J. Bmmechanm Vol. 26. No. 8. pp 929-943. 1993. Printed in Great Britain

1X.21 -9290/93 $6.00+.00 0 1993 Pergamon Press Ltd

NUMERICAL SIMULATION OF LEAFLET FLEXURE IN BIOPROSTHETIC VALVES MOUNTED ON RIGID AND

EXPANSILE STENTS

S. KRUCINSKI,* I. VESELY,~$§ M. A. DOKAINISH* and G. CAMPBELLS~II *Department of Mechanical Engineering, McMaster University, Hamilton; TDepartment of Electrical Engineering; $Medical Biophysics, University of Western Ontario; §The John P. Robarts Research

Institute; and )/The University Hospital, London, Ontario, Canada

Abstract-Recent studies suggest that flexural stresses induced during the opening phase may be responsible for much of the mechanical failures of bioprosthetic heart valves. Sharp leaflet bending is promoted by the mounting of valves on rigid stents that do not mimic the systolic expansion of the natural aortic root. We, therefore, hypothesized that flexural stresses could be significantly reduced by incorporating a flexible or expansile supporting stent into the valve design. Using our own non-linear finite element code (INDAP) and the pre- and post-processor modules of a commercial finite element package (PATRAN), we simulated the opening and closing beha.iour a trileaflet bovine pericardial valve. The leaflets of this valve were assumed to be of uniform thickness. with a non-linear elastic behaviour adapted from experimentally obtained bending stiffness data. Our simulations have shown that during maximal systolic valve opening, sharp curvatures are induced in the leaflets near their commissural attachment to the supporting stent. These areas of sharp flexure experience compressive stresses of similar magnitude to the tensile stresses induced in the leaflets during valve closure. By incorporating a stent with posts that pivot about their base, such that a 10% expansion at the commissures is realized, we were able to reduce the compressive commissural stressing from 250 to 150 kPa. This was a reduction of40%. Conversely, a simple pliable stent with stent posts that deflect inward and outward under load did not achieve a significant reduction of compressive stresses. This numerical analysis, therefore, supports the theory that (i) high flexural and compressive stresses exist at sites of sharp leaflet bending and may promote bioprosthetic valve failure, and (ii) that proper design of the supporting stent can significantly reduce such flexural stresses.

INTRODUCTION

Bioprosthetic heart valve implants have proven clinic- ally successful over the short term, but their long-term performance has been disappointing (Bortolotti et al., 1987: Gallo et al., 1986, 1988; Magilligan et al., 1985; Teoh ef al., 1989). Bovine pericardial valves, in par- ticufar. have suffered from poor durability, usually performing significantly worse than porcine xenog- rafts (Bortolotti et al., 1987; Thiene et al., 1986; Wheatley et ul., 1987). The degenerative failure of these bioprostheses results mainly from severe calci- fication and from leaflet tearing at points of their attachment to the supporting frame or stent. Indeed, many of the early failures of pericardial valves can be attributed to design flaws that induced high stresses in the valve cusps (Trowbridge et al., 1988). While the calcification problem may be eventually solved through biological and chemical intervention (Golomb et al., 1987; Pathak et al., 1990; Vesely et al., 1991a), mechanical failure caused by faulty design could be reduced immediately through appropriate modifications to the stent or cusp geometry.

Until recently, it was generally believed that the majority of structural failures of bioprosthetic valves

Received in jnal form 28 December 1992. Address correspondence to: Ivan Vesely, The John P.

Robarts Research Institute, P.O. Box 5015. London, Ontario, Canada N6A 5K8.

resulted from tensile stress developed during valve closure. Numerous studies have suggested, however, that cuspal degeneration is mediated through flexural stressing and compressive buckling (Gabbay er al., 1988; Thubrikar et al., 1986; Trowbridge and Crofts, 1987; Trowbridge et al., 1988; Vesely and Boughner, 1989; Vesely et al., 1988). Pathologic studies, in par- ticular, have correlated the sites of cuspal tears to areas that experience high flexural stresses, likely caused by deficiencies in valve design (Ishihara et nl., 198 1; Pomar et al., 1984; Stein et al., 1985; Walley and Keon, 1987; Walley et al., 1987). Design flaws in failed bioprostheses may be readily apparent in retrospect, but the prospective evaluation of new bioprosthetic valve designs is much more difficult. Mechanical test- ing can assess the performance of leaflet materials and stent designs (Lee et al., 1984; Pereira er a[., 1990; Song et al., 1990; Vesely, 1991; Vesely and Boughner, 1989, 1990; Vesely and Noseworthy, 1992; Vesely et al., 1990a), but cannot predict how the valve will function as a whole. Mathematical modelling, on the other hand, has the potential to become a very useful tech- nique in the study and evaluation of new prosthetic valve designs. Over the years, numerical modelling has made significant contributions to understanding the loading of natural and prosthetic heart valves during the closing phase (Cataloglu et al., 1977; Chris- tie and Medland, 1982; Hamid et al., 1986; Rousseau et al., 1988). More recently, one group has simulated leaflet flexure during valve opening in a two-dimen- sional model of a pericardial valve (Huang et al., 1990)

930 S. KRUCINSKI et al.

and in a three dimensional bileaflet valve (Black et al., 1991). This work has demonstrated the presence of significant flexural stresses during valve opening.

Our previous research into valve leaflet mechanics (Vesely et al., 1988, 1990b, 1991b, 1992), and math- ematical modelling by others (Hamid et al., 1985, 1986), suggests that the patterns of leaflet stress in the prosthetic valves can be influenced by stent design. We have, therefore, hypothesized that if the natural expansion of the aortic root is duplicated in a biopros- thetic valve, the resultant flexural stresses induced during valve opening could be significantly reduced. A numerical modelling technique was, therefore, developed to test this hypothesis. Specifically, we were interested in determining to what extent the opening curvatures and the associated flexural stresses could be modified with the use of pliable and expansile mounting frames.

METHODS

In finite element methods, the volume of an ana- lysed structure is subdivided into small but finite subvolumes called elements. If these elements are of finite thickness and curved, they can approximate the shape of the valve very well. In our approach, we have chosen 20-node isoparametric brick elements for both the valve leaflets and the supporting stent. The shape of the valve leaflet was adapted from existing clinically available pericardial valves, as described by Quijano (1989). The leaflets were symmetrical, with a top free- edge angle of 17.5” and a bottom angle of 36.5”. The shape of the cusp was spherical, with a flattened coaptation region (Fig. 1). Coaptation height was 6.4 mm at the commissures, and zero at the centre, where all three cusps came together. The leaflets were assumed to be 0.5 mm thick, and the connection between the stent and the leaflets was assumed to be perfect, in the sense that nodal displacements at the attachment surface were identical for the leaflet and the stent.

Our modelling approach utilized non-linear con- tinuum mechanics to describe the highly non-linear material characteristics of the valve leaflets, and the large deformations that occur during valve opening. We have used true three-dimensional brick elements, rather than shell elements, for accurately simulating leaflet stresses (as suggested by Black et al., 1991), and have paid particular attention to simulating contact between the cusps during valve opening and closure to model valve coaptation (Black et al., 1991). We have also relied heavily on experimentally acquired data of flexural stiffness to model properly the valve opening process and realisticatty simulate the magni- tude of flexural stresses.

The finite element model was deformed by a time- varying physiological pressure waveform, incremen- tally applied to both the inflow and outflow surfaces of the valve leaflets. The deformed shape of the pros-

thetic valve was solved iteratively after each incremen- tal increase in applied pressure. The equilibrium shape of the valve at time t was characterized by the principle of virtual displacements referred to the in- itial unstressed configuration, as given by the relation

+ ‘fT6(uT- U,)dS. r J sc

In equation (1) Sij and Eij denote components of the second Piolla-Kirchoff stress tensor and the Lag- range strain tensor, respectively. The f: and fi are components of the externally applied body and sur- face force vectors due to the hydrostatic force, and the 6Ui symbolize the ith component of the virtual dis- placement vector that deforms the mesh. The last two terms on the right-hand side define an additional constraint due to the contact condition, which pre- vents any point in space to be in the interior of more than one body. The use of such a contact algorithm enabled realistic simulations of the leaflets in contact both during opening and closure. Rather than con- straining the finite element mesh not to deform past a given point, as done by Hamid et al. (1986), Christie and Medland (1982) and Huang et al. (1990), we used a Lagrangian multipliers approach to simulate con- tact, as first described by Chaudhary and Bathe (1985). In this approach, the distance between con- tacting bodies is measured by a gap function. As any points of the body come into contact, the value of the gap function for the points decreases and eventually reaches zero when contact is established. The Lagran- gian multipliers are solved at every node in the vicin- ity of the coaptation region, but the coaptation region does not have to be known exactly beforehand. If the nodes slide, the ratio of the tangential to normal force is determined by the coefficient of friction and the nodal forces are adjusted appropriately. A coefficient of friction of zero was selected to simulate the absence of friction at the leaflet interface. The material for the valve leaflets, the pericardium, was assumed to be of uniform thickness and isotropic. While some investi- gators have reported that pericardium is mildly an- isotropic in its extensibility (Crofts and Trowbridge, 1988; Lee et al., 1989), there is no evidence that pericardial valves are constructed to accommodate this mild anisotropy in a systematic way. The material model also assumed that the pericardium was elastic in the sense that during a closed deformation cycle the strain energy is stored and released so that no net work is done on the body. The strain energy function of the material can, therefore, be expressed in terms of invariants of tensor E,

w= W(II E ,I~ E V ~ ~ E), (2)

Numerical simulation of leaflet 931

and the stress-strain relationship as

S, = a(.&)“, (3)

where S,=(31;,) ‘I2 and E,=2(1;,/3)“‘, and are referred to as effective stress and effective strain, respectively. The terms 1 is and I;, represent the second invariants of stress and strain deviators. The strain potential function associated with equation (2), under multiaxial stress, has the form

- - W1E112E.13E)= W1E,&E)+XW3E), (4a)

where

@=[a/(a+l)]E:+ (4b)

and

i2E=12e-2(1,,-i). (4c)

The last term in equation (4a) reflects the near incom- pressibility of the material (Cescotto and Fonder, 1979). The positive constant 1 represents the bulk modulus of the material. The material parameters appearing in equation (3) were estimated as a=650 and n =4.0, from curves obtained experimentally (Fig. 2). and modified as explained below.

A realistic evaluation of valve leaflet deformation requires a relationship between stress (S) and strain (E) tensors that is based on experimental evidence. A large volume of experimental data on the stress-strain behaviour of pericardium is available from the literature (Pereira et al., 1990; Trowbridge and Crofts, 1986; van Noort et al., 1982), and values for parameters a and n can be fitted relatively well. Experimental data at very low strains and loads, how- ever, are imprecise. In fact. most experimentalists be- gin measuring the elastic response of test strips only

above loads of 0.5 g (Pereira et al., 1990). For a 5 mm wide specimen 0.5 mm thick, a 0.5 g load corresponds to a stress of 1.9 kPa. If data below this value are neglected, the length of the ‘toe’ region of the stress-strain curve can be greatly underestimated. Indeed, the measured extensibility of soft tissues va- ries between laboratories (Lee et al., 1989; Trowbridge and Crofts, 1986; van Noort et al., 1982). The extensi- bility of the tissue and the general shape of the stress-strain curve at stresses lower than 2 kPa is, therefore, unlikely to be very precise. This lack of good experimental data may not be important for modelling valve closure but it becomes a serious prob- lem for modelling cusp flexure.

Because we wished to simulate leaflet bending usmg realistic flexural mechanics, we chose to incorporate direct experimental measurements of bending stiff- ness. These data were previously acquired by us for glutaraldehyde-treated tissues (Vesely ef al.. 1988). and were used to calculate an effective elastic modulus which could be used to augment the gap in the stress-strain curves of pericardium at low strains. The bending stiffness of a 5 mm wide, 0.5 mm thick strip of glutaraldehyde-treated aortic valve tissue was found to be 12 nNm* (Vesely et al., 1988). Using the simple beam theory as a first approximation, the effective elastic modulus of such a material would be 230 kPa. The elastic modulus calculated at 5% strain from published data can range from a low of 200 kPa for Pereira et al. (1990), to a high of 1.0 MPa for van Noort et al. (1982). The use of such stress-strain relationships, in our view, would greatly overestimate the bending stiffness of the material at strains less than 5% and produce erroneous results in simulating leaf- let flexure. Our modified stress-strain response, based in part on experimental data of van Noort er irl. (1982), gives an elastic modulus of 325 kPa at 5%

1 ,

. Experimental Data -.__ Fitted Function

0.8

0.2

0.1 Strain

9.

Fig. 2. The stressstrain curve used for the modelling of the bovine pericardium. A power approximation fitted to the experimental data obtained from van Noort et al. (1982). and the shape of the curve was modified in the extremely low strain portion (below 5% strain) to reflect the very low bending stiffness of the

material.

932 S. KRUCINSKI et nl

strain, and drops exponentially towards zero strain such that at 2% strain the elastic modulus is only 20.8 kPa. By comparison, the lowest elastic modulus that can be calculated from published stress-strain curves is roughly100 kPA at 2% strain (Pereira et al., 1990). We found that this refined material model, based on experimentally obtained bending stiffness data, enabled the valve to open at physiological pres- sures, while a direct implementation of uniaxial stress-strain data produced a valve that did not open at all. It should be noted that Huang’s two-dimen- sional model (Huang et al., 1990), the only other finite element simulation that modelled complete valve opening, also used elastic moduli much lower than those observed experimentally with uniaxial tensile tests.

The physiological pressure waveform used for this simulation was adapted from a Medical Physiology reference book (Ganong, 1983) (Fig. 3). The peak pressure gradient was 87 mmHg (11.6 kPa) in dias- tale, 2.0 mmHg (0.27 kPa) in systole, the pressures were incremented in 10 ms steps, and the solution was iterated to convergence prior to each successive step for 1: cycles.

Three examples of stent designs were simulated to examine the effects of rigid, pliable and expansile stents on the distribution of leaflet stresses. A rigid stent was simulated by setting all nodal displacements of the stent to zero. This effectively prevented any leaflet motion along the line of attachment between the stent and the valve cusps. A pliable stent was simulated by setting the material of the stent to be linear and isotropic, with a Young’s modulus of 1.0 GPa and a Poisson’s ratio of 0.40. This material model roughly approximated a stent made of high- density polyethylene. For the third configuration, the stent posts were prescribed to pivot outward during systolic valve opening. The bottoms of the posts were

Pressure Waveform

“0 150 300 450 600 750

The in milliseconds

Fig. 3. Plot of the time-varying pressures simulated on the ventricular and aortic surfaces of the prosthetic valve. A maximal pressure gradient of 2 mmHg was assumed dur-

ing the opening phase of the valve.

hinged at the rigid base of the stent, and the tops of the stent posts moved radially outward, away from the vertical axis of symmetry of the valve. This is a very practical design that could be duplicated by affixing the tops of the stent posts to the recipient aorta. The amount of outward radial movement was set at lo%, to reflect the expansion observed experi- mentally (Vesely et al., 1990b). No inward motion of the stent $osts was permitted in diastole since, in this simulation, we wanted to examine only the outward motion of the stent.

RESULTS

Rigid scent

In the closed position, a tensile major principal stress of 350 kPa was observed in the commissural region [Fig. 4(a)]. Such stresses are induced at the commissures due to the tension that is generated in the valve leaflets in the closed position, and is consis- tent with all previous mathematical simulations of closed valves (Black et al., 1991; Christie and Med- land, 1982; Hamid et al., 1985; Rousseau et al., 1988). In the fully open configuration, a compressive major principal stress of 250 kPa was observed, again in the commissural region. This compressive stress resulted from the sharp circumferential bending of the leaflet at that location [Fig. 4(b)]. Compressive stresses induced by leaflet flexure are, therefore, of magnitude comparable to tensile stresses and likely contribute to material failure through a compressive buckling phe- nomenon (Vesely et al., 1988).

Pliable stent

In this simulation, the stent posts deflected inward by 3.6% of the top radius (0.37 mm) during diastole and outward by 4.9% (0.5 mm) during systole. Be- cause of the inward deflection of the pliable stent posts in systole, significant compressive stresses were induced near the central coaptation area [Fig. 5(a)], This was accompanied by some wrinkling of the leaf- let free edge [Fig. 5(b)]. During opening, this stent design produced compressive stress at the commis- sural region very similar to that of the rigid stent [Fig. 5(a)]. This simulation has, therefore, demonstrated that our very simple flexible stent design not only disturbed the coaptation process but also restrained the outward radial motion of the valve cusps. Such a stent design, therefore, offers no appreciable reduc- tion in compressive stresses during the opening phase of this type of valve.

Pivoting stent posts

The results of this numerical simulation showed that an outward movement of the stent posts, equiva- lent to a 10% radial expansion of the commissural diameter, considerably reduced the circumferential leaflet curvatures at the commissures when the valve

Fig

1. O

bliq

ue

and

outfl

ow

view

s of

the

mod

el

peric

ardi

al

valv

e us

ed

in t

he f

inite

el

emen

t sim

ulat

ions

. Th

e ge

omet

ry

of t

he v

alve

w

as

appr

oxim

ated

fro

m

publ

ished

in

form

atio

n on

a s

ize

25 A

(2

4.34

mm

ba

se

diam

eter

, 20

.7 m

m

top

diam

eter

) Ca

rpen

tier-E

dwar

ds

bovi

ne

peric

ardi

al

xeno

graf

t. O

ther

th

an

the

size

and

appr

oxim

ate

geom

etry

, no

ot

her

desig

n fe

atur

es

wer

e du

plic

ated

.

Clo

sed

Rig

id S

tent

Ope

n Fi

g. 4

(a).

Stre

ss m

ap o

f a v

alve

with

a r

igid

ste

m i

n th

e cl

osed

and

ope

n co

nfig

urat

ions

. A

map

of

the

maj

or p

rinci

pal

stres

ses

is su

perim

pose

d on

one

hal

f of

the

unde

form

ed l

eafle

t for

the

sak

e of

cla

rity.

The

rig

ht e

dge

of th

e le

afle

t is

the

line

of s

ymm

etry

. Re

d de

note

s th

e co

mpr

essiv

e st

ress

es th

at a

re p

oten

tially

da

mag

ing

to

biop

rosth

etic

tis

sues

. Not

e th

e hi

gh c

once

ntra

tion

of co

mpr

essiv

e str

ess

at th

e re

gion

of t

he c

omm

issur

es

in th

e op

en c

onfig

urat

ion.

Th

e ste

nt h

as z

ero

stres

s sjn

ce it

was

as

sum

ed t

o be

per

fect

ly r

igid

and

, th

eref

ore,

did

not

defo

rm.

The

colo

ur b

ar d

enot

es s

tress

es i

n kP

a.

Rigid Stent

Fig. 4(b). An oblique view of the whole valve in the closed and open configuration. Note the high reverse flexing of the leaflets in the open configuration and the sharp curvatures near the stent posts.

Clo

sed

Plia

ble

Sten

t

Ope

n Fi

g. 5

(a).

Stre

ss m

ap o

f a v

alve

with

a p

liabl

e ste

nt i

n th

e cl

osed

and

ope

n co

nfig

urat

ions

. Th

is va

lve

had

a ste

m t

hat d

efle

cted

onl

y by

the

actio

n of

the

leaf

lets

push

ing

the

posts

out

war

d. N

ote

the

high

con

cent

ratio

n of

com

pres

sive

stres

s ne

ar t

he c

entra

l co

apta

tion

regi

on, d

ue to

the

inw

ard

mov

emen

t of

the

stent

pos

ts du

ring

valv

e cl

osur

e. A

lso n

ote

that

suc

h a

plia

ble

stent

did

not

sig

nific

antly

red

uce

the

com

pres

sive

com

miss

ural

str

esse

s in

the

open

con

figur

atio

n.

The

defle

ctio

n of

the

stem

in

dias

tole

cr

eate

d th

e hi

gh t

ensil

e an

d co

mpr

essiv

e str

esse

s.

Pliable Stent

Fig. 5(b). An oblique view of the whole valve in the closed and open configuration. Note the sharp curvatures near the stent posts that were not eliminated by this type of stent design.

93-l

Piv

otin

g Stent

Posts

Closed

Ope

n Fi

g. 6

(a).

Stre

ss m

ap o

f a v

alve

with

piv

otin

g ste

nt p

osts.

In

this

simul

atio

n, t

he s

tent

pos

ts w

ere

pres

crib

ed t

o pi

vot

outw

ard

abou

t th

e ba

se s

uch

that

the

sep

arat

ion

of th

e co

mm

issur

es i

ncre

ased

by

10%

dur

ing

systo

le.

Not

e th

e de

crea

se i

n co

mpr

essiv

e str

ess

at t

he c

omm

issur

es,

com

pare

d to

the

pre

viou

s tw

o de

signs

. Si

nce

the

stent

was

pr

escr

ibed

to

mov

e on

ly d

urin

g sy

stole

, it

beha

ves

like

a rig

id s

tent

in

dias

tole

and

is,

ther

efor

e, no

t str

esse

d.

Pivotiner Stent Posts

Fig. 6(b). An oblique view of the whole valve in the closed and open configuration. Note the wide leaflet excursion, and the much more gentle free-edge curvatures of this stent design.

939

Sim

ula

tion

R

eal V

alve

Fig.

7.

Com

paris

on i

mag

es o

f our

fin

ite e

lem

ent

simul

atio

n of

a p

eric

ardi

al v

alve

in th

e op

en c

onfig

urat

ion

and

a pe

ricar

dial

val

ve o

pen

in a

pul

se d

uplic

ator

sy

stem

. N

ote

the

simila

rity

of

the

flexu

re p

atte

rn

at t

he f

ree

edge

. The

im

age

on t

he r

ight

has

bee

n re

prin

ted

by k

ind

perm

issio

n of

Sor

in B

iom

edic

a.

Numerical simulation of leaflet 941

was fully open [Fig. 6(b)]. The amount of compres- sive stressing of the valve leaflets during opening decreased as well [Fig. 6(a)]. Compared with a major principal stress of 250 kPa for a rigid stent, this simu- lation produced a major principal stress of 150 kPa, a reduction of 40%. This reduction in compressive stress, however, was not accompanied by an increase in tensile stress at the centre of the cusp free edge, as would be expected if the radial movement of the stem posts produced leaflet tension.

DISCUSSION

The relatively good durability of some biopros- thetic valves, and the premature failure of others, is likely related to design parameters. The early failure of pericardial valves employing an alignment stitch at the commissures is a typical example of a design feature that proved to be catastrophic (Walley and Keon, 1987; Walley et al., 1987). We feel that another detrimental feature of all stent mounted tissue valves is the inability of the supporting stent to expand radially with the recipient aortic root. The natural aortic root has clearly been shown to expand during the ejection phase of the cardiac cycle (Brewer et al., 1976; Thubrikar et al., 1980; Vesely et al., 1990b), yet no existing valve mounting frame has addressed this phenomenon. Presently. most supporting stents have pliable stent posts that aim to reduce shock loading on the leaflets during valve closure. No stent design, to our knowledge, has attempted to reduce the com- missural flexing that damages valve leaflets (Vesely et al., 1988). and contributes to leaflet tearing in the absence of calcification (Ishihara et al., 1981: Pomar et al.. 1984; Stein et al., 1985). We have, therefore, simulated the flexure patterns that occur during valve opening with our three-dimensional modelling capa- bility, and have examined how varying the design of the stent can affect leaflet stresses.

This model, like most numerical analyses of biop- rosthetic valves. has three primary limitations: (i) it does not mimic the interaction of fluid flow with tissue deflection; (ii) it is an elastic model that does not take into account the viscous nature of the valve tissue; and (iii) it employs a solid mechanics approximation of a composite material that has poorly characterized constituents. The principal advantage of this model over works published to date, however, is that it simulates flexure patterns in three-dimensions during complete valve opening. The ideal simulation that could duplicate the complexity of the leaflet motion during valve opening would require a coupling of the fluid flow during the ejection phase to the motion of the valve leaflets in a mutually dependent way. Such a coupled simulation. lo our knowledge, has never been done on heart valves because of the difficulty of the problem, and the large computational power re- quired to do so. An uncoupled simulation that would deflect the leaflets by a prescribed flow field was not

possible because sufficient experimental data are un- available in the literature, and because such an ap- proach could not simulate stent-leaflet interactions. Because of their high pliability, the systolic leaflet motion is likely affected the most by pressure fluctu- ations resulting from unsteady fluid flow through the open valve. Our modelling approach, therefore, can- not simulate the subtle undulations and flutter of the valve leaflets during systole. Nevertheless, since final leaflet curvatures are ultimately the factor that gener- ates the pattern of three-dimensional stressing in the open configuration, it is reasonable to simulate the opening phase in the best way possible without using a solid-fluid interaction, as long as the patterns of leaflet curvature compare well with experimental data. Indeed, the orifice view of our simulated valve with a rigid stent compares very well with that of a pericardial valve functioning in a pulse duplicator (Fig. 7). If the patterns of the free-edge curvature are similar between the model and the real valve, and the model utilizes a reasonable material description, it is safe to assume that the patterns of leaflet stress are also realistic.

The only major difference between the circumferen- tial curvatures produced by our model and those of the real pericardial valve is at the centre of the coapta- tion region. We assumed the zero-stress-state config- uration to be that shown in Fig. 1. In reality, this is not the case. In the manufacture of the prosthesis, the pericardial leaflets are actually pushed together and bent into the shape shown in Fig. 1. The valve is. therefore, pre-stressed even when it is at zero pressure, and will open to a more circular shape without the central peak. This initial pre-stress, however, is limited to the central portion of the free edge and does not affect the curvatures induced elsewhere in the leaflets. Our model also had somewhat more radial bending, but that did not affect the circumferential curvatures at the free edge. This model, therefore, promises to be a good analytical instrument for evaluating the curva- tures and induced flexural stresses in prospective bioprosthetic valve designs, and for providing an in- sight into the mechanical behaviour of biological tis- sues.

The generated stress maps (Figs 4-6) indicate that flexural stresses are, indeed, induced near the commis- sures and within the free margin during valve opening. The magnitude of these compressive stresses is com- parable to those presented by Huang et al. (1990) with a two-dimensional simulation, and to those of Black et al. (1991), who modelled a bicuspid pericardial valve. We have shown previously that such compres- sive deformations damage bioprosthetic valve mater- ial, likely through a compressive buckling process (Vesely and Boughner, 1989). This numerical simula- tion, therefore, further supports the hypothesis that an important mode of mechanical failure of tissue valves is flexural damage induced during valve opening (Vesely and Boughner, 1989; Vesely et al., 1988). In-

942 S. KRUCINSKI et al.

deed, if one examines the patterns of tears on ex- planted pericardial heterografts and porcine xenog- rafts (Gabbay et al., 1988; Ishihara et al., 1981; Stein et al, 1985; Wheatley et al., 1987), there is a high correla- tion between the sites of tears and the location of stress concentrations produced by our simulation.

These simulations have also demonstrated that valves with stent posts that pivot about their attach- ment to the base can significantly reduce leaflet tlexure and the associated leaflet stresses. While addi- tional iterative studies need to be performed to deter- mine the optimal amount of expansion that can re- duce compressive stresses the most, a 10% radial expansion at the commissures appears to be adequate. Conversely, a simple, pliable stent support that relies on the compressive forces generated within the valve cusps to deflect the stent posts outward does not provide a sufficient amount of stress reduction. While it can be argued that a stent constructed from a more pliable material would have deflected more and could have performed better in systole, a more pliable stent would have collapsed under diastolic loads. In our simulation, the stent posts deflected inward by 0.37 mm, increasing the free-edge angle from 17.5 to 19”. A greater stent post flexibility would have de- formed the leaflets to such an extent that leaflet coap- tation during diastole would be compromised. With this engineering exercise, we have shown that one type of stent design can significantly reduce commissural stressing, while another equally plausible design can- not. It is, therefore, clear that detailed mathematical simulations of preliminary valve designs can provide valuable contributions to the development of new bioprosthetic valves. Perhaps, the best stent would be one that functions in harmony with the patient’s aor- tic root. Ideally, the tops of the stent posts should be fastened to the recipient aortic root such that the expansion of the aorta during ventricular contraction will pull the tops of the stent posts outward with it. Such motion will enable the reduction of leaflet stres- ses during valve opening and may reduce the inci- dence of leaflet tearing at the flexure points.

Because of the inherent simplicity of pericardial tissue relative to aortic valves leaflets, we chose to investigate the pericardial valve first. Such an ap- proach, however, does not need to be limited to pericardial valves. Our primary interest lies in the porcine aortic valve xenograft, with its more physiolo- gical construction. The mechanism that we propose to reduce flexural stresses at the commissures already exists in natural aortic valves. The commissures of

aortic valves naturally move outward during systolic valve opening as the root expands (Brewer et al., 1976; Thubrikar et al., 1980; Vesely et al., 1990b). It is, therefore, logical to assume that the mechanism of stress reduction proposed for the pericardial valve can be readily applied to porcine aortic valve xenografts as well. It should be noted, however, that because of the much greater complexity of the natural aortic valve (Vesely and Noseworthy, 1992) many of the

assumptions applicable to pericardial valves, such as material isotropy and geometrical symmetry, cannot be applied. The concept of pivoting stent posts, how- ever, is the rational application of a mechanism that exists in natural aortic valves. If such a stent design can significantly reduce flexural stresses in pericardial valves, it is likely to perform equally well in porcine xenografts.

Acknowledgement-This research was supported in part by a Strategic Grant from the Natural Sciences and Engineering Research Council of Canada, and by the Heart and Stroke Foundation of Ontario. Dr Vesely is a Research Scholar of the Foundation. The authors are grateful to Dr D. R. Bough- ner, from the University of Western Ontario, for his advice and discussion during the course of this work.

REFERENCES

Black, M. M., Howard, I. C., Huang, X. and Patterson, E. A. (1991) A three-dimensional analysis of a bioprosthetic heart’valve. J. Biomechanics 24, 793-801. _

Bortolotti. U.. Milano. A.. Thiene. G.. Guerra. F.. Mazzucco. A., Valente; M., Talenti, E. and Gallucci, V. (1987) Early mechanical failures of the Hancock pericardial xenograft. J. 7’horacic Cardiovasc. Surg. 94, 200-207.

Brewer, R. J., Deck, J. D. and Copohe, B. (1976) The dynamic aortic root: its role in aortic valve function. J. Thoracic Cardiovasc. Surg. 72, 413.

Cataloglu, A., Clark, R. E. and Goulds, P. L. (1977) Refined stress analysis of human aortic heart valves. J. Bio- mechanics 10, 153-158.

Cescotto, S. and Fonder, G. (1979) A finite element approach for large strains of nearly incompressible rubber-like materials. J. Solids Struct. 15, 589-605.

Chaudhary, H. B. and Bathe, K. J. (1985) A solution method for static and dynamic analysis of the three-dimensional contact problems with friction. Comput. Struct. 24, 855-873.

Christie, G. W. and Medland, I. C. (1982) A non-linear finite stress analysis of bioprosthetic heart valves. In Finite Ele- ments in Biomechanics (Edited by Gallagher, R. H.). Wiley, New York.

Crofts, C. E. and Trowbridge, E. A. (1988) The tensile strength of natural and chemically modified bovine pericardium. J. biomed. Mater. Res. 22, 89-98.

Gabbay, S., Kadam, P., Factor, S. and Cheung, T. K. (1988) Do heart valve bioprostheses degenerate for metabolic or mechanical reasons? J. Thoracic Cardiovasc. Surg. 95, 208-2 15.

Gallo, I., Nistal, E., Arbe, E. and Artinano, E. (1988) Com- parative study of primary tissue failure between porcine (Hancock and Carpentier-Edwards) and bovine pericar- dial (Ionescu-Shiley) bioprostheses in the aortic position at five- to nine-year follow-up. Am. J. Cardiol. 61,812-g 16.

Gallo, I., Nistal, F. and Artinano, E. (1986) Six to ten-year follow-up on patients with the Hancock cardiac biopros- theses. J. Thoracic Cardiovasc. Surg. 92, 14-20.

Ganong, W. F. (1983) Review of Medical Physiology. Lange Medical Publications, Los Altos, California.

Golomb, G., Dixon,M., Smith, M. S. and Schoen, F. J., et al. (1987) Controlled-release drug delivery of diphosphonates to inhibit bioprosthetic heart valve calcification: release rate modulation with silicone matrices via drug solubility. J. Pharm. Sci. 76,271-276.

Hamid, M. S., Sabbah, H. N. and Stein, P. D. (1985) Finite element evaluation of stresses on closed leaflets of biopros- thetic heart valves with flexible stents. Finite Elements Anal. Des. 1, 213-225.

Numerical simulation of leaflet 943

Hamid, M. S.. Sabbah, H. N. and Stein, P. D. (1986) Influ- Thubrikar, M. J., Aouad, J. and Nolan, S. P. (1986) Patterns ence of stem height upon stresses on the cusps of closed of calcific deposits in operatively excised stenotic or purely bioprosthetic valves. J. Biomechanics 19, 759-769. regurgitant aortic valves and their relation to mechanical

Huang, X., Black. M. M., Howard, I. C. and Patterson, E. A. stress. Am. J. Cardio!. 54, 304-308. (1990) A two dimensional finite element analysis of a biop- Thubrikar, M., Nolan, S. P., Bosher, L. P. and Deck, J. D. rosthetic heart valve. J. Biomechanics 23, 753-762. (1980) The cyclic changes and structure of the base of the

Ishihara, T., Ferrans. V. J., Boyce, S. W., Jones, M. and aortic valve. Am. Heart J. 99, 217-224. Roberts, W. C. (1981) Structure and classification of cuspal Trowbridge, E. A. and Crofts, C. E. (1986) The standardisa- tears and perforations in porcine bioprosthetic cardiac tion of gauge length: its influence of the relative extensibil- valves implanted in patients. Am. J. Cardiol. 48, 665-678. ity of natural and chemically modified pericardium.

Lee, J. M., Boughner, D. W. and Courtman, D. W. (1984) J. Biomechanics 19, 1023-1033. The glutaraldehyde-stabilized porcine aortic valve xeno- Trowbridge, E. A. and Crofts, C. E. (1987) Pericardial hetero- graft. II. Etfect of fixation with or without pressure on the graft valves: an assessment of leaflet stresses and their tensile viscoelastic properties of the leaflet material. implications for heart valve design. J. biomed. Engng 9, J/. hiomed. Mater. Res. 18. 79-98. 345-355.

Lee, J. M., Haberer, S. A. and Boughner. D. R. (1989) The bovine pericardial xenograft: I. Effect of fixation in al- dehydes without constraint on the tensile viscoelastic properties of bovine pericardium. J. hiomed. Mater. Res. 23, 457475.

Magilligan, D. J.. Lewis. J. W., Tilley, B. and Peterson, E. (1985) The porcine bioprosthetic valve. Twelve years later. J. Thoracic Cardiovasc. Stag. 89, 4999507.

Noort. R. van. Yates, S. P., Martin, T. R. P., Barker, A. T. and Blact. M. M. (1982) A study of the effects of glutaral- dehyde and formaldehyde on the mechanical behaviour of bovine pericardium. Biomaterials 3, 21-26.

Pathak, Y. V.. Boyd. J., Levy, R. J. and Schoen. F. J. (1990) Prevention of calcification of glutaraldehyde pretreated bovine pericardium through controlled release polymeric implants: studies of Fe3+, A13+, protamine sulphate and levamisole. Biomaterials 11, 718-723.

Pereira. C. A.. Lee. J. M. and Haberer, S. A. (i990) Effects of alternative crosslinking methods on the low strain rate viscoelastic properties of bovine pericardial bioprosthetic material. J. hiomed. Mater. Res. 24, 345361.

Pomar. J. L., Bosch, X., Chaitman, B. R., Pelletier, C. and Grodin. C. M. (1984) Late tears in leaflets of porcine bioprostheses in adults. .4nn. Thoracic Surg. 37, 78883.

Quijano, R. C. (1989) Engineering and design considerations of the Carpentier-Edwards pericardial valve. In Proceed- ings of‘ the Carpentier-Edwards Pericardial Bioprostheses Mini-Symposium. pp. 2740. Luzern, Switzerland.

Rousseau. E. P. M., van Steenhoven, A. A. and Janssen, J. D. (1988) A mechanical analysis of the closed Hancock heart valve prosthesis. J. Biomerhanics 21, 545-562.

Song, T., Vesely, I. and Boughner, D. R. (1990) ElTects of dynamic fixation on the shear behaviour of porcine xenog- raft valves. Biomaterials II, 191-196.

Stein, P. D., Kemp. S. R., Riddle, J. M., Lee, M. W., Lewis, J. W. and Magilligan, D. J. (1985) Relation of calcification to torn leaflets of spontaneously degenerated porcine biop- rosthetic valves. Ann. Thoracic Surg. 40, 175-180.

Teoh, K. H., Ivanov, J.. Weisel, R. D., Darcel, I. C. and Rakowski, H. (1989) Survival and bioprosthetic valve fail- ure. Ten-year follow-up. Circulation 80 (Suppl. I). 18-115.

Thiene. G., Laborde. F., Valvente. M., Bical, 0.. Talenti, E., Bortolotti. U. and Gallix, P. (1986) Experimental evalu- ation of porcine-valved conduits processed with a cal- cium-retarding agent (Tb). J. Thoracic Cardiooasc. Stag. 91.215-224.

Trowbridge, E. A., Lawford, P. V., Crofts, C. E. and Roberts, K. M. (1988) Pericardial heterografts: why do these valves fail? J. Thoracic Surg. 95, 577-585.

Vesely, I. (1991) Analysis of the Medtronic INTACT bioprosthetic valve: effects of ‘zero pressure’ fixation. .I. Thoracic Cardiovasc. Surg. 101, 90-99.

Vesely, I. and Boughner, D. R. (1989) Analysis of the bending behaviour of porcine xenograft leaflets and of natural aortic valve material: bending stiffnes, neutral axis and shear measurements. J. Biomechanics 22, 655-671.

Vesely, I., Boughner, D. R. and Song, T. (1988) Tissue buck- ling as a mechanism of bioprosthetic valve failure. Ann. Thoracic Surg. 46, 302-308.

Vesely, I., Gonzalez-Lavin, L., Graf, D. and Boughner, D. R. (1990a) Mechanical testing of cryopreserved aortic allografts: comparison with xenografts and fresh tissue. J. Thoracic Cardiovasc. Surg. 99, il9-123.

Veselv. I.. Krucinski. S. and Camnbell. G. (1992) Microm- echanics and mathematical mbdelling: an inside look atbioprosthetic valve function. J. Cardiac Surg. 7, 85-95.

Vesely, I., Menkis, A. and Campbell, G. (1990b) A com- puterized system for video analysis of the aortic valve. IEEE Trans. biomed. Engng 37, 925-929.

Vesely, I., Menkis, A. H., Rutt, B. and Campbell, G. (1991b) Aortic valve/root interactions in porcine hearts: implica- tions for bioprosthetic valve sizing. J. Cardiac Surg. 6, 482489.

Vesely, I. and Noseworthy, R. (1992) Micromechanics of the fibrosa and ventricularis of aortic valve leaflets. J. Biomechanics 25, 101-113.

Vesely, I., Noseworthy, R. and Wilson, G. (1991a) Develop- ment of a hybrid xenograft/autograft aortic valve biop- rosthesis. Can. J. Card& 7; (Sup& A), 83A.

Walley. V. M. and Keon, W. J. (1987) Patterns of failure in Ionescu-Shiley bovine pericardial bioprosthetic valves. J. Thoracic Cardiovasc. Surg. 93, 925-933.

Wallev, V. M.. Bedard. P.. Brais. M. and Keon. W. J. (1987) Valie failure caused by cusp tears in’ low-profile Ionescu-Shiley bovine pericardial bioprosthetic valves. J. Thoracic Cardioeasc. Surg. 93, 583-586.

Wheatley, D. J., Fisher, J., Reece, I. J., Spyt, T. and Breeze. P. (1987) Primary tissue failure in pericardial heart valves. J. Thoracic Cardiovasc. Surg. 94, 367-374.