Figure 1.
Spectrum of rostral proportions in marine tetrapods.
Dorsal view of various skulls, showing the spectrum of rostral proportions in (from top) crocodilians, odontocetes, plesiosaurs, ichthyosaurs and thalattosuchians. Skulls are resized to equivalent width at the back of the skull and for each group longirostrine taxa are on the right, brevirostrine on the left. Taxa shown are Caiman latirostris (A), Gavialis gangeticus (B), Feresa attenuata (C), Platanista gangetica (D), Leptocleidus capensis (E), Dolichorhynchops osborni (F), Temnodontosaurus eurycephalus (G), Ophthalmosaurus icenicus (H), Suchodus brachyrhynchus (I), Steneosaurus gracilirostris (J). Scale bars = 10 cm. Based on photos by CRM of specimen BMNH 86.10.4.2 (A), BMNH 1935.6.4.1 (B), BMNH 1897.6.30.1 (C) and USNM 504917 (D), after Cruichshank [60] (E), after O’Keefe [61] (F) based on fossil specimen BMNH R1157 illustrated by Owen [62] (G), after Motani [63] (H), after Andrews [64](I), after Mueller-Töwe [65](J).
Figure 2.
Range of skull shape in crocodilians.
Specimens are scaled to approximately the same width and arranged from most longirostrine to most brevirostrine. Left: cranium and mandible in lateral view, Centre left: dorsal view of mandible, Centre right: Cranium in ventral view, Right: species name and specimen number.
Figure 3.
Mandibular symphysis length vs mandible length in extant crocodilians.
X axis plots the ratio of mandibular length to width, giving a size-controlled proxy for the spectrum of brevisrostral to longirostral morphology. Y axis is the proportion of symphyseal length to mandibular length. Values shown are natural logarithms. (A), data for 82 specimens of crocodilian, data measured from photographs of museum skulls; regression line is based upon mean values for each species. (B), data points as for (A), with data points ordered by width in each species and connected by lines. In effect, this plot shows the allometric trajectory of ML/W for each species, with the smallest animals on the right and largest on the left of each species plot; i.e. as animals increase in size, head width increases as a proportion of head length. Within each species, the symphyseal length (as a proportion of mandible length) remains consistent. (C), Regression lines for alligatorids, non-tomistomine crocodylids, Gavialis, and Tomistoma.
Figure 4.
Second moments of area for beam models.
Second moments of area correspond to the geometry of long and short symphysis crocodilians. (A) shows the beam approximation of mandibles with long and short symphyseal lengths. (B) shows the change in second moment of area (length4) for long and short symphyseal beam models; these were calculated at discrete locations from the tip (anterior) of each mandible, as a conceptual illustration of the differences in second moments of area between the two morphologies. Corresponding locations are shown with dotted lines and the Y axis is a uniform arbitrary scale throughout. (C) shows (from top) the loading regimes associated with shaking, biting and twisting; where red arrows represent forces and black crosses represent restraints.
Figure 5.
From top left: Crocodylus intermedius, Tomistoma schlegelii, Mecistops cataphractus, Crocodylus moreletii, Crocodylus novaeguineae, Crocodylus johnstoni, Osteolaemus tetraspis.
Table 1.
Specimen scan information.
Figure 6.
Manual correction of diffraction artefacts in Crocodylus intermedius scan.
Left: scan data before correction. Right: scan data after correction. See text for explanation.
Figure 7.
Quality of isosurface models and error quantification.
The mask (shown in blue) represents the segmented/selected voxels that will be used to create isosurfaces. The three different contour qualities represent the 3D approximation of the mask and will form the isosurface. Contour error is the measured distance between the isosurface contour and the mask it was generated from (lower left of image).
Table 2.
Calculation and standardisation of error in the 3D models.
Figure 8.
Mesh optimisation and solid mesh generation.
Mesh optimisation and solid mesh generation was performed using Harpoon (SHARC). The left images show the complex internal geometry captured from isosurface generation. The middle column shows removal of complex internal geometry whilst still retaining important geometrical features. Images at right show the final solid mesh.
Table 3.
Mesh resolution for ‘complex’ FE models.
Figure 9.
Linear measurements and landmarks for mandible.
(A), linear measurements of mandible; (B), landmark locations. See text for explanation.
Table 4.
Landmark characterisation.
Figure 10.
Variations for beam models #1.
Model variations used to explore relationship between strain and linear variables in the first set of beam models. Abbreviations are defined as follows: (CL, CSL; VA, VW) – Constant length and symphyseal length, variable angle and width. (CL, CW; VSL, VA) – Constant length and width, variable symphyseal length and angle. (CA, CW; VSL, VL) – Constant angle and width, variable symphyseal length and length. (CSL, CW; VL, VA) – Constant symphyseal length and width, variable length and angle.
Figure 11.
Beam models showing axes, restraints and loads.
From top; shows loads and restraints for biting, shaking and twisting respectively. In all three cases models are fully restrained (rotation and translation) at the most posterior points of the beam model. Loads are all placed at the most anterior point of the beam model.
Table 5.
Dimensions for beam models #1.
Table 6.
Dimensions for beam models #2.
Figure 12.
Reptile version of ‘dry skull method’ in a crocodile skull.
Skull of Mecistops cataphractus, showing: (A), temporal (red) and pterygoid (yellow) muscle vectors; temporal vector is oriented vertically with the skull aligned horizontally, pterygoid vector runs between a point that is half of the cranial height at the postorbital bar, to the ventral surface of the mandible directly below the jaw joint. (B), calculation of the cross sectional area (CSA) for the temporal muscles; the outline maps the extent of the adductor chamber defined from osteological boundaries, viewed normal to the relevant vector. (C), calculation of CSA for pterygoid muscles; the outline is drawn normal to the vector. Outlines in B and C also show centroids, used for calculation of inlevers (see Thomason [15], McHenry [29]).
Table 7.
Jaw muscle groups in crocs.
Table 8.
Beam pretensions used for functional muscle groups.
Figure 13.
Bite points for bite, shake and twist.
Teeth used in simulating front, mid and back bite points are shown in orange. Crocodylus intermedius (A), Osteolaemus tetraspis (B), Crocodylus novaeguineae (C), Crocodylus moreletii (D), Crocodylus johnstoni (E), Mecistops cataphractus (F), Tomistoma schlegelii (G).
Table 9.
Material properties for elements used in each model.
Figure 14.
The problem definition used to determine the equations of motion that describe the feeding behaviour associated with shaking a prey item. This motion is considered to be harmonic; since the skull oscillates about a neutral axis in a set period of time (); in our case this period is 0.25 seconds – i.e at a frequency (
) of 4 full cycles per second. Left: the equations of motion associated with shaking, where
is angular displacement,
is angular velocity and is angular acceleration. Maximum angular acceleration (
) occurs each time the skull changes direction; in our case (radians/sec2), where a positive value indicates counter clockwise acceleration and a negative value indicates a clockwise acceleration. Right: the range of motion for a crocodile shaking a prey item. Bottom right: shows the equations used to calculate the maximum force (
) exerted on the skull as a result of shaking a prey item of mass (
) – approximately 2.55 kg in the M. cataphractus example shown here. Here
denotes linear acceleration (in the direction of force
) and
denotes the distance to the centre of rotation. For our calculations
is calculated as the perpendicular distance from the jaw hinge axis to the centre of mass of the prey item (outlever length) – approx. 297 mm in M. cataphractus.
Figure 15.
The problem definition used to determine the equations of motion that describe the feeding behaviour associated with twisting a prey item. Bottom Left: the range of motion for a crocodile twisting a prey item. Bottom right: the equations used to calculate the Torque generated by a crocodile of mass () as a result of twisting about its own axis with a prey item held in its jaws. Torque is the produce of moment of inertia (
) about the animals long axis and the angular acceleration (
) – which is assumed to be constant. Moment of inertial is calculated using mass (
) and radius (
); in our calculations mass is approximated as fifty times the mass of the skull (approx. 40 kg in the M. cataphractus example shown here), while radius is approximated as skull width (approx. 152 mm in M. cataphractus). Initial angular velocity (
) is zero since in this case the twist is being made from a standing start.
denotes the angular displacement of the twist in radians (
or 360 degrees in this case), while
denotes the time taken to complete the rotation –0.5 seconds.
Figure 16.
Values of strain from complex FE models.
Shows mean, 50%, 75%, 90%, 95%, 99% and 100% strain values for taxon used in this study. 95% strain represents the largest elemental value of strain in the model if the highest 5% of all values are ignored. 100% strain is the maximum elemental strain in the model and likely represents constraint artefacts caused by boundary conditions. Taxon abbreviations: Ot, Osteolaemus tetraspis; Cm, Crocodylus moreletii; Cng, Crocodylus novaeguineae; Ci, Crocodylus intermedius; Cj, Crocodylus johnstoni; Mc, Mecistops cataphractus; Ts, Tomistoma schlegelii.
Figure 17.
Principal component 1 (PC1) versus principal component 2 (PC2) from geometric morphometric analysis Taxon abbreviations: Ot, Osteolaemus tetraspis; Cm, Crocodylus moreletii; Cng, Crocodylus novaeguineae; Ci, Crocodylus intermedius; Cj, Crocodylus johnstoni; Mc, Mecistops cataphractus; Ts, Tomistoma schlegelii.
Table 10.
Length, Symphyseal Length, Angle and Width for each of the mandibles used in this study.
Figure 18.
Quantification of Principal Component 1 (PC1).
Wireframe (left) of mandible from dorsal and lateral perspectives illustrates the change in shape along PC1 axis. Note the longer symphyses at higher PC1 values. The chart in the centre shows the value of each morphological variable (e.g. symphyseal length) at a given PC value, as a percentage of the maximal value for that morphological variable. Specimens are plotted according to their respective PC1 values (centre right). Phylogram (right) shows poor correlation of specimen PC1 scores with phylogeny. Phylogeny based upon the results of Erickson and colleagues [47].
Figure 19.
Quantification of Principal Component 2 (PC2).
Wireframe (left) of mandible from dorsal and lateral perspectives illustrates decreasing mandible robustness with increasing PC2 values. The chart in the centre shows the value of each morphological variable (e.g. symphyseal length) at a given PC value, as a percentage of the maximal value for that morphological variable. Specimens are plotted according to their respective PC2 values (centre right). Phylogram (right) shows poor correlation of specimen PC2 scores with phylogeny. Phylogeny based upon the results of Erickson and colleagues [47].
Figure 20.
Bite force estimates for high resolutions FEMs.
Estimates of bite force generated by the high resolution FEMs, plotted against outlever length (distance from jaw hinge axis to bite point). Charts to right show natural logarithm transformed data. (A) and (B) show results from models at ‘natural’ sizes, (C) and (D) show results from models rescaled to the volume of the M. cataphractus model. Note the strong correlation between volume-scaled bite force and outlever (D). Front, mid, and rear bites for each FEM are shown. Taxon abbreviations: O.t, Osteolaemus tetraspis; C.ng, Crocodylus novaeguineae; C.i, Crocodylus intermedius; C.j, Crocodylus johnstoni; M.c, Mecistops cataphractus; T.s, Tomistoma schlegelii; C.m, Crocodylus moreletii.
Table 11.
Bite force estimates for natural sized and volume rescaled models.
Figure 21.
Strain for simple beam models #1.
Strain in the first set of simple beam models, plotted against morphological variables (from top) length, symphyseal length, angle, and width, for biting (left), shaking (middle) and twisting (right) loads. Note the strong correlation between bite and overall length, shake and symphyseal length, and twist and angle. Data is plotted as natural logarithms of linear measurements (mm) and angles (degrees). Model abbreviations are as follows: (CL-CSL-VA-VW) – Constant length and symphyseal length, variable angle and width. (CL-CW-VSL-VA) – Constant length and width, variable symphyseal length and angle. (CA-CW-VSL-VL) – Constant angle and width, variable symphyseal length and length. (CSL-CW-VL-VA) – Constant symphyseal length and width, variable length and angle.
Figure 22.
Stress contour plots for beam models.
Stress contour plots for beam model based on M. cataphractus for biting (A), shaking (B), and twisting (C) loading regimes. The models are shown from lateral (left), anterior (middle) and dorsal (right) views. The regions of high tensile (reds) and compressive (blues) stresses are shown. Deformations are exaggerated to better illustrate the structural response to loads. The general pattern of strain is similar for all beam models.
Figure 23.
Strain for simple beam models #2.
Strain in the second set of simple beam models, plotted against morphological variables (from top) length, symphyseal length, angle, and width, for biting (left), shaking (middle) and twisting (right) loads. Note the strong correlation between bite and overall length, shake and symphyseal length, and twist and angle. Data is plotted as natural logarithms of linear measurements (mm) and angles (degrees). Dimensions of the beam models are based upon the volume rescaled versions of the high resolution FEMs for the corresponding species. Taxon abbreviations: O.t, Osteolaemus tetraspis; C.ng, Crocodylus novaeguineae; C.i, Crocodylus intermedius; C.j, Crocodylus johnstoni; M.c, Mecistops cataphractus; T.s, Tomistoma schlegelii; C.m, Crocodylus moreletii.
Table 12.
Comparison of morphological variables for predicting shake strain in a simplified beam representation of a crocodile mandible.
Table 13.
Comparison of morphological variables for predicting twist strain in a simplified beam representation of a crocodile mandible.
Figure 24.
Strain plots for volume scaled FEMs.
Strain plots for volume scaled FEMs under biting, shaking, and twisting loads to show details of strain patterns. Top: biting load case plotted with a maximum strain limit of 0.001 (left) and 0.003 (right); the latter limit shows the position of the peak strains, and the former gives best comparison between the different load cases. Bottom left: shaking load case plotted with a maximum strain limit of 0.001. Bottom right: twisting load case plotted with a maximum strain limit of 0.001. Taxa: A, Tomistoma schlegelii; B, Mecistops cataphractus; C, Crocodylus johnstoni; D, Crocodylus intermedius; E, Crocodylus novaeguineae; F, Crocodylus moreletii; G, Osteolaemus tetraspis.
Figure 25.
Strain plot response to equal biting and twisting loads.
Direct comparison of mandible response to equal biting and shaking loads at the most anterior bite point (front). Strain magnitude is higher under the biting loads; the difference is noticeable for longirostrine (A–C) and mesorostrine (D–F) taxa. Taxon labels: A, Tomistoma schlegelii; B, Mecistops cataphractus; C, Crocodylus johnstoni; D, Crocodylus intermedius; E, Crocodylus novaeguineae; F, Crocodylus moreletii; G, Osteolaemus tetraspis.
Figure 26.
Peak mandibular strain (95% values).
Peak mandibular strain (95% values) plotted against morphometric variables (from top) length, symphyseal length, angle, width, and PC1 score for biting (left), shaking (middle) and twisting (right) loads. Note that strain in biting correlates strongly with overall length and very poorly with both angle and width, whilst in shaking strain has reasonable correlations with both symphyseal length and PC1. Contrary to beam predictions strain in twisting correlated strongly with symphyseal length and very poorly with angle. Data is plotted as natural logarithms of linear measurements (mm) and angles (degrees). Taxon: O.t, Osteolaemus tetraspis; C.ng, Crocodylus novaeguineae; C.i, Crocodylus intermedius; C.j, Crocodylus johnstoni; M.c, Mecistops cataphractus; T.s, Tomistoma schlegelii; C.m, Crocodylus moreletii.
Table 14.
Comparison of morphological variables predicting bite strain for hi-res FEMs.
Table 15.
Comparison of morphological variables predicting shake strain for hi-res FEMs.
Table 16.
Comparison of morphological variables predicting twist strain for hi-res FEMs.
Figure 27.
Peak strain under twist loads for beam and FE models.
Peak strain under twist loads plotted against symphyseal length for beam (left) and FE (right) models. Note the relationship between symphyseal length and strain predicted by beam models is inverted in the complex FE models; additionally, beam models fail to predict ranked order under twisting. Data is plotted as natural logarithms of linear measurements (mm).Taxon abbreviations are as follows: O.t, Osteolaemus tetraspis; C.ng, Crocodylus novaeguineae; C.i, Crocodylus intermedius; C.j, Crocodylus johnstoni; M.c, Mecistops cataphractus; T.s, Tomistoma schlegelii; C.m, Crocodylus moreletii.
Table 17.
Ranked performance of beam and FE models.
Figure 28.
Strain in biting loads for TeT and NoLLC.
Left: Strain response of mandibles when subject to equal bite force (TeT), plotted against length for (from top) front, mid and back bites. Right: Strain response of mandibles at maximal bite force (NoLLC), plotted against length for (from top) front, mid and back bites. In the TeT load cases, muscle forces are adjusted so that all models experience the same bite force as the M. cataphractus model for each bite point; with the exception of the Osteolaemus model, this has little effect on the qualitative pattern of results, with longirostrine taxa exhibiting higher strain in TeT and NoLLC load cases. Data is plotted as natural logarithms of linear measurements (mm).
Figure 29.
Comparison of FEM predictions and in vivo measurements of bite force.
Natural logarithms of FEM predicted bite force (red squares) and in vivo bite force (blue diamonds), plotted against body mass. Bite force is for rear bites, in vivo bite force data from Erickson [47]. For the FEMs, body mass was calculated from skull volume using the equation log10 body mass = log10 (skull volume x 0.9336+1.9763) using data from McHenry [29]. Slopes of regression lines are similar, but the difference in intercept means that the in vivo bite force is a factor of approximately 1.6 times the FEM predicted bite force for crocodilians of a given mass.