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Journal of Biomechanics 41 (2008) 2539–2550

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Numerical simulation of the dynamics of a bileaflet prosthetic heartvalve using a fluid–structure interaction approach

Matteo Nobilia, Umberto Morbiduccib, Raffaele Ponzinic, Costantino Del Gaudiod,Antonio Balduccid, Mauro Grigionid, Franco Maria Montevecchib, Alberto Redaellia,�

aDepartment of Bioengineering, Politecnico di Milano, P.za Leonardo da Vinci, 32, 20133 Milano, ItalybDepartment of Mechanics, Politecnico di Torino, Turin, Italy

cCILEA, Interuniversity Consortium, Milan, ItalydCardiovascular Bioengineering Unit, Technology and Health Department, Istituto Superiore di Sanita, Italy

Accepted 6 May 2008

Abstract

The main purpose of this study is to reproduce in silico the dynamics of a bileaflet mechanical heart valve (MHV; St Jude

Hemodynamic Plus, 27mm characteristic size) by means of a fully implicit fluid–structure interaction (FSI) method, and experimentally

validate the results using an ultrafast cinematographic technique. The computational model was constructed to realistically reproduce the

boundary condition (72 beats per minute (bpm), cardiac output 4.5 l/min) and the geometry of the experimental setup, including the valve

housing and the hinge configuration.

The simulation was carried out coupling a commercial computational fluid dynamics (CFD) package based on finite-volume method

with user-defined code for solving the structural domain, and exploiting the parallel performance of the whole numerical setup. Outputs

are leaflets excursion from opening to closure and the fluid dynamics through the valve.

Results put in evidence a favorable comparison between the computed and the experimental data: the model captures the main features

of the leaflet motion during the systole. The use of parallel computing drastically limited the computational costs, showing a linear

scaling on 16 processors (despite the massive use of user-defined subroutines to manage the FSI process). The favorable agreement

obtained between in vitro and in silico results of the leaflet displacements confirms the consistency of the numerical method used, and

candidates the application of FSI models to become a major tool to optimize the MHV design and eventually provides useful

information to surgeons.

r 2008 Elsevier Ltd. All rights reserved.

Keywords: Prosthetic heart valve; Computer modelling; Finite-volume method; Dynamic analysis; Cardiovascular biofluid mechanics; Fluid–structure

interaction; Parallel processing

1. Introduction

Artificial heart valves are designed to replicate thefunction of the natural valves of human heart. Since theirintroduction about 50 years ago (DeWall et al., 2000; Gottet al., 2003), the study of their fluid mechanics became animportant research field in bioengineering. While thedesign of the early prototypes was driven by a quantitativeanalysis of the overall flow field downstream of the valve,

e front matter r 2008 Elsevier Ltd. All rights reserved.

iomech.2008.05.004

ing author. Tel.: +392 239 93 375; fax: +39 2 239 93 360.

ess: alberto.redaelli@polimi.it (A. Redaelli).

more recent studies focused their attention on the localfluid dynamics occurring near the valve housing (Leo et al.,2002, 2006; Kelly, 2002) and in proximity of the leaflets,which are areas commonly associated with thrombosis (Yinet al., 2004).The geometrical complexity of prosthetic mechanical

heart valves (MHVs), joined with the nonlinear, dynamicnature of the interplay between the pulsatile flowingblood, the complex ventricular and aortic anatomies, andthe valve mobile components, renders the study of flowsthrough MHVs a challenging issue for even the mostadvanced computational and experimental techniques

ARTICLE IN PRESSM. Nobili et al. / Journal of Biomechanics 41 (2008) 2539–25502540

(Yoganathan et al., 2004, 2005; Grigioni et al., 2004). Inparticular, limitations in performing fully three-dimen-sional quantitative observations of the local flow field byany of the currently available laboratory techniques likeparticle image velocimetry and laser Doppler anemometry,that are present even in controlled, relatively simplified invitro conditions, have suggested the numerical method asan additional instrument for understanding the dynamicsof flowing blood.

In this context the use of computational fluid dynamics(CFD) has gained relevance for heart valve assessment(Alemu and Bluestein, 2007; Dumont et al., 2007; Dasiet al., 2007; Guivier et al., 2006; Bluestein et al., 2000).Indeed, as computational resources become more power-ful, and data handling algorithms become more sophisti-cated, numerical analysis has became an increasinglyimportant part of biomechanics research, allowing themodelling of even more realistic geometries, and achieve-ment of high space–time resolution.

Two different approaches are currently used for compu-tational investigations. The first approach neglects theinterplay between leaflets and blood, focusing on the peakand near peak systole phase of the cardiac cycle. In thisresearch framework, Grigioni et al. (2005a) performed anumerical simulation on a model of mechanical valveplaced in a physiological aortic root shaped model,furnishing indications about the role of Valsava sinus onthe fluid dynamic downstream of the valve at mid systole.Yokoyama et al. (2006) demonstrated that curved leafletconfiguration generates a Venturi effect that is able toreduce the boundary layer separation zones and stabilizethe leaflets’ position. Ge et al. (2003) provided an insightinto the complex hemodynamics of a bileaflet MHV at themaximum Reynolds number, putting in evidence that highgrid densities are mandatory, in order to fully characterizelocal aspects of the flow field downstream of the prostheticvalve.

Alternatively, the investigation of blood–leaflet interplaythrough the application of fluid structure interaction (FSI)models moves into the direction of providing more in-depth insight into the MHVs hemodynamics, including thevalve opening and closure assessment. More in depth, therationale that moves research is that development ofnumerical methods/strategies to predict even more realisticbiofluid–structure interaction phenomena related to im-planted devices could:

�

provide useful responses to surgeons (choice of the bestimplantation strategy); � reduce costs to manufacturers (physical modelling can

be a very effective and accurate method of research andtesting, but it can be very expensive and time consumingdue to the numerous tests that must be conducted);

� lead to a better comprehension of those physiopatholo-

gical processes for which a multiphysics, multiscaleapproach is needed (and for which computer modellingis the strategy of election).

A hybrid approach is the one proposed by Lai et al.(2002), who studied the closure phase of a prosthetic heartvalve using a prescribed leaflet motion in a simplified flowdomain. Very recently, direct numerical simulation (DNS)of pulsatile flow through a bileaflet MHV mounted in anidealized axisymmetric aorta geometry (with a suddenexpansion modelling the aortic sinus region) was carriedout by prescribing leaflet motion from the experimentaldata (Dasi et al., 2007). However a prescribed kinematicsavoids the complex phenomena of fluid–leaflet interactionrequired for an accurate simulation of the valve function.Therefore FSI algorithms have been developed to predictthe motion of the leaflets, which is the direct consequenceof the momentum applied by the fluid on them (Penroseand Staples, 2002; Vierendeels et al., 2005; Nobili et al.,2007).FSI studies generally focus on a specific phase of the

valve dynamics and introduce geometrical simplificationsbecause of the complexity of the problem and the com-putational costs. Redaelli et al. (2004) analyzed the openingphase of a bileaflet heart valve under low flow rates andvalidated the leaflet motion experimentally. Cheng et al.(2004) presented a three-dimensional unsteady flow analy-sis past a bileaflet valve prosthesis in the mitral positionduring the closing phase, incorporating the leaflet motionin their model. The whole cardiac cycle was simulated byDumont et al. (2005) in laminar condition, applyinga symmetry assumption to the flow. Lately Pelliccioniet al. (2007) presented a bidimensional study of MHVdynamics in laminar flow condition by applying the lattice-Boltzmann method.The aim of this study is the investigation, in silico, of the

bileaflet mechanical valve dynamics during the wholesystolic phase. For this purpose, a fully three-dimensional,realistic model consisting of the MHV plus aortic root,mimicking the physical model embedded in a mock loop(Scotten et al., 2002; Grigioni et al., 2003; Redaelli et al.,2004), was implemented.A previously developed implicit FSI method (Nobili

et al., 2007) was parallelized (16 nodes) and DNS wasperformed. The ALE method was used for modelling theleaflets and fluid motion. Indeed the ALE approach, whichcombines the use of the classical Lagrangian and Eulerianreference frames, is used largely in the analysis of FSIsystems (Donea et al., 2004), with the inclusion of MHVs(Cheng et al., 2004).The results of the dynamics of the valve were validated

with an experimental counterpart, measuring valve leafletkinematics in vitro with the ultrafast cinematographictechnique (Barbaro et al., 1997).

2. Materials and methods

The kinematics of a St. Jude bileaflet MHV (27mm tissue annulus

diameter, maximum opening angle 851) was investigated using both

experimental and computational approaches. In particular, a pulsatile,

open loop mock circulatory system (MCS) was used, properly modified in

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Fig. 2. Driving waveform corresponding to physiological flow condition

applied to experimental and numerical simulations.

M. Nobili et al. / Journal of Biomechanics 41 (2008) 2539–2550 2541

order to allow visualization of the valvular function. In parallel, numerical

simulations were carried out in a computational domain designed to

closely mimic the in vitro experimental setup. In both cases, the valve

dynamics was simulated in physiological flow condition and with realistic

geometry (Valsalva sinus, valve hinges).

2.1. Experimental method

The MCS-VSI (Vivitro Systems Inc., Canada, Fig. 1) is equipped with a

piston-in-cylinder pump head pulse duplicator, and a physical model of

the left heart system made of: (i) a hydraulic chamber containing a

transparent and compliant polyurethane sac mimicking the left ventricle;

(ii) an open reservoir that mimics the atrial chamber; (iii) a simplified glass

model of the aortic root, with three hemispheres placed at 1201 symmetry

mimicking the sinuses of Valsalva.

A detailed description of the MCS can be found in Grigioni et al. (2003)

and in Redaelli et al. (2004). A mechanical no-leakage valve was inserted

in the mitral site of the physical model. This was done in order to avoid

interferences in the dynamics of the aortic MHV under investigation, due

to the dynamics of the mitral valve (not present in the in silico model). A

monitoring system (accurately described in Redaelli et al., 2004, Grigioni

et al. 2003) allowed acquiring and analyzing ventricular, atrial, and aortic

pressure, together with aortic and mitral flow rate. The MCS was driven

by imposition of a pump drive waveform resembling an instantaneous

ventricular volume. A driving waveform corresponding to physiological

flow conditions (a cardiac output of 4.5 l/min) was set (Fig. 2). A mean

arterial pressure of 100mmHg and a heart beat of 72 beats per minute

(bpm) was prescribed. A blood analogue, 35% water–glycerol solution

(3.7� 10�3 Pa s dynamic viscosity), was used as test fluid.

The study of leaflet kinematics was performed using ultrafast

cinematographic technique, which allows recording the movement of

characteristic points of the leaflets during the working cycle of the MHV.

To reach this aim, the mock circulatory loop was modified to allow vision

of the valvular function in the aortic site by means of a thin glass window

(to guarantee against optical distortions) at the top of the aortic site,

through which a high-speed video camera was oriented (Kodak Ektapro,

which records 239� 192 pixels images at a frequency fc ¼ 1000 frames/s).

Time resolution was gained by splitting the camera screen into 12 parallel

slices, thus grabbing the image of the central piece of the prosthetic valve

at a rate of 12,000 frames/s. This allowed recording the position of the

Fig. 1. Schematic of the pulsatile, open loop, mock circulatory system

(MCS)-VSI (Vivitro Systems Inc., Canada). The MCS consists of a pump

system (PS) controlled by PC for waveform generation and data

acquisition and a physical model of the left heart system. The left

ventricle chamber (LVC) consists of a left ventricle model made of

polyurethane, a glass model for the aortic root, and an entrapping air

system for the aortic compliance. The atrium consists of an open ceiling

reservoir (OCR). The left ventricle and the atrium are connected by a

circuit with a variable peripheral resistance (VPR). The St. Jude HP valve

is located in the aortic valve site (AVS); the aortic flow measurements were

performed by an electromagnetic flow meter (FM).

innermost points at the internal left and right leaflet edge, thus measuring

the width of the gap between the leaflets (B Datum): the knowledge of this

sole measured component of the displacement is sufficient to fully describe

leaflet dynamics for St. Jude HP valve model, due to the design features of

the coupling between the housing ring and the leaflet, which allows only a

pure rotational motion to the leaflet (rigid motion). In this way, it was

possible to follow in time leaflet motion in detail. For each experimental

section, 10 cycles were acquired and averaged (Ektapro recording

synchronized with the cardiac cycle). Exhaustive details about the leaflet

motion analysis can be found in Barbaro et al. (1997) and Redaelli et al.

(2004).

2.2. Numerical simulation

2.2.1. Numerical model

A detailed three-dimensional model was shaped on the geometry of the

experimental setup (Fig. 3). The aortic and ventricular conduits were

constructed from the two-dimensional MCS technical drawings. Only the

region located in between the sections hosting the MCS pressure sensors

was taken into account (domain of interest) and, as in the MCS, the

Valsalva sinuses were modelled as three semispheres (1201 symmetry)

placed downstream of the valve site (Fig. 3).

The geometry of the valve, including the valve housing and the hinges

geometry, was constructed from data provided by St. Jude Medical.

Leaflets were scaled by a factor 0.98, thus allowing us to put three rows of

cells in between the closed leaflets and the valve housing, and to attain an

high-quality mesh during the whole cardiac cycle.

To minimize the influence from outlet boundary conditions, a straight

flow extension (10 cm length) was added 14.5 cm downstream of the aortic

duct.

The valve region was discretized with tetrahedral elements according to

the moving deforming mesh module requirements (described in the

following section), and the mesh was locally refined to accommodate for

the leaflet movement. In the remaining domain (aortic and ventricular

ducts) a hexahedral mesh was used in order to limit the numerical

diffusion. The total number of cells used in the simulation was equal to 2.1

millions (1.2 milion tetrahedral and 900,000 hexahedral).

The dimension of the grid was set according to the assumption that

large and intermediate eddies are the ‘‘energy containing eddies’’

(Kolmogorov, 1941a, b), and are the structures responsible for leaflet

motion, Hence, we set a grid density that allows for solving the Taylor

microscale (calculated at peak systole, considering the diameter of the

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Fig. 3. Configuration of the computational domain: (a) zoom into the St Jude valve site and (b) top view of the valve in closed position that clarify the

orientation of the leaflet with respect to the Valsalva sinuses (black arrow pointing to the leaflet tip).

M. Nobili et al. / Journal of Biomechanics 41 (2008) 2539–25502542

straight conduit at the sinutubular region as the integral length scale),

which represents the length scale falling in between the large scale eddies

and the small scale eddies (further details in the Discussion section).

The grid density was increased around the leaflets and in the whole

aortic root, where the three jet-like configurations of the velocity field and

the vortex shedding phenomena characterize the flow field.

The models and the mesh generation were performed by using software

GAMBIT (Ansys Inc., USA). The simulations were performed adopting a

control-volume-based technique (finite-volume method) to solve the

Navier–Stokes equations (FLUENT general-purpose fluid dynamic code,

Ansys Inc., USA).

To match the rheology of the test fluid used in the experimental

campaign, blood was modelled as an isotropic, homogenous, incompres-

sible and Newtonian fluid (with a 0.0037Pa s viscosity).

The aortic walls were assumed rigid, and no slip (i.e., zero velocity)

conditions were set at the walls. At the inlet, we assigned a flow rate that

corresponds to the one prescribed in the experimental section, with the

inflow boundary condition set in terms of velocity profiles: a flat inflow

velocity profile for the axial velocity and zero transverse velocity

components were assigned at the ventricular duct inlet.

To solve the flow field we applied a DNS. In this approach the flow field

is solved directly from the Navier–Stokes equations and no averaging or

turbulence modelling is applied.

The cardiac cycle (0.8 s) was discretized with 4000 time steps,

corresponding to a physical time step of 0.2ms. Two cycles were simulated

in order to remove the dependence of the opening phase from the initial

condition and guarantee cycle invariance.

2.2.2. Fluid structure interaction

An accurate solution of the Navier Stokes equations for deforming

meshes is provided by the use of the ALE formulation, which makes it

possible to include grid velocities in the momentum and continuity

equation of the fluid domain (Le Tallec and Mouro, 2001).

The ALE description conjugates Lagrangian and Eulerian features.

The computational grid is neither moved with the boundary (Lagrangian)

or held fixed (Eulerian). Rather, it is moved in some arbitrarily specified

way to give a continuous reconfiguration capability. Because of this

freedom in moving the computational mesh offered by the ALE

description, greater distortions of the continuum can be handled better

than would be allowed by a purely Lagrangian method, with more

resolution than that afforded by a purely Eulerian approach. The

partitioned approach was used to simulate the interplay between leaflets

and blood. This strategy preserves the fluid and the structural solvers as

separate. Both parts are alternately integrated in time and the interaction

is taken into account by the boundary conditions of both the solvers. As a

direct consequence there exists an intrinsic time lag between the

integration of the fluid and the structure, which can be avoided by

repeating the interaction until both the solution consistently produce the

same result.

The general scheme of the coupling procedure is shown in Fig. 4.

As mentioned above, the fluid domain was solved using the finite-

volume method computational code Fluent (Ansys Inc., USA), which

provides a number of features well suited to handle the specific problem of

rotating boundaries. We used a spring-based moving, deforming mesh

module, which allows a robust mesh deformation handling by assuming

that the mesh element edges behave like an idealized network of

interconnected springs. In order to maximize the influence of the boundary

node displacements on the motion of the interior nodes, no damping was

applied to the springs (spring constant factor ¼ 0).

To preserve the quality of the mesh during the valve motion, the

maximum admissible skewness of the computational cells was set equal to

0.7. The Fluent remeshing algorithm was adopted to properly treat

degenerated cells, which agglomerates cells that violate the skewness

criterion, and locally remeshes the agglomerated cells. If the new cells

satisfy the skewness criterion, the mesh is locally updated with the new

cells (with the solution interpolated from the old cells); otherwise, the new

cells are discarded (FLUENT Users Manual, 2007).

In this work, the moving deforming mesh module was used in

conjunction with two user-defined subroutines, named MDM and center

of gravity motion (cg_motion), respectively; at the beginning of each step

the first one calculates and updates the kinematics of the leaflets on the

basis of the moment applied to the leaflet, which is calculated by the

second subroutine at the end of the time step, once the time step

convergence has been achieved. An iterative call to the fluid solver is

performed by an external subroutine in order to update the solution of

the fluid dynamic field and achieve the convergence of the FSI cycle,

until the difference between the external momentum divided by the inertia

of the fluid (calculated by MDM) and the angular acceleration (imposed

by cg_motion) is not below a threshold value (e ¼ 500 s�2).

More in detail, since the valve leaflet is rigid body in rotation on a fixed

pivot, the angular position is the only degree of freedom and leaflet

dynamics can be calculated using:

I €W ¼Mp þMt

where I is the angular inertia, equal to 8.75� 10�9Kgm2, €W the angular

acceleration of the leaflet, Mp the torque applied on the leaflet external

surface by the pressure field, and Mt is the moment generated by shear

stresses.

The acceleration value for the subsequent iteration within the generic

time step i is updated through an under-relaxation scheme (Le Tallec and

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Fig. 4. Implicit coupling procedure scheme of the interaction between the valve leaflet and the blood flow: W is the angular position of the leaflet position, i

the time step number, k the number of FSI iteration within the same time step, o the under-relaxation coefficient, and e the FSI convergence threshold.

Two loops are annealed: the loop that checks for the convergence of the FSI procedure (inner loop), and the temporal advancement loop that makes the

simulation proceed to the next time step starting from the acceleration of the leaflet calculated at the end of the previous one ( €Westimation).

M. Nobili et al. / Journal of Biomechanics 41 (2008) 2539–2550 2543

Mouro, 2001) as

€Wkþ1

i ¼ €Wk

i þ wMkþ1

pi þMkþ1tii

I� €W

k

i

!

where k is the iteration index and w (set equal to 0.01) is the under-

relaxation factor, which plays the role of damping changes in the

acceleration produced during each iteration. Starting from the accelera-

tion obtained in Eq. (2), the velocity and the displacement of the leaflet

were calculated using the Newmark method. More details about the

coupling procedure can be found in Nobili et al. (2007).

3. Results

Fig. 5a shows the angular valve displacement obtainedfrom numerical and experimental simulations. For the sakeof clarity, among the measured cycles (10 cycles), four ofthem are reported as representative of the whole data set(Fig. 5a). At the early acceleration phase, the experimentalmeasurements put in evidence a highly repeatable kine-matics of the leaflets. Accordingly, during the openingphase, the average of the displacement of the leaflets overall the measured cycles is representative of the phenomenonunder investigation. The numerical results showed amaximum delay, with respect to the average experimentalcurve, of about 4.1ms (within the 72SD) after about 8ms.Afterward the experimental variability in the leaflets’

position increases and becomes particularly pronouncedduring the valve closure.This different behavior exhibited by the MHV at closure

in vitro, under the same experimental conditions, is widelydocumented in literature and could be ascribed to theunsteadiness of mock loops, or to cyclic variations existingin the pulsatile flow (Kleine et al., 1998). In order to make aconsistent comparison between numerical and experimen-tal results, the mean closing time was used. The experi-mental simulations displayed a mean closing time equal to0.3324 s (Texp), a value that is very close to the numericalresult of 0.34 s (Tnum), which again falls in a range of twotimes the standard deviation (std ¼ 0.0058 s).The comparison between experimental and numerical

transvalvular pressure drop waveforms is depicted inFig. 6. In the numerical simulation, aortic and ventricularpressures were calculated by integrating pressure values atsections corresponding to the ones where pressure wasmeasured in the experimental session.Experimental and numerical transvalvular pressure

drops agree satisfactorily. Differences between them canbe ascribed to the role played during the experiments by thesystemic and the aortic root compliance, which have notbeen taken into account in the in silico model. Hence, thenumerical model represents a system more rigid than the

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Fig. 5. Comparison between experimental and numerical results: (a) angular displacement and (b) zoom to the opening phase.

Fig. 6. Comparison of numerical and experimental pressure drops

through the valve during the systolic phase. Pressure was evaluated at a

distance of 2.5 diameters upstream and downstream of the valve site.

M. Nobili et al. / Journal of Biomechanics 41 (2008) 2539–25502544

experimental one. This speculation is sustained by the factthat the numerical simulation put in evidence maximumabsolute values for the numerical transvalvular pressuredrop, higher than that of the experimental setup (about6.6mmHg difference).

The temporal evolution of the axial velocity through thevalve is shown in Fig. 7. Four time instants representativeof the cardiac cycle are considered, referring to the centrallongitudinal plane.

In accordance with consolidated results on St. JudeMHVs (see for example Yoganathan et al., 2004) a triplejet pattern characterizes the forward flow, the major part ofwhich emerges from the two side orifices: the growingmagnitude of the three jets outgoing from the valve duringthe acceleration phase is clearly visible as well as therecirculation areas in the sinus region and the evolution ofthe vortices shed by the leaflets. The maximum value of1.38m/s is observed at peak systole (Fig. 7 panel 2) in thejets emerging from the side orifices of the prosthetic device.

This can be better appreciated in Fig. 8, depicting(magnified view of the flow field, with respect to Fig. 7)the velocity vector field superimposed on the velocity colorcontour at peak systole.During the deceleration phase, the flow is more evenly

distributed downstream of the valve plane (Fig. 7 panel 3),and it emerges from central orifice with a speed higher thanduring the acceleration phase. This loss of organizationin the flow field during the deceleration phase is due toinertial effects, in a manner that is typical for intermittentturbulent flows in the transitional range (Yin et al., 2004).The closing phase (Fig. 7 panel 4) takes place when the flowis completely inverted.Fig. 9 shows the vorticity distribution in the same

transversal plane used for the axial velocities and for thesame instants. As the valve opens and the flow rate grows,the vorticity magnitude increases in the valve region. Themaximum values of vorticity are reached all around theleaflet surface, near the valve ring, and in the transitionalareas between the three jets. The presence of a recirculatingregion, pointed out in the vorticity map, is clearly evidentwithin the sinus of Valsalva and in the opposite location, inaccordance with the inversion of the velocity values in thisregion, depending on design features of the MHVs and alsoon the shape of the aortic root geometry.In order to elucidate the intricate structure of the

pulsatile flow through the valve, the computed results areanalyzed from a Lagrangian viewpoint. Particle tracescan offer a four-dimensional (both in space and time)visualization of flow patterns and have proven to be usefultools to interrogate complex flow fields in vessels (Marklet al., 2004; Steinman, 2000; Morbiducci, et al., 2007) and,in general, to reveal global organization of blood flow(Buonocore and Bogren, 1999). Hence, a framework of theevolving flow field is obtained by emitting masslessparticles 2mm upstream of the valve site. Renderedtrajectories, color coded with the residence time (startingfrom the instant of injection, at peak systole) are shown inFig. 10, where the particle traces selected on the basis oftheir high helical content (according to Grigioni et al.,

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Fig. 7. Sequential presentation of the axial velocity contour plot at four different instants of the cardiac cycle: opening phase (1); peak systole (2);

decelaration phase (3); and closing phase (4).

Fig. 8. Velocity vector plot of the flow past the valve at peak systole

(midplane cross-section).

M. Nobili et al. / Journal of Biomechanics 41 (2008) 2539–2550 2545

2005b and Morbiducci et al., 2007, who recently usedhelicity-related quantities to depict important topologicalfeatures of blood flow) are depicted: the panel A allows usto appreciate the highly three-dimensional structure of thecomputed flow field. In particular, the Lagrangian visua-lization put in evidence (Fig. 10, panel B) three-dimen-sional vortices with higher helical content in sinus II andsinus III (due to the orientation of the leaflet), and helicalstructures in flow channels developing downstream of eachsinus (arrowheads).

4. Discussion

In recent years, due to growing computational power,and the need to investigate with greater accuracy thehemodynamic performance of prosthetic heart valves, theuse of FSI modelling has been introduced in the numericalsimulations of these devices. In this study we included thefully nonlinear FSI effects between the leaflet motion andblood flow, use of realistic geometry and experimentalvalidation of leaflet dynamics.Many improvements were done with respect to our pre-

vious work (Redaelli et al., 2004). Indeed, the investigation

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Fig. 9. Contour plot of the vorticity magnitude at five different instants of the cardiac cycle: opening phase (1); peak systole (2); decelaration phase (3);

and closing phase (4).

Fig. 10. Three-dimensional visualization of the traces of the particles emitted on a surface 2mm upstream of the valve site (panel A). The rendered

trajectories are color coded with the residence time (starting from the instant of injection, at peak systole), and selected in terms of their helical content

(Morbiducci et al., 2007). The Lagrangian visualization put in evidence (panel B) three-dimensional vortices with higher helical content in sinus II and

sinus III (due to the orientation of the leaflet), and helical structures in flow channels developing downstream of each sinus (arrowheads).

M. Nobili et al. / Journal of Biomechanics 41 (2008) 2539–25502546

was extended from the opening phase to the whole systoleand the flow rate was increased to physiological values.Concerning the numerical model, the simplifications at thegeometry of the hinges and at the configuration of theValsalva sinus were removed, thus leading to a more

detailed evaluation of the behavior of the valve. Moreover,the coupling between leaflets and blood was changed fromexplicit in time to implicit in order to remove the time delayin leaflet response due to the intrinsic numerical limits ofthe explicit scheme. As recently stated (Nobili et al., 2007),

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implicit coupling suppresses physical inaccuracy at smallspace/time scale or, equivalently at high derivatives.

To validate the numerical prediction of the dynamics ofthe leaflets, we compared the computed leaflet displace-ment with the experimental data obtained from a MCS.The remarkable agreement between leaflet displacement,shown in Fig. 2, demonstrates the capability of implicitcoupling proposed herein to capture the interactionbetween the leaflet movement and blood flow. The valveopening motion was notably consistent from cycle to cycle,perhaps due to the stable nature of accelerating flow. Onthe contrary, leaflet kinematics measurements show thatthe position of the leaflets exhibits a more pronouncedvariability from cycle to cycle during the valve closurephase: our experimental results are similar to the highlytime resolved PIV measurements by Kaminsky et al.(2007a) and Dasi et al. (2007).

Compared to this study last we moved in a differentdirection. We checked for a spatial grid sufficiently refinedto reproduce the in vitro observed leaflet dynamics,modelling a more realistic geometry for the aortic rootand for the valve, and choosing a real FSI approach to theproblem. Thereby, the over 2 millions of elements used todiscretize the computational domain represent a tradeoffbetween accuracy and computational costs.

However, Fig. 5–10 clearly testify that the proposednumerical scheme is able to capture the relevant structuresin the flow field downstream of the MHV, in accordancewith the state-of-art in silico and in vitro observations(Browne et al., 2000; Grigioni et al., 2000; Yoganathanet al., 2004, 2005; Dumont et al., 2007; Kaminsky et al.,2007b).

Temporal and local information on the axial velocityand vorticity magnitude of the blood flowing through theMHV was obtained in the current study. The presence ofthree sinuses of Valsalva in the aortic root makes the regionimmediately downstream of the aortic valve nonsymmetric,with remarkable consequences in the flow domain. Therendered trajectories elucidate the rich three-dimensionalstructure of the flow field: flow visualization based on traceselection was applied to permit extraction of salient flowfeatures data simplification, and compression.

The current unsteady numerical calculation has highcomputational costs. For the completion of calculation forone cardiac cycle, 790 h on a single processor (Xeon3.2GHz, 8Gb RAM) were taken. The use of parallelcomputing has considerably reduced the computationalcosts, showing a linear scaling up to 16 processors aspreviously observed by other authors (Yue et al., 2004;Wang et al., 2007) despite the use of external subroutine tomanage the FSI process.

Although numerical computational techniques arepowerful tools in the design and virtual assessment ofMHVs, they still have limitations. The presence of a gapseveral orders of magnitude smaller than the aortadiameter (1.118 mm vs 27mm ) represent a remarkableproblem for the discretization of the computational

domain (high cell size variation and large cell numberneeded) that actually could not be handled by the CAEsoftware. Moreover, FSI requires physical separation ofthe fluid domain from the structural domain, dictating theinclusion of a gap between the closed valve leaflets andtheir housing. Accordingly, the clearance gaps wereachieved by slightly reducing the size of the leaflets to98% of their actual size, yielding a central gap of 98.633 mm(original size 1.118 mm). This leads to an overestimation ofthe leakage flow through the gaps and the consequentunderestimation of the related shear stress levels, aspointed out by Dumont et al. (2007). Hence, the leakageflow was not analyzed in the present work.However, scaling leaflet dimensions could directly affect

their motion due to the reduced value of moment of inertia(the angular position of the leaflet is a function of itsmoment of inertia, see Eq. (1)). The consequence for thiscould be that the dynamics of scaled leaflets could bedifferent from the real ones. The values of the moment ofinertia used in our study for the scaled leaflet is equal to8.75� 10�9 kgm2, and it was calculated using a referencevalue for the density of 2200 kgm�3, according to the dataprovided by St. Jude Medical for pyrolitic carbon. Themoment of inertia of the leaflets of real dimension, for thesame density value given by the manufacturer, is equal to9.94� 10�9 kgm2. The application of a scaling factor of0.98 leads also to a 4% leaflet total area reduction. Hence,the 98% of leaflet scaling, even if only 98%, is likelypartially responsible for the differences in leaflet dynamicsbetween numerical and experimental results, in particularduring the opening phase (Fig. 5 left panel): after the first10ms (where the inertial effect of the fluid columnimpinging on the valve is predominant), the numericalleaflet dynamics is more rapid than the real leafletkinematics, in consequence of the lower moment of inertiaof the modelled leaflets. The role played by the loweredmoment of inertia for leaflets dynamics in silico at closureis less clear, due to the variability in the closing behaviorexhibited by the valve in the experimental setup (Fig. 5right panel).Concerning the full nonlinear FSI, uncertainties due to

lack of specific knowledge of two important parametersaffecting the motion of the leaflets were undertaken. Inparticular, we neglected modelling the compliance of theaortic root (which contributes to the difference observed intransvalvular pressures), which is generally modelled invitro by an entrapping air system in the physical modelrepresenting our experimental counterpart, and the frictionforces due to the presence of the hinge mechanism(idealized in our model), which is valve specific and noteasy to assess. Notwithstanding such parameters not beingtaken into account in the simulation, the modelled leafletsdynamics resembles the cycle-to-cycle variation measuredin vitro, as shown in Fig. 4.As previously mentioned, during the construction of the

computational grid for DNS implementation, we aimed ata tradeoff between accuracy and computational costs.

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Actually the resolution of the grid is another limitation ofthe present investigation.

The range of scales that need to be accurately representedin a computation is dictated by the physics. According tocommon literature on MHV fluid dynamics, we consideredthe diameter of the straight conduit at the sinutubular region(D ¼ 27mm) as the length scale of larger eddies in theinvestigated flow field, characterized by a maximumReynolds number value equal to 6000, and a mean Reynoldsnumber value equal to 3775. Hence, the Kolmogorov scaleZ ¼ D Re�3/4 of the investigated flow field results in theorder of 0.04 and 0.06mm for the maximum and the meanReynolds, respectively. This represents an important indica-tion for the dimension of the grid, although it has beendemonstrated that the achievement of dissipative lengthscale in terms of grid size is not needed to have an accuratesolution (Moin and Mahesh, 1998; Moser & Moin, 1987;Spalart, 1988; Rogers et al., 1987).

Indeed, the numerical grid determines the scales that arerepresented in the simulation: while in the gap region andclose to the surfaces of the leaflet we set a computationalcell mean dimension equal to 0.03mm, the spatialdiscretization inside the straight conduit downstream ofthe sinutubular region is 0.49mm, i.e., one order magni-tude greater than the Kolmogorov scale.

This choice is dictated, since DNS, even though is themost accurate way of simulating turbulent flow, unfortu-nately, is also the most expensive way. Hence, theprediction of such flows at a reduced computational costwith a suitably developed turbulence model is a widely usedpractice in studying flows through prosthetic valves. Thereis a great variety of turbulence models available throughcommercial codes for computational fluid dynamics,especially the ones employing the Reynolds averagedNavier–Stokes (RANS) approach. However, these two-equation models suffer from severe limitations. As alsopointed out by Varghese et al., (2008), ‘‘most of theseturbulence models have been developed using knowledge ofsimple classes of well-behaved two-dimensional flows’’.

In virtue of the computational resources needed for arealistic FSI simulation of the turbulent flow through aprosthetic valve, we beg the question whether it isappropriate to consider a grid size greater than theKolmogorov length scale to capture the effects of featuresof a through-mechanical valve flow field involved in leafletdynamics. Features that can be thought as the onescharacterizing the integral scale of the flow.

Very recently Dasi et al. (2007) used the DNS approach,which is the same approach adopted in the presentinvestigation, to depict the transient flow past a MHVand the wake dynamics. Dasi and colleagues were able touse an highly resolved spatial grid (about 9.7 millions ofnodes), which solves the Kolmogorov scale. However, theprice to be paid by Dasi et al. (2007) was neglecting the FSIand oversimplifying the geometry.

The good agreement of the results obtained in our studywith experiments in terms of leaflet dynamics supports the

hypothesis that the dynamics of the leaflets is mainlydependent on the integral scale of the flow, because eventhough the Kolmogorov length scale is not resolved (i.e.,turbulence is not sustained at the dissipative scale), the flowfield is resolved down to the inertial subrange, wheremotions are determined by inertial effects. Accordingly, therelevant length scales would go from the integral one downto the Taylor microscale l ¼ D(10/Re)1/2, which falls inbetween the large scale eddies and the small scale eddiesand is equal to 1.1mm for the investigated flow field, atpeak Reynolds (i.e., the double of the cell size downstreamof the sinutubular region).We believe that the FSI approach, coupled with the

decision to go the DNS route, even if not fully resolving theKolmogorov scale at this stage of the investigation, ispromising. In the future, we plan to combine thequantitative depiction of flow fields and stresses withinthe MHV with blood damage accumulation models(Grigioni et al., 2005c; Nobili et al., 2008) in order toevaluate the hemolytic and the thrombogenic potency ofthe MHV. Moreover we are currently working on a morecomprehensive analysis of the helical flow structure down-stream of the valve by using the helical flow index proposedby Morbiducci et al. (2007). These calculations will requirea finer computational mesh compared to the one used inthe present study (increasing the grid density, thus movingtoward the Kolmogorov scale) in order to enhanceresolution in the region downstream of the leaflets, wherea chaotic flow emerges in the second part of the systole,as very recently assessed both in vitro and in silico by Dasiet al. (2007).In conclusion, the favorable agreement obtained between

in vitro and numerical results of the leaflet displacementssuggests the application of FSI models as a major tool toinvestigate and/or solve problems related to the implanta-tion of MHVs, and to optimize their design.

Conflict of interest

Disclosures and freedom of investigation

The research on FSI applied to prosthetic heart valveswas not supported financially by the St. Jude company.The devices were not donated by the company; they wereacquired at the regular market cost in Italy. None of theauthors has a financial agreement with the company StJude. None of the authors has conflicts of interests in thestudy.All authors had full control of the design of the study,

methods used, outcome parameters, analysis of the data,and production of the written report.

References

Alemu, Y., Bluestein, D., 2007. Flow-induced platelet activation and

damage accumulation in a mechanical heart valve: numerical studies.

Artificial Organs 31 (9), 677–688.

ARTICLE IN PRESSM. Nobili et al. / Journal of Biomechanics 41 (2008) 2539–2550 2549

Barbaro, V., Grigioni, M., Daniele, C., Boccanera, G., 1997. Reconstruc-

tion of closing phase kinematics by motion analysis for a prosthetic

bileaflet valve. In: Power, H., Brebbia, C.A., Kenny, JM. (Eds.),

Simulations in Biomedicine IV. Computational Mechanics Publica-

tions. WIT Press, Southampton, Boston, pp. 349–358.

Bluestein, D., Rambod, E., Gharib, M., 2000. Vortex shedding as a

mechanism for free emboli formation in mechanical heart valves.

Journal of Biomechanical Engineering 122 (2), 125–134.

Browne, P., Ramuzat, A., Saxena, R., Yoganathan, A.P., 2000.

Experimental investigation of the steady flow downstream of the St.

Jude bileaflet heart valve: a comparison between laser Doppler

velocimetry and particle image velocimetry techniques. Annals of

Biomedical Engineering 28 (1), 39–47.

Buonocore, M.H., Bogren, H.G., 1999. Analysis of flow patterns using

MRI. International Journal of Cardiac Imaging 15, 99–103.

Cheng, R., Lai, Y.G., Chandran, K.B., 2004. Three-dimensional

fluid–structure interaction simulation of bileaflet mechanical heart

valve flow dynamics. Journal of Heart Valve Disease 12, 772–780.

Dasi, L.P., Ge, L., Simon, H.A., Sotiropoulos, F., Yoganathan, A.P.,

2007. Vorticity dynamics of a bileaflet mechanical heart valve in an

axisymmetric aorta. Physics of Fluids 19 (067105), 17.

DeWall, R.A., Qasim, N., Carr, L., 2000. Evolution of mechanical heart

valves. Annals of Thoracic Surgery 69 (5), 1612–1621.

Donea, J., Huerta, A., Ponthot, J.-Ph., Rodriguez-Ferran, A., 2004.

Arbitrary Lagrangian–Eulerian methods. In: Erwin, Stein, Ren0e de

Borst, Thomas, J.R., Hughes (Eds.), Fundamentals. Encyclopedia of

Computational Mechanics, vol. 1. Wiley, New York.

Dumont, K., Vierendeels, J.A., Segers, P., Van Nooten, G.J., Verdonck,

P.R., 2005. Predicting ATS open pivot heart valve performance with

computational fluid dynamics. Journal of Heart Valve Disease 14 (3),

393–399.

Dumont, K., Vierendeels, J., Kaminsky, R., van Nooten, G., Verdonck, P.,

Bluestein, D., 2007. Comparison of the haemodynamic and thrombo-

genic performance of two bileaflet mechanical heart valves using a CFD/

FSI model. Journal of Biomechanical Engineering 129 (4), 558–565.

Fluent Inc., 2007. FLUENT Users Manual, 2007. FLUENT Software

Package, ver. 6.3.26, Fluent Inc., Lebanon, NH.

Ge, L., Jones, S.C., Sotiropoulos, F., Healy, T.M., Yoganathan, A.P.,

2003. Numerical simulation of flow in mechanical heart valves: grid

resolution and the assumption of flow symmetry. Journal of

Biomechanical Engineering 125 (5), 709–718.

Gott, V.L., Alejo, D.E., Cameron, D.E., 2003. Mechanical heart valves: 50

years of evolution. Annals of Thoracic Surgery 76 (6), S2230–S2239.

Grigioni, M., Daniele, C., D0Avenio, G., Barbaro, V., 2000. On the

monodimensional approach to the estimation of the highest Reynolds

shear stress in a turbulent flow. Journal of Biomechanics 33 (6),

701–708.

Grigioni, M., Daniele, C., Romanelli, C., Morbiducci, U., D0Avenio, G.,

Del Gaudio, C., Barbaro, V., 2003. Pathological patient in protocol

definition for bench testing of mechanical cardiac support systems.

International Journal of Artificial Organs 26 (1), 64–72.

Grigioni, M., Daniele, C., D0Avenio, G., Morbiducci, U., Del Gaudio, C.,

Abbate, M., Di Meo, D., 2004. Innovative technologies for the

assessment of cardiovascular medical devices: state of the art

techniques for artificial heart valves testing. Expert Review of Medical

Devices 1 (1), 89–101.

Grigioni, M., Daniele, C., Del Gaudio, C., Morbiducci, U., Balducci, A.,

D’Avenio, G., Barbaro, V., 2005a. Three-dimensional numerical

simulation of flow through an aortic bileaflet valve in a realistic model

of aortic root. Asaio Journal 51 (3), 176–183.

Grigioni, M., Daniele, C., Morbiducci, U., Del Gaudio, C., D’Avenio, G.,

Balducci, A., Barbaro, V., 2005b. A mathematical description of blood

spiral flow in vessels: application to a numerical study of flow in

arterial bending. Journal of Biomechanics 38 (7), 1375–1386.

Grigioni, M., Morbiducci, U., D’Avenio, G., Di Benedetto, G., Del

Gaudio, C., 2005c. Proposal for a new formulation of the power law

mathematical model for blood trauma prediction. Biomechanics and

Modeling in Mechanobiology 4 (4), 249–260.

Guivier, C., Deplano, V., Pibarot, P., 2006. New insights into the

assessment of the prosthetic valve performance in the presence of

subaortic stenosis through a fluid–structure interaction model. Journal

of Biomechanics 40 (10), 2283–2290.

Kaminsky, R., Morbiducci, U., Rossi, M., Scalise, L., Verdonck, P.,

Grigioni, M., 2007a. Time resolved PIV technique allows a high

temporal resolution measurement of mechanical prosthetic aortic

valves fluid dynamics. International Journal of Artificial Organs 30 (2),

153–162.

Kaminsky, R., Dumont, K., Weber, H., Schroll, M., Verdonck, P., 2007b.

PIV validation of blood–heart valve leaflet interaction modelling.

International Journal of Artificial Organs 30 (7), 640–648.

Kelly, S., 2002. Computational fluid dynamics insights in the design of

mechanical heart valves. Artificial Organs 26 (7), 608–613.

Kleine, P., Perthel, M., Nygaard, H., Hansen, S.B., Paulsen, P.K., Riis,

C., Laas, J., 1998. Medtronic Hall versus St. Jude Medical mechanical

aortic valve: downstream turbulences with respect to rotation in pigs.

Journal of Heart Valve Disease 7, 548–555.

Kolmogorov, A.N., 1941a. Local structure of turbulence in an incom-

pressible fluid for very large Reynolds numbers. Doklady Akademii

Nauk SSSR 30, 299–303.

Kolmogorov, A.N., 1941b. Energy dissipation in locally isotropic

turbulence. Doklady Akademii Nauk SSSR 32, 19–21.

Lai, Y.G., Chandran, K.B., Lemmon, J.A., 2002. Numerical simulation of

mechanical heart valve closure fluid dynamics. Journal of Biomecha-

nics 35 (7), 881–892.

Le Tallec, P., Mouro, J., 2001. Fluid structure interaction with large

structural displacements. Computer Methods in Applied Mechanics

and Engineering 190, 3039–3067.

Leo, H.L., Simon, H.A., Dasi, L.P., Yoganathan, A.P., 2006. Effect of

hinge gap width on the microflow structures in 27mm bileaflet

mechanical heart valves. Journal of Heart Valve Disease 15 (6),

800–808.

Leo, H.L., He, Z., Ellis, J.T., Yoganathan, A.P., 2002. Microflow fields in

the hinge region of the CarboMedics bileaflet mechanical heart valve

design. Journal of Thoracic and Cardiovascular Surgery 124 (3),

561–574.

Markl, M., Draney, M.T., Hope, M.D., Levin, J.M., Chan, F.P., Alley,

M.T., Pelc, N.J., Herfkens, R.J., 2004. Time-resolved 3-dimensional

velocity mapping in the thoracic aorta. Visualization of 3-directional

blood flow patterns in healthy volunteers and patients. Journal of

Computer Assisted Tomography 28, 459–468.

Moin, P., Mahesh, K., 1998. Direct numerical simulation: a tool in

turbulence research. Annual Review of Fluid Mechanics 30, 539–578.

Morbiducci, U., Ponzini, R., Grigioni, M., Redaelli, A., 2007. Helical flow

as fluid dynamic signature for atherogenesis in aortocoronary bypass.

A numeric study. Journal of Biomechanics 40 (3), 519–534.

Moser, R.D., Moin, P., 1987. The effects of curvature in wall-bounded

turbulent flows. Journal of Fluid Mechanics 175, 479–510.

Nobili, M., Passoni, G., Redaelli, A., 2007. Two fluid–structure

approaches for 3D simulation of St. Jude Medical bileaflet valve

opening. Journal of Applied Biomaterials and Biomechanics 5 (1),

49–59.

Nobili, M., Sheriff, J.F., Morbiducci, U., Redaelli, A., Jesty, J., Bluestein,

D., 2008. Platelet activation due to hemodynamic shear stresses:

damage accumulation model and comparison to in vitro measure-

ments. Asaio Journal 54 (1), 64–72.

Pelliccioni, O., Cerrolaza, M., Herrera, M., 2007. Lattice Boltzmann

dynamic simulation of a mechanical heart valve device. Mathematics

and Computers in Simulation 75 (1–2), 1–14.

Penrose, J.M.T., Staples, C.J., 2002. Implicit fluid–structure coupling for

simulation of cardiovascular problems. International Journal for

Numerical Methods in Fluids 40 (3-4), 467–478.

Redaelli, A., Bothorel, H., Votta, E., Soncini, M., Morbiducci, U., Del

Gaudio, C., Balducci, A., Grigioni, M., 2004. 3-D simulation of the

SJM bileaflet valve opening process: fluid–structure interaction study

and experimental validation. Journal of Heart Valve Disease 13,

804–813.

ARTICLE IN PRESSM. Nobili et al. / Journal of Biomechanics 41 (2008) 2539–25502550

Rogers, M.M., Moin, P., 1987. The structure of the vorticity field in

homogeneous turbulent flows. Journal of Fluid Mechanics 176, 33–66.

Scotten, L.N., Walker, D.K., Dutton, J.W., 2002. Modified Gorlin

equation for the diagnosis of mixed aortic valve pathology. Journal of

Heart Valve Disease 11, 360–368.

Spalart, P.R., 1988. Direct numerical simulation of a turbulent boundary

layer up to Rey ¼ 1410. Journal of Fluid Mechanics 187, 61–98.

Steinman, D.A., 2000. Simulated pathline visualization of computed

periodic blood flow patterns. Journal of Biomechanics 33 (5), 623–628.

Varghese, S.S., Frankel, S.H., Fischer, P.F., 2008. Modeling transition to

turbulence in eccentric stenotic flows. Journal of Biomechanical

Engineering 130 (1), 014503.

Vierendeels, J., Dumont, K., Dick, E., Verdonck, P., 2005. Analysis and

stabilization of fluid–structure interaction algorithm for rigid-body

motion. AIAA Journal 43 (12), 2549–2557.

Wang, C., Pekkan, K., de Zelicourt, D., Horner, M., Parihar, A.,

Kulkarni, A., Yoganathan, A.P., 2007. Progress in the CFD modeling

of flow instabilities in anatomical total cavopulmonary connections.

Annals of Biomedical Engineering 35 (11), 1840–1856.

Yin, W., Alemu, Y., Affeld, K., Jesty, J., Bluestein, D., 2004. Flow-

induced platelet activation in bileaflet and monoleaflet mechanical

heart valves. Annals of Biomedical Engineering 32 (8), 1058–1066.

Yoganathan, A.P., He, Z., Casey Jones, S., 2004. Fluid mechanics of heart

valves. Annual Review of Biomedical Engineering 6, 331–362.

Yoganathan, A.P., Chandran, K.B., Sotiropoulos, F., 2005. Flow in

prosthetic heart valves: state-of-the-art and future directions. Annals

of Biomedical Engineering 33 (12), 1689–1694.

Yokoyama, Y., Medart, D., Hormes, M., et al., 2006. CFD simulation of

a novel bileaflet mechanical heart valve prosthesis: an estimation of the

Venturi passage formed by the leaflets. International Journal of

Artificial Organs 29 (12), 1132–1139.

Yue, X., Hwang, F.N., Shandas, R., Cai, X.C., 2004. Simulation of

branching blood flows on parallel computers. Biomedical Sciences

Instrumentation 40, 325–330.