Comparative hemodynamics in an aorta with bicuspid and trileaflet valves

Transcription

1 Theor. Comput. Fluid Dyn. DOI /s ORIGINAL ARTICLE Anvar Gilmanov Fotis Sotiropoulos Comparative hemodynamics in an aorta with bicuspid and trileaflet valves Received: 14 February 2015 / Accepted: 24 August 2015 Springer-Verlag Berlin Heidelberg 2015 Abstract Bicuspid aortic valve (BAV) is a congenital heart defect that has been associated with serious aortopathies, such as aortic stenosis, aortic regurgitation, infective endocarditis, aortic dissection, calcific aortic valve and dilatation of ascending aorta. There are two main hypotheses to explain the increase prevalence of aortopathies in patients with BAV: the genetic and the hemodynamic. In this study, we seek to investigate the possible role of hemodynamic factors as causes of BAV-associated aortopathy. We employ the curvilinear immersed boundary method coupled with an efficient thin-shell finite-element formulation for tissues to carry out fluid structure interaction simulations of a healthy trileaflet aortic valve (TAV) and a BAV placed in the same anatomic aorta. The computed results reveal major differences between the TAV and BAV flow patterns. These include: the dynamics of the aortic valve vortex ring formation and break up; the large-scale flow patterns in the ascending aorta; the shear stress magnitude, directions, and dynamics on the heart valve surfaces. The computed results are in qualitative agreement with in vivo magnetic resonance imaging data and suggest that the linkages between BAV aortopathy and hemodynamics deserve further investigation. Keywords Bicuspid valve Trileaflet valve Fluid structure interaction Immersed boundary method Finite element Thin shell Rotation-free approach 1 Introduction and background A healthy aortic valve is trileaflet (TAV) consisting of three leaflets: the left coronary cusp (LCC), the right coronary cusp (RCC) and the non-coronary cusp (NCC) (see Fig. 1). Bicuspid aortic valve (BAV) is a lifethreatening congenital heart defect that occurs in approximately 1 2 % of the population [1,2]. BAV arises during valvulogenesis when two of the three leaflets of the aortic valve are fused together. With reference to Fig. 1, there exist three types of BAV anatomies: (1) fusion of the RCC and LCC (referred to as R-L) with the raphe positioned between the left (L) and right (R) coronary sinuses; (2) fusion of the RCC and NCC (referred to as R-N); and (3) fusion of the LCC and NCC (referred to as L-N) [3]. The morbidity rate associated with the development and progression of the BAV disease is higher than all other congenital heart conditions combined [4,5]. BAV patients have a higher prevalence of various aortopathies, including aortic Communicated by Jeff D. Eldredge. Anvar Gilmanov Fotis Sotiropoulos (B) Saint Anthony Falls Laboratory, Department of Civil, Environmental and Geo-Engineering, College of Science and Engineering, University of Minnesota, Minneapolis, MN, USA fotis@umn.edu Tel.: +1(612) Anvar Gilmanov gilmanov.anvar@gmail.com Tel.: +1(612)

2 A. Gilmanov, F. Sotiropoulos Fig. 1 Types and orientations of trileaflet (TAV, left) and bicuspid (BAV, right) valves used in the simulations. RC is the right coronary sinus, LC is the left coronary sinus, NC is the non-coronary sinus, RCC is the right coronary cusp, LCC is the left coronary cusp, and NCC is the non-coronary cusp. Presented orientation of BAV is called R-L and occurs in about 70 % of patients with BAV [29]. The BAV eccentricity index is defined as ε = d 1 /d 2 = 1.25 [39] stenosis, aortic regurgitation, infective endocarditis and aortic dissection [5], dilatation of ascending aortic andarchofaorta[6]. However, the linkages between BAV and the higher prevalence of such aortopathies are not fully understood. In fact this issue is still debated in the literature [5 7], and two main hypotheses are at the center of this debate: genetic factors [8 10] and hemodynamic factors [5,11 13]. Alongside with two competing theories, some studies [14,15] have also suggested that BAV aortopathy is not caused by genes or hemodynamics alone, but it may rather be related to both genes and hemodynamics. Although the genetic hypothesis has been prevalent for a long time, there is increasing evidence that hemodynamic factors may also play a significant role as a cause of aortopathy in patients with BAV disease [5,16]. For example, Bissell et al. [17] showed links between flow patterns and aortic dilation by using cardiovascular magnetic resonance (CMR). Nathan et al. [18] employed electrocardiogram-gated computed tomographic angiography (ECG-gated CTA) and finite-element (FE) analysis to show that the magnitude of wall shear stress (WSS) on the aorta wall for patients with BAV tends to be higher in comparison with patients with TAV. Barker et al. [9] demonstrated that BAVs cause significant changes in the large-scale flow patterns in the ascending aorta when compared with the flowfields associated with TAVs. Computational fluid dynamics methods have advanced in recent years to be able to carry out fluid structure interaction (FSI) simulations of cardiovascular flows [19 24] and have the potential to provide insights into BAV hemodynamics. A first attempt to explore hemodynamic factors as possible causes to calcification of BAV was reported by Chandra et al. [25]. They employed 2D model to simulate FSI of TAV and BAV with blood flow and argued that their numerical simulations yielded insights in favor of the hemodynamic hypothesis. However, the highly three-dimensional anatomy of a BAV and the ascending aorta render results from 2D FSI studies questionable insofar as their clinical relevance is concerned. Multiscale finite-element simulations (organ scale, tissue scale and cell scale [21]) of the BAV and TAV tissue were reported by Weinberg and Mofrad [21]. They used the commercial finite-element LS-DYNA code to carry out FSI simulations and found that the leaflets of a BAV do not open as smoothly and undergo more flexure relative to the normal TAV. The developed multiscale model for tissue of a valve is a strong point of this approach, but flow patterns reported by these authors appear simple and suggest low numerical resolution, thus raising questions about the clinical relevance of their findings. Jermihov et al. [26] performed a dynamic finite-element (FE) structural analysis of the valves in the aortic position in order to compute the strain and stress distribution on the TAV and BAV models. Their simulations showed that the valve orifice area in the fully open position was significantly reduced in BAV compared to that for the TAV. Such reduction can lead to significant changes in strain and stresses. This model, however, does not consider the stresses imparted by the flow on the tissue. It appears from the above review of the literature that while previous computational studies have made progress and provided some evidence in support of hemodynamic factors as possible cause for BAV-related aortopathies, computational methods that can simulate the coupled FSI of anatomically realistic BAV and TAV in anatomic aortas at physiologic conditions and high resolution have yet to be used to tackle this problem. Therefore, the main contribution of our paper is to take the first step in this direction by reporting FSI simulations of both TAV and BAV in a 3D anatomic aorta and analyzing the computed results to elucidate important differences between the aortic hemodynamics induced by each valve type. We employ our previously developed and validated curvilinear immersed boundary (CURVIB) method [27] coupled with an efficient, rotation-free thin-shell FE formulation [28] to carry out FSI simulations of a TAV and a BAV placed in the same

3 Comparative hemodynamics in an aorta anatomic aorta. The specific BAV morphology we consider in this work is the R-L type (Fig. 1, right) because this case occurs in approximately 70 % of the patients with BAV [29]. We analyze and discuss the simulated flowfields induced by the TAV and the BAV to reveal significant differences in the near-valve and ascending aorta flow patterns between the two valve types. We argue that based on our findings the hemodynamic hypothesis for BAV-related aortopathy deserves further serious investigation. The paper is organized as follows. In Sect. 2, we describe the governing equations for the fluid and solid structure and discuss FSI algorithm for solving these equations. In Sect. 3, we represent problems and give their computational details. In Sect. 4, we discuss the numerical results and discuss the TAV and BAV flowfields. Finally, in Sect. 5, we summarize the major findings of this work. 2 Governing equations and numerical method We consider FSI of a deformable body Ω s submerged in an incompressible fluid occupying a volume Ω f bounded by Ω f as shown in Fig. 2. The leaflets of a heart valve are represented as three-dimensional surfaces and modeled as thin flexible shells with stretched and bending stiffness [28]. The fluid and solid domains intersect at the interface: Γ fsi = Ω s Ωf. We use bold symbols for vectors and bold symbols with two underlines for tensors and matrices. The regular and italic symbols are reserved for scalar and vector/tensor components, respectively. 2.1 The equations for the fluid domain The fluid boundary consists of three non-overlapping parts: Ω f = Γf D Γ N f Γ fsi. Here, Γf D and Γf N are the stationary boundaries in which Dirichlet and/or Neumann boundary conditions are specified. As mentioned above, Γ fsi is the interface between the fluid domain and the solid domain, i.e., the moving interface the configuration of which needs to be determined by solving the FSI problem. The equations govern the motion of Newtonian incompressible fluid in a domain Ω f as the Navier Stokes and continuity equations, which read in vector/tensor notation as follows: dv ρ f = σ f in Ω f, dt v = 0 in Ω f. (1) In the above equations, ρ f is the density of the fluid, d/dt is the material or Lagrangian time derivative, v is the fluid velocity vector and σ f is the stress tensor. The above equations are subjected to various boundary conditions for the velocity v on the various segments comprising fluid boundary. For example, on the Dirichlet portion of the boundary Γf D, Dirichlet boundary conditions are specified: v = v on Γf D (2) On the Neumann segment of the boundary, a stress boundary condition of the following form may be applied: σ f n f = t f on Γf N (3) where v and t f are known functions and n f is the normal unit vector to the Γf N boundary. Fig. 2 Definition of the computational domains for a FSI problem with a thin structure Ω s immersed in a background fluid domain Ω f. The solid body is considered as a thin shell, and Γ fsi is the interface between the fluid and solid domain intersection

4 A. Gilmanov, F. Sotiropoulos 2.2 The equations for the solid domain The momentum equations for the solid part, formulated in the current configuration, have the following form: d u ρ s = σ s + ρ s b, (4) dt where ρ s is the density of the material, σ s is the Cauchy stress tensor, u is the displacement of a material point, u = du/dt is the velocity of the material point and b is the body force per unit mass. The boundary of the solid structure can be represented as sum of non-overlapping parts Ω s = Γs D Γ N s Γ fsi, where the indices D and N denote boundaries with Dirichlet and Neumann conditions, respectively: u = u on Γs D σ s n s = t s on Γs N (5) where Γs D and Γs N represent the portions of the surface of the body in its current configuration where Dirichlet and Neumann conditions are applied, respectively, t s is a traction vector acting on the surface, n s is a unit normal to the boundary and u is the velocity prescribed on the surface. For FSI problems, additional boundary conditions must be implemented on the Γ fsi : u = v on Γ fsi, σ s n s = t f, on Γ fsi, (6) where Γ fsi is part of the moving structure surface the configuration of which needs to be determined by solving the FSI problem and t f = σ f n f is a traction vector which acts on this part of surface from the fluid in which σ f and n f are the stress tensor and surface normal unit vector from the fluid. 2.3 FSI algorithm for solving the fluid and structure equations In this section, we give only a short description of FSI algorithm we employ to solve the TAV/BAV interaction with unsteady blood flow in a real aorta. More detailed description of various aspects of the numerical method can be found in the following references. The fluid solver is based on the CURVIB approach of Ge and Sotiropoulos [27], which uses the hybrid stagger/non-staggered approach originally proposed by Gilmanov and Sotiropoulos [30] to solve the governing equations in generalized curvilinear grids with complex immersed boundaries [27]. For the structural solver, we use the FE model proposed by Stolarski et al. [28], which approximates the shell curvature tensor, associated with a main element and three surrounding elements, using a complete quadratic polynomial in a moving with the shell rectilinear coordinate system attached to that element. This approach permits accurate approximations in the evaluation of the element curvature tensor. Most importantly, since the above approach is used only to compute the bending strains within the element calculation of membrane strains is based on the flat geometry of the element non-physical membrane locking is automatically avoided. For the derivation and the details of our method, the reader is referred to Stolarski et al. [28]. We use the sharp interface immersed boundary method of Gilmanov and Sotiropoulos [30], Ge and Sotiropoulos [27] and Borazjani et al. [31], which is able to deal with the arbitrarily complex, moving/deformable solid bodies in a curvilinear background grid where the fluid equations are solved. The effect of the moving body is represented by reconstructing boundary conditions at the grid points that are closest to and in the exterior of the solid surface, which we refer to as the immersed boundary (IB) nodes. The boundary conditions that needed to be reconstructed at the IB nodes are the three velocity components, which are reconstructed by interpolation [30] along the local normal direction. The governing equations of the fluid (Eq. 1) and solid (Eq. 4) domains as well as the continuity conditions on the interface constitute a modularly partitioned FSI problem, which can be solved by coupling together two independent solvers: the fluid solver F and the solid solver S. We use the conventional Dirichlet Neumann partition [32] to couple the system of fluid solid equations. This means that the fluid equations are solved by enforcing Dirichlet boundary condition, while the solid equations are solved by prescribing the load on the interface Γ fsi [33]. To facilitate the description of the FSI algorithm, let us assume, without loss of generality,

5 Comparative hemodynamics in an aorta that the pressure and velocity fields v n, p n for the fluid alongside with the displacements and velocities of the solid structure u n, u n are known at time step t n. In our fixed-point iteration, the fluid solver F uses displacements and velocity of the solid structure u n+1, u n+1 and gives new fluid velocity v n+1 and pressure field p n+1 by solving Eqs. (1). The solid solver S in turn uses the so-updated fluid velocities and pressure field to advance the solution of displacements and velocity of the solid structure u n+1, u n+1. Sub-iterations (l) are implemented every time step to satisfy the coupled system of fluid solid equations (Eqs. 1, 4) and advance the solution to time step t n+1, where index l is the number of fixed-point iteration and all variables at level l = 0are at the previous time step t n. The sub-iterations continue until an appropriate norm of the error of the flow and structural variables between levels l and l + 1 has been reduced to a desired tolerance, and the above equations have been satisfied at level t n+1. The above procedure is generally described as a strongly coupled FSI algorithm and ensures that the continuity of the stress at the fluid structure interface is satisfied within the desired convergence threshold. For mass ratio problems of order one (ρ f /ρ s 1), which arise in simulations of heart valves, the Aitken nonlinear relaxation technique is also implemented to accelerate the convergence of the strongly coupled FSI algorithm [31,34]. The convergence tolerance for the structural solver in the above algorithm is of order Note that we enforce the convergence of both the displacement and velocity fields of the structure and thus we employ a strict requirement for convergence. For more details, the reader is referred to [35] where we demonstrated the capabilities of the proposed CURVIB FE FSI algorithm by applying it to simulate several problems of increasing complexity, all involving FSI with thin flexible structures. Two sets of problems with available experimental and computational data were chosen to validate our new method in that study [35]: (1) the oscillations of a cantilever mounted on the wake of a square cylinder; and (2) the flapping dynamics of an inverted flag. Comparisons of our simulations and previously published simulations and available experimental data show excellent agreement [35]. 3 Test cases and computational details The leaflets of the aortic heart valves we consider herein are modeled as a thin shell using the rotation-free FE formulation of Stolarski et al. [28]. We use a model of a prosthetic polymeric aortic valve with isotropic material and the Neo-Hookean constitutive equation. The geometric and material characteristics of the valve are specified from values available in the literature to correspond to actual prosthetic polymeric valve [36] and are as follows: valve diameter d 0 = m, leaflet thickness h 0 = m, Young s modulus E = 1 MPa, Poisson s coefficient ν = 0.35 and density ρ s = kg/m 3.AsshowninFig.3, thevalve is placed in an anatomic aorta, which has been reconstructed from patient-specific MRI data this is the same aorta geometry used in [24] for the simulations of a mechanical bileaflet heart valve. Fig. 3 Computational domain for the FSI simulations of a TAV or BAV in an anatomic aorta

6 A. Gilmanov, F. Sotiropoulos Fig. 4 Physiologic incoming flow waveform specified at the inlet of the aorta The pulsatile flow wave form we prescribe as the inflow boundary condition at the inlet of the aorta domain is shown in Fig. 4. The corresponding heart beat is equal to 70 bpm, which gives a period of the cardiovascular cycle T = s. The valve diameter d 0 is used as the characteristic length scale, and the peak systolic velocity of U 0 = 0.8 m/s is used as the velocity scale. Using the viscosity of blood μ = Pa s and blood density ρ f = 1050 kg/m 3 gives a peak systolic Reynolds number Re = 6000 and Womersley number α = 18.3, which are well within the physiologic range [36 38]. The characteristic timescale is equal to T 0 = d 0 /U 0 = s, and thus, the non-dimensional period of cardiac cycle is T = T/T 0 = 0.857/ = non-dimensional time units. The non-dimensional time step for the simulations is set equal to Δ t = 0.005, which corresponds to discretizing the cardiac cycle with N time = T / t = 5530 computational time steps. Since the density ratio for this problem is of order one (ρ f /ρ s 1), the strong coupling FSI iteration with the Aitken nonlinear relaxation technique [31] is required for stable and robust simulations. In all subsequently presented simulations, 4 10 strong coupling iterations are sufficient to reduce the residuals by 9 orders of magnitude. The overall computational setup is shown in Fig. 3 and consists of (a) the anatomic aorta, (b) the flexible trileaflet/bicuspid prosthetic heart valve, (c) the rigid valve supported structure and (d) the valve housing. A curvilinear boundary-fitted grid is used to discretize aorta domain with , in the two transverse and streamwise directions, respectively. This grid is identical to that used in [24] to discretize the same aorta geometry but with a mechanical bileaflet heart valve. The leaflets are discretized with 476 triangle elements in the case of TAV and by 491 triangle elements for the BAV. We simulate the trileaflet valve with the common orientation of its leaflets toward the sinuses (RCC right coronary cusp; LCC left coronary cusp; and NCC non-coronary cusp, Fig. 1, left) and the BAV for the most often case occurring in BAV patients: type R-L (Fig. 1, right). The calculated value of eccentricity index forabavisdefinedasε = d 1 /d 2 (Fig. 1, right). For the case we simulate herein ε = 1.25, which renders the BAV, we study eccentric [39]. We will show that such eccentricity can have a major effect in the flow by changing the direction of jet flow from the valve during the acceleration phase of systole to impact on the wall of the aorta. The flow wave shown in Fig. 4, which corresponds to the systolic phase of the cardiac cycle during which the aortic valve opens and closes, is used to specify time-dependent Dirichlet conditions for the velocity at the inlet. At the outlet of the aorta, zero-gradient Neumann condition is applied for all three velocity components ( v/ n = 0) along with a correction of the so-resulting velocity field to enforce global mass conservation. No-slip and no-flux boundary conditions are applied on all solid surfaces. 4 Results and discussion 4.1 Comparison of TAV and BAV kinematics A clinically relevant quantity that depends on the structural properties, anatomic factors and dynamics of the aortic heart valve and can impact the hemodynamics of blood flow is the so-called aortic orifice area (AOA). As implied by its name, the AOA is a direct or indirect measure of the size of the orifice generated as the valve opens. Following by Garcia et al. [40], one can define three different AOA-type quantities: (1) the geometric

7 Comparative hemodynamics in an aorta orifice area (GOA), (2) the effective orifice area (EOA) and (3) the Gorlin area (GA). The GOA represents the real orifice surface and can be estimated directly from MRI data. This quantity does not directly relate to valve hemodynamics and is rarely used in clinical practice. The EOA, on the other hand, is defined as the minimum cross-sectional area of the aortic valve jet. As such, this quantity is directly linked with the valve hemodynamics and is used in clinical practice as an estimator of severity of aortic stenosis (AS) [40]. The relationship between GOA and EOA is defined as a ratio C L = EOA/GOA, which is called the contraction coefficient and is always <1. The value of C L depends on the shape of the open leaflets and at the peak of systole is equal to C L [40]. Finally, the GA is defined based on the flow rate Q sys (ml/s) and the transvalvular pressure gradient TVP = Δp sys (mmhg) at peak systole, as follows: GA = Q sys 50. (7) Δpsys In the 2D FSI model of Chandra et al. [25], the GOA for a TAV was calculated as an approximation to an idealized equilateral triangular orifice, while the GOA of the BAV was calculated as an approximation to an elliptical orifice. Such approximation was necessitated by the 2D nature of these simulations. In our work, we use the results of our 3D FSI simulations to calculate the GOA of the two valves directly and without any simplifying assumptions by triangulating the real orifice area and computing the GOA as the sum of triangles. In Fig.5, we show samples of the TAV and BAV orifices and the discretization for calculating their respective GOA. In Fig. 6a, time history of the so-computed GOAs for the TAV and BAV is plotted, and it is consequently not equal to zero (GOA 0 TAV = 0.6cm2,GOA 0 BAV = 0.2cm2 ) at start of systole t = 0. The rate of opening of the BAV during the initial phase (t < 0.05 s) is significantly greater than that of the TAV. The former undergoes a steep opening phase starting at t s. The BAV response should be juxtaposed with the more gradual opening of the TAV, the orifice of which is opening far more gradually reaching GOA of about 3 cm 2 at peak systole (0.25 s). The GOA of the BAV, on the other hand, is considerably less (1.7 cm 2 ) at peak systole, which clearly indicates the presence of aortic stenosis for the BAV case. The very rapid initial opening of the BAV leads to significant increase in the mean ejection velocity V m (t) which is defined as the instantaneous average of the velocity over the GOA. One can see this important trend in Fig. 6b, which plots the temporal variation of the mean ejection velocity for each valve type. It is evident that, while the GOA of the BAV does not change during the very early phase of systole (t < t 1 ), the velocity increases rapidly until about 1.8 m/s. After the GOA of the BAV started increasing, the mean velocity decreases to 1 m/s and then again starts rising but at smaller rate than during the initial phase t < t 1. The value of the mean velocity reaches its maximum 2.6m/s, which isconsidered ratherhighfrom theclinicalstandpointas itfurther indicates the occurrence of mild/moderate aortic stenosis [41 43]. The mean velocityof the TAV, on the other hand, accelerates more gradually, and the maximum value does not exceed 1.7m/s. The GA can also be readily estimated from our simulations. The peak systole flow rate in our simulations is Q sys = 25 L/min = 416.7mL/s, and the transvalvular pressure gradients for TAV and BAV are Δp sys TAV = Fig. 5 Triangulations of orifices used to calculate the effective opening area of the valve. NCC is the non-coronary cusp, RCC is the right coronary cusp, and LCC is the left coronary cusp. Δ i are triangles discretizing the valves orifices. Surface of orifice is equal to S orif = i Δ i

8 A. Gilmanov, F. Sotiropoulos Fig. 6 Time history of a effective orifice area and b mean velocity for TAV (solid)andbav(dashed-dot-dot). Dashed-dot lines indicate flow rate at the inlet of the aorta (corresponding axes on the right side) Table 1 Velocities, areas and TVP of TAV and BAV at the peak of systole V m (m/s) V max (m/s) GOA (cm 2 ) EOA (cm 2 ) GA(cm 2 ) Δp (mmhg) TAV BAV mmhg and Δp sys BAV = 44.4 mmhg, respectively. Using these values in Eq. (7), we obtain GA TAV = 2.58 cm 2 and GA BAV = 1.25 cm 2 for the TAV and BAV, respectively. The various AOA estimates for the simulated cases are summarized in Table 1. NotethatV max in Table 1 is the jet maximum velocity at peak systole, which was extracted from the simulated flowfields. Also to calculate the EOA, we used CL= 0.7[40]. Note that based on clinical recommendations [41], the various parameters calculated for the BAV in Table 1 (V max, GA, Δp) indicate moderate aortic stenosis. 4.2 Wall shear stresses characteristics on the TAV and BAV leaflets To quantify the flow-imparted forces on the TAV and BAV leaflets, we employ herein the well-known and clinically relevant quantities of the wall shear stress magnitude (WSS), direction (limiting streamlines or LSL) [45] and the oscillatory shear index (OSI) [44] that quantifies the action of shear (or friction) forces on the endothelial cells. High WSS has been linked to endothelial injury, while low WSS and high OSI values can lead to atherosclerotic lesions [45] and other complications [46]. Yet, these quantities are difficult if not possible to obtain in vivo, and computational studies provide the only viable alternative for comparing these quantities in anatomic geometries and for various valve anatomies and pathologies. One of the first numerical simulations showing the differences in the WSS and LSL between the ventricular and aortic sides of a TAV was done by Ge and Sotiropoulos [46] for straight aorta and with the valve moving with prescribed kinematics (no FSI).

9 Comparative hemodynamics in an aorta Fig. 7 Contours of WSS (in Pa) and limiting streamlines (LSL) (white lines with arrows, indicating direction of a blood flow) on the TAV (left)andbav(right) leaflet surfaces from the ventricular side at that start of systole (t a = s) and peak systole (t b = 0.23 s) Here, we apply the same analysis as that reported by Ge and Sotiropoulos [46] but for a TAV and a BAV placed in an anatomic aorta and with FSI to simulate the coupled interaction of the valve leaflets with the blood flow. Figures 7 and 8 show instantaneous WSS contours and limiting streamlines on the ventricular (Fig. 7) and aortic (Fig. 8) sides of the TAV and BAV leaflets at two instants in time: early systole and peak systole. It is evident from these figures that the valve anatomy has a profound impact on the magnitude and topology of the shear stress field on the leaflets. On the ventricular side of the leaflets (Fig. 7), the LSL for the TAV suggest a linear and well-organized flow along the valve leaflets in agreement with the earlier findings of Ge and Sotiropoulos [46]. A strikingly different picture emerges for the BAV, however, for which the LSL suggest patterns of near leaflet motion growing in complexity as peak systole is approached. For both valves, the WSS increases with the incoming flow rate and regions of high WSS appear at the leaflet tips the LCC and RCC tips for the TAV and the RL-cusp and NCC for the BAV. Similar differences between the two valves are also observed in Fig. 8 for the aortic side of the leaflets. Of interest in this figure is that for the TAV while the complexity of the LSL increases, especially at peak systole, the overall shear stress patterns are more organized than those observed in the simulations of Ge and Sotiropoulos [46]. We note, however, that in Ge and Sotiropoulos [46] the TAV was placed in a straight aorta with a simplified axisymmetric sinus. Therefore, our present result emphasizes the importance of aortic anatomy on the shear stress patterns on the valve leaflets. We also note that the WSS on the aortic side for both valves is approximately three times smaller than on the ventricular side (WSS < 0.5 Pa), which is consistent with the clinical observations of the propensity of atherosclerotic lesions to develop on the aortic side of the leaflets [45]. The striking differences between the TAV and BAV wall shear stress fields are further underscored in Fig. 9, which plots the oscillatory shear index (OSI) on the ventricular (a) and aortic (b) side of the leaflets, respectively. The OSI is defined as follows [47]: τ OSI = 1 WSS dt τ, (8) WSS dt 0

10 A. Gilmanov, F. Sotiropoulos Fig. 8 Contours of WSS (in Pa) and limiting streamlines (LSL) (white lines with arrows, indicating direction of a blood flow) on the TAV (left) andbav(right) leaflet surfaces from the aortic side at that start of systole (t a = s) and peak systole (t b = 0.23 s) where WSS represents the instantaneous wall shear stress vector and τ = 0.23s is the time from the start of systole until peak systole. On the ventricular side, we find that the level of the OSI for the TAV is 10 times lower than for the BAV. On the aortic side, the levels of the OSI for the TAV are less than 0.08 and for the BAV this value is higher than 0.2. In summary, the comparisons of the wall shear stress field magnitude (WSS), direction (LSL) and dynamics (OSI) lead to the conclusion that both sides of the BAV leaflets experience a much more complex flow environment than of the TAV leaflets. This finding could suggest that the BAV leaflets may have a higher propensity than the TAV leaflets for developing atherosclerotic lesions. 4.3 Comparison of TAV and BAV hemodynamics The significant differences in the opening kinematics of the TAV and BAV we uncovered in the previous section also suggest major differences in the hemodynamic patterns induced by each valve. More specifically, the aortic stenosis caused by the BAV and the associated high-velocity jet has the potential to impinge on the aortic wall and increase the wall shear stress in the ascending aorta. Increased shear stresses and associated turbulence in the blood flow have been postulated to play an important role BAV-related diseases and cause dilatation and dissection [7]. Our simulations provide clear evidence that the BAV has a profound impact in the ascending aorta hemodynamics. The calculated flowfields for one simulated systolic cardiac cycle are shown in Fig. 10 for both the TAV and BAV. Contours of instantaneous velocity magnitude are plotted in this figure on a plane passing through the center of the aorta. In Fig. 10, one can see the major differences between the TAV and BAV hemodynamics and the interaction of the jets they give rise to with the ascending aorta walls. The BAV jet is very strong and strikes the aorta wall with much higher velocity than for the TAV case. The TAV jet is more organized and appears laminar during the early stages of its formation in contrast to the BAV jet, which appears to transition to turbulence early on in the diastolic phase (Fig. 10a, right column). Furthermore, the BAV is seen to engage in a very strong action with the ascending aorta wall starting from time t c s (Fig. 10b, right column).

11 Comparative hemodynamics in an aorta Fig. 9 Contours of the OSI and limiting streamline patterns (white lines with arrows, indicating direction of a blood flow) on the leaflet surfaces from the ventricular (a) and aortic (b) sides at peak of systole The differences in the AOA and rate of opening of the two valves should be expected to lead to major differences in the near-valve shear-layer roll up and vortex formation processes. This is shown in Fig. 11, which depicts iso-surfaces of the Q-criterion [48] for the TAV and BAV at the same instants in the cardiac cycle. The TAV exhibits the initial three-lobed vortex ring structure, which is seen to deform in an non-planar manner with the lobes near the commissures bending forward as systole advances. The vortex ring structure of the BAV, on the other hand, is more complex characterized with small-scale structures from the early systole phase. This should be associated with the higher overall velocities of the BAV jet and the more intense shear layer that develops for this case. As a result, the BAV jet is seen to break up into small-scale turbulence at approximately halfway through the accelerating phase of systole and at a time instant when the TAV vortex ring is still laminar and organized (Fig. 12b, c). Therefore, our results clearly suggest that the BAV jet transitions much earlier to turbulence than the TAV jet. Once again, the higher velocities of the BAV jet and the much larger acceleration of its leaflets during opening are very much consistent with the so-observed faster transition to turbulence. To enable qualitative comparisons of our computed results with recent MRI data [9,49], we plot in Fig. 13 instantaneous streamlines at various instants in time for the TAV (left column) and BAV (right column). The streamlines in this figure are colored with velocity magnitude. Note that while instantaneous streamlines in an unsteady flow do not have a physical meaning (Lagrangian analysis and pathlines will be required for flow visualization that can be realized in the actual flow), they are used extensively in visualizing 4D MRI data [9,13,17,49,50] and can provide a general view of the 3D structure of the instantaneous flow. The results in Fig. 13 show that the formation and advance of the BAV jet take place much faster than the TAV jet. Initially (t < s), both the BAV and TAV jets are oriented in the vertical direction, but the velocity magnitude for the BAV is higher than for the TAV jet. The roll up of the shear layers emanating from the valve leaflets and the associated large-scale vortex formation is clearly evident in Fig. 13, especially for the BAV for which the roll up of streamlines is more intense and clearly visible. At the subsequent time t b, the BAV jet is seen to impact the aorta wall transporting the high-velocity valve jet on the aorta wall and increasing the wall shear stress. This feature is not observed in the TAV flow. The BAV jet is clearly seen to change its direction, which

12 A. Gilmanov, F. Sotiropoulos Fig. 10 Comparison of instantaneous contours of velocity magnitude on a plane through the aorta during systolic phase showing the opening process of the TAV (left)andbav(right). The red dot in the inset of each figure identifies the corresponding instant during the cardiac cycle: t a = (s), t b = (s), t c = (s) (color figure online) is shown by the black arrow in Fig. 13. Such changing of the jet direction arises from the eccentricity of the BAV (ε = 1.25) and the highly asymmetric opening of the BAV leaflets, because the NC-cusp has much bigger displacements in comparison with the RL-cusp (Fig. 7, right column). On the contrary at the same instant in the cycle, the TAV vortex ring has just started becoming evident by the twist of the instantaneous streamlines. Furthermore, the direction of the TAV jet has remained approximately the same as during the earlier instant in time (Fig. 13b). Figures 14 and 15 illustrate the flow patterns in the aorta at later times (Fig. 14a, b t a = 0.12 s, t b = 0.16 s; Fig. 15a, b t a = 0.19 s, t b = 0.21 s). The direction of the TAV jet remains unchanged until the peak of systole is reached, and the largest velocity magnitudes are observed inside the jet and not on the aorta wall. This flow pattern is in direct contrast to the BAV flowfield, which has already collided with the aortic wall, increased dramatically the wall shear stresses at the point of impingement and given rise to a right-handed helical flow

13 Comparative hemodynamics in an aorta Fig. 11 Comparison of instantaneous iso-surfaces of the Q-criterion [48] at various instants in time during the acceleration phase of systole: t a = s, t b = s, t c = s. The red dot in the inset of each figure identifies the corresponding instant during the cardiac cycle (color figure online) within the aorta. Figures 13, 14 and 15 further show the formation of the large-scale vortex within the aortic sinuses, which has also been revealed in recent 4D MRI experiments [49]. The flow patterns we described above are qualitatively consistent with the 4D MRI flow in vivo measurements of Barker at al.[9]. More specifically, Barker et al. [9] reportedthat for the TAV case the direction ofthe aortic valve jet remains nearly constant from the start of systole as the flow advances in the ascending aorta and does not impact the aorta wall. This feature of the flow is also evident in our simulations of the TAV case shown in Figs. 13, 14,and15. For the BAV case, however, the in vivo results of Barker et al. [9] clearly show that the valve jet changes its direction during early systole propagating in a direction opposite to the RL-cusp toward the right anterior region of the ascending aorta wall and ultimately impinging on the wall. This important feature of the in vivo flow is in excellent qualitative agreement with our simulation results and elucidates the mechanism via which the BAV increases the shear stress on the ascending aorta wall. Another feature of the flow that is clearly evident in the instantaneous streamlines constructed from the MRI measurements of Barker et al. [9] is the development of a strong right-handed helical flow patterns in the ascending aorta, which is also readily apparent in our simulations (see Figs. 13, 14, 15).

14 A. Gilmanov, F. Sotiropoulos Fig. 12 Comparison of instantaneous iso-surfaces of the Q-criterion [48] at various instants in time during the acceleration phase of systole: t a = s, t b = s, t c = s. The red dot in the inset of each figure identifies the corresponding instant during the cardiac cycle (color figure online) 5Conclusion We have reported the first high-resolution FSI simulation of physiologic blood flow in an anatomic aorta with TAV and BAV in order to conduct a systematic investigation of the leaflet kinematics and hemodynamics of the two types of valves. We showed that there are profound differences in both regards between the two valves. The BAV in particular was shown to create mild aortic stenosis by reducing the aortic valve effective orifice area and increasing considerably the velocity magnitude through the valve as compared to the TAV. The distinguishing feature of the BAV compared with the TAV is that the leaflets of the former are exposed, on both sides, to a spatially more complex and temporally more dynamic shear stress environment than the latter. These differences were quantified in terms of the magnitude (WSS), topology (LSL) and dynamics (OSI) of the shear stress field and suggest that BAV leaflets may be at higher risk for developing atherosclerotic lesions than TAV leaflets. The differences in valve kinematics, shape and eccentricity were shown to give

15 Comparative hemodynamics in an aorta Fig. 13 Computed instantaneous streamlines colored with velocity magnitude contours for the TAV (left) andbav(right) at times: t a = s and t b = s. The black arrow indicates the direction of the aortic valve jet flow for each case rise to significantly different hemodynamics both near the valve and in the ascending aorta. For the BAV case, the aortic valve vortex ring is seen to grow in complexity rapidly and ultimately break into turbulence much sooner during the accelerating phase of systole than for the TAV case. Another striking difference between the two valves is the fact that, while for the TAV case the valve jet preserves its direction and does not impinge on the aortic wall, the BAV jet undergoes a significant change in orientation causing it to impinge on the ascending aorta wall. This impingement is associated with the eccentricity of the BAV and provides the mechanism via which the BAV increases the shear stress on the wall of the ascending aorta. Our computational findings are in a good qualitative agreement with 4D MRI in vivo measurements. Essentially all the large-scale flow features in the ascending aorta observed in vivo are also found in our simulated flowfields. These include: the change in direction of the aortic jet for a BAV; the impingement of the BAV jet on the aortic wall; and the growth of a right-handed helical flow in the ascending aorta. Both our simulations and the in vivo data suggest, therefore, that a BAV has a profound effect on the large-scale hemodynamics in the ascending aorta when compared to a healthy TAV. Of particular interest in this regard is the significant increase in the WSS on the aorta wall caused by the BAV jet impingement on the wall of the ascending aorta. This finding points to the conclusion that hemodynamic factors cannot be excluded as a

16 A. Gilmanov, F. Sotiropoulos Fig. 14 Computed instantaneous streamlines colored with velocity magnitude contours for the TAV (left) andbav(right) at times: t a = 0.12 s and t b = 0.16 s. The black arrow indicates the direction of the aortic valve jet flow for each case possible cause for BAV-related aortopathies. The extent and precise nature of this connection, however, need to be further investigated via comprehensive studies integrating numerical simulations, in vivo measurements and clinical studies. A limitation of our computational model stems from the constitutive model we have employed for the valve leaflets, which does not correspond to biological tissue but is rather representative of polymeric materials. This limitation notwithstanding, however, our results have clearly illustrated the major impact of a BAV and a TAV in the ascending aorta hemodynamics. The fact that the large-scale flow features we have simulated herein are in good agreement with in vivo measurements further suggests that parameters such as valve shape and eccentricity, which are accounted for in the present study, may be dominant. Nevertheless, future studies seeking to investigate the wall shear stress fields on the valve leaflets should consider FSI models with constitutive relations appropriate for simulating the highly nonlinear and hyperelastic response of biological tissues. We are currently in the process of extending our computational methodology to incorporate such models, and we will be reporting results seeking to elucidate the effects of tissue properties on hemodynamics in future publications. In our simulations, we have not solved so far the contact problem of leaflets interaction, and consequently, we cannot consider closing

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