Biomechanics of Felid Skulls: a Comparative Study Using Finite Element Approach

Home , Thylacosmilus

Biomechanics of felid skulls: A comparative study using finite element approach

Uphar Chamoli

Supervisor: Dr. Stephen Wroe

Submitted in partial fulfilment of the requirements for the degree of

Master of Philosophy

School of Biological, Earth and Environmental Sciences,

Faculty of Science,

The University of New South Wales

August 2011

CERTIFICATE OF ORIGINALITY

I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, nor material which to a substantial extent has been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere is explicitly acknowledged in the thesis.

I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project's design and conception or in style, presentation and linguistic expression is acknowledged.

Signature ......

Date......

Dedicated to my parents

and my deep rooted faith in Karma

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A major component of this thesis is published in Journal of Theoretical Biology

x Chamoli, U. & Wroe, S., 2011. Allometry in the distribution of material properties and geometry of the felid skull: Why larger species may need to change and how they may achieve it. Journal of Theoretical Biology 283(1):217- 226.

Some of the other methodologies used in this project, OR developed during the course of this project are also published in following journals

x Attard, M., Chamoli, U., Ferrara, T., Rogers, T., and Wroe, S., 2011. Skull mechanics and implications for feeding behaviour in a large marsupial carnivore guild: the thylacine, Tasmanian devil and spotted-tailed quoll. Journal of Zoology. doi:10.1111/j.1469-7998.2011.00844.x

x Wroe, S., Ferrara, T. L., McHenry, C. R., Curnoe, D., Chamoli, U., 2010. The craniomandibular mechanics of being human. Proceedings of the Royal Society B: Biological Sciences, 277: 3579-3586.

ABSTRACT

Shape and scale-related effects on biomechanical construct of organisms depend strongly on the properties and distribution of materials of which they are built, with an inherent requirement for avoiding failure over the entire lifetime. The cat family

(Felidae) has been considered morphologically and behaviourally conservative, and hence an appropriate focus for investigations into the role of allometry. Here I apply three-dimensional (3D) finite element analysis (FEA) to models representing the skulls of seven extant felid species in order to (1) more fully assess their biomechanical performance; and (2) to predict allometric trends regarding overall geometry and relative distributions of cortical and cancellous bone. Results derived from incorporating material properties distribution for cortical and cancellous bone in the finite element models (FEMs) largely support the contention that mechanical behaviour in the felid skull is conservative across species. A negative allometric trend between cortical bone volume and total skull bone volume, and positive allometry between total skull bone volume and skull surface area were also observed. Further mathematical modelling using beam mechanics suggests that these allometric trends reflect a need for larger species to respond to physical challenges associated with increased size, and, that changes in skull shape, bone composition, or a combination of both, may be required to accommodate these challenges. I conclude that as felids become larger, overall skull bone volume relative to surface area increases by adding less dense and more compliant cancellous bone. This brings an overall saving in mass and a reduced burden on metabolism to produce biologically expensive cortical bone, without compromising much on the overall stiffness. In a further study, I constructed

3D FEMs of two extinct sabretooth predators (Smilodon fatalis and Thylacosmilus atrox) to investigate functional convergence. Relative to the conical-toothed Panthera pardus, predicted jaw muscle driven bite forces in both were low, but their skulls appeared well-adapted to resist forces generated by cervical muscles. Although findings for S. fatalis are consistent with an extension of 'normal' biting behaviour, estimated jaw adductor driven bite forces for T. atrox considered with evidence for a major translational component, suggest that it was more specialised.

ACKNOWLEDGEMENTS

I feel very honoured to have the opportunity to work on this project in the

Computational Biomechanics Research Group’s lab in the School of Biological, Earth and Environmental Sciences at UNSW. I am deeply indebted to my advisor Dr. Stephen

Wroe for instilling in me the love and passion for Science, and for his unwavering support and encouragement all through my MPhil. Thank you Steve for the academic, moral and financial support – without any of these, I wouldn’t have ever conceived of completing this project.

I am very grateful to all my brilliant, generous colleagues in the group for sharing their intellectual diversity, and for stimulating discussions and free exchange of ideas throughout the duration of this project, particularly William Parr, Toni Ferrara, Marie

Attard, Peter Aquilina, Natalie Rogers and Naomi Tsafnat. Thanks to Assoc. Prof. Phil

Clausen of University of Newcastle, Callaghan for help with engineering basics, Prof.

Timothy Rowe of University of Texas, Austin and Prof. Lawrence Witmer of Ohio

University, Athens for access to CT scan data. Thanks to Assoc. Prof. David Cohen,

Head of School-BEES, for providing additional funding to help me write this thesis.

My gratitude for the entire Nodiyal family for assimilating me into their family will always be deep and long. A special thanks to kids – how relaxing it is to come home after a tiring day at work and talk to kids – loved you for your innocence. Thank you to

Neeta and Tanu for putting up with my constant complaining nature, and for having those mind-boggling maths and physics discussions. Lastly, big thanks to my brother and sister for having faith in me and my potential!

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TABLE OF CONTENTS

Certificate of Originality...... (ii)

Abstract...... (v)

Acknowledgement...... (vii)

List of Abbreviations...... (xii)

List of Softwares used...... (xiii)

List of Figures...... (xiv)

List of Tables...... (xvii)

List of Appendices...... (xviii)

Chapter 1 Introduction

1.1. Finite Element Analysis and Biology...... 2 1.2. Comparing Biomechanical Performance in Extant Felid Skulls...... 4 1.3. Allometry in Material Properties distribution and Geometry of Extant Felid Skulls...... 6 1.4. Bone: A Composite Structure...... 9 1.5. A study of convergence in the biomechanical performance of two extinct mammalian sabretooths Smilodon fatalis and Thylacosmilus atrox...... 12 1.6. Main Aims...... 18

Chapter 2 Methods and Materials

2.1 Digital Reconstruction of Felid Skulls using Computer Tomography (CT) Scan data...... 19 2.2 Finite Element Modelling Protocols in Strand7...... 22 2.2.1 Model Pre-processing...... 22 2.2.1.1 Modelling Major Muscle Sub-groups...... 23 viii

2.2.1.2 Jaw Hinge Mechanism...... 25 2.2.1.3 Gape Angle and Hinge Rotation Co-ordinate System...... 26 2.2.1.4 Estimation of Temporalis and Masseteric Muscle Force: Dry Skull Method...... 26 2.2.1.5 Muscle Fibres Attachment, Pre-tension and Distribution...... 30 2.2.1.6 Assigning Material Properties to Beam and Brick Elements...... 33 2.2.1.7 Boundary conditions for simulating bilateral canine and unilateral carnassial bite case...... 35 2.2.1.8 Occipital Link Constraints...... 37 2.2.1.9 Quantitative Estimation of Cortical and Cancellous Bone Volume using CT Intensity Data...... 38 2.2.1.10 Controlling Size differences ...... 40 2.2.1.11 Assembling Homogeneous Finite Element Models (FEMs)...... 41 2.2.2. Model Solving...... 42 2.2.2.1 Linear Static Analysis...... 44 2.2.2.2 Batch Solver...... 45

2.2.3. Model Post-processing...... 46 2.2.3.1 Visual Post-processing Plots...... 47 2.2.3.2 Peek Entity Results...... 47 2.2.3.3 List Results...... 48 2.2.3.4 Model Summary...... 49 2.3. The Allometry Signal...... 49 2.4. Statistical Analyses...... 53 2.3.1 Mean Brick VM stress...... 53 2.3.2 Two- factor ANOVA on Mean VM Stress Data...... 54 2.3.3 Root Mean Square (RMA) Regression Analysis...... 55 2.3.4 Confidence Intervals for Slopes of Allometry Graphs...... 56

2.5 Finite element modelling of sabretooth predators Smilodon fatalis and Thylacosmilus atrox...... 57

2.5.1 Body mass estimates...... 59

2.5.2 Theoretical maximal gape angle...... 60

2.5.3 Centres of sabretooth arcs ...... 61

2.5.4 Digitally 'stitching' the volumetric meshes ...... 63

2.5.5 Head-depressors reconstruction...... 64

Chapter 3 Results

3.1 Comparative Biomechanical Performance in Extant Felid Skulls...... 66 3.1.1 Felid FEMs Scaled to the same Surface Area and same Jaw Muscle Force as A. jubatus...... 66 3.1.2 Felid FEMs Scaled to the same Surface Area and same Output Bite Force as A. Jubatus...... 70 3.2 Homogeneous and Heterogeneous FEMs...... 74 3.3 Allometry in the distribution of Material Properties and Geometry of the Felid Skulls...... 77 3.3.1 Allometry in the Relative Proportion of Cortical Bone Volume...... 78 3.3.2 Allometry between the Skull Bone Volume and Skull Surface Area...81 3.4 Comparing Biomechanical Performance between Extinct Sabre-toothed Felids Thylacosmilus atrox, Smilodon fatalis and Extant Felid Panthera pardus 3.4.1 Jaw Musculature driven Canine Bite and its variation with Gape...... 83 3.4.2 Cervical Muscles driven Canine Bite...... 92

Chapter 4 Fundamental Challenges Associated with Increasing Size: Beam Analogies

4.1 Self Weight Beam Analogy...... 95

4.2 Composite Beam Analogy...... 100

Chapter 5 Discussion and Conclusions

5.1 Biomechanical conservatism and allometric trends in extant felid skulls..110

5.2 Biomechanical comparisons between the extinct sabretooth predators S. fatalis, T. atrox and the extant conical tooth felid P. pardus...... 115

References...... 119

Appendices...... 132

LIST OF ABBREVIATIONS

ANOVA Analysis of Variance

CT Computer Tomography

FEA Finite Element Analysis

FEM Finite Element Model

GPa Giga-Pascals

Kgs Kilo-grams

KPa Kilo-Pascals m Metre mm Millimetres

N Newtons

RMA Reduced Major Axis

TMJ Temporomandibular joint

VM Von-Mises

F. s. lybica Felis silvestris lybica

L. pardalis Leopardus pardalis

N. nebulosa Neofelis nebulosa

A. jubatus Acinonyx jubatus

P. pardus Panthera pardus

P. concolor Puma concolor

P. leo Panthera leo

S. fatalis Smilodon fatalis

T. atrox Thylacosmilus atrox

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LIST OF SOFTWARES USED

Strand7 - version 2.4.1 Strand7 Pty Ltd., Sydney

Mimics- version 13.1.0.70 Materialise, Leuven, Belgium

Matlab- version R2007b Math Works, Natick MA

Wolfram Mathematica 7.01 Wolfram Research, Champaign IL

Microsoft Office suite 2007 Microsoft Corporation, Richmond WA

Abode Photoshop CS2 Adobe Systems Incorporated, San Jose CA

GEUP 3 - version 6.1 GEUP.net, Santa Cruz de Tenerife, Spain

Rhinoceros3D version 4.0 McNeel North America, Seattle WA

Irfan view version 4.28 Copyright of Irfan Skiljan, Graduate of Vienna

Institute of Technology

Screen Garb Pro - version 1.8 Traction Software, UK

EndNote X3 Thomson Reuters

xiii

LIST OF FIGURES

Figure 1.1 A hollow and a solid pipe having the same amount of material and subjected to the same loads and constraints...... 7

Figure 1.2 The internal structure of a human long bone...... 10

Figure 1.3 Composite beam structure...... 11

Figure 2.1 An example of a single slice from the CT scan of a Panthera leo skull... 21

Figure 2.2 Pre-processed finite element model of Acinonyx jubatus showing muscle origin and insertion areas...... 24

Figure 2.3 Ventral view and dorsoposterior view of a skull: Thomason's dry skull method ...... 28

Figure 2.4 Screen-grab of the GEUP work window showing the 50 sided muscle X- sectional area polygon and the reference triangle...... 29

Figure 2.5 Pre-processed finite element model of Acinonyx jubatus showing muscle trusses...... 32

Figure 2.6 Transverse CT scan slices of three felids F. s. lybica, P. pardus and P. leo through the posterior-most carnassial tooth...... 39

Figure 2.7 Simple linear regression models of Y on X...... 56

Figure 2.8 Theoretical maximal gape angle in (a) S. fatalis, (b) T. atrox, and (c) P. pardus ...... 61

Figure 2.9 Centres of sabre-canines in (a) S. fatalis, and (b) T. atrox...... 62

Figure 2.10 Pre-processed FE model of S. fatalis showing neck muscle reconstruction and attachment web...... 65

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Figure 3.1 Estimated bite force (N) in the FEMs of different felid skulls. Models were scaled to the same surface area and used the same muscle force recruitment as A. jubatus...... 67

Figure 3.2 Von-Mises (VM) stress distribution in the FEMs of different felid skulls. Models were scaled to the same surface area and recruited the same net jaw muscle force as A. jubatus...... 68

Figure 3.3 Estimated jaw muscle recruitment (N) in the FEMs of different felid skulls. Models were scaled to the same surface area and generated the same output bite force as A. jubatus...... 71

Figure 3.4 Von-Mises (VM) stress distribution in the FE models of different felid skulls. Models were scaled to the same surface area and muscle force recruitment was adjusted to generate the same bite force as A. jubatus...... 73

Figure 3.5. VM stress distribution in homogeneous and heterogeneous unscaled FE models of F. s. lybica and P.leo...... 75

Figure 3.6 Graph showing RMA regression equation between log-transformed cortical bone volume and Skull bone volume...... 80

Figure 3.7 Graph showing RMA regression equation between log-transformed skull bone volume and skull surface area...... 82

Figure 3.8. VM stress distribution at theoretical maximal gape angle in S. fatalis, T. atrox, P. pardus during a jaw muscle driven bilateral canine bite....84

Figure 3.9 Smilodon fatalis: Variations in canine bite reaction force, jaw muscle recruitment , mean VM stresses with changing gape...... 86

Figure 3.10 Thylacosmilus atrox: Variations in canine bite reaction force, jaw muscle recruitment , mean VM stresses with changing gape...... 88

Figure 3.11 Panthera pardus: Variations in canine bite reaction force, jaw muscle recruitment, mean VM stresses with changing gape...... 89

Figure 3.12 VM stress distribution in jaw adductor driven bite with bite reaction force adjusted for body mass in S. fatalis, T. atrox, P. pardus...... 91

Figure 3.13 VM stress distributions in cervical musculature driven bite with bite reaction force adjusted for body mass in S. fatalis, T. atrox, P. pardus.92

Figure 4.1. Gain coefficient (β) plotted against thickness ratio (n) for a two property composite beam...... 106

Figure 4.2. Percentage weight savings (η) plotted against thickness ratio (n) for a two property composite beam...... 107

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LIST OF TABLES

Table 2.1 Hounsfield units (HU) range for cortical and cancellous bone in

CT scans...... 34

Table 2.2 Volumes of cortical and cancellous bone (mm3) in different felid skull specimens used in this study...... 40

Table 2.3 Total skull bone volume and total skull surface area of different felid skull FEMs...... 52

Table 3.1 Mean VM stress values (MPa) in different regions of the felid skull FEMs during a bilateral canine bite - models were scaled to the same surface area and recruited the same net jaw muscle force...... 69

Table 3.2 Mean VM stress in the skull of different felid skull FEMs. Models were scaled to the same surface area and same output bite force as A . jubatus...... 72

Table 3.3 Mean and standard deviation values of brick VM stresses in homogeneous and heterogeneous FE models of P. leo and F. s. lybica...... 77

Table 3.4 Cortical bone volume, total bone volume and proportion of cortical bone by volume in different felid skull FEMs...... 79

Table 3.5 Skull bone volume, skull surface area and thickness coefficient of different felid FEMs...... 81

Table 3.6 Body mass adjusted canine bite force results for a jaw adductor driven bite at maximum gape...... 90

Table 3.7 Body mass scaled bite force results for a cervical musculature driven bite at maximum gape ...... 93

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LIST OF APPENDICES

Appendix 1 Matlab code for plotting the graph between gain coefficient (β) and

thickness ratio (n) for a two property composite beam...... 132

Appendix 2 Matlab code for plotting the graph between percentage weight savings

(η) and thickness ratio (n) for a two property composite beam...... 133

Appendix 3 Strand7 pre-processing inputs for Felis silvestris lybica...... 134

Appendix 4 Strand7 pre-processing inputs for Leopardus pardalis...... 135

Appendix 5 Strand7 pre-processing inputs for Neofelis nebulosa...... 136

Appendix 6 Strand7 pre-processing inputs for Acinonyx jubatus...... 137

Appendix 7 Strand7 pre-processing inputs for Puma concolor...... 138

Appendix 8 Strand7 pre-processing inputs for Panthera pardus...... 139

Appendix 9 Strand7 pre-processing inputs for Panthera leo...... 140

Appendix 10 Strand7 pre-processing inputs for Smilodon fatalis ...... 141

Appendix 11 Strand7 pre-processing inputs for Thylacosmilus atrox...... 142

Appendix 12 Strand7 pre-processing inputs for Panthera pardus modelled for

comparisons with the sabretooth predators...... 143

xviii

1. INTRODUCTION

Linking the relationship between the form of an object and its intended function, and the need to optimise the use of materials for ‘design’ are the most fundamental of challenges facing engineers, designers and architects. The same broad challenges are widely thought to hold true in the evolution of living organisms. However an important difference exists between machines and organisms: Machines have a specific final purpose for which they are designed, whereas organisms are a result of long series of stepwise, incremental changes over vast tracts of time. Nonetheless, Darwinian theory predicts strong correlation between an organism’s structure, material composition and its function.

Thus, Darwinian theory gave rise to functionalism, the theory that the characteristics of living organisms are adapted to perform useful functions, and that the best way to understand the mechanical behaviour of a biological structure is to try and understand how it might be or have been used to perform specific functions for the organism. In particular, differences in morphology between vertebrate skulls, of both extinct and extant species, have intrigued researchers for centuries. Vertebrate skulls are adapted to perform various functions. In addition to housing the brain and other sensory organs, it plays an important role in acquiring and processing the food before passing it onto the gut for digestion. These functional demands are reflected in skull's structure

(Shipman et al. 1985). Understanding the functional morphology of the vertebrate skull has been the basis of many evolutionary studies, as it can reveal important information regarding the dietary preference of the species (Attard et al., 2011).

1.1 Finite element analysis and biology

Unlike most engineering structures, biological structures such as the skull have complicated geometries which cannot be simply represented by standard mathematical shapes. For problems involving complicated geometries, loadings, and multiple material properties, it is generally difficult to investigate the relationships between form, function and optimal use of materials using traditional analytical mathematical approaches. Analytical solutions are those given by a mathematical expression that yield the values of the desired unknown at any location, and are thus valid for an infinite number of locations in a structure.

For problems whose analytical solutions are difficult to obtain, we need to rely on numerical methods such as finite element analysis (FEA). Unlike analytical methods, where exact solutions of partial differential equations are sought, FEA is a powerful computational technique that converts the problem into a system of simultaneous algebraic equations, solutions of which yield approximate values of the unknowns at a discrete (finite) number of points in the continuum. This process of modelling an object by dividing it into a system of smaller elements of known geometry (called finite elements), interconnected at points common to two or more elements (nodes) is called discretisation. There are three basic steps involved in finite element modelling of any structure.

1. Model pre-processing: which involves creating the model geometry, applying forces and constraints, assigning material properties to different parts of the geometry and meshing (transformation of the geometric model into a set of discrete finite elements)

2. Solving: The FE solver assembles all the simultaneous equations resulting from the applied forces and constraints on the model and solves them for nodal displacements.

3. Post-processing and validation: involves interpreting the results obtained from the

FE solver and comparing them to the experimental data.

Initially developed for use in the aerospace industry, FEA has been used by engineers for decades to predict the stability of man-made structures such as dams and bridges and to optimise the use of materials for design. One major advantage of FEA over more classical techniques is its ability to predict stresses and strains not only across the entire surface of a structure but also throughout its internal geometry. When adequately validated, FEA can accurately predict the most likely sites of material failure. Considering the diverse and complex range of shapes known for biological structures, FEA is a very effective computational tool to gain a more precise understanding of their functional roles. Increasingly (Ross, 2005; Richmond et al.,

2005; Panagiotopoulou, 2009; Rayfield, 2005; Rayfield, 2007; Kupczik, 2008; Moreno et al., 2008; Wroe et al., 2010; Chamoli & Wroe 2011) FEA based on computer tomography (CT) scan data is being applied in biology, medical and earth sciences to address questions related to morphology, function and evolution. In addressing paleo- sciences questions, where the fossil specimen often cannot be mechanically tested,

FEA has the further benefit of being a non-destructive technique. In the present study,

I have used FEA based on computer tomography (CT) scan data to examine a number of hypotheses of broad significance to the understanding of form and function in the vertebrate skull. These are largely centred on the use of the felid skull for reasons discussed below.

1.2 Comparing biomechanical performance in extant felid skulls

Among terrestrial mammals, members of the cat family (Felidae) are amongst the most specialised towards a purely carnivorous lifestyle. Despite great differences in size between species, all cat species (felids), living and extinct are hyper-carnivores

(Valkenburgh, 1988; Van Valkenburgh and Ruff, 1987) with the great majority of their diets comprised of vertebrate meat, mostly from killed as opposed to scavenged prey.

Excepting lions (Panthera leo), adult felids are typically solitary ambush predators (Ewer,

1973) and the success of a hunt depends on the ability of a single predator to capture, restrain and kill its prey. Occasionally male cheetahs (Acinonyx jubatus), are known to hunt in groups. Larger prey may be tackled, but the efficiency of co-operative hunting is thought to be lower, i.e. predators are more easily spotted by prey (Schaller, 1972;

Packer, Scheel and Pusey, 1990).

The exact hunting modus operandi and prey choice in extant felids varies between species (Radloff and Du Toit, 2004; Hayward and Kerley, 2005). Felids generally use any of the three techniques to subdue their prey: (1) a throat bite, effectively compressing the trachea leading to suffocation; (2) a muzzle bite, covering the prey’s nostril with their mouth, also leading to suffocation; (3) a very precise nape bite to the back of the neck, where the canines push between vertebrae of the neck and disrupt the spinal cord (Therrien, 2005). Of these, the first and second techniques are typically used by large cats to kill large prey. This does not require particularly long canine teeth, but does require powerful enough jaw musculature to effectively restrict the airway of a large animal for a sustained period. The third technique is typically used to kill relatively small prey. However, a few taxa such as the jaguar (Panthera onca) and

4 cheetah are known to use variations on typical felid hunting and killing techniques.

Panthera onca is the only felid known to kill by biting through the temporal bones of its prey, effectively piercing the skull with its canines (Emmons, L., 1987). Acinonyx jubatus is unusual among felids in that it runs down typically small-medium sized prey over longer distances at high speeds and transfers momentum to the prey with its front paws. Once its prey has been grounded, A. jubatus preferentially delivers a throat bite that acts as a ‘clamp stranglehold’ (Ewer, 1973; Turner and Anton, 1997).

Craniomandibular morphology in all felids is highly derived for predatory activities

(Christiansen, 2008). A number of previous studies (Buckland-Wright, 1978;

Christiansen, 2007; Wroe and Milne, 2007; Christiansen, 2008) have found correlations between craniodental morphology and feeding behaviour in felids. Some have focused on particular features such as canine shape (Christiansen, 2007) and a growing number have addressed craniomandibular biomechanics more broadly by applying beam theory (Therrien, 2005; Thomason, 1991; Wroe et al., 2005) or FEA (McHenry et al.,

2007; Slater and Van Valkenburgh, 2009; Wroe, 2008). Despite great differences in size between species, both the morphology and predatory behavior of felids is generally considered to be conservative in that there is relatively little variation regarding either the shape of craniomandibular structures, or how they are applied (Christiansen, 2008;

Turner and Anton, 1997).

In the present study, the biomechanical performance of seven extant felid skulls (Felis silvestris lybica, Leopardus pardalis, Neofelis nebulosa, Acinonyx jubatus, Panthera pardus, Puma concolor and Panthera leo) was compared using CT based finite element modelling under similar conditions of loading and constraints. Here my approach was

5 to analyse skull structure by taking bite force and stresses developed during biting and feeding into consideration. To disentangle the effects of shape and size variations on biomechanical performance, all FE models were scaled to the same size following previously published protocols (Dumont et al., 2009). Bite reaction force, jaw muscle force recruited and von-Mises (VM) stresses were used as metrics to compare biomechanical performances across species.

1.3 Allometry in material properties distribution and geometry of extant felid skulls.

When a force is applied to an object, the deformation and stresses induced are not only a function of object’s geometry, these also depend on the property of the materials that comprise the object and how they are distributed within it. To date, comparative functional and evolutionary studies have usually been concerned with the biomechanical implications of skeletal shape. Such studies often pay little attention to differences between species regarding the distribution of material properties within the structure and how these may influence biomechanical performance, yet allometry may play an important role here (Strait et al., 2010). In biology, allometry refers to how characteristics associated with an organism scale relative to each other if other factors are held constant. Allometric scaling in cranial bone thickness has been reported in a recent study on primates (Strait et al., 2010), in which the authors concluded that the relative thickness of cortical bone in the crania of extant primates decreased with increasing body mass. Thickness of the cortical bone contributes to the cross-sectional area, which in turn influences the rigidity of the skull structure. The findings of Strait et al.

(2010) suggest a direction for further investigations into the role of allometric distribution of materials in vertebrate craniomandibular mechanics. Because felids are considered 6 morphologically and behaviourally conservative, they are an appropriate focus for such allometry studies. The first allometric investigation in the present study follows from variations observed in the volume proportion of cortical and cancellous bone materials in different felid skulls.

Shape (geometry) clearly has an important role in influencing the distributions and magnitudes of stresses induced in a structure when forces are applied to it. The example below (fig. 1.1) shows how (when using the same amount of material) a solid and a hollow pipe will have different second moments of inertia*, and hence different bending strengths.

Solid pipe Hollow pipe

Cross-sectional area – 10 cm2 10 cm2

Second moment of inertia – 7.957 cm4 23.868 cm4

Longitudinal bending stress - 100 % 30.305%

Figure 1.1 A hollow and a solid pipe having the same amount of material and subjected to the same loads and constraints

* Second moment of Inertia (I) is the property of a beam cross-section to resist bending or deformation

Thus in this instance, when the materials are distributed further away from the neutral axis of bending, the longitudinal bending stresses are reduced by ~ 70%. Clearly, through an efficient distribution of materials, as well as changes in shape, the stresses and deformations in a structure can be significantly reduced. The magnitude of induced stresses in a structure dictate its propensity to fail. When the stresses within a structure exceeds its fracture limit (which is a material property and is constant for a given material), the structure fails.

A recent study of felid skull biomechanics by Slater and Van Valkenburgh (2009) identified allometric trends regarding shape, but assumed that the felid skull was comprised of a single material with uniform properties. In this study, the authors concluded that in felids a positive allometric trend exists between cranial rigidity and body mass, and that the crania of larger felids were able to achieve increased cranial rigidity by increasing the skull bone volume relative to the surface area. However due to a small specimen size (n=3), the authors could not test their hypothesis statistically.

Also, only crania of the specimens were modelled and mandibles were not included in the analysis.

The second allometric investigation in the present study overcomes both the above limitations, by considering a relatively large sample size (n=7), and modelling the cranium and mandible as one articulated unit. Although a sample size of seven may be considered small for most statistical analyses, this is one of the most comprehensive

3D FE based comparative studies conducted on any taxon to-date. Total skull bone volume and skull surface area were measured from the Finite Element Models (FEMs) of seven extant felid skulls. Because visual inspection of CT images pointed towards a

8 negative allometric relationship between the thickness and skull size, regression analysis was then performed between Log-transformed skull bone volume and skull surface area to more fully examine the proposal of Slater and van Valkenburgh (2009).

1.4 Bone: A composite structure

Bone is a living, dynamic connective tissue that provides mechanical integrity for both locomotion and protection. It is a composite, viscoelastic, anisotropic material (Turner et al., 1995) which gets its strength from organic fibres (collagen) and inorganic crystals (hydroxyapatite: Ca10(Po4)6(OH)2) (Einhorn, T.A., 1992). At a macro-structural level, bone is classified into two major categories: cancellous bone (also known as trabecular or spongy bone) and cortical bone (also known as compact bone). Cortical bone is denser and has a relatively high resistance to bending and torsion. Cancellous bone is less dense and more elastic than cortical bone (Rho et al. 1993). The basic materials that comprise cancellous and cortical bone are the same (mineral and fibres).

Differences between cortical and cancellous bone are largely structural, with cancellous bone being more porous. The mechanical properties are influenced greatly by the degree of porosity, which is much less in cortical bone (5% to 10%) as opposed to cancellous bone (75% to 95%) - hence the ‘spongy’ appearance. Bone is a constantly remodelling material and its internal architecture is influenced by the mechanical stresses associated with its functions. The mechanical properties of bone vary not only according to the nature of force applied on it, but also to the direction and rate of application of these forces.

Figure 1.2 The internal structure of a human long bone (source: www.merriam-webster. com/concise/bone)

The principles of Wolff's law (Wolff, 1982) are based on the concept that there is a strong correlation between the patterns of trabecular alignment and the direction of the principle stresses estimated during normal functions. Under normal physiological conditions, the structure-function relationship, together with its role in maintaining mineral homeostasis in the body, strongly suggest that bone is a material of optimal

'design', i.e., it is spent in a parsimonious fashion. Bone can also be described as a composite structure (Currey, 2002) in which the cancellous bone is sandwiched between outer cortical bone layers. The strength and properties of the composite material may be quite different from that of the individual phases, and often the composite is stronger than either phase (Currey, 2002).

Materials can break under either brittle or ductile models of fracture. A truly brittle material in tension will simply cleave catastrophically from some pre-existing flaw within the material. A ductile material deforms plastically under high stress and undergoes large deformation before failing. Cortical bone alone behaves as a brittle material under compressive loading (Yu, Zhao et al. 2011), and has a low modulus of toughness. However, a composite structure of cortical and cancellous bone has enhanced modulus of toughness, and is better at resisting the growth of micro- fractures/cracks induced during normal loading. The increase in strength and toughness depends on the volume proportion of the component phases in the composite. Such an arrangement can also make the structure lighter without compromising its effective stiffness.

a/2 b a/2

L c

Cross-section b

E1: modulus of elasticity of the inner compliant material

E2: modulus of elasticity of the outer stiff material

Figure 1.3 Composite beam made from two different material

It may also reduce the burden on metabolism partly because the cortical bone is metabolically expensive to produce and maintain (Parfitt, A.M.,1987), and partly because more muscular effort is required to sustain or move heavier structures. In order to elucidate the biomechanical significance of cancellous and cortical bone volume proportions in felid skulls, I developed an explanatory model using a two property composite beam. Applying classical beam mechanics, the variation in effective stiffness with the volume proportions of the two phases was determined.

E2 > E1; E2 is analogous to cortical bone, E1 is analogous to cancellous bone material.

Due to density differences in the two materials, there was an effective weight savings with the increased proportion of lighter material in the composite beam. The variation in weight savings was also deduced.

1.5 A study of convergence in the biomechanical performance of two extinct mammalian sabertooths Smilodon fatalis and Thylacosmilus atrox.

With their very large to enormous canine teeth, mammalian sabretooth carnivores were among the most extraordinary predators of the Cenazoic. The question of just how the unusual skull and canine morphology of the sabretooths, unknown among extant carnivores, was used to kill prey has intrigued palaeontologists for decades, particularly regarding the iconic American sabrecat S. fatalis (Christiansen, 2010, 2007;

McHenry et al., 2007; Therrien, 2005; Anton et al., 2004; Bryant, 1996; Bryant and

Russel, 1995; Van Valkenburgh and Ruff, 1987; Akersten, 1985; Emerson and Radinsky,

1980; Martin, 1980; Kurten, 1952; Simpson, 1941; Mathew, 1901, 1910; Warren,

1853). 12

Sabretooth morphology has appeared at least four times among Mammalia: in the carnivoran Felidae and Nimravidae, in the extinct order Creodonta, and in the

Marsupialia (Thylacosmilidae). Among these mammalian sabretooths, the placental S. fatalis and the marsupial T. atrox were perhaps the two most highly specialised species. Sabretooth morphology is often raised as a classic example of convergent evolution, since it appeared independently in several evolutionary lineages. Different theories have been proposed by various researchers regarding the predatory behaviour of sabretooths. Most previous analyses have focused on the function of the upper canines and structural attributes of the cranium and lower jaw associated with the necessarily large maximum gape, but there is still no common consensus amongst researchers regarding the function of their highly derived skull and dentition.

Warren (1853) was one of the first to study the behavioural implications of the sabretooth felid Smilodon. He concluded that Smilodon used their elongated canines on the prey in a stabbing mode, followed by a cut and tear action as the predator’s head was pulled backwards. Matthew (1901, 1910) introduced the “stabbing theory”, which supported and elaborated on Warren’s idea that the upper canines in Smilodon were used for stabbing. He proposed that these sabretooth felids used their neck flexing and head-depressing muscles to power a downward stroke or stab, driving the upper canines into the flesh of the prey and causing death through massive blood loss.

The mandible clearly had no role to play in this type of attack and he argued that, assisted by clear morphological adaptations for a wider gape, the predator would have kept the mandible out of play during the stabbing action.

Simpson (1941) further refined Matthew’s theory and proposed a combined action of the head depressor and neck-flexor musculature, augmented by the inertia of the predator’s leaping body, to generate the needed force for stabbing. Miller (1969) and

Schultz et al. (1970), who considered Smilodon to be active predators that used their canines in a stabbing mode, speculated that these carnivores must have used their powerful forelimbs to immobilize their victims while the long sabres were used to stab the prey animal. But some authors such as Marinelli (1938) or Bohlin (1940) have rejected the stabbing theory for Smilodon, arguing that the sabres were poorly adapted for this kind of killing behaviour and must have served primarily as slicing devices. Their interpretations were primarily based on functional and anatomical arguments, i.e., that the long, flattened upper canines were not strong enough to withstand violent lateral forces which would have occurred in the stabbing of large struggling prey. This interpretation was later supported in another study by

Bicknevicius and Van Valkenburgh (1996).

Kurten (1952) introduced an elegant model of sabretooth jaw mechanics, wherein he pointed towards the shortened carnassial to jaw joint distance and more vertically oriented temporalis muscles as compensations to minimise a previously inferred reduction of bite strength in sabretooths. He also inferred that the masseter fibers were more vertically oriented in sabretooths, as additional compensation for a weakened temporalis. From analysis of drawings of skulls with gaping jaws, Kurten concluded that sabretooths despite their large gapes were unlikely to adopt a deep canine penetration mode of killing. Rather he viewed sabretooth canines as a specialisation for making long shallow slashes into medium to small sized prey.

Regarding Smilodon, perhaps the most widely accepted interpretation was given by

Akersten (1985), popularly known as the canine-shear bite hypothesis. As with many previous interpretations, Akersten considered that the upper canines were used to deliver a bite that caused death through loss of blood. However, Akersten proposed that the mandible played an important, active role in killing by providing anchorage to resist force generated by the downward movement of upper canines, which was powered partly by the jaw adductors but augmented by the atlanto-mastoid musculature of the neck. Support for Akersten’s model was provided in another study

(Bryant, 1996) of jaw adductor mechanics in Smilodon, suggesting that application of neck musculature was necessary to compensate for a relatively low out force developed by the jaw adductors at wide gapes. In addition, a recent study (Anton and

Galobart, 1999) has shown that the cervical morphology of sabretooth felids reflects adaptation for strong muscular control of motions of the neck, including extension and lateral rotation as would be necessary in a canine ‘shear’ bite, rather than the exclusive emphasis on efficient neck flexion required by the ‘stabbing’ theory. A comprehensive model of bite mechanics in sabercats, including the forces and vector angles involved at different gapes has been presented in a recent study by Christiansen (2010). The author in this study concluded that the 'head-flexors' contribution would have been important during the initial stages of the killing bite in sabercats.

Riggs (1934) thought it probable that the habits of the marsupial Thylacosmilus atrox and its methods of killing prey were similar to that of highly specialised sabretooth felids such as S. fatalis. But despite similarities in canine morphology, there are some clear differences between the marsupial and placental. Miller (1969) suggested that S.

15 fatalis was able to use the incisors either in conjunction with or independently of the upper canines, the animal using them to ingest meat in a manner similar to many extant carnivores. However the incisors are either missing in T. atrox, or greatly reduced. Schulz et al. (1970) pointed to similarities between the North American nimravid sabretooth Barbourofelis and the South American Thylacosmilus, noting that these animals represented an excellent example of morphological convergence in evolution. They, however, also suggested that details of the stabbing action must have differed, arguing that in Barbourofelis the elongated canines are parallel and were probably embedded simultaneously in the prey, whereas in Thylacosmilus the upper canines diverge ventrally and could not have been embedded simultaneously into the prey without being spread apart. This was contradicted by Marshall (1976) who pointed that the upper canines of the T. atrox are deeply rooted in the maxillary, and hence there is a little chance that these would spread. Marshall also pointed out that that the tooth blade is relatively thin transversely and excess lateral strain would likely result in breakage and for this reason the mandibular flange evolved to protect the canines from damage when the mouth was closed. Churcher (1985) suggested that the maximum gape in T .atrox (~ 102 )̊ was higher than that in any placental sabretooth species. Turnbull (1976) argued that the neck musculature in T. atrox was probably not only powerful, but also capable of controlling the head and canine blades with precision, otherwise there would be a risk of angled or transverse blow, breaking the greatly elongated and transversely thin canines. Recent detailed analyses (Argot,

2004a,b) of T. atrox have concluded that the species relatively long neck was capable of strong head flexion.

McHenry et al. (2007) were the first to apply a computational approach, finite element analysis, to simulate the biomechanical performance of a sabre-toothed predator in their comparison of the skulls of S. fatalis and Panthera leo under both intrinsic and extrinsic loading scenarios. McHenry et al. concluded that the bite force generated using the jaw muscles was relatively weak in S. fatalis, nearly one-third that of a lion

(P. leo) of comparable size and that the skull was relatively poorly adapted to bear the extrinsic forces generated by struggling prey.

Using finite element analysis as the primary investigative tool, the central question that I intend to test in this study is the degree to which morphological similarities in two of the most highly derived but distantly related mammalian sabretooths, S. fatalis and T. atrox, might also imply functional similarity. I will further compare their biomechanical performance to that of an extant conical toothed felid, Panthera pardus.

Maximal gape angle is an important variable in the assessment of sabretooth killing behaviour and this was determined using an analytical approach in all three FEMs. If

Akersten’s (1985) hypothesis is correct, then the canines should be well adapted to take the rotational forces about the temporomandibular joint (TMJ), or in other words the canine’s arc centre should be at or near the TMJ such that the bending stresses are minimised. The efficiency of jaw musculature in the generation of bite force at the canines was also compared at different gapes. In a separate analysis, the two major head depressing muscles: Sternomastoideus and Obliquus capitus were also reconstructed in these FEMs to investigate the potential role of neck musculature in augmenting the bite force at canines. In this analysis, the head-depressing muscle

17 forces were adjusted to generate the same body-mass scaled bite force at canines.

Stresses generated in the cranium, mandible, tooth-roots, tooth-crowns were also compared.

1.6 Main aims

The main aims of this project are: a) To compare the biomechanical performance in seven extant felid skulls using CT scan data and Finite Element Analysis. b) To investigate any allometric trends with respect to material property distribution or geometry of the felid skulls? c) To examine how differences in the proportion of cortical to cancellous bone might influence biomechanical performance? d) To determine how any differences in proportions of cortical to cancellous bone be related to fundamental challenges associated with increasing size? e) To investigate the degree to which the morphological similarities between two highly specialised but unrelated sabretooth carnivores (Smilodon fatalis and

Thylacosmilus atrox) might also imply functional similarities? And further compare their biomechanical performance with a conical toothed felid.

2. MATERIALS AND METHODS

2.1 Digital reconstruction of felid skulls based on computer tomography (CT) scan data

In finite element modelling, achieving geometrical accuracy of the model is imperative to simulate the given problem closer to realism. It is not difficult to capture the desired geometrical accuracy with common engineering structures, as they are typically based on parametrically controlled drawings. However in case of biological structures such as the vertebrate skull, the process is complicated by the often very irregular geometries within the specimen. In the fields of comparative biomechanics and biomedicine, models are generally based on computer tomography (CT) scan data (fig.2.1). These can be digitally 'stitched' with appropriate software to generate a three dimensional digital model. For the purposes of the present study, CT scan data for seven extant felid species, namely Felis silvestris lybica (African Wildcat), Leopardus pardalis

(ocelot), Neofelis nebulosa (clouded leopard), Acinonyx jubatus (cheetah), Puma concolor (puma), Panthera pardus (leopard) and Panthera leo (lion) were obtained from the digital archives of Digimorph (www.digimorph.org). Digimorph is a University of Texas at Austin initiative to build and maintain a collection of high resolution X-ray computer tomography data of biological specimens. All specimens used were adult male representatives of their species (see Appendix 3- 9).

The CT scan data for each specimen were imported into image processing software

Mimics (v. 13.1.0 Materialise, Belgium), which converted 2-D pixel data from CT stacks into a three dimensional array of voxels. A voxel is the 3-D equivalent of a pixel, the third dimension being the CT slice thickness. Each voxel was associated with a CT 19 attenuation value measured in Hounsfield units (HU). The contrast enhancement tool was used to set the minimum and maximum HU range in each specimen. For a specific

HU range, all voxels below the minimum value were displayed black and the brightness of voxels increased with HU value. Using the 'Thresholding' tool, a 3-D mask was then created designating the voxels in the array that formed the object of interest.

Surface meshes were then generated using the Mimics remesher tool. Maximum triangle edge length was adjusted in each specimen to produce surface meshes with comparable number of surface triangles. The surface mesh was then exported as a .STL

(stereo-lithography) file from Mimics. Using previously published protocols (McHenry et al., 2007; Wroe, 2008; Wroe et al, 2007ab) the STL file was imported into a blank

Strand7 file (FE software used in this study, v.2.4.1, Strand7 Pty Ltd, Australia). The surface triangles in STL mesh became plate elements in Strand7. The surface mesh was then cleaned of any overlapping or free plate elements by using the clean surface mesh tool. Once the surface mesh was fixed, the automatic solid meshing tool was used to fill the surface mesh with four noded tetrahedral elements. In Strand7 these tetrahedral elements are referred to as 'bricks'. Since the surface meshes in all the specimens had comparable number of triangles, the volumetric mesh also contained broadly comparable number of tetrahedral elements. Higher order 10-noded brick elements are also available in Strand7, and would have theoretically produced more accurate results. However, differences between 4-noded and 10-noded tetrahedral elements converge with increasing model size. A difference of around 10% might be expected between models containing 250,000 brick elements (Dumont et al., 2005) and given that the FEMs in this study had at least 2.5 times more elements than this,

20 the results should be well within 10% of those computed from 10-noded tetrahedral elements.

Figure 2.1: An example of a single slice from the CT scan of a Panthera leo (lion) skull. Different grey-scale values in the bitmap correspond to different densities (as measured by CT attenuation), with black indicating the lowest density and white the highest.

Material properties were assigned to the volumetric mesh based on CT attenuation data following previously established protocols (McHenry et al. 2007; Wroe et al. 2010;

Chamoli & Wroe 2011). A uniform heterogeneous 8 material property distribution based on grey values was assigned to the model. The final Strand7 model included the geometry and material properties distribution and was ready to be pre-processed.

2.2 Finite element modelling protocols in Strand7

2.2.1 Model assembly and pre-processing

Finite element analysis is a powerful structural analysis tool, but only appropriate usage can provide real insight into the questions addressed. A sound understanding of the capacities and limitations of the tools available in the FE software is essential to the realistic modelling of 'real world' problems. Many factors contribute to the decisions made when building FE models, including the purpose of the analysis, the level of results accuracy required and the computational 'cost'. In simple terms, a finite element model is composed of an array of interconnected points in space called nodes, which are used to represent and simulate the behaviour of a physical system.

The task of constructing a finite element model is referred to as Model Pre-processing.

This typically involves:

x Defining geometric and loading characteristics of the model.

x Defining the material characteristics.

x Dividing the structure into various elements and nodes.

In the present study, the model geometry and material properties distribution were based on CT scan images (as described in section 2.1), and the 3-D model obtained from the last step was ready to be pre-processed in Strand7. The pre-processing protocols used in assigning various loads and constraint characteristics to the FEMs, and test various hypotheses are described in detail here

2.2.1.1 Modelling major muscle subgroups

A total of eight muscle subgroups representing major jaw adducting muscle divisions were modelled in the FEMs of felid skulls: Temporalis superficialis, Temporalis profundus, Temporalis zygomaticus, Masseter superficialis, Masseter profundus,

Zygomatico-mandibularis, Pterygoideus internus and Pterygoideus externus. Surface plates were first created on the FEMs by tessellating the tetrahedral bricks. These plates were then coloured to depict the origin and insertion regions for different muscle subgroups (fig. 2.2). To colour plates in Strand7, eight different plate properties were first created representing the muscle subgroups, and each was assigned a unique colour code. Plates were then selected to map the approximate anatomical location of the muscle subgroup in the FEM. These were then assigned to the respective property type. Once coloured, the plates depicting the origin and insertion areas for a particular muscle type/division were placed in a separate group.

The approximate anatomical locations for different muscle groups in felids were taken from Turnbull (1970b).

Figure2.2 The figure above shows a pre-processed finite element model of a cheetah (Acinonyx jubatus). Different coloured regions in the skull depict the origin and insertion areas for different muscle subgroups.

2.2.1.2 Jaw –Hinge mechanism

For a given felid skull specimen, once the Nastran (.nas) files for cranium and mandible were imported into Strand, the three dimensional orientation of mandible with respect to cranium was adjusted to include appropriate clearance (~ 1-3 mm) between the temporomandibular joint (TMJ) articulating surfaces. The jaw mechanism or masticatory apparatus is the driving mechanism behind biting activity. It plays an important role in determining the bite force and stress distribution within the skull structure. Previous studies have used a variety of techniques to model the jaw mechanism including constraining a single node against displacement at each TMJ or articular surface, effectively creating an axis of rotation for the skull (Slater and van

Valkenburgh, 2009; Strait et al., 2005). These techniques have their own implications as the mandible is not modelled as part of the skull structure, and the model therefore does not account for the effect of jaw movement on the stresses induced in the skull.

To overcome this problem of joint articulation, a suitable hinge mechanism was used to simulate normal jaw operation following previously established protocols (Wroe et al., 2008; Wroe et al., 2010; Attard et al., 2011; Chamoli and Wroe 2011). The plates in the condyle and cotyle region were first selected and tessellated to create a network of beams. This was done to avoid stress singularities at the points of attachment of jaw hinge to the cotyle and condyle. The jaw-hinge mechanism used rigid links and beams to connect the cranium and mandible. The rigid link provided a stiff connection between the articulating surfaces, and a point on the axis of rotation to which the hinge beam was connected. The beam provided the pivot or hinge in the joint. To

25 provide adequate stiffness to the hinge beam, material properties of structural steel

(SS 4100-1999) were assigned to it.

2.2.1.3 Gape angle and hinge rotation coordinate system

After the jaw-hinge was created, a user defined cylindrical co-ordinate system was assigned to it. The longitudinal axis of this cylindrical co-ordinate system was kept the same as long axis of the jaw-hinge. The node common to the rigid link and beam in the jaw-hinge was used as a pivotal node to define the cylindrical coordinate system. Since this is a comparative study, a uniform gape angle of 35 degrees was used in all seven felid skull FEMs (Bourke et al., 2008). The gape angle was measured laterally between the upper medial incisor, temporomandibular joint and the lower medial incisor.

Starting from a zero gape angle position, the cranium was rotated by 17.5 degrees in the anticlockwise sense about the jaw-hinge cylindrical coordinate axis. The mandible was then rotated by 17.5 degrees in the clockwise sense to achieve a 35 degree gape angle.

2.2.1.4 Dry –skull method and estimation of Temporalis and Masseteric muscle force

The maximum force that can be generated by skeletal muscle is proportional to its maximal cross-sectional area. I adapted the dry-skull method (Thomason, 1991) to measure the cross-sectional area of the two major jaw adductors: temporalis and masseteric muscles. In this analytical approach, osteological (fig. 2.3) landmarks were first estimated from areas on the cranium bounding the spaces filled in by these major 26 jaw adducting muscles. To find the maximum cross-sectional areas (CSA) of temporalis and masseteric muscles from felid FEMs, geometrical data from the outer contours of shaded areas (fig. 2.3 A&B) were first extracted. User defined Cartesian co-ordinate systems were created to represent the temporalis and masseteric muscle cross-section planes. Each muscle contour was approximated by 50 control points, and in-plane two dimensional co-ordinates for these control points were obtained using the

'Whiteboard' tool in Strand7. Solid geometry software GEUP3 was then used to plot the control points for each of these muscle contours. These were then connected to give a 50 sided polygon (fig. 2.4). The area of this polygon was measured to give an estimate of the cross-sectional area of each muscle type. There is a minor functional glitch in the 'measure-area' tool in GEUP3. The measured area of the polygon depends on the extent of magnification, and gives different values if the image is zoomed in or out. To overcome this difficulty in measurement, a reference triangle of known dimensions and area was also created in the same plot, and the area of the polygon was obtained for the correct dimensions and area of this reference triangle (fig.2.4).

Force estimates for the temporalis and masseteric muscles were made by multiplying

CSA by 300kPa, the maximum longitudinal stress developed in mammalian muscles.

(Weijs and Hillen, 1985).

Figure 2.3 (A) Ventral view - the shaded light blue area represents the estimated cross-sectional area for Masseteric muscles. (B) Dorsoposterior view - the shaded light blue area represents the estimated cross-sectional area for temporalis muscles. (C) In lateral view, marker dots green and red were used to define the cross-sectional plane for masseteric muscles, and blue and blue were used to define the cross-sectional plane for the temporalis muscles. The resultant masseteric muscle force (M) acts normal to the masseteric plane (A) through the centroid of the area (yellow star), and the resultant temporalis muscle force (T) acts normal to the temporalis plane ( through the centroid of the area (yellow star). Image source: Wroe et al., 2005

Figure 2.4 Screen-grab of the GEUP work window showing the 50 sided muscle X- sectional area polygon and the reference triangle. The area of the muscle X-sectional polygon was measured for the correct area of the reference triangle.

2.2.1.5 Muscle fibre attachment/Muscle pretension, numbers and distribution

Muscle fibres in the FEMs were modelled as truss elements. A truss element is capable of transmitting only axial forces. The element only has three translational degrees of freedom at each node and behaves like a pin jointed beam element (Using Strand7, ed.3, 2010). In each model, a total of 200 truss elements (100 on either side of the skull) were distributed among different muscle subgroups on the basis of their origin and insertion areas. The 'Model Summary' tool in Strand7 gave a comprehensive summary of the plate surface area occupied by different muscle subgroups. From this data, the percentage area occupied by different muscle subgroups was calculated, which was then used to calculate the number of muscle-fibre truss element representatives for each muscle subgroup (for calculation details see Appendix 3-9) .

The temporalis muscle force calculated in the previous step using the 'dry-skull' method was equally distributed among all the temporalis muscle truss elements, which included Temporalis superficialis, Temporalis zygomaticus and Temporalis profundus. Likewise the Masseteric muscle force was equally distributed amongst the

Masseteric muscle truss elements (Masseter superficialis, Masseter profundus and

Zygomatico-mandibularis). Pterygoideus internus and Pterygoideus externus muscle trusses were not assigned any pre-tension, and only served as balancing muscles during physiological loading. To find the diameter of individual temporalis truss elements, the combined cross-sectional area of the temporalis muscle trusses was equated to the maximum cross-sectional area for the temporalis muscle obtained using GEUP 3 in the previous step. The same procedure was adopted to find the diameter of individual masseteric truss elements.

Connecting cranium and mandible using muscle trusses

Plates around the muscle origin, and the corresponding insertion points were first selected and tessellated to create a network of beam elements. This was primarily done to minimize artificial stress singularities, and also to achieve a more uniform distribution of forces. These beam elements were assigned the material properties of structural steel (SS 4100-1998). Origin and insertion points were then connected using the corresponding muscle truss type. The origin and insertion points for muscle trusses were selected in such a way to ensure that the muscle trusses were always above the temporomandibular joint axis (TMJ), broadly simulating fibre orientation and maximizing the jaw closing torque. Once created, the muscle trusses were assigned a tensile force using 'Pre-load' tool in Strand7 (fig. 2.5). The three dimensional orientation of muscle trusses was made as symmetric as possible on both sides of the skull. In order to avoid any modelling artefacts in the comparisons, the same modelling protocols were used to model muscle architecture in all the FEMs.

A limitation of modelling the muscle fibres using this, and most other methods, is that it does not fully consider any inter-specific differences in jaw muscle pennation angles and assumes it to be the same for different species. But as the focus of this study is comparative, it is relative not the absolute values that are important.

Complete model summaries including the muscle origin and insertion areas, muscle truss numbers, pre-tension, diameter and material properties data for all seven felid

FEMs are provided in the Appendix section.

Figure 2.5 The figure shows a pre-processed finite element model of cheetah (A. jubatus). Lines running from cranium to mandible represent muscle trusses. The arrows on the truss elements depict muscle pretension.

2.2.1.6 Assigning material properties to the beam and brick elements

The formulation of a finite element structural analysis problem requires that the behaviour of the elements used in modelling be fully described. The mathematical relationship between the response and the action applied to a material is called the constitutive equation of the material. The stress –strain relationship, or Hooke’s law, is an example of such a constitutive relationship. All materials deform when forces are applied on them. If the deformation is reversible, i.e., on removing forces the material regains its original dimensions and shape; the behaviour is called elastic deformation.

If the deformation in the material is permanent, the behaviour is called plastic deformation. In general, biological materials such as bone exhibit material anisotropy, which refers to the variation in material properties with the loading orientation.

Material anisotropy of bone is an additional challenge for modelling not just in terms of computational burden, but also in terms of getting the experimental data needed as an input in FE modelling.

To reduce the level of computational complexity in the FEMs in this study, we assigned isotropic material properties to the brick and beam elements. In an isotropic material, the elastic properties are independent of the loading orientation. In most of the linear static finite element analyses, three physical quantities are needed to completely specify material properties of the constituent elements: Young’s modulus of elasticity

(E), Poisson’s ratio (ν) and density (ρ). Young’s modulus (E) relates stress to the strain,

Poisson’s ratio (ν) gives a relationship between longitudinal and lateral strain and density and (ρ) is a measure of mass per unit volume.

Cortical and cancellous bone material properties were assigned to the brick elements in the heterogeneous FEMs. The distribution of cortical and cancellous bricks was based on the CT density data measured in Hounsfield units (HU). Although all the skulls specimens used in this study were scanned in the same machine, it is nonetheless difficult to be completely confident that settings were consistent without full calibration. To overcome this difficulty, HU ranges were deduced for the two bone material types, by careful visual examination of CT scan slices for each felid specimen

(Table 2.1). Using the frequency distribution data (grey-values histogram) between brick material types and HU in Mimics, and the HU range for cortical and cancellous bone, bricks in each FEM were selected and assigned cortical and cancellous bone material properties. In homogeneous FEMs, cortical bone material properties were assigned to all the brick elements throughout the volume of the solid.

Table 2.1 Hounsfield units (HU) range for cortical and cancellous bone in CT scans

HU (cancellous bone) HU (cortical bone) F. silvestris lybica 443-2448 2448-4080 L. pardalis 1215-2423 2423-4080 N. nebulosa 22-1424 1424-3056 A. jubatus 496-2341 2341-4080 P. concolor 93-1796 1796-3056 P. pardus 337-1619 1619-3056 P. leo 549-2235 2235-4080

Muscle trusses were assigned isotropic material properties from the literature

(McHenry et al., 2007). A detailed description of the material properties for brick, truss 34 and beam elements for all the FEMs used in the present study is also provided in the

Appendix section.

2.2.1.7 Boundary conditions for simulating bilateral canine and unilateral carnassial bite case

Boundary conditions are nodal constraints that prevent rigid body movement and thus help in maintaining a static equilibrium in the model when muscular loads are applied to it. In Strand7 all nodes have six degrees of freedom: three rotational and three translational (Using Strand7, ed.3, 2010). Degrees of freedom (DOFs) define the number of independent sets of displacements or rotations that can be used to completely specify the change in position or orientation of an element or a node. Each of the three degrees of freedom corresponds to one of the axes of the current coordinate system, either global Cartesian or any other User defined coordinate system

(UCS). Any degree of freedom can have one of the three fundamental restraint types: free, fixed or a specified value of displacement/rotation, i.e. an enforced displacement or rotation.

Special care must be taken in deciding which nodes/elements, and also which degrees of freedom on these nodes/elements are to be constrained or applied, as these can have serious implications on the final results. In this study, I aimed to simulate two of the most common feeding/biting behaviours in felids: a bilateral canine bite and a unilateral carnassial bite. A bilateral canine bite occurs in killing behaviour when the predator is biting into its prey using the upper and lower canines, with all jaw muscles

35 firing. The sum of the reactionary forces experienced by the tip of canines gives an estimate for the theoretical maximal bite force achievable under these conditions. To simulate this biting case in the FEMs, one node at each canine tip was restrained in all degrees of freedom. Restraining all degrees of freedom might be methodologically preferable but it is biologically unrealistic, as it represents a scenario when the animal is biting down on an impenetrable object. This would concentrate the reactionary force on individual nodes and may give rise to stress artefacts. To minimize these artefacts and more evenly distribute the stresses, plates near the restrained canine tips were selected and tessellated to create a network of beams (Clausen at al. 2008,

Moreno et al. 2008, Wroe et al. 2007a).

Another approach to model a bite case more realistically is to use the “Brick-face support” tool in Strand7. Instead of restraining the node at the canine tips from any possible movement, the face support attribute can be assigned to the bricks near canine tips, which on application of muscular forces generates a reactive pressure directly proportional to the displacement of the brick face. The proportionality constant is known as modulus of sub-grade reaction and is a direct measure of stiffness of the support structure. This tool can be particularly useful in comparing the stresses generated in the skull and the teeth when biting/chewing the hard and soft tissues of prey. But in the absence of experimental data for the modulus of sub-grade reaction for hard and soft tissues, this approach could not be implemented in the present study. To model a unilateral carnassial bite, nodes on the posterior most carnassial tooth (on both upper and lower jaw) on the right side of the skull were restrained using the same protocols used for a bilateral canine bite. This simulates the

36 predator's behaviour in chewing/slicing using the carnassial teeth on one side of the skull, with all the jaw muscles firing simultaneously.

2.2.1.8 Occipital link constraints

The occipital condyle is a protrusion on the back of the cranium that forms a joint with the first cervical vertebra, enabling the head to move relative to the neck. Extant felids may variably use their neck and head depressing muscles to augment the killing bite, thus moving their head relative to the vertebra. The main focus of the present study is to compare the biomechanical performance of different felid skulls using jaw musculature alone, and hence in this context the relative movement between the head and the neck is not important. The effects of head-neck movement were removed by creating a rigid link in the occipital condyle region (Wroe et al. 2007a;

Wroe et al. 2010). This rigid link was then split into two halves. The centre node was fixed in all six degrees of freedom and acted as an anchor point for the cranium. In order to avoid stress artefacts in the region where the occipital link was attached to the cranium, the plates around the attachment nodes were selected and tessellated to create a network of beams.

To examine the potential role of the neck musculature in contributing to bite force, a separate study was performed comparing the neck-muscles driven bite in extinct sabre-toothed carnivores Thylacosmilus atrox, Smilodon fatalis and the extant felid

Panthera pardus. The centre node on the occipital link was restrained from movement in all degrees of freedom except for rotation about its long axis. Bite reaction forces for

37 a neck-muscle driven bite at the canine tips were compared with those from a jaw- muscle driven bite.

2.2.1.9 Quantitative estimation of cortical and cancellous bone volume based on CT intensity data

On visual inspection (fig. 2.6) of CT scan data, it was apparent that there was a variation in the volume proportion of cortical and cancellous bone in felid skull specimens. Qualitatively, a decrease in the proportion of cortical bone volume was apparent with increasing skull size. This parsimonious distribution of bone in the skull of larger felids, where most of the stiff cortical bone was distributed on the outside and more compliant cancellous bone served as a filler material, was similar to a

'sandwich beam' construct. In a typical sandwich beam (Bauchau and Craig, 2009), relatively light filler material is sandwiched between two layers of outer stiff material, enhancing its overall stiffness and also reducing its mass. To quantitatively assess proportional differences in cortical and cancellous bone volume between species, two separate solids for each specimen were generated: an original one including both cortical and cancellous bone, and a second one in which the cancellous bone was excluded using the 'Segmentation' tool in Mimics 13.2.

Figure 2.6 Transverse CT scan slices of three felids through the posterior-most carnassial tooth. (a) F. s. lybica; (b) P. pardus; (c) P. leo. Intensity was measured (Table 3.6.1) in Hounsfield units.

Hounsfield unit ranges deduced previously for each specimen (Table 2.1) were used to segment out cancellous bone from the specimens. The volume of these two separate solids was measured using the 'Model Summary' tool in Strand7. For each specimen, volume of cancellous bone (Table 2.2) was calculated by subtracting the volume of cortical bone from the total skull bone volume.

Table 2.2 shows the calculated volumes of cortical and cancellous bone (mm3) in different felid skull specimens used in this study.

Cortical bone Cancellous bone

Volume(mm^3) Volume (mm^3)

F. s. lybica 3.343×104 6.597×103

L. pardalis 3.499×104 8.308×103

N. nebulosa 2.270×105 6.805×104

A. jubatus 2.284×105 9.747×104

P. concolor 2.963×105 1.017×105

P. pardus 3.951×105 1.482×105

P. leo 1.604×106 1.127×106

2.2.1.10 Controlling size differences

Size profoundly impacts on the biomechanical performance of organisms (Biewener,

2005; Dumont et al., 2009). In a comparative finite element study of biological structures, the biomechanical performance can be truly compared only when the differences in size and shape are disentangled. To control for the differences in size between different felid skull FEMs, and compare the differences in biomechanical performance arising due to shape variations alone, recently published protocols on scaling by Dumont et al. (2009) were used.

Since mean VM stress was used as a metric to compare the biomechanical performance, all the FEMs were arbitrarily scaled to the same surface area and loaded with the same total muscle force as A. jubatus. In a comparative context the choice of 40 taxon to which other species are scaled is immaterial (Dumont et al., 2009). The dimensional scaling factor (k) for scaling individual FEMs was calculated using

ௌ௨௥௔௙௖௘௔௥௘௔ሺிாெሻ k = ට ௌ௨௥௔௙௖௘௔௥௘௔ሺ஺Ǥ௝௨௕௔௧௨௦ሻ

For a given felid skull FEM, the same dimensional scaling factor (k) was used to scale the model along all three Cartesian coordinate axes. To further compare the muscle force recruitment needed to generate the same bite force and also to compare the associated stresses, simulations were run with FEMs scaled to the same surface area and scaled to the same bite force as A. jubatus.

2.2.1.11 Assembling homogeneous and heterogeneous FEMs

Depending on material properties, bone is broadly classified into two types of osseous tissue, cortical and cancellous bone. In the present study, the heterogeneous FEMs included the material properties distribution for both cortical and cancellous bone, whereas homogeneous FEM assumed only a single material property for cortical bone throughout the volume. Whether it is necessary to incorporate cortical and cancellous bone material distribution in comparative FE studies may depend on the question to be asked, but the use of heterogeneous models certainly has the potential to yield greater precision (Strait et al. 2005, Panagiotopoulou et al., 2010b). In a comparative context such as this, incorporating cortical and cancellous bone material properties in

FEMs is likely to be of particular importance where there are clear differences in the volume proportion of cortical and cancellous bone between specimens. To compare

41 the differences in VM stress distribution between homogeneous and heterogeneous

FEMs, two homogeneous FEMs representing the largest (P.leo) and smallest (F.s. lybica) felid were also modelled. These species were selected because they represented the two extremes with respect to the differences in both size and the relative proportion of cortical and cancellous bone by volume and consequent variation in the overall stiffness between the homogeneous and heterogeneous models.

By volume, the skull of F.s. lybica was ~ 84% cortical bone and ~16% cancellous bone, and that of P.leo was ~ 59% cortical bone and 41% cancellous bone. The contention was that with an increase in cortical bone volume proportional, the FE model tends to approach homogeneity in its material composition and thus more closely resembles the homogeneous model.

2.2.2 Model Solving

Once the pre-processing on the model was complete, the model was submitted to one of the Strand7 solvers. Before submitting any model to the solver, it is good practice to check all the model parameters and review any modelling assumptions. The solvers store the results of each solution into a file which normally has the same name as the model file but with an extension that identifies it as a solution file of a particular type

(e.g. the LSA file contains Linear Static Analysis results).

Finite element analysis involves the assembly and solution of a matrix defining the relationship between force and displacement, also called the stiffness matrix. Finite

42 element stiffness matrices are generally symmetric and banded. In a banded matrix the non-zero entries are clustered together near the diagonal of the matrix. In this context 'symmetric' means that the upper triangle of the matrix is an image of the lower triangle. The bandwidth, together with the number of equations, is a measure of the size of the matrix. Whilst the number of equations is fixed for any given model, the bandwidth can be minimized by reordering the node numbers. The bandwidth has a strong impact on the time taken to solve the model and the amount of memory and disk space required. To improve solver performance, Strand7 includes (Using Strand7, ed.3, 2010) three methods of minimizing the bandwidth:

1) The Geometry method

2) The Tree method

3) The Sparse method

Different models are suited to different bandwidth minimization strategies. Generally models that have a dominant length direction in the X, Y or Z axis are suited to the geometry mode. Most other models are suited to the tree mode. Models where the geometry 'branches' off in all directions are also suited to the tree mode. The

Bandwidth tab of the solver dialog box provides a means for visualizing the actual shape of the global stiffness matrix. This can be used to make an informed decision about the method to use. The best method to use depends firstly on the amount of physical memory (RAM) available and secondly on the amount of disk space. The

Average Node Jump value corresponds to the amount of time spent reading and writing the results to the disk. A lower value of Average Node Jump will mean lesser computational time and lower disk space needed to write the results. The Maximum

Memory Index corresponds to the RAM requirement and a lower value of this means that the corresponding method will require the least amount of RAM to generate the result file.

2.2.2.1 Linear Static Analysis

For a given analysis task, the characteristics of the question and the purpose of the analysis will dictate which solver to use. The choice of appropriate solvers depends on such considerations as the loading conditions, the response characteristics, and the aspect of the response that one is interested in capturing. Loading can be classified into two broad categories – static loads and dynamic loads. Static loads refer to those that are applied to the structure slowly so that no dynamic/inertia effect is excited, and therefore structure responds in a static manner. Dynamic loads often refer to loads which are suddenly/repeatedly applied to the structure and cause a time- dependent structural response.

In this, as in most comparative biomechanical studies, a linear static analysis solver was used for all models. The main reason being because of the difficulties in obtaining input data for dynamic solves. In a linear static analysis, the following assumptions

(Strand7 Theoretical Manual, ed. 1, 2005) are made:

1. All the materials of the structure remain linearly elastic. This requires the

displacements of the structure be linearly proportional to the applied loads and

that when the loads are removed, the structure returns to its original shape.

2. The displacements/deflections are negligible. This assumption implies that the

equilibrium equations are established with respect to the original geometry

without referring to the deformed geometry. 44

3. The boundary conditions are pre-defined and will not change after the load is

applied

4. The loading is static

This implies that the loads are slowly applied to the structure and does not

consider dynamic/inertia effects.

Mathematically the linear static equation can be written as

K U = P , where

K = global stiffness matrix

U= unknown nodal displacement vector

P = global equivalent nodal load vector

The global stiffness matrix K is a constant matrix, which makes the displacement vector

U, a linear function of the applied load vector P. The solver yields a solution for the nodal displacements U using the stiffness matrix K and load vector P. Based on this solution, element results (strains, stresses, strain energy densities, etc.) are then calculated.

2.2.2.2 Batch Solver

The batch solver tool in Strand7 prepares the solver files for individual FEMs and adds them to the job queue, such that when the batch solver is selected, models in the job queue are solved one by one. While writing the batch file for a FEM, the location of the result file can be specified and the batch solver will save the results at this pre-defined location. Also, before writing the batch file it must be ensured that under 'Results' tab in the solver, boxes for the desired output quantities are checked. The batch solver tool is particularly useful when more than one FEM is to be solved on one workstation. 45

Once the pre-processing was complete for all felid skull FEMs, batch files were written and submitted to the batch solver.

2.2.3 Model post-processing

Once the finite element model has been solved, the results of the analysis are extracted and interpreted. This phase of the analysis is called Model post-processing.

Post-processing should begin with a thorough check for problems that may have occurred during the solution. Strand7 solvers provide a log file (.LSL file), which contains information about any warnings or errors that may have occurred during the solution. The log file also contains information on the size of stiffness matrix, and model solution time. In Strand7 the maximum size of the stiffness matrix can be up to

32GB (gigabytes). Once the solution is verified to be free of any numerical problems, the quantities of interest may be examined. This begins with loading the results file

(.LSA file for linear static analysis solves) in the original Strand7 model file (.St7 file).

The choice of post-processing tools depends on the desired output variables. Strand7 provides a number of post-processing tools, including dynamic and animation capabilities. The results settings option allows displaying the analysis results in a graphical form such as colour coded contour plots, with the provisions for selecting the range for the desired results. For the purpose of this study, four main post-processing tools were used:

x Visual post-processing plots

x Peek entity results

x List results

x Model summary

2.2.3.1 Visual post-processing plots

Visual post-processing tools were used for making qualitative inspections of the results. Von-Mises (VM) stress distribution in the brick elements on the model surface and within the geometry were the main results of interest in the present study. Bone is a relatively elastic material which fails under a ductile model of fracture and VM stress, a function of principal stresses σ1, σ2, σ3, is a good predictor of failure in ductile materials (Nalla et al., 2003; Tsafnat and Wroe, 2011). Therefore mean VM stress was used here as a metric to compare the structural strength of felid skull FEMs under different loading conditions. Colour-coded plots for the VM stress distributions were generated for all the FEMs. The 'Histogram' tool under the 'Results settings' option was also used to generate the frequency distribution spread of VM stress for the entire model geometry. The colour coded plots can also be exported from Strand7 as “.jpeg" or ".bmp" files. It is worth noting that the displayed colour-coded VM stress results only depict the surface stresses in the model, and reveal no information regarding the state of stress within the model geometry. However, the stresses within the model can be dynamically observed by selecting the 'all-faces' button under the entity display option for bricks.

2.2.3.2 Peek entity results

The 'Peek entity' result option in Strand7 is used to investigate specific results at a single entity (node, beam plate or brick), such as the Cartesian components of the 47 nodal displacements. The peek tool is also useful in finding the location of an entity having maximum or minimum value of an active output variable (displacement/ reaction/stress/strain etc.). In this study, bite reaction forces at the constrained nodes were obtained using the peek tool. The reaction forces at the constrained nodes under a UCS axis system were noted. Only the tangential component (FT) of the force was of interest, as only this was responsible for producing the jaw closing torque about the

TMJ axis.

2.2.3.3 List results

The 'Result listings' option in Strand7 displays analysis results such as displacements and stresses in a tabular format. This data can then be exported as a text (.txt) file for further processing. In this study, for quantitative comparisons of VM stresses within and across species, the brick VM stress data was exported using the "listings" tool.

Brick VM stress data were exported in separate text files for different regions of the skull in all the FEMs. For exporting the data "Combined values" was selected as the coordinate system option. Under the "Groups" tab, only the group containing the brick elements under investigation should be highlighted. In the "Columns" tab, only the VM stress option needs to be highlighted. This would display only two columns of data, one containing the brick numbers and other the VM stresses for the corresponding bricks. "Print -preview" option on the top-left of the 'Results viewer' window was clicked to export this data as a text file.

2.2.3.4 Model Summary

The Model Summary tool in Strand7 is used to summarise various aspects of the model including Bill of Materials, Centre of Mass, Local and Global Mass Moments of Inertia.

This tool was particularly useful for extracting geometrical and material properties distribution information from the models. The following data were extracted for each of the felid skull FEMs using this tool:

1) Origin and insertion (plate) areas for different muscle sub-groups

2) Total skull surface area (plates) and skull bone volume (bricks)

3) Combined volume of cortical bone material bricks.

2.3 The Allometry Signal

In biology, allometric equations are valuable statistical tools with which to understand the principles and connections between variables that might otherwise remain obscure. Allometric equations are mathematical descriptions of how two biological characteristics associated with an organism scale relative to each other, all other things being equal and can be expressed as follows:

b y = a.x (1)

This equation contains two important numerical terms: the proportionality coefficient a and the exponent b; x and y represents the characteristic variables being studied. A

Log-transformation of this equation produces the linear equation

Log (y) = Log (a) + b Log(x) (2)

Using the Log-transformed data points (Log(x), Log(y)) for the characteristic variables being studied, a regression line is calculated. The regression line represents the statistical-best-fit for the original Log-transformed data. Increasing the number of data points will increase confidence in inferences drawn from the analysis. The proportionality coefficient a is obtained from the y intercept of the Logarithmic plot.

The slope of the logarithmic plot gives the exponent b for the above allometric equation. The exponent b can take on different values and can be either positive or negative, depending on the function being considered. A small difference in the value of the exponent b on a logarithmic plot may represent a sizable magnitude difference when expressed arithmetically.

How do we interpret allometric equations? To draw meaningful inference from allometric equations, one must understand the hypothetical isometric condition between the two characteristic variables being studied. Isometry refers to geometrical and material similarity between two objects of different sizes. In geometrically similar

(isometric) bodies, all corresponding linear dimensions are related in the same proportion, and all corresponding surfaces have areas related in the same proportion squared, and volumes in the same proportion cubed.

length)2) ן Surface area

length)3) ן Volume

Surface area) 3/2) ןVolume

So an isometric equation between the volume and surface area of two geometrically similar objects will have an exponent of 3/2. The allometric equation is an indirect

50 measure of deviation from the isometry condition between the characteristic variables. For an allometric equation between volume and surface area, if the exponent b is greater than 3/2, the relationship is called positively allometric, which essentially means that with an increase in surface area, volume scales at a faster rate compared than for the isometric condition. If the exponent b is less than 3/2, the relationship is called negatively allometric. Before comparison with the isometric equation, statistical analysis must be performed to determine confidence intervals for the values of exponent b. Allometric equations are useful for estimating the characteristic variables within the data points, but cannot be used for extrapolations beyond the range of the data on which they are based (Knut – Schmidt Nielsen,1984).

Allometry in the material properties distribution and geometry in felid skulls

Visual inspection of CT scan slices of the felid skulls specimens in this study pointed towards two possible allometric trends. The first trend was related to the distribution of cortical and cancellous bone materials in the skull, in which the volume proportion of cortical bone was found to decrease with increasing skull size. In order to investigate any allometric relationship between cortical bone volume and total skull bone volume, the two characteristic variables were obtained for all the seven extant felid skulls using techniques discussed in section 2.2.1.9 , and regression analysis was performed.

The second trend was a previously suggested relationship in which the relative thickness of the skull was thought to increase with increasing skull size (Slater & van

Valkenburgh 2009). This trend was indirectly investigated in the present study by

51 quantifying the volumes and surface areas for all felid skull FEMs in order to confirm whether the volume increased at a faster rate than the surface area (as this would necessarily be accompanied by a net increase in the overall thickness of the structure).

Thus a positive allometric relationship between the volume and surface area would also imply positive allometry between the skull thickness and skull size.

Table 2.3 Total skull bone volume and total skull surface area of different felid skull FEMs

Total skull bone volume Total skull surface (mm^3) area (mm^2)

F. s. lybica 4.000×104 3.258×104

L. pardalis 4.330×104 3.741×104

N. nebulosa 2.950×105 1.214×105

A. jubatus 3.260×105 1.421×105

P. concolor 3.980×105 1.461×105

P. pardus 5.430×105 1.652×105

P. leo 2.730×106 4.261×105

2.4 Statistical Analyses

Uncertainties in the analysis of biological data can arise during the process itself or during the observation phase, the latter is commonly called sampling error. To draw any useful inferences from the biological data, associated uncertainties are often viewed in the light of various statistical treatments. Statistical analyses or procedures help in quantifying the uncertainties associated with the data in terms of probabilities.

The role of statistics in analyzing the data is of further importance in a comparative study such as this, where the absolute values are insignificant and the data can only be interpreted in a comparative sense. In the sections that follow, various statistical procedures and software tools that were used in this study to test hypotheses and draw inferences are discussed in detail.

2.4.1 Mean brick von-Mises (VM) stress

VM stress data for the brick elements of interest were exported as .txt files from

Strand7 (see section 2.2.3.3). Text files were then imported into a Microsoft Excel

(v.2007) file using its 'Text import wizard' tool. While importing the data, the 'Fixed width' option was selected as the 'File-type'. Once the data were imported into the

Excel file, it was sorted in an increasing order using the 'A-Z sort tool'. The top 0.1% of the stressed bricks were not considered in the analyses, as they were likely to represent stress artefacts in the FE model. Using in-built features in Excel, average, maximum, minimum and standard deviation values were obtained from the data for the remaining 'brick' elements. The average values thus calculated were the measure of mean brick VM stress data for the regions denoted in the model. Most statistics based decisions rely on identifying the distribution of the variable around the mean 53 value. Standard deviation values gave a rough estimate of the spread of brick VM stress around the mean value.

2.4.2 Two-factor without replication ANOVA on mean VM brick stress data.

The analysis of variance (ANOVA) is a useful statistical technique that allows testing of whether the differences observed in the sample means of two or more populations are statistically significant. This approach tests null and alternate hypothesis at defined levels of significance (α). The level of significance is a direct measure of confidence ((1-

α)*100) in the conclusions drawn. The basic requirements for ANOVA are that each population should have a normal distribution of the variable being analysed and that all populations should have the same standard deviation.

In this study, two-factor without replications ANOVA was used to test whether the brick stress variations observed within different regions of the FEM for a species, and across the species were statistically significant. The null hypothesis was that there are no differences in the mean values of brick VM stress and that differences observed were attributable to sampling error. The alternate hypothesis was that there was some difference in the mean VM brick stress values within each species and also across the species. The p value obtained from ANOVA was used to accept or reject the null hypothesis for a chosen level of significance. If the p value was greater than the level of significance (α), the null hypothesis was accepted. If not the alternate hypothesis was accepted. The data analysis toolkit in Microsoft Excel (v.2007) was used to perform two-factor without replication ANOVA. The data for mean VM brick stress for four different regions of the skull (mandible, zygomatic arch, face and rostrum, and rest of the cranium) were imported into Excel for all seven felid skull FEMs. 54

2.4.3 Reduced major axis (RMA) regression analysis

Regression analysis is a statistical tool used to investigate the causal effect of a predictor variable (X) upon a response variable (Y). Simple linear regression analysis involves writing the equation of a line of best fit (regression line) for the given data points (X, Y). With most biological data, there is an error variability associated with both variables X and Y. In the present study the main aim of regression analysis was not prediction, but to describe the true nature of the relationship between Y and X.

Consequently, reduced major axis (RMA) regression criteria were used in finding the relationship between X and Y. In RMA regression, the sum of areas of the triangle formed by vertical and horizontal lines from each observation to the line of best fit is minimized (fig.2.7).

In the present study, simple linear regression models were used to find the relationship between characteristic variables in the allometric investigations. Code written by W. Parr in Mathematica (v. 7.0 Wolfram Research Inc., Champaign) was used to perform RMA regression between the following characteristic variables to test hypotheses describing allometric relationships:

1) Log-transformed cortical bone volume (Y) and total skull bone volume (X).

2) Log- transformed total skull bone volume (Y) and total skull surface area (X).

1 2

Figure 2.7 Simple linear regression models of Y on X. (1) minimizing the distance along y-axis in ordinary least square (OLS) method, (2) minimizing the perpendicular distance from the observation point to the line of best fit in major-axis (MA) method, (3) Reduced major axis (RMA) method minimizes the area of triangle formed by horizontal and vertical extensions drawn from the observation point, and the line of best fit.

A correlation coefficient (r), which is a measure of the degree of variation in Y that can be explained by the linear relationship with X, was also determined for both the regression equations.

2.4.4 Confidence intervals for the slopes of allometry equations

The slopes of the regression lines estimated using RMA analysis are the most important parameters for testing allometric hypotheses developed in this study. Based on a confidence level of C%, confidence intervals (CI) for the slopes of regression lines

56 were calculated. CI uses sample data to predict a range within which the parameter being analysed is most likely to fall. The interval has the form

estimate േ margin of error

The margin of error depends on the value of C. Using code written by W. Parr in

Mathematica, 95% confidence intervals were calculated for slopes of both the regression equations, before comparing them to the isometric condition. Theoretically, this means that there is a 95% chance that the slopes of the regression lines lie in the confidence intervals calculated using sample data.

2.5 Finite element modelling of sabretooth predators Smilodon fatalis and

Thylacosmilus atrox

The 'canine shear-bite' hypothesis proposed by Akersten (1985) is the most widely accepted hypothesis explaining killing behaviour of the highly specialised sabretooth cat, Smilodon. Akersten argued that Smilodon killed their prey by means of a modified type of bite in which the mandible had a supporting role to play for the opposing action of the upper jaw and canines. The forces driving penetration of the upper canines were provided by some combination of the head flexing action of the atlanto- mastoid musculature, as well as the jaw adductors. If the interpretation of Akersten

(1985) for Smilodon holds true, then assuming that the sabre-teeth follow the path of least resistance to minimise transverse stresses when rotational forces are applied about temporomandibular joint (TMJ) axis, the centre of rotation of the arc of upper canines should be at or near the TMJ. This interpretation of killing behaviour in

Smilodon is essentially an extension of 'normal' felid killing behaviour. However, at

57 wide gapes in particular it has been argued that the jaw musculature of Smilodon would have been unable to generate significant bite reaction forces at the canines

(Bryant, 1996) and therefore the respective roles of jaw adducting and cervical musculature in both Smilodon and other sabre-toothed carnivores has received considerable attention (Bryant, 1996; Anton and Galobart, 1999; McHenry et al., 2007;

Christiansen 2010; Anton et al. 2004).

Smilodon fatalis and its cogenerics, the smaller S. gracilis and larger S. populator, are regarded as the most specialised of felid sabretooths. However, with enormous upper canines that are even larger relative to the size of its skull, another contender for most specialised sabretooth carnivore of all-time is the South American marsupial

Thylacosmilus atrox. The degree to which similarity in form might represent similarity in function (and hence functional convergence) between these placental and marsupial predators is the overarching question I aim to examine here. Using 3D finite element modelling approaches, the specific questions I address are:

1. Is the centre of the arc of rotation about the canines in these two species at or near the TMJ?

2. How does bite force vary with gape angle and how does it compare between species?

3. How effectively might the major head depressing musculature augment bite reaction forces?

4. Are stresses in the skull and canine teeth of these two sabretooths comparable with those developed in a conical toothed extant felid Panthera pardus?

New analytical approaches detailed below were developed to estimate the centre of the sabre-tooth arcs and theoretical maximum gape angles. Separate solves were run to estimate the amount of head-depressing muscle force and jaw-adductor muscle force needed to generate body-mass scaled canine bite force in the two sabretooth carnivores and P. pardus. 3D models were generated from computed tomography

(serial X-ray) data. The upper canines (including roots) and cranium were generated as separate models. A new approach was developed to digitally 'stitch' the volumetric meshes of the upper canines and the cranium. Finite element modelling protocols described in the preceding sections (2.1-2.4) were used for model pre-processing.

2.5.1 Body mass estimates

The postcranial skeleton of Smilodon fatalis is extremely robust and far more massive than that of any extant cat (Wroe, Anton & Lowry 2008). Recent body mass estimates based on postcranial data suggest that it was up to around 280 kg, comparable to the very largest living felid subspecies, the Siberian tiger (Christiansen 2005). Thus, estimates based on cranial dimensions, deduced on the basis of regression data from living felids, almost certainly underestimate its body weight. The body mass estimate for the specimen of S. fatalis included in this study was generated using the approach applied by McHenry et al. (2007), i.e., geometric similitude was assumed between the specimen used in our study (FMNH P 12418) and the only specimen of S. fatalis for which a near complete skeleton is known (LACM PMS 1-1). That is, body mass for this near complete specimen was calculated on the basis of proximal limb bone minimum circumference data and a 2/3rd power relationship was then assumed between the 59 basal skull length and the body mass. The body mass estimate for (FMNH P 12418) using this approach was ~ 259 Kgs.

The body mass estimate for T. atrox was obtained directly from the literature (Argot,

2004a) and was ~ 82 Kgs. Argot used Anyonge’s (1993) regression equations between body mass and femoral measurements. Body mass estimates based on proximodistal length and circumference of the femur at midshaft were also obtained from Anyonge’s equation and the two estimates were averaged. For P. pardus, the body mass estimate was obtained using Van Valkenburgh’s (1990) regression relationship between body mass and skull length, and was ~ 68 Kgs.

2.5.2 Theoretical maximal gape angle

Theoretical maximum gape angles in S. fatalis, T. atrox and P. pardus were determined on the basis of surface STLs generated from the FEMs. The TMJ articulating surfaces, i.e. the condyle and the cotyle surfaces, were extruded by 1 mm to simulate cartilage covering the joint. Actual joint cartilage thickness was not known for these specimens, but this is likely a conservative estimate for animals of this size. An Iterative Closest

Point (ICP) registration process (Besl and McKay, 1992) was then employed to fit the cartilage surfaces together. By selecting the regions of the cartilage layer that are in contact during the maximum gape, a rough maximum gape angle was determined.

However the cartilage layers still overlap to some extent at this position. The mandible was then moved until the cartilage layers just touched. In the present study, a 2D gape angle measured between the upper medial incisor, TMJ and lower medial incisor was

60 used. The theorecal maximum gape angle obtained using this process was 87.1 ͦ for S. fatalis, 105.8 ͦ for T. atrox, and 72.6 ͦ for P. pardus.

Figure 2.8 Theoretical maximal gape angle in (a) S. fatalis, (b) T. atrox, and (c) P. pardus

2.5.3 Centres of sabretooth arcs

To find the centres of the sabretooth arcs, STL surface meshes were used as above.

The average of the inner and outer arc of the sabreteeth was assumed to be the arc of best-fit. The centres of the canines are posterior-ventrally displaced from the TMJ in S.

61 fatalis, whereas they are anterio-ventrally displayed in T. atrox. The relative separation between the centre and TMJ is more in T. atrox than S. fatalis. The average (on left and right hand side of the skull) separation between the TMJ and the centre of the sabrecanines in S. fatalis was 58.9 mm, and in T. atrox was 75.3 mm. This separation was normalised by dividing it with their respective skull lengths. The normalised separation between the TMJ and the centre of sabrecanines was 0.189 in S. fatalis and

0.343 in T. atrox.

Figure 2.9 Blue and red dots show that centres of sabre-canines in (a) S. fatalis, and (b) T. atrox.

2.5.4 Generation and digital 'stitching' of the volumetric meshes

Bone density data is rarely preserved in fossil specimens meaning that distinctions cannot be drawn between cancellous and cortical bone. Consequently, in the present study, as with the great majority of previous FEA based analyses incorporating fossil species, FEMs were assigned a single material property for bone, i.e., as for wholly cortical models used in the above section. It is noted that differences in size between the crania used here are relatively small compared to the differences observed in extant species (see Appendix). It is certainly possible, if not likely that our results may slightly overestimate the stiffness of the S. fatalis cranium relative to that of the other two, as it was the largest of the three.

However, preservation of the canine teeth including their roots was sufficient to allow distinction from the cranium. For each specimen, separate meshes were generated in

Mimics (vers. 13.1.0) for the sabre teeth and crania. These meshes were exported as

Nastran (.nas) files and solid meshed in Strand7 as above. The main challenge was that despite being in anatomically correct position in the FEMs, the upper canines were not digitally connected to the crania, and hence any force or constraint applied to them would not have transmitted its influence to the rest of the skull.

To overcome this problem, 'bricks' representing the upper canines and skull were assigned to separate groups in Strand7. Only cranial bricks were then selected and tessellated to create surface plates. Using the select region tool in Strand7, plates representing the upper canine root-cavity were then selected and a volumetric mesh was generated representing the canine tooth roots. Using the Clean-mesh tool, the was digitally stitched to the cranium. Bricks representing the upper canine tooth roots 63 in the original solid were then deleted. The canine tooth roots were connected to the tooth crowns using a network of rigid links. The upper canines including the tooth roots were assigned material properties for dentine and the surface 'bricks' of the canine crowns were assigned properties for enamel (see Appendix 10-12).

2.5.5 Head-depressors reconstruction

Akersten's (1985) 'canine-shear-bite' hypothesis emphasised the potential importance of head-depressing musculature in Smilodon fatalis in generating bite reaction forces at wide gapes. In the present study the two major head depressors, Obliquus capitus and Sternomastoideus, were reconstructed in the FEMs of S. fatalis, T. atrox and P. pardus based on known mastoid anatomy (Anton et al. 2004; Argot 2004a,b ). The cranium and the mandible were first rotated about the TMJ to the theoretical maximal gape angle. Rigid links were then used to create an ellipse (major and minor axis in mm: 40 and 26 for S. fatalis, 36 and 25 for T. atrox, and 30 and 20 for P. pardus) and a circle (radius in mm: 30 for S. fatalis, 26 for T. atrox, and 23 for P. pardus) that served as attachment 'webs' for the head-flexors. The circle and ellipse web were kept perpendicular to each other, and their dimensions were kept proportional to skull length. The centre node in each web was fixed in all six degrees of freedom.

Figure 2.10 Pre-processed FE model of S. fatalis showing neck muscle reconstruction and attachment web. Muscle trusses in dark red represents Obliquus capitus and in light coral represents Sternomastoideus.

Forty truss elements were used to simulate the action of Sternomastoideus muscles and thirty elements were used for the Obliquus capitus. In S. fatalis each head- depressing truss element was assigned a pretension of 25N. Body mass scaled canine bite force output was then estimated for T. atrox and P. pardus using S. fatalis as a reference following the approach of Wroe et al. (2005). The head-flexor muscle recruitment needed to generate this bite force was deduced from the FE solves for T. atrox and P. pardus. Jaw adductors were not recruited. 65

3. RESULTS

3.1. Comparative biomechanical performance in extant felid skulls

The biomechanical performance of extant felid skull FEMs were compared under different loading scenarios. In the context of this study, I define biomechanical performance as an animal’s ability to convert the jaw muscle recruitment into bite reaction forces, and its ability to bear associated stresses. To isolate the differences in morphology from size differences, and study the differences in biomechanical performance attributable to morphological differences alone, all the FEMs were scaled to the same size. Scaling protocols used and the results obtained from each scaling type are described in detail below.

3.1.1 FEMs scaled to the same surface area and recruiting the same net jaw muscle

force as A. jubatus.

In this analysis, the size variation between different felid skulls was controlled by scaling them to the same surface area following previously published protocols by

Dumont et al. (2009). All the FEMs were then loaded with the same net jaw muscle recruitment and similar constraints. For models scaled to the same surface area and using the same muscle force recruitment as in A. jubatus, the highest bite force at canines (fig. 3.1) was generated in A. jubatus (689N) and the lowest was in L. pardalis

(403N). In unilateral bites at the carnassial tooth, the highest bite force was observed in P. concolor (1089N) and the lowest in L. pardalis (808N).

Figure 3.1 Estimated bite force (N) in the FEMs of different felid skulls under similar loading conditions. Models were scaled to the same surface area and used the same muscle force recruitment as A. jubatus.

Irrespective of the specimen or the load case, the mandible was consistently more stressed than the cranium (fig.3.2), but the stresses in the zygomatic arch region were comparable to that in the mandible. The highest mean VM stress in the mandible

67 for both the load cases was observed in A. jubatus and the lowest in L. pardalis (Tab.

3.1).

Figure 3.2 Von-Mises (VM) stress distribution in the FE models of different felids during a bilateral canine bite. Models were scaled to the same surface area and recruited the same net jaw muscle force as A. jubatus.

It was also observed that in all felids, VM stress distributions in the crania were not uniform and certain regions were consistently more stressed than others (fig.3.2).

Therefore, the mean values of VM stress in the cranium would not have captured the stress variations between different felids. To study the stress variations in different regions of the cranium within and between different felid species, cranial bricks were grouped into three different regions: 1) zygomatic arch; 2) face and rostrum; 3) rest of the cranium. The highest mean VM stress in all three regions of the cranium was again observed in A. jubatus (Tab. 3.1). Visual post-processing plot (fig.3.2) and two-factor

ANOVA results suggest that for any given felid FEM, there was a highly significant variation (p<0.001) in the mean VM stress distribution within different regions of the

Mandible Zygomatic arch Rostrum Cranium rest F. s. lybica 3.924 2.544 0.539 0.583 L. pardalis 3.684 4.098 0.515 0.699 N. nebulosa 7.712 3.729 0.683 1.193 A. jubatus 9.170 6.194 0.904 1.343 P. concolor 6.456 4.242 0.500 0.901 P. pardus 5.671 4.923 0.579 1.304 P. leo 3.873 1.780 0.614 0.688 skull. However no significant variation (p = 0.056) in the mean VM stress distribution was observed across species.

Table 3.1 Mean VM stress values (MPa) in different regions of the felid skull FEM during a bilateral canine bite - models were scaled to the same surface area and recruited the same net jaw muscle force. MPa = Mega-Pascals.

3.1.2 FEMs scaled to the same surface area and the same output bite force as A. jubatus.

In this analysis the FEMs were again scaled to the same surface area in order to remove size differences, but the jaw muscle recruitment was adjusted to generate the same output bite force. In a relative sense, this analysis was used to compare the efficiency of the jaw musculature in different FEMs, and the associated stresses induced in the skull for the same output bite force. When models were scaled to the same surface area, and muscle recruitment in the jaw was adjusted to generate the same output bite force as A. jubatus (~ 690N in bilateral canine and 976N in unilateral carnassial bite), the highest muscle recruitment (fig. 3.3) for the bilateral canine bite case was observed in L. pardalis and the lowest was in A. jubatus. This suggests that among other felids, A. jubatus had the most efficient system for generating a bilateral canine bite.

Depending on acceleration and speed over greater distances to capture its prey than other felids, this greater biomechanical efficiency in A. jubatus may have been selected for because it permits a reduction in the mass of musculature and bone required to achieve a comparable result. For a unilateral carnassial bite, the highest muscle recruitment was again observed in L. pardalis, and the lowest was observed in P. pardus.

To capture the stress variations within each FEM, the bricks in the model were grouped under four regions of interest: the mandible; zygomatic arch; rostrum, and rest of the cranium. Table 3.2 shows mean VM stress values in different regions of the skull during a bilateral canine bite. The highest mean VM stress in the jaw was observed in A. jubatus and the lowest was in P. leo. The stresses in the mandible of A. jubatus and N. nebulosa in this bite case were quite comparable.

3500

3000

2500

2000

Bilateral canine 1500 Unilateral carnassial Muscle force recruited (N) recruited force Muscle 1000

500

Figure 3.3 Estimated jaw muscle recruitment (N) in the FEMs of different felids under two different loading conditions. Models were scaled to the same surface area and generated the same output bite force as A. jubatus.

In the cranium, the zygomatic arch region was consistently more stressed in all the simulations, while the other two regions (rostrum and cranium rest) maintained low stresses. A. jubatus again had the highest mean VM stress in the "rostrum" and "cranium rest" regions. This suggests a possible trade-off between bite force, mass and a low safety factor in the skull of A. jubatus.

Table 3.2 Mean VM stress in the skull of different felid FEMs during a bilateral canine bite - models were scaled to the same surface area and same output bite force as A . jubatus.

Mandible Zygomatic arch Rostrum Cranium rest F. s. lybica 5.359 3.475 0.736 0.796 L. pardalis 5.905 7.006 0.881 1.195 N. nebulosa 9.014 3.787 0.693 1.212 A. jubatus 9.170 6.194 0.904 1.343 P. concolor 7.778 5.110 0.602 1.085 P. pardus 7.311 6.346 0.746 1.681 P. leo 4.591 2.110 0.728 0.816

Qualitative comparisons of the stress distributions in different felid FEMs (Fig. 3.4), and

quantitative estimation of the variance in mean VM stress using two-factor ANOVA

suggests a highly significant variation (p<0.001) in the stress distribution within different

regions of the skull. However no significant variation (p = 0. 092) in the mean VM stress

distribution was observed across species. As with the ANOVA results from the previous

scaling type (section 3.1.1), these results indicate that if the size variations in different

felid skulls are removed, there is no statistically significant variation in the biomechanical

performance by this measure. These findings broadly support the contention that the

felids considered in this study are morphologically conservative, and the differences in

biomechanical performance are mostly attributable to the size differences amongst

species.

Figure 3.4 Von-Mises (VM) stress distribution in the FE models of different felids during a bilateral canine bite. Models were scaled to the same surface area and muscle force recruitment was adjusted to generate the same bite force as A. jubatus.

3.2. Homogeneous and heterogeneous finite element models

Bony structures are heterogeneous in nature, i.e. the spatial distribution of bone materials is not uniform throughout the volume. Bone density is correlated with its stiffness and hence a non-uniform spatial density distribution will have implications for the overall stiffness of the structure. In the present study, I define homogeneous FEMs as those having material homogeneity, i.e., being composed of only one type of material. Heterogeneous FEMs are those in which cancellous and cortical bone material properties were incorporated into the models based on CT attenuation data.

To distinguish possible differences in biomechanical performance between homogeneous and heterogeneous models of biological structures, homogeneous FEMs for the smallest felid F. s. lybica and the largest felid P.leo were assembled. These homogeneous FEMs comprised only one material property (cortical bone). The rationale behind selecting F. s. lybica and P. leo was that these species represented the two extremes with respect to differences in the relative proportions of cortical and cancellous bone volume in the skull, and hence likely extremes in consequent variation of overall stiffness between the homogeneous and heterogeneous FEMs. The loading conditions and constraints were kept exactly the same as for homogeneous and heterogeneous models. Results derived from these homogeneous FEMs were compared to the heterogeneous solves.

On visual inspection (fig. 3.5) of post-processing plots, it was observed that the differences in VM stress distribution between homogeneous and heterogeneous models were more profound in P. leo than F. s. lybica. This was possibly because the skull specimen of P. leo had the lowest proportion of cortical bone among felids in this

74 study. This is reflected in the large stiffness difference between the homogeneous and heterogeneous FEM of P.leo.

Figure 3.5. VM stress distribution in the unscaled FE models of F. s. lybica [ a), b)] and P. leo [c), d)] during a bilateral canine bite. Models b) & d) are homogeneous (cortical bone material property) and a) & c) are heterogeneous (cortical and cancellous bone material properties)

On the other hand, the differences in VM stress distribution were minor in F. s. lybica.

These were mostly concentrated in the mandible, unlike P.leo where differences in

75 both cranium and mandible were obvious. This is likely because F. s. lybica had the highest proportion of cortical bone volume in the skull amongst other felids, and hence the differences between the heterogeneous and homogeneous models were relatively small. It is worth noting that the above visual plots depict only the surface stresses and reveal no information regarding the internal brick stresses. To qualitatively assess the variation in stress between homogeneous and heterogeneous models of F. s. lybica and P. leo, the bricks in the FEM were grouped under four regions of interest, as discussed in section 3.1.1.

Table 3.3 shows the mean and standard deviation (s.d.) values of brick VM stresses for homogeneous and heterogeneous FEMs during a bilateral canine bite. There may not be a significant difference in the mean VM stress values between the heterogeneous and homogeneous models, but because of higher proportion of cancellous bone in

P. leo, the standard deviation values were consistently higher compared to those for F. s. lybica. These results suggest that even with a much higher (~ 41%) volume proportion of cancellous bone, the skull of P.leo is able to maintain its overall stiffness and perhaps minimise weight through efficient distribution of materials.

Table 3.3 Mean and standard deviation (s.d.) values of brick VM stresses for different regions of skull in homogeneous and heterogeneous FE models of P. leo and F. s. lybica in a bilateral canine bite – (Hom. – homogeneous, Het. – heterogeneous)

Brick VM stress (MPa)

(mean ± s.d.)

Mandible Zygomatic Arch Face and rostrum Cranium rest

Hom. 6.235 ± 2.859 2.497 ± 1.296 0.870 ± 0.397 1.057 ± 0.569 P. leo

Het. 6.417 ± 4.092 3.061 ± 2.485 1.058 ± 1.028 1.218 ± 1.094

Hom. 2.902 ± 1.470 1.892 ± 1.029 0.480 ± 0.303 0.490 ± 0.321 F.s. lybica 3.103 ± 1.857 1.933 ± 1.397 0.504 ± 0.387 0.514 ± 0.413 Het.

3.3. Allometry in the distribution of material properties and geometry of the felid skulls

Allometric scaling in bone thickness has been reported in a recent study on primates

(Strait et al., 2010), in which the authors concluded that the thickness of cortical bone in the crania of extant primates decreased with increasing body mass. Thickness of the cortical bone contributes to the cross-sectional area, which in turn influences the rigidity of the skull structure. But thickness alone does not describe how bones deform in response to applied loads, inter-specific variation in material properties may also be important. For example, a thin bone that is relatively stiffer may deform to the same degree as a thick bone that is more compliant (Wang et al., 2006). There may be a fine balance between bone thickness and stiffness, and the structural strength of a bone

77 might be increased through either developing thicker yet less stiff or thinner but denser bone (Strait et al., 2010).

Strait et al. (2010) findings opened a direction for further investigations into the role of allometric distribution of materials in understanding the craniomandibular mechanics of other taxa. Because felids are considered morphologically and behaviourally conservative, they are an appropriate focus for such allometry studies. The first allometric investigation in this study was related to the variations observed in the volume proportion of cortical bone in felid skulls. Using Strand7 tools, the cortical and cancellous bone volumes were quantified in all seven felid skull FEMs.

Another recent allometry study on felid skull shape by Slater and Van Valkenburgh

(2009) suggested a positive allometry between cranial rigidity and body mass. In this study, the authors concluded that the larger felids were able to achieve increased cranial rigidity by increasing the skull bone volume relative to the surface area. This study, however, did not differentiate between cortical and cancellous bone materials when considering the skull bone volume. The second allometric investigation in the present study is related to the geometry of the felid skulls. Again using Strand7 tools, the total skull bone volume and total skull surface area were quantified for all seven felid skull FEMs.

3.3.1 Allometry in the distribution of material properties in extant felid skulls.

From visual inspection of CT scan images (fig. 2.9.1) it appeared that the volume proportion of the stiffer cortical bone decreased with increasing skull size in felids.

Using a novel approach described in section 2.6 and 2.9, the cortical and cancellous bone volumes were quantified in all seven felid skull FEMs. Table 3.4 shows that the 78 volume percentage of cortical bone in different felid FEMs decreased with increasing skull size and body size in general.

Table 3.4 Cortical bone volume, total bone volume and proportion of cortical bone by volume in different felid skull FEMs

Cortical bone Vol. Total skull bone vol. % cortical bone by

(mm^3) (mm^3) volume 3.343×104 6.597×103 F. s. lybica 83.518 3.499×104 8.308×103 L. pardalis 80.812 2.270×105 6.805×104 N. nebulosa 76.936 2.284×105 9.747×104 A. jubatus 70.089 2.963×105 1.017×105 P. concolor 74.447 3.951×105 1.482×105 P. pardus 72.722 1.604×106 1.127×106 P. leo 58.733

To establish the allometric relationship between cortical bone volume and total skull bone volume, the two characteristic variables were first log-transformed and root mean square (RMA) regression analysis was then performed. Graph 3.6 shows the regression equation and the correlation coefficient between the log-transformed characteristic variables.

1.00E+07 Log10 (cortical bone volume) = 0.926 Log10 (skull bone volume) + 0.263, R² = 0.998

1.00E+06

1.00E+05

Different felid specimens Cortical bone volume, Cortical bone volume, mm^3

1.00E+04 1.00E+04 1.00E+05 1.00E+06 1.00E+07

Skull bone volume, mm^3

Figure 3.6 Graph showing RMA regression equation between log-transformed cortical bone volume and Skull bone volume.

The slope of the linear regression equation thus derived was 0.926. Strong correlation was found between log-transformed cortical bone and total skull bone volume (r =

0.998). A 95% Confidence Interval (CI) was also determined for the slope of this regression line.

95% CI: 0.884-0.968, p<0.001

These results show that the slope of regression line is significantly less than the slope for the isometric condition (slope=1) between the two characteristic variables, suggesting a negative allometric relationship between the two.

4.4.2 Allometry in the geometry of extant felid skulls

Slater and Van Valkenburgh (2009) concluded that larger felids achieve greater skull strength by increasing skull bone volume relative to the surface area. However due to a small sample size (n=3), the authors could not test their hypothesis using a statistical approach. Moreover, they only considered crania and did not include the mandibles of different felid species in their analysis. The present study overcomes both the above limitations by including a relatively bigger sample size (n=7) and also considering mandibles in the finite element modelling. In the context of this study, thickness coefficient (t) defined as the ratio of skull bone volume to the skull surface area, was used to compare the overall variation in bone thickness in different felid skulls. Table

3.5 shows the thickness coefficient (t) for different felid skull FEMs in the present study.

Table 3.5 Skull bone volume, skull surface area and thickness coefficient for different felid FEMs.

Skull surface area Thickness Skull bone vol. (mm^3) (mm^2) coefficient (t)

F. s. lybica 4.000×104 3.258×104 1.228 L. pardalis 4.330×104 3.741×104 1.157 N. nebulosa 2.950×105 1.214×105 2.430 A. jubatus 3.260×105 1.421×105 2.294 P. concolor 3.980×105 1.461×105 2.724 P. pardus 5.430×105 1.652×105 3.287

P. leo 2.730×106 4.261×105 6.407

To investigate any possible allometric relationship between total skull bone volume and total skull surface area, another regression analysis was performed between the

Log-transformed values of the two characteristic variables.

Log10 (skull bone volume) = 1.648 Log10 (surface area) - 2.891, R² = 0.994 2.50E+06 3

2.50E+05

Different felid specimens

Skull bone volume, mm 2.50E+04 2.50E+04 2.50E+05 2.50E+06 Surface area, mm2

Figure 3.7 Graph showing RMA regression equation between log-transformed skull bone volume and skull surface area.

The slope of the linear regression equation in this case was 1.648. Strong correlation was found between the log-transformed skull bone volume and skull surface area

(r=0.996). A 95% Confidence Interval for the slope of the regression line was also determined.

95% CI: 1.501- 1.794, p <0.001

The results thus obtained shows that the slope of the linear regression equation just exceeds the slope for isometry condition (slope =1.5), suggesting a weak positive allometric relationship between the two characteristic variables. The weak positive allometry could possibly be because of the small sample size for a rigorous statistical 82 treatment. Further work including a bigger sample size will be necessary to fully assess the nature of this allometric relationship.

3.4. Comparing biomechanical performance between the sabretooth predators

Smilodon fatalis and Thylacosmilus atrox, and the extant conical tooth felid Panthera pardus

3.4.1 Jaw musculature driven canine bite and its variation with gape angle

The theoretical maximal gape angle based on the point of disarticulation of jaw joint in

S. fatalis was approximately 87.5 .̊ This was slightly less than determined by Emerson and Radinsky (1980) who suggested that the maximum gape in S. fatalis was between

90-95 ̊, and considerably less than the maximal gape of 100 ̊ assumed by Bryant (1996).

The biomechanical implicaons of variaon in gape angle were determined for each of the three species, starng from a minimum gape of 15 ̊, at 25 ̊ increments up to the theoretical maximal. Bite reaction force, jaw muscle recruitment, mean VM stresses in the cranium, mandible, upper canine-crowns and in the tooth-roots were also quantified at different gapes. The force generated by vertebrate skeletal muscle fibres varies with the degree of fibre stretching, with the maximum force generally generated at the rest length (Harris and Warshaw, 1991, Rassier et al., 1999, Ferrara et al. 2011).

However, in the Masseter of Felis, maximum tension may be developed at wider gapes

(MacKenna and Turker 1978) and variability has been documented between species in other mammalian taxa (Eng et al. 2009). Clearly determining the muscle fibre force- tension relationship for jaw adductors in large extant predators would be difficult, and 83 impossible for extinct species. Consequently this relationship was not considered in the present study and maximum tension was assumed at all gape angles.

Figure 3.8. VM stress distribution at theoretical maximal gape angle in (a) S. fatalis, (b) T. atrox, (c) P. pardus, during a bilateral canine bite. Only jaw muscles were recruited.

Figure 3.9a shows that with an increase in gape angle, both the canine bite reaction force and jaw muscle recruitment decreases. However, the bite force declines much more steeply than does jaw muscle recruitment. These results are consistent with the previous contention that in S. fatalis, the jaw adductors alone may not have produced

84 sufficient force to drive the upper canines through the hide of large prey, especially at larger gapes (Bryant, 1996, McHenry et al., 2007, Wroe et al., 2005). Little difference was observed in mean VM stresses in the tooth-roots across different gapes. The canine-crowns experienced lowest mean VM stress at maximal gape whereas the cranium experienced the highest.

6000

4966 5000 4612 4257 4000 3785

3000 Force (N) 2000 1408 1053 991 1000 519 a) 0 G15 G 40 G65 G_max (87.5)

Jaw muscle Canine Bite

8.000

7.000 6.76 6.159 6.136 6.000 5.456

5.000 4.088

4.000 3.816 3.287 3.000 2.593 Von mises stress (MPa) stress Von mises 2.000 1.074 1.056 0.823 1.000 0.815 0.570 0.585 0.468 0.552

b) 0.000 G15 G 40 G65 G_max (87.5)

Canine-crowns Tooth roots Cranium Mandible

Figure 3.9 Smilodon fatalis: a) Variations in canine bite reaction force and jaw muscle recruitment with changing gape, b) Variations in mean VM stresses in different regions of the skull with changing gape.

At approximately 106 degrees, the theoretical maximal gape angle in T. atrox was considerably higher than that observed in S. fatalis. My estimate for T. atrox was slightly larger than the maximum gape of 102 predicted̊ by Churcher (1985). The bite force results (Fig. 3.10a) suggest that the efficiency of the jaw musculature in generating bite force at the canines decreases considerably with increasing gape, and at maximum gape the output bite force is ~38N for ~ 636N of jaw muscle force recruitment. However, the minimum values for mean VM stresses in different regions of the skull were also observed at maximum gape (fig.3.10b).

2500

2080 2000 1947 1814

1500 1326 Force (N) 1000

a) 636 585 500 414 274 93 38 0 G15 G40 G65 G90 G_max(105.8)

Jaw muscle Canine Bite

2.500 2.221 2.112 2.109 2.045 2.000 1.536

1.500 1.429 0.973 1.000 0.954 0.707 0.669 Von mises stress (MPa) stress mises Von 0.636 0.633 0.546 0.500 0.365 0.364 0.346 0.292 b) 0.2486 0.147 0.077 0.000 G15 G40 G65 G90 G_max(105.8)

Canine-crowns Tooth roots Cranium Mandible

Figure 3.10 Thylacosmilus atrox: a) Variations in canine bite reaction force and jaw muscle recruitment with changing gape, b) Variations in mean VM stresses in different regions of the skull with changing gape.

The theoretical maximal gape angle in P. pardus was the lowest among three species

(~72.6 degrees), which was again close to Emerson & Radinsky's (1980) prediction of maximum gape in modern felids. The efficiency of jaw musculature in generating the bite force was considerably higher in P. pardus compared to the other two across different gapes (Fig.3.11a). The lowest mean VM stresses in different regions of the skull were again observed at maximum gape (Fig.3.11b). Overall the conversion of jaw muscle force to canine bite force across different gapes was observed to be highest in

P. pardus and lowest in T. atrox. Consequently, the mean VM stresses in the cranium, mandible , tooth-roots and canines were highest in P. pardus and lowest in T. atrox.

3000 2750

2500 2408

1952 2000 a) 1476 1500 1222

Force (N) 984 1000 641 484 500

0 G15 G 40 G 65 G_max (72.6) Jaw muscle Bite reaction

9.000 8.226 8.005 7.865 8.000 6.808 7.000 6.412

6.000 5.722 5.180 5.000 4.219 4.000

3.000 Von mises stress (MPa) stress Von mises 1.808 1.702 1.654

2.000 1.577 1.397 1.333 1.245 1.000 0.908

0.000 G15 G 40 G 65 G_max (72.6) Canine-crowns Tooth roots Cranium Mandible b)

Figure 3.11 Panthera pardus: a) Variations in canine bite reaction force and jaw muscle recruitment with changing gape, b) Variations in mean VM stresses in different regions of the skull with changing gape.

The body mass estimate for T. atrox was directly obtained from the literature and was

~82 Kgs (Argot,2004a). For S. fatalis geometrical similitude was assumed with the specimen used by McHenry et al. (2007) in their study and the estimated mass was

~259Kgs . For P. pardus, estimated body mass was derived using previously published protocols(Van Valkenburgh, 1990), and was ~68 Kgs. A near 2/3 power relationship between body mass and bite reaction force has been proposed for vertebrates (Huber et al. 2005). This follows from the observation that maximal muscle force is proportional to muscle cross-sectional area, whereas body mass is directly proportional to volume (Schmidt-Nielsen, 1984). When the canine bite reaction force was adjusted to the body mass using the 2/3rd power relationship between body mass and bite force, the jaw muscle recruitment in T. atrox was ~24010N and in P. pardus was ~1658N to achieve comparable bite force outputs. The stresses in the sabretooths were quite similar in this case, but P. pardus was significantly less stressed.

Table 3.6 Body mass adjusted canine bite force results for a jaw adductor driven bite at maximum gape. * Muscle forces were back calculated from FE models that gave the body mass scaled bite force output at canines. S. fatalis was arbitrarily chosen as a reference.

Mean VM stress (MPa) Canine bite Body mass Jaw muscle force (N)- estimates driven bite Rest of 2/3rd Tooth Canine- (Kgs) force (N) Mandible the power rule root crowns Cranium 3785 S. fatalis 259 519 0.552 2.593 6.136 1.074 (reference) T. atrox 82 24010* 241 0.486 2.311 6.05 1.577 P. 68 1658* 212 0.397 1.848 2.981 0.584 pardus

Figure 3.12 VM stress distribution in response to jaw adductor driven bite with bite reaction force adjusted for body mass (a) S. fatalis, (b) T. atrox, and (c) P. Pardus. S. fatalis was arbitrarily chosen as a reference for bite force output for T. atrox and P. pardus

3.4.2 Cervical muscles driven canine bite

To investigate the degree to which the head depressing muscles augment the bite force at upper canines at maximal gape, and the capacity of the skull to bear associated stresses, an additional set of solves were run which included a canine bite driven solely by head depressing musculature.

Figure 3.13 VM stress distributions in response to forces generated by cervical musculature to achieve bite reaction forces adjusted for body mass (a) S. fatalis,(b) T. atrox, and (c) P. pardus.

Assuming a 2/3rd power relationship between body mass and the bite force

output, and considering S. fatalis as a reference, scaled bite forces were deduced for

T. atrox and P. pardus. I note that the choice of reference species here is arbitrary and

immaterial in the comparative context in which it is used here. Head depressing

muscle forces required to generate these scaled bite forces were then estimated from

FE results. Mean VM stresses in the tooth roots and the canines were also compared.

Table 3.7 Body mass scaled bite force results for a cervical musculature driven bite. * Muscle forces were back calculated from FE models that gave the body mass scaled bite force output at canines. S. fatalis was arbitrarily used as a reference and head depressing muscle force used in the model was a hypothetical value (1750).

Mean VM stress (MPa) Body mass Head depressing Canine bite estimates muscle force (N) force (N) Tooth Canine- Man Rest of the (Kgs) root crowns dible Cranium S. fatalis 259 1750 (assumed) 269 0.193 0.585 0.161 0.446

T. atrox 82 1547* 125 0.190 0.440 0.171 0.450 P. 68 1076* 110 0.269 0.969 0.153 0.574 pardus

The above table shows that for body mass scaled bite force in T. atrox, the mean VM

stresses in the sabre-canines were comparable to that in S. fatalis. But in P. pardus, the

stresses in canines and also in the cranium (Tab.3.7) were significantly higher

compared to the other two species.

4. FUNDAMENTAL CHALLENGES ASSOCIATED WITH INCREASING SIZE

In the classic book on importance of animal size, Scaling: Why is animal size so important? by Knut Scmidt-Nielsen, the author explains the structural and functional implications of change in size among otherwise similar organisms. Similarity, structures of different sizes, whether biological or man-made, can be broadly classified into three different types: geometrical similarity, stress similarity and elastic similarity.

Geometrical similarity refers to a simple scaling in linear dimensions without any change in material composition or shape. The stress similarity model assumes a constant stress level in the structure with increasing size. Elastic similarity model requires structures to exhibit same elasticity, i.e. the deformations in the structure increase in direct proportion to the size. To achieve stress or elastic similarity in structures, a change in shape or material property distribution or a combination of both is essential.

As biological structures increase in size, maintaining geometrical similarity can pose fundamental challenges on metabolism and physiology. The first allometric relationship observed in the felid skulls considered in the present study is related to the distribution of bone material properties. With increasing size the relative proportion of cortical bone was found to decrease, which indicates a possible stress/elastic similarity model. The second allometric trend observed was linked to skull shape. Positive allometry was observed between total skull bone volume and skull surface area, which resulted in a net increase in bone thickness with increasing size. In theoretical terms, this net increase in skull thickness resulted in an increase in

94 second moment of inertia, which is a measure of the property of a cross-section to resist bending or deflection.

In order to better understand the nature of these challenges associated with increasing size, two analogies were developed using classical beam mechanics. These analogies provided an explanatory model to elucidate the biomechanical significance of the allometric trends observed in shape and material properties distribution in the felid skull specimens included in the present study.

3.1 Self weight beam analogy

This analogy examined a possible explanatory model for why cortical bone volume allometry might occur, by investigating the influence of geometrical scaling and consequent increase in mass, on mechanical behaviour of a symmetric homogeneous isotropic beam. In this analogy, the effects of geometrically scaling the beam without changing the material composition were determined in terms of maximum deflection and stresses induced (Popov, E.P., 2004).

Deflection and stresses induced in a symmetric homogeneous isotropic beam bending due of its self weight alone

y y

x z

Neutral axis X-section view

L a

L: length of the beam

a: width

b: thickness

g: acceleration due to gravity

ρ: density of beam material

E: Young’s modulus of beam material

I: second moment of inertia of the beam about the neutral axis

Theory: Volume of the beam =

Weight = ɏ

Weight per unit length ଴ ൌɏ (1)

Using the conditions for static equilibrium;

σ = 0 and σ = 0 (F = Force, M = Moment)

୛ ୐ yields R = R = బ A B ଶ

(2)where RA and RB are the reaction forces from the roller supports placed at both

ends

If x represents the distance from the left end of the beam

Shear force (V) = WO (L/2 – x) (3)

Internal bending moment (M) = WO x (L-x)/2 (4)

Maximum internal bending moment will be at x = L/2

2 Mmax = WOL /8

Bending stress at top and bottom surface of the beam (σ)

ெୠ σ = (5) ଶ୍

Second moment of inertia (I) of a rectangular section beam about the neutral axis

ୟୠయ I = (6) ଵଶ

Substituting I, M in eq. (5)

ଷ஡୥௫ሺ୐ି௫ሻ ɐൌ (7) ୠ

Bending stresses will be maximum at x=L/2 97

ଷ஡୥୐మ σ = max ସୠ

Using moment-curvature relationship to find the deflection of the beam from neutral axis

ௗమ௬ ெ = (8) ௗ௫మ ୉୍ substituting M in eq.(8)

మ ௗ ௬ ௐ೚௫ሺ௅ି௫ሻ = (9) ௗ௫మ ଶாூ

Boundary conditions: x=0, y=0 and x=L, y=0 (no deflection at left and right end of the beam)

Integrating eq.(9) twice and substituting the above boundary conditions,

ௗమ௬ ୛ ௫ሺ୐ି௫ሻ ׬ሺ׬ ݀ݔሻ݀ݔ = ׬ሺ׬ ౥ ݀ݔሻ݀ݔ ௗ௫మ ଶ୉୍

୛ ௫ሺଶ୐௫మି௫యି୐యሻ y = బ (10) ଶସ୉୍

Maximum deflection will occur at x=L/2

ିହ୛ ୐ర y = బ max ଷ଼ସ୉୍

substituting Wo ,I in the above equation yields

ିଵହ஡୥୐ర y = (11) max ଽ଺୉ୠమ

Maximum deflection and longitudinal bending stress at the mid section of the beam

ିଵହ஡୥୐ర ଷ஡୥୐మ y = ; σ = max ଽ଺୉ୠమ max ସୠ

If this beam is geometrically scaled by a factor of k without changing the material properties, such that anew = k a; b new = k b; L new = k L

We observe that the deflection scales by a factor of k2 and longitudinal bending stress by the same factor k.

2 ynew = k yold ; σnew = k σold

However the ultimate failure stress is a material property, i.e., for a given material it is usually invariant. This implies that as an engineering structure increases in size, in order to avoid failure and preserve functionality, there is an inherent requirement for a change in shape or material properties distribution, or some combination of both.

These same fundamental principles are applicable to both man made and biological structures as they increase in size.

3. 2 Composite beam analogy

Above I have shown that with increasing size, a biological structure must alter its shape, material composition, or both in order to meet inherent challenges associated with change in size. The most simple means of overcoming this problem would be for the structure to become more massive, resulting in a relative increase in mass.

However, this would have obvious implications regarding mobility and metabolism. As previously noted from observation of CT data of the felid skulls included in this study, it is apparent that the distribution of materials changes with increasing size, as well as shape.

Bone is broadly classified into two types of osseous tissue, dense cortical and less dense cancellous bone. Cancellous bone is typically 'sandwiched' between cortical bone layers. I have developed the following analogy to examine the influence of variations in cortical and cancellous bone volume proportions on stiffness and mass in the felid skull (Chamoli and Wroe, 2011). Effective stiffness and second moment of inertia of a symmetric rectangular section composite beam made of two isotropic homogeneous materials (E1 and E2) were determined using classical beam mechanics.

Typically in composite sandwich structure analysis, for example Bauchau and Craig

(2009), E1 is set to zero as the core material is foam with the external cladding either a composite or metallic material. Here I include the structural effects of the core material also.

100

a/2 b a/2

L c

Cross-section b

E1: modulus of elasticity of the inner compliant material

E2: modulus of elasticity of the outer stiff material

E2 > E1; E2 is analogous to cortical bone, E1 is analogous to cancellous bone,

Thickness ratio (n) n = a/b (12)

a is the total thickness of outer layer and b is the total thickness of inner layer

Total thickness of the beam (t) t = a+b (13)

Using a = n.b from equation (12) yields

b = t/(n+1) ; a = n.t/(n+1)

Longitudinal Strain (Є) = change in length/ original length

Longitudinal Strain (Є) Є= െݕȀߩ where ߩ denotes the radius of curvature of the beam

Longitudinal Stress (σ) σ = E Є

101

ି௬ா (σ = = -ݕߢ (ૂ= 1/ρ ఘ

Using the condition for static equilibrium on any transverse section of the beam,

1. Sum of forces acting on a cross-section is zero σ ܨ = 0

0 = ܣ׬ ߪ݀ substituting σ in the above equation

ሺ௔ା௕ሻȀଶ ൌ Ͳ ܣ݀ ݕሻܧെ ሺߢ ׬ିሺ௔ା௕ሻȀଶ

ሺ௔ା௕ሻȀଶ ૂ ൌ Ͳ ܣ݀ ݕሻܧെߢ ሺ (׬ିሺ௔ା௕ሻȀଶ (- = constant

௕Ȁଶ ି௕Ȁଶ ሺ௔ା௕ሻȀଶ ܣ݀ ݕ ܧ ܣ݀ ݕ ܧ ܣ݀ ݕ ܧ ׬ି௕Ȁଶ ଵ +׬ିሺ௔ା௕ሻȀଶ ଶ +׬௕Ȁଶ ଶ = 0

Solving the three integrals separately yields

௕Ȁଶ ௕Ȁଶ ݕ݀ݕ ܣ݀ ݕ ܧ (I. ׬ି௕Ȁଶ ଵ = c E1 ׬Ȃ௕Ȁଶ = 0 (dA = c. dy

ି௕Ȁଶ ሺ௔ା௕ሻȀଶ ܣ݀ ݕ ܧ ܣ݀ ݕ ଶ + ׬௕Ȁଶ ଶܧ II. ׬ିሺ௔ା௕ሻȀଶ

ି௕Ȁଶ ሺ௔ା௕ሻȀଶ ݕ݀ݕ ݕ݀ݕ { cE2 {׬Ȃሺ௔ା௕ሻȀଶ +׬௕Ȁଶ =

102

In the cross-sectional view of the beam, the x-axis was assumed to be the horizontal axis of symmetry and y-axis the vertical axis of symmetry. The above calculations show that in a composite beam bending along the y axis, the neutral plane passes through the horizontal plane of symmetry.

2. Sum of moments across any cross-section is zero σ ܯ = 0

Substituting internal bending moment (Mz) and distance of beam fibres from the neutral axis(y) in the above equation

0 =ܣ݀ Mz + ׬ ݕߪ

ଶ 0 = ܣ݀ ݕܧ Mz – ૂ ׬

௕Ȁଶ ି௕Ȁଶ ሺ௔ା௕ሻȀଶ ܣ݀ ݕଶ ܧ ܣ݀ ݕଶ ܧ ܣ݀ ݕଶ ܧ ૂ [ Mz = [ ׬ି௕Ȁଶ ଵ + ׬ିሺ௔ା௕ሻȀଶ ଶ +׬௕Ȁଶ ଶ

If the effective stiffness of the composite beam is denoted by Eeff, and the effective second moment of inertia by Ieff

Mz = ૂ [ Eeff . Ieff]

௕Ȁଶ ି௕Ȁଶ ሺ௔ା௕ሻȀଶ ܣ݀ ݕଶ ܧ ܣ݀ ݕଶ ܧ ܣ݀ ݕଶ ܧ (Eeff . Ieff = [ ׬ି௕Ȁଶ ଵ + ׬ିሺ௔ା௕ሻȀଶ ଶ + ׬௕Ȁଶ ଶ ] (14

Solving the above integrals separately, yields

௕Ȁଶ ௕Ȁଶ ݕଶ ݀ݕ ܧൌܿ ܣ݀ ݕଶ ܧ I. ׬ି௕Ȁଶ ଵ ଵ ׬Ȃ௕Ȁଶ

ா ௖௕య ൌ భ ଵଶ

103

ି௕Ȁଶ ሺ௔ା௕ሻȀଶ ܣ݀ ݕଶ ܧ ܣ݀ ݕଶ ܧ II. ׬ିሺ௔ା௕ሻȀଶ ଶ +׬௕Ȁଶ ଶ

ି௕Ȁଶ ሺ௔ା௕ሻȀଶ ݕଶ ݀ݕ ܧܿ ݕଶ ݀ݕ ܧൌܿ ଶ ׬Ȃሺ௔ା௕ሻȀଶ + ଶ ׬௕Ȁଶ

௖ா ൌ మ ሺܽଷ ൅ܾଷ ൅͵ܾܽሺܽ൅ܾሻ െܾଷሻ ଵଶ

௖ா ൌ మ ൫ܽଷ ൅͵ܾܽሺܽ൅ܾሻ൯ ଵଶ

combining I. and II. yields

ܧ ܾܿ͵ ௖ா E . I ൌ ͳ మ ൫ܽଷ ൅͵ܾܽሺܽ൅ܾሻ൯ eff eff ͳʹ + ଵଶ substituting a = n b in the above equation

௖௕య E . I ൌ ሾܧ ൅ܧ ሺ݊ଷ ൅͵݊ଶ ൅͵݊ሻ (15) eff eff ଵଶ ଵ ଶ

య ௖௕ ଷ ଶ Ieff ൌ ; Eeff ൌ ൅ ሺ݊ ൅͵݊ ൅͵݊ሻ ଵଶ ଵ ଶ

Comparing the magnitude of longitudinal strains in the composite beam to a homogeneous isotropic beam made from cortical bone material alone.

Єcor = - My/Ecor Icor

Єcomp = - My/ Eeff Ieff

Gain coefficient (β) β = Єcor / Єcomp

104

ు ౙ౗౤ାሺ୬యାଷ୬మାଷ୬ሻ ు β = ౙ౥౨ (16) ሺ୬ାଵሻయ

Here 1/β is the strain increment factor in the composite beam compared to the cortical beam of same absolute dimensions. Fig. 4.1 shows a graph between gain coefficient and thickness ratio for three different values of young’s modulus for cortical and cancellous bone. The graph shows that as the thickness ratio increases, the gain coefficient (β) and the strain increment factor (1/β) both approaches unity which supports the contention that there is essentially no difference in mechanical behaviour of the composite beam and cortical beam when the thickness ratios are high (n>3). With increasing thickness ratios, the composite beam tends to approach homogeneity in material composition and more closely resembles the cortical beam.

105

Figure 4.1. Gain coefficient (β) plotted against thickness ratio (n) for a two property composite beam. Three different pairs of modulus of elasticity (E in GPa) values for cortical and cancellous bone were considered. I. Ecor = 13.7 and Ecan = 0.689 (source

Cook et al.), II. Ecor = 20.7 and Ecan= 14.8(source Rho et al., 1993), III. Ecor = Ecan (dummy set assuming no difference in the material properties of cortical and cancellous bone)Because of the density (ρ) differences between cortical and cancellous bone materials (Rho et al., 1995), there will an effective weight savings (η) in the composite beam compared to the cortical beam.

106

Weight of the cortical beam (Wcort) Wcort = ctL (17)

Weight of the composite beam (Wcomp)

Wcomp = cbLρcan + caLρcor (18)

Figure 4.2. Percentage weight savings (η) plotted against thickness ratio (n) for a two property composite beam. Three different pairs of density (ρ in Kg/m3) values for cortical and cancellous bone were considered. I. ρcor = 2092.30 and ρcan = 1056.84

(source Cook et al., 1982 and Rho et al.1995), II. ρcor = 2596.15 and ρcan = 2142.30

(source Rho et al., 1993 and 1995), III. ρcor = ρcan (dummy set assuming no difference in the material properties of cortical and cancellous bone)

107

3 where ρcan and ρcor are the densities (Kg/m ) of the cancellous and cortical bone material respectively

௖௧௅஡ ௖௅௡௧஡ W = ܿܽ݊ + ೎೚ೝ comp ௡ାଵ ௡ାଵ

% weight savings (η) in the composite beam of same absolute dimensions

ௐ೎೚ೝିௐ೎೚೘೛ η = X 100 ௐ೎೚ೝ

ଵ଴଴ ஡ η = ሺͳെ ౙ౗౤ሻ (19) ௡ାଵ ஡ౙ౥౨

Fig. 4.2 shows the variation in weight savings (η) with the thickness ratio (n) for three different sets of material properties for cortical and cancellous bone. With increase in thickness ratio, the percentage weight savings (η) decreases, but this partly helps in checking the strain increment factor (1/β) in the composite beam.

Conclusion

Optimally designed large engineering structures maximise strength to weight ratio, partly to avoid excessive deformations due to self weight, and partly to save on materials. The allometric trends observed in the shape and material property distribution in the felid skulls included in this study points towards the same fundamental rationale. The above composite beam analogy demonstrates how, by varying the proportions of cancellous and cortical bone tissue in their skeletal structures, larger animals may achieve a higher strength to weight ratio and save considerably on materials. 108

Another reason for the observed allometric trend in the cortical bone volume proportion could be related to the brittle nature of cortical bone. Under compressive loading cortical bone alone behaves as a brittle material (Yu, Zhao et al. 2011), and has a low modulus of toughness. This means that a bony structure made from cortical bone material alone is more likely to fail by brittle fracture without any prior warning.

A composite structure made from stiff cortical and more compliant cancellous bone will have an enhanced modulus of toughness and is more likely to fail under a ductile model of fracture, where the material undergoes serious deformation before failing.

Thus, in addition to conveying an increase in strength relative to weight, the incorporation of a higher proportion of cancellous bone in the skull may act to dampen high loads and reduce the likelihood of material failure.

109

5. DISCUSSION AND CONCLUSIONS

5.1 Biomechanical conservatism and allometric trends in extant felid skulls

The stress distribution and bite force results obtained from heterogeneous FEMs are broadly consistent with the proposition that the extant felid skull is biomechanically conservative, i.e., if the size differences are controlled across species, then biomechanical efficiency does not vary greatly between species as measured by the indicators used in the present study. Here I define biomechanical efficiency as an animal's ability to convert applied muscle force into bite force and the capacity to bear associated stresses.

In heterogeneous models scaled to the same surface area, maximum predicted bite forces do not vary greatly where uniform muscle forces are applied, and stress distributions are similar in these FEMs (fig. 3.1 and fig. 3.2). However, results of two- factor ANOVA (p = 0.056) derived from heterogeneous models do indicate that variation approaches significance where a uniform muscle force is applied to the scaled models. This may be an artefact of the small sample size and still more extensive studies will be needed to confirm this, but it may also be that the efficiency of biting in A. jubatus largely contributes to this finding.

The bilateral canine or killing bite of the cheetah is considerably more efficient than that of other taxa considered in the present study (fig.3.1), suggesting that it might achieve bite forces comparable to those of other cats using less muscle force. The results also suggest that if A. jubatus did apply the maximum muscle forces available to

110 it, then it would be operating within lower safety margins than other cats (Tab.3.1). As with all felids A. jubatus is essentially an ambush predator, but it is highly specialized for speed and acceleration to a degree beyond that of other cats. This greater biomechanical efficiency may have been selected for because it permits a reduction in the mass of musculature and bone required to achieve a comparable result. This contention is further supported by the results derived from the FEMs scaled to the same surface area and generating the same bite force output(fig.3.3). The jaw adductor force needed to produce same bite force at canines was minimum for A. jubatus. However the skull of A. jubatus was most stressed among other felids (fig. 3.4 and Tab. 3.2).

Differences in stress distribution between homogeneous and heterogeneous models are considerably greater in P. leo, where the skull contains a lower proportion of cortical bone (fig. 3.5). The skull of F. s. lybica had the highest proportion of cortical bone amongst extant felids considered in the present study (Tab. 2.2). With an increase in cortical bone volume proportion, the heterogeneous FE model of F. s. lybica tends to approach homogeneity in material distribution and hence more closely resembles the homogeneous model. The standard deviation values for the mean VM stress were relatively higher in the heterogeneous model of P. leo, possibly because of a lower proportion of cortical bone volume and hence greater deviation from material homogeneity (Tab. 3.3).

Support for the hypothesis that the biomechanical performance of the cranium and mandible of extant felids is conservative across species, with the possible exception of

A. jubatus, further strengthens the case for use of this taxon as a means to better

111 understand skull allometry, and three clear allometric trends are suggested by the results:

1. The relative proportion of cortical to cancellous bone tends to decrease with increasing size among felids. Thus, by volume, the skull of the smallest species, F. s. lybica is 84% cortical and 16% cancellous bone, whereas that of the largest, P. leo, is

59% cortical and 41% cancellous bone.

2. The finding that relative cortical bone volume decreases with increasing size is consistent with the proposal of Strait et al. (2010) that skull rigidity decreases with increasing size for primates, but not with the findings of Slater and van Valkenburgh

(2009) who concluded that stiffness increases with size for felids. Strait et al. (2010) drew their conclusions regarding trends in rigidity based on assessment of differences in cortical bone thickness and concluded that comparisons may be easiest to interpret where the species included encompass a limited size range. The FEM analysed by

Slater and van Valkenburgh (2009) did not distinguish between cortical and cancellous bone. Because the results show allometry regarding proportions of cortical and cancellous bone, and relatively high mean VM stress values in heterogeneous FEMs of specimens where the proportion of cortical bone is low, I posit that consideration of the role of cancellous bone may be important, at least where size differences between specimens are great.

3. Overall skull bone volume relative to surface area of the felid skull increases with increasing size. This is manifest in a consequent increase in cross-sectional area and hence second moment of inertia.

112

It is possible that changes in the relative proportions of cortical and cancellous bone and stress distributions identified here may be adaptive responses to differences in relative prey sizes between larger and smaller cat species. Larger felids tend to take relatively larger prey, often larger than their own body size (Radloff and Du Toit, 2004;

Hayward and Kerley, 2005) and relatively larger prey could pose significant mechanical challenges to the skull of the predator, inducing relatively higher stresses. However, I consider it unlikely that differences in relative prey size could fully explain the relationship between cortical/cancellous bone proportions and size in felids suggested by these results. This is because, with the possible exception of A. jubatus, the mid- sized cats included in the present study also commonly prey on taxa that are large relative to their own body masses, and, yet the proportion of cortical bone is consistently lower in mid-sized cats than in the larger P. leo. Moreover, the cheetah, which developed the highest stresses of any felid included in our analyses, forms a sister group with P. concolor which lies between lynx and leopard cat lineages (Johnson et al., 2006) and it seems unlikely that our findings are strongly influenced by phylogeny.

Results from the first beam analogy (Chapter 4.1) suggest that the underlying reasons for the allometry observed in the present study may be more fundamental, representing adaptation to accommodate the fact that stress and displacement increase with increasing size, whereas the failure point of bone does not.

Consequently, in order to limit the increasing stresses and deformations that accompany increased size, some adjustment needs to be made either in the distribution of material properties, the shape of the structure, or both.

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In engineering, optimally designed large structures maximise static stiffness for a given mass. Our composite beam analogy demonstrates that this may be achieved in the felid skull by adding a relatively thin layer of stiffer cortical bone material around an increased volume of more compliant and less dense cancellous bone material. There is an overall increase in skull bone volume relative to the surface area with increasing skull size and consequent increase in cross-sectional area and second moment of inertia, which effectively brings down stresses in the structure. Distribution of cortical bone away from the neutral axis further aids in reducing the stresses in the structure.

Results here further suggest that through such a mechanism considerable weight savings (fig.4.2) can be achieved without necessarily compromising on the effective stiffness.

I conclude that as vertebrate species become larger, increasing both skull bone volume relative to surface area and the proportion of cancellous bone in the skull is, at least in part, a response to universal physical challenges that accompany increasing size.

Although the proportion of cortical bone diminishes, the fact that it is distributed on internal and external skull surfaces is significant. The results of the present study suggest that skull rigidity may tend to decrease with increasing size. However, by increasing bone volume in the skull relative to the surface area, cross-sectional area and second moment of inertia are also increased, and this, at least in part, acts to bring down stresses in the skull. This suite of allometric traits may allow savings in mass while to some extent countering losses in effective stiffness. It may also reduce the burden on metabolism, partly because cortical bone is biologically more expensive to produce and maintain than cancellous bone (Parfitt, 1987), and partly because more

114 muscular effort is required to sustain or move heavier structures. Validation including a wide range of vertebrate taxa will clearly be needed to fully test this hypothesis, but if correct, then as suggested by Strait et al. (2010) future comparative FE analyses will need to consider these trends, at least where differences in body size are great.

5.2 Biomechanical comparisons between the extinct sabretooth predators Smilodon fatalis, Thylacosmilus atrox and the extant conical tooth felid Panthera pardus.

In absolute terms jaw muscle force recruitment was higher in S. fatalis compared to T. atrox and P. pardus across different gapes and bite force generated at the canines was observed to be highest in S. fatalis at different gape angles. For all species there was a decrease in the jaw muscle force recruitment with increasing gape resulting in a decline in the bite force output. However the decline in the bite force output was much steeper compared to the jaw muscle recruitment. This was possibly because with increasing gape, the out-lever of the jaw muscle fibres decreases and so does their efficiency in generating jaw adducting rotational forces around the TMJ. The decline was highest for T. atrox which only generated ~38N of canine bite force while recruiting ~636N of jaw adducting muscle force at maximum gape (fig. 3.10a). Mean

VM stresses in different regions of the skull were observed to be highest in P. pardus and lowest in T. atrox.

When the jaw adductor driven bite force was adjusted for body mass, the estimated jaw muscle force recruitment in T. atrox was ~24000N (Tab. 3.6). This is a highly unlikely value for jaw muscle recruitment for an animal of this size, as the

115 cross-sectional area of muscle fibres needed to generate this force would be disproportionate to the skull dimensions of T. atrox. Interestingly, the stresses generated in the skull for this muscle recruitment in T. atrox were comparable to that in S. fatalis, but were significantly higher compared to P. pardus (Tab. 3.6). These results are in broad agreement with previous studies (Bryant,1996; McHenry et al.,2007; Christiansen, 2010) suggesting that at wide gapes, the jaw adductor driven bite would have been less efficient in sabre-toothed predators when compared to extant conical toothed felids.

For the body mass scaled cervical muscles driven bite, the estimated cervical muscle force recruitment in T. atrox was ~1547N and in P. pardus was ~1076N, assuming a hypothetical value of 1750N for S. fatalis. Qualitative (fig.3.13) and quantitative (Tab.

3.7) estimation of VM stresses show that the skull of P. pardus experienced the highest stresses amongst the three species and was thus the least well-adapted to resist forces induced by cervical musculature. The stresses in T. atrox were lower than that in S. fatalis, which points towards the possibility that, for its body mass, the cervical musculature in T. atrox could have been relatively large compared to both S. fatalis and particularly P. pardus. This is in agreement with Turnbull's (1976) argument that the neck musculature in T. atrox was especially powerful, even compared to other sabre-toothed carnivores, and was capable of controlling the head and canine blades with precision, minimising the risk of transverse blow to its long slender upper canines.

In short, compared to P. pardus, at wide gapes the sabretooth predators S. fatalis and

T. atrox were not well adapted to generate high jaw muscle driven bite forces, but better adapted to sustain forces from the cervical musculature.

116

Another difference observed in the skull morphology of the sabretooths was in the centre point of sabre-arcs. The two centres of the sabre-arcs were only slightly posterior-ventrally displaced from the TMJ in S. fatalis, whereas distance from the TMJ in T. atrox was much greater and anterio-ventrally (fig.2.9).

If the upper canines were to resist only the rotational forces around TMJ, the centre of the arcs would be located at or very close to the TMJ in order for the principal forces to run through the entire length of canines. These findings suggest that the two sabretooth predators would have used their skull and canines differently in the killing bite. The fact that the centre of the tooth arcs in S. fatalis is very close to the TMJ is consistent with Akersten's (1985) thesis wherein the mandible is held against the prey to act as a brace while a combination of cervical and jaw closing musculature is applied to rotate the cranium around the TMJ.

This is not so for T. atrox. Here the much greater offset of the canine arc centre point from the TMJ in T. atrox suggests that considerable translation of the skull and canines, as well as rotation, was deployed in the killing bite. The centre point of the sabre-arcs would be at the mean of the dynamic centre of rotation in such a case, meaning that a range of cervical and postcranial muscles would have been deployed in order to maintain the transmission of principal forces through the entire length of canines as they were driven into the prey. I conclude that it was therefore unlikely that T. atrox used the mandible as a brace in a fashion similar to that hypothesised for S. fatalis.

In conclusion, results show that jaw muscle driven maximal bite forces at maximum gape in both sabretooths were relatively low and that the crania of both species were relatively well-adapted to resist the forces generated by head depressing cervical 117 muscles. This suggests that cervical muscles played an important role in developing sufficient reaction forces at the canines to penetrate the thick hides of large prey.

However, these similarities aside there are major differences between the marsupial and placental sabretooths.

Findings for S. fatalis are consistent with Akersten's canine shear bite hypothesis, which is essentially an extension of 'normal' biting in felids, but even relative to S. fatalis, the predicted jaw adductor driven bite forces at wide gapes in T. atrox are extremely low, while its cranium is even better adapted to resist neck driven forces.

Considered together with the evidence for a major translational component for the skull during a killing bite, I suggest that the mandible was not used as a brace and that the role of the jaw closing muscles in the killing bite was minimal. Thus, the killing bite of T. atrox was not a 'natural' extension of a 'normal' mammalian biting action, but a more highly specialised bite in which the neck muscles were dominant, with the mandible and jaw closing muscles playing no major role.

118

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APPENDICES

Appendix 1 MATLAB code for plotting the graph between gain coefficient ( β) and thickness ratio (n) for a two property composite beam.

% GAIN COEFFICIENT BETA & THICKNESS RATIO % Using three different sets of elasticity values (GPa) for cortical and cancellous bone Ecor(1)=13.7; Ecan(1)=0.689; Ecor(2)=20.7; Ecan(2) =14.8; Ecor(3)=1; Ecan(3)=1; N=0:0.001:6; % Creating a zero matrix beta = zeros(3,6001); for j =1:1:3; for i = 1:1:6001; beta(j,i)= ((Ecan(j)/Ecor(j))+(N(i)^3+3*N(i)^2+3*N(i)))/((N(i)+1)^3); end; end; plot(N,beta)

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Appendix 2 MATLAB code for plotting the graph between percentage weight savings (η) and thickness ratio (n) for a two property composite beam.

% WEIGHT SAVINGS NEETA & THICKNESS RATIO % Using three different sets of density values (Kg/m3) for cortical and cancellous bone dcor(1)=2092.30; dcan(1)=1056.84; dcor(2)=2596.15; dcan(2)=2142.30; dcor(3)=1; dcan(3)=1; N=0:0.001:6; % Creating a zero matrix neeta=zeros(3,6001); for j =1:1:3; for i = 1:1:6001; neeta(j,i)= (100/(N(i)+1))*(1- (dcan(j)/dcor(j))); end; end; plot(N,neeta)

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Strand7 inputs for the FEMs used in this study Appendix 3

African Wildcat (Felis silvestris lybica) - LACM 14480

Force /Muscle area (KPa)) 300 Cranial length (premaxilla-lambdoid),mm 90.03

One sided X-sectional area 324.06 Cranial width at Zygomatic arch, mm 66.23

Temporalis (mm^2) One sided X-sectional Mandible length (anterior dentary to condyle), 223.39 60.97 area - Masseter (mm^2) mm

Mandible width at condyles, mm 57.60 One sided Temporalis 97.22 Total skull bone volume (mm^3) 4.00E+04 muscle force (N) One sided Masseteric 67.02 Surface area (mm^2) 3.25E+04 muscle force(N) Total muscle force (N) 164.24 Number of brick elements in FE model 622821 Truss Truss elements on either Muscle pennation areas (mm^2) % area occupied Force/beam (N) diameter side of the skull (mm) Temporalis superficialis 1563.14 25.28 25 1.45 2.48 Temporalis profundus 1845.02 29.84 30 1.45 2.48 Temporalis zygomaticus 736.05 11.91 12 1.45 2.48 Masseter superficialis 438.66 7.10 7 2.39 3.19 Masseter profundus 814.02 13.17 13 2.39 3.19 Zygomatico mandibularis 498.28 8.06 8 2.39 3.19 Pterygoideus internus 203.98 3.30 3 balancing beam 3.50 Pterygoideus externus 83.20 1.35 2 balancing beam 3.50

Total 6182.34 100.00 100 Young's modulus Density Poisson’s Brick material properties# (GPa) (T/mm^3) ratio Cortical bone 13.7 2.09E-09 0.30 Cancellous bone 0.689 1.06E-09 0.30

Beam material Young's modulus Density Poisson’s Diameter properties (MPa) (T/mm^3) ratio (mm) Muscle trusses 1.00E-01 1.01E-09 0.30 table above Occipital beams 5.00 Cotyle beams 0.50 Condyle beams Structural steel (Strand7 material 0.50 library SS4100-1998) Hinge beam 5.00 Origin and Insertion 0.50 beams * Weijs and Hillen, 1985 #Cook et al., 1982 and Rho et al., 1995

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Appendix 4

Ocelot (Leopardus pardalis) - LACM 25127

Force /Muscle area (KPa)* 300 Cranial length (premaxilla - lambdoid),mm 99.86

One sided X-sectional area - 286.76 Cranial width at Zygomatic arch,mm 65.93 Temporalis (mm^2)

One sided X-sectional area - Mandible length (anterior dentary to 247.79 64.04 Masseter (mm^2) condyle),mm

Mandible width at condyles,mm 61.28 One sided Temporalis muscle 86.03 Total skull bone volume (mm^3) 4.330E+04 force (N) One sided Masseteric muscle 74.34 Surface area (mm^2) 3.740E+04 force(N) Total muscle force (N) 160.37 Number of brick elements in FE model 720497

Truss elements on Truss Muscle pennation areas (mm^2) % area occupied either side of the Force/beam (N) diameter skull (mm) Temporalis superficialis 2437.52 31.63 32 1.16 2.22 Temporalis profundus 2595.58 33.68 34 1.16 2.22 Temporalis zygomaticus 611.37 7.93 8 1.16 2.22 Masseter superficialis 715.19 9.28 9 3.38 3.79 Masseter profundus 470.47 6.11 6 3.38 3.79 Zygomatico mandibularis 532.88 6.92 7 3.38 3.79 Pterygoideus internus 229.20 2.97 3 balancing beam 3.50 Pterygoideus externus 113.47 1.47 1 balancing beam 3.50

Total Area 7705.68 100 100

Young's Density Poisson’s Brick material properties# modulus (GPa) (T/mm^3) ratio Cortical bone 13.7 2.09E-09 0.30 Cancellous bone 0.689 1.06E-09 0.30

Young's Density Poisson’s Beam material properties Diameter (mm) modulus (MPa) (T/mm^3) ratio Muscle trusses 1.00E-01 1.01E-09 0.30 table above Occipital beams 5.00 Cotyle beams 0.50 Structural steel (Strand7 material Condyle beams 0.50 library SS4100-1998) Hinge beam 5.00 Origin and Insertion beams 0.50 * Weijs and Hillen, 1985 #Cook et al., 1982 and Rho et al., 1995

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Appendix 5

Clouded leopard(Neofelis nebulosa) - USNM 282124

Force /Muscle area 300 Total cranial length (premaxilla - lambdoid),mm 192.87 (KPa)* One sided X-sectional area - 2613.19 Cranial width at Zygomatic arch,mm 136.18

Temporalis (mm^2) One sided X-sectional area - Masseter 1544.29 Mandible length (anterior dentary to condyle),mm 141.11 (mm^2) Mandible width at condyles,mm 123.04 One sided Temporalis 783.96 Total skull bone volume (mm^3) 2.95E+05 muscle force (N) One sided Masseteric 463.29 Surface area (mm^2) 1.21E+05 muscle force(N) Total muscle force (N) 1247.24 Number of brick elements in FE model 1097331

Truss Truss elements on Muscle pennation areas (mm^2) % area occupied Force/beam (N) diameter either side of the skull (mm) Temporalis superficialis 6681.07 22.65 23 11.36 6.94 Temporalis profundus 9621.85 32.61 32 11.36 6.94 Temporalis zygomaticus 4154.44 14.08 14 11.36 6.94 Masseter superficialis 3109.14 10.54 11 17.16 8.53 Masseter profundus 2417.66 8.19 8 17.16 8.53 Zygomatico mandibularis 2445.87 8.29 8 17.16 8.53 Pterygoideus internus 846.34 2.87 3 balancing beam 3.50 Pterygoideus externus 226.86 0.77 1 balancing beam 3.50

Total Area 29503.23 100 100

Young's Density Poisson’s Brick material properties# modulus (GPa) (T/mm^3) ratio Cortical bone 13.7 2.09E-09 0.30 Cancellous bone 0.689 1.06E-09 0.30

Beam material Young's Density Poisson’s Diameter (mm) properties modulus (MPa) (T/mm^3) ratio Muscle trusses 1.00E-01 1.01E-09 0.30 table above Occipital beams 5.00 Cotyle beams 0.50 Condyle beams Structural steel (Strand7 material 0.50 library SS4100-1998) Hinge beam 5.00 Origin and Insertion 0.50 beams * Weijs and Hillen, 1985 #Cook et al., 1982 and Rho et al., 1995

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Appendix 6

Cheetah (Acinonyx jubatus) - FMNH 29635

Total cranial length (premaxilla - Force /Muscle area (KPa)* 300 195.66 lambdoid),mm

One sided X-sectional area - 1709.87 Cranial width at Zygomatic arch,mm 135.15 Temporalis (mm^2)

One sided X-sectional area - Mandible length (anterior dentary to 1195.72 136.02 Masseter (mm^2) condyle),mm

Mandible width at condyles,mm 116.56 One sided Temporalis muscle 512.96 Total skull bone volume (mm^3) 3.26E+05 force (N) One sided Masseteric muscle 358.72 Surface area (mm^2) 1.42E+05 force(N) Total muscle force (N) 871.68 Number of brick elements in FE model 807235

Truss elements on Truss Muscle pennation areas (mm^2) % area occupied either side of the Force/beam (N) diameter skull (mm) Temporalis superficialis 6378.27 22.01 22 7.77 5.74 Temporalis profundus 9653.30 33.32 33 7.77 5.74 Temporalis zygomaticus 3289.33 11.35 11 7.77 5.74 Masseter superficialis 3012.64 10.40 11 12.81 7.37 Masseter profundus 2153.78 7.43 8 12.81 7.37 Zygomatico mandibularis 2699.64 9.32 9 12.81 7.37 Pterygoideus internus 1110.68 3.83 4 balancing beam 3.50 Pterygoideus externus 677.72 2.34 2 balancing beam 3.50

Total Area 28975.36 100.00 100

Young's modulus Density Poisson’s Brick material properties# (GPa) (T/mm^3) ratio Cortical bone 13.7 2.09E-09 0.30 Cancellous bone 0.689 1.06E-09 0.30

Young's modulus Density Poisson’s Beam material properties Diameter (mm) (MPa) (T/mm^3) ratio Muscle trusses 1.00E-01 1.01E-09 0.30 table above Occipital beams 5.00 Cotyle beams 0.50 Structural steel (Strand7 material Condyle beams 0.50 library SS4100-1998) Hinge beam 5.00 Origin and Insertion beams 0.50 * Weijs and Hillen, 1985 #Cook et al., 1982 and Rho et al., 1995

137

Appendix 7

Puma (Puma concolor)- LACM 87430

Force /Muscle area 300 Total cranial length (premaxilla - lambdoid),mm 205.54 (KPa)*

One sided X-sectional area - 2348.52 Cranial width at Zygomatic arch,mm 142.88 Temporalis (mm^2)

One sided X-sectional area - 1445.69 Mandible length (anterior dentary to condyle),mm 137.87 Masseter (mm^2)

Mandible width at condyles,mm 129.95 One sided Temporalis 704.56 Total skull volume (mm^3) 3.98E+05 muscle force (N) One sided Masseteric 433.71 Surface area (mm^2) 1.46E+05 muscle force(N) Total muscle force (N) 1138.26 Number of brick elements in FE model 895835

Truss Truss elements on Force/beam Muscle pennation areas (mm^2) % area occupied diameter either side of the skull (N) (mm) Temporalis superficialis 6666.10 19.98 20 10.07 6.54 Temporalis profundus 11544.20 34.60 34 10.07 6.54 Temporalis zygomaticus 5311.90 15.92 16 10.07 6.54 Masseter superficialis 3327.70 9.97 10 16.06 8.26 Masseter profundus 2552.60 7.65 8 16.06 8.26 Zygomatico mandibularis 2894.45 8.67 9 16.06 8.26 balancing Pterygoideus internus 803.00 2.41 2 3.50 beam balancing Pterygoideus externus 269.12 0.81 1 3.50 beam Total Area 33369.07 100 100

Young's modulus Density Poisson’s Brick material properties# (GPa) (T/mm^3) ratio Cortical bone 13.7 2.09E-09 0.30 Cancellous bone 0.689 1.06E-09 0.30

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Appendix 8

Leopard (Panthera pardus) - LACM 11704

Force /Muscle area (KPa)* 300 Total cranial length (premaxilla - lambdoid),mm 238.2

One sided X-sectional area - 3002.29 Cranial width at Zygomatic arch,mm 154.75 Temporalis (mm^2) One sided X-sectional area - Mandible length (anterior dentary to 1580.24 160.31 Masseter (mm^2) condyle),mm Mandible width at condyles,mm 137.82 One sided Temporalis 900.69 Total skull bone volume (mm^3) 5.43E+05 muscle force (N) One sided Masseteric 474.07 Surface area (mm^2) 1.65E+05 muscle force(N) Total muscle force (N) 1374.76 Number of brick elements in FE model 782498

Truss Truss elements on Muscle pennation areas (mm^2) % area occupied Force/beam (N) diameter either side of the skull (mm) Temporalis superficialis 6140.17 15.40 15 13.86 7.67 Temporalis profundus 14294.81 35.85 36 13.86 7.67 Temporalis zygomaticus 5657.99 14.19 14 13.86 7.67 Masseter superficialis 4902.18 12.30 12 16.93 8.48 Masseter profundus 3326.70 8.34 8 16.93 8.48 Zygomatico mandibularis 3170.73 7.95 8 16.93 8.48 Pterygoideus internus 1782.84 4.47 5 balancing beam 3.50 Pterygoideus externus 594.98 1.49 2 balancing beam 3.50

Total Area 39870.40 100 100

Young's Density Poisson’s Brick material properties# modulus (GPa) (T/mm^3) ratio Cortical bone 13.7 2.09E-09 0.30 Cancellous bone 0.689 1.06E-09 0.30

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Appendix 9

Lion (Panthera leo)- MVZ 117849

Force /Muscle area (KPa)* 300 Total cranial length (premaxilla - lambdoid),mm 391.74

One sided X-sectional area - 6778.49 Cranial width at Zygomatic arch,mm 263.74 Temporalis (mm^2)

One sided X-sectional area - Mandible length (anterior dentary to 5110.09 273.34 Masseter (mm^2) condyle),mm Mandible width at condyles,mm 215.67 One sided Temporalis 2033.55 Total skull bone volume (mm^3) 2.73E+06 muscle force (N) One sided Masseteric 1533.03 Surface area (mm^2) 4.26E+05 muscle force(N) Total muscle force (N) 3566.57 Number of brick elements in FE model 985522

Truss Truss elements on Muscle pennation areas (mm^2) % area occupied Force/beam (N) diameter either side of the skull (mm) Temporalis superficialis 24184.24 23.39 23 30.35 11.35 Temporalis profundus 32806.53 31.73 32 30.35 11.35 Temporalis zygomaticus 12261.97 11.86 12 30.35 11.35 Masseter superficialis 11611.45 11.23 11 51.10 14.73 Masseter profundus 9485.95 9.17 9 51.10 14.73 Zygomatico mandibularis 10414.34 10.07 10 51.10 14.73 Pterygoideus internus 1613.37 1.56 2 balancing beam 3.50 Pterygoideus externus 1026.46 0.99 1 balancing beam 3.50

Total Area 103404.31 100 100

Young's Density Poisson’s Brick material properties# modulus (GPa) (T/mm^3) ratio Cortical bone 13.7 2.09E-09 0.30 Cancellous bone 0.689 1.06E-09 0.30

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Appendix 10

Smilodon fatalis - FMNH P 12418

Total cranial length (premaxilla - Force /Muscle area (KPa)* 300 310.74 lambdoid),mm One sided X-sectional area - 4481.28 Cranial width at Zygomatic arch,mm 196.91 Temporalis (mm^2) One sided X-sectional area - Mandible length (anterior dentary to 3796.17 203.98 Masseter (mm^2) condyle),mm Mandible width at condyles,mm 163.28 One sided Temporalis Cranial bone volume (mm^3) 1291913 1344.38 muscle force (N) Mandible bone volume (mm^3) 254445 One sided Masseteric 1138.85 Surface area (mm^2) 3.26E+05 muscle force(N) Total muscle force (N) 2483.24 Number of brick elements in FE model 1643322

Truss elements Truss Jaw muscle pennation areas (mm^2) % area occupied on either side Force/beam (N) diameter of the skull (mm) Temporalis superficialis 8175.48 16.78 17 19.77 9.16 Temporalis profundus 18119.12 37.18 37 19.77 9.16 Temporalis zygomaticus 6950.105 14.26 14 19.77 9.16 Masseter superficialis 4988.14 10.24 10 39.27 6.46 Masseter profundus 5030.212 10.32 11 39.27 6.46 Zygomatico mandibularis 3890.397 7.98 8 39.27 6.46 Pterygoideus internus 1061.358 2.18 2 balancing beam 7.00 Pterygoideus externus 515.6937 1.06 1 balancing beam 7.00 Truss elements on Truss diameter Head-depressing muscle pennation areas (mm^2) either side of the Force/beam (N) (mm) skull Sternomastoideus 1926.791 40 25 5 Obliquus capitus 6688.694 30 25 5 Young's modulus Brick material properties# Density (T/mm^3) (GPa) Cranium and Mandible 21.734 1.86E-09 Dentine 32.704 2.526E-09 Enamel 38.575 2.861E-09

Young's modulus Beam material properties Density (T/mm^3) Diameter (mm) (MPa) Muscle trusses 1.00E-01 1.01E-09 table above Occipital beams 5.00 Cotyle beams 0.50 Structural steel (Strand7 material library Condyle beams SS4100-1998) 0.50 Hinge beam 5.00 Origin and Insertion beams 0.50 * Weijs and Hillen, 1985 #Cook et al., 1982 and Rho et al., 1995

141

Appendix 11

Thylacosmilus atrox - FMNH P 14531

Total cranial length (premaxilla - Force /Muscle area (KPa)* 300 219.141 lambdoid),mm One sided X-sectional area - 1717.47 Cranial width at Zygomatic arch,mm 139.305 Temporalis (mm^2) One sided X-sectional area - Mandible length (anterior dentary to 1748.08 192.851 Masseter (mm^2) condyle),mm Mandible width at condyles,mm 132.842 One sided Temporalis Cranial bone volume (mm^3) 853363 515.24 muscle force (N) Mandible bone volume (mm^3) 285662 One sided Masseteric 524.42 Surface area (mm^2) 2.66E+05 muscle force(N) Total muscle force (N) 1039.67 Number of brick elements in FE model 1643322

Truss elements on Truss Force/beam Jaw muscle pennation areas (mm^2) % area occupied either side of the diameter (N) skull (mm) Temporalis superficialis 4013.01 12.96 13 9.37 6.31 Temporalis profundus 9091.46 29.36 29 9.37 6.31 Temporalis zygomaticus 4055.56 13.10 13 9.37 6.31 Masseter superficialis 5424.58 17.52 17 12.79 7.37 Masseter profundus 4527.18 14.62 15 12.79 7.37 Zygomatico mandibularis 2722.71 8.79 9 12.79 7.37 balancing Pterygoideus internus 2.53 3 7.00 784.875 beam balancing Pterygoideus externus 1.11 1 7.00 343.757 beam Truss elements on Truss Head-depressing muscle pennation areas (mm^2) either side of the Force/beam (N) diameter skull (mm) Sternomastoideus 786.45 40 25 5 Obliquus capitus 2897.2 30 25 5 Young's modulus Brick material properties# Density (T/mm^3) (GPa) Cranium and Mandible 21.734 1.86E-09 Dentine 32.704 2.526E-09 Enamel 38.575 2.861E-09

Young's modulus Beam material properties Density (T/mm^3) Diameter (mm) (MPa) Muscle trusses 1.00E-01 1.01E-09 table above Occipital beams 5.00 Cotyle beams 0.50 Structural steel (Strand7 material library Condyle beams 0.50 SS4100-1998) Hinge beam 5.00 Origin and Insertion beams 0.50 * Weijs and Hillen,1985 #Cook et al., 1982 and Rho et al., 1995

142

Appendix 12

Leopard (Panthera pardus) - LACM 11704 , remodelled for comparisons with sabretooths using protocols used for modelling sabretooths.

Total cranial length (premaxilla - Force /Muscle area (KPa)* 300 216.974 lambdoid),mm One sided X-sectional area - 3002.29 Cranial width at Zygomatic arch,mm 155.839 Temporalis (mm^2) One sided X-sectional area - Mandible length (anterior dentary to 1580.24 159.145 Masseter (mm^2) condyle),mm Mandible width at condyles,mm 137.753 One sided Temporalis Cranial bone volume (mm^3) 358241 900.69 muscle force (N) Mandible bone volume (mm^3) 112470 One sided Masseteric muscle 474.07 Surface area (mm^2) 1.65E+05 force(N) Total muscle force (N) 1374.76 Number of brick elements in FE model 1631130 Truss elements on Truss Jaw muscle pennation areas (mm^2) % area occupied either side of the Force/beam (N) diameter skull (mm) Temporalis superficialis 12296.8 32.60702308 33 14.39 7.82 Temporalis profundus 6117.92 16.22268872 16 14.39 7.82 Temporalis zygomaticus 4920.82 13.04837774 13 14.39 7.82 Masseter superficialis 5589.89 14.82252881 15 23.53 9.99 Masseter profundus 4172.99 11.06538134 11 23.53 9.99 Zygomatico mandibularis 3065.64 8.129057499 8 23.53 9.99 Pterygoideus internus 1087.61 2.883979928 3 balancing beam 7.00 Pterygoideus externus 460.451 1.220962884 1 balancing beam 7.00 Truss elements on Head-depressing muscle pennation areas Truss diameter either side of the Force/beam (N) (mm^2) (mm) skull Sternomastoideus 1273.25 40 25 5 Obliquus capitus 2672.62 30 25 5 Young's Brick material properties# Density (T/mm^3) modulus (GPa) Cranium and Mandible 21.734 1.86E-09 Dentine 32.704 2.526E-09 Enamel 38.575 2.861E-09 Young's Beam material properties Density (T/mm^3) Diameter (mm) modulus (MPa) Muscle trusses 1.00E-01 1.01E-09 table above Occipital beams 5.00 Cotyle beams 0.50 Structural steel (Strand7 material library Condyle beams 0.50 SS4100-1998) Hinge beam 5.00 Origin and Insertion beams 0.50 * Weijs and Hillen,1985 #Cook et al., 1982 and Rho et al., 1995

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