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Confined and unconfined stress relaxation of cartilage: appropriateness of a transversely isotropic analysis
Journal of Biomechanics 32 (1999) 1125}1130
Technical Note
Con"ned and uncon"ned stress relaxation of cartilage: appropriateness of a transversely isotropic analysis Predrag M. BursacH , Toby W. Obitz, Solomon R. Eisenberg, Dimitrije StamenovicH * Department of Biomedical Engineering, Boston University, 44 Cummington Street, Boston, MA 02215, USA Received 31 March 1998; accepted 4 May 1999
Abstract Previous studies have shown that stress relaxation behavior of calf ulnar growth plate and chondroepiphysis cartilage can be described by a linear transverse isotropic biphasic model. The model provides a good "t to the observed uncon"ned compression transients when the out-of-plane Poisson's ratio is set to zero. This assumption is based on the observation that the equilibrium stress in the axial direction (p ) is the same in con"ned and uncon"ned compression, which implies that the radial stress p "0 in con"ned X compression. In our study, we further investigated the ability of the transversely isotropic model to describe con"ned and uncon"ned stress relaxation behavior of calf cartilage. A series of con"ned and uncon"ned stress relaxation tests were performed on calf articular cartilage (4.5 mm diameter, &3.3 mm height) in a displacement-controlled compression apparatus capable of measuring p and p . In X equilibrium, p '0 and p in con"ned compression was greater than in uncon"ned compression. Transient data at each strain were X "tted by the linear transversely isotropic biphasic model and the material parameters were estimated. Although the model could provide good "ts to the uncon"ned transients, the estimated parameters overpredicted the measured p . Conversely, if the model was constrained to match equilibrium p , the "ts were poor. These "ndings suggest that the linear transversely isotropic biphasic model could not simultaneously describe the observed stress relaxation and equilibrium behavior of calf cartilage. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Axial stress; Radial stress; Elastic moduli; Poisson's ratio
1. Introduction Early biphasic models of cartilage described the solid component of the tissue as homogenous and linearly isotropic (Mow and Lai, 1980; Mow et al., 1984). Although this approach successfully explains the observed stress relaxation behavior of cartilage in con"ned compression, it does not adequately explain the observed stress relaxation in uncon"ned compression. The response to a ramp-strain input in uncon"ned compression exhibits a high peak-to-equilibrium stress ratio followed by a fast stress decay (Brown and Singerman, 1986). Fits of the isotropic biphasic model to uncon"ned stress relaxation data substantially underestimate the peak stress and radial permeability (Armstrong et al., 1984; Brown and Singerman, 1986).
* Corresponding author. Tel: 001-617-353-5902; fax: 001-617-3536766. E-mail address: [email protected] (D. StamenovicH )
Cohen et al. (1998) advanced a biphasic model of tissue mechanics that incorporates a linear transversely isotropic solid phase to explain observed stress relaxation behavior. They obtained a good "t to the uncon"ned compression transient of calf ulnar growth plate and chondroepiphysis cartilage when the out-of-plane Poisson's ratio (l ) was set to zero. They based the assumpX tion l "0 on their observation that the equilibrium X axial stress (p ) was the same in con"ned and uncon"ned X compression (Cohen et al., 1998), which implies that the radial con"ning stress p "0. However, recent "ndings by Khalsa and Eisenberg (1997) show that equilibrium p '0 for con"ned compression of adult bovine patellar groove cartilage over a wide range of applied strains (5}26%). In this study, we investigated the ability of the linear transversely isotropic biphasic model to describe calf articular cartilage in uncon"ned and con"ned compression with a single set of material parameters when p '0, and no assumptions regarding parameter values. The ability to fully characterize tissue behavior under those
0021-9290/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 9 9 ) 0 0 1 0 5 - 0
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two loading paradigms is important since they may be considered limiting cases of physiological loading. Con"ned and uncon"ned stress relaxation tests were performed on the same specimen, and p as well as p in X con"ned compression were measured. Material parameters were estimated by "tting the model to these data. We found that the transversely isotropic model could not simultaneously describe all of the observed behavior.
2. Materials and methods Experiment: Fresh hind legs from three calves (3}4 week old) were obtained from a local abattoir. Full thickness 7 mm diameter cartilage/bone plugs were excised from the patellar groove region, rinsed in phosphate bu!ered saline with protease inhibitors (PBS#PI) (Schinagl et al., 1996) and frozen (!203C, 1}7 days). Plugs were thawed at room temperature for 1 h in PBS#PI just prior to testing. After removing&0.5 mm from the articular surface, 4.5 mm diameter and &3.3 mm thick plane parallel cartilage disks were prepared. Con"ned and uncon"ned compression tests were conducted on the same specimen in PBS#PI, using a displacement controlled apparatus previously described (Khalsa and Eisenberg, 1997). Radial stress p was mea sured in con"ned compression by a transducer in the side wall of the con"ning chamber. Axial stress p was meaX sured by a transducer mounted under the chamber. The platen used for con"ned compression was polished, porous, sintered stainless steel (40 lm pores). For uncon"ned compression, the platen was polished plexiglass. The axial direction of compression coincides with the z-axis of a cylindrical coordinate system. A series of ramp-stress relaxation tests were performed (0.115 lm/s displacement rate for con"ned and uncon"ned compression). Initially, a &3% ramp-strain (all strains de"ned relative to unloaded thickness) was applied to the specimen and the tissue was allowed to relax until the change in stress did not exceed 3 kPa in 16 min. This criterion was based on the signal-to-noise ratio of the apparatus and was used to assess mechanical equilibrium throughout this study. The initial strain was used as a tare load and was not considered in subsequent data analysis. Following the initial strain, a series of &3% ramp strains were applied, followed by stress relaxation to equilibrium, until a total strain of &15% was reached. Corresponding p in con"ned and uncon"ned comX pression and p in con"ned compression were recorded and stored for analysis. Nine specimens were tested. Each specimen was "rst subjected to a con"ned compression series followed by an uncon"ned compression series of measurements. Between these tests, the specimen was allowed to relax in PBS#PI solution at room temperature for 1.5 h.
Model and analysis: A biphasic material is initially compressed at constant strain rate e for time t , and then allowed to relax at constant strain, e t . The correspon ding stress relaxation response p (t) in con"ned compresX sion for an isotropic material is (Mow et al., 1980): e d 2e d e\LR OY p (t)"H e t# ! X 3k n pk X X L for 0(t(t and
(1)
2e d e\LR OY!e\LR\R OY p (t)"H e t ! X pk n X L for t't , where
(2)
d q , , (3) pH k X t is time, H is the aggregate modulus, k is the axial X permeability and d is sample thickness. The identical solution (Eqs. (1)}(3)) is obtained for a transversely isotropic material. For uncon"ned compression, p (t) relationships for X a transversely isotropic material can be found in Armstrong et al. (1984) and Cohen et al. (1998). These equations are recast below in terms of H , the axial Young's modulus (E ), in-plane Poisson's ratio (l ), lateral X F modulus (j) and radial permeability (k ): e a p (t)"E e t# (j#E !H ) X X X jk 1 e\?L R O ; ! for 0(t(t 8 a[(1#q)a/4!q] L L L (4) and
e a p (t)"E e t # (j#E !H ) X X jk X e\?L R O !e\?L R\R O ; for t't , a[(1#q)a/4!q] L L L where
(5)
a H !E X, q , (6) jk 1#q 1!l E F X , q, (7) 1#l H F a is sample radius and a are roots of the characteristic L equation J (x)!0.5(1#1/q)xJ (x)"0, with J (x) and J (x) being the zero- and "rst-order Bessel functions, respectively. Elastic moduli were obtained as the ratio of the change in the equilibrium stress to the corresponding change in
P.M. Bursac& et al. / Journal of Biomechanics 32 (1999) 1125}1130
axial strain. The axial and radial moduli in con"ned compression represent H and j, respectively, whereas the axial modulus in uncon"ned compression represents E . X Stress relaxation data were "tted to the appropriate p (t) relationship using a global least square error optiX mization procedure (Csendes, 1988). For each strain in con"ned compression, Eqs. (1) and (2) were used with the corresponding equilibrium H to determine k . Stress X relaxation data for uncon"ned compression were "tted in two ways. First, for each strain, corresponding equilibrium H and E for a given specimen were substituted X into Eqs. (4) and (5) to determine j, l and k (similar to F Cohen et al., 1998). Second, the equilibrium j was used together with H and E to constrain Eqs. (4) and (5), and X l and k were determined. Twenty-"ve terms were used F in the sums in Eqs. (1) and (2), and 15 in Eqs. (4) and (5) in each specimen. The statistical signi"cance of the dependance of parameters on the applied strain was assessed using a one way
1127
ANOVA. Statistically signi"cant di!erences between equilibrium moduli were assessed by the paired t-test. In both tests, di!erences with p(0.05 were considered signi"cant.
3. Results Stress relaxation data for con"ned and uncon"ned compression were characterized by high peak-to-equilibrium stress ratios and fast stress decay (Figs. 1a and b). In con"ned compression, p 'p for all strain increments X (Fig. 1a). The average (n"9) equilibrium H and E decreased X with increasing strain (p)9;10\), whereas j exhibited no systematic strain dependence (Table 1). These data imply strain-softening (see appendix). Although H 'E X over the entire strain range, this di!erence was signi"cant (p)0.04) only for the last two strain increments. Both H and E were consistent with values in the literature X (Athanasiou et al., 1991).
Fig. 1. Representative stress relaxation data obtained for a series of ramp-strain increments (&3%) from con"ned (a) and uncon"ned (b) compression tests on the same specimen. Data for con"ned compression (a) include both axial (top curve) and radial stresses (bottom curve).
Table 1 Average aggregate modulus (H ), Young's modulus (E ) and lateral modulus (j) $SD calculated at each strain increment from con"ned and X uncon"ned equilibrium portions of stress relaxation curves for n"nine specimens. The lateral modulus (j estimated) $ SD obtained at each strain increment from the best "ts of the transverse isotropic model (Eqs. (4) and (5)) to transient uncon"ned compression data with equilibrium values of H and E as constraining parameters (n"9). Isotropic Poisson's ratio (l) calculated either from the average equilibrium values of H and j or H and X E . Transversely isotropic out-of-plane Poisson's ratio (l ) calculated from the average equilibrium values of H , E and j. Uncertainty intervals for X X X l and l are given XP Measured
Estimated
Strain (%)
H (MPa)
E X (MPa)
j (MPa)
j (MPa)
l(H , j)
l(H , E ) X
l (H , E , j) XP X
5.59$0.13 8.30$0.08 11.1$0.06 13.9$0.05
1.72$0.72 1.23$0.40 0.97$0.20 0.79$0.12
1.50$0.50 1.09$0.43 0.74$0.28 0.68$0.13
0.40$0.28 0.39$0.43 0.37$0.26 0.39$0.21
1.29$0.78 1.03$0.45 0.82$0.40 0.86$0.47
0.19$0.13 0.24$0.21 0.28$0.15 0.33$0.12
0.22$0.36 0.21$0.37 0.29$0.15 0.23$0.16
0.27$1.11 0.18$0.78 0.30$0.46 0.13$0.24
Signi"cant trend p(0.05.
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Interpreting these moduli in terms of isotropy, Poisson's ratio (l) was calculated at each strain as l"j/(H #j) or from 2H l#(H !E )l!(H !E )"0 (Table 1). X X Computed values were not signi"cantly di!erent and agreed with previously reported values (Mow et al., 1989; Khalsa and Eisenberg, 1997). When interpreted in terms of transverse isotropy, l "(H !E )/2j was calculated X X (Table 1). Values of l were also not signi"cantly di!erX ent from the values of l. The best "t of Eqs. (1) and (2) to the con"ned compression transients with H constrained provided good corre spondence to the data (Fig. 2a). The best "t of Eqs. (4) and (5) to uncon"ned compression transients with H and E constrained also provided a good correspond X ence to the data (Fig. 2b). However, the values of j obtained from the "t were 2}3 times greater ( p)0.021) than the measured equilibrium values (Table 1). When equilibrium values of H , E and j constrained the "t, X
Fig. 2. Representative con"ned (a) and uncon"ned (b) axial stress relaxation data (open circles) together with best "ts. The data are from the 4th strain increment (&9}12%) on the same specimen. Solid lines represent the best "t of the transversely isotropic biphasic model (a) to con"ned stress relaxation data with the aggregate modulus (H ) con strained from the equilibrium measurements, and (b) to uncon"ned stress relaxation data with H and the Young's modulus (E ) con X strained from equilibrium measurements. Dashed line (b) represents the best "t of the biphasic model to uncon"ned stress relaxation data with H , E and the lateral modulus (j) constrained from equilibrium X measurements.
a consistently poor correspondence to the data was obtained (Fig. 2b). The axial permeability k obtained from the con"ned X compression "ts and the radial permeability k obtained from the uncon"ned compression "ts with H and E constrained were in general of order 10\ m/N s, X consistent with values from the literature (Mow et al., 1984).
4. Discussion The most signi"cant "nding of this study is that the linear biphasic transversely isotropic model could not simultaneously describe the observed stress relaxation and equilibrium behavior of calf articular cartilage. Two major inconsistencies between the model and experimental data lead to this conclusion. First, when "tted to the uncon"ned transient data using two equilibrium moduli (H and E ) as constraining parameters, the X model provided good "ts (Fig. 2b). However, the values of j obtained from these "ts were 2}3 times greater than the values determined from direct measurements in equilibrium (Table 1). Second, when "tted to uncon"ned transient data using three equilibrium moduli as constraining parameters (H , E and j), the model provided X poor "ts (Fig. 2b). The above conclusion is contrary to the results of the previous application of the transversely isotropic model to the stress relaxation behavior of calf cartilage (Cohen et al., 1998). This could be due to the di!erent source of cartilage used in the respective studies (3}4 week calf patellar groove vs. 4 month calf ulnar growth plate and chondroepiphysis). More importantly, Cohen et al. (1998) set l "0 based on their observation that the X equilibrium p in con"ned and uncon"ned compression X did not di!er signi"cantly. This is equivalent to making p "0 in con"ned compression. We measured p directly in our study, and found it to be greater than zero throughout the tested range. This latter measurement is important, because it provided an additional measure by which the transversely isotropic model could be tested. Another interesting "nding from our analysis is that the isotropic Poisson's ratio l and the transversely isotropic Poisson's ratio l determined directly from the X equilibrium elastic moduli did not di!er signi"cantly (Table 1) and were consistent with optical measurements of Jurvelin et al. (1997). This suggests that these measurements are not able to distinguish between isotropic and transversely isotropic behavior of articular cartilage. Friction artifacts at the platen}tissue interface were minimized by choosing specimens with high aspect (thickness-to-diameter) ratio (&0.73). Such e!ects could contribute to a high peak-to-equilibrium stress ratio and fast stress relaxation during uncon"ned compression. However, the "nite element simulation of uncon"ned
P.M. Bursac& et al. / Journal of Biomechanics 32 (1999) 1125}1130
compression (Spilker et al., 1990) with high aspect ratio specimens (0.8) showed that a totally adhesive platentissue interface has a negligible e!ect on the shape of the stress relaxation curve. Furthermore, visual observation of specimens lubricated with synovial #uid having an aspect ratio of &0.55 showed no signi"cant friction e!ects (Jurvelin et al., 1997). Thus, we do not believe that friction had a major impact on our uncon"ned stress relaxation curves. In view of the poor performance of the linear transversely isotropic model of the cartilage solid phase, alternative approaches should be sought. Cohen et al. (1998) hypothesized that the tissue anisotropy observed in uncon"ned compression is due to di!erent tissue sti!nesses in tension and compression (i.e., radial deformation produces tensile strains whereas the axial deformation produces compressive strains). However, this intuitively appealing bimodular tension}compression paradigm is not theoretically tenable when modeling the solid component of the tissue as a single phase continuum. Curnier et al. (1991) showed that the mechanical behavior of a bimodular elastic continuum (i.e., one which exhibits di!erent linear stress}strain behaviors in tension and compression) during uncon"ned compression is described only by compressive elastic moduli. However, incorporating nonlinear stress}strain relationships into a single-phase continuum model of the solid can e!ectively yield di!erent incremental behaviors in tension and compression. Another approach is to use composite models. Soulhat et al. (1999) have proposed a nonhomogeneous composite model of the solid phase to explain the stress relaxation in uncon"ned compression. The model predicts a high peak-toequilibrium stress ratio and a fast stress decay. The same authors developed a composite-based "nite element model with nonlinear elastic collagen "brils (Soulhat et al., 1998) that also predicts the observed stress relaxation response. In summary, a linear biphasic transversely isotropic model is not able to simultaneously describe the observed equilibrium and stress relaxation behavior of calf articular cartilage. Thus, a constitutive equation that incorporates a single phase linear continuum solid may not be adequate for describing the observed mechanical behavior of cartilage in uncon"ned compression. Models that include material nonlinearities or focus on the composite nature of the cartilage solid phase and on nonlinear properties of its constituents seem to o!er alternative approaches.
Acknowledgements This study was supported by National Aeronautical Space Administration Grant NAG9-836 and National Heart, Lung and Blood Institute, HL-33009.
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Fig. 3. Con"ned compression axial stress-strain curve for calf articular cartilage obtained from one sample.
Appendix A Our observation of strain-softening of calf articular cartilage in both con"ned and uncon"ned compression (Table 1) appears to be at odds with previous observations of linear stress}strain behavior (Eisenberg and Grodzinsky, 1985; Khalsa and Eisenberg, 1997) and strain-hardening (Kwan et al., 1990; Ateshian et al., 1997) of adult articular cartilage. We examined the stress-strain behavior of our samples in con"ned compression over the strain range 0}40% (Fig. 3). Again, we observed strain-softening for the smaller strains, followed by strain-hardening for the larger strains. The latter observation is consistent with previously reported strain-hardening behavior.
References Armstrong, C.G., Lai, W.W., Mow, V.C., 1984. An analysis of the uncon"ned compression of articular cartilage. ASME Journal of Biomechanical Engineering 106, 165}173. Athanasiou, K.A., Rosenwasser, M.P., Buckwalter, J.A., Malinin, T.I., Mow, V.C., 1991. Interspecies comparison of in situ intrinsic mechanical properties of distal femoral cartilage. Journal of Orthopaedic Research 9, 330}340. Ateshian, G.A., Warden, W.H., Kim, J.J., Grelsamer, R.P., Mow, V.C., 1997. Finite deformation biphasic material properties of bovine articular cartilage from con"ned compression experiments. Journal of Biomechanics 30, 1157}1164. Brown, T.D., Singerman, R.J., 1986. Experimental determination of the linear biphasic constitutive coe$cients of human fetal proximal femoral chondroepiphysis. Journal of Biomechanics 19, 597}605. Cohen, B., Lai, W.M., Mow, V.C., 1998. A transversely isotropic biphasic model for uncon"ned compression growth plate and chondroepiphysis. ASME Journal of Biomechanical Engineering 120, 491}496. Csendes, T., 1988. Nonlinear parameter estimation by global optimization - e$ciency and reliability. Acta Cybernetica 8, 361}370.
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Curnier, A., He, Q.-C., Zysset, P., 1991. Conewise linear elastic materials. Journal of Elasticity 37, 1}38. Eisenberg, S.R., Grodzinsky, A.J., 1985. Swelling of articular cartilage and other connective tissues: electromechanochemical forces. Journal of Orthopaedic Research 3, 148}159. Jurvelin, J.S., Buschmann, M.D., Hunziker, E.B., 1997. Optical and mechanical determination of Poisson's ratio of adult bovine humeral articular cartilage. Journal of Biomechanics 30, 235}241. Khalsa, P.S., Eisenberg, S.R., 1997. Compressive behavior of articular cartilage is not completely explained by proteoglycan osmotic pressure. Journal of Biomechanics 30, 589}594. Kwan, M.K., Lai, W.M., Mow, V.C., 1990. A "nite deformation theory for cartilage and other soft hydrated connective tissues * I. Equilibrium results. Journal of Biomechanics 23, 145}155. Mow, V.C., Lai, W.M., 1980. Recent developments in synovial joint biomechanics. SIAM Review 22, 275}317. Mow, V.C., Holmes, M.H., Lai, W.M., 1984. Fluid transport and mechanical properties of articular cartilage: a review. Journal of Biomechanics 17, 377}394. Mow, V.C., Kuei, S.C., Lai, W.M., Armstrong, C.G., 1980. Biphasic creep and stress relaxation of articular cartilage in compression:
theory and experiments. ASME Journal of Biomechanical Engineering 102, 73}84. Mow, V.C., Gibbs, M.C., Lai, W.M., Zhu, W.B., Athanasiou, K.A., 1989. Biphasic indentation of articular cartilage II. A numerical algorithm and an experimental study. Journal of Biomechanics 22, 853}861. Schinagl, R.M., Ting, M.K., Price, J.H., Sah, R.L., 1996. Video microscopy to quantitate the inhomogeneous equilibrium strain with articular cartilage during con"ned compression. Annals of Biomedical Engineering 24, 500}512. Soulhat, J., Buschmann, M.D., Shirazi-Adl, A., 1999. A "bril-networkreinforced biphasic model of cartilage in uncon"ned compression. ASME Journal of Biomechanical Engineering 121, 340}347. Soulhat, J., Buschmann, M. D., Shirazi-Adl, A., 1998. Non-linear cartilage mechanics in uncon"ned compression. In: Transactions of the 44th Annual Meeting of the Orthopaedic Research Society, Ed. G.B.J. Andersson, ORS, New Orleans, LA, p. 226. Spilker, R.L., Suh, J.-K., Mow, V.C., 1990. E!ects of friction on the uncon"ned compressive response of articular cartilage: a "nite element analysis. ASME Journal of Biomechanical Engineering 112, 138-1.
Technical Note
Con"ned and uncon"ned stress relaxation of cartilage: appropriateness of a transversely isotropic analysis Predrag M. BursacH , Toby W. Obitz, Solomon R. Eisenberg, Dimitrije StamenovicH * Department of Biomedical Engineering, Boston University, 44 Cummington Street, Boston, MA 02215, USA Received 31 March 1998; accepted 4 May 1999
Abstract Previous studies have shown that stress relaxation behavior of calf ulnar growth plate and chondroepiphysis cartilage can be described by a linear transverse isotropic biphasic model. The model provides a good "t to the observed uncon"ned compression transients when the out-of-plane Poisson's ratio is set to zero. This assumption is based on the observation that the equilibrium stress in the axial direction (p ) is the same in con"ned and uncon"ned compression, which implies that the radial stress p "0 in con"ned X compression. In our study, we further investigated the ability of the transversely isotropic model to describe con"ned and uncon"ned stress relaxation behavior of calf cartilage. A series of con"ned and uncon"ned stress relaxation tests were performed on calf articular cartilage (4.5 mm diameter, &3.3 mm height) in a displacement-controlled compression apparatus capable of measuring p and p . In X equilibrium, p '0 and p in con"ned compression was greater than in uncon"ned compression. Transient data at each strain were X "tted by the linear transversely isotropic biphasic model and the material parameters were estimated. Although the model could provide good "ts to the uncon"ned transients, the estimated parameters overpredicted the measured p . Conversely, if the model was constrained to match equilibrium p , the "ts were poor. These "ndings suggest that the linear transversely isotropic biphasic model could not simultaneously describe the observed stress relaxation and equilibrium behavior of calf cartilage. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Axial stress; Radial stress; Elastic moduli; Poisson's ratio
1. Introduction Early biphasic models of cartilage described the solid component of the tissue as homogenous and linearly isotropic (Mow and Lai, 1980; Mow et al., 1984). Although this approach successfully explains the observed stress relaxation behavior of cartilage in con"ned compression, it does not adequately explain the observed stress relaxation in uncon"ned compression. The response to a ramp-strain input in uncon"ned compression exhibits a high peak-to-equilibrium stress ratio followed by a fast stress decay (Brown and Singerman, 1986). Fits of the isotropic biphasic model to uncon"ned stress relaxation data substantially underestimate the peak stress and radial permeability (Armstrong et al., 1984; Brown and Singerman, 1986).
* Corresponding author. Tel: 001-617-353-5902; fax: 001-617-3536766. E-mail address: [email protected] (D. StamenovicH )
Cohen et al. (1998) advanced a biphasic model of tissue mechanics that incorporates a linear transversely isotropic solid phase to explain observed stress relaxation behavior. They obtained a good "t to the uncon"ned compression transient of calf ulnar growth plate and chondroepiphysis cartilage when the out-of-plane Poisson's ratio (l ) was set to zero. They based the assumpX tion l "0 on their observation that the equilibrium X axial stress (p ) was the same in con"ned and uncon"ned X compression (Cohen et al., 1998), which implies that the radial con"ning stress p "0. However, recent "ndings by Khalsa and Eisenberg (1997) show that equilibrium p '0 for con"ned compression of adult bovine patellar groove cartilage over a wide range of applied strains (5}26%). In this study, we investigated the ability of the linear transversely isotropic biphasic model to describe calf articular cartilage in uncon"ned and con"ned compression with a single set of material parameters when p '0, and no assumptions regarding parameter values. The ability to fully characterize tissue behavior under those
0021-9290/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 9 9 ) 0 0 1 0 5 - 0
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P.M. Bursac& et al. / Journal of Biomechanics 32 (1999) 1125}1130
two loading paradigms is important since they may be considered limiting cases of physiological loading. Con"ned and uncon"ned stress relaxation tests were performed on the same specimen, and p as well as p in X con"ned compression were measured. Material parameters were estimated by "tting the model to these data. We found that the transversely isotropic model could not simultaneously describe all of the observed behavior.
2. Materials and methods Experiment: Fresh hind legs from three calves (3}4 week old) were obtained from a local abattoir. Full thickness 7 mm diameter cartilage/bone plugs were excised from the patellar groove region, rinsed in phosphate bu!ered saline with protease inhibitors (PBS#PI) (Schinagl et al., 1996) and frozen (!203C, 1}7 days). Plugs were thawed at room temperature for 1 h in PBS#PI just prior to testing. After removing&0.5 mm from the articular surface, 4.5 mm diameter and &3.3 mm thick plane parallel cartilage disks were prepared. Con"ned and uncon"ned compression tests were conducted on the same specimen in PBS#PI, using a displacement controlled apparatus previously described (Khalsa and Eisenberg, 1997). Radial stress p was mea sured in con"ned compression by a transducer in the side wall of the con"ning chamber. Axial stress p was meaX sured by a transducer mounted under the chamber. The platen used for con"ned compression was polished, porous, sintered stainless steel (40 lm pores). For uncon"ned compression, the platen was polished plexiglass. The axial direction of compression coincides with the z-axis of a cylindrical coordinate system. A series of ramp-stress relaxation tests were performed (0.115 lm/s displacement rate for con"ned and uncon"ned compression). Initially, a &3% ramp-strain (all strains de"ned relative to unloaded thickness) was applied to the specimen and the tissue was allowed to relax until the change in stress did not exceed 3 kPa in 16 min. This criterion was based on the signal-to-noise ratio of the apparatus and was used to assess mechanical equilibrium throughout this study. The initial strain was used as a tare load and was not considered in subsequent data analysis. Following the initial strain, a series of &3% ramp strains were applied, followed by stress relaxation to equilibrium, until a total strain of &15% was reached. Corresponding p in con"ned and uncon"ned comX pression and p in con"ned compression were recorded and stored for analysis. Nine specimens were tested. Each specimen was "rst subjected to a con"ned compression series followed by an uncon"ned compression series of measurements. Between these tests, the specimen was allowed to relax in PBS#PI solution at room temperature for 1.5 h.
Model and analysis: A biphasic material is initially compressed at constant strain rate e for time t , and then allowed to relax at constant strain, e t . The correspon ding stress relaxation response p (t) in con"ned compresX sion for an isotropic material is (Mow et al., 1980): e d 2e d e\LR OY p (t)"H e t# ! X 3k n pk X X L for 0(t(t and
(1)
2e d e\LR OY!e\LR\R OY p (t)"H e t ! X pk n X L for t't , where
(2)
d q , , (3) pH k X t is time, H is the aggregate modulus, k is the axial X permeability and d is sample thickness. The identical solution (Eqs. (1)}(3)) is obtained for a transversely isotropic material. For uncon"ned compression, p (t) relationships for X a transversely isotropic material can be found in Armstrong et al. (1984) and Cohen et al. (1998). These equations are recast below in terms of H , the axial Young's modulus (E ), in-plane Poisson's ratio (l ), lateral X F modulus (j) and radial permeability (k ): e a p (t)"E e t# (j#E !H ) X X X jk 1 e\?L R O ; ! for 0(t(t 8 a[(1#q)a/4!q] L L L (4) and
e a p (t)"E e t # (j#E !H ) X X jk X e\?L R O !e\?L R\R O ; for t't , a[(1#q)a/4!q] L L L where
(5)
a H !E X, q , (6) jk 1#q 1!l E F X , q, (7) 1#l H F a is sample radius and a are roots of the characteristic L equation J (x)!0.5(1#1/q)xJ (x)"0, with J (x) and J (x) being the zero- and "rst-order Bessel functions, respectively. Elastic moduli were obtained as the ratio of the change in the equilibrium stress to the corresponding change in
P.M. Bursac& et al. / Journal of Biomechanics 32 (1999) 1125}1130
axial strain. The axial and radial moduli in con"ned compression represent H and j, respectively, whereas the axial modulus in uncon"ned compression represents E . X Stress relaxation data were "tted to the appropriate p (t) relationship using a global least square error optiX mization procedure (Csendes, 1988). For each strain in con"ned compression, Eqs. (1) and (2) were used with the corresponding equilibrium H to determine k . Stress X relaxation data for uncon"ned compression were "tted in two ways. First, for each strain, corresponding equilibrium H and E for a given specimen were substituted X into Eqs. (4) and (5) to determine j, l and k (similar to F Cohen et al., 1998). Second, the equilibrium j was used together with H and E to constrain Eqs. (4) and (5), and X l and k were determined. Twenty-"ve terms were used F in the sums in Eqs. (1) and (2), and 15 in Eqs. (4) and (5) in each specimen. The statistical signi"cance of the dependance of parameters on the applied strain was assessed using a one way
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ANOVA. Statistically signi"cant di!erences between equilibrium moduli were assessed by the paired t-test. In both tests, di!erences with p(0.05 were considered signi"cant.
3. Results Stress relaxation data for con"ned and uncon"ned compression were characterized by high peak-to-equilibrium stress ratios and fast stress decay (Figs. 1a and b). In con"ned compression, p 'p for all strain increments X (Fig. 1a). The average (n"9) equilibrium H and E decreased X with increasing strain (p)9;10\), whereas j exhibited no systematic strain dependence (Table 1). These data imply strain-softening (see appendix). Although H 'E X over the entire strain range, this di!erence was signi"cant (p)0.04) only for the last two strain increments. Both H and E were consistent with values in the literature X (Athanasiou et al., 1991).
Fig. 1. Representative stress relaxation data obtained for a series of ramp-strain increments (&3%) from con"ned (a) and uncon"ned (b) compression tests on the same specimen. Data for con"ned compression (a) include both axial (top curve) and radial stresses (bottom curve).
Table 1 Average aggregate modulus (H ), Young's modulus (E ) and lateral modulus (j) $SD calculated at each strain increment from con"ned and X uncon"ned equilibrium portions of stress relaxation curves for n"nine specimens. The lateral modulus (j estimated) $ SD obtained at each strain increment from the best "ts of the transverse isotropic model (Eqs. (4) and (5)) to transient uncon"ned compression data with equilibrium values of H and E as constraining parameters (n"9). Isotropic Poisson's ratio (l) calculated either from the average equilibrium values of H and j or H and X E . Transversely isotropic out-of-plane Poisson's ratio (l ) calculated from the average equilibrium values of H , E and j. Uncertainty intervals for X X X l and l are given XP Measured
Estimated
Strain (%)
H (MPa)
E X (MPa)
j (MPa)
j (MPa)
l(H , j)
l(H , E ) X
l (H , E , j) XP X
5.59$0.13 8.30$0.08 11.1$0.06 13.9$0.05
1.72$0.72 1.23$0.40 0.97$0.20 0.79$0.12
1.50$0.50 1.09$0.43 0.74$0.28 0.68$0.13
0.40$0.28 0.39$0.43 0.37$0.26 0.39$0.21
1.29$0.78 1.03$0.45 0.82$0.40 0.86$0.47
0.19$0.13 0.24$0.21 0.28$0.15 0.33$0.12
0.22$0.36 0.21$0.37 0.29$0.15 0.23$0.16
0.27$1.11 0.18$0.78 0.30$0.46 0.13$0.24
Signi"cant trend p(0.05.
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Interpreting these moduli in terms of isotropy, Poisson's ratio (l) was calculated at each strain as l"j/(H #j) or from 2H l#(H !E )l!(H !E )"0 (Table 1). X X Computed values were not signi"cantly di!erent and agreed with previously reported values (Mow et al., 1989; Khalsa and Eisenberg, 1997). When interpreted in terms of transverse isotropy, l "(H !E )/2j was calculated X X (Table 1). Values of l were also not signi"cantly di!erX ent from the values of l. The best "t of Eqs. (1) and (2) to the con"ned compression transients with H constrained provided good corre spondence to the data (Fig. 2a). The best "t of Eqs. (4) and (5) to uncon"ned compression transients with H and E constrained also provided a good correspond X ence to the data (Fig. 2b). However, the values of j obtained from the "t were 2}3 times greater ( p)0.021) than the measured equilibrium values (Table 1). When equilibrium values of H , E and j constrained the "t, X
Fig. 2. Representative con"ned (a) and uncon"ned (b) axial stress relaxation data (open circles) together with best "ts. The data are from the 4th strain increment (&9}12%) on the same specimen. Solid lines represent the best "t of the transversely isotropic biphasic model (a) to con"ned stress relaxation data with the aggregate modulus (H ) con strained from the equilibrium measurements, and (b) to uncon"ned stress relaxation data with H and the Young's modulus (E ) con X strained from equilibrium measurements. Dashed line (b) represents the best "t of the biphasic model to uncon"ned stress relaxation data with H , E and the lateral modulus (j) constrained from equilibrium X measurements.
a consistently poor correspondence to the data was obtained (Fig. 2b). The axial permeability k obtained from the con"ned X compression "ts and the radial permeability k obtained from the uncon"ned compression "ts with H and E constrained were in general of order 10\ m/N s, X consistent with values from the literature (Mow et al., 1984).
4. Discussion The most signi"cant "nding of this study is that the linear biphasic transversely isotropic model could not simultaneously describe the observed stress relaxation and equilibrium behavior of calf articular cartilage. Two major inconsistencies between the model and experimental data lead to this conclusion. First, when "tted to the uncon"ned transient data using two equilibrium moduli (H and E ) as constraining parameters, the X model provided good "ts (Fig. 2b). However, the values of j obtained from these "ts were 2}3 times greater than the values determined from direct measurements in equilibrium (Table 1). Second, when "tted to uncon"ned transient data using three equilibrium moduli as constraining parameters (H , E and j), the model provided X poor "ts (Fig. 2b). The above conclusion is contrary to the results of the previous application of the transversely isotropic model to the stress relaxation behavior of calf cartilage (Cohen et al., 1998). This could be due to the di!erent source of cartilage used in the respective studies (3}4 week calf patellar groove vs. 4 month calf ulnar growth plate and chondroepiphysis). More importantly, Cohen et al. (1998) set l "0 based on their observation that the X equilibrium p in con"ned and uncon"ned compression X did not di!er signi"cantly. This is equivalent to making p "0 in con"ned compression. We measured p directly in our study, and found it to be greater than zero throughout the tested range. This latter measurement is important, because it provided an additional measure by which the transversely isotropic model could be tested. Another interesting "nding from our analysis is that the isotropic Poisson's ratio l and the transversely isotropic Poisson's ratio l determined directly from the X equilibrium elastic moduli did not di!er signi"cantly (Table 1) and were consistent with optical measurements of Jurvelin et al. (1997). This suggests that these measurements are not able to distinguish between isotropic and transversely isotropic behavior of articular cartilage. Friction artifacts at the platen}tissue interface were minimized by choosing specimens with high aspect (thickness-to-diameter) ratio (&0.73). Such e!ects could contribute to a high peak-to-equilibrium stress ratio and fast stress relaxation during uncon"ned compression. However, the "nite element simulation of uncon"ned
P.M. Bursac& et al. / Journal of Biomechanics 32 (1999) 1125}1130
compression (Spilker et al., 1990) with high aspect ratio specimens (0.8) showed that a totally adhesive platentissue interface has a negligible e!ect on the shape of the stress relaxation curve. Furthermore, visual observation of specimens lubricated with synovial #uid having an aspect ratio of &0.55 showed no signi"cant friction e!ects (Jurvelin et al., 1997). Thus, we do not believe that friction had a major impact on our uncon"ned stress relaxation curves. In view of the poor performance of the linear transversely isotropic model of the cartilage solid phase, alternative approaches should be sought. Cohen et al. (1998) hypothesized that the tissue anisotropy observed in uncon"ned compression is due to di!erent tissue sti!nesses in tension and compression (i.e., radial deformation produces tensile strains whereas the axial deformation produces compressive strains). However, this intuitively appealing bimodular tension}compression paradigm is not theoretically tenable when modeling the solid component of the tissue as a single phase continuum. Curnier et al. (1991) showed that the mechanical behavior of a bimodular elastic continuum (i.e., one which exhibits di!erent linear stress}strain behaviors in tension and compression) during uncon"ned compression is described only by compressive elastic moduli. However, incorporating nonlinear stress}strain relationships into a single-phase continuum model of the solid can e!ectively yield di!erent incremental behaviors in tension and compression. Another approach is to use composite models. Soulhat et al. (1999) have proposed a nonhomogeneous composite model of the solid phase to explain the stress relaxation in uncon"ned compression. The model predicts a high peak-toequilibrium stress ratio and a fast stress decay. The same authors developed a composite-based "nite element model with nonlinear elastic collagen "brils (Soulhat et al., 1998) that also predicts the observed stress relaxation response. In summary, a linear biphasic transversely isotropic model is not able to simultaneously describe the observed equilibrium and stress relaxation behavior of calf articular cartilage. Thus, a constitutive equation that incorporates a single phase linear continuum solid may not be adequate for describing the observed mechanical behavior of cartilage in uncon"ned compression. Models that include material nonlinearities or focus on the composite nature of the cartilage solid phase and on nonlinear properties of its constituents seem to o!er alternative approaches.
Acknowledgements This study was supported by National Aeronautical Space Administration Grant NAG9-836 and National Heart, Lung and Blood Institute, HL-33009.
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Fig. 3. Con"ned compression axial stress-strain curve for calf articular cartilage obtained from one sample.
Appendix A Our observation of strain-softening of calf articular cartilage in both con"ned and uncon"ned compression (Table 1) appears to be at odds with previous observations of linear stress}strain behavior (Eisenberg and Grodzinsky, 1985; Khalsa and Eisenberg, 1997) and strain-hardening (Kwan et al., 1990; Ateshian et al., 1997) of adult articular cartilage. We examined the stress-strain behavior of our samples in con"ned compression over the strain range 0}40% (Fig. 3). Again, we observed strain-softening for the smaller strains, followed by strain-hardening for the larger strains. The latter observation is consistent with previously reported strain-hardening behavior.
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