# Effect of three-dimensional valve shape on the hemodynamics of aortic stenosisThree-dimensional echocardiographic stereolithography and patient studies

## Author + information

- Received October 29, 2001
- Revision received June 6, 2002
- Accepted July 2, 2002
- Published online October 16, 2002.

## Author Information

- Dan Gilon, MD, FACC
^{*},^{*}(gilond{at}cc.huji.ac.il), - Edward G Cape, PhD
^{†}, - Mark D Handschumacher, BS
^{*}, - Jae-Kwan Song, MD, FACC
^{‡}, - Joan Solheim
^{‡}, - Michael VanAuker, BS
^{†}, - Mary Etta E King, MD, FACC
^{*}and - Robert A Levine, MD, FACC
^{*}

- ↵*
**Reprint requests and correspondence:**

Dr. Dan Gilon, Cardiac Ultrasound Laboratory, VBK508, Massachusetts General Hospital, 32 Fruit Street, Boston, Massachusetts, USA 02114.

## Abstract

**Objectives** This study tested the hypothesis that the impact of a stenotic aortic valve depends not only on the cross-sectional area of its limiting orifice but also on three-dimensional (3D) valve geometry.

**Background** Valve shape can potentially affect the hemodynamic impact of aortic stenosis by altering the ratio of effective to anatomic orifice area (the coefficient of orifice contraction [Cc]). For a given flow rate and anatomic area, a lower Cc increases velocity and pressure gradient. This effect has been recognized in mitral stenosis but assumed to be absent in aortic stenosis (constant Cc of 1 in the Gorlin equation).

**Methods** In order to study this effect with actual valve shapes in patients, 3D echocardiography was used to reconstruct a typical spectrum of stenotic aortic valve geometrics from doming to flat. Three different shapes were reproduced as actual models by stereolithography (computerized laser polymerization) with orifice areas of 0.5, 0.75, and 1.0 cm^{2} (total of nine valves) and studied with physiologic flows. To determine whether valve shape actually influences hemodynamics in the clinical setting, we also related Cc (= continuity/planimeter areas) to stenotic aortic valve shape in 35 patients with high-quality echocardiograms.

**Results** In the patient-derived 3D models, Cc varied prominently with valve shape, and was largest for long, tapered domes that allow more gradual flow convergence compared with more steeply converging flat valves (0.85 to 0.90 vs. 0.71 to 0.76). These variations translated into differences of up to 40% in pressure drop for the same anatomic area and flow rate, with corresponding variations in Gorlin (effective) area relative to anatomic values. In patients, Cc was significantly lower for flat versus doming bicuspid valves (0.73 ± 0.14 vs. 0.94 ± 0.14, p < 0.0001) with 40 ± 5% higher gradients (p < 0.0001).

**Conclusions** Three-dimensional valve shape is an important determinant of pressure loss in patients with aortic stenosis, with smaller effective areas and higher pressure gradients for flatter valves. This effect can translate into clinically important differences between planimeter and effective valve areas (continuity or Gorlin). Therefore, valve shape provides additional information beyond the planimeter orifice area in determining the impact of valvular aortic stenosis on patient hemodynamics.

The hemodynamic impact of valvular stenosis is determined primarily by the effective orifice area (A_{eff}) seen by flow, which differs from anatomic orifice area (A_{anat}). This distinction has been recognized in mitral stenosis but assumed to be absent in aortic stenosis, with little investigation to date. Understanding these differences is important in clinical applications now that anatomic planimetry is more feasible and common in patients with aortic stenosis; this understanding is required to relate anatomic measures to A_{eff} by the continuity or Gorlin equation, which will in turn provide physicians with a consistent classification system for evaluating severity of stenosis (1). Therefore, we explored the relation between A_{anat} and A_{eff} for aortic stenosis, using three-dimensional (3D) echocardiography (2–24) to provide actual valve geometries derived from patients for precise testing in the controlled in vitro environment. The transition from patient data to flow models was made by laser stereolithography, which reproduces 3D shapes as solid objects (25,26). Using these techniques, we addressed the hypothesis that, for a given two-dimensional limiting A_{anat}, overall 3D leaflet geometry importantly modifies the A_{eff} seen by flow and the resulting pressure loss, which determines the work load imposed on the left ventricle. The influence of valve shape on A_{eff} was then confirmed in the clinical setting by comparing A_{eff} and A_{anat} in patients with a spectrum of valve shape.

## Basic principles

The effect of valve shape on flow can be expressed by the coefficient of orifice contraction, which is the ratio of A_{eff} at the vena contracta (the smallest cross-sectional area encountered by flow) to the A_{anat}. Flow, directed to converge toward a narrow orifice, will continue to converge beyond it until interaction with surrounding fluid causes divergence (27–33). A relatively flat valve produces steep flow convergence both toward and beyond the orifice, leading to a smaller A_{eff} and coefficient of contraction (Cc) (Fig. 1). This is important because, for a given flow rate and anatomic area, a smaller effective area would predict a higher velocity and therefore a higher pressure gradient (PG), which is proportional to velocity squared (34,35). In contrast, a tube has no flow convergence toward or beyond the orifice, giving a larger A_{eff} and a contraction coefficient of 1.0 and, therefore, a lower velocity and PG for the same flow rate and anatomic area. A gradually doming valve would lie between these extremes. Also, the Gorlin equation is widely assumed to convert effective to A_{anat} (with the original aim of surgical validation), but uses a constant contraction coefficient of 1.0 for aortic stenosis (36); if this coefficient varies with valve shape (37), so would the relation between Gorlin and anatomic areas. To date, these concepts could not be tested with the actual 3D shapes of valves in patients; 3D echocardiography with stereolithography now permits this.

## Methods

### Patient-derived 3D valve models

A spark gap (audible sound) transducer locating system (4,9,10,13,14) with respiratory and electrocardiographic gating was used in patients to reconstruct three leaflet geometries spanning the range typically seen in patients with significant aortic stenosis (Fig. 2) : a relatively immobile valve with the flattest leaflet geometry, as seen in calcific aortic stenosis; an intermediate, funnel-like valve with less mobile leaflets; and a doming valve with more mobile leaflets, characteristic of bicuspid valves. The aortic valve was scanned with a 3.5 MHz transducer (Hewlett-Packard, Andover, Massachusetts) in parasternal short- and long-axis or rotated apical views to visualize the entire leaflet surface. Two-dimensional images and their spatial locations were automatically combined by computer and digitally encoded (13). The ventricular surfaces of the aortic leaflets were traced and reconstructed from 24 intersecting views at a consistent point in the respiratory and cardiac cycles to reflect maximal systolic leaflet opening. The traces were then fitted to form a continuous 3D leaflet surface using a previously validated polyhedral algorithm (13).

For each of the three valve shapes derived from patients, three exact models were constructed by stereolithography and machined to have A_{anat} of 0.5, 0.75, and 1.0 cm^{2}. Stereolithography (3D Systems, Valencia, California), an industrial process for constructing prototypes from computer designs, uses laser light to induce polymerization of a liquid substrate to form successive solid layers (25,26). The A_{anat} was confirmed by planimetry of direct video images of the orifices. Each of these nine valve models was then studied using an aqueous glycerin solution (33%) with physiologic viscosity and density; three instantaneous steady flow rates of 3 to 18 l/min were used for each orifice area to provide a physiologic range of orifice velocities from 1 to 3 m/s (based on anatomic area) for all valves. Flow rates were provided by a calibrated rotameter pump and confirmed by timed volumetric collection. Effective orifice area was calculated as flow rate divided by continuous wave Doppler orifice velocity (the maximal velocity at the vena contracta). The Cc was calculated as effective divided by A_{anat}. The PG was measured directly using a manometer. Gorlin valve area was calculated as flow rate divided by a constant (44.5) times the square root of the mean PG, based on the originally assumed contraction coefficient of 1.0 (27,28).

#### Clinical studies

We aimed to determine whether the hemodynamic effects of 3D aortic valve shape can actually be demonstrated in the clinical setting, and tested the hypothesis that the shape of a stenotic valve affects its contraction coefficient. This hypothesis can be tested noninvasively using cardiac ultrasound and Doppler to measure the A_{anat} by planimetry in patients with suitable high-quality images and to calculate effective orifice area by the continuity equation.

We addressed this hypothesis retrospectively with transthoracic echocardiography because transesophageal echocardiography studies are not routinely performed to assess stenotic aortic valve area. We reviewed 240 consecutive patients with significant aortic stenosis, defined as continuity area of ≤1 cm^{2}, and selected 35 with especially high-quality echocardiographic studies to permit planimetry of the limiting orifice area. These included 14 patients with doming bicuspid valves and 21 with flat tricuspid aortic valves (mean age = 35 ± 18 years vs. 78 ± 8 years, p < 0.0001; 8 men and 6 women vs. 14 men and 7 women, p = NS). Doming was defined by restricted leaflet tip motion with preserved mobility of the leaflet bodies, making each leaflet concave toward the left ventricle (Fig. 2). In the flat valve shape, the leaflets appeared individually flat, with a more triangular systolic configuration in the long-axis view. The A_{anat} was obtained by planimetry of the inner margins of the limiting orifice in mid-systole in a parasternal short-axis view (average of three measurements). Peak systolic flow rate across the valve was obtained as the product of subvalvular area at the level of the aortic annulus, calculated as (π/4) × diameter squared, multiplied by peak subvalvular velocity by pulsed Doppler. The A_{eff} at peak flow rate was calculated as flow rate divided by peak orifice velocity by continuous-wave Doppler. The contraction coefficient was then calculated as the ratio of effective to anatomic areas.

#### Data and statistical analysis

##### Patient-derived 3D models

Two-way analysis of variance was used to test for significant differences among Cc as a function of both 3D shape and orifice area.

To test for the relation between PG and valve shape for valves of different anatomic areas at different flow rates, each PG was first normalized to that for a doming valve of the same A_{anat} and flow rate (Q); these normalized values were then evaluated as a function of valve shape, expressed by the squared ratio of (Cc for the doming valve/Cc for the valve being tested). Because of the reciprocal relation between PG and Cc^{2}: PG = 4v^{2} by Bernoulli (v = velocity) = 4(Q/A_{anat} × Cc)^{2} by continuity. This relation between normalized PG and Cc^{2} ratios was tested by linear regression analysis. Also, because area (Gorlin)/area (anat) = Cc (actual)/Cc (assumed), and Cc was assumed to be constant and equal to 1 by Gorlin, we tested the relation of area (Gorlin)/area (anat) to Cc (actual) by linear regression analysis.

##### Clinical studies

Anatomic area, effective area, and contraction coefficients in the patient groups with doming and flat valves were compared by *t* test. To determine whether differences in effective area translate into differences in PG, the peak PG was normalized for its determinants other than valve shape (flow rate and anatomic area) and compared between the two groups: PG = 4v^{2} = 4Q^{2}/area (eff)^{2}, so PG (normalized for flow rate) = PG/Q^{2}; PG = 4v^{2} = 4Q^{2}/Cc^{2} area (anat)^{2}, so PG (normalized for flow rate and anatomic area) = (PG/Q^{2}) × area (anat)^{2}.

## Results

### Patient-derived 3D models

#### Valve shapes

Figures 3A and 3B show the flattest reconstructed valve viewed from the side with the surface fitted to the leaflets in yellow, and then the corresponding stereolithography model. Figures 3C and 3D show the intermediate doming valve as traces and bulging surface, with the corresponding model shape. Figures 3E and 3F show the valve with the longest dome (compare Fig. 2 bottom) and its associated model, with the elliptical bicuspid valve orifice at the apex of the dome.

##### Cc

Figure 4 shows the coefficients of contraction for the valve shapes and orifice sizes studied. Analysis of variance showed significant differences in the coefficient among the different valve shapes and anatomic areas (p < 0.0001). The Cc was smallest for the flattest valve shape, because the steep flow convergence proximal to the limiting orifice causes flow to continue to converge steeply beyond it, most like a flat plate. Coefficients were largest for the tapered shapes, because the more gradual flow convergence within the funnel produces less contraction beyond the orifice, closer to the limit of 1.0 for a straight tube. The longest dome (Fig. 3E) favored the most gradual flow convergence over the longest path, and therefore the highest contraction coefficients. Also, for each shape, the coefficient increased as the orifice size increased (the size of the inlet being constant), making the entire structure more like a tube with less steep flow convergence. The values in Figure 4 represent the mean for each valve over the three flow rates studied, because there was minimal variation of coefficient with flow rate over the range of flows studied; linear regression showed no significant relation between coefficient and flow rate for each valve studied.

##### PGs and Gorlin areas

Direct pressure measurements confirmed that these variations in contraction coefficient translated into corresponding changes in PG. For each A_{anat} and flow rate, the PG was highest for the flattest valve, which had the smallest effective area and, therefore, the highest velocity and PG. The PGs for the flattest valves were 40 ± 5% higher relative to the most domed valves for the same anatomic area and flow rate; gradients were 14 ± 2.6% higher for the intermediate versus most domed valves (all p < 0.0001). This ratio of PG to the PG of a corresponding dome correlated well with the squared ratio of the contraction coefficients, expressing the differences in valve shape (y = 0.94x + 0.69, r = 0.99, SEE = 0.021). The calculated Gorlin valve areas derived from these pressures also had a variable relation to the actual anatomic areas, spanning a 20% range, because the original Gorlin equation assumes a constant contraction coefficient. The ratio of Gorlin to anatomic area correlated with the Cc, because the basic Gorlin equation is designed to provide an A_{eff} (y = 1.05 x + 0.06, r = 0.89, SEE = 0.04).

##### Clinical studies

In the entire group of 35 patients, there was no significant difference in A_{anat} between those with flat and domed valves (0.96 ± 0.24 cm^{2} vs. 0.95 ± 0.25 cm^{2}); A_{eff}, however, was significantly smaller for the flat valves than the doming ones, averaging 0.69 ± 0.17 cm^{2} versus 0.86 ± 0.13 cm^{2} (p < 0.001). Therefore, the contraction coefficients were significantly smaller in patients with flat versus doming valves, with an average of 0.73 ± 0.14 versus 0.94 ± 0.14 (p < 0.001). Of note, the contraction coefficients for the flat and domed stereolithography models lay within the range of values for the corresponding clinical study groups: 0.71 to 0.76 for the flattest models and 0.85 to 0.90 for the most doming. The smaller effective areas in patients with flat valves translated into higher peak PGs, expressed as gradients normalized for flow rate (10.0 ± 4.7 vs. 5.8 ± 2.1 mm Hg per flow rate [in cm^{3}/s] squared for the flat vs. the doming valves, p = 0.004), and normalized for both flow rate and A_{anat} to express the effect of shape alone (8.1 ± 3.9 vs. 4.7 ± 1.7 [mm Hg/(cm^{3}/s)^{2}] × (cm^{2})^{2} for flat vs. doming valves, p = 0.005).

## Discussion

The results of this study demonstrate that the impact of a stenotic aortic valve on pressures and flows in the heart depends not only on the cross-sectional area of the limiting orifice but also on the 3D geometry of the valve leaflets proximal to the orifice.

This geometry directs the pattern of flow convergence toward and beyond the orifice and, therefore, determines the relation between the A_{anat} and the A_{eff}, which fundamentally influences flow. Patients with flat valves and steeper flow convergence have smaller A_{eff} than those with more gradually tapering domed valves for the same anatomic area and flow rate: corresponding pressure losses are increased by 40% or more, generating increased ventricular work loads. There is corresponding variation of the calculated Gorlin area relative to the anatomic area, reflecting the basic derivation of the Gorlin as an A_{eff}.

### Aortic versus mitral stenosis

Although this effect of valve shape on contraction coefficient has been recognized in mitral stenosis (25), it has long been assumed to be unimportant for aortic stenosis, with a constant contraction coefficient of 1 used in the Gorlin equation. This study demonstrates these effects are, in fact, important, especially for the flattest aortic valves, for which the contraction coefficient falls into the range seen with doming mitral valves.

These effects of valve shape on flow have been established in vitro, where precise and direct measurements of anatomic orifice areas and flow rates can be made using actual geometries from patients derived by 3D echo and reproduced by stereolithography. These findings have been confirmed in correlative clinical studies.

#### Clinical implications: planimeter versus effective areas

In patients, this understanding should be of value in comparing planimeter anatomic versus effective valve areas by the continuity or Gorlin equations (1,27,38). These measures are evaluated relative to cutoff values commonly used to demarcate significant and critical stenosis in routine clinical practice. Reconciling differences between measures of anatomic and effective area could be important, therefore, in providing a unified and consistent evaluation relevant to the clinical presentation. Bernard et al. (1), for example, found clinically important overestimation by transesophageal echocardiography planimetry of valve areas compared with Gorlin and Doppler effective areas in patients (1). This overestimation was greater for smaller valves (<0.9 cm^{2}), which in clinical experience tend to be flatter. The authors state that these discrepancies are “unacceptable in clinical practice, as the need for surgery is decided on a very narrow range of aortic valve area values” (1). Our patient studies similarly show important differences in effective area (0.69 vs. 0.86 cm^{2}) for comparable anatomic areas but different shapes. This may be particularly important with increasing prevalence of calcific (flat) stenosis with age, making it prudent not to exclude severe stenosis by planimetry in the face of small Gorlin or continuity areas; cutoffs for severity classification may also need to be reconsidered in that context. In addition, these observations could potentially and partially explain differences in clinical presentation in patients with the same anatomic areas but different valve shapes, and are also important in view of the prognostic value of aortic valve area (39).

#### Implications for fluid flow

The need for such studies has been pointed out by Gorlin himself, who asked whether “flow lines approach the valve smoothly as through a funnel, or abruptly as toward a window in a wall,” suggesting that such variations might “alter the orifice constant, perhaps drastically ” (37). (Such variations, for example, may have been reflected in the change from the original mitral stenotic contraction coefficient of 0.7 to the subsequent 0.85 [27,28] possibly because of differences among patient groups). Flow dependence of the contraction coefficient (40) was not apparent over the flow rates studied, consistent with the findings that flow dependence is limited to low flows (41,42) or negligible (43).

Another way of stating the results is that for a given anatomic orifice area and pressure gradient, doming valves permit a higher cardiac output than flat ones. This is analogous to the observations of Grayburn et al. (44) that slitlike aortic insufficiency orifices typical of bicuspid and degenerative disease permit more regurgitant flow (larger A_{eff}) than central rheumatic or circular orifices for the same anatomic orifice and PG.

The adverse effect of flat shape on pressure losses could also in principle be magnified because the smaller vena contracta, in relation to the aortic root, can increase turbulent losses and decrease pressure recovery (45,46).

This study also demonstrates the value of stereolithography based upon 3D echocardiographic imaging (25,26) as a technique for providing hemodynamic insights of importance for clinical evaluation.

#### Study limitations and future directions

The 3D technique has been extensively validated in vitro and in vivo (9,13,17,18), and the resolution of spark gap localization is <1 mm (10,13); variability has been reduced in this study by respiratory registration. Although the aortic valve deforms as it opens, the fixed models used capture funnel shape at the time of maximal leaflet distension for all valves, roughly corresponding to the time of peak velocity and gradient, to isolate the effect of valve shape at that moment. Future studies could examine deformable models and any second-degree effects of pulsatile flow on the evolution of the contraction coefficient that might modulate the main geometric effects described, but not alter them, given the absence of significant effects of flow rate on the contraction coefficient over the range of flows studied. It is interesting to speculate that some of the improvement in valve area seen with dobutamine in patients with low-flow states (47,48) may reflect a further benefit of improved valve shape: increased leaflet tip excursion could change shape from flat toward funnel, increasing contraction coefficient and effective area. As in the case of mitral valve prolapse (9), although investigation requires full 3D studies, the resulting insights ultimately have the potential to be applied on the basis of correlated two-dimensional images. Finally, it was not the intent of this study to examine all possible valve shapes, but rather selected ones covering the spectrum seen in patients.

In conclusion, therefore, the coefficient of orifice contraction for stenotic aortic valves is affected by the variations in leaflet geometry seen in patients and reconstructed by 3D echocardiography, translating into important differences in the pressure drop induced by these valves. Three-dimensional valve shape, therefore, provides additional information beyond the planimeter orifice area in determining the impact of aortic stenosis on patient hemodynamics. This effect can translate into clinically important differences between planimeter and effective valve areas (continuity or Gorlin). Stereolithography, using data generated by 3D echocardiography, can allow us to address such uniquely 3D questions using actual information from patients to provide insight into the relation among structure, pressure, and flows in the heart, relevant to the clinical setting.

## Footnotes

☆ Supported in part by grants R01 HL53702, HL38176, and K24 HL64697 of the National Institutes of Health, Bethesda, Maryland (Dr. Levine). Dr. Gilon was a Research Fellow from Hadassah University Hospital and Medical School, Jerusalem, and was supported in part by a grant from the American Physicians Fellowship for Medicine in Israel. On the basis of this study, Dr. Gilon received a Young Investigator Award of the American College of Cardiology. Dr. Levine was an Established Investigator of the American Heart Association, Dallas, Texas. Dr. Gilon is presently affiliated with the Heart Institute, Hadassah University Hospital, Faculty of Medicine, Hebrew University, in Jerusalem, Israel.

- Abbreviations
- A
_{anat} - anatomic orifice area
- A
_{eff} - effective orifice area
- Cc
- coefficient of contraction
- PG
- pressure gradient
- 3D
- three-dimensional
- Q
- flow rate

- Received October 29, 2001.
- Revision received June 6, 2002.
- Accepted July 2, 2002.

- American College of Cardiology Foundation

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