Validation of a 3D computational fluid-structure interaction model simulating flow through an elastic aperture

Abstract

This work presents a validation of a fluid-structure interaction computational model simulating the flow conditions in an in vitro mock heart chamber modeling mitral valve regurgitation during the ejection phase during which the trans-valvular pressure drop and valve displacement are not as large. The mock heart chamber was developed to study the use of 2D and 3D color Doppler techniques in imaging the clinically relevant complex intra-cardiac flow events associated with mitral regurgitation. Computational models are expected to play an important role in supporting, refining, and reinforcing the emerging 3D echocardiographic applications. We have developed a 3D computational fluid-structure interaction algorithm based on a semi-implicit, monolithic method, combined with an arbitrary Lagrangian-Eulerian approach to capture the fluid domain motion. The mock regurgitant mitral valve corresponding to an elastic plate with a geometric orifice, was modeled using 3D elasticity, while the blood flow was modeled using the 3D Navier-Stokes equations for an incompressible, viscous fluid. The two are coupled via the kinematic and dynamic conditions describing the two-way coupling. The pressure, the flow rate, and orifice plate displacement were measured and compared with numerical simulation results. In-line flow meter was used to measure the flow, pressure transducers were used to measure the pressure, and a Doppler method developed by one of the authors was used to measure the axial displacement of the orifice plate. The maximum recorded difference between experiment and numerical simulation for the flow rate was 4%, the pressure 3.6%, and for the orifice displacement 15%, showing excellent agreement between the two.

1 Introduction

Mitral regurgitation (MR) is a disorder of the heart in which the mitral valve fails to close completely during ventricular systole. This causes blood to flow back from the left ventricle into the left atrium. See Figure 1. MR can lead to atrial arrhytmias, pulmonary artery hypertension, congestive heart failure, and death. The decision to proceed with surgical valve repair or replacement is based on an assessment of symptoms and valve regurgitation severity. Although echocardiography is the primary tool to assess MR, the accurate and reproducible quantification of regurgitant volume is an ongoing challenge. Rapid improvements in imaging technology, including single beat 3D color Doppler imaging, are providing new tools to face this challenge. Computational models of mitral valve regurgitant flows are expected to play important role in supporting, refining, and reinforcing the emerging 3D echocardiographic applications [29]. Once validated, a computational model provides detailed, point-wise information about the quantities that are used in echocardiographic assessment of MR, thereby providing information that can be used to tune, and refine the already existing protocols, or design new protocols for the emerging new technologies. The quantities used in the echocardiographic assessment of MR such as the proximal isovelocity surface area (PISA), the regurgitant jet area, vena contracta, or the Coanda effect, can all be constructed using computational simulations at a fraction of the cost of the imaging modalities (e.g., echo and MRI or CT) that would have to be used to provide the information as detailed as that obtained using computer simulations.

Figure 1.

(a) Anatomy of the heart showing the mitral valve. (b) Echocardiographic image of central regurgitant jet flowing from the left ventricle (LV) to the left atrium (LA).

In this work we validate a computational model of fluid-structure interaction (FSI) against experiments performed in an in vitro mock heart chamber shown in Figure 2, developed at the Methodist DeBakey Heart & Vascular Center [22] to study the use of 2D and 3D color Doppler techniques in imaging the clinically relevant complex intra-cardiac flow events associated with MR [22, 23]. The FSI model simulates the interaction between a mock regurgitant valve and pulsatile flow in the mock heart chamber connected to a flow loop driven by a pulsatile Left Ventricular Assist Device. The flow conditions generated in the heart chamber mimic the systolic ventricular flow with MR during which the trans-valvular pressure drop and valve displacement are relatively small. The mock regurgitant valves, consisting of an elastic plate with a geometric orifice, were created to reflect the effective regurgitant orifice area clinically encountered in patients with significant mitral regurgitation (0.3 – 0.4cm²). The geometry of the orifice (arc or slot shaped) was designed to reflect the known geometry of those clinical situations when MR volume is most difficult to quantify using standard echo Doppler methods. In patients with functional MR the leaflets are pulled apart in systole due to adverse remodeling of the left ventricle. This tethering of the leaflets creates an arc-shaped coaptation defect which significantly limits the standard application of 2D color Doppler quantification techniques (2D PISA or vena contracta diameter) which rely on assumptions of circular orifice geometry. Likewise, in patients with primary leaflet pathology (such as prolapse) the effective flow orifice is more slot-shaped than circular with similar limitations to commonly employed Doppler techniques for flow quantification.

Figure 2.

Experimental set-up. Arrows indicate flow direction.

In our computational FSI model the flexible plate with an orifice, simulating the mock regurgitant valve, shown in Figure 3, was modeled using 3D linear elasticity, while the pressure-driven fluid flow was modeled using the Navier-Stokes equations for an incompressible, viscous, Newtonian fluid. Our computational method is based on a semi-implicit, monolithic Finite Element Method (FEM), recently proposed in [28, 2, 30, 3]. To track the motion of the mock mitral valve and the surrounding fluid, an Arbitraty Lagrangian Eulerian (ALE) method was adopted. Three different flow scenarios were simulated, ranging from mild (15 ml/beat) to moderate (30 ml/beat) mitral regurgitation. Local flow conditions near the orifice were measured and compared with the results of the computational model. The three physical quantities that determine the solution to the underlying fluid-structure interaction problem (the pressure, the flow rate, and the structure displacement) were measured. In-line flow meter was used to measure the flow, pressure transducers were used to measure the pressure on both sides of the orifice, and a Doppler method developed by one of the authors in [17] was used to measure the axial displacement of the orifice plate. The maximum recorded difference between the experimental measurements and the results of the numerical simulations was 4% for the flow rate, 3.6% for the pressure, and the maximum difference in the displacement was 15%, showing excellent agreement between the two.

Figure 3.

Structure mesh for (a) flat plate with rectangular orifice; (b) bulged plate with arc-shaped orifice; (c) side view of (b).

This work presents one of the few studies in which computational FSI results are compared with experimental measurements in the context of cardiovascular simulations. We mention [4, 5, 26] for a validation of 1D reduced models, [9, 11, 12, 33, 20] for a validation of 2D simulations, and [24, 31, 8, 15] for a validation of 3D models describing blood flow-valve interaction. Among the most popular computational FSI methods in this area are the Immersed Boundary method [27], the space-time approach [1, 34, 35], the Fictitious Domain method [16], the Coupled Momentum method [14], the Lattice-Boltzmann method [21], and the Level Set method [7]. To track the motion of the fluid domain, versions of the Arbitrary Lagrangian-Eulerian (ALE) approach [19, 13] have been used, see, e.g., [12, 24, 31, 8, 15]. Although a great variety of algorithms for the numerical simulation of FSI problem in hemo-dynamics exists in the literature (see, e.g., [28] for a review), there are still many problems that remain open.

The FSI model presented in this manuscript shows superb performance with respect to the experimental measurements, and it presents a first step in our program to combine computational fluid dynamics (CFD) studies with experimental models of mitral regurgitation to support the development of the emerging imaging technologies for the assessment of severity of MR.

2 Experimental set-up

The experimental set up consisted of a pulsatile circulatory flow loop containing a mock heart chamber, as shown in Figure 2. The flow loop was designed to achieve up to 7 L/min forward flow. To mimic blood viscosity (0.035 poise) 30% glycerin was added to water with corn starch (less than 1%) to serve as ultrasound scattering particles. Total flow rate was assessed using an ultrasonic flow meter (Transonic Systems, Ithaca, NY, USA). Using constant pump volume displacement and frequency, flow volume tailored to experimental need was directed into the regurgitant limb by increasing downstream resistance within the circulatory loop. Regurgitant volume per cardiac cycle was estimated by dividing the flow rate reading from the flow meter (ml/min) by the stroke frequency per minute.

The regurgitant loop incorporated an imaging chamber composed of two acrylic cylinders partitioned by an elastic divider plate containing a geometric orifice, as shown in Figure 2. The divider plate was made of silicone medical rubber, whose mechanical properties were obtained from manufacturer specifications, and are reported in Table 1. Two deformable orifices differing in size and shape were used in experiments: a 0.35 cm² rectangle (2.2 cm × 0.16 cm) on a flat silicon divider plate, and a 0.4 cm² arc (width 0.16 cm; radius 1.43 cm) on a bulged divider plate, see Figure 3.

Table 1.

Fluid and structure parameters.

Fluid parameters
Density: ρf = 1 0 g/cm3	Viscosity: μ = 0.035 poise
Structure parameters
Density: ρs = 1.25 g/cm3	Thickness: hs = 0.15 cm
Young modulus: E = 6 · 107 dyne/cm2	Poisson ratio: ν = 0.49

The motion of displacement of the orifice was measured by Doppler methods [17]. Briefly, a 10 MHz ultrasound transducer was held at one end of the flow chamber with the sound beam pointed at the divider plate near the orifice. The Doppler was set to process the echoes from the orifice plate and display its displacement during the pumping cycle as shown in Figures 4(c), 5 (c), and 6(c), with a resolution of 20 μm.

Figure 4.

Experiment 1: Comparison between measured (in dashed-line) and simulated (in solid line) pressure at transducer 1 (a), pressure at transducer 2 (b), axial displacement at the orifice (c), and flow rate (d). The legend in (b) refers to the four subfigures.

Figure 5.

Experiment 2: Comparison between measured (in dashed-line) and simulated (in solid line) pressure at transducer 1 (a), pressure at transducer 2 (b), axial displacement at the orifice (c), and flow rate (d). The legend in (b) refers to the four subfigures.

Figure 6.

Experiment 3: Comparison between measured (in dashed-line) and simulated (in solid line) pressure at transducer 1 (a), pressure at transducer 2 (b), axial displacement at the orifice (c), and flow rate (d). The legend in (b) refers to the four subfigures.

Pressure transducers (Merit Medical, South Jordan, UT, USA) were attached and positioned on either side of the divider plate via short fluid-filled lines to record peak chamber pressure and the transorifice pressure difference. Additionally, inflow and outflow pressures were considered for all experiments.

Three experiments were considered corresponding to three different flow rates: 15 ml/beat (mild MR), 25 ml/beat (mild-to-moderate MR), and 35 ml/beat (moderate MR). The rectangular orifice was used in the experiment with 35 ml/beat, while the arc-shaped orifice was used in the experiments with 15 and 25 ml/beat.

Incorporated into the imaging chamber were ultrasound windows at standard cardiac anatomic position and distance to the flow orifice, mimicking the apical and parasternal clinical imaging windows. A 2D spectral Doppler transducer (2 to 4 MHz,) and a hand-held 3D-Color Doppler transducer (X4, Sonos 7500, Philips Medical Systems, Andover, MA) can be used to assess transorifice flow from an apical equivalent view parallel to regurgitant flow and from a parasternal equivalent view perpendicular to flow.

3 The CFD Model

3.1 The Model Equations

The equations modeling the fluid and structure are defined on domains that depend on time and are not know a priori, due to the two-way fluid-structure coupling. ${\mathrm{\Omega }}_t^f$ will be denoting the fluid domain corresponding to the rigid chamber minus the elastic plate containing an orifice, and ${\mathrm{\Omega }}_t^s$ the elastic structure corresponding to the elastic plate. The common boundary between ${\mathrm{\Omega }}_t^f$ and ${\mathrm{\Omega }}_t^s$ is the fluid-structure interface Σ_t.

The fluid problem is governed by the incompressible Navier-Stokes equations

$$\begin{array}{cc}{\rho }_f{\partial }_t\mathit{u}+{\rho }_f\mathit{u}·\nabla \mathit{u}-\nabla ·{\mathit{\sigma }}_f={\mathit{f}}_f& \mathrm{in}{\mathrm{\Omega }}_t^f\times \left(0,T\right),\end{array}$$ (1)

$$\begin{array}{cc}\nabla ·\mathit{u}=0& \mathrm{in}{\mathrm{\Omega }}_t^f\times (0,T),\end{array}$$ (2)

where ρ_f is the fluid density, u is the fluid velocity, σ_f the Cauchy stress tensor, and f_f the body force. For Newtonian fluids:

$${\mathit{\sigma }}_f(\mathit{u},p)=-p\mathbf{I}+2{\mu }_f{\mathit{\epsilon }}_f(\mathit{u}),$$ (3)

where p is the fluid pressure, μ_f is the fluid dynamic viscosity coefficient, and ${\mathit{\epsilon }}_f(\mathit{u})=\frac{1}{2}(\nabla \mathit{u}+{(\nabla \mathit{u})}^T)$ is the strain rate tensor. The Reynolds, Strouhal, and Womersely numbers will be employed to describe the flow conditions generated by the experiments:

$$\mathit{Re}=\frac{{\rho }_f\mathit{LU}}{\mu },\mathit{Sr}=\frac{L}{\tau U},\alpha =\sqrt{2\pi (\mathit{Sr}·\mathit{Re})}.$$

Here, L is the characteristic length (the hydraulic diameter of the orifice), U is the characteristic velocity (the root mean square of the peak velocity at the orifice), and τ is the characteristic time (period of the flow pulse).

The time-dependent motion of the 3D elastic structure (the orifice plate) is modeled by the elastodynamics equations:

$$D_{\mathit{tt}}\mathit{\eta }-\frac{1}{\rho }\nabla ·{\mathit{\sigma }}_s={\mathit{f}}_s\mathrm{in}{\mathrm{\Omega }}_t^s\times (0,T).$$ (4)

Here η is the structure displacement, f_s is the body force, and D_t denotes the material derivative. Equation (4) is supplemented with the linearly elastic constitutive equations

$${\mathit{\sigma }}_s(\mathit{\eta })=\mu (\nabla \mathit{\eta }+{(\nabla \mathit{\eta })}^T)+\lambda (\nabla ·\mathit{\eta })\mathbf{I}$$ (5)

specifying a relationship between the Cauchy stress tensor σ_s and displacement η for the Saint-Venant Kirchhoff elastic model with linear strain-displacement relationship. Here μ and λ are the Lamé constants describing the properties of the linearly elastic structure.

The fluid and structure are coupled at the interface Σ_t via: the kinematic coupling condition describing continuity of velocity (the no-slip boundary condition):

$$\mathit{u}={\partial }_t\mathit{\eta }\mathrm{on}{\sum }_t\times (0,T),$$

and the dynamic coupling condition describing the balance of contact forces:

$${\mathit{\sigma }}_s·{\mathit{n}}_s+{\mathit{\sigma }}_f·{\mathit{n}}_f=0\mathrm{on}{\sum }_t\times (0,T).$$

Here, n_s denotes the outward normal to the structure domain boundary, and n_f denotes the outward normal to the fluid domain boundary (n_f = −n_s on Σ_t).

To numerically approximate equations on moving domains, we adopt an ALE approach in the context of a FEM approximation of the problem [18, 25, 30]. The ALE method deals effciently with the deformation of the mesh, especially near the fluid-structure interface, and with the issues related to the approximation of the time-derivatives ∂u/∂t ≈ (u(tⁿ⁺¹) − u(tⁿ))/Δt which, due to the fact that Ω_t depends on time, is not well defined since the values u(tⁿ⁺¹) and u(tⁿ) correspond to the values of u defined at two different domains. ALE approach is based on introducing an (arbitrary, invertible, smooth) mapping A_t defined on a single, fixed, reference domain ${\mathrm{\Omega }}_0^f$

$${\mathit{A}}_t:{\mathrm{\Omega }}_0^f\to {\mathrm{\Omega }}_t^f,{\mathit{A}}_t:x_0\mapsto x.$$

In our case we define A_t via the harmonic extension to the interior of the fluid domain of the values of η at the interface

$${\mathit{A}}_t(x_0)=x_0+\mathrm{Ext}(\mathit{\eta }(x_0,t)\mid {\sum }_0),x_0\in {\mathrm{\Omega }}_0^f.$$ (6)

Here Ext(η(x₀, t)∣ Σ₀) denotes the harmonic extension operator in the reference domain. Using this mapping, one can map the current domain ${\mathrm{\Omega }}_t^f$ onto the reference domain ${\mathrm{\Omega }}_0^f\left(\mathrm{via}{A}_t^{-1}\right)$, perform the time-differentiation on the reference domain, and then map everything back to the current domain ${\mathrm{\Omega }}_f^t$. During this process, the time derivative of the mapping A_t needs to be taken into account through the simple chain rule so that

$$\frac{\partial \mathit{u}}{\partial t}{\mid }_{x_0}=\frac{\partial \mathit{u}}{\partial t}{\mid }_x+\mathit{w}·{\nabla }_x\mathit{u},\mathrm{where}\mathit{w}=\frac{\partial {\mathit{A}}_t}{\partial t}.$$ (7)

Velocity w is often called the domain velocity or the mesh velocity.

With this approach, our moving-boundary problem in (strong) ALE form is given by the following: Find velocity u, pressure p and displacement η such that

$$\mathrm{FLUID}\{\begin{array}{llll}{\partial }_t\mathit{u}{\mid }_{x_0}+(\mathit{u}-\mathit{w})·\nabla \mathit{u}-\frac{1}{{\rho }_f}\nabla ·{\sigma }_f& =& {\mathit{f}}_f& \mathrm{in}{\mathrm{\Omega }}_t^f\times (0,T),\\ \hfill \nabla ·\mathit{u}& =& 0& \mathrm{in}{\mathrm{\Omega }}_t^f\times (0,T),\end{array}$$

$$\mathrm{STRUCTURE}\{D_{\mathit{tt}}\mathit{\eta }-\frac{1}{\rho }\nabla ·{\mathit{\sigma }}_s={\mathit{f}}_s,\mathrm{in}{\mathrm{\Omega }}_t^s\times (0,T),$$ (9)

$$\mathrm{COUPLING}\{\begin{array}{cccc}\hfill \mathit{u}& =& {\partial }_t\mathit{\eta },& \mathrm{on}{\sum }_t\times (0,T),\\ {\mathit{\sigma }}_s·n_s+{\mathit{\sigma }}_f·n_f& =& 0,& \mathrm{on}{\sum }_t\times (0,T),\end{array}$$ (10)

where σ_f and σ_s are given by (3) and (5), respectively. Problem (8)-(10) is supplemented with initial and boundary data.

3.2 Numerical algorithm

To solve this problem numerically, we employed a FSI solver which is based on a semi-implicit, monolithic FEM algorithm, developed in [28, 2, 30, 3]. The algorithm is called semi-implicit because the fluid-structure coupling is treated implicitly while the nonlinearities of the fluid-structure problem are treated explicitly. The algorithm is called monolithic because the fluid and the structure are solved simultaneously in a monolithic fashion.

A key feature of a numerical FSI algorithm is the way how the fluid-structure coupling is implemented. We have imposed the kinematic coupling condition in the strong sense, and the dynamic coupling condition in the weak sense.

The spatial discretization of the problem was performed using a single finite element partition of the entire domain, which implies matching grids at the fluid-structure interface. The structure equations were reformulated in terms of velocity instead of displacement. The same finite element spaces were then used for the fluid and structure velocities. In this context, the fluid-structure coupling conditions are easily implemented. To circumvent the discrete inf-sup condition, we used a stabilization technique called Orthogonal Subgrid Scale [10]. Due to the application of this technique, the same finite element interpolation space can be used for both the pressure and velocity unknowns. In particular, for the numerical simulation of the problem discussed in Section 4, we used ℙ₁ finite elements for all the unknowns.

The time discretization was performed by using the Implicit Euler Scheme for both the fluid and structure equations.

To deal with the nonlinearities in the problem, first notice that they are of two kind: the nonlinearity in the convective term in equation (8), and the nonlinearity due to domain motion. Our approach utilizes one fixed-point algorithm to deal with both [3]. First the problem is linearized, and then only one fixed point iteration is performed per time step as it was proved that this approach does not endanger stability [25], or spoil accuracy [3].

The fully discretized and linearized FSI problem leads to a (monolithic) linear system for the fluid velocity, fluid pressure, and structure velocity. To solve it, we used the GMRES algorithm [32] preconditioned in two steps: first a diagonal scaling of the FSI matrix is applied, and then the resulting system is preconditioned by an incomplete LU factorization (ILUT preconditioner). Preconditioning is needed since the conditioning of the FSI system matrix is degenerated due to a large difference in the magnitude of fluid and structure entries [2]. An important feature of the ILUT-GMRES method for FSI problem is that, unlike other FSI algorithms, its performance does not deteriorate when the structure density approaches the fluid one [6], as is the case in the hemodynamics applications.

4 Comparison: Numerical Results vs. Experiment

Three numerical experiments were performed to test the FSI solver against experimental measurements. The three flow conditions correspond to three different regurgitant volumes ranging from mild to moderate MR: 15, 25, and 35 ml/beat. Two different orifice shapes were considered: a rectangular orifice and an arc-shaped orifice, as shown in Figure 3. The corresponding flow parameters described by the Reynolds, Strouhal, and Womersley numbers are listed in Table 2.

Table 2.

Flow parameters.

Experiment	1	2	3
Reynolds number	2250	2340	3020
Strouhal number	0.0014	0.0036	0.003
Womersley number	4.4	7.2	7.5
Divider plate	flat	bulged	bulged
Orifice shape	rectangular	arc	arc
Regurgitant volume	35 ml/beat	15 ml/beat	25 ml/beat

The flexible plate containing a geometric orifice is immersed in the fluid, while it is fixed to the rigid walls of the chamber at the plate perimeter. The values of the plate’s structure parameters are shown in Table 1. The chamber contains an ultrasound window at an angle on one side, as shown in Figure 2. The corresponding computational domain is shown in Figure 2(d). The no-slip boundary condition was prescribed at the walls of the chamber, while the inlet and outlet boundary conditions were the fluid velocity and normal stress, respectively. The inlet velocity was obtained from the measured flow rate, while the outlet normal stress was obtained from the measured pressure modified by a (small) contribution due to the velocity gradient. The initial conditions were taken to be u(x, 0) = 0, p(x, 0) = 0, while the structure was assumed to be in the reference configuration and with initial velocity equal to zero.

Space discretization of the computational domain was achieved using a mesh of tetrahedra. In order to capture the fine flow structure in different parts of the domain, unstructured non-uniform meshes were employed. Meshes with different level of refinement were tested. The results reported in Figures 4, 5, and 6 are mesh-indepedent within 2% for the range of flow conditions under consideration. More precisely, the results reported in this paper were obtained with the mesh size for which the relative L²-error in the simulated quantities was less than 2% when compared with the results obtained with a coarser mesh. In particular, we used the meshes obtained by setting the mesh diameter to 0.02 cm at the orifice, 0.03 cm in the structure volume (i.e., 1/5 of the structure thickness), and 0.5 cm in the fluid volume, resulting in approximately 17000 nodes and 93000 tetrahedra. The time step was set to 10⁻²s.

Pressure was recorded at two locations: the inlet chamber (Transducer 1), and the outlet chamber (Transducer 2). See Figure 2.

For every experiment in Table 2, the pressure at Transducer 1 and 2, the flow rate, and the structure displacement at the orifice (axial component) were acquired and filtered from noise. The sensitivity of the pressure transducers was +/− 1 mmHg, the Transonic Flow meter +/− 5%, and the Doppler method used to capture structure displacement 20 μm. The comparisons between the measured and simulated quantities are reported in Figures 4, 5, 6 for experiment 1, 2, and 3, respectively. The 3D displacement of the elastic orifice plate is shown in Figure 7. To quantitate the agreement, we report in Table 3 the relative difference in the L²-norm between the measurements and the numerical results. We find more than 85% agreement for each quantity in each experiment. The discrepancy in the peak orifice displacement reported in Figure 4(c) can likely be attributed to the location of the probe which was pointing directly through the orifice as the systolic peak when the orifice size is stretched the most.

Figure 7.

Structure displacement at time (a) t = 0.22 s, (b) t = 0.33 s, and (c) t = 0.44 s for experiment 3. The legend shows displacement magnitude in mm.

Table 3.

Relative L² difference between measured and computed pressure at transducer 1 (p₁), pressure at transducer 2 (p₂), displacement at the orifice (η∣_orifice), and flow rate for every experiment. The number in parenthesis is the percentage of agreement.

Experiment	p1	p2	η∣orifice	flow rate
1	0.035 (96.5%)	0.016 (98.4%)	0.15 (85%)	0.04 (96%)
2	0.011 (98.9%)	0.002 (99.8%)	0.015 (98.5 %)	0.017 (98.3%)
3	0.014 (98.6%)	0.036 (96.4%)	0.026 (97.4 %)	0.013 (98.7%)

5 Discussion and Conclusions

In this work we validated a computational FSI algorithm against experiments performed at a mock heart chamber simulating the flow conditions associated with MR. This is a first step in our program to reconstruct, using computer simulations, the quantities relevant for the echocardiographic assessment of MR, such as the Proximal Isovelocity Surfaces, streamlines through the orifice and Vena Contracta, regurgitant jet, Coanda effect, etc. As an example we show in Figure 8 the Proximal Isovelocity Surfaces obtained using 3D computer simulations (subfigure (a)) and 3D Color Doppler (subfigure (b)), for an arc-shaped orifice, viewed from a side. The proximal isovelocity surface (40 cm/s), located under the regurgitant orifice, is shown in yellow on both figures. The corresponding flow conditions and streamlines are shown in Figure 9. A future goal is to combine the detailed flow information from computer simulations with the echocardiographic images of the same flow conditions to understand the limitations, and improve the use, of 2D and 3D color Doppler techniques in imaging and assessing the severity of MR.

Figure 8.

A 2D slice of Proximal Isovelocity Surface captured using numerical simulations (a) and 3D color Doppler (b). The Proximal Isovelocity Surface corresponding to v = 40 cm/s, depicted in yellow on both figures, is located under the orifice, in the flow convergence zone.

Figure 9.

A snap-shot of the streamlines in the flow chamber for the flow conditions shown in Figure 8.