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TECHNICAL PAPERS: Fluids/Heat/Transport

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

[+] Author and Article Information
D. Mokhbi Soukane

Department of Mechanical Engineering,  Ecole Polytechnique, Montréal, Québec, Canada

Department of Mechanical Engineering,  Ecole Polytechnique, Montréal, Québec, Canada

J. P. Urban

Physiological Laboratory,  Oxford University, Oxford, England

1

Corresponding author. Fax: 514- 340 4176; e-mail: abshir@meca.polymtl.ca

J Biomech Eng 127(7), 1121-1126 (Jul 29, 2005) (6 pages) doi:10.1115/1.2073674 History: Received December 02, 2004; Revised June 09, 2005; Accepted July 29, 2005

## 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.

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## Figures

Figure 1

Axisymmetric finite element mesh of the intervertebral disc composed of the anulus and nucleus regions with boundary supplies from the end-plate and the anulus periphery. Due to the symmetry about the disc horizontal mid-plane, only half the disc is analyzed.

Figure 2

Measured variation of pH versus lactic acid concentration in the nucleus region considered in the coupled analyses (24)

Figure 3

Comparison of lactic acid concentrations computed by the developed in-house program and ABAQUS ; (a) along the radius at z=0 showing also the variation of the lactic acid production versus lactic acid concentration taken for this case, (b) along the z axis at r=0

Figure 4

Variation of oxygen concentration computed by the coupled and uncoupled models with constant pH levels of 5.4, 6.0, and 6.6; (a) along the radius at z=0, (b) along the z axis at r=0

Figure 5

Variation of lactic acid concentration computed by the coupled and uncoupled models with constant oxygen tension of 5, 10, and 15 kPa; (a) along the radius at z=0, (b) along the z axis at r=0

Figure 6

Computed results in the disc in the coupled model (see Table 1 for input data); (a) oxygen concentration in kPa, (b) lactic acid concentration in nmol∕mm3, (c) pH level

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