Finite Elements in BIOMECHANICS

Finite Elements in BIOMECHANICS Edited by R. H. Gallagher B. R. Simon P. C. Johnson J. F. Gross University of Arizona, Tucson, Arizona

A Wiley—Interscience Publication

JO H N W ILEY & SONS Chichester

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Copyright © 1982 by John Wiley & Sons Ltd. All rights reserved. No part of this book may be reproduced by any means, nor transmitted, nor translated into a machine language without the written permission of the publisher.

Library o f Congress Cataloging in Publication Data: Main entry under title: Finite elements in biomechanics. (Wiley series in numerical methods in engineering) ‘A Wiley-Interscience publication.’ Includes index. 1. Human mechanics—Mathematics—Congresses. 2. Finite element method—Congresses. I. Gallagher, Richard H. II. Series. QP303. F47 ISBN

612'.76

0 471 09996 1

81-13084 AACR2

British Library Cataloguing in Publication Data: Finite elements in biomechanics.—(Wiley series in numerical methods in engineering) 1. Human mechanics 1. Gallagher, R. H. 612\76 QP303 ISBN 0 471 09996 1 Typeset by Macmillan India Ltd, Bangalore and printed by Vail-Ballou Press, Inc., Binghamton, U.S.A.

Contributing Authors

K. -N. A n

Orthopaedic Biomechanics Laboratory, Mayo Clinic, Rochester, MN 55907, U.S.A.

G. Bergm ann

Orthopaedische Klinik und Poliklinik der Freien Universitaet Berlin, D-1000 Berlin, 33, Germany

M. P. Bieniek

Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027, U.S.A.

T. D. Brown

Department of Orthopaedic Surgery, University of Pittsburgh, PA 15261, U.S.A.

E. Y. S. C hao

Orthopaedic Biomechanics Laboratory, Mayo Clinic, Rochester, MN 55901, U.S.A.

G. W. C hristie

Theoretical and Applied Mechanics Department, The University of Auckland, New Zealand

I. C. C larke

Department of Orthopaedics, Orthopaedic Hospital — University of Southern California, Los Angeles, CA 90007, U.S.A.

A. B. F erguson

Department of Orthopaedic Surgery, University of Pittsburgh, PA 15261, U.S.A.

F. H. F u

Department of Orthopaedic Surgery, University of Pittsburgh, PA 15261, U.S.A.

vi

Contributing Authors

T. A. GRUEN

Department o f Orthopaedics, Orthopaedic Hospital University o f Southern California, Los Angeles, CA 90007, U.S.A.

W. C. H a y es

Orthopaedic Biomechanics Laboratory, Beth Israel Hospital and Harvard Medical School, Boston, MA 02215, U.S.A.

R. R. H osey

Center for Materials Research, College o f Engineering, University o f Iowa, Iowa City, I A 52242, U.S.A.

R. HUISKES

Laboratory for Experimental Orthopaedics, Univer­ sity o f Nijmegen, The Netherlands

J. D. J a n ssen

Division o f Applied Mechanics, Department o f Mec­ hanical Engineering, Eindhoven University o f Technology, The Netherlands

A. D. K a r a k a p l a n

Department o f Civil Engineering and Engineering Mechanics, Columbia University, New York, N Y 10027, U.S.A.

D. W. K elly

Civil Engineering Department, University o f Wales, Swansea, SA2 8PP, U.K.

R. K o e l b e l

Orthopaedische Klinik und Poliklinik der Freien Universitaet Berlin, D-1000 Berlin, 33, Germany

G. C. L ee

Faculty o f Engineering and Applied Science, State University o f New York at Buffalo, Amherst, NY 14260, U.S.A.

B. M. L ev in e

Department o f Orthopaedic Surgery, University of Pennsylvania, Philadelphia, PA 19104, U.S.A.

Y. K. Liu

Center for Materials Research, College of Engineering, University o f Iowa, Iowa City, I A 52242, U.S.A.

I. C. M e d l a n d

Theoretical and Applied Mechanics Department, The University o f Auckland, New Zealand

Contributing Authors

vii

Y. C. PAO

Department o f Engineering Mechanics, University o f Nebraska-Lincoln, NE 68588, U.S.A.

G. T. RAB

Department o f Orthopaedics, University o f California —Davis School o f Medicine, Sacramento, CA 95817, U.S.A.

S. R a m a s w a m y

Orthopaedic Biomechanics Laboratory, Beth Israel Hospital and Harvard Medical School, Boston, MA 02215, U.S.A.

A. R o h l m a n n

Orthopaedische Klinik und Poliklinik der Freien Universitaet Berlin, D-1000 Berlin, 33, Germany

E. R y b ic k i

Department o f Mechanical Engineering, University o f Tulsa, Tulsa, OK 74104, U.S.A.

A. S a r m ie n t o

Department o f Orthopaedics, Orthopaedic Hospital — University o f Southern California, Los Angeles, CA 90007, U.S.A.

W . H . S h ie l d s

Oceanic Division, Westinghouse Corporation, Baltimore, MD 21404, U.S.A.

R. S k a l a k

Department o f Civil Engineering and Engineering Mechanics, Columbia University, New York, N Y 10027, U.S.A.

T. J. SLOOFF

Laboratory for Experimental Orthopaedics, Univer­ sity o f Nijmegen, The Netherlands

B. S n y d e r

Department o f Orthopaedic Surgery, University o f Pennsylvania, Philadelphia, PA 19104, U.S.A.

R. L. S p il k e r

Department o f Materials Engineering, University o f Illinois at Chicago Circle, Chicago, IL 60680, U.S.A.

R. R. T a r r

Department o f Orthopaedics, Orthopaedic Hospital — University o f Southern California, Los Angeles, CA 90007, U.S.A.

H. TO z e r e n

Department o f Engineering Sciences, Middle East Technical University, Ankara, Turkey

Vlll

Contributing Authors

N . T. T se n g

Faculty o f Engineering and Applied Sciences, State University o f New York at Buffalo, Amherst, N Y 14260, U.S.A.

D . L. V a w t e r

School o f Engineering Science and Mechanics, Georgia Institute o f Technology, Atlanta, GA 30332, U.S.A.

M. E. W ay

Biomedical Engineering Program, Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A.

S. O. W il l e

Institutt for Informatikk, University o f Oslo, Blindern, Oslo, 3, Norway

O. C. ZlENKIEWICZ

Civil Engineering Department, University o f Wales, Swansea, SA2 8PP, U.K.

Finite Elements in Biomechanics Edited by R. H. Gallagher, B. R. Simon, P. C. Johnson, and J. F. Gross © 1982, John Wiley & Sons, Ltd

Chapter 9

A Non-linear Finite Element Stress Analysis o f Bioprosthetic H eart Valves G. W. Christie and I. C. Medland

9.1

IN TRO D U CTIO N

When one of the natural valves of the heart begins to display a significant degree of dysfunction, the cardiac surgeon will generally replace it with one of a large range of commercially available prosthetic devices. Here we present an analysis of the static mechanics of one class of such devices, that is bioprosthetic valves modelled on the basic anatomy of the natural trileaflet aortic valve. The term ‘bioprosthetic’, while not universally accepted, is widely used in the medical literature to distinguish between the purely prosthetic valves such as ‘ball-incage’ or ‘tilting disc’ devices, and the wide assortment of trileaflet designs in which the leaflet material is of biological origin. Fundamentally, the bioprosthetic valve consists of three approximately equal tissue leaflets sutured to a tricornute frame called a stent. The stent itself may be rigid or slightly flexible (Reis et a I 1971; Thomson and Barratt-Boyes, 1977) and always has a tight cloth covering, usually of Dacron, to provide a firm substrate for suturation as well as protecting the stent material from direct contact with the blood with its attendent risks of thromboembolism. A description of the assembly of a Hancock porcine xenograft bioprosthesis may be found in Zuhdi et al. (1974) and further details are given by Cohn and Collins (1979). The leaflet material most commonly encountered consists of an entire porcine aortic valve preserved in a solution of gluteraldehyde although the use of antibiotic-treated human aortic valves (allografts) and various non-differentiated connective tissues such as bovine pericardium is also common. The tissue of natural aortic valves displays a significant degree of elastic anisotropy with the fibres of the collagen network preferentially aligned in the circumferential rather than radial directions within each leaflet (Clark and Finke, 1974; Broom, 1978). Whereas human tissues only require to be treated with antibiotic solutions having a negligible effect on its elastic properties, all the non­ human tissues are treated with a gluteraldehyde solution which acts as 153

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Finite Elements in Biomechanics

preservative, an antibiotic, and greatly reduces the antigenicity of the preserved tissue. However, treatment with gluteraldehyde has a profound effect on the compliance of the tissue, the degree of which depends on the state of stress the tissue was in at the time fixation took place (Broom and Thomson, 1979). The fluid mechanics of heart valves has been studied by, amongst others, Bellhouse and Talbot (1969), Lee and Talbot (1979), Peskin (1972), van Streenhoven and van Dongen (1979), and Swanson and Clark (1973). However, for the purposes of this study, only the mechanics of a statically loaded valve will be considered, and hence the position of the valve within the heart (aortic, mitral, pulmonary, or tricuspid) is of no consequence except in so far as the magnitude of the total pressure will be different in each case. For this reason the total pressure load applied during all calculations reported in this work was taken to be 16 kPa, which is approximately the normal pressure for the mitral position and the maximum pressure the bioprosthesis is likely to sustain in a normal individual. For hypertensive recipients, however, the applied pressure could exceed this value by more than 50 %. With natural valves, as wear and damage take place in the tissue the damaged components are continually replaced to maintain the full structural integrity of the material. With the non-viable tissue of the bioprosthetic valves, this process cannot take place and a process of steady degradation of the tissue structure starting from the time of implantation has been observed to occur (Ferrans, Spray, Billingham, and Roberts, 1978). Although no experimental studies have been undertaken as yet, it seems to be most likely that the rate of structural degradation or fatigue of the tissue components is closely related to the stress the tissue is required to sustain. Biochemical processes are also an important factor but it must be recognized that any biochemical process which acts to weaken the tissue structure will tend to make it even more susceptible to damage by mechanical forces. It must therefore be considered advantageous for the recipient to receive a bioprosthesis which is so designed that the stress in the tissue components of the valve is as low as possible. Before this state can be achieved a detailed knowledge of the mechanics, both static and dynamic, of these structures is essential. Certainly, the nature of the failure of the bioprosthesis is such that the progress of its deterioration can be traced clinically and a suitable moment chosen for the implantation of a new device. In this respect the bioprosthesis offers considerable advantages over the catastrophic and generally fatal mode of failure of the prosthetic valves. Nevertheless, the risk of an early, post-operative death is significantly increased on reoperation (Rossiter et al., 1979) so the effective, long-term survival rate of the recipient should be significantly enhanced by reducing the stress levels in the first implant. Binet and associates (1965) were the first to implant a bioprosthetic valve in a human and since that time considerable advances and diversification in design has taken place. The first porcine xenografts in the late 1960s were fixed in formaldehyde with highly unfavourable results, most failing with leaflet

Finite Element Stress Analysis o f Bioprosthetic Heart Valves

155

perforation in less than one year. Gluteraldehyde was found to give much more satisfactory long-term results. Numerous long-term follow-up studies of porcine xenografts (Stinson, Griepp, Oyer, and Shumway, 1977; Carpentier et al., 1976), heterologous tissue valves (Ionescu et al., 1974), and allografts (Roche, BarrattBoyes, Heng and Rutherford, 1977) have shown that the expected lifetimes for these valves appears to be in the range of five to ten years. Results from accelerated cycling experiments have been reported by Clark et al., (1978) for whole valves and by Broom (1977,1980) for uniaxial stretching of fresh and treated heart valve tissue. Such tests provide information on the mechanism of the fatigue process but unreliable estimates of valvular longevity in vivo. To the knowledge of the authors, no theoretical work has previously been reported on bioprosthetic valves. Analyses of stress in unstented, natural human (Cataloglu, Clark, and Gould, 1977) and natural porcine (Chong and Missirilis, 1978) aortic valves have been presented, but both differ substantially from the method adopted for this analysis. The approach used for this work has been to develop a ‘model’ for the mechanical behaviour of the heart valve, based on anatomical observations of the structure of the leaflet material. The first model treated the leaflets as a thin isotropic membrane such as would be the case for many heterologous tissue valves. The second model refined this by allowing the tissue to have anisotropic elastic properties as observed experimentally in natural, differentiated aortic valve tissue. This was achieved by embedding hyperelastic fibres into the isotropic membrane and aligning them circumferentially in accordance with the data given by Clark and Finke (1974). Since some of the parameters for the models were either not available or not reliably known, numerical ‘experiments’ were conducted using the finite element program holding all parameters constant except one. By varying the remaining free parameter over assumed physiological limits, its effect on the mechanics of the whole valve could be estimated. In what follows, the foundation of the theory of thin membrane structures is outlined, together with a brief discussion of the choice of strain-energy function. The formulation of the finite element equilibrium equations is given and the boundary conditions used are justified. Results are presented for both the isotropic and anisotropic models which show the anisotropic properties of natural aortic valve tissue substantially improve the effectiveness of the seal formed between two adjacent leaflets. 9.2

D E FIN IT IO N S O F STRESS AND STRAIN TENSORS

The leaflets of aortic heart valves exhibit both large displacements and finite strains when subjected to hydrostatic pressure loading. As observed by Swanson and Clark (1973), bending effects are probably negligible compared with the in­ plane forces so the assumption of purely membrane behaviour provides a good mathematical description. In addition, we assume the existence of a strain-energy

156

Finite Elements in Biomechanics

density function, W, which characterizes the elastic properties of the tissue and gives the strain energy per unit of the undeformed volume. Such materials have been termed hyperelastic. A detailed account of a general theory for the finite deformation of hyperelastic membranes has been given by Green and Adkins (1960), and it is on their work that the current formulation is based. In what follows the usual tensor notation has been used with summation assumed over repeated indices unless otherwise stated; Greek indices take the values 1, 2 while Latin indices take the values 1, 2, 3 with upper- and lower-case symbols representing quantities in the deformed and undeformed states respectively. The reference state is taken to be the undeformed state, so the GreenSt. Venant tensor, yij9 provides a consistent definition of strain and the corresponding symmetric stress tensor, r u, is called the second Piola-Kirchhoff tensor. A system of convected coordinates, 0, may be defined such that 0a lie in the middle surface of the membrane and 03 is directed normal to the surface. The covariant metric tensors associated with the deformed and undeformed states are Gij and gijy while those associated with the respective middle surfaces are Aap and aap. Green and Adkins (1960) have shown that the Green-St. Venant strain tensor may be defined as y*i> = 2 ( '4*o- a «/>)’

?33 = 2 ^ 2 _ 1 )

(9.1)

where A is the extension ratio normal to the middle surface. The remaining terms, ya3, are zero because 03 is normal to the middle surface and hence Ga3 = g 3.0 mm, the fibre elements become the dominant load-bearing structural component and produced marked changes in the mechanical be­ haviour of the valve. 9.8.1

Distribution of stress in anisotropic leaflets

The change in distribution of stress for rf = 0.1 mm was most pronounced in the vicinity of the commissures, as may be seen in Figure 9.10, were there has been a reduction in the stress by a factor of ten. A stress reduction takes place over the entire leaflet but the change away from the commissures is not as pronounced. The reason for the very large reduction at the commissures lies in the fact that the stress field is almost uniaxial at these points whereas elsewhere there is a significant biaxial component. The fibre elements are, of course, ideal for transmitting uniaxial forces and so can be expected to produce the greatest benefits at points in the leaflet where such forces are present.

Finite Element Stress Analysis o f Bioprosthetic Heart Valves

9.8.2

171

Change in shape for anisotropic leaflets

As the fibre radii are increased from zero up to the maximum considered here of 3.0 mm, several major changes in the mechanical response of the leaflet surface to load were noted. Firstly, the downward displacement of all points in the leaflet was retarded. Secondly, there was a corresponding movement of the leaflet towards the coaptation interface which increased with increasing fibre radii. This situation contrasts sharply with that previously noted for isotropic leaflets where there was no noticeable movement towards the coaptation interface and hence, little dependence of coaptive area on applied pressure. Figure 9.11 shows the quasi-static motion o f the leaflet under pressure (0.01 kPa to 16 kPa) for r j — 3.0 mm. Because the net movement of the leaflet is towards the coapating surface in the reinforced membrane, the area of coaptation was found to increase with both pressure and fibre radii as is shown in Figure 9.12. The discrete jumps in the area are the result o f a new element coming into contact with the coaptive boundary while the gradual rise on the remaining portions of the curves is due to the stretching o f elements, already in contact, under applied pressure. In the isotropic case, no new elements came into contact with the coaptive boundary but with increasing degrees of circumferential reinforcement a substantial rise was found in the area of contact between adjacent leaflets. In fact, for the relaxed shape used in this model, the final coaptive area achieved appears

172

Finite Elements in Biomechanics z (mm)

Figure 9.11 Displacement of line BD for an aniso­ tropic leaflet (r^= 3.0 mm) as a function of pressure

Applied

Figure 9.12

pressure (kPa)

Coaptive area as a function of pressure

to level off at about a factor o f two greater than that o f the isotropic membrane. Note that some o f the membrane is already in contact in its relaxed configuration and were it not for this, it appears reasonable to suppose that the isotropic membrane would have formed no coaptive surface.

Finite Element Stress Analysis o f Bioprosthetic Heart Valves

173

This result lends strong credence to modelling natural aortic valve leaflets as a fibre-reinforced membrane structure. Leaflets with almost any degree of cir­ cumferential reinforcement can be expected to have significantly greater coaptive surface area than the corresponding isotropic leaflet. 9.8.3

O rientation of the stress field in anisotropic leaflets

From the previous model, the stress field in isotropic leaflets was found to be predominantly aligned in the circumferential direction at least at points not close to the leaflet boundary and, furthermore, this alignment was virtually inde­ pendent of the applied pressure load. For anisotropic leaflets, this simple pattern of behaviour changes progressively as the circumferential reinforcement increases. Figure 9.13 shows the orientation of the stress field in the membrane elements relative to the circumferential direction at the leaflet belly for a range of rf values. For small values (rf ^ 0.1), the alignment is almost the same as in the isotropic case. For intermediate values (0.50 < rf < 2.0), there is a strong dependence of stress alignment on pressure, the alignment being radial at low pressures and circumferential at high pressures. For high values (rf ^ 3.0 mm), the stress field remains radially aligned over the entire range of pressures considered.

Applied

pressure

(kPa)

Figure 9.13 Orientation of the membrane stress field relative to the fibre direction in the leaflet belly as a function of pressure

Figure 9.14 gives a plot of membrane stress field alignment taken at a pressure of 16 kPa along the same fibre traversing the membrane from the central belly to the boundary at the base of the stent post. It may be observed that the change in alignment with fibre radii is greatest in the centre of the leaflet and becomes progressively less significant towards the rigid boundary.

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Finite Elements in Biomechanics

Figure 9.14 Orientation of the membrane stress field relat­ ive to the fibre direction across the leaflet at P — 16 kPa

9.8.4

Force transmission to the stent frame

The distribution of load which the structure will be required to sustain is very relevant to the design of stent frames. The leaflet transmits the applied pressure load exerted by the blood on to the supporting structure, be it a stent or the aortic root as in natural valves. In terms of the static global equilibrium of the bioprosthesis, all forces exerted on the valve leaflets must ultimately be exactly balanced by reaction forces exerted by the boundary. Although each leaflet exerts force on its neighbouring leaflets via the coapting surface, these are exactly balanced by the symmetry requirement so it will suffice to examine the reaction forces only at nodes on the stent boundary. Figure 9.15 gives the magnitudes of the nodal force vector components on the stent boundary as a function of the global