The Biomechanics Behind Extreme Osteophagy in Tyrannosaurus rex

Most carnivorous mammals can pulverize skeletal elements by generating tooth pressures between occluding teeth that exceed cortical bone shear strength, thereby permitting access to marrow and phosphatic salts. Conversely, carnivorous reptiles have non-occluding dentitions that engender negligible bone damage during feeding. As a result, most reptilian predators can only consume bones in their entirety. Nevertheless, North American tyrannosaurids, including the giant (13 metres [m]) theropod dinosaur Tyrannosaurus rex stand out for habitually biting deeply into bones, pulverizing and digesting them. How this mammal-like capacity was possible, absent dental occlusion, is unknown. Here we analyzed T. rex feeding behaviour from trace evidence, estimated bite forces and tooth pressures, and studied tooth-bone contacts to provide the answer. We show that bone pulverization was made possible through a combination of: (1) prodigious bite forces (8,526–34,522 newtons [N]) and tooth pressures (718–2,974 megapascals [MPa]) promoting crack propagation in bones, (2) tooth form and dental arcade configurations that concentrated shear stresses, and (3) repetitive, localized biting. Collectively, these capacities and behaviors allowed T. rex to finely fragment bones and more fully exploit large dinosaur carcasses for sustenance relative to competing carnivores.

Because trace evidence for extreme osteophagy is best documented in T. rex, we focus the present study on this taxon's capacity to comminute bone. To do so requires a compound understanding of: (1) feeding behaviour; (2) tooth crown morphology; (3) forces applied through the teeth to create crack-initiating contact pressures (force/ area 8 ); and (4) consideration of how bones were contacted by individual and adjacent series of teeth. Of these factors, only the aforementioned trace evidence of feeding behaviour (see above) and adult bite forces for T. rex have been examined. However, with regard to bite-force values, disparate estimates have been reported. Specifically, Erickson and colleagues 25 used an indentation simulation on cow pelves to estimate the force required to create a single T. rex bite mark during post-mortem feeding. This served to test the hypothesis that the taxon possessed structurally weak teeth. Although not assumed to be produced during maximal-force biting 25,27 , a conservative value of 13,400 N (3,013 pounds [lb]) was obtained. At the time this was the highest experimentally derived bite force for any animal, suggesting the teeth were robust by neontological standards 25 . Because the contralateral teeth were similarly engaged during biting 25,27 , the bite may have approached 26,800 N 27 (6,025 lb). Meers 28 subsequently used body-size scaling of bite-force values from a diversity of relatively small extant mammals and reptiles and derived an estimated maximal bite force of 253,123 N (56,907 lb) for an adult T. rex (MOR 555). Alternatively, Therrien et al. 29 30 that was then doubled to account for the contralateral adductor musculature. The authors deduced a taxon-representative, maximal bite-force value of 300,984 N (67,667 lb). Finally, Bates and Falkingham 31 estimated maximal bite forces for BHI 3033, using: (1) a computed tomography (CT) rendition of the cranium; (2) various musculoskeletal architectural configurations based on generalized functional groups for both squamate reptiles and crocodylians; (3) a mammalian appendicular muscle stress value; and (4) a biomechanical model predicting recoil on initial tooth impact that was assumed to occur in crocodylians. (Note: this is inconsistent with real-time, in vivo bite force readings 32 ). The study arrived at maximum-force estimates ranging from 35,000-57,000 N (7,869-12,815 lb).
Because of the broad range of previous bite-force estimates for T. rex and an absence of data regarding applications of load and tooth pressures, we conducted a multifactorial examination of the biomechanics by which T. rex pulverized bone. Specifically we: (1) directly examined the crania and dentitions of specimens (n = 7) using articulated fossils, high-resolution museum-grade casts, and CT data spanning the entire known adult size range for the taxon; (2) characterized the contact areas of the prominent maxillary tooth crowns used to fracture bones during feeding 8, 33 ; (3) reconstructed the three-dimensional (3-D), clade specific (Sauria: Archosauria) muscle architecture using Extant Phylogenetic Bracketing (i.e., inferring muscle configurations based on T. rex osteology and jaw adductors in Crocodylia-archosaurian sister clade to Dinosauria, and Neornithes-living theropod dinosaurs [34][35][36] (Fig. 2); (4) determined muscle forces using an experimentally validated, extant archosaurian jaw adductor muscle model 35 ; (5) size-scaled muscle forces and quantified specimen-specific lever mechanics of each jaw to estimate individual bite-force capacities; (6) deduced pressure generation as the teeth penetrated bones 33  (7) considered the shear stress failure properties of bone to determine how dental and palatal contact configurations (Figs 3 and 4) facilitated skeletal element fragmentation in a manner consistent with T. rex bite marks and coprolitic evidence.

Results
We found that adult T. rex skull lengths in our sample range from 111.5 to 136.5 cm (BHI 4100 and LACM 23844, respectively) ( Table 1). Skull widths range from 59.2 to 90.2 cm (BHI 4100 and FMNH PR 2081, respectively) ( Table 1). (Notably, FMNH PR 2081 has been reported as the largest specimen for the taxon 37 ; however, we instead found that LACM 23844 has the longest and, marginally, the second widest skull). Based on the minimum reliable measurement of tooth contact areas (at 1 mm from the crown apex) 8 [9,298 psi]) for at least 25 mm of crown height in nearly all maxillary teeth. One of the largest T. rex individuals (FMNH PR 2081) maintained such pressures up to (and presumably beyond) the 37 mm indentation maximum utilized in this study 15 .
Analysis of dental-arcade configurations from T. rex skulls shows that its palatal and dental anatomy would have promoted: (1) fractures during biting that spanned between the mesial and distal carinae of adjacent teeth due to localized stress concentrations (Fig. 3); and (2) numerous three-and four-point loading configurationsclassic means by which the tensional and shear weaknesses of beams (including bones) are exploited in mechanical and orthopaedic engineering with non-opposing loading points 4,5,40 . Three-point arrangements likely occurred: (1) between consecutive, large teeth along the dental arcade and the opposing tooth crown (

Discussion
Our findings, coupled with evidence of T. rex carcass utilization from bite marks, explain how this taxon along with other large North American tyrannosaurids comminuted bone in the absence of dental occlusion. The maximum adult T. rex bite forces (18,014-34,522 N; 4,050-7,761 lb) reported here for seven specimens spanning the adult size range for the taxon (see Table 1) are each moderately to considerably lower than previous estimates (35,000-300,984 N 28, 29, 31 ; 7,869-67,667 lb). We suspect the differences stem primarily from previous models not implementing archosaurian-specific, jaw-closing musculature and force generation as well as not utilizing experimentally validated neontological models 35 . Nonetheless, the values we estimate are still prodigious. Adductor forces introduced tooth pressures substantially higher than the ultimate shear stress of cortical bone, even at great depth, allowing deep penetration of impacted bones. Tooth penetration served to drive open cracks (engendered first by localized fractures at tooth contact points), using broadly expanding tooth crowns 41 . Carinae accentuated these stresses and directed crack propagation towards adjacent teeth, resulting in high-pressure fracture arcades as cracks from the broadest and most procumbent teeth intersected during biting (Fig. 3). Together the dental and palatal anatomy also provided for three-and four-point loading configurations that facilitated localized and whole-element bone shear (Fig. 4). (Although not testable in our modelling, catastrophic explosion of some bones, particularly smaller elements or those with thin cortices, may have also occurred due to the introduction of strain energy densities exceeding the limits of bone 4 ). Following fracture, repetitive and localized carnivoran-like biting (evidenced from bite marks; Fig. 1) served to accentuate fine-scale fragmentation, expose bone surfaces, and liberate marrow for rapid digestion by low pH stomach acids 22 .
The few osteophagous reptiles capable of driving cracks through bones, such as adult crocodylians 8,33 and tyrannosaurids, have force-resistant, thecodont dentitions. However, because of their characteristically offset dental rows, reptiles tend to generate a mechanical couple while biting (e.g., opposing but equal forces acting in parallel around a single axis; for an illustration see pages 19-20 in Cochran 5 ), which can rotate isolated bones or those within carcasses and, potentially, load tooth crowns in unexpected ways. Such loads may induce reaction forces that can cause permanent structural failure [41][42][43] . Unexpected loads are counteracted by possessing semi-conical crowns with high, transverse-plane area moments of inertia. Such teeth are capable of sustaining comparable loads from any direction 30,33,44 , prolonging their functionality until replacement (e.g., over a year for large adult crocodylians 24,45 and ~777 days for T. rex 24 ) in these polyphodont taxa 24 . Taken together with the aforementioned prodigious bite forces, tooth pressures, localized biting, and absence of mammal-like, precise dental occlusion, our findings indicate that the extensive fragmentation of bone practiced by large tyrannosaurids was directly facilitated by their elongate, semi-conical, carinated, rooted, and polyphyodont dental arcades.
It is intriguing that the maximum tooth pressures shown here for T. rex overlap tightly with those reported for large adult crocodylians (e.g., A. mississippiensis, C. porosus) that are also capable of fracturing bone during feeding 8, 33 (although not sequentially). Even though extant crocodylians are considerably smaller than adults of  Table S2], respectively), using teeth with relatively thin enamel shells 16,46 (e.g., A. mississippiensis and T. rex mean ± standard error of enamel thicknesses sampled along the crown are 237 ± 6 and 223 ± 30 microns, respectively; GME unpublished data). In the case of crocodylians, the enamel shell is only slightly stronger than the tooth pressures that are typically endured during feeding (e.g., tooth safety factors-how mechanically overbuilt a structure is versus its function 4 -range from 1.0-1.4 8,33 ), and apical tip spalls 43 are structurally similarly to those documented previously in T. rex 16 . In the context of our integrative analysis, this functional convergence suggests that: (1) the performance capacities elucidated by this study are realistic; and (2) T. rex tooth crowns would be unlikely to sustain bite forces that are substantially greater 28,29 than those reported here. Notably, juvenile crocodylians with smaller and less robust dentitions are incapable of rupturing large bones 33 , which is also consistent with crown morphologies and bite marks from juvenile T. rex that similarly do not show evidence of bone removal. Instead these consist of only shallow punctures and scores 20 . Expansion of our protocol throughout ontogeny will help to elucidate at what size and age 25, 47 this taxon's capacity for bone fragmentation first occurred.
The collective results of this taxon's biomechanical and physiological feeding capacities allowed these large-bodied theropods to uniquely exploit large bones from dinosaur carcasses-known to include giant horned-dinosaurs (e.g., Triceratops 17, 18 ), duck-billed hadrosaurids (e.g., Edmontosaurus 15,19 ) and even other T. rex 20 -that could not be consumed otherwise by contemporary carnivores. Tyrannosaurus rex, therefore, was able to derive sustenance from bones of prey 15 and scavenged carcasses 27 , much like extant grey wolves 1-3 and spotted hyenas 1,3,48 . Overall, our study shows how meaningful understanding of unusual behaviours and physical capacities not seen together in living animals can be determined through multifaceted, cross-disciplinary approaches. This research adds to a growing body of literature 49-51 that illustrates how sophisticated feeding capacities-analogous to those of modern mammals and their immediate ancestors-were first achieved in Mesozoic archosaurs.

Methods
Specimen Examination. We examined fossil specimens, high-resolution museum-grade casts, computed tomography (CT) data, and professional photographs of the skulls, jaws, and dentitions of seven adult T. rex specimens ( . Adulthood in these individuals is based on corroborating information from craniofacial osteology 52 , overall size, and a mass-age growth curve 47 . We documented variation in head size using measures (linear distance to the nearest millimetre [mm; here and throughout our protocol]) of head width across the quadrates and head length from the anterior surface of the premaxillae to the posterior superior margin of the parietal bones. Individual variation in the lever mechanics of each skull was accounted for by measuring the linear distances between the quadrate-articular joint, and (1) the anteroposterior midpoint for osteological correlates of each jaw adductor muscle insertion along the lower jaw (i.e., "anatomical in-levers, " sensu 35 ); and (2) the midpoints for the first premaxillary alveolus (P1), the third, fourth, and fifth maxillary alveoli (M3, M4, M5, respectively), and the most distal maxillary alveolus (variably M11 or M12) on the left and right sides of the skulls (out-levers).

Characterization of Tooth Contact Areas.
Moulds were made of M3, M4, and M5 using fast-set silicon moulding putty (Knead-a-Mold ® , Townsend Atelier, Chattanooga, TN, USA) on the right and left sides of all specimens for which teeth were fully erupted (the right M5 of BHI 4100 and left M4 of MOR 980 were not fully erupted and not used). These tooth crowns are the longest in the T. rex jaw and would, therefore, be the first to engage tissues in isolation during biting (and were determined to be responsible for the bite marks modelled previously 17,25 ). When present, M5 is typically the longest although either M3 or M4 may act to initiate tooth indentation when M5 is missing, broken, or beginning to erupt.
High-resolution epoxy replicas of the teeth were then made (Epoxyset #145-20005, Allied High Tech Products, Inc., Rancho Dominguez, California, USA). Crown heights and cross-sectional areas along each cast from the tooth apex towards the root of the crown were measured following 8,33 . Cross-sectional measurements of conical and lenticular tooth crowns can serve as surrogates for realized tooth contact area (as demonstrated by Gignac and Erickson 33 ), which sums the total indenter surface area that is in contact with indented tissues and perpendicular to the application of bite force through the long axis of the tooth. These measurements were ultimately used for estimating pressures generated along each tooth crown (see below) 8, 33 .

3-D Muscle Reconstruction.
The actual, fully-articulated cranium, stereolithography files of individual skull bones (provided by BHI), and an articulated, high-resolution 14.45% scale replica rendered from the CT scans of T. rex specimen BHI 3033 were examined for making adductor muscle reconstructions. To further examine the adductor chambers and relationships of muscle attachment points, a micro-CT (μCT) scan of the articulated replica was undertaken at the Microscopy and Imaging Facility of the AMNH (2010 GE phoenix v|tome|x s240 high-resolution microfocus CT system; General Electric, Fairfield, Connecticut, USA). A standard X-ray scout image was obtained prior to scanning to confirm specimen orientation and define the scan volume. The scan was performed at 170 kilovolts [kV] and 145 micro-amps [μA], using a 0.1 mm copper filter, air as the background medium, and a tungsten target. The specimen was scanned at an isometric voxel size of 111.97 micrometres [μm] ( = 774.88 μm at life-sized dimensions), and slices were assembled on an HP z800 workstation (Hewlett-Packard, Palo Alto, California, USA) running VG Studio Max (Volume Graphics GmbH, Heidelberg, Germany). The specimen image stack was imported into Avizo Lite 9.0 (FEI Co., Hillsboro, Oregon, USA), where the skull and lower jaw were reconstructed separately. The lower jaw was abducted from the skull around the quadrate-articular joint to a standardized gape of 20° (measured from the anteroinferior margin of the premaxilla to the center of the quadrate-articular joint to the anterosuperior margin of the dentary). The skull and lower jaw were then resampled into one volume as a single material at an effective voxel size of 2.3513 mm (2,351.3 μm) to reduce file size and memory consumption for adductor muscle model generation as well as to scale the digital model to life-size dimensions. Although coarser, this abducted model retained bone-surface details necessary for use as an osteological scaffolding to reconstruct the jaw adductor musculature in three dimensions.
The eight adductor muscles (Musculus adductor mandibulae externus medialis, M. adductor mandibulae externus profundus, M. adductor mandibulae externus superficialis, M. adductor mandibulae posterior, M. pseudotemporalis complex, M. intramandibularis, M. pterygoideus dorsalis, and M. pterygoideus ventralis) that make up the archosaur jaw-closing system, based on extant Crocodylia and Aves 34,35 , were rendered on both right and left sides (see Fig. 2). Based on comparisons to gross dissections 35 and diffusible iodine-based contrast-enhanced CT (diceCT) specimens 53,54 , the adductor muscle origins and insertions listed in Holliday 55 (see Table 4 and in-text discussion) for the above jaw-closing muscles were regionalized in BHI 3033, and polygon volumes connecting those regions were rendered in Avizo Lite. Deviations in our model from the muscle attachments discussed by Holliday 55 include the following: the two portions of M. pseudotemporalis were represented as a single muscle belly for model simplification that originated along the anteromedial, medial, and posterior portions of the upper temporal fenestra 35 , laterosphenoid 35,55 and epipterygoid 55 ; M. add. mand. ext. profundus originated along the anterolateral, lateral, and posterolateral surfaces of the upper temporal fenestra; and M. pterygoideus dorsalis did not extend anteriorly past the orbits, as is the case for most birds 34 but not for modern crocodylians 34,35,55 . In addition, the presence of a crocodylian-like, distinct M. intramandibularis in T. rex is unclear because it is hypothesized to have been fused to the M. pseudotemporalis musculature during the evolution of Avemetatarsalia (see Holliday 55 for a detailed assessment). In this scenario the cartilaginous sesamoid (i.e., cartilago transiliens) that joined M. pseudotemporalis to M. intramandibularis was lost, resulting in a continuous muscle belly where there had once been two. In crocodylians the cartilago transiliens leaves a shallow fossa along the superiomedial surface of the mandible, adjacent to the pterygoid flange. Particularly well-preserved T. rex specimens such as FMNH PR 2081 (e.g., right mandible) may show faint evidence of such a depression (P.M.G., personal observation). Regardless, musculature attaching to the lower jaw and mandibular fossa in the position of the crocodylian M. intramandibularis and inferior avian M. pseudotemporalis 34,55 was necessary for jaw adduction. Here we retained the crocodylian muscle topology based on trace evidence of this sesamoid cartilage. Lastly, M. pterygoideus ventralis was interpreted to wrap around the posteroinferior margin of the mandible and insert along the lateral surface of the lower jaw, inferior to the dorsolateral crest of the surangular (as depicted in Fig. 7C, left of Holliday 55 ; also see Fig. 2C).

Estimated Muscle Forces.
Contractile forces for each adductor muscle were derived for BHI 3033 following a validated, free-body model analogue developed by Gignac and Erickson 35 for A. mississippiensis. All muscles were assumed to have negligible pennation 55 , following the common fascicle configurations of both crocodylian 35 and bird 34 jaw adductor muscles. The exception to this is the uniquely pennate M. pterygoideus ventralis of crocodylians. The evolution of this muscle in eusuchians 56 for high bite-force generation at the expense of adductor mandibulae and temporalis musculature (e.g., primary force generators in birds and other terrestrial amniotes) 34,55 promoted stealthy prey-capture behaviors at the water's edge 35 . Modern crocodylians utilize this jaw adductor configuration for a substantially different feeding strategy than that inferred for T. rex. Without an a priori biological reason for assuming crocodylian-like pennation in M. pterygoideus ventralis, we modelled this muscle with a parallel fiber arrangement in T. rex.
To estimate physiological cross-sectional areas (PCSA) left-and right-side volumes for each muscle were (1) averaged, (2) divided by the density of archosaur skeletal muscle (1.056 gram [g]/cm 3 ) 35 Table S1). As demonstrated by Gignac and Erickson 35 muscle length can serve as a proxy for fascicle length in parallel-fibered muscles when statically modelled. Length measurements (Supplementary Table S1) were made as the 3-D linear distance between the centroids of each muscle's origin and insertion. Each muscle was assigned an archosaur-specific muscle stress of 32.4 N/cm 2 that was empirically determined from A. mississippiensis jaw adductor musculature 35 . (These values are not known for jaw muscles in Aves, which also lack teeth). Muscle forces were modelled assuming a tetanic contraction of 100% Scientific RepoRts | 7: 2012 | DOI:10.1038/s41598-017-02161-w (Supplementary Table S1), which is broadly consistent with the maximum muscle recruitment values quantified by Cleuren et al. 57 for Caiman crocodilus and comparable to Gignac and Erickson 35 for A. mississippiensis through modelling of empirically derived bite-force values.
Jaw closure in T. rex is orthal with bite forces acting exclusively along the vertical (i.e., Y-oriented) axis of the teeth. Therefore, the dorsad-only component of each tetanic muscle force (Supplementary Table S1) was derived to estimate muscle moments at insertion points following the iterative Pythagorean Theorem approach of Gignac and Erickson 35 . This technique facilitates the ready transformation of tetanic muscle forces into muscle moments based on the anatomical in-lever lengths 35 unique to each skull (Supplementary Table S2). The muscle-specific vertical components of contractile force for left and right-side jaw adductor muscle reconstructions in BHI 3033 were averaged (Supplementary Table S1) for use in estimating size-specific, Y-axis contractile muscle forces in the other adult T. rex specimens.
Specimen Specific Bite-force Estimates. We used the tetanic muscle forces derived for BHI 3033 and bite-force relevant anatomical differences 35 across our sample of skulls as the basis for estimating bite-force performance in the six-other adult T. rex specimens (Supplementary Table S2). Among living carnivorous archosaurs such as crocodylians, head width across the quadrates is a strong indicator of body size differences because it correlates strongly with body mass (Pearson's product moment correlation = 0.99, n = 35) 56 . Therefore, we compared head width across the quadrates for our T. rex sample and linearly scaled the BHI 3033 vertical components of muscle force for each jaw adductor muscle to the values appropriate for BHI 4100, FMNH PR 2081, LACM 23844, MOR 008, MOR 980, and RTMP 81.6.1 based on these widths (Supplementary Table S2). Linear scaling (e.g., instead of a power curve) was used because the seven adult T. rex specimens occupy only a 34% range of total possible ontogenetic variation in head width, and linear versus power best-fit curves of muscle forces plotted against head width are not statistically distinguishable across such relatively small spans of body size in adult individuals of living animal models with low sample size (e.g., A. mississippiensis 35 ).
For each specimen, we then multiplied the scaled vertical component of muscle force into that muscle's (right and left-side averaged) anatomical in-levers to produce muscle moments (Supplementary Table S2). The moments were summed, doubled (to account for the contralateral side jaw adductor musculature), and divided by each of the out-lever distances for P1, M3, M4, M5, and M11/12 tooth positions on the right and left sides to produce maximum bite-force estimations at the most mesial (P1) and distal (M11/12) tooth positions as well as for the most procumbent teeth (M3, M4, M5) (Supplementary Table S2).

Tooth Pressures.
Pressures along the tooth crowns of each individual were calculated by dividing the maximum estimated individual bite force at each tooth position by the tooth's cross-sectional area measured at 1, 7, 13, 19, 25, 31 and 37 mm from the apex (Supplementary Table S2). These distances provided for the minimum cross-sectional area that could be measured reliably (1 mm), maximum indentation depth known for an adult T. rex bite mark (~37.5 mm) 15,17 , and a series of regular intervals in between. Only tooth casts of pristine (undamaged) or nearly pristine (slightly worn) teeth were used in the analysis. These values were then compared to the ultimate shear stress of cortical bone 4,39 . (Note: of the 31 tooth crowns we cast and measured, the following [see also Supplementary Table S2]  Tooth and Palatal Contact Characterizations. The distribution and spacing of adult T. rex tooth crowns and palatal architecture were taken into consideration with regard to how they contributed to this animal's ability to remove/comminute bones during biting. Specifically, we interpreted the loading of individual teeth (Fig. 3) as well as the collective damage invoked on bones by simultaneous indenting of both the upper and lower dentition in conjuction with potential palatal contacts (Figs 1, 3 and 4).