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

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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.

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Notes

  1. The erect position is considered as the resting position of the spine in scoliotic spine models, i.e. it is assumed that there is no load on the spine model and r of all vertebrae is zero [5, 21, 32].

  2. In multi-body models of the scoliotic spine, to simulate the lateral bending positions, a force is typically exerted on the uppermost vertebra in the spine model in the erect position [12].

  3. 504 displacements = 14 vertebrae from L3 to T2 × 2 positions × 18 patients; note that L4 had no displacement according to definition of G.

  4. The bending X-rays are taken while the patients perform maximum voluntary bending movements to the right/left sides, implying small difference between R and r.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Block EA, #02-17, National University of Singapore, 1 Engineering Drive 2, Singapore, 117576, Singapore

    Athena Jalalian, Francis E. H. Tay & Soheil Arastehfar

  2. Department of Orthopedic Surgery, National University of Singapore, Singapore, Singapore

    Gabriel Liu

Authors
  1. Athena Jalalian
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  2. Francis E. H. Tay
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  3. Soheil Arastehfar
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  4. Gabriel Liu
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Corresponding author

Correspondence to Athena Jalalian.

Appendix

Appendix

Period of tangent functions is influential to the fitting of such a function to data. The catch is that the data may not show where the vertical asymptotes will fall precisely, and thus, the period cannot be identified directly. To estimate the asymptotes, the range of the available data on the horizontal axis is considered as the initial value for the period of the tangent, and then, both sides of the range are extended until the best fitting is achieved. In our case, the rotational displacements from the erect to right/left bending positions are considered as the initial value for the period. This traces back to the fact that the patients perform maximum voluntary bending movements to the right/left sides, implying that the vertebrae may rotate almost as much as their displacement limits (the vertical asymptotes). Therefore, to identify R Right/Left, we studied the discrepancy between the fitted tangent functions and the displacement data acquired from the lateral bending X-rays against the extension to [r Right, r Left] of the vertebrae (Fig. 8). To do the study, the minimization problem expressed by (5) was solved for 0, 10, 20, 30, and 40 % enlargements of r Right/Left. According to RMSEs plotted in Fig. 8, the enlargement by 20 % resulted in the smallest discrepancies among the considered enlargement percentages. Therefore, R Right/Left was estimated by 1.2 × r Right/Left.

Fig. 8
figure 8

The effect of enlargement of the range of the rotational displacements on the approximation of the load–displacement data by the tangent function

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Jalalian, A., Tay, F.E.H., Arastehfar, S. et al. A new method to approximate load–displacement relationships of spinal motion segments for patient-specific multi-body models of scoliotic spine. Med Biol Eng Comput 55, 1039–1050 (2017). https://doi.org/10.1007/s11517-016-1576-8

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