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Research Papers

An Alternative Method to Characterize the Quasi-Static, Nonlinear Material Properties of Murine Articular Cartilage

[+] Author and Article Information
Alexander Kotelsky

Department of Biomedical Engineering,
University of Rochester,
207 Goergen Hall, Box 270168,
Rochester, NY 14627
e-mail: akotelsk@ur.rochester.edu

Chandler W. Woo

Department of Biomedical Engineering,
University of Rochester,
207 Goergen Hall, Box 270168,
Rochester, NY 14627
e-mail: cwoo8@u.rochester.edu

Luis F. Delgadillo

Department of Biomedical Engineering,
University of Rochester,
207 Goergen Hall, Box 270168,
Rochester, NY 14627
e-mail: ldelgadi@ur.rochester.edu

Michael S. Richards

Department of Surgery,
School of Medicine and Dentistry,
University of Rochester Medical Center,
601 Elmwood Avenue, Rm 2.4153,
Rochester, NY 14627
e-mail: Michael_Richards@urmc.rochester.edu

Mark R. Buckley

Department of Biomedical Engineering,
University of Rochester,
207 Goergen Hall, Box 270168,
Rochester, NY 14627
e-mail: mark.buckley@rochester.edu

1Corresponding author.

Manuscript received December 14, 2016; final manuscript received September 19, 2017; published online October 31, 2017. Assoc. Editor: James C. Iatridis.

J Biomech Eng 140(1), 011007 (Oct 31, 2017) (9 pages) Paper No: BIO-16-1515; doi: 10.1115/1.4038147 History: Received December 14, 2016; Revised September 19, 2017

With the onset and progression of osteoarthritis (OA), articular cartilage (AC) mechanical properties are altered. These alterations can serve as an objective measure of tissue degradation. Although the mouse is a common and useful animal model for studying OA, it is extremely challenging to measure the mechanical properties of murine AC due to its small size (thickness < 50 μm). In this study, we developed novel and direct approach to independently quantify two quasi-static mechanical properties of mouse AC: the load-dependent (nonlinear) solid matrix Young's modulus (E) and drained Poisson's ratio (ν). The technique involves confocal microscope-based multiaxial strain mapping of compressed, intact murine AC followed by inverse finite element analysis (iFEA) to determine E and ν. Importantly, this approach yields estimates of E and ν that are independent of the initial guesses used for iterative optimization. As a proof of concept, mechanical properties of AC on the medial femoral condyles of wild-type mice were obtained for both trypsin-treated and control specimens. After proteolytic tissue degradation induced through trypsin treatment, a dramatic decrease in E was observed (compared to controls) at each of the three tested loading conditions. A significant decrease in ν due to trypsin digestion was also detected. These data indicate that the method developed in this study may serve as a valuable tool for comparative studies evaluating factors involved in OA pathogenesis using experimentally induced mouse OA models.

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References

Neogi, T. , 2013, “ The Epidemiology and Impact of Pain in Osteoarthritis,” Osteoarthritis Cartilage, 21(9), pp. 1145–1153. [CrossRef] [PubMed]
Kleemann, R. U. , Krocker, D. , Cedraro, A. , Tuischer, J. , and Duda, G. N. , 2005, “ Altered Cartilage Mechanics and Histology in Knee Osteoarthritis: Relation to Clinical Assessment (ICRS Grade),” Osteoarthritis Cartilage, 13(11), pp. 958–963. [CrossRef] [PubMed]
Setton, L. A. , Elliott, D. M. , and Mow, V. C. , 1999, “ Altered Mechanics of Cartilage With Osteoarthritis: Human Osteoarthritis and an Experimental Model of Joint Degeneration,” Osteoarthritis Cartilage, 7(1), pp. 2–14. [CrossRef] [PubMed]
Armstrong, C. G. , and Mow, V. C. , 1982, “ Variations in the Intrinsic Mechanical Properties of Human Articular Cartilage With Age, Degeneration, and Water Content,” J. Bone Jt. Surg. Am., 64(1), pp. 88–94. [CrossRef]
Wilusz, R. E. , Zauscher, S. , and Guilak, F. , 2013, “ Micromechanical Mapping of Early Osteoarthritic Changes in the Pericellular Matrix of Human Articular Cartilage,” Osteoarthritis Cartilage, 21(12), pp. 1895–1903. [CrossRef] [PubMed]
Rivers, P. A. , Rosenwasser, M. P. , Mow, V. C. , Pawluk, R. J. , Strauch, R. J. , Sugalski, M. T. , and Ateshian, G. A. , 2000, “ Osteoarthritic Changes in the Biochemical Composition of Thumb Carpometacarpal Joint Cartilage and Correlation With Biomechanical Properties,” J. Hand Surg. Am., 25(5), pp. 889–898. [CrossRef] [PubMed]
Glasson, S. S. , Blanchet, T. J. , and Morris, E. A. , 2007, “ The Surgical Destabilization of the Medial Meniscus (DMM) Model of Osteoarthritis in the 129/SvEv Mouse,” Osteoarthritis Cartilage, 15(9), pp. 1061–1069. [CrossRef] [PubMed]
Culley, K. L. , Dragomir, C. L. , Chang, J. , Wondimu, E. B. , Coico, J. , Plumb, D. A. , Otero, M. , and Goldring, M. B. , 2015, “ Mouse Models of Osteoarthritis: Surgical Model of Posttraumatic Osteoarthritis Induced by Destabilization of the Medial Meniscus,” Methods Mol. Biol., 1226, pp. 143–173. https://www.ncbi.nlm.nih.gov/pubmed/25331049 [PubMed]
David, M. A. , Smith, M. K. , Pilachowski, R. N. , White, A. T. , Locke, R. C. , and Price, C. , 2016, “ Early, Focal Changes in Cartilage Cellularity and Structure Following Surgically Induced Meniscal Destabilization in the Mouse,” J. Orthop. Res., 35(3), pp. 537–547. [CrossRef] [PubMed]
Kamekura, S. , Hoshi, K. , Shimoaka, T. , Chung, U. , Chikuda, H. , Yamada, T. , Uchida, M. , Ogata, N. , Seichi, A. , Nakamura, K. , and Kawaguchi, H. , 2005, “ Osteoarthritis Development in Novel Experimental Mouse Models Induced by Knee Joint Instability,” Osteoarthritis Cartilage, 13(7), pp. 632–641. [CrossRef] [PubMed]
McNulty, M. A. , Loeser, R. F. , Davey, C. , Callahan, M. F. , Ferguson, C. M. , and Carlson, C. S. , 2012, “ Histopathology of Naturally Occurring and Surgically Induced Osteoarthritis in Mice,” Osteoarthritis Cartilage, 20(8), pp. 949–956. [CrossRef] [PubMed]
Saamanen, A. K. , Salminen, H. J. , Dean, P. B. , De Crombrugghe, B. , Vuorio, E. I. , and Metsaranta, M. P. , 2000, “ Osteoarthritis-like Lesions in Transgenic Mice Harboring a Small Deletion Mutation in Type II Collagen Gene,” Osteoarthritis Cartilage, 8(4), pp. 248–257. [CrossRef] [PubMed]
Xu, L. , Flahiff, C. M. , Waldman, B. A. , Wu, D. , Olsen, B. R. , Setton, L. A. , and Li, Y. , 2003, “ Osteoarthritis-like Changes and Decreased Mechanical Function of Articular Cartilage in the Joints of Mice With the Chondrodysplasia Gene (Cho),” Arthritis Rheum., 48(9), pp. 2509–2518. [CrossRef] [PubMed]
Holt, D. W. , Henderson, M. L. , Stockdale, C. E. , Farrell, J. T. , Kooyman, D. L. , Bridgewater, L. C. , and Seegmiller, R. E. , 2012, “ Osteoarthritis-like Changes in the Heterozygous Sedc Mouse Associated With the HtrA1-Ddr2-Mmp-13 Degradative Pathway: A New Model of Osteoarthritis,” Osteoarthritis Cartilage, 20(5), pp. 430–439. [CrossRef] [PubMed]
Cao, L. , Youn, I. , Guilak, F. , and Setton, L. A. , 2006, “ Compressive Properties of Mouse Articular Cartilage Determined in a Novel Micro-Indentation Test Method and Biphasic Finite Element Model,” ASME J. Biomech. Eng., 128(5), pp. 766–771. [CrossRef]
Berteau, J. P. , Oyen, M. , and Shefelbine, S. J. , 2016, “ Permeability and Shear Modulus of Articular Cartilage in Growing Mice,” Biomech. Model. Mechanobiol., 15(1), pp. 205–212. [CrossRef] [PubMed]
Batista, M. A. , Nia, H. T. , Onnerfjord, P. , Cox, K. A. , Ortiz, C. , Grodzinsky, A. J. , Heinegard, D. , and Han, L. , 2014, “ Nanomechanical Phenotype of Chondroadherin-Null Murine Articular Cartilage,” Matrix Biol., 38, pp. 84–90. [CrossRef] [PubMed]
Doyran, B. , Tong, W. , Li, Q. , Jia, H. , Zhang, X. , Chen, C. , Enomoto-Iwamoto, M. , Lu, X. L. , Qin, L. , and Han, L. , 2016, “ Nanoindentation Modulus of Murine Cartilage: A Sensitive Indicator of the Initiation and Progression of Post-Traumatic Osteoarthritis,” Osteoarthritis Cartilage, 25(1), pp. 108–117. [CrossRef] [PubMed]
Azadi, M. , Nia, H. T. , Gauci, S. J. , Ortiz, C. , Fosang, A. J. , and Grodzinsky, A. J. , 2016, “ Wide Bandwidth Nanomechanical Assessment of Murine Cartilage Reveals Protection of Aggrecan Knock-In Mice From Joint-Overuse,” J. Biomech., 49(9), pp. 1634–1640. [CrossRef] [PubMed]
Nia, H. T. , Gauci, S. J. , Azadi, M. , Hung, H. H. , Frank, E. , Fosang, A. J. , Ortiz, C. , and Grodzinsky, A. J. , 2015, “ High-Bandwidth AFM-Based Rheology is a Sensitive Indicator of Early Cartilage Aggrecan Degradation Relevant to Mouse Models of Osteoarthritis,” J. Biomech., 48(1), pp. 162–165. [CrossRef] [PubMed]
Stolz, M. , Gottardi, R. , Raiteri, R. , Miot, S. , Martin, I. , Imer, R. , Staufer, U. , Raducanu, A. , Duggelin, M. , Baschong, W. , Daniels, A. U. , Friederich, N. F. , Aszodi, A. , and Aebi, U. , 2009, “ Early Detection of Aging Cartilage and Osteoarthritis in Mice and Patient Samples Using Atomic Force Microscopy,” Nat. Nanotechnol., 4(3), pp. 186–192. [CrossRef] [PubMed]
Moore, A. C. , DeLucca, J. F. , Elliott, D. M. , and Burris, D. L. , 2016, “ Quantifying Cartilage Contact Modulus, Tension Modulus, and Permeability With Hertzian Biphasic Creep,” ASME J. Tribol., 138(4), p. 0414051. [CrossRef]
Buckley, M. R. , Bonassar, L. J. , and Cohen, I. , 2013, “ Localization of Viscous Behavior and Shear Energy Dissipation in Articular Cartilage Under Dynamic Shear Loading,” ASME J. Biomech. Eng., 135(3), p. 31002. [CrossRef]
Henak, C. R. , Ross, K. A. , Bonnevie, E. D. , Fortier, L. A. , Cohen, I. , Kennedy, J. G. , and Bonassar, L. J. , 2016, “ Human Talar and Femoral Cartilage Have Distinct Mechanical Properties Near the Articular Surface,” J. Biomech., 49(14), pp. 3320–3327. [CrossRef] [PubMed]
Robinson, D. L. , Kersh, M. E. , Walsh, N. C. , Ackland, D. C. , de Steiger, R. N. , and Pandy, M. G. , 2016, “ Mechanical Properties of Normal and Osteoarthritic Human Articular Cartilage,” J. Mech. Behav. Biomed. Mater., 61, pp. 96–109. [CrossRef] [PubMed]
Silverberg, J. L. , Barrett, A. R. , Das, M. , Petersen, P. B. , Bonassar, L. J. , and Cohen, I. , 2014, “ Structure-Function Relations and Rigidity Percolation in the Shear Properties of Articular Cartilage,” Biophys. J., 107(7), pp. 1721–1730. [CrossRef] [PubMed]
Guterl, C. C. , Gardner, T. R. , Rajan, V. , Ahmad, C. S. , Hung, C. T. , and Ateshian, G. A. , 2009, “ Two-Dimensional Strain Fields on the Cross-Section of the Human Patellofemoral Joint Under Physiological Loading,” J. Biomech., 42(9), pp. 1275–1281. [CrossRef] [PubMed]
Park, S. , Costa, K. D. , Ateshian, G. A. , and Hong, K. S. , 2009, “ Mechanical Properties of Bovine Articular Cartilage Under Microscale Indentation Loading From Atomic Force Microscopy,” Proc. Inst. Mech. Eng., Part H, 223(3), pp. 339–347. [CrossRef]
Chen, X. , Zhou, Y. , Wang, L. , Santare, M. H. , Wan, L. Q. , and Lu, X. L. , 2016, “ Determining Tension-Compression Nonlinear Mechanical Properties of Articular Cartilage From Indentation Testing,” Ann. Biomed. Eng., 44(4), pp. 1148–1158. [CrossRef] [PubMed]
Mow, V. C. , Kuei, S. C. , Lai, W. M. , and Armstrong, C. G. , 1980, “ Biphasic Creep and Stress Relaxation of Articular Cartilage in Compression? Theory and Experiments,” ASME J. Biomech. Eng., 102(1), pp. 73–84. [CrossRef]
Ateshian, G. A. , Warden, W. H. , Kim, J. J. , Grelsamer, R. P. , and Mow, V. C. , 1997, “ Finite Deformation Biphasic Material Properties of Bovine Articular Cartilage From Confined Compression Experiments,” J. Biomech., 30(11–12), pp. 1157–1164. [CrossRef] [PubMed]
Schinagl, R. M. , Gurskis, D. , Chen, A. C. , and Sah, R. L. , 1997, “ Depth-Dependent Confined Compression Modulus of Full-Thickness Bovine Articular Cartilage,” J. Orthop. Res., 15(4), pp. 499–506. [CrossRef] [PubMed]
Wong, B. L. , and Sah, R. L. , 2010, “ Mechanical Asymmetry During Articulation of Tibial and Femoral Cartilages: Local and Overall Compressive and Shear Deformation and Properties,” J. Biomech., 43(9), pp. 1689–1695. [CrossRef] [PubMed]
Han, L. , Frank, E. H. , Greene, J. J. , Lee, H. Y. , Hung, H. H. , Grodzinsky, A. J. , and Ortiz, C. , 2011, “ Time-Dependent Nanomechanics of Cartilage,” Biophys. J., 100(7), pp. 1846–1854. [CrossRef] [PubMed]
Elliott, D. M. , Narmoneva, D. A. , and Setton, L. A. , 2002, “ Direct Measurement of the Poisson's Ratio of Human Patella Cartilage in Tension,” ASME J. Biomech. Eng., 124(2), pp. 223–228. [CrossRef]
Griffin, D. J. , Vicari, J. , Buckley, M. R. , Silverberg, J. L. , Cohen, I. , and Bonassar, L. J. , 2014, “ Effects of Enzymatic Treatments on the Depth-Dependent Viscoelastic Shear Properties of Articular Cartilage,” J. Orthop. Res., 32(12), pp. 1652–1657. [CrossRef] [PubMed]
Chen, A. C. , Nguyen, T. T. , and Sah, R. L. , 1997, “ Streaming Potentials During the Confined Compression Creep Test of Normal and Proteoglycan-Depleted Cartilage,” Ann. Biomed. Eng., 25(2), pp. 269–277. [CrossRef] [PubMed]
Bonassar, L. J. , Frank, E. H. , Murray, J. C. , Paguio, C. G. , Moore, V. L. , Lark, M. W. , Sandy, J. D. , Wu, J. J. , Eyre, D. R. , and Grodzinsky, A. J. , 1995, “ Changes in Cartilage Composition and Physical Properties Due to Stromelysin Degradation,” Arthritis Rheum., 38(2), pp. 173–183. [CrossRef] [PubMed]
Armstrong, C. G. , Lai, W. M. , and Mow, V. C. , 1984, “ An Analysis of the Unconfined Compression of Articular Cartilage,” ASME J. Biomech. Eng., 106(2), pp. 165–173. [CrossRef]
Schneider, C. A. , Rasband, W. S. , and Eliceiri, K. W. , 2012, “ NIH Image to ImageJ: 25 Years of Image Analysis,” Nat. Methods, 9(7), pp. 671–675. [CrossRef] [PubMed]
Maas, S. A. , Ellis, B. J. , Ateshian, G. A. , and Weiss, J. A. , 2012, “ FEBio: Finite Elements for Biomechanics,” ASME J. Biomech. Eng., 134(1), p. 011005. [CrossRef]
Wergedal, J. E. , Sheng, M. H. , Ackert-Bicknell, C. L. , Beamer, W. G. , and Baylink, D. J. , 2005, “ Genetic Variation in Femur Extrinsic Strength in 29 Different Inbred Strains of Mice is Dependent on Variations in Femur Cross-Sectional Geometry and Bone Density,” Bone, 36(1), pp. 111–122. [CrossRef] [PubMed]
Madden, R. , Han, S. K. , and Herzog, W. , 2013, “ Chondrocyte Deformation Under Extreme Tissue Strain in Two Regions of the Rabbit Knee Joint,” J. Biomech., 46(3), pp. 554–560. [CrossRef] [PubMed]
Simon, W. H. , 1970, “ Scale Effects in Animal Joints—Part I: Articular Cartilage Thickness and Compressive Stress,” Arthritis Rheum., 13(3), pp. 244–256. [CrossRef] [PubMed]
Hughes, L. C. , Archer, C. W. , and ap Gwynn, I. , 2005, “ The Ultrastructure of Mouse Articular Cartilage: Collagen Orientation and Implications for Tissue Functionality. A Polarised Light and Scanning Electron Microscope Study and Review,” Eur. Cells Mater., 9, pp. 68–84. [CrossRef]
Glasson, S. S. , Chambers, M. G. , Van Den Berg, W. B. , and Little, C. B. , 2010, “ The OARSI Histopathology Initiative—Recommendations for Histological Assessments of Osteoarthritis in the Mouse,” Osteoarthritis Cartilage, 18(Suppl. 3), pp. S17–S23. [CrossRef] [PubMed]
Blaker, C. L. , Clarke, E. C. , and Little, C. B. , 2017, “ Using Mouse Models to Investigate the Pathophysiology, Treatment, and Prevention of Post-Traumatic Osteoarthritis,” J. Orthop. Res., 35(3), pp. 537–547. [CrossRef] [PubMed]

Figures

Grahic Jump Location
Fig. 1

Flow chart of the experimental approach for independent measurements of drained Poisson's ratio (ν) and solid matrix Young's modulus (E). Characterization of these quasi-static material properties includes computational and experimental methods. Arrows indicate the sequence of steps.

Grahic Jump Location
Fig. 2

Schematic representation of the experimental setup. Controlled loads (Fapplied) of 0.1 N, 0.2 N, or 0.3 N were applied on the anterior side of the murine distal femur. Cartilage on the posterior medial femoral condyles was imaged with confocal microscope as it was compressed against the cover glass.

Grahic Jump Location
Fig. 3

Representative sagittal cross sections (left) and spatially varying thickness maps (right) obtained from confocal z-stacks of murine cartilage prior to compression (top row) and after application of 0.3 N (bottom row). Thickness maps were used to quantify cartilage deformation.

Grahic Jump Location
Fig. 4

Representative finite element model of murine cartilage on the medial femoral condyle. The tissue was modeled as an ideal hemispherical shell governed by a neo-Hookean, hyperelastic constitutive relationship. A rigid wall (flat plate) contact was applied with a prescribed displacement to compress the bottom (outer) surface in the positive z-direction. The top surface represents the cartilage–bone interface, which was constrained from any rotation and translation. The FEM was used to determine drained Poisson's ratio (ν) and solid matrix Young's modulus (E) through iFEA.

Grahic Jump Location
Fig. 5

Representative confocal micrographs of murine AC surface with 21 spots photobleached onto the extracellular matrix (a) 1 min after unloading (F = 0 N) and (b) 5 min after loading to F = 0.3 N. These micrographs were used to quantify compression-induced lateral strains between the photobleached spots from the center of compression (the center of the grid). (c) Representative FEM simulation (view from the bottom surface) used to determine drained Poisson's ratio (ν) by matching experimentally and computationally quantified lateral strains. The highlighted nodes in the simulation are along the photobleached grid pattern in the experiments.

Grahic Jump Location
Fig. 6

(a) Comparison of the experimentally measured, spatially varying lateral strains in a representative specimen (circles) to FEM simulation results for ν = 0, ν = 0.21, ν = 0.35, and ν = 0.49 (triangles). The closest match between experimental and FEM results was observed for ν = 0.21 in this representative dataset. (b) Representative SSE curve calculated for a range of simulated drained Poisson's ratios (ν = 0–0.49 in increments of 0.01). SSE is the square root of the sum of squared error between the simulated lateral strains (in the FEM) and the experimentally calculated lateral strains. ν was determined based on the global minima of the curve (ν = 0.21 for the representative dataset).

Grahic Jump Location
Fig. 7

Representative images of (a) posterior and (b) anterior sides of a murine femur. Prior to imaging, the femur was placed posterior side down onto a sheet of paper with ink. Thus, the contact points on the posterior condyles and on the bone are ink-stained and visible in image (a). Additionally, a flat weight with ink was placed onto the anterior side of the femur, enabling visualization of the weight–femur contact point (i.e., the point of force application) in image (b). (c) Posterior side of the femur with superimposed coordinates of contact forces from the weight (Fapplied), from the glass onto the femoral condyles (F2 and F3) and from the glass onto the bone (F1). These coordinates are calculated relative to a reference point (ref) point on the tape used to secure the specimen onto the cover glass.

Grahic Jump Location
Fig. 8

(a) Relative boundary displacements measured for each loading group in control and trypsin-treated specimens. Increased displacements for a given force in trypsin-treated specimens suggest decreased stiffness in comparison with controls. (b) Reaction forces measured on the articular surface of the medial condyle for each loading group in control and trypsin-treated specimens. As expected, the fraction of the applied force transmitted onto the medial condyle was independent of treatment. (c) Force versus displacement for both treatment groups. These curves are qualitatively consistent with a nonlinear mechanical response. Statistical analyses of these data are in (a) and (b). In all figure panels, data are mean±standard deviation. * denotes p ≤ 0.05 versus control. (d) FEM-predicted behavior of reaction force versus displacement for representative specimens from the control (solid lines) and trypsin-treated (dashed lines) groups comparing every load level to the unloaded state. Incremental analysis comparing every load level to the previous loaded state was also performed (see Supplemental Material 4, which is available under the “Supplemental Materials” tab for this paper on the ASME Digital Collection).

Grahic Jump Location
Fig. 9

(a) Solid matrix Young's modulus E of murine femoral AC in control and trypsin-treated specimens (n = 9/group) estimated for applied loads of 0.1 N, 0.2 N and 0.3 N. In all loading groups, trypsin treatment significantly reduced E. (b) Drained Poisson's ratio ν in control and trypsin-treated specimens. ν was also reduced by trypsin treatment. In both panels, data are mean + standard deviation. Brackets denote p ≤ 0.05 for pairwise comparisons between loads; * denotes p ≤ 0.05 versus control.

Grahic Jump Location
Fig. 10

(a) Solid matrix Young's modulus E of murine femoral AC in control and trypsin-treated specimens (n = 9/group) estimated for applied load of 0.025 N. Trypsin treatment significantly reduced E. (b) Drained Poisson's ratio ν in control and trypsin-treated specimens (n = 9/group). ν was also reduced by trypsin treatment. (c) AFM-based nano-indentation experiment performed on murine femoral AC inthe control group (n = 3/group). AFM- and confocal microscope-based measurements yielded similar values of E. Each data point represents averaged Young's moduli obtained at multiple locations of murine AC per condyle. The data are mean + standard deviation. * denotes p ≤ 0.05 versus control.

Grahic Jump Location
Fig. 11

Analysis of the uniqueness of iFEA measurements for (a) solid matrix Young's modulus E and (b) drained Poisson's ratio ν based on one- and two-parameter fits. Every data point represents a solution (output parameter) of the same representative iFEA based on different initial guesses for iterative optimization. Initial guesses for ν in two-parameter fits were 0–0.49, whereas ν for one-parameter fits was set to its measured value (0.21). Initial guesses for the solid matrix Young's modulus E varied between 0.1 and 10 MPa for both approaches. These data indicate that only the one-parameter approach converged to a unique solution for E.

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