Abstract
Introduction Sagittal balance is a combination of a balance function (T1 maintained vertically over S1) that partially constrains the spine, the passive constraints provided by soft tissues and the active constraints – muscle force and gravity. Normal standing posture is likely to be the posture of minimum muscle activity and soft tissue energy. Observed deviation from this position would require muscle action. A mathematical model describing spinal balance without muscle activity is described.
Methods The spine was modeled as a series of articulations between the hip and T1 that were controlled by a third degree polynomial ‘spring’ function that approximates the force displacement curves as measured by Panjabi et al. T1 was constrained to remain over S1. Geometric data imported from the erect radiograph of a 34 female without back pain was used to set the zero point for the stiffness functions. All spring functions except the hip function were identical. The system was then perturbed by changing the rest disc space (or hip) angles. An initial smoothing function was used to ‘distribute’ this perturbation amongst several adjacent vertebrae as a guess. The model then minimized the total soft tissue energy to find the new position by treating the system as a series of damped rotational spring – mass constructs. Minimization was achieved using Euler’s method to solve a system of second order nonlinear ordinary differential equations. The iterations were run until oscillations ceased. The model was then perturbed by creating a series of kyphotic deformities at multiple levels and the results were observed.
Results Most perturbations converged to a minimum solution almost instantly. With the hip fixed, it was found that kyphotic deformities in the lower and mid lumbar spine led to compensatory lordosis at most other levels – particularly at the apex of the thoracic kyphosis. The spine tended to straighten and lengthen (possibly causing a rise in the centre of mass of the body). This tendency was substantially mitigated by allowing the hip joint to move. By trial and error, a spring function with of one tenth of the stiffness allowed the centre of gravity to move minimally and the compensatory lordosis occurred at segments closer to the induced kyphosis. When an apical thoracic kyphosis was applied with a fixed hip, the spine shortened with compensation being mostly by lordosis in the upper lumbar spine. When the hip was allowed to flex the tendency was for some of the compensation to occur at hip and for the spine to shorten further. The compensatory lordosis that developed at the level above an induced lumbar kyphosis could be partially corrected by applying a flexion moment. However as there is no muscle that is capable of applying such a moment over a single segment an alternative approach suggested that the hyperlordosis could be reduced by applying an extension moment to multiple segments above the hyperlordotic level.
Discussion Sagittal Spinal balance is complex. A minimum energy stiffness model may lead to further understanding of spinal balance. The prototype model suggests that the hip joint may have a role in preventing excessive lengthening (with a rise in the centre of gravity) of the spine. The model predicts extensor muscle contraction more than one level above a lumbar kyphosis.
The abstracts were prepared by Assoc Prof Bruce McPhee. Correspondence should be addressed to him at the Division of Orthopaedics, The University of Queensland, Clinical Sciences Building, Royal Brisbane Hospital, Herston, Brisbane, 4029, Australia.
