A new method to approximate load–displacement relationships of spinal motion segments for patient-specific multi-body models of scoliotic spine

Abstract

Load-displacement relationships of spinal motion segments are crucial factors in characterizing the stiffness of scoliotic spine models to mimic the spine responses to loads. Although nonlinear approach to approximation of the relationships can be superior to linear ones, little mention has been made to deriving personalized nonlinear load-displacement relationships in previous studies. A method is developed for nonlinear approximation of load-displacement relationships of spinal motion segments to assist characterizing in vivo the stiffness of spine models. We propose approximation by tangent functions and focus on rotational displacements in lateral direction. The tangent functions are characterized using lateral bending test. A multi-body model was characterized to 18 patients and utilized to simulate four spine positions; right bending, left bending, neutral, and traction. The same was done using linear functions to assess the performance of the proposed tangent function in comparison with the linear function. Root-mean-square error (RMSE) of the displacements estimated by the tangent functions was 44 % smaller than the linear functions. This shows the ability of our tangent function in approximation of the relationships for a range of infinitesimal to large displacements involved in the spine movement to the four positions. In addition, the models based on the tangent functions yielded 67, 55, and 39 % smaller RMSEs of Ferguson angles, locations of vertebrae, and orientations of vertebrae, respectively, implying better estimates of spine responses to loads. Overall, it can be concluded that our method for approximating load-displacement relationships of spinal motion segments can offer good estimates of scoliotic spine stiffness.

Computational modelling of the scoliotic spine: A literature review

“…When the IVD stiffness is included, it is often a linear and lumped parameter (Table 3), the impact the non‐linearity may have has been studied in greater detail with healthy MBMs. This is only suitable for a limited RoM and could result in inaccurate predictions when simulating surgeries where large rotations and loads are present 136 . Jalalian et al developed a method to characterise a non‐linear stiffness for the IVD using a 2D model.…”

“…It was also able to provide accurate predictions for shapes not used in the characterisation of the stiffness. While limited to only being in 2D they suggest the method could easily be expanded to 3D if suitable data is available 136 …”

“…This is only suitable for a limited RoM and could result in inaccurate predictions when simulating surgeries where large rotations and loads are present. 136 Jalalian et al developed a method to characterise a non‐linear stiffness for the IVD using a 2D model. The non‐linear stiffness resulted in better shape predictions for all motions, the improvement being more pronounced for larger motions.…”

“…While limited to only being in 2D they suggest the method could easily be expanded to 3D if suitable data is available. 136 …”

“…Many scoliotic models are validated by simulating the surgical intervention performed and then comparing predicted curvature to the post‐operative curvature 13 , 15 , 70 , 72 , 73 , 74 , 81 , 84 , 106 , 127 , 128 , 129 , 131 , 132 , 133 , 139 or by simulating a bending test and comparing the predicted curvature to the measured curvature from radiographic images. 69 , 85 , 96 , 134 , 135 , 136 , 137 , 140 Predicted Cobb angles within 5° of the clinically measured values are often predicted and are considered acceptable as it corresponds to the clinical accuracy. 155 …”