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On the modeling of the intervertebral joint in multibody models for the spine

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Abstract

The need to develop feasible computational musculoskeletal models of the spine has led to the development of several multibody models. Central features in these works are models for the ligaments, muscles, and intervertebral joint. The purpose of the present paper is to show how experimental measurements of joint stiffnesses can be properly incorporated using a bushing element. The required refinements to existing bushing force functions in musculoskeletal software platforms are discussed and further implemented using a SpineBushing element specific to the intervertebral joint. Four simple lumbar spine models are then used to illustrate the accompanying improvements. Electronic supplemental material for this article includes a complementary review of formulations of stiffness matrices for the intervertebral joint.

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Notes

  1. An example of a stiffness matrix with the aforementioned symmetries can be seen in Eq. (24) below.

  2. The reader is also referred to Sect. 5.1 of the Online Resource for further details on the specifications of Δ y A and Δ β.

  3. In OpenSim, the generalized force due to the bushing element \(\mathsf{F}^{B}_{1} = - \mathsf{F}^{B}_{2}\) is exerted on the bushing frame \(\mathbb{B}_{2}\) and an additional transformation applied to account for the shift from \(\mathbb{B}_{2}\) to \(\mathbb{B}_{1}\). Consequently, the generalized force acting on the frame \(\mathbb{B}_{1}\) is not equal and opposite to that exerted on the frame \(\mathbb{B}_{2}\) by the bushing element. Rather, the force and moment vectors exerted on \(\mathbb{B}_{1}\) by the bushing element are given by

    $$\mathbf{F}^{B}_1 = -\mathbf{F}^{B}_2, \qquad \mathbf{M}^{B}_1 = -\mathbf{M}^{B}_2 + \bigl( \mathbf{x}_B^2 - \mathbf{x}_B^1 \bigr) \times\mathbf{F}_B^1. $$

    We will refrain from using this convention.

  4. Recall that in both of these studies, the motion and loads at the vertebral centers of geometry (≈ vertebral center of mass) were used to determine the elements of K E.

  5. The coefficients associated with their reported stiffness matrices were determined by performing highly controlled motion in one direction, measuring the ensuing forces and moments exerted at the geometric centers of the vertebrae, and then using a least-squares fit to the experimental data by the method specified in [35].

  6. It is important to note that the computed muscle forces depend on the optimization routine employed.

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Acknowledgements

The authors thank Professor Scott Delp and the members of the OpenSim team for their generous technical support with this software. Maurice Curtin’s work was supported by Enterprise Ireland and a bursary from University College Dublin, while the four other coauthors’ work was partially supported by the National Science Foundation of the United States under Grant No. CMMI 0726675.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA, 94720-1740, USA

    Miguel Christophy, Nur Adila Faruk Senan & Oliver M. O’Reilly

  2. School of Electrical, Electronic and Communications Engineering, University College Dublin, Belfield, Dublin, 4, Ireland

    Maurice Curtin

  3. Department of Orthopaedic Surgery, University of California at San Francisco, San Francisco, CA, 94143-0514, USA

    Jeffrey C. Lotz

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  1. Miguel Christophy
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Correspondence to Oliver M. O’Reilly.

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Electronic supplementary material for On the modeling of the intervertebral joint in multibody models for the spine: Review of Stiffness Matrices for the Intervertebral Joint (PDF 200 kB)

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Christophy, M., Curtin, M., Faruk Senan, N.A. et al. On the modeling of the intervertebral joint in multibody models for the spine. Multibody Syst Dyn 30, 413–432 (2013). https://doi.org/10.1007/s11044-012-9331-x

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