Biomechanics and Modeling in Mechanobiology

, Volume 18, Issue 4, pp 953–968 | Cite as

The combined impact of tissue heterogeneity and fixed charge for models of cartilage: the one-dimensional biphasic swelling model revisited

  • Václav KlikaEmail author
  • Jonathan P. Whiteley
  • Cameron P. Brown
  • Eamonn A. Gaffney
Original Paper


Articular cartilage is a complex, anisotropic, stratified tissue with remarkable resilience and mechanical properties. It has been subject to extensive modelling as a multiphase medium, with many recent studies examining the impact of increasing detail in the representation of this tissue’s fine scale structure. However, further investigation of simple models with minimal constitutive relations can nonetheless inform our understanding at the foundations of soft tissue simulation. Here, we focus on the impact of heterogeneity with regard to the volume fractions of solid and fluid within the cartilage. Once swelling pressure due to cartilage fixed charge is also present, we demonstrate that the multiphase modelling framework is substantially more complicated, and thus investigate this complexity, especially in the simple setting of a confined compression experiment. Our findings highlight the importance of locally, and thus heterogeneously, approaching pore compaction for load bearing in cartilage models, while emphasising that such effects can be represented by simple constitutive relations. In addition, simulation predictions are observed for the sensitivity of stress and displacement in the cartilage to variations in the initial state of the cartilage and thus the details of experimental protocol, once the tissue is heterogeneous. These findings are for the simplest models given only heterogeneity in volume fractions and swelling pressure, further emphasising that the complex behaviours associated with the interaction of volume fraction heterogeneity and swelling pressure are likely to persist for simulations of cartilage representations with more fine-grained structural detail of the tissue.


Cartilage modelling Heterogeneity Swelling pressure Compaction 



V.K. is grateful for support from the International Mobility of Researchers—MSCA-IF in Czech Technical University grant CZ.02.2.69/0.0/0.0/17_050/0008025 funded by The Ministry of Education, Youth and Sports (MEYS) of the Czech Republic, as well as support from the Mathematical Institute at the University of Oxford.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Václav Klika
    • 1
    • 2
    Email author
  • Jonathan P. Whiteley
    • 3
  • Cameron P. Brown
    • 4
    • 5
  • Eamonn A. Gaffney
    • 2
  1. 1.Department of Mathematics, FNSPECzech Technical University in PraguePragueCzech Republic
  2. 2.Mathematical InstituteUniversity of OxfordOxfordUK
  3. 3.Department of Computer ScienceUniversity of OxfordOxfordUnited Kingdom
  4. 4.Botnar Research Centre, NDORMSUniversity of OxfordOxfordUK
  5. 5.MERF, CPMEQueensland University of TechnologyBrisbaneAustralia

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