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

- 25 Downloads

## Abstract

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.

## Keywords

Cartilage modelling Heterogeneity Swelling pressure Compaction## Notes

### Acknowledgements

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.

## References

- Ateshian G (2007) On the theory of reactive mixtures for modeling biological growth. Biomech Model Mechanobiol 6(6):423–445Google Scholar
- Ateshian G, Rajan V, Chahine N, Canal C, Hung C (2009) Modeling the matrix of articular cartilage using a continuous fiber angular distribution predicts many observed phenomena. J Biomech Eng 131(6):061,003Google Scholar
- Athanasiou K, Darling E, Hu J, DuRaine G, Reddi A (2013) Articular Cartilage. CRC Press, Boca RatonGoogle Scholar
- Barabadi B, Nathan R, Jen KP, Wu Q (2009) On the characterization of lifting forces during the rapid compaction of deformable porous media. J Heat Transf Trans ASME 131(10):101006Google Scholar
- Batchelor GK (2011) Field, forces and flows in biological systems. Cambridge University Press, CambridgeGoogle Scholar
- Bennethum LS, Cushman JH (2002) Multicomponent, multiphase thermodynamics of swelling porous media with electroquasistatics: II. Constitutive theory. Transp Porous Media 47(3):337–362MathSciNetGoogle Scholar
- Biot M (1941) General theory of three-dimensional consolidation. J Appl Phys 12(2):155–164zbMATHGoogle Scholar
- Broom N, Oloyede A (1993) Experimental-determination of the subchondral stress-reducing role of articular-cartilage under static and dynamic compression. Clin Biomech 8(2):102–108Google Scholar
- Brown CP, Houle MA, Popov K, Nicklaus M, Couture CA, Laliberté M, Brabec T, Ruediger A, Carr AJ, Price AJ et al (2014) Imaging and modeling collagen architecture from the nano to micro scale. Biomed Opt Express 5(1):233–243Google Scholar
- Buschmann M, Grodzinsky A (1995) A molecular model of proteoglycan-associated electrostatic forces in cartilage mechanics. J Biomech Eng 117(2):179–92Google Scholar
- Chen Y, Chen X, Hisada T (2006) Non-linear finite element analysis of mechanical electrochemical phenomena in hydrated soft tissues based on triphasic theory. Int J Numer Methods Eng 65(2):147–173zbMATHGoogle Scholar
- DiSilvestro MR, Suh JKF (2001) A cross-validation of the biphasic poroviscoelastic model of articular cartilage in unconfined compression, indentation, and confined compression. J Biomech 34(4):519–525Google Scholar
- Durst F, Ray S, Unsal B, Bayoumi O (2005) The development lengths of laminar pipe and channel flows. ASME J Fluids Eng 127:1154–1160Google Scholar
- Grodzinsky A (2000) An introduction to fluid dynamics. Garland Science, LondonGoogle Scholar
- de Groot SR, Mazur P (1984) Non-equilibrium thermodynamics. Dover Publications, New YorkzbMATHGoogle Scholar
- Gu W, Lai W, Mow V (1998) A mixture theory for charged-hydrated soft tissues containing multi-electrolytes: passive transport and swelling behaviors. J Biomech Eng 120(2):169–180Google Scholar
- Gurtin M, Fried E, Anand L (2010) The mechanics and thermodynamics of continua. Cambridge University Press, CambridgeGoogle Scholar
- Hodge W, Fijan R, Carlson K, Burgess RG, Harris WH, Mann RW (1986) Contact pressures in the human hip joint measured in vivo. Proc Natl Acad Sci 83:2879–2883Google Scholar
- Hou J, Holmes M, Lai W, Mow V (1989) Boundary conditions at the cartilage-synovial fluid interface for joint lubrication and theoretical verifications. J Biomech Eng 111(1):78–87Google Scholar
- Huyghe J, Janssen J (1997) Quadriphasic mechanics of swelling incompressible porous media. Int J Eng Sci 35(8):793–802zbMATHGoogle Scholar
- Julkunen P, Wilson W, Isaksson H, Jurvelin J, Herzog W, Korhonen R (2013) A review of the combination of experimental measurements and fibril-reinforced modeling for investigation of articular cartilage and chondrocyte response to loading. Comput Math Methods Med. https://doi.org/10.1155/2013/326150 Google Scholar
- Klika V (2014) A guide through available mixture theories for applications. Crit Rev Solid State Mater Sci 39(2):154–174Google Scholar
- Klika V, Gaffney EA, Chen YC, Brown CP (2016) An overview of multiphase cartilage mechanical modelling and its role in understanding function and pathology. J Mech Behav Biomed Mater 62:139–157Google Scholar
- Kozeny J (1927) Ueber kapillare Leitung des Wassers im Boden. Sitzungsber Akad Wiss 136(10)Google Scholar
- Lai WM, Hou J, Mow VC (1991) A triphasic theory for the swelling and deformation behaviors of articular cartilage. J Biomech Eng 113(3):245–258Google Scholar
- Lang GE, Stewart PS, Vella D, Waters SL, Goriely A (2014) Is the Donnan effect sufficient to explain swelling in brain tissue slices? J R Soc Interface 11(96):20140,123Google Scholar
- Lemon G, King JR, Byrne HM, Jensen OE, Shakesheff KM (2006) Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory. J Math Biol 52(5):571–594MathSciNetzbMATHGoogle Scholar
- Lu X, Mow V (2008) Biomechanics of articular cartilage and determination of material properties. Med Sci Sports Exerc 40(2):193–199. https://doi.org/10.1249/mss.0b013e31815cb1fc Google Scholar
- MacMinn CW, Dufresne ER, Wettlaufer JS (2016) Large deformations of a soft porous material. Phys. Rev. Appl. 5(4):044,020Google Scholar
- Manzano S, Armengol M, Price AJ, Hulley PA, Gill HS, Doblaré M, Doweidar MH (2016) Inhomogeneous response of articular cartilage: a three-dimensional multiphasic heterogeneous study. PLoS One 11(6):e0157,967Google Scholar
- Manzano S, Manzano R, Doblaré M, Doweidar MH (2015) Altered swelling and ion fluxes in articular cartilage as a biomarker in osteoarthritis and joint immobilization: a computational analysis. J R Soc Interface 12(102):20141,090Google Scholar
- 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(1):73–84Google Scholar
- Mow V, Kwan M, Lai W, Holmes M (1986) A finite deformation theory for nonlinearly permeable soft hydrated biological tissues. In: Schmid-Schönbein GW, Woo SL, Zweifach BW (eds) Frontiers in biomechanics. Springer, New York, pp 153–179Google Scholar
- Mow VC, Mansour JM (1977) The nonlinear interaction between cartilage deformation and interstitial fluid flow. J Biomech 10(1):31–39Google Scholar
- Murakami T, Yarimitsu S, Nakashima K, Sakai N, Yamaguchi T, Sawae Y, Suzuki A (2015) Biphasic and boundary lubrication mechanisms in artificial hydrogel cartilage: a review. Proc Inst Mech Eng Part H J Eng Med 229(12, SI):864–878Google Scholar
- Pierce D, Ricken T, Holzapfel G (2013) A hyperelastic biphasic fibre-reinforced model of articular cartilage considering distributed collagen fibre orientations: continuum basis, computational aspects and applications. Comput Methods Biomech Biomed Eng 16(12):1344–1361Google Scholar
- Raphael B, Khalil T, Workman VL, Smith A, Brown CP, Streuli C, Saiani A, Domingos M (2017) 3D cell bioprinting of self-assembling peptide-based hydrogels. Mater Lett 190:103–106Google Scholar
- Rossetti L, Kuntz L, Kunold E, Schock J, Müller K, Grabmayr H, Stolberg-Stolberg J, Pfeiffer F, Sieber S, Burgkart R et al (2017) The microstructure and micromechanics of the tendon-bone insertion. Nat Mater 16(6):664Google Scholar
- Soltz MA, Ateshian GA (1998) Experimental verification and theoretical prediction of cartilage interstitial fluid pressurization at an impermeable contact interface in confined compression. J Biomech 31(10):927–934Google Scholar
- Wilson W, Huyghe J, Van Donkelaar C (2006) A composition-based cartilage model for the assessment of compositional changes during cartilage damage and adaptation. Osteoarthr Cartil 14(6):554–560Google Scholar
- Wilson W, Huyghe J, Van Donkelaar C (2007) Depth-dependent compressive equilibrium properties of articular cartilage explained by its composition. Biomech Model Mechanobiol 6(1–2):43–53Google Scholar
- Wilson W, Van Donkelaar C, Huyghe J (2005) A comparison between mechano-electrochemical and biphasic swelling theories for soft hydrated tissues. J Biomech Eng 127(1):158–165Google Scholar
- Wilson W, Van Donkelaar C, Van Rietbergen B, Huiskes R (2005) A fibril-reinforced poroviscoelastic swelling model for articular cartilage. J Biomech 38(6):1195–1204Google Scholar