RESEARCH ARTICLE

Ontogeny of bite force in a validated biomechanical model of theAmerican alligatorKaleb C Sellers1 Kevin M Middleton1 Julian L Davis2 and Casey M Holliday1

ABSTRACTThree-dimensional computational modeling offers tools with which toinvestigate forces experienced by the skull during feeding and otherbehaviors American alligators (Alligator mississippiensis) generatesome of the highest measured bite forces among extant tetrapods Aconcomitant increase in bite force accompanies ontogeneticincreases in body mass which has been linked with dietarychanges as animals increase in size Because the flattened skull ofcrocodylians has substantial mediolaterally oriented musclescrocodylians are an excellent model taxon in which to explore therole of mediolateral force components experienced by the feedingapparatus Many previous modeling studies of archosaur cranialfunction focused on planar analysis ignoring the mediolateralaspects of cranial forces Here we used three-dimensionallyaccurate anatomical data to resolve 3D muscle forces Usingdissection imaging and computational techniques we developedlever and finite element models of an ontogenetic series of alligatorsto test the effects of size and shape on cranial loading and comparedestimated bite forces with those previously measured in vivo inA mississippiensis We found that modeled forces matched in vivodata well for intermediately sized individuals and somewhatoverestimated force in smaller specimens and underestimated forcein larger specimens suggesting that ontogenetically static muscularparameters and bony attachment sites alone cannot account for allthe variation in bite force Adding aponeurotic muscle attachmentswould likely improve force predictions but such data are challengingto model and integrate into analyses of extant taxa and are generallyunpreserved in fossils We conclude that anatomically accuratemodeling of muscles can be coupled with finite element and leveranalyses to produce reliable reasonably accurate estimate biteforces and thus both skeletal and joint loading with known sources oferror which can be applied to extinct taxa

KEY WORDS Crocodylia Biomechanics Feeding Finite elementanalysis Modeling

INTRODUCTIONDeveloping computational models that accurately estimate biologicaland biomechanical functions remains difficult for researchersinterested in complex systems such as the feeding apparatus ofvertebrates Even modeling superficially simple systems such as theakinetic two-part skull of crocodylians poses challenges given the

complex 3D nature of the bones and the jaw muscles that actuatethem Nonetheless accurately characterizing the biomechanicalperformance of the feeding apparatus is critical to ourunderstanding of the evolution of the skull Many modelingtechniques have been advanced in recent years such as finiteelement analysis (FEA) multi-body dynamic analysis and BoneLoad(Davis et al 2010) but the predictions generated by these techniquesoften remain to be validated by comparisons with in vivo dataValidated modeling may be used to carry out in silico experimentalinvestigations into the performance of the feeding apparatus in extanttaxa Moreover the use of validated models is critical to studies ofform and function in fossil taxa

The feeding apparatus of extant crocodylians produces thehighest measured bite forces among extant tetrapods (Ericksonet al 2003) Counter-intuitively crocodylians do not display thetypical morphology of other hard-biting tetrapods such as tegulizards hyenas or Tyrannosaurus in which the skull is dorsallyheightened expanding the attachment area of temporal muscles andresisting dorsoventral bending of the rostrum (Molnar 1998Metzger and Herrel 2005 Tseng and Stynder 2011 Schaerlaekenet al 2012) Instead crocodylians evolved a dorsoventrallyflattened skull which is hypothesized to be an adaptation foraquatic ambush predation (Iordansky 1973) This configuration isassociated with a suite of biomechanical modifications of thefeeding apparatus The consequences of this flattened skull includea shift in temporal muscles toward more mediolateral orientations aswell as increased resistance to mediolateral bending and axialtorsion of the cranium (Busbey 1995) The discordance between themorphology of the crocodylian skull and that of most animals withhard bites suggests that study of the former group can yield valuableinsights into the anatomical determinants of feeding performance

The morphology and performance of the crocodylian feedingapparatus have been the focus of numerous investigations over thepast century The early comparative and functional morphologicalstudies of Iordansky (1964) and Langston (1973) set the stage formore quantitative biomechanical investigations of the crocodylianfeeding apparatus such as the functionalmorphological investigationsof Sinclair and Alexander (1987) Busbey (1989 1995) and Cleurenet al (1995) These early works used 2D static equilibrium leveranalyses to estimate the magnitude and orientation of bite force andjoint forces in the parasagittal plane More recently advances incomputational power have facilitated in silico quantitative studies offeeding biomechanics FEA provides tools to computationallyinvestigate stress and strain (Daniel and McHenry 2001 Metzgeret al 2005) and reaction forces (Porro et al 2011 2013) under awiderange of loading conditions Researchers have applied this techniqueto investigate how feeding behavior loads the rostrum in comparativesamples of extant crocodylians (McHenry et al 2006 Pierce et al2008) comparing among archosaurs (Rayfield et al 2007Rayfield and Milner 2008) and extinct crocodylomorphs such asthalattosuchians (Pierce et al 2009) Using multi-body dynamicsReceived 11 January 2017 Accepted 15 March 2017

1Department of Pathology and Anatomical Sciences University of Missouri M263Medical Sciences Building Columbia MO 65212 USA 2Department ofEngineering University of Southern Indiana 2030 Business and EngineeringCenter 8600 University Boulevard Evansville IN 41172 USA

Author for correspondence (kcsty5mailmissouriedu)

KCS 0000-0002-3588-9562

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copy 2017 Published by The Company of Biologists Ltd | Journal of Experimental Biology (2017) 220 2036-2046 doi101242jeb156281

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Bates and Falkingham (2012) compared estimates of bite force ofTyrannosaurus with estimates from various tetrapods including alarge individual of Alligator mississippiensis These authors foundthat peak (impact) bite forces in A mississippiensis matchedpreviously reported maximum in vivo dataNumerous in vivo studies have provided crucial data and

advanced our understanding of the performance of the feedingapparatus in crocodylians Researchers using electromyography(EMG) and X-ray cineradiography found that all major groups ofadductor muscles were active during crushing bites but showed thatthe pterygoideus muscles were inactive during holding bites (VanDrongelen and Dullemeijer 1982 Busbey 1989 Cleuren and deVree 1992 Cleuren et al 1995) Metzger et al (2005) measuredin vivo strain in the cranium during biting and found that these databroadly agreed with computational predictions Most recently Porroet al (2013) found that in vivo strain magnitudes in the mandiblesgenerally surpass predictions from FEAIn vivo bite force recordings using force transducers are an

invaluable source of data for further understanding the ontogeneticscaling and comparative biomechanics of the crocodylian feedingapparatus Erickson and colleagues conducted a series of studiesthat measured bite force in a growth series of American alligator(Erickson et al 2003 Gignac and Erickson 2015) and extantcrocodylian species (Erickson et al 2012 2014) Across a sizerange of 315 to 4055 cm total length these studies found a positiveallometric relationship (b) between maximum bite force and avariety of body size proxies such as total length (b=262isometry=2) snoutndashvent length (b=259 isometry=2) mass(b=079 isometry=0667) and head length (b=275 isometry=2)Erickson et al (2003) hypothesized that this positively allometricincrease in bite force may be responsible for intraspecific nichepartitioning in A mississippiensis Furthermore with the exceptionof Gavialis gangeticus all extant crocodylians of a given size haveequivalent bite forces These studies have provided a solid in vivobasis against which in silico predictions may be tested

Gignac and Erickson (2016) compared estimates ofA mississippiensis bite force derived from static bite force modelingwith their previously published in vivo findings The modelingtechniques employed by the authors reliably calculate bite force inA mississippiensis These authors used photographs of dissections toprecisely measure muscular physiological cross-sectional area(PCSA) Although this method is successful in calculating bite forcein A mississippiensis it relies on access to cadaveric specimens andas such is not applicable to the fossil record

Although most previous modeling studies investigated the effectsof muscle force on cranial forces in the skulls of crocodylians fewrelied on anatomically detailed muscular attachment geometry from

List of symbols and abbreviations3D LM three-dimensional lever mechanicsAins area of muscle insertion (mandibular attachment)Aor area of muscle origin (cranial attachment)FB bite force vectorFEA finite element analysisFM muscle force magnitudeFM muscle force vectorlf fiber lengthlM muscle lengthmAMEM M adductor mandibulae externus medialismAMEP M adductor mandibulae externus profundusmAMES M adductor mandibulae externus superficialismAMP M adductor mandibulae posteriormDM M depressor mandibulaemPSTp M pseudotemporalis profundusmPSTs M pseudotemporalis superficialismPTd M pterygoideus dorsalismPTv M pterygoideus ventralisMJJA magnitude of moment about jaw joint axisPCSA physiological cross-sectional arearB position vector of bite forcerM position vector of muscle force vectorTspecific specific tension of muscleuJJA unit vector of jaw joint axisVM muscle volumeθ pennation angle

AL 031 skull length=48 cm

AL 622 skull length=99 cm

AL 612 skull length=203 cm

AL 024 skull length=269 cm

AL 700 skull length=333 cm

AL 008 skull length=454 cm

Fig 1 Left lateral (left) and caudal (right) views of modeled Alligatormississippiensis specimens All models are scaled to the same skull length

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an ontogenetic series of animals Of course some studies of extantcrocodylian biomechanics have benefited from dissection ofcadaveric specimens to inform muscle force reconstructions(Porro et al 2011 Gignac and Erickson 2016) but thesemethods are not immediately applicable to questions of cranialfunction in extinct crocodyliforms As such accurate methods formuscle modeling in fossil taxa should be pursued to betterunderstand the evolution of this system Thus despite the longhistory of functional analyses of crocodylian skulls ourunderstanding of the biomechanical environment of the feedingapparatus requires improvementHere we tested the hypothesis that bite force predictions using

digital models of skull morphology myology and 3D computationalmodeling will be consistent with in vivo data Specifically wehypothesized that our model will generate bite forces consistentwith the positively allometric relationship previously reported inin vivo bite force data from an ontogenetic series ofAmississippiensis(Erickson et al 2003) Bite force was calculated using 3D levermechanics and FEA Although finite element software was used tocreate models map muscle attachment sites and interface withBoneLoad in order to distribute muscle forces we used FEA tocalculate bite force we did not investigate stress or straindistributions or deformation in the present study Barringbiologically unrealistic deformation forces calculated with FEAshould converge with results obtained with 3D lever mechanics

MATERIALS AND METHODSSpecimens and model constructionFive frozen unpreserved specimens of A mississippiensis Daudin1802 were obtained from Rockefeller Wildlife Refuge (GrandChenier LA USA) and a single dry skull was obtained from aprivate collector Skull lengths ranged from 48 to 454 cm (Fig 1Table 1) corresponding to total body lengths of approximately 38ndash326 cm (Woodward et al 1995) The smallest individual was CTscanned at the University of Missouri Biomolecular Imaging Center(Siemens Inveon MicroCT Siemens Medical Solutions USA IncMalvern PA USA) the largest individual was scanned at theUniversity of Missouri School of Medicine Department ofRadiology (Siemens Somatom Definition Scanner SiemensMedical Solutions USA Inc) All other animals were scanned atthe University of Missouri School of Veterinary Medicine (GELightSpeed VCT CT scanner GE Medical Milwaukee WI USA)Stacked images were manually segmented in Avizo 9

(Visualization Sciences Group SAS Merignac France Fig 2A)and three-dimensional models of skeletal anatomy were created(Fig 2B) Using Geomagic Studio 13 (Geomagic Inc ResearchTriangle Park NC USA) models were aligned to world axes andcleaned to remove features that unnecessarily increasedcomputational time Meshes were constructed with four-nodedtetrahedra in Strand7 (G1D Computing Pty Ltd Sydney Australia

Fig 2C) Four-noded tetrahedral bricks were used to construct finiteelement models Finally muscle attachments were lsquomappedrsquo ontothe models using Strand7 following the methods of Grosse et al(2007 Fig 2D) All models had at least 500000 elements to ensuremodels behaved convergently (McCurry et al 2015 Table S1) Thedimensions of the models are x is positive in the left lateraldirection y is positive in the dorsal direction and z is positive in therostral direction All models were tested at 5 deg of gape Finiteelement models are available online (httpsosfiojmpck) Wemodeled unilateral left-sided static crushing bites at the mostcaudally located maxillary tooth where bite force is theoreticallyhighest and where Erickson et al (2003) measured bite forceMuscles were assumed to contract maximally as reported byprevious EMG studies (Busbey 1989 Cleuren et al 1995)

Muscle modelingCalculations of bite force using both 3D lever mechanics (3D LM)and FEA require either direct measurements or accurate estimationsof the force of muscular contraction Muscles generate force inproportion to PCSA (Gans 1982) PCSA is a function of musclevolume fiber length (in terms of fractions of total muscle length)and muscle pennation (Gans 1982) Because cross-sectional area ischallenging to measure volume is divided by fiber length to

CT data

Segment

3D model

Clean manipulateand mesh

Finite element modelMapped finiteelement model

Bone load

3D lever mechanics

Finite element analysis

A B

Mapattachments

CD

E

F

G

Fig 2 Workflow of model creation and analysis (A) Raw CT scans weresegmented manually (B) Manual segmentation data were used to generate a3D model (C) Models were cleaned manipulated to the same gape andmeshed to create finite element models (D) Bony muscle attachment siteswere mapped to surfaces of the model (E) The computational packageBoneLoad (Davis et al 2010) was used to distribute muscle forces acrossattachment sites Bite point is indicated with a vertical arrow 3D levermechanics (F) and finite element analysis (G) were used to calculate biteforces

Table 1 Alligator mississippiensis specimens skull length scan dataand model data

Specimen

Skulllength(cm)

Pixel size(mm)

Interslicespacing (mm)

Tetrahedron no(skull)

AL 031 48 0083374 0083374 612061AL 622 99 016 05 1242107AL 612 203 025 05 988762AL 024 269 0429689 0625 967293AL 700 333 051 05 1313622AL 008 454 0570313 06 613219

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an ontogenetic series of animals Of course some studies of extantcrocodylian biomechanics have benefited from dissection ofcadaveric specimens to inform muscle force reconstructions(Porro et al 2011 Gignac and Erickson 2016) but thesemethods are not immediately applicable to questions of cranialfunction in extinct crocodyliforms As such accurate methods formuscle modeling in fossil taxa should be pursued to betterunderstand the evolution of this system Thus despite the longhistory of functional analyses of crocodylian skulls ourunderstanding of the biomechanical environment of the feedingapparatus requires improvementHere we tested the hypothesis that bite force predictions using

digital models of skull morphology myology and 3D computationalmodeling will be consistent with in vivo data Specifically wehypothesized that our model will generate bite forces consistentwith the positively allometric relationship previously reported inin vivo bite force data from an ontogenetic series ofAmississippiensis(Erickson et al 2003) Bite force was calculated using 3D levermechanics and FEA Although finite element software was used tocreate models map muscle attachment sites and interface withBoneLoad in order to distribute muscle forces we used FEA tocalculate bite force we did not investigate stress or straindistributions or deformation in the present study Barringbiologically unrealistic deformation forces calculated with FEAshould converge with results obtained with 3D lever mechanics

MATERIALS AND METHODSSpecimens and model constructionFive frozen unpreserved specimens of A mississippiensis Daudin1802 were obtained from Rockefeller Wildlife Refuge (GrandChenier LA USA) and a single dry skull was obtained from aprivate collector Skull lengths ranged from 48 to 454 cm (Fig 1Table 1) corresponding to total body lengths of approximately 38ndash326 cm (Woodward et al 1995) The smallest individual was CTscanned at the University of Missouri Biomolecular Imaging Center(Siemens Inveon MicroCT Siemens Medical Solutions USA IncMalvern PA USA) the largest individual was scanned at theUniversity of Missouri School of Medicine Department ofRadiology (Siemens Somatom Definition Scanner SiemensMedical Solutions USA Inc) All other animals were scanned atthe University of Missouri School of Veterinary Medicine (GELightSpeed VCT CT scanner GE Medical Milwaukee WI USA)Stacked images were manually segmented in Avizo 9

(Visualization Sciences Group SAS Merignac France Fig 2A)and three-dimensional models of skeletal anatomy were created(Fig 2B) Using Geomagic Studio 13 (Geomagic Inc ResearchTriangle Park NC USA) models were aligned to world axes andcleaned to remove features that unnecessarily increasedcomputational time Meshes were constructed with four-nodedtetrahedra in Strand7 (G1D Computing Pty Ltd Sydney Australia

Fig 2C) Four-noded tetrahedral bricks were used to construct finiteelement models Finally muscle attachments were lsquomappedrsquo ontothe models using Strand7 following the methods of Grosse et al(2007 Fig 2D) All models had at least 500000 elements to ensuremodels behaved convergently (McCurry et al 2015 Table S1) Thedimensions of the models are x is positive in the left lateraldirection y is positive in the dorsal direction and z is positive in therostral direction All models were tested at 5 deg of gape Finiteelement models are available online (httpsosfiojmpck) Wemodeled unilateral left-sided static crushing bites at the mostcaudally located maxillary tooth where bite force is theoreticallyhighest and where Erickson et al (2003) measured bite forceMuscles were assumed to contract maximally as reported byprevious EMG studies (Busbey 1989 Cleuren et al 1995)

Muscle modelingCalculations of bite force using both 3D lever mechanics (3D LM)and FEA require either direct measurements or accurate estimationsof the force of muscular contraction Muscles generate force inproportion to PCSA (Gans 1982) PCSA is a function of musclevolume fiber length (in terms of fractions of total muscle length)and muscle pennation (Gans 1982) Because cross-sectional area ischallenging to measure volume is divided by fiber length to

CT data

Segment

3D model

Clean manipulateand mesh

Finite element modelMapped finiteelement model

Bone load

3D lever mechanics

Finite element analysis

A B

Mapattachments

CD

E

F

G

Fig 2 Workflow of model creation and analysis (A) Raw CT scans weresegmented manually (B) Manual segmentation data were used to generate a3D model (C) Models were cleaned manipulated to the same gape andmeshed to create finite element models (D) Bony muscle attachment siteswere mapped to surfaces of the model (E) The computational packageBoneLoad (Davis et al 2010) was used to distribute muscle forces acrossattachment sites Bite point is indicated with a vertical arrow 3D levermechanics (F) and finite element analysis (G) were used to calculate biteforces

Table 1 Alligator mississippiensis specimens skull length scan dataand model data

Specimen

Skulllength(cm)

Pixel size(mm)

Interslicespacing (mm)

Tetrahedron no(skull)

AL 031 48 0083374 0083374 612061AL 622 99 016 05 1242107AL 612 203 025 05 988762AL 024 269 0429689 0625 967293AL 700 333 051 05 1313622AL 008 454 0570313 06 613219

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iology

estimate cross-sectional area as defined in Eqn 1 (Sacks and Roy1982)

PCSA frac14 VM

lf cosethuTHORN eth1THORN

where VM is muscle volume lf is the fiber length of the muscle and θis the angle of pennation Fiber length and pennation data are fromPorro et al (2011) As a goal of this study was to validate a methodwith applicability to the fossil record we chose to calculate PCSAby estimating muscle volume from the surface area of attachmentsites rather than from measured muscle volume Muscle volumeswere therefore estimated by treating each muscle as a frustum acone with its apex cut off parallel with its base Eqn 2 defines thevolume of a frustum

VM frac14 lM3 ethAor thorn Ains thorn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAor Ains

pTHORN eth2THORN

where lM is the length of the muscle Aor is the surface area of theorigin of the muscle and Ains is the surface area of its insertionMuscle attachment sites were mapped onto finite element models inStrand7 Because of small variations in mesh construction muscleattachment areas were not always perfectly symmetrical but neverdiffered by more than 5 Dissections and reference to the literature(Iordansky 1964 2000 Schumacher 1973 Busbey 1989Holliday and Witmer 2007 Holliday et al 2013) guided musclemapping The ratio between PCSA and force produced is specifictension defined in Eqn 3

FM frac14 PCSA Tspecific eth3THORNwhere FM is muscle force and Tspecific is specific tension Specifictension data value is from Porro et al (2011)All muscle terminology follows Holliday and Witmer (2007) In

the present study muscles modeled were M adductor mandibulaeexternus superficialis (mAMES) M adductor mandibulaeexternus medialis (mAMEM) M adductor mandibulae externusprofundus (mAMEP) M adductor mandibulae posterior (mAMP)M pseudotemporalis superficialis (mPSTs) M pseudotemporalisprofundus (mPSTp) M pterygoideus dorsalis (mPTd)M pterygoideus ventralis (mPTv) and M depressor mandibulae(mDM) In extant crocodylians most cranial muscles havesubstantial mediolateral components (mAMEM mAMEPmPSTp) rostrocaudal components (mPTd) or both (mAMESmAMP mPSTs mPTv mDM Fig 3)In this study muscle force was distributed over the surface area of

attachment of the origin rather than modeled as a single vector Eachface of a tetrahedral element belonging to a muscle origin bore aportion of the total force directed at the centroid of the muscleinsertion The computational toolkit BoneLoad version 7 (Daviset al 2010) was used for distributing muscle forces across attachmentsites and to calculate moments about axes BoneLoad was originallyused in modeling bite forces in phyllostomid bats and its predictionsare well supported by in vivo measurements (Davis et al 2010Santana et al 2010) BoneLoad uses the geometry of muscleattachments (Fig 3) and magnitude of muscle forces to automate thecalculation of moments about an axis of rotation and distribute thesemuscular forces (Fig 2E) These muscular force distributions werethen used in both 3D lever analysis (Fig 2F) and FEA (Fig 2G)

3D LMLever systems transmit force by the rotational tendency of anelement about an axis An input force acting at a distance from this

axis imparts a moment of force (a measure of rotational tendency)around the axis A second object at a distance from the axis willexperience an output force resisting this rotational tendency In thefeeding apparatus cranial muscles provide the input force forrotation and the food item experiences the output force which isrealized as bite force The perpendicular distance from the muscleforce vector to the axis of rotation is the moment arm of the muscleEqn 4 describes the calculation of moments about the jaw jointaxis (JJA)

MJJA frac14 uJJA ethrM FMTHORN eth4THORNwhere MJJA is the moment about the jaw joint axis uJJA is the unitvector describing the JJA (defined as the vector passing through themiddle of the joint surfaces of each articular bone) rM locates themuscle insertion (and thus the muscle force vector) relative to one ofthe jaw joints and FM is the vector describing the magnitude andorientation of muscle force

Output forces in lever systems act perpendicularly to the planecontaining the axis of rotation and the output moment arm Eqn 5describes the relationship of moments about the jaw joint axis andbite force

MJJA frac14 uJJA ethrB FBTHORN eth5THORNwhere rB locates the bite point relative to a jaw joint and FB describesthe magnitude and orientation of bite force Other variables are as inEqn 4 By performing these calculations for each muscle the totalmoment about the jaw joint axis was calculated Bite force is thenthe quotient of total moments and the perpendicular distance fromthe bite point to the jaw joint axis Fig 4 illustrates the calculation ofbite force using lever mechanics for a single muscle

mAMESmAMEMmAMEP

mAMPmPSTsmPSTp

mPTdmPTvmDM

C D

A B

E F

Fig 3 Left lateral (top) caudal (middle) and dorsal (bottom) views ofmuscle attachments (ACE) Attachment sites on bony morphology(BDF) Skeleton removed to show attachment sites alone Muscleabbreviations are as in Materials and methods

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FEABite force was also calculated using FEA In FEA the userconstrains the degrees of freedom at specified nodes The FEAsoftware then calculates the force required at each constraint tomaintain equilibrium Methods described by Strait et al (2005)were followed to assign constraints A single node at the tip of thecaudal tooth and a single node in the middle of the articular surfaceof each quadrate bone were constrained in all three translational andall three rotational degrees of freedom Material properties ofalligator mandibular cortical bone were assigned to all elements ofthe FEM following Zapata et al (2010) The scope of this projectprohibited the inclusion of cranial sutures in the models further thematerial properties of sutures in A mississippiensis (or indeed thematerial properties of cranial sutures in any reptile) are unknownPorro et al (2011) found that although including sutures in finiteelement models affects stress and strain distributions in the alligatormandible reaction forces including bite force were not dramaticallyaffected Although we were not investigating stress and straindistributions in the skull in the present study we would expectartificial concentrations of stress and strain near bite points andmuscle attachments (Curtis et al 2013) For these reasons cranialsutures were not included here Because FEA provides forceorientations the component of force in each dimension is reportedin addition to overall magnitudes (mediolateral Fx dorsoventral Fyrostrocaudal Fz total Fsum)

Statistical analysisBiomechanical models are useful only insofar as they produceconsistent results that are at least broadly comparable with in vivodata To validate this method bite forces calculated using both 3D

LM and FEAwere compared with in vivo bite force data reported byErickson et al (2003) Erickson and colleagues measured maximumin vivo bite force in an ontogenetic series of A mississippiensisusing force transducers To assess how bite forces calculated in thisstudy correspond to in vivo data ordinary least squares regressionwas conducted on bite force calculated with both FEA and 3D LMagainst skull length using R (httpwwwR-projectorg) BecauseErickson et al (2003) did not report skull lengths we used thepublished relationship of skull length against snoutndashvent length(Woodward et al 1995) to calculate skull lengths for individuals inthe study of Erickson et al (2003) Ordinary least squares regressionis justified over standardized major axis regression becausealthough skull lengths were presumably not measured withouterror the error is likely to be low and the ratio of this error to theerror in either directly measured or estimated bite force is also lowTo compare slopes of regressions of log-transformed bite force onlog-transformed skull length between Erickson et alrsquos (2003) dataand our results we used a linear model with data source (in vivoFEA and 3D LM) skull length and the interaction term Thisanalysis of covariance model allows each source of bite force data tohave a separate slope while allowing comparison between slopes

In the case of significantly different slopes between modeled andmeasured bite force data we used the JohnsonndashNeyman techniqueto determine the region in which there is no significant difference inslope (Johnson and Neyman 1936 White 2003) The JohnsonndashNeyman technique compares two regressions and provides upperand lower values of the independent variable between which slopesdo not significantly differ We used the JohnsonndashNeyman techniqueto compare in vivo data with both FEA and 3D LM forces All codefor analysis is available online (httpsosfiojmpck)

rBBite point

x

y

FM

rM

rJJA

rJJA

FMrM

rJJA

rJJA

FBrB

z

y

Bite point

A

D

B

F

C

E

Fig 4 3D lever analysis Muscleattachment colors are as in Fig 3 Left leftlateral view Right rostral view (AB) Muscleattachments (CD) Calculation of momentabout jaw joint axis (JJA) Attachment ofadductor mandibulae posterior (mAMP) ishighlighted FM muscle force rMperpendicular vector from muscleattachment site to axis of rotation rJJA vectorbetween the two jaw joints (EF) Calculationof output (bite) force FB bite force rBperpendicular vector from axis of rotation tobite point See Eqns 4 and 5

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FEABite force was also calculated using FEA In FEA the userconstrains the degrees of freedom at specified nodes The FEAsoftware then calculates the force required at each constraint tomaintain equilibrium Methods described by Strait et al (2005)were followed to assign constraints A single node at the tip of thecaudal tooth and a single node in the middle of the articular surfaceof each quadrate bone were constrained in all three translational andall three rotational degrees of freedom Material properties ofalligator mandibular cortical bone were assigned to all elements ofthe FEM following Zapata et al (2010) The scope of this projectprohibited the inclusion of cranial sutures in the models further thematerial properties of sutures in A mississippiensis (or indeed thematerial properties of cranial sutures in any reptile) are unknownPorro et al (2011) found that although including sutures in finiteelement models affects stress and strain distributions in the alligatormandible reaction forces including bite force were not dramaticallyaffected Although we were not investigating stress and straindistributions in the skull in the present study we would expectartificial concentrations of stress and strain near bite points andmuscle attachments (Curtis et al 2013) For these reasons cranialsutures were not included here Because FEA provides forceorientations the component of force in each dimension is reportedin addition to overall magnitudes (mediolateral Fx dorsoventral Fyrostrocaudal Fz total Fsum)

Statistical analysisBiomechanical models are useful only insofar as they produceconsistent results that are at least broadly comparable with in vivodata To validate this method bite forces calculated using both 3D

LM and FEAwere compared with in vivo bite force data reported byErickson et al (2003) Erickson and colleagues measured maximumin vivo bite force in an ontogenetic series of A mississippiensisusing force transducers To assess how bite forces calculated in thisstudy correspond to in vivo data ordinary least squares regressionwas conducted on bite force calculated with both FEA and 3D LMagainst skull length using R (httpwwwR-projectorg) BecauseErickson et al (2003) did not report skull lengths we used thepublished relationship of skull length against snoutndashvent length(Woodward et al 1995) to calculate skull lengths for individuals inthe study of Erickson et al (2003) Ordinary least squares regressionis justified over standardized major axis regression becausealthough skull lengths were presumably not measured withouterror the error is likely to be low and the ratio of this error to theerror in either directly measured or estimated bite force is also lowTo compare slopes of regressions of log-transformed bite force onlog-transformed skull length between Erickson et alrsquos (2003) dataand our results we used a linear model with data source (in vivoFEA and 3D LM) skull length and the interaction term Thisanalysis of covariance model allows each source of bite force data tohave a separate slope while allowing comparison between slopes

In the case of significantly different slopes between modeled andmeasured bite force data we used the JohnsonndashNeyman techniqueto determine the region in which there is no significant difference inslope (Johnson and Neyman 1936 White 2003) The JohnsonndashNeyman technique compares two regressions and provides upperand lower values of the independent variable between which slopesdo not significantly differ We used the JohnsonndashNeyman techniqueto compare in vivo data with both FEA and 3D LM forces All codefor analysis is available online (httpsosfiojmpck)

rBBite point

x

y

FM

rM

rJJA

rJJA

FMrM

rJJA

rJJA

FBrB

z

y

Bite point

A

D

B

F

C

E

Fig 4 3D lever analysis Muscleattachment colors are as in Fig 3 Left leftlateral view Right rostral view (AB) Muscleattachments (CD) Calculation of momentabout jaw joint axis (JJA) Attachment ofadductor mandibulae posterior (mAMP) ishighlighted FM muscle force rMperpendicular vector from muscleattachment site to axis of rotation rJJA vectorbetween the two jaw joints (EF) Calculationof output (bite) force FB bite force rBperpendicular vector from axis of rotation tobite point See Eqns 4 and 5

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iology

RESULTSBite force typically refers only to the compressive (ie dorsoventral)components of force acting on a food item as this is the component offorce that will do work on food FEA calculates forces acting in alldimensions but lever mechanics calculates only forces actingperpendicular to the plane containing the axis of rotation and thepoint of force application Similarly the bite force transducers used byErickson et al (2003)measured only forces acting perpendicular to thelong axis of the cantilever In both cases forces are dorsoventrallyoriented The results of FEA presented below therefore only includethe magnitude of dorsoventral force and statistical analyses wereperformed on only the dorsoventral component of bite force whichwas typically sim90 of total bite force

Model construction and muscle modelingRegression of muscle length volume and force against skull lengthshowed that these parameters scaled isometrically in all muscleswith the exception of mAMEP and mPSTp in which muscle

volume and force scaled with slight negative allometry (slopeestimates of 159 and 152 respectively see Table 2 and Table S1)Fig 5 shows reconstructed muscle force and the proportion eachmuscle contributes to total muscle force Note that mAMP andmPTd together account for approximately two-thirds of muscleforce in our model However our methods likely underestimate theforce of mPTv see Discussion

3D LM and FEABite force estimates ranged from 493 N in the smallest individual(both methods) to 3460 N in the largest individual (3D LM) Biteforce estimation with 3D LM and FEA yielded nearly identicalresults (Table 3) Magnitudes of total bite forces calculated withFEA and 3D LM differed by lt6Whereas bite forces calculated inintermediately sized individuals matched in vivo datawell bite forcein larger and smaller individuals diverged from in vivo data withlower force estimates in larger individuals and higher forceestimates in smaller individuals relative to in vivo data Thepercentage contribution of a muscle to bite force is not necessarilythe same as its percentage contribution to total muscle force (Fig 5Tables 4 and 5) because muscles vary in attachment site geometryin the crocodylian adductor chamber For comparisons with in vivobite force data we only considered the dorsoventral component ofbite force However the conditions of static equilibrium demandthat forces be balanced in all three dimensions Therefore our FEAalso calculated rostrocaudal and mediolateral components of biteforce (Table 6) Bite points experienced medially and rostrallyacting forces in addition to dorsoventral force

Statistical analysisRegression of bite force against skull length showed that both 3DLM- and FEA-calculated bite forces do not significantly differ fromisometry (3D LM 95 confidence interval 164ndash206 FEA 95confidence interval 163ndash205) By contrast in vivo data fromErickson et al (2003) showed positive allometry (95 confidenceinterval 251ndash261) Application of the JohnsonndashNeyman techniqueon both sources of calculated bite force data against in vivo results ofErickson et al (2003) revealed that both samples had a region ofnon-significant difference of slopes For bite force calculated with3D LM median values of lower and upper skull length in the regionwhere slopes were not significantly different were 98 to 179 cmrespectively Between these sizes 3D LM predicts a slope that doesnot significantly differ from in vivo data For bite force calculatedwith FEA median values of skull length were 95ndash174 cm (Fig 6)Between these sizes FEA predicts a slope that does not significantlydiffer from in vivo data

DISCUSSIONBiomechanical modeling offers researchers powerful tools withwhichto test hypotheses of feeding performance of extant and extinct taxaIn vivo bite force data of wild crocodylians are challenging to obtainand in vivo measurements are obviously not possible in extinct taxamaking computational modeling necessary to explore patterns of formand function in the group Accurate computationalmethods canmodelbiting under varying conditions of tooth contact gape and musclerecruitment and thus modern computational methods are an excellentoption for investigating the relationship between morphology biteforce and resulting cranial forces

Validation with in vivo bite force dataThe two biomechanical modeling techniques used in this paperproduce results consistent with each other Like other validated

20

mAMESmAMPmPTd

mAMEMmPSTsmPTv

mAMEPmPSTpmDM

15

10

5

T

otal

forc

e

0

10 20 30 40

15

10

5 T

otal

mom

ents

0

10 20Skull length (cm)

30 40

Fig 5 Proportion eachmuscle contributes to total muscle force (top) andtotal moment about the jaw joint and therefore bite force (bottom)Muscleattachment colors are as in Fig 3 Note that mDM opens the jaw and thereforeis not included in the bottom panel Although mPTd is consistently thestrongest muscle mPSTs contributes the most to bite force in the smaller twospecimens

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iology

models (Davis et al 2010 Santana et al 2010) our alligatormodels faithfully perform within an acceptable range of in vivobite force The good performance of the model furtherdemonstrates the utility of high-fidelity muscle inputs Howeverbite force in most individuals differed somewhat from predicted invivo bite forces from similarly sized animals These differencesbetween in silico and in vivo techniques shed light on thechallenges of modeling complex feeding function and cranialbiomechanics Causes for this mismatch may be divided into (1)differences between modeled bites and in vivo bites and (2)submaximal model performanceFirst the mechanism of bite force production in our models

differs from the direct measurements of Erickson et al (2003)Whereas we modeled static crushing bites Erickson et al (2003)included some unknown amount of momentum the maximum forceduring lsquoaggressive snappingrsquo bites Therefore the peak forcereported by these authors likely included some degree of impactforce resulting from rapid mandibular deceleration Daniel andMcHenry (2001) suggested that lsquodynamic loading due to rapiddecelerationrsquo likely plays a role in maximal forces experienced bythe skull Because the present study modeled static crushing biteswhich have no momentum contribution from impact forcesestimated maximum bite forces are presumably below peak forcesexperienced by the cranium

Second the modeling techniques employed here underestimatethe force of mPTv because models do not adequately captureaponeurotic muscle attachments Crocodylians have a complextendinous skeleton among the adductor mandibulae andpterygoideus muscle bellies (Iordansky 1964 2000Schumacher 1973 Busbey 1989) that is challenging to modelTraditional tomography techniques fail to image these tendonswith enough reliability to create digital models Because thetendinous attachments of mPTv are missing (lsquoU-tendonrsquo ofIordansky 1964 lsquopterygoideus-tendon aponeurosesrsquo ofSchumacher 1973 lsquoposterior pterygoid tendonrsquo of Busbey1989) we underestimate the total surface area of muscleattachment and thus the total estimated force mPTv accountsfor approximately one-third of A mississippiensis jaw musclemass (Busbey 1989 Cleuren et al 1995) but only about 3 ofthe total muscle force in our model (Table 2) Moreover themethods employed in this study place force vectors directlybetween attachment sites mPTv originates on the edge of thepterygoid flange courses caudoventrolaterally then passes aroundthe ventral border of the mandible to attach on the lateral surfaceof the angular bone Force vectors oriented from muscle origin toinsertion will therefore pass through the body of the mandibleresulting in erroneously collinear forces between the cranial andmandibular attachments of this muscle Although BoneLoad

Table 2 Alligator mississippiensis jaw muscle forces calculated by estimating PCSA by frustum muscle modeling

Muscle

Force

AL 03148 cm

AL 62299 cm

AL 612203 cm

AL 024269 cm

AL 700333 cm

AL 008454 cm

(N) () (N) () (N) () (N) () (N) () (N) ()

L mAMES 545 248 326 432 728 331 154 301 287 346 665 348R mAMES 539 245 316 420 705 321 160 313 299 361 612 320L mAMEM 417 190 125 166 311 141 452 0883 120 144 272 142R mAMEM 420 191 121 161 322 147 463 0905 117 141 298 156L mAMEP 230 105 555 0737 152 0693 353 0690 343 0414 845 0443R mAMEP 226 103 513 0681 152 0691 331 0648 333 0402 832 0436L mAMP 316 144 110 146 309 141 659 129 1150 138 2720 142R mAMP 300 136 107 142 300 137 652 128 1040 126 2710 142L mPSTs 117 531 345 458 859 391 197 385 361 435 698 366R mPSTs 113 513 345 458 879 400 197 386 351 422 672 352L mPSTp 0420 0191 122 0161 397 0181 448 0088 880 0106 108 0057R mPSTp 0432 0197 114 0151 447 0204 492 0096 964 0116 133 0069L mPTd 425 193 131 174 425 194 1020 199 1560 188 3640 191R mPTd 400 182 135 179 427 194 1030 201 1590 192 3820 200L mPTv 671 305 293 390 685 311 172 337 323 390 701 367R mPTv 623 283 300 397 750 341 177 347 332 401 694 363L mDM 776 353 241 257 105 383 330 518 340 411 674 353R mDM 753 343 255 272 111 407 326 513 336 406 729 381

Alligator specimens are listed at the top with skull length given below (increasing from left to right) Force is given in absolute values (N) and as a percentage oftotal force Muscle abbreviations are as in Materials and methods L left R right PCSA physiological cross-sectional area

Table 3 Summary of A mississippiensis bite forces calculated with 3D LM and FEA

Specimen Skull length (cm)Lever mechanicsbite force (N) FEA bite force (N) Difference In vivo bite force (N) Error

AL 031 48 493 493 0 167 195AL 622 99 150 146 270 122 213AL 612 203 443 421 509 882 minus510AL 024 269 938 913 270 1913 minus516AL 700 333 1500 1470 202 3440 minus568AL 008 454 3460 3420 116 8070 minus574

Note finite element analysis (FEA) force presented here is dorsoventral force only see Table 6 for bite force in all dimensions 3D LM three-dimensional levermechanics

2042

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ofEx

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entalB

iology

accounts for muscle fibers that pull on a curved surface of bone(lsquomuscle wrappingrsquo sensu Grosse et al 2007) it does not accountfor subsequent changes to the course of a muscle fibers such asoccur when a muscle passes around a bone or other structure Wesuggest that the former phenomenon (muscle forces distributedacross a curved attachment surface) be called lsquomuscle tractionrsquo andthe latter anatomical phenomenon be called lsquomuscle wrappingrsquoTo better reconstruct muscle function accurately futureneontological studies could use contrast-enhanced CT imagingof soft tissues (Gignac et al 2016) to facilitate the inclusion ofthe tendinous skeleton and associated musculature as well asproperly orient force vectors in wrapping (Moazen et al 2008Groumlning et al 2013) However tendons are rarely preserved inthe fossil record leaving studies of muscle function in extinct taxato rely on inferential methods As this method was developed toapply to fossil crocodylomorphs and other vertebrates tendinousattachments were not included in muscle attachments

Because mPTv makes up a sizeable proportion of jaw musclemass (Busbey 1989 Cleuren et al 1995) errors in modeling thismuscle may be particularly deleterious to model fidelity Toexplore the effects of altering the magnitude and orientation ofmPTv force on model performance we used our largest specimento calculate bite force under three additional scenarios To orientthe force vector of mPTv more correctly we used DiceCT-basedscans to determine the angles of insertion of mPTv on the lateralsurface of the articular bone We then oriented muscle force alongthis adjusted vector rather than towards the musclersquos cranialattachment site (Fig 7) To account for the underestimated PCSAof mPTv we scaled muscle force magnitude by the ratio of muscleforce calculated for mPTv by Gignac and Erickson (2016) over thatof our own mPTv for equivalently sized animals (approximately525 times) Gignac and Erickson (2016) dissected cadavericspecimens to calculate PCSA This is inapplicable to fossil taxabut presumably yields more accurate PCSA data We then

Table 4 Muscle moments about jaw joint axis calculated with 3D LM

Muscle

MJJA (N m)

AL 03148 cm

AL 62299 cm

AL 612203 cm

AL 024269 cm

AL 700333 cm

AL 008454 cm

L mAMES 0033 0372 144 447 922 285R mAMES 0037 0355 148 505 892 256L mAMEM 0025 0111 0625 108 307 117R mAMEM 0025 0107 0592 123 303 965L mAMEP 0025 0102 0498 151 170 672R mAMEP 0025 0098 0460 149 171 630L mAMP 0114 0837 494 158 249 108R mAMP 0127 0755 450 155 239 961L mPSTs 0164 0801 360 110 241 681R mPSTs 0154 0829 363 116 235 662L mPSTp 0003 0017 0111 0158 0355 0746R mPSTp 0003 0016 0117 0180 0400 0885L mPTd 0141 0809 624 164 303 119R mPTd 0128 0755 483 169 305 120L mPTv 0027 0247 128 398 108 295R mPTv 0025 0241 142 399 142 315L mDM minus0025 minus0139 minus0692 minus408 minus630 minus162R mDM minus0024 minus0162 minus0910 minus395 minus629 minus186

Alligator specimens are listed at the top with skull length given below (increasing from left to right) Muscle abbreviations are as in Materials and methods L leftR right

Table 5 Contribution of each muscle to MJJA (and thus bite force) calculated with 3D LM

Muscle

AL 03148 cm

AL 62299 cm

AL 612203 cm

AL 024269 cm

AL 700333 cm

AL 008454 cm

Proportion ( total moment)

L mAMES 310 576 403 405 438 391R mAMES 352 550 414 458 424 352L mAMEM 235 172 175 0977 146 160R mAMEM 237 166 165 111 144 132L mAMEP 234 159 139 137 0807 0922R mAMEP 234 152 128 135 0816 0864L mAMP 108 130 138 143 118 149R mAMP 120 117 126 141 113 132L mPSTs 155 124 101 100 114 935R mPSTs 146 128 102 105 112 909L mPSTp 0295 0264 0310 0143 0169 0102R mPSTp 0296 0255 0328 0163 0190 0121L mPTd 134 125 174 149 144 163R mPTd 121 117 135 153 145 165L mPTv 253 383 357 361 512 405R mPTv 240 373 397 362 674 432

Note mDM is not included in this calculation

2043

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iology

combined these orientation and magnitude variables Bite force forthe 454 cm-long specimen was originally 3200 N ReorientingmPTv force raised bite force to 3377 N while scaling its force by525 raised bite force to 4160 N Combined these effects resultedin a bite force of 4515 N Both models in which mPTv force wasscaled up resulted in substantially higher bite force Of course bothsources of correctional data would be unavailable without access tofresh cadaveric specimens Because a primary goal of this studywas to develop a method with applicability to the fossil record wepresent these results but do not incorporate them into the broaderworkflow or statistical analysis in this paper

The prospect of 3D musculoskeletal cranial biomechanicsAn integrative understanding of the feeding apparatus requires anaccurate comprehensive characterization of muscular inputs andtheir concomitant impacts on joint and cranial function 3Dcomputational analysis of musculoskeletal behavior is timeintensive however these techniques allow researchers toinvestigate performance in conditions and numbers that cannotbe replicated in vivo Researchers are able to visualize muscleresultants and cranial forces in three dimensions correlate muscleinputs with cranial performance and uncover loadingenvironments of key cranial structures such as jaw joints andsutures This modeling workflow can assess each musclersquoscontribution to various cranial forces characterize joint reactionforce magnitude and orientation to better understand the grossanatomical and microanatomical adaptations joints have to loadingenvironment and investigate intracranial joints secondarycraniomandibular joints and dual joint systems The jaw jointminimally resists all dorsoventrally oriented input forces that donot contribute to bite force As such it plays a key role inmodulating and dispersing forces in the feeding apparatus We

would expect that the jaw joint will have a tissue composition thatis well suited to its loading regime Therefore biomechanicalstudies and histological investigations can provide reciprocalilluminations into how joint morphology and compositioncorrelate with loading environment

Evolution of the crocodylian skullThe 3D modeling approaches used in this study are well suited forapplication to the fossil record Because these techniques use thearea of muscle attachment site rather than the weight of dissectedmuscles to estimate PCSA they can be used to make accurateestimates of muscle force moments about axes and reaction forcesin extinct animals Effective use of osteological correlates (Holliday2009) and the extant phylogenetic bracket (Witmer 1995 Hollidayand Witmer 2007) can constrain reconstructions of muscleattachment location size and shape Extant relatives can alsoinform the reconstruction of myological parameters such aspennation angle and fiber length

The evolution of the modern crocodylian skull involvedsubstantial changes to the skull (Langston 1973 Busbey 1995)In contrast to the platyrostral skulls of crocodylians the earliestmembers of crocodylian-line archosaurs had oreinirostral skullssuch as the rauisuchian Postosuchus (Chatterjee 1985) and thesphenosuchian Sphenosuchus (Walker 1990) In these animals theskull is dorsoventrally deep and mediolaterally narrow Thequadrates were dorsoventrally oriented (Walker 1990) and atleast in sphenosuchians were still not rigidly sutured to thebraincase (Langston 1973) A mobile suspensorium alongside anopen palatobasal joint (Langston 1973 Busbey 1995) has beeninterpreted as evidence that these early ancestors of crocodylians

Table 6 Components of A mississippiensis bite force calculated withFEA

Specimen Fx (N) Fy (N) Fz (N) Fsum (N)

AL 031 minus546 493 134 513AL 622 minus807 146 507 155AL 612 minus816 421 122 438AL 024 minus154 913 272 965AL 700 minus282 1470 403 1550AL 008 minus469 3420 704 3520

x y and z refer to force in the mediolateral dorsoventral and rostrocaudaldirection Fsum is total force All data are for low gape

Skull length (cm)

Bite

forc

e (N

)

In vivo3D LMFEA

1000030001000

200

50

10

5 15 25 35 45 55

Fig 6 Regression analysis of modeled and in vivo bite forces againstskull length The gray box indicates the region in which there is no significantdifference in slope between calculated and in vivo bite force In vivo dataErickson et al 2003 Modeled data 3D LM three-dimensional levermechanics FEA finite element analysis

B

D

Articular

mPTvmPTv

A

C

Fig 7 Reorientation of mPTv force with DiceCT data Solid arrowrepresents the original attachment-based orientation of mPTv force dashedarrow shows the reoriented force (A) Caudal view of 454 cm skull lengthspecimen Vertical line represents the parasagittal slice shown in CTransparent portion of arrow illustrates how original orientation of mPTv forceclips through mandible (B) Left lateral view of the same specimen Verticalline represents the axial slice shown in D (C) Parasagittal slice through mPTv(D) Axial slice through mPTv Note the muscle wrapping around thearticular bone

2044

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were kinetic (Walker 1990) The evolution of the crown grouptherefore involved substantial reorientation of adductor musclesalong with a reduction in kinetic potential The transformation fromoreinirostry to platyrostry presumably required crocodylians toevolve higher mass or pennation of adductor muscles to achieveequivalent bite forces to their fossil ancestorsThe biomechanical origins of the pterygoid buttress are also

poorly understood In crocodylians the pterygoid buttressarticulates with the medial surface of the mandible in what hasbeen described as an lsquoopenrsquo or lsquoslidingrsquo joint (Schumacher 1973)Some researchers have hypothesized that it braces the mandibleagainst lsquoreverse-wishboningrsquo (Iordansky 1964 Schumacher 1973Busbey 1995) Porro et al (2011) included the pterygoid buttress asa constraint of the mandible Recently the pterygoid buttress hasbeen suggested to represent a key innovation underlying thecrocodylian feeding apparatus (Holliday et al 2015) Althoughmediolaterally acting muscle forces will cancel out whensymmetrically recruited they will load structures on which theyact In crocodylians and other taxa with substantial mediolateralcomponents to muscle force these forces are likely to be a primarysource of stress in the skullThe loss of cranial kinesis along with the elaboration of the

pterygoid flange into a novel craniomandibular articulation representkey features of crocodylian evolution (Holliday et al 2015) Indeedmany of the hallmark features of the crocodylian skull including anexpanded retroarticular process laterally attaching pterygoideusmuscles a bony secondary palate and broad scarf joints followed theevolution of the pterygoid buttress The methods used in this paperwill be applied to assess the kinetic status of fossil relatives ofcrocodylians and to investigate the role of the pterygoid buttress inthe feeding apparatus of these derived archosaurs

ConclusionsThis study is one of the first to use both 3D LM and FEA toinvestigate the production of bite force in an ontogenetic series ofA mississippiensis The use of anatomically accurate muscleattachments is key to the success of the models and the goodagreement between the two methods lends support to thesetechniques The modeling techniques in this study can be used toassess the effect of changing muscle size and orientation during theevolution of the modern crocodylian skull Key features of thecrocodylian skullmay have permitted novelmuscularmorphologiesThese methods will allow researchers to test hypotheses linkingbony features such as the loss of kinesis secondary palate scarfjoints and the pterygoid buttress with muscular innovations such asgenerally enlarged adductor mass laterally inserting mPTv and theextensive cranial tendinous skeleton

AcknowledgementsWe thank Ruth Elsey and the staff of Rockefeller Wildlife Refuge for providingspecimens We thank the University of Missouri Biomolecular Imaging Center theUniversity of Missouri School of Medicine Department of Radiology and theUniversity of Missouri School of Veterinary Medicine for scanning specimens Wethank Betsy Dumont Larry Witmer Laura Porro and Kent Vliet for helpfuldiscussions We thank two anonymous reviewers whose comments greatlyenhanced the quality and clarity of the manuscript

Competing interestsThe authors declare no competing or financial interests

Author contributionsConceptualization KCS and CMH Methodology KCS KMM JLD andCMH Software KCS KMM JLD and CMH Formal Analysis KCS andKMM Investigation KCS Resources KMM and CMH DataCuration KCSandCMHWriting - Original Draft KCS and CMH Writing - ReviewandEditing

KCS KMM JLD and CMH Visualization KCS KMM and CMHProject Administration KCS and CMH Funding Acquisition KCS KMMJLD and CMH

FundingThis research was supported by the National Science Foundation (IOS 1457319 andEAR 1631684) the University of Missouri Research Board the University of MissouriResearch Council and the University of Missouri Department of Pathology andAnatomical Sciences

Data availabilityData are available from Open Science Framework httpsosfiojmpck

Supplementary informationSupplementary information available online athttpjebbiologistsorglookupdoi101242jeb156281supplemental

ReferencesBates K T and Falkingham P L (2012) Estimating maximum bite performance

in Tyrannosaurus rex using multi-body dynamics Biol Lett 8 660-664Busbey A B (1989) Form and function of the feeding apparatus of Alligator

mississippiensis J Morphol 202 99-127Busbey A B (1995) The structural consequences of skull flattening in

crocodilians In Functional Morphology in Vertebrate Paleontology (J JThomason) pp 173-192 Cambridge Cambridge University Press

Chatterjee S (1985) Postosuchus a new thecodontian reptile from the triassic oftexas and the origin of tyrannosaurs Philos Trans R Soc Lond 309 395-460

Cleuren J and de Vree F (1992) Kinematics of the jaw and hyolingual apparatusduring feeding in Caiman crocodilus J Morphol 212 141-154

Cleuren J Aerts P andDeVree F (1995) Bite and joint force analysis inCaimancrocodilus Belg J Zool 12 79-94

Curtis N Jones M E H Evans S E OrsquoHiggins P and Fagan M J (2013)Cranial sutures work collectively to distribute strain throughout the reptile skullJ R Soc Interface 10 1-9

Daniel W J T and McHenry C (2001) Bite force to skull stress correlationmdashmodelling the skull of Alligator mississippiensis In Crocodilian Biology AndEvolution (ed G C Grigg F Seebacher and C Franklin) pp 135-143 ChippingNorton NSW Surrey Beatty and Sons

Davis J L Santana S E Dumont E R andGrosse I R (2010) Predicting biteforce in mammals two-dimensional versus three-dimensional models J ExpBiol 213 1844-1851

Erickson G M Lappin A K and Vliet K A (2003) The ontogeny of bite-forceperformance in American alligator (Alligator mississippiensis) J Zool 260317-327

Erickson G M Gignac P M Steppan S J Lappin A K Vliet K ABrueggen J D Inouye B D Kledzik D and Webb G J W (2012) Insightsinto the ecology and evolutionary success of crocodilians revealed through bite-force and tooth-pressure experimentation PLOS ONE 7 e31781

Erickson G M Gignac P M Lappin A K Vliet K A Brueggen J D andWebb G J W (2014) A comparative analysis of ontogenetic bite-force scalingamong Crocodylia J Zool 292 48-55

Gans C (1982) Fiber architecture and muscle function Exerc Sports Sci Rev 10160-207

Gignac P M and Erickson G M (2015) Ontogenetic changes in dental form andtooth pressures facilitate developmental niche shifts in American alligatorsJ Zool 295 132-142

Gignac P M and Erickson G M (2016) Ontogenetic bite-force modeling ofAlligator mississippiensis implications for dietary transitions in a large-bodiedvertebrate and the evolution of crocodylian feeding J Zool 299 229-238

Gignac P M Kley N J Clarke J A Colbert MW Morhardt A C Cerio DCost I N Cox P G Daza J D Early C M et al (2016) Diffusible iodine-based contrast-enhanced computed tomography (diceCT) an emerging tool forrapid high-resolution 3-D imaging of metazoan soft tissues J Anat 228889-909

Groning F Jones M E H Curtis N Herrel A OrsquoHiggins P Evans S Eand Fagan M J (2013) The importance of accurate muscle modelling forbiomechanical analyses a case study with a lizard skull J R Soc Interface 1020130216

Grosse I R Dumont E R Coletta C and Tolleson A (2007) Techniques formodeling muscle-induced forces in finite element models of skeletal structuresAnat Rec 290 1069-1088

Holliday C M (2009) New insights into dinosaur jaw muscle anatomy Anat Rec292 1246-1265

Holliday C M and Witmer L M (2007) Archosaur adductor chamber evolutionintegration of musculoskeletal and topological criteria in jaw muscle homologyJ Morphol 268 457-484

2045

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accounts for muscle fibers that pull on a curved surface of bone(lsquomuscle wrappingrsquo sensu Grosse et al 2007) it does not accountfor subsequent changes to the course of a muscle fibers such asoccur when a muscle passes around a bone or other structure Wesuggest that the former phenomenon (muscle forces distributedacross a curved attachment surface) be called lsquomuscle tractionrsquo andthe latter anatomical phenomenon be called lsquomuscle wrappingrsquoTo better reconstruct muscle function accurately futureneontological studies could use contrast-enhanced CT imagingof soft tissues (Gignac et al 2016) to facilitate the inclusion ofthe tendinous skeleton and associated musculature as well asproperly orient force vectors in wrapping (Moazen et al 2008Groumlning et al 2013) However tendons are rarely preserved inthe fossil record leaving studies of muscle function in extinct taxato rely on inferential methods As this method was developed toapply to fossil crocodylomorphs and other vertebrates tendinousattachments were not included in muscle attachments

Because mPTv makes up a sizeable proportion of jaw musclemass (Busbey 1989 Cleuren et al 1995) errors in modeling thismuscle may be particularly deleterious to model fidelity Toexplore the effects of altering the magnitude and orientation ofmPTv force on model performance we used our largest specimento calculate bite force under three additional scenarios To orientthe force vector of mPTv more correctly we used DiceCT-basedscans to determine the angles of insertion of mPTv on the lateralsurface of the articular bone We then oriented muscle force alongthis adjusted vector rather than towards the musclersquos cranialattachment site (Fig 7) To account for the underestimated PCSAof mPTv we scaled muscle force magnitude by the ratio of muscleforce calculated for mPTv by Gignac and Erickson (2016) over thatof our own mPTv for equivalently sized animals (approximately525 times) Gignac and Erickson (2016) dissected cadavericspecimens to calculate PCSA This is inapplicable to fossil taxabut presumably yields more accurate PCSA data We then

Table 4 Muscle moments about jaw joint axis calculated with 3D LM

Muscle

MJJA (N m)

AL 03148 cm

AL 62299 cm

AL 612203 cm

AL 024269 cm

AL 700333 cm

AL 008454 cm

L mAMES 0033 0372 144 447 922 285R mAMES 0037 0355 148 505 892 256L mAMEM 0025 0111 0625 108 307 117R mAMEM 0025 0107 0592 123 303 965L mAMEP 0025 0102 0498 151 170 672R mAMEP 0025 0098 0460 149 171 630L mAMP 0114 0837 494 158 249 108R mAMP 0127 0755 450 155 239 961L mPSTs 0164 0801 360 110 241 681R mPSTs 0154 0829 363 116 235 662L mPSTp 0003 0017 0111 0158 0355 0746R mPSTp 0003 0016 0117 0180 0400 0885L mPTd 0141 0809 624 164 303 119R mPTd 0128 0755 483 169 305 120L mPTv 0027 0247 128 398 108 295R mPTv 0025 0241 142 399 142 315L mDM minus0025 minus0139 minus0692 minus408 minus630 minus162R mDM minus0024 minus0162 minus0910 minus395 minus629 minus186

Alligator specimens are listed at the top with skull length given below (increasing from left to right) Muscle abbreviations are as in Materials and methods L leftR right

Table 5 Contribution of each muscle to MJJA (and thus bite force) calculated with 3D LM

Muscle

AL 03148 cm

AL 62299 cm

AL 612203 cm

AL 024269 cm

AL 700333 cm

AL 008454 cm

Proportion ( total moment)

L mAMES 310 576 403 405 438 391R mAMES 352 550 414 458 424 352L mAMEM 235 172 175 0977 146 160R mAMEM 237 166 165 111 144 132L mAMEP 234 159 139 137 0807 0922R mAMEP 234 152 128 135 0816 0864L mAMP 108 130 138 143 118 149R mAMP 120 117 126 141 113 132L mPSTs 155 124 101 100 114 935R mPSTs 146 128 102 105 112 909L mPSTp 0295 0264 0310 0143 0169 0102R mPSTp 0296 0255 0328 0163 0190 0121L mPTd 134 125 174 149 144 163R mPTd 121 117 135 153 145 165L mPTv 253 383 357 361 512 405R mPTv 240 373 397 362 674 432

Note mDM is not included in this calculation

2043

RESEARCH ARTICLE Journal of Experimental Biology (2017) 220 2036-2046 doi101242jeb156281

Journal

ofEx

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iology

combined these orientation and magnitude variables Bite force forthe 454 cm-long specimen was originally 3200 N ReorientingmPTv force raised bite force to 3377 N while scaling its force by525 raised bite force to 4160 N Combined these effects resultedin a bite force of 4515 N Both models in which mPTv force wasscaled up resulted in substantially higher bite force Of course bothsources of correctional data would be unavailable without access tofresh cadaveric specimens Because a primary goal of this studywas to develop a method with applicability to the fossil record wepresent these results but do not incorporate them into the broaderworkflow or statistical analysis in this paper

The prospect of 3D musculoskeletal cranial biomechanicsAn integrative understanding of the feeding apparatus requires anaccurate comprehensive characterization of muscular inputs andtheir concomitant impacts on joint and cranial function 3Dcomputational analysis of musculoskeletal behavior is timeintensive however these techniques allow researchers toinvestigate performance in conditions and numbers that cannotbe replicated in vivo Researchers are able to visualize muscleresultants and cranial forces in three dimensions correlate muscleinputs with cranial performance and uncover loadingenvironments of key cranial structures such as jaw joints andsutures This modeling workflow can assess each musclersquoscontribution to various cranial forces characterize joint reactionforce magnitude and orientation to better understand the grossanatomical and microanatomical adaptations joints have to loadingenvironment and investigate intracranial joints secondarycraniomandibular joints and dual joint systems The jaw jointminimally resists all dorsoventrally oriented input forces that donot contribute to bite force As such it plays a key role inmodulating and dispersing forces in the feeding apparatus We

would expect that the jaw joint will have a tissue composition thatis well suited to its loading regime Therefore biomechanicalstudies and histological investigations can provide reciprocalilluminations into how joint morphology and compositioncorrelate with loading environment

Evolution of the crocodylian skullThe 3D modeling approaches used in this study are well suited forapplication to the fossil record Because these techniques use thearea of muscle attachment site rather than the weight of dissectedmuscles to estimate PCSA they can be used to make accurateestimates of muscle force moments about axes and reaction forcesin extinct animals Effective use of osteological correlates (Holliday2009) and the extant phylogenetic bracket (Witmer 1995 Hollidayand Witmer 2007) can constrain reconstructions of muscleattachment location size and shape Extant relatives can alsoinform the reconstruction of myological parameters such aspennation angle and fiber length

The evolution of the modern crocodylian skull involvedsubstantial changes to the skull (Langston 1973 Busbey 1995)In contrast to the platyrostral skulls of crocodylians the earliestmembers of crocodylian-line archosaurs had oreinirostral skullssuch as the rauisuchian Postosuchus (Chatterjee 1985) and thesphenosuchian Sphenosuchus (Walker 1990) In these animals theskull is dorsoventrally deep and mediolaterally narrow Thequadrates were dorsoventrally oriented (Walker 1990) and atleast in sphenosuchians were still not rigidly sutured to thebraincase (Langston 1973) A mobile suspensorium alongside anopen palatobasal joint (Langston 1973 Busbey 1995) has beeninterpreted as evidence that these early ancestors of crocodylians

Table 6 Components of A mississippiensis bite force calculated withFEA

Specimen Fx (N) Fy (N) Fz (N) Fsum (N)

AL 031 minus546 493 134 513AL 622 minus807 146 507 155AL 612 minus816 421 122 438AL 024 minus154 913 272 965AL 700 minus282 1470 403 1550AL 008 minus469 3420 704 3520

x y and z refer to force in the mediolateral dorsoventral and rostrocaudaldirection Fsum is total force All data are for low gape

Skull length (cm)

Bite

forc

e (N

)

In vivo3D LMFEA

1000030001000

200

50

10

5 15 25 35 45 55

Fig 6 Regression analysis of modeled and in vivo bite forces againstskull length The gray box indicates the region in which there is no significantdifference in slope between calculated and in vivo bite force In vivo dataErickson et al 2003 Modeled data 3D LM three-dimensional levermechanics FEA finite element analysis

B

D

Articular

mPTvmPTv

A

C

Fig 7 Reorientation of mPTv force with DiceCT data Solid arrowrepresents the original attachment-based orientation of mPTv force dashedarrow shows the reoriented force (A) Caudal view of 454 cm skull lengthspecimen Vertical line represents the parasagittal slice shown in CTransparent portion of arrow illustrates how original orientation of mPTv forceclips through mandible (B) Left lateral view of the same specimen Verticalline represents the axial slice shown in D (C) Parasagittal slice through mPTv(D) Axial slice through mPTv Note the muscle wrapping around thearticular bone

2044

RESEARCH ARTICLE Journal of Experimental Biology (2017) 220 2036-2046 doi101242jeb156281

Journal

ofEx

perim

entalB

iology

were kinetic (Walker 1990) The evolution of the crown grouptherefore involved substantial reorientation of adductor musclesalong with a reduction in kinetic potential The transformation fromoreinirostry to platyrostry presumably required crocodylians toevolve higher mass or pennation of adductor muscles to achieveequivalent bite forces to their fossil ancestorsThe biomechanical origins of the pterygoid buttress are also

poorly understood In crocodylians the pterygoid buttressarticulates with the medial surface of the mandible in what hasbeen described as an lsquoopenrsquo or lsquoslidingrsquo joint (Schumacher 1973)Some researchers have hypothesized that it braces the mandibleagainst lsquoreverse-wishboningrsquo (Iordansky 1964 Schumacher 1973Busbey 1995) Porro et al (2011) included the pterygoid buttress asa constraint of the mandible Recently the pterygoid buttress hasbeen suggested to represent a key innovation underlying thecrocodylian feeding apparatus (Holliday et al 2015) Althoughmediolaterally acting muscle forces will cancel out whensymmetrically recruited they will load structures on which theyact In crocodylians and other taxa with substantial mediolateralcomponents to muscle force these forces are likely to be a primarysource of stress in the skullThe loss of cranial kinesis along with the elaboration of the

pterygoid flange into a novel craniomandibular articulation representkey features of crocodylian evolution (Holliday et al 2015) Indeedmany of the hallmark features of the crocodylian skull including anexpanded retroarticular process laterally attaching pterygoideusmuscles a bony secondary palate and broad scarf joints followed theevolution of the pterygoid buttress The methods used in this paperwill be applied to assess the kinetic status of fossil relatives ofcrocodylians and to investigate the role of the pterygoid buttress inthe feeding apparatus of these derived archosaurs

ConclusionsThis study is one of the first to use both 3D LM and FEA toinvestigate the production of bite force in an ontogenetic series ofA mississippiensis The use of anatomically accurate muscleattachments is key to the success of the models and the goodagreement between the two methods lends support to thesetechniques The modeling techniques in this study can be used toassess the effect of changing muscle size and orientation during theevolution of the modern crocodylian skull Key features of thecrocodylian skullmay have permitted novelmuscularmorphologiesThese methods will allow researchers to test hypotheses linkingbony features such as the loss of kinesis secondary palate scarfjoints and the pterygoid buttress with muscular innovations such asgenerally enlarged adductor mass laterally inserting mPTv and theextensive cranial tendinous skeleton

AcknowledgementsWe thank Ruth Elsey and the staff of Rockefeller Wildlife Refuge for providingspecimens We thank the University of Missouri Biomolecular Imaging Center theUniversity of Missouri School of Medicine Department of Radiology and theUniversity of Missouri School of Veterinary Medicine for scanning specimens Wethank Betsy Dumont Larry Witmer Laura Porro and Kent Vliet for helpfuldiscussions We thank two anonymous reviewers whose comments greatlyenhanced the quality and clarity of the manuscript

Competing interestsThe authors declare no competing or financial interests

Author contributionsConceptualization KCS and CMH Methodology KCS KMM JLD andCMH Software KCS KMM JLD and CMH Formal Analysis KCS andKMM Investigation KCS Resources KMM and CMH DataCuration KCSandCMHWriting - Original Draft KCS and CMH Writing - ReviewandEditing

KCS KMM JLD and CMH Visualization KCS KMM and CMHProject Administration KCS and CMH Funding Acquisition KCS KMMJLD and CMH

FundingThis research was supported by the National Science Foundation (IOS 1457319 andEAR 1631684) the University of Missouri Research Board the University of MissouriResearch Council and the University of Missouri Department of Pathology andAnatomical Sciences

Data availabilityData are available from Open Science Framework httpsosfiojmpck

Supplementary informationSupplementary information available online athttpjebbiologistsorglookupdoi101242jeb156281supplemental

ReferencesBates K T and Falkingham P L (2012) Estimating maximum bite performance

in Tyrannosaurus rex using multi-body dynamics Biol Lett 8 660-664Busbey A B (1989) Form and function of the feeding apparatus of Alligator

mississippiensis J Morphol 202 99-127Busbey A B (1995) The structural consequences of skull flattening in

crocodilians In Functional Morphology in Vertebrate Paleontology (J JThomason) pp 173-192 Cambridge Cambridge University Press

Chatterjee S (1985) Postosuchus a new thecodontian reptile from the triassic oftexas and the origin of tyrannosaurs Philos Trans R Soc Lond 309 395-460

Cleuren J and de Vree F (1992) Kinematics of the jaw and hyolingual apparatusduring feeding in Caiman crocodilus J Morphol 212 141-154

Cleuren J Aerts P andDeVree F (1995) Bite and joint force analysis inCaimancrocodilus Belg J Zool 12 79-94

Curtis N Jones M E H Evans S E OrsquoHiggins P and Fagan M J (2013)Cranial sutures work collectively to distribute strain throughout the reptile skullJ R Soc Interface 10 1-9

Daniel W J T and McHenry C (2001) Bite force to skull stress correlationmdashmodelling the skull of Alligator mississippiensis In Crocodilian Biology AndEvolution (ed G C Grigg F Seebacher and C Franklin) pp 135-143 ChippingNorton NSW Surrey Beatty and Sons

Davis J L Santana S E Dumont E R andGrosse I R (2010) Predicting biteforce in mammals two-dimensional versus three-dimensional models J ExpBiol 213 1844-1851

Erickson G M Lappin A K and Vliet K A (2003) The ontogeny of bite-forceperformance in American alligator (Alligator mississippiensis) J Zool 260317-327

Erickson G M Gignac P M Steppan S J Lappin A K Vliet K ABrueggen J D Inouye B D Kledzik D and Webb G J W (2012) Insightsinto the ecology and evolutionary success of crocodilians revealed through bite-force and tooth-pressure experimentation PLOS ONE 7 e31781

Erickson G M Gignac P M Lappin A K Vliet K A Brueggen J D andWebb G J W (2014) A comparative analysis of ontogenetic bite-force scalingamong Crocodylia J Zool 292 48-55

Gans C (1982) Fiber architecture and muscle function Exerc Sports Sci Rev 10160-207

Gignac P M and Erickson G M (2015) Ontogenetic changes in dental form andtooth pressures facilitate developmental niche shifts in American alligatorsJ Zool 295 132-142

Gignac P M and Erickson G M (2016) Ontogenetic bite-force modeling ofAlligator mississippiensis implications for dietary transitions in a large-bodiedvertebrate and the evolution of crocodylian feeding J Zool 299 229-238

Gignac P M Kley N J Clarke J A Colbert MW Morhardt A C Cerio DCost I N Cox P G Daza J D Early C M et al (2016) Diffusible iodine-based contrast-enhanced computed tomography (diceCT) an emerging tool forrapid high-resolution 3-D imaging of metazoan soft tissues J Anat 228889-909

Groning F Jones M E H Curtis N Herrel A OrsquoHiggins P Evans S Eand Fagan M J (2013) The importance of accurate muscle modelling forbiomechanical analyses a case study with a lizard skull J R Soc Interface 1020130216

Grosse I R Dumont E R Coletta C and Tolleson A (2007) Techniques formodeling muscle-induced forces in finite element models of skeletal structuresAnat Rec 290 1069-1088

Holliday C M (2009) New insights into dinosaur jaw muscle anatomy Anat Rec292 1246-1265

Holliday C M and Witmer L M (2007) Archosaur adductor chamber evolutionintegration of musculoskeletal and topological criteria in jaw muscle homologyJ Morphol 268 457-484

2045

RESEARCH ARTICLE Journal of Experimental Biology (2017) 220 2036-2046 doi101242jeb156281

Journal

ofEx

perim

entalB

iology

combined these orientation and magnitude variables Bite force forthe 454 cm-long specimen was originally 3200 N ReorientingmPTv force raised bite force to 3377 N while scaling its force by525 raised bite force to 4160 N Combined these effects resultedin a bite force of 4515 N Both models in which mPTv force wasscaled up resulted in substantially higher bite force Of course bothsources of correctional data would be unavailable without access tofresh cadaveric specimens Because a primary goal of this studywas to develop a method with applicability to the fossil record wepresent these results but do not incorporate them into the broaderworkflow or statistical analysis in this paper

The prospect of 3D musculoskeletal cranial biomechanicsAn integrative understanding of the feeding apparatus requires anaccurate comprehensive characterization of muscular inputs andtheir concomitant impacts on joint and cranial function 3Dcomputational analysis of musculoskeletal behavior is timeintensive however these techniques allow researchers toinvestigate performance in conditions and numbers that cannotbe replicated in vivo Researchers are able to visualize muscleresultants and cranial forces in three dimensions correlate muscleinputs with cranial performance and uncover loadingenvironments of key cranial structures such as jaw joints andsutures This modeling workflow can assess each musclersquoscontribution to various cranial forces characterize joint reactionforce magnitude and orientation to better understand the grossanatomical and microanatomical adaptations joints have to loadingenvironment and investigate intracranial joints secondarycraniomandibular joints and dual joint systems The jaw jointminimally resists all dorsoventrally oriented input forces that donot contribute to bite force As such it plays a key role inmodulating and dispersing forces in the feeding apparatus We

would expect that the jaw joint will have a tissue composition thatis well suited to its loading regime Therefore biomechanicalstudies and histological investigations can provide reciprocalilluminations into how joint morphology and compositioncorrelate with loading environment

Evolution of the crocodylian skullThe 3D modeling approaches used in this study are well suited forapplication to the fossil record Because these techniques use thearea of muscle attachment site rather than the weight of dissectedmuscles to estimate PCSA they can be used to make accurateestimates of muscle force moments about axes and reaction forcesin extinct animals Effective use of osteological correlates (Holliday2009) and the extant phylogenetic bracket (Witmer 1995 Hollidayand Witmer 2007) can constrain reconstructions of muscleattachment location size and shape Extant relatives can alsoinform the reconstruction of myological parameters such aspennation angle and fiber length

The evolution of the modern crocodylian skull involvedsubstantial changes to the skull (Langston 1973 Busbey 1995)In contrast to the platyrostral skulls of crocodylians the earliestmembers of crocodylian-line archosaurs had oreinirostral skullssuch as the rauisuchian Postosuchus (Chatterjee 1985) and thesphenosuchian Sphenosuchus (Walker 1990) In these animals theskull is dorsoventrally deep and mediolaterally narrow Thequadrates were dorsoventrally oriented (Walker 1990) and atleast in sphenosuchians were still not rigidly sutured to thebraincase (Langston 1973) A mobile suspensorium alongside anopen palatobasal joint (Langston 1973 Busbey 1995) has beeninterpreted as evidence that these early ancestors of crocodylians

Table 6 Components of A mississippiensis bite force calculated withFEA

Specimen Fx (N) Fy (N) Fz (N) Fsum (N)

AL 031 minus546 493 134 513AL 622 minus807 146 507 155AL 612 minus816 421 122 438AL 024 minus154 913 272 965AL 700 minus282 1470 403 1550AL 008 minus469 3420 704 3520

x y and z refer to force in the mediolateral dorsoventral and rostrocaudaldirection Fsum is total force All data are for low gape

Skull length (cm)

Bite

forc

e (N

)

In vivo3D LMFEA

1000030001000

200

50

10

5 15 25 35 45 55

Fig 6 Regression analysis of modeled and in vivo bite forces againstskull length The gray box indicates the region in which there is no significantdifference in slope between calculated and in vivo bite force In vivo dataErickson et al 2003 Modeled data 3D LM three-dimensional levermechanics FEA finite element analysis

B

D

Articular

mPTvmPTv

A

C

Fig 7 Reorientation of mPTv force with DiceCT data Solid arrowrepresents the original attachment-based orientation of mPTv force dashedarrow shows the reoriented force (A) Caudal view of 454 cm skull lengthspecimen Vertical line represents the parasagittal slice shown in CTransparent portion of arrow illustrates how original orientation of mPTv forceclips through mandible (B) Left lateral view of the same specimen Verticalline represents the axial slice shown in D (C) Parasagittal slice through mPTv(D) Axial slice through mPTv Note the muscle wrapping around thearticular bone

2044

RESEARCH ARTICLE Journal of Experimental Biology (2017) 220 2036-2046 doi101242jeb156281

Journal

ofEx

perim

entalB

iology

were kinetic (Walker 1990) The evolution of the crown grouptherefore involved substantial reorientation of adductor musclesalong with a reduction in kinetic potential The transformation fromoreinirostry to platyrostry presumably required crocodylians toevolve higher mass or pennation of adductor muscles to achieveequivalent bite forces to their fossil ancestorsThe biomechanical origins of the pterygoid buttress are also

poorly understood In crocodylians the pterygoid buttressarticulates with the medial surface of the mandible in what hasbeen described as an lsquoopenrsquo or lsquoslidingrsquo joint (Schumacher 1973)Some researchers have hypothesized that it braces the mandibleagainst lsquoreverse-wishboningrsquo (Iordansky 1964 Schumacher 1973Busbey 1995) Porro et al (2011) included the pterygoid buttress asa constraint of the mandible Recently the pterygoid buttress hasbeen suggested to represent a key innovation underlying thecrocodylian feeding apparatus (Holliday et al 2015) Althoughmediolaterally acting muscle forces will cancel out whensymmetrically recruited they will load structures on which theyact In crocodylians and other taxa with substantial mediolateralcomponents to muscle force these forces are likely to be a primarysource of stress in the skullThe loss of cranial kinesis along with the elaboration of the

pterygoid flange into a novel craniomandibular articulation representkey features of crocodylian evolution (Holliday et al 2015) Indeedmany of the hallmark features of the crocodylian skull including anexpanded retroarticular process laterally attaching pterygoideusmuscles a bony secondary palate and broad scarf joints followed theevolution of the pterygoid buttress The methods used in this paperwill be applied to assess the kinetic status of fossil relatives ofcrocodylians and to investigate the role of the pterygoid buttress inthe feeding apparatus of these derived archosaurs

ConclusionsThis study is one of the first to use both 3D LM and FEA toinvestigate the production of bite force in an ontogenetic series ofA mississippiensis The use of anatomically accurate muscleattachments is key to the success of the models and the goodagreement between the two methods lends support to thesetechniques The modeling techniques in this study can be used toassess the effect of changing muscle size and orientation during theevolution of the modern crocodylian skull Key features of thecrocodylian skullmay have permitted novelmuscularmorphologiesThese methods will allow researchers to test hypotheses linkingbony features such as the loss of kinesis secondary palate scarfjoints and the pterygoid buttress with muscular innovations such asgenerally enlarged adductor mass laterally inserting mPTv and theextensive cranial tendinous skeleton

AcknowledgementsWe thank Ruth Elsey and the staff of Rockefeller Wildlife Refuge for providingspecimens We thank the University of Missouri Biomolecular Imaging Center theUniversity of Missouri School of Medicine Department of Radiology and theUniversity of Missouri School of Veterinary Medicine for scanning specimens Wethank Betsy Dumont Larry Witmer Laura Porro and Kent Vliet for helpfuldiscussions We thank two anonymous reviewers whose comments greatlyenhanced the quality and clarity of the manuscript

Competing interestsThe authors declare no competing or financial interests

Author contributionsConceptualization KCS and CMH Methodology KCS KMM JLD andCMH Software KCS KMM JLD and CMH Formal Analysis KCS andKMM Investigation KCS Resources KMM and CMH DataCuration KCSandCMHWriting - Original Draft KCS and CMH Writing - ReviewandEditing

KCS KMM JLD and CMH Visualization KCS KMM and CMHProject Administration KCS and CMH Funding Acquisition KCS KMMJLD and CMH

FundingThis research was supported by the National Science Foundation (IOS 1457319 andEAR 1631684) the University of Missouri Research Board the University of MissouriResearch Council and the University of Missouri Department of Pathology andAnatomical Sciences

Data availabilityData are available from Open Science Framework httpsosfiojmpck

Supplementary informationSupplementary information available online athttpjebbiologistsorglookupdoi101242jeb156281supplemental

ReferencesBates K T and Falkingham P L (2012) Estimating maximum bite performance

in Tyrannosaurus rex using multi-body dynamics Biol Lett 8 660-664Busbey A B (1989) Form and function of the feeding apparatus of Alligator

mississippiensis J Morphol 202 99-127Busbey A B (1995) The structural consequences of skull flattening in

crocodilians In Functional Morphology in Vertebrate Paleontology (J JThomason) pp 173-192 Cambridge Cambridge University Press

Chatterjee S (1985) Postosuchus a new thecodontian reptile from the triassic oftexas and the origin of tyrannosaurs Philos Trans R Soc Lond 309 395-460

Cleuren J and de Vree F (1992) Kinematics of the jaw and hyolingual apparatusduring feeding in Caiman crocodilus J Morphol 212 141-154

Cleuren J Aerts P andDeVree F (1995) Bite and joint force analysis inCaimancrocodilus Belg J Zool 12 79-94

Curtis N Jones M E H Evans S E OrsquoHiggins P and Fagan M J (2013)Cranial sutures work collectively to distribute strain throughout the reptile skullJ R Soc Interface 10 1-9

Daniel W J T and McHenry C (2001) Bite force to skull stress correlationmdashmodelling the skull of Alligator mississippiensis In Crocodilian Biology AndEvolution (ed G C Grigg F Seebacher and C Franklin) pp 135-143 ChippingNorton NSW Surrey Beatty and Sons

Davis J L Santana S E Dumont E R andGrosse I R (2010) Predicting biteforce in mammals two-dimensional versus three-dimensional models J ExpBiol 213 1844-1851

Erickson G M Lappin A K and Vliet K A (2003) The ontogeny of bite-forceperformance in American alligator (Alligator mississippiensis) J Zool 260317-327

Erickson G M Gignac P M Steppan S J Lappin A K Vliet K ABrueggen J D Inouye B D Kledzik D and Webb G J W (2012) Insightsinto the ecology and evolutionary success of crocodilians revealed through bite-force and tooth-pressure experimentation PLOS ONE 7 e31781

Erickson G M Gignac P M Lappin A K Vliet K A Brueggen J D andWebb G J W (2014) A comparative analysis of ontogenetic bite-force scalingamong Crocodylia J Zool 292 48-55

Gans C (1982) Fiber architecture and muscle function Exerc Sports Sci Rev 10160-207

Gignac P M and Erickson G M (2015) Ontogenetic changes in dental form andtooth pressures facilitate developmental niche shifts in American alligatorsJ Zool 295 132-142

Gignac P M and Erickson G M (2016) Ontogenetic bite-force modeling ofAlligator mississippiensis implications for dietary transitions in a large-bodiedvertebrate and the evolution of crocodylian feeding J Zool 299 229-238

Gignac P M Kley N J Clarke J A Colbert MW Morhardt A C Cerio DCost I N Cox P G Daza J D Early C M et al (2016) Diffusible iodine-based contrast-enhanced computed tomography (diceCT) an emerging tool forrapid high-resolution 3-D imaging of metazoan soft tissues J Anat 228889-909

Groning F Jones M E H Curtis N Herrel A OrsquoHiggins P Evans S Eand Fagan M J (2013) The importance of accurate muscle modelling forbiomechanical analyses a case study with a lizard skull J R Soc Interface 1020130216

Grosse I R Dumont E R Coletta C and Tolleson A (2007) Techniques formodeling muscle-induced forces in finite element models of skeletal structuresAnat Rec 290 1069-1088

Holliday C M (2009) New insights into dinosaur jaw muscle anatomy Anat Rec292 1246-1265

Holliday C M and Witmer L M (2007) Archosaur adductor chamber evolutionintegration of musculoskeletal and topological criteria in jaw muscle homologyJ Morphol 268 457-484

2045

RESEARCH ARTICLE Journal of Experimental Biology (2017) 220 2036-2046 doi101242jeb156281

Journal

ofEx

perim

entalB

iology

were kinetic (Walker 1990) The evolution of the crown grouptherefore involved substantial reorientation of adductor musclesalong with a reduction in kinetic potential The transformation fromoreinirostry to platyrostry presumably required crocodylians toevolve higher mass or pennation of adductor muscles to achieveequivalent bite forces to their fossil ancestorsThe biomechanical origins of the pterygoid buttress are also

poorly understood In crocodylians the pterygoid buttressarticulates with the medial surface of the mandible in what hasbeen described as an lsquoopenrsquo or lsquoslidingrsquo joint (Schumacher 1973)Some researchers have hypothesized that it braces the mandibleagainst lsquoreverse-wishboningrsquo (Iordansky 1964 Schumacher 1973Busbey 1995) Porro et al (2011) included the pterygoid buttress asa constraint of the mandible Recently the pterygoid buttress hasbeen suggested to represent a key innovation underlying thecrocodylian feeding apparatus (Holliday et al 2015) Althoughmediolaterally acting muscle forces will cancel out whensymmetrically recruited they will load structures on which theyact In crocodylians and other taxa with substantial mediolateralcomponents to muscle force these forces are likely to be a primarysource of stress in the skullThe loss of cranial kinesis along with the elaboration of the

pterygoid flange into a novel craniomandibular articulation representkey features of crocodylian evolution (Holliday et al 2015) Indeedmany of the hallmark features of the crocodylian skull including anexpanded retroarticular process laterally attaching pterygoideusmuscles a bony secondary palate and broad scarf joints followed theevolution of the pterygoid buttress The methods used in this paperwill be applied to assess the kinetic status of fossil relatives ofcrocodylians and to investigate the role of the pterygoid buttress inthe feeding apparatus of these derived archosaurs

ConclusionsThis study is one of the first to use both 3D LM and FEA toinvestigate the production of bite force in an ontogenetic series ofA mississippiensis The use of anatomically accurate muscleattachments is key to the success of the models and the goodagreement between the two methods lends support to thesetechniques The modeling techniques in this study can be used toassess the effect of changing muscle size and orientation during theevolution of the modern crocodylian skull Key features of thecrocodylian skullmay have permitted novelmuscularmorphologiesThese methods will allow researchers to test hypotheses linkingbony features such as the loss of kinesis secondary palate scarfjoints and the pterygoid buttress with muscular innovations such asgenerally enlarged adductor mass laterally inserting mPTv and theextensive cranial tendinous skeleton

AcknowledgementsWe thank Ruth Elsey and the staff of Rockefeller Wildlife Refuge for providingspecimens We thank the University of Missouri Biomolecular Imaging Center theUniversity of Missouri School of Medicine Department of Radiology and theUniversity of Missouri School of Veterinary Medicine for scanning specimens Wethank Betsy Dumont Larry Witmer Laura Porro and Kent Vliet for helpfuldiscussions We thank two anonymous reviewers whose comments greatlyenhanced the quality and clarity of the manuscript

Competing interestsThe authors declare no competing or financial interests

Author contributionsConceptualization KCS and CMH Methodology KCS KMM JLD andCMH Software KCS KMM JLD and CMH Formal Analysis KCS andKMM Investigation KCS Resources KMM and CMH DataCuration KCSandCMHWriting - Original Draft KCS and CMH Writing - ReviewandEditing

KCS KMM JLD and CMH Visualization KCS KMM and CMHProject Administration KCS and CMH Funding Acquisition KCS KMMJLD and CMH

FundingThis research was supported by the National Science Foundation (IOS 1457319 andEAR 1631684) the University of Missouri Research Board the University of MissouriResearch Council and the University of Missouri Department of Pathology andAnatomical Sciences

Data availabilityData are available from Open Science Framework httpsosfiojmpck

Supplementary informationSupplementary information available online athttpjebbiologistsorglookupdoi101242jeb156281supplemental

ReferencesBates K T and Falkingham P L (2012) Estimating maximum bite performance

in Tyrannosaurus rex using multi-body dynamics Biol Lett 8 660-664Busbey A B (1989) Form and function of the feeding apparatus of Alligator

mississippiensis J Morphol 202 99-127Busbey A B (1995) The structural consequences of skull flattening in

crocodilians In Functional Morphology in Vertebrate Paleontology (J JThomason) pp 173-192 Cambridge Cambridge University Press

Chatterjee S (1985) Postosuchus a new thecodontian reptile from the triassic oftexas and the origin of tyrannosaurs Philos Trans R Soc Lond 309 395-460

Cleuren J and de Vree F (1992) Kinematics of the jaw and hyolingual apparatusduring feeding in Caiman crocodilus J Morphol 212 141-154

Cleuren J Aerts P andDeVree F (1995) Bite and joint force analysis inCaimancrocodilus Belg J Zool 12 79-94

Curtis N Jones M E H Evans S E OrsquoHiggins P and Fagan M J (2013)Cranial sutures work collectively to distribute strain throughout the reptile skullJ R Soc Interface 10 1-9

Daniel W J T and McHenry C (2001) Bite force to skull stress correlationmdashmodelling the skull of Alligator mississippiensis In Crocodilian Biology AndEvolution (ed G C Grigg F Seebacher and C Franklin) pp 135-143 ChippingNorton NSW Surrey Beatty and Sons

Davis J L Santana S E Dumont E R andGrosse I R (2010) Predicting biteforce in mammals two-dimensional versus three-dimensional models J ExpBiol 213 1844-1851

Erickson G M Lappin A K and Vliet K A (2003) The ontogeny of bite-forceperformance in American alligator (Alligator mississippiensis) J Zool 260317-327

Erickson G M Gignac P M Steppan S J Lappin A K Vliet K ABrueggen J D Inouye B D Kledzik D and Webb G J W (2012) Insightsinto the ecology and evolutionary success of crocodilians revealed through bite-force and tooth-pressure experimentation PLOS ONE 7 e31781

Erickson G M Gignac P M Lappin A K Vliet K A Brueggen J D andWebb G J W (2014) A comparative analysis of ontogenetic bite-force scalingamong Crocodylia J Zool 292 48-55

Gans C (1982) Fiber architecture and muscle function Exerc Sports Sci Rev 10160-207

Gignac P M and Erickson G M (2015) Ontogenetic changes in dental form andtooth pressures facilitate developmental niche shifts in American alligatorsJ Zool 295 132-142

Gignac P M and Erickson G M (2016) Ontogenetic bite-force modeling ofAlligator mississippiensis implications for dietary transitions in a large-bodiedvertebrate and the evolution of crocodylian feeding J Zool 299 229-238

Gignac P M Kley N J Clarke J A Colbert MW Morhardt A C Cerio DCost I N Cox P G Daza J D Early C M et al (2016) Diffusible iodine-based contrast-enhanced computed tomography (diceCT) an emerging tool forrapid high-resolution 3-D imaging of metazoan soft tissues J Anat 228889-909

Groning F Jones M E H Curtis N Herrel A OrsquoHiggins P Evans S Eand Fagan M J (2013) The importance of accurate muscle modelling forbiomechanical analyses a case study with a lizard skull J R Soc Interface 1020130216

Grosse I R Dumont E R Coletta C and Tolleson A (2007) Techniques formodeling muscle-induced forces in finite element models of skeletal structuresAnat Rec 290 1069-1088

Holliday C M (2009) New insights into dinosaur jaw muscle anatomy Anat Rec292 1246-1265

Holliday C M and Witmer L M (2007) Archosaur adductor chamber evolutionintegration of musculoskeletal and topological criteria in jaw muscle homologyJ Morphol 268 457-484

2045

RESEARCH ARTICLE Journal of Experimental Biology (2017) 220 2036-2046 doi101242jeb156281

Journal

ofEx

perim

entalB

iology

Holliday C M Tsai H P Skijan R J George I D and Pathan S (2013) A 3Dinteractive model and atlas of the jaw musculature of Alligator mississippiensisPLOS ONE 8 e62806

Holliday C M Sellers K C Vickaryous M K Ross C F Porro L BWitmer L M and Davis J L (2015) The functional and evolutionarysignificance of the crocodyliform pterygomandibular joint Integr Comp Biol 55suppl 1 e81

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