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Three-dimensional macro-scale assessment of regional and temporal wall shear stress characteristics on aortic valve leaflets K. Caoa, M. BukaČb & P. Sucoskya a Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, USA b Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, USA Published online: 08 Jul 2015.

To cite this article: K. Cao, M. BukaČ & P. Sucosky (2015): Three-dimensional macro-scale assessment of regional and temporal wall shear stress characteristics on aortic valve leaflets, Computer Methods in Biomechanics and Biomedical Engineering, DOI: 10.1080/10255842.2015.1052419

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Three-dimensional macro-scale assessment of regional and temporal wall shear stress characteristics on aortic valve leaflets

K. Cao a , M. Bukač

b and P. Sucosky

a *

a Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, USA;

b Department of Applied and

Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, USA

(Received 30 September 2014; accepted 14 May 2015)

The aortic valve (AV) achieves unidirectional blood flow between the left ventricle and the aorta. Although hemodynamic stresses have been shown to regulate valvular biology, the native wall shear stress (WSS) experienced by AV leaflets remains largely unknown. The objective of this study was to quantify computationally the macro-scale leaflet WSS environment using fluid–structure interaction modeling. An arbitrary Lagrangian–Eulerian approach was implemented to predict valvular flow and leaflet dynamics in a three-dimensional AV geometry subjected to physiologic transvalvular pressure. Local WSS characteristics were quantified in terms of temporal shear magnitude (TSM), oscillatory shear index (OSI) and temporal shear gradient (TSG). The dominant radial WSS predicted on the leaflets exhibited high amplitude and unidirectionality on the ventricularis (TSM . 7.50 dyn/cm

2 , OSI , 0.17, TSG . 325.54 dyn/cm

2 s) but low amplitude and

bidirectionality on the fibrosa (TSM , 2.73 dyn/cm 2 , OSI . 0.38, TSG , 191.17 dyn/cm

2 s). The radial WSS component

computed in the leaflet base, belly and tip demonstrated strong regional variability (ventricularis TSM: 7.50–22.32 dyn/cm 2 ,

fibrosa TSM: 1.26–2.73 dyn/cm 2 ). While the circumferential WSS exhibited similar spatially dependent magnitude

(ventricularis TSM: 1.41–3.40 dyn/cm 2 , fibrosa TSM: 0.42–0.76 dyn/cm

2 ) and side-specific amplitude (ventricularis TSG:

101.73–184.43 dyn/cm 2 s, fibrosa TSG: 41.92–54.10 dyn/cm

2 s), its temporal variations were consistently bidirectional

(OSI . 0.25). This study provides new insights into the role played by leaflet–blood flow interactions in valvular function and critical hemodynamic stress data for the assessment of the hemodynamic theory of AV disease.

Keywords: aortic valve; arbitrary Lagrangian Eulerian approach; computational modeling; hemodynamics; fluid–structure interaction; wall shear stress

1. Introduction

The aortic valve (AV) achieves unidirectional blood flow

from the left ventricle to the aorta. It functions in a

complex mechanical environment including pulsatile flow,

cyclic pressure, bending and tensile stresses (Thubrikar

1990). While those mechanical stresses are required for

proper valvular function, abnormal mechanical signals are

presumed to play a role in disease development (Butcher

et al. 2008; Merryman 2010; Balachandran et al. 2011).

In particular, abnormalities in fluid wall shear stress

(WSS) magnitude and frequency promote valvular

calcification (Sucosky et al. 2009; Hoehn et al. 2010;

Sun et al. 2012; Sucosky and Rajamannan 2013; Sun et al.

2013; Sucosky 2014). Despite the compelling evidence for

the involvement of hemodynamic stress abnormalities in

valvular disease, the investigation of causality between

abnormal stresses and pathological states requires the

quantification of the local degree of hemodynamic

abnormality experienced by the leaflets relative to a

reference physiologic hemodynamic state, which remains

largely unknown.

Experimental flow studies in prosthetic and native

valves have reported a wide range of leaflet WSS. Laser

Doppler velocimetry (LDV) measurements carried out in a

polyurethane trileaflet valve indicated a maximum WSS of

79 dyn/cm 2 near the tip of the leaflets (Weston et al. 1999).

Particle image velocimetry measurements in native

porcine valves suggested a maximum fluid shear stress

of 30 dyn/cm 2 in the shear layer extending from the tip of

the leaflets (Seaman et al. 2014). LDV measurements on

prosthetic AV leaflets subjected to pulsatile flow suggested

the existence of a high WSS ranging between 251 and þ71 dyn/cm2 on the leaflet ventricularis (i.e., surface facing the left ventricle) and a low WSS ranging between

0 and þ20 dyn/cm2 on the leaflet fibrosa (i.e., surface facing the aorta) (Yap, Saikrishnan, and Yoganathan 2012).

While those studies have provided important prelimi-

nary insights into the global characteristics of the valvular

WSS environment, the quantitative assessment of the

native leaflet WSS requires: (1) the availability of a highly

resolved near-wall velocity field, (2) the identification of

the exact location of the leaflet surface, and (3) the

determination of the three velocity components of the flow

in a three-dimensional (3D) domain. Computational

approaches have been developed in an effort to satisfy

those requirements. Fluid dynamics simulations with

q 2015 Taylor & Francis

*Corresponding author. Email: [email protected]

Computer Methods in Biomechanics and Biomedical Engineering, 2015

http://dx.doi.org/10.1080/10255842.2015.1052419

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prescribed leaflet kinematics demonstrated the existence

of a side-specific leaflet WSS environment characterized

by low magnitude and unsteadiness on the fibrosa and high

magnitude and unidirectionality on the ventricularis

(Sacks and Yoganathan 2007; Ge and Sotiropoulos

2010). However, the lack of coupling between the leaflets

and the blood flow prevented the reliable estimation of the

regional leaflet WSS characteristics. Fully coupled fluid–

structure interaction (FSI) models accounting for the

transfer of momentum between the leaflets and blood flow

have also been designed. Two-dimensional (2D) and 3D

valve models based on a fictitious domain method

captured the leaflet and flow dynamics under subphysio-

logic flow (De Hart et al. 2000; De Hart, Baaijens, et al.

2003) or during the systolic phase (De Hart, Peters, et al.

2003). Another 2D FSI formulation based on a sharp

interface technique was proposed to capture the leaflet

deformation under physiologic pulsatile flow conditions

(Vigmostad et al. 2010). Dynamic FSI simulations of an

aortic root using a commercial explicit finite element

solver enabled the characterization of the structural stress

and strain on the leaflets (Sturla et al. 2013). To date, only

two FSI studies have quantified the regional leaflet WSS.

The first thorough characterization of the side-specific

regional leaflet WSS was conducted using a 2D FSI model

implementing an arbitrary Lagrangian–Eulerian (ALE)

approach (Chandra et al. 2012). This study confirmed the

side-specificity of the leaflet WSS, the existence of

significant differences in WSS pulsatility and magnitude

between the base, belly and tip of the cusps, and predicted

a maximum WSS of 71 dyn/cm 2 near the tip of the leaflets.

More recently, 3D FSI simulations in valve models with

symmetric and porcine-specific collagen fiber alignments

reported peak-systolic WSS of 280 and 420 dyn/cm 2 on the

leaflet fibrosa and ventricularis, respectively (Marom et al.

2013), which are substantially larger than the levels

reported by other studies.

The increasing evidence for the involvement of fluid

stresses in valvular disease motivates the detailed

assessment of the native leaflet WSS environment, which

is still lacking. Therefore, the aim of this study was to

design a 3D FSI valve model to investigate the macro-

scale temporal and regional WSS characteristics on AV

leaflets under physiologic flow conditions.

2. Materials and methods

2.1 AV geometry and computational grid

The 3D AV geometry was designed in Solid Edge

(Siemens PLM Software, Plano, TX, USA). The

geometrical model consisted of three identical leaflets

attached to an aortic root and housed inside the aortic

sinuses (Figure 1(a)). Two straight-tube extensions

(length: 12 mm) were added to the inlet section

(ventricular extension) and outlet section (aortic exten-

sion) of the aortic root for computational stability. The

dimensions of the aortic root, sinus and leaflets (Figure 1

(b)) were obtained from published human AV data

(Thubrikar 1990; De Hart, Peters, et al. 2003) and are

reported in Table 1. The cusp profile was generated with a

non-uniform thickness varying between 0.18 mm in the

belly region and 2.06 mm in the nodulus of Aranti (Grande

et al. 1998) (Table 2). Given the focus of the study on the

characterization of the macro-scale regional leaflet WSS,

the leaflet microtopology and local roughness were not

accounted for. While AV leaflets are not dimensionally

identical, the geometrical model was designed with a

symmetric trileaflet structure. As a result, the simulations

were performed in only a 608 section of the full geometry

Figure 1. AV geometrical model: (a) 3D views of the model structural features; and (b) dimensional characteristics.

Table 1. Aortic root geometric parameters (all dimensions in mm).

ra rv ds hl hs ta ts tv

12.5 12.5 6.0 14.0 22.1 1.5 1.5 1.5

Table 2. Leaflet thickness (all dimensions in mm).

Attachment edge Belly

Coaptation area

Free margin

Nodulus of

Aranti

1.16 0.18–0.58 0.68–1.29 1.53 2.06

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(see Figure 1(b)) and the initial stress-free state was

assumed to be in the coapted leaflet position. The

geometry was imported in the commercial software

ANSYS 15.0 (ANSYS, Inc., Canonsburg, PA, USA). The

fluid domain was meshed using tetrahedral elements and

the structural domain was meshed with second-order

tetrahedral elements in order to prevent shear locking

issues. Mesh inflation was implemented to increase the

spatial resolution of the fluid domain in the boundary layer

near the leaflet surface. A mesh sensitivity analysis was

performed based on the peak WSS value predicted in the

belly of the leaflet ventricularis. The initial coarse fluid

mesh consisted of 92,421 elements in the 608 section. The progressive increase in mesh density by 31% (120,753

elements), 65% (199,815 elements) and 75% (348,951

elements) produced an 8.1%, 2.3% and 1.3% increase in

peak WSS, respectively. Therefore, the first mesh

refinement was considered sufficiently resolved to capture

the macro-scale valvular hemodynamics. The final

computational grid generated in the 608 section consisted of 29,686 elements in the AV structure (i.e., leaflets, aortic

sinus and aortic wall) and 120,753 elements in the fluid

domain. Those parameters permitted to achieve a spatial

resolution of 200 mm in the fluid domain near the leaflet.

2.2 FSI approach

The accurate assessment of the leaflet WSS environment

required the precise identification of the leaflet surface and

the prediction of the interactions between the pulsatile

blood flow and the compliant leaflets. Those requirements

were addressed via the adoption of an ALE FSI approach

and its implementation through the system coupling

module of the commercial software ANSYS. Briefly, the

fluid (blood) domain was discretized using a moving grid

fitted to the moving structure boundary (i.e., leaflets and

aortic wall) and the motion of the fluid grid was accounted

for by formulating the flow governing equations relative to

the moving grid (Donea et al. 1982). At each time step, the

fluid mesh was updated to conform to the moving solid

boundary and the flow and structure equations were solved

implicitly and iteratively until convergence was reached.

The governing equations consisted of the momentum and

continuity equations in their ALE forms for the fluid

domain and the momentum equation for the structure, as

previously described (Chandra et al. 2012; Atkins et al.

2014). Those equations were solved along with three

coupling conditions enforcing continuity of displace-

ments, continuity of velocities and compatibility of

traction at the fluid–structure interface.

The Navier–Stokes and continuity equations were

solved using a finite volume method and a pressure-based

coupled solver. Pressure–velocity coupling was achieved

by deriving an additional condition for pressure by

reformatting the continuity equation. The implicit flow

solver used a spatially second-order upwind scheme and a

second-order temporal discretization. The convergence

criterion for the continuity and momentum equations was

set at 10 26

. The structural solver employed an implicit

direct displacement-based finite element method with no

material damping. The system of equations was solved

using the sparse direct solver and the equilibrium

equations were solved by the Newton–Raphson method.

The convergence criterion was determined by multiplying

the reference value determined by ANSYS based on

external forces by a tolerance of 0.5%.

One cardiac cycle (0.86 s) was discretized in 1720 time

steps, corresponding to a constant physical time step of

0.5 ms. At each time step, a remeshing technique was

employed to update the low-quality cells generated by

large structural displacements in the fluid domain. The

remeshing criteria were based on cell skewness (maximum

skewness: 0.8) and minimum and maximum cell lengths

(minimum length: 100 mm; maximum length: 600 mm). The fluid and structure problems were sub-iterated until

Figure 2. Leaflet material model and flow boundary condition: (a) comparison between the leaflet stress–strain curve measured experimentally (adapted from Missirlis and Chong 1978) and that predicted by the Mooney–Rivlin hyperelastic model; and (b) dynamic pressure waveform imposed at the model inlet.

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the relative error between two consecutive force and

displacement approximations attained 10 23

. The average

number of coupling iterations required for convergence

was around 20 at each time step.

2.3 Constitutive models

The leaflet material behavior is nearly incompressible but

also non-linear due to the progressive locking of the

collagen fibers, which increases material stiffness under

increased applied load (Billiar and Sacks 2000). There-

fore, the leaflet tissue was modeled using a three-

parameter incompressible Mooney–Rivlin model

described by the following strain energy function:

W ¼ C10ð�I1 2 3Þ þ C01ð�I2 2 3Þ þ C11ð�I1 2 3Þð�I2 2 3Þ; ð1Þ

where C10, C01 and C11 are material constants, and I1 and

I2 are the principal invariants of the Cauchy–Green

deformation tensor. The calibration of the model with

respect to published mechanical test data on porcine valve

tissue (Missirlis and Chong 1978) yielded

C10 ¼ 32,823 Pa, C01 ¼ 2955.1 Pa and C11 ¼ 585,790 Pa (R

2 ¼ 0.999). The resulting stress–strain curve is shown in Figure 2(a). The aortic wall was approximated as a

nearly incompressible, linear elastic and isotropic material

(density: 1080 kg/m 3 ; elastic modulus: 2 MPa; Poisson’s

ratio: 0.45) (Lantz et al. 2011). Blood was modeled as an

incompressible, homogeneous and Newtonian fluid

(density: 1050 kg/m 3 ; dynamic viscosity: 0.0035 Pa s).

The Newtonian viscous formulation was shown to

generate less than 6% error in fibrosa WSS relative to a

more realistic power-law model (see ‘Newtonian viscous

model justification’ in Supplemental Material).

2.4 Flow and boundary conditions

Valvular flow was simulated by imposing a near-

physiologic transvalvular pressure waveform ranging

between 294 and þ6 mmHg as a traction condition at the inlet of the fluid domain. The pressure waveform

reflected a normal heart rate of 70 beats per minute and a

physiologic diastolic/systolic ratio of 2:1 (Figure 2(b)).

Leaflet and aortic wall surfaces in contact with blood flow

were treated as fluid–structure interfaces satisfying the no-

slip condition. Fixity conditions were prescribed at the

inlet and outlet of the structural domain to prevent

longitudinal motion. The vertical plane bisecting the

leaflet (plane 1 in Figure 1(b)) was treated as a symmetry

plane while the other vertical plane bounding the structural

and fluid domains (plane 2 in Figure 1(b)) was treated as a

frictionless contact surface and a symmetry plane,

respectively. The simulations were run over three cardiac

cycles to achieve temporal convergence and all results

were extracted from the last period.

2.5 Hemodynamic characterization

Global valvular hemodynamics was assessed in terms of

velocity and total WSS fields. In addition, the temporal

variations of the regional radial and circumferential WSS

components on the leaflets were obtained by averaging the

nodal WSS over circular regions (3 mm diameter) centered

in the base, belly and tip of the leaflets (see Figure 1(a)).

The WSS waveforms predicted at each site on the leaflet

fibrosa and ventricularis were further characterized in

terms of the temporal shear magnitude (TSM),

TSM ¼ 1 T

ðT 0

jtjdt ð2Þ

oscillatory shear index (OSI),

OSI ¼ 1 2

1 2

ðT 0

t dt

���� ����= ðT 0

jtj dt � �� �

ð3Þ

and temporal shear gradient (TSG),

TSG ¼ 1 T

ðT 0

›t

›t

���� ����dt ð4Þ

where t is the local WSS and T is the cardiac period (Ku 1997).

3. Results

3.1 Global flow and leaflet dynamics

The flow rate predicted at the outlet section of the model

(Figure 3) is characterized by a strong forward flow over

the systolic phase (duration: 280 ms; peak flow rate: 24.3 l/

min), a brief and moderate reverse flow immediately

following valve closure (duration: 60 ms; minimum flow

Figure 3. Temporal volume flow rate variations predicted at the model outlet over one cardiac cycle.

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rate: 24.5 l/min) followed by a long period of no flow until the end of the cycle (duration: 520 ms). Those

characteristics result in a cardiac output of 4.3 l/min, which

falls within the physiologic range (3.7–5.4 l/min) reported

in vivo (Stalder et al. 2011; Barker et al. 2012),

demonstrating the ability of the inlet pressure condition

and material model to simulate physiologic valvular

hemodynamics.

Snapshots of the leaflet profile and flow velocity vector

field computed at early systole (t ¼ 0 ms), mid-systole (t ¼ 150 ms), peak systole (t ¼ 220 ms), early diastole (t ¼ 300 ms) and mid-diastole (t ¼ 480 ms) are shown in Figure 4. Animations of the flow velocity field in 3D

isometric and vertical sectional views are also provided in

the supplemental online material (see movie1.mp4 and

movie2.mp4). Between early and mid-systole

(t , 150 ms), the positive pressure drop imposed across the valve contributes to the partial opening of the leaflets

and generates a forward jet through the valve orifice

(maximum velocity: 1.26 m/s) (Figure 4(a)). During that

phase, the leaflets maintain a streamlined profile and the

tip region aligns tangentially to the flow (Figure 4(b)).

At peak systole (t ¼ 220 ms), the flow rate attains its maximum value (24.3 l/min), which forces the leaflets in

the fully open position (Figure 4(a)). The peak systolic

Reynolds number attains 6262, which is in good

agreement with the physiologic value of 6000 reported

in the literature (Chandran et al. 2007; Dasi et al. 2009).

The increased jet momentum (maximum velocity: 1.14m/s)

pushes the leaflets toward the aortic wall, isolating the flow

located in the sinus region. As a result, the peak-systolic

sinus flow becomes nearly stagnant (maximum velocity:

0.12m/s) (Figure 4(b)). Following the deceleration phase

(t ¼ 300 ms), the rapid decrease in transvalvular pressure drop initiates valvular closure, which generates in turn

a flow reversal toward the aortic sinus (maximum

Figure 4. Leaflet deformation and global flow characteristics: snapshots of the leaflet profile and flow velocity vector field computed at early systole, mid-systole, peak systole, early diastole and late diastole in (a) the vertical section of the model; and (b) the full geometry.

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velocity: 0.64m/s) and through the valve orifice (Figure 4

(a)). The progressive decrease in ventricular pressure and

increase in aortic pressure generates a decreased but

essentially uniform leaflet radius of curvature (Figure 4(b)).

At mid-diastole (t ¼ 480 ms), the persistence of a negative transvalvular pressure drop results in initial leaflet contact,

coaptation and complete valve closure (Figure 4(a)). The

flow is dominated by low-velocity vortical structures

developing between the leaflet fibrosa and the sinus wall

(maximum velocity: 0.09 m/s) (Figure 4(b)). Those

structures progressively dissipate before the beginning of

the next cycle.

3.2 Total WSS distribution

Predictions of the total WSS distribution on the aortic and

ventricular sides of the leaflet are shown in Figure 5.

Animations of the WSS fields on the ventricularis and

fibrosa over one cardiac cycle are also provided in the

supplemental online material (see movie3.mp4 and

movie4.mp4). At the beginning of the cardiac cycle

(t ¼ 0 ms), the nearly stagnant flow imposed near the ventricular and aortic leaflet surfaces generates essentially

uniform and low-magnitude WSS distributions (ventricu-

laris WSS: 0.54dyn/cm 2 ; fibrosa WSS: 0.15dyn/cm

2 ). The

acceleration phase is associated with a progressive increase in

WSS on the ventricular surface, which reaches a maximum at

peak systole (61.4dyn/cm 2 ). In contrast, the near absence of

flow in the sinus region at peak systole subjects the leaflet

fibrosa to a much lower WSS level (14.6dyn/cm 2 ). At the

beginning of diastole (t ¼ 300ms), the progressive reduction

in flow rate is accompanied by a substantial decrease in WSS

on the ventricularis (14.3dyn/cm 2 ), while the increased flow

vorticity and recirculation in the sinus region results in an

overall increase in fibrosa WSS (17.9dyn/cm 2 ). At mid-

diastole (t ¼ 480ms), the near stagnant blood flow subjects both surfaces to low WSS levels (ventricularis WSS:

1.22dyn/cm 2 ; fibrosa WSS: 1.15dyn/cm

2 ).

3.3 Radial and circumferential WSS characteristics

The temporal variations of the radial and circumferential

WSS predicted in the base, belly and tip regions of the

leaflet are shown in Figure 6. The qualitative inspection of

the radial WSS reveals the existence of a mostly

unidirectional (i.e., positive) and pulsatile waveform on

the ventricularis and a bidirectional (i.e., alternatively

positive and negative) oscillatory waveform on the fibrosa.

The qualitative comparison of the waveforms predicted in

the base, belly and tip regions indicates marked

differences in radial WSS magnitude and oscillation

frequency, suggesting the spatial dependence of the leaflet

WSS environment. In contrast, the circumferential WSS

component does not exhibit as much spatial variability and

is consistently one order-of-magnitude lower than the

radial component. Regardless of the cusp region, the

circumferential WSS predicted on the ventricularis and

fibrosa exhibit more similarities and feature high-

frequency oscillations.

Those qualitative observations are supported by the

quantitative analysis of the OSI, TSM and TSG (Table 3).

First, the ventricularis experiences a more unidirectional

Figure 5. Total leaflet WSS: snapshots of the regional WSS distribution on (a) the ventricularis; and (b) the fibrosa at early systole, mid- systole, peak systole, early diastole and late diastole.

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Table 3. Radial and circumferential WSS characteristics predicted in the base, belly and tip of the leaflet (TSM is expressed in dyn/cm 2 ,

TSG is expressed in dyn/cm 2 s).

Ventricularis Fibrosa

Base Belly Tip Base Belly Tip

Radial OSI 0.07 0.17 0.06 0.41 0.38 0.47 Radial TSM 7.79 7.50 22.32 1.26 2.20 2.73 Radial TSG 352.54 321.81 406.69 104.29 191.17 185.87 Circumferential OSI 0.25 0.27 0.35 0.38 0.48 0.27 Circumferential TSM 1.41 1.62 3.40 0.51 0.42 0.76 Circumferential TSG 101.73 129.06 184.43 46.52 41.92 54.10

Figure 6. Radial and circumferential leaflet WSS: temporal variations of the radial and circumferential WSS predicted in the base, belly and tip of the leaflets over one cardiac cycle.

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(OSI , 0.17) and more pulsatile (TSG . 325.54 dyn/ cm

2 s) radial WSS than the fibrosa (OSI . 0.38;

TSG , 191.17 dyn/cm 2 s). The TSM predictions also

evidence the side-specificity of the radial WSS magnitude.

The radial WSS magnitude predicted on the ventricularis

(TSM . 7.50 dyn/cm 2 ) is systematically higher than that

computed along the fibrosa (TSM , 2.73 dyn/cm 2 ).

Consistent with the description of the WSS field, the

largest radial WSS magnitudes are experienced by the tip

of the leaflet. On the ventricularis, the radial WSS TSM in

the tip region is 2.9-fold and 3.0-fold larger than that

computed in the base and belly region, respectively.

A similar trend is captured along the fibrosa for which the

radial WSS TSM in the tip region is 2.2-fold and 1.2-fold

larger than that computed in the base and belly region,

respectively.

The circumferential WSS is essentially oscillatory

regardless of the leaflet surface (ventricularis OSI: 0.25–

0.35; fibrosa OSI: 0.27–0.48), while its temporal

variations present more side-specificity (ventricularis

TSG: 101.73–184.43 dyn/cm 2 s; fibrosa TSG: 41.92–

54.10 dyn/cm 2 s). Lastly, the circumferential WSS magni-

tude exhibits the same regional dependence as the radial

component. The TSM predicted in the leaflet tip is 2.4-fold

and 2.1-fold larger than in the base and belly of the

ventricularis, respectively, and 1.5-fold and 1.8-fold larger

than in the base and belly of the fibrosa, respectively.

4. Discussion

In this study, we performed fully coupled transient ALE

FSI simulations to characterize the flow, leaflet dynamics

and regional leaflet WSS in an idealized 3D AV geometry.

The results demonstrate the side- and site-specificity of the

leaflet WSS environment and the existence of important

magnitude and directional differences between the radial

and circumferential leaflet WSS components.

Our 3D FSI model was able to simulate valvular flow

and leaflet dynamics at a near physiologic cardiac output

of 4.3 l/min (peak systolic flow rate: 24.3 l/min) and peak-

systolic Reynolds number of 6262. The global systolic and

diastolic valvular flow features captured by our model are

consistent with previous in vitro (Bellhouse and Talbot

1969; Yoganathan et al. 2004; Leo et al. 2006; Seaman

et al. 2015) and computational (De Hart, Baaijens, et al.

2003; Weinberg et al. 2009; Ge and Sotiropoulos 2010;

Chandra et al. 2012) reports indicating the three-

dimensionality of the flow, the strong coupling between

the leaflets and the surrounding blood flow, and the

existence of a central jet emanating from the valve orifice

and vortical structures developing in the sinus cavity.

The temporal and regional analyses of the leaflet WSS

environment revealed the dominance of the radial over the

circumferential WSS component. The predominant radial

WSS was high and essentially unidirectional on the

ventricularis but one order-of-magnitude lower and

bidirectional on the fibrosa, which is also consistent with

previous in vivo (Kilner et al. 2000), in vitro (Yap,

Saikrishnan, Tamilselvan, et al. 2012; Yap, Saikrishnan,

and Yoganathan 2012) and computational (Ge and

Sotiropoulos 2010; Chandra et al. 2012) reports.

In addition, the model isolated the spatial variability of

the radial WSS magnitude caused by the change in relative

alignment between the leaflet and the flow from the leaflet

base to the tip. On the ventricular leaflet surface, the

temporal variations of the radial WSS component during

systole, which consist of half a sine wave, are qualitatively

similar to those previously reported by LDV measure-

ments (Yap, Saikrishnan, and Yoganathan 2012) and 2D

FSI simulations (Chandra et al. 2012). The maximum

radial WSS predicted at peak systole near the base and

belly of the leaflet (46–70 dyn/cm 2 ) is also in good

agreement with the range (64–71 dyn/cm 2 ) measured

using LDV in the middle of the leaflet (Yap, Saikrishnan,

and Yoganathan 2012). The maximum instantaneous radial

WSS predicted at peak systole on the leaflet tip (93 dyn/

cm 2 ) also falls within the range (79–100 dyn/cm

2 )

reported by previous in vitro and computational studies

(Weston et al. 1999; Ge and Sotiropoulos 2010; Chandra

et al. 2012), and the location of maximum WSS predicted

by our model (leaflet tip) matches that predicted by our

previous 2D FSI model (Chandra et al. 2012).

A significant innovation brought by the present model

is the implementation of the relatively simple ALE

approach to model the dynamics of valvular flow and the

large deformations of the leaflets. Although this technique

is typically limited to problems involving small structural

deformations, its application to the modeling of the

leaflet–blood interactions under physiologic conditions

was made possible by the mesh adaptation techniques

recently developed in commercial software. While

remeshing introduces uncertainties associated with the

interpolation of the solution from a source grid to a newly

generated grid, the implementation of this method in

ANSYS permitted to track accurately the position of the

blood–leaflet interface while solving for the flow

equations on a well-conditioned computational grid. This

approach differs from previous FSI modeling efforts,

which implemented fictitious domain and immersed

boundary methods to characterize global valvular

hemodynamics on a fixed Eulerian fluid grid (De Hart,

Baaijens, et al. 2003; Nicosia et al. 2003; Weinberg and

Kaazempur Mofrad 2007). While those studies provided

invaluable insights into the complex leaflet–blood flow

interactions, they were not able to provide accurate

estimates of the near-wall spatial velocity gradients.

Lastly, the regional and temporal WSS characteriz-

ation conducted in this study may have some important

implications for the elucidation of valvular disease.

K. Cao et al.8

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Ex vivo studies conducted in our laboratory have

demonstrated the ability of WSS magnitude and frequency

alterations on the fibrosa to trigger acute inflammatory,

remodeling and osteogenic processes similar to those

marking the early stage of calcific AV disease.

Specifically, a WSS environment characterized by twice

the physiologic TSM has been shown to be particularly

conducive to inflammation and paracrine signaling, while

a WSS environment characterized by half or twice the

physiologic WSS oscillation frequency is more conducive

to ECM degradation (Hoehn et al. 2010; Sun et al. 2013).

Similarly, a decrease in OSI (from 0.5 to 0) on the fibrosa

has been shown to trigger pro-inflammatory events

(Sucosky et al. 2009). Supported by those results, we

anticipate that the local degree of abnormality in (1) WSS

magnitude; (2) WSS directionality; and (3) WSS

oscillation frequency on valve leaflets may be used as

predictors of the local tissue vulnerability to disease.

However, it is important to point out that, while those

studies have provided a clear evidence of the sensitivity of

leaflet tissue to its hemodynamic environment and the

involvement of WSS abnormalities in AV disease, they

quantified WSS abnormalities relative to an idealized

reference state consisting of purely pulsatile and

oscillatory WSS waveforms or waveforms obtained from

2D flow simulations. The detailed stress analysis

performed in the present study provides a new

hemodynamic benchmark for the assessment of local

hemodynamic stress abnormalities on AV leaflets, which

will be instrumental to the elucidation of the mechano-

etiology of valvular disease.

5. Limitations

Several limitations could potentially be addressed in future

research.

While valvular flow is known to be in transition to

turbulence at peak systole, the simulations were based on a

laminar flow model. Although the investigation of the

native valvular hemodynamics would clearly benefit from

an approach such as direct numerical simulations capable

of resolving turbulence explicitly at the highest level of

fidelity, current computational capabilities prevent the

implementation of such methodology to fully coupled FSI

simulations through a compliant trileaflet valve geometry.

A laminar flow model provides an acceptable compromise

to capture the near-native hemodynamic characteristics in

the aortic root during most of the cardiac cycle.

The valve geometry was assumed symmetrical in two

ways: (1) the three leaflet-sinus units were assumed

identical; and (2) each unit was assumed symmetrical with

respect to the axial plane passing through the nodulus of

Arantius. First, it is important to note that the coronary

leaflets and sinuses are known to differ from their

non-coronary counterparts and those differences have

been shown to influence the local biomechanics of the

aortic root (Grande et al. 1998; Marom et al. 2013).

In addition, the prescription of symmetry conditions,

which prevents flow development in the circumferential

direction, may have resulted in the underestimation of the

circumferential leaflet WSS. Simulations performed under

steady peak-systolic flow in a symmetric (608) and a full 3D (3608) valve model indicated less than 6% difference in WSS and only a 10% difference in radial-to-circumfer-

ential WSS ratio between the two models (see ‘Symmetric

boundary condition justification’ in Supplemental

Material), suggesting the limited impact of the symmetry

boundary condition on the flow and WSS predictions.

The uniform pressure condition prescribed at the model

inlet did not reflect the native asymmetry of the flow

emanating from the left ventricle (Zhou et al. 1993). In this

context, the system modeled in the present study is more

similar to the configuration typically found in organ culture

systems for tissue-engineering applications, in which

physiologic flow is defined in terms of the global

transvalvular pressure gradient and cardiac output. Future

computational studies are needed to investigate the potential

impact of left-ventricular outflow skewness on leaflet WSS.

While the Mooney–Rivlin model did not account for

the anisotropy of the leaflet material, this simplification is

not expected to affect significantly the hemodynamic and

WSS predictions. In fact, during systole, valve leaflets

typically experience strains below 10% (Thubrikar 1990;

Weinberg and Kaazempur Mofrad 2007) and behave

essentially as an isotropic material (Billiar and Sacks

2000). During diastole, while the progressive locking of

the collagen fibers increases material stiffness along the

circumferential direction, the valve is mostly or fully

closed during this phase, which results in negligible leaflet

WSS levels. Therefore, while the anisotropy of the leaflet

material could potentially alter the leaflet curvature during

coaptation, its impact on the regional leaflet WSS is

expected to be low.

Lastly, while the present model did not include the

coronary circulation, the large amount of blood flow

migrating through the coronary ostia during diastole has

been shown to impact the diastolic sinus hemodynamics

(Moore and Dasi 2014) and is expected to alter in turn the

WSS distribution on the leaflet fibrosa. FSI simulations

integrating systemic and coronary circulations in a more

realistic asymmetric AV geometry are currently underway

in our laboratory.

6. Conclusion

This computational study implemented a 3D ALE FSI

approach to characterize the dynamic temporal and

regional WSS characteristics on the surface of AV leaflets

Computer Methods in Biomechanics and Biomedical Engineering 9

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throughout the cardiac cycle. This model provides new

insights into the role played by leaflet-blood flow

interactions in valvular function and a new reference

hemodynamic state, which may become instrumental to

the assessment of the hemodynamic theory of AV disease.

Acknowledgements

The authors thank Andrew McNally (University of Notre Dame) for his technical assistance with the model and Samantha Atkins (University of Notre Dame) for providing manuscript editing assistance.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Funding

This work was supported by the National Science Foundation under Career Grant [grant number CMMI-1148558]; and the American Heart Association under Grant [grant number 14PRE18940010].

Supplemental data

Supplemental data for this article can be accessed http://dx.doi. org/10.1080/10255842.2015.1052419.

References

Atkins SK, Cao K, Rajamannan NM, Sucosky P. 2014. Bicuspid aortic valve hemodynamics induces abnormal medial remodeling in the convexity of porcine ascending aortas. Biomech Model Mechanobiol. 13(6):1209–1225. doi:10. 1007/s10237-014-0567-7.

Balachandran K, Sucosky P, Yoganathan AP. 2011. Hemody- namics and mechanobiology of aortic valve inflammation and calcification. Int J Inflam. 2011:1–15. doi:10.4061/2011/ 263870.

Barker AJ, Markl M, Bürk J, Lorenz R, Bock J, Bauer S, Schulz- Menger J, von Knobelsdorff-Brenkenhoff F. 2012. Bicuspid aortic valve is associated with altered wall shear stress in the ascending aorta. Circ Cardiovasc Imaging. 5(4):457–466. doi:10.1161/CIRCIMAGING.112.973370.

Bellhouse BJ, Talbot L. 1969. The fluid mechanics of the aortic valve. J Fluid Mech. 35(4):721–735. doi:10.1017/ S0022112069001406.

Billiar KL, Sacks MS. 2000. Biaxial mechanical properties of the natural and glutaraldehyde treated aortic valve cusp – Part I: experimental results. J Biomech Eng. 122(1):23–30. doi:10. 1115/1.429624.

Butcher JT, Simmons CA, Warnock JN. 2008. Mechanobiology of the aortic heart valve. J Heart Valve Dis. 17:62–73.

Chandra S, Rajamannan NM, Sucosky P. 2012. Computational assessment of bicuspid aortic valve wall-shear stress: implications for calcific aortic valve disease. Biomech Model Mechanobiol. 11(7):1085–1096. doi:10.1007/ s10237-012-0375-x.

Chandran K, Rittgers S, Yoganathan A. 2007. Biofluid mechanics: the human circulation. Boca Raton, FL: CRC Press/Taylor and Francis.

Dasi LP, Sucosky P, de Zelicourt D, Sundareswaran K, Jimenez J, Yoganathan AP. 2009. Advances in cardiovascular fluid mechanics: bench to bedside. Ann N Y Acad Sci. 1161(1):1–25. doi:10.1111/j.1749-6632.2008.04320.x.

De Hart J, Baaijens FP, Peters GW, Schreurs PJ. 2003. A computational structure interaction analysis of a fiber- reinforced stentless aortic valve. J Biomech. 36(5):699–712. doi:10.1016/S0021-9290(02)00448-7.

De Hart J, Peters GWM, Schreurs PJG, Baaijens FPT. 2000. A two-dimensional fluid–structure interaction model of the aortic valve. J Biomech. 33(9):1079–1088. doi:10.1016/ S0021-9290(00)00068-3.

De Hart J, Peters GWM, Schreurs PJG, Baaijens FPT. 2003. A three-dimensional computational analysis of fluid– structure interaction in the aortic valve. J Biomech. 36(1):103–112. doi:10.1016/S0021-9290(02)00244-0.

Donea J, Giuliani S, Halleux JP. 1982. An arbitrary Lagrangian– Eulerian finite element method for transient dynamic fluid– structure interactions. Comput Methods Appl Mech Eng. 33(1–3):689–723. doi:10.1016/0045-7825(82)90128-1.

Ge L, Sotiropoulos F. 2010. Direction and magnitude of blood flow shear stresses on the leaflets of aortic valves: is there a link with valve calcification? J Biomech Eng. 132(1):014505. doi:10.1115/1.4000162.

Grande KJ, Cochran RP, Reinhall PG, Kunzelman KS. 1998. Stress variations in the human aortic root and valve: the role of anatomic asymmetry. Ann Biomed Eng. 26(4):534–545. doi:10.1114/1.122.

Hoehn D, Sun L, Sucosky P. 2010. Role of pathologic shear stress alterations in aortic valve endothelial activation. Cardiovasc Eng Technol. 1(2):165–178. doi:10.1007/s13239-010-0015-5.

Kilner PJ, Yang GZ, Wilkes AJ, Mohiaddin RH, Firmin DN, Yacoub MH. 2000. Asymmetric redirection of flow through the heart. Nature. 404(6779):759–761. doi:10.1038/ 35008075.

Ku DN. 1997. Blood flow in arteries. Annu Rev Fluid Mech. 29(1):399–434. doi:10.1146/annurev.fluid.29.1.399.

Lantz J, Renner J, Karlsson M. 2011. Wall shear stress in a subject specific human aorta – influence of fluid–structure interaction. Int J Appl Mech. 03(4):759–778. doi:10.1142/ S1758825111001226.

Leo HL, Dasi LP, Carberry J, Simon HA, Yoganathan AP. 2006. Fluid dynamic assessment of three polymeric heart valves using particle image velocimetry. Ann Biomed Eng. 34(6):936–952. doi:10.1007/s10439-006-9117-5.

Marom G, Peleg M, Halevi R, Rosenfeld M, Raanani E, Hamdan A, Haj-Ali R. 2013. Fluid–structure interaction model of aortic valve with porcine-specific collagen fiber alignment in the cusps. J Biomech Eng. 135(10):101001–101006. doi:10. 1115/1.4024824.

Merryman WD. 2010. Mechano-potential etiologies of aortic valve disease. J Biomech. 43(1):87–92. doi:10.1016/j. jbiomech.2009.09.013.

Missirlis YF, Chong M. 1978. Aortic valve mechanics – Part I: material properties of natural porcine aortic valves. J Bioeng. 2:287–300.

Moore B, Dasi LP. 2014. Spatiotemporal complexity of the aortic sinus vortex. Exp Fluids. 55(7):1770. doi:10.1007/s00348- 014-1770-0.

Nicosia MA, Cochran RP, Einstein DR, Rutland CJ, Kunzelman KS. 2003. A coupled fluid–structure finite element model of the aortic valve and root. J Heart Valve Dis. 12:781–789.

K. Cao et al.10

D ow

nl oa

de d

by [

N ew

Y or

k U

ni ve

rs it

y] a

t 22

:2 0

13 J

ul y

20 15

Sacks MS, Yoganathan AP. 2007. Heart valve function: a biomechanical perspective. Philos Trans R Soc London Ser B Biol Sci. 362(1484):1369–1391. doi:10.1098/rstb. 2007.2122.

Seaman C, Akingba AG, Sucosky P. 2014. Steady flow hemodynamic and energy loss measurements in normal and simulated calcified tricuspid and bicuspid aortic valves. J Biomech Eng. 136(4)–11. doi:10.1115/1.4026575.

Seaman C, McNally A, Biddle S, Jankowski L, Sucosky P. 2015. Generation of simulated calcific lesions in valve leaflets for flow studies. J Heart Valve Dis. 24:1–11.

Stalder AF, Frydrychowicz A, Russe MF, Korvink JG, Hennig J, Li K, Markl M. 2011. Assessment of flow instabilities in the healthy aorta using flow-sensitive MRI. J Magn Reson Imaging. 33(4):839–846. doi:10.1002/jmri.22512.

Sturla F, Votta E, Stevanella M, Conti CA, Redaelli A. 2013. Impact of modeling fluid–structure interaction in the computational analysis of aortic root biomechanics. Med Eng Phys. 35(12):1721–1730. doi:10.1016/j.medengphy. 2013.07.015.

Sucosky P. 2014. Hemodynamic mechanisms of bicuspid aortic valve calcification and aortopathy. In: Rajamannan N, editor. Molecular biology of valvular heart disease. London: Springer; p. 81–94.

Sucosky P, Balachandran K, Elhammali A, Jo H, Yoganathan AP. 2009. Altered shear stress stimulates upregulation of endothelial VCAM-1 and ICAM-1 in a BMP-4- and TGF- b1-dependent pathway. Arterioscler Thromb Vasc Biol. 29(2):254–260. doi:10.1161/ATVBAHA.108.176347.

Sucosky P, Rajamannan NM. 2013. Bicuspid aortic valve disease: from bench to bedside. In: Rajamannan N, editor. Cardiac valvular medicine. London: Springer; p. 17–21.

Sun L, Chandra S, Sucosky P, Aikawa E, Sun L, Chandra S, Sucosky P. 2012. Ex vivo evidence for the contribution of hemodynamic shear stress abnormalities to the early pathogenesis of calcific bicuspid aortic valve disease. PLoS One. 7(10):e48843. doi:10.1371/journal.pone.0048843.

Sun L, Rajamannan NM, Sucosky P, Aikawa E, Sun L, Rajamannan N, Sucosky P. 2013. Defining the role of fluid

shear stress in the expression of early signaling markers for calcific aortic valve disease. PLoS One. 8(12):e84433. doi:10.1371/journal.pone.0084433.

Thubrikar M. 1990. The aortic valve. Boca Raton, FL: CRC Press.

Vigmostad SC, Udaykumar HS, Lu J, Chandran KB. 2010. Fluid–structure interaction methods in biological flows with special emphasis on heart valve dynamics. Int J Numer Method Biomed Eng. 26(3–4):435–470. doi:10.1002/cnm. 1340.

Weinberg EJ, Kaazempur Mofrad MR. 2007. Transient, three- dimensional, multiscale simulations of the human aortic valve. Cardiovasc Eng. 7(4):140–155. doi:10.1007/s10558- 007-9038-4.

Weinberg EJ, Schoen FJ, Mofrad MRK, Capogrossi M. 2009. A computational model of aging and calcification in the aortic heart valve. PLoS One. 4(6):e5960. doi:10.1371/journal. pone.0005960.

Weston MW, LaBorde DV, Yoganathan AP. 1999. Estimation of the shear stress on the surface of an aortic valve leaflet. Ann Biomed Eng. 27(4):572–579. doi:10.1114/1.199.

Yap CH, Saikrishnan N, Tamilselvan G, Yoganathan AP. 2012. Experimental measurement of dynamic fluid shear stress on the aortic surface of the aortic valve leaflet. Biomech Model Mechanobiol. 11(1–2):171–182. doi:10.1007/s10237-011- 0301-7.

Yap CH, Saikrishnan N, Yoganathan AP. 2012. Experimental measurement of dynamic fluid shear stress on the ventricular surface of the aortic valve leaflet. Biomech Model Mechanobiol. 11(1–2):231–244. doi:10.1007/s10237-011- 0306-2.

Yoganathan AP, He Z, Casey Jones S. 2004. Fluid mechanics of heart valves. Annu Rev Biomed Eng. 6(1):331–362. doi:10. 1146/annurev.bioeng.6.040803.140111.

Zhou YQ, Faerestrand S, Matre K, Birkeland S. 1993. Velocity distributions in the left ventricular outflow tract and the aortic anulus measured with Doppler colour flow mapping in normal subjects. Eur Heart J. 14(9):1179–1188. doi:10.1093/ eurheartj/14.9.1179.

Computer Methods in Biomechanics and Biomedical Engineering 11

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Y or

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y] a

t 22

:2 0

13 J

ul y

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