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In vivo data demonstrates that strain is not distributed uniformly on the surface of the primate skull during feeding. However, in vivo studies are unable to identify or track changes in stress and strain throughout the whole structure. Finite element (FE) analysis, a powerful engineering tool long used to predict the performance of man-made devices, has the capacity to track stress/strain in three dimensions (3-D) and, despite the time-consuming nature of model generation, FE has become an increasingly popular analytical device among biomechanists. Here, we apply the finite element method using sophisticated computer models to examine whether 3-D stress and strain distributions are nonuniform throughout the primate skull, as has been strongly suggested by 2-D in vivo strain analyses. Our simulations document steep internal stress/strain gradients, using models comprising up to three million tetrahedral finite elements and 3-D reconstructions of jaw adducting musculature with both cranium and mandible in correct anatomical position. Results are in broad concurrence with the suggestion that few regions of the hominid cranium are clearly optimized for routine feeding and also show that external stress/strain does not necessarily reflect internal distributions. Findings further suggest that the complex heterogeneity of bone in the skull may act to dissipate stress, but that consequently higher strain must be offset by additional strain energy. We hypothesize that, despite energetic costs, this system may lend adaptive advantage through enhancing the organism's ability to modify its behavior before reaching catastrophic failure in bony or dental structures. Anat Rec, 290:1248–1255, 2007. © 2007 Wiley-Liss, Inc.
In vivo studies of the primate skull have demonstrated strong strain gradients in the surface of the primate cranium during feeding, leading to the conclusion that the skull may not be strongly optimized for mastication (Hylander et al., 1991; Hylander, 1997). However, in vivo analyses cannot directly address the question of whether, or to what degree, similarly nonuniform distributions of stress/strain may occur within the structure, and the number of gauges that can be applied within a given area can be limiting, even with respect to the documentation of external strain (Dumont et al., 2005; Rayfield, 2007). As an alternative, beam theory, although applicable to materially homogeneous structures, such as the midshaft cortex of long bones, cannot accurately predict mechanical behavior in more complex heterogeneous structures that incorporate both cortical and cancellous bone (Thomason, 1995), as is found in the crania and mandibles of primates and many other vertebrates.
Finite element (FE) analysis is routinely used to predict the mechanical behavior of man-made structures. In the life sciences, the ability of FE to facilitate nondestructive analyses of mechanical behavior under controlled and easily replicated conditions lends it promise as a valuable source of data to researchers in fields ranging from the prediction of feeding ecology in living and fossil species, to the optimization of prosthetic devises. FE allows the investigator to map stress/strain distributions throughout three-dimensional structures (Thomason, 1995). However, despite very notable advances (Rayfield et al., 2001; Rayfield, 2004, 2007; Dumont et al., 2005; Strait et al., 2005; Tizzard et al., 2005; McHenry et al., 2006), the considerable potential of FE analysis in biology has been constrained by the time-consuming nature of model generation (Rayfield, 2007). Achieving sufficient resolution to incorporate the variable material properties of bone (Dumont et al., 2005; Rayfield, 2007) has also been problematic, yet investigations involving FE modeling of the cranium of Macaca fascicularis have shown that allowance for differences in bone properties might strongly impact on the accuracy of results (Strait et al., 2005). However, in this instance properties were assigned manually, and sudden shifts between regional boundaries may not have been realistic (Strait et al., 2005). Three-dimensional reconstruction of muscle remains another challenge, with crania and mandibles typically treated separately and forces reduced to single vectors, producing high point loads that can confound interpretation of results (Dumont et al., 2005).
Our aim in the present study has been to investigate internal stress/strain gradients in the primate skull and compare differences between simulations that assume a single uniform material property (homogeneous), and models that incorporate multiple material properties for cortical and cancellous bone (heterogeneous). An additional objective has been to develop protocols that improve model realism while reducing assembly time. Models presented here comprise two to three million 3-D “brick” finite elements, produced using a relatively rapid and largely automated method that allows the assignment of variable material properties for cortical and cancellous bone, as well as tooth enamel. Further advances over previous models include treatment of the skull and mandible as a single articulated mechanism, and more accurate simulation of the 3-D architecture of jaw adducting musculature, spreading loads across muscle origin/insertion points and minimizing the confounding influence of point loads. These simulations are based on computer assisted tomography (CT) from a common chimpanzee, Pan troglodytes, (Fig. 1A,B), long recognized as our closest living relative together with Pan paniscus, the bonobo (Begun, 1992).
Figure 1. The 3,877,678 heterogeneous brick element model of Pan troglodytes. A,B: Comparison of a computer assisted tomography slice (A) with a slice through same region of finite element model (B). C: Model before addition of musculature. D: Areas for muscle origins and insertions for major jaw adductors and placement of trusses used to simulate muscle actions.
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MATERIALS AND METHODS
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- MATERIALS AND METHODS
- LITERATURE CITED
Both homogeneous (single material property) and heterogeneous models (seven material properties) were constructed using CT data for a common chimpanzee cranium and mandible comprising 478 transaxial slices separated by 0.5-mm intervals from the University of Austin Digital Morphology Web site (digimorph.org). In the original surface mesh, maximum and minimum triangle edge lengths were kept at a 1:3 ratio (0.1 geometric error) to minimize differences between triangle dimensions, which can lead to major discrepancies in brick element size in the final solid and thereby introduce artifacts. Solid meshing was performed with Strand7 Finite Element software (Vers. 2.3). Models were assembled using 3-D low-order four-noded tetrahedral “brick” elements (tet4). Tet4 based models can produce less accurate results than those built from higher order elements; however, differences diminish with increasing brick element number. Differences of around 10% have been recorded in comparisons between tet4 and higher order ten-noded (tet10) models of <252,000 brick elements (Dumont et al., 2005). It is likely that our use of tet4 elements is to some degree mitigated by the use of high brick element numbers and associated increases in geometric accuracy.
For the simulation of bilateral loads (i.e., boundary conditions and, hence, the distribution of stress and strain identical on both sides of the skull), we used half skull and mandible models that comprised 1,938,839 brick elements. All nodes at the midline were fixed relative to the transverse axis. In unilateral biting, boundary conditions differ between the two sides of the skull and distributions of stress and strain are asymmetrical. Consequently, we used a full skull model to simulate unilateral biting through mirroring the heterogeneous half skull and lower jaw. Total 3-D brick number in this instance (3,877,678) exceeded the current computational limits of the software. The mandible was removed giving a brick element total of 3,023,365 with the trusses simulating jaw musculature (see below) retained and fixed in the position of insertion, thereby allowing bite simulation for the cranium. We modeled the temporomandibular joint using a hinged beam linked to both upper and lower jaws.
Simulations of bilateral bites at the upper and lower second molars and second incisors, and unilateral bites at the second molars were performed (Figs. 2, 3). For bilateral bites, two half models inclusive of both upper and lower jaws were used and a half model was mirrored to simulate a unilateral bite. The first of the half models was homogeneous, wherein it was assumed that the entire skull comprised cortical bone only. Remaining simulations were heterogeneous, with seven material properties applied (Fig. 1A). Mean brick element stress and strain were compared in five regions of interest; the brow-ridge, postorbital bar, zygomatic arch, cranium, and mandible (Table 1).
Figure 2. A–H: Stress (Von Mises) distributions in frontal views top row (A–D) and cross-sectional views bottom row (E–H) in four simulations of maximal static bites in Pan troglodytes: bilateral bite at the second molars in homogeneous model (A,E), bilateral bite at the second molars in heterogeneous model (B,F), bilateral bite at the second incisors in heterogeneous model (C,G), and unilateral bite at the second molars in heterogeneous model (D,H). MPa, mega pascals.
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Figure 3. A–H: Strain (Von Mises) distributions in frontal views top row (A–D) and cross-sectional views bottom row (E–H) in four simulations of maximal static bites in Pan troglodytes: bilateral bite at the second molars in homogeneous model (A,E), bilateral bite at the second molars in heterogeneous model (B,F), bilateral bite at the second incisors in heterogeneous model (C,G), and unilateral bite at the second molars in heterogeneous model (D,H).
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Table 1. Mean von Mises brick element stresses and strains for regions of interest in four static bite simulations in the skull of Pan troglodytesa
| ||Ho Bilat M2||He Bilat M2||He Bilat I2||He Unilat M2|
|Cranial brick stress (VM)||3.969E−01||3.738E−01||7.070E−01||5.062E−01|
|Mandibular brick stress (VM)||1.489E+00||1.155E+00||1.545E+00||N/A|
|Brow-ridge brick stress (VM)||2.614E−01||2.169E−01||6.679E−01||2.238E−01|
|Brow-ridge brick stress R (VM)||N/A||N/A||N/A||1.901E−01|
|Brow-ridge brick stress L (VM)||N/A||N/A||N/A||2.575E−01|
|Postorbital bar brick stress (VM)||6.336E−01||6.180E−01||8.082E−01||7.681E−01|
|Postorbital bar brick stress R (VM)||N/A||N/A||N/A||6.841E−01|
|Postorbital bar brick stress L (VM)||N/A||N/A||N/A||8.511E−01|
|Zygomatic arch brick stress (VM)||7.999E+00||5.342E+00||5.594E+00||5.890E+00|
|Zygomatic arch brick stress R (VM)||N/A||N/A||N/A||5.885E+00|
|Zygomatic arch brick stress L (VM)||N/A||N/A||N/A||5.894E+00|
|Cranial brick strain (VM)||2.052E−05||9.899E−05||1.319E−04||1.212E−04|
|Mandibular brick strain (VM)||7.699E−05||1.824E−04||2.169E−04||N/A|
|Brow-ridge brick strain (VM)||1.351E−05||2.587E−05||1.218E−04||2.706E−05|
|Brow-ridge brick strain R (VM)||N/A||N/A||N/A||2.324E−05|
|Brow-ridge brick strain L (VM)||N/A||N/A||N/A||3.088E−05|
|Postorbital bar brick strain (VM)||3.273E−05||6.587E−05||8.432E−05||5.291E−05|
|Postorbital bar brick strain R (VM)||N/A||N/A||N/A||5.080E−05|
|Postorbital bar brick strain L (VM)||N/A||N/A||N/A||5.502E−05|
|Zygomatic arch brick strain (VM)||4.135E−04||8.297E−04||8.684E−04||6.111E−04|
|Zygomatic arch brick strain R (VM)||N/A||N/A||N/A||6.108E−04|
|Zygomatic arch brick strain L (VM)||N/A||N/A||N/A||6.114E−04|
For the homogeneous model, all brick elements were assigned a single set of material properties for cortical bone (Young's modulus of Elasticity [Y] = 27.0 GPa; Poisson's ratio = 0.4; density = 2,190 Kg/m3). For heterogeneous models, six additional material properties were assigned on the basis of density values (Rho et al., 1995; Schnider et al., 1996; Fig.. 1A,B). These ranged from property 1 (lowest density), with values intermediate between free spaces and low-density tissue (Y = 1.5 GPa; Poisson's ratio = 0.4; density = 250 Kg/m3), to property 7, simulating enamel (Y = 38.6 GPa; Poisson's ratio = 0.4; density = 2,860 Kg/m3).
Calculations of muscle forces were based on a dry skull method using estimates for cross-sectional area (Thomason, 1991; Wroe et al., 2005; Christiansen and Wroe, 2007) adjusted for application to hominids (O'Conner et al., 2005). In each simulation, the 3-D architecture and actions of the musculature were approximated using pretensioned trusses, beam finite elements that carry axial loads only (Fig. 1C). The number and diameters of the truss elements assigned to each muscle group were determined on the basis of muscle origin and insertion areas (Fig. 1C). Pretension values for each truss were adjusted accordingly. These were 6.5 N, 18.7 N, and 2.9 N, respectively, for each of 29 temporalis, 10 masseter, and 5 medial pterygoid trusses. Although the effect of the lateral pterygoid with respect to the power stroke is negligible, an additional four unloaded trusses were inserted to simulate its potential stabilizing influence.
Regions of particular interest with respect to feeding mechanics in the hominid skull include the zygomatic arches, supraorbital torus (brow-ridge) and orbital bar (Hylander et al., 1991; Hylander, 1997; O'Conner et al., 2005). These regions were selected within the models and analyzed statistically using a program written in RGui by K. Moreno.
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Estimated unilateral muscle forces for the primary jaw adductors (temporalis, masseteric, and medial pterygoid) were 189, 187, and 15 Newtons (N), respectively. Total maximal bilateral muscle force was 782 N.
Comparison of the homogeneous and heterogeneous models biting at the second molars reveals broadly similar distributions of von Mises stress and strain (Figs. 2, 3). In both homogeneous and heterogeneous simulations, mean brick element stresses and strains within the brow-ridge (Table 1) are considerably less than means for the cranium as a whole (67% and 27%, respectively) and tiny relative to those in the zygomatic arch (<4%).
However, although broad surface distributions are similar, the distributions of stress and strain through cross-sections of these two models demonstrate strong internal gradients, and furthermore, that correspondence between internal and external distributions of stress and strain are not necessarily tight. For example, high internal brick element von Mises stresses of up to 5.3 MPa evident in cross-sections of the heterogeneous model (Fig. 2) at the anteromedial aspect of the face are much higher than in overlaying external elements (2.7–3.1 MPa).
Evaluation of the homogeneous and heterogeneous models further shows that, for each region of interest, mean and maximum stresses were higher in the homogeneous model (comprising cortical bone only), while mean and maximum strains were higher in the heterogeneous structure (Figs. 2, 3; Table 1). Distinctions between the two are best exemplified in comparison of displacements, which are much higher in the heterogeneous model (see Fig. 4).
Figure 4. A–L: Von Mises stress (A–D), strain (E–H), and displacement (I–L) in lateral views in four simulations of maximal static bites in Pan troglodytes. A,E,I: The bilateral bite at the second molars in a homogeneous model. B,F,J: The bilateral bite at the second molars in a heterogeneous model. C,G,K: The bilateral bite at the second incisors in a heterogeneous model. D,H,L: Unilateral bite at the second molars in heterogeneous model.
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Simulation of bilateral incisive biting demonstrates differences with respect to the distribution of stresses and strains relative to biting at the second molars. In the heterogeneous models of these behaviors, cross-sections show that internal stresses are relatively high anteromedial to the orbit in an incisive bite (Fig. 2). However, as with bilateral biting at the molars, mean stresses and strains in the brow-ridge remain well below those of the cranium as a whole (Table 1). In unilateral molar biting, we found that stress and strain were slightly higher in the balancing than working side (Figs. 2, 3; Table 1).
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Surface strain patterns correspond qualitatively with external in vivo data taken from other primate species, which have suggested that, when biting, strain is minimal in the brow-ridge and highest in the zygomatic arches and lower jaws (Hylander, 1997; Ravosa et al., 2000; Ross and Metzger, 2004). There is further correspondence in that our models reveal an increasing gradient of stress anteriorly for the zygomatic arch (Hylander, 1997).
Our finding that, in unilateral molar biting, stress and strain were slightly higher in the balancing than working side, is contrary to empirically derived results. This is likely because our model assumes that maximal bite forces are applied on both sides, whereas empirical data demonstrate that the working-side masseter usually exerts more force than does that of the balancing-side during the power stroke (Hylander, 1997). Our interpretation is further supported by in vivo evidence attesting to a reduction in the discrepancy between the two sides as increasingly tough foods are chewed (Hylander, 1997).
Our results demonstrate the potential value of 3-D FE studies that incorporate the variable material properties of bone, strongly suggesting that internal stress/strain distributions are nonuniform and moreover, that the external distributions of surface stress/strain do not necessarily reflect internal distributions. We conclude that externally derived data may not accurately reflect distributions throughout the entire skull, and these findings offer empirical support for the argument of Thomason (1995), that is, that FE analyses are required to fully investigate the behavior of structures that incorporate cancellous bone.
These findings remain in general agreement with the proposal that much of the facial anatomy of primates may not be strongly influenced by its role in routine food processing (Hylander, 1997). Cranially, the most informative regions with respect to feeding mechanics are the zygomatic arches. Overall, our finding that the mandible appears more clearly optimized for feeding than the cranium is in agreement with the conclusions of Preuschoft and Witzel (2002) and Ross and Metzger (2004). However, we note that to date only jaw adducting (intrinsic) musculature has been considered in either in vivo or FE studies of stress/strain during feeding in the primate skull. It has been shown that extrinsic forces generated by cervical musculature can be relatively high in predatory mammals in the dispatch of small prey (Preuschoft and Witzel, 2005; Wroe et al., In Press) and Pan troglodytes is known to kill and eat other vertebrates (Anderson and Kitchener, 1983; Boesch, 1994).
Our finding that strain and displacement were considerably greater in the heterogeneous as opposed to the homogeneous models may have important implications. Mean brick element strain energy stored in the heterogeneous mandible was 7.2 times higher than that stored in the homogeneous simulation. Historically, it has been argued that the advantage of incorporating lighter cancellous bone centers on mass reduction and consequent energy savings to the organism. However, our results show that additional work is required to achieve the same ends in the more realistic model accommodating cancellous as well as cortical bone. This additional energy must be generated by muscle.
The storage and appropriate release of energy has been well demonstrated in other biological systems, particularly with respect to tendons in locomotion, where 90% or more of strain energy can be recovered in passive elastic recoil within the power stroke (Alexander, 1984; Bullimore and Burn, 2005). Our results suggest that cancellous bone may also play some role in storage and release of energy for long bones in locomotion. However, there is little obvious advantage to such energy storage in biting because there is no means to recoup such energy from the bone for another power stroke in closing the jaws, and opening the jaws is largely accomplished by gravity (Turnbull, 1970).
Results suggest that cancellous bone may assist in dissipating stress, as has been proposed for sutures (Rayfield, 2007). However, our results indicate that weight savings gained through the use of lighter cancellous bone are to some degree offset by the requirement for additional, energetically expensive musculature. Considered holistically, the primary advantages associated with heterogeneous construction may not simply be direct optimization to produce a minimal mass of bone tissue, rather, there may be a trade-off between the materials and energy invested in the bony and muscular components of the system.
Structures constructed entirely of compact bone are stiffer with higher yield points. They are, however, also more brittle, and less deformation over a shorter time span will presage ultimate failure (Currey, 2004). Theoretically, the greater elasticity imparted through the incorporation of cancellous bone translates into a lower yield point, but also a less brittle structure that permits greater deformation over a longer period before reaching ultimate failure. Moreover, tooth breakage is common in primates (Cuozzo and Yamashita, 2006), and greater elasticity in surrounding bone may reduce the likelihood of tooth failure following accidental occlusion with hard materials.
Mechanoreceptors within the periodontal ligament are known to inhibit jaw adductors and trigger a jaw opening response as reactions to biting on unexpectedly resistant, potentially damaging foods (Anderson et al., 1970; Dessem et al., 1988). We suggest that a system that experiences higher displacements and strains will facilitate greater opportunity for feedback through the nervous system and, hence, a greater capacity for the organism to modify behavior and avoid catastrophic failure in bones and teeth. This line of reasoning might also apply to sutures.
Whether a chimpanzee could develop sufficient bite force to seriously damage parts of its own cranium or mandible remains untested. That primates can generate sufficient force to damage their own teeth appears more certain. More broadly, the mechanism described here may be of greater importance in dissipating stress and facilitating modulation of behavior among carnivorous species. Structural damage is relatively common in predators, where powerful killing bites to soft tissue in prey can result in unexpected contact with bone (Van Valkenburgh, 1988).
We believe that the approaches and models described here represent several steps forward in computer simulation of the vertebrate skull. These include (1) a relatively rapid method for the incorporation of variable properties for bone, which allows for more realistic modeling of structural behavior; (2) the addition of a temporomandibular joint that facilitates more accurate reconstruction of the 3-D architecture of muscle and collation of data from both cranium and mandible in correct anatomical relationship to one another, and minimizes the introduction of artifacts through point loading; (3) a program to compute mean brick stress and strain in regions of interest for very large models that permits statistical comparisons; and (4) overall, our protocols allow rapid assembly and solution of models on standard desktop computers. The solid mesh used here was generated in 1 day, and the complete model assembled from CT data within a week.
There are, however, many areas in which further improvements can be made. Validation, based on investigations of both specimens in vivo and using materials testing systems is required to fully assess the material properties of both cortical and cancellous bone. Our joint reconstructions are rigid and do not account for elastic properties of connective soft tissue, and, although our muscle simulations better describe 3-D architecture and allow for distribution of forces across relevant structures, they do not fully account for the influence of muscle pennation. Our loadings are further simplifications of actual chewing behavior in that all muscles are maximally activated in unison. In reality, muscle recruitment can be variable within and between working and balancing sides (Ross and Metzger, 2004). Sutures may also influence the distribution of stress/strain (Rayfield, 2005; Kupczik et al., 2007). In the few FE analyses that have addressed this issue, sutures have been introduced manually into homogeneous models, either as breaks in the mesh, or by assignment of different material properties to bricks along sutures (Rayfield, 2004; Kupczik et al., 2007). In the models presented here, the role of sutures is not strictly addressed. However, some well-defined sutures visible in the CT data, are partially captured in our FE models as incompletely defined regions comprising brick elements assigned high elasticity. Relatively minor additional increases in brick element number for heterogeneous FE meshes such as these may facilitate accurate modeling of the role of suture morphology.
Work is under way to improve modeling in each of these areas. With the development of methodologies and procedures allowing relatively rapid generation of more realistic skull models, it will be possible to simulate and compare a wide range of hominid material, fossil and extant, facilitating further detailed mechanically based investigations into relationships between anatomy and behavior. These techniques may also have practical potential in biomedical fields, including prosthetics, orthodontics, and the design of safety equipment.