A H Hakki, A S Iskandrian, C E Bemis, D Kimbiris, G S Mintz, B L Segal and C BriceA simplified valve formula for the calculation of stenotic cardiac valve areas.

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A Simplified Valve Formula for the Calculationof Stenotic Cardiac Valve Areas

A-HAMID HAKKI, M.D., ABDULMASSIH S. ISKANDRIAN, M.D., CHARLES E. BEMIS, M.D.,DEMETRIOS KIMBIRIS, M.D., GARY S. MINTZ, M.D., BERNARD L. SEGAL, M.D.,

AND CLAUDIA BRICE, B.A.

SUMMARY We have simplified the Gorlin formula and have compared our measurements of the aortic ormitral valve area, using the original Gorlin formula and the simplified valve formula in 100 consecutive pa-tients. The valve area was measured by the simplified formula as cardiac output (I/min) divided by the squareroot of pressure differences across the valve.

In patients with aortic stenosis of varying severity there was excellent correlation between the originalGorlin formula and the simplified formula (r = 0.96, y = 0.99x + 0.01, SEE = ± 0.10,p < 0.001). The cor-relation was unchanged when the peak gradient was used instead of the mean gradient in the simplified for-mula. Excellent correlation was also seen in patients with mitral stenosis of varying severity (r = 0.94, y =0.97x - 0.02, SEE = ± 0.19; p < 0.001). The simplicity of the formula makes it easy to memorize and use.

THE GORLIN FORMULA, introduced more thanthree decades ago, has been useful in measuring thestenosis of cardiac valves.1 That formula states thatthe valve area (cm2) is equal to the flow across thevalve (ml/sec) divided by the product of two constantsand square root of pressure difference across the valve.One of the constants is the discharge coefficient that isan empirical constant with an assumed arbitrary valueof 1 for the aortic valve and 0.7 for the mitral valve.The second constant is 44.5, which is equal to thesquare root of twice the gravity acceleration factor(980 cm/sec/sec). The flow across the valve is equal tothe cardiac output (ml/min) divided by the product ofthe heart rate (beats/min) and the systolic ejectionperiod or diastolic filling period (sec/beat).

In 1972, Cohen and Gorlin revised the originalformula and suggested the use of 0.85 for the mitralvalve (instead of 0.7) as the discharge coefficient.2

Because the original formula is cumbersome andtime-consuming, it is rarely used by cardiologists whoare not involved with hemodynamic measurements.We have simplified this formula, and our results byboth the original and the simplified formulas in 100patients with either aortic stenosis or mitral stenosisare the subject of this report.

Materials and Methods

We selected 60 consecutive patients with aorticstenosis (35 men and 25 women, ages 22-74 years;mean age 59 years) and 40 patients with mitralstenosis (nine men and 31 women, ages 22-76 years;mean age 55 years) who were evaluated at our institu-tion for this study. Each patient underwent combinedleft- and right-heart catheterization by standard tech-niques. Simultaneous pressures were recorded across

From the LikofT Cardiovascular Institute, Hahnemann MedicalCollege and Hospital, Philadelphia, Pennsylvania.

Address for correspondence: Abdulmassih S. Iskandrian, M.D.,Hahnemann Medical College and Hospital, 230 North BroadStreet, Philadelphia, Pennsylvania 19102.

Received April 17, 1980; revision accepted September 15, 1980.Circulation 63, No. 5, 1981.

the aortic valve on equisensitive transducers by meansof either transaortic or transseptal left ventricularcatheterization. Similarly, simultaneous pressureswere recorded across the mitral valve by measuringthe left ventricular and pulmonary artery wedgepressures. The pulmonary wedge pressure was con-firmed in each patient by characteristic wave form orby oximetry. Cardiac output was measured by theFick method, and oxygen consumption was measuredin most patients.

Each patient also underwent left ventriculography,aortography and selective coronary arteriography bystandard techniques. The aortic or mitral valve areawas calculated in each patient by the original Gorlinformula. The systolic ejection period was measuredfrom the aortic pressure tracings, from the beginningof the ejection to the dicrotic notch. The diastolic fill-ing period was measured between the crossover pointsof the pulmonary artery wedge and the left ventric-ular pressure tracings. The heart rate was calculatedat the time of cardiac output measurement by count-ing the RR cycles over a 60-second interval. The peakaortic gradient was measured as a simple peak-to-peak gradient. The peaks were not necessarily at theexact time during systole. The mean pressuredifference across the aortic or mitral valve was mea-sured by planimetry. We used the same cardiac outputin both the original Gorlin and the simplified for-mulas.The aortic or mitral valve area (cm2) was measured

by the simplified formula as the cardiac output (1/min)divided by the square root of the pressure differencesacross the valve. For the aortic valve, we used eitherthe peak or the mean pressure difference across thevalve in the simplified formula, but for the mitralvalve, we used only the mean pressure difference.We performed the statistical correlation by means

of Pearson product moment correlation and the t test.

ResultsAortic Stenosis

The hemodynamic data for patients with aorticstenosis are shown in table 1.

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TABLE 1. Hernodyanmic Data in GO Patients with Aortic Stenosis

Aortic valve gradient(mm Hg)

Peak Meaw

60 6740 3642 3620 2220 2098 8364 60

125 101120 9383 66

104 6940 4540 3150 5070 6898 9890 8195 7445 5790 9530 3520 2060 4750 4390 7127 2745 2640 4520 2047 5840 4226 2620 13120 10415 1785 7517 2680 8245 46110 10885 6755 4890 9055 4670 5818 2035 4990 8035 4040 43135 9845 46120 9560 6170 5660 45100 7912 9.530 1640 39

CO(1/min)

4.82.54.94.65.84.13.33.74.38.04.52.74.74.26.35.53.95.33.22.63.42.71.94.75.26.32.93.26.64.34.24.94.63.94.87.8.5.26.46.13.23.95.84.97.38.07.08.45.64.06.95.54.44.03.25.44.54.76.65.59.1

HR(beats/min)

65-

120807966647553775857957683841088073608183888870828570687986849272689248

1146,576589876102797.5887,576777665608466928O746278

SEP(sec/beat)

0.340.250.190.240.290.330.310.330.370.290.320.320.250.310.270.310.240.280.260.400.290.220.310.280.320.290.210.270.300.290.260.300.280.320.36n..()0.330.250.330.340.260.250.280.230.280.320.260.310.360.300.340.280.380.270.340.290.280.320.420.29

OriginalGorlinformnula0.600.490.811.111.260.460.490.340.510.990.640..500.810.560.770.480.370.610.510.260.55)0.740.220.660.621.1.50.720.5)71.630.660.661.101.110.371.070.731.430.560.940.270.710.770.551.041.071.461.200.620.531.020.490.800.410.420.710.570.522.041.191.445

AVA (cm2)Simplified formulaMean Peak

gradient gradient

0.59 0.620.42 0.400.82 0.760.96 1.021.23 1.230.45 0.410.43 0.420.37 0.330.44 0.390.99 0.880.54 0.440.40 0.430.84 0.740.59 0.590.76 0.750.56 0.560.43 0.410.61 0.540.43 0.480.27 0.280.57 0.620.61 0.610.27 0.240.72 0.670.62 0.551.22 1.220.58 0.440.48 0.511.48 1.480.57 0.630.65 0.661.23 1.231.27 1.020.38 0.351.16 1.240.90 0.851.01 1.260.70 0.710.89 0.900.31 0.310.47 0.420.84 0.790.52 0.521.08 0.991.05 0.961.56 1.651.20 1.430.63 0.590.63 0.681.05 1.090.56 0.480.63 0.660.41 0.370.41 0.420.72 0.640.68 0.590.53 0.472.15 1.911.38 1.001.46 1.44

Abbreviations: CO = cardiac output; HE, = heart rate; AVA

The correlation between the aortic valve areas mea-sured by the original Gorlin formula and thesimplified formula was excellent (r = 0.96, y = 0.99x+ 0.01, SEE = + 0.10, p < 0.001) (fig. 1). There was

= aortic valve area; SEP = systolic ejection period.

also an excellent correlation between the originalformula and the simplified formula when the peakgradient was used in the simplified formula instead ofthe mean gradient (r = 0.96, y = 0.97x - 0.006).

Pt

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

--

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VOL 63, No 5, MAY 1981

2.0F

E 1.60a-

> 1.2CO

-0. 0.8

CX 0.4

I

?

I

a a a a a A _---~~~~~an

AORTIC VALVE AREA (cm2) 0

a~~~~~~~

0

a 110~~~~~0 0

0

N=60, - wr=0.96

. P< 0.001Y= 0.99x+0.01

0.4 0.8 1.2 1.6 2.0

Original Gorlin FormulaFIGURE 1. Correlation between aortic valve areacalculated by the original Gorlin and the simplified valveformulas. The mean pressure difference was used in bothformulas.

Further, there was an excellent correlation betweenthe two simplified formulas, version one using themean and version two using the peak gradient (r -0.98). The mean gradient in 32 of the 60 patients wasless than the peak-to-peak aortic gradient. In the re-maining 28 patients, the mean gradient was equal toor slightly greater than the peak gradient. A peak-to-peak aortic gradient less than the mean aortic grad-ient has been explained by other investigators inpatients with porcine xenograft tissue valves on thebasis of leaflet inertia.3

In 32 patients with aortic valve areas less than 0.7cm2 (simplified formula), the error of estimate was0-0.24 cm2 (0.06 ± 0.05, mean ± SD). In 17 patientswith aortic valve areas of 0.7-1.1 cm2 (simplified for-mula), the error of estimate was 0-0.42 cm2 (0.08 ±0.1 1). In 14 of these 17 patients, the error of estimatewas less than 0.17 cm2. In the remaining 11 patientswith aortic valve areas greater than 1.1 cm2 (simplifiedformula), the error of estimate was 0-0. 19 cm2 (0. 10 ±0.06). In all 11 patients, the variation was less than0.19 cm2.The reason for this remarkable correlation may be

explained by the results shown in figure 2. The productof systolic ejection period times heart rate times con-stant times 44.5 is very close to 1 X 103 over a widerange of valve areas included in this study (y = 0.02x+ 1.02, SEE = ± 0.13, p < 0.001). Thus, althoughthese variables were deleted in the simplified formula,the results were comparable to those determined bythe original Gorlin formula.

Mitral Stenosis

The hemodynamic data for patients with mitralstenosis are shown in table 2.The correlation between the mitral valve areas

2.0o

1.61FLO

I

cc

1.2

0.81

p0 40014 0

* 700g. W^

0.41

LiE -.

N=60Y=-0.02x+1 .02

0.4 0.8 1.2 1.6 2.0

Aortic Valve Area (cm2)FIGURE 2. Correlation between the product of systolicejection period (SEP) times heart rate (HR) times 44.5 andthe aortic valve area, as measured by the original Gorlin for-m ula.

calculated by the original Gorlin formula and thesimplified formula is shown in figure 3. The correla-tion between the two formulas was excellent (r = 0.94,y = 0.97x - 0.02, SEE = ± O.l9,p < 0.001). In 13 pa-tients with mitral valve areas less than 1.0 cm2(simplified formula), the error of estimate was0.01-0.32 (0.14 + 0.11). In another 13 patients withvalve areas of 1-1.49 cm2 (simplified formula), theerror of estimate was 0.01-0.49 cm2 (0.12 00.14); in12 of these patients the error of estimate was less than0.29 cm2. In eight patients with valve areas of 1.5-1.99cm2 (simplified formula), the error of estimate was0.06-0.39 cm3 (0.19 ± 0.13). In the remaining six pa-tients with valve areas greater than 2.0 cm2 (simplified

3.2r

2.81(o0

LL()

CO

a)n-.

E0c

2.41

2.0O

1.61

1.21

0.81

0.4

cma- -a m

MITRAL VALVE AREA (cm2)0

0

a~~~~~~~

0

0a

* 0

a* 0* N=400 r=0.94

a P<0.001Y=0.97x-0.

0.4 0.8 1.2 1.6 2.0

t.02

2.4 2.8 3.2

Original Gorlin FormulaFIGURE 3. Correlation between mitral valve areacalculated by the original Gorlin and the simplified valveformulas.

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SIMPLIFIED VALVE FORMULA/Hakki et al.

TABLLE 2. Hemodynamic Data in Patients with Mitral Stenosis

Pt

1

2

3456789

10

11

121314

15161718192021222324

25262728293031

323334

353637

383940

Abbreviations:valve area.

Meangradient(mm Hg)

713

101614

910

131911

10

14237

21139

13148

713

7915145

11

71722

12

1510

12

716141612

CO = c,

CO(1/mill)

6.82.5

5.54.02.8

6.35.14.1

3.73.56.83.26.54.33.97.2

6.1

3.85.54.5

3.43.34.4

3.25.33.6

6.55.13.41.52.74.73.22.63.93.53.21.84.45.6

HR(beats/min)

677950577586887594526984727581797481

80

7984627285718659707881

12264

100

1181086.5108

756895

DFP(sec/beat)

0.530.280.770.540.450.350.420.410.300.670.520.380.470.340.450.360.370.320.410.380.390.430.430.350.430.300.600.490.330.380.280.500.240.34

0.200.440.220.410.480.29

MVA (cm2)OriginalGorlin Sinformula foi

2.261.01

1.46 1

1.05 1

0.80 c

2.23 2

1.38 1

1.90 1

0.97 C0.98 1

1.950.87 C1.29 1

1.98 1

0.75 c2.26

2.401.29 1

1.44 11.66 1

1.29 11.13 C1.76 1

1.17 11.46 11.19 c

2.611.40 1

1.55 1

0.39 c

0.53

1.37 ]1.11

0.661.60 11.54 1

1.10 c

0.51 c

1.091.93

ardiac outpuit; HR = heart rate; DFP = diastolic filling period;

nplifiedrmula

2.52

0.691.761.000.762.081.581.840.851.06

2.17

0.861.361.590.862.012.05

1.071.471.581.31).931.691.08

1.380.972.901.511.280.370.581.36).830.811.11L.350.810.491.1.01.64

MVA = mitral

formula), the error of estimate was 0.15-0.35 cm2(0.25 0.07). The product of the diastolic fillingperiod times heart rate times 31 in relation to mitralvalve areas is shown in figure 4 (y = -0.02x + 0.98,SEE = 0.14, p < 0.001). This product, as in patients

with aortic stenosis, was close to 1 X 103. We alsomeasured the mitral valve area using the simplifiedformula from the hemodynamic data provided byGorlin and Gorlin' (39 measurements in 24 patients),and compared our results with valve areas measured

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VOL 63, No 5, MAY 1981

N=40Y=-0.02x+(

*.

2.0

2.0p.0398 E0

U-

0.98 Ea)

. _E

'D

0

2.8

1.6

1.21.

0.8

0.4

0

MITRAL VALVE AREA (cm2)

0

1 ~~~~~00

*/ 0

S 00

00O

0

* **N=390 ,r-=0.96* P< 0.00o1

Y=0.81x+0.18

0.4 0.8 1.2

Original Gorlin Formula

by these investigators using the original formula (fig.5). Again, the correlation was excellent (r = 0.96, y =

0.81x + 0.18, SEE = ± 0.13, p < 0.001).Although physiologic interventions that would pro-

duce variations in such factors as the cardiac output,heart rate, systolic ejection period or diastolic fillingperiod were not available because of the retrospectivenature of this study, analysis of the data provided intables 1 and 2 shows considerable variation in severalof these measurements in the resting condition. Thus,in patients with aortic stenosis the peak-to-peak aorticgradient was less than 50 mm Hg in 28 patients, 50-99mm Hg in 24 patients and equal to or greater than 100mm Hg in eight patients. The heart rate was less than80 beats/min in 35 patients and greater than 80beats/min in 25 patients. In fact, the heart rate in ninepatients was less than 60 beats/min or more than 100

FIGURE 5. Correlation of mitral valve area as described inthe original study by Gorlin and Gorlin and the valve area

measured by the simplified valve formula.

beats/min. The systolic ejection period was less than0.3 sec/beat in 31 patients and more than 0.3 sec/beatin 29 patients. The cardiac output was less than 51/min in 36 patients and more than 5.0 1/min in 24patients. Similarly, in patients with mitral stenosis theheart rate was less than 80 beats/min in 23 patientsand more than 80 beats/min in 17 patients, and ninepatients had heart rates less than 60 beats/min ormore than 100 beats/min. The diastolic filling periodvaried widely and was less than 0.3 sec/beat or greaterthan 0.5 sec/beat in 12 patients. The cardiac outputwas less than 4.0 1/min in 20 patients and more than4.0 1/min in 20 patients.

Moreover, we calculated the mitral valve areas at

TABWLE 3. Hernodynamic Data at Rest and During Excrcise in Patients Stdlidl by Gorlin and Gorlin

Mitral valve -MVA (cm2) MVA (cm2)gradient C( HE I)FB Original Gorlin Simplified(mm Hg) (1/min) (beats/min) (sec/beat) formula formula

Pt Rest Ex I2est Ex IBest lEx Rest E1,x R)est Fix Rest Ex

1 14 36 9.4 11.1 100 120 0.31 0.22 2.6 2.2 2,5 1.92 16 41 6.5 9.4 72 110 0.47 0.225 1.6 1.6 1.6 1.53 14 25 .5 5.9 100 142 0.32 0.16 1. ) 1.7 1.5 1.24 19 41 4.4 5.7 70 108 0.52 0.30 0.9 0.9 1.0 0.9

5 12 23 3.2 4.8 96 93 0.40 0.41 0.8 0.9 0.9 1.06 21 46 3.9 4.1 80 94 0.30 0.30 0.7 0.7 0.9 0.6

7 23 30 3.3 4.1 84 70 0.44 0.48 0.6 0.7 0.7 0.88 23 30 3.2 4.0 105 136 0.36 0.24 0.5 0.7 0.7 0.79 6 8 2.9 3.5 71 75 0.152 0.48 1.1 1.2 1.2 1.2

10 17 30 2.3 2.8 56 136 0.77 0.25 0.4 0.5 0.6 0.5

Abbreviations: CO = cardiac output; Hl - heart rate; DFP diastolic filling period; MVA = mitral valve area; Ex -exercise.

2.8 Bi

2.4 .

U(Y)COx

Ir

2.0

1.2

0.4

a

m m- a m a

0.4 1.2

Mitral Valve Area (cm2)FIGURE 4. Correlation of the product of diastolic fillingperiod (dfp) times heart rate (HR) times 31 and the mitralvalve area, as measured by the original Gorlin formula.

1.6 2.0 2.4

-- - "~r

a a also" a a a a a 1 ia 1 LWORJ

CIRCULATION1054

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SIMPLIFIED VALVE FORMULA/Hakki et al.

rest and during exercise in 10 patients from the hemo-dynamic data provided by Gorlin and Gorlin1 (table 3)and compared our results with Gorlin's originalresults. We found an excellent correlation (r = 0.96, y= 0.89x + 0.04, SEE = ± 0.16, p < 0.001).

Discussion

Since 1951 the hemodynamic evaluation of theseverity of valvular stenoses has relied on the estima-tion of valve orifice areas using the hydraulic equa-tion of Gorlin and Gorlin.' This equation incorporatescardiac output, heart rate, systolic ejection period (ordiastolic filling period), an empirical constant,acceleration of gravity factor and the pressuredifference across the stenotic valve. Gorlin and Gorlinfound that when they used this formula in 11 patientswith mitral stenosis, the calculated valve area differedby less than 0.2 cm2 from the actual valve areameasured at surgery or autopsy.Our results show that a simplified version of the

original Gorlin formula using the cardiac output andthe pressure difference across the valve can be used tomeasure reliably the severity of aortic or mitralstenosis (figs. 1 and 3). Moreover, in patients with aor-tic stenosis, we noted little error in estimating theseverity of stenosis when the peak pressure differencewas used instead of the mean pressure differenceacross the valve. Under most circumstances, it takes atight stenosis to generate a pressure gradient and

small variations may in fact have little clinicalsignificance. Physiologic interventions that wouldproduce changes in several factors used in valve areameasurements may have been important to prove theusefulness of our formula but unfortunately were notavailable. However the mitral valve areas calculatedwith the simplified valve formula correlated well with10 duplicate measurements obtained at rest and dur-ing exercise by Gorlin and Gorlin' (table 3). Also, theresting hemodynamic data in our patients (tables Iand 2) show considerable variation in all of the abovefactors.There is no question that the Gorlin formula has

been extremely useful in the evaluation of patientswith valvular heart disease. However, physicians whoare not involved on a daily basis in hemodynamicmeasurements find the simplified formula easy toremember and to use. Most important, the accuracyof valve area determined by means of the simplifiedformula is not reduced.

References

l. Gorlin R, Gorlin SG: Hydraulic formula for calculation of thearea of the stenotic mitral valve, other cardiac valves, and cen-tral circulatory shunts. Am Heart J 41: 1, 1951

2. Cohen MV, Gorlin R: Modified orifice equation for the calcula-tion of mitral valve area. Am Heart J 84: 839, 1972

3. Chaitman BR, Bonan R, Lepage G, Tubau JF, David PR,Hydra I, Grondin CM: Hemodynamic evaluation of theCarpentier-Edwards porcine xenograft. Circulation 60: 1170,1979

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