ofthese i-low (4) and area of the valve. dynamics predict that the shape

Grant 13-532-867 from the American Heart Asso iate, Inc.. Nredham. Massachusetts. Manuscript received July 26, 1989; revised manuscript received October 25. 1989. accepted November 22, 19&Y. Address for reorints: James D. Thon;as, IX?. Cardiac Ultrasound Eaboratory. Massachusetts General Hospita!, Zero Emerson Place. 2nd floor. 2F. Boston, Massachusetts 02114. 01990

by the American College of Cardiology

time in ~i~l~seco~ds)

been shown to contain

ve area, 3) aring gradu restrictive

factors.

entric ng an

a

41 I?/

orifices). 073s1097/w53.50

FLACHSKAMPFETAL. FACTORStNFLUENClNGEFFECTlVEVALVEAREA

1174

ve area. The by two approximated flowthrough a restrictive orifice can be equations: the continuifyequation

where q = flow rate, AeR= effective orifice area and v = flow velocity; and the simpl$ed Bernoulli equation Ap = ‘/7pv2.

P!

where Ap = the instantaneous pressure gradient across the orifice, p = the fluid density of blood and v = flow velocity. The reason for introducing opposed PO the geometric

an effective orifice area as “trae” area is that the observed

real flow is always lower than the ow calculated by the Bernoulli equation and the “true” area. This difference is caused by the phenomenon of flow contraction distal to a narrowing, leading to the so-called vena contracta, and a viscous loss of velocity between the orifice and the vena contracta. The factor relating the true area to the effective area is called the discharge coefficient(cu):

Figure1. Schematicdrawingof the flowmodel.See text fordetails. pressuregradient:&A = left atria! APO= initialatrioventricular chamber;LV = left ventricularchamber; orifice. The Gorlin constant

therefore

carres~o~~s

to c#L4.

The Gorlin formula can easily be derived from these two fundamental relations by rearrangingequations 1 and 2 into expressions for v and equating them as v = qlAeRandv = (2Ap/~)~.

In this equation, Bp is in metric units (dyne/cm*), whereas clinicallyit usually is expressed in mm g. Thus, substituting this conversion (1 mm Hg = I.333 dynelcm2) and p = 1.05 g/m!, the expression becomes

e

the empiric factor ient CD,as done in

the correctly defined

v = (2 x 1,333Ap/1.05)~ = (2,539Ap)”= 50.4fip,

and for effective area we obtain A,, = ql(50.4fip). Thisformula differsfrom the Gorlin formula only in the constant 50.4. However, this formula calculates

numeric

effectiveorifice area. The differencein relation to the Gorlin formula, which calculates true area, therefore results from the implicit inclusion of the discharge coetkient into the Gorlin constant. True area can be calculated as A muc =

ql(cbs0.4~p).

I31

To measure the effective area (and implicitly the discharge coefficient!,we used a previously described in vitro mode! in which gravity mimics the dynamic forces involved in transmitra! flow (8.9). This mode! consists of a Plexiglas box divided into an “atrium” and “ventricle” by a removable vertical Plexiglas septum (Fig. 1). A mouat in the “septum” permits installation of a variety of orifices. Pressure is m n each side of the orifice by t Statham transducers. Additionally, sured w ectromagnetic flow probe attached to a Statham SP2202flow meter!. After preamplification,the data are digitized at 20to 100Hz by a converting board (QT 2801A,Data Translation) and fed into an 80286-

is ca

*Note that “eccentricity” in mathematics and en~iaee~ng has a different meaning. Consider an ellipse with long radius a and short radios b: for the sake o~s~~p~i~ity, eccentricity (ECC) here is the ratio of long to short radius: KC = a/b. so that ECC = 1 describes a circ!e and KC>1 a stretched slit-like ellipse. Mathematically. eccentricity (e) denotes the quotient E = x&z=--&, which takes on values between 0 (a variables can be converted by ECC =

JACC Vol. 15. No. 5 pd 1 :Bm=m

FLACHSKAMPF ET AL. FACTORS INFLUENCINGEFFECTIVE VALVE AREA

1176

Flaw km/ssd

200

150 100 50 0

of the flow model with a circular 1 cm’ valve dient and flow tracings are displayed with a line) for the flow decay and the parabotic fit t for details) for pressure decay.

Qata

from

at least

Jive

runs fir

each

o$ce

were

aver-

analysis and calculations were execute software written in the the microcomputer using cust cmillan SoftwareComAsyst programmingenvironm

aged.

The

iple

data

stepwise

hear

regression

analysis

was

a

per-

with the discharge coefficient as a dependent variable. Size and eccentricity were tested separately and co relation to bined as ems of the the disch circular orifices with and without the “nozzle” were compared by a paired r test.

sample run of the in vitro model is

ing exponential decay would be expected if the discharge coefficient varied with flow rate, also resulted in a v od but lower correlation coefficient(3 = 0.9594, p < 1). The goodness of the parabolic fit to the pressure decay curve was s greater than that of a ponentiall for all tested: 8 = 0.9997t 0 (mean f ) as compared with ? = 0.985 f 0.009 for the exponential fit (p c differencebetween the F highly significant(p < 0.

because of the added degree of freedom in the general expression. Thus, the constrained parabolacorrespondingto equation 4 is the best descriptor of pressure decay. This indicatesthat over the range of flowrates encountered in our experiments the discharge coefficientwas indeed stable.

assumption of a linear well; the additional d

dratic form only minimally improved the correlation, but at

trends were noticed: 1) the discharge coefficient diminishes with decreasing orifice size; 2) increasing eccentricity is

Eccentricity

21

3:3

51

0.71 .72 0.74 0.76 0.77 0.77

.70 .71 0.73 0.74 0.75 0.74

0.71 0.73 0.73 s

Area G.xn*~ .3 0.5 1. 1. 2. 2.

0.81 0.83

( = abm

rea IcmP

1178

JAW Apri

FLACHSKAMPFETAL. FACTORSINFLUENCINGEFFECTIVEVALVEAREA

in the centerof the flow. This effect is described by the coefficient of velocity, which is the ratio of mean to maximalvelocity. Unfortunately, the coefficientsofcontmtjon &. velocity are difficult to measure separately and so are usually combined into the discharge coefficient, which comparesthe real flow through an orifice to the predicted flow for a given pressure gradient if no contraction or viscous 10~soccucs(12). A gradually tapering inlet acts as a nozzle (conversely. a gradually expanding outlet acts as a diffuser)and minimizesthe decrease in pressure by allowing a smooth transition into the higher velocity in the stenosis. Dischargecoefficientshave been observed experimentallyto range from 0.6 for sharp-edged orifices to >O.9 for ideal nozzles (13).

.5 80

&an that

Note that our model

The coejicients of discharge measured in the current study accordingly lay between 0.68 and 0.93. These coeffi-

cients are somewhat higher than velocity potential theory would predict for two major reasons: I) the emptying chamber is finite and the obstruction to flow may not be as abrupt as in the theoretic case where the streamlines originate at infinity;and 2) the receiving chamber also has finite boundaries, which may allow some recovery of pressure from the hi kinetic energy within the vena contracta. The net pressure loss across the orifice is therefore less, and the effectiveorifice appears to be greater than in the absence of pressure recovery (14). ow. We observed an excellent fit to a parabolic pressure-time relation and to a linear flow decay. This observation indicatesthat pressure gradient varies with the square of the Bow rate, in agreement with fundamental and longstanding hydrodynamic notions, and further implies a constant discharge coeacient throughout the range of flows encountered in our experiments. These data differ from an earlier study (4) t the discharge coefficient(co) to vary with the the pressure gradient, leading to a correction of the Gorlin formula with the form A,,, = q/(k’Ap)+ h, where k’ and b are empirical constants. The authors (4) obtained fitted values of k’ and h using 1) in vitro model data of bioprostheses, 2) catheterization data from patients with bioprostheses, and 3) the original hemodynamic data from Gorlin (1). This study (4) raises several points with respect to our Owndata. First, if co were to vary with I& (thatis, if q were proportional to Ap), we would have expected to see exponential pressure and flow decay in our in vitromodel, rather than the parabolic and linear curves observed. Sec.and, in the cited study (4), the fitted values for k’ and h for the in vitro data we? 1.92and -0.226, respectively; for the in vivo ‘data80.3 and 1.20,respectively; and for the original Gorlindata6.84 and 0.08, respectively (a surprising amount of variabihty among data sets). Note that our observations and those of others (15)represent refinements to the Gorhn constant that are relatively small in magnitude, given the wide range oforifice geometries studied. We would expecta

cry low flow rates, where

ow dependency of the disc

cross-sectional area is t-eaterthan in a circular one. ow, where viscosity is the 0

ing flow contraction to dominate viscous effects orifices. We noted that th charge coefficient increased average of 8.9% as 0 e area was augmented from 2.5 cm’. This also may be explained by visco because the pe~meter/area ratio is ~~m0stthe large for a 0.3 cm* orifice as it is for a 2.5 cm2 o Finally, changing the inlet geometry to a nozzle improve the discharge coeflcient by a mean of 8.8% for the circular

orifices, leading to discharge coefficients 0.93. This effect was remarkably similar (Fig. 6). Anatomically, the mitral valve inlet leads to a gradual tapering of cross-sectional area, rather than present-

JACC VQI. 85. No. 5 :I 173-80 April I

I. Gorlin R. Gorlin SJ.

ic formula for calculation oi the area of lbe rdiac valves, and central circulatory shunts.

alhete~~rization and A~~~o~ra~~y. Lea & Febiger, 1980: 125-X.

~i~i~ade~~bia:

odihed orifice equation for the calc~lalion Hydraulic eslima~io~

*Note that this is a quadratic equation of the form Ax2 + which has the solutions x = (-5 ? -)/2A.

of

ofste~otic

1180

JACC Apri

FLACHSKAMPF ET AL. FACTORS INFLUENCING EFFECTIVE VALVE AREA

orifice area: a correction of the Gorlin formula. Circulation 1985:7l: 1170-8. 5. Reid CL, McKay CR. Chandratatna PAN, Kawanishi DT. Rahimtoola SH. Mechanisms of increase in mitral valve area and influence of anatomic features in double-balloon, catheter balloon valvuloplasty in adults with rheumatic mitral stenosis: a Doppler and two-dimensional echocardiographic study. Circulation 1987;67:628-36. 6. Wilkins GT, Weyman AE. Abascal VM. Block PC. Palacios IF. Percutaneous mitral valvotomy: an analysis of echocardiographic variables related to outcome and :he rnczhanismof dilatation. Br Heart J 1988:M): 299-308. 7. Hatle L, Angelsen B, Tromsdal A. Noninvasive assessment of atrioventricular pressure half-time by Doppler ultrasound. Circulation 1979;60: 1096-104. 8. Thomas JD, Wilkins GT, Choong CYP, et al. Inaccuracy of mitral pressure half-time immediately after percutaneous mitral valvotomy. Circulation 1988;78:980-93.

5 so

12. Batchelor GK. An ~n?r~uction to Fluid Dynamics. Cambridge: bridge University Press, 1%7:493-7. 13. Blevins RD. Applied Flu Nostrand Reinhold, 1984

Hanttbook. New York Van

14. Levine RA. Jimoh A, Cape EC. ilkm S, Yoganathan AP, AE. Pressure recovery distal to nosis: potential cause of “overestimation” by Doppler echocardiography. J Am Colt Cardiol 1989zl3:706-15. 15. Segal J. Lemer DJ, Miller C. S, Alderman EA, Popp should Doppler-determined valve area be better then the Gorli Variation in hydraulic constants in low flow states. 3 Am Colt Cardiol 1987:9:1294-305. 16. Chambers JB. Cochrane T, 51 11. Jackson GJ. The Gorlin formula validated against directly obser rifice area in porcine mitral bioprostheses. J Am Coll Cardiol 1989;13:348-53.

9. Thomas JD, Weyman AE. Fluid dynamics model of mitral valve Row: description with in vitro validation. J Am Colt Cardiol 1989:13:221-33.

17. Gabbay S. &Queen DM. Yellin EL, hydrodynamic comparison of mitral valve prostheses at hi& Sow rates. J Thorac Cardiovasc Surg 1978;76:771-87.

IO. Thomas JD, Weyman AE. Doppler mitral pressure half-time: a clinical tool in search of theoretical justificaiion. J Am Coll Cardiol 1987:lO: 923-9.

18. Carabello BA. Advances in the hemodynamic assessment of stenotic cardiac valves. J Am Colt Cardiol 1987;10:912-9.

II. Press WH, Flannery BP, Teukolsky SA. Vetterling WT. Numerical Recipes: The Art of Scientific Computing. Cambridge: Cambridge University Press. 1986~523-8.

19. O’Shea JP, Thomas JD. Abascal V , et al. The Gorlin formula: is the traditional method of calculating the pressure gradient accurate? (abstr). J Am Coll Cardiol 1989;13(suppl A):73A.

Influence of orifice geometry and flow rate on effective valve area: an in vitro study.

Fluid dynamics suggests that orifice geometry is a determinant of discharge properties and, therefore, should influence empiric constants in formulas ...
2MB Sizes 0 Downloads 0 Views