Vertebral Bodies, End Plates, and Facet Cartilages

Even though bone is a poroelastic, anisotropic, and viscoelastic structure, in reality,¹⁶ vertebral cortical shell and cancellous bone usually are considered as linear elastic isotropic or transversely isotropic. Dorsal bony elements most commonly are modeled as linear elastic and isotropic (Tables 22-1 to 22-3). Researchers have either neglected the end plate due to its low elasticity module or modeled it as a composite structure (i.e., bony and cartilaginous layers). End plates most commonly were modeled as linear elastic and isotropic bone with a cartilaginous layer (Table 22-4). Cartilage layers also are added to the facets in most modeling paradigms (Table 22-5).

Intervertebral Disc: Anulus Fibrosus and Nucleus Pulposus

The intervertebral disc plays a crucial role in the movement and load-bearing functions of the spinal segments. It probably is the most intricate part of the functional spinal unit (i.e., vertebra-disc-vertebra complex) in finite element modeling, due to the composite structure of the anulus fibrosus and fluid-like behavior of the nucleus pulposus. The anulus fibrosus is formed by a matrix, anulus ground substance, and direction-dependent fibers. Even though there are a few homogeneous models for the anulus fibrosus, more realistic composite models that include matrix and angled fibers have been widely accepted and used (Tables 22-6 to 22-8). Anulus fibers are most commonly modeled with bilinear link, truss, or spring elements, and, in a few cases, with reinforced membrane elements. The material modeling of fibers is almost independent of the modeling technique.

Ligaments

Ligaments are uniaxial structures that do not bear compressive loads. Like the anulus fibers, ligaments are modeled with link, truss, spring, or membrane elements. In general, it is assumed that linear, multilinear, or nonlinear elastic constitutive laws apply for these tissues (Table 22-9).

Articular Facet Joints

The facet joint plays an important role in load bearing and restriction of motion in segmental movements. It functions in tandem with the intervertebral disc in load transfer between adjacent vertebral bodies.^17,18 The articulating structure must be modeled with realistic attributes and proper procedures in computational modeling of spinal segments because it has a significant effect on the quality and quantity of the motion. The articulation characteristics and relative motion depend on many factors, including cartilage layers, gap distance, the condition of the intervertebral disc, loading type, and geometric features of the articulating surface.

Kumaresan et al.¹⁹ compared different techniques for modeling of cervical facet joints. They reconstructed the joint capsule using four different methods: slide-line, contact plane, hyperelastic, and fluid models. The slide-line and contact plane models lacked the synovial fluid and synovial membrane. In the slide-line model, interaction was defined by using slide-line elements (gap elements), whereas in the contact plane model it was achieved by defining a contact plane between the cartilage surfaces. A friction coefficient of 0.01 was assigned for both models. The other models, hyperelastic and fluid, did not have elements for contact modeling; instead, they included synovial fluid between the cartilage layers modeled using noncompressible hyperelastic solid elements and hydrostatic noncompressible fluid elements, respectively. They concluded that fluid modeling of the facet joint matched both actual facet joint anatomy and its function better than the other three models. Similarly, Wheeldon et al.²⁰ used full anatomic parts, such as facet joint cartilage, facet joint synovial fluid, and facet joint membrane, for modeling of the articular facet joints.

Shirazi-Adl et al.^17,18 viewed the articulation process as a general moving contact problem and assigned 1 mm for the initial gap between the articulating surfaces. Guan et al.^7,21 used 3D surface-to-surface contact to simulate the articulation phenomenon and assigned the initial gap between the surfaces according to CT images. Three-dimensional eight-noded compression-only gap elements were used to simulate the articulation between the facets in a study by Goel et al.²² Based on the CT images, they assigned the value of 0.45 mm for the initial gap between the surfaces.

The facet joints also are modeled with 3D gap contact elements with cartilage layers modeled using a parameter of “softened contact,” which exponentially adjusts force transfer between facet surfaces, according to the size of the gap.^9,23–27 An initial gap of 0.5 mm was assigned, and at full closure, the joint was assumed to have the same stiffness as the surrounding bone material.

Schmidt et al.^28,29 modeled the facet joints as nonlinear, with frictionless contact. They assigned the value of 0.6 mm for the initial gap between the cartilage layers. In other studies,^30,31 the facet joints were simulated using frictionless surface-to-surface contact elements in combination with the penalty algorithm with a normal contact stiffness of 200 N/mm, and the initial gap between the cartilage layers assumed to be 0.4 mm.

Some researchers viewed the facet joints as a 3D contact problem with friction.^8,32–34 To allow random motion of the surfaces, such as separation, sliding, and rotation, they defined a finite sliding interaction. For the friction characteristic, they chose a classic isotropic Coulomb friction model and assigned a relatively high friction coefficient of 0.1 as a worst-case scenario.

Lu et al.³⁵ modeled the facet joints using sliding contact elements and assumed the contact pressure to be between 0 and 5000 megapascals (MPa), depending on the gap size. Pressure has been considered to be 0 MPa at a gap size of 0.5 mm and 5000 MPa when the gap is closed completely.