Analysis of nonlinear coupled diffusion of oxygen and lactic acid in intervertebral discs.

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.

[1]  S. Holm,et al.  Factors Influencing Oxygen Concentration Gradients in the Intervertebral Disc: A Theoretical Analysis , 1991, Spine.

[2]  S Holm,et al.  Nutrition of the intervertebral disc: solute transport and metabolism. , 1981, Connective tissue research.

[3]  S. Rajasekaran,et al.  ISSLS Prize Winner: A Study of Diffusion in Human Lumbar Discs: A Serial Magnetic Resonance Imaging Study Documenting the Influence of the Endplate on Diffusion in Normal and Degenerate Discs , 2004, Spine.

[4]  Jill P. G. Urban,et al.  The effect of extracellular pH on matrix turnover by cells of the bovine nucleus pulposus , 2003, European Spine Journal.

[5]  O. Zienkiewicz.  The Finite Element Method In Engineering Science , 1971 .

[6]  Ruth M Ripley,et al.  Metabolism of the Intervertebral Disc: Effects of Low Levels of Oxygen, Glucose, and pH on Rates of Energy Metabolism of Bovine Nucleus Pulposus Cells , 2005, Spine.

[7]  K. Huebner.  The finite element method for engineers , 1975 .

[8]  S Holm,et al.  Nutrition of the intervertebral disc: effect of fluid flow on solute transport. , 1982, Clinical orthopaedics and related research.

[9]  H. Gruber,et al.  Recent Advances in Disc Cell Biology , 2003, Spine.

[10]  Manohar M. Panjabi,et al.  Clinical Biomechanics of the Spine , 1978 .

[11]  Keita Ito,et al.  Fluid flow and convective transport of solutes within the intervertebral disc. , 2004, Journal of biomechanics.

[12]  A. Ejeskär,et al.  Oxygen tension measurements in the intervertebral disc. A methodological and experimental study. , 1979, Upsala journal of medical sciences.

[13]  D. McNally.  BIOMECHANICS OF THE INTERVERTEBRAL DISC - DISC PRESSURE MEASUREMENTS AND SIGNIFICANCE , 1995 .

[14]  A R Hargens,et al.  Intervertebral disc nutrition. Diffusion versus convection. , 1986, Clinical orthopaedics and related research.

[15]  A. Nachemson,et al.  Factors involved in the nutrition of the human lumbar intervertebral disc: cellularity and diffusion of glucose in vitro. , 1975, Journal of anatomy.

[16]  Sally Roberts,et al.  Degeneration of the intervertebral disc , 2003, Arthritis research & therapy.

[17]  Edward C. Benzel,et al.  Biomechanics of the Spine , 2005 .

[18]  J. Urban,et al.  Nutrition of the Intervertebral Disc , 2004, Spine.

[19]  C. Peter Winlove,et al.  Oxygen and Lactate Concentrations Measured in Vivo in the Intervertebral Discs of Patients With Scoliosis and Back Pain , 1998, Spine.

[20]  A. Nachemson,et al.  In vitro diffusion of dye through the end-plates and the annulus fibrosus of human lumbar inter-vertebral discs. , 1970, Acta orthopaedica Scandinavica.

[21]  R. Moore,et al.  The relationship between height, shape and histological changes in early degeneration of the lower lumbar discs , 1998, European Spine Journal.

[22]  A Shirazi-Adl,et al.  On the fibre composite material models of disc annulus--comparison of predicted stresses. , 1989, Journal of biomechanics.

[23]  H. Tsuji,et al.  Water Diffusion Pathway, Swelling Pressure, and Biomechanical Properties of the Intervertebral Disc During Compression Load , 1989, Spine.

[24]  J A Buckwalter,et al.  Aging and degeneration of the human intervertebral disc. , 1995, Spine.

[25]  K. A. Jackson.  Mathematics of Diffusion , 2005 .

[26]  S. Bibby,et al.  Effect of nutrient deprivation on the viability of intervertebral disc cells , 2004, European Spine Journal.

[27]  J. Urban,et al.  Effects of low oxygen concentrations and metabolic inhibitors on proteoglycan and protein synthesis rates in the intervertebral disc , 1999, Journal of orthopaedic research : official publication of the Orthopaedic Research Society.

[28]  H. Horner,et al.  2001 Volvo Award Winner in Basic Science Studies: Effect of Nutrient Supply on the Viability of Cells From the Nucleus Pulposus of the Intervertebral Disc , 2001, Spine.

[29]  A. Nachemson,et al.  Correlation between lactate levels and pH in discs of patients with lumbar rhizopathies , 1968, Experientia.

[30]  A Shirazi-Adl,et al.  Finite Element Study of Nutrient Diffusion in the Human Intervertebral Disc , 2003, Spine.

[31]  A. Nachemson,et al.  Carbohydrate metabolism and concentration profiles of solutes in the canine lumbar intervertebral disc. , 1982, Acta physiologica Scandinavica.