رفتار ساختاری بخش های حرکتی ستون فقرات کمری انسان

The objectives of this study were to obtain linearized stiffness matrices, and assess the linearity and hysteresis of the motion segments of the human lumbar spine under physiological conditions of axial preload and fluid environment. Also, the stiffness matrices were expressed in the form of an ‘equivalent’ structure that would give insights into the structural behavior of the spine. Mechanical properties of human cadaveric lumbar L2-3 and L4-5 spinal motion segments were measured in six degrees of freedom by recording forces when each of six principal displacements was applied. Each specimen was tested with axial compressive preloads of 0, 250 and 500 N. The displacements were four slow cycles of ±0.5 mm in anterior–posterior and lateral displacements, ±0.35 mm axial displacement, ±1.5° lateral rotation and ±1° flexion-extension and torsional rotations. There were significant increases with magnitude of preload in the stiffness, hysteresis area (but not loss coefficient) and the linearity of the load-displacement relationship. The mean values of the diagonal and primary off-diagonal stiffness terms for intact motion segments increased significantly relative to values with no preload by an average factor of 1.71 and 2.11 with 250 and 500 N preload, respectively (all eight tests p<0.01). Half of the stiffness terms were greater at L4-5 than L2-3 at higher preloads. The linearized stiffness matrices at each preload magnitude were expressed as an equivalent structure consisting of a truss and a beam with a rigid posterior offset, whose geometrical properties varied with preload. These stiffness properties can be used in structural analyses of the lumbar spine.

The mechanical function of the spine is the summation of the behavior of its individual motion segments, where a motion segment is a structural unit of the spine consisting of two vertebrae and the intervening soft tissues (Fig. 1). Motion segment behavior is a key component of biomechanical analyses of the spine, including analyses of spinal loading (Stokes and Gardner-Morse, 2001), dynamics of injury (Kasra et al., 1992; Pankoke et al., 2001), spinal stability (Bergmark, 1989; Cholewicki and McGill, 1996; Gardner-Morse et al., 1995; Gardner-Morse and Stokes, 1998), and simulations of surgery (Aubin et al., 2003; Stokes and Gardner-Morse, 1993).Isolated tests of motion segment behavior in individual degrees of freedom provide information that is specific to those degrees of freedom and cannot be generalized to three-dimensional analyses because of the interaction between degrees of freedom, called ‘coupling’ (Panjabi et al., 1976). A 6×6 stiffness or flexibility matrix is needed to describe how forces displace a vertebra relative to its fixed neighbor (Panjabi et al., 1976). In general a 6×6 stiffness matrix has 36 terms. The number of independent stiffness matrix terms is reduced to 21 by consideration of matrix symmetry required by conservation of energy if the material properties are linear. Matrix symmetry results in complementary pairing of off-diagonal terms hence k12=k21, etc., in Fig. 2. Sagittal plane symmetry requires that nine of the 21 terms are zero (terms for forces expected to be zero for displacements within the sagittal plane, e.g. no lateral force associated with axial compression, hence k13 and k31=0, etc.). This leaves 12 nonzero stiffness terms ( Fig. 2). Goel (1987) defined a simplified stiffness matrix for the motion segment with six diagonal terms (relating co-linear displacements or rotations) and two “primary” off-diagonal terms, assuming that the motion segment had beam-like behavior. The two primary off-diagonal terms relate the anterior–posterior (A–P) shear forces to the applied flexion-extension rotations (or the complementary flexion-extension moments to A–P shear displacements) and lateral shear forces with lateral bending rotations (or the complementary lateral bending moments to lateral shear displacements). This beam-like behavior requires that the motions segment's axis system be aligned with its structural axis.Since the motion segment has two vertebrae, each having six degrees of freedom, it requires a 12×12 stiffness matrix. This matrix can be derived from the 6×6 matrix for one vertebral center moving relative to a fixed adjacent vertebra, by using the principle of force equilibrium, compatible with the specified distance between the two vertebral centers (Gardner-Morse et al., 1990). A shear beam with a rigid A–P offset was proposed by Gardner-Morse et al. (1990) as an approximate representation of an experimental stiffness matrix. This ‘equivalent’ beam has seven independent parameters, compared with up to 12 terms in the experimental stiffness matrix (Fig. 2). In this paper we propose an extension of that method, that also includes a truss element, thus permitting a closer approximation to the experimental data. There are several limitations of the existing human motion segment experimental stiffness data, such that they probably do not accurately represent in vivo behavior. Most reported data do not include all six degrees of freedom (Berkson et al., 1979; Nachemson et al., 1979; Schultz et al., 1979), were obtained without physiological levels of axial compression and were performed with the specimen not surrounded by physiological isotonic fluid (e.g. Panjabi et al., 1976). Physiological axial compressive preload is known to increase stiffness by a factor of two or more (Edwards et al., 1987; Janevic et al., 1991; Gardner-Morse and Stokes, 2003) and may reduce the amount of load-displacement nonlinearity (Janevic et al., 1991; Gardner-Morse and Stokes, 2003). Discs in a physiological saline bath have greater hydration than discs that are just exposed to saline spray and wrap (Pflaster et al., 1997), and this increased hydration affects the disc biomechanics (Race et al., 2000; Costi et al., 2002). The increased hydration may also increase the repeatability of the load-displacement behavior with slow cyclic loading (Gardner-Morse and Stokes, 2003). This paper reports the stiffness matrix and other properties of human lumbar motion segments tested with slow-rates of displacement, to obtain the quasi-static stiffness response of the motion segments, and with small displacements, to approximate the assumption of linear load-displacement behavior. The purposes of the study were: 1. Quantify the effects of 0, 250 and 500 N axial compressive preload on the motion segment stiffness matrix, and on the hysteresis and linearity of the load-displacement relationship. The effect of preload was determined for intact motion segments and in isolated intervertebral discs. 2. Examine whether the stiffness matrix terms correlated with physical dimensions of the motion segments, and compare the behavior of L2-3 and L4-5 segments. 3. Analyze the derived stiffness matrices at the three magnitudes of axial compressive preload to obtain the geometrical properties of an equivalent structure that included a truss and a beam element with a rigid posterior offset.

The load-displacement behavior of human lumbar motion segments was found to depend on the magnitude of the axial compressive preload. Preload increased the stiffness, linearity of the load-displacement behavior, and hysteresis area, but not loss coefficient. These findings indicate that motion segment stiffness, linearity and hysteresis measured without preloads underestimate the in vivo values in all degrees of freedom. These effects were present in both intact motion segments and in isolated intervertebral discs. The stiffness values and changes with axial compressive preload found in this study were similar to previous studies (Edwards et al., 1987; Janevic et al., 1991; Gardner-Morse and Stokes, 2003). The increase in hysteresis with preload was similar to that found in porcine motion segments (Gardner-Morse and Stokes, 2003). However, this effect of preload was not evident in the loss coefficient. Loss coefficient is the hysteresis area normalized by the maximum strain energy. Because of the fluid shifts that occurred when specimens equilibrated to changes in preload, both the water content (and volume) of the disc, as well as its loading state may have produced the observed changes in mechanical behavior. All of the motion segment stiffness matrix terms that were expected to be zero as a result of sagittal plane symmetry had empirical values not significantly different from zero. Additionally, the stiffness term associated with coupling between axial compression and A–P shear was not significantly different from zero. The significant terms in the experimental stiffness matrices correspond to non-zero terms in the stiffness matrix of a beam with a rigid posterior offset. For the stiffness terms that differed between the two anatomical levels with 250 and 500 N preloads, L4-5 was stiffer than L2-3. Since these differences were not significant in the isolated discs, these differences were attributed to the posterior elements. No correlations of motion segment stiffnesses with physical dimensions were found, as was also reported by Berkson et al. (1979). The equivalent structure consisting of a truss and a shear beam with a rigid posterior offset was found to be an accurate representation of the linearized stiffness matrix under the tested conditions of slow and small displacements. When the stiffness matrices of the ‘equivalent’ structures were calculated, the terms were within ±0.57SD of the measured stiffness means. This structure provides insights into the physical behavior of the motion segment. While many properties showed large changes with increasing preload, it was notable that the rigid offset δ was almost constant with increasing preload. In intact motion segments the rigid offset δ represents a posterior displacement (relative to the vertebral center) of the effective structural axis of the motion segment. It was not evident for disc-only specimens, hence it was attributed to the posterior elements. The addition of a truss allows the equivalent structure to match the coupling between axial displacement/force and flexion-extension moment/rotation exactly (k16 in Fig. 2). Gardner-Morse et al. (1990) found that a shear beam without offsets was a good match to the thoracic flexibility matrices in Panjabi et al. (1976), suggesting that the posterior elements have a greater effect in lumbar than in thoracic motion segments. All of the motion segment stiffnesses except for the axial stiffness decreased with the removal of posterior elements. A large decrease in torsional stiffness with the removal of the posterior elements, and lesser decreases in lateral bending and flexion and extension stiffness have been reported previously, but using greater ranges of motion than used here (Schultz et al., 1979; Stokes, 1988; Posner et al., 1982). The observed differences in stiffness between flexion-extension and A–P shear were compatible with the expected stiffening effects of facet joint engagement and these differences were not observed after facet removal. The nonlinear behavior in compression-tension did not change with the removal of the posterior elements, so it was presumably due to the disc. Panjabi et al. (1976) presented two stiffness matrices, one for positive and one for negative displacements. This approach defines the diagonal stiffness terms for both displacement directions, but does not include all permutations of the off-diagonal terms for differing displacement directions. The main limitation of this approach to represent the motion segment as a stiffness matrix was that it required an assumption of linear load-displacement behavior. The high R2 values suggest that this is a reasonable assumption over the range of displacements tested here. The displacement rate used in these tests was slow, to simulate quasi-static loadings. The apparent stiffness is expected to increase at faster displacement rates. The low temperature (∼4°C) may have increased the measured stiffness relative to physiological values. The linearized stiffness matrices at each preload magnitude can be used in structural analyses of the lumbar spine. Alternatively, these stiffness matrices can be expressed as an equivalent structure consisting of a truss and a beam with a rigid posterior offset, with cross-sectional properties that vary with preload. These equivalent structures can be employed in structural spinal analyses, and the changes in spinal stiffness with preload can be used to update spinal stiffness properties as shown in Stokes and Gardner-Morse (2003).