The Intervertebral Joint as a Stiffness Matrix: Theory, Practice, and Application

Abstract

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.