Impact of assuming a circular orifice on flow error through elliptical regurgitant orifices: computational fluid dynamics and in vitro analysis of proximal flow convergence

Grounded in hydrodynamic theory, proximal isovelocity surface area (PISA) is a simplistic and practical technique widely used to quantify valvular regurgitation flow. PISA provides a relatively reasonable, though slightly underestimated flow rate for circular orifices. However, for elliptical orifices frequently seen in functional mitral regurgitation, PISA underestimates the flow rate. Based on data obtained with computational fluid dynamics (CFD) and in vitro experiments using systematically varied orifice parameters, we hypothesized that flow rate underestimation for elliptical orifices by PISA is predictable and within a clinically acceptable range. We performed 45 CFD simulations with varying orifice areas 0.1, 0.3 and 0.5 cm2, orifice aspect ratios 1:1, 2:1, 3:1, 5:1, and 10:1, and peak velocities (Vmax) 400, 500 and 600 cm/s. The ratio of computed effective regurgitant orifice area to true effective area (EROAC/EROA) against the ratio of aliasing velocity to peak velocity (VA/Vmax) was analyzed for orifice shape impact. Validation was conducted with in vitro imaging in round and 3:1 elliptical orifices. Plotting EROAC/EROA against VA/Vmax revealed marginal flow underestimation with 2:1 and 3:1 elliptical axis ratios against a circular orifice (< 10% for 8% VA/Vmax), rising to ≤ 35% for 10:1 ratio. In vitro modeling confirmed CFD findings; there was a 8.3% elliptical EROA underestimation compared to the circular orifice estimate. PISA quantification for regurgitant flow through elliptical orifices produces predictable, but generally small, underestimation deemed clinically acceptable for most regurgitant orifices.


Introduction
Hydrodynamic theory predicts that inviscid f luid approaches a point orifice on a planar surface with a spherically symmetric, inward radial profile. By conservation of mass, flow accelerates as it approaches the orifice with a velocity that is inversely proportional to the squared distance from the orifice, forming isovelocity contours that are hemispherical shells. The proximal isovelocity surface area (PISA) method, first described by Recusani and colleagues in 1991, is based on this simple assumed flow profile and is a guideline-recommended method for assessing valve regurgitation [1][2][3][4]. Previous work, however, has demonstrated that the PISA method is subject to a number of technical and operator errors. Even under idealized conditions, it slightly underestimates the flow rate through finite, non-point orifices, but this error is accepted in general practice [5,6].
In the case of mitral regurgitation (MR), accurate quantitation is paramount given epidemiologic analysis showing worsened cardiovascular morbidity and mortality as effective regurgitant orifice area (EROA) increases in both organic and functional MR [7,8]. In primary MR the shape of the regurgitant orifice is usually close to circular due to focal leaflet degeneration, whereas in secondary MR the shape tends to be more elongated and elliptical with leakage along the closure line due to tethering of the valve apparatus [2,9,10]. Three-dimensional (3D) reconstructions of the flow convergence region reveal isovelocity contours that resemble a hemiellipsoid rather than a hemisphere for functional MR and even for some cases of organic MR [11]. As a consequence, underestimation of regurgitant flow may occur when the standard hemispheric two-dimensional (2D) PISA method is used, especially when the ratio of the ellipse major and minor radii are greater than 1.5 [9,10,12,13]. These studies bring into question the validity of the hemispheric PISA method to assess functional MR, particularly in the setting of especially elongated orifices [2,14].
Numerous efforts have been made to show the superiority of using hemicylindrical and hemiellipsoidal PISA methods for elongated orifices, yet they require multiple orthogonal measurements in long and short axes that are time-consuming, technically challenging, and subject to their own measurement errors [15][16][17][18][19][20][21][22][23]. Three-dimensional (3D) PISA using 3D echocardiography (echo) has shown promising results but is still limited by a need for considerable operator manipulation to acquire images, poor spatial resolution, and cumbersome image processing using proprietary software [19,22,23]. For these reasons, the standard 2D PISA method is still widely used in the clinical workflow of a busy echo lab or for instantaneous assessments during percutaneous interventions. Knowing the degree of underestimation by the standard PISA method for elliptical orifices would thus be useful for clinicians to acquire a quantitative sense of how much underestimation is involved in their assessment. To that end, prior in vitro and computational modeling of flow convergence in round finite orifices has shown that far from the orifice (where the aliasing contour is typically measured), the flow field closely approximates that of a point-like orifice, with significant deviation seen only in the immediate vicinity of the orifice [6]. We hypothesized that a similar phenomenon might reduce the anticipated underestimation of elliptical orifices, bolstering confidence in the hemispheric PISA method for functional MR.
Computational fluid dynamics (CFD) is a central tool in fluid mechanics that employs numerical methods to compute complex fluid flow in a wide variety of geometries and boundary conditions. The objective of this study was to apply CFD modeling to precisely simulate flow field upstream from the orifice and parametrically vary orifice sizes and shapes to test the hypothesis that the compounded effect of underestimation of flow for finite, elliptic orifices is relatively small compared to circular orifices when using the standard PISA. Focused in vitro analysis of flow through circular and elliptical orifices was used to confirm the CFD findings. In short, this study was undertaken to define, under optimal conditions, what impact an elliptical orifice has on PISA quantification in hopes of guiding use in general practice.

Computational fluid dynamics
The three-dimensional flow field near a finite elliptical outflow orifice was computed using the open-source incompressible CFD solver Nalu. It uses a control-volume finite element method to simulate flow field by solving the continuity and Navier-Stokes equations [24]. The flow domain was modeled as a cylinder with a height of 10 cm and a radius of 10 cm, large enough so that inflow boundary conditions have a negligible impact on flow near the orifice. An elliptical orifice with a prescribed major-to-minor diameter ratio and a prescribed area, A O , is located at the center of the base of the cylinder, representing the regurgitant valve. Note that throughout this work, a distinction is made between the effective regurgitant orifice area (EROA), which is the ratio of the flow rate through the orifice compared to the maximum velocity, and the actual geometrical area of the orifice A O . The geometry and computational mesh for the domain were created using the commercial pre-processing software Cubit (formerly Trelis, v15.1, Coreform, Utah, USA). For each geometry, a mesh of quadrilateral elements was generated that was fine enough to resolve both the complex flow field near the orifice and the boundary layer at the wall. Total mesh size varied with orifice geometry, but all meshes contained between 110,000 and 190,000 elements while over one million elements were tested to explore mesh discretization errors (see Supplement 2.1 for full details). A typical geometry and mesh are shown in Figs. 1 and 2A. Figure 2B shows the resulting vertical velocity component of an idealized flow with the element boundaries hidden. For boundary conditions, low velocity (< 0.5 cm/s on average for the highest flow simulation) was prescribed at the sides and top of the domain to generate the proper flow rate through the orifice (see Supplement for full details). A zero relative pressure condition was prescribed at the orifice outflow with along zero flow velocity at the base of the cylinder (i.e., noslip condition) representing the valve leaflets.
Blood was modeled as a Newtonian fluid with density 1.05 g/cm 3 and viscosity 3.0 centipoise, whereas it is known to be a shear-thinning non-Newtonian fluid. However, the impact of viscosity itself on the flow formation is negligible as shown in a previous CFD study by Rodriguez et al. [6] They showed that even with a viscosity 100-fold greater than the physiological value of blood only produces a little change (12.5%) in velocity distribution in the orifice upstream. It is only in a thin boundary layer near the wall that viscosity plays a dominant role but, even in this region, the blood can be approximated as a Newtonian fluid because of high shear rates in the boundary layer near the orifice, which further reduces the impact of viscosity. Flow was assumed to be laminar, which is generally observed in converging flow fields because of the low shear-rate, accelerating flow. Because our focus was solely upstream of the valve, we did not model the downstream jet, which would have required a turbulence model to capture the effects of sub-grid scale velocity fluctuations.
We simulated 3 areas A O (0.1, 0.3, and 0.5 cm 2 , representing mild, moderate and severe MR as specified in the guidelines [4]), 3 maximal velocities (V max = 400, 500 and 600 cm/s), corresponding to a reasonable clinical range of LV to LA pressure differences of 64 mmHg, 100 mmHg, and 144 mmHg, respectively), and 5 different ellipse axis ratios (1:1, 2:1, 3:1, 5:1 and 10:1, spanning the range of observed orifice shapes [9]). All combinations of these parameters were numerically evaluated, resulting in 45 separate simulations. For each, the steady state flow field was computed instead of pulsatile flow as the Womersley number calculated using the average orifice diameter of 0.8 cm, heart rate of 60 bpm and kinematic viscosity of 3 cP was calculated as 0.06, much lower than unity, indicating that the flow field at any given instant can be considered to be in a steady state. Imposing pulsatile flow condition produced < 0.6% maximal difference in velocity value compared to the steady state simulation, as shown in Supplementary Figure S3. On the centerline normal to the orifice, the axial velocity component V A was sampled and used to calculate the flow rate, Qc, that would be predicted by the PISA method, (Qc = 2πz 2 V A where z is the axial distance from the orifice plane and V A is the velocity at the point, equivalent to the aliasing velocity of the PISA method).

In vitro model
In vitro modeling was done to compare PISA characteristics from the CFD to echo. We constructed a chamber out of Plexiglas as seen in Fig. 3, similar to one previously described [25]. The chamber was divided into a left ventricular side (8 cm × 8 cm × 57 cm) and a left atrial side (8 cm × 6 cm × 57 cm). On the bottom of the chamber is an insert for different orifices. For this interrogation, we analyzed 0.5 cm 2 orifices, one was circular and the other was elliptical with a 3:1 aspect ratio. Water with cornstarch (to provide ultrasound contrast) at room temperature was used as a working fluid, which was pumped from the left atrium side to the left ventricle side to generate a pressure head (h in Fig. 3) and then abruptly released. This created a flow through the orifice driven by gravity with a predictable linear deceleration in velocity (and flow rate, accordingly) until the pressure gradient equilibrated [26].

Echo analysis
Images were obtained with a commercially available GE Vivid E95 echo machine (GE Medical Systems, Milwaukee, WI). Color Doppler and continuous wave Doppler imaging was performed with a 2.2 MHz transducer. Two different water heights were used (h = 187 and 275 mm) to achieve various initial pressure gradients for the round and elliptical orifices. The transducer was rotated 90° to access long and short axis views of the elliptical orifice. Images were analyzed in EchoPAC (GE Medical Systems, Milwaukee, WI). Maximal velocities of flow through the orifice measured by using continuous wave Doppler spectrum as shown in Fig. 4. After baseline velocity shifting to set V A to 16 cm/s, the PISA radius, r, which is the distance from the vena contracta to the aliasing contour, was measured as shown in Fig. 5. EROA was calculated by the PISA method as described above. All measurements were within 1-3 frames after first appearance of the PISA hemisphere to assure that the flow velocity was maximum.   Figure 6 A shows predicted isovelocity contour shells converging on a finite elliptical orifice with a 5:1 long to short axis diameter ratio. In general, the velocity falls approximately as the inverse square of the distance from the orifice, approaching zero in the periphery. These contour shells appear hemi-elliptic when close to the finite orifice but assume a more hemispheric shape away from the orifice. The velocity is higher in the vicinity of the orifice but decreases substantially moving away from the orifice and approaches zero at the periphery. Figure 6 also displays the predicted isovelocity contours expressed as a percentage of maximum velocity, for finite circular (Panel B) and elliptical (Panel C) orifices with cross sectional and en-face views of each orifice type. Noteworthy in the en-face views is that the isovelocity contours appear hemispheric far from the orifice for both orifice shapes but flatten into hemispheroids and hemi-ellipsoids respectively for the circular and elliptical orifices close to the orifice. Figure 7 shows the hemispherical flow rates as would be computed with PISA scaled by true orifice flow rate. This value is also equivalent to the effective regurgitant orifice area scaled by the true value, also represented in Fig. 6. Curves are shown for the circular and 5:1 elliptical orifices for all values of orifice area and flow rate. Computed flow rate is plotted against distance from the orifice scaled by the orifice length scale, taken as the diameter of a circular orifice with area equal to A O . The data from the nine simulations for each orifice shape (3 areas × 3 driving pressures) collapse onto a single curve, as would be expected for steady, incompressible flow, allowing us to focus our analysis on the impact of orifice shape alone. Note that application of PISA at 2 orifice diameters distant yields almost perfect quantitation for a circular orifice (~ 98%) but misses 5% more for the ellipse at this distance.

Results
With the percent underestimation of flow shown to be insensitive to true orifice area and flow velocity, percent underestimation was next calculated for various ellipses by altering the axis ratios. Figure 8 shows the percent underestimation of flow rate relative to a circular orifice for various orifice aspect ratios, i.e., Q C /Q circ , where Q circ is the flow rate that would be computed using PISA for a circular hole. This relative underestimation is plotted against the ratio of measured contour velocity to the velocity at the orifice, a ratio which is approximately equal to the flow underestimation for the circular hole [6]. For example, assuming an aliasing contour of V A = 40 cm/s with a V max = 5 m/s jet at the orifice (V A /V max = 8%), an approximate 8% flow error is expected for a circular orifice [6]. According to Fig. 8, elliptical orifice aspect ratios of 2 and 3, which are in the physiological range, exhibit an additional underestimation of this flow by approximately 3% and 8% respectively. Greater relative underestimation was seen for higher elliptical orifice aspect ratios.

In vitro model results
The "true" effective regurgitant orifice area EROA was calculated based on the continuity equation (q = Av) and simplified Bernoulli's equation (in metric units or 4v 2 for pressure in mmHg and velocity in m/sec). The true EROA was 0.462 and 0.435 cm 2 for water height of 187 and 275 mm, respectively. The maximum velocities showed excellent agreement with theoretical peak velocity as shown in Table 1 validating continuous wave Doppler acquisition. Table 2 shows true EROA and measured EROA for the circular and the elliptical orifice. EROA of with a water height of 275 mm was always smaller compared to a water height of 187 mm, perhaps reflecting some variation in the coefficient of contraction with V max . The main finding of the in vitro experiments was that there was indeed systematic underestimation of the EROA of the elliptical orifice relative to that of the circular orifice, but the difference was less than 10%, regardless of whether the PISA radius was measured in short or long axis, averaging 8.3% (Fig. 5; Table 2). These findings support the overall CFD results, showing small, predictable reduction in EROA for elliptical orifices vs. circular ones.

Discussion
The 2020 guideline on native valve regurgitation from the American Society of Echocardiography details a number of suggested methods for assessing and quantifying MR [4]. Measuring the area of the regurgitant jet is time-honored but fraught with limitations due to instrumentation factors, jet morphology (central vs. wall-hugging), and sensitivity to blood pressure. Volumetric methods using conservation of mass are theoretically sound, subtracting two stroke volumes from each other, e.g., one through the mitral valve (such as the difference between LV end-diastolic and endsystolic volumes), the other through an unaffected valve (such as the left ventricular outflow tract). Unfortunately, they require multiple measurements with intrinsic errors that propagate and ultimately compound by subtracting one large number from another. For example, Aurich et al., using 3D analysis tools report consistent large underestimation of LV volumes vs. cardiac magnetic resonance imaging (end diastolic volume, 3D echo − CMR measurements: −62 ± 54 mL; ESV, −20 ± 49 mL). Since errors add as the root sum square, the predicted LV stroke volume error would be −42 ± 73 mL [27]. Similarly, in a highly controlled analysis of LVOT stroke volume using direct 3D flow quantitation (in Fig. 8 Percent underestimation of flow rate relative to a circular orifice for elliptical orifices (axis ratios of 1, 2, 3, 5 and 10) against the aliasing velocity (V A ) normalized the maximal velocity (V max ). The hashed line at a flow error of 0.08 indicates a 3 to 8% underestimation for elliptical orifices with axis ratios of 2 and 3, respectively, relative to a circular orifice. D maj and D min refer to the major and minor axis diameters of an elliptical orifice, respectively  comparison to CMR), the limits of agreement were 0.7 ± 18 mL; the usual 2D LVOT estimation had an error of −11 ± 40 mL [28]. Subtracting LVOT SV from LV SV (and adjusting the error) shows this difference, the estimated regurgitant volume, to be −43 ± 75 mL for the 3D LVOT estimate and −31 ± 83 mL for 2D LVOT SV. Even in this best-case scenario, these error ranges are too wide to be useful. The PISA method offers potential advantages over these other approaches. Based on fluid dynamic principles (the continuity equation) it offers a truly quantitative approach to instantaneous flow rate and regurgitant orifice area. It also has the advantage of being a simpler method than volumetric approaches with multiple input parameters, and requires only a single measurement (distance to first aliasing contour) to estimate regurgitant flow and an additional one (peak continuous wave Doppler velocity of the jet) to estimate EROA. Moreover, simpler approaches have been validated requiring mere seconds to implement, making it well suited for the clinical workflow of a busy echo lab or for instantaneous assessments in the midst of percutaneous interventions [29]. However, there are a number of limitations with PISA, even in the setting of circular orifices, such as flattening of isovelocity contours as they approach the orifice due to finite orifice size, leading to progressive underestimation [3], and distortions in the hemispheric flow convergence due to surrounding cardiac structure or a flail leaflet [30][31][32]. While errors introduced by these limitations are predictable and may be corrected, concerns have been raised that application of the standard hemispheric 2D PISA formula to assess functional MR, that are often elliptical in orifice shape as confirmed by us [9] and others [10,12,13], will result in an unacceptable underestimation of true regurgitant flow, which prompted this study.
Using CFD, we demonstrated that there is relatively little impact on calculated flow rate and regurgitant orifice area when the standard PISA approach was applied to elliptical orifices in comparison to similarly sized circular orifices. Indeed, we observed only a small 3 to 8% underestimation of flow rate relative to a circular orifice for ellipse axis ratios between 2 and 3 under typical conditions (V A /V max ~8%), a difference that is unlikely to alter grading of MR and lead to missed clinical events. While isovelocity contours are elliptical in shape in the immediate vicinity of the orifice, as Fig. 6 C shows, they rapidly become axisymmetric as one goes away from the orifice. For a graphic illustration of this, recall that the panels 8B and C are orthogonal cuts through the long and short axes of a 3:1 ellipse, but the displayed contours differ by only about 1.5:1 as the contours rapidly become hemispheric remote from the valve.
Choosing the optimal V A for the PISA method involves tradeoffs. An aliasing contour of a very low V A will show little impact from the orifice shape but may instead be adversely impacted by the surrounding geometry and flow out the LVOT. Conversely, using too high V A eliminates the concerns about flow constraint but will underestimate the flow due to hemiellipsoidal contour shape. In most cases, we have found that a V A that is around 8% of V max is a reasonable compromise. For the typical 5 m/sec V max jet (corresponding to 100 mmHg LV-LA pressure difference), this translates to an aliasing contour at V A = 40 cm/s and has the added advantage of allowing use of the simplified PISA formula, whereby EROA is given simply by (PISA radius) 2 /2, an approach that allows MR quantitation to be extended to busy clinical and interventional labs with very little additional time requirements [29]. As shown in Fig. 8, applying the standard PISA formula with V A = 8% V max , leads to flow underestimation (relative to a circular orifice) of 3%, 8%, and 17% for aspect ratios of 2:1, 3:1, and 5:1, respectively. This is in addition to the approximately 8% underestimation caused by the finite circular orifice itself, which is generally neglected but could easily be corrected. These observations thus are at the lower range of flow underestimation reported in the literature, which have ranged from 12 to 50% [13,33,34]. The results of this study should encourage the clinical use of the hemispheric PISA formula for functional MR, while acknowledging a certain amount of predictable underestimation. In a clinical setting, the MR orifice shape can be approximated from vena contracta width in the apical two-chamber and long-axis views (or 3D imaging). If this ratio is less than 3:1, then the simple PISA approach can likely be used. Beyond 3:1 ratio, some correction based on these results may be reasonable (e.g., 17% for 5:1 orifice), but these are less commonly encountered in clinical practice.
The main strengths of this study are the use of CFD to systematically investigate the effects of a variety of input conditions and elliptical geometry configurations (the ratio of the ellipse major and minor radii) in a clinically applicable scenario. There are prior studies that have utilized more advanced CFD techniques to simulate 3D MR flow fields with more realistic regurgitant orifice geometry, but are limited in practicality. For example, Qin et al., have created a numerical patient-specific left heart model based on computerized tomography images and adjusted papillary muscles to generate five different functional MR cases which had elongated orifices [23]. They simulated regurgitant flow field using fluid-structure interaction modeling and found that hemicylindrical and hemiellipsoidal PISA improve MR flow rate prediction compared to the hemispheric approach, while 3D PISA outperformed all 2D methods. Jamil et al., developed an ultrasound-based CFD technique that simulates 3D flow field in the upstream of patient-specific MR orifice models obtained from 3D TEE B-mode [21]. They have demonstrated that 3D PISA provides more accurate flow rate than 2D PISA while both approaches are sensitive to V A which is in alignment with our findings using a simple CFD analysis. While these advanced CFD techniques are 1 3 certainly useful to compare the performance of a variety of PISA methods in realistic MR anatomy, the shape of the MR orifice is highly individualized. Therefore, a large sample size is required to have acceptable statistical power, which can be challenging due to the high computational cost associated with these advanced CFD techniques. Furthermore, to model the MR jet in the left atrium requires a turbulent flow model with complex boundary conditions, greatly limiting the likelihood that these methods will be used clinically. In our study, the focus was to provide the degree of inherent underestimation of the standard PISA formula when applied to elliptical orifices. Therefore, a simple well-validated laminar flow CFD solver was sufficient to determine theoretical errors in PISA. Furthermore, our CFD findings were complimented by in vitro experiments that again only showed a difference in EROA of circular and elliptical orifices of 10% or less under expected clinical conditions.
We used a simple gravity-driven orifice flow phantom that allows for modeling similar flow conditions used in our CFD since the main purpose of in vitro investigation was to validate the CFD findings. Papolla et al., have evaluated the accuracy of EROA when using different geometric assumptions in PISA for a variety of MR orifice shapes using a whole left heart flow model [19]. Similar to previous CFD findings, they noted that 3D PISA provides the most accurate EROA while using hemicylindrical and hemiellipsoidal geometric assumptions in 2D PISA can compensate the underestimation by the standard PISA in elongated orifices.
Replacing the hemisphere with hemicylinder or hemiellipsoid geometry in PISA is thought to improve absolute estimates of EROA [15-20, 22, 35]. Hopmeyer et al., developed a curve-fitting algorithm to extract the radius of hemiellipsoidal isovelocity shells in three directions from 2 orthogonal planes, to better assess surface area of the flow convergence region, but the method has not been adopted clinically as it requires consistently high quality images with direct digital output and software updates that are not available in current echo software [15]. A simpler approach has been proposed that measures the width of long and short axes of elliptical flow convergence in addition to the height (i.e., standard PISA radius) to approximate hemicylinder or hemiellipsoidal isovelocity surface area [19,22]. However, color Doppler underestimates the actual width of isovelocity surfacea area due to angular deviation of inward flow velocity vectors from the ultrasound beamline [36]. Moreover, these methods require an imaging plane to be aligned with the long axis of the orifice (for hemicylinder) or two orthogonal planes in long and short axes (for hemiellipsoid) that may be time-consuming and technically challenging and thus, the incremental clinical gain from these more complex assessments may be low and therefore are infrequently used in daily practice. The results of this study will help to extend prior simplified PISA methods [29,37] to a wider range of orifice geometry, answering the need for efficiency in a busy echo lab.
Real time three-dimensional echo has been under investigation for over a decade as a means by which to better quantify MR with clinical validation in vitro and in vivo studies using electromagnetic probes or MRI as comparator gold standards [11,18,34,[38][39][40]. Previously, these methods were limited by considerable operator manipulation for image acquisition; however, Thavendiranathan and colleagues (2013) have proposed a fully automated technique that offers to simplify and expedite quantification in three dimensions, though it has not had widespread uptake [28]. Further research and clinical validation studies are needed before such methods can be incorporated into future guidelines as a recommended means to quantify MR.
There are limitations to our study, including that it only investigates single orifice mitral regurgitant systems. Future directions of research would include using CFD to model double orifice MR, as may be observed after mitral valve clipping, or slit-like MR. In addition, we did not include in vivo data in this study, preferring to focus on the well-controlled numerical and in vitro environment before venturing into the much less controlled clinical situation where a valid reference standard may be lacking.
In conclusion, we demonstrated that the added error in using a traditional hemispherical assumption for calculating PISA is relatively minor for non-circular orifices routinely seen in clinical practice. Given the high time demands of an echo lab for quantification of valvular heart disease, additional complicated techniques to quantify MR may come at a price of expediency and not translate to clinically meaningful changes in management recommendations. Here we show the utility of parametric CFD experiments, confirmed by an in vitro model, to justify a simplified PISA approach appropriate to the workflow demands of the clinic.