parr et al 2012 integrating gmm & fea

8/10/2019 Parr Et Al 2012 Integrating GMM & FEA

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Toward integration of geometric morphometrics and computationalbiomechanics: New methods for 3D virtual reconstruction andquantitative analysis of Finite Element Models

W.C.H. Parr a,n, S. Wroe a,b, U. Chamoli a, H.S. Richards b, M.R. McCurry b, P.D. Clausen b, C. McHenry b,c

a School of Biological, Earth and Environmental Sciences, University of New South Wales, Sydney, NSW 2052, Australiab School of Engineering, University of Newcastle, NSW 2308, Australiac School of Biomedical Science, Monash University, Clayton, VIC 3800, Australia

a r t i c l e i n f o

Article history:

Received 17 May 2011

Received in revised form

11 January 2012

Accepted 16 January 2012Available online 10 February 2012

Keywords:

Finite Element Analysis

Geometric morphometrics

Varanid

Warping

Landmark point

a b s t r a c t

The ability to warp three-dimensional (3D) meshes from known biological morphology to fit other

known, predicted or hypothetical morphologies has a range of potential applications in functional

morphology and biomechanics. One of the most challenging of these applications is Finite Element

Analysis (FEA), a potentially powerful non-destructive tool in the prediction of mechanical behaviour.

Geometric morphometrics is another typically computer-based approach commonly applied in

morphological studies that allows for shape differences between specimens to be quantified and

analysed. There has been some integration of these two fields in recent years. Although a number of

shape warping approaches have been developed previously, none are easily accessible. Here we present

an easily accessed method for warping meshes based on freely available software and test the

effectiveness of the approach in FEA using the varanoid lizard mandible as a model. We further present

new statistical approaches, strain frequency plots and landmark point strains, to analyse FEA results

quantitatively and further integrate FEA with geometric morphometrics. Using strain frequency plots,strain field, bending displacements and landmark point strain data we demonstrate that the mechanical

behaviour of warped specimens reproduces that of targets without significant error. The influence of

including internal cavity morphology in FEA models was also examined and shown to increase bending

displacements and strain magnitudes in FE models. The warping approaches presented here will

be useful in a range of applications including the generation and analysis of virtual reconstructions,

generic models that approximate species means, hypothetical morphologies and evolutionary inter-

mediaries.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Methods for warping of three-dimensional (3D) meshes from

known biological structures to fit other known, predicted, orhypothetical target morphologies have been developed for a

range of purposes. These include the virtual reconstruction of

missing or deformed geometry in fossil specimens (Gunz et al.,

2009), the generation of generic models to approximate species

means (Parr et al., 2011a; Parr, 2009), the reduction of time

needed to produce Finite Element Models (Sigal et al., 2008,2010;

Stayton, 2009) and the development of hypothetical morpholo-

gies and evolutionary intermediaries (OHiggins et al., 2010).

Mesh warping may also have applications in the design andtesting of prosthetic devices. However, to date, all mesh warping

approaches known to us have been developed using purpose

specific code (Gunz et al., 2009;Sigal et al., 2010), software that is

not freely available (Sigal et al., 2008), or by moving individual

vertices (Wroe et al., 2010), which is both time consuming and

potentially less reproducible.

In recent years the application of Finite Element Analysis (FEA)

to biological structures has invigorated the field of functional

morphology (Wroe et al., 2010;Bourke et al., 2008;Ferrara et al.,

2011; Fry et al., 2009; Moreno et al., 2008; Panagiotopoulou,

2009; Rayfield et al., 2001; McHenry et al., 2006; Strait et al.,

2009, 2010; Wroe et al., 2007a, 2007b, 2008; Wroe, 2008). By

applying simulation methods such as Finite Element Modelling

(FEM) morphologists are using biomechanical principals to infer

Contents lists available at SciVerse ScienceDirect

journal homepage: www.elsevier.com/locate/yjtbi

Journal of Theoretical Biology

0022-5193/$- see front matter& 2012 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jtbi.2012.01.030

n Corresponding author. Tel.: 0061 405658783.

E-mail addresses: [email protected],

[email protected] (W.C.H. Parr), [email protected] (S. Wroe),

[email protected] (U. Chamoli), [email protected] (H.S. Richards),

[email protected] (M.R. McCurry),

[email protected] (P.D. Clausen),

[email protected] (C. McHenry).

Journal of Theoretical Biology 301 (2012) 114


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function from form in biological structures. However, due to the

time consuming nature of FEM assembly, sample sizes are

typically tiny, frequently with species being represented by a

single specimen and only two or three species considered.

Missing regions or deformation of specimens present common

obstacles to the application of Finite Element Analysis (FEA),

particularly in palaeontological studies. Successful FEA requiresintact, coherent mesh models. For FEA of species for which

complete 3D data are not available, there is the potential to

deform existing 3D meshes of morphologically similar individuals

or species to match the preserved morphology of the study

(target) specimen using geometric morphometric (GMM) princi-

ples and methods. This idea has been explored and applied in

some recently published studies (Gunz et al., 2009; Sigal et al.,

2008; Stayton 2009; OHiggins et al., 2010; Wroe et al., 2010;

Strait et al., 2009;McHenry, 2009).

Although GMM and FEA are two of the most productive

techniques used in the research fields of functional morphology,

computational biomechanics and virtual anthropology, questions

have been raised over how these two fields might be more fully

integrated (Weber et al., 2011). FEA returns detailed informa-tion about the biomechanics of individual biological structures,

whereas GMM allows shape differences across a sample to be

quantified and analysed. The problem then arises in integrating

the FEA approach, based on single specimens, with the GMM

approach, typically based on samples of numerous specimens.

Limitations inherent to the generation of FEMs make it difficult if

not impossible to control element numbers and positions in FEMs

of different specimens. Consequently, the specific elements

between different models are not homologous, making it difficult

to compare the results between two or more different specimens

in a meaningful way. Typically, mean element stresses and/or

strains, or strain energies are reported for the whole, or selected

parts of the model (Strait et al., 2009;Wroe et al., 2007a;Slater

et al., 2010; Chamoli and Wroe, 2011). In the present study wefurther aim to develop new statistical tools to help interpret the

results of FEMs in greater depth, and to try and further integrate

these two fields of GMM and FEA.

1.1. Aims

To develop:

A method for the warping of biological structures that is

simple to apply and freely accessible, based on a geometric

morphometric (GMM) approach. GMM uses discrete, reliably

identifiable landmark points placed in homologous positions

on specimens to quantify morphology and shape differences

between specimens. For this study we focus on varanoid lizard

(seeSection 2.2) mandibles as our biological structures. These

mandibles are morphologically complex enough to test our

method without being so large as to result in extended

solution times for the FEA software.

Protocols for ensuring consistent FEA results that are achieved

by warping and solid meshing techniques.

Statistical methods that enable more robust quantitative

analysis of high resolution FEMs.

Using the GMM concept of landmark homology (spatial

and true homology), we aim to develop a method of creat-

ing homologous data sets that enable the direct comparison

of FEA results of FEMs that contain different element numbers.

To assess:

How closely FEMs produced by warping are able to reproduce

the behaviour of FEMs of original target meshes?

How much accuracy may be lost in FEA by using meshes with

no internal cavities (which are easier to warp)?

1.2. Hypotheses

The specific hypotheses we test are as follows:

1. For each mandible specimen, initial models with internal geo-

metry (cavity models) will demonstrate higher strain values and

increased bending magnitudes compared to solid (no-cavity)

mandible models. This hypothesis is based on theoretical beam

mechanics that predict increased second moments of inertia in

annular (hollow) cross-section tubes compared to circular (solid)

cross section tubes (seesupporting information).

2. Mandible models with different initial morphologies will demon-

strate similar biomechanical behaviours in terms of FEA predicted

strain distributions, strain magnitudes, and bending displace-

ments when they have been warped to similar shapes. This is

hypothesised for both cavity and no-cavity model warps.

2. Materials and method

2.1. Materials

Our modelling is based on data from varanoid lizards; five

species of varanid, and one lanthanotid. Species selected were

determined largely by the availability of CT data. Traditionally

regarded as morphologically conservative, varanoids are widely

considered a model group for ecological studies (Pianka, 1995).

Despite this conservatism, both body size and feeding ecology

vary considerably within the clade. Species included in the

present study comprise the earless monitor,Lanthanotus borneensis,

a poorly understood species thought to be a generalist feeder on

invertebrate prey (Losos and Greene, 1988;Das, 2003) with a totallength (TL) of up to 55 cm (Pianka, 2004); the Komodo dragon

(Varanus komodoensis), a more specialized predator of relatively

large vertebrates (TL 3.0 m) (Auffenberg, 1981); the sand goanna,

Varanus gouldii (TL 1.5 m), a generalist predator (Thompson, 2004;

Pianka, 1994; Shine, 1986) the ridge-tailed goanna Varanus

acanthurus (TL 70 cm), thought to be principally insectivorous

(Losos and Greene, 1988; Dryden, 2004); the crocodile monitor

Varanus salvadorii (TL 2.5 m), an aboreal species that preys upon

invertebrates, lizards, birds, and mammals (Allison, 1982; Horn,

2004); and the savannah monitor, Varanus exanthematicus (TL

70 cm), which feeds on terrestrial arthropods and molluscs

(Bennett, 2002, 2004;Cisse, 1972). Varanus gouldii was selected as

the focal point of the study because it is intermediate in size, and,

within the context of the Varanidae, its skull morphology andecology are unspecialised. The relative sizes and phylogenetic

relationships of the species used in this study are shown in Fig. 1.

2.2. Methods

2.2.1. Segmentation and reconstruction of CT data

Meshes for theV. acanthurus, V. gouldii, V. exanthematicus and

L. borneensis specimens were generated from scans provided by

the University of Texas High-Resolution X-ray Computed Tomo-

graphy Facility (UTCT;www.digimorph.org). For theV. komodoensis

and V. salvadorii specimens, CT scans were taken on a Toshiba

Aquilion 64 slice medical scanner at the Newcastle Mater Hospital.

Medical scanners have lower resolutions than the UTCT scanner:

counteracting that loss of resolution, those two specimens are muchlarger and thus the voxel sampling rate, as a proportion of skull, is

comparable. CT data were segmented in MIMICS (Materialise, 2010)

W.C.H. Parr et al. / Journal of Theoretical Biology 301 (2012) 1142


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to produce surface meshes, which were exported from MIMICS

as.PLY files. For each specimen, two meshes were created by

manually segmenting the scan data; a mesh wherein the Meckelian

canal was included as an internal surface (hereafter termed cavity

models), and a mesh wherein the Meckelian canal was closed at the

adductor fossa and in-filled (no-cavity models). Our modelling did

not include bonebone sutures or jaw kinesis as this would have

been very time consuming and our primary objective was not to

assess the role of sutures, or model kinetic jaw behaviour, but to

develop accurate warping protocols for biological shapes. The

meshes were (1) rotated to their principal components (anterior

posterior length to the x-axis, medio-lateral width to the y-axis,

dorsalventral height to thez-axis), and (2) scaled to the maximum

length of theV. gouldiimodel.

2.2.2. 3D mesh warping using template optimisation

The method of warping a 3D mesh to the shape of another

target 3D mesh using a subset of landmarks and slid semiland-

marks is termed template optimisation and is accurate in

reproducing the target meshs shape (Ruto, 2009). The template

optimisation technique we present here comprises two steps;

(1) assigning homologous 3D landmarks, pseudo landmarks and

slid semilandmarks to the meshes, and (2) warping the meshes

using the GMM thin plate spline function. Both steps were carriedout in Landmark (Wiley et al., 2005;Ghosh et al., 2009;Institute

for Data Analysis and Visualisation IDAV, 2010). This software is

freely available at: http://www.idav.ucdavis.edu/research/Evo

Morph. The MIMICS.PLY files were imported into Landmark. Subse-

quently, landmarks were placed upon each surface mesh using the

curve function, which assigns slid semilandmarks along a user

defined curve, and the point function, which assigns landmarks to

user defined point locations (seeTable 1for anatomical description

of point and curve positions). Curves were chosen to be spatially

homologous between different specimens and to describe a natu-

rally occurring curve within the bone (with no inflections); each

curve is constructed from two user defined landmark or pseudo

landmark points, with additional slid semilandmarks automatically

generated for each curve (Appendix Fig. A1). For each part of the

mandible, curves were used to mark the medial, lateral, dorsal and

ventral edges. To determine the optimal number of landmarks for

performing the warps, a separate sensitivity analysis was conducted

using 5 different landmark configurations and densities ranging

from using 92 single landmark points to using a combination of 20

single landmarks in addition to 540 pseudo landmarks and slid

semilandmarks. Accuracy of warps was judged by warping the

V. gouldii mesh to the target shape of theL. borneesis mesh and

comparing the bending displacement in the warp with that in the

originalL. borneesis. We achieved the most accurate warping results

using 20 single landmarks and 24 curves (12 of which had 8 sliding

semilandmarks and 2 pseudo landmarks, 12 of which had 18 sliding

semilandmarks and 2 pseudo landmarks), giving a total of 380 pointsfor the mesh of each mandible (Table 1, Fig. 2, Appendix Fig. A2).

For the shape analysis of these landmarks, the 3D points were

Fig. 1. Relative sizes and phylogenetic context of the varanoids included in this study. Phylogeny follows (Ast, 2001); mandibles are shown in dorsal view, scaled to the

same centroid size; lateral view of skulls are show absolute size of specimens used (scale bar50 mm).

W.C.H. Parr et al. / Journal of Theoretical Biology 301 (2012) 114 3


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Procrustes superimposed (rotated, translated and scaled) before being

entered into a PC analysis. The differing tooth counts in the varying

species were maintained through the warping process for the cavity

and no-cavity models, with an additional set of no-cavity models

generated with all of the teeth removed except for the distal pair(no-cavity toothless models). No landmarks were placed on the teeth,

as their number and relative position varied between different

species, so the warping process did not warp the teeth shape directly.

Changes in tooth shape of theV. gouldiimodel occurred as a result of

changes in the mandible shape (e.g. length) when warped to different

target shapes.

Our assessment of the accuracy of the warping process inreproducing biomechanical behaviour of target models, for the

final models used in the experiment, was based on warping the

Table 1

Anatomical description of single landmark and curve (pseudolandmark and slid semilandmark) point placement on varanid mandibles.

Description Curves

(number of

landmarks)

Description

S1 Dorso-lateral apex of posterior surface of retro-articularprocess (left) [S7]

C1 (10) Dorsal margin of coronoid; from coronoid processposteriorly to dentary-coronoid contact anteriorly (left) [C2]

S2 Dorso-medial apex of posterior surface of retro-articular

process (left) [S6]

C2 (10) Dorsal margin of coronoid; from coronoid process

posteriorly to dentary-coronoid contact anteriorly

(right) [C1]

S3 Lat er al apex of ant erior rim of glenoid f os sa (left ) [S1 0] C 3 ( 20 ) Ventr al margin of post erior ramus ; fr om vent ral apex of

posterior surface of retroarticular process posteriorly to

dentary-splenial-angular contact anteriorly (right) [C4]

S4 Antero-medial apex of anterior rim of glenoid fossa (left) [S8] C4 (20) Ventral margin of posterior ramus; from ventral apex of

posterior surface of retroarticular process posteriorly to

dentary-splenial-angular contact anteriorly (left) [C3]

S5 Postero-medial apex of anterior rim of glenoid fossa (left) [S9] C5 (20) Ventral margin of dentary; from dentary-angular contact

posteriorly to mandibular symphysis anteriorly (right) [C6]

S6 Dorso-medial apex of posterior surface of retro-articular

process (right) [S2]

C6 (20) Ventral margin of dentary; from dentary-angular contact

posteriorly to mandibular symphysis anteriorly (left) [C6]

S7 Dorso-lateral apex of posterior surface of retro-articular

process (right) [S1]

C7 (20) Lateral margin of tooth row (right) [C8]

S8 Antero-medial apex of anterior rim of glenoid fossa (right) [S4] C8 (20) Lateral margin of tooth row (left) [C7]

S9 Postero-medial apex of anterior rim of glenoid fossa (right) [S5] C9 (20) Lateral margin of posterior ramus; from anterior rim of

glenoid fossa posteriorly to surangular-dentray contact

anteriorly (left) [C10]

S10 Lateral apex of anterior rim of glenoid fossa (right) [S3] C10 (20) Lateral margin of posterior ramus; from anterior rim of

glenoid fossa posteriorly to surangular-dentray contact

anteriorly (right) [C9]

S11 Anterior most apex of dentary (left) [S12] C11 (10) Lateral margin of coronoid; from coronoid process

posteriorly to coronoid-dentary contact anteriorly (left)

[C12]

S12 Anterior most apex of dentary (right) [S11] C12 (10) Lateral margin of coronoid; from coronoid process

posteriorly to coronoid-dentary contact anteriorly (right)

[C11]

S13 Dorsal margin of adductor fossadorsal apex (left) [ S17] C13 (10) Medial bo rder of posterior Mecke lian fora men; fro m

posterior-medial process of coronoid anteriorly to anterior

rim of glenoid fossa posteriorly (left) [C14]

S14 Dorsal margin of adductor fossaposterior most extent (left) [S18] C14 (10) Medial border of posterior Meckelian foramen; from

posterior-medial process of coronoid anteriorly to anterior

rim of glenoid fossa posteriorly (right) [C13]S15 Dorsal margin of adductor fossaanterior most extent (left) [S19] C15 (10) Medial edge of dorsal surface of retro-articular process

(right) [C17]

S16 Anterior most projection of angular (medial surface) (left) [S20] C16 (10) Lateral edge of dorsal surface of retro-articular process

(right) [C18]

S17 Dorsal margin of adductor fossad ors al apex ( right ) [S1 3] C 17 ( 10 ) Med ial edge of d or sal s urf ace of retro-art icular p rocess ( left)

[C15]

S18 Dorsal margin of adductor fossaposterior most extent (right) [S14] C18 (10) Lateral edge of dorsal surface of retro-articular process (left)

[C16]

S19 Dorsal margin of adductor fossaanterior most extent (right) [S15] C19 (20) Ventro-lateral margin of dentary; directly ventral to tooth

row, from dentary-surangular contact posteriorly to

mandibular symphysis anteriorly (right) [C20]

S20 Anterior most projection of angular (medial surface) (right) [S16] C20 (20) Ventro-lateral margin of dentary; directly ventral to tooth

row, from dentary-surangular contact posteriorly to

mandibular symphysis anteriorly (left) [C19]

C21 (10) Dorso-lateral border of posterior Meckelian foramen; from

lateral apex of anterior rim of glenoid fossa posteriorly to

posterior medial process of coronoid anteriorly (right) [C22]C22 (10) Dorso-lateral border of posterior Meckelian foramen; from

lateral apex of anterior rim of glenoid fossa posteriorly to

posterior medial process of coronoid anteriorly (left) [C21]

C23 (20) Medial margin of tooth row; from posterior most extent of

tooth row posteriorly to posterior limit of mandibular

symphysis anteriorly (left) [C24]

C24 (20) Medial margin of tooth row; from posterior most extent of

tooth row posteriorly to posterior limit of mandibular

symphysis anteriorly (right) [C23]



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V. gouldii mesh to the target shape of the other species and

comparing bending displacements and Von Mises (VM) strain

readings of the warped models to their targets in FEA. These tests

were repeated for cavity (11 models), no-cavity (11 models) and

no-cavity toothless (11 models), giving a total of 33 models for

use in the subsequent FEA, 15 of which were produced by

warping.

Several different methods of scaling 3D models are used in the

current FE literature. Recently, scaling 3D surface mesh models by

surface area has been suggested by Dumont et al. (2009). This

method is potentially problematic if some of the specimens

contain cavities, which increase the surface area of the model,

and some of the models have no internal cavities. Centroid size is

typically used to assess the size of specimens when they are

represented by landmarks (Parr et al., 2011a, 2011b; Rohlf and

Slice 1990). However, centroid size varies according to the

number of landmarks used. In this study, we conducted a sensi-

tivity analysis to establish the optimal number of landmarks toachieve accurate warps. Centroid size would not be comparable

across the sample where different warps are achieved using

different numbers of landmarks. The FE analysis consists of a

cantilever bending analysis. Therefore, the distance between

restraint and load (the moment arm) is key to determining the

overall bending displacement. Thus, the models were all scaled to

the length of theV. gouldii model.

2.2.3. Solid meshing of mandible surface meshes

The unwarped and warped surface meshes were exported

from Landmark as.STL files and imported into Harpoon; version

4.1(a) (Sharc, 2010) to be solid-meshed (as low order tetrahedral

elements). To check the accuracy of using tetrahedral brick

elements, we performed a convergence analysis on a solid beam.

The beam was given a radius computed as the average radius of

the mandible of V. gouldii (2 mm) and the length was set at

50 mm, which was the distance between the restraint and load

points in the V. gouldii model. Both ends of the beams surface

were tessellated, with one central node being restrained (one

end) and loaded with 5 N (the other end). The beam was solid

meshed at different mesh densities until the bending displace-

ment of the beam models converged. The solid brick elementswere assigned the same bone material properties as in the FEMs

(see below). The theoretical bending displacement for the beam

was calculated as the target displacement for the FEMs to con-

verge at (1.26 mm) (see supplementary information). Our beam

FEMs converged with tetrahedral element numbers over 200,000

on a displacement of 1.23 mm, showing that Strand7 can produce

accurate FEA results using low order tetrahedral elements when

meshed at sufficiently high resolution (AppendixFig. A3).

To establish the optimal solid mesh density for the scaled

mandible models, a convergence analysis on the number of brick

elements in the models was conducted (Appendix Fig. A3B).

Bending displacement results converged when models were

comprised of over 1 million elements. The meshing functions

were set so that internal elements were of similar size to theexternal elements. No-cavity models were solid meshed easily.

However, accurate solid meshing of cavity models proved more

difficult, as the meshing algorithm tended to fill the cavities. This

required the mesh resolution to be adjusted so that the cavities

were represented accurately in the different models (element

numbers for each of the models are listed in Table 2).

Fig. 2. Landmark and semilandmark template. V. gouldii surface mesh model with

all 20 single landmarks and 24 curves (with slid semi-landmarks shown) placed

on the 3D surface mesh model (see Table 1 for description of landmarks).

(A) Dorsalventral view, (B), medio-lateral view (C) ventraldorsal view. Table 2

Number of brick elements in each of the models entered into the FE analyses.

Mod el Num ber of element s

Cavity No-cavity No-cavity toothless

L ba 1,201,810 1,225,125 1,618,872

V ab 1,250,692 1,051,243 962,451

V ec 1,172,976 1,158,030 1,568,298

Vgd 1,449,671 1,072,637 1,007,145

V ke 1,269,685 1,060,328 997,925

V sf 1,055,084 1,034,668 1,188,340

Vg-L bg 1,460,497 1,228,406 1,653,658

Vg-Va 1,117,285 1,103,828 1,010,878

Vg-V e 1,479,362 1,039,109 1,643,113

Vg-Vk 1,053,534 1,282,578 996,179

Vg-V s 1,173,903 1,135,577 1,066,247

a L b; Lanthanotus borneesis.b Va; Varanus acanthurus.c V e; Varanus exanthamaticus.d Vg; Varanus gouldii.e

Vk; Varanus komodoensis.f V s; Varanus salvadorii.g The symbol- indicates warped to, i.e.Vg-L bVaranus gouldiiwarped to

the target mesh shape ofLanthanotus borneesis.



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2.2.4. Finite element analysis

FEA was performed using Strand7 version 2.4.1 (Strand7,

2010). The solid meshed models were exported as NAS files

from Harpoon to Strand7. To solve the FE models we used both

iterative and direct sparse linear static schemes with geometry

and AMD (respectively) node ordering algorithms. We found

identical results between the methods provided that the iterationlevel was sufficient to ensure convergence of the iterative geo-

metry solve. The direct sparse AMD was significantly faster for

our models, so we used this algorithm. The elements of the solid

models were assigned the following material properties: modulus

13,145 MPa, Poissons ratio0.4, density1.593109 T/mm3.

Youngs modulus was assigned on the basis of mean density, based

on equations published by Rho et al. (1995)for human bone. Density

was determined from X-Ray attenuation values (Houndsfield units)

as described previously (McHenry et al., 2007). The value for

Poissons ratio follows published values (Vogel, 2003). Elements

making up the surface of the temporomandibular joint (TMJ) were

selected and their surfaces tessellated to rigid links to minimise the

generation of artefacts through point loadings (McHenry, 2009;

McHenry et al., 2007). A single node restraint was then positionedin the middle of each articulation. This restraint prevented rotation

and translation movements in thex, y, zaxes. The forces generated

at these restraints were spread via the rigid links covering the

surface of the TMJ articulations. Surface bricks comprising the distal

most teeth from either side of the jaw were also tessellated to links

in order to spread applied loads. A 5 N load was applied to a single

node on the surface of each of the two front teeth (total 10 N load

applied in the direction-Z; Appendix Fig. A4). In this way the

maximum bending displacements and strain patterns of the mand-

ible models could be tested. This loading regime was used to test our

method of warping mandible morphologies rather than to simulate

realistic bite function.

2.2.5. Strain frequency plots

Strain frequency was plotted as follows; brick element Von

Mises (VM) strain data were exported from the Strand7 post-

processor as ASCII text files. The strain data were analysed using

code written in Mathematica (vs 7.0). As discussed above,

traditionally the range of approaches that can be applied in

comparisons of FEA results between different models has been

limited because different FEMs are made up of different numbers

of elements. Further, there is no strict homology in position

within the FEM of elements, so stress/strain/displacement results

for brick x in one FEM is incomparable to brickx in another FEM.

2.2.6. Landmark point strains

To allow for direct comparison of results of FEAs of differentFEMs, and to try and further integrate the fields of FEA with

GMM, we took the mean VM strain values of the closest 10 bricks,

in terms ofx , y , zcoordinates, to each landmark. This effectively

collapses the large and un-comparable data sets of the FEMs

to 380 3D landmark points and their associated VM strain

values. As these landmark points are homologous (being either

biologically homologous, representing homologous biological

structures, or spatially homologous, being for example 2/10ths

of the distance between point A and B in all models), VM

Fig. 3. PC analysis of template landmarks and semilandmarks. PC1 against PC2 scores for PC analysis of standardised original and warped landmarks. Note that as only the

landmarks, rather than the whole 3D models, are used in the PC analysis, the warps, which are created by matching the template landmarks to their targets landmark

positions, score almost identical PC scores to their targets. PC1 extreme positive and negative shapes are shown.

Table 3

Maximum FE model displacements (mm) for the original (un-warped) models

cavity, no-cavity and no-cavity toothless models, for each of the six speciesanalysed.

Species Cavity No-cavity No-cavity toothless

L b 0.612625 0.581379 0.67285

V a 1.63511 1.5093 1.54116

V e 0.455135 0.421291 0.481933

Vg 1.96704 1.69971 1.8022

V k 3.22508 2.19008 2.39123

V s 2.59192 2.33023 2.37768

Abbreviations as inTable 2.

Table 4

Maximum model displacements (mm) for the warped models cavity, no-cavity

and no-cavity toothless, for each of the warped models analysed.

Species Cavity No-cavity No-cavity toothless

Vg-L b 0.779618 0.614537 0.701655

Vg-V a 1.41368 1.34722 1.40311

Vg-V e 0.53987 0.453739 0.466255

Vg-V k 3.31423 2.35244 2.34285

Vg-V s 3.18295 2.71277 2.32214




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3.1. Principal components analysis

Principal Component (PC) 1 identifies the main axis of shape

variation between the different jaw models (Fig. 3). PC1 captured

81.3% of the shape variance, with PC2 capturing 7.9% of the shape

variance present in the varanoid mandibles. PC 3 only accounted for

6.0% of shape variation. The first five PCs captured 99.99% of varanoidmandible shape variation. In our sample, the shape variation identi-

fied by PC1 are between V. exanthematicus andL. borneensis (both

relatively shorter, wider and deeper mandibles) and V. salvadorii (a

longer, narrower and shallower mandible), as shown in Fig. 3. PC1

scores for original and warped toothless specimens were correlated

with bending displacements in the FEAs (r0.94, significant at

p0.01) showing that PC1 identifies shape characters related to

robusticity. PC2 identifies shape characters related to the relative

positioning of the posterior point of the coronoid process, the relative

angle of the retro-articular process and the relative extension of the

anterior most apex of the dentary (see Table 1 and Fig. 2 for

positioning of landmarks). PC scores for V. gouldiimodels warped to

target model shapes varied by 0.0002 or less.

3.2. FEA maximum bending displacements

Bending displacements of warped meshes resembled targets

more than original meshes for no-cavity, cavity and no-cavity

toothless model warps (Tables 3 and 4,Fig. 4). Warped no-cavity

models are more accurate in reproducing target displacement

behaviour than cavity models, and no-cavity toothless models are

more accurate than no-cavity models with teeth (Fig. 4). Warped

models that incorporated internal cavities demonstrated greater

maximum bending displacements in the FEAs than the warped

no-cavity models (Tables 3 and 4).

3.3. Von Mises (VM) strain fields

Visual plots of VM strain fields are consistent with the patterns

from maximum displacement data (Fig. 5). No-cavity models are a

close match for no-cavity targets. The cavity warps also showed

strain fields that were more similar to their target than their

originals. However, visual plots are hard to quantify. Note that the

method of using single point restraints and loads spread via rigid

links was successful in not creating high strain anomalies on

single bricks (Fig. 5).

3.4. Strain frequency plots

Strain frequency plots indicate that warping works well for the

V. gouldii/L. borneensis and V. gouldii/V. exanthematicus pairs, for

cavity, no-cavity and particularly no-cavity toothless meshes; the

frequency distribution of the warps closely matched those for the

Fig. 5. Strain fields for the no-cavity V. gouldii mesh warped to the shape of other species. The original V. gouldii model is in the centre of the figure, the original target

models are outer most, the warped (from original to target morphology) V. gouldiimodels are inbetween. Note the similarities in strain fields of the warped V. gouldiimodel

to its target models.



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targets, and were more different from their source models (Fig. 6).

The frequency plots also show that, for these pairs, the source and

target models had very different strain frequency plots. The

pattern is different for the other pairs. ForV. gouldii/V. acanthurus,

V. gouldii/V. salvadorii and V. gouldii/V. komodoensis warps the

strain frequency plots did not clearly match the targets more

closely than they did the source. For the no-cavity and no-cavity

toothless warps this is even more apparent. Also clear is that for

V. acanthurus, V. komodoensis and V. salvadorii, the distributions

for the source and targets are much closer than for V. exanthe-

maticus and L. borneensis. Comparing the strain frequency plots

for the cavity models with those of the no-cavity models; the

cavity models show peaks that are longer (extended to the right)

and the peak of the strain frequency is shifted to the right (Fig. 6).

Both of these signify more instances of higher strain occurrence in

the cavity models than the no-cavity models.

3.5. Landmark point strains

Landmark point strains for no-cavity toothless models areshown in Fig. 7. Qualitative assessment of these models shows

that (a) they are broadly similar to the strain fields shown inFig. 5

and, (b) support the results of the strain frequency analysis in that

the V. gouldii/V. exanthematicus and V. gouldii/L. borneesis warps

show reduced instances of high strains. Percentage differences in

mean landmark point strains ofV. gouldii/original target model,

warp/target and V. gouldii/warp showed the largest differences

between V. gouldii and target models for V. Exanthematicus

(264.1%), L. borneesis (240.7%) and V. acanthurus (23.3%) targets,

but only small percentage differences (5.1%, 3.4% and 3.6% %,

respectively) between warps and these same targets (Table 5).

Percentage differences between warps/targets forV. komodoensis

and V. salvadorii also showed only small differences (5.7% and

1.4%, respectively, negative percentages showing that mean

strains for the targets were larger than those for the warps).

However, for these comparisons the originalV. gouldii and target

models showed only small percentage differences in landmark

point strains as well (8.0% and 6.4%, respectively,Table 5).

4. Discussion

The main aims of this study were to develop a method of

accurately warping 3D models and to develop additional

Fig. 6. Strain frequency plots. Von Mises strain FEA results. Von Mises strain frequency plots for cavity, no-cavity and no-cavity toothless original and warped models. (For

interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).



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Fig. 7. Landmark point strain plots. Mean Von Mises strain values for the closest 10 element centroids to each landmark are calculated. This mean Von Mises strain value is

coloured coded at each 3D landmark coordinate point. Colours are rebased against the maximum Von Mises strain value experienced in this sample (occurs in V. salvadorii).

Cool colours (blues) signify low strain, hot colours (red) signify high strain. Note the v isual similarities between original landmark point strain plots and those of theV. gouldii

model when warped to other varanoid targets. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)



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statistical tools for analysing FEA results. We have shown that our

method of warping 3D models from an original shape to that of a

different target shape can produce FEMs that accurately repro-

duce two commonly applied measures of mechanical behaviour

(bending displacement and VM strain distribution) in the

target model.

Further, we have introduced two new approaches that areapplicable to FEA results:

1) strain frequency, which can be used on FEMs characterised by

different numbers of elements and,

2) Landmark point strains that allow comparisons of VM strain

values at homologous points in different FEMs. This technique

applies the GMM principals of landmark homology to the

analysis of FEMs.

For each of the six original (un-warped) mandibles, the cavitymodels exhibited greater displacements and greater VM strain

magnitudes than the no-cavity and no-cavity toothless models

(seeTable 3), therefore hypothesis 1 is accepted. That is, for each

jaw specimen, initial models with internal geometry (cavity

models) will demonstrate higher strain values and increased

bending magnitudes over solid (no-cavity) jaw models. Although

the internal cavities increased the strains (Fig. 6) and bending

displacements (Tables 3 and 4), Fig. 8shows that the pattern of

strain distribution through the inside of the jaw bones and around

the cavities remains similar in the cavity and no-cavity models.

Table 3shows the displacements obtained from the analyses

of the warped models. In all cases the displacements of the

warped models are more similar to the displacements of their

target jaw models than their original jaw models (Fig. 4). Thus,

hypothesis 2 is also accepted: jaw models with different initial

morphology, and consequently different biomechanical proper-

ties, will demonstrate similar biomechanical properties (strain

and bending displacement patterns), when they have been

warped to similar shapes.

Within the PC shape space, the warped models match the

shape of the target models exactly (Fig. 3). This is because in

template optimisation the warp is performed on the subset of the

original 3D mesh, the landmarks and semilandmarks making up

the template. Error in reproducing target shape of the whole 3D

mesh models during the warping process arises from the tem-

plate of landmarks either not being extensive enough, or from

Table 5

Percentage differences in mean landmark point strains between Vgouldii and

target, Vgouldii and warp, warp and target FE results.

Specimen comparison % Difference in mean

landmark point strain

Vg, L b 240.7

Vg, Vg-L b 229.5

Vg-L b, L b 3.4Vg, Va 23.3

Vg, Vg-Va 19.1

Vg-Va, Va 3.6

Vg, V e 264.1

Vg, Vg-V e 246.4

Vg-V e, V e 5.1

Vg, Vk 8.0

Vg, Vg-Vk 2.5

Vg-Vk, V k 5.7

Vg, V s 6.4

Vg, Vg-V s 7.9

Vg-V s, V s 1.4


Fig. 8. Internal VM strain distribution; cavity vs no-cavity. Cross section view showing VM strain distribution calculated by the FE analysis through the cavity (A) and no

cavity (B)V. gouldiijaw models. Note that although the magnitude of the strains may be greater in the cavity model (also see Table 1), the distribution pattern of the strains

are little affected by the presence of the cavities.



12/14


13/14

This study introduces a new method for the warping of surface

models using a GMM approach and a new method with which to

test the effects of shape differences between models in FEA.

Further, the introduction of landmark point strains (equally

applicable to stress data), allows the reduction of large data sets

common to FEA results, without information loss. Direct qualita-

tive and quantitative comparisons can be made between homo-

logous points on different FEMs. These are potentially powerful

techniques representing a further synthesis of the fields of morpho-

metrics and FEA. We believe that the warping and statistical

approaches presented here may be useful in a range of applica-

tions. These include the generation and analysis of virtual

reconstructions, generic models that approximate species means,

hypothetical morphologies and evolutionary intermediaries, and

prosthetic devices.

5. Conclusions

We introduce a method for warping of 3D mesh geometry

based on freely available software and new methods (strain

frequency plots and landmark point strains) for the interpretation

of FEA results. In applying these approaches to FEMs based

on warped and target surface meshes for models of varanoid

mandibles, with and without internal cavities included, we found

that initial models with internal cavities exhibited greater bend-

ing displacements and VM strain magnitudes than models with-

out jaw cavities. This was also largely true for jaw models warped

to a different target jaw shape.These warping and statistical procedures were successful in

matching VM strains and bending displacements of warped

models to their targets, i.e., no significant differences were found

between warped and target jaw models for displacement and VM

strain behaviours. VM strain field data, as typically used to assess

FEA results in previous studies, are difficult to analyse quantita-

tively. We introduce landmark point strains as a way to quantita-

tively compare homologous points across an FEM sample and

used this method to show that VM strain results between the

warp and target FEMs were very similar. VM strain frequencies

showed that cavity models, as well as cavity warps, exhibit higher

strains and a greater proportion of high strains than no-cavity

models.

Acknowledgements

This work was partly funded by an Endeavour Award Post

Doctoral Research Fellowship (2359_2011) to WCHP, an Australian

Research Council Discovery Project Grant (DP0986471) to CRM, and

Discovery Project (DP0666374 and DP0987985) and University of

New South Wales Internal Strategic Initiatives and Gold Star Grants

to S.W. We thank Eleanor Cunningham (Newcastle Mater Hospital)

for CT scanning, Ross Sadlier and Cecile Beatson (Australian

Museum) for access to specimens, and Matthew Colbert and Jessie

Maisano (University of Texas) for access to CT scan data taken by the

Digital Morphology Lab (www.digimorph.org). Andrew and Helen

Parr are thanked for financial support during this work. The authors

declare no conflicts of interest related to this work.

Appendix A

SeeFigs. A1A4.

Fig. A4. Finite Element Model (FEM) in Strand7. (A) overview of the solid meshed V. gouldii mandible (blue) with restraints spread over the temporomandibular joints articular

surfaceby rigid links (proximal light blue/green lattice; close up in C). Also shownare the two5 N loads applied to the front teeth (A,B). Theloadswerespread over the surfaceof the

teeth using rigid links (close up in B). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. A3. Element number convergence analyses. (A) Convergence analysis on a solid meshed beam (radius 2 mm). (B) Convergence analysis for the no-cavity V. gouldii

model (with teeth).



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Appendix B. Supplementary material

Supplementary data associated with this article can be found

in the online version atdoi:10.1016/j.jtbi.2012.01.030.

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