3D design of Tricuspid Aortic valve in Inventor

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SIMULATION OF NATIVE, DISEASED, AND PROSTHETIC HEART VALVE FUNCTION

Computational comparison of regional stress and deformation characteristics in tricuspid and bicuspid aortic valve leaflets

K. Cao1 and P. Sucosky2,*,†

1Department of Aerospace and Mechanical Engineering, University of Notre Dame, 365 Fitzpatrick Hall, Notre Dame, IN 46556, USA

2Department of Mechanical and Materials Engineering, Wright State University, 3640 Colonel Glenn Highway, Dayton, OH 45435, USA

SUMMARY

The bicuspid aortic valve (BAV) is the most common congenital valvular defect and a major risk factor for secondary calcific aortic valve disease. While hemodynamics is presumed to be a potential contributor to this complication, the validation of this theory has been hampered by the limited knowledge of the mechan- ical stress abnormalities experienced by BAV leaflets and their dependence on the heterogeneous BAV fusion patterns. The objective of this study was to compare computationally the regional and temporal fluid wall shear stress (WSS) and structural deformation characteristics in tricuspid aortic valve (TAV), type-0, and type-I BAV leaflets. Arbitrary Lagrangian–Eulerian fluid-structure interaction models were designed to simulate the flow and leaflet dynamics in idealized TAV, type-0, and type-I BAV geometries subjected to physiologic transvalvular pressure. The regional leaflet mechanics was quantified in terms of temporal shear magnitude (TSM), oscillatory shear index (OSI), temporal shear gradient (TSG), and stretch. The sim- ulations identified regions of WSS overloads and increased WSS bidirectionality (174% increase in tempo- ral shear magnitude, 0.10 increase in OSI on type-0 leaflets) in BAV leaflets relative to TAV leaflets. BAV leaflets also experienced larger radial deformations than TAV leaflets (4% increase in type-0 BAV leaflets). Type-I BAV leaflets exhibited contrasted WSS environments marked by WSS overloads on the non- coronary leaflet and sub-physiologic WSS levels on the fused leaflet. This study provides important insights into the mechanical characteristics of BAV leaflets, which may further our understanding of the role played by hemodynamic forces in BAV disease. Copyright © 2016 John Wiley & Sons, Ltd.

Received 19 October 2015; Revised 22 March 2016; Accepted 20 April 2016

KEY WORDS: aortic valve; bicuspid aortic valve; fluid-structure interaction; hemodynamics; stress; stretch

1. INTRODUCTION

The aortic valve achieves unidirectional blood flow from the left ventricle to the aorta. It opens dur- ing systole following ventricular contraction to allow blood ejection and closes during diastole to prevent backflow. The normal tricuspid aortic valve (TAV) consists of three nearly identical leaflets (or cusps) attached to the base of the aortic root and located in front of three hemispherical sinuses. The leaflets are typically referred to as the left-coronary, right-coronary, and non-coronary leaflets, according to their anatomic position relative to the coronary ostia. In 1–2% of the general popula- tion [1], the aortic valve forms with only two leaflets, giving rise to a bicuspid aortic valve (BAV) anatomy. The BAV is the most common congenital cardiac defect and covers a spectrum of clinical

*Correspondence to: Philippe Sucosky, Department of Mechanical and Materials Engineering, Wright State University, 3640 Colonel Glenn Hwy, 257 Russ Engineering Center, Dayton, OH 45435, USA.

†E-mail [email protected]

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING Int. J. Numer. Meth. Biomed. Engng. (2017); e02798 Published online 24 June 2016 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cnm.2798

presentations depending on the pattern of leaflet fusion. Based on the classification system devel- oped by Sievers [2], the type-0 BAV (7% of all BAVs) consists of two nearly identical and symmet- ric leaflets housed in two sinuses. The more prevalent type-I BAV (88%) features two cusps of unequal size (resulting from the fusion of two of the three leaflets) and a fibrous raphe at the location of congenital fusion [2–4] but maintains the same three-lobed aortic sinus as in the TAV [2, 5]. The type-I BAV exists in different morphologic phenotypes, depending on the location of the raphe [6–8]. While 71% of type-I BAVs result from the fusion between the left-coronary and right-coronary leaflets (L-R subtype), 15% feature right-coronary and non-coronary cusp fusion (R–N subtype) and 3% present with non-coronary and left-coronary cusp fusion (N–L subtype) [2]. The BAV malformation has emerged as the first indication for surgical valve replacement and as

a major risk factor for secondary valvulopathies and aortopathies such as calcific aortic valve dis- ease (CAVD) [1, 6, 9] and aortic dilation [10–12]. Although similar pathologies also develop in the normal TAV, their progression in the BAV is accelerated and more severe [13, 14]. While the apparent correlation between the presence of a BAV and the high incidence of those valvular and vascular disorders has historically supported a genetic etiology, the modulation of those pathol- ogies by the type of leaflet fusion pattern has increasingly pointed to mechanical stresses as an alternate pathway [15–19]. Support for this hemodynamic etiology stems from the known sensitiv- ity of leaflet and aortic tissue to their surrounding mechanical stress environment and the recent evidence of the potential of structural and fluid stresses to promote precursor biological events to valvular calcification and aortic dilation. Although previous studies have provided important insights into the possible role of tensile strains and fluid wall shear stress (WSS) in the development of BAV disease [20–28], the validation of the hemodynamic etiology requires the characterization of (i) the regional stress abnormalities experienced by BAV leaflets and (ii) their dependence on the BAV morphotype, which, to date, has only been partially elucidated. Clinical and in vitro studies implementing echocardiography [29–31], magnetic resonance imag-

ing [32–34] and particle-image velocimetry [35–37] have revealed the existence of increased energy loss and abnormal downstream helical flow patterns in BAVs. Computational studies have also been performed to characterize BAV mechanics at higher temporal and spatial resolutions. A dynamic three-dimensional (3D) finite element (FE) model of a type-I L-R BAV with a realistic geometry revealed the existence of structural stress overloads in BAV leaflets [38]. Another FE structural analysis aimed at computing the strain and stress distributions on the leaflets of a TAV and four BAV morphotypes indicated the existence of a reduced valve orifice area in BAVs and the strong dependence of the leaflet mechanical stress and strain states on the BAV geometry [39]. Steady computational fluid dynamics (CFD) simulations in idealized 2D BAV models revealed that both the pressure drop through the valve and the severity of valvular dysfunction cor- related with systolic jet eccentricity [40]. While those FE and CFD analyses have isolated some important biomechanical alterations due to the presence of a BAV, they treated the structural and fluid mechanics aspects separately and did not account for the transfer of momentum between the compliant leaflets and the surrounding blood flow. To address this limitation, a more sophisticated fluid-structure interaction (FSI) analysis per-

formed in a realistic TAV geometry computed the tissue strains and stresses and demonstrated the necessity of coupling fluid and structural dynamics in aortic valve simulations [41]. The first FSI study aimed at comparing TAV and BAV mechanics was conducted on symmetric TAV and type- 0 BAV geometries using the commercial software LS-DYNA [42]. Although the study was able to quantify valve deformation and leaflet stretch at the cell, tissue, and organ levels, it did not provide information on the regional fluid WSS on the leaflets. FSI simulations implementing the arbitrary Lagrangian–Eulerian (ALE) approach have also been carried out to quantify the impact of the BAV morphologic characteristics on valvular disease progression in terms of systolic flow, leaflet motion, and stress/strain distribution in a BAV and normal/stenotic TAV geometries [43]. The sim- ulations demonstrated the presence of bending strain overloads on BAV leaflets during systole. Several subsequent studies focused on the determination of the fluid WSS environment on TAV

and BAV leaflets. A 2D FSI model developed in our laboratory and based on a commercial imple- mentation of the ALE method demonstrated the existence of WSS magnitude and pulsatility abnor- malities on type-I BAV leaflets, under physiologic flow conditions [44]. In a follow-up study, the 2D

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flow characteristics were used to investigate the impact of different type-I BAV morphotypes in a 3D aorta model and demonstrated the existence of asymmetric WSS distributions on the BAV proximal ascending aorta and their dependence on the type of BAV cusp fusion [45]. Another FSI model using a fixed-grid technique, but operating at a sub-physiologic Reynolds number predicted a maximum WSS of 980 and 1430 dyn/cm2 on the fibrosa and ventricularis of the non-coronary leaflet, respec- tively [46]. More recently, we developed a 3D ALE FSI model to predict valvular flow and leaflet dynamics in an idealized TAV geometry subjected to physiologic transvalvular pressure [47]. The dynamic temporal and regional WSS characteristics predicted on the leaflet surface throughout the cardiac cycle were consistent with previous in vitro measurements [48, 49]. While those studies have characterized the fundamental hemodynamic differences between

TAVs and BAVs, the impact of the most common BAV morphotypes (i.e., type-0 and type-I [50]) on the regional leaflet hemodynamic stresses and dynamic deformations under physiologic flow has not been thoroughly described. This characterization is critical toward the assessment of the hemodynamic theory of BAV disease and the demonstration of causality between mechanical abnormalities and the expression of BAV complications. In an effort to address this need, the objective of this study was to compare the local fluid WSS and structural strain patterns experienced by TAV, type-0, and type-I BAV leaflets using 3D FSI modeling.

2. METHOD

2.1. Valve geometries and mesh generation

The 3D TAV, type-0 BAV, and type-I BAV geometries were designed in SOLIDWORKS 2014 (Dassault Systemes, Inc, Vélizy-Villacoublay, France). Consistent with our previous valve model [47], the TAV geometry consisted of three identical leaflets attached to a three-lobed aortic root (Figure 1(a)). Two straight-tube extensions (length: 12mm) were added to the inlet section (ventric- ular extension) and outlet section (aortic extension) of the aortic root to enhance computational stability. Dimensional parameters for the aortic root (Table I(a)), sinus, and leaflets (Table I(b)) were obtained from published human aortic valve data [51, 52]. The type-0 BAV geometry

Figure 1. Geometrical models: (a) tricuspid aortic valve; (b) type-0 bicuspid aortic valve (BAV); and (c) type-I BAV. Insets show the base, belly, and tip regions on each leaflet.

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consisted of two symmetric leaflets and sinuses (Figure 1(b)). The type-I BAV was modeled with one normal leaflet (non-coronary leaflet) and one larger leaflet mimicking fusion between the two coronary leaflets (Figure 1(c)). In all models, the reference state consisted of the valve in a partially open configuration (i.e., in transition between the coapted state and the opening state) and was con- structed by leaving a small gap (TAV: 3.0mm; type-0 BAV: 0.6mm; type-I BAV: 1.5mm) be- tween the free edges of the leaflets. This strategy was adopted to prevent the separation of the fluid domain into two domains and to facilitate computational convergence during the first time step. Based on previous morphological reports [2, 5], the non-coronary leaflet and its sinus were modeled to span a circumferential angle of 150° (Figure 2(a)). The left-coronary and right-coronary leaflets were assumed to be identical, and the raphe was placed along the free edge of these two leaf- lets. Four anatomical landmarks were considered for the creation of the fused leaflet: the commis- sure point (P1) located on the leaflet commissure plane at the intersection between the leaflet free edge and the aortic sinus, the annulus point (P2) located at the intersection between the leaflet at- tachment line and the valve annulus, the tip point (P3) located at the uppermost position in the valve symmetry plane, and the raphe point (P4) located at the intersection between the raphe and the aor- tic sinus (Figure 2(b)). The vertical distances between P2 and P1 (h1 =18mm) and P2 and P3 (h2 =14mm) were selected based on the literature [51, 53]. Because of the limited availability of anatomical data on fused BAV leaflets, the exact locations of the annulus point and the raphe point were defined to match qualitatively previous clinical reports [2, 5, 54]. The annulus point was positioned at a circumferential angle of 50° relative to the commissure point, and the raphe point was set at a vertical distance h3 =12mm above the valve annulus. The leaflet attachment line (L1), centerline (L2), and free edge (L3) were then generated by fitting spline curves through those points. This method resulted in a discontinuity in the circumferential curvature of the fused leaflet at the vertical symmetry plane, which is consistent with the presence of the fibrous protruding raphe at that location [5, 50, 54–57]. Following the creation of all construction lines, the leaflet surfaces were generated using the surface-loft function, and the 3D leaflet volume was obtained by knitting the leaflet surfaces together. The design of the non-coronary leaflet followed similar steps but only required the definition of three anatomical landmarks: the commissure point (P5), the annulus point

Table I(a). Aortic root and leaflet dimensional parameters (all dimensions in mm).

ra rv ds hs h1 h2 h3 ta ts tv

12.5 12.5 6.0 22.1 18.0 14.0 12.0 1.5 1.5 1.5

Table I(b). Regional 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

Figure 2. Anatomical landmarks used for the reconstruction of the type-I bicuspid aortic valve leaflets: (a) top view of the fused and non-coronary leaflets; (b) side-view of the fused leaflet; and (c) side-view of the

non-coronary leaflet.

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(P6), and the tip point (P7). While the commissure and tip points were defined similarly to their counterparts in the fused leaflet, the annulus point of the non-coronary leaflet was located on the valve symmetry plane (Figure 2(c)). Based on the angular symmetry of the TAV model and the planar symmetry of the two BAV

geometries, the simulations were run in a 60°, 90°, and 180° section of the TAV, type-0 BAV, and type-I BAV, respectively. While the implementation of symmetric geometries effectively prevents flow development along the circumferential direction, this simplification was shown to have a limited impact on the macro-scale hemodynamics [47] and was therefore considered acceptable for the purpose of the present study. The fluid domain was meshed using tetrahedral elements and the structure domain was meshed with second-order tetrahedral elements. The mesh located within 1mm from the FSI interface was refined to reach a final cell size of 200μm. A mesh sensitivity analysis was performed to determine the minimum number of cell elements enabling numerical convergence. Specifically, three mesh sizes were tested for each model by increasing the initial mesh size (TAV: 92,421 elements; type-0 BAV: 275,080 elements; type-I BAV: 470,621 elements) by 30% and 65% (Table II). The relative errors in peak ventricularis WSS between the finest and the intermediate meshes were below 3% for all models. Thus, the first mesh refinement was considered sufficient to ensure grid independence in all models. The final computational grid characteristics are reported in Table II.

2.2. Fluid-structure interaction framework

The FSI framework implemented in this study has been described in details in our previously pub- lished work [27, 45, 47]. Briefly, the governing equations for the fluid domain consisted of the momentum and continuity equations in their ALE forms. The governing equation for the structure (i.e., aortic wall and leaflets) was the momentum equation expressed in a Lagrangian coordinate sys- tem. In addition, three coupling conditions were considered to enforce continuity of displacements, continuity of velocities (no-slip), and compatibility of traction at the fluid-structure interface. The two-way coupled FSI simulations were performed in the commercial software ANSYS 15.0 (Canonsburg, PA, USA) using the dynamic mesh technique. The Navier–Stokes and continuity equa- tions were solved using a finite-volume method in Fluent. The pressure-based coupling solver was employed 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�6. The structural solver employed an implicit direct displacement-based finite element method in ANSYS Mechanical. The system of equations was solved using the sparse direct solver, and the equi- librium equations were solved by the Newton–Raphson method. The convergence criterion was deter- mined by multiplying the reference value determined by ANSYS based on external forces by a tolerance of 0.5%. The simulations were run over three cardiac cycles, and each cardiac cycle (0.86s) was discretized with a constant time step size of 0.5ms. At each time step, the fluid and structure problems were sub-iterated until the relative error between two consecutive force and displacement approxima- tions attained 10�3. The TAV, type-0 BAV, and type-I BAV simulations required 25, 35, and 60days

Table II. Mesh sensitivity analysis (dependence of peak ventricularis WSS on fluid grid size) and computational grid characteristics.

TAV Type-0 BAV Type-I BAV

Grid size (elements)

Peak WSS (dyn/cm2)

Grid size (elements)

Peak WSS (dyn/cm2)

Grid size (elements)

Peak WSS (dyn/cm2)

Initial grid 92,421 44.1 275,080 63.3 470,621 67.4 1st Refinement 120,753 61.7 357,620 73.8 615,892 75.6 2nd Refinement 199,815 63.1 453,910 75.1 781,709 76.3 Final fluid grid 120,753 357,620 615,892 Final structural grid 29,686 39,762 51,853

BAV, bicuspid aortic valve; TAV, tricuspid aortic valve; WSS, wall shear stress.

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of computing time, respectively, on a Dell Precision T7500 workstation (dual quad-core 3.20GHz Intel Xeon processors, 48GB DDR3 SDRAM).

2.3. Boundary conditions

Blood flow was simulated by imposing a physiologic transvalvular pressure waveform as a traction condition at the inlet of the fluid domain (Figure 3(a)), while maintaining a constant zero gage pres- sure at the outlet of the fluid domain. The inlet and outlet of the structural domain were fixed in the longitudinal and circumferential directions to prevent translation and rotation. The external surface of the aortic root was allowed to deform radially. A symmetry boundary condition was applied to the vertical plane bisecting the leaflet. Frictionless conditions were prescribed between adjacent leaflets to simulate leaflet contact during coaptation. These conditions, which were enforced each time the distance between two leaflets became smaller than 100μm, permitted to effectively prevent flow through the small orifice formed between the leaflets and to physically consider the leaflets in contact. Leaflet and aortic wall surfaces in contact with blood flow were treated as fluid-structure interfaces satisfying the no-slip condition.

2.4. Constitutive models

The demonstrated similarity in fiber organization in TAV and BAV leaflets [58] justified the imple- mentation of the same material formulation, which consisted of an incompressible Mooney–Rivlin hyperleastic model. The strain energy function W was defined as

W ¼ C10 I 1 � 3 � �

þ C01 I 2 � 3 � �

þ C11 I 1 � 3 � �

I 2 � 3 � �

; (1)

where Ī1 and Ī2 are the principal invariants of the Cauchy–Green deformation tensor and C10, C01, and C11 are material parameters calibrated with respect to in vitro tensile test data on porcine leaflets [59]. The incompressibility constraint was imposed via a Lagrangian multiplier-based mixed u-P formulation [60]. In this formulation, Ī1 and Ī2 were made volume-insensitive to model the near incompressibility of the leaflet material [46, 61, 62]. The values of the material parameters are reported in Table III, and the comparison between the model and the in vitro data is shown in Figure 3(b) [47]. The aortic sinus and the ventricular and aortic extensions were modeled as isotropic, nearly incompressible, linear elastic materials (density: 2000kg/m3; elastic modulus: 2MPa; Poisson’s ratio: 0.45) [63]. Blood was approxi- mated as an incompressible, homogeneous, and Newtonian fluid (density: 1050kg/m3; dynamic viscos- ity: 0.0035Pa.s). This approximation was shown to generate less than 6% error in the prediction of the WSS on the leaflet fibrosa relative to a non-Newtonian power-law model [47].

Figure 3. Fluid-structure interaction model setup: (a) inlet flow boundary condition and (b) leaflet material model (in vitro porcine data obtained from Missirlis and Chong [59]).

Table III. Mooney–Rivlin material parameters for porcine leaflet properties.

Parameters C10 C01 C11

Values (kPa) 32.823 2.955 585.790

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2.5. Mechanical characterization

Valve opening was characterized in terms of the geometric orifice area (GOA) (i.e., minimal cross- sectional area of the valve orifice at peak systole). This metric was approximated as the area of the region bounded by the leaflets and contained within the plane passing through the lowest points on the leaflet free-edges. Global valvular hemodynamics were characterized in terms of velocity and total WSS fields, while the global leaflet mechanical state was investigated in terms of the equiva- lent von Mises strain field, which represents the overall 3D strain state of the material:

εeq ¼ ffiffiffi 2

3 ε1 � ε2ð Þ2 þ ε2 � ε3ð Þ2 þ ε3 � ε1ð Þ2

h i1=2 ; (2)

where εi are the principal engineering strains. Stress and deformation patterns were also characterized locally by discretizing each leaflet into three sub-regions (i.e., base, belly, and tip), each aligned along the circumferential direction, and spanning one third of the total radial length of the leaflet (see insets in Figure 1). In each region, the temporal characteristics of the radial WSS (τ) were further analyzed over one cardiac cycle (T) in terms of the temporal shear magnitude (TSM),

TSM ¼ 1 T ∫ T

0 τj jdt; (3)

temporal shear gradient (TSG),

TSG ¼ 1 T ∫ T

0 ∂τ ∂t

�� dt; (4)

and oscillatory shear index (OSI),

OSI ¼ 1 2

1 � ∫ T

0 τ dt ��

�� . ∫ T

0 τj jdt � ��

: (5)

A similar analysis was performed to quantify the regional leaflet deformations in terms of the radial and circumferential stretch ratios,

λr ¼ lr Lr

(6)

and

λc ¼ lc Lc

; (7)

respectively, where l and L are the final and initial lengths, respectively, of a line element aligned along the radial (r) or circumferential (c) direction.

3. RESULTS

3.1. Flow field and leaflet dynamics

Although the three valve models were subjected to a similar transvalvular pressure gradient, the type-0 and type-I BAV models generated cardiac outputs 12% and 9% lower than the TAV model, respectively (TAV: 4.3L/min; type-0 BAV: 3.8L/min; type-I BAV 3.9L/min; Figure 4), suggesting the intrinsic degree of stenosis of the BAV anatomy. Snapshots of the leaflet profiles and flow velocity vector fields computed during the acceleration phase (t=100 ms), at peak systole

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(t=190ms), during the deceleration phase (t=260 ms) and during diastole (t=450ms) are shown in Figure 5. A video of the transient flow velocity field captured in each model is also provided in Supporting Information (Video_1_SuppInfo.mp4). During the acceleration phase, the pressure drop across the valve initiates the opening of the leaflets in all models. As compared with the TAV and type-0 BAV models which generate symmetric orifice jets, the type-I BAV results in a jet skewed toward the non-coronary leaflet. In all models, the GOA attains a maximum value at peak systole (TAV GOA: 3.7cm2, type-0 BAV GOA: 2.2cm2, and type-I BAV GOA: 2.3cm2) when the valve is subjected to the maximum transvavular pressure gradient. Consistent with the degree of stenosis captured in both BAV models, those models generate higher orifice jet velocities (type-0 BAV: 2.0m/s; type-I BAV: 1.9m/s) than the TAV (1.1m/s). During the deceleration phase, while the re- duction in flow momentum contributes to the progressive closure of the leaflets in all models, the reduction in GOA captured in the TAV model (22% decrease vs. peak-systolic GOA) is not as ex- tensive as that predicted in the BAV models (type-0 BAV: 63% decrease vs. peak-systolic GOA, type-I BAV: 54% decrease vs. peak-systolic GOA). The skewness of the type-I BAV orifice jet is exacerbated during the deceleration phase due to the asymmetric and asynchronous closure of the leaflets. During diastole, the negative pressure drop imposed across each model produces a backflow, which forces the leaflets to coapt while generating strong vortices in the aortic sinuses.

3.2. Global wall shear stress distribution

Predictions of the total WSS distribution on the ventricularis and fibrosa of all leaflet models are shown in Figures 6 and 7, respectively. Videos of the transient WSS fields captured on the ventricularis and fibrosa of all models are also provided in Supporting Information (Video_2_SuppInfo.mp4 and Video_3_SuppInfo.mp4, respectively). The WSS distributions on the ventricularis of all leaflets exhibit similar spatial distribution patterns throughout the cardiac cycle (Figure 6). The base and belly regions are exposed to low WSS levels, while high WSS levels are concentrated in the tip region. The intrinsic degree of stenosis in the BAV model, which translated into a higher orifice jet velocity magnitude than in the TAV, subjects the ventricularis of the BAV leaflets to higher WSS levels than the ventricularis of the TAV leaflets (up to 185% difference at peak systole). While the fibrosa exhibits the same gradual increase in WSS level from the base region to the tip region, the magnitudes are systematically lower than those captured on the ventricularis because of the nearly stagnant flow in the aortic sinus (Figure 7). In addition, relative to the ventricularis, the regional WSS distribution on the leaflet fibrosa exhibits more heterogeneity and is more dependent on the valve morphology and the specific phase of the cardiac cycle.

3.3. Temporal wall shear stress characteristics

The temporal variations of the radial WSS predicted in the base, belly, and tip regions of the leaflets in all models are shown in Figure 8. The qualitative inspection of the WSS variations suggests the existence of a pulsatile and mostly unidirectional (i.e., positive) WSS on the ventricularis and an oscillatory and bidirectional (i.e., alternatively positive and negative) WSS on the fibrosa. The com- parison of the results predicted in the three sub-regions demonstrates significant spatial variations in

Figure 4. Volume flow rate variations predicted in all models over one cardiac cycle.

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WSS magnitude, pulsatility, and directionality across the ventricularis and fibrosa. The average radial WSS magnitude on each leaflet region during the acceleration phase, at peak systole, during the deceleration phase, and during diastole is shown in Figure 9. The temporal and regional characteristics of the leaflet radial WSS are qualitatively similar to those of the total WSS described earlier. Those qualitative observations are supported by the quantitative analysis of the TSM, TSG,

and OSI (Table IV). Regardless of the valve anatomy and leaflet surface, the TSM increases from the base to the tip. BAV anatomies systematically subject the belly and tip of the leaflet ventricularis to higher WSS levels (12.3 <TSM <66.5 dyn/cm2) than the TAV (5.8 <TSM <19.9 dyn/cm2). Additionally, while the models predict similar WSS levels on the fibrosa of TAV and type-0 BAV leaflets, the fibrosa of the non-coronary and fused type-I BAV leaflets is subjected to systematically higher and lower WSS, respectively (fibrosa TSM: >3.8 dyn/cm2 on non-coronary type-I BAV leaflet, <1.3 dyn/cm2 on fused type-I BAV leaflet), than the fibrosa of the TAV leaflets (0.8 <TSM <3.5 dyn/cm2). The ventricularis TSG results suggest that BAV anatomies generate less intense WSS variations in the leaflet base than the TAV (up to 49% difference in TSG). This trend is inverted in the tip region in which all BAV leaflets experience higher TSG than the TAV leaflets (up to 200% difference). This trend is not maintained in the belly region in which the type-0 BAV generates a 25% increase in TSG but the type-I BAV generates up to 45% lower TSG than the TAV. On the fibrosa,

Figure 5. Snapshots of the leaflet profiles and flow velocity vector fields computed during the acceleration phase, at peak systole, during the deceleration phase, and during diastole.

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with the exception of the base region of the non-coronary type-I BAV leaflet which experi- ences a 38% increase of TSG relative to the TAV, the results are less dependent on the valve morphology as all BAV leaflet regions are subjected to less intense WSS variations than their TAV counterparts. On the ventricularis, while the base region of the TAV and non-coronary type-I BAV leaflets experiences nearly unidirectional WSS (OSI <0.14), the same region in the type-0 BAV and fused type-I BAV leaflets experiences bidirectional WSS (OSI >0.30). The belly and tip regions are subjected to nearly unidirectional WSS (OSI <0.06) in all models. On the fibrosa, with the exception of the base of the non-coronary type-I BAV leaflet that is subjected to nearly unidirectional WSS (OSI <0.03), all other leaflet models experience a bidirectional WSS (OSI >0.14) across the surface.

3.4. Equivalent strain distribution

Equivalent von Mises strain distributions captured on TAV and BAV leaflets at four phases of the cardiac cycle are shown in Figure 10. Videos of the transient von Mises strain fields captured in each leaflet model are also provided in supplemental online mate- rial (Video_4_SuppInfo.mp4). During systole, because all leaflets are in their open posi- tion and are subjected to low transvalvular pressure, they exhibit essentially identical strain fields characterized by uniform and low strain levels over their entire surface. In contrast, during diastole, the rapid increase in transvalvular pressure, which leads to the coaptation of the leaflets, is accompanied by a tremendous increase in strain in all leaflet models. Additionally, the spatial strain distribution exhibits significant variability across the leaflet, as indicated by the progressive increase captured from the base to the tip re- gion. While the leaflet strain distributions are qualitatively similar in the TAV and BAV models, the valve anatomy affects the specific locations of strain overloads. In the TAV

Figure 6. Snapshots of the total wall shear stress (WSS) distributions on the leaflet ventricularis computed during the acceleration phase, at peak systole, during the deceleration phase, and during diastole. BAV,

bicuspid aortic valve; TAV, tricuspid aortic valve.

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leaflets, high strains are concentrated in the tip region but only near the commissures. In the type-0 and fused type-I BAV leaflets, high strains are present over the full extent of the coaptation region (i.e., tip region). Lastly, the non-coronary type-I BAV leaflet exhibits less spatial heterogeneity and generates regions of high strains in the tip region but only near the axis of symmetry of the leaflet.

Figure 7. Snapshots of the total wall shear stress (WSS) distributions on the leaflet fibrosa computed during the acceleration phase, at peak systole, during the deceleration phase, and during diastole. BAV, bicuspid

aortic valve; TAV, tricuspid aortic valve.

Figure 8. Temporal variations of regional radial wall shear stress (WSS) magnitude over one cardiac cycle. BAV, bicuspid aortic valve; TAV, tricuspid aortic valve.

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Figure 9. Average radial wall shear stress (WSS) magnitude by leaflet region during the acceleration phase, at peak systole, during the deceleration phase, and during diastole. BAV, bicuspid aortic valve; TAV,

tricuspid aortic valve.

Table IV. Side-specific regional radial WSS characteristics on TAV, type-0, and type-I BAV leaflets.

Ventricularis Fibrosa

Base Belly Tip Leaflet average Base Belly Tip

Leaflet average

TSM (dyn/cm2)

TAV Leaflets 6.00 5.75 19.93 10.56 0.83 1.74 3.52 2.03 Type-0 BAV

leaflets 3.41 17.03 66.45 28.96 1.49 2.10 2.94 2.18

NC type-I BAV leaflet

6.30 12.27 44.67 21.08 3.79 3.12 5.38 4.10

F Type-I BAV leaflet

1.18 8.92 36.20 15.43 0.74 0.31 1.29 0.78

TSG (dyn/cm2�s)

TAV Leaflets 187.27 264.89 390.78 280.98 61.22 179.29 240.28 160.26 Type-0 BAV

leaflets 179.52 331.30 1174.49 561.77 57.41 108.72 153.22 106.45

NC Type-I BAV leaflet

151.94 237.55 550.64 313.38 84.66 152.02 221.45 152.71

F type-I BAV leaflet

95.06 145.79 481.88 240.91 23.49 22.74 84.40 43.54

OSI TAV Leaflets 0.06 0.06 0.05 0.06 0.27 0.38 0.14 0.26 Type-0 BAV

leaflets 0.40 0.04 0.04 0.16 0.24 0.39 0.39 0.34

NC Type-I BAV leaflet

0.14 0.05 0.01 0.07 0.03 0.49 0.29 0.27

F type-I BAV Leaflet

0.30 0.02 0.02 0.11 0.20 0.48 0.28 0.32

NC: non-coronary; F: fused; BAV, bicuspid aortic valve; TAV, tricuspid aortic valve; WSS, wall shear stress; OSI, oscillatory shear index; TSM, temporal shear magnitude; TSG, temporal shear gradient.

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3.5. Radial and circumferential stretch characteristics

The transient regional stretch levels captured in the radial and circumferential directions are plotted in Figure 11. Regardless of the valve morphotype, stretch is higher in the radial direc- tion than in the circumferential direction and increases from the base to the tip region of the leaflet. During systole, the leaflets are under radial compression (radial stretch ratio <1) and circumferential tension (circumferential stretch ratio >1). The radial and circumferential stretch ratios increase rapidly during early diastole, then plateau at their peak value following coapta- tion (TAV leaflet: 1.19 radially and 1.13 circumferentially; type-0 BAV leaflet: 1.24 radially and 1.16 circumferentially; non-coronary type-I BAV leaflet: 1.22 radially and 1.12 circumferentially; fused type-I BAV leaflet: 1.20 radially and 1.13 circumferentially). The unloading phase driven by the rapid decrease in transvavular pressure magnitude at the end of diastole is accompanied by an equally rapid decrease in stretch back to the reference state value. The average radial and circumferential stretch predicted in each leaflet region during the acceleration phase, at peak systole, during the deceleration phase, and during diastole are shown in Figure 12. Regardless of the valve anatomies, both the radial and circumferential stretch increase temporally from the acceleration phase to diastole and spatially from the base to the tip region. Regional maximum stretch values predicted in the base, belly, and tip of the leaflets are reported in Table V. Similar to the trend exhibited by the average stretch levels, the maximum stretch levels increase from the base to the tip region, and the maximum radial stretch is higher than the maximum circumferential stretch.

Figure 10. Snapshots of the leaflet equivalent von Mises strain distributions computed during the acceler- ation phase, at peak systole, during the deceleration phase, and during diastole. BAV, bicuspid aortic valve;

TAV, tricuspid aortic valve.

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4. DISCUSSION

In this study, we implemented an ALE FSI approach to quantify the temporal and regional differ- ences in flow, leaflet dynamics, WSS, and deformation fields in 3D representations of a TAV, type-0 BAV, and type-I BAV. The results reveal the existence of intrinsic stenosis, high jet velocity, WSS, and stretch abnormalities in BAVs and the dependence of those abnormalities on the BAV morphotype.

Figure 12. Average stretch magnitude by leaflet region during the acceleration phase, at peak systole, during the deceleration phase, and during diastole. BAV, bicuspid aortic valve; TAV, tricuspid aortic valve.

Figure 11. Temporal variations of regional radial and circumferential stretch ratios over one cardiac cycle. BAV, bicuspid aortic valve; TAV, tricuspid aortic valve.

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4.1. Tricuspid aortic valve versus bicuspid aortic valve mechanics

The present study suggests the existence of important mechanical abnormalities in otherwise normal (i.e., non-calcified) type-0 and type-I BAVs. While the models were all subjected to the same transvalvular pressure, the BAVs were intrinsically stenotic. They generated smaller GOAs and lower cardiac outputs than the TAV. The degree of stenosis was particularly apparent at peak systole during which the type-0 and type-I BAV jet velocities were 82% and 73% higher, respec- tively, than in the TAV. The existence of lower cardiac outputs in BAV patients has already been reported in clinical studies [64, 65], while the increased energy loss though the BAV has been doc- umented in vitro [37, 66]. The qualitative inspection of the flow fields also isolated the eccentricity and skewness of the type-I BAV orifice jet, which is a defining feature of this morphotype [67–70]. An interesting result revealed by the simulations is the difference in vorticity dynamics between the TAV and BAV anatomies. While the TAV and type-0 BAV generated symmetrical vortical struc- tures in the aortic sinuses during the deceleration phase, the type-I BAV generated asymmetrical vortex patterns with larger strength on the fused leaflet side than on the non-coronary leaflet side. This vortex asymmetry resulted in the asymmetrical and asynchronous closure of the type-I BAV leaflets and may be responsible for the intrinsic degree of regurgitation frequently exhibited by this valvular phenotype [3]. The flow alterations captured through the BAVs also subjected the leaflets to substantial mechan-

ical abnormalities characterized by:

• WSS overloads on both leaflet surfaces (174% and 100% increase in TSM on the ventricularis of type-0 BAV and type-I BAV leaflets, respectively; 7% and 102% increase in TSM on the fibrosa of type-0 BAV and type-I BAV leaflets, respectively)

• Side-specific alterations in temporal WSS variations (100% and 12% increase in TSG on the ventricularis of type-0 BAV and type-I BAV leaflets, respectively; 34% and 73% decrease in TSG on the fibrosa of type-0 BAV and type-I BAV leaflets, respectively)

• Increased WSS oscillation and bidirectionality on both leaflet surfaces (0.10 and 0.05 increase in OSI on the ventricularis of type-0 BAV and type-I BAV leaflets, respectively; 0.08 and 0.06 increase in OSI on the fibrosa of type-0 BAV and type-I BAV leaflets, respectively)

• Increased radial deformation (4% and 3% increase in type-0 BAV and type-I BAV leaflets, respectively)

The WSS environments captured by our models are both qualitatively and quantitatively similar to those reported previously in vitro. The ventricularis WSS predicted in our models exhibits the same change in direction at the end of systole as that measured using laser Doppler velocimetry on a tri-leaflet polymeric valve [48]. The peak-systolic WSS levels captured in the belly and tip regions of the TAV leaflets (61 and 108dyn/cm2, respectively) are of the same order of magnitude as those reported exper- imentally or computationally at a point or over several nodes (belly: 64–71dyn/cm2 [48]; tip: 79–100dyn/cm2 [44, 71, 72]). The regional leaflet WSS captured by the present 3D valve models also exhibits the same gradual increase from the base to the tip region reported by our previous 2D models

Table V. Regional stretch characteristics on TAV, type-0, and type-I BAV leaflets.

base belly tip leaflet average

Maximum radial stretch TAV leaflets 1.09 1.18 1.19 1.15 Type-0 BAV leaflets 1.13 1.21 1.24 1.19 NC type-I BAV leaflet 1.12 1.20 1.22 1.18 F Type-I BAV leaflet 1.11 1.19 1.20 1.17

Maximum circumferential stretch TAV leaflets 1.09 1.13 1.13 1.12 Type-0 BAV leaflets 1.08 1.14 1.16 1.13 NC Type-I BAV leaflet 1.10 1.12 1.12 1.11 F Type-I BAV leaflet 1.09 1.11 1.13 1.11

NC: non-coronary; F: fused; BAV, bicuspid aortic valve; TAV, tricuspid aortic valve; WSS, wall shear stress.

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[44] but is systematically higher (up to 215% increase in leaflet-averaged TSM). This demonstrates the 3D of valvular flow and justifies the need for 3D geometries in valve models. The predictions of the dynamic leaflet stretch are also in good agreement with previous reports. The

present results indicate the predominance of the radial stretch relative to the circumferential stretch in all valve models, which has also been observed in vitro on porcine leaflets [73]. Regardless of the valve morphology, the analysis of the leaflet deformation characteristics indicates a contracted state along the radial direction during systole and a taut state during diastole, which is similar to the features pre- dicted by previous TAV and type-0 BAV FSI models [42]. Lastly, the temporal characteristics of the radial and circumferential stretch waveforms, which indicated relatively low stretch levels during sys- tole (<1.05) and early diastolic stretch elevation followed by sustained elevated stretch during coapta- tion (>1.10), are qualitatively and quantitatively similar to results reported by previous in vitro measurements and computational simulations [42, 73, 74]. Importantly, the existence of quantifiable differences in the stretch environments of TAV and BAV leaflets suggests the important impact of the BAV anatomy and hemodynamics on the mechanical stretch abnormalities observed on the leaflets.

4.2. Type-0 versus type-I bicuspid aortic valve leaflets mechanics

The type-0 and type-I BAVs exhibited similar performance characteristics in terms of cardiac out- put (0.2% difference), GOA (0.5% difference) and maximum jet velocity (0.5% difference). How- ever, the stress and deformation environments captured in BAV leaflets were found to be strongly dependent on the BAV morphotype. The ventricularis of type-0 BAV leaflets experiences system- atically higher WSS levels, larger WSS temporal variations, and more pronounced WSS bidirectionality than the ventricularis of type-I BAV leaflets. Similar observations can be made of the radial and circumferential stretch levels, which are systematically elevated (up to 20% higher) on type-0 BAV leaflets relative to type-I BAV leaflets. Lastly, the asymmetry of the type-I BAV was shown to generate leaflet-specific WSS abnormalities. The fibrosa of the non-coronary type-I BAV leaflet experiences elevated WSS magnitude and temporal variations relative to the type-0 BAV leaflets. In contrast, the fibrosa of the fused type-I BAV leaflet is subjected to a sub-physiologic WSS environment relative to TAV and type-0 BAV leaflets.

4.3. Potential clinical impact

The mechanical characteristics captured by the models designed in this study provide some new insights into BAV function that may help elucidate the potential role played by hemodynamics in BAV calcification. Previous in vitro studies have established the sensitivity of valvular tissue to fluid WSS and stretch. In particular, abnormalities in WSS magnitude, directionality, and frequency of oscillation have been shown to trigger inflammatory, remodeling, and paracrine processes, which typically precede calcification [15, 16, 20, 21, 23, 24, 75]. Similarly, elevated stretch levels have been shown to inhibit the contractile phenotype of valvular interstitial cells and to increase in turn extracellular matrix synthesis [25, 26, 76–78]. While the impact of a mechanical environment com- bining abnormally elevated WSS and stretch, such as that captured in the non-coronary type-I BAV leaflet, needs to be further investigated, those ex vivo results tend to suggest that BAV mechanics may be particularly conducive to CAVD. Leaflet calcification has been shown to be more severe, more rapid, and to follow a faster pro-

gression in type-I BAVs than in any other morphotype [79]. In addition, in the type-I BAV, the fused leaflet has been identified as the most vulnerable to CAVD [7]. An interesting parallel can be drawn between those observations and the heterogeneity of the mechanical environment captured in this study on type-I BAV leaflets. The contrasted WSS magnitude, directionality, and temporal variations predicted on the fused and non-coronary leaflets may be responsible for the leaflet-specific susceptibility to CAVD development. A recent study conducted in our laboratory attempted to isolate the impact of BAV flow on

valvular biology [22]. By characterizing the WSS environments in 2D representations of a TAV and a type-I BAV and by subjecting fresh porcine leaflets to those environments in a shear stress bioreactor, the study revealed the ability of type-I BAV hemodynamic stresses to promote valvular calcification. The temporal and regional WSS and stretch characteristics provided by the 3D TAV

e02798 (16 of 21) K. CAO AND P. SUCOSKY

and BAV models developed in the present study will enable the design of improved mechanobiological studies through the implementation of more realistic mechanical signals mim- icking more closely the native valvular hemodynamics. The possible demonstration of a hemody- namic pathway of BAV calcification may lead to the development of different diagnosis and therapeutic modalities aimed at (i) detecting BAV patient subpopulations most prone to CAVD, (ii) normalizing BAV flow before it causes irreversible biological changes, or (iii) blocking the mechanically induced biological cascade before calcification has reached a point of no return.

4.4. Assumptions justification and model limitations

• Symmetry assumption: The native asymmetry of the aortic root and the coronary arteries were not considered in the present study. A previous FE model of a patient-specific TAV has dem- onstrated that the native differences in size and shape of the three leaflets and sinuses could result in asymmetric stress distributions on the leaflets [80]. Studies on full valve geometries with coronary arteries are currently underway in our laboratory. In addition, while TAV and type-0 BAV simulations were performed in smaller models (60° section in the TAV, 90° sec- tion in the type-0 BAV) than type-I BAV simulations (180° section) due to the respective sym- metry exhibited by each valve, the implementation of more complex 180° TAV and type-0 BAV models was found to generate less than 6% and 4% difference in leaflet-averaged WSS predictions, respectively, suggesting the limited impact of the symmetry boundary condition on the flow.

• Raphe material formulation: The raphe in the type-I BAV was modeled using the same material properties as the leaflets. While the presence of fibrous tissue in the raphe may alter the leaflet dynamics, the raphe mechanical properties have not been characterized.

• Leaflet material formulation: The simulations approximated the leaflet material as isotropic in order to avoid mesh distortion issues in the structure domain. While the WSS and strain fields predicted by our models are in good agreement with those predicted by models that accounted for tissue anisotropy [42, 71, 72], this apparent agreement is not sufficient to validate the strain results as it may be the result of different artifacts (thickness of leaflet, linearized modulus). The inclusion of the tissue anisotropy in the model is necessary to improve the assessment of the native leaflet dynamics and structural mechanics. In addition, the use of similar material models for TAV and BAV leaflets is justified by the recent demonstration of similar fiber ar- rangements in those leaflets [58]. This observation along with the particular focus of the pres- ent models on the prediction of the macro-scale valvular biomechanics justify the implementation of the same material formulation for both TAV and BAV leaflets as a first approximation.

• Laminar flow assumption: Turbulence effects were not accounted for in the models. It is impor- tant to note that Reynolds averaged Navier–Stokes-based turbulence models available in com- mercial solvers have been primarily developed and tuned for fully developed turbulence and are therefore not suitable for transitional and relaminarizing flows such as that through the valve [81–83]. In addition, such models are typically too dissipative and their use to simulate valvular flow would lead to incorrect hemodynamic predictions [84, 85]. At the other end of the spectrum, direct numerical simulations would permit to resolve turbulence explicitly at the highest level of fidelity, but this approach requires a computational grid able to resolve all the scales and is therefore not practical for FSI valve problems [81]. Lastly, while large- eddy simulations constitute a compromise between those two approaches, they still require high mesh densities and small time steps, which complicates their use for valvular FSI simula- tions [83]. In this context, it can be argued that the benefits of predicting turbulent effects gen- erated at peak systole through the valve are outweighed by the computational costs and challenging implementation of this approach in the context of a fully coupled FSI model. Therefore, while modeling blood flow as laminar may cause inaccuracy at peak systole, this simplification permitted to capture the near-native hemodynamic characteristics in the aortic root during most of the cardiac cycle, at a much lower computational cost than direct numerical simulations.

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• Leaflet contact treatment: The implementation of an ALE approach required a small geometri- cal gap to be left between the leaflets during coaptation in order to prevent the separation of the fluid domain into two domains. While this requirement prevented the geometrical coaptation of the leaflets, the frictionless contact condition enforced when the leaflets came in close proxim- ity (<100μm) and the wall boundary condition imposed on the fluid face bounded by the free edges of the leaflets permitted to virtually treat the leaflets as in contact and to effectively pre- vent flow through this gap. As a result, this gap is expected to have little or no effect on the global flow and leaflet dynamics.

5. CONCLUSIONS

This comparative computational study aimed at characterizing the key mechanical differences between TAV, type-0 BAV, and type-I leaflets. The models demonstrate that (i) type-0 and type-I BAV leaflets experience abnormal hemodynamic stresses relative to TAV leaflets and (ii) the degree and regional characteristics of those abnormalities in BAVs are morphotype-dependent and leaflet-dependent. Those results provide important insights into the mechanical characteristics of BAV leaflets and their dependence on BAV morphotype, which may further our understanding of the role played by hemodynamic forces in BAV disease.

ACKNOWLEDGEMENTS

This work was supported by the National Science Foundation under Grants CMMI-1148558 and CMMI-1550144; and the American Heart Association under Grant 14PRE18940010.

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