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Current Directions in Biomedical Engineering

Joint Journal of the German Society for Biomedical Engineering in VDE and the Austrian and Swiss Societies for Biomedical Engineering

Editor-in-Chief: Dössel, Olaf

Editorial Board: Augat, Peter / Buzug, Thorsten M. / Haueisen, Jens / Jockenhoevel, Stefan / Knaup-Gregori, Petra / Kraft, Marc / Lenarz, Thomas / Leonhardt, Steffen / Malberg, Hagen / Penzel, Thomas / Plank, Gernot / Radermacher, Klaus M. / Schkommodau, Erik / Stieglitz, Thomas / Urban, Gerald A.

CiteScore 2018: 0.47

Source Normalized Impact per Paper (SNIP) 2018: 0.377

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Biomechanical parameter determination of scaffold-free cartilage constructs (SFCCs) with the hyperelastic material models Yeoh, Ogden and Demiray

T. Reuter / I. Ponomarev
Published Online: 2015-09-12 | DOI: https://doi.org/10.1515/cdbme-2015-0106


The biomechanical properties are crucial indicators for the functional characterization of cartilaginous tissue. In this contribution native articular cartilage and three-dimensional scaffold-free cartilage constructs (SFCCs) are characterized by hyperelastic material models (Yeoh, Ogden and Demiray). SFCCs were developed for the therapy of damaged articular cartilage. The normalized error (NE) of fit and experiment is in the range of 0.04 and 0.13. The material model Yeoh with two parameters yields the best fit. The stress-like parameterc 20 is 0.489 MPa for native cartilage, 0.120 MPa and 0.041 MPa for SFCCs produced from mesenchymal stem cells and chondrocytes, respectively. The significance of the fits and the derived parameters are presented and evaluated.

Keywords: Hyperelastic Material Models; Cartilage; Scaffold-free Cartilage Constructs; Biomechanical Simulation

1 Introduction

Articular cartilage and SFCCs are investigated and characterized by biomechanical, biochemical and histological analysis. The biomechanical properties contain a variety of relevant information for the functional characterization of cartilaginous tissue [1]. For this purpose the modulus of elasticityE is commonly evaluated at a defined pressure applied to the cartilaginous tissue. However, the modulus of elasticityE cannot describe the entire strain behavior, because of the non-linear, non-homogeneous and anisotropic structure of cartilaginous tissue. On this account, the hyperelastic material models Yeoh, Ogden and Demiray are utilized for the characterization of the non-linear stress–strain–behavior of cartilage and SFCCs. Hence, within the scope of this article isotropic hyperelastic material models are fitted to the experimentally determined stress–strain–curves.

2 Methods

In this section the experimental approach and the hyper-elastic material models Yeoh, Ogden and Demiray are explained. Furthermore, the method to evaluate model quality is described.

2.1 Experimental approach

The stress–strain–curves of native cartilage and SFCCs were determined experimentally using a material testing machine INSTRON® 4466 (100 N). The uniaxial compression of the specimens was effected at a testing velocity of v = 1 mm/s, with an initial load of F = 0.1 N and a strain of 30 %.

The experiment was carried out on four native cartilage specimens, on five SFCCs-MSC produced from mesenchymal stem cells and five SFCCs-Chondro produced from chondrocytes. The native cartilaginous tissue was obtained from the equine knee joint (Trochlea ossis femoris). The SFCCs were produced according to patented technology [2] and implanted in the equine knee joint (Trochlea ossis femoris). After one-year-long implantation time the SFCCs were extracted for biomechanical characterization. In Figure 1 native cartilaginous tissue and SFCC are depicted. For the modelling averaged values were used.

2.2 Hyperelastic material models

For the characterization of the stress–strain–behavior the following isotropic incompressible hyperelastic material models were utilized: Yeoh [3], Ogden [4] and Demiray [5].

SFCC (left) and native cartilage (right).
Figure 1

SFCC (left) and native cartilage (right).

The material model Yeoh was calculated with equation 1


where Ψ is the strain energy density, I1 is the first invariant of the right Cauchy-Green tensor C and ci0 are stress-like material parameters (defining the stiffness of material). For a unique solution c30 had to be fixed on 0.01 MPa.

The parameters of the model Ogden were calculated with equation


where λ1, λ2 and λ3 are the principle stretch ratios, μr are stress-like material parameters (defining the stiffness of material) and αr are dimensionless material parameters. Applying the one-term formulation, the material parameters μ2 and μ3 were fixed on 1 MPa and α2 and α3 on zero, respectively.

The material model Demiray was calculated with equation


where a is the stress-like material parameter (defining the stiffness of material) and b is the dimensionless material parameter (defining progression of stress–strain–curve). The material models shown here are commonly used to describe the stress–strain–behavior of rubber-like solids. However, the model Demiray was also applied to characterize the stress–strain–behavior of soft tissue [5]. The calculations were carried out with the software Hyperfit® [6].

For the evaluation of model quality the normalized error (NE) of fit and experiment was calculated with equation


where ye is the experimental (observed) value, ym is the model (theoretical) value,i is the data-point index, n is the number of data-points and ye¯ is the mean value of the experimental values. For a perfect fit NE equals zero [6, 7].

3 Results

In Figures 24 the material model fits of Yeoh, Ogden and Demiray at the averaged experimental stress–strain– curves of native cartilage and SFCCs-MSC and SFCCsChondro are depicted.

It is evident that stress–strain–behavior is non-linear. The stress behavior of native cartilage and SFCCs-MSC and SFCCs-Chondro are nearly equal up to 10 % strain. From 10 % strain the stress values of the native cartilage increase stronger than those of the SFCCs. At 30 % strain the stress value of native cartilage is six times and three times higher than those of SFCCs-Chondro and SFCCs-MSC, respectively.

Stress–strain–curve experimental data (red, green and black) and material model fit Yeoh (blue).
Figure 2

Stress–strain–curve experimental data (red, green and black) and material model fit Yeoh (blue).

Stress–strain–curve experimental data (red, green and black) and material model fit Ogden (blue).
Figure 3

Stress–strain–curve experimental data (red, green and black) and material model fit Ogden (blue).

Stress–strain–curve experimental data (red, green and black) and material model fit Demiray (blue).
Figure 4

Stress–strain–curve experimental data (red, green and black) and material model fit Demiray (blue).

In Table 1 the hyperelastic material models are summarized with the respective normalized errors (NE) of experiment and fit as well as the number of parameters necessary for the calculation. The model Yeoh yielded the best fit for the characterization of the non-linear stress–strain– behavior. The normalized error (NE) of these models lies below 0.05. The models Ogden and Demiray are not suited for native cartilage.

The calculated parameters for native cartilage and SFCCs-MSC and SFCCs-Chondro, respectively, are listed in Tables 24. The results show, that the native cartilage has a higher stiffness than the SFCCs. This is illustrated particularly by the parametersc 20 (Yeoh),µ 1 (Ogden) und a (Demiray). Furthermore, the parameters calculated from the model Yeoh and Demiray show that the SFCCs-MSC have a higher stiffness than SFCCs-Chondro. The parameters calculated from the model Ogden show contrarious behavior and no significant differences in the calculated stiffness between native cartilage and SFCCs.

Table 1

Number of parameters and the quality (NE) of the models for the characterization of native cartilage and SFCCs.

Table 2

Calculated material parameters Yeoh.

Table 3

Calculated material parameters Ogden.

4 Conclusion

In this contribution isotropic incompressible hyperelastic material models (Yeoh, Ogden and Demiray) were utilized for the characterization of non-linear stress–strain– behavior of native cartilage and SFCCs. Particularly the model Yeoh (two parameters) with a normalized error of 0.04 is suitable for this purpose. The number of parameters can be increased to three for the models Yeoh and Ogden. Thereby the quality of the fit would be improved [7]. However, negative fit parameters may occur, which are not allowed physically [8]. SFCCs may be described with the model Demiray, but for native cartilage the model is insufficient. The stiffness mismatch between native cartilage and SFCCs resulted from the incomplete maturing process of the SFCCs. Thus the SFCCs can be used for the mathematical modelling of cartilaginous tissue as intermediate stages or for the development of a mathematical arthrosis model. Furthermore, the parameters obtained from the model may be used for FEM simulations. The models used so far regarded the monophasic state, i.e. the time-depended behavior (relaxation and creep) was not considered.

Further research will be extended to material models of the biphasic or triphasic theory [9]. Additionally, the material behavior of the native cartilage and SFCCs will be examined under cyclic load (stress softening, hysteresis, viscoelastic analysis) [10, 11].

Table 4

Calculated material parameters Demiray.


The authors would like to thank Petra Prokop for translation and editing.

Funding: This study is part of a research project supported by the German Ministry of Economy and Technology: INNO-WATT Reg. Nr.: IW091034.


  • [1]

    Goldsmith AAJ, Hayes A, Clift CE. Application of finite elements to the stress analysis of articular cartilage. Med. Eng. Phys. 1996; 18(2):89-98. Google Scholar

  • [2]

    Ponomarev I, Wilke I. Manufacturing process of three dimensional tissue structures and structures obtainable thereby. EP 1 550 716 B1, 2004. 

  • [3]

    Yeoh OH. Characterization of elastic properties of carbon black filled rubber vulcanizates. Rubber Chemistry and Technology 1990; 63(5):792-805. Google Scholar

  • [4]

    Ogden RW. Large deformation isotropic elasticity: on the correlation of theory and experiment for in-compressible rubberlike solids. Proceedings of the Royal Society 1972; Vol. A (326):565-584. Google Scholar

  • [5]

    Demiray H. A note on the elasticity of soft biological tissues. Journal of Biomechanics 1972; 5:309-311. Google Scholar

  • [6]


  • [7]

    Ponomarev I, Reuter T, Kammel A, Hauspurg C, Genzel A, Beck M, Hoffmann P, Barnewitz D. Biomechanical Characterisation of Scaffold-Free Cartilage Constructs with Hyperelastic Material Models. Biomedical Engineering / Biomedizinische Technik. ISSN (Online) 1862-278X, ISSN (Print) 0013-5585, . CrossrefGoogle Scholar

  • [8]

    Hu T, Desai JP. Characterization of soft-tissue material properties: Large deformation analysis. Medical Simulation, 2004, pp. 28-37. Google Scholar

  • [9]

    Lai WM, Hou JS, Mow VC. A triphasic Theory for the Swelling and Deformation Behaviors of Articular Cartilage. J Biomechanical Eng. 1991; 113:245-258. Google Scholar

  • [10]

    Fung YC. Biomechanics: Mechanical Properties of Living Tissues. New York, N.Y., Springer-Verlag, 1981. 

  • [11]

    Reuter T, Ponomarev I. Biomechanical Characterisation of Scaffold-free Cartilage Constructs with Dynamic Mechanical Analysis. Biomed Tech 2014; 59 (s1), ISSN 0013-5585, e-ISSN 1862-278X, ©2014 by Walter de Gruyter •Berlin •Boston. DOI 10.1515/bmt-2014-4416. CrossrefGoogle Scholar

About the article

Published Online: 2015-09-12

Published in Print: 2015-09-01

Author’s Statement

Conflict of interest: Authors state no conflict of interest. Material and Methods: Informed consent: Informed consent has been obtained from all individuals included in this study. Ethical approval: The research related to human use has been complied with all the relevant national regulations, institutional policies and in accordance the tenets of the Helsinki Declaration, and has been approved by the authors’ institutional review board or equivalent committee.

Citation Information: Current Directions in Biomedical Engineering, Volume 1, Issue 1, Pages 442–445, ISSN (Online) 2364-5504, DOI: https://doi.org/10.1515/cdbme-2015-0106.

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