Measurement and calibration of the nucleus position and its cross-sectional area ratio to increase the accuracy of finite element analysis

Background: As a widely used biomechanical research method, finite element analysis (FEA) is a significant tool for investigating the pathogenesis of disc degenerative diseases and optimizing of spine surgical methods. However, the definitions of the relative nucleus position and its cross-sectional area ratio do not conform to a uniform standard, thus affecting the accuracy (ACC) of the FEA. Hence, this study aimed to determine a precise definition of the relative nucleus position and its cross-sectional area ratio to increase ACC of following FEA studies. Methods: The lumbar relative nucleus position and its cross-sectional area ratio were measured from magnetic resonance imaging data, and then calibrated and validated via FEA. Imaging data from patients without disc degeneration were recruited. The L4-L5 nucleus and disc cross-sectional areas and the distances between the edges of the annulus and nucleus were measured; the ratios between these values were calculated as P1 and P2, respectively. The FEA model was constructed using these measured values, and the relative nucleus position was calibrated by estimating the differences in the range of motions (ROMs) between the model, wherein the ligaments, facet joints and nucleus were supressed, and an in vitro study. Then, ACC were re-estimated in the model with all non-bony structures to validate the measured and calibrated indicators. Results: The interobserver homogeneity is acceptable, and the measured P1 and P2 values are 1.22 and 38%, respectively. Furthermore, an ACC of up to 99% was attained for the model under flexion–extension conditions when the calibrated P1 value (1.62) was used, with a model validation of greater than 90% attained under all loading conditions. Conclusion: The measured and calibrated relative nucleus position and its cross-sectional area ratio increase the ACC of the FEA model, and can therefore be used in subsequent studies. we present an approach to accurately define such indicators in a uniform manner and increase the accuracy of following FEA studies. We first analysed magnetic resonance imaging (MRI) data of non-degenerative discs to measure the key indicators. We then constructed a 3D lumbar

greater than 90% attained under all loading conditions. Conclusion: The measured and calibrated relative nucleus position and its cross-sectional area ratio increase the ACC of the FEA model, and can therefore be used in subsequent studies.

Background
Biomechanical deterioration is a key trigger for lumbar disc degenerative diseases (DDDs) 1-5. Therefore, optimised finite element analysis (FEA), which is a widely used biomechanical research method, is an ideal approach for the accurate investigation of DDD pathogenesis and treatments6-15. A series of model validation methods have been implemented by comparing the differences in the range of motion (ROM), which is a key index for lumbar motility and stability, between the FEA results and those from widely cited in vitro studies7, 10, 11, 14-18. The accuracy (ACC) of the FEA models have been improved via model calibrations that used annulus and bone structures19, 20. However, inaccurate definitions of the relative nucleus position and its cross-sectional area still negatively affect the computational ACC of FEA studies.
To clarify this issue, the anterior and posterior parts of the annulus can be approximated as two different-sized quadrangular annuli on a specific plane. The O point is set as the pivot point of the applied torque; R1 and R2 are the distances between the anterior and posterior parts perpendicular of the torque axis centre and edges of the nucleus, respectively; and M1 and M2 are the forward and backward torques, respectively. Material mechanics theory indicates that changes in R1 and R2 origined from different nucleus position and cross-sectional areas will induce changes in M1 and M2, leading to ROM variations ( Fig. 1). Furthermore, this change will extend to different sections if the force on the two-dimensional plane structure is extended to three-dimensional space.
However, such methods have not been defined in a uniform manner in previous FEA studies. For example, Tang et al modelled the centre of the nucleus as being slightly posterior to the centre of the disc and occupying 43% of the total disc volume9. Hendrik et al modelled the centre of the nucleus as being 3.5 mm towards the posterior side, with a size that was approximately 44% of the total disc area20. The lumbar model of Masni et al determined that the nucleus accounted for 33% of the disc volume21, whereas this rate increased to 50% in the model of Claudia et al22. Furthermore, numerous FEA studies are available in addition to the above-mentioned studies that do not address such issues. Given that individual differences are inevitable among the discs in different models, the accurate definition of the relative nucleus position should be a ratio (such as a ratio of the anterior and posterior edges of the annulus to the relative nucleus edges) rather than an ambiguous description (e.g. 'slightly posterior') or a specific value (e.g. 3.5 mm).
Herein, we present an approach to accurately define such indicators in a uniform manner and increase the accuracy of following FEA studies. We first analysed magnetic resonance imaging (MRI) data of non-degenerative discs to measure the key indicators. We then constructed a 3D lumbar model to calibrate and validate our measured values. An extensive literature search indicates that no similar studies have been published to date.

MRI measurements of the indicators Collection of MRI data
Approval for this study was obtained from the ethics committees of Jiangsu Province Hospital on Integration of Chinese and Western Medicine. Three observers, two senior spine surgeons and a seasoned musculoskeletal radiologist reviewed the lumbar MRI (Discovery MR750 3.0T, GE Healthcare; Chicago, Illinois, USA) data in our hospital over the past three years. The L4-L5 disc was selected owing to the high incidence rate of DDD. The T2-weighted imaging in the sagittal plane was selected to assess the grades of disc degeneration based on Pfirrmann et al23, with Grade I and II degeneration considered indicative of a normal disc and included in this study 24. The MRI data were independently reviewed, and they were included in the analysis only when all observers confirmed that the disc was a normal disc. To avoid the influence of gender difference, only male MRI data were collected for the following FEA model was constructed based on the imaging data from a 24 years old male volunteer. The kappa statistic was used to analyse the included imaging data to ensure homogeneity in the interobserver classification.

MRI measurements
The included MRI data have been checked using the imaging workstation. The distances from the anterior and posterior edges of the annulus to the nucleus edges are set as D1 and D2, respectively.
The cross-sectional areas of the nucleus and disc are set as A1 and A2, respectively. The ratios between the mean values of D1 and D2 and A1 and A2 are set as P1 and P2, respectively. These indicators (D1, D2, A1 and A2) were measured from the MRI data. Cronbach's α reliability statistic was conducted to ensure the homogeneity of the measured values. The relative nucleus position and its cross-sectional area ratio were represented as P1 and P2, which were used in the following FEA study.
Calibration and validation of the FEA model Construction of the normal model The L4-L5 segment 3D model was constructed using high-resolution computed tomography imaging data. 6,8,9,12,14 The reconstructed bone structures served as the template, which was layered along the long axis. Standard curves were used to fit its contours, with the external contour of the new model overlapping the template to the maximum extent. The constructed non-bony structures were also fitted using standard curves, and the facet joint gap was set as 0.5 mm6, 9, 13. The centriod of the outlines of annulus and the inferior surface of L4 were defined as a same point for the accurate placement of annulus and following calibration of annulus average radius. Six different ligaments and a capsule of facet joints were constructed during the FEA preprocessing phase11, 12, 18, 21, 25, 26 (Fig. 2). The definition of the relative nucleus position and its cross-sectional area ratio were confirmed according to P1 and P2, and the outer contour of the nucleus was obtained using the same ratio reduction of the disc contour to ensure that the intervertebral disc and nucleus have the same central point for easy adjustment during the calibration process.

Boundary and loading conditions
Tetrahedral and hexahedral elements with different sizes were selected during the mesh generation, with smaller sizes and mesh refinement being used in areas that experienced serious mesh distortion.
The contact type of each surface, excluding facet cartilage, was set as bounded, and the cartilagecartilage contact was defined as frictionless8, 9, 14, 18, 26 31. Six degrees of freedom were rigidly fixed under the inferior of L5, and moments were applied to the superior of L46, 12, 16, 21. The following equation was applied to mimic the compressive load on the superior of L1 that was transferred from the superior of L4.

Model calibration
The calibration process was accomplished by adjusting the value of P1 via which is relatively easy adjustment in the model construction process, then estimating the ROMs under a moment of 10 Nm and flexion-extension conditions, and finally by comparing our FEA model results from a widely cited in vitro study19, 20.The nucleus, ligaments and facet joints were supressed during this process.  Table 1 show the calibration algorithm and list the material properties, respectively.  Table 2).

Calibration of the relative nucleus position
The definition of ACC is shown in Figure 3, with the sensitivity of the data adjustment tested prior to calibration. We found that the ACC under flexion-extension was up to 99% when the P1 value was calibrated to 1.62. This value was confirmed and used during the model validation process (Table   3 and Figure 4).

Model validation
The validated model was modified using the calibrated relative nucleus position. We found that the ACC was higher than 90% under all of loading conditions including flexion-extension, left-right lateral bending and left-right axial rotation, and was relatively higher under the flexion-extension condition (Table 4 and Figure 5).

Discussion
We explored several methods that defined the relative nucleus position and its cross-sectional area ratio, but there were still several issues with regard to our FEA models9, 20, 21, 27, 28. The data from previous studies could not be directly used in our model construction owing to individual differences and the lack of uniform standards. For example, the previously published nucleus location, wherein its centre was located 3.5 mm towards the posterior of the disc 20, yielded a D2 value of less than 2 mm, which is obviously less than the minimum value measured in our study. Furthermore, the application of this value to the models constructed using imaging data from short volunteers may result in an impossible situation wherein part of nucleus lies outside the disc's boundary. The lack of uniform definition methods also leads to repeated attempts at model construction and validation.
Finally, we can only define the relative nucleus position as being 'slightly posterior' to the centre of the disc, such that the lowest ACC of our model was lower than 70% and which needs to be improved furtherly, even though the model validation process was verified in our published studies. Therefore, the calibration of the relative nucleus position and the investigation of a reliable ratio to define the above indicators are vital for improving ACC of FEA, a widely used research method in the investigation of pathogenesis of DDD and optimization of spine surgical methods 6, 9, 13, 25, 28.
The reduced T2 signal in the MRI data is closely related to disc degeneration and is generally selected to measure such a pathological change23, 29. The homogeneity test is important for ensuring the ACC and credibility of the study as it is based on subjective observer measurements23, 30 Notably, the nucleus itself was supressed during the model calibration process in the current study, even though the measured and calibrated values were closely associated with the nucleus. This is because R1 and R2 have an important impact on both M1 and M2 and their resultant ROMs. The variation in the ROMs is more likely due to the change in the approximately quadrangular annulus areas caused by the change of relative nucleus position and its cross-sectional area rather than the nucleus itself. The nucleus, ligaments and facet joints were therefore suppressed during the model calibration to investigate this factor individually. The FEA study results indicated that the retrodisplacement of the nucleus improved the ACC during the calibration process. Furthermore, the posterior structures overlapped in the model calibration under the extension condition. This phenomenon is not indicated by the computation error; rather, it is caused by the omitted contact types between the bone structures. Therefore, spatial positions are independently calculated when facet cartilages have been supressed (Fig. 4).
Differences still exist between the ROMs in the current FEA study and the widely cited in vitro study27; these differences may be due to the supressed structures on the ROMs, even though the measurement and calibration of the relative nucleus position and its cross-sectional area ratio increase the ACC indeedly. This defect may also affect the ACC because the definition of the ligaments also lack a standard method. Furthermore, there are no published in vitro ROMs values that have been computed from models with the ligaments, facet joints and nucleus removed under lateral bending and axial rotation conditions; hence, we are unable to calibrate the FEA model under these loading conditions. This defect can provide a good explanation for the relatively lower ACC under lateral bending and axial rotation conditions. Nevertheless, we believe that the difference in the model validation is acceptable because the lowest ACC value considered in the current study is greater than 90% and the difference is far less than one standard deviation and than which in previously published studies6, 9, 10, 13, 15, 18, 26, 28. Therefore, the measured P2 and calibrated P1 values in our subsequent FEA studies can be used to increase ACC.
The current study still faces some limitations. The ratio measurement of nucleus position is based on the cross-sectional areas in a specific two dimentional plane rather than the three dimentional volume, such that the models do not capture differences in the lumbar lordotic angle. For example, obvious changes in the disc volume can be observed in the models with the same cross-sectional area ratio and different lordotic angles. Furthermore, the definition of the ligaments was accomplished based on our observer measurements and do not conform to a uniform standard, even though ligaments play a significant role of the maintenance of lumbar stability6, 11, 31,and are a key index in the ROMs. Therefore, the definition of ligaments should be investigated and calibrated in future studies to further develop more accurate FEA models.

Conclusion
The measured and calibrated relative nucleus position (P1 = 1.62) and its cross-sectional area ratio (P2 = 38%) could increase the ACC of the FEA model indeedly in our study, and such values can therefore be used in subsequent studies concerning the investigation of pathogenesises of DDD and optimization of spine surgical methods.      Calibration algorithm for the relative nucleus position.

Left-Right Axial Rotation
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