Computational Modeling of the Spine

Summary of Key Points

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The reduced costs and time, and the ability to simulate sophisticated loading conditions and vary certain parameters (such as treatment strategies) in a perfectly controlled analysis environment, are the major advantages of finite element analysis.
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Finite element analysis can determine the stress or strain response of complex spinal structures and kinetic and kinematic changes in the spinal segment under imposed conditions.
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Various spinal disorders, degeneration, fusion surgeries, or motion preservation procedures can be simulated in finite element models, and their effect on the spinal motion and load sharing can be estimated.
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The quality of the information obtained in finite element analysis depends on the accuracy of the assumptions (i.e., material properties and loan and boundary conditions) made during modeling.

An anatomic experimental approach has been the gold standard for the biomechanical evaluation of spinal structures. In vitro biomechanical investigations commonly use cadavers, animal carcasses, or synthetic models. Among these options, cadaveric investigations produce the most reliable and realistic results. However, despite its high clinical relevance, fresh-frozen cadaveric spinal specimens may be expensive (> $1000 in the United States) or difficult to acquire. Animal models such as calves, sheep, and pigs, on the other hand, are relatively inexpensive (~$100). Animal models offer two main advantages. First, bovine, ovine, and porcine spine models exhibit gross anatomic structures (e.g., facets, processes, ligaments) similar to those of the human spine, which permits the convenient application of the instrumentation used in surgery. Second, between-subject variability is minimal with animal models, which reduces the statistical sample size demand. Synthetic models usually involve polyurethane foam surrogates, which are available in any density and geometry, ranging from blocks to any anatomic shape. They are inexpensive (< $100). In addition, synthetic models can be produced homogeneously, presenting little or no interspecimen or intraspecimen variation. Although synthetic models can represent the human anatomy better than animal models and have less specimen-related variability, they are considered the least clinically relevant models because they cannot match the material properties of the natural bone.

Some investigators, facing the shortcomings of all of these approaches, turn to computational techniques. With the advances in computer and imaging technologies, finite element models of spinal segments or vertebrae can be developed and validated rapidly and inexpensively. Finite element analysis, a mathematical method performed by fictively dividing structures into simpler substructures such as triangles or cubes, can determine the stress or strain response of complex structures under imposed conditions by performing calculations on the simpler substructures. Once the anatomic model is introduced into the finite element analysis software, kinetic and kinematic analyses, as well as thermal, flow, and time-related problems, can be solved.

Computational techniques, especially finite element analysis, have provided substantial help for researchers in the area of musculoskeletal biomechanics. The reduced costs and time (in most cases) and the ability to simulate sophisticated loading conditions and vary certain parameters (such as treatment strategies) in a perfectly controlled analysis environment are the major advantages of finite element analysis. Finite element modeling will become much more pervasive with developments in the hardware and software technology that provide the ability to build complex and comprehensive models that truly reconstruct real-life structures and conditions of the spine.

This chapter examines common finite element modeling procedures (i.e., obtaining three-dimensional [3D] models of spinal segments, determining structural and material properties, and assigning contact, loads, and boundary conditions) and some biomechanical applications.

Finite Element Models

Even though the finite element method has been applied to structural engineering problems since the 1950s, it was first applied to the biomechanics of the spine in the 1970s. With improvements in numerical techniques and computer technology, its use in the field has gained momentum and it has become an integral part of spine biomechanics that is viewed as a complementary and insightful analysis ( Fig. 35-1 ).

Modeling of Spinal Elements

Finite element modeling of spine segments or vertebrae, similar to other engineering solutions, involves the transfer of the physical sample into the 3D reconstructed form in the computer environment, assignment of structural and material properties, definition of contact properties, and application of loads and boundary conditions. Although the finite element modeling procedure is essentially the same in most of the studies, many different techniques and properties are used in these steps.

Three-Dimensional Reconstruction and Structural Modeling

In the finite element method, geometric accuracy of the model has a significant effect on problem solving and can cause marked differences in the results.

Several methods have been commonly used to model spinal units, including touch probe digitizer, laser scanner, computed tomography (CT), and magnetic resonance imaging (MRI). Even though precise, highly accurate geometric models can be obtained using the touch probe digitizer and laser scanner methods, noninvasive techniques such as CT and MRI often are preferred because of their ability to retrieve data on internal features of the specimens—and even from live subjects—with ease. Some techniques also combine some of the methods mentioned earlier for the reconstruction of spinal units.

In thorough finite element models, vertebral bodies usually are modeled in several sections, such as cortical shell, cancellous core, dorsal bony elements, bony end plates, and cartilage layers, due to their unique material properties.

In almost all studies, intervertebral discs are formed by three significant parts: the nucleus pulposus, the anulus fibrosus, and the cartilaginous end plates. The anulus is further divided into subcomponents such as ground substance and anulus fibers. Anulus ground substance is modeled as several layers surrounding the nucleus. The anulus fibers are angled 30 degrees and 150 degrees with respect to the transverse plane in adjacent layers. Some studies have used more realistic approaches in the design of these fibers by considering the increase in fiber angle from ventral to dorsal, and also fiber angle variation between anulus layers motivated by the histologic findings.

After a solid model of the spine is built, each of its components is carefully divided into simpler substructures called elements for mathematical calculation purposes. Generally, hard tissues such as cortical shell, trabecular core, and dorsal elements are composed of tetrahedral (pyramid) and pentahedron (wedge) elements. With soft tissues, such as the anulus ground substance or cartilage layers, it is necessary to use hexahedral (rectangular prism) elements due to low-elasticity modulus and high Poisson ratio properties. In most studies, bilinear link or spring elements (cables), which are only capable of sustaining tension, are employed for ligaments and anulus fibers.

Material Definition

Material definition is one of the most sensitive and intrinsic parts of finite element modeling of the spine. Due to variability within and among test specimens, different researchers have defined many material properties for the different components the spine.

Vertebral Bodies, End Plates, and Facet Cartilages.

Even though bone is a poroelastic, anisotropic, and viscoelastic structure, in reality, 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 35-1 to 35-3 ). 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 35-4 ). Cartilage layers also are added to the facets in most modeling paradigms ( Table 35-5 ).

TABLE 35-1

Material Properties Considered for the Cortical Shell

Model	E [MPa]	v	References
Linear elastic, isotropic	5000–15,000	0.2–0.3

TABLE 35-2

Material Properties Considered for the Trabecular Core

Model	E [MPa]	v	G [MPa]	References
Linear elastic, isotropic	100–500	0.2–0.3	41.7

TABLE 35-3

Material Properties Considered for the Posterior Elements

Model	E [MPa]	v	G [MPa]	References
Linear elastic, isotropic	3000–6000	0.25–0.3	1400

TABLE 35-4

Material Properties Considered for the End Plates

Model	E [MPa]	v	References
Bony, linear elastic, isotropic	500–12,000	0.3–0.4
Cartilaginous, linear elastic, isotropic	23.8–25	0.3–0.4

TABLE 35-5

Material Properties Considered for the Facet Cartilage Layers

Model	E [MPa]	v	References
Linear elastic, isotropic	0.7–35	0.1–0.4

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 35-6 to 35-8 ). 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.

TABLE 35-6

Material Properties Considered for the Nucleus Pulposus

Model	E [MPa]	v	Other
Incompressible fluid	—	—	—
Poroelastic	1.5	0.49	Poroelasticity k _x= k _y= 1.82e-16 k _z= 1.561e-04 mm ⁴/Ns
Fluid-like solid, linear elastic, isotropic	0.2–3.4	0.499	—
Incompressible, hyperelastic, Mooney-Rivlin	—	0.499	Mooney-Rivlin c ₁= 0.12 c ₂= 0.03–0.09

TABLE 35-7

Material Properties Considered for the Anulus Ground Substance

Model	E [MPa]	v	Other
Linear elastic, isotropic	3.15–4.2	0.4–0.45	G [MPa]1.6
Hyperelastic, neo-Hookean	—	—	Neo-Hookean c ₁₀= 0.3448 D ₁= 0.3
Hyperelastic, Mooney-Rivlin	≈=4.2	0.45	Mooney-Rivlin c ₁= 0.18–0.56 c ₂= 0.045–0.14
Poroelastic	Neo-Hookean	—	Neo-Hookean c ₁₀= 0.25 MPaK = 0.67 MPa	Poroelasticity k ₀= 7.5e-16 m ⁴/Nse ₀= 2.3M = 8.5
Poroelastic	Hyperfoam, nonlinear	—
Poroelastic	2.56–12.29	0.35
Poroelastic	2.5	0.17

TABLE 35-8

Material Properties Considered for the Anulus Fibers

Model	E [MPa]	v	Specifications
Nonlinear elastic	Nonlinear curve	—	Ratio of E through layers 1.0–0.9–0.75–0.65
Nonlinear elastic	Nonlinear	—	Ratio of stiffness through layers 1.0–0.9–0.75–0.65
Linear, elastic	60–500	0.3–0.35	—

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 35-9 ).

TABLE 35-9

Material Properties Considered for Ligaments in the Literature

Model	ALL	PLL	LF	ITL	CL	ISL	SSL
Nonlinear	Nonlinear Stress–Strain Curve
CS area (mm ²)	24–49.1	14.4–22.2	71.7–75	4–12	36–130.9	40–49.2	30–70.3
Linear
E [MPa]	20	70	50	50	20	28	28
v	0.3	0.3	0.3	0.3	0.3	0.3	0.3
CS area (mm ²)	38	20	60	10	40	35.5	25.5

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. 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 and associates 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 and colleagues 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 and coworkers viewed the articulation process as a general moving contact problem and assigned 1 mm for the initial gap between the articulating surfaces. Guan and colleagues 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 and coworkers. 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. 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 and colleagues 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, 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. 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 and associates 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.

Loads and Boundary Conditions

Application of loads and boundary conditions varies according to the aim of the study. Generally, loading types used in the computational analysis of the spine are simple static or incrementally altering quasi-static loads. In general, the inferior end plate of the lowest vertebra is fixed rigidly, and loads (or displacements) are applied to the superior end plate of the uppermost vertebral body.

In most cases it is easy to simulate tension or compression loads. It is more complicated to apply moments, add muscle forces and upper body weight effects, or realize coupled motions.

Shirazi-Adl and colleagues and Polikeit and colleagues applied a linearly varying distribution of axial loads to the top of the upper vertebra to simulate pure bending moments, with no net axial force. Kumaresan and colleagues and Wheeldon and colleagues succeeded in producing pure moments by using a force couple acting on the rigid plate attached to the superior vertebra.

Sharma and associates simulated pure flexion-extension moments using a pair of concentrated axial loads on top of the upper vertebra. They also studied the effect of anteroposterior shear during sagittal motion by a horizontal force applied at the center of the upper vertebral body. Two equal and opposite forces at the nodes at the edge of the sagittal or medial periphery of the upper vertebral body were applied to produce a force couple yielding a pure moment.

Researchers have used the follower load technique, suggested by Patwardhan and coworkers, extensively since the early 2000s to simulate the upper-body weight and the stabilizing effects of muscle forces seen in vivo without inducing any intervertebral rotations.

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