Quantification of Flow Rates and Flow Volumes in Valve Regurgitation Using 3-D High Frame-Rate Ultrasound

Valve regurgitation is a cardiac condition caused by the incomplete closure of a cardiac valve. Untreated, this condition may result in cardiac failure. Regular monitoring of this condition is essential in guiding the decision process for surgical intervention. Current guidelines recommend a multi-parametric assessment of valve regurgitation using echocardiography, which is both time consuming and heavily dependent on the experience of the examiner. Several methods have been proposed to provide quantitative markers to facilitate the assessment of valve regurgitation, most notably the Proximal Isovelocity Surface Area (PISA) method and methods based on the quantification of the total Regurgitant Volume (RVol) from the power of backscattered blood signal. In this work, we propose a framework based on trans-thoracic 3-D high frame-rate acquisitions for the simultaneous estimation of the jet cross-sectional area and jet velocity directly at the jet core, which are then combined to estimate the instantaneous flow rate and RVol patients with aortic or mitral insufficiency. We compare two methods for the segmentation of the jet cross-sectional area from the power Doppler signal. Validation on simulated data indicates good segmentation accuracy for the best method (<inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> = 0.97, <inline-formula> <tex-math notation="LaTeX">${R}^{{2}}$ </tex-math></inline-formula> = 0.91). Validation on recordings from a flow phantom shows good agreement (<inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> = 1.2, <inline-formula> <tex-math notation="LaTeX">${R}^{{2}}$ </tex-math></inline-formula> = 0.88) with an external flow rate meter. Clinical feasibility of the method is also shown in a patient with mitral regurgitation.


I. INTRODUCTION
V ALVE regurgitation is a cardiac condition caused by the incomplete closure of a cardiac valve. Untreated, this condition leads to a cascade of events that may result in cardiac failure. Regular monitoring of this condition is essential in the decision process for surgical intervention [1], [2].
Current recommendations suggest an integrative, multi-parametric assessment of valve regurgitation using echocardiography [3], [4]. The procedure currently involves the evaluation of valve morphology and a variety of quantitative and semi-quantitative indices based on Color Flow, Pulsed Wave (PW) Doppler and Continuous Wave (CW) Doppler. As a result, valve regurgitation assessment is heavily dependent on the examiner's experience and can be a cumbersome procedure. The multitude of recommended markers could be regarded as a sign of the complexity of valve disorders. Arguably, it is also a consequence of the lack of a reliable quantitative method.
Among the available quantitative markers for valve regurgitation assessment, the total Regurgitant Volume (RVol) and EROA are the most commonly used. An established method VOLUME 3, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ for the assessment of RVol and EROA is the Proximal Isovelocity Surface Area (PISA) method [3], [5], [6], [7], [8], [5], [9], [10], [11], [12], [13], [14], [15], [16]. When using the PISA method, the orifice is assumed to be circular and planar, mass is conserved, and both source and sink chambers are sufficiently large. With these assumptions fulfilled, it can be shown that flow accelerates towards the orifice forming isovelocity hemispheres. The instantaneous flow rate can then be estimated from 2-D Color Flow by measuring the diameter of the aliased velocity region. The Effective Regurgitant Orifice Area (EROA) is estimated by dividing the instantaneous flow rate by the peak velocity estimated from CW Doppler. Finally, the Regurgitant Volume (RVol) is calculated as the velocity time integral over the regurgitation phase multiplied with the estimated EROA.
More recently, a number of methods combining transesophageal 3-D Color Flow and more refined assumptions on the flow field have been proposed. In the 3-D Field Optimization Method (3-D FOM) [35] the flow field in the convergence region, still assumed as hemispheric, is estimated by performing an automated frame-by-frame fit between modelled and observed Color Flow velocities. 3-D FOM achieved good agreement with an external flow meter in an experimental setup [36] and with 2-D PISA in a clinical study [37]. In a later development by Militaru et al. [38] assumptions on the shape of the flow convergence regions were removed by using a flow model based on a simplified version of the Navier-Stokes equations. In recent work by Singh et al. [39] the flow model was further refined by providing a 3-D segmentation of the valve geometry from B-Mode as boundary conditions. Although promising, these methods currently rely on transesophageal recordings, which limits their daily use in the clinics.
Other research studies attempted the direct quantification of EROA and Rvol at the jet core [40], [41], [42]. Assuming laminar flow and constant hematocrit, the volume of blood within a sample volume of an ultrasound beam is proportional to the back-scattered power from red blood cells after clutter filtering. After performing a calibration procedure, the backscattered blood signal at the jet core can be used to estimate the jet Cross Sectional Area (CSA). Once the jet CSA is known, the instantaneous flow rate and RVol can be calculated by multiplying the jet CSA by the Velocity Time Integral (VTI) from CW Doppler. This method was investigated in vivo using single beam, focused, high PRF (HPRF) acquisitions [43], [44], [45].
One known limitation of this method is the inability to evaluate jets larger than the beam CSA. Although reducing the active aperture can be used to expand the ultrasound beam at the cost of signal strength, the application of the method remains precluded to severe insufficiencies, as well as non-severe insufficiencies with slit-like orifice. After discarding orifice sizes exceeding the beam width, studies have shown good agreement with MRI [18], [44]. Multi-beam, focused, 3-D high PRF acquisitions have been investigated using matrix array transducers, to overcome the limitations of single-beam acquisitions, also showing good agreement with MRI [46], [47], [48].
In this work, we propose a trans-thoracic acquisition setup and a processing chain to estimate the jet CSA, the instantaneous flow rate and the total Regurgitant Volume (RVol) directly at the jet core, in patients with aortic or mitral insufficiency. As first suggested by Avdal et al. [49], we propose to estimate the flow rate by first segmenting the cross-sectional power Doppler image of the jet. By using unfocused beams, 3-D Doppler data can be acquired continuously rather than using packets, enabling the simultaneous estimation of blood velocities and jet cross-sectional area. Two methods for the segmentation of the jet CSA are proposed. Moreover, two transmit beam configurations and their impact on segmentation performance are investigated. The pipeline is validated using simulated data and experimental recordings from a jet phantom. Finally, clinical feasibility is shown in a patient with mitral regurgitation.
The acquisition setup and methods are described in section II. The validation on experimental and simulated data, and the clinical feasibility study are described in Section III. The results are presented in Section IV and discussed in Section V.

II. METHODS
In the proposed method, the instantaneous flow rate Q is estimated from continuous, unfocused 3-D ultrasound recordings, by numerically estimating the following integral over a cross sectional slice of the jet parallel to the transducer surface.
where v(x, y) is the axial velocity estimate at coordinates (x, y). The velocity is estimated from the PW Doppler spectrum at coordinates (x, y) using the procedure explained in Section II-B. denotes the integration support, which is segmented from the power Doppler signal of the jet, as explained in Section II-D. A schematic representation of  Schematic depiction of a jet (gray cylinder) imaged by two transducers, with surfaces T 1 and T 2 . As the beam-to-flow angle θ increases (blue plane), the cross sectional area of the sliced cylinder increases by a factor 1/ cos θ . On the other hand, the estimated phase shift from PW Doppler decreases by a factor cos θ . The instantaneous flow rate, being the product of the two, is therefore angle independent.
the flow rate estimation in Eq. (1) is shown in Fig. 1. Finally, the total regurgitant volume (RVol) is obtained by numerical integration of the instantaneous flow rate Q over the regurgitation period T whereas the jet CSA is estimated as the area of As illustrated for the case of a cylindrical jet in Fig. 2, as the insonification angle θ varies, the velocity estimated from PW Doppler varies by a factor cos θ, whereas the observed crosssectional area varies by 1/ cos θ. Therefore, the flow rate

A. ACQUISITION SETUP
Ultrasound data were recorded using a GE E95 scanner (GE Vingmed AS, Horten), equipped with a GE 4Vc-D matrix transducer (0.16 mm azimuth pitch, 0.23 mm elevation pitch, 100 azimuth elements, 60 elevation elements, 2.8 MHz center frequency). The software was locally modified to enable high frame-rate acquisitions. Two transmit configurations were evaluated for this application, one using weakly focused beams, and one using diverging beams. The focal points for the transmitted beams were 30 cm in front of the transducer and 40 cm behind it, along the central axis.
Beam profiles for the two configurations are shown in Fig. 3. Moreover, high-PRF mode was enabled to prevent aliasing, at the cost of spatial ambiguity. A summary of the acquisition parameters is given in Table 1. For every recording, 1 second of IQ channel data were stored for off-line processing. Chan-VOLUME 3, 2023 nel data were beamformed using the UltraSound ToolBox (USTB). To avoid erroneous calculations of the jet area under the presence of recruited flow, beamformed data were clutter filtered to preserve velocities above 70% of the maximum velocity in the PW Doppler spectrum. PW Doppler spectra and power Doppler were generated using the same temporal observation window for every voxel at a manually defined depth. Cross-sectional jet area and flow rates were estimated using the approaches described in Section II. A summary of the post-processing parameters is given in Table 2.

B. BLOOD VELOCITY ESTIMATION
In this framework we used the mean velocity to quantify the flow rate. Rather than using a lag-1 autocorrelation estimator, we estimated the mean velocity from the PW Doppler spectrum asv where v max is the maximum velocity envelope in the spectrum, estimated as the highest velocity value with intensity above an user-defined threshold above the noise floor, based upon visual inspection of the maximum velocity envelope trace, and bw is the bandwidth of the PW Doppler spectrum, estimated as the number of velocity samples above the same power threshold used for the identification of v max . A depiction of the estimated maximum velocity and bandwidth in a PW spectrum is given in Fig. 1, panel C.

C. DEPTH ESTIMATION
Because of potential movement of the valve throughout the regurgitation phase, it is necessary to follow the movement of the jet core. This is achieved in a two-step approach. First, the maximum velocity envelope from the PW Doppler spectrum is traced in every voxel. Second, the position of the jet core is estimated as the centroid of the volume that contains the 75% highest velocities. Finally, the jet cross-section is segmented according to Section II-D along a plane perpendicular to the transmit beam direction and passing by the estimated centroid depth.

D. CROSS-SECTIONAL AREA ESTIMATION
We investigated two methods for segmenting the crosssectional area of a jet from its power Doppler signal.

1) −3 dB THRESHOLD
In the 1-D case, assuming the spatial invariance of the Point Spread Function (PSF) and a large object size compared to the PSF, the −3 dB power level coincides with the contour of the object. A visual depiction is given in Fig. 4. Under the aforementioned assumptions, the area enclosed within the −3 dB contour will provide an estimate of the jet cross sectional area.

2) ITERATIVE RECONSTRUCTION METHOD
The shape and position of the object is estimated using an iterative approach that maximizes the 2-D normalized crosscorrelation (NCC) between the observed power Doppler image and a synthetic power Doppler that the estimated object would generate given the current imaging conditions where N is the total number of pixels in each of the two images, f and g are the observed and synthetic power Doppler images respectively, and µ and σ denote mean and standard deviation of the two images. The maximum for the normalized cross correlation is found using the direct search method described in [50], which does not require numerical evaluation of the gradient of the cost function. A simulation approach based on averaging the power of the backscattered signal from randomly distributed scatterers would take a considerable amount of time to be feasible for practical purposes. The proposed method instead generates a power Doppler depiction of a given object by numerical integration of the power of the pulse-echo response for each pixel where g denotes the power Doppler image of the object, x and y are the power Doppler image coordinates, denotes the object surface, and P h is the power of the pulse-echo 32 VOLUME 3, 2023 response, calculated as where h() denotes the pulse-echo response, calculated using Field II. Numerical integration is performed along fast time t.
The pulse-echo response is evaluated at coordinates (w, v), which should define a finer grid than (x, y) for accurate results. It should be noted that, assuming dynamic receive focusing, the pulse-echo response must be calculated for every (x, y) in the power Doppler image. The power of the pulse-echo response is computed before the iterative reconstruction method takes place and is specific to the combination of transducer geometry, impulse response, imaging depth and acquisition setup. Throughout this work we used the acquisition parameters presented in Section II-A to generate the pulse-echo response used in the method. Provided that the imaging parameters do not change during acquisition, the same pulse-echo response power can be reused. Furthermore, only elliptical objects are modelled. This choice reduces the number of parameters explored by the optimization algorithm to five: major axis a, minor axis b, orientation θ and center coordinates (x 0 , y 0 ). A schematic representation of the method is given in Fig. 5.

III. VALIDATION A. SIMULATIONS
A validation based on simulated data was performed with two purposes in mind. First, to verify that the fast power Doppler simulator could replicate results from a conventional simulation approach such as Field II. Second, to benchmark the iterative reconstruction approach against thresholding. The methods described in Section II-D to segment the jet and estimate the cross-sectional area were tested on a set of 100 images, each representing the cross-sectional power Doppler of a jet. The source objects were modelled as ellipses, with geometric properties representative of valve insufficiencies with different severity. RF channel data of each object were simulated in Field II [51], using the GE 4Vc-D transducer geometry and the acquisition parameters given in Table 1. The objects were populated with point scatterers according to a uniform random distribution to achieve and approximate density of 10 scatterers within the -6dB contour of the Point Spread Function. This condition was chosen to ensure Gaussian distributed channel data. Channel data were beamformed using the UltraSound ToolBox (USTB). Finally, the power Doppler image of each object was generated by averaging 100 independent realizations to simulate a long observation window and achieve low variance in the final power Doppler image.

B. PHANTOM EXPERIMENT
The experimental setup used to validate the flow rate estimation algorithm is shown in Fig. 6. The setup consisted of two fluid reservoirs, one acting as source and the other   water with a ratio of 5 g/L to replicate the scattering properties of blood. The height difference between source and drain reservoirs determined the pressure drop across the phantom and could be adjusted to achieve different flow rates. Three height differences were tested, 65 cm, 100 cm, and 135 cm. The phantom allowed the user to replace the orifice model to validate the method against different orifice sizes. In this work we performed recordings from circular orifices with 0.15 cm 2 , 0.25 cm 2 , 0.35 cm 2 and 0.45 cm 2 cross-sectional area, representing valve insufficiencies of different severity. The wide acoustic window allowed the user to record data at different insonification angles. We performed recordings at 0 • , 30 • , 40 • and 50 • beam-to-flow angle, which should cover even the more challenging cases in clinical practice. For each combination of angle and orifice size, three different flow rates values were achieved by placing the source reservoir at predefined heights. The imposed pressure difference remained the same for the different orifice models. Consequently, each orifice model resulted in different ranges of achieved flow rates, with the smaller orifice achieving lower flow rates due to higher flow resistance.

C. CLINICAL FEASIBILITY ANALYSIS
Feasibility of the estimation method and the acquisition setup described in Section II were tested in a patient with mitral regurgitation. The study was approved by the Regional Committee for Medical and Health Research Ethics (REK) with protocol number 2017/900, and the patient provided informed written consent before the recording took place. Channel data were recorded by an expert echocardiographer using a GE E95 scanner equipped with a GE 4Vc-D matrix transducer (GE Vingmed AS, Horten). The scanner was locally modified to enable high frame-rate acquisitions. The transmit focus was placed 40 cm behind the transducer to achieve diverging beams, and high-PRF mode was enabled to prevent aliasing. The Pulse Repetition Frequency (PRF) was approximately 12 kHz. Around 2 seconds of channel data were stored for offline processing. The instantaneous flow rate was estimated using the approaches explained in Section II, and the total regurgitant volume (RVol) was estimated by numerical integration of the flow rate curves. An expert echocardiographer estimated EROA and flow rate from the same patient using the 2-D PISA method. The total regurgitant volume (RVol) was estimated by multiplying the estimated EROA from 2-D PISA by the velocity time integral (VTI) from CW Doppler. The clinical analysis was performed using ECHOPAC (version 204, GE Vingmed AS, Horten).

A. IN SILICO VALIDATION
In Fig. 7 the power Doppler image simulated using Field II is compared against the power Doppler image simulated using the iterative algorithm presented in Section II-D. The estimated normalized cross correlation in the example was 0.99. The reference and estimated objects are shown, as well as the −3 dB segmentation result from the power Doppler image. It can be noticed that the −3 dB segmentation suggests a smaller cross-sectional area than the iterative method, which matches the reference object more closely. It can be also noticed that the power Doppler image simulated by integration of the pulse-echo response (lower panel) appears similar to the reference (upper panel), which was instead simulated using Field II by averaging independent realizations from randomly distributed scatterers. Visual comparison between the simulation methods highlights minor differences in the sidelobes but not in the main lobe. In Fig. 8 the area estimates using a −3 dB threshold and the iterative reconstruction method are compared on a set of 100 objects mimicking the cross-section of insufficiency jets of varying severity. Results indicate that using a slightly diverging beam on transmit yields better correlation and reduced bias, compared to a weakly focused transmit beam. Similarly, the iterative reconstruction method yields better overall fit (y 0 = 0.00, β = 0.97, R 2 = 0.91) compared to a −3 dB threshold (y 0 = 0.08, β = 0.58, R 2 = 0.71). Moreover, the presence of a non zero intercept y 0 in the −3dB threshold approach suggests that this method may inherently overestimate small orifices. Fig. 9 shows estimated flow rates using the flow phantom setup and weakly focused waves, versus reference values from a transit-time flow meter. The width of the error bars displays the standard deviation over 50 repeated measurements. Results indicate that the overall fit using this setup is relatively low, but the iterative reconstruction method achieves better correlation (R 2 = 0.51) compared to a −3 dB threshold (R 2 = 0.34). On the other hand, the increased width of the error bars indicates that the iterative method hardly converges to a stable estimate. Moreover, the regression line for the iterative reconstruction method is closer to the identity line both in terms of slope (β = 0.76 vs β = 0.69, respectively) and intercept (y 0 = 36.92 vs y 0 = 44.43, respectively), although differences are minor.

B. PHANTOM EXPERIMENT
Corresponding results using diverging beams are shown in Fig. 10. The correlation between estimated and reference flow rates is higher than when using weakly focused beams. The results also indicate that the iterative reconstruction method yields better correlation (R 2 = 0.88) compared to a −3 dB threshold (R 2 = 0.64). Moreover, results indicate a substantially improved agreement between the regression line and the identity line, both in terms of slope (β = 1.13 vs β = 0.76, respectively) and intercept (y 0 = 5.78 vs y 0 = 34.59, respectively). Fig. 11 shows the results processed from a high framerate recording of a patient with mitral regurgitation. The segmented jet cross-sectional area using a −3dB threshold and the iterative reconstruction method are shown at three instants. The flow rate curves estimated using the approaches  Table 1.

C. CLINICAL FEASIBILITY
explained in Section II are also shown. The estimated RVol was 45 ml using a −3 dB threshold and 48 ml using the iterative method. We also estimated the equivalent EROA value by dividing the estimated instantaneous flow rate by the maximum velocity envelope from the PW Doppler spectrum (shown using an orange line in Fig. 11). The equivalent EROA varied between 0.20 cm 2 and 0.38 cm 2 in the regurgitant phase, with an average value of 0.26 cm 2 . Fig. 12 presents the results from a 2-D PISA analysis along with a standard 2-D Color Flow image of the mitral valve and a CW Doppler spectrum from the same patient. The estimated RVol from 2-D PISA, averaged over several measurements, was 40 ml, whereas the estimated EROA was 0.2 cm 2 .

V. DISCUSSION
In this work we have presented a method for the quantification of jet cross-sectional area, instantaneous flow rate and total regurgitation volume in aortic and mitral regurgitation. The method combines trans thoracic 3-D high framerate acquisitions and unfocused beams to perform jet area segmentation and velocity estimation at the jet core from a single recording. The proposed method does not require VOLUME 3, 2023 a preliminary calibration procedure as in [18], [43], [44], and [45] and does not require Color Flow recordings at the inflow region as in the PISA method. Moreover, flow rate estimation is performed autonomously at multiple frames throughout the regurgitation phase.

A. ACQUISITION SETUP
Results from both in silico validation in Fig. 8 and phantom experiments in Fig. 9 and Fig. 10 indicate that using weakly diverging waves yields better correlation between estimated and reference flow rates than using weakly focused waves. Notably, the area of large orifices are systematically underestimated when using weakly focused beams and a −3dB threshold in Fig. 8. These results can be explained by the improved field uniformity achieved by diverging beams, as shown in Fig 3. Increased transmit beam uniformity ensures a more uniform PSF, which directly benefits the accuracy of both jet segmentation methods.
The results so far suggest that diverging beams should be used in a clinical setup. On the other hand, given the same excitation voltage, diverging beams achieve lower intensity, as shown in Fig. 3. The lower beam intensity results in weaker backscattered signal and limited penetration, which could limit the applicability of the method in patients with challenging imaging conditions. These considerations suggest that the trade off between beam uniformity and penetration depth should be carefully investigated with the aid of simulation programs and water tank measurement.
Finally, enabling high-PRF was necessary to prevent aliasing of the highest jet velocities. It should be noted that depth ambiguity, a side effect of this acquisition modality, does not constitute a problem in this application, because the high velocities of interest (v > 1 m/s) can only be related to insufficiency jets.

B. DEPTH ESTIMATION
As reported in Section II-C, the flow rate is estimated at the depth of the centroid of the volume containing the 75% highest velocities from PW Doppler. Due to possible asymmetric velocity distributions at the convergence region, this approach can potentially miss the vena contracta. However, because of the principle of conservation of mass, the proposed flow rate estimation method is not sensitive to the imaging depth, provided that it is possible to separate the jet core from the surrounding recruited flow by means of a high-pass filter.

C. AREA ESTIMATION
Of the investigated segmentation strategies, the iterative reconstruction method delivered more consistent and less biased estimates. However, the study pointed out a few limitations that should be discussed. One limitation of the iterative method is that the jet cross section is modelled as an ellipse. This assumption was made to reduce the number of parameters explored by the direct solver and facilitate convergence. However, this assumption may be not applicable in some cases, for example when fused valve cusps generate triangular orifices. In this case the iterative method may provide wrong area estimates or even fail to converge. One solution could be to define the orifice geometry using a linear combination of parametric functions such as B-splines or Bezier curves. On the other hand, the increased number of parameters and their weaker relation with the shape of the final power Doppler image may hinder convergence or increase the chance of meeting local minima. Further work could focus on investigating a new trade-off between generality and robustness, as well as extending the iterative method to cope with bifurcated jets.
Another limitation of the iterative reconstruction method is that it relies on a fast power Doppler simulator that does not take into account the flow characteristics of the jet, but only its spatial extent. The effects of flow recruitment, turbulence and the interaction of the backscattered signal with clutter filtering were not taken into consideration. Analysis of both experimental and clinical data highlighted the importance of suppressing velocity components not associated with the jet core, which can lead to an overestimation of the jet cross-sectional area. These velocity components are most likely associated with flow recruitment or flow picked up by secondary sample volumes in high-PRF acquisitions.
Furthermore, the fast power Doppler simulator is based on the assumptions of long observation windows, which may be weaker in a clinical setting due to movement of the regurgitant jet. Future work could aim at investigating how the flow regime affects the power Doppler depiction of a jet from an orifice of known size, and could be conducted by coupling Computational Fluid Dynamics (CFD) simulations with ultrasound simulations. The study could as well provide the foundation to investigate the optimal clutter filtering strategy and choice of the observation window length.
In silico and phantom results both indicate that the method used for area estimation is critical for the performance of flow rate estimation. Recognising the limitations of the iterative method discussed above, and that there is still potential for improvement in the correspondence between estimated and reference values, the results highlight that segmentation methods based on spatially variant PSF should be preferred to methods based on the assumption of spatially invariant PSF.

D. VELOCITY ESTIMATION
The proposed method uses the mean velocity for the estimation of the instantaneous flow rate. Although the experimental results shown in Fig. 9 and 10 indicate consistent flow rate estimates with reference flow rate measurements, it is not known whether using a mean velocity estimator will perform equally well in patients, due to the possible differences in the flow regimes. Further validation using a CFD model could help investigate the validity of the proposed method in more realistic flow regimes.
It should be noted that the proposed mean velocity estimator is based on the PW Doppler spectrum. The choice of this velocity estimator over a lag-1 autocorrelation estimator was motivated by a number of reasons. First, the autocorrelation estimator can be strongly biased if clutter is not properly filtered, whereas proposed estimator would yield at most a bias equal to half the clutter bandwidth. Second, due to the high bandwidth of the PW Doppler spectra in this application, maximum velocities above twice the Nyquist limit would render velocity estimation impossible. The aliasing properties of the autocorrelation method are therefore not advantageous in this application.

E. CLINICAL FEASIBILITY
The clinical feasibility example shown in Fig. 11 suggests that the proposed method may be used in patients. However, it also revealed challenges related to limited image quality, which may require fine tuning of the processing parameters for every patient to ensure correct tracing of the maximum velocity envelope and tracking of the jet position. Moreover, for this method to be feasible in a clinical routine, the echocardiographer may have to reduce the transmit frequency to prevent aliasing, because of the high velocities associated with the application. The drawback of this measure would be reduced lateral and axial resolution, which already is a limiting factor for the correct estimation of the jet crosssectional area. Future work could investigate more robust methods to estimate the maximum velocity envelope as well a more robust method to track the displacement of the jet core.
Comparison with 2-D PISA estimates from the same patient, shown in Fig. 12, showed good Rvol and EROA agreement (48 ml and 0.26 cm 2 compared to 40 ml and 0.2 cm 2 from 2-D PISA). It should be noted that PISA is based on the assumption that blood flows with the maximal velocity in the entire EROA, whereas for the proposed method, the area and mean velocity is estimated in the cross-section of interest and used for flow rate estimation. This means that the proposed method may use different cross-sections for flow rate estimation, but also that the CSA used in the proposed method is not directly related to the EROA derived from PISA. Typically, the former will be larger, depending on the velocity gradients in the cross-section of interest. It should also be noted that PISA measurements are dependent on instantaneous flow rate estimates at selected frames, whereas the proposed framework can account for variations in flow rates and EROA values during the regurgitant phase. For the recording in Fig. 11, the estimated flow rate peaks between 450 ms and 550 ms, when a potential bifurcation is visible in the 3-D color Flow depiction of the jet, shown in Fig. 13. This bifurcation is not detected by the area estimation methods and may thus contribute to a slight overestimation of both regurgitant volume and EROA.

VI. CONCLUSION
In this work we have presented a framework for the estimation of flow rates and flow volumes in aortic and mitral regurgitations. The method combines high frame-rate acquisitions and unfocused beams for the simultaneous estimation of blood velocities and cross-sectional area directly at the jet core. The two estimates are then combined to quantify the instantaneous flow rate. We found that using diverging beams yielded improved estimation accuracy compared with using weakly focused waves, probably because of the more uniform distribution of the ultrasound energy in the field of view. We also compared two different approaches to segment the jet cross-sectional areas, a thresholding approach and an iterative method. A validation study performed on simulated power Doppler images shows that the iterative method estimates the underlying target size with reduced bias and better consistency compared to a −3 dB threshold (β = 0.97, R 2 = 0.91 and β = 0.8, R 2 = 0.6, respectively), due to the ability to account for the non uniformity of the PSF. Results are confirmed from recordings of a jet phantom, which show that the iterative method delivers more consistent flow rate estimates compared to a −3 dB threshold (β = 1.2, R 2 = 0.88 and β = 1.16, R 2 = 0.44, respectively), but also highlight an overestimation bias that should be further investigated. Finally, clinical feasibility was shown in a patient with mitral regurgitation.