## Abstract

The duration of diastole can be defined in terms of mechanical events. Mechanical diastolic duration (MDD) is comprised by the phases of early rapid filling (E wave), diastasis, and late atrial filling (A wave). The effect of heart rate (HR) on diastolic duration is predictable from kinematic modeling and known cellular physiology. To determine the dependence of MDD of each phase and the velocity time integral (VTI) on HR, simultaneous transmitral Doppler flow velocities and ECG were recorded during supine bicycle exercise in healthy volunteers. Durations, peak values, and VTI using triangular approximation for E- and A-wave shape were measured. MDD, defined as the interval from the start of the E wave to end of the A wave, was fit as an algebraic function of HR by MDD = *B*_{MDD} + *M*_{LMDD}·HR + *M*_{IMDD}/HR, derivable from first principles, where *B*_{MDD} is a constant, and *M*_{LMDD} and *M*_{IMDD} are the constant coefficients of the linear and inverse HR dependent terms. Excellent correlation was observed (*r*^{2} = 0.98). E- and A-wave durations were found to be very nearly independent of HR: 100% increase in HR generated only an 18% decrease in E-wave duration and 16% decrease in A-wave duration. VTI was similarly very nearly independent of HR. Diastasis duration closely tracked MDD as a function of HR. We conclude that the elimination of diastasis and merging of E and A waves of nearly fixed durations primarily govern changes in MDD. These observations support the perspective that E- and A-wave durations are primarily governed by the rules of mechanical oscillation that are minimally HR dependent.

- heart rate
- E wave
- A wave

cardiovascular and exercise physiologists and cardiologists are keenly aware of heart rate (HR) dependence of physiological indexes used to assess cardiac function. Cardiovascular function indexes obtained from a single cardiac cycle usually involve the duration of a component of the cycle and can be referenced either to electrical or mechanical events.

Selected pump-function indexes rely on stroke volume, typically based on systolic ejection, and significant work has been done to analyze the HR dependence of the duration of systole (1, 4, 5, 16, 28, 29). However, assessment of the HR dependence of diastole and its phases have received little attention. The systolic time interval has been used as an index of cardiac function (28), and earlier work on the duration of systole as a function of HR has included both electrical and mechanical events. In electrocardiography, the corrected QT (QT_{c}) interval for HR in terms of the electrocardiographic RR (RR = 60/HR) interval is given by Bazett's formula as QT_{c} = QT/, where QT and RR are measured intervals from the ECG (4, 28). A dimensionally accurate form of QT based on physiological and thermodynamic arguments has been shown to depend on HR in an algebraic form given by (1) where *C*_{1} and *C*_{2} denote the intercept and slope of the linear relation between QT and HR (1, 5, 16). Often, electromechanical systole (QS_{2}) is used; QS_{2} is defined by the interval from the Q wave on the ECG to the second heart sound (S_{2}) on the simultaneous phonocardiographic recording. QS_{2} also follows a linear relation to HR (5).

Diastolic duration (DD) has previously been assessed as cardiac cycle duration (RR) minus electromechanical systole (QS_{2}). Prior work concerning DD has been motivated primarily by consideration of diastolic myocardial perfusion time rather than the duration of mechanical events and has been used to assess the HR-dependent effects of pharmacological agents (6, 22, 24). In the absence of a first principles-based-derived expression, the experimental observations of the percentage of the RR interval that diastole occupies, i.e., %D = (RR − QS_{2})/RR, has previously been empirically fit using a second-order polynomial in HR (6).

To our knowledge, there has been no prior study in humans that sought to characterize DD as a function of HR, specifically in terms of its mechanical phases. The three phases of filling, early rapid filling, diastasis, and late atrial filling combine to form the mechanical duration of diastole (MDD). Isovolumic relaxation, which is often included as part of diastole, is ignored in the current analysis because all valves are closed and because it is essentially akinetic. The three mechanical phases are defined by the mitral inflow patterns they create: specifically, the Doppler E wave for early rapid filling; an interval between flows for diastasis; and the Doppler A wave for late filling due to atrial systole.

Diastolic function has previously been analyzed via kinematic modeling. The mechanical analog of the suction pump attribute of the heart is a simple harmonic oscillator (SHO), whose kinematic predictions for the contour of the E and A waves as a function of time have been validated experimentally in various physiological (13, 14, 17) and pathophysiological settings (8, 20, 25). Specifically, early rapid filling (E wave) is initiated by mechanical suction (dP/dV < 0), which in kinematic terms is equivalent to the release of the stretched spring of a previously displaced SHO. The relevance to heart rate dependence resides in the kinematics, in that the oscillator's motion after its release is independent of when or how fast it was originally displaced; in other words, the model treats filling as independent of the timing or duration of the preceding systole, i.e., HR. Hence, the kinematic modeling approximation predicts that, to the extent that filling follows the rules of mechanical oscillation, the duration of the E and A waves should be essentially HR independent.

Cellular physiological experiments also make predictions regarding the duration of ejection (force generation) or filling (force decay or relaxation). Previous work (11, 12) has shown that Ca^{2+} (currents) concentrations and associated relaxation rates are essentially independent of the pacing rate, which in the experimental setting, is the equivalent of HR. The duration of force generation (contraction), i.e., systole, was also observed to be weakly dependent on, or essentially be independent of pacing rate (11, 12). These observations at the cellular level, when extrapolated to the whole organ level, imply that cardiac relaxation behaves nearly independent of HR.

In this study, we determined the relationships between HR and MDD and its three phases: early rapid filling, diastasis, and late atrial filling. Because the RR interval is defined by the sum of systolic and diastolic durations, we anticipated that our results would be in concert with previous studies regarding systolic duration. Our measurements also permitted us to validate the extent to which the predicted algebraic expression for MDD as a function of HR agrees with measurements and to independently validate the kinematic model-based prediction that E- and A-wave durations should be essentially HR independent.

## GLOSSARY

MVA_{eff}

- MDD
- Mechanical diastolic duration: E wave start to A wave end, ms
- VTI
- Velocity time integral, cm
- E
_{DUR} - Duration of E wave, ms
- AT
- Acceleration time of E wave, ms
- DT
- Deceleration time of E wave, ms
- E
_{peak} - Peak E-wave velocity, cm/s
*D*_{diastasis}- Duration of diastasis interval: E wave end to A wave start, ms
- A
_{DUR} - Duration of A wave, ms
- A
_{peak} - Peak A-wave velocity, cm/s
- SV
- Stroke volume, ml
- MVA
_{eff} - Effective mitral valve area, cm
^{2} - HR
- Heart rate, beats/min (bpm)
- QT
- Electrophysiological systole, ms
*C*_{1}- Intercept of regression relation for electrophysiological systole, ms
*C*_{2}- Slope of regression relation for electrophysiological systole, ms/bpm
- QS
_{2} - Electromechanical systole, ms
*B*_{P}- Intercept of regression relation for given parameter
*P*, {*P*} - M
_{L}_{P} - Linear dependence (slope) on HR of regression relation for given parameter
*P*, {*P*}/bpm - M
_{I}_{P} - Inverse dependence on HR of regression relation for given parameter
*P*, {*P*}·bpm

## METHODS

### Data Acquisition

Transmitral Doppler echocardiography was performed on 25 subjects (13 female) ranging in age from 20 to 32 yr (Table 1) during supine bicycle exercise. Subjects represented a cross-section of normal healthy volunteers of varying fitness. Informed consent was obtained in accordance with Washington University Medical Center Human Studies Committee criteria. Transmitral pulsed-Doppler flow velocities were obtained by an experienced sonographer using a clinical echocardigraphic imaging system (Acuson Sequoia; Mountain View, CA) equipped with a 2-MHz transducer in accordance with standard ASE criteria (26). The ultrasound sweep speed was set to 100 mm/s. The transducer was placed apically, and the sample volume was located at the tips of the mitral valve leaflets. Simultaneous ECG limb lead II was recorded and displayed on the Doppler images. Subjects exercised with a standard supine bicycle ergonometer (Lode MRI Ergonometer; Groniengen, The Netherlands), which was placed on a standard hospital bed. This allowed subjects to maintain a relatively static torso position, simplifying echocardiographic transmitral Doppler image acquisition. At least 25 cardiac cycles were first recorded at rest to allow for baseline measurement; all subjects had normal transmitral E and A waves with diastasis during this period. Image acquisition continued as bicycle resistance was incrementally advanced to generate an increase in HR. HR was increased until Doppler velocity data became too noisy to analyze, usually between 120 and 140 beats/min. E and A waves were recorded continuously on a VHS tape and were digitized off-line for analysis using a custom video editing station.

### Data Analysis

HR was determined using the simultaneous ECG recording on the digitized echocardiographic frame via HR = 60/RR. The MDD was defined from the start of the E wave to the end of the A wave. The E- and A-wave contours were approximated as triangles and conventional Doppler-based diastolic function parameters: E_{DUR}, AT, DT, A_{DUR}, and E_{peak,} and A_{peak} were measured (Fig. 1) (9). Because mechanical diastasis was eliminated at high HRs, the E-wave duration was determined using a straight-line extension of the deceleration portion of the E wave to baseline. The *D*_{diastasis} was defined as the interval between the end of the E wave and start of the A wave. This convention defines *D*_{diastasis} as positive when the E and A waves are separated (as traditionally measured); when the E and A waves are merged, *D*_{diastasis} remains continuous and takes on negative values. Additionally, the VTI under the E and A waves was calculated using the formula for the area of a triangle when *D*_{diastasis} was positive; otherwise, A-wave VTI was calculated by computing the area above the contribution of the E wave (23).

### Mechanical Diastolic Duration

Electrophysiological diastolic duration (DD) is related to systole by: DD = RR − QT, where QT is defined as in *Eq. 1* as previously studied (16). In the current study, we define and algebraically model MDD as (2) where RR was transformed to an inverse HR term (1/HR). *B* and *M* denote the intercept and slope of a linear regression relation, respectively, of the form *P* = *M*·*X* + *B*. *M*_{LP} and *M*_{IP} denote the magnitude of linear (*M*_{LP}·R) or inverse dependence (*M*_{IP}/HR) for the given parameter *P*. *B*_{MDD} is related to *C*_{1} and *M*_{LMDD} is related to *C*_{2} from *Eq. 1*. *M*_{IMDD} is included to account for additional effects such as isovolumic relaxation, which is not explicitly included in MDD. The HR dependence of the VTI was modeled as having a linear form.

### Phases of Mechanical Diastole

Because previous kinematic modeling and cellular physiology suggest no significant HR dependence for E- and A-wave durations, it leaves diastasis as the remaining, and necessarily dominant, phase by which modulation of the entire duration of diastole can be achieved. In this study we used a linear model of the form *P* = *B* + *M*_{LP} × HR to assess the extent to which any of our measured parameters or intervals may be HR dependent.

#### Early rapid filling phase (E wave).

The measured E_{DUR}, AT, DT, and the E_{peak} as a function of HR were all analyzed using linear regression.

#### Diastasis.

*D*_{diastasis} was fit (analyzed) using the form of *Eq. 2* to confirm the role of diastasis in modulating MDD. A linear fit was also used to facilitate comparison of HR dependence with the other linearly fit phases of filling.

#### Late atrial filling phase (A wave).

The measured A_{DUR} was fit linearly. Because in the absence of diastasis (*D*_{diastasis} < 0), we expected the contribution of the E wave would be significant due to superposition (23); therefore, we established separate fits for A_{peak} in the presence and in the absence of diastasis.

### Statistics

Analysis was performed using a commercially available statistics package (SPSS 7.5 for Windows, SPSS; Chicago, IL). Multiple linear regression analysis was used to analyze the data to allow for multiple parameters, including dependence on HR and 1/HR, subject-to-subject variability, and gender differences (27). The inter- and intraobserver variation of E- and A-wave feature measurements was <5%. All data are expressed as means ± SE; *P* value < 0.05 was considered statistically significant.

## RESULTS

Results are listed in Table 2. Gender was not found to be associated with any dependent parameter. All regression relations yielded *P* < 0.01.

### Mechanical Diastolic Duration and VTI

Figure 2 shows the very strong agreement between data and model given by *Eq. 2*. Significant correlation (*r*^{2} = 0.98) between MDD and HR was observed. The VTI had an extremely weak correlation (*r*^{2} = 0.12) to HR.

### Early Rapid Filling Phase

On the basis of the shallow slope of the linear regression between E_{DUR} and HR, a weak dependence on HR was observed (Fig 3). The standard AT and DT measurements also yielded a weak dependence on HR as expected from E_{DUR} = AT + DT. The model yielded small *r*^{2} values based on the small slope. We found E_{peak} to be essentially independent of HR (*r*^{2} < 0.05) and report the average value as 96 ± 19 cm/s.

### Diastasis

The graph of *D*_{diastasis} as a function of HR was clearly curvilinear. When fit using *Eq. 2*, which includes HR terms that are both linear and inverse, high correlation (*r*^{2} = 0.89) was observed. This was expected because the 1/HR (*M*_{Idiastasis} dependent) term in *Eq. 2* had a stronger correlation to *D*_{diastasis} than the linear term (*M*_{Ldiastasis} dependent). The data and fit are shown in Fig 3. To facilitate comparison to the other phases of diastole, we also fit *D*_{diastasis} assuming a linear HR dependence and found, as expected, a lower correlation (*r*^{2} = 0.78). Accordingly, residual analysis and the lower *r*^{2} value confirm that assuming linear HR dependence yields a poorer fit than does the nonlinear model (*Eq. 2*).

## DISCUSSION

In this study, we analyzed the HR dependence of the duration of diastole and its phases in healthy normal volunteers. Our conceptual perspective regarding the entire duration of diastole was motivated by previous work employing physical and thermodynamic arguments, which we used to derive the relationship of MDD to HR as *Eq. 2* (16). We found that, within the range of HRs measured, *Eq. 2* provides an excellent fit (*r*^{2} = 0.98) to measured data. Baseline values of peak velocity and DT fall within a previously reported normal range (21); exercise determined systolic time intervals (QT) fall within an established range of normal values (6).

From the known physiology of cardiac electromechanical coupling, all of the mechanical events we characterize have well-described electrical precursors. The use of *Eq. 1* to derive *Eq. 2* allows us to compare our mechanical findings with previous electrophysical and electromechanical results, and we find our results to be in close concordance with those results (1, 5, 16). Our *B*_{MDD} (−549 ± 42 ms) corresponds with the negative of reported value of *C*_{1}, whereas *M*_{LMDD} (2.13 ± 0.24 ms/bpm) correlates with the negative of *C*_{2}. ECG-based QT interval determination has previously reported *C*_{1} approximately 520 ms and *C*_{2} approximately −1.9 ms/bpm (1, 5, 16). Because our data are based on mechanical events rather than electrical, we expect, and find, that our values more closely agree with studies of QS_{2}, which report *C*_{1} approximately 541 ms and *C*_{2} approximately −2.1 ms/bpm (5). The results also match the relation of %D × RR based on diastolic perfusion time (6).

Turning our attention from MDD to the phases of diastole, we note that kinematic modeling suggests that the individual phases of filling can be viewed as being governed by harmonic oscillator motion, which accurately predicts E- and A-wave contours (13, 14, 17). Kinematic modeling (18, 19) idealizes the physiology by treating the recoil process as mechanical oscillations unrelated to the overall contraction-recoil cycle frequency, i.e., HR, and as such, makes testable prediction that the E- and A-wave durations should be independent of HR. By determining the HR dependence of E- and A-wave durations via linear regression, we were also able to assess the extent to which E- and A-wave durations were indeed HR independent.

We found that the duration of the early rapid and late atrial filling phases showed only a small change with an increase in HR, as anticipated from kinematic modeling (13, 14, 17). Specifically, E_{DUR} shortens by about 18% as HR increases by 100%. The slight HR dependence of E-wave duration can be understood in terms of the known ATP dependence of cardiac myocyte relaxation. Published data shows that this Ca^{2+}-modulated relaxation process is minimally HR dependent (11, 12). The A_{DUR} also showed only a 16% decrease with a 100% increase in HR, which is in concert with the known weak HR dependence of the PR interval. The weak dependence of A_{DUR} (−0.454 ± 0.049 ms/bpm) on HR is slightly stronger than the previously reported change in the PR interval (−0.287 ms/bpm) as a function of HR; however, the previous study considered a narrower range of HR than the current study (3).

The observed significant change in MDD is associated with a change in *D*_{diastasis}. With the use of a linear fit, diastasis duration has a HR dependence more than seven times as strong as E_{DUR} (−6.82 ± 0.28 vs. −0.957 ± 0.115 ms/bpm) or more than 10 times as strong as A_{DUR}. Furthermore, the low correlation observed for E_{DUR} and A_{DUR} (*r*^{2} = 0.29 and 0.20, respectively) suggest weak dependence on HR for these two filling phases. However, when *D*_{diastasis} is fit using *Eq. 2,* it follows the measured HR trend more accurately (*r*^{2} = 0.89) and illustrates the importance of the (nonlinear) 1/HR term in the full expression for MDD. Because MDD, consisting of E_{DUR} + *D*_{diastasis} + A_{DUR}, is very well fit by *Eq. 2*, and because E_{DUR} and A_{DUR} are weakly, linearly HR dependent, the observed close fit (*r*^{2} = 0.89) of *D*_{diastasis} by *Eq. 2* is reassuring and anticipated. Thus suggesting that changes in the duration of diastasis, not a change in E- or A-wave duration, dominates changes in MDD.

The analyses of peak flow rates indicate that, during supine bicycle exercise, E_{peak} is essentially independent of HR. A_{peak} clearly increases with exercise and is seen to be HR dependent and is known to rise rapidly as it merges with the deceleration portion of the E wave (15). In assessing this relationship, we took into consideration the effects of merging E and A waves, as suggested by Meisner et al. (23). Their analysis suggests that A-wave amplitude should not be measured from baseline but that the flow due to atrial systole should only consider the “booster-pump” contribution in excess relative to the contribution of the E wave (23). The doubling of slope of A_{peak} versus HR, from the region of separate E and A waves (*D*_{diastasis} > 0) to where E- and A-waves merge (*D*_{diastasis} < 0) supports these arguments and indicates that the A_{peak} should not be measured relative to baseline but rather relative to the E-wave velocity at the commencement of the A wave, as suggested by Appleton et al. (2). In contrast, E_{peak} remains essentially constant until the A wave completely merges with and becomes indistinguishable from the E wave. The shortening of MDD thus requires that average diastolic velocity increase throughout the range of observed HRs.

The area under the E and A waves (VTI) remains essentially constant during supine bicycle exercise. The regression relation showed slight increase in VTI; however, the low correlation (*r*^{2} = 0.12) suggests only weak dependence on HR. With the assumption of a constant effective mitral valve area (MVA_{eff}), it implies that stroke volume (SV) remains similarly constant via SV = VTI·MVA_{eff}. However, within the HR range of our study, SV is known to increase significantly and later plateau at HRs above our range (30). This observed increase in SV as a function of HR with an essentially constant VTI can be explained by an increase in MVA_{eff} with an increase in HR (7). This is likely associated with the increase in average diastolic transmitral flow velocity. Further independent validation of this predicted flow-dependent increase in effective mitral valve area at changing HRs may be achieved by echocardiography, cardiac MRI, or similar cardiac output measurements.

Whereas the early rise in A_{peak} (HR < 85) suggests an exercise-related change in load, multiple measures and minimal scatter in the data indicates that respiration-induced variation does not appear to have a significant effect.

### Limitations

The main limitation in this study relates to the ability to obtain quality echocardiographic images during exercise. The increased motion of the thorax, as a result of increased respiration due to exercise, can temporarily alter the geometric relationship between the transducer-generated sample volume and the flow. This is obviated somewhat by averaging measurements as well as by sonographer experience. In addition, the supine (rather than upright) position of the subject made transducer positioning and image acquisition more reliable. Because our focus was on transmitral flow during increasing HR, we did not specifically monitor the workload for each volunteer. In addition, it is possible that E- and A-wave durations may be affected by whether HR is increasing (exercise) or decreasing (cool down or recovery period). We did not analyze subjects during the cool-down period or in an upright posture. Details of E and A waves were not reliably discernable above 120 bpm due to noise and resolution limitations. Because a change in HR from 50–60 bpm (at rest) to ∼120–140 bpm (with exercise) was deemed to be adequate based on image acquisition constraints, subjects were not asked to exercise to their maximum HR according to HR_{max} = 220 − age (10). Achieving this limit could assist in resolving possible nonlinear and other effects at high HRs; however, obtaining quality echo images at these high HRs becomes a significant and potentially unachievable challenge.

Because our intent is to consider diastole in terms of its mechanical or kinematic components, such as flow, we have not explicitly included isovolumic relaxation. Its inclusion may have a slight effect on our measured fitting parameters. The inclusion of isovolumic relaxation time, which is minimally HR dependent, would essentially add a constant (shift) to the data. It would not alter our MDD relation other than generate a different numerical constant for the fit. Because it is essentially a constant offset, it would not change the significant finding that it is the change in the duration of diastasis that modulates the change in the duration of diastole. From the concordance of our parameters with those obtained in previous studies of QT, QS_{2} duration, and diastolic perfusion time (1, 5, 6, 16), we believe that the effects are insignificant.

In conclusion, in normal, healthy volunteer subjects performing supine bicycle exercise, the observed duration of diastole as a function of HR in the 50–60 to 120–140 bpm range is extremely well fit (*r*^{2} = 0.98) by the derived algebraic expression for MDD. The duration of diastasis, its shortening, and eventual disappearance as HR increases is the dominant factor accounting for the shortening of MDD. Compared with the duration of diastasis, the mechanical filling phases associated with the E and A waves are essentially HR independent. These observations support the kinematic modeling-based perspective that early rapid filling (E waves) and late atrial filling (A-waves) are mechanical oscillations that are minimally HR dependent.

## GRANTS

This study was supported in part by the Heartland Affiliate of the American Heart Association (Dallas, TX), the Whitaker Foundation (Roslyn, VA), National Heart, Lung, and Blood Institute (Grants HL-54179 and HL-04023), and the Alan A. and Edith L. Wolff Charitable Trust (St. Louis, MO).

## Acknowledgments

We thank our sonographer Peggy Brown for expert data acquisition. The authors acknowledge helpful discussions with Andrew Bowman and Amy Nichols during manuscript preparation.

## Footnotes

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

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- Copyright © 2004 by the American Physiological Society