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Abstract
An in-house finite element program is developed to solve the nonlinear coupled diffusion equations governing the transport of the solutes in the human intervertebral discs considering both axisymmetric and 3D geometries. Because of the strong nonlinearity of the source terms, a pseudo transient approach with a backward integration scheme is employed to improve convergence. The supply sources are assumed at the outer annulus periphery and disc endplates above nucleus and inner annulus regions with the region above the outer annulus remaining in all cases completely impermeable. The models represent distinct regions; cartilaginous end-plates, nucleus, inner annulus and outer annulus. The solute diffusivities are assumed isotropic within each region but vary from a region to another as is the case for cell densities and water content.
The model is initially validated by comparison of its results with those obtained by a commercially-available package program. The importance of coupling between consumption/production-concentration rates with the pH level is then examined. The effects of changes in the endplate exchange area (EA) adjacent to the nucleus and/or the inner annulus on the transport of nutrients are subsequently determined by altering the porosity or equivalently the relative diffusivity between 100% (completely permeable) and 0% (completely impermeable). Moreover, changes in the disc geometry as well as tissue diffusivities under static compression loading are studied assuming overall fluid losses of 11% and 20%. Alterations in solute diffusion following a central endplate fracture as seen in Schmorl's node are also investigated. Finally, effects of increases in cell metabolic rates by 25%, 50% or 100% following growth factor injection and of alterations in the lumbar posture (kyphotic or lordotic) by ±2° or ±4° on extreme values of nutrient and metabolite concentrations and their spatial locations are studied.
Oxygen tension as well as glucose concentration decreased with distance from the source of supply at the end-plates and annulus outer periphery, falling to a minimum at the disc center where the distance from blood supply is greatest. Inversely, the lactic acid concentration was highest at the center of the disc and lower at the source supply regions. The simulations indicated that the coupling influenced the oxygen and lactic acid concentrations throughout the disc, in particular the gradient of concentrations along the disc mid-height at the nucleus-annulus boundary where the solutes reached their most critical values; minimum for the oxygen tension and maximum for the lactate.
The endplate disruptions (calcifications and fractures) and mechanical loads substantially influenced the distribution of nutrients throughout the disc as well as the magnitude and location of critical concentrations; maximum for the lactic acid and minimum for oxygen and glucose. Computations also demonstrated a non-linear dependence of species concentrations on exchange area of the endplates; results pointed to a critical threshold below which the disc nutrition is disrupted significantly. In this respect, the glucose appears to be the critical solute for the survival of the disc cells.
In a degenerated disc, where the nutrient supply is already disrupted, the effects of metabolic stimulation are computed to be severe where growth factor injection may further accelerate disc degeneration rather than reversing it. Simulations also indicated that a kyphotic posture associated with forward flexion increased oxygen and glucose concentrations in the intervertebral disc whereas these concentrations fell under backward flexion postures (i.e., extension). These relative differences would further magnify had a flexed posture been compared directly with an extended rather than a neutral posture.
Even though the trend of solute concentrations was predicted to remain the same in both axisymmetric and 3D model studies, the amplitude and position of the extreme concentrations were substantially modified demonstrating the importance in realistic representation of the disc 3D geometry. Results also suggest that for realistic estimates of nutrient and metabolite gradients across the disc, it is important to take into account the coupling between the rates of synthesis and overall local metabolite/nutrient concentrations. (Abstract shortened by UMI.)
