ELSEVIER 13504533(95)00075-5

Med. Eng. Phys. Vol. 18, No. 6, -152-462. pp. 1996 Copyright 0 1996 Elsevier Science Ltd for IPEMB

Printed in Great Britain. All rights reserwd 1350-4533/96 $15.00 + 0.011

The integrated design of mechanical bi-leaflet prosthetic heart valves

Tim David and C. H. Hsu

Department of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK

Received 25 October 1994; accepted 24 May 1995

ABSTRACT A flow model utilizing an irrotational, inviscid algon’thm of vortex-ring elements simulating the leafleets and

source/sink elements simulating the aortic root coupled with a boundaly layer model has been developed to model the

internal flow phenomena of bi&a$et mechanical heart valves implanted in the aortic root. The inviscid repeser-

tation evaluates the aerodynamic lift, the induced drag, the pitching moment and flow velocity along the leafleet

surface thus providing data for evaluating the boundary-layer thickness, the shear stress and flow separation point

by the boundary layer theory. Full integration with the geom&y enables immediate updates of thejow solution when

changes in geometry have been made. R is shown that the effects of the internal flow domain model are necessary in

the correct evaluation of lifi. and drag for subsequent dynamic analysis. The enuironmal presented provides for the

ability to produce significant and immediate design changes so that crucial decisions may be made whilst still within

the software design lo@. New designs are shown along with data for the improved flow model. Copyright 0 1996 Elseuier Science Ltd for IPEMB.

Keywords: Integrated design, mechanical bi-leaflet, prosthetic heart valves

Med. Eng. Phys., 1996, Vol. 18, 452-462, September

INTRODUCTION

Providing a quick 3-D CFD/design environment to initially model the flow through mechanical heart valves is crucial to the initial stages of the design of prosthetic valves, for both flexible and rigid occluders. This is primarily owing to the requirement that a large number of design vari- ants may be needed to be analysed; at each stage a large amount of computational time will be spent in solving the complete Navier-Stokes equa- tions for the flowfield throughout the complex 3D domain. Hence design to product cycle times are large and initial complex analysis can be coun- ter-productive.

The natural paradigm for an integrated design system is to allow the designer complete control over the essential processes. That is he/she must be able to alter their designs, see the effect and make decisions accordingly, most importantly these effects and the corresponding decisions should be seen and made quickly in the early stages of design. Certainly complex fluids (or other types of) analysis will be required at some stage during the design process but this will be several stages downstream in the cycle. Consider- able numbers of design variants will need to be inspected before choosing a small number for further development. This stage of concept design is essential in minimizing the design life cycle.

Fluid flow analysis of rigid occluder mechanical heart valves have predominantly been attempted using highly complex computational models incorporating full viscous effects’, and in some cases extremely complex geometries*. Dynamic analysis has, however, utilized very simple fluids models, notably flat plate aerodynamic theory (utilising incompressible, inviscid flow), and con- strained the leaflet to operate in an infinite medium such as that used by Prabhu and Hwang’, Reif et aL4 and Cheon and Chandran5. The first analysis, as noted above, requires huge computer resources and large timescales to complete. The second, although offering vital data, has been modelled without influence from either aortic walls or the other leaflet (if bi-leaflet valves are being analysed) , constraining the valve geometry to a single form (that of a flat plate) and no vis- cous component. Increasing the complexity of both geometry and fluids model does not neces- sarily incur large computational or time penalties. More importantly integrating geometry and fluids models can provide an interactive environment where data on the valve concept design can be viewed and immediately acted upon.

This work concentrates on the aortic valve since it is the most commonly replaced valve. The pri- mary function of the aortic valve is to permit ‘smooth’ flow between ventricle and ascending aorta and it seems prudent to introduce a fluids

boundaries. These are the prescribed curves in (x,y,z) space X( U,U) and their derivatives with respect to either u or V, X,, X,. Figure I shows the design environment with the network editor space and several widget panels from which the designer can interactively change geometry and fluid boundary conditions. In addition the derived surface(s) is also shown in an independent ren- der window.

For this work we present the boundary con- ditions which formulate a basic valve leaflet sur- face.

Figure 2 shows the coordinate system and boundary conditions for the basic model. The x direction is that of the main blood flow, the y direction essentially models the valve pivoting axis with the .z direction orthogonal to both x and J. Dirichlet conditions in vector component form (x,y,.z) are listed in Table 1. The leaflet is described initially, in plan, by a semi-circle of radius R (although this is not a restriction since any curve

could be used) with the left and right quadrants associated with the u = 0 and u = 1 boundaries. The U= 1 boundary maps to a singular point at the trailing edge and the V= 0 boundary (modelling the leading edge) is an ellipse in the y,.z plane, this is used to ensure roughly equivalent blood flow through all three orifices of the valve. Additionally a simple cubic curve in the x,.z plane is placed on both U= 0,l boundaries with the height above the x axis at the trailing edge given as a design parameter A. This can be thought of as essentially modelling the opening angle of the valve but also allowing the leading edge to be par- allel to the oncoming flow. This has distinct advantages to the valve design in terms of reduc- ing the onset of flow separation as discussed in the results and discussion sections.

Neumann (tangent) boundary conditions are also required along the four U,U boundaries and these are also given in vector component form in Table 1. Here the constants k,,!zJzJ,k4 are used to provide simple interactive geometry changes with- out having to alter the full boundary conditions at every step. For this case 8 (in the (x,~) plane) is measured from the 2’ axis in a clockwise direction. I;@re 2 also shows a wire frame image of the resultant surface for the given boundary con- ditions.

model at an early stage without compromising the ability to range over a large number of designs within a small period of time.

In order to achieve this ‘quick design’ philo- sophy, the authors” have developed a design environment using Iris ExplorerTM (software orig- inally available with Silicon Graphics hardware platforms) where valve geometries may be easily, accurately and quickly defined at the workstation. Additionally we have integrated the fluid flow analysis with the geometry by simulating an internal inviscid and incompressible flow such that viscous effects are constrained to lie in an infinitesimal layer at the body surface. These vis- cous effects may be modelled using vortex or doublet sheets either forming part of the body surface or lying in a ‘force-free’ position within the fluid domain. This type of mathematical model is numerically solved utilizing the well- known panel method of Hess and Smith’ and Katz and Plotkir?. For this work a coupled boundary layer model using theory developed by Thwaite? and Stratford”’ is also used in simulating the flow close to the leaflet surfaces where viscous forces exerted on the blood become important. The results emanating from the coupled fluids model can be directly related to important design criteria such as flow separation, drag reduction and dynamic response (i.e. how fast will the leaflet open/close, will it remain open during systole etc.). This enables the model to be used from the very first stages of design, an especially important phase. A more detailed explanation of the full software environment can be found in David and Hsu”.

THEORY AND MODELLING

The generation of the free form surfaces model- ling the valve geometry (in this case the leaflets) can be accomplished using the solution to a parti- cular set of mth order elliptic partial differential equations, initiated by Bloor and Wilson”, in the independent variables u and v given by the gen- eral form

l>:yl (X) = F( 21,79. (1‘)

Here X is the solution vector given by

x = (x(2L,v),y(?~,7~),2(2L,v)). (2)

I,:::,,(X) the mth order differential operator and F( IL,U) a vector valued function. However for the purposes of explanation and the present appli- cation we use the simpliflcd bi-harmonic form as SllOWIl helo\\

(3)

‘(1’ is a parameter which essentially scales the 7’ co-ordinate. The solution to equation (3). X, is a surf’ace parametrized in terms of the indepcndenr variables u and 7'. To complete the model and in Fact to provide the appropriate ‘design handles’, WC require boundal? conditions for the ahovcs fourth order equation for each of the follr ( U,V)

THE FLOW MODEL AND AERODYNAMIC LOADS

It is assumed that as a first stage dcsisn tool the fluids model should hc reasonably simple thus enabling good interaction with the surface gener- ator and givillg quick graphical results. In a simi- lar manner with the previously referenced work it [vi11 1,~ 11ecwsa1-y 10 model not onl>, the solid body of‘ the wtl\.c but also the lifting c.omponent and tlic profilcb dr-ag crc.at~~l bv the J&YS immersion in the Illlid. \,t’t. c~hoose a11 irrotational and incom- l)rc+sible nlodcl since this provides loading, ch-dg,

pl“‘s4”t‘~‘ ;ind sinlplc vclocit!’ data from which tlc3igtl c~li;m~c~ ('all l,c ;tssessetl htrt we also

453

Mechanical bi-leaflet prosthetic heart valves: T. David and C. H. Hsu

Figure 1 Network editor and resulting surface

inclu the 7

Fo teria pote

q=- With

v*q =

we 0

tde a coupled boundary layer model to assess riscous component of the design. lr a given velocity vector q the irrotational cri-

provides the definition of a scalar velocity ntial 4 such that

-V$. (4)

v%$ = 0. (

Using Green’s Identity elementary solutions the LaPlace equation may be distributed over finite discretization of the bounding surfa throughout the domain. Linear combinations these elementary solutions can be found whi

the continuity of mass satisfy the boundary conditions at infinity viz.

q 0, (5) v4 - 0, r - co. (

btain the well known LaPlace equation The Neumann boundary condition on the so:

6)

to . a .ce of ch

7)

lid

454

Table 1 Dirichlet conditions in wctw component form

Dirichlet Urllnlallll

surface represents the fact that there is no net flow across solid boundaries and is given h)

V(+ + $-*)a = II,, (8)

where n is the unit normal vector to the surface and V,, is a prescribed velocity normal to the boundary. We refer the reader to Hess and Smith” and Katz and Plotkin’ for a full derivation of the relevant theory. Essentially the panel method dis- tributes a finite number of elementary singular- ities on the boundary of the fluid domain and a set of algebraic equations is obtained by invoking either the Dirichlet or the Neumann boundan condition or both. This set of equations is in dense matrix form and is solved using conju$atc gradient methods as shown by David and Lewis”.

The advantage of using panel methods is that a solution for the velocities on the body surface and hence the pressure distribution can be obtained without solving for the flowfield throughout the domain. Additionally full inte- gration of the fluids model with the geometry is provided by the fact that the panel vertices arc exactly those generated by the intersection of tht co-ordinate lines u = constant and u = constant mapped onto the resulling surface. 1x1 addition

the conjugate gradient method can be fully paral- lelised relatively easily using al+gorithms developed by David and Blyth”’ so providing a fast compu- tational algorithm for potential flow.

In order to represent the flow of blood through valves implanted in the aortic root we are required to generate an internal fluid domain. This internal flow model is accomplished by rep- resenting the ascending aorta as an axi-symmetric closed converging tube (including the constric- tion and expansion modelling the sinus) with the surface normal vector pointing into the interior of the tube. Inflow and outflow mass fluxes into the domain may be prescribed as normal velo- cities on groups of inlet and exit panels.

Sink/source elements are utilized to model the internal duct which forms the aortic root, inlet surface and outlet surface because of the non-lift- ing nature of these elements.

In contrast to the aortic root, the valve leaflet has to be modelled by utilizing lifting elements such as the doublet or vortex ring. Furthermore, valves are normally manufactured with a high chord to thickness ratio in order to reduce the blood damage to a minimum and allow relatively quick action opening characterisric-s. Hence we

455

Mechanical bi-leaflet prosthetic heart valves: T. David and C. H. Hsu

choose a ‘lifting surface’ model with vortex-ring elements panelling the surface of the leaflet. Only a single surface need be generated at this time to fully describe the valve leaflet. In addition, with regard to the effect of the downstream wake we may model this by distributed vortex-ring elements parallel to the local flow direction. Figure 3 shows the internal flow domain and the associ- ated leaflet.

The solution of the vortex-ring algorithm requires that Nplanar quadrilateral panels be gen- erated along the surface of the lifting component. The singularity element used for this portion of the surface is a ring-vortex, of strength I whose position is such that the leading segment of the vortex ring is at the quarter chord point of each panel and the collocation point lies at the three- quarter chord point. The position of the vortex- ring with respect to the panel geometry is determ- ined from the Z-D case which ensures the Kutta condition is satisfied for the simple flat plate model.

The resultant lift, L, on the valve can be evalu- ated by summing the lift owing to each bound vor- tex i.e. L = liLii where L, is the lift owing to the vortex ring element formed from the intersection of the lines u = ui, V= v. in the leaflet geometry model. Using the Kutta-joukowski theorem the L, can be evaluated as

L, = pa 64 l-q, (9)

where U, is the uniform undisturbed free steam velocity evaluated at the position of the leading edge taking into account the increase in speed owing to the constriction of the aorta. The lifting force coefficient C,, is evaluated as

(10)

with S the total leaflet surface area. In addition the moment M exerted on the valve leaflet by the fluid force can be evaluated by

Here 4 is a characteristic length scale which for this application is the valve diameter. From an engineering viewpoint the total drag D of the valve includes the induced drag and the viscous drag that is

D = 0, + DC). (13)

The induced drag is produced owing to the downwash/upwash Wi. induced by the wakes of both leaflets. yii is evamated by integrating all the trailing vortex influences so that the induced drag is defined as

Q = m, = cpw, @ rij (14)

The pressure difference across the thin vortex sheet can be evaluated by using the lift and panel area so that

Ap,, = bL, q AS,’

where ASi. is the ijth panel area. It should not be assumed that the flow and cor-

responding forces exerted on the valve are sym- metric, in fact in normal circumstances the asym- metric case would very probably be the norm. This is owing to the production of swirl formed from the ventricle contraction. Hence we calculate the forces/moments about the pivot axis since this will provide crucial design data for the pivot mechanism. We have already calculated the total moment and formulating the rolling moment about the x axis MR and yawing moment about the .Z direction M, is simply the corresponding components of the total moment vector, M.

This model can therefore simulate both the increased lift forces owing to the internal flow and the ‘ground effect’ phenomenon of the other leaflet, a significant and important improvement on the nast analvses of Reif et aL4 and Cheon and ChandGan’ using potential theory. In addition the solution algorithm is defined so that very little extra computational work is required for the increase in complexity of the model. The environ- ment is such that geometric changes to a parti- cular design are available ‘immediately’, and in tests done by the group the solution takes no longer than a few seconds for the models pre- sented here. Clearly the increased available pro- cessor power will reduce these times considerably.

M=CM,=Cr@ L,

r = radius vector from pivot axis to panel (11)

whilst the moment coefficient Cm,P is defined as

BOUNDARY LAYER MODEL

Complex 3D CFD analysis of flow through bi- leaflet valves by King et al’ has shown that the position of the flow separation point on the suc- tion side of the leaflets is important in determin- ing the size of the recirculation zones that exist and are convected downstream behind the leaflets. The work of Gross et aLI has shown this to occur experimentally. These recirculation zones are undesirable from a design viewpoint

Figure 3 Internal flow domain and the associated leaflet since they can provide mechanisms for the formu-

456

lation of thrombi and platelet activation.‘” In addition, the wall shear stress evaluated along the leaflet surface which can cause blood cell damage has become crucial in valve leaflet designs. We give evaluations of the separation point along the centre line of the leaflets and the shear stress using a coupled boundary-layer model. The boundary layer equation along the streamwise direction q is given as

77R = 770 - (25)

A value recommended for all-around use is

(53-51

Thus, the separation point T,~.,, is given by

au au dU a’,u t-+v-=u-tv-

aq at dq sly

Here U is the mainstream velocity profile, u is lLll

the streamwise coordinate boundary layer profile and ,$ a local coordinate orthogonal to q. This can EFFECTIVE ORIFICE ARFA AND PRESSURE be rewritten as LOSS

a(dh.2) + iJ(v(u-2~)) dU 2

arl x -+ (u-u)-=-715

drt at’* (17)

Integrating the above across the boundary layer we obtain

(18)

with 6, the displacement thickness and 8 the momentum thickness.

In order to predict the separation point along the leaflets, we rearrange the above as

7 ,,, CJtI d0 PU’ -= PU

--- -- t --y (2+H), v dq

(1%

here H= S,/% is the shape factor and U’ = dU/dx. The momentum integral can be rewritten by introducing a parameter defined by

0”lP Y = -----. (20) 71

Thus, the integral equation could be expressed as

Using an estimation of the thickness of the wake from both separation points on the leading edge of the leaflet and the stiffening ring we are able to calculate an effective orifice area and hence a reasonably accurate estimation of the pressure loss incurred by the fluid in passing through the valve. We first find the growth of the wake from the simple formula found in Batchelor16

Gw

Here 71 is the dynamic viscosity, U is the free stream velocity in the aorta and (x - q,) is the downstream distance measured from the separ- ation point %. For the cases presented below the distance downstream is terminated at the trailing edge of the leaflet. The wake is treated as an effec- tive blockage to forward flow. An area for both the wake growth from the stiffening ring and leaflet separation (AK,AI) is evaluated which is used to find the effective area. Using the conservation of mass a mean velocity at the trailing edge, U,, of the leaflet is found. The incompressible Bernoulli equation then provides the required pressure loss by comparing with the upstream velocity.

(21) RESULTS

Thwaites” proposed a simple linear fit for Fwith

F(c) = 0.45 - 6.05,

so that

8” II - = 0.45 w IP dr/. 71 0

(22)

(23)

In a separate work Stratford’” suggested that the pressure coefficient (;;, at the separation point, must satisq

’ 2 (?q,J’ CI, drl =r constant. ( 1

“c!, (24)

Even with the small number of parameters describing; the basic leaflet a large number of dif- ferent shipes and configuration; may be derived, thus we restrict ourselves at this juncture to the variation of a small subset thought to be of import- ante. This ensures that we gain an insight into both the internal flow model and the variation in aerodynamic characteristics. We present below aerodinamic data as a function of the boundary conditions of the pde modelling the geometry. We use the basic leaflet shape as defined above. Firstly we use the derived model to cornDare with simple potential theory for a will-known geometry.

Here Y,~ is an artificial origin such that the true momentum thickness at qlo equals the value of 8

AERODYNAMIC FORCES

which would grow in a B&i& layer over the dis- E;igur~ 4 compares the lift coefficient for the flat tance q. - r],. From the Thwaites approximation, we obtain

plate, infinite domain theory used by Reif et al.’ (shown as the linear function with gradient 23~)

457

Mechanical bi-leajlet pros&hetic heart valves: T. David and C. H. Hsu

- cL I --- Reif(1990)

0 P. M.

773 a ( rad )

I A =0.3

“/3 0 ( Tad )

St Jude valve Figure 6 Drag coefficient as a function of angle (Y for A = 0.1, 0.2,

Figure 4 Lift coefficient for St Jude valve leaflet simulation 0.3

with the present model simulating a St Jude valve leaflet inside the aorta. This simulation is evalu- ated by creating a flat plate of the same dimen- sions as a St Jude valve using the pde algorithm with associated boundary conditions. The angle (Y represents the angle subtended by the leaflet from the mainstream blood flow,. thus, low values of (Y represent fully open condmons and vice versa. It is clear that the lift is higher for the internal model as expected owing to the increased strength of singularity elements maintaining the zero normal flow condition along the aortic wall and the ground effect from the other leaflet. The variation of lift coefficient as a function of angle (Y for various values of A are shown in Figure 5. As A increases the lift coefficient increases at constant values of opening angle 8. This is owing to the effective increase in camber of the leaflet. The drag coefficient (induced drag only) as a function of opening angle for various values of A is shown in Figure 6. For comparison the results of Reif et aL4 are shown for the St Jude simulation in F&ZAW 7. This indicates that although the lift is higher for the presented internal flow model the drag has been reduced. This may be explained by the ‘ground effect’ since the downwash created by the

I CDi

J

- Reif(1990)

1.5 0 P. M.

rad )

St Jude valve

Figure 7 Drag coefficient for St Jude valve leaflet simulation

‘mirror’ leaflet reduces the downwash of the mod- elled leaflet and hence its own induced drag.

PRESSURE DROP

Table 2 shows a comparison of pressure loss between the presented model, as described in the theory section above, and the experimental find- ings of Fisher” and Butterfield et aZ.18 for pulsatile

Table 2 Comparison of pressure loss

/ A =0.3

A =0.2

A =O.l J,. 76 rr/3 Q ( +ad )

Valve type Volume Fisher” Butterfield’s Wake flow rate Nm-’ Nmm2 model (I/mm) Nm-?

St Jude (19 mm)

St Jude (21 mm)

St Jude (23 mm)

9 956 571 12 1554 1012 15 2471 1581

9 380 308 12 683 547 15 1063 855 12 431 352 18 1162 797 21 1461 1089

Figure 5 Life coefficient as a function of angle a for A = 0.1,0.2,0.3

458

flow whilst Table 3 shows comparisons with that of Yoganathan’” for steady flow conditions. In both cases a St Jude valve has been simulated for vari- ous aortic diameters.

SHEAR STRESS AND SEPARATION POINT EVALUATION

&~re 8 shows the shear-stress, T,, evaluated as a function of the nondimensionalized streamwise coordinate at the leaflet surface along the centre streamline for various values of A. l$gures 9-11 show the separation points evaluated along the centre streamline and parallel lines across the leaflet span for various values of A. Here the effect of an increase in h is to produce the onset of sep- aration closer to the leading edge of the valve, a not unexpected result owing to the higher curva- ture at the leading section. This will produce a larger recirculation zone behind the valve and increase the probability of clotting in the blood owing to slow moving fluid. In the CFD exper- iments carried out by King et nZ.’ the separation point is, if not exactly very close to, the leading edge for simulations of flat plate type leaflets. The results above show that by curving the leaflet to allow a leading section of the surface to be initially parallel to the main blood flow effectively reduces the onset of separation, an important design para- meter. The values of both Dirichlet and Neumann boundary conditions are as for Table 1.

Shear stress measurements obtained with ill z&-o studies for the flow downstream of the St

Jude heart valve have been published by many investigators’“-“‘, whilst there are none, to the authors knowledge on the measurement of leaflet surface shear stress owing to difficulties in exper- imental techniques. Table 4 shows experimental values of shear stress in the bulk fluid from”‘-“” for a 25 mm St Jude bi-leaflet valve.

It should be noted here that firstly those results g.iven by Hanle used a blood analogue fluid whose viscosity was significantly lower (lo-” kg- m-’ s-‘) than the other authors (3.5 x lo-:’ kg m-l s-‘) and secondly the final four results are for turbulent conditions which would enhance the maximum total shear stress as shown above. The coupled boundary layer thee? as presented in this works gives a maximmn lammar shear stress of 103 nm-” using the higher viscosity. This compares reason- ably well with the figures given in Tabk 4, especially considering the assumptions of laminar flow in the model.

Mechanical btleajlet prosthetic heart valves: T. David and C. H. Hsu

I 0.124

h =0.2

Figure 10 Separation points along the centre streamline for A = 0.2

Figure 11 Separation points along the centre streamline for A = 0.3

Table 4

Author Flow conditions Max. shear stress Nrn-?

Hanle (1989)” Hanle ( 1989)2? Yoganathan et al. (1982)‘” Yoganathan et al. ( 1986)23 Woo et al. (1985)“’ Walker et al. (1992)”

Steady 76 Pulsatile 74

Steady 60 Pulsatile 200 Pulsatile 160 Pulsatile 200

FORCES/MOMENTS ACTING ABOUT THE PIVOTING AXIS

The forces acting about the pivoting point are the lift and induced drag whilst the moments are the pitching moment, rolling moment and yawing moment (asymmetric loads). Asymmetric surfaces/loads can be generated by changing the magnitude of the Neumann boundary condition on u = 0 and u = 1. Table 5 shows the effectiveness of the asymmetric loads generated by changing the magnitude of either X,(O,v) or XU(l,v), here these are represented by the parameters K, and & as defined in Table I. As the value of Izr increases this causes a small elevation of the surface in the neighbourhood of the trailing edge u = 0 thus

Table 5 Effectiveness of asymmetric loads

2.0 1.0 0.0147 0.003 2.5 1.0 0.0227 0.0055 3.0 1.0 0.0313 0.0086 3.5 1.0 0.0407 0.0121 4.0 1.0 0.0505 0.0159 1.0 2.0 -0.0147 0.003 1.0 2.5 -0.0227 0.0055 1.0 3.0 -0.0313 0.0086 1.0 3.5 -0.0407 0.0121 1.0 4.0 -0.0505 0.0159

increasing the related vorticity strength, and therefore producing the positive rolling moment and yawing moment. On the contrary, an opposite effect is produced by increasing the value of 4 (X,( 1,~)). A similar phenomenon would occur when asymmetric flow profiles are produced by asymmetric ventricle contraction.

DISCUSSION

The comparison of the presented model with that of the analysis of Reif et aL4 shows that the inclusion of the internal domain is crucial to the correct evaluation of lift and drag exerted by the leaflet. In addition the mirror effect induced by the other leaflet in bi-leaflet designs is an important phenomena since it will show the increased velocity through the central orifice between the two leaflets and will also effect the opening and closing characteristics. In fact the separation distance of the two leaflets will provide an additional design parameter in that smaller leaflet separation induces higher lift coefficients during the systolic phase of the cardiac cycle ther- eby ensuring that the leaflets remain opening dur- ing both acceleration and part of the deceleration phase of systole, however it also seems reasonable to ensure a roughly equivalent blood mass flux through each of the orifices and hence the design presented provides an elliptic end profile of the valve to allow for this, although no quantitative calculations have been made at this time.

Flow separation can be assessed through the use of a simple coupled boundary layer analysis and provide the designer with necessary information concerning the profile of the leaflet during sys- tole. In the complex CFD analysis by Ring et aZ.’ flow separation occurred almost immediately at the leading edge for all models investigated. This was owing to the geometry of the leading edge and its orientation to the mainstream flow. By allowing the valve to have curvature the leading edge can be placed parallel to the mainstream and reduce the onset of separation, thus weaken- ing the recirculator-y eddies occurring downstre- am.

As can be seen, the wake model underestimates the pressure drop by about 25% overall when compared to experimental results. However, it does provide important trends which can easily be treated as design criteria when forming an ‘opti-

460

mum’ design of valve. The underestimation of pressure drop can be associated with a variety of reasons. Firstly the wake would continue to grow and hence provide an increase in the pressure drop as the distance downstream is increased. At this time we have no hard evidence to suggest a better stopping distance than that of the trailing edge of the leaflet. Secondly we have not included the additional pressure gradient required to over- come the time-dependent acceleration of the fluid. This is to be treated in future developments. Thirdly the calculation for the effective area is a relatively simple one and could be increased in complexity by allowing the wake growth to be evaluated at several sites along the leaflet span. The laminar shear stress estimations are encour- aging considering the laminar assumptions and that there exists no estimates of the shear stress on the leaflet itself. The more complex conditions encountered downstream in the bulk fluid where all measurements have been taken would be expected to be higher as shown. It should be noted that the maximum values occur close to the leading edge of the leaflet where flow acceleration and velocity are at their highest.

Since the presented fluids model is that of a thin vortex sheet positioned along the leaflet cam- ber and immersed in a duct of varying cross-sec- tion the above evaluations of shear stress and sep- aration point ignore the thickness of the leaflets, that is the high pressure surface is not necessarily a streamline one and therefore will not provide the correct potential flow for a boundary layer simulation. Therefore it may be necessary to improve the flow model to include the capability of simulating the closed geometry of the leaflets, ~8. the source/doublet distribution approach.

It is clear from the description of the software environment that given data on the forces exerted on the valve mechanism the dynamic character- istics of the valve, i.e. opening and closing times, can be evaluated through the introduction and integration of a differential equation solver mod- elling the dynamic phenomena. Pressure drop estimations can be made on the basis of the avail- able information and trends can be seen easily. Additionally, the geometry changes occurring through the redesign of other parts of the valve such as the sewing or stiffening ring can have a radical effect on the flow phenomena. These changes can be easily incorporated into the above model and work is continuing in this area.

CONCLUSIONS

A flow model utilizing an irrotational inviscid algorithm of vortex-ring elements simulating the leaflets and source/sink elements simulating the aortic root coupled with a boundary layer model has been developed to model the internal flow phenomena of bi-leaflet mechanical heart valves implanted in the aortic root. The inviscid rep- resentation evaluates the aerodynamic lift, the induced drag, the pitching moment and flow velo- city along the leaflet surface thus providing data for evaluating the boundary-layer thickness and

the shear stress by the boundary layer theory. Full integration with the geometry enables immediate updates of the flow solution when changes in geometry have been made. Variation of the pde boundary conditions can provide a variety of dif- ferent leaflet geometries some of which have been analysed in this paper. Future work will concen- trate on producing new innovative valve designs which have ‘optimum’ parametrizations.

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