The leaflets of the mitral valve interact with the mitral jet and significantly impact diastolic flow patterns, but the effect of mitral valve morphology and kinematics on diastolic flow and its implications for left ventricular function have not been clearly delineated. In the present study, we employ computational hemodynamic simulations to understand the effect of mitral valve leaflets on diastolic flow. A computational model of the left ventricle is constructed based on a high-resolution contrast computed-tomography scan, and a physiological inspired model of the mitral valve leaflets is synthesized from morphological and echocardiographic data. Simulations are performed with a diode type valve model as well as the physiological mitral valve model in order to delineate the effect of mitral-valve leaflets on the intraventricular flow. The study suggests that a normal physiological mitral valve promotes the formation of a circulatory (or “looped”) flow pattern in the ventricle. The mitral valve leaflets also increase the strength of the apical flow, thereby enhancing apical washout and mixing of ventricular blood. The implications of these findings on ventricular function as well as ventricular flow models are discussed.

The mitral valve (MV) is unique among cardiac valves in that it has only two leaflets, and they are highly asymmetric, with the anterior leaflet being roughly 1.5 times longer than the posterior leaflet. While it is known that leaflets of the mitral valve (both natural and prosthetic) interact with the mitral jet and in doing so, significantly impact diastolic flow patterns,1,2 no study has clearly delineated the effect of natural mitral valve morphology and kinematics on diastolic flow and its implications for left ventricular function. While some past studies have focused on the dynamics of mitral valve leaflets (MVLs)3–5 and others have studied flow-leaflet interaction in highly idealized (e.g., simple pipe flow) flows,6–8 a few9,10 studies have combined realistic intraventricular flow with physiologically representative MV leaflets. Insights into these effects of the MV on the diastolic flow and ventricular function would enhance our basic understanding of the functional morphology of this critical component of the heart and would also be useful for understanding the implications of surgical interventions such as leaflet clipping,11 leaflet plication,12 and mitral valve replacement.

The objective of the present paper is to systematically investigate the effect of MVLs and their motion on diastolic flow patterns in a normal human ventricle using computational fluid dynamics (CFD) modeling. The left ventricle (LV) model in the current study is derived from a high-resolution contrast computed tomography (CT) scan of a normal human heart, and the ventricular flow is simulated by solving the incompressible Navier-Stokes equations. In order to clearly delineate the effect of the mitral valve leaflets on the flow, we consider two mitral valve models: a diode mitral valve (DMV) model, such as the ones employed in Refs.13,15 where the mitral valve has no leaflets, is represented by a flow (diastole) or no-flow (systole) boundary condition imposed at inlet of the mitral inflow and a physiological mitral valve (PMV) model, which consists of two leaflets, the geometry and kinematics of which are based on available data on MV anatomy and kinematics. The comparative study approach employed here, coupled with the ability to precisely control all other conditions and to compute a variety of flow metrics, is only possible with this kind of in-silico model and is expected to provide new insights into the function of the mitral valve.

The current study also makes a significant contribution to CFD modeling of cardiac flows, an activity that has grown rapidly in the last decade. The vast majority of previous CFD studies have employed highly simplistic models of the mitral valve,2,13–21 and the effect of this simplification on model prediction has never been assessed. Our comparative study suggests the importance of incorporating more detailed physiological information (leaflet shapes and motion in particular) in the mitral valve model on the prediction of such models and provides guidance for more realistic CFD models of ventricular flows.

A relatively simple model of the LV is constructed based on the three-dimensional LV geometry extracted from a high-resolution, multi-detector contrast CT scan of a normal human ventricle as shown in Fig. 1(a). The end systolic volume (ESV) of the ventricle model is 39 ml, and the mitral annulus has an elliptic shape with major and minor axes of 2.4 cm and 2 cm, respectively (see Fig. 1(b)). The shape of the PMV is based on the detailed anatomical measurements of Ranganathan et al.22 As shown in Fig. 1(b), the lengths of anterior and posterior leaflets are set to LAL = 2 cm and LPL = 0.8 cm, respectively. It is noted that the PMV is a prescribed kinematics model and not a fluid-structure interaction (FSI) model. FSI models of natural mitral valves are highly involved3–5,23,24 and introduce complexities and uncertainties of non-linear deformation and constitutive properties. Since the objective here is not to examine valve dynamics but to assess the effect of mitral valves on the intraventricular flow, a prescribed kinematics model of the mitral valve based on physiological data should provide sufficient fidelity for the current objective. The motion of mitral valve leaflets during diastole is modeled by prescribing the opening angles (angles between the leaflet and the mitral annulus plane) for anterior and posterior leaflets, ϕAL and ϕPL, respectively as indicated in Fig. 1(a). The opening angles at different cardiac phases are measured from the echocardiographic data as depicted in Fig. 1(c).

FIG. 1.

(a) Left ventricle and mitral valve model; left: contrast CT scan data of the normal human heart. The left heart is highlighted middle: LV model reconstructed from contrast CT scan data, right: simplified computational model, LA: left atrium, AO: aorta, LVOT: LV outflow tract, AL: anterior leaflet, PL: posterior leaflet, ϕAL, ϕPL: AL, PL opening angles. (b) Mitral valve geometry, top: 3D shape, bottom: “unrolled” valve geometry, LAL, LPL: AL, PL lengths. (c) Measurements of mitral valve leaflet opening angles with respect to the mitral annulus plane using the echocardiogram data. (d) Flow rate wave form through LV, AT: E-wave acceleration time, DT: deceleration time, DIA: diastasis duration, Adur: A-wave duration, SYS: systole duration. (e) Time profiles of mitral valve opening angles.

FIG. 1.

(a) Left ventricle and mitral valve model; left: contrast CT scan data of the normal human heart. The left heart is highlighted middle: LV model reconstructed from contrast CT scan data, right: simplified computational model, LA: left atrium, AO: aorta, LVOT: LV outflow tract, AL: anterior leaflet, PL: posterior leaflet, ϕAL, ϕPL: AL, PL opening angles. (b) Mitral valve geometry, top: 3D shape, bottom: “unrolled” valve geometry, LAL, LPL: AL, PL lengths. (c) Measurements of mitral valve leaflet opening angles with respect to the mitral annulus plane using the echocardiogram data. (d) Flow rate wave form through LV, AT: E-wave acceleration time, DT: deceleration time, DIA: diastasis duration, Adur: A-wave duration, SYS: systole duration. (e) Time profiles of mitral valve opening angles.

Close modal

The heart rate is assumed to be 60 beats per minute (BPM) and the flow rate, Q(t) (plotted in Fig. 1(d)) through the LV is based on published data.25 The wave forms for the early (E-) and atrial (A)-waves are based on a simple harmonic oscillator (SHO) model which is verified with Doppler echocardiographic data,26,27 and the E/A ratio (EAR) is set to 1.5. The peak flow rates for E- and A-waves are 292 and 195 ml/s, respectively, and this results in a stroke volume (SV) of 62 ml and ejection fraction (EF) of 61%. The above numbers correspond to a typical normal adult left ventricle, and further details regarding the flow rate are given in Appendix  A. The volume flow rates into and out of the ventricle are driven by a prescribed expansion and contraction of the LV, respectively, and the motion of endocardial surface is prescribed such that it satisfies the given volume flow rate condition.28 The piecewise continuous sinusoidal functions (described in Appendix  B) are fit to the mitral valve opening angles at different cardiac phases in order to generate the smooth temporal variation shown in Fig. 1(e).

The motion of the leaflets has a correspondence with the phases of the cardiac cycle; the largest opening angle (90°) coincides with the peaks of the E-wave and the A-wave. The leaflets do not close completely during diastasis but maintain a deflection of about 45°. During the simulation, the anterior and posterior valve leaflets are folded along the axis parallel to the edge-lines shown in Fig. 1(b) as per the opening angles prescribed above, thereby providing a realistic representation of actual mitral valves. It is noted that during systole, the mitral annulus is completely closed and has little effect on the systolic flow. In the current simulations, the systolic condition is therefore modeled simply by removing the leaflets and imposing a no-flow condition at the mitral annulus.

The blood in the ventricle is assumed to be a Newtonian fluid and its dynamics is governed by the incompressible Navier-Stokes equations

U = 0 , U t + ( U ) U + 1 ρ 0 P = ν 0 2 U ,
(1)

where U is a velocity vector, P is a pressure, and ρ0 and ν0 are the density and kinematic viscosity of the blood, respectively. The discretized equations are solved on the nonbody-conformal Cartesian grid and the complex, moving boundaries are treated by a sharp-interface immersed boundary method.29 The solver has been successfully applied to cardiac flows in previous studies,15,28,30 and a recent validation study (will be published soon) for the model ventricle31 confirmed that our computational modeling procedure could accurately produce all the key features of the velocity and vorticity fields observed in intraventricular flows.

In the present study, the endocardial wall and the mitral valve leaflet surfaces are discretized with a total of 37 138 and 7200 triangular elements, respectively. The domain size for the Cartesian volume grid is set to 6.5 cm × 6 cm × 11 cm, and covered by a total of 128 × 128 × 256 (4.2 million) grid points. This resolution is chosen based on the previous grid convergence study for the LV flow simulations.28 The time step size used is 1 × 10−4 s which results in the maximum CFL (Courant-Friedrichs-Lewey) number of 0.6. The flow Reynolds number based on the average peak E-wave velocity and mitral annulus diameter is 4370, and the Womersley number based on the mitral annulus radius and heart rate is 14. These non-dimensional parameters are in the range15 considered typical for a healthy adult human heart. The simulations are performed on a parallel computer using 128 central processing unit (CPU) cores and the computation for one cardiac cycle takes about 24 h. Flow simulations are performed for two successive cardiac cycles for each case, and the comparison between the two cases is based on the second cardiac cycle.

A variety of metrics are used to quantify the flow in the left ventricle. Vortical structures such as vortex rings are identified via the second invariant of the velocity gradients32 

q 2 = 1 2 Ω i j Ω i j S i j S i j ,
(2)

where Ωij and Sij are vorticity and shear strain tensors.

In order to investigate blood mixing and washout in the LV, we employ “virtual ventriculography,” which is performed by simultaneously solving the following passive scalar transport equation for the contrast agent:

C t + ( U ) C = D 2 C ,
(3)

where C is the normalized contrast agent concentration, D is the diffusion coefficient which is set to 1e − 9 m2/s,33 and the velocity field is obtained from the hemodynamic simulation. The above equation is solved by a fully explicit, four-stage Runge-Kutta method, and the convection term is discretized by a second-order upwind scheme. The time evolution of the contrast agent is then used to examine blood transport and mixing in the ventricle.

Lagrangian coherent structures (LCS)34 are finding ever-increasing use in the analysis of intraventricular flows,35 and here we compute “attraction” LCS by backward time evolution of fluid particle motion34 over an interval of 0.05 s. The finite time Lyapunov exponent (FTLE) defined by

FTLE( x ) = 1 T ln d Φ t + T ( x ) d x
(4)

is used for the LCS. In Eq. (4), T is the time interval and Φt+T is the flow map for the time evolution of flow particles: x ( t ) x ( t + T ) .

The circulatory movement of the flow in the ventricle is quantified via the quantity circulation (Γ) which is defined

Γ = C U d l = S × U d S ,
(5)

where × U is the vorticity and S is the area of integration.

Figure 2 shows vortex structures during the early filling (E-wave) phase of the cardiac cycle and the figure shows the formation, pinch-off, and propagation of the E-wave vortex ring. For the DMV model, one can clearly see the formation of a nearly circular vortex ring at the mitral annulus (t = 0.1 s, E-wave peak). This vortex ring pinches off and propagates toward apex (t = 0.15 s) and starts to disintegrate as it reaches the middle of the ventricle (at ∼t = 0.2 s). By t = 0.25 s, the disintegration of the vortex ring is complete but the vortex structures are confined to the top two-thirds of the ventricle.

FIG. 2.

Three-dimensional vortical structure visualized by iso-surfaces of the second invariant of velocity gradients and colored by axial (normal to the mitral annulus plane) velocity. Negative velocity indicates toward apex. (a) DMV and (b) PMV.

FIG. 2.

Three-dimensional vortical structure visualized by iso-surfaces of the second invariant of velocity gradients and colored by axial (normal to the mitral annulus plane) velocity. Negative velocity indicates toward apex. (a) DMV and (b) PMV.

Close modal

With the PMV model, the initial vortex ring like structure that is formed at t = 0.1 s is significantly distorted due to the non-axisymmetric leaflets. However, at about t = 0.15 s, the vorticity has reorganized into a clear ring structure that has propagated nearly to the center of the ventricle. At t = 0.2 s, the vortex ring has propagated past the center of the ventricle and starts to disintegrate. As with the DMV model, the process of disintegration is complete by t = 0.25 s but the vortex structures have nearly reached the apex by this time.

The time evolution of the contrast (i.e., virtual ventriculography) and the LCS on the long-axis plane are shown in Fig. 3, and the different propagation speeds of the contrast for the DMV and PMV is clearly evident. At t = 0.1 s, the DMV case shows a roll-up of vortices at the mitral annulus, while with the PMV, the vortex rolls up at the tip of MVLs. Since the leaflets in the PMV model are asymmetric (anterior leaflet is longer than posterior leaflet), the resulting vortex ring at t = 0.15 s for this case is also asymmetric, but the one for the DMV case is nearly symmetric. Also note that, at t = 0.15 s, the vortex ring for DMV case still remains in the basal portion of the ventricle, while with the PMV, the vortex ring is already located at the middle of ventricle.

FIG. 3.

Time evolution of the contrast agent concentration (color contour) and the attraction LCS (line contour) on the long-axis plane for the DMV (a) and PMV (b) cases. For the LCS, the contour of normalized, finite time Lyapunov exponent, FTLE/FTLEmax = 0.6 is plotted. MVLs are highlighted with green lines for the PMV case.

FIG. 3.

Time evolution of the contrast agent concentration (color contour) and the attraction LCS (line contour) on the long-axis plane for the DMV (a) and PMV (b) cases. For the LCS, the contour of normalized, finite time Lyapunov exponent, FTLE/FTLEmax = 0.6 is plotted. MVLs are highlighted with green lines for the PMV case.

Close modal

The characteristics of blood transport are also quite different for DMV and PMV during this course of time. At t = 0.1 s, the material-front for the PMV case extends over more than half of the ventricle, but with the DMV model, the front covers only the upper 35% of the ventricle. Note that total amount of contrast inside the ventricle is the same for both cases. For the PMV case, the contrast agent initially remains in the region under the mitral annulus in the horizontal direction and propagates down to the apex. For the DMV case, however, one can see that the contrast agent spreads more in the horizontal direction due to the early vortex roll-up at the mitral annulus and the absence of the mitral leaflets. The same behavior is observed at t = 0.15 s, but the PMV case also shows the horizontal spread of the contrast agent caused by the clear vortex roll-up in the lower half of the ventricle.

At t = 0.2 s, the vortical pattern with the PMV becomes significantly asymmetric as depicted by the LCS; the vortex located towards the septal wall becomes dominant and is positioned at the center of ventricle. This vortex redirects the incoming mitral jet to the lateral wall side and also promotes a large-scale clockwise (CW) rotational flow inside the LV. The partial closing of the large anterior leaflet further directs the jet in the direction of the lateral wall, thereby enhancing the asymmetry of the jet. The virtual ventriculography also confirms this flow pattern produced by the PMV; at t = 0.2 s, the contrast reaches the apical region and due to the strong clock-wise rotational flow, moves upward on the septal side; at the end of E-wave (t = 0.25 s), the contrast fills almost the entire ventricle. On the other hand, the contrast fails to reach the apex by the end of E-wave for the DMV case. The vortex ring structures for the A-wave are also plotted at t = 0.6 s and one can see the same trend as with the E-wave, i.e., that the presence of mitral valve leaflets enhances the asymmetry of the vortical structures.

The above results indicate that the propagation speed of the mitral jet is affected by the mitral valve configuration. In order to quantify the propagation speed, the spatio-temporal velocity profile is assessed along the normal to the mitral annulus in the similar manner to a color M-mode echocardiogram36 and plotted in Fig. 4 for both cases. The line along which the velocity profiles are extracted is shown in Fig. 4(a). Figures 4(b) and 4(c) are the spatio-temporal profiles of the velocity component directed along the line shown in Fig. 4(a), and they clearly show the propagation of the velocity wave generated by mitral inflow. The positive velocity indicates the direction toward the apex, and the distance is measured from the mitral annulus location. The velocity wave front propagation speed is quantified by tracking the peak of the velocity wave and also plotted in Figs. 4(b) and 4(c). The spatio-temporal trajectory of the velocity wave peak exhibits a linear slope which represents the propagation speed. Interestingly, the slope changes at around t = 0.1 s when the mitral jet transits from acceleration to deceleration. These phenomena have also been observed from the in-vivo study of Stewart et al.37 For the PMV case, the wave front propagation speeds for the E-wave are 73 cm/s and 29 cm/s in the acceleration and deceleration phases, respectively. For the DMV case, the corresponding speeds are 66 cm/s and 23 cm/s, respectively, which are about 13%-20% slower than the PMV case. This difference in the propagation speed between the PMV and DMV cases results in different penetration depths of the mitral jet as shown in Figs. 4(b) and 4(c) and also affects the transport of the atrial blood volume to the apex as shown in Fig. 3.

FIG. 4.

Spatio-temporal velocity profile along the mitral annulus center line. The profiles of velocity component normal to the mitral annulus are extracted along the line normal to the mitral annulus plane on the long-axis plane (a). The spatio-temporal profiles are plotted in the similar fashion with the color M-mode echocardiogram for the DMV (b) and PMV (c) cases. Black lines indicate the track of the peak point of velocity waves.

FIG. 4.

Spatio-temporal velocity profile along the mitral annulus center line. The profiles of velocity component normal to the mitral annulus are extracted along the line normal to the mitral annulus plane on the long-axis plane (a). The spatio-temporal profiles are plotted in the similar fashion with the color M-mode echocardiogram for the DMV (b) and PMV (c) cases. Black lines indicate the track of the peak point of velocity waves.

Close modal

The other key difference in the diastolic flow pattern between the DMV and PMV cases observed in Fig. 3 is the septal-lateral asymmetry of the flow. In order to quantify this, we have monitored the temporal change in velocity at several locations on the long-axis plane; these locations are: (a) 3 cm below (to the apical) from the septal edge of the mitral annulus, (b) 3 cm below from the mitral center, (c) 3 cm below from the lateral edge of the mitral annulus, and (d) at the center of mitral annulus. These locations are indicated in Fig. 5(a). Note that this temporal velocity profile assessment is similar to the pulsed wave Doppler analysis38 in clinical practice.

FIG. 5.

Assessments of temporal velocity profiles. The velocity component normal to the mitral annulus is monitored at four different locations ((a), see text for the detailed description) for DMV (b) and PMV (c) cases.

FIG. 5.

Assessments of temporal velocity profiles. The velocity component normal to the mitral annulus is monitored at four different locations ((a), see text for the detailed description) for DMV (b) and PMV (c) cases.

Close modal

The monitored velocity profiles are plotted in Figs. 5(b) and 5(c) for the DMV and PMV cases, respectively. Velocity profiles at (b) and (d) show the propagation of the velocity wave along the mitral centerline, and one can see that the arrival of the E-wave peak at the location (b) is further delayed for the DMV case than the PMV case, which confirms the slower wave propagation speed for the DMV case. Similarity between the velocity profiles at (a) and (c) represents the septal-lateral symmetry of the flow pattern. For the DMV case, velocities at both (a) and (c) are mostly positive, and the profiles have a similar shape. The correlation coefficient between velocity profiles at (a) and (c) is found to be 0.73, which indicates that the overall flow pattern is quite symmetric in the septal-lateral direction. For the PMV case, on the other hand, the velocity values at (a) is mostly negative, while the velocity at (c) is positive. The correlation coefficient between two profiles is only 0.06, which implies that the flow pattern for the PMV case is highly asymmetric. The opposing directions of flow at locations (a) and (c) indicate the overall clock-wise circulation of the flow.

In order to assess the effect of the MVL on the overall flow pattern inside the LV, the velocity fields are averaged over one cardiac cycle and plotted in Fig. 6. As can be seen in Fig. 6(a), which shows contours of velocity magnitude and in-plane velocity vectors, the DMV case exhibits two regions of high mean velocity, one near the mitral exit and one located near the outflow tract. These are clearly associated with diastole and systole, respectively. The velocity in other areas of the LV including the lateral and septal endocardium and the LV apex is quite low. The PMV case also shows the highest velocity near the mitral annulus and the outflow tract but in contrast to the DMV case, also shows a nearly continuous region of high speed flow that extends from the mitral inflow, along the lateral, apical, and septal walls to the outflow tract.

FIG. 6.

Flow field averaged over a cardiac cycle. (a) and (b) Time averaged velocity magnitude and vectors. (a) DMV, (b) PMV. (c) and (d) Time averaged vorticity and stream trace. (c) DMV, (d) PMV.

FIG. 6.

Flow field averaged over a cardiac cycle. (a) and (b) Time averaged velocity magnitude and vectors. (a) DMV, (b) PMV. (c) and (d) Time averaged vorticity and stream trace. (c) DMV, (d) PMV.

Close modal

The vorticity and stream traces corresponding to the average velocity on the long-axis plane are presented in Figs. 6(c) and 6(d). The DMV case exhibits two elongated counter rotating vortices that extend from the mitral annulus, and these are associated with the propagation of the symmetric vortex ring as describe previously. The strongest area of the CW component of vorticity is located adjacent to the outflow tract and associated with an abrupt redirection of flow from the mitral annulus to the outflow tract. The region of counter-clockwise (CCW) vorticity is also relatively strong and extends along the lateral wall nearly halfway across the length of the ventricle.

For the PMV model, the vortical structures exhibit a large-scale asymmetry; the region of CW vorticity is significantly larger than the CCW vorticity region, and the CW vorticity region is also positioned at the center of ventricle. This dominant, centrally located CW vorticity is associated with a large scale CW circulation of the flow which is evident from the stream traces. An unbroken layer of CCW vorticity is also found on the lateral, apical, and septal walls of the ventricle, this is in contrast to the DMV case, and signifies the formation of a strong boundary layer on these walls. This is further indication of the strong CW flow that penetrates into every region of the ventricle for the PMV case.

In order to compare the strength of CW circulation between the DMV and PMV cases, the circulation is calculated for the area indicated by the red-dotted box in Figs. 6(c) and 6(d). The circulation values are found to be 65 cm2/s and 105 cm2/s for the DMV and PMV cases, respectively, and thus the CW circulation in the PMV case is about 40% stronger than the DMV case.

The above investigations reveal that the MVLs affect the mitral inflow propagation, subsequent LV penetration depth of the inflow, and overall circulatory flow pattern. These factors directly affect the “washout” of blood in the ventricle, and although the overall ventricular washout is primarily determined by the EF,17,28 local washout is more closely associated with the particular flow patterns that develop in the ventricle. The mitral inflow propagation speed and the LV penetration depth in particular affect the degree of apical washout. Poor apical washout could result in blood flow stasis and subsequent thrombus formation38–40 since flow stasis allows sufficient time for blood-tissue interaction and the bio-chemical reactions in the coagulation cascade.41 

In order to investigate the effect of different flow patterns caused by the MVL on the apical washout, we resort again to virtual ventriculography. To track the apical blood volume, the contrast pool is initially localized in the apical region as shown in Fig. 7(a), with an initial normalized concentration of 1.0. The transport and mixing of the contrast agent by the blood flow are then simulated, and the statistics of the contrast concentration such as peak, average, and standard deviation are calculated.

FIG. 7.

Contrast agent distribution on the long-axis plane (color contours of normalized concentration). (a) Initial distribution (beginning diastole). (b) End-systole (DMV). Peak concentration: 0.89, volume average: 0.057, and deviation: 0.10. (c) End-systole (PMV). Peak concentration: 0.19, volume average: 0.057, and deviation: 0.016.

FIG. 7.

Contrast agent distribution on the long-axis plane (color contours of normalized concentration). (a) Initial distribution (beginning diastole). (b) End-systole (DMV). Peak concentration: 0.89, volume average: 0.057, and deviation: 0.10. (c) End-systole (PMV). Peak concentration: 0.19, volume average: 0.057, and deviation: 0.016.

Close modal

Figures 7(b) and 7(c) show the distribution of contrast agent at the end systole after one cardiac cycle for the DMV and PMV cases, respectively. Note that, at the end systole, the total amount of contrast in the LV is nearly the same for both cases, and the volume-averaged concentration is also the same (value equal to 0.057). For the DMV case, however, the contrast agent is mostly concentrated along the septal wall of LV, and the highest concentration is found at the apical region with the maximum normalized concentration of 0.89. The standard deviation of concentration is about 0.10 which is about two times higher than the average value. This indicates poor apical washout and mixing caused by weak apical flow for the DMV case. On the other hand, for the PMV case, the contrast agent is effectively washed out from the apical region and mixed well over the entire LV. The peak concentration is 0.19, and the standard deviation is only about 0.016, which indicate good quality of apical washout and mixing.

In the present study, the effect of MVL and its motion on the intraventricular flow pattern is investigated by performing CFD simulations of flow in a model of a normal human LV with two radically different models of the mitral valve. The computational results show that the inclusion of a physiological mitral valve promotes the generation of a strong circulatory (or “looped”) flow pattern in the LV. This flow pattern has a significant effect on the local washout of ventricular blood from the apical region.

The vortex ring generated by the mitral jet during diastolic filling is one of the most well-known features of intraventricular flow, and the presence of MVL obviously affects the formation of this vortex ring. As shown in Figs. 2 and 3, for the PMV model, the vortex ring rolls up at the tips of the MVLs instead of the mitral annulus. This agrees with the observation of Bellhouse & Bellhouse from their seminal experimental study42 and also the recent phase contrast MRI (pcMRI) study of Charonko et al.43 The resulting vortex ring is therefore positioned at the middle of ventricle and helps the mitral inflow reach the apical region. Without the MVL, the vortex ring forms at the mitral annulus and begins its downward propagation from this initial location. Due to the moderately high Reynolds number (Re ≈ 4000) of the mitral jet and the interaction of the ring with the ventricular wall, the vortex ring becomes unstable and rapidly disintegrates before it reaches the apex. Consequently, as shown in Fig. 6(a), the apical flow without the MVLs is very weak. Also, as we have quantitatively investigated in Fig. 4, the non-physiological configuration of MV decreases the propagation speed of velocity wave and the penetration depth of mitral inflow.

The virtual ventriculography and LCS investigated in Fig. 3 clearly show the effect of MVL on the evolution of vortical flow pattern and transport of the atrial blood into the ventricle. The evolution of LCS for the PMV case in Fig. 3(b) shows that the vortex dominated flow and the blood volume transported by E-wave fill almost the entire ventricle at the end of the E-wave and this is in general agreement with the pattern for a healthy ventricle described in-vivo by Töger et al.35 The contrast agent transport pattern for the PMV shown in Fig. 3(b) also agrees with the in-vivo contrast echo study for the normal ventricle performed by Beppu et al.39 

Circulatory or “looped” flow in the LV which redirects the flow from mitral to outflow tract is another feature of intraventricular flows that is well-established from in-vivo visualizations.44 Several computational studies have shown that this rotational flow pattern results from the asymmetric development of the vortex ring.2,18,28 Such asymmetric vortex ring development however requires the imposition of some asymmetry on the mitral jet. In some previous computational studies where MVLs were absent, asymmetry has been introduced by adjusting the inflow direction2,28 or the location of mitral annulus.18 The current results show that the MVLs play an important role in the development of asymmetry in the vortex ring, which are quantitatively confirmed in Fig. 5. As shown in Figs. 3(b) and 6(b), the MVLs deflect the flow towards the lateral wall and this rapidly diminishes the strength of the vortex ring in the region adjacent to the lateral wall. This deflection of flow by the MVL towards the lateral wall has also been noted by Mihalef et al.10 in their patient-specific heart model. The pcMRI data of Charonko et al.43 also showed that the mitral inflow is deflected toward the lateral wall during deceleration phase of E-wave. The remaining portion (clockwise rotating component in the long-axis view) of the vortex ring then fills the ventricle and generates an overall clockwise circulating flow as shown in Fig. 6(d).

This stream pattern showing smooth redirection of LV flow also agrees with the in-vivo echocardiographic results of Hong et al.45 and Faludi et al.,1 and other in-vivo LV flow visualization results reported in Sengupta et al.46 The cycle-averaged vorticity showing the positive core at the center of the LV surrounded by negative vorticity is also quite similar to the measurement of Faludi et al.1 for the normal subject. Without the MVLs, on the other hand, the stream traces show an abrupt flow redirection which is mainly caused by the cardiac phase transition from diastole to systole. The strong rotational flow pattern with MVL also increases the apical flow strength as shown in Fig. 6.

A looped flow pattern in the LV has in the past been associated with the improved hydrodynamic efficiency of the LV.2,18,44 However, other studies13,28 have shown that the viscous dissipation resulting from different intraventricular flow patterns is negligibly small compared to the pressure work done on the LV flow; thus the effect of intraventricular flow patterns on the hydrodynamic efficiency of LV is not significant. Instead, it has been postulated that the most significant effect of diastolic flow patterns is on mixing and washout of blood in the ventricle.15,28 As mentioned in Sec. III, this has implications for thrombogenesis in the LV,38–40 since diminished washout, especially in the apical regions will allow sufficient time for interaction between the blood and the endocardium and the bio-chemical reactions associated with the platelet activation and coagulation cascade.41 In the current study, the effect of the mitral valve on blood mixing and washout has been examined using “virtual” ventriculography (Fig. 7). Given the weak apical flow in the DMV case, the ventricular blood in the apical region exhibits only weak washout and poor mixing as shown in Fig. 7(b). On the other hand, for the PMV case, the large-scale circulatory flow in the LV effectively washes out the blood pool in the apical region and mixes it with the other blood in the LV.

The current study also has important implications for computational modeling of ventricular flows. In most of previous computational studies, the effect of mitral valve is modeled in a highly simplified ways such as prescribing the mitral inflow velocity profile,2,16,18 variation of mitral annulus area,17,19–21 or diode type opening and closing.13–15 A reduced order model using the immersed surface has also been proposed47 and it has been shown that the inclusion of the leaflet surface leads to more realistic ventricular flow pattern. The present study also suggests that incorporating a mitral valve model based on the physiology would make the ventricular flow modeling more accurate and predictive.

While the left-ventricular and PMV model employed here incorporates many of the key features of physiologically realistic ventricles, it is worth noting some of these limitations of the current model. First, the motion of the leaflets in the PMV model is prescribed and not produced from FSI modeling. Thus, the leaflet structure and the flow are one-way coupled. An accurate FSI model of the leaflets would generate more complex features such as higher frequency leaflet “flutter” and leaflets curvature, as well as the timing of mitral valve opening and closing. One example of a two-way coupled, FSI for the cardiac flow and valve leaflets is the work done by McQueen and Peskin.9 However, the challenge in FSI modeling of the mitral valve is the complexity and uncertainty associated with constitutive modeling of the leaflets,3–5,8,23,24 as well as the difficulty of modeling large-scale deformation and leaflet contact. Thus, a two-way coupling between the leaflets and the flow does not necessarily imply a higher modeling fidelity with regard to the hemodynamics.

The present leaflet model is hinged at the mitral annulus and does not present any curvature in the axial direction and this might have some secondary effects on the flow. Also, the motion of the leaflets is assumed to be in-phase with the transmitral flow wave form. It has been reported that the mitral valve Doppler peak precedes the E-wave velocity peak by about 20 ms on average,48 which is only about 8% of the E-wave duration and less than 1% of the total cardiac cycle. Thus, an in-phase motion might be reasonable given all the other modeling assumption, although this needs to be verified by additional studies.

There is also the possibility of developing more accurate, image-based leaflet models that capture the intricate motions mentioned above, but the temporal and/or spatial resolution of existing imaging modalities (phase contrast MRI, cardiac CT, and echocardiography) may not be sufficient to allow such reconstruction at the current time. The other limitations of this study are the LV shape and endocardial motion employed, which although based on physiological data are nevertheless simplified. It is however expected that the above simplifications in modeling the LV motion as well as the leaflets in the PMV model do not significantly diminish the main objective of the current study, which is to assess the dominant effects of the inclusion of MV leaflets in LV simulations.

In this study, the effect of mitral valve leaflets and their motion on the dynamics of flow in the left ventricle has been investigated. A comparison of the results for a simple diode-mitral valve and a physiological mitral valve model shows that the valve leaflets influence the location of vortex ring formation and promote the formation of looped flow in the left ventricle. The mitral valve leaflets also affect flow propagation velocity and the strength of the apical flow as well as the local washout of ventricular blood. These effects have direct implications for the ability of a healthy ventricle to reduce flow statis in the apical regions and avoid left-ventricular thrombogenesis. Given these significant effects on diastolic flow patterns, it is strongly recommended that CFD models of left-ventricular flows include reasonably realistic representations of the mitral valve.

This research is supported by NSF through Grant Nos. IOS-1124804 and IIS-1344772. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by NSF Grant No. TG-CTS100002.

Temporal parameters for the flow rate specification shown in Fig. 1(d) are based on the data by Chung et al.25 for the heart rate of 60 BPM, and given by AT = 0.1, DT = 0.156, DIA = 0.205, Adur = 0.139, and SYS = 0.3 (s). We assumed that the peak systole time is at the one-third of SYS. The flowrate wave forms for E-wave (QE), A-wave (QA), and systole (QS) are obtained from the harmonic oscillator model26,27 and given by Eqs. (A1), (A2), and (A3), respectively,

Q E ( t ) = A E exp ( α E t ) sin π t A T + D T ,
(A1)
Q A ( t ) = A A sin ( ω A t ) ω A t exp ( ω A t ) ,
(A2)
Q S ( t ) = A S exp ( α S t ) sin π t S Y S ,
(A3)

where αE and αS are defined by

α E = π A T + D T cot π A T A T + D T , α S = π SY S cot π SY S p SY S ,
(A4)

where SYSp = SYS/3 and ωA is obtained from the solution of QA (Adur) = 0. Amplitude coefficients, AE, AA, and AS are related with the SV and EAR by

A E π ( A T + D T ) exp α E ( A T + D T ) + 1 α E 2 ( A T + D T ) 2 + π 2 + A A exp ( A dur ω A ) A dur ω A exp ( A dur ω A ) cos ( A dur ω A ) + 1 ω A 2 = S V ,
(A5)
A E exp ( α E A T ) sin π A T A T + D T = E A R A A 1 . 7 ω A exp ( 1 . 7 ) + sin 1 . 7 ω A ,
(A6)
A S π SY S exp α S SY S + 1 α S 2 S Y S 2 + π 2 = S V .
(A7)

The mitral valve leaflet Doppler echocardiography49 shows that the leaflet motion wave form is “M” shaped and resembles the transmitral flow wave form and we have assumed a similar profile for the opening of the leaflets. Data collected in-vivo48 also indicate that the leaflet opening precedes the transmitral flow rate only by about 20 ms and this is connected with the very low inertia of the leaflets. In light of this in-vivo data and also with the purpose of avoiding the introduction of an additional parameter in our model, we have assumed that the leaflet motion is in-phase with the transmitral flow wave form. The temporal functions of the opening angle of mitral valve leaflets, ϕi, where i = AL or PL shown in Fig. 1(e) are given by

ϕ i ( t ) = ϕ i , 0 + ( ϕ i , f ϕ i , 0 ) sin π t 2 t E P 0 t t E P ϕ i , f + ( ϕ i , m ϕ i , f ) 1 2 1 cos π t t E P t E E t E P t E P t t E E ϕ i , m t E E t t A S ϕ i , m + ( ϕ i , f ϕ i , m ) 1 2 1 cos π t t A S t A P t A S t A S t t A P ϕ i , f + ( ϕ i , 0 ϕ i , f ) 1 2 1 cos π t t A P t E D t A P t A P t t E D ,
(B1)

where ϕi,0 is the initial angle just after the separation of two leaflets from the coaptation, ϕi,f is the angle at the fully opened position, ϕi,m is the angle during diastasis, and tEP = 0.1, tEE = 0.256, tAS = 0.461, tAP = 0.539, and tED = 0.65 (s) correspond to the times of peak E-wave, end E-wave, beginning A-wave, peak A-wave, and end diastole, respectively. In the present study, we set ϕAL,0 = 24.78°, ϕPL,0 = 70.6°, ϕAL,f = ϕPL,f = 90°, ϕAL,m = 45°, and ϕPL,m = 70.6°. These angles are based on the measurements from the echocardiogram as depicted in Fig. 1(c). The measured angles are slightly modified to be fitted to the present computational model of the left ventricle.

1.
R.
Faludi
,
M.
Szulik
,
J.
D’hooge
,
P.
Herijgers
,
F.
Rademakers
,
G.
Pedrizzetti
, and
J.-U.
Voigt
, “
Left ventricular flow patterns in healthy subjects and patients with prosthetic mitral valves: An in vivo study using echocardiographic particle image velocimetry
,”
J. Thorac. Cardiovasc. Surg.
139
(
6
),
1501
1510
(
2010
).
2.
G.
Pedrizzetti
,
F.
Domenichini
, and
G.
Tonti
, “
On the left ventricular vortex reversal after mitral valve replacement
,”
Ann. Biomed. Eng.
38
(
3
),
769
773
(
2010
).
3.
V.
Prot
,
B.
Skallerud
,
G.
Sommer
, and
G. A.
Holzapfel
, “
On modelling and analysis of healthy and pathological human mitral valves: Two case studies
,”
J. Mech. Behav. Biomed. Mater.
3
,
167
177
(
2010
).
4.
M.
Stevanella
,
F.
Maffessanti
,
C. A.
Conti
,
E.
Votta
,
A.
Arnoldi
,
M.
Lombardi
,
O.
Parodi
,
E. G.
Caiani
, and
A.
Redaelli
, “
Mitral valve patient-specific finite element modeling from cardiac MRI: Application to an annuloplasty procedure
,”
Cardiovasc. Eng. Technol.
2
,
66
76
(
2011
).
5.
Q.
Wang
and
W.
Sun
, “
Finite element modeling of mitral valve dynamic deformation using patient-specific multi-slices computed tomography scans
,”
Ann. Biomed. Eng.
41
(
1
),
142
153
(
2013
).
6.
D. R.
Einstein
,
K. S.
Kunzelman
,
P. G.
Reinhall
,
M. A.
Nicosia
, and
R. P.
Cochran
, “
Non-linear fluid-coupled computational model of the mitral valve
,”
J. Heart Valve Dis.
14
(
3
),
376
385
(2005). Available at https://www.icr-heart.com/?cid=1552.
7.
B. E.
Griffith
,
X.
Luo
,
D. M.
Mcqueen
, and
C. S.
Peskin
, “
Simulating the fluid dynamics of natural and prosthetic heart valves using the immersed boundary method
,”
Int. J. Appl. Mech.
1
(
1
),
137
177
(
2009
).
8.
P. N.
Wattona
,
X. Y.
Luob
,
M.
Yinc
,
G. M.
Bernaccad
, and
D. J.
Wheatleyd
, “
Effect of ventricle motion on the dynamic behavior of chorded mitral valves
,”
J. Fluids Struct.
24
(
1
),
58
74
(
2008
).
9.
D. M.
Mcqueen
and
C. S.
Peskin
, “
A three-dimensional computer model of the human heart for studying cardiac fluid dynamics
,”
ACM SIGGRAH Comput. Graphics
34
(
1
),
56
60
(
2000
).
10.
V.
Mihalef
,
R.
Ionasec
,
P.
Shuarma
,
B.
Georgescu
,
I.
Voigt
,
M.
Suehling
, and
D.
Comaniciu
, “
Patient-specific modeling of whole heart anatomy, dynamics, and hemodynamics from four-dimensional cardiac CT images
,”
J. R. Soc. Interface Focus
1
,
286
296
(
2011
).
11.
K.
Tiroch
,
M.
Vorpahl
, and
M.
Seyfarth
, “
Novel mitral clipping technique overcoming extreme atrial dilatation
,”
Catheter. Cardiovasc. Interv.
84
,
606
609
(
2013
).
12.
E. N.
Feins
,
H.
Yamauchi
,
G. R.
Marx
,
F. P.
Freudenthal
,
H.
Liu
,
P. J.
Del Nido
, and
N. V.
Vasilyev
, “
Repair of posterior mitral valve prolapse with a novel leaflet plication clip in an animal model
,”
J. Thorac. Cardiovasc. Surg.
147
(
2
),
783
790
(
2014
).
13.
H.
Watanabe
,
S.
Sugiura
, and
T.
Hisada
, “
The looped heart does not save energy by maintaining the momentum of blood flowing in the ventricle
,”
Am. J. Physiol.: Heart Circ. Physiol.
294
,
H2191
H2196
(
2008
).
14.
H.
Watanabe
,
S.
Sugiura
,
H.
Kafuku
, and
T.
Hisada
, “
Mutiphysics simulation of left ventricle filling dynamics using fluid-structure interaction finite element method
,”
Biophys. J.
87
,
2074
2085
(
2004
).
15.
X.
Zheng
,
J. H.
Seo
,
V.
Vedula
,
T.
Abraham
, and
R.
Mittal
, “
Computational modeling and analysis of intracardiac flows in simple models of the left ventricle
,”
Euro. J. Mech.-B/Fluids
35
,
31
39
(
2012
).
16.
F.
Domenichini
,
G.
Pedrizzetti
, and
B.
Baccani
, “
Three-dimensional filling flow into a model left ventricle
,”
J. Fluid Mech.
539
,
179
198
(
2005
).
17.
T.
Doenst
,
K.
Spiegel
,
M.
Reik
,
M.
Markl
,
J.
Henning
,
S.
Nitzsche
,
F.
Beyersdorf
, and
H.
Oertel
, “
Fluid-dynamics modeling of the human left ventricle: Methodology and application to surgical ventricular reconstruction
,”
Ann. Thorac. Surg.
87
,
1187
1195
(
2009
).
18.
G.
Pedrizzetti
and
F.
Domenichini
, “
Nature optimizes the swirling flow in the human left ventricle
,”
Phys. Rev. Lett.
95
,
108101
(
2005
).
19.
N. R.
Saber
,
A. D.
Gosman
,
N. B.
Wood
,
P. J.
Kilner
,
C. L.
Charrier
, and
D. N.
Firmin
, “
Computational flow modeling of the left ventricle based on in vivo MRI data: Initial experience
,”
Ann. Biomed. Eng.
29
,
275
283
(
2001
).
20.
N. R.
Saber
,
N. B.
Wood
,
A. D.
Gosman
,
R. D.
Merrifield
,
G.
Yang
,
C. L.
Charrier
,
P. D.
Gatahouse
, and
D. N.
Firmin
, “
Progress towards patient-specific computational flow modeling of the left heart via combination of magnetic resonance imaging with computational fluid dynamics
,”
Ann. Biomed. Eng.
31
,
42
52
(
2003
).
21.
T.
Schenkel
,
M.
Malve
,
M.
Markl
,
B.
Jung
, and
H.
Oertel
, “
MRI-based CFD analysis of flow in a human left ventricle methodology and application to a healthy heart
,”
Ann. Biomed. Eng.
37
,
505
515
(
2009
).
22.
N.
Ranganathan
,
J. H. C.
Lam
,
E. D.
Wigle
, and
M. D.
Silver
, “
Morphology of human mitral valve: II. The valve leaflets
,”
Circulation
41
,
459
467
(
1970
).
23.
C. H.
Lee
,
R.
Amini
,
R. C.
Gorman
,
J. H.
Gorman 3rd
, and
M. S.
Sacks
, “
An inverse modeling approach for stress estimation in mitral valve anterior leaflet valvuloplasty for in-vivo valvular biomaterial assessment
,”
J. Biomech.
47
(
9
),
2055
2063
(
2014
).
24.
M. K.
Rausch
,
N.
Famaey
,
T. O.
Shultz
,
W.
Bothe
,
D. C.
Miller
, and
E.
Kuhl
, “
Mechanics of the mitral valve: A critical review, an in vivo parameter identification, and the effect of prestrain
,”
Biomech. Model. Mechanobiol.
12
(
5
),
1053
1071
(
2013
).
25.
C. S.
Chung
,
M.
Karamanoglu
, and
S. J.
Kovács
, “
Duration of diastole and its phases as a function of heart rate during supine bicycle exercise
,”
Am. J. Physiol.: Heart Circ. Physiol.
287
,
H2003
H2008
(
2004
).
26.
A. F.
Hall
and
S. J.
Kovács
, “
Automated method for characterization of diastolic transmitral Doppler velocity contours: Early rapid filling
,”
Ultrasound Med. Biol.
20
(
2
),
107
116
(
1994
).
27.
A. F.
Hall
,
J. A.
Aronovitz
,
S. P.
Nudelman
, and
S. J.
Kovács
, “
Automated method for characterization of diastolic transmitral Doppler velocity contours: Late atrial filling
,”
Ultrasound Med. Biol.
20
(
9
),
859
869
(
1994
).
28.
J. H.
Seo
and
R.
Mittal
, “
Effect of diastolic flow pattern on the function of the left ventricle
,”
Phys. Fluids
25
,
110801
110821
(
2013
).
29.
R.
Mittal
,
H.
Dong
,
M.
Bozkurttas
,
F. M.
Najjar
,
A.
Vargas
, and
A.
von loebbecke
, “
A versatile sharp interface method for incompressible flows with complex boundaries
,”
J. Comput. Phys.
227
(
10
),
4825
4852
(
2008
).
30.
J. H.
Seo
,
V.
Vedula
,
T.
Abraham
, and
R.
Mittal
, “
Multiphysics computational models for cardiac flow and virtual cardiography
,”
Int. J. Numer. Methods Biomed. Eng.
29
(
8
),
850
869
(
2013
).
31.
S.
Fortini
,
G.
Querzoli
,
S.
Espa
, and
A.
Cenedese
, “
Three-dimensional structure of the flow inside the left ventricle of the human heart
,”
Exp. Fluids
54
,
1609
(
2013
).
32.
J.
Jeong
and
F.
Hussain
, “
On the identification of vortex
,”
J. Fluid Mech.
285
,
69
94
(
1995
).
33.
T.
Kim
,
A. Y.
Cheer
, and
H. A.
Dwyer
, “
A simulated dye method for flow visualization with a computational model for blood flow
,”
J. Biomech.
37
,
1125
1136
(
2004
).
34.
S. C.
Shadden
and
C. A.
Taylor
, “
Characterization of coherent structures in the cardiovascular system
,”
Ann. Biomed. Eng.
36
(
7
),
1152
1162
(
2008
).
35.
J.
Töger
,
M.
Kanski
,
M.
Carlsson
,
S. J.
Kovács
,
G.
Söderlind
,
H.
Arheden
, and
E.
Heiberg
, “
Vortex ring formation in the left ventricle of the heart: Analysis by 4D flow MRI and Lagrangian coherent structures
,”
Ann. Biomed. Eng.
40
(
12
),
2652
2662
(
2012
).
36.
M. W.
Sessoms
,
J.
Lisauskas
, and
S. J.
Kovács
, “
The left ventricular color M-mode Doppler flow propagation velocity V(p): In vivo comparison of alternative methods including physiologic implications
,”
J. Am. Soc. Echocardiogr.
15
(
4
),
339
348
(
2002
).
37.
K. C.
Stewart
,
J. C.
Charonko
,
C. L.
Niebel
,
W. C.
Little
, and
P. P.
Vlachos
, “
Left ventricular vortex formation is unaffected by diastolic impairment
,”
Am. J. Physiol.: Heart Circ. Physiol.
303
(
10
),
H1255
H1262
(
2012
).
38.
B. J.
Delemarre
,
C. A.
Visser
,
H.
Bot
, and
A. J.
Dunning
, “
Prediction of apical thrombus formation in acute myocardial infarction based on left ventricular spatial flow pattern
,”
J. Am. Coll. Cardiol.
15
(
2
),
355
360
(
1990
).
39.
S.
Beppu
,
S.
Izumi
,
K.
Miyatake
,
S.
Nagata
,
Y. D.
Park
,
H.
Sakakibara
, and
Y.
Nimura
, “
Abnormal blood pathways in left ventricular cavity in acute myocardial infarction. Experimental observations with special reference to regional wall motion abnormality and hemostasis
,”
Circulation
78
(
1
),
157
164
(
1988
).
40.
R.
Delewi
,
F.
Zijlstra
, and
J. J.
Piek
, “
Left ventricular thrombus formation after acute myocardial infarction
,”
Heart
98
,
1743
1749
(
2012
).
41.
B.
Furie
and
B. C.
Furie
, “
Mechanisms of thrombus formation
,”
N. Engl. J. Med.
359
(
9
),
938
949
(
2008
).
42.
B. J.
Bellhouse
and
F. H.
Bellhouse
, “
Fluid mechanics of mitral valve
,”
Nature
224
,
615
616
(
1969
).
43.
J. J.
Charonko
,
R.
Kumar
,
K.
Stewart
,
W. C.
Little
, and
P. P.
Vlachos
, “
Vortices formed on the mitral valve tips aid normal left ventricular filling
,”
Ann. Biomed. Eng.
41
(
5
),
1049
1061
(
2013
).
44.
P. J.
Kilner
,
G.
Yang
,
A. J.
Wilkes
,
R. H.
Mohiaddin
,
N.
Firmin
, and
H.
Ycoub
, “
Asymmetric redirection of flow through the heart
,”
Nat. Lett.
404
,
759
761
(
2000
).
45.
G.-R.
Hong
,
G.
Pedrizzetti
,
G.
Tonti
,
P.
Li
,
Z.
Wei
,
J. K.
Kim
,
A.
Baweja
,
S.
Liu
,
N.
Chung
,
H.
Houle
,
J.
Narula
, and
M. A.
Vannan
, “
Characterization and quantification of vortex flow in the human left ventricle by contrast echocardiography using vector particle image velocimetry
,”
J. Am. Coll. Cardio. Imaging
1
(
6
),
705
717
(
2008
).
46.
P. P.
Sengupta
,
G.
Pedrizzetti
,
P. J.
Kilner
,
A.
Kheradvar
,
T.
Ebbers
,
G.
Tonti
,
A. G.
Fraser
, and
J.
Narula
, “
Emerging trends in CV flow visualization
,”
JACC Cardiovasc. Imaging
5
(
3
),
305
316
(
2012
).
47.
M.
Astorino
,
J.
Hamers
,
S. C.
Shadden
, and
J.-F.
Gerbeau
, “
A robust and efficient valve model based on resistive immersed surfaces
,”
Int. J. Numer. Methods Biomed. Eng.
28
,
937
959
(
2012
).
48.
A. W.
Bowman
,
P. A.
Frihauf
, and
S. J.
Kovács
, “
Time-varying effective mitral valve area: Prediction and validation using cardiac MRI and Doppler echocardiography in normal subjects
,”
Am. J. Physiol.: Heart Circ. Physiol.
287
(
4
),
H1650
H1657
(
2004
).
49.
A. S.
Omran
,
A. A.
Arifi
, and
A. A.
Mohamed
, “
Echocardiography of the mitral valve
,”
J. Saudi Heart Assoc.
22
(
3
),
165
170
(
2010
).