Finite volume method is used in the simulation. As shown in Figure 2, the computational meshes of the models created using the CAD software Gambit 2.2 (ANSYS, Inc., Canonsburg, PA) were all unstructured tetrahedron grids. The commercially available computational fluid dynamics (CFD) code, FLUENT 6.2 (ANSYS, Inc., Canonsburg, PA) was used in the numerical simulation. Discretization of the pressure and momentum at each control volume was in a second-order scheme, and temporal discretization was performed using the second-order implicit time integration scheme. The iterative process of computation was terminated when the residual of continuity and velocity were all less than the convergence criterion 1.0 × 10−5.
We used the size function in Gambit and the grid adaptation technique in FLUENT to refine the meshes in the computational domain, especially in areas where the velocity gradient was steep, until the computational results reached grid independence. In this study, the average WSS SYMBOL and the length (L) of the six flow recirculation zones induced by the stents were utilized to asses the grid independence. It showed that grid independence was achieved at 1,100,000-1,200,000 cells for the whole model. Figure 3, A and B shows the differences in the two parameters between the coarser mesh and the finer one when grid independence was achieved.
As for time-step independence, we used 250 time steps per pulse cycle based on the work by Dwyer et al.,32 who found that 240 time steps per pulse cycle could give very good results with high resolution. In this study, the average value of SYMBOL (SYMBOL) over the six disturbed flow zones induced by the stents was used to asses the cyclic independence. The computation showed that cyclic independence was achieved at the 4th cycle. Figure 3C shows the differences in SYMBOL among the 3rd, 4th, and 5th cycles.
Figure 4 presents the flow patterns at the positions of S (A-A) and S (B-B) for the normal flow and the swirling flows with different inlet helical strength of the swirling flow under steady-state flow conditions. As evident from the figure, an apparent spiral or swirling flow was created in the swirling flow models. The numerical simulation revealed that when compared with the normal flow model, the swirling flows significantly altered the velocity profiles at both S (A-A) and S (B-B). Different from the flow in the normal flow model, the maximums of the axial velocity profiles shifted away from the tube center. The simulation also revealed that the spiral flows were attenuating progressively along the tube, leading to significant reductions in the strengths of the swirl flows at S (B-B). As the strength of the spiral flow decreased, the maximum axial velocity returned to the tube center at S (B-B).
Effect of Inlet Helicity on the Length of Flow Recirculation Zones in the Stent.
Figure 5, A and B shows the effect of inlet helicity density(H) on the length (L) of the six flow recirculation zones indicated in Figure 1 and their average value (
) in the stent. As evident from the figures, L and L̄ decreases with increasing inlet H, indicating that swirling flow can indeed suppress the flow disturbance in the stent, and the stronger the swirling flow, the shorter the flow recirculation zones in the stent. The flow simulation also revealed that when the inlet H dropped to 3.5 m/s2, the value of L̄ had almost no difference with the averaged length of the flow recirculation zones in the normal flow stent.
Effect of Inlet Helicity on WSS in the Flow Recirculation Zones.
Figure 6A shows the effect of inlet H on the average WSS (SYMBOL) in each of the flow recirculation zones. As a comparison, the average WSS in the model artery without stent in place was also calculated for the four different inlet H (Figure 6B). As shown in the figures, the average wall shear stresses in the disturbed flow zones induced by the stent are considerably lower than those in the model artery without stent. The results show that the swirling flow can not only reduce the lengths of the flow recirculation zones, hence suppressing the flow disturbance in the stent, but also enhance the average WSS in each of the flow recirculation zones (Figure 6A). SYMBOL in each of the recirculation zones increases with increasing inlet H. Again, the flow simulation revealed that when inlet H dropped to 3.5 m/s2, SYMBOL in the swirling flow stent had no significant difference with that in the normal flow stent.
Pressure Drop (Δp) Across the Stent.
Figure 7 shows pressure drops of the stent with swirling flow with different inlet helicity densities. The flow simulation showed that Δp of the stent with swirling flow was 164, 167, and 173 Pa for inlet H = 3.5, 6.5, and 14.5 m/s2, respectively, whereas for the stent with normal flow, Δp was 163 Pa. The flow simulation therefore indicated that though a stent with swirling flow had beneficial effects on hemodynamics of the stent, it could lead to a higher pressure drop across the stent when compared with a stent with normal flow.
Effect of Inlet Helicity on the Average Value of OSI in the Stent.
Figure 8A shows the local value of oscillating shear index (SYMBOL) in each disturbed flow zone of the stent. SYMBOL was calculated using the method described by Morbiducci et al.22 and with respect to an area of 0.1 cm2 immediately distal to each ring of the stent. Figure 8B shows the average value of SYMBOL (SYMBOL) over the six disturbed flow zones denoted by P1, P2, P3, P4, P5, and P6,
The pulsatile flow simulation showed that when compared with the flow in the stent under the normal flow condition (H = 0 m/s2), the swirling flow reduced SYMBOL and SYMBOL in all the disturbed flow zones. The higher the H, the higher the reduction in SYMBOL and SYMBOL.
Determination of Minimum Inlet Helicity
Based on all the results obtained from the flow simulations, it can be concluded that too small H would have almost no beneficial impact on the hemodynamic performance of endovascular stents. For instance, an H of 3.5 m/s2 can only produce approximately 3% reduction in L̄ with no significant change in SYMBOL when compared with the normal flow stent. Therefore, it has been decided that the minimum swirling flow strength required should be approximately 6.5 m/s2, which can reduce 16% of L̄ 44% of SYMBOL and enhance 46% of SYMBOL when compared with the normal flow stent. The computation showed that the reduction in helicity from the inlet to the first ring stent was negligibly small because the distance between the inlet and the first ring stent was only 1 mm. For instance, when the inlet helicity was 6.5 m/s2, the helicity at first ring stent was 6.3 m/s2, only 3% reduction in helicity.
Studies have shown that because of the structure of the endovascular stent, blood flow will be inevitably disturbed in the stent.2,25 These disturbed flow zones are always characterized by vortices with low WSS/high OSI and therefore play an important role in the development of in-stent restenosis.6,24 Therefore, eliminating or suppressing disturbed flow zones in a stent might be a solution to the restenosis problem of endovascular stents. But the question now is how to achieve this target with the inherent functional structure of the stent.
In this study, we propose to apply the swirling flow mechanism of the human ascending aorta to the design of the stent to eliminate/suppress the flow disturbance. We believe that in this way, the genesis and development of arterial restenosis might be suppressed. To test this hypothesis, we used a simplified ideal circular ring stent model similar to the one used by Seo et al.11 and numerically analyzed the flow patterns in the stent.
The numerical study demonstrated that by intentionally creating swirling flow in the stent, the flow disturbance was indeed suppressed and the average WSS in the stent was enhanced with a significantly reduced OSI, depending on the strength of the swirling flow created. The numerical simulation showed that though swirling flows had a beneficial effect on the hemodynamic performance of endovascular stents, the strength of the swirling flow required to suppress the flow disturbance must be larger than a minimal value. The minimal strength of swirling flow required for the present stent model was determined to be 6.5 m/s2.
Now that swirling flow created in an endovascular stent can improve its hemodynamic performance, the question is whether it is feasible to design such a stent with the structure that can not only fulfill the task of a conventional stent but also automatically induce blood flow to rotate in the stent. The work by Seo et al.11 demonstrated that a stent consisting of three successive 360-degree spirals with no rings at its downstream end exhibited the flow characteristics of a helical flow. Their flow simulation showed that with this kind of structure, no flow recirculation occurred in the stent. Therefore, designing endovascular stents with the feature of helical flows is not impossible.
Here, it should be mentioned that the minimal strength of swirling flow required to suppress the flow disturbance in a stent is not a fixed value. It should depend on the structure of the stent concerned. Therefore, the minimum swirling flow strength is a very important parameter that has to be taken into consideration in the design of a swirling flow stent. In addition, the present flow simulation study showed that because of the attenuation in the strength of swirling flow along the stent, its beneficial effect on the hemodynamic performance of the stent was compromised significantly in the distal part of the stent. Therefore, this study indicates that a stent with a build-in structure that possesses intrinsic characteristics to automatically induce swirling flow in the stent is better than a stent with a front swirling flow inducer in terms of hemodynamics.
It should be mentioned that the axisymmetric model used in the study was unrealistic because no commercial stent is purely axisymmetric. Nevertheless, as a preliminary study, the numerical simulation theoretically substantiated our hypothesis and provided us with useful information for the design of swirling flow endovascular stents.
In this article, although it is premature to make a conclusion based solely on a CFD study and animal experiments have to be carried out to validate the swirling flow hypothesis, we have proposed a new direction to improve the performance of the endovascular stent.
Supported by Grants-in-Aid from the National Natural Science Foundation of China (No. 10632010, 30670517, 10772054).
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