The objective of this study is to analyze how different crown morphologies and different root lengths respond to stresses generated by the bite forces of Albertosaurus sarcophagus and Tyrannosaurus rex. Six well-preserved teeth of A. sarcophagus from the Albertosaurus bonebed in Dry Island Park (southern Alberta) were selected to study their biomechanics, and T. rex teeth were included for comparison. The three-dimensional (3-D) models were obtained through computerized tomography (CT) scanning and 3-D digitizing. Finite element analyses were performed in Strand7[R]. Bite forces for Albertosaurus and Tyrannosaurus were calculated based on cranial and jaw proportions. The results were viewed with the Tresca yield criterion. The ratios of shear stresses observed along the mesio-distal versus labio-lingual axes of all models allows the identification of similar stress distribution patterns in the upper and lower jaws of Albertosaurus and the upper jaws of Tyrannosaurus, with a higher amount of shear along the mesio-distal axis occurring in the mid-maxillary teeth. The dentary teeth of Tyrannosaurus, however, show a different stress distribution pattern, with a higher amount of shear occurring along the labio-lingual axis of the mid-dentary teeth. These differences in jaw mechanics suggest that the function of teeth in the lower jaw of Tyrannosaurus shifted a few positions to compensate different proportions in the dentary that cause the anterior dentary teeth to be aligned with the largest maxillary teeth in Tyrannosaurus. These results suggest that heterodonty in these groups is different and that tooth form and function are sensitive to jaw proportions.
Cette etude a pour but d'analyser la reponse de differentes formes de couronne et de differentes longueurs de racine aux contraintes induites par les forces de morsure d'Albertosaurus sarcophagus et de Tyrannosaurus rex. Six dents d'A. sarcophagus bien presences provenant du lit a ossements d'Albertosaurus dans le parc de Dry Island (sud de l'Alberta) ont ete choisies pour en etudier la biomeecanique; des dents de T. rex ont eegalement ete etudieees a des fins de comparaison. Des modeles tridimensionnels des dents ont ete obtenus par tomographie assistee par ordinateur et par numerisation 3-D. Des analyses par elements finis ont ete; effectuees avec le logiciel Strand7MD. Le calcul des forces de morsure d'Albertosaurus et de Tyrannosaurus est base sur les proportions du crane et des machoires. Les resultats sont presentes avec le critere de resistance de Tresca. Les rapports entre les contraintes de cisaillement observeees le long des axes meesiodistal, d'une part, et labiolingual, d'autre part, de tous les modeles revelent des motifs semblables de repartition des contraintes dans les machoires superieures et inferieures d'Albertosaurus et dans les machoires superieures de Tyrannosaurus, un cisaillement plus important etant observe le long de l'axe meesiodistal des dents mediomaxillaires. Les dents dentaires de Tyrannosaurus presentent toutefois un motif de repartition des contraintes different, un cisaillement plus important etant observe le long de l'axe labiolingual des dents meediodentaires. Ces differences sur le plan de la mecanique des machoires laissent croire a un deecalage de quelques positions de la fonction des dents de la machoire infeerieure de Tyrannosaurus en raison des proportions differentes de l'os dentaire qui font que, chez Tyrannosaurus, les dents dentaires anterieures s'alignent avec les plus grandes dents maxillaires. Ces resultats portent a croire que l'heterodontie de ces groupes differe et que la forme et la fonction des dents dependent des proportions des machoires.
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Tyrannosaurid teeth have been simplistically referred to as dull smooth blades (Abler 1992). They function by concentrating large forces onto small areas. Tyrannosaurids did indeed have high bite forces, estimated as being up to 13400 N (Erickson et al. 1996). Additionally, these bite forces were often applied to bone (Erickson et al. 1996) or teeth, especially during feeding (Erickson et al. 1996; Molnar 1998) and intra-specific face-biting (Tanke and Currie 2000), and this frequently resulted in worn or broken tips. Nevertheless, the wear patterns observed in tyrannosaurid teeth usually do not indicate tooth-to-tooth contact inside the mouth (Molnar 1998).
Tyrannosaurid heterodonty is well documented (Currie et al. 1990; Molnar 1998; Smith 2005) and their teeth can generally be grouped into at least three classes (Smith 2005). This variation in tooth morphology suggests that there are different functions for each region in the mouth. The anterior portion of the jaws of a tyrannosaurid has teeth that are slightly curved posteriorly and have characteristic D-shaped cross sections. Robust, curved, and tall teeth characterize the middle portion of the jaws. Finally, the posterior region of the jaws has small, strongly curved, and labio-lingually compressed teeth.
The jaw position of a tooth in a tyrannosaurid, therefore, dictates the curvature of the tooth. This is because the "line of action" of a tooth (Rieppel 1979), or the direction it moves relative to the food that is being processed in the animal's mouth, depends on its position relative to the jaw hinge (D'Amore 2009). For a more efficient bite, the apex of a tooth needs to contact the food first, focusing the force onto a smaller area for puncturing the food (D'Amore 2009).
Mechanical models made of metal have been used to simulate tyrannosaurid bites (Abler 1992) and test the performance of teeth. But little is known about how the tooth itself responds to such stresses and how the different morphologies found within one specimen react to various situations involved in biting motions. Mazzetta et al. (2004) performed stress analyses on a tooth of Giganotosaurus carolinii. They used a three-dimensional (3-D) model generated by a computerized tomography (CT) scanner and simulated forces in four different directions. In that experiment, the authors were able to estimate the amount of force tolerated by that tooth and inferred the type of prey that Giganotosaurus would have preferred.
However, Giganotosaurus does not feature the same variation on tooth morphology as seen in tyrannosaurids. According to Smith (2005), Tyrannosaurus appears to have a higher degree of heterodonty than Albertosaurus. Some differences in the skull proportions of these taxa could influence the degree of morphological variation in their teeth. The snout of Albertosaurus is more elongated and narrower than in Tyrannosaurus and the maxillary teeth of Tyrannosaurus are larger relative to the skull than those of Albertosaurus, so that Tyrannosaurus has been referred to as a "saber-toothed dinosaur" (Molnar and Farlow 1990). Farlow et al. (1991) analyzed various theropod teeth, and their statistical analyses also showed that some maxillary teeth of Tyrannosaurus are indeed disproportionately tall. However, even though this significant difference in tooth proportions between Albertosaurus and Tyrannosaurus has been noticed, little has been inferred regarding the functional aspects of teeth and consequences to feeding behavior differences between them, as well as how their heterodonty differs. The two taxa studied in this project, therefore, show a great potential for exploring these differences between two largesized predators from the Late Cretaceous of North America, which usually end up having their feeding behaviors generalized to the family level (Farlow et al. 1991; Abler 1992). Form and function inferences can be made from the dentition, and there is a significantly large amount of specimens available for such studies.
In this project, 3-D models of teeth from Albertosaurus sarcophagus and Tyrannosaurus rex are compared. These models were based on six tooth specimens of Albertosaurus and six casts of a Tyrannosaurus specimen ("Stan", BHI 3033) from the Black Hills Institute. The taxa chosen for this study represent two groups of tyrannosaurids: the albertosaurines (Albertosaurus) and tyrannosaurines (Tyrannosaurus). The comparison between these groups adds a functional aspect to the analyses, in addition to the study of biomechanical differences that occur in the dentition within these taxa. This study combines detailed morphological and finite element analyses (FEA) to examine (1) the effects of heterodonty on tyrannosaurid tooth function, and (2) how the heterodonty is affected by the distinct tooth proportions observed in these two groups of tyrannosaurids. The influence of different sized roots is also analyzed in the models that represent specimens with this structure preserved. These analyses will test the hypothesis that distinct patterns of heterodonty occur in Tyrannosaurus and Albertosaurus so that different tooth functions are observed for specific tooth positions, and that, consequently, the root/crown proportions in the teeth of these two groups are different.
BHI, Black Hills Institute, Hill City, South Dakota, USA; RSM, Royal Saskatchewan Museum, Regina, Saskatchewan, Canada. TMP, Royal Tyrrell Museum of Palaeontology, Drumheller, Alberta, Canada; UALVP, University of Alberta, Laboratory of Vertebrate Paleontology, Edmonton, Alberta, Canada.
Materials and methods
The models were based on a total of 12 isolated tooth specimens (Table 1). The Albertosaurus sarcophagus specimens from the Royal Tyrrell Museum of Palaeontology were collected from the Albertosaurus bonebed (TMP locality L2204) in the Horseshoe Canyon Formation in Dry Island Buffalo Jump Provincial Park, Alberta (Fig. 1). The Tyrannosaurus rex models were based on casts of the specimen BHI 3033 ("Stan") from the Black Hills Institute (Fig. 2).
Most specimens were CT-scanned with an I-Cat Classic scanner at the Dentistry and Pharmacy Centre at the University of Alberta. The cone-beam CT scans were taken with a 0.4 mm voxel size. The models were transferred to the software Mimics[c] v.12.11, in which a 3-D mesh compatible with FEA was created. Some of the specimens were too large for the CT scanner used in this project and were, therefore, digitized using a MicroScribe[R] MX System that samples multiple points on the surface of a 3-D object. This system allows the collection of point clouds using the Immersion[R] Corporation MicroScribe[R] Utility Software 184.108.40.206. The point clouds are then transferred to the software Rhinoceros[R] v.4.0, which converts them into 3-D meshes that can be used for FEA. The FEA for all models were done with Strand7[c] v.7.2.3.
Bite force estimates were made for Albertosaurus and Tyrannosaurus with the method used by McHenry (2009) for Kronosaurus queenslandicus. The measurements were taken from the skulls of the specimens RTMP 81.10.1 (Albertosaurus) and BHI 3033 (Tyrannosaurus). The cross-sectional area of bite muscles through the subtemporal fenestra was estimated as 260.2 [cm.sup.2] for Albertosaurus and 772.6 [cm.sup.2] for Tyrannosaurus. The jaw proportions necessary for calculating forces at the different tooth positions include the "in lever" (distance from the jaw articulation to the center of the jaw muscle insertions) and the "out lever" (distance from the centre of jaw muscle insertions to specific positions along the tooth row). The angle between the muscle line of pull and the dentary bone was estimated to be approximately 45[degrees]. Based on these measurements, the bite force calculations were done as follows. The concentric specific tension (as a muscle shortens) is conservatively equivalent to 20 N [cm.sup.-2] (Bamman et al. 2000; Snively and Russell 2007a)in muscles with simple fiber architecture. This specific tension multiplied by the cross sectional area gives the muscle force ([F.sub.y]). The total vertical force ([F.sub.in]) applied by the temporal muscles to its point of attachment (in this case, to the jaw) is given by the following formula:
[F.sub.in] = sin[alpha] x [F.sub.y], in which [alpha] = 45[degrees], the muscle's angle of pull relative to the vertical.
The overall line of pull for each of the temporal muscles is in the same sagittal plane as its insertion on the mandible, so medial or lateral components of the force were judged to be insignificant for calculating the [F.sub.in].
After [F.sub.in] is known, it is possible to calculate the bite forces for each part of the jaw using the following formula:
[l.sub.in] x [F.sub.in] = [l.sub.out] x [F.sub.out], in which [l.sub.in] and [l.sub.out] are, respectively, the "in lever" and the "out lever," measured previously in centimetres, and [F.sub.in] and [F.sub.out] are, respectively, the concentric force applied by the muscle to its point of attachment and the bite force at specific point of the jaw.
Four material and structural performance properties dictate how a 3-D object will react to the forces applied to it. The elastic or Young's modulus is a ratio of stress to strain and is, thus, a measure of stiffness. The Young's modulus value used in the analyses is 2.5 e10 Pascals (Pa). It is based on the value measured in human teeth (Kinney et al. 1996). Poisson's ratio (transverse versus axial strain) describes how a structure deforms perpendicularly to the direction of force, by bulging transversely under compression and thinning under tension. The Poisson's ratio used in the analyses is 0.31, which is the same as in human teeth (Rees and Jacobsen 1997). The density--2100 kg [m.sup.-3])--assigned to the models is also that of human dentine (Johansson et al. 1945). The choice of dentine material properties for the model reflects the predominance of that material in theropod teeth. These teeth have a characteristically thin layer of enamel (Stokosa 2005), and the influence of this thin layer with different material properties on tooth mechanics can be tested at a later stage.
The models were kept the same size as the specimens, so that scale effects could also be considered. The bite forces calculated for Albertosaurus and Tyrannosaurus were applied to the tips of the models according to their position in the jaws. The forces were applied at a 45[degrees] angle, as a vector of the X and Y axes (Fig. 3). The X axis represents the mesio-distal axis of the tooth, Y represents apico-basal, and Z represents labio-lingual. All models had their roots restrained, simulating the ligaments present in the jaws. One exception is the Albertosaurus anterior dentary tooth specimen (TMP 1998.63.11), which had no root preserved, and the model was, therefore, restrained at its base.
The results were viewed with the Tresca yield criterion, which indicates how close a given material is to failure. This yield criterion, also known as maximum shear-stress criterion, approaches objects in such a way that instead of showing how tension and compression are gotten from certain stresses, it shows how forces pull or push at right angles to each other, causing shear. In that case, materials will fail whenever their molecules slide past each other (also known as "slip" in engineering). This concept is widely used in material mechanics, especially for shear-related phenomena in ductile metals, and more information about how it is calculated is detailed elsewhere (Boresi and Schmidt 2003). The stress values in this analysis, therefore, indicate shear, and the scale in all models was set to an upper level of 300 Megapascals (MPa), which is the yielding point for dentine (Currey 2002). Higher shear stresses would suggest failure of the material.
The specimens with roots each had the total tooth height and root height measured. These measurements were then used to calculate the proportion of the root size in relation to the total tooth size in each specimen. A t-statistic test was done to compare the root proportions between Albertosaurus and Tyrannosaurus. The critical value (p) was calculated with an [alpha] = 0.05. The null hypothesis (Ho) for the test is that there is no significant difference between the root proportions in Albertosaurus and Tyrannosaurus.
The estimated bite forces for an adult specimen of Albertosaurus were 1536 N for the anterior teeth, 2143 N for the middle teeth, and 3413 N for the posterior teeth. For an adult Tyrannosaurus, the bite forces were estimated at 5880 N (anterior teeth), 8178 N (middle teeth), and 13 876 N (posterior teeth).
The resulting Tresca stresses after the calculated forces were applied to the 3D models are shown on Figs. 4 and 5. The scale maxima were set to 300 MPa, the yielding stress of dentine (Currey 2002). Some of the models show stresses superior to that and any off-scale values are shown in the models as white areas.
Tooth measurements and shear stresses for all models are given in Table 2. The maximum stresses were measured along the X-Y plane. Shear stresses were also measured along the Z axis so that the labio-lingual width of each tooth could be taken into consideration. A ratio between the stresses measured in the X-Y plane and Z axis was calculated. Higher ratios indicate higher stresses along the mesiodistal and apico-basal axes (usually a combination of both, because of the 45[degrees] angle of the force applied to the models), as opposed to labio-lingual stresses.
The root proportions and the stress ratios for X-Y/Z are plotted on a graph in Fig. 6. The graph shows that the roots of Albertosaurus teeth are generally shorter than in Tyrannosaurus, but that the shear stress along the X and Y axes are highest for the mid-maxillary and mid-dentary teeth of Albertosaurus.
A pattern can be observed for both Albertosaurus and Tyrannosaurus regarding the distribution of shear stresses in the maxillary teeth. Both taxa show the highest X-Y/Z stress ratios in the mid-maxillary teeth, followed by the premaxillary and posterior maxillary teeth (the last two show little stress ratio differences).
In the dentary teeth, however, Albertosaurus and Tyrannosaurus teeth behave differently. Even though the anterior dentary tooth for Albertosaurus is not shown (Fig. 6) because the root length is unknown, the X-Y/Z stress ratio was still measured on the crown (Table 2). The ratios for Albertosaurus are highest in the mid dentary teeth, while for Tyrannosaurus the same ratios are highest in the anterior dentary and posterior dentary teeth.
The distribution of root proportions in Albertosaurus and Tyrannosaurus is also different. Although both taxa have the relatively tallest roots in their premaxillary teeth (and in the case of Tyrannosaurus also in the anterior dentary tooth) followed by the posterior maxillary teeth, the next tallest roots differ in these taxa. For Albertosaurus, the third tallest root is the one in the posterior dentary, and the shortest roots are found in the mid-maxillary and mid-dentary, which have both the same root/crown proportions. For Tyrannosaurus, the third tallest roots are the ones in the mid-maxillary and mid-dentary (also with equal root/crown proportions), and the relatively shortest root is found in the posterior dentary tooth.
A t-test was done to compare all the root height percentages of Albertosaurus and Tyrannosaurus. There is no significant difference between the average root height proportions in these two groups, which does not allow the rejection of the null hypothesis. However, although both taxa have similar overall root proportions, the differences in the way these proportions are distributed along the tooth positions are still informative and allow for some biomechanical interpretations.
Discussion and conclusions
The bite forces estimated for Tyrannosaurus in this project are comparable to the values obtained by Erickson et al. (1996). Whereas the methods in this paper include skull and jaw proportions, Erickson et al. (1996) calculated the bite forces for this taxon based on bite marks found in bones. The bite forces for Albertosaurus have not been previously estimated and additional calculations using the same methods by Erickson et al. (1996) would test the results obtained here.
The Albertosaurus specimens used in the analyses in this paper pose a problem regarding the application of bite forces on them. The forces were estimated based on an adult specimen. Some of the tooth specimens are of sizes expected for adults (for example TMP 1999.50.86), but others are more likely to have belonged to juveniles (for example TMP 2004.56.19). This becomes evident (Fig. 4) where some of the models show large off-scale areas (representing failure of the material, due to exceedingly large shear stresses caused by large forces).
Because of the problems related to different sizes of teeth in the sample (as also noted by Buckley et al. 2010), the use of proportions was preferred over the use of absolute values for the comparative analyses. When dividing the stress values measured along the X-Y plane by the stress value measured along the Z axis, the resulting proportion is less dependant on how big or how small the tooth is, even though the proportions of the teeth could also play a roll in stress distribution. Nevertheless, it makes comparisons between taxa with significant size differences more informative. The same logic is applied to the root measurements. It is obvious that the absolute values for root height in Tyrannosaurus are larger than these of Albertosaurus, but the objective of this study is to compare the proportions of teeth to test the influence of root height in tooth biomechanics. This will help to better understand the heterodonty observed in these taxa and how their tooth morphologies differ, as opposed to how their absolute sizes differ.
Another reason for material failure could be the angle that the forces were applied. In this study, all models had the forces applied in a 45[degrees] angle (as shown in Fig. 3), but this angle caused high enough stresses along the X-Y plane in some models to cause the material to fail even in the mesial portion of the tooth, near the base (Fig. 5), in the case of the posterior dentary tooth of Tyrannosaurus. This suggests that different tooth positions have different biting angles, and that the distance from the jaw hinges influences tooth shape, as also observed by D'Amore (2009). Additional studies on tooth wear would help to learn more details about biting angles and forces used in the different regions of the jaws.
The 3-D models used in this study helped to visualize two things: (1) how shear stress is distributed in tyrannosaurid teeth along the different jaw positions, and (2) how different root heights are distributed along the jaw. The Tresca stresses measured along the X-Y plane were the highest in all models. Some shear stress was also observed along the Z axis. The results showed similar patterns for Albertosaurus and Tyrannosaurus in the maxillary teeth, but opposite patterns for shear stress in the dentary teeth of Tyrannosaurus.
In Albertosaurus, the highest shear stresses along the X-Y plane in the mid-maxillary teeth indicate that these teeth are more efficient than premaxillary and posterior maxillary teeth in avoiding labio-lingual (Z) shear. Most of the stress is deflected to the mesio-distal--apico-basal (X-Y) plane on the mid-maxillary teeth. The premaxillary and posterior maxillary teeth, however, were more efficient at sustaining forces along the X-Y plane, because some of the stress is deflected to the Z axis. In summary, mid-maxillary teeth in Albertosaurus are suited to endure labio-lingual stresses, whereas premaxillary and posterior maxillary teeth in that taxon are capable of resisting higher mesio-distal--apicobasal stresses. The same pattern is observed in the dentary teeth of Albertosaurus and the upper jaw of Tyrannosaurus. It is important to point out, however, that caution should be used when interpreting the results for the anterior dentary tooth of Albertosaurus. This specimen did not have the root preserved and, therefore, the model had to be restrained at the base of the crown. Nevertheless, the general stress distribution on that model is still informative, and the stress values obtained are similar to the ones obtained for the premaxillary tooth in Albertosaurus. An additional model with the root preserved would reinforce this analysis.
For the dentary teeth in Tyrannosaurus, the exact opposite of what is described for Albertosaurus is observed: the middentary models can endure high mesio-distal--apico-basal stresses, whereas anterior and posterior dentary teeth are more suited to resist labio-lingual stresses.
In addition the root proportion distribution among different tooth positions in Albertosaurus and Tyrannosaurus is also distinct in these two genera. Whereas Albertosaurus has the relatively shortest roots in the mid-maxillary and mid-dentary teeth, Tyrannosaurus has the relatively shortest roots in the posterior dentary teeth. These results suggest a shift in high mechanical stress points (associated with longer roots) in the lower jaws of these taxa to a more anterior position in Tyrannosaurus.
As mentioned earlier, tyrannosaurid dinosaurs are characterized by teeth that are slightly curved posteriorly and have characteristic D-shaped cross sections in the anterior portion of their jaws; robust, curved, and tall teeth in the middle portion of the jaws; and small, strongly curved, and labiolingually compressed teeth in the posterior region of the jaws. Based on the analyses done here, the upper and lower jaws of Albertosaurus have anterior teeth suited for pulling on prey, to remove large pieces of meat. The middle teeth could be employed when capturing struggling prey or during feeding if lateral movements of the jaws are required. The posterior teeth are located at the position of the jaw capable of maximum bite force, and that force could be employed in securing food firmly and pulling, in a "grab and hold" fashion in which the teeth and jaw bones act like a clamp, rather than for crushing bones or other tough materials. The same pattern is observed in the upper jaws of Tyrannosaurus.
When the lower jaw of Tyrannosaurus is analyzed, a different pattern is identified. The anterior and posterior teeth are compatible with lateral movements of the jaw during feeding, whereas the mid-dentary teeth seem more suited for pulling on carcasses. Craniocervical feeding dynamics in tyrannosaurids suggest powerful lateral movements during feeding (Snively and Russell 2007a, 2007b), consistent with large lateral semicircular canals (Witmer and Ridgely 2009).
The increased bending resistance observed in both taxa would facilitate behaviors, such as holding struggling prey or rapid head movements, to remove large pieces of meat from carcasses. Tyrannosaur tooth resistance to lateral bending has also been suggested by Farlow et al. (1991) and Snively et al. (2006). The differences in stress distribution in the upper and lower jaws of Tyrannosaurus show that heterodonty developed differently in this taxon when compared with Albertosaurus, in concordance to the hypothesis, because the strongest axes of dentary teeth are distributed differently in these taxa.
When taking into consideration that Tyrannosaurus has an overbite (E. Snively, personal communication 2010), causing the anterior dentary teeth to align with the first few maxillary teeth (as opposed to the premaxillary teeth), this difference in heterodonty seems to compensate for different jaw proportions in these two taxa. The fact that in Albertosaurus the anterior dentary teeth are somewhat aligned with the premaxillary teeth suggests that, in Tyrannosaurus, the function of the dentary teeth shifted a few tooth positions to compensate for that disparity. The tooth mechanics, therefore, seems similarly regionalized in both taxa, even though the heterodonty and tooth function is distributed differently when considering tooth position alone.
It is important to note, however, that the variety of wear facets occurring in shed and in situ teeth (personal observation) indicates that although some teeth are stronger in one axis than the other, they were probably still employed in different ways than the optimal ways predicted in this analysis. A more detailed description of wear patterns for both Albertosaurus and Tyrannosaurus would be a valuable tool to compare the results obtained in this study.
The new techniques introduced here show potential for further dentition comparisons of closely related taxa, and for understanding the forms and functions of simple structures. Tyrannosaurid teeth indeed had potential for performing different functions in different regions of the jaws, as suggested by some authors (Currie et al. 1990; Molnar 1998; Smith 2005).
The author would like to thank Dr. E. Snively, for helpful comments at early stages of this project, and Dr. P.J. Currie for comments on the manuscript. Additional thanks to the crew at the T. rex Discovery Centre in Eastend, Saskatchewan, for lending some of the materials used for comparisons in this study. Thanks to the Vertebrate Paleontology Laboratory at the University of Alberta, and Manuel Lagravere at the Dentistry and Pharmacy Centre at the University of Alberta for the CT scans. Thanks to the reviewers and Associate Editor Hans-Dieter Sues for their comments and suggestions on the manuscript. Special thanks to all volunteers who cooperated with such hard work in the wonderful, yet challenging fieldwork at the Albertosaurus bonebed.
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Received 6 April 2010. Accepted 15 June 2010. Published on the NRC Research Press Web site at cjes.nrc.ca on 14 September 2010.
Paper handled by Associate Editor H.-D. Sues.
M. Reichel. Department of Biological Sciences, University of Alberta, Edmonton, AB T6G 2E9, Canada (e-mail: email@example.com).
(1) This article is one of a series of papers published in this Special Issue on the theme Albertosaurus.
Table 1. List of specimens used for building the 3-D models. Specimen No. Taxon Institution TMP Albertosaurus Royal Tyrrell Museum 2001.45.28 sarcophagus of Paleontology, AB TMP Albertosaurus Royal Tyrrell Museum 1999.50.67 sarcophagus of Paleontology, AB TMP Albertosaurus Royal Tyrrell Museum 2004.56.19 sarcophagus of Paleontology, AB TMP Albertosaurus Royal Tyrrell Museum 1998.63.11 sarcophagus of Paleontology, AB TMP Albertosaurus Royal Tyrrell Museum 1999.50.86 sarcophagus of Paleontology, AB TMP Albertosaurus Royal Tyrrell Museum 1999.50.158 sarcophagus of Paleontology, AB UALVP Tyrannosaurus rex University of Alberta, 48586.21 AB UALVP Tyrannosaurus rex University of Alberta, 48586.9 AB UALVP Tyrannosaurus rex University of Alberta, 48586.17 AB UALVP Tyrannosaurus rex University of Alberta, 48586.29 AB UALVP Tyrannosaurus rex University of Alberta, 48586.2 AB UALVP Tyrannosaurus rex University of Alberta, 48586.30 AB Method of data Specimen No. Jaw position acquisition TMP Premaxillary CT-scanner 2001.45.28 TMP Left mid-maxillary CT-scanner 1999.50.67 TMP Right posterior CT-scanner 2004.56.19 maxillary TMP Left anterior dentary CT-scanner 1998.63.11 TMP Left mid-dentary MicroScribe1 1999.50.86 Digitizer TMP Left posterior dentary CT-scanner 1999.50.158 UALVP Right premaxillary 1 CT-scanner 48586.21 UALVP Right maxillary 7 MicroScribe1 48586.9 Digitizer UALVP Left maxillary 11 CT-scanner 48586.17 UALVP Right dentary 1 MicroScribe1 48586.29 Digitizer UALVP Right dentary 6 MicroScribe1 48586.2 Digitizer UALVP Left dentary 13 CT-scanner 48586.30 Taxon Tooth position Root Total tooth length (mm) length (mm) Albertosaurus Premaxillary 79 106.3 Albertosaurus Mid-maxillary 66 103.2 Albertosaurus Posterior maxillary 19.5 28.6 Albertosaurus Anterior dentary ? ? Albertosaurus Mid-dentary 66 103.2 Albertosaurus Posterior dentary 39 60 Taxon Tooth position Percentage Maximum shear stress of root along XY plane (MPa) Albertosaurus Premaxillary 0.74 3818.3 Albertosaurus Mid-maxillary 0.64 4342.3 Albertosaurus Posterior maxillary 0.68 11293.3 Albertosaurus Anterior dentary ? 4436.2 Albertosaurus Mid-dentary 0.64 858.1 Albertosaurus Posterior dentary 0.65 3292.9 Taxon Tooth position Maximum shear stress Ratio XY/Z along Z axis (MPa) stress Albertosaurus Premaxillary 1994 1.86 Albertosaurus Mid-maxillary 868.3 5.00 Albertosaurus Posterior maxillary 7462.7 1.51 Albertosaurus Anterior dentary 1901.9 2.33 Albertosaurus Mid-dentary 132.8 6.46 Albertosaurus Posterior dentary 1099.5 2.99 Note: Specimens from the University of Alberta are casts of BHI 3033 (Stan) from the Black Hills Institute, South Dakota. Table 2. Root height and Tresca stresses measurements in Albertosaurus and Tyrannosaurus. Root length Total tooth Taxon Tooth position (mm) length (mm) Albertosaurus Posterior dentary 39 60 Tyrannosaurus Premaxillary 102 135 Tyrannosaurus Mid-maxillary 135 195 Tyrannosaurus Posterior maxillary 74 105 Tyrannosaurus Anterior dentary 115 154 Tyrannosaurus Mid-dentary 145 209 Tyrannosaurus Posterior dentary 42 64 Maximum shear Percentage stress along X-Y Taxon Tooth position of root plane (MPa) Albertosaurus Posterior dentary 0.65 3292.9 Tyrannosaurus Premaxillary 0.75 24485.6 Tyrannosaurus Mid-maxillary 0.69 1401.2 Tyrannosaurus Posterior maxillary 0.7 331773.8 Tyrannosaurus Anterior dentary 0.75 1152.1 Tyrannosaurus Mid-dentary 0.69 557.9 Tyrannosaurus Posterior dentary 0.66 41036.7 Maximum shear stress along Z Ratio X-Y/Z Taxon Tooth position axis (MPa) stress Albertosaurus Posterior dentary 1099.5 2.99 Tyrannosaurus Premaxillary 12415.6 1.97 Tyrannosaurus Mid-maxillary 553.7 2.53 Tyrannosaurus Posterior maxillary 161715 2.05 Tyrannosaurus Anterior dentary 335 3.44 Tyrannosaurus Mid-dentary 237 2.35 Tyrannosaurus Posterior dentary 12315 3.33