Off‐axis response due to mechanical coupling across all six degrees of freedom in the human disc

The kinematics of the intervertebral disc are defined by six degrees of freedom (DOF): three translations (Tz: axial compression, Tx: lateral shear, and Ty: anterior‐posterior shear) and three rotations (Rz: torsion, Rx: flexion‐extension, and Ry: lateral bending). There is some evidence that the six DOFs are mechanically coupled, such that loading in one DOF affects the mechanics of the other five “off‐axis” DOFs, however, most studies have not controlled and/or measured all six DOFs simultaneously. Additionally, the relationships between disc geometry and disc mechanics are important for evaluation of data from different sized donor and patient discs. The objectives of this study were to quantify the mechanical behavior of the intervertebral disc in all six degrees of freedom (DOFs), measure the coupling between the applied motion in each DOF with the resulting off‐axis motions, and test the hypothesis that disc geometry influences these mechanical behaviors. All off‐axis displacements and rotations were significantly correlated with the applied DOF and were of similar magnitude as physiologically relevant motion, confirming that off‐axis coupling is an important mechanical response. Interestingly, there were pairs of DOFs that were especially strongly coupled: lateral shear (Tx) and lateral bending (Ry), anterior‐posterior shear (Ty) and flexion‐extension (Rx), and compression (Tz) and torsion (Rz). Large off‐axis shears may contribute to injury risk in bending and flexion. In addition, the disc responded to shear (Tx, Ty) and rotational loading (Rx, Ry, and Rz) by increasing in disc height in order to maintain the applied compressive load. Quantifying these mechanical behaviors across all six DOF are critical for designing and testing disc therapies, such as implants and tissue engineered constructs, and also for validating finite element models.

axial torsion. [18][19][20] Most studies do not control or measure all six DOFs simultaneously, meaning that the off-axis DOF are ignored. Importantly, there is evidence that the six DOFs are mechanically coupled, such that loading in one DOF affects the mechanics of the other five "off-axis" DOFs. 21 For example, applying axial compression increases stiffness in the rotational DOFs (Rx, Ry, Rz). 22,23 In this study, we sequentially tested the disc's response to cyclic loading in each of the six DOFs, using a state-of-the-art hexapod system to measure both same-axis and off-axis responses. 9,24,25 We hypothesized that applied loading in each DOF would generate coupled off-axis motions (translations and rotations) in the other five DOFs, and we tested this hypothesis by correlating, for each DOF, the applied motion vs measurements of the resulting five off-axis motions.
Relationships between disc geometry and disc mechanics, including coupled off-axis responses, are important for evaluation of data from different sized donor and patient discs, interpretation of finite element models, and development of patient-specific models. Geometry, such as height and width, is known to affect the same-axis mechanical response of a disc segment when loading is applied to a DOF. 26,27 For example, in a computational parametric simulation, disc height correlated with range of motion in pure moment flexionextension (Rx), lateral bending (Ry), and axial torsion (Rz). 27 In this study, we hypothesized that coupled off-axis motions for each DOF are also, like the same-axis mechanical response, correlated with disc geometry.
Therefore, the objectives of this study were to quantify the mechanical behavior of the intervertebral disc in all six degrees of freedom (DOFs), measure the coupling between each applied motion in each DOF and the resulting off-axis motions, and test the hypothesis that disc geometry influences these mechanical behaviors.

| Specimens
Human lumbar intervertebral disc bone-disc-bone segments (n = 8) from five spines (three male and two female; age 49 ± 9 years, range  were dissected with posterior elements removed and graded for degeneration (Levels: L1-L2 (n = 1), L2-L3 (n = 1), L3-L4 (n = 2), L4-L5 (n = 4)). To limit variation from degeneration, only mild to moderately degenerated discs (Pfirrmann grades 2 to 3) were used. The disc's anterior-posterior width (W AP ), lateral width (W LAT ), and disc height were determined from lateral and coronal X-ray images. The disc's cross-sectional area was calculated from A = 0.84 * W AP * W LAT. 28 The disc's aspect ratio was calculated as W LAT  Three k-wires were inserted halfway into the sides of each vertebra, 120 apart, and the vertebrae were potted into custom cups using polymethylmethacrylate (bone cement). During potting, the lordotic angle measured from the x-ray images was maintained using an alignment device and plastic wedges as previously described. 9 The potted bone-disc-bone segment specimen was then mounted in a 6DOF hexapod robot with a 37 C 0.15 M PBS bath for mechanical testing.

| Mechanical testing
Segments were tested using a state-of-the-art hexapod system that simultaneously controls and measures all six DOFs. 9,24,25 The protocol consisted of an axial compression preload followed by six sequential tests, subsequently called "DOF tests," each of which applied cyclic loading to one target DOF ( Figure 2). A novel hybrid control system was used, for details of the apparatus and control system, please see previous publications. 9,24,25,29 In each DOF test, prescribed cyclic loading was applied to the target DOF in position or load control, and the off-axis DOFs were load controlled in real time to maintain (approximately) zero off-axis forces and moments. 29 This adaptive approach to minimizing off-axis forces and moments allows the disc's center of rotation to change during the protocol, which is an advantage over displacement controlled tests that fix the center of rotation to one location, potentially causing supra-physiologic and damaging loads. 21,29 The preload consisted of axial compression with off-axis forces and moments controlled to zero. It was held overnight for 12 hours before the DOF tests started. The preload was chosen to produce a nucleus pulposus (NP) pressure of 0.2 MPa, which mimicked physiologic conditions and reduced axial displacement creep during the subsequent DOF tests. The preload force was calculated as F = P*A 1:5 , where P is the desired NP pressure (0.2 MPa), A is the disc crosssectional area, and 1.5 is an empirically derived factor. 24 The axial preload was maintained during through the DOF tests except (as described below) when axial force was cyclically varied in the Tz test.
Six "DOF tests", each of which applied cyclic loading to a specific DOF and controlled the off-axis moments and forces to be zero (hybrid control), were performed in the sequence Tx, Ty, Rz, Ry, Rx, and Tz ( Figure 2). As testing sequence can influence disc mechanics, the specific order used was chosen to minimize disc volume lost throughout testing. 25 Each DOF test was separated from the others by a 30 minute recovery period (held in the preload state). Each DOF test consisted of a three blocks of five loading cycles at fast, medium, then slow cycle frequencies. In the Tx, Ty, Rz, Ry, and Rx tests, the blocks' frequencies were 0.5, 0.05, and 0.005 Hz, and the target DOF was sinusodially loaded in displacement or rotation control. In the Tz test, the blocks' frequencies were 1, 0.1, and 0.01 Hz, and Tz was loaded with haversine cycles in force control. Loading in Tz caused significant axial creep, which is why the Tz test was done in force control. An additional 30 minute recovery period was also inserted after the Tz medium rate block to allow some recovery of axial creep before the slow rate block started. The Tx and Ty tests had a slightly modified hybrid control scheme to prevent the test fixtures from colliding. In the Tx test, Ry rotation was controlled at 0 , and in the Ty test, Rx rotation was controlled at 0 . Overall, this mechanical testing sequence ( Figure 2) was designed to minimize disc volume loss (water exudation) during the protocol. 25 The cycle amplitudes in each DOF test were chosen to be physiologically reasonable [30][31][32] : Tx = ±0.6 mm, Ty = ±0.6 mm, Rz = ±3 , Ry = ±4 , Rx = ±3 , and Tz = −0.2 MPa to −1.1 MPa.

| Data analysis
For each DOF test, stiffness in the applied DOF (same-axis stiffness) was calculated from the load-displacement curve of the applied DOF in the final loading cycle (slow rate), as the first 4 cycles were considered preconditioning. Stiffnesses were calculated by linear regression between 70% and 90% of the maximum force or moment. Potential correlations between stiffness and each geometry parameter (height, lateral width, anterior-posterior width, aspect ratio) were determined using Pearson correlation.
To determine the relationship (coupling) between each applied DOF and its off-axis responses, the final loading cycle (a slow rate cycle) for the applied DOF was plotted against its corresponding offaxis motions. To evaluate the strength of this correlation, a secondorder polynomial regression (r 2 ) was performed with significance set at P < 0.05. For the purposes of this study, data from the slow tests was used because the hexapod system controller exhibited poor control at the fast rate (future instrument improvements will address this).
Since the applied loading and hence the off-axis responses in each DOF test were cyclic, each off-axis response was also characterized by the offset of its oscillations and the amplitude between its value at the start of the DOF test and the value about which it oscillated during the DOF test ( Figure 3). The offset arises because the disc moves to a new reference position when a new cyclic loading pattern is applied. The amplitude and offset of each off-axis response was determined relative to a line fit through the oscillitory off-axis displacement/rotation vs time curve (Figure 3, red line). The offset is the FIGURE 3 Representative response of Tx (lateral shear) throughout the entire test protocol. Large off-axis responses to each DOF test can be observed. The expanded response of Tx to Rx loading (inset) shows how the offset and amplitude were calculated from the line that was fit through the oscillation (solid red line). The offset is the intercept of the line with the start of loading. The amplitude is the average distance from the line to the peak and valley of the cyclic response over the five applied cycles for each loading rate, calculated by subtracting the fit line from the experimental data FIGURE 2 Schematic for mechanical testing protocol. Testing consisted of an overnight 0.2 MPa compression preload followed by Degree of Freedom (DOF) tests in which 5 cycles each at 0.5, 0.05, and 0.005 Hz (fast, medium, and slow rates) were performed for each DOF, followed by a 30 minute recovery period. To maintain off-axis zero force and moment conditions, the off-axes displace and rotate in response to loading. DOF test order was Tx (lateral shear), Ty (anterior-posterior shear), Rz (axial torsion), Ry (lateral bending), and Rx (flexion-extension). Finally, the Tz (axial compression) test was performed with 5 cycles at 1, 0.1, and 0.01 Hz with recovery after 0.1 Hz as well as after the 0.01 Hz test intercept of this line with the start of loading. The amplitude is the average distance from the line to the peak and valley of the oscillation over the five applied cycles. To determine whether there was a significant off-axis response, the offset and amplitude were compared to zero using a t-test. To determine the effect of geometry, each geometry parameter was correlated against each off-axis offset and amplitude.

| Same-axis mechanical response in the applied DOF
A representative same-axis load-displacement (or moment-rotation) curve for each DOF test's slow rate cycles is shown in Figure 4 and average same-axis stiffnesses are reported in Table 1. Correlations of same-axis stiffness with geometry were evaluated ( Figure 5, Table 1).

| Off-axis mechanical response
The six DOF response for a representative specimen throughout the entire testing protocol is shown in Figure 6. During the 12-hour preload (0.2 MPa axial compression), there was a large axial creep Tz = −0.70 mm (Table 1). A small amount of creep occurred in the lateral shear Tx = 0.10 mm, A-P shear Ty = 0.21 mm, and torsion Rz = 0.74 axes (Table 1). Similarly, throughout the duration of testing the segment continued to creep in axial displacement (Tz, Figure 6C), with small amounts of additional creep in torsion (Rz, Figure 6F), lateral shear (Tx, Figure 6A) and A-P shear (Ty, Figure 6B). There was no creep in flexion-extension (Rx, Figure 6D) and lateral bending (Ry, Figure 6E), as these were controlled to be zero.
The off-axis responses were large, with magnitudes similar to the applied DOF loadings (Figure 6). For example, lateral shear (Tx), was controlled to +/− 0.6 mm, but when the slow rate lateral bending (Ry) loading was applied, the coupled displacement in lateral shear ranged from +/−0.7 mm ( Figure 6A). Similar effects, where the coupled off-axis motion is greater than or equal to the applied motion, can be seen for all of the directions ( Figure 6B-E), except perhaps torsion Rz which is less than 1 ( Figure 6F). Notably, the response in the axial direction was qualitatively different than that of the lateral and anterior-posterior directions ( Figure 6C). In the Tz DOF test, applied compressive loading caused the disc to decrease in height, but loading of the other DOFs caused disc height to increase in order to keep the axial force at its controlled preload value ( Figure 6C). That is, when the disc was loaded in all other DOFs the disc pressure increases, 22 and in order to maintain the applied preload, the axial compression decreases.
The average off-axis responses were plotted against each applied DOF and tested for correlations (Figure 7). The off-axis response was nonlinear and significantly correlated (P < 0.001) to applied displacements and rotations for all combinations and at both slow and medium rates ( Figure 7B). Off-axis rotations that were controlled to be 0 (Ry during the Tx test, and Rx during the Ty test) were not included; these are shaded gray in Figure 7A. The strength of the correlations ranged from r 2 ≈ 1 to ≈ 0 ( Figure 7B). Off-axis axial FIGURE 4 Representative same-axis response of the final cycle for the slow rate in each DOF test. Top: the translation degrees of freedom (force vs displacement, A-C), Bottom: the rotation degrees of freedom (moment vs rotation, D-F) compression (Tz) was strongly correlated to all applied DOFs (red box in Figure 7A, r 2 > 0.8 in Figure 7B). The other off-axis translations (Tx, Ty) tended to be especially strongly coupled to applied rotation (Rx, Ry, and Rz tests). For example, the off-axis Tx response correlated with applied Ry and Rz loading with r 2 values of 0.99 and 0.98, respectively. In addition, we observed hysteresis in the off-axis rotations that tended to be larger than in the off-axis translations. The pattern of correlations between applied DOF loading and off-axis responses was similar between the slow and medium loading rates, but with decreased r 2 values for the medium loading rate ( Figure 7B).

| Off-axis amplitudes and offsets
The off-axis responses in each DOF test was decomposed into an offset and amplitude ( Figure 8). Many off-axis offsets and amplitudes were large; all significantly differed from zero. A large Tz offset (0.19 mm) was observed in response to an applied axial torsion (Rz). Some off-axis amplitudes were comparable to the applied amplitudes chosen for the DOF test protocols (Figure 8, blue dashed line). As in the off-axis correlations (Figure 7), Tx and Ry, Ty and Rx, and Tz and Rz were found to be particularly strongly coupled. The   Table 1 off-axis lateral shear (Tx) amplitude (0.62) was strongly coupled with applied lateral bending (Ry) and was similar to the amplitude applied in the Tx test (0.6 mm). Similarly, the off-axis A-P shear These observations support the hypothesis that the disc's off-axis displacements and rotations depend on (are coupled to) motion in the applied DOF.

| Relationship between off-axis mechanical response and disc geometry
The off-axis mechanical response in each DOF test was hypothesized to depend on disc geometry. Disc height, A-P width, and lateral width were found to be significantly correlated with some, but not all, of the off-axis offset and amplitude response variables (Figure 9). Aspect ratio was not significantly correlated with any off-axis response. More offsets than amplitudes were significantly correlated with disc geometry. Lateral width had a particularly large number of significant correlations. Interestingly, lateral width and A-P width tended to have opposed effects on the off-axis offsets and amplitudes (opposite sign correlation coefficients).

| DISCUSSION
This study quantified the mechanical behavior of the intervertebral disc in all six degrees of freedom, measured the coupling between applied loading and off-axis motion, and evaluated the potential role of disc geometry in these mechanical behaviors. All off-axis displace- shears may contribute to injury risk in bending and flexion. Importantly, the large coupled motions observed in this study were caused by the disc alone, as the facet joints were removed. In addition, the disc responded to shear (Tx, Ty) and rotational loading (Rx, Ry, and Rz) by increasing in disc height in order to maintain the applied compressive load. This increase in disc height may be caused by an increase in intradiscal pressure during multiaxial loading. 22 The average same-axis stiffnesses in this study were comparable to those from those reported in literature for tests in lateral shear, A-P shear, compression, and torsion. 8    This study demonstrated that disc geometry correlates with the same-axis stiffness in all six DOFs and with some off-axis responses.
In particular, the disc's aspect ratio (lateral width/A-P width) was a significant indicator of rotational and shear stiffnesses, whereas disc height was significantly correlated with compressive stiffness. In addition, lateral width was significantly correlated to each off-axis offset.
The correlation between lateral width and stiffness and off-axis offsets may be related to the fact that lateral width is the largest dimension of the disc. It is the largest contributor to both the second area moment about z (resistance to torsion) and to the moment arm that resists lateral bending. The observed correlations demonstrate that geometry affects off-axis DOF coupling and therefore contributes to the forces and moments generated in the disc during physiological loading. While not explicitly addressed here, the effect of spinal level and potential difference in geometry due to lordosis angles may also effect off-axis responses. The relationship between geometry and mechanics is important for the validation and interpretation of finite element models. Geometry is a known indicator of disc mechanics 27 and typically one geometry is used to represent the disc when developing a finite element model. When reviewing a model's predictions for multiaxial stiffnesses, one should put its simulated stiffness in the context of the model's geometry and the available data relating mechanics to disc geometry.
The discs used in this study were isolated discs without posterior elements, demonstrating that the isolated disc has large coupled motions even without the facets. Facets have a role in spine mechanics and may produce different off-axis effects at the spine scale. Existing literature includes no information on the impact of facets on offaxis responses, and there is conflicting reports regarding their impact on same-axis stiffness. In position control tests, discs with facets had 60%-100% higher compressive and rotational stiffness than those without, 33 whereas in hybrid control tests, discs with facets showed no discernible difference in stiffness compared to isolated bone-discbone segments. 25 Additional experiments with facets are needed to determine the relative role of the disc and the facets in six DOF mechanical responses. Another limitation of this study was the use of grade 2 to 3 mild/moderately degenerated discs. It remains unknown whether the coupled response measured here would be different with more advanced degeneration.
A strength of this study is the hybrid control approach used in the mechanical testing, where the applied DOF is position controlled (except for axial compression Tz) while the off-axis is force-and FIGURE 9 Heat maps showing correlations of disc geometry with offset (A) and amplitude (B) for each axis in each DOF test for the slow rate.
Offsets were strongly correlated with geometry (top row), particularly lateral width, while amplitudes were generally not correlated with geometry (bottom row). * denotes P < 0.05, # denotes P < 0.1, and N/A denotes same-axis loading and fixed loading (e.g., Rx = 0 during the Ty test) be useful to support future computational modeling work, which in turn would support simulation of between-axis coupling in a wider variety of loading conditions.
The measured off-axis coupling and their relationships with geometry are important to improve the design and evaluation of disc implants. Conventional design and evaluation of implants are limited to 1-2 DOFs 40 but this study showed that large off-axis effects are generated during disc loading and that they are related to the disc's geometry. If not accounted for, these may contribute to implant failure, expulsion, or reduced function. Moreover, these results suggest the need for a patient-specific approach to implant design in order to account for the correlation in mechanics with geometry. For example, bending in discs that have a larger lateral width may increase AF stresses and/or cause expulsion of the implant. Moreover, if an implant is designed without considering the off-axis loads, the implant itself may fatigue due to large loads generated by the coupled off-axes. Lastly, when a disc undergoes discectomy (changing the disc structure) or is replaced with an implant (changing the material), the stiffness in each of the six DOFs and their coupled off-axis responses will be altered.
These factors should be considered in the design and evaluation of disc implants and other treatments.
In conclusion, this study quantified strong mechanical coupling between DOFs in six DOF hybrid-control mechanical tests of the intervertebral disc and demonstrated the mechanical coupling that varied with disc geometry. This multiaxial coupling and its interaction with geometry should be addressed in the design and evaluation of disc implants. This will likely require increased usage of multiaxial test equipment and consideration of patient-specific disc geometry. Combined control and measurement of all six DOFs is needed to account for the multiaxial loads and their coupling experienced by the in vivo disc, and hybrid control multiaxial testers are well-suited for this application.