Analysis of Nonlinear Coupled Diffusion of Oxygen and Lactic Acid in Intervertebral Discs

Abstract

The transport of oxygen and lactate (i.e., lactic acid) in the human intervertebral disc was investigated accounting for the measured coupling between species via the pH level in the tissue. Uncoupled cases were also analyzed to identify the extent of the effect of such coupling on the solute gradients across the disc. Moreover, nonlinear lactic production rate versus lactic concentration and oxygen consumption rate versus oxygen concentration were considered. The nonlinear coupled diffusion equations were solved using an in-house finite element program and an axisymmetric model of the disc with distinct nucleus and anulus regions. A pseudotransient approach with a backward integration scheme was employed to improve convergence. Coupled simulations influenced the oxygen concentration and lactic acid concentration throughout the disc, in particular the gradient of concentrations along the disc mid-height to the nucleus-anulus boundary where the solutes reached their most critical values; minimum for the oxygen tension and maximum for the lactate. Results suggest that for realistic estimates of nutrient and metabolite gradients across the disc, it could be important to take into account the coupling between the rates of synthesis and overall local metabolite∕nutrient concentration.