The excitation force measurements as well as their Welch power spectral density are represented in Fig. 3. It can be seen that the frequencies between 0 and 350 Hz are dominant. The raw measurements such as strains (µε) and accelerations (g) for a typical dataset are shown in Fig. 4. The magnitude of these measurements both for FE and LRLB in group I are higher than other groups. In addition, strain magnitudes measured in 4R+4TC group for FE were among the lowest between groups. However, to investigate whether the intensity is critical or not, statistical tests were performed as described in the following section. Fig. 5 shows the frequency response functions (FRF) for the excitation loads and acceleration responses. The uncertainties and variability in these specimens are high due to nonlinearities in hard-soft tissue of the spine. The peak resonance frequencies and damping ratios over the frequencies range were estimated using "modalfrf" and "modalfit" functions in MATLAB.
3.1 Statistical results of the resonance frequencies, damping ratios and absolute maximum strains
The statistical data for the three groups are summarized in Fig. 6 and Table 1. The statistical analysis demonstrated that the most significant differences concern resonance frequencies (RFs) and absolute maximum strains (AMS) (p < .05). For the FE motion, damping ratios at mod II are not different between groups (F = .112, p = .894). For the LRLB motion, there are also no significant differences between the groups' parameters such as 1st (F =.441, p = .644) and 2nd damping ratio (F = 1.029, p = .361).
Post-Hoc multi Tukey test was performed for multi comparison of groups. The details of the statistical analysis are given in Table 2-3. The results showed that RFs in the 2nd and 3rd groups are significantly higher than in the 1st group both for FE and LRLB motions. (p < .05). Compared with the 2R specimen, 4R+2TC and 4R+4TC constructs significantly increased the 1st RF in FE (17% and 18% improvement, respectively; both p <.05) and LRLB motion (11% and 24% improvement, respectively; both p < .05). In addition, 4R+2TC and 4R+4TC constructs significantly reduced AMS compared with 2R configuration for FE (80% and 92% reduction, respectively;both p < .05) and LRLB motion (77% and 85% reduction, respectively; both p < .05). Although the average 1st RF for FE motion in 4R+4TC construct is higher than 4R+2TC, there is no significant difference between them (p > .05). Furthermore, while the AMS in the 4R+4TC construct group is lower than in the 4R+2TC group, no significant differences were observed. These results suggest that the additional ARs involve a significant effect on dynamic response of the constructed spine compared with the standard 2R configuration. In addition, while the additional TCs improve the 1st RF in LRLB, they have not a similar significant involvement in FE motion.
The RFs and AMS were also evaluated using receiving operating characteristics (ROC) analysis to identify how well the ARs and TCs influence the dynamic stability of the instrumented spine. The ROC curves are typically established by plotting true positive rate (TPR) against false positive rate (FPR) at various thresholds. A curve close to the TPR axis, the classification score rises, while it is close to the FPR axis the score declines. Therefore, the best possible classification is yielded at a point of (0,1). The performance of the curve is assessed via area under curve (AUC). The best performance is observed for AUC = 1, and the worst is for AUC = 0.5. Generally, an AUC of 0.5 is considered "no separation" between groups, 0.6 to 0.7 is "sufficient", 0.7 to 0.8 is "acceptable", 0.8 to 0.9 is "very good", and higher than 0.9 is "outstanding". Considering the first RF value (in FE motion) between 2R and 4R+2TC construct groups, the AUC value is 0.86 (Fig. 7). This result shows that the 1st RF in group I and II are separated with a score of "very good". However, once evaluating the 1st RF for the 4R+2TC and 4R+4TC groups in FE motion, the AUC is 0.58. This result indicates that the 1st RF values in group II and III could not be separated. The ROC results for the rest parameters are represented in Fig. 7. The overall ROC curves showed that the dynamic characteristic parameters are discriminated in group I vs. II, and group I vs. III constructs, while they are not discriminated in group II vs. III in FE motion. However, group III data are separated both from group I and II in LRLB, which shows that the additional transverse connectors influence the dynamic integrity of the constructed spine in lateral bending.
3.2 Correlation between the parameters of nondestructive dynamic and quasi-static eccentric loading tests
To determine the correlation level between dynamic and quasi-static characteristic data, we utilized eccentric loading tests for constructed spines that were investigated with nondestructive dynamic test setup. The quasi-static eccentric loading test was performed with a machine (Autograph/Shimadzu, Japan) at a low loading speed of 1.66 mm/s. First, the constructed specimen that its lower part was installed in polyester paste was fixed. Then, a compression force acted at a distance of 48 mm from the principal centroidal axis. The sample was equipped with an orientation sensor (BNO055, Bosch, Sensortec, Germany) to monitor the RoM. The construct was loaded quasi statically, and the data were acquired until the bending moment reached 7.5 Nm with a sampling rate of 20 Hz. [1,18].
Regression analysis showed that a linear dependence exists between the parameters of dynamic and quasi-static test results (Fig. 8). Positive significant correlation is observed between AMS and RoM in FE (R = 0.749, p = 0.00, "strong") and LRLB (R = 0.691, p = 0.001, "strong"). In addition, the 1st RF correlated negatively with RoM in FE (R = -0.606, p = 0.008, "strong") and LRLB (R = -0.622, p = 0.006, "strong") directions. The results confirm that a decrease in longitudinal rod strain or an increase in the resonance frequency is associated with a rise in structural integrity of the spinal construction.