Numerical Investigation of the E ﬀ ects of Prosthetic Aortic Valve Design on Aortic Hemodynamic Characteristics

: The superior performance of single-point attached commissures (SPAC) molded valve design has been validated by several numerical, in vitro and in vivo animal studies. However, the impacts of the SPAC molded valve design on aortic hemodynamic environments are yet to be investigated. In this study, multiscale computational models were prepared by virtually implanting prosthetic aortic valves with SPAC tubular, SPAC molded and conventional designs into a patient-speciﬁc aorta, respectively. The impacts of the valve designs on e ﬀ erent ﬂow distribution, ﬂow pattern and hemodynamic characteristics in the aorta were numerically investigated. The results showed that despite the overall ﬂow phenomena being similar, the SPAC tubular valve exhibited a suboptimal performance in terms of higher spatially averaged wall shear stress (SAWSS) in ascending aorta (AAo), higher helix grade, stronger secondary ﬂow mean secondary velocity in descending aorta, as well as more complex vortex distribution. The results from the current study extend the understanding of hemodynamic impacts of the valve designs, which would further beneﬁt the optimization of the prosthetic aortic valve.


Introduction
Aortic valve diseases are estimated to affect more than 12 million people globally, and surgical aortic valve replacement (SAVR) is one of the primary management strategies among patients with severe conditions [1][2][3]. More than 200,000 SAVRs are performed annually worldwide [4].
The stentless bioprosthetic aortic valves have been widely used in SAVR, which are not only advanced in terms of being free of anticoagulation after implantation, but also exhibit superior hemodynamics performances comparable to the native aortic valve [5,6]. However, the implantation of stentless valves is technically more difficult [7]. Therefore, a stentless aortic valve prosthesis with an easy implantation technique is of great interest.
The single-point attached commissure (SPAC) implantation technique, which sutures the commissure to the aortic wall at the level of the sinotubular junction (STJ) at only three single points  Figure 1b illustrated the implant approaches as well as leaflet designs of the SPAC tubular prosthesis, SPAC molded prosthesis and conventional prosthesis. Among these, the leaflets of the SPAC tubular valve followed the geometry of a cylinder at the fully open configuration; the leaflets of conventional valve leaflet followed the design of Thubrikar et al. [27]; The SPAC molded valve shared similar geometric characteristics with the conventional valve, whereas the gaps between leaflets were merged for the feasibility of the SPAC technique.

Valve Geometry
The structural deformations of all the three valve designs were analyzed under time-varying physiological pressure loading over a full cardiac cycle by using non-linear finite element code ABAQUS (ABAQUS, Inc., Pawtucket, RI, USA) [12]. Compared with the SPAC molded valve and the conventional valve, the SPAC tubular valve exhibited a faster opening in the early acceleration stage of systole as well as a smaller maximum effective orifice area (EOA) at the acceleration peak. These morphological differences of valves within a cardiac cycle could result in altered local and downstream hemodynamic characteristics. Thus, the deformed leaflet configurations of the three valve designs at early acceleration (t 1 : 0.067 s and t 2 : 0.068 s), mid acceleration (t 3 : 0.085 s), acceleration peak (t 4 : 0.090 s) and mid-deceleration (t 5 : 0.218 s) were derived from finite element method (FEM) analyses ( Figure 1c) and virtually implanted into the aortic root under the supervision of an experienced cardiac surgeon (Figure 1d), respectively. For the conventional valve, the scalloped leaflet attachment line was sutured to the aortic root. For both of the SPAC valves, only three single points were sutured to the aorta at the STJ level as well as suturing the valve base to the aorta annular level. After the virtual implantation, the aortic roots were connected to the aorta model by using SolidWorks software (Dassault Systems S.A., Paris, France). Finally, 15 models were prepared for the numerical simulation.

Numerical Methods
The open-source CFD package SimVascular (http://simvascular.github.io/) was used as the pre-processor and solver in this study. Blood was assumed as an incompressible Newtonian fluid. The density and dynamic viscosity of the blood were set to 1060 kg/m 3 and 0.004 Pa·s, respectively. The arterial wall was assumed as the non-slip rigid wall. The 3D flow domain was governed by steady-state Navier-Stokes (N-S) equations (Equations (1) and (2)).
The volumetric flow rates derived from the time-varying physiological profile ( Figure 2) at the five time points were specified on the proximal end of the aortic root, which was mapped onto the inlet plane by using a plug velocity profile [28]. The specific flow rates at t 1 to t 5 were 32 mL/s, 36 mL/s, 204 mL/s, 350 mL/s and 154 mL/s, respectively.

Numerical Methods
The open-source CFD package SimVascular (http://simvascular.github.io/) was used as the preprocessor and solver in this study. Blood was assumed as an incompressible Newtonian fluid. The density and dynamic viscosity of the blood were set to 1060 kg/m 3 and 0.004 Pa·s, respectively. The arterial wall was assumed as the non-slip rigid wall. The 3D flow domain was governed by steadystate Navier-Stokes (N-S) equations (Equations (1) and (2)).
The volumetric flow rates derived from the time-varying physiological profile ( Figure 2) at the five time points were specified on the proximal end of the aortic root, which was mapped onto the inlet plane by using a plug velocity profile [28]. The specific flow rates at t1 to t5 were 32 mL/s, 36 mL/s, 204 mL/s, 350 mL/s and 154 mL/s, respectively. At the outlets, the resistance of distal vascular systems was considered by coupling 0D lumped parameter models (LPMs) with the 3D aorta model at the efferent arteries. As illustrated in Figure 3, the R1, R2, R3, R4 represent the resistance of vascular systems distal to the BA, LCCA, LSA and descending aorta (DAo), respectively. The specific values of resistances were calculated according to Murry's law [28]. The compliance components of the 0D models were neglected due to the simulations being conducted under steady flow.
In the coupling scheme, the 0D LPMs with resistance elements were coupled with the 3D model at the outlet boundaries, including the outlets of descending aorta (DAo), BA, LCCA and LSA. A "tight coupling" algorithm [29][30][31] was used to transfer the data at the coupling interfaces. At the beginning of the coupling, an initial pressure is applied as the initial condition. Afterward, the areaaveraged flow rate at the coupling interface of the 3D CFD model is computed and sent to the LPM to calculate the corresponding cross-sectional static pressure for the next iteration until the convergence is reached for the current time step. At the outlets, the resistance of distal vascular systems was considered by coupling 0D lumped parameter models (LPMs) with the 3D aorta model at the efferent arteries. As illustrated in Figure 3, the R 1 , R 2 , R 3 , R 4 represent the resistance of vascular systems distal to the BA, LCCA, LSA and descending aorta (DAo), respectively. The specific values of resistances were calculated according to Murry's law [28]. The compliance components of the 0D models were neglected due to the simulations being conducted under steady flow.
In the coupling scheme, the 0D LPMs with resistance elements were coupled with the 3D model at the outlet boundaries, including the outlets of descending aorta (DAo), BA, LCCA and LSA. A "tight coupling" algorithm [29][30][31] was used to transfer the data at the coupling interfaces. At the beginning of the coupling, an initial pressure is applied as the initial condition. Afterward, the area-averaged flow rate at the coupling interface of the 3D CFD model is computed and sent to the LPM to calculate the corresponding cross-sectional static pressure for the next iteration until the convergence is reached for the current time step. All the models were discretized by using unstructured 4-node tetrahedral elements with three layers of prism boundary layer. Mesh independence analyses were conducted on the conventional valve model at t4 with eight different element numbers (Table 1). The comparison showed that the differences in flow rates at efferent ends as well as spatially averaged WSS (SAWSS) on ascending aorta were smaller than 2% between cases 7 and 8. Thus, the element number of 34.5 million was adopted in the discretization of all the models. The y + < 2 was achieved by adjusting the initial thickness as well as the growth rate of the boundary layer at this mesh setting, which is capable of capturing the near-wall flow features.

Data Analysis
The simulation results were post-processed in ParaView software (Kitware, Inc., Clifton Park, NY, USA). The flow fields of the studied cases were visualized by using streamline, and the volumetric efferent flow rates at BA, LCCA, LSA and DAo were calculated. In addition, the following critical hemodynamic characteristics associated with aortic lesions were compared. All the models were discretized by using unstructured 4-node tetrahedral elements with three layers of prism boundary layer. Mesh independence analyses were conducted on the conventional valve model at t 4 with eight different element numbers (Table 1). The comparison showed that the differences in flow rates at efferent ends as well as spatially averaged WSS (SAWSS) on ascending aorta were smaller than 2% between cases 7 and 8. Thus, the element number of 34.5 million was adopted in the discretization of all the models. The y + < 2 was achieved by adjusting the initial thickness as well as the growth rate of the boundary layer at this mesh setting, which is capable of capturing the near-wall flow features.

Data Analysis
The simulation results were post-processed in ParaView software (Kitware, Inc., Clifton Park, NY, USA). The flow fields of the studied cases were visualized by using streamline, and the volumetric efferent flow rates at BA, LCCA, LSA and DAo were calculated. In addition, the following critical hemodynamic characteristics associated with aortic lesions were compared.
WSS: the WSS, a frictional force on the arterial wall exerted by the blood flow, was calculated from Equation (3): where τ w is the WSS, µ is the dynamic viscosity, u is the velocity and y is the distance to the wall. Helix: the helix quantitatively describes the rotation degree of helical flow along the longitudinal axis of an arterial segment in a grading scale of 1 to 3, which represents rotation degree less than 180 • , greater than 180 • and greater than 360 • , respectively [32].
Mean secondary velocity: the secondary flow strength in the ascending aorta, aortic arch and descending aorta (slice 1 to 5 in Figure 3) were quantitatively analyzed by using mean secondary velocity, which is defined as the ratio of the cross-sectional mean of the secondary velocities to the cross-sectional mean of the axial velocities at each slice [33].
Q-criterion: to compare the strength of vortex quantitatively, the Q-criterion was introduced [34]. It defines the vortex as the second invariant of the velocity gradient tensor in the flow field with a positive value, which is expressed as Equation (4): where ||Ω|| 2 represents the vorticity magnitude, and ||S|| 2 denotes the strain-rate magnitude. Figure 4 shows the efferent flow rates of BA, LCCA, LSA and DAo at each time point. The maximum percentage difference in efferent flow rates among the three designs is less than 2%.

The Impacts of Valve Designs on the Efferent Flow
Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 22 WSS: the WSS, a frictional force on the arterial wall exerted by the blood flow, was calculated from Equation (3): where is the WSS, is the dynamic viscosity, is the velocity and is the distance to the wall. Helix: the helix quantitatively describes the rotation degree of helical flow along the longitudinal axis of an arterial segment in a grading scale of 1 to 3, which represents rotation degree less than 180°, greater than 180° and greater than 360°, respectively [32].
Mean secondary velocity: the secondary flow strength in the ascending aorta, aortic arch and descending aorta (slice 1 to 5 in Figure 3) were quantitatively analyzed by using mean secondary velocity, which is defined as the ratio of the cross-sectional mean of the secondary velocities to the cross-sectional mean of the axial velocities at each slice [33].
Q-criterion: to compare the strength of vortex quantitatively, the Q-criterion was introduced [34]. It defines the vortex as the second invariant of the velocity gradient tensor in the flow field with a positive value, which is expressed as Equation (4): where ‖Ω‖ represents the vorticity magnitude, and ‖S‖ denotes the strain-rate magnitude.

The Impacts of Valve Designs on the Flow Pattern
In the aorta, identical flow patterns were observed among the three models ( Figure 5). During the early acceleration phase (t 1 to t 2 ), fairly organized flows dominated the entire aorta in all cases. With the increasing of flow velocity, obvious right-handed helical flows started to appear in the immediate vicinity of the valves at t 3 . The helix at the acceleration peak (t 4 ) in the ascending aorta was 1.2, 1.4 and 1.2 for SPAC molded valve, SPAC tubular valve and conventional valve, respectively.

The Impacts of Valve Designs on the Flow Pattern
In the aorta, identical flow patterns were observed among the three models ( Figure 5). During the early acceleration phase (t1 to t2), fairly organized flows dominated the entire aorta in all cases. With the increasing of flow velocity, obvious right-handed helical flows started to appear in the immediate vicinity of the valves at t3. The helix at the acceleration peak (t4) in the ascending aorta was 1.2, 1.4 and 1.2 for SPAC molded valve, SPAC tubular valve and conventional valve, respectively.    The averaged jet velocity (Vmean) and maximum jet velocity (Vmax) in the plane that was 10 mm downstream of the commissure level (slice 1) of all cases are listed in Table 2. Because of the faster opening, the SPAC tubular valve exhibited a lower Vmean as well as Vmax than that of the other two designs at the early acceleration phase. On the contrary, the SPAC molded and conventional design showed lower Vmean and Vmax at acceleration peak (t4) compare with the SPAC tubular design, which is due to the larger maximum EOAs. The averaged jet velocity (V mean ) and maximum jet velocity (V max ) in the plane that was 10 mm downstream of the commissure level (slice 1) of all cases are listed in Table 2. Because of the faster opening, the SPAC tubular valve exhibited a lower V mean as well as V max than that of the other two designs at the early acceleration phase. On the contrary, the SPAC molded and conventional design showed lower V mean and V max at acceleration peak (t 4 ) compare with the SPAC tubular design, which is due to the larger maximum EOAs.
The velocity distributions in slice 1 at t 1 to t 5 were illustrated in Figure 7. None of the three cases showed an eccentric flow pattern.

Wall Shear Stress (WSS)
The WSS distributions on the entire aorta of each case are illustrated in Figure 8. All the cases showed similar WSS distribution patterns at corresponding time points.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 22 The specific values of SAWSS and low WSS areas on the entire aorta are listed in Table 3. Increased SAWSS as well as decreased low WSS area were observed in all three cases during the acceleration phase. The low WSS areas (<0.4 Pa) [35] on the aorta with SPAC tubular and SPAC molded designs was 64.29% and 16.39% smaller than that with the conventional at acceleration peak (t4), respectively.

SAWSS (Pa)
Low WSS Areas (cm 2 ) The specific values of SAWSS and low WSS areas on the entire aorta are listed in Table 3. Increased SAWSS as well as decreased low WSS area were observed in all three cases during the acceleration phase. The low WSS areas (<0.4 Pa) [35] on the aorta with SPAC tubular and SPAC molded designs was 64.29% and 16.39% smaller than that with the conventional at acceleration peak (t 4 ), respectively.  Table 4 further lists the SAWSS magnitudes of all the cases on different segments of the aorta at the acceleration peak (t 4 ). On the AAo segment, the SAWSS of SPAC tubular valve case is 20.24% and 29.63% higher than that of the SPAC molded valve and conventional valve, respectively. In contrast, the percentage difference of SAWSS on the aortic arch and DAo are within 11% among the cases. The highest SAWSS was observed in the aortic arch segment in all the cases  Figure 9 showed the velocity vector distributions of secondary flow in the ascending aorta (slice 1), aortic arch (slice 2) and descending aorta (slice 4) of all three cases at the acceleration peak (t 4 ). The visualized results showed that different valve designs mainly affect the secondary flow distribution patterns in the ascending aorta (slice 1).  Table 4 further lists the SAWSS magnitudes of all the cases on different segments of the aorta at the acceleration peak (t4). On the AAo segment, the SAWSS of SPAC tubular valve case is 20.24% and 29.63% higher than that of the SPAC molded valve and conventional valve, respectively. In contrast, the percentage difference of SAWSS on the aortic arch and DAo are within 11% among the cases. The highest SAWSS was observed in the aortic arch segment in all the cases  Figure 9 showed the velocity vector distributions of secondary flow in the ascending aorta (slice 1), aortic arch (slice 2) and descending aorta (slice 4) of all three cases at the acceleration peak (t4). The visualized results showed that different valve designs mainly affect the secondary flow distribution patterns in the ascending aorta (slice 1).  Table 5. Obvious quantitative differences of both metrics in aorta were observed between the SPAC tubular valve case and the rest two valve designs. The maximum and averaged secondary flow velocity in slices 1, 3 and 5 of the SPAC tubular valve, SPAC molded valve, and conventional valve are listed in Table 5. Obvious quantitative differences of both metrics in aorta were observed between the SPAC tubular valve case and the rest two valve designs. The relative differences of the metrics to the conventional case are listed in Table 6.  Figure 10 illustrates the strength of secondary flow in the aorta at the acceleration peak (t 4 ). The secondary flow strength increased along the ascending aorta and the strongest secondary flow occurred near the aortic arch in all three cases. The relative differences of the metrics to the conventional case are listed in Table 6.   The specific values of the mean secondary velocity of all cases as well as the relative differences to the conventional case at t4 are listed in Table 7. Although the mean secondary velocities in the aortic arch of both SPAC cases are around 10% higher than the conventional valve case, its magnitude in the AAo and DAo of the SPAC molded valve case is identical to that of the case of the conventional valve. The specific values of the mean secondary velocity of all cases as well as the relative differences to the conventional case at t 4 are listed in Table 7. Although the mean secondary velocities in the aortic arch of both SPAC cases are around 10% higher than the conventional valve case, its magnitude in the AAo and DAo of the SPAC molded valve case is identical to that of the case of the conventional valve.  Figure 11 shows the vortex development processes in the ascending aorta by plotting the 3D isosurface of the Q-criterion. During the early systolic phase (t 2 ), a ring-like vortex structure presented in the vicinity of the free edges and separated into the three small vortex regions near the STJ level of all three valves. Among these, the SPAC tubular valve showed a lower vortex strength. At the peak of systole acceleration (t 4 ), the ring-like vortex structure broke up immediately downstream of the free edges and dominated the ascending aorta. Although no obvious distinctions were observed in the aortic arch and DAo among the cases, vortices with non-identifiable patterns appeared in the SPAC tubular valve.

Q-Criterion
Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 22  Figure 11 shows the vortex development processes in the ascending aorta by plotting the 3D isosurface of the Q-criterion. During the early systolic phase (t2), a ring-like vortex structure presented in the vicinity of the free edges and separated into the three small vortex regions near the STJ level of all three valves. Among these, the SPAC tubular valve showed a lower vortex strength. At the peak of systole acceleration (t4), the ring-like vortex structure broke up immediately downstream of the free edges and dominated the ascending aorta. Although no obvious distinctions were observed in the aortic arch and DAo among the cases, vortices with non-identifiable patterns appeared in the SPAC tubular valve.

Discussion
The introduction of the SPAC tubular valve into clinical practice has greatly eased technical requirements and simplified implantation procedures. On this basis, the SPAC molded valve resembling the native leaflet geometry was proposed by Goetz et al. [36]. Benefiting from a longer free edge and a larger EOA, the SPAC molded valve design has shown comparable structural dynamic behavior with the conventional approach, which is superior to its tubular counterpart [12]. Promising results from in vitro and in vivo animal trails further proved the feasibility and reliability of SPAC molded valve under physiological conditions [13,36]. In this study, the effects of the SPAC tubular valve, SPAC molded valve and conventional valve on the aortic flow were investigated by using 0D/3D coupled numerical models.

Discussion
The introduction of the SPAC tubular valve into clinical practice has greatly eased technical requirements and simplified implantation procedures. On this basis, the SPAC molded valve resembling the native leaflet geometry was proposed by Goetz et al. [36]. Benefiting from a longer free edge and a larger EOA, the SPAC molded valve design has shown comparable structural dynamic behavior with the conventional approach, which is superior to its tubular counterpart [12]. Promising results from in vitro and in vivo animal trails further proved the feasibility and reliability of SPAC molded valve under physiological conditions [13,36]. In this study, the effects of the SPAC tubular valve, SPAC molded valve and conventional valve on the aortic flow were investigated by using 0D/3D coupled numerical models.

Modeling Simplifications
Constrained by the computational resource, capability of the numerical algorithm as well as the availability of data, several simplifications have been made in the current study.
First of all, only steady-state CFD analyses were performed in this study. In the numerical investigations of prosthetic aortic valves, the use of FEM, CFD and fluid-structure interaction (FSI) techniques have been reported. Among these, FSI simulations are advanced in providing the valvular hemodynamics characteristics and strain/stress on leaflets simultaneously, and have started to merge in recent years . The most popular FSI algorithms for valvular simulations are arbitrary Lagrangian-Eulerian (ALE) [59], immersed boundary (IB) [60] and smoothed particle hydrodynamics (SPH) [61]. However, the extremely computation-intensive nature of the FSI method has constrained its application in dealing with large-scale models, which would take weeks to months to solve [62][63][64]. Compared with FSI simulation, the steady-state CFD simulation is advanced in providing a fast prediction of non-temporal flow characteristics with ease and has been widely used in the studies of valvular and aortic hemodynamics [65][66][67][68][69][70][71][72]. Despite the fact that the blood flow in the vascular system was pulsatile under physiological conditions, several numerical [73,74] and in vitro [75][76][77] studies have suggested that steady-state simulation can predict the non-temporal related flow characteristics at the corresponding point of the pulsatile flow profile. In addition, it has been reported that the WSS distribution patterns derived from the steady flow simulations are similar to the time-averaged WSS (TAWSS) from the unsteady analyses [67,78,79].
Secondly, the volumetric flow rates were imposed at the inlet and mapped onto the inlet plane by using an idealized parabolic velocity profile [28]. Morbiducci et al. concluded that the inlet boundary condition profile could drastically affect the helical flow pattern in the aorta and suggested imposing an in vivo velocity profile at the inlet to accurately predict the individual flow phenomena in patient-specific analyses [80]. This observation was further supported by the study of Youssefi et al. [81]. Nevertheless, the idealized inlet profiles, such as parabolic and plug, are still commonly used in the related studies [82][83][84] due to the limited availability of in vivo data. As the scope of this preliminary study is not to predict the patient-specific hemodynamics metrics, but to compare the impacts of valve structures on aortic flow, the use of the same idealized inlet flow profile among the cases at each time point could be a reasonable baseline.
Moreover, the laminar flow condition was assumed in this study due to the constriction of the solver. Although the laminar assumption has been widely used in the flow simulations in the aortic valve and aorta [42,52,80,85,86], its inaccurate estimation of flow parameters within the transitional and turbulent flow regime region should be noted in the individualized evaluations [87].

Helical Flow
Previous studies have demonstrated that the complex morphology and flow condition in the aorta lead to right-handed helical flow in AAo as well as left-handed helical flow in DAo among healthy subjects [14,16,32,88,89]. Morbiducci et al. further investigated the development of helical flow in the aorta during the systolic phase by using in vivo four-dimensional phase contrast magnetic resonance imaging (4D PC-MRI), and concluded that the helical flow pattern in the aorta could optimize the blood flow in the aorta by avoiding excessive energy dissipation and reducing flow instabilities [14,15]. In addition, the helical flow also affects the aortic mass transportation. Several studies on the low-density lipoprotein (LDL) transportation in the aorta reported that the helical flow environment would reduce the luminal surface LDL concentration in the aortic arch [90,91]. Meanwhile, enhanced oxygen transportation as well as suppressed activation of blood components have been associated with the helical flow in arteries [18,92,93]. Such phenomena indicate that the physiological helical flow plays an important role in protecting the aorta form atherogenesis and thrombosis [16]. However, abnormally high grades of helical flow could elevate the potential risk of aortic wall remodeling and lead to aortic dilation [94]. Clinical evidence showed that the aortic insufficient and eccentric jet caused by a non-physiological aortic valve structure are the major causes of high helix grade [19,88,89]. The result from this study showed that in all of the three cases, right-handed helical flows appeared in the ascending and aortic arch at the very early stage of the acceleration phase and dominated the entire aorta at the systolic peak. These flow patterns agree well with the previous clinical observations of aorta flow in healthy subjects. Even though the helix grades of all cases are within the normal range at the acceleration peak (1.1 ± 0.1 to 2.03 ± 0.67) [32,88], the SPAC tubular valve presents a 16.7% higher helix than that of the rest of the cases due to its smaller EOA. In addition to the higher helix grade, the smaller EOA of the SPAC molded valve also caused higher V mean and V max in the ascending aorta at the systolic peak. Also noted is that highly disturbed inter-sinus flow existed in the both SPAC cases, which is not observed in the conventional valve. This phenomenon is mainly due to the non-isolated sinuses of the SPAC approaches, and the distal flow in the aorta was not affected by this localized disturbance.

Secondary Flow
Besides the helical flow, the secondary flow has been recognized as another essential aortic flow phenomenon caused by the curvature structure of aorta [95]. However, some recent studies have shown that the SAVR could affect the secondary flow strength due to the differed geometric characteristics of prosthetic valves [96,97]. In this study, we observed obvious secondary flow in the upper ascending aorta and aortic arch during the systolic phase in all the three cases investigated. Compared with the SPAC tubular and conventional valve, a lower mean secondary velocity in AAo as well as a higher mean secondary velocity in DAo was found in the case of the SPAC tubular valve. These phenomena are likely related to the higher jet velocity downstream of the SPAC tubular valve [97].

WSS Distribution
Regarding the WSS, the higher jet velocity induced by constrained EOA that would result in an increased WSS in the ascending aorta have been reported by studies that investigated the effect of valve geometry on aortic hemodynamics [23,97,98]. The results from the current study agree well with the previous observations. For the SPAC tubular valve case, the SAWSS in the AAo segment was 20.34% and 29.64% higher than that of the SPAC molded valve and conventional valve, respectively. Due to the existence of helical flow and secondary flow, the SAWSS magnitudes among the three cases gradually became consistent along the flow direction. In the distal aortic arch, the differences of SAWSS among three cases were less than 2%.

Vortex Distribution
The presence of the vortex ring in the opening phase of the aortic valve, which is caused by the shear layer shed of the fluid at the tip of leaflets, has been reported by several in vitro and numerical studies as well [99][100][101]. In the present study, the formation and dissipation of vortex rings were observed in all three valve designs during the opening stage. The vortex rings of all the valves were bending towards the flow direction with lobes near the commissures at the early stage of acceleration and separated into small-scale vortices at the acceleration peak, which agrees well with the results from FSI studies on prosthetic aortic valves [37,102]. The vortex distribution in the SPAC molded valve case is similar to that in the case with a conventional valve, and a more complex vortex structure within the valve section was observed in the SPAC tubular valve at the acceleration peak. This distinctive vortex pattern could be attributed to the smaller EOA of the SPAC tubular valve [12].

Limitations and Future Works
The current study has some inherent limitations. Firstly, despite the fact that the prediction of an accurate WSS level is not within the scope of this study, it should be noted that the rigid wall as well as laminar flow assumption may lead to an inaccurate estimation of WSS [103,104]. In addition, the steady-state simulation precluded the evaluation of time-dependent parameters such as time-averaged WSS (TAWSS), relative resident time (RRT) as well as oscillatory shear index (OSI), and is incapable of reflecting the vortex dynamics within the entire cardiac cycle. Moreover, the use of idealized flow profiles may not be able to reflect the helical flow patterns in physiological conditions. Thus, more comprehensive studies that include the FSI simulations with in vivo boundary conditions should be conducted in the next stage of the investigation.

Conclusions
The hemodynamic impacts of three prosthetic aortic valve designs were numerically investigated in this study. The preliminary results showed that the hemodynamic performance of the SPAC molded valve is similar to that of the conventional valve in terms of aortic flow characteristics. The findings from this study can expand the understanding of the influence of prosthetic aortic valve design on aortic hemodynamic characteristics and benefit the optimization of prosthetic valve design.