Planar biaxial extension of the lumbar facet capsular ligament reveals significant in-plane shear forces

Abstract

The lumbar facet capsular ligament (FCL) articulates with six degrees of freedom during spinal motions of flexion/extension, lateral bending, and axial rotation. The lumbar FCL is composed of highly aligned collagen fiber bundles on the posterior surface (oriented primarily laterally between the rigid articular facets) and irregularly oriented elastin on the anterior surface. Because the FCL is a capsule, it has multiple insertion sites across the lumbar facet joint, which, along with its material structure, give rise to complicated deformations in vivo. We performed planar equibiaxial mechanical tests on excised healthy cadaveric lumbar FCLs (n=6) to extract normal and shear reaction forces, and fit sample-specific two-fiber-family finite element models to the experimental force data. An eight-parameter anisotropic, hyperelastic model was used. Shear forces at maximum extension (mean values of 1.68N and 3.01N in the two directions) were of comparable magnitude to the normal forces perpendicular to the aligned collagen fiber bundles (4.67N) but smaller than normal forces in the fiber direction (16.11N). Inclusion of the experimental shear forces in the model optimization yielded fits with highly aligned fibers oriented at a specific angle across all samples, typically with one fiber population aligned nearly horizontally and the other at an oblique angle. Conversely, models fit to only the normal force data resulted in a broad range of fiber angles with low specificity. We found that shear forces generated through planar equibiaxial extension aided the model fit in describing the anisotropic nature of the FCL surface.

FIGURE 1

Lumbar facet capsular ligament (FCL) preparation for planar biaxial mechanical testing. (a) Schematic of the posterior lumbar spinal column from the third lumbar vertebra (L3) to the fifth lumbar vertebra (L5). The FCL (not shown) spans between the rigid superior and inferior articular facets (S/IAF) that bilaterally flank the spinous processes (SP) (b) Cadaveric L4–L5 motion segment in same view as (a) with posterior musculature removed to expose the bilaterial facet joint capsules. (c) The capsule was resected from the motion segment, and the thin surface membrane was removed. (d) Some trabecular bone was removed to create a planer orientation. The collagen fiber bundles were preferentially oriented between the remaining rigid articular facets (BONE), and free ligament ends (LIG) were along the perpendicular axis. (e) The sample was loaded into grips connected to load cells in the planar biaxial setup. The fibers were aligned on the horizontal axis (Axis 1) and the ligament tabs on the vertical axis (Axis 2). Normal forces were measured from all four load cells, and shear forces were measured through load cells attached to the grips labeled INFERIOR and IAF (L4). TP: transverse process; SP: spinous process; IAF: inferior articular facet; SAF: superior articular facet

FIGURE 2

Planar biaxial mechanical tests give (a) normal force- and (b) shear force-stretch curves for n = 6 samples. (a) Along the fiber axis (N1, dashed), forces reached a larger peak magnitude than did those along the ligament axis (N2, solid). (b) Peak shear forces (pooled S1 and S2) were lower than both peak N1 forces and N2 forces. Peak shear forces of S2 (solid) were of lower magnitude than those of S1 (dashed).

FIGURE 3

Model fit (lines) to a representative experimental (a) normal force- and (b) shear force-displacement data. The model accurately captures the sign of the normal and shear forces, but cannot capture the full non-linearity of S2. (c) Optimal parameter values for the model fits for all optimization simulations (n = 6) with standard deviations. The parameters α and β had the smallest distribution of values. The primary fiber angle (θ₁) was distinctly negative in value and was highly aligned (b₁), whereas the secondary fiber angle (θ₂) had a larger distribution and less strength of alignment (b₂).

FIGURE 4

Goodness of fit for each parameter obtained by dividing parameter values by their respective 95% confidence region width is shown for the (a) parameters fit with shear and normal force data and (b) parameters fit with only normal force data. Good parameter estimates have a value greater than one. (a) The shear-and-normal-force model fit gave the fiber angles (θ₁, θ₂) with high confidence. The fiber distribution parameter (b₁) falls below zero because the upper 95% confidence bound was calculated as infinite for half of the samples, generating a large confidence region width. The low confidence for b₁ suggests that once the fibers are strongly aligned, there is little effect if they become even more strongly aligned. (b) The normal-force-only model fit the fiber angles (θ₁, θ₂) with a low-level of confidence and instead, specified the matrix parameter (C₁) with a high confidence.

FIGURE 5

Surface displacement maps showing displacement vectors and shear (E_XY) strains from a representative experiment along with the predictions from the model fit to the force data. Experiment and model fit are both at 12% equibiaxial extension. The RMS error for this representative sample is 5.29% shear strain.

FIGURE 6

Comparison of the shear force model fit (Row 1) and normal force model fit (Row 2) for a representative sample. (Col a) The primary fiber angles (bold) from the optimizations fit to the opposite sign, and both secondary fiber angle fits are relatively horizontally aligned. (Col b) Despite the opposite primary fiber angle sign, both models capture the normal force fits as expected, but (Col c) the normal force model does not match the sign of the shear forces.