## 2 Materials and Methods
### Material Properties  
For the LBM, isotropic, homogeneous, and linear elastic material properties from the literature were assigned to the bony elements [16; 34; 35] (Figure 2a). The moduli of elasticity and the Poisson's ratios of the bony elements for the LBM are presented in Table 1.  
In the PSM, locally defined material properties were applied to model bone heterogeneity, as the mechanical properties of the bone tissues were assigned to each element based on the QCT scans. Hydroxyapatite calibration phantoms (Hitachi Presto, Hitachi Medical Corporation, Tokyo, Japan) were used to determine the relationship between the Hounsfield Units (HU) and elastic moduli. \[\varrho_{\text{QCT}}=5.2\times 10^{-3}+8\times 10^{-4}\times HU\Big{[}\frac{g}{cc} \Big{]} \tag{1}\]\[\varrho_{QCT}=\varrho_{ash} \tag{2}\] \[\varrho_{ash}=\varrho_{app}\times 0.6\] (3) \[\varrho_{app}=8.667\times 10^{-3}+1.333\times 10^{-3}\times HU\left[ \frac{\delta}{cc}\right]\] (4) \[E=4730\times\varrho_{app}^{1.56}\left[\frac{\delta}{cc}\right] \tag{5}\]  
Figure 1: Workflow of the FE model development. (**a**) The geometry of the LBM and the PSM was defined by segmentation based on QCT scans. The smoothed and remeshed surface mesh was used for the volume meshing. For the LBM, the applied material properties were taken from the literature, while for the PSM, the bone material was assigned according to the HUs, using multiple sets of equations. (**b**) The FE model of the intervertebral disc includes the cartilaginous endplate, the nucleus pulposus, the annulus fibrosus ground substance, and the collagen fibres. (**c**) Ligaments are illustrated on the L4–L5 FE model in sagittal and posterior views. First, the HU was converted to equivalent bone mineral density (\(\rho_{QCT}\)) by linear calibration using calibration phantoms (1) with known densities of 50, 100, 150, and 200 mg/cm\({}^{3}\)[36]. Equivalent bone mineral density was considered to be equal to ash density (\(\rho_{ash}\)) (2) [11; 37; 38], while the apparent density (\(\rho_{\text{app}}\)) was calculated as 60% of ash density (3) [11; 39]. The relationship between apparent density and HU (4) was calculated employing the Equations (1)-(3). The isotropic elastic modulus (\(E\)) was calculated from apparent density based on a relationship from the literature [40]. To assign the calculated elastic moduli to the elements, the volume meshes were imported into Materialise Mimics software. Elements were placed in ten equal-sized groups based on the average HU value of the voxels contained by an element [41]. Elements with HU < 0 were placed in an eleventh group, representing fat- and marrow-like materials with lower density [42]. The median HU of each group was set as the representative value and was assigned to all elements in the group. The Poisson's ratio of all bony tissues in the PSM was set to be 0.3 [37; 38; 39] (Figure 2b). The material properties of the soft tissues were identical in the two models. The cartilaginous endplates were assigned with linear elastic material property taken from the literature [27]. NP and AF ground substance was modelled with an incompressible 2-parameter hyperelastic Mooney-Rivlin formulation. The mechanical behaviour of the annulus' collagenous fibres was represented by a nonlinear stress-strain curve from Shirazi-Adl [35]. Since the outer layers of the IVD are stiffer than the inner layers, the collagen fibres were weighted by scalar factors defined for each layer (Figure S1) [43]. The facet cartilage was modelled with a hyperelastic Neo-Hooke formulation [27]. Tension-only, nonlinear uniaxial spring elements [44] were used to simulate the ligament behaviour (Appendix A). The material properties used in the current study are summarised in Table 1.