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Keywords:

  • mandible;
  • computed tomography;
  • image reconstruction;
  • finite-element method

Abstract

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. Acknowledgements
  7. LITERATURE CITED

Finite-element modeling provides a full-field method for describing the stress environment of the skull. The utility of finite-element models, however, remains uncertain given our ignorance of whether such models validly portray states of stress and strain. For example, the effects of boundary conditions that are chosen to represent the mechanical environment in vivo are largely unknown. We conducted an in vitro strain gauge experiment on a fresh, fully dentate adult mandible of Macaca fascicularis to model a simplified loading regime by finite-element analysis for purposes of model validation. Under various conditions of material and structural complexity, we constructed dentate and edentulous models to measure the effects of changing boundary conditions (force orientation and nodal constraints) on strain values predicted at the gauge location. Our results offer a prospective assessment of the difficulties encountered when attempting to validate finite-element models from in vivo strain data. Small errors in the direction of load application produce significant changes in predicted strains. An isotropic model, although convenient, shows poor agreement with experimental strains, while a heterogeneous orthotropic model predicts strains that are more congruent with these data. Most significantly, we find that an edentulous model performs better than a dentate one in recreating the experimental strains. While this result is undoubtedly tied to our failure to model the periodontal ligament, we interpret the finding to mean that in the absence of occlusal loads, teeth within alveoli do not contribute significantly to the structural stiffness of the mandible. © 2005 Wiley-Liss, Inc.

Finite-element analysis (FEA) is the method of choice for theoretical analysis of the mechanical behavior of complex shapes in biology. The approach of FEA approximates real geometry using a large number of smaller simple geometric elements (e.g., triangles, bricks, tetrahedrons). Since complex shapes defy simple mathematical solution (i.e., in terms of engineering formulas), FEA simplifies a problem by analyzing multiple simple elements of known shapes with established mathematical solutions. These multiple solutions are in the end combined together to depict states of stress and strain through the entire structure.

The last half of the 20th century saw increasing interest in testing and analysis of bone biomechanics (Cowin, 2001), including increased use of finite-element (FE) methods. Many theoretical and experimental studies have sought to quantify the distribution of stresses and strain in the mandible (Knoell, 1977; Hylander, 1984; Bouvier and Hylander 1996; Daegling and Hylander, 1997, 1998; Dechow and Hylander, 2000), and several finite-element analyses of human mandibles have been developed. One of the first mandibular finite-element models was developed over 30 years ago (Gupta et al., 1973). This half-mandible model was symmetrical about the symphysis. The authors attempted to study the stress distribution and the deformation that occur during biting. A model designed 4 years later offered improved resolution through the inclusion of the full mandibular dentition (Knoell, 1977). The use of computed tomography (CT) scans to develop a three-dimensional geometric model was pioneered by Hart and Thongpreda (1988). Hart et al. (1992) offered further refinements of better geometry, mesh density, and realistic material properties assignment. The work of Korioth et al. (1992) represents the first attempt to include the periodontal ligament and dental structures in combination with cortical and cancellous bone in a whole mandible model. More recently, model designs have incorporated a voxel-based approach using CT data (Vollmer et al., 2000).

All these studies attempted to portray the stress-strain behavior of the mandibular bone, but the degree to which they succeeded in doing so remains unknown. The lack of information about mandibular material properties, the uncertainty of correct load distribution or assigning the proper boundary conditions all compromised these finite-element studies to some extent (Korioth and Versluis, 1997). Recent studies on the primate skull have emphasized the limited utility of unvalidated models (Ross and Patel, 2003; Strait et al., 2003).

In this article, we explore some of the important issues involved in model validation. We proceed from the assumption that issues of geometric reconstruction are largely solved and focus instead on the impact of decisions made concerning assignment of material properties and specification of boundary conditions. We constructed a finite-element model of a Macaca fascicularis mandible after recording in vitro strain data from this specimen under controlled loading conditions. In effect, our validation of the finite-element model is an attempt to reconstruct the conditions of this in vitro experiment.

The design of a finite-element model proceeds by first obtaining a geometric model and then converting that precise geometry into a finite-element model. The geometric model can be obtained through direct or indirect methods, i.e., by reconstruction of a 3D model from a stack of CT scan images or from a cloud of coordinate points or by using the dimensions of the bone to build an approximate model with a computer-aided design system (Gupta et al., 1973; Knoell, 1977; Meijer et al., 1993). The model obtained in this latter (indirect) way is in fact an idealized model, an approximation of the real object. Reconstruction from CT scans usually generates an improved virtual model because simplifying assumptions of geometry are avoided (Fütterling et al., 1998; Hart and Thongpreda, 1988; Keyak et al., 1990; Hart et al., 1992; Korioth et al., 1992; Hollister et al., 1994; van Rietbergen et al., 1995; Lengsfeld et al., 1998; Vollmer et al., 2000). Obtaining geometry by CT is the preferred method since it offers more accuracy than reconstructions based on planar radiographs. The advantage of CT scanning is that it gathers multiple images of the object from different angles and then combines them together to obtain a series of cross-sections.

Comparisons between the experimental and finite-element analyses presented here represent our continuing attempt to validate the model originally described by Daegling et al. (2003). In the current study, we aim to improve existing modeling efforts by introducing new parameters: designing a less stiff edentulous model, imposing more realistic boundary conditions, and incorporating heterogeneity and transverse isotropy into the models.

MATERIALS AND METHODS

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. Acknowledgements
  7. LITERATURE CITED

Experimental Strain Analysis

The experimental strain analysis was performed on a fresh mandible of a 6-year-old adult female macaque. Prior to excision of the mandible, the skull was wrapped in gentamicin-doped saline-soaked gauze and frozen at −20°C. The mandible was thawed and cleaned using conventional techniques (scalpel, scissors) and stored in soaked gauze when not being tested or scanned.

A rectangular rosette strain gauge was bonded to the lateral aspect of the mandibular corpus below the left second molar and surface bone strain data were obtained from mechanical testing of the mandible. The tests were performed using MTS 858 MiniBionix Test System (Eden Prairie, MN) in displacement mode with a 407 Controller. The metal fixture that restrained the mandible during the procedure was composed from a base that supported the mandible at each angle along its base and a roller that prevented movement at the condyles (Fig. 1). The mandible was subjected to a vertical occlusal load (70 N) applied to the left central incisor. [The experiment was concluded when visual inspection of the raw strains of the individual elements (expressed as voltages) suggested that the shear strain was at or close to 1,000 microstrain (μϵ). The load at this point was 70 N, and the actual calculated shear strain was 976 μϵ.] From the raw strain data from the individual gauge elements, principal strain magnitude and direction were calculated (Fig. 2).

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Figure 1. Experimental strain analysis. The specimen was a fresh mandible of Macaca fascicularis. A rectangular rosette strain gauge was bonded to the lateral aspect of the mandibular corpus below the left second molar. The metal fixture that restrained the mandible during the procedure was composed from an aluminum base that supported the mandible at each angle and a steel roller that prevented movement at the condyles. The mandible was subjected to a vertical occlusal load (70 N) applied to the left central incisor.

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Figure 2. Rectangular rosette strain gauge. From the raw strain data from the individual gauge elements, principal strain magnitude and direction were calculated using the formulas presented.

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Finite-Element Analysis

The geometric model of the mandible was obtained through volumetric reconstruction from CT scans. The mandible was scanned in a series of parasagittal planes. The thickness of each slice was 0.63 mm, giving a total of 90 cross-sections through the mandible.

Each cross-section was converted from a Digital Imaging and Communications in Medicine (DICOM) file to a Bitmap (BMP) file. Each scan was segmented to obtain digitized contours (Fig. 3). The volume (three-dimensional geometry) of the mandible was reconstructed from 90 digitized contours using commercial software (SURFdriver, Kailua, HI; Fig. 4). Because of the limits of spatial resolution in conventional CT, the soft tissue interface between the teeth and alveolar bone (the periodontal ligament) was not adequately visualized and consequently was not modeled. The dentate model was developed from a stack of 90 digitized outer contours and therefore lacks a high degree of anatomical detail.

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Figure 3. Digitized CT cross-sections. The geometric model of the mandible was obtained through volumetric reconstruction from CT scans. The mandible was scanned and 90 cross-sections were obtained and 2D digitized contours collected from each (left and right panels).

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Figure 4. Geometric mandible model: wireframe (without the outer shell, left top panel) and surface models (with the outer shell, left bottom panel). The volume (three-dimensional geometry) of the mandible was reconstructed from a stack of 90 digitized outer contours.

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The role of teeth as load-bearing structures in the absence of occlusal loads is ambiguous (Daegling et al., 1992; Daegling and Hylander 1994a, 1994b). Since most finite-element models ignore the periodontal ligament (Gupta et al., 1973; Knoell, 1977; Hart and Thongpreda, 1988; Hart et al., 1992; Strait et al., 2003), this question is not easily addressed with reference to single models. We thus investigated the possibility that in the absence of occlusal loads, the teeth contribute insignificantly to structural stiffness and strength of the mandible. For this reason, we developed a second mandible model without teeth. The edentulous model was reconstructed from the same CT cross-sections used for reconstructing the dentate model. Teeth were “extracted” (not digitized) and gaps were created where the tooth roots were visualized. While the comparison of the dentate and edentulous models does not resolve the functional role of the periodontal ligament in distributing occlusal versus remotely situated loads, it is used here to ascertain the structural contribution of the teeth in the typical mandibular finite-element model, i.e., where the periodontal ligament is ignored.

Both models were imported into the MSC Patran finite-element analysis package (MSC Software, Santa Ana, CA; Fig. 5). The models were transformed into solids and then meshed with a tetrahedral mesh using quadratic elements. The dentate model had 13,616 quadratic tetrahedral elements. A convergence test was conducted to verify the mesh and minimize the number of finite elements used, consequently reducing calculation times and memory requirements. The edentulous model had 9,735 quadratic tetrahedral elements.

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Figure 5. Dentate and edentulous models. Both were reconstructed from the same 90 CT cross-sections, but tooth crown and root structures were not digitized for the edentulous model. Both models were imported into the MSC Patran finite-element analysis package and were transformed into solids and then meshed with a tetrahedral mesh using quadratic elements. The dentate model had 13,616 quadratic tetrahedral elements (right top panel). The edentulous model had 9,735 quadratic tetrahedral elements (right bottom panel).

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The FEA was designed to replicate as close as possible the conditions from the experimental setup. Thus, the models were subjected to a vertical occlusal load of 70 N, which was applied to the left central incisor (this tooth was preserved in the edentulous models) and the models were constrained bilaterally at condyles and angles. Strain analyses were performed in order to determine the principal strains at the strain gauge site and compare these values with the experimental data (Fig. 6).

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Figure 6. Prediction of surface strains from the finite-element models. Surface strains were averaged over five elements corresponding to strain gauge location. The models were subjected to a vertical occlusal load of 70 N applied to the left central incisor and were constrained bilaterally at condyles and angles (top panel). Strain analyses were performed in order to determine the principal strains corresponding to the strain gauge site and compare these values with the experimental data (bottom panel).

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In an attempt to improve previous modeling efforts (Daegling et al., 2003), variations in boundary condition and material properties parameters were explored. In that study, we used a homogeneous isotropic model with condyles and angles completely constrained in three planes. The maximum and minimum principal strains (288 and −225 μϵ) and their ratio (1.28) showed poor agreement with experimental values (Table 1). More realistic boundary conditions were investigated by decreasing the number of nodes constrained, altering the degrees of freedom and changing the force orientation. Initially, we fully constrained the model such that the “virtual” mandible was not deflecting in the angular region, which served (in retrospect) to overconstrain the model. Simulations were performed by reducing the nodal constraints bilaterally below the condyles and at gonion from 25 to 12, 6, 3, and 1 at each location in successive iterations. To account for the deflection of the specimen during the experiment, simulations were performed by reducing the number of degrees of freedom at condyles and angles in the transverse direction. Three analyses were performed (isotropic case) for each model: altering the nodal constraints at the right condyle and right base, at the right condyle and left base, and changing the nodal constraints simultaneously in the right condyle, left condyle, and right base.

Table 1. Experimental and theoretical principal strain data*
 ϵ1 (μϵ)ϵ2 (μϵ)ϵ12
  • *

    Principal strains and the principal strain ratios are calculated from the lateral corpus. The reported values are for a homogeneous isotropic model, fully constrained bilaterally at condyles and angles (four nodes). The models were subjected to a vertical occlusal load of 70 N, applied to the left central incisor.

  • a

    Daegling et al. (2003).

  • b

    In the current study, predicted strains were determined by averaging strains from five elements while in the 2003 study the strains were taken from a single element. The coordinate systems used in the 2003 study (a global system) and the present study (a local system) also differed slightly.

Experiment755−2213.41
FE dentate modela288−2251.28
FE dentate modelb231−2470.93
FE edentulous modelb528−1743.03

We also investigated the influence on model principal strain values of variation in orientation of the external force applied at the incisor. Four analyses were performed using the homogeneous isotropic model in which four different inclination angle values (θ = 0°, 5°, 10°, 20°) were considered. The force vector was tilted in the frontal plane in an attempt to replicate the conditions of the experiment, given the likely deflection of the specimen.

We also investigated the effects on model behavior of spatial variation and directional dependence of elastic properties. Based on material properties assignment, three models were developed: an isotropic homogeneous model, an isotropic heterogeneous model (with material properties assigned based on CT density), and a transverse isotropic heterogeneous model (using regional assignment of material properties, with each region of the mandible having a different orientation of the local material axes). Material properties assignment is reported in Table 2.

Table 2. Material properties assignment*
MaterialYoung's modulus (GPa)Poisson's ratioShear modulus (GPa)Number of elements
DentateEdentulous
  • *

    The isotropic heterogeneous model had three sets of material properties assigned based on CT density using BoneMat software. The transverse isotropic heterogeneous model considered the mandible to have regional dependence of material properties, with three regions of the mandible having a different orientation of the local material axes.

Isotropic homogeneous150.3 136169735
Isotropic heterogeneousEA = 15.4νA = 0.3 A: 12353A: 8428
 EB = 9.08νB = 0.3 B: 65B: 62
 EC = 3.7νC = 0.3 C: 1198C: 1245
Transverse isotropic heterogeneousE1,2cortical = 13ν12 = 0.3G12 = 5123538428
 E3cortical = 17ν23 = 0.229G23 = 6.91  
 Etrabecular = 1.5ν = 0.3 12631307

The simplest assignment was an isotropic homogeneous model, that is, all model elements had the same material properties assigned, independent of direction [Young's modulus (E) = 15 GPa; Poisson's ratio (ν) = 0.3]. Elastic properties assigned to the isotropic homogeneous model were estimated from micromechanical tests performed on the specimen in another study (Rapoff et al., 2003) completed after the strain experiment and the CT scanning.

The second model, the isotropic heterogeneous model, was developed using Bonemat software (Zannoni et al., 1998; Taddei et al., 2004). The Bonemat program assigns material properties based on CT density. The material properties are automatically calculated assuming a linear relationship between CT numbers and apparent bone density and a power relationship between apparent density and Young's modulus. Three different isotropic materials were assigned to three groups of elements (A, B, C). The three groups of elements were determined automatically, based on the material property derived from the CT density of the tissue, as stored in the CT scan data. The model made from the three groups of elements was consequently isotropic but heterogeneous (EA = 15.4 GPa, EB = 9.08 GPa, EC = 3.7 GPa, and νA = νB = νC = 0.3). Material properties were assigned using MSC Patran according to a local coordinate system: the three-axis was defined as following the length of the mandible, with the one-axis mediolaterally oriented and two-axis superoinferiorly oriented.

The third model, the heterogeneous transversely isotropic model, was developed by assigning two sets of material properties for cortical and trabecular bone. A few studies have succeeded in assigning directionally dependent material properties, with each region of the mandible having a different orientation of the local material axes. Hart and Thongpreda (1988) and Hart et al. (1992) defined three main mandibular regions: the left mandible, symphysis, and right mandible. Similar mandibular regions were defined by Korioth et al. (1992): the symphysis, gonion region, and the rest of the mandible. We similarly defined three regions: right (posterior corpus and ramus), anterior corpus, and left (posterior corpus and ramus). One local coordinate system was defined for each region to be generally aligned with the natural curvilinear axis around the mandible (Fig. 7). The groups of elements defined previously (A, B, and C) were each further divided in three subgroups: right, symphyseal, and left regions. MSC Patran was used to assign material properties for cortical and trabecular bone to each region, according to their local coordinate system. Using the same groups of elements determined in the previous case, two materials were defined: group A, representing the cortical bone, and groups B and C, representing the trabecular bone. This model combined material information data obtained through micromechanical testing (Rapoff et al., 2003). For cortical bone, the material properties were assigned specific to the frontal plane and in the longitudinal direction depending on the region under consideration (E1,2cortical = 13 GPa, G12 = 5 GPa, E3cortical = 17 GPa, G23 = 6.91 GPa, ν12 = 0.3, and ν23 = 0.229). The trabecular region was modeled as isotropic (Etrabecular = 1.5 GPa and νtrabecular = 0.3).

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Figure 7. Heterogeneous transverse isotropic model showing specification of local material axes for three regions: right region (posterior corpus and ramus), symphyseal region (anterior corpus), and left region (posterior corpus and ramus). One local coordinate system was designed for each region, which followed the longitudinal axis from one condyle to the other. MSC Patran was used to assign material properties to each region according to their local coordinate system.

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The method used for recording and evaluating the principal strain data generated by the finite-element models consists of averaging principal strain values of a node common to the neighboring elements, following similar methods used in the past (Lengsfeld et al., 1998; Remmler et al., 1998; Coleman et al., 2002; Guo et al., 2002). The strain gauge location on the virtual mandible is subject to a small but undetermined error with respect to the location of the strain gauge on the real specimen. Rather than rely on single element values, we chose to calculate an averaged strain value. The five elements situated at the gauge location were found and their principal strain values for these elements were averaged. The principal strains from the various models were determined from these same elements in all simulations.

RESULTS

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. Acknowledgements
  7. LITERATURE CITED

The experimental principal strains and the principal strain ratio for lateral regions of the mandible are presented in Table 1. The maximum principal strain found was 755 μϵ and the minimum principal strain was −221 μϵ, yielding a ratio of maximum to minimum principal strain of 3.41. The principal strains and the principal strain ratios for this location obtained using the FEA (homogeneous isotropic model, fully constrained bilaterally at condyles and angles) are also reported in Table 1. The principal strain ratio value obtained from the edentulous model is more compatible to the experimental values.

Nodal Constraints

The absolute number of constrained nodes has the predictable effect of influencing model stiffness. An equal number of nodes were constrained on each side, at condyles and angles (Fig. 8). The greater number of nodes constrained results in a stiffer model and consequently produces lower principal strains (Fig. 9, Table 3). Reduction of nodal constraints from 25 nodes to 1 node at each location has predictably large effects, resulting in a more than 50% increase in principal strain magnitudes. The dentate model yields a principal strain ratio near 1.0 throughout these iterations while the edentulous model fits the experimental data better in all cases.

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Figure 8. Variation in the number of constrained nodes in finite-element models. Shown are the two extremes simulated: 1 node (left panel) vs. 25 nodes (right panel) (only condylar constraints are shown here). During different FEA iterations, an equal number of nodes were constrained on each side, at condyles and angles. Simulations were performed by reducing the nodal constraints bilaterally below the condyles and at gonion from 25 to 12, 6, 3, and 1 at each location to determine the influence of the number of constrained nodes on principal strain results.

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Figure 9. Effect of nodal constraint on predicted maximum principal strain values. The absolute number of constrained nodes has the predictable effect of influencing model stiffness. The greater number of nodes constrained results in a stiffer model and consequently produces lower principal strains.

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Table 3. Effect of nodal constraint on principal strain values*
Number of constraintsDentate modelEdentulous model
ϵ1 (μϵ)ϵ2 (μϵ)ϵ12ϵ1 (μϵ)ϵ2 (μϵ)ϵ12
  • *

    All simulations were performed using the homogeneous isotropic model, constrained bilaterally at condyles and angles. The models were subjected to a vertical occlusal load of 70 N, applied to the left central incisor. These simulations did not allow for relaxation of constraint in any direction. Experimental strain data: ϵ1 = 755 μϵ, ϵ2 = −221 μϵ, ϵ12 = 3.41.

25171−1720.99340−1073.17
12189−1801.05343−1083.17
6193−2110.91377−1272.96
3205−2390.85429−1632.63
1231−2470.93528−1743.03

Force Direction

Changing the orientation of the applied force has a large impact on model principal strains (Fig. 10). As the inclination of the applied force deviates from a purely sagittal orientation to having a progressively greater lateral component, the principal strain ratio values increase due to the more rapid increase in the maximum principal strain (Table 4). Both dentate and edentulous models respond similarly to these perturbations. Altering occlusal point load directions by as little as 10° alters maximum principal strains by 11–18%.

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Figure 10. Alteration of direction of the applied force. Simulations were performed in which four different inclination angle values (θ = 0°, 5°, 10°, 20°) were considered. The force vector was tilted in the frontal plane in an attempt to replicate the conditions of the experiment.

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Table 4. Influence of force orientation on principal strain values*
Inclination angleDentate modelEdentulous model
ϵ1 (μϵ)ϵ2 (μϵ)ϵ12ϵ1 (μϵ)ϵ2 (μϵ)ϵ12
  • *

    Each iteration employs an occlusal load of 70 N applied to the left central incisor. These analyses were performed using the homogeneous isotropic model, fully constrained bilaterally at condyles and angles. Inclination of the occlusal force vector lies within the frontal plane. Experimental strain data: ϵ1 = 755 μϵ, ϵ2 = −221 μϵ, ϵ12 = 3.41.

231−2470.93528−1743.03
253−2570.98561−1813.09
10°274−2671.02591−1893.12
20°311−2801.11638−2003.19

Degrees of Freedom

Further FE analyses were performed by relaxing successively the degrees of freedom at condyles and angles in an attempt to simulate the deflection of the specimen during the experiment (Fig. 11). Relaxation of the degrees of freedom in the transverse direction causes a significant increase in principal strain values (Table 5). In particular, allowance for translation in the transverse direction on the right side of the edentulous model (the right condyle and the right base) yields strain results generally congruent with the experimental data.

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Figure 11. Relaxation of boundary conditions. To account for the deflection of the specimen during the experiment, simulations were performed by relaxing successively the degrees of freedom at condyles and angles in the transverse direction. In this image, the anterior aspect of the condylar necks and the right angle are totally constrained, while the left angle is free to deflect in the transverse direction.

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Table 5. Influence of the degrees of freedom on principal strain values*
Degree of relaxationDentate modelEdentulous model
ϵ1 (μϵ)ϵ2 (μϵ)ϵ12ϵ1 (μϵ)ϵ2 (μϵ)ϵ12
  • *

    All analyses use the homogeneous isotropic model (70 N applied to the left central incisor). Relaxation of the degrees of freedom is restricted to the transverse direction in all cases. Experimental strain data: ϵ1 = 755 μϵ, ϵ2 = −221 μϵ, ϵ12 = 3.41.

Fixed231−2470.93528−1743.03
R condyle, R base439−2431.80768−2433.16
R condyle, L condyle520−4911.051311−7461.75
R condyle, R base, L condyle648−4211.531396−6072.29

Material Properties Assignment

Principal strain values are influenced by different assignment of spatial variation (homogeneity vs. heterogeneity) and directional dependence (isotropy vs. orthotropy) of elastic properties in both dentate and edentulous models (Table 6). The decision to employ isotropic and homogeneous models results in different errors depending on the analysis of the dentate versus edentulous case. Introducing structural heterogeneity and directional dependence of material behavior by regional assignment of material properties acts to decrease strains in the dentate model, while this same alteration results in an increase in principal strains in the edentulous model (Table 6). In the case of this particular mandible, the choice of material properties assignment and structural simplicity introduce large differences in predicted strains on the order of 10%. In the edentulous model, introduction of heterogeneity and directional dependence actually increases departure of the predicted strain ratio values from the experimentally observed value. This is also true of the dentate model, although this model is always in marked disagreement with experimental values regardless of the nature of perturbations.

Table 6. Influence of material properties assignment on principal strain values*
MaterialDentate modelEdentulous model
ϵ1 (μϵ)ϵ2 (μϵ)ϵ12ϵ1 (μϵ)ϵ2 (μϵ)ϵ12
  • *

    The models were subjected to an occlusal load of 70 N, applied to the left central incisor, and were fully constrained bilaterally at condyles and angles. Experimental strain data: ϵ1 = 755 μϵ, ϵ2 = −221 μϵ, ϵ12 = 3.41.

  • a

    Without directional dependence of material behavior throughout mandible.

  • b

    With directional dependence of material behavior throughout mandible.

Isotropic homogeneousa231−2470.93528−1743.03
Isotropic heterogeneousa193−2230.86569−1902.99
Transverse isotropic heterogeneousb136−1690.80576−2152.67

DISCUSSION

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. Acknowledgements
  7. LITERATURE CITED

The application of finite-element analysis to skeletal mechanics has changed since this method was first introduced. Initially, this method was used to investigate questions of structural failure (Huiskes and Hollister, 1993). Currently, this method is successfully used in the context of understanding the biomechanical behavior of the bone under physiological loading conditions. Despite advances in techniques and hypothesis specification, the accuracy and reliability of using finite-element analysis to address functional morphological questions are incompletely established.

It is widely appreciated that the geometry of the finite-element model plays a crucial role in obtaining accurate results. In the present case, the geometry of the dental alveoli rather than the teeth themselves is more informative of actual behavior. Our edentulous model's better match with the experimental data suggests that in the absence of occlusal loads, teeth do not function as load-bearing structures. Our dentate model is always too stiff, that is, the strain values obtained from it are extremely low when compared with the experimental strain values. The periodontal ligament was not modeled and consequently no interface existed between the teeth and alveoli. This increases the structural stiffness of the model and at least partially explains the low strain values obtained. The strain results obtained from analyzing the edentulous model are more congruent with the experimental strain data not only because the edentulous model is less stiff, but also because the removal of teeth lowers the neutral axis in bending (which readily explains the better fit of the principal strain ratio values to those observed in the experiment). An increase in strain values can be explained by tooth removal given reduction in the overall amount of material resisting deformation.

Correct boundary conditions are also critical for finite-element model validation. Constraining the finite-element model excessively, as we did initially, produces inaccurately low principal strain values. In contrast, applying minimal constraints produces strain results more compatible with the experimental strain data, even if single-node constraints are not biologically reasonable. If the purpose of the experimental tests was to provide benchmarks for simulation validation, then the loading and constraints most reproducible in the simulations should be used in the tests even if the tests do not represent physiologic conditions. Whether our model constraints are at all appropriate for in vivo caseloads is arguable and also irrelevant; the point of the above analysis is to establish a baseline for understanding the sensitivity of the whole mandible model to changes in boundary conditions. The main difficulty in correlating strain data with finite-element model predictions arises from the difficulty in recreating identical parameters in experimental and virtual contexts. A multitude of variables act simultaneously during a strain gauge experiment (e.g., orientation and magnitude of the applied load, nature of constraints, material property variation, and geometric subtleties of the real specimen), whereas in a finite-element analysis these variables are necessarily subject to a number of simplifying assumptions. This discrepancy might lead to different results and interpretations of the stress-strain behavior of the mandibular bone (Hylander et al., 1998; Dechow and Hylander, 2000). Our objective was to portray the difficulties encountered when attempting to validate finite-element models from an in vitro context where some parameters are ostensibly under investigator control. For the in vivo context, the specification of appropriate boundary conditions and material properties assignment will be more difficult.

Relaxing the finite-element model by decreasing the number of nodes constrained or by altering the degrees of freedom in the constrained nodes has, as expected, a large impact on the strain results. During the strain experiment, the mandible was restrained by a symmetrical steel fixture; however, mandibles are not perfectly symmetrical structures. We assumed initially that the mandible was totally constrained in three dimensions below the condyles and at the angles during the experiment. Deflection of the specimen during the experiment, however, suggests movement occurred in the transverse direction at the constraint locations.

Modest errors in specifying the direction of the applied force in the FE simulation can have substantial impact on principal strain values, and the critical point here is that this is true even at locations remote to the point of load application. As the mandible deflected during the experiment, the loading environment became modified such that a horizontal force component was introduced. Because of the oblique orientation of the mandibular angles on the supporting rod during the experiment, a horizontal reaction force was induced at those points. Because of the relationships between force and displacement, and displacement and strain, modifying the direction of the applied load produces a novel load case—in the present context, in terms of the relative contributions of bending and torsion to recorded and predicted strains. Certainly, one factor preventing us from achieving better validation is that we do not know the precise magnitude of this horizontal force component introduced when the specimen deflected.

Spatial variation (homogeneity vs. heterogeneity) and directional dependence (isotropy vs. orthotropy) of elastic properties greatly influence principal strain values, sufficiently so that the convenience of using isotropic models entails a significant cost in model accuracy. The first analysis considered an isotropic homogeneous model obtained by assigning the same material property to all elements, independent of direction. The second analysis was performed on an isotropic heterogeneous mandible model in which three different isotropic properties were assigned to three different groups of elements. The third analysis employed a transverse isotropic heterogeneous model with spatial variation and directional dependence of elastic properties. As expected, introduction of heterogeneity and transverse isotropy in the less stiff edentulous model increases principal strain values because approximately 14% of the finite elements are assigned a relatively low modulus corresponding to the trabecular region. As in all the previous simulations, the edentulous model performs better than the dentate one. Incorporating heterogeneity and transverse isotropy into the finite-element models improves the congruence of experimental and theoretical values. This will not be obvious in cases where the effects of load direction and boundary conditions have not been satisfactorily controlled.

In the absence of precise information about the material properties of the mandible or because of the need for simplicity, finite-element mandible models are idealized as isotropic, linearly elastic solids (Gupta et al., 1973; Knoell, 1977; Meijer et al., 1993). The material properties assigned to the models are therefore directionally independent, consequently eliminating one of the most cumbersome steps in the process of obtaining a realistic finite-element model, that of assigning directionally dependent material properties.

Multiple studies performed to determine the elastic properties of the mandible show how complex the mandibular bone is. The elastic properties vary directionally between different regions in the mandible (Dechow and Hylander, 2000). Developing a model without directional dependence of elastic properties, although very convenient and significantly less time-consuming, will provide only a first approximation of the strain field in a loaded mandible.

The material properties for the second type of finite-element analysis, the isotropic heterogeneous mandible model, were assigned according to a single local coordinate system (the three-axis in the longitudinal direction and the other two axes oriented in the frontal plane) without variation throughout the mandible. The chosen coordinate system facilitates recording the principal strains from the region of interest where the strain gauge was attached—the lateral aspect of the left basal corpus, below the second molar. Although the second model is heterogeneous, it is an isotropic model and therefore unrealistic.

The material properties for the third type of finite-element analysis, the heterogeneous transverse isotropic model, were assigned according to three local coordinate systems corresponding to three regions in the mandible: right region, symphysis, and left region. Introducing structural heterogeneity and directional dependence of material behavior produces an increase in principal strains in the edentulous model as it can be seen in Table 6.

We have presented a simple case of using strain data from a single point (i.e., one strain gauge) to validate a model. Utilizing multiple strain gauge sites would allow for mapping strain gradients, which would offer additional insight into the influences of the modeling parameters. This would not be an option in the in vivo context due to experimental limitations or size and geometry of the specimen (Hylander et al., 1998; Dechow and Hylander, 2000).

Determination of appropriate boundary conditions is as critical as recreation of precise geometry for finite-element model validation. We hypothesize that improved depiction of dental and periodontal structures in the current model will lead to improved congruence between the experimental data and the finite-element recreation of those data. Modeling physiological loads and constraints is likely to be the greatest future challenge in physical anthropology for successful application of finite-element methods for modeling in vivo mechanical behavior.

Acknowledgements

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. Acknowledgements
  7. LITERATURE CITED

The authors thank the Southwest Foundation for Biomedical Research for providing the specimen used in this study.

LITERATURE CITED

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. Acknowledgements
  7. LITERATURE CITED
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