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Non-invasive estimation of pressure gradients in regurgitant jets: an overdue consideration

Alessandro Giardini , Theresa A. Tacy
DOI: http://dx.doi.org/10.1093/ejechocard/jen156 578-584 First published online: 1 January 2008


Aims This investigation sought to discern the relative accuracy of Doppler predictions of pressure drops in regurgitant jets across a broad spectrum of conditions, using an in vitro pulsatile flow model.

Methods and results We studied the accuracy of Doppler pressure gradients derived from regurgitant jet peak velocities using the simplified Bernoulli equation (SBE) using an in vitro flow model of atrio-ventricular valve regurgitation. We observed overall a good correlation (r = 0.89, P < 0.0001) with actual pressure gradient, when there is normal fluid viscosity and the jet is free of wall interaction. However, we observed various degrees of underestimation of pressure gradient by Doppler when regurgitant chamber size was reduced (P = 0.0003), when fluid viscosity was increased (P < 0.0001), or in the presence of wall interaction (P < 0.0001). Chamber compliance had no effect on the accuracy of pressure gradient prediction (P = 0.36). Significant underestimation error in pressure gradient prediction by Doppler of up to 43.2% was observed.

Conclusion When jet impingement or wall interaction are present, or when viscosity is increased, caution should be used in applying the SBE to a regurgitant jet, as significant underestimation in pressure gradient prediction may occur.

  • Doppler
  • Regurgitant jet
  • Pressure gradient


The prediction of ventricular pressure from the velocity of a regurgitant jet escaping from that chamber is an important aspect of the quantitative echocardiographic exam.1 This prediction is based on an assumed complete conversion from potential to kinetic energy. The mechanical energy balance equation says, in essence, that energy is never lost, only converted, and as applied to flow across a restrictive orifice, it states that potential energy is converted to kinetic energy. Potential energy is that harnessed by the high-pressure chamber, whereas kinetic energy is that within the high-velocity jet. If one assumes perfect conversion from potential to kinetic energy, one can measure the pressure drop across the restrictive orifice by using the simplified Bernoulli equation [SBE; ΔP = 4 × (Velocity of the regurgitant jet)2].2 This application assumes that there are negligent thermal or viscous energy losses. This simple mathematical relationship is widely accepted and applied in cardiovascular applications.

There are, however, many physical factors which impact flow conditions and result in imperfect transformation of potential to kinetic energy. In these settings, the accuracy of the SBE is adversely affected and can result in either an overestimation or underestimation of a pressure gradient, depending upon which physical factors are involved.3 Investigations into Doppler accuracy have been plentiful, yet have been applied exclusively to stenotic jets.410 In clinical practice, the regurgitant jet is often regarded as error-free. A belief in a fortuitous and perfect cancellation of errors is often cited as the reason for this belief.

However, the regurgitant jet often experiences conditions that impact the relation between SBE-predicted and actual pressure drop. For instance, in atrio-ventricular valve regurgitation, interaction with the atrial wall, alterations in atrial compliance, and viscosity deviations from normal should all have an impact on the accuracy of the SBE-predicted gradient by Doppler.

This investigation sought to discern the relative accuracy of Doppler predictions of pressure drops in regurgitant jets across a broad spectrum of conditions, using an in vitro pulsatile flow model. We hypothesized that conditions commonly experienced by regurgitant jets, such as regurgitant chamber size, wall interaction, viscosity, and altered atrial compliance, could result in an inaccurate prediction of pressure drop when the SBE was employed.



To address this hypothesis under controlled conditions, a pulsatile flow model of atrio-ventricular valve regurgitation was used. This in vitro model provided for varying fluid viscosity, regurgitant chamber size, interaction between the regurgitant jet and the lateral wall, and regurgitant chamber compliance. The pulse simulator consisted of the following components (Figure 1).

  1. An atrial reservoir consisting of a polymethyl methacrylate crystal-clear box. The material used is particularly suitable for in vitro studies given the high transparency to ultrasound waves. The inflow was located on a lateral wall, and the outflow was located on a wall adjacent to the inflow, at the same height. A bovine aortic valve was mounted in the outflow to create an atrio-ventricular valve. A 4.0 mm hole was punched into one of the valve leaflets to create a regurgitant orifice. A 6 F pressure port was present on the lateral wall opposite to and at the same level as the inflow.

  2. A ventricular pumping chamber consisting of a compressible bulb. Ventricular contraction was accomplished by compression of the bulb using an external air source. Solenoid valves under computer control opened and closed the inlet and exhaust ports of the ventricular chamber. The timing of the solenoid valves was computer-controlled. A 6 F pressure port was present on the ventricular side of the atrio-ventricular valve, at the same level as the atrial pressure port.

  3. An aortic flow chamber consisting of a valve and an arterial compliance section. Total aortic flow was measured by an external-clamp Doppler flow-meter (model T110, Transonic Systems Inc., NY, USA).

  4. A resistance section distal to the aortic compliance section, consisting of capillary tubing in parallel. By occluding different proportion of the tubes, the peripheral resistance could be varied in a reproducible fashion.

Figure 1

Fluid loop model. AV, atrio-ventricular; FP, flow probe.

Flow conditions

The following conditions were varied: atrial chamber size, atrial chamber compliance, presence of lateral wall jet impingement, peripheral resistance, and fluid viscosity. Four different atrial chamber sizes were studied (6, 4.5, 3, and 1.5 cm of length). Atrial chamber size was modified by direct immersion of a plastic bulkhead into the atrial chamber proximally to the regurgitant valve (Figure 2A). To evaluate the impact of wall interaction on regurgitant jet velocity, an appropriately shaped plastic bulkhead was placed in continuity with the valve ring (Figure 2B). Absence of wall interaction was evaluated by removal of the lateral bulkhead. Two different conditions of atrial compliance were considered: low atrial compliance was modelled by sealing the roof of the regurgitant chamber, high atrial compliance by leaving the roof open.

Figure 2

Modification of regurgitant chamber size and of wall interaction in the model. Atrial chamber size was modified by direct immersion of a plastic bulkhead into the atrial chamber 6, 4.5, 3, and 1.5 cm proximally to the regurgitant valve (A). To evaluate the impact of wall interaction on regurgitant jet velocity, an appropriately shaped plastic bulkhead was placed in continuity with the valve ring (B). AV, atrio-ventricular.

For each experimental condition, the pulse simulator was filled with a solution of 10, 30, and 50% glycerin by volume, to obtain a range of fluid viscosities from ∼1 to 5 cPs. For each condition, 0.5 g of cornstarch was added to the glycerin solution to improve Doppler velocity profile. Three different and interchangeable sections representing resistance levels of 2, 10, and 20 mmHg/L/min were fashioned to simulate after-load ranging from pulmonary to systemic arterial conditions. Filling pressure of the atrial chamber was between 5 and 7 mmHg for all conditions. A total of 144 different conditions were studied. For each experimental flow condition, three samples were analysed and the average of these was used in subsequent analysis.

The pulse rate was 70 bpm, and the flow rate was maintained between 3 and 5 L/min for all conditions studied.

Data acquisition

The solenoid valve and pressure transducers were interfaced to a Macintosh PC running customized software (LabView, National Instruments, Austin, TX, USA) through a 16 bit analogue-to-digital converter, which allowed for simultaneous control of ventricular ejection as well as recordings of atrial and ventricular pressures.

In the atrial and ventricular chambers, instantaneous pressures were recorded with fluid-filled catheters connected to disposable pressure transducers (Merit Medical Systems Inc., UT, USA), once a steady state was achieved. The actual peak atrio-ventricular pressure gradient was measured from these tracings.

For each flow condition, continuous wave Doppler recordings of regurgitant jet flow velocity were obtained with the use of an ultrasound machine (Acuson Sequoia, Siemens Medical Solutions, USA), equipped with a 3.0 MHz transducer. The continuous wave Doppler beam was aligned parallel to flow to avoid error in velocity measurements. Continuous wave Doppler was preferred to pulsed wave Doppler for the high flow velocities to be measured, and because of the independence of peak velocity measurement from the position of the sample volume along the pathway of the regurgitant jet. The spectral Doppler image was digitally transferred to a personal computer running Acuson KinetDx DS3000 software (Siemens Medical Solutions), for analysis and storage purposes. With the use of KineticDx software, the predicted peak gradient of the Doppler signal was calculated using the SBE.

The difference between the actual and the Doppler predicted peak atrio-ventricular pressure gradient was calculated for each condition. The percentage of error of over- or underestimation of Doppler-estimated pressure gradient was calculated as: error% = [(SBE − predicted pressure gradient/actual pressure gradient)−1] × 100.

Data analysis

The per cent error in Doppler prediction of the actual pressure gradient was calculated. The logarithm of the per cent error (LPE) data was calculated to transform the data to a normal distribution. The LPE therefore is the outcome variable.

The impact of each of the four variables (wall interaction, chamber compliance, fluid viscosity, and chamber size) on LPE was assessed. The Mann–Whitney test was used to assess the impact of each dichotomous variable (wall interaction and chamber compliance) on LPE; the variables for each were coded as 0 (no wall interaction, low compliance) or 1 (wall interaction, high compliance). Viscosity conditions of 1, 3, and 5 cPs were coded as 1, 3, and 5; the four chamber sizes were coded as 1.5, 3, 4.5, and 6, and each treated as continuous variables. The Kruskal–Wallis test with Dunn's multiple comparison post-test was used to analyse the impact of these continuous variables on LPE.

The impact of the four variables on LPE was further evaluated by multiple linear regression. The variables were coded as described above. The regression analysis calculated a P-value, an r value, and coefficient for each independent variable. To assess the impact of each condition tested on accuracy of Doppler pressure gradient prediction, the fold effect was calculated for each independent variable as: exp (coefficient).

Percentage effect of each independent variable was calculated as=100 × [exp (coefficient)−1]. The interaction between two factors was calculated as net effect, using the equation=100 × [exp (coefficienta×coefficientb)−1]. To analyse the possible interaction between different parameters, multiple linear regression analysis was repeated using a model considering multiplicative effects. A P-value of <0.05 was considered statistically significant.


Actual pressure gradients measured across the atrio-ventricular valve ranged from 70 to 182 mmHg for different flow conditions. Pressure drops predicted by the SBE ranged from 40 to 204 mmHg. As expected, there was a significant correlation between actual and Doppler-predicted pressure gradients (r = 0.89, P < 0.0001; Figure 3).

Figure 3

Correlation between actual and Doppler-predicted pressure gradient across the regurgitant valve. PG, pressure gradient.

We observed a progressive underestimation of pressure gradient by Doppler when regurgitant chamber size was reduced from 6 to 1.5 cm (P = 0.0003; Figure 4), when fluid viscosity was increased (P < 0.0001; Figure 5), or in the presence of wall interaction (P < 0.0001; Figure 6). No significant difference in LPE was observed when chamber compliance was reduced (P = 0.36; Figure 7). The progressive Doppler underestimation of pressure gradient following decrease in regurgitant chamber size was particularly evident when physiologic viscosity (3 cPs) was considered (P = 0.0048; Figure 8).

Figure 4

Effect of chamber size on log error % in the overall study conditions. Values were compared with the Kruskal–Wallis test with Dunn's multiple comparison post-test.

Figure 5

Effect of increasing fluid viscosity on log error %. Values were compared with the Kruskal–Wallis test with Dunn's multiple comparison post-test. cPs, centipoises.

Figure 6

Effect of wall interaction on log error %. Values were compared with Mann–Whitney test.

Figure 7

Effect of atrial compliance on log error % in the overall study conditions. Values were compared with Mann–Whitney test.

Figure 8

Effect of chamber size on log error % in the 3 cPs simulation only. Values were compared with the Kruskal–Wallis test with Dunn's multiple comparison post-test. cPs, centipoises.

Multiple linear regression also showed a significant negative association between fluid viscosity and LPE (P < 0.0001; Table 1). Wall interaction resulted in underestimation of the pressure gradient by Doppler (P < 0.0001), whereas atrial compliance (P = 0.26), and regurgitant chamber size (P = 0.29) did not appear to be relevant.

View this table:
Table 1

Results of multiple linear regression between log % error and different flow conditions

Independently, chamber size had no effect; however, when its effect was considered with fluid viscosity or with wall interaction, atrial size seems to have an impact on the Doppler pressure gradient estimation. Indeed, the interaction between wall impingement and chamber size was significant (r = −0.46, P < 0.0001), as was the combination of fluid viscosity and chamber size (r = −0.26, P = 0.003), both of which appeared to be significantly associated with an underestimation effect.

No effect of atrial compliance was noted when its interaction with fluid viscosity or with wall interaction was considered (r = 0.26, P = 0.23; and r = −0.11, P = 0.3, respectively).

Wall interaction showed a fold effect on LPE of 0.76, and a percentage effect equal to –24.3% (Table 1). Fluid viscosity showed a fold effect on LPE of 0.57, and a percentage effect of –43.2%, meaning that as viscosity increases, so does Doppler underestimation of the pressure gradient.


Our study demonstrates that flow conditions that could be encountered in clinical practice impact the accuracy of Doppler-derived estimation of pressure gradients in regurgitant jets. In an in vitro model of atrio-ventricular valve regurgitation, significant pressure gradient underestimation occurred when fluid viscosity was high and when wall interaction with the jet was present.

Much is known about the factors that impact the accuracy of Doppler estimation of gradients in stenotic jets. These include: the relative importance of terms neglected in simplifying the Bernoulli equation, pressure recovery effects,7,11 fluid viscosity,3 jet eccentricity,12 and others.13 Although there are differences between stenotic and regurgitant jets, many fluid dynamics principles apply to both kinds of jets. Yet, in clinical practice, the Doppler prediction of the peak gradient of a regurgitant jet has been widely assumed to be accurate, possibly due to an assumed cancellation of error. However, this accuracy has not been empirically tested under a variety of flow conditions.

As expected, in our model, we observed an overall correlation between actual and Doppler-predicted pressure gradients (P < 0.0001), with a coefficient of 0.95, suggesting the presence of an overall underestimation of ∼5%. This finding is in accordance with data from clinical studies.14 However, a favourable correlation does not translate into accurate pressure gradient prediction for various conditions seen in clinical practice.

Effect of wall interaction and atrial size

Our data demonstrate that when there is an interaction between the regurgitant jet and the atrial wall, Doppler-predicted pressure gradient seems to underestimate the actual pressure gradient, especially when the size of the atrial chamber is small. Doppler-catheter discrepancies under these flow conditions appear to particularly evident in the case of increased fluid viscosity. Similar haemodynamic conditions are commonly encountered in the echocardiographic examination of children with congenital heart disease who typically have small cardiac structures and who may have eccentric regurgitant orifices and increased haematocrit.

Limitations relative to the Bernoulli principle itself can explain the underestimation of pressure drop by Doppler observed for the conditions characterized by small atrial chamber size and presence of wall interaction, especially if fluid viscosity is increased. In clinical practice, regurgitant jets from atrio-ventricular valves are usually considered as free jets. A free jet is defined as a jet issuing into a relatively stagnant environment where the cross-sectional area of the jet is less than one-fifth of the cross-sectional area of the region of chamber into which it is flowing, and it develops free from influence of external or chamber boundaries (i.e. no wall effects).13,15 From engineering studies, we know that as a free jet leaves a nozzle into a receiving chamber, a turbulent shear layer develops between the receiving chamber fluid and the inflow jet stream boundary.1618 The shear layer will eventually consume the core of the jet, from which point the jet becomes fully developed or freely turbulent.16,18 All sides of the jet are equally affected, and although the width of the jet experiences instantaneous changes, the jet expands symmetrically and mean jet width will also grow over time in a balanced fashion, resulting in a symmetrical jet flow.13,18 It is also known that as the jet intrudes into the receiving chamber, vortex motion entrains surrounding fluid.18 These vortices envelope and drag in pockets of stagnant fluid, increasing jet mass, and decreasing flow velocity, and jet kinetic energy transforms into jet stream expansion.1820 Under these flow conditions, viscous losses related to boundary layers are minimal and pressure drop across the regurgitant valve can be described by the reduced form of the Navier–Stokes equations, known as SBE. However, under experimental conditions, as in clinical practice, a regurgitant jet might experience significant lateral or distal interaction with chamber walls. When a surface is placed beside the nozzle, stagnant flow entrainment through the jet′s large-scale vortex structures is inhibited on the surface side of the jet, creating an asymmetrical shear layer between the jet stream and the surface.1821 Viscous forces acting from the surface in the direction opposite to the jet flow retard the flow adjacent to the surface.18,22 Because mass entrainment is retarded on the surface side, and because flow momentum must be conserved throughout the jet, flow velocities on the surface side of the jet will be higher than the velocities on the free side.18 However, since flow velocity at the surface must be zero because of viscosity and the no-slip condition, spatial transverse velocity gradients will be significant, leading to high shearing forces and increased viscous effects that ‘pull’ the jet flow towards the surface.18,23 Under these conditions, viscous losses not accounted for by Bernoulli equation will lead to underestimation of pressure drop by Doppler ultrasound, especially if fluid viscosity is increased. Such jet–wall interactions have been extensively studied in different settings and represent the fluido-dynamic bases of the Coanda effect.18 In the present experiment, when atrial chamber size was increased or wall interaction was removed, the pressure gradient predicted by the SBE appeared to estimate well the actual PG. Indeed, under these conditions, no interaction between regurgitant jet and boundary layers can develop, viscous losses are negligible, and the SBE is applicable.

Effect of fluid viscosity

The range of fluid viscosities studied in the present experiment ranged from 1 cPs (simulating anaemia) to 5 cPs (simulating polycythaemia). An intermediate fluid viscosity of 3 cPs was used to simulate the normal blood viscosity. In the present study, increased fluid viscosity was associated with a Doppler underestimation of actual pressure gradient, whereas reduction of viscosity to 1 cPs was associated with an overestimation error by Doppler. Fluid viscosity of 3 cPs was associated with the most accurate Doppler prediction of actual pressure drop. The SBE is derived from the more general Navier–Stokes equations, assuming viscous/turbulent losses to be negligible. Therefore, when viscous losses are not negligible, because of increased fluid viscosity or jet interaction with boundary layers, SBE-predicted pressure drop is expected to underestimate actual pressure gradient.13 However, the Bernoulli equation, even in its extensive form, does not explain the occurrence of overestimation by Doppler. Therefore, to account for the Doppler overestimation of actual pressure gradient observed during low fluid viscosity conditions, additional fluid-dynamic phenomena must be considered, in particular a pressure recovery effect. Pressure recovery has been extensively studied in aortic valve stenosis to explain the common occurrence of Doppler overestimation of actual pressure drop.3,58

Both stenotic and regurgitant jets are characterized by a laminar core just distal to the orifice from which the jet emanates. In a stenotic jet, the region where the laminar core is at its smallest diameter is referred to as the vena contracta, which is also the location of highest velocity, and the site of the Doppler detection of peak jet velocity. After the vena contracta, the jet expands into the receiving chamber, some of the kinetic energy of the jet is dissipated, and some is returned to potential energy (pressure). Thus, the overall pressure drop becomes less than the one predicted by Doppler, since some pressure was ‘recovered’ distal to the vena contracta site. The magnitude of pressure recovery in stenotic jets is impacted by the anatomy of the orifice and the receiving chamber. There is less pressure recovery (and therefore more accurate Doppler prediction of gradients), when there is a large receiving chamber, and/or a small stenotic orifice.10 A regurgitant jet is usually defined in fluid dynamics as a ‘free jet’ with a regurgitant orifice cross-sectional area <20% of the cross-sectional area of the receiving chamber.13 Thus, one expects less pressure recovery (more accuracy) of Doppler-predicted pressure gradient in regurgitant jets when compared with stenotic jets. However, when regurgitant chamber size is reduced, free jet conditions that apply to the regurgitant jet may be lost, and the jet becomes a confined jet.

As previously reported in stenotic jets,3 an approach based on the Reynolds number may be able to reconciliate discrepancies due to simplification of the Bernoulli equation and those related to pressure recovery. The Reynolds number is a dimensionless quantity representing the ratio of inertial to viscous forces.3 The Reynolds number embodies viscous forces in its denominator, so that when the Reynolds number is low (viscosity is high), viscous forces are important by definition. Therefore, at low Reynolds numbers, the viscous term, which is deleted from the SBE, would be an important cause of underestimation. With increasing Reynolds numbers, viscous forces would be less important whereas inertial forces would increase, and pressure recovery effects would be relatively predominant, resulting in overestimation. Intermediate Reynolds numbers (intermediate viscosity) would lead to an accurate Doppler estimate of pressure drop by reciprocal cancellation of viscous and pressure recovery effects.3

These general considerations seem to apply also to regurgitant jets under particular haemodynamic conditions, like the one observed for small atrial chamber size and wall interaction. Under these circumstances, the regurgitant jet becomes ‘confined’. For a ‘confined’ jet, Doppler-catheter discrepancies in the estimation of pressure drops are regulated by Reynolds number and can be represented by the following equation:

Embedded Image

As described above, for high fluid viscosity (5 cPs), inertial forces responsible for pressure recovery are negligible, whereas viscous losses become prevalent and lead to Doppler underestimation of actual pressure gradient. For low fluid viscosity conditions (1 cPs), viscous losses become negligible, inertial forces increase, leading to the occurrence of a pressure recovery effect and overestimation of actual pressure drop by Doppler. For intermediate fluid viscosity (3 cPs), reciprocal cancellation of viscous and pressure recovery effects would lead to an accurate prediction of actual pressure drop by the SBE.

Clinical implications

Doppler ultrasound has become a widely used method for non-invasive estimation of pressure gradient in regurgitant atrio-ventricular valves, especially for the estimation of right ventricular systolic pressures trough Doppler interrogation of tricuspid regurgitant jets. The results of the present study suggest that particular attention has to be paid in the estimation of right ventricular systolic pressure in neonates and infants. Indeed, all the flow conditions found to alter the accuracy of SBE in the estimation of actual pressure drops, like abnormal fluid viscosity related to polycythaemia and small regurgitant chambers (atria) with great potential for wall interaction, can be encountered in the paediatric age group. The SBE underestimation related to these flow conditions might be even more significant in paediatric patients with right ventricular outflow tract obstruction or those with pulmonary hypertension. Indeed, the observed 43.2% underestimation of the actual pressure drop by Doppler would magnify the absolute underestimation (in mmHg) in those patients who have high right ventricular systolic pressure.

Study limitations

In vitro flow modelling is well suited to this type of investigation due to the ability of the investigator to control and vary flow conditions in order to determine the effects of various parameters, such as chamber compliance, regurgitant chamber size, and wall interaction in the present study. However, studies employing in vitro flow modelling are only as applicable as the modelling is physiologic. In this investigation, the model analysed extreme conditions regarding atrial compliance (very low compliance and high compliance), and wall interaction (wall interaction absent or present). Regurgitant chamber size and fluid viscosity could, however, be modulated across a range of physiologic values. Intermediate values of atrial compliance and of wall interaction may be present in vivo; therefore, any potential effect observed between different patients is directly related to their individual differences.

In the present in vitro model of atrio-ventricular valve regurgitation, a single circular regurgitant orifice of 4 mm in diameter was studied. In vivo characteristics of the regurgitant orifice in terms of size, shape, and number may vary. Previous computational models have shown that the haemodynamic behaviour of a double orifice regurgitant valve does not differ from that of a valve with a single orifice of same total area and that pressure drops are not influenced by the configuration of the valve.24 However, for the same flow conditions, different sizes of the regurgitant orifice might have produced slightly different results. Indeed, a larger size of the regurgitant orifice would directly increase the Reynolds number, leading to an increase of inertial forces and enhancement of pressure recovery effects.13

Conflict of interest: none declared.


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