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A biphasic visco-hyperelastic damage model for articular cartilage: application to micromechanical modelling of the osteoarthritis-induced degradation behaviour

  • Dongxu Liu
  • Songyun MaEmail author
  • Marcus Stoffel
  • Bernd Markert
Original Paper
  • 16 Downloads

Abstract

Osteoarthritis-induced microstructural and compositional changes of articular cartilage affect its load-bearing capacity and the damage resistance. The aim of the present study is to analyse effects of the osteoarthritis-induced microstructural degradation on the damage behaviour of articular cartilages. A poro-visco-hyperelastic damage model is proposed within the theoretical framework of continuum mechanics to describe the deformation and damage behaviour of collagen fibrils and highly hydrated proteoglycan matrix in articular cartilages. An integral-type nonlocal algorithm is employed to overcome the mesh dependence of simulation results involving strain localization. 3D computational models for a normal cartilage and two osteoarthritic cartilages with different degeneration levels are developed to study the degradation of the damage resistance of articular cartilages. In addition, the present simulations take into account the alterations of collagen fibril networks as well as compositional changes of cartilage constituents at different osteoarthritic stages. The material parameters of the constitutive model are identified by comparing the computational results to unconfined compression tests. The simulation results of spherical indentation tests show that damage in the articular cartilage with high-stage osteoarthritis is much more significant than that in the normal cartilage under identical loadings. The proposed computational methods can be used for studying the relationship between the damage behaviour and the complex morphology of the collagen fibril networks in biomaterials.

Keywords

Articular cartilage Osteoarthritis Damage behaviour Porous media Poro-visco-hyperelastic Nonlocal modelling 

Notes

Acknowledgements

The first author would like to acknowledge the research funding from China Scholarship Council (201606890022).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. Andrade F, César de Sá J, Andrade Pires F (2011) A ductile damage nonlocal model of integral-type at finite strains: formulation and numerical issues. Int J Damage Mech 20(4):515–557CrossRefGoogle Scholar
  2. Andrade F, de Sá JC, Pires FA (2014) Assessment and comparison of non-local integral models for ductile damage. Int J Damage Mech 23(2):261–296CrossRefGoogle Scholar
  3. Antons J, Marascio M, Nohava J, Martin R, Applegate L, Bourban P, Pioletti D (2018) Zone-dependent mechanical properties of human articular cartilage obtained by indentation measurements. J Mater Sci Mater Med 29(5):57CrossRefGoogle Scholar
  4. Arokoski J, Jurvelin J, Vaatainen U, Helminen H (2000) Normal and pathological adaptations of articular cartilage to joint loading. Scand J Med Sci Sports 10(4):186–198CrossRefGoogle Scholar
  5. Bachrach NM, Mow VC, Guilak F (1998) Incompressibility of the solid matrix of articular cartilage under high hydrostatic pressures. J Biomech 31(5):445–451CrossRefGoogle Scholar
  6. Balzani D, Ortiz M (2012) Relaxed incremental variational formulation for damage at large strains with application to fiber-reinforced materials and materials with truss-like microstructures. Int J Numer Meth Eng 92(6):551–570MathSciNetzbMATHCrossRefGoogle Scholar
  7. Berger L, Bordas R, Kay D, Tavener S (2017) A stabilized finite element method for finite-strain three-field poroelasticity. Comput Mech 60(1):51–68MathSciNetzbMATHCrossRefGoogle Scholar
  8. Bluhm J (2002) Modelling of saturated thermo-elastic porous solids with different phase temperatures. In: Ehlers W, Bluhm J (eds) Porous media. Springer, Berlin, pp 87–118zbMATHCrossRefGoogle Scholar
  9. Canal Guterl C, Hung CT, Ateshian GA (2010) Electrostatic and non-electrostatic contributions of proteoglycans to the compressive equilibrium modulus of bovine articular cartilage. J Biomech 43(7):1343–1350CrossRefGoogle Scholar
  10. Canty EG (2005) Procollagen trafficking, processing and fibrillogenesis. J Cell Sci 118(7):1341–1353CrossRefGoogle Scholar
  11. Changoor A, Nelea M, Méthot S, Tran-Khanh N, Chevrier A, Restrepo A, Shive M, Hoemann C, Buschmann M (2011) Structural characteristics of the collagen network in human normal, degraded and repair articular cartilages observed in polarized light and scanning electron microscopies. Osteoarthr Cartil 19(12):1458–1468CrossRefGoogle Scholar
  12. Choi K, Kuhn J, Ciarelli M, Goldstein S (1990) The elastic moduli of human subchondral, trabecular, and cortical bone tissue and the size-dependency of cortical bone modulus. J Biomech 23(11):1103–1113CrossRefGoogle Scholar
  13. Clark JM (1990) The organisation of collagen fibrils in the superficial zones of articular cartilage. J Anat 171:117–130Google Scholar
  14. Clark JM (1991) Variation of collagen fiber alignment in a joint surface: a scanning electron microscope study of the tibial plateau in dog, rabbit, and man. J Orthop Res 9(2):246–257CrossRefGoogle Scholar
  15. DiSilvestro M, Suh J (2001) A cross-validation of the biphasic poroviscoelastic model of articular cartilage in unconfined compression, indentation, and confined compression. J Biomech 34(4):519–525CrossRefGoogle Scholar
  16. Doyran B, Tong W, Li Q, Jia H, Zhang X, Chen C, Enomoto-Iwamoto M, Lu X, Qin L, Han L (2017) Nanoindentation modulus of murine cartilage: a sensitive indicator of the initiation and progression of post-traumatic osteoarthritis. Osteoarthr Cartil 25(1):108–117CrossRefGoogle Scholar
  17. Fathi F, Hatefi Ardakani S, Fatemi Dehaghani P, Mohammadi S (2017) A finite strain integral-type anisotropic damage model for fiber-reinforced materials: application in soft biological tissues. Comput Methods Appl Mech Eng 322:262–295MathSciNetCrossRefGoogle Scholar
  18. Ferreira J, Parente M, Jabareen M, Jorge RN (2017) A general framework for the numerical implementation of anisotropic hyperelastic material models including non-local damage. Biomech Model Mechanobiol 16(4):1119–1140CrossRefGoogle Scholar
  19. García JJ, Cortés DH (2007) A biphasic viscohyperelastic fibril-reinforced model for articular cartilage: formulation and comparison with experimental data. J Biomech 40(8):1737–1744CrossRefGoogle Scholar
  20. Gottardi R, Hansen U, Raiteri R, Loparic M, Düggelin M, Mathys D, Friederich NF, Bruckner P, Stolz M (2016) Supramolecular organization of collagen fibrils in healthy and osteoarthritic human knee and hip joint cartilage. PLoS ONE 11(10):e0163552CrossRefGoogle Scholar
  21. Han L, Grodzinsky AJ, Ortiz C (2011) Nanomechanics of the cartilage extracellular matrix. Annu Rev Mater Res 41(1):133–168CrossRefGoogle Scholar
  22. He B, Wu J, Chim S, Xu J, Kirk T (2013) Microstructural analysis of collagen and elastin fibres in the kangaroo articular cartilage reveals a structural divergence depending on its local mechanical environment. Osteoarthr Cartil 21(1):237–245CrossRefGoogle Scholar
  23. Henao-Murillo L, Ito K, van Donkelaar CC (2018) Collagen damage location in articular cartilage differs if damage is caused by excessive loading magnitude or rate. Ann Biomed Eng 46(4):605–615CrossRefGoogle Scholar
  24. Heuijerjans A, Wilson W, Ito K, van Donkelaar C (2017) The critical size of focal articular cartilage defects is associated with strains in the collagen fibers. Clin Biomech 50:40–46CrossRefGoogle Scholar
  25. Hollander AP, Heathfield TF, Webber C, Iwata Y, Bourne R, Rorabeck C, Poole A (1994) Increased damage to type II collagen in osteoarthritic articular cartilage detected by a new immunoassay. J Clin Investig 93(4):1722–1732CrossRefGoogle Scholar
  26. Holzapfel GA, Gasser TC (2001) A viscoelastic model for fiber-reinforced composites at finite strains: continuum basis, computational aspects and applications. Comput Methods Appl Mech Eng 190(34):4379–4403CrossRefGoogle Scholar
  27. Hosseini S, Wilson W, Ito K, Van Donkelaar C (2014) A numerical model to study mechanically induced initiation and progression of damage in articular cartilage. Osteoarthr Cartil 22(1):95–103CrossRefGoogle Scholar
  28. Hui Mingalone CK, Liu Z, Hollander JM, Garvey KD, Gibson AL, Banks RE, Zhang M, McAlindon TE, Nielsen HC, Georgakoudi I, Zeng L (2018) Bioluminescence and second harmonic generation imaging reveal dynamic changes in the inflammatory and collagen landscape in early osteoarthritis. Lab Investig 98(5):656–669CrossRefGoogle Scholar
  29. Hwang W, Li B, Jin L, Ngo K, Schachar N, Hughes G (1992) Collagen fibril structure of normal, aging, and osteoarthritic cartilage. J Pathol 167(4):425–433CrossRefGoogle Scholar
  30. Inamdar SR, Knight DP, Terrill NJ, Karunaratne A, Cacho-Nerin F, Knight MM, Gupta HS (2017) The secret life of collagen: temporal changes in nanoscale fibrillar pre-strain and molecular organization during physiological loading of cartilage. ACS Nano 11(10):9728–9737CrossRefGoogle Scholar
  31. Jirásek M (2007) Nonlocal damage mechanics. Rev Eur Génie Civ 11(7–8):993–1021CrossRefGoogle Scholar
  32. Julkunen P, Wilson W, Jurvelin JS, Rieppo J, Qu C, Lammi MJ, Korhonen RK (2008) Stressrelaxation of human patellar articular cartilage in unconfined compression: prediction of mechanical response by tissue composition and structure. J Biomech 41(9):1978–1986CrossRefGoogle Scholar
  33. Kaliske M, Rothert H (1997) Formulation and implementation of three-dimensional viscoelasticity at small and finite strains. Comput Mech 19(3):228–239zbMATHCrossRefGoogle Scholar
  34. Klika V, Gaffney EA, Chen Y, Brown CP (2016) An overview of multiphase cartilage mechanical modelling and its role in understanding function and pathology. J Mech Behav Biomed Mater 62:139–157CrossRefGoogle Scholar
  35. Korhonen RK, Laasanen MS, Töyräs J, Lappalainen R, Helminen HJ, Jurvelin JS (2003) Fibril reinforced poroelastic model predicts specifically mechanical behavior of normal, proteoglycan depleted and collagen degraded articular cartilage. J Biomech 36(9):1373–1379CrossRefGoogle Scholar
  36. Lane L, Bullough P (1980) Age-related changes in the thickness of the calcified zone and the number of tidemarks in adult human articular cartilage. J Bone Joint Surg Br 62(3):372–375CrossRefGoogle Scholar
  37. Lewis JL, Johnson SL (2001) Collagen architecture and failure processes in bovine patellar cartilage. J Anat 199(4):483–492CrossRefGoogle Scholar
  38. Liukkonen MK, Mononen ME, Klets O, Arokoski JP, Saarakkala S, Korhonen RK (2017) Simulation of subject-specific progression of knee osteoarthritis and comparison to experimental follow-up data: data from the osteoarthritis initiative. Sci Rep 7(1):1–14CrossRefGoogle Scholar
  39. Ma S, Yuan H (2015) Computational investigation of multi-axial damage modeling for porous sintered metals with experimental verification. Eng Fract Mech 149:89–110CrossRefGoogle Scholar
  40. Ma S, Scheider I, Bargmann S (2016a) Anisotropic constitutive model incorporating multiple damage mechanisms for multiscale simulation of dental enamel. J Mech Behav Biomed Mater 62:515–533CrossRefGoogle Scholar
  41. Ma S, Scheider I, Bargmann S (2016b) Continuum damage modeling and simulation of hierarchical dental enamel. Modell Simul Mater Sci Eng 24(4):45014CrossRefGoogle Scholar
  42. Ma S, Zhou B, Markert B (2018) Numerical simulation of the tissue differentiation and corrosion process of biodegradable magnesium implants during bone fracture healing. ZAMM J Appl Math Mech/Z Angew Math Mech 98(12):2223–2238MathSciNetCrossRefGoogle Scholar
  43. Mäkelä J, Cooper B, Korhonen R, Grinstaff M, Snyder B (2018) Functional effects of an interpenetrating polymer network on articular cartilage mechanical properties. Osteoarthr Cartil 26(3):414–421CrossRefGoogle Scholar
  44. Mansfield J, Bell J, Winlove C (2015) The micromechanics of the superficial zone of articular cartilage. Osteoarthr Cartil 23(10):1806–1816CrossRefGoogle Scholar
  45. Men Y, Jiang Y, Chen L, Zhang C, Ye J (2017) On mechanical mechanism of damage evolution in articular cartilage. Mater Sci Eng C 78:79–87CrossRefGoogle Scholar
  46. Mente P, Lewis J (1994) Elastic modulus of calcified cartilage is an order of magnitude less than that of subchondral bone. J Orthop Res 12(5):637–647CrossRefGoogle Scholar
  47. Mononen M, Julkunen P, Töyräs J, Jurvelin J, Kiviranta I, Korhonen R (2011) Alterations in structure and properties of collagen network of osteoarthritic and repaired cartilage modify knee joint stresses. Biomech Model Mechanobiol 10(3):357–369CrossRefGoogle Scholar
  48. Mononen ME, Tanska P, Isaksson H, Korhonen RK (2016) A novel method to simulate the progression of collagen degeneration of cartilage in the knee: data from the osteoarthritis initiative. Sci Rep 6(1):21415CrossRefGoogle Scholar
  49. Mow VC, Holmes MH, Michael Lai W (1984) Fluid transport and mechanical properties of articular cartilage: a review. J Biomech 17(5):377–394CrossRefGoogle Scholar
  50. Mow VC, Ratcliffe A, Robin Poole A (1992) Cartilage and diarthrodial joints as paradigms for hierarchical materials and structures. Biomaterials 13(2):67–97CrossRefGoogle Scholar
  51. Natarajan V, Madhan B, Tiku ML (2015) Intra-articular injections of polyphenols protect articular cartilage from inflammation-induced degradation: suggesting a potential role in cartilage therapeutics. PLoS ONE 10(6):e0127165CrossRefGoogle Scholar
  52. Nickien M, Thambyah A, Broom N (2013) How changes in fibril-level organization correlate with the macrolevel behavior of articular cartilage. Wiley Interdiscip Rev Syst Biol Med 5(4):495–509CrossRefGoogle Scholar
  53. Párraga Quiroga J, Wilson W, Ito K, van Donkelaar C (2017) The effect of loading rate on the development of early damage in articular cartilage. Biomech Model Mechanobiol 16(1):263–273CrossRefGoogle Scholar
  54. Peña E (2011a) A rate dependent directional damage model for fibred materials: application to soft biological tissues. Comput Mech 48(4):407–420MathSciNetzbMATHCrossRefGoogle Scholar
  55. Peña E (2011b) Prediction of the softening and damage effects with permanent set in fibrous biological materials. J Mech Phys Solids 59(9):1808–1822MathSciNetzbMATHCrossRefGoogle Scholar
  56. Pierce DM, Ricken T, Holzapfel GA (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):1344–1361CrossRefGoogle Scholar
  57. Pierce DM, Unterberger MJ, Trobin W, Ricken T, Holzapfel GA (2016) A microstructurally based continuum model of cartilage viscoelasticity and permeability incorporating measured statistical fiber orientations. Biomech Model Mechanobiol 15(1):229–244CrossRefGoogle Scholar
  58. Richard F, Villars M, Thibaud S (2013) Viscoelastic modeling and quantitative experimental characterization of normal and osteoarthritic human articular cartilage using indentation. J Mech Behav Biomed Mater 24:41–52CrossRefGoogle Scholar
  59. Ricken T, Dahmen U, Dirsch O (2010) A biphasic model for sinusoidal liver perfusion remodeling after outflow obstruction. Biomech Model Mechanobiol 9(4):435–450CrossRefGoogle Scholar
  60. Saarakkala S, Julkunen P, Kiviranta P, Mäkitalo J, Jurvelin J, Korhonen R (2010) Depth-wise progression of osteoarthritis in human articular cartilage: investigation of composition, structure and biomechanics. Osteoarthr Cartil 18(1):73–81CrossRefGoogle Scholar
  61. Schinagl RM, Gurskis D, Chen AC, Sah RL (1997) Depth-dependent confined compression modulus of full-thickness bovine articular cartilage. J Orthop Res 15(4):499–506CrossRefGoogle Scholar
  62. Sophia Fox AJ, Bedi A, Rodeo SA (2009) The basic science of articular cartilage: structure, composition, and function. Sports Health 1(6):461–468CrossRefGoogle Scholar
  63. Stender ME, Regueiro RA, Klisch SM, Ferguson VL (2015) An equilibrium constitutive model of anisotropic cartilage damage to elucidate mechanisms of damage initiation and progression. J Biomech Eng 137(8):081010CrossRefGoogle Scholar
  64. Stender ME, Carpenter RD, Regueiro RA, Ferguson VL (2016) An evolutionary model of osteoarthritis including articular cartilage damage, and bone remodeling in a computational study. J Biomech 49(14):3502–3508CrossRefGoogle Scholar
  65. Thambyah A, Broom N (2007) On how degeneration influences load-bearing in the cartilagebone system: a microstructural and micromechanical study. Osteoarthr Cartil 15(12):1410–1423CrossRefGoogle Scholar
  66. Torzilli P, Grigiene R, Borrelli J, Helfet D (1999) Effect of impact load on articular cartilage: cell metabolism and viability, and matrix water content. J Biomech Eng 121(5):433–441CrossRefGoogle Scholar
  67. Van der Voet A (1997) A comparison of finite element codes for the solution of biphasic poroelastic problems. Proc Inst Mech Eng H 211(2):209Google Scholar
  68. Vanden Berg-Foels W, Scipioni L, Huynh C, Wen X (2012) Helium ion microscopy for high-resolution visualization of the articular cartilage collagen network. J Microsc 246(2):168–176CrossRefGoogle Scholar
  69. Waffenschmidt T, Polindara C, Menzel A, Blanco S (2014) A gradient-enhanced large-deformation continuum damage model for fibre-reinforced materials. Comput Methods Appl Mech Eng 268:801–842MathSciNetzbMATHCrossRefGoogle Scholar
  70. Wen C, Wu C, Tang B, Wang T, Yan C, Lu W, Pan H, Hu Y, Chiu K (2012) Collagen fibril stiffening in osteoarthritic cartilage of human beings revealed by atomic force microscopy. Osteoarthr Cartil 20(8):916–922CrossRefGoogle Scholar
  71. Wilson W, van Donkelaar C, van Rietbergen R, Huiskes R (2005) The role of computational models in the search for the mechanical behavior and damage mechanisms of articular cartilage. Med Eng Phys 27(10):810–826CrossRefGoogle Scholar
  72. Wilson W, Driessen N, van Donkelaar C, Ito K (2006a) Prediction of collagen orientation in articular cartilage by a collagen remodeling algorithm. Osteoarthr Cartil 14(11):1196–1202CrossRefGoogle Scholar
  73. Wilson W, Huyghe J, van Donkelaar C (2006b) A composition-based cartilage model for the assessment of compositional changes during cartilage damage and adaptation. Osteoarthr Cartil 14(6):554–560CrossRefGoogle Scholar
  74. 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):43–53CrossRefGoogle Scholar
  75. Wilusz R, Zauscher S, Guilak F (2013) Micromechanical mapping of early osteoarthritic changes in the pericellular matrix of human articular cartilage. Osteoarthr Cartil 21(12):1895–1903CrossRefGoogle Scholar
  76. Workman J, Thambyah A, Broom N (2017) The influence of early degenerative changes on the vulnerability of articular cartilage to impact-induced injury. Clin Biomech 43:40–49CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Dongxu Liu
    • 1
  • Songyun Ma
    • 1
    Email author
  • Marcus Stoffel
    • 1
  • Bernd Markert
    • 1
  1. 1.Institute of General MechanicsRWTH Aachen UniversityAachenGermany

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