## Abstract

We previously found the frequency distribution of the left ventricular (LV) effective afterload elastance (E_{a}) of arrhythmic beats to be nonnormal or non-Gaussian in contrast to the normal distribution of the LV end-systolic elastance (E_{max}) in canine in situ LVs during electrically induced atrial fibrillation (AF). These two mechanical variables determine the total mechanical energy [systolic pressure-volume area (PVA)] generated by LV contraction when the LV end-diastolic volume is given on a per-beat basis. PVA and E_{max} are the two key determinants of the LV O_{2} consumption per beat. In the present study, we analyzed the frequency distribution of PVA during AF by its χ^{2}, significance level, skewness, and kurtosis and compared them with those of other major cardiodynamic variables including E_{a} and E_{max}. We assumed the volume intercept (V_{0}) of the end-systolic pressure-volume relation needed for E_{max} determination to be stable during arrhythmia. We found that PVA distributed much more normally than E_{a} and slightly more so than E_{max} during AF. We compared the χ^{2}, significance level, skewness, and kurtosis of all the complex terms of the PVA formula. We found that the complexity of the PVA formula attenuated the effect of the considerably nonnormal distribution of E_{a} on the distribution of PVA along the central limit theorem. We conclude that mean (SD) of PVA can reliably characterize the distribution of PVA of arrhythmic beats during AF, at least in canine hearts.

- end-systolic elastance
- effective afterload elastance
- frequency distribution
- normality

many groups, including ours, investigated the statistical characteristics of cardiodynamic variables during ventricular arrhythmia under atrial fibrillation (AF) in basic and clinical studies (1–5, 7–13, 15, 23, 24). We recently found that the frequency distribution of the left ventricular (LV) effective afterload elastance (E_{a}; Fig. 1) (16, 21) is nonnormal or non-Gaussian, with considerable skewness and kurtosis in canine hearts under AF, in contrast to the LV contractility index (E_{max}; Fig. 1), with relatively normal or Gaussian frequency distribution (11, 12, 24). Therefore, the mean (SD) of E_{a} alone could not reliably represent its nonnormal distribution, with considerable skewness and kurtosis, and may lead to an incorrect evaluation of the LV mechanical performance during AF (11, 12).

We previously found that E_{max} and E_{a} are the two major determinants of the total mechanical energy of LV contraction that is measurable as the systolic pressure-volume area (PVA; Fig. 1) (18, 19). The PVA is the sum of the LV external mechanical work (EW) and the elastic potential energy (PE; Fig. 1). Then we wondered whether normally or nonnormally the PVA determined by the combination of the nonnormally distributing E_{a} and the normally distributing E_{max} would distribute. If PVA distributes normally, its mean (SD) could fully characterize its distribution. However, if PVA distributes nonnormally, its mean (SD) alone could not fully characterize its distribution.

In the present study, we investigated the frequency distribution of PVA and compared it with that of other major cardiodynamic variables including E_{max} (16, 18, 19) and E_{a} (16, 21) during AF in the canine in situ ejecting heart. The present experimental data showed that PVA distributed much more normally, with less skewness and kurtosis, than E_{a} and even slightly more normally than E_{max}. The PVA is determined predominantly by E_{max} and E_{a} when the end-diastolic volume (EDV) of each arrhythmic beat is given (16, 18). Here, E_{max} is primarily determined by the preceding beat intervals (24), E_{a} is determined by the ventricular-aortic coupling in each arrhythmic beat (16), and EDV is determined by the beat intervals, E_{max}, and E_{a} of the preceding beats. We found the normal distribution of PVA to be attributable to the complexity of the PVA formula as a function of E_{max} and E_{a}. The present result indicates that the mean (SD) of PVA of arrhythmic beats of the ventricle could reasonably characterize the distribution of PVA in the same manner as E_{max}.

## 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 Japan Physiological Society and the American Physiological Society. Six mongrel dogs (6–12 kg) were anesthetized with pentobarbital sodium (25 mg/kg iv) after premedication with ketamine hydrochloride (50 mg/kg im) and intubated in each experiment. The anesthesia was maintained by fentanyl (100 μg/h per dog iv) as usual in our laboratory (11, 12, 24).

The chest of the dog was opened midsternally. A 3-F catheter-tipped micromanometer was inserted into the LV from the apex to measure the LV pressure (LVP). A 7-F eight-electrode conductance catheter (Webster Laboratories, Baldwin Park, CA) was introduced into the LV through an apical stab and placed along the ventricular long axis to measure LV volume (LVV) continuously. The method for measuring LVV with this catheter was described in detail previously (11, 12, 24).

Briefly, the catheter measured continuously the time-varying electrical conductance [*G*_{i}(*t*)] of the five segments (*i* = 1–5) of blood in the LV cavity. The LV total blood volume was then continuously calculated from the five segmental *G*_{i}(*t*) (*i* = 1–5) after calibration of blood conductivity in the sampling cuvette. Our custom-made signal conditioner-processor (SI Medicotech) was used to convert the segmental *G*_{i}(*t*) to LV conductance volume. The parallel conductance (*G*_{p}) due to the conductance of the LV wall and the surrounding tissues and fluid was obtained by the standard hypertonic saline-dilution method. A constant offset volume (V_{c}) was calculated from *G*_{p}. The absolute LVV was obtained by subtracting V_{c} from the LV conductance converted from the segmental *G*_{i}(*t*).

We determined LV volume-axis intercept (V_{0}) of the end-systolic pressure-volume (P-V) relation (ESPVR) by clamping the inferior vena cava during 10–20 regular beats, as described elsewhere (11, 12, 14, 16, 18, 21, 24). The clamp gradually decreased LVP and LVV and shifted the shrinking LV P-V loop toward the origin of the P-V diagram. We drew a straight line tangential to the left-upper or end-systolic corners of these P-V loops and extrapolated the line to the volume axis to obtain its V_{0} (11, 12, 24).

We assumed that this V_{0} was reasonably stable, even in individual arrhythmic beats, because there was no direct method to obtain V_{0} in every arrhythmic beat with a variable E_{max}. We also knew that V_{0} decreased and increased by a few milliliters with considerable increases and decreases, respectively, in E_{max} in regular beats (20). We had also confirmed that these changes in V_{0} could not significantly affect E_{max} and PVA values to the extent that they mislead the cardiac mechanoenergetics (16).

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 (11, 12, 24). AF started on and stopped off the stimulation. We produced AF for 2 min and recorded LVP and LVV during the latter 1 min at sampling intervals of 3 ms in a computer.

#### PVA.

The systolic PVA quantifies the total mechanical energy generated by a ventricular contraction (18, 19). Figure 1 illustrates PVA in the ventricular P-V diagram. PVA consists of two areas: the EW (rectangular area) within the P-V loop and the elastic PE (triangular area) between the end-systolic P-V or E_{max} line and the end-diastolic P-V curve on the origin side of the P-V loop.

We calculated PVA of each contraction by summing all the PVA increments per 3 s from end diastole to end systole. Each 3-ms (Δ*t*) PVA increment (ΔPVA) corresponds to the narrow triangular area scanned by a line connecting the instantaneous P-V data point drawing the P-V loop and V_{0} on the volume axis (16) (Fig. 1). We obtained EW by summing ΔPVA from the onset of the present contraction to the onset of the next contraction (16). We obtained PE by subtracting EW from PVA (16).

We further obtained the contractility index, E_{max}, of each contraction as the slope of the ESPVR (Fig. 1) (16, 18, 19). We also obtained the effective E_{a} of each contraction as the slope of the line connecting the end-systolic P-V point on the left upper corner of the P-V loop and the EDV on the LVV axis (Fig. 1) (16, 18–21).

We know that PVA is formulated approximately by *Eq. A10* as explained in the appendix. *Equation A10* indicates that PVA is a complex function of E_{max}, E_{a}, and EDV.

#### Statistics.

We performed the basic statistical analyses of PVA as well as E_{max}, E_{a}, beat interval (RR), EW, PE, EDV, and stroke volume (SV) of all the arrhythmic beats sampled in an arbitrary 1 min in each heart. We studied their frequency distributions and performed χ^{2} test of their normality, evaluating it by its *P* value (Table 1; see Figs. 4 and 5) (6, 17).

We also obtained two dimensionless measures of departure from the normality or Gaussianity of the frequency distributions, i.e., skewness and kurtosis, as in our previous study (Table 1; see Figs. 4 and 5) (11, 12). The normal distribution has zero skewness and zero kurtosis. A positive skewness indicates a leftward shift of the peak frequency, and a negative skewness indicates its rightward shift. A positive kurtosis indicates a sharper peak, and a negative kurtosis indicates a dull peak or even two separate peaks.

We analyzed correlations among E_{max}, E_{a}, EDV, and PVA and obtained their correlation coefficients (*r*) and coefficients of determination (*r*^{2}; Table 2). We also obtained autocorrelation coefficients of EDV, E_{max}, E_{a}, and PVA with a lag of only one beat interval, e.g., correlation coefficient between EDV_{i} and EDV_{i + 1} where *i* was increased from 1 to the one beat before the last beat in the sampled 1 min (Table 3).

We compared the basic statistics of E_{max}E_{a}^{2}, E_{max}^{2}E_{a}, E_{max}^{2}E_{a} + E_{max}E_{a}^{2}/2, (EDV − V_{0})^{2}, (E_{max} + E_{a})^{2}, E_{max} + E_{a}, and (E_{max}^{2}E_{a} + E_{max}E_{a}^{2}/2)/(E_{max} + E_{a})^{2}, which are the variously complex terms of *Eq. A10*, to calculate PVA from EDV, E_{max}, and E_{a} (see appendix). We also compared the basic statistics between PVA calculated by *Eq. A10* and the measured PVA (see Fig. 6).

We obtained moving-averaged PVA values by increasing the number of moving-averaged beats from 1 to 50 (see Fig. 7).

In these statistical analyses, we used Microsoft Office 2001 Excel and StatView version 4.5. Values are means (SD).

## RESULTS

Figure 2 shows a representative 3-s segment of the continuous traces of LVP and LVV during AF. It contains nine arrhythmic beats separated by various beat intervals and featured by all varying peak LVPs and end-diastolic and end-systolic LVVs. All six hearts showed similar arrhythmic beats during AF.

Figure 3 shows the same set of LV P-V loops of the four consecutive arrhythmic beats during AF and their PVA as well as E_{max} and E_{a} (16, 18–21). These correspond to the first four beats in Fig. 2. Figure 3*A* illustrates PVA of the arrhythmic beat with the highest contractility (largest E_{max}) and largest SV of the four beats. PVA consists of the sum of the rectangular (shaded) area within the P-V loop and the triangular (hatched) area between the ESPVR line and the EDPVR curve on the origin side of the P-V loop of this contraction. The former area represents EW performed by the LV during this contraction. The latter area represents PE that was generated by this contraction in the LV wall but was not converted to EW (18, 19). Figure 3*B* illustrates PVA of the arrhythmic beat with the lowest contractility (smallest E_{max}) and the smallest SV of the four beats. In this beat, EW was nearly zero.

Figure 4 shows a representative set of the frequency distributions of PVA and E_{a} in one heart (*heart 1* in Table 1). PVA distributed much more normally than E_{a}. We chose E_{a} to compare with PVA, because we had found E_{a} to have a considerable nonnormal distribution (11). The χ^{2}, its *P* value, skewness, and kurtosis quantify the difference of the frequency distribution between PVA and E_{a}. The mean, median, and mode scattered less for PVA than for E_{a}. The other hearts showed similar results (Table 1). For PVA, no *P* value of χ^{2} test was <0.05; the smallest *P* value was 0.162. Neither skewness (*P* > 0.661) nor kurtosis (*P* > 0.116) was significantly different from 0.

Figure 5 compares means (SD) of χ^{2}, its *P* value, skewness, and kurtosis of not only E_{a} and PVA but also other cardiodynamic variables (RR, E_{max}, EW, PE, SV, and EDV) observed in all six hearts. The order of the cardiodynamic variables on the abscissas was the increasing order of χ^{2} values from left to right. E_{a} had the largest χ^{2}, skewness, and kurtosis and the smallest *P* value for χ^{2} test. On the other hand, PVA as well as EDV and SV had much smaller χ^{2}, skewness, and kurtosis and larger *P* value for χ^{2} test. For E_{max}, χ^{2} and *P* values were intermediate but skewness and kurtosis were small.

The one-way repeated-measures analysis of variance (ANOVA) showed statistical significance of *F*-test for χ^{2}, its *P* value, skewness, and kurtosis among the cardiodynamic variables on the abscissas. The inset *P* values for χ^{2} test on the solid horizontal arrows in Fig. 5 indicate that Bonferroni's *t*-test and least significant difference (LSD) method showed statistical significance between the variables bridged by the arrows. The *P* values on the dotted arrows indicate statistical significance only by LSD.

The results in Fig. 5, *A* and *B*, indicate that PVA, as well as E_{max}, PE, EW, SV, and EDV, had significantly smaller χ^{2} values with significantly higher *P* values for χ^{2} test than E_{a}. RR had intermediate χ^{2} and *P* values between E_{a} and either E_{max} or PVA. PVA as well as EDV, SV, and PE had significantly smaller χ^{2} values with significantly higher *P* values for χ^{2} test than E_{a} and RR. The *P* values for χ^{2} test of PVA was >0.4 in five hearts and 0.1 in one heart (Table 1), negating the null hypothesis of the nonnormal distribution of PVA and, hence, supporting its normal distribution. However, the *P* values for χ^{2} test of E_{a} were <0.05 in three hearts, supporting the null hypothesis of the nonnormal distribution of E_{a} and, hence, negating its normal distribution. The other variables had *P* values for χ^{2} test >0.1. This negated the null hypothesis of their nonnormal distributions.

Moreover, the results in Fig. 5, *C* and *D*, indicate that PVA, as well as EDV, SV, EW, E_{max}, and RR, had much smaller skewness and kurtosis than E_{a} and PE. PVA, as well as EW and E_{max}, had virtually zero skewness on average. PVA, as well as SV, EW, and E_{max}, had small negative kurtosis on average. Although we did not include end-systolic pressure (ESP) and end-systolic volume (ESV) in Fig. 5, their small χ^{2}, skewness, and kurtosis were comparable to those of PVA and EDV and much smaller than those of E_{a}.

Figure 6 compares means (SD) of χ^{2}, its *P* value, skewness, and kurtosis of the various complex terms of *Eq. A10* to determine PVA from EDV, E_{max}, and E_{a} (see appendix) in all six hearts. *Terms 1–7* (Fig. 6) are arranged in decreasing order of χ^{2} values from left to right. *Term 8* is the PVA calculated by *Eq. A10*, and *term 9* is the measured PVA.

The calculated PVA (*term 8*) had relatively small χ^{2} and large *P* (>0.155, all insignificant). However, its skewness and kurtosis were closest to zero. These statistical values were comparable to those of the measured PVA (*term 9*). The measured PVA data used here are the same as those used in Fig. 5.

The results in Fig. 6 indicate that E_{max}E_{a}^{2} (*term 1*) had the most nonnormal distribution with the largest χ^{2}, the smallest *P* value (<0.05 only in 2 hearts), and the largest skewness and kurtosis. However, *term 7* (E_{max}^{2}E_{a} + E_{max}E_{a}^{2}/2)/(E_{max} + E_{a})^{2} had the smallest χ^{2}, the largest *P* value (>0.772), and small, but not the smallest, skewness and kurtosis, although it contains multiple E_{a} values.

Table 2 shows *r* and *r*^{2} values among E_{max}, E_{a}, and EDV in *Eq. A10* and the calculated PVA. They were positively or negatively correlated significantly in all six hearts. The *r*^{2} values indicate that the variation of any of E_{max}, E_{a}, EDV, and PVA was attributable to those of the other three variables by 30–80%. This result indicates that E_{max}, E_{a}, EDV, and PVA were variably, although not fully, independent of each other.

Table 3 shows the autocorrelation coefficients (*r*) of EDV, E_{max}, E_{a}, and PVA with a lag of only one beat. Most *r* values were insignificant. Therefore, these variables changed on a per-arrhythmic-beat basis, rather randomly, independent of their respective values of the last beat in most of the six hearts.

Figure 7 shows that the percent coefficient of variation (CV) of moving-averaged PVA decreased with an increasing number of moving-averaged beats in the six hearts. The percent CV of ∼35% on average of raw PVA values during AF decreased rapidly to ∼10% and further to ∼5% as a result of an increase in the number of moving-averaged beats to 20 and 40 beats, respectively.

## DISCUSSION

The present results have evidently shown for the first time the normality or Gaussianity of the frequency distribution of PVA of arrhythmic beats in canine LVs under electrically induced AF (Figs. 4 and 5). This normality existed despite the significantly nonnormal or non-Gaussian frequency distribution of E_{a} of the same arrhythmic beats. Here, E_{a} is one of the major determinants of PVA. Because PVA is a measure of the total mechanical energy of each ventricular contraction (18, 19), its normal distribution indicates that the ventricular total mechanical energy of each arrhythmic beat distributes normally during AF, despite the considerably nonnormally distributing E_{a} (11, 12, 24).

The present results have also shown that although E_{a} (Fig. 4) and E_{max}E_{a}^{2} (Fig. 6) in the formula (*Eq. A10*) to calculate PVA distributed nonnormally, the other terms of the formula distributed normally to the comparable extent of the calculated PVA as well as the measured PVA. Of these, E_{max} and EDV have been determined by the onset of each contraction as the results of the cardiodynamic variables of the preceding arrhythmic beats (22, 24). E_{a} is defined as ESP/SV in each contraction, where ESP and SV are determined as the result of the ventricular-aortic interaction during each contraction (14, 16, 21).

Therefore, PVA is determined as the consequence of a complex interaction of various cardiodynamic variables with normal or nonnormal frequency distributions (3, 4, 7, 9–13, 24). The normal distribution of PVA, therefore, seems principally attributable to the central limit theorem or the law of large numbers (6). This theorem assumes all the component variables to be mutually independent. However, the component variables, i.e., EDV, E_{max}, and E_{a}, of PVA correlated mutually by 30–80% in terms of *r*^{2} (Table 2). This may partly account for some, although not significant, residual χ^{2} of PVA, despite its virtually zero skewness and kurtosis (Table 1, Fig. 6). The present study for the first time revealed the normal or Gaussian frequency distribution of PVA during AF and its underlying mechanism, although in the LV of normal canine hearts.

The present study has also shown that the mean of PVA per beat during AF could be estimated reliably with a reasonably small (10–5%) CV of the mean by averaging PVA values of 20–40 beats, respectively (Fig. 7). This indicates that the mean (SD) of PVA of only 20–40 beats could give us a reasonably reliable estimation of the frequency distribution of PVA of arrhythmic beats during AF, at least in normal canine hearts. This suggests that only 7–15 s are necessary to estimate reasonable mean (SD) of PVA under AF. This rather rapid convergence of the SD of moving-averaged PVA seems partly attributable to the largely insignificant autocorrelation of PVA between the present and preceding beats as the result of the similarly insignificant autocorrelation of the components (EDV, E_{max}, and E_{a}) of PVA (Table 3).

Although we analyzed neither LV myocardial Ca^{2+} handling nor cross-bridge cycling in the present study, we know that an increased EDV increases Ca^{2+} handling and cross-bridge cycling according to the Starling law of the heart (16, 18). An increased E_{max} manifests an LV contractility enhanced with increased Ca^{2+} handling and recruitability of cross-bridge cycling (16, 18). Here, myocardial Ca^{2+} handling is determined by its restitution and potentiation over the last two preceding beats (24), whereas cross-bridge cycling is primarily determined by EDV, E_{max}, and E_{a} (16, 18). E_{a} can directly and simply be related to neither myocardial Ca^{2+} handling nor cross-bridge cycling, because it is defined as ESP/SV (11, 21), where ESP and SV are determined by EDV, E_{max}, and total peripheral resistance (16, 21). These cardiac system-element interrelations suggest that PVA of an arrhythmic beat is determined in a complex manner by not only element, but also system, factors. Although we do not know the statistical characteristics of all the system-element factors involved in PVA determination, we could speculate that the central limit theorem may serve for the normal distribution of PVA.

An obvious limitation of the present study is the use of normal canine hearts with electrically induced AF. It is possible that statistical characteristics of PVA during AF in human diseased hearts are different from those we have found in the present study (1–4, 10). This warrants future studies to investigate the statistical characteristics of PVA in various pathological hearts under artificial or spontaneous AF. Their statistical characteristics may reflect the pathophysiological conditions of the heart.

One more limitation of the present study is the assumption of a constant V_{0} under arrhythmia with a widely varying E_{max}. Fortunately, our previous studies showed that changes in V_{0}, if any, with a varying E_{max} would be relatively small, e.g., a few milliliters within a physiologically working range of E_{max} (16, 20). Any change in E_{max} affects PVA via PE (Figs. 1 and 3, *Eqs. A1* and *A3* in appendix). For example, even a 2-ml decrease in V_{0} with an increase in E_{max} from 2.5 to 10 mmHg/ml could increase PE only by an additional triangular area bound by the end-systolic P-V point and the new 2-ml base on the volume axis in the P-V diagram, though not shown graphically. When the end-systolic pressure increases representatively from 50 to 100 mmHg with the increased E_{max} (Figs. 2 and 3), the decreased V_{0} with the increased E_{max} would increase PVA by only 50–100 mmHg/ml. This increase in PVA seems too small to deviate the frequency distribution of PVA from normality (e.g., Fig. 4*A*).

We therefore conclude that the total mechanical energy generated by arrhythmic beats during AF can reliably be characterized by the mean (SD) of PVA in the normal canine LV. The normal distribution of PVA during AF seems theoretically attributable to the complexity of the PVA formula as a function of E_{a}, E_{max}, and EDV of individual arrhythmic beats.

## APPENDIX

Figure 1 helps in our understanding of the following relations of PVA to the other cardiodynamic variables, including E_{a} and E_{max}, in the LV P-V diagram (16).

First, by definition, (A1) These EW and PE are formulated as follows (A2) (A3) Here, we assumed that LVP during ejection was constant at ESP and the end-diastolic P-V relation was virtually zero below ESV. This assumption does not disadvantageously affect the magnitude of PVA, except in the presence of aortic stenosis or regurgitation.

ESV and ESP are formulated by the following simultaneous equations (A4) (A5) Solving simultaneous *Eqs. A4* and *A5* yields (A6) (A7) Substituting *Eqs. A6* and *A7* into *Eqs. A2* and *A3* yields (A8) (A9) Substituting *Eqs. A8* and *A9* into *Eq. A1* yields (A10) *Equation A10* indicates that PVA is a complex function of E_{max}, E_{a}, and EDV, all of which change beat-by-beat with variable magnitudes of SD, skewness, and kurtosis (Fig. 5). The complexity of the PVA equation (*Eq. A10*) includes additions, multiplications, squares, and divisions of E_{max}, E_{a}, and EDV.

## GRANTS

This work was partly supported by Ministry of Education, Culture, Sports, Science, and Technology (Japan) Scientific Research Grants 13558113, 13770350, 13854030, 13878185, 13878192, 14380405, 15650095, 15659186, 15700330, and 16659057 and Ministry of Health, Labour, and Welfare (Japan) Cardiovascular Diseases Research Grant 14A-1.

## Footnotes

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