2.1 Vectors, Moment Arms, Bending Moments, and Axes of Rotation

Forces applied to the spine can be broken down into component vectors. A vector is defined here as a force oriented in a fixed and well-defined direction in three-dimensional space (Fig. 2.1).

A force vector may act on a lever (moment arm), causing a bending moment. The bending moment applied to a point in space causes rotation, or a tendency to rotate, about an axis. This axis, regarding the spine, is termed the instantaneous axis of rotation (IAR). The word instantaneous is included to emphasize that the axis of rotation changes (or rather moves), depending on the loads and forces applied. To establish an easily defined and reproducible coordinate system, the standard Cartesian coordinate system has been applied to the spine. In this system, there are three axes: the x, y, and z axes. Several definitions of the x, y, and z axes have been used. To simplify terminology, the terms rostral, caudal, ventral, dorsal, right, and left are used here. Rotational and translational movements can occur about these axes. This results in 12 potential movements about the IAR: two translational movements along each of the three axes (one in each direction) and two rotational movements around each of the axes (one in each direction). These potential movements may also be considered in terms of degrees of freedom; thus, six degrees of freedom exist about each IAR (Fig. 2.2).

For our purposes, the IAR is the axis about which each vertebral segment rotates at any given instant. This is, by definition, the center of the coordinate system (in the plane perpendicular to the IAR) for each motion segment. When a spinal segment moves, there is an axis passing through, or close to, the vertebral body that does not move; this is the axis about which the vertebral body rotates (the IAR). Its location is variable, depending on the intrinsic curvature of the spine, as well as other factors.^3,​4 Multiple factors, such as degenerative disease, fractures, ligamentous injuries, instrumentation and/or fusion placement, and segmental motion, can affect the position of the IAR (Fig. 2.3). For example, the IAR is affected by the extent of degeneration and deformation in the spondylotic spine with a lysis of the pedicle (spondylolysis). The IAR migrates rostrally as the extent of the pars defect advances and the wedge deformity progresses.⁵

In a sense, the IAR is a fulcrum. For example, if the spine is flexed, all points ventral to the IAR come closer together and all points dorsal to the IAR move farther apart (Fig. 2.4). It is important to note that these considerations are very important clinically. For example, both cervical spine flexion and extension can decompress the neuroforamina, depending on the location of the IAR. In the case in which the IAR is located in the region of the facet joint, flexion causes neuroforaminal compression and extension causes neuroforaminal decompression. The surgeon can determine and dictate such. For example, if the surgeon desires to achieve decompression of the neuroforamina via extension of the spine, the axis about which the motion segment is extended (IAR) must be dorsal to the neuroforamina—perhaps in the region of the facet joints (Fig. 2.5).⁶

The location of the IAR at the time of spinal column failure dictates the resultant failure type. Hoshikawa et al observed such in a porcine model. They observed that the motion axis of fracture (MAF), essentially equivalent to the IAR at the time of failure, contributed to flexion–distraction injuries, even when vertical compressive loads were applied.⁷ Such an observation should cause one to consider carefully the location of the MAF or IAR when evaluating spine trauma patients. Such could aid in the clinical decision-making process.

The IAR should be considered dynamic. As spinal movement occurs, the IAR of each involved spinal segment moves. The IAR is derived, in the clinical situation, from dynamic radiographs (i.e., flexion and extension radiographs).

The IAR, however, depends on the method of determination. In the present context, the IAR is theoretical. In the spine exposed to excessive loads at the moment of failure, the IAR is often located within the vertebral body’s confines in the sagittal plane (see Chapter 5). This assumes that the vertebral body pivots about a point within, ventral to, or dorsal to its confines (Fig. 2.6a).⁴

Another theoretical assumption (or clinical situation) is that flexion or extension, as elicited by adjacent vertebral body flexion or extension about an intervertebral disc, results in positioning of the IAR in the region of the intervertebral disc (Fig. 2.6b). Therefore, the location of the IAR depends on the theoretical foundation on which its definition is based and the manner in which it is determined. The IAR can be determined as by White and Panjabi (Fig. 2.6c).⁸ The center of rotation (COR), as applied by Smith and Femie, is similar to the IAR.⁹ Its determination and clinical application, however, present problems similar to those encountered with the IAR (Fig. 2.6d).^9,​10

When rotation is superimposed on translation, the resultant component of movement described by the translational movement vector is called the helical axis of motion (HAM) (Fig. 2.6e). It is oriented in the direction of the translational movement. A screw motion can be defined, in part, by this parameter. It must be emphasized that the determination of each of the axes described here is subject to error.

The concepts of the moment arm and the bending moment are critical to the understanding of spinal biomechanics. The moment arm associated with a spinal implant is defined as that “imaginary lever” that extends from a point (IAR) to the position of application of force to the spine (perpendicular to the direction of the applied force). This is true regardless of the nature of the force’s application, whether natural (e.g., from ligaments) or extrinsic (e.g., via instrumentation constructs). The bending moment (M) is defined as the product of the force (F) applied to the lever arm and the length of the lever arm (D):

where M = bending moment, F = applied force, and D = the perpendicular distance from the force vector to the IAR (moment arm; see Fig. 2.6). The bending moment is effectively the torque applied by the force (circular force). More correctly, torque is a moment with magnitude. The application of a bending moment results in a concentration of stress, such that failure becomes more likely, as depicted in Fig. 2.7a, b.

In the pages that follow, some of the discussion and associated illustrations address concepts related to the bending moment. In these illustrations, the bending moment is portrayed as a curved arrow, with the center of the arc being the IAR.

2.2 Paradoxical Spinal Motion

Paradoxical spinal motion is the unexpected and potentially untoward segmental spinal movement that occurs during the application of flexion, extension, or rotation stresses to the involved spinal segment and adjacent segments. It occurs in two circumstances: (1) in cases of segmental spine instability and (2) in cases in which stabilization techniques (spinal implants or external splints) are used that limit motion between two nonadjacent vertebrae with at least two intervertebral discs located between the termini of the implant or splint (Fig. 2.8). In the case of segmental spine instability, paradoxical movement can occur at the index and adjacent levels. In the case of stabilization techniques, the suspension of vertebral body segments between rigidly immobilized segments allows segmental muscular attachments to cause segmental movement in a paradoxical manner (snaking). Snaking is a characteristic type of movement of spinal segments in response to external force applications. With such movement, the sum of the movements of individual spinal motion segments is greater than the overall spinal movement observed.¹¹ In some clinical circumstances, it can be objectively assessed.³ In these cases, it can be quantified by measuring the overall movement between the rigidly immobilized rostral and caudal components. This is subtracted from the sum of the absolute values of each of the individual intervening segmental movements (see Fig. 2.8).¹¹

The paradoxical motion phenomenon may become significant with external spinal splinting or in cases in which an instrumented spine is not instrumented at every segmental level (e.g., only at the termini of the construct). In either case, movement of the suspended spinal segments can occur (between the extremes of the fixation). Paradoxical motion and related phenomena are difficult to predict or to reproduce in the laboratory.¹²