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Last Lumbar Facet and Pedicle Orientation in Orthograde Primates

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PAGE 1

LAST LUMBAR FACET AND PEDICLE ORIENTATION IN ORTHOGRADE PRIMATES By DORION AMANDA KEIFER A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS UNIVERSITY OF FLORIDA 2005

PAGE 2

Copyright 2005 by DORION AMANDA KEIFER

PAGE 3

For my mother and father and, of course, Mishka.

PAGE 4

ACKNOWLEDGMENTS I would like to thank, first and foremost, my committee, Drs. David Daegling and Michael Warren, for their support and guidance. The thoughtful insights of my committee made me think of this problem in new ways and were essential to the finished document. I would also like to thank Dr. Kenneth Mabry, Gary Sawyer, and Anita Caltabiano, from the Department of Anthropology, and Jean Spence, Teresa Pacheco, and Patricia Brunauer, from the Department of Mammalogy, at the American Museum of Natural History, for both the use of their collections as well as their encouraging words. A special thank you goes to Dr. Susan Anton at New York University for the use of her equipment; without her the data could not have been collected. I would like to thank Dr. Anthony Falsetti for the use of his equipment as well. The extra practice made all the difference. I thank the University of Florida Department of Anthropology staff, Karen Jones, Rhonda Riley, Patricia King, and LeeAnn Martin. I thank Shanna E. Williams for her artistic talent as well as her laughter. I would also like to thank Joseph, Jenna, and Maxine Coplin who moved me into their home for two months during data collection, as well as for their unwavering support. I thank Kristine Hulse, Jack Lackman, Linda Stanley and Mike Stanley, good neighbors and great friends. I would also like to thank Sandra Campbell, Sharon Milton-Simmons, Kelley Paulling, Jennifer Brookins, Mary Eckert, Renee Smith, Tina Sporer, and Barbara Young at the University of Florida of Florida, Department of Otolaryngology. They are the best coworkers anyone could ask for and they encouraged me every step of the way. I thank Dr. Jack Sedwick who has iv

PAGE 5

been so supportive in this endeavor. Finally, I would like to thank my family: My mother and father, Pat and Orion Keifer, as well as my sisters Rachel McLaren, Megan Keifer, my brothers Orion (Paul) Keifer, Seth Keifer, and Chris McLaren, and last, but not least, my niece Maccayla Keifer. v

PAGE 6

TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES...........................................................................................................ix ABSTRACT.........................................................................................................................x CHAPTER 1. INTRODUCTION........................................................................................................1 The Two-Column Model of Force Transmission.........................................................3 Anthropological Research on the Primate Spine..........................................................7 Goals of the Current Project.......................................................................................12 2. PRINCIPLES OF FUNCTIONAL MORPHOLOGY................................................14 Phylogeny and Phylogenetic Constraint.....................................................................14 Allometry....................................................................................................................15 Ontogeny and Evolutionary Developmental Biology.................................................15 Bone Biology and Behavior........................................................................................16 3. RESEARCH DESIGN AND DATA COLLECTION PROCEDURES.....................19 Taxonomic Sample.....................................................................................................19 Specimen Inclusion Criteria.......................................................................................21 Data Collection Instrumentation.................................................................................22 Data Collection Procedures........................................................................................23 Force Calculation: A Static Free Body Model..........................................................26 Statistical Testing........................................................................................................30 4. DATA TRANSFORMATION...................................................................................33 vi

PAGE 7

Statistical Analysis and Results.........................................................................................46 Combining Mixed Sex Samples.................................................................................46 Mean Directions and Mean Resultant Lengths...........................................................49 Pair-wise Testing........................................................................................................51 DISCUSSION....................................................................................................................55 Pair-wise Testing........................................................................................................55 Pair-wise Comparisons and Phylogeny...............................................................56 Pair-wise Comparisons and Allometry................................................................57 Pair-wise Comparisons and Bone Biology and Behavior...................................57 Force Transmission in the Posterior Elements...........................................................59 Feasibility of the Model..............................................................................................60 Vector Concentrations (MRL)....................................................................................61 Facet Shape.................................................................................................................61 CONCLUSIONS................................................................................................................63 MAJOR AXES-MINOR AXIS RATIO............................................................................66 ANGLE ERROR CALCULATION (DEGREES).............................................................68 RAW VECTOR DATA TABLES.....................................................................................70 LIST OF REFERENCES...................................................................................................78 BIOGRAPHICAL SKETCH.............................................................................................84 vii

PAGE 8

LIST OF TABLES Table page 1. Sample description.........................................................................................................21 2. Data transformation coordinate systems.......................................................................36 3. Pooling data, Gorilla sample.........................................................................................48 4. Descriptive statistics......................................................................................................49 5. Pair-wise results.............................................................................................................52 viii

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LIST OF FIGURES Figure page 1. Potential forces acting on the right pedicle.....................................................................6 2. Facet shape....................................................................................................................25 3. Free body diagram.........................................................................................................28 4. Normal vectors to the superior facets, right and left......................................................29 5. Normal vectors to the inferior facets, right and left.......................................................29 6. Axial vectors to the pedicles, right and left...................................................................29 7. Major axis vectors of the superior facets, right and left...............................................31 8. Major axis vectors of the inferior facets, right and left.................................................31 9. Major axis vectors of the pedicle, right and left............................................................32 ix

PAGE 10

Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Arts LAST LUMBAR FACET AND PEDICLE ORIENTATION IN ORTHOGRADE PRIMATES By Dorion Amanda Keifer May 2005 Chair: David J. Daegling Major Department: Anthropology The spine is critical for locomotor behavior. The spine is the foundational structure of the trunk and plays a key role in trunk position and stability. The spine also anchors major muscles of the forelimb and hindlimb, connecting them via a flexible column. Further, the spine articulates with the hindlimb, transmitting associated locomotor forces to the trunk. Despite the certain role of the spine in locomotion, the functional significance of vertebral variation is unestablished. Morphological analyses of the primate spine indicate that the last lumbar vertebra posterior element morphology potentially correlates with locomotion. The calculation of the forces in the posterior elements of the last lumbar vertebra is necessary to demonstrate this connection. For the future development of a model for the calculation of these forces facet and pedicle orientation in the last lumbar vertebra were measured for four orthograde primates: Homo sapiens, Gorilla gorilla, Pan troglodytes, and Pongo pygmaeus.

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Facet and pedicle orientation were described via six unit vectors for both the right and left side: normal to the superior and inferior facets, major axis of the superior and inferior facets, as well as axial to, and the major axis of, the pedicle. These vectors were calculated with respect to a vertebral reference axis, as well as the gravity vector when the spine is in orthograde posture. Descriptive statistics were calculated for all four species. Pair-wise tests were conducted among humans, gorillas, and chimpanzees. The facet normal and pedicle axial vectors are essential for the development of a biomechanical model to estimate force direction in vertebral posterior elements. The statistical analysis of these vectors show they are very consistent within species, indicating that the mean vectors can be used for the future development of a biomechanical model. The pair-wise tests demonstrated that the species have statistically different mean directions. The facet major axis vectors were calculated, as they may indicate the direction of the greatest range of motion in the facets. Generally, the descriptive statistics demonstrated that there was fairly large within species variability of these vectors. The pair-wise tests for equal mean direction indicated that, with a few exceptions, the species have statistically different mean directions with respect to these vectors. The pedicle major axis vectors were calculated as they potentially indicate how the pedicle is oriented to resist bending force. The statistical data indicated that the orientation of these vectors is very consistent for the great apes tested. In contrast in humans, these vectors have greater variability though the functional significance of this is not clear. The intra-species pair-wise tests indicated that each species had a statistically different mean direction.

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CHAPTER 1 INTRODUCTION The study of the evolution of the relationship between form and function, or functional morphology, has long been of interest to anthropological researchers. Within functional morphology, the terms form and function are heuristic devices that allow the separation of phenomena that are inextricably linked within an organism. In general, form refers to shape and size, while function refers to how this form moves through space and how the form interacts with its surroundings. However, these two definitions do not have any real context unless they are applied to a specific problem involving an organism or group of organisms, (Wainwright, 1988). One such application explores the relationship between skeletal form and type of locomotion, including associated posture. Extant adult skeletal morphologies are not exclusively the product of body size but also of long histories of evolutionary changes, developmental processes, and daily activities. It is not easy to determine the extent of the individual contributions of these factors to the adult form, as they are interrelated parts of the total functional-morphological complex of an individual organism. Therefore, it is important to apply myriad techniques and perspectives to a specific problem. The relationship between skeletal form and locomotion in extinct and extant primates has been a focus of study within the field of anthropology. Historically, these studies have centered on lower limb and pelvic morphology as well as the primate forelimb and shoulder girdle (Demes et al., 2000; Duncan et al., 1994; Heinrich et al., 1993; Hunt, 1991; Kimura, 2003; Latimer and Lovejoy, 1990a, 1990b; Lovejoy, 1975, 1

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2 1979,1988; Ohman et al., 1997; Schmitt, 1994, 2003; Susman et al., 1984; Susman and Demes, 1994; Tuttle, 1985). However, these studies lacked an in-depth exploration of the spine as part of the functional complex of limb movement, ignoring the fact that [a]mong mammals, structural differences of the lower precaudal spine correspond with contrasts between species in columnar function and positional behavior (Sanders 1995: 97). Recently, there have been a number of studies that address this gap in knowledge (Sanders, 1995, 1998; Shapiro, 1991, 1993, 1995; Shapiro and Johnson, 1998; Shapiro et al., 2001; Shapiro and Simons 2002; Velte 1987). These studies have revealed that the primate spine is morphologically conservative, despite quite dramatic differences in locomotor behavior. However, studies that compared the absolute and relative size and shape ratios of the posterior elements and pedicles of the primate spine have demonstrated morphological differences between bipedal and quadrupedal orthograde primates. This has lead to the speculation, though justifiably guarded, that these differences may represent differences in force transmission patterns in the posterior elements due to specific locomotor behavior. This conclusion is based upon traditional morphometric techniques that use size and shape ratios as proxy values for force magnitude and direction. This proves useful for locating possible areas where bone morphology may reflect function. An underlying assumption of previous primate spine morphometric studies is the assumption that the size of a given vertebral element is related to the magnitude of the forces acting upon that element (Davis, 1961: 337). However, this assumption may not always be reliable. Therefore, to examine this

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3 relationship further, it is important to understand how force is transmitted and to calculate the magnitude of those forces using biomechanical principles. The Two-Column Model of Force Transmission Although there is a lack of biomechanically based studies that focus on non-human primate spines, the biomechanics of the human spine has been, and continues to be, the focus of a great number of studies. These studies were conducted for clinical (orthopaedics and orthopaedic surgery), occupational (ergonomics, proper lifting, etc.), military (human tolerance in operational and crash situations), and public safety purposes (seatbelt and other restraints, transportation safety, amusement rides). Given the large number of studies, it is impractical to exhaustively list all of them. Therefore, focus will be on research that is pertinent to the development of the two-column model of force transmission in the spine. The two-column model of force transmission is a general description of force transmission in the spinal column. The spine is comprised of two compressive columns called the anterior and posterior columns. The anterior column includes the vertebral bodies, intervertebral disks, and associated ligaments, while the posterior column is comprised the articular facets and the lamina and associated ligaments. The pedicles are the bony bridges that connect the two columns. These data will be presented prior to the discussion of past anthropological research, as it provides an analytical framework in which to evaluate the anthropological data, as well as defines crucial terminology. The two-column pattern of force transmission is now widely accepted in the literature. However, prior to the 1960s, it was assumed that the pedicles and posterior elements did not transmit any significant force as they are smaller than the vertebral

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4 bodies. They were believed to have three main purposes, to resist lateral rotation, to serve as muscle attachment sites, and to protect the spinal cord from trauma. The first researcher to conclude that vertebral bodies could not be the sole means of force transmission was P.R. Davis in 1961. His conclusion was based upon a visual examination of human spinal morphology. Davis noted the area of the lumbar centra consistently increases in size from lumbar one to lumbar four. However, lumbar five, the last lumbar vertebra in humans, has a smaller lumbar centra than the vertebra above it. Davis concluded that if the vertebral bodies were the sole means of force transmission, the last lumbar centra would have to be larger than the one above it as the compressive force is cumulative. Based upon this observation, he concluded that the last lumbar vertebral body was subjected to less compressive force than the one above and posited that the compressive force was somehow resisted by the neural arch, which is comprised of the lamina and pedicles. In 1974, Prasad and colleagues conducted a study to measure the role of the articular facets during acceleration. The researchers used several techniques to measure the load bearing capability of the facets in cadaveric studies. To qualitatively measure facet load, they attached strain gauges to the pedicles and lamina of their subject vertebrae. Quantitative measures of facet load were obtained using an intervertebral load cell that was designed to fit under the disk or vertebral body. The transducer was capable of measuring both axial force and the eccentricity of the axial force with respect to its geometric center. Cadavers were then accelerated in such as way as to simulate travel conditions experienced by U.S. Air Force pilots. Load cells were also mounted to the chair and restraint system. From this analysis, Prasad and colleagues proved both

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5 qualitatively and quantitatively that the articular facets were involved in load transmission. However, they were unable to calculate the facet load. In 1980, the work of Adams and Hutton filled in the information gap. They proved experimentally the significant role that articular facets play in forces transmission and also provided a measure of load transmission. Using green cadaver vertebral sections consisting of two vertebrae and the intervening disk, positioned as they would be in vivo, the researchers subjected the sections to compressive loads and various angles and measured deformation for acute loads and bone compression for constant loads. The load borne by the facets was shown to be posture dependent. They found facets withstand the highest force load when the body is in a standing, upright position. In this posture, facets transmit 16% of the total force load. This is the generally accepted number in the literature: however, some studies have placed this figure as high as 23% in the fifth (last) lumbar in humans (Pal and Routal, 1987). In 1989, El-Bohy and colleagues cadaveric studies demonstrated that the mechanism of posterior element load transmission was via facet/lamina contact by directly measuring contact pressure in the joints under various loading conditions using motion segments consisting of three vertebrae and their intervertebral disks. They measured contact pressure by placing a transducer between the inferior facets and the lamina of the vertebra below. Their results verified that loads passing through the facets are transmitted via contact between the facets and lamina. Previous research confirms that the posterior elements are involved in force transmission. However, the magnitude, direction, and type of force acting on the individual posterior elements and pedicles are not known. There has been some

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6 speculation as to the loading conditions of the pedicles. One hypothesis is that the pedicles are subjected to bending stresses as a result of muscles action on the spinous or transverse processes or alternately from the posterior elements (Bogduk and Twomey, 1987). It has also been suggested that the pedicles may be subjected to tension from the facets locking to prevent the vertebrae from sliding forward (Bogduk and Twomey, 1987). Finally, it has also been hypothesized that the pedicles may be subjected to compressive axial loads from the vertebral body to the lamina (Pal and Routal, 1986,1987). However, none of these hypotheses has been proven. There are four potential forces acting upon the pedicles: torsion, compression, bending and shear (Figure 1). In this figure compression is shown. Alternately, this force could be tension. The torsion is depicted as clockwise, however it may be counter clockwise. Similarly the bending and shear is depicted as an upward force. However, it may also be a downward motion. Figure 1. Potential forces acting on the right pedicle.

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7 Anthropological Research on the Primate Spine Three research studies were instrumental in the development of the current research design. Comparative studies on the primate spine indicated that biomechanical analysis of the posterior elements is important for understanding their role in posture and locomotion. The results of these three works and their relevance to the current problem are discussed in detail in the following section. An early functional morphology study addressing the primate spine from a comparative perspective was Margaret Velte's (1987) dissertation that developed and described a biomechanical model for the anthropoid spine. Velte focused completely on the morphology of anterior vertebral elements, which are comprised of the vertebral centra, intervertebral disks, and associated ligaments. Her sample included Alouatta, Ateles, Cebus, Cercopithecus, Homo, Gorilla, Pan, and Papio, as well as some fossil material. The fossil material will not be discussed as it is beyond the scope of the current project. Further, the discussion will focus on results for the species included in the current study, Homo, Gorilla, and Pan. Velte modeled individual vertebral bodies as short, deep beams. Using strength values of trabecular bone, she calculated the tensile and compressive bending moments and the shear force at failure. The shear and the bending moments were calculated for both dorsoventral flexion and lateral flexion axes. Her results indicate that shear and tensile stresses intensify along both axes. However, her data indicates that compressive stresses are reduced along the lateral flexion axis in cercopithecoids and along the dorsoventral flexion axis in hominoids. Velte concluded that the vertebral centra of Homo, Gorilla, and Pan are strongest in compression. Velte postulated that these data

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8 indicate the hominoid pattern of short, broad vertebral centra is an adaptation to orthograde posture. Liza Shapiro (1991,1993) also addressed the primate spine from a comparative perspective. Her extant sample also contained a large number of species including Aloutta, Ateles, Cebus, Gorilla, Homo (pygmy and non-pygmy), Hylobates, Indri, Pan, Papio, Pongo, Propithecus, and Varecia. Again, this discussion focuses on the species included in the present study, Homo, Gorilla, Pan, and Pongo. Like Velte, Shapiro measured and evaluated vertebral centra. However, she also focused on the posterior elements and the pedicles. Shapiro's goal was to define aspects of vertebral morphology that were uniquely human. In order to accomplish this goal, she recorded various vertebral measurements: vertebral body (centra) area (calculated as an ellipse from the measured ventrodorsal and mediolateral distances at midpoint of the body), pedicle area (product of the length and width), pedicle shape (ratio of the width to the length), lamina area (product of the width and ventrodorsal thickness), and lamina shape (ratio of the lamina ventrodorsal thickness to the width). These metrics were selected to reflect the relative loads passing through the anterior and posterior columns. Finally, she measured the angulation of the superior facet (medial interfacetal width lateral interfacetal width/2 times the facet width). From the results of these measurements and calculations, Shapiro concluded, as compared to the great apes (Gorilla, Pan, and Pongo), humans have large vertebral centra (body) areas for their body weight. Shapiro speculated that this large body area may be related to increased compressive force associated with bipedalism. She did note that, like Velte, her studies indicated that humans and great apes have shorter, broader

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9 vertebral bodies as compared to other primates. As mentioned, human vertebral body areas increase in size from the first to the penultimate lumbar vertebra; however, at the last lumbar level, the vertebral body area reduces in size. Once thought to be a uniquely human pattern, Shapiros results indicate this pattern was found in nearly all the primates in her sample. For both pedicle area and pedicle shape, her ANOVA tests indicated that Homo, Gorilla, Pan, and Pongo each have statistically different mean values. There were no statistically significant within species differences between male and female and therefore, for the all species, the sexes were pooled. Although each species had a unique mean for both pedicle area and pedicle shape, all four species trend toward increased pedicle area from the penultimate to last lumbar vertebrae. Shapiro speculated that this increased pedicle area is a function of the reduced vertebral body area of the last lumbar. Counter to speculation, this pattern is also seen in primates that do not have a reduced vertebral body area of the last lumbar vertebra. Shapiro also discerned that pedicle area is highly correlated with body size, with the notable exception of the last lumbar vertebra. At this level humans have a larger pedicle area than would be expected for their size. The results of pedicle shape (ratio of width to height) analysis indicate that, for humans, the increase in pedicle area at the last lumbar level occurs due to an increase in pedicle width relative to pedicle height. The Pan, Pongo, and Gorilla samples did not exhibit a pedicle shape change between the penultimate and last lumbar vertebrae. However, when compared with non-hominoids in her sample, the great apes and humans have short, wide pedicles. On one hand, the human and great ape samples have similar trends in the lumbar region, with humans expressing these trends more dramatically. This would suggest that

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10 this pattern is more likely related to postural behavior. On the other hand, the ANOVA test indicated the species have statistically different mean values, suggesting that each species has a unique posterior element morphology. The ANOVA tests of lamina shape (ratio of ventrodorsal thickness and width) again indicate humans, gorilla, chimpanzees, and orangutan have unique mean values. As with the pedicle ANOVA test, males and females were pooled. Shapiros results indicate that humans have wider lamina than the great apes. Further, human laminae become progressively wider, relative to thickness from lumbar one to the last lumbar. The gorilla, chimpanzee, and orangutan laminae to do not display this change. Shapiro (1991) indicates that the laminae are an important aspect of posterior element force transmission because compressive force moves down the vertebral column via facet/lamina contact. The increasing width of lamina in humans may therefore be a reflection of increased compressive forces due to bipedal locomotion. Interestingly, despite great apes orthograde posture, they do not share a similar pattern. Finally, Shapiro (1991) measured facet angulation of the superior facets. The ANOVA test again indicated that the four groups have statistically different superior facet angulation means. As with the other ANOVA tests, males and females were not found to have any statistically significant differences and were pooled. Despite statistically different mean values, general trends were found. For example, gorillas and chimpanzees demonstrated similar patterns, with superior facet angles becoming more acute from lumbar-one to the last lumbar, while humans facet angulations becomes more oblique. Orangutans appear to have a mean facet angle value at the last lumbar level that is identical to humans (i.e. laterally oriented), though facet

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11 angles appear to follow the chimpanzee and gorilla pattern until the last lumbar level. Humans follow a pattern in which the superior facet angle becomes more oblique. The angulations of the superior facets provide information about the lateral rotation permitted in the spine. The gorilla and chimpanzee facets seem oriented to restrict lateral rotation. The increasingly oblique angulations in humans have been posited to help resist forward displacement of the vertebrae as the last lumbar vertebra in humans is angled forward. Finally, the functional significance of the orangutan facet angulation pattern is not currently understood. William Sanders (1995,1998) also conducted comparative morphological studies of the primate spine. The goal of his analysis was to use an extant primate sample to describe Australopithecine spinal biomechanics. He had a diverse sample, however I will only focus on the species relevant to the present study. Two of his measures are significant to the current problem, facet spacing and orientation and pedicular robustness and cross-section area. With respect to facet angulation, Sanders(1995;1998) data supported Shapiros (1991;1991) finding. Like Shapiro, Sanders only documented facet angulation for the superior facets. However, Sanders (1995;1998) also highlighted facet spacing. Gorillas, chimpanzees, and orangutans have superior facet spacing that narrows slightly from the first to last lumbar vertebrae. In humans, the spacing between the superior facets increases from the first to last lumbar vertebrae. Sanders (1998) suggested that this pattern is important for spinal stability as more widely spaced facets create a more stable base.

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12 Sanders (1995;1998) also measured pedicle width and length to calculate pedicle area and robustness. Sanders data indicate that ape pedicles are robust, shorter and wider, when compared to monkey pedicles. According to Sanders (1995;1998), muscle force accounts for part of the robustness as the transverse process of hominoids are rooted on the pedicles. However, Sanders (1995;1998) also documents scaling trends that indicate bending stresses as a critical factor for pedicular dimensions. From the first to last lumbar vertebra, great ape pedicles increase in width and decrease in length. Human vertebrae also follow this pattern; however, the last lumbar vertebra in humans has an extremely wide pedicle. These data are in agreement with Shapiro (1991;1993). The results of the above research studies are not straightforward. On one hand, ANOVA tests indicate that for measures defined by the study each species had a statistically different mean value. However, the results also indicate that humans and great apes share generalized patterns of vertebral morphology. To further explore the functional morphology of the posterior elements it is necessary to develop a methodology to calculate the magnitude of stresses and strains on these elements. It is also desirable to measure and compare previously unmeasured variables. The current project addresses both of these issues. Goals of the Current Project The goal of this project is to document last lumbar facet and pedicle orientation in orthograde primates. These orientations were analyzed from two perspectives. First, descriptive statistics were calculated. Second, pair-wise tests for equal mean direction were conducted. The results indicate whether vectors means can be used to develop biomechanical models for force calculation in the last lumbar posterior elements. The results also indicate how orientation vectors vary among orthograde species with diverse

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13 locomotor behavior. If current and future research on the posterior element morphology demonstrates that the morphology of the elements reflects postural and locomotor behavior, it will provide another method for elucidating postural and locomotor behavior in the fossil record.

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CHAPTER 2 PRINCIPLES OF FUNCTIONAL MORPHOLOGY Anthropological researchers rely heavily on skeletal material to make many inferences, such as taxonomy, evolutionary relationships, life history, and behavior. The current projects focus is the extent to which locomotor and positional behavior can be inferred from the posterior element morphology of the last lumbar vertebrae. The theoretical paradigm of functional morphology relies heavily on physics and engineering principles. Unlike engineered objects, however, the biological form is a product of many interrelated factors and processes: evolutionary history, growth and size, locomotor and postural behaviors, and so forth. There are four major areas in which the simple form follows function axiom of functional morphology is called into question. These major areas are phylogeny and phylogenetic constraint, allometry, ontogeny and evolutionary biology, and bone biology and behavior. However, the reader is cautioned that these categories, though artificially separated, are interrelated. Phylogeny and Phylogenetic Constraint The current form of any biological organism is constrained by their evolutionary, genetic history. The significance of this in terms of the current, and all, functional morphological studies is that aspects of the current morphology may not have functional significance. Rather they are simply a reflection of common ancestry and subsequent phylogenetic constraint. In order to elucidate which aspects of a species' morphology are likely related to phylogenetic constraint, a comparative sample is necessary. In this 14

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15 study, a comparative sample of four closely related orthograde species with diverse locomotor repertoires was included. Allometry Allometry refers to changes in morphology that are not necessarily related to change in function but rather are "by products" of body size. In the present study, the methodology does not remove the factor of size in a meaningful way. Size is removed in the sense that the data are presented as unit vectors. Unit vectors do not, by definition, give any information regarding size as the vectors are divided by their length, resulting in all vectors having a length of one. Consequently, unit vectors only describe direction. Correcting for size is not appropriate in this case, as the goal is to determine orientation. The sample does include an extremely sexually dimorphic species, Gorilla gorilla as well as two similarly sized species, Homo sapiens and Pan troglodytes. Pair wise comparisons for equal mean direction were conducted between the male and female Gorilla vectors, as well as between Homo and Pan. The results of these tests will give some information regarding the dependence of the measured unit vectors on body size. Ontogeny and Evolutionary Developmental Biology Developmental processes affect the adult form in ways that may not be a reflection of function but rather are the results of canalization and developmental stability. The concepts of [c]analization and developmental stability refer to the tendency of developmental processes to follow a particular trajectory despite external and internal perturbation, (Hallgrimsson et al., 2002: 131). Currently, there is no information in the literature that documents the extent to which posterior element vertebral morphology is constrained by canalization and developmental stability. This information is necessary to understand the functional morphology of the vertebral column, and it is expected that this

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16 research will be conducted in the future. However, the reader is cautioned that the results of the current study were interpreted without the benefit of this information. Bone Biology and Behavior Studies of bone biology and bone behavior consider bones response to the internal and external environment. The theory that the distribution of strain trajectories engendered through functional activity is responsible for the development and maintenance of trabecular alignment and cancellous bone density within a bone is the commonly accepted modern conceptualization of Wolffs law (Biewener et al., 1996: 1). This theory is very attractive in terms of functional morphological analyses because if it is correct, bone morphology can unproblematically be interpreted as a reflection of function. However, it is important to note, as Cullinane and Einhorn (2002) pointed out there are compelling critiques of theory (Bertram and Schwartz, 1991; Bienwener et al., 1996; Fyrie and Carter, 1986, Pearson and Lieberman, 2004) as well as scholarly debates about the mechanisms of the bones response to the mechanical environment (Martin, 2000; Mullender and Huiskes, 1995; Turner and Pavalko, 1998). Generally, interpretations support that, in the course of normal loading during daily activities, the skeleton accumulates microscopic damage. The field of bone biology currently supports the theory that this microdamage triggers a remodeling response within the skeleton (Yerby and Carter, 2000). It is at this point, however, that dissension arises within the literature; bone tissue is not consistent or uniform in its response to stresses and strains. For example, a study that used a canine model resected the subjects radii and allowed locomotion with the ulna supporting all the weight. The results of this study demonstrated a lack of uniform skeletal response as some of the subjects suffered fatigue fractures of the ulna, while others developed massive hypertrophy of the ulna

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17 (Chamay and Tschantz, 1972). However, the results of this study are difficult to interpret. In the canine model the radius is much larger than the ulna. Therefore, the ulna was subjected to loading that would never be encountered outside of the research setting. In a follow-up study, Carter and associates (1981) also used a canine model, but the ulna rather than the radius was resected. In this study, they found that bone formation in the radii was negligible. This would seem to indicate that the radius of the canine is overbuilt with respect to the forces that it must withstand. To add another piece to this already difficult puzzle, fractures can occur at the upper levels of normal activity. For example, pars defects, fractures of the pars inarticularis, are a very common condition in humans, especially in the last lumbar vertebra (Burkus, 1988; Kip et al., 1994; Sermon and Spengler, 1981). Contrary to Wolffs Law, however, this shear fracture is not caused by a large acute stress to the par inarticularis. Rather this fracture occurs due to repetitive loading and subsequent fatigue failure. This fracture is often found in athletes. A clear understanding of the cellular process by which bone repairs itself as well as bone as a biological material is essential to understanding how bone responds to the mechanical environment. However, it is not currently understood. The adult skeleton is in a constant state of deterioration and reformation by the osteoclasts and osteoblasts in trabecular surfaces and in Haversian systems. In simplified terms, when a microcrack forms in bone, osteoclasts activate on the crack surface resorbing the damaged bone. This activates the osteoblasts, which stimulate new bone growth (Brukner et al., 1999; Mundy, 1999; Schaffler, 2000). Colloquially, this is known as the drill and fill process. The intuitive and pervading opinion, until recently, was that the formation of microcracks is a

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18 mechanical process in which microscopic fissures in the bone are formed faster than they can be repaired by the body. However, recent research indicates that stress fractures are a response to a positive feedback mechanism. The mechanism of increased usage stimulates bone turnover, which results in focally increased bone remodeling. It is theorized that increased bone porosity and bone mass result, weakening the bone (Brukner et al., 1999; Schaffler, 2000). In other words contra, to the current understanding of Wolffs Law, this theory indicates that when skeletal material is subjected to repetitive stresses and strains, it is ultimately weakened, not strengthened by the remodeling process. Fortunately, more and more anthropological researchers are interested in moving beyond the Wolffs Law paradigm. Currently, the two most influential perspectives that move beyond this paradigm are bone developmental genetics and biomechanics (Pearson and Lieberman, 2004). The goal of this project is to move beyond morphological studies on the primate spine heavily reliant on Wolffs Law and develop biomechanically based research to further our understanding of spinal functional morphology.

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CHAPTER 3 RESEARCH DESIGN AND DATA COLLECTION PROCEDURES Taxonomic Sample This research further explores the extent to which postural and locomotor behavior affects the morphology of the last lumbar posterior elements. The last lumbar vertebra was chosen as previous research has correlated statistical differences between species, suggesting that its morphology is most likely to reflect locomotor patterns (Shapiro, 1993). In order to determine if stresses and strains resulting from postural and locomotor behavior affect the morphology of the posterior elements, it is necessary to compare species with similar postural behavior but different locomotor repertoires. Four orthograde species with diverse locomotor behaviors were chosen for the study: Gorilla gorilla, Pan troglodytes, Pongo pygmaeus, and Homo sapiens. Locomotor behavior cannot accurately be described using gross categories such as knuckle-walker. Despite the fact that primates generally have a dominant form of locomotion, they engage in diverse locomotor behaviors that should not be ignored. Outlined below are the locomotor behaviors of the orthograde species included in this study. In the adult gorilla, terrestrial knuckle walking accounts for 94% of the distance traveled by mountain gorillas. They rarely engage in climbing (vertical) and engage in bipedalism, leaping, brachiation, or bridging even less frequently (Shapiro, 1991; see also Tuttle and Watts, 1985). Lowland gorillas appear to be more arboreal than mountain gorillas (Shapiro, 1991; see also Dixon, 1981; Fleagle, 1988). However, they were 19

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20 included in the gorilla sample because they feed, rest, and sleep on the ground where they move by quadrupedalism, (Shapiro, 1991: 17). Pan troglodytes most frequently engage in knuckle-walking quadrupedalism. This locomotor behavior accounts for 86% of their locomotor activity. However, this species also engages in, from most to least frequent, quadrumanous climbing and scrambling (11%), arm swinging and bipedalism (1%), and leaping (< 1%). Only 16% of this species locomotion is arboreal (Shapiro, 1991; see also Doran, 1989). Pongo pygmaeus is the most arboreal of all the species included in the study. Quadrumanous scrambling is the most important locomotor behavior for this species. However, they also engage in the following behaviors, listed from most to least frequent: brachiation, tree swaying, quadrupedal walking, and climbing, (Shapiro, 1991, see also Sugardjito, 1982; Sugardjito and van Hooff, 1986). Homo sapiens are the most orthograde of all the species included in the sample. They are capable of a variety of locomotor behaviors, such as walking, running, climbing, swimming, and hanging. However, bipedal walking is their single most important form of locomotion. Table 1 describes the taxonomic sample for the study. As can be seen from the table the gorilla sample is comprised of 17 adult specimens of known sex. The chimpanzee sample is comprised of 18 adults specimens. There were 6 male specimens and 12 of unknown sex. The orangutan sample is quite small and consists of 4 male specimens and 1 specimen of unknown sex. The human sample consists of 19 adult specimens, 17 males and 2 females. The specimens were from an older population ( x = 57.47 years, sd = 11.57years).

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21 Table 1. Sample description. Taxonomic Designation Sex Total Gorilla Sample Gorilla gorilla gorilla Male 7 Gorilla gorilla gorilla Female 6 Gorilla gorilla beringei Male 2 Gorilla sp indent Male 1 Gorilla sp indent Female 1 Total 17 P an tro g lod y tes Sample Species Designation Sex Total P an troglodytes Male 6 P an troglodytes Unknown 12 Total 18 P on g o p yg maeus Sample Taxonomic Designation Sex Total P ongo pygmaeus Male 4 P ongo pygmaeus Unknown 1 Total 5 H omo sapiens Sample Taxonomic Designation Sex Race Decade of Life Total H omo sapiens Male White Fifth 4 H omo sapiens Male White Sixth 4 H omo sapiens Male White Seventh 3 H omo sapiens Male White Eighth 3 H omo sapiens Male White N inth 1 H omo sapiens Male Black Fifth 1 H omo sapiens Male Black Sixth 1 H omo sapiens Female White Fifth 1 H omo sapiens Female Black Seventh 1 Total 19 Specimen Inclusion Criteria Specimens were collected from the American Museum of Natural History, Departments of Mammalogy and Anthropology. With the exception of two Gorilla specimens, all of the non-human specimens were collected from the Department of Mammalogy. The human specimens were collected from the Department of Anthropologys Morphology Collection.

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22 Only adult specimens were included in the study. Adulthood was defined by epiphyseal fusion and dental criteria. Further, for the non-human primates only wild caught specimens were included. Before collecting data, each specimen was carefully checked for breakage and pathology. Given the limited number of specimens available, a vertebra was not discarded outright for data collection if pathology, breakage, or cut marks were present. If the pathology was very limited data was not collected from the predetermined points on areas where the pathology was present. Similarly, if the vertebrae had breakage or were cut, data was not collected from the area with the breakage or cut marks. The Pongo pygmaeus sample posed a special challenge. It was noted by Schultz (1941) that even though Pongo pygmaeus can be said to reach adulthood by dental criteria it is possible that the epiphyses can remain unfused. In the present study, dental criteria were used to define orangutan adults, however specimens were rejected if the last lumbar vertebra showed incomplete epiphyseal fusion. This criterion severely limited the Pongo pygmaeus sample size. Data Collection Instrumentation Digitized three-dimensional coordinates of predetermined points were taken on a Microscribe model 3DX (Immersion Corporation, 801 Fox Lane, San Jose, California 95131). At the beginning of each data collection session, Accuracy Check #1 and Accuracy Check #2, two calibration procedures recommended by the manufacturer were performed. Accuracy Check #1 describes the Microscribe position prior to turning it on. The Microscribe performs a self calibration based upon the position at start up. Prior

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23 to each measurement session, the Microscribe was positioned carefully to correspond with the home position prior to turning it on. Accuracy Check #2 required the program MSTest. This test procedure ensures that the joint encoders are working properly. MSTest gives joint angles for various stylus positions, which must be matched up to charts detailing the expected angles. Finally, the Microscribe 3DX has a moveable coordinate system. The X and Y axis are determined by the position of the swiveling arm at start up. Therefore, the Microscribe was positioned consistently throughout the measurement procedure. The data points were imported directly into a Microsoft Excel spreadsheet via Inscribe software. Linear measurements were made using a Mitutoyo digital caliper, model 573-225-10 (Mitutoyo American Corporation, 965 Corporate Boulevard, Aurora Illinois 60504). The digital caliper was zeroed and internally calibrated, in accordance with the operating manual, before each use. Data Collection Procedures The posterior element morphology of the vertebrae samples was documented via thirty-eight landmark points. The superior and inferior vertebral centra (body) surfaces were documented via five points each: the most anterior point at the midline, the most posterior point at the midline, the most lateral point in the mid-coronal plane on the right side, the most lateral point in the mid-coronal plane on the left, and the center point. Each facet had 5 landmark points for a total of 20 points. The center point was defined as an average of height and width. The four additional points were defined as the articular surface end points of the minor and major axes. Finally, the pedicles were defined by four points each: on the superior surface, at the midline, on the inferior surface, at the

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24 midline, on the medial surface, in the midtransverse plane, and on the lateral surface, in the midtransverse plane. For the purpose of replicability and consistency, each point was located and marked. The measurements required to locate the landmarks were documented using a Mitutoyo digital caliper (accurate to 0.01mm), when possible. Some points were recorded by sight, as accurate measurements were not possible with the calipers. This occurred when the element of interest was so small that accurate measurements could not be obtained. When this occurred, following the detailed measuring procedure would be sufficient to produce consistent measurements from observer to observer as the vertebral elements involved are extremely small. All measurements were taken at least two times and were accepted if the two values were within 0.05mm. The two accepted values were then averaged. The procedures used to locate the landmark points for each vertebral element are outlined below. The vertebral body landmark points were located first. The center point was calculated using the measured values of vertebral width at the widest lateral point and anterior-posterior depth at the centerline of lateral symmetry. Once this point was located, a perpendicular set of axes was drawn on the vertebral body using a straight edge, aligned with the anterior-posterior centerline. The four additional points were located at the points were these axes intersected the edge of the vertebra superior vertebral body. The facets points were located by first measuring the facet height and width. It should be noted that the author found that facet morphology varied greatly; however, the morphology could be divided into five general types as shown in Figure 1. The center

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25 was located by taking half the value of both the measured height and width, and positioned in accordance with Figure 2. Once the center point was located, the center point as well as the facet shape category was used to define the major axis of the facet. Once the major axis was located, the minor axis was defined as the line perpendicular to the major axis, passing through the center point. The intersections of these axes with the edge of the facet defined the four additional points. The focus of this project is the major axis orientation. However the extent to which the major axis is larger than the minor axis may be of interest to the reader. Therefore, this value was calculated as the base ten logarithm of the ratio of the major to minor axis and is included in Appendix 1. Figure 2. Facet shape. This figure shows the five major facet shapes that were seen during data collection, as well as the how the minor and major axes were drawn for each shape. From left to right, Type I, Lateral Oval, Type II, Square, Type III, Diagonal Oval, Type IV, Pear, and Type V, Kidney. The pedicle landmark points were defined for the axial center of the pedicle, between the vertebral anterior body and the lamina. The specific points were defined as the middle point of each the superior, lateral, and inferior surface, again using height and width measurements. The medial point had to be located visually as the calipers did not fit into the neural canal. In order to improve the accuracy and consistency a method for locating the medial point was developed. The midpoint of the medial pedicle was located for both the superior and inferior surface. A line was then drawn to connect the two

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26 points. This line was then used as a guide to locate the medial point. These points define the plane perpendicular to the axial line. Defining the exact center of the superior-inferior plane is not as critical as defining the lateral-medial center, which was accomplished using the above methodology. Once all the landmark points were measured and marked, each vertebra was placed on a ring-stand, anterior side down, and superior side away from the ring-stand stanchion and secured by dental wax. The specimen cannot be moved during data collection as all landmarks were measured with respect to the origin of the Microscribe The relative position of the Microscribe allowed access to all the landmark points on the vertebra to be measured without moving the vertebra. The axes of the Microscribe was approximately aligned with the natural axes of the vertebra, such that the x, y and z axes of the Microscribe approximated the superior-inferior, anterior-posterior and lateral axes of the vertebra, respectively. All vertebrae were measured with a similar orientation to the digitizer. On all specimens, the landmark points were recorded in the same order. Further, for each specimen the landmark points were measured at least two times in order to calculate an error rate. Force Calculation: A Static Free Body Model Analysis of the forces within the various elements within the vertebra is dependent on all external forces acting on the vertebra. From mechanics, these forces are depicted using a static free body diagram. The static free body diagram is an accepted and widely used methodology in biomechanics (see Panjabi and White, 2001 and Ozkaya and Nordin 1998, for fundamental biomechanics techniques). A two dimensional static free body diagram, as depicted in Figure 2, has been developed by the author to as a simplified starting point for the determination of the forces within the vertebral posterior elements.

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27 As can be seen from the figure, the vertical force on the vertebra from the body above is supported by the forces from the inferior disk and the inferior facets. To determine how the force is transmitted, it is necessary to document the size and orientation of the load bearing surfaces and the elements of interest for calculations of the load. Sizes of various vertebral elements have been extensively documented as previously discussed. Orientation of the facets and pedicles can be documented via three-dimensional vectors, which is key for determining the angles in the static free body method. It is important to stress, however, that the static free body model is a simplified starting point that operates under certain assumptions. First, the model assumes that the force on the facets can be accurately modeled as normal to the center of the facet. The initial model does not contain any information regarding muscle force or ligaments. The work of Frank Holdsworth (1963) demonstrated that the posterior element ligaments are essential to stability in the spine. Finally, the accuracy of the model is dependent upon the accurate knowledge of the vertebra's in vivo position. Fortunately, a study conducted by Albert Schultz and colleagues validated a biomechanical model for the calculation of load on the lumbar spine. In this study they compared values predicted by a biomechanical model to intradiscal pressure and myoelectric signals. They also correlated the effects of intraabdominal pressure on spinal loading. Their results indicate that a biomechanical model is "valid in the [loading] situations that were examined," (Schultz et al., 1982: 717). They also measured intraabodominal pressure and concluded, "the intraabdominal pressured were not large and seldom had a major influence on the overall mechanics of the trunk," (Schultz et al., 1982: 720). Given these conclusions, a biomechanical model that takes into account

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28 vertebral geometry and muscle action should be an accurate predictor of the loads borne by the spine. Figure 3. Free body diagram. This figure depicts a two-dimensional free body diagram. This is a simplified depiction of forces acting on the inferior facets. However, the figure clearly demonstrates the necessity of facet and pedicle orientation for force calculation. (Source: Drawn by Author). The vectors needed to develop the free body model are described below: 1. The unit vector normal to superior facet. This is the vector normal to the center of the superior facet, divided by the vector length (Figure 4). 2. The unit vector normal to inferior facet. This is the vector normal to the center of the inferior facet, divided by its length (Figure 5). 3. The unit vector in the axial direction of pedicle. This vector describes the axial direction of the pedicle vector, divided by its length (Figure 6).

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29 Figure 4. Normal vectors to the superior facets, right and left (Source: Shanna E. Williams.) Figure 5. Normal vectors to the inferior facets, right and left (Source: Shanna E. Williams). Figure 6. Axial vectors to the pedicles, right and left (Source: Shanna E. Williams)

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30 The above vectors can be calculated for individual specimens. However, it is desirable to develop general species models. In order for such a generalized model to be possible, it is necessary that for each vector a species mean direction and mean resultant length is found. This is the measure of concentration, analogous to a standard deviation, which describes vector concentration as a value between zero and one. A value of one indicates a high concentration, i.e. all vectors are parallel and a value of zero indicates no concentration and randomly orientated vectors. The mean resultant length is an extremely important measure with respect to the development of species models. A low concentration would indicate that there is great within species variation with respect to vector direction. This in turn indicates that the mean value lacks any real meaning. A high concentration indicates that there is a true species mean, which can be used to develop species specific models. Statistical Testing The second goal of this project is to analyze the mean and variance of facet and pedicle orientation. The vectors that were calculated for the static free body diagram were also subjected to statistical analysis, as these vectors are crucial to the calculation of stresses and strains in the vertebrae. However, additional vectors were calculated, as they may be biomechanically significant. These vectors were the superior and inferior major axis vectors and the pedicle major axis vectors. The facets potentially allow sagittal, lateral, and axial rotation. The major axis of the facet may indicate the direction in which the facets allow the most rotation. The pedicle is the bony bridge between two compressive columns and it has been hypothesized to be subject to bending moments. The major axis indicates the direction

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31 on which the pedicle is oriented to resist the most bending force. However, it has not been conclusively proven that the pedicles are subject to bending forces. 4. Major axis of the superior facet. This is the vector with the larger magnitude between either the superior-inferior axis vector of the superior facet or the lateral-medial axis vector of the superior facet. The vector is then divided by its length in order to transform it into a unit vector. (Figure 7) 5. Major axis of inferior facet. This is the vector with the larger magnitude between either the superior-inferior axis vector of the inferior facet or the lateral-medial axis vector of the facet. The vector is then divided by its length in order to transform it into a unit vector. (Figure 8). 6. Major axis of pedicle at midpoint. This is the vector with the larger magnitude between either the superior-inferior axis vector at the midpoint of the pedicle or the lateral-medial axis vector at the midpoint of the pedicle. The vector is then divided by its length in order to transform it into a unit vector. (Figure 9). Figure 7. Major axis vectors of the superior facets, right and left. (Source: Shanna Williams). Figure 8. Major axis vectors of the inferior facets, right and left. (Source: Shanna E. Williams)

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32 Figure 9. Major axis vectors of the pedicle, right and left. (Source: Shanna E. Williams) The calculation and analysis of the above vectors will further our understanding of the relationship of posterior element morphology and posture and locomotion. Further, it is a first step in the development of a model to calculate actual stresses and strains in the posterior elements. This information will enrich our knowledge with respect to the relationship of posterior element morphology to load bearing.

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CHAPTER 4 DATA TRANSFORMATION In order to document the orientation of the facets and pedicles, it is necessary to record each vertebras morphology, as expressed by the 38 landmark points, in three-dimensions. Fortunately, recent advances in computer technology have made this possible and have given anthropological researchers a new tool with which to study the skeletal form. The digitizer, in this case a Microscribe 3DX, records x, y, and z coordinates with respect to its own origin. Using the information provided by the digitizer, the researcher must transform the data into useful information. In this study, the goal, of course, is to transform the coordinate data into unit vectors that describe facet and pedicle orientation with respect to a common coordinate system. This chapter describes in detail how a coordinate system for the vertebra was selected and how the measured quantities were transformed to the coordinate system. The transformation process relies on three concepts from vector algebra; determining a vector from two points in space, the concept of a unit vector and, a vector multiplication operation (cross product). Given point 1 in space with coordinates (x 1 ,y 1 ,z 1 ) and point 2 with coordinates (x 2 ,y 2 ,z 2 ) the vector a, from point 1 to point 2 is found as follows: 212121a(xx)i(yy)j(zz)k Where i,j,andk are unit vectors in the x, y and z directions, respectively. 33

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34 Substituting in a x a y and a z for (x 2 x 1 ), (y 2 y 1 ), and (z 2 z 1 ), the vector becomes: xyza(a)i(a)j(a)k Because i,j,andk are all mutually orthogonal, Pythagorean Theorem can be used to determine the length of vector a between point 1 and point 2, as follows: 222xyzaaaa Vectors and their multiples all have the same direction. If one were to multiply a vector by the ratio of one to its length, the vector would have the same direction but would have a length of one. This is the definition of a unit vector. Mathematically, it can be expressed as follows: yxzunitaaaaaijaaaa k The third vector algebra concept, the cross product, can be expressed in two ways as follows: axbnabsin Where: a= vector a with magnitude a a b = vector b with magnitude b n = vector normal (perpendicular to both vectors a and b with magnitude 1) = angle between vectors a and b. The cross product can also be calculated using the determinant, as follows:

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35 xyzxyzijkaXbaaabbb ()()(yzzyzxxzxyyxaXbababiababjababk ) Where: a= vector with magnitude a a b = vector b with magnitude b i,j,andk = vectors with magnitude 1 in the x, y and z directions respectively a x a y and a z = components of vector a in the x, y and z directions respectively b x b y and b z = components of vector b in the x, y and z directions respectively The first mathematical expression shows that the magnitude of the cross product is equal to the product of three variables, the magnitude of vector a the magnitude of vector b, and the sine of the angle between vector a and vector b. Of note, directional vectors are not necessarily tied to a particular point in space, that is all parallel vectors have the same directionality; however, an important consequence of the cross product calculation is that vector and vector a b are visualized as coplanar. The family of parallel planes containing the two vectors can be uniquely defined by the normal vector perpendicular to both vector a and vector b The second expression of the cross product gives the vector in its coordinate form. As stated previously, our data is in the form of x, y, and z coordinates. The coordinates can also be thought of as a vector from the origin to the coordinate of the point. Therefore, vectors from the origin can be expressed as their x, y, and z coordinates which

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36 are represented as a x a y and a z for vector a and b x b y and b z for vector b. These values can be used in the cross product calculation. The goal of the data transformation procedure was to determine three-dimensional coordinates of the measured points with respect to a defined coordinate system based on the morphology of the specimen. The initial step in this process was to choose the specimen-referenced axes. The axes that were chosen has the x direction in the anterior-posterior (anterior positive) direction, the y direction was in the right to left lateral (left positive) direction, and the z direction was in the superior-inferior (vertical) direction (up positive) of the main body of the vertebra. Several translations and rotations of one coordinate system to the next were required, beginning with the coordinate system of the Microscribe and ending with the gravity vector referenced coordinate system. Several intermediate coordinate systems were also required. The various coordinate systems are summarized in Table 2. Table 2. Data transformation coordinate systems. Coordinates Nomenclature Change from previous coordinate system X,Y, & Z Microscribe referenced Data as originally recorded X,Y, & Z Intermediate Axes changed to approximate specimen referenced axes X 1 ,Y 1 & Z 1 Intermediate Move origin to approximate sample referenced origin X 2 ,Y 2 & Z 2 Intermediate Rotated about Z 1 axis by the first Euler angle X 3 ,Y 3 & Z 3 Intermediate Rotated about X 2 axis by the second Euler angle X 4 ,Y 4 & Z 4 Intermediate Rotated about Z 3 axis by the third Euler angle x, y, & z Sample referenced Translate origin to center of inferior body x 1 y 1 & z 1 Gravity referenced Rotated about y axis by inferior body-horizontal angle The Microscribe records points with respect to its internal reference axes. Therefore, the specimen being digitized cannot be moved with respect to the digitizer reference during the digitizing process. In order to collect all the points of interest to the

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37 study, it was necessary to have the anterior surface of the body in the approximate direction of the digitizers Z axis, while the superior surface approximated the digitizers Y axis and the lateral portions of the vertebra approximated the digitizers X axis. The first step was to switch the axes nomenclature from that of the Microscribe to that approximating the vertebral axes. The resulting renamed vector increases in the Z direction corresponding to increases in the superior direction and increases in the X direction corresponding to increases anteriorially. This was accomplished by changing the (X, Y, Z) nomenclature to the approximate vertebral reference axis (X, Y, and Z), as follows: X-axis values for the approximate vertebral coordinate system were set as the negative of the measured Z values Y-axis values for the vertebral coordinate system were set as the measured X values. Z-axis values for the vertebral coordinate system were set as the negative of the measured Y values. The next step for aligning the specimens was to reference the data to an origin point on the vertebra. For this study, the center point of the inferior body was chosen. Though this designation is somewhat arbitrary, this point was chosen for ease of data interpretation. Problematically, however, the inferior surface of a vertebra is asymmetrical and is not a flat surface. It was necessary to correct for these limitations when locating the origin point in the three directions. For the y-direction, the origin was selected as a point halfway between the measured left and right landmark points. In the x direction the origin point selected was half way between the ventral and dorsal landmark points. The remaining point, in the z direction, was defined as the average of the four measured landmark points (left, right, ventral and dorsal). The appropriate center coordinate was subtracted from the data in the X,Y,Z coordinate system to determine the

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38 coordinates in the new X 1 Y 1 Z 1 intermediate coordinate system. This essentially translated the origin from that defined by the Microscribe to a point that approximates the center of the inferior vertebral centra (body). The asymmetry of the vertebra may introduce a slight error with respect to the coordinate system. However, subsequent rotations will not produce additional error. Once the data were translated to the approximate vertebral coordinate system and referenced to the approximate vertebral origin, it was then necessary to rotate the data. This step is very important because although every effort was made to consistently line up the individual specimens with the axis of the Microscribe rotational misalignment of a few degrees was unavoidable and is almost certainly inconsistent from sample to sample. To correct the misalignment, the data was rotated to correspond to the selected vertebral reference system using Euler angles. The x convention for Euler rotation is used. The Euler angle rotation formula requires three distinct rotations. First, the coordinates are rotated about the Z-axis, then about the X-axis, and about the Z-axis a second time. For the first rotation, the data were referenced to an intermediate axis (X 2 Y 2 and Z 2 ). The data were rotated an angle about the Z 1 -axis such that the X 2 axis is contained in both the X 1 -Y 1 plane of the old coordinate axes and the x-y plane of the vertebra referenced axis. The data are then referenced to a second intermediate coordinate axes system (X 3 Y 3 and Z 3 ). For this rotation, the vertebra is rotated an angle about the X 2 axis such that the X 3 Y 3 plane of the intermediate coordinate axes system coincides with the x-y plane of the vertebra reference system, or stated another way, such that Z 3

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39 coincides with the z-axis of the vertebra reference system Finally, the data are referenced to the vertebra reference system (x, y, and z) by rotating an angle about the Z 3 axis to bring the Y 3 axis parallel with the y-axis in the vertebral reference axis. In order to use the Euler Angle Rotation procedure, it is necessary to calculate the Euler angles (, ,and ). Fortunately, the vectors necessary for this calculation can be derived directly from the data taken and the three vector algebra concepts noted above. Recall that one property of the cross product calculation is that it can be used to determine a vector that is perpendicular to both vectors used in the cross product. Further, the angle between two vectors can be determined using the cross product calculation and the arcsine function. The first Euler angle is the angle between the X 1 -axis and the intersection of the X 1 -Y 1 plane of the X 1 Y 1 and Z 1 coordinate system and the x-y plane of the vertebra referenced system. In some preliminary data reduction, the x-y plane of the vertebra coordinate system (inferior body plane) was approximated as containing the y direction vector formed by subtracting the coordinates left edge from the right edge of the inferior body and the x direction vector, formed by subtracting the coordinates of the posterior edge from the anterior edge coordinate of the inferior body. This definition was problematic, because the main bodies of the specimen contained some wedging, not only in the anterior posterior direction but also in the lateral direction. The consequence was that the z axis did not pass through the center of the anterior body plane. To minimize the variability between specimens, the x-y plane was defined by the z axis, which by definition is perpendicular to both the x and y axes. The z direction was determined by subtracting the inferior body center from the superior body center. This definition is also

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40 attractive because the vertebral body is a compressive member, with the force being transmitted by the pressure in the vertebral discs. The superior and inferior forces would be centered within the respective body. The cross product of the z-axis and the normal vector to the X 1 -Y 1 plane is perpendicular to both normal vectors and therefore contained in both planes. In other words, the cross product of the normal vectors is in the direction of the intersection of the two planes. The sine of angle is determined by taking the arcsine of the cross product of a unit vector in the X 1 direction and a unit vector in the direction of the intersection of the X 1 -Y 1 plane and the x-y plane. The data are transformed to the new coordinate system (X 2 Y 2 and Z 2 ) using the following matrix: cossin0sincos000 1 Note that only the X 2 and Y 2 coordinates are changed from the previous X 1 and Y 1 coordinates, the Z 2 coordinates remain the same as the Z 1 coordinates. That is because the rotation is about the Z 1 -axis. Note the sine and cosine functions for the rotation angle The second Euler angle is the angle between the X 2 -Y 2 plane and the x-y plane. It is also the angle between the normal vector to the X 2 -Y 2 plane and the normal vector to the x-y plane. The normal vector to the X 2 -Y 2 plane is the Z 2 axis. The normal vector to the x-y plane has to be recalculated because of the rotation, but is still the coordinates of the inferior body center subtracted from the superior body center. Using unit vectors, the cross product of the two normal vectors produces a vector perpendicular to the two

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41 vectors with a magnitude of the sine of the angle between the two vectors. The arcsine of the magnitude of the cross product gives the angle The data are transformed to the new coordinate system (X 3 Y 3 and Z 3 ) using the following transformation matrix: 1000cossin0sincos Angle the third Euler angle, is the angle between the X 3 axis and the x-axis (or between theY 3 axis and the y axis). As noted previously, the x-axis is approximated by the vector between the posterior and anterior inferior body of the vertebra and the y-axis is approximated by the vector between the right and left edge of the inferior body of the vertebra. These two definitions however, do not necessarily produce perpendicular vectors. For the purposes of this thesis, angle is defined as the angle between theY 3 axis and the y-axis. The y-axis vector is determined using the coordinates of the right and left inferior body measurements and then is made into a unit vector. The cross product of the unit vector in the y direction and the unit vector in the Y 3 direction gives a vector perpendicular to both vectors and of length equal to the sine of the angle between the two vectors (angle ). The angle is found using the arcsine function. The data are transformed to the vertebral referenced axis (X 4 Y 4 and Z 4 ) using the following matrix. cossin0sincos000 1 Finally, a linear translation of the X 4 Y 4 and Z 4 data to the x, y and z axis was required to re-center the origin of the axis system between the left and right and dorsal

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42 and ventral edges of the inferior body plane. Of note, the rotations could have been performed prior to the first translation, avoiding a second translation; however, making the translation first facilitated checking the rotation process by looking at coordinate data. The small change in the center of the vertebra, with respect to the origin of the axis, occurred due to the asymmetry of the vertebrae and the shifting of the data points during the three rotations. This produced the coordinates of the 38 reference points in the vertebral referenced coordinates (x, y, z). Once the measured coordinates were referenced to the vertebral coordinate system, it was then possible to calculate the unit vectors of interest using very simple calculations: The superior-inferior axis vector at the midpoint of the pedicle was calculated by subtracting the inferior coordinates from the superior coordinates. The lateral-medial axis vector at the midpoint of the pedicle was calculated by subtracting the medial coordinates from the lateral coordinates. The superior-inferior axis vector of the superior facet was calculated by subtracting the inferior coordinates from the superior coordinates. The lateral-medial axis vector of superior facet was calculated by subtracting the medial coordinates from the lateral coordinates. The superior-inferior axis vector of inferior facet was calculated subtracting the inferior coordinates from the superior coordinates. The lateral-medial axis of inferior facet was calculated by subtracting the medial coordinates from the lateral coordinates. The normal vector to the superior facet was determined from the cross product of the superior-inferior axis vector of superior facet and the lateral-medial axis vector of superior facet. Recall one of the properties of the cross product calculation is that it can be used to find the vector perpendicular to the plane of two vectors. The normal vector to the inferior facet was determined from the cross product of the superior-inferior axis vector of inferior facet and the lateral-medial axis vector

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43 of inferior facet The axial direction vector of the pedicle was determined from the cross product of superior-inferior axis vector at the midpoint of pedicle and the lateral-medial axis vector at the midpoint of the pedicle. The following unit vectors were calculated: The major axis of pedicle at its midpoint. This is the vector with the larger magnitude between either the superior-inferior axis vector at the midpoint of the pedicle and the lateral medial axis vector at the midpoint of the pedicle. The vector was then divided by its length in order to transform it into a unit vector. The major axis of the superior facet. This is the vector with the larger magnitude between either the superior-inferior axis vector of the superior facet and the lateral medial axis vector of the superior facet. The vector was then divided by its length in order to transform it into a unit vector. The major axis of inferior facet. This is vector with the larger magnitude between either the superior-inferior axis vector of the inferior facet and the lateral medial axis vector of the facet. The vector was then divided by its length in order to transform it into a unit vector. The unit vector normal to superior facet. This is the vector normal to the center of the superior facet, divided by its length. The unit vector normal to inferior facet. This is the vector normal to inferior facet, divided by its length. The unit vector in the axial direction of pedicle. This vector describes the axial direction of the pedicle vector, divided by its length. The vectors calculated via this data transformation were referenced to the vertebra itself. However, this is not necessarily reflective of the vertebras in vivo position. For the purpose of the statistical analysis, it is important to calculate these vectors with respect to the gravity vector as well for comparative information. In this study, an orthograde posture was chosen. This position was chosen as the literature contained some information regarding the vertebral position for this posture.

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44 Radiologically, the lumbosacral junction is well documented in humans. From radiological evidence the angle ( ), the forward rotation of the last lumbar vertebra can be approximated. In this position the human last lumbar vertebra is rotated forward approximately thirty degrees from the horizontal (Yochum and Rowe, 2005). Unfortunately, this angle is not well documented for non-human primates. Therefore, this angle was approximated using indirect evidence. AH Schultz (1961) documented the exact vertebral curvature of the non-human specimens by making molds of the abdominal cavity of eviscerated specimens. His diagrams demonstrate a kyphotic curvature in the thoracic region. However, in lower lumbar region, the diagrams indicate that the spine is held perpendicular to the ground and therefore the last lumbar would not be significantly rotated. The human coordinate system (x 1 y 1 and z 1 ) was rotated about the y-axis to the approximate vertebral position with respect to the gravity vector using the following matrix: cos0sin010sin0cos The precision to which landmark points are measured is of great interest. Errors can be induced by the operator (intraoperator error) or by the measuring instrument (instrument error). To minimize the combined intraoperator and instrument error, the landmark points were taken twice on each specimen. While individual points can be compared, the calculated directions are based on several vectors calculated by finding the difference between two measured points. Therefore, the vectors were calculated in real time for both sets of measurements and the direction of the results were compared. In

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45 order to calculate angle error between the vectors found from the two sets of data, the cross product calculation was used. Recall that from the cross product, the sine of the angle between the two vectors can be found. The unit normal vectors calculated from the two sets of data were calculated then the cross product was taken. The angle between the two vectors is the arcsine of the length of the normal vector. This is expressed in radians, which are easily converted to degrees by multiplying the radians by 180 and dividing by pi. The data transformation Excel spread was programmed to calculate this value for six of the unit directionality vectors. The data points were rejected if the angle of difference between the measurements exceeded five degrees. The intraoperator error values were calculated for six of the 12 vectors calculated (Appendix B). The unit vectors were the vectors normal the facets and the vectors axial to the pedicles. These vectors were calculated as they represent the highest errors values possible. Recall that the normal and axial vectors were calculated from four data points and the major axis vectors are calculated from two of the four same landmark points. Please see the Appendix B for a detailed description of the accepted error rates.

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CHAPTER 5 STATISTICAL ANALYSIS AND RESULTS Directional data cannot be analyzed using traditional univariate or multivariate statistics. Fortunately, there are methods available for the analysis of this type of data. To enrich our understanding of the posterior element morphology, both descriptive and comparative statistical methodologies were used to analyze the vector data. Combining Mixed Sex Samples Recall from Chapter Three that the species samples are mixed sex. Pooling the males and females of a species increases sample size. However, pooling males and females can be problematic, especially for a species with extreme sexual dimorphism, as this can result in a misleading mean (or Mean Direction) and a large standard deviation (or Mean Resultant Length), making the data insensitive to pair-wise testing. Therefore, before pooling the males and females, it is desirable to determine that they have equal mean directions. However, this is not always possible as was the case with three of the four species included in this project. The Pongo pygmaeus and Pan troglodytes samples were composed of specimens of both known and unknown sex. Therefore, it was not possible to determine if the males and females had equal mean directions. When reviewing the results, it is important to keep in mind that these species were pooled without the benefit of the test of equal means. The test for equal male and female means for the Homo sapiens sample was not conducted for a different reason. In the human sample, the sex of each specimen is 46

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47 known, however, the sample is overwhelmingly male (17 males, 2 females). A statistician was consulted and it was determined that a test to determine if the males and females can be pooled is not meaningful given the extremely small female sample size. However, the reader again is cautioned that if the sample was more balanced, this test would not only be appropriate, it would be required. Fortunately, the sexes of all specimens are known for the most sexually dimorphic species included in the sample, Gorilla gorilla. Further, this sample is fairly balanced. Therefore, the test of equal means direction was conducted. In order to confidently pool cross-sex specimens, the sex specific samples are required to have not only equal mean direction, but also equal concentration. The equal mean direction hypothesis was tested using corrected version of the Likelihood Ratio Test given in Mardia and Jupp (2000). This corrected test was developed by Presnell and Rumcheva (2005). It is corrected in the sense that instead of referring the test statistic to a chi-square distribution, it is referred an F distribution. This has a better performance in terms of Type I error. The hypothesis of equal mean directions is rejected for large values of the test statistic. The results are reported in the data tables as p-value 1. Equal concentration was verified using a pair-wise specific test found in Mardia and Jupp (2000). The results are reported in the data tables 5-1 as p-value 2. The null hypotheses of equal mean direction and equal mean concentration were not rejected if the p-value > 0.05. Table 3 describes the statistical results of the male vs. female gorilla tests. Descriptive statistics, mean direction (MD) and mean resultant length (MRL) are also listed in the table. A detailed description of these values is given in the next section.

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48 Table 3. Pooling data, Gorilla sample. Superior Facet Left Right Major Axis Normal Major Axis Normal Sample Size (M,F) (9,7) (-9.7) (10,7) (10,7) MD -0.15, 0.67, 0.73 -0.59, -0.66, 0.46 -0.19, -0.75, 0.63 -0.53, 0.69, 0.48 Gorilla M MRL 0.78 0.95 0.86 0.98 MD -0.22, 0.49, 0.84 -0.56, -0.75, 0.36 -0.35, -0.58, 0.74 -0.56, 0.75, 0.36 Gorilla F MRL 0.86 0.98 0.84 0.98 P-value1 0.6485 0.3721 0.5968 0.1652 P-value2 0.2831 0.0975 0.8084 0.3849 Inferior Facet Left Right Major Axis Normal Major Axis Normal Sample Size (M,F) (7,7) (7,7) (10,7) (10,7) MD -0.02, 0.18, 0.98 0.58, 0.77, -0.26 0.12, -0.29, 0.95 0.48, -0.80, -0.35 Gorilla M MRL 0.76 0.97 0.74 0.95 MD 0.20,0.045, 0.98 0.54,0.81, -0.23 -0.01, -0.43, 0.90 0.52, -0.81, -0.28 Gorilla F MRL 0.82 0.97 0.74 0.96 P-value1 0.6543 0.7903 0.7874 0.7258 P-value2 0.3222 0.4943 0.5638 0.3223 Pedicle Left Right Major Axis Axial Direction Major Axis Axial Direction Sample Size (M,F) (8,7) (8,7) (10,7) (10,7) MD 0.27, -0.29, 0.92 -0.87, 0.33, 0.38 0.30,0.14,0.94 -0.93, -0.19, 0.32 Gorilla M MRL 0.99 0.98 0.99 0.97 MD 0.20, -0.25, 0.95 -0.85, 0.43, 0.31 0.20, 0.21, 0.96 -0.93, -0.28, 0.25 Gorilla F MRL 0.99 0.97 0.99 0.96 P-value1 0.2864 0.3725 0.122 0.4635 P-value2 0.4241 0.8339 0.4717 0.7506 The results in Table 3 indicate that for all the vectors, male and female gorillas did not reject the null hypotheses of equal mean concentration. The test for equal mean direction indicates that the null hypothesis was not rejected for all the vectors tested. Therefore, the male and female specimens were pooled.

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49 Mean Directions and Mean Resultant Lengths The descriptive statistics reported in Table 4 give information regarding mean values and the variability about that mean. In the case of directional data, the statistical data is expressed as mean direction (MD) and mean resultant length (MRL). The mean direction (MD) is a unit vector, resulting from the means of the x, y, and z coordinates. These values are divided by the mean resultant length (MRL) so that the vector becomes a unit vector. The mean resultant length (MRL) is a measure of concentration that is analogous to a standard deviation. The mean resultant length is obtained by applying Pythagorean Theorem (length = square root of the sum of the squares of the three mean values) to the mean direction vector, prior to its conversion to a unit vector. Descriptive statistics are given for all four species of the sample. The first goal of this project is to test the feasibility of the development of species-specific vector based models for calculating force magnitude in the posterior vertebral elements. In order to develop such a model, there must be tolerable within-species diversity with respect to the axial and normal vectors calculated. The mean directions (MD) and mean resultant lengths (MRL) for each species are listed in the tables below. The major axis vectors, while not calculated for the first goal of this project, were included. Table 4. Descriptive statistics. Superior Facet Left Right Vector Major Axis Normal Major Axis Normal Sample Size (19,16,18,5) (19,16,18,5) (16,17,18,5) (16,17,18,5) MD -0.36,0.77,0.52 -0.62, -0.62,0.48 -0.43, -0.77,0.47 -0.56,0.64,0.53 H omo MRL 0.84 0.97 0.89 0.98 MD -0.18,0.59,0.79 -0.58, -0.70,0.42 -0.33, -0.67,0.67 -0.55,0.72,0.43 Gorilla MRL 0.82 0.96 0.86 0.98

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50 Table 4. Continued Superior Facet Left Right Vector Major Axis Normal Major Axis Normal Sample Size (19,16,18,5) (19,16,18,5) (16,17,18,5) (16,17,18,5) MD -0.14,0.43,0.89 -0.45, -0.83,0.34 0.08, -0.33,0.94 -0.34,0.88,0.33 P an MRL 0.84 0.99 0.95 0.99 MD 0.35,0.31,0.89 -0.64, -0.61,0.46 0.25, -0.59,0.77 -0.72,0.45,0.54 P ongo MRL 0.97 0.96 0.94 0.97 Inferior Facet Left Right Vector Major Axis Normal Major Axis Normal Sample Size (19,14,18,5) (19,14,18,5) (19,17,18,5) (19,17,18,5) MD 0.85, -0.21,0.49 0.51,0.69, -0.50 0.83,0.03,0.55 0.47, -0.71, -0.52 H omo MRL 0.67 0.98 0.58 0.98 MD 0.09,0.11,0.99 0.56,0.79, -0.24 0.07, -0.35,0.93 0.49, -0.81, -0.32 Gorilla MRL 0.78 0.97 0.74 0.96 MD 0.53,0.02,0.85 0.39,0.89, -0.25 0.54, -0.07,0.84 0.31, -0.92, -0.25 P an MRL 0.90 0.99 0.92 0.99 MD 0.56,0.15,0.81 0.51,0.70, -0.49 0.48, -0.25,0.84 0.63, -0.56, -0.54 P ongo MRL 0.99 0.98 0.99 0.96 Pedicle Left Right Vector Major Axis Axial Direction Major Axis Axial Direction Sample Size (17,15,18,2) (17,15,18,2) (17,17,18,4) (17,17,18,4) MD 0.51,0.84, -0.20 -0.76,0.55,0.36 0.56, -0.82,0.15 -0.83, -0.48,0.28 H omo MRL 0.82 0.98 0.56 0.98 MD 0.24, -0.27,0.93 -0.86,0.37,0.35 0.26,0.17,0.95 -0.93, -0.23,0.29 Gorilla 0.99 0.97 0.98 0.97 MD 0.37, -0.28,0.88 -0.80,0.39,0.46 0.38,0.18,0.91 -0.86, -0.29,0.42 P an MRL 0.99 0.99 0.99 0.99 MD 0.61,0.59,0.53 -0.80,0.53,0.29 0.27, -0.15,0.95 -0.89, -0.35,0.30 P ongo MRL 0.57 0.99 0.69 0.98 The descriptive statistics, MD and MRL values, are given in Table 4. With respect to the superior facet vectors, the left major axis vectors indicate relatively low concentration for Homo, Gorilla, and Pan while the Pongo vectors have high concentrations. Interestingly, for the right major axis vectors, Homo and Gorilla have

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51 low concentrations while Pan and Pongo have high concentrations. For the left and right normal vectors, all species have high concentration values. The results for the inferior facet vectors indicate that for the left and right major axis vectors, Homo and Gorilla have low concentrations. The Pan and Pongo major axis vectors have high concentration. The left and right normal vector data for all species indicates very high concentration values. For the pedicles, the descriptive statistics that the left and right major axis vectors have low concentrations for Homo and Pongo. The MRL values for Gorilla and Pan indicate that these vectors have a high concentration. The right and left normal vectors have high MRL values for all four species. Interestingly, only humans have low concentration for all the pedicle major axis vectors. It is interesting to note that for the superior and inferior facet major axis vectors, the chimpanzee and orangutan vectors have high concentrations. Comparatively, humans and gorillas have relatively low concentrations. Pair-wise Testing Pair-wise tests were conducted in lieu of an ANOVA. Preliminary testing indicated that pair-wise testing was a more appropriate for determining mean directional differences. These preliminary tests demonstrated that it was likely that all species had statistically different mean values. Therefore, four-way ANOVA testing would not give results with a satisfactory level of detail. Further the preliminary testing indicated that for the between-species pair-wise testing, a test that assumes equal concentration was not appropriate. A test that does not assume equal concentration, also based upon the von Mises-Fisher distribution (Mardia and Jupp 2000), was performed. A total of 60 pair-wise tests were conducted. The

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52 orangutan data were not included in the pair-wise tests as the sample size was quite small (n=2-5) and the results would not be meaningful. Pair-wise tests were conducted with both the vertebral referenced system and the data rotated to account for the in-vivo gravity vector. Tables 5 lists the results for the pair-wise analyses for the superior facet, inferior facet, and pedicle vectors. The null hypothesis of equal mean direction was not rejected for p-values <0.05. Five pair-wise tests were conducted for each sample, Gorilla vs. unrotated Homo, Gorilla vs. rotated Homo, Gorilla vs. Pan, Pan vs. unrotated Homo, and Pan vs. rotated Homo. Table 5. Pair-wise results. Test of Equal Mean Direction, Superior Facet Vectors Left Major Axis Normal Major Axis Normal Gorilla vs. Homo (Sample Size) (16,19) (16,19) (17,16) (17,16) p-value 0.0570 0.2074 0.1866 0.0708 Gorilla vs. Rotated Homo (16,19) (16,19) (17,16) (17,16) p-value 0.2099 <0.0001 0.2398 <0.0001 Gorilla vs. Pan (16,18) (16,18) (17,18) (17,18) p-value 0.4338 0.0026 <0.0001 <0.0001 Pan vs. Homo (18,19) (18,19) (18,16) (18,16) p-value 0.0005 <0.0001 <0.0001 <0.0001 Pan vs. Rotated Homo (18,19) (18,19) (18,16) (18,16) p-value 0.0089 <0.0001 <0.0001 <0.0001 Inferior Facet Vectors Left Right Major Axis Normal Major Axis Normal Gorilla vs. Homo (Sample Size) (14,19) (14,19) (17,19) (17,19) p-value <0.0001 <0.0001 0.0004 0.0015 Gorilla vs. Rotated Homo (14,19) (14,19) (17,19) (17,19) p-value <0.0001 <0.0001 <0.0001 <0.0001 Gorilla vs. Pan (14,18) (14,18) (17,18) (17,18) p-value 0.0139 0.0017 0.0022 0.0005

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53 Table 5. Continued Test for Equal Mean Direction Inferior Facet Vectors Left Right Major Axis Normal Major Axis Normal Pan vs. Homo (Sample Size) (18,19) (18,19) (18,19) (18,19) p-value 0.0125 <0.0001 0.1306 <0.0001 Pan vs. Rotated Homo (18,19) (18,19) (18,19) (18,19) p-value <0.0001 <0.0001 <0.0001 <0.0001 Pedicle Vectors Left Right Major Axis Axial Major Axis Axial Gorilla vs. Homo (15,17) (15,17) (17,17) (17,17) p-value <0.0001 <0.0001 <0.0001 <0.0001 Gorilla vs. Rotated Homo (15,17) (15,17) (17,17) (17,17) p-value <0.0001 0.0011 <0.0001 <0.0001 Gorilla vs. Pan (Sample Size) (15,18) (15,18) (17,18) (17,18) p-value 0.0005 <0.0001 0.0013 0.0093 Pan vs. Homo (18,17) (18,17) (18,17) (18,17) p-value <0.0001 <0.0001 <0.0001 <0.0001 Pan vs. Rotated Homo (18,17) (18,17) (18,17) (18,17) p-value <0.0001 <0.0001 <0.0001 <0.0001 With respect to the superior facet vectors, the vectors generally have statistically different mean directions. There are exceptions. The Gorilla vs. unrotated Homo test indicates that the right and left major axis and normal vectors do not have statistically different mean directions (p-values, 0.0570, 0.2074, 0.1866, and 0.0708 respectively). The pair-wise tests for Gorilla and rotated Homo indicate that the right and left superior facet major axis vectors do not have statistically different mean directions (p-values 0.2099 and 0.2398 respectively). Finally, the Gorilla vs. Pan pair-wise tests indicate that for the left superior facet major axis vector they do not have statistically different mean directions (p-value 0.4338).

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54 For the inferior facet vectors, all tests except one indicate that each species has statistically different mean directions. The exception is the Pan vs. unrotated Homo right inferior facet major axis vector test. The pair-wise test indicated that for this vector, humans and chimpanzees do not have statistically different mean directions. With respect to the pedicle vectors, the results indicate that the species have statistically different mean directions (p<0.05).

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CHAPTER 6 DISCUSSION Pair-wise Testing An overwhelming majority of pair-wise tests indicate the each species vector had a statistically different mean direction. However, a minority of the pair-wise tests indicated did not reject the null hypothesis. With respect to the Homo and Gorilla pair-wise tests, the superior facet vector results indicated that when the human vectors were not rotated to the gravity vector, the right and left major axis and normal vectors did not have statistically different mean directions. This would suggest that, with respect the major axis vectors, Homo and Gorilla species have a similar direction of motion for the superior facets. The equal mean normal vectors indicate that Homo and Gorilla have a similar pattern of force transmission in the superior facets. However, these results are problematic; when the human spine is held vertically to the ground, the last lumbar is rotated forward 30 degrees from the horizontal. When the same pair-wise test was conducted with the rotated Homo vectors, the major axis vectors indicate equal mean directions with an even higher p-value (p-value 0.0570 and 0.1866 vs. 0.2099 and 0.2398). However, the normal vectors no longer have equal mean directions (MD), suggesting they have different force transmission patterns in the superior facets. These results are difficult to interpret. It would seem to indicate that despite a dramatic change in human vectors orientation, humans and gorillas have statistically non-divergent superior facet major axis orientation, which in turn may suggest that the superior facets allow the greatest rotation in the same plane. However, the functional interpretation of 55

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56 this is not yet clear. More research, such as the calculation of facet curvature, is necessary to prove the assumption that the major axis facet vectors have a functional significance. The Pan vs. Gorilla pair-wise tests indicate that for the left superior facet major axis vectors, they do not have statistically different mean directions. It is surprising that the right major axis vector does not indicate an equal mean direction as well, (p-values 0.4338 and <0.0001 respectively). Interestingly, this asymmetrical pattern is echoed in the pair-wise results for the inferior facet major axis vectors of Pan and unrotated Homo vectors. However, the side is reversed. In this case the left major vectors indicate that they do not have equal mean directions, while the right indicate equal mean direction, (p-values of 0.0125 and 0.1306 respectively). The assumption that these vectors are measuring function has not been proven. These results may indicate that they are not accurately capturing function as intended. These asymmetrical results may also be the result of Type I error. Pair-wise Comparisons and Phylogeny The pair-wise tests were conducted in three closely related species, humans, chimpanzees, and gorillas. The phylogenetic relationship between these species is still contested. However, molecular (Koop et al., 1989; Ruvolo, 1997; Satta et al., 2000 Salem et al., 2003), morphological (Begun, 1992; Shoshani et al., 1996), and morphometric (Lockwood et al., 2004) studies suggest that humans and chimpanzees are more closely related to each other than either is to gorillas. If unit vector orientation was dependent upon phylogenetic constraint one would expect that one of two outcomes: all species would have statistically non-divergent mean directions or alternatively humans and chimpanzees would have statistically non-divergent mean direction, while gorillas

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57 would have statistically different mean directions. However, the results indicate, overwhelmingly, that each species has statistically different mean values for the vectors calculated, suggesting that facet and pedicle orientation are not a consequence of phylogenetic constraint. Pair-wise Comparisons and Allometry In order to address allometry, the sample included males and females of a highly dimorphic species (Gorilla gorilla) as well as two species that have similar body size (Homo sapiens and Pan troglodytes). The Gorilla test for equal means for males and females indicate that the vectors calculated have a small dispersal and statistically non-divergent mean directions. The Homo sapiens and Pan troglodytes pair-wise tests indicate that despite a similarity in body size, these vectors are statistically different between the species. While this is not conclusive, the results suggest that these vectors are insensitive to body size alone. Pair-wise Comparisons and Bone Biology and Behavior To investigate whether bone size and shape are indicative of load transmission, it is necessary to calculate the forces that these elements are subject to during normal loading. To better understand the force data, when it is obtained, pair-wise analyses were conducted for the force vectors. When considered with respect to a biomechanical model, three vectors in the analysis may be indicative of force transmission patterns. They are the pedicle major axis and axial vectors as well as the vectors that are normal to the superior and inferior facets. A simplified two-dimensional free body diagram indicates that the mean normal and axial vectors can be used as a starting point for developing models for force

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58 calculation in the posterior elements. The pair-wise comparison results indicate that for all of these vectors, each species has a statistically different mean direction. This would suggest that each species does in fact have different force transmission patterns in the posterior elements. It is somewhat surprising that the gorilla and chimpanzee vectors are statistically different as the most frequently engaged locomotor behavior for both species is knuckle-walking. However, it is not totally surprising, as ANOVA tests from Shapiro (1991,1993) demonstrated statistically different mean values for the parameters she measured. Perhaps these data indicate that facet and pedicle orientation is responsive to the species total locomotor/postural repertoire, not simply orthograde posture or the most frequently engaged in locomotion. Further, the descriptive statistics indicate the vectors had very high concentrations. This may suggest that facet and pedicle orientation is highly constrained. Current, there have been no research studies to determine the extent of morphological canalization of the primate spine. Clearly, this research is a necessary component to a function morphological analysis of the spine as it gives important information with respect to extent that development and function affect morphology. The pedicle major axis vector may also be an important indicator of force transmission patterns. The pedicles connect two compressive columns and they are assumed to be involved in force transmission and subject to stresses and strains. The mechanism of force transmission through the pedicles is not understood and consequently the nature of the stresses and strains are not understood. However, it has been suggested that one possibility is that the pedicle is being subjected to bending stresses. The long, or major axis, of the pedicle has the ability to resist the most bending force. As would be

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59 expected from the facet normal and pedicle axial results, the pair-wise tests indicate that each species has a statistically different mean direction with respect to this vector. Of note, the descriptive statistics demonstrate that Gorilla and Pan mean pedicle major axis vectors have high concentration values. However, the human pedicle major axis vectors have very low concentrations indicating that humans lack the uniformity of major axis direction that was found for the non-human primates. As the exact mechanism of force transmission is not currently known, it is not clear whether this measure has functional significance. The work of Shapiro (1991; 1993) and Sanders (1995; 1998) both demonstrated that human pedicles, especially at the last lumbar are short and wide. In humans, the minor axes-major axes ratio data for the pedicles (Appendix A) indicate that there is a general trend for lateral and superior axes to be of fairly similar in length. This would indicate that, in cross section, human pedicles may be more accurately described as round, rather than elliptical. This would make defining the major axis more difficult may explain the low concentration values. More calculation of forces acting upon the pedicle is necessary to determine if it is subjected to significant bending forces. Force Transmission in the Posterior Elements The exact manner of force transmission with individual posterior elements is not clearly understood. The most current model, the two column model of force transmission, states that the vertebral bodies and disks form one compressive column, while the posterior elements comprise a second compressive column. The pedicles are a bony bridge between the two columns. It has been hypothesized that the pedicles are subjected to significant bending stresses as a result of muscles action on the spinous or transverse processes or alternately from the posterior elements (Bogduk and Twomey,

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60 1987). Alternately, Bogduk and Twomey also suggested that the pedicles may be subjected to tension from the facets locking to prevent the vertebrae from sliding forward (Bogduk and Twomey, 1987). It has also been hypothesized that the pedicles may be subjected to compressive axial loads from the vertebral body to the lamina (Pal and Routal, 1986, 1987). There are four potential forces acting in the pedicle, torsion, compression, bending and shear. The development of a biomechanical model of the pedicles and posterior elements will help determine the extent to which these forces are significant. Sanders (1998) also hypothesized that facet spacing is important for the resistance of compressive force in the posterior elements. He suggested that the more widely spaced facets that he demonstrated in humans may create a more stable base. Intuitively this hypothesis is satisfactory. However, the development of an accurate biomechanical model is necessary to prove if this is a correct interpretation of this aspect of posterior element morphology. Feasibility of the Model With respect to the development of species models for the calculation of stresses, the results are very encouraging. The vectors necessary for the current simplified model, normal and axial, all have very high mean resultant lengths. However, the model does not incorporate the effects of muscle attachments and the subsequent stresses and strains caused by muscle action. It should be possible to calculate that loads borne by the lumbar spine, if the muscle forces can be accurately modeled and the position of the vertebrae during diverse locomotor behavior is known. More research is needed document the angles of the vertebra with respect to the gravity vector or alternately a methodology needs to be developed that can estimate these angles. This research may

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61 include estimation from radiographic information or possibly estimating the angles using known biological parameters (arm, leg, and trunk length, weight, and documented positional and locomotor behavior). However, it would be necessary to develop a measure of accuracy for such estimations. Vector Concentrations (MRL) The concentration values (MRLs) are extremely important measures of within sample variability. The measure of concentration (MRL) is a number from 0-1. A value of one indicates that the vectors are parallel and a value of zero indicates that the vectors are randomly oriented. When interpreting the validity of a mean value, it is necessary to have an approximate cut-off value of for concentration. The cut off value is subjective, and varies from project to project. However, if we had a sample of ten vectors and five had an equal lateral mean direction and five had equal mean directions perpendicular to the lateral vectors, the concentration (MRL) value would be ~0.70. For the purpose of building the current project a high degree of precision is required and the values were considered to have high concentration for values of ~0.95 and above. Facet Shape This research indicates that the species included in the sample have 5 major facet shape types as outlined in Figure 8. Thus far, no other study has addressed the issue of facet shape. It is important to note that these facet types were defined qualitatively, in the interest of having a consistent major axis, and by default, minor axis designations for each shape. It is not clear, based on this research, if these shapes are exclusive or if they correlate with other factors. More work is necessary to quantitatively document facet shape including facet curvature, which may lead to an understanding of correlation with other factors. Further it has been hypothesized that the facet major axis vectors may

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62 indicate a preferred direction of motion. Analysis of facet curvature will give additional information as to whether this measure is a reasonable approximation of the direction of facet rotation. A Fourier analysis would be appropriate for this study as orientation is an important factor.

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CHAPTER 7 CONCLUSIONS The results of this study further confirm that the lumbar spine, specifically the last lumbar vertebra, is fertile ground for future functional morphological studies. The morphology of the spinal column is critical to postural and locomotor behavior. One important goal in a functional morphological analysis on the spine is to determine if there is a relationship between the size and shape of a bony element and the loads it must withstand. The current research indicates that generalized species models can be developed. The vectors that are key to the development of these models, normal and axial, were demonstrated to have high concentrations and therefore can form the basis for valid models for each species. Further, these high concentration values may indicate that this measure of facet and pedicle orientation is not sensitive to individual behavior. Rather, the orientation is very consistent within species. Pair-wise tests indicate that the human, gorillas, and chimpanzees have unique mean directions for these vectors, with the exception of the unrotated human and gorilla. This test demonstrated that these species have statistically equal mean directions. Previously undocumented vertebral measures, the major axes of the facets and pedicles, were also addressed. The facets joints allow, or alternatively constrict, motion within the spinal column. The direction of this vector may indicate the plane in which the facet allows the most rotation. The statistical analyses overwhelmingly indicate that humans, gorilla and chimpanzees have unique mean directions with respect to these vectors. There were a few exceptions. For both the unrotated and rotated human vectors 63

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64 and gorilla vectors, pair-wise tests indicate these two species have statistically non-divergent mean directions. Another exception was the left superior facet major axis vectors of gorilla and chimpanzees. The left, though notably not the right, vectors of these species have non-divergent mean directions. Finally, the unrotated human and chimpanzee pair-wise test of the right inferior major axis vectors, though not the left, indicate that these two species have statistically non-divergent mean directions. The functional implications of these data are not readily transparent. Interestingly, the results of the major axis facet analyses demonstrate a low concentration of the facet major axes vectors. The low concentrations indicate within species variability for this measure. This seems to suggest that facet shape is responsive and plastic in the individual, though it is not clear if it is responsive to locomotor and postural behaviors. The pedicle major axis vectors were measured, as this directional vector may indicate how the pedicle is oriented to resist the most bending force. Results of the statistical analyses of pedicle major axis vectors are also very interesting in that they indicate the species have statistically different mean directions with respect to this measure. Finally, with regard to concentration, gorillas and chimpanzees have extremely high concentrations. However, human major axis pedicle vectors have extremely low concentration values. These results may indicate that the major axis vector is not measuring function as intended. The minor-axis major axis ratio calculation (Appendix A) confirms that there is a general trend in humans for the medial and lateral axes to be fairly equal in length. Further, the shape of the pedicle may not be ideally suited, as is assumed, to a specific type of force. The trend in orthograde primates to have short, wide

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65 pedicles may indicate that the response to increased compressive loads is constrained by function or neurological considerations. Finally, the results of this analysis demonstrate that, given certain assumptions, it is possible to develop species-specific models to calculate forces in the spinal column.

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APPENDIX A MAJOR AXES-MINOR AXIS RATIO The tables below list the major axis-minor axis ratio for each specimen. The value is calculated via the formula log (superior-inferior axis) log(lateral-medial axis). This value is the base ten logarithm of the ratio of the major to minor axis. A positive value indicated that the superior-inferior axis is longer, while a negative value indicated that the lateral-medial axis is the longer of the two axes. Homo major axis-minor axis ratio. Homo Right Pedicle Left Pedicle Right Sup. Facet Left Sup. Facet Right Inf. Facet Left Inf. Facet 1 0.154004028 0.122777501 -0.076485421 -0.083315591 -0.036990687 0.085841076 2 -0.168245483 -0.024223181 -0.034702602 0.045502107 -0.035288941 0.200988587 3 -0.022722562 -0.003593294 -0.10217809 -0.137450031 0.133044418 0.143808298 4 NA NA 0.191864849 -0.181905303 0.108410102 0.144166515 5 -0.19587025 -0.008572706 -0.099681284 -0.10488135 0.139241109 0.19679049 6 -0.008060033 -0.049878814 0.189004265 0.090310548 0.136960776 -0.073634987 7 -0.205698218 -0.120912741 -0.176895834 -0.059726442 -0.111932747 -0.115985709 8 0.011222033 0.029822045 -0.083507239 -0.047776674 0.211911927 0.006849671 9 0.165669964 -0.017378339 -0.061136929 0.066483798 0.209886421 0.234550164 10 -0.14308604 -0.148556922 -0.029226677 -0.16739032 0.115877559 0.222156199 11 -0.023937285 -0.108195036 -0.019557449 -0.150507035 0.14289058 0.162482517 12 0.021455712 -0.01708491 -0.048078851 -0.047295453 -0.120152347 -0.13369963 13 NA NA -0.054751394 -0.062539143 0.111309967 0.31360054 14 -0.153090849 -0.18451456 -0.009755018 -0.013468729 0.036861323 0.105409743 15 -0.157696731 -0.1701001 NA -9.11852E-05 0.05109364 -0.058043173 16 -0.093144815 -0.124482014 -0.183166616 -0.162388367 -0.194479701 0.02441287 17 > 0.001 -0.065732342 NA -0.228174563 0.118806321 0.076225337 18 0.017244966 -0.03146184 0.064152452 0.081666506 0.209628657 0.078556068 19 0.038259755 -0.047534837 NA -0.078808292 0.073729481 0.082819371 Gorilla major axis-minor axis ratio. Gorilla Right Pedicle Left Pedicle Right Sup. Facet Left Sup. Facet Right Inf. Facet Left Inf. Facet 1 0.060167877 0.067643681 -0.064164788 -0.217178634 -0.222365457 0.038108348 2 0.127629537 0.223340381 0.130056818 0.218542269 0.056772197 0.112606279 3 0.375613123 0.320078963 -0.357718716 -0.182414896 -0.114841461 0.043987822 4 0.222651315 0.246796519 0.256098669 0.145279049 0.167937031 0.159292774 5 0.287470785 0.182472506 0.138466694 0.121439514 -0.003561981 0.081388742 66

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67 Gorilla. Continued. Gorilla Right Pedicle Left Pedicle Right Sup. Facet Left Sup. Facet Right Inf. Facet Left Inf. Facet 6 0.12936712 0.145883044 -0.084745501 -0.039511551 0.074280664 0.120548935 7 0.141641702 0.222122531 -0.127360023 -0.134062605 0.026483505 NA 8 0.081080411 0.119080867 -0.192925955 -0.27067006 0.081999694 -0.082457784 9 0.119551448 0.164047131 0.251696202 -0.226917138 -0.159622372 -0.204039531 10 0.137548209 NA 0.224990253 0.079112759 0.321164729 0.26279744 11 0.101727584 0.148862339 -0.121686273 -0.146663289 0.047584429 -0.02817221 12 0.118749878 0.084090499 -0.131024216 -0.137255376 -0.158394163 -0.09308975 13 0.138983315 NA 0.165514375 NA 0.123303367 NA 14 0.137558074 0.223620517 -0.088671365 0.08403637 -0.227318767 0.190659229 15 0.134192423 0.086933583 -0.124312935 0.136050062 0.03654971 0.067166727 16 0.300352283 0.216250665 -0.203508887 -0.010258393 0.029210435 NA 17 0.248338172 0.176788136 -0.147019214 -0.060164388 0.166190544 0.058113367 Pan minor axes-major axes ratio. Pan Right Pedicle Left Pedicle Right Sup. Facet Left Sup. Facet Right Inf. Facet Left Inf. Facet 1 0.22183874 0.280385713 0.131302338 0.141002239 0.144568675 0.106252181 2 0.255433552 0.200764578 0.124194868 0.102431458 0.061718878 0.165303483 3 0.405821594 0.341866563 0.155701514 0.154558942 0.057895504 0.145964859 4 0.326159647 0.281167648 0.039289209 -0.091357267 0.194611474 0.283394472 5 0.259558909 0.212682574 0.085998391 -0.024546424 0.219947716 0.231392834 6 0.106969764 0.103704398 0.110323302 0.217555913 0.000779009 -0.044905124 7 0.18342086 0.199781293 0.154620771 0.121204753 0.154995693 0.281804273 8 0.253017076 0.262167926 -0.016198363 -0.036284862 0.208380764 0.199520161 9 0.28797178 0.189091318 0.15263867 0.090480153 0.040798182 -0.162966207 10 0.184171769 0.252979075 0.10504127 0.038045831 0.013012161 0.047561334 11 0.084186624 0.171771641 0.063849173 -0.136923196 0.111160804 0.023174003 12 0.383823739 0.345946783 0.097182104 0.096102229 -0.088161714 0.049324453 13 0.277177855 0.259971814 0.190514199 0.072523392 0.17743898 0.061610458 14 0.151779711 0.192508569 0.086232445 0.033836483 0.189028532 0.215425648 15 0.210597829 0.210260155 0.14357679 0.10951732 0.18732476 0.141981253 16 0.321865888 0.317752038 0.122224054 0.146863685 0.27884123 0.166475019 17 0.257802873 0.222618699 0.052256937 -0.09097867 0.131903682 0.161802253 18 0.272826335 0.307301703 -0.080063049 -0.049644983 0.110340009 0.143839923 Pongo minor axes-major axes ratio. Pongo Right Pedicle Left Pedicle Right Sup. Facet Left Sup. Facet Right Inf. Facet Left Inf. Facet 1 0.092599566 NA 0.062045738 0.027134765 0.06229625 0.151568419 2 -1.382393424 -1.354590371 0.070912757 0.02605175 0.100561634 0.18084306 3 0.381737882 -1.195793895 0.211296274 0.241418164 0.162105728 0.044230445 4 -0.062836729 -0.048311536 0.30678293 0.266221028 0.173352109 0.200495819 5 0.135336062 0.252095092 0.234525199 0.104844507 0.161262349 0.323901507

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APPENDIX B ANGLE ERROR CALCULATION (DEGREES) Homo. Angle Error Calculations Mean Std. dev. n= N ormal to Superior Facet Righ t 1.38239 0.882528 16 N ormal to Superior Facet Lef t 1.694435 0.677398 19 N ormal to Inferior Facet Righ t 1.503308 0.777949 19 N or m al to Inferior Facet Left 1.369888 0.788105 19 Axial Direction of Pedicle Right 2.326571 1.198554 17 Axial Direction of Pedicle Left 2.047828 1.010862 17 All 1.712785 0.944116 107 Gorilla. Angle Error Calculations Mean Std. dev. n= N ormal to Superior Facet Right 1.517525 0.886495 17 N ormal to Superior Facet Lef t 2.195902 1.407393 16 N ormal to Inferior Facet Righ t 2.579414 1.632107 17 N ormal to Inferior Facet Lef t 1.768057 1.152024 14 Axial Direction of Pedicle Right 2.368777 1.098082 17 Axial Direction of Pedicle Left 1.664009 0.992631 15 All 2.028797 1.257412 96 Pan. Angle Error Calculations Mean Std. dev. n= N ormal to Superior Facet Righ t 1.496232 1.033072 18 N ormal to Superior Facet Lef t 2.067263 0.978035 18 N ormal to Inferior Facet Right 1.666827 1.058175 18 N ormal to Inferior Facet Lef t 3.829464 1.758387 18 Axial Direction of Pedicle Right 1.939254 0.585871 18 Axial Direction of Pedicle Left 2.229736 0.786581 18 All 1.879014 0.973882 108 Pongo. Angle Error Calculations mean Std. dev. n= N ormal to Superior Facet Righ t 1.258144 0.407892 5 N ormal to Superior Facet Lef t 1.676243 0.662998 5 N ormal to Inferior Facet Righ t 2.905199 1.387219 5 N ormal to Inferior Facet Lef t 2.034397 0.886031 5 68

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69 Pongo. Continued P ongo pygmaeus Angle Error Calculations mean Std. dev. n= Axial Direction of Pedicle Right 1.437808 1.166763 5 Axial Direction of Pedicle Left 1.507351 0.693615 4 All 1.813392 1.018929 29

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APPENDIX C RAW VECTOR DATA TABLES Tables 8-1 through 8-4 contain the raw vector data. Key: A. = axial AMNH#-American Museum of Natural History Inventory Number Dir. = direction Inf. = inferior L. = left MA = major axis N. = normal Ped.= pedicle Sup. = superior Yo = years old Homo. R. = right Raw Vector Data AMNH# 98-46, 82 yo, Wt. Male AMNH# 98.49, 74yo, Wt. Male AMNH# 98-52, 60yo, Wt. Male Description of Vector x y z x y z x y z MA of Ped. R. 0.217647 0.25191 0.94296 0.31027 -0.7929 -0.52448 0.40255 -0.85729 -0.32095 MA of Ped. L. 0.139378 -0.3266 0.93484 0.49412 0.78444 -0.37484 0.37423 0.81527 -0.44191 MA of Sup. Facet R. -0.485039 -0.8393 0.24558 -0.5126 -0.7847 0.34858 -0.67793 -0.66194 0.31977 MA of Sup. Facet L. -0.485757 0.78904 0.37611 0.39183 0.55654 0.73262 -0.53428 0.68735 0.49202 MA of Inf. Facet R. -0.117477 -0.7618 0.63709 -0.21229 -0.8701 0.44491 0.79758 0.08984 0.59648 MA of Inf. Facet L. 0.874694 -0.1862 0.44749 0.71831 -0.6354 0.28338 0.89112 -0.17631 0.41812 N to Sup. Facet R. -0.67373 0.53768 0.50694 -0.65182 0.61989 0.43688 -0.51864 0.73894 0.43009 N to Sup. Facet L. -0.587696 -0.6133 0.52768 -0.81089 -0.1673 0.56077 -0.47226 -0.72547 0.50066 N to Inf. Facet R. 0.401513 -0.6232 -0.67113 0.6214 -0.4716 -0.62569 0.44479 -0.75555 -0.48095 N to Inf. Facet L. 0.456574 0.62634 -0.63185 0.59404 0.34813 -0.7252 0.38613 0.77864 -0.4946 A. Dir. Of Ped. R. -0.909646 -0.2979 0.28953 -0.80119 -0.5151 0.30466 -0.87403 -0.46417 0.14361 A. Dir. Of Ped. L. -0.864222 0.42075 0.27584 -0.62879 0.62019 0.46903 -0.88839 0.45184 0.08125 AMNH# 98-102, 67yo, Wt. Male AMNH# 98-119, 44yo, Wt. Male AMNH# 98-125, 50yo, Wt. Male Description of Vector x y z x y z x y z MA of Ped. R. 0.34507 -0.8814 -0.32258 0.38664 -0.8305 -0.40099 0.45885 -0.76618 -0.4499 MA of Ped. L. 0.594566 0.78605 -0.16917 0.51345 0.77332 -0.37196 0.53888 0.78943 -0.29395 MA of Sup. Facet R. -0.405535 -0.7452 0.52939 0.18703 -0.4657 0.86498 -0.62033 -0.70966 0.33402 MA of Sup. Facet L. -0.367225 0.72564 0.58188 -0.07188 0.51237 0.85575 -0.58076 0.75949 0.29307 MA of Inf. Facet R. 0.984399 0.16947 0.04731 0.76035 0.05166 0.64745 -0.71522 -0.56723 0.4083 MA of Inf. Facet L. 0.898105 -0.4298 0.09324 -0.54579 0.7152 0.43658 -0.49445 0.75331 0.43364 N to Sup. Facet R. -0.591147 0.65554 0.46991 -0.60849 0.63634 0.47414 -0.56898 0.70028 0.43113 70

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71 Homo. Continued AMNH# 98-102, 67yo, Wt. Male AMNH# 98-119, 44yo, Wt. Male AMNH# 98-125, 50yo, Wt. Male Description of Vector x y z y Raw Vector Data x z x y z N to Sup. Face t L. -0.486314 -0.6831 0.54491 -0.44223 -0.7854 0.4331 -0.64243 -0.64869 0.40801 N to Inf. Facet R. 0.175937 -0.9451 -0.27526 0.4765 -0.7218 -0.50199 0.41403 -0.81454 -0.40634 N to Inf. Facet L. 0.432632 0.82535 -0.36281 0.55787 0.69892 -0.44754 0.57304 0.65763 -0.48903 A. Dir. Of Ped. R. -0.694192 -0.471 0.54431 -0.85747 -0.4838 0.17519 -0.75952 -0.601 0.24886 A. Dir. Of Ped. L -0.673776 0.6019 0.42865 -0.68503 0.63044 0.36508 -0.70487 0.61364 0.35582 AMNH# 98-59, 70yo, Wt. Male AMNH# 98-135, 52yo, Wt. Male AMNH#, 98-147, 50yo, Wt. Male Description of Vector x y z x y z x y z MA of Ped. R. 0.04487 0.20633 0.97745 -0.03115 0.20715 0.97781 0.35 -0.85939 -0.37277 MA of Ped. L. 0.165192 -0.2801 0.94565 0.18434 0.98094 -0.06153 0.41043 0.85439 -0.31868 MA of Sup. Facet R. -0.529928 -0.6982 0.48128 -0.63135 -0.6836 0.3662 -0.85258 -0.45569 -0.25582 MA of Sup. Facet L. -0.336754 0.79087 0.511 0.61893 0.01023 0.78538 -0.66688 0.74373 -0.04634 MA of Inf. Facet R. 0.91116 0.39277 0.12458 0.75998 0.63875 0.12012 0.925 0.36061 -0.11974 MA of Inf. Facet L. 0.823772 -0.1404 0.54925 0.85869 -0.4262 0.28456 0.95697 -0.29019 0.00202 N to Sup. Facet R. -0.467238 0.71401 0.52142 -0.38263 0.68531 0.61963 -0.45482 0.40596 0.79267 N to Sup. Facet L. -0.57803 -0.602 0.55085 -0.51364 -0.7512 0.41456 -0.55544 -0.45466 0.69626 N to Inf. Facet R. 0.406732 -0.8089 -0.42462 0.55388 -0.5398 -0.63391 0.207 -0.7425 -0.63706 N to Inf. Facet L. 0.442936 0.76414 -0.46894 0.5125 0.71625 -0.47365 0.23877 0.78339 -0.57384 A. Dir. Of Ped. R. -0.907985 -0.3996 0.12603 -0.97357 -0.2277 0.01723 -0.89528 -0.42397 0.13683 A. Dir. Of Ped. L. -0.780696 0.54878 0.29892 -0.9821 0.18137 -0.05082 -0.77604 0.51078 0.36995 AMNH# 98-174, 56yo, Bk. Male AMNH# 98-217, 71yo, Wt. Male AMNH# 98-246, 47yo, Bk. Male Description of Vector x y z x y z x y z MA of Ped. R. 0.383857 -0.8284 -0.40792 0.29838 0.05687 0.95275 NA NA NA MA of Ped. L. 0.466618 0.82837 -0.30996 0.52961 0.81866 -0.22204 NA NA NA MA of Sup. Facet R. -0.516952 -0.7387 0.43258 -0.2245 -0.9577 0.18016 -0.55718 -0.72727 0.40079 MA of Sup. Facet L. -0.543826 0.69103 0.47617 -0.31002 0.92426 0.22279 -0.90047 0.30067 -0.31425 MA of Inf. Facet R. 0.891714 0.30689 0.33266 -0.68401 -0.7292 -0.02061 0.82642 0.23214 0.51297 MA of Inf. Facet L. 0.849028 -0.5207 0.08964 -0.32292 0.87031 0.37185 0.8704 -0.24779 0.42545 N to Sup. Facet R. -0.578245 0.67395 0.45981 -0.74864 0.28785 0.59723 -0.52629 0.68261 0.50701 N to Sup. Facet L. -0.576365 -0.72 0.38658 -0.78236 -0.3812 0.49258 -0.37877 -0.89725 0.22689 N to Inf. Facet R. 0.452189 -0.6353 -0.62599 0.6311 -0.5774 -0.51806 0.49522 -0.7332 -0.46603 N to Inf. Facet L. 0.487857 0.70746 -0.51137 0.59644 0.4922 -0.63403 0.38579 0.88012 -0.27667 A. Dir. Of Ped. R. -0.812599 -0.5129 0.27686 -0.85026 -0.4377 0.29241 NA NA NA A. Dir. Of Ped. L. -0.663831 0.55959 0.49617 -0.72722 0.57297 0.37796 NA NA NA AMNH# 98-295, 42yo, Wt. Male AMNH# 98-320, 45yo, Wt. Male AMNH# 98-324, 49yo, Wt. Male Description of Vector x y z x y z x y z MA of Ped. R. 0.43254 -0.8085 -0.39912 0.44223 -0.8059 -0.39373 0.39673 -0.86678 -0.30218 MA of Ped. L. 0.496524 0.78714 -0.36588 0.43473 0.83522 -0.33678 0.54002 0.77625 -0.32528 MA of Sup. Facet R. -0.467581 -0.8167 0.3381 -0.332 -0.8498 0.40951 MA of Sup. Facet L. -0.805779 0.51244 -0.29685 -0.05671 0.82818 0.55759 -0.36091 0.679 0.6393 MA of Inf. Facet R. 0.809175 0.51062 0.2907 -0.70969 -0.7037 0.03348 0.90868 0.41132 0.07151 MA of Inf. Facet L. 0.687704 -0.7234 0.06188 0.78275 -0.4032 0.47403 0.87976 -0.47426 0.03319 N to Sup. Facet R. -0.622415 0.57579 0.53015 -0.69561 0.51378 0.50216 N to Sup. Facet L. -0.591365 -0.7231 0.35696 -0.72195 -0.4198 0.55007 -0.62607 -0.68448 0.37354 N to Inf. Facet R. 0.573977 -0.5811 -0.57691 0.62129 -0.6476 -0.44121 0.3795 -0.74241 -0.55209

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72 Homo. Continued Raw Vector Data AMNH# 98-295, 42yo, Wt. Male AMNH# 98-320, 45yo, Wt. Male AMNH# 98-324, 49yo, Wt. Male Description of Vector x y z x y z x y z N to Inf. Facet L. 0.640971 0.56494 -0.51962 0.60571 0.66852 -0.4315 0.45298 0.81498 -0.36141 A. Dir. of Ped. R. -0.727338 -0.5745 0.37544 -0.77275 -0.5652 0.28884 -0.83245 -0.47845 0.27948 A. Dir. of Ped. L. -0.654664 0.61637 0.43761 -0.70478 0.54834 0.45012 -0.62585 0.62877 0.46148 AMNH# 98-327, 63yo, Wt. Male AMNH# 98-333, 58yo, Wt. Male AMNH# 98-99, 47yo, Wt. Female Description of Vector x y z x y z x y z MA of Ped. R. 0.277084 0.16413 0.94672 0.19631 0.36932 0.90833 NA NA NA MA of Ped. L. 0.377997 0.83091 -0.4083 0.38899 0.86122 -0.32709 NA NA NA MA of Sup. Facet R. 0.331514 -0.3078 0.89183 NA NA NA 0.18258 -0.4973 0.84815 MA of Sup. Facet L. 0.210123 0.47817 0.85276 -0.46559 0.82031 0.33214 -0.08619 0.76112 0.64286 MA of Inf. Facet R. 0.917028 0.35515 0.18146 0.58915 -0.0683 0.80514 0.87051 0.27316 0.40939 MA of Inf. Facet L. 0.848145 -0.5017 0.1701 0.81179 -0.2149 0.54298 0.75483 -0.36555 0.54462 N to Sup. Facet R. -0.535314 0.71702 0.44645 NA NA NA -0.08149 0.85203 0.51712 N to Sup. Facet L. -0.770911 -0.4554 0.44532 -0.69258 -0.5714 0.44031 -0.71052 -0.49928 0.49586 N to Inf. Facet R. 0.395497 -0.7512 -0.52852 0.47233 -0.7794 -0.4117 0.42671 -0.83339 -0.35126 N to Inf. Facet L. 0.517115 0.71431 -0.47154 0.51166 0.70985 -0.48406 0.64054 0.58956 -0.49207 A. Dir. of Ped. R. -0.80536 -0.4977 0.322 -0.77382 -0.5106 0.37484 NA NA NA A. Dir. of Ped. L. -0.76068 0.53013 0.37461 -0.8076 0.48962 0.32872 NA NA NA AMNH# 98-315, 65yo Bk. Female Description of Vector x y z MA of Ped. R. 0.448193 -0.7965 -0.40585 MA of Ped. L. 0.472769 0.80337 -0.36206 MA of Sup. Facet R. NA NA NA MA of Sup. Facet L. -0.402228 0.73841 0.54127 MA of Inf. Facet R. 0.757824 0.28566 0.5866 MA of Inf. Facet L. -0.35049 0.79254 0.49903 N to Sup. Facet R. NA NA NA N to Sup. Facet L. -0.621201 -0.6544 0.43112 N to Inf. Facet R. 0.644962 -0.4639 -0.60732 N to Inf. Facet L. 0.552185 0.60524 -0.57339 A. Dir. of Ped. R. -0.639858 -0.6029 0.47657 A. Dir. of Ped. L. -0.649357 0.59537 0.47314 Gorilla. Raw Vector Data AMNH# 90289, Male AMNH# 167336, Male AMNH# 167335, Male Description of Vector x y z x y z x y z MA of Ped. R. 0.123004 0.08752 0.98854 0.10109 0.12651 0.9868 0.29847 0.11771 0.94713 MA of Ped. L. 0.206535 -0.2949 0.93296 0.14711 -0.2724 0.95087 0.25925 -0.4447 0.85734 MA of Sup. Facet R. -0.555758 -0.7772 0.29523 -0.46005 -0.8231 0.33309 -0.14254 -0.80451 0.57657 MA of Sup. Facet L. -0.646692 0.55173 0.52668 -0.33819 0.79438 0.50457 -0.13565 0.41819 0.89818 MA of Inf. Facet R. -0.822841 -0.5682 -0.01124 0.4056 0.02639 0.91367 0.44876 -0.15836 0.87951 MA of Inf. Facet L. 0.473646 -0.2999 0.82807 0.14267 0.0073 0.98974 NA NA NA N to Sup. Facet R. -0.664696 0.62868 0.40366 -0.74093 0.56258 0.36677 -0.69033 0.49826 0.52457 N to Sup. Facet L. -0.531501 -0.8212 0.20765 -0.73432 -0.5581 0.38644 -0.58373 -0.76624 0.2686

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73 Gorilla. Continued. Raw Vector Data AMNH# 90289, Male AMNH# 167336, Male AMNH# 167335, Male Description of Vector x y z x y z x y z N to Inf. Facet R. 0.551469 -0.7936 -0.25712 0.45616 -0.8721 -0.17732 0.43927 -0.81798 -0.37141 N to Inf. Facet L. 0.674831 0.72775 -0.12239 0.52585 0.84661 -0.08205 NA NA NA A. Dir. of Ped. R. -0.950125 -0.2773 0.14277 -0.90886 -0.3917 0.14333 -0.89962 -0.29671 0.32038 A. Dir. of Ped. L. -0.866873 0.38703 0.31423 -0.9409 0.25795 0.21946 -0.82083 0.36634 0.43822 AMNH# 90290, Male AMNH# 201460, Male AMNH# 81651, Male Description of Vector x y z x y z x y z MA of Ped. R. 0.206148 0.24282 0.94791 0.40205 0.05745 0.91381 0.57253 0.15358 0.80538 MA of Ped. L. 0.22276 -0.3029 0.92663 0.39388 -0.3348 0.85604 NA NA NA MA of Sup. Facet R. -0.670153 -0.5171 0.5325 0.30906 -0.4469 0.83948 0.48306 -0.59782 0.63973 MA of Sup. Facet L. -0.563011 0.57459 0.59402 -0.51366 0.74089 0.43272 0.59325 0.67882 0.43274 MA of Inf. Facet R. 0.343504 -0.0753 0.93613 -0.56101 -0.6477 0.51555 0.4168 -0.43828 0.79636 MA of Inf. Facet L. -0.78381 0.45357 0.42417 -0.14997 0.47247 0.8685 0.24969 0.25685 0.93364 N to Sup. Facet R. -0.391525 0.85576 0.33819 -0.47883 0.68953 0.54339 -0.30567 0.56952 0.76303 N to Sup. Facet L. -0.510736 -0.807 0.29652 -0.49136 -0.6675 0.55952 -0.5715 -0.02345 0.82026 N to Inf. Facet R. 0.177614 -0.9736 -0.14343 0.51316 -0.7608 -0.39734 0.52647 -0.5978 -0.60454 N to Inf. Facet L. 0.44211 0.88723 -0.13177 0.44599 0.81631 -0.36706 0.57709 0.73478 -0.35648 A. Dir. of Ped. R. -0.932207 -0.2458 0.26569 -0.81059 -0.4418 0.38441 -0.76339 -0.25848 0.59197 A. Dir. of Ped. L. -0.732501 0.57522 0.3641 -0.78974 0.35326 0.50152 NA NA NA AMNH# 214103, Male AMNH# 54089, Male AMNH# 54090, Male Description of Vector x y y x y z x y z MA of Ped. R. 0.332068 0.1249 0.93495 0.30302 0.14124 0.94246 0.40936 0.02153 0.91212 MA of Ped. L. 0.247312 -0.1598 0.95566 NA NA NA 0.49471 -0.14547 0.8568 MA of Sup. Facet R. -0.40226 -0.6227 0.67119 0.00195 -0.595 0.8037 -0.71041 -0.63294 0.30775 MA of Sup. Facet L. 0.018459 0.72651 0.68691 NA NA NA 0.8147 -0.34713 0.46449 MA of Inf. Facet R. -0.820166 -0.2488 0.51518 0.36189 -0.1297 0.92316 0.50503 0.24162 0.82859 MA of Inf. Facet L. -0.665585 0.5112 0.54376 NA NA NA 0.59929 -0.46842 0.64918 N to Sup. Facet R. -0.44516 0.77364 0.4509 -0.53632 0.67771 0.50306 -0.49568 0.76039 0.41965 N to Sup. Facet L. -0.689308 -0.4884 0.53508 NA NA NA -0.53841 -0.75029 0.38364 N to Inf. Facet R. 0.177021 -0.9667 -0.18506 0.40616 -0.8694 -0.28132 0.85814 -0.2433 -0.4521 N to Inf. Facet L. 0.473052 0.8525 -0.22241 NA NA NA 0.80051 0.34567 -0.48958 A. Dir. of Ped. R. -0.937192 -0.0685 0.34202 -0.95085 0.11102 0.28908 -0.90374 0.1468 0.40213 A. Dir. of Ped. L. -0.930957 0.23424 0.2801 NA NA NA -0.8658 0.00285 0.50039 AMNH# 99.1/1577, Male AMNH# 54327, Female AMNH# 167340, Female Description of Vector x y z x y z x y z MA of Ped. R. 0.184267 0.30092 0.93568 0.25612 0.30784 0.91632 0.13296 0.07039 0.98862 MA of Ped. L. 0.187111 -0.37 0.90998 0.06217 -0.2735 0.95985 0.19553 -0.25347 0.94738 MA of Sup. Facet R. -0.61885 -0.6417 0.45301 0.07148 -0.4098 0.90939 -0.54127 -0.74053 0.39829 MA of Sup. Facet L. -0.307662 0.64625 0.69836 -0.22834 0.70307 0.67347 -0.63225 0.53689 0.55858 MA of Inf. Facet R. 0.624217 -0.1385 0.76888 0.67167 0.03049 0.74022 -0.81519 -0.55679 0.15951 MA of Inf. Facet L. NA NA NA 0.46998 0.09141 0.87793 0.53933 -0.14749 0.82908 N to Sup. Facet R. -0.485773 0.76586 0.42129 -0.50795 0.76969 0.38674 -0.66442 0.66698 0.33717 N to Sup. Facet L. -0.436482 -0.748 0.49993 -0.5637 -0.6595 0.49733 -0.50014 -0.83345 0.23499 N to Inf. Facet R. 0.435759 -0.7551 -0.48979 0.45684 -0.8036 -0.38143 0.51867 -0.82435 -0.22679 N to Inf. Facet L. NA NA NA 0.46851 0.81712 -0.33589 0.37836 0.92201 -0.0821

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74 Gorilla. Continued Raw Vector Data AMNH# 99.1/1577, Male AMNH# 54327, Female AMNH# 167340, Female Description of Vector x y z x y z x y z A. Dir. of Ped. R. -0.959499 -0.1513 0.23762 -0.76732 -0.5118 0.3864 -0.90645 -0.39476 0.15002 A. Dir. of Ped. L. -0.850838 0.40194 0.3384 -0.74409 0.62825 0.22724 -0.85697 0.42554 0.29072 AMNH# 167339, Female AMNH# 167337, Female AMNH# 81652, Female Description of Vector x y z x y z x y z MA of Ped. R. 0.034384 0.18516 0.98211 0.18626 0.17048 0.9676 0.31645 0.17666 0.93202 MA of Ped. L. -0.003492 -0.2988 0.95432 0.33053 -0.257 0.90814 0.3329 -0.23795 0.91245 MA of Sup. Facet R. -0225022 -0.4334 0.87265 0.52927 0.0509 0.84693 -0.47442 -0.73073 0.49088 MA of Sup. Facet L. 0.008223 0.27295 0.96199 0.40538 0.1992 0.89218 -0.46925 0.63876 0.60975 MA of Inf. Facet R. 0.298767 -0.2569 0.91911 -0.69438 -0.4488 0.56255 0.3445 -0.1082 0.93253 MA of Inf. Facet L. 0.410187 0.02381 0.91169 -0.15653 0.14635 0.97677 -0.93425 0.30394 0.18653 N to Sup. Facet R. -.422773 0.85035 0.31333 -0.72611 0.54354 0.4211 -0.57304 0.67966 0.45792 N to Sup. Facet L. -.568668 -0.7926 0.22003 -0.78832 -0.418 0.4515 -0.45294 -0.76684 0.45475 N to Inf. Facet R. 0.274102 -0.8994 -0.34046 0.45621 -0.8791 -0.13814 0.30934 -0.92478 -0.22157 N to Inf. Facet L. 0.493566 0.83482 -0.24387 0.5116 0.85796 -0.04656 0.25688 0.93638 -0.23916 A. Dir. of Ped. R. -.822847 -0.5525 0.13297 -0.97024 -0.1232 0.20847 -0.86766 -0.34325 0.35966 A. Dir. of Ped. L. -.745638 0.63669 0.19659 -0.68358 0.59826 0.4181 -0.85958 0.32126 0.39739 AMNH# 54091, Female AMNH# 99.1/1578, Female Description of Vector x y z x y z MA of Ped. R. 0.046485 0.19536 0.97963 0.37877 0.3591 0.85299 MA of Ped. L. 0.132641 -0.2112 0.9684 0.35765 -0.1945 0.91337 MA of Sup. Facet R. -0.697589 -0.6704 0.25295 -0.71269 -0.4339 0.5512 MA of Sup. Facet L. 0.338823 0.11258 0.93409 -0.74712 0.5054 0.43174 MA of Inf. Facet R. -0.235297 -0.8761 0.42092 0.3751 -0.0257 0.92663 MA of Inf. Facet L. 0.355317 -0.0754 0.9317 0.4526 -0.0829 0.88785 N to Sup. Facet R. -0.617502 0.74155 0.26228 -0.32785 0.90069 0.28509 N to Sup. Facet L. -0.523832 -0.8021 0.28669 -0.40422 -0.8611 0.30848 N to Inf. Facet R. 0.872306 -0.3813 -0.30604 0.61135 -0.7445 -0.26814 N to Inf. Facet L. 0.802694 0.53536 -0.26282 0.74361 0.58461 -0.32447 A. Dir. of Ped. R. -0.997251 -0.0476 0.05681 -0.92408 0.09577 0.37002 A. Dir. of Ped. L. -0.946888 0.26175 0.18679 -0.92985 0.01635 0.36758 Pan. Raw Vector Data, Pan Sample AMNH# 51377, Male AMNH# 51278, Male AMNH# 174861, Male Description of Vector x y z x y z x y z MA of Ped. R. 0.437118 0.279958 0.854723 0.385527 0.167147 0.907431 0.341573 0.157836 0.926507 MA of Ped. L. 0.380015 -0.24063 0.893133 0.317779 -0.24027 0.917217 0.317087 -0.35364 0.879997 MA of Sup. Facet R. 0.236742 -0.20905 0.948816 0.036721 -0.36084 0.931903 0.058365 -0.28476 0.956821 MA of Sup. Facet L. 0.183441 0.225498 0.956818 0.023484 0.279007 0.960002 -0.16172 0.395314 0.904198 MA of Inf. Facet R. 0.629024 -0.01281 0.777281 0.759219 0.015771 0.650644 0.515149 0.000552 0.8571 MA of Inf. Facet L. 0.506402 0.01236 0.862209 0.690194 -0.10721 0.715638 0.558144 -0.07621 0.826237 N to Sup. Facet R. -0.45975 0.836215 0.298956 -0.34446 0.870813 0.350763 -0.37202 0.883214 0.285544 N to Sup. Facet L. -0.504355 -0.81387 0.288505 -0.30254 -0.91326 0.272822 -0.45914 -0.84119 0.285647 N to Inf. Facet R. 0.20357 -0.96226 -0.18061 0.311 -0.88698 -0.3414 0.301933 -0.93601 -0.18087

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75 Pan. Continued Raw Vector Data AMNH# 51377, Male AMNH# 51278, Male AMNH# 174861, Male Description of Vector x y z x y z x y z N to Inf. Facet L. 0.320591 0.925524 -0.20156 0.358357 0.909816 -0.20932 0.333991 0.932175 -0.13964 A. Dir. of Ped. R. -0.846491 -0.19309 0.496154 -0.92158 0.021486 0.387583 -0.83146 -0.40885 0.376183 A. Dir. of Ped. L. -0.852023 0.284802 0.439255 -0.85708 0.340913 0.386249 -0.76193 0.457529 0.458408 AMNH# 51379, Male AMNH# 51202, Male AMNH# 51392, Male Description of Vector x y z x y z x y z MA of Ped. R. 0.307012 0.266302 0.913689 0.317965 0.15459 0.935414 0.264879 0.221105 0.93859 MA of Ped. L. 0.325353 -0.24053 0.914489 0.321483 -0.25284 0.912536 0.340492 -0.37802 0.860908 MA of Sup. Facet R. 0.216414 -0.32202 0.921665 0.100986 -0.28317 0.953739 0.022741 -0.28273 0.958929 MA of Sup. Facet L. 0.078224 0.305897 0.948846 0.466869 0.192864 0.863039 -0.88553 0.31371 -0.34266 MA of Inf. Facet R. 0.367143 -0.1811 0.912365 0.541253 -0.02866 0.840371 0.695159 -0.02214 0.718515 MA of Inf. Facet L. 0.519744 0.107518 0.84753 0.568858 0.004627 0.822423 0.704297 -0.02241 0.709552 N to Sup. Facet R. -0.101073 0.931574 0.349218 -0.40142 0.865544 0.299485 -0.21273 0.935843 0.28097 N to Sup. Facet L. -0.254778 -0.91402 0.315675 -0.44836 -0.78957 0.418989 -0.44352 -0.7904 0.422562 N to Inf. Facet R. 0.288291 -0.91041 -0.29672 0.259326 -0.94501 -0.19925 0.250903 -0.92919 -0.27138 N to Inf. Facet L. 0.349932 0.878219 -0.32601 0.318589 0.920669 -0.22554 0.325162 0.898674 -0.29437 A. Dir. of Ped. R. -0.814204 -0.42359 0.397042 -0.87976 -0.3197 0.351881 -0.85035 -0.40542 0.335481 A. Dir. of Ped. L. -0.840999 0.36851 0.396133 -0.82156 0.404699 0.401563 -0.70215 0.506734 0.500208 AMNH# 81854, Unk. AMNH# 51381, Unk. AMNH# 51394, Unk. Description of Vector x y z x y z x y z MA of Ped. R. 0.550067 0.233764 0.801736 0.422411 0.23281 0.875996 0.47507 0.066673 0.877419 MA of Ped. L. 0.535515 -0.14146 0.832594 A. Dir. of Ped. R. -0.890747 -0.26507 0.369197 -0.93132 -0.17919 0.317078 -0.851 -0.35375 0.388147 0.460736-0.37385 0.80496 0.365075 -0.35178 0.86196 MA of Sup. Facet R. 0.070312 -0.33421 0.9398740.092389-0.51697 0.8510030.280848 -0.49518 0.822141 MA of Sup. Facet L. 0.416969 0.2703710.867777-0.55517 0.7122990.429434-0.56567 0.6388670.521408 MA of Inf. Facet R. 0.414836 -0.06111 0.9078420.604408-0.05841 0.7945310.535665 -0.18963 0.822862 MA of Inf. Facet L. 0.448901 0.21927 0.8662610.570545-0.00607 0.8212440.610939 -0.0586 0.789506 N to Sup. Facet R. -0.318183 0.8854750.338666-0.33179 0.7898320.515831-0.49847 0.6567610.565855 N to Sup. Facet L. -0.596845 -0.63861 0.485754-0.54625 -0.70161 0.457558-0.5266 -0.76644 0.367786 N to Inf. Facet R. 0.389974 -0.88952 -0.23807 0.355493-0.87274 -0.33459 0.337957 -0.84487 -0.41471 N to Inf. Facet L. 0.520409 0.723908-0.45292 0.4309570.853448-0.29309 0.49759 0.804077-0.32537 A. Dir. of Ped. R. -0.784343 -0.18504 0.592086-0.84717 -0.24225 0.472891-0.78825 -0.41096 0.458017 A. Dir. of Ped. L. -0.724142 0.4303760.538883-0.75992 0.3023970.575397-0.75176 0.4347510.495828 AMNH# 90189, Unk. AMNH# 90191, Unk. AMNH# 167341, Unk. Description of Vector x y z x y z x y z MA of Ped. R. 0.330696 0.1792550.9265570.2645010.2657180.9270560.375816 0.1060390.920607 MA of Ped. L. 0.284985 -0.35495 0.89039 0.171031-0.43069 0.8861450.384584 -0.27468 0.881277 MA of Sup. Facet R. 0.492565 -0.17921 0.851625-0.15154 -0.33697 0.929239-0.53407 -0.5213 0.665589 MA of Sup. Facet L. 0.374429 0.0331890.926661-0.02959 0.3313680.943038-0.45348 0.5138350.728239 MA of Inf. Facet R. 0.697442 -0.06356 0.7138170.339338-0.04085 0.9397770.399262 -0.1596 0.902838 MA of Inf. Facet L. -0.677869 0.49778 0.5410260.275515-0.00883 0.9612560.540192 -0.01432 0.84142 N to Sup. Facet R. -0.193346 0.9315790.30786 -0.38121 0.8872950.259595-0.40845 0.8483960.336737 N to Sup. Facet L. -0.33851 -0.92549 0.169926-0.60944 -0.75379 0.245748-0.40688 -0.84632 0.343784 N to Inf. Facet R. 0.180364 -0.94843 -0.26067 0.37246 -0.91157 -0.17411 0.36183 -0.87737 -0.31511 N to Inf. Facet L. 0.490501 0.8543920.817439 -0.17153 0.555666 -0.15175 0.202823 0.972597 -0.11366

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76 Pan. Continued Raw Vector Data AMNH# 90189, Unk. AMNH# 90191, Unk. AMNH# 167341, Unk. Description of Vector x y z x y z x y z A. Dir. of Ped. L. -0.829607 0.373966 0.41461 -0.80685 0.454951 0.376846 -0.89148 0.137182 0.431793 AMNH# 167342, Unk. AMNH # 167343, Unk. AMNH# 167346, Unk. Description of Vector x y z x y z x y z MA of Ped. R. 0.362846 0.109064 0.925445 0.524503 0.185839 0.83088 0.168865 0.247432 0.954076 MA of Ped. L. 0.450944 -0.15561 0.878883 0.473631 -0.42566 0.771031 0.274117 -0.24688 0.929468 MA of Sup. Facet R. 0.586252 -0.12097 0.801046 0.090002 -0.35277 0.931371 0.003336 -0.09244 0.995712 MA of Sup. Facet L. 0.511312 0.012676 0.859302 0.141493 0.392407 0.908843 -0.54215 0.423273 0.725889 MA of Inf. Facet R. 0.814039 0.079754 0.575309 0.926036 0.129184 0.354638 0.103358 0.033597 0.994077 MA of Inf. Facet L. -0.305927 0.43667 0.846007 0.859116 -0.21065 0.466419 0.777663 -0.19955 0.596171 N to Sup. Facet R. -0.30701 0.881863 0.357859 -0.21465 0.906319 0.364024 -0.39259 0.915654 0.086326 N to Sup. Facet L. -0.550089 -0.76339 0.338582 -0.38765 -0.8228 0.415607 -0.4335 -0.88092 0.189896 N to Inf. Facet R. 0.267955 -0.93038 -0.25017 0.282479 -0.86038 -0.4242 0.384448 -0.9231 -0.00877 N to Inf. Facet L. 0.378159 0.871241 -0.31295 0.388925 0.861102 -0.32747 0.405398 0.883967 -0.23293 A. Dir. of Ped. R. -0.857752 -0.349 0.377436 -0.73734 -0.38881 0.552416 -0.89123 -0.37507 0.255013 A. Dir. of Ped. L. -0.790336 0.387948 0.4742 -0.67694 0.38409 0.627875 -0.86229 0.364838 0.351211 AMNH# 54330, Unk. AMNH# 174860, Unk AMNH# Unk., Unk. Description of Vector x y z x y z x y z MA of Ped. R. 0.473025 0.056875 0.879211 0.321498 0.106226 0.940933 0.48209 0.225415 0.846627 MA of Ped. L. 0.431014 -0.03341 0.901726 0.333581 -0.28208 0.899529 0.46998 -0.29634 0.831444 MA of Sup. Facet R. -0.039516 -0.27455 0.960761 0.238388 -0.16465 0.957112 -0.43005 -0.46848 0.771738 -0.121767 0.44288 0.888274 -0.39422 0.863268 -0.5284 0.660935 0.532879 MA of Inf. Facet R. -0.640971 -0.56817 0.516085 0.597732 0.115512 0.79333 0.634349 -0.08663 0.768177 MA of Inf. Facet L. 0.743024 -0.17555 0.645832 0.519948 -0.03566 0.853453 0.675502 -0.09531 0.731172 N to Sup. Facet R. -0.400064 0.885429 0.236566 -0.45193 0.8535090.367283 0.259386 -0.29999 0.880404 N to Sup. Facet L. -0.370872 -0.85041 0.373163 -0.27316 -0.93708 0.21742 -0.45555 -0.75037 0.47897 N to Inf. Facet R. 0.536677 -0.81244 -0.22788 0.219861 -0.97524 -0.02365 0.281122 -0.89981 -0.33363 N to Inf. Facet L. 0.457564 0.837474 -0.29879 0.158522 0.985801 -0.05538 0.412996 0.870379 -0.26809 A. Dir. of Ped. R. -0.871405 -0.11705 0.476397 -0.84069 -0.42526 0.335256 -0.83663 -0.16842 0.521238 A. Dir. of Ped. L. -0.828614 0.380983 0.410184 -0.83378 0.357002 0.42115 -0.66368 0.502373 0.554209 MA of Sup. Facet L. 0.315211 Pongo. Raw Vector Data AMNH# 61586, Male AMNH# 238487, Male AMNH# 140246, Male Description of Vector x y z x y z x y z MA of Ped. R. 0.08109 0.166519 0.982698 NA NA NA 0.111743 0.319078 0.941118 MA of Ped. L. NA NA NA NA NA NA NA NA NA MA of Sup. Facet R. 0.256731 -0.79678 0.547027 0.284882 -0.71853 0.634471 0.322088 -0.08601 0.942794 MA of Sup. Facet L. 0.400035 0.129814 0.90726 0.413573 0.222408 0.882889 0.147248 0.136135 0.979686 MA of Inf. Facet R. 0.565543 -0.2728 0.778292 0.538296 -0.18348 0.822541 0.297263 -0.22971 0.926752 MA of Inf. Facet L. 0.465343 -0.01891 0.884928 0.447483 0.179745 0.876043 0.541475 0.208685 0.814405 N to Sup. Facet R. -0.757461 0.185666 0.625924 -0.76236 0.231406 0.60437 -0.63264 0.721301 0.281935 N to Sup. Facet L. -0.665756 -0.63917 0.385004 -0.6676 -0.58529 0.460163 -0.40088 -0.89727 0.184936 N to Inf. Facet R. 0.745713 -0.23391 -0.62386 0.763734 -0.30643 -0.56816 0.461009 -0.81546 -0.34999 N to Inf. Facet L. 0.67332 -0.45527 0.656516 -0.34004 0.644576 0.614207 0.236317 0.891866 -0.38565

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77 Pongo. Continued Raw Vector Data AMNH# 61586, Male AMNH# 238487, Male AMNH# 140246, Male Description of Vector x y z x y z x y z A. Dir. of Ped. R. -0.909372 -0.39124 0.141335 NA NA NA -0.78734 -0.54939 0.279751 A. Dir. of Ped. L. NA NA NA NA NA NA NA NA NA AMNH# 28252, Male AMNH# 28253, Unk. Description of Vector x y z x y z MA of Ped. R. 0.40951 0.150059 -0.96389 -0.21999 0.058964 0.910398 MA of Ped. L. 0.261747 0.432109 0.831914 -0.34814 -0.15328 0.952888 MA of Sup. Facet R. 0.388142 -0.093279 -0.67987 0.727377 -0.506 0.770269 MA of Sup. Facet L. 0.284891 0.431487 0.361134 0.826681 0.640942 0.712763 MA of Inf. Facet R. 0.479223 0.51462 -0.30385 0.801773 -0.2302 0.846966 MA of Inf. Facet L. 0.637168 0.683651 0.095752 0.7235 0.298844 0.710429 N to Sup. Facet R. -0.65829 -0.65802 0.590363 0.46742 0.432709 0.615967 N to Sup. Facet L. -0.61944 -0.751262 -0.36347 0.550904 -0.44436 0.647176 0.440951 0.57948 -0.56596 -0.58643 -0.77122 -0.45911 N to Inf. Facet L. 0.446346 0.507396 0.650199 -0.5655 0.608383 -0.65624 A. Dir. of Ped. R. -0.88338 -0.91495 -0.2197 0.338523 -0.22368 0.411843 -0.7947 -0.803535 0.530423 0.270154 0.526022 0.302908 N to Inf. Facet R. A. Dir. of Ped. L.

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83 Tuttle RH (1985) Ape footprints and Laetoli impressions: A response to SUNY claims. In Hominid Evolution Past, Present and Future. Tobia PV (ed.). Alan R. Liss, New York. Tuttle RH and Watts DP (1985) The positional behavior and adaptive complexes of Pan gorilla. In Primate Morphophysiology, Locomotor Analyses and Human Bipedalism. Kondo S (ed.). University of Tokyo Press, Tokyo. Velte MJ (1987) A New Biomechanical Model of the Vertebral Column in Anthropoid Primates. PhD Dissertation, University of Chicago Wainwright SA (1988) Form and function in organisms. American Zoology. 28: 671-680. Yerby SA and DR Carter (2000) Bone fatigue and stress fractures. In Musculoskeletal Fatigue and Stress Fractures. Burr DB and Milgrom C (eds.). CRC Press, New York. Yocum TR and Rowe LJ (2005) Essentials of Skeletal Radiology, Third Edition, Vol.1. Lippincott, Williams, and Wilkins, Philadelphia.

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BIOGRAPHICAL SKETCH Dorion Amanda Keifer was born in Idaho Falls, Idaho, on April 20th, 1979. She received her bachelors degree in anthropology from New York University in 2000.


Permanent Link: http://ufdc.ufl.edu/UFE0010491/00001

Material Information

Title: Last Lumbar Facet and Pedicle Orientation in Orthograde Primates
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0010491:00001

Permanent Link: http://ufdc.ufl.edu/UFE0010491/00001

Material Information

Title: Last Lumbar Facet and Pedicle Orientation in Orthograde Primates
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0010491:00001


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LAST LUMBAR FACET AND PEDICLE ORIENTATION IN ORTHOGRADE
PRIMATES















By

DORION AMANDA KEIFER


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF ARTS

UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by

DORION AMANDA KEIFER

































For my mother and father and, of course, Mishka.















ACKNOWLEDGMENTS

I would like to thank, first and foremost, my committee, Drs. David Daegling and

Michael Warren, for their support and guidance. The thoughtful insights of my

committee made me think of this problem in new ways and were essential to the finished

document. I would also like to thank Dr. Kenneth Mabry, Gary Sawyer, and Anita

Caltabiano, from the Department of Anthropology, and Jean Spence, Teresa Pacheco, and

Patricia Brunauer, from the Department of Mammalogy, at the American Museum of

Natural History, for both the use of their collections as well as their encouraging words.

A special thank you goes to Dr. Susan Anton at New York University for the use of her

equipment; without her the data could not have been collected. I would like to thank Dr.

Anthony Falsetti for the use of his equipment as well. The extra practice made all the

difference. I thank the University of Florida Department of Anthropology staff, Karen

Jones, Rhonda Riley, Patricia King, and LeeAnn Martin. I thank Shanna E. Williams for

her artistic talent as well as her laughter. I would also like to thank Joseph, Jenna, and

Maxine Coplin who moved me into their home for two months during data collection, as

well as for their unwavering support. I thank Kristine Hulse, Jack Lackman, Linda

Stanley and Mike Stanley, good neighbors and great friends. I would also like to thank

Sandra Campbell, Sharon Milton-Simmons, Kelley Paulling, Jennifer Brookins, Mary

Eckert, Renee Smith, Tina Sporer, and Barbara Young at the University of Florida of

Florida, Department of Otolaryngology. They are the best coworkers anyone could ask

for and they encouraged me every step of the way. I thank Dr. Jack Sedwick who has









been so supportive in this endeavor. Finally, I would like to thank my family: My mother

and father, Pat and Orion Keifer, as well as my sisters Rachel McLaren, Megan Keifer,

my brothers Orion (Paul) Keifer, Seth Keifer, and Chris McLaren, and last, but not least,

my niece Maccayla Keifer.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv


L IST O F TA B LE S ...... .. .......................... .................... ........ .............. viii


LIST OF FIGURES ......... ......................... ...... ........ ............ ix


A B STR A C T ................................................. ..................................... .. x


CHAPTER


1. IN TR OD U CTION ............................................... .. ......................... ..

The Two-Column Model of Force Transmission.......................................................3
Anthropological Research on the Primate Spine........................................................7
G oals of the C current P project .......................................................................... .. .... 12

2. PRINCIPLES OF FUNCTIONAL MORPHOLOGY..............................................14

Phylogeny and Phylogenetic Constraint .............. ..............................................14
A llo m etry .................. ................... ............ .......... .... ..................... 15
Ontogeny and Evolutionary Developmental Biology ..........................................15
B one B iology and B ehavior......................................................................... .... .... 16

3. RESEARCH DESIGN AND DATA COLLECTION PROCEDURES...................19

T ax o n o m ic S am p le .............................................. .. .......................... ..................... 19
Specim en Inclusion C riteria ............................................... ............................ 21
D ata Collection Instrum entation........................................... .......................... 22
D ata Collection Procedures .................................... .... ........................... 23
Force Calculation: A Static Free Body M odel .................................. ............... 26
Statistical T testing ....................................................30

4. DATA TRANSFORMATION ..................................................33









Statistical A naly sis and R results .............................................................. .....................46

Com bining M ixed Sex Sam ples ........................................ ........................... 46
Mean Directions and Mean Resultant Lengths................................ ...............49
P air-w ise T estin g ................................................................5 1

DISCUSSION ............. .............................................. ..... .............. 55

Pair-wise Testing ................................................ 55
Pair-wise Comparisons and Phylogeny ................................ .. ..... ....... 56
Pair-wise Comparisons and Allometry.................... .............................57
Pair-wise Comparisons and Bone Biology and Behavior .................................57
Force Transmission in the Posterior Elements ................................. ............... 59
Feasibility of the M odel ......... ...... ................................ ..... .. ..................... 60
V ector Concentrations (M RL) ....................... .. ............................... ............... 61
Facet Shape .....................................................61

CON CLU SION S....... .... ............................ ............ ............................. .. 63


MAJOR AXES-MINOR AXIS RATIO ........................................ ........................ 66


ANGLE ERROR CALCULATION (DEGREES)................ .................................68


RAW VECTOR DATA TABLES .............................................................................. 70


L IST O F R E FE R E N C E S ....................................................................... ... ................... 78


B IO G R A PH IC A L SK E TCH ..................................................................... ..................84
















LIST OF TABLES

Table p

1. Sam ple description ............. .................................... .. ....... .. ...... .... 21

2. Data transform ation coordinate system s ............................................ ............... 36

3. Pooling data, G orilla sam ple. ............. ............ ................................... ................... 48

4. D descriptive statistics. ............................................ ... .... ........ ......... 49

5. P air-w ise results. ..........................................................................52
















LIST OF FIGURES

Figure pge

1. Potential forces acting on the right pedicle................. ........ ...............6

2 F acet sh ap e ...........................................................................2 5

3. F ree body diagram .......... ............ ..... .. ...... .... .... ................ .............. 28

4. Normal vectors to the superior facets, right and left.................... ........................... 29

5. Normal vectors to the inferior facets, right and left................... ..................................29

6. A xial vectors to the pedicles, right and left ........................................ .....................29

7. Major axis vectors of the superior facets, right and left.. .............................................31

8. Major axis vectors of the inferior facets, right and left................................................31

9. Major axis vectors of the pedicle, right and left ........................................................32
















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Arts

LAST LUMBAR FACET AND PEDICLE ORIENTATION IN ORTHOGRADE
PRIMATES

By

Dorion Amanda Keifer

May 2005

Chair: David J. Daegling
Major Department: Anthropology

The spine is critical for locomotor behavior. The spine is the foundational structure

of the trunk and plays a key role in trunk position and stability. The spine also anchors

major muscles of the forelimb and hindlimb, connecting them via a flexible column.

Further, the spine articulates with the hindlimb, transmitting associated locomotor forces

to the trunk. Despite the certain role of the spine in locomotion, the functional

significance of vertebral variation is unestablished.

Morphological analyses of the primate spine indicate that the last lumbar vertebra

posterior element morphology potentially correlates with locomotion. The calculation of

the forces in the posterior elements of the last lumbar vertebra is necessary to

demonstrate this connection. For the future development of a model for the calculation of

these forces facet and pedicle orientation in the last lumbar vertebra were measured for

four orthograde primates: Homo sapiens, Gorilla gorilla, Pan troglodytes, and Pongo

pygmaeus.









Facet and pedicle orientation were described via six unit vectors for both the right

and left side: normal to the superior and inferior facets, major axis of the superior and

inferior facets, as well as axial to, and the major axis of, the pedicle. These vectors were

calculated with respect to a vertebral reference axis, as well as the gravity vector when

the spine is in orthograde posture. Descriptive statistics were calculated for all four

species. Pair-wise tests were conducted among humans, gorillas, and chimpanzees.

The facet normal and pedicle axial vectors are essential for the development of a

biomechanical model to estimate force direction in vertebral posterior elements. The

statistical analysis of these vectors show they are very consistent within species,

indicating that the mean vectors can be used for the future development of a

biomechanical model. The pair-wise tests demonstrated that the species have statistically

different mean directions.

The facet major axis vectors were calculated, as they may indicate the direction of

the greatest range of motion in the facets. Generally, the descriptive statistics

demonstrated that there was fairly large within species variability of these vectors. The

pair-wise tests for equal mean direction indicated that, with a few exceptions, the species

have statistically different mean directions with respect to these vectors.

The pedicle major axis vectors were calculated as they potentially indicate how the

pedicle is oriented to resist bending force. The statistical data indicated that the

orientation of these vectors is very consistent for the great apes tested. In contrast in

humans, these vectors have greater variability though the functional significance of this is

not clear. The intra-species pair-wise tests indicated that each species had a statistically

different mean direction.














CHAPTER 1
INTRODUCTION

The study of the evolution of the relationship between form and function, or

functional morphology, has long been of interest to anthropological researchers. Within

functional morphology, the terms form and function are heuristic devices that allow the

separation of phenomena that are inextricably linked within an organism. In general,

form refers to shape and size, while function refers to how this form moves through space

and how the form interacts with its surroundings. However, these two definitions do not

have any real context unless they are applied to a specific problem involving an organism

or group of organisms, (Wainwright, 1988). One such application explores the

relationship between skeletal form and type of locomotion, including associated posture.

Extant adult skeletal morphologies are not exclusively the product of body size but also

of long histories of evolutionary changes, developmental processes, and daily activities.

It is not easy to determine the extent of the individual contributions of these factors to the

adult form, as they are interrelated parts of the total functional-morphological complex of

an individual organism. Therefore, it is important to apply myriad techniques and

perspectives to a specific problem.

The relationship between skeletal form and locomotion in extinct and extant

primates has been a focus of study within the field of anthropology. Historically, these

studies have centered on lower limb and pelvic morphology as well as the primate

forelimb and shoulder girdle (Demes et al., 2000; Duncan et al., 1994; Heinrich et al.,

1993; Hunt, 1991; Kimura, 2003; Latimer and Lovejoy, 1990a, 1990b; Lovejoy, 1975,









1979,1988; Ohman et al., 1997; Schmitt, 1994, 2003; Susman et al., 1984; Susman and

Demes, 1994; Tuttle, 1985). However, these studies lacked an in-depth exploration of

the spine as part of the functional complex of limb movement, ignoring the fact that

amongog mammals, structural differences of the lower precaudal spine correspond with

contrasts between species in columnar function and positional behavior" (Sanders 1995:

97).

Recently, there have been a number of studies that address this gap in knowledge

(Sanders, 1995, 1998; Shapiro, 1991, 1993, 1995; Shapiro and Johnson, 1998; Shapiro et

al., 2001; Shapiro and Simons 2002; Velte 1987). These studies have revealed that the

primate spine is morphologically conservative, despite quite dramatic differences in

locomotor behavior. However, studies that compared the absolute and relative size and

shape ratios of the posterior elements and pedicles of the primate spine have

demonstrated morphological differences between bipedal and quadrupedal orthograde

primates. This has lead to the speculation, though justifiably guarded, that these

differences may represent differences in force transmission patterns in the posterior

elements due to specific locomotor behavior. This conclusion is based upon traditional

morphometric techniques that use size and shape ratios as proxy values for force

magnitude and direction. This proves useful for locating possible areas where bone

morphology may reflect function. An underlying assumption of previous primate spine

morphometric studies is the assumption that the size of a given vertebral element is

related to the magnitude of the forces acting upon that element (Davis, 1961: 337).

However, this assumption may not always be reliable. Therefore, to examine this









relationship further, it is important to understand how force is transmitted and to calculate

the magnitude of those forces using biomechanical principles.

The Two-Column Model of Force Transmission

Although there is a lack of biomechanically based studies that focus on non-human

primate spines, the biomechanics of the human spine has been, and continues to be, the

focus of a great number of studies. These studies were conducted for clinical

(orthopaedics and orthopaedic surgery), occupational (ergonomics, proper lifting, etc.),

military (human tolerance in operational and crash situations), and public safety purposes

(seatbelt and other restraints, transportation safety, amusement rides). Given the large

number of studies, it is impractical to exhaustively list all of them. Therefore, focus will

be on research that is pertinent to the development of the two-column model of force

transmission in the spine. The two-column model of force transmission is a general

description of force transmission in the spinal column. The spine is comprised of two

compressive columns called the anterior and posterior columns. The anterior column

includes the vertebral bodies, intervertebral disks, and associated ligaments, while the

posterior column is comprised the articular facets and the lamina and associated

ligaments. The pedicles are the bony bridges that connect the two columns. These data

will be presented prior to the discussion of past anthropological research, as it provides

an analytical framework in which to evaluate the anthropological data, as well as defines

crucial terminology.

The two-column pattern of force transmission is now widely accepted in the

literature. However, prior to the 1960's, it was assumed that the pedicles and posterior

elements did not transmit any significant force as they are smaller than the vertebral









bodies. They were believed to have three main purposes, to resist lateral rotation, to

serve as muscle attachment sites, and to protect the spinal cord from trauma.

The first researcher to conclude that vertebral bodies could not be the sole means of

force transmission was P.R. Davis in 1961. His conclusion was based upon a visual

examination of human spinal morphology. Davis noted the area of the lumbar centra

consistently increases in size from lumbar one to lumbar four. However, lumbar five, the

last lumbar vertebra in humans, has a smaller lumbar centra than the vertebra above it.

Davis concluded that if the vertebral bodies were the sole means of force transmission,

the last lumbar centra would have to be larger than the one above it as the compressive

force is cumulative. Based upon this observation, he concluded that the last lumbar

vertebral body was subjected to less compressive force than the one above and posited

that the compressive force was somehow resisted by the neural arch, which is comprised

of the lamina and pedicles.

In 1974, Prasad and colleagues conducted a study to measure the role of the

articular facets during acceleration. The researchers used several techniques to measure

the load bearing capability of the facets in cadaveric studies. To qualitatively measure

facet load, they attached strain gauges to the pedicles and lamina of their subject

vertebrae. Quantitative measures of facet load were obtained using an intervertebral load

cell that was designed to fit under the disk or vertebral body. The transducer was capable

of measuring both axial force and the eccentricity of the axial force with respect to its

geometric center. Cadavers were then accelerated in such as way as to simulate travel

conditions experienced by U.S. Air Force pilots. Load cells were also mounted to the

chair and restraint system. From this analysis, Prasad and colleagues proved both









qualitatively and quantitatively that the articular facets were involved in load

transmission. However, they were unable to calculate the facet load.

In 1980, the work of Adams and Hutton filled in the information gap. They proved

experimentally the significant role that articular facets play in forces transmission and

also provided a measure of load transmission. Using "green" cadaver vertebral sections

consisting of two vertebrae and the intervening disk, positioned as they would be in vivo,

the researchers subjected the sections to compressive loads and various angles and

measured deformation for acute loads and bone compression for constant loads. The load

borne by the facets was shown to be posture dependent. They found facets withstand the

highest force load when the body is in a standing, upright position. In this posture, facets

transmit 16% of the total force load. This is the generally accepted number in the

literature: however, some studies have placed this figure as high as 23% in the fifth (last)

lumbar in humans (Pal and Routal, 1987).

In 1989, El-Bohy and colleagues' cadaveric studies demonstrated that the

mechanism of posterior element load transmission was via facet/lamina contact by

directly measuring contact pressure in the joints under various loading conditions using

motion segments consisting of three vertebrae and their intervertebral disks. They

measured contact pressure by placing a transducer between the inferior facets and the

lamina of the vertebra below. Their results verified that loads passing through the facets

are transmitted via contact between the facets and lamina.

Previous research confirms that the posterior elements are involved in force

transmission. However, the magnitude, direction, and type of force acting on the

individual posterior elements and pedicles are not known. There has been some









speculation as to the loading conditions of the pedicles. One hypothesis is that the

pedicles are subjected to bending stresses as a result of muscles action on the spinous or

transverse processes or alternately from the posterior elements (Bogduk and Twomey,

1987). It has also been suggested that the pedicles may be subjected to tension from the

facets locking to prevent the vertebrae from sliding forward (Bogduk and Twomey,

1987). Finally, it has also been hypothesized that the pedicles may be subjected to

compressive axial loads from the vertebral body to the lamina (Pal and Routal,

1986,1987). However, none of these hypotheses has been proven. There are four

potential forces acting upon the pedicles: torsion, compression, bending and shear (Figure

1). In this figure compression is shown. Alternately, this force could be tension. The

torsion is depicted as clockwise, however it may be counter clockwise. Similarly the

bending and shear is depicted as an upward force. However, it may also be a downward

motion.







Torsion




o VCompression


Bending and Shear


Figure 1. Potential forces acting on the right pedicle.









Anthropological Research on the Primate Spine

Three research studies were instrumental in the development of the current

research design. Comparative studies on the primate spine indicated that biomechanical

analysis of the posterior elements is important for understanding their role in posture and

locomotion. The results of these three works and their relevance to the current problem

are discussed in detail in the following section.

An early functional morphology study addressing the primate spine from a

comparative perspective was Margaret Velte's (1987) dissertation that developed and

described a biomechanical model for the anthropoid spine. Velte focused completely on

the morphology of anterior vertebral elements, which are comprised of the vertebral

centra, intervertebral disks, and associated ligaments. Her sample included Alouatta,

Ateles, Cebus, Cercopithecus, Homo, Gorilla, Pan, and Papio, as well as some fossil

material. The fossil material will not be discussed as it is beyond the scope of the current

project. Further, the discussion will focus on results for the species included in the

current study, Homo, Gorilla, and Pan.

Velte modeled individual vertebral bodies as short, deep beams. Using strength

values of trabecular bone, she calculated the tensile and compressive bending moments

and the shear force at failure. The shear and the bending moments were calculated for

both dorsoventral flexion and lateral flexion axes. Her results indicate that shear and

tensile stresses intensify along both axes. However, her data indicates that compressive

stresses are reduced along the lateral flexion axis in cercopithecoids and along the

dorsoventral flexion axis in hominoids. Velte concluded that the vertebral centra of

Homo, Gorilla, and Pan are strongest in compression. Velte postulated that these data









indicate the hominoid pattern of short, broad vertebral centra is an adaptation to

orthograde posture.

Liza Shapiro (1991,1993) also addressed the primate spine from a comparative

perspective. Her extant sample also contained a large number of species including

Aloutta, Ateles, Cebus, Gorilla, Homo (pygmy and non-pygmy), Hylobates, Indri, Pan,

Papio, Pongo, Propithecus, and Varecia. Again, this discussion focuses on the species

included in the present study, Homo, Gorilla, Pan, and Pongo.

Like Velte, Shapiro measured and evaluated vertebral centra. However, she also

focused on the posterior elements and the pedicles. Shapiro's goal was to define aspects

of vertebral morphology that were uniquely human. In order to accomplish this goal, she

recorded various vertebral measurements: vertebral body centraa) area (calculated as an

ellipse from the measured ventrodorsal and mediolateral distances at midpoint of the

body), pedicle area (product of the length and width), pedicle shape (ratio of the width to

the length), lamina area (product of the width and ventrodorsal thickness), and lamina

shape (ratio of the lamina ventrodorsal thickness to the width). These metrics were

selected to reflect the relative loads passing through the anterior and posterior columns.

Finally, she measured the angulation of the superior facet (medial interfacetal width -

lateral interfacetal width/2 times the facet width).

From the results of these measurements and calculations, Shapiro concluded, as

compared to the great apes (Gorilla, Pan, and Pongo), humans have large vertebral centra

(body) areas for their body weight. Shapiro speculated that this large body area may be

related to increased compressive force associated with bipedalism. She did note that,

like Velte, her studies indicated that humans and great apes have shorter, broader









vertebral bodies as compared to other primates. As mentioned, human vertebral body

areas increase in size from the first to the penultimate lumbar vertebra; however, at the

last lumbar level, the vertebral body area reduces in size. Once thought to be a uniquely

human pattern, Shapiro's results indicate this pattern was found in nearly all the primates

in her sample.

For both pedicle area and pedicle shape, her ANOVA tests indicated that Homo,

Gorilla, Pan, and Pongo each have statistically different mean values. There were no

statistically significant within species differences between male and female and therefore,

for the all species, the sexes were pooled. Although each species had a unique mean for

both pedicle area and pedicle shape, all four species trend toward increased pedicle area

from the penultimate to last lumbar vertebrae. Shapiro speculated that this increased

pedicle area is a function of the reduced vertebral body area of the last lumbar. Counter

to speculation, this pattern is also seen in primates that do not have a reduced vertebral

body area of the last lumbar vertebra. Shapiro also discerned that pedicle area is highly

correlated with body size, with the notable exception of the last lumbar vertebra. At this

level humans have a larger pedicle area than would be expected for their size. The results

of pedicle shape (ratio of width to height) analysis indicate that, for humans, the increase

in pedicle area at the last lumbar level occurs due to an increase in pedicle width relative

to pedicle height. The Pan, Pongo, and Gorilla samples did not exhibit a pedicle shape

change between the penultimate and last lumbar vertebrae. However, when compared

with non-hominoids in her sample, the great apes and humans have short, wide pedicles.

On one hand, the human and great ape samples have similar trends in the lumbar

region, with humans expressing these trends more dramatically. This would suggest that









this pattern is more likely related to postural behavior. On the other hand, the ANOVA

test indicated the species have statistically different mean values, suggesting that each

species has a unique posterior element morphology.

The ANOVA tests of lamina shape (ratio of ventrodorsal thickness and width)

again indicate humans, gorilla, chimpanzees, and orangutan have unique mean values. As

with the pedicle ANOVA test, males and females were pooled. Shapiro's results indicate

that humans have wider lamina than the great apes. Further, human laminae become

progressively wider, relative to thickness from lumbar one to the last lumbar. The

gorilla, chimpanzee, and orangutan laminae to do not display this change.

Shapiro (1991) indicates that the laminae are an important aspect of posterior

element force transmission because compressive force moves down the vertebral column

via facet/lamina contact. The increasing width of lamina in humans may therefore be a

reflection of increased compressive forces due to bipedal locomotion. Interestingly,

despite great apes' orthograde posture, they do not share a similar pattern.

Finally, Shapiro (1991) measured facet angulation of the superior facets. The

ANOVA test again indicated that the four groups have statistically different superior

facet angulation means. As with the other ANOVA tests, males and females were not

found to have any statistically significant differences and were pooled.

Despite statistically different mean values, general trends were found. For

example, gorillas and chimpanzees demonstrated similar patterns, with superior facet

angles becoming more acute from lumbar-one to the last lumbar, while humans facet

angulations becomes more oblique. Orangutans appear to have a mean facet angle value

at the last lumbar level that is identical to humans (i.e. laterally oriented), though facet









angles appear to follow the chimpanzee and gorilla pattern until the last lumbar level.

Humans follow a pattern in which the superior facet angle becomes more oblique.

The angulations of the superior facets provide information about the lateral rotation

permitted in the spine. The gorilla and chimpanzee facets seem oriented to restrict lateral

rotation. The increasingly oblique angulations in humans have been posited to help resist

forward displacement of the vertebrae as the last lumbar vertebra in humans is angled

forward. Finally, the functional significance of the orangutan facet angulation pattern is

not currently understood.

William Sanders (1995,1998) also conducted comparative morphological studies of

the primate spine. The goal of his analysis was to use an extant primate sample to

describe Australopithecine spinal biomechanics. He had a diverse sample, however I will

only focus on the species relevant to the present study. Two of his measures are

significant to the current problem, facet spacing and orientation and pedicular robustness

and cross-section area.

With respect to facet angulation, Sanders'(1995;1998) data supported Shapiro's

(1991;1991) finding. Like Shapiro, Sanders only documented facet angulation for the

superior facets. However, Sanders (1995;1998) also highlighted facet spacing. Gorillas,

chimpanzees, and orangutans have superior facet spacing that narrows slightly from the

first to last lumbar vertebrae. In humans, the spacing between the superior facets

increases from the first to last lumbar vertebrae. Sanders (1998) suggested that this

pattern is important for spinal stability as more widely spaced facets create a more stable

"base."









Sanders (1995;1998) also measured pedicle width and length to calculate pedicle

area and robustness. Sanders' data indicate that ape pedicles are robust, shorter and

wider, when compared to monkey pedicles. According to Sanders (1995;1998), muscle

force accounts for part of the robustness as the transverse process of hominoids are rooted

on the pedicles. However, Sanders (1995;1998) also documents scaling trends that

indicate bending stresses as a critical factor for pedicular dimensions. From the first to

last lumbar vertebra, great ape pedicles increase in width and decrease in length. Human

vertebrae also follow this pattern; however, the last lumbar vertebra in humans has an

extremely wide pedicle. These data are in agreement with Shapiro (1991;1993).

The results of the above research studies are not straightforward. On one hand,

ANOVA tests indicate that for measures defined by the study each species had a

statistically different mean value. However, the results also indicate that humans and

great apes share generalized patterns of vertebral morphology. To further explore the

functional morphology of the posterior elements it is necessary to develop a methodology

to calculate the magnitude of stresses and strains on these elements. It is also desirable to

measure and compare previously unmeasured variables. The current project addresses

both of these issues.

Goals of the Current Project

The goal of this project is to document last lumbar facet and pedicle orientation in

orthograde primates. These orientations were analyzed from two perspectives. First,

descriptive statistics were calculated. Second, pair-wise tests for equal mean direction

were conducted. The results indicate whether vectors means can be used to develop

biomechanical models for force calculation in the last lumbar posterior elements. The

results also indicate how orientation vectors vary among orthograde species with diverse






13


locomotor behavior. If current and future research on the posterior element morphology

demonstrates that the morphology of the elements reflects postural and locomotor

behavior, it will provide another method for elucidating postural and locomotor behavior

in the fossil record.














CHAPTER 2
PRINCIPLES OF FUNCTIONAL MORPHOLOGY

Anthropological researchers rely heavily on skeletal material to make many

inferences, such as taxonomy, evolutionary relationships, life history, and behavior. The

current project's focus is the extent to which locomotor and positional behavior can be

inferred from the posterior element morphology of the last lumbar vertebrae. The

theoretical paradigm of functional morphology relies heavily on physics and engineering

principles. Unlike engineered objects, however, the biological form is a product of many

interrelated factors and processes: evolutionary history, growth and size, locomotor and

postural behaviors, and so forth. There are four major areas in which the simple "form

follows function" axiom of functional morphology is called into question. These major

areas are phylogeny and phylogenetic constraint, allometry, ontogeny and evolutionary

biology, and bone biology and behavior. However, the reader is cautioned that these

categories, though artificially separated, are interrelated.

Phylogeny and Phylogenetic Constraint

The current form of any biological organism is constrained by their evolutionary,

genetic history. The significance of this in terms of the current, and all, functional

morphological studies is that aspects of the current morphology may not have functional

significance. Rather they are simply a reflection of common ancestry and subsequent

phylogenetic constraint. In order to elucidate which aspects of a species' morphology are

likely related to phylogenetic constraint, a comparative sample is necessary. In this









study, a comparative sample of four closely related orthograde species with diverse

locomotor repertoires was included.

Allometry

Allometry refers to changes in morphology that are not necessarily related to

change in function but rather are "by products" of body size. In the present study, the

methodology does not remove the factor of size in a meaningful way. Size is removed in

the sense that the data are presented as unit vectors. Unit vectors do not, by definition,

give any information regarding size as the vectors are divided by their length, resulting in

all vectors having a length of one. Consequently, unit vectors only describe direction.

Correcting for size is not appropriate in this case, as the goal is to determine orientation.

The sample does include an extremely sexually dimorphic species, Gorilla gorilla as well

as two similarly sized species, Homo sapiens and Pan troglodytes. Pair wise

comparisons for equal mean direction were conducted between the male and female

Gorilla vectors, as well as between Homo and Pan. The results of these tests will give

some information regarding the dependence of the measured unit vectors on body size.

Ontogeny and Evolutionary Developmental Biology

Developmental processes affect the adult form in ways that may not be a reflection

of function but rather are the results of canalization and developmental stability. The

concepts of"[c]analization and developmental stability refer to the tendency of

developmental processes to follow a particular trajectory despite external and internal

perturbation," (Hallgrimsson et al., 2002: 131). Currently, there is no information in the

literature that documents the extent to which posterior element vertebral morphology is

constrained by canalization and developmental stability. This information is necessary to

understand the functional morphology of the vertebral column, and it is expected that this









research will be conducted in the future. However, the reader is cautioned that the results

of the current study were interpreted without the benefit of this information.

Bone Biology and Behavior

Studies of bone biology and bone behavior consider bone's response to the internal

and external environment. The theory that "the distribution of strain trajectories

engendered through functional activity is responsible for the development and

maintenance of trabecular alignment and cancellous bone density within a bone" is the

commonly accepted modern conceptualization of Wolff s law (Biewener et al., 1996: 1).

This theory is very attractive in terms of functional morphological analyses because if it

is correct, bone morphology can unproblematically be interpreted as a reflection of

function. However, it is important to note, as Cullinane and Einhorn (2002) pointed out

there are compelling critiques of theory (Bertram and Schwartz, 1991; Bienwener et al.,

1996; Fyrie and Carter, 1986, Pearson and Lieberman, 2004) as well as scholarly debates

about the mechanisms of the bones response to the mechanical environment (Martin,

2000; Mullender and Huiskes, 1995; Turner and Pavalko, 1998).

Generally, interpretations support that, in the course of normal loading during daily

activities, the skeleton accumulates microscopic damage. The field of bone biology

currently supports the theory that this microdamage triggers a remodeling response

within the skeleton (Yerby and Carter, 2000). It is at this point, however, that dissension

arises within the literature; bone tissue is not consistent or uniform in its response to

stresses and strains. For example, a study that used a canine model rejected the subjects'

radii and allowed locomotion with the ulna supporting all the weight. The results of this

study demonstrated a lack of uniform skeletal response as some of the subjects suffered

fatigue fractures of the ulna, while others developed massive hypertrophy of the ulna









(Chamay and Tschantz, 1972). However, the results of this study are difficult to

interpret. In the canine model the radius is much larger than the ulna. Therefore, the ulna

was subjected to loading that would never be encountered outside of the research setting.

In a follow-up study, Carter and associates (1981) also used a canine model, but the

ulna rather than the radius was rejected. In this study, they found that bone formation in

the radii was negligible. This would seem to indicate that the radius of the canine is

overbuilt with respect to the forces that it must withstand.

To add another piece to this already difficult puzzle, fractures can occur at the

upper levels of normal activity. For example, pars defects, fractures of thepars

inarticularis, are a very common condition in humans, especially in the last lumbar

vertebra (Burkus, 1988; Kip et al., 1994; Sermon and Spengler, 1981). Contrary to

Wolff's Law, however, this shear fracture is not caused by a large acute stress to the par

inarticularis. Rather this fracture occurs due to repetitive loading and subsequent fatigue

failure. This fracture is often found in athletes.

A clear understanding of the cellular process by which bone repairs itself as well as

bone as a biological material is essential to understanding how bone responds to the

mechanical environment. However, it is not currently understood. The adult skeleton is

in a constant state of deterioration and reformation by the osteoclasts and osteoblasts in

trabecular surfaces and in Haversian systems. In simplified terms, when a microcrack

forms in bone, osteoclasts activate on the crack surface resorbing the damaged bone. This

activates the osteoblasts, which stimulate new bone growth (Brukner et al., 1999; Mundy,

1999; Schaffler, 2000). Colloquially, this is known as the "drill and fill" process. The

intuitive and pervading opinion, until recently, was that the formation of microcracks is a









mechanical process in which microscopic fissures in the bone are formed faster than they

can be repaired by the body. However, recent research indicates that stress fractures are a

response to a positive feedback mechanism. The mechanism of increased usage

stimulates bone turnover, which results in focally increased bone remodeling. It is

theorized that increased bone porosity and bone mass result, weakening the bone

(Brukner et al., 1999; Schaffler, 2000). In other words contra, to the current

understanding of Wolff s Law, this theory indicates that when skeletal material is

subjected to repetitive stresses and strains, it is ultimately weakened, not strengthened by

the remodeling process.

Fortunately, more and more anthropological researchers are interested in moving

beyond the Wolff s Law paradigm. Currently, the two most influential perspectives that

move beyond this paradigm are bone developmental genetics and biomechanics (Pearson

and Lieberman, 2004). The goal of this project is to move beyond morphological studies

on the primate spine heavily reliant on Wolff s Law and develop biomechanically based

research to further our understanding of spinal functional morphology.














CHAPTER 3
RESEARCH DESIGN AND DATA COLLECTION PROCEDURES

Taxonomic Sample

This research further explores the extent to which postural and locomotor

behavior affects the morphology of the last lumbar posterior elements. The last lumbar

vertebra was chosen as previous research has correlated statistical differences between

species, suggesting that its morphology is most likely to reflect locomotor patterns

(Shapiro, 1993). In order to determine if stresses and strains resulting from postural and

locomotor behavior affect the morphology of the posterior elements, it is necessary to

compare species with similar postural behavior but different locomotor repertoires. Four

orthograde species with diverse locomotor behaviors were chosen for the study: Gorilla

gorilla, Pan troglodytes, Pongopygmaeus, and Homo sapiens. Locomotor behavior

cannot accurately be described using gross categories such as "knuckle-walker." Despite

the fact that primates generally have a dominant form of locomotion, they engage in

diverse locomotor behaviors that should not be ignored. Outlined below are the

locomotor behaviors of the orthograde species included in this study.

In the adult gorilla, terrestrial knuckle walking accounts for 94% of the distance

traveled by mountain gorillas. They rarely engage in climbing (vertical) and engage in

bipedalism, leaping, brachiation, or bridging even less frequently (Shapiro, 1991; see also

Tuttle and Watts, 1985). Lowland gorillas appear to be more arboreal than mountain

gorillas (Shapiro, 1991; see also Dixon, 1981; Fleagle, 1988). However, they were









included in the gorilla sample because "they feed, rest, and sleep on the ground where

they move by quadrupedalism," (Shapiro, 1991: 17).

Pan troglodytes most frequently engage in knuckle-walking quadrupedalism.

This locomotor behavior accounts for 86% of their locomotor activity. However, this

species also engages in, from most to least frequent, quadrumanous climbing and

scrambling (11%), arm swinging and bipedalism (1%), and leaping (< 1%). Only 16% of

this species' locomotion is arboreal (Shapiro, 1991; see also Doran, 1989).

Pongopygmaeus is the most arboreal of all the species included in the study.

Quadrumanous scrambling is the most important locomotor behavior for this species.

However, they also engage in the following behaviors, listed from most to least frequent:

brachiation, tree swaying, quadrupedal walking, and climbing, (Shapiro, 1991, see also

Sugardjito, 1982; Sugardjito and van Hooff, 1986).

Homo sapiens are the most orthograde of all the species included in the sample.

They are capable of a variety of locomotor behaviors, such as walking, running,

climbing, swimming, and hanging. However, bipedal walking is their single most

important form of locomotion.

Table 1 describes the taxonomic sample for the study. As can be seen from the

table the gorilla sample is comprised of 17 adult specimens of known sex. The

chimpanzee sample is comprised of 18 adults specimens. There were 6 male specimens

and 12 of unknown sex. The orangutan sample is quite small and consists of 4 male

specimens and 1 specimen of unknown sex. The human sample consists of 19 adult

specimens, 17 males and 2 females. The specimens were from an older population (x =

57.47 years, sd = 11.57years).









Table 1. Sample description.
Taxonomic Designation Sex Total
Gorilla Sample
Gorilla gorilla gorilla Male 7
Gorilla gorilla gorilla Female 6
Gorilla gorilla beringei Male 2
Gorilla sp indent Male 1
Gorilla sp indent Female 1
Total 17
Pan troglodytes Sample
Species Designation Sex Total
Pan troglodytes Male 6
Pan troglodytes Unknown 12
Total 18
Pongo pygmaeus Sample
Taxonomic Designation Sex Total
Pongo pygmaeus Male 4
Pongo pygmaeus Unknown 1
Total 5
Homo sapiens Sample
Taxonomic Designation Sex Race Decade of Life Total
Homo sapiens Male White Fifth 4
Homo sapiens Male White Sixth 4
Homo sapiens Male White Seventh 3
Homo sapiens Male White Eighth 3
Homo sapiens Male White Ninth 1
Homo sapiens Male Black Fifth 1
Homo sapiens Male Black Sixth 1
Homo sapiens Female White Fifth 1
Homo sapiens Female Black Seventh 1
Total 19

Specimen Inclusion Criteria

Specimens were collected from the American Museum of Natural History,

Departments of Mammalogy and Anthropology. With the exception of two Gorilla

specimens, all of the non-human specimens were collected from the Department of

Mammalogy. The human specimens were collected from the Department of


Anthropology's Morphology Collection.









Only adult specimens were included in the study. Adulthood was defined by

epiphyseal fusion and dental criteria. Further, for the non-human primates only wild

caught specimens were included.

Before collecting data, each specimen was carefully checked for breakage and

pathology. Given the limited number of specimens available, a vertebra was not

discarded outright for data collection if pathology, breakage, or cut marks were present.

If the pathology was very limited data was not collected from the predetermined points

on areas where the pathology was present. Similarly, if the vertebrae had breakage or

were cut, data was not collected from the area with the breakage or cut marks. The

Pongopygmaeus sample posed a special challenge. It was noted by Schultz (1941) that

even though Pongopygmaeus can be said to reach adulthood by dental criteria it is

possible that the epiphyses can remain unfused. In the present study, dental criteria were

used to define orangutan adults, however specimens were rejected if the last lumbar

vertebra showed incomplete epiphyseal fusion. This criterion severely limited the Pongo

pygmaeus sample size.

Data Collection Instrumentation

Digitized three-dimensional coordinates of predetermined points were taken on a

Microscribe model 3DX (Immersion Corporation, 801 Fox Lane, San Jose, California

95131). At the beginning of each data collection session, Accuracy Check #1 and

Accuracy Check #2, two calibration procedures recommended by the manufacturer were

performed.

Accuracy Check #1 describes the Microscribe position prior to turning it on.

The Microscribe performs a "self calibration" based upon the position at start up. Prior









to each measurement session, the Microscribe was positioned carefully to correspond

with the "home" position prior to turning it on.

Accuracy Check #2 required the program MSTest. This test procedure ensures that

the joint encoders are working properly. MSTest gives joint angles for various stylus

positions, which must be matched up to charts detailing the expected angles.

Finally, the Microscribe 3DX has a "moveable" coordinate system. The X and Y

axis are determined by the position of the swiveling arm at start up. Therefore, the

Microscribe was positioned consistently throughout the measurement procedure.

The data points were imported directly into a Microsoft Excel spreadsheet via

Inscribe software. Linear measurements were made using a Mitutoyo digital caliper,

model 573-225-10 (Mitutoyo American Corporation, 965 Corporate Boulevard, Aurora

Illinois 60504). The digital caliper was zeroed and internally calibrated, in accordance

with the operating manual, before each use.

Data Collection Procedures

The posterior element morphology of the vertebrae samples was documented via

thirty-eight landmark points. The superior and inferior vertebral centra (body) surfaces

were documented via five points each: the most anterior point at the midline, the most

posterior point at the midline, the most lateral point in the mid-coronal plane on the right

side, the most lateral point in the mid-coronal plane on the left, and the center point. Each

facet had 5 landmark points for a total of 20 points. The center point was defined as an

average of height and width. The four additional points were defined as the articular

surface end points of the minor and major axes. Finally, the pedicles were defined by

four points each: on the superior surface, at the midline, on the inferior surface, at the









midline, on the medial surface, in the midtransverse plane, and on the lateral surface, in

the midtransverse plane.

For the purpose of replicability and consistency, each point was located and

marked. The measurements required to locate the landmarks were documented using a

Mitutoyo digital caliper (accurate to 0.01mm), when possible. Some points were

recorded by sight, as accurate measurements were not possible with the calipers. This

occurred when the element of interest was so small that accurate measurements could not

be obtained. When this occurred, following the detailed measuring procedure would be

sufficient to produce consistent measurements from observer to observer as the vertebral

elements involved are extremely small. All measurements were taken at least two times

and were accepted if the two values were within 0.05mm. The two accepted values were

then averaged. The procedures used to locate the landmark points for each vertebral

element are outlined below.

The vertebral body landmark points were located first. The center point was

calculated using the measured values of vertebral width at the widest lateral point and

anterior-posterior depth at the centerline of lateral symmetry. Once this point was

located, a perpendicular set of axes was drawn on the vertebral body using a straight

edge, aligned with the anterior-posterior centerline. The four additional points were

located at the points were these axes intersected the edge of the vertebra superior

vertebral body.

The facets points were located by first measuring the facet height and width. It

should be noted that the author found that facet morphology varied greatly; however, the

morphology could be divided into five general types as shown in Figure 1. The center









was located by taking half the value of both the measured height and width, and

positioned in accordance with Figure 2. Once the center point was located, the center

point as well as the facet shape category was used to define the major axis of the facet.

Once the major axis was located, the minor axis was defined as the line perpendicular to

the major axis, passing through the center point. The intersections of these axes with the

edge of the facet defined the four additional points. The focus of this project is the major

axis orientation. However the extent to which the major axis is larger than the minor axis

may be of interest to the reader. Therefore, this value was calculated as the base ten

logarithm of the ratio of the major to minor axis and is included in Appendix 1.







v- i -i "



Figure 2. Facet shape. This figure shows the five major facet shapes that were seen
during data collection, as well as the how the minor and major axes were
drawn for each shape. From left to right, Type I, Lateral Oval, Type II,
Square, Type III, Diagonal Oval, Type IV, Pear, and Type V, Kidney.

The pedicle landmark points were defined for the axial center of the pedicle,

between the vertebral anterior body and the lamina. The specific points were defined as

the middle point of each the superior, lateral, and inferior surface, again using height and

width measurements. The medial point had to be located visually as the calipers did not

fit into the neural canal. In order to improve the accuracy and consistency a method for

locating the medial point was developed. The midpoint of the medial pedicle was located

for both the superior and inferior surface. A line was then drawn to connect the two









points. This line was then used as a guide to locate the medial point. These points define

the plane perpendicular to the axial line. Defining the exact center of the superior-

inferior plane is not as critical as defining the lateral-medial center, which was

accomplished using the above methodology.

Once all the landmark points were measured and marked, each vertebra was placed

on a ring-stand, anterior side down, and superior side away from the ring-stand stanchion

and secured by dental wax. The specimen cannot be moved during data collection as all

landmarks were measured with respect to the origin of the Microscribe. The relative

position of the Microscribe allowed access to all the landmark points on the vertebra to

be measured without moving the vertebra. The axes of the Microscribe was

approximately aligned with the natural axes of the vertebra, such that the x, y and z axes

of the Microscribe approximated the superior-inferior, anterior-posterior and lateral axes

of the vertebra, respectively. All vertebrae were measured with a similar orientation to

the digitizer. On all specimens, the landmark points were recorded in the same order.

Further, for each specimen the landmark points were measured at least two times in order

to calculate an error rate.

Force Calculation: A Static Free Body Model

Analysis of the forces within the various elements within the vertebra is dependent

on all external forces acting on the vertebra. From mechanics, these forces are depicted

using a static free body diagram. The static free body diagram is an accepted and widely

used methodology in biomechanics (see Panjabi and White, 2001 and Ozkaya and Nordin

1998, for fundamental biomechanics techniques). A two dimensional static free body

diagram, as depicted in Figure 2, has been developed by the author to as a simplified

starting point for the determination of the forces within the vertebral posterior elements.









As can be seen from the figure, the vertical force on the vertebra from the body above is

supported by the forces from the inferior disk and the inferior facets. To determine how

the force is transmitted, it is necessary to document the size and orientation of the load

bearing surfaces and the elements of interest for calculations of the load. Sizes of various

vertebral elements have been extensively documented as previously discussed.

Orientation of the facets and pedicles can be documented via three-dimensional vectors,

which is key for determining the angles in the static free body method. It is important to

stress, however, that the static free body model is a simplified starting point that operates

under certain assumptions. First, the model assumes that the force on the facets can be

accurately modeled as normal to the center of the facet. The initial model does not

contain any information regarding muscle force or ligaments. The work of Frank

Holdsworth (1963) demonstrated that the posterior element ligaments are essential to

stability in the spine. Finally, the accuracy of the model is dependent upon the accurate

knowledge of the vertebra's in vivo position.

Fortunately, a study conducted by Albert Schultz and colleagues validated a

biomechanical model for the calculation of load on the lumbar spine. In this study they

compared values predicted by a biomechanical model to intradiscal pressure and

myoelectric signals. They also correlated the effects of intraabdominal pressure on spinal

loading. Their results indicate that a biomechanical model is "valid in the [loading]

situations that were examined," (Schultz et al., 1982: 717). They also measured

intraabodominal pressure and concluded, "the intraabdominal pressured were not large

and seldom had a major influence on the overall mechanics of the trunk," (Schultz et al.,

1982: 720). Given these conclusions, a biomechanical model that takes into account











vertebral geometry and muscle action should be an accurate predictor of the loads borne


by the spine.


Vertical Force


Facet Normal Force


Disk Nornmal Force


Facet Normal
Force


Vertical Force


Diuk Normal
Force


Figure 3. Free body diagram. This figure depicts a two-dimensional free body diagram.
This is a simplified depiction of forces acting on the inferior facets. However,
the figure clearly demonstrates the necessity of facet and pedicle orientation
for force calculation. (Source: Drawn by Author).

The vectors needed to develop the free body model are described below:


1. The unit vector normal to superior facet. This is the vector normal to the center of
the superior facet, divided by the vector length (Figure 4).


2. The unit vector normal to inferior facet. This is the vector normal to the center of
the inferior facet, divided by its length (Figure 5).


3. The unit vector in the axial direction of pedicle. This vector describes the axial
direction of the pedicle vector, divided by its length (Figure 6).























Figure 4. Normal vectors to the superior facets, right and left (Source: Shanna E.
Williams.)


Figure 5. Normal vectors to the inferior facets, right and left (Source: Shanna E.
Williams).


Figure 6. Axial vectors to the pedicles, right and left (Source: Shanna E. Williams)









The above vectors can be calculated for individual specimens. However, it is

desirable to develop general species models. In order for such a generalized model to be

possible, it is necessary that for each vector a species mean direction and mean resultant

length is found. This is the measure of concentration, analogous to a standard deviation,

which describes vector concentration as a value between zero and one. A value of one

indicates a high concentration, i.e. all vectors are parallel and a value of zero indicates no

concentration and randomly orientated vectors. The mean resultant length is an

extremely important measure with respect to the development of species models. A low

concentration would indicate that there is great within species variation with respect to

vector direction. This in turn indicates that the mean value lacks any real meaning. A

high concentration indicates that there is a true species mean, which can be used to

develop species specific models.

Statistical Testing

The second goal of this project is to analyze the mean and variance of facet and

pedicle orientation. The vectors that were calculated for the static free body diagram

were also subjected to statistical analysis, as these vectors are crucial to the calculation of

stresses and strains in the vertebrae. However, additional vectors were calculated, as they

may be biomechanically significant. These vectors were the superior and inferior major

axis vectors and the pedicle major axis vectors. The facets potentially allow sagittal,

lateral, and axial rotation. The major axis of the facet may indicate the direction in which

the facets allow the most rotation.

The pedicle is the bony bridge between two compressive columns and it has been

hypothesized to be subject to bending moments. The major axis indicates the direction









on which the pedicle is oriented to resist the most bending force. However, it has not

been conclusively proven that the pedicles are subject to bending forces.

4. Major axis of the superior facet. This is the vector with the larger magnitude
between either the superior-inferior axis vector of the superior facet or the lateral-medial
axis vector of the superior facet. The vector is then divided by its length in order to
transform it into a unit vector. (Figure 7)


5. Major axis of inferior facet. This is the vector with the larger magnitude between
either the superior-inferior axis vector of the inferior facet or the lateral-medial axis
vector of the facet. The vector is then divided by its length in order to transform it into a
unit vector. (Figure 8).


6. Major axis of pedicle at midpoint. This is the vector with the larger magnitude
between either the superior-inferior axis vector at the midpoint of the pedicle or the
lateral-medial axis vector at the midpoint of the pedicle. The vector is then divided by its
length in order to transform it into a unit vector. (Figure 9).


Figure 7. Major axis vectors of the superior facets, right and left. (Source: Shanna
Williams).


Figure 8. Major axis vectors of the inferior facets, right and left. (Source: Shanna E.
Williams)


























Figure 9. Major axis vectors of the pedicle, right and left. (Source: Shanna E. Williams)

The calculation and analysis of the above vectors will further our understanding of

the relationship of posterior element morphology and posture and locomotion. Further, it

is a first step in the development of a model to calculate actual stresses and strains in the

posterior elements. This information will enrich our knowledge with respect to the

relationship of posterior element morphology to load bearing.














CHAPTER 4
DATA TRANSFORMATION

In order to document the orientation of the facets and pedicles, it is necessary to

record each vertebra's morphology, as expressed by the 38 landmark points, in three-

dimensions. Fortunately, recent advances in computer technology have made this

possible and have given anthropological researchers a new tool with which to study the

skeletal form. The digitizer, in this case a Microscribe 3DX, records x, y, and z

coordinates with respect to its own origin. Using the information provided by the

digitizer, the researcher must transform the data into useful information. In this study,

the goal, of course, is to transform the coordinate data into unit vectors that describe facet

and pedicle orientation with respect to a common coordinate system. This chapter

describes in detail how a coordinate system for the vertebra was selected and how the

measured quantities were transformed to the coordinate system.

The transformation process relies on three concepts from vector algebra;

determining a vector from two points in space, the concept of a unit vector and, a vector

multiplication operation (cross product).

Given point 1 in space with coordinates (xi,yi,zi) and point 2 with coordinates

(x2,y2,Z2) the vector from point 1 to point 2 is found as follows:

S= (x2- x)i + (y,- y)j + (z2- Z)k

Where 1, J, and k are unit vectors in the x, y and z directions, respectively.









Substituting in ax, ay and az for (x2 xl), (y2 yi), and (z2 z1), the vector

becomes:

= (ax)T + (ay)j +(az)k

Because i, j, and kare all mutually orthogonal, Pythagorean Theorem can be used

to determine the length of vector a between point 1 and point 2, as follows:

S (ax)2 +(ay)2 (az)2

Vectors and their multiples all have the same direction. If one were to multiply a

vector by the ratio of one to its length, the vector would have the same direction but

would have a length of one. This is the definition of a unit vector. Mathematically, it can

be expressed as follows:

n~- d + -a + az
a a + Ya ja

The third vector algebra concept, the cross product, can be expressed in two ways

as follows:

Sx b= fid b( in


Where: d = vector d with magnitude IC

b= vector b with magnitude H

n= vector normal (perpendicular to both vectors a and b with magnitude 1)

0= angle between vectors a and b.

The cross product can also be calculated using the determinant, as follows:










i jk
a Xb a- a, a,



X b = (ab -a b )i + (a b ab )j + (a b, -abx)k


Where: = vector d with magnitude P

S= vector b with magnitude b


i, j, and k = vectors with magnitude 1 in the x, y and z directions respectively

ax, ay, and az, = components of vector a in the x, y and z directions respectively

bx, by, and bz, = components of vector b in the x, y and z directions respectively

The first mathematical expression shows that the magnitude of the cross product is

equal to the product of three variables, the magnitude of vector d, the magnitude of

vector b and the sine of the angle between vector d and vector b Of note, directional

vectors are not necessarily tied to a particular point in space, that is all parallel vectors

have the same directionality; however, an important consequence of the cross product

calculation is that vector d and vector b are visualized as coplanar. The family of

parallel planes containing the two vectors can be uniquely defined by the normal vector

perpendicular to both vector d and vector b .

The second expression of the cross product gives the vector in its coordinate form.

As stated previously, our data is in the form of x, y, and z coordinates. The coordinates

can also be thought of as a vector from the origin to the coordinate of the point.

Therefore, vectors from the origin can be expressed as their x, y, and z coordinates which









are represented as ax, ay, and az for vector a and bx, by, and bz for vector b These

values can be used in the cross product calculation.

The goal of the data transformation procedure was to determine three-dimensional

coordinates of the measured points with respect to a defined coordinate system based on

the morphology of the specimen. The initial step in this process was to choose the

specimen-referenced axes. The axes that were chosen has the x direction in the anterior-

posterior (anterior positive) direction, the y direction was in the right to left lateral (left

positive) direction, and the z direction was in the superior-inferior (vertical) direction (up

positive) of the main body of the vertebra.

Several translations and rotations of one coordinate system to the next were

required, beginning with the coordinate system of the Microscribe and ending with the

gravity vector referenced coordinate system. Several intermediate coordinate systems

were also required. The various coordinate systems are summarized in Table 2.

Table 2. Data transformation coordinate systems.
Coordinates Nomenclature Change from previous coordinate system
X,Y, & Z Microscribe referenced Data as originally recorded
X,Y, & Z Intermediate Axes changed to approximate specimen referenced axes
X1,Y1, & Z1 Intermediate Move origin to approximate sample referenced origin
X2,Y2, & Z2 Intermediate Rotated about Zi axis by the first Euler angle q
X3,Y3, & Z3 Intermediate Rotated about X2 axis by the second Euler angle 0
X4,Y4, & Z4 Intermediate Rotated about Z3 axis by the third Euler angle Vf
x, y, & z Sample referenced Translate origin to center of inferior body
xl, yl, & zi Gravity referenced Rotated about y axis by inferior body-horizontal angle c)

The Microscribe records points with respect to its internal reference axes.

Therefore, the specimen being digitized cannot be moved with respect to the digitizer

reference during the digitizing process. In order to collect all the points of interest to the









study, it was necessary to have the anterior surface of the body in the approximate

direction of the digitizer's Z axis, while the superior surface approximated the digitizer's

Y axis and the lateral portions of the vertebra approximated the digitizer's X axis. The

first step was to switch the axes nomenclature from that of the Microscribe to that

approximating the vertebral axes. The resulting renamed vector increases in the Z

direction corresponding to increases in the superior direction and increases in the X

direction corresponding to increases anteriorially. This was accomplished by changing

the (X, Y, Z) nomenclature to the approximate vertebral reference axis (X, Y, and Z), as

follows:

* X-axis values for the approximate vertebral coordinate system were set as the
negative of the measured Z values
* Y-axis values for the vertebral coordinate system were set as the measured X
values.
* Z-axis values for the vertebral coordinate system were set as the negative of the
measured Y values.

The next step for aligning the specimens was to reference the data to an origin point

on the vertebra. For this study, the center point of the inferior body was chosen. Though

this designation is somewhat arbitrary, this point was chosen for ease of data

interpretation. Problematically, however, the inferior surface of a vertebra is

asymmetrical and is not a flat surface. It was necessary to correct for these limitations

when locating the origin point in the three directions. For the y-direction, the origin was

selected as a point halfway between the measured left and right landmark points. In the x

direction the origin point selected was half way between the ventral and dorsal landmark

points. The remaining point, in the z direction, was defined as the average of the four

measured landmark points (left, right, ventral and dorsal). The appropriate center

coordinate was subtracted from the data in the X,Y,Z coordinate system to determine the









coordinates in the new X1, Y1, Zi intermediate coordinate system. This essentially

translated the origin from that defined by the Microscribe to a point that approximates

the center of the inferior vertebral centra (body). The asymmetry of the vertebra may

introduce a slight error with respect to the coordinate system. However, subsequent

rotations will not produce additional error.

Once the data were translated to the approximate vertebral coordinate system and

referenced to the approximate vertebral origin, it was then necessary to rotate the data.

This step is very important because although every effort was made to consistently line

up the individual specimens with the axis of the Microscribe, rotational misalignment of

a few degrees was unavoidable and is almost certainly inconsistent from sample to

sample. To correct the misalignment, the data was rotated to correspond to the selected

vertebral reference system using Euler angles. The "x convention" for Euler rotation is

used.

The Euler angle rotation formula requires three distinct rotations. First, the

coordinates are rotated about the Z-axis, then about the X-axis, and about the Z-axis a

second time.

For the first rotation, the data were referenced to an intermediate axis (X2, Y2, and

Z2). The data were rotated an angle 4 about the Zi-axis such that the X2 axis is contained

in both the X1-Y1 plane of the old coordinate axes and the x-y plane of the vertebra

referenced axis. The data are then referenced to a second intermediate coordinate axes

system (X3, Y3, and Z3). For this rotation, the vertebra is rotated an angle 0 about the X2

axis such that the X3- Y3 plane of the intermediate coordinate axes system coincides with

the x-y plane of the vertebra reference system, or stated another way, such that Z3









coincides with the z-axis of the vertebra reference system Finally, the data are

referenced to the vertebra reference system (x, y, and z) by rotating an angle Y about the

Z3 axis to bring the Y3 axis parallel with the y-axis in the vertebral reference axis.

In order to use the Euler Angle Rotation procedure, it is necessary to calculate the

Euler angles (4, 0, and y). Fortunately, the vectors necessary for this calculation can be

derived directly from the data taken and the three vector algebra concepts noted above.

Recall that one property of the cross product calculation is that it can be used to

determine a vector that is perpendicular to both vectors used in the cross product.

Further, the angle between two vectors can be determined using the cross product

calculation and the arcsine function.

The first Euler angle q is the angle between the X1-axis and the intersection of the

XI-Y1 plane of the X1, Y1, and Zi coordinate system and the x-y plane of the vertebra

referenced system. In some preliminary data reduction, the x-y plane of the vertebra

coordinate system (inferior body plane) was approximated as containing the y direction

vector formed by subtracting the coordinates left edge from the right edge of the inferior

body and the x direction vector, formed by subtracting the coordinates of the posterior

edge from the anterior edge coordinate of the inferior body. This definition was

problematic, because the main bodies of the specimen contained some wedging, not only

in the anterior posterior direction but also in the lateral direction. The consequence was

that the z axis did not pass through the center of the anterior body plane. To minimize

the variability between specimens, the x-y plane was defined by the z axis, which by

definition is perpendicular to both the x and y axes. The z direction was determined by

subtracting the inferior body center from the superior body center. This definition is also









attractive because the vertebral body is a compressive member, with the force being

transmitted by the pressure in the vertebral discs. The superior and inferior forces would

be centered within the respective body. The cross product of the z-axis and the normal

vector to the X1-Y1 plane is perpendicular to both normal vectors and therefore contained

in both planes. In other words, the cross product of the normal vectors is in the direction

of the intersection of the two planes. The sine of angle 4 is determined by taking the

arcsine of the cross product of a unit vector in the X1 direction and a unit vector in the

direction of the intersection of the X1-Y1 plane and the x-y plane.

The data are transformed to the new coordinate system (X2, Y2, and Z2) using the

following matrix:

Scos sin O'
sin cos q 0
0 0 1

Note that only the X2 and Y2 coordinates are changed from the previous X1 and Yi

coordinates, the Z2 coordinates remain the same as the Zi coordinates. That is because

the rotation is about the Zi-axis. Note the sine and cosine functions for the rotation angle



The second Euler angle 0 is the angle between the X2-Y2 plane and the x-y plane.

It is also the angle between the normal vector to the X2-Y2 plane and the normal vector to

the x-y plane. The normal vector to the X2-Y2 plane is the Z2 axis. The normal vector to

the x-y plane has to be recalculated because of the rotation, but is still the coordinates of

the inferior body center subtracted from the superior body center. Using unit vectors, the

cross product of the two normal vectors produces a vector perpendicular to the two









vectors with a magnitude of the sine of the angle between the two vectors. The arcsine of

the magnitude of the cross product gives the angle 0.

The data are transformed to the new coordinate system (X3, Y3, and Z3) using the

following transformation matrix:

1 0 0
0 cos 0 sin 0
0 -sin cos 0

Angle \, the third Euler angle, is the angle between the X3 axis and the x-axis (or

between theY3 axis and the y axis). As noted previously, the x-axis is approximated by

the vector between the posterior and anterior inferior body of the vertebra and the y-axis

is approximated by the vector between the right and left edge of the inferior body of the

vertebra. These two definitions however, do not necessarily produce perpendicular

vectors. For the purposes of this thesis, angle x, is defined as the angle between theY3

axis and the y-axis. The y-axis vector is determined using the coordinates of the right

and left inferior body measurements and then is made into a unit vector. The cross

product of the unit vector in the y direction and the unit vector in the Y3 direction gives a

vector perpendicular to both vectors and of length equal to the sine of the angle between

the two vectors (angle y). The angle x is found using the arcsine function. The data are

transformed to the vertebral referenced axis (X4, Y4, and Z4) using the following matrix.

Scos y sin g O
-sin f cos/ 0
0 0 1

Finally, a linear translation of the X4, Y4, and Z4 data to the x, y and z axis was

required to re-center the origin of the axis system between the left and right and dorsal









and ventral edges of the inferior body plane. Of note, the rotations could have been

performed prior to the first translation, avoiding a second translation; however, making

the translation first facilitated checking the rotation process by looking at coordinate data.

The small change in the center of the vertebra, with respect to the origin of the axis,

occurred due to the asymmetry of the vertebrae and the shifting of the data points during

the three rotations. This produced the coordinates of the 38 reference points in the

vertebral referenced coordinates (x, y, z).

Once the measured coordinates were referenced to the vertebral coordinate system,

it was then possible to calculate the unit vectors of interest using very simple

calculations:

* The superior-inferior axis vector at the midpoint of the pedicle was calculated by
subtracting the inferior coordinates from the superior coordinates.

* The lateral-medial axis vector at the midpoint of the pedicle was calculated by
subtracting the medial coordinates from the lateral coordinates.

* The superior-inferior axis vector of the superior facet was calculated by subtracting
the inferior coordinates from the superior coordinates.

* The lateral-medial axis vector of superior facet was calculated by subtracting the
medial coordinates from the lateral coordinates.

* The superior-inferior axis vector of inferior facet was calculated subtracting the
inferior coordinates from the superior coordinates.

* The lateral-medial axis of inferior facet was calculated by subtracting the medial
coordinates from the lateral coordinates.

* The normal vector to the superior facet was determined from the cross product of
the superior-inferior axis vector of superior facet and the lateral-medial axis vector
of superior facet. Recall one of the properties of the cross product calculation is
that it can be used to find the vector perpendicular to the plane of two vectors.

* The normal vector to the inferior facet was determined from the cross product of
the superior-inferior axis vector of inferior facet and the lateral-medial axis vector









of inferior facet

* The axial direction vector of the pedicle was determined from the cross product of
superior-inferior axis vector at the midpoint of pedicle and the lateral-medial axis
vector at the midpoint of the pedicle.


The following unit vectors were calculated:

* The major axis of pedicle at its midpoint. This is the vector with the larger
magnitude between either the superior-inferior axis vector at the midpoint of the
pedicle and the lateral medial axis vector at the midpoint of the pedicle. The
vector was then divided by its length in order to transform it into a unit vector.

* The major axis of the superior facet. This is the vector with the larger magnitude
between either the superior-inferior axis vector of the superior facet and the lateral -
medial axis vector of the superior facet. The vector was then divided by its length
in order to transform it into a unit vector.

* The major axis of inferior facet. This is vector with the larger magnitude between
either the superior-inferior axis vector of the inferior facet and the lateral medial
axis vector of the facet. The vector was then divided by its length in order to
transform it into a unit vector.

* The unit vector normal to superior facet. This is the vector normal to the center of
the superior facet, divided by its length.

* The unit vector normal to inferior facet. This is the vector normal to inferior facet,
divided by its length.

* The unit vector in the axial direction of pedicle. This vector describes the axial
direction of the pedicle vector, divided by its length.

The vectors calculated via this data transformation were referenced to the vertebra

itself. However, this is not necessarily reflective of the vertebra's in vivo position. For

the purpose of the statistical analysis, it is important to calculate these vectors with

respect to the gravity vector as well for comparative information. In this study, an

orthograde posture was chosen. This position was chosen as the literature contained

some information regarding the vertebral position for this posture.









Radiologically, the lumbosacral junction is well documented in humans. From

radiological evidence the angle (0 ), the forward rotation of the last lumbar vertebra can

be approximated. In this position the human last lumbar vertebra is rotated forward

approximately thirty degrees from the horizontal (Yochum and Rowe, 2005).

Unfortunately, this angle is not well documented for non-human primates. Therefore,

this angle was approximated using indirect evidence. AH Schultz (1961) documented the

exact vertebral curvature of the non-human specimens by making molds of the abdominal

cavity of eviscerated specimens. His diagrams demonstrate a kyphotic curvature in the

thoracic region. However, in lower lumbar region, the diagrams indicate that the spine is

held perpendicular to the ground and therefore the last lumbar would not be significantly

rotated.

The human coordinate system (xi, yi and zi) was rotated about the y-axis to the

approximate vertebral position with respect to the gravity vector using the following

matrix:

Scos ) 0 sin o)
0 1 0
-sino 0 cos o)

The precision to which landmark points are measured is of great interest. Errors

can be induced by the operator (intraoperator error) or by the measuring instrument

(instrument error). To minimize the combined intraoperator and instrument error, the

landmark points were taken twice on each specimen. While individual points can be

compared, the calculated directions are based on several vectors calculated by finding the

difference between two measured points. Therefore, the vectors were calculated in real

time for both sets of measurements and the direction of the results were compared. In









order to calculate angle error between the vectors found from the two sets of data, the

cross product calculation was used. Recall that from the cross product, the sine of the

angle between the two vectors can be found. The unit normal vectors calculated from the

two sets of data were calculated then the cross product was taken. The angle between the

two vectors is the arcsine of the length of the normal vector. This is expressed in radians,

which are easily converted to degrees by multiplying the radians by 180 and dividing by

pi. The data transformation Excel spread was programmed to calculate this value for six

of the unit directionality vectors. The data points were rejected if the angle of difference

between the measurements exceeded five degrees. The intraoperator error values were

calculated for six of the 12 vectors calculated (Appendix B). The unit vectors were the

vectors normal the facets and the vectors axial to the pedicles. These vectors were

calculated as they represent the highest errors values possible. Recall that the normal and

axial vectors were calculated from four data points and the major axis vectors are

calculated from two of the four same landmark points. Please see the Appendix B for a

detailed description of the accepted error rates.














CHAPTER 5
STATISTICAL ANALYSIS AND RESULTS

Directional data cannot be analyzed using traditional univariate or multivariate

statistics. Fortunately, there are methods available for the analysis of this type of data.

To enrich our understanding of the posterior element morphology, both descriptive and

comparative statistical methodologies were used to analyze the vector data.

Combining Mixed Sex Samples

Recall from Chapter Three that the species samples are mixed sex. Pooling the

males and females of a species increases sample size. However, pooling males and

females can be problematic, especially for a species with extreme sexual dimorphism, as

this can result in a misleading mean (or Mean Direction) and a large standard deviation

(or Mean Resultant Length), making the data insensitive to pair-wise testing. Therefore,

before pooling the males and females, it is desirable to determine that they have equal

mean directions. However, this is not always possible as was the case with three of the

four species included in this project.

The Pongopygmaeus and Pan troglodytes samples were composed of specimens of

both known and unknown sex. Therefore, it was not possible to determine if the males

and females had equal mean directions. When reviewing the results, it is important to

keep in mind that these species were pooled without the benefit of the test of equal

means.

The test for equal male and female means for the Homo sapiens sample was not

conducted for a different reason. In the human sample, the sex of each specimen is









known, however, the sample is overwhelmingly male (17 males, 2 females). A

statistician was consulted and it was determined that a test to determine if the males and

females can be pooled is not meaningful given the extremely small female sample size.

However, the reader again is cautioned that if the sample was more balanced, this test

would not only be appropriate, it would be required.

Fortunately, the sexes of all specimens are known for the most sexually dimorphic

species included in the sample, Gorilla gorilla. Further, this sample is fairly balanced.

Therefore, the test of equal means direction was conducted.

In order to confidently pool cross-sex specimens, the sex specific samples are

required to have not only equal mean direction, but also equal concentration. The equal

mean direction hypothesis was tested using corrected version of the Likelihood Ratio

Test given in Mardia and Jupp (2000). This corrected test was developed by Presnell and

Rumcheva (2005). It is corrected in the sense that instead of referring the test statistic to

a chi-square distribution, it is referred an F distribution. This has a better performance in

terms of Type I error. The hypothesis of equal mean directions is rejected for large

values of the test statistic. The results are reported in the data tables as p-value 1.

Equal concentration was verified using a pair-wise specific test found in Mardia

and Jupp (2000). The results are reported in the data tables 5-1 as p-value 2. The null

hypotheses of equal mean direction and equal mean concentration were not rejected if the

p-value > 0.05.

Table 3 describes the statistical results of the male vs. female gorilla tests.

Descriptive statistics, mean direction (MD) and mean resultant length (MRL) are also

listed in the table. A detailed description of these values is given in the next section.










Table 3. Pooling data, Gorilla sample.
Superior Facet
Left Right
Major Axis Normal Major Axis Normal
Sample Size (M,F) (9,7) (-9.7) (10,7) (10,7)
Gorilla M MD -0.15, 0.67, 0.73 -0.59, -0.66, 0.46 -0.19, -0.75, 0.63 -0.53, 0.69, 0.48
MRL 0.78 0.95 0.86 0.98
Gorilla F MD -0.22, 0.49, 0.84 -0.56, -0.75, 0.36 -0.35, -0.58, 0.74 -0.56, 0.75, 0.36
MRL 0.86 0.98 0.84 0.98
P-valuel 0.6485 0.3721 0.5968 0.1652
P-value2 0.2831 0.0975 0.8084 0.3849
Inferior Facet
Left Right
Major Axis Normal Major Axis Normal
Sample Size (M,F) (7,7) (7,7) (10,7) (10,7)
Gorilla M MD -0.02, 0.18, 0.98 0.58, 0.77, -0.26 0.12, -0.29, 0.95 0.48, -0.80, -0.35
MRL 0.76 0.97 0.74 0.95
Gorilla F MD 0.20,0.045, 0.98 0.54,0.81, -0.23 -0.01, -0.43, 0.90 0.52, -0.81, -0.28
MRL 0.82 0.97 0.74 0.96
P-value 1 0.6543 0.7903 0.7874 0.7258
P-value2 0.3222 0.4943 0.5638 0.3223
Pedicle
Left Right
Major Axis Axial Direction Major Axis Axial Direction
Sample Size (M,F) (8,7) (8,7) (10,7) (10,7)
Gorilla M MD 0.27, -0.29, 0.92 -0.87, 0.33, 0.38 0.30,0.14,0.94 -0.93, -0.19, 0.32
MRL 0.99 0.98 0.99 0.97
Gorilla F MD 0.20, -0.25, 0.95 -0.85, 0.43, 0.31 0.20, 0.21, 0.96 -0.93, -0.28, 0.25
MRL 0.99 0.97 0.99 0.96
P-valuel 0.2864 0.3725 0.122 0.4635
P-value2 0.4241 0.8339 0.4717 0.7506



The results in Table 3 indicate that for all the vectors, male and female gorillas did

not reject the null hypotheses of equal mean concentration. The test for equal mean

direction indicates that the null hypothesis was not rejected for all the vectors tested.


Therefore, the male and female specimens were pooled.









Mean Directions and Mean Resultant Lengths

The descriptive statistics reported in Table 4 give information regarding mean

values and the variability about that mean. In the case of directional data, the statistical

data is expressed as mean direction (MD) and mean resultant length (MRL). The mean

direction (MD) is a unit vector, resulting from the means of the x, y, and z coordinates.

These values are divided by the mean resultant length (MRL) so that the vector becomes

a unit vector. The mean resultant length (MRL) is a measure of concentration that is

analogous to a standard deviation. The mean resultant length is obtained by applying

Pythagorean Theorem (length = square root of the sum of the squares of the three mean

values) to the mean direction vector, prior to its conversion to a unit vector. Descriptive

statistics are given for all four species of the sample.

The first goal of this project is to test the feasibility of the development of species-

specific vector based models for calculating force magnitude in the posterior vertebral

elements. In order to develop such a model, there must be tolerable within-species

diversity with respect to the axial and normal vectors calculated. The mean directions

(MD) and mean resultant lengths (MRL) for each species are listed in the tables below.

The major axis vectors, while not calculated for the first goal of this project, were

included.

Table 4. Descriptive statistics.
Superior Facet
Left Right
Vector Major Axis Normal Major Axis Normal
Sample Size (19,16,18,5) (19,16,18,5) (16,17,18,5) (16,17,18,5)
Homo MD -0.36,0.77,0.52 -0.62, -0.62,0.48 -0.43, -0.77,0.47 -0.56,0.64,0.53
MRL 0.84 0.97 0.89 0.98
Gorilla MD -0.18,0.59,0.79 -0.58, -0.70,0.42 -0.33, -0.67,0.67 -0.55,0.72,0.43
MRL 0.82 0.96 0.86 0.98










Table 4. Continued
Superior Facet
Left Right
Vector Major Axis Normal Major Axis Normal
Sample Size (19,16,18,5) (19,16,18,5) (16,17,18,5) (16,17,18,5)
Pan MD -0.14,0.43,0.89 -0.45, -0.83,0.34 0.08, -0.33,0.94 -0.34,0.88,0.33
MRL 0.84 0.99 0.95 0.99
Pongo MD 0.35,0.31,0.89 -0.64, -0.61,0.46 0.25, -0.59,0.77 -0.72,0.45,0.54
MRL 0.97 0.96 0.94 0.97
Inferior Facet
Left Right
Vector Major Axis Normal Major Axis Normal
Sample Size (19,14,18,5) (19,14,18,5) (19,17,18,5) (19,17,18,5)
Homo MD 0.85, -0.21,0.49 0.51,0.69, -0.50 0.83,0.03,0.55 0.47, -0.71, -0.52
MRL 0.67 0.98 0.58 0.98
Gorilla MD 0.09,0.11,0.99 0.56,0.79, -0.24 0.07, -0.35,0.93 0.49, -0.81, -0.32
MRL 0.78 0.97 0.74 0.96
Pan MD 0.53,0.02,0.85 0.39,0.89, -0.25 0.54, -0.07,0.84 0.31, -0.92, -0.25
MRL 0.90 0.99 0.92 0.99
Pongo MD 0.56,0.15,0.81 0.51,0.70, -0.49 0.48, -0.25,0.84 0.63, -0.56, -0.54
MRL 0.99 0.98 0.99 0.96
Pedicle
Left Right
Vector Major Axis Axial Direction Major Axis Axial Direction
Sample Size (17,15,18,2) (17,15,18,2) (17,17,18,4) (17,17,18,4)
Homo MD 0.51,0.84, -0.20 -0.76,0.55,0.36 0.56, -0.82,0.15 -0.83, -0.48,0.28
MRL 0.82 0.98 0.56 0.98
Gorilla MD 0.24, -0.27,0.93 -0.86,0.37,0.35 0.26,0.17,0.95 -0.93, -0.23,0.29
0.99 0.97 0.98 0.97
Pan MD 0.37, -0.28,0.88 -0.80,0.39,0.46 0.38,0.18,0.91 -0.86, -0.29,0.42
MRL 0.99 0.99 0.99 0.99
Pongo MD 0.61,0.59,0.53 -0.80,0.53,0.29 0.27, -0.15,0.95 -0.89, -0.35,0.30
MRL 0.57 0.99 0.69 0.98

The descriptive statistics, MD and MRL values, are given in Table 4. With respect

to the superior facet vectors, the left major axis vectors indicate relatively low

concentration for Homo, Gorilla, and Pan while the Pongo vectors have high

concentrations. Interestingly, for the right major axis vectors, Homo and Gorilla have









low concentrations while Pan and Pongo have high concentrations. For the left and right

normal vectors, all species have high concentration values.

The results for the inferior facet vectors indicate that for the left and right major

axis vectors, Homo and Gorilla have low concentrations. The Pan and Pongo major axis

vectors have high concentration. The left and right normal vector data for all species

indicates very high concentration values.

For the pedicles, the descriptive statistics that the left and right major axis vectors

have low concentrations for Homo and Pongo. The MRL values for Gorilla and Pan

indicate that these vectors have a high concentration. The right and left normal vectors

have high MRL values for all four species. Interestingly, only humans have low

concentration for all the pedicle major axis vectors.

It is interesting to note that for the superior and inferior facet major axis vectors,

the chimpanzee and orangutan vectors have high concentrations. Comparatively, humans

and gorillas have relatively low concentrations.

Pair-wise Testing

Pair-wise tests were conducted in lieu of an ANOVA. Preliminary testing

indicated that pair-wise testing was a more appropriate for determining mean directional

differences. These preliminary tests demonstrated that it was likely that all species had

statistically different mean values. Therefore, four-way ANOVA testing would not give

results with a satisfactory level of detail.

Further the preliminary testing indicated that for the between-species pair-wise

testing, a test that assumes equal concentration was not appropriate. A test that does not

assume equal concentration, also based upon the von Mises-Fisher distribution (Mardia

and Jupp 2000), was performed. A total of 60 pair-wise tests were conducted. The










orangutan data were not included in the pair-wise tests as the sample size was quite small

(n=2-5) and the results would not be meaningful. Pair-wise tests were conducted with

both the vertebral referenced system and the data rotated to account for the in-vivo

gravity vector.

Tables 5 lists the results for the pair-wise analyses for the superior facet, inferior

facet, and pedicle vectors. The null hypothesis of equal mean direction was not rejected

for p-values <0.05. Five pair-wise tests were conducted for each sample, Gorilla vs.

unrotated Homo, Gorilla vs. rotated Homo, Gorilla vs. Pan, Pan vs. unrotated Homo, and

Pan vs. rotated Homo.

Table 5. Pair-wise results.
Test of Equal Mean Direction,
Superior Facet Vectors
Left
Major Axis Normal Major Axis Normal
Gorilla vs. Homo (Sample Size) (16,19) (16,19) (17,16) (17,16)
p-value 0.0570 0.2074 0.1866 0.0708

Gorilla vs. Rotated Homo (16,19) (16,19) (17,16) (17,16)
p-value 0.2099 <0.0001 0.2398 <0.0001

Gorilla vs. Pan (16,18) (16,18) (17,18) (17,18)
p-value 0.4338 0.0026 <0.0001 <0.0001

Pan vs. Homo (18,19) (18,19) (18,16) (18,16)
p-value 0.0005 <0.0001 <0.0001 <0.0001

Pan vs. Rotated Homo (18,19) (18,19) (18,16) (18,16)
p-value 0.0089 <0.0001 <0.0001 <0.0001
Inferior Facet Vectors
Left Right
Major Axis Normal Major Axis Normal
Gorilla vs. Homo (Sample Size) (14,19) (14,19) (17,19) (17,19)
p-value <0.0001 <0.0001 0.0004 0.0015

Gorilla vs. Rotated Homo (14,19) (14,19) (17,19) (17,19)
p-value <0.0001 <0.0001 <0.0001 <0.0001

Gorilla vs. Pan (14,18) (14,18) (17,18) (17,18)
p-value 0.0139 0.0017 0.0022 0.0005










Table 5. Continued
Test for Equal Mean Direction
Inferior Facet Vectors
Left Right
Major Axis Normal Major Axis Normal
Pan vs. Homo (Sample Size) (18,19) (18,19) (18,19) (18,19)
p-value 0.0125 <0.0001 0.1306 <0.0001

Pan vs. Rotated Homo (18,19) (18,19) (18,19) (18,19)
p-value <0.0001 <0.0001 <0.0001 <0.0001
Pedicle Vectors
Left Right
Major Axis Axial Major Axis Axial
Gorilla vs. Homo (15,17) (15,17) (17,17) (17,17)
p-value <0.0001 <0.0001 <0.0001 <0.0001

Gorilla vs. Rotated Homo (15,17) (15,17) (17,17) (17,17)
p-value <0.0001 0.0011 <0.0001 <0.0001

Gorilla vs. Pan (Sample Size) (15,18) (15,18) (17,18) (17,18)
p-value 0.0005 <0.0001 0.0013 0.0093

Pan vs. Homo (18,17) (18,17) (18,17) (18,17)
p-value <0.0001 <0.0001 <0.0001 <0.0001

Pan vs. Rotated Homo (18,17) (18,17) (18,17) (18,17)
p-value <0.0001 <0.0001 <0.0001 <0.0001

With respect to the superior facet vectors, the vectors generally have statistically

different mean directions. There are exceptions. The Gorilla vs. unrotated Homo test

indicates that the right and left major axis and normal vectors do not have statistically

different mean directions (p-values, 0.0570, 0.2074, 0.1866, and 0.0708 respectively).

The pair-wise tests for Gorilla and rotated Homo indicate that the right and left superior

facet major axis vectors do not have statistically different mean directions (p-values

0.2099 and 0.2398 respectively). Finally, the Gorilla vs. Pan pair-wise tests indicate that

for the left superior facet major axis vector they do not have statistically different mean

directions (p-value 0.4338).






54


For the inferior facet vectors, all tests except one indicate that each species has

statistically different mean directions. The exception is the Pan vs. unrotated Homo right

inferior facet major axis vector test. The pair-wise test indicated that for this vector,

humans and chimpanzees do not have statistically different mean directions.

With respect to the pedicle vectors, the results indicate that the species have

statistically different mean directions (p<0.05).














CHAPTER 6
DISCUSSION

Pair-wise Testing

An overwhelming majority of pair-wise tests indicate the each species vector had a

statistically different mean direction. However, a minority of the pair-wise tests indicated

did not reject the null hypothesis. With respect to the Homo and Gorilla pair-wise tests,

the superior facet vector results indicated that when the human vectors were not rotated to

the gravity vector, the right and left major axis and normal vectors did not have

statistically different mean directions. This would suggest that, with respect the major

axis vectors, Homo and Gorilla species have a similar direction of motion for the superior

facets. The equal mean normal vectors indicate that Homo and Gorilla have a similar

pattern of force transmission in the superior facets. However, these results are

problematic; when the human spine is held vertically to the ground, the last lumbar is

rotated forward 30 degrees from the horizontal. When the same pair-wise test was

conducted with the rotated Homo vectors, the major axis vectors indicate equal mean

directions with an even higher p-value (p-value 0.0570 and 0.1866 vs. 0.2099 and

0.2398). However, the normal vectors no longer have equal mean directions (MD),

suggesting they have different force transmission patterns in the superior facets. These

results are difficult to interpret. It would seem to indicate that despite a dramatic change

in human vectors orientation, humans and gorillas have statistically non-divergent

superior facet major axis orientation, which in turn may suggest that the superior facets

allow the greatest rotation in the same plane. However, the functional interpretation of









this is not yet clear. More research, such as the calculation of facet curvature, is

necessary to prove the assumption that the major axis facet vectors have a functional

significance.

The Pan vs. Gorilla pair-wise tests indicate that for the left superior facet major

axis vectors, they do not have statistically different mean directions. It is surprising that

the right major axis vector does not indicate an equal mean direction as well, (p-values

0.4338 and <0.0001 respectively). Interestingly, this asymmetrical pattern is echoed in

the pair-wise results for the inferior facet major axis vectors of Pan and unrotated Homo

vectors. However, the side is reversed. In this case the left major vectors indicate that

they do not have equal mean directions, while the right indicate equal mean direction, (p-

values of 0.0125 and 0.1306 respectively). The assumption that these vectors are

measuring function has not been proven. These results may indicate that they are not

accurately capturing function as intended. These asymmetrical results may also be the

result of Type I error.

Pair-wise Comparisons and Phylogeny

The pair-wise tests were conducted in three closely related species, humans,

chimpanzees, and gorillas. The phylogenetic relationship between these species is still

contested. However, molecular (Koop et al., 1989; Ruvolo, 1997; Satta et al., 2000

Salem et al., 2003), morphological (Begun, 1992; Shoshani et al., 1996), and

morphometric (Lockwood et al., 2004) studies suggest that humans and chimpanzees are

more closely related to each other than either is to gorillas. If unit vector orientation was

dependent upon phylogenetic constraint one would expect that one of two outcomes: all

species would have statistically non-divergent mean directions or alternatively humans

and chimpanzees would have statistically non-divergent mean direction, while gorillas









would have statistically different mean directions. However, the results indicate,

overwhelmingly, that each species has statistically different mean values for the vectors

calculated, suggesting that facet and pedicle orientation are not a consequence of

phylogenetic constraint.

Pair-wise Comparisons and Allometry

In order to address allometry, the sample included males and females of a highly

dimorphic species (Gorilla gorilla) as well as two species that have similar body size

(Homo sapiens and Pan troglodytes). The Gorilla test for equal means for males and

females indicate that the vectors calculated have a small dispersal and statistically non-

divergent mean directions. The Homo sapiens and Pan troglodytes pair-wise tests

indicate that despite a similarity in body size, these vectors are statistically different

between the species. While this is not conclusive, the results suggest that these vectors

are insensitive to body size alone.

Pair-wise Comparisons and Bone Biology and Behavior

To investigate whether bone size and shape are indicative of load transmission, it

is necessary to calculate the forces that these elements are subject to during normal

loading. To better understand the force data, when it is obtained, pair-wise analyses were

conducted for the "force" vectors.

When considered with respect to a biomechanical model, three vectors in the

analysis may be indicative of force transmission patterns. They are the pedicle major

axis and axial vectors as well as the vectors that are normal to the superior and inferior

facets.

A simplified two-dimensional free body diagram indicates that the mean normal

and axial vectors can be used as a starting point for developing models for force









calculation in the posterior elements. The pair-wise comparison results indicate that for

all of these vectors, each species has a statistically different mean direction. This would

suggest that each species does in fact have different force transmission patterns in the

posterior elements. It is somewhat surprising that the gorilla and chimpanzee vectors are

statistically different as the most frequently engaged locomotor behavior for both species

is knuckle-walking. However, it is not totally surprising, as ANOVA tests from Shapiro

(1991,1993) demonstrated statistically different mean values for the parameters she

measured. Perhaps these data indicate that facet and pedicle orientation is responsive to

the species' total locomotor/postural repertoire, not simply orthograde posture or the most

frequently engaged in locomotion. Further, the descriptive statistics indicate the vectors

had very high concentrations. This may suggest that facet and pedicle orientation is

highly constrained. Current, there have been no research studies to determine the extent

of morphological canalization of the primate spine. Clearly, this research is a necessary

component to a function morphological analysis of the spine as it gives important

information with respect to extent that development and function affect morphology.

The pedicle major axis vector may also be an important indicator of force

transmission patterns. The pedicles connect two compressive columns and they are

assumed to be involved in force transmission and subject to stresses and strains. The

mechanism of force transmission through the pedicles is not understood and consequently

the nature of the stresses and strains are not understood. However, it has been suggested

that one possibility is that the pedicle is being subjected to bending stresses. The long, or

major axis, of the pedicle has the ability to resist the most bending force. As would be









expected from the facet normal and pedicle axial results, the pair-wise tests indicate that

each species has a statistically different mean direction with respect to this vector.

Of note, the descriptive statistics demonstrate that Gorilla and Pan mean pedicle

major axis vectors have high concentration values. However, the human pedicle major

axis vectors have very low concentrations indicating that humans lack the uniformity of

major axis direction that was found for the non-human primates. As the exact

mechanism of force transmission is not currently known, it is not clear whether this

measure has functional significance. The work of Shapiro (1991; 1993) and Sanders

(1995; 1998) both demonstrated that human pedicles, especially at the last lumbar are

short and wide. In humans, the minor axes-major axes ratio data for the pedicles

(Appendix A) indicate that there is a general trend for lateral and superior axes to be of

fairly similar in length. This would indicate that, in cross section, human pedicles may be

more accurately described as round, rather than elliptical. This would make defining the

major axis more difficult may explain the low concentration values. More calculation of

forces acting upon the pedicle is necessary to determine if it is subjected to significant

bending forces.

Force Transmission in the Posterior Elements

The exact manner of force transmission with individual posterior elements is not

clearly understood. The most current model, the two column model of force

transmission, states that the vertebral bodies and disks form one compressive column,

while the posterior elements comprise a second compressive column. The pedicles are a

bony bridge between the two columns. It has been hypothesized that the pedicles are

subjected to significant bending stresses as a result of muscles action on the spinous or

transverse processes or alternately from the posterior elements (Bogduk and Twomey,









1987). Alternately, Bogduk and Twomey also suggested that the pedicles may be

subjected to tension from the facets locking to prevent the vertebrae from sliding forward

(Bogduk and Twomey, 1987). It has also been hypothesized that the pedicles may be

subjected to compressive axial loads from the vertebral body to the lamina (Pal and

Routal, 1986, 1987). There are four potential forces acting in the pedicle, torsion,

compression, bending and shear. The development of a biomechanical model of the

pedicles and posterior elements will help determine the extent to which these forces are

significant.

Sanders (1998) also hypothesized that facet spacing is important for the resistance

of compressive force in the posterior elements. He suggested that the more widely

spaced facets that he demonstrated in humans may create a more stable base. Intuitively

this hypothesis is satisfactory. However, the development of an accurate biomechanical

model is necessary to prove if this is a correct interpretation of this aspect of posterior

element morphology.

Feasibility of the Model

With respect to the development of species models for the calculation of stresses,

the results are very encouraging. The vectors necessary for the current simplified model,

normal and axial, all have very high mean resultant lengths. However, the model does

not incorporate the effects of muscle attachments and the subsequent stresses and strains

caused by muscle action. It should be possible to calculate that loads borne by the

lumbar spine, if the muscle forces can be accurately modeled and the position of the

vertebrae during diverse locomotor behavior is known. More research is needed

document the angles of the vertebra with respect to the gravity vector or alternately a

methodology needs to be developed that can estimate these angles. This research may









include estimation from radiographic information or possibly estimating the angles using

known biological parameters (arm, leg, and trunk length, weight, and documented

positional and locomotor behavior). However, it would be necessary to develop a

measure of accuracy for such estimations.

Vector Concentrations (MRL)

The concentration values (MRLs) are extremely important measures of within

sample variability. The measure of concentration (MRL) is a number from 0-1. A value

of one indicates that the vectors are parallel and a value of zero indicates that the vectors

are randomly oriented. When interpreting the validity of a mean value, it is necessary to

have an approximate cut-off value of for concentration. The cut off value is subjective,

and varies from project to project. However, if we had a sample of ten vectors and five

had an equal lateral mean direction and five had equal mean directions perpendicular to

the lateral vectors, the concentration (MRL) value would be -0.70. For the purpose of

building the current project a high degree of precision is required and the values were

considered to have high concentration for values of -0.95 and above.

Facet Shape

This research indicates that the species included in the sample have 5 major facet

shape types as outlined in Figure 8. Thus far, no other study has addressed the issue of

facet shape. It is important to note that these facet types were defined qualitatively, in the

interest of having a consistent major axis, and by default, minor axis designations for

each shape. It is not clear, based on this research, if these shapes are exclusive or if they

correlate with other factors. More work is necessary to quantitatively document facet

shape including facet curvature, which may lead to an understanding of correlation with

other factors. Further it has been hypothesized that the facet major axis vectors may






62


indicate a preferred direction of motion. Analysis of facet curvature will give additional

information as to whether this measure is a reasonable approximation of the direction of

facet rotation. A Fourier analysis would be appropriate for this study as orientation is an

important factor.














CHAPTER 7
CONCLUSIONS

The results of this study further confirm that the lumbar spine, specifically the last

lumbar vertebra, is fertile ground for future functional morphological studies. The

morphology of the spinal column is critical to postural and locomotor behavior. One

important goal in a functional morphological analysis on the spine is to determine if there

is a relationship between the size and shape of a bony element and the loads it must

withstand. The current research indicates that generalized species models can be

developed. The vectors that are key to the development of these models, normal and

axial, were demonstrated to have high concentrations and therefore can form the basis for

valid models for each species. Further, these high concentration values may indicate that

this measure of facet and pedicle orientation is not sensitive to individual behavior.

Rather, the orientation is very consistent within species. Pair-wise tests indicate that the

human, gorillas, and chimpanzees have unique mean directions for these vectors, with the

exception of the unrotated human and gorilla. This test demonstrated that these species

have statistically equal mean directions.

Previously undocumented vertebral measures, the major axes of the facets and

pedicles, were also addressed. The facets joints allow, or alternatively constrict, motion

within the spinal column. The direction of this vector may indicate the plane in which the

facet allows the most rotation. The statistical analyses overwhelmingly indicate that

humans, gorilla and chimpanzees have unique mean directions with respect to these

vectors. There were a few exceptions. For both the unrotated and rotated human vectors









and gorilla vectors, pair-wise tests indicate these two species have statistically non-

divergent mean directions. Another exception was the left superior facet major axis

vectors of gorilla and chimpanzees. The left, though notably not the right, vectors of

these species have non-divergent mean directions. Finally, the unrotated human and

chimpanzee pair-wise test of the right inferior major axis vectors, though not the left,

indicate that these two species have statistically non-divergent mean directions. The

functional implications of these data are not readily transparent.

Interestingly, the results of the major axis facet analyses demonstrate a low

concentration of the facet major axes vectors. The low concentrations indicate within

species variability for this measure. This seems to suggest that facet shape is responsive

and plastic in the individual, though it is not clear if it is responsive to locomotor and

postural behaviors.

The pedicle major axis vectors were measured, as this directional vector may

indicate how the pedicle is oriented to resist the most bending force. Results of the

statistical analyses of pedicle major axis vectors are also very interesting in that they

indicate the species have statistically different mean directions with respect to this

measure. Finally, with regard to concentration, gorillas and chimpanzees have extremely

high concentrations. However, human major axis pedicle vectors have extremely low

concentration values. These results may indicate that the major axis vector is not

measuring function as intended. The minor-axis major axis ratio calculation (Appendix

A) confirms that there is a general trend in humans for the medial and lateral axes to be

fairly equal in length. Further, the shape of the pedicle may not be ideally suited, as is

assumed, to a specific type of force. The trend in orthograde primates to have short, wide






65


pedicles may indicate that the response to increased compressive loads is constrained by

function or neurological considerations. Finally, the results of this analysis demonstrate

that, given certain assumptions, it is possible to develop species-specific models to

calculate forces in the spinal column.
















APPENDIX A
MAJOR AXES-MINOR AXIS RATIO

The tables below list the major axis-minor axis ratio for each specimen. The value

is calculated via the formula log (superior-inferior axis) log(lateral-medial axis). This

value is the base ten logarithm of the ratio of the major to minor axis. A positive value

indicated that the superior-inferior axis is longer, while a negative value indicated that the

lateral-medial axis is the longer of the two axes.

Homo maior axis-minor axis ratio.
Homo Right Pedicle Left Pedicle Right Sup. Facet Left Sup. Facet Right Inf. Facet Left Inf. Facet
1 0.154004028 0.122777501 -0.076485421 -0.083315591 -0.036990687 0.085841076
2 -0.168245483 -0.024223181 -0.034702602 0.045502107 -0.035288941 0.200988587
3 -0.022722562 -0.003593294 -0.10217809 -0.137450031 0.133044418 0.143808298
4 NA NA 0.191864849 -0.181905303 0.108410102 0.144166515
5 -0.19587025 -0.008572706 -0.099681284 -0.10488135 0.139241109 0.19679049
6 -0.008060033 -0.049878814 0.189004265 0.090310548 0.136960776 -0.073634987
7 -0.205698218 -0.120912741 -0.176895834 -0.059726442 -0.111932747 -0.115985709
8 0.011222033 0.029822045 -0.083507239 -0.047776674 0.211911927 0.006849671
9 0.165669964 -0.017378339 -0.061136929 0.066483798 0.209886421 0.234550164
10 -0.14308604 -0.148556922 -0.029226677 -0.16739032 0.115877559 0.222156199
11 -0.023937285 -0.108195036 -0.019557449 -0.150507035 0.14289058 0.162482517
12 0.021455712 -0.01708491 -0.048078851 -0.047295453 -0.120152347 -0.13369963
13 NA NA -0.054751394 -0.062539143 0.111309967 0.31360054
14 -0.153090849 -0.18451456 -0.009755018 -0.013468729 0.036861323 0.105409743
15 -0.157696731 -0.1701001 NA -9.11852E-05 0.05109364 -0.058043173
16 -0.093144815 -0.124482014 -0.183166616 -0.162388367 -0.194479701 0.02441287
17 > 0.001 -0.065732342 NA -0.228174563 0.118806321 0.076225337
18 0.017244966 -0.03146184 0.064152452 0.081666506 0.209628657 0.078556068
19 0.038259755 -0.047534837 NA -0.078808292 0.073729481 0.082819371

Gorilla major axis-minor axis ratio.
Gorilla Right Pedicle Left Pedicle Right Sup. Facet Left Sup. Facet Right Inf. Facet Left Inf. Facet
1 0.060167877 0.067643681 -0.064164788 -0.217178634 -0.222365457 0.038108348
2 0.127629537 0.223340381 0.130056818 0.218542269 0.056772197 0.112606279
3 0.375613123 0.320078963 -0.357718716 -0.182414896 -0.114841461 0.043987822
4 0.222651315 0.246796519 0.256098669 0.145279049 0.167937031 0.159292774
5 0.287470785 0.182472506 0.138466694 0.121439514 -0.003561981 0.081388742











Gorilla. Continued.
Gorilla Right Pedicle Left Pedicle Right Sup. Facet Left Sup. Facet Right Inf. Facet Left Inf. Facet
6 0.12936712 0.145883044 -0.084745501 -0.039511551 0.074280664 0.120548935
7 0.141641702 0.222122531 -0.127360023 -0.134062605 0.026483505 NA
8 0.081080411 0.119080867 -0.192925955 -0.27067006 0.081999694 -0.082457784
9 0.119551448 0.164047131 0.251696202 -0.226917138 -0.159622372 -0.204039531
10 0.137548209 NA 0.224990253 0.079112759 0.321164729 0.26279744
11 0.101727584 0.148862339 -0.121686273 -0.146663289 0.047584429 -0.02817221
12 0.118749878 0.084090499 -0.131024216 -0.137255376 -0.158394163 -0.09308975
13 0.138983315 NA 0.165514375 NA 0.123303367 NA
14 0.137558074 0.223620517 -0.088671365 0.08403637 -0.227318767 0.190659229
15 0.134192423 0.086933583 -0.124312935 0.136050062 0.03654971 0.067166727
16 0.300352283 0.216250665 -0.203508887 -0.010258393 0.029210435 NA
17 0.248338172 0.176788136 -0.147019214 -0.060164388 0.166190544 0.058113367

Pan minor axes-major axes ratio.
Pan Right Pedicle Left Pedicle Right Sup. Facet Left Sup. Facet Right Inf. Facet Left Inf. Facet
1 0.22183874 0.280385713 0.131302338 0.141002239 0.144568675 0.106252181
2 0.255433552 0.200764578 0.124194868 0.102431458 0.061718878 0.165303483
3 0.405821594 0.341866563 0.155701514 0.154558942 0.057895504 0.145964859
4 0.326159647 0.281167648 0.039289209 -0.091357267 0.194611474 0.283394472
5 0.259558909 0.212682574 0.085998391 -0.024546424 0.219947716 0.231392834
6 0.106969764 0.103704398 0.110323302 0.217555913 0.000779009 -0.044905124
7 0.18342086 0.199781293 0.154620771 0.121204753 0.154995693 0.281804273
8 0.253017076 0.262167926 -0.016198363 -0.036284862 0.208380764 0.199520161
9 0.28797178 0.189091318 0.15263867 0.090480153 0.040798182 -0.162966207
10 0.184171769 0.252979075 0.10504127 0.038045831 0.013012161 0.047561334
11 0.084186624 0.171771641 0.063849173 -0.136923196 0.111160804 0.023174003
12 0.383823739 0.345946783 0.097182104 0.096102229 -0.088161714 0.049324453
13 0.277177855 0.259971814 0.190514199 0.072523392 0.17743898 0.061610458
14 0.151779711 0.192508569 0.086232445 0.033836483 0.189028532 0.215425648
15 0.210597829 0.210260155 0.14357679 0.10951732 0.18732476 0.141981253
16 0.321865888 0.317752038 0.122224054 0.146863685 0.27884123 0.166475019
17 0.257802873 0.222618699 0.052256937 -0.09097867 0.131903682 0.161802253
18 0.272826335 0.307301703 -0.080063049 -0.049644983 0.110340009 0.143839923

Pongo minor axes-major axes ratio.
Pongo Right Pedicle Left Pedicle Right Sup. Facet Left Sup. Facet Right Inf. Facet Left Inf. Facet
1 0.092599566 NA 0.062045738 0.027134765 0.06229625 0.151568419
2 -1.382393424 -1.354590371 0.070912757 0.02605175 0.100561634 0.18084306
3 0.381737882 -1.195793895 0.211296274 0.241418164 0.162105728 0.044230445
4 -0.062836729 -0.048311536 0.30678293 0.266221028 0.173352109 0.200495819
5 0.135336062 0.252095092 0.234525199 0.104844507 0.161262349 0.323901507
















APPENDIX B
ANGLE ERROR CALCULATION (DEGREES)


Homo.
Angle Error Calculations Mean Std. dev. n=
Normal to Superior Facet Right 1.38239 0.882528 16
Normal to Superior Facet Left 1.694435 0.677398 19
Normal to Inferior Facet Right 1.503308 0.777949 19
Normal to Inferior Facet Left 1.369888 0.788105 19
Axial Direction of Pedicle Right 2.326571 1.198554 17
Axial Direction of Pedicle Left 2.047828 1.010862 17

All 1.712785 0.944116 107

Gorilla.
Angle Error Calculations Mean Std. dev. n=
Normal to Superior Facet Right 1.517525 0.886495 17
Normal to Superior Facet Left 2.195902 1.407393 16
Normal to Inferior Facet Right 2.579414 1.632107 17
Normal to Inferior Facet Left 1.768057 1.152024 14
Axial Direction of Pedicle Right 2.368777 1.098082 17
Axial Direction of Pedicle Left 1.664009 0.992631 15

All 2.028797 1.257412 96

Pan.
Angle Error Calculations Mean Std. dev. n=
Normal to Superior Facet Right 1.496232 1.033072 18
Normal to Superior Facet Left 2.067263 0.978035 18
Normal to Inferior Facet Right 1.666827 1.058175 18
Normal to Inferior Facet Left 3.829464 1.758387 18
Axial Direction of Pedicle Right 1.939254 0.585871 18
Axial Direction of Pedicle Left 2.229736 0.786581 18

All 1.879014 0.973882 108

Pongo.
Angle Error Calculations mean Std. dev. n=
Normal to Superior Facet Right 1.258144 0.407892 5
Normal to Superior Facet Left 1.676243 0.662998 5
Normal to Inferior Facet Right 2.905199 1.387219 5
Normal to Inferior Facet Left 2.034397 0.886031 5










Pongo. Continued
Pongo pygmaeus Angle Error Calculations mean Std. dev. n=
Axial Direction of Pedicle Right 1.437808 1.166763 5
Axial Direction of Pedicle Left 1.507351 0.693615 4

All 1.813392 1.018929 29


















APPENDIX C
RAW VECTOR DATA TABLES

Tables 8-1 through 8-4 contain the raw vector data.

Key:
A. = axial
AMNH#-American Museum of Natural History Inventory Number
Dir. = direction
Inf. = inferior
L. = left
MA = major axis
N. = normal
Ped.= pedicle
R. = right
Sup. = superior
Yo = years old


Homo.
Raw Vector Data
AMNH# 98-46, 82 yo, Wt. Male AMNH# 98.49, 74yo, Wt. Male AMNH# 98-52, 60yo, Wt. Male
Description of Vector x y z x y z x y z
MAof Ped. R. 0.217647 0.25191 0.94296 0.31027 -0.7929 -0.52448 0.40255 -0.85729 -0.32095
MAof Ped. L. 0.139378 -0.3266 0.93484 0.49412 0.78444 -0.37484 0.37423 0.81527 -0.44191
MA of Sup. Facet R. -0.485039 -0.8393 0.24558 -0.5126 -0.7847 0.34858 -0.67793 -0.66194 0.31977
MA of Sup. Facet L. -0.485757 0.78904 0.37611 0.39183 0.55654 0.73262 -0.53428 0.68735 0.49202
MA of Inf. Facet R. -0.117477 -0.7618 0.63709 -0.21229 -0.8701 0.44491 0.79758 0.08984 0.59648
MA of Inf. Facet L. 0.874694 -0.1862 0.44749 0.71831 -0.6354 0.28338 0.89112 -0.17631 0.41812
N. to Sup. Facet R. -0.67373 0.53768 0.50694 -0.65182 0.61989 0.43688 -0.51864 0.73894 0.43009
N. to Sup. Facet L. -0.587696 -0.6133 0.52768 -0.81089 -0.1673 0.56077 -0.47226 -0.72547 0.50066
N. to Inf. Facet R. 0.401513 -0.6232 -0.67113 0.6214 -0.4716 -0.62569 0.44479 -0.75555 -0.48095
N. to Inf. Facet L. 0.456574 0.62634 -0.63185 0.59404 0.34813 -0.7252 0.38613 0.77864 -0.4946
A. Dir. OfPed. R. -0.909646 -0.2979 0.28953 -0.80119 -0.5151 0.30466 -0.87403 -0.46417 0.14361
A. Dir. OfPed. L. -0.864222 0.42075 0.27584 -0.62879 0.62019 0.46903 -0.88839 0.45184 0.08125
AMNH# 98-102, 67yo, Wt. Male AMNH# 98-119, 44yo, Wt. Male AMNH# 98-125, 50yo, Wt. Male
Description of Vector x y z x y z x y z
MAof Ped. R. 0.34507 -0.8814 -0.32258 0.38664 -0.8305 -0.40099 0.45885 -0.76618 -0.4499
MAofPed. L. 0.594566 0.78605 -0.16917 0.51345 0.77332 -0.37196 0.53888 0.78943 -0.29395
MA of Sup. Facet R. -0.405535 -0.7452 0.52939 0.18703 -0.4657 0.86498 -0.62033 -0.70966 0.33402
MA of Sup. Facet L. -0.367225 0.72564 0.58188 -0.07188 0.51237 0.85575 -0.58076 0.75949 0.29307
MA of Inf. Facet R. 0.984399 0.16947 0.04731 0.76035 0.05166 0.64745 -0.71522 -0.56723 0.4083
MA of Inf. Facet L. 0.898105 -0.4298 0.09324 -0.54579 0.7152 0.43658 -0.49445 0.75331 0.43364
N. to Sup. Facet R. -0.591147 0.65554 0.46991 -0.60849 0.63634 0.47414 -0.56898 0.70028 0.43113













Homo. Continued
Raw Vector Data
AMNH# 98-102, 67yo, Wt. Male AMNH# 98-119, 44yo, Wt. Male AMNH# 98-125, 50yo, Wt. Male
Description of Vector x y z x y z x y z
N. to Sup. Facet L. -0.486314 -0.6831 0.54491 -0.44223 -0.7854 0.4331 -0.64243 -0.64869 0.40801
N. to Inf. Facet R. 0.175937 -0.9451 -0.27526 0.4765 -0.7218 -0.50199 0.41403 -0.81454 -0.40634
N. to Inf. Facet L. 0.432632 0.82535 -0.36281 0.55787 0.69892 -0.44754 0.57304 0.65763 -0.48903
A. Dir. OfPed. R. -0.694192 -0.471 0.54431 -0.85747 -0.4838 0.17519 -0.75952 -0.601 0.24886
A. Dir. OfPed. L -0.673776 0.6019 0.42865 -0.68503 0.63044 0.36508 -0.70487 0.61364 0.35582
AMNH# 98-59, 70yo, Wt. Male AMNH# 98-135, 52yo, Wt. Male AMNH#, 98-147, 50yo, Wt. Male
Description of Vector x y z x y z x y z
MAof Ped. R. 0.04487 0.20633 0.97745 -0.03115 0.20715 0.97781 0.35 -0.85939 -0.37277
MAofPed. L. 0.165192 -0.2801 0.94565 0.18434 0.98094 -0.06153 0.41043 0.85439 -0.31868
MA of Sup. Facet R. -0.529928 -0.6982 0.48128 -0.63135 -0.6836 0.3662 -0.85258 -0.45569 -0.25582
MA of Sup. Facet L. -0.336754 0.79087 0.511 0.61893 0.01023 0.78538 -0.66688 0.74373 -0.04634
MA of Inf. Facet R. 0.91116 0.39277 0.12458 0.75998 0.63875 0.12012 0.925 0.36061 -0.11974
MA of Inf. Facet L. 0.823772 -0.1404 0.54925 0.85869 -0.4262 0.28456 0.95697 -0.29019 0.00202
N. to Sup. Facet R. -0.467238 0.71401 0.52142 -0.38263 0.68531 0.61963 -0.45482 0.40596 0.79267
N. to Sup. Facet L. -0.57803 -0.602 0.55085 -0.51364 -0.7512 0.41456 -0.55544 -0.45466 0.69626
N. to Inf. Facet R. 0.406732 -0.8089 -0.42462 0.55388 -0.5398 -0.63391 0.207 -0.7425 -0.63706
N. to Inf. Facet L. 0.442936 0.76414 -0.46894 0.5125 0.71625 -0.47365 0.23877 0.78339 -0.57384
A. Dir. OfPed. R. -0.907985 -0.3996 0.12603 -0.97357 -0.2277 0.01723 -0.89528 -0.42397 0.13683
A. Dir. OfPed. L. -0.780696 0.54878 0.29892 -0.9821 0.18137 -0.05082 -0.77604 0.51078 0.36995
AMNH# 98-174, 56yo, Bk. Male AMNH# 98-217, 71yo, Wt. Male AMNH# 98-246, 47yo, Bk. Male
Description of Vector x y z x y z x y z
MAofPed. R. 0.383857 -0.8284 -0.40792 0.29838 0.05687 0.95275 NA NA NA
MAofPed. L. 0.466618 0.82837 -0.30996 0.52961 0.81866 -0.22204 NA NA NA
MA of Sup. Facet R. -0.516952 -0.7387 0.43258 -0.2245 -0.9577 0.18016 -0.55718 -0.72727 0.40079
MA of Sup. Facet L. -0.543826 0.69103 0.47617 -0.31002 0.92426 0.22279 -0.90047 0.30067 -0.31425
MA of Inf. Facet R. 0.891714 0.30689 0.33266 -0.68401 -0.7292 -0.02061 0.82642 0.23214 0.51297
MA of Inf. Facet L. 0.849028 -0.5207 0.08964 -0.32292 0.87031 0.37185 0.8704 -0.24779 0.42545
N. to Sup. Facet R. -0.578245 0.67395 0.45981 -0.74864 0.28785 0.59723 -0.52629 0.68261 0.50701
N. to Sup. Facet L. -0.576365 -0.72 0.38658 -0.78236 -0.3812 0.49258 -0.37877 -0.89725 0.22689
N. to Inf. Facet R. 0.452189 -0.6353 -0.62599 0.6311 -0.5774 -0.51806 0.49522 -0.7332 -0.46603
N. to Inf. Facet L. 0.487857 0.70746 -0.51137 0.59644 0.4922 -0.63403 0.38579 0.88012 -0.27667
A. Dir. OfPed. R. -0.812599 -0.5129 0.27686 -0.85026 -0.4377 0.29241 NA NA NA
A. Dir. Of Ped. L. -0.663831 0.55959 0.49617 -0.72722 0.57297 0.37796 NA NA NA
AMNH# 98-295, 42yo, Wt. Male AMNH# 98-320, 45yo, Wt. Male AMNH# 98-324, 49yo, Wt. Male
Description of Vector x y z x y z x y z
MAof Ped. R. 0.43254 -0.8085 -0.39912 0.44223 -0.8059 -0.39373 0.39673 -0.86678 -0.30218
MAofPed. L. 0.496524 0.78714 -0.36588 0.43473 0.83522 -0.33678 0.54002 0.77625 -0.32528
MA of Sup. Facet R. -0.467581 -0.8167 0.3381 -0.332 -0.8498 0.40951
MA of Sup. Facet L. -0.805779 0.51244 -0.29685 -0.05671 0.82818 0.55759 -0.36091 0.679 0.6393
MA of Inf. Facet R. 0.809175 0.51062 0.2907 -0.70969 -0.7037 0.03348 0.90868 0.41132 0.07151
MA of Inf. Facet L. 0.687704 -0.7234 0.06188 0.78275 -0.4032 0.47403 0.87976 -0.47426 0.03319
N. to Sup. Facet R. -0.622415 0.57579 0.53015 -0.69561 0.51378 0.50216
N. to Sup. Facet L. -0.591365 -0.7231 0.35696 -0.72195 -0.4198 0.55007 -0.62607 -0.68448 0.37354
N. to Inf. Facet R. 0.573977 -0.5811 -0.57691 0.62129 -0.6476 -0.44121 0.3795 -0.74241 -0.55209













Homo. Continued
Raw Vector Data
AMNH# 98-295, 42yo, Wt. Male AMNH# 98-320, 45yo, Wt. Male AMNH# 98-324, 49yo, Wt. Male
Description of Vector x y z x y z x y z
N. to Inf. Facet L. 0.640971 0.56494 -0.51962 0.60571 0.66852 -0.4315 0.45298 0.81498 -0.36141
A. Dir. of Ped. R. -0.727338 -0.5745 0.37544 -0.77275 -0.5652 0.28884 -0.83245 -0.47845 0.27948
A. Dir. of Ped. L. -0.654664 0.61637 0.43761 -0.70478 0.54834 0.45012 -0.62585 0.62877 0.46148
AMNH# 98-327, 63yo, Wt. Male AMNH# 98-333, 58yo, Wt. Male AMNH# 98-99, 47yo, Wt. Female
Description of Vector x y z x y z x y z
MAof Ped. R. 0.277084 0.16413 0.94672 0.19631 0.36932 0.90833 NA NA NA
MAof Ped. L. 0.377997 0.83091 -0.4083 0.38899 0.86122 -0.32709 NA NA NA
MA of Sup. Facet R. 0.331514 -0.3078 0.89183 NA NA NA 0.18258 -0.4973 0.84815
MA of Sup. Facet L. 0.210123 0.47817 0.85276 -0.46559 0.82031 0.33214 -0.08619 0.76112 0.64286
MA of Inf. Facet R. 0.917028 0.35515 0.18146 0.58915 -0.0683 0.80514 0.87051 0.27316 0.40939
MA of Inf. Facet L. 0.848145 -0.5017 0.1701 0.81179 -0.2149 0.54298 0.75483 -0.36555 0.54462
N. to Sup. Facet R. -0.535314 0.71702 0.44645 NA NA NA -0.08149 0.85203 0.51712
N. to Sup. Facet L. -0.770911 -0.4554 0.44532 -0.69258 -0.5714 0.44031 -0.71052 -0.49928 0.49586
N. to Inf. Facet R. 0.395497 -0.7512 -0.52852 0.47233 -0.7794 -0.4117 0.42671 -0.83339 -0.35126
N. to Inf. Facet L. 0.517115 0.71431 -0.47154 0.51166 0.70985 -0.48406 0.64054 0.58956 -0.49207
A. Dir. of Ped. R. -0.80536 -0.4977 0.322 -0.77382 -0.5106 0.37484 NA NA NA
A. Dir. ofPed. L. -0.76068 0.53013 0.37461 -0.8076 0.48962 0.32872 NA NA NA
AMNH# 98-315, 65yo Bk. Female
Description of Vector x y z
MAof Ped. R. 0.448193 -0.7965 -0.40585
MAofPed. L. 0.472769 0.80337 -0.36206
MA of Sup. Facet R. NA NA NA
MA of Sup. Facet L. -0.402228 0.73841 0.54127
MA of Inf. Facet R. 0.757824 0.28566 0.5866
MA of Inf. Facet L. -0.35049 0.79254 0.49903
N. to Sup. Facet R. NA NA NA
N. to Sup. Facet L. -0.621201 -0.6544 0.43112
N. to Inf. Facet R. 0.644962 -0.4639 -0.60732
N. to Inf. Facet L. 0.552185 0.60524 -0.57339
A. Dir. of Ped. R. -0.639858 -0.6029 0.47657
A. Dir. of Ped. L. -0.649357 0.59537 0.47314


Gorilla.
Raw Vector Data
AMNH# 90289, Male AMNH# 167336, Male AMNH# 167335, Male
Description of Vector x y z x y z x y z
MAof Ped. R. 0.123004 0.08752 0.98854 0.10109 0.12651 0.9868 0.29847 0.11771 0.94713
MAof Ped. L. 0.206535 -0.2949 0.93296 0.14711 -0.2724 0.95087 0.25925 -0.4447 0.85734
MA of Sup. Facet R. -0.555758 -0.7772 0.29523 -0.46005 -0.8231 0.33309 -0.14254 -0.80451 0.57657
MA of Sup. Facet L. -0.646692 0.55173 0.52668 -0.33819 0.79438 0.50457 -0.13565 0.41819 0.89818
MA of Inf. Facet R. -0.822841 -0.5682 -0.01124 0.4056 0.02639 0.91367 0.44876 -0.15836 0.87951
MA of Inf. Facet L. 0.473646 -0.2999 0.82807 0.14267 0.0073 0.98974 NA NA NA
N. to Sup. Facet R. -0.664696 0.62868 0.40366 -0.74093 0.56258 0.36677 -0.69033 0.49826 0.52457
N. to Sup. Facet L. -0.531501 -0.8212 0.20765 -0.73432 -0.5581 0.38644 -0.58373 -0.76624 0.2686













Gorilla. Continued.

Raw Vector Data
AMNH# 90289, Male AMNH# 167336, Male AMNH# 167335, Male
Description of Vector x y z x y z x y z
N. to Inf. Facet R. 0.551469 -0.7936 -0.25712 0.45616 -0.8721 -0.17732 0.43927 -0.81798 -0.37141
N. to Inf. Facet L. 0.674831 0.72775 -0.12239 0.52585 0.84661 -0.08205 NA NA NA
A. Dir. ofPed. R. -0.950125 -0.2773 0.14277 -0.90886 -0.3917 0.14333 -0.89962 -0.29671 0.32038
A. Dir. of Ped. L. -0.866873 0.38703 0.31423 -0.9409 0.25795 0.21946 -0.82083 0.36634 0.43822
AMNH# 90290, Male AMNH# 201460, Male AMNH# 81651, Male
Description of Vector x y z x y z x y z
MAof Ped. R. 0.206148 0.24282 0.94791 0.40205 0.05745 0.91381 0.57253 0.15358 0.80538
MAof Ped. L. 0.22276 -0.3029 0.92663 0.39388 -0.3348 0.85604 NA NA NA
MA of Sup. Facet R. -0.670153 -0.5171 0.5325 0.30906 -0.4469 0.83948 0.48306 -0.59782 0.63973
MA of Sup. Facet L. -0.563011 0.57459 0.59402 -0.51366 0.74089 0.43272 0.59325 0.67882 0.43274
MA of Inf. Facet R. 0.343504 -0.0753 0.93613 -0.56101 -0.6477 0.51555 0.4168 -0.43828 0.79636
MA of Inf. Facet L. -0.78381 0.45357 0.42417 -0.14997 0.47247 0.8685 0.24969 0.25685 0.93364
N. to Sup. Facet R. -0.391525 0.85576 0.33819 -0.47883 0.68953 0.54339 -0.30567 0.56952 0.76303
N. to Sup. Facet L. -0.510736 -0.807 0.29652 -0.49136 -0.6675 0.55952 -0.5715 -0.02345 0.82026
N. to Inf. Facet R. 0.177614 -0.9736 -0.14343 0.51316 -0.7608 -0.39734 0.52647 -0.5978 -0.60454
N. to Inf. Facet L. 0.44211 0.88723 -0.13177 0.44599 0.81631 -0.36706 0.57709 0.73478 -0.35648
A. Dir. of Ped. R. -0.932207 -0.2458 0.26569 -0.81059 -0.4418 0.38441 -0.76339 -0.25848 0.59197
A. Dir. ofPed. L. -0.732501 0.57522 0.3641 -0.78974 0.35326 0.50152 NA NA NA
AMNH# 214103, Male AMNH# 54089, Male AMNH# 54090, Male
Description of Vector x y y x y z x y z
MAof Ped. R. 0.332068 0.1249 0.93495 0.30302 0.14124 0.94246 0.40936 0.02153 0.91212
MAof Ped. L. 0.247312 -0.1598 0.95566 NA NA NA 0.49471 -0.14547 0.8568
MA of Sup. Facet R. -0.40226 -0.6227 0.67119 0.00195 -0.595 0.8037 -0.71041 -0.63294 0.30775
MA of Sup. Facet L. 0.018459 0.72651 0.68691 NA NA NA 0.8147 -0.34713 0.46449
MA of Inf. Facet R. -0.820166 -0.2488 0.51518 0.36189 -0.1297 0.92316 0.50503 0.24162 0.82859
MA of Inf. Facet L. -0.665585 0.5112 0.54376 NA NA NA 0.59929 -0.46842 0.64918
N. to Sup. Facet R. -0.44516 0.77364 0.4509 -0.53632 0.67771 0.50306 -0.49568 0.76039 0.41965
N. to Sup. Facet L. -0.689308 -0.4884 0.53508 NA NA NA -0.53841 -0.75029 0.38364
N. to Inf. Facet R. 0.177021 -0.9667 -0.18506 0.40616 -0.8694 -0.28132 0.85814 -0.2433 -0.4521
N. to Inf. Facet L. 0.473052 0.8525 -0.22241 NA NA NA 0.80051 0.34567 -0.48958
A. Dir. ofPed. R. -0.937192 -0.0685 0.34202 -0.95085 0.11102 0.28908 -0.90374 0.1468 0.40213
A. Dir. ofPed. L. -0.930957 0.23424 0.2801 NA NA NA -0.8658 0.00285 0.50039
AMNH# 99.1/1577, Male AMNH# 54327, Female AMNH# 167340, Female
Description of Vector x y z x y z x y z
MAof Ped. R. 0.184267 0.30092 0.93568 0.25612 0.30784 0.91632 0.13296 0.07039 0.98862
MAof Ped. L. 0.187111 -0.37 0.90998 0.06217 -0.2735 0.95985 0.19553 -0.25347 0.94738
MA of Sup. Facet R. -0.61885 -0.6417 0.45301 0.07148 -0.4098 0.90939 -0.54127 -0.74053 0.39829
MA of Sup. Facet L. -0.307662 0.64625 0.69836 -0.22834 0.70307 0.67347 -0.63225 0.53689 0.55858
MA of Inf. Facet R. 0.624217 -0.1385 0.76888 0.67167 0.03049 0.74022 -0.81519 -0.55679 0.15951
MA of Inf. Facet L. NA NA NA 0.46998 0.09141 0.87793 0.53933 -0.14749 0.82908
N. to Sup. Facet R. -0.485773 0.76586 0.42129 -0.50795 0.76969 0.38674 -0.66442 0.66698 0.33717
N. to Sup. Facet L. -0.436482 -0.748 0.49993 -0.5637 -0.6595 0.49733 -0.50014 -0.83345 0.23499
N. to Inf. Facet R. 0.435759 -0.7551 -0.48979 0.45684 -0.8036 -0.38143 0.51867 -0.82435 -0.22679
N. to Inf. Facet L. NA NA NA 0.46851 0.81712 -0.33589 0.37836 0.92201 -0.0821













Gorilla. Continued
Raw Vector Data
AMNH# 99.1/1577, Male AMNH# 54327, Female AMNH# 167340, Female
Description of Vector x y z x y z x y z
A. Dir. ofPed. R. -0.959499 -0.1513 0.23762 -0.76732 -0.5118 0.3864 -0.90645 -0.39476 0.15002
A. Dir. of Ped. L. -0.850838 0.40194 0.3384 -0.74409 0.62825 0.22724 -0.85697 0.42554 0.29072
AMNH# 167339, Female AMNH# 167337 Female AMNH# 81652, Female
Description of Vector x y z x y z x y z
MAofPed. R. 0.034384 0.18516 0.98211 0.18626 0.17048 0.9676 0.31645 0.17666 0.93202
MAofPed. L. -0.003492 -0.2988 0.95432 0.33053 -0.257 0.90814 0.3329 -0.23795 0.91245
MA of Sup. Facet R. -0225022 -0.4334 0.87265 0.52927 0.0509 0.84693 -0.47442 -0.73073 0.49088
MA of Sup. Facet L. 0.008223 0.27295 0.96199 0.40538 0.1992 0.89218 -0.46925 0.63876 0.60975
MA of Inf. Facet R. 0.298767 -0.2569 0.91911 -0.69438 -0.4488 0.56255 0.3445 -0.1082 0.93253
MA ofInf. Facet L. 0.410187 0.02381 0.91169 -0.15653 0.14635 0.97677 -0.93425 0.30394 0.18653
N. to Sup. Facet R. -.422773 0.85035 0.31333 -0.72611 0.54354 0.4211 -0.57304 0.67966 0.45792
N. to Sup. Facet L. -.568668 -0.7926 0.22003 -0.78832 -0.418 0.4515 -0.45294 -0.76684 0.45475
N. to Inf. Facet R. 0.274102 -0.8994 -0.34046 0.45621 -0.8791 -0.13814 0.30934 -0.92478 -0.22157
N. to Inf. Facet L. 0.493566 0.83482 -0.24387 0.5116 0.85796 -0.04656 0.25688 0.93638 -0.23916
A. Dir. of Ped. R. -.822847 -0.5525 0.13297 -0.97024 -0.1232 0.20847 -0.86766 -0.34325 0.35966
A. Dir. ofPed. L. -.745638 0.63669 0.19659 -0.68358 0.59826 0.4181 -0.85958 0.32126 0.39739
AMNH# 54091, Female AMNH# 99.1/1578, Female
Description of Vector x y z x y z
MAof Ped. R. 0.046485 0.19536 0.97963 0.37877 0.3591 0.85299
MAof Ped. L. 0.132641 -0.2112 0.9684 0.35765 -0.1945 0.91337
MA of Sup. Facet R. -0.697589 -0.6704 0.25295 -0.71269 -0.4339 0.5512
MA of Sup. Facet L. 0.338823 0.11258 0.93409 -0.74712 0.5054 0.43174
MA of Inf. Facet R. -0.235297 -0.8761 0.42092 0.3751 -0.0257 0.92663
MA of Inf. Facet L. 0.355317 -0.0754 0.9317 0.4526 -0.0829 0.88785
N. to Sup. Facet R. -0.617502 0.74155 0.26228 -0.32785 0.90069 0.28509
N. to Sup. Facet L. -0.523832 -0.8021 0.28669 -0.40422 -0.8611 0.30848
N. to Inf. Facet R. 0.872306 -0.3813 -0.30604 0.61135 -0.7445 -0.26814
N. to Inf. Facet L. 0.802694 0.53536 -0.26282 0.74361 0.58461 -0.32447
A. Dir. of Ped. R. -0.997251 -0.0476 0.05681 -0.92408 0.09577 0.37002
A. Dir. ofPed. L. -0.946888 0.26175 0.18679 -0.92985 0.01635 0.36758


Pan.

Raw Vector Data, Pan Sample
AMNH# 51377, Male AMNH# 51278, Male AMNH# 174861, Male
Description of Vector x y z x y z x y z
MAofPed. R. 0.437118 0.279958 0.854723 0.385527 0.167147 0.907431 0.341573 0.157836 0.926507
MAofPed. L. 0.380015 -0.24063 0.893133 0.317779 -0.24027 0.917217 0.317087 -0.35364 0.879997
MA of Sup. Facet R. 0.236742 -0.20905 0.948816 0.036721 -0.36084 0.931903 0.058365 -0.28476 0.956821
MA of Sup. Facet L. 0.183441 0.225498 0.956818 0.023484 0.279007 0.960002 -0.16172 0.395314 0.904198
MA of Inf. Facet R. 0.629024 -0.01281 0.777281 0.759219 0.015771 0.650644 0.515149 0.000552 0.8571
MA of Inf. Facet L. 0.506402 0.01236 0.862209 0.690194 -0.10721 0.715638 0.558144 -0.07621 0.826237
N. to Sup. Facet R. -0.45975 0.836215 0.298956 -0.34446 0.870813 0.350763 -0.37202 0.883214 0.285544
N. to Sup. Facet L. -0.504355 -0.81387 0.288505 -0.30254 -0.91326 0.272822 -0.45914 -0.84119 0.285647
N. to Inf. Facet R. 0.20357 -0.96226 -0.18061 0.311 -0.88698 -0.3414 0.301933 -0.93601 -0.18087













Pan. Continued
Raw Vector Data
AMNH# 51377, Male AMNH# 51278, Male AMNH# 174861, Male
Description of Vector x y z x y z x y z
N. to Inf. Facet L. 0.320591 0.925524 -0.20156 0.358357 0.909816 -0.20932 0.333991 0.932175 -0.13964
A. Dir. ofPed. R. -0.846491 -0.19309 0.496154 -0.92158 0.021486 0.387583 -0.83146 -0.40885 0.376183
A. Dir. ofPed. L. -0.852023 0.284802 0.439255 -0.85708 0.340913 0.386249 -0.76193 0.457529 0.458408
AMNH# 51379, Male AMNH# 51202, Male AMNH# 51392, Male
Description of Vector x y z x y z x y z
MAofPed. R. 0.307012 0.266302 0.913689 0.317965 0.15459 0.935414 0.264879 0.221105 0.93859
MAofPed. L. 0.325353 -0.24053 0.914489 0.321483 -0.25284 0.912536 0.340492 -0.37802 0.860908
MA of Sup. Facet R. 0.216414 -0.32202 0.921665 0.100986 -0.28317 0.953739 0.022741 -0.28273 0.958929
MA of Sup. Facet L. 0.078224 0.305897 0.948846 0.466869 0.192864 0.863039 -0.88553 0.31371 -0.34266
MA of Inf. Facet R. 0.367143 -0.1811 0.912365 0.541253 -0.02866 0.840371 0.695159 -0.02214 0.718515
MA of Inf. Facet L. 0.519744 0.107518 0.84753 0.568858 0.004627 0.822423 0.704297 -0.02241 0.709552
N. to Sup. Facet R. -0.101073 0.931574 0.349218 -0.40142 0.865544 0.299485 -0.21273 0.935843 0.28097
N. to Sup. Facet L. -0.254778 -0.91402 0.315675 -0.44836 -0.78957 0.418989 -0.44352 -0.7904 0.422562
N. to Inf. Facet R. 0.288291 -0.91041 -0.29672 0.259326 -0.94501 -0.19925 0.250903 -0.92919 -0.27138
N. to Inf. Facet L. 0.349932 0.878219 -0.32601 0.318589 0.920669 -0.22554 0.325162 0.898674 -0.29437
A. Dir. ofPed. R. -0.814204 -0.42359 0.397042 -0.87976 -0.3197 0.351881 -0.85035 -0.40542 0.335481
A. Dir. ofPed. L. -0.840999 0.36851 0.396133 -0.82156 0.404699 0.401563 -0.70215 0.506734 0.500208
AMNH# 81854, Unk. AMNH# 51381, Unk. AMNH# 51394, Unk.
Description of Vector x y z x y z x y z
MAofPed. R. 0.550067 0.233764 0.801736 0.422411 0.23281 0.875996 0.47507 0.066673 0.877419
MAofPed. L. 0.535515 -0.14146 0.832594 0.460736 -0.37385 0.80496 0.365075 -0.35178 0.86196
MA of Sup. Facet R. 0.070312 -0.33421 0.939874 0.092389 -0.51697 0.851003 0.280848 -0.49518 0.822141
MA of Sup. Facet L. 0.416969 0.270371 0.867777 -0.55517 0.712299 0.429434 -0.56567 0.638867 0.521408
MA of Inf. Facet R. 0.414836 -0.06111 0.907842 0.604408 -0.05841 0.794531 0.535665 -0.18963 0.822862
MA ofInf. Facet L. 0.448901 0.21927 0.866261 0.570545 -0.00607 0.821244 0.610939 -0.0586 0.789506
N. to Sup. Facet R. -0.318183 0.885475 0.338666 -0.33179 0.789832 0.515831 -0.49847 0.656761 0.565855
N. to Sup. Facet L. -0.596845 -0.63861 0.485754 -0.54625 -0.70161 0.457558 -0.5266 -0.76644 0.367786
N. to Inf. Facet R. 0.389974 -0.88952 -0.23807 0.355493 -0.87274 -0.33459 0.337957 -0.84487 -0.41471
N. to Inf. Facet L. 0.520409 0.723908 -0.45292 0.430957 0.853448 -0.29309 0.49759 0.804077 -0.32537
A. Dir. ofPed. R. -0.784343 -0.18504 0.592086 -0.84717 -0.24225 0.472891 -0.78825 -0.41096 0.458017
A. Dir. ofPed. L. -0.724142 0.430376 0.538883 -0.75992 0.302397 0.575397 -0.75176 0.434751 0.495828
AMNH# 90189, Unk. AMNH# 90191, Unk. AMNH# 167341, Unk.
Description of Vector x y z x y z x y z
MAofPed. R. 0.330696 0.179255 0.926557 0.264501 0.265718 0.927056 0.375816 0.106039 0.920607
MAofPed. L. 0.284985 -0.35495 0.89039 0.171031 -0.43069 0.886145 0.384584 -0.27468 0.881277
MA of Sup. Facet R. 0.492565 -0.17921 0.851625 -0.15154 -0.33697 0.929239 -0.53407 -0.5213 0.665589
MA of Sup. Facet L. 0.374429 0.033189 0.926661 -0.02959 0.331368 0.943038 -0.45348 0.513835 0.728239
MA of Inf. Facet R. 0.697442 -0.06356 0.713817 0.339338 -0.04085 0.939777 0.399262 -0.1596 0.902838
MA ofInf. Facet L. -0.677869 0.49778 0.541026 0.275515 -0.00883 0.961256 0.540192 -0.01432 0.84142
N. to Sup. Facet R. -0.193346 0.931579 0.30786 -0.38121 0.887295 0.259595 -0.40845 0.848396 0.336737
N. to Sup. Facet L. -0.33851 -0.92549 0.169926 -0.60944 -0.75379 0.245748 -0.40688 -0.84632 0.343784
N. to Inf. Facet R. 0.180364 -0.94843 -0.26067 0.37246 -0.91157 -0.17411 0.36183 -0.87737 -0.31511
N. to Inf. Facet L. 0.490501 0.854392 -0.17153 0.555666 0.817439 -0.15175 0.202823 0.972597 -0.11366
A. Dir. ofPed. R. -0.890747 -0.26507 0.369197 -0.93132 -0.17919 0.317078 -0.851 -0.35375 0.388147













Pan. Continued
Raw Vector Data
AMNH# 90189, Unk. AMNH# 90191, Unk. AMNH# 167341, Unk.
Description of Vector x y z x y z x y z
A. Dir. ofPed. L. -0.829607 0.373966 0.41461 -0.80685 0.454951 0.376846 -0.89148 0.137182 0.431793
AMNH# 167342, Unk. AMNH # 167343, Unk. AMNH# 167346, Unk.
Description of Vector x y z x y z x y z
MAofPed. R. 0.362846 0.109064 0.925445 0.524503 0.185839 0.83088 0.168865 0.247432 0.954076
MAofPed. L. 0.450944 -0.15561 0.878883 0.473631 -0.42566 0.771031 0.274117 -0.24688 0.929468
MA of Sup. Facet R. 0.586252 -0.12097 0.801046 0.090002 -0.35277 0.931371 0.003336 -0.09244 0.995712
MA of Sup. Facet L. 0.511312 0.012676 0.859302 0.141493 0.392407 0.908843 -0.54215 0.423273 0.725889
MA of Inf. Facet R. 0.814039 0.079754 0.575309 0.926036 0.129184 0.354638 0.103358 0.033597 0.994077
MA of Inf. Facet L. -0.305927 0.43667 0.846007 0.859116 -0.21065 0.466419 0.777663 -0.19955 0.596171
N. to Sup. Facet R. -0.30701 0.881863 0.357859 -0.21465 0.906319 0.364024 -0.39259 0.915654 0.086326
N. to Sup. Facet L. -0.550089 -0.76339 0.338582 -0.38765 -0.8228 0.415607 -0.4335 -0.88092 0.189896
N. to Inf. Facet R. 0.267955 -0.93038 -0.25017 0.282479 -0.86038 -0.4242 0.384448 -0.9231 -0.00877
N. to Inf. Facet L. 0.378159 0.871241 -0.31295 0.388925 0.861102 -0.32747 0.405398 0.883967 -0.23293
A. Dir. of Ped. R. -0.857752 -0.349 0.377436 -0.73734 -0.38881 0.552416 -0.89123 -0.37507 0.255013
A. Dir. of Ped. L. -0.790336 0.387948 0.4742 -0.67694 0.38409 0.627875 -0.86229 0.364838 0.351211
AMNH# 54330, Unk. AMNH# 174860, Unk AMNH# Unk., Unk.
Description of Vector x y z x y z x y z
MAofPed. R. 0.473025 0.056875 0.879211 0.321498 0.106226 0.940933 0.48209 0.225415 0.846627
MAofPed. L. 0.431014 -0.03341 0.901726 0.333581 -0.28208 0.899529 0.46998 -0.29634 0.831444
MA of Sup. Facet R. -0.039516 -0.27455 0.960761 0.238388 -0.16465 0.957112 -0.43005 -0.46848 0.771738
MA of Sup. Facet L. -0.121767 0.44288 0.888274 -0.39422 0.315211 0.863268 -0.5284 0.660935 0.532879
MA of Inf. Facet R. -0.640971 -0.56817 0.516085 0.597732 0.115512 0.79333 0.634349 -0.08663 0.768177
MA ofInf. Facet L. 0.743024 -0.17555 0.645832 0.519948 -0.03566 0.853453 0.675502 -0.09531 0.731172
N. to Sup. Facet R. -0.400064 0.885429 0.236566 -0.45193 0.853509 0.259386 -0.29999 0.880404 0.367283
N. to Sup. Facet L. -0.370872 -0.85041 0.373163 -0.27316 -0.93708 0.21742 -0.45555 -0.75037 0.47897
N. to Inf. Facet R. 0.536677 -0.81244 -0.22788 0.219861 -0.97524 -0.02365 0.281122 -0.89981 -0.33363
N. to Inf. Facet L. 0.457564 0.837474 -0.29879 0.158522 0.985801 -0.05538 0.412996 0.870379 -0.26809
A. Dir. ofPed. R. -0.871405 -0.11705 0.476397 -0.84069 -0.42526 0.335256 -0.83663 -0.16842 0.521238
A. Dir. ofPed. L. -0.828614 0.380983 0.410184 -0.83378 0.357002 0.42115 -0.66368 0.502373 0.554209


Pongo.

Raw Vector Data
AMNH# 61586, Male AMNH# 238487, Male AMNH# 140246, Male
Description of Vector x y z x y z x y z
MAofPed. R. 0.08109 0.166519 0.982698 NA NA NA 0.111743 0.319078 0.941118
MAof Ped. L. NA NA NA NA NA NA NA NA NA
MA of Sup. Facet R. 0.256731 -0.79678 0.547027 0.284882 -0.71853 0.634471 0.322088 -0.08601 0.942794
MA of Sup. Facet L. 0.400035 0.129814 0.90726 0.413573 0.222408 0.882889 0.147248 0.136135 0.979686
MA of Inf. Facet R. 0.565543 -0.2728 0.778292 0.538296 -0.18348 0.822541 0.297263 -0.22971 0.926752
MA of Inf. Facet L. 0.465343 -0.01891 0.884928 0.447483 0.179745 0.876043 0.541475 0.208685 0.814405
N. to Sup. Facet R. -0.757461 0.185666 0.625924 -0.76236 0.231406 0.60437 -0.63264 0.721301 0.281935
N. to Sup. Facet L. -0.665756 -0.63917 0.385004 -0.6676 -0.58529 0.460163 -0.40088 -0.89727 0.184936
N. to Inf. Facet R. 0.745713 -0.23391 -0.62386 0.763734 -0.30643 -0.56816 0.461009 -0.81546 -0.34999
N. to Inf. Facet L. 0.67332 0.656516 -0.34004 0.644576 0.614207 -0.45527 0.236317 0.891866 -0.38565













Pongo. Continued
Raw Vector Data
AMNH# 61586, Male AMNH# 238487, Male AMNH# 140246, Male
Description of Vector x y z x y z x y z
A. Dir. ofPed. R. -0.909372 -0.39124 0.141335 NA NA NA -0.78734 -0.54939 0.279751
A.Dir. of Ped. L. NA NA NA NA NA NA NA NA NA
AMNH# 28252, Male AMNH# 28253, Unk.
Description of Vector x y z x y z
MAofPed. R. 0.40951 0.150059 -0.96389 -0.21999 0.058964 0.910398
MAofPed. L. 0.261747 0.432109 0.831914 -0.34814 -0.15328 0.952888
MA of Sup. Facet R. 0.388142 -0.093279 -0.67987 0.727377 -0.506 0.770269
MA of Sup. Facet L. 0.284891 0.431487 0.361134 0.826681 0.640942 0.712763
MA of Inf. Facet R. 0.479223 0.51462 -0.30385 0.801773 -0.2302 0.846966
MA of Inf. Facet L. 0.637168 0.683651 0.095752 0.7235 0.298844 0.710429
N. to Sup. Facet R. -0.65829 -0.65802 0.590363 0.46742 0.432709 0.615967
N. to Sup. Facet L. -0.61944 -0.751262 -0.36347 0.550904 -0.44436 0.647176
N. to Inf. Facet R. 0.440951 0.57948 -0.56596 -0.58643 -0.77122 -0.45911
N. to Inf. Facet L. 0.446346 0.507396 0.650199 -0.5655 0.608383 -0.65624
A. Dir. of Ped. R. -0.88338 -0.91495 -0.2197 0.338523 -0.22368 0.411843
A. Dir. of Ped. L. -0.7947 -0.803535 0.530423 0.270154 0.526022 0.302908















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BIOGRAPHICAL SKETCH

Dorion Amanda Keifer was born in Idaho Falls, Idaho, on April 20th, 1979. She

received her bachelor's degree in anthropology from New York University in 2000.