where τv was turbulent viscous shear stress and μ(SijSij) was dynamic viscosity of the fluid.
According to Tennekes and Lumley,25 the turbulent dissipation rate was ϵ=2ν (sijsij). In our study, because we already obtained the turbulent dissipation rate, we could directly calculate the turbulent viscous shear stress as
The high-speed CCD camera showed that the opening process took approximately 60 msec for the SJM valve and 70 msec for the TRI valve. During the opening phase, the two leaflets of the SJM valve took 60.5 ± 2.6 msec and 59.8 ± 2.4 msec to open, and we applied 60 msec as the opening time. The three leaflets of the TRI valve took 55.7 ± 6.8 msec, 59.9 ± 7.4 msec, and 62.4 ± 7.8 msec to open, and we applied 70 msec as the opening time. There was no apparent difference between the two test valves at the opening phase. During the opening phase, each leaflet of the TRI valve and the SJM valve would flutter, and this phenomenon might cause the leaflets to start closing at different times. However, this phenomenon was stochastic, and no specific leaflet would close first; sometimes all leaflets might also close at the same time. During the closing phase, the two leaflets of the SJM valve took 30.1 ± 2.2 msec and 29.7 ± 3.1 msec to close, and we applied 30 msec as the closing time. The three leaflets of the TRI valve took 43.1 ± 3.7 msec, 42.6 ± 2.5 msec, and 43.3 ± 4.2 msec to close, and we applied 50 msec as the closing time. As mentioned earlier, because each leaflet might start to close at different times and the time difference between each leaflet might reach 5–10 msec, the real closure time should be considered slightly longer to account for the variations. Corresponding to the pressure waveforms and flow rates, the TRI valve starts to close earlier than the SJM valve. This finding can be explained by the different closing mechanisms between the two test valves.
When the SJM valve is fully opened, there is an angle of 5° between the valve leaflets and the flow. As the flow decreases, the reverse flow due to the pressure gradient between the aorta and the left ventricle pushes on the valve leaflets to initiate closure. On the other hand, the mechanism of TRI valve closure is different. Because the angle between the valve leaflets and the direction of the flow is almost 0 when the TRI valve is fully opened, the reverse flow cannot effectively contribute to push the leaflets closed. The leaflets are mainly pushed by the vortices in the aortic sinus. In addition, during initial valve closure, the flow velocity through the orifice of the TRI valve is 0.9 meter/sec, whereas the velocity through the SJM valve is only approximately 0.5 meter/sec, further supporting that the TRI valve closure does not primarily depend on reverse flow.
According to Yoganathan et al.,26,27 the effective orifice area (EOA), performance index (PI), and regurgitant volume can be applied to evaluate valve efficacy. Effective orifice area is an index of how well a valve design uses its primary or internal stent orifice area. It is defined as
, where Qrms is the root mean square of the systolic flow rate (cm3/s), 51.6 is the gravitational acceleration constant, and Δp is the mean systolic pressure drop (mm Hg). A larger EOA corresponds to a smaller pressure drop and, therefore, a smaller energy loss. However, EOA is also a function of the valve size. Another measure of a valve's resistance to forward flow is the PI, which is the ratio of the EOA to the valve sewing ring area. Regurgitant volume is the total volume of fluid through the valve per beat owing to the retrograde flow. It is equal to the sum of the closing volume and leakage volume.
On the basis of these definitions, we calculated the EOA, PI, and regurgitant volume from the flow waveforms over 30 cardiac cycles. For the SJM valve, the EOA was 4.12 ± 0.40 cm2, PI was 0.72 ± 0.07, and regurgitant volume was 4.51 ± 0.78 ml/beat. According to Yoganathan et al.,26,27 the EOA was 4.09 cm2, PI was 0.71, and regurgitant volume was 10.8 ml/beat for the SJM 27 valve, which were similar to our results. For the TRI valve, the EOA was 4.21 ± 0.47 cm2, PI was 0.74 ± 0.08, and regurgitant volume was 3.88 ± 1.22 ml/beat. This indicates that the TRI valve has a smaller energy loss and regurgitation than the SJM valve.
Figure 4 shows the ensemble phase average velocity profile of the flow fields. The maximum velocities in the flow fields of the SJM and TRI valves were 1.53 meter/sec and 2.09 meter/sec, respectively. The peak Reynolds number based on peak bulk velocity and pipe diameter was approximately Re = 7500. Higher shear stresses were distributed over the higher velocity gradient regions in the flow fields. For the SJM valve, the flow separated into three jet flows when the valve was fully opened. Because there were no-slip boundary conditions at the leaflet surfaces and the tube walls, both sides of each jet flow would appear as larger velocity gradient regions. When the TRI valve was fully opened, the flow separated into a major central orifice flow and three minor jet flows across the minor orifices between the leaflets and valve housing. The higher velocity gradient regions were distributed at the sides of the aortic sinus, so the locations of higher shear stresses can be estimated based on the velocity profile.
After calculating RNS-X, RNS-Z, and RSS, we could change the reference coordinates to find the actual maximum values of RSSs, as shown in Figure 5. The maximum values of the PRSS-maj appeared at the peak flow phase D for both valves, at 71.1 N/m2 for the SJM valve and 93.0 N/m2 for the TRI valve. The PRSS-maj of neither valve alone exceeds the threshold for RBC damage of 150 N/m2.
Figure 6 shows the Kolmogorov length scale fields. Although there was a value of Kolmogorov length scale at every measurement point, only the small scale vortices would damage RBCs and were of main interest. As the RBC diameter approximates 8–10 μm, we only plotted scales smaller than 100 μm to discriminate the regions of influence. For the SJM valve, the minimum Kolmogorov length scale was 27 μm, which is approximately 3–4 times the diameter of RBCs. They distributed along the two sides of the jet flows and the downstream aortic sinus. In addition, the magnitudes of the Kolmogorov length scales were more significant near the valve orifice, and they were also distributed along the wake regions of the leaflets. For the TRI valve, the minimum Kolmogorov length scale was 25 μm, which is similar with the SJM valve. They were also distributed along the wake regions of the valve leaflets.
On the basis of the turbulent kinetic energy dynamic equilibrium, we calculated the turbulent dissipation rate and then used Equation 4 to calculate the turbulent viscous shear stress (TVSS) (Figure 7). The maximum turbulent viscous shear stresses were approximately 11.43 N/m2 for the SJM valve and 13.32 N/m2 for the TRI valve at peak flow phase. For both valves, the TVSS was mainly distributed along the two side jets and the downstream aortic sinus. Furthermore, the magnitudes of the TVSS were larger near the valve orifice and were also distributed along the wake regions of the valve leaflets.
The velocities, vorticities, RNSs, and RSSs in the flow fields are summarized in Table 2. The maximum velocity occurred at the phase of peak flow and then decreased with the flow. At phase F, although the flow rate was negative, the maximum velocity maintained a positive value because of the existence of vortices in the flow field. Bellhouse and Talbot28 studied the closure mechanism of the human aortic valve, and their results indicated that there was a vortex within each aortic sinus and these vortices would benefit the closure of the valve leaflets. Three quarters of the valve's closure was accomplished during forward flow, requiring only very little reverse flow to seal it. This same kind of closure mechanism occurs with the TRI MHV in our study. The high-speed CCD camera shows the TRI valve beginning to close earlier than the SJM bileaflet valve at phase E (Figure 2). In addition, the flow velocity was maintained at 0.9 meter/sec through the TRI valve and was greater than that of the SJM valve. This also showed that the closure mechanisms were different between the TRI valve and the SJM bileaflet valve, and the TRI valve only required very little reversed flow to seal it.
Lim et al.29 measured a porcine bioprosthetic aortic valve. They applied the fluid with a dynamic viscosity of 3.5 cP and a density of 1.01 g/cm3. The heart rate was 72 bpm, and cardiac output was 5 L/min. Their results showed that the valve developed a triangular velocity profile with peak velocities of up to 1.8 meter/sec at peak systole phase. At the same phase (phase C), our results showed that the maximum velocity of the TRI valve was similar in magnitude at 2.09 meter/sec. However, Lim et al.29 indicated that the maximum RSS was 346.9 N/m2, whereas our result with the TRI valve was 93 N/m2. Although our TRI valve was made with titanium alloy and its motion was not as soft as the bioprosthetic valve, the flow field of the TRI valve was still similar to that of the bioprosthetic valve.
The maximum vorticity appeared at the peak flow phase C, and the magnitude of vorticity across the SJM valve was larger than the TRI valve. Because there were three jet flows through the SJM valve orifice, the regions of high vorticity were distributed on both sides of the jet flows (Figure 8). The regions of large vorticity corresponded with the wake flow regions downstream the valve leaflets, indicating that the flow might slow down or even stop in these regions. However, there was no distribution of high vorticity at the central flow downstream the TRI valve, meaning the flow velocity through the TRI valve orifice was larger than the SJM valve.
PRNS-maj and shear stresses increased with the flow rate, and the maximum values appeared at deceleration phase D. Although the magnitudes of these two kinds of shear stresses were both larger across the TRI valve than the SJM valve, they still did not exceed the threshold for RBC damage. At phase F during valve closure, PRNS-maj and PRSS-maj suddenly increased for the SJM valve, but this was not observed for the TRI valve. It is likely due to the SJM leaflets closing rapidly and producing a large fluctuation at the instant of the valve closure. Ge et al.19 studied the flow fields across the SJM valve in detail by two-dimensional DPIV measurements and two-dimensional and three-dimensional numerical simulations. Their results showed that RSS increased with the flow rate, and the values were 6 N/m2 at early acceleration and 50 N/m2 at peak systole. These values were similar to our results without modifying the coordinates.
At peak flow phase through the SJM and TRI valves, the minimum values of the Kolmogorov length scale were 27 and 25 μm, and the Kolmogorov time scales were approximately 0.24 and 0.21 msec, respectively. Liu et al.9 applied the laser Doppler velocimeter to measure the flow fields across a SJM valve and analyzed with the turbulent energy spectrum. They showed that the Kolmogorov length scale was 29 μm at peak phase, which was very similar to our results in magnitude. The locations of these values were also similar to our results. Ge et al.19 applied DPIV under the assumption that flow fields were isotropic and homogenous turbulence. They used the equation
to calculate the turbulent dissipation rate and then calculated the Kolmogorov length scale. They showed the Kolmogorov length scale was 47 μm across the SJM valve at peak systole. Quinlan and Dooley18 considered that the Kolmogorov length scale of 46 μm across the SJM valve was underestimated. Ellis et al.30 indicated that the Kolmogorov length scale was 7 μm in hinge flow. Based on these studies, it is obvious that there are different Kolmogorov length scales due to different experimental loops and methods of analysis. Therefore, it is safe to say that an accurately calculated Kolmogorov length scale across a SJM valve is currently unknown and at best is generalized within a proper range of values. In conclusion, the Kolmogorov length scales across a SJM valve were approximately 20–70 μm in forward flows but might be smaller than 10 μm in backward flows.
On the basis of the assumption of dynamic equilibrium, we could use the turbulent dissipation rate to calculate TVSS from Equation 4 by the concept of Jones,17 the results of which are listed in Table 3. The TVSS increased with the flow rate, and the maximum value occurred at peak flow phase C, then it decreased subsequently. However, the TVSS suddenly spiked at the instant of the valve closure (phase F) for the SJM valve, which was similar to the PRSS-maj. All turbulent viscous shear stresses were smaller than 15 N/m2. Ge et al.19 applied DPIV measurement data into the turbulent equations to directly calculate viscous shear stresses. Similarly, their results showed that the viscous shear stress was 12 N/m2 at peak systole, and all values were smaller than 15 N/m2 at each phase.
In summary, our results showed that PRSS-maj were all smaller than 100 N/m2 for both valves, and the TVSS were smaller than 15 N/m2. As the magnitudes of shear stresses did not differ substantially, the closing velocity of the valve leaflets should be considered as a more important factor.
We used a high-speed CCD camera to capture the valve motions to study the closure behaviors of the SJM valve and a new TRI valve. We applied the large eddy simulation and sub-grid-scale model to overcome the insufficient spatial resolution of DPIV and evaluate the turbulent dissipation rate in the flow field. Then, we used the dissipation rate to calculate Kolmogorov length scales and TVSS.
Our results showed that the process of TRI valve closure was obviously slower than the SJM bileaflet valve, but the PRSS-maj and the TVSS were both a little higher for the TRI valve. However, the shear stresses spiked again at the instant of SJM leaflet impact, which was not observed for the TRI valve. As the magnitudes of shear stresses for both valves did not exceed the threshold of RBC damage, the closing velocity of the valve leaflets should be considered as a more important factor in evaluating the valve efficiency and contribution to cavitation. From our results, because the closing velocity of the TRI valve was slower than the SJM bileaflet valve, this new design might reduce the risks of thrombosis and thromboembolism.
The authors thank the Division of Medical Engineering, National Health Research Institutes, Taiwan, for providing their technical assistance and primary laboratory equipment.
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