Where SYMBOL and p represent the fluid velocity vector and the pressure, respectively. ρ and μ are the density and dynamic viscosity of blood, respectively.
The inlet velocity was set 0.144 m/s for the steady flow simulation, and the corresponding Reynolds number based on graft diameter and blood viscosity was 260. A native coronary waveform was specified as the inflow boundary for pulsatile flow simulation,21 its maximum Reynolds number during the cycle is 612. Assuming a heart rate of 80 beats/min, the period of each cycle is 0.75 seconds.
The export boundary condition was defined as outflow. The graft and vessel wall were assumed to be rigid and nonslip.
Grid refinement studies were carried out for both configurations under steady flow condition. The WSS distributions along the artery floor were presented at three different meshes in Figure 2. The maximum relative errors between the medium and fine meshes are <3%. Therefore, the medium grid were considered satisfactory and adopted for the following investigation.
The study on time step for unsteady flow was carried out, and finally, each pulse cycle was divided into 300 time steps of size 2.5 ms. In addition, four cardiac cycles were performed for each simulation to eradicate any start-up effects and achieve stable results.
The flow visualization and analysis were completed by the commercial CFD software Fluent 6.0, which was based on the finite volume method, default Segregate Implicit 3D Solver was adopted. Discretization of the equations involved a second order upwind differencing scheme. SIMPLEC was adopted for the pressure velocity correction, and the residual error convergence threshold was set as 1e-5.
The WSS contour maps in the complete ABG models (Figure 3 A and B) show that the WSS magnitude in a helical-type ABG was not only increased in its graft segment but also at its anastomosis region, especially near its suture line and on its artery bed. The WSS distributions along the axial lines of artery beds (Figure 2) hold similar general characters for two ABG models and gradually became to consistent with each other when distally away from the distal anastomosis. However, quantitatively, the WSS values of the helical-type ABG were remarkably larger than those in the traditional-type ABG.
The WSS contour maps on host artery beds where the horizontal axis of the map represents the axial distance along the host vessel, whereas the vertical axis represents the circumferential distance along the host vessel wall, which were presented in Figure 3, C and D. In addition, it was indicated that the WSS was symmetrically distributed in the traditional-type ABG, whereas asymmetrically in the helical-type ABG. The low-WSS region (smaller than 0.6 Pa) in the helical-type ABG was proximally displaced, which indicated elevated shear stress levels in the occluded host vessel region, when compared with the traditional-type model.
Figure 4 shows the axial components of velocity vectors superimposed the contour maps of velocity magnitude in the symmetry planes, and the recirculation regions were zoomed in. It was observed that the helical-type ABG improved the axial velocity at its distal anastomosis while reduced the size of stagnation and recirculation flow zone, when compared with the traditional-type ABG.
Three cross-sections at the distal anastomosis were chosen to show the cross-flow streamlines and flow velocity contours: C1 represents the mean section; C3 represents the toe section; and C2 represents the mean section of C1 and C3. Symmetrical flow was kept in the traditional-type ABG. Also, two Dean Vortices were visible at its C3 section, and stagnation zones near the suture line appeared on its C1 and C2 sections. The out-of-plane geometry of helical graft broke the flow symmetry, and a stronger clockwise vortex accompanying with a weaker counter-clockwise vortex were observed at all selected sections, which helped to avoid flow stagnation. When flowing downstream, the vortex cores in helical-type ABG moved to the centerline of its host artery, and the strength of two vortices gradually became equal (Figure 5).
The local values of OSI at the distal anastomosis were presented in Figure 6. When compared with the traditional model, the helical graft significantly reduced the region size of high OSI, decreased the maximum OSI at the distal anastomosis by 58.4%, in contrast to a decrease of 5.2% in the occluded region.
To have a better view of the effect of helical bypass on pressure drop, 11 slices of the configurations as shown in Figure 7 were chosen for the calculation of the area-averaged pressures at four moments: a) systolic acceleration; b) maximum velocity; c) minimum velocity; and d) diastole. The pressure drop varied with time and reached maximum at systole, whereas minimum at diastole. Generally, the helical bypass did increase the pressure drop compared with the traditional-type ABG, and the percentage increase varied with cycle. The maximum percentage increase of 20.6% happened at moment of maximum velocity, and the pressure drop along helical bypass was 1.5 mm Hg. Noteworthily, the pressure drops along two ABGs at systole were consistent with each other.
Figure 8 presented the views of the evolution of particle sets emitted at different time moments. Color coding of the particle traces was used to display the instantaneous positive (negative) value of the LNH. Although because of the geometry alteration, 3D flow was induced at the proximal anastomosis in the traditional-type ABG, few flow mixing happened, and the helicity of near-wall flow was quite low. However, as the flow moved through the helical graft, the momentum of the fluid caused it to follow the graft wall curvature, thus near-wall flow was kept highly 3D, and flow mixing was significantly improved. The rotation of flow continued after reentering the downstream host artery.
Figure 9 presented the HFI values calculated over traces of particle sets emitted at the four time instants Tj, over the time interval T es − T j, and the percent distribution of particle traces with respect to the hfik, i.e., the average LNH of each trajectory of particles. Helical-type ABG exhibited hfik percent distributions with low degree of asymmetry, whereas traditional-type ABG presented high asymmetry, especially at time T3 and T 4. All the trajectories had flat distribution except that the one in traditional-type ABG emitted at T 4. The HFIs of helical-type ABG were increased, when compared with the traditional-type ABG at all moments, and the percentage increase reached 28% at time T 4. In addition, the maximum hfi k in the helical-type ABG was 0.831, in contrast to 0.643 in the traditional-type ABG.
The aim of this investigation is to find out whether a helical-type ABG is hemodynamically beneficial to ABG patency, when compared with the traditional-type ABG, and the investigation focused on the distal anastomosis region, which is most prone to IH. Although no quantitative relationship between the hemodynamic parameters and IH is found, there is, however, a trend relationship between the local flow field and IH. It has been demonstrated that IH tends to occur preferentially in regions of low time averaged shear stress and long particle residence time.22 It has also been suggested a connection between the spatial gradient and sharp temporal variation of WSS and regions where IH preferentially develops.23
The geometric features of an ABG directly affected its hemodynamic performance: 1) the loop shape of the traditional graft is semicircular in contrast to the “M-shaped” loop of helical graft. Consequently, the WSS and OSI profiles along the helical loop were approximately diametrically opposed, with the high and low zones rotating with axial position; 2) The insertion of the helical graft was rotated slightly counter clockwise relative to the traditional graft when looking from downstream end (Figure 1). Consequently, symmetric flow was kept at the distal anastomosis of the traditional-type ABG, whereas asymmetric flow patterns and WSS distribution emerged at the anastomosis of the helical-type ABG. On the whole, the geometry of the helical-type ABG is out of plane in contrast to the in-plane geometry of the traditional-type ABG. Consequently, three-dimensional flow and flow swirling were significantly strengthened in the helical-type ABG. The increased flow helicity especially of the near-wall flow and strong secondary flow facilitated the mixing of particle between the near-wall region and core of the flow in the helical-type ABG, which brought the high-momentum fluid to the surface and retarded the onset of flow separation. Furthermore, the forward streaming blood curled by the swirling flow in the helical-type ABG filled the space left by the flow separation at the inner wall, which helped to eliminate stagnation flow regions and mechanical trauma to blood cells.
In this study, the level of shear stress involved was on the order of 2 dynes/cm2, which was much lower than 50 dynes/cm2, the threshold to activate platelets and induce platelet aggregation. In this case, the elevated WSS may lead to a reduction in the concentration of platelets near the vessel wall possibly through a mechanism of the Magnus effect24,25 that induces forces on platelets and pushes them away from the vessel wall. In addition, the swirling flow-induced in-plane mixing would further lower the concentration of platelets near the wall and suppressed the interaction of platelets with the wall of ABG.
As indicated by the study on individual helical grafts, the magnitude of the pressure drop along a helical graft was considerably increased compared with a traditional one, which might prevent its medical application. However, according to the present investigation, although the helical-type ABG increased the pressure drop compared with the traditional one, its maximum percentage increase was <21%, which is still within physiological sense and can be overcome by optimal geometry design.
In addition, compared with cylindrical tube, the structural property of a helical tube improved resistance to kinking when subjected to bending moments, which helps to guarantee its graft geometry in vivo,9 and the graft artery junction of a helical-type ABG is similar to the traditional type, which will not increase operation difficulty and artery injury.
In a summary, desirable high WSS, low OSI, and increased flow swirling were achieved in the helical-type ABG. We, therefore, may expect that a helical-type ABG may decrease the likelihood of platelets adhesion, clot formation, and thrombosis in the graft surface and reduce the possibility of IH at the distal anastomosis.
The graft and vessel wall in the present investigation were assumed to be rigid and nonslip, which is not true in vivo. However, previous studies indicate that the wall elasticity may be of considerable significance in transport mechanisms but of somewhat lesser importance as far as the gross features of the flow is concerned.26 This study helped to increase our understanding of the flow mechanism in helical-type ABG and provided an important basis for its clinical applications and theory support.
Supported by the National Natural Science Foundation of China (10925208 and 11072162) and “863” High Technology Project.
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