Comparative cranial biomechanics reveal that Late Cretaceous tyrannosaurids exerted relatively greater bite force than in early‐diverging tyrannosauroids

Tyrannosaurus has been an exemplar organism in feeding biomechanical analyses. An adult Tyrannosaurus could exert a bone‐splintering bite force, through expanded jaw muscles and a robust skull and teeth. While feeding function of adult Tyrannosaurus has been thoroughly studied, such analyses have yet to expand to other tyrannosauroids, especially early‐diverging tyrannosauroids (Dilong, Proceratosaurus, and Yutyrannus). In our analysis, we broadly assessed the cranial and feeding performance of tyrannosauroids at varying body sizes. Our sample size included small (Proceratosaurus and Dilong), medium‐sized (Teratophoneus), and large (Tarbosaurus, Daspletosaurus, Gorgosaurus, and Yutyrannus) tyrannosauroids, and incorporation of tyrannosaurines at different ontogenetic stages (small juvenile Tarbosaurus, Raptorex, and mid‐sized juvenile Tyrannosaurus). We used jaw muscle force calculations and finite element analysis to comprehend the cranial performance of our tyrannosauroids. Scaled subtemporal fenestrae areas and calculated jaw muscle forces show that broad‐skulled tyrannosaurines (Tyrannosaurus, Daspletosaurus, juvenile Tyrannosaurus, and Raptorex) exhibited higher jaw muscle forces than other similarly sized tyrannosauroids (Gorgosaurus, Yutyrannus, and Proceratosaurus). The large proceratosaurid Yutyrannus exhibited lower cranial stress than most adult tyrannosaurids. This suggests that cranial structural adaptations of large tyrannosaurids maintained adequate safety factors at greater bite force, but their robust crania did not notably decrease bone stress. Similarly, juvenile tyrannosaurines experienced greater cranial stress than similarly‐sized earlier tyrannosauroids, consistent with greater adductor muscle forces in the juveniles, and with crania no more robust than in their small adult predecessors. As adult tyrannosauroid body size increased, so too did relative jaw muscle forces manifested even in juveniles of giant adults.


| INTRODUCTION
Biomechanical analyses have assessed the feeding function and ecological role of dinosaurs, especially theropods (Carpenter, 2002;Erickson & Olson, 1996;Farlow, 1976;Farlow & Holtz, 2002;Gignac et al., 2010;Rayfield, 2004;Sakamoto (2006Sakamoto ( , 2010)).Theropod dinosaurs have been subject to various analyses addressing their feeding function, offering numerous perspectives about the evolution of feeding behavior in theropod clades.Early small-bodied theropods such as Eodromaeus and Coelophysis were general carnivores in their environments (Martinez et al., 2011).Upon extinction of terrestrial carnivorous pseudosuchians, theropods occupied the vacancy and became the new dominant predators, ranging toward powerful predators such as Allosaurus and Tyrannosaurus (Ezcurra et al., 2017;Whiteside et al., 2010).Here, we apply comparative biomechanics to the evolution, ontogeny, and ecology of feeding in the Tyrannosauroidea, a longeval and diverse clade of predatory dinosaurs.
The purpose of this study is to evaluate the feeding function in Tyrannosauroidea through muscle force reconstruction and finite element analysis (FEA).In addition to estimating bite force and cranial stresses in adult tyrannosauroids of differing sizes, we incorporated Tyrannosaurus specimens at different ontogenetic stages to address possible functional correlations between similarly sized adult tyrannosauroids and Tyrannosaurus ontogeny.
To assess feeding function and infer the feeding ecologies of tyrannosauroids, we applied biomechanical analyses to test specific hypotheses.

| HYPOTHESES
We compared the level of stress present in the finite element models to test the following hypotheses.
Hypothesis 2. As tyrannosauroid body size increased, the level of cranial stress decreased, which is to be expected because linearly larger structures are exponentially more resistant to forces.We infer skull size as a proxy for body mass as in crocodylians (Gignac & O'Brien, 2016;O'Brien et al., 2019), although negative allometry in tyrannosauroid subclades would contradict this assumption.

| Finite element analysis model creation
Calculating the jaw muscle forces and performing FEA on models of tyrannosauroid crania enabled interpretation of the cranial performance in Tyrannosauroidea, and how cranial morphology of these theropods evolved.
We applied FEA to the 3D models of tyrannosauroid crania.FEA is a powerful tool that models the mechanical response of a complex structure to forces to evaluate its stress and strain (Bright, 2014;Rayfield, 2007;Rowe & Snively, 2021).The application of FEA to modern animals allows us to assess their mechanical feeding capabilities in the context of feeding behavior, and to use extant models as an analog to the mechanical feeding behavior of extinct animals (Cost et al., 2019;Rayfield, 2007;Tseng, 2013;Tseng & Flynn, 2015).FEA enables us to evaluate and comprehend how the skeletal structures of living and extinct animals functioned and their roles throughout animal groups' evolutionary history (Rayfield, 2007).
To maximize our comparative sample of tyrannosauroids, we obtained model surface data through varying methods, including CT (computerized tomography) and surface scans of articulated casts, and digital model reconstructions based on primary observation and literature.All models were compromised in that they lacked endocranial cavities and internal pneumaticity, which would underestimate stresses in the pertinent regions.The stress results therefore serve as hypotheses subject to further testing and refinement, as with box-modeling methods of Rahman and Lautenschlager (2016), fossil specimen scans that capture internal detail, and sensitivity comparisons of solid models versus those with full internal cavities (Rowe & Rayfield, 2022).Errors are likely to be precise between our models, at least in the endocrania of comparably sized subjects.To maximize precision between models obtained though different methods, we slightly smoothed intracranial joints between models reconstructed from articulated separate elements (Daspletosaurus and Gorgosaurus) to better match surfaces in whole-cranium-based digital models and scanned casts.
The cranium model sources for the Tyrannosaurus specimens (adult and juvenile) consisted of either highresolution casts that were CT-scanned or surface scanned to process into a 3D surface mesh (e.g., USNM 555000, BMRP 2002.4.1), or rigorous 3D reconstructions that were modeled after original specimens (FMNH PR 2081).CT scans of the cast of BMRP 2002.4.1 and 1/3 scale sculpture of FMNH PR 2081 were at 0.7 mm resolution, at O'Bleness Memorial Hospital (Athens, Ohio) and Foothills Hospital (Calgary, Alberta).The sculpture of FMNH PR 2081 was corrected for the specimen's taphonomic distortion with reference to other T. rex specimens.The scan data were imported into visualization and analysis software, Avizo, for surface refinement and preparation for FEA.With Avizo, we made refinements to the highresolution surface scanned cranium of an adult T. rex (USNM 555000), such as cleaning the surfaces filling in empty gaps to reflect Tyrannosaurus' real-life skeletal anatomy.For example, we modeled a maxillary fenestra (a fenestra or hole which pierces the maxilla anterior to the antorbital fenestra), a feature that is present in theropod dinosaurs, through Avizo's "Scan Surface to Volume" feature and its "Segmentation Editor" (Rauhut & Fechner, 2005)."Scan Surface to Volume" converts surfaces into slice data, enabling internal modeling of data from surface scans which may include unprepared, "solid" regions of the original specimen.We resampled the new volume data to replicate a "smooth" CT scan, to about 1 mm voxel size.
Similarly, the surface scanned data of the cranium of Teratophoneus was imported into Avizo to refine it for FE (finite element) modeling.The nasal airway and maxillary fenestrae of Teratophoneus were hollowed out in Avizo, and the palate was refined and reconstructed after the morphology of Tyrannosaurus (FMNH PR 2081).
The models of Gorgosaurus and Daspletosaurus were based on individually sculpted bones which were articulated into models of the complete crania.The bones were modeled as symmetrical, for idealized reconstructions within the span of likely individual variation and were based on taphonomically undistorted specimens.These models were combined into continuous surfaces in Meshmixer, and their joints smoothed as described above.
Because the holotype skull of Yutyrannus was flattened in lateral view, its 3D geometry was partly modeled with reference to the model of Proceratosaurus, given that the two tyrannosauroids are phylogenetically classified as proceratosaurids.The holotype of Proceratosaurus bradleyi lacks the dorsal portion of its cranium and missing cranial elements (e.g., crest) were modeled after the small proceratosaurid, Guanlong; this makes our Proceratosaurus model a composite proceratosaurid.The digitally sculpted models of the aforementioned tyrannosauroids were aimed to be accurate as possible, with their reconstructions based on published references and comparisons with later members (Tyrannosaurus).Two of us (Xu and Li) described the holotypes and paratypes of Dilong, Yutyrannus, and the proceratosaurid Guanlong, and approved the proportions and detail of the reconstructed morphologies.Digital models of Yutyrannus and Proceratosaurus are subject to revision but are the best interpretations at the moment.
From the refined surface models for all taxa, we applied Avizo's "prepare generate tetra grid" before generating a solid tetrahedral mesh.This method created biologically realistic models of the crania, with lowaspect-ratio triangles necessary for minimizing strain artifacts.From the surface mesh, we produced tetrahedral solid meshes of the crania with at least 200,000 nodes and 1 million elements each.
The surface scan of Raptorex was incompatible with production of a volumetric mesh, with errors triangle overlap we were unable to resolve with Avizo (even through Scan Surface to Volume) or Blender.We created a model of Raptorex for meshless structural mechanics in the commercial software program, Simsolid, which is forgiving of such errors.The method works on geometries without the need for discretization (internal meshing), and retains fine surface detail that often requires simplification, and therefore deviations from geometric accuracy, in traditional FEA of biological structures.The same methods for muscle force estimates, constraints, and material properties were applied to the Raptorex analysis as for the traditional FEA of other tyrannosauroids.Details of modeling and solution for the meshless Raptorex analysis will be revealed in a future study (Snively et al., 2023).(Gignac & Erickson, 2017;Holliday, 2009) where forces were applied onto Tyrannosaurus rex model in Strand 7. Nodes were sampled on homologous areas of all specimens.Names of the muscles are listed in Table 2.

| Muscle force reconstruction
The muscle forces of the theropod taxa were estimated based on previous muscle reconstructions of archosaurs (Figure 1; Table S2; Gignac & Erickson, 2016, 2017;Holliday, 2009;Lautenschlager, 2013).Identification of muscle origination and insertion were based on osteological correlates (Figure 1; Table 2; Cost et al., 2019;Holliday, 2009;Witmer, 1995).Our reconstruction of the origin of the muscle m.pterygoideus dorsalis was anteriorly restricted following Gignac & Erickson (2017).The muscle likely originated posteriorly from the pterygoid as well (Cost et al., 2019;Holliday, 2009).Differences in stress distributions will be revealing with future analyses comparing the reconstructions.Our estimates of overall force magnitudes converge with those of Cost et al. (2019).
These jaw muscle forces were scaled to those previously estimated for a specimen of Tyrannosaurus rex (Gignac & Erickson, 2017;Rowe & Snively, 2021).Specifically, the muscle forces were scaled from the subtemporal fenestrae of the adult Tyrannosaurus rex specimen BHI 3033, as jaw muscles penetrate the subtemporal fenestra and insert onto the mandible (Gignac & Erickson, 2017;Rowe & Snively, 2021;Sakamoto, 2010).We used a ratio of areas of the subtemporal fenestrae and multiplied that by the force of BHI 3033 (Gignac & Erickson, 2017;Rowe & Snively, 2021).We applied the same ratio to individual muscles, assuming that individual muscles' areas were scaled the same as the area of the whole subtemporal fenestra.
Although this division of forces enabled consistency between models, it is likely and even certain that the animals varied in the muscles' relative sizes and contributions to force.For example, the shorter post-temporal bar in Tarbosaurus compared with Tyrannosaurus suggests a relatively smaller m.adductor mandibulae externus superficialis, which originates from this structure.However, the entire postorbial region of the skull is anteroposteriorly shorter in Tarbosaurus, perhaps equalizing the relative forces of individual temporal muscles.Given the escalating permutations, equivalent relative proportions of muscle force is a reasonable baseline assumption, testable with future detailed muscle reconstructions for individual taxa.
We used ImageJ (Schindelin et al., 2012) to directly measure the ventral area of the adductor chamber and calculate the jaw muscle forces (Figure 2).Using figures of tyrannosaur skulls from previous studies, we measured the length of the scale bar in pixels and set the scale in length and units specified in each figure.When measuring the scale bar, the image size was increased, to reduce margins of error.After the scale was set, the tyrannosaur skull was measured, beginning from the posterior end (quadrate) to the anterior or terminal end (premaxilla).This provided the actual length of the theropod's skull.We checked the measurements against those in publications with theropod cranial lengths.Following the measurements, we imported ventral views of the theropods' crania into ImageJ and set the scale based on the recorded measurements of the theropods from lateral view.Our reasoning for using the cranium's ventral view was to reflect the anatomy and direction of the jaw muscles' attachments as the jaw muscles passed through the adductor chamber (Gignac & Erickson, 2017).Using The jaw muscles of tyrannosauroids with their attachments based on inferences of muscle presence (Cost et al., 2019;Gignac & Erickson, 2017;Holliday, 2009;Witmer, 1995).
This calculation assumes that each respective muscle's anatomical cross-sectional area has the same contribution to total subtemporal fenestra area in all of the tyrannosauroid specimens, and that other contributors to PCSA are constant.The scaling is testable by reconstructing volumes of individual muscles in each specimen, using methods of Gignac & Erickson (2017), Lautenschlager (2013), andCost et al. (2019).To calculate the muscle force components, we trigonometrically measured the distances and angles of the jaw muscles' origin/insertion, based on our own models and previous studies as seen in Tables S1-S3 (Gignac & Erickson, 2017;Rowe & Snively, 2021).All estimated forces for jaw muscles and total jaw muscle force of the tyrannosauroids are reported in Table 3, and vertical, lateral, and longitudinal muscle force components for each tyrannosauroid specimens are listed in Tables S4-S12.

| Parameters and procedures for FEA
The FE models were imported into Strand7.We applied a Poisson's ratio and elastic moduli as material properties to estimate the object's resistance to elastic deformation and distortion under mechanical loading (Askeland & Phulé, 2006;Greaves et al., 2011).We considered Alligator bone as an appropriate model for material properties, because crocodylians are closely related to dinosaurs (Porro et al., 2011).We assumed similar average bone density in the theropods' crania to that of a hard-biting Alligator.We therefore assumed similar stiffness and Poisson's ratios to Alligator's (Porro et al., 2011) because these properties are proportional to density (Table S13).Lower densities and stiffness, possible in lighter-skulled birds, would result in linearly greater strain but would not greatly affect stresses (Strait et al., 2005) in similarly shaped structures.
We selected nodes in areas of the crania based on reconstructed muscle attachments and the force component magnitudes were divided by the number of nodes to obtain a force per node (Figure 1; Gignac & Erickson, 2017;Holliday, 2009;Lautenschlager, 2013;Rowe & Snively, 2021).We then assigned constraints at the quadrates (the cranial component of the jaw articulation) and at the anterior maxillary teeth, preventing free body motion and simulating a bite at this tooth position.
To compare the stresses and magnitudes displayed by the tyrannosaur skulls, we interpreted von Mises stress that was present in the crania (Rayfield, 2007).In the chosen stress-level color scale, red indicates high stress and blue indicates low stress.A large concentration of red and yellow suggests high regional stress and a large concentration of blue and teal suggest lower stress.We interpret differing stress concentrations as potentially informative about the feeding function of tyrannosauroids.
The jaw muscle forces of the theropods were calculated by measuring the area of subtemporal fenestra of the theropod crania via the measurement software ImageJ.Here, the areas of the subtemporal fenestrae in the Tyrannosaurus rex (USNM 555000) are outlined in orange in ventral view.
Tyrannosaurus juveniles show expectedly low jaw muscle forces (BMRP 2002.4.1; 9167 N) compared with adults (FMNH PR 2081; 63,322 N) (Figure 4; Table 3) Adult Tyrannosaurus such as FMNH PR 2081 and USNM 555000 were capable of exerting high total jaw muscle forces, 63,322 and 60,788 N, respectively, allowing them to apply high tooth forces when biting.Other large robust-snouted tyrannosaurids showed total jaw muscle forces that were lower than that of an adult Tyrannosaurus (Figure 4).This included Tarbosaurus, Daspletosaurus, and Gorgosaurus having total jaw muscle forces of 11,905, 14,360, and 10,912 N, respectively; Daspletosaurus possessed the highest total jaw muscle force.The total jaw muscle forces of the robust-snouted tyrannosaurids were higher than the total jaw muscle force of a late-stage juvenile Tyrannosaurus (9167 N).Medium-sized tyrannosaurines such as Teratophoneus showed total jaw muscle force of 5232 N with a correspondingly lower adductor chamber area than a juvenile Tyrannosaurus.Early-stage juvenile tyrannosaurines such as Raptorex (1156 N) and a juvenile Tarbosaurus (623 N) showed high total jaw muscle forces than small, earlydiverging tyrannosauroids (Proceratosaurus, 390 N and Dilong 337 N).Both Raptorex and the juvenile Tarbosaurus  possessed a wider adductor chamber than small, earlydiverging tyrannosauroids.This would suggest that juvenile tyrannosaurines could exert a higher bite force, a trend that would continue throughout their ontogeny (Figures 5 and  6).The large early-diverging tyrannosauroid Yutyrannus had a total jaw muscle force of 2405 N, which was lower than a juvenile Tyrannosaurus and other large, robustsnouted tyrannosaurids.The large proceratosaurid possessed a skull length (60 cm) that was shorter than that of large, robust-snouted tyrannosaurids (Tyrannosaurus, 139 cm; Daspletosaurus, 102 cm), but comparable to the skull length of a late-stage juvenile Tyrannosaurus (67 cm).

| FEA stresses
Using color-coding of von Mises stresses, we examined the cranial performance of the 3D models and the stress magnitudes present, with red being an indicator for high stress and blue an indicator for low stress.Statistical quantification of such stress differences will be ideal, as with the pioneering multivariate approaches of Marcé-Nogué et al. (2017).Here, we restrict our interpretation to broad-scale comparisons of stress in homologous cranial regions, because uncertainties in biological input values (muscle morphology and PCSA) recommend cautious global interpretation over subtle, statistically nuanced local variation in stress.
Ontogenetically the level of stress is rather high in the juvenile condition of Tyrannosaurus but decreases upon transitioning into the adult condition (Figure 5).Both of the smaller juvenile tyrannosaurines, Raptorex and juvenile Tarbosaurus, exhibit high cranial stress, although the level of stress magnitude is higher in the former (Figure 6).Both the adult Tyrannosaurus and adult Tarbosaurus exhibit moderately low cranial stress, although the stress concentrations occur at different locations of the cranium (nasals and lacrimals of Tyrannosaurus, and the nasals and maxilla of Tarbosaurus; Figure 6).
Although tyrannosauroids (Figure 7) exhibit varying levels of cranial stress, stress concentrations occur at consistent areas of the cranium such as the nasals, lacrimal, F I G U R E 3 Adductor chamber area regressed against skull length.Juvenile tyrannosaurines such as Raptorex and juvenile Tarbosaurus and Tyrannosaurus exhibit a larger adductor chamber area than similarly sized tyrannosauroids such as Dilong, Proceratosaurus, and Teratophoneus.Adult Tyrannosaurus possessed a relatively greater adductor chamber area than other large tyrannosauroids.Daspletosaurus has a relatively larger adductor chamber area than the Mongolian tyrannosaurine Tarbosaurus, and correspondingly greater bite force for its skull length.Similarly, juvenile Tyrannosaurus and Teratophoneus have much greater adductor chamber areas than Yutyrannus at similar skull lengths.Relatively larger adductor chamber area of tyrannosaurids indicate a relatively higher bite force than earlier tyrannosauroids.pterygoids, and quadrate.Small, early tyrannosauroids and juvenile tyrannosaurines show higher von Mises stress than the low values present in robust-snouted tyrannosauroids (Tyrannosaurus, Yutyrannus).
The proceratosaurid Proceratosaurus (NHM R 4860) shows primarily low cranial stress in the maxilla, lacrimals, postorbitals, quadrates, quadratojugals, and parietals (Figure 7 and Figure S1).Low stress occurs in the crest of Proceratosaurus, along with high stress being present in the palate and pterygoid of Proceratosaurus.Proceratosaurus is similar in size to Raptorex and Dilong but shows overall lower cranial stress.
The large proceratosaurid Yutyrannus (ZCDM V5000) shows low von Mises stress distributions throughout its cranium such as the maxilla, lacrimals, frontals, and parietals, and quadratojugal (Figure 7 and Figure S2).Similar to other tyrannosauroids, high palatal stress is present in Yutyrannus.The rugose nasals of Yutyrannus show primarily low stress, similar to the crest of Proceratosaurus.
The early pantyrannosaur Dilong (IVPP 14243) shows similar cranial stress distributions to the adult Tyrannosaurus USNM 555000.Low stress occurs at the nasallacrimal suture, lacrimal, parietals, and quadratojugal (Figure 7 and Figure S3).High stress is present at the palate and pterygoids.The palatal stress of Dilong is higher than that in Proceratosaurus, but lower than in Raptorex.
The albertosaurine Gorgosaurus (TCMI 2001.89.1) exhibits stress magnitude distribution analogous to that of Teratophoneus (Figure 7 and Figure S4).Relatively lower stress magnitudes are present in the nasals, nasallacrimal suture, lacrimal bar, maxilla, quadratojugal, squamosals, and vomer.Moderately low stress does occur at the quadrate and palate.Gorgosaurus exhibits lower cranial stresses than the similarly sized tyrannosaurine Daspletosaurus.Similar to other large tyrannosaurids, Gorgosaurus exhibits high palatal stress in contrast to the lower palatal stress in Teratophoneus.
The North American tyrannosaurid, Daspletosaurus (NMC [CMN] 8506) exhibits higher stress magnitudes than the similarly sized albertosaurine, Gorgosaurus (Figure 7 and Figure S5).Higher stress magnitudes are present in the nasals, nasal-maxillary suture, and the palate.Daspletosaurus exhibits higher palatal stress in its pterygoid flanges than Gorgosaurus.Low cranial stress magnitudes are present in the squamosal, lacrimal, postorbital, and quadrate.
Teratophoneus (BYU 8120), an early robust-snouted tyrannosaurine, exhibits lower stress magnitudes than a similarly sized, late-stage juvenile Tyrannosaurus (Figure 7 and Figure S6).Lower stress magnitudes are predominantly present in the anterior portion of the cranium (e.g., nasals and maxilla).Moderately low stress occurs at the jugals, squamosals, postorbitals, quadrates, and pterygoid flanges.Low stress magnitudes occur in the palate and pterygoid of Teratophoneus.
The juvenile Tarbosaurus (MPC-D 107/7) displays stress magnitudes that occur at areas of the cranium similar to those in adult Tarbosaurus, although the juvenile condition shows higher cranial stress than the adult condition (Figure 7 and Figure S7).Moderately low stress is present throughout the nasals, maxillae, lacrimal bars, quadrates, and quadratojugals.Relatively high stress is present in the pterygoid and palate of the juvenile Tarbosaurus.The juvenile Tarbosaurus shows higher cranial stress than Raptorex in the nasals and parietal, whereas palatal stress in Raptorex is higher than the juvenile Tarbosaurus.Raptorex also exhibits a wider adductor chamber than the juvenile Tarbosaurus, which contributes to Raptorex's high jaw muscle forces.
The Asian tyrannosaurine Tarbosaurus (ZPAL MgDl/4) displays overall lower cranial stress than in other  tyrannosaurids, with relatively high stress magnitudes in the palate, the inferior portion of the jugal, the superior portion of the maxilla, and the posterior portion of the quadrate (Figure 7 and Figure S8).In contrast to Tyrannosaurus and other North American tyrannosaurids, stress is prominent in the maxillae, nasals, and maxillary-nasal connection of Tarbosaurus.North American tyrannosaurids have stress concentrations occurring at the nasals, lacrimals, and nasal-lacrimal connection.In Raptorex (LH PV18), an early-stage juvenile (Fowler et al., 2011), high cranial stress is present at the lacrimal, postorbital, jugal, quadratojugal, quadrate, and nasallacrimal suture (Figure 7 and Figure S9).High stress magnitudes are present in the palate of Raptorex.It should be noted that both pterygoideus muscles originate from the palate, with m. pterygoideus dorsalis originating at the palatine.High stress occurs at the nasal-lacrimal connection, similar to the juvenile Tarbosaurus (Figure 7 and Figure S7).
The juvenile Tyrannosaurus (BMRP 2002.4.1) displays lower stress magnitudes than Raptorex.Low stress magnitudes are present in the nasals, maxillae, postorbitals, jugals.High peak stress magnitudes are present in the squamosals, quadrates, quadratojugals, and at the pterygoid flanges (Figure 7 and Figure S10).Similar to Raptorex, high stress magnitudes are present in the palatine and at the nasal-lacrimal connection of the late-stage juvenile Tyrannosaurus.
In an adult Tyrannosaurus (USNM 555000), the cranium exhibits stress concentrations similar to other large robust-snouted tyrannosaurids (Figure 7 and Figure S11).Areas of the cranium that show low stress concentrations include the nasals, squamosals, maxillae, quadrates, quadratojugals, and at the nasal-lacrimal suture (Figures 5, 7 and Figure S11).High stress is present at the posterior portion of the quadrate and quadratojugal, as well as at the palatine.High stress occurs at the palatine of the adult Tyrannosaurus.
The senescent adult Tyrannosaurus (FMNH PR 2081) exhibits stress patterns in regions of the cranium (e.g., squamosals, nasals, and nasal-lacrimal suture) that are consistent with those in the adult Tyrannosaurus (USNM 555000) and a late-stage juvenile Tyrannosaurus (Figure 5 and Figure S12).High stresses occur at the posterior portion of the quadrate and pterygoid flanges that are also analogous to stresses in the adult Tyrannosaurus (USNM 555000) and a late-stage juvenile Tyrannosaurus.

| Hypothesis summaries
With Hypothesis 1, we expected gracile-snouted tyrannosauroids (juvenile tyrannosaurines and small early tyrannosauroids) to show higher cranial stress than robust-snouted tyrannosauroids.The von Mises results of our tyrannosauroid crania appear to support Hypothesis 1. Juvenile tyrannosaurines (Raptorex and juvenile Tarbosaurus) show higher cranial stress than small early tyrannosauroids such as Dilong and Proceratosaurus.Levels of muscle force relative to skull size, including relatively greater forces in tyrannosaurids than in earlier tyrannosaurines (see Hypothesis 3 discussion) governed stresses more than snout shape and robustness.This implies that there is not an absolute correlation with cranial morphology and cranial performance (von Mises stress), as predicted with Hypothesis 1.
With Hypothesis 2, we expected the level of cranial stress to decrease as tyrannosauroid body size increased.This did not consistently occur.Small tyrannosauroids such as Proceratosaurus showed low cranial stress, in contrast to the high cranial stress magnitudes in larger tyrannosauroids (Gorgosaurus, juvenile and most adult tyrannosaurines).Adult Tarbosaurus and Teratophoneus exhibited lower stress magnitudes than other similarly sized tyrannosaurines.Stresses did decrease as expected from juvenile to adult Tarbosaurus.These results largely contradict Hypothesis 2.
For Hypothesis 3, we predicted juvenile tyrannosaurines (Raptorex, and juvenile Tarbosaurus, and Tyrannosaurus) would exert higher jaw muscle forces than similarly sized tyrannosauroids (Dilong, Proceratosaurus, and Teratophoneus), given the wide adductor chamber of juvenile tyrannosaurines versus the narrow adductor chamber area of other tyrannosauroids.When scaled proportionally, the calculated jaw muscle forces of juvenile tyrannosaurines were higher than similarly sized tyrannosauroids.Juvenile tyrannosaurines also exhibit higher von Mises stresses than similarly sized tyrannosauroids; Dilong and Teratophoneus showing lower cranial stress than Raptorex and juvenile Tyrannosaurus.The enlarged adductor chamber area and high jaw muscle forces indicate that juvenile tyrannosaurines could exert a higher bite force at the teeth (and experienced concomitantly greater cranial stress) than similarly sized tyrannosauroids, consistent with Hypothesis 3.

| Comparative feeding function of similarly-sized tyrannosauroids and Tyrannosaurus rex: correspondences are not exact
Ontogenetically the skull of Tyrannosaurus transitioned from a gracile-snouted juvenile to a robust-snouted, robust adult.Further analyses have evaluated the dramatic metamorphosis of Tyrannosaurus skull morphology including the theropod's skull roof, snout, and mandible (Bates & Falkingham, 2012;Carr, 2020;Rowe & Snively, 2021).The cranium of Tyrannosaurus experienced more changes than the mandible (Carr, 2020).The mandibular ramus experienced an increase in height, along with an expansion of the angular, surangular, and dentary (Carr, 2020;Monteiro & Soares, 1997;Rowe & Snively, 2021).The cranium of juvenile Tyrannosaurus was low relative to its length, while the adult condition possessed a relatively taller cranium with inflated cranial protuberances such as expanded lacrimal crests, postorbitals, and prominent nasal bumps (Carr, 2020).The palatal bones that serve as attachment points of the m.pterygoideus ventralis in an adult Tyrannosaurus show significant enhancement, supporting the forcefulness of the adductor jaw musculature (Carr, 2020;Cost et al., 2019;Gignac & Erickson, 2017;Rowe & Snively, 2021).
Because FEA reveals the consequences of dramatic changes in the skull morphology and jaw muscle performance in Tyrannosaurus, this study in parallel highlights the correlations and contrasts in the feeding function of similarly sized tyrannosauroids and Tyrannosaurus (juvenile Tyrannosaurus and Teratophoneus; adult Tyrannosaurus and Tarbosaurus).
Small tyrannosauroids such as Raptorex, juvenile Tarbosaurus, Dilong, and Proceratosaurus show cranial stress concentrations in similar areas of the crania such as the quadrate, qaudratojugal, parietals, and squamosals.The juvenile tyrannosaurines display higher cranial and palatal stress than Dilong and Proceratosaurus, a consequence of the juveniles' wider adductor chamber and higher jaw muscle force (especially in Raptorex).This suggests that tyrannosaurines as early-stage juveniles, especially Tyrannosaurus, were possibly adapted at exerting relatively great muscular forces which would continue throughout its ontogeny (Peterson et al., 2021;Peterson & Daus, 2019;Rowe & Snively, 2021).Similar stress magnitudes in larger Tyrannosaurus specimens maintained similar safety factors, despite relatively escalated bite forces.
The late-stage juvenile Tyrannosaurus, BMRP 2002.4.1 exhibits higher cranial stress than Teratophoneus, a robust-snouted tyrannosaurine.Teratophoneus shows lower cranial stress, because the late-stage juvenile Tyrannosaurus exhibits a higher bite force than Teratophoneus, similarly to Raptorex versus other small tyrannosauroids.The adductor chamber of BMRP 2002.4.1 is wider than Teratophoneus.This enabled Tyrannosaurus juveniles to exert relatively higher bite force than Teratophoneus.Although they perforated bone, juvenile Tyrannosaurus lacked the ability to splinter bone as seen in the adult condition (Gignac & Erickson, 2017;Peterson et al., 2021;Peterson & Daus, 2019).
Larger tyrannosauroids including an adult Tyrannosaurus (FMNH PR 2081 and USNM 555000) exhibit high jaw muscle forces and relatively low cranial stress magnitudes in the nasals and squamosals, although high stress does occur at the palate.Like adult Tyrannosaurus, Daspletosaurus and Gorgosaurus exhibited higher stress and jaw muscle force than adult Tarbosaurus and Yutyrannus.
The FEA results and muscle force analyses support analogous stress magnitudes between similarly sized tyrannosauroids and different aged individuals of Tyrannosaurus.The jaw muscle forces, however, suggest that Tyrannosaurus at different ontogenetic stages was capable at exhibiting higher bite forces than other similarly sized tyrannosauroids.

| Inferred feeding behavior and paleoecology of early-diverging Tyrannosauroids (Dilong and Proceratosaurids)
The early-diverging pantyrannosaur Dilong (Brusatte & Carr, 2016) shows low cranial stress and a low jaw muscle force in contrast to juvenile tyrannosaurines.This suggests that the diet of Dilong may have consisted of smaller prey such as mammals, reptiles, and small maniraptoriformes (Chang et al., 2017).Analyses on the braincase and inner ear of Dilong show that the small pantyrannosaur possessed good agility and balance but lacked heightened sense of smell of larger tyrannosauroids (Bistahieversor and Tyrannosaurus) (Kundr at et al., 2018;McKeown et al., 2020).The fusion of nasals in Dilong suggest that they may have been capable of reducing stress when biting, an attribute which would become more pronounced in later tyrannosauroids (Rauhut et al., 2010;Rayfield, 2004;Snively et al., 2006;Xu et al., 2004).
Similarly, to Dilong, the small proceratosaurid Proceratosaurus displays low cranial stress.Similar to juvenile tyrannosaurids, the teeth of Proceratosaurus were ziphodont or flattened, and the animal possessed a relatively slender craniomandibular morphology (Rauhut et al., 2010).This suggests that Proceratosaurus and other small proceratosaurids (Guanlong) likely fed on smaller prey.Moderate stress was present in the posterior portion of the cranial crest of Proceratosaurus, indicating that cranial crests may have acted as a stress sink and buttress while biting, in addition to the crests serving as display features (Rauhut et al., 2010).Fused nasals are also present in proceratosaurids.It has been hypothesized that proceratosaurids (Proceratosaurus) and other early tyrannosauroids (Dilong) may have employed puncture-pull style feeding (Erickson & Olson, 1996;Rauhut et al., 2010;Rayfield, 2004), and this can be further tested (Snively & Russell, 2007).The craniomandibular morphology in small proceratosaurids indicate that early tyrannosauroids possessed specialized cranial and dental adaptations unique among theropods (Rauhut et al., 2010).
Large proceratosaurids such as Yutyrannus possessed a deeper skull and lacked the large, elaborate cranial crests in contrast to small proceratosaurids (Guanlong) (Rauhut et al., 2010;Xu et al., 2006Xu et al., , 2012)).Similar to the reconstructed nasals of Proceratosaurus, the nasals of Yutyrannus exhibited low stress, and may support that the crests of Yutyrannus would have been capable of resisting high forces, especially with larger prey.Specimens of Yutyrannus at different growth stages (juveniles and adults) were recovered (Xu et al., 2012).Yutyrannus may have exhibited niche partitioning with juveniles pursuing smaller prey (Psittacosaurus and maniraptoriformes) and adults feeding on larger prey (Dongebititan) (Therrien et al., 2021;Xing et al., 2012;Zhou, 2006).Sauropod material was recovered in the quarry where the Yutyrannus specimens were found, suggesting that sauropods were a potential food source for the large proceratosaurid (Xing et al., 2012;Xu et al., 2012).

| Inferred feeding behavior and paleoecology of other Tyrannosaurines
Daspletosaurus shared its environment with a variety of ornithischian dinosaurs and the large albertosaurine Gorgosaurus (Eberth, 1997;Farlow & Pianka, 2002;Russell, 1970).The two tyrannosaurids have been hypothesized as having separate niches in which Gorgosaurus fed on more gracile prey (hadrosaurids) and Daspletosaurus fed on ceratopsians (Farlow & Pianka, 2002;Russell, 1970).Hadrosaur bones also were recovered in the stomach area of Daspletosaurus, suggesting that the tyrannosaurine did not have a specific prey preference (Varricchio, 2001).The two tyrannosaurids appear to have had differing postcranial builds with Gorgosaurus being more lightly built and Daspletosaurus being more robust, consistent with Daspletosaurus having a broader muzzle and Gorgosaurus having a lower snout (Snively et al., 2006).
While there have not been recorded feeding traces of the early tyrannosaurine Teratophoneus, it can be inferred that Teratophoneus would have fed upon similarly sized hadrosaurids and ceratopsids similarly to Tyrannosaurus and Gorgosaurus, and experienced ontogenetic niche partitioning (Holtz, 2021;Therrien et al., 2021;Zanno & Sampson, 2005).Teratophoneus was the only large terrestrial predator of its environment.Bonebeds of Teratophoneus specimens at different ages support the possibility of gregarious behavior in tyrannosaurids such as Albertosaurus and Daspletosaurus (Titus et al., 2021).
The consequences of these differences are unclear and remain to be tested.Differing stress distributions may reflect that the two large tyrannosaurines fed on different prey.Bite marks attributed to Tarbosaurus have been recovered on the bones of hadrosaurids, titanosaurs, and the large ornithomimosaur Deinocheirus (Bell et al., 2013;Hurum & Sabath, 2003;Owocki et al., 2019).Although Tarbosaurus likely experienced ontogenetic niche partitioning through growth, high bite force in the juveniles suggests the capability for engaging more robust or resistant prey than likely in similarly sized early tyrannosauroid adults (Holtz, 2021;Maleev, 1955Maleev, , 1974;;Therrien et al., 2021).

| Inferred feeding behavior and paleoecology of Tyrannosaurus
The results present a profile of Tyrannosaurus ontogeny, transitioning from small, generalized carnivores to largebodied predators.Given their small and gracile skull, juvenile Tyrannosaurus were likely incapable of delivering bite forces as great as those of an adult Tyrannosaurus when scaled to the same size.
Given their size and reinforced skull morphology, the larger adult Tyrannosaurus served as the apex predators of their environment.Adult Tyrannosaurus preyed upon a variety of herbivorous dinosaurs such as ceratopsians and hadrosaurs, with healed bite marks attributed to adult tyrannosaurids along with dinosaur bones being recovered in tyrannosaurid coprolites (fossil feces) (Carpenter, 1997;Chin et al., 1998;DePalma et al., 2013;Happ, 2008;Rothschild & DePalma, 2013;Varricchio, 2001).Stress results are consistent in Tyrannosaurus with cranial features such as fused nasals, immobile cranial joints, and robust cranial proportions that allowed tyrannosauroids to resist high compressive forces and display low shear stress.Features such as these enabled Tyrannosaurus to engage in predation involving extensive tooth-bone contact, and effective osteophagy in Tyrannosaurus (Gignac & Erickson, 2017).

| CONCLUSION AND FUTURE DIRECTIONS
The presented results are consistent with adaptations in tyrannosauroids such as fused nasals, cranial crests, akinetic cranial joints, a secondary palate, and expanded adductor chambers that allowed them to deliver powerful bite forces and resist high loadings when feeding (Brusatte et al., 2010;Brusatte & Carr, 2016;Coombs, 1978;Farlow et al., 1991;Holtz, 1994Holtz, , 1995;;Molnar, 1991;Osborn, 1906;Rowe & Snively, 2021;Snively et al., 2004Snively et al., , 2006Snively et al., , 2019;;Snively & Russell, 2002, 2003;Therrien et al., 2021).This study focused on broadly assessing the evolution of feeding function in Tyrannosauroidea, especially with an emphasize of early diverging tyrannosauroids and other large tyrannosaurids.Early tyrannosauroids such as Proceratosaurus and Dilong show low cranial stress, suggesting a lower bite force.This serves as a major contrast to the cranial performance of later-diverging juvenile tyrannosaurines (Raptorex and juvenile Tarbosaurus) which exhibited a higher bite force and higher cranial stress.
The cranial morphologies of tyrannosauroids suggest different feeding adaptations, such as robustsnouted crania adapted for feeding on large prey and gracile-snouted crania adapted for smaller prey (Rowe & Snively, 2021).The current findings offer perspective into further evaluating the feeding function of alioramins, such as Qianzhousaurus and Alioramus.These tyrannosauroids had longirostrine skulls and gracile hindlimbs.It has been hypothesized that alioramins lacked a powerful bite force and were unable to engage in "puncture pull" style feeding compared to robust-snouted tyrannosaurids (Foster et al., 2022;Rayfield, 2004Rayfield, , 2005aRayfield, , 2005b)).Alioramins have a relatively short and narrow postorbital region suggesting lower jaw muscle forces.They also possessed a notably gracile dentary (Foster et al., 2022) suggesting reduced capacity to absorb high bite forces compared with a more robust dentary in other tyrannosaurids with similar skull lengths (Rowe & Snively, 2021).
This study promises fruitful replication of mechanical feeding analyses in other under-studied theropod clades, such as Allosauroidea, Spinosauridae, and Abelisauridae.The inclusion of non-tyrannosauroid theropods will facilitate a detailed comparative analysis of tyrannosauroid biology and evolution with other theropods.
ImageJ's "polygon tool," we traced the left and right adductor chambers separately.The physiological cross-sectional area (PCSA) reflected and incorporated the muscle properties of the animal (e.g., volume, length; Zatsiorsky & Prilutsky, 2012).The total muscle force was the sum of both the left and right adductor chambers.To calculate the individual jaw muscle forces of other tyrannosaurs (Equation1), we used as the baseline the forces fromGignac & Erickson (2017: supporting information)  for an adult Tyrannosaurus BHI 3033, and scaled force magnitudes for the other tyrannosaurs by ratios of subtemporal fenestra areas, multiplied by an isometric force per area of vertebrate muscle, 31.5 N/cm 2(Rowe & Snively, 2021) Individual and total jaw muscle forces (N) for the tyrannosauroid specimens.

F
I G U R E 4 Charted muscle forces of the tyrannosauroid specimens show that adult Tyrannosaurus exhibited by far the highest total muscle force.Other adult tyrannosaurids (Gorgosaurus and Daspletosaurus) show total muscle forces that are either equal to or higher than a late-stage juvenile Tyrannosaurus.Small early-diverging tyrannosauroids (Dilong and Proceratosaurus) exhibit lower muscle forces that are lower than the juvenile tyrannosaurids Raptorex and Tarbosaurus.F I G U R E 5 Legend on next page.

F
I G U R E 5 Comparative von Mises stresses in tyrannosaurine crania, with cooler colors indicating lower proximity to yield (damaging) stress, and hotter colors indicating closer proximity to yield ($80 MPa in shear).Lateral profiles (top to bottom) of Raptorex kriegsteini (LH PV18), juvenile Tyrannosaurus (BMRP 2002.4.1), adult Tyrannosaurus (USNM 555000), and senescent adult Tyrannosaurus (FMNH PR 2081) crania showing the absolute stress and performance of the Tyrannosaurus at various ontogenetic stages.The large adult Tyrannosaurus (FMNH PR 2081) has proportionally greater jaw muscle force than the smaller adult yet lacks a relatively taller cranium, and does not have appreciably lower stress despite its greater size.The greatest visualized stress is 20 MPa for all crania; white indicates stress greater than 20 MPa.Raptorex shows greater palatal stress than in the larger tyrannosaurine specimens.

F
I G U R E 6 Ontogenetic comparisons of and adult Tyrannosaurus (a), Raptorex (b), and an adult and juvenile Tarbosaurus (c and d, respectively).The adult Tyrannosaurus and Tarbosaurus display differing stress magnitudes at the jugal and squamosal (see text).Raptorex displays lower cranial stress in the nasals and maxilla than the juvenile Tarbosaurus, although Raptorex shows higher stress in the palate and postorbital than the juvenile Tarbosaurus.The greatest color-visualized stress is 20 MPa; white areas indicate stresses greater than this value.All crania maintained adequate safety factors.

F
I G U R E 7 Tyrannosauroids display varying cranial morphologies with differing levels of cranial stress.Greater stresses are typically concentrated in the nasals, lacrimals, quadrate, and postorbitals.Note the cooler colors and lower von Mises stresses in earlier tyrannosauroids, consistent with relatively lower bite forces.
Tyrannosaurus and other tyrannosauroid specimens used for finite element analyses.
T A B L E 1