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Elastic anisotropy of articular cartilage is associated with the microstructures of collagen fibers and chondrocytes
Journal of Biomechanics 35 (2002) 931–942
Elastic anisotropy of articular cartilage is associated with the microstructures of collagen fibers and chondrocytes J.Z. Wu1, W. Herzog* Human Performance Laboratory, Faculty of Kinesiology, Department of Mechanical Engineering, Faculty of Engineering, The University of Calgary, Calgary, Alta., Canada Accepted 5 March 2002
Abstract Chondrocyte shape and volumetric concentration change as a function of depth in articular cartilage. A given chondrocyte shape produces different effects on the global material properties depending on the structure of the collagen fiber network. The shape and volumetric concentration of chondrocytes in articular cartilage appear to be related to the mechanical stability of the matrix. The present study was aimed to investigate, theoretically, the effects of the structural arrangement of the collagen fiber network, and the shape and distribution of chondrocytes, on the global material behavior of articular cartilage. Articular cartilage was assumed to be a four-phasic composite comprised of a matrix (associated with the properties of the proteoglycan structure), vertically and horizontally distributed collagen fibers, and spheroidal inclusions representing chondrocytes. A solution for composite materials was used to estimate the global, effective material properties of cartilage. Only the elasticity of the solid phase was investigated in the present study. Our simulations suggest that a soft, spheroidal cell inclusion in a fiber-reinforced proteoglycan matrix affects the material properties differently depending on the shape of the spheroidal inclusions. If the long axis of the inclusions is parallel to the collagen fibers, as in the deep zone, the soft inclusions increase the stiffness of the composite in the fiber direction, and reduce the stiffness of the composite in the direction normal to the fibers. Furthermore, we found that Young’s modulus normal to the contact surface increases from the superficial to the deep zone in articular cartilage by a factor of 10–50, a finding that agrees well with experimental observations. Our analysis suggests that the combination of proteoglycan matrix, fiber orientation, and shape of chondrocytes are intimately related and are likely adapted to optimize the mechanical stability and load carrying capacity of the structure. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Cartilage mechanics; Cell mechanics; Collagen fibers; Chondrocytes; Composite; Elastic modulus; Inclusions; Theoretical model; Transversely isotropic
1. Introduction Articular cartilage serves mainly as a load-bearing medium in joints; thus, the structure of cartilage is ‘‘customarily designed’’ to carry high stresses. Articular cartilage is a poroelastic material consisting of a fluid component (75–80% wet weight) and a solid matrix (20–25% wet weight) (Stockwell, 1979). The solid phase of articular cartilage is mainly composed of collagen *Corresponding author. Human Performance Laboratory, Faculty of Kinesiology, The University of Calgary, 2500 University Drive N.W., Calgary, Alta., Canada T2N 1N4. Tel.: +1-403-220-3438; fax: +1-403-284-3553. E-mail address: [email protected] (W. Herzog). 1 Current address: National Institute for Occupational Safety & Health, Morgantown, WV 26505, USA.
(65%), proteoglycans (25%), glycoproteins, chondrocytes (o10%) and lipids (o10%) (Eyre, 1990). Under physiological conditions, the proteoglycans are negatively charged and produce a swelling pressure that depends on the saline concentration of the fluid. At equilibrium, the swelling pressure is counteracted by the external load and the structural elements in the solid matrix, mainly the collagen fibers. Since collagen fibers, chondrocytes, and the other components in the solid phase have different and distinctly directional material properties, articular cartilage is a composite with anisotropic material properties. The collagen fibers form a three-dimensional network that is adapted to the mechanical loading in the joint (Aspden and Hukins, 1981; Minns and Steven, 1977). Based on the cellular density, cellular morphology, and
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collagen fiber arrangement, articular cartilage can be divided into three distinct, morphological zones. In the superficial zone (10–20% of the total thickness), collagen fibers are oriented parallel to the articular surface; in the middle zone (40–60% of the total thickness), there is no preferred orientation for the collagen fibers; and in the deep zone (30% of the total thickness), the collagen fibers are approximately perpendicular to the articular surface. The volumetric concentration of the collagen fibers increases from a range of 16%–31% in the superficial zone to a range of 14–42% in the deep zone (e.g., Hedlund et al., 1993; Langsjo et al., 1999). The diameter of collagen fibers increases from about 35 nm in the superficial zone to about 50 nm in the deep zone (Langsjo et al., 1999). Collagen fibers have a high stiffness compared to the surrounding cartilage matrix. Young’s modulus of collagen has been reported to vary from 2 to 40 MPa for articular cartilage (Pins et al., 1997) to 2:9 GPa for tendon (Sasaki and Odajima, 1996). It is well accepted that chondrocytes play an important role in cartilage adaptation and degeneration. The mechanical environment of chondrocytes can stimulate and regulate the biosynthetic activities in the cell, which, in turn, influences cartilage adaptation. In normal cartilage, chondrocytes change shape across the thickness (Guilak, 1995; Guilak et al., 1995; Clark et al., 1999): chondrocytes are typically flattened in the surface zone, spherical in the middle zone, and arranged in columns in the deep zone. The distribution of chondrocytes in cartilage is not uniform: the average cell volumetric concentration increases from the deep zone to the surface zone by a factor of about three (Wong et al., 1997). Chondrocytes differ in material properties from the surrounding matrix. According to the measurements by Jones et al. (1997) and Shin and Athanasiou (1997), the elastic modulus of chondrocytes is smaller by three orders of magnitude and the permeability of chondrocytes is greater by five orders of magnitude than the corresponding values of the extracellular matrix. Because of the great difference in material properties between proteoglycan matrix, cells and collagen fibers, cartilage is not a uniform material. In order to include these structural, non-uniform effects, cartilage has been represented using a transversely isotropic material (e.g., Garcia et al., 1998; Donzelli et al., 1999). However, in these theoretical models, the global material properties have not been related to the microstructure of the tissue. Soulhat et al. (1999) and Li et al. (1999) investigated theoretically the effects of the collagen fiber network on anisotropy of cartilage properties. In their models, the effect of volumetric concentration of collagen fibers, the structure of the fiber network, and the distributed chondrocytes were not included. Since the material anisotropy of cartilage is likely caused by microstruc-
tural variations in the tissue, i.e., the distribution of collagen fibers and chondrocytes, the description of cartilage using a uniform, transversely isotropic model is not appropriate. Bursac et al. (1999) found that simulations based on a uniform, transversely isotropic, biphasic cartilage model cannot fit the stress response curves of confined and unconfined compression tests. The purpose of the present study was to investigate, theoretically, the effects of the structural arrangement of the collagen fiber network, and the shape and distribution of chondrocytes, on the global material behavior of articular cartilage. Articular cartilage was assumed to be a four-phasic composite comprised of a proteoglycan matrix, vertically and horizontally distributed collagen fibers, and spheroidal inclusions representing chondrocytes. A solution for composite materials (Qiu and Weng, 1990) was used to estimate the global, effective material properties of cartilage. Only the elasticity of the solid phase was investigated in the present study.
2. Methods 2.1. Homogenized transversely isotropic cartilage model 2.1.1. Transversely isotropic moduli for composite For convenience of numerical simulation, the heterogeneous, anisotropic cartilage was approximated as transversely isotropic in our model. Assuming axis 1 to be symmetric and plane 2–3 isotropic, a transversely isotropic relation using Hill’s (Hill, 1964) notation is expressed as þ s33 Þ ¼ kðe22 þ e33 Þ þ l 0 e11 ; ¼ lðe22 þ e33 Þ þ ne11 ;
1 2ðs22
s11
s22 s33 ¼ 2mðe22 e33 Þ; s23 ¼ 2me23 ; s12 ¼ 2pe12 ;
s13 ¼ 2pe13 ;
ð1Þ
where k is the plane-strain bulk modulus, l and l 0 are the associated cross-moduli, n is the modulus for longitudinal uniaxial straining, and m and p are the shear moduli. The stiffness modulus tensor has to be symmetric, therefore, we further have l ¼ l 0 : Conventional Young’s modulus, E11 ; and Poisson’s ratio, n12 ; were obtained from Eq. (1) in a test under uniaxial loading (i.e., s11 a0; s22 ¼ s33 ¼ 0) as l2 E11 ¼ n ; k
n12 ¼ n13 ¼
1l : 2k
ð2Þ
Conventional Young’s modulus and Poisson’s ratio in the 2–3 plane, E22 and n23 ; respectively, and Poisson’s ratio, n21 ; were obtained in another uniaxial loading test,
J.Z. Wu, W. Herzog / Journal of Biomechanics 35 (2002) 931–942
s22 a0 and s11 ¼ s33 ¼ 0; as 4mkE11 ; E22 ¼ E33 ¼ n kE11 þ n m kE11 n m lð1 n23 Þ n23 ¼ n32 ¼ : ; n21 ¼ n31 ¼ kE11 þ n m n
f CLA ¼
N1 X
933
2cr pr =f ðrÞ ;
r¼1
ð3Þ
Comparing Hill’s notation (Eq. (1)) with the generalized Hooke’s law, it is obvious that the conventional shear moduli, m12 ; m13 ; and m23 ; are E22 : ð4Þ m12 ¼ m13 ¼ p; m23 ¼ m ¼ 2ð1 þ n23 Þ
gCLA ¼
N 1 X
cr ðlr d ðrÞ nr gðrÞ Þ=l ðrÞ ;
r¼1
hCLA ¼
N 1 X
cr ðlr cðrÞ 2kr hðrÞ Þ=l ðrÞ ;
r¼1
c
CAI
¼
N 1 X
! ðrÞ
cr c =l
ðrÞ
=l CAI ;
r¼1
2.1.2. Composite with N 1 spheroidal inclusions For a N-phasic material with one elastic matrix and N 1 elastic spheroidal inclusions (Fig. 1), Qiu and Weng (1990) derived a solution for equivalent elastic moduli using Mori and Tanaka’s (1973) method. Qiu and Weng’s (1990) solution is valid for a non-dilute concentration of inclusions and, can be used for cartilage layers with large volumetric concentrations of collagen fibers and chondrocytes. In their solution (Qiu and Weng, 1990), inclusions were considered to be aligned and spheroidal; each inclusion may have different, transversely isotropic material properties; the aspect ratio of the inclusions, a ¼ b=a; can be varied from 0 to N; and the inclusions are considered to be bound perfectly to the matrix. Therefore, Qiu and Weng’s (1990) solution can be used to include the effects of horizontally distributed fibers ða ¼ 0Þ; vertically distributed fibers ða-NÞ; and chondrocytes shaped from oblate, spherical, to prolate (Fig. 1). The effective, transversely isotropic elastic moduli for a N-phasic composite is expressed as (Qiu and Weng, 1990)
d
CAI
N 1 X
¼
e
CAI
¼
N 1 X
f
CAI
¼
N 1 X
g
CAI
¼
N 1 X
h
CAI
¼
N 1 X
l
CAI
¼
N 1 X
2p ¼ f CLA f CAI ;
ð5Þ CAI
CLA
CAI
CLA
CAI
where c ; c ; d ; d ; e ; e ; f ; f ; gCLA ; gCAI ; hCLA ; hCAI are functions of the material properties of the matrix and inclusions, and the aspect ratios of each inclusion: c
¼
2cr ðkr d
d CLA ¼
ðrÞ
lr g Þ=l ;
cr ðnr cðrÞ 2lr gðrÞ Þ=l ðrÞ ;
r¼1
eCLA ¼
N 1 X r¼1
ðrÞ
ðrÞ
ðrÞ
ðrÞ
ðrÞ
cr g =l
=l CAI ; !
cr h =l
=l CAI ;
cr d =l
! X
X
cr gðrÞ =l ðrÞ
cr cðrÞ =l ðrÞ
X
cr hðrÞ =l ðrÞ ;
n r n0 ½S1111 2n0 S1122 E0 2ðlr l0 Þ þ ½ð1 n0 ÞS1122 n0 S1111 ; E0
eðrÞ ¼ 1 þ
2ðmr m0 Þ S2323 ; m0
f ðrÞ ¼ 1 þ
2ðpr p0 Þ S1212 ; p0
2ðkr k0 Þ ½ð1 n0 ÞS1122 n0 S1111 E0 lr l0 þ ½S1111 2n0 S1122 ; E0
gðrÞ ¼ 1 þ 2cr mr =eðrÞ ;
! ðrÞ
d ðrÞ ¼ 1 þ
r¼1 N 1 X
;
2ðkr k0 Þ ½ð1 n0 ÞðS2222 þ S2233 Þ E0 2ðlr l0 Þ 2n0 S2211 þ ½ð1 n0 ÞS1122 n0 S1111 ; E0
2m ¼ eCLA eCAI ;
ðrÞ
!1
r¼1
2
n ¼ d CLA d CAI þ 2gCLA hCAI ;
ðrÞ
cr =f
ðrÞ
;
r¼1
cðrÞ ¼ 1 þ
N 1 X
cr =e
r¼1
l 0 ¼ hCLA d CAI þ cCLA hCAI ;
CLA
!1 ðrÞ
r¼1
where
CLA
=l CAI ;
r¼1
l ¼ gCLA cCAI þ d CLA gCAI ;
CAI
cr d =l
ðrÞ
r¼1
2k ¼ cCLA cCAI þ 2hCLA gCAI ;
CLA
! ðrÞ
ð6Þ
J.Z. Wu, W. Herzog / Journal of Biomechanics 35 (2002) 931–942
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Fig. 1. Cartilage modeled as a composite with aligned, spheroidal inclusions.
Fig. 2. Effective elastic properties in the surface zone with thin-flattened cells ða ¼ 1=10Þ as a function of collagen volumetric fraction: (a) E11 =E0 ; (b) E22 =E0 ; (c) n23 ; (d) n12 :
nr n0 ½S2211 n0 ðS2222 þ S2233 Þ E0 lr l0 þ ½ð1 n0 ÞðS2222 þ S2233 Þ 2n0 S2211 ; E0
hðrÞ ¼ 1 þ
l ðrÞ ¼ cðrÞ d ðrÞ 2gðrÞ hðrÞ ;
ð7Þ
where E0 and n0 are Young’s modulus and Poisson’s ratio of the matrix, respectively; cr ðr ¼ 1; 2; y; N 1Þ is the volumetric concentration of the rth inclusion; lr ; nr ; kr ; mr ; pr ðr ¼ 1; 2; y; N 1Þ are the material properties of the rth inclusions based on Hill’s notation (Eq. (1)); Sijkl are components of Eshelby’s tensor (Eshelby, 1957),
J.Z. Wu, W. Herzog / Journal of Biomechanics 35 (2002) 931–942
and are defined as 1 3a2 1 1 2n0 þ 2 S1111 ¼ 2ð1 nj0Þ a 1 3a2 1 2n0 þ 2 g ; a 1
1 1 1 2n0 þ 2 2ð1 n0 Þ a 1 1 3 1 2n0 þ g; 2ðn0 1Þ 2ða2 1Þ
S1122 ¼ S1133 ¼
S2323 3 a2 2 8ð1 n0 Þ a 1 1 9 1 2n0 þ g; 4ðn0 1Þ 4ða2 1Þ
S2222 ¼ S3333 ¼
S2233
1 a2 ¼ S3322 ¼ 4ð1 n0 Þ 2ða2 1Þ 3 1 2n0 þ g ; 4ða2 1Þ
1 a2 S2211 ¼ S3311 ¼ 2ð1 n0 Þ a2 1 1 3 ð1 2n0 Þ þ 2 þ g; 4ðn0 1Þ a 1
935
1 a2 ¼ 4ð1 n0 Þ 2ða2 1Þ 3 þ 1 2n0 g ; 4ða2 1Þ
1 a2 þ 1 1 2n0 S1212 ¼ S1313 ¼ 4ð1 n0 Þ 2ða2 1Þ 1 3ða2 þ 1Þ 1 2n0 2 g ; 2 a 1
ð8Þ
where g is a function of the aspect ratio of the inclusions a ½aða2 1Þ1=2 cosh1 a ; g¼ 2 ða 1Þ3=2 a ¼ a=b > 1 ðprolateÞ; a ½cos1 a að1 a2 Þ1=2 ; g¼ ð1 a2 Þ3=2 a ¼ a=bo1 ðoblateÞ: ð9Þ
Fig. 3. Effective elastic properties in the surface zone with flattened cells ða ¼ 1=2Þ as a function of collagen volumetric fraction: (a) E11 =E0 ; (b) E22 =E0 ; (c) n23 ; (d) n12 :
J.Z. Wu, W. Herzog / Journal of Biomechanics 35 (2002) 931–942
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It can be seen that a ¼ 1 (spherical inclusion) is a singularity in Eq. (9). The components of Eshelby’s tensor (Eshelby, 1957) at a ¼ 1 are given as S1111 ¼ S2222 ¼ S3333 ¼
If the thermal dynamic restrictions (11) are satisfied, the cell–collagen–matrix composite will be mechanically stable. In the following numerical simulations, condition (11) is used to check the mechanical stability of the cartilage composite.
S1122 ¼ S2233 ¼ S3311
2.2. Numerical tests
S1212 ¼ S2323 ¼ S3131
7 5n0 ; 15ð1 n0 Þ 5n0 1 ; ¼ 15ð1 n0 Þ 4 5n0 : ¼ 15ð1 n0 Þ
ð10Þ
2.1.3. Mechanical stability of the cell–collagen–matrix composite Neglecting heat transfer, the thermal dynamic restriction of the constitutive relation requires ds : de > 0; which means that the mechanical work must be positive during the deformation. In terms of the transversely isotropic elastic material, the thermal dynamic restrictions are expressed as E11 ; E22 ; m12 ; m23 > 0; jn23 jo1;
jn12 joðE11 =E22 Þ1=2 ;
1 n223 2n12 n23 2n12 n21 n23 > 0:
ð11Þ
We used Qiu and Weng’s (1990) solution to estimate the homogenized elastic modulus of cartilage. Cartilage was represented by a four-phasic composite, composed of one matrix (volumetric concentration cm ), one cell inclusion (volumetric concentration cc ), and two collagen fiber inclusions, i.e., vertical and horizontal fiber inclusions (volumetric concentration cf ¼ cfv þ cfh ). The horizontal and vertical fiber inclusions were represented by using two extremes of the aspect ratio of spheroidal inclusions, afh ¼ 0 and afv -N; respectively. The effective elastic material properties in the superficial layer, in the middle layer, and in the deep layer of the articular cartilage were calculated. In the superficial layer, collagen fibers were assumed to be parallel to the contact surface ðcf ¼ cfh Þ; and the shape of the chondrocytes was assumed to change from flat to
Fig. 4. Effective elastic properties in the surface zone with spherical cells ða ¼ 1Þ as a function of collagen volumetric fraction: (a) E11 =E0 ; (b) E22 =E0 ; (c) n23 ; (d) n12 :
J.Z. Wu, W. Herzog / Journal of Biomechanics 35 (2002) 931–942
spherical (ac ¼ 1=10; 1/2, and 1); in the middle zone, 1/3 of the fibers were assumed vertical to the contact surface and 2/3 of the fibers were assumed parallel to the contact surface [i.e., cfh ¼ ð2=3Þcf and cfv ¼ ð1=3Þcf ]. The shape of the chondrocytes was assumed to vary around spherical (ac ¼ 1=2; 1, and 3/2). In the deep zone, collagen fibers were assumed to be vertical to the contact surface ðcf ¼ cfv Þ; and the shape of the chondrocytes was assumed to change from spherical to columnar (ac ¼ 1; 2, and 10). The proteoglycan matrix, collagen fibers, and chondrocytes were assumed to be isotropic. Young’s moduli and Poisson’s ratios for cartilage matrix, collagen fibers, and chondrocytes were taken from published experimental data (e.g., Athanasiou et al., 1991; Shin and Athanasiou, 1997; Pins et al., 1997). They were: E0 ¼ 0:20 MPa; n0 ¼ 0:05; Ef ¼ 10:0 MPa; nf ¼ 0:30; and Ec ¼ 1:0 kPa; nc ¼ 0:40; respectively. The transversely isotropic material parameters, l; n; k; m; and p; are related to the isotropic material parameters by l ¼ K 23 m; n ¼ K þ 43 m; and m ¼ p ¼ m with K ¼ E=½3ð1 2nÞ and m ¼ E=½2ð1 þ nÞ being the conventional bulk modulus and shear modulus, respectively. The effective elastic moduli in the vertical and horizontal directions, E11 and E22 ; i.e., the directions
937
parallel and normal to the contact surface, respectively, the shear moduli, m23 and m12 ; and Poisson’s ratios, n23 and n12 ; were predicted as a function of the volumetric concentration of the collagen fibers. The aspect ratio of the cells was assumed to be 1/10, 1/2, and 1 (i.e., the shape of the cells varied from flattened to spherical) in the surface zone (Figs. 2, 3 and 4), 1/2 and 1 in the middle zone (Figs. 5 and 6), and 1, 2, and 10 (i.e., the shape of the cells varied from spherical to column like) in the deep zone (Figs. 7 and 8). The calculations were stopped if the thermal dynamic restrictions (11) were violated, i.e., the material became mechanically unstable.
3. Results In the surface zone, changes in the cell volumetric fraction of thin-flattened cells ðac ¼ 1=10Þ have virtually no effect on the horizontal elastic modulus, E22 ; (Fig. 2b) and the major shear modulus, m23 (results not shown). However, E22 and m23 decrease greatly as the volumetric concentration of the cells increases when the shape of the cells is round, i.e., ac is close to 1 (Figs. 3b and 4b). The vertical elastic modulus, E11 ; and the shear
Fig. 5. Effective elastic properties in the middle zone with flattened cells ða ¼ 1=2Þ as a function of collagen volumetric fraction: (a) E11 =E0 ; (b) E22 =E0 ; (c) n23 ; (d) n12 :
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Fig. 6. Effective elastic properties in the middle zone with spherical cells ða ¼ 1Þ as a function of collagen volumetric fraction: (a) E11 =E0 ; (b) E22 =E0 ; (c) n23 ; (d) n12 :
modulus, m12 (results not shown), decrease greatly with increasing cell volumetric fraction, independent of the shape of the chondrocytes (Figs. 2a, 3a, and 4a). When ac > 1=2; Poisson’s ratio n23 becomes negative for cc > 10% (Figs. 3c and 4c). In the middle zone, the material is stable when the shape of the chondrocytes is between mildly flattened and spherical, i.e., ac is between 1/2 and 1, (Figs. 5 and 6); the material becomes unstable when the cell shape becomes column like (i.e., ac > 3=2; results not shown). When the shape of the cells is spherical, a small volumetric concentration will result in a reduction in E22 (Fig. 6b). In the deep zone, spherical-like ðac ¼ 1; 2Þ cell inclusions cause instability of the material, i.e., E11 and E22 become negative (Figs. 7a, b), or the thermal dynamic restrictions (11) are not satisfied (Fig. 7). The existence of column-like cells reinforces slightly the material, i.e., E11 and E22 increase as the volumetric concentration increases (Figs. 8a, b). A large cell volumetric concentration (e.g., > 15%) will result in a material unstability (Fig. 8b), i.e., the thermal dynamic restrictions (11) will be violated. The shape of the cell inclusions has little influence on the shear moduli, m23 and m12 (results not shown).
4. Discussion and conclusion Collagen fibers are structural elements in articular cartilage. The structure of the collagen network is thought to be related to the mechanical stability of the tissue (Aspden and Hukins, 1981; Minns and Steven, 1977). The surface layer of cartilage is like the wall of a pressure vessel that is designed to withstand the swelling pressure in the tissue; the collagen fibers are oriented to achieve an optimal tangential stiffness of the tissue. In the deep layer, collagen fibers are oriented, likely, to optimize (a) the ties to the underlying calcified tissue and (b) the stiffness normal to the contact surface. The intermediate layer allows for a change in the orientation of the collagen fibers without discontinuity. Chondrocytes have different shapes and different volumetric concentrations in the different layers of articular cartilage. A given chondrocyte shape produces different effects on the global material properties, depending on the structure of the collagen fiber network. The shape and volumetric concentration of chondrocytes in articular cartilage appear to be related to the mechanical stability of the matrix. The present study was intended to investigate the dependence of the material properties of articular cartilage on the combination of the collagen
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Fig. 7. Effective elastic properties in the deep zone with spherical cells ða ¼ 1Þ as a function of collagen volumetric fraction: (a) E11 =E0 ; (b) E22 =E0 ; (c) n23 ; (d) n12 :
fiber network structure and the distributed chondrocyte structure. Our analysis suggests that chondrocyte shape and volumetric fraction influence material properties and mechanical stability of articular cartilage considerably. Our simulations suggest that a soft, spheroidal inclusion in a fiber-reinforced structure, such as articular cartilage with collagen fibers and chondrocytes, gives different material properties depending on the shape of the spheroidal inclusions. If the long axis of the inclusions is parallel to the collagen fibers, as in the deep zone, the soft inclusions increase the stiffness of the composite in the fiber direction, and reduce the stiffness of the composite in the direction normal to the fibers (Figs. 2a, b and 8a,b). This result is caused in this transversely isotropic material because the soft inclusions reduce the stiffness parameters, k; l and n; non-uniformly; as a consequence, Young’s modulus in the direction of the long axis of the inclusions, E11 ; is reinforced (Eq. (2)). In an isotropic matrix, stiffness parameters, k; l and n; would be reduced uniformly by the soft inclusions, and Young’s moduli in both directions, E11 and E22 ; would be reduced. Our analysis suggests that fiber orientation and shape of the chondrocytes in the surface and deep
zones in articular cartilage are intimately related and are likely adapted to optimize the mechanical stability and load carrying capacity of the structure. A comparison of the predicted Young’s modulus normal to the contact surface from the superficial zone (Fig. 2) with that from the deep zone (Fig. 8) shows an increase by a factor of 10–50 (Fig. 9a). This prediction agrees well with experimental measurements by Schinagl et al. (1997) who reported that the compressive elastic modulus increases with depth in bovine articular cartilage, from 0:08 MPa in the surface layer to 2:10 MPa in the deep layer (i.e., by a factor of about 26). According to our predictions, the tangential (in the direction parallel to the contact surface) Young’s modulus for a typical articular cartilage increases from the deep to the surface zone by 50–150% (Figs. 2, 8 and 9b). We have not yet found experimental measurements in the literature for comparison with this prediction. Our prediction that Young’s modulus in the surface layer and parallel to the contact surface, E22 ; is approximately 5–20 times of that normal to the contact surface, E11 (Fig. 2), agrees well with experimental measurements (Mow et al., 2000). Mow et al.
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J.Z. Wu, W. Herzog / Journal of Biomechanics 35 (2002) 931–942
Fig. 8. Effective elastic properties in the deep zone with narrow column-like cells ða ¼ 10Þ as a function of collagen volumetric fraction: (a) E11 =E0 ; (b) E22 =E0 ; (c) n23 ; (d) n12 :
(2000) determined the tensile Young’s modulus using an indentation test. They found that Young’s moduli parallel and normal to the contact surface are approximately 3.33–6:44 MPa and 0.33–0:36 MPa; respectively. In our analysis, collagen fibers were assumed to have an isotropic material property with the same stiffness in tension and compression. This assumption differs from that proposed by Soulhat et al. (1999) who used a material model of collagen fibers with zero stiffness in compression, because of the structural instability of thin, isolated rod-like fibers in compression. However, such an interpretation ignores the fact that collagen fibers are not isolated but are embedded in matrix material that provides stability to the collagen fibers, similar to the stability provided by cement to steel rods in reinforced concrete structures. Our modelling is vindicated by the observations that the compressive modulus (normal to the contact surface) in articular cartilage reaches its maximum in the deep zone (Schinagl et al., 1997) where the collagen fibers are oriented normal to the contact surface. This prediction could not be made in previous models (e.g., Soulhat et al., 1999; Li et al., 1999), except by changing the
material properties of the cartilage model without a direct structural analog. Here, the structural analog directly provides the non-uniformities in the material properties as observed in experiments. Our results indicate that the matrix is mechanically stable even when the volumetric concentration of chondrocytes reaches 20% in the surface zone (Fig. 2). In the deep zone, the matrix becomes mechanically unstable when the volumetric concentration of chondrocytes is over 10% (Fig. 8). This result suggests that, in order to maintain mechanical stability of the articular cartilage, the volumetric concentration of chondrocytes should decrease from the surface to the deep zone, and should not exceed certain limiting values. This prediction agrees well with the experimental observation that cell volumetric concentrations in articular cartilage increase from the deep to the surface zone by a factor of about three (Wong et al., 1997). In numerical approximations, the stiffness matrix of the composite material may turn out to be asymmetric when the number of inclusions is more than two (Qiu and Weng, 1990). In order to be thermodynamically consistent, we enforced the stiffness matrix to be
J.Z. Wu, W. Herzog / Journal of Biomechanics 35 (2002) 931–942
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Acknowledgements This study was supported by The Alberta Heritage Foundation for Medical Research, the Medical Research Council of Canada, and The Arthritis Society of Canada.
References
Fig. 9. The present model predicts that Young’s modulus of the articular cartilage normal to the contact surface [(a) E11 =E0 ] increases and that parallel to the contact surface [(b) E22 =E0 ] decreases from the surface to the deep layer.
symmetric by using the average value of l and l 0 in the present investigation. Articular cartilage is a biphasic material composed of a solid, elastic phase and a fluid phase (Mow et al., 1980). The homogenized elastic property obtained in the present study can be included into poroelastic models to account for the effects of collagen fibers and chondrocytes. The oriented and distributed collagen fibers and chondrocytes might have effects on the hydrostatic permeability of articular cartilage which were not included in the present approach. The homogenization approach used in the present study (Qiu and Weng, 1990) is based on infinitesimal deformation theory. A correlation of theoretical and experimental studies on cartilage suggests that the infinitesimal theory is adequate for local compressive strains of up to around 20–30% (Mow et al., 1980; Kwan et al., 1990). For large deformations, hyperelastic constitutive models (Holmes 1986; Holmes and Mow, 1990) have been used instead of the linear elastic relations for the solid phase.
Aspden, R., Hukins, D., 1981. Collagen organization in articular cartilage, determined by X-ray diffraction, and its relationship to tissue function. Proceedings of the Royal Society, London, Series B 212, 299–304. Athanasiou, K.A., Rosenwasser, M.P., Buckwalter, J.A., Malinin, T.I., Mow, V.C., 1991. Interspecies comparisons in in situ intrinsic mechanical properties of distal cartilage. Journal of Orthopaedic Research 9, 330–340. Bursac, P., Obitz, T., Eisenberg, S., Stamenovic, D., 1999. Confined and unconfined stress relaxation of cartilage: appropriateness of a transversely isotropic analysis. Journal of Biomechanics 32, 1125–1130. Clark, A., Herzog, W., Matyas, J., Barclay, L., 1999. Chondrocyte deformation resulting from in-situ compression of articular cartilage. In Proceedings of the XVII Congress of the International Society for Biomechanics (ISB’99). Calgary, Canada, pp. 339. Donzelli, P., Spilker, R., Mow, V., 1999. Contact analysis of biphasic transversely isotropic cartilage layers and correlations with tissue failure. Journal of Biomechanics 32, 1037–1047. Eshelby, J., 1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society, London, Series A 241, 376–396. Eyre, D., 1990. Structure and function of the cartilage collagen: role of type IX in articular cartilage. In: Brandt, K.D. (Ed.), Cartilage Changes in Osteoarthritis. Indianapolis: Ciba-Geigy, pp. 12–16. Garcia, J., Altiero, N., Haut, R., 1998. Approach for the stress analysis of transversely isotropic biphasic cartilage under impact load. ASME Journal of Biomechanical Engineering 120, 608–613. Guilak, F., 1995. Compression-induced changes in the shape and volume of the chondrocyte nucleus. Journal of Biomechanics 28, 1529–1541. Guilak, F., Ratcliffe, A., Mow, V., 1995. Chondrocyte deformation and local tissue strain in articular cartilage: a confocal microscopy study. Journal of Orthopaedic Research 13, 410–421. Hedlund, H., Mengarelli-Widholm, S., Reinholt, F., Svensson, O., 1993. Stereological studies on collagen in bovine articular cartilage. Acta Pathologica, Microbiologica et Immunologica Scandinavica (APMIS) 101, 133–140. Hill, R., 1964. Theory of mechanical properties of fiber-strengthened materials: I. elastic behaviour. Journal of Mechanics and Physics of Solids 12, 199–212. Holmes, M., 1986. Finite deformation of soft tissue: analysis of a mixture model in uni-axial compression. ASME Journal of Biomechanical Engineering 108, 372–381. Holmes, M., Mow, V., 1990. Nonlinear characteristics of soft gels and hydrated connective tissues in ultrafiltration. Journal of Biomechanics 23, 1145–1156. Jones, W., Ting-Beall, H., Lee, G., Kelley, S., Hochmuth, R., Guilak, F., 1997. Mechanical properties of human chondrocytes and chondrons from normal and osteoarthritic cartilage. Transactions of the Orthopaedic Research Society 22 (1), 199. Kwan, M., Lai, W., Mow, V., 1990. A finite deformation theory for cartilage and other soft hydrated connective tissues. Journal of Biomechanics 23, 145–155.
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Langsjo, T., Hyttinen, M., Pelttari, A., Kiraly, K., Arokoski, J., Helminen, H., 1999. Electron microscopic stereological study of collagen fibrils in bovine articular cartilage: volume and surface densities are best obtained indirectly (from length densities and diameters) using isotropic uniform random sampling. Journal of Anatomy 195, 281–293. Li, L., Soulhat, M., Buschmann, M., Shirazi-Adl, A., 1999. Nonlinear analysis of cartilage in unconfined ramp compression using a fibril reinforced poroelastic model. Clinical Biomechanics 14, 673–682. Minns, R., Steven, F., 1977. The collagen fibril organization in human articular cartilage. Journal of Anatomy 123, 437–457. Mori, T., Tanaka, K., 1973. Average stress in matrix and average elastic energy of material with misfitting inclusions. Acta Metallurgica 21, 571–574. Mow, V.C., Kuei, S.C., Lai, W.M., Armstrong, C.G., 1980. Biphasic creep and stress relaxation of articular cartilage: theory and experiment. ASME Journal of Biomechanical Engineering 102, 73–84. Mow, V.C., Good, P.M., Gardner, T.R., 2000. A new method to determine the tensile properties of articular cartilage using the indentation test. Proceedings of the Orthopaedic Research Society 1, 103. Pins, G., Huang, E., Christiansen, D., Silver, F., 1997. Effects of static axial strain on the tensile properties and failure mechanisms of self-
assembled collagen fibers. Journal of Applied Polymer Science 63, 1429–1440. Qiu, Y., Weng, G., 1990. On the application of Mori-Tanaka’s theory involving transversely isotropic spheroidal inclusions. International Journal of Engineering Science 28, 1121–1137. Sasaki, N., Odajima, S., 1996. Stress–strain curve and Young’s modulus of a collagen molecule as determined by the X-ray diffraction technique. Journal of Biomechanics 29, 655–658. Schinagl, R., Gurski, D., Chen, A., Sah, R., 1997. Depth-dependent confined compression modulus of full-thickness bovine articular cartilage. Journal of Orthopaedic Research 15, 499–506. Shin, D., Athanasiou, K., 1997. Biomechanical properties of the individual cell. Transactions of the Orthopaedic Research Society 22 (1), 352. Soulhat, J., Buschmann, M., Shirazi-Adl, A., 1999. A fibril-networkreinforced biphasic model of cartilage in unconfined compression. ASME Journal of Biomechanical Engineering 121, 340–347. Stockwell, R., 1979. Biology of Cartilage Cells. Cambridge University Press, Cambridge. Wong, M., Wuethrich, P., Buschmann, M., Eggli, P., Hunziker, E., 1997. Chondrocyte biosynthesis correlates with local tissue strain in statically compressed adult articular cartilage. Journal of Orthopaedic Research 15, 189–198.
Elastic anisotropy of articular cartilage is associated with the microstructures of collagen fibers and chondrocytes J.Z. Wu1, W. Herzog* Human Performance Laboratory, Faculty of Kinesiology, Department of Mechanical Engineering, Faculty of Engineering, The University of Calgary, Calgary, Alta., Canada Accepted 5 March 2002
Abstract Chondrocyte shape and volumetric concentration change as a function of depth in articular cartilage. A given chondrocyte shape produces different effects on the global material properties depending on the structure of the collagen fiber network. The shape and volumetric concentration of chondrocytes in articular cartilage appear to be related to the mechanical stability of the matrix. The present study was aimed to investigate, theoretically, the effects of the structural arrangement of the collagen fiber network, and the shape and distribution of chondrocytes, on the global material behavior of articular cartilage. Articular cartilage was assumed to be a four-phasic composite comprised of a matrix (associated with the properties of the proteoglycan structure), vertically and horizontally distributed collagen fibers, and spheroidal inclusions representing chondrocytes. A solution for composite materials was used to estimate the global, effective material properties of cartilage. Only the elasticity of the solid phase was investigated in the present study. Our simulations suggest that a soft, spheroidal cell inclusion in a fiber-reinforced proteoglycan matrix affects the material properties differently depending on the shape of the spheroidal inclusions. If the long axis of the inclusions is parallel to the collagen fibers, as in the deep zone, the soft inclusions increase the stiffness of the composite in the fiber direction, and reduce the stiffness of the composite in the direction normal to the fibers. Furthermore, we found that Young’s modulus normal to the contact surface increases from the superficial to the deep zone in articular cartilage by a factor of 10–50, a finding that agrees well with experimental observations. Our analysis suggests that the combination of proteoglycan matrix, fiber orientation, and shape of chondrocytes are intimately related and are likely adapted to optimize the mechanical stability and load carrying capacity of the structure. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Cartilage mechanics; Cell mechanics; Collagen fibers; Chondrocytes; Composite; Elastic modulus; Inclusions; Theoretical model; Transversely isotropic
1. Introduction Articular cartilage serves mainly as a load-bearing medium in joints; thus, the structure of cartilage is ‘‘customarily designed’’ to carry high stresses. Articular cartilage is a poroelastic material consisting of a fluid component (75–80% wet weight) and a solid matrix (20–25% wet weight) (Stockwell, 1979). The solid phase of articular cartilage is mainly composed of collagen *Corresponding author. Human Performance Laboratory, Faculty of Kinesiology, The University of Calgary, 2500 University Drive N.W., Calgary, Alta., Canada T2N 1N4. Tel.: +1-403-220-3438; fax: +1-403-284-3553. E-mail address: [email protected] (W. Herzog). 1 Current address: National Institute for Occupational Safety & Health, Morgantown, WV 26505, USA.
(65%), proteoglycans (25%), glycoproteins, chondrocytes (o10%) and lipids (o10%) (Eyre, 1990). Under physiological conditions, the proteoglycans are negatively charged and produce a swelling pressure that depends on the saline concentration of the fluid. At equilibrium, the swelling pressure is counteracted by the external load and the structural elements in the solid matrix, mainly the collagen fibers. Since collagen fibers, chondrocytes, and the other components in the solid phase have different and distinctly directional material properties, articular cartilage is a composite with anisotropic material properties. The collagen fibers form a three-dimensional network that is adapted to the mechanical loading in the joint (Aspden and Hukins, 1981; Minns and Steven, 1977). Based on the cellular density, cellular morphology, and
0021-9290/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 0 2 ) 0 0 0 5 0 - 7
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J.Z. Wu, W. Herzog / Journal of Biomechanics 35 (2002) 931–942
collagen fiber arrangement, articular cartilage can be divided into three distinct, morphological zones. In the superficial zone (10–20% of the total thickness), collagen fibers are oriented parallel to the articular surface; in the middle zone (40–60% of the total thickness), there is no preferred orientation for the collagen fibers; and in the deep zone (30% of the total thickness), the collagen fibers are approximately perpendicular to the articular surface. The volumetric concentration of the collagen fibers increases from a range of 16%–31% in the superficial zone to a range of 14–42% in the deep zone (e.g., Hedlund et al., 1993; Langsjo et al., 1999). The diameter of collagen fibers increases from about 35 nm in the superficial zone to about 50 nm in the deep zone (Langsjo et al., 1999). Collagen fibers have a high stiffness compared to the surrounding cartilage matrix. Young’s modulus of collagen has been reported to vary from 2 to 40 MPa for articular cartilage (Pins et al., 1997) to 2:9 GPa for tendon (Sasaki and Odajima, 1996). It is well accepted that chondrocytes play an important role in cartilage adaptation and degeneration. The mechanical environment of chondrocytes can stimulate and regulate the biosynthetic activities in the cell, which, in turn, influences cartilage adaptation. In normal cartilage, chondrocytes change shape across the thickness (Guilak, 1995; Guilak et al., 1995; Clark et al., 1999): chondrocytes are typically flattened in the surface zone, spherical in the middle zone, and arranged in columns in the deep zone. The distribution of chondrocytes in cartilage is not uniform: the average cell volumetric concentration increases from the deep zone to the surface zone by a factor of about three (Wong et al., 1997). Chondrocytes differ in material properties from the surrounding matrix. According to the measurements by Jones et al. (1997) and Shin and Athanasiou (1997), the elastic modulus of chondrocytes is smaller by three orders of magnitude and the permeability of chondrocytes is greater by five orders of magnitude than the corresponding values of the extracellular matrix. Because of the great difference in material properties between proteoglycan matrix, cells and collagen fibers, cartilage is not a uniform material. In order to include these structural, non-uniform effects, cartilage has been represented using a transversely isotropic material (e.g., Garcia et al., 1998; Donzelli et al., 1999). However, in these theoretical models, the global material properties have not been related to the microstructure of the tissue. Soulhat et al. (1999) and Li et al. (1999) investigated theoretically the effects of the collagen fiber network on anisotropy of cartilage properties. In their models, the effect of volumetric concentration of collagen fibers, the structure of the fiber network, and the distributed chondrocytes were not included. Since the material anisotropy of cartilage is likely caused by microstruc-
tural variations in the tissue, i.e., the distribution of collagen fibers and chondrocytes, the description of cartilage using a uniform, transversely isotropic model is not appropriate. Bursac et al. (1999) found that simulations based on a uniform, transversely isotropic, biphasic cartilage model cannot fit the stress response curves of confined and unconfined compression tests. The purpose of the present study was to investigate, theoretically, the effects of the structural arrangement of the collagen fiber network, and the shape and distribution of chondrocytes, on the global material behavior of articular cartilage. Articular cartilage was assumed to be a four-phasic composite comprised of a proteoglycan matrix, vertically and horizontally distributed collagen fibers, and spheroidal inclusions representing chondrocytes. A solution for composite materials (Qiu and Weng, 1990) was used to estimate the global, effective material properties of cartilage. Only the elasticity of the solid phase was investigated in the present study.
2. Methods 2.1. Homogenized transversely isotropic cartilage model 2.1.1. Transversely isotropic moduli for composite For convenience of numerical simulation, the heterogeneous, anisotropic cartilage was approximated as transversely isotropic in our model. Assuming axis 1 to be symmetric and plane 2–3 isotropic, a transversely isotropic relation using Hill’s (Hill, 1964) notation is expressed as þ s33 Þ ¼ kðe22 þ e33 Þ þ l 0 e11 ; ¼ lðe22 þ e33 Þ þ ne11 ;
1 2ðs22
s11
s22 s33 ¼ 2mðe22 e33 Þ; s23 ¼ 2me23 ; s12 ¼ 2pe12 ;
s13 ¼ 2pe13 ;
ð1Þ
where k is the plane-strain bulk modulus, l and l 0 are the associated cross-moduli, n is the modulus for longitudinal uniaxial straining, and m and p are the shear moduli. The stiffness modulus tensor has to be symmetric, therefore, we further have l ¼ l 0 : Conventional Young’s modulus, E11 ; and Poisson’s ratio, n12 ; were obtained from Eq. (1) in a test under uniaxial loading (i.e., s11 a0; s22 ¼ s33 ¼ 0) as l2 E11 ¼ n ; k
n12 ¼ n13 ¼
1l : 2k
ð2Þ
Conventional Young’s modulus and Poisson’s ratio in the 2–3 plane, E22 and n23 ; respectively, and Poisson’s ratio, n21 ; were obtained in another uniaxial loading test,
J.Z. Wu, W. Herzog / Journal of Biomechanics 35 (2002) 931–942
s22 a0 and s11 ¼ s33 ¼ 0; as 4mkE11 ; E22 ¼ E33 ¼ n kE11 þ n m kE11 n m lð1 n23 Þ n23 ¼ n32 ¼ : ; n21 ¼ n31 ¼ kE11 þ n m n
f CLA ¼
N1 X
933
2cr pr =f ðrÞ ;
r¼1
ð3Þ
Comparing Hill’s notation (Eq. (1)) with the generalized Hooke’s law, it is obvious that the conventional shear moduli, m12 ; m13 ; and m23 ; are E22 : ð4Þ m12 ¼ m13 ¼ p; m23 ¼ m ¼ 2ð1 þ n23 Þ
gCLA ¼
N 1 X
cr ðlr d ðrÞ nr gðrÞ Þ=l ðrÞ ;
r¼1
hCLA ¼
N 1 X
cr ðlr cðrÞ 2kr hðrÞ Þ=l ðrÞ ;
r¼1
c
CAI
¼
N 1 X
! ðrÞ
cr c =l
ðrÞ
=l CAI ;
r¼1
2.1.2. Composite with N 1 spheroidal inclusions For a N-phasic material with one elastic matrix and N 1 elastic spheroidal inclusions (Fig. 1), Qiu and Weng (1990) derived a solution for equivalent elastic moduli using Mori and Tanaka’s (1973) method. Qiu and Weng’s (1990) solution is valid for a non-dilute concentration of inclusions and, can be used for cartilage layers with large volumetric concentrations of collagen fibers and chondrocytes. In their solution (Qiu and Weng, 1990), inclusions were considered to be aligned and spheroidal; each inclusion may have different, transversely isotropic material properties; the aspect ratio of the inclusions, a ¼ b=a; can be varied from 0 to N; and the inclusions are considered to be bound perfectly to the matrix. Therefore, Qiu and Weng’s (1990) solution can be used to include the effects of horizontally distributed fibers ða ¼ 0Þ; vertically distributed fibers ða-NÞ; and chondrocytes shaped from oblate, spherical, to prolate (Fig. 1). The effective, transversely isotropic elastic moduli for a N-phasic composite is expressed as (Qiu and Weng, 1990)
d
CAI
N 1 X
¼
e
CAI
¼
N 1 X
f
CAI
¼
N 1 X
g
CAI
¼
N 1 X
h
CAI
¼
N 1 X
l
CAI
¼
N 1 X
2p ¼ f CLA f CAI ;
ð5Þ CAI
CLA
CAI
CLA
CAI
where c ; c ; d ; d ; e ; e ; f ; f ; gCLA ; gCAI ; hCLA ; hCAI are functions of the material properties of the matrix and inclusions, and the aspect ratios of each inclusion: c
¼
2cr ðkr d
d CLA ¼
ðrÞ
lr g Þ=l ;
cr ðnr cðrÞ 2lr gðrÞ Þ=l ðrÞ ;
r¼1
eCLA ¼
N 1 X r¼1
ðrÞ
ðrÞ
ðrÞ
ðrÞ
ðrÞ
cr g =l
=l CAI ; !
cr h =l
=l CAI ;
cr d =l
! X
X
cr gðrÞ =l ðrÞ
cr cðrÞ =l ðrÞ
X
cr hðrÞ =l ðrÞ ;
n r n0 ½S1111 2n0 S1122 E0 2ðlr l0 Þ þ ½ð1 n0 ÞS1122 n0 S1111 ; E0
eðrÞ ¼ 1 þ
2ðmr m0 Þ S2323 ; m0
f ðrÞ ¼ 1 þ
2ðpr p0 Þ S1212 ; p0
2ðkr k0 Þ ½ð1 n0 ÞS1122 n0 S1111 E0 lr l0 þ ½S1111 2n0 S1122 ; E0
gðrÞ ¼ 1 þ 2cr mr =eðrÞ ;
! ðrÞ
d ðrÞ ¼ 1 þ
r¼1 N 1 X
;
2ðkr k0 Þ ½ð1 n0 ÞðS2222 þ S2233 Þ E0 2ðlr l0 Þ 2n0 S2211 þ ½ð1 n0 ÞS1122 n0 S1111 ; E0
2m ¼ eCLA eCAI ;
ðrÞ
!1
r¼1
2
n ¼ d CLA d CAI þ 2gCLA hCAI ;
ðrÞ
cr =f
ðrÞ
;
r¼1
cðrÞ ¼ 1 þ
N 1 X
cr =e
r¼1
l 0 ¼ hCLA d CAI þ cCLA hCAI ;
CLA
!1 ðrÞ
r¼1
where
CLA
=l CAI ;
r¼1
l ¼ gCLA cCAI þ d CLA gCAI ;
CAI
cr d =l
ðrÞ
r¼1
2k ¼ cCLA cCAI þ 2hCLA gCAI ;
CLA
! ðrÞ
ð6Þ
J.Z. Wu, W. Herzog / Journal of Biomechanics 35 (2002) 931–942
934
Fig. 1. Cartilage modeled as a composite with aligned, spheroidal inclusions.
Fig. 2. Effective elastic properties in the surface zone with thin-flattened cells ða ¼ 1=10Þ as a function of collagen volumetric fraction: (a) E11 =E0 ; (b) E22 =E0 ; (c) n23 ; (d) n12 :
nr n0 ½S2211 n0 ðS2222 þ S2233 Þ E0 lr l0 þ ½ð1 n0 ÞðS2222 þ S2233 Þ 2n0 S2211 ; E0
hðrÞ ¼ 1 þ
l ðrÞ ¼ cðrÞ d ðrÞ 2gðrÞ hðrÞ ;
ð7Þ
where E0 and n0 are Young’s modulus and Poisson’s ratio of the matrix, respectively; cr ðr ¼ 1; 2; y; N 1Þ is the volumetric concentration of the rth inclusion; lr ; nr ; kr ; mr ; pr ðr ¼ 1; 2; y; N 1Þ are the material properties of the rth inclusions based on Hill’s notation (Eq. (1)); Sijkl are components of Eshelby’s tensor (Eshelby, 1957),
J.Z. Wu, W. Herzog / Journal of Biomechanics 35 (2002) 931–942
and are defined as 1 3a2 1 1 2n0 þ 2 S1111 ¼ 2ð1 nj0Þ a 1 3a2 1 2n0 þ 2 g ; a 1
1 1 1 2n0 þ 2 2ð1 n0 Þ a 1 1 3 1 2n0 þ g; 2ðn0 1Þ 2ða2 1Þ
S1122 ¼ S1133 ¼
S2323 3 a2 2 8ð1 n0 Þ a 1 1 9 1 2n0 þ g; 4ðn0 1Þ 4ða2 1Þ
S2222 ¼ S3333 ¼
S2233
1 a2 ¼ S3322 ¼ 4ð1 n0 Þ 2ða2 1Þ 3 1 2n0 þ g ; 4ða2 1Þ
1 a2 S2211 ¼ S3311 ¼ 2ð1 n0 Þ a2 1 1 3 ð1 2n0 Þ þ 2 þ g; 4ðn0 1Þ a 1
935
1 a2 ¼ 4ð1 n0 Þ 2ða2 1Þ 3 þ 1 2n0 g ; 4ða2 1Þ
1 a2 þ 1 1 2n0 S1212 ¼ S1313 ¼ 4ð1 n0 Þ 2ða2 1Þ 1 3ða2 þ 1Þ 1 2n0 2 g ; 2 a 1
ð8Þ
where g is a function of the aspect ratio of the inclusions a ½aða2 1Þ1=2 cosh1 a ; g¼ 2 ða 1Þ3=2 a ¼ a=b > 1 ðprolateÞ; a ½cos1 a að1 a2 Þ1=2 ; g¼ ð1 a2 Þ3=2 a ¼ a=bo1 ðoblateÞ: ð9Þ
Fig. 3. Effective elastic properties in the surface zone with flattened cells ða ¼ 1=2Þ as a function of collagen volumetric fraction: (a) E11 =E0 ; (b) E22 =E0 ; (c) n23 ; (d) n12 :
J.Z. Wu, W. Herzog / Journal of Biomechanics 35 (2002) 931–942
936
It can be seen that a ¼ 1 (spherical inclusion) is a singularity in Eq. (9). The components of Eshelby’s tensor (Eshelby, 1957) at a ¼ 1 are given as S1111 ¼ S2222 ¼ S3333 ¼
If the thermal dynamic restrictions (11) are satisfied, the cell–collagen–matrix composite will be mechanically stable. In the following numerical simulations, condition (11) is used to check the mechanical stability of the cartilage composite.
S1122 ¼ S2233 ¼ S3311
2.2. Numerical tests
S1212 ¼ S2323 ¼ S3131
7 5n0 ; 15ð1 n0 Þ 5n0 1 ; ¼ 15ð1 n0 Þ 4 5n0 : ¼ 15ð1 n0 Þ
ð10Þ
2.1.3. Mechanical stability of the cell–collagen–matrix composite Neglecting heat transfer, the thermal dynamic restriction of the constitutive relation requires ds : de > 0; which means that the mechanical work must be positive during the deformation. In terms of the transversely isotropic elastic material, the thermal dynamic restrictions are expressed as E11 ; E22 ; m12 ; m23 > 0; jn23 jo1;
jn12 joðE11 =E22 Þ1=2 ;
1 n223 2n12 n23 2n12 n21 n23 > 0:
ð11Þ
We used Qiu and Weng’s (1990) solution to estimate the homogenized elastic modulus of cartilage. Cartilage was represented by a four-phasic composite, composed of one matrix (volumetric concentration cm ), one cell inclusion (volumetric concentration cc ), and two collagen fiber inclusions, i.e., vertical and horizontal fiber inclusions (volumetric concentration cf ¼ cfv þ cfh ). The horizontal and vertical fiber inclusions were represented by using two extremes of the aspect ratio of spheroidal inclusions, afh ¼ 0 and afv -N; respectively. The effective elastic material properties in the superficial layer, in the middle layer, and in the deep layer of the articular cartilage were calculated. In the superficial layer, collagen fibers were assumed to be parallel to the contact surface ðcf ¼ cfh Þ; and the shape of the chondrocytes was assumed to change from flat to
Fig. 4. Effective elastic properties in the surface zone with spherical cells ða ¼ 1Þ as a function of collagen volumetric fraction: (a) E11 =E0 ; (b) E22 =E0 ; (c) n23 ; (d) n12 :
J.Z. Wu, W. Herzog / Journal of Biomechanics 35 (2002) 931–942
spherical (ac ¼ 1=10; 1/2, and 1); in the middle zone, 1/3 of the fibers were assumed vertical to the contact surface and 2/3 of the fibers were assumed parallel to the contact surface [i.e., cfh ¼ ð2=3Þcf and cfv ¼ ð1=3Þcf ]. The shape of the chondrocytes was assumed to vary around spherical (ac ¼ 1=2; 1, and 3/2). In the deep zone, collagen fibers were assumed to be vertical to the contact surface ðcf ¼ cfv Þ; and the shape of the chondrocytes was assumed to change from spherical to columnar (ac ¼ 1; 2, and 10). The proteoglycan matrix, collagen fibers, and chondrocytes were assumed to be isotropic. Young’s moduli and Poisson’s ratios for cartilage matrix, collagen fibers, and chondrocytes were taken from published experimental data (e.g., Athanasiou et al., 1991; Shin and Athanasiou, 1997; Pins et al., 1997). They were: E0 ¼ 0:20 MPa; n0 ¼ 0:05; Ef ¼ 10:0 MPa; nf ¼ 0:30; and Ec ¼ 1:0 kPa; nc ¼ 0:40; respectively. The transversely isotropic material parameters, l; n; k; m; and p; are related to the isotropic material parameters by l ¼ K 23 m; n ¼ K þ 43 m; and m ¼ p ¼ m with K ¼ E=½3ð1 2nÞ and m ¼ E=½2ð1 þ nÞ being the conventional bulk modulus and shear modulus, respectively. The effective elastic moduli in the vertical and horizontal directions, E11 and E22 ; i.e., the directions
937
parallel and normal to the contact surface, respectively, the shear moduli, m23 and m12 ; and Poisson’s ratios, n23 and n12 ; were predicted as a function of the volumetric concentration of the collagen fibers. The aspect ratio of the cells was assumed to be 1/10, 1/2, and 1 (i.e., the shape of the cells varied from flattened to spherical) in the surface zone (Figs. 2, 3 and 4), 1/2 and 1 in the middle zone (Figs. 5 and 6), and 1, 2, and 10 (i.e., the shape of the cells varied from spherical to column like) in the deep zone (Figs. 7 and 8). The calculations were stopped if the thermal dynamic restrictions (11) were violated, i.e., the material became mechanically unstable.
3. Results In the surface zone, changes in the cell volumetric fraction of thin-flattened cells ðac ¼ 1=10Þ have virtually no effect on the horizontal elastic modulus, E22 ; (Fig. 2b) and the major shear modulus, m23 (results not shown). However, E22 and m23 decrease greatly as the volumetric concentration of the cells increases when the shape of the cells is round, i.e., ac is close to 1 (Figs. 3b and 4b). The vertical elastic modulus, E11 ; and the shear
Fig. 5. Effective elastic properties in the middle zone with flattened cells ða ¼ 1=2Þ as a function of collagen volumetric fraction: (a) E11 =E0 ; (b) E22 =E0 ; (c) n23 ; (d) n12 :
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J.Z. Wu, W. Herzog / Journal of Biomechanics 35 (2002) 931–942
Fig. 6. Effective elastic properties in the middle zone with spherical cells ða ¼ 1Þ as a function of collagen volumetric fraction: (a) E11 =E0 ; (b) E22 =E0 ; (c) n23 ; (d) n12 :
modulus, m12 (results not shown), decrease greatly with increasing cell volumetric fraction, independent of the shape of the chondrocytes (Figs. 2a, 3a, and 4a). When ac > 1=2; Poisson’s ratio n23 becomes negative for cc > 10% (Figs. 3c and 4c). In the middle zone, the material is stable when the shape of the chondrocytes is between mildly flattened and spherical, i.e., ac is between 1/2 and 1, (Figs. 5 and 6); the material becomes unstable when the cell shape becomes column like (i.e., ac > 3=2; results not shown). When the shape of the cells is spherical, a small volumetric concentration will result in a reduction in E22 (Fig. 6b). In the deep zone, spherical-like ðac ¼ 1; 2Þ cell inclusions cause instability of the material, i.e., E11 and E22 become negative (Figs. 7a, b), or the thermal dynamic restrictions (11) are not satisfied (Fig. 7). The existence of column-like cells reinforces slightly the material, i.e., E11 and E22 increase as the volumetric concentration increases (Figs. 8a, b). A large cell volumetric concentration (e.g., > 15%) will result in a material unstability (Fig. 8b), i.e., the thermal dynamic restrictions (11) will be violated. The shape of the cell inclusions has little influence on the shear moduli, m23 and m12 (results not shown).
4. Discussion and conclusion Collagen fibers are structural elements in articular cartilage. The structure of the collagen network is thought to be related to the mechanical stability of the tissue (Aspden and Hukins, 1981; Minns and Steven, 1977). The surface layer of cartilage is like the wall of a pressure vessel that is designed to withstand the swelling pressure in the tissue; the collagen fibers are oriented to achieve an optimal tangential stiffness of the tissue. In the deep layer, collagen fibers are oriented, likely, to optimize (a) the ties to the underlying calcified tissue and (b) the stiffness normal to the contact surface. The intermediate layer allows for a change in the orientation of the collagen fibers without discontinuity. Chondrocytes have different shapes and different volumetric concentrations in the different layers of articular cartilage. A given chondrocyte shape produces different effects on the global material properties, depending on the structure of the collagen fiber network. The shape and volumetric concentration of chondrocytes in articular cartilage appear to be related to the mechanical stability of the matrix. The present study was intended to investigate the dependence of the material properties of articular cartilage on the combination of the collagen
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Fig. 7. Effective elastic properties in the deep zone with spherical cells ða ¼ 1Þ as a function of collagen volumetric fraction: (a) E11 =E0 ; (b) E22 =E0 ; (c) n23 ; (d) n12 :
fiber network structure and the distributed chondrocyte structure. Our analysis suggests that chondrocyte shape and volumetric fraction influence material properties and mechanical stability of articular cartilage considerably. Our simulations suggest that a soft, spheroidal inclusion in a fiber-reinforced structure, such as articular cartilage with collagen fibers and chondrocytes, gives different material properties depending on the shape of the spheroidal inclusions. If the long axis of the inclusions is parallel to the collagen fibers, as in the deep zone, the soft inclusions increase the stiffness of the composite in the fiber direction, and reduce the stiffness of the composite in the direction normal to the fibers (Figs. 2a, b and 8a,b). This result is caused in this transversely isotropic material because the soft inclusions reduce the stiffness parameters, k; l and n; non-uniformly; as a consequence, Young’s modulus in the direction of the long axis of the inclusions, E11 ; is reinforced (Eq. (2)). In an isotropic matrix, stiffness parameters, k; l and n; would be reduced uniformly by the soft inclusions, and Young’s moduli in both directions, E11 and E22 ; would be reduced. Our analysis suggests that fiber orientation and shape of the chondrocytes in the surface and deep
zones in articular cartilage are intimately related and are likely adapted to optimize the mechanical stability and load carrying capacity of the structure. A comparison of the predicted Young’s modulus normal to the contact surface from the superficial zone (Fig. 2) with that from the deep zone (Fig. 8) shows an increase by a factor of 10–50 (Fig. 9a). This prediction agrees well with experimental measurements by Schinagl et al. (1997) who reported that the compressive elastic modulus increases with depth in bovine articular cartilage, from 0:08 MPa in the surface layer to 2:10 MPa in the deep layer (i.e., by a factor of about 26). According to our predictions, the tangential (in the direction parallel to the contact surface) Young’s modulus for a typical articular cartilage increases from the deep to the surface zone by 50–150% (Figs. 2, 8 and 9b). We have not yet found experimental measurements in the literature for comparison with this prediction. Our prediction that Young’s modulus in the surface layer and parallel to the contact surface, E22 ; is approximately 5–20 times of that normal to the contact surface, E11 (Fig. 2), agrees well with experimental measurements (Mow et al., 2000). Mow et al.
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Fig. 8. Effective elastic properties in the deep zone with narrow column-like cells ða ¼ 10Þ as a function of collagen volumetric fraction: (a) E11 =E0 ; (b) E22 =E0 ; (c) n23 ; (d) n12 :
(2000) determined the tensile Young’s modulus using an indentation test. They found that Young’s moduli parallel and normal to the contact surface are approximately 3.33–6:44 MPa and 0.33–0:36 MPa; respectively. In our analysis, collagen fibers were assumed to have an isotropic material property with the same stiffness in tension and compression. This assumption differs from that proposed by Soulhat et al. (1999) who used a material model of collagen fibers with zero stiffness in compression, because of the structural instability of thin, isolated rod-like fibers in compression. However, such an interpretation ignores the fact that collagen fibers are not isolated but are embedded in matrix material that provides stability to the collagen fibers, similar to the stability provided by cement to steel rods in reinforced concrete structures. Our modelling is vindicated by the observations that the compressive modulus (normal to the contact surface) in articular cartilage reaches its maximum in the deep zone (Schinagl et al., 1997) where the collagen fibers are oriented normal to the contact surface. This prediction could not be made in previous models (e.g., Soulhat et al., 1999; Li et al., 1999), except by changing the
material properties of the cartilage model without a direct structural analog. Here, the structural analog directly provides the non-uniformities in the material properties as observed in experiments. Our results indicate that the matrix is mechanically stable even when the volumetric concentration of chondrocytes reaches 20% in the surface zone (Fig. 2). In the deep zone, the matrix becomes mechanically unstable when the volumetric concentration of chondrocytes is over 10% (Fig. 8). This result suggests that, in order to maintain mechanical stability of the articular cartilage, the volumetric concentration of chondrocytes should decrease from the surface to the deep zone, and should not exceed certain limiting values. This prediction agrees well with the experimental observation that cell volumetric concentrations in articular cartilage increase from the deep to the surface zone by a factor of about three (Wong et al., 1997). In numerical approximations, the stiffness matrix of the composite material may turn out to be asymmetric when the number of inclusions is more than two (Qiu and Weng, 1990). In order to be thermodynamically consistent, we enforced the stiffness matrix to be
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Acknowledgements This study was supported by The Alberta Heritage Foundation for Medical Research, the Medical Research Council of Canada, and The Arthritis Society of Canada.
References
Fig. 9. The present model predicts that Young’s modulus of the articular cartilage normal to the contact surface [(a) E11 =E0 ] increases and that parallel to the contact surface [(b) E22 =E0 ] decreases from the surface to the deep layer.
symmetric by using the average value of l and l 0 in the present investigation. Articular cartilage is a biphasic material composed of a solid, elastic phase and a fluid phase (Mow et al., 1980). The homogenized elastic property obtained in the present study can be included into poroelastic models to account for the effects of collagen fibers and chondrocytes. The oriented and distributed collagen fibers and chondrocytes might have effects on the hydrostatic permeability of articular cartilage which were not included in the present approach. The homogenization approach used in the present study (Qiu and Weng, 1990) is based on infinitesimal deformation theory. A correlation of theoretical and experimental studies on cartilage suggests that the infinitesimal theory is adequate for local compressive strains of up to around 20–30% (Mow et al., 1980; Kwan et al., 1990). For large deformations, hyperelastic constitutive models (Holmes 1986; Holmes and Mow, 1990) have been used instead of the linear elastic relations for the solid phase.
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