2.1 General BMHV flow characteristics

Early studies of BMHV flow characteristics showed that the flow was characterized by a triple-jet structure that emerged from the three orifices of the valve and by vortices that formed in the sinuses, between the lateral jets and the aortic wall (Bruss et al. Reference Bruss, Reul, Van Gilse and Knott1982; Yoganathan et al. Reference Yoganathan, Chaux, Gray, Robertis and Matloff1982; Chandran Reference Chandran1985). It was suggested that the reverse flows associated with these vortices were responsible for flow in the coronary arteries, which depart from two of the three sinuses in the aortic root (de Tullio et al. Reference de Tullio, Cristallo, Balaras and Verzicco2009). Shear layers and flow separation were observed downstream of the valve housing and the leaflet tips (Dasi et al. Reference Dasi, Ge, Simon, Sotiropoulos and Yoganathan2007; de Tullio et al. Reference de Tullio, Cristallo, Balaras and Verzicco2009). Dasi et al. (Reference Dasi, Ge, Simon, Sotiropoulos and Yoganathan2007) further characterized the vorticity field past a BMHV in an axisymmetric channel using planar particle image velocimetry (PIV) as well as direct numerical simulations (DNS). They found that, during the early stages of systole, the overall flow pattern was organized and repeatable from cycle to cycle, but, as the flow accelerated, von Kármán vortices rolled off of the leaflet tips and were advected downstream, causing significant cycle-to-cycle vorticity variations. Shortly following the flow peak and at the onset of deceleration, the organized vortices were broken up into non-coherent turbulence. Borazjani et al. (Reference Borazjani, Ge and Sotiropoulos2010) performed a fluid–structure interaction simulation of the flow past a BMHV in an anatomically realistic aorta geometry with prescribed physiological inlet flow conditions, whereas Le & Sotiropoulos (Reference Le and Sotiropoulos2013) expanded this work by including a model of the left ventricle. Most recently, Yun et al. (Reference Yun, Dasi, Aidun and Yoganathan2014a ,Reference Yun, Dasi, Aidun and Yoganathan b ) modelled the flow past a BMHV in an axisymmetric tube using the lattice-Boltzmann method, which resolved motions with sizes close to the Kolmogorov microscale and revealed fine vortical structures not previously found.

2.2 Indicators of blood damage

Blood cells can experience damage resulting from impact with incompatible foreign surfaces or by excessive mechanical loading (Johnston, Marzec & Berstein Reference Johnston, Marzec and Berstein1975). High levels of fluid stress imposed by prosthetic valves may cause hemolysis (the destruction of red blood cells) and, more commonly for BMHVs, activate platelets, which may initialize thrombosis (Bluestein, Rambod & Gharib Reference Bluestein, Rambod and Gharib2000). The level of damage to blood cells and platelets depends on both the magnitude of, and the exposure time to, the stress. It has been found that there is a threshold stress level below which a cell will not be damaged. For hemolysis, this threshold has been found experimentally to be between 150 and $400~\text{N}~\text{m}^{-2}$ for exposure times of the order of minutes, which are deemed to be relatively long (Ge et al. Reference Ge, Dasi, Sotiropoulos and Yoganathan2008). Platelets are much more sensitive to shear stress than red blood cells; shear stress levels of $10~\text{N}~\text{m}^{-2}$ have been found to cause platelet activation and initiate thrombosis (Ge et al. Reference Ge, Dasi, Sotiropoulos and Yoganathan2008).

Turbulent stresses have long been associated with blood damage (Smith et al. Reference Smith, Blick, Coalson and Stein1972; Stein & Sabbah Reference Stein and Sabbah1974). Also referred to as Reynolds stresses, the turbulent stresses are the components of the stress tensor $\unicode[STIX]{x1D70F}_{ij}^{\ast }=\unicode[STIX]{x1D70C}\langle u_{i}u_{j}\rangle$ , where $\unicode[STIX]{x1D70C}$ is the fluid density, $u_{i},i=1,2,3$ is the velocity fluctuation along the $x_{i}$ Cartesian axis and angle brackets denote phase averaging. Yoganathan et al. (Reference Yoganathan, Woo and Sung1986) used two-dimensional (2-D) laser Doppler velocimetry (LDV) to measure the Reynolds shear stress past a BMHV and found that high levels occurred at locations of high velocity gradients, particularly at locations immediately downstream of the valve leaflets. Ge et al. (Reference Ge, Dasi, Sotiropoulos and Yoganathan2008) argued that the turbulent stress is in fact a ‘pseudo stress’ and that it is actually the viscous shear stresses that cause damage to blood constituents. The viscous shear stress tensor is defined as

(2.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}_{ij}=\unicode[STIX]{x1D707}\left(\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{j}}+\frac{\unicode[STIX]{x2202}u_{j}}{\unicode[STIX]{x2202}x_{i}}\right), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D707}$ is the fluid viscosity. In agreement with Ge et al. numerous researchers have reported the viscous shear stress measured past a BMHV as the sole parameter relevant to blood damage (Dumont et al. Reference Dumont, Vierendeels, Kaminsky, Van Nooten, Verdonck and Bluestein2007; Borazjani et al. Reference Borazjani, Ge and Sotiropoulos2010; Wu et al. Reference Wu, Yun, Fallon, Hanson, Aidun and Yoganathan2010; Jun, Saikrishnan & Yoganathan Reference Jun, Saikrishnan and Yoganathan2013). Other researchers have reported a combined stress, determined as the sum of the Reynolds shear stress and the viscous shear stress (Bluestein, Li & Krukenkamp Reference Bluestein, Li and Krukenkamp2002; Balducci et al. Reference Balducci, Grigioni, Querzoli, Romano, Daniele, D’Avenio and Barbaro2004; Xenos et al. Reference Xenos, Girdhar, Alemu, Jesty, Slepian, Einav and Bluestein2010). Although logic dictates that turbulent stresses cannot directly damage blood elements, clinical and experimental evidence shows that turbulent stress level is better correlated with cell damage than viscous shear stress level, which is smaller by orders of magnitude (Kameneva et al. Reference Kameneva, Burgreen, Kono, Repko, Antaki and Umezu2004). To explain this dichotomy, it was suggested by Antiga & Steinman (Reference Antiga and Steinman2009) that, although turbulent stresses do not physically act on cells, the associated turbulent fluctuations cause an increase in the local instantaneous viscous shear stress that the cells experience. These authors estimated by an order of magnitude analysis that the true viscous shear stress experienced by cells in a turbulent flow would be intermediate between the nominal viscous shear stress and the Reynolds shear stress. In addition to high levels of fluid stress, the presence of regions of slow flow and recirculation may also increase the risk of thrombosis, as these regions cause prolonged contact between potentially activated platelets and proteins involved in coagulation. For example, Bluestein et al. (Reference Bluestein, Rambod and Gharib2000) suggested that the vortices that are shed from the leaflet tips during flow acceleration provide favourable conditions for thrombosis: they cause platelets that have been exposed to high shear stresses – and may have become activated as a result – to become trapped alongside proteins involved in coagulation, cultivating the formation of thrombi. In view of past experience, it seems appropriate for current studies of BMHV to examine the distributions of both viscous shear stresses and turbulent stresses as possible indicators of relative risk to blood integrity.

2.3 Effects of BMHV orientation

Several studies have shown that BMHV valve orientation affects markedly the flow characteristics downstream of the valve. Akutsu et al. (Reference Akutsu, Saito, Imai, Suzuki and Cao2008) investigated the effect of BMHV orientation within an anatomic-like sinus in a pulse duplicator. The sinus consisted of three cusps, without coronary arteries, and converged to a straight tube at the end of the cusps. Using PIV, they examined two orientations each for three BMHV designs: the St Jude Medical (SJM), the On-X and the Edwards MIRA valve. They found that, for all three valves, the forward flow and the sinus recirculation flow were stronger when the valve symmetry plane bisected the non-coronary cusp compared to when the valve was rotated by $90^{\circ }$ from that. In a subsequent study, Akutsu, Matsumoto & Takahashi (Reference Akutsu, Matsumoto and Takahashi2011) measured the velocity past the same three BMHV models in a channel with a triple sinus and with outlets representing the coronary arteries. They measured the velocity in the coronary arteries with an ultrasonic flow meter and found that the coronary flow rate correlated with the strength of recirculating flow in the sinuses; for two of the three models, the coronary flow was higher when the valve symmetry plane was aligned with one of the three sinuses than that when it was rotated by $90^{\circ }$ from that.

Kleine et al. (Reference Kleine, Perthel, Nygaard, Hansen, Paulsen, Riis and Laas1998) implanted SJM bileaflet and Medtronic Hall tilting disc valves into four healthy pigs (each valve model in two pigs), using a device that allowed rotation of the valve without reopening the aorta. They measured the normal Reynolds stress using pulsed Doppler ultrasonography and found that the turbulence level downstream of the tilting disc valve was significantly lower when the major orifice was oriented towards the right-posterior aortic wall, which experienced the highest velocities during ejection. The SJM bileaflet valve also had lower turbulence when one orifice was oriented towards this region, however the effect of valve orientation was not as strong. Subsequently, Kleine et al. (Reference Kleine, Scherer, Abdel-Rahman, Klesius, Ackermann and Moritz2002) investigated the effect of valve orientation on left coronary flow and, in addition to the Medtronic Hall and SJM valves, also tested the Medtronic Advantage bileaflet valve. They measured the flow rate in the left anterior descending coronary artery for two orientations of each valve and found that, for the bileaflet valves, the left coronary flow was higher when one of the lateral orifices was aligned with the right coronary cusp than for the orientation perpendicular to it. This finding indicated that, because the majority of coronary flow occurs during diastole while the valve is closed, the valve orientation would affect the downstream flow field even during that phase. These authors also reported that the coronary flow rate was less sensitive to valve orientation for the SJM valve model than for the two other models.

Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2010) used a curvilinear-immersed boundary fluid–structure interaction solver to simulate the flow past a BMHV within an anatomically shaped aorta for three valve orientations which differed from each other by 45 degrees. They found that the valve orientation did not affect significantly the shear stress distribution in the aorta, but affected the symmetry of leaflet closure; the leaflets closed most synchronously when the valve was implanted with its symmetry plane parallel to the plane of curvature of the ascending aorta, and they closed most asynchronously when implanted perpendicular to the plane of curvature; they attributed this effect to the curvature-induced pressure gradient.

Figure 2 illustrates the fact that the BMHV orientations recommended by previous authors for optimizing various parameters varied widely. The orientation for maximising coronary flow found by Akutsu et al. (Reference Akutsu, Matsumoto and Takahashi2011) was close to the one recommended by Kleine et al. (Reference Kleine, Scherer, Abdel-Rahman, Klesius, Ackermann and Moritz2002), namely with one lateral orifice facing the right coronary cusp, or the line of symmetry intersecting the non-coronary cusp; this differed from the orientation recommended by Kleine et al. (Reference Kleine, Perthel, Nygaard, Hansen, Paulsen, Riis and Laas1998) for minimising turbulence, in which one of the valve’s lateral orifices faced the right-posterior wall. On the other hand, Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2010) found that valve orientation had a negligible effect on the viscous stresses past the valve, but recommended that the valve be oriented in line with the direction of aorta curvature for more synchronous valve kinematics and potentially less regurgitant flow; the latter orientation differs from those recommended by other authors.

Figure 2.

‘Best’ BMHV orientations, as recommended by previous studies, with respect to the aortic root geometry; lines indicate the corresponding valve planes of symmetry; abbreviations as in figure 1.

The previous results suggest that optimal valve operation with respect to different flow parameters would correspond to different valve orientations; for example, one orientation may be best for reducing fluid stresses, while another may be best for increasing coronary flow. In this case, a surgeon would be required to choose the best orientation for each patient’s specific condition. Each of the studies discussed in this section had inherent limitations. The model of Akutsu et al. (Reference Akutsu, Matsumoto and Takahashi2011) did not include the aorta’s curvature; to our knowledge, there has not yet been an experimental study of a BMHV within a fully anatomical model of the aorta. Measurements of turbulent stresses made by Kleine et al. (Reference Kleine, Perthel, Nygaard, Hansen, Paulsen, Riis and Laas1998) were limited to several discreet points. That study, as well as their later one of coronary flow (Kleine et al. Reference Kleine, Scherer, Abdel-Rahman, Klesius, Ackermann and Moritz2002), used only two animals for each valve model and had no measure of the flow patterns inside the aorta to explain the results. The model of Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2010) did not include the coronary arteries and coronary flow.

The present study attempts to overcome some of these limitations. An experimental study of BMHV orientation was made within an anatomical aorta model, under controlled and repeatable physiological flow conditions. The effects of valve orientation on turbulent stress, viscous stress and coronary flow were determined from 2-D PIV measurements made on various planes in three orientations with respect to the aorta geometry. The flow patterns past the valve were examined and used to explain the results.

6.1 Planar velocity maps

Figure 9.

Phase-averaged velocity $q_{xz}$ on the $x$ – $z$ centreplane at phases B (a,c,e) and C (b,d,f), for normal (a,b), aligned (c,d) and inclined (e,f) valve orientations. Red dashed lines indicate the $x$ – $z$ centreplane, viewed from downstream.

The phase-averaged velocity maps on the $x$ – $z$ centreplane at mid-acceleration (phase B) and peak flow (phase C) are shown in figure 9 for the three valve orientations.

For the normal valve orientation, the $x$ – $z$ plane shows the three jets (see also figure 8), which at mid-acceleration (phase B) extended into the aortic root. At this phase, the lateral jets were more developed and stronger than the central jet and their axes were inclined towards the sinus walls, with the lateral jet on the $x^{+}$ side deflecting off the channel wall above the sinus; recirculation vortices are seen to form between the lateral jets and the sinus walls.

For the aligned valve orientation, the $x$ – $z$ plane was parallel to the valve symmetry plane and contained the profile of the central jet. At mid-acceleration, this jet was seen to emerge past the valve leaflets into the aortic root. The velocity in the sinuses was very low – lower than the value for the normal orientation at this phase. At peak flow (phase C), the profile of the central jet was very narrow. Its axis was aligned with the $z$ -axis up to approximately $z/D=1.25$ , beyond which it curved slightly following the aorta curvature. At this location, a second velocity peak, lower in magnitude than the peak of the central jet, formed near the concave channel wall. This double-peaked profile was similar to the one that developed past the valve on the $y$ – $z$ plane for the normal orientation, as seen in figure 8.

For the inclined valve orientation, the $x$ – $z$ centreplane was oblique to the valve symmetry plane and its central orifice and intersected the corners of the valve’s lateral orifices, as illustrated by figure 6(c). The flow issuing from these corners is shown on the $x$ – $z$ plane as the narrower, shorter jets located on either side of the central jet profile (figure 9).

Figure 10.

Phase-averaged axial velocity profiles on the $x$ – $z$ centreplane at (a) $z/D=0.65$ and (b) $z/D=1.60$ for normal, inclined and aligned valve orientations at peak flow.

Figure 10 shows profiles of the mean axial velocity $\langle W\rangle$ in the $x$ -direction for the three valve orientations at $z/D=0.65$ and 1.60. The profiles at $z/D=0.65$ , which was approximately two-thirds down the length of the sinus, reveal that, for all three valve orientations, the reverse velocity in the right coronary sinus (on the $x^{-}$ side) was much stronger than in the left coronary sinus. Because this asymmetry appears to be independent of valve orientation, it is likely caused by the aorta curvature or other asymmetries in the anatomical geometry. The aligned valve orientation did, however, result in slightly stronger reverse flow in the left sinus, compared to the two other orientations. Further downstream, at $z/D=1.60$ , the triple jets of the normal orientation were less pronounced than at $z/D=0.65$ , as they merged and spread; they were also slightly skewed toward the $x^{-}$ side as a result of the aorta’s curving toward the opposite direction.

The phase-averaged velocity maps for the three valve orientations on the $y$ – $z$ centreplane at phases B and C are shown in figure 11. As was shown in figure 8, the $y$ – $z$ centreplane contained the cross-section of the central jet for the normal orientation. For the aligned valve orientation, the $y$ – $z$ plane intersects all three jets. At phase B for this orientation, the lateral jet on the $y^{+}$ side appears to be deflected off of the channel wall as was observed for the normal valve orientation at the same phase (as shown by the $x$ – $z$ plane in figure 9). At phase C, the axes of the three jets were aligned with the $z$ -axis and all jets appeared to be symmetric.

For the inclined orientation, the map on the $y$ – $z$ plane shows oblique cross-sections of the three jets, which were similar to those shown for the aligned orientation on this plane. In contrast to the velocity maps on the $x$ – $z$ plane for this orientation, in which the central jet was relatively wide and the lateral jets were very narrow, the jet profiles on the $y$ – $z$ plane were all comparable in width.

At mid-acceleration, the recirculating flow in the non-coronary sinus was stronger for the aligned and inclined orientations, where it was adjacent to the lateral jets, than for the normal orientation, where it was adjacent to the central jet. This pattern, however, was reversed at peak flow, when the sinus flow was stronger for the normal orientation than for the two other orientations.

Figure 11.

Phase-averaged velocity $q_{yz}$ on the $y$ – $z$ centreplane at phases B (a,c,e) and C (b,d,f) for normal (a,b), aligned (c,d) and inclined (e,f) valve orientations. A circular area located at the top of the non-coronary sinus was masked from viewing because this area was optically blocked by the right coronary artery, which exited the model there.

6.2 Turbulent stress

The maximum turbulent stress $\unicode[STIX]{x1D70F}_{max}^{\ast }$ is shown in figure 12 for the three valve orientations at peak flow on the $x$ – $z$ and $y$ – $z$ centreplanes. In general, regions of high turbulent stress occurred at the outer edges of the lateral jets and, to a lesser degree, between the lateral jets and the central jet (shown in planes perpendicular to the valve symmetry plane) as well as at the sides of the central jet (shown in planes parallel to the valve symmetry plane).

For each valve orientation, average values of $\unicode[STIX]{x1D70F}_{max}^{\ast }$ at peak flow were calculated by taking into account the measurements on both the $x$ – $z$ and $y$ – $z$ planes within the ranges $z/D=0.2$ to 1.5, $x/D=-0.8$ to 0.83 ( $x$ – $z$ planes) and $y/D=-0.85$ to 0.7 ( $y$ – $z$ planes). The average value for the normal valve orientation was 21 % higher than for the aligned one (18.5 Pa versus 15.3 Pa). The average value for the inclined orientation was 16.9 Pa, however, because the measurement planes were not orthogonal to the valve’s symmetry as was the case for the two other orientations, the turbulent stress levels for the inclined orientation cannot not be compared directly to those for the normal and aligned orientations.

Figure 12.

Turbulent stress $\unicode[STIX]{x1D70F}_{max}^{\ast }$ at peak flow for valve orientations: normal (a,b), aligned (c,d) and inclined (e,f), in the $x$ – $z$ centreplane (a,c,e) and the $y$ – $z$ centreplane (b,d,f). Red arrows connect the planes that were perpendicular to the valve’s symmetry plane and the planes that were parallel to it. The region for the inclined orientation marked with cross-hatching was believed to be strongly affected by error, attributed to loss of optical access, and was not considered in the analysis.

More specifically, the turbulent stress in the direction perpendicular to the valve’s symmetry plane was higher for the normal than for the aligned orientation at the outer edges of the lateral jets, with the corresponding peak value being 10 % higher (92 Pa versus 84 Pa). These regions of relatively high stress were also somewhat more extensive for the normal orientation. The more significant increase in turbulent stress for the normal orientation compared to the aligned one, however, was on planes parallel to the valve’s symmetry plane at the edges of the central jet and, even more so, in the spaces between the central jet and the channel walls.

For the inclined valve orientation, two streaks with relatively high turbulent stress levels are shown on the $x$ – $z$ plane. These streaks corresponded to flow emerging from the corners of the valve’s lateral orifices. The turbulent stress in these regions was even higher than that at the outer edges of the lateral jets for the normal orientation (peak values of 205 Pa versus 92 Pa). Both streaks extended to the end of the sinuses. The high stress level in these streaks is attributed to separation from the valve housing and the leaflets, which were very close together on this plane.

Figure 13.

Averaged $\unicode[STIX]{x1D70F}_{max}$ at peak flow for valve orientations: normal (a,b), aligned (c,d) and inclined (e,f), in the $x$ – $z$ centreplane (a,c,e) and the $y$ – $z$ centreplane (b,d,f). Red arrows connect the planes that were perpendicular to the valve’s symmetry plane and the planes that were parallel to it.

Measurements near the base of the aortic root just past the valve on the $y$ – $z$ plane for the inclined orientation are believed to be strongly affected by error, attributed to loss of optical access, and therefore will not be considered in the analysis. The $y$ – $z$ plane reveals a turbulent stress distribution similar to that measured for the aligned orientation on the same plane, but the stresses were slightly lower: excluding the masked region, the average and peak stresses were, respectively, 16 % and 25 % lower than those for the aligned orientation.

6.3 Viscous shear stress

Figure 13 shows the phase-averaged viscous shear stress $\unicode[STIX]{x1D70F}_{max}$ at peak flow for the three valve orientations on the $x$ – $z$ and $y$ – $z$ centreplanes. Normal to the valve’s plane of symmetry (the $x$ – $z$ plane for the normal orientation and the $y$ – $z$ plane for the aligned orientation), high levels of viscous shear stress existed at the shear layers of the jets; such shear layers formed between the lateral jets and the valve housing, between the lateral jets and the valve leaflets and between the central jet and the leaflets.

To show the distributions of the instantaneous levels of shear stress, histograms were generated from the instantaneous measurements for both the normal and aligned valve orientations. The measurements from both the $x$ – $z$ and $y$ – $z$ centreplanes were used to generate the histogram for each orientation, which was then normalized by the area beneath them to obtain the probability density function (PDF), denoted $f(\unicode[STIX]{x1D70F}_{max})$ . It has been previously shown that viscous shear stress values as low as 10 Pa can cause damage to platelets (Ge et al. Reference Ge, Dasi, Sotiropoulos and Yoganathan2008). The PDFs, shown in figure 14, indicate that shear stress levels exceeding this threshold occurred more frequently for the normal valve orientation than for the aligned one; the percentages of measurements that exceeded 10 Pa were 1.7 % and 1.3 % for the normal and aligned valve orientations, respectively.

Figure 14.

Normalized histogram of $\unicode[STIX]{x1D70F}_{max}$ for normal and aligned valve orientations from the $x$ – $z$ and $y$ – $z$ centreplanes (a) with a zoomed-in view about the threshold for platelet damage (b).

A PDF was not made for the inclined valve orientation because, as was explained for the turbulent stress measurements, the measurement planes were not orthogonal to the valve’s symmetry as was the case for the other two orientations, thereby making a direct comparison of the results inappropriate.

Maps of phase-averaged viscous shear stress (figure 13) indicate that the viscous stress was higher for the normal orientation than for the aligned orientations on planes both perpendicular and parallel to the valve’s symmetry plane. Perpendicular to the valve’s symmetry plane the peak shear stress level at the base of the shear layers was 7 % higher for the normal orientation than for the aligned orientation (8.8 Pa versus 8.2 Pa).

From the maps of phase-averaged viscous shear stress for the inclined orientation (figure 13), streaks of high viscous shear stress were shown on the $x$ – $z$ centreplane, corresponding to the flow emitted from the corners of the lateral orifices. These stresses, however, were not as high as at the outer shear layers of the lateral jets, shown on planes normal to the valve symmetry plane. In other words, the flow from these corners exhibited extremely high levels of turbulence and moderately high levels of viscous shear stress.

8.1 Effects of valve orientation on flow distribution

It was found that, on the $x$ – $z$ centreplane, the reverse flow in the left coronary sinus was greater for the aligned valve orientation than for the normal and inclined orientations. This may be because in the normal orientation the lateral jets spread outward into the sinuses, leaving less room for recirculation. Comparatively, in the aligned orientation the valve’s central orifice was roughly adjacent to the left sinus – the central jet was narrower than the lateral jets and left more room for sinus recirculation.

From the measurements on the $y$ – $z$ plane for the normal orientation, it was shown that the low velocity region of the non-coronary sinus extended far into the ascending aorta. In vivo, this non-physiological low flow region could foster thrombus growth by increasing the contact time between potentially activated platelets and proteins involved in coagulation. Furthermore, by altering the viscous shear stress distribution at the adjacent vessel walls, it may promote arterial disease (Paszkowiak & Dardik Reference Paszkowiak and Dardik2003). For the aligned and inclined orientations, however, the field of view was limited to the aortic root and did not include the ascending aorta. As a result, it was not possible to determine the extent of the low velocity region for these valve orientations. It was interesting to note, however, that the velocity in the non-coronary sinus was even lower for both of these orientations than for the normal one. One may postulate that, because the lateral velocity components of the three jets were very small at peak flow, the low flow region would extend fairly deeply into the ascending aorta. As was shown in figures 9 and 11, however, the fluid velocities near the channel walls were typically higher on planes perpendicular than on planes parallel to the valve’s plane of symmetry, due to the spread of the lateral jets. As a result, the low velocity region (measured on the $y$ – $z$ centreplane) would not likely extend as far for the aligned and inclined orientations as it did for the normal valve orientation.

8.2 Effects of valve orientation on turbulent and viscous stresses

The turbulent stresses were on average 21 % higher for the normal valve orientation than for the aligned orientation. Additionally, the normal valve orientation resulted in 31 % more occurrences of viscous shear stresses that were high enough to potentially activate platelets. These results suggest that the aligned orientation may be preferable over the normal one from the viewpoint of minimizing the stresses acting on blood elements, and reducing the potential for thrombosis.

The previous suggestion is in contrast with the conclusion of Borazjani et al. (Reference Borazjani, Ge and Sotiropoulos2010) that the valve orientation had no significant effect on the viscous shear stress downstream of the valve. These authors, however, also recommended the aligned orientation, but on the grounds that it resulted in the most synchronous valve closure and lowest leaflet rebound, which they attributed to the near uniformity of pressure on the leaflets for the aligned orientation; for other orientations, the two leaflets were unequally affected by the pressure gradient imposed by the aorta curvature. Borazjani et al. (Reference Borazjani, Ge and Sotiropoulos2010) also postulated that the aligned configuration would be less susceptible to intermittent regurgitation.

Kleine et al. (Reference Kleine, Perthel, Nygaard, Hansen, Paulsen, Riis and Laas1998) found that, from measurements made in two pigs, the Reynolds normal stresses were lowest when a BMHV was oriented with one lateral orifice toward either the posterior wall or the right-posterior wall of the aorta; these orientations were close to, respectively, the aligned and inclined orientations of the present study. This result was fairly consistent with that of the present study.

It would be of interest to compare these findings with clinical outcomes for varying BMHV orientations, however, to our knowledge, such data have not yet been published. Furthermore, thromboembolic rates may be a poor indicator of valve performance, because they can be highly influenced by patient risk factors and antithrombotic management (Sotiropoulos et al. Reference Sotiropoulos, Le and Gilmanov2016).

Figure 12 shows that the largest differences in turbulent stress for the two orientations occurred in the shear layers in the wake of the valve housing. Specifically, the turbulent stress at the outside edge of the $x^{-}$ lateral jet in the normal orientation was much higher and extended over a larger area than in any of the shear layers for the aligned orientation. This was likely a consequence of the lateral jet impingement on the concave wall. Conversely, when the valve was in the aligned orientation, there was more space between the jets and the concave wall, and the jets did not impinge on it.

Another significant difference in the turbulent stresses was shown on the valve symmetry plane: in the normal orientation, high levels of turbulent stress developed in the space between the central jet and the non-coronary sinus (on the $y$ – $z$ centreplane). This did not occur for the aligned orientation, for which the turbulent stresses were very low on the valve symmetry plane (the $x$ – $z$ centreplane). Figure 19 shows the velocity projections on the valve symmetry planes for the normal and aligned orientations at peak flow and $z/D=0.40$ . For the normal orientation, the velocity profile on the $y^{-}$ side of the central jet protruded, forming a local velocity peak between the central jet and the reverse flow in the non-coronary sinus. This resulted in a steep velocity gradient between this protrusion and the reverse flow, which increased the production of turbulence there. In the aligned orientation, on the other hand, the velocity transition between the central jet and the reverse sinus flows was much more gradual. The asymmetric protrusion of the velocity profile observed for the normal orientation may have resulted from the asymmetry of the sinuses with respect to the central jet on the valve symmetry plane. In contrast, for the aligned orientation, the channel walls of the left and right sinuses were nearly symmetric about the central jet profile.

Figure 19.

Velocity projections on the planes parallel to the valve symmetry plane for normal (orange arrows) and aligned (blue arrows) valve orientations at peak flow and $z/D=0.40$ ; as a scale for velocity magnitudes, the figure shows a black arrow with a length equivalent to $1~\text{m}~\text{s}^{-1}$ .

Coronary artery disease has been found to be an appreciable cause of morbidity following aortic valve replacement, while 34 % of patients having valve replacement for aortic valve stenosis also have significant coronary artery disease (Exadactylos, Sugrue & Oakley Reference Exadactylos, Sugrue and Oakley1984). Previous studies have found that the formation and development of atherosclerotic plaques can be predicted by deviations in the physiological patterns of wall shear stress; specifically, complex flows that result in low level or oscillating wall shear stress have been found to be pro-atherogenic (Gibson et al. Reference Gibson, Diaz, Kandarpa, Sacks, Pasternak, Sandor, Feldman and Stone1993; Friedman, Krams & Chandran Reference Friedman, Krams and Chandran2010). Such flows, which can be found in bifurcating and curved arteries (Friedman et al. Reference Friedman, Krams and Chandran2010), may also occur in the flow separation region observed at the downstream side of the right coronary ostium. To reduce the risk of plaque formation there, it is desirable to decrease the size and duration of this flow separation. The area of flow separation appeared largest for the normal valve orientation and smallest for the inclined valve orientation, indicating that the inclined orientation would cause the least risk for plaque development there.

8.3 Effects of valve orientation on right coronary flow

As was previously documented (Yoganathan, He & Jones Reference Yoganathan, He and Jones2004), the lateral jets emerged earlier than the central jet during flow acceleration. This was also apparent when comparing the coronary flows on the $x$ – $z$ plane for the different valve orientations (figure 15). Consequently, the portion of the sinus vortex that was adjacent to the lateral jet advanced earlier than the portion that was adjacent to the central jet. As a result, when a lateral valve orifice was oriented toward the coronary artery (i.e. in the normal orientation), the sinus vortex blocked the coronary ostium earlier than for the other valve orientations. This caused the coronary flow rate during flow acceleration to be much lower than for the other orientations.

Figure 20.

Sketch of inferred vortex ring shape without the effects of valve orientation.

At mid-acceleration, the aligned orientation resulted in the highest coronary flow rate. In addition to the delayed advance of the sinus vortex, the coronary flow for this orientation was augmented by the high velocity fluid of the jet adjacent to it, part of which rolled over the sinus vortex into the artery. For the inclined orientation, the coronary flow also benefited from the delayed sinus vortex; however, the adjacent fluid emitted from the valve was not as fast and, consequently, the coronary flow rate was slightly lower than for the aligned orientation.

During flow deceleration, the coronary flow was highest for the inclined valve orientation. This appears to have been a direct result of the size, shape and orientation of the sinus vortex. For both the normal and aligned orientations, the sinus vortex next to the coronary ostium was confined by the adjacent lateral jet, which was aligned with the axial direction. For the inclined orientation, however, the fluid velocity in the aortic root had a significant transverse component directed away from the coronary ostium. As a result, the sinus vortex was confined by a lesser amount, and it entrained more fluid from the channel centre. Additionally, this caused the long axis of the vortex to be inclined more than for the other orientations, which permitted the fluid from it that entered the coronary artery to be more in line with the artery’s axis. Consequently, there was a weaker flow separation at the artery entrance, and a larger portion of the artery contained forward moving fluid.

The measurements presented here reveal information about the three-dimensional shape and time evolution of the sinus vortex ring. The vortex had a larger cross-sectional diameter on the $x$ – $z$ plane at $y/D=0.165$ , where the right coronary sinus was deepest, than on the $x$ – $z$ centreplane, which did not extend as far into the sinus. This was expected as it is presumed that the vortex would occupy the space between the valve housing and the channel wall, which is larger in the depths of each sinus than at the commissures between them. We can infer that the sinus ring would therefore be thicker in all three sinuses and thinner at the locations between them, taking on a trimodal shape similar to the perimeter of the aortic root. This inferred shape is depicted in figure 20. The cross-sectional diameter of the sinus ring was further affected by the valve geometry; it was larger in parts that were adjacent to the central orifice than in parts that were adjacent to the lateral orifices, where it was confined by the outward-spreading lateral jets. The largest vortex ring diameter was thus achieved when the valve’s line of symmetry was in line with the right coronary ostium (namely, the inclined orientation); in this orientation the portion of the vortex ring located within the right sinus was also adjacent to the valve’s central orifice. As we saw in § 7, this configuration resulted in the largest forward flow volume in the contiguous right coronary artery. Furthermore, the measurements illustrated that the sinus ring was advected downstream earlier in parts adjacent to the lateral orifices than parts adjacent to the central orifice. This would likely cause the trimodal-shaped ring to bend in a ‘V-shape’ about the valve’s plane of symmetry, in a fashion similar to the one reported by Dasi et al. (Reference Dasi, Ge, Simon, Sotiropoulos and Yoganathan2007).

It must be noted that the results from the measurements of coronary flow have limited relevance to physiological conditions. In vivo, the majority of coronary flow occurs during diastole, because the contraction of the heart during systole compresses the coronary arteries, increasing their flow resistance. During diastole, while the bulk flow rate in the aorta is nearly zero, the heart muscle is relaxed and the coronary arteries dilate, causing blood to flow through them, a process made possible by arterial compliance which serves as a flow reserve. This effect was not accounted for in the present mock circulatory apparatus; to do so, an active, timed resistance would have had to be applied at the outlets of the coronary arteries, which would have been considerably difficult to implement in our set-up and so has been deferred to a specific future investigation. Because in this study we applied a passive resistance that was constant over the course of the cycle, the majority of the coronary flow occurred during systole, rather than during diastole. The reverse flow in the coronaries at the end of systole may be attributed to the local adverse pressure gradient that builds up at this time. This reverse flow would likely have been accentuated by imperfect sealing of the BMHV during valve closure (Dasi et al. Reference Dasi, Murphy, Glezer and Yoganathan2008), which would expose the fluid at the coronary inlets to the suction upstream of the valve. Moreover, gravity may have likely contributed to the draining of the coronaries, as our model was oriented such that the anatomical frontal plane was vertical.

Despite this limitation, our work has provided some useful and novel information. First, we have confirmed that the size, shape and space–time evolution of the vortex ring depends strongly on BMHV orientation, and described details of this dependence. Moreover, we have demonstrated that, in the absence of intramyocardial pressure, the coronary flow rate depends strongly on the size, shape and space–time evolution of the sinus vortex ring. These observations may be considered towards predicting the dependence of coronary flow on BMHV orientation when physiological, time-varying resistances are imposed. We may plausibly assume that, even when subjected to physiological resistance, the sinus vortex would still have some, albeit somewhat lessened, influence on the coronary flow. The present results reveal the complex interactions between the flow in the aortic root with those in the coronary arteries without the added effect of vascular resistance caused by intramyocardial pressure. They provide a first step in understanding the physiological flow interactions and may be used in future studies, which account for the time-varying vascular resistance, as a basis for comparison in order to understand those effects. Such studies with more realistic coronary flow are being considered for future work.

Although the flow rate in the left coronary artery was not measured, it may be presumed that the inclined valve orientation, which was optimal for the right coronary flow, would not be optimal for the flow in the left coronary artery. When the valve’s symmetry plane is aligned with the right coronary ostium (i.e. for the inclined orientation), one of the valve’s lateral orifices would be positioned next to the left coronary ostium – for the right coronary artery, this arrangement resulted in the lowest coronary flow rate. One may then speculate that the aligned valve orientation would be optimal for the total flow in both coronary arteries; this suggestion may be justified by the facts that the aligned orientation resulted in the second highest right coronary flow rate (after the inclined orientation), and a symmetric positioning of the ostia with respect to the valve symmetry plane, which means that the right and left arterial flows would be equally affected. This hypothesis is supported by the centreplane velocity measurements presented in § 6.1, which showed that the aligned orientation resulted in the strongest reverse flow in the left coronary sinus (figure 10) and consequently the strongest left coronary flow. In case, however, the flow in one coronary artery was for some reason deficient, it could potentially be boosted by implantation of the valve so that its symmetry plane would be aligned with the axis of that artery.

Kleine et al. (Reference Kleine, Scherer, Abdel-Rahman, Klesius, Ackermann and Moritz2002) found, in contrast, that the flow in the left coronary artery of pigs was highest when the BMHV was oriented with one lateral orifice toward the right coronary cusp. This orientation was comparable to our ‘normal’ orientation which resulted in the lowest right coronary flow, but may have positioned the central orifice towards the left coronary, which would be consistent with the present findings. This shows further evidence of a need to test the effect of BMHV orientation on the flows in both coronary arteries simultaneously. Furthermore, an analysis of clinical outcomes for varying BMHV orientations from a large patient sample would be valuable for validating these findings.