Direct and indirect measures of the permeability of the nucleus pulposus

1. Direct and indirect measures of the permeability of the nucleus pulposus Dr Phil Riches Bioengineering Unit

2. Permeability • Permeability governs the relative movement of solid and fluid within porous media. • The intervertebral disc resists high loads by creating a hydrostatic pressure through a low permeability. • The rate at which fluid flow out of the disc, fluid pressure is lost, and hence the deflation of the disc, is governed by permeability. • During periods of rest, the disc swells and the rate of fluid uptake is also governed by permeability. • Since the disc is avascular, fluid uptake is also associated with nutrient uptake.

3. Osmotic pressure (≈ swelling stress) • Due to the presence of negatively charged proteoglycans, the disc has a high osmotic pressure, p. • π contributes to disc swelling during rest Resistance to compressive stress by augmenting the hydrostatic pressure. convective transport of large nutrients. • Fluid flows from regions of low p to regions of high p. • p increases with compression

4. Altered biosynthesis with loading Iatridis et al., J.Bone Joint Surg. Am., 88, 41-46, 2006

5. Determining the permeability of IVD tissue • There are two ways: • Direct permeation tests • Fluid induced deformation • Indirect determination via fitting a mathematical model to experimental data • How good is the model?

6. Questions • Can osmotic pressure effects be seen in permeability data from indirect measurement? • Is permeability affected by the disc’s swelling pressure? • Do ramp and hold phases have different parameters? • p should augment stress relaxation and hinder loading. Apparent permeability should be increased in former. • Is the direct method any better? • Can fluid flow induced deformation be analysed?

7. Indirect methodology

8. Large strain model of confined compression • Holmes J. Biomech. Eng. 108, 372-381, 1986. • U - displacement with respect to undeformed state • Z - is undeformed axial dimension • l - is the stretch ratio (= 1 + strain) • HA0, b, s0are model parameters to be found Initial swelling stress

9. Permeability models, k(l) • A few different models exist, with two common ones being: • The following model is proposed: Lai et al., 1980 Holmes and Mow, 1990

10. TO MTS POROUS PLATENS SAMPLE Heneghan and Riches, J. Biomech., 41, 2411-2416, 2008 Experimental Methods • 24 Bovine coccygeal samples • 10mm diameter • 1130 ± 140μm thick • Samples were tested in 0.15 M, 3.0 M and 6.0M NaCl, to vary s0 • External salt solution can negate osmotic effects After 2 hours equilibrating with external salt solution, a ramp-hold 20% compression was applied at a strain rate of 2mm/s. Displacement, stress and time recorded.

11. Direct methodology • The models were fitted in 3 steps • Initial swelling stress, s0 • HA0 and b were fitted using equilibrium data • k0 and M were determined from time dependent data using the Nelder-Mead simplex method • Data were fitted to both loading and stress relaxation phases. Parameters compared between phases.

13. Data Analysis • Preliminary graphs indicated that a linear relationship existed between Ln(k0) and Ln(s0). • An ANCOVA was used to assess the effects of Ln(s0), loading/stress relaxation and model on Ln(k0) and M. • The statistical model explained 80% of the variation in Ln(k0) and 50% of the variation in M.

15. After adjustment for the covariate…

19. Results & discussion • An increase in Ln(s0) pressure decreased Ln(k0) (p < 0.001) • Loading and stress relaxation phases resulted in different k0 and M values, and a significant interactions also existed with phase and Ln(s0) (all p < 0.001). • If the model was representative of the tissue, then these differences should not exist.

20. Conclusion • Before the model can be used reliably for permeability determination of IVD (and cartilage?) tissue, either • The constitutive equations for the large strain theory need to be amended to include osmotic pressure as a driving force for fluid flow, or • A triphasic large strain theory needs to be developed (the third phase being the negative ion phase)

21. Direct methodology

22. Heneghan & Riches, J. Biomech. 41, 903-906, 2008 Direct permeabilitymeasurement Hydraulic permeability values vary from 2x10-15 m4/Ns at l = 1, to 3x10-16 m4/Ns at l = 0.35 DP = 30kPa; v≈ 0.06 m/s

23. Modelling direct permeation • Using the aforementioned model, we can theoretically analyse the permeation experiment • Viscous drag associated with fluid flow will deform the tissue. • Two boundary conditions exist • Solid is fixed to top platen (clamped) • Boundary of tissue is free to deform due to viscous drag (unclamped)

25. Modelling conclusions • The model suggests: • Localised variation in strain at equilibrium • With applied strain, tissue becomes more homogenous, but a 20% difference in k still exists at l = 0.3. • Actual k0 may be three times that measured by direct permeation due to fluid flow induced deformation

28. Conclusion • Theoretical modelling predicts significant fluid flow induced deformation below 20% compression. • This can be “seen” in experimental data, although inconclusively. • Osmotic effects are not readily apparent from this experiment since external salt solution was constant at 0.15M NaCl

30. Overall conclusion • No method currently provides a gold standard measurement of permeability • Direct method will always suffer from flow induced deformation, unless very slow pumps are used. • Models need to be developed to incorporate osmotic effects into a large strain formulation.

31. Acknowledgements • Paul Heneghan

32. Indirect methodology • Models of the disc predict high localised strain, near the boundary of load application • Material properties need to be determined to high compressive strain to validate models. • Large strain theory is required

35. Permeation Internal reorganisation Fluid pressure gradient Fluid flow - Osmotic pressure gradient

36. Ramp phase Consolidation Fluid pressure gradient Fluid flow - Osmotic pressure gradient

37. Hold phase Internal reorganisation Fluid pressure gradient Fluid flow - Osmotic pressure gradient

38. Expansion phase Internal reorganisation Fluid pressure gradient Fluid flow - Osmotic pressure gradient