Finite deformation biphasic material properties of bovine articular cartilage from confined compression experiments

J. Bmtrchanm. c 1997 Published

Vol 30. Nos. I I/ 12. pp I I i7-1164. 1997 by Elsewr Scwxx Ltd All rights reserved Printed m Great Britain (HE-9?90!97 %I 7.00 + 00

PII: SOO21-9290(97)000936-6

FINITE

DEFORMATION OF BOVINE FROM CONFINED G. A. Ateshian,*

Orthopaedic

BIPHASIC MATERIAL PROPERTIES ARTICULAR CARTILAGE COMPRESSION EXPERIMENTS

W. H. Warden,

J. J. Kim,

R. P. Grelsamer

and

V. C. Mow

Research Laboratory, Department of Mechanical Engineering, Columbia 500 West 120th St. SW Mudd 220, New York. NY 10027-6699, U.S.A.

University,

Abstract--In 1990, Holmes and Mow [Journal ofBiomechanics 23, 1145-l 1561 developed a hyperelastic biphasic theory to describe finite deformation behaviors of articular cartilage. To date, however, no experimental finite deformation studies have been made to assess the ability of this constitutive model to describe its finite deformation behaviors (e.g. kinetic creep and stress-relaxation, and equilibrium responses). The objectives of this study are: (1) to investigate whether this hyperelastic biphasic theory can be used to curve-fit the finite deformation compressive stress-relaxation behavior of the tissue, and from this procedure, to calculate its material coefficients; and (2) to investigate whether the theory. together with the calculated material coefficients, can accurately predict the outcome of an independent creep experiment followed by cyclical loading of the tissue. To achieve these objectives, circular cylindrical cartilage plugs were tested in confined compression in both stress-relaxation and creep experiments. Results demonstrated that curve-fits of the stress-relaxation experiments produced nonlinear generalized correlation coefficients of r2 = 0.99 k 0.02 (mean + standard deviation); theoretical predictions of the creep test differed on average by 10.0% k 2.0% relative to experimental results. When curve-fitting the creep experiments as well, it was found that the permeability coefficients differed from those obtained from the experiments (k,,,, = 2.2 k 0.8 x lo- ” m4 N 1 s-’ stress-relaxation and M,, = 0.4 & 0.8 vs ko,,, = 2.7 + 1.5 x IO-l5 m4 N-’ s-‘, and M,, = 2.2 + 1.0); these differences may be attributed to imprecisions in the curvefitting procedure stemming from the low sensitivity of the stress-relaxation and creep behaviors to large variations of M in the permeability function. Advantages and limitations of this theoretical model are presented in the text. ‘C: 1997 Published by Elsevier Science Ltd. All rights reserved Kr~~~~ords:

Cartilage;

Biphasic

theory:

Finite

deformation;

NOMENCLATURE

Confined

Greek P

left Cauchy-Green strain tensor initial cartilage thickness aggregate modulus under zero strain (material parameter for finite deformation theory) aggregate modulus (material parameter for infinitesimal deformation theory) identity tensor three invariants of B cartilage permeability function (constant for infinitesimal deformation theory) cartilage permeability under zero strain (material parameter for finite deformation theory) k, as obtained from creep experiments k. as obtained from stress-relaxation experiments cartilage diffusive drag constant nonlinear permeability coefficient (material parameter for finite deformation theory) M as obtained from creep experiments M as obtained from stress-relaxation experiments number of data points for equilibrium results number of data points for transient results cartilage interstitial fluid pressure applied stresses time variable and constants axial deformation of solid matrix in material reference frame solid and fluid phase velocities ramp speed for the stress-relaxation test axial coordinate

v

compression.

letters compressive-stiffening coefficient (non-dimensional material parameter for finite deformation theory) solid content under zero strain solid and fluid contents, @ + 4’ = 1 solid matrix stretch non-dimensional material parameter for finite deformation theory (reduces to Poisson’s ratio in infinitesimal theory) angular frequency of sinusoidal load momentum exchange force elastic stress solid and fluid stresses

INTRODUCTION

In most studies of the compressive properties of articular cartilage, only infinitesimal strain theories have been used to analyze experimental data (e.g. Armstrong and Mow, 1982; Athanasiou et al., 1991; Hayes et al., 1972; Hirsch, 1994; Holmes et al., 1985; Hori and Mockros, 1976; Jurvelin et al., 1987; Kempson et al., 1971; Mow et al., 1989). Material properties calculated under these conditions have been very useful in studies characterizing various tissues and in quantifying changes due to diseases or injuries (Mow et al., 1992). However, physiologic joint loads and stresses are usually quite high (e.g. Ahmed and Burke, 1983; Andriacchi and Mikosz, 1991; Brown and Shaw, 1984; Hodge et al., 1989; Huberti and Hayes, 1984; Paul, 1967), with cartilage contact stresses generally ranging from 2 to 12 MPa, or possibly higher. Recent theoretical contact studies have demonstrated that a

Received in final form 27 Mny 1997. *Address correspondence to: Dr G. A. Ateshian, Department of Mechanical Engineering, Columbia University, 500 West 120th St. SW Mudd 220, Mail Code 4703, New York, NY 10027-6699, USA. 1157

G. A. Ateshian

i IiS

sign&cant portion (e.g. 80-95%) of contact stresses are actually supported by interstitial fluid pressurization under the majority of physiological loading conditions, hence shielding the solid matrix of cartilage from excessive stresses and strains (e.g. Ateshian et a!., 1994, 1995); nevertheless, the portion of stresses applied on the solid matrix may still be in the range of 0.2-2.5 MPa or greater. Since the compressive modulus of cartilage is in the range of 0.5-l MPa according to infinitesimal theories (e.g. Armstrong and Mow, 1982; Athanasiou et al., 1991; Jurvelin et al., 1987), it is evident that cartilage could easily be subject to finite deformation under many physiological situations. Because of interstitial fluid pressurization however, the tissue strains may not necessarily be excessive (e.g. < 50%), as suggested in part by the in situ measurements of Armstrong and coworkers (1979) who found cartilage engineering strains up to 14% only, in the human hip joint. Various forms of the finite deformation biphasic theory have been developed to describe cartilaginous tissues (e.g. Cohen et al., 1992, 1997; Holmes, 1986; Holmes and Mow, 1990; Kwan et nl., 1990). These theories have various strain-energy and strain-dependent permeability functions, and are mathematically valid under various material assumptions (e.g. see Trusdell and Nell, 1965). To date, to our knowledge, no studies have been made to assess the ability of any of these theories to successfully describe both the kinetic (i.e. creep and stress-relaxation) and the equilibrium behaviors of the tissue. In this study, our hypothesis is that the finite deformation biphasic theory developed by Holmes and Mow (1990) can describe both the kinetic and equilibrium compressive behaviors of cartilage under high strain fields. As this study represents the first set of tests of this theory, it only addresses the one-dimensional analysis of confined compression. Thus, the objectives of this study are: (1) to investigate whether this finite deformation biphasic theory can be used to curve-fit the compressive stressrelaxation behavior (kinetic and equilibrium) of articular cartilage, and from this procedure, to calculate the finite deformation biphasic material coefficients; and (2) to investigate whether the theory, together with the calculated material coefficients, can correctly predict the outcome of a creep experiment followed by cyclical loading of the tissue. MATHEMATICAL

THEORY

Biphasic model

A finite deformation biphasic theory for intrinsically incompressible mixtures was first formulated by Mow and coworkers (1980). Later, Kwan and coworkers (1990) provided a finite deformation constitutive law for the nonlinear response of the tissue solid phase; this theory was refined by Holmes (1986), and again later by Holmes and Mow (1990).t In these models, the solid phase of cartilage was assumed to be a hyperelastic and isotropic

e/ (I/

medium. Most recently, Cohen and coworkers (19971 extended the finite deformation biphasic theory to account for a specific form of tissue anisotropy as well. In the present study. we have used the isotropic finite deformation theory of Holmes and Mow (1990), as well a:, additional results of Cohen and coworkers (1997) to describe and predict our experimental studies. The biphasic theory, when applied to articular cartilage, assumes that the tissue consists of an intrinsically incompressible solid phase representing collagen, proteoglycans, chondrocytes, and other quantitatively minor glycoproteins, and an intrinsically incompressible fluid phase representing interstitial water and dissolved electrolytes; the fluid phase has been measured to range between 65 and 90% of the tissue by weight (e.g. Maroudas, 1975; Mankin and Thrasher, 1975; Armstrong and Mow, 1982). The continuity equation for a biphasic medium is V.(fp

+ fjfuf) = 0.

11)

Under quasi-static conditions, the momentum equations for the solid and fluid phases of the tissue are given by V~a”+It=o,

(24

V.a’-71=0,

W

where IL is the momentum exchanged between the phases due to friction, as fluid particles flows pass the solid phase. In articular cartilage, it has been shown that this term is much more significant than the frictional dissipation between fluid particles (i.e. viscosity) which gives rise to viscous stresses within the fluid (Hou et al., 1992). Following Holmes and Mow (1990), and Cohen and co-workers (1997), the Cauchy stresses and momentum exchange for the finite deformation biphasic theory are given by us = - cppl + be,

(34

uf = - c#l’pI,

(3b)

7l = K(u’ - US)+ pvp.

(3c)

The diffusion drag constant K is related to the permeability k through k = (4f)2/K (Lai and Mow, 1980). The principal components of the isotropic elastic stress ue are related to the principal components of stretch lbl, &, i,, by the following constitutive relations (Cohen et al., 1997): i

-3v

l-v

+

where i, j, k permutate

over 1,2,3, and @ is given by

@=exp

-3) + &

/? E(Jl ii

LJ2 - 3) - W3)

I)

. (5)

tThe dates of these publications appear to be inconsistent. Due to the delay in the publication of the Kwan et al. (1990) paper, the later Holmes (1986) paper appeared in print prior to the publication of the Kwan and coworker (1990) paper.

It is therefore the objective of the present study to test whether these constitutive relations, equations (4) and (5), can describe the response of cartilage under finite deformation - for both stress relaxation or creep. Here.

Finite

deformation

material

properties

HA0 represents the aggregate modulus and v is a nondimensional coefficient which would reduce to the solidphase Poisson’s ratio under infinitesimal strains; the compressive-stiffening coefficient /I is a non-dimensional constant which measures the sensitivity of be to large strains (Holmes and Mow, 1990). The quantities Ji, Jz, J3 are the invariants of the left Cauchy-Green strain tensor B of the solid phase.f Along principal directions of stretch, the components of B are all zero except for the three diagonal entries lb:, iz, and 25. In these directions, the invariants J1 = tr B, J2 = i[(tr B)’ tr B2] and J3 = det B can be easily determined. In finite deformation, large strains can significantly alter the tissue porosity. Consequently, the solid and fluid fractions and the permeability are related to the tissue dilatation (Holmes and Mow, 1990) i.e.:

of bovine

articular

1159

cartilage

not influence the solution (Mow et (II., 1980). The governing equations for finite deformation of the tissue, i.e. equations (l)-(3), can be simplified considerably to produce the following one-dimensional non-linear partial differential equation as shown by Holmes (1986):

abea2u -__

ILau

h 2

iii. az2=ki7t3

Here, 8 is related to 1&through equation (7) and k is related to 1 through equation (6) where J3 = 12. The initial condition for this equation is U(Z, 0) =O. Since inertial effects are neglected, the inertial reference frame has been located at the midplane of the cartilage cylindrical specimen to take advantage of the resulting midplane symmetry; thus the boundary conditions are U(h/2, t) =O,

(64 (6b)

where 40 and k. are respectively the solid content and permeability of the tissue in the absence of strain, and M is the non-dimensional permeability coefficient weighting the exponential functional dependence on strain. The relations of equations (6a, b) are physical constraints, while that of equation (6~) is a constitutive permeability function postulated for articular cartilage. Thus, the material behavior of an isotropic hyperelastic biphasic tissue is completely determined by the five material coefficients I&,. /I, V,k,, and M. This hyperelastic biphasic theory for finite deformation reduces to the infinitesimal deformation linear biphasic theory of Mow et ul. (1980). ANALYSIS OF THE ONE-DIMENSIONAL COMPRESSION EXPERIMENT

CONFINED

In this section, these general equations are reduced to the one-dimensional configuration to describe the experiments conducted in this study. A circular cartilage plug confined within a cylindrical chamber with impermeable and smooth walls, and loaded axially by rigid porouspermeable loading platens undergoes one-dimensional deformation. Under these conditions, the principal stretches reduce to i,i = 1, & = 1, and A3 = 1, and the axial stress-stretch law can be derived from equations (4) and (5): eBc+‘) where i = 1 + aU/aZ, U(Z, t) is the axial deformation expressed in a material (Lagrangian) reference frame, Z the axial material coordinate, and t the time. Note that v no longer explicitly appears in the constitutive equation since in one-dimensional analysis, this coefficient does

$Basic descriptions theories can be found.

of strain tensors used in finite e.g. in Truesdell and Noll (1965).

deformation

U(0, t) = U,(t)/2,

(94

for stress-relaxation

cre(O,t) = a;(t),

experiment, (9b)

for creep experiment,

(9c)

where h is the initial thickness of cartilage, U,(t) is a prescribed surface displacement and a:(t) is a prescribed surface stress. In this study, the stress-relaxation experiments were conducted by applying a constant ramp displacement at the specimen surface for to seconds followed by a constant displacement. i.e. Vof, b(f)

=

Odtdto,

v i

(10)

t 0

07

t

>

to.

while the creep experiments consisted of applying a constant stress at the specimen surface for t, seconds followed by a sinusoidal stress, i.e., a;(t) =

so7

0 d t d t1,

(11) i so + s1 sin 0X, t > t1 (s1 < so). The system of partial differential equations, equation (8), along with the boundary conditions in equations (9)-( 1l), were solved numerically by the method of finite differences. Central difference discretization in space and backward discretization in time were used to yield an implicit scheme for the dependent variable U. The resulting non-linear tri-diagonal system of equations was solved at each time step using the Newton-Raphson method. MATERIALS

AND METHODS

Specimenpreparation and testing

Seventeen cylindrical osteochondral plugs were harvested from the glenoid surface of five fresh skeletally mature bovine shoulder joints. The articular cartilage was then dissected from the subchondral bone with a scalpel. Each cartilage disc was subsequently placed on freezing stage of a sledge microtome (Leitz, Rockleigh, NJ, U.S.A.), immersed in M-l embedding matrix (Lipshaw, Detroit, MI, U.S.A.) and frozen to -20°C for sectioning. Serial slices 10 pm thick were removed from the bottom surface until it became parallel with the articular surface. After washing off the M-l matrix, and equilibrating the tissue for 30 min in physiological saline, a trephine (dia = 6.35 mm) was then used to remove

G. A. Ateshian

1 IhO

rt till.

micrometer

/ I--

load cell

time dl;p;lluegrent

Fig. 2. Displacement profile for stress-relaxation tests: Five consecutive ramp displacements, of duration to and rate V,, followed by constant displacements of Vote are applied to achieve 10% engineering strain (i.e. li,r,/h = 0.10) at each increment, for a total of 50% strain.

porous indenter

original thickness. The stress acting on the tissue was obtained by dividing the measured force with the known area of the specimen (31.7 mm”). At the end of this loading sequence, whose total duration averaged 9530 + 860 s, the tissue was unloaded down to the tare load, and allowed to recover until the tissue displacement rate was less than 0.01 pm s-l; the recovery period averaged 1380 Ifr 240 s. Following tissue recovery, the creep test was initiated by applying a step compressive stress so = 0.19 MPa onto the tissue. This stress was maintained until the surface creep displacement rate was again less than 0.01 pm s-i (duration: 1640 f 250 s). Following this, an oscillatory stress of amplitude s1 = 0.15 MPa was applied, as described in equation (11) at an angular frequency of o = 0.03 rad s- ‘, for a duration of 2000 s. The entire experiment was conducted for each sample on the same day, and lasted 15 510 + 120 s on average.

bath cartilage specimen

fre

translation stage Fig. 1. Schematic of a confined compression loading apparatus; the cylindrical disc of cartilage is loaded between two rigid porouspermeable filters.

a cylindrical disc from the tissue sample; specimens were kept frozen at - 20°C until the day of testing. The plugs (h = 1.41 i 0.31 mm)? were tested in a previously described loading apparatus (Guilak et al., 1989a), first under displacement control (stress-relaxation test) then under load control (creep test). Each thawed cartilage plug was placed inside a smooth confining cylindrical chamber with impervious side wall and porous bottom surface. The chamber and its surroundings were submerged within a physiological saline solution containing enzyme inhibitors (2 mM ethylenediamine tetra-acetic acid, 5 mM benzamidine, 10 mM N-ethylmaleimide, 1 mM phenylmethyl-sulfonyl fluoride, 0.15 M NaCl) at room temperature. The top surface of the specimen was loaded in compression through a porous filter attached to a load cell and an LVDT (Fig. 1). Prior to the actual tests, a pre-conditioning load of 14.7 N was applied on the tissue for 60 s and removed, followed with a tare load of 0.26 N for 900 s. In the stress-relaxation test which followed, the surface displacement Uo(t) of the specimen was then prescribed incrementally in the form of five consecutive sequences of a linear ramp displacement each of duration to at a rate of V. = 0.25 pm s-l, followed by a constant displacement over approximately 1400 s (Fig. 2). The time to was prescribed such that the equilibrium strain at the end of each ramp was increased by approximately lo%, for a total equilibrium strain of approximately 50% (to = 520 t- 80 s); to varied from specimen to specimen due to variations in the specimen’s TrThroughout deviation.

this paper

the symbol

k

denotes,

mean * standard

DETERMINATION

OF MATERIAL BY CURVE-FITTING

PARAMETERS

The experimental results for the kinetic stress-relaxation and equilibrium behaviors were used to calculate the four material coefficients of the one-dimensional theory using non-linear regression analysis. These coefficients are the aggregate modulus (HAO, the compressive-stiffening coefficient (/I), and the non-linear strain-dependent permeability coefficients (k,, M). Water content was measured in all specimens and was found to be 81.1 + 2.0%; because of these relatively tight results, the solid content in the absence of strain was assumed to be 20% for all specimens in this study (40 = 0.2). To obtain, HA0 and /I we first curve-fitted the equilibrium stress vs stretch (ae vs /2) data using the hyperelastic constitutive law defined by equation (7) and a non-linear least-squares regression procedure. Next, to obtain k. and M, numerical solutions of equation (8), and boundary conditions (9a) and (9b), along with the calculated material coefficients HA0 and /3, were curve-fitted to the kinetic stress-relaxation data. Again, a non-linear leastsquares criterion was used. I/ 11As noted above, the remaining material coefficient in this theory, Poisson’s ratio r, cannot be determined from this test since it does not influence the one-dimensional tissue response.

Finite

deformation

material

properties

of bovine

articular

Following these curve-fitting procedures, the calculated material parameters were used to predict the creep response of the tissue under the loading conditions provided in equation (11). This theoretical prediction was plotted against the corresponding experimental finite deformation creep results, without adjusting any of the material coefficients, as an a posteriori check of the predictive ability of the hyperelastic biphasic theory. As an alternative, the parameters k,, and M were also curvefitted from the creep results, and compared to the corresponding values obtained from the stress-relaxation curve-fits. RESULTS

The specimens were found to recover by 98.7 f 0.7% of their initial thickness at the end of the stress-relaxation experiment, prior to initiating the creep experiment. A typical finite deformation stress-relaxation response is shown in Fig. 3(a), and the corresponding equilibrium stress-stretch curve for the same specimen is shown in Fig. 3(b). The mean and standard deviations for all curve-fitted material coefficients, as obtained from the stress-relaxation results, were: HA0 = 0.40 + 0.14 MPa, a = 0.35 + 0.29, ko,,, = 2.7 k 1.5 x lo-l5 m4 N-’ s-l, and MS, = 2.2 f 1.0. For all specimens, the curve-fits of the stress-relaxation response showed very good agreement between theory and experiment, as assessed by the nonlinear generalized correlation coefficient of r2 = 0.99 -t 0.02 for all specimens [e.g. Fig. 3(a)].

time (s) Fig. 4. Experimental and predicted creep responses for the specimen whose material properties were obtained from the stress--relaxation experiment of Fig. 3.

time (s) Fig. 5. Example of sponses for another based on material periment, does not

(a)

time (s)

~11 0.90 (b)

I .oo

stretch, h

Fig. 3. (a) A typical stress-stretch curve of experimental and curvefitted data obtained from kinetic results; (b) Experimental and curvefitted equilibrium response for the same specimen.

1161

cartilage

experimental, predicted, and curve-fitted creep retypical specimen. The predicted response, which is properties obtained from the stress-relaxation exdiffer appreciably from the curve-fitted response.

Two typical comparisons of experimental data and predicted creep responses are shown in Figs 4 and 5. For all 17 specimens, the predictions of the creep response compared very consistently with the experimental results. The quality of the creep predictions was assessed by measuring the relative error between theory and experiment at each time step, and evaluating the root-mean square (rms) value of this relative error over all time steps. For all specimens, this rms value averaged 10.0 * 2.0%. Alternatively, instead of predicting the creep response using the material properties curve-fitted from the stress-relaxation test, the values of k. and M were obtained by fitting the creep data (keeping the same values from the equilibrium reOf HA0 and p as determined sponses of the stress-relaxation test). The results from these alternative curve-fits produced k. cr = 2.2 + 0.8x10-‘5m4N-‘s-‘andMC,=0.4+0.8,\;lithanonlinear correlation coefficient of r2 = 1.47 &- 0.10 and rms relative error of 8.1 + 1.5%; these values of ko,,, and MC, were found to differ statistically from ko,,, and MS, determined by the stress-relaxation curve-fits, using a paired two-tailed Student t-test (p = 0.02). Figure 5 demonstrates the tissue deformation response from a typical creep experiment, in addition to the curve-fitted response (using ko,,, and MC,) and the theoretically

G. A. Ateshian

1 If,

er ~1

predicted response using the material properties obtained from the stress-relaxation curve-fits (liO.,r and M,,). 3

HA,=SlOpe

at origin

DISCUSSION

The first objective of this study was to investigate whether the hyperelastic biphasic theory of Holmes and Mow (1990) can successfully be curve-fitted to the kinetic and equilibrium compressive behaviors of articular cartilage under finite deformation stress-relaxation. The one-dimensional confined compression experiment was employed as a first step because it yields a numerically tractable governing equation, and therefore it is amenable for curve-fitting analyses which are computationally intensive. This first objective is the most basic step in demonstrating that the proposed theory can indeed be used to describe the response of articular cartilage under finite deformation; clearly, failure to achieve this objective would obviate the need for further tests. The results of this study demonstrate that the curvefitting procedures employed for the stress-relaxation data were quite successful, e.g. based on the non-linear generalized correlation coefficient, thus achieving the goal of this first objective. The four material parameters that influence the confined compression response were obtained two at a time, from two different but related experimental curves derived from the stress-relaxation experiments. In relation to previous studies using the linear biphasic theory, the present results are consistent with those determined under infinitesimal loading conditions. Since infinitesimal biphasic properties of bovine glenoid cartilage have not been reported previously in the literature, a reasonable though imperfect comparison can be made with the findings of Athanasiou and coworkers (199 1) who reported values of HA = 0.47 + 0.15 MPa and k = 1.4 & 0.6 x 10-l’ m4N-’ SC’ for bovine femoro-trochlear cartilage, for an average tissue thickness of h = 1.38 k.O.19 mm. On average however, the hyperelastic biphastc theory is expected to yield an aggregate modulus which is slightly lower than that predicted by the infinitesimal theory. This can be explained from the equilibrium stress-stretch relation in equation (7) which shows that the slope of the stress-stretch curve at i. = 1 is equal to HAO. Thus, the aggregate modulus in the finite deformation biphasic theory is measured in the limit when there is unit stretch (zero strain) in the tissue. At any other value of /?, the slope of the stress-stretch curve becomes greater than

HA0 CFk. WI.

In the infinitesimal theory, however. the slope HA of the stress-stretch (or stress-strain) curve remains constant. In some applications of the biphasic theory, the infinitesimal theory has been used with experimental compressive strains up to 20% (2 = 0.8). To fit the resulting slightly non-linear experimental stress-stretch curve using the linear theory would yield a slope HA which represents an average of the slopes in the range 0.8 < i < 1, and thus is greater than the slope HA0 at 1. = 1. It can be easily calculated that if the experimental data indeed follow equation (7) say with HA,, = 0.40 MPa and /? = 0.35, then using the linear theory to curve-fit those data in the range 0.8 < i ,< 1 yields an

compressive strain

-I

compressive strain Fig. 6. Schematic representation of HA0 (aggregate modulus) and k, (permeability) at zero strain from the finite deformation theory, and HA (aggregate modulus) and k (permeability) from the infinitesimal strain theory.

aggregate modulus of HA = 0.44 MPa, which represents a 10% increase over the finite deformation HA0 in this case. Hence the present finite deformation results are consistent with prior infinitesimal strain results. Similarly, the permeability k calculated from experiments using the linear infinitesimal strain theory is expected to be lower than k. obtained from the non-linear finite deformation theory [Fig. 6(b)]. This is attributed to the fact that k,, represents the permeability in the limit when there is no strain in the tissue, while the permeability function decreases with increasing compressive strain as described in equation (6~). On the other hand, k from the linear theory is assumed to remain constant during compression; this constant value represents some average permeability over the range of applied tissue strains (Lai and Mow, 1980). The second objective of this study was to investigate whether the proposed finite deformation biphasic model can correctly predict the outcome of a creep experiment followed by cyclical loading of the tissue. This is a demanding and rigorous test of the predictability of the finite deformation biphasic theory. Encouraging results include the observed consistency among all specimens, and the relatively small error between experimental and theoretically predicted creep displacements (about 10% on average). However, when the permeability coefficients were determined from the creep experiments, they were found to be significantly different from the corresponding values determined from the stress-relaxation experiand MC,= ments (b. cr =2.2+0.8x10~‘5m4N-‘s-1 0.4*0.8 vs ko.,,=2.7& 1.5x 10-15m4N-1s-1, and n/l,, = 2.2 k 1.0). Under more favorable conditions, these values should not have been different. Despite these differences however, the rms error in the creep predictions (10% on average) was not much greater than the rms error in creep curve-fits (8% on average), a fact which is further apparent when comparing the curve-fitted and predicted creep curves for a typical specimen (Fig. 5).

Finite deformation material properties of bovine articular cartilage

This finding can be interpreted in two ways: (a) the constitutive model for the permeability, equation (6c), is not appropriate for articular cartilage; or (b) the indirect determination of the permeability coefficients from creep or stress-relaxation experiments does not yield precise values of those coefficients. While the first of these interpretations is certainly a legitimate concern, it remains that direct permeation experiments (e.g. Lai and Mow, 1980) strongly support the modeling assumption embodied in equation (6c), i.e. that cartilage permeability decreases exponentially as a function of tissue compressive dilatation. From the direct permeation experiments of Mansour and Mow (1976) on bovine femoro-trochlear cartilage, Lai and Mow (1980) obtained permeability coefficients of k0 = 1.7 x lo- is m4 N-l s-l and M =4.3. Their results compare favorably with our permeability coefficients from stress-relaxation tests. This experimental evidence from direct permeability measurements suggests that the second interpretation offered above is a more likely explanation for the findings of this study; since creep and stressrelaxation experiments are used to measure permeability indirectly, it is possible that their lack of sensitivity to relatively large changes in M may explain the differences observed in the curve-fits of the creep and stress-relaxation experiments. Indeed, using straightforward sensitivity analysis [i.e. the ratio of dk/k to aM/M from equation (6c)], it can be seen that the permeability li changes by 10-l 2.5% only, as M changes by lOO%, for 0.4 d M < 2.2 and 2.2 x lo- I5 < k. < 2.7 x ~0-15m4N~1 s-1, and assuming an average stretch of j, = 0.7. Another manifestation of this insensitivity can be seen in the value of k obtained from equation (6~) when using mean values for ko, M and 1 as follows: For stressrelaxation, the mean value, over all specimens, of the time-averaged stretch over the entire duration of the experiment was jLsr= 0.75; for the creep experiments, the equivalent mean value was i,,, = 0.70. Substituting &,, as well as the mean values of kO,srand M,, as described above, into equation (6~) produces k,, = 0.79 x 10-‘sm4N-1s-l. similarly, substituting A,, and the mean values of k,,,si and M,, as described above, into the same equation, results in k,, = 0.78 x lo- ’ 5 m4 N- ’ s- ‘. The parameters k,, and k,, can be thought of as average permeabilities over the entire duration of the stress-relaxation and creep experiments, respectively; clearly, these values are very similar, despite the differences observed between k,,,,, MS, and kOIFr,MC,. The significance of these findings is that the results of this study are only partly conclusive. While it was demonstrated that the hyperelastic biphasic theory of Holmes and Mow (1990) can fit the experimental data well, it seems that the determination of the permeability coefficients k,, M from stress-relaxation or creep experiments may not be very precise. However, the imprecision noted

in the present study is somewhat alleviated by the good agreement observed in the ‘average’ value of the permeability (k,, and k,,) between the two types of experiments. The issue of imprecision under certain types of testing configurations does not invalidate the theory, though it poses practical difficulties terial parameters are sought.

when

accurate

ma-

1163

It is well known that theoretical models of physical phenomena cannot be proved to be correct, but only disproved when they fail in their predictions within their expected range of application. Hence, it is not really possible to conclude from this study that the proposed finite deformation biphasic model is ‘correct’, only that it succeeded with mitigated results in this first extensive set of experimental finite deformation tests under the configuration of confined compression. Only with the accumulation of a wide arrary of data will it be possible to tell whether this (or any) constitutive law can withstand the test of time, or whether it fails under the most common testing configurations. It is already possible to anticipate some of the limitations and strengths of the proposed hyperelastic biphasic model for articular cartilage. We believe that the principal limitation of the theory is that it uses an isotropic description for the cartilage solid matrix, when it is known already that cartilage is in fact anisotropic. However, three points need to be made in support of our choice of an isotropic model: (1) in confined compression, anisotropic and isotropic models predict the same material behavior; what would differ would be the magnitude of the material parameters when the tissue is tested along different orientations; hence, the current compression study can also be viewed as one of several steps toward assessing the anisotropic tissue properties. (2) indeed, anisotropic models of cartilage involve far more material constants than isotropic models, providing significantly greater experimental and theoretical challenges for determining those properties from multiple tests on the same tissue sample. (3) In view of the fact that this is the first extensive experimental study of the finite deformation biphasic response of cartilage in the literature, and considering the issues addressed in (1) and (2) above, we find it reasonable to start our analysis of the finite deformation of cartilage with an isotropic model in compression, with the view of extending this model in future studies to account for tissue anisotropies and inhomogeneities. In contrast to the above limitation, the finite deformation biphasic model poses no restrictions on the amount of strain developed inside or load applied onto the tissue, or the rate at which displacement or load that can be prescribed at the tissue surface. In previous studies, such restrictions had to be satisfied when using infinitesimal strain theories (e.g. Holmes et ul., 1985). In future studies, tissue anisotropy and inhomogeneity (e.g. see review by Mow et al., 1992) may be taken into consideration in order to further improve the predictive ability of the cartilage biphasic finite deformation constitutive theory. Testing methods which improve the precision of the curve-fitted permeability coefficients may also be investigated. Acknowledgements-This work was supported by grants from the Orthopaedic Research and Education Foundation, the Whitaker Foundation, the National Institutes of Health (AR-41913) and the National Science Foundation (ASC-93-18184). REFERENCES Ahmed, A. M. and Burke, D. L. (1983) In-vitro measurement of static pressure distribution in synovial joints - part I: Tibia1 surface of the knee. Journal of Biomechanical Engineering ASME, 105, 216-225.

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