INTRODUCTION

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ATRIAL FIBRILLATION (AF) has recently attracted more interest in cardiology and cardiac surgery (4). AF produces ventricular irregular (absolute) arrhythmia and decreases cardiac output (3, 11). The decreased cardiac output seems to be partly caused by decreased ventricular end-diastolic volume (11). However, the contribution of depressed ventricular contractilities to the decreased cardiac output remains to be fully elucidated (5-8, 26). Recently, we found that the average contractility (E_max; end-systolic pressure-volume ratio, which is a load-independent index of contractility, see METHODS) of individual arrhythmic beats in AF was comparable to the E_max of regular beats at the average arrhythmic heart rate in the canine left ventricle (LV) (26). Moreover, the E_max of each arrhythmic beat was reasonably predictable from the ratio (RR1/RR2) of the preceding beat interval (RR1) to the beat interval immediately preceding RR1 (RR2) (26). However, this interesting finding has not yet been accounted for by the restitution and potentiation phenomena of myocardial contractility (6-8).

We hypothesized that the experimentally observed E_max-RR1/RR2 relation would be a manifestation of the basic characteristics of the mechanical restitution and potentiation phenomena. In fact, we found that E_max was positively correlated with RR1 and negatively correlated with RR2 (26). Although similar correlations have been documented (6-8), no previous studies had been done with a load-independent index of contractility such as E_max (26). The E_max-RR1/RR2 relation reminded us of the mechanical restitution and potentiation mechanisms as functions of the premature and postextrasystolic beat intervals (27).

We therefore investigated whether the mechanical restitution and potentiation curves could account for the experimentally observed E_max-RR1/RR2 relation of irregularly arrhythmic beats. We performed a computer simulation using the comprehensive equation that Yue et al. (27) established to describe the mechanical restitution and potentiation curves. The irregular arrhythmia was simulated by randomized beat intervals. We obtained results that reasonably simulated the E_max-RR1/RR2 relation during absolute arrhythmia (26), supporting our hypothesis.

	METHODS

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We used the following equation in the simulation

(1)

where nE_max is normalized contractility (dimensionless) of an irregularly arrhythmic beat of interest immediately after two consecutive beat intervals, RR1 (in s) and RR2 (in s), as schematically shown in Fig. 1A. Here, E_max of the arrhythmic beat (marked by arrow in Fig. 1A) was normalized to E_max of the regular beat at the average arrhythmic beat rate. G is an amplitude constant (dimensionless); H is a plateau level constant (dimensionless); t₁ and t₂ are refractory periods (in s) of the restitution and potentiation, respectively; and  is a time constant (in s) common to both restitution and potentiation.

View larger version (22K): [in this window] [in a new window]   Fig. 1.   Experimentally observed curves and definitions of variables. A: irregular arrhythmia during experimentally produced atrial fibrillation in excised cross-circulated canine hearts. Atrium was continuously stimulated by 20-Hz alternative current. LVP, left ventricular isovolumic pressure; ECG, left ventricular epicardial bipolar electrocardiogram; Emax, contractility index of a beat of interest, obtained as peak isovolumic pressure divided by isovolumic volume minus unstressed volume (V0). V0 was obtained in regular beats by decreasing ventricular volume until peak isovolumic pressure became zero. RR1, preceding beat interval; RR2, beat interval immediately preceding RR1; RR3-6, beat intervals preceding RR2. B: postextrasystolic potentiation of the first postextrasystolic beat (PES1) following an extrasystole (ES). PESI, first postextrasystolic beat interval, which is equivalent to RR1 as viewed from PES1; ESI, extrasystolic beat interval, which is equivalent to RR2 as viewed from PES1; RI, regular beat intervals.

Essentially the same equation as Eq. 1 had been proposed by Yue et al. (27) as a model of postextrasystolic potentiation (PESP) in the canine LV. They intended to describe the potentiated contractility of the first postextrasystolic beat (PES1) following the extrasystole (ES) produced artificially after a stable series of regular beats, as shown in Fig. 1B. The regular beat intervals (RI) were switched to an extrasystolic beat interval (ESI) and then to a postextrasystolic beat interval (PESI). Yue et al. (27), however, did not study the second and later postextrasystolic beats.

PESP usually decays to the regular beat stable level over 5-7 beats (1, 9, 13, 14, 17, 18, 24). This might imply that Eq. 1, based on the model of Yue et al. (27), would be inappropriate for the present study because any irregular beat in AF may be influenced by not only RR1 and RR2 but also the beat intervals preceding RR2 (RR3-6). However, our previous study provided evidence that only RR1 and RR2 strongly correlated, but RR3-6 correlated little, with E_max of the beat of interest (26). This seems to underlie the clinical observation that beat-to-beat variation of LV stroke volume and maximal rate of pressure development in AF was largely accounted for by RR1 and RR2 (6-8, 15). Therefore, we considered it reasonable to adopt Eq. 1 in the present study.

The first term of Eq. 1 describes the magnitude of the potentiation of PES1 as a decreasing function of ESI (Fig. 2A) (27). The second term describes the magnitude of the restitution as an increasing function of PESI (Fig. 2B; Ref. 27). Figure 2C draws a family of the PES1 restitution curves (Eq. 1) as the product of those two curves with ESI as a parameter. Yue et al. (27) showed explicitly that this family of theoretical curves reasonably simulated their experimentally observed curves.

View larger version (20K): [in this window] [in a new window]   Fig. 2.   Mechanical restitution and potentiation curves given by Eq. 1. A: postextrasystolic potentiation curve given by G · exp[(RR2  t1)/] + H term in Eq. 1, where G is amplitude constant, H is plateau level constant, t1 and t2 are refractory periods of restitution and potentiation, respectively, and  is time constant. Curve indicates normalized contractility of PES1 as a function of ESI. Unity normalized contractility corresponds to contractility of preceding regular beats. G = 1.7 and H = 1, which are their mean values in excised, cross-circulated canine LV (Ref. 27). B: postextrasystolic restitution curve given by 1  exp [(RR1  t2)/] term in Eq. 1. Curve indicates normalized contractility of PES1 as a function of PESI. Unity normalized contractility corresponds to contractility of preceding regular beats. C: a family of postextrasystolically potentiated restitution curves described by Eq. 1 as product of postextrasystolic potentiation and restitution curves (A and B, respectively) with ESI as a parameter. Each curve indicates normalized contractility of PES1 as a function of PESI at ESI = 0.3, 0.4, 0.6, or 1.5 s.

G and H in Eq. 1 were fixed as 1.7 and 1.0, respectively, in Figs. 2-7, unless otherwise specified. These G and H values were the average values of the data experimentally obtained and documented by Yue et al. (27). Refractory periods t₁ and t₂ in Eq. 1 were assumed to be variable with RR2 = 0.2 + 0.1 (RR2  0.3)^0.5 s, unless otherwise specified, on the basis of the data of Yue et al. Time constant  was fixed to 0.18 s for both restitution and potentiation, as shown by Yue et al. (27). E_max of each arrhythmic beat was normalized by the E_max value calculated by Eq. 1 with RR1 = RR2 = average arrhythmic beat interval = RI. This normalization is reasonable because neither ES nor PESP occurs when both ESI (=RR2) and PESI (= RR1) are equal to RI. Therefore, unity nE_max means the contractility that is the same as that of the regular beat equal to the average arrhythmic rate.

To simulate irregular arrhythmia, we irregularly changed R-R intervals using a random function in the Microsoft Excel version 5.0b software. The range of R-R interval changes is specified in the respective cases.

	RESULTS

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Figure 3 shows simulation results of a representative set of changes in RR1 and RR1/RR2 as a function of beat number (Fig. 3, A and B, respectively) and the correlogram between RR1 and RR2 (Fig. 3C) over 100 consecutive arrhythmic beats. Although RR2 changes are not shown, they were essentially the same as the RR1 changes except that the beat number was lagged by one; any RR1 was RR2 in the next beat by definition. The changes in RR1 and RR2 were irregular by mathematical randomization, as seen by the lack of significant correlation in the correlogram, which is called a Lorenz plot (Refs. 10, 12; Fig. 3C). As the result, RR1/RR2 changed widely and randomly. We judged this arrhythmia to have reasonably simulated the irregular arrhythmia in our previous experimental AF (26), although correlation between RR1 and RR2 may not always be nil in reality (10, 12). However, this difference would not have influenced the consequent results (see DISCUSSION). In Fig. 3, the range of RR1 or RR2 was 0.3-0.9 s with a mean of 0.6 s. Similar results were obtained for other ranges including 0.3-0.5 s (mean 0.4 s) and 0.4-2.3 s (mean 1.35 s) as described in Fig. 7.

View larger version (24K): [in this window] [in a new window]   Fig. 3.   Data obtained by computer simulation. RR1 was given as 0.3 + 0.6 RR(i), where RR(i) is a randomizing function giving a random value each time between 0 and 1. A: RR1 over 100 consecutive arrhythmic beats. B: RR1/RR2 ratio over same 100 arrhythmic beats. RR2 in a beat is RR1 in the previous beat. C: no correlation between RR1 and RR2. This correlogram is called a Lorenz plot.

Figure 4 shows simulation results using the same RR1 and RR2 data shown in Fig. 3. The correlogram between nE_max of the irregular arrhythmic beats and their RR1 (Fig. 4A) shows a positive and significant correlation with a correlation coefficient (r) of 0.469 (P < 0.05). The correlogram between nE_max of the irregular arrhythmic beats and their RR2 (Fig. 4B) shows a negative and significant correlation, with r = 0.657 (P < 0.05). Figure 4C shows the sequential changes in nE_max over this series of 100 irregular arrhythmic beats. These contractility changes were random in a similar manner to RR1, RR2, or RR1/RR2 shown in Fig. 3, A and B.

View larger version (33K): [in this window] [in a new window]   Fig. 4.   Simulation results of varied normalized contractility over same 100 consecutive arrhythmic beats as in Fig. 3. A: positive correlation between normalized contractility and RR1. B: negative correlation between normalized contractility and RR2. C: changes in normalized contractility over same 100 arrhythmic beats. D: positive correlation between normalized contractility and RR1/RR2. Relation passed closely at RR1/RR2 = 1.

Despite these random changes, the correlogram between the normalized contractilities of these arrhythmic beats and their RR1/RR2 (Fig. 4D) showed a significant positive correlation (r = 0.955, P < 0.001). Moreover, the data points at or near RR1/RR2 = 1 fell on or close to unity nE_max, as we reported recently in canine LV (26).

Figure 5 illustrates the potentiation curve, G · exp[(RR2  t₁/)] + H, as a function of RR2; the restitution curve, 1 exp[(RR1  t₂/)], as a function of RR1; and their product as a function of RR1 = RR2 (hence RR1/RR2 = 1). This product is the PESP as a function of RR1 = RR2. G, H, t₁, t₂, and  for these simulated curves were the same as those used for the standard case shown in Figs. 3 and 4. RR1 and RR2 changed between 0.3 and 0.9 s around a mean of 0.6 s. When RR1 = RR2 = 0.6 s, there were no restitution and potentiation and hence nE_max after RR1 = RR2 = 0.6 s was unity as shown by the horizontal line at the height of unity nE_max. The solid curve shows nE_max of an arrhythmic beat as a function of RR1 = RR2. It passed through the unity level at RR1 = RR2 = 0.6 s. It increased as RR1 = RR2 increased from 0.2 to 0.4 s and rolled off thereafter at or very close to the unity level. This indicates that normalized contractilities of arrhythmic beats were equal or close to unity at any RR1 = RR2 over its wide range between 0.4 and at least 1.2 s. We confirmed the generality of this characteristic as described below.

View larger version (21K): [in this window] [in a new window]   Fig. 5.   Normalized contractility and its components as a function of beat interval (RR1, RR2). Restitution and potentiation curves are shown as functions of RR1 and RR2, respectively, and their product curve as a function of RR1 = RR2 (hence RR1/RR2 = 1).

Figure 6 shows six graphs, similar to Fig. 5, with different G values ranging from 0.5 to 3 at intervals of 0.5. While G increased from 0.5 to 1.5 (Fig. 6, A-C), the nE_max curve of arrhythmic beats increased to unity with increasing RR1 = RR2 from 0.2 to 0.6 s. While G further increased from 2 to 3 (Fig. 6, D-F), the nE_max curve more steeply increased to unity although the curve slightly overshot with increasing RR1 = RR2 from 0.2 to 0.6 s and then gradually decayed below unity. Taken together, nE_max of arrhythmic beats was always equal or very close to unity as long as the values for RR1 = RR2 moved between 0.3 and 0.9 s over the wide range of G. The representative G = 1.7, which was obtained physiologically (27) and hence used in our simulation, fell within the range that yielded unity nE_max for a wide range of RR1 = RR2.

View larger version (26K): [in this window] [in a new window]   Fig. 6.   Normalized contractility and its components as a function of beat interval (RR1, RR2) for different values of G in potentiation term of equation. G was increased from 0.5 (A) to 1.0, 1.5, 2, 2.5, and 3.0 (B-F). Each panel draws restitution and potentiation curves as functions of RR1 and RR2, respectively, and their product curve as a function of RR1 = RR2 (in s) (hence RR1/RR2 = 1). H = 1; t1 = t2 = 0.2 s;  = 0.18 s.

Figure 7, A-C, shows simulation results similar to Figs. 5 and 6 with different ranges and means of RR1 = RR2; they are 0.4-2.3 (mean 1.35); 0.3-0.9 (mean 0.6), which was the standard range; and 0.3-0.5 s (mean 0.4 s), respectively. G was always fixed at the standard value of 1.7. In Fig. 7, A-C, nE_max of arrhythmic beats was equal to or very close to unity within the given range of RR1 = RR2.

View larger version (28K): [in this window] [in a new window]   Fig. 7.   Correlograms between normalized contractility of arrhythmic beat and RR1/RR2 (A, C, and E) and normalized contractility and its components as a function of RR1 and RR2 (B, D, and F) for different working ranges of RR1 and RR2. RR1 was given by 0.3 + 2 RR(i) s in A and B, 0.3 + 0.9 RR(i) s in C and D, and 0.3 + 0.2 RR(i) s in E and F. B, D, and F show restitution and potentiation curves as functions of RR1 and RR2, respectively, and their product curve as a function of RR1 = RR2 (hence RR1/RR2 = 1).

Figure 7, D-F, shows correlograms of nE_max with RR1/RR2. When the range of RR1 = RR2 was the widest in Fig. 7D, the relation was curved and scattered above RR1/RR2 = 1 but still passed by the unity nE_max at and around RR1/RR2 = 1. When the range of RR1 = RR2 was the narrowest in Fig. 7F, the relation was almost linear and passed through the unity nE_max at RR1/RR2 = 1. When the range of RR1 = RR2 was intermediate in Fig. 7E, the relation was curvilinear but passed by the unity nE_max at or near RR1/RR2 = 1. The case in Fig. 7F resembled our previous experimental result in the isovolumically contracting LV (26).

Figure 7, D-F, also shows that nE_max at an RR1/RR2 other than 1, e.g., 1.5, scattered widely and their mean levels decreased with lengthening of the average RR from 0.4 to 1.35 s or with decreasing average heart rate from 150 to 45 beats/min. Therefore, the uniqueness of the nE_max-RR1/RR2 relation and its linearity increased with shortening of the average RR or increasing average heart rate. This occurred because Eq. 1 is not a function only of the RR1/RR2 ratio but also of their absolute values.

	DISCUSSION

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The present results have shown that Eq. 1 can reasonably well simulate our previous observation in experimental AF (26). This indicates that the underlying mechanism of the reasonably linear relation between nE_max and RR1/RR2 that we observed experimentally (26) would primarily be a manifestation of the well-known mechanical restitution and potentiation or more generally the interval-force relation (2, 25, 27). Moreover, the results have also shown that the beat intervals (RR3-6) preceding RR1 and RR2 little affect E_max of an arrhythmic beat of interest in AF.

The present simulation has also shown that, at any range and mean values of arrhythmic beat intervals, their mean nE_max virtually coincides with the unity nE_max of regular beats whose RI is equal to mean RR1 (= mean RR2). This is interesting because we simply fixed G at 1.7 and H at 1.0 in the standard cases of simulation where RI was fixed at 0.6 s (Figs. 3-6). These G and H values were taken from the report in which RI was kept at 0.46 s (27). Although we changed G widely (Fig. 6), nE_max not only at the given RI of 0.6 s but also at other RR1 (= RR2) values fell on or very close to the unity nE_max.

It should be noted that the heavy solid curves in Figs. 5, 6, and 7, A-C, are a function of RR1 and RR2 in a specific condition such as RR1 = RR2 but are not a function of arbitrarily changing RR1 and RR2. The plotted data points in Fig. 7, D-F, indicate that nE_max variously deviates from unity when RR1 and RR2 change independently of each other and hence RR-to-RR2 ratio deviates from unity. Figure 7, D-F, also indicates that scattering of nE_max at a given RR1/RR2 decreases with narrowing of the range of RR1 and RR2. Because the average R-R was ~0.35 s and the range of the R-R was 0.25-0.60 s in our previous experiments (26), Fig. 7F seems closest to reality. Whether and how closely Fig. 7, D and E, simulates reality remain to be experimentally studied.

The high correlation between nE_max and RR1/RR2 (26) is outstanding among documented correlations between a variety of cardiodynamic and beat interval variables, which are generally low whether significant or insignificant (4, 6-8, 13, 14). This most unique nE_max-RR1/RR2 relation is now found to be a manifestation of the mechanical restitution and potentiation that are known to be the basic characteristics of myocardial contraction (13, 27).

An interesting use of Eq. 1 would be a simulation of LV pump performance in AF in the cardiovascular system model (19, 20). Such a simulation would facilitate better understanding of the factors (range and mean of arrhythmic heart rate, changes in venous return, end-diastolic volume, afterload pressure, etc.) that have been suspected to be responsible for decreased cardiac output in AF (3, 11).

Equation 1 was based on physiological experiments on normal canine hearts (27). Qualitatively the same mechanical restitution and potentiation phenomena have been obtained in human hearts (6-8, 13, 14). Therefore, we expect that essentially the same results as the present simulation would occur in human hearts. However, we do not know yet whether the same results as the present simulation would occur even in pathological hearts. The generality of the E_max-RR1/RR2 relation remains to be studied in patients.

We only assumed E_max to be a function of RR1 and RR2. Therefore, we could simulate isovolumic contractions. However, we already know that E_max is slightly affected by ejecting activation and deactivation (16, 21-23). This mechanism may partly underlie the scattering of the nE_max-RR1/RR2 relation in ejecting contractions (26). In this respect, the generality of the E_max-RR1/RR2 relation also remains to be studied in patients.

Although we attribute the contractility-RR1/RR2 relation to the mechanical restitution and potentiation, we do not discuss any deeper intracellular mechanism here (25, 27). This aspect is beyond the scope of the present simulation.

A limitation of the present result may exist in its application to irregular beats in diseased hearts, in which intraventricular conduction pathway is abnormal. For such an application, one should first study empirically whether the relatively unique relation found in our previous study (26) exists between nE_max and RR1/RR2. One should also confirm the results observed by Yue et al. (27) in the normal canine heart model, on which Eq. 1 is based, in such diseased hearts.

We conclude that the relatively unique contractility-RR1/RR2 relation that we recently discovered (26) could be reasonably well simulated mathematically by a combination of the known mechanical restitution and potentiation. This strongly suggests that these basic myocardial contractile properties are primarily responsible for the beat-to-beat changes in LV contractility of irregular arrhythmic beats in AF.