## Abstract

The pressure-velocity relationship across the normal mitral valve is approximated by the Bernoulli equation ΔP = ½ ρΔ*v*
^{2} + *M* · d*v*/d*t*, where ΔP is the atrioventricular pressure difference, ρ is blood density,*v* is transmitral flow velocity, and *M* is mitral inertance. Although *M* is indispensable in assessing transvalvular pressure differences from transmitral flow, this term is poorly understood. We measured intraoperative high-fidelity left atrial and ventricular pressures and simultaneous transmitral flow velocities by using transesophageal echocardiography in 100 beats (8 patients). We computed mean mitral inertance (*
*) by *
* = ∫(ΔP−½ · ρ*v*
^{2})d*t*/∫(d*v*/d*t*)d*t*and we assessed the effect of the inertial term on the transmitral pressure-flow relation. *
* ranged from 1.03 to 5.96 g/cm^{2} (mean = 3.82 ± 1.22 g/cm^{2}). ΔP calculated from the simplified Bernoulli equation (ΔP = ½ · ρ*v*
^{2}) lagged behind (44 ± 11 ms) and underestimated the actual peak pressures (2.3 ± 1.1 mmHg). *
* correlated with left ventricular systolic pressure (*r* = −0.68,*P* < 0.0001) and transmitral pressure gradients (*r* = 0.65, *P* < 0.0001). Because mitral inertance causes the velocity to lag significantly behind the actual pressure gradient, it needs to be considered when assessing diastolic filling and the pressure difference across normal mitral valves.

- mitral valve
- Doppler echocardiography

the quantification of transvalvular pressure gradients has been one of the most important clinical applications of Doppler echocardiography (5). On the basis of the physical principle of conservation of energy, the Bernoulli equation is used to calculate transvalvular pressure gradients from observed blood velocity through the valve. In its complete form, the Bernoulli equation consists of the following four terms: *1*) convective, which relates the drop in pressure to the rise in kinetic energy as blood velocity increases on passing through the valve; *2*) inertial, which specifies the pressure drop needed to accelerate the mass of blood through the valve; *3*) viscous, which relates the pressure loss because of viscous drag along the walls; and *4*) gravitational, which relates to the effects of gravitational vector forces (10). For most stenotic and regurgitant orifices of clinical relevance, the inertial, viscous, and gravitational components of the Bernoulli equation are negligible compared with the convective component, and the equation is reduced to the simple form ΔP = 4*v*
^{2}, where ΔP is the transvalvular pressure gradient and *v* is velocity through the valve. The relationship has been validated in many clinical settings and is routinely used in daily practice (1, 5).

On the other hand, when flow through a nonrestrictive orifice, such as the normal mitral valve, is considered, the situation is more complex. Because of the larger mass of blood passing through the valve, a significant pressure difference is required to accelerate it, and the inertial term of the Bernoulli equation is no longer negligible. Furthermore, the change in blood velocity passing through the valve is not nearly so great as passage through a restrictive orifice, and so the convective term is much smaller than it would be otherwise. Viscous components are still not important for nonrestrictive orifice flow because the blood is in contact with the walls only briefly, even in situations of impaired left ventricular (LV) function, where flow or transmitral gradients may be minimal (10). Thus quantification of the pressure drop across the normal mitral valve with Doppler echocardiography requires careful assessment of the contribution of the inertial term in the Bernoulli equation.

Although previous authors have pointed out the importance of the inertial term in the complete Bernoulli equation (4, 11), there are strikingly little data on this term, especially in humans. Moreover, effects of the inertial term on the transmitral pressure-flow relation have not been known. Therefore, the purpose of the study was twofold: *1*) to determine the inertial term in humans, and*2*) to assess the effect of this term on the difference between the absolute pressure difference and the estimated pressure difference (ΔP = 4*v*
^{2}) from observed transmitral velocity curves.

## METHODS

### Theoretical Basis of Mitral Inertance

For transmitral flow along a streamline between the points where velocity and pressure are measured in the left atrium (A) and the left ventricle (V), the fall in pressure is governed by the Bernoulli equation for unsteady flow (neglecting viscous losses)
Equation 1where p_{A} and *v*
_{A} are pressure and velocity, respectively, within the left atrium; p_{V} and*v*
_{V} are similar measurements taken within the left ventricle at the tips of the mitral leaflet;*v*(*s*, *t*) is blood velocity at distance*s* along the streamline and time *t* in diastole, and ρ is blood density (1.05 g/cm^{3}) (2, 8). The integral term in *Eq. 1
* could be more readily evaluated if a simpler expression for *v*(*s*, *t*) were available. To accomplish this, we assume that the geometry of the left atrium, mitral valve, and left ventricle is relatively constant throughout diastole (2, 7-9). In this case, the spatial velocity profile along the streamline would have a similar shape throughout diastole, differing only in absolute magnitude (Fig.1). We call this spatial profile λ(*s*), which rises from near 0 in the atrium to 1 at the tips of the mitral leaflets, where velocity is maximum, before falling again in the ventricle. We can then obtain*v*(*s*, *t*) by multiplying λ(*s*) by *v*
_{M}(*t*), the temporal velocity at the vena contracta, where velocity is the highest along the streamline. Because this corresponds to the usual inflow velocity measured in clinical echocardiography as well as the ventricular velocity (*v*
_{V}), we will subsequently write *v* rather than *v*
_{M} or*v*
_{V}. λ(s) [or*v*(*s*, *t*)/*v*
_{M}(*t*)] then is seen to be a dimensionless number between 0 and 1, allowing us to rewrite the inertial term
Equation 2as
where we take advantage of the fact that λ(*s*) is independent of time and *v*(*t*) is independent of space, to separate the derivative from the integral term, simplifying the analysis considerably. With these substitutions, we can rewrite*Eq. 1
* as
Inasmuch as λ(*s*) is dimensionless, its integral over the streamline has the dimension of length, and we can identify it as the effective length (*L*) of the blood column being accelerated through the mitral valve (Fig. 1). We identify ρ*L* as the mitral inertance (*M*) and can rewrite*Eq. 1
* in the following form
Equation 3where ΔP is transmitral pressure gradient (2,7-9).

### Determination of Mean Mitral Inertance

Considering the proximal flow is relatively slow in the left atrium, we can neglect the proximal velocity,*v*
_{A}, and we describe Δ(*v*
^{2}) as simply *v*
^{2}. By rewriting *Eq. 3
*, *M* can be obtained as
When the numerator and denominator are small, relatively minor measurement errors will result in large errors in calculating*M*. To minimize this, we compute mean mitral inertance (*
*) by calculating integrals of each term during flow acceleration during early diastole as
Equation 4Thus, by measuring transmitral flow velocity and left atrial (LA) and LV pressures simultaneously, we can calculate*
* in humans.

### Study Population

We studied eight patients undergoing open heart surgery. Three patients had LV assist device (LVAD) implantation, four patients had coronary artery bypass grafting, and one patient had mitral valve repair for torn chordae. There were 6 males and 2 females, aged 34 to 74 yr (mean 53 ± 7 yr). All were in sinus rhythm and none had mitral stenosis. The study protocol was approved by the Institutional Review Board of the Cleveland Clinic Foundation and informed consent was obtained from all patients.

### Measurements

We measured high-fidelity LA and LV pressures by using a dual micromanometer-tipped catheter (Mikro-Tip model SPC-751, Millar Instruments; Houston, TX) in the operating room. After calibrating relative to atmospheric pressure, we inserted the catheter through the right upper pulmonary vein and passed across the mitral valve, enabling simultaneous measurements of LA and LV pressures. Pressure waveforms were digitized at 1,000 Hz and transferred to a Pentium-based personal computer (PC). LV systolic, minimum, end-diastolic pressures, and LA and LV crossover pressure in early diastole were determined. Intraoperative transesophageal echocardiography was performed in the standard manner with a commercially available echocardiographic system (Sonos OR and Sonos 1500, Hewlett-Packard; Andover, MA) and a 5-MHz phased-array transducer. This echocardiograph was specifically chosen because it does not introduce a time delay in the Doppler output, which would have limited the ability to obtain the synchronized hemodynamic and Doppler signals. Transmitral flow velocity was recorded simultaneously with pressure measurements by using pulsed Doppler echocardiography in either the basal four-chamber or LV long-axis views, locating a sample volume at the mitral tip and stored to optical disk in TIFF format. A timing signal was generated and stored simultaneously with both pressures and Doppler ultrasound to ensure temporal alignment of pressure waveforms and Doppler velocity spectra.

### Calculation of *
*

The data processing and analysis were performed with the use of a Pentium-based PC equipped with a commercially available scientific software package (LabView, version 4.0, National Instruments; Austin, TX). An additional library of image processing tools (Ultimage Concept VI, Graftek; Voisins-Le-Bretonneux, France) was used for image display, manipulation, and processing. Pressure waveforms and Doppler transmitral flow velocity spectra in identical beats were aligned by using a timing marker. This timing marker allowed for a beat-by-beat analysis of available individual beats hence eliminating the need to average the results of multiple, and potentially unsynchronized, beats. From digitized pressure waveforms, instantaneous pressure gradients between the left atrium and ventricle during diastole were calculated (ΔP). By tracing Doppler velocity spectra, instantaneous pressure gradients calculated with the use of the simplified Bernoulli equation (½ · ρ*v*
^{2}) and instantaneous acceleration of transmitral flow (d*v*/d*t*) were obtained. *
* during flow acceleration during early diastole was then computed with*Eq. 4
*.

### Comparison of Measured and Calculated Pressure Gradient Curve

Because the instantaneous ΔP curve derived from high-fidelity catheter measurements and the pressure gradient curve calculated from measured Doppler velocity (½ · ρ*v*
^{2}) were different in magnitude and phase, we compared them in terms of peak pressure difference and phase lag. The pressure difference was measured between the actual peak ΔP and the predicted peak ΔP by the simplified Bernoulli equation. The phase lag was measured as the time difference between the timing of the actual peak ΔP and the calculated peak ΔP. We then determined the effects of *M* on the pressure difference and the phase lag.

### Statistical Analysis

Data are expressed as means ± SD. Least-squares linear regression analysis was used to correlate two variables, specifically the relation between pressure underestimation and phase lag on*
*, and the relationship of *
* to LV maximal, minimal, and end-diastolic pressures. Comparison of unpaired measurements between two conditions (e.g., pre- and post-LVAD insertion) was done by using unpaired *t*-testing. We considered results significant when *P* was < 0.05.

## RESULTS

### Hemodynamics

We analyzed 100 beats from 40 runs obtained from eight patients. The hemodynamic and clinical characteristics are summarized in Table1. Because these beats were recorded intraoperatively, their loading conditions were variable; 74 beats were measured before, and 26 beats were measured after cardiopulmonary bypass. After LVAD insertion, LV systolic pressure was extremely low because of significant ventricular unloading by the LVAD. Thus LV systolic pressure ranged from 33 to 123 mmHg and LV end-diastolic pressure ranged from 11 to 30 mmHg.

### Mitral Inertance

*
* ranged from 1.03 to 5.96 g/cm^{2} with a mean of 3.82 ± 1.22 g/cm^{2}. Figure2 is a schematic example of the ΔP curve measured by a dual-tip catheter (a solid line) aligned with a calculated pressure difference curve from observed transmitral flow velocity by using the simplified Bernoulli equation (a dotted line). As flow accelerates during early diastole, the calculated ΔP curve lags behind the actual ΔP curve and the peak pressure is smaller than that on the actual pressure curve. Mean pressure underestimation was 2.3 ± 1.1 mmHg (ranging from 2 to 4.8) and mean phase lag was 44 ± 11 ms (ranging from 19 to 81). There were significant correlations between these differences and *
* (*r*= 0.85 for pressure underestimation, *r *= 0.78 for time lag, both *P* < 0.0001, Fig.3).

### Effects of Hemodynamics and Disease States on

There was a significant correlation between *
* and LV systolic pressure (*r *= −0.68, *P* < 0.0001). However, no significant correlation was found between*
* and other intracardiac pressures (*r *= 0.02, *P* = 0.80 for LV end-diastolic pressure, *r*= 0.01, *P* = 0.89 for LV minimum pressure, *r*= 0.33, *P* = 0.021 for LA and LV crossover pressure in early diastole). A significant correlation was noted between*
* and actual transmitral ΔP measured by a high-fidelity catheter (*r *= 0.65, *P* < 0.0001). To confirm the consistency of *
* on a beat-to-beat basis under constant hemodynamic states, we calculated coefficient of variation and mean ± SD of measured*
* in each run. The beat-to-beat variability was small with an average coefficient of variation of 12% and mean ± SD of 0.46 g/cm^{2}, suggesting that *
* changed little under identical hemodynamics. *
* derived from 50 beats before LVAD insertion in three patients who had device implantation, was significantly higher than that derived from 39 beats derived from other patients (coronary artery bypass grafting and mitral valve repair, 4.34 ± 0.83 vs. 2.54 ± 0.82 g/cm^{2}, *P* < 0.0001). When*
* was compared in patients before (50 beats) and after (11 beats) LVAD implantation, there was no significant difference (4.34 ± 0.83 vs. 4.67 ± 1.63 g/cm^{2},*P* = 0.324).

## DISCUSSION

### Previous Studies

Doppler echocardiography has enabled the noninvasive assessment of transvalvular pressure gradient for most stenotic or regurgitant orifices by using the simplified Bernoulli equation (1,5). However, for normal orifices, this advantage of Doppler echocardiography has not been fully taken because inertance cannot be ignored (10). Although Yellin (11) and Pasipoularides (4) have identified the inertance as a critical factor to predict pressure gradient from observed velocity, there have been few attempts to qualify it. Flachskampf et al. (2) has determined the value of *M* in an experimental model and investigated the physical determinants of the inertial term in the complete Bernoulli equation. However, no studies have attempted to quantify the *M* in beating hearts of humans. We determined the value of *
* in humans for the first time by comparing the pressure difference derived from a high-fidelity catheter and that from Doppler transmitral flow velocity.

### Effect of on Diastolic Pressure-Flow Relation

In vitro experiments have shown *M* to be on the order of three times the valve diameter (2). Thus for a stenotic valve, the inertance would be small and the velocity would track the actual pressure gradient closely (subject to conservation of energy, ΔP = ½ · ρΔ*v*
^{2}). In contrast, for a normal valve, the inertial effects are larger and the velocity lags farther behind the actual pressure gradient than in the case of the stenotic valve because pressure energy is being expended to overcome inertia rather than being converted solely to kinetic energy (10). As expected from this theoretical framework, we found that during flow acceleration in early diastole, the calculated pressure difference curve obtained from observed transmitral flow velocity significantly lagged behind the actual pressure difference curve. Also, the peak of the calculated pressure difference curve was lower than that of the actual pressure curve. The major finding of the present study showed that measured *M* showed a linear relationship to both the phase delay and the pressure difference. Thus*M* significantly affected the diastolic pressure-flow relation.

### Factors Affecting *M*

Flachskampf et al. (2) has demonstrated in an in vitro model in which a known pressure gradient was applied to orifices and conduits, that inertance depended on the orifice diameter and the conduit length and was not significantly related to pressure gradients. The study suggests that *M* depends mainly on the mitral apparatus geometry (mitral valve diameter and the length of mitral apparatus) but not significantly on loading conditions. On the other hand, we found that *M* had significant relationships with LV systolic pressure and transmitral pressure gradients. Also, we found that patients with profound heart failure who required LVAD insertion showed larger values of inertance than other patients. It may be a hasty conclusion that both studies are contradictory of each other. Changes in loading conditions or disease states may cause changes in mitral apparatus geometry. For example, patients with profound heart failure may have a dilated mitral annulus with low LV systolic pressure, both producing high *M*. At the same time, however, a dilated ventricle may decrease the length of mitral apparatus by preventing full opening of the mitral valve because of tethered mitral leaflets, which decreases *M*. These hemodynamic and geometric effects may counterbalance in patients undergoing LVAD insertion, resulting in unchanged *M* before and after the assist device. In the present study, none of the patients had mitral stenosis but three patients had a dilated mitral annulus in association with a dilated left atrium. The value of *
* in the present study ranged from 1.03 to 5.96 g/cm^{2}. This wide range of inertance might be explained by a variety of mitral apparatus geometry and by a variety of loading conditions. Further study is necessary to separate the effects of mitral apparatus geometry and loading conditions on *M*.

### Clinical Implications

We determined the value of *
* in beating hearts of humans for the first time. Because of *M*, the flow velocity curve was delayed on average 44 ms after the actual pressure difference curve, and the simplified Bernoulli equation underestimated pressure difference by ∼2.3 mmHg for normal mitral orifices. Thus *M*cannot be neglected when assessing pressure difference across the normal mitral valve by Doppler echocardiography. *M* may be calculated noninvasively by color M-mode Doppler echocardiography (8). By obtaining the spatiotemporal velocity characteristics that can be abstracted from the color M-mode profile, the different components of the unsteady Bernoulli equation can be solved for and used to derive the discrete convective and nonconvective components of transmitral flow. Our present method, using both invasive pressures and Doppler velocity spectra, would be useful to validate values of *M* obtained by color M-mode Doppler echocardiography.

If *M* is thus accurately and easily obtained in a routine clinical situation, we could accurately assess transmitral pressure gradient in diastole. Moreover, we could assess more precisely diastolic function beyond that assessed by such simple transmitral flow parameters as the E-to-A ratio. It may further be possible to determine the time constant of LV relaxation (τ) noninvasively (9). The initial acceleration of transmitral flow is mainly determined by the rate of growth in the transmitral pressure gradient (dΔP/d*t*), which has been shown to be approximately equal to LA pressure divided by τ. In situations where LA pressure is known, such as in the intensive care unit or the operating room, we could estimate dΔP/d*t* and thereby τ with a knowledge of transmitral flow and *M*. In the more common situation, where LA pressure is not directly available, it may still be possible to estimate diastolic function by utilizing isovolumic relaxation time. A previous study (7) from our laboratory has demonstrated that the isovolumic relaxation time is a function of LA pressure and τ. By measuring transmitral flow and isovolumic relaxation time, we could know both LA pressure and τ simultaneously. Combined with earlier work (3) relating the slope of mitral deceleration to chamber compliance, this may allow a truly quantitative assessment of LV diastolic function by noninvasive means.

Whereas the accurate determination of *M* in routine clinical scenarios may not be practical, our findings demonstrate some of the hemodynamic determinants of this critical parameter of LV diastolic filling. With this understanding, clinicians may better apply noninvasive quantitative echocardiographic techniques, such as mitral inflow Doppler velocity profiles, to better understand diastolic function and the effects of various medical and surgical interventions.

### Limitations

We calculated *
* with pulsed Doppler echocardiography by using a sample volume at the mitral tip level. Pulsed Doppler echocardiography can detect velocity at a specific point and does not always detect the highest velocity through the valve. Theoretically, *M* should be determined from the highest velocity through the valve. In addition, whereas the effective mitral valve area has been shown to play a significant role in determining*M*, it was not measured in this study. Although this information would have provided additional clinical insight, the ability to obtain continuous and accurate effective mitral valve area measurements on a beating and moving heart by direct planimetry is a significant limitation of transesophageal echocardiography.

A significant limitation in applying the principles of fluid dynamics to transmitral flow is the assumption of an invariant mitral valve area during diastole. Clinically, it is known that as the valve opens early in diastole and closes at the end of diastole, dynamic changes occur in the three-dimensional structure of the mitral annulus and the area of the valve orifice. Unfortunately, transesophageal echocardiographic techniques limit the ability to accurately measure these complex geometric changes. Hopefully, with advances in real-time three-dimensional echocardiography, these dynamic changes can be obtained with simultaneous pressure data to further understand these relationships. Nevertheless, our model of transmitral flow can serve as a foundation to build on with advances in imaging techniques and further understanding of the fluid dynamics of transmitral flow.

Another limitation of the present study is the small number of patients with a limited range of cardiac pathology. A future study of many patients with various cardiac conditions would be warranted to further clarify hemodynamic and geometric determinants of *M*.

In conclusion, application of the complete Bernoulli equation to calculate the transmitral pressure difference throughout diastole requires knowledge of mitral valve inertance. This study provides for the first time an estimate of *M* in humans. Because mitral inertial force causes the velocity to lag significantly behind the actual pressure difference, it needs to be taken into account in the assessment of pressure difference across the normal mitral valve and LV diastolic function by Doppler echocardiography.

## Acknowledgments

This study was supported in part by a grant from the Uehara Memorial Foundation (to S. Nakatani), National Heart, Lung, and Blood Institute Grant 1R01-HL-56688, and National Aeronautics and Space Administration Grant NCC9-60 (to J. Thomas).

## Footnotes

Present address of S. Nakatani: National Cardiovascular Center, Osaka, Japan.

Present address of P. M. Vandervoort: Heart Center Limburg, Genk, Belgium.

Address for reprint requests and other correspondence: J. D. Thomas, Dept. Cardiology, The Cleveland Clinic Foundation, Desk F15, 9500 Euclid Ave., Cleveland, OH 44195 (E-mail:thomasj{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.

- Copyright © 2001 the American Physiological Society