Direct and indirect measures of the permeability of the nucleus pulposus

Direct and indirect measures of the permeability of the

nucleus pulposus

Dr Phil RichesBioengineering Unit

Phil Riches, Meditech, 6 November 2008

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.


Osmotic pressure (≈ swelling stress)

Due to the presence of negatively charged proteoglycans, the disc has a high osmotic pressure, .

π contributes todisc swelling during rest

Resistance to compressive stress by augmenting the hydrostatic pressure.

convective transport of large nutrients.

Fluid flows from regions of low to regions of high .

increases with compression


Altered biosynthesis with loading

Iatridis et al., J.Bone Joint Surg. Am., 88, 41-46, 2006


Determining the permeability of IVD tissueThere are two ways:

Direct permeation tests– Fluid induced deformation

Indirect determination via fitting a mathematical model to experimental data– How good is the model?


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? 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?


INDIRECT METHODOLOGY


Large strain model of confined compressionHolmes J. Biomech. Eng. 108, 372-381, 1986.

U - displacement with respect to undeformed stateZ - is undeformed axial dimension - is the stretch ratio (= 1 + strain)HA0, , s0 are model parameters to be found

kk

seH

Z

Ukv

t

U

Ae

e

0

)1(12

2

0

2

2

0

21

2

1

Initial swelling stress


Permeability models, k()A few different models exist, with two common ones being:

The following model is proposed:)1(

0)( Mekk

]2/)1([

0

00

21

Mekk

Lai et al., 1980

Holmes and Mow, 1990

0

01

0 11

M

Me

e

kk


24 Bovine coccygeal samples

10mm diameter

1130 ± 140 μm 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

POROUS PLATENS

TO MTS

SAMPLE

Experimental Methods

Heneghan and Riches, J. Biomech., 41, 2411-2416, 2008

After 2 hours equilibrating with external salt solution, a ramp-hold 20% compression was applied at a strain rate of 2m/s.

Displacement, stress and time recorded.


Direct methodology

The models were fitted in 3 steps

– Initial swelling stress, s0

– HA0 and 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.



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.



After adjustment for the covariate…





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.


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)


DIRECT METHODOLOGY


Direct permeabilitymeasurement

Hydraulic permeability values vary from2x10-15 m4/Ns at = 1, to3x10-16 m4/Ns at = 0.35

Heneghan & Riches, J. Biomech. 41, 903-906, 2008

P

vhk

P = 30kPa; v ≈ 0.06 m/s


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)



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 = 0.3.

– Actual k0 may be three times that measured by direct permeation due to fluid flow induced deformation




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



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.


Acknowledgements

Paul Heneghan


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




Fluid pressure gradientFluid flow- Osmotic pressure gradient

Permeation

Internal reorganisation



Ramp phase

Consolidation



Hold phase




Expansion phase





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Direct and indirect measures of the permeability of the nucleus pulposus