open circles) and work efficiency (solid squares, dimensionless) on the ordinate against LV O2 consumption (VO2) on the abscissa in one canine heart under electrically induced atrial fibrillation

Full size image

The EW efficiency corresponds to the slope of the line connecting the respective EW − VO2 data (open circle) to the origin, but not to the positive VO2 intercept, indicating the PVA-independent VO2. As the result, the EW efficiency tended to saturate as VO2 increased. Similar EW efficiency-VO2 relations were observed in all the other hearts.

Figure 3 shows the frequency distribution histogram of the EW efficiency of the same data used in Fig. 2. The histogram obviously appeared to skew rightward as compared to the normal distribution curve, although χ2 was not statistically significant (P > 0.05). Similar distributions of the EW efficiency were observed in all the other hearts.

Fig. 3
figure 3

Frequency distribution histogram of external work (EW) efficiency of arrhythmic beats under atrial fibrillation in a canine heart skewed to the higher efficiency and exceeded the mean EW efficiency of regular beats

Full size image

Figure 3 evidently shows that the mode EW efficiency of the arrhythmic beats was comparable to the constant EW efficiency of the regular beats and the maximum EW efficiency was variably greater than the stable EW efficiency of the regular beats (P < 0.01, paired t test).

Table 1 compares the EW efficiency data of the arrhythmic beats under AF and regular beats in each of the six hearts. Paired comparisons between them also showed that the maximum efficiency was significantly greater than the efficiency of regular beats in every heart as well as on average and the mode efficiency was comparable to the regular beat efficiency.

Table 1 Comparison of external work efficiency between arrhythmic and regular beats under atrial fibrillation in six hearts
Full size table

However, the mean efficiency was significantly smaller than the regular beat efficiency in every heart as well as on average (Fig. 3; Table 1).

Discussion

This is the first study to have revealed the statistical characteristics of the distribution of EW efficiency of individual arrhythmic beats under electrically induced AF in non-failing hearts, though canine hearts. The mode EW efficiency (15% on average) of the LV arrhythmic beats under electrically induced AF was comparable to the mean EW efficiency (15%) of the regular beats in the six normal canine in situ hearts (Table 1). Moreover, the maximum EW efficiency (18% on average) under AF was greater than the mean EW efficiency of the regular beats in these hearts (Table 1).

As Fig. 2 indicates, EW can be expressed as a linear function as

$$ {\text{EW}} = {p}\,{V \text{O}}_{2} + {q}. $$
(3)

Here, p is the slope of the VO2-EW relation (p > 0) and q is the extrapolated EW axis intercept of the relation (q < 0). So, the EW efficiency can be expressed as

$$ {\text{EW}}/{V \text{O}}_{2} = {p} + {q}/{V \text{O}}_{2}. $$
(4)

As seen in Fig. 2, the EW efficiency showed the hyperbolic relation. This hyperbolic relation means the higher EW efficiency at the beat consuming the more oxygen. The short RRs in both regular tachycardia and AF, both decreasing the contractile force and shortening the filling time, decrease PVA, EW, and VO2 per beat. Thus, this hyperbolic relation between VO2 and EW efficiency indicates that the shorter RR causes the lower EW efficiency and may account for the reason of the beneficial effect of the rate control therapy for patients suffering from AF [16, 17].

The mean EW efficiency (13%) of the arrhythmic beats was, however, significantly smaller than the stable EW efficiency (15%) of the regular beats (Table 1). This confirms the already known experimental and clinical findings [18–21].

These mean, mode, and maximum efficiencies of the arrhythmic beats were also within the normal working ranges found not only in our previous studies on normal canine hearts [1, 22, 23], but also in other studies [24].

There are some limitations in this study. We were unable to compare EW/VO2 at the same heart rate between the irregular beats under AF and the regular beats under the regular atrial pacing. The latter was 22% lower than the former. This was simply because the LV started to have pulsus alternans already at a pacing rate before reaching the mean arrhythmic heart rate under AF. Therefore, we cannot yet state that the present results would hold if the pacing rate is matched with the mean arrhythmic heart rate under AF.

An obvious limitation of the present study existed in the LV volumetry with the conductance catheter. The LV conductance changed even in the isovolumic contraction and relaxation phases due to the deformation of the LV cavity, as seen in Fig. 1. Therefore, even when no SV and hence no EW is produced in a relatively weak contraction, the P–V loop is a slightly open loop, but not a vertical line with zero width. For this reason, the minimum EWs even of weak arrhythmic beats presumably without SV were not zero as shown in Figs. 1 and 2. Therefore, the minimum EW efficiency was not zero as shown by the leftmost slightly high histogram bar in Fig. 3.

To evaluate the effect of this volume error, we zeroed the efficiency values of the leftmost histogram bar and reanalyzed the frequency distribution. As the result, the minimum EW efficiency could be zero, and the mean value would decrease in Table 1. Although this correction could increase the absolute value of the negative skewness and the positive kurtosis and augment the non-normality, the mode and maximum values in Table 1 remained unchanged.

Another limitation is that the present study was performed in normal, but not failing hearts. Therefore, the present finding cannot simply be extrapolated to variably failing hearts.

Taken together, only the mode, but not the mean, minimum, and maximum of the left ventricular EW efficiency of arrhythmic beats under AF remained comparable to the stable efficiency at the regular tachycardia, which was, however, slightly lower than the mean arrhythmic tachycardia in the normal canine hearts. We reasonably accounted for these statistical findings mentioned above by the ratio of the normally distributing EW of each beat under AF to the respective VO2 linearly correlating with EW, but always containing a considerable PVA-independent VO2 consisting of both basal metabolic VO2 and excitation-contraction coupling VO2. These cardiac energetic properties should be recognized to elucidate the mechanisms of the decreased cardiac performance under AF.