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Open Access Publications from the University of California

Scholarly Works (3 results)

  • Thesis
  • Peer Reviewed
UC Berkeley Electronic Theses and Dissertations (2011)

The relationship between the motion of a pair of vertebras and the ensuing loads exerted by the intervertebral joint have been studied extensively. In the realm of small and quasi-static motion, this relationship is approximately linear and can be modeled using a 6 × 6 stiffness matrix. Efforts to determine the elements of these stiffness matrices as well as to apply them in models of the joint have been underway since this joint model was first proposed in the early 1970s. Meanwhile, a considerable amount of theoretical work has been conducted in identifying linear mappings that relate increments in force and moment components acting on a system of rigid bodies to the infinitesimal motions that produced them. Most prominent among these is the Cartesian stiffness matrix parameterization. Alongside these developments, advancements in musculoskeletal software modeling has led to the developments of various software platforms capable of performing the underlying computations necessary to study musculoskeletal structures. The motion of articulating bodies in this case is defined using joints. In the case of the spine, bushing elements - similar to the rubber bushings employed in vehicular dynamics - are a popular alternative. Unfortunately, numerous difficulties exist in adapting these bushing elements to mimic the stiffness matrix model of the intervertebral joint.

In this dissertation, we connect the inter-related subjects above to pave the way for a more comprehensive model of the intervertebral joint. To do this, we derive the Cartesian stiffness matrix associated with the joints of the spine using arguments based on energetics consideration and the Euler angle parameterization of rotations. We next show how this Cartesian stiffness matrix can be related to elements of the experimentally measured stiffness matrix of the intervertebral joint. Finally, we apply the resulting stiffness matrices in a model of the lumbar spine developed in OpenSim using a stiffness matrix plugin we created specific to the intervertebral joint.

    Cover page: The Intervertebral Joint as a Stiffness Matrix: Theory, Practice, and Application
    • Article
    • Peer Reviewed
    UC Berkeley Previously Published Works (2010)
    Several Cartesian stiffness matrices for a single rigid body subject to a conservative force field are developed in this paper. The treatment is based on energetic arguments and an Euler angle parameterization of the rotation of the rigid body is employed. Several new representations for the stiffness matrix are obtained and the relation to other works on Cartesian stiffness matrices and Hessians is illuminated. Additional details are presented with respect to determining the Cartesian stiffness matrix for a pair of rigid bodies, as well as for a system of rigid bodies constrained to a plane.
      Cover page: On Cartesian stiffness matrices in rigid body dynamics: an energetic perspective
      • Article
      • Peer Reviewed
      UC Berkeley Previously Published Works (2013)
      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. © 2012 Springer Science+Business Media Dordrecht.
        Cover page: On the modeling of the intervertebral joint in multibody models for the spine
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