Finite element algorithm for lumbar spine modeling incorporating physical nonlinearity of material

Kurochka K.S., Komarov V.V., Tsitko E.L., Panarin K.A., Semenchenya T.S.
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This paper presents the development and verification of a finite element algorithm for modeling nonlinear deformation in the functional spinal unit (FSU) of the human lumbar spine. The study is motivated by the increasing prevalence of musculoskeletal disorders and the need for improved predictive methods for spinal motion segment behavior under various loading conditions and pathologies.

Current modeling approaches often employ simplified geometry and linear deformation laws (Hooke’s law) for all spinal elements, including the intervertebral disc. However, these models inadequately represent human anatomy and the physical properties of the intervertebral disc, which comprises the annulus fibrosus and nucleus pulposus exhibiting pronounced nonlinear and viscoelastic characteristics.

To address these limitations, the authors propose a mathematical model wherein vertebrae are treated as linearly elastic, heterogeneous bodies, while the intervertebral disc is modeled according to nonlinear deformation theory. An anatomically accurate 3D reconstruction of the lumbar spine was generated from computed tomography (CT) images. Volumetric tetrahedral finite elements were employed for discretization. The intervertebral disc behavior is described using thin plate bending theory based on Kirchhoff-Love assumptions.

The algorithm’s key feature is a two-step solution procedure: an initial linear elastic solution is obtained, followed by the calculation of additional nodal forces for disc elements, enabling efficient nonlinear analysis without stiffness matrix modification.

Model verification was performed using two geometric representations of the lumbar spine: a simplified version comprising parallelepipeds and cylinders, and a patient-specific realistic model derived from CT data. Numerical simulation results demonstrate that incorporating nonlinear properties of the intervertebral disc substantially alters both deformation patterns and magnitudes compared to purely elastic models, producing nonlinear segments on stress-strain curves. The discrepancy between results obtained with the proposed algorithm and those from Comsol software did not exceed 10%, confirming the model’s validity. This algorithm shows promise for investigating lumbar spine biomechanics under normal conditions, degenerative changes, and post-surgical states.

Keywords: finite element method, convergence, lumbar spine, vertebrae, intervertebral disc
Citation in English: Kurochka K.S., Komarov V.V., Tsitko E.L., Panarin K.A., Semenchenya T.S. Finite element algorithm for lumbar spine modeling incorporating physical nonlinearity of material // Computer Research and Modeling, 2026, vol. 18, no. 4, pp. 974-988
Citation in English: Kurochka K.S., Komarov V.V., Tsitko E.L., Panarin K.A., Semenchenya T.S. Finite element algorithm for lumbar spine modeling incorporating physical nonlinearity of material // Computer Research and Modeling, 2026, vol. 18, no. 4, pp. 974-988
DOI: 10.20537/2076-7633-2026-18-4-973-988
URL: http://crm-en.ics.org.ru/journal/article/3754/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

Copyright © 2026 Kurochka K.S., Komarov V.V., Tsitko E.L., Panarin K.A., Semenchenya T.S.

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