Abstract
At present, there are two main numerical approaches that are frequently used to simulate the mechanical behaviour of the human spine. Researchers with a continuum-mechanical background often utilise the finite-element method (FEM), where the involved biological soft and hard tissues are modelled on a macroscopic (continuum) level. In contrast, groups associated with the science of human movement usually apply discrete multi-body systems (MBS). Herein, the bones are modelled as rigid bodies, which are connected by Hill-type muscles and non-linear rheological spring-dashpot models to represent tendons and cartilaginous connective tissue like intervertebral discs (IVD). A possibility to benefit from both numerical methods is to couple them and use each approach, where it is most appropriate. Herein, the basic idea is to utilise MBS in simulations of the overall body and apply the FEM only to selected regions of interest. In turn, the FEM is used as homogenisation tool, which delivers more accurate non-linear relationships describing the behaviour of the IVD in the multi-body dynamics model. The goal of this contribution is to present an approach to couple both numerical methods without the necessity to apply a gluing algorithm in the context of a co-simulation. Instead, several pre-computations of the intervertebral disc are performed offline to generate an approximation of the homogenised finite-element (FE) result. In particular, the discrete degrees of freedom (DOF) of the MBS, that is, three displacements and three rotations, are applied to the FE model of the IVD, and the resulting homogenised forces and moments are recorded. Moreover, a polynomial function is presented with the discrete DOF of the MBS as variables and the discrete forces an moments as function values. For the sake of a simple verification, the coupling method is applied to a simplified motion segment of the spine. Herein, two stiff cylindrical vertebrae with an interjacent homogeneous cylindrical IVD are examined under the restriction of purely elastic deformations in the sagittal plane.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
de Zee M, Hansen L, Andersen TB, Wong C, Rasmussen J, Simonsen EB (2003) On the development of a detailed rigid-body spine model. In: Doblare M, Cerrolaza M, Rodrigues H (eds) Proceedings of international congress on computational bioengineeringm, Zaragosa
de Zee M, Hansen L, Wong C, Rasmussen J, Simonsen EB (2007) A generic detailed rigid-body lumbar spine model. J Biomech 40: 1219–1227
Eberlein R, Holzapfel GA, Fröhlich M (2004) Multi-segment FEA of the human lumbar spine including the heterogeneity of the anulus fibrosus. Comput Mech 34: 147–165
Ehlers W (1993) Constitutive equations for granular materials in geomechanical context. In: Hutter K (ed) Continuum mechanics in environmental sciences and geophysics, CISM courses and lectures No. 337. Springer, Wien, pp 313–402
Ehlers W (2002) Foundations of multiphasic and porous materials. In: Ehlers W, Bluhm J (eds) Porous media: theory, experiments and numerical applications. Springer, Berlin, pp 3–86
Ehlers W, Karajan N, Markert B (2006) A porous media model describing the inhomogeneous behaviour of the human intervertebral disc. Mater Sci Eng Technol 37: 546–551
Ehlers W, Karajan N, Markert B (2009) An extended biphasic model for charged hydrated tissues with application to the intervertebral disc. Biomech Model Mechanobiol 8: 233–251
Eipper G (1998) Theorie und Numerik finiter elastischer Deformationen in fluidgesättigten Porösen Medien. Dissertation, Bericht Nr. II-1 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart
Esat V, Acar M (2007) A multi-body model of the whole human spine for whiplash investigations. In: 20th enhanced safety of vehicles conference: innovations for safety: opportunities and challenges, Lyon
Esat V, Acar M (2009) Viscoelastic finite element analysis of the cervical intervertebral discs in conjunction with a multi-body dynamic model of the human head and neck. Proc Inst Mech Eng Part H J Eng Med 223: 249–262
Esat V, Lopik DW, Acar M (2005) Combined multi-body dynamic and fe models of human head and neck. In: Gilchrist MD (ed) IUTAM symposium on impact biomechanics: from fundamental insights to applications, Springer, The Netherlands, pp 91–100
Frijns AJH, Huyghe JM, Janssen JD (1997) A validation of the quadriphasic mixture theory for intervertebral disc tissue. Int J Eng Sci 35: 1419–1429
Gardner-Morse MG, Stokes IA (2003) Physiological axial compressive preloads increase motion segment stiffness, linearity and hysteresis in all six degrees of freedom for small displacements about the neutral posture. J Orthop Res 21: 547–552
Gardner-Morse MG, Stokes IA (2004) Structural behavior of human lumbar spinal motion segments. J Biomech 37: 205–512
Günther M, Ruder H (2003) Synthesis of two-dimensional human walking: a test of the λ-model. Biol Cybern 8(2): 89–106
Hansen L, de Zee M, Rasmussen J, Andersen TB, Wong C, Simonsen EB (2006) Anatomy and biomechanics of the back muscles in the lumbar spine with reference to biomechanical modelling. Spine 31: 1888–1899
Hassanizadeh SM, Gray WG (1987) High velocity flow in porous media. Transp Porous Media 2: 521–531
Holzapfel GA, Schulze-Bauer CAJ, Feigl G, Regitnig P (2005) Mono-lamellar mechanics of the human lumbar anulus fibrosus. Biomech Model Mechanobiol 3: 125–140
Hsieh AH, Wagner DR, Cheng LY, Lotz JC (2005) Dependence of mechanical behavior of the murine tail disc on regional material properties: a parametric finite element study. J Biomech Eng 127: 1158–1167
Huynh K, Gibson I, Lu W, Jagdish B (2010) Simulating dynamics of thoracolumbar spine derived from LifeMOD under haptic forces. World Acad Sci Eng Technol 64: 278–285
Karajan N (2009) An extended biphasic description of the inhomogeneous and anisotropic intervertebral disc. Dissertation, Bericht Nr. II-19 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart
Karajan N (2012) Multiphasic intervertebral disc mechanics: theory and application. Arch Comput Methods Eng 19: 261–339
Klisch SM, Lotz JC (2000) A special theory of biphasic mixtures and experimental results for human annulus fibrosus tested in confined compression. ASME J Biomech Eng 122: 180–188
Lanir Y (1987) Biorheology and fluid flux in swelling tissues I. Bicomponent theory for small deformations, including concentration effects. Biorheology 24: 173–187
Markert B (2008) A biphasic continuum approach for viscoelastic high-porosity foams: comprehensive theory, numerics and application. Arch Comput Methods Eng 15: 371–446
Monteiro N (2009) Analysis of the intervertebral discs adjacent to interbody fusion using a multibody and finite element co-simulation. Dissertation, Instituto Superior Técnico, Lisboa
Monteiro NM, Silva MP, Folgado JO, Melancia JP (2011) Structural analysis of the intervertebral discs adjacent to an interbody fusion using multibody dynamics and finite element costimulation. Multibody Syst Dyn 25: 245–270
Moroney SP, Schultz AB, Miller JAA, Andersson GBJ (1988) Load-displacement properties of lower cervical spine motion segments. J Biomech 21: 769–779
Natarajan RN, Lavender SA, An HA, Andersson GB (2008) Biomechanical response of a lumbar intervertebral disc to manual lifting activities: a poroelastic finite element model study. Spine 33: 1958–1965
Panjabi MM (1973) Three-dimensional mathematical model of the human spine structure. J Biomech 6: 671–680
Panjabi MM, Brand RA Jr, White A III (1976) Three-dimensional flexibility and stiffness properties of the human thoracic spine. J Biomech 9: 185–192
Rohlmann A, Zander T, Schmidt H, Wilke HJ, Bergmann G (2006) Analysis of the influence of disc degeneration on the mechanical behaviour of a lumbar motion segment using the finite element method. J Biomech 39: 2484–2490
Ruder H, Gruber K, Hospach F, Ruder M, Subke J, Widmayer K (1994) Die Dynamik der Körpermassen: Einfache Modelle zur Simulation von Körperbewegungen und Aufprallvorgängen. In: Oehmichen M, König H (eds) Biomechanik-Rekonstruktion. Vol. 8 aus Rechtsmedizinische Forschungsergebnisse, Schmidt-Röhmhild, Lübeck, pp 63–84
Schmidt H, Heuer F, Drumm J, Klezl Z, Claes L, Wilke HJ (2007) Application of a calibration method provides more realistic results for a finite element model of a lumbar spinal segment. Clin Biomech 22: 377–384
Schmidt H, Heuer F, Claes L, Wilke HJ (2008) The relation between the instantaneous center of rotation and facet joint forces—a finite element analysis. Clin Biomech 23: 270–278
Schmitt S (2006) Über die Anwendung und Modifikation des Hill’schen Muskelmodells in der Biomechanik. Dissertation, Theoretische Astrophysik am Institut für Astronomie und Astrophysik, Eberhard Karls Universität Tübingen
Schröder Y, Wilson W, Huyghe JM, Baaijens FPT (2006) Osmoviscoelastic finite element model of the intervertebral disc. Eur Spine J 15: 361–371
Shirazi-Adl A (1994) Nonlinear stress analysis of the whole lumbar spine in torsion-mechanics of facet articulation. J Biomech 27: 289–299
Shirazi-Adl A (2006) Analysis of large compression loads on lumbar spine in felxion and torsion using a novel wrapping element. J Biomech 39: 267–275
Skempton AW (1960) Significance of Terzaghi’s concept of effective stress (Terzaghi’s discovery of effective stress). In: Bjerrum L, Casagrande A, Peck RB, Skempton AW (eds) From theory to practice in soil mechanics. Wiley, New York, pp 42–53
Stokes IA, Gardner-Morse M, Churchill D, Laible JP (2002) Measurement of a spinal motion segment stiffness matrix. J Biomech 35: 517–521
Wang J, Ma ZD, Hulbert GM (2003) A gluing algorithm for distributed simulation of multibody systems. Nonlinear Dyn 34: 159–188
White AA, Panjabi MM (1990) Clinical biomechanics of the spine. 2. Lippincott Williams, Philadelphia
Yao H, Gu WY (2007) Three-dimensional inhomogeneous triphasic finite-element analysis of physical signals and solute transport in human intervertebral disc under axial compression. J Biomech 40: 2071–2077
Zander T, Krishnakanth P, Bergmann G, Rohlmann A (2010) Diurnal variations in intervertebral disc height affect spine flexibility, intradiscal pressure and contact compressive forces in the facet joints. Comput Methods Biomech Biomed Eng 13: 551–557
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Karajan, N., Röhrle, O., Ehlers, W. et al. Linking continuous and discrete intervertebral disc models through homogenisation. Biomech Model Mechanobiol 12, 453–466 (2013). https://doi.org/10.1007/s10237-012-0416-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10237-012-0416-5