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Circulation. 1996;94:452-459

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(Circulation. 1996;94:452-459.)
© 1996 American Heart Association, Inc.

Articles

Insights From Three-dimensional Echocardiographic Laser Stereolithography

Effect of Leaflet Funnel Geometry on the Coefficient of Orifice Contraction, Pressure Loss, and the Gorlin Formula in Mitral Stenosis

Dan Gilon, MD; Edward G. Cape, PhD; Mark D. Handschumacher, BS; Leng Jiang, MD; Charles Sears; Joan Solheim; Eleanor Morris, RDCS; John T. Strobel, BS; Stockton M. Miller-Jones, PhD; Arthur E. Weyman, MD; Robert A. Levine, MD

the Cardiac Ultrasound Laboratory, Massachusetts General Hospital, Harvard Medical School, Boston, Mass (D.G., M.D.H., L.J., E.M., A.E.W., R.A.L.); Children's Hospital, School of Medicine and Engineering, University of Pittsburgh, Pa (E.G.C., J.T.S.); Hewlett-Packard Co, Andover, Mass (S.M.M.-J.); and Santin Engineering, Peabody, Mass (C.S., J.S.).


*    Abstract
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 *Abstract
down arrowIntroduction
 down arrowMethods
 down arrowResults
 down arrowDiscussion
 down arrowReferences
 
Background Three-dimensional echocardiography can allow us to address uniquely three-dimensional scientific questions, for example, the hypothesis that the impact of a stenotic valve depends not only on its limiting orifice area but also on its three-dimensional geometry proximal to the orifice. This can affect the coefficient of orifice contraction (Cc=effective/anatomic area), which is important because for a given flow rate and anatomic area, a lower Cc gives a higher velocity and pressure gradient, and Cc, routinely assumed constant in the Gorlin equation, may vary with valve shape (60% for a flat plate, 100% for a tube). To date, it has not been possible to study this with actual valve shapes in patients.

Methods and Results Three-dimensional echocardiography reconstructed valve geometries typical of the spectrum in patients with mitral stenosis: mobile doming, intermediate conical, and relatively flat immobile valves. Each geometry was constructed with orifice areas of 0.5, 1.0, and 1.5 cm2 by stereolithography (computerized laser polymerization) (total, nine valves) and studied at physiological flow rates. Cc varied prominently with shape and was larger for the longer, tapered dome (more gradual flow convergence proximal and distal to the limiting orifice): for an anatomic orifice of 1.5 cm2, Cc increased from 0.73 (flat) to 0.87 (dome), and for an area of 0.5 cm2, from 0.62 to 0.75. For each shape, Cc increased with increasing orifice size relative to the proximal funnel (more tubelike). These variations translated into important differences of up to 40% in pressure gradient for the same anatomic area and flow rate (greatest for the flattest valves), with a corresponding variation in calculated Gorlin area (an effective area) relative to anatomic values.

Conclusions The coefficient of contraction and the related net pressure loss are importantly affected by the variations in leaflet geometry seen in patients with mitral stenosis. Three-dimensional echocardiography and stereolithography, with the use of actual information from patients, can address such uniquely three-dimensional questions to provide insight into the relations between cardiac structure, pressure, and flows.


Key Words: echocardiography • mitral valve • stenosis • pressure


*    Introduction
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 up arrowAbstract
 *Introduction
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 down arrowResults
 down arrowDiscussion
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Recent advances in three-dimensional (3D) echocardiography1 2 3 4 5 6 7 8 9 10 11 12 can potentially allow us to address uniquely 3D questions of scientific interest. One such question addresses the hypothesis that the impact of a stenotic 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 stenotic valve proximal to the orifice. In particular, we proposed that for a given anatomic orifice area and flow rate (cardiac output), 3D leaflet geometry can importantly affect the maximal velocity and pressure gradient induced by the stenosis. This effect can be expressed in terms of the coefficient of orifice contraction of the valve, defined as the effective orifice area at the vena contracta (the smallest cross-sectional area encountered by flow) divided by the anatomic area; it reflects the principle that flow, directed to converge toward a narrow orifice, will continue to converge beyond the orifice until its con-vergence is blunted by interaction with surrounding fluid.13 14 15 16 17 18 19 The steeper the proximal convergence (for example, from the proximal chamber toward an orifice in a flat plate), the steeper the distal contraction (Cc{cong}0.6); in contrast, for a prolonged tube, the collimated flow has no distal contraction (Cc=1.0) (Fig 1Down). This effect is important because, for a given anatomic area and flow rate, a lower coefficient gives a smaller effective orifice area and therefore a higher maximal velocity (=flow rate/area) and, by Bernoulli's equation, a higher pressure gradient (proportional to velocity squared20 21 ). A greater amount of the initial pressure head driving fluid forward is therefore lost. Also, this coefficient is assumed constant in the Gorlin equation as routinely applied,22 which had the original aim of correcting effective area back to anatomic for validation13 ; if this coefficient were to vary with 3D funnel shape,23 there would be a variable relation between calculated and anatomic areas. Previous studies, for example, have shown that Cc varies with the two-dimensional (2D) shapes of orifices seen in aortic insufficiency24 and with orifice type in engineering models of stenosis.17 18 19 25



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  		Figure 1. Flow convergence beyond the orifice: dependence on the convergence pattern established proximal to the orifice for idealized geometries (see text). A, Flat shape; B, dome; and C, tube. Ao indicates anatomic orifice area; Ae, effective orifice area; and Cc, coefficient of orifice contraction.

To date, however, it has not been possible to study these concepts with the actual 3D shapes of valves in patients. The purpose of this study was therefore to study the effect of 3D shape on the coefficient of contraction, pressure losses, and Gorlin areas in patients with mitral stenosis by applying 3D echocardiographic reconstruction to obtain valve geometries and stereolithography, a process of computerized laser-induced polymerization, to reproduce these shapes as actual models.


*    Methods
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 up arrowAbstract
 up arrowIntroduction
 *Methods
down arrowResults
 down arrowDiscussion
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Stage 1
A spark gap (audible sound) transducer locating system3 8 9 10 26 27 28 29 30 31 32 33 34 with respiratory and ECG gating was used in patients to reconstruct three leaflet geometries selected to span the range typically seen in patients with significant mitral stenosis and valve area <=1.5 cm2: a mobile, doming valve, as appreciated from leaflet motion in the parasternal long-axis view (Fig 2Down); an intermediate, more conical and funnel-like valve with less mobile leaflets; and an immobile valve with the flattest leaflet geometry. (Please see reconstruction and figures below for illustrations of valve geometries.) These shapes reflect the spectrum of rheumatic deformity, which begins with thickening of the distal leaflet tips only and commissural fusion, allowing for mobility of the leaflet bodies and doming, and progresses to involve more of the leaflet surface to produce immobile, flattened leaflets.35 36 Patients were scanned with the 3.5-MHz transducer of a Hewlett-Packard Sonos 1500 scanner with software allowing simultaneous display of respiratory traces from a nasal thermistor and an ECG trace along with 2D images. The mitral valve was scanned in the series of parasternal short- and long-axis views or rotated apical views that optimally visualized the leaflet surface circumferentially. As previously reported, 2D images and their spatial location were automatically combined by a 386-series computer and recorded on videotape.26 Subsequently, 24 video frames were digitally encoded and traced to reconstruct the mitral leaflets at a consistent point in respiration (quiet end expiration) and in the cardiac cycle (maximal leaflet distention in early diastole). Traces of the mitral valve and left atrial surface were combined and fitted to form a continuous 3D surface using a polyhedral algorithm,26 previously validated to provide accurate volume measures from the best weighted fit to the actual traces along 800 latitude and longitude grid points.10 26 27 29 30 31 32 33 34 The portion of the surface underlying the mitral valve traces was then identified by the computer for subsequent analysis (Fig 3Down, A through C, and Fig 4Down, A and C).



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  		Figure 2. Diastolic parasternal long-axis view of a doming mitral valve with atrial borders traced in color. LA indicates left atrium; LV, left ventricle.






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  		Figure 3. Reconstructed stenotic mitral valve with a relatively flat overall leaflet shape. A, View of mitral valve and left atrium from the side, with the orifice (white) to the left; the surface fitted to the leaflets is shown in yellow, and the left atrium is represented by the curved purple traces. B, Angled view from the left ventricular perspective of the same valve (original traces in orange and green; fitted surface in yellow). C, Valve surface alone viewed from the left ventricular perspective. D, Corresponding stereolithography model, with the anterior leaflet region above, the posterior region below, and an elliptical orifice.






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  		Figure 4. A, Reconstructed surface of an intermediate funnel-shaped valve viewed from the side (anterior leaflet above). B, Corresponding stereolithography model. C, Reconstructed surface of a doming mitral valve (compare Fig 2Up): angled view from the left ventricular perspective, with the large doming anterior leaflet coming toward the viewer. D, Corresponding stereolithography model (commissural fusion represented by lack of orifice opening at the sides).

Stage 2
For each of the three valve geometries, three exact polymeric models were constructed by laser stereolithography and machined to have elliptical anatomic orifice areas of 0.5, 1.0, and 1.5 cm2, for a total of nine valves (Figs 3D, 4B, and 4DUpUpUp). Stereolithography (3D Systems), an industrial process for constructing prototypes from computer designs, uses laser light to induce polymerization of a liquid substrate to form successive solid layers. Anatomic orifice area was confirmed by planimetry of direct video images of the orifices.

Stage 3
Each of these nine valves was then studied in steady flow state using an aqueous glycerin solution (33%) with physiological viscosity and density; three instantaneous flow rates of 2 to 18 L/min were used for each anatomic orifice area to provide a physiological range of expected orifice velocities from 0.7 to 2 m/s (based on anatomic area) for all valves (that is, 2, 4, and 6 L/min for the 0.5-cm2 orifices, increasing by a factor of 2 to 3 for the other orifices that were 2 to 3 times larger).

Measurements and Definitions
Desired flow rates were provided by a rotameter pump, calibrated 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 coefficient of contraction was calculated as effective area divided by anatomic orifice area. Pressure gradient was measured directly using a manometer interfaced to the flow model. Gorlin valve area was calculated as flow rate divided by a constant (44.5x the assumed contraction coefficient) times the square root of the mean pressure gradient, based on the original contraction coefficient of 0.713 or the later modified value of 0.85.14

Data and Statistical Analysis
Two-way ANOVA was used to test for significant differences among coefficients of contraction as a function of both 3D shape and orifice area. To test for the relation between pressure gradient and valve shape for valves of different anatomic areas and exposed to different flow rates, each pressure gradient was first normalized to that for a doming valve of the same anatomic orifice area (Aanat.) and flow rate (Q); these normalized values were then plotted against the squared ratio of (Cc for the doming valve/Cc for the valve being tested), a function of valve shape, because of the reciprocal relation in principle between pressure gradient (PG) and Cc2: PG=4v2 by Bernoulli (v=velocity)=4(Q/Aanat.xCc)2 by continuity.

This relation between pressure gradient and Cc2 ratios was then tested by linear regression analysis. Also, because area (Gorlin)/area (anat.)=Cc (actual)/Cc (assumed), we plotted the calculated Gorlin areas normalized to the anatomic values against actual Cc (since the assumed Cc is a constant) and tested their relation by linear regression analysis.


*    Results
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 up arrowAbstract
 up arrowIntroduction
 up arrowMethods
 *Results
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 down arrowReferences
 
Coefficient of Contraction
Fig 5Down shows the coefficients of contraction for the three valve shapes studied, with the orifice sizes along the left-hand column. ANOVA showed significant differences in the coefficient among both the different valve shapes and the different anatomic areas (P<.0001). The coefficient of contraction was smallest for the flattest valve shape, for which the steep flow convergence proximal to the limiting orifice causes flow to continue to converge beyond it, most like a flat plate. Coefficients were largest for the tapered domes, for which the more gradual flow convergence within the funnel produces less contraction beyond the orifice, closer to the limit of 1 for a straight tube. The intermediate, more funnel-like valve shape had intermediate 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 than a restrictive orifice in a flat plate. The values in Fig 5Down represent the mean values for each valve over the three flow rates studied, since there was minimal variation of coefficient with flow rate over the range of flows studied: when the coefficient for each flow rate was normalized to the mean for each of the nine valves and plotted versus flow rate, linear regression showed no significant relation: y=0.0002x+0.98, r=.28, SEE=0.05.



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  		Figure 5. Coefficient of orifice contraction (indicated as Cc).

Pressure Gradients
Direct pressure measurements confirmed that these variations in contraction coefficient translated into corresponding changes in pressure gradient (Fig 6Down, A through C). For example, the back row of bars in Fig 6ADown shows that, for a constant anatomic orifice area of 1.5 cm2 and a constant flow rate, the pressure gradient was highest for the flattest valve, which had the smallest vena contracta and therefore the highest velocity. The same was observed for the 1-cm2 and 0.5-cm2 areas (Fig 6Down, B and C). Comparison of the pressure gradient for any valve with that of a dome (Fig 6DDown) showed that for the same anatomic area and flow rates, variation in 3D valve shape could produce varying pressure gradients that were up to 40% higher for the flattest valves. This ratio of pressure gradient to the pressure gradient of a corresponding dome related to the squared ratio of the contraction coefficients for the different valve shapes (y=0.89x+0.11, r=.98, SEE=0.037).



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  		Figure 6. Pressure gradient (A, 1.5 cm2; B, 1.0 cm2; C, 0.5 cm2) versus flow rate, demonstrating effect of three-dimensional funnel shape (left to right) for each anatomic orifice area: highest gradient for the flattest valve (greatest distal convergence, therefore, highest velocity). D, Pressure gradient (PG), normalized to that for a doming valve at the same anatomic orifice area and flow rate, versus the reciprocal ratio of contraction coefficients (Cc) squared (dome/valve studied).

Gorlin Areas
The calculated Gorlin areas (Fig 7Down) also had a variable relation to the actual anatomic areas, spanning a 20% range, using either the original Gorlin equation or the subsequent modification, both of which incorporate a constant coefficient. The Gorlin to anatomic area ratio correlated with the coefficient of contraction, as it would for an effective orifice area (for the original equation, y=0.79x+0.45, r=.80, SEE=0.05; for the modified equation, y=0.70x+0.40, r=.80, SEE=0.04).



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  		Figure 7. Relation between Gorlin area (normalized to anatomic) and contraction coefficient (Cc), using the original Gorlin constant13 (upper line) or the subsequent modification14 (lower line).


*    Discussion
up arrowTop
 up arrowAbstract
 up arrowIntroduction
 up arrowMethods
 up arrowResults
 *Discussion
down arrowReferences
 
The results of this study demonstrate that the 3D geometry of the leaflet funnel proximal to the limiting orifice of stenotic mitral valves importantly affects the contraction coefficient and therefore the pressure gradient across the valve for any given anatomic orifice area and flow rate. The need for such studies has been pointed out by Gorlin,23 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."23 Such variations also may be reflected in the change from the mitral contraction coefficient of 0.7 (Reference 13) to the subsequent 0.85 (Reference 14), possibly reflecting differences among patient groups. The results of this study also demonstrate one potential mechanism by which some patients with an anatomic mitral valve area of 1.0 can be asymptomatic, whereas others may be dyspneic with the same valve area37 and stroke volume but different left atrial pressures.

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 al24 that slitlike aortic insufficiency orifices typical of bicuspid and degenerative disease permit more regurgitant flow (larger effective orifice areas) than central rheumatic or circular orifices for the same anatomic orifice and pressure gradient. The findings in our study, in the broader sense, are also consistent with the preliminary report of Tsuji et al38 that a converging flow field is affected by the relationship between the size of the limiting anatomic orifice and that of a larger portion of the valvular nozzle.

The results of the current study provide greater insight into the fundamental strength of the Gorlin equation, which provides a measure of effective orifice area that tracks directly with the contraction coefficient (Fig 7Up). The Gorlin equation is basically the continuity equation, substituting the square root of the pressure gradient for velocity. Therefore, although the relation between the Gorlin and anatomic areas may vary with valve shape, nevertheless, regardless of the constant used, the Gorlin calculations will vary directly with the effective orifice area, which is hemodynamically and physiologically most meaningful.19 39 40 41 The flow dependence of the contraction coefficient and therefore the Gorlin constant42 was not apparent over the range of flow rates studied, consistent with the observations of Voelker et al39 and Segal et al25 that flow-related effects are important mainly at low flows, as well as the results of Grayburn et al,24 Flachskampf et al19 (no variation), and Daboin et al43 (mild effect with scatter).

Three-dimensional Echocardiography
Because such studies require 3D objects, they demonstrate the value of a technique such as 3D echocardiography44 45 46 47 48 that can provide actual valve shapes from beating patient hearts and permit the use of stereolithography to convert these shapes into models in order to address the scientific questions under controlled conditions and with direct measurements of critical variables.

Study Limitations
The 3D technique has been extensively validated in vitro and in vivo,8 26 30 31 32 33 34 with the resolution of spark gap localization checked to be <1 mm9 26 ; variability has been reduced in this study by respiratory registration. Although the mitral valve deforms as it opens, the fixed models used are designed to capture funnel shape at a comparable time in the cycle for all valves, particularly at the time of maximal leaflet distention, roughly corresponding to that of peak velocity and gradient, to isolate the effect of valve shape at that moment. Future studies could examine deformable models and any effects that pulsatile flow might have on the evolution of the contraction coefficient. Nevertheless, such second-order effects will not eliminate or reduce the primary geometric effects described, particularly without significant effects of flow rate on the contraction coefficient in the range of flows studied. 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. Fig 8Down displays two shape indexes plotted for 87 consecutive patients with valve areas <=1.5 cm2 drawn from our echocardiographic database: a doming index (curved dome versus flat cone, similar to an index of Reid et al36 ) and a tapering index, representing rapid tapering from inlet to orifice by higher values. The valves studied (squares) and models built cover major portions of this clinical spectrum. Of note is that as in the case of mitral valve prolapse,8 although investigation requires full 3D studies, the resulting insights ultimately have the potential to be applied on the basis of correlated 2D images. Future studies also could use the same technique to explore differences in aortic stenosis between flat degenerative versus congenitally doming bicuspid valves.49



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  		Figure 8. Top, Spectrum of leaflet funnel shape indexes in 87 consecutive patients with significant mitral stenosis studied by two-dimensional echocardiography, showing location of reconstructed valves (black squares). Bottom, Shape indexes: D1, inlet diameter; D2, outlet diameter; L, axial length of funnel; Lc, curved length of anterior leaflet; and Lh, chord (hypotenuse) length.

Conclusions
The coefficient of orifice contraction for stenotic mitral valves is importantly affected by the variations in leaflet geometry seen in patients and reconstructed by 3D echocardiography. These variations translate into important differences of up to 40% in the pressure drop induced by these valves; for any given anatomic area and flow rate, the pressure loss is greatest for the flattest valves and least for the more gradually tapering domes. There is a corresponding variation in calculated Gorlin area relative to anatomic area, reflecting its fundamental derivation from the continuity equation as an effective orifice area. Stereolithography, with data generated by 3D echocardiography, can therefore allow us to address such uniquely 3D questions using actual information from patients to provide insight into the relation between structure, pressure, and flows in the heart.


*    Acknowledgments
 
This study was supported in part by grant RO1-HL-53702 of the National Institutes of Health, Bethesda, Md, and by a donation from Bernard L. Adams, Holyoke, Mass. Dr Gilon is a research fellow from Hadassah University Hospital and Medical School, Jerusalem, Israel, and was supported in part by a grant from the American Physicians Fellowship for Medicine in Israel. Dr Levine is an Established Investigator of the American Heart Association, Dallas, Tex, with funds supplied in part by its Massachusetts Affiliate, Needham, Mass.


*    Footnotes
 
Reprint requests to Robert A. Levine, MD, Cardiac Ultrasound Laboratory, VBK508, Massachusetts General Hospital, 32 Fruit St, Boston, MA 02114.

Received September 25, 1995; revision received January 17, 1996; accepted January 22, 1996.


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 up arrowIntroduction
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 up arrowResults
 up arrowDiscussion
 *References
 

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