Each year, in the United States alone, more than 570,000 coronary artery bypass graft procedures are performed, leading to a great demand for small caliber (usually <6 mm) vascular grafts.1 These small caliber vascular grafts are also widely used for peripheral vascular surgery.2 Although arterial grafts have shown promise in large diameter (usually larger than 6 mm) applications, the occurrence of acute thrombus formation and intimal hyperplasia (IH), has limited the patency rate of small diameter grafts for clinical use.3
The local flow field and in particular the wall shear stress (WSS) markedly influences vascular biology and development of disease.4 Intimal hyperplasia, the abnormal progressive thickening of the innermost layer of the artery wall,3 was found to develop preferentially in low WSS or stagnation regions.5 Based on these findings, researchers have been trying various ways to improve the hemodynamic performance of the implanted grafts. A conventional graft has been found commonly to be two-dimensional, favoring extremes of low WSS including stagnation flow.6 Recently, the use of helical geometry for arterial grafts was proposed and some preliminary investigation was conducted on their performances. From the viewpoint of fluid mechanics, a helical geometry is capable of achieving three-dimensionality and physiological- type swirling flow which might also lead to low fluid velocity and shear stress region with flow separation, instability and even stagnation.
The beneficial role played by helical flow was assessed. Stonebridge and Brophy7 suggested that a spiral configuration of blood flow might protect the arterial wall from damage by reduction of laterally directed forces. Houston et al.8 found that an absence of aortic helical flow might be a predictor of renal impairment deterioration in patients with renal artery stenosis. In the recent study by Caro et al.9 found that, although the conventional straight graft was occluded by thrombus along approximately its distal half, the swirl graft was patent throughout its length, and there were only near-diametrically opposed ribbons of pathology which is inconsistent with the WSS distribution predicted by their computational fluid dynamics (CFD) study.
The CFD study of Caro only presented the WSS distribution and in-plane flow mixing. However, other hemodynamic parameters, such us velocity magnitude, fluid recirculation, and pressure drops, are also important to the development of IH. In addition, the geometrical parameters of a helical graft such as helical pitch and amplitude can affect the flow pattern and the shear rate distribution of the graft, hence affecting its hemodynamic performance.
Encouraged by the preliminary studies of Caro et al. and aimed at getting more insight into the mechanical mechanism of the helical graft, the present study conducted a detailed comparison in terms of flow pattern and shear rate distribution between a helical graft and a conventional one. Furthermore, to optimize the design of helical grafts, detailed geometrical parameter studies in terms of Dean Number, helical pitch, and helical amplitude were conducted.
A sketch of helical graft is shown in Figure 1A where R is the helical amplitude, 2πk denotes the helical pitch, and D, the internal diameter. Cross sections like A-A in Figure 1A which are normal to centerline of the helical graft are set as shown in Figure 1B, where slice 1 corresponds to the starting of the helical part, and slice 9 corresponds to the ending of the graft. Slices 10, 11, and 12 are the sections in the downstream straight part.
Dean number, a dimensionless number giving the ratio of the viscous force acting on a fluid flowing in a curved tube to the centrifugal force, is often taken as an important parameter in helical problems.10 With higher Dean numbers, flow can separate along the inner curve:
where ρ and μ represent, respectively, the density and dynamic viscosity of blood, and vint is the inlet velocity.
Helical Flow Quantification
Helicity, the extent to which crockscrew like motion occurs, is defined as:
where V is the velocity vector and ω is the vorticity vector.
If a parcel of moving fluid undergoes solid body motion about an axis parallel to the direction of motion, it will have helicity. If the rotation is clockwise when viewed from ahead of the body, the helicity will be positive, or, it will be negative. Considering Equation 2 and remarks on helical flows, Morbiducci et al.11 defined the basic quantity: the normalized helicity.
By the definition, normalized helicity physically represents the angle between velocity and vorticity and its value range from −1 to 1. As such, normalized helicity can be a useful indicator of how the velocity vector field is oriented with respect to the vorticity vector field for a given flow field. When normalized helicity equals one in the module, it indicates that the flow is highly 3D, although the flow is purely axial, or circumferential, when it becomes zero.
All numerical simulations were performed at a Reynolds number of 500, based on the measurements of flow and vessel diameter D for the porcine A-V access grafts.12 At such a rather low Re number, the flow in the grafts should remain laminar. In addition, as a preliminary study, the simulations were carried out under steady flow conditions, which is different from in vivo pulsatile flow conditions. But as a common assumption in the hemodynamic investigation of blood flow,13 it captures qualitatively the average effect of flow and can also reflect, to a certain extent, the essence of the investigated problem. The governing equations for incompressible laminar steady flow motion are given as
ū and p represent, respectively, the fluid velocity vector and the pressure. ρ and μ are the density and dynamic viscosity of blood. Because the shear rate in most part of the flow field in the present study was over 100 s−1, blood was assumed to be isotropic, homogeneous, incompressible, and Newtonian with a constant dynamic viscosity of 3.5 × 10−3 kg/m/s and a density of 1050 kg/m3. Due to the fact that the graft radius is small in comparison with its length, the gravitational term was neglected, as is conventionally done.14
The boundary conditions are: 1) Inlet: a uniform inflow velocity profile for the axial velocity component and a zero transverse velocity component was used: vinlet = 0.278 m/s. 2) Outlet: outflow condition was set at the graft outlet. 3) The vessel wall was 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, for example, transport mechanisms, but of somewhat lesser importance as far as the gross features of the flow are concerned.15 4) To guarantee the flow was fully developed, straight grafts with length of 20D were set at both ends of the helical graft.
The geometry and mesh generation were built using Gambit 2.0. Sections of the nonstructural hexahedron grids overlaying the model are presented in Figure 2. The mesh-grid was built dividing the fluid domain into approximately 434,280 hexahedral cells. To establish grid independent solution, a finer mesh consisting of approximately 751,420 cells was tested. Very similar flow fields were achieved on the two meshes, the maximum relative errors of the velocity and wall shear stress magnitudes were 1.2% and 3.5%, respectively. Therefore, the original grid was considered satisfactory.
The package Fluent was adopted for the flow visualization and analysis in the present study. Discretization of the equations at each control volume involved a second order upwind differencing scheme and the residual error convergence threshold was set as 1e-5. The resulting system of algebraic equations was solved iteratively using a procedure based on the SIMPLE algorithm.
To make a fair comparison, the internal diameter, and the total helical pitch length of all grafts, were set as D = 6 mm and L = 40 D = 240 mm, respectively.
Comparison Between the Helical Graft and the Conventional Graft
A four turn helical graft was first undertaken for the comparison with the conventional straight graft. The helix pitch and amplitude, defined in terms of the internal diameter D, were 10D and 0.5D, respectively.
Wall Shear Stress Distribution
The contour map of WSS is presented in Figure 3A. It was found that near-diametrically opposed 200% higher and 50% lower than the corresponding value of a conventional graft WSS zones were observed to seem alternately along the helical conduit, which is consistent with Caro's finding.9 To have a more quantitative look, the normalized average WSS of each cross section was shown in Figure 3B. It is clear that at section 2, due to the sharp change of geometry, a sharp increase of the WSS was observed, which hit almost 190% of the straight graft WSS. As the helical graft developed, the WSS decreased 15% and then kept almost stable in the remaining helical part. On average, the WSS level in most parts of helical graft was 170% higher than the straight graft WSS. At section 10, located in the downstream straight part, WSS was decreased by almost 30% over the value at the exit of the helical part, and then the WSS continued to decrease in the straight part downstream. But at the outlet, the WSS level was still 5% higher than in the straight graft.
Normalized Helicity Analysis and Flow Behavior
As is known for a conventional straight graft, the flow velocity is only a function of radius r, and is basically 2D (velocity is axial and vorticity is normal to velocity), which favor extremes of wall shear, including flow stagnation. To have a clear view of the flow field in a helical graft, the section view of slices normal to centerline of the helical graft are presented in Figure 4, where the red color represents the flow is highly 3D (ψ ≥0.6) while green color means the 3D feature is very weak (ψ ≤0.1). It is observed that the sudden geometry change at slice 2 induced a sharp change in the distribution of fluid velocity, a second flow developed. In addition, although 3D effects diverse in section, the normalized helicity is always greater than zero (ψ >0). And as the helix goes further, the swirling flow becomes more apparent. The flow simulation also showed that after the end of the helix, rotation of flow continues but weakens progressively along the straight part of the graft. However, it is interesting to notice that a strong second flow region is not always consistent with high 3D flow features which represents that in some regions the flow keeps recirculation instead of going forward and the flow recirculation will elevate particle residence time.
For the convenience of the comparison of flow velocity between straight graft and helical graft, we defined any area with velocity >0.4 m/s as a high velocity zone, whereas any area with velocity <0.1 m/s as a low velocity zone. As such the low velocity region covers 18% of a cross section while the high velocity region covers 28% for a conventional straight graft. (Figure 5) However, it was found that in a helical graft the low velocity zone only covered about 13% and the high velocity zone covered about 16% of the section—a ratio spread markedly lower than the corresponding ratio for the straight graft, which indicates a more uniformly distributed velocity profile in a helical graft. After the helical part ended, the velocity level progressively returned to its straight graft level.
Pressure Drop Distribution
For a fully developed laminar flow in the straight graft, the pressure depends on the axis z only and the pressure drop is linear to the axis z. However, in a helical graft, the pressure is a function of all coordinates. In Figure 6 the pressure drops, based on the minimum pressure or the maximum pressure, respectively, in a cross section were presented. It is very interesting to notice that along the helical graft, the pressure drop still revealed the same characteristic as in the straight graft, namely, it is linearly proportional to the z axis, but its slope is larger than the one of the downstream straight part, which revealed that the flow resistance in the helical graft increased compared with a conventional straight one. In addition, it is more intriguing that the two pressure drop lines were parallel to each other, which indicates that the pressure difference between the minimum and maximum pressure of each cross section kept constant along the helical graft.
Parameter Study of Dean Number
In this section, two helical graft geometries were chosen for the investigation of the Dean Number.
Graft 1: 4-turn helical graft. 2πk = 60 mm, R = 3 mm, D = 6 mm, Re = 500, thus the graft curvature k = 0.09 and dean number De = 300; Graft 2: 6-turn helical graft. 2πk = 40 mm, R = 1.26 mm, D = 6 mm, Re = 500, thus the graft curvature k = 0.09 and dean number De = 300.
In Figure 7A, the normalized average WSS distribution of the cross sections (shown in Figure 1B) was shown for two grafts. It is clear that although the two grafts have the same Dean number, their WSS is totally different. The four turn graft greatly improved the WSS level compared to the straight graft although the six turn graft brought little change to the WSS. It indicates that the same Dean number but different geometry leads to totally different hemodynamic performance in a graft, which is also reflected from the low velocity zone ratio of the two grafts (Figure 7B).
Parameter Study of Helical Pitch and Amplitude
To clarify the effect of the helicity level, two-turn, four-turn, and six-turn helical grafts, which have the same amplitudes 0.5D, were taken as a group for the investigation. To study the effect of helical amplitude, three grafts with the same helical pitch (four turns) but with different amplitudes (0.25D, 0.5D, and 1D) were investigated.
The hemodynamic performances of cross sections are presented in Figures 8 and 9 for comparison. As evident from the figures, a shorter helical pitch or larger amplitude enhanced the WSS and three dimensionality effects in the helical graft, and decreased the low and high velocity zone ratios. However, the improvement in hemodynamic was nonlinear. The improvement rate not only slowed down but resulted in more serious localization of low velocity area accompanying larger pressure drop as the pitch further shortened or amplitude increased. It is also interesting to notice that the pressure difference between minimum and maximum pressure of each cross section remained constant along the helical graft, and the difference increased as the pitch shortened or amplitude increased. Compared with pitch change, from Figures 8 and 9, it is fair to say that the amplitude change had less effect on hemodynamic performance in the helical part, but neither one led to much difference in hemodynamic of the downstream straight part.
The present numerical study aims to get more insight into the local flow fields of the helical grafts. Our results show that the helical structure of the graft indeed can create physically three dimensional swirling flow and lead to more uniformly distributed flow field which is believed to enhance fluid/wall mass transport in noncurved conduits by virtue of mixing.9 Compared with the conventional graft, the WSS is enhanced an average of 170% in the helical graft and remains 5% higher in the downstream straight part. We believe that the enhanced wall shear rate and flow velocity may impede the staying and adherence of platelets and leukocytes to the surface of the graft, reducing the possibility of thrombosis formation.
However, benefits aside, in the helical graft of this kind we have also predicted increased pressure drop in helical graft which revealed more flow resistance at a fixed flow rate that may lead to hemodynamic failure. Also with the flow redistributed, flow areas with low velocity will concentrate on one corner, which might possibly suffer from flow stagnation, inhabitation thus vulnerable to IH and thrombosis. In addition, when the helix suddenly starts the straight tube, the sharp change in geometry might lead to a dramatic increase in shear stress which is detrimental to cells. These kinds of flow areas are inherent in helical graft structures of this kind.
To overcome these problems, optimal structural design of the helical graft is needed. Our parameter study on Dean Number indicates that, even at the same Dean number but with different geometry, the hemodynamic performances of two grafts are totally different. Then the effect of the helical amplitude and pitch were investigated and it was found that increased helical amplitude or shortened helical pitch enhances not only the average WSS and three dimensionality effects, which is beneficial to the connective washing of the vessel walls, but flow swirling and uniform distribution as well.9 However, the benefit of shear stress increase and swirling flow of a helical graft is accompanied by enlarged pressure drop and localized low velocity zone. In addition, too high a shear stress is associated with excessive arterial wall strain, which may induce initial endothelial lesions and inflammatory activation with possible platelet aggregation and acute thrombosis.16 It should be noted that a helical graft of this kind, unlike a conventional graft, has a contorted configuration. A seriously contorted graft has problems keeping its mechanism solidity after implantation. Moreover, too high helical amplitude may not be applicable clinically.
Considering the aforementioned shortcomings of this kind of helical graft, we believe a better structure should be incorporated in the grafts, which can create swirling flow with more uniform shear stress distribution in the grafts, without inducing low fluid velocity areas, to provide smooth and even washing of the vessel by the flowing blood.
Although some assumptions have been made, the current study can still shed some light on helical grafts and increase our understanding of the flow mechanism in helical flow grafts, which is useful in the structural design of swirling flow vascular devices.
Supported by the National Nature Science Foundation of China (No.10527001, 10632010, 10672015).
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