Computational strategies toward the modelling of the intervertebral disc.
PhD thesis, University of Glasgow.
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Lumbar back pain has considerable socio-economical impacts, motivating a recently increasing interest from the research community. Yet, mechanisms triggering pain are not fully understood and this considerably hinders the development of efficient treatments and therapies. The objective of this thesis is to participate to the general understanding of the biomechanics of the spine through the development of computational strategies for the intervertebral disc.
The intervertebral disc is a complex structure mainly comprised of the nucleus pulposus and the annulus fibrosus. The nucleus pulposus is the gelatinous core of the disc, which consists of a charged and hydrated extra-cellular matrix and an ionised interstitial fluid. It is enclosed in the annulus fibrosus which is formed by concentric layers of aligned collagen fibre sheets, oriented in an alternating fashion.
A biphasic swelling model has been derived using mixture theory for soft, hydrated and charged tissues in order to capture the salient characteristics of the disc's behaviour. The model fully couples the solid matrix under finite deformations with the ionised interstitial fluid. The nucleus is assumed to behave isotropically while the effects of the collagen fibres in the annulus fibrosus are accounted for with a transversely isotropic model. The fixed negative charges of the proteoglycans, which induce an osmotic pressure responsible for the swelling capabilities of the disc, are constitutively modelled under the simplifying Lanir hypothesis.
A Newton-Raphson solver was specifically built to solve the resulting nonlinear system of equations, together with a verification procedure to ensure successful implementation of the code. This was first reduced to the one dimensional case in order to demonstrate the appropriateness of the biphasic swelling model. The three dimensional model exhibited numerical instabilities, manifesting in the form of non-physical oscillations in the pressure field near boundaries, when loads and free-draining boundary conditions are simultaneously applied. As an alternative to considerable mesh refinement, these spurious instabilities have been addressed using a Galerkin Least-Square formulation, which has been extended for finite deformations. The performance and limitations of the GLS framework, which drastically reduces the pressure discrepancies and prevents the oscillations from propagating through the continuum, are demonstrated on numerical examples. Finally, the current state of the model's development is assessed, and recommendations for further improvements are proposed.
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