TENSION-COMPRESSION NONLINEARITY AND INTRINSIC VISCOELASTICITY OF ARTICULAR CARTILAGE SOLID
MATRIX
*Huang, C (A-NIH); *Soltz, M A. (A-NIH); *Kopacz, M; *Mow, V C.; +*Ateshian, G A. (A-NIH)
+*Columbia University, New York, NY. 500 W. 120th St, MC 4703, New York, NY 10027, 212-854-8602, Fax: 212-854-3304, [email protected]
∞
0
and
g (t ) = 1 + c [ Ei (t τ 2 ) − Ei (t τ1 ) ] ,
3
3


σ e (E) = ∑ λ1 [A a : E]tr ( A a E)A a + ∑ λ2tr ( A a E) A b  + 2µ E .
a =1 
b =1

In these equations, E is the infinitesimal strain tensor, Aa are texture tensors,
λ1 [ A a : E] = λ−1 if A a :E < 0 and λ + 1 otherwise, and the material properties
of the solid matrix are given by H + A , H − A , λ 2 , µ, c,τ 1 , τ 2 ( H ± A = λ ±1 + 2 µ ).
For each cartilage sample, CCS was used to extract the compressive aggregate
modulus, H − A , and axial permeability kz. UCS was used to obtain initial
estimates of H + A , λ 2 and radial permeability kr, and UCF was used to obtain
initial estimates of c,τ 1, τ 2 . Simultaneous curvefitting of UCS and UCF was
then used to refine these estimates. The UCD experimental results were
compared to the model’s prediction using the same material properties.
RESULTS: Representative curvefits of the slow and fast unconfined
compression stress-relaxation tests are presented in Figure 1, with the model’s
prediction of the experimental dynamic unconfined compression results given
in Figure 2. Mean and standard deviation of all material properties obtained
from these tests were: H + A = 8.55±2.77 MPa, H − A = 0.53±0.20 MPa, λ 2 =
0.29±0.14 MPa, c=0.54±0.24, τ1=0.82±0.53 s, τ2=153±96 s, kr= 1.1±0.51 ×10-
0418
1.0
0.5
8
4
0.0
0
1000
2000
0
3000
Time (sec)
(a)
0.01
0.1
1
10
100
1000
Time (sec)
(b)
Figure 1: Experimental and theoretical curvefits of unconfined compression
stress-relaxation with (a) slow (UCS), and (b) fast (UCF) strain rates.
20.0
Dynamic Amplitude (MPa)
where
∂σ e
g (t − τ )
[E(τ )] dτ
∂τ
UCF
Theoretical curvefit
12
UCS
Theoretical curvefit
1.5
Load (N)
e
2.0
50
10.0
5.0
40
30
20
10
0
0.0
10
Theoretical prediction
UCD
60
Theoretical prediction
UCD
15.0
Phase Angle (Deg)
σ (t ) = g (t )σ [E(0) ] + ∫
ve
m4/N.s, kz= 1.5±1.0 ×10-15 m4/N.s; a paired t-test analysis finds that
H + A ≠ H − A (p<0.0001) while c ≠ 0 . Nonlinear coefficients of determination
for the curvefits of experimental data were r2=0.950±0.034 (CCS),
r2=0.964±0.034 (UCS), r2=0.998±0.002 (UCF), and for the prediction of
experimental dynamic loading data r2=0.950±0.051 (UCD).
DISCUSSION: The findings of this study support the hypothesis that a
combination of intrinsic viscoelastic and tension-compression nonlinearity of
the solid matrix response is necessary to adequately describe cartilage
mechanics, as attested by the high r2 values of the model’s curvefits and
predictions. Intrinsic viscoelasticity effects are evident from the non-zero
value of the material constant c, while tension-compression nonlinearity is
evident from the large difference between the moduli H + A and H − A . Adding
the QLV viscoelasticity model to the CLE tension-compression model can
produce better predictions of the higher-frequency dynamic loading response
than achieved in previous studies [5]. Compared to studies which only model
the solid matrix intrinsic viscoelasticity of a biphasic material [4,9,12], the
current analysis produces a set of material constants more consistent with
tensile and compressive studies of articular cartilage[7,8,10,13]. Further
investigations of this combined model can be performed under additional
testing conditions.
ACKNOWLEDGMENTS: National Institutes of Health, NIAMS, AR46532
and AR43628.
REFERENCES
[1] Ateshian, G.A., Soltz M.A., 1999, Trans Orthop Res Soc, 24:158. [2]
Cohen B, Lai WM, Mow VC, 1998, J Biomech Eng, 120:491-496. [3] Curnier
A, He Q-C, Zysset P, 1995, J Elasticity 37:1-38, 1995. [4] DiSilvestro MR,
Zhu Q, Suh J-K, 1999, Bioengng Conf, BED-42:105-106. [5] Fortin M,
Soulhat J, Shirazi-Adl A, Hunziker EB, Buschmann MD, J Biomech Eng
122:189-195. [6] Fung YC, 1981, Springer-Verlag, New York. [7] Huang CY, Stankiewicz A, Ateshian GA, Flatow EL, Bigliani LU, Mow VC, 1999,
Bioengng Conf, BED 42:469-470. [8] Kempson GE, Freeman MA, Swanson
SA, 1968, Nature, 220:1127-1128. [9] Mak AF, 1986, Biorheology, 23:37183. [10] Mow VC, Kuei SC, Lai WM, Armstrong CG, 1980, J Biomech
Engng, 102:73,. [11] Soulhat J, Buschmann MD, Shirazi-Adl A, 1999, J
Biomech Eng, 121:340-347. [12] Suh J-K, DiSilvestro MR, 1999, J Appl
Mech, 66:528-535. [13] Woo SL-Y, Simon BR, Kuei SC, Akeson WH, 1980,
J Biomech Eng 102:85-90.
15
Load (N)
INTRODUCTION: Recent studies of articular cartilage mechanics have
focused on two competing hypothesized mechanisms which can describe the
response of the tissue in unconfined compression. One school of thought, to
which we have strongly adhered, proposes that the large disparity in the
tensile and compressive moduli of the tissue (which may differ by up to two
orders of magnitude [7]), is a dominant mechanism for describing cartilage
response [1,2,11]. A competing approach has attributed a more significant
role to the intrinsic viscoelasticity of the collagen-proteoglycan solid matrix
[4,9,12]. Both approaches have produced good agreement between theory and
experiments in unconfined compression at select loading frequencies or strain
rates, suggesting that additional testing configurations should be used to help
resolve this scientific question. Using our recently developed biphasic-CLE
model to describe the tension-compression nonlinearity of cartilage [1], we
found that the transient response of cartilage to uniaxial tensile loading (a
common testing configuration [8,13]) cannot be predicted qualitatively from
this model, unless intrinsic solid matrix viscoelasticity is also incorporated in
the analysis. Based on these theoretical findings, the hypothesis of the current
study is that a biphasic model of cartilage, which accounts for both intrinsic
viscoelasticity and tension-compression nonlinearity, is necessary to describe
the general response of articular cartilage. Up to four testing configurations
are employed on bovine cartilage cylindrical samples to verify this hypothesis.
MATERIALS AND METHODS: Ten full-thickness cartilage cylindrical
plugs (diam.=4.78mm) were harvested from 6-10 months old bovine
glenohumeral joints, microtomed by ~200 microns in the deep zone (final
thickness h=1.11±0.16 mm), and stored at -20° C. On two consecutive days
of testing, the specimen was thawed then mounted on a custom cartilage
loading device for undergoing each of two experiments; a tare load of 0.89 N
was first applied on the sample and maintained until equilibrium was achieved
(~3,000 s). On one day, a confined compression stress-relaxation test (CCS)
was performed (applied strain = 5%, ramp strain rate =1.25×10-4 sec-1); the
specimen was then allowed to recover. An unconfined compression stressrelaxation test was also performed, using the same applied strain and strain
rate (UCS). On the other day, an unconfined compression stress-relaxation
test was performed with applied strain = 5% and a faster ramp displacement
rate of 1 mm/s (UCF); for four of the specimens, this was followed by
dynamic loading (UCD) at a sinusoidal displacement amplitude of 8 µm and
frequencies of 1, 0.5, and 0.1 Hz (10 cycles each), and 0.05, 0.01, 0.005,and
0.001 Hz (5 cycles each).
A biphasic model [9] is employed in this analysis, with a solid-matrix
constitutive stress-strain relation combining Fung’s quasi-linear viscoelastic
model [6], with the conewise-linear elasticity model of Curnier et al. [1,3],
-7
10
-6
10
-5
10
-4
10
-3
10
-2
Frequency (Hz)
(a)
10
-1
1
10
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
Frequency (Hz)
(b)
10
-1
1
10
Figure 2: Experimental and theoretical prediction of unconfined compression
dynamic loading: (a) amplitude response, (b) phase angle response.
Poster Session - Cartilage Mechanics - Hall E
47th Annual Meeting, Orthopaedic Research Society, February 25 - 28, 2001, San Francisco, California