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A method to estimate the elastic properties of the extracellular matrix of articular cartilage
ARTICLE IN PRESS
Journal of Biomechanics 37 (2004) 401–404
Short communication
A method to estimate the elastic properties of the extracellular matrix of articular cartilage Salvatore Federicoa, Walter Herzogb,*, John Z. Wub, Guido La Rosaa b
a Dipartimento di Ingegneria Industriale e Meccanica, Facolta" di Ingegneria, Universita" degli Studi di Catania, Catania, Italy Faculty of Kinesiology, Human Performance Laboratory, The University of Calgary, 2500 University Drive NW, Calgary, Canada AB T2N 1N4
Received 7 October 2002; accepted 8 July 2003
Abstract In this work we propose a method to estimate the elastic properties of the extracellular matrix of articular cartilage, once the elastic properties of the chondrocytes and the whole tissue are known. The influence of the elastic properties of the tissue and the cell concentration on the estimated elastic properties of the matrix are investigated. r 2003 Published by Elsevier Ltd. Keywords: Articular cartilage; Matrix; Chondrocyte; Homogenization procedure; Osteoarthritis
1. Introduction Articular cartilage is a multi-phasic composite material. One of the simplest models describes articular cartilage as a composite made of cells (chondrocytes), an extracellular matrix (consisting of proteoglycans and collagen fibres), and water. For numerical calculations, and comparisons to macroscopic experimental data (e.g. compression tests), it is opportune to apply a homogenization procedure in order to obtain a mixture that represents the average properties of the tissue. In the work of Wu et al. (1999), the homogenized elastic properties of articular cartilage were calculated starting from the estimated elastic properties of the matrix and the measured properties of isolated cells. The matrix properties were approximated by values from whole tissue tests (Athanasiou et al., 1991). In fact, while it is possible to measure the properties of the cells alone, it has not been possible to obtain a cartilage sample containing matrix only and no cells. Here, we address the problem of calculating the elastic properties of the cartilage matrix, using the procedure of Wu et al. (1999), and changing the initial conditions of the differential problem in which the unknowns are two *Corresponding author. Tel.: +1-403-220-8525; fax: +1-403-2843553. E-mail address: [email protected] (W. Herzog). 0021-9290/$ - see front matter r 2003 Published by Elsevier Ltd. doi:10.1016/S0021-9290(03)00280-X
independent elastic constants of the homogenized mixture.
2. Methods Wu et al. (1999) showed that two independent elastic constants (shear modulus m and bulk modulus K) can be found for a uniform and dilute solution of spherical cells (of elastic constants mc and Kc ) within a homogeneous matrix (of elastic constants mm and Km ) using Eshelby’s Principle (Eshelby, 1956; Christensen, 1991): m mm ð15Km þ 20mm Þðmc mm Þmm ¼ ; c ð6Km þ 12mm Þmc þ ð9Km þ 8mm Þmm
ð1aÞ
K Km ð3Km þ 4mm ÞðKc Km Þ ¼ ; 3Kc þ 4mm c
ð1bÞ
where c is the concentration of cells within the matrix. According to McLaughlin (1977) and Norris (1985), the concentration of cells can be expressed as a function of the parameter G : cðGÞ ¼ 1 eG ; with G ¼ 0 and G-N representing the states of 0% and 100% cellvolumetric concentration, respectively. For dilute solutions (i.e. when the parameter G takes a value g-0), the concentration can be expanded with the Taylor formula: cðgÞ ¼ 1 eg Dg:
ð2Þ
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A non-dilute solution with a certain concentration cðGÞ and shear and bulk moduli equal to mðGÞ and KðGÞ; respectively, can be considered as mechanically equivalent to a homogeneous matrix having moduli equal to mm and Km : Further, a non-dilute solution of concentration cðG þ gÞ (and moduli mðG þ gÞ and KðG þ gÞ) can be thought of as an ‘equivalent homogeneous matrix’ (equivalent to a solution of concentration cðGÞ; with moduli mðGÞ and KðGÞ), to which cells were added for a concentration equal to cðgÞ: With these considerations, one can substitute c with cðgÞDg; m and K; respectively, with mðG þ gÞ and KðG þ gÞ; mm and Km ; respectively, with mðGÞ and KðGÞ in Eqs. (1a) and (1b), and they become mðG þ gÞ mðGÞ g ð15KðGÞ þ 20mðGÞÞðmc mðGÞÞmðGÞ ¼ ð3aÞ ð6KðGÞ þ 12mðGÞÞmc þ ð9KðGÞ þ 8mðGÞÞmðGÞ KðG þ gÞ KðGÞ ð3KðGÞ þ 4mðGÞÞðKc KðGÞÞ ¼ : g 3Kc þ 4mðGÞ
ð3bÞ
Considering the limit g-0 on the left-hand side of both Eqs. (3a) and (3b), one obtains the derivatives m0 and K 0 calculated in G; and thus, the following system of non-linear differential equations: ð15K þ 20mÞðmc mÞm m0 ¼ ; ð6K þ 12mÞmc þ ð9K þ 8mÞm ð3K þ 4mÞðKc KÞ K0 ¼ ð4Þ 3Kc þ 4m that can be solved numerically if the elastic constants of the cells (mc and Kc ) are known. Wu et al. (1999), set the initial conditions such that, at zero cell concentration, m and K must have the values of the matrix:
In this way, after finding the functions m and K; it is possible to obtain their values at G ¼ 0 (that means c ¼ 0; i.e., pure matrix): mð0Þ ¼ mm and Kð0Þ ¼ Km ; the elastic moduli of the matrix. The differential system (4), with the initial conditions (6), was solved numerically with the commercial software Mathematica 4.0 and the values for the shear modulus mð0Þ ¼ mm and bulk modulus Kð0Þ ¼ Km for the matrix were obtained for several values of the actual cell-volumetric concentration ct involved in the initial conditions (6), from 0% to 30%, that safely covers experimental observations in a variety of different species (unpublished observations). The outputs mm and Km were converted into the normalized matrix Poisson’s ratio nm =nt and the normalized matrix Young’s modulus Em =Et : In order to study the sensitivity of nm =nt and Em =Et with respect to the values of the elastic constants of the tissue, they were calculated parametrically. The elastic constants, Et and nt ; that appear in the initial conditions (6) were defined as follows: nt ¼ a n# t 8aA½0:25; 3:5 ; Et ¼ b E# t 8bA½0:25; 5 :
ð7Þ
here n# t ¼ 0:098 and E# t ¼ 0:686 MPa are average experimental elastic constants of articular cartilage from human lateral condyles (Athanasiou et al., 1991). As for the properties of the cells, that are input data for the differential system (4), they were assumed to be constant throughout the calculations and taken as nc ¼ 0:370 and Ec ¼ 1:160 kPa (Shin and Athanasiou, 1997). The constants n and E for tissue and cells can be converted into a suitable input for Eqs. (4)–(6) by the general relations m ¼ E=2ð1 þ nÞ and K ¼ E=3ð1 2nÞ:
mð0Þ ¼ mm ; Kð0Þ ¼ Km :
ð5Þ
This assumption was acceptable, because there are no data about the elastic moduli of the matrix. However, the values used by Wu et al. (1999) for mm and Km are the tissue moduli mt and Kt ; respectively, determined experimentally by Athanasiou et al. (1991). Since the elastic moduli of the tissue are known from experiments, one can instead calculate the elastic moduli of the pure matrix. The unknowns of the problem then become mm and Km ; and the initial conditions (5) should be changed to mðGt Þ ¼ mt ¼ Et =2ð1 þ nt Þ; KðGt Þ ¼ Kt ¼ Et =3ð1 2nt Þ;
ð6Þ
where Gt is such that cðGt Þ ¼ ct is the actual cell concentration in the tissue and nt and Et are the Poisson’s ratio and the Young’s modulus of the tissue, respectively.
3. Results In Figs. 1–4 the behaviour of nm =nt and Em =Et as a function of the cell concentration ct ; parameterized by a and b; is shown. The normalized Poisson’s ratio of the matrix is weakly affected by variations in the Young’s modulus of the tissue (Fig. 2), but is greatly influenced by variations of the Poisson’s ratio of the tissue (Fig. 1). In Fig. 1, it is possible to see how nm =nt (and, as a consequence, nm ) can take negative values, which corresponds to mechanical instability of the tissue for those combinations of cell concentration and tissue and cell elastic properties. In contrast to the normalized Poisson’s ratio, the normalized Young’s modulus of the matrix is not very sensitive to variations of the elastic properties of the tissue (Figs. 3 and 4).
ARTICLE IN PRESS S. Federico et al. / Journal of Biomechanics 37 (2004) 401–404
Fig. 1. Normalized Poisson’s ratio of the matrix; aA½0:25; 3:5 ; b ¼ 1 (kept constant).
Fig. 2. Normalized Poisson’s ratio of the matrix; a ¼ 1 (kept constant), bA½0:25; 5 :
The behaviour of nm =nt and Em =Et shows an ‘inversion point’ with the parameter a (related to the Poisson’s ratio of the tissue, see Eq. (7)). The slope of nm =nt as a function of the cell concentration becomes positive for a greater than E2 (Fig. 1). As already evidenced, Em =Et is not greatly affected by variations of a and b; and it is always positive. Note that the slope gets smaller for values of a greater than E2.5. 4. Conclusions Figs. 1–4 illustrate the dependence of the mechanical properties of articular cartilage on the cell concentra-
403
Fig. 3. Normalized Young’s modulus of the matrix; aA½0:25; 3:5 ; b ¼ 1 (kept constant).
Fig. 4. Normalized Young’s modulus of the matrix; a ¼ 1 (kept constant), bA½0:25; 5 :
tion. Since the Young’s modulus of the cells is about three orders of magnitude less than the Young’s modulus of the tissue, and cell concentration can reach values close to 30% (unpublished observations), the matrix must be very stiff to keep the tissue stiffness similar to stiffness observed in articular cartilage with a low cell concentration. It has been impossible to directly measure the elastic moduli of the matrix of articular cartilage in experiments, because cartilage samples always contain chondrocytes. An indirect approach may include the measurement of the moduli of isolated matrix molecules, but such an
ARTICLE IN PRESS 404
S. Federico et al. / Journal of Biomechanics 37 (2004) 401–404
approach does not include the structural effects produced by the specific arrangement and interactions of the matrix molecules within the tissue. The present method gives a simple and immediate way to estimate the mechanical properties of the matrix of articular cartilage, once the properties of the cells and the tissue are known from experiment. The main limitation of this technique is that it features a homogenization procedure that starts from average tissue properties; therefore, only the average properties of the matrix can be found. Likely, the mechanical properties of the matrix of articular cartilage vary with the depth from the articular surface due to the nonuniform concentration and shape of chondrocytes, and the distribution and arrangement of collagen fibres (Clark et al., 2003; Wu and Herzog, 2002). Therefore, the current approach is useful when the average mechanical behaviour of articular cartilage is of interest. For the detailed local stress and strain within articular cartilage exposed to mechanical loading, a more refined approach is required.
Acknowledgements The Canadian Institute of Health Research (CIHR), the Arthritis Society of Canada.
Universita" degli Studi di Catania, Dipartimento di Scienze Chimiche, Cluster 26 P14 WP2.
References 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. Christensen, R.M., 1991. Mechanics of composite materials. Krieger, Malabar, FL, USA. Clark, A.L., Barclay, L.D., Matyas, J.R., Herzog, W., 2003. In-situ chondrocyte deformation with physiological compression of the feline patellofemoral joint. Journal of Biomechanics 36, 553–568. Eshelby, J.D., 1956. The continuum theory of lattice defects. In: Seitz, F., Turnbull, D. (Eds.), Progress in Solid State Physics, Vol. 3. Academic Press, New York, pp. 79. McLaughlin, R., 1977. A study of the differential scheme for composite materials. International Journal of Engineering Science 15, 237–244. Norris, A.N., 1985. A differential scheme for the effective moduli of composites. Mechanical Material 4, 1–16. Shin, D., Athanasiou, K.A., 1997. Biomechanical properties of the individual cell. Transactions of the Orthopaedic Research Society 22 (1), 352. Wu, J.Z., Herzog, W., 2002. Elastic anisotropy of articular cartilage is associated with the micro-structures of collagen fibers and chondrocytes. Journal of Biomechanics 35, 931–942. Wu, J.Z., Herzog, W., Epstein, M., 1999. Modelling of location- and time-dependent deformation of chondrocytes during cartilage loading. Journal of Biomechanics 32, 563–572.
Journal of Biomechanics 37 (2004) 401–404
Short communication
A method to estimate the elastic properties of the extracellular matrix of articular cartilage Salvatore Federicoa, Walter Herzogb,*, John Z. Wub, Guido La Rosaa b
a Dipartimento di Ingegneria Industriale e Meccanica, Facolta" di Ingegneria, Universita" degli Studi di Catania, Catania, Italy Faculty of Kinesiology, Human Performance Laboratory, The University of Calgary, 2500 University Drive NW, Calgary, Canada AB T2N 1N4
Received 7 October 2002; accepted 8 July 2003
Abstract In this work we propose a method to estimate the elastic properties of the extracellular matrix of articular cartilage, once the elastic properties of the chondrocytes and the whole tissue are known. The influence of the elastic properties of the tissue and the cell concentration on the estimated elastic properties of the matrix are investigated. r 2003 Published by Elsevier Ltd. Keywords: Articular cartilage; Matrix; Chondrocyte; Homogenization procedure; Osteoarthritis
1. Introduction Articular cartilage is a multi-phasic composite material. One of the simplest models describes articular cartilage as a composite made of cells (chondrocytes), an extracellular matrix (consisting of proteoglycans and collagen fibres), and water. For numerical calculations, and comparisons to macroscopic experimental data (e.g. compression tests), it is opportune to apply a homogenization procedure in order to obtain a mixture that represents the average properties of the tissue. In the work of Wu et al. (1999), the homogenized elastic properties of articular cartilage were calculated starting from the estimated elastic properties of the matrix and the measured properties of isolated cells. The matrix properties were approximated by values from whole tissue tests (Athanasiou et al., 1991). In fact, while it is possible to measure the properties of the cells alone, it has not been possible to obtain a cartilage sample containing matrix only and no cells. Here, we address the problem of calculating the elastic properties of the cartilage matrix, using the procedure of Wu et al. (1999), and changing the initial conditions of the differential problem in which the unknowns are two *Corresponding author. Tel.: +1-403-220-8525; fax: +1-403-2843553. E-mail address: [email protected] (W. Herzog). 0021-9290/$ - see front matter r 2003 Published by Elsevier Ltd. doi:10.1016/S0021-9290(03)00280-X
independent elastic constants of the homogenized mixture.
2. Methods Wu et al. (1999) showed that two independent elastic constants (shear modulus m and bulk modulus K) can be found for a uniform and dilute solution of spherical cells (of elastic constants mc and Kc ) within a homogeneous matrix (of elastic constants mm and Km ) using Eshelby’s Principle (Eshelby, 1956; Christensen, 1991): m mm ð15Km þ 20mm Þðmc mm Þmm ¼ ; c ð6Km þ 12mm Þmc þ ð9Km þ 8mm Þmm
ð1aÞ
K Km ð3Km þ 4mm ÞðKc Km Þ ¼ ; 3Kc þ 4mm c
ð1bÞ
where c is the concentration of cells within the matrix. According to McLaughlin (1977) and Norris (1985), the concentration of cells can be expressed as a function of the parameter G : cðGÞ ¼ 1 eG ; with G ¼ 0 and G-N representing the states of 0% and 100% cellvolumetric concentration, respectively. For dilute solutions (i.e. when the parameter G takes a value g-0), the concentration can be expanded with the Taylor formula: cðgÞ ¼ 1 eg Dg:
ð2Þ
ARTICLE IN PRESS 402
S. Federico et al. / Journal of Biomechanics 37 (2004) 401–404
A non-dilute solution with a certain concentration cðGÞ and shear and bulk moduli equal to mðGÞ and KðGÞ; respectively, can be considered as mechanically equivalent to a homogeneous matrix having moduli equal to mm and Km : Further, a non-dilute solution of concentration cðG þ gÞ (and moduli mðG þ gÞ and KðG þ gÞ) can be thought of as an ‘equivalent homogeneous matrix’ (equivalent to a solution of concentration cðGÞ; with moduli mðGÞ and KðGÞ), to which cells were added for a concentration equal to cðgÞ: With these considerations, one can substitute c with cðgÞDg; m and K; respectively, with mðG þ gÞ and KðG þ gÞ; mm and Km ; respectively, with mðGÞ and KðGÞ in Eqs. (1a) and (1b), and they become mðG þ gÞ mðGÞ g ð15KðGÞ þ 20mðGÞÞðmc mðGÞÞmðGÞ ¼ ð3aÞ ð6KðGÞ þ 12mðGÞÞmc þ ð9KðGÞ þ 8mðGÞÞmðGÞ KðG þ gÞ KðGÞ ð3KðGÞ þ 4mðGÞÞðKc KðGÞÞ ¼ : g 3Kc þ 4mðGÞ
ð3bÞ
Considering the limit g-0 on the left-hand side of both Eqs. (3a) and (3b), one obtains the derivatives m0 and K 0 calculated in G; and thus, the following system of non-linear differential equations: ð15K þ 20mÞðmc mÞm m0 ¼ ; ð6K þ 12mÞmc þ ð9K þ 8mÞm ð3K þ 4mÞðKc KÞ K0 ¼ ð4Þ 3Kc þ 4m that can be solved numerically if the elastic constants of the cells (mc and Kc ) are known. Wu et al. (1999), set the initial conditions such that, at zero cell concentration, m and K must have the values of the matrix:
In this way, after finding the functions m and K; it is possible to obtain their values at G ¼ 0 (that means c ¼ 0; i.e., pure matrix): mð0Þ ¼ mm and Kð0Þ ¼ Km ; the elastic moduli of the matrix. The differential system (4), with the initial conditions (6), was solved numerically with the commercial software Mathematica 4.0 and the values for the shear modulus mð0Þ ¼ mm and bulk modulus Kð0Þ ¼ Km for the matrix were obtained for several values of the actual cell-volumetric concentration ct involved in the initial conditions (6), from 0% to 30%, that safely covers experimental observations in a variety of different species (unpublished observations). The outputs mm and Km were converted into the normalized matrix Poisson’s ratio nm =nt and the normalized matrix Young’s modulus Em =Et : In order to study the sensitivity of nm =nt and Em =Et with respect to the values of the elastic constants of the tissue, they were calculated parametrically. The elastic constants, Et and nt ; that appear in the initial conditions (6) were defined as follows: nt ¼ a n# t 8aA½0:25; 3:5 ; Et ¼ b E# t 8bA½0:25; 5 :
ð7Þ
here n# t ¼ 0:098 and E# t ¼ 0:686 MPa are average experimental elastic constants of articular cartilage from human lateral condyles (Athanasiou et al., 1991). As for the properties of the cells, that are input data for the differential system (4), they were assumed to be constant throughout the calculations and taken as nc ¼ 0:370 and Ec ¼ 1:160 kPa (Shin and Athanasiou, 1997). The constants n and E for tissue and cells can be converted into a suitable input for Eqs. (4)–(6) by the general relations m ¼ E=2ð1 þ nÞ and K ¼ E=3ð1 2nÞ:
mð0Þ ¼ mm ; Kð0Þ ¼ Km :
ð5Þ
This assumption was acceptable, because there are no data about the elastic moduli of the matrix. However, the values used by Wu et al. (1999) for mm and Km are the tissue moduli mt and Kt ; respectively, determined experimentally by Athanasiou et al. (1991). Since the elastic moduli of the tissue are known from experiments, one can instead calculate the elastic moduli of the pure matrix. The unknowns of the problem then become mm and Km ; and the initial conditions (5) should be changed to mðGt Þ ¼ mt ¼ Et =2ð1 þ nt Þ; KðGt Þ ¼ Kt ¼ Et =3ð1 2nt Þ;
ð6Þ
where Gt is such that cðGt Þ ¼ ct is the actual cell concentration in the tissue and nt and Et are the Poisson’s ratio and the Young’s modulus of the tissue, respectively.
3. Results In Figs. 1–4 the behaviour of nm =nt and Em =Et as a function of the cell concentration ct ; parameterized by a and b; is shown. The normalized Poisson’s ratio of the matrix is weakly affected by variations in the Young’s modulus of the tissue (Fig. 2), but is greatly influenced by variations of the Poisson’s ratio of the tissue (Fig. 1). In Fig. 1, it is possible to see how nm =nt (and, as a consequence, nm ) can take negative values, which corresponds to mechanical instability of the tissue for those combinations of cell concentration and tissue and cell elastic properties. In contrast to the normalized Poisson’s ratio, the normalized Young’s modulus of the matrix is not very sensitive to variations of the elastic properties of the tissue (Figs. 3 and 4).
ARTICLE IN PRESS S. Federico et al. / Journal of Biomechanics 37 (2004) 401–404
Fig. 1. Normalized Poisson’s ratio of the matrix; aA½0:25; 3:5 ; b ¼ 1 (kept constant).
Fig. 2. Normalized Poisson’s ratio of the matrix; a ¼ 1 (kept constant), bA½0:25; 5 :
The behaviour of nm =nt and Em =Et shows an ‘inversion point’ with the parameter a (related to the Poisson’s ratio of the tissue, see Eq. (7)). The slope of nm =nt as a function of the cell concentration becomes positive for a greater than E2 (Fig. 1). As already evidenced, Em =Et is not greatly affected by variations of a and b; and it is always positive. Note that the slope gets smaller for values of a greater than E2.5. 4. Conclusions Figs. 1–4 illustrate the dependence of the mechanical properties of articular cartilage on the cell concentra-
403
Fig. 3. Normalized Young’s modulus of the matrix; aA½0:25; 3:5 ; b ¼ 1 (kept constant).
Fig. 4. Normalized Young’s modulus of the matrix; a ¼ 1 (kept constant), bA½0:25; 5 :
tion. Since the Young’s modulus of the cells is about three orders of magnitude less than the Young’s modulus of the tissue, and cell concentration can reach values close to 30% (unpublished observations), the matrix must be very stiff to keep the tissue stiffness similar to stiffness observed in articular cartilage with a low cell concentration. It has been impossible to directly measure the elastic moduli of the matrix of articular cartilage in experiments, because cartilage samples always contain chondrocytes. An indirect approach may include the measurement of the moduli of isolated matrix molecules, but such an
ARTICLE IN PRESS 404
S. Federico et al. / Journal of Biomechanics 37 (2004) 401–404
approach does not include the structural effects produced by the specific arrangement and interactions of the matrix molecules within the tissue. The present method gives a simple and immediate way to estimate the mechanical properties of the matrix of articular cartilage, once the properties of the cells and the tissue are known from experiment. The main limitation of this technique is that it features a homogenization procedure that starts from average tissue properties; therefore, only the average properties of the matrix can be found. Likely, the mechanical properties of the matrix of articular cartilage vary with the depth from the articular surface due to the nonuniform concentration and shape of chondrocytes, and the distribution and arrangement of collagen fibres (Clark et al., 2003; Wu and Herzog, 2002). Therefore, the current approach is useful when the average mechanical behaviour of articular cartilage is of interest. For the detailed local stress and strain within articular cartilage exposed to mechanical loading, a more refined approach is required.
Acknowledgements The Canadian Institute of Health Research (CIHR), the Arthritis Society of Canada.
Universita" degli Studi di Catania, Dipartimento di Scienze Chimiche, Cluster 26 P14 WP2.
References 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. Christensen, R.M., 1991. Mechanics of composite materials. Krieger, Malabar, FL, USA. Clark, A.L., Barclay, L.D., Matyas, J.R., Herzog, W., 2003. In-situ chondrocyte deformation with physiological compression of the feline patellofemoral joint. Journal of Biomechanics 36, 553–568. Eshelby, J.D., 1956. The continuum theory of lattice defects. In: Seitz, F., Turnbull, D. (Eds.), Progress in Solid State Physics, Vol. 3. Academic Press, New York, pp. 79. McLaughlin, R., 1977. A study of the differential scheme for composite materials. International Journal of Engineering Science 15, 237–244. Norris, A.N., 1985. A differential scheme for the effective moduli of composites. Mechanical Material 4, 1–16. Shin, D., Athanasiou, K.A., 1997. Biomechanical properties of the individual cell. Transactions of the Orthopaedic Research Society 22 (1), 352. Wu, J.Z., Herzog, W., 2002. Elastic anisotropy of articular cartilage is associated with the micro-structures of collagen fibers and chondrocytes. Journal of Biomechanics 35, 931–942. Wu, J.Z., Herzog, W., Epstein, M., 1999. Modelling of location- and time-dependent deformation of chondrocytes during cartilage loading. Journal of Biomechanics 32, 563–572.