Osmoviscoelastic finite element model of the intervertebral disc

Abstract

Intervertebral discs have a primarily mechanical role in transmitting loads through the spine. The disc is subjected to a combination of elastic, viscous and osmotic forces; previous 3D models of the disc have typically neglected osmotic forces. The fibril-reinforced poroviscoelastic swelling model, which our group has recently developed, is used to compute the interplay of osmotic, viscous and elastic forces in an intervertebral disc under axial compressive load. The unloaded 3D finite element mesh equilibrates in a physiological solution, and exhibits an intradiscal pressure of about 0.2 MPa. Before and after axial loading the numerically simulated hydrostatic pressure compares well with the experimental ranges measured. Loading the disc decreased the height of the disc and results in an outward bulging of the outer annulus. Fiber stresses were highest on the most outward bulging on the posterior-lateral side. The osmotic forces resulted in tensile hoop stresses, which were higher than typical values in a non-osmotic disc. The computed axial stress profiles reproduced the main features of the stress profiles, in particular the characteristic posterior and anterior stress which were observed experimentally.

Appendix

Abaqus has a biphasic model of the form:

$${\text{Momentum}}\;{\text{balance}}:\quad \vec{\nabla }{\mathop { * \varvec{\upsigma}_{e} - \vec{\nabla }}\limits }p = 0$$

$${\text{Mass}}\;{\text{balance}}:\quad \vec{\nabla } * \dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {u} } + \vec{\nabla }\vec{q}=0$$

$${\text{Darcy's}}\;{\text{law}}:\quad \vec{q} = - k\vec{\nabla }p$$

With Dirichlet boundary conditions: [u] =0 and [p] =0.

Here ${\varvec{\upsigma}}_{\mathbf{e}}$ is the effective stress, p the fluid pressure, $\vec{q}$ the fluid flux, k the permeability and u the displacements.

The swelling behavior through the Donnan osmotic theory was included into the biphasic theory as follows:

$${\text{Mass}}\;{\text{balance}}:\quad \vec{\nabla } * \dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{u} }_{s} - \vec{\nabla } * k\vec{\nabla }{\left({p - \Delta \pi } \right)} = 0$$

$$\hbox{substitution}:\quad\vec{\nabla} * {\left({{\varvec{\upsigma}}_{{\rm non-fibrillar}} + \rho _{c} {\sum\limits_{{\rm all\;fibrils}\,{\mathbf i}} {\sigma^{i}_{\rm f} {\kern 1pt} \vec{e}^{i}_{\rm f} {\kern 1pt} \vec{e}^{i}_{\rm f} } }} \right)} - \vec{\nabla}{\left({\mu ^{\rm f} + \Delta \pi } \right)} = 0$$

$$\vec{\nabla } * \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {u}_{\rm s} - \vec{\nabla } * k\vec{\nabla }\mu ^{f} = 0$$

$$\mu ^{f} = {\left({p - \Delta \pi } \right)}$$

Here μ^f is the chemical potential and Δπ is the osmotic pressure given by Eq. 4 with Dirichlet boundary conditions: [u] = 0 and [μ^f] = 0.