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Determine the equilibrium mechanical properties of articular cartilage from the short-term indentation response
Journal of Biomechanics 48 (2015) 176–180
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Short communication
Determine the equilibrium mechanical properties of articular cartilage from the short-term indentation response Xingyu Chen, Brandon K. Zimmerman, X. Lucas Lu n Department of Mechanical Engineering, University of Delaware, Newark, Delaware
art ic l e i nf o
a b s t r a c t
Article history: Accepted 31 October 2014
Indentation testing is widely used to evaluate the mechanical properties of articular cartilage. However, most curve-fitting solutions for indentation analysis require the deformation data of cartilage at the equilibrium state, which often takes the tissue hours to reach. The lengthy testing time reduces the efficiency of indentation, increases the chance of tissue deterioration, and prevents in vivo applications. To overcome these limitations, a novel technique based on principal component analysis (PCA) was developed in this study, which can predict the full indentation creep curve based on the first few minutes’ deformation history and the principal components. The accuracy of this technique was confirmed using the indentation data from 40 temporomandibular joint condylar cartilage samples and 17 bovine knee joint samples. The mechanical properties determined by biphasic curve-fitting using predicted and experimental data are in good agreement, with the difference between the two less than 5%. For TMJ and knee cartilages, it is found that any number of full tests beyond eight will not lead to any increase larger than 1% in the accuracy, indicating a low sample number required for prediction. In addition, the principal components of indentation creep curves are consistent for the same type of cartilage tested with identical protocols, but significantly different between two distinct cartilages. Therefore PCA may also represent a new method to compare the mechanical behaviors of different cartilages, as it avoids the assumptions associated with mechanical constitutive models and relies purely on the experimental data. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Articular cartilage Indentation Creep test Biphasic theory Principal component analysis Mechanical properties
1. Introduction Indentation testing is widely used to determine the mechanical properties of articular cartilage (Mow et al., 2005). Since the tissue is left untouched on the bone and tested in its natural state without disturbing the structure and pre-stress in the solid matrix, indentation serves as a major technique for the in situ or potentially in vivo evaluation of cartilage mechanical properties. As the testing device only records the history of tissue deformation or force response, mechanical properties of the tissue have to be obtained by analyzing the experimental data with a proper constitutive model for cartilage. Hayes et al. (Hayes et al., 1972) developed an indentation solution based on linear elastic theory, which correlates the Young's modulus and Poisson's ratio with indentation force and deformation by a closed-form equation. However, one of the two properties, usually Poisson's ratio, has to be assumed a priori to determine the other property using this equation (Hoch et al., 1983; Mow et al., 1989). Later a technique n Correspondence to: Cartilage Bioengineering Laboratory Department of Mechanical Engineering University of Delaware 130 Academy Street SPL 126 Newark, DE 19716. Tel.: þ (302) 831 3581; fax: þ(302) 831 3619. E-mail address: [email protected] (X.L. Lu).
http://dx.doi.org/10.1016/j.jbiomech.2014.10.036 0021-9290/& 2014 Elsevier Ltd. All rights reserved.
utilizing multiple creep tests on a sample with different-sized indenter tips was developed which can extract both elastic properties of cartilage (Jin and Lewis, 2004). Another widely used indentation solution was developed by Mow et al. based on biphasic theory and the Hayes elastic solution (Mow et al., 1980; Mak et al., 1987; Mow et al., 1989), which can simultaneously determine the aggregate modulus, shear modulus and permeability of the tissue using a single indentation creep test (Athanasiou et al., 1991). With the advent of finite-element software and advanced porous media theories, complex constitutive models such as triphasic mixture theory are also used to analyze indentation data (Le and Fleming, 2008; Lu et al., 2010). Most of these analytical solutions for indentation require the equilibrium deformation data for curve-fitting, since the deformation-load correlation at equilibrium stage (with no fluid flow) can be defined by the closed-form Hayes solution (Hayes et al., 1972; Mow et al., 1989; Jin and Lewis, 2004; Lu et al., 2004). Due to the significant viscoelastic behavior of cartilage, however, it may take up to several hours for the tissue to reach a final steady state, where the characteristic time of creep is defined as t¼a2/Hank (a ¼indenter tip size, Ha ¼tissue modulus, and k¼ hydraulic permeability) (Spilker et al., 1992; Bae et al., 2006; Han et al., 2011). Such a long testing time reduces the efficiency of indentation and
X. Chen et al. / Journal of Biomechanics 48 (2015) 176–180
hinders in vivo applications. For example, creep testing of five regions on a small animal joint could take a dozen hours (Lu et al., 2009), which considerably increases the possibility of tissue degeneration during testing. The primary objective of this study is to develop a data processing technique using principal component analysis (PCA) which can predict the full indentation creep curve based on the transient data obtained in the first few minutes of indentation testing. The accuracy of the prediction is verified using experimental data from two types of articular cartilage - bovine knee cartilage and condylar cartilage from the porcine temporomandibular joint (TMJ). The mechanical properties determined by biphasic theory based on predicted curves are compared with those from full experimental data in order to validate the accuracy of this method.
2. Materials and methods
177
and matrix PC. 2
t^k þ 1 6^ 6 tk þ 2 6 6 ⋮ 4 t^ n
2 PC 1k þ 1 PC 2k þ 1 7 6 PC PC 2k þ 2 1k þ 2 7 6 76 7 6 ⋮ 5 4 PC 1n PC 2n 3
⋯ ⋱ ⋯
32 c^ 1 6 PC mk þ 2 7 76 c^ 2 76 76 ⋮ ⋮ 54 PC mn c^ m
PC mk þ 1
3 7 m 7 7 or tðk ^ þ 1 : nÞ ¼ ∑ c^ i PC i ðk þ1 : nÞ 7 i¼1 5
ð4Þ
Thus PCA of the full creep curves from a small group of samples can generate the principal component matrix, and then the long-term creep data of the other samples can be predicted by their short-term response using this matrix. 2.3. Accuracy of predicted data To test the accuracy of the predicted curve, eight sets of full indentation data were randomly selected for each type of cartilage to generate the corresponding principal components, and the first 10 min’ data from the unselected samples were used to predict their long-term response with PCA. The predicted curves were directly compared with the actual long-term experimental data. Moreover, the mechanical properties (aggregate modulus, Poisson's ratio, permeability) were obtained for both the predicted curve and the actual experimental data using a biphasic curve-fitting program (Mow et al., 1989). The agreement between the two sets of mechanical properties was then examined (Martin Bland and Altman, 1986).
2.1. Indentation testing Indentation creep tests were performed on condylar cartilage from porcine TMJ and bovine knee cartilage, as described in previous studies (Lu et al., 2004; Lu et al., 2009). Briefly, seventeen 2 cm 2 cm rectangular cartilage-bone blocks were harvested from the trochlear groove of mature bovine knee joints. Samples were mounted onto a step-loading indentation device equipped with a rigid flat-ended porous–permeable indenter tip (ϕ¼ 2.1 mm). At the start of the creep test, a 50 mN tare load was applied for 0.5 h, followed by a 200 mN step load for another 1 h to generate the creep data. Additionally, eight TMJs were harvested from mature porcine heads, and five regions (anterior, posterior, central, medial, and lateral) on the condylar head were indented by a custom-built micro-indenter with a porouspermeable indenter tip (ϕ ¼1.6 mm), by the same loading protocol detailed for bovine cartilage (Lu et al., 2009).
2.2. Principal component analysis The creep displacement of each sample was first resampled at 1 Hz by linear interpolation and denoted as an n 1 vector t. Vectors from m samples were further combined into an m n matrix. Principal component analysis (PCA) (Jackson, 1991) was then conducted on this matrix without centering, which generated m principal components. Each obtained principal component is an n 1 unit vector, denoted as PC i . Based on the PCA definition, the creep curve (vector t) can be decomposed by the principal component matrix PC as 3 2 PC PC 21 11 t1 6 t2 7 6 PC 12 PC 22 6 7 6 6 7¼6 ⋮ 4 ⋮ 5 6 4 PC 1n PC 2n tn 2
⋯ ⋱ ⋯
3 PC m1 2 c1 PC m2 7 c2 76 76 6 ⋮ 7 54 ⋮ PC mn cm
3 7 m 7 7 or t ¼ ∑ ci PC i where ci ¼ t U PC i 5 i¼1
ð1Þ
Here ci is the two norm of vector t's projection on PC i . We now hypothesize that the principal components are consistent for the same type of cartilage tested with an identical protocol. Therefore the creep curves of the other samples, which are not initially included in the m samples for PCA, can also be decomposed by the above principal components PC. To verify this assumption, we performed PCA on 50 different combinations of five indentation creep curves that were randomly selected from either bovine knee joint samples or TMJ samples, i.e., 50 PCA for each type of cartilage. The variances of 50 obtained PCs at each time point (n total points) were calculated to determine the consistency of PCs. Based on the PCA consistency assumption, the short-term creep displacement ^ of a new sample can be decomposed by the principal component matrix PC as (t) 2
3 2 PC 11 PC 21 t^ 1 6 ^ 7 6 PC PC 22 12 6 t2 7 6 6 76 6 ⋮ 7 6 ⋮ 4 5 4 PC 1k PC 2k t^ k
⋯ ⋱ ⋯
32 c^ 1 6 PC m2 7 76 c^ 2 76 6 ⋮ ⋮ 7 54 PC mk c^ m
PC m1
3. Results To understand the effectiveness of PCA for predicting cartilage indentation data, PCA of all the creep curves were performed to obtain the contribution to the total variance of each principal component (PC) (Fig. 1). The first PC alone contributes 98.5% and 99.8% to the total variance for TMJ and knee cartilage, respectively. The first and second PCs contribute over 99.5% of the variance for both tissues. Thus, in the following PC consistency analysis, only the first two PCs were presented, as the third and higher PCs contribute little to the total variance. For each type of tissue, average and standard deviation of the first two PCs from 50 analyzed groups are shown in Fig. 2. The standard deviations are close to 0 for PC1 at all points, i.e., PC1 remains constant for any five randomly selected indentation curves. The standard deviations of PC2 are larger than PC1, but PC2 explains only 1.4% and 0.1% of variance for TMJ and knee cartilages, respectively. Therefore, it can be concluded that the principal components are consistent for the same type of cartilage. In contrast, indentation curves of TMJ and knee cartilage have drastically different PCs in terms of magnitude and distribution over time (Fig. 2). Two typical experimental creep curves for each cartilage are plotted in Fig. 3 together with the PCA prediction. The first 10 min’ data and the PCs based on eight creep curves are able to provide an accurate prediction of the long-term indentation responses for both types of cartilage. The average difference of equilibrium deformation
3 7 m 7 7 or tð1 ^ : kÞ ¼ ∑ c^ i PC i ð1 : kÞ; k o n: 7 i¼1 5
ð2Þ
^ : kÞ contains only k components (k 5n) since it represents only Here vector tð1 the first k seconds of a creep curve. PC i ð1 : kÞ denotes the first k components of PC i . If the shortened principal component matrix is denoted as B, the coefficient vector c^ can be calculated as 1 ^ : kÞ c^ ¼ BT B BT tð1
ð3Þ
Note that c^ is the coefficient vector with m components. The long-term creep ^ þ 1 : nÞ, can be estimated using c^ deformation of the sample after k seconds, tðk
Fig. 1. Contribution of principal components to the total variance of indentation creep curves. The first principal component (PC1) explains 98.5% and 99.8% of the total variance for TMJ cartilage and knee joint cartilage, respectively. The third principal component contributes less than 0.5%.
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between experiment and PCA is 4:5 7 1:1% for bovine knee cartilage and 5:1 7 1:3% for porcine TMJ cartilage. This difference was also determined for varying numbers of creep curves used to obtain the PCs (Fig. S1). For both types of cartilage, PCs from 11 or more creep curves generate similar prediction errors. The mechanical properties determined by biphasic curvefitting based on the experimental and predicted data are summarized in Fig. 4. The magnitudes of all properties are consistent with those reported in literature (Athanasiou et al., 1991, Lu et al., 2004, Lu et al., 2009). One sample t-test showed that the parameters from the two methods are not statistically identical. However, the data points are clustered around the line Y¼ X in each plot, and the 2 2 coefficient of determination, R2 1 ∑ yi f i =∑ yi y , is close to 1 for all three mechanical properties, displaying excellent agreement between both data sets. The errors (mean 7 standard deviation) of the estimated mechanical properties of TMJ condylar cartilage are 3:3% 7 3:3%, 2:0 72:0% and 3:2% 7 3:2% for aggregate modulus, permeability and shear modulus, respectively. For bovine knee joint cartilage the corresponding errors are 2:2% 7 3:1%, 7:4% 712:4% and 3:1% 74:3%; respectively. The mean difference of the aggregate moduli is close to zero, and the 95% confidence interval of the differences (mean 7 2 SD) is less than 20% of the average, showing excellent agreement between the two data sets (Fig. 5). A linear regression analysis showed that the error is not dependent on the magnitude of the aggregate modulus (Martin Bland and Altman, 1986).
features of the creep curves rather than the absolute values of equilibrium deformation. The results further showed that the predicted creep curves match the actual experimental data well, and the mechanical properties determined from the two sets of curves agree with each other. The first PC, explaining over 95% of the variance, is highly dependent on the ultrastructure and composition of cartilage, while the stiffness of the tissue is mainly represented by the coefficient vector c. The explained variance only implies the indentation data can be well-recovered in the respect of the Frobenius norm, but the lost information remains critical in determining the mechanical properties. Furthermore, the length of base data is more critical in minimizing error than the number of full tests (Fig. S1). When 600s base data is used, the error from four full tests is much lower than the combination of 20 full tests and 300s base data. Therefore, in most testing situations, PCs from five full tests should be able to provide reasonable prediction of the creep curves. If high precision is required, a longer base data (e.g., 5 extra minutes) can be more efficient in reducing the errors than increasing the number of full tests. Meanwhile, for both knee cartilage and TMJ condylar cartilage in our experiments, the increase in accuracy slows as the number of full tests exceeds five.
4. Discussion PCA, for the first time, is employed to analyze cartilage indentation creep curves for two types of cartilage with different ultrastructure and mechanical properties (Mow et al., 2005; Lu et al., 2009). The creep deformation of cartilage under indentation can be accurately decomposed by PCs, and the first two PCs contribute over 99.5% of the variance. More importantly, the PCs are consistent for the same type of cartilage tested with identical protocols, which provides the theoretical foundation to predict the full deformation curve using PCA based on the transient data of the first few minutes. It is important to note that the PCs are normalized unit vectors. They are dependent on the transient
Fig. 3. PCA predicted indentation creep curves of two samples for each type of cartilage, using the experimental data in the first 600 seconds. The corresponding full experimental curves are also plotted for comparison.
Fig. 2. Average principal components of 50 random combinations of 5 indentation creep curves for (a) TMJ condylar cartilage and (b) knee joint cartilage. The first principal component (PC1) of any five creep curves remains highly consistent with low standard deviations at all time. The second principal component (PC2) is not as consistent as PC1.
X. Chen et al. / Journal of Biomechanics 48 (2015) 176–180
179
Fig. 4. Comparison of mechanical properties determined by indentation curve-fitting using experimental and PCA predicted curves, (a-b) aggregate modulus, ! (c-d) permeability, and (e-f) shear modulus. Line Y ¼ X is plotted for reference. R2 value is calculated as R2 1 ∑ðyi f i Þ2 =∑ðyi yÞ2 :
Fig. 5. The difference of aggregate modulus determined from predicted and experimental data is plotted against their average value. For most samples, the difference is smaller than 5% of the average value, and the mean of the difference is close to zero ( 0.003 Mpa for TMJ and 0.012 Mpa for knee). Linear regression showed no significant correlation between the errors and average modulus.
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Conflict of interest statement All authors state that they have no conflicts of interest.
Acknowledgment DOD W81XWH-13-1-0148 and Musculoskeletal Transplant Foundation.
Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jbiomech.2014.10.036. References Athanasiou, K.A., Rosenwasser, M.P., Buckwalter, J.A., Malinin, T.I., Mow, V.C., 1991. Interspecies comparisons of in situ intrinsic mechanical properties of distal femoral cartilage. J. Orthoped. Res. 9, 330–340. Bae, W.C., Lewis, C.W., Levenston, M.E., Sah, R.L., 2006. Indentation testing of human articular cartilage: effects of probe tip geometry and indentation depth on intra-tissue strain. J. Biomech. 39, 1039–1047. Han, L., Frank, E.H., Greene, J.J., Lee, H.Y., Hung, H.H., Grodzinsky, A.J., Ortiz, C., 2011. Time-dependent nanomechanics of cartilage. Biophys. J. 100, 1846–1854. Hayes, W., Keer, L., Herrmann, G., Mockros, L., 1972. A mathematical analysis for indentation tests of articular cartilage. J. Biomech. 5, 541–551.
Hoch, D.H., Grodzinsky, A.J., Koob, T.J., Albert, M.L., Eyre, D.R., 1983. Early changes in material properties of rabbit articular cartilage after meniscectomy. J. Orthoped. Res. 1, 4–12. Jackson, J.E., 1991. A user's guide to principal components. John Wiley & Sons, New York. Jin, H., Lewis, J.L., 2004. Determination of Poisson's ratio of articular cartilage by indentation using different-sized indenters. J. Biomech. Eng. 126, 138–145. Le, N.A.T., Fleming, B.C., 2008. Measuring fixed charge density of goat articular cartilage using indentation methods and biochemical analysis. J. Biomech. 41, 715–720. Lu, X., Mow, V., Guo, X., 2009. Proteoglycans and mechanical behavior of condylar cartilage. J. Dent. Res. 88, 244–248. Lu, X.L., Sun, D.D., Guo, X.E., Chen, F.H., Lai, W.M., Mow, V.C., 2004. Indentation determined mechanoelectrochemical properties and fixed charge density of articular cartilage. Annal Biomed. Eng. 32, 370–379. Lu, X.L., Wan, L.Q., Edward Guo, X., Mow, V.C., 2010. A linearized formulation of triphasic mixture theory for articular cartilage, and its application to indentation analysis. J. Biomech. 43, 673–679. Mak, A.F., Lai, W.M., Mow, V.C., 1987. Biphasic indentation of articular cartilage–I. Theoretical analysis. J. Biomech. 20, 703–714. Martin Bland, J., Altman, D., 1986. Statistical methods for assessing agreement between two methods of clinical measurement. The Lancet 327, 307–310. Mow, V., Gibbs, M., Lai, W., Zhu, W., Athanasiou, K., 1989. Biphasic indentation of articular cartilage—II. A numerical algorithm and an experimental study. J. Biomech. 22, 853–861. Mow, V., Kuei, S., Lai, W., Armstrong, C., 1980. Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments. J. Biomech. Eng. 102, 73–84. Mow, V.C., Gu, W.Y., Chen, F.H., 2005. Structure and function of articular cartilage and meniscus. In: Mow, V.C., Huiskes, R. (Eds.), Basic Orthopaedic Biomechanics and Mechano-Biology. Lippincott Williams & Wilkins, Philadelphia, pp. 181–258. Spilker, R.L., Suh, J.K., Mow, V.C., 1992. A finite element analysis of the indentation stress-relaxation response of linear biphasic articular cartilage. J. Biomech. Eng. 114, 191–201.
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Short communication
Determine the equilibrium mechanical properties of articular cartilage from the short-term indentation response Xingyu Chen, Brandon K. Zimmerman, X. Lucas Lu n Department of Mechanical Engineering, University of Delaware, Newark, Delaware
art ic l e i nf o
a b s t r a c t
Article history: Accepted 31 October 2014
Indentation testing is widely used to evaluate the mechanical properties of articular cartilage. However, most curve-fitting solutions for indentation analysis require the deformation data of cartilage at the equilibrium state, which often takes the tissue hours to reach. The lengthy testing time reduces the efficiency of indentation, increases the chance of tissue deterioration, and prevents in vivo applications. To overcome these limitations, a novel technique based on principal component analysis (PCA) was developed in this study, which can predict the full indentation creep curve based on the first few minutes’ deformation history and the principal components. The accuracy of this technique was confirmed using the indentation data from 40 temporomandibular joint condylar cartilage samples and 17 bovine knee joint samples. The mechanical properties determined by biphasic curve-fitting using predicted and experimental data are in good agreement, with the difference between the two less than 5%. For TMJ and knee cartilages, it is found that any number of full tests beyond eight will not lead to any increase larger than 1% in the accuracy, indicating a low sample number required for prediction. In addition, the principal components of indentation creep curves are consistent for the same type of cartilage tested with identical protocols, but significantly different between two distinct cartilages. Therefore PCA may also represent a new method to compare the mechanical behaviors of different cartilages, as it avoids the assumptions associated with mechanical constitutive models and relies purely on the experimental data. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Articular cartilage Indentation Creep test Biphasic theory Principal component analysis Mechanical properties
1. Introduction Indentation testing is widely used to determine the mechanical properties of articular cartilage (Mow et al., 2005). Since the tissue is left untouched on the bone and tested in its natural state without disturbing the structure and pre-stress in the solid matrix, indentation serves as a major technique for the in situ or potentially in vivo evaluation of cartilage mechanical properties. As the testing device only records the history of tissue deformation or force response, mechanical properties of the tissue have to be obtained by analyzing the experimental data with a proper constitutive model for cartilage. Hayes et al. (Hayes et al., 1972) developed an indentation solution based on linear elastic theory, which correlates the Young's modulus and Poisson's ratio with indentation force and deformation by a closed-form equation. However, one of the two properties, usually Poisson's ratio, has to be assumed a priori to determine the other property using this equation (Hoch et al., 1983; Mow et al., 1989). Later a technique n Correspondence to: Cartilage Bioengineering Laboratory Department of Mechanical Engineering University of Delaware 130 Academy Street SPL 126 Newark, DE 19716. Tel.: þ (302) 831 3581; fax: þ(302) 831 3619. E-mail address: [email protected] (X.L. Lu).
http://dx.doi.org/10.1016/j.jbiomech.2014.10.036 0021-9290/& 2014 Elsevier Ltd. All rights reserved.
utilizing multiple creep tests on a sample with different-sized indenter tips was developed which can extract both elastic properties of cartilage (Jin and Lewis, 2004). Another widely used indentation solution was developed by Mow et al. based on biphasic theory and the Hayes elastic solution (Mow et al., 1980; Mak et al., 1987; Mow et al., 1989), which can simultaneously determine the aggregate modulus, shear modulus and permeability of the tissue using a single indentation creep test (Athanasiou et al., 1991). With the advent of finite-element software and advanced porous media theories, complex constitutive models such as triphasic mixture theory are also used to analyze indentation data (Le and Fleming, 2008; Lu et al., 2010). Most of these analytical solutions for indentation require the equilibrium deformation data for curve-fitting, since the deformation-load correlation at equilibrium stage (with no fluid flow) can be defined by the closed-form Hayes solution (Hayes et al., 1972; Mow et al., 1989; Jin and Lewis, 2004; Lu et al., 2004). Due to the significant viscoelastic behavior of cartilage, however, it may take up to several hours for the tissue to reach a final steady state, where the characteristic time of creep is defined as t¼a2/Hank (a ¼indenter tip size, Ha ¼tissue modulus, and k¼ hydraulic permeability) (Spilker et al., 1992; Bae et al., 2006; Han et al., 2011). Such a long testing time reduces the efficiency of indentation and
X. Chen et al. / Journal of Biomechanics 48 (2015) 176–180
hinders in vivo applications. For example, creep testing of five regions on a small animal joint could take a dozen hours (Lu et al., 2009), which considerably increases the possibility of tissue degeneration during testing. The primary objective of this study is to develop a data processing technique using principal component analysis (PCA) which can predict the full indentation creep curve based on the transient data obtained in the first few minutes of indentation testing. The accuracy of the prediction is verified using experimental data from two types of articular cartilage - bovine knee cartilage and condylar cartilage from the porcine temporomandibular joint (TMJ). The mechanical properties determined by biphasic theory based on predicted curves are compared with those from full experimental data in order to validate the accuracy of this method.
2. Materials and methods
177
and matrix PC. 2
t^k þ 1 6^ 6 tk þ 2 6 6 ⋮ 4 t^ n
2 PC 1k þ 1 PC 2k þ 1 7 6 PC PC 2k þ 2 1k þ 2 7 6 76 7 6 ⋮ 5 4 PC 1n PC 2n 3
⋯ ⋱ ⋯
32 c^ 1 6 PC mk þ 2 7 76 c^ 2 76 76 ⋮ ⋮ 54 PC mn c^ m
PC mk þ 1
3 7 m 7 7 or tðk ^ þ 1 : nÞ ¼ ∑ c^ i PC i ðk þ1 : nÞ 7 i¼1 5
ð4Þ
Thus PCA of the full creep curves from a small group of samples can generate the principal component matrix, and then the long-term creep data of the other samples can be predicted by their short-term response using this matrix. 2.3. Accuracy of predicted data To test the accuracy of the predicted curve, eight sets of full indentation data were randomly selected for each type of cartilage to generate the corresponding principal components, and the first 10 min’ data from the unselected samples were used to predict their long-term response with PCA. The predicted curves were directly compared with the actual long-term experimental data. Moreover, the mechanical properties (aggregate modulus, Poisson's ratio, permeability) were obtained for both the predicted curve and the actual experimental data using a biphasic curve-fitting program (Mow et al., 1989). The agreement between the two sets of mechanical properties was then examined (Martin Bland and Altman, 1986).
2.1. Indentation testing Indentation creep tests were performed on condylar cartilage from porcine TMJ and bovine knee cartilage, as described in previous studies (Lu et al., 2004; Lu et al., 2009). Briefly, seventeen 2 cm 2 cm rectangular cartilage-bone blocks were harvested from the trochlear groove of mature bovine knee joints. Samples were mounted onto a step-loading indentation device equipped with a rigid flat-ended porous–permeable indenter tip (ϕ¼ 2.1 mm). At the start of the creep test, a 50 mN tare load was applied for 0.5 h, followed by a 200 mN step load for another 1 h to generate the creep data. Additionally, eight TMJs were harvested from mature porcine heads, and five regions (anterior, posterior, central, medial, and lateral) on the condylar head were indented by a custom-built micro-indenter with a porouspermeable indenter tip (ϕ ¼1.6 mm), by the same loading protocol detailed for bovine cartilage (Lu et al., 2009).
2.2. Principal component analysis The creep displacement of each sample was first resampled at 1 Hz by linear interpolation and denoted as an n 1 vector t. Vectors from m samples were further combined into an m n matrix. Principal component analysis (PCA) (Jackson, 1991) was then conducted on this matrix without centering, which generated m principal components. Each obtained principal component is an n 1 unit vector, denoted as PC i . Based on the PCA definition, the creep curve (vector t) can be decomposed by the principal component matrix PC as 3 2 PC PC 21 11 t1 6 t2 7 6 PC 12 PC 22 6 7 6 6 7¼6 ⋮ 4 ⋮ 5 6 4 PC 1n PC 2n tn 2
⋯ ⋱ ⋯
3 PC m1 2 c1 PC m2 7 c2 76 76 6 ⋮ 7 54 ⋮ PC mn cm
3 7 m 7 7 or t ¼ ∑ ci PC i where ci ¼ t U PC i 5 i¼1
ð1Þ
Here ci is the two norm of vector t's projection on PC i . We now hypothesize that the principal components are consistent for the same type of cartilage tested with an identical protocol. Therefore the creep curves of the other samples, which are not initially included in the m samples for PCA, can also be decomposed by the above principal components PC. To verify this assumption, we performed PCA on 50 different combinations of five indentation creep curves that were randomly selected from either bovine knee joint samples or TMJ samples, i.e., 50 PCA for each type of cartilage. The variances of 50 obtained PCs at each time point (n total points) were calculated to determine the consistency of PCs. Based on the PCA consistency assumption, the short-term creep displacement ^ of a new sample can be decomposed by the principal component matrix PC as (t) 2
3 2 PC 11 PC 21 t^ 1 6 ^ 7 6 PC PC 22 12 6 t2 7 6 6 76 6 ⋮ 7 6 ⋮ 4 5 4 PC 1k PC 2k t^ k
⋯ ⋱ ⋯
32 c^ 1 6 PC m2 7 76 c^ 2 76 6 ⋮ ⋮ 7 54 PC mk c^ m
PC m1
3. Results To understand the effectiveness of PCA for predicting cartilage indentation data, PCA of all the creep curves were performed to obtain the contribution to the total variance of each principal component (PC) (Fig. 1). The first PC alone contributes 98.5% and 99.8% to the total variance for TMJ and knee cartilage, respectively. The first and second PCs contribute over 99.5% of the variance for both tissues. Thus, in the following PC consistency analysis, only the first two PCs were presented, as the third and higher PCs contribute little to the total variance. For each type of tissue, average and standard deviation of the first two PCs from 50 analyzed groups are shown in Fig. 2. The standard deviations are close to 0 for PC1 at all points, i.e., PC1 remains constant for any five randomly selected indentation curves. The standard deviations of PC2 are larger than PC1, but PC2 explains only 1.4% and 0.1% of variance for TMJ and knee cartilages, respectively. Therefore, it can be concluded that the principal components are consistent for the same type of cartilage. In contrast, indentation curves of TMJ and knee cartilage have drastically different PCs in terms of magnitude and distribution over time (Fig. 2). Two typical experimental creep curves for each cartilage are plotted in Fig. 3 together with the PCA prediction. The first 10 min’ data and the PCs based on eight creep curves are able to provide an accurate prediction of the long-term indentation responses for both types of cartilage. The average difference of equilibrium deformation
3 7 m 7 7 or tð1 ^ : kÞ ¼ ∑ c^ i PC i ð1 : kÞ; k o n: 7 i¼1 5
ð2Þ
^ : kÞ contains only k components (k 5n) since it represents only Here vector tð1 the first k seconds of a creep curve. PC i ð1 : kÞ denotes the first k components of PC i . If the shortened principal component matrix is denoted as B, the coefficient vector c^ can be calculated as 1 ^ : kÞ c^ ¼ BT B BT tð1
ð3Þ
Note that c^ is the coefficient vector with m components. The long-term creep ^ þ 1 : nÞ, can be estimated using c^ deformation of the sample after k seconds, tðk
Fig. 1. Contribution of principal components to the total variance of indentation creep curves. The first principal component (PC1) explains 98.5% and 99.8% of the total variance for TMJ cartilage and knee joint cartilage, respectively. The third principal component contributes less than 0.5%.
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between experiment and PCA is 4:5 7 1:1% for bovine knee cartilage and 5:1 7 1:3% for porcine TMJ cartilage. This difference was also determined for varying numbers of creep curves used to obtain the PCs (Fig. S1). For both types of cartilage, PCs from 11 or more creep curves generate similar prediction errors. The mechanical properties determined by biphasic curvefitting based on the experimental and predicted data are summarized in Fig. 4. The magnitudes of all properties are consistent with those reported in literature (Athanasiou et al., 1991, Lu et al., 2004, Lu et al., 2009). One sample t-test showed that the parameters from the two methods are not statistically identical. However, the data points are clustered around the line Y¼ X in each plot, and the 2 2 coefficient of determination, R2 1 ∑ yi f i =∑ yi y , is close to 1 for all three mechanical properties, displaying excellent agreement between both data sets. The errors (mean 7 standard deviation) of the estimated mechanical properties of TMJ condylar cartilage are 3:3% 7 3:3%, 2:0 72:0% and 3:2% 7 3:2% for aggregate modulus, permeability and shear modulus, respectively. For bovine knee joint cartilage the corresponding errors are 2:2% 7 3:1%, 7:4% 712:4% and 3:1% 74:3%; respectively. The mean difference of the aggregate moduli is close to zero, and the 95% confidence interval of the differences (mean 7 2 SD) is less than 20% of the average, showing excellent agreement between the two data sets (Fig. 5). A linear regression analysis showed that the error is not dependent on the magnitude of the aggregate modulus (Martin Bland and Altman, 1986).
features of the creep curves rather than the absolute values of equilibrium deformation. The results further showed that the predicted creep curves match the actual experimental data well, and the mechanical properties determined from the two sets of curves agree with each other. The first PC, explaining over 95% of the variance, is highly dependent on the ultrastructure and composition of cartilage, while the stiffness of the tissue is mainly represented by the coefficient vector c. The explained variance only implies the indentation data can be well-recovered in the respect of the Frobenius norm, but the lost information remains critical in determining the mechanical properties. Furthermore, the length of base data is more critical in minimizing error than the number of full tests (Fig. S1). When 600s base data is used, the error from four full tests is much lower than the combination of 20 full tests and 300s base data. Therefore, in most testing situations, PCs from five full tests should be able to provide reasonable prediction of the creep curves. If high precision is required, a longer base data (e.g., 5 extra minutes) can be more efficient in reducing the errors than increasing the number of full tests. Meanwhile, for both knee cartilage and TMJ condylar cartilage in our experiments, the increase in accuracy slows as the number of full tests exceeds five.
4. Discussion PCA, for the first time, is employed to analyze cartilage indentation creep curves for two types of cartilage with different ultrastructure and mechanical properties (Mow et al., 2005; Lu et al., 2009). The creep deformation of cartilage under indentation can be accurately decomposed by PCs, and the first two PCs contribute over 99.5% of the variance. More importantly, the PCs are consistent for the same type of cartilage tested with identical protocols, which provides the theoretical foundation to predict the full deformation curve using PCA based on the transient data of the first few minutes. It is important to note that the PCs are normalized unit vectors. They are dependent on the transient
Fig. 3. PCA predicted indentation creep curves of two samples for each type of cartilage, using the experimental data in the first 600 seconds. The corresponding full experimental curves are also plotted for comparison.
Fig. 2. Average principal components of 50 random combinations of 5 indentation creep curves for (a) TMJ condylar cartilage and (b) knee joint cartilage. The first principal component (PC1) of any five creep curves remains highly consistent with low standard deviations at all time. The second principal component (PC2) is not as consistent as PC1.
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Fig. 4. Comparison of mechanical properties determined by indentation curve-fitting using experimental and PCA predicted curves, (a-b) aggregate modulus, ! (c-d) permeability, and (e-f) shear modulus. Line Y ¼ X is plotted for reference. R2 value is calculated as R2 1 ∑ðyi f i Þ2 =∑ðyi yÞ2 :
Fig. 5. The difference of aggregate modulus determined from predicted and experimental data is plotted against their average value. For most samples, the difference is smaller than 5% of the average value, and the mean of the difference is close to zero ( 0.003 Mpa for TMJ and 0.012 Mpa for knee). Linear regression showed no significant correlation between the errors and average modulus.
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Conflict of interest statement All authors state that they have no conflicts of interest.
Acknowledgment DOD W81XWH-13-1-0148 and Musculoskeletal Transplant Foundation.
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