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Multiphasic Intervertebral Disc Mechanics: Theory and Application

Abstract

The human spine is the flexible support structure of our body. Its geometric shape is a result of the human evolutionary history, where especially the lumbar spine area (L1–L5) is most at risk of causing discomfort resulting from mechanical stresses. Herein, the intervertebral discs (IVD) between the vertebral bodies are the most susceptible elements, as these avascular structures have to provide the flexibility. It is widely accepted that the IVD are often the trigger for back pain. In the context of biomechanical research, it is therefore important to develop a model for the human lumbar spine with particular focus on the IVD.

The objective of the presented work is to subject the IVD of the lumbar spine to continuum-biomechanical research. Herein, a three-dimensional (3-d) finite-element model is developed that allows to estimate the influence of lumbar spine motion, i.e., bending, torsion and compression, on the resulting stress-field inside the IVD. Following this, the model can be utilised to detect inappropriate or excessive loading of the spine. The theoretical description of the IVD is based on a multi-phase continuum approach in the framework of the well-known “Theory of Porous Media” (TPM). This is a natural choice resulting from the avascular composition of the IVD. In general, IVD tissue is categorised as charged hydrated material with mechanical and electro-chemical internal coupling mechanisms. In order to capture these couplings, the underlying model incorporates an extracellular matrix (ECM) with fixed negative charges, which is saturated by a mixture of a liquid solvent and ions. Following the basic concept of the TPM, a volumetric averaging process is prescribed leading to volume fractions for the pore space and the solid skeleton as well as molar concentrations for the ion species in the pore fluid.

In detail, the IVD exhibits a gelatinous core known as the nucleus pulposus (NP), and an onion-like surrounding structure consisting of anisotropic crosswise fibre-reinforced lamellae, the annulus fibrosus (AF). Both regions are seamlessly merging into each other and consist of mostly collagen fibres of varying strength and direction as well as proteoglycans with adhering negative charges. As a result of these fixed negative charges and the fact that the interstitial fluid carries positively and negatively charged ions, a model is created that describes the mutually coupled behaviour of solid deformation and fluid flow. To illustrate the resulting coupled swelling and shrinkage process, it is sufficient to recall that the size of the human body is reduced by roughly 2 centimetres during the waking phase of the day. This change in height is triggered by mechanical loads stemming from the body weight, which squeezes the interstitial fluid out of the IVD, thereby losing altitude. Simultaneously, an electro-chemical imbalance is generated, which is compensated during the nocturnal resting phase, where the interstitial fluid is driven back into the IVD due to osmotic effects.

In summary, the presented paper provides a review on IVD mechanics and can be understood as a compilation of relevant information giving valuable guidance to researchers starting to work in this challenging field. In this regard, the paper opens with anatomical and chemical fundamentals of IVD tissue with a particular focus on the inherent inhomogeneities. This is followed by an introduction to the continuum-mechanical fundamentals including an overview of the TPM as well as the inelastic non-linear kinematics and the balance equations of the resulting porous continua. Thereafter, the constitutive modelling process is illustrated with a particular focus on the thermo-dynamically consistent modelling process of the intrinsic viscoelasticity of the ECM as well as the inhomogeneous characteristics of the embedded collagen fibres and the viscous pore fluid. The resulting system of coupled partial differential equations is then numerically discretised using the finite-element method which finally allows the simulation of deformation processes of the IVD using either single- or multi-processor machines. Moreover, an automated computation scheme is presented to systematically capture the inhomogeneities of the IVD. The paper is closed with several sample applications, which embrace the capabilities of the presented computational model on the one hand and give advice for further validation in terms of the applied material parameters.

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Notes

  1. 1.

    Nuclei pulposi from young subjects absorb fluid equal to 95 % of their original weight when soaked in saline, while those from adult and aged subjects absorbed the equivalent of 55 % and 25 % of their weight, respectively, cf. Naylor [116] and qualitatively in [157].

  2. 2.

    In vivo means “in the living organism” and is contrary to ex vivo standing for “out of the body”.

  3. 3.

    Clayey silt has a hydraulic conductivity of about 10−8 m/s, while IVD tissue reaches down to a value of about 10−11 m/s.

  4. 4.

    Porous media Adaptive Nonlinear finite element solver based on Differential Algebraic Systems (http://www.get-pandas.com).

  5. 5.

    Mesh, Multigrid and More of C. Wieners (http://www.mathematik.uni-karlsruhe.de/~wieners).

  6. 6.

    Calculations were executed on the Beowulf Linux cluster of the Institute of Applied Mechanics (Chair of Continuum Mechanics) at the Universität Stuttgart. It consists of 98 Opteron CPU (2.2 GHz, 47 dual boards and one quad board), 104 GB RAM (at least 1 GB/CPU) and two Gigabit networks, which share the load stemming from the system administration and the data transfer of the parallel computations.

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Acknowledgements

I thank the German Research Foundation (DFG) for funding the research project “Diffusions- und Strömungsprozesse in der anisotropen menschlichen Bandscheibe” under grant number ‘Eh 107/16’.

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Karajan, N. Multiphasic Intervertebral Disc Mechanics: Theory and Application. Arch Computat Methods Eng 19, 261–339 (2012). https://doi.org/10.1007/s11831-012-9073-1

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