Presented at: The 2nd Annual Advances in Conservative Health Care Conference, Lombard, IL, Oct., 1983
Treating the lumbar spine as a deformable, slightly buckled column, the curvature produced by axial loading is calculated. The fundamental relation used is the principle of conservation of energy, i.e. that the work done on the spine during the process of buckling is stored as strain energy in the intervertebral joints. The work done by an axial load is equal to that load multiplied by the vertical distance it moves during the buckling. The strain energy stored in a vertebral joint is proportional to both the angle of bending that occurs and the stiffness of the joint.
For calculation of the stress and strain present in an intervertebral joint subjected to extension, a simple model of the disc was generated. All of the stiffness of the vertebral joints in bending was attributed to the elasticity of the disc. The disc was assumed to be of circular cross-section, and to deform symmetrically in flexion type movements. During extension of the modeled vertebral joint, it is predicted that the posterior half of the disc would be compressed, that the anterior half would be elongated and that at the central axis of bending there is no deformation. Using this approach the expression for deformation of a section of the disc located Y away from the central axis is:
Assuming the disc material is uniformly distributed between rigid vertebral bodies, the stress due to some bending moment, M, was approximated by the Elastic Flexure Formula for bending in beams:
Since the force working on the model spine is acting along the length of the spine, the actual moments needed in the calculation of stress at each segment are unknown. However, the moment acting at a joint can be inferred from the stiffness of the joint and the deformation that occurs there. Data are available in the literature which relate deformation of cadaver spinal segments to the moments required to produce such deformations. These data may be used to obtain an approximation for stiffness. Using such an approximation for stiffness in extension (after Shultz, 1978), the stress can be expressed in terms of the angle of deformation the disc undergoes:
In an ideal system where energy is conserved, the strain energy is balanced by work done by the buckling force. The work done by an axial force working on the spine is the magnitude of that force multiplied by the change in vertical height of the spine as it buckles. Treating the spine as a series of incompressible segments which bend equally at each joint, the change in height may be found by simple geometric considerations. Setting the strain energy equal to the work done, the final derived relation for the angle of bending at each joint of a spine of original height, H, due to a buckling load, P, is:
Given a certain spinal shape it may be possible to predict what compressive forces are acting in a portion of the spine. One observation made from this work is that if spinal buckling is uniform, i.e. that all segments have equal bending angles between them, then the stress is uniformly distributed along the column. In other words, the intervertebral disc under the greatest stress would be the most deformed. Further, the results suggest that the joints tend to increase in stiffness at greater bending angles.
The findings of this work are limited by the fact that only small angles of deformation were used and that the experimental data used shows large interspecimen variability. Also, since the experimental data are gathered from the spines of cadavers, there is no information on the contribution of muscle tension to the stiffness of the joints. The muscles may play a key role in maintaining or reducing spinal curvatures; muscles may be responsible for increasing the compression between vertebrae considerably. Some measure of the tension in the paraspinal muscles is necessary for a more complete assessment of the structural component of spinal curvature.