# Comparison of Valvular Resistance, Stroke Work Loss, and Gorlin Valve Area for Quantification of Aortic Stenosis

## An In Vitro Study in a Pulsatile Aortic Flow Model

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## Abstract

*Background *Valvular resistance and stroke work loss have been proposed as alternative measures of stenotic valvular lesions that may be less flow dependent and, thus, superior over valve area calculations for the quantification of aortic stenosis. The present in vitro study was designed to compare the impacts of valvular resistance, stroke work loss, and Gorlin valve area as hemodynamic indexes of aortic stenosis.

*Methods and Results *In a pulsatile aortic flow model, rigid stenotic orifices in varying sizes (0.5, 1.0, 1.5, and 2.0 cm^{2}) and geometry were studied under different hemodynamic conditions. Ventricular and aortic pressures were measured to determine the mean systolic ventricular pressure (LVSP_{m}) and the transstenotic pressure gradient (ΔP_{m}). Transvalvular flow (F_{m}) was assessed with an electromagnetic flowmeter. Valvular resistance [VR=1333 · (ΔP_{m}/F_{m})] and stroke work loss [SWL=100 · (ΔP_{m}/LVSP_{m})] were calculated and compared with aortic valve area [AVA=F_{m}/(50√ΔP_{m})]. The measurements were performed for a large range of transvalvular flows. At low-flow states, flow augmentation (100→200 mL/s) increased calculated valvular resistance between 21% (2.0-cm^{2} orifice) and 66% (0.5-cm^{2} orifice). Stroke work loss demonstrated an increase from 43% (2.0 cm^{2}) to 100% (1.0 cm^{2}). In contrast, Gorlin valve area revealed only a moderate change from 29% (2.0 cm^{2}) to 5% (0.5 cm^{2}). At physiological flow rates, increase in transvalvular flow (200→300 mL/s) did not alter calculated Gorlin valve area, whereas valvular resistance and stroke work loss demonstrated a continuing increase. Our experimental results were adopted to interpret the results of three clinical studies in aortic stenosis. The flow-dependent increase of Gorlin valve area, which was found in the cited clinical studies, can be elucidated as true further opening of the stenotic valve but not as a calculation error due to the Gorlin formula.

*Conclusions *Within the physiological range of flow, calculated aortic valve area was less dependent on hemodynamic conditions than were valvular resistance and stroke work loss, which varied as a function of flow. Thus, for the assessment of the severity of aortic stenosis, the Gorlin valve area is superior over valvular resistance and stroke work loss, which must be indexed for flow to adequately quantify the hemodynamic severity of the obstruction.

I t is assumed that in mitral or aortic stenosis the valve orifice remains fixed irrespective of the magnitude of transvalvular pressure gradient or flow.^{1} Thus, a flow-dependent increase in calculated valve area, which was found in some clinical studies in aortic^{2} ^{3} and mitral stenoses,^{1} ^{4} had been interpreted to reflect a flow dependence of the Gorlin formula rather than an actual increase in the orifice area.

Valvular resistance^{3} ^{4} ^{5} ^{6} ^{7} ^{8} and stroke work loss^{9} ^{10} have been proposed as alternative measures of stenotic valvular lesions that may be less flow dependent than valve area calculations based on the Gorlin formula.^{11}

To determine the validity of valvular resistance and stroke work loss in the quantification of aortic stenosis, we performed a hemodynamic study in a well-controlled pulsatile aortic flow model. Valvular resistance and stroke work loss were assessed under changing hemodynamic conditions and compared with calculations of valve area according to the Gorlin formula.

## Methods

### Experimental Model

All experiments were performed in an electrohydraulic, computer-controlled pulsatile flow model simulating the left side of the human circulatory system (Fig 1⇓). This circuit has been previously described.^{12} ^{13} ^{14} ^{15} The test fluid consisted of a 60% water–40% glycerol solution (viscosity, 3.6 cP; density, 1.06 g/mL). Heart rate was held constant at 70 beats per minute. Piezoelectric catheter-tip transducers (Sensodyne) were used to record the instantaneous pressures in the ventricular apex and aorta. The aortic pressure was measured 130 mm distal to the aortic valve plane. Each pressure signal was averaged over 64 consecutive beats to achieve highly reproducible results. The variation of subsequent measurements was within 1 mm Hg. The transstenotic flow rate was measured with an electromagnetic flowmeter (Zapeda Instruments) positioned in the aorta.

Different types of stenotic orifices were mounted in aortic position of this pulsatile flow circuit (Fig 2⇓): circular central orifices of 0.5, 1.0, 1.5, and 2.0 cm^{2}, representing severe to mild aortic stenosis, and stenoses with identical orifice areas of 0.5 cm^{2} but with variable configurations, simulating different types of adult aortic stenosis^{16} ; these included circular, slitlike, and Y-shaped orifices with abrupt narrowing and circular orifices with a long (2.0 cm) and a short (0.5 cm) nozzle.

The flow rate was increased stepwise from a basal systolic flow of 50 mL/s to 200 mL/s (0.5 cm^{2}), 300 mL/s (1.0 cm^{2}), and 400 mL/s (2.0 cm^{2}). At each flow rate, pressure and flow data were recorded simultaneously.

### Calculations

Mean systolic pressure gradient (ΔP_{m}) was obtained by averaging the integrated differences between the simultaneously recorded pressure curves over the systolic time period. Mean systolic flow (F_{m}) was derived from the electromagnetic flow curve and expressed in mL/s.

For calculation of valvular resistance, ΔP_{m} was divided by F_{m}. To express resistance in metric units (dynes · s · cm^{−5}), the conversion factor for pressure has to be included (1 mm Hg=1333 dynes · cm^{−2}):

Stroke work loss (SWL) was calculated as the ratio of the mean transstenotic pressure gradient to the mean systolic ventricular pressure (LVSP)^{10} :

Functional aortic valve area (AVA_{F}) was calculated according to a modified version of the Gorlin formula:

In addition, the effective aortic valve area (AVA_{EFF}) within the vena contracta was calculated as the following:

where AA is cross-sectional area of the aorta (see “Appendix”).

To adopt these experimental results for a valid interpretation of clinical studies in aortic stenosis, the Reynolds number (Re) was determined (see “Appendix”):

### Statistical Analysis

All pooled clinical data were expressed as mean±SD. The significance of differences between paired measurements was assessed with Student’s paired *t* test.

## Results

### Aortic Valve Area

At a normal cardiac output of 5 L/min, which corresponds to a systolic flow of 200 mL/s, the effective valve areas were 0.37 cm^{2} (0.5 cm^{2}), 0.69 cm^{2} (1.0 cm^{2}), 0.90 cm^{2} (1.5 cm^{2}), and 1.07 cm^{2} (2.0 cm^{2}). For the 0.5-cm^{2} orifices, calculated valve area was dependent on the geometry of the stenosis and ranged from 0.35 cm^{2} (Y-shaped orifice) to 0.42 cm^{2} (circular orifice with long nozzle) (see Tables 1⇓⇓ and 2⇓).

For all stenotic models, valve area increased with increasing flow and Reynolds number. The flow depen-dence of valve area was maximal for the least severe obstruction (Fig 3⇓). Within the low range of flow (100→200 mL/s), valve area increased by 24% (2.0 cm^{2}), 20% (1.5 cm^{2}), 13% (1.0 cm^{2}), and 6% (0.5 cm^{2}). Further increase in flow (200→300 mL/s) had only a small effect on calculated valve area (see Table 2⇑).

The transition from an increasing to a nonincreasing phase of valve area was dependent on the ratio of Reynolds number to the effective valve area. When Re/AVA_{EFF} exceeded 10 000, calculated valve area reached a plateau and was independent of the actual flow (Fig 3⇑).

### Valvular Resistance

At a standard cardiac output of 5 L/min, valvular resistance ranged from 64 (2.0 cm^{2}) to 639 dynes · s · cm^{−5} (0.5 cm^{2}). For a constant orifice size of 0.5 cm^{2}, valvular resistance was 693 (Y shaped), 635 (slitlike), 546 (short nozzle), and 515 (long nozzle) dynes · s · cm^{−5}.

Within the low range of transvalvular flow (100→200 mL/s), the increase in valvular resistance ranged from 21% (2.0-cm^{2} orifice) to 66% (0.5-cm^{2} orifice). Flow augmentation from 200 to 300 mL/s resulted in an additional increase in valvular resistance of 20% (2.0-cm^{2} orifice) to 51% (1.0-cm^{2} orifice) (Fig 4⇓, Table 2⇑).

### Stroke Work Loss

Stroke work loss at 5 L/min was 10% (2.0 cm^{2}), 14% (1.5 cm^{2}), 22% (1.0 cm^{2}), and 49% (0.5 cm^{2}). This index demonstrated a considerable increase (43% to 100%) during flow augmentation at low-flow states (100→200 mL/s). The further increase in flow (200→300 mL/s) resulted in an additional increase in stroke work loss: 20% (2.0 cm^{2}) to 43% (1.0 cm^{2}) (Table 2⇑, Fig 5⇓).

## Discussion

In several studies,^{3} ^{4} ^{6} ^{7} ^{8} the accuracy of the Gorlin formula for the assessment of aortic stenosis has been challenged. Thus, stroke work loss has been claimed to be a measure of stenotic valvular lesions that obviates the assessment of transvalvular flow and appears to remain more stable than Gorlin valve area.^{9} ^{10} Alternatively, valvular resistance, the simple ratio of transvalvular pressure gradient to flow, has been proposed as a valid parameter for quantification of aortic stenosis.^{3} ^{5} ^{6} ^{7} ^{8} Calculated aortic valve area has been found to increase significantly more than valvular resistance either during exercise^{2} or after the application of nitroprusside^{17} or dobutamine.^{3} With the assumption that in calcific aortic stenosis the anatomic valve area is flow independent and remains unchanged under varying hemodynamic conditions, the stability of valvular resistance has been interpreted as an argument favoring valvular resistance over valve area calculations for assessment of the severity of aortic stenosis.^{6} ^{7} ^{8} ^{18} However, a flow-mediated increase in true anatomic valve area could also explain why valvular resistance varies less than valve area. Whether a flow-dependent increase in calculated valve area is due to an actual change in anatomic valve area or due to the Gorlin formula cannot be clarified under clinical conditions but requires exploration under well-controlled in vitro conditions.

In the present study, rigid stenotic orifices were mounted in a pulsatile aortic flow model to evaluate the impacts of valvular resistance, stroke work loss, and Gorlin valve area as hemodynamic indicators in aortic stenosis. At low-flow states, flow augmentation (100→200 mL/s) increased valve area less than stroke work loss and valvular resistance. During further increase in transvalvular flow (200→300 mL/s), calculated aortic valve area remained constant, whereas valvular resistance and stroke work loss demonstrated a continuing increase (Fig 6⇓). For all stenotic orifices used in this setting, a square-law dependence of pressure gradient on flow was found (Fig 7⇓).

This supports the concept underlying the Gorlin formula (that pressure gradient is proportional to the square of flow) but is an argument against the assessment of valvular resistance (simple ratio of pressure gradient to flow) or stroke work loss for assessment of aortic stenosis.

It is well known from the study of fluid dynamics that the coefficient of velocity, which represents the viscous losses at the inlet of the stenosis, varies with flow at low Reynolds numbers but remains constant at larger Reynolds numbers.^{19} ^{20} In the present study, the Reynolds number exceeded 2200 in all cases. Under these circumstances, viscous losses can be neglected. Thus, the documented increase in effective valve area at low-flow states cannot be interpreted as a flow-dependent increase in the coefficient of velocity (c_{V})^{21} but rather expresses an increase in the coefficient of orifice contraction (c_{C}; the ratio of effective valve area and anatomic valve area) (“improved streamlines of flow as the velocity of pulsatile flow increases”^{22} ). In contrast to our results, Cannon et al^{23} noticed a linear relation between transvalvular pressure gradient and flow, supporting the concept of valvular resistance. However, Cannon et al restricted their experimental study to low-flow states because their circulation model did not allow ventricular pressures of >150 mm Hg. In concordance with our results, Flachskampf et al^{24} found no appreciable change in valve area with flow. However, these authors used a steady-state model and did not specifically address low-flow rates. Finally, Segal et al^{21} performed an in vitro study to evaluate the effect of flow on Gorlin valve area. They also found a flow dependence of the Gorlin formula at low-flow states and, thus, considered Doppler-determined valve area to be more accurate than Gorlin valve area. However, in this study only very small stenotic orifices—0.06 and 0.34 cm^{2}—were used. In these unphysiological models of aortic stenosis, viscous losses may become significant (“spray phenomenon”) and prohibit the application of the simplified Bernoulli equation as a prerequisite of the Gorlin formula.^{20} ^{25}

Our experimental data were adopted to interpret the results of three clinical studies in aortic stenosis that demonstrated a flow-dependent increase in calculated valve area either during exercise^{2} or after administration of dobutamine^{3} or nitroprusside.^{17} The analysis of the pooled data from these three studies revealed that mean Gorlin valve area increased significantly, by 17% (0.72±0.26→0.84±0.31 cm^{2}) (Table 3⇑). Our experimental data are of help in determining whether this increase is due to the Gorlin formula or represents an actual increase in the valve orifice. Calculation of the Reynolds number (Re) and effective aortic valve area (AVA_{EFF}) revealed that the ratio of Re to AVA_{EFF} exceeded 10 000 before flow augmentation in almost all of these patients (Table 3⇑, Fig 8⇓). Our experimental results suggest that beyond this flow range, no further increase in the contraction coefficient can be expected. Thus, it can be concluded that the flow-dependent increase of valve area, which was found in these clinical studies, does not represent a “computational artifact of the Gorlin formula”^{2} or a “flow dependence of the Gorlin formula”^{3} but rather is due to true further increase in actual valve area in the individual patients.

### Study Limitations

For the present study, a newtonian fluid was used, whereas blood is a nonnewtonian medium with velocity-dependent viscosity. However, because all flow conditions are in the turbulent range, nonnewtonian viscosity effects can be neglected.

Pressure measurements within the vena contracta are necessary for calculation of effective aortic valve area according to the Gorlin formula.^{26} These pressure measurements alter the hemodynamic conditions at the stenosis and thus were abandoned in the present study. Under consideration of pressure recovery,^{27} ^{28} effective valve area was calculated. It has previously been shown that this calculation is reliable under well-controlled conditions.^{15} ^{29}

Because the jet velocity was not directly measured, the Reynolds number was derived from pressure measurements according to the simplified Bernoulli equation. The simplified Bernoulli equation (ie, Torricelli’s law) assumes that there are negligible viscous losses upstream of the vena contracta. This is expected to be valid when the Reynolds number is sufficiently large (>1500). Several experimental studies confirmed the reliability of the Bernoulli equation within this flow range.^{20} ^{25}

In the present study, effective aortic valve area was assessed. No attempt was undertaken to calculate anatomic valve area, which necessitates the inclusion of an empirically derived discharge coefficient. In correspondence, the flow dependence of the true or anatomic valve area, which was found in patients with aortic stenosis,^{2} ^{3} ^{17} has not been directly measured but rather calculated from the pressure and flow measurements. However, we agree with Dumesnil and Yoganathan^{30} that the effective area is the more important variable to consider clinically because it is directly related to pressure and flow. It better reflects the patient’s clinical and hemodynamic status than the anatomic area.

For calculation of valve area, the square root of the mean pressure gradient was used instead of the more correct root-mean-square gradient. Although this simplification may lead to an underestimation of effective valve area by ≤10%,^{31} we adopted the original version of the Gorlin formula to make a valid comparison of the experimental data and the clinical results.

### Conclusions

Within the physiological range of transvalvular flow, valvular resistance and stroke work loss were flow dependent, whereas calculated aortic valve area remained constant. This is due to the squared relation between pressure gradient and flow, which was confirmed in this pulsatile flow model.

In aortic stenosis, calculation of valve area according to the Gorlin formula is superior over valvular resistance or stroke work loss, which must be indexed for flow to quantify the hemodynamic severity of the obstruction.

Calculation of the Reynolds number and valve area revealed that the flow-dependent increase in calculated valve area, which has been described in the cited clinical studies in aortic stenosis, does not represent a computational error of the Gorlin formula but is generally due to an actual increase of orifice size.

In low-flow, low-gradient situations, the Gorlin formula may overestimate the actual severity of aortic stenosis. Because the effect of increasing flow on actual orifice area cannot be predicted by the measurements at rest,^{32} maneuvers to increase cardiac output may be necessary to fully characterize the severity of aortic stenosis.

## Appendix A

### Calculation of Reynolds Number

The Reynolds number (Re) is calculated as follows:

where v is cross-sectional averaged fluid velocity, d is diameter of the vena contracta, and kinematic viscosity υ=3.4 · 10^{−2} cm^{2}/s.

The continuity equation yields:

Under the assumption that the orifice is circular, AVA_{EFF} can be calculated as follows:

Thus, for calculation of the Reynolds number, v and d (Equation 1) are replaced:

### Calculation of Effective Aortic Valve Area

For the valid assessment of valve area according to the Gorlin formula, the maximal transstenotic pressure gradient (P_{V}−P_{X}) is necessary.^{26} However, in the present study, pressure was not measured within the vena contracta at the site of minimal aortic pressure X but rather at the site A distal to the region of pressure recovery. Thus, the maximal pressure gradient (P_{V}−P_{X}) has to be derived from the measured pressure gradient or total pressure loss (P_{V}−P_{A}). The difference between P_{X} and P_{A} corresponds to the amount of pressure recovery, which can be calculated according to the simplified Bernoulli equation and the momentum equation^{15} :

With the simplified Bernoulli equation (Equation 6) of Equation 5 can be rearranged:

It yields the relation between the maximum pressure gradient (P_{V}−P_{X}) and the total pressure loss (P_{V}−P_{A}):

The Gorlin formula is a special version of the continuity equation (Equation 2), which uses the simplified Bernoulli equation to replace the velocity term (in mL/s):

The combination of Equations 8 and 9 gives:

Thus, when the Gorlin formula is calculated using (P_{V}−P_{A}) instead of (P_{V}−P_{X}), a functional aortic valve area (AVA_{F}) will be assessed. Between AVA_{EFF} and AVA_{F}, the following relation exists:

C is replaced according to Equation 5, and thus Equation 12 can be rearranged as follows:

Under the assumption AVA_{EFF}<<AA, the equation yields: 2 · AVA_{EFF}^{2}/AA^{2}≈AVA_{EFF}^{2}/AA^{2}. Thus, Equation 13 can be simplified as follows:

Thus, AVA_{EFF} can be calculated from AVA_{F} and AA (6.15 cm^{2} in this circuit):

## Footnotes

Presented in part at the 65th Scientific Sessions of the American Heart Association, November 1992, New Orleans, La.

- Received April 12, 1994.
- Revision received August 12, 1994.
- Accepted September 23, 1994.

- Copyright © 1995 by American Heart Association

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- Comparison of Valvular Resistance, Stroke Work Loss, and Gorlin Valve Area for Quantification of Aortic StenosisWolfram Voelker, Helmut Reul, Georg Nienhaus Ing, Thomas Stelzer Ing, Bernhard Schmitz Ing, Anselm Steegers Ing and Karl R. KarschCirculation. 1995;91:1196-1204, originally published February 15, 1995http://dx.doi.org/10.1161/01.CIR.91.4.1196
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- Comparison of Valvular Resistance, Stroke Work Loss, and Gorlin Valve Area for Quantification of Aortic StenosisWolfram Voelker, Helmut Reul, Georg Nienhaus Ing, Thomas Stelzer Ing, Bernhard Schmitz Ing, Anselm Steegers Ing and Karl R. KarschCirculation. 1995;91:1196-1204, originally published February 15, 1995http://dx.doi.org/10.1161/01.CIR.91.4.1196

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