Role of Bone Compliance

In biomechanical response studies of various joint systems, the bony structures are much stiffer than the remaining tissues and, hence, have occasionally been considered rigid bodies. This consideration suggests that the joint laxity is primarily due to connective soft tissues rather than the bony structures. The rigid simulation of bony elements is also motivated in part by the relative ease in modeling and the cost efficiency of the analysis, particularly in a nonlinear response study. A number of biomechanical studies have modeled bony structures as rigid bodies [45, 46, 70-74]. In the lumbar spine, loads are transmitted from one segment to the adjacent one via soft tissues and bony structures. The latter parts are, however, much stiffer than the former parts and, hence, are expected to play a smaller role in joint flexibility via their internal deformations. Our previous model studies have indicated the deformability of the bony elements and the need for their modeling as deformable solids and not rigid bodies [33, 53]. These studies, however, did not determine the extent by which the vertebral compliance influences the joint biomechanics.

Detailed identification of the role of vertebral compliance in joint biomechanics is essential in areas such as prosthetic replacement of segmental elements, in vitro experimental studies, and in vivo measurements of joint displacements through bony posterior elements. Changes in bone material properties are also known to occur with aging, remodeling, and osteoporosis [75-77]. The joint biomechanics, as well as degenerative processes, therefore, could be affected by changes in the structure and density of the bony vertebrae. Moreover, rigid simulation of bony elements, if found reliable in yielding accurate results, significantly reduces the size of the numerical problem to be solved and, hence, allows for the cost-efficient modeling of more complex musculoskeletal systems.

In order to investigate the role of bone compliance in mechanics of motion segments, five models with different representation of bony elements are developed and analyzed under various loads. The modeling of facet joints, intervertebral discs, and ligaments remains identical in these models. The vertebrae of the motion segment are simulated as follows:

1. Bony elements are assumed to be deformable with realistic isotropic material properties; i.e., modulus of elasticity of E = 12000 MPa for the cortical bone, E = 100 MPa for the cancellous bone, and E = 3500 MPa for the bony posterior elements [33, 53].

2. Bony elements are all assumed to be significantly stiffer with homogeneous isotropic properties where E = 26000 Mpa.

3. Each vertebra is modeled as a single rigid body.

4. Each vertebra is modeled as a collection of two rigid bodies attached by two deformable beam elements. The rigid bodies represent the anterior vertebral body and posterior bony elements while deformable beam elements are placed at and oriented along the centroid of pedicles. These beams are expected to somewhat account for the deformability of posterior bony elements. After a number of trials, structural properties of these beams are taken as modulus of elasticity E = 3500 MPa, initial length L = 15 mm, initial cross-sectional area A = 50 mm2, and moments of inertia Iy = 275 mm4, Iz = 150 mm4, and Jx = 500 mm4, where local rigidly moving axes x, y, and z represent the longitudinal and two cross-sectional principal axes, respectively.

5. Bony material properties of Case (1) are reduced by a factor of 5 to model a marked reduction in bone mechanical properties associated with loss of bone density, for example.

The finite element mesh for all cases with deformable vertebrae is similar to that shown in Fig. 1.6 while that for Case (4) is depicted in Fig. 1.12.

Under axial compression forces up to 5000 N, the predicted axial displacements for various vertebral models and boundary conditions are shown in Fig. 1.13. The segmental axial stiffness increases as the coupled sagittal rotation (TY) is restrained, a trend that further continues when the sagittal translation (DX) is also constrained. The foregoing stiffening effect is due to the articulation at the facets that tends to cause coupled flexion in the unconstrained segment. Use of a rigid body for the whole vertebra (Case 3) is seen to considerably stiffen the segment, whereas the presence of a deformable beam connecting two rigid bodies (Case 4) tends to partially correct the overstiffness due to the rigid modeling of vertebrae. As for the facet forces, not shown here, Cases 1 and 3 yield nearly the same results. The facet forces increase as the coupled motions are constrained and as the vertebral compliance is neglected [55].

During flexion moments, an increase in bone stiffness markedly increases the segmental rotational stiffness and tensile forces in supra/interspinous ligaments. The disc pressure, facet contact forces, and forces in disc fibers are decreased. During extension moments, stiffer bone increases the sagittal stiffness and facet contact forces but decreases the disc pressure. During axial torques, stiffer bone noticeably increases the rotational rigidity. Reverse trends are computed as the bone properties reduce. The predicted segmental rotation under flexion, extension, and torsion moments are shown in Fig. 1.14 for various models of bony vertebrae. Detailed results for various cases at 60 N-m axial torque are listed in Table 1.1, indicating that stiffer bone increases the segmental rigidity and facet contact forces but decreases the disc pressure and forces in disc fibers. Reverse trends are predicted as bone mechanical properties are reduced. The use of deformable beam elements in addition to rigid bodies is found to yield results comparable with those computed with realistic material properties for bony vertebrae.

Alteration in the relative stiffness of bony elements noticeably affects the joint biomechanical response in terms of both the gross response and the state of stress and strain in various components. The extent of change depends on the magnitude and type of applied loads. The results of this investigation suggest that changes in bone properties associated with the aging, remodeling, and osteoporosis could have marked effects on mechanics of the human spine. Alteration in the stress distribution due to changes in

DEFORMABLE BEAM ELEMENT

DEFORMABLE BEAM ELEMENT

Bone Remodeling Due Stress
RIGID LOWER VERTEBRAL BODY

(A) SAGITTAL SECTION

FIGURE 1.12 Two cross-sections showing the finite element model of the motion segment with each vertebra represented as a collection of two rigid bodies attached by two deformable beam elements (model D).

bone properties could initiate a series of action and reaction that may accelerate the process of remodeling and segmental degeneration.

Presence of adjacent vertebrae in a multisegmental model in which ligaments and facet joints of neighboring segments apply opposite forces on the posterior elements of the vertebra in between could diminish the extent of the above predicted changes only if the opposing forces are of nearly the same magnitude. This, however, has been found not to be the case in our model studies of the entire lumbar spine subjected to single moments [45, 46]. Under axial compression force, due primarily to facet articulation, the vertebrae are found to experience rotations at the posterior elements different from those at the anterior body. In compression loads, the difference is much larger in the sagittal plane at the L5 vertebra. Under 800 N axial compression force, the posterior elements of the L5 vertebra rotate 1.4° in flexion in comparison with the L5 anterior body. This difference further increases in a more lordotic posture, in the presence of right axial torque and when the L5-S1 nucleus fluid content is lost. The increase of compression load to 2800 N substantially increases the foregoing difference in rotation at the L5 level to 4.1°; that is, while the L5 anterior body is restrained in sagittal rotation, the L5 posterior elements rotate 4.1° in flexion. Such marked differences in rotations point to the level of stress at the

FIGURE 1.13 Effect of vertebral modeling and boundary conditions on the axial response in axial compression force. A: vertebrae with realistic material properties; C: each vertebra as a single rigid body; D: each vertebra as a collection of two rigid bodies attached by two deformable beam elements; TY: coupled sagittal rotation of upper vertebral body; DX: coupled sagittal translation of upper vertebral body.

FIGURE 1.13 Effect of vertebral modeling and boundary conditions on the axial response in axial compression force. A: vertebrae with realistic material properties; C: each vertebra as a single rigid body; D: each vertebra as a collection of two rigid bodies attached by two deformable beam elements; TY: coupled sagittal rotation of upper vertebral body; DX: coupled sagittal translation of upper vertebral body.

Segmental Translation

FIGURE 1.14 Effect of vertebral modeling on the segmental response under various moments. A: vertebrae with realistic material properties; B: vertebrae with material properties increased; C: each vertebra as a single rigid body; D: each vertebra as two rigid bodies attached by two deformable beams; E: vertebrae with reduced material properties.

FIGURE 1.14 Effect of vertebral modeling on the segmental response under various moments. A: vertebrae with realistic material properties; B: vertebrae with material properties increased; C: each vertebra as a single rigid body; D: each vertebra as two rigid bodies attached by two deformable beams; E: vertebrae with reduced material properties.

posterior bony elements, particularly in the pars interarticularis and pedicles. These stresses are caused by the facet contact forces as the posterior ligaments are negligibly loaded in neutral postures. In view of the effect of fatigue and creep on bone failure properties [78, 79], prolonged neutral postures, especially

Table 1.1 Predicted Results for Various Cases Under 60 N-m

Axial Torque

Case

A

B

C

D

E

Axial rotation (deg)

5.6

3.9

3.6

4.8

9.0

Coupled rotations (deg):

Lateral

0.7

0.4

0.2

0.6

1.4

Flexion

1.2

1.0

0.9

1.2

0.9

Disc pressure (MPa)

1.02

0.94

0.94

1.05

1.05

Total facet force (N)

892

1140

1153

883

638

Total fiber force (N):

Innermost

462

455

454

473

454

Outermost

1627

1332

1308

1600

2077

Note: A: realistic deformable material properties for vertebrae; B: much stiffer properties for bony vertebrae; C: a single rigid body for each vertebra; D: each vertebra as a collection of two rigid bodies attached by two deformable beams; and E: reduced properties for vertebrae.

Note: A: realistic deformable material properties for vertebrae; B: much stiffer properties for bony vertebrae; C: a single rigid body for each vertebra; D: each vertebra as a collection of two rigid bodies attached by two deformable beams; and E: reduced properties for vertebrae.

in degenerated discs, could play a role in the pathomechanics of spondylolysis [80-82]. The foregoing results also indicate the likely error involved in the extrapolation of results of in vivo measurements through external systems attached to the spine (usually by insertion of pins to bony spinous processes) directly to corresponding intervertebral motions [83 84].

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