Dr Kornelius Kupczik, Department of Human Evolution, Max Planck Institute for Evolutionary Anthropology, Deutscher Platz 6, 04103 Leipzig, Germany. T: +49 (341) 3550 756; F: +49 (341) 3550 399; E: Kornelius.firstname.lastname@example.org
Crucial to the interpretation of the results of any finite element analysis of a skeletal system is a test of the validity of the results and an assessment of the sensitivity of the model parameters. We have therefore developed finite element models of two crania of Macaca fascicularis and investigated their sensitivity to variations in bone material properties, the zygomatico-temporal suture and the loading regimen applied to the zygomatic arch. Maximum principal strains were validated against data derived from ex vivo strain gauge experiments using non-physiological loads applied to the macaque zygomatic arch. Elastic properties of the zygomatic arch bone and the zygomatico-temporal suture obtained by nanoindentation resulted in a high degree of congruence between experimental and simulated strains. The findings also indicated that the presence of a zygomatico-temporal suture in the model produced strains more similar to experimental values than a completely separated or fused arch. Strains were distinctly higher when the load was applied through the modelled superficial masseter compared with loading an array of nodes on the arch. This study demonstrates the importance of the accurate selection of the material properties involved in predicting strains in a finite element model. Furthermore, our findings strongly highlight the influence of the presence of craniofacial sutures on strains experienced in the face. This has implications when investigating craniofacial growth and masticatory function but should generally be taken into account in functional analyses of the craniofacial system of both extant and extinct species.
Here we present a validation study of finite element models (FEMs) of two macaque crania and assess the sensitivity of these models to elastic properties of bone, the presence or absence of a suture and the mode of load application.
In this study we aim to validate and investigate the sensitivity of FEMs of two crania of the crab-eating macaque Macaca fascicularis focusing on the infraorbital region and the zygomatic arch. We are particularly concerned with the effects of (1) the variation of bone material properties, (2) the complexity of the morphology involved (i.e. the presence or absence of representation of morphological features such as sutures in the FEM) and (3) the nature of loading of the zygomatic arch on the predicted strain magnitudes observed in the FEM both locally within the zygomatic arch and the infraorbital region and globally throughout the cranium. The results of the FEA are validated against data derived from ex vivo strain gauge experiments using non-physiological loads applied to the macaque zygomatic arch.
It has been observed that the specification of the material properties (i.e. magnitude, direction, spatial variation) in the model has significant implications on the results of an FEA (Marinescu et al. 2005; Richmond et al. 2005; Strait et al. 2005). Although solving an FEM with heterogeneous, orthotropic elastic properties predicts the most congruent strain results when compared with experimental data (Marinescu et al. 2005; Strait et al. 2005), for the sake of simplicity the materials involved in the present FEA are modelled with homogeneous, isotropic and linear elastic properties. In order to enable the validation of the FEA strain results, elastic properties of selected regions in the bone of the zygomatic arch were obtained using a nanoindention technique (see below). Moreover, the effect of changes in bone stiffness on the distribution of strains in the zygomatic arch and the circumorbital region are investigated by sequential alteration in the FEA of the experimentally determined Young's modulus of elasticity (E, GPa) of bone and these effects are compared with our strain gauge data.
The zygomatic arch has been chosen as it represents a discrete morphological region of the cranium of importance in terms of investigating craniofacial biology and as a key component in the FEM (e.g. Hylander & Johnson, 1997; Rafferty et al. 2000; Witzel et al. 2004). The attachment areas of the superficial and deep heads of the masseter muscle are easily identifiable and the arrangement and orientation of the fibres makes this muscle easier to model than, for instance, the temporalis muscle – attaching as it does to the curved parietal bone. Moreover, the presence of the zygomatico-temporal suture is of interest from both a functional and a developmental point of view (Herring, 1972; Jaslow, 1990; Hylander & Johnson, 1997; Mao, 2002; Mao et al. 2003; Rafferty et al. 2003; Rayfield, 2005). In vivo strain gauge experiments have shown that the zygomatic arch of M. fascicularis experiences a strain gradient, the largest strains occurring at the anterior end of the arch (Hylander & Johnson, 1997). As the superficial masseter muscle attaches along the anterior portion of the zygomatic arch anterior to the zygomatico-temporal suture, the force will be concentrated anteriorly and hence strains are expected to be larger in this region.
The zygomatic arch resembles to some extent a beam with fixed ends, to which a uniform but off-centre load has been applied (Hylander & Johnson, 1997). Under this model bending moments are consequently largest in the anterior portion of the arch, resulting in relatively larger strains in this region and lower strains towards the posterior arch (Hylander & Johnson, 1997). However, as these authors also noted such a beam model would not take into account the zygomatico-temporal suture connecting the zygomatic processes of the zygomatic and temporal bones. It is known from research on miniature pigs that flexible cranial sutures are subjected to large deformations and limit the strains that can develop in the delicate bones of the face during dynamic loading (Herring & Teng, 2000; Rafferty et al. 2003). Flexible sutures are also said to act as shock-absorbers, being able to absorb more impact energy than bone (Buckland-Wright, 1978; Jaslow, 1990). Moreover, Sun et al. (2004) showed that the magnitude and polarity of strains in some cranial sutures of pigs change with increasing age. This is particularly relevant to solving FEMs of juvenile vs. adult macaques. In terms of bending moments and induced stresses and strains in the zygomatic arch our model is like a beam with a built-in (cantilevered) constraint at one end and a flexible joint at the other. In immature individuals with a patent suture the bending moment of the anterior zygomatic arch will be higher than in older individuals with more mineralized and thus stiffer sutures. It is thus hypothesized that strains in the anterior zygomatic arch in particular decrease with gradual obliteration of the suture. This is tested by comparing experimental strain values with those computed from FEMs with zygomatic arches that are (1) disconnected, (2) fully fused and (3) connected by a zygomatico-temporal suture of varying elastic properties.
A further issue to be addressed here is the modelling of the muscle forces involved. Commonly, force vectors of a certain magnitude and direction are simply applied to the surface nodes of the FEM. The selection of these nodes should reflect the attachment site and the distribution and the concentration of the muscle fibres (Richmond et al. 2005; Ross et al. 2005). One question of interest is how the strain field is influenced by the distribution of the applied forces. It can be hypothesized that the more focussed, centralised application of loads, the more complex and potentially larger the loading and induced stresses and strains experienced by the bone will be. We investigate this by conducting FEAs with varying arrangements of distributed load while keeping the total force applied to the zygomatic arch constant. An alternative approach is to incorporate the muscle itself into the FEM and load the zygomatic arch indirectly via the modelled muscle. This has the advantage of capturing the exact geometry of the muscle and might be expected to result in predicted strains that are more congruent with experimental data.
Material and methods
We selected whole cadaveric heads of two adult male M. fascicularis for this study. The study animals (referred to here as MAC-14 and MAC-17) had been used previously in dental caries experiments unrelated to the present study (Smith & Beighton, 1986, 1987). The exact history of preservation of the cadaveric heads is uncertain, although it is known that they have been kept in formaldehyde, 70% industrial methylated spirit and other aqueous solutions at various times. MAC-14 is a relatively small specimen with the third molars congenitally missing, while MAC-17 is a large adult with a complete dentition. The zygomatico-temporal suture is clearly defined and unfused in MAC-14, whereas in the more mature MAC-17 specimen the suture is less visible and appears more mineralised.
Experimental strain analysis
In both specimens the skin and the underlying soft tissue were removed to reveal the bone and the masticatory muscles. The superficial masseter was dissected on one side of each of the heads (MAC-14: left-hand side; MAC-17: right-hand side) and detached from its insertion site on the mandible sufficiently to permit independence when loads were applied to it. Dehydration was minimized by applying a glycerine/water solution onto the soft tissue and bone and by keeping the specimens in a sealed container in a fridge when not under experimental employment. Prior to the strain gauging experiments the exposed bone was degreased with isopropanol (M-Line GC-6 Degreaser, Vishay Measurements Group, Basingstoke, UK) and abraded with pumice powder. Six 120 ± 0.5-Ω rosette strain gauges wired in a three-wire quarter-bridge circuit (TML FRA-1–11, Tokyo Sokki Kenkyujo, Tokyo, Japan) were bonded with cyanoacrylate along the zygomatic arch of each specimen, from the infra-orbital region to the posterior zygomatic arch (gauges IO and ZA1–ZA5), three gauges being placed anterior to the suture and two posterior to it (Fig. 1). The strain gauges were connected to a Vishay 5100B Wheatstone bridge amplifier with an excitation voltage of 0.5 V (Vishay Micro-Measurements). The heads were then placed on a specially constructed test rig and constrained by means of a rubber plug fixing the foramen magnum. The second molars and the first molars of MAC-17 and MAC-14, respectively, rested on a rubber-coated bar. Each of the dissected superficial masseters (working side) and the gonion region of the mandible on the balancing side were perforated to allow passage and attachment of nylon-coated strings (1 mm diameter, Berkley Outdoor Technologies Group, Spirit Lake, IA, USA). We conducted the loading via the strings in line with the orientation of the superficial masseters, by manually placing weights at the ends of the strings up to a maximum of 15 N for the smaller MAC-14 specimen and 20 N for the MAC-17 specimen. Simultaneously, in each of the specimens an equal load was applied to the string on the balancing side jaw also in line with the orientation of the superficial masseter. We recorded the orientation of the working side masseter by taking three-dimensional (3D) coordinates with a MicroScribe GTX digitizing system (Immersion Corporation, San Jose, CA, USA) in order to derive the force vectors for the FEA. Maximum principal strains (expressed as microstrain µɛ) under static loads were recorded as the strain most readily comparable with the FE results. Five consecutive strain readings at the strain peak were taken per loading, and loadings were repeated four times within 15 min of the experiment to minimize the effects of dehydration and atmospheric change.
Nanoindentation of zygomatic arch cortical bone and zygomatico-temporal suture
Although the fixation history of the specimens is unknown it is likely to have had an effect on mechanical properties. Thus, to establish the elastic properties of the cortical bone and the zygomatico-temporal suture of the fixed specimens we investigated the mechanical properties of the bone using an instrumented nanoindenter. Following the strain gauge experiments, the zygomatic arch from the balancing side of each specimen was cut both at the anterior (zygomatic) end and at the posterior (temporal) end of the arch. Subsequently, the isolated bones were sectioned with an Isomet 5000 linear precision saw (Buehler Ltd, Lake Bluff, IL, USA) at two locations along the arch of MAC-14 (anteriorly and through the zygomatico-temporal suture) and three locations along the arch of MAC-17 (anteriorly, through the zygomatico-temporal suture and posteriorly). The three samples were then embedded in epoxy cold-mounting resin (Epothin, Buehler Ltd) before the surfaces were ground (Phoenix Beta Vector, Buehler Ltd) and polished (ChemoMat synthetic polishing pad and Mastermet silica polishing suspension, Buehler Ltd). Using a nano-hardness tester with a Berkovitch diamond indenter (CSM Instruments S.A., Peseux, Switzerland), we measured the Young's elastic modulus at 16 locations along the perimeter of each cross-sectional sample, using a load of 100 mN applied for 5 s. In addition, four nanoindentation measurements were taken through the zygomatico-temporal suture of the two specimens. The mean of the E of bone and suture for each of the sections of MAC-14 and MAC-17 are summarized in Table 1. The grand mean for bone is 9.1 ± 2.8 GPa (MAC-14) and 11.0 ± 3.0 GPa (MAC-17), while it is 1.9 ± 1.7 and 7.7 ± 3.4 GPa for the suture of MAC-14 and MAC-17, respectively.
Table 1. Mean, standard deviation (SD) and maximum and minimum Young's elastic moduli (in GPa) for cortical bone of the zygomatic arch (MAC-14: two sections, MAC-17: three sections) and for the zygomatico-temporal suture in MAC-14 and MAC-17. The values are based on 16 nanoindentation measurements for each of the bone sections (anterior, suture and posterior) and four and five for the suture tissue of MAC-14 and MAC-17, respectively
Zygomatico- temporal suture
CT scanning and 3D reconstruction
Coronal computed tomography (CT) scans were taken of MAC-14 on a Philips Mx8000 scanner (helical mode, 0.3 mm slice thickness, 120 kV, 228 mA). MAC-17 was scanned on an X-Tek HMX 160 scanner (X-Tek Systems Ltd, Tring, UK) at 123 kV, 87 µA and with a 0.2-mm Cu filter. The voxel size was 0.23 mm. Prior to scanning, the specimen MAC-17 was deliberately dehydrated for 14 days at room temperature in order to assist in the visualization of the masticatory muscles. We subsequently used the visualization software Amira (Mercury Computer Systems, Inc., San Diego, CA, USA) to segment the bone and teeth (treating enamel and dentine as a single material) and, by extracting the surfaces, reconstructed these materials in three dimensions. The components of the periodontal ligament were not taken into account as the limited resolution of the CT scans did not allow for an accurate reconstruction. In MAC-14 a clearly defined zygomatico-temporal suture was also segmented as a discrete object, thus separating the zygomatic from the temporal bone (Fig. 1A). In MAC-17 this suture was partly fused and hence only weakly indicated in the segmentation. Moreover, in MAC-17 the superficial masseter was segmented and reconstructed in three dimensions (Fig. 1B). Both surface models were then converted into a volumetric tetrahedral grid to be imported into ANSYS Mechanical (ANSYS, Inc., Canonsburg, PA, USA).
Finite element modelling
The FEMs derived from the CT data used quadratic, tetrahedral elements (ANSYS type SOLID 92). The MAC-14 model consists of 178 427 elements, while the larger MAC-17 model including the superficial masseter comprises 232 564 elements. Given the size difference, this resulted in a similar mesh density for the two samples. All materials (i.e. bone, teeth, muscle and suture) were modelled as linear elastic and isotropic (see E values for each of the materials used below) with a Poisson's ratio of 0.3 in all cases.
In both models we modelled the boundary conditions for the foramen magnum and occlusal surfaces of the relevant molars by fully constraining the corresponding nodes in these areas. In MAC-14, ten nodes on the inferior border of the left zygomatic arch were selected and a total load of 15 N applied. The loading conditions of the zygomatic arch of MAC-17 are described in detail below. In addition, the equivalent surface nodes on the opposite zygomatic arch were fully constrained. This idealization of the balancing side forces was considered justifiable as it was remote from the primary region of interest. Three series of finite element analyses were conducted by varying: (1) the E of the bone material (both models); (2) the presence, absence and E of the zygomatico-temporal suture (MAC-14 only); and (3) the method of load application (MAC-17 only). The sensitivity of the FEMs to these parameters was assessed and the parameters which resulted in predicted strains concurring with those measured experimentally were identified.
(1) Variation of bone elastic properties
We varied the Young's modulus of the bone material using values measured from the nanoindentation study of the bone (see Table 1). A range of values were noted and the highest, lowest and average value were used for the modelling, with further fine adjustment to identify the value that produced results similar to those recorded experimentally. The teeth were assigned an elastic modulus E of 70 GPa based on a mean of published values for enamel and dentine (Cuy et al. 2002; Kinney et al. 2003).
(2) Variations of the zygomatico-temporal suture
The defined zygomatico-temporal suture in the MAC-14 specimen consisted of 420 tetrahedral elements. Modelling was initially conducted to represent the two extremes of fusion and complete disconnection. We modelled the suture either with the same properties as the surrounding bone (E = 9.1GPa) or omitted it from the model completely, thus separating the zygomatic and temporal portions of the zygomatic arch. We subsequently modelled the suture with E-values derived from the literature (1.20 ± 0.20 MPa in 8-week-old rabbits; Radhakrishnan & Mao, 2004) and also from the nanoindentation results (see Table 1), with further variation conducted to provide agreement with the experimental results.
(3) Variation of load application
In MAC-17 we modelled the zygomatic arch loading either by direct application of the load vectors to nodes along the zygomatic arch or by application of the load vectors to the modelled superficial masseter muscle. Where direct loading was used, the load of 20 N was uniformly distributed between 10, 20 and 41 nodes along the superficial masseter bone scar on the zygomatic arch (Fig. 2). These loading conditions represented a linear distribution (ten nodes) to an area distribution covering the full attachment site of the muscle (41 nodes).
In the case of the muscle loading in the MAC-17 model, the load was applied to attempt to replicate the experimental load conditions in a number of ways. In the absence of a literature value for the bulk elastic modulus of the superficial masseter, we experimented by varying E and Poisson's ratio. A value for E of 1 GPa and Poisson's ratio of 0.3 was found to deviate least from the experimentally determined values. Thus, in loading scenarios 1–3 the load of 20 N was distributed between all the nodes of a cross-section of the muscle through inferior, central and superior parts of the muscle (Fig. 3). In loading scenario 4, a closer replication of experimental conditions was attempted by loading a line of nodes through the depth of the muscle at the approximate location where the loading string penetrated the muscle (Fig. 3).
We compared the results from the finite element modelling in each case to the experimental results. This was done by calculating the mean maximum principal strain of three superficial nodes in the model coinciding with each of the six strain gauge locations on the skulls (see Fig. 1). The overall difference between the strains obtained experimentally at all six locations and those obtained in the FEA was expressed as the Euclidean distance calculated for each of the models. This was computed as the square root of the sum of squared differences between experiment and model. The minimum Euclidean distance values obtained identified the modelling parameters which best fit the experimental results. No statistical treatment of sensitivity beyond this was feasible given the small number of models solved.
Variation of bone elastic properties
Figure 4(A,B) compares the mean maximum principal strains recorded for five strain gauges (MAC-14 – with the suture modelled using optimum E from the studies of Fig. 5, see below) and six strain gauges (MAC-17) in the loading experiments with the results of the FE modelling while taking into account a range of bone E values. The strain gauge ZAL3 for MAC-14 turned out to be faulty, and hence the results were excluded from the subsequent analyses. In MAC-14, at all bone stiffness levels the largest maximum principal strains occur in the anterior zygomatic arch (ZAL1) with rapidly decreasing strains posteriorly (Fig. 4A; except ZAL4, E = 4.0 GPa). Both the experimental and the modelled strains in MAC-17 are more uniformly distributed and peak in the middle of the anterior zygomatic arch for E = 11–17 GPa (ZAR-2, Fig. 4B).
In general, the strains decrease with an increase in the stiffness of bone in both models. The Euclidean distances for MAC-14 indicate that the difference between experimental and FE strains is lowest when modelled with the mean E of bone derived from nanoindentation (9.1 GPa). By contrast, for MAC-17 it was found that E = 14 GPa yields a lower Euclidean distance than the average value obtained by nanoindentation (11.0 GPa). The largest discrepancy between the strains in the FEM and in the experiments in MAC-17 is found for location IOR (c. 75 µɛ see Fig. 4B).
Variations of the zygomatico-temporal suture
Figure 5(A) illustrates the effects on the maximum principal strains in MAC-14 when the zygomatic arch is fully fused (i.e. the E of the zygomatico-temporal suture equals that of bone), completely disconnected and connected by a zygomatico-temporal suture. The bone elastic modulus is kept constant in all three models (E = 9.1 GPa). If the arch is separated, the strains are markedly higher in the infraorbital region and on the anterior portion of the zygomatic arch, whilst a decrease is observed at these sites when the arch is completely fused. The Euclidean distance shows that modelling the zygomatic arch with a suture with E = 0.0025 GPa results in strain values that are most similar to the experimental ones. If the stiffness of the suture is either increased (E = 1.9 GPa; the mean obtained by our nanoindentation study) or decreased (E = 0.0012 GPa; Radhakrishnan & Mao, 2004) the differences in strain magnitude between the model and the experiments become larger (Fig. 5B). It is worth noting that the difference in Euclidean distance between complete arch fusion (E = 9.1 GPa, Fig. 5A), E = 1.9 GPa and E = 0.8 GPa (minimum value from nanoindentation) is negligible.
Variation of load application to the zygomatic arch
Figure 6 shows the strain magnitudes in the six locations of MAC-17 when the zygomatic arch is loaded either through the superficial masseter muscle or by directly loading 10, 20 or 41 nodes of the zygomatic arch. The zygomatic arch experiences relatively higher strains when loaded through the superficial masseter muscle as opposed to being loaded directly, although the variation of strains between both types of loading is less obvious in the infraorbital region (IOR). Among the muscle-loaded models it is the loading at the most superior level of the masseter muscle that is associated with the relatively lowest strains. If the zygomatic arch is loaded directly, the strains generally decrease with an increase in the number of selected nodes (10–41) although at gauge locations IOR and ZAR1 the model with 41 nodes exhibits slightly higher strains than do the other two models. As already shown in Fig. 4(B), the difference between the modelled and experimental strains is largest for IOR.
Local vs. global effects of strain distribution
Figure 7 illustrates the effects of the presence or absence of the zygomatic suture in MAC-14 and Fig. 8 shows the effects of the mode of load application on the global strain distribution in the skull. In the case of a fused zygomatic arch, strains of high magnitude (i.e. > 164 µɛ) are only present in the anterior end of the zygomatic process and at the base of the zygomatic process of the temporal (Fig. 7A). By contrast, if the zygomatic arch is modelled with the suture present, the relatively highest maximum principal strains are observed in the zygoma covering parts of the infraorbital region and extending into the lateral orbital margin (Fig. 7B). In MAC-17, the variation of the loading conditions has less of an impact on regions in the face adjacent to the zygomatic arch (Fig. 8A,B). The relatively highest strains (> 273 µɛ) are limited to the anterior portion of the zygomatic process when the arch is loaded directly (Fig. 8A). By contrast, when the arch is loaded via the superficial masseter, these peak strains occur both in the zygomatic arch and in the root of the zygomatic process (Fig. 8B).
The main objective of this study is to validate and assess the sensitivity of FEMs of the macaque cranium to variations in bone material properties, the presence or absence of the zygomatico-temporal suture and the loading regimen applied to the zygomatic arch.
To this end we carried out three studies to assess the sensitivity of the principal strain distribution to variations in these parameters. The first examined the effects of varying the material properties of the bone and was carried out in conjunction with direct measurement of the material properties of the experimental specimens. The second examined the effects of presence or absence and varying material properties of the zygomatico-temporal suture. This aimed to test the hypothesis that strains in the zygomatic arch decrease with gradual obliteration of the suture. The third investigated the effects of varying the ways in which loads are applied to the zygomatic arch. The influence of the distribution of the applied force on the predicted strain magnitudes in the model was ascertained. It was hypothesised that a more focussed, centralised application of loads on the zygomatic arch would result in a larger bending moment and hence maximum principal strains. Additionally, we experimented with applying loads through a simplified (i.e. isotropic, linear elastic) model of the superficial masseter in order to see whether this led to strains that are more congruent with experimental data. Finally, we plotted strain maps in order to evaluate local vs. global effects of the above experiments. The validity of the different models was evaluated by examining maximum principal strains in the infraorbital region and the zygomatic arch and comparing values measured by experimental strain gauge analysis with those obtained through FEA simulation.
Variation of bone elastic properties
The results of Fig. 4 indicate that variations in E have marked effects on the observed magnitudes of the maximum principal strain as measured at each strain gauge site. Unfortunately, our experiments suffered from a strain gauge malfunction (ZAL3, MAC-14). This gave extraordinarily high strain readings, which we were able to demonstrate were erroneous in subsequent experiments with a replacement gauge that gave much better agreement with the modelling results. However, because of uncertainties about alterations to bone material properties due to dehydration in the intervening period we have omitted the data obtained for this site.
The Euclidean distances between successive experiments indicate that using values of E close to experimentally determined values results in a high degree of congruence between experiment and model. The result in MAC-14 was optimal when using the experimentally obtained value (E = 9.1 GPa) while that in MAC-17 was very close (using the experimentally determined value of 11 GPa), but an iteratively determined value of E = 14 GPa provided an even closer match. In MAC-17 very close agreement between experimental and modelled strains is obtained at this value for E at all locations except IOR (c. 75 µɛ discrepancy; see Fig. 4B). In this specimen this gauge was placed on the maxilla with the zygomatico-maxillary suture between it and the load while in MAC-14 the equivalent gauge was on the zygomatic. Thus, in MAC-17 it is highly likely that the omission of this suture from the model underlies this result (see below).
Irrespective of the bone Young's modulus used, maximum principal strains peak in the anterior portion of the zygomatic arch and then fall off sharply towards the posterior end (MAC-14; Fig. 4A). These findings confirm results of in vivo experiments of adult macaques by Hylander & Johnson (1997), who also described a steep strain gradient in the zygomatic arch. The situation is somewhat different in both the experimental and the modelling results of MAC-17 where such a strain gradient is less pronounced (Fig. 4B). This can probably be attributed to the fact that the zygomatico-temporal suture was more fused in this specimen and hence not clearly defined in our FEM (see also below).
The mean E values of the zygomatic arch obtained by the nanoindentation study do not compare well with those used in previous FE studies of the macaque cranium. Strait et al. (2005, their table 2) used an E value of 12.5 GPa for the posterior zygomatic arch (based on unpublished data for Macaca mulatta by Wang & Dechow), whilst E = 20.8 GPa was used for the anterior zygomatic arch. This lack of correspondence between our measurements of E (see Table 1) and those of other workers is little surprising given the uncertain preservation history of our material. Several studies have suggested that formalin fixation changes, to some extent, the material properties of bone (Reilly & Burstein, 1974; Zioupos et al. 2000). In particular, it was shown that fixation results in a decrease in bone stiffness of pigs after 2 weeks of preservation (Dechow & Huynh, 1994), although Currey et al. (1995) showed that fixing bone from an 18-month-old bovine in a 10% formalin mixture for 3 h and subsequent buffering for 3 days has only a limited effect on the elastic modulus in tension. While FEA should be accompanied by mechanical testing of the materials involved, the anisotropic and heterogeneous behaviour of bone as well as the type (cortical or cancellous), the provenance (i.e. species and anatomical region) and the preservation should ideally be considered as well (Dechow & Hylander, 2000; Zioupos et al. 2000; Peterson & Dechow, 2003; Schwartz-Dabney & Dechow, 2003; Marinescu et al. 2005; Strait et al. 2005). For example, Peterson & Dechow (2003) found that the zygomatic process of the temporal bone has the highest anisotropy of all sites. We did not take this into account in this study because of the complexity and size of the model and yet obtained a remarkably close match between experimental and modelling results. It will be of interest in future studies further to explore the sensitivity of FEMs to variations in the direction and spatial variation of material properties over a more substantial anatomical area.
Variations of the zygomatico-temporal suture
As noted in the introduction, the zygomatic arch has been likened to a beam with fixed ends that is subjected to an off-centre but uniform load, although the zygomatico-temporal suture is likely to modulate the ways in which loads are borne (Hylander & Johnson, 1997). Consequently, in our FEMs we introduced the zygomatico-temporal suture such that the zygomatic arch possessed a built-in constraint at one end and a flexible joint at the other. The effects on the maximum principal strains in MAC-14 of variations in the zygomatico-temporal suture were therefore examined in order to investigate the role of this suture in the FEM.
In these experiments the zygomatic arch is either fully fused, disconnected or connected by a suture (with E = 0.0025 GPa). Our results (Fig. 5A) indicate that when the arch is fused the strains are markedly lower in the infraorbital region (IOL) and on the anterior portion of the zygomatic arch (ZAL1), whereas strains increase when the arch is completely disconnected. By contrast, introducing a suture into the model results in predicted strains most similar to experimental values.
A second set of experiments examined the effects of varying the stiffness of the suture (between E = 1.9 GPa and E = 0.0012 GPa). The results (Fig. 5B) indicate that a value of 0.0025 GPa, close to that reported in the literature, gives the best overall fit to the experimentally determined zygomatic and facial strains. This is not consistent with the estimate of E = 1.9 GPa from nanoindentation and serves to illustrate the difficulties we experienced in sectioning the bone and suture while maintaining low temperatures to avoid drying out of the material.
Our study supports the view that sutures are decisive for the accuracy of skeletal FEMs, at least with respect to the cranium (Rayfield, 2005; Wang et al. 2006) and that the elastic properties of the suture have profound effects on estimated strain magnitudes. This is consistent with experimental findings, which show that the strains in facial bones are limited while the interposed sutures experience large deformations (Herring et al. 1996; Herring & Teng, 2000). Clearly this issue is most significant when dealing with subadult material as sutures tend to fuse gradually with age in larger mammals (Sun et al. 2004). Under parasagittal bending the infant/juvenile zygomatic arch with a relatively unmineralized suture will initially experience a relatively steep anterior–posterior strain gradient due to large bending moments at the anterior end (cf. Fig. 5B; E = 0.001 GPa). With a gradual fusion of the suture the arch will resemble more of a beam with two built-in ends and consequently the bending moments and hence strains decrease anteriorly (cf. Fig. 5B; E = 1.9 GPa).
The material properties of sutures are highly variable within a single cranium and between species. Thus, the cranial sutures of neonate and young rats and pigs have Young's moduli ranging between 0.64 and 171.5 MPa (Margulies & Thibault, 2000; McLaughlin et al. 2000; Tanaka et al. 2000). This might have implications for primate FEMs if the FE model uses sutural elastic properties taken from non-primate material. However, the estimation of the material properties of sutures is not straightforward as our attempt has shown. Therefore, it might be reasonable where experimental data are available to estimate the Young's modulus iteratively from the FE model – as indeed we have done in this study. This approach is, however, only applicable in limited situations and would not be possible for fossils. In any case our findings underline the importance of sutures and their mechanical properties in building FEMs of crania.
Variation of load application to the zygomatic arch
A further experiment examined the effects of different approaches to loading of the zygomatic arch. In the first approach we applied loads to the arch directly, varying the distribution of the loads however by selecting varying numbers of model nodes for their application. In the second approach loads were applied via a simplified model of the superficial masseter. The results indicate that direct loadings consistently produce results that are closer to the experimentally determined values. Interestingly, loading the arch through a linear array of ten nodes shows the best agreement with experimental strains, although differences compared with the models with a higher number of selected nodes are relatively small. The findings also corroborate the hypothesis that a more centralized load application results in relatively larger maximum principal strains in the zygomatic arch (Fig. 6).
The effect on the model's results of loading the zygomatic arch through the masseter muscle is very surprising, particularly as the magnitude of the predicted strains is significantly higher (Fig. 6) when the muscle is included. It is not clear why this is the case. Possibly the muscle is providing an additional torsional load to the arch and it might also add leverage if the force vectors are not exactly in line with the long axis of the muscle, as suggested by an anonymous reviewer. Alternatively, it may indicate that the superficial masseter transfers forces to the zygomatic arch via a limited number of tendinous connections, most like the arrangement of loads in Fig. 2(A). The failure of the muscle loading simulations to achieve results close to experimental values likewise indicates the inadequacy of modelling muscle as an isotropic linear elastic material. In future models, if muscles are to be used to apply loads it will be necessary to include more accurate information on the muscle's elasticity as well as anisotropies and non-linearities to account for true muscle architecture.
Local vs. global effects of strain distribution
Finally we examined the full FEM of each macaque cranium under varying loadings and conditions as in the preceding experiments. FEA has the advantage of allowing strain maps of the whole to be estimated and visualized although, as we have shown here, there are many variables that can undermine the accuracy of the FEM. In producing these maps our aim is to evaluate local vs. global effects of the above experiments.
From Figs 7 and 8 it is clear that the effects we describe above in relation to the areas local to strain gauges obtain more generally. Thus, the presence of a zygomatico-temporal suture affects the strains experienced over a wide area (Fig. 7A vs. Fig. 7B) including the anterior part of the zygomatic and infraorbital regions. The effect of direct vs. ‘muscle’ loading of the zygomatic arch (Fig. 8A vs. Fig. 8B) is more confined to within the zygomatic region. However, within this region the effects are substantial.
In detail, the effects of varying these diverse parameters are significant in that estimates of absolute values of strain at specific loci are markedly affected. However, in general all loading regimens show that the zygomatic arch manifests relatively large strains, most concentrated anteriorly and falling away in magnitude with distance. Whether this level of generality is acceptable is entirely dependent on the question at hand; but our data indicate that reasonable approximations of how loads are handled in the cranium might be obtained for regions not amenable to strain gauging in living species and in fossil material as long as attention is paid to the modelling of sutures and muscle attachments. If accuracy is required then the details of suture morphology, muscle attachment, applied loads and mechanical properties become of too great significance to be ignored.
We gratefully acknowledge the financial support of the Leverhulme Trust (F/00224/I). We also thank Sue Taft, Neil Curtis, Lee Page, Andrea Cardini, Sam Cobb, Callum Ross, Fred Spoor, Chris Dean and staff at Hull Royal Infirmary for their advice and assistance. We are also grateful to Daniel Lieberman and three anonymous reviewers for their helpful comments during the preparation of the manuscript.