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BIOMAT., MID. DEV., ART. ORG., 6 ( 3 ) , 225-244 ( 1 9 7 8 )
AORTIC VALVE MECHANICS PART 11: A STRESS ANALYSIS OF THE PORCINE AORTIC VALVE LEAFLETS IN DIASTOLE M. Chong, M.Eng. a n d Y.F. Missirlis, Ph.D. Department o f Engineering Physics McMaster University, Hamilton, Ontario, Canada ABSTRACT
The s t r e s s analysis of the porcine a o r t i c valve l e a f l e t s in diastole a t 80 mm Hg pressure in-vitro i s presented. Incorporation of local geometrical asymmetry, material inhomogeneity, anistropy and non-linearity are applied. The s t r e s s theory used i s a modified form of the thin membrane stress theory for a homogeneous linearly e l a s t i c a n d orthotropic lamina. Modifications are mde so t h a t the Hooke's law constitutive equations of stress may be applied t o the inhomogeneous, non-1 inearly e l a s t i c and orthotropic t h i n (membrane) a o r t i c valve l e a f l e t s . Stress calculations are made on the premise t h a t the valve i s in pre-transi tion ( i .e. low e l a s t i c modulus) in the circumferent i a l direction and post-transition ( i .e. high e l a s t i c modulus) in the radial direction. I t i s shown that aCIR < 1 gm/mm2, and for most of the noncoronary l e a f l e t , 0 < uRAD < 30 gm/nm2. The areas of highest stress concentrations are in the areas of mutual leafl e t coaptation near the Node of Arantii. A progressive increase of radial stresses from the sinus-annulus edge toward the node i s observed. INTRODUCTION
Substitute a o r t i c valves o f various types and designs have been in use for some two decades. Numerous patients have undergone aorti c valve rep1 acements using ei ther rigid mechanical protheses (e.g. caged ball and disk, t i l t i n g and pivoting disk), or biological tissue valves (e.g. homografts, xenografts, fascia l a t a , dura 225 Copyright 0 1978 by Marcel Dekker, Inc. All Rights Reserved. Neither this work nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher.
226
CHONG, ENG, AND MISSIRLIS
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mater and pericardial constructed valves), Both types a1 though acceptable nave proven t o be s t i l l unsatisfactory: the former because of t h e i r h i g h thromboembolic risks require continual use of anticoagulant which carries i t s own substantial risk; a n d the l a t t e r have shown questionable durability in vivo (2,4,23). I t i s believed t h a t a prosthetic flexible t r i l e a f l e t valve would be the ideal a o r t i c valve substitute (13,17). In order t o obtain information which could be useful t o the design of such a valve, detailed analysis of the structure and mechanical propert i e s of the natural human aortic valve have been conducted (9,10, 1 7 ) . Others have investigated the strength aspect of the valve l e a f l e t s by means of s t r e s s analyses of the l e a f l e t s with the valve being subjected t o simulated s t a t i c diastolic pressures ( 5 , 6 , 7, 8, 11, 12, 13, 17, 18). All of the above s t r e s s analyses have been based on several common b u t questionable assumptions: 1 ) local surface geometrical symmetry 2 ) material homogeneity 3) isotropy. Those assumptions have generally been acknowledged t o be n o t r e a l i s t i c , (13, 18), b u t nevertheless these researchers have used them i n t h e i r analytical methodologies. Also the previous analyses have assumed that the valve i s in the post-transition s t a t e in diastole. Cataloglu e t a l . ( 6 ) made this assumption stating that " l e a f l e t stresses generally f a l l in the linear s t i f f phase of the stress-strain curve"; thus, "the much higher post-transition e l a s t i c modulus i s used". However, the referenced stress-strain curve and reported stresses do not seem t o substantiate the assumption. Comparison of observed strains from an in-vitro experiment of a porcine aortic valve under simulated diastolic pressures with strains observed for microtensile stress-strain experiments on porcine a o r t i c valve l e a f l e t s , seem t o suggest t h a t there i s a definite likelihood of pre-transition (i.e., low e l a s t i c modulus) s t a t us for the circumferential strains (19). Radial strains appear characteristically t o be post-transition (i .e. , high e l a s t i c modu-
AORTIC VALVE MECHANICS.
I1
l u s ) , b u t t r a n s i t i o n (1.e. likely.
227
i n t e r m e d i a t e e l a s t i c modulus) i s a l s o
The v a r i o u s phases a r e i l l u s t r a t e d i n F i g u r e 1.
T h i s paper i s a r e p o r t of a s t r e s s a n a l y s i s on a p o r c i n e a o r t i c v a l v e w i t h considerations o f t h e above f o u r assumptions i.e., Artif Cells Blood Substit Immobil Biotechnol Downloaded from informahealthcare.com by Dalhousie University on 12/01/12 For personal use only.
g e o m e t r i c a l asymnetry, m a t e r i a l i n h o m g e n e i ty, m a t e r i a l a n i s o t r o p y ( o r t h o t r o p y i s assumed) and m a t e r i a l non-1 i n e a r i ty.
METHOD OF ANALYSIS T h i n Membrane S t r e s s Theory The s t r e s s t h e o r y used i s t h a t d e s c r i b e d b y Hooke's Law f o r a homogeneous, l i n e a r l y e l a s t i c , and o r t h o t r o p i c l a m i n a (14).
Figu r e 2 d e p i c t s t h e s t r e s s and s t r a i n v a r i a b l e s f o r a t y p i c a l lamina. W i t h t h e t h i n membrane s h e l l and z e r o shear s t r e s s assumptions, t h e Hooke's Law c o n s t i t u t i v e e q u a t i o n s o f s t r e s s a r e :
OA: pre-transition
AB : transition BC : post-transition N
E E
\
0 Y
?
LA II
b c
cn cn w
E
l-
cn
0 FIG. 1 S t r e s s - s t r a i n t e r m i n o l o g y used i n t h i s paper.
CHONG, ENG, AND M I S S I U I S
228
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TRN
thin membrane assumption I
t / R o r < 1 depending on l e a f l e t locale; e.g. uR/uC > 1 i n the l e a f l e t ' s center and < 1 a t the coaptation and sinus-annulus edges. The others reported uR/uC < 1 (5,6,7,11). The source of t h i s discrepancy stems from
-
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AORTIC VALVE MECHANICS.
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241
t h e radial p o s t - t r a n s i t i o n and circumferential p r e - t r a n s i t i o nassumption applied i n our a n a l y s i s ; i . e . E R,pos d E C , pre : 10% 1 . In other words, the s t r e s s a n a l y s i s i s rendered strongly materiallydependent. In c o n t r a s t , M i s s i r l i s and Armeniades (18) used ER, post /E C ,post = 0 . 3 and t h e o t h e r s assumed t h e i s o t r o p i c condit i o n of E R, post /E C, post = 1.0. These figures therefore render t h e calculated s t r e s s e s t o be primarily dependent on the geometry i n puts. Spatial d i s t r i b u t i o n of s t r e s s e s over t h e e n t i r e l e a f l e t has a l s o been reported by Clark e t a l . (11) and Cataloglu e t a l . ( 6 ) i n the form of i s o - s t r e s s maps f o r a d i a s t o l i c pressure of about Both showed t h a t f o r a d i a s t o l i c pressure of about 80 mm.Hg. 2 80 mm.Hg., l o c a l i z a t i o n of higher s t r e s s e s ( > 15 gm/mm ( 6 ) ; 2 > 40 gm/m ( 1 1 ) ) a r e s i t u a t e d near t h e node o f Arantii and in a region o f f l e a f l e t c e n t e r toward t h e annulus edge. These s t r e s s e s were however predominantly circumferential. Our r e s u l t s f o r t h e non-coronary l e a f l e t show no such l o c a l i z a t i o n s ; i n s t e a d , a progressive increase in ( r a d i a l ) s t r e s s e s away f r o m the annulus edge is observed. A s i m i l a r progressive increase i s reported by M i s s i r l i s and Armediades' (18) a n a l y t i c a l s t r e s s a n a l y s i s . An important consideration i n t h e evaluation of t h e calcul a t e d stresses K i t h regard t o a o r t i c valve design i s t h e followi n g : how r e p r e s e n t a t i v e of t h e in-vtvo s i t u a t i o n a r e the calcul a t e d s t r e s s e s i f t h e s t r e s s e s are determined using geometry and material constant i n p u t s obtained from s t a t i c o r q u a s i - s t a t i c inv i t r o loading schemes. The in-vivo s t r a i n r a t e of the l e a f l e t s b e i n g approximately 15,000 %/min (17) is much l a r g e r than the s t r a i n r a t e s used in obtaining our data. For instance, the whole valve experiment used t o obtain the geometry data ( i . e . , s t r a i n s ) employed an approximate loading s t r a i n r a t e of l O % / m i n . , and the microtensile s t r e s s - s t r a i n experiments used t o obtain t h e material p r o p e r t i e s ( i . e . , E and E ) employed loading s t r a i n r a t e s ranging from 15%-45%/min. I t has been reported t h a t biological t i s s u e becomes l e s s e x t e n s i b l e ( i . e . smaller s t r a i n s and t r a n s i t i o n
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CHONG, ENG, AND MISSIRLIS
( 1 , 16, 2 0 ) . Reported e f f e c t s consistent. Missirlis (17) L i m and Boughner (16) workt h e e l a s t i c modulus i s essent i a l l y independent o f strain r a t e . Lake (15) however reported a 10% increase in the e l a s t i c modulus of e l a s t i n f o r each decade i n crease in i n i t i a l strain r a t e , and Van Brocklin and E l l i s ( 2 2 ) and Abrahams (1) observed an increase i n modulus with s t r a i n r a t e f o r tendons which i s predominantly collagen. Thus, a probable i m p l i c a t i o n i s t h a t the calculated i n - v i t r o s t r e s s e s a r e smaller than t h e in-vivo s t r e s s e s . Recently, Brewer e t a l . ( 3 ) made a s i m i l a r conclusion a f t e r observing t h a t i n - v i v o circumferential s t r a i n s f o r a dog's a o r t i c valve a r e only approximately 2% in the course o f a f u l l cardiac cycle.
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s t r a i n s ) w i t h increasing s t r a i n r a t e on t h e e l a s t i c moduli have been less working with a o r t i c valve t i s s u e and ing w i t h chordae tendinae found t h a t
SUMMARY AND CONCLUSIONS A method o f conducting a stress analysis of (porcine) a o r t i c valve l e a f l e t s which incorporates t h e e f f e c t s of local surface
geometry data, and of local inhomogeneous and a n i s o t r o p i c (orthotropy is assumed) material p r o p e r t i e s i s presented. A1 though t h e sampling of data is i n s u f f i c i e n t t o base any firm conclusions a s t o the magnitudes of s t r e s s e s , the procedure has shown t h a t these p a r t i c u l a r incorporations a r e e s s e n t i a l f o r a v a l i d a n a l y s i s . E l a s t i c moduli used i n the stress a n a l y s i s a r e c l e a r l y inhomogeneous and o r t h o t r o p i c . I t is f u r t h e r acknowledged t h a t t h e cons t a n t i s o t r o p i c Poisson's r a t i o o f w = 0.3 assumed f o r the analys i s i s a major l i m i t a t i o n , and t h a t i t warrants f u r t h e r i n v e s t i g a t i on. Of crucial importance is the recognition of t h e need t o re-
solve the question as t o what status the in-vitro valve is i n a t diastolic pressures. The data seem t o suggest p r e - t r a n s i t i o n ( i . e . , low s t r e s s and small e l a s t i c moduli) s t a t u s f o r the circumferential strains and p o s t - t r a n s i t i o n ( i . e . , higher stress and l a r g e r e l a s t i c moduli) for t h e r a d i a l s t r a i n s .
AORTIC VALVE MECHANICS.
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In designing a f l e x i b l e t r i l e a f l e t a o r t i c valve one should r e a l i z e t h a t the determination o f maximum stress magnitudes, o r a l s o , f o r t h a t matter, t h e s t r e s s p a t t e r n s a r e necessary but not s u f f i c i e n t design c r i t e r i a . One must be reminded t h a t the l e a f l e t s a r e not r i g i d structures f o r which maximum stress d e t e r minants would be s u f f i c i e n t . For this reason then, i t is suggested t h a t knowledge of how and when d i f f e r e n t regions of the valve l e a f l e t s undergo t r a n s i t i o n and p o s t - t r a n s i t i o n ( i f they do a t a l l ) would c o n t r i b u t e v a s t l y t o f u t u r e designs o f t r i l e a f l e t valves f o r which d u r a b i l i t y , f a t i g u e and f l e x i b i l i t y a r e probably o f g r e a t e r importance than maximum strength alone. REFERENCES 1. Abrahams, M . , Med. & Biol. Eng., 5, 433 (1967). 2.
Donahoo, J.S. and Gott, V . L . ,
Brawley, R . K . ,
Am. J . C a r d i o l . ,
35, 855 (1975).
3.
Brewer, R.J., Mentzer J r . , R.M., Deck, J.D., R i t t e r , R.C., Tref i l , J.S., & Nolan, S . P . , J. Thorac. Cardiovasc. S u r g . , z . 645 (1977).
35, 843
4.
Bonchek, L.F.,
and S t a r r , A . , Am. J . Cardiol.,
5.
Cataloglu, A., 347 (1975).
Gould, P . L . ,
6.
Cataloglu, A., Gould, P.L. and Clark, R.E., J . Enging. Mech, Div. h. SOC. Cly. Engs., E 2 - ( E M ) , 135 (1976).
7.
Chen, M., Rossow, M.P., Gould, P.L. and Clark, R.E., Math. Modelling 1977, 1,218 (1977).
8.
Chong, K.P., Wieting, D.W., Hwang, N.H.C., and Kennedy, J.H., Biomat., Med. Dev., Art. Org., 1,307 (1973).
9.
Clark, R.E.,
10.
& Clark, R.E.,
(1975).
J . Biomechanics,
Appl.
J. Thorac. Cardiovasc. S u r g . , 6 6 , 202 (1973).
Clark, R.E. and Finke, E.J., J . Thorac. Cardiovasc. Surg.,
67, 792 (1974).
11.
Clark, R.E., Karara, H . M . , Cataloglu, A. and Gould, P . L . , Trans. Amer. SOC. Artif. I n t . Organs, 11,71 (1975).
12.
Ghista, D.N.,
Med. & Biol. Eng.,
14,122
(1976).
8.
Artif Cells Blood Substit Immobil Biotechnol Downloaded from informahealthcare.com by Dalhousie University on 12/01/12 For personal use only.
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13.
Gould, P.L., Clark, R.E.,
14.
Jones, R.M., i n Mechanics o f Composite Materials,p, Book Co., Washington (1974).
15.
Lake, L.W., (1973).
16.
Lim, K.O.
Cataloglu, A., Dhatt, G., Chatto adhyay, A. and J. Comput. S t r u c t . , 3, 377 (1973p.
Ph.D. t h e s i s , Rice U n i v e r s i t y , Houston, Texas and Boughner, D.R.
53, 330 (1974).
, Can.
J. P h y s i o l .
, Pharmacol. ,
17.
M i s s i r l i s , Y.F., (1973).
18.
M i s s i r l i s , Y.F. (1976).
and Armeniades, C.D.,
19.
M i s s i r l i s , Y.F.
and Chong, M.,
20.
Rigby, J.B., H i r a i , N . , Spikes, J.D. and Eyring, H., P h y s i o l . , 43, 265 (1959).
21.
Swanson, W.,
22. 23.
45, S c r i p t a
Ph.D. Thesis, Rice U n i v e r s i t y , Houston, Texas
and Clark, R.E.,
J. Bioeng.,
Van B r o c k l i n , J.D. and E l l i s , D.G., Am. J . C a r d i o l . ,
35,
Archs.,
9,
477
I n press (1978).
C i r c u l a t i o n Res.,
46, 369 (1965). Wallace, R.B.,
J. Biomechanics,
35,
J. Gen.
871 (1974).
Phys. Med. Rehabil.,
866 (1975).
BIOMAT., MID. DEV., ART. ORG., 6 ( 3 ) , 225-244 ( 1 9 7 8 )
AORTIC VALVE MECHANICS PART 11: A STRESS ANALYSIS OF THE PORCINE AORTIC VALVE LEAFLETS IN DIASTOLE M. Chong, M.Eng. a n d Y.F. Missirlis, Ph.D. Department o f Engineering Physics McMaster University, Hamilton, Ontario, Canada ABSTRACT
The s t r e s s analysis of the porcine a o r t i c valve l e a f l e t s in diastole a t 80 mm Hg pressure in-vitro i s presented. Incorporation of local geometrical asymmetry, material inhomogeneity, anistropy and non-linearity are applied. The s t r e s s theory used i s a modified form of the thin membrane stress theory for a homogeneous linearly e l a s t i c a n d orthotropic lamina. Modifications are mde so t h a t the Hooke's law constitutive equations of stress may be applied t o the inhomogeneous, non-1 inearly e l a s t i c and orthotropic t h i n (membrane) a o r t i c valve l e a f l e t s . Stress calculations are made on the premise t h a t the valve i s in pre-transi tion ( i .e. low e l a s t i c modulus) in the circumferent i a l direction and post-transition ( i .e. high e l a s t i c modulus) in the radial direction. I t i s shown that aCIR < 1 gm/mm2, and for most of the noncoronary l e a f l e t , 0 < uRAD < 30 gm/nm2. The areas of highest stress concentrations are in the areas of mutual leafl e t coaptation near the Node of Arantii. A progressive increase of radial stresses from the sinus-annulus edge toward the node i s observed. INTRODUCTION
Substitute a o r t i c valves o f various types and designs have been in use for some two decades. Numerous patients have undergone aorti c valve rep1 acements using ei ther rigid mechanical protheses (e.g. caged ball and disk, t i l t i n g and pivoting disk), or biological tissue valves (e.g. homografts, xenografts, fascia l a t a , dura 225 Copyright 0 1978 by Marcel Dekker, Inc. All Rights Reserved. Neither this work nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher.
226
CHONG, ENG, AND MISSIRLIS
Artif Cells Blood Substit Immobil Biotechnol Downloaded from informahealthcare.com by Dalhousie University on 12/01/12 For personal use only.
mater and pericardial constructed valves), Both types a1 though acceptable nave proven t o be s t i l l unsatisfactory: the former because of t h e i r h i g h thromboembolic risks require continual use of anticoagulant which carries i t s own substantial risk; a n d the l a t t e r have shown questionable durability in vivo (2,4,23). I t i s believed t h a t a prosthetic flexible t r i l e a f l e t valve would be the ideal a o r t i c valve substitute (13,17). In order t o obtain information which could be useful t o the design of such a valve, detailed analysis of the structure and mechanical propert i e s of the natural human aortic valve have been conducted (9,10, 1 7 ) . Others have investigated the strength aspect of the valve l e a f l e t s by means of s t r e s s analyses of the l e a f l e t s with the valve being subjected t o simulated s t a t i c diastolic pressures ( 5 , 6 , 7, 8, 11, 12, 13, 17, 18). All of the above s t r e s s analyses have been based on several common b u t questionable assumptions: 1 ) local surface geometrical symmetry 2 ) material homogeneity 3) isotropy. Those assumptions have generally been acknowledged t o be n o t r e a l i s t i c , (13, 18), b u t nevertheless these researchers have used them i n t h e i r analytical methodologies. Also the previous analyses have assumed that the valve i s in the post-transition s t a t e in diastole. Cataloglu e t a l . ( 6 ) made this assumption stating that " l e a f l e t stresses generally f a l l in the linear s t i f f phase of the stress-strain curve"; thus, "the much higher post-transition e l a s t i c modulus i s used". However, the referenced stress-strain curve and reported stresses do not seem t o substantiate the assumption. Comparison of observed strains from an in-vitro experiment of a porcine aortic valve under simulated diastolic pressures with strains observed for microtensile stress-strain experiments on porcine a o r t i c valve l e a f l e t s , seem t o suggest t h a t there i s a definite likelihood of pre-transition (i.e., low e l a s t i c modulus) s t a t us for the circumferential strains (19). Radial strains appear characteristically t o be post-transition (i .e. , high e l a s t i c modu-
AORTIC VALVE MECHANICS.
I1
l u s ) , b u t t r a n s i t i o n (1.e. likely.
227
i n t e r m e d i a t e e l a s t i c modulus) i s a l s o
The v a r i o u s phases a r e i l l u s t r a t e d i n F i g u r e 1.
T h i s paper i s a r e p o r t of a s t r e s s a n a l y s i s on a p o r c i n e a o r t i c v a l v e w i t h considerations o f t h e above f o u r assumptions i.e., Artif Cells Blood Substit Immobil Biotechnol Downloaded from informahealthcare.com by Dalhousie University on 12/01/12 For personal use only.
g e o m e t r i c a l asymnetry, m a t e r i a l i n h o m g e n e i ty, m a t e r i a l a n i s o t r o p y ( o r t h o t r o p y i s assumed) and m a t e r i a l non-1 i n e a r i ty.
METHOD OF ANALYSIS T h i n Membrane S t r e s s Theory The s t r e s s t h e o r y used i s t h a t d e s c r i b e d b y Hooke's Law f o r a homogeneous, l i n e a r l y e l a s t i c , and o r t h o t r o p i c l a m i n a (14).
Figu r e 2 d e p i c t s t h e s t r e s s and s t r a i n v a r i a b l e s f o r a t y p i c a l lamina. W i t h t h e t h i n membrane s h e l l and z e r o shear s t r e s s assumptions, t h e Hooke's Law c o n s t i t u t i v e e q u a t i o n s o f s t r e s s a r e :
OA: pre-transition
AB : transition BC : post-transition N
E E
\
0 Y
?
LA II
b c
cn cn w
E
l-
cn
0 FIG. 1 S t r e s s - s t r a i n t e r m i n o l o g y used i n t h i s paper.
CHONG, ENG, AND M I S S I U I S
228
Artif Cells Blood Substit Immobil Biotechnol Downloaded from informahealthcare.com by Dalhousie University on 12/01/12 For personal use only.
TRN
thin membrane assumption I
t / R o r < 1 depending on l e a f l e t locale; e.g. uR/uC > 1 i n the l e a f l e t ' s center and < 1 a t the coaptation and sinus-annulus edges. The others reported uR/uC < 1 (5,6,7,11). The source of t h i s discrepancy stems from
-
Artif Cells Blood Substit Immobil Biotechnol Downloaded from informahealthcare.com by Dalhousie University on 12/01/12 For personal use only.
AORTIC VALVE MECHANICS.
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t h e radial p o s t - t r a n s i t i o n and circumferential p r e - t r a n s i t i o nassumption applied i n our a n a l y s i s ; i . e . E R,pos d E C , pre : 10% 1 . In other words, the s t r e s s a n a l y s i s i s rendered strongly materiallydependent. In c o n t r a s t , M i s s i r l i s and Armeniades (18) used ER, post /E C ,post = 0 . 3 and t h e o t h e r s assumed t h e i s o t r o p i c condit i o n of E R, post /E C, post = 1.0. These figures therefore render t h e calculated s t r e s s e s t o be primarily dependent on the geometry i n puts. Spatial d i s t r i b u t i o n of s t r e s s e s over t h e e n t i r e l e a f l e t has a l s o been reported by Clark e t a l . (11) and Cataloglu e t a l . ( 6 ) i n the form of i s o - s t r e s s maps f o r a d i a s t o l i c pressure of about Both showed t h a t f o r a d i a s t o l i c pressure of about 80 mm.Hg. 2 80 mm.Hg., l o c a l i z a t i o n of higher s t r e s s e s ( > 15 gm/mm ( 6 ) ; 2 > 40 gm/m ( 1 1 ) ) a r e s i t u a t e d near t h e node o f Arantii and in a region o f f l e a f l e t c e n t e r toward t h e annulus edge. These s t r e s s e s were however predominantly circumferential. Our r e s u l t s f o r t h e non-coronary l e a f l e t show no such l o c a l i z a t i o n s ; i n s t e a d , a progressive increase in ( r a d i a l ) s t r e s s e s away f r o m the annulus edge is observed. A s i m i l a r progressive increase i s reported by M i s s i r l i s and Armediades' (18) a n a l y t i c a l s t r e s s a n a l y s i s . An important consideration i n t h e evaluation of t h e calcul a t e d stresses K i t h regard t o a o r t i c valve design i s t h e followi n g : how r e p r e s e n t a t i v e of t h e in-vtvo s i t u a t i o n a r e the calcul a t e d s t r e s s e s i f t h e s t r e s s e s are determined using geometry and material constant i n p u t s obtained from s t a t i c o r q u a s i - s t a t i c inv i t r o loading schemes. The in-vivo s t r a i n r a t e of the l e a f l e t s b e i n g approximately 15,000 %/min (17) is much l a r g e r than the s t r a i n r a t e s used in obtaining our data. For instance, the whole valve experiment used t o obtain the geometry data ( i . e . , s t r a i n s ) employed an approximate loading s t r a i n r a t e of l O % / m i n . , and the microtensile s t r e s s - s t r a i n experiments used t o obtain t h e material p r o p e r t i e s ( i . e . , E and E ) employed loading s t r a i n r a t e s ranging from 15%-45%/min. I t has been reported t h a t biological t i s s u e becomes l e s s e x t e n s i b l e ( i . e . smaller s t r a i n s and t r a n s i t i o n
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( 1 , 16, 2 0 ) . Reported e f f e c t s consistent. Missirlis (17) L i m and Boughner (16) workt h e e l a s t i c modulus i s essent i a l l y independent o f strain r a t e . Lake (15) however reported a 10% increase in the e l a s t i c modulus of e l a s t i n f o r each decade i n crease in i n i t i a l strain r a t e , and Van Brocklin and E l l i s ( 2 2 ) and Abrahams (1) observed an increase i n modulus with s t r a i n r a t e f o r tendons which i s predominantly collagen. Thus, a probable i m p l i c a t i o n i s t h a t the calculated i n - v i t r o s t r e s s e s a r e smaller than t h e in-vivo s t r e s s e s . Recently, Brewer e t a l . ( 3 ) made a s i m i l a r conclusion a f t e r observing t h a t i n - v i v o circumferential s t r a i n s f o r a dog's a o r t i c valve a r e only approximately 2% in the course o f a f u l l cardiac cycle.
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s t r a i n s ) w i t h increasing s t r a i n r a t e on t h e e l a s t i c moduli have been less working with a o r t i c valve t i s s u e and ing w i t h chordae tendinae found t h a t
SUMMARY AND CONCLUSIONS A method o f conducting a stress analysis of (porcine) a o r t i c valve l e a f l e t s which incorporates t h e e f f e c t s of local surface
geometry data, and of local inhomogeneous and a n i s o t r o p i c (orthotropy is assumed) material p r o p e r t i e s i s presented. A1 though t h e sampling of data is i n s u f f i c i e n t t o base any firm conclusions a s t o the magnitudes of s t r e s s e s , the procedure has shown t h a t these p a r t i c u l a r incorporations a r e e s s e n t i a l f o r a v a l i d a n a l y s i s . E l a s t i c moduli used i n the stress a n a l y s i s a r e c l e a r l y inhomogeneous and o r t h o t r o p i c . I t is f u r t h e r acknowledged t h a t t h e cons t a n t i s o t r o p i c Poisson's r a t i o o f w = 0.3 assumed f o r the analys i s i s a major l i m i t a t i o n , and t h a t i t warrants f u r t h e r i n v e s t i g a t i on. Of crucial importance is the recognition of t h e need t o re-
solve the question as t o what status the in-vitro valve is i n a t diastolic pressures. The data seem t o suggest p r e - t r a n s i t i o n ( i . e . , low s t r e s s and small e l a s t i c moduli) s t a t u s f o r the circumferential strains and p o s t - t r a n s i t i o n ( i . e . , higher stress and l a r g e r e l a s t i c moduli) for t h e r a d i a l s t r a i n s .
AORTIC VALVE MECHANICS.
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In designing a f l e x i b l e t r i l e a f l e t a o r t i c valve one should r e a l i z e t h a t the determination o f maximum stress magnitudes, o r a l s o , f o r t h a t matter, t h e s t r e s s p a t t e r n s a r e necessary but not s u f f i c i e n t design c r i t e r i a . One must be reminded t h a t the l e a f l e t s a r e not r i g i d structures f o r which maximum stress d e t e r minants would be s u f f i c i e n t . For this reason then, i t is suggested t h a t knowledge of how and when d i f f e r e n t regions of the valve l e a f l e t s undergo t r a n s i t i o n and p o s t - t r a n s i t i o n ( i f they do a t a l l ) would c o n t r i b u t e v a s t l y t o f u t u r e designs o f t r i l e a f l e t valves f o r which d u r a b i l i t y , f a t i g u e and f l e x i b i l i t y a r e probably o f g r e a t e r importance than maximum strength alone. REFERENCES 1. Abrahams, M . , Med. & Biol. Eng., 5, 433 (1967). 2.
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