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Biomechanics of the mammalian jaw 2001

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BIOMECHANICS OF THE MAMMALIAN JAW by FUTANG ZHANG B Med (Dent) , Shandong Medical Univers i ty, 1988 M Med (Dent) , Shandong Medical Univers i ty, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Depar tment of Oral Health Sc iences) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA 28 August 2001 © Futang Zhang , 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada ABSTRACT Mammal ian jaw b iomechanics are not fully understood. They can be studied by dif ferent approaches, including, but not l imited to, jaw cross- sect ional measurements , stress and strain analys is , and computer model ing. Five studies compr ise this thesis: In the first study, three cross-sect ions were examined with high- resolut ion computed tomography (CT) in the human jaw. A l though the cross-sect ional areas var ied among the three locat ions, the cross- sect ional masses were homogeneous, suggest ing uni form shear rigidity. Despite s imi lar i t ies in shape among the three cross-sect ions, cort ical bone th ickness and density var ied, indicating regional loading condit ions may be determinants in the cross-sect ional des ign. The second study tested two hypotheses. The first postulated that symphysea l stress and strain are s imi lar in pigs and humans . The second proposed that the symphysea l or ientat ion in the pig j aw keeps the stress and strain level within a funct ional range. Individual musc le lever a rms , cross-sect ional moments of inert ia, symphysea l centro ids, and mean muscle tens ions were considered in the pig and human jaws. The est imated stress and strain levels were marked ly s imi lar for pigs and humans with their symphyses in normal " funct iona l " or ientat ions. However, the est imated strain for the pig mandib le was h igher than the reported max imum funct ional strain when the symphys i s was in a s imulated "upr ight" or ientat ion. In the fol lowing two studies, pig and human jaw mass propert ies were est imated f rom CT scans. The mass and geometr ic centers were close in both pig and human mandib les, and consistent ly located at the last molar region, suggest ing imaging methods reveal ing 3D shape alone can be used to est imate mass propert ies. Jaw mass and moment s of inertia could also be predicted by s imple d imens iona l measurements of the jaw. Dynamic model ing of individual jaws is, therefore, possible. The sensit iv i ty of mass propert ies in dynamic model ing was conf i rmed in a previously publ ished dynamic human jaw mode l . In the final study, the respect ive mass propert ies were est imated by CT for each half of a pig jaw split into two halves, and rejoined with a rigid link. Dorsoventra l shear, media l and lateral t ransverse bendings were predicted in the pig jaw symphys i s dur ing a uni lateral chewing stroke. The predict ion supported the hypothes is that the pig symphysea l or ientat ion is essent ia l to keep symphysea l s t resses and stra ins within funct ional levels. TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iv LIST OF FIGURES viii LIST OF TABLES xi ACKNOWLEDGEMENTS xiii 1 Introduct ion 1 1.1 Introduct ion to the thesis 1 1.2 Mandibular form and funct ion 3 1.2.1 Form and funct ion of the mamma l i an jaw 3 1.2.2 Research models 4 1.3 Mammal ian jaw b iomechanics 5 1.3.1 Material propert ies of bone 6 1.3.1.1 Stress and strain under normal load 6 1.3.1.2 Shear ing stress and strain 9 1.3.1.3 Strength and sti f fness 10 1.3.2 B iomechanica l design of the mandibu lar corpus 10 1.3.2.1 Bending 11 1.3.2.2 Tors ion 13 1.3.2.3 Mandibular corpus cross-sect ional form 16 1.3.2.4 Cross-sect ional measurements and the i r s ignif icance... . 18 1.3.2.5 Mandibular corpus loading and stress and strain 26 1.3.3 B iomechanica l s ignif icance of the jaw symphys i s 32 1.3.3.1 Symphysea l form and function 32 1.3.3.2 Symphysea l loading and stress and strain 33 1.3.4 Model ing jaw b iomechanics 37 1.3.4.1 Stat ic jaw model ing 38 1.3.4.2 Dynamic jaw model ing 38 1.4 Final c omment 40 2 S ta tement of the problem 42 Chapter 1 46 3 Cross-sect ional b iomechanics of the human mandib le 47 3.1 Abstract 47 3.2 Introduct ion 48 3.3 Materials and methods 53 3.3.1 Computed tomography 53 3.3.2 Image processing 54 3.3.3 Cross-sect ional measurements 56 3.3.4 Stat ist ical Analys is 60 3.4 Results 61 3.4.1 Binary cross-sect ions 61 3.4.2 Graysca le cross-sect ions 61 3.4.3 Cort ical th ickness, densi ty and rigidity index 64 3.4.4 Bone isometry 66 3.4.5 Predict ions by ideal models 66 3.5 Discussion 66 3.5.1 Error of method 66 3.5.2 Signi f icance of cross-sect ional measurements 68 3.5.3 Cross-sect ional design of the human mandib le 72 3.5.3.1 The molar region 73 3.5.3.2 The symphys i s 77 3.5.3.3 The canine region 77 3.5.4 Model ing the mandibu lar corpus 78 3.6 S u m m a r y and Conclus ions 79 Chapter II 82 4 Symphysea l Mechanics in Pig and Human Mandibles 83 4.1 Abstract 83 4.2 Introduct ion 84 4.3 Materials and methods 92 4.3.1 Computed tomography 92 4.3.2 Image convers ion and preparat ion 93 4.3.3 Cross-sect ional measurements 94 4.3.4 Stress and strain calculat ions 96 4.3.5 Stat ist ical analys is 101 4.4 Results 101 4.4.1 Cross-sect ional measurements 102 4.4.2 Symphysea l stress and strain 102 4.5 Discussion 105 4.5.1 Symphysea l cross-sect ions 105 4.5.2 St resses and stra ins. 107 4.5.3 Final Commen t on Wishboning 109 4.6 Conc lus ions 110 Chapter III 113 5 Mass Propert ies of the Pig Mandible 114 5.1 Abstract 114 5.2 Introduct ion 115 5.3 Materials and methods 117 5.3.1 Material preparat ion 117 5.3.2 CT scanning 118 5.3.3 Image processing 118 5.3.4 Mass propert ies calculat ion 119 5.4 Results 124 5.5 Discussion 129 Chapter IV 143 6 Mass Propert ies of the Human Mandible and the i r Functional Signi f icance 144 6.1 Abstract 144 6.2 Introduct ion 145 6.3 Materials and methods 146 6.3.1 Mass property est imat ion 146 6.3.2 Predict ion of mass and moments of inertia 148 6.3.3 Jaw model 148 6.4 Results 151 6.4.1 Cal ibrat ion 151 6.4.2 Est imated masses and mean bone densi ty 151 6.4.3 Mass and geometr ic centers 154 6.4.4 Moments of inertia 154 6.4.5 Mass and moments of inertia predict ions 154 6.4.6 Model sensit iv i ty to mass propert ies 158 6.5 Discussion 164 Chapter V 170 7 Dynamic mechanics in the pig mandibu lar symphys i s 171 7.1 Abstract 171 7.2 Introduct ion 172 7.3 Materials and methods 175 7.3.1 Model generat ion 175 7.3.2 S imulat ions 179 7.4 Results 179 7.4.1 Incisor point mot ion 179 7.4.2 Muscle tension 182 7.4.3 Forces at the artif icial food bolus, tooth and jo ints 184 7.4.4 Tr i-axial symphysea l forces 184 7.4.5 Tr i-axial symphysea l torques 187 7.5 Discussion 190 7.5.1 The model 190 7.5.2 Predicted shear 190 7.5.3 Predicted t ransverse bending 191 7.6 S u m m a r y 192 8 Genera l d iscuss ion 194 8.1 Jaw cross-sect ional mechanics 195 8.1.1 Ideal cross-sect ional models 196 8.1.2 Open vs. c losed jaw models 197 8.1.3 Eff iciency - how does bone structure meet mechanica l and funct ional needs? 197 8.1.4 Bone mechanica l propert ies, densi ty and CT numbers 198 8.1.5 Bone mechanica l propert ies, bone compos i t ion and organizat ion 200 8.2 Jaw mass propert ies 201 8.2.1 Signi f icance of bone density, jaw d imens ions , and j aw mass propert ies 201 8.2.2 Signi f icance of mass propert ies in dynamic model ing 202 8.2.3 Signi f icance of the imaging method in dynamic model ing . 203 8.3 Symphysea l b iomechanics 203 8.3.1 Stress and strain s imi lar i ty 203 8.3.2 Symphysea l or ientat ion 204 8.4 Bone growth, model ing, and the mechanica l env i ronment 205 8.5 Signi f icance of dynamic models for s tudy ing jaw b iomechanics 207 8.6 Personal comment on comput ing 207 8.7 Future direct ions 208 9 Genera l s ummary and conclusions 211 10 Bib l iography 212 11 Append ix 222 11.1 RIC - Raw Image Conver ter 222 11.1.1 Purpose of this program 222 11.1.2 Main interface 222 11.1.3 Core a lgor i thm 223 11.2 Ca l image - Calculate Image 226 11.2.1 Purpose of this program 226 11.2.2 Main interface 227 11.2.3 Core a lgor i thms 227 11.2.3.1 C++ code for mass propert ies calculat ion 228 11.2.3.2 C++ code for cross-sect ional measurements 231 12 Publ icat ions 235 12.1 Recent publ icat ions 235 12.2 Prospect ive publ icat ions 235 LIST OF FIGURES Figure 1.1 Beam bending theory 12 Figure 1.2 Best designs for resisting bending 14 Figure 1.3 Stress pattern of a cross-section under torsion 15 Figure 1.4 A better design for resisting torsion is to move material from the center to the outer surface to create a circular hollow section 17 Figure 1.5 Diagram of a cross-section with variable wall thickness under torsional load 22 Figure 1.6 Cross-section of a thin-walled tube 30 Figure 1.7 A multi-cell thin-walled tube can be modeled as an extension of a single-cell thin-walled tube 31 Figure 1.8 A growth series of Macaca fascicularis mandibles illustrating ontogenetic changes in symphyseal curvature 34 Figure 3.1 Suggested mechanism of wishboning in the human mandible 51 Figure 3.2 Diagram showing the three reslicing planes seen from above 55 Figure 3.3 Three typical cross-sections at the canine, the symphysis and the first molar 57 Figure 3.4 Definition of regional cortical thickness 58 Figure 3.5 Possible stress distribution caused by tooth-loading 75 Figure 3.6 Equivalence of canine, first molar and symphysis cross-sections of an edentulous mandible 76 Figure 3.7 Torsion flow in a section containing a tooth 80 Figure 4.1 A growth series of Sus scrofa mandibles illustrating ontogenetic changes in symphysis, and for comparison, an adult modern human mandible 90 Figure 4.2 Typical cross-sections for pig and human mandibular symphyses 95 Figure 4.3 Diagram of the median axis method used to construct the circle containing the radius of curvature of the flexure 97 Figure 4.4 Suggested mechanism of wishboning and patterns of stress in the pig mandible 111 Figure 5.1 Diagram of the coordinate system used to express moments of inertia of the jaw 123 Figure 5.2 Lateral and horizontal views of a voxel-based, reconstructed dry mandible with calculated mass center and geometric center locations 127 Figure 5.3 Distribution of mass center locations relative to the mid-condylar point infradentale line for 10 pig mandibles 128 Figure 5.4 Estimated mass with bone marrow, Estimated mass, wet weight and dry weights plotted against mass descriptor 132 Figure 5.5 Moments of inertia with marrow plotted against moment of inertia descriptors . 133 Figure 6.1 Coordinate system used to express moments of inertia 149 Figure 6.2 Calibration curve for the phantom used in the present study 152 Figure 6.3 Estimated mass with marrow, estimated mass, wet weight, and dry weight plotted against mass descriptor 157 Figure 6.4 Moments of inertia plotted against the moment of inertia descriptors 160 Figure 6.5 Incisor-point motion from tooth contact to the jaw's rest position, plotted against time. The curves represent mass properties used by Koolstra and van Eijden (1995) and median values from the present study 161 Figure 6.6 Incisor-point motion from tooth contact to the jaw's rest position, plotted against time. The curves represent mass center locations 10 mm / anterior and 10 mm posterior to the original 162 Figure 6.7 Incisor-point motion during maximum jaw opening, plotted against time. The curves represent mass properties used by Koolstra and van Eijden (1995) and median values from the present study. 163 Figure 7.1 Frontal and lateral views of the A D A M S wireframe dynamic model. 178 Figure 7.2 Conventions used to express symphyseal forces and torques 180 Figure 7.3 Incisor point motion during a simulated right-sided, 0.5 second chewing cycle 181 Figure 7.4 Muscle tensions expressed in time during simulated right-sided chewing 183 Figure 7.5 Predicted forces expressed in time 185 Figure 7.6 Tri-axial symphyseal forces expressed in time 186 Figure 7.7 Tri-axial symphyseal torques expressed in time 188 Figure 7.8 Joint reaction forces and bolus reaction force when artificial food bolus is being crushed 193 LIST OF TABLES Table 3.1 Area, moment of area, bending and cortical indices for the binary total and cortical cross-sections 62 Table 3.2 Area and mass, MGSV, second moment of area and second moment of mass, bending and mass bending indices, and cortical and mass cortical indices of the grayscale total and cortical cross-sections 63 Table 3.3 MGSV, cortical thickness and cortical rigidity index of the lingual, facial and basal cortices 65 Table 3.4 Ratios between area and moments of inertia predicted elliptical models and the respective actual measurements 67 Table 4.1 Area, mass, MGSV, second moments of area, second moments of mass, area and mass bending indices, area and mass cortical indices, and t- test results for pig and human jaw symphyses 103 Table 4.2 Estimated stresses and strains for pig mandibles with their symphyses in their normal, and simulated upright orientations, and for human mandibles 104 Table 5.1 Descriptive statistics of the dry and wet weights, estimated masses without and with bone marrow, calculated mean bone densities without and with bone marrow, and the ratios between these variables 126 Table 5.2 Differences between calculated mass center and geometric center without and with bone marrow, and the anteroposterior anatomical locations of the mass center 130 Table 5.3 Moments of inertia without and with bone marrow and the ratios between the two groups 131 Table 5.4 Error distribution in landmark definition for 10 repeated measurements in pig jaw #10 137 Table 6.1 Measured jaw weights, estimated masses, and calculated mean bone densities and M B D with marrow, and the ratios between these variables. 153 Table 6.2 Differences between geometric and mass centers without marrow, and with marrow ; 155 Table 6.3 Moments of inertia without marrow, moments of inertia with marrow and the ratios between them 156 Table 6.4 Coefficients of determination for power function curve fits between estimated mass, estimated mass with marrow, wet and dry weights, moments of inertia, and all descriptors for dentate human mandibles 159 ACKNOWLEDGEMENTS I am extremely grateful to Dr. Alan Gordon Hannam for his scientific guidance and inspiration that have made this project such a rewarding experience. One could not ask for a better mentor and teacher, and I only hope that some of his brilliance has rubbed off onto me. I would also like to thank Drs. Virginia Diewert and Colin Price for their invaluable instruction, constructive criticism and guidance throughout my candidacy, from the initial planning of the proposal to the final stages of thesis compilation. I want to express my sincere thanks to Drs. Christopher Charles Peck and Geerling Evert Jan Langenbach for their scientific assistance. I appreciate the availability of Dr. Susan W. Herring as the academic consultant during my whole study period. I am also grateful to Dr. Douglas Waterfield for his valuable advice towards my graduate studies. I appreciate all the helps from the department, especially those from Ms. Vicki Beretanos Koulouris, the wonderful graduate secretary. I have been very lucky to enlist the expertise of Ms. Joy Scott, who has provided direction and support for many of the computational aspects of my experiments. I have been very fortunate to have family and friends who have always been a constant source of encouragement and support. Special thanks go to Nancy and Rick, whom I have relied upon very much. Finally, I would like to thank those who have developed today's computers. This thesis would not exist without this technology. 1 INTRODUCTION 1.1 INTRODUCTION TO THE THESIS The complex i ty of the b iomechanica l behavior of the mast icatory sys tem chal lenges researchers. During the power st roke of mast icat ion, the jaw interacts with var ious muscle forces, grav i ty, react ion forces f rom the occlusion and the temporomand ibu la r jo ints. All these forces demand a mechanica l ly opt imized sys tem. Unfortunately, the interact ions are not fully understood. Our knowledge of the jaw b iomechanics is l imited, and our current understanding is not based so much on cause-effect re lat ionships, but most ly der ived f rom associat ions, interpretat ions and assumpt ions . For a load-bear ing sys tem like the jaw, stress and strain analys is is a key method to comprehend mechanica l s ignif icance, and in vivo surface strain analys is of cortical bone reflects the funct ional env i ronment (Bouvier and Hylander, 1981a, 1981b; Daegl ing and Hylander, 1997, 2000; Dechow and Hylander, 2000; Herr ing e t al., 1996; Herr ing and Teng, 2000; Hylander, 1977, 1979a, 1984, 1986; Hy lander e t a / . , 1987, 1998; Hylander and Crompton, 1986; Hylander and Johnson, 1997a, b; Liu and Herr ing, 2000; Mikic and Carter, 1995; Ravosa e t a/., 2000) . Understandably, this methodology has l imitat ions, and is not avai lable in l iving humans . A l ternat ive approaches not direct ly involv ing l iving mater ia l include the b iomechanica l analys is of jaw cross-sect ions based on physical and engineer ing principles (Daegl ing, 1989, 1993; Daegl ing et a/., 1992; Daegl ing and Gr ine, 1991; Daegl ing and Hylander, 1998; Daegl ing and Jungers, 2000) , and stress and strain est imat ion der ived f rom jaw cross- sect ional and musculoske leta l informat ion (Hylander, 1985; V inyard and Ravosa, 1998) . Mathemat ica l model ing (both stat ic and dynamic) founded on jaw morphometr i c and funct ional data (Curt is et al., 1999; Hannam, 1994; Hannam et al., 1997; Koolstra et al., 1988; Koolstra and van Ei jden, 1995, 1997a, b; Korioth etal., 1992; Kor ioth and Hannam, 1994a; Langenbach and Hannam, 1999; Ng, 1994; Peck etal., 2000) also offers insight into jaw biomechanica l behavior, des ign, and stress and strain condit ions. Though numerous invest igat ions have been conducted in the above areas, they are incomplete. For example , cross-sect ional b iomechanica l ana lyses have not taken regional bone densit ies into account, and have been l imited to molar sect ions only; stress and stra in est imat ions have been based on scal ing factors, and thus cannot provide comparab le data to in vivo strain studies; the mass propert ies (i.e., mass , mass center, and moments of inertia) incorporated by dynamic j aw models have been crude and gener ic; and dynamic models have not been used to predict jaw internal forces and torques, which could lead to better understanding of the j aw loading condit ions. This thes is probed these areas in order to cast more l ight on t hem. Three d imens iona l (3D) computed tomography (CT) was employed to analyze the cross-sect ional b iomechanics in the human mandib le . Focus was directed on whether the human mandibu lar corpus cross-sect ions at different locations behave homogeneous ly , and how regional var iat ions in bone density and cortical th ickness within each cross-sect ion are related to regional loading condit ions. Second, the jaw's cross-sect ional data , plus j aw musc le and morphometr i c data , were used to predict the stress and strain magni tudes along the l ingual surface of the mandibu lar symphys i s in both pigs and humans. Attent ion was placed on the hypotheses that the pig jaw symphys i s is uniquely or iented so as to increase its abi l i ty to resist high wishboning stress and stra in, and that stress and strain s imi lar i ty is mainta ined across mamma l i an orders (in this case pigs and humans) . Third, 3D CT was used aga in , this t ime to est imate the jaw's mass and inertial propert ies in pigs and humans . One issue addressed here was whether these mass propert ies might be est imated or predicted with less invasive methods than CT so that this highly radiat ion-dependent method could be replaced with other morphologica l tools. A second issue was assess ing the s ignif icance mass propert ies have in dynamic model ing. Finally, the method was used to est imate mass propert ies in a pig jaw artif icial ly segmented into two halves, and later jo ined with a rigid link in a dynamic work ing model of the pig mast icatory sys tem. By analyz ing the forces and torques t ransmit ted through the symphys i s link, it was possible to predict the funct ional demands placed at the pig symphys i s dur ing a uni lateral chewing stroke. In summary , the work introduced severa l new exper imenta l approaches appl icable to the study of human and other mamma l i an jaw b iomechanics, and provided some addit ional insight into the structural and funct ional interact ions which take place in the mamma l i an mast icatory sys tem. 1.2 MANDIBULAR FORM AND FUNCTION 1.2.1 Form and function of the mammalian jaw The mamma l i an jaw sys tem is the funct ional unit of the body pr imari ly responsible for mast icat ion. The mandib le is a major part of the jaw sys tem. It is suspended below the maxi l la by musc les , l igaments and other soft t issues to provide the mobi l i ty involved in funct ions which include suck l ing, swal lowing, speech, and most important ly the incision and chewing of food. This sys tem also plays a major rule in tast ing and breath ing. Unl ike its repti l ian precursor, the complex i ty of the mamma l i an mandib le has been great ly reduced. It consists of a s ingle bone, the dentary, rather than a series of bones (Dechow and Car l son, 1997) . Large var iat ions, however, exist in the mamma l i an j aw s ize, art icu lar and musc le fo rm, which may be presumed to have b iomechanica l consequences. 1.2.2 Research models Numerous exper imenta l an imals have been used to study human jaw structure and funct ion. These include rodents and carn ivores (Bouvier and Hylander, 1984; L ieberman and Crompton , 2000; Ot ten, 1987; Weijs, 1973, 1975; Wei js and Dantuma, 1975), rabbits and ungulates (Herr ing, 1972, 1976, 1977; Herring and Teng, 2000; Langenbach et al., 1992; Langenbach and Weijs, 1990), and anthropoid pr imates (Hylander, 1979a, b; Hy lander etal., 2000) . The resemblance of the miniature pig mandib le (Sus scrofa) to the human jaw makes the pig a useful an imal model for funct ional studies of the mast icatory sys tem (Strôm et al., 1986) . The pig and human mandibu lar anatomies , occlusions, movements , and loading patterns are actual ly quite s imi lar (Herr ing, 1995) . The pig j aw musc les are also s imi lar to the human jaw musc les (Herr ing et al., 1993; Strôm et al., 1986). The pig mandib le differs f rom the human mandib le however, in that: A) it is relat ively larger in overal l s ize; B) it is more prognathic; C) - 4 - the two mandibu lar corpora and dental arches are long and parallel to each other rather than divergent; D) the funct ional occlusal plane is a lmost paral lel to the lower border of the mandib le; E) there are large d ias temata between incisors and canines and between canines and premolars; and F) the symphys i s is large and or iented dif ferently. 1.3 MAMMALIAN JAW BIOMECHANICS In the text "Vector Mechanics for Engineers: stat ics and dynamics" , Beer and Johnston (1988) state "Mechanics may be def ined as that sc ience which descr ibes and predicts the condit ions of rest or mot ion of bodies under the act ion of forces". We can extend the above definit ion and define b iomechanics as mechanics appl ied to biological bodies. Mechanics is d iv ided into three parts: the mechan ics of rigid bodies, the mechanics of deformable bodies, and the mechan ics of f luids. In the bioengineer ing l i terature, the mandib le has been treated e i ther as a rigid body (Koolstra and van Ei jden, 1997a, b; Langenbach and Hannam, 1999; Peck e r a / . , 2000) or a deformable one (Chen e r a / . , 1998; Chen and Xu , 1994; Korioth et al., 1992; Kor ioth, 1997; Korioth and Hannam, 1994a; Korioth and Vers lu is, 1997). The b iomechanica l behavior of bone t issue and of the mandibu lar bone as a whole has been thoroughly rev iewed recently (see van Ei jden, 2000) . Here, only those points most re levant to the thesis are d iscussed. Due to the relat ive scarcity of b iomechanica l data on pig and human mandib les, much of the information presented in this l i terature review was obta ined f rom pr imate studies. This informat ion may not be entirely appl icable to pigs and humans. 1.3.1 Material properties of bone While bone is a hard t issue composed main ly of hydroxyapat i te , in which the crysta ls are very stiff and strong, it is also a l iving organ, a compos i te of col lagen f ibers and hydroxyapat i te . The a r rangement of the col lagen f ibers and hydroxyapat i te determines its mechanica l propert ies. Bone is not isotropic. It is stiffest longitudinal ly, less stiff tangent ia l ly and least stiff in a direct ion normal to the bone's surface (Dechow et al., 1993; Dechow and Hylander, 2000) . The strength of bone also differs under different stress reg imens (van Ei jden, 2000) . Despite the above reservat ions, bone is often treated as being s imi lar to many engineer ing mater ia ls , not least al lowing its stresses and stra ins to be ana lyzed in much the same way as the methods used in engineer ing structural analys is (Fung, 1981) . 1.3.1.1 Stress and strain under normal load Stress is " the internal force exerted by e i ther of two adjacent parts of a body upon the other across an imagined plane of separa t ion" (Roark and Young, 1975) . It is def ined as force per unit cross-sect ional area or intensity of the forces distr ibuted over a g iven sect ion (Beer and Johnston, 1981), i.e. P o = — Equation l . i where a denotes stress; P and A are the load and the cross-sect ional area, respect ively. The unit of stress is Newton per square meter (N /m 2 ) or Pascal (Pa). Practical ly, stress is expressed as MPa ( = 1 0 6 Pascal) or GPa ( = 1 0 9 Pascal). When the load is normal to the cross-sect ion, it is cal led normal stress. If the normal stress is directed toward the part on which it acts, it is cal led compress ive stress; if the normal stress is directed away from the part on which it acts, it is tensi le stress. The m a x i m u m stress that the bone can susta in is cal led ul t imate stress and is a measure of bone st rength. Bone is weaker in tension than in compress ion . For example , the ul t imate tensi le stress for the human femoral cort ical bone is 124 MPa whi le the u l t imate compress ive stress is 170 MPa (Fung, 1981) . The ul t imate compress ive strength for the pig mand ibu lar compact bone is reported to be 120 MPa (Robertson and Smi th , 1978) . Stra in is a measure of a body's deformat ion under stress. It is expressed as the ratio of the total deformat ion over the total length, i.e. S e = — Equation 1.2 where e denotes stra in; 5 and L are the amount of deformat ion ( length change) and the total length, respect ively. Stra in is a d imens ion less quant i ty. As a convent ion, strain is depicted as ue. For example , a 1,000 ue is 0 . 1 % deformat ion. Stretch is a tensi le s t ra in, and shortening is a compress ive stra in. The funct ional strain level for human cortical bone is reported to be below 3,000 ue and the max imum strain that the human cortical bone can susta in is reported to be 6,300 ue (Hylander, 1985) . It has been suggested that certain load-bear ing skeleta l e lements exper ience s imi lar strain magni tudes dur ing habitual dynamic loading. This concept of dynamic strain s imi lar i ty, which has been proposed to be determined by the mater ia l propert ies of bony t issue (Rubin and Lanyon, 1984) , has been observed in a large number of dif ferent vertebrates f rom birds to horses, d isregarding their s ignif icant di f ferences in body size and locomotor behavior (Bert ram and Biewener, 1988; Biewener, 1982; Biewener etal., 1983, 1986; Lanyon and Rubin, 1984; Rubin and Lanyon, 1984; V inyard and Ravosa, 1998) . When bone is compressed or tensed, not only will it deform in that direct ion, but it will also deform in a perpendicu lar direct ion. The first strain is cal led pr imary strain and the second is cal led secondary stra in. The ratio of the secondary strain over pr imary strain is Poisson's ratio, and a measure of the abil ity of a structure to resist deformat ion in a direction perpendicular to that of the appl ied load. v Equation 1.3 £x where v is Poisson's ratio; ey and ex are the secondary and pr imary strains, respect ively. Poisson's ratio for human mandibu lar cortical bone has been reported to be 0.27-0.41 (Dechow et al., 1993), i.e. a 1% pr imary strain will cause 0 .27 -0 .41% strain in a direct ion perpendicular to the load. The stress strain relat ionship fol lows Hooke's law if the deformat ion can be complete ly recovered when stress is re leased. This deformat ion is cal led elast ic deformat ion. The ratio between the stress and strain is the Young's or elast ic modulus. r- (7 t = — Equation 1.4 where E is the elast ic modulus. Young's modulus is a measure of the abil ity of bone t issue to resist deformat ion in the direct ion of the appl ied load. Young's modulus is posit ively related to bone mineral density (Currey, 1984a) . Because strain has no unit, its unit inherits the stress's unit Pa and is often expressed as GPa . Since bone is an anisotropic - 8 - mater ia l , the elast ic modulus differs according to appl ied load direct ions. The elast ic modul i for macaque mandibu lar buccal cortex are reported to be 9.0 GPa in the direct ion normal to the bone surface, 15.9 GPa in the tangent ia l direct ion to the bone surface, and 21.0 GPa in the longitudinal d irect ion. They are a little h igher for the l ingual cortex in macaque (Dechow and Hylander, 2000) . 1.3.1.2 Shearing stress and strain When forces are parallel to the plane of cross-sect ion, they yield shear ing stress. These forces are cal led shear ing forces. The average shear ing stress in the sect ion can be depicted as P T - — Equation 1.5 where T is the shear ing stress; P is the shear load. The amount of deformat ion occurr ing under shear ing force is the shear ing stra in. The relat ionship between the shear ing stress and strain in the elast ic range also fol lows the Hooke's law, i.e. Equation 1.6 where G is the shear modulus; y i s the shear ing stra in. The shear modulus is also cal led the rigidity modulus and is a measure of the abil ity of a structure to resist shear stress in a given plane. Bone is especial ly weak in shear. The shear modulus tends to be one third to half of the va lue of the elast ic modulus (van Ei jden, 2000) . For examp le , the shear modulus for macaque mandibu lar buccal cortex is only 3.8-7.0 GPa depending on the direct ions (Dechow and Hylander, 2000) . The shear rigidity of bone cross-sect ion can be raised by increasing the absolute amount of bony mater ia l (vo lume and/or densi ty) in a cross-sect ion. 1.3.1.3 Strength and stiffness Both strength and sti f fness are important propert ies of bone. Strength is related to stress and stiffness to stra in. The stress at which bone yields is cal led the yield strength. This is the point that separates elast ic from plastic deformat ion. The max imum stress that bone can susta in def ines the u l t imate s t rength. The breaking strength is the stress corresponding to bone break. Bone can be treated as brittle mater ia l and there is little dif ference between the ult imate strength and breaking strength (Beer and Johnston, 1981; van Ei jden, 2000) . The va lue of u l t imate strength of bone t issue depends on the type of stress. Bone has lowest va lue for u l t imate shear s t rength, middle va lue for u l t imate tensi le s t rength, and the highest va lue for u l t imate compress ive strength (van Ei jden, 2000) . St i f fness is the abil ity of bone t issue to resist deformat ion within the l inear range. Therefore, it is expressed as the modu lus of elast ic i ty. As ment ioned earl ier, bone is stiffest in its longitudinal d i rect ion, less stiff in its tangent ia l direct ion and least stiff in the direct ion normal to its surface. 1.3.2 Biomechanical design of the mandibular corpus A complete review on methods used in mand ibu lar stress and strain studies can be found e lsewhere (Daegl ing and Hylander, 2000) . Here focus is placed on the methods that infer stress and strain patterns through study of the mandibu lar cross-sect ional s ize, shape and bone distr ibut ion, i.e. the b iomechanica l des ign. First, two more basic concepts regarding engineer ing considerat ions in structural des ign are introduced. 1.3.2.1 Bending When a beam is loaded under pure bending, it undergoes a stress gradient, i.e. tensi le stress grows f rom zero at the centered neutral surface to max imum at the beam surface on one side and compress ive stress increases f rom zero at the beam neutral surface to max imum on the other (Figure 1.1, p l 2 ) . The neutral surface is also cal led the neutral axis. The bending stress at any given point in the sect ion (either compress ion or tens ion) can be g iven as Equation 1.7 where M is the bending moment; c is the d istance f rom the neutral surface; J is the cross-sect ional moment of inertia (see Equat ion 1.10, p20) with respect to the axis perpendicular to the bending moment . Therefore, it is not difficult to deduce that the amount of stress can be decreased with an increased cross-sect ional moment of inert ia. There are two ways to increase the cross-sect ional momen t of inert ia. One is to add more mater ia l . An adverse effect of th is approach is the increase in body weight. If mater ia l economy is the main concern, this is not a good remedy. Another method is to redistr ibute the mater ia l by mov ing the mater ia l f rom the less-stressed center to the outer surface, and thereby increases the external d imens ion of the beam. Expans ion of the external d imens ion also increases the distance f rom the neutral surface to the external surface l inearly, but the second momen t of inertia is a funct ion of the square distance. This will eventua l ly increase the cross-sect ional abi l ity to resist bending. As the actual mass remains the Figure 1.1 Beam bending theory. A: bending moment couple (M and M'); B: stress pattern in the cross-section. There is no stress at along neutral axis or surface. Both compressive and tensile stresses increase towards the beam surface. Based on van Eijden (2000). same, this is an eff icient, robust, and economic des ign. Because bending stress demonst ra tes a gradient with the highest stress at the surface and a gradual reduct ion towards the center, it demands that the mater ia l strength is also highest on the surface and lower towards the center. The abil ity to resist bending in one direct ion can be increased more eff iciently if the d imens ion in the plane, of bending is en larged more than the d imens ion orthogonal to that plane (Figure 1.2, p l 4 ) . 1.3.2.2 Torsion When a torque is appl ied to a c ircular beam, the beam will twist. The f ibers towards the surface twist more than those towards the center. The stress pattern in any given cross-sect ion is shown in Figure 1.3 ( p l 5 ) . The tors ional shear stress at any given point in the sect ion can be descr ibed as T = ~J~ Equation 1.8 where T is the torsional shear ing stress; T is the appl ied torque; p is the distance f rom the beam center; and J is the cross-sect ional polar moment of inertia (see Polar moment of inert ia, p20) . Therefore, to increase the abi l ity to resist tors ion, the best way is to raise the polar moment of inert ia. As in bending, there are two ways to increase the polar moment of inertia in a cross-sect ion. One is to add more mater ia l to the sect ion, with the adverse effect of increasing the mass of the beam. The second, and the best, method to increase polar moment of inert ia is to move the mater ia l f rom the less-stressed center, to the outer surface (to increase the external d imens ion of the beam). Expans ion of the external Figure 1.2 Best designs for resisting bending (by moving material from the center to the outer surface to create hollow sections). A: the original solid circular section; B: increased ability to resist transverse bending; C: increased ability to counter vertical bending. Figure 1.3 Stress pattern of a cross-section under torsion. Torsional shear stress increases from the center to the surface. dimens ion also increases the distance f rom the center to the external surface l inearly, but the polar moment of inert ia is a funct ion of the radius squared. This will eventual ly increase the cross-sect ional abi l ity to resist tors ion. As the actual mass remains unchanged, this is an efficient, robust, and economic des ign. Because the tors ional stress is in a gradient with the highest stress at the surface and the stress gradual ly reducing towards the center, it demands that the mater ia l strength is also highest on the surface, and gradual ly reducing towards the center. The best shape to resist tors ional shear stress is a c ircular sect ion, i.e. a hollow sect ion retaining the circular ity (Figure 1.4, p l 7 ) . 1.3.2.3 Mandibular corpus cross-sectional form Studies of the mandibu lar corpus cross-sect ional fo rm can provide insight into the b iomechanica l des ign of the mandib le (Daegl ing, 1989) . There are a number of ways to achieve a mechanica l ly robust mandib le, though how the mandib le is des igned may depend upon species. Pr imate studies (Dechow and Hylander, 2000; Hylander, 1979b) suggest that the mandib le is built so as to use its bony mater ia l more economica l ly than a solid rod of s imi lar rigidity, i.e. it is a hollow sect ion with the densest mater ia l d istr ibuted along the surface (cortical bone). Daegl ing (1989) carr ied out what seems to be the first s tudy on the " internal des ign" of the mandibu lar corpus. He used CT to examine mandibu lar cross-sect ions of Pan, Pongo, Gori l la Homo and two fossil spec imens of Paranthropus at the first and second molars. The mandibu lar corpus cross-sect ions were measured as cort ical and total sect ions. One of the signif icant f indings in this study was that while the fossil homin ids did not differ s ignif icantly f rom extant homino ids in the relat ive contr ibut ion of compact bone to total subper iostea l area, the  shape of the robust austra lopithecine mandib le was fundamenta l ly different f rom that of modern hominoids in te rms of its abi l i ty to resist t ransverse bending and tors ion (Daegl ing, 1989) . It has been suggested that the mandibu lar corpus might behave as an open and/or c losed sect ion under load (Hylander, 1979a; Sm i th , 1983) . Previous studies (Daegl ing, 1989; Korioth et al., 1992) support neither idea because the corpus is actual ly a combinat ion of both (i.e. open for sect ions through the teeth and closed for sect ions between the teeth) . Moreover, it seems that neither a so l id- nor hol low-el l ipse model adequate ly descr ibes the mechanica l behavior of the mand ibu lar corpus cross-sect ion, s ince neither model predicts cross-sect ional area and moments of inertia with acceptable accuracy (Daegl ing, 1989) . Due to the structural complex i ty or irregular ity of the mandibu lar corpus cross-sect ion, it seems an invest igat ion which includes regional cortical densi ty (Daegl ing etal., 1992; Daegl ing and Hylander, 1998) and cortical th ickness might provide a better understand ing of corpus b iomechanics. Moreover, s ince the reported first and second molar sect ions are quite s imi lar in their cross-sect ional fo rms (Daegl ing, 1989), one would need to know whether this s imi lar i ty appl ies to mand ibu lar cross-sect ions at other locations, e.g. the canine and symphys i s regions. 1.3.2.4 Cross-sectional measurements and their significance 1.3.2.4.1 Cross-sectional area Since the stress is def ined as internal res istance provided by a unit area (Beer and Johnston, 1981; Mott, 1996), cross-sect ional area is one of the most important measurements in mater ia l mechan ics for counter ing stress under normal or shear ing load (Hearn, 1997; Mott, 1996; van Ei jden, 2000) . Unfortunately, this parameter does not reflect the distr ibut ion of mater ia l . It is obvious that the differential redistr ibut ion of the bony mater ia l is equal ly important for changes in mandibu lar mechanica l propert ies as modif icat ion of the amount of compact bone uti l ized (Daegl ing and Gr ine, 1991) . For example , adding more cortical bone at the center of the mandibu lar corpus does not have the same effect as adding the same bone to the per iphery. Because the mandibu lar corpus cross-sect ion is not regular, the best way to calculate the area is by calculus, i.e. d iv id ing the whole section into sma l l , but equal-s ized, square e lements . The area of a cross-sect ion can be calculated as A = jdA Equation 1.9 where A is the cross-sect ional area; dA is the area of each e lement. The unit can be expressed as c m 2 for the mandibu lar cross-sect ion. 1.3.2.4.2 Second moment of inertia Second moment of inertia or area is a measure of the distr ibut ion of bone around a part icular axis. By deposit ing bone as far as possible f rom the neutral axis of the cross-sect ion, the moment of area can be increased without an increase in the actual amount of mater ia l . In a cross-sect ion with a large cross-sect ional moment of a rea , stress can be kept relat ively low. Hence, an increase in the cross-sect ional moment of area is more opt imal to susta in heavy bending loads (van Ei jden, 2000) . The second moment of area is ax is-dependent . For any g iven cross- sect ion, unl imited numbers of second moments of inertia can be calculated because the number of axes is infinite. In real ity, it is usual to choose a pair of orthogonal axes (usual ly denoted as x, y ) . For a cross¬ - 19 - sect ion with an ovoid shape like the mandibu lar corpus, it is intuit ive to choose the major and minor axes. Therefore, the second momen t of area with respect to the major axis (Iy, vert ical axis in corpus sect ion) is a measure of its abi l ity to resist facio-l ingual bend ing, and the second moment of area with respect to the minor axis (Ix, t ransverse) is a measure of its abi l ity to counter sagittal bending. Second moments of area must also be calculated by calculus for irregular sect ion such as the mandibu lar corpus. h = \y2dA Equation 1.10 where Ix, Iy are the second moments of area with respect to the x-, and y-axis , respect ively; x, y are the location of each e lement in the coordinate sys tem with its origin at the centro id. The units for Ix, Iy are c m 4 . Actual calculat ion of the second moments of inert ia needs their translat ion f rom the original image matr ix origin to the centroid by the paral le l-axis theorem (Beer and Johnston, 1981; Beer and Johnston, 1988) because the centroid is unknown initial ly. 1.3.2.4.3 Polar moment of inertia The polar moment of inertia or area, (which is actual ly the sum of the above two second moments of area), takes into account not only the amount of cortical bone area, but also the disposit ion of the cortical bone with respect to the center of mass . The further the bone t issue is deposi ted f rom the center of mass, the larger the polar momen t of area. The larger the polar moment of area, the smal ler is the induced shear stress and the larger the abil ity of the bone to resist tors ional load. Although this parameter has been used in l i terature to indicate the abil ity of a mand ibu lar cross-sect ion in tors ion (Daegl ing, 1989; van Ei jden, 2000) , according to Gere and T imoshenko (Gere and T imoshenko , 1990), the tors ion theory in formula (Equat ion 1.8, p l 3 ) is appl icable to solid or hollow bars of c ircular cross-sect ion only. Such shapes are not actual ly appl icable for the mamma l i an mandib le. The danger of using Equation 1.8 ( p l 3 ) to ana lyze mand ibu lar tors ion is obvious. In the cross-sect ion as i l lustrated in Figure 1.5 (p22), the formula predicts stress at location A is less than stress at location B because A is c loser to the center than B. This is in confl ict with exper imenta l results reported by Daegl ing and Hy lander (1998) . 1.3.2.4.4 Cortical index The cort ical index (a ratio between cortical and total areas) is a measure of the relat ive amount of cort ical bone to the total bone. Daegl ing (1989) used the cortical area (all area enc losed by the cortical outl ine jo ined at the a lveolar marg ins by a one m m thick cap, i.e. a hollow beam) and the total subper iosteal area (all area enc losed by the periosteal border to the a lveolar marg ins, with a stra ight line connect ing those marg ins , i.e. a solid beam). By def init ion, these areas do not take into account the densi ty and possible porosity of the cortical and cancel lous bone, so they do not measure true bony mater ia l . It seems that a higher va lue, i.e. relat ively more cortical bone, might indicate a stronger cross-sect ion. This may be mis lead ing, however, as a solid sect ion of compact bone has a cortical index equal ing unity. This is not the ideal des ign for the mandibu lar corpus. A low cortical index may indicate a more economica l use of mater ia l and therefore is a measure of robust ic i ty (Daegl ing and Gr ine, 1991) . The reported cort ical indices for Figure 1.5 Diagram of a cross-section with variable wall thickness under torsional load. According to formula (Equation 1.8, pi3), TA < TB , which conflicts with experimental data. ' ' hominoid molar cross-sect ions vary between 0.29-0.54 (Daegl ing, 1989; Daegl ing and Gr ine, 1991). Whether this large var iat ion indicates how efficiently different hominoids use mater ia l in their mand ibu lar corpora var ies great ly or not remains unknown. It is possible to define another cortical index which takes such things as bone density and porosity into cons iderat ion, and to measure true bony mater ia l . The cortical area should be the bony cortex only, and the total area should include all bony mater ia l in the sect ion. In both area measurements , areas not fil led by bone (i.e. porosity) would be exc luded. For example , if there is a hole in the cortex, it should not be counted as part of the cortical area. For a hollow sect ion, a high va lue of this cortical index would indicate less trabeculat ion, and vice ve rsa . Therefore, this index is also an indirect measure of the degree of t rabecu lat ion. 1.3.2.4.5 Bending index The bending index (Iy/Ix, when Iy<Ix) is a shape indicator, because size is e l iminated (Daegl ing, 1989) . A low value signif ies increased abil ity to resist bending stress about the short axis, with the loss of the abi l ity to resist bending stress about the long axis. It is also a tors ional rigidity index because if the size is constant, a bending index of unity indicates a rounded cross-sect ion, which is the best des ign to susta in tors ional stress (van Ei jden, 2000) . Therefore, there are two b iomechanica l consequences of high bending index va lues: an enhanced res istance to t ransverse bending rigidity and a more efficient shape for tors ional r igidity (Daegl ing and Gr ine, 1991) . This index, however, cannot be used to compare absolute bone rigidity. This bending index for the homino id mandibu lar corpus has been reported to vary between 0.30-0.69 at the molar sect ions, in which the fossil hominoids demonst rate highest bending indices (Daegl ing and Gr ine, 1991) . The authors interpret this as structural response to e levated tors ional moments . 1.3.2.4.6 Bone density Another important factor, yet one which has not been taken into account in the previous studies (though ment ioned by Daegl ing er a/., 1992, and Daegl ing and Hylander, 1998) , is regional bone density. It is common ly recognized that bone mineral densi ty is a cons istent predictor of bone strength (Currey, 1984a; Hobatho e r a / . , 1997; Mart in and Ishida, 1989; S tens t rom e r a / . , 2000) , especial ly for cancel lous bone (Rho e r a / . , 1995) . The high l inearity between the CT graysca le va lue and bone mineral densi ty (Cheng et ai., 1995; Lampmann et al., 1984; Zhang e r a / . , 2001a) encourages the use of CT grayscale va lues to represent regional bone densi ty. When CT grayscale va lues are inc luded, cross-sect ional mass , second and polar moments of mass can be est imated in proport ion. Cross-sect ional mass seems to be a better var iab le indicat ing beam uniformity in the mandibu lar corpus, for the relat ive amount of mater ia l use by one cross-sect ion can be easi ly compared to another within the same mandib le. It is understandable that CT grayscale va lues vary among different CT machines, and among different scans. Even a rout inely cal ibrated CT machine does not guarantee a grayscale va lue cons istency between different scans. Therefore, care must be taken when compar ing spec imens scanned in different mach ines and/or sess ions, i.e. a reliable cal ibrat ion phantom should a lways be included in the CT scan. 1.3.2.4.7 The centroid of a cross-section In case of a homogeneous cross-sect ion, the centroid or the center of gravity or the center of mass can be calculated as cy - i \y* Equation 1.11 where Cx, Cy are the x and y coordinates of centro id, respect ively. In case of a heterogeneous cross-sect ion like the mand ibu lar corpus, the centroid can be calculated as where M denotes the total cross-sect ional mass; dm is the mass of each e lement. When the second or polar moment of inertia is d iscussed, (if not speci f ied), it should be with respect to the centroid instead of the origin of the image matr ix, because the centroid is independent upon the cross- sect ional posit ion and or ientat ion in the image matr ix . The centroid for all the cross-sect ions of a beam forms the centroidal ax is. In a unif ied straight beam, the centroidal axis coincides with the neutral ax is. In a curved beam with heterogeneous mater ia l l ike the mandib le, however, the neutral axis does not coincide with the centroidal axis and is not determined immediate ly (Beer and Johnston, 1981) . S ince the two axes may actual ly be very close to each other, the centroidal axis has been used to approx imate the neutral axis in pr imate jaw studies m m Equation 1.12 (Hylander, 1985) . 1.3.2.4.8 Variation in bone density and cortical thickness If the assumpt ion , that the stress pattern is l inked to the distr ibut ion of cort ical bone in a mandibu lar corpus cross-sect ion, is va l id , then the regional di f ference in bone density and „ cort ical th ickness in the mandibu lar cross-sect ion should be taken into account. If we assume that regional bone rigidity is a funct ion of the cortical th ickness t imes the mean density of that region as demonstrated by the fo rmula , Rigidity = K x Cortical Thickness x Density Equation 1.13 where K is a constant, the product of the cort ical th ickness and mean CT graysca le va lue would be a cortical rigidity index (CRI) . This index could be used to compare the cortical r igidit ies among cort ices at different locations within a cross-sect ion or cross-sect ions in the same mandib le . It must be stressed that this may not be a truly l inear index, because the relat ionship between bone mineral densi ty and bone mechanica l property may not be l inear (Lang er a/., 1997; Rho er a/., 1995; Stens t rom e r a / . , 2000) . 1.3.2.5 Mandibular corpus loading and stress and strain 1.3.2.5.1 Sagittal bending of the mandible Sagitta l bending of the mandibu lar corpus is due to the vert ical components of bite force, muscle force and condy lar and symphysea l reaction forces act ing in the tangent ia l plane (the orthogonal plane to both the longitudinal and t ransverse planes) of the mand ibu lar corpus (Weijs, 1989) . Dur ing uni lateral bit ing, sagittal bending occurs on both sides of the mandibu lar corpora. On the work ing-s ide, sagittal bending causes tens ion along the lower border and compress ion a long the a lveolar side of the mandibu lar corpus. A reverse bending momen t which tenses the a lveolar processes and compresses the lower border of the mandib le occurs on the balancing-side (Hylander, 1979b; Kor ioth etal., 1992; van Ei jden, 2000; Weijs, 1989) . The ideal structural form for responding to this part icular load is a relat ively deep corpus in the mo lar region, which increases the cross-sect ional second moment of inertia with respect to the t ransverse axis of the corpus (Daegl ing and Gr ine, 1991) . In this respect, the prehistoric Polynesian " rocker" mandib les (Houghton, 1977, 1978) seem wel l-designed for such a loading condit ion. Sagitta l bending of the corpus also induces shear ing stresses along the entire length of the mandib le (Weijs, 1989) . Shear ing forces attain their largest va lues in the mandibu lar region between bite force and musc le force on the work ing-s ide, and in region between musc le force and jo int force on the balancing-side (van Ei jden, 2000) . Shear ing stress is inversely proport ional to the cross-sect ional area of the mandibu lar corpus, i rrespect ive of its shape. Hence a certain amount of bony mater ia l should be present along the entire mand ibu lar corpus. As the shape of the mandibu lar corpus sect ion is somewhat el l ipt ical, both solid and hollow el l ipse models have been proposed (Smi th , 1983) . And because of the extens ive t rabecular bone in some cross-sect ion, a semi-so l id model has also been ment ioned (Hylander, 1985) . S ince the role of tooth is undetermined, both open and c losed mode ls have been considered (Kor ioth et al., 1992). However, accord ing to the formula (Equat ion 1.7, p l l ) , for cross-sect ions to resist sagittal bending, all of the above models may be val id as long as the second moment of inertia truly reflects the mater ia l d istr ibut ion. 1.3.2.5.2 Transverse bending of the mandible Medial t ransverse bending occurs dur ing the jaw opening phase and lateral t ransverse bending occurs dur ing the jaw closing phase (Hylander, 1985; Hy lander and Johnson, 1994) . Both lateral bending moments become largest at the symphys is . Thus the stresses caused by them are quite low at the molar region of the mandibu lar corpus but grow larger towards the symphys i s in a l inear manner (Daegl ing and Gr ine, 1991) . 1.3.2.5.3 Mandibular torsion Torsion of the mandibu lar corpus occurs on the work ing-s ide during the power stroke of mast icat ion and on both s ides dur ing the power stroke of ingest ion. In both cases, the twist ing tends to evert the lower border of the mandib le and invert the a lveolar process. On the work ing- side, this specif ic load is partial ly reduced by the resultant mast icatory bite force which tends to twist the corpus in an opposite direct ion (Hylander, 1979b) . These twist ing moments are highest in the molar regions (Daegl ing and Gr ine, 1991) . To counter this kind of stress effect ively, both the cross-sect ional shape and bone distr ibut ion are cr it ical. A c ircular hollow sect ion with the max imum possible external d imens ion is the ideal des ign. The bending index is a measure of the cross-sect ional c ircular ity. A value approaching unity indicates better c ircular i ty. The fact that mandibu lar corpus sect ions for modern hominoids are not as c ircular as the fossil spec imens of Paranthropus supports the hypothes is that the Paranthropus mandib le is more robust in resist ing tors ion (Daegl ing, 1989; Daegl ing and Gr ine, 1991) . Under tors ional load, the mandibu lar corpus can be mode led as s ingle- or mult i-cel l th in-wal led tubular members (Cook and Young, 1985; Ugural and Fenster, 1987) depending on the intensity of the t rabeculat ion. A single-cel l th in-wal led tubular member may be adequate to model cross- sect ions with less t rabeculat ion. A mult i-cel l th in-wal led tubu lar member might be appropr iate for the mandibu lar corpus cross-sect ion with extens ive t rabecu lar struts and co lumns. For a th in-wal led tubular member , the tors ional shear fol lows a constant shear f low (f) throughout the shel l , i.e. f - tt Equation 1.14 where 7 is the shear ing stress for a location and t is the wall th ickness at that locat ion. S ince f i s constant, the largest shear stress occurs where the th ickness of the tube is smal lest and vice versa . As d iscussed earl ier, use of a tors ion formula (Equat ion 1.8, p l 3 ) causes confl icts with exper imenta l results. The calculat ion of shear stress can be per formed accord ing to the fol lowing fo rmula: x = ——— Equation 1.15 2tAm where T is the appl ied torque; Am is the area bounded by the center l ine of the cross-sect ion (dashed line in Figure 1.6, p30) . A mult i-cel l th in-wal led tube can be ana lyzed by a s imple extens ion of the one-cel l ana lys is (Figure 1.7, p31) . Figure 1.6 Cross-section of a thin-walled tube. The highest shear stress occurs where the thickness of the tube is smallest. The dashed line is the centerline of the cross-section and Am is the area bounded by the center line. T is the torsional stress and t is the wall thickness. *3 U Figure 1.7 A multi-cell thin-walled tube can be modeled as an extension of a single- cell thin-walled tube. It is assumed shearing stresses are directed as shown, the shear flow yields titi = T2t2 + 13*3. This diagram shows a simple example of two cells, but the number of cells is not limited to two. 1.3.3 Biomechanical significance of the jaw symphysis 1.3.3.1 Symphyseal form and function The mandibu lar symphys i s remains unfused throughout life in most mamma l i an species. Fused symphyses only occur in some specif ic taxa including anthropoid pr imates and many art iodacty ls. The funct ional advantages of fused and unfused mandibu lar symphyses in mamma l s have been rev iewed recently by L ieberman and Crompton (L ieberman and Crompton , 2000) . The unfused symphys i s , by al lowing independent inversion and evers ion of the two halves of the mandib le before and dur ing the mast icatory power stroke, enables the steep occluding surfaces of oppos ing teeth in some mamma l s to match dur ing mast icat ion (Hylander, 1979b; Kal len and Gans, 1972; L ieberman and Crompton, 2000; Oron and Crompton , 1985; Scapino, 1981) . In contrast, the fused symphys i s s t rengthens and stiffens the jaw, reducing its risk of structural fai lure as a result of lateral t ransverse bending or "w ishbon ing" , and f rom the dorsoventra l shear stresses which occur dur ing uni lateral mast icat ion (Hylander, 1984; Hylander et al., 2000; Ravosa, 1996; Ravosa and Hylander, 1993; Ravosa and S imons , 1994) . Mamma l s producing predominant ly vert ica l ly-or iented occlusal forces tend to have unfused symphyses (which can transfer dorsal ly-direct ly forces with equal eff ic iency as in fused symphyses through their interdigitat ing rugosit ies), whi le mamma l s producing mainly t ransverse ly-or iented occlusal forces tend to have fused symphyses (L ieberman and C rompton , 2000) . Symphysea l fusion accompanies deve lopment of a funct ional occ lus ion. Part ia l ly-fused symphyses are common in juven i le an imals , and complete fusion often takes place with the erupt ion of the permanent first molars (Ravosa, 1999) . In the miniature pig, adult- l ike t ransverse mast icatory movements develop after wean ing, and the mandib le is subject to wishboning s imi lar to that in anthropoids (Huang e r a / . , 1994) . The shape and size of the Cercopithec ine symphys i s appear to be al lometr ic with body size and mandibu lar dental arch width, i.e. symphysea l width and length scale posit ively a l lometr ica l ly with body size, and negat ively with mandibu lar dental arch width (Hylander, 1985) . S ince the width of the symphys i s increases more rapidly than its length, there is also a change in symphysea l shape with increased body size (Figure 1.8, p34) . These changes are bel ieved to mainta in funct ional equivalence in bone stresses and strains across taxa and ontogeny (Hylander, 1985; V inyard and Ravosa, 1998) . During wishboning, for examp le , tensi le strains at the l ingual border of the pr imate symphys i s can approach 2,000 ue, wel l-below 3,000 ue, when structural fai lure is possible, i.e. adapt ive remodel ing alone cannot occur fast enough to cope with funct ional demands in an imals which chew vigorous ly every day (Bouv ier and Hylander, 1981a; Hannam e t a / . , 1997; Hylander, 1979b) . There is a general agreement that the form is l inked to funct ion. It has been suggested that the super ior and inferior tori common ly found in anthropoid pr imates funct ion to resist the effects of wishboning of the mandibu lar corpora (Hylander, 1984) . However, due to large var iat ion of the shape of the symphys i s , using symphysea l morpho logy as a marker in species identi f icat ion, or in systemat ic a rguments is problemat ic (Daegl ing, 1993; Daegl ing and Jungers, 2000) . 1.3.3.2 Symphyseal loading and stress and strain If the purpose of the symphysea l fusion were to s t rengthen and stiffen the jaw, thereby reducing its risk of structural fai lure f rom the high stresses and strains dur ing funct ion (Hylander, 1984; Hy lander e t a/., Figure 1.8 A growth series of Macaca fascicularis mandibles illustrating ontogenetic changes in symphyseal curvature. Shown from left to right are a juvenile with M i erupting, a "subadult" with M2 erupting, an adult female, and an adult male. Inspection of this ontogenetic series indicates that the macaque mandible gets relatively longer during growth, while mandibular arch width becomes relatively narrower. This suggests a postnatal increase in symphyseal curvature—a pattern duplicating the interspecific allometric changes in curvature. From Vinyard and Ravosa (1998). 2000; Ravosa, 1996; Ravosa and Hylander, 1993; Ravosa and S imons , 1994), stress and strain analys is would be an appropr iate method to reveal funct ional adaptat ion in the mamma l i an symphys i s . Severa l patterns of stress have been postulated to occur in the pr imate symphys i s dur ing funct ion. These stresses have been reviewed in detai l e lsewhere (see Hylander, 1984). In brief, tens ion occurs along the l ingual and/or infero-l ingual side of the symphys i s and compress ion occurs along the facial and/or supero-facial side of the symphys i s dur ing wishboning and/or twist ing of the mandibu lar corpora. Wishboning is due to 1) the force f rom the deep masseter musc le on the balancing-s ide at the very end of the power stroke, 2) the lateral components of the bite force on the work ing-s ide, 3) probable t ransverse components to work ing-s ide j aw closing muscle forces (Hylander, 1984; Hylander and Johnson, 1994) , and 4) the reaction force appl ied to the media l pole of the condy lar head by the media l wall of the condy lar fossa (Figure 4.4, p i l l ) . Twist ing of the jaw occurs on the work ing-s ide dur ing the power stroke of mast icat ion and on both sides dur ing the power stroke of ingest ion due to the mast icatory musc le forces (Hylander, 1979a , b). Wishboning can create high tensi le stresses and stra ins on the l ingual surface of the symphys i s . In pr imates, these tensi le s t resses are about 2.5 t imes larger than the compress ive stresses on the labial surface (Hylander, 1985; Ravosa and S imons , 1994) . They can be resisted by synostos is (Ravosa, 1996, 1999; V inyard and Ravosa, 1998) , bony enhancement (e.g. super ior and inferior t ransverse tor i ) , and increased horizontal or ientat ion of the symphys i s . These features are often seen in pr imates (Daegl ing, 1993; Hylander, 1984, 1985; Ravosa and S imons , 1994) . Tension occurs along the facial surface of the symphys i s and compress ion occurs along the symphysea l l ingual surface dur ing media l t ransverse bending of the mandibu lar corpora. Medial t ransverse bending of the corpora is due mainly to the bi lateral contract ion of the lateral pterygoid musc les (Hylander, 1985) . Tors ion occurs along the t ransverse axis of the symphys i s dur ing powerful chewing when the work ing-s ide of the mandib le is depressed and the balancing-side corpus is e levated (see Hylander, 1984) . Severa l patterns of shear ing stress also occur in the symphys i s . Dorsoventra l shear is caused by the balancing-s ide j aw musc le force and due to the downward and upward movements , respect ive ly, of the work ing-s ide and balancing-side mandibu lar corpora (Hylander, 1975, 1977, 1979a, b; Ravosa, 1996, 1999; Ravosa and S imons , 1994) . Anteroposter ior shear is due to the balancing-s ide tempora l i s having the tendency to pull the balancing-side dentary in a poster ior direct ion relat ive to the work ing-s ide dentary during the power stroke (Beecher, 1977) . A l though this is observed in unfused symphyses , it is l ikely true in fused symphys i s too (see Hylander, 1984) . Despite the complex i ty of the stresses occurr ing at the symphysea l region, the most important stresses are the tens ion and compress ion caused by wishboning, and dorsoventra l shear. The cross-sect ional area of bone and symphysea l shape affect the jaw's res istance to these stresses (Hylander, 1984, 1985), and an adequate cross-sect ional area of bone in the plane of stress is needed to resist dorsoventra l shear. In contrast, both the cross-sect ional area of bone and symphysea l shape are signif icant in order to counter stress effect ively dur ing symphysea l wishboning (Hylander, 1984, 1985). Whi le in vivo bone strain studies provide a faithful depict ion of the true in vivo mechanica l env i ronment (Daegl ing and Hylander, 2000) , there are some l imitat ions. An alternat ive approach is to use the theor ies normal ly employed by mechanics of mater ia ls . This seems appeal ing. Unfortunately, many assumpt ions have to be made here; e.g. assuming the symphysea l cross-sect ion is of regular shape, and made of uniform mater ia l , assuming the muscle force is constant or a l lometr ic to body weight, assuming the jaw length scales to the musc le lever a rm , and assuming dental arch width represents the radius of mand ibu lar curvature. If any of these assumpt ions are not true, the est imated stress and strain are quest ionable. Even so, postulates can be made, expla ined or defended (Vinyard and Ravosa, 1998) . One approach might be to apply as much individual morphological and cross-sect ional data as is avai lable and compare the est imated stress and strain to the in vivo exper imenta l results. Success here would complement in vivo exper iments and offer stress and strain informat ion where in vivo approaches are imposs ib le. 1.3.4 Modeling jaw biomechanics The data col lected f rom direct human and an imal exper iments are often incomplete, though mathemat ica l models can use incomplete data to provide hypothet ical va lues for the miss ing var iab les (Hannam, 1994) . Models al low postulates to be demonst ra ted , exp la ined, defended or a l tered as required. They invite informed specu lat ion, for different scenar ios can be constructed to explore new ideas, deve lop novel hypotheses, and gain insight into the consequences of sys tem var iables (Hannam er a/., 1997) . Plausible models can mimic j aw funct ion and b iomechanics in the virtual env i ronment , and al low alterat ion of var iables. This is usual ly diff icult or impossib le to ach ieve in vivo. 1.3.4.1 Static jaw modeling An easy way to s imulate jaw, art icular and occlusal funct ion is to assume that the biological structures are r igid. In the stat ic s i tuat ion (e.g. during tooth c lenching), equi l ibr ium theory can be invoked to solve any bi- (two d imens iona l model) or tr i-axial ( three-d imens ional model) forces act ing within an arbitrary coordinate sys tem, including unknown forces and torques created at locations of interest. The known inputs can be occlusal forces, and the unknown outputs can be musc le tens ions and condylar forces (or vice versa) , for it is ax iomat ic that all forces and torques in a c losed, stat ic sys tem must be zero. Based on this theory, var ious models have been developed (Greaves, 1978; Koolstra er a/., 1988; Korioth e r a / . , 1992; Korioth and Hannam, 1994b; van Eijden e r a / . , 1988) . The most advanced stat ics models are three d imens iona l f inite e lement (FE) models in which regional physical propert ies are ass igned to each group of e lements to represent different t issues. These have been loaded by s imulated musc le tens ions to demonst rate de format ion, stress and strain patterns in the mandib le, and the differential loading patterns on the mandibu lar condyle (e.g. Korioth and Hannam, 1994b; Beek e r a / . , 2001) . Evolut ion of these models involves compar ing actual surface strains recorded on exc ised mandib les with stra ins predicted by the model (Korioth e r a / . , 1992) . 1.3.4.2 Dynamic jaw modeling The main l imitat ion of stat ic model ing is its inabi l i ty to s imulate j aw dynamics . Three-d imens iona l dynamic models of the human mast icatory sys tem have been developed by only a few groups very recent ly e.g. those by Koolstra and van Eijden (1995, 1997a, b), by Langenbach and Hannam (1999) , and by Peck er a/. (2000) . These mode ls work according to rigid body mechanics, a l though they can mimic compress ion and - 38 - distort ions in the temporomand ibu la r jo int and musc les with "energy- s torage" components such as spr ing-damper ana logues (Peck, 1999) . They incorporate a great deal of informat ion with respect to jaw muscle morpho logy and propert ies, musc le tension and t im ing, dental occ lus ion, and jaw physical propert ies. The models are promis ing because they can use complex mathemat ica l integrat ion and convergence a lgor i thms to predict j aw mot ion, the result ing reaction forces between parts, and der ivat ives of these va lues (Hannam etal., 1997) . In contrast to stat ic models, dynamic j aw models require specif icat ion of the jaw's mass propert ies including mass, mass center and moments of inert ia. These can be difficult to est imate in biological t i ssues (Braune and Fischer, 1988) and the methods can be invas ive and prone to errors (Braune and Fischer, 1988; Koolstra and van Ei jden, 1995, 1997b) . For example , to calculate the jaw's moments of inert ia, Koolstra and van Eijden (1995, 1997b) cut an excised female cadaver jaw into cubic- cent imeter blocks of t issue. In related studies, Hannam et al. (1997) , Langenbach and Hannam (1999) , Peck et al. (2000) assumed mass propert ies predicted by an FE model of the human jaw deve loped ear l ier (Korioth e t al., 1992) . The FE model was constructed f rom CT images, and included e lements with t issue propert ies specif ic for dif ferent jaw regions. The funct ional signif icance of these mass propert ies in dynamic model ing remains unclear, a l though it has been suggested the moments of inertia are less signif icant than mass center locat ion in a s imulated jaw-c los ing movements (Koolstra and van Ei jden, 1995) . CT imaging is useful for mass-property calculat ion because x-ray l inear attenuat ion disc loses regional mineral densit ies (Cheng et al., 1995; Lampmann et al., 1984; Wi l l iams et al., 1980), which account for much of the jaw's mass . Thus individual pixels with dif ferent intensity va lues, distr ibuted non-uni formly in the imaged mandib le , can be ass igned densit ies reflecting mineral content, mak ing it possible to est imate the jaw's mass propert ies (Smi th e r a / . , 1995) . To calculate mass propert ies f rom 3D CT, the CT graysca le va lue must first be converted into equivalent BMD. This can be fulfi l led with a cal ibrat ion phantom. Phantom studies have shown that the relat ionship between CT graysca le value and phantom equiva lent BMD is a lmost l inear (r=0.99 for K H 2 P 0 4 solut ions; Cheng er a/., 1995; Lampmann er a/., 1984) . Therefore the convers ion f rom CT graysca le va lue to BMD is possible. Because CT is invas ive, it would be useful to explore less invasive approaches for est imat ing these mass propert ies in l iving humans. Candidate 3D imaging modal i t ies include magnet ic resonance imaging (MRI), 3D optical surface scanning and other 3D surface digit iz ing methods (Smi th er a/., 1995) . MRI seems more appropr iate than others for it d isc loses both surface and internal structures. It does not however, image bone, nor reflect mineral densi ty. Methods reveal ing jaw shape alone (l ike MRI) will not be val id unless it is c lear that the jaw's mass and geometr ic centers coincide. Jaw mass and moments of inertia also seem al lometr ic with its d imens ions. If a consistent relat ionship could be shown between t hem, it would be possible to est imate jaw mass and moment s of inert ia by s imple, direct measurements . 1.4 FINAL COMMENT Previous approaches in the study of j aw b iomechanics , such as analyses of cross-sect ional shape and size, theoret ica l stress and stra in, and mathemat ica l model ing employ principles adopted f rom physics and/or the mechanics of mater ia ls . A l though these principles are quite sol id, the biological mater ia ls do not usual ly have the propert ies of engineer ing mater ia ls . Therefore, theoret ical ana lyses need to be va l idated, where possible, by exper imenta l results. In other words, theoret ical ana lyses at best only complement exper imenta l studies. S ince in vivo exper imenta l studies are present ly l imited in scope, and are l ikely to remain so, the approaches are interdependent. The exchange of informat ion between exper imenta l and theoret ical studies is the way of scientif ic research; when based on avai lable theoret ical and exper imenta l data , current hypotheses can be defended and novel hypotheses can be postulated and tested. 2 STATEMENT OF THE PROBLEM Since the b iomechanics of the mamma l i an jaw are not ful ly understood, exper imenta l and theoret ical studies can be used to comp lement each other. Elevat ing the value of one above the other is of dubious benefit. For example , even if in vivo bone strain studies are cons idered "gold standards", their interpretat ion depends upon assumpt ions regarding physical principles and the propert ies of a loaded beam. Converse ly cross-sect ional shape and size ana lyses, stress and stra in est imat ion, and mathemat ica l models are obviously theoret ica l , though they can incorporate more physical and engineer ing principles. Cross-sect ional shape and size ana lyses previously carr ied out on hominoid mandib les have only focused on the mo lar region, yet provided valuable b iomechanica l informat ion l inked to in vivo and in vitro stress and strain in the post-canine corpus. The b iomechanica l behavior of other mandibu lar corpus regions remains unclear. Fur thermore, the bone- density contr ibut ion to corpus cross-sect ions is unknown, and regional var iat ions in bone density and cortical th ickness are not def ined. Without such informat ion, model ing the whole mand ibu lar corpus is very difficult. A l though in vivo bone strain studies have provided insight into the symphysea l stress and strain patterns in non-human pr imates, it is imposs ib le to apply this methodology in humans . A lso, the l i terature has been devoid of in vivo bone strain informat ion in the pig symphys i s , despite the fact this is a preferred an imal model for the study of the human mast icatory sys tem. The dist inct morphologica l character ist ics of the pig and human jaw symphyses however, encourage study of their respect ive stress and strain patterns. Dynamic human jaw models require specif icat ion of mass propert ies, some of which have been necessar i ly der ived f rom invasive measurements . Speci f icat ion of mass propert ies would seem desirable for successful dynamic models s imulat ing the human jaw funct ion, especial ly when this might conceivably involve l iving subjects and subjects with miss ing jaw f ragments . Biomedical computed tomograph ic imaging offers a solut ion here, for it discloses regional bone densi ty at high resolut ion. MRI is even more promis ing because it is less- invas ive and can unvei l j aw shapes, though it is unable to disclose bone density. Previous dynamic jaw models have been used to study jaw movements (Koolstra and van Ei jden, 1995, 1997b) , musc le funct ions (Koolstra and van Ei jden, 1996, 1997a; Langenbach and Hannam, 1999; Peck, 1999; Peck et al., 2000) and temporomand ibu la r jo int funct ions (Peck, 1999; Peck et al., 2000) . One of the advantages of dynamic jaw models is their abi l ity to accept var ious structura l and funct ional parameters to mimic s i tuat ions difficult or imposs ib le to study in vivo. They seem an ideal way to study forces and torques related to stress and strain in artif icial ly created jo ints since the models are usual ly run in mathemat ica l ly and physical ly proscr ibed env i ronments . In the present study, therefore, the fol lowing work ing hypotheses were proposed: 1. In humans , the densest cortical bone is found in sect ions with the least cortical area, and cross-sect ional mass is uni form throughout the entire mand ibu lar corpus and symphys i s . Conf i rmat ion of this hypothes is would suggest that cross-sect ional shear of the corpus is the main loading state of the human mandib le . 2. Regional cort ical rigidit ies (i.e. th ickness and dens i ty) in human jaw cross-sect ions differ with respect to locations, and are assoc iated with local loading condit ions. Conf i rmat ion of this hypothes is would relate bone regional rigidity to current hypotheses of j aw loading condit ions, and suggest that model ing mandibu lar corpus cross-sect ion requires specif icat ion of these dif ferences. 3. The dist inct shape and or ientat ion of the pig jaw symphys i s compared to that in man are adaptat ions to resist concentrated wishboning stresses and stra ins caused by strong musc le tens ions, long lever a rms and large symphysea l curvatures; these are important structural features to keep stresses and stra ins within the funct ional to lerance that bone t issue susta ins. Conf i rmat ion of these hypotheses would contr ibute further ev idence to exist ing notions of stra in s imi lar i ty, and improve comprehens ion of the structural and funct ional adaptat ions in mamma l s . 4. Mass propert ies of the mamma l i an j aw can be est imated with computed tomography. Bone density is uneven throughout the mandib le, but mass is d istr ibuted symmetr i ca l ly with respect to the geometr ic center, and mass and moments of inert ia are posit ively a l lometr ic with the jaw d imens ions. Conf i rmat ion of these hypotheses in pig and human mandib les would make approaches less- invasive than CT practical for future est imat ion, and widen the possibi l i ty for individual dynamic model ing in l iving subjects ( including humans , pigs and other extant mamma l s ) . 5. Dynamic models of jaw b iomechanics can be used to predict forces and torques passing through the mand ibu lar symphys i s dur ing s imulated normal funct ion. These forces and torques change in t ime and reflect the complex loading condit ions induced by changing muscle contract ions. Conf i rmat ion of these hypotheses would reinforce the suggest ion that the pig mandibu lar symphys i s is uniquely des igned to accommodate the symphysea l loading, and encourage deve lopment of s imi lar dynamic models in humans. Addit ional ly, fur ther modif icat ion of the model a long s imi lar l ines would be feasible to predict loading patterns in other regions of the jaw, such as the mo lar and canine.  3 CROSS-SECTIONAL BIOMECHANICS OF THE HUMAN MANDIBLE 3.1 ABSTRACT Cross-sect ional analys is of the human mand ibu lar corpus faci l i tates the understanding of its b iomechanica l behavior. Previous studies have focused on the post-canine region only, and have not included the effect regional bone density may have on cross-sect ional mechanics . In this study, eight dry adult human mandib les were scanned with computed tomography (CT). Each mandib le was resl iced digital ly to obtain cross-sect ions at the first molar, canine, and symphys i s . Binary and grayscale total and cortical sect ions at each location were segmented . The cortical sect ion was further segmented into l ingual, facial and basal aspects. CT grayscale va lues were used as indices for regional bone density. The cross-sect ional area and mass , second moments of area and mass were measured . Cort ical and bending indices were also ca lcu lated. Paired t-tests (with Bonferroni 's correct ion) were used to disclose signif icant di f ferences among cross- sect ions at the three locations, and among the three regions of the cortical sect ions. Though cross-sect ional areas var ied among the three locations, their masses were s imi lar, suggest ing uni form shear rigidity. The distr ibut ion pattern of the cortical bone for each cross-sect ion seems des igned to withstand the specif ic stress pattern at that locat ion. S ince sagittal bending and torsion are the main stresses at the molar region, the cross-sect ions follow a hollow ovoidal shape with its long axis or iented a lmost vert ical ly. In the symphysea l region, wishboning is the main source of stress, and this sect ion had the most robust bone on its l ingual aspect. The canine region represented a transit ion between the molar and symphys i s . In addit ion to sagittal bending, tooth- loading seems associated with basal bone robust ic i ty. The bending indices indicated little shape di f ferences among the three locations. The high grayscale cortical indices at the molar sect ion signif ied less t rabeculat ion. This study suggests , when model ing the mandibu lar corpus, cortical bone distr ibut ion, regional bone densi ty, and trabeculat ion all need to be taken into account. 3.2 INTRODUCTION During the power stroke of mast icat ion, the mandib le is subjected to forces produced by the jaw musc les and grav i ty, react ion forces at the temporomand ibu la r jo ints, and react ion forces at teeth . These forces generate stresses and strains along the mand ibu lar corpus and symphys i s . The stresses can be ana lyzed indiv idual ly by means of principles borrowed from the mechanics of mater ia ls . A number of stress- inducing condit ions can occur in the mand ibu lar corpus and symphys i s dur ing funct ion. They include bending, tors ion, and shear of the corpus and the symphys i s . Sagitta l bending of the corpus is due to the vert ical components of bite force, musc le force and condylar and symphysea l react ion forces act ing in the tangent ia l plane of the mand ibu lar corpus (i.e. an orthogonal plane to both the longitudinal and t ransverse planes; Weijs, 1989) . During uni lateral bit ing, sagittal bending occurs on both s ides of the mandib le . On the work ing-s ide, sagittal bending causes tension along the lower border, and compress ion along the a lveo lar side of the mandibu lar corpus. A reverse bending moment which tenses the a lveolar processes and compresses the lower border of the mandib le occurs on the balancing-side (Hylander, 1979b; Kor ioth et al., 1992; van Ei jden, 2000; Weijs, 1989) . The ideal structure responding to this part icular load is a relat ively deep corpus in the mo lar region, which increases the cross-sect ional second moment of inertia with respect to the t ransverse axis of the corpus (Daegl ing and Gr ine, 1991) . Sagit ta l bending of the corpus also induces shear ing stresses along the entire length of the mandib le (Weijs, 1989) . Shear ing forces attain the largest va lues in the mandibu lar region between bite force and musc le force on the working-s ide and in region between musc le force and jo int force on the balancing-side (van Ei jden, 2000) . Shear ing stress is inversely proport ional to the cross-sect ional area of the mandibu lar corpus, irrespect ive of its shape. Hence a m in imum crit ical amount of bony mater ia l is required along the ent ire mand ibu lar corpus. Medial t ransverse bending of the corpus occurs dur ing the jaw opening phase and lateral t ransverse bending of the corpus occurs during the jaw closing phase (Hylander, 1985; Hy lander and Johnson, 1994) . Both lateral bending moments become largest at the symphys i s . Thus the stresses caused by them are quite low at mo lar region of the mandibu lar corpus but grow larger towards the symphys i s in a l inear manner (Daegl ing and Gr ine, 1991) . Tors ion of the mandibu lar corpus occurs on the work ing-s ide dur ing the power stroke of mast icat ion, and on both s ides of the j aw dur ing the power stroke of ingest ion. In both cases, the twist ing tends to evert the lower border of the mandib le, and inverts the a lveolar process. On the work ing-s ide, this effect is part ial ly reduced by the resultant mast icatory bite force which tends to twist the corpus in opposite direct ion (Hylander, 1979b) . These twist ing moment s are highest in the molar regions (Daegl ing and Gr ine, 1991) . To counter this kind of stress effectively, both the cross-sect ional shape and bone distr ibut ion are cr it ical, and a circular hollow sect ion with a max imum possible external d imens ion is ideal. Medial t ransverse bending of the symphys i s is due main ly to the bi lateral contract ion of the lateral pterygoid musc les dur ing the jaw opening phase of the chewing cycle (Hylander, 1985) . Contract ion of the media l pterygoid musc les may also contr ibute to this effect dur ing the jaw closing phase (Hylander and Johnson, 1994) . Lateral t ransverse bending (or wishboning) of the symphys i s is main ly due to force f rom the deep masseter musc le on the balancing-s ide at the end of the power stroke, and to lateral components of bite force on the work ing-s ide, t ransverse components in the work ing-s ide j aw closing muscle forces (Hylander, 1984; Hylander and Johnson, 1994) , and possibly the react ion force appl ied to the media l pole of the work ing- side condy lar head by the medial wall of the condy lar fossa (Figure 3.1, p51). Medial t ransverse bending of the j aw symphys i s produces tension along the facial surface and compress ion along the l ingual surface in the symphys i s (Hylander, 1984, 1985) . Wishboning creates tensi le stresses on the l ingual surface and compress ive stress on the facial surface of the symphys i s . In a curved beam like the mandib le , these stresses are not l inear and the tensi le stresses on the concave s ide are h igher than the compress ive stresses on the convex side (Figure 3.1, p51) . In pr imates, these tensi le stresses can be 2.5 t imes larger than compress ive stresses (Hylander, 1985; Ravosa and S imons , 1994) . They can be resisted by synostos is (Ravosa, 1996, 1999; V inyard and Figure 3.1 Suggested mechanism of wishboning in the human mandible. The main active forces are the balancing-side deep masseter (Fmb) and probably the transverse component to the working-side jaw closing muscle force (Fmw). Reaction forces from occlusion (Fb) and medial condylar pole (Fc) are the passive forces. Fb and Fc act in opposite direction of Fm. The force resultant tends to bend the mandible in its plane of curvature, causing tension on its lingual side and compression on its facial side. All force vectors indicate directions only. Their magnitudes are unknown. Based on Hylander and Johnson (1994). Ravosa, 1998) . Medial t ransverse bending of the symphys i s causes reversed wishboning effect and the induced stresses are relat ively low compared to the wishboning stresses (Hylander, 1985) . The cross- sect ional area of bone and symphysea l shape affect the jaw's resistance to these wishboning stresses (Hylander, 1984, 1985) . Symphysea l dorsoventra l shear is caused by the balancing-s ide jaw muscle force and due to the downward and upward movements , respect ively, of the working-s ide and balancing-s ide mand ibu lar corpora (Hylander, 1975, 1977, 1979a, b; Ravosa, 1996, 1999; Ravosa and S imons , 1994) . Adequate cross-sect ional area of bone in the plane of stress is needed to resist dorsoventra l shear. There are a number of ways to ach ieve a mechanica l ly robust mandib le. The two ext remes are to add cortical bone within the endosteal marg ins whi le external cross-sect ional d imens ions remain constant, and to increase the corpus d imens ion without adding addit ional compact bone. Whi le the former is ineff icient because mater ia l is added in regions where bending and tors ional s t resses are low, the latter is eff icient in te rms of mater ia l cost. How the mandib le is des igned in nature may depend upon species. Pr imate studies (Dechow and Hylander, 2000; Hylander, 1979b) have demonst ra ted that the mandib le is built so as to use its bony mater ia l more economica l ly than a solid rod of s imi lar r igidity. It is possible to quanti fy the amount of s t ress-bear ing mater ia l in the mandibu lar corpus cross-sect ions through the nondestruct ive technique of computed tomography (CT) (Daegl ing, 1989; Daegl ing and Gr ine, 1991) . This method also makes it feasible to est imate regional bone density. However, previous studies involv ing hominoid mandib les (Daegl ing, 1989; Daegl ing e r a / . , 1992; Daegl ing and Gr ine, 1991; Daegl ing and Hylander, 1998) have been focused on the cross- sectional b iomechanics of molar region only, and have not included bone density in their ana lyses. The importance of bone densi ty in such studies has been emphas ized previously (Daegl ing er a/., 1992; Daegl ing and Hylander, 1998) . In the present study, we evaluated the b iomechanica l s ignif icance of modern human mandib les at three sites including the first molar, the canine and the symphys is . Specif ical ly, we tested the fol lowing hypotheses: that the densest bone occurs at cross-sect ions with the least cortical a rea; that the cross-sect ional mass is uni form throughout the whole mand ibu lar corpus; and that regional cort ical rigidity (th ickness and density) in a cross-sect ion differs with respect to different regions, and is assoc iated with local loading condit ions. Conf i rmat ion of these postulates would improve our understand ing of human jaw b iomechanics, and contr ibute to the more appropr iate models of the human mandibu lar corpus. 3.3 MATERIALS AND METHODS 3.3.1 Computed tomography The exper iments were carr ied out on eight osseous spec imens with adult dent i t ions. They were selected f rom an exist ing col lect ion of modern human mandib les, in which the gender and precise age of each spec imen were unknown. Use of this archival mater ia l compl ied with the requ i rements of The Univers i ty of Brit ish Co lumbia 's Ethical Review Commi t tee . All spec imens were placed in a plast ic box f ixed with wooden spacers and remained in water dur ing CT scann ing. High resolut ion CT sl ices with 512x512 pixels and voxe l s ize of 0.43 x 0.43 x 1 m m 3 were obta ined with a Toshiba Xpress SX scanner (kV=100 , mA=150 ; Toshiba Corp., Tokyo, Japan) at The Univers i ty of Brit ish Co lumbia . A l though the criteria for posit ioning the mandib les in the plastic box were theoret ical ly not cr it ical, we arranged all mandib les to be imaged in the coronal plane to save space, i.e. they over lapped anteroposter ior ly without contact. 3.3.2 Image processing The original raw images were s igned big end ian 16-bit data. Because the image processing program we used (3Dviewnix, Univers i ty of Pennsylvania Medical Center, Phi ladelphia, PA) did not accept minus 16-bit numbers , we wrote a dedicated PC program (RIC - Raw Image Converter , Craniofacial Laboratory, The Univers i ty of Brit ish Co lumbia , avai lable onl ine at http://condor.dent istry.ubc.ca, or see Append ix , p222) to convert the original 16-bit images into 8-bit raw images. The 8-bit image fi les were then imported into 3Dviewnix running on an SGI Indigo Extreme workstat ion (Si l icon Graph ics Inc., Mountain V iew, CA) . Each mandib le was segmented and saved as a single fi le. Cross-sect ional sl ices at the left first molar ( M l ) , the left canine (CA), and the symphys i s (SY) regions were obta ined by resl ic ing. The resl icing plane was or iented so that it was located at the center of the structure paral lel to its long axis, and perpendicu lar to the facial surface at that region (Figure 3.2, p55) . This procedure ensured a true cross-sect ion at each location. Each slice was saved as uncompressed Microsoft Windows (Microsoft Corp., Redmond, WA) b i tmap file for further process ing on a desktop PC (Dual Pent ium III 450 MHz) .  Commerc ia l software (Paint Shop Pro 7, Jasc Sof tware, Inc., Eden Prairie, MN) was used to complete further segmentat ion . Each sect ion was first or iented so that its anatomica l major axis was vert ical in the image matr ix . Four groups of segmented images were created. They included grayscale total sect ions, grayscale cort ical sect ions, binary total sect ions and binary cortical sect ions. For graysca le sect ions, inhomogenei t ies in cortical bone and porosit ies in cancel lous bone were left intact; for binary sect ions, the enclosed areas were fil led with white pixels. The cortical sect ions did not include any cancel lous bone or marrow spaces. This was accompl ished by means of a f reehand select ion tool , which enabled all internal s t ructures ( including t rabecular br idges) to be removed. The resultant sect ions thus provided e i ther open or c losed hollow models . To obtain binary cortical sect ions, we selected the edges of the cortex contour, deleted the cortical contents, and converted this area to pure black, then inverted the cortex. This ensured that the cortex was a unique white area. To obtain binary total sect ions, we took the binary cort ical sect ions, capped the cross-sect ions with a single pixel line across the top marg in of the alveol i (see Daegl ing, 1989) and per formed the same trac ing and invert ing method descr ibed above. These final total sect ions were closed sect ions with unique white contents (Figure 3.3, p57) . To reveal regional di f ferences within each cortex, the graysca le cortex was further segmented into l ingual, facial and basal aspects. The l ingual and facial aspects were split by the major ax is , whi le the basal cortex was considered to be the lower part below a horizontal line at the upper edge of the basal cortex (Figure 3.4, p58) . 3.3.3 Cross-sectional measurements A B D C A S Y c « M 1 0 • Figure 3.3 Three typical cross-sections at the canine (CA), the symphysis (SY) and the first molar (Ml). Vertically, A: original sections; B: grayscale total sections; C: grayscale cortical sections; D: binary cortical sections; E: binary total sections. For all sections, left side is lingual and right is facial. Figure 3.4 Definition of regional cortical thickness. L i , L2 and L3 represent the three levels for the facial and lingual cortical measurements, from which the mean cortical thicknesses were calculated; L4 denotes the location where the basal cortical thickness was measured. L4 is also the line that separated the facial and lingual cortices and L3, the line where the basal cortex was detached. For each cross-sect ion, the cross-sect ional area and second moments of area (Ix, Iy) were measured , and the cort ical index (a ratio between cortical area and total area, CI) , and the bending index (BI, Iy/Ix) were also calculated. For each graysca le cross-sect ion, the mean grayscale va lue (MGSV) , cross-sect ional mass , and second moments of mass ( Ixm, Iym) were measured , and the mass cortical index (CIM) and the mass bending index (BIM, Iym/Ixm) were then ca lcu lated. All measurements were per formed digital ly at the pixel level by a custom program (Cal image - Calculate Image, Craniofacia l Laboratory, The Univers ity of Brit ish Co lumb ia; avai lable f rom http://condor.dent istrv.ubc.ca, or see Append ix , p226) . This program batch-processed all image fi les, and output the results in c o m m a - del imited text formats, which were then imported into Microsoft Excel® (Microsoft Corp. , Redmond, WA) . We wrote this program with Bor land C++ Bui lder 5.0 (Impr ise Corp., Scot ts Va l ley, CA) . All area and second moment calculat ions were made accord ing to convent ional formulae (Beer and Johnston, 1988) . To assess regional var iat ions within each graysca le cort ical sect ion, we calculated the MGSV, and measured the mean cortical bone th ickness in each of the three regions. The mean cortical bone th ickness was measured on the binary cort ical cross-sect ion. We div ided this d istance (from below the tooth root apex towards the upper edge of the basal cortex) into two port ions, and per formed three measurements , each represent ing the horizontal th ickness of the cortex at that locat ion. A mean value was calculated to represent the mean cortical bone th ickness for each side. The cort ical th ickness at the basal aspect was measured f rom the point where the major axis intersects with the basal cortex. The th ickness mult ip l ied by its corresponding MGSV provided the cortical r igidity index (CRI), considered an indicator for cortical bone rigidity (Figure 3.4, p58) . 3.3.4 Statistical Analysis Since the image processing included subject ive segmentat ion operat ions, we carr ied out an error study, in which we al lowed two persons to perform the same segmentat ion processes accord ing to the same cr i ter ia. We then calculated the number of pixels included in each image, and appl ied paired t-tests on the two sets of images. For operator one, the mean number of pixels was 1201 (SD 264) , whi le for operator two, it was 1192 (SD 229) . There was no stat ist ical dif ference between the two samples (P>0.05) . S ince we were interested in compar ing the b iomechanica l propert ies of the cross-sect ions at the three regions ( M l , CA and SY) , and as each site was represented by a group, paired t-tests with Bonferroni 's inequal i ty correct ions (B-method) were used to indicate any stat ist ical ly signif icant dif ference (at the 5% level) for each measured or calculated parameter . This correct ion reduced the chance for false posit ive results. The same test was used to detect regional di f ferences between the l ingual, facial and basal aspects of the graysca le cortical sect ion. To test whether cortical bone distr ibut ion was isometr ic , one sample t-tests were used for di f ferences between the geometr i c centers of binary cortical and total cross-sect ions versus zero. Finally, the areas and moments of inertia predicted by ideal sol id and hollow el l ipse models (i.e. with the ca lculated vert ical and horizontal d imens ions represent ing their respect ive majo r and minor axes, and mean cortical th ickness represent ing the uni form wall th ickness) were compared by means of paired t-tests to the measured binary total and cortical areas and moments of inert ia. All stat ist ical ana lyses were carr ied out with S P S S 8 (SPSS Inc., Chicago, IL). 3.4 RESULTS 3.4.1 Binary cross-sections Descr ipt ive stat ist ics for the areas, second moment s of areas, cortical indices, bending indices, and B-method paired t-test results are presented in Table 3.1 (p62). There were no stat ist ical ly signif icant di f ferences between cross-sect ions at CA and M l , or between CA and SY for all area and moment measurements . However, the cross- sect ions at M l differed f rom SY in total area, and in total Ix. In each case, cross-sect ions at M l were greater than at SY. Dif ferences were also found between cortical indices for cross-sect ions at M l and SY. The cross-sect ions at M l had the least relat ive cort ical bone. The bending indices for both cortical and total sect ions were s imi lar through the entire mandibu lar corpus. 3.4.2 Grayscale cross-sections Table 3.2 (p63) provides the results of measurements , and B- method paired t-tests for the graysca le cross-sect ions. Unl ike the binary cross-sect ions, more dif ferences were d isc losed. First, MGSV of M l was the greatest, and there was no MGSV dif ference between CA and SY. Though the total area at M l was the least, its total mass , and Table 3.1 Area (cm2), moment of area (cm4), bending and cortical indices for the binary total and cortical cross-sections. Also shown here are the paired t test (B-method) p values. Blank spaces indicate non-significant comparisons. Abbreviations: CA, canine; M l , first molar; SY, symphysis. Ix and Iy, moments of inertia around x and y axes respectively; BI, bending index; CI, cortical index. C A M l SY Paired t test Mean SD Mean SD Mean SD CAvs. M l CAvs. SY M l vs. SY Total Area 2.83 0.37 2.98 0.48 2.63 0.40 0.03 Ix 1.44 0.55 1.52 0.56 1.16 0.44 0.03 iy 0.36 0.11 0.37 0.13 0.31 0.09 BI 0.26 0.08 0.25 0.06 0.28 0.09 Cortical Area 1.78 0.45 1.53 0.21 1.64 0.27 Ix 1.10 0.63 0.86 0.34 0.90 0.36 iy 0.30 0.10 0.30 0.11 0.27 0.08 BI 0.31 0.10 0.36 0.09 0.31 0.09 CI 0.63 0.11 0.52 0.03 0.63 0.06 <0.01 Table 3.2 Area (cm 2) and mass (cm 2), MGSV, second moment of area (cnv») and second moment of mass (cmt), bending and mass bending indices, and cortical and mass cortical indices of the grayscale total and cortical cross-sections. Also shown here are the paired t test (B-method) p values. Blank spaces indicate non-significant comparisons. Abbreviations: CA, canine; M l , first molar; SY, symphysis. Ix and Iy, moments of inertia around x and y axes respectively; BI, bending index; CI, cortical index; Ixm and lym, mass moments of inertia around x and y axes respectively; BIM, mass bending index; CIM, mass cortical index; MGSV, mean grayscale value. C A M l SY Paired t test Mean SD Mean SD Mean SD CAvs. M l CAvs. SY M l vs. SY Total Area 3.17 0.52 2.4 0.44 3.17 0.47 <0.01 <0.01 Mass 125.2 25.8 110.7 20.9 117.6 19.3 M G S V 39.26 3.18 46.07 1.74 37.08 2.65 <0.01 <0.01 Ix 1.77 0.84 1.23 0.53 1.66 0.59 0.03 0.03 iy 0.57 0.14 0.45 0.15 0.5 0.14 Ixm 71 37.1 60.39 26.4 53.86 20.2 l y m 18.25 6.62 19.93 7.58 15.92 5.32 BI 0.35 0.1 0.38 0.09 0.32 0.11 B I M 0.28 0.1 0.34 0.1 0.31 0.1 Cortical Area 2.43 0.51 1.9 0.24 2.29 0.32 0.03 <0.01 Mass 105.5 27.9 102.7 16.1 89.13 16.4 0.01 M G S V 43.13 3.76 53.91 3.79 38.87 3.22 <0.01 <0.01 Ix 1.62 0.83 1.12 0.4 1.46 0.5 0.04 iy 0.53 0.13 0.42 0.13 0.46 0.13 Ixm 65.36 38.4 58.11 24 47.79 18.4 0.03 lym 17.07 6.25 19.43 7.28 14.59 4.9 BI 0.36 0.1 0.39 0.09 0.34 0.11 B I M 0.3 0.11 0.34 0.1 0.32 0.09 CI 0.76 0.06 0.8 0.07 0.72 0.04 C I M 0.84 0.07 0.93 0.04 0.76 0.05 0.02 0.03 <0.01 CO second moments of inertia approached those of the other two sect ions. A s imi lar pattern was seen for cortical sect ions, except for cortical mass and second moment of mass with respect to the t ransverse axis (Ixm) at SY (which showed an opposite relat ionship to those at M l ) . The bending, and mass bending indices were s imi lar for the total and cortical cross-sect ions. While no cortical index di f ferences were revealed among the three sect ions, their mass cort ical indices differed between each pair. 3.4.3 Cortical thickness, density and rigidity index Table 3.3 (p65) summar i zes the means, s tandard deviat ions, and paired t test results for MGSV, cortical th ickness, and the cortical rigidity index for the l ingual, facial and basal regions of the graysca le cortex. For all cross-sect ions, MGSV was signif icantly h igher at the basal aspect than the facial and l ingual aspects. Whi le facial and l ingual MGSVs did not differ in CA and M l sect ions, MGSV in the l ingual aspect was higher than that at the facial aspect for SY sect ion. The cortical th ickness of the basal aspect was greatest in the CA and M l sect ions. In M l , the cortical th ickness was equal for both facial and l ingual aspects. However, it was th icker on the l ingual aspect in both CA and SY sect ions. The basal CRI was the greatest for both CA and M l sect ions. It was, however, a lmost equal to the l ingual CRI in the SY sect ion. There were no CRI di f ferences between the facial and l ingual regions at M l , or between the l ingual and basal regions at SY. Table 3.3 MGSV, cortical thickness and cortical rigidity index of the lingual, facial and basal cortices. Also shown are paired t test p values (B-method) between lingual and facial, lingual and basal, and facial and basal cortices at the three locations. Abbreviations are in the text. Blank spaces indicate non- significant comparisons. Abbreviations: CA, canine; M i , first molar; SY, symphysis; MGSV, mean grayscale value. M G S V Thickness CRI Mean SD P Mean SD P Mean SD P C A Lingual 39.1 1.9 3.74 0.67 146.05 28.05 Facial 39.6 5.6 2.3 0.35 92.08 23.43 Basal 53.2 3.2 4.54 0.74 242.02 44.38 Facial vs. lingual <0.01 <0.01 Lingual vs. basal <0.01 0.01 <0.01 Facial vs. basal <0.01 <0.01 <0.01 M l Lingual 46.7 2.3 2.68 0.24 125.15 13.03 Facial 45.2 4.8 2.6 0.42 118.2 26.48 Basal 60.6 4.2 4.14 0.69 252.12 53.26 Facial vs. lingual Lingual vs. basal <0.01 <0.01 <0.01 Facial vs. basal <0.01 <0.01 <0.01 SY Lingual 40.4 3 3.64 0.69 148.35 39.23 Facial 33.3 2.8 2.47 0.26 81.9 9.39 Basal 46.1 4.5 3.17 0.72 148.3 44.59 Facial vs. lingual <0.01 0.01 0.01 Lingual vs. basal 0.02 Facial vs. basal <0.01 0.02 3.4.4 Bone isometry The dif ferences between the cortical geometr ic centers, and the total geometr ic centers at CA, M l and SY were 1.30±0.62 m m , 1.96±0.41 m m , and 0.65+0.59 m m , respect ively. T-tests against zero showed they all differed f rom zero (p<0.05) , though it seemed SY was more isometr ic than others. 3.4.5 Predictions by ideal models Ratios between areas and moments of inertia predicted by ell iptical models, and the respect ive actual measurements , are shown in Table 3.4 (p67) . T-tests against unity indicated most rat ios differed f rom unity except for cortical Iy at CA. 3.5 DISCUSSION 3.5.1 Error of method There was l ikely min imal error in our area and moments of area calculat ions because they were performed at the pixel level; however, the number of pixels involved in such calculat ion is cr i t ical , and would have been affected by subject ive segmentat ion . We did not trace the images on paper as has been done previously (Daegl ing, 1989; Daegl ing and Gr ine, 1991), but the segmentat ion process was nevertheless arbi trary. For example , when we removed a tooth f rom the cross-sect ion, we manipulated the br ightness and contrast of the moni tor to opt imize edge def init ion, and to min imize errors. To keep the original graysca le va lues intact, however, we did not perform any image color operat ions which would have altered the original graysca le Table 3.4 Ratios between area and moments of inertia predicted elliptical models and the respective actual measurements. T tests were performed against a constant of unity. Abbreviations: CA, canine; M i , first molar; SY, symphysis. Ix and Iy, moments of inertia around x and y axes respectively. Total Cortical Area Ix iy Area Ix iy C A Mean 1.18 1.18 1.40 0.62 0.79 1.00 SD 0.14 0.20 0.31 0.02 0.06 0.18 P 0.01 0.04 0.01 <0.01 <0.01 0.97 M l Mean 0.94 0.83 0.89 0.59 0.70 0.65 SD 0.03 0.04 0.06 0.05 0.08 0.05 P <0.01 <0.01 <0.01 <0.01 <0.01 <0.01 SY Mean 1.12 1.11 1.29 0.56 0.66 0.83 SD 0.06 0.11 0.11 0.06 0.10 0.06 P <0.01 0.02 <0.01 <0.01 <0.01 <0.01 values. We def ined the edge for cortical and cancel lous bone in the same way. Also, our inter-operator error for segmentat ion was low. To min imize automat ic edge detect ion errors in Jasc Pa intShop Pro, we selected a to lerance value of 20 for all cross-sect ions, i.e. it was reproducible. The select ion of the resl icing plane used to obtain our cross- sect ions was arbi trary, and also subject to human error. The mult ip le-compar ison problem (Fisher and van Bel le, 1993) could have been signif icant if no correct ion had been carr ied out. This problem occurs when many statist ical procedures are being appl ied to the same data . It was for this reason we per formed Bonferroni 's inequal ity correct ions on our paired t-test results, to min imize the chances of incorrect ly reject ing the null hypotheses. 3.5.2 Significance of cross-sectional measurements Since stress is def ined as internal resistance provided by a unit area (Mott, 1996), cross-sect ional area is one of the most important measurements in the mater ia l mechanics for counter ing normal (direct axial) and shear stresses (Hearn, 1997; Mott, 1996; van Ei jden, 2000) . Our data suggest that though cross-sect ional area var ies among the three sect ions, their total masses remain surpr is ing ly uniform throughout the corpus, i.e. in regions with smal le r areas such as the molars, denser bone is required (Table 3.2, p63) , match ing a previous f inding that mand ibu lar apparent densi ty is negat ive ly corre lated with the cross-sect ional area (Kingsmi l l and Boyde, 1998) . This might lead to an immedia te conclusion that the mandibu lar corpus is uni form in its abi l i ty to resist normal and shear stresses. The reason denser„bone has better shear rigidity can be expla ined theoret ica l ly. It has been reported that highly mineral ized bone has high Young's modulus (Currey, 1984a) , and the bone's shear modulus lies between one third to half of the elast ic modulus (van Ei jden, 2000) . The area parameter , however, does not reflect the distr ibut ion of mater ia l , and axial loads (i.e. dur ing anteroposter ior shear) are not the main sources of stress. It is obvious that the differential distr ibut ion of bony mater ia l is more important to changes in mandibu lar mechanica l propert ies than modif icat ion in the amount of compact bone (Daegl ing and Gr ine, 1991) . For example , adding more cortical bone in the center of the mandibu lar corpus does not have the same effect as adding the same bone to the per iphery. The second moment of area is a measure of the distr ibut ion of bone around a part icular axis. By distr ibut ing bone as far as possible f rom the neutral axis of the cross-sect ion, the momen t of area can be increased without an increase in mater ia l . In a cross-sect ion with a large cross-sect ional moment of area, stress can be kept relat ively low. Hence, an increase in the cross-sect ional momen t of area is more opt imal to susta in heavy bending loads (van Ei jden, 2000) . For a cross-sect ion with an ovoidal shape like the mand ibu lar corpus, its abi l ity to resist bending about its minor axis (i.e., facio- l ingual axis) is greater than about its major axis (i.e., superoinfer ior ax i s ) . This is the main bending load during mast icat ion. Our data are cons istent with this assumpt ion (Table 3.1, p62, and Table 3.2, p63) . The cort ical index (ratio between cortical and total areas) is a measure of the relat ive amount of cortical bone to the total bone. Different kinds of cortical indices have different mean ings . The cortical index reported by Daegl ing (1989) used the cort ical area (the entire area enclosed by the cortical outl ine jo ined at the a lveo lar marg ins by a one m m thick cap, i.e. a hollow beam) and the total subper iost ium area (the area enclosed by the periosteal border to the a lveolar marg ins, with a straight line connect ing those marg ins , i.e. a solid beam). By def init ion, both areas do not take into account the densi ty or the possible porosity of cortical and cancel lous bone. It would seem that a higher va lue, i.e. relat ively more cortical bone, might indicate a stronger cross-sect ion. This may be mis lead ing, because a solid sect ion of compact bone has a cortical index equal ing unity. This is, of course, not the opt imal des ign for the mandibu lar corpus for it is not efficient, or robust, or economica l . A low cortical index might also indicate a more economica l use of mater ia l , and is a measure of robustic ity (Daegl ing and Gr ine, 1991). In this case, the mo lar sect ion of the human mandib le seems more robust than the canine and symphys i s (Table 3.1, p62) . However, our binary cort ical index for the molar sect ion was a little h igher than that publ ished previous ly (0.50 vs. 0.40, the latter being obtained f rom five female and f ive male human mandib les; Daegl ing, 1989) . This may have been due to the different trac ing methods used in the two studies (digital vs . paper) . The indices were even higher for the canine and symphys i s sect ions. The cort ical indices (ratios between graysca le cort ical and total cross-sect ional areas or masses) we introduced in the present study reflected the degree of cross-sect ional t rabeculat ion. It is obv ious that the va lues of these indices should be higher than that of the binary cortical index. Our grayscale cortical indices at the three locations were s imi lar. However, their mass cortical indices were different, the highest va lue occurr ing at the molar sect ion, and the lowest at the symphys i s . In other words, the symphys i s had more trabeculat ion than the other two locations (Table 3.2, p63) . The bending index is a shape indicator, because the size factor is e l iminated (Daegl ing, 1989), and a low value signif ies an increased abil ity to resist bending stress about the short axis, with the loss of abi l ity to resist bending stress about the long axis. It is also a tors ional rigidity index, because if the size is constant, a bending index of unity indicates a rounded cross-sect ion, which is an ideal des ign to susta in tors ional stress (van Ei jden, 2000) . Therefore, there are two b iomechanica l consequences of a high bending index: an enhanced resistance to t ransverse bending rigidity and a more eff ic ient shape for tors ional rigidity (Daegl ing and Gr ine, 1991) . This index cannot be used to compare absolute bone rigidity. Our data revealed no bending index di f ferences between any two cross-sect ions e i ther binary or graysca le, total or cort ical . The molar cross-sect ion appears to be more circular, suggest ing its tors ional rigidity is increased. One can easi ly conclude, however, that shape di f ferences among the three locations are min imal (Table 3.2, p63) . The graysca le va lue is an indicator of bone dens i ty. The absolute va lues themse lves are not useful for compar ison between two different CT scans unless these have been ca l ibrated. However, the var iat ion in the va lue for individual CT scans indicates bone densi ty var iat ion, because there is a l inear relat ionship between graysca le va lue and bone physical densi ty ( Lampmann e r a / . , 1984; Zhang e r a / . , 2001a) . With a cal ibrat ion phantom, true densi ty approx imat ion is possible for each pixel , and the mass for the mandib le can also be est imated (Zhang er a/., 2001a) . It is common ly recognized that bone mineral densi ty is a cons istent predictor of bone strength for cortical bone (Currey, 1984a; Lang er a/., 1997; Martin and Ish ida, 1989; S tens t rom er a/., 2000) . Therefore, all dens i ty-weighted measurements in this study appear va l id . These include the mass , second moments of mass , mass bending index for both the cortical and total cross-sect ions, and the mass cort ical index. The inconsistency of the results for binary and graysca le cross-sect ional measurements in Table 3.1 (p62) and Table 3.2 (p63) veri f ies that bone regional densi ty needs to be taken into cons iderat ion. We used our cortical rigidity index to compare cort ices at different regions. The regional di f ferences in cortical bone densit ies found in the human mandibu lar cross-sect ion encouraged us to explore a new parameter combin ing both cortical th ickness and regional bone density. We supposed the cortical bone increased its rigidity in dif ferent ways, one was by th ickness, where space was not an issue (e.g. at the symphys i s ) , or where the kind of received stress demanded it. Unfortunately, the relat ionship between bone densi ty and bone mechanica l propert ies is controvers ia l (Carter and Hayes, 1976; Currey, 1984a; Mart in and Ishida, 1989), a l though denser bone tends to be more rigid (Carter and Hayes, 1976; Lang e t a / . , 1997; Turner, 1989; van Ei jden, 2000) . Nevertheless, we suggest our cort ical rigidity index is at least posit ively related to regional cort ical r igidity. 3.5.3 Cross-sectional design of the human mandible In genera l , the corpus cross-sect ion is ovo ida l . A l though it appears to be anatomica l ly "hol low" in that the cortical bone is d istr ibuted only at the per iphery of the sect ion, extens ive trabeculat ion is rout inely found within the interior of the corpus, suggest ing the corpus may not behave as a hollow beam during funct ion (Daegl ing, 1989) . Trabeculat ion is bel ieved to counter or d iss ipate s t resses in bone (Currey, 1984b; Lanyon, 1974). A l though the cross-sect ional masses were quite homogeneous in - 72 - the three locations, their bone distr ibut ions var ied , suggest ing specif ic des igns may be required for different loading condit ions. 3.5.3.1 The molar region There are three stress-bear ing load reg imens other than vert ical shear in this region: sagittal bending, tors ion and t ransverse bending. Whi le t ransverse bending requires a t ransverse ly increased d imens ion, it causes very low stress in the molar region (Daegl ing and Gr ine, 1991) . Sagit ta l bending calls for a corpus sect ion with a relat ively larger vert ical d imens ion, and it is high in this area (Wei js, 1989) . In this case, bending index should differ f rom unity. Our reported bending indices for this region were only 0.25-0.39 (Table 3.1, p62, and Table 3.2, p63) . Tors ion requires c ircular cross-sect ional f o rm. In such a fo rm, the polar moment of inertia is a determinant of its abi l i ty to resist tors ion (Daegl ing, 1989; van Ei jden, 2000) . The polar momen t of inertia takes into account not only the amount of cortical bone area , but also the disposit ion of the cortical bone with respect to the center of mass . In the human mandib le however, truly c ircular fo rms do not exist. For a thin-wal led tube of arbitrary cross-sect ional shape with var iab le wal l - th ickness (l ike the mandibu lar cross-sect ion) subject to tors ion, a model based on shear flow theory may be more suitable, i.e. one in which the product of shear stress and wal l - th ickness at any location is constant. The largest shear stress occurs where the wal l - th ickness of the tube is smal lest (Gere and T imoshenko, 1990) . A mand ibu lar cross-sect ion may be modeled as either single-cel l or mult i-cel l th in- wal led tubular membe r depending on the dens i ty of t rabeculat ion (see Figure 1.6, p30, and Figure 1.7, p31) . Even so, the c loser to c ircular a sect ion is, the better its abil ity to resist tors ion. The bending index is also a measure of the degree of circular ity, and a tru ly c i rcular sect ion has a bending index of unity. The molar cortical sect ion tended to have the highest bending index of the three sect ions (Table 3.1, p62, and Table 3.2, p63) conf i rms this. Both bending and tors ion require hollow des igns, and the binary cortical index (0.52±0.03) indicated the molar sect ion conformed to this (Table 3.1, p62) . There was no difference in rigidity between the facial and l ingual cort ices in the molar sect ion. Both densi ty and cort ical th ickness were s imi lar. However, the basal cortex was not only the densest , but also the thickest, result ing in a cortical r igidity index twice as high as those in the facial or l ingual regions (Table 3.3, p65) . Whi le th is may be attr ibuted to the high tension or compress ion caused by sagittal bending in this region (Weijs, 1989), here we propose a tooth- loading hypothes is. Bite force is the main react ion force exerted on mandibu lar corpus, and the direct ion of this force var ies dur ing funct ion. These dynamic forces have t ransverse components which bend the facial and l ingual cortex in a way reminiscent of the t ransverse bending seen in the mandibu lar symphys i s . Stress concentrat ions can occur in the basal cortex (Figure 3.5, p75), but unl ike the mand ibu lar symphys i s , part of this bending stress may be diss ipated by the t rabecu lar br idges between the facial and l ingual cort ices (when they ex is t ) . Our impress ion of an associat ion between tooth- loading and basal cort ical rigidity has been reinforced by a CT-scanned edentu lous mandib le we have stud ied, in which the th ickness of the basal cortex was reduced, approach ing those of the facial and l ingual cort ices (Figure 3.6, p76) . The idea also corre lates with previous observat ions that basal bone Tooth loads Compressive stress Figure 3.5 Possible stress distribution caused by tooth-loading. The transverse components of tooth forces bend the facial and lingual cortices. The bending results in compressive stress along the convex surface and tensile stress along the concave surface of the basal cortex. Vector components indicate probable directions only. Figure 3.6 Equivalence of canine (left), first molar (middle) and symphysis (right) cross-sections of an edentulous mandible. For all three sections, left side is lingual and right is facial. See text for significance of their forms. height correlates with a lveolar bone height in human mandib les (Kingsmi l l and Boyde, 1998). 3.5.3.2 The symphysis Wishboning produces higher tensi le stress a long the l ingual cortex than the compress ive stress along facial s ide (Hylander, 1985) . This requires more rigid cortical bone on the l ingual s ide. Our data indicated that in this region, the l ingual cortex was denser and th icker than its facial counterpart (Table 3.3, p65) . Twist ing of the bi lateral mandibu lar corpora generates tension along the infero-l ingual aspect, and incisor bit ing induces vert ical bending (for review, see Hylander, 1984). A l though these hypotheses have most ly been based on anthropoid pr imates, they are probably true for humans . Both loading patterns require rigid cortical bone in the basal reg ion. Our data support this by demonstrat ing the basal cort ical bone was equal ly rigid to the l ingual cortex (Table 3.3, p65) . The edentu lous mandib le also demonst rated a reduct ion in cortical th ickness in the basal aspect after loss of tooth loads, though not in its l ingual aspect (Figure 3.6, p76) . 3.5.3.3 The canine region The loading pattern in this region has been infrequent ly studied a l though it seems to be an area of stress concentrat ion second to the symphys i s . Our study suggests this was a transi t ional area between the molar and symphys i s ; whi le the basal cortex was the most r igid, the rigidity of the l ingual cortex superseded its facial counterpart (Table 3.3, p65) . We suggest this is due to a combinat ion of sagittal bending, tors ion and t ransverse bending. Sagit ta l bending may occur dur ing molar or incisor biting (van Ei jden, 2000) . Twist ing of the corpus can extend to this area and wishboning of the mand ibu lar corpora increases here (Daegl ing and Gr ine, 1991) . It is not surpr is ing therefore that this corner structure is assoc iated with stress concentrat ion. S ince the edentulous mandib le signif ied a reduct ion in cortical th ickness in its basal aspect after tooth loss, tooth- loading may also be a factor (Figure 3.6, p76) . In genera l , our data indicate cortical bone densi ty increases f rom the a lveolar r idge to the basal part, f rom the anter ior region to the molar area, and f rom the facial side to the l ingual aspect, though it might be too premature to draw a final conclus ion because our sample was sma l l . Nevertheless, a study on pig regional bone densi ty indicates the same density pattern (Powell er a/., 1973) , and a macaque study (Dechow and Hylander, 2000) also shows the l ingual cortex is s t ronger than its facial s ide counterpart . 3.5.4 Modeling the mandibular corpus Based on the above d iscuss ion, it seems cort ical bone distr ibut ion and density should be taken into account when model ing the mandibu lar corpus. Neither s imple sol id, nor s imple hol low ell iptical models are appropr iate (Table 3.4, p67) . This f inding is cons istent with previous reports (Daegl ing, 1989; Daegl ing and Hylander, 1998) . Whether trabeculat ion needs to be considered may depend on the regions invo lved. The high grayscale cortical indices for the molar region indicated this region was the least t rabecu lated, whi le the symphys i s was the most (Table 3.2, p63) . We postulate there is a balance between b iomechanica l design and funct ional demand in the molar region. A comparat ive ly large internal space is needed for the mult ip le tooth roots and mandibu lar nerves and vesse ls here, whereas in the symphysea l area, this demand is min ima l . Thus a high mass cortical index in the molar region (94%) may imply economica l space use. The strength of t rabecular bone is much less than that of cortical bone (van Ei jden, 2000) , but these t rabecular br idges, albeit with less- dense cortical bone, may provide adequate and equal eff ic iency, especial ly when regions need to respond to dif ferent loading patterns than those at the molar region. Whether the mandibu lar corpus should be mode led as an open and closed sect ion remains controvers ia l . A l though it seems open because teeth are not part of the mandibu lar bone, an open sect ion only possesses a smal l fract ion of the rigidity of a c losed sect ion in resist ing tors ion (Daegl ing e r a / . , 1992). The mandibu lar sect ion may never be open because teeth seem an integral part of the sect ion, and tors ion can flow from one side of the a lveolar process to the other through them (Figure 3.7, p80) . For sect ions between the teeth , the top of the sect ion is a lways l inked by a lveolar bone. 3.6 SUMMARY AND CONCLUSIONS One a im in engineer ing design is to provide the m a x i m u m stress- bearing abi l ity with m in imum cost in mater ia l and space. The human mandib le appears des igned to withstand a var iety of funct ional loads with the least possible mater ia l and space. Its s t ress-bear ing cortical bone is d istr ibuted at the per iphery, mak ing it suitable for resist ing tors ion and bending. Moreover, the corpus is larger in its vert ical d imens ion, a design which counters high sagittal bending stresses. The var iat ion in cortical rigidity within each cross-sect ion also reflects sound des ign. At the symphys i s and canine, more cort ical bone is found along the l ingual s ide, a suitable strategy to resist w ishbon ing. Figure 3.7 Torsion flow in a section containing a tooth. The tooth is supposed to act as a bridge transmitting torsion from one side to the other. Based on Daegling et al. (1992). T is the applied torque. For cross-sect ions receiving high tooth loads, the basal cortex is more robust, providing high resistance to bending of the facio- l ingual cortex. For the mandibu lar corpus as a whole, the overal l abi l i ty to resist vert ical shear stress appears homogeneous.  4 SYMPHYSEAL MECHANICS IN PIG AND HUMAN MANDIBLES 4.1 ABSTRACT Monkey studies suggest that the fused mand ibu lar symphys i s prevents structural fai lure f rom lateral t ransverse bending, or wishboning, attr ibuted to forces generated by the deep masseter musc le late in the chewing cycle. High symphysea l tensi le s t resses and strains at the symphys i s are related to increased symphysea l curvature, strong jaw musc les, e longated momen t a rms, and decreased symphysea l width in the plane of bend ing. Increases in symphysea l width anteroposter ior ly raise the second momen t of inert ia, and lessen stresses and stra ins. Here, we compared symphysea l mechanics in two mamma l s with dist inct ly different j aw shapes, s izes and symphysea l character ist ics (pigs and humans) . We wished to determine whether induced stress and strain remain s imi lar between these mamma l i an orders, and in part icular, the role of symphysea l or ientat ion, if any, in this process. The exper iments were carr ied out on 10 age-matched pig (Sus scrofa) mandib les ( inc luding six l iving an imals) , and eight modern dentate human jaws . CT and MR imaging were used to der ive re levant bone and musc le parameters , including cross-sect ional moments of inert ia, centroids, moment a rms , and radii of curvature. Est imated symphysea l stresses and stra ins for pigs (8.18 MPa for stress; 818.37 ue for strain) and humans (8.21 MPa for stress; 820.92 us for strain) were marked ly s imi lar, and within the funct ional range previously reported for the pr imate symphys i s (i.e. below 2000 ue). Exper imenta l upright reorientat ion of the pig symphys i s increased its musc le- induced strain to 2258.63 UE, above the highest funct ional strain reported for macaques. The results suggest funct ional equiva lence in stress and strain levels across these mamma l i an orders, and emphas ize the importance of symphysea l or ientat ion in the pig. 4.2 INTRODUCTION The funct ional advantages of unfused and fused mandibu lar symphyses in mamma l s have been reviewed recent ly by L ieberman and Crompton (2000) . The unfused symphys i s , by al lowing independent invers ion and evers ion of the two halves of the mandib le before and dur ing the mast icatory power stroke, enables the steep occluding surfaces of opposing teeth in some mamma l s to match during mast icat ion (Hylander, 1979b; Kal len and Gans, 1972; L ieberman and Crompton , 2000; Oron and Crompton , 1985; Scapino, 1981). In contrast, the fused symphys i s s t rengthens and sti f fens the jaw, reducing its risk of structural fai lure as a result of lateral t ransverse bending ("wishboning"), and f rom dorsoventra l shear stress occurr ing dur ing uni lateral mast icat ion (Hylander, 1984; Hy lander er a/., 2000; Ravosa, 1996; Ravosa and Hylander, 1993; Ravosa and S imons , 1994) . Some mamma l s producing predominant ly vert ical occlusal forces have unfused symphyses , which, despi te the i r mobi l i ty, can transfer dorsal ly-direct ly forces through interdigitat ing rugosit ies. It has been suggested mamma l s with main ly t ransverse ly-or iented occlusal forces tend to have fused symphyses (L ieberman and Crompton , 2000) , though se lendont art iodactyls do not d isplay fus ion, nor do rodents, which have noted t ransverse components in their chewing strokes. Wishboning seems mainly due to de layed act iv i ty in the balancing¬ - 84 - side deep masseter muscle at the end of the mast icatory power stroke, and results in separat ion and t ransverse lateral bending of the two mandibu lar corpora (Hylander, 1975; Hylander e r a / . , 1987; Hylander et a/., 1998, 2000; Hylander and Johnson, 1994; Ravosa, 1999) . Balancing-s ide musc le act ivat ion also encourages upward and downward movements of the balancing and work ing-s ide mandibu lar corpora respect ively, producing dorsoventra l shear stress (Hylander, 1975, 1977, 1979a , b; Ravosa, 1996, 1999; Ravosa and S imons , 1994). Jaw musc le act iv ity in baboons, macaques , owl monkeys and galagos (Hy lander e r a / . , 2000; Hylander and Johnson, 1994) suggests there is an associat ion between wishboning and a need for symphysea l fusion because galagos (with an unfused symphys i s ) do not exhibit the deep masseter act iv i ty character ist ic of w ishbon ing. The forces involved in this form of bending are bel ieved to include lateral ly- directed components of bite force on the work ing-s ide, oppos ing the balancing-side muscle tens ions. Residual tens ions in some re lax ing, work ing-s ide jaw adductors may also contr ibute (Hy lander er a/., 2000; Hylander and Johnson, 1994). In humans, lateral ly-directed force f rom the balancing-s ide masseter l ikely exceeds that of any media l contr ibut ion f rom the medial pterygoid of the same side. Dif ferences in t iming between these musc les dur ing the late intercuspal phase of the chewing cycle argue for the deep masseter as a pr imary contr ibutor to wishboning (Hannam and Wood, 1981) . Symphysea l fusion accompanies deve lopment of a funct ional occ lus ion. Part ia l ly-fused symphyses are c ommon in juven i le an imals , and complete fusion often takes place with the erupt ion of the permanent first molars (Ravosa, 1999) . In the min iature pig, adult- l ike t ransverse mast icatory movements develop after wean ing , and the mandib le is subject to wishboning s imi lar to that in anthropoids (Huang e r a / . , 1994) . Wishboning can create high tensi le stresses and stra ins on the l ingual surface of the symphys i s . In pr imates, these tensi le stresses are two to three t imes larger than compress ive s t resses on the labial surface (Hylander, 1985; Ravosa and S imons , 1994) . They can be resisted by synostos is (Ravosa, 1996, 1999; V inyard and Ravosa, 1998), bony enhancement (e.g. t ransverse tor i) , and increased horizontal or ientat ion of the symphys i s , features often seen in pr imates (Daegl ing, 1993; Hylander, 1984, 1985; Ravosa and S imons , 1994) . The cross-sect ional area of bone, and symphysea l shape determine the jaw's res istance to wishboning (Hylander, 1984, 1985) , and an adequate cross-sect ional area in the plane of stress is needed to resist dorsoventra l shear. Cross-sect ional shape and regional bone density can both be revealed by high-resolut ion computed tomography (CT). This is a useful technique for analys ing homin id mandib les (Daegl ing, 1989), not least because it provides true cross-sect ions at any locat ion, and al lows the segmentat ion of teeth f rom cortical and cancel lous bone. The shape and size of the Cercopithec ine symphys i s appear to be al lometr ic with body size and the width of the mand ibu lar dental arch, i.e. symphysea l width anteroposter ior ly, and its length superoinfer ior ly, scale posit ively a l lometr ic with body size, and negat ive ly with mand ibu lar arch width (Hylander, 1985) . S ince the width of the symphys i s increases more rapidly than its length, there is also a change in symphysea l shape with increasing body s ize. These changes are bel ieved to mainta in funct ional equiva lence in bone stresses and strains across taxa and ontogeny (Hylander, 1985; V inyard and Ravosa, 1998) . During wishboning, for example , tensi le stra ins at the l ingual border of the pr imate symphys i s remain under 2,000 ue, wel l - below 3,000 ue when structural fai lure is possible. This kind of safety factor seems desirable, s ince it has been proposed adapt ive remodel l ing may be insuff icient to cope with funct ional demands in v igorously chewing an imals (Bouvier and Hylander, 1981a; Hylander, 1979b) . Physical ly, the mandib le is assumed to behave like a curved beam, in which the max imum tensi le bending stress ( a m a x ) induced at its symphysea l surface is proport ional to the bending force (F) and the moment a rm (L). The bending force is related to the cross-sect ional size of the balancing deep masseter musc le, and its momen t a rm to the distance between this musc le and the symphys i s . The bending stress induced in a hollow beam is proport ional to the distance between the outer surface of the beam and its centroidal axis. The radius of the (unstressed) neutral axis depends on the mater ia l , the shape of its cross-sect ion, the curvature, and whether the beam is un i form. When a beam shaped like the mandib le is bent lateral ly, its neutral axis dev iates towards the concave side of the centroidal axis (Figure 4.4, p i l l ) . Here, the distance f rom the axis to the surface of the sect ion is denoted as c. Any induced stress is inversely proport ional to the second moment of inertia (I) of the sect ion with respect to an axis perpendicular to the plane of curvature . A stress concentrat ion factor (K) can be der ived, which differs for concave and convex surfaces, and is usual ly found exper imenta l ly (Mott, 1996) . It depends on the ratio between the radius of curvature (R) and the distance f rom the surface to the centroidal axis (R/c). For R/c va lues between 1.2 and 10.0, K ranges between 3.0 and 1.0 for a concave surface. K va lues for the convex surface of hollow ell iptical sect ion are 2.3 to 5.2 t imes smal ler than those for the corresponding concave surface (Hylander, 1985; Mott, 1996; Roark and Young, 1975) . The max imum stress along the concave or convex surface can be expressed as K x F x Lxc . o-max = -T Equation 4.1 The max imum strain (e) on the l ingual surface of the symphys i s can be calculated as £ =  amax Equation 4.2 where E is the modulus of elast ic ity (Young's modu lus) . Recent ly, V inyard and Ravosa (Vinyard and Ravosa , 1998) used some of these physical principles to es t imate relat ive stress magni tudes in pr imate symphyses . Apply ing a formula introduced by Hylander (1985) , they scaled a nominal bite force of 238 N (see Hylander, 1979a) by body mass to est imate musc le bending-forces in different species, the assumpt ion here being bite force, musc le cross- sect ional s ize and body mass scale proport ional ly. The bending moment -a rm in this case was est imated by mand ibu lar length (considered proport ional to the musc le lever a rm) . The non- d imens iona l correct ion factor K was calculated by R/c, where the curvature R was est imated by the mandibu lar arch width, and the distance f rom the centroidal axis to the l ingual surface of the symphys i s (c) by the width of the symphys i s . Because a oc ̂ F^^2i^ > (3 )(b) where a and b represent the symphysea l width and length respect ively, the relat ive magn i tudes of the stresses on the l ingual surfaces of the symphyses were calculated, then compared among species and species of dif ferent ages. V inyard and Ravosa's (1998) results support the idea that changes in symphysea l form dur ing ontogeny, and across species, mainta in funct ional equiva lence in stress levels at the papionin symphys i s . This useful s tudy did not est imate induced strain however, and was l imited by its use of bite force and body mass to es t imate lateral bending forces. Bite forces are the result of synerg ist ic act iv i ty in a number of different, b i lateral ly-coact ivated, jaw-c los ing musc les , some of which have smal l lateral components , whi le others have marked media l ly-d irected ones. Moreover, the lateral or ientat ion of the muscle pr imari ly impl icated in wishboning (the deep masseter) var ies with craniofacial f o rm, depending on the spatial a r rangement between the mandibu lar ramus and zygomat ic arch. The magn i tude (i.e. the max imum possible tension) of this lateral component (the vector responsible for t ransverse bending) thus depends upon the deep masseter 's or ientat ion, and is proport ional to the deep masseter 's cross-sect ional area and degree of act ivat ion ( length changes in this musc le being insignif icant dur ing the late intercuspal phase of the chewing cyc le). Finally, the moment a rm of any lateral force component is best measured from the muscle 's line of act ion (e.g. the midl ine d istance f rom its insert ion site to the symphysea l center. These individual var iables are mult ip l ied dur ing the est imat ion of any t ransverse bending moment . The pig mandib le differs marked ly in shape and size f rom its human counterpart (Figure 4 .1 , p90) . S ince it is long and narrow, it has the propensity for high stress concentrat ions at its symphys i s , i.e. pig Figure 4.1 A growth series of Sus scrofa mandibles (rows one to three) illustrating ontogenetic changes in symphysis, and for comparison, an adult modern human mandible (bottom row). From top to bottom, the pig jaws are aged 250 days, 131 days and 29 days. The mandibles are shown from above (left) and laterally (right). With increasing age, the symphyseal width increases more rapidly than mandibular length and arch width, and the symphysis orients more horizontally. In contrast, the human mandible is shorter, and has a less-curved symphysis which is almost perpendicular to the occlusal plane. mandib les are l ikely to have higher K va lues than human mandib les. Like that in cercopithecines (Hylander, 1985; V inyard and Ravosa, 1998), the pig's symphysea l width appears to be posit ively a l lometr ic with mand ibu lar length, and its width seems to increase more rapidly than its height dur ing growth (unpubl ished observat ions, see Figure 4 .1 , p90) . Other c lear dif ferences between pig and human mandib les of b iomechanica l s ignif icance include the relat ive posit ions and or ientat ions of the mandibu lar ramus and zygomat i c arch (affecting the line of act ion of the deep masseter musc le) , c losing j aw musc le cross-sect ional s izes (affecting bending forces) and the incl ination of the mandibu lar symphys i s (affecting its res istance to wishbon ing) . These craniofacial di f ferences encouraged us to test the proposit ion that the form of the pig symphys i s , like its pr imate counterpart , reduces the l ikel ihood of structural fai lure due to w ishbon ing, quite aside f rom other funct ional advantages the form might have. We expected the physical principles present ly expla in ing var iat ions in form within pr imate species might well expla in some of the di f ferences between orders, at least in pigs and humans. A second, more genera l , reason for compar ing symphysea l b iomechanics in these two morphologica l ly-dist inct jaws is current use of the pig as a model for s tudying funct ion and dysfunct ion in the human jaw and its art iculat ion (Herr ing, 1995; S t rom er a/., 1986) . Understand ing b iomechanica l s imi lar i t ies or di f ferences in structure and funct ion between pig and human jaws would be beneficial in other exper imenta l sett ings. In this study, we used CT imaging to ana lyze the mand ibu lar cross- sect ions, and extended the analyt ical approach used previously by V inyard and Ravosa (1998) to include the jaw's cross-sect ional moments of inert ia, centroidal axes, cross-sect ional s izes of the deep masseter musc les , the lateral component of their d irect ion vectors, and the moment a rms from their l ines of act ion. We combined this informat ion with the stress-concentrat ion factor K, j aw curvature, musc le tens ions and lever a rms, to est imate the symphysea l stress and strain caused by wishboning. Two specif ic hypotheses were tested; first, that the shape and or ientat ion of the pig symphys i s are opt imized to resist the effects of wishboning; second, that s imi lar levels of stress and strain in the pig and human symphys i s are induced by wishboning, despite their di f ferences in fo rm, and remain within a safe funct ional range, thus min imiz ing the risk of structural fai lure. 4.3 MATERIALS AND METHODS 4.3.1 Computed tomography Images of the pig mandib les were taken f rom four dry osseous spec imens, and six l iving an imals (Sus scrofa, aged around eight months) all with mixed dentit ions i.e. with only the first permanent molars erupted. The dry jaw spec imens included two males and two females, and the l iving pigs compr ised three males and three females. This mater ia l was obtained f rom the Univers i ty of Wash ington , and the exper imenta l procedures were approved by the An ima l Care Commi t tee of that Univers i ty. The dry spec imens were placed in a plastic box f ixed with wooden spacers, and remained in water dur ing CT scann ing. High resolut ion CT sl ices with 512x512 pixels and voxe l s izes 0.49 x 0.49 x 1 m m (kV=120, mA=150) were obta ined with a Toshiba Xpress SX scanner (Toshiba Corp., Tokyo, Japan) at the Univers i ty of Brit ish Co lumbia . All mandib les were imaged axia l ly, i.e. coronal planes of sect ion perpendicular to the occlusal plane, proceeding anteroposter ior ly along the corpora. The scans on l iving animals were carr ied out at the Univers i ty of Wash ington with a Genera l Electric High Advantage Tomograph ic Unit (Mi lwaukee, WI) . These scans were also axial to the animal 's head, and provided sl ices at one m m interval (field of v iew 240 x 240 m m , 512x512 pixels, pixel s izes of 0.47 x 0.47 m m , kV=120, mA=180) . The human jaw spec imens included eight dry, adult, dentate mandib les of unknown age and gender. These scans were per formed at the Univers i ty of Brit ish Co lumbia in the same manner as that descr ibed for the dry pig jaws. 4.3.2 Image conversion and preparation The image data were in a s igned, big-endian 16-bit format. We used a custom program (RIC - Raw Image Converter , Craniofacial Laboratory, The Univers ity of Brit ish Co lumb ia; avai lable f rom http://condor.dent istrv.ubc.ca) to convert these into 8-bit images, then per formed manua l segmentat ion of the hard-t issue profi les with 3DViewnix (Univers i ty of Pennsylvania Medical Center , Phi ladelphia, PA) on an SGI Indigo Extreme computer (Si l icon Graph ics Inc., Mountain View, CA) . Midl ine resl ic ing revealed true symphysea l cross-sect ions. The reformatted sl ices were saved as uncompressed b i tmap fi les for further process ing on a desktop computer (Dual Pent ium III 450 MHz). Commerc ia l software (Paint Shop Pro 7, Jasc Sof tware, Inc., Eden Prairie, MN) was used to segment teeth, cort ical and cancel lous bone. The cross-sect ions were re-or iented so each long ax is was vert ical relat ive to the image matr ix. Each dental outl ine was t raced, and the teeth (crowns and roots) were removed, leaving only bone (i.e. "total" cross-sect ions) . Finally, we traced the outl ine of cancel lous bone, and - 93 - deleted it to obtain cortical cross-sect ions. These were retained as grayscale images, preserving their regional bone densi ty. The human symphysea l sect ions were treated the same way (Figure 4.2, p95) . 4.3.3 Cross-sectional measurements Quant i tat ive calculat ions for each sect ion included its cross- sect ional area (Area), mean grayscale va lue (MGSV) , cross-sect ional mass (Mass) , second moments of inertia of its area (Ix, Iy) and of its mass ( Ixm, l y m ) . Bending indices were calculated f rom the ratios of Iy/Ix (BI) and Iym/Ixm (BIM). These calculat ions were made for the total and the cortical sect ions. Cort ical indices were also calculated f rom the ratios of cortical area to total area (CI) and cort ical mass to total mass (CIM). The area and mass calculat ions were related to the abi l ity of a sect ion to resist shear. MGSV provided an index of relat ive mean bone density for the cross-sect ion. The second moments of inertia quanti f ied the abil ity of a sect ion to counter bending with respect to one of its two orthogonal axes. The bending index provided a measure of shape with respect to its res istance to bending and tors ion, and the cortical index was a measure of the eff ic iency of the cross-sect ional des ign (defined as the abi l ity to resist the most stress with the least mater ia l ) . All cross-sect ional measurements were made using a dedicated program (Ca l image - Calculate Image, Craniofac ia l Laboratory, The Univers i ty of Brit ish Co lumb ia; avai lable f rom http://condor.dent istrv.ubc.caV It also ca lculated the max imum horizontal and vert ical cross-sect ional d imens ions, i.e. the bounding box, and the mass center (centroid) of each sect ion relat ive to its bounding box (expressed as the distance f rom the centroid to the most poster ior l ingual surface of the symphys i s ) . ^ Ô Ù ù 0 Figure 4.2 Typical cross-sections for pig and human mandibular symphyses. Sections in the first row are pig symphyseal sections, showing the original section, total section without teeth, cortical section, total section in its normal orientation, and cortical section in its normal orientation, respectively. Sections in the second row are the corresponding human sections. In both cases, the facial surfaces are to the left. Seen here are the two symphyses oriented quite differently and more cortical bone distributed on their lingual sides. W indicates symphyseal width measurement, and H indicates symphyseal height measurement. 4.3.4 Stress and strain calculations Est imates of the max imum stress along the l ingual surface of the mandibu lar symphys i s were der ived f rom a formula (Equat ion 4 .1 , p88). The correct ion factor K (derived f rom R/c) was obta ined f rom a convers ion table for hollow ell iptical sect ions (Roark and Young, 1975) . The negat ive a l lometry of the mandibu lar arch width decreased the R/c (i.e. increased K) for the concave surface of the symphys i s (Hylander, 1985). Previously, dental arch width, i.e. the d istance between the left and right mand ibu lar third molars, has been used to est imate the d iameter of curvature (Hylander, 1985), but we were unable to employ this approach because our pigs were adolescents. Therefore, we used a median axis method descr ibed by St raney (1990) , and Daegl ing (1993) (see Figure 4.3, p97) . Lines were drawn to approx imate each long axis of the left and right corpora. A third line was constructed through the centroid of the symphys i s , perpendicu lar to the midsagitta l plane. The three l ines intersected forming an open trapezo id, a circle was constructed internal ly tangent ia l to its s ides, and the radius of this circle was taken to be the radius of symphysea l curvature. The centroid was used to approx imate the c va lue. We also used this approach when measur ing the human mandib les . The deep masseter (known as zygomat i comand ibu lar i s , ZM , in pigs, Herring er a/., 1993) was assumed the pr imary bending force. Accordingly, Magnet ic Resonance (MR) images f rom four of the l iving pigs were obta ined as part of another study at the Univers i ty of Washington with a S i gma MR scanner (Genera l Electric Medical Sy s t em, Mi lwaukee, WI) using Spin-Echo sequences (TR/TE 11.1/2 ms, matr ix 2 5 6 x 1 9 2 , FOV 2 4 x 2 4 cm , slice intervals 1.5 m m ) . Sect ions were obta ined f rom the masseter musc les, below the zygomat i c arch, parallel to the occlusal plane, and through the roots of the maxi l lary molars. In each case, the cross-sect ional area of one whole Figure 4.3 Diagram of the median axis method used to construct the circle containing the radius of curvature of the flexure. A circle is constructed so that it is tangential to the three sides of the open trapezoid. A represents a line passing through the centroid of the symphyseal section, which is also the tangential point. B and C indicate the long axes of the left and right mandibular corpora. R indicates the radius of curvature. muscle was measured , and then halved to represent the deep masseter (clearly dist inguishing the ZM musc le f rom superf ic ial masseter is difficult with MRI). This mean area for the four an imals was 5.91 c m 2 (range 4.72-6.59 c m 2 ) , represent ing the deep masseter 's cross-sect ional s ize in all subsequent calculat ions for the pig. Also, coronal MR sl ices 1.5 m m thick were made midway through the s igmoid notch (i.e. between the coronoid and condy lar processes) to disclose masseter muscle angulat ions in three l iving an ima ls . The upper border was traced f rom the media l surface of the zygomat i c arch to the muscle 's insert ion in the masseter ic fossa below the s igmoid notch, and the lower border f rom the bottom of the arch to the insert ion of the muscle on the mandib le 's externa l obl ique ridge (this border was marked by a vis ible in t ramuscu lar aponeuros is) . The angles formed by these two muscle borders relat ive to the occlusal plane were bisected to produce a resultant line of act ion for the deep masseter . Its average angle for the three an imals was 46 degrees relat ive to the occlusal plane (range 43-51 degrees) and this va lue was used in all subsequent calculat ions for the pig. No direct measurements were made of the human deep masseter 's cross-sect ional s ize or or ientat ion, since average data were a lready avai lable. We assumed the muscle had a mean cross-sect ional area of 2.04 c m 2 (Langenbach and Hannam, 1999) and a mean frontal plane angulat ion of 57 degrees (Kor ioth e r a / . , 1992) . S ince max imum tension in a mamma l i an skeleta l musc le is proport ional to its cross-sect ional s ize (Hannam and Wood, 1989; Langenbach and Hannam, 1999; Peck etal., 2000; Sasak i etal., 1989; Zajac, 1989) , we mult ip l ied the masseters ' cross-sect ional areas in both instances by 40 N/cm 2 to est imate their m a x i m u m possible tens ions (Hannam, 1997; Langenbach and Hannam, 1999; Peck e r a / . , 2000; van Eijden and Raadsheer, 1992) , and converted these to lateral force vectors by means of the musc les ' mean coronal angulat ions. The respect ive moment a rms were measured f rom CT-der ived (voxel-based) surface reconstruct ions of the entire j aws . In each case, a line was drawn between the centers of the rami (def ined by the midpoints between the s igmoid notch and lower border, and the anter ior and poster ior borders respect ive ly). The momen t a rm was def ined as the midl ine distance f rom this line to the symphysea l centroid, paral lel to the occlusal plane. Different methods can be used to express the mandib le 's plane of curvature, e.g. the occlusal plane, and the lower border of the mandib le . In the pig mandib le, there is little di f ference between these as the lower border is a lmost parallel to the occlusal plane, but in the human jaw, there can be an 18 degree dif ference between them (Sadowsky, 1995) . Here we chose the occlusal plane, s ince wishboning occurs at the end of the power stroke, i.e. near max imum intercuspat ion (Hylander er a/., 1987; Hylander and Johnson, 1994) . Definit ion of symphysea l length (or height) and width with respect to wishboning is related to the plane of curvature. Hy lander (Hylander, 1985) def ined symphysea l length as the max imum distance f rom the midl ine crest of the mandibu lar incisor a lveolus to the most inferior port ion of the mandibu lar symphys i s . Symphysea l width was taken as the max imum d imens ion of the symphys i s in the sagittal plane, perpendicular to symphysea l length. This def init ion suits the human symphys i s because its long axis is a lmost perpendicu lar to the funct ional occlusal plane (our definit ion of the plane of curvature for wishboning). However, it is inappropriate for the pig symphys i s , which is marked ly incl ined anteroposter ior ly. We cons idered a more appropr iate definit ion of symphysea l width here was the max imum dimens ion of the symphys i s with respect to the occlusal plane, and that of its height as the max imum dimens ion in a plane perpendicu lar to this. Here we use the word "he ight" to replace " length" . When we measured these d imens ions programmat ica l ly , we used the horizontal and vert ical d imens ions of the bounding box in the image matr ix to represent the width and height, respect ively; the sect ion was or iented so that the angle between the long axis of the sect ion and the horizontal d imens ion of the image matr ix conformed to the angle between the long axis of the symphys i s and the funct ional occlusal plane (Figure 4.2, p95) . This angle was measured f rom the reconstructed whole-jaw image. We used s imi lar cr iter ia to define human jaw widths and heights. The second moment of inertia with respect to the axis perpendicu lar to the plane of curvature, takes into account the relat ive amounts and distr ibut ion of cortical bone, bone mass d istr ibut ion, and the shape and size of the cross-sect ions. The cortical cross-sect ional moment of inertia was calculated with respect to the centroid and the vert ical axis, i.e. Iy, and used in Equation 4.1 (p88). The parameter c (the distance f rom the l ingual surface to the centroidal axis) was calculated programmat ica l ly . The centroid calculat ion took into account the shape and size of the cortical sect ion and the distr ibut ion of bone mass. Genera l ly , the compact bone is reported to have a Young's elast ic modulus between 10 to 20 GPa (van Ei jden, 2000) . This is very s imi lar to that in humans (11.3 to 19.4 GPa; Dechow er a/., 1993) and macaque (9.0 to 21.0 GPa; Dechow and Hylander, 2000) . We assumed the same for the pig mandib le. S ince cortical bone is weaker in tens ion, we used the lower va lue (10 GPa) to est imate st ra in. The stress and strain calculat ions were made for the cortical bone cross-sect ions on the assumpt ion the symphys i s was a hol low, el l iptical sect ion. To assess the signif icance of symphysea l or ientat ion in the pig, we also calculated its max imum tensi le stress and stra in when the symphys i s was s imulated as having an upright or ientat ion. Because the centroid changed, the R/c ratio, K va lue and force moment a rm were adjusted accordingly. These cr iter ia were also appl ied to the human mandib le , a l though here each long axis was a lmost perpendicular to the funct ional occlusal plane, and no s imulat ions were made with an altered symphysea l or ientat ion. 4.3.5 Statistical analysis Two-sample t-tests were used to test the di f ferences between pig and human jaw symphysea l measurements . One way ANOVA was used to determine if there were dif ferences in es t imated stresses and strains among the pig mandib le with its symphys i s in a normal or ientat ion, the pig mandib le with its symphys i s in the s imulated upright or ientat ion, and the human mandib le . Further compar i sons between groups were performed with Tukey's honest ly s ignif icant di f ference test. The signif icance level was set to 0.05. All stat ist ical ana lyses were carr ied out with SPSS for Windows 8.0 (SPSS Inc., Chicago, I l l inois). 4.4 RESULTS 4.4.1 Cross-sectional measurements For comparat ive analys is, the cross-sect ional shapes of the pig and human symphyses were or iented so their principal axes were al igned the same way. Most cross-sect ional measurements for the pig sect ions were signif icantly greater than those for humans (Table 4 .1 , p l 0 3 ) . These parameters included the areas, and second moment s of areas for total and cortical bone cross-sect ions. Whi le the relat ive amounts of cort ical bone in the respect ive samples (i.e. the cort ical indices) were quite s imi lar, the bending indices in humans were signif icantly greater than those in pigs. 4.4.2 Symphyseal stress and strain The est imated stresses and strains were s imi lar in the pig and human mandib les (Table 4.2, p l 0 4 ) . The strains were all within the range previously reported for pr imates, i.e. below 2000 ue. When force on the pig symphys i s was s imulated in its upr ight or ientat ion however, the est imated stresses and strains were about three t imes greater, i.e. the strain exceeded the highest va lue of 2000 ue reported for the macaque (Hylander, 1985) . The cross-sect ional moment s of inertia for the normal pig and human symphyses in Table 4.2 ( p l 0 4 ) differed f rom those in Table 4.1 ( p l 0 3 ) because the latter were ca lculated for sect ions with their principal long axes or iented upr ight in the image matr ices. The radii of curvature for the pig and human mandib le were unexpected ly s imi lar (19.40±2.63 m m and 20.13±2.36 m m Table 4.1 Area (cm2), mass (cm2), mean grayscale value (MGSV), second moments of area (Ix, Iy in cnvQ, second moments of mass (Ixm, lym in cnv»), area and mass bending indices (BI, BIM), area and mass cortical indices (CI, CIM), and t-test results for pig and human jaw symphyses. The symbol (-) indicates no comparison available. Pig Human t test Mean SD Mean SD P value Cortical Area 6.59 1.70 2.29 0.32 <0.01 Mass 387.78 146.78 89.13 16.44 - M G S V 57.30 10.09 38.87 3.22 - Ix 17.12 7.31 1.46 0.50 - iy 2.20 1.06 0.46 0.13 <0.01 Ixm 760.40 394.36 47.79 18.35 - lym 109.60 59.99 14.59 4.90 - BI 0.13 0.03 0.34 0.11 <0.01 B I M 0.14 0.03 0.32 0.09 <0.01 Total Area 8.68 1.70 3.17 0.47 <0.01 Mass 480.54 165.50 117.61 19.25 - M G S V 54.31 11.69 37.08 2.65 - Ix 21.03 7.97 1.66 0.59 <0.01 iy 2.57 1.21 0.50 0.14 <0.01 Ixm 918.55 421.80 53.86 20.16 - lym 127.94 68.44 15.92 5.32 - BI 0.12 0.03 0.32 0.11 <0.01 B I M 0.14 0.03 0.31 0.10 <0.01 CI 0.74 0.07 0.72 0.04 0.37 C I M 0.80 0.08 0.76 0.05 0.22 Table 4.2 Estimated stresses and strains for pig mandibles with their symphyses in their normal, and simulated upright orientations, and for human mandibles. The table includes means, standard deviations, minimum and maximum values. Abbreviations and units: R, the radius of curvature in mm; c, distance from the centroidal axis to the most posterior lingual surface of the symphysis section in mm; K, a correction factor for curved beam stress calculation; D M , mean deep masseter force in N ; L, lever arm in mm; I, second moment of inertia with respect to the vertical axis in cm4; a, calculated stress in MPa; e, calculated strain, in ue. The symbol (-) indicates the variable has constant value. The symbol (*) indicates differences between groups (P<0.01). R c K D M F L I a e Pig normal Mean 19.40 20.77 3.03 164.22 118.08 16.41 8.18 818.37 SD 2.63 2.18 - - 10.86 6.87 1.96 195.84 M i n 14.00 17.80 - - 99.86 8.60 4.78 477.86 Max 22.00 24.30 - - 129.32 30.19 11.19 1118.82 Pig upright Mean 19.40 11.93 2.03 164.22 109.20 2.20 22.59 2258.68 SD 2.63 1.60 0.41 - 10.62 1.06 9.04 903.73 M i n 14.00 9.67 1.54 - 90.80 0.89 14.29 1428.82 Max 22.00 14.59 3.00 - 121.49 4.38 41.89 4188.61 Human Mean 20.13 8.22 1.54 44.44 71.47 0.51 8.21 820.92 SD 2.36 0.69 - - 4.34 0.13 1.27 127.28 M i n 16.00 7.34 - - 66.29 0.32 6.94 693.72 Max 23.00 9.30 - - 77.60 0.71 10.41 1041.46 Tukey's test* Pig normal vs. pig upright Human vs. pig upright Pig normal vs. human a <0.01 <0.01 >0.05 E <0.01 <0.01 >0.05 respect ive ly) , but K va lues for the pig mandib les were about double those found in the human sample . 4.5 DISCUSSION 4.5.1 Symphyseal cross-sections Al though our pigs were juven i les , their mandib les were larger than the human spec imens. The absolute size of the pig symphys i s seems important for resist ing shear ing stresses. Like monkeys , three shear ing forces may occur here dur ing funct ion. Dorsoventra l shear is created by vert ical components of musc le force on the balancing-s ide dur ing uni lateral molar bit ing, and by non-mid l ine incisai forces (Beecher, 1977; Hylander, 1984), though the latter may be more important in pr imates than in pigs. The strong adductor musc les in the latter an imals are a l ikely source of dorsoventra l shear stress. Anteroposter ior shear is due to the balancing-side tempora l i s having the tendency to pull the balancing-side dentary in a poster ior direct ion relat ive to the work ing-s ide dentary during the power st roke (Beecher, 1977) . Addit ional ly, shear ing forces can be tors ional (i.e. twist ing about the t ransverse axis of the symphys i s when the balancing-s ide corpus is e levated and working-s ide is depressed (see Hylander, 1984) . Whi le the fo rmer two s imply require suff ic ient bone in the symphysea l cross-sect ion to be resisted sat isfactor i ly, tors ional shear is best resisted by a c ircular cross-sect ional shape (Daegl ing, 1989; Daegl ing e r a / . , 1992; Daegl ing and Gr ine, 1991; Daegl ing and Hylander, 1998) . Though the pig symphysea l sect ion is far f rom c ircular ( indicated by a mean bending index around 0.13, see Table 4 .1 , p l 0 3 ) , its resistance to tors ional shear may also be helped by its internal t rabeculat ion (Daegl ing, 1989; Hylander, 1979b) which could d iss ipate shear f low. Shear ing stresses in the human symphys i s might be expected to be smal ler than those in the pig due to the relat ive size of each jaw and its musc les. Whi le the pig might be predicted to have a robust symphys i s g iven its s ize and feeding habits (pigs root forceful ly with the i r long snouts) , this alone does not expla in the preferential distr ibut ion of cort ical bone in the infero-l ingual part of the symphysea l cross-sect ion. In the upright sect ion, there appears to be more cortical bone anteroinfer ior ly, but when the sect ion is v iewed in its normal anatomica l or ientat ion, the bone is th ickest posteroinfer ior ly, as it is in the normal upr ight human jaw (Table 4.2, p l 0 4 ) . As the pig l ikely incurs s igni f icant stress concentrat ions along the l ingual aspect of its symphys i s , our f indings favor the general hypothes is proposed for pr imates that more bone is needed here to counter bending (Hylander, 1984) . For a cross-sect ion to resist bending, shape is important . The ideal des ign is a hol low structure with its long axis in the plane of bending (Hylander, 1979b) . The bending index for the pig symphys i s is around 1 3 % (Table 4 .1 , p l 0 3 ) , corresponding to its long and thin shape. This form is clearly not des igned to resist bending in the plane of the short axis, but the pig symphys i s is very resistant to bending in the plane of its long axis. The strong jaw musc les, long musc le lever a rm , high K va lue, and the absence in the pig of t ransverse bony tori or s imian shelf, appear to require both the asymmet r i c d istr ibut ion of cort ical bone, and horizontal or ientat ion of the sect ion's long axis. We have ment ioned that as pigs grow, their symphyses seem to or ient more horizontal ly relat ive to the occlusal plane (see Figure 4 .1 , p90) since symphysea l width increases faster than its height. Therefore, dur ing growth, any increase in symphysea l anteroposter ior width, and especial ly in symphysea l or ientat ion, would seem important to help resist increasing wishboning stresses. Addit ional ly, th is or ientat ion presumably benefits rooting behavior. 4.5.2 Stresses and strains Al though the pig deep masseter was 3.70 t imes larger in cross- sect ional s ize than its human counterpart, its lever a rm was 1.65 t imes longer, and it had a K va lue twice that of the human jaw, expect ing to create high wishboning stresses, the high second momen t of inertia in the more hor izontal ly or iented pig symphys i s compensated for t hem, and kept the stresses and strains within the same genera l ranges as those in the human jaw (Table 4.2, p l 0 4 ) . The predicted stresses and strains add support to the proposit ion that if the pig symphys i s were or iented more vert ica l ly, it would be more vu lnerab le to fai lure as a consequence of wishboning (Table 4.2, p l 0 4 ) . Whi le the est imated strains in the normal pig symphys i s (477.86 to 1118.82 UE) fell within an expected funct ional range (e.g. below 2000 UE), the strains est imated with the s imulated upright symphys i s could reach 4189 .61 UE (mean value 2258 .68 UE). These are well above the highest funct ional strain measured in the macaque symphys i s (Hylander, 1985), and approach the 3000 UE va lue cons idered vulnerable for bone. Our est imat ions may in fact be lower than those actual ly occurr ing in the pig. In addit ion to its contr ibut ion to wishboning, the superf ic ial masseter everts the lower border of the mandib le and inverts the a lveolar process, caus ing tens ion along the lower border of the symphys i s and compress ion on its a lveolar s ide. The bi lateral occ lus ion of pigs may make the symphysea l strain even worse, as evers ion would be occurr ing on both s ides. S ince the pig symphys i s is obl iquely or iented, its posteroinfer ior aspect will thus undergo tens ion, whi le its superoanter ior aspect will undergo compress ion. This tension would super impose upon any due to wishboning, result ing in more tensi le stress (and therefore strain) than we have est imated here. Furthermore, there are shear stresses at the symphys i s , and the bone's shear rigidity modu lus is only one third of its elast ic modulus (Dechow e r a / . , 1993; Dechow and Hylander, 2000; van Ei jden, 2000) . Presumably, the symphys i s might be expected to funct ion with an added safety factor, a l lowing the pig to accommodate its wide var iety of diets (harder foods are assoc iated with higher act ivity levels of the jaw-c los ing musc les, Herr ing, 1977; Herr ing and Scapino, 1973; Huang e r a / . , 1993). Our results suggest relat ionships between dynamic stress and induced strain in the fused mamma l i an symphys i s may be mainta ined across orders, s ince they appear s imi lar for pig and human (818.37 ue vs. 820 .92 ue) even though the shapes, s izes, j aw musc les and funct ions in the two mamma l i an examples differ widely. The results comp lement the hypothesis that the mater ia l propert ies of their const i tuent bone t issue seem to be s imi lar in an ima ls over a wide range of body weight (Rubin and Lanyon, 1984) . Our use of mean data to est imate musc le force, but individual measurements for skeletal t issue could expla in some of the var iance in est imated stresses and stra ins. A l though we selected pigs of s imi lar ages, their mand ibu lar s izes nevertheless var ied (the dif ference between the m in imum and max imum lever a rms was as high as 30 m m ; cf. 10 m m for the human sample; see Table 4.2, p l 0 4 ) . The cross-sect ional moments of inertia also var ied (the max imum- to - min imum ratio was 3.51 for the normal pig symphyses , and 4.92 for the upright symphyses; cf. 2.22 for the human jaws; see Table 4.2, p l 0 4 ) . The var iance may also have lessened if indiv idual ized muscle data had been used in the pig sample . Even so, one should not expect this var iance to exceed those f rom in vivo measurements , s ince the latter reflect all our anatomica l var iables plus addit ional exper imenta l and physiological factors. We speculate that pigs have not developed upright symphyses , nor s t rengthened them with super ior and/or inferior tori (l ike cercopithecines) due to their feeding requ i rements . Pigs root, and do not usual ly rely on incisai bit ing. In the macaque, the super ior torus and the inferior s imian shelf represent a balance between the need for symphysea l strength and vert ical incisai funct ion in a long-jawed an ima l . In the shorter human jaw, the upright, cort ical ly-reinforced symphys i s seems adequate to meet funct ional demands . In the pig, the cort ica l ly-re inforced, hor izontal ly-or iented symphys i s effect ively resists high wishboning stresses in a long, powerful j aw shaped for root ing. 4.5.3 Final Comment on Wishboning Wishboning has been attr ibuted to force f rom the balancing deep masseter musc le at the end of the power stroke, lateral components of bite force on the work ing-s ide, and t ransverse components of work ing- side jaw-c los ing musc le forces (Hylander, 1984, 1985; Hy lander et al., 1987; Hylander and Johnson, 1994) . Medial pterygoid musc le forces reduce these wishboning stresses ("reverse" w ishbon ing) . In pigs, dur ing the late mast icatory power stroke, when the balancing-s ide deep masseter reaches its peak, and the work ing-s ide superf ic ial masseter and the working-s ide medial pterygoid remain act ive late in the stroke, the lateral ly-directed component of work ing-s ide (and perhaps the balancing-s ide in pigs) bite force would appear to be an important force contr ibut ing to wishbon ing. If so, one would expect wishboning to be reduced if the teeth were flat. We suggest the potential contr ibut ion of the working-s ide art iculat ion seems to have been largely neglected here. If work ing-s ide teeth can react to the lateral pull of the balancing-side musc les, so can the super ior and media l parts of the working-s ide condylar fossa, especia l ly as the work ing condyle is f i rmly embedded at the end of the power stroke by residual e levator musc le act iv ity. Wishboning is enhanced when any lateral components of resist ive force are directed opposite the balancing-side musc les whether these forces occur at the teeth or the work ing-s ide art iculat ion (see Figure 4.4, p i l l ) . 4.6 CONCLUSIONS Symphysea l wishboning occurs in the mandib les of pigs and humans. The pig symphys i s has larger areas of total cross-sect ion, cortical cross-sect ion and second moment than its human counterpart (though the relat ive amounts of cort ical bone in both cases are s imi lar) , and it has a lower bending index. Despite these di f ferences, and obvious diss imi lar i t ies in jaw form and musc le morpho logy between pigs and humans , t ransverse bending forces f rom the respect ive balancing-s ide deep masseter create s imi lar amounts of symphysea l stress and strain (less than 2,000 ue). This is accounted for by the more hor izontal ly-or iented symphys i s in the pig. The absolute size of the pig symphys i s seems important for reducing shear ing stresses, Figure 4.4 Suggested mechanism of wishboning and patterns of stress in the pig mandible. The main active forces are the balancing-side deep masseter (Fmb) and most likely the transverse component to the working-side jaw closing muscle force (Fmw, see text). Reaction forces from occlusion (Fb) and medial condylar pole (Fc) are the passive forces. Fb and Fc act in an opposite direction to Fmb when the major tooth loads are on the working-side. The force resultant tends to bend the mandible in its plane of curvature causing tension on its lingual side and compression on its facial side. All force vectors indicate directions only. Their magnitudes are unknown. R is the radius of curvature and c is the distance from the centroidal axis to the symphyseal lingual surface. The shaded area represents the stress distribution pattern across the symphysis: tensile and compressive stresses are indicated to the left and right of the midline, respectively. Due to the curvature of the symphysis, tensile stresses on the lingual side increase nonlinearly at a faster rate compared with compressive stresses on the facial side. Stress magnitudes are unknown. CA and NA represent the centroidal and neutral axes, respectively. In this curved beam, they do not coincide as in a uniform straight beam. Based on Hylander and Johnson (1994). - I l l - while the preferential distr ibut ion of cort ical bone posteroinfer ior ly, combined with symphysea l or ientat ion, apparent ly compensates for the pig's large masseters and long mandib le, which otherwise would demand some form of symphysea l buttress ing (e.g. in the form of l ingual tori) to reduce the possibi l i ty of structural fai lure during funct ion. The relat ionship between induced stress and stra in seems to be mainta ined across mamma l i an orders (in this case, pigs and humans) as might be expected given the b iomechanica l principles involved, and is consistent with the hypothes is that funct ional equiva lence, i.e. s imi lar i ty in dynamic stress and stra in in the fused symphys i s , is c ommon to many mamma l s .  5 MASS PROPERTIES OF THE PIG MANDIBLE 5.1 ABSTRACT Speci f icat ion of mass propert ies is an essent ia l step when model ing jaw dynamics , but obtaining them can be difficult. Here, we used three- d imens iona l computed tomography (CT) to est imate jaw mass , mean - bone densi ty, anatomica l locations of the mass and geometr ic centers, and moments of inertia in the pig jaw. High-resolut ion CT scans were performed at one m m slice intervals on spec imens submerged in water. The mean est imated jaw mass was 1 2 % greater than the mean wet weight, and 3 3 % more than the mean dry weight. Putat ive bone marrow accounted for an extra 1 3 % of mass . There was a posit ive correlat ion between est imated mean bone density and age. The mass center was consistent ly in the midl ine, near the last molar . The mean distance between the mass center and geometr ic center was sma l l , especial ly when bone marrow was taken into account (0.58±0.21 m m ) , suggest ing mass distr ibut ion in the pig jaw is a lmost symmetr i ca l with respect to its geometr ic center. The largest moment of inertia occurred around each mandible 's superoinfer ior axis, and the smal les t around its anteroposter ior ax is. Bone marrow contr ibuted an extra 9 % to the moments of inertia in all three axes. Linear re lat ionships were found between the actual mass and a mass descr iptor (product of the bounding vo lume and mean bone dens i ty) , and between the moment s of inertia and moment of inertia descr iptors (product of the mass descr iptor and two orthogonal d imens ions forming the bounding box) . The study suggests imaging modal i t ies reveal ing 3-dimensional j aw shape may be adequate for est imat ing the bone mass propert ies in pigs. 5.2 INTRODUCTION Dynamic models of musculoskeleta l b iomechanics are a useful way to study structural and funct ional interact ions in the mamma l i an mast icatory sys tem (Hannam e r a / . , 1997; Koolstra and van Ei jden, 1995, 1997a, b; Otten, 1987) . When dr iven with funct ions s imulat ing motor dr ive to var ious j aw musc les, their act ive and passive musc le tens ions produce realist ic jaw mot ions, and generate art icu lar and dental react ion forces. S ince their propert ies can be changed easi ly, models provide a f lexible env i ronment for ana lyz ing var iat ions in craniofacial morpho logy, muscu lar and art icular d isorders, prosthet ic addit ions, and s imulated surgical a l terat ions to the mast icatory sys tem. Also, they can be used to expla in known assoc iat ions, or predict new ones. Virtual models, however, require specif icat ion of the jaw's mass propert ies (e.g. its mass, mass center, and moments of inert ia), and these can be diff icult to est imate in biological t i ssues (Braune and Fischer, 1988). In a study of the human head, Smi th er a/. (1995) used three different b iomedical imaging modal i t ies to compute the center of gravi ty and moments of inert ia, and assumed the head was uni formly dense. Koolstra and van Eijden (1995, 1997b) used cub ic-cent imeter blocks of t issue exc ised f rom a female cadaver jaw to calculate its moments of inert ia, and assumed the mass distr ibut ion of the preparat ion was homogeneous throughout the mandib le. In related studies, Hannam er a/. (1997) , and Langenbach and Hannam (Langenbach and Hannam, 1999) ass igned mass propert ies predicted by a F in i te-e lement model of the human jaw deve loped ear l ier by Korioth er a/. (1992) . The FE model was constructed f rom computed tomograph ic (CT) images, and included e lement with t issue-propert ies specif ic for dif ferent j aw regions. CT imaging is useful for mass-property calculat ion because x-ray l inear attenuat ion disc loses regional mineral densit ies ( Lampmann etal., 1984; Wi l l iams e t al., 1980), which account for much of the jaw's mass . Thus individual pixels with different intensity va lues, d istr ibuted non-uni formly in the imaged mandib le, can be ass igned densit ies ref lect ing mineral content, mak ing it possible to est imate the jaw's mass propert ies (Smi th etal., 1995) . Whi le CT has l imited appl icat ion in humans due to its radiat ion cost, it is feasible in non-human mamma l s like pigs, which are often employed as exper imenta l an imal models for studying human j aw funct ion (Herr ing, 1995; Herr ing er al., 1996; Teng and Herr ing, 1998) . In the present report, we used this approach to est imate the pig jaw's mass , mass center, and moments of inert ia, s ince none of these propert ies have been reported previously, and we needed them to develop a dynamic model of the pig mast icatory sys tem. In part icular, we were interested in how mass propert ies changed with age and jaw size. Var iat ions in regional bone density would be expected to affect the location of the mass center with respect to the pig jaw's geometr ic center, which is determined by shape, but is unaffected by dif ferences in regional densi ty. A large difference between these two centers would conf i rm that regional densit ies would have to be taken into account each t ime pig j aw mass propert ies were es t imated. A smal l di f ference, however, could s impl i fy pre-model ing procedures, because it is eas ier to es t imate the jaw's geometr ic center than its mass center, and methods other than CT imaging are avai lable for doing this. With the same goal of s impl i f icat ion in mind, we hypothes ized that the jaw's mass and inertial propert ies could be sat isfactor i ly est imated by using s imple physica l descr iptors such as mean j aw densi ty, and three orthogonal d istances def ining its s ize. If so, it would be unnecessary to CT scan every pig j aw used for dynamic s imulat ion. 5.3 MATERIALS AND METHODS 5.3.1 Material preparation The exper iments were carr ied out on 10 osteological spec imens selected f rom an exist ing col lection of min iature pig mandib les (Sus scrofa). We used dry spec imens because we were pr imar i ly interested in the contr ibut ion of mineral ized t issue to growing j aws of different shapes and s izes. In part icular, we wished to define the relat ive magn i tudes of inertial moments around their mass centers, s ince they may be unique to the pig jaw, g iven its morphology. Since we a imed for op t imum edge- resolut ion with CT scanning, we considered the added presence of any invest ing soft t issues (per iosteum and attached g ingiva) would degrade rather than enhance image qual i ty. By def ining the internal marrow spaces however, we were able to est imate the added effect of a putat ive marrow component . The sample compr ised four male and five female j aws f rom Char les River, plus one female Purdue Minipig mandib le. We selected spec imens aged 29-250 days to include different j aw sizes and stages of tooth deve lopment (see Table 5.1, p l 2 6 ) . Use of this archival mater ia l compl ied with the requ i rements of The Univers i ty of Brit ish Co lumbia 's Commi t tee on An ima l Care. The j aws were weighed dry, and then re-weighed after hydrat ion for 48 hours. Prior to imaging, they were separated with wooden spacers, and placed in a s ingle, water-f i l led, plastic container. The spec imens remained underwater dur ing imaging to opt imize resolut ion of the bone interface (Daegl ing, 1989) and to min imize the vo lume-averag ing error. A cal ibrat ion phantom was included to permit calculat ion of an equat ion express ing bone mineral density (BMD) as a funct ion of pixel va lue ( Lampmann er al., 1984) . It consisted of four tubes of K H 2 P 0 4 solut ion at different concentrat ions (0.05 g / cm 3 , 0.15 g / cm 3 , 0.25 g / c m 3 and 0.50 g / cm 3 ) . 5.3.2 CT scanning Computed tomography was performed with a Tosh iba Xpress SX scanner (Toshiba Corporat ion, Tokyo, Japan) operat ing at 120 kV and 150 mA. Four sequences yie lded 350 near-axia l s l ices at consecut ive one m m intervals. Each slice had a 2 5 0 x 2 5 0 m m field of v iew made up of 5 1 2 x 5 1 2 pixels, each measur ing 0 . 49x0 .49 m m . The images were imported digital ly f rom the scanner to a UNIX-based workstat ion (SGI Indigo Ext reme, Si l icon Graphics Inc., Mountain V iew, CA) . S ingle 8-bit fi les were created and f i ltered so that any structure equal in densi ty to or less dense than water was exc luded to provide image backgrounds of uniform densi ty and to s imulate the wet bone without bone marrow. 5.3.3 Image processing A commerc ia l image-process ing program (3DVIEWNIX, Univers i ty of Pennsy lvania Medical Center, Phi ladelphia, PA) was used for image segmentat ion , jaw-sur face reconstruct ion, landmark identi f icat ion and measurement . We also wrote a dedicated program (Ca l image - Calculate Image, Craniofac ia l Laboratory, The Univers i ty of Brit ish Co lumbia; avai lable f rom http://condor.dentistrv.ubc.ca') and used a desktop mic rocomputer (Pent ium 200 MHz MMX) to perform specif ic image matr ix operat ions. These two programs were run interchangeably. 5.3.4 Mass properties calculation The three-d imens iona l (3D) image was cons isted of voxe ls located relat ive to the scanner 's coordinate sys tem. We assumed the vo lume (v) of each voxel was: v = w x h x d Equation 5.1 where w, h, and d were the width, height, and depth (slice th ickness) of a voxe l , respect ively. S ince it has been shown that CT grayscale va lues (GSV) vary l inearly with BMD (Lampmann er a/., 1984), we used the fol lowing equat ion to calculate BMD for each voxe l : BMD = a + bx GSV Equation5.2 where a and b are two constants represent ing the intercept and slope of the l inear equat ion, respect ively. The mass of each voxel was then determined by: dm = BMD x v Equation 5.3 where dm is each voxel 's mass . Because the voxe l s ize for each CT image set is constant, the relat ive magni tude of each voxel 's mass will be represented by BMD and in turn by the CT graysca le va lue. With calculus, the total mass (M) can be easi ly ca lculated as: M = J d m Equation 5.4 and the three coordinates of the mass center (Cx, Cy, Cz) as: — x \xxdm M J 1 y x dm Equation 5.5 X M C . = — x f z x dm 2 M J where x, y, z are the three coordinates of each voxe l relat ive to the image matr ix origin (usual ly the front-left-top point in a r ight-hand coordinate sys tem) . Assuming dm in Equation 5.3 ( p l l 9 ) to be constant, we obtain the three coordinates of the geometr ic center. The d istance between the two centers (CD) can be calculated as: CD =| MC - GC I Equation 5.6 where MC and GC are the mass and geometr ic centers as 3D vectors. The total vo lume (I/) of the jaw is jus t a product of each voxel 's vo lume and the total number of voxels . Therefore, the mean bone density (MBD) can be calculated as: MBD = M / V Equation 5.7 This is the est imated mean bone density of the ent ire j aw. Calculat ions of the jaw moments of inertia f rom the image matr ix depended upon the jaw's original or ientat ion in the scanner, and consequent ly in the image matr ix. These are calculated as: Ixx = \(y2+z2)xdm Iyy = \(x2 +z2)xdm Izz = \(x2 + Y2) x dm Equation 5.8 Ixy = Jx x y x dm x zx dm where Ixx, Iyy, Izz are the three moments of inert ia with respect to the three axes in the image matr ix coordinate sy s t em, respect ively; Ixy, Iyz, Izx are the products of inertia with respect to the image matr ix sys tem. As a convent ion, however, it is best to express the jaw's moments of inertia relat ive to some anatomica l reference. Therefore, moments of inertia t ransformat ion was necessary. The t ransformat ion of moments of inertia needed two steps. First was a translat ion f rom the image matr ix coordinate sys tem to the mass center by paral le l-axis theo rem: I XX Mxr 2 y y yy Mxr2 Mxr2 Equation 5.9 Ixy = Ixy ~CxxCyxM = lyZ -Cy XCZ X M zx -Cz xCx x M where r is the magni tude of the center of mass expressed as a 3D vector relat ive to the original image matr ix or ig in. This step trans lates the moments of inertia f rom the image matr ix to the mass center. The second step was a rotation according to the anatomica l ly def ined coordinate sys tem. Three unit vectors represent ing the or ientat ion of the new coordinate sys tem must be obtained f rom direct morphometr i c measurements . The rotation was completed by: loi. - Ixx X Ax + Iyy X Xf, + Izz X X\ - 2 X Ixy X Xx X Xy - - Equation 5.10 — 2 x IyZ x Ay x Xz — 2 x Izx x Xz x Xx where XXl Xy, and Xz are the three components of each unit vector represent ing the three new axes, and IOL is the momen t of inertia with respect to the new axis. Calculat ion of mass propert ies was performed programmat ica l ly by Ca l image. Moments of inertia were expressed in the anatomica l coordinate sys tem as i l lustrated in Figure 5.1 ( p l 2 3 ) . The program also inserted an artif icial "ma rke r " voxel of known intensity at the mass center. When the image was reconstructed as a 3D object, the marke r could be visual ized and measured with respect to each jaw's ana tomy. To est imate the potential contr ibut ion of bone marrow space to the jaw's mass propert ies, we segmented the non-minera l i zed, marrow component in each CT sect ion, ass igned these pixels a dens i ty va lue of one g / cm 3 , and recalculated the mass propert ies. The densi ty ass igned to the marrow component was based on a cal ibrated t issue density of 1.003±0.034 g / cm 3 , which we measured f rom cal ibrated CT scans of the marrow space in the mandib le of a l iving pig. Thus, we assumed the mean density of all non-minera l f luid and cel lu lar components in the marrow space was very close to one g / cm 3 . Figure 5.1 Diagram of the coordinate system used to express moments of inertia of the jaw. The origin was located at the mass center. The X-axis was directed transversely, the Y-axis superoinferiorly in the midsagittal plane, and the Z-axis anteroposteriorly in the midsagittal plane, with the X-Z plane parallel to the occlusal plane. The fol lowing var iables were used to determine how effectively "g loba l" , rather than CT-der ived descr iptors could be used to predict the jaw's mass and inertial propert ies: • Jaw width (WD), def ined as the horizontal d istance between the two lateral condy lar poles • Jaw height (HD), def ined as the vert ical d istance f rom the tip of the coronoid process to the lower border of the ramus • Jaw length (LD), def ined as the horizontal d istance between the tip of the central incisor and the poster ior border of the ramus • Jaw Vo lume (VD) , def ined as the product of WD, HD and LD • Jaw Mass (MD), def ined as the product of VD and MBD • Three moments of inertia (Ixx D, Iyy D, and Izz D ), each def ined by the product of MD and two orthogonal descr iptors descr ib ing moments of inertia with respect to the third axis The descr iptors for jaw width, height and length represented l inear measurements determined by the respect ive bounding box of the bone vo lume, i.e. they def ined the anatomica l l imits of the spec imens in each d imens ion. Descr iptors normal ly used for cepha lometr i c measurements in humans (e.g. j aw length) were inappropr iate in the pig s ince they do not include the l imits of t issue contr ibut ion to mass property es t imat ion. Regress ion curves descr ib ing the relat ionships between the descr iptors and the specif ic mass propert ies der ived f rom the CT data were then fitted to the data. 5.4 RESULTS As expected, the phantom-der ived data revealed a l inear relat ionship between pixel va lues and mineral densi ty (correlat ion coeff ic ient 0.999). The equat ion descr ib ing the relat ionship was BMD = 0.997 + 0.013 x PixelValue. As this funct ion was obtained after all pixel va lues less than those for water were set to zero, the procedure was in effect sel f-val idat ing (with a pixel va lue of zero, the BMD was 0.997, very close to the density of water) . Est imated masses with and without bone marrow, the measured weights, and mean bone densit ies with and without bone marrow, are shown in Table 5.1 ( p l 2 6 ) . Overa l l , the est imated j aw mass was 1 2 % greater than the wet weight (mean EM/WW=1.12±0.05; coeff ic ient of var iat ion 4 . 64%) , and 4 9 % greater than the dry weight (mean EM/DW=1.49±0.13; coeff icient of var iat ion 8 .72%) . The inclusion of bone marrow added 1 3 % to the mass est imated without marrow (mean EMM/EM = 1.13±0.06). The est imated mass with marrow was 6 8 % greater than the measured dry weight of the jaw (mean EMM/DW=1.68±0.23). When bone marrow was included in the calculat ions, the mean bone density decreased by 6%, for the bone marrow was less dense than minera l ized bone. The mean bone density (which included dent in and enamel) was greater in the older an imals , but there was no c lear relat ionship between mean bone density and gender. The mass center was a lways located near the last-erupted tooth (DP4 or first molar in this sample, Figure 5.2, p l 2 7 ) . Relat ive to the dent i t ion, mass center posit ions in young mandib les were further forward than in older an imals (Figure 5.3, p l 2 8 ) . When mass center was calculated relat ive to the normal ized, mid-sagitta l d istance between the intercondylar point and infradentale, there was no sys temat i c pattern in the sample as a whole (Figure 5.3, p l 2 8 ) . Inclusion of s imulated bone marrow in the calculat ions altered mass center locat ions by a mean distance of 1.03±0.29 m m , and geometr ic center locat ions by 1.74±0.49 m m , but in no systemat ic direct ion (Figure 5.3, p l 2 8 ) . Differences Table 5.1 Descriptive statistics of the dry and wet weights (DW, WW), estimated masses without and with bone marrow (EM, EMM), calculated mean bone densities without and with bone marrow (MBD, MBDM), and the ratios between these variables. The table also contains each animal's gender and age. All mass measurements are in g, and all density measurements are in g/cm3. [aw # Sex Age D W WW E M E M M E M M / E M M B D M B D M M B D M / M B D E M / D W E M M / D W E M / W W E M M / W W 1 F 29 5.28 7.04 8.55 10.63 1.24 1.41 1.31 0.93 1.62 2.01 1.21 1.51 2 M 35 11.16 18.60 19.59 23.84 1.22 1.38 1.29 0.93 1.76 2.14 1.05 1.28 3 F 77 45.71 62.60 68.09 77.63 1.14 1.56 1.46 0.94 1.49 1.70 1.09 1.24 4 F 86 46.65 64.90 73.57 82.91 1.13 1.55 1.46 0.94 1.58 1.78 1.13 1.28 5 M 115 79.16 98.95 113.01 123.22 1.09 1.70 1.61 0.95 1.43 1.56 1.14 1.25 6 F 131 67.42 85.81 94.20 104.53 1.11 1.64 1.54 0.94 1.40 1.55 1.10 1.22 7 M 175 114.83 147.64 154.95 169.78 1.10 1.73 1.63 0.94 1.35 1.48 1.05 1.15 8 F 186 155.46 190.23 215.16 230.01 1.07 1.77 1.69 0.95 1.38 1.48 1.13 1.21 9 F 230 116.15 146.19 168.64 184.09 1.09 1.73 1.63 0.94 1.45 1.58 1.15 1.26 10 M 250 180.91 217.09 255.83 280.02 1.09 1.74 1.63 0.94 1.41 1.55 1.18 1.29 Mean 131.40 82.27 103.91 117.16 128.67 1.13 1.62 1.53 0.94 1.49 1.68 1.12 1.27 SD 77.27 58.80 70.08 81.02 87.03 0.06 0.14 0.14 0.01 0.13 0.23 0.05 0.09 Adj.* 104.61 114.88 1.45 1.36 1.33 1.50 1.00 1.13 * Adjusted value: since the phantom overestimated all masses by 12%, the adjusted values are those divided by 1.12. Figure 5.2 Lateral (above) and horizontal (below) views of a voxel-based, reconstructed dry mandible with calculated mass center (MC) and geometric center (GC) locations. Figure 5.3 Distribution of mass center (MC) locations relative to the mid-condylar point infradentale (MCP-Id) line for 10 pig mandibles. The MCP-Id line length is normalized to unity. The central region of the MCP-Id axis has been expanded for clarity (indicated by broken MCP-Id axis). Numbers for each plot indicate specimens from Table 5.1 (pi26). between the mass center and geometr ic center locat ions were smal l , especial ly when the bone marrow space was inc luded. Table 5.2 ( p l 3 0 ) shows the mean l inear d istances between mass center and geometr ic center were 1.15±0.30 m m without bone marrow, and only 0.58±0.30 m m with marrow. The moment of inertia was smal lest around the jaw's anteroposter ior z-axis, and largest around the vert ical y-ax is . Inclusion of bone marrow increased the moments of inertia around each axis by about 9 % (Table 5.3, p l 3 1 ) . Regress ion analys is revealed virtual ly l inear re lat ionships between est imated masses , measured weights and the general mass descr iptor (R 2 =0.9945 , 0.9930, 0 .9851, and 0.9909 for EMM, EM, WW and DW, respect ively, Figure 5.4, p l 3 2 ) . There were also l inear re lat ionships between the three moments of inertia and their respect ive moments of inertia descr iptors (R 2 =0.9978 , 0.9964 and 0.9854 for Ixx D, Iyy D, and Izz D respect ively, Figure 5.5, p l 3 3 ) . L inear regress ion equat ions are also presented in Figure 5.4 ( p l 3 2 ) and Figure 5.5 ( p l 3 3 ) . 5.5 DISCUSSION X-rays are attenuated according to the density of the structure, and the degree of beam attenuat ion in a CT image is expressed in Hounsf ield units or CT numbers ( Lampmann er al., 1984) . A l though the distr ibut ion of this at tenuat ion mainta ins an a lmost l inear re lat ionship with structural densi ty (mak ing it possible to measure t issue densi ty indirect ly f rom the CT image; Lampmann er al., 1984) , the accuracy of measurement is affected by intr insic art i facts. One of these is the result of vo lume averag ing. If the slice th ickness is too thick, or the pixel s ize too large to Table 5.2 Differences between calculated mass center and geometric center without and with bone marrow (CD, CDM in mm), and the anteroposterior anatomical locations of the mass center. Abbreviation: DP4, deciduous fourth premolar; M l , permanent first molar. Jaw # C D C D M Anteroposterior mass center location 1 1.08 0.77 Mesial cusp DP4 2 0.72 0.28 Middle cusp of DP4 3 1.47 0.95 Distal cusp of DP4 4 1.02 0.37 Distal cusp of DP4 5 1.07 0.38 Mesial cusp of Ml 6 1.44 0.75 Mesial cusp of Ml 7 0.71 0.55 Mesial cusp of Ml 8 1.10 0.53 Distal cusp of Ml 9 1.48 0.63 Mesial cusp of Ml 10 1.44 0.54 Distal cusp of Ml Mean 1.15 0.58 SD 0.30 0.21 Table 5.3 Moments of inertia without and with bone marrow (Ixx, Iyy, Izz, IxxM, IyyM, IzzM) and the ratios between the two groups. Units for all moments of inertia are g-cm2. Jaw # Ixx IxxM IxxM/Ixx Iyy IyyM IyyM/Iyy Izz IzzM IzzM/Izz 1 23.97 27.28 1.14 33.52 38.89 1.16 12.64 15.11 1.20 2 102.65 117.75 1.15 123.00 142.36 1.16 64.32 75.28 1.17 3 662.31 752.19 1.14 906.98 1030.46 1.14 446.51 499.49 1.12 4 892.14 962.35 1.08 1154.79 1253.82 1.09 520.87 565.14 1.08 5 1519.55 1617.49 1.06 2069.83 2203.69 1.06 843.30 890.54 1.06 6 1218.20 1315.04 1.08 1414.10 1527.74 1.08 638.71 688.31 1.08 7 2817.68 3004.58 1.07 3513.63 3747.36 1.07 1449.34 1526.26 1.05 8 4644.09 4846.24 1.04 5185.03 5431.21 1.05 2465.34 2559.35 1.04 9 3190.93 3404.54 1.07 3594.85 3835.34 1.07 1855.17 1969.05 1.06 10 7230.04 7667.65 1.06 8626.23 9169.59 1.06 3346.58 3541.30 1.06 Mean 1.09 1.09 1.09 SD 0.04 0.04 0.05 Figure 5.4 Estimated mass with bone marrow (EMM), estimated mass (EM), wet weight (WW) and dry weights (DW) plotted against the mass descriptor (MD). Figures also include the regression equations and coefficients of determination (R2). Ml against Ml descriptor • Ixx M. = 0.0212x+ 63.994 R 2 = 0.9978 • Iyy M. = 0.0117x+ 73.297 R 2 = 0.9964 A I Z Z M. y = 0.0056x+145.79 R 2 = 0.9854 100000 200000 300000 400000 500000 600000 700000 800000 900000 Ixx D, Iyy D, Izz D (g.cm2) Figure 5.5 Moments of inertia with marrow (IxxM, IyyM, IzzM) plotted against moment of inertia descriptors (Ixx D, Iyy D, Izz D). Figures also include the regression equations and coefficients of determination (R 2). include the smal lest region scanned, the area will be averaged with adjacent ones, and the CT number will not reflect the actual densi ty of the region. This happens frequent ly in bone due to its porous structure. In l iving subjects, these smal l spaces are fi l led with soft-t issue containing cells and f luid, whi le in dry bone they are fi l led with air. The error increases when the bone is vo lume-averaged with air. Se lect ing smal ler voxel s izes can min imize the artifact, but this requires h igher imaging resolut ion than the resolution used in our study. A second source of error is due to beam harden ing. S ince the scanned structure absorbs more low-energy photons than h igh-energy ones, the attenuat ion coeff ic ient becomes non-l inear (i.e. decreases) as the th ickness of the structure increases (Crawley, 1990) . Whi le the th ickness of the object cannot be adjusted, mathemat ica l correct ions are usual ly made in the scanner 's image reconstruct ion process (Crawley, 1990). Dual-energy CT scanning a ims to min imize the beam hardening effect by scanning the structure at two different kV levels, which involves addit ional radiat ion. Dual-energy x-ray absorpt iometry (DEXA) is widely used cl inical ly to measure BMD (Gr ier er al., 1996; Jergas er al., 1995; Koo er al., 1995; Mazess er al., 1990) . The main l imitat ion of this technique is the inabil ity to separate cortical f rom cancel lous bone. Furthermore, DEXA converts a three-d imens iona l structure into a two- d imens iona l image, and measures "area l" dens i ty rather than true vo lumetr ic densi ty (Gr ier er al., 1996) . Thus, DEXA's precis ion depends on consistent subject posit ioning. We bel ieve s ingle-energy CT scanning was an appropr iate method in the present context because it has better precis ion than DEXA (Crawley, 1990; Lindh er al., 1996) and the measurements are three-d imens iona l . S ince it produces very thin sl ices with smal l pixel s izes, it is suitable for measur ing regional bone densit ies. Addit ional ly, the correlat ion between CT numbers and BMD-equiva lent phantoms is known to be quite high, with correlat ion coeff icient over 0.99 (0.999 in the present case) for both hydroxyapat i te and potass ium dihydrogen phosphate solut ions ( Lampmann e ra / . , 1984). There are other potential sources of error in the present study. First, we could have chosen an incorrect threshold at which to f i l ter out the image background and bone marrow space. Ideal ly, everyth ing less dense than water should be set to a zero pixel va lue, mak ing the intercept of the cal ibrat ing equat ion exact ly the densi ty of water. Normal ly , this is impract ical because pixel va lues are not consecut ive numbers . For example , in this data set, when we set the pixel va lue threshold to 56, we obtained an intercept of 0.997. We could have used a threshold of 57, in which case the intercept would have been 1.017. Second , our choice of threshold did not guarantee removal of pixel "noise" f rom the very low- value single pixels around the border of each mandib le , some of which could have affected the calculated moments of inert ia if they were located far f rom the center of mass . We min imized this error by careful manual segmentat ion of each image before any calculat ions were made. A third source of error may have occurred due to the med ium we selected to fill the dry bone. We used water because it more c losely approx imates the Hounsfield unit va lue for soft t issue, and provides a c lear interface between bone and any background med ium (Daegl ing, 1989; Snyder and Schneider, 1991) . If we had used air for example , the bony outer marg ins and cancel lous bone would have been vo lume-averaged with air, result ing in lower Hounsf ield unit va lues. This would have s igni f icant ly under- est imated bone densi ty and mass. The fourth source of error was the vo lume of the assumed bone marrow space and the dens i ty we ass igned it. The dry mandib le had lost its bone marrow, and its space was fil led with water. If some of the marrow spaces had been smal le r than the voxel s ize (0.24 m m 3 ) , they would have been vo lume-averaged with bone. Finally, our landmark measurements on the reconstructed images may not have been accurate. Theoret ical ly, the m in imum error is determined by voxel s ize, because software interpolat ions dur ing the 3D reconstruct ion process are art i f ic ial. If an anatomica l l andmark actual ly falls on a surface between two voxels , only one can be se lected. Table 5.4 ( p l 3 7 ) shows the var iance in four typical l andmarks that were re- measured 10 t imes in one spec imen. The landmarks include the right condylar lateral pole, infradentale, mass center, and right molar point (def ined as the tip of the mid-facial cusp of DP4). The var iance, though smal l , differs according to axis, and might have affected our measurements of landmarks for mandib les with di f ferent or ientat ions. S imi lar errors have been discussed previously (R ichtsmeier e r a / . , 1995). Notwithstanding the above reservat ions, the ratios between est imated mass and the actual wet and dry weights were quite cons istent for the sample as a whole (1.12±0.05 and 1.49±0.13 respect ive ly) even when s imulated bone marrow was included (1.27±0.09 and 1.68±0.23 respect ive ly) , conf i rming the est imated va lues were a lways greater than the actual j aw weights. These relat ionships remained constant, even though the sample compr ised different ages and genders . The two youngest mandib les contr ibuted most to the var iance, most l ikely due to their internal morphology (which included a much larger proport ion of marrow space) . Data f rom the eight more-mature an ima ls suggests the var iance in est imated mass with respect to measured weights (with and without bone marrow inclusion) is indeed quite low. Overest imat ion of dry-bone mass for CT scanning with uniform phantom cal ibrators has been reported previously. For examp le , Cheng er Table 5.4 Error distribution in landmark definition for 10 repeated measurements in pig jaw #10. Data include mean landmark coordinates (x, y, z), their standard deviations and standard errors for right condylar lateral pole (RCLP), infradentale (Id), mass center (MC) and right molar point (RMP). Units are in mm. Mean SD S E R C L P x -11.36 0.32 0.10 R C L P y 4.00 0.35 0.11 R C L P 2 14.97 0.41 0.13 Idx -15.43 0.14 0.04 Id y 29.39 0.27 0.09 I d z -94.26 0.25 0.08 M C x -7.89 0.09 0.03 M C y 18.49 0.21 0.07 M C z 8.64 0.56 0.18 R M P x 48.19 0.09 0.03 R M P y -29.37 0.38 0.12 R M P z -84.67 0.68 0.22 al. (1995) , who used K H 2 P O 4 as bone-standard so lut ion, reported a 1 5 % overest imat ion of ash-apparent densi ty for cow bone. In the present case, a consistent 1 2 % overest imat ion in pig mand ibu lar bone mass is therefore not surpr is ing. Most l ikely, it is attr ibutable to the uni form density of the solut ion used for cal ibrat ion (compared with the inhomogeneous structure of bone), and to the vo lume averag ing of bone and water that occurs when pixel s izes exceed the bone components within t hem. It is difficult to est imate which of these two factors had the most profound effect in this instance. Theoret ical ly, sma l le r pixel s izes than those used here might have lessened the overes t imat ion . It seems appropr iate to compare masses es t imated f rom the CT scans with the wet weights of the same jaws . In the fo rmer instance, though the imaged bone was immersed in water dur ing imag ing, all pixels containing water alone were exc luded in the initial analys is (i.e. correct ions for marrow space were made later); in the latter case, we drained all free water f rom the marrow spaces and canals immediate ly prior to we igh ing. We could not, however, ensure that all the very smal l spaces among the porous bone were free of water. S imi lar ly , when we appl ied the fi lter, we could not exc lude the water f rom those spaces smal ler than the voxel s ize, i.e. the water was vo lume-averaged with bone, and would have resulted in higher pixel va lues than pure water. Thus we expected the est imated masses to be c losest to the wet weights, and we conclude that the 1 2 % overest imat ion we found represents an error attr ibutable to the physical homogene i ty of the cal ibrat ing solut ion and the imaging process. A 1 2 % reduct ion in the mass es t imated by CT scanning apparent ly provides a good est imate of the pig jaw's wet weight without marrow, especial ly in mature an imals . Compar i son of the est imated mass including (theoret ical) marrow with the wet weight of the same jaws indicated an overest imat ion of 2 7 % . Since the 1 2 % reduct ion referred to above would still apply to the minera l ized component when living an imals are imaged, a useful work ing f igure to est imate wet weight plus marrow might be to reduce the est imated mass by 1 3 % . It should be noted however that true marrow density was not verif ied direct ly in our study, and may not actual ly conform to the va lue we ass igned it (though we bel ieve it was c lose). Though DEXA expresses bone area density in g / c m 2 (a planar measurement ) , we expressed density in g / cm 3 . Our results show that mean bone density increases with age f rom 1.38 g / c m 3 to 1.74 g / cm 3 . As the oldest pig in our sample was actual ly quite young (250 days) , our max imum density of 1.74 g / c m 3 might be expected to be less than that reported by Martin and Ishida (Mart in and Ishida, 1989) for adult bovine femur (2.01 g / c m 3 wet, and 1.80 g / cm 3 dry, respect ive ly) . The lack of any relat ionship between mean bone density and gender may s imply be that our sample was too smal l to reveal one if it was present. Whi le it would have been ideal to compare our est imat ions of the jaw's moments of inertia with direct measurements , this is not an easy task. It is difficult, for example , to balance pig mandib les across kni fe-edges in order to apply known tr i-axial torques, and to record induced jaw mot ion accurate ly with acce lerometers. Approaches l ike this have their own signif icant errors. One is then faced with the task of compar ing and interpret ing data f rom two different sources, both of which have presumed, but unknown and unprovable errors. In the present case, we were reassured by our abil ity to predict the jaw's mass successful ly in spec imens of such different shapes and sizes, s ince this calculat ion would have been inf luenced by errors in ass igned density, and its d istr ibut ion. If it can be assumed that our voxe l -based predict ions of mass were indeed acceptable, our method for subsequent calculat ion of the jaw's moments of inertia is arguably the most accurate we know, for it depends solely on the abi l ity of CT-scanning to express the d imens ions of the j aw by means of voxe ls measur ing sub-cubic m m . Once voxel masses are known, specif icat ion of their relative d istances f rom a c ommon origin is the only remain ing requirement, and we consider this method an accurate way of doing this. Because the pig mandib le is U-shaped and long anteroposter ior ly, its larger moments of inertia would be expected to occur with respect to its t ransverse x-axis and its superoinfer ior y-ax is , and its smal lest moment of inertia with respect to the anteroposter ior z-axis, as conf i rmed in the present study. The fact that the moment around the y-ax is was a lways the largest may be specif ic to the pig jaw. Different combinat ions of moments probably exist in different mamma l s , and our observat ion highl ights the care needed when extrapolat ing propert ies l ike these to humans. The different moments of inertia seen in young and old mandib les in our study can be attr ibuted direct ly to var iat ions in jaw mass and size. These dif ferences appear s izeable enough to require attent ion when j aw b iomechanics are modeled at dif ferent stages of growth. The extent to which inertial propert ies (and errors in the i r specif icat ion) affect jaw mot ion in s imulat ion studies of the pig remains unclear, a l though it is obv ious va lues must be ass igned in order to predict the mot ion of any body in space. In a work ing env i ronment that includes high act ive and pass ive muscle tens ions (as well as other constra in ing forces such as art icular and tooth loads) the s ignif icance of va lues used to descr ibe moments of inertia has to be balanced against the jaw's accelerat ion dur ing normal funct ion. Since this is not part icular ly high, it is possible that the jaw's dynamics are l imited more by external constra ints than by its inherent inert ia, but this can only be establ ished in subsequent dynamic model ing studies of the pig mast icatory sys tem. Because dental enamel is denser than bone, we supposed the location of mass center would be related to the number, s ize and location of the teeth, and in fact found it was consistent ly located near the last-erupted tooth. A l though the younger mandib les seemed to have their mass centers further forward, there was no systemat ic pat tern. This, aga in, may be attr ibuted to our smal l sample size. S ince the pig mandib le compr ises t issues of dif ferent density, we expected its mass center would differ cons iderably f rom its geometr ic center (which is shape-dependent) , but they were quite c lose, even in j aws of dif ferent s izes; the l inear d istance between them actual ly decreased to an average of 0.58 m m when we included s imulated marrow in our calculat ions. A change in the mass center would be expected with the addit ion of hypothet ical bone marrow, but it is less obv ious why geometr ic center would also alter, s ince it is def ined solely by jaw shape. In our study, jaw shape was def ined by both its internal and external bony boundar ies. The internal boundary obvious ly a l tered when the jaw was " f i l l ed" with marrow, causing a shift in the geometr i c center as well as mass center. The smal l dif ference between mass center and geometr ic center may be specif ic to the pig (i.e. uniquely determined by the way t issues of dif ferent densi ty are distr ibuted) but would be representat ive of a c ommon mamma l i an t rend. This quest ion can only be answered by further exper iment . Our results, however, offer the opt ion of est imat ing the mass center by measur ing the geometr ic center a lone, at least in the pig. Being shape-der ived, geometr ic center may be eas ier to obtain by less- invasive means than CT scanning, and if future exper iments show that mass center and geometr ic center are also c lose in the human jaw, magnet ic resonance (MR) imaging might be a practical way to est imate mass center when model ing l iving humans. Finally, our study suggests a mass descr iptor (the product of mean bone density and three s imple l inear measurements of j aw size and shape) can be used to predict the jaw's mass . S ince we have provided est imates of average j aw density in the present study, we can make a reasonable est imate of mass in a specif ic pig by using these va lues, and three l inear measurements obta ined either by direct mensurat ion , or f rom var ious jaw images. Taken together, this method for predict ing j aw mass , and the equat ions l inking the descr iptors for moments of inert ia with calculated moments of inert ia, offer a s imple way of est imat ing the mass propert ies needed for model ing l iving pigs (at least in young adults of dif ferent s izes) without the need for CT scanning. A separate s tudy of the age-dens i ty relat ionship would however be useful in pigs, especia l ly in very young and very old an ima ls . It should be feasible to carry out a s imi lar study to ours on human mandib les, not only to determine their mass propert ies, but also to ver i fy whether these too can be predicted, with acceptable accuracy, by MR imaging or by direct measurements of the jaw. If so, it might be possible to model mass propert ies, and consequent ly jaw dynamics , in l iving subjects.  6 MASS PROPERTIES OF THE HUMAN MANDIBLE AND THEIR FUNCTIONAL SIGNIFICANCE 6.1 ABSTRACT Realist ic computer s imulat ion of mast icatory sys tem dynamics requires specif icat ion of the jaw's mass propert ies. Recent ly, we est imated these in the pig, and suggested imaging modal i t ies with uni form representat ion of bone densi ty may be adequate to perform this task (Zhang et al., 2001a , see Chapter III, p l l 4 ) . Here, we wished to determine if this is true for the human jaw, since it differs morphologica l ly f rom that in the pig. We also wished to determine the sensit iv i ty of an exist ing dynamic jaw model to these mass propert ies dur ing postural rest arid jaw opening. High-resolut ion CT scans were per formed on 13 osseous spec imens. Cal ibrat ion phantoms were used to convert CT numbers to mineral density. The mean est imated jaw mass was 1 3 % greater than the mean wet weight for the adult dentate mandib les, and 1 5 % greater for the whole sample . Putative bone marrow accounted for an extra 9 % of mass. The mean bone densit ies for adult dentate mand ib les were very consistent (1.72±0.02 g / cm 3 ) . The mass and geometr i c centers were close (mean l inear difference 0.57±0.32 m m ) . The largest moment of inertia (MI) occurred around each jaw's supero infer ior ax is , and the smal lest around its t ransverse axis. Bone marrow added an extra 7-9% to M is around the three axes. Linear re lat ionships were found between the actual mass and a mass descr iptor (bounding vo lume x mean bone dens i ty) , and between M is and three MI descr iptors (mass descr iptor x two orthogonal d imens ions of the bounding box). Dynamic model ing with median inertial va lues suggests while mass and mass center are critical aspects in model ing jaw dynamics , the moments of inert ia are low, and less inf luential . 6.2 INTRODUCTION Dynamic models are a useful way to observe muscu loske leta l structure and funct ion in the human mast icatory sys tem (Hannam er a/., 1997; Koolstra and van Ei jden, 1995, 1997a, b; Langenbach and Hannam, 1999; Otten, 1987) . S ince they permit rapid changes in craniofacial fo rm, musc le propert ies and muscle dr ive, these models provide a construct for expla in ing force interact ions among the musc les, jo ints and teeth. Also, they are a potential ly beneficial way to study the b iomechanics of c l in ical ly-relevant condit ions including deve lopmenta l abnormal i t ies, musculoske leta l d isorders, surgical intervent ions and rep lacement prostheses. The extent to which dynamic jaw models are sens i t ive to mass propert ies ass igned to them is not ful ly understood. These propert ies can be difficult to est imate in l iving subjects (Braune and Fischer, 1988) , and even in exc ised human t issue (Koolstra and van Ei jden, 1995) . Changes in the center of gravi ty alter jaw and condy lar veloc i t ies dur ing j aw- c los ing, though var iat ions in the moments of inert ia are reported to have little effect (Koolstra and van Ei jden, 1995) . A re levant factor here is musc le damp ing , which has been shown to affect j aw mot ion in l iving subjects and kinetic jaw models (Peck er a/., 2000) . Muscle damping is l ikely to be part icular ly effective dur ing jaw open ing, when both act ive and passive musc le tensions are generated. Recently, we used computed tomography (CT) to es t imate mass propert ies of the pig jaw (Zhang er al., 2001a , see Chapter III, p l l 4 ) . With suitable correct ion factors, the method provides a good approx imat ion of the jaw's wet weight, including s imulated bone marrow. The pig jaw's mass center and geometr ic center a lmost coincide, and there are l inear relat ionships between the jaw's actual mass and a general mass "descr iptor" (def ined by j aw vo lume and mean bone dens i ty) , and also between its moments of inertia and moment s of inertia "descr iptors" (def ined by the mass "descr iptor" and the overal l d imens ions of the j aw) . The f indings suggest the dens i ty distr ibut ion of the pig mandib le is relat ively homogeneous around its geometr i c center. Here, we est imated the mass propert ies of the human j aw the same way. We considered if the mass and geometr ic centers also coincided in humans , and if s imple d imensional descr iptors could be used to est imate mass propert ies (as in the pig), then s imple non- invas ive methods might be used to est imate jaw mass propert ies in l iving subjects . We then studied the effect of modi fy ing these mass propert ies, and those reported by Koolstra and van Eijden (1995) , in an exist ing musc le -damped model of jaw-open ing (Peck e r a / . , 2000) . We were part icular ly interested in the sensit iv i ty of the model to alterat ions in the mass , mass center, and moments of inertia when the jaw was at postural rest, and dur ing act ive opening to max imum gape. 6.3 MATERIALS AND METHODS 6.3.1 Mass property estimation Mass propert ies were est imated in 13 arch ived human mandib les of unknown gender. The sample included eight j aws with adult dent i t ions, two with mixed dent it ions, two with dec iduous dent i t ions, and one edentulous jaw. Use of this mater ia l compl ied with the requ i rements of The Univers i ty of Brit ish Co lumbia 's Ethical Review Commi t tee . Detai ls of the CT scanning, image processing and calculat ion of the propert ies have been reported e lsewhere (Zhang er al., 2001a , see Chapter III, p i 14). In brief, the jaws were weighed dry, and re-weighed after hydrat ion for 48 hours. They were submerged in water during imaging to opt imize resolut ion of the bone interface (Daegl ing, 1989), and to min imize vo lume-averag ing errors. Cal ibrat ion phantoms containing K H 2 P O 4 solut ions at concentrat ions of 0.05, 0.15, 0.25 and 0.50 g / c m 3 were used to express bone minera l dens i ty (BMD) as a funct ion of pixel va lue ( Lampmann er al., 1984) . Coronal scans at one m m intervals (field of v iew 220 x 220 m m , pixel s izes 0.43 x 0.43 mm) were obta ined with a Toshiba Xpress SX scanner (Toshiba Corporat ion, Tokyo, Japan) operat ing at 100 kV and 150 mA. The images were converted to single 8-bit files and f i ltered so that structures equal in density to, or less dense than, water were exc luded; thus the image background was pure black and disclosed wet bone without bone marrow (Zhang et al., 2001a , see Chapter III, p l l 4 ) . We used a commerc ia l program (3DVIEWNIX 1.2, Univers i ty of Pennsy lvan ia Medical Center, Phi ladelphia, PA) for segmentat ion, jaw surface reconstruct ion, landmark identif ication and measurement . Another program (Ca l image - Calculate Image, Craniofacia l Laboratory, The Univers i ty of Brit ish Co lumb ia; see http://condor.dent istrv.ubc.ca or the Append ix , p226) per formed image matr ix operat ions and mass property calculat ions (Zhang er al., 2001a , see Chapter III, p i 14). Moments of inertia were referenced to an anatomica l coordinate sys tem with its x-ax is d irected t ransverse ly f rom left to right (v iewed frontal ly), its y-axis directed superoinfer ior ly, and its z-axis anteroposter ior ly. The x-z plane was paral lel to the dental occlusal plane, and the y-z plane was mid-sagitta l (Figure 6.1, p l 4 9 ) . To est imate the contr ibut ion of bone marrow, we segmented the non-minera l ized component in each CT sect ion, ass igned the selected pixels a density of one g / cm 3 (Zhang e r a / . , 2001a , see Chapter III, p l l 4 ) , and recalculated the mass propert ies. 6.3.2 Prediction of mass and moments of inertia Here, we used the s imi lar general descr iptors as those in our pig study. They included jaw width (distance between the two lateral condylar poles, WD) , jaw height (vert ical d istance f rom the tip of the coronoid process to the lower border of the mandib le , HD), j aw length (distance between front edge of the central incisor and the posteroinfer ior point of the condyle on one side, LD), all represent ing the bounding box d imens ions, jaw vo lume ( V D = W D x H D x L D ) , j aw mass (product of VD and mean bone densi ty, MD), and jaw moments of inert ia (product of MD and two orthogonal d imens ions of the bounding box, IxD, IyD and IzD). We also included an addit ional cephalometr ic " t o ta l " j aw length descr iptor (TLD) v iz . the d istance between condyl ion and gnath ion. Regress ion curves were then f itted to plots descr ib ing relat ionships between these descr iptors and the mass propert ies der ived f rom the CT images. 6.3.3 Jaw model The three-d imens iona l , dynamic model of the j aw has been descr ibed in detai l e lsewhere (Langenbach and Hannam, 1999; Peck e r a / . , 2000) . It included re levant musculoske leta l geometry (Baron and Debussy, 1979) and musc le propert ies (Otten, 1987; van Eijden e r a / . , 1995, 1996, 1997; van Eijden and Raadsheer, 1992) . Jaw mot ion was shaped by the act ive and passive tens ions of 16 cran iomandibu lar actuators (represent ing Figure 6.1 Coordinate system used to express moments of inertia. The transverse axis is represented by x, the superoinferior, by y, and the anteroposterior, by z. The axes z, y lie in the mid-sagittal plane, and x, z are parallel to the occlusal plane. major musc le groups) , by reaction forces in the two temporomand ibu la r jo ints , and by gravi ty. The muscle subgroups were s imulated with Hi l l- type, f lexible, s ingle- l ine actuators with f iber and tendon components (Zajac, 1989) . Each actuator was lightly damped (10 Nsm" 1 ) to prevent h igh-frequency, internal osci l lat ion. The actuators ' pass ive tens ions permitted an inter-incisal j aw gape of 50 m m when a 5 IM external opening force was appl ied to the jaw for 0.5-1.0 sec (for detai ls, see Peck er a/., 2000) . This gape was also attained when jaw-open ing was dr iven by actuators s imulat ing digastr ic and lateral pterygoid musc le co- act ivat ion bi lateral ly (max imum act ive tens ions 11.6 N and 16.8 N, respect ive ly) . The jaw's art iculat ion with the c ran ium was modeled with paired, canted, el l ipsoidal condyles rotat ing and sl iding against fr ict ionless curv i l inear surfaces. The mandib le was considered a rigid body within a vert ical gravi tat ional field of 9.8 m/s 2 (i.e. the head was assumed to be held in an upright posture). The model was des igned with commerc ia l software (ADAMS; MDI, Ann Arbor, MI) using mixed , non-l inear, differential and a lgebraic equat ions to compute its dynamics (van den Bogert and Nigg, 1999) i.e. numerica l integrat ion of component accelerat ions enabled calculat ion of their velocit ies and posit ions. An iterat ive, two-phase predictor-corrector technique involv ing user-def ined to lerances produced solut ions which were rejected when they did not converge. Two vers ions of the model were tested, one which used the median mass property data for adult dentate j aws obta ined in the present study, and another in which the mass propert ies reported by Koolstra and van Eijden (1995; mass , 440 g; Ix, 2900 g . cm 2 ; Iy, 8600 g . cm 2 ; Iz, 6100 g .cm 2 ) were inserted. In both models, mot ion of the midl ine incisor point (a region of unrestra ined jaw mot ion) was measured when resting posture and max imum jaw opening were s imu la ted . In addi t ion, the respect ive centers of mass were moved systemat ica l ly one cm anter ior ly, posterior ly, super ior ly and inferiorly, and the moments of inert ia of both models were arbitrar i ly ha lved. 6.4 RESULTS 6.4.1 Calibration As shown in Figure 6.2 ( p l 5 2 ) , the cal ibrat ion data revealed a l inear relat ionship between pixel va lues and mineral dens i ty (correlat ion coeff icient 0.995) i.e. BMD = 0.012 x Pixel Va lue + 1.005 6.4.2 Estimated masses and mean bone density Table 6.1 ( p l 5 3 ) i l lustrates the est imated masses with and without putat ive bone marrow, the measured weights, and the mean bone densit ies with and without bone marrow. Overa l l , the es t imated jaw mass was 1 5 % greater than the wet weight (mean EM/WW=1.15±0.04; coeff icient of var iat ion, CV 4 .39%) , and 3 8 % greater than the dry weight (mean EM/DW=1.38±0.17; CV 12 .32%) . The inclusion of putat ive bone marrow added 9 % more mass (mean EMM/EM = 1.09±0.05). The est imated mass with marrow was 5 1 % greater than the measured dry weight of the j aw (mean EMM/DW=1.51±0.26). When the less-dense bone marrow was included in the calculat ions, the mean bone density decreased by 4 % (MBDM/MBD=0.96±0.01). The mean bone density (which included tooth dent in and enamel) was less in the younger and edentulous mandib les, but very s imi lar in the adult dentate mandib les Phantom curves — — P i g mandibles — x — H u m a n mandibles y= 0.013x+0.997 y = 0.012x+1.005 0 5 10 15 20 25 30 35 40 45 50 Pixel value Figure 6.2 Calibration curve for the phantom used in the present study. For comparison, data are also shown for the previous study in the pig (Zhang et al, 200ia, see Chapter III, pii4). Table 6.1 Measured jaw weights (dry weights, DW; wet weight, WW), estimated masses (estimated mass, EM; estimated mass with marrow, EMM), and calculated mean bone densities (MBD) and MBD with marrow (MBDM), and the ratios between these variables. All weights and masses measurements are in g, and all density measurements are in g/cm3. Adults Adults + Children Entire sample Mean SD Mean SD Mean SD D W 80.18 14.46 62.61 28.75 59.76 29.39 WW 90.85 16.08 72.03 30.98 69.36 31.18 E M 102.32 18.33 81.69 34.30 78.93 34.31 E M M 108.90 19.02 87.34 35.77 84.97 35.30 E M M / E M 1.07 0.02 1.08 0.03 1.09 0.05 M B D 1.72 0.02 1.67 0.09 1.65 0.12 M B D M 1.65 0.03 1.60 0.09 1.58 0.12 M B D M / M B D 0.96 0.01 0.96 0.01 0.96 0.01 E M / D W 1.28 0.02 1.35 0.12 1.38 0.17 E M M / D W 1.36 0.04 1.45 0.16 1.51 0.26 E M / W W 1.13 0.02 1.14 0.03 1.15 0.04 E M M / W W 1.20 0.04 1.23 0.05 1.25 0.09 (1.72±0.02 g / cm 3 , CV 1.16%). 6.4.3 Mass and geometric centers The mass centers, and the dif ferences between the mass and geometr ic centers, are shown in Table 6.1 ( p l 5 3 ) . In the adult dentate mandib les, the mass centers lay between the second and third molars when third molars were present; otherwise, they were near the last erupted tooth. They a lways lay within the upper one third of the distance f rom the dental occlusal surface to the inferior mand ibu lar border. The mean l inear dif ference between the mass and geometr i c centers was smal l (0.57±0.32 m m ) , the inclusion of s imulated marrow having little effect (0.65±0.27 m m , p>0.05) . 6.4.4 Moments of inertia The smal lest moment of inertia occurred around the jaw's t ransverse, and the greatest around its superoinfer ior (vert ical) ax is . Added "bone marrow" increased the moment of inertia by 8% around the t ranverse axis, and by 9 % around the vert ical and longitudinal axes (Table 6.3, p l 5 6 ) . 6.4.5 Mass and moments of inertia predictions In the dentate mandib les, there were v irtual ly l inear re lat ionships among est imated masses , measured weights and the general mass descr iptor (R 2 =0.9646 , 0.9689, 0.9771 and 0.9710 for EMM, EM, WW and DW, respect ively, Figure 6.3, p l 5 7 ) . There were also l inear re lat ionships among the three moments of inertia and their respect ive moments of inertia descr iptors (R 2 =0.9664 , 0.9259 and 0.9475 for IxD, IyD, and IzD Table 6.2 Differences between geometric and mass centers without marrow (CD, mm), and with marrow (CDM, mm). Also included are the horizontal and vertical mass center locations relative to the teeth (HMCL), and in the vertical dimension (VMCL, expressed as percentage of the distance from the dental occlusal plane to the inferior border of the mandible). A: adult; M: mixed dentition; D: deciduous dentition; E: edentulous; M l , M2, and M3, permanent first, second and third molars; Dm2, deciduous second molar. Paired t-test p >o.05 for CD and CDM. # Age C D C D M H M C L V M C L 10 A 0.55 0.79 Mesiobuccal cusp of M3 0.30 4 A 0.30 0.44 Distal of M2 0.33 3 A 0.48 0.48 Distal of M2 0.33 11 A 0.15 0.72 Distobuccal Cusp M2 0.33 9 A 0.31 0.35 Mesiobuccal cusp of M3 0.30 5 A 0.47 0.61 Distobuccal cusp of M2 0.40 2 A 0.41 0.46 Distobuccal cusp of M2 0.37 1 A 0.75 0.60 Distobuccal cusp of M2 0.36 8 M 0.99 0.99 Distobuccal Cusp of Ml 0.33 7 M 1.34 1.31 Distal of DM2 0.30 6 D 0.46 0.59 Distal of DM2 0.35 12 D 0.44 0.35 Distobuccal cusp of DM2 0.41 13 E 0.82 0.79 Anterior border of ramus 0.13 Mean 0.57 0.65 0.33 SD 0.32 0.27 0.07 Table 6.3 Moments of inertia without marrow (Ix, Iy, Iz), moments of inertia with marrow (IxM, IyM, IzM) and the ratios between them (IxR, IyR, IzR). All moment of inertia measurements are in g.cm2. Adults Adults + Children Entire sample Mean SD Mean SD Mean SD Ix 776.65 235.61 570.98 358.83 555.62 347.99 IxM 825.69 238.71 608.16 375.2 595.88 361.95 IxR 1.07 0.03 1.07 0.03 1.08 0.05 iy 1482.03 421.59 1108.41 650.77 1071.76 636.92 IyM 1581.74 439.62 1186.15 686.38 1155.53 666.37 IyR 1.07 0.03 1.08 0.03 1.09 0.05 Iz 1250.08 356.12 922.92 562.92 893.67 549.18 IzM 1337.97 364.77 990.48 592.61 966.46 573.95 IzR 1.07 0.03 1.08 0.03 1.09 0.05 • EMM • EM WW DW y = 0.1256x09526 y = 0.0921x0'9876 y = 0.0641x1 0 2 1 6 y = 0.0259x11315 R2 = 0.9737 R2 = 0.97 R2 = 0.9788 R2 = 0.9653 0 1 1 1 1 1 1 1 1 1 80 280 480 680 880 1080 1280 1480 1680 MD Figure 6.3 Estimated mass with marrow (EMM), estimated mass (EM), wet weight (WW), and dry weight (DW) plotted against the mass descriptor (MD). In each case, a power function, and a linear function have been fitted to the data. respect ively, see Figure 6.4, p l 6 0 ) . Table 6.4 ( p l 6 0 ) includes coeff ic ients of determinat ion when the descr iptors were used to predict EM, EMM, WW, DW, Ix, Iy, and Iz. In each case, TLD had the largest coeff icient of determinat ion. 6.4.6 Model sensitivity to mass properties There was a marked difference in the magn i tude of inter- incisal separat ion when the smal ler mass propert ies represent ing the median of our adult dentate sample , and the larger propert ies reported by Koolstra and van Eijden (1995) were compared with the j aw rest ing in a vert ical gravitat ional field (Figure 6.5, p l 6 1 ) . Most of the mot ion f rom tooth contact to the rest posit ion occurred within 250 msec, the heav ier of the two jaws opening about 10 m m further than the l ighter. In both cases, halv ing the respect ive moments of inertia (without changing the mass) had no effect on the incisor-point t ime-d isp lacement curves. Movement of the mass centers anter ior ly and posterior ly had a profound effect on the final rest ing posit ions (Figure 6.6, p l 6 2 ) , whereas movemen t super ior ly or inferiorly had no effect. In the heavier jaw, anter ior movemen t of the mass center caused the descending jaw to overshoot its final resting posit ion. When dr iven by jaw-opening musc le "act ivat ion", both mode ls reached 50 m m inter- incisal gape. Initial ly however, the heav ier j aw opened wider than the l ighter, sustain ing an increased gape throughout the first 500 msec. The l ighter jaw took a lmost 200 msec before open ing signif icantly. In both cases, the final 10-15 m m was reached quick ly (around 600 msec) at which t ime both act ive opening and pass ive closing "musc le" tens ions were nea r -max imum (Figure 6.7, p l 6 3 ) . Table 6.4 Coefficients of determination (R2) for power function curve fits between estimated mass (EM), estimated mass with marrow (EMM), wet and dry weights (WW, DW), moments of inertia (Ix, Iy, Iz), and all descriptors for dentate human mandibles. These include total length descriptor (TLD), length descriptor (LD), width descriptor (WD), height descriptor (HD), volume descriptor (VD), mass descriptor (MD), moments of inertia descriptors (IxD, IyD, IzD). T L D W D H D L D V D M D IxD IyD IzD E M 0.98 0.81 0.91 0.94 0.94 0.97 E M M 0.98 0.81 0.92 0.94 0.95 0.97 WW 0.98 0.82 0.92 0.95 0.95 0.98 D W 0.97 0.81 0.89 0.95 0.95 0.98 Ix 0.99 0.83 0.94 0.96 0.97 0.99 0.99 iy 0.98 0.84 0.93 0.93 0.96 0.98 0.97 Iz 0.99 0.85 0.95 0.96 0.98 0.99 0.98 • Ix • ly A Iz y = 0 . 0 0 2 4 X 1 0 4 6 5 R 2 = 0 .9866 y = 0 . 0 0 3 6 X 1 0 2 6 4 R 2 = 0.9651 y = 0 . 0 0 3 4 x 1 0 5 1 4 R 2 = 0.9787 y = 0 .0044x - 30.042 y R 2 = 0 .9699 = 0 .0054X - 54.697 y R 2 = 0 .9332 = 0 . 0067X- 52.566 R 2 = 0.9447 2500 2000 A • 1500 Ix , ly , 1000 5 0 0 "* 0 1 1 1 80 5 0 0 8 0 1 0 0 0 8 0 150080 200080 2 5 0 0 8 0 3 0 0 0 8 0 3 5 0 0 8 0 400080 IxD, ly D, IzD Figure 6.4 Moments of inertia (Ix, ly, and Iz) plotted against the moment of inertia descriptors (IxD, IyD, and IzD). In each case, a power function and a linear function have been fitted to the data. ••— median x koolstra 0 100 200 300 400 500 600 700 800 time (ms) Figure 6.5 Incisor-point motion from tooth contact to the jaw's rest position, plotted against time. The curves represent mass properties used by Koolstra and van Eijden (1995) and median values from the present study. In both cases, halving the respective moments of inertia did not affect the curves. -•— median_mc_post -•— median_mc_ant koolstra_mc_post —x— koolstra_mc_ant 30 0 200 400 600 800 time (ms) Figure 6.6 Incisor-point motion from tooth contact to the jaw's rest position, plotted against time. The curves represent mass center locations 10 mm anterior and 10 mm posterior to the original. Data are shown for mass properties used by Koolstra and van Eijden (1995) and median values from the present study. In both cases, moving mass centers superiorly and inferiorly had no effect. -•—median - -------koolstra 60 r time (ms) Figure 6.7 Incisor-point motion during maximum jaw opening, plotted against time. The curves represent mass properties used by Koolstra and van Eijden (1995) and median values from the present study. 6.5 DISCUSSION Sources of error in the est imat ion of mass propert ies by CT scanning have been descr ibed e lsewhere (Zhang er al., 2001a , see Chapter III, p l l 4 ) , and all ex isted in the present study. They include l imitat ions in the resolut ion of the CT scans, the threshold va lues emp loyed , the med ium fil l ing the dry bone (water), t issue segmentat ion , the three-d imens iona l (3D) reconstruct ion process, and landmark ident i f icat ion. The lower kV value (100 kV) we used here generated much less heat than what we used in the previous study (120 kV, Zhang e r a / . , 2001a , see Chapter III, p l l 4 ) . It also produced a sl ightly different cal ibrat ion curve than that obta ined in the pig study. A low kV produces photons of low max imum energy (i.e., a "sof t " x-ray beam); since more photon energy is absorbed (especial ly in t issues with high atomic density; see Morgan, 1983) , the CT number increases, and a g iven pixel va lue will represent a lower densi ty than that obta ined with a higher kV. Overest imat ion of bone mass by CT scanning with uni form cal ibrators has been reported previously. Cheng er al. (1995) , using K H 2 P 0 4 as a bone-standard, reported a 1 5 % overest imat ion of ash-apparent densi ty for cow bone. In our pig study, we reported a 1 2 % overest imat ion of jaw bone mass (Zhang er al., 2001a , see Chapter III, p l l 4 ) . In the adult dentate mandib les here, the est imated mass was 1 3 % greater than the wet weight, whi le in the pooled sample of dentate jaws , it was 1 4 % greater, and in the entire sample , it was 1 5 % greater. This is l ikely related to the porosity of the different jaws. Whi le the EM/WW ratio was highly cons istent for the adult dentate mandib les , its var iat ion progress ive ly increased with the addit ion of the mixed dent i t ion, the dec iduous dent i t ion, and the edentulous jaw. The latter was the most porous, and was overest imated by 2 3 % . A second reason for overest imat ion may have been the lower kV. A l though K H 2 P 0 4 solut ions are often employed as cal ibrat ion phantoms (Cheng er al., 1995; Lampmann e r a / . , 1984; Zhang et al., 2001a) , the i r dens i ty ranges do not necessar i ly coincide with those in the imaged bone (solut ion densit ies typical ly range f rom 1.05 to 1.50 g / cm 3 , whereas the mean bone density can approx imate 1.7g/cm 3 ) . High solut ion concentrat ions can reach their saturat ion points, and tr igger heterogeneity. Dur ing imag ing, lower- energy photons are preferential ly absorbed by the harder t issues (e.g. bone) due to beam-harden ing effects, and the resultant attenuat ion coeff ic ients become non-l inear (Morgan, 1983) . The same photons, however, are hard enough to pass cal ibrat ion solut ions with l inear at tenuat ion. Whi le our lower kV value may thus have produced a less- rel iable cal ibrat ion curve, we bel ieve the error is sma l l . In any event, the results are consistent with those reported by Cheng et al. (1995) , and the previous study (Zhang et al., 2001a , see Chapter III, p l l 4 ) . In dynamic model ing, true specif icat ion of the mandib le 's mass and mass center is not a tr ivial undertak ing. The effect ive mass const i tutes the total instantaneous mass of all hard and soft t issue being moved , and could conceivably include the tongue (Langenbach and Hannam, 1999). Though neither the pig nor human jaw is regu lar ly-shaped, it is possible to define the mass centers of i rregular objects by direct exper iment , e.g., by suspending the j aw in different or ientat ions. In a di f ferent approach, Koolstra and van Eijden (1995) sect ioned a cadaver j aw into e lements , assumed the mass distr ibut ion was homogeneous , and used e lement locations to calculate the mass center. A drawback of all direct approaches, however, is their inappl icabi l i ty to l iving subjects. The present imaging technique is essent ia l ly a modi fed vers ion of the general method used by Koolstra and van Eijden (1995, 1997b) . The X-ray beam and the smal l e lement size both permit a ref ined est imat ion of regional bone density. The mean difference found between the mass and geometr i c centers was less than that we reported in the pig (Zhang er al., 2001a , see Chapter III, p l l 4 ) , and the addit ion of s imulated bone marrow made no dif ference stat ist ical ly (Table 6.1, p l 5 3 ) . We conclude that e i ther the human mandib le has relat ively less bone-marrow space, and/or the marrow space is more evenly distr ibuted in humans than in young pigs. S ince the mass center in the adult dentate human mandib le lies between the second and third molars (or at the last molar if there are no third molars) on the midsagitta l plane, and is about one third of the distance f rom the occlusal surface of teeth to the lower border of the mandib le, in most cases it could be approx imated to within a few m m by l inear measurement of convent ional radiographic images. A l though moments of inertia can be est imated by suspending a body and measur ing its osci l lat ions, Koolstra and van Eijden (1995) used the equal ly-s ized pieces f rom their sect ioned jaw and integral calculus to calculate the moments of inertia f rom the e lements ' masses and their locations. In related studies, Hannam er al. (1997) , Langenbach and Hannam (1999) , and Peck er al. (2000) ass igned mass propert ies predicted f rom a f in i te-e lement model of the human jaw deve loped by Korioth er al. (1992) . The latter was der ived f rom CT scann ing, and included e lements with propert ies specif ic to dif ferent j aw regions. These two approaches are somewhat s imi lar to the one used here. Smi th e r a / . (1995) indicated a close approx imat ion of mass propert ies can be made by ass igning associated density to voxe ls compr is ing the structure; but it remains difficult to val idate calculated moment s of inert ia, and the use of inaccurate or imprec ise physical methods does not make sense. S ince, in the present context, the momen t of inertia for each e lement is the product of its mass and the squared d istance f rom the center of the e lement to the mass center, any momen t of inertia est imated for each scanned e lement is theoret ica l ly val id provided the e lement 's mass itself is va l id . As the total moment of inert ia equals the sum of the moments of inertia of all const i tuent e lements , the val id ity of the total moment of inertia calculated ul t imately depends on that of the total mass ca lcu lated. Thus, a 1 3 % overest imat ion of the mass for the adult dentate human mandib le would affect its moment s of inertia s imi lar ly in all three axes. The proport ional magni tudes of the moment s of inert ia we have descr ibed are intuit ively predictable. As in the pig, the largest moment of inertia occurred around the human jaw's superoinfer ior ax is , due to the contr ibut ion of its largest anteroposter ior, and second- largest t ransverse d imens ions. The smal lest moment occurred around its anteroposter ior axis, due to the contr ibut ion of its smal lest vert ical and second-smal lest t ransverse d imens ions. In our sample , the mean jaw width was sl ightly greater than the mean length (data not reported), and the smal lest moment of inertia would be expected to occur around the t ransverse axis. Like others (Cheng er al., 1995; Smi th er al., 1995) , we assumed var iat ions in the jaw's physical density would have a major effect on its mass propert ies. We found the mean bone densi ty in the adult dentate human mandib le to be quite consistent. In the pig, we reported l inear relat ionships between jaw mass and a general mass descr iptor (Zhang er al., 2001a , see Chapter III, p l l 4 ) , and here we found the same (albeit with different constants) . While there were strong corre lat ions among the actual mass and all d imens iona l descr iptors, TLD proved the best predictor (Table 6.4, p l 5 9 ) . G iven the human jaw's relat ively- homogenous density, it is perhaps unsurpr is ing that a momen t of inertia can be predicted with d imensional descr iptors, because it is a funct ion of the mass and the sum of the squares of two orthogonal d imens ions. Since mass is proport ional to an object's vo lume and dens i ty, it is a funct ion of the d imens ions cubed. Thus, the moment of inertia is a funct ion of the object 's densi ty, and its d imens ions raised to the fifth power, and moments of inertia might be expected to be predictable f rom three- d imensional sca lar measurements of the jaw. The general s imi lar i ty in the proport ions among mass and moments of inertia when our data are compared to the more-direct est imat ions made by Koolstra and van Eijden (1995) also lends credence to the idea of total mass and jaw d imens iona l descr iptors being pr imary determinants of mass propert ies. The Koolstra and van Eijden's (1995) j aw mass of 440g (which included all at tached soft t issue) was about four t imes heav ier than the median dentate j aw mass in our sample (about 105 g) , yet the proport ions among its mass propert ies are about the same as ours. It seems therefore, for model ing purposes, the dens i ty of human mandib le can be considered homogeneous, and that a non- invas ive imaging technique such as MR (which does not reveal bone densi ty, but which can reveal bony contours) might be adequate for est imat ing j aw mass propert ies in l iving humans. If so, dynamic models of individual j aws would be practical in normal humans , and in cases like facial a symmet ry , part ial ly resected mandib les, etc. The segmentat ion technique we have descr ibed could be used with any 3D imaging method, al lowing mass-proper ty est imat ion for skeletal parts. As an example , in unreported exper iments , we have sect ioned the mandib le into components , est imated the i r respect ive mass propert ies, and used these in dynamic models to measure the forces and torques t ransmi t ted through junct ions between the reassembled parts. The dependence of the resting posture of dynamic jaw models on at least some inertial propert ies is to be expected. A heavy mandib le must reach a lower posit ion than a light one, though it is less obv ious how long it will take to reach it. V iscous damping by the musc les and other t issues surrounding the jaw can be expected to affect its speed in response to induced forces, including musc le contract ion. Koolstra and van Eijden (1995) used a jaw-c los ing model dr iven by the media l pterygoid musc les, and damped it by apply ing friction at the center of grav i ty. Their f indings on changing the mass center and moments of inert ia, coupled with our observat ions here, conf irm dynamic jaw models remain sensi t ive to the specif icat ion of both their masses and mass centers even when musc le tensions are present. Much depends upon what is cons idered the jaw's true mass , e.g. how much related soft t issue should be inc luded, and whether or not this should include the tongue which we ighs about 50-60g. Taken together, the studies suggest errors of severa l m m or more in any direct ion when mass centers are specif ied will affect jaw-open ing and jaw-c los ing predict ions to and f rom max imum gape. Errors in the anteroposter ior direct ion will affect the jaw's rest ing posture, though deviat ions in vert ical direct ion are un important here. Our demonstrat ion that moments of inertia have little effect on j aw opening and resting posture comp lements s imi lar f indings dur ing j aw closing (Koolstra and van Ei jden, 1995) . Both studies thus infer these in some latitude when specifying the jaw's moments of inert ia, if its dynamics are modeled in the midl ine. Gravi tat ional accelerat ion, v iscous damp ing , and the generat ion of musc le tens ions seem suff ic ient to ensure these relat ively low moments do not play a signif icant role in shap ing free jaw movements .  7 DYNAMIC MECHANICS IN THE PIG MANDIBULAR SYMPHYSIS 7.1 ABSTRACT During funct ion, var ious b iomechanica l events occur at the mamma l i an j aw symphys i s . Previously, these have been studied in the static env i ronment , or by direct recording of surface bone stra ins. So far however, it has not been possible to demonst ra te direct ly the forces and torques pass ing through the symphys i s in assoc iat ion with dynamica l ly changing musc le tens ions. Recently, dynamic models have been used to study j aw b iomechanics in humans and pigs. Here, we modif ied a previously publ ished dynamic pig jaw model to measure the forces and torques at the symphys i s , and related these to s imulated mast icatory musc le tens ions, bite, jo int and food bolus forces. The model was based on an individual pig's musculoskeleta l structure and included specif ic mass propert ies for each half of the mandib le. An artif icial rigid jo int was created at the symphys i s , al lowing measurements of the tr i-axia l forces and torques pass ing through it. An artif icial food bolus was placed at the right fourth dec iduous premolars (DP4) dur ing s imulated r ight-s ided chewing. The model successful ly predicted three prev ious ly postulated loading patterns at the symphys i s . Dorsoventra l shear occurred when the lower teeth hit the artif icial food bolus. It was assoc iated with ba lanc ing- side jaw adductor forces, and reaction forces f rom the work ing-s ide food bolus. Medial t ransverse bending occurred dur ing j aw open ing, and was associated with bilateral tensions in the lateral pterygoid and digastr ic musc les. Lateral t ransverse bending occurred at the late stage of the power stroke, and was assoc iated with the act ions of the deep and superf ic ial masseters . The largest predicted force was dorsoventra l shear force, and the largest torque was a "w i shbon ing" torque about the superoinfer ior ax is. We suggest dynamic model ing offers a new and powerful method for studying jaw b iomechanics, especia l ly when the parameters involved are difficult or impossib le to measure in vivo. 7.2 INTRODUCTION The b iomechanics of the mamma l i an mandibu lar symphys i s have been studied extens ive ly in vivo by strain gauge measurements in non-human pr imates (Hylander, 1979a, 1984, 1985) , e lec t romyograph ic and c ineradiographic recordings in non-human pr imates and mamma l s with unfused symphyses (Hy lander e r a / . , 1998, 2000; Hy lander and Johnson, 1994; L ieberman and Crompton , 2000) and by morpholog ica l ana lyses in a wide range of mamma l s (Daegl ing, 1993; Daegl ing and Jungers, 2000; L ieberman and Crompton , 2000; Ravosa, 1996, 1999; Ravosa and S imons , 1994) . In addi t ion, the relat ionship between symphysea l stress and strain has been est imated in a number of pr imate spec ies (V inyard and Ravosa, 1998) , and in pigs and humans (Chapter II, p83) . In genera l , the unfused symphys i s , by al lowing independent inversion and evers ion of the two halves of the mandib le before and dur ing the mast icatory power stroke, enables the steep occ luding surfaces of oppos ing teeth in some mamma l s to match dur ing mast icat ion (Hylander, 1979b; Kal len and Gans, 1972; L ieberman and C rompton , 2000; Oron and Crompton , 1985; Scapino, 1981) . The fused symphys i s s t rengthens and stiffens the jaw, reducing its risk of structural fai lure as a result of lateral t ransverse bending, and dorsoventra l shear st resses occurr ing dur ing uni lateral mast icat ion (Hylander, 1984; Hy lander er a/., 2000; Ravosa, 1996; Ravosa and Hylander, 1993; Ravosa and S imons , 1994) . In mamma l s with fused symphyses , high stresses and stra ins can occur dur ing funct ion. Dorsoventra l shear is created by the upward component of the balancing-side jaw-musc le force, and the downward component of the bite point reaction force dur ing uni lateral biting (Hylander, 1979a, 1984, 1985) . Wishboning occurs at the very end of the power stroke, i.e. after the initial occurrence of m a x i m u m intercuspat ion, and is assoc iated with the late peak act iv ity of the balancing-s ide deep masseter coupled with the rapid decl ine in the act iv i ty of the ba lanc ing- side media l pterygoid and superf ic ial masseter (Hy lander and Johnson, 1994) . Whi le the lateral component of balanc ing-s ide deep masseter is cons idered the pr imary mast icatory musc le force assoc iated with symphysea l w ishbon ing, the opposi te ly-directed lateral component of the work ing-s ide bite point reaction force and residual act iv i ty f rom the work ing-s ide superf ic ial masseter may also contr ibute (Hylander, 1984, 1985; Hy lander and Johnson, 1994) . Previously (Chapter II, p83) , we postulated that in addit ion to these contr ibut ions, the super ior and medial parts of the work ing-s ide condylar fossa may play a role, especia l ly as the work ing condyle is loaded at the end of the power st roke by residual e levator musc le act iv i ty (see Figure 4.4, p i l l ) . Wishbon ing produces compress ive stress and strain on the facial aspect and high tensi le stress and strain on the l ingual aspect of the symphys i s (Hylander, 1984, 1985) . A third loading pattern associated with the power st roke is media l t ransverse bending, which occurs dur ing the opening phase and has been postulated to be caused main ly by the bi lateral contract ion of the lateral pterygoid musc les, producing a reversed wishboning effect (Hylander, 1984, 1985) . This form of loading causes compress ion on the l ingual aspect and tens ion on the facial aspect of the symphys i s . One l imitat ion of previous morphologica l ana lyses and stress and strain est imat ions is their considerat ion in stat ic s i tuat ions only. For example , V inyard and Ravosa's (1998) approaches, and our own in Chapter II (p83) est imated max imum stress and stra in occurr ing along the l ingual surface of the symphys i s as a consequence of tens ion inferred f rom the morpho logy of deep masseter only. Even previous in vivo studies, though measur ing dynamic strains, have l imitat ions. One cannot sample surface stra ins at more than a few sites in the pr imate mandib le without compromis ing the structural and funct ional integrity of the mast icatory sys tem (Daegl ing and Hylander, 2000) . Also, the interpretat ion of such strain data is not ent irely unambiguous . Recent ly, dynamic jaw models have been uti l ized to study jaw musculoske leta l mechanics in humans (Hannam er a/., 1997; Koolstra and van Ei jden, 1995, 1996, 1997a, b; Langenbach and Hannâm, 1999; Peck e r a / . , 2000) , and in pigs (Langenbach e r a / . , 1999) . These dynamic jaw models incorporate large amounts of structural and funct ional data including musc le act ive and passive tens ions, jo int react ion forces, occlusal forces and the jaw's mass propert ies, many of which change dynamica l ly dur ing funct ion. A major advantage of the models is their abi l ity to predict changing muscle tens ions in real t ime, parameters which are present ly imposs ib le to record in vivo. Dynamic models thus have the unique potential to reveal internal forces and torques induced by mult ip le, changing musc le tens ions. Previously, we descr ibed a method, based on computed tomography (CT) for obtain ing mass propert ies of the pig and human mandib les (Zhang er a/., 2001a , b). Speci f icat ion of mass , mass center and other inertial propert ies is an essential step in dynamic mode l ing . In the current study, we used this method to est imate the mass propert ies for a pig mandib le artif ic ial ly div ided into right and left ha lves, each with dist inct mass propert ies, and constructed a dynamic model in which the two halves were jo ined with a rigid link at the symphys i s . This made it possible to measure the tr i-axial dynamic forces, and torques passing through the link as a result of musc le act iv ity dur ing chewing; thus, we were able to est imate the resultant forces and torques l ikely to be t ransmit ted across the symphys i s dur ing funct ion, i.e. the forces and torques which presumably require a specif ic symphysea l morpho logy to withstand them. 7.3 MATERIALS AND METHODS 7.3.1 Model generation The original model of the pig jaw has been reported previously (Langenbach er al., 1999) and was based on a dynamic model of the human jaw (Langenbach and Hannam, 1999) . It was des igned with a commerc ia l software package (ADAMS 10.0, Automat i c Dynamic Analys is of Mechanical Sys tems; Mechanical Dynamics Inc., Ann Arbor , MI). Briefly, the model incorporated the muscu lar and skeletal morpho logy f rom an anaesthet ized female miniature pig (Sus scrofa, 8 months) . CT with a cal ibrat ion phantom was performed to obtain its skeleta l structure and mass propert ies (see Zhang e r a / . , 2001a , see Chapter III, p l l 4 ) . Muscle cross-sect ional s izes and lines of act ions were obta ined through magnet ic resonance imaging on a separate occas ion. The cr iter ia for des ignat ing musc le a t tachment sites were based on anatomica l descr ipt ions by Herr ing and Scapino (1973) , the reconstructed musc le images, and bone surfaces with known musc le at tachments . Three parts were ass igned to the tempora l i s musc le (anter ior tempora l i s ; middle tempora l i s ; and poster ior tempora l i s ) , two parts to the masseter (superf ic ial masseter , and deep masseter ) , one part to the media l pterygo id, one part to the lateral pterygoid, and one part to the digastr ic musc le . Each muscle included f iber and tendon components . The f iber/tendon length ratios were def ined according to Herring and Scapino (1973) and Anapol and Herr ing (1989) . The model was dr iven with musc le act iv i ty patterns based on e lect romyograph ic data. Muscle funct ion was s imulated as descr ibed previously by Langenbach and Hannam (1999) . Brief ly, mot ion of the mandib le was produced by act ive musc le tens ions generated by 'contract ing ' musc le f ibers. Each act ive musc le tens ion was determined by the product of the muscle 's cross-sect ional a rea , a constant of 40 N/cm 2 (Weijs and Hil len, 1985) and a specif ied level of act ivat ion (0 -1 , where unity represents 1 0 0 % act ivat ion). This va lue, expressed in Newtons (N), was scaled according to the muscle 's instantaneous length and shortening velocity by means of length-tension and ve loc i ty-tens ion curves (see Langenbach and Hannam, 1999) . Any pass ive musc le tens ion induced by damping or stretch was then added to this act ive tens ion. Passive stretch tens ions were only present for lengths beyond the opt imal musc le length, taken as the musc le length at an interincisal d istance of 30 m m (for a detai led descr ipt ion of the damping forces and the musc le or tendon tens ions, see Langenbach and Hannam, 1999) . Segmentat ion of the mandib le f rom CT images has been descr ibed e lsewhere (Zhang et al., 2001a , see Chapter III, p i 14). Divis ion of the mandib le at the symphys i s was accompl ished midsagi t ta l ly through the entire image set. Each half of the mandib le was saved as a separate file for mass propert ies calculat ion with a customized program (Cal image - Calculate Image, Craniofacial Laboratory, The Univers i ty of Brit ish Co lumb ia; see http://condor.dent istry.ubc.ca or Append ix , p226) . The model thus consisted of two independent masses represent ing the split lower jaw, which were rel inked with a rigid pin-joint placed at the center of the symphys i s and or iented orthogonal ly to the dental occlusal plane (Figure 7.1, p l 7 8 ) . The jaw's mot ions relat ive to the grounded cran ium were shaped by var ious forces at dif ferent sites on the jaw, including gravi ty, reaction forces at the temporomand ibu la r jo ints , dental occlusal react ion forces, food bolus resistance force, and act ive and passive musc le tens ions. Condy lar guidance was s imulated with a horizontal p lane. Under loading, the condylar center could indent this plane (the react ion force increased exponent ia l ly to reach 1000 N at 0.25 m m compress ion) , but rotat ions and trans lat ions on the plane were fr ict ionless. All musc le actuators l inked the mandib le to the c ran ium, except the digastr ic, which ran between the mandib le and a grounded part equiva lent to the hyoid bone. The locat ions of three mandibu lar bite points (buccal cusp tip locations of the bi lateral dec iduous fourth premolar, DP4, and mid- inc isor) in the dental arch were obta ined f rom the 3D CT reconstruct ion of the mandib le. Reaction forces at these bite points were assumed to be perpendicu lar to the occlusal plane, and were generated when the j aw reached the dental intercuspal posit ion, where the interocclusal contact force at each bite point increased exponent ia l ly to reach 2000 N with 0.25 m m inter- occlusal compress ion . The model was des igned to accommodate a food bolus on the work ing-s ide at the DP4 bite point. The bolus had a compress ive resistance which depended on its th ickness (equivalent to the distance separat ing the dental arches at that locat ion). It was three m m thick, and "sof t-edged", so that its res istance increased step-wise over the first 1.5 m m of compress ion, to reach a max imum of 60 N. Forces less than 60 N (or of insuff icient durat ion) resulted in incomplete bolus compress ion . Figure 7.1 Frontal (A) and lateral (B) views of the ADAMS wireframe dynamic model. Muscle action lines are described elsewhere (Langenbach et al, 1999). The rigid joint linking the two halves of the mandible is indicated by the lock icon located at the center of the symphysis. 7.3.2 Simulations The plausibi l i ty of a pig dynamic model has been demonst rated previously (Langenbach e r a / . , 1999), i.e. the predicted j aw mot ion is an acceptable analogue of publ ished average data for m a x i m u m opening, latero-deviat ion, and t iming of the different parts of the pig chewing cycle (i.e. for open ing, closing and power s t roke) . In the present study, we ran the s imulat ion for one r ight-sided chewing cycle over 0.5 seconds. Output predict ions included incisai point movement in vert ica l , hor izonta l , and anteroposter ior d imens ions, tens ions of the 16 jaw musc les , reaction forces at the work ing-s ide bite points, react ion forces at the work ing-s ide temporomand ibu la r jo int and tr i-axial forces and torques pass ing through the symphys i s . All were t ime-re lated dynamic measurements . The convent ions used to express symphysea l forces and torques are i l lustrated in Figure 7.2 ( p l 8 0 ) . 7.4 RESULTS 7.4.1 Incisor point motion Figure 7.3 ( p l 8 1 ) shows the shape of the s imulated chewing cycle. The predicted jaw mot ion was reminiscent of that in previous ly publ ished character ist ics of pig chewing (cf. Herr ing, 1976) . V iewed frontal ly, j aw opening began in the mid l ine. Af ter the first 10 m m of gape, the j aw deviated f rom the midl ine towards the work ing-s ide. Max imum opening (33 mm) was fol lowed by fast c losure of the jaw combined with a further lateral deviat ion (7 mm) of the j aw. When the artif icial food bolus was reached, the jaw moved back to the midl ine, and closed s lowly. Vert ical j aw mot ion stopped when the teeth came into Figure 7.2 Conventions used to express symphyseal forces and torques. Forces exerted by the right corpus on the left are positive when they are in the same direction as the axes. Arrows around each axis indicate the directions of positive torques exerted by the right corpus on the left. -10mm 0 Rright i Figure 7.3 Incisor point motion during a simulated right-sided, 0.5 second chewing cycle. A: X-Y plot representing the frontal view of incisor point motion. B: Z-Y plot representing the lateral view of incisor point motion. Al l scales are in mm. Arrows indicate direction of motion, including the next stroke to the contralateral side. See text for full description. contact, result ing in a horizontal sl ide through the mid l ine towards the balancing-s ide. The r ight-sided cycle took 0.33 seconds before the next cycle began, which started on the balancing side of the midl ine (Figure 7.3, p l 8 1 ) . The opening trajectory of this cycle was ent irely on the balancing-s ide of the midl ine. 7.4.2 Muscle tension Figure 7.4 ( p l 8 3 ) shows the musc le tens ions expressed in t ime for the s imulated chewing stroke. The lateral pterygoid and digastr ic musc les on both s ides init iated the cycle. The lateral pterygoid musc le on the work ing-s ide reached its peak tension ear l ier (about 0.02 seconds) than its balancing-s ide counterpart . The lateral pterygoid musc les reached max imum tension ear l ier than digastr ics on both s ides. This was the point where fast opening began (the first vert ical l ine). When the j aw reached max imum gape (33 m m ) , the tension of lateral pterygoid musc les reduced to zero. The digastr ic musc le tens ions d isappeared when the closing phase began. When jaw opening reached two. thirds of its m a x i m u m gape, passive tens ions were produced in the superf ic ial and deep masseters . These passive tens ions turned into act ive tens ions at the beginning of the closing phase (the second vert ical l ine). As soon as the teeth hit the artif icial bolus (the third vert ical l ine), all adductor musc les , as well as the lateral pterygoid, were act ive. On the work ing-s ide, whi le the deep masseter reached max imum tension immediate ly after the teeth began to crush the bolus, other adductors and lateral pterygoid musc le reached their max imum tens ions later, i.e. when the bolus was a lmost complete ly crushed, and the lower teeth contacted the upper teeth (the fourth vert ical l ine). On the balancing-side, the middle and poster ior tempora l i s Right 40N AT MT PT SM DM MP LP DG 0.3 0.4 0.5 40mm Time (sec) 1 Left i i i i / i i i u— A" \ V 0.1 0.2! 0.3 0.4 C Time (s< k) Figure 7.4 Muscle tensions expressed in time during simulated right-sided chewing. Data are shown for the right and left sided muscles. The lowest curves are the corresponding incisor point motions in the vertical dimension. The dotted lines in each figure (from left to right) represent fast jaw opening, maximum jaw opening, onset of bolus crush, and the onset of tooth-to-tooth contact in intercuspation. Abbreviations: AT, anterior temporalis; MT, middle temporalis; PT, posterior temporalis; SM, superficial masseter; DM: deep masseter; MP, medial pterygoid; LP, lateral pterygoid; DG, digastric. muscles, superf ic ial masseter , and media l pterygoid musc le all reached their max imum tens ions ear l ier than the deep masseter and lateral pterygoid. The latter two musc les reached their m a x i m u m tens ions late, i.e. dur ing the intercuspal sl ide to the midl ine. 7.4.3 Forces at the artificial food bolus, tooth and joints Figure 7.5 ( p l 8 5 ) i l lustrates the forces expressed in t ime at the food bolus, work ing-s ide tooth point (DP4), work ing-s ide temporomand ibu la r jo int and balancing-side temporomand ibu la r jo int. React ion forces at both jo ints commenced with opening, and reached smal l peaks jus t before max imum gape. These forces increased steeply when the tooth hit the artif icial bolus. The balancing-side jo int force reached max imum short ly after, and the work ing-s ide jo int force reached m a x i m u m after initial tooth contact occurred. The reaction force at the food bolus began to increase when struck by the lower teeth, increased steeply in about 0.01 seconds, and after a s lower phase of 0.02 seconds reached its max imum; this peak force cont inued for about 0.03 seconds, then decreased sharply when intercuspat ion occurred. Tooth force was produced and reached its max imum immediate ly after tooth contact was made . All these forces d isappeared before the next cycle began. 7.4.4 Tri-axial symphyseal forces Figure 7.6 ( p l 8 6 ) demonst rates the tr i-axia l symphysea l forces expressed in t ime. During j aw opening, the artif icial jo int represent ing the symphys i s underwent compress ion along its t ransverse X-axis (i.e. the left s ide corpus was compressed by the right s ide corpus in the occlusal plane; cf. Figure 7.2, p l 8 0 ) . This compress ion coinc ided with peak tension in the lateral pterygoid and digastr ic musc les (cf. Figure 7.4, Figure 7.5 Predicted forces expressed in time. They include bolus, working-side bite point (DP4), working-side temporomandibular joint (WSJ) and balancing-side temporomandibular joint (BSJ) forces. The lowest curve is the corresponding incisor point motion in the vertical dimension. The dotted lines (from left to right) represent maximum jaw opening, onset of bolus crush, and the onset of tooth-to-tooth contact in intercuspation. 40N Force X Force Y Force Z 0.1 0.2 0.3 0.4 0.5 -40mm Time (sec) Figure 7.6 Tri-axial symphyseal forces expressed in time. The lowest curve is the corresponding incisor point motion in the vertical dimension. The dotted lines (from left to right) represent maximum jaw opening, onset of bolus crush, and the onset of tooth-to-tooth contact in intercuspation. p l 8 3 ) . When the jaw reached max imum open ing, symphysea l tens ion was induced (i.e. the left side corpus was tensed by the right side corpus; cf. Figure 7.2, p l 8 0 ) , and stayed a lmost constant unti l the lower teeth hit the artif icial bolus. This tension reached max imum dur ing the early stage of bolus crushing and then decreased. These effects were related to peak act ive tension in the closing musc les, most notably the superf ic ial and deep masseters (cf. Figure 7.4, p l 8 3 ) . A negat ive shear force began along the supero infer ior Y-axis when the food bolus was hit, and increased steeply. This shear was related to the reaction force appl ied on the working-s ide DP4, which tended to lower the work ing-s ide corpus, and jaw adductor forces on the balancing-s ide, which lifted the balancing-side corpus. The shear force cont inued until the bolus was complete ly crushed, and began to decrease when the lower teeth made contact with the upper teeth (cf. Figure 7.5, p l 8 5 ) . A smal l shear force along the anteroposter ior Z-axis was caused by different t iming in the tempora l is , media l pterygoid, and masseter musc les, and as a result of musc le forces on the work ing-s ide exceeding those on the balancing-side after tooth contact c ommenced . The main contr ibutor was the working-s ide superf ic ial masseter due to its anter ior ly directed force component (cf. Figure 7.4, p l 8 3 ) . 7.4.5 Tri-axial symphyseal torques Figure 7.7 ( p l 8 8 ) shows the symphysea l torques around the three axes (see Figure 7.2, p l 8 0 , for convent ions) . Two smal l posit ive torques (approx imate ly 250 N-mm) occurred around the t ransverse X-ax is . The first occurred at about 0.1 seconds. The balancing-s ide j aw tended to be twisted more than the work ing-s ide due to the balanc ing-s ide lateral TX — T Y — T Z N mm -40mm Time (sec) Figure 7.7 Tri-axial symphyseal torques expressed in time. The lowest curve is the corresponding incisor point motion in the vertical dimension. The dotted lines (from left to right) represent maximum jaw opening, onset of bolus crush, and the onset of tooth-to-tooth contact in intercuspation. TX, TY and TZ are torques about the X, Y, and Z axes, respectively. pterygoid tens ion exceeding that of the work ing-s ide lateral pterygoid, so the twist ing force exerted on the left s ide symphys i s by the right side was in a c lockwise direct ion (cf. Figure 7.2, p l 8 0 ) . The second posit ive torque commenced with the onset of bolus crush ing, reached peak after the bolus was complete ly crushed, and was due to the bolus react ion force coupl ing with the balancing-side jaw lifting forces (Figure 7.4, p l 8 3 ) . The torques around Y-axis were induced by twist ing related to the t ransverse bending of the mandibu lar corpora. The initial negat ive torque was assoc iated with the act iv ity of the two jaw openers when the medial t ransverse bending occurred. The magni tude of th is torque was about - 1500 N-mm. This torque changed direct ion when pass ive tension of superf ic ial and deep masseter (wishboning) began to increase and quickly reached about 750 N-mm the moment the j aw reached max imum opening. The torque stayed a lmost constant until the lower teeth began to crush the food bolus. It then rose steeply to a m a x i m u m of over 2500 N-mm dur ing the first one third of the bolus crushing phase, fol lowed by a sharp drop in magni tude. The peak torque coinc ided with the max imum tens ions of the superf ic ial and deep masseter musc les on both s ides, and the peak bolus react ion force. The torque reduced to about 200 N-mm after tooth contact occurred, and started to decl ine before the next cycle (cf. Figure 7.4, p l 8 3 ) . The torques around Z-axis were the results of asymmetr i ca l j aw opening and closing muscle forces, which tended to turn the mandib le about its anteroposter ior Z-axis. They coincided with the two t ransverse torques, and were the consequences of the same musc le contract ion patterns. 7.5 DISCUSSION 7.5.1 The model The dynamic model assumed the mandib le was two rigid structures l inked by a f ixed jo int placed central ly at the mand ibu lar symphys i s . Conceptual ly , it was equiva lent to two rigid beams ( independent of their cross-sect ional forms) l inked at a point through which all forces and torques were t ransmi t ted . Whi le the model was incapable of predict ing stress and strain within or between its components , it was able to reveal the env i ronment in which the symphys i s must work. The assumpt ion was that the design of the corpora provides a high degree of rigidity in the intact an ima l ; thus, any symphysea l link, whatever its f o rm, would have to be des igned to cope with the resultant total forces and torques demonst rated by the mode l . The model s impl i f ied some of the pig's muscu loske leta l propert ies. For example , the pig superf ic ial masseter is large, and shows differential act ivit ies (Herr ing er al., 1989), yet here we treated it as a single component because the detai led cross-sect ional data for its components were unknown. Also, using a single pin-point jo int to represent the symphys i s was another s impl i f icat ion. It would be more ideal perhaps to use two or more of such jo ints , in which case facial and l ingual side connect ions might separate compress ive and tensi le forces respect ively, more like a real symphys i s dur ing wishboning. This would make the model more complex. 7.5.2 Predicted shear The dynamic model suggested dorsoventra l shear is the main dynamic loading pattern in the pig symphys is . This shear force starts to increase when react ion force on the lower teeth couples with balancing-s ide jaw adductor forces (Figure 7.6, p l 8 6 ) . The predict ion accords with the hypothes is that like that in pr imates (Hylander, 1979a , 1984, 1985), the pig jaw symphys i s undergoes dorsoventra l shear dur ing uni lateral mast icat ion. The magni tude of this shear force was the largest among all forces passing through the symphys is . To resist this large shear ing force, a large symphysea l cross-sect ional area would be necessary, as suggested by the f indings in Chapter II (p83) . Our model also predicted a smal l amount of anteroposter ior shear occurr ing at the very end of the power stroke, i.e. when the lower jaw moves back to the midl ine. It tends to shear the work ing-s ide jaw forward and the balancing-side jaw backward (Figure 7.6, p l 8 6 ) . In a c inef luorographic analys is of jaw movements in ga lagos with unfused symphys i s , Beecher (1977) noted that work ing-s ide j aw frequent ly moves anter ior ly relat ive to the balancing-side jaw. He attr ibuted this anteroposter ior shear to the balancing-s ide poster ior tempora l i s pull ing the balancing-s ide mandib le backwards, whi le the work ing-s ide corpus is s imul taneous ly pul led forward by work ing-s ide mast icatory force. 7.5.3 Predicted transverse bending The dynamic model predicted media l t ransverse bending assoc iated with j aw opening. The initial bending was expressed as compress ion , and started with the contract ion of lateral pterygoid and digastr ic musc les on both s ides, and reached a max imum when the lateral pterygoid musc les reached their peak tens ions (Figure 7.4, p l 8 3 and Figure 7.6, p l 8 6 ) . The effect is s imi lar to one reported in pr imate studies (Hylander, 1984, 1985), coincident with the second largest torque (Figure 7.7, p l 8 8 ) . Lateral t ransverse bending (Hylander and Johnson, 1994) was also predicted by the current mode l . The predicted max imum torque (about the superoinfer ior axis) exceeded 2,500 N-mm, well beyond the largest torque dur ing media l t ransverse bending (Figure 7.7, p l 8 8 ) . In addit ion to the masseter contr ibut ion, the model assoc iated wishboning with bite and art icular forces. As the artif icial food bolus was crushed in a supero- media l direct ion by the lower tooth, the react ion force f rom the food bolus had a lateral ly directed component . S ince the pig has a relat ively flat art icular fossa with a mediolateral cant, wishboning here would also have been inf luenced by jo int reaction forces on both s ides (cf. Figure 7.5, p l 8 5 and see Figure 7.8, p l 9 3 ) . To counter this wishboning torque effect ively, not only the size, but also the shape and cortical bone distr ibut ion of the pig j aw symphys i s are important (Hylander, 1984). A large cross-sect ional momen t of inertia with respect to the axis perpendicular to the bending plane is required (Hylander, 1985; van Ei jden, 2000) . This was accompl i shed by a horizontal ly or iented symphys i s as seen in pigs. The results in this study offered further ev idence for the hypothes is that the pig j aw symphysea l or ientat ion is an adaptat ion to counter concentrated wishboning stresses dur ing funct ion (see Chapter II, p83) . 7.6 SUMMARY Dynamic models predict parameters which are diff icult to measure in vivo. The val id ity of the current sp l i t -symphys is dynamic model was supported by its plausible jaw mot ion when dr iven with musc le act ivat ion patterns based on exper imenta l data. Dynamic model ing offers a l ternat ive and powerful methods for s tudy ing mamma l i an jaw b iomechanics. This study provides addit ional informat ion regarding the forces and torques which the symphys i s is cal led upon to resist dur ing dynamic funct ion. The size, shape, and or ientat ion of the pig jaw symphys i s appear ideal to accompl ish this. Joint reaction forces on the condyles Tooth react ion force f rom bolus crush Direction of tooth movement Figure 7.8 Joint reaction forces and bolus reaction force when the artificial food bolus is being crushed. Vectors show approximate direction only. 8 G E N E R A L D I S C U S S I O N The exper iments reported in Chapters I through V support the hypotheses proposed in the s tatement of the prob lem. In human mandib les, the densest bone occurred at the mo lar sect ions, which had the least cortical area. Despite the di f ferences in areas, the cross-sect ional masses were homogeneous. It seems the human mandib le is uni form with respect to its abi l ity to resist shear ing stresses over the entire mand ibu lar corpus and symphys i s (see Chapter I, p47) . The fact that cortical bone density and th ickness var ied within each cross-sect ion can be expla ined by local loading condit ions, i.e. regional loading may be a determinant factor in human mand ibu lar cross-sect ional des ign. These regional var iat ions should be taken into account when model ing the effect of jaw cross-sect ions on j aw b iomechanics (see Chapter I, p47) . Despite their dist inct jaw lengths, musc le forces, and symphysea l curvatures, the pig and human jaw symphyses apparent ly undergo s imi lar stresses and stra ins. The results f rom Chapter II (p83) support the dynamic strain s imi lar i ty hypothes is observed in a large range of different vertebrates. The pig jaw symphysea l shape and or ientat ion seem to be an adaptat ion to resist concentrated wishboning stresses and stra ins; if the symphys i s is or iented vert ical ly, the es t imated stresses and strains are high enough to cause possible symphysea l structure fai lure (see Chapter II, p83) . The mass propert ies in the pig and human mandib les can be est imated by CT. S ince the mass and geometr ic centers coinc ided in the pig and human mandib les , less- invasive methods than CT reveal ing 3D jaw shape alone might be used to est imate mass propert ies. A lso, we found there were l inear relat ionships between jaw mass and mass descr iptor, and between moments of inertia and moments of inertia descr iptors. These descr iptors were obta ined by s imple jaw d imens iona l measurements . This indicates j aw mass and moments of inertia can be est imated with s imple non- invas ive, direct measurements . S ince the jaw mass center was consistent ly located at the last molar region in both pig and human mandib les i rrespect ive of age, mass est imat ion is a s imple step in dynamic j aw model ing. A l though moments of inert ia are not very sensi t ive determinants of mot ion in dynamic mode l ing, the mass and mass center locat ions are signif icant (see Chapters III, p i 14 and IV, p l 4 4 ) . Dynamic models offer a powerful approach to s tudy j aw internal forces and torques related to loading condit ions. The three main symphysea l loading patterns were all successful ly predicted by our dynamic pig model . A lso, the model predicted condit ions that are diff icult or imposs ib le to measure by in vivo exper iments (see Chapter V, p l 7 1 ) . 8.1 JAW CROSS-SECTIONAL MECHANICS The beam theory of the jaw b iomechanics requires def init ion of cross- sect ions. The entire mamma l i an j aw ( including the dentary, i.e. mand ibu lar corpora and symphys i s , the condyles, and the rami , Iwasaki e r a / . , 1997; Nickel and McLachlan, 1994) undergoes var ious loads during funct ion, and these loads demand a b iomechanica l ly opt imized jaw sys tem. Unfortunately, our understanding of the loads is l imited, and the examinat ion of j aw cross-sect ions cannot just i fy any opt imizat ion theory without their speci f icat ion. Instead, such an analys is s imply offers clues to understand ing, according to current knowledge and general assumpt ions . The study in Chapter I (p47) provided such hints. 8.1.1 Ideal cross-sectional models Al though apparent ly ovoidal , j aw cross-sect ions cannot be modeled as s imple el l iptical shapes. The results reported do not support e i ther a solid el l iptical model or a hollow ell iptical mode l . However, a thin-wal led tubular structure with different wal l - th icknesses would seem an appropr iate model under tors ion. In such a mode l , the stress distr ibut ion does not fol low the normal torsion formula (Equat ion 1.8, p l 3 ) in which the polar moment of inertia plays the important role. The shape and regional dif ference in cortical th ickness and density seem to be assoc iated with local loading, especial ly bend ing. For example , the results showed the corpus was larger in its vert ical d imens ion (the bending index was less than unity), a des ign suited to counter high sagittal bending. This bending incurs high compress ive stress along the lower border of the corpus, and the results suggest this was the area with the densest and th ickest cortical bone. The strong basal cortex seems also well des igned for tooth loads. In the symphys i s sect ion, because wishboning causes the most stress on the l ingual s ide cortex, this region has a high cortical rigidity index. In the canine sect ion, the combinat ion of bending and wishboning requires more cortical bone at the basal and l ingual aspects. An interest ing f inding was provided by the edentu lous mandib le, where the bone distr ibut ion changed so that cort ical bone was more evenly distr ibuted around the entire sect ion in the molar region (perhaps due to the reduct ion in heavy mast icatory force), whi le in the canine and symphys i s regions, the l ingual cortex cont inued to be thick (perhaps because lateral t ransverse bending stil l occurred there) . To precisely model the jaw cross-sect ions, shape and regional cortical th ickness both need to be taken into cons iderat ion. 8.1.2 Open vs. closed jaw models The mandibu lar corpus sect ion has been proposed to be " open " in that the teeth are separate structures f rom the bony sect ion. However, the open model theory has been proved untenable theoret ica l ly and exper imenta l ly (Daegl ing er al., 1992) . Under tors ional load, an open sect ion only possesses a smal l fract ion of the rigidity of a c losed sect ion, and the teeth, per iodontal l igaments and alveol i apparent ly have a stiffening effect on the sect ion. The present work does not provide direct support for this hypothes is. However, analys is of the edentu lous mandib le indirectly suggests the mandibu lar corpus acts more as a c losed than a open sect ion. The problem of uti l iz ing s imple open or c losed models is related to the s impl ic i ty of these models, because nei ther takes into account factors such as regional bone densi ty, cort ical th ickness, the health of the per iodont ium and trabeculat ion (Daegl ing er al., 1992; Daegl ing and Gr ine, 1991) . Therefore, the solut ion to the problem requires the deve lopment of more sophist icated model ing cr i ter ia. Finite e lement analys is is l ikely to provide the most product ive future approach in this context. 8.1.3 Efficiency - how does bone structure meet mechanical and functional needs? Based on the f indings that denser bone occurs in region with less area (see Table 3.2, p63), it was suggested there was a compromise between b iomechanica l opt ions and funct ional needs in the mo lar site, for here, a large space is required to accommodate molar roots, and the inferior alveolar nerve and vessel complex. Also b iomechanica l ly , a min imal amount of bony mater ia l is necessary to resist shear ing forces exist ing along the whole mandibu lar corpus. Though the prob lem could be solved by increasing the occupied space, i.e. increasing the cross-sect ional area by extending its external d imens ion, this has other consequences, e.g. the increase in the lower jaw size may cause dispar i ty between the lower and upper jaws . It has been suggested that the mandib le is des igned to use bone economica l ly (Hylander, 1979b) . It is postu lated, for the same reason, that the jaw uses space economica l ly . 8.1.4 Bone mechanical properties, density and CT numbers The densi ty revealed by CT numbers is close to the apparent density, for each voxel in the CT image matr ix represents a smal l vo lume of bone. The CT number is taken f rom the mean attenuat ion of this bulk bone, which includes Havers ian canals, marrow spaces, and other voids smal l enough to be contained within the vo lume. Therefore, it includes both porosity, and the degree of minera l izat ion. Apparent ly , the CT number is main ly determined by bone minera l izat ion. The relat ionship between bone minera l izat ion and mechanica l propert ies however is not highly consistent. A l though it is general ly agreed there is a posit ive corre lat ion between t hem, large var iat ions have been reported, especia l ly for compact bone. Vose and Kubala (1959) suggested bone mechanica l propert ies are funct ions of its mineral content, and have been supported by Ascenz i and Bonucci (1968) and Currey (1984a) . S imi lar ly , Car ter and Hayes (1976) found that the compress ive strength and sti f fness of bone are power funct ions of its apparent density. In a more recent study, S tens t rom et al. (2000) found a signif icant correlat ion between bone minera l densi ty (BMD) and all mechanica l parameters and therefore suggested BMD is a cons istent predictor of bone strength for cortical bone but not for cancel lous bone, where trabecular th ickness is of more va lue. Other studies (Lang er al., 1997; Martin and Ishida, 1989) however, have indicated bone mineral izat ion and BMD alone are general ly poor predictors of cort ical bone strength, a l though they are better for t rabecular bone. The best est imates of strength have been obtained with CT, which is capable of account ing for 9 0 % of the st rength var iabi l i ty in a s imple in vitro test (see Mart in, 1991) . Phantom studies support a l inear relat ionship between phantom densit ies and CT numbers (Cheng et al., 1995; Lampmann et al., 1984; Zhang er al., 2001a , see Chapter III, p l l 4 ) . A poor relat ionship has also been reported for CT numbers and cortical bone (Rho et al., 1995) . There are two possible explanat ions for this anomaly . One is that the phantoms used in these studies are usual ly potass ium d ihydrogen phosphate solut ions. Even for the highest concentrat ion used in these studies e.g. 0.50 g / c m 3 (Zhang er al., 2001a , see Chapter III, p l l 4 ) , the densi ty of solut ion is still not comparab le to that of compact bone, i.e. l inearity has been assumed to extend to include the densi ty range of compact bone. Another explanat ion may be that because compact bone is so dense, some CT mach ines or f i lter a lgor i thms s imply treat it as the max imum CT number of uni form density (a good analogy is made when a bone fi lter is appl ied to ename l , and enamel is seen to be apparent ly of uni form dens i ty) . This may expla in why the same study found good relat ionships between CT numbers and cancel lous bone densit ies (Rho er al., 1995) . Hence, the CT numbers may be a cortical bone densi ty index only when the appropr iate f i lter is appl ied. The raw CT number usual ly uses part of the s igned 16-bit integer which covers f rom -32768 to 32767 . This range should be large enough to represent the entire range f rom the lowest densi ty (i.e. air) to the highest density (i.e. enamel) in l iving mater ia l . When converted to 8-bit integers (usual ly f rom 0 to 255) by an inappropr iate fi lter, the l inearity alters, especial ly in the high density bone range. The program, we used in our studies (3Dv iewnix , Univers i ty of Pennsylvania Medical Center, Phi ladelphia, PA), whi le powerful in most aspects, does not accept negat ive numbers in the CT database. We previously used a command- l ine program to perform the convers ion and found it was very difficult to set the low and high bounds accurate ly. The resultant CT image could indicate either not enough discrete numbers (when the select ion range was too large) or the high dens i ty compact bone t issue became too uni form (when the select ion range was too smal l ) . For this reason, we wrote a dedicated program (RIC - Raw Image Converter , Craniofacia l Laboratory, The Univers i ty of Brit ish Co lumbia , avai lable onl ine at http://condor.dentistrv.ubc.ca') to per form the fi lter v isual ly (for detai ls of this program see RIC - Raw Image Converter , p222, in the Append ix) . The cortical bone rigidity index (CRI) we introduced ear l ier (see Chapter I, p47) may not be l inearly related to cort ical r igidity. This index, a combinat ion of the cortical bone density and th ickness, will arguably account for more of the cortical var iabi l i ty than any single var iab le. Lang er al. (1997) found whi le BMD alone accounts for only 4 8 - 7 7 % of the var iabi l i ty in cort ical bone strength, a combinat ion of BMD and geometry var iables can expla in up to 9 3 % of the var iance. The var iances among the three aspects we measured met our expectat ions. However, they did not provide insight into the mechanica l contr ibut ion of the tooth, per iodontal l igament, or a lveolar process. 8.1.5 Bone mechanical properties, bone composition and organization The mechanica l propert ies of bone depend not only on the bone's macroscopic structure, i.e. its shape and size, but also on the mechanica l propert ies of the mater ia l with in. The latter is a s sumed to depend on the composi t ion (porosity and mineral izat ion) and organizat ion (trabecular or cortical bone archi tecture, and col lagen f iber or ientat ion) of the bone. The methods previously used (Daegl ing, 1989; Daegl ing er al., 1992; Daegl ing and Gr ine, 1991; Daegl ing and Hylander, 1998) only provide information regarding macroscopic structure. The present work included some composi t ional e lements, i.e. mineral dens i ty and porosity. A good example of the signif icance of bone mater ia l propert ies is the observat ion that the bone's elast ic modul i differ direct ional ly (Dechow et al., 1993; Dechow and Hylander, 2000; van Ei jden, 2000) . Unfortunate ly, CT cannot reveal such informat ion. 8.2 JAW MASS PROPERTIES 8.2.1 Significance of bone density, jaw dimensions, and jaw mass properties It has been suggested that true distr ibut ion of bone mass should be taken into account when determin ing the moment s of inertia in the human tibia (Cheng er al., 1995) and in the human head (Smi th er al., 1995) . For this purpose, CT has been proposed (Smi th et al., 1995) . The present f indings (Chapters III, p i 14 and IV, p l 4 4 ) indicate this is not necessary for e i ther the pig or human mandib le, s ince the centers def ined by bone mass and vo lume are close, and this di f ference (around one mm) does not make a signif icant difference in dynamic model ing (see Chapter IV, p l 4 4 ) . Therefore, regional bone density does not play an important role in determinat ion of j aw mass propert ies in pigs and humans , and possibly in other mamma l s , too. In contrast, jaw' d imens ions are important determinants of mass propert ies. L inear re lat ionships between j aw mass and a mass descr iptor and between moment s of inertia and moments of inertia descr iptors were demonst ra ted . These descr iptors are determined by j aw d imens ions. For example , our smal lest pig mandib le had a mass (dry weight) of 5.28 g and a mean bone densi ty of 1.41 g / cm 3 , whi le our largest pig jaw had a mass of 180.91 g and a mean bone density of 1.74 g / c m 3 (see Table 5.1, p l 2 6 ) . Clear ly, the mass dif ference is not main ly attr ibuted to the densi ty but to the size of the jaw. The moment of inertia is a funct ion of mass and two orthogonal d imens ions. It is therefore also determined main ly by j aw d imens ions. The relat ive magn i tude of moments of inertia for the j aw can be est imated qual i tat ively by its d imens ions. For examp le , s ince the pig jaw is longest anteroposter ior ly, the moment of inert ia with respect to this axis should be the smal lest . Depending on the s izes in the other two d imens ions, the moment of inertia is largest with respect to the smal lest d imens ion. 8.2.2 Significance of mass properties in dynamic modeling The signif icance of the mass propert ies in dynamic model ing were tested with a previously developed human jaw mode l . Jaw mass and anteroposter ior mass center locations were important var iables in s imulated j aw opening, whi le the moments of inert ia a l lowed larger var iat ions in express ion, complement ing s imi lar f indings reported dur ing jaw closing (Koolstra and van Ei jden, 1995) . Grav i tat iona l acce lerat ion, v iscous damp ing , and the generat ion of musc le tens ions seem suff ic ient to ensure these relat ively low moments do not play a major role in shaping free jaw movements . 8.2.3 Significance of the imaging method in dynamic modeling The imaging method descr ibed here not only al lows mass est imat ion for normal human mandib les; it is also appl icable in cases such as facial a symmet ry , and partial ly resected mandib les, where j aw mass centers may not coincide with vo lumetr ic centers. The segmentat ion technique al lows mass property est imat ion for v irtual ly any skeleta l part. The pig model (Chapter V, p l 7 1 ) is an example of this appl icat ion. The method is also appl icable to any CT- imaged physical mater ia ls with known densit ies, e.g. it has been used with good precis ion to est imate mass propert ies in an artif icial denture mode l . 8.3 SYMPHYSEAL BIOMECHANICS 8.3.1 Stress and strain similarity Fused ossif ied symphyses have to wi thstand bending, tors ional and shear ing stresses during var ious mast icatory tasks (Hylander, 1984) . Wishboning seems to be the most important load. Under w ishbon ing, the l ingual side of the symphys i s can undergo tensi le stress 2.5 t imes higher than the compress ive stress on the facial s ide. Because bone is weaker in tension than compress ion, it seems reasonable that more bone is needed on this s ide. In Chapter I (p47) , regional di f ferences among the l ingual, basal and facial cort ices were measured . The results c learly showed that the l ingual cortex was stiffer than its facial counterpart in the human mandibu lar symphys i s . The or ientat ion of the pig j aw symphys i s should be taken into considerat ion when evaluat ing the bone distr ibut ion. In Chapter II (p83) , it was found more bone was depos i ted on the infero- l ingual aspect of the pig symphys i s when it was or iented funct ional ly (i.e. normal) . This was also attr ibuted to the need for res istance to wishboning, because wishboning would tend to bend the pig mandib le in its funct ional occlusal plane, rather than in a plane perpendicu lar to the long axis of the symphysea l sect ion. The stress and strain s imi lar i ty between the pig and human symphyses may not be surpr is ing because both the pig and human jaws use the same mater ia l . Despite var iat ions in pig and human bone (Fung, 1981), their compos i t ion is s imi lar, and their propert ies lie within the same range. It then fol lows that the funct ional stress and stra in may be s imi lar in mamma l s with fused ossif ied symphyses . 8.3.2 Symphyseal orientation The dist inct dif ference between the pig and human symphyses is their or ientat ion. Whi le the human jaw symphys i s or ients a lmost vert ical ly relat ive to the funct ional occlusal plane, the pig j aw symphys i s is angled more hor izontal ly. There is a tendency that as pigs grow, this or ientat ion becomes more horizontal (see Chapter II, p83) . This pattern is very s imi lar to that found in pr imates (Hylander, 1985; V inyard and Ravosa, 1998) . The present study supports the hypothes is that th is or ientat ion of the pig jaw symphys i s funct ions to mainta in the stress and strain caused by wishboning within an acceptable range. The results suggest an upr ight ly-or iented pig symphys i s cannot wi thstand the high stress and strain caused by deep masseter tens ion. The reason that changing the symphysea l or ientat ion alone increases its abi l ity to resist lateral bending is due to the increase of the cross-sect ional moment s of inertia with respect to the axis perpendicular to the curvature of bend ing. The results in Chapter V ( p l 7 1 ) also conf i rmed this hypothes is in a dynamic pig chewing mode l , in that the torque related to wishboning was the largest among the three predicted tr i-axial torques. It may not be necessary for the human jaw symphys i s to orient horizontal ly like the pig jaw because the human jaw has relat ively smal l adductor musc les, short lever a rms and low degrees of symphysea l curvature. The vert ica l ly-deeper human jaw symphys i s seems related to sagittal bending, which can occur dur ing incisor bit ing. Incis ion is very uncommon in pigs. However, they use their long snouts to root forceful ly. This requires a hor izontal ly-or iented symphys i s . There seems to be a compromise between incisai funct ional needs and symphysea l strength in the long-jawed pr imates, in which super ior tori and/or inferior s imian shelves are deve loped (Daegl ing, 1993; Hylander, 1984; Hylander, 1985). These observat ions, as well as those of others, support the idea that wishboning is a major determinant of symphysea l form and funct ion. 8.4 BONE GROWTH, MODELING, AND THE MECHANICAL ENVIRONMENT While basic skeleta l morphology is main ly determined genet ical ly, its final mass and architecture are modulated by adapt ive mechan i sms sensit ive to the mechanica l env i ronment . Relat ionships between the mechanica l env i ronment and the form of the ske leton have long been recognized (Forwood, 2001) . The results of the mechan ica l env i ronment can be expressed at organ, t issue, cel lular and molecu lar levels (Carter et al., 1998) . Organ level mechanica l s ignals can be character ized in te rms of loading history, which includes the vary ing effects of such quant i t ies as forces (i.e. musc le act ive and passive tens ions, pass ive tens ions f rom var ious l igaments, reaction forces f rom jo ints and dental occ lus ion, and any accidental force not belonging to the a forement ioned) , movements , and deformat ions. T issue level mechanics can be expressed in the mater ia l propert ies of bone t issue, such as elast ic and shear modul i . The other two levels deal with such th ings as cell pressure, cell shape changes, oxygen tens ions and cytoskeleton damage or d isrupt ions (for review, see Carter e r a / . , 1998) . Changes in skeletal form as a result of changes in musc le funct ion have long been observed (see Miller, 1991) . Hall and Herr ing (1990) demonst ra ted that paralys is of av ian embryos reduces skeleta l growth by reducing the loads imposed on the bones by musc le contract ion, changes that represent a l terat ions in the mechanica l env i ronment of the ske leton. Mechanical s ignals inf luence bone growth, model ing and remodel ing act iv it ies. Appl ied mechanica l loads can effect adaptat ions in both cortical and cancel lous bone (Forwood, 2001) . It is bel ieved that tensi le forces on the per iost ium are osteogenic, whereas compress ive loads lead to resorpt ion (Teng and Herr ing, 1998) . The idea that the facial ske leton is opt imized for counter ing or diss ipat ing mast icatory forces, invokes an "opt imal strain env i ronment" theory (Rubin er a/., 1994) . Accord ing to this theory, dur ing chewing and bit ing, there should be relat ively high and near uni form amounts of bone strain throughout the facial ske leton. Counter-ev idence however, has been col lected by in vivo strain studies (Hy lander er al., 1991; Hylander and Johnson, 1997b) in that not all facial bones are especia l ly des igned so as to min imize bone t issue and max imize st rength. The morphology of certa in facial bones does not necessar i ly have any importance or special re lat ionship to routine and habitual cycl ical mechanica l loads assoc iated with chewing or bit ing, or in other words, bone format ion here seems to be determined purely genet ical ly. 8.5 SIGNIFICANCE OF DYNAMIC MODELS FOR STUDYING JAW BIOMECHANICS Dynamic stress and strain patterns have been studied by in vivo approaches (Bouvier and Hylander, 1981a; Herr ing er al., 1996; Herr ing and Mucci, 1991; Herring and Teng, 2000; Hylander, 1977, 1979a, 1984; Hylander e r a / . , 1998; Hylander and Bays, 1979; Hy lander and Johnson, 1997a, 1997b; Liu and Herr ing, 2000; Rafferty and Herr ing, 1999; Teng and Herr ing, 1998) . It is difficult though, to relate loading condit ions inferred by this approach to muscle act iv it ies, bite and jo int forces in vivo. The dynamic pig model (Chapter V, p l 7 1 ) however, demonst ra ted this possibi l i ty. For example , dur ing the peak tension and torque in relation to wishboning, the model could correlate all musc le tens ions, bite and jo int forces, mak ing possible to associate the act iv it ies of work ing-s ide deep and superf ic ial masseters , bi lateral jo int react ion forces, balancing-side deep and superf ic ial masseters , work ing-s ide react ion bite force, and wishboning (see Figure 4.4, p i l l and Figure 7.8, p l 9 3 ) . With this dynamic mode l , it was possible to detect a smal l degree of anteroposter ior shear, which has only been possible previous ly to observe in mamma l s with unfused symphyses such as galagos (by. c inef luorographic analys is; Beecher, 1977) . Therefore, dynamic models can provide powerful tools for studying jaw dynamic mechan ics . 8.6 PERSONAL COMMENT ON COMPUTING I would not have been able to complete this thes is wi thout a computer . The work required calculat ion of cross-sect ional measurements and mass propert ies which would not have been so smooth if I had not been the programmer . In fact, the first step was difficult. I started to write this program with Microsoft Visual Basic 4.0 (Microsoft Corp. , Redmond, WA) because it was said to be the easiest language for non-profess ional p rogrammers . Later I became interested in C++ and was immediate ly subdued by its power. I rewrote this program in Microsoft V isual C++ 5.0 (Zhang er al., 2001a , see Chapter III, p i 14). The current vers ion of this program (Ca l image - Calculate Image, Craniofac ia l Laboratory, The Univers i ty of Brit ish Co lumb ia; avai lable f rom http://condor.dentistry.ubc.ca') was written in Bor land C++ Bui lder 5.0 (Impr ise Corp. , Scotts Val ley, CA) because this is a package with both power and s impl ic i ty. Original ly, I used a command- l ine program to convert the raw CT images f rom s igned 16-bit to unsigned 8-bit, which is suitable for work under 3DViewnix. It was very painful to select the appropr iate f i lter by trial and error. I decided to write such a program with a user interface, and carry out the convers ion under v isual inspect ion. This program was finally named RIC (Raw Image Converter , Craniofac ia l Laboratory, The Univers i ty of Brit ish Co lumbia , avai lable onl ine at http://condor.dent istrv.ubc.ca). The included CD-ROM (copyr ight © 2001 The Univers i ty of Brit ish Co lumbia) contains my whole thes is (in portable document format, PDF, and HTML format) . The CD-ROM compl ies with the Microsoft Windows ® AutoRun protocol . Both my programs, plus all source codes can be found there. I can customize these programs to fit special needs. 8.7 FUTURE DIRECTIONS The study of mamma l i an jaw b iomechanics inc ludes many aspects, and the present work touched only a few. Although my studies est imated the contr ibut ion of regional bone density and regional cortical th ickness to cross-sect ional mechan ics , these, and previous ones (Daegl ing, 1989; Daegl ing er al., 1992; Daegl ing and Gr ine, 1991; Daegl ing and Hylander, 1998) used smal l sample s izes. Thus the conclus ions may be premature. S imi la r studies with larger samples might conf i rm the conclusion that denser bone occurs at cross- sect ions where the area is smal ler , and that the human mand ibu lar cross- sect ional mass is homogeneous along the whole mand ibu lar corpus and symphys i s . My observat ion that a dense and thick basal cortex is related to normal tooth loading may seem interest ing, but it needs larger samples of both dentate and edentu lous mandib les for fur ther defense of this proposit ion. A l though my stress and strain est imat ion involved more individual data than previously reported (Vinyard and Ravosa, 1998) , I was unable to obtain individual musc le data for each of the spec imens. As noted in the d iscuss ion (see p l 0 7 ) , this may expla in part of the large var iat ion in the results. Understandably, repeat ing such a study in humans would not improve the results. It is certainly possible to perform a further study with l iving pigs using these imaging and est imat ion methods . S imi lar studies are also possible in other mamma l s such as monkeys , which are also good an imal models (Herr ing, 1995) . Future studies could focus on incorporat ing as much individual data as possible and hypotheses could be tested on more mamma l species, especial ly those for which there is in vivo bone strain data in the symphys i s , e.g., macaque mandib les . Also, an in vivo bone strain study would be useful to per form for the pig jaw symphys i s , and would be a nice val idat ion of my current stress and strain est imat ions. Val idated est imat ions in pigs and other mamma l s would make human est imat ions more l ikely to be true, a l though these may never be possible to conf i rm. Previously, dynamic models have been used to study musc le and jo int b iomechanics dur ing j aw opening and closing movemen t s (Koolstra and van Ei jden, 1995, 1997a, b; Peck er a/., 2000) , and dur ing chewing (Langenbach and Hannam, 1999) . Here, I per formed the f irst s tudy of the pig j aw symphys i s . A l though l imited, this model successfu l ly predicted the three main postulated symphysea l loading condit ions, i.e. dorsoventra l shear, media l t ransverse bending, and lateral t ransverse bending. A s imi lar model might be expected in the human jaw for which dynamic models a l ready exist, and where little is known about the loading condit ions in the symphys i s . It would also be appeal ing to build macaque dynamic j aw models to study their symphysea l loading condit ions, for which more in vivo data are avai lable. S imi la r mode ls could be constructed to study loading condit ions in other areas such as the molar and canine regions in pigs, humans and macaques . Current dynamic models assume r igid-body condit ions only, whi le current finite e lement (FE) models are l imited to stat ic s imulat ions. Dynamic FE models are appeal ing but l imited by the present comput ing power. As computers become faster, dynamic FE models will be the pr imary choice for such study of j aw b iomechanics. 9 G E N E R A L S U M M A R Y A N D C O N C L U S I O N S In the current studies, a number of methods were used to invest igate j aw b iomechanics. These approaches required the use of basic physical and engineer ing principles. A l though l imited, appl icat ion of these principles in mamma l i an jaw b iomechanics seems promis ing . The jaw cross-sect ional study supp lemented exist ing studies on jaw cross-sect ions by including regions represent ing the entire mand ibu lar corpus and symphys i s , and by assess ing contr ibut ion of bone distr ibut ion and density to jaw b iomechanics. The stress and strain est imat ions of the jaw symphys i s comp lemented previous in vivo bone stra in studies, and provided more ev idence in support of current hypotheses. The study also suggests a future for individual dynamic jaw models in l iving humans . The jaw's cross-sect ional shape, s ize and cort ical bone distr ibut ion reflect its b iomechanica l des ign. In the cross-sect ional study of the human mandib le, the cross-sect ional mass was uni form across the mandibu lar corpus and symphys i s . This suggests the human jaw funct ions as a uni form curved beam resist ing shear ing stress. The differential cort ical bone distr ibut ion among the molar, canine and symphys i s cross-sect ions suggests stress and stra in may be important factors regulat ing model ing and remodel ing processes in mand ibu lar bone. The stress and strain est imat ions in the pig and human symphyses support the conclus ions that stress and strain s imi lar i ty ex ists across mamma l i an orders, and that pig j aw symphysea l or ientat ion is an important des ign factor for mainta in ing funct ional equiva lence and dynamic stra in. The f inding that jaw mass propert ies can be est imated by direct measurements is especial ly useful if and when individual dynamic model ing in l iving humans becomes an issue. The successfu l appl icat ion of a dynamic model in predict ing jaw loading patterns suggests dynamic models may be powerful tools for studying j aw b iomechanics . 10 B I B L I O G R A P H Y Anapol F and Herring SW. 1989. Length-tension relationships of masseter and digastric muscles of miniature swine during ontogeny. J Exp Biol 143:1-16. Ascenzi A and Bonucci E . 1968. The compressive properties of single osteons. Anat Rec 161:377-391. Baron P and Debussy T. 1979. 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Abstract 11 A P P E N D I X 11 .1 R I C - RAW IMAGE CONVERTER 11.1.1 Purpose of this program RIC (short for Raw Image Converter) was des igned to convert grayscale 16-bit raw image files into 8-bit raw image fi les or b i tmap fi les. The process is carr ied out under direct v isual contro l . Common ly , the input data are computed tomograph ic (CT) or magnet ic resonance (MR) images in single or mult i-s l ice formats . RIC supports big endian and little endian input data formats, e i ther s igned or uns igned. Multiple s ingle-sl ice fi les, and single mult i-s l ice files can be taken into the program. Output fi les can be generated as either 8-bit raw image fi les, or b i tmap fi les. There are two options when saving raw and b i tmap fi les. One can save the current sl ice, or a selected sub-set of s l ices. An a l ternat ive opt ion al lows one to save the whole set of s tacked sl ices. These opt ions are provided by a menu and shortcut tool button sys tem. 11.1.2 Main interface Here is the main interface of this p rogram. For comple te use of the program, see its help manual on the accompan ied CD-ROM. C:\Chris-sinner\007\I.017 - RIC - Raw Image Converter File View Selection Tools Help y B P , t | X Open Raw Bitmap LPan Tools About j Exit ~Data format — C Little-endian unsignedC Big-endian unsigned C Little-endian signed (* Big-endian signed Intensity of interest (101)" IB ' 10711 Meanff): 68; SD: 19.3(1 to 18049) Selection (X): 4 to 921 (917) F Crop range Zoom ÏJJ3 ID 17 "Slice control- Image size: Slice index: J j 1 Selection: Prefix chris width B from 256 height 256 1 _d 17 45 d to 45 d Auto 8 start 001 Slice 101 low 1 101 high IF! C:\Chris-sinner\007\l.016 0 752 —• C:\Chris-sinner\007\l.018 4 921 ; C:\Chris-sinner\007\l.019 4 921 C:\Chris-sinner\007\l.020 4 921 C:\Chris-sinner\007\l.021  4 921 • jshow output image here No error Lowest Sel 0 Highest Sel 921 Max Range 921 Â 11.1.3 Core algorithm The core a lgor i thm of the program is the convers ion f rom 16-bit pixels to 8-bit pixels. This was done by a component cal led TRawImage, which will take a 16-bit raw image buffer and convert it into an 8-bit raw image buffer and display the 8-bit image as bi tmap on the screen. The fol lowing code is the C++ routine for this convers ion only. Full source codes for this program are over 500 thousand l ines, and are avai lable on the CD-ROM. Copyright © 2001 The University of British Columbia. All rights reserved. The University of British Columbia owns the intellectual property rights, including copyright of the source code. i n t len=Width*Height; //width and height of the image i f (!Data8) Data8=new Byte[len]; //create 8bit buffer ZeroMemory(Data8, l e n ) ; / / i n i t i a l i z e the buffer i n t range=UBound-LBound+l; //user selected lower and upper bounds i f (FDataFormat==dfLEU){ //unsigned l i t t l e endian format fo r (int i=0; i<len; ++){ unsigned short* u=(unsigned short*)Datal6+i; i f (*u<LBound) continue; i f (*u>UBound ScSc IFCropRange) Data8[i]= 255; else Data8 [i] = ((*u)-LBound)*255/range; //actual conversion } } else i f (FDataFormat==dfLES){ //signed l i t t l e endian format fo r (int i=0; i<len; ++){ short* s=(short*)Datal6+i; unsigned short u= (*s)+32768 ; i f (*s<LBound) continue; i f (*s>UBound && !FCropRange) Data8 [i]=255; else Data8 [i] = (*s-LBound)*255/range; //actual conversion } } else i f (FDataFormat==dfBEU){ //unsigned big-endian format for (int i=0; i<len; ++){ unsigned short* u=(unsigned short*)Datal6+i; unsigned short v=Convert(u); //Convert i s a routine to swap //the two bytes that represent the 16-bit integer. i f (v<LBound) continue; i f (v>UBound && IFCropRange) Data8[i]=255; else Data8[i]=((v)-LBound)*255/range; //actual conversion } } else i f (FDataFormat==dfBES){ //signed l i t t l e endian format f o r (int i=0; i<len; ++){ short* s=(short*)Datal6+i; short t=Convert(s); unsigned short u=t+32768; i f (t<LBound) continue; i f (t>UBound && IFCropRange) Data8[i]=255; else Data8[i]=(t-LBound)*255/range; //actual conversion } } 11.2 CALIMAGE - CALCULATE IMAGE 11.2.1 Purpose of this program This program was des igned to calculate mass propert ies form 3D CT images, and also to perform cross-sect ional measurements f rom single sl ice images. For mass propert ies calculat ion, the program takes a number of mul t i - sl ice 3D CT image fi les (they must be raw image fi les or 3DViewnix IMO image fi les) and separate landmark files (with the same names but ".Imk" as extens ion) as input fi les, and it calculates the mass , mass center, geometr ic center and moments of inertia with respect to the landmark def ined coordinate sys tem. If no landmark file counterparts are found in the image source directory, the program will ca lculate the mass , mass and geometr ic centers the same way, but it will ca lculate the moments of inertia with respect to the image matr ix coordinate sys t em. For cross-sect ional measurements , the program takes mult ip le s ingle- slice images (can be either raw image fi les or b i tmap image files or 3DViewnix IMO files each containing a single image sl ice) as input source and calculate their cross-sect ional areas, masses , centro ids of area, centroids of mass , second moments of inertia with respect to their image matr ix coordinate sys tems. In both cases, the user is al lowed to specify a cal ibrat ion curve by which Ca l image converts pixel va lues into real densi t ies. The output is c omma del imited text which can be easi ly imported into Microsoft Excel for further process ing. 11.2.2 Main interface Here is the main interface of this program. For complete use of the program, see its help manual on the accompanied CD-ROM. '"^Calimage File Edit Calculate Tools Help . ; jnjxj D G» • M A in 6 X New Open Save Close Mass Area P2V RIC Exit Input file" fir Open Open 2D image(s) to calculate Image matrix- Rows Columns Slices 512 [5Ï2~ Voxel— Width Height Depth 0.49 0.49 Equation — Density = |ï~ X Pixel Value Filter (inclusive) ~ Lower bound [ï~ Upper Bound 255 rMark Center?- C Yes y Doit X Dismiss Calimage (c) 1997-2001 The University of British Columbia • hm-m1rhm01-001.... • hm-m1rhm02-001.... I_J • hm-m1rhm03-001.... - • i hm-m1rhm04-001... M Current file: 12 jJJ Current slice 1 11.2.3 Core algorithms The core a lgor i thms are the calculat ion of mass propert ies and cross- sect ional measurements . Full source codes for this program are mil l ions of l ines, and are avai lable on the CD-ROM. Here, only the codes for actual calculat ion are g iven. The calculat ion is based on formulae presented in the Introduct ion ( p i ) and Chapter III ( p l l 4 ) . 11.2.3.1 C++ code for mass properties calculation Copyright © 2001 The University of British Columbia. All rights reserved. The University of British Columbia owns the intellectual property rights, including copyright of the source code. //Funciton: CalcMassProperties //Parameter: image f i l e name //Return value: none void f a s t c a l l TCalimage::CalcMassProperties(AnsiString fn) { TFileStream* fs;//define a f i l e stream try{ fs=new TFileStream(fn,fmOpenRead);//try to open the f i l e } catch (...){ GlobalWarning("Can not open "+fn); return; / / i f f a i l s , return } / / i f f i l e i s opened s u c c e s s f u l l y / / f i r s t c a l c u l a t e the header s i z e i n t header=fs->Size-ipData.rows*ipData.columns*ipData.slices; fs->Seek(header,soFromBeginning);//go to the f i r s t p i x e l p o s i t i o n i n t r, c, s; //loop integers double totalX=0, totalY=0, totalZ=0; //mass weighted t o t a l x, y, z double totalGX=0, totalGY=0, totalGZ=0; / / f o r geometric center i n t vn=0; //calculated number of voxels double totalMass=0; //calculated t o t a l mass double lxx=0, lyy=0, lzz=0; //moments of i n e r t i a double lxy=0, lxz=0, lyz=0; //product of i n e r t i a //create the read buffer of size r; unsigned char* rb=new unsigned char[ipData.rows]; //precalculate these variables to improve performance double x2, y2, z2;//square current x, y, z coordinates of the voxel double xp, yp, zp;//current x, y, z p o s i t i o n of the voxel center double w=ipData.width*ipData.height*ipData.depth/1000 ;//volume of //each voxel i n cm3 //loop and ca l c u l a t e f o r (s=0; s<ipData.slices; s++)//loop each s l i c e { zp=(s+0.5)*ipData.depth; z2=zp*zp;//square the current p o s i t i o n on Z-axis for (c=0; c<ipData.columns;c++) { yp=(c+0.5)*ipData.height ; y2=yp*yp; //sqare the current p o s i t i o n on Y-axis fs->Read(rb,ipData.rows); //read a row f o r (r=0;r<ipData.rows;r++) { i f (rb[r]==0) continue; //ignore p i x e l s of value 0 to //improve performance i f (rb[r]<ipData.lowerbound || rb[r]>ipData.upperbound) continue; //out of range, ignore vn++; //increase the c a l c u l a t e d number of voxels xp=(r+0.5)*ipData.width; x2=xp*xp; //square the current p o s i t i o n on X-axis double m= (rb [r] *ipData. slope+ipData. intercept) *w; //g totalMass+=m; //calculated mass //mass weighted t o t a l x, y, z totalGX+=r+0.5; totalGY+=c+0.5; totalGZ+=s+0.5; totalX+=(r+0.5)*m; totalY+=(c+0.5)*m ; totalZ+=(s+0.5)*m; //Moments of i n e r t i a with respect to image matrix o r i g i n Ixx+=m*(y2+z2); //g.mm2 Iyy+=m*(x2+z2); Izz+=m*(x2+y2); //product of i n e r t i a Ixy+=xp*yp*m; //g.mm2 Ixz+=xp*zp*m; Iyz+=yp*zp*m; } } } delete f s ; delete []rb;//delete buffer //Calculate center of mass double cx, cy, cz; cx=ipData.width*totalX/totalMass; cy=ipData.height*totalY/totalMass; cz=ipData.depth*totalZ/totalMass; double gx,gy,gz; gx=ipData.width*totalGX/vn; gy=ipData.height*totalGY/vn,- gz=ipData.depth*totalGZ/vn; double cdx, cdy, cdz, cd; cdx=cx-gx; cdy=cy-gy; cdz=cz-gz; cd=sqrt(cdx*cdx+cdy*cdy+cdz*cdz); / / t r a n s l a t i n g mass center to image matrix o r i g i n , //and moments of i n e r t i a are //converted to Mis with respect to the mass center //the o r i g i n a l unit of MI i s g.mm2, convert i t to g. Ixx=(Ixx-totalMass*(cy*cy+cz*cz))/100, Iyy=(Iyy-totalMass*(cx*cx+cz*cz))/100, Izz=(Izz-totalMass*(cx*cx+cy*cy))/100, cm2 double lx=0, ly=0, lz=0; / / i f no landmark data, ignore the following i f (mpData.p!=0) { Ixy=(Ixy-totalMass*(cx*cy))/100 Ixz=(Ixz-totalMass*(cx*cz))/100 Iyz=(Iyz-totalMass*(cy*cz))/100 //Reorient moments of i n e r t i a //Points i n landmark database are / / l . Id-infradentale //2. Left premolar //3. Left molar //4. r i g h t molar 1/5. r i g h t premolar 1/6. r i g h t condylar pole 111. l e f t condylar pole 1/8. mass center / / F i r s t , LM-LPM(P3-P2) x LM-RM(P3-P4), get the y-axis //Second, y-axis x MidCondylar-ID, obtain x-axis //Third, x-axis x y-axis to achieve z axis //Forth, rotate the moments of i n e r t i a Vector3D Ox, Oy, Oz; Oy=Normalize(CrossProd(mpData.p[2]-mpData.p[3],mpData.p[4]- mpData.p[3])); Ox=Normalize(CrossProd(mpData.p[1]- (mpData.p[6]+mpData.p[7]),0y)); Oz=Normalize(CrossProd(Ox, Oy)); / / f i n n a l moments of i n e r t i a Ix=Ixx*Ox.x*Ox.x+Iyy*Ox.y*Ox.y+Izz*Ox.z*Ox.z-2*Ixy*0x.x*0x. 2*Iyz*0x.y*0x.z-2*Ixz*0x.x*0x.z; Iy=Ixx*Oy.x*0y.x+Iyy*0y.y*0y.y+Izz*0y.z*Oy.z-2*Ixy*0y.x*Oy. 2*Iyz*0y.y*0y.z-2*Ixz *0y.x*Oy.z; Iz=Ixx*Oz.x*Oz.x+Iyy*Oz.y*Oz.y+Izz*Oz.z*Oz.z-2*Ixy*0z.x*Oz. 2*Iyz*0z.y*Oz.z-2*Ixz*0z.x*Oz.z; } / / F i l l mpData mpData.duration=elapsed; mpData.vn=vn; //calculated voxels mpData.volume=FormatFloat("0.00", vn*w); / / t o t a l volumecm3 mpData.mass=FormatFloat("0.00",totalMass);//mass mpData.mbd=FormatFloat("0.00",totalMass/(vn*w)); //mean bone //density g/cm3 mpData.mcx=FormatFloat("0.00", cx); //mass center mpData.mcy=FormatFloat("0.00", cy); mpData.mcz=FormatFloat("0.00", cz) ; mpData.gcx=FormatFloat("0.00", gx); //geometric center mpData.gcy=FormatFloat("0.00", gy); mpData.gcz=FormatFloat("0.00", gz); mpData.cd=FormatFloat("0.00", cd);//two center d i f f e r e n c e mpData.ixx=FormatFloat("0.00", Ixx);//moments of i n e r t i a mpData.iyy=FormatFloat("0.00", Iyy); mpData.izz=FormatFloat("0.00", Izz); mpData.Ix=FormatFloat("0.00", Ix);//moments of i n e r t i a mpData.Iy=FormatFloat("0.00", Iy); mpData.Iz=FormatFloat("0.00", Iz) ; //mark center i f ( !ipData.markcenter) return; i n t vx, vy, vz; vz=cz/ipData.depth+0.5 ; vx=cx/ipData.width+0.5 ; vy=cy/ipData.height+0.5 ; try{ fs=new TFileStream(fn,fmOpenWrite); } catch (...){ GlobalWarning("Can not open "+fn); return; / / i f f a i l s , return } fs->Seek(header+(vz-1)*ipData.rows*ipData.columns+(vy- 1)*ipData.rows+vx,soFromBeginning); char wr[1]={2}; fs->Write(wr,1); delete f s ; //output r e s u l t s mpData.Output(REdit->Lines, 1); //data } 11.2.3.2 C++ code for cross-sectional measurements Copyright © 2001 The University of British Columbia. All rights reserved. The University of British Columbia owns the intellectual property rights, including copyright of the source code. //Funciton: CalcAreaProperties //Parameter: image f i l e name //Return value: none void f a s t c a l l TCalimage::CalcAreaProperties(AnsiString fn) { //read bitmap or raw i n t rows=ipData.rows, columns=ipData.columns; i n t size=rows*columns; Byte* buff; //define a buffer to hold data i f (Lowercase(ExtractFileExt(fn))==".bmp") buff=ReadBitmap(fn, s i z e ) ; else buff=ReadRaw(fn, s i z e ) ; i f (size !=rows*columns) { GlobalWarning("The s p e c i f i e d s i z e i s d i f f e r e n t from the bitmap image"); return; } i n t r, c; //loop integer double totalX=0, totalY=0; //mass weighted t o t a l x, y double totalGX=0, totalGY=0; / / f o r geometric center i n t pn=0; //calculated number of p i x e l s double totalMass=0; //calculated t o t a l areal mass double lxm=0, lym=0; //mass moments of i n e r t i a double lx=0, ly=0;//area moments of i n e r t i a //precalculate these variables to improve performance double x2, y2;//square current x, ycoordinates of the p i x e l double xp, yp;//current x, yp o s i t i o n of the p i x e l center double ps=ipData.width*ipData.height;//pixel s i z e i n mm2 double Xmin=10000, Ymin=10000, Xmax=0, Ymax=0; in t cb; //loop and ca l c u l a t e f o r (c=0; c<columns; c+ + ) { yp=(c+0.5)*ipData.height; //mm y2=yp*yp; //sqare the current p o s i t i o n on Y-axis mm2 fo r (r=0;r<rows;r++) { cb=c*columns+r; i f (buff[cb]==0) continue; //ignore p i x e l s of value 0 //to improve performance i f (buff[cb]<ipData.lowerbound || buff[cb]>ipData.upperbound) continue; //out of range, ignore pn++; //increase the cal c u l a t e d number of voxels xp= (r+ 0.5)* ipData.width; i f (Ymin>yp) Ymin=yp; i f (Ymax<yp) Ymax=yp; i f (Xmin>xp) Xmin=xp; i f (Xmax<xp) Xmax=xp; x2=xp*xp; //square the current p o s i t i o n on X-axis double m=(buff[cb]*ipData.slope+ipData.intercept)*ps; //areal mass mg totalMass+=m; //calculated mass //mass weighted t o t a l x, y totalGX+=r+0.5; totalGY+=c+0.5; totalX+=(r+0.5)*m; totalY+=(c+0.5)*m; //Moments of i n e r t i a with respect to image matrix o r i g i n Ixm+=m*y2; //mass moments of i n e r t i a , mm4 Iym+=m*x2; Ix+=ps*y2;//area moments of i n e r t i a , mm4 Iy+=ps*x2; } } delete []buff ;//delete buffer //Calculate center of mass double cx, cy; cx=ipData.width*totalX/totalMass; cy=ipData.height*totalY/totalMass; double gx,gy; gx=ipData.width*totalGX/pn,• gy=ipData.height*totalGY/pn; double cdx, cdy, cd; cdx=cx-gx; cdy=cy-gy; cd=sqrt(cdx*cdx+cdy*cdy); double major=Ymax-Ymin; double minor=Xmax-Xmin; / / t r a n s l a t i n g mass center to image matrix o r i g i n , //and moments of i n e r t i a are //converted to Mis with respect to the mass center //convert to cm4 or g.cm2 Ixm=(Ixm-totalMass*cy*cy) ; Iym=(Iym-totalMass*cx*cx); Ix=(Ix-ps*pn*gy*gy); Iy=(Iy-ps*pn*gx*gx); / / F i l l apData apData.pn=pn; //calculated p i x e l s apData.area=FormatFloat("0.00",ps*pn/100);//total area i n cm2 apData.major=FormatFloat("0.00",major); //major axis apData.minor=FormatFloat("0.00",minor); //minor axis apData.mass=FormatFloat("0.00",totalMass/100);//mass i n cm2 apData.mbd=FormatFloat("0.00",totalMass/(pn*ps)); //mean //grayscale value apData.mcx=FormatFloat("0.00", cx); //mass center apData.mcy=FormatFloat("0.00", cy); apData.gcx=FormatFloat("0.00", gx); //geometric center apData.gcy=FormatFloat("0.00", gy); apData.cd=FormatFloat( "0 .00", cd);//two center d i f f e r e n c e apData.ix=FormatFloat( "0 .00",Ix /10000); //mi i n cm4 apData.iy=FormatFloat ( "0 .00 " , Iy /10000); apData.jo=FormatFloat("0.00" , (Ix+Iy)/10000); apData.ixm=FormatFloat("0.00",Ixm /10000); apData.iym=FormatFloat("0.00",Iym /10000); apData.jom=FormatFloat ("0.00" , (Ixm+Iym)/10000); apData.Xmin=FormatFloat("0.00", Xmin); apData.Ymin=FormatFloat("0.00", Ymin),- apData.Xmax=FormatFloat("0.00", Xmax); apData.Ymax=FormatFloat("0.00", Ymax); apData.cx=FormatFloat( "0 .00", cx-Xmin); apData.cy=FormatFloat( "0 .00", cy-Ymin); //output r e s u l t s apData.Output(REdit->Lines, 1) 1 2 P U B L I C A T I O N S 12 .1 RECENT PUBLICATIONS Zhang F, Langenbach GEJ, Hannam AG , Herr ing SW. 1999. Est imat ion of bone mass propert ies by CT in pig mandib les. Journal of Dental Research, 78 (special issue): 440 . Zhang F, Langenbach GEJ, Hannam AG , Herr ing SW. 2001 . Mass propert ies of the pig mandib le. Journal of Dental Research 80(1) : 327¬ 335. Zhang F, Peck CC, Hannam AG . 2001 . Mass propert ies of the human mandib le. Journal of Dental Research 80 (special i ssue): 270 . 12 .2 PROSPECTIVE PUBLICATIONS Zhang F, Peck CC, Hannam AG . 2001 . Mass propert ies of the human mandib le . Journal of B iomechanics. (Current ly in review) Zhang F, Hannam AG . 2001 . Cross-sect ional b iomechan ics of the human mandib le . (Submit ted to Amer i can Journal of Physical Anthropology) Zhang F, Herr ing SW, Hannam AG . 2001 . Symphysea l mechanics in pig and human mandib les. (Submit ted to Amer i can Journal of Physical Anthropo logy) Zhang F, Langenbach GEJ, Herring SW, Hannam A G . 2001 . Dynamic mechanics in the pig mandibu lar symphys i s . (To be submit ted to Arch ives of Oral Biology) Langenbach GEJ , Zhang F, Herr ing SW, Hannam AG . 2001 . Reconstruct ion of the pig's jaw sys tem and the predict ion of its mast icatory b iomechanics. (In preparat ion) Hannam AG and Zhang F. 2001 . Jaw structure and funct ion in the virtual env i ronment . (In preparat ion) Zhang F and Hannam AG . 2001 . Three-d imens iona l surface reconstruct ion, morphometr i c measurement , and mesh construct ion of pig and human jaws . (In preparat ion)

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