# Turbulent/Viscous Interactions Control Doppler/Catheter Pressure Discrepancies in Aortic Stenosis

## The Role of the Reynolds Number

## Jump to

## Abstract

*Background* Despite good correlation between Doppler and catheter pressure drops in numerous reports, it is well known that Doppler tends to apparently overestimate pressure drops obtained by cardiac catheterization. Neither (1) simplification of the Bernoulli equation nor (2) pressure recovery effects can explain this dilemma when taken alone. This study addressed the hypothesis that a Reynolds number–based approach, which characterizes (1) and (2), provides a first step toward better agreement of catheter and Doppler assessments of pressure drops.

*Methods and Results* Doppler and catheter pressure drops were studied in an in vitro model designed to isolate the proposed Reynolds number effect and in a sheep model with varying degrees of stenosis. Doppler pressure drops in vitro correlated with the directly measured pressure drop for individual valves (*r*=.935, .960, .985, .984, .989, and .975) but with markedly different slopes and intercepts. A Bland-Altman type plot showed no useful pattern of discrepancy. The Reynolds number was successful in collapsing the data into the profile proposed in the hypothesis. Parallel results were found in the animal model.

*Conclusions* Apparent overestimation of net pressure drop by Doppler is due to pressure recovery effects, and these effects are countered by both viscous effects and inertial/turbulent effects. Only by reconciliation of discrepancies by use of a quantity such as Reynolds number that embodies the relative importance of competing factors can the noninvasive and invasive methods be connected. This study shows that a Reynolds number–based approach accomplishes this goal both in the idealized in vitro setting and in a biological system.

Doppler ultrasound has become a widely used method for noninvasive assessment of valvular stenosis.^{1} ^{2} ^{3} ^{4} ^{5} Pressure drops are typically calculated by insertion of a maximal velocity, V, into the simplified Bernoulli equation (pressure drop=4V^{2}). Despite good correlation between Doppler and catheter pressure drops in numerous reports, an interesting problem reported in the literature has been apparent overestimation by Doppler. Apparent overestimation by Doppler cannot be fully explained by the widely acknowledged difference between peak-to-peak and instantaneous pressure drops (peak-to-peak refers to comparison of peak pressures proximal and distal to the obstruction). Because of obligatory displacement of the catheter from the vena contracta of the stenotic jet, catheter measurement of pressure drop and Doppler estimation of pressure drop are fundamentally different quantities and can still differ significantly when instantaneous catheter pressure drops are used. We will therefore use the words “apparent overestimation” and “discrepancy” in this article in place of the commonly used “overestimation” and “error.”

Apparent overestimation has been observed for aortic stenosis in adults^{6} and children^{7} and across subvalvular obstructions.^{8} The common but largely informal knowledge that overestimation occurs has led to development of “mental correction factors” and variable utility of the Doppler pressure drop for clinical decision making between centers. Therefore, to take full advantage of the noninvasive nature of Doppler measurements, it is an important clinical goal to understand Doppler-catheter discrepancies, because they now occur unpredictably.

It is a logical first step to link Doppler/catheter discrepancies solely to (1) the simplification of the Bernoulli equation^{4} or (2) pressure recovery effects.^{9} ^{10} Unfortunately, these two effects are not explicitly linked, they can compete with one another, and their relative importance is not represented by commonly measured quantities in an echocardiographic examination. This study describes the first steps of an approach to reconciling Doppler and catheter pressure drops that relies on a classic fluid dynamic quantity, the Reynolds number, which embodies the ratio of inertial to viscous forces in the flow field. In the context of aortic stenosis, this quantity characterizes the relative importance of (1) and (2) above. The purpose of this study is to validate the concept by use of in vitro and in vivo models, providing a foundation for ultimate clinical application.

## Theory

### Bernoulli Applicability

The simple explanation that the clinical version of the Bernoulli equation is oversimplified does not account for the commonly observed apparent overestimation of pressure drops by Doppler. That is, in proceeding from Bernoulli's complete mechanical energy balance, ΔP=ρ(V_{2}^{2}−V_{1}^{2})/2+acceleration effects+viscous effects, to the simple form that is applied clinically, ΔP=4V_{2}^{2}, the acceleration and viscous terms are deleted, as is the velocity proximal to the stenosis. Neglecting acceleration and viscous terms would cause underestimation, in contrast to the commonly observed problem of apparent overestimation. To stay within the context of the Bernoulli equation, deletion of the proximal velocity would be the only possibility for explaining apparent overestimation. However, for aortic stenosis, in the absence of subaortic obstruction, velocities in the outflow tract are much smaller than those in the stenotic jet and can appropriately be deleted.^{4} Therefore, an analysis of deleted terms in the simplified Bernoulli equation cannot fully explain apparent overestimation of pressure drops by Doppler.

### Pressure Recovery

Alternatively, pressure recovery effects have been described previously as a source for Doppler/catheter discrepancies.^{9} ^{10} It is well known from fluid dynamic principles that pressure falls to a minimum in the vena contracta of a jet emerging from a constriction and that this minimum in pressure is accompanied by a maximum in velocity. In an echocardiographic examination of aortic stenosis, it is this maximum velocity that is obtained by continuous-wave Doppler and used to calculate a pressure drop by the simplified Bernoulli equation. Distal to the vena contracta, however, flow expands, velocity falls, and pressure rises to a level representing the overall energy loss due to flow through the obstruction. (“Energy loss” refers to kinetic or potential energy that has been converted into heat and is therefore irrecoverable. Use of this term hereafter implies this meaning only, and not actual loss of energy, which would not be possible because total energy must be conserved.) The pressure drop corresponding to this overall energy loss is what is measured in most catheterization examinations, because pressure recovers within a short distance of the vena contracta, and it is a virtual impossibility to maintain a catheter within the vena contracta, since it cannot be visualized. Because flow always expands distal to the vena contracta, pressure recovery is a fundamental cause of apparent overestimation when Doppler and catheter pressure drops are compared. However, the extent of apparent overestimation cannot be fully explained simply by expansion of flow from the vena contracta into a larger chamber, since the distal chamber is always larger than the stenotic valve, but Doppler and catheter pressure drops sometimes agree or Doppler will underestimate. Therefore, an analysis of pressure recovery based on the ratio of areas of the receiving chamber and the stenosis cannot fully explain apparent overestimation of pressure drops by Doppler.

### Reynolds Number Reconciliation of Deleted Terms and Pressure Recovery

How can discrepancies due to simplification of the Bernoulli equation and those due to pressure recovery be reconciled by use of quantities available in an echocardiographic examination? We have developed a mathematical/physical explanation of discrepancies that combines Bernoulli simplification and pressure recovery into a single model. Validation of this model will be a first step toward true reproduction of catheter data by Doppler. The flow phenomena described above would in theory produce a plot with the structure shown in Fig 1⇓, which shows apparent overestimation by the simplified Bernoulli equation, expressed as a percentage, versus Reynolds number. The Reynolds number, N_{Re}, is a dimensionless quantity representing the ratio of inertial to viscous forces, N_{Re}=ρVd/μ=inertial forces/viscous forces, where ρ is fluid density, V is velocity, d is effective diameter, and μ is viscosity. The four quantities can be expressed in any consistent set of units such that Reynolds number is dimensionless: for example, density in g/cm^{3}, velocity in cm/s, diameter in cm, and viscosity in poise (g/cm·s). This quantity should normalize the pattern of discrepancies as follows (and referring to Fig 1⇓): The Reynolds number embodies viscous forces in its denominator, so that when the Reynolds number is low, viscous forces are important by definition. Therefore, at low Reynolds numbers, the viscous term, which is deleted from the simplified Bernoulli equation, would be an important cause of underestimation (zone 1, Fig 1⇓). With increasing Reynolds numbers, viscous forces would be less important, and pressure recovery effects would be relatively more predominant, resulting in apparent overestimation (zone 2). Further increases in Reynolds number beyond the peak or plateau of apparent overestimation would be associated with conversion of energy into nonrecoverable forms as turbulent shearing between the stenotic jet and the relatively stagnant parajet region increases. Less pressure recovery is possible under these conditions, and therefore, apparent overestimation by Doppler would be reduced. In this latter regime of flow, the hypothetical curve should descend back toward zero, as shown in zone 3 of Fig 1⇓.

It is important to note the spatial locations of these phenomena. Viscous losses, for example, occur primarily upstream and within the stenosis, whereas pressure recovery effects are controlled by flow factors distal to the obstruction. In fact, the “viscous effect” term in the Bernoulli equation above, which is the commonly written form in the cardiology literature, also includes the irrecoverable losses due to turbulent shearing in the distal jet, which are not related to viscosity at all. Nevertheless, when true viscous losses or irrecoverable losses due to turbulence occur, the fundamental phenomenon of pressure recovery (and overestimation) will be restrained (zone 1 or 3).

It was the purpose of this study to show that the proposed model of Fig 1⇑, which is based on a single quantity representing the balance of relevant forces, holds for aortic stenosis. Specifically, we addressed the hypothesis that apparent overestimation of net pressure drop by Doppler is due to pressure recovery effects, and these effects are countered by both viscous effects on the low end of a Reynolds number scale and inertial/turbulent effects on the high end of the scale. We selected both in vitro and in vivo (animal) models to address this hypothesis; the former to maintain precise control of flow conditions and the latter to demonstrate that this complex explanation would actually manifest itself in a biological system.

## Methods

### In Vitro

Noting that pressure recovery effects are controlled not only by turbulent/viscous considerations but also by receiving-chamber geometry considerations, for the in vitro portion of the study we selected a model design that isolates the phenomena addressed in our hypothesis. Voelker and colleagues^{11} have published a pressure recovery index that potentially links the Doppler prediction of maximal pressure drop to the overall pressure drop by use of an area-based correction factor shown in the following equation as (1−C): P_{v}−P_{A}=(ρ/2)V_{x}^{2}(1−C), where C=2[(A_{x}/A_{A})−(A_{x}^{2}/A_{A}^{2})], P_{v} is pressure in the left ventricle, P_{A} is pressure in the aorta, V_{x} is velocity in the vena contracta, A_{x} is vena contracta area, and A_{A} is area of the aorta. The correction factor (1−C) suggested by Voelker et al includes only the ratio of effective orifice area in the vena contracta, A_{x}, to aortic area, A_{A}. As (1−C) approaches unity, more overall energy loss would occur, pressure recovery should not be appreciable, and Doppler predictions should coincide with the overall pressure drop (if receiving-chamber size were the only factor controlling the discrepancy). To isolate the Reynolds number effect for the in vitro portion of this study, we used an oversized distal chamber with a diameter of 30 cm (Fig 2⇓) and severe degrees of aortic stenosis (effective areas of 0.07 to 0.42 cm^{2} determined by the method described below), precluding the need for Voelker pressure recovery correction factors [since (1−C)=1]. This model design therefore removes significant pressure recovery based on the Voelker index and leaves only overestimation or underestimation resulting from a changing balance of inertial and viscous forces. This simplified model allows us to perform the following specific test: a plot of Doppler overestimation versus Reynolds number (1) should be flat and average 0% if chamber size is the only effect, or (2) should follow the pattern in Fig 1⇑, which would support our hypothesis.

The commissures of bovine pericardial valves (25 mm) were sutured to mimic aortic stenosis. A pulsatile flow regime was used with a 33% aqueous glycerin solution as the blood analog fluid. Pulse frequency was 70 bpm, with systole occupying 35% of the cycle. Stroke volumes were varied from 15 to 100 cm^{3} with a standard Harvard pulsatile pump. Valves were sutured to produce six degrees of stenosis, producing a total of 55 hemodynamic stages, with peak continuous-wave velocities ranging from 1.4 to 7.3 m/s. Velocities were measured with a Vingmed 775 Doppler system interfaced to a Macintosh IIci computer with Echopac software (Vingmed Sound, A/S). Peak Reynolds numbers were calculated from the above equation, with a diameter calculated from an area obtained by dividing the stroke volume by the velocity-time integral over systole. A circular geometry was assumed to obtain the effective diameter. Peak Reynolds numbers ranged from 2426 to 8940. Chamber pressures were measured with Millar pressure transducers, which were also interfaced to a computer. Pressure taps were located a distance of 3.25 cm upstream and downstream from the valve to ensure that static and recovered pressures were measured in each chamber. Peak instantaneous pressure drops ranged from 7.5 to 210 mm Hg.

### In Vivo

Anesthesia was induced in four juvenile sheep with sodium pentobarbital (25 mg/kg IV) and maintained with 1% to 2% isoflurane with oxygen; the animals were ventilated via an endotracheal tube with a volume-cycled ventilator. Bilateral transverse thoracotomies were performed. Aortic stenosis was created by banding the ascending aorta at the top of the sinuses of Valsalva to approximately coincide with the location of the leaflet tips, because the commissures extend to the sinotubular junction. Orifice diameters ranged from 0.09 to 1.6 cm (see below for measurement details). A Swan-Ganz catheter was positioned in the main pulmonary artery inserted via the femoral vein. Another catheter was positioned in the right common femoral artery for monitoring systemic arterial pressure and arterial blood gases. These catheters were interfaced with a physiological recorder (ES 2000, Gould Inc) by use of fluid-filled pressure transducers (model PD23 ID, Gould Statham). Arterial blood gases and pH were maintained within physiological ranges. Simultaneous aortic and left ventricular pressures were obtained from intracavity manometer-tipped catheters (model SPC-350, Millar Instruments, Inc) positioned transmurally and interfaced with the same recorder. For each experiment, the distal catheter was positioned to ensure measurement of fully recovered pressure. It was advanced from a distal position until a noticeable deviation of pressure appeared and signal noise increased, indicating that the stenotic jet had been reached. It was then withdrawn carefully until these phenomena disappeared and an overall peak instantaneous pressure drop was established. Care was taken to avoid kinetic pressure effects. All hemodynamic data were recorded at paper speeds of 250 mm/s. Four consecutive cardiac cycles were analyzed for each hemodynamic determination. A hydrostatic standard was used for calibration of all pressure recordings. After baseline measurements, varying degrees of severity of stenosis (peak pressure drops ranging from 8 to 150 mm Hg) were produced by alteration of the degree of constriction and preload with whole-blood transfusion. Cardiac outputs ranged from 0.975 to 4 L/min. Insensible fluid loss and associated electrolyte disturbances exacerbated by the open thoracotomy were monitored by frequent determinations of serum electrolyte and hematocrit; aberrations were avoided by continuous infusions of lactated Ringer's solution and 5% dextrose in water supplemented with potassium and calcium, as necessary. Hematocrit varied minimally among the four sheep: 36±4% throughout the experiments, providing a relatively consistent viscosity between the in vivo studies. The value of viscosity was taken as 3.5 cP for all calculations. A total of 30 hemodynamic states (4 to 11 per animal) were obtained (including 1 nonstenotic baseline for each sheep). Simultaneous continuous-wave Doppler–determined velocities (0.55 to 6.0 m/s) were recorded with a Vingmed 775 system interfaced to a Macintosh IIci computer, permitting direct digital transfer of the ultrasound data. A 2.5-MHz transducer was positioned epicardially so as to obtain optimal spectral displays and audio interrogational signals. The peak Reynolds number was calculated from the defining equation given above, with a stenotic diameter that was measured as follows. The circumference of the aortic constriction was measured directly in the open-chest sheep during data acquisition. This value was corrected for wall thickness, which was measured postmortem, and converted into an effective diameter. Peak Reynolds numbers for the in vivo studies ranged from 1350 to 7476.

### Statistical Analysis

For the in vitro studies, peak Doppler-predicted pressure drop was correlated with peak instantaneous catheter pressure drop by regression analysis. Apparent overestimation by Doppler was calculated for both the in vitro and in vivo studies with the instantaneous catheter pressure drop used as the standard, and the pattern of discrepancy between Doppler-predicted and catheter pressure drops was assessed with a Bland-Altman–type graph.^{12} Overestimation was also plotted versus peak Reynolds number and compared with Fig 1⇑.

## Results

### In Vitro

The mean Doppler pressure drop was 73±53 mm Hg, and the mean catheter pressure drop was 62±52 mm Hg. Doppler pressure drops correlated with but overestimated the directly measured pressure drop for each individual valve with correlation coefficients of *r*=.935, .960, .985, .984, .989, and .975 (Fig 3⇓). Doppler pressure drops exceeded the catheter pressure drops by a maximum of 64% (mean, 23±17%; range, −9% to 64%). Mixing the valves (all 55 stages) in a Bland-Altman plot (Fig 4 top⇓) demonstrates our inability to readily characterize apparent overestimation.

Transforming the data into the Reynolds number domain (Fig 4 bottom⇑) collapsed the data into the pattern suggested by theory (compare Figs 4 bottom and 1⇑⇑). Overestimation occurred in most cases as postulated. Four of the 55 points showed slight underestimation, and these were partitioned on either side of the Reynolds number scale (two at N_{R}<3000 and two at N_{R}>8000) in accordance with our hypothesis. Moving through increasing Reynolds numbers between 3000 and 8000, a smooth increase in the discrepancy is observed, followed by a section of decreasing discrepancies as turbulence grows and produces irrecoverable energy loss (removing pressure-recovery effects). The mean Reynolds number for the in vitro studies was 5549±1809.

### In Vivo

The mean Doppler pressure drop was 70±32 mm Hg, and the mean catheter pressure drop was 62±32 mm Hg. Doppler pressure drops for each animal overestimated the catheter pressure drop by up to 75% for all stenotic valve conditions (mean, 20±22%; range, −11% to 75%). These numbers in the biological system are very similar to those in the in vitro model. Mixing the in vivo data in a Bland-Altman plot (Fig 5 top⇓) demonstrates our inability to characterize the discrepancies by traditional means.

Plotting the discrepancies versus Reynolds number (Fig 5 bottom⇑) collapsed the data into a pattern suggested by theory and by the in vitro results. Apparent overestimation occurred according to the pattern postulated in the theoretical section of this article and as shown in the idealized in vitro studies. One point was especially interesting, showing virtual agreement (1.9% overestimation) at a high Reynolds number of 7476, although the bulk of data points have already descended back to zero at a Reynolds number of 5500. It is useful to observe that this single point remaining near zero as Reynolds number increases beyond 5500 is exactly as postulated theoretically: that Doppler catheter discrepancies descend to zero at very high Reynolds numbers, then remain there as Reynolds number is increased further (zone 3, Fig 1⇑). The critical maximum range of Reynolds numbers, ≈5500, is smaller than that for the in vitro studies (≈8000), as expected because of lack of the oversized chamber. The mean Reynolds number for the in vivo studies was 3893±1371.

## Discussion

It is well known that peak pressure drops estimated by Doppler ultrasound often overestimate the values obtained by cardiac catheterization. Discrepancies have been attributed to (1) oversimplification of the Bernoulli equation and (2) pressure recovery effects. Until now, these effects have not been directly linked, since the Bernoulli equation, simplified or not, is applied between a point in the outflow tract and the vena contracta. In contrast, pressure recovery effects are controlled primarily by phenomena distal to the vena contracta. The ultimate degree of apparent overestimation or underestimation of catheter pressure drop by Doppler is dependent in a complex way on both. This study proposes a Reynolds number–based approach that simultaneously considers deficiencies in the simplified Bernoulli equation and turbulence in the stenotic jet, which modulate the extent of pressure recovery. We confirmed our hypothesis using (1) in vitro modeling with an oversized distal chamber to remove all previously demonstrated pressure-recovery effects associated with chamber size and (2) an animal model to demonstrate the principles in a biological system.

A postulated pattern for Doppler/catheter discrepancies was described and shown graphically in Fig 1⇑. Most of the peak flows studied in vitro and in vivo fell into zones 2 and 3, showing apparent overestimation of the overall pressure drop by Doppler. As flow reexpands distal to the obstruction, pressure recovers as shown in Fig 6 top⇓, and the overall pressure drop becomes less than that estimated by Doppler. As Reynolds numbers increase further, more permanent loss of energy occurs, shifting the solid line in Fig 6 top⇓ down toward the dotted line and reducing the discrepancies as shown in Fig 6 bottom⇓. This latter regimen of flow is represented by zone 3 in Fig 1⇑, in which the apparent overestimation begins to descend back toward zero. Figs 4 bottom and 5 bottom⇑⇑ confirm this hypothesis for in vitro and biological data, respectively. In addition to demonstrating the role of Reynolds number, another clear result from these data is that the relationship between Doppler and catheter pressure drops cannot be modeled on the basis of receiving chamber size alone, as has been suggested,^{11} because the data in Fig 4 bottom⇑ would fall on a horizontal line if that were the case. Further work in patients with aortic stenosis is necessary to determine the more precise thresholds of zones 1, 2, and 3, since these data were obtained from an idealized in vitro model that isolates the effect of Reynolds number and from a sheep model that does not exactly mimic valvular aortic stenosis in humans. (Likewise, it would be potentially misleading to report a specifically fitted equation to Figs 4 bottom⇑ and/or 5 bottom.) Differences in the methods for determining effective orifice diameter (in vitro, from an area calculated by stroke volume divided by time-velocity integral; in vivo, by direct measurement) may explain variability between Reynolds number values for similar flow conditions.

It is a first-order point that apparent overestimation is basic to any stenosis, because pressure recovery will always occur to some extent. The more subtle point is that when apparent agreement occurs, it is due to factoring in of viscous forces on the low end of the Reynolds scale or incremental obliteration of pressure recovery due to irrecoverable turbulent losses on the high end of the scale. Furthermore, if the viscous term is very significant (rare for aortic stenosis), we can see underestimation by the simplified Bernoulli equation, even though pressure recovery is still occurring beyond the stenosis. It is not surprising that considerable confusion exists regarding Doppler/catheter agreement, since apparent overestimation is basic, due to pressure recovery, but apparent agreement can occur for two reasons that appear to be in contrast (high viscous effects versus high inertial effects).

These phenomena are not unique to the valves in this study. Hidden evidence supporting our explanation exists in some previously published work. The concept is nicely supported by the clinical studies of Bengur and colleagues,^{7} for example, who studied aortic stenosis in children and provided a comprehensive summary of their data. Although these investigators were attempting to predict aortic valve area, if we compare peak pressure drops by echo with simultaneous catheter measurements (which were provided by the authors), we see apparent overestimation in 41 of 42 patients by straight numerical comparison. When mean pressure drops were compared, however, only 67% were overestimated and 33% were equal or underestimated. This reduction of apparent overestimation for mean pressure drops versus peak pressure drops fits well with our model. Averaging over a beat includes low Reynolds number points from zone 1 (see Fig 1⇑) at the beginning and end of the cardiac cycle, causing apparent overestimation to be partially canceled, exactly canceled (producing agreement), or overcome (causing underestimation due to lack of the viscous term in the Bernoulli equation). Therefore, although low Reynolds number points in zone 1 typically do not appear when peak pressure drops are considered (Figs 4 bottom and 5 bottom⇑⇑), they play an important role in modulation of agreement of mean pressure drops because they counteract the basic overestimation due to pressure recovery. Peak or mean pressure drops may be used in the clinical setting. The concepts presented in this article and supported in the clinical data of Bengur et al highlight the fact that mean Doppler pressure drops consist of an integrated pressure over the course of a beat and that the extent of agreement or disagreement with catheter values will vary within the beat. Apparent agreement of mean pressure drops should be viewed with caution, because it must be due to a fortunate cancellation of overestimation and underestimation points within the beat.

### Instantaneous Versus Peak-to-Peak Pressure Drops

Discussions of “overestimation” of catheter pressure drops by Doppler often lead to a distinction between peak-to-peak and instantaneous pressure drops. Peak-to-peak pressure drops are obtained by comparison of the peak proximal pressure with the peak distal pressure. Since there is some time delay between these peaks, owing to the noninfinite wave speed of the pressure pulse in compliant vessels, this measure will be less than the peak instantaneous pressure drop by catheter. Although this concept is important and certainly exists, it is not sufficient to explain commonly observed discrepancies. The comprehensive clinical data from Bengur and colleagues^{7} were obtained by use of instantaneous pressure drops, as were both the in vitro and in vivo data in this study.

### Clinical Implications

Doppler ultrasound has become a widely used method for noninvasive assessment of valvular stenosis. The pressure drop obtained at cardiac catheterization, however, remains the gold standard used in surgical decision making. It is common knowledge that pressure drops estimated by Doppler in the echocardiography laboratory are often much lower when measured subsequently by catheter. These often dramatic decreases in pressure drop cannot be fully explained by changes in the hemodynamic state of the patient. Indeed, when simultaneous echo-catheter studies are performed, apparent overestimation by Doppler is still found (Reference 7 and the present study, for example). The present study demonstrates that overestimation is basic because of the concept of pressure recovery and that it is modulated by forces embodied in the Reynolds number. This study will support subsequent investigations that may be designed to allow Doppler prediction of the catheter pressure drop, since (1) it is the catheter pressure drop that is cited in the accumulated natural history data on aortic stenosis and (2) the overall pressure drop, rather than the maximal drop estimated by continuous-wave Doppler and the Bernoulli equation, most directly reflects ventricular stress and associated quantities such as work. The Reynolds number does not completely reconcile Doppler and catheter pressure drops, as reflected by the scatter in our data. A complete reconciliation will probably include proximal and distal chamber geometry effects, factors that are becoming more accessible with the advent of three-dimensional echocardiography. Estimation of the Reynolds number for use in the clinical setting will rely on any number of proposed methods for calculating valve area (continuity, stroke-volume/time-velocity integral, Gorlin equation); as these methods improve, so will the usefulness of the Reynolds number. One might ask: if the valve area is already available, then why is Reynolds number correction of the pressure drop needed? Because, despite the variation of pressure drop with flow, only pressure (which produces systolic stress) can be used to assess the elevated workload placed on the ventricle by a stenotic valve. Although assessment of area is very useful, increased work most directly reflects the physiological impact of aortic stenosis.

### Study Limitations

The hematocrits of blood from the four sheep studied were relatively consistent (see “Methods”). Although the methods used to produce volume changes may have produced some change in the hematocrit from point to point, this effect was minimized by the use of whole-blood transfusion. Furthermore, variable viscosity (which would be produced by variable hematocrit) would affect the contribution of the viscous term in the Bernoulli equation, and these results for aortic stenosis showed that viscous effects were of minimal importance (Figs 4 bottom and 5 bottom⇑⇑ do not descend to zone 1, as shown in Fig 1⇑). Variations in viscosity produced by the hematocrits of 36±4% would also have a minimal effect on calculated Reynolds numbers.

The method for maintaining the distal catheter in the fully recovered pressure zone is the best possible choice, given the complex nature of pressure recovery phenomena. It was important to position the catheter for each experiment because, for different flow rates and Reynolds numbers, the separation region in the vicinity of the valve may change size and envelop the catheter if care is not taken.

The Womersley number is another dimensionless parameter that includes the frequency of pulsation and its effect on inertial and viscous forces.^{13} Future studies that systematically vary heart rate may allow introduction of this more complex parameter to produce even better reconciliation of Doppler and catheter pressure drops.

We chose to create the aortic stenosis using a banding method so that the severity of stenosis could be adjusted within the period of the acute animal study. The potential for such a band to act as a peripheral stenosis rather than a valvular one is dependent on the distance from the aortic valve to the band. Peripheral stenosis is characterized by a time delay of the pressure pulse transmitted from the ventricle to the obstruction and by wave reflections from the distal vasculature that return to a peripheral stenosis before a valvular one. Placement of the band at the level of the leaflet tips acts as a valvular stenosis in a spatial sense and was the best model to confirm this basic concept while minimizing the number of animals required.

### Conclusions

Apparent overestimation of net pressure drop by Doppler is due to pressure recovery effects, and these effects are countered by both viscous effects on the low end of a Reynolds number scale and inertial/turbulent effects on the high end of the scale. The principles of physics that govern flow through a stenosis clearly show that Doppler and catheter pressure drops should not agree except at the points at which the curve in Fig 1⇑ coincides with zero. Attempts to force Doppler and catheter pressure drops to agree in one's mind, whether the catheter pressure drops are peak-to-peak or instantaneous, would be highly inadvisable, because uniform agreement contradicts the laws of physics. Only by reconciliation of discrepancies by use of a quantity that embodies the relative importance of competing factors can the noninvasive and invasive methods be connected. This study shows that a Reynolds number–based approach makes this basic connection both in the idealized in vitro setting and in a biological system, providing a basis for clinical studies in patients in which parabolic correction factors may be developed.

## Acknowledgments

This work was supported in part by funds from the American Heart Association, Pennsylvania Affiliate, and the Research Advisory Committee of Children's Hospital of Pittsburgh.

- Received February 26, 1996.
- Revision received July 3, 1996.
- Accepted July 11, 1996.

- Copyright © 1996 by American Heart Association

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- Turbulent/Viscous Interactions Control Doppler/Catheter Pressure Discrepancies in Aortic StenosisEdward G. Cape, Michael Jones, Izumi Yamada, Michael D. VanAuker and Lilliam M. Valdes-CruzCirculation. 1996;94:2975-2981, originally published December 1, 1996https://doi.org/10.1161/01.CIR.94.11.2975
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- Turbulent/Viscous Interactions Control Doppler/Catheter Pressure Discrepancies in Aortic StenosisEdward G. Cape, Michael Jones, Izumi Yamada, Michael D. VanAuker and Lilliam M. Valdes-CruzCirculation. 1996;94:2975-2981, originally published December 1, 1996https://doi.org/10.1161/01.CIR.94.11.2975Permalink: