SEARCH

SEARCH BY CITATION

Keywords:

  • crocodiles;
  • skull biomechanics;
  • finite-element analysis;
  • feeding;
  • hydrodynamics;
  • aquatic/marine tetrapods;
  • comparative modeling

Abstract

  1. Top of page
  2. Abstract
  3. Previous Work on Biomechanics of Crocodile Skulls
  4. Aims of Study
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSIONS
  9. Acknowledgements
  10. LITERATURE CITED
  11. Appendices: Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D

This article reports the use of simple beam and finite-element models to investigate the relationship between rostral shape and biomechanical performance in living crocodilians under a range of loading conditions. Load cases corresponded to simple biting, lateral head shaking, and twist feeding behaviors. The six specimens were chosen to reflect, as far as possible, the full range of rostral shape in living crocodilians: a juvenile Caiman crocodilus, subadult Alligator mississippiensis and Crocodylus johnstoni, and adult Caiman crocodilus, Melanosuchus niger, and Paleosuchus palpebrosus. The simple beam models were generated using morphometric landmarks from each specimen. Three of the finite-element models, the A. mississippiensis, juvenile Caiman crocodilus, and the Crocodylus johnstoni, were based on CT scan data from respective specimens, but these data were not available for the other models and so these—the adult Caiman crocodilus, M. niger, and P. palpebrosus—were generated by morphing the juvenile Caiman crocodilus mesh with reference to three-dimensional linear distance measured from specimens. Comparison of the mechanical performance of the six finite-element models essentially matched results of the simple beam models: relatively tall skulls performed best under vertical loading and tall and wide skulls performed best under torsional loading. The widely held assumption that the platyrostral (dorsoventrally flattened) crocodilian skull is optimized for torsional loading was not supported by either simple beam theory models or finite-element modeling. Rather than being purely optimized against loads encountered while subduing and processing food, the shape of the crocodilian rostrum may be significantly affected by the hydrodynamic constraints of catching agile aquatic prey. This observation has important implications for our understanding of biomechanics in crocodilians and other aquatic reptiles. Anat Rec Part A, 288A:827–849, 2006. © 2006 Wiley-Liss, Inc.

Crocodilians have long been used as a model for interpreting the paleobiology of fossil species. The range of overall head shape and dentition in living taxa is reasonably broad (Mook,1921), and observed correlations between these anatomical features and feeding ecology in crocodilians (Iordansky,1973; Langston,1973; Aoki,1977; Van Drongelen and Dullemeijer,1982; Busbey,1989) have formed the bases for hypotheses of feeding paleoecology in a large array of reptiles, including dinosaurs, phytosaurs, ichthyosaurs, mosasaurs, champsosaurs, plesiosaurs, and fossil crocodilians (e.g., Massare,1987; Taylor,1987; Aoki,1989; Taylor and Cruickshank,1993; Hua et al.,1994; Sereno et al.,1998).

Many investigations into the functional morphology of these taxa have employed quantitative morphometric techniques as a basis for comparisons between fossil and extant forms. However, the use of morphometrics in identifying correlations between anatomical shape and ecology becomes complex when considering fossil groups, such as ichthyosaurs and plesiosaurs, that are only distantly related to extant “model” reptile groups. The differing phylogenetic histories make identification of homologous landmarks difficult or impossible, and the fossil group frequently combines features reminiscent of the model group with anatomical novelties (autapomorphies) or convergences on completely separate living taxa. For example, the skull of the large pliosauroid plesiosaurs is indeed reminiscent of the skull in some members of the genus Crocodylus, but pliosaurs also have cranial features reminiscent of some lizards, aquatic birds, toothed whales, and even seals. It is difficult, a priori, to identify which morphological features have trophic significance and which might thus form the basis for a comparative paleoecological analysis using morphometric data.

Some recent studies have used biomechanical analysis of skull shape to investigate links between morphology and ecology: bite force (a property affected by a complex of morphological factors) has been shown to correlate with maximum prey size in terrestrial mammalian and reptilian carnivores (Meers,2002; Wroe et al.,2005). An understanding of the underlying mechanics of the crocodilian skull may shed light on which aspects of crocodilian head shape are most useful in interpreting the paleoecology of fossil crocodilians, pliosaurs, and other marine reptiles. Our long-term goal is to identify those features of the skull's shape that most effectively account for variation in feeding behavior among living crocodiles by modeling components of feeding behavior and evaluating the mechanical performance of the skull. We report here on some preliminary findings of this work that have important implications for our understanding of crocodilian skull biomechanics.

Previous Work on Biomechanics of Crocodile Skulls

  1. Top of page
  2. Abstract
  3. Previous Work on Biomechanics of Crocodile Skulls
  4. Aims of Study
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSIONS
  9. Acknowledgements
  10. LITERATURE CITED
  11. Appendices: Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D

Busbey (1995) used beam theory to evaluate the mechanical consequences of rostral shape in living and extinct crocodilians. He identified that, consistent with mechanical principles, oreinirostral (tall-snouted) skulls are better able to cope with bending forces in the dorsoventral plane, while platyrostral (flat-snouted) skulls are better able to resist forces in the mediolateral plane. As an explanation for why living crocodilians have a platyrostral skull (and are thus not efficient at resisting dorsoventral bending loads during feeding), Busbey discussed five possibilities as follows.

(i) Suction feeding.

The broad skull may facilitate “gape-and-suck” feeding, of the sort seen in some living chelonians and hypothesized for some extinct “labyrinthodonts.” Busbey rejected this explanation because the behavior has not been observed in living crocodilians.

(ii) Crypsis.

The flattened skull may allow crocodilians to remain cryptic near the water surface.

(iii) Hydrodynamics.

The low cross-section of the platyrostral skull reduces drag during lateral movement of the head. Rapid lateral head sweeps are reported to be an important behavior for crocodilians feeding on small agile aquatic prey.

(iv) Twist feeding (rolling).

Busbey (1995) stated that “a rostrum with low sides is more effective in withstanding compression and axial torsion during rolling than would an oreinirostral snout.” Twist feeding, where the prey (or part thereof) is held in the mouth while the crocodile rolls its entire body about its craniocaudal axis, is an important technique of subduing and processing prey for many crocodilians.

(v) Sutures.

Busbey also listed the suggestion by Bolt (1974) that a flattened rostrum may “minimize the moment of sutures relative to mediolateral forces.”

Of these, Busbey proposed that the hydrodynamic and twist feeding arguments could serve as explanations for the evolution of platyrostry in crocodilians. Although Busbey discussed the importance of hydrodynamic constraints operating during feeding, he also emphasized the evolution of twist feeding behaviors as a proximal explanation. Among modern reptilian and mammalian large carnivores, the twist feeding [Taylor (1987): equivalent in sense to the term “rolling” in Busbey (1995)] behavior is common in crocodilians (Pooley and Gans,1976) but is largely absent in noncrocodilians (Busbey,1995). Busbey suggested that several morphological features were important in the ability of the skull to withstand the torsional loads imposed by this behavior: the extensive sutural overlapping (scarf joints) of bones in the posterior part of the rostrum; increased thickness of the rostral bones; the development of the closed bony secondary palate; the enlargement of the lateral pterygoid flanges, which act as medial braces for the mandibular rami, resisting movement of the mandible in the transverse plane; and the dorsoventrally flattened snout of the platyrostral skull.

Using beam theory, Busbey showed that the possession of a closed palate (secondary palate) in modern crocodilians provides a mechanical reinforcement of the skull that at least partially counteracts the increased tendency of the platyrostral snout to bend during dorsoventral loading. The closed palate is also mechanically very important in effectively resisting torsional and compressive forces during twist feeding, and Busbey strongly supported the earlier contention by Langston (1973) that the evolution of the closed palate in crocodilians is linked to the mechanics of feeding behavior, rather than simply serving to separate the airway from the mouth (which had been the traditional view).

However, Busbey did not offer an explanation as to why twist feeding should necessarily lead to a platyrostral shape as opposed to a tubular shape, where rostral height and width are equivalent. The contention that a platyrostral skull represents an optimal mechanical solution to torsional loads conflicts with traditional engineering principles, as illustrated in Figure 1, which shows the maximum von Mises stress induced into beams of three different shapes of the same cross-sectional area under the action of a pure torsional load. For the same maximum stress, a beam of cylindrical cross-section can handle almost double the torque compared to a rectangular beam with a 5:1 aspect ratio. From a mechanical viewpoint, for a given sectional area, a platyrostral snout (where width exceeds height) would perform no better in resisting torsional loads than an oreinirostral snout (where height exceeds width), and each of these would be inferior to a tubular snout design.

thumbnail image

Figure 1. Maximum stress in a solid cylinder, square and 5:1 aspect ratio rectangle of the same cross-sectional area subject to a pure torque. The cylinder carries less stress than the other shapes for a given torque.

Download figure to PowerPoint

Aims of Study

  1. Top of page
  2. Abstract
  3. Previous Work on Biomechanics of Crocodile Skulls
  4. Aims of Study
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSIONS
  9. Acknowledgements
  10. LITERATURE CITED
  11. Appendices: Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D

While four of the features of crocodilian skull anatomy listed by Busbey (1995)—scarf joints, thickened rostral bones, closed palate, and pterygoid flanges—can obviously help resist torsional loadings, a simple two-dimensional analysis suggests that platyrostry per se does not confer any obvious mechanical advantages to loads of this type. To allow for the possibility that the three-dimensional shape of the crocodilian rostrum allows resistance to torsional loads that cannot be explored in a two-dimensional analysis, we used three-dimensional beam and finite-element models to investigate two of the key points raised by Busbey's 1995 study: does the platyrostral shape of the skull in crocodilians help resist the torsional loadings encountered during twist feeding, and how useful are simple beam models in investigating the mechanical behavior of complex structures such as a crocodilian rostrum?

We used a comparative approach, constructing models from six crocodilian specimens that, between them, encompass a wide range of rostral shapes, from extremely platyrostral (Alligator mississippiensis) to the most oreinirostral living crocodilian (Paleosuchus palpebrosus). We included specimens representing intermediate stages of platyrostry, Melanosuchus niger, an adult Caiman crocodilus, and a juvenile C. crocodilus, as well as a longirostrine taxon (Crocodylus johnstoni). Mechanically, the rostrum can be thought of as a cantilevered beam (Rafferty et al.,2003; Metzger et al.,2005), and as such the proximal sections are of particular importance in determining its structural characteristics. Ultimately, we compared results from beam and finite-element models of each specimen to test for variation due to modeling technique and formulated two specific hypotheses: A. the most platyrostral taxon will give the best mechanical performance under loads simulating twist feeding behaviors; B. qualitatively, the mechanical performance of the six rostral finite-element models should match the predictions of simple beam theory models based on the dimensions of the proximal end of the rostral beam.

MATERIALS AND METHODS

  1. Top of page
  2. Abstract
  3. Previous Work on Biomechanics of Crocodile Skulls
  4. Aims of Study
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSIONS
  9. Acknowledgements
  10. LITERATURE CITED
  11. Appendices: Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D

Daniel and McHenry (2001) constructed a finite-element model of an Alligator mississippiensis skull based on the high-resolution X-ray computed topography (CT) scans produced by the Digital Morphology (DigiMorph) Group at the University of Texas (http://www.digimorph.org). At the time the model was constructed, there were no techniques readily available for the automated conversion of CT data into a 3D solid model, which can be meshed for use in a finite-element analysis. Consequently, the 1,453 node mesh of the Alligator skull was constructed manually in STRAND6 finite-element software (Fig. 2). That mesh resolution should be appropriate to the hypotheses being tested has been emphasized, particularly during the initial development of functional models (Fastnacht et al.,2002; Jenkins et al.,2002; Snively and Russell,2002; Rayfield,2004; Richmond, et al.,2005; Strait et al.,2005). Because we were interested in the qualitative effects of gross shape on mechanical behavior, we manually constructed six new models, each with a c. 2,500 elements mesh (Table 1). As the hypotheses that we wished to test (logically developed from Busbey's 1995 analysis) concerned the shape of the rostrum, the new models were of rostra only; this removed the need to model muscle forces accurately in the rear part of the skull (Jenkins et al.,2002) and allowed us to maximize the extent of the comparative analysis.

thumbnail image

Figure 2. Finite-element model of Alligator mississippiensis constructed manually in STRAND6 from CT scan data in 1998 (Daniel and McHenry,2001).

Download figure to PowerPoint

Table 1. Models and specimens used in this study
ModelNodes (unrefined mesh, refined mesh)TaxonCommon nameSpecimen no.Head length (mm)Skull length (mm)
  1. See text for discussion. Head length, tip of snout–posterior end of lower jaw; Skull length, tip of snout–posterior apex of occipital condyle. A refined mesh involves sub-dividing the original plate elements into smaller elements. Institutional abbreviations: TMM, Texas Memorial Museum; FMNH, Field Museum of Natural History; BMNH, British Museum of Natural History; AMNH, American Museum of Natural History.

A.f1434, 1434Alligator mississippiensisAmerican alligatorTMM M-983217 
A.r599, 599Alligator mississippiensisAmerican alligatorTMM M-983217 
A.m169, 2827Alligator mississippiensisAmerican alligatorTMM M-983217 
C.c(j)196, 2877Caiman crocodilus scleropsspectacled caimanFMNH 73711136 
C.c(a)196, 2915Caiman crocodilus scleropsspectacled caimanBMNH 1933.5.10.2 237
M.n196, 2938Melanosuchus nigerblack caimanBMNH unnumbered 456
P.p196, 2884Paleosuchus palpebrosusCuvier's dwarf caimanAMNH 93812 150
Cr.j170, 2226Crocodylus johnstonifreshwater crocodileTMM M-6807169 

Taxa

Crocodilian material held in Australian museum collections are dominated by the two native species and most extant taxa are not represented: the choice of crocodilian taxa was therefore partly determined by availability of suitable scans from the DigiMorph data library, including Alligator mississippiensis, Caiman crocodilus sclerops, and Crocodylus johnstoni. In each instance, the scan data were taken from juvenile to subadult animals. In addition to these three models, we morphed the Caiman crocodilus model to produce three additional models that were dimensionally comparable to an adult Caiman crocodilus sclerops, a Melanosuchus niger, and a Paleosuchus palpebrosus (Table 1).

Finite-Element Model Construction 1: Direct Use of CT Data

The rostrum in these models was defined as the portion of the snout anterior to a transverse line passing through the mediolaterally narrowest part of the median intraorbital bar (formed mainly by the frontals). This line usually passed through the midpoint (in the rostral-caudal axis) of the orbits (Fig. 3). From this posterior boundary to the tip of the snout inclusive, a total of 18 transverse slices were digitized. For each of the Alligator, Caiman, and Crocodylus data sets, the exact position of the slices were determined by the position of rostral landmarks deemed a priori to be important (Fig. 3A, Table 2). The CT scans corresponding to each slice were taken from the requisite DigiMorph data set, and a minimum number of important geometric features were isolated and assigned as nodes. For each side of the rostrum, between three and nine nodes were identified per slice, depending on the structural complexity (of topology and overall shape) of the rostrum at that slice. The nodes were then used to define the corner nodes of low-order three-node and four-node plate elements, from which the geometry of the model was constructed in STRAND7.

thumbnail image

Figure 3. A: Transverse slices used to construct finite-element model of Caiman crocodilus sclerops, based on CT scan data from a juvenile individual. Vertical lines on whole skull images (left) show the location of the 18 slices. Note that the caudal-most line (slice 18), which passes more or less through the middle of the orbits, is the line used to define the rear of the rostrum in the FE models. CT scans corresponding to the 18 slices are shown at right. CT scan data copyrighted by Digital Morphology Group, University of Texas (DigiMorph). B: Dorsal and lateral photographs of the adult Caiman crocodilus sclerops specimen used to morph the juvenile caiman finite-element model shown in Figure 4. The vertical lines identify the anatomical landmarks corresponding to the 18 transverse slices used to build the finite-element model of the juvenile specimen (Fig. 3A). The Cartesian coordinates of the dorsal, lateral, and ventral margins of the rostrum in the adult specimen were used to morph all nodes in the finite-element model of the juvenile caiman into a new model that superficially resembled the adult.

Download figure to PowerPoint

Table 2. Anatomical landmarks used to locate transverse slices from which finite element meshes were created
SliceAlligator mississippiensisCaiman crocodilusCrocodylus johnstoni
  1. See text and Figure 3. Numbers in square brackets indicate position of landmarks used in morphometric analysis and to construct beam models (see text).

1Tip of snoutTip of snoutTip of snout
2Alveolus of 3rd premaxillary toothAlveolus of 2nd premaxillary toothAlveolus of 2nd premaxillary tooth, at front of external nares
3Alveolus of 4th premaxillary tooth, near front of external naresAlveolus of 3rd premaxillary tooth, near front of external naresHalf-way between the 3rd and 4th premaxillary teeth, at the maximum width of the external nares
4[4] Alveolus of 5th premaxillary tooth, in line with the front of the ossified nasal ‘septum’[4] Half-way between the 3rd and 4th premaxillary teeth, at the maximum width of the external nares[4] Alveolus of the 4th premaxillary tooth, at the maximum width of the external nares and at the maximum width across the premaxillae
5[3] At the suture between premaxilla and maxilla, on the labial margin of the jawAlveolus of 4th premaxillary toothAlveolus of 5th premaxillary tooth
6Alveolus of the 2nd maxillary tooth, in line with the rear margin of the external naresHalfway between the 4th and 5th premaxillary alveoli, in line with the front of the ossified nasal ‘septum’[3] At the suture between premaxilla and maxilla, on the labial margin of the jaw
7Half-way between the 2nd and 3rd maxillary alveoliIn line with the rear margin of the external naresHalf-way between the 1st and 2nd maxillary alveoli
8Half-way between the 3rd and 4th maxillary alveoli[3] At the suture between premaxilla and maxilla, on the labial margin of the jawHalf-way between the 2nd and 3rd maxillary alveoli
9[2] Alveolus of the 4th maxillary tooth (maxillary fang)Alveolus of the 2nd maxillary toothAlveolus of the 3rd maxillary tooth
10Half-way between 5th and 6th maxillary alveoli[2] Alveolus of the 4th maxillary tooth (maxillary fang)Alveolus of the 4th maxillary tooth
11Half-way between 5th and 6th maxillary alveoli, and in the line of the rearmost medio-lateral constriction of the maxillae (2nd maxillary waist)Alveolus of the 5th maxillary tooth[2] Alveolus of the 5th maxillary tooth (maxillary fang)
12A quarter-way between the 8th and 9th maxillary alveoliAlveolus of the 6th maxillary tooth, and in the line 2nd maxillary waistAlveolus of the 7th maxillary tooth
13Alveolus of the 10th maxillary tooth (also the largest tooth posterior to the 2nd maxillary waist)Alveolus of the 8th maxillary toothHalf-way between the 8th and 9th maxillary alveoli
14Alveolus of the 11th maxillary tooth–in line with the anterior edge of the palatal foramenaAlveolus of the 9th maxillary tooth (also the largest tooth posterior to the 2nd maxillary waist)Half-way between the 10th and 11th maxillary alveoli
15[1] Half-way between the 12th and 13th maxillary alveoli–in line with the anterior edge of the orbits[1] Alveolus of the 10th maxillary tooth–in line with the anterior edge of the orbitsHalf-way between the 12th and 13th maxillary alveoli
16Point of inflection along anterio-medial margin of orbitsAnteriormost point of inflection along anterio-medial margin of orbits[1] In line with the anterior edge of the orbits
17Narrowest (medio-laterally) part of inter-orbital barSecond point of inflection along anterio-medial margin of orbitsPoint of inflection along anterio-medial margin of orbits
18Just behind narrowest part of inter-orbital barNarrowest (medio-laterally) part of inter-orbital barNarrowest (medio-laterally) part of inter-orbital bar
thumbnail image

Figure 4. Finite-element models of crocodilian rostra. A.m, Alligator mississippiensis; C.c(j), Caiman crocodilus (juvenile); Cr.j, Crocodylus johnstoni; C.c(a), Caiman crocodilus (adult); M.n, Melanosuchus niger; P.p, Paleosuchus palpebrosus.

Download figure to PowerPoint

This process produced three models corresponding to the rostra of a subadult Alligator mississippiensis (referred to as the A.m model), a juvenile Caiman crocodilus [C.c(j) model], and a subadult Crocodylus johnstoni (Cr.j model; Fig. 4). These models were of a near-identical level of resolution and a similar but not identical topology.

Scaling

In each of the new models, the thickness of each plate was determined by measuring the cross-section of skull bone on the corresponding CT slice using ImageJ image analysis software. Because we wanted to examine the effect of shape on the mechanics of the rostrum, we attempted to reduce as far as possible the effects of model size by scaling each model to the same width at the back of the rostrum. The width chosen was 95 ± 1 mm (including plate thickness); this was the width of the rostrum in the scanned Alligator mississippiensis specimen. In the absence of data on the actual scaling relationship between bone thickness and skull width in crocodilians, bone thickness might theoretically scale by anything between the first and second power of skull width, but each of these extremes seems unlikely and as a compromise we scaled plate thicknesses in the models by a factor of √2 times the linear scaling factor between the specimens.

Finite-Element Model Construction 2: Morphed Variants of Caiman Model

The juvenile Caiman crocodilus model was used to generate models of three other caimans: an adult Caiman crocodilus sclerops [C.c(a) model], a large adult Melanosuchus niger (M.n), and an adult Paleosuchus palpebrosus (P.p). The landmarks used to define the 18 slices in the juvenile Caiman crocodilus model were used to identify the corresponding slices in dorsal and lateral photographs of each specimen (Fig. 3B), and the Cartesian coordinates of the points defining the dorsal, ventral, and lateral margins of the rostrum at each slice were extracted. For each slice, the differences in X, Y, and Z coordinates between these points on the model and the photographed specimen gave the proportions by which the shape varied, and these proportions were used to transform the Cartesian coordinates of all the nodes, internal and external, in the juvenile Caiman crocodilus model. The result was a new version of the caiman mesh, topologically identical to the original, but of a shape that corresponded very well to the photographed specimen (Fig. 4). The new models were then scaled to the same width as the original variant of the model. Plate thicknesses were held constant for all of the caiman models.

Axes

In all the models, the rostral-caudal axis was set as the x-axis, the mediolateral as the y-axis, and the dorsoventral as the z-axis. Direction of positive values in each axis is shown in Figure 4. Point (0, 0, 0) was set at the tip of the snout (the anteriormost extent of the premaxillae, slightly dorsal to the jaw margin). Axis (x, 0, 0)—nominally, the central-line axis—was defined as a line between the tip of the snout and the posterior apex of the occipital condyle in the specimens on which the finite-element models were based.

Material Properties

Following Daniel and McHenry (2001), elastic isotropic properties were assumed for the skull material with the following values: E (Young's modulus of elasticity) = 10 GPa, ν (Poisson's ratio) = 0.4, and ρ (density) = 2,300 kg/m3.

Loading Conditions

Busbey (1995) listed three behavioral categories used by crocodilians to catch, subdue, and process prey: simple biting, or jaw adduction; head shaking, through pitching (rotation in the vertical sagittal plane) or yawing (rotation in the horizontal frontal plane) of the head with the prey held in the mouth; and rolling, with the prey held tightly in the mouth and the whole body rotated about the cranial-caudal axis (also called twist feeding).

We used this list to determine three loading regimes (each comprising of several specific load cases) to be applied to the finite-element models (Table 3). Here forces were applied to the appropriate nodes to simulate biting.

Table 3. Loading conditions used in this study
Load caseLoading regimebite pointbite typeforcesloads
verticallateraldorso-ventral bendingmedio-lateral bendingtorsion
right sideleft sideright sideleft side
  1. See text for discussion.

1simple bitemidasymmetric100 N    
2simple bitemidsymmetric100 N100 N    
3simple biteanteriorsymmetric100 N100 N    
4simple biteanteriorasymmetric100 N    
5simple biteposteriorasymmetric100 N    
6head shakeanteriorsymmetric100 N100 N100 N100 N 
7head shakemidasymmetric100 N 100 N 
8head shakemidsymmetric100 N100 N100 N100 N 
9twist feedinganteriorasymmetric200 N100 N   
10twist feedingmidasymmetric200 N100 N   
Simple biting.

We simulated simple bites in the middle of the tooth row (load cases 1 and 2), at the tip of the jaw (load cases 3 and 4), and at the posterior cheek teeth (load case 5). Bilaterally symmetrical bites produce only dorsoventral bending loads (load cases 2 and 3), while unilateral bites produce dorsoventral bending loads together with some torsional loads (load cases 1, 4 and 5).

Lateral head shaking.

This is where the head is yawed rapidly while holding the prey tightly in the mouth. In this loading regime, the rostrum is subjected to lateral bending moments and dorsoventral bending moments. If the prey is being held asymmetrically in the mouth, the rostrum will also be exposed to torsional forces. We simulated lateral head shaking with the prey held symmetrically at the tip of the jaws (load case 6) and the middle of the jaws (load case 8), and for prey being held asymmetrically in the middle of the jaws (load case 7). We did not consider shaking through pitching of the head because the loads in that behavior are qualitatively similar to the loads encountered during simple biting.

Twist feeding.

This is where the prey is held in the mouth and the entire head of the crocodilian is rolled around the anteroposterior axis. During this behavior, the rostrum is exposed to significant dorsoventral bending and torsional forces. We simulated twist feeding with the prey being held symmetrically in the tip of the jaws (load case 9) and the middle of the jaws (load case 10).

Loads were applied to individual tooth positions, with four nodes sharing the load equally at each tooth loaded. Anterior bites were simulated at either the third or fourth premaxillary teeth, while mid bites were simulated at either the third, fourth, or fifth maxillary teeth, depending on taxon. Posterior bites were simulated at a point in the maxillary tooth row level with the anterior margin of the palatal foramina (Fig. 5).

thumbnail image

Figure 5. Position of load points simulating cheek, mid, and tip bites, shown on the A.f model in palatal view. For each bite position, loads were divided and spread equally about the most ventral four nodes of the respective alveoli. White arrows indicate loads in the dorsoventral axis: on the left tooth row, loads are purely dorsoventral (e.g., unilateral bites, normal loading regime). On the right tooth row, the black arrows indicate transverse loads that are coupled with the dorsoventral loads in the lateral shake loading regime. Black angled bars at rear of skull indicate restraints.

Download figure to PowerPoint

The magnitude of the forces applied in the various load cases are arbitrary, although the vertical components are consistent with the magnitude of bite forces that have been measured in similar-sized Alligator mississippiensis and Caiman crocodilus (Cleuren et al.,1995; Erickson et al.,2003). No attempt was made to vary the forces according to moment from the jaw joint.

Boundary Conditions

Three different boundary conditions ranging from completely restrained to relatively free were applied to each rostrum finite-element model. In the most-restrained boundary conditions, nodes at the rear of the rostrum were prevented from translating in all directions, while in the intermediate boundary conditions, each cheek bar was completely constrained and nodes on the medial part of the skull were prevented from moving in the x-axis only. In the least-restrained boundary, the cheek bars were sufficiently restrained to prevent rigid body motion from occurring.

Although testing the finite-element models under these different set of boundary conditions provides a basic sensitivity analysis for the models, it does not identify which set of restraints most closely matches the manner in which the rostrum is actually braced by the posterior part of the skull. To examine this question, we included the two versions of the Alligator mississippiensis mesh constructed by Daniel and McHenry (2001) in the analysis (Table 1). The full-skull version (A.f model) was a mesh of the entire cranium; a rostrum version (A.r) was based on that same mesh, but included only the rostral portion of skull as described above. Each of these models was scaled to the same width at the back of the rostrum as described above. Note that the mesh from the previous study was of a different topology to those created in this study, so that the mesh in the A.r model was different to the A.m model, even though they were each based on the same data. The A.f. model was constrained identically in each set of boundary conditions: with all nodes in the occipital region, as well as the posteroventral edges of the pterygoid flanges, fixed in all three axes to prevent rigid body motion. The restraints on the A.r model in the different boundary conditions matched those for the other rostral models as described above. By including these models and using the criteria that the most realistic set of boundary conditions would be those in which the mechanical behavior of the A.r model resembled that of the A.f model, we hoped to identify which set of restraints most closely matches the manner by which the posterior skull braces the rostrum in crocodilian skulls.

Morphometrics and Rostral Sections

Measurements were taken from photographs of the specimens on which the finite-element models were based, at four positions: (1) at the front of the orbits, (2) at the position of the largest maxillary alveolus (the maxillary fang), (3) at the maxilla-premaxilla suture on the labial surface of the jaw margin, and (4) at the widest extent of the external nares (Table 2).

Positions 1, 2, and 4 were used by Busbey (1995) as part of his morphometric analysis of crocodilian rostral shape. In addition, position 3 is the site of a constriction between the anterior and mid regions of the rostrum in crocodylids, which we deemed to be of potential structural importance. For the rostral section at each of these positions, rostral width and height were recorded; in addition, the distance of each position from the rear of the rostrum was taken. For each rostrum, these measurements were divided by the width of the model at the rear of the rostrum (as defined above) to produce scaled dimensionless measurements of rostral proportions (Appendix A).

Beam Models

Using the morphometric data described above, six simple beam models were constructed in STRAND7 with hollow elliptical sections and wall thickness set to 3% of beam width (the figure measured at the posterior-most slice of the Alligator CT scan). As we were modeling the rostrum as a cantilevered beam (Rafferty et al.,2003; Metzger et al.,2005), we assigned the posterior part of the rostrum the greater importance in determining the shape of the simple beam model: the dimensions of each beam were set as the average of the rostral dimensions at the front of the orbits and at the largest maxillary tooth (i.e., the fourth maxillary tooth in alligatorids, the fifth in crocodylids) for each respective specimen (Appendix A). Each beam model was then subjected to loads equivalent to the 10 load cases used on the 6 finite-element models. The mechanical performance of the beam models were ranked for each load case.

Measurements of Mechanical Performance

Two measures of mechanical performance were taken for each model: strain and displacement. Each of these measurements has advantages and disadvantages. Strain is commonly used as a measure of skeletal biomechanical performance in vivo and is a biologically relevant proxy for measuring the capacity of skeletal elements to resist forces. The strain readings reported here are the highest von Mises strain on the outside surface of the plate elements in each finite-element model. However, in the displacement form of the finite-element method, the displacements of nodes are determined first, with strains within the elements determined from the element's shape functions and the spatial displacements. This makes measurement of strain vulnerable to artifacts in low-resolution meshes. Conversely, a drawback of using displacement is that it is a function of the stiffness of a structure and the moment (force magnitude × distance from restraining point) of the load; this consideration becomes important in situations where the structures being compared are of differing lengths, as in this study. A second limitation of using displacement as a measure of mechanical performance is that it is of questionable biological importance. These points are considered further in the discussion of the results below.

For each of the strain and displacement data sets, performance was assessed by ranking the data for each model. This facilitated comparison between the two data types and between the results for the beam and finite-element models.

RESULTS

  1. Top of page
  2. Abstract
  3. Previous Work on Biomechanics of Crocodile Skulls
  4. Aims of Study
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSIONS
  9. Acknowledgements
  10. LITERATURE CITED
  11. Appendices: Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D

Morphometrics

Morphometric data for the six new rostral models are presented in Figure 6. The Crocodylus johnstoni (Cr.j) was notably narrower than the other five models, which did not vary much in rostral width. There was more variation in rostral height, with the Paleosuchus palpebrosus (P.p) model being tallest in the rear and mid portions of the rostrum (sections 1 and 2), the Melanosuchus niger (M.n) and Alligator mississippiensis (A.m) models being tallest just behind the nares (section 3), and the C.c(a) (Caiman crocodilus adult) model tallest at the front of the snout. The A.m model was very flat at section 1, and while the Cr.j model was flattest at the largest maxillary tooth (section 2), the A.m and C.c(j) (Caiman crocodilus juvenile) models were relatively flat at this point also. Rostral length was greatest in the Cr.j model, but quite similar among the other five models (Fig. 6C).

thumbnail image

Figure 6. Visual plot of morphometric data from the six specimens on which the models were based. Charts show rostral height (A) and width (B) for each specimen at sections 1 to 4. C shows distance from location of each section to the back of the rostrum. All measurements are shown as proportions of the width at the rear of the rostrum. For descriptions of the landmarks defining each section, see text and Table 2.

Download figure to PowerPoint

Busbey (1995) discussed platyrostry in terms of simple rostral height, but did not consider to which part of the rostrum the term should apply. Given the mechanical importance of the proximal part of the rostrum, we designated the skulls that were flattest in this region to be the most platyrostral. Thus, the model with the lowest height (i.e., most platyrostral) was the alligator (A.m), followed by the C.c(j) and the Cr.j models. The least platyrostral (i.e., most oreinirostral) was the P.p model, followed by M.n and C.c(a) models.

Beam Models

Rankings based on maximum (von Mises) strain for the six beam models for each load case are shown in Table 4; models with the best mechanical performance (i.e., least strain magnitude) are shown ranked first. The ranked order of the models was consistent for all load cases (Table 4), with the M.n model showing the lowest strain and the Cr.j model the highest. The only variation in the ranked order of the models was in load case 5, where the C.c(a) and C.c(j) models reversed places.

Table 4. Ranked results of mechanical performance in beam models, based upon (A) strain and (B) displacement data
A.
load caseA.mC.c(j)C.c(a)M.nP.pCr.j
  1. See Tables 5,7 for abbreviations.

1534126
2534126
3534126
4534126
5543126
6534126
7534126
8534126
9534126
10534126
mean5.03.13.91.02.06.0
mode534126
variance0.000.100.100.000.000.00
B.
load caseA.mC.c(j)C.c(a)M.nP.pCr.j
1543126
2543126
3543216
4543216
5541226
6543216
7543126
8543126
9543216
10543126
mean5.04.02.81.51.66.0
mode543126
variance0.000.000.400.280.270.00
6Y135246
7Y235146
8Y235146
Table 5. Model rankings: strain data
lcB.p.typeA.mC.c(j)C.c(a)M.nP.pCr.jA.f-A.r
  1. Rankings based on maximum strain for the six finite element models (columns A.m to ‘Cr.j’), and differences between maximum strain readings of the old (Daniel & McHenry,2001) mesh full skull and rostral models (column ‘A.f-A.r’, far right). For ranked data models, lowest strains are given highest rank. Differences between A.f and A.r are shown in microstrain–note that in this case the mean is of the absolute values. A.f, Alligator full skull model (old mesh); A.r, Alligator rostral model (old mesh), all other models new (this study) mesh models of rostrum; A.m, Alligator mississippiensis; C.c(j), Caiman crocodilus (juvenile), C.c(a), C. crocodilus (adult), M.n, Melanosuchus niger, P.p, Paleosuchus palpebrosus; Cr.j, Crocodylus johnstoni; l.c., load case; B.p., Bite position; symm, symmetrical bites; asymm, asymmetrical bites.

least restrained boundary
1midasymm462351110
2midsymm256341215
3tipsymm24513685
4tipasymm345125−30
5cheekasymm623541310
6tipsymm245136−285
7midasymm643241−50
8midsymm36513285
9tipasymm146235−295
10midasymm26435170
mean3.14.54.42.23.62.9153.5
mode245131 
variance3.01.61.81.70.95.2 
intermediate boundary
1midasymm54321590
2midsymm5342160
3tipsymm345216100
4tipasymm54312630
5cheekasymm642214250
6tipsymm345116−280
7midasymm632351−80
8midsymm5612430
9tipasymm354126−60
10midasymm443126−60
mean4.54.13.21.72.04.995.0
mode543216 
variance1.40.81.70.52.03.0 
most restrained boundary
1midasymm645321215
2midsymm643412235
3tipsymm523416100
4tipasymm534216−170
5cheekasymm554312250
6tipsymm534216−300
7midasymm653214−10
8midsymm642315100
9tipasymm533216−300
10midasymm65143295
mean5.53.83.22.91.34177.5
mode643216 
variance0.31.11.30.80.54.2 

Finite-Element Models

Strain data.

Qualitatively, the maximum von Mises strain for the three Alligator models (A.f = full-skull model; A.r = rostral model; and A.m) varies between the three sets of boundary conditions. As stated above, the A.f model was constrained identically under all three sets of boundary conditions, and thus maximum strain did not vary with boundary conditions in this model (Appendix C). Overall, the figures for maximum strain in the A.r model was most similar to that in the A.f model under the intermediate boundary conditions (Table 5).

Maximum strain in the A.m model tended to be higher than in the A.r model under the least and most-restricted boundary conditions, but under the intermediate boundary conditions, strain magnitude in one was not consistently higher than in the other (Appendix C).

Rankings based on maximum von Mises strain for the six new rostral models are shown in Table 5 (for raw data, see Appendix C). Again, the results varied slightly between boundary conditions. In all these models, variance in the rankings between load cases was lower in the most- than in the least-restrained boundary conditions (Table 5). However, for three of the models, variance was lowest of all under the intermediate boundary conditions. Under the intermediate boundary conditions, the M.n model had the highest mean rank (although its mode rank was 2), followed closely by the P.p and then, in order, the C.c(a), C.c(j), A.m, and Cr.j models. Mean rankings under the most-restrained were similar, but the M.n and P.p models swapped positions, as did the A.m and Cr.j models. Under the least-restrained boundary conditions, the order was somewhat different, with the M.n model having the highest mean rank but followed by the Cr.j and then the A.m models.

Within each set of boundary conditions, the variation in rankings can be summarized by bite position (i.e., tip, mid, or cheeks bites), bite type (symmetrical or asymmetrical bites), or loading regime (simple, lateral shake, or twist bites; Table 6). As with overall levels of variation, variability in each of these was greatest under the least-restricted boundary conditions.

Table 6. Summary of effects of bite position, bite type, and loading regime upon mean rankings for FE models across different boundary positions
 A.mC.c(j)C.c(a)M.nP.pCr.j
  1. Note that n varies for each value of mean shown (refer to Table 5).

least restrained boundary
Bite position      
 tip2.04.05.31.32.85.5
 mid3.45.44.02.44.21.2
 cheek6.02.03.05.04.01.0
Bite type      
 symm2.34.85.31.53.33.8
 asymm3.74.33.82.73.82.3
Load regime      
 simple3.44.24.22.63.62.8
 lateral shake3.74.74.31.33.33.0
 twist1.55.05.02.54.03.0
intermediate boundary
Bite position      
 tip3.54.34.31.31.56.0
 mid5.04.02.62.02.64.2
 cheek6.04.02.02.01.04.0
Bite type      
 symm4.04.33.81.81.85.3
 asymm4.84.02.81.72.24.7
Load regime      
 simple4.83.83.41.81.25.4
 lateral shake4.74.32.72.03.33.3
 twist3.54.53.51.02.06.0
most restrained boundary
Bite position      
 tip5.02.83.52.51.06.0
 mid6.04.42.83.21.62.8
 cheek5.05.04.03.01.02.0
Bite type      
 symm5.53.33.03.31.04.8
 asymm5.54.23.32.71.53.5
Load regime      
 simple5.43.63.83.21.23.4
 lateral shake5.74.03.02.31.05.0
 twist5.54.02.03.02.04.0

The effect of bite position was most pronounced under these conditions in the long-snouted Cr.j model, which ranked 1 and 2 for mid and cheek bites, but 5 and 6 for tip bites (Table 5). Under the intermediate and most-restricted boundary conditions, the Cr.j model also ranked last in all tip bites and did slightly better in mid and cheek bites, although not to the same degree (Table 6). With the A.m model, bite position showed the opposite effect, with this model doing better in tip bites than in the other positions, although the difference between the mean rankings for the different bite positions was less marked than in the case of the Cr.j model. Under the least-restrained boundary conditions, the C.c(j), M.n, and P.p models also did better in tip bites, but bite position appeared to have the opposite effect on the C.c(a) model. Again, these differences were less pronounced under the intermediate and most-restricted boundary conditions. The ranks for the single load case simulating a cheek bite were similar to the mean rankings for tip and mid bites under the intermediate and most-restrained boundary conditions, but were quite different under the least-restrained boundary conditions.

Overall, bite type did not show a strong effect on the mean rankings (Table 6), although, as with bite position, its effect was greatest under the least-restrained boundary conditions. Likewise, loading regime appeared to make little difference in the performance of each model; however, under the least-restrained (and, to a lesser degree, the intermediate) boundary conditions, the A.m model ranked higher in the twist feeding regime than it did in the simple biting or lateral shake regimes, while the Cr.j model performed noticeably better in the lateral shake than the other loading regimes.

Displacement data.

Qualitatively, the models' maximum displacement varied only slightly between boundary conditions (Appendix D). Unlike the pattern seen with the strain data, displacement in the A.r model was equally similar to that of the A.f model under the intermediate and the least-restrained boundary conditions (Fig. 7A). Displacement in the A.m model was consistently higher than in the A.r model (Appendix D). For the new models, simple ranking of the total displacement data suggests that the P.p model performed the best under all combinations of load case and boundary conditions (Table 6). The average ranking of the models is largely reflected in regressions of displacement/F against moment arm (Fig. 7B); however, by taking the effects of moment into account, the latter plot suggests that the C.c(a) model performed consistently better than the C.c(j), and that for its length the A.m model performed worse than the Cr.j (Table 3, Appendix D).

thumbnail image

Figure 7. Displacement divided by force plotted against moment arm for (A) the A.f and A.r models and (B) the other six finite-element models. Data points are shown for the intermediate boundary conditions. Solid lines show power regressions. A: Heavy solid line (labeled A.f) is the regression for the A.f. model. Regression lines for the A.r model are shown for the least-restrained (l), intermediate (i), and the most-restrained (m) boundary conditions. Note that the regression lines for the A.r model under the least-restrained and intermediate boundary conditions overlie each other. B: Regression lines for the six other models are shown under the intermediate boundary conditions only.

Download figure to PowerPoint

DISCUSSION

  1. Top of page
  2. Abstract
  3. Previous Work on Biomechanics of Crocodile Skulls
  4. Aims of Study
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSIONS
  9. Acknowledgements
  10. LITERATURE CITED
  11. Appendices: Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D

In the beam models, variation in rankings between load cases and loading regimes was very low for the strain data. Qualitatively, the results based on strain matched those based on displacement, the principle difference between the two data sets being the relative mechanical performance of the M.n and P.p models, resulting in slightly higher total variance in the displacement data set (Table 7).

Table 7. Rankings based upon maximum displacement
l.c.B.p.typeA.mC.c(j)C.c(a)M.nP.pCr.jA.f-A.r
  1. Displacement measured in mm. D(XYZ), total displacement; D(Y), displacement in Y axis. Data in A.f-A.r column is difference in absolute displacement (in mm: i.e. not ranked) between these models–for this column the mean is of the absolute values. Abbreviations as for Table 5.

least restrained boundaryD(XYZ)
1midasymm5342160.112
2midsymm5342160.223
3tipsymm5342160.291
4tipasymm534216−0.023
5cheekasymm5234160.052
6tipsymm5342160.014
7midasymm5243160.117
8midsymm5243160.283
9tipasymm5342160.440
10midasymm5243160.413
mean5.02.63.92.51.06.00.197
mode534216 
variance0.000.270.100.500.000.00 
D(Y)
6tipsymm534216 
7midasymm635412 
8midsymm634215 
intermediate boundaryD(XYZ)
1midasymm5342160.109
2midsymm5342160.226
3tipsymm5432160.294
4tipasymm543216−0.014
5cheekasymm6423150.051
6tipsymm5432160.019
7midasymm5342160.115
8midsymm5342160.278
9tipasymm5432160.441
10midasymm5342160.411
mean5.13.53.42.11.05.90.196
mode534216 
variance0.100.280.490.100.000.10 
D(Y)
6tipsymm543216 
7midasymm645213 
8midsymm634215 
most restrained boundaryD(XYZ)
1midasymm5342160.121
2midsymm5342160.234
3tipsymm5432160.306
4tipasymm543216−0.005
5cheekasymm6523140.053
6tipsymm5432160.030
7midasymm5342160.136
8midsymm5342160.296
9tipasymm5432160.459
10midasymm5342160.424
mean5.13.63.42.11.05.80.206
mode534216 
variance0.100.490.490.100.000.40 
D(Y)
6tipsymm534216 
7midasymm643125 
8midsymm463125 

The displacement data for the finite-element models suggest that sensitivity to boundary conditions was low, i.e., the ranked results did not vary much between boundary conditions. As with the beam models, variation between load cases was low and there was no appreciable difference in the relative performance of the models between loading regimes. The load case in which rankings most varied from the mode was load case 5, which was the cheek bite, and it may be that minor differences in the position of the simulated bite point between models were exaggerated by the shortened moment arm of this load (load case 5 was also the only variation from the mode in the strain data for the beam models).

Because displacement is a function of the rostrum's stiffness and the moment of the load, in order to use displacement data to assess the stiffness of the rostrum, the effect of moment must first be taken into account. Ranked results based on total displacement are unable to do this (Table 7). Given, however, that variations in ranked displacement data between load cases were low, a chart of displacement divided by total vertical force and plotted against moment arm can be used to show the relative mechanical performance of each model (Fig. 7B). Lines of regression relating to each model allow the relative performance of the models to be visualized. As noted above, this chart suggests that, although absolute displacement was higher in the Cr.j model, when the moment is taken into account, the A.m model actually showed the worst mechanical performance. It also suggests that, taking moment into account, the C.c(a) model was consistently stronger than the C.c(j) model—an observation that is not clear from the ranked data—although the performance of these two models did show some sensitivity to boundary conditions, being closer under the least-restrained than under the other two sets of boundary conditions.

Strain is the preferred measurement of mechanical performance, although, as noted previously, it is vulnerable to artifacts in low-resolution meshes. Compared to the strain data for the beam models, the rankings for the finite-element models showed greater levels of variation between load cases (Table 5); however, patterns in this variation were difficult to distinguish. Qualitatively, the ranked results were similar under the intermediate and most-restrained boundary conditions and were furthermore similar to the rankings seen in the displacement data (Table 8): across all three loading regimes, the P.p and M.n models performed best, followed by the C.c(a) and C.c(j) models, with the A.m and Cr.j models performing worst.

Table 8. Summary of ranked results for beam and finite element models, according to data type and across boundary conditions
ModelData typeboundaryA.mC.c(j)C.c(a)M.nP.pCr.jTotal variance
  1. For abbreviations see Table 5.

BeamDisplacement 5431260.1
 Strain 5431260.95
FEA least5342160.87
 Displacementinter5432160.97
  most5432161.58
  least36514214.2
 Straininter5431269.4
  most6432158.2

Only under the least-restrained boundary conditions did the ranked results differ qualitatively from this pattern. On the basis of mean rank, the M.n model performed best but was followed by the Cr.j and A.m models (Table 5). There was also a clear instance of a model ranking higher in one loading regime than in the others (the A.m model in the twist feeding regime). Aggregate variance was also highest for this part of the data set (Table 5).

The effects of bite position and bite type were most pronounced under the least-restrained boundary conditions (Table 6). The Cr.j model was most affected by bite position, and given the large moment incurred by the bite forces applied to the tip of this longirostrine form, it is not surprising that it performed worse in tip bites than in bites at the mid and cheek parts of the tooth row. The C.c(a) model had the next longest rostrum and it too fared worse in tip bites. The cheek bites were only represented by one load case (load case 5), and for each model, the rank for this matched the overall mean rankings reasonably well under the intermediate and most-restrained boundary conditions, but again was in less agreement under the least-restrained boundary conditions. That the A.m model performed notably worse in the unilateral bites than it did in the bilateral bites under the least-restrained boundary conditions is interesting, given its better result in the twist loading regime and the fact that both these and the unilateral bites induce torsional loads.

For strain data under the least-restrained set of boundary conditions, the relatively good result of the A.m model in the twist feeding loading regime (in which it ranked highest with a mean rank of 1.5) contrasts with its result under the other boundary conditions, i.e., intermediate (ranked third with a mean rank of 3.5) and most restrained (ranked last with a mean rank of 5.5) (Table 6). As noted above, that result also contrasts with the mechanical performance of the model in asymmetric bites of other loading regimes, in which it ranked fourth, third, sixth, and sixth (load cases 1, 4, 5, 7, respectively) under the least-restrained boundary conditions. In asymmetrical bites, the skull is subjected to torsional and bending loads that are qualitatively similar to the loads encountered in the twist feeding regime, and in combination with the qualitative difference between the strain and displacement results this may raise the possibility that the strain data under the least-restrained boundary conditions includes significant artifacts.

Sensitivity to Boundary Conditions

The issue of relevant boundary conditions is of great importance to the interpretation of the data. As detailed above, we sought to identify which set of the boundary conditions most closely replicates restraint of the rostrum by the postorbital skull by comparing the results from the A.f and A.r models. The boundary conditions where the results from these models were most similar should logically be the most realistic.

The consistency of the displacement data across the boundary conditions offered little insight into this issue; the difference between the A.f and A.r models was of similar magnitude between all boundary conditions (Fig. 7A). This suggests that, for these models at least, displacement data are not particularly sensitive to boundary conditions. Whether this holds true more generally remains to be seen.

The strain data, however, displayed greater sensitivity to boundary conditions and suggests that the performance of the A.f and A.r models were most similar under the intermediate set of boundary conditions, under which the aggregate difference in maximum strain between the two models was appreciably lower than the other sets of boundary conditions (Table 5). Interestingly, the qualitative patterns in the strain data under the intermediate boundary conditions were congruent with those in the displacement data, perhaps suggesting that this part of the data set may have been subject to fewer artifacts.

Hypothesis A: The Most Platyrostral Model Will Give the Best Mechanical Performance Under Loads Simulating Twist Feeding Behaviors

For the strain data under the intermediate set of boundary conditions, the most platyrostral model—the A.m model—ranked equal third (mean rank, 3.5) in the twist feeding loading regime. The other two platyrostral models—the Cr.j and C.c(j) models—ranked sixth and fifth, respectively. The strongest models were the M.n and P.p, respectively, with the other oreinirostral model—C.c(a)—taking equal third place. As such, these results do not offer support for the hypothesis that platyrostral skulls perform best in twist feeding.

The displacement data offer even less support. In the twist feeding regime, the three platyrostral models occupy fourth, fifth, and sixth place whether ranked according to absolute displacement (Table 7) or the regression of displacement divided by force and plotted against moment (Fig. 7B).

A weaker form of the hypothesis, that the more platyrostral models perform better in twist feeding than in other regimes, is not supported by the displacement data set. However, for the strain data, the A.m model ranked slightly higher in the twist feeding regime (mean rank of load cases 9 and 10 was 3.5, which was the equal third placed aggregate rank) than in the simple bite and head shake regimes. The two other platyrostral models, C.c(j) and Cr.j, did not show improved rankings in the twist feeding regime (Table 5).

The asymmetrical simple bites are mechanically similar to the twist feeding regime in that they impose torsional and dorsoventral bending moments on the skull. For the strain data, the A.m model ranked slightly lower in unilateral simple bites compared with bilateral simple bites (Table 5), a pattern that does not suggest this model performs well under torsional loads.

Even though the least-restrained set of boundary conditions did not appear to be the most realistic, they did produce the only result contrary to this general pattern, where the most platyrostral model exhibited the best mechanical performance under the twist feeding loading regime (although this result was not matched by its performance under unilateral loading regimes). Although we consider this result most likely to be an artifact of the mesh, until this has been demonstrated, we must accept it at face value. In summary, therefore, we find that the hypothesis was not supported by most of the data, although it was supported by the least-restrictive boundary conditions.

Hypothesis B: Qualitatively, the Mechanical Performance of the Six Finite-Element Models Should Match the Predictions of Their Performance Based on Simple Beam Theory Models

The results from the beam models evaluated using strain data show that there was very little difference in rankings between loading regimes/load cases. The average ranking of the models (from best to worst) was M.n, P.p, C.c(j), C.c(a), A.m, Cr.j, and in all but load case 5 (cheek bite), maximum strain in the Cr.j model was considerably greater than the other models (Table 5, Appendix B).

Accepting that the intermediate boundary conditions were the most relevant, these results can be compared with the results from the finite-element (FE) models. In the FE models, variability between load cases was higher in the finite-element than in the beam model results, but was nevertheless not high in absolute terms. This variability in rankings appeared to have a weak correlation with bite type (bilateral vs. unilateral) and bite position, rather than with loading regime. The ranking (from best to worst), based on mean rank, for the FE models was M.n (1.7), P.p (2.0), C.c(a) (3.2), C.c(j) (4.1), A.m (4.5), and Cr.j (4.9)—very close to the order of averaged rankings from the beam model results (Table 8). Strain in the Cr.j model was significantly higher than the other models for tip bites, but not in mid bites.

Qualitatively, the results based on the very simple beam theory models agreed well with the results from the FE models. This suggests that many basic features of the biomechanical performance of the crocodilian rostrum can be usefully studied using even basic beam theory models and brings to mind Thomason's (1991) summary of his study of carnivoran skull biomechanics: “I am drawn to the conclusion that modeling the skull of these long-faced mammals as a beam works, despite the number of assumptions and sources of error in the method; perhaps the errors cancel one another” (p. 2332).

Summary of Results

Overall, the data presented here do not support the hypothesis that platyrostral skulls are well suited to help resist the torsional loads encountered when a crocodilian twist feeds. Although a small part of our data set (which may reflect artifacts in the models) might suggest that the platyrostral A.m. model displayed greater resistance to torsion under certain sets of boundary conditions, we suggest that a reasonable reading of these results is as follows: comparison of the mechanical behavior of various crocodilian rostra, using finite-element and simple beam models, agrees substantially with predictions based on mechanical first principles and indicates that rostra with the largest second moment of area and conforming most closely to a tubular section will exhibit the greatest strength under torsional loads. However, further work regarding the possibly unique performance found in Alligator mississippiensis is justified to ensure more comprehensive understanding of this taxon.

Mechanical first principles also suggest that the more oreinirostral models will be the strongest under dorsoventral bending moments, and this is strongly supported by the data. Because the skulls of extant crocodilians are essentially all platyrostral (i.e., wider than tall), the most oreinirostral crocodilian models will also be those that conform most closely to a tubular cross-section. Thus, the overall mechanical performance of the oreinirostral models agrees with prediction based on mechanical first principles: the P.p, M.n, and C.c(a) models ranked highest in both simple bites and twist feeding bites.

The results from the beam models and the finite-element models used in this study were largely congruent. Both the beam and the finite-element models used were relatively simple and of low resolution, though likely of sufficient resolution to allow the inferences drawn here.

Biological Significance of Results

With the above provisos in mind, what are the implications of these results for our understanding of the biological consequences of rostral shape in crocodilians? As we have seen, in trying to explain the evolution of platyrostry in crocodilians, Busbey (1995) considered two possibilities: that platyrostry confers an increased resistance to torsional loading on the skull, and that platyrostry decreases the hydrodynamic resistance to lateral head movements during feeding. Our results suggest that the former argument is, in isolation, problematic. The models that best resisted torsional forces were the least platyrostral, the M.n and P.p models, based on Melanosuchus and Paleosuchus, respectively. But even Paleosuchus palpebrosus, considered (along with Osteolaemus) to have the tallest rostrum among living crocodilians, has a snout shape that is flatter than the rostrum in most varanid lizards, theropod dinosaurs, and carnivoran mammals.

Does the Crocodilian Skull Represent a Functional Compromise Between Hydrodynamics and Structural Mechanics?

Nearly all crocodilians typically strike aquatic prey items by means of a lateral sweeping motion of the head, either by ambush or through trapping of prey items (Pooley and Gans,1976; Diefenbach,1979; Schaller and Crawshaw,1982; Thorbjarnarson,1990,1993; personal observation, M.B.M.). The extensive cervical musculature is used to accelerate the feeding apparatus at a much greater rate than would a forward leap of the entire body, particularly given the drag that would be induced in an aquatic environment. Assuming that increased speed results in a greater number of successful strikes, then crocodilians may be expected to optimize rostral morphology for lateral movement of the head and rostrum.

Two options for increasing the speed of the attack are thus possible. Specifically, a longirostrine morphology can be adopted, the simple physics of which result in greater angular acceleration at the end of the snout and thus increased speed in the region where approximately 45% of prey are captured [Thorbjarnarson (1990), in reference to Gavialis gangeticus]. However, simple elongation of the rostrum would necessarily result in increased drag in isometric forms, thus necessitating the narrow rostra typical of longirostrine species (Gavialis gangeticus, Tomistoma schlegelii, Crocodylus johnstoni, and Crocodylus cataphractus). An alternative or complement to simple elongation of the rostrum is platyrostry, which may result in a remarkably low lateral profile as seen in Alligator mississippiensis. A low lateral profile offers less hydrodynamic resistance during lateral movement, allowing the head to be moved sideways faster for a given effort and to generate less drag while doing so. For a review of the theoretical issues involved with feeding in aquatic predators, see Taylor (1987).

Logically, these modifications of form (platyrostry and longirostry) must significantly increase the efficiency with which small agile prey such as fish can be captured, an important part of the diet in nearly all crocodilian taxa, even for mature adults of large species such as Crocodylus porosus and C. niloticus (Cott,1961; Allen,1974). Indeed, some evidence exists to this effect. Schaller and Crawshaw (1982) were the first to study strike behavior (their “fishing behavior”) in a crocodilian, Caiman crocodilus and found a 15.9% efficiency rate (catches/strikes) in somewhat unusual conditions. Later, Thorbjarnarson (1993) examined the same species, but subdivided the strike techniques into several subcategories, with an average efficiency of approximately 7.2% for typical stationary and cross-posture snaps. The only available comparative data comes from the largely piscivorous (Whitaker and Basu,1983) and decidedly longirostrine Gavialis gangeticus, which exhibited 34.4% efficiency in seminatural conditions during which only lateral sweeping strikes were observed (Thorbjarnarson,1990). Interestingly, G. gangeticus may also exhibit an elongated cervical region as an adaptation to this form of feeding (Whitaker and Basu,1983).

The relative importance of lateral acceleration of the rostrum and head is obviously correlated with the relative importance of swift aquatic prey in the diets of any given crocodilian species. The data regarding crocodilian food habits, however, is plagued with the same problems as most literature built on stomach contents (e.g., secondary ingestion, size bias in sampling). Nonetheless, it is possible to draw some conclusions from available data. For instance, Monteiro et al. (1997) suggest that the relatively short rostrum characteristic of Caiman latirostris evolved in response to the importance of slow prey (Ampularid gastropods) in the diet of that species (Diefenbach,1979), though the likely mechanism in this case was presumably lack of selection for the maintenance of the primitive relatively long rostrum for Caiman and Melanosuchus (Brochu,1999). Data regarding the diets of several taxa relevant to the present study are available—Alligator mississippiensis, Caiman crocodilus, Melanosuchus niger, Paleosuchus trigonatus (included here in absence of detailed data on P. palpebrosus), and Crocodylus johnstoni—and allow some speculation about the relative importance of the lateral sweep among these taxa. Fish, for example, make up a large proportion of the diet in Caiman crocodilus, but were not found in any of 25 specimens of Melanosuchus niger examined (Da Silveira and Magnusson,1999), with the later species relying principally on terrestrial invertebrates at small size classes. Similarly, fish constitute a large percentage of the diet of the American alligator at relatively small sizes, though at larger size classes, they become less significant. For example, in very large (3.0+ m) animals, mammals may constitute 81.4% of diet by mass, while fish only constitute 15.1% (Wolfe et al.,1987), consistent with Dodson's (1975) suggestion, based on growth studies of the skull in this species, that allometric changes in cranial structure have important effects on prey choice among crocodilians [this was also discussed by Hutton (1987) regarding Crocodylus niloticus]. Magnusson et al. (1987), however, suggest that diet among Amazonian crocodilians is a function of habitat selection, rather than a function of cranial mechanics. Still, they attribute the relatively large amount of terrestrial vertebrates eaten by Paleosuchus trigonatus to selection for terrestrial prey based on the fact that even small individuals show a bias toward terrestrial vertebrate prey, while other species only begin to acquire terrestrial vertebrates at larger body sizes. Consequently, the relatively tall rostrum in this taxon is consistent with an apparent dietary preference for terrestrial vertebrates. Overall, then, it is apparent that rostral structure of crocodilian crania may be significantly constrained by hydrodynamic factors that affect prey capture. Clearly, further work is needed on prey speeds and hydrodynamic properties of crocodilian crania and strike kinematics.

The factors involved can be visualized by plotting the total drag moment versus skull length (scaled to body size) for a range of crocodilian taxa. Here we assume the snout of the crocodile presents a rectangular profile to resist motion through the fluid during a lateral sweep of the head. Assuming a constant coefficient of drag along the whole length of the snout, this basic “bluff body” analysis shows that drag moment increases quadratically with rostral length (Fig. 8). Variation in rostral height between taxa is of secondary importance to rostral length in determining the total drag. This approach is crude and considers only rostral height and length—it makes no allowance for streamlined surfaces and is likely to overestimate actual drag in longirostrine forms—but it does show the potential costs of sweeping an elongated skull rapidly sideways through water. In particular, it illustrates the selection pressures operating on crocodilians to reduce drag.

thumbnail image

Figure 8. Drag moment induced by the rostra of various crocodilian specimens, scaled to biquadrate width (qq) to remove the effects of body size. Drag moment is a measure of the resistance encountered when rotating an object about a fulcrum through a fluid medium: in this case, when a crocodile sweeps its snout sideways through water by flexing the head about the cranial-cervical joint. Measurements were taken from photographs of museum specimens and include the finite-element models from this study. Drag moment was calculated using a “bluff body” drag calculation, which considers only the dorsoventral profile of the rostrum and models it as a flat vertical wall; it does not consider rostral width. The rostrum was divided into four sections: section 1, lying between the front of the orbits and the maxillary fang; section 2, between the maxillary fang and the maxilla-premaxilla suture on the jaw margin; section 3, between the maxillapremaxilla suture and the widest point of the external nares; and section 4, between the widest point of the external nares and the rostral tip. For each section, the drag (d) was calculated by the equation d = 1/2 ρA(rω)2CD, where ρ is the density of water (1,000 kg/m3), CD is the drag coefficient (= 1), r is the median radius (i.e., distance from cranial-cervical joint) for each section, ω is the angular velocity in radians/sec (set to n radians, or 180°, per second for all specimens equivalent to a 45° sweep in a quarter of a second), and A is the area of the profile (calculated as the average height of the section times the length of the section). Summing the product of drag and median radius for each section gave the total drag moment (DM = Σ dr). As can be seen, drag moment scaled quadratically with rostral length, and the variations in rostral height between specimens had a secondary effect. This approach is likely to overestimate the drag moment acting on actual skulls, but provides an indication of the potential drag cost of using an elongate rostrum to catch aquatic prey with a lateral sweep of the jaws.

Download figure to PowerPoint

Can a functional model that considers both hydrodynamic and structural mechanics offer any insight into the question of skull biomechanics in crocodiles? A comprehensive approach of this kind would require much better ecological data and dynamic/mechanical models than are currently available, but in the interests of stimulating discussion, we suggest that under such an approach living crocodilians may be classified into four ecomorphological groups as follows.

(A) species with elongated rostra (longirostrine), such as Gavialis gangeticus, Tomistoma schlegelii, Crocodylus cataphractus, and C. johnstoni, are understood to be specialist piscivores. In these animals, the increased length of the snout theoretically allows the tip of the jaws to be moved very rapidly through the water as a result of lateral flexion of the head, neck, and cranial portion of the body. Inevitably, for a given angle and rate of lateral flexion, the tips of the jaws in a longirostrine crocodile move faster and through a greater distance than do the tips of the jaws in a crocodile with a shorter snout. The consequent advantages for an animal hunting small agile prey would appear to be obvious. However, the hydrodynamic consequences of an elongated rostrum are also significant. For a given region of the snout, the greater its velocity through the water, the more drag it incurs; in addition, the greater the distance from the pivot (e.g., the cranial-cervical joint), the greater the resulting drag moment acting on the skull. This means that, as the snout is elongated, drag increases to the square of the linear increase in length (assuming isometric forms) and so energetic cost of sweeping the head rapidly through the water also increases with positive allometry. Longirostrine taxa counteract this by reducing the height of the rostrum relative to body size, which minimizes pressure drag. The smaller pressure wave preceding the snout margin during the lateral sweep may also reduce the extent to which the predator stimulates an escape response in the prey. However, even a snout with a low profile will cause significant drag if it is broad, and so longirostrine taxa also reduce snout width, which minimizes friction drag during sideways movement of the snout. The result is a structure that is mechanically vulnerable to loading due to both its length (which increases the moment of any forces acting on the rostrum) and its small sectional area (which reduces the capacity of the rostrum to resist bending and torsional forces). With some important exceptions, longirostrine crocodiles are therefore restricted to taking small prey.

(B) most living crocodilians are mesorostrine, that is, the snout is still long compared with most reptiles, but is shorter than in the longirostrine skull. The shorter snout has two important mechanical consequences: the loading moments acting on the skull during feeding are reduced (linearly), and the drag moment acting on the rostrum is reduced (quadratically). The reduction in rostral length therefore allows the mesorostrine crocodile to handle larger forces during feeding and thus target larger prey. In addition, the decrease in drag moment due to rostral length provides the opportunity to increase the sectional area of the rostrum without exceeding the level of drag moment that acts on longirostrine crocodiles. An increased sectional area will further increase the capacity of the rostrum to handle bending and torsional loads and thus significantly increase the maximum size of prey that can be targeted. However, pressure drag (controlled by rostral height during sideways movement of the snout) is more important than friction drag (controlled by rostral width during sideways movement). As long as fish are an important part of the diet for mesorostrine crocodilians and they are caught by a lateral sweep of the head, it follows that crocodilians can increase the sectional area of their rostrum and minimize the increase in drag costs by increasing rostral width rather than rostral height. The result is the characteristic platyrostral skull of most living crocodilians. This shape is still efficient at catching fish, although not as efficient as longirostrine crocodiles, but has the advantage of being able to handle larger prey. Mesorostrine platyrostral crocodilians are therefore less specialized on a diet of small fishes than are longirostrine taxa—they are generalist predators that can, by virtue of the large size of adults in some taxa (e.g., C. niloticus, C. porosus), take large terrestrial prey.

Within the living mesorostrine crocodilians, there is a spectrum of rostral shape, from narrower and taller to broader and flatter (although all are essentially platyrostral). The two ends of the spectrum may be characterized as follows. (1) relatively tall and narrow, e.g., Crocodylus rhombifer, C. porosus, C. niloticus, and Melanosuchus niger. Although by no means oreinirostral, this shape is better suited to resisting the dorsoventral bending and torsional forces incurred when feeding on large prey, but will induce greater drag during the lateral head sweep and may thus be less adept at catching agile aquatic prey. (2) flat and broad, e.g., Alligator mississippiensis and Crocodylus siamensis. The lower lateral profile of the skull induces less pressure drag during the lateral head sweep, making this shape better suited to catching agile aquatic prey. The reduction in rostral height reduces the extent to which this shape can resist dorsoventral bending and torsional forces, making it less suited for handling large prey, although the increased width of the rostrum partially counteracts this, and this shape remains better suited to handling medium-sized prey than are the longirostrine crocodilians.

(3) living species of crocodilian with short snouts (brevirostrine) are few, although many fossil crocodilians can be placed in this category. Among extant species, Paleosuchus palpebrosus and Osteolaemus tetraspis are the most obvious examples, and Paleosuchus trigonatus, Alligator sinensis, and Caiman latirostris can also be placed within this category. From the shortness of the snout, it may be postulated that small agile aquatic prey are of reduced importance in their diet, compared with the longer-snouted crocodilians, and qualitative descriptions of their prey base (Ross and Magnusson,1989) suggest that small aquatic invertebrates and terrestrial prey are particularly important components of diet in these taxa. In the case of P. palpebrosus in particular, we predict that it has been able to develop a relatively oreinirostral snout by substantially reducing the importance of agile aquatic prey in its diet, thereby reducing the degree to which the skull is constrained by the consequent hydrodynamics considerations. However, more detailed information on this species' ecology is required to fully test this suggestion.

Whether these biomechanical categories accurately explain the range of trophic ecology in living crocodilians is not presently known, but this question could be investigated by integrating fluid dynamics and finite-element models with quantitative data on diet (work underway by the authors). If such a model, based on hydrodynamics in combination with structural mechanics, proves useful in relating skull biomechanics with feeding ecology of living crocodilians, then it may be applied to other predatory aquatic tetrapods, including fossil forms.

Circumventing Mechanical Constraint of Platyrostry?

It is thus possible that the broad rostra of forms such as Alligator only helps in resisting torsional loads given the low profile of the snout enforced by hydrodynamic factors (Busbey,1995; Metzger et al.,2005). The mechanical consequences of platyrostry were emphasized by Busbey (1995): compared to an oreinirostral skull, a platyrostral skull will be exposed to higher bending loads as a result of jaw adduction. As discussed above, Busbey suggested that various features of the crocodilian rostrum may serve to reduce the resulting strain in the platyrostral skull: the extensive scarf joints, thickness of the bones, and the closed secondary palate. Busbey's interpretation of the mechanical consequences of platyrostry has been supported by Metzger and colleagues, who found that strain magnitudes measured in the rostra of juvenile Alligator mississipiensis were much greater than has been reported in vivo for various mammals (Ross and Metzger,2004; Metzger et al.,2005). Of the potential strain-reducing features listed by Busbey (1995), the large extent of scarf joints in crocodilians has been commented on by several authors (Jenkins et al.,2002; Metzger et al.,2005); being areas of extensive overlap between skull bones, scarf joints are thought to be important in providing resistance to shear and bending stress.

Preuschoft and Witzel (2002) contrasted the elongate platyrostral skull in crocodilians with the shorter, more oreinirostral skulls of other carnivorous reptiles such as dinosaurs and lizards. They noted that, unlike other archosaurs, crocodilians lack antorbital fenestrae and have adductor muscles that attach well in front of the posteriormost teeth in both the upper and lower jaws, suggesting that both features could be understood as strategies for reducing strain under dorsoventral bending (they also noted that the resultants of the muscles might also serve to reduce joint reaction forces). Each of these may be added to Busbey's list of features that may reduce strain in the high loads experienced by platyrostral skulls.

Iordansky (1973) noted that the cranial osteoderms, which adhere to the skull roof and produce the cranial sculpturing characteristic of adult mesorostrine crocodilians, are aligned with the direction of split lines and the trabeculae within the rostrum; each of these are thought to indicate the major load pathways within the skull, a conclusion supported by finite-element analysis (FEA) (Daniel and McHenry,2001). Oslov (1939) suggested that this cranial osteodermic relief increased mechanical strength in tetrapods generally, and Iordansky (1973) emphasized this explanation with respect to crocodilians, adding that the pronounced preorbital ridges in several meso- and brevirostrine taxa may also help increase the strength of the platyrostral skull.

Methodological Issues

FE has been employed to model biological structures for more than 2 decades (for summary, see Ross,2005), and recent studies have made use of high-resolution meshes (Rayfield et al.,2001; Dumont et al.,2005; Rayfield,2005; Ross et al.,2005; Strait et al.,2005) and the validation of FE models against in vivo/in vitro data (Metzger et al.,2005; Strait et al.,2005). In the context of our study, the work of Metzger et al. (2005) is of particular relevance; by using the Alligator skull mesh generated by Daniel and McHenry (2001) and comparing the FE model with in vivo strain data from two subadult Alligator mississippiensis and predictions based on simple beam models, the authors were able to evaluate the accuracy of beam and FE models in predicting in vivo patterns of strain during biting. They found no significant similarities between the in vivo results and the predictions based on FE or beam models. Noting that other studies had been able to demonstrate agreement between in vivo results and FE models of the primate Macaca fascicularis (Ross et al.,2005; Strait et al.,2005), Metzger et al. (2005) suggested that a number of factors may have reduced the accuracy of the Alligator FE model: mesh resolution, accuracy of material properties, the accuracy of jaw muscle forces, scaling issues, and noninclusion of sutural joints.

The accuracy of FE models is dependent on the accuracy of the material properties included in the model, and the importance of deriving this data from material that is anatomically and phylogenetically relevant to the structure being modeled has been demonstrated (Strait et al.,2005). Unfortunately, there is scant data for the material properties of reptilian bone, and the Alligator skull model used by Daniel and McHenry (2001) and Metzger et al. (2005) used properties measured from bovine femora. The well-known anisotropic properties of bone further complicate this point. The Alligator model assumed that the cranial bones were homogeneous, whereas the cranial skeleton includes bone of varying properties at different regions around the skull (Peterson and Dachow,2003) in addition to the obvious differences between cortical and cancellous bone tissue.

Similarly, the reconstruction of muscle forces acting on the skull in the FEA is also important for model accuracy. The Macaca model used by Ross et al. (2005) and Strait et al. (2005) used detailed data on muscle activity measured directly by EMG from several individual Macaca, which undoubtedly contributed to the accuracy of their model. The muscle forces used by Metzger et al. (2005) were also measured by EMG, but from a different species of alligatorid, Caiman crocodilus (Cleuren et al.,1995). If the point made by Strait et al. (2005) concerning the need for material properties to be based on the species under study also holds true for measurements of muscle forces, then this may be another reason for the lack of congruence between the FE model and the in vivo data reported by Metzger et al. (2005).

The Alligator FE model used by Metzger et al. (2005) was based on an individual that was significantly larger than the animals from which their in vivo data was measured. Since it is unlikely that rostral wall thickness scales isometrically with rostral length, the simple isometric scaling of the FE model to match the size of the experimental animals was identified by Metzger et al. (2005) as a potential source of inaccuracy for their model. Data on the scaling relationships of cranial bones in crocodilians are still lacking, a problem that affected our study also.

The contention that sutures are of significance to skull biomechanics is supported by a body of experimental work, and sutures appear to experience strain magnitudes that are an order of magnitude higher than those within individual bones (Herring and Teng,2000; Rafferty et al.,2003). Based on morphology, it has been suggested that rostral sutures are of particular importance to crocodilian skulls (Busbey,1995), and Metzger et al. (2005) highlighted the possibility that, as the bone strains they measured in two juvenile Alligator were higher than has been measured in other vertebrates, the sutural strains in crocodilian skulls may be higher still. So far, only Rayfield (2004,2005) has modeled sutures within a finite-element analysis, although it is likely that the inclusion of sutures within an FE model will have important effects on the mechanical behavior of the model.

Our primary interest is the ecomorphology and evolutionary biology of aquatic carnivores, and comparative techniques have always been, and are likely to remain, important tools in investigating questions of this sort. Previously, only two analyses have examined FE models of more than one species (Rayfield,2004; Dumont et al.,2005), while the present study compared six models from five species. In all cases, data on the modeling parameters—material properties, muscle forces, and in vivo strain—were not available for the majority of species modeled. As discussed above, analyses of high-resolution models are sensitive to any lack of data of these factors, but the measurement of material properties and muscle forces requires a large amount of experimental work (and is in any case impossible for extinct taxa). It is conceivable that we will soon reach the point where our ability to collect such data will always lag behind our ability to construct complex meshes. However, qualitative analyses based on low-resolution meshes are far less vulnerable to such uncertainties and remain useful for testing many functional/evolutionary hypotheses, a point emphasized by numerous authors (Fastnacht et al.,2002; Jenkins et al.,2002; Snively and Russell,2002; Rayfield,2004; Richmond et al.,2005; Strait et al.,2005).

In this context, the primary difference between the use of the Alligator mesh by Metzger et al. (2005) and the present study is that Metzger and colleagues used it quantitatively, while it was here used qualitatively. The fact the mesh is not of sufficient quality to predict in vivo strain data accurately does not preclude its value in examining more general functional hypotheses. The functional hypothesis examined in the present study concerns the relationship of gross rostral shape with mechanical strength. By scaling all models to the same size, and ignoring the effects of muscle forces, material anisotropy/heterogeneity, and suture morphology, we are in effect simply assuming that all these factors affect the taxa in our study equally, leaving gross shape as the only difference between the models. Given the known morphological variation among living crocodilians, particularly within the alligatorids that formed the main focus of our study, these do not appear to be unreasonable assumptions, although they remain untested. By making these assumptions, we can explore general relationships between rostral shape and biomechanical function. Future work, using more detailed models, will be able to explore whether variation in these other modeling parameters ameliorates or exacerbates the basic patterns so identified. As such, our approach is typical of many attempts to examine evolutionary patterns from a biomechanical perspective using comparative techniques (Thomason,1991; Busbey,1995; Plotnick and Baumiller,2000; Preuschoft and Witzel,2002; Rayfield,2004; Wroe et al.,2005). However, low-resolution meshes are vulnerable to strain artifacts, and as our own results in this study illustrate, care must be taken to evaluate the results in light of this possibility.

Because comparative biologists are unlikely to restrict their interest to those animals for which all modeling parameters have been experimentally measured, low-resolution models are likely to be important to comparative studies using FEA for the foreseeable future. It is also likely that beam models will remain an important part of any comparative biomechanical analysis (Thomason,1991; Snively et al.,2006). While FE models can be used to describe the pattern of mechanical performance in organic structures, the very complexity that allows them to perform this role largely prevents any insight into underlying mechanical processes. Thus, complex models lend themselves to descriptive interpretation, but can be difficult to interpret mechanistically. With our study, the variation in mechanical performance between models seemed to correlate with the degree of platyrostry, but this result on its own does not indicate which aspects of rostral morphology are of primary importance in explaining the variation. Comparison with beam models suggests that the section of the posterior part of the rostrum is important (although the lack of alternative beam models in our study precludes definitive conclusions at this stage).

Similarly, in an instance where several mechanical analogies can be made of the rostrum, the FE models do not in themselves indicate which may be a more accurate representation of the biomechanics. Comparing biological structures with mechanical analogues is an essential part of the biomechanical approach, allowing the mechanical behavior of biological form to be investigated using well-understood mechanical principles (Kolston,2000; Plotnick and Baumiller,2000; Hildebrand and Goslow,2001; Vogel,2003). The crocodilian rostrum can be represented by various beam models (Busbey,1995), for instance, as a simple conical beam with an ellipsoid section (Fig. 9A), or instead as a compound structure composed of subparallel beam elements (the tooth rows), which are braced by dorsal (skull roof) and ventral (closed palate) plates (Fig. 9B). Which analogy is a more accurate representation can be evaluated by comparing beam models of each with FE models. The use of the beam models to identify biomechanically informative components of skull shape can then provide a foundation for morphometric analyses, which can in turn explore the broad patterns of the morphology within and between large taxonomic/ecomorphological groupings.

thumbnail image

Figure 9. Two possible beam models for the crocodilian rostrum. A: Simple model with the rostral section idealized as an ellipse. B: Section with the tooth rows modeled as beam elements connected by dorsal (skull roof) and ventral (closed palate) plates.

Download figure to PowerPoint

CONCLUSIONS

  1. Top of page
  2. Abstract
  3. Previous Work on Biomechanics of Crocodile Skulls
  4. Aims of Study
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSIONS
  9. Acknowledgements
  10. LITERATURE CITED
  11. Appendices: Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D

Busbey (1995) discussed the mechanical consequences of platyrostry in crocodilians with respect to resisting the torsional loads induced by twist feeding behaviors. The results from this study fail to support the hypothesis that platyrostry evolved in crocodilians in response to those torsional loadings. If this conclusion is validated by further studies, it leaves the alternative hypothesis identified by Busbey—that platyrostry in crocodilians evolved in response to the hydrodynamics acting on the rostrum during feeding on agile aquatic prey—as the more plausible explanation. The crocodilian rostrum has been popularly viewed as a structure that is mechanically optimized against the stresses resulting from the behaviors that crocodiles use to subdue and process large prey. The truth may be instead that the crocodilian rostrum is structurally constrained by the technique crocodiles use to acquire their smallest food items.

Acknowledgements

  1. Top of page
  2. Abstract
  3. Previous Work on Biomechanics of Crocodile Skulls
  4. Aims of Study
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSIONS
  9. Acknowledgements
  10. LITERATURE CITED
  11. Appendices: Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D

The authors thank Tim Rowe and staff at DigiMorph for use of CT scan data; Callum Ross for organizing the FEA workshop at the seventh International Congress of Vertebrate Morphology (ICVM); Betsy Dumont, Ian Grosse, Ian Jenkins, Keith Metzger, Karen Moreno, Holger Preuschoft, Emily Rayfield, Eric Snively, David Strait, and Ulrich Witzel for enlightenment and discussion on various aspects of biomechanics and FEA. Specimens were examined during visits to University of California Museum of Paleontology, the Natural History Museum (London), the American Museum of Natural History, and the U.S. National Museum by C.R.M., which were funded by the Lerner-Gray Grants for Marine Research at the American Museum of Natural History, the Australian Vice Chancellors Committee, and the Australian Geographic Society. The authors thank the staff of these organizations. They are especially grateful to Tim Hamley, Tony Tucker, C. Andrew Ross, Art Busbey, Peter Brazaitis, and Ewan Fordyce for discussion and encouragement, and Sarah Johnston, Inez Johnston, and Steve Wroe for logistical support. Comments from Callum Ross, Art Busbey, and several anonymous reviewers greatly improved earlier versions of this manuscript. Funds from the Anatomical Record and the ICVM allowed C.R.M. to present this research at ICVM7. Further assistance was received from ARC Discovery Project DP00666374 (to S. Wroe). Finally, the authors wish to express their thanks to the Inter-Library Loans Services at their respective institutions.

LITERATURE CITED

  1. Top of page
  2. Abstract
  3. Previous Work on Biomechanics of Crocodile Skulls
  4. Aims of Study
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSIONS
  9. Acknowledgements
  10. LITERATURE CITED
  11. Appendices: Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
  • Allen GR. 1974. The marine crocodile, Crocodylus porosus from Ponape, Eastern Caroline Islands with notes on food habits of crocodiles from the Palau Archipelago. Copeia 2: 553.
  • Aoki R. 1977. The globular teeth of the dwarf crocodile. Nippon Herpetol J 8: 1518.
  • Aoki R. 1989. The jaw mechanics in the heterodont crocodilians. In: MatsuiM, HikidaT, GorisRC, editors. Current herpetology in East Asia. Kyoto: Second Japan-China Herpetological Symposium. p 1721.
  • Bolt JR. 1974. Evolution and functional interpretation of some suture patterns in Paleozoic labyrinthodont amphibians and other lower tetrapods. J Paleontol 48: 434458.
  • Brochu CA. 1999. Phylogeny, systematics, and historical biogeography of Alligatoroidea. Soc Vertebr Paleontol Memoir 6: 9100.
  • Busbey AB. 1989. Form and function of the feeding apparatus of Alligator mississippiensis. J Morphol 202: 99127.
  • Busbey AB. 1995. The structural consequences of skull flattening in crocodilians. In: ThomasonJJ, editor. Functional morphology in vertebrate paleontology. Cambridge: Cambridge University Press. p 173192.
  • Cleuren J, Aerts P, De Vree F. 1995. Bite force analysis in Caiman crocodilus. Belg J Zool 125: 7994.
  • Cott HB. 1961. Scientific results of an inquiry into the ecology and economic status of the Nile crocodile (Crocodylus niloticus) in Uganda and Northern Rhodesia. Trans Zool Soc Lond 29: 211357.
  • Daniel WJT, McHenry C. 2001. Bite force to skull stress correlation: modelling the skull of Alligator mississippiensis. In: GriggGC, SeebacherF, FranklinCE, editors. Crocodilian biology and evolution. Chipping Norton: Surrey Beatty. p 135143.
  • Da Silveira R, Magnusson WE. 1999. Diets of spectacled and black caiman in the Anavilhanas Archipelago, Central Amazonia, Brazil. J Herpetol 33: 181192.
  • Diefenbach COC. 1979. Ampullarid gastropod: staple food of Caiman latirostris? Copeia 162163.
  • Dodson P. 1975. Functional and ecological significance of relative growth in Alligator. J Zool Lond 175: 315355.
  • Dumont ER, Piccirillo J, Grosse IR. 2005. Finite-element analysis of biting behavior and bone stress in the facial skeleton of bats. Anat Rec 283A: 319330.
  • Erickson GM. Lappin AK, Vliet KA. 2003. The ontogeny of bite-force performance in American alligator (Alligator mississippiensis). J Zool Lond 260: 317327.
  • Fastnacht M, Hess N, Frey E, Weiser H. 2002. Finite element analysis in vertebrate paleontology. Senckenbergiana Lethaea 82: 195206.
  • Herring SW, Yeng S. 2000. Strain in the braincase and its sutures during function. Am J Phys Anthropol 112: 575593.
  • Hildebrand M, Goslow G. 2001. Analysis of vertebrate structure, 5th ed. New York: John Wiley and Sons.
  • Hua S, Vignaud P, Pennetier E, Pennetier G. 1994. A skeleton of Steneosaurus obtusidens Andrews 1909 from the Upper Callovian of Villers-sur-Mer (Calvados, France) and the problem of definition of the Teleosauridae possessing blunt teeth. Comp Rend Acad Sci Ser II Sci Terre Planetes 318: 15571562.
  • Hutton JM. 1987. Growth and feeding ecology of the Nile crocodile Crocodylus niloticus at Ngezi, Zimbabwe. J Animal Ecol 56: 2538.
  • Iordansky NN. 1973. The skull of the Crocodilia. In: GansC, editor. Biology of the reptilia, vol. 4. New York: Academic Press. p 201262.
  • Jenkins I, Thomason JJ, Norman DB. 2002. Primates and engineering principles: applications to craniodental mechanisms in ancient terrestrial predators. Senckenbergiana Lethaea 82: 223240.
  • Kolston PJ. 2000. Finite-element modelling: a new tool for the biologist. Phil Trans R Soc Lond A 358: 611631.
  • Langston W Jr. 1973. The crocodilian skull in historical perspective. In: GansC, editor. The biology of the reptilia, pt. 4D. New York: Academic Press. p 263289.
  • Magnusson WE, Da Silva EV, Lima AP. 1987. Diets of Amazonian crocodilians. J Herpetol 21: 8595.
  • Massare JA. 1987. Tooth morphology and prey preference of Mesozoic marine reptiles. J Vert Paleont 7: 121137.
  • Meers MB. 2002. Maximum bite force and prey size of Tyrannosaurus rex and their relationships to the inference of feeding behavior. Hist Biol 16: 112.
  • Metzger KA, Daniel WJT, Ross CF. 2005. Comparison of beam theory and finite element analysis with in vivo bone strain data from the alligator cranium. Anat Rec 283A: 331348.
  • Monteiro LR, Cavalcanti MJ, Sommer HJS III. 1997. Comparative ontogenetic shape changes in the skull of Caiman species (Crocodylia, Alligatoridae). J Morphol 231: 4362.
  • Mook CC. 1921. Skull characteristics of recent crocodilia, with notes on the affinities of the recent genera Cont. No. 10. Bull Am Mus Nat Hist 44: 126268.
  • Orlov JA. 1939. On mechanics of skull in lower Tetrapoda. Dokl Akad Nauk USSR 25: 452459.
  • Peterson J, Dachow PC. 2003. Material properties of the human cranial vault and zygoma. Anat Rec 274A: 785797.
  • Plotnick RE, Baumiller TK. 2000. Invention by evolution: functional analysis in paleobiology. Paleobiology 26: 305323.
  • Pooley AC, Gans C. 1976. The Nile crocodile. Sci Am 234: 114124.
  • Preushscoft H, Witzel U. 2002. Biomechanical investigations on the skulls of reptiles and mammals. Senckenbergiana Lethaea 82: 207222.
  • Rafferty K, Herring SW, Marshall CD. 2003. Biomechanics of the rostrum and the role of facial sutures. J Morphol 257: 3344.
  • Rayfield EJ, Norman DB, Horner CC, Horner JR, May Smith P, Thomason JJ, Upchurch P. 2001. Cranial form and function in a larger theropod dinosaur. Nature 409: 10331037.
  • Rayfield EJ. 2004. Aspects of comparative cranial mechanics in the theropod dinosaurs Coelophysis, Allosaurus and Tyrannosaurus. Zool J Linn Soc 144: 309316.
  • Rayfield EJ. 2005. Using finite element analysis to investigate suture morphology: a case study using large carnivorous dinosaurs. Anat Rec 283A: 349365.
  • Richmond BG, Wright BW, Grosse I, Dechow P, Ross CF, Spencer MA, Strait DS. 2005. Finite element analysis in functional morphology. Anat Rec 283A: 259274.
  • Ross CA, Magnusson WE. 1989. Living crocodilians. In: RossCA, GarnettS, editors. Crocodiles and alligators. New York: Facts on File. p 5873.
  • Ross CF, Metzger KA. 2004. Bone strain gradients and optimization in vertebrate skulls. Ann Anat 186: 387396.
  • Ross CF. 2005. Finite element analysis in vertebrate biomechanics. Anat Rec 283A: 253258.
  • Ross CF, Patel BA, Slice DE, Strait DS, Dechow PC, Richmond BG, Spencer MA. 2005. Modeling masticatory muscle force in finite element analysis: sensitivity analysis using principal coordinates analysis. Anat Rec 283A: 288299.
  • Schaller GB, Crawshaw PG Jr. 1982. Fishing behavior of Paraguayan Caiman (Caiman crocodilus). Copeia 1982: 6672.
  • Sereno PC, et al. 1998. A long-snouted predatory dinosaur from Africa and the evolution of spinosaurids. Science 282: 2981302.
  • Snively E, Russell A. 2002. The tyrannosaurid metatarsus: bone strain and inferred ligament function. Senckenbergiana Lethaea 82: 3542.
  • Snively E, Henderson DM, Phillips DS. 2006. Fused and vaulted nasals of tyrannosaurid dinosaurs: implications for cranial strength and feeding mechanics. Acta Palaeontol Polonica (in press).
  • Strait DS, Wang Q, Dechow PC, Ross CF, Richmond BG, Spencer MA, Patel BA. 2005. Modeling elastic properties in finite-element analysis: how much precision is needed to produce an accurate model? Anat Rec 283A: 275287.
  • Taylor MA. 1987. How tetrapods feed in water; a functional analysis by paradigm. Zool J Linn Soc 91: 171195.
  • Taylor MA, Cruickshank ARI. 1993. Cranial anatomy and functional morphology of Pliosaurus brachyspondylus (Reptilia: Plesiosauria) from the Upper Jurassic of Westbury, Wiltshire. Phil Trans R Soc Lond B 341: 399418.
  • Thomason JJ. 1991. Cranial strength in relation to estimated biting forces in some mammals. Can J Zool 69: 23262333.
  • Thorbjarnarson JB. 1990. Notes on the feeding behavior of the Gharial (Gavialis gangeticus) under semi-natural conditions. J Herpetol 24: 99100.
  • Thorbjarnarson JB. 1993. Fishing behavior of spectacled caiman in the Venezuelan Llanos. Copeia 1993: 11661171.
  • Van Drongelen W, Dullemeijer P. 1982. The feeding apparatus of Caiman crocodilus; a functional-morphological study. Anat Anz 151: 337366.
  • Vogel S. 2003. Comparative biomechanics: life's physical world. Princeton, NJ: Princeton University Press.
  • Whitaker R, Basu D. 1983. The gharial (Gavialis gangeticus): a review. J Bombay Nat Hist Soc 79: 531547.
  • Wolfe JL, Bradshaw DK, Chabreck RH. 1987. Alligator feeding habits: New data and a review. Northeast Gulf Sci 9: 18.
  • Wroe S, McHenry CR, Thomason JJ. 2005. Bite Club: comparative bite force in big biting mammals and the prediction of predatory behavior in fossil taxa. Proc R Soc Lond Ser B 272: 619625.

Appendices: Appendix A

  1. Top of page
  2. Abstract
  3. Previous Work on Biomechanics of Crocodile Skulls
  4. Aims of Study
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSIONS
  9. Acknowledgements
  10. LITERATURE CITED
  11. Appendices: Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
Table  . Morphometric data
  fOrb (1)mx5t (2)mx-pmx (3)nos (4)av(fOrb+mx5t)2nd moment area (x104 mm4) of beam models
  1. All measurements scaled to width at the back of the rostrum. fOrb (1): Section 1, taken at the font of the orbits. mx5t (2): Section 2, taken at the maxillary fang (i.e. the largest maxillary alveolus). mx-pmx (3): Section 3, taken at the maxilla-premaxilla suture on the jaw margin. nos (4): Section 4, taken at the widest extent of the external nares (which roughly corresponds to the 3rd premaxillary alveolus). av(fOrb+mx5t): Mean of dimensions at sections 1 and 2, used to establish the width and height of the beam models.

height (z axis)A.m0.200.210.210.210.211.86
 C.c(j)0.360.210.130.170.283.74
 C.c(a)0.400.260.170.280.335.22
 M.n0.360.310.210.240.335.63
 P.p0.480.320.190.240.406.89
 Cr.j0.360.160.140.160.262.11
width (y axis)A.m0.930.820.570.410.8822.13
 C.c(j)0.910.780.590.460.8421.81
 C.c(a)0.890.660.490.520.7717.34
 M.n0.940.790.590.450.8725.84
 P.p0.840.700.550.410.7719.37
 Cr.j0.840.530.300.340.6910.00
length (x axisA.m0.210.781.061.10  
 C.c(j)0.250.781.031.20  
 C.c(a)0.190.861.131.31  
 M.n0.220.741.011.16  
 P.p0.240.941.061.20  
 Cr.j0.271.171.832.08  

Appendix B

  1. Top of page
  2. Abstract
  3. Previous Work on Biomechanics of Crocodile Skulls
  4. Aims of Study
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSIONS
  9. Acknowledgements
  10. LITERATURE CITED
  11. Appendices: Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
Table  . Raw data for beam models : B.1: Strain data (in microstrain).
 A.mC.c(j)C.c(a)M.nP.pCr.j
mid388269316199246650
mid7765376333984931299
tip10948269646236292310
tip5474134823123151155
cheek10486705963150
tip116192710987157622639
mid412301361228298742
mid8246027214565961484
tip1641123914459359443464
mid11638069495977391949
B.2: Displacement data (in mm).
 A.mC.c(j)C.c(a)M.nP.pCr.j
mid0.1180.0660.0620.0380.0490.469
mid0.2240.1310.1240.0760.0980.939
tip0.4090.2640.2480.1590.1432.434
tip0.2040.1320.1230.0790.0721.217
cheek0.0110.0080.0040.0040.0040.029
tip0.4100.2680.2570.1620.1522.487
mid0.1180.0670.0650.0390.0520.480
mid0.2360.1330.1300.0780.1040.959
tip0.6130.3970.3700.2380.2153.651
mid0.3530.1970.1870.1140.1461.408
6y0.0340.0450.0740.0350.0510.512
7y0.0100.0110.0190.0080.0170.099
8y0.0200.0230.0370.0170.0350.197

Appendix C

  1. Top of page
  2. Abstract
  3. Previous Work on Biomechanics of Crocodile Skulls
  4. Aims of Study
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSIONS
  9. Acknowledgements
  10. LITERATURE CITED
  11. Appendices: Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
Table  . Raw strain data for F.E. Models (values in microstrain)
lcFbpbtA.fA.rA.mC.c(j)C.c(a)M.nP.pCr.j
least restrained
1100midasymm940830630745510560670390
2200midsymm1000785870112011709201060750
3200tipsymm13001215147017901910145015102000
4100tipasymm85088090098010056608301005
5100cheekasymm450140350225250320300200
6200tipsymm11501435150016202055145515702120
7100midasymm800850730580540470580410
8200midsymm9008158501225995735850803
9300tipasymm19002195211027902920225023352850
10300midasymm15001430143017051550148016501125
intermediate
1100midasymm940850750600550450400750
2200midsymm1000100010509409607506501300
3200tipsymm13001200150015901620130012002000
4100tipasymm8508209208507505006501000
5100cheekasymm450200300200150150120200
6200tipsymm11501430141517001720120012002150
7100midasymm800880800600500600640450
8200midsymm9009009001080720800850830
9300tipasymm19001960190024502350160018002800
10300midasymm15001560140014001150104010701800
fully restrained
1100midasymm940725755500510430400390
2200midsymm1000765990730700730660680
3200tipsymm13001200141012301270130011701900
4100tipasymm85010209007307505605001020
5100cheekasymm450200300300180165106140
6200tipsymm1150145015301310147010708502120
7100midasymm800810750520400365350410
8200midsymm900800990690610640590750
9300tipasymm19002200223019701970154014602855
10300midasymm1500140515001200980115010601035

Appendix D

  1. Top of page
  2. Abstract
  3. Previous Work on Biomechanics of Crocodile Skulls
  4. Aims of Study
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSIONS
  9. Acknowledgements
  10. LITERATURE CITED
  11. Appendices: Appendix A
  12. Appendix B
  13. Appendix C
  14. Appendix D
Table  . Displacement data for F.E. Models (values in mm)
lcFbpbtA.fA.rA.mC.c(j)C.c(a)M.nP.pCr.j
least restrainedD(XYZ)
1100midasymm0.2800.1680.3740.1820.2440.1710.1310.580
2200midsymm0.4760.2530.5600.3330.4640.3250.2531.160
3200tipsymm0.9010.6100.9370.7490.8010.5190.3923.216
4100tipasymm0.3550.3780.5230.3640.4120.2620.1991.715
5100cheekasymm0.0640.0120.1230.0480.0480.0490.0443.520
6200tipsymm0.6300.6150.9500.7520.8060.5230.3953.257
7100midasymm0.3100.1930.4630.1770.2320.1860.1420.585
8200midsymm0.4980.2150.6270.2990.4250.3400.2621.174
9300tipasymm1.3780.9381.4201.1231.2020.7790.5874.824
10300midasymm0.7320.3190.8830.4300.6110.4950.3801.739
D(Y)
6200tipsymm0.1840.0970.1360.0980.1080.0880.0790.546
7100midasymm0.1010.0690.2260.0860.0970.0900.0660.081
8200midsymm0.1530.0740.1930.0920.0950.0860.0840.188
intermediate - D(XYZ)
1100midasymm0.2800.1710.3420.1480.1510.0900.0600.519
2200midsymm0.4760.2500.5470.2740.2850.1700.1091.039
3200tipsymm0.9010.6070.9180.6750.5650.3120.2073.036
4100tipasymm0.3550.3690.4990.3260.2920.1590.1061.623
5100cheekasymm0.0640.0130.0830.0200.0120.0140.0110.020
6200tipsymm0.6300.6110.9230.6780.5690.3200.2153.062
7100midasymm0.3100.1950.3980.1390.1440.1110.0860.522
8200midsymm0.4980.2210.5870.2410.2580.1830.1231.046
9300tipasymm1.3780.9361.3851.0120.8480.4680.3104.553
10300midasymm0.7320.3200.8460.3420.3590.2560.1641.558
D(Y)
6200tipsymm0.1840.0920.1140.0880.0850.0730.0590.443
7100midasymm0.1010.0660.1560.0630.0650.0560.0540.059
8200midsymm0.1530.0720.1380.0810.0840.0700.0690.132
fully restrainedD(XYZ)
1100midasymm0.2800.1590.2760.1360.1420.0850.0570.473
2200midsymm0.4760.2420.4880.2530.2710.1620.1040.945
3200tipsymm0.9010.5960.8360.6400.5420.2980.1992.882
4100tipasymm0.3550.3600.4360.3080.2790.1510.1021.544
5100cheekasymm0.0640.0110.0480.0160.0100.0120.0090.014
6200tipsymm0.6300.5990.8390.6430.5450.3030.2062.910
7100midasymm0.3100.1730.2930.1210.1310.1010.0790.476
8200midsymm0.4980.2030.4980.2170.2410.1710.1150.954
9300tipasymm1.3780.9181.2530.9600.8130.4470.2994.324
10300midasymm0.7320.3080.7440.3120.3380.2430.1561.418
D(Y)
6200tipsymm0.1840.0780.1070.0660.0790.0560.0490.407
7100midasymm0.1010.0490.0640.0380.0530.0170.0430.061
8200midsymm0.1530.0550.0820.0500.0640.0500.0530.131