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: v inlet = 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 3 A. 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, R e = 500, thus the graft curvature k = 0.09 and dean number D e = 300; Graft 2: 6-turn helical graft. 2πk = 40 mm, R = 1.26 mm, D = 6 mm, R e = 500, thus the graft curvature k = 0.09 and dean number D e = 300.
In Figure 7 A, the normalized average WSS distribution of the cross sections (shown in Figure 1 B) 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 7 B).
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).
1. Mitchell SL, Niklason LE: Requirements for growing tissue-engineered vascular grafts. Cardiovasc Pathol
12: 59–64, 2003.
2. Eagle K, Guyton RA, Davidoff R, et al
: ACC/AHA guidelines for coronary artery bypass graft surgery. J Am Coll Cardiol
341: 1262–1347, 1999.
3. Cole JS, Watterson JK, Reilly MJ: Numerical investigation of the hemodynamic at a patched arterial bypass anastomosis. Med Eng Phys
24: 393–401, 2002.
4. Derdeyn CP, Carpenter DA, Videen TO, et al
: Patterns of infarction in Hemodynamic failure. Cerebrovas Dis
24: 11–19, 2007.
5. Giordana S, Sherwin SJ, Peiró J, et al
: Automated classification of peripheral distal by-pass geometries reconstructed from medical data. J Biomech
38: 47–62, 2005.
6. Losi P, Lombardi S, Briganti E, et al
: Luminal surface microgeometry affects platelet adhesion in small-diameter synthetic grafts. Biomaterials
25: 4447–4455, 2004.
7. Stonebridge PA, Brophy CM: Spiral laminar flow in arteries? Lancet
338: 1360–1361, 1991.
8. Houston JG, Gandy SJ, Milne W, et al
: Spiral laminar flow in the abdominal aorta: A predictor of renal impairment deterioration in patients with renal artery stenosis? Nephrol Dial Transpl
19: 1786–1791, 2004.
9. Caro CG, Nick JC, Watkins N: Preliminary comparative study of small amplitude helical and conventional EPTFE arteriovenous shunts in pigs. J R Soc Interface
2: 261–266, 2005.
10. Wang CY, Bassingthwaighte JB: Blood flow in small curved tubes. J Biomech Eng
125: 910–914, 2003.
11. Morbiducci U, Ponzini R, Grigioni M, Redaelli A: Helical flow as fluid dynamics signature for atherogenesis risk in aortocoronary bypass. A numerical study. J Biomech
40: 519–534, 2007.
12. Rotmans JL, Velema E, Verhagen HJ, et al
: Rapid arteriovenous graft failure due to intimal hyperplasia: A porcine, bilateral, carotid arteriovenous graft model. J Surg Res
113: 161–171, 2003.
13. Fan Y, Xu Z, Jiang W, et al
: An S-type bypass can improve the hemodynamics in the bypassed arteries and suppress intimal hyperplasia along the host artery floor. J Biomech
41: 2498–2505, 2008.
14. Payne SJ: Analysis of the effects of gravity and wall thickness in a model of blood flow through axisymmetri
c vessels. Med Biol Eng Comput
42: 799–806, 2004.
15. Jou LD, Berger SA: Numerical simulation of the flow in the carotid bifurcation. Theor Comp Fluid Dyn
10: 239–248, 1998.
Copyright © 2009 by the American Society for Artificial Internal Organs
16. Altman R: Risk factors in coronary atherosclerosis atheroinflammation: The meeting point. Thrombosis J
1: 4, 2003.