Vascular reconstruction: CFD predictions of bypass graft haemodynamics

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1 Vascular reconstruction: CFD predictions of bypass graft haemodynamics J.S. Cole 1, J.K. Watterson 1 & M.J.G. O Reilly 2 1 School of Mechanical and Aerospace Engineering, The Queen s University of Belfast, United Kingdom. 2 Vascular Surgical Department, Belfast City Hospital, United Kingdom. Abstract Anastomotic intimal hyperplasia is a major cause of failure of arterial bypass grafts. The focal nature of this disease suggests that its development is influenced by local haemodynamics. Numerical simulations of pulsatile, non-newtonian blood flow through typical human femorodistal bypass models were performed in order to improve understanding of bypass flows and identify any associations between the flow features and the sites most susceptible to the disease. In general, the flow at the distal anastomosis is characterised by extensive recirculation, flow separation and reversal, complicated three-dimensional flow paths and an unusual wall shear stress distribution. It is concluded that progression of intimal hyperplasia will be encouraged by elevated fluid particle residence times and adverse shear stresses in the vicinity of the junction. A notable dependence of the flow patterns on aspects of the bypass geometry, including the graft attachment angle and the ratio of the diameters of the graft and recipient vessel, was demonstrated. The addition of a Taylor patch at the distal anastomosis was predicted to control the junction flow beneficially and produce a less critical shear stress distribution on the floor. The improved flow through the patched anastomosis should contribute to its observed, enhanced performance. Elements of the haemodynamics are worsened when a vein cuff is included at the distal anastomosis. The flow is characterised by an expansive, low momentum recirculation within the cuff, and high shear stresses and large spatial gradients in the shearing force on the artery floor. The benefits associated with the cuffed grafts may be related primarily to the presence of venous material at the anastomosis. 1 Introduction Cardiovascular disease, which encompasses disease of the heart and circulatory system, is the principal cause of death in the United Kingdom. In a typical year, cardiovascular disease is responsible for more than 235,000 deaths [1]. This represents about 40% of all deaths in the United Kingdom. The foremost underlying contributor to cardiovascular disease is atherosclerosis, a degenerative disease of medium and large arteries. This disorder is characterised by the patchy thickening of the artery wall due to the accumulation of fatty material. The disease process occurs doi: / /01

2 2 REPAIR AND REDESIGN OF PHYSIOLOGICAL SYSTEMS Figure 1: Schematic of arterial bypass system showing typical distribution of anastomotic intimal hyperplasia. over a lifetime and results in the gradual narrowing of the blood vessel. Severe atherosclerosis reduces the blood supply because of the arterial narrowing a critical condition. An arterial bypass procedure is a common and well-established treatment for atherosclerotic vessels. This entails the surgical attachment of a bypass graft from above the area of narrowing to a relatively disease-free vessel below this site. A new route is thus created, enabling an adequate supply of blood to the part of the artery beyond the blockage. The bypass graft may be of a prosthetic material, or comprise a segment of vein harvested from the patient s leg. Unfortunately, the effectiveness of the bypass is compromised in the medium and long term by the development of anastomotic intimal hyperplasia. This abnormal, progressive thickening of the innermost layer of the artery wall is observed to occur predominantly at the distal anastomosis of a bypass system, as depicted in Figure 1, being especially prominent at the heel, toe and along the suture line where the graft is fixed to the recipient vessel, and on the artery floor opposite the junction [2]. Intimal hyperplasia causes the gradual narrowing of the vessel lumen, restricting the flow of blood, and is a major factor responsible for bypass graft failure [3]. Eventually, the graft may become totally occluded or a blood clot may form. Mechanisms for intimal hyperplasia are unclear, but it is believed that many interrelated issues are involved [3 5]. These include a defect or injury to the endothelium (the inner lining of the artery wall) at anastomotic sites, complicated interactions between elements of the blood and the vessel wall, and a mismatch in the mechanical properties of the graft and artery [6, 7]. Significantly, the focal nature of the intimal hyperplasia indicates that local haemodynamic factors may play a key role in its development [8]. Recent work performed by our group aims to provide, via numerical modelling, definitive descriptions of the pulsatile, non-newtonian flow of blood through realistic models of human femoral artery bypass graft models. The objectives of the study are to describe the flow behaviour in a typical, conventional femorodistal bypass graft model to seek correlations between aspects of the flow and the localised development of intimal hyperplasia to examine the dependence of the flow patterns on various geometrical variables of the bypass configuration, including the anastomotic angle and the ratio of the diameters of the bypass graft and the recipient artery to analyse the flow characteristics in the special case of a Taylor patch bypass graft anastomosis to investigate the flow through a cuffed bypass graft

3 VASCULAR RECONSTRUCTION: CFD PREDICTIONS OF BYPASS GRAFT HAEMODYNAMICS 3 With such knowledge, it may be possible subsequently to modify the design of bypass prostheses, or to adjust the surgical technique, in such a way that the graft artery junction will be less susceptible to disease. 2 Conventional femorodistal bypass 2.1 Model geometry The model host artery, representative of a human femoral artery, has an internal diameter of 8 mm and is assumed to be fully occluded. The bypass graft is a circular tube of internal diameter 6 mm, symbolising a synthetic graft. The graft is of length 14 cm and follows the arc of a circle, making an angle of 20 with the host artery at both junctions. It is believed that, due to physical constraints, this is the smallest angle at which the surgeon can suture the graft to the artery. This geometry is termed the conventional model. The artery and graft walls are assumed to be rigid. It is claimed that such an idealisation can be made when only local flow patterns in short segments of large arteries are of interest [9, 10]. These conditions prevail in the current study. The fully structured grid, containing approximately 97,000 cells, overlaying the bypass geometry is presented in Figure 2. The host artery is continued 25 artery diameters upstream from the proximal anastomosis. This extended inflow channel would provide for the desired fully developed flow profiles on arrival at the anastomosis, despite the condition of a timedependent, uniform velocity profile specified at the inlet boundary. The outlet boundary was placed at approximately 12 artery diameters downstream from the distal junction, at sufficient distance so as not to affect the computed junction flow behaviour. It had previously been established that grid-independent solutions would result using this mesh. Additional computations for Newtonian and non-newtonian pulsatile flows were obtained on a finer mesh consisting of approximately 250,000 cells. Very similar flow fields were achieved on the two meshes, and the agreement between predicted wall shear stress distributions was generally better than 95%. Therefore, the original grid was considered satisfactory. 2.2 Numerical model and flow conditions The density of the blood is 1050 kg/m 3. A power law was employed to account for the non- Newtonian, shear-thinning nature of blood: μ eff 1 kγ n Figure 2: Mesh overlaying the conventional bypass model.

4 4 REPAIR AND REDESIGN OF PHYSIOLOGICAL SYSTEMS where μ eff is the effective viscosity, γ is the magnitude of the local shear rate, the constants k = kg/ms n 2 and n = This relation is a good fit to the dynamic viscosity curve for human blood with a haematocrit of 45% [11, 12]. Blood flow through the femoral artery bypass has the attributes of three-dimensional, timedependent, incompressible, isothermal, laminar flow. The governing equations for such a flow are u = 0 Continuity equation ρ u ρ( u ) u p ( μeff u) Navier-Stokes equations t where u is the velocity vector, ρ is the density, t is the time and p is the pressure. The commercial Computational Fluid Dynamics (CFD) flow solver FLUENT 4 [13], based on the finite volume method [14], was used to obtain a numerical solution to the governing equations. Flow simulations were conducted under representative physiological conditions. A timedependent velocity, calculated from the femoral artery volume flow pulse [15] (Figure 3), was applied at the inflow boundary. The mean flow rate of 2.25 ml/s, analogous to a mean flow velocity of 4.48 cm/s, is comparable to measured femoral artery bypass flow rates. The mean femoral artery Reynolds number, based on arterial diameter and a reference blood viscosity of kg/m/s, is 107. (The reference viscosity is the viscosity of the blood at high shear rates when its behaviour approaches that of a Newtonian fluid.) The maximum Reynolds number during the cycle is 830. Assuming a typical heart rate of 75 beats per min, the period of each cycle is 0.8 s. FLUENT s OUTLET condition was stipulated at the artery outflow boundary, requiring that the gradients in the flow variables (except pressure) be zero in the flow direction at this site. Such a condition is acceptable, the boundary having been located sufficiently far downstream from the junction and the associated junction flow disturbances. The no-slip condition was applied at all walls. volume flow rate/ml/s Sys. Accel. Max. Flow Sys. Decel. Diastole time/s Figure 3: Femoral artery volume flow pulse. (Labels indicate times for which results are plotted.)

5 VASCULAR RECONSTRUCTION: CFD PREDICTIONS OF BYPASS GRAFT HAEMODYNAMICS 5 Each pulse cycle was divided into 320 time steps of size 2.5 ms. The computation of 1.3 cycles was necessary in order to eradicate any start-up effects and achieve repeatability between flow patterns computed in successive cycles. Extensive validation of this numerical model has been accomplished [16]. 2.3 Flow behaviour and wall shear stress distribution The flow patterns within the bypass system exhibited very complicated temporal and spatial variations during the cardiac cycle. Since disease is much more likely to develop at the distal anastomosis, this paper concentrates on reporting the flow features at that site. During the acceleration phase of the cardiac cycle, local flow disturbances are low (Figure 4a). The characteristic features of separation at the heel (H) and a stagnation point (SP) on the artery floor opposite the junction are present. A zone of low momentum, recirculating fluid (R) is contained between the junction and the blockage in the artery. Separation does not yet arise at the toe (T) and the velocity distribution across the channel downstream from the junction is symmetric. At the maximum flow rate (Figure 4b), the fluid exiting the graft tends to continue across the channel and impacts on the far wall of the artery. Separation at the toe is encouraged, the floor stagnation point drifts downstream, while the recirculation (R) over the heel is enlarged. The junction flow disturbances are greatly magnified throughout the deceleration phase. The blood flow at the junction is increasingly driven across the vessel in the direction of the far wall. The separation region develops downstream from the toe while the recirculatory motion, formerly located close to the heel, becomes more elongated in shape and is drawn downstream to sit opposite the centre of the junction. Figure 4: Velocity vectors in the symmetry plane of the bypass model at various times during the cardiac cycle. (a) Systolic acceleration, (b) maximum flow, (c) systolic deceleration and (d) diastole.

6 6 REPAIR AND REDESIGN OF PHYSIOLOGICAL SYSTEMS With further deceleration, the bypass flow is completely reversed and the stagnant flow distal to the toe moves back towards the junction (Figure 4c). The expansive recirculation progresses towards the interface between the graft and artery. The local flow patterns are less dramatic during the remainder of the cardiac cycle when the flow returns to the primary direction and the flow velocities are much lower than those occurring earlier (Figure 4d). The skewing of the flow within the bypass graft, and as the fluid turns through the junction, generates secondary motions which are essential to the flow topology and which influence the residence times of blood elements in the vicinity of the junction. Secondary flow is particularly noticeable during the deceleration phase. Thus, it was demonstrated that the fluid advances downstream along complicated helical paths during this phase of the cardiac cycle. Wall shear stress is a crucial variable associated with intimal thickening. Figure 5 plots the shear stress distribution along the far wall of the host artery at different times during the cycle. The most notable spatial and temporal variations in the shearing force occur opposite the junction. In this region, large shear stress magnitudes are detected during the early part of the cycle as the powerful systolic flow issuing from the bypass impacts on the artery floor. The negative shear stresses are related to the zone of recirculating fluid confined between the mainstream and the occlusion, with fluid in this region passing upstream along the far wall of the vessel. Moreover, the local shear stress alters sharply over a narrow section of the endothelium, of about three artery diameters in length, where the flow divides about the floor stagnation point. This gives rise to elevated spatial gradients of wall shear stress so that the artery wall is subjected to a stretching action. During the later part of the cycle, wall shear stress levels are low and show little variation along the wall due to the depressed flow rate and associated more quiescent flow behaviour. Further downstream in the artery, at all times, the wall shear stress attains a constant value as fully developed flow is achieved. Between the junction and the occlusion, the shear stress is continually very low due to the stagnant flow in that region. 5 4 wall shear stress/pa s 0.12s 0.16s 0.24s 0.44s Opp.Heel Opp.Toe -4-5 distance along artery/mm Figure 5: Shear stress distributions along the artery wall opposite the distal anastomosis at various times during the cycle. Points opposite the heel and toe are denoted by and, respectively.

7 2.4 Implications for disease VASCULAR RECONSTRUCTION: CFD PREDICTIONS OF BYPASS GRAFT HAEMODYNAMICS 7 It is increasingly acknowledged that vascular processes are influenced by the local haemodynamics. In vivo experiments have demonstrated that augmenting the blood flow rate, and hence wall shear stress, inhibits the proliferation of intimal thickening in polytetrafluoroethylene (PTFE) grafts [17 19]. The existence of flow disturbance at the graft artery junction has been highlighted [20]. In vitro and numerical studies have confirmed that the junction flow patterns are strongly dependent on the local geometry [21 25]. Correlations have been made between the anastomotic regions susceptible to disease and the features of flow separation, recirculation, elevated particle residence times and the unsteady wall shear stress distribution. It was shown that under exercise conditions, the distal junction flow contained more vigorous axial and secondary motions which amplified wall shear stress magnitudes and reduced the areas affected by high particle residence times [26]. It is relevant that arteries establish a diameter which, under normal flow conditions, results in a mean fluid dynamic wall shear stress in the range 1 2 Pa [27]. Intimal thickening could be an adaptive response of the live arteries to the disturbed flow conditions in an attempt to restore a wall shear stress level within the normal range [19, 28]. The current study has identified several important characteristics of the junction haemodynamics including extensive recirculation (especially at the distal junction), flow separation and reversal, complicated flow paths, elevated fluid particle residence times and an unusual unsteady floor shear stress distribution. Here, associations between these aspects of the disturbed flow and the preferential sites for disease are considered. It has been advocated that the intimal thickening at the heel, toe and suture line arises from injury to the vessel wall induced by the surgical procedure, and is promoted by the compliance mismatch between the graft material and the artery [8]. The flow conditions at the distal anastomosis may also favour the progression of intimal hyperplasia at the heel and toe. In the vicinity of the heel, the flow within the occluded segment of the artery exhibits a large, low momentum recirculation. The residence times of blood elements in this region will be augmented, thus increasing the likelihood of adhesion of platelets and leucocytes to the endothelium, and leading to the stimulation of intimal thickening. It is probable that intimal thickening at the toe of the anastomosis is promoted in the like manner. Fluid particle residence times are elevated when flow separation arises just downstream from the toe during the flow deceleration phase, while the subsequent migration of the vortical structure towards the graft artery interface is followed by stagnation in the graft upstream from the toe, facilitating interaction between the blood and the thrombogenic prosthetic surface of the graft. Haemodynamic events surely fulfil a more prominent role in the development of intimal hyperplasia on the floor of the artery opposite to, and isolated from, the distal anastomosis. Some have linked this intimal thickening to flow stagnation and low shear stress on the floor [21]. However, this study has identified that the distal floor stagnation point travels rapidly downstream. Moreover, strong flow alongside the floor, in both the downstream and upstream directions, was demonstrated during the early part of the cycle along with the related, relatively high wall shear stress levels. This flow behaviour, and the lack of a concentrated low shear region, suggests that a low shear mechanism for intimal thickening, involving the aggregation of platelets and leucocytes, is unlikely at this location. The spatial gradient of shear stress implies that the endothelial cells on the floor are experiencing a stretching force, the magnitude of which varies over the cardiac cycle. This cyclic, unnatural, sharp stretching of the endothelial cells, caused by the changing spatial gradients of the wall shear stress, may promote intimal hyperplasia, possibly due to the activation of cellular reactions or through endothelial deformation or injury [29, 30].

8 8 REPAIR AND REDESIGN OF PHYSIOLOGICAL SYSTEMS 3 Anastomotic angle One important geometrical variable in the system is the anastomotic angle. For typical endto-side and side-to-end bypass configurations, the suturing technique and the graft material impose a lower limit of about 20 on the anastomotic angle. This section examines the effect of anastomotic angle on the bypass flow features. Computations of the pulsatile, non-newtonian flow of blood through typical femorodistal bypass models having anastomotic angles of 30 and 45 have been performed. The flow fields are contrasted with that observed in the conventional model. 3.1 Flow behaviour and wall shear stress distribution During the systolic acceleration phase, flow separation at the toe becomes more prominent with increasing anastomotic angle as the fluid must turn more sharply into the arterial direction. The effect of the anastomotic angle on the flow behaviour is even more pronounced at the maximum flow rate (Figure 6). While separation at the toe is just commencing in the smallest angle model (Figure 4b), large zones of separated flow extending at least halfway across the symmetry plane of the vessel are observed in the 30 and 45 cases. The dissimilar shapes of the separation regions are of note it is clear that the fluid exiting the graft impinges more directly on the artery floor as the anastomotic angle is augmented. In all configurations, during systolic deceleration, the toe separation grows in magnitude, while the recirculation zone next to the heel edges distally. The flow pattern during diastole is not strongly affected by anastomotic angle. In all cases, there is no toe separation and the velocity profile across the distal artery is largely uniform. It was observed that the cross-flow patterns are affected by anastomotic angle; increasing the angle strengthens the secondary flows. The general character of the distributions of shear stress on the wall opposite the distal junction is similar for all three cases considered. However, some important differences exist (Figure 7). The maximum shear stress and the gradient of shear stress increase with increasing anastomotic angle. This is consistent with the observed flow patterns. At higher angles, the fluid coming from the graft is forced further across the recipient artery in the direction of the Figure 6: Velocity vectors in the symmetry planes of the bypass models at maximum flow rate. (a) 30 model, (b) 45 model.

9 VASCULAR RECONSTRUCTION: CFD PREDICTIONS OF BYPASS GRAFT HAEMODYNAMICS wall shear stress/pa degrees 30 degrees 20 degrees Opp. Heel Opp. Toe -4-6 distance along artery/mm Figure 7: Shear stress distributions along the artery wall opposite the distal anastomosis at maximum flow rate. Points opposite the heel and toe are denoted by and, respectively. far wall. The inflated velocities close to the wall are responsible for the augmented shear rates and stresses. Moreover, the spatial gradient in the shearing force during systole acts over a narrower section of the wall in the larger angled geometries. 3.2 Implications for disease It has been observed that the influence of increased anastomotic angle is to increase any flow separations (and, therefore, residence times) and raise wall shear stresses. It would seem appropriate therefore to consider how anastomotic angle may affect the development of intimal hyperplasia. Earlier, it was speculated that the low momentum recirculation within the host artery close to the heel causes local blood particle residence times to increase. It was concluded that such conditions would improve the probability of adhesion of platelets and leucocytes to the endothelium, leading to the stimulation of intimal thickening. The nature of the flow in the vicinity of the heel remains relatively unchanged even when the anastomotic angle is raised. Accordingly, it is believed that intimal hyperplasia will proceed at that site in a similar manner and to a similar degree in the different angled models. In contrast, it has been shown that the flow in the toe region of the distal anastomosis is affected by the anastomotic angle. The separation region at the toe is more extensive and prevails for a longer time when the graft attachment angle is augmented. Furthermore, secondary motion distal to the junction is enhanced in the larger angled models at this time, with some fluid elements being directed over the side walls and upstream within the separation region. In such cases, the accumulation of blood particles in the neighbourhood of the toe would be favoured, and the progression of intimal thickening may be encouraged.

10 10 REPAIR AND REDESIGN OF PHYSIOLOGICAL SYSTEMS A correlation was observed between the part of the arterial floor subject to intimal hyperplasia and the local adverse shear stress distribution. It has been proposed that the cyclic, sudden stretching effect experienced by the endothelial cells opposite the distal anastomosis, represented by the time-varying spatial gradients of shear stress, promotes intimal hyperplasia. This investigation has indicated that, when the anastomotic angle is greater, not only are the disadvantageous shear stress gradients amplified, but also the unnatural stretching action is concentrated within a narrower section of the floor. Thus, the evidence suggests that a more severe response will occur on the artery floor when the anastomotic angle is raised. Finally, it should be noted that increasing the anastomotic angle causes the cross-sectional area and the boundary length of the anastomosis to decrease. The latter will provide a smaller site for the growth of diseased material at the anastomosis. In such a situation, development of intimal thickening will have a more detrimental impact with regard to restricting the flow through the system, and a more rapid occlusion and failure of the bypass graft may be expected. 4 Graft/artery diameter ratio The ratio of the diameters of the host artery and the bypass graft is another significant geometrical variable. The surgeon may choose a graft of diameter greater than, equal to, or less than that of the host artery. This selection is based on a number of factors, including the location of the recipient artery, its size, the quality of the outflow, and the patient s exercise potential. There is clear evidence implying that oversized grafts should, in general, be avoided [19, 31, 32]. This is because, in the absence of severe occlusive disease, arterial blood flow rates are controlled by the peripheral resistance, rather than by vessel size. Therefore, in oversized grafts the flow velocity is reduced, encouraging thrombosis and subsequent failure of the bypass. However, it is also advised that undersized grafts cause the flow to be restricted [19, 31]. This section considers the flow through a typical femorodistal bypass system in which the host artery and bypass graft possess identical diameters of 8 mm. The predicted flow patterns are contrasted with those computed in the conventional model. 4.1 Flow behaviour and wall shear stress distribution The major difference between the corresponding flow fields concerns the velocity of the blood within the graft. Obviously, when the diameter of the prosthesis is increased from 6 to 8 mm, a reduction in the average velocity of the fluid contained therein and an associated decrease in its momentum must result, since the flow rate is unchanged. Moreover, the sudden deceleration observed at the distal junction in the conventional model, and linked to the expansion in the cross-sectional area on passing from the graft to the artery, does not ensue when the graft and artery have equal diameters. These factors ensure that, over the systolic phase, the anastomotic flow is remarkably less disturbed in the present structure as compared to the conventional bypass model. At maximum flow rate, the more forceful fluid of the conventional model exits the graft and travels across the channel to impinge on the far wall (Figure 8). In contrast, there is a much smoother negotiation of the junction in the current model separation at the toe is delayed while the flow profiles in the distal artery exhibit minimal departure from a symmetric nature. As systolic deceleration commences, a narrow separation region develops distal to the toe. Subsequently, this zone grows in magnitude but remains considerably less extensive across the symmetry plane of the artery than that featured in the conventional model.

11 VASCULAR RECONSTRUCTION: CFD PREDICTIONS OF BYPASS GRAFT HAEMODYNAMICS 11 Figure 8: Velocity vectors in the symmetry planes of the bypass models at maximum flow rate. (a) Graft, artery same diameter, (b) conventional model. 4 3 wall shear stress/pa distance along artery/mm Figure 9: Shear stress distributions along the artery wall opposite the distal anastomosis at maximum flow rate. Graft and artery of equal diameter (solid line), conventional model (dashed line). Points opposite the heel and toe are denoted by and, respectively. Figure 9 highlights some important discrepancies between the current wall shear stress distribution across from the distal anastomosis and that predicted for the conventional bypass model. These differences are most evident along the floor of the artery during systole. When the wider graft is attached, the variation of the shearing force along the floor is significantly diminished. The peak shear stress, detected across from the toe at maximum flow rate, is reduced. The negative shear stress values, associated with the recirculation opposite the junction, are notably less extreme. The dissimilarities can be accounted for by recognising that the blood passing through the narrower graft possesses the greater momentum. Upon returning to the recipient artery at the distal anastomosis, there is enhanced motion across the channel in that case, and thereby an augmented shearing force is imparted along the far wall as the blood flows downstream. The spatial gradients of shear stress along the floor of the host artery during systole are decreased when the broader diameter graft is utilised. Therefore, under the current bypass arrangement, the undesirable stretching effect experienced by the endothelial cells of the artery floor during the systolic phase is less critical.

12 12 REPAIR AND REDESIGN OF PHYSIOLOGICAL SYSTEMS 4.2 Implications for disease The calculations have demonstrated that the flow characteristics at the distal anastomosis are strongly dependent on the ratio of the diameters of the graft and recipient artery. Therefore, it is wise to examine how the altered haemodynamic environment may influence the development of anastomotic intimal hyperplasia. As in the conventional model, the quiescent flow behaviour between the distal junction and the occlusion throughout the cardiac cycle will ensure that the residence times of any elements of blood that circulate within this region will be elevated. This should enhance the probability of the adhesion of platelets and leucocytes to the arterial wall at the heel, where the endothelium has been damaged by the surgical procedure, and thereby precipitate localised intimal thickening. Flow separation in the artery just downstream from the toe is significantly curtailed when the larger graft is employed. The washing of the fluid back and forth alongside the toe should beneficially reduce the likelihood of accumulation of blood particles in that locality. The preferential occurrence of intimal hyperplasia on the floor of the artery facing the distal anastomosis has been linked to the local, unnatural shear stress distribution. In the current bypass system, the shear stress gradients present during systole are remarkably less steep as compared to those prevailing in the conventional model. Thus, when the wider graft is attached, the stretching effect applied to the endothelial cells of the floor is not as critical, suggesting that the unwanted response of the artery wall will be less severe. It is also clear that flow disturbance at the distal anastomosis is significantly reduced when the bypass graft and host artery are of equal diameter the blood is directed more smoothly across the junction throughout systole, flow profiles within the distal channel are less strongly skewed towards the floor, flow separation and reversal downstream from the toe are less expansive and secondary motion in the recipient vessel is weakened. Another relevant issue regarding the graft/host artery diameter ratio and the associated future performance of the bypass is highlighted by in vivo experiments [19, 31, 32]. In addition to anastomotic intimal hyperplasia at the suture line, it was observed that a thin intimal lining consisting of protein, platelets and blood cell fragments grows over the inner surface of the PTFE graft as part of the healing process. The use of a wider prosthesis resulted in an increased amount of intimal hyperplasia at the anastomotic suture line and a thicker intimal layer over the graft s surface; these effects possibly regulated by the shear stress levels experienced at the wall of the graft. It was therefore surmised that an optimal graft/recipient artery diameter ratio exists a judicious choice of graft diameter could contribute to the restricting of intimal thickening at the anastomosis and along the walls of the graft, although the diameter must not be too small, causing the flow to be impeded. In conclusion, the use of the 8 mm graft may promote a slightly thicker extent of intimal hyperplasia at the suture line of the distal anastomosis. However, the bypass haemodynamics are notably improved and the shear stress distribution along the artery floor is less adverse in that case. It is therefore proposed that, of the two different conditions analysed, patency rates will be better when the graft diameter matches that of the recipient artery. 5 Taylor patch Experience has shown that autologous vein grafts give better results than prosthetic grafts when bypassing to the small vessels below the knee. However, sufficient healthy vein is not always available. Therefore, in an attempt to overcome the problem of narrowing at the graft outlet, Taylor et al [33] have advocated a modification to the conventional surgical technique for

13 VASCULAR RECONSTRUCTION: CFD PREDICTIONS OF BYPASS GRAFT HAEMODYNAMICS 13 femorodistal PTFE grafts, entailing the incorporation of a small segment of vein, known as a patch, at the distal anastomosis. This resulted in a decrease in intimal hyperplasia and improved patency rates, particularly when bypassing to the lower limb. The surgeons speculated that the gains might derive from intrinsic biological properties of the live vein, or be due to the vein bridging the compliance mismatch between the artery and the prosthetic graft. This section considers whether a haemodynamic advantage is also promoted by the revised geometry of the Taylor patch anastomosis. 5.1 Flow behaviour and wall shear stress distribution The addition of the patch at the distal anastomosis produces small but significant changes to the local flow field. It is apparent that the effect of the Taylor procedure is to reduce gradually the average velocity of the blood as it approaches the distal anastomosis, since the cross-sectional area of the bypass is steadily becoming larger. This prevents the sudden deceleration experienced by the fluid returning to the host vessel in the conventional model. For example, the influence of the Taylor design is pronounced at the maximum flow rate (Figure 10a). In that case, the decrease in the momentum possessed by fluid particles arriving at the distal anastomosis ensures that the blood is guided more smoothly through the junction. In contrast, the fluid exiting the conventional graft tends to continue across the channel due to its higher momentum and therefore, impacts more strongly on the far wall of the artery, and encourages separation at the toe. The anastomotic flow disturbances are greatly magnified throughout the systolic deceleration phase. The blood flow at the anastomosis is increasingly driven across the vessel in the direction of the far wall and separation develops distal to the toe. However, the onset of separation is later, and the zone of separation is less extensive, both across the lumen and in the downstream direction, in the Taylor model (Figure 10b). A similar, less dramatic behaviour exists in the two cases during diastole. Figure 10: Velocity vectors in the symmetry planes of the Taylor (upper plot) and conventional (lower plot) bypass models. (a) Maximum flow rate, (b) systolic deceleration.

14 14 REPAIR AND REDESIGN OF PHYSIOLOGICAL SYSTEMS wall shear stress/pa (a) maximum flow rate distance along artery/mm (b) systolic deceleration wall shear stress/pa distance along artery/mm Figure 11: Comparison of the shear stress distributions along the floor of the recipient artery for the Taylor (solid line) and conventional (dashed line) bypass models. Points opposite the heel and toe are denoted by and, respectively. (a) Maximum flow rate, (b) systolic deceleration. The general character of the shear stress distributions along the wall opposite the distal anastomosis is similar for the two systems, with the most notable spatial and temporal variations in the shearing force occurring opposite the anastomosis (Figure 11). However, it is clear that the addition of the Taylor patch affects the shear stress distribution, especially along the artery floor during systole. Firstly, the maximum shear stresses, exerted across from the toe of the anastomosis, are less extreme in the Taylor bypass. The large negative shear stresses, associated with the recirculation opposite the junction, are also weaker in the modified system. This is consistent with the observed flow patterns. The adjusted geometry of the Taylor model causes the momentum of the blood to be reduced on approach to the distal anastomosis. Therefore, in that case, as the blood flows into the recipient artery, motion across the channel is not as prevalent as in the conventional model, and a reduced shearing force is applied along the far wall. The enlarged interface between the graft and artery in the Taylor construction ensures that the heel vortex is less constricted against the vessel floor, accounting for the decreased shear stress magnitudes observed in that locality during systole. Furthermore, the steep spatial gradients in the shear stress, present along the artery floor throughout systole, are dampened when the vein patch is added to the conventional bypass system. Thus, the unnatural stretching of the endothelium opposite the junction during systole is less severe in the Taylor model. 5.2 Implications for disease In practice, the inclusion of a vein patch into the distal anastomosis of PTFE grafts leads to greatly improved long-term patency rates. The results of this numerical study have shown that the modification of the anastomotic geometry associated with the addition of the vein patch clearly influences the local flow behaviour. Therefore, it can be considered how the altered haemodynamic environment may contribute to the enhanced success rates of the Taylor bypass procedures.

15 VASCULAR RECONSTRUCTION: CFD PREDICTIONS OF BYPASS GRAFT HAEMODYNAMICS 15 Since the flow close to the heel is relatively unaffected by the incorporation of the vein patch, it is expected that a similar development of intimal hyperplasia will occur at the heel in the conventional and Taylor bypass models. In the conventional model, fluid particle residence times are elevated when flow separation arises just distal to the toe during systolic deceleration, while the migration of the vortical structure towards the toe during late systole is followed by stagnation on the graft s hood, facilitating interaction between the blood and the prosthetic surface. The progression of intimal hyperplasia at the toe should be alleviated in the Taylor model in which flow separation is reduced and where the thrombogenic graft surface has been replaced by a vein patch. With the Taylor adaptation, it is expected that the advancement of intimal hyperplasia on the artery floor will be inhibited, since a more favourable shear stress distribution prevails in that case. A bonus with the Taylor construction is the enlarged interface between the bypass and the recipient artery. Not only is it likely that the extent of intimal hyperplasia at the junction will be curtailed on account of the improved flow field, but furthermore, the intimal thickening will provide less of a restriction to the flow due to the wider opening between graft and artery. Indeed, the benefits of the Taylor model outlined above are due to the altered anastomotic geometry. Since the vein patch section was modelled as a rigid wall in the simulation, it could be that the increased success of the Taylor construction is not entirely related to the properties of the live vein. This creates the possibility of manufacturing a graft with one end already shaped in the form of a Taylor patch, and thereby also making the surgical procedure simpler. However, it must be remembered that the mismatch in the mechanical properties of a prosthetic graft and the artery is most probably one of the factors promoting intimal hyperplasia. Therefore, it would be vital to ensure that any haemodynamic gains associated with the Taylor geometry were not reversed by the use of a prosthetic material. 6 Cuffed anastomoses The addition of a vein cuff or collar at the distal anastomosis of below-knee prosthetic grafts also seems to improve patency, when compared with conventional end-to-side anastomoses [34 36]. While the presence of the vein cuff does not prevent the development of intimal hyperplasia, there is evidence that the intimal hyperplasia is located mainly at the graft cuff junction, leaving the more critical heel, toe and floor regions of the narrow recipient artery relatively spared [37 40]. This redistribution of the disease may account for the better results associated with the cuffed technique, but a mechanism for this behaviour remains unclear. This section examines what haemodynamic changes are promoted by the cuff. 6.1 Model geometry and flow conditions Two new geometries, consisting of intersecting straight tubes, were created to model the conventional and cuffed bypass anastomoses. They are representative of a femoropopliteal bypass system. In the conventional model, the recipient artery has a diameter of 4 mm and is assumed to be fully occluded, proximal to the distal anastomosis. The prosthetic bypass is of 6 mm diameter. The cross section of the graft alters gradually along its length, changing from a circular to an elliptical shape, to fit the smaller artery at the anastomosis. The anastomotic angle is 30. A similar geometry was constructed for the cuffed bypass model (Figure 12). A Miller cuff of height 10 mm and of elliptical cross section was inserted between the artery and graft. Again, all vessel walls are assumed to be rigid.

16 16 REPAIR AND REDESIGN OF PHYSIOLOGICAL SYSTEMS occluded region floor recipient artery vein cuff blood flow graft Figure 12: Symmetry plane of the cuffed bypass graft model. 400 Peak Systole volume flow rate/ml/min Sys. Accel. Sys. Decel. Diastole time/s Figure 13: Volume flow pulse in the graft. The simulation was performed under representative, physiological flow conditions [37]. As before, the non-newtonian, shear-thinning nature of blood was accounted for by a power law. A time-dependent velocity, calculated from the flow pulse (Figure 13), was specified in the graft. The mean flow rate of 180 ml/min corresponds to a graft inflow mean velocity of 10.6 cm/s or a mean Reynolds number (based on graft diameter and a reference blood viscosity of kg/m/s) of 191. The Reynolds number varies between 79 and 401 during the cycle. The pulse rate is 75 cycles/min. 6.2 Flow behaviour and wall shear stress distribution The flow patterns occurring at the cuffed anastomosis are significantly different to those observed at the conventional anastomosis. During systolic acceleration, flow separation arises at the graft heel (GH) in the cuffed bypass (Figure 14a). On passing into the recipient artery, blood turns along tightly curved paths

17 VASCULAR RECONSTRUCTION: CFD PREDICTIONS OF BYPASS GRAFT HAEMODYNAMICS 17 Figure 14: Velocity vectors in the symmetry planes of the cuffed (upper plot) and conventional (lower plot) bypass models. (a) Systolic acceleration, (b) systolic deceleration. and a separation region develops at the cuff toe (CT). Very stagnant flow exists around the cuff heel (CH) while relatively low fluid speeds exist at the graft toe (GT). By peak systole, a large, low momentum recirculation has formed along the cuff proximal wall, extending into the recipient artery. The velocity distribution in the recipient artery is strongly skewed against the floor. In contrast, a much less disturbed flow prevails throughout early systole in the conventional bypass model (Figure 14a). Separation at the toe is prevented as the blood accelerates into the small artery. A zone of slowly recirculating fluid resides over the heel in the occluded segment of the artery. The recirculation in the vein cuff becomes more substantial during the systolic deceleration phase, reaching more than halfway across the symmetry plane at the cuff artery junction (Figure 14b). Consequently, the bypass flow is increasingly swept towards the cuff toe while the stagnation point (SP) on the artery floor, marking the position of flow division, drifts downstream opposite the anastomosis throughout this phase. The small recirculation remains next to the graft toe, and a notable separation is sustained as the blood enters the artery at the cuff toe. In the conventional system, the fluid continues to proceed more smoothly through the anastomosis. The zone of slowly recirculating fluid displays only a limited growth during this time. In the early part of diastole, the recirculation in the cuff becomes weaker and less extensive. This feature subsequently disappears, leaving a relatively stagnant flow in the neighbourhood of the cuff proximal wall and cuff heel throughout the rest of the cycle. A quiescent flow behaviour is maintained throughout diastole in the conventional bypass. As in the cuffed model, increased stagnation in the locality of the heel is predicted. Of course, the three-dimensional nature of the flow must again be emphasised. In general, the secondary flow in the distal artery intensifies over systole but is diluted during diastole.

18 18 REPAIR AND REDESIGN OF PHYSIOLOGICAL SYSTEMS (a) peak systole 10 (b) systolic deceleration 10 wall shear stress/pa wall shear stress/pa Opp.Heel Opp.Toe Opp.Heel Opp.Toe distance along artery/mm distance along artery/mm Figure 15: Comparison of the shear stress distributions along the floor of the recipient artery for the cuffed (solid line) and conventional (dashed line) bypass models. (a) Peak systole, (b) systolic deceleration. The addition of the vein cuff produces some clear differences between the corresponding shear stress distributions in the vicinity of the anastomosis (Figure 15). Typically, shear stress levels opposite the toe are augmented in the cuffed bypass model. In that case, the fluid must turn more sharply on entering the artery; a more skewed velocity distribution results across the vessel with higher fluid speeds and thus, higher shear rates acting along the floor. Moreover, the steep spatial gradients in the shear stress, exerted over the floor, are more extreme in the modified system. This is a consequence of the flow being more restricted on its passage through the anastomosis due to the presence of the large vortex within the cuff. In both bypass models, the section of artery close to the heel is continually exposed to very low shear stresses due to the proximity of relatively stagnant fluid. 6.3 Implications for disease There is evidence that better results are obtained for below-knee prosthetic grafts when a vein cuff or collar is interposed at the end-to-side anastomosis. One recent study [34] indicated that the two-year patency rates for cuffed and conventional grafts to the below-knee popliteal artery were 52 and 29%, respectively. It has been advocated that these improved results are associated with an altered distribution of intimal hyperplasia at the distal anastomosis in the cuffed grafts, intimal hyperplasia develops within the vein cuff, especially at the graft cuff interface, leaving the recipient artery relatively spared [37, 38]. This redistribution of the disease away from the narrow artery to the more accommodating cuff region should delay any restenosis and occlusion in the artery. However, it is not clear how the cuff produces these benefits, although mechanical [41, 42], haemodynamic [37, 43], impedance [44], and biological [33, 45] effects have been considered. This study has examined whether the local haemodynamics will influence the progression of intimal hyperplasia and contribute to the enhanced performance of the cuffed grafts. Shear stresses on the cuff proximal wall and in the artery close to the heel are expected to be very small due to low momentum recirculation alongside. Such conditions are known to promote intimal thickening [8, 27, 46]. Therefore, the local flow patterns may account, at least in part, for the observed focal occurrence of intimal hyperplasia at the graft heel. Furthermore,

19 VASCULAR RECONSTRUCTION: CFD PREDICTIONS OF BYPASS GRAFT HAEMODYNAMICS 19 flow separation within the artery, just distal to the toe, is more significant in the cuffed graft model. This is not surprising, since the blood must turn more sharply through the junction in that case. However, this feature, in addition to secondary motions in the artery, should augment the residence times of fluid elements in the proximity of the toe, increasing the likelihood of adhesion of platelets and leucocytes to the local endothelium. Intimal hyperplasia may also be encouraged in the like manner at the graft toe where a small recirculation exists through systolic deceleration. A more adverse shear stress distribution, focused on a narrower section of the artery floor opposite the junction, is predicted when the cuff is present. These observations suggest that aspects of the anastomotic haemodynamics are worsened when a cuff is employed. It is recognised that this study considers the special case in which the recipient vessel proximal to the anastomosis is totally occluded. The ratio of the flow division between the proximal and distal segments of the recipient artery has a major bearing on the junction flow behaviour, particularly in the cuff heel region [21, 45]. There is enhanced washing of the fluid alongside the cuff heel when a fraction of the flow proceeds into the proximal artery [37, 45, 47]. Therefore, perhaps the suitability and effectiveness of a cuffed graft will depend on the outflow in the artery. However, it is unusual that the cuff toe stays relatively free from disease despite the clear evidence of the unfavourable flow separation as the blood enters the artery. It is concluded that, rather than the local haemodynamics, the presence of venous material at the anastomosis is the main contributor to the improved performance of the cuffed grafts. The findings of other investigations may support this conclusion. There is uncertainty regarding whether the advantages of the vein cuff arise from its mechanical properties [41, 42, 48, 49]. But, the insertion of the cuff provides for a technically less flawed anastomosis to the artery [34, 37], and the artery is protected from distortion at the junction [41]. Linton and Wilde [50] advise that a technically perfect anastomosis is vital to secure lasting patency, even of a vein graft. It is also possible that intrinsic biological properties of the vein [33], or the attendance of a natural, less thrombogenic surface close to the cuff artery junction [45], are important issues. These results have implications for the construction of PTFE grafts, pre-shaped to resemble the cuffed bypass configuration [37, 51]. Although the developers of such grafts recognise that PTFE grafts stimulate a more severe hyperplastic response at the distal anastomosis than do vein grafts [8], they contend that one mode of action of interposition vein cuffs is to optimise the anastomotic haemodynamics, reducing the influence of the undesirable low shear rate zones in the artery at the critical heel and toe regions, and thereby inhibiting the progression of intimal hyperplasia. Thus, since the local flow patterns are strongly affected by the geometry of the anastomosis, rather than the material from which it is fabricated, it is argued that an appropriately shaped PTFE graft may be applicable [37, 51]. This approach is dependent on possible haemodynamic benefits outweighing the disadvantages of directly suturing PTFE to the small artery, while any biological gains promoted by the vein cuff will also be lost. However, in view of the anastomotic flow behaviour revealed in the current study, caution is urged with regard to the use of pre-shaped PTFE cuff grafts. 7 Conclusions Numerical studies of the non-newtonian, pulsatile flow of blood through realistic models of human femorodistal bypass grafts have been performed. Arterial reconstructions of various designs have been investigated.