Effect of neglecting passive spinal structures: a quantitative investigation using the forward-dynamics and inverse-dynamics musculoskeletal approach

Purpose: Inverse-dynamics (ID) analysis is an approach widely used for studying spine biomechanics and the estimation of muscle forces. Despite the increasing structural complexity of spine models, ID analysis results substantially rely on accurate kinematic data that most of the current technologies are not capable to provide. For this reason, the model complexity is drastically reduced by assuming three degrees of freedom spherical joints and generic kinematic coupling constraints. Moreover, the majority of current ID spine models neglect the contribution of passive structures. The aim of this ID analysis study was to determine the impact of modelled passive structures (i.e., ligaments and intervertebral discs) on remaining joint forces and torques that muscles must balance in the functional spinal unit. Methods: For this purpose, an existing generic spine model developed for the use in the demoa software environment was transferred into the musculoskeletal modelling platform OpenSim. The thoracolumbar spine model previously used in forward-dynamics (FD) simulations provided a full kinematic description of a flexion-extension movement. By using the obtained in silico kinematics, ID analysis was performed. The individual contribution of passive elements to the generalised net joint forces and torques was evaluated in a step-wise approach increasing the model complexity by adding individual biological structures of the spine. Results: The implementation of intervertebral discs and ligaments has significantly reduced compressive loading and anterior torque that is attributed to the acting net muscle forces by −200% and −75%, respectively. The ID model kinematics and kinetics were cross-validated against the FD simulation results. Conclusion: This study clearly shows the importance of incorporating passive spinal structures on the accurate computation of remaining joint loads. Furthermore, for the first time, a generic spine model was used and cross-validated in two different musculoskeletal modelling platforms, i.e., demoa and OpenSim, respectively. In future, a comparison of neuromuscular control strategies for spinal movement can be investigated using both approaches.


Forces and torques acting on the spinal motion segment
The musculoskeletal (MSK) system of the spine comprises bones (e.g., vertebrae) connected by joints, muscles, and passive structures such as ligaments and the intervertebral disc (IVD) that significantly contribute to the total net joint forces and torques. Thus, with respect to the general equation of motion in multibody dynamics [1], the force f i (q,q) and torque τ i (q,q) acting on a joint i can be calculated according to Eqs 1, 2.
where q,q are the vectors of generalised joint coordinates and velocities, respectively; F MTU m (q,q) and F LIG n (q) are the vectors of musculotendon and ligament forces for all muscles m and all ligaments n spanning the intervertebral joint i; R MTU m,i (q) R LIG n,i (q) are the corresponding muscle and ligament moment arms, and R MTU m,i (q)F MTU m (q,q) and R LIG n,i (q)F LIG n (q) are the resulting torque contributions to the joint load; F IVD i (q) and T IVD i (q) are the IVD force and torque vectors of IVD torques; and F E i (q,q) and T E i (q,q) are the vectors of external forces and torques acting onto the joint, which only contain gravitational loads in this study. Note, in general, IVD and ligament forces do also depend onq. In this study, however, mechanical damping of these two passive tissue elements was neglected. Fig. 1 shows all forces and torques acting on a spinal motion segment under the assumption of a six degree of freedom (DOF) joint. Note, in inverse-dynamics (ID) analysis the contribution of muscles ( Fig. 1: red) is neglected.
Considering the standard "plain" model stripped by all IVD and ligament forces, the force equilibrium would require muscles to produce a positive (pushing) force according to represent the gravitational forces of the summed segment and vertebral body (VB) weights, respectively, which are located superior to joint i. For the torque balance, the muscles would need to produce a positive (posterior) net torque, i.e., by applying a negative (pulling) force, to counteract the negative (anterior) torque created by the VB T VB i and segment weight T S Given muscles can only act in compression through muscle contraction, this inconsistency is typically solved by constraining the translational DOFs in intervertebral joints and neglecting the force balance.
On the other hand, considering the "all elements" model, the modelled passive elements in sum are decompressive, and can, thus, compensate for the pulling forces of muscles.

ID analysis using the "intrinsic IVD" model
IVD contribution to the generalised axial joint force F y,i and torque in the sagittal plane T z,i was evaluated in two steps. First, the level-specific offset force F IVD,0 in axial direction was applied using the "intrinsic IVD" model. Second, the six DOF stiffness matrix was included in the "full IVD" model, see Table 2 and Figure 3 in the main text for reference. The results from the "intrinsic IVD" model ID analysis are shown in Fig. 2. With respect to the "plain" model, the inclusion of the IVD offset forces uniformly reduced the constant compressive loading of the "plain" model on average by −101% between L1/2 and L5/S1 such that 0 N > F y,i > −7 N . This is expected, given the IVD offset forces were estimated from the weight of cumulated VB and segment masses located proximally to each joint. Applied as a constant unidirectional force along the local axial direction of each IVD, F IVD,0 did not have an effect on the remaining anterior joint torques T z,i . The weight of the cumulated VB and segment masses pointing downwards create a negative (compressive) joint force (F VB + F S ) and a negative (anterior) joint torque (T VB + T S ) with respect to the axis convention indicated. The IVD bushing element counteracts the anterior torque of the motion segment by introducing a positive (posterior) torque T IVD . In addition to T IVD , the total IVD force F IVD (see Eq. 3 in the main text) is applied such that −F IVD,0 ≈ F S + F VB . The IVD forces (F IVD,stiff : blue, F IVD,0 : purple) are applied at the intervertebral joint position with opposite signs onto adjacent VBs. The ligaments (green) and muscles (red) together further counteract the anterior torque such that −F IVD,stiff ≈ F LIG + F MTU,back + F MTU,abd by pulling on the superior vertebra downwards. Note, given a six DOF joint accounting for translational movements is used, no joint reaction forces are present. The load created by the upper body with respect to the joint is carried by surrounding tissue. Note that depicted force arrows are not to scale.  Table 2 in the main text with the generalised axial force F y,i (left) and torque in the sagittal plane T z,i (right) for all lumbar levels i.