f

Can vortices in the flow across mechanical heart valves contribute to

cavitation?

I. Av rahami I M. Rosenfeld I S. Einav I M. Eichler 2 H. Reul 2

1Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel 2Helmholtz Institute, Aachen, Germany

Abstract--Cavitat ion in mechanical heart valves is tradit ionally attributed to the hammer effect and to squeeze and clearance f low occurring at the moment of valve closure. In the present study, an addit ional factor is considered--the contribu- tion of vortex flow. Using a computat ional f luid dynamics analysis of a 2D model of a t i l t ing disk mitral valve, we demonstrate that vortices may form in the vicinity of the inf low side of the valve. These vortices roll up from shear layers emanating from the valve tips during regurgitation. A signif icant decrease in the pressure at the centre of the vortices is found. The contribution of the vortex to the total pressure drop at the instant of closure is of the order of 70mmHg. Adding this figure to the other pressure drop sources that reach 670mmHg, it might be that this is the deciding factor that causes the drop in blood pressure below vapour pressure. The total pressure drop near the upper tip (750mmHg) is larger than near the lower tip (670 mmHg), indicating a preferential location for cavitation inception, in agreement with existing experimental findings.

Keywords--Mechanical heart valves, Flow field, Cavitation, Vortices, Navier-Stokes equations, CFD

Med. Biol. Eng. Comput., 2000, 38, 93-97

J

1 In t roduct ion

BASED ON the finding of pitting and erosion marks on the surface of several failed mitral mechanical heart valves (MHV), cavitation was suspected to be a possible cause of valve failure (GRAF et al., 1991). it is a physical phenomenon in which bubbles are formed in the flow field when the pressure decreases below the saturated vapour pressure. The bubbles explode a few microseconds after being generated, forming high-speed jets that may damage the valve surface (visible by erosion marks) or be deleterious to the blood (haemolysis).

The conditions leading to in-vitro cavitation were studied by numerous investigators (GRAF et al., 1994; KAFESJIAN et al., 1994; GARRISON et al., 1994; SHU et al., 1994; LEE et al., 1996; CHANDRAN and ALURI, 1997). These in-vitro observa- tions showed that the preferred regions of cavitation bubbles are on the inflow side of the valve, at the tip of the major orifice of tilting disk valves ( 12 o'clock position) and in the peripheral edges of bileaflet valves. Cavitation bubbles were identified experimentally only during the very short period of the final stage of the valve closure. Single disk valves were found to be more susceptible to the generation of cavitation bubbles than bileaflet valves (SHU et al., 1994).

Even though the flow conditions necessary for in-vitro generation of cavitation are more severe than those found in physiological conditions, they do not rule out the possible existence of a comparable in-vivo phenomenon (GRAF et al.,

Correspondence should be addressed to Dr M. Rosenfeld; e-mail: rosenf@eng.tau.ac.il

First received 9 March 1999 and in final form 24 August 1999

© IFMBE:2000

Medical & Biological Engineering & Computing 2000, Vol. 38

1991). Assuming the hammer effect to be the primary cavita- tion generation mechanism (GRAF et al., 1991 ), several experi- mental studies (GRAF et al., 1994; SHU et al., 1994; LEE et al., 1996) attempted to correlate cavitation to maximum d P / d t , where P is the pressure in the left ventricle and t is the time. Thresholds were suggested ( d P / d t > 2000 mmHg/s) for several valve types above which cavitation was likely to appear. In more recent studies on implanted MHV, it was shown that the d P / d t loading calculated only during the closure time is a more meaningful quantity for predicting cavitation (CHANDRAN et al., 1998).

Cavitation can also appear in regions of high velocity that are characterised by large pressure drops. During the closing of the valve, the blood velocity can reach very high values. The maximum velocity was found experimentally to be in the range o f 2 -5 m/s (C HEON and C HANDRAN, 1994 ) because the flow is squeezed into a very narrow passage, in addition to being subjected to the high closing velocity of the valve itself (BLUESTEIN et al., 1994; MAKHIJANI et al., 1996). The generation of cavitation in squeezed flows was studied experi- mentally by Wu et al. (1994) and LEE et al. (1996) and numerically by BLUESTEIN et al. (1994) and MAKHIJANI et al. (1996).

The flow in the clearance between the valve leaflet and the housing was studied numerically by LEE and CHANDRAN (1994, 1995). They imposed an experimentally measured pressure drop on a steady 2D model of the clearance flow for clearance sizes of 0.05 and 0.10 mm. Their pressure measure- ments showed a relatively large pressure drop between the entrance and exit of the clearance for ~ 0.5 ms at the moment of valve closure. The numerical simulations also resulted in relatively large pressure reduction within the clearance.

93

Another possible mechanism that might promote the genera- tion of cavitation bubbles through the increase of the velocity and decrease of the pressure is vortex formation, it is well known that pressure decreases towards the centre of vortices. The stronger the vortex, the lower will be the pressure at the centre. Indeed, vortices were observed experimentally near the inflow side of the valve at the instant of valve closure, leading GARRISON et al. (1994) and SNECKENBERGER et al. (1996) to raise the possibility that these vortices might contribute to the generation of cavitation bubbles. However, there is no quanti- tative evidence to support this speculation.

The purpose of the present work is to evaluate the possible contribution of the vortices created near the inflow side of the valve to the generation of cavitation bubbles, employing numerical simulation of simplified models. Using a computa- tional fluid dynamic (CFD) analysis of a simplified two- dimensional (2D) model, we aimed to demonstrate that vortices generated on the inflow side of MHV may contribute signifi- cantly to the pressure drop and thus to the formation of cavitation bubbles.

2 Numerical model

While the cavitation in MHV is a local phenomenon that lasts only a few microseconds, the pressure and velocity fields are determined by global flow processes. Thus, an accurate simulation of cavitation in MHV must incorporate the appro- priate global and local spatial and time scales. This is not a simple task since the flow across MHV is a very complex phenomenon. The flow is time dependent and three dimen- sional and takes place within a complex geometry with flexible walls, it is characterised by a relatively high Reynolds number, and the motion of the valve is coupled to the flow. Consequently, this task is beyond current computational capabilities. To make the problem tractable, simplifications should be introduced.

it is believed that the shape of the geometry has only a secondary effect on cavitation because of its local nature. Therefore, to simplify the geometrical setup, we used a 2D model consisting of a generic flat valve within a straight channel to simulate the flow field in the vicinity of the mechanical valve. The 2D computational domain is shown in Fig. la. The valve is placed inside a channel of height H = 2.7 cm. The hinge of the valve is located at a distance of H from the upstream boundary and approximately one-third of the distance from the lower edge of the valve. The upstream portion of the channel represents the atrium, while the region downstream of the valve is on the left ventricle side.

Y ' ~ 38H ,

z a

b

0

~" -40- _OO >=

-60-

-80 0.45

Fig. 2

30

0 0.49 0.50 o.:~ o.:~7 o.:~8

time, S

Inlet axial velocity ( ) and vah,e motion ( - - - ) in closing phase. Eah,e motion is coupled to flow fieM

Navier-Stokes equations were employed, assuming an unsteady, laminar, incompressible and Newtonian flow. No- slip and no-penetration boundary conditions were specified on the upper and bottom walls of the channel and on the valve. A typical physiological mitral waveform was imposed on the upstream boundary. Fig. 2 shows the incoming velocity wave- form in the closing phase only, insofar as the present work is focused on this period of time. The analysis, however, was performed for the whole cardiac cycle in order to resolve the phenomena as a whole. The Reynolds numbers that were based on the peak and mean incoming velocity were Re = 4700 and 800, respectively, and the beat rate was fixed at 75 bpm. Stress- free conditions were imposed on the downstream boundary. The valve motion was determined by the fluid flow using a weak fluid structure interaction method (ROSENFELD et al., 1998). The resulting motion of the valve in the closing phase is shown in Fig. 2.

A typical mesh is shown in Fig. lb (every second node is shown in each direction for the purpose of clarity). Mesh points were clustered around the valve and along the walls. A moving mesh approach was used during the opening and closing of the valve, and remeshing was employed whenever necessary to avoid excessive mesh distortion. A commercial finite element package was used to solve the Navier-Stokes equations.* Mesh and time-step independence tests verified that a mesh with 20 000 mesh nodes and 1280 time-steps per cardiac cycle result in an accurate solution (ROSENFELD et al, 1998).

3 Results and discussion

In the closing phase of the cycle, as the gap size between the leaflet and the valve housing decreased, the velocity of the fluid in the narrow passages increased and reached a maximum value very close to the moment of closure. At the same time, the pressure drop across the valve increased and also reached a maximum value. Therefore, for the rest of the present study, we focus on these several milliseconds prior to the closure of the valve.

The global flow field in the vicinity of the valve near the instant of closure (t = 0.488 s as marked in Fig. 2) is shown in Fig. 3 by the instantaneous streamlines. At the time shown, the opening angle was 0 = 60 and the flow through the clearances between the tips of the valve and the enclosing walls was directed upstream (Fig. 2). in addition to the large vortices observed on both sides of the valve, small vortices could be found in the vicinity of both tips on the inflow side (Fig. 4).

Fig. 1 (a) Computational domain and (b) O.'pical mesh. Even; second node is plotted in each direction for clarity *FIDAP, Fluent Inc., Evanston, IL, USA

94 Medical & Biological Engineering & Computing 2000, Vol. 38

Fig. 3 Pattern o f streamlines in the vicinity o f vah,e (t= 0.488s) Fig. 5 Pressure field in the viciniO: o f vah,e (t= 0.488s)

Fig. 4

~ x a

t t ~

; ~ L l l

b

~lociO; field near (a) upper and (b) lower tips" o f vah,e (t = 0.488 s)

The large vortical structures had earlier been shown to be a result ofvorticity waves (ROSENFELD et al., 1998) and they are of interest in the study of the global flow.

An accurate simulation of cavitation bubbles, including the computation of the two-phase flow, is beyond the scope of the present study. The possibility of bubble inception was therefore deduced from the pressure field. Cavitation was found in regions where the absolute pressure decreased to the order of the vapour pressure ( 4 0 m m H g = - 7 2 0 m m H g gauge pres- sure). Therefore, the present study relied on the pressure field to detect regions prone to cavitation.

The pressure field for the same time of t = 0.488 is shown in Fig. 5. There was a large pressure drop between the outflow and inflow sides of the valve. The largest pressure drops were

found across the upper and lower gaps. The pressure was relatively uniform on the inflow and outflow sides of the valve; the pressure variations were of the order of -4-50mmHg as opposed to a difference of approximately 800 mmHg between the two sides of the valve (Fig. 5). However, as Fig. 5 indicates, the site of minimum pressure was not on the inflow side of either one of the valve tip surfaces, but farther away in the vicinity of the regions where small vortices were observed (Fig. 3). insofar as the main objective of the present study is to evaluate the significance of these regions in the creation of cavitation bubbles, the rest of this paper focuses on the inflow vicinity of the upper and lower tips of the valve.

Figs 4a and 4b show the velocity field in the vicinity of the upper and lower valve tips, respectively, for the same moment as depicted in Fig. 3. The geometry of the upper and lower clearances was different: on the upper side, the flow went through a converging channel (nozzle) while, on the bottom side, the flow went through a diverging channel (diffuser). Moreover, the flow rate through the upper gap was larger than the one through the lower gap. These differences have significant implications on both the velocity and pressure fields. One important consequence is the development of a stronger shear layer from the upper tip because of the larger velocity difference there. While each shear layer rolls up into a vortex in the inflow side of the leaflet, the upper vortex is stronger, because of the stronger shear layer that feeds it.

In our fluid-structure interaction calculations, the upper valve tip velocity was found to be 1.3 m/s near the closure, in reasonable agreement with the experimental results of Guo et al. (1994) and NAEMURA et al. (1997), who, at closure, measured a tip velocity of 0 .8 -2m/s and 1-3 m / s , respec- tively. The maximum fluid velocity in our calculation was 8.9 m / s , which is in reasonable agreement with the numerical results of MAKHIJANI et al. (1994) and the experimental results of CHEON and CHANDRAN (1994), which were 4 -10m/s and 3-5 re~s, respectively.

The difference in the geometry of the upper and lower clearances affected the pressure field as well (Figs 6a and 6b). The nozzle-shaped upper clearance created a large pres- sure drop (674mmHg) within the gap, while the pressure increased by only 6mmHg on the lower diffuser-like passage. However, the pressure at the exit (the inflow side) was lower (669mmHg) on the bottom because of the large pressure drop that took place at the entrance (the pressure at that point decreased from 65 mmHg to -675 mmHg, Fig. 6b). Consequently, the pressure drop across both tips was very large, marginally reaching the vapour pressure value.

Medical & Biological Engineering & Computing 2000, Vol. 38 95

b

Fig. 6 Pressure field near (a) upper and (b) lower tips of valve (t = 0.488 s)

in our calculation, the loading rate index (dP/dt)cLas defined by LEE and CHANDRAN (1994) was

( ~ t ) c L 2 l ~ " ( t ) d t _ 800 mmHg/s

a value that is clearly in the range of that of 700-2300 mmHg/s obtained by CHANDRAN et al. (1998) in vivo. in other words, the present model may closely represent a comparable physio- logical condition.

in many publications, the flow inside the narrow gaps that form between the leaflet and housing of the value is referred to as either a clearance flow (LEE and CHANDRAN, 1994, 1995) or as a squeeze flow (BLUESTEIN et al., 1994, CHEON and CHANDRAN, 1994, MAKIHANJI et al., 1996). In both cases, a relatively large pressure drop is obtained due to the large velocity through the narrow passages. In the clearance flow cases, pressure drops as large as 677 mmHg were found (LEE and CHANDRAN, 1994), very similar to our present finding, in a numerical simulation of a squeeze flow between a bileaflet valve and the housing, BLUESTEIN et al. (1994) found a pressure drop of 500 mmHg, which is of the same order of magnitude as the present finding of 674 mmHg.

When there are vortices near the upper and lower tips of the valve, an additional pressure drop can be expected, indeed, Fig. 6 demonstrates that the pressure in the centre of the upper and lower vortices was -748 and - 6 7 2 m m H g , which is an additional drop of 138 and 3mmHg from the exit values of - 6 1 0 and - 6 6 9 m m H g , respectively. Thus, these vortices existing only during regurgitation and strong enough only at the moment of near closure can create favourable conditions for the generation of cavitation bubbles, even in cases when the pressure drop due to other sources is not large enough to do so.

96

600,

O) 400'

E 0_-

200'

\ \

\

0 29 33 37 41 45 49

~, ~m

Fig. 7 Dependence on upper gap size e of squeeze flow and upper tip vortex contributions to pressure drop. A P,.o,.t~x: . . . . A P s q . . . . . .

To decrease the likelihood of cavitation, it would appear that designs that minimise regurgitation or the generation of strong inflow tip-vortices should be preferred.

We demonstrated that the upper tip is more prone to cavitation because of the stronger vortex there. Our findings are in agreement with many experimental observations (KAFESJIAN et al., 1994, GRAF et al., 1994, CHANDRAN et al., 1994, CHANDRAN and ALURI, 1997), which have found this as being the preferred location for the generation of cavitation bubbles (the ' 12 o'clock position').

Fig. 7 shows the pressure drop attributed to clearance flow and to the vortex flow against the upper gap width e (Fig. 1 ). it should be noted that e varies with time due to the motion of the valve. For large values of e, the only contribution to pressure drop is due to the clearance flow. As the gap size e decreases and the vortex strength increases, the contribution of the vortex pressure drop increases, both absolutely and relatively to the overall pressure drop.

Experimental support of the existence of vortices was provided in the study of CHANDRAN and ALURI (1997). They measured a lower pressure at a distance of 1.5mm from the inflow surface of the valve as opposed to at a distance of 1 mm or 3 mm, demonstrating that there is a minimum in the pressure away from the valve, in our 2D calculations, the minimum was located at a distance of 0.7 mm from the upper tip of the valve.

4 Concluding remarks

Two contributions to the pressure drop that may lead to cavitation were considered in the present study. The first is due to the large flow velocity in the narrow clearance between the valve tip and the upper wall. The second, assessed for the first time in the present study, is due to the development of strong tip vortices in the inflow part of the valve during the back flow phase of the cardiac cycle.

The CFD analysis of a 2D approximation employed in the present simulation, the simplifications of the geometry and the approximations made in the physical modelling affect the results. The simulation of a full 3D representation may give different quantitative results. However, in the present study, we succeeded in demonstrating that the vortices in the vicinity of the valve tips could be a factor in the inception of cavitation, in addition to other factors such as clearance flow, squeeze f low and water hammer effects. The pressure drop due to the the tip vortices is relatively small in comparison to the other sources (10-20% in the present case), yet it might be meaningful for the creation of cavitation bubbles. We speculate that similar

Medical & Biological Engineering & Computing 2000, Vol. 38

conclusions can be made for more realistic 3D cases, al though the quantitative results (such as the relative contr ibut ion o f the vortices to the total pressure drop) may differ.

Acknowledgments--This research was supported by a grant from G.I.E, the GermaxMsraeli Foundation for Scientific Research and Development.

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