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W. MILTON SWANSON and RICHARD E. CLARKPressure

Dimensions and Geometric Relationships of the Human Aortic Value as a Function of

Print ISSN: 0009-7330. Online ISSN: 1524-4571Copyright © 1974 American Heart Association, Inc. All rights reserved.is published by the American Heart Association, 7272 Greenville Avenue, Dallas, TX 75231Circulation Research

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Dimensions and Geometric Relationships of the Human

Aortic Value as a Function of Pressure

W. Milton Swanson and Richard E. Clark

ABSTRACTIn a continuing effort to develop improved prosth etic h ear t valves, a redefini-

tion of the anatomy of the human aortic valve as a function of stress wasundertaken. Dimensions and geometric relationships of the hum an aortic valveas a function of intraaortic pressure between 0 and 120 mm Hg were obtainedfrom a series of silicone rubber valve casts. The axial length of the valve regionwas found to vary negligibly with press ure, but significant variations ingeometry and angular dimensions were seen. The leaflet atta chm ent annulusforms an ellipse at the plane of intersec tion with th e cylindrical surfacepassing from the left ven tricular tra ct through the aorta. Deductions fromstress considerations for the measured geometry indicate that the loadedleaflet is a section of a cylindrical surface. The equation for this developedsurface was obtained, and a prosthetic design was determined using averagevalues at 100 mm Hg. The leaflet is developable onto a plane with a cutrequired along part of the junction line between the initially cylindrical partand the plane coapting surfaces. Optimum valve shape mandates a base anglebetween the cylindrical leaflet and the center axis of 70° a =20-22°, where a isthe leaflet angle).

KEY WORDS aortic valve structu releaflet sha pe and d imensionsprosthetic valve design

aortic modulusstresses in valve leaflets

• An accurate definition of the geometry of

the aortic valve is necessary prior to develop-

ment and fabrication of a prosthetic valve.

As part of a program to determine the geom-

etry and stru ctu re of the human aortic

valve, silicone rubber molds were cast under

pressure. Measurements considered to be im-

portant were made and analyzed. Prelimi-

nary studies in our laboratory have demon-

strated the sensitivity of in-plane stresses to

the geometry of this structure during dias-

tole and systole (1). Previous investiga tions

by Wood et al. (2) and Sauvage et al. (3)

utilized pig hearts and a freezing technique

under pressure. Recently, Mercer et al. (4)

have investigated the geometry of the hu-

man aortic leaflet via a molding technique at100 mm Hg of pressure. The present paper is

a report on our 2-year investigation of the

geometry and proportionalities of the human

aortic valve from which important design

conclusions can be drawn. Accurate knowl-

From the Departments of Mechanical Engineering andCardiothoracic Surgery, Washington University, St.Louis, M issouri 63130.

This work was supported in part by U- S. Public HealthService Grant HL-13803 from the National Heart andLung Institute .

Received January 31, 1974. Accepted for publication

August 8, 1974.Circulation R esearch VoL 35 December 197k

edge of valve and sinus region geometry is

required for flow calculations yielding infor-

mation on leaflet motion during opening and

closing (5).

Methods

Fresh human hearts were obtained at autopsy,stored at 4°C, and used within 1-3 days afterdeath. The specimens consisted of two to threediameters of aorta beyond the sinuses of Valsalvaand one-half to one diameter of tissue on the leftventricle side. The aor ta was held with th reehemostats hung on ring-stand hook arms. Then,40-50 ml of low-viscosity room temperature-vul-canizing silicone rubber (RTV GE-11) was pre-pared. The coronary ar ter ies w ere at first tied off,bu t it was late r found t ha t coronary leak age couldbest be eliminated by plugging them with siliconerubber beads, 4-5 mm in diameter. Part of thesilicone rubber was injected into the sinus pocketswith a 20-ml syringe and a 6-cm tube extension toallow filling from the bottom up to eliminate airpockets. When the preparation was nearly full, agrooved stopper with a 5-cm length of glass tubein it was slowly pushed into the aorta, filling thetube. The aorta was secured to the stopper withumbilical tape around the groove. The remainingsilicone rubber was poured into a large reservoirsyringe connected to the stopper tube with a shortpiece of flexible tubing, and the reservoir syringewas then suspended on a ring s tand. A tubethrough a s topper in the top of the reservoirsyringe was connected thr ough a T-tube to a

87 1

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872 SWANSON, CLARK

standard sphygmomanometer. The pressure wasgradually increased to the desired casting pres-sure and maintained during th e cure. A jar ofsaline was placed around the aorta to maintain amoist condition and a tem pe ra tu re of 37°C. Thisslightly warm temperature also accelerated thecure. A period of about 2 hours was required for aminimum stable-dimension c ure. The cast wasremoved, and the procedure was repeated at thenext pressure. With this technique, five or sixcasts could be made in 1 day. The d eterio ratingeffect of the casting procedure on the elasticproperties of the ao rta was determined by repeat-ing the preparation of a cast at 20 mm Hg. Thischeck cast was made on th ree series (series 5 and6 of Table 2 and one other) after the last cast atmaximum pressure had been made. No significantdimensional variations were found. One serieswas also repeated at 50 mm Hg, and no significantvaria tion s were noted. The series 8 casts are

shown in Figure 1.

Data on subject aortas are presented in Table 1.The significant dimensions recorded in Tables 2-4were measured with vernier calipers to the near-est 0.1 mm. In some critical cases, several sets ofreadings were taken for one dimension and aver-aged to determine repeatability. The variationsobtained were usually within 0.2 mm or about 1%.Dimensions involving the three separate sinusesand leaflets were average for the three. The non-coronary sinus was usually, but not always, thesmallest.

A profile trac ing of the sinus region in a planeperpendicular to the center axis was made andplanimetered to obtain the maximum sinus areafrom _yvhich an equ iva lent area-averaged diame-ter, d,, was determined. Each cast tracing wasplanimetered ten times to get an acc urate meas-

TABLE 1

Valve Origin

Valve series

45

6T»

Age

(years)

293929

M4f

Sex

FMFFM

urement. The ten measurements usually did notvary by more tha n 1%. When the variation waslarger, more readings were taken. The circum-scribing sinus diameter, dsm, was also recorded .

Nomenclaturec = C o a p t a t i o n ( F i g . 2 ).d = D i a m e t e r ( m m ) (F i g . 2).E = E l a s t i c m o d u l u s ( d yn e s / c m * ).E d = E l a s t i c m o d u l u s b a s e d on a o r t i c d i a m e -

t r a l s t r a i n : E,, = A p / A da /d a ) ( d y n e s /cm

2) .

f = F r e e e d g e ( c m ) ( F i g . 2).

h = Height from ven tricu lar tra ct base

plane to top of annulus fibrosis (Fig.

2).I = Length (cm) (Fig. 2).p = Press ure (dynes/cmJ).x, y = Coordinates.

a = Leaflet angle (Figs. 2, 9).= Angles (Fig. 9). f = Free edge angle (Fig. 2).a = Stress (dynes/cm

2).

FMURE 1

a: Series 8 valve molds, 0 to 100 mm Hg. b: 80-mm Hg mold mated with left ventricle mold.

Circulation Research VoL 35 December 1971







AORTIC VALVE DIMENSIONS AND GEOMETRY 873

SUBSCRIPTS

a = —

m =

s =V

Aorta.Center.Maximum.

Sinus of Valsalva.Ventricular tract.

SUPERSCRIPTS

* = Dimensionless q uan tity referred to dv.# = Dimens ionless q uan t i ty re fe rred to

that quantity at zero pressure.

Results

The valves listed in Tables 1-4 were chosen

for evaluation. Previous series were develop-

mental and had too few casts to yield signifi-

cant data.

The dimensions obtained from cast meas-

urements are listed in Tables 2-4. Figure 2illustrates the valve region and shows repre-

sentative dimensions. Graphic illustrations

of dimensionless variations with pressure

are presented in Figures 3-5. Dimensions

were reduced to dimensionless variables (de-

noted by asterisks) with respect to the diam-

eter, dv, of the left ventricle immediately

below the aortic valve for two reasons: dv is

the flow inlet diameter, and it varies very

little with pressure. The average variation of

dv with pressure from 20 mm Hg to 120 mm

Hg was approximately 10% (Fig. 3).

For calculating the flow conditions

through the valve region (5), the sinus di-

mensions d^*, dsm*, and s* are significant

along with the leaflet length €c* and the

center and maximum coaptation dimensions

Q* and Cm*. dv was measured as the diameter

of an indentation ring formed at the annularattachment (Fig. 2). The inlet flow diameter

is slightly smaller than dv by the thickness of

TABLE 2

Dimensional Quantities

Series

4

5

6

7

8

AverageAverage at 100

mm Hg

P(m m Hg)

0

2040

60

0

20

20

40

60

80

0

20

20

50

80

0

2040

70

10 0

12 0

0

20

40

60

80

10 0

(1 )

d v

(mm)

23.6

25.227.228.022.222.622.222.022.622.623.026.626.824.826.023.5

24.025.026.626.827.023.024.625.024.825.025.0

(2 )

d .

(mm)

19.7

21.623.127.517.818.118.519.921.021.420.123.524.826.528.918.2

19.721.224.024.927.023.025.226.627.030.130.7

(3 )

h

(mm)

20.0

19.320.319.815.115.715.715.715.615.719.520.019.0

19.519.517.8

17.817.618.018.518.017.317.717.317.417.717.8

(4)

<t>

51

4645

37

43

44

43

41

36

39

51

4 5

42

39

34

52

4944

41

39

32

34

32

31

29

25

23

40

32

(5 )

a

23

1920

23

15

15

15

17

20

20

15

17

16

22

24

9

1013

13

15

17

23

23

24

25

27

29

20

22

(6)

E,, x 1 0-3

(dynes/cm1)

2.22.2

2.4

5.3

5.2

5.2

5.0

5.3

1.6

1.6

2.1

2.4

3.33.1

3.2

3.3

3.4

3.0

3.3

3.6

3.8

4.0

3.3

3.5

See text for abbreviations.

Circulation Research, Vol. 35 , December I97i









TABLE 4

Dimensionless Quantities Relative to Zero Pressure Value

Series

4

5

6

7

8

P

(m m Hg)

0

20

40

60

0

20

20

40

60

80

0

20

20

5080

0

20

40

70

10 0

12 0

0

20

40

60

80

10 0

(1 )

d v *

1.001.071.151.241.001.020.990.991.021.021.001.161.17

1.081.131.001.021.061.131.141.151.001.071.091.081.091.09

(2 )

d . '

1.001.091.161.391.001.011.031.111.181.201.001.171.23

1.321.441.001.081.161.321.371.481.001.091.151.171.311.33

3)

h *

1.000.971.020.991.001.041.041.041.031.041.001.030.97

1.001.001.001.000.991.011.041.011.001.021.001.011.021.03

(4 )

d . '

1.001.121.231.301.001.021.021.061.081.101.001.261.28

1.361.481.001.081.131.221.261.271.001.061.131.151.201.23

(5 )

d . n '

1.001.131.231.351.001.051.031.081.131.141.001.251.27

1.341.481.001.061.121.161.191.251.001.081.161.161.201.22

(6 )

fr*

1.001.091.121.171.001.041.041.031.071.041.001.131.10

1.101.141.001.021.041.101.131.141.001.051.021.041.071.08

7)

e

1.000.991.011.081.001.041.051.051.041.051.001.101.12

1.131.201.001.111.161.161.161.231.001.011.031.011.071.07

See text for abbrev i a t i ons .

ferred to as pressure modulus in ref. 6: Ed =Ap/(da/da). The range 1.6 x 10

5 < Ert < 5.3 x

105 dynes/cm1

in Table 2 is in the range ofpublished data for pig aortas (Ed = 2 x 10

s

dynes/cm2 [6]) and for the femoral artery (2 x105

< i^ < 6 x 106 dynes/cm

2 [3, 7]). This

relatively large range of values for a physio-logical property is not unusual . The modulusde te rmined in th is ma nner m ight includeinaccuracies because of the method of deter-mining the strain Ad

a/d

a. The uncertainty for

Ed varied from about 100% a t 20 mm Hg toabout 10% at 80 mm Hg. The max imum localdata variation for Ed values calculated froma smoothed curve of Ap vs. (da* - 1) was 15%.More accurate means determined using spe-cial s t ra in tes t ing apparatus (8) give tru emoduli and yield values of the circumferen-tial modulus E for the aorta of about 4 x 10

6

dynes/cm2 at 10% strain. Values of Ed were

converted to E by m ultiplying by d/2t, whichwas a bout 10 (where t is the wall thickness).The average resulting E of approximately 5

x 10s

dynes/cm2

was close to published da taCirculation Research, VoL 35, December 1S7J,

SYMBOL SERIES

1.4

oD

A

•

•

mY2Lm

0 40 80 120

PRESSURE-p mm Hg)FIGURES

Relative dimensional variation of inlet diameter, djwith

pressure, p.








876 SWANSON, CLARK

5 1.0LLJ

I I

ZxLU H-

uos

L U

0.8 h

0.6

0.4 0 40 80 120

PRESSURE-p (mm Hg)FIGURE 4

Dimensionless leaflet center length, tr*, as a function ofpressure, p.

(8) for this type of comparison. This modulus

is not constant but increases with increasing

load. The marked nonlinear behavior of aor-

tic leaflets gives values of E from 2 x 105 to 6

x 107 dynes/cm

2 (9, 10).

The width of the coaptation at the center,c,., decreased with incre asing pre ssu re as a

consequence of a greater increase in diame-

ter than in length. The coaptation surface

then peeled back as the diameter of the

leaflet supporting structure increased with

increasing pressure. Also, because of the

small rate of increase of dv with pressure, the

sinuses bulged out over this base diameter

producing an increase in the angle of the

lower leaflet surface, a and a decrease in the

angle of the free edge, <f>, as indicated in

Table 2

(columns 5 and 4,

respectively) and in

Figures 6 and 7. The free edge length, f,

increased only slightly at the expense of a

decrease in coaptation width (c,. and 0 , and

it increased as a increased and decreased

(i.e., da increase d m ore th an dv). This

behavior was also evid ent from sectio ns

through one of the sinuses at three different

pressures (Fig. 7a). Sauvage et al. (3) also

indicated a slight increase in f with pressure;

the ir dimensionless r esu lts (3, Figs. 1-14) are

quite similar to ours and give f* =0.58 at 100

mm Hg for pig hearts compared with f* =0.62

for human hearts. They indicated that <£

decreases from 34° to 24° when pressures are

increased from 80 mm Hg to 120 mm Hg with

<f> = 28° at 100 mm Hg compared with our value

of 0 =32° at 100 mm Hg. These results are as

close as can be expected considering the dif-

ference of species. The most obvious signifi-

cant structural or geometric difference is in

the configuration of the sinuses of Valsalva

(6, Figs. 1-6). In addition to the difference in

species, freezing also produces a variation

effect on tissue properties (9).

Uncertainties based on errors from re-

peated measurements of basic qua ntities

were 5 for f*, <j>, and c*, 4 for a 3% for <?c*,

and 2 for other dimensionless quantities.

These values are close to the maximum rela-

tive variations of data points from smoothedcurves.

Since th e leaflets meet at an angle at the

noduli Aranti and since th e pressure loading

is balanced across the coapting surfaces,

there can be no stress along the free edge or

in t he coapting surfaces in the central region

except for the compressive stress equal to

the pressure.

LEAFLET STRUCTURE

Striations on the surfaces of the casts ad-

jacent to the leaflets indicated a fibrous

structure across the leaflet on the aortic

'0 40 80 120

PRESSURE p (mmHg)FIGURES

Dimensionless overall vertical height, h* , as a function ofpressure, p.

Circulation Research Vol. 35 December 197U









oz6 0

u 40

40 80 120

PRESSURE-p (mm Hg)FIGURE a

Bottom leaflet surface a ngle, a, variation with pressure, p.

(b )FMURE7

a: Sections through a sinus vertical center plane at SO 60and 80 mm Hg. b: Cylindrical sections at SO, 50, an d 80mm Hg . Tick marks indicate leaflet attachment points.

surface (Fig. 8). These striations were in aplane perpendicular to the axis of the cylin-drical leaflet surface and extended from oneattachment to the other. Mating casts in the

left ventricle side had a smooth surface adja-cent to the leaflets. The leaflets are essen-tially thin flexible membranes, and they tendto form a cylindrical surface between theirpoints (or lines) of main support. Sectionsthrough the leaflet profiles along the stria-tion lines are shown in Figure 7b.

Since the leaflets end at a free edge in asection through the coaptation zone, therecan be no radial stress component in them.This conclusion is also corroborated by thefact that the radial profiles were essentiallystr aig ht (no significant definable cu rv atu re

in the radial direction [Figs. 8 and 9]). Theonly load stress component is then the cir-cumferential stress carried by the circumfer-ential collagen fiber structure.

LEAFLET THICKNESS

Leaflet molds were made on the series 8aortic molds. The fibrous structure in thecylindrical portions and in the coapting sur-faces closely resembled that indicated in Fig-ure s 8 and 10. As the molding press ure wasincreased, the coapting leaflet thickness de-creased. Measurements made on series 8

casts gave a 30% decrease in average thick-Circulation Research, VoL 35, December 1971,

ness measu red a t the m idpoint of th e coapt-ing surfaces from 0.48 mm to 0.32 mm (Fig.11). Variations were large from one leaflet toanother on the same valve at a given pres-

sure. At 100 mm Hg, thickness varied from0.22 mm to 0.4 mm.

OVERALL STRUCTURE

The valve structure consists of thin flexi-ble sheets (the leaflets) freely suspended be-tween the attachments, forming interleafletseals along the coaptation zone.

Details of an idealized valve structure areshown in Figure 9. Figure 9a is a view look-ing from the left ventricle side. The load-carrying collagen fibers appear in the angledview of the bottom side of the leaf-

let as ellipses. The attachment annulus linewhich forms the three-way intersection ofthe leaflets with the sinus and ventriculartract walls projects into a circle (the leftventr icular out le t t ract d iameter) in th isview. A section in the plane of the circulararc through the leaflet is shown in Figure 9c.The leaflet contour b in Figure 9c is one-third of a circle. The adjoining sinus contouris also nearly circular. The leaflet and sinuscurvatures are parallel at their l ine of at-tachment intersection with the left ventricu-lar outflow tract wall yielding a load-stress

balance, as indicated by the arrows in Figure









FIGURES

Views of aortic valve, a: View from left ventricle, b : Sideview iv plane of attachment line, c: Section throughcontour b of Figure Ha.

attachment line section is jus t the one-third

chord of the dv circle or 0.866 relative to dv.

The relative length of the elliptical contour

is 1/4 of the major axis length or 0.45. The

contours of the attachment annulus come

together at the top in a vertical short com-

missure section as indicated in Figure 10a, c,

and d. The center coaptation also turns up to

the vertical (Fig. 10a).

The platform projection of the leaflet in

Figure lOe is obtained from laying out chord

lengths on the projections from 10a as calcu-lated from arc len gths from 10c. The leaflet

contour chord length is obtained from the

cylindrical leaflet surface contour in Figure

10c and is just x=dv0/2 or x*=6/2, where 0 is the

angle out from the center along the leaflet arc

whose radius is dv/2. The longitudinal coordi-

nate obtained from projection onto the plane

perpendicular to the axis of the leaflet is

_ dr l - cosfl)V 2tany

Substituting 6 = 2x* into this expi-ession for ygives

Circulation Research Vol. J.5. December 1971

tany

for the equation of the leaflet contour. Fig-ure lOf is a layout of the vertical coapting

plane surfaces as the projection from Figure

10a and b. The bottom curves are parts of

ellipses formed by the intersection of the

coaptation planes with the leaflet cylinder

section. The two coapting planes from Figure

lOf are reconstructed on Figure lOe to be

coincident at the commissure attachment

point and tangent to the top curve of the

cylindrical leaflet section there.

The light internal lines of Figure 10 are

represen tative of the collagen load-carrying

fiber bundles. The load in the top point of thecylindrical section (Fig. lOe) is carried by the

fibers running down through the coaptation

zones (two fiber lines are illustrated). The

parts of the coaptation surfaces above the

lines to the point are unstressed (except for

the compressive pressure loading).

The entire geometry is essentially deter-

mined by the angle a (since 3 = y). The

comm issure heig ht (0.37) and the ce nte r

coaptation h eigh t (0.17) do not affect the

final shape of the cylindrical part of the

leaflet or its load-carrying ability and stress.

The size of the split necessary to allow a

development of the coapting surfaces with

the cylindrical surface with contiguity at the

attachment lines shown in Figure lOe is also

determined by a. The value 20° < a < 25°

minimizes this separation and the size of the

slit window where the fold line diverges.

If the leaflet thickness were uniform, the

stress would be cr = pd/2t, since it is a th in

mem brane (t < < d) with a uniform radius

(cylindrical section). The maximum value of

the pressure, p, is about 100 mm Hg at valve

closure. The maximum membrane stress isthen on the order of 27 x 10

5 dynes/cm

2 for a

leaflet thickness of 0.5 mm.

Discussion

The design geometry derived from valve

cast measurements is an averaged repre-

sentative geometry for the aortic heart

valve. The simple cylindrical geometry of the

load-carrying part of the leaflet gives a uni-

form stress resultant (load per unit thick-

ness) equal to pd/2.

This derived simplified geometry does nottake into account variations from one leaflet








880 SWANSON, CLARK

FIGURE 10

a: Side -profile through attachment plane, b: View from left ventricular side, c: Sectionthrough cylindrical leaflet profile, d: P rojected attachment line profile, c: Developedsurface of leaflet, f: Planar layout of coapting surfaces. Dimensions are relative to dr* =

1.0.

40 80 120PRESSURE-mm Hg

FIGURE 11

Coapting leaflet thickness as a function of pressure.

to another, specifically observed variationsin attachment line geometry between coro-nary and noncoronary leaflets. These varia-tions were not included because of the objec-tive of obtaining a simplified geometry thatcould be fabricated for clinical installationand because the variations were not largeenough to be considered as physiologicallys ignif icant for pros thet ic valve ins tal la-tion.

The leaflet and attachment load stressesare max imum a t va lve c losu re . Bend ingstresses during the folding wave motion dur-ing valve opening are an order of magnitudesmaller than the static load stresses followingvalve closure (11).

Since the free edge is always unloaded orunstressed, its length should not change sig-nificantly. As the diameter at the top of the

commissure increases with pressure, the freeCirathtion Raearch VoL 35 December 197i








88 SWANSON CLARK

kg). This downward force is balanced primar-ily by the distributed pressure force on thecurved wall of the aortic arch.

References

1. GOULD PL CATALOGLU A DH TT G CHATTOPA

DHYAY A, CLARK R E: Stress analysis of th e hu-

man aortic valve. National Symposium on Com-

puterized Structural Analysis and Design G«orgeWashington University 1972

2. WOOD SJ, ROBEL SB, SAUVAGE LR : Technique for

study of heart valves. J Thorac Cardiovasc Surg46:369- 385 , 1963

3. S A U V A G E LR , V I G G E R S RF , B E R G E R K, R O B E L SB,

SAWYER PN WOOD SJ: Pro sthetic Repla cemen t of

the Aortic Valve. Springfield Illinois Charles C

Thomas 1972

4. MERCER JL BENEDECTY M BAHNSON HT: Geomet ryand construction of th e aortic leaflet. J ThoracCardiovasc Surg 65:511-518 1973

5. SWANSON WM CLARK RE: Aortic valve leaflet mo-

tion during systole. Circ Res 32:42-48 1973

6. MOZERSKY DJ , SUMNER DS: Transcuta neous meas-

urement of th e elastic properties of the humanfemoral artery. Circulation 46:948-955 1972

7. A R N D T J O , KOBERG: Die Druc k-Dur chme sser-Bez ie-

hung der in takten A. Femoralis des wachen

Menschen un d ihre Beein flussung durch Nora-drenalin-Infusionen. Pfluegers Arch 318:130-1461970

8. MINNS RJ , SODIN PD : Role of the fibrous compo-nents an d ground subst ance in the mechanicalproperties of biological tissues: A preliminary in-

vestigation. J Biomechanics 6:153-165 19739. CLARK RE: Stress-strain characteristics of fresh and

frozen human aortic an d mitral leaflets and chor-dae tendineae. J Thorac Cardiovasc Surg 66:202-208, 1973

10. CLARK RE, BUTTERWORTH GAM: Charac te riza tionof the mechanics of human aortic an d mitral valveleaflets. Surg Forum 22:134-135 1971

11. SWANSON WM CLARK RE: Motion and stresses in

aortic valve leaflet during systole. American Soci-ety of Mechanical Engin eers Paper 72-WA/BHF-51972

12. CLARK RE, FlNKE EH: Scanning an d light micro-scopy of human aortic leaflets in stressed and

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