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Towards in vitro MRI based analysis of spinal cord injury Ming, Kevin 2008

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Towards In Vivo MRI Based Analysis of Spinal Cord Injury by Kevin Ming B. Sc., Queen's University, 2005  A THESIS SUMBII 1 ED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Master of Applied Science  in  THE FACULTY OF GRADUATE STUDIES  (Electrical & Computer Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (VANCOUVER) May 2008  © Kevin Ming, 2008  Abstract A novel approach for the analysis of spinal cord deformation based on a combined technique of non-invasive imaging and medical image processing is presented. As opposed to traditional approaches where animal spinal cords are exposed and directly subjected to mechanical impact in order to be examined, this approach can be used to quantify deformities of the spinal cord in vivo, so that deformations — specifically those of myelopathy-related sustained compression — of the spinal cord can be computed in its original physiological environment. This, then, allows for a more accurate understanding of spinal cord deformations and injuries. Images of rat spinal cord deformations, acquired using magnetic resonance imaging (MRI), were analyzed using a combination of various image processing methods, including image segmentation, a versor-based rigid registration technique, and a B-spline-based non-rigid registration technique. To verify the validity and assess the accuracy of this approach, several validation schemes were implemented to compare the deformation fields computed by the proposed algorithm against known deformation fields. First, validation was performed on a synthetically-generated spinal cord model data warped using synthetic deformations; error levels achieved were consistently below 6% with respect to cord width, even for large degrees of deformation up to half of the dorsal-ventral width of the cord (50% deflection). Then, accuracy was established using in vivo rat spinal cord images warped using those same synthetic deformations; error  levels achieved were also consistently below 6% with respect to cord width, in this case for large degrees of deformation up to the entire dorsal-ventral width of the cord (100% ii  deflection). Finally, the accuracy was assessed using data from the Visible Human Project (VHP) warped using simulated deformations obtained from finite element (FE) analysis of the spinal cord; error levels achieved were as low as 3.9% with respect to cord width. This in vivo, non-invasive semi-automated analysis tool provides a new framework through which the causes, mechanisms, and tolerance parameters of myelopathy-related sustained spinal cord compression, as well as the measures used in neuroprotection and regeneration of spinal cord tissue, can be prospectively derived in a manner that ensures the bio-fidelity of the cord.  iii  ^  Table of Contents ii  ^Abstract ^ Table of Contents ^  iv  List of Figures ^  vii  Acknowledgements ^  .xi  Dedication ^  xii  1 Introduction and Background ^  1  1.1^Thesis Objectives ^  1  1.2^Motivation and Statement of Problem ^  2  1.3^Overview of SCI and Myelopathy ^  2  1.4^Medical Imaging Approaches to SC Studies ^  6  1.5^Medical Image Analysis Approaches to Spinal Studies ^ 10 1.6^Introduction to Image Registration ^  13  1.6.1^Registration Criteria ^  13  1.6.2^Registration Framework^  15  1.6.3^Spatial Transformation^  15  1.6.4^Image Interpolation ^  17  1.6.5^Similarity Metric ^  17  1.6.6^Parameter Optimization ^  18  2 Image Acquisition and Modeling ^  20  2.1^Ethics Approval ^  21  2.2^Overview of Anatomical Terms of the Rat Spine ^  21  2.3^Rat Imaging ^  23  2.3.1^In Vivo Rat MRI ^ iv  24  2.3.2^Ex Vivo Rat MRI ^ 2.4^Synthetic MR Spinal Cord Model Image ^  28 31  2.5^Visible Human Project Spinal Cord Model Image ^ 32 3 Deformation Analysis and Validation ^  37  3.1^Cord Segmentation^  38  3.2^Deformation Analysis via Image Registration ^  39  3.3^Rigid Image Registration ^  39  3.3.1^Versor Rigid Transform ^  40  3.3.2^Linear Interpolation ^  41  3.3.3^Mattes Mutual Information Metric ^  42  3.3.4^Custom Versor Optimizer ^  45  3.4^Non-rigid Image Registration ^  45  3.4.1^B-spline Deformable Transform ^  46  3.4.2^B-spline Interpolation ^  50  3.4.3^Mean Squares Metric ^  51  3.4.4^Limited Memory Broyden Fletcher Goldfarb Shannon Minimization with Simple Bounds Optimizer ^ 52 3.5^General Validation Scheme ^  53  3.6^Warping and Validation on Synthetic MR Spinal Cord Model Image ^ 55 3.7^Warping and Validation on In Vivo MR Rat Spinal Cord Image ^ 57 3.8^Warping and Validation on VHP Spinal Cord Model Image ^ 58 4 Results and Discussion ^  61  4.1^Results for Validation on Synthetic MR Spinal Cord Model Image ^ 61 4.2^Results for Validation on In Vivo MR Rat Spinal Cord Image ^ 64 4.3^Results for Validation on VHP Spinal Cord Model Image ^ 67  ^ ^  4.4^Analysis of Rat Spinal Cord Deformation ^  71  4.5^Limitations of the MRI Approach ^  74  5 Conclusions and Future Work ^  ^5.1^Thesis Contributions ^ 5.1.1^List of Publications ^ 5.2^Future Work ^ Bibliography  ^  Appendix A - Statement of Co-Authorship ^ Appendix B - Imaging Ethics Certificate  vi  76  76 77 78 79 .88 ^ 89  List of Figures Figure 1.1: a) The different sections of the vertebral column (illustration courtesy of http://www.sci-recovery.org ) [7]. b) The spinal cord in relation to the vertebra (illustration courtesy of http://www.sci-recovery.org) [7]. c) Cross-section diagram of the spinal cord. 4 Figure 1.2: A diagram corresponding the different levels of spinal cord to different parts of the anatomy, and how damage to different levels of spinal cord relates to paralysis and loss of sensation to different parts of the human body (illustration courtesy of http://www.sci-recovery.org ) [7]. 6 Figure 1.3: The registration framework used in this project [29]. ^ 15 Figure 1.4: Rigid transformation is a subset of affine transformation, which is in turn a subset of non-rigid or deformable transformation. ^ 17 Figure 2.1: An overview of the various kinds of acquired and model images, and their purposes, used through the thesis ^ 21 Figure 2.2: Diagram illustrating terms of anatomical location. ^ 22 Figure 2.3: The spinal cord in relation to the spine (illustration courtesy of http://www.sci-recovery.org) [7] and cross-section diagram of the spinal cord with additional features compared to Figure 1.1. Green ring illustrates the position of dura relative to the cord; and yellow ring illustrates the position of CSF relative to the cord. 22 Figure 2.4: Quadrature birdcage RF coil. ^  25  Figure 2.5: In vivo rat imaging setup. ^  26  Figure 2.6: Diagram illustrating the process of inflating the custom-made inflation device in order to create deformation in the spine. a) Before inflation. b) After injecting water into balloon with a syringe, thereby inflating it to create a change in posture of the spine. 27 Figure 2.7: Sample slices of the in vivo rat spinal cord data image in the non-deformed state. Note locations of the spinal cord, nerve roots, and vertebra. ^ 27 Figure 2.8: Diagram illustrating the process of setup to deform the spine for ex vivo imaging. Note that 'flexion' refers to bending of the rat towards the ventral direction... 29 Figure 2.9: Diagram illustrating the different components of the ex vivo imaging setup and their positioning relative to one anther ^ 29 Figure 2.10: Custom-made wooden apparatus in which the rat was fixated in place for ex vii  vivo imaging. Note the location of the quadrature surface coil. ^ 30  Figure 2.11: Sample slices of the ex vivo rat T9-T11 spinal cord data image in the nondeformed state. Note locations of the spinal cord, CSF/dura layer, 'butterfly', vertebra, blood vessel, and nerve roots. 31 Figure 2.12: Sample slices of the synthetic MR spinal cord model image. ^ 32 Figure 2.13: Sample slices of the segmented VHP male subject C4-C6 spinal cord. The slices, from left to right, correspond to the C4, C5, and C6 levels of the spine, respectively. 34 Figure 2.14: Diagram illustrating loading conditions for the case of distraction in the FE analysis of a human spinal cord, as described in [32]. Note the location of analysis on the spine (red box), C4-C6 vertebrae (dark blue), and spinal cord (light blue). 35 Figure 2.15: Diagram illustrating loading conditions for the case of contusion in the FE analysis of a human spinal cord, as described in [32]. Note the location of analysis on the spine (right red box), C4-C6 vertebrae (dark blue), spinal cord (light blue), indentor (right red box). 35 Figure 2.16: Diagram illustrating loading conditions for the case of dislocation in the FE analysis of a human spinal cord, as described in [32]. Note the location of analysis on the spine (red box), C4-C6 vertebrae (dark blue), and spinal cord (light blue) 36 Figure 3.1: Flow diagram illustrating the components involved during a spinal cord deformation analysis as proposed in this thesis. ^ 37 Figure 3.2: The process of manual spinal cord segmentation to create a binary mask Note the manual region-filling process (red). ^ 38 Figure 3.3: a) Scalar, S, is defined as the quotient of two parallel vectors. b) Versor, V, is defined as the quotient of two non-parallel vectors equal in length [36] ^ 40 Figure 3.4: The versor represented as an arc with direction on a unit sphere. Because the identity of a versor does not change with translation in space, the direct arc can move freely along the perimeter of the circle in the sphere and still represent the same versor [36] 41 Figure 3.5: In Parzen windowing, the density function of an image can be constructed by super-positioning a kernel function K(s), in this case a Gaussian function, on the elements of the set S, which is randomly sampled from the image [38]. 44 Figure 3.6: This diagram illustrates the deformable registration process, where a deformation field that takes the fixed image F to the moving image M is computed, though the moving image is warped to resemble the fixed image at the output of the registration. 46 viii  Figure 3.7: B-splines of degrees a) 0, b) 1, c) 2, and d) 3 [40]. ^ 48 Figure 3.8: Diagram illustrating the process of transforming an image using a B-splinebased transformation in order to match a moving image (spinal cord in background) to a fixed image. Thick line represent grid lines, thin lines represent voxels, and red arrows represent the displacement that each knot (red dots) undergoes during an iteration of transformation. 49 Figure 3.9: Flow diagram illustrating the general validation process of the proposed deformation algorithm. Note that the 'Analysis' component represents the process illustrated in Figure 3.1. 55 Figure 3.10: Flow diagram illustrating the validation process using the synthetic MR SC model image and synthetic warping schemes. Note that the 'Analysis' component represents the process illustrated in Figure 3.1. 56 Figure 3.11: Diagram illustrating the process of synthetically warping (`bending') a spinal cord, in order to be used for validation. 56 Figure 3.12: Diagram illustrating the process of synthetically warping (`contortion') a spinal cord, in order to be used for validation. 57 Figure 3.13: Flow diagram illustrating the validation process using the in vivo MR rat spinal cord image and synthetic warping schemes. Note that the 'Analysis' component represents the process illustrated in Figure 3.1. 57 Figure 3.14: Flow diagram illustrating the validation process using the VHP spinal cord model image and FEA-based warping schemes. Note that the 'Analysis' component represents the process illustrated in Figure 3.1 60 Figure 4.1: Visualization of the synthetic spinal cord model before (yellow) and after (green) (a) Warp 1 (`bending'), and (b) Warp 2 (`contortion'). Corresponding deformation fields at 3 locations are overlaid (blue arrows). 62 Figure 4.2: Accuracy of measured deformation, represented as median error between the known and calculated deformation fields. Various levels of deformation of the synthetic spinal cord model with Warp 1 (`bending') and Warp 2 (`contortion') are shown. Both the error and deformation extent are measured relative to the cord width. 64 Figure 4.3: Visualization of MR image of the in vivo rat spinal cord before (green) and after (blue) (a) Warp 1 (`bending'), and (b) Warp 2 (`contortion'). Corresponding deformation fields at 3 locations are overlaid (red arrows). Vertebrae are only shown for the non-deformed position for clarity. 65 Figure 4.4: Accuracy of measured deformation, represented as median error between the known and calculated deformation fields. Various levels of deformation of the in vivo spinal cord data with Warp 1 (`bending') and Warp 2 (`contortion') are shown. Both the ix  error and deformation extent are measured relative to the cord width. ^ 66 Figure 4.5: a) Visualization of the FEA-based distraction deformation node points used to apply deformation to the VHP spinal cord model image for validation, with the spinal cord before (yellow) and after (green) deformation. b) VHP spinal cord model image before (yellow) and after (green) applying the distraction deformation. Corresponding deformation fields at 3 locations are overlaid (red arrows). 68 Figure 4.6: a) Visualization of the FEA-based contusion deformation node points used to apply deformation to the VHP spinal cord model image for validation, with the spinal cord before (yellow) and after (green) deformation. b) VHP spinal cord model image before (yellow) and after (green) applying the contusion deformation. Corresponding deformation fields at 3 locations are overlaid (red arrows). 69 Figure 4.7: a) Visualization of the FEA-based dislocation deformation node points used to apply deformation to the VHP spinal cord model image for validation, with the spinal cord before (yellow) and after (green) deformation. b) VHP spinal cord model image before (yellow) and after (green) applying the dislocation deformation. Corresponding deformation fields at 3 locations are overlaid (red arrows). 70 Figure 4.8: Visualization of MR image of the in vivo rat spine: a) Visualization of the original, non-deformed spinal cord (yellow) and the cord produced from the registration (green). b) The spinal cord including the vertebrae before (green) and after (blue) a real deformation. Corresponding deformation fields at 3 locations are overlaid (red arrows). Vertebrae are only shown for the non-deformed position for clarity. 72 Figure 4.9: Visualization of MR image of the ex vivo rat spinal cord: a) Visualization of the original, non-deformed spinal cord (yellow) and the cord produced from the registration (green). b) The spinal cord before (yellow) and after (green) a real deformation. Corresponding deformation fields at 3 locations are overlaid (red arrows).73  Acknowledgements I would like to thank my supervisors Dr. Rafeef Abugharbieh and Dr. Peter A. Cripton for their continued support and guidance. I would also like to thank my collaborators Dr. Piotr Kozlowski and Andrew Yung for their expertise and assistance with animal imaging; and Dr. Wolfram Tetzlaff for forming the initial ideas of this study. I would also like to thank my colleagues from both Biomedical Signal and Imaging Computing Laboratory (BiSICL) and Injury Biomechanics Laboratory (IBL), for all their help in my project and the wonderful memories over the past two years. Special thanks are given to Claire F. Jones for her contributions with the mechanical procedures, setups, and general assistance with regards to the animal imaging; and Carolyn Y. Greaves for providing and contributing expertise to her finite element analysis deformation model. Finally, I would like to thank my parents for their love and encouragement.  KEVIN MING  The University of British Columbia  May 2008  xi  In loving memory of my Grandmother.  xii  1 Introduction and Background This chapter presents the objective of this work and highlights the novelty as compared to traditional studies of spinal cord injury, the motivation behind the desire to study spinal cord injury and myelopathy, and the accomplishments. A brief introduction to spinal cord injury and myelopathy-related pathology will also be given. A literature review of the state-of-the-art in spinal cord studies using imaging and image analysis techniques will then be presented. Finally, a brief introduction to the idea of image registration, registration framework, and the registration components used in this thesis will be provided.  1.1 Thesis Objectives Research in the area of spinal cord injury (SCI) has traditionally been carried out through biomechanical testing where animal spinal cords are exposed and subjected to mechanical injury [1],[2],[3],[4]. Such approaches are widely used in studies of neuroprotection and regeneration of the spinal cord. However, a major limitation of this is the need to surgically expose the cord, which changes the physiological environment and mechanical boundary conditions of the spinal cord. Compared to traditional biomechanical-based research, an alternative approach based on magnetic resonance imaging (MRI) and image based deformation analysis for the non-invasive study and quantitative assessment of spinal cords, in vivo, without exposure is proposed. It is demonstrated that the proposed approach constitutes a novel advancement as it allows for the study of deformations — particularly those of myelopathy-related sustained  1  compression — of the spinal cord in its natural physiological environment.  1.2 Motivation and Statement of Problem Among the different neurological disorders and impairments, SCI is perhaps the most devastating in terms of social and economical impacts. SCI can often result in paralysis, which significantly decreases the quality of life of not only those suffering injury but also the family and friends of those affected. It also results in many lost opportunities in terms of education, career, and social interactions. It is estimated that there are 36,000 Canadians living with SCI [5]. Of those, approximately 80% of all SCI around the world occurs to individuals under the age of 30, many of whom living a normal lifespan [5]. Economically speaking, it costs $121,600 (2002 $CDN) in health care per person with a complete SCI, and $42,100 per person with an incomplete injury during the first year of injury. In the subsequent 5 years, annual costs are $5,400 and $2,800 for persons with complete and incomplete SCIs, respectively [6]. Over an individual's lifetime, the health care cost can range from $1.25 million for a low thoracic paraplegic to $5 million for a high cervical quadriplegic such as the late actor Christopher Reeve, who required continuous ventilator support and 24/7 care [5]. Therefore, if the understanding of SCI can be bettered, improved techniques to heal, protect, or even prevent SCI can be devised.  1.3 Overview of SCI and Myelopathy The human spine (as known as the vertebral column) is made up of the boney vertebrae that are held together by the soft intervertebral discs, allowing movement of the spine. It 2  serves to protect the thin, tubular bundle of nerves collectively called the spinal cord, which extends from the brain to mainly the peripheries of the body such as arms and legs; the space in the vertebrae through which the spinal cord passes is called the spinal canal. The purpose of the spinal cord is to provide a passageway through which the brain communicates sensory and motor signals with the rest of the body. Like the brain, the spinal cord is composed of the grey matter (GM) and white matter (WM). The GM pattern within WM in the spinal cord is also sometimes referred to as the 'butterfly'. The GM is responsible for responding to sensory or motor stimuli and the processing of other sensory information; the WM is responsible for transmitting that information to and from different areas of the GM. Together, GM and WM allow the transmission and reception of signals between the brain, spinal cord, and rest of the body. The bundles of nerves extending from the spinal cord, protruding out of the vertebral columns, and branching out to nerves in the rest of the body for communication are called the nerve roots.  3  Figure removed due to copyright permission. Figure removed due to copyright permission. The human spine. Cross-section of the human spine. http://www.sci-recovery.org http://www.sci-recovery.org  (b) White Grey Matter  Matter^'Butterfly  (a)  ^  (c)  Figure 1.1: a) The different sections of the vertebral column (illustration courtesy of http://www.sci-recovery.org) [7]. b) The spinal cord in relation to the vertebra (illustration courtesy of http://www.sci-recovery.org) [7]. c) Cross-section diagram of the spinal cord.  SCI refers to those injuries that pertain to the damage or impairment of the cord, leading to loss of sensation, motor functions, and/or other bodily functions. SCIs can either be traumatic or non-traumatic. Traumatic SCIs typically result from a sudden, traumatic blow to the spine causing fractures, dislocations, crushing, or compression of one or more vertebrae. The causes include, but are not limited to, motor vehicle accidents, falling from large heights, or sports-related injuries. SCI can also be resulting from external wounds that penetrate the spine and cuts into the cord. Non-traumatic SCIs are those injuries due to infections or chronic conditions, for example.  4  Myelopathy is a general term used to describe any pathological changes or  sensory-motor functional disorders in the spinal cord. It can the result of a growth or tumor compressing against the cord, bone spurs where one or more vertebral columns protrude into the spinal canal, external damage resulting from SCI, or degeneration of the cord itself. One form of spinal degeneration, which is especially prevalent in individuals over the age of 50 [8], is spinal stenosis, whereby the cord experiences a constriction or narrowing. The effects range from pain and loss of balance, to loss of sensation and loss of bowel or bladder control. Current treatment is for the patient to undergo decompression surgery in order to relieve pressure on the spinal cord at the site of stenosis. Spinal cord compression, or more specifically referred to as malignant or metastatic spinal cord compression (MSCC), refers to the scenario where the cord is compressed by a tumor or any other lesion [9]. About 90% of patients have pain and up to 50% may be unable to walk and have sensory and/or bladder/bowel dysfunction [10]. It is also a major cause of morbidity in patients with cancer [11]. While treatment options are available, such as radiation therapy and surgical removal of obstruction, the outcome is often poor [10]. Regardless of the cause, spinal cord impairment deals damage to the nerve fibres corresponding to not only those muscles and nerves at the site of harm, but also those below, leading to either partial or total paralysis. The severity of disability depends on the level at which the damage occurs. Damage at the upper or lower back level may cause partial or total loss of leg movement, total loss of torso movement, and sometimes the loss of bowel and bladder control — referred to as paraplegia. Damage at the neck level  5  may cause partial or total loss of arm movement, total loss of torso and leg movement, and sometimes the loss of the ability to breathe due to paralysis of chest muscles — referred to as paraplegia. The loss of sensation at and below the site of damage is shared between paraplegics and quadriplegics. Other effects include severe pain, deformity, and additional degeneration of spinal cord, which may lead to further complications. Damages to the spinal cord can also be classified as either complete or incomplete. 'Complete' refers to those where the patient has total loss of sensory and motor functions below the site of damage, whereas 'incomplete' refers to those where the patient has only partial loss of sensory and motor functions below the site of damage. Figure removed due to copyright permission. The human spine and correlation between the spinal cord and the human body http://www.sci-recovery.org Figure 1.2: A diagram corresponding the different levels of spinal cord to different parts of the anatomy, and how damage to different levels of spinal cord relates to paralysis and loss of sensation to different parts of the human body (illustration courtesy of http://www.sci-recovery.org) [7].  1.4 Medical Imaging Approaches to SC Studies In addition to those studies whereby the spinal cord is studied mechanically, medical imaging techniques — specifically MRI — is becoming a prevalent tool for studying the cord. The ability to penetrate deep into the body and examine the internal anatomy in detail has made imaging a powerful tool in the field of medicine. And due to its ability to image soft tissues such as those composing the spinal cord, MRI has become the ideal modality for the study of spinal cord and spinal cord-related impairment. However, most current MRI studies of the cord involve laminectomy — surgical expose the cord — prior to 6  applying mechanical insult to the open vertebrae. Bilgen et al. studied the spatial and temporal evolution of hemorrhage in controlled SCI of rats [12]. They also examined the use of dynamic contrast-enhanced MRI for studying injury progression [13], and assessed the post-injury blood-spinal cord barrier permeability of rats [14]. Gareau et al. used magnetization transfer imaging techniques to study compression injuries in rat spinal cords [15]. Behr et al. established the feasibility and evaluated the performance of high-resolution MRI of rat spinal cord at 17.6T [16]. More recently, [17], MR images were acquired on injured rat spinal cords using a 17.6 T scanner in order to establish feasibility studying pathological changes in injured spinal cords. The images were acquired two to 58 days post injury on two ex vivo and 5 in vivo specimens, with one uninjured control case. Laminectomy was performed at T10 level of the cords in order to expose the dorsal side to contusion impact. Results demonstrated the feasibility of this approach towards the qualitative study of contusion SCI via visualization of structures changes in the cord. In [18] Bilgen et. al. tracked anatomical, pathological, structural, and functional changes of a contusion in mice. Its effects were observed on 8 specimens over the period of 1, 7, 14, and 28 days post injury. In this case the imaging was also preceded by laminectomy of the cord, around the T11 level, to expose it for mechanical impact. And like the previous study, this study has also demonstrated the ability of MRI to track changes damaged spinal cord.  7  Yukawa et. al. in [19] examined the possibility of correlating MR image intensity values of SCI in human patients with patient age, duration of disease, postoperative severity of myelopathy, and recovery rates. The study was performed across 104 patients with cervical compressive myelopathy, whose spinal cords were imaged before and after corrective surgery on C3-C7. It was found that higher preoperative image intensities around the area of injury corresponded to those patients who were older, had longer durations of the disease, the injury was more severe, and that the postoperative recovery time was longer. Diffusion tensor imaging (DTI), a special functionality of MRI, is a relatively new tool for the study of spinal cord and related impairments. This technique is based on the idea of measuring the continuous random motion (i.e. diffusion) of molecules in fluid systems by measuring the amount of impedance to that motion in a particular direction (i.e. reduction in the diffusion coefficient), which may be due to structural or impairmentinduced resistance. It, then, enables the visual reconstruction of WM fiber tracts in either the brain or the spinal cord. Elshafiey et al. in [20] established the feasibility of characterizing the diffusion behaviour of water molecules in rat spinal cords using DTI. In turn, they proposed to apply this to the understanding of SCIs. More recent advances in DTI spinal cord studies include the one conducted by [21], which suggested that a specific parameter derived from DTI can be used to assess the severity of acute and slowly progressive spinal cord compression in patients. Statistical analysis was performed across 15 clinical symptoms of spinal cord compression, with 2 acute and 13 slowly progressive. Eleven healthy 8  volunteers were also selected for comparison purposes. Both standard and DT imaging were performed using a 1.5T scanner. Of the three assessment techniques used — signal intensity derived from the standard MRI, and the apparent diffusion coefficient (ADC) and fractional anisotropy (FA) parameters derived from the DTI — FA demonstrated the highest sensitivity with statistical significance for detecting abnormalities in the both acute and slowly progressive spinal cord compression. Loy et al. demonstrated in [22] the ability of DTI to study severity of SCI in mice with SCI. As with the studies described previously involving animals, laminectomy was performed before applying a mild, moderate, or severe contusive SCI to 20 female mice. A surface coil, covering the T11-T13 vertebrae, was also used to improve imaging signals. It was found that during the acute phase of injury, the relative anisotropy (RA) provided excellent contrast between the GM and WM for all injury levels. But more importantly, axial diffusivity, a DTI-derived statistic, could be use to assess the severity of injury with good histological correspondence. The imaging approaches described above, however, is not sufficient. Though they provide the means of visualizing the spinal cord non-invasively, they are only able to provide qualitative rather than quantitative analysis of deformation, which lacks objectivity and does not give insight into the impairment-inducing event. In addition, even when limited quantitative analyses were proposed, such as [18],[19],[22], they were typically represented in the form of generalized statistics and measures that attempted to encapsulate the severity or mechanism of injury in one or a few variables. Furthermore, all previous studies were primarily focused on the injury progression after a mechanical 9  insult was produced on the cord via an open spinal column. One widely-acknowledged limitation of this approach is the need to surgically alter the spinal canal to expose the cord [4]. This changes the mechanical and biological environment of the cord by direct alteration of the bony canal, dura, and the cerebrospinal fluid (CSF) layer.  1.5 Medical Image Analysis Approaches to Spinal Studies With advances in medical imaging come emergence and improvement of medical image analysis techniques. The two predominant medical image analysis techniques are image segmentation — the process of extracting features of interest within an image — and image registration — the process of finding the spatial transform that maps points from one image to the corresponding points in another image [23]. An overview of some of these techniques for applications in spinal studies is provided below. Schmit and Cole [24] quantified morphological changes in the spinal cord for patients suffering from chronic SCI. Images of both neurologically intact and spinal cord injured human subjects were acquired using MRI to be studied. A seeded region growing segmentation technique was performed on the images to extract the spinal cord in order for the cross-sectional areas of the cords to be calculated. It was found that those cords with SCI exhibited much smaller cross-sectional areas than those of the neurologically intact subjects. Similarly, cervical spinal cord atrophy was quantified using cross-sectional area in [25]. As in the case above, segmentation of the spinal cord was performed, and the surface of the cord was modeled using a B-spline parametric surface approach. The 10  parametric surface provided a means to define a medial axis down the section of the cord, from which orthogonal cross-sections can be derived, thus allowing the calculation of cross-section areas along the cord. In [26], Mathias et. al. presented a method to quantify pathological changes in the spinal cord associated with multiple sclerosis, using methods of texture analysis. By studying the statistical relationships between voxel intensity values of spinal cord images, it was demonstrated that textural differences exist between images of control subjects and patient subjects. This suggests that textures analysis can be used to detect pathological changes and perhaps correlate spinal cord image texture with disability. In [27], Yuan et. al. studied in vivo human cervical spinal cord during head flexion. MR images of the spinal cord of volunteers, before and after flexion, were acquired using the 'tagged MRI' technique. Originally developed to track cardiac motion, tagged MRI overlays a series equally-spaced, parallel tag lines on the image for measurement. Using these tag lines, the study was able to manually compute the deformation and displacement of the cord in flexion. It was found that the cervical cord elongates and displaces considerably during head flexion. More recently, in [28], du Bois d'Aische et. al. proposed a registration framework to track motion in multimodal medical images of the cervical spine for use in radiation oncology. Segmentation of the vertebrae is followed by an overall matching of images of the spine before and after motion. An articulated model of the spinal column, composed of elements having limited degrees of freedom, is then applied to the segmented images in order for further, more accurate registration. Validation was performed by 11  locating landmarks defined on the images before and after registration and computing the distance between corresponding landmarks. It was shown that this approach is effective in tracking motion in images of the neck acquired using different imaging modalities. Including the work described above, the number of researchers utilizing image analysis techniques to study the spine and spinal cord is still relatively limited to the best of our knowledge. And while the studies reviewed above have made some promising advances in this area of research, [24],[25][26] tended to focus on the examination of the spinal cord after damage has been done, concentrating on quantifying the pathology or atrophy. In the case of [27], where manual analysis of cord deformation was performed, the examination process was time-consuming and the results reported can only be applied to a very specific case of non-injurious cord deformation. And while [28] demonstrated the effectiveness of their approach in tracking motion in the spine, the technique is only applicable to the bony, rigid vertebrae, whose range of motion is very limited — reducing analysis complexity — compared to the spinal cord, which can deform quite freely. This thesis presents a novel non-invasive approach for the analysis of dynamic deformations in rat spinal cords. This analysis examines deformities in the cord, either injurious or pathological in nature, and provides a solution as to how the deformity might have occurred. The understanding gained through this approach allows for potential derivation of the causes, mechanisms, and tolerance parameters of spinal cord deformations, as well as the measures used in neuroprotection and regeneration of spinal cord tissue. And since this is all accomplished in vivo and non-invasively it can be ensured that the physiological and biomechanical properties of the spinal cord and its 12  environment are retained.  1.6 Introduction to Image Registration Image registration is the process of finding the spatial transform that maps points from one image to the corresponding points in another image [23]. In this application it is used to match the image of the spinal cord before and after a deformation or compression, from which the deformation field describing the deformity can be derived. 1.6.1 Registration Criteria In general, registration algorithms can be classified using the criteria through which images are matched, and the three schemes are: landmark-based, segmentation-based, and intensity-based. Landmark-based registration uses landmarks — small, distinctive features or fiducial markers, to identify and match corresponding locations between the fixed and moving image. The selection of landmarks is usually performed manually by the user, and they can be as simple as points to as complex as intersection of lines. Because the number of landmarks is quite small compared to the total number of voxels in the image, landmark-based registrations are typically quite fast computationally. However, the drawback is need for user input for locating landmarks, which is prone to inconsistency and reproducibility issues, thus reducing accuracy of the registration. Segmentation-based registration uses binary structures — curves, surfaces, or volumes — obtained through a pre-process of segmentation to match images. In this approach, the segmented structure of the first image can either be registered to 1) the 13  segmented structure of the second image, or 2) the entire, non-segmented second image. In the latter case, the boundary of the segmented structure is typically required to match the corresponding boundary of the non-segmented image. Due to the increase in complexity of the features to be matched, segmentation-based approaches are computationally more demanding and thus slower than landmark-based approaches, but with improved accuracy due to the same reason. Yet, the accuracy of this scheme is highly-dependent on the segmentation that precedes the registration, as differences between the two images could cause mis-registration. Intensity-based methods operate directly on the image intensity by comparing corresponding intensity values between the two input images. They are more flexible compared to the landmark-based and segmentation-based approaches because they utilize all the information in the image and do not require user input or pre-processing, making the intensity-based approaches more automatic. On the other hand, computational cost is increased due to the fact that the entire image content is used, making intensity-based approaches unsuitable for time-constrained applications [29]. In contrast to the first two approaches, where the registration typically involves minimizing distances between physical features such as landmarks or segmented boundaries, intensity-based registration seeks to minimize the cost function that measures similarity in regions of interest between intensity values of the two input images. While computationally slower than both the landmark-based approach and the segmentation-based approach, intensity-based registration offers the highest accuracy of matching in all three schemes because it takes into account all of the information available in the images.  14  1.6.2 Registration Framework In the registration scheme used in this work, an intensity-based registration process was adopted, whose functionalities are divided into several components: input images, spatial transformation, image interpolation, similarity metric, and parameter optimization. Except for the input images, each of these functionalities can be accomplished by a variety of different techniques. This suggests modularity in the structural nature of the registration process, which allows different components to be used for each of the functionalities to suit different registration problems. The modular nature in the structure of the registration process is depicted in Figure 1.3. The input images included the fixed image, F, and the moving image, M. The goal of registration is to find the spatial mapping that will map voxels in the fixed image to the corresponding voxels in the moving image, though the mapping is applied in reverse at the output of the registration process such that it will bring the moving image into alignment with the fixed image; this output then becomes the registered result. Figure removed due to copyright permission. The ITK registration framework http://www.itk.org Figure 1.3: The registration framework used in this project [29].  1.6.3 Spatial Transformation A registration problem is usually defined by the type of spatial transform it uses to map the points in one image space to the image space of the second image. The transform is typically defined by a set of parameters, and the goal of registration is to optimize these parameters with respect to the registration criterion used to measure the similarity 15  between fixed and moving images. Therefore in general, more parameters mean more degrees of freedom, which means increased difficulty in registration due to a more difficult optimization problem. On the other hand, though, more degrees of freedom means the images can move more freely, which could result in improved registration results. The spatial transformation in any registration problem, either 2-dimensional (2D) or 3-dimensional (3D), can be grouped into three categories: rigid, affine, or deformable. Rigid transformations are those that allow only translations and rotations. Affine transformations, sometimes also classified generally as rigid transforms, are those that allow shearing and scaling in addition to translations and rotations. Deformable transformations are those that allow free-form mappings such that the objects in the image are not limited to a set pattern of movement. As such the registration problem can yield many solutions, which is why deformable registration problems typically specify a certain constraint in the optimization (e.g. minimal energy) to limit the number of solution spaces. In a hierarchical sense, rigid transformation is a subset of affine transformation; affine transformation is in turn a subset of deformable transformation. This idea is illustrated in Figure 1.4.  16  Non-rigid/Deformable Transformation  Figure 1.4: Rigid transformation is a subset of affine transformation, which is in turn a subset of non-rigid or deformable transformation.  1.6.4 Image Interpolation  When mapping points from one image space to another image space interpolation is needed because the mapped points will generally fall on a non-grid location in the moving image, requiring a means to obtain image intensities at these locations. For the intensity-based registration approaches, image interpolation is even more crucial as the registration works directly with the image intensities. This means that the interpolation affects the smoothness of the optimization search space. And because interpolations are performed thousands of times in a single optimization cycle, one needs to select an scheme that optimizes for computational cost and ease of optimization result. 1.6.5 Similarity Metric  The metric component represents the similarity measure that determines how well the fixed image matches a transformed moving image after each iteration of the spatial transform. This allows for a quantitative measure of the criterion by which the registration is to terminate, when a suitable spatial transform has been found that matches the moving image to fixed image. That is, this measure is what the optimizer performs on 17  over the search space of the transform parameters.  1.6.6 Parameter Optimization As explained in Section 1.6.3 Spatial Transform, registration problems can be thought of as optimization problems, where the registration criterion — a measure of similarity between the fixed and moving image — is considered as the cost function to be minimized over the search space spanned by the spatial transformation parameters. Beginning with an initial set of parameters, the optimization procedure iteratively searches for an ideally global minimum solution by evaluating the value of the cost function at different locations of the search space. Within the registration framework through which the registration is performed, the same idea applies: The optimizer component optimizes the quantitative measure provided by the metric component with respect to the parameters of the transform component. Starting with an initial set of parameters, the optimizer iteratively searches for the optimal solution to the registration problem by evaluating the metric at different locations in the transform parameter space. Note that when optimizing for the registration parameters, special attention needs to be paid to the scale of the parameters, since transformation parameters may have drastically different dynamic ranges. For example, in the rigid registration case, where the optimization is specified by translations and rotations, a unit of change in the displacement (in mm) is quite different from a unit of change in the angle (in radians).  18  In fact, the change in angle has a much greater impact on the optimization process than that of the change in displacement. A simple implementation of rescaling is to multiply or divide the metric gradient by weights chosen to balance the parameters.  19  2 Image Acquisition and Modeling Due to the high level of anatomical details (i.e. presence of subject-specific features and irregularities) and the realism with which the cord deforms, both of which are difficult to replicate in digital, synthetic cases, images of both in vivo and ex vivo rat spinal cord were acquired. They, then, were used for purposes of 1) validation, and 2) simulation of analysis in human spinal cord deformation. In this chapter the various procedures, setups, hardware, protocol and limitations involved in acquiring the images will be discussed. First in vivo imaging of the rat spinal cord before and after deformation will be considered. The ex vivo imaging of the rat spinal cord, before and after a deformation case, will then be discussed. Finally, the SC model images developed, and the corresponding finite element analysis adopted, in order to assess the accuracy of this deformation analysis tool, will be considered.  20  Acquired Data Real Deformation (Hyperfiexion)  Ex Vivo Rat SC MR Image  1 In Vivo Rat SC MR Image  L____J Model Data  Real Deformation (Custom-made Inflation Device)  Analysis of Real Cord Deformation  I Synthetic Warping (Bspline) Valdiation  1 FEA-based VHP SC \ I Warping (Thin Model Image —1-11" Plate Spline)  L _ _  Figure 2.1: An overview of the various kinds of acquired and model images, and their purposes, used through the thesis.  2.1 Ethics Approval All images acquired for this thesis were obtained using rats and protocols approved by the appropriate Institutional Review Boards and Ethics Boards of the University of British Columbia (UBC). Scientific and ethical reviews were also conducted by the UBC High Field MRI Research Centre. Please refer to the Appendix section for details.  2.2 Overview of Anatomical Terms of the Rat Spine Before rat imaging details are discussed, an overview of the rat spinal anatomy is in order. As a precursor, however, Figure 2.2 below illustrates the anatomical terms of 21  location used in this thesis in order to help identity the positions or directions of the various anatomical features.  Caudal  Figure 2.2: Diagram illustrating terms of anatomical location.  Figure 2.3 below highlights the anatomical features of the spine and spinal cord, and their relative positions that will be referred to in subsequent sections. Note that these features are shared between both rats and humans. Figure removed due to copyright permission. Cross-sections of the human spine and spinal cord http://www.sci-recovery.org Figure 2.3: The spinal cord in relation to the spine (illustration courtesy of http://www.sci-recovery.org) [7] and cross-section diagram of the spinal cord with additional features compared to Figure 1.1. Green ring illustrates the position of dura relative to the cord; and yellow ring illustrates the position of CSF relative to the cord.  In addition to those features described in Section 1.3above, the spinous processes are fin-shaped structures at the dorsal side of the bony vertebrae. The dura mater, or simply dura, is a tough, inflexible membrane surrounding the spinal cord, providing basic protection of the cord. The cerebrospinal fluid (CSF), within the spinal canal, is a layer of clear bodily fluid that serves as buffer between the spinal cord and the exterior 22  membranes (e.g. dura). It also provides basic mechanical and immunological protection for the cord.  2.3 Rat Imaging The ability to examine organs and features deep within the body in great details, without any surgical procedures, makes medical imaging a powerful tool for studying anatomy. MRI, a modality that is predominantly used to image the water content in soft tissues, is the perfect and arguably the only suitable modality for imaging the spinal cord, as it is a mass of soft tissues consisting of mostly water [1]. In this study two types of rat spinal cord images were acquired: in vivo and ex vivo. Unlike ex vivo imaging where spinal cord tissues breakdown after euthanasia, which  can significantly affect its mechanical properties 2 hours post mortem [30], in vivo imaging ensures that the physiological and biological properties of the cord are retained for the entire duration of imaging. Furthermore, rigor mortis, the process whereby a cadaverous body stiffens and makes changing its posture more difficult, does not arise during in vivo imaging. Yet, in vivo rat spinal cord imaging is restricted in many ways compared to its ex vivo counterpart. For one, the deformation allowed in the in vivo cord is very limited  because it must be non-injurious, to ensure the wellbeing of the animal. In addition, one has to ensure that the duration in which the rat can be imaged is carefully controlled as prolonged anesthesia can be harmful to the animal — a non-issue in the ex vivo case. Additionally, unlike ex vivo scans, in vivo imaging suffers from motion artifacts arising 23  from animal breathing. And in order to minimize these motion artifacts in vivo acquisitions are respiration-triggered, prolonging overall scan times. Ex vivo imaging, in contrast, has the advantage of longer allowed acquisition time due to the lack of need for respiration-triggering and issues regarding prolonged animal anesthesia, enabling the improvement of imaging signals and thus image quality. Finally, it should be noted that current MRI technology does not allow the imaging of traumatic spinal cord injury, where the injury event takes place in the order milliseconds as opposed to minutes to tens of minutes required for imaging. However, with this approach, MRI can be used for the analysis of myelopathy-related sustained compression of spinal cord deformations. 2.3.1 In Vivo Rat MRI In vivo MR scans of the rat spinal cord, captured before and after applying a specified  small, non-injurious deformation, were obtained. The images were acquired on a Bruker Biospec 7T MRI scanner (Ettlingen, Germany) using a 120 mm i.d. gradient coil (fixed to scanner) and 70 mm i.d. quadrature birdcage RF coil (Figure 2.4). The procedures prior to imaging, with photograph of the setup (Figure 2.5), are as follows: 1) Anesthetize rat. 2) Position rat on the bottom half of a cylindrical plastic casing, and insert nose and mouth into anesthesia feeder. Note that a surface coil has been built into the bottom casing to enhance signal reception and transmission. 3) Attach heart rate monitor to chest of animal. 4) Place custom-made inflation device underneath lumbo-sacral region of animal 24  spine. The device is a balloon made of rubber with rubber tubing attached to the open end so that water can be injected, via a syringe, to inflate the balloon. This allowed the posture of the spine to be changed in situ (Figure 2.6). 5) A heating pad to keep the animal warm during the imaging session is then placed over the animal. 6) Close cylindrical plastic casing by placing the top half on top of bottom half. 7) Insert the full plastic casing setup into the birdcage coil. 8) Insert the entire birdcage coil setup into scanner. 9) Begin imaging.  Cylindrical Plastic Casing Heating Pad  Figure 2.4: Quadrature birdcage RF coil.  25  Cylindrical Plastic Casing (Top Half)  Heating Pad  Rubber Tubing  Heart Rate Monitor  fi  Custom-made Inflation Device  Anesthesia Feeder  Surface Coil^Cylindrical Plastic Casing (Bottom Half)  Figure 2.5: In vivo rat imaging setup.  Empty Balloon Rubber Tubing  (a)  26  Water from Syringe  Filled Balloon Rubber Tubing  (b) Figure 2.6: Diagram illustrating the process of inflating the custom-made inflation device in order to create deformation in the spine. a) Before inflation. b) After injecting water into balloon with a syringe, thereby inflating it to create a change in posture of the spine.  A heavily T1-weighted 3D FLASH (Fast Low Angle SHot) scan was acquired with slab direction parallel to the lumbar spine axis (TE,/TR = 3/15 ms, 50x50x25mm field of view, 0.195x0.195x0.390 mm resolution, acquisition time approximately 24 minutes with respiratory triggering). An identical scan with the same orientation and position was acquired after inflating the custom-made device to generate sustained cord compression.  Figure 2.7: Sample slices of the in vivo rat spinal cord data image in the nondeformed state. Note locations of the spinal cord, nerve roots, and vertebra. 27  Given that the birdcage RF coil is only 70 mm in diameter, and that the entire plastic casing setup occupies most of that space, the degree and types of deformation to which the rat could be deformed was very limited. Moreover, in consideration for the safety and well-being of the animal, the deformation applied to the rat spinal cord in vivo was further limited to a non-injurious one. In order to allow the imaging of a nearinjurious cord deformation, ex vivo imaging of the rat spinal cord was decided. 2.3.2 Ex Vivo Rat MRI The procedures for the ex vivo imaging, with figures illustrating the setup and process, are as follows: 1) Euthanize rat, procedures of which were in accordance with animal ethics. 2) For the deformation, the rat was placed in a hyperflexion scenario at the T9-T12 region by tying all four limbs together and posture held in place with wedges (Figure 2.8). Note that the image of the cord after deformation was acquired first because to reduce imaging complexities, as complexities of the deformed cord setup are higher relative to the non-deformed case.  3) The setup was position on a quadrature surface coil in such a way that the flexion was placed directly on top of the coil for optimal signal reception and transmission. The coil was fixed to a cylindrical half shell plastic casing, which in turn was fixed in place within a custom-made wooden framework (Figure 2.9 illustrates positioning of the setup and Figure 2.l0shows the wooden framework). 4) The entire setup was then inserted into the scanner for imaging.  28  Limbs tied^ Flexion from together  •  11)1t  Orib4 Figure 2.8: Diagram illustrating the process of setup to deform the spine for ex vivo imaging. Note that 'flexion' refers to bending of the rat towards the ventral direction. Deformation Setup  Quadrature Surface Coil  Cylindrical Half Shell Plastic Casing  Custom-made Wooden Framework  Figure 2.9: Diagram illustrating the different components of the ex vivo imaging setup and their positioning relative to one anther. 29  Figure 2.10: Custom-made wooden apparatus in which the rat was fixated in place for ex vivo imaging. Note the location of the quadrature surface coil.  The imaging was performed with the same scanner as that used for the in vivo rat imaging. Localization scans were performed to prescribe the axial slice geometry through the deformed spinal cord. Two-dimensional spin echo axial RARE (Rapid Acquisition with Relaxation Enhancement) images were acquired at high resolution (1b/TR = 15.6/2500 ms, RARE factor = 8; # averages = 32, 0.1x0.1x0.5 mm resolution, fat suppression on, total acquisition time 42min40sec). The entire setup was then removed from the scanner and the rat was put into the neutral position by untying the limbs and removing the wedges. Rigor mortis was observed at this point and care was taken to ensure that any undesirable shifts in the unflexing process was minimal. That is, caution was used when untying the limbs, removing the wedges, and applying small amounts of 30  pressure to both the cranial and caudal ends of the cadaver such that it returned to neutral. The setup was then placed back into the same position of the scanner and the exact same slice prescription and settings were used for the scan.  Figure 2.11: Sample slices of the ex vivo rat T9-T11 spinal cord data image in the non-deformed state. Note locations of the spinal cord, CSF/dura layer, 'butterfly', vertebra, blood vessel, and nerve roots.  One constraint arising from this setup is due to the fact that in order to ensure optimal imaging quality, the location of spinal cord flexion must be positioned as closely to the quadrature surface coil as possible. Coupled with the fact that the bore of the magnet where the custom-made wooden framework, and in turn the deformed cadaver, is to be inserted has very limited spacing (the spacing inside the bore has a diameter of 120 mm), the types of deformations that can be imaged is limited.  2.4 Synthetic MR Spinal Cord Model Image As a first step toward validating the spinal cord deformation analysis tool, see Chapter 3, a synthetic MR model data of the spinal cord was generated. To generate the synthetic image volume, four nerve roots, two dorsal and two ventral, running orthogonal to the cord at regular intervals, acted as gross control points. Based on scans of the rat spinal  31  cord, shown in Figure 2.7 and Figure 2.11, a synthetic model image was developed to incorporate the following features inside the cord: the 'butterfly', a blood vessel on the dorsal aspect running along the length of the cord, and a layer of CSF/dura surrounding the spinal cord. The same grey-scale contrast scheme (e.g. GM has a higher grey-scale value than WM) is also seen in human spinal cords. The synthetic cord image, shown in Figure 2.12, was simulated with a spacing of 0.1x0.1x0.4 mm and signal-noise-ratio (SNR) of 42.0 dB, which was estimated to the be optimal level of signal quality achievable during in vivo rat imaging for this study. The mean dorsal-ventral width was a constant 4.40 mm in the synthetic model, which approximately corresponds to the width of a rat spinal cord at the C5 level (rat C5 level width 4.3 mm [31]). Blood.Vesse  Figure 2.12: Sample slices of the synthetic MR spinal cord model image.  2.5 Visible Human Project Spinal Cord Model Image As further means of validating the spinal cord deformation analysis approach, see Chapter 3, another spinal cord model image was generated, this time using photographs of cryosection images provided by the National Library of Medicine's Visible Human Project (VHP) male subject (Figure 2.13), cervical vertebrae four to six. The 3D image of the VHP human spinal cord, reconstructed using the photographs of 2D cryosection images, was created using 0.1x0.1x1.0 mm spacing according to VHP specifications. Note that for the VHP male subject spinal cord the mean dorsal-ventral spinal cord width was measured digitally, based on the photographs and its known spacing, to be 32  9.35 mm (range 8.58-9.99 mm; standard deviation 0.37 mm), across 61 slices of the data image (cord length of 61 mm). The cord was manually segmented from the surrounding features using the user-guided approach to be described in Section 3.1  33  Cord Segmentation. Figure removed due to copyright permission. Cryosection image of the male subject spinal cord M. Ackerman, "The visible human project," Proceedings of the IEEE 86, pp. 504-511, March 1998 Figure 2.13: Sample slices of the segmented VHP male subject C4-C6 spinal cord. The slices, from left to right, correspond to the C4, C5, and C6 levels of the spine, respectively.  In order to utilize the VHP SC model image (Figure 2.13) for a more sophisticated validation, a model of the human spine and spinal cord based on finite element (FE) analysis was adopted, developed to study clinical injury mechanisms of the cord and its surrounding structures [32]. Finite element analysis is a computer-aided simulation technique used to approximate the behaviour of materials, whose properties are not fully understood, under applied forces. This is done by assuming that the material in question is composed of many small elements — or finite elements — of different materials with known properties so that, when forced is applied to the object in question, the behaviour of the object can be approximated by the total behaviour of these finite elements. Finite element analyses often require finding solutions to partial differential equations as well as integral equations. The approach to finding these solutions is usually based on approximation, with the underlying principle being that of minimum energy principles or similar concepts, which limit the number of solutions. The geometry of the FE model was in fact based on the same photographs used to generate the human SC model image shown in Figure 2.13. The development of the FE model was done in ANSYS [33] where the three vertebrae and spinal cord, the dura mater, and the ligaments and discs were modeled with brick, shell, and cable elements, 34  respectively. Three types of spine loading conditions were modeled using FE analysis in [32]: distraction, contusion, and dislocation. Figure 2.14, Figure 2.15, and Figure 2.16 below illustrate each of these loading conditions. Figure removed due to copyright^Figure removed due to copyright permission.^ permission. The human spine^  Spine distraction  http://www.sci-recovery.org  C.Y. Greaves, M.S. Gadala, and T.R. Oxland, "A Three Dimensional Finite Element Model of the Cervical Spine with Spinal Cord: an investigation of three mechanisms," Annals of Biomedical Engineering, vol. 36(6), pp. 396-405, 2008.  Figure 2.14: Diagram illustrating loading conditions for the case of distraction in the FE analysis of a human spinal cord, as described in [32]. Note the location of analysis on the spine (red box), C4-C6 vertebrae (dark blue), and spinal cord (light blue).  Figure removed due to copyright permission.  Figure removed due to copyright permission.  The human spine  Spine contusion  http://www.sci-recovery.org  C.Y. Greaves, M.S. Gadala, and T.R. Oxland, "A Three Dimensional Finite Element Model of the Cervical Spine with Spinal Cord: an investigation of three mechanisms," Annals of Biomedical Engineering, vol. 36(6), pp. 396-405, 2008.  Figure 2.15: Diagram illustrating loading conditions for the case of contusion in the FE analysis of a human spinal cord, as described in [32]. Note the location of analysis on the spine (right red box), C4-C6 vertebrae (dark blue), spinal cord (light blue), indentor (right red box).  35  Figure removed due to copyright permission.  Figure removed due to copyright permission.  The human spine  Spine dislocation  http://www.sci-recovery.org  C.Y. Greaves, M.S. Gadala, and T.R. Oxland, "A Three Dimensional Finite Element Model of the Cervical Spine with Spinal Cord: an investigation of three mechanisms," Annals of Biomedical Engineering, vol. 36(6), pp. 396-405, 2008.  Figure 2.16: Diagram illustrating loading conditions for the case of dislocation in the FE analysis of a human spinal cord, as described in [32]. Note the location of analysis on the spine (red box), C4-C6 vertebrae (dark blue), and spinal cord (light blue).  36  3 Deformation Analysis and Validation In this chapter the various image processing techniques used to analyze deformities in the spinal cord will be discussed. A description of validation schemes used for assessing the accuracy of the deformation analysis tool then follows. First, image segmentation as a pre-processing step will be discussed. The general image registration approach and the idea of a registration framework will then be examined. Next, both the rigid and non-rigid image registration techniques in the context of this study will be discussed, respectively, providing background on the concepts as needed. Subsequently, the general outline of procedures involved in validation will be considered. The synthetic warping schemes that have been developed as initial validation tools and applied on the synthetic MR SC model image will then be examined. Following this, the details of applying the synthetic warping schemes to the in vivo rat data are discussed. Finally, the utilization of a finite element analysis-based image warping approach is analyzed. Analysis Fixed Image  It■ Segmentation  Registered Result  ^.1 Rigid Registration  Moving Image  Non-rigid Registration Computed Deformation Field  Segmentation  Figure 3.1: Flow diagram illustrating the components involved during a spinal cord deformation analysis as proposed in this thesis.  37  3.1 Cord Segmentation Image segmentation is the process of extracting features of interest within an image. For this study, the spinal cord was first segmented from the surrounding anatomy (e.g. vertebrae and epidural contents) to facilitate subsequent registration of the cord images, depicting the cord undergoing deformation. By excluding non-relevant regions of the anatomy, the registration speed was increased. This also circumvented interference with the registration process from features outside of the spinal cord thus accuracy was improved. For the images, spinal cord segmentation was performed using a user-guided, manual region-filling tool in MRIcro [34] to create image masks over the region of interest (ROI), i.e. spinal cord, on a slice-by-slice basis for each of the 3D images (Figure 3.2). The 3D binary masks were then overlaid on the original rat data, and after performing a simple voxel-based multiplication in MATLAB, 3D images of the segmented cords with black backgrounds was obtained. Note that the creation of the binary masks was performed manually on the basis of intensity gradients between the cord and vertebrae, guided by anatomical atlases and the user's knowledge of the spinal anatomy. Figure removed due to copyright permission. Cryosection images of the male subject spinal cord and segmentation M. Ackerman, "The visible human project," Proceedings of the IEEE 86, pp. 504-511, March 1998 Figure 3.2: The process of manual spinal cord segmentation to create a binary mask Note the manual region-filling process (red).  38  3.2 Deformation Analysis via Image Registration As was described in Section 1.5, image registration is the process of finding the spatial transform that maps points from one image to the corresponding points in another image [23]. For this application it is used to match the image of the spinal cord before and after a deformation or compression, from which the deformation field describing the deformation can be derived. In this study, image registration was implemented using an open-source software system, the National Library of Medicine Insight Segmentation and Registration Toolkit (1TK) [35]. The spinal cord, being a mass of very soft tissue (Young's modulus of 0.26 MPa with variation of ±0.03 MPa [1]), is flexible and does not move as a rigid body, thus necessitating a non-rigid registration process. However, the cord can undergo rotation and/or translation, combined with deformation, in response to certain external loading modes. The registration process is thus initialized by rigidly registering image pairs for global cord alignment, using a mutual-information metric, prior to performing non-rigid registration to account for local changes.  3.3 Rigid Image Registration Rigid image registration, as described in 1.6.3 Spatial Transform, is used to map transformations of translational or rotational nature. In this section, the various components that were used to describe the rigid registration framework suited for spinal cord deformation studies will be described.  39  3.3.1 Versor Rigid Transform A versor-based rigid 3D transform [35] was used as the spatial transform component in the proposed rigid registration framework because it was found to be the most robust (capable of accounting for both rotation and translation), provided the most overlap between the fixed and moving images, and one of the least computationally intensive (requiring only 6 parameters, 3 for rotations and 3 for translations) rigid transforms when compared against other rigid transforms. The versor represents the orientation of one vector relative to another, and is defined as the quotient — or the orientation change — between two equally-length but nonparallel vectors. It is because of this definition that versors provide a natural representation of rotations in 3D space. The figure in Figure 1.5 illustrates the ideas of the definitions of scalar and versor operators. Figure removed due to copyright^Figure removed due to copyright permission.^ permission. Scalar representation  Versor representation  L. Ng and L. Ibanez, "Quaternions," in Insight into Images: Principles and Practice for Segmentation, Registration, and Image Analysis, T.S. Yoo, Ed., Wellesley, Massachusetts: A K Peters, 2004, pp. 243 (a)  L. Ng and L. Ibanez, "Quaternions," in Insight into Images: Principles and Practice for Segmentation, Registration, and Image Analysis, T. S . Yoo, Ed., Wellesley, Massachusetts: A K Peters, 2004, pp. 243 (b)  Figure 3.3: a) Scalar, S, is defined as the projection of two parallel vectors. b) Versor, V, is defined as the quotient of two non-parallel vectors equal in length [36].  An intuitive representation of a versor is one that maps its elements onto the surface of a unit sphere, illustrated in Figure 3.4. A versor is fully defined by a directed arc traced on the sphere surface: By placing the tails of the two vectors A and /3 in the 40  centre of the sphere a plane that contains the two vectors can be identified. The circle that intersects the sphere surface with the plane then yields the directed arc, in such a way that the angle defining the arc is the angle between the two vectors. Much like how a vector can translate freely in space without losing its identity, this arc can also translate freely along the surface of the circle without losing its identity. Figure removed due to copyright permission. Versor concept L. Ng and L. Ibanez, "Quaternions," in Insight into Images: Principles and Practice for Segmentation, Registration, and Image Analysis, T.S. Yoo, Ed., Wellesley, Massachusetts: A K Peters, 2004, pp. 243 Figure 3.4: The versor represented as an arc with direction on a unit sphere. Because the identity of a versor does not change with translation in space, the direct arc can move freely along the perimeter of the circle in the sphere and still represent the same versor [36].  This idea can be extended to the case such that the rotation is described around an arbitrary point in space, as opposed to an origin. This new transformation, then, defines a rigid transform as it incorporates both rotations and translations. 3.3.2 Linear Interpolation  For a transformation as simple and linear as rotations and translations, a linear image interpolation scheme [35] was used because it is more robust than other simpler interpolators (e.g. nearest neighbour interpolation), and more computationally effective than other needlessly-sophisticated techniques (e.g. B-spline interpolation). In linear interpolation, the image being interpolated is assumed to be piece-wise linear, i.e. its intensity varies linearly between grid positions. The resultant, interpolated  41  image is spatially continuous but not differentiable. The interpolation is computed as a weighted sum of the 2' 1 neighbours of the pixel being considered: M(x)=  (3.1)  where M(x) represents the intensities of the interpolated or moving image at position x, w i represents the weights, and n is the dimension of the image space. The weights are computed using the distance between the pixel considered and its neighbours such that  w, _  i(x)k -(x,)k  1)  •  (3.2)  3.3.3 Mattes Mutual Information Metric Because rigid registration serves to globally align the fixed and moving images, it is important that the similarity measure, or metric, used is robust enough to account for the large translational, rotational, or other intensity discrepancies arising from imaging complications (e.g. signal drop-off and motion artifacts) between the two images. The Mattes mutual information metric [37] was used for this purpose. Mutual information is an information theoretic entity that allows one to measure  the amount of information gained in one random variable given that the amount of information in another random variable is known. In the case of registration, the image intensities of the fixed and moving images represent the random variables. The major advantage of using mutual information is that the actual form of dependency between the two random variables (or images in this case) does not need to be known or specified. Thus, it is robust even for the case where a complex relationship exists between the two  42  images. Mutual information is defined in terms of entropies. Let (A) = — p A (a)logp A (a)da ^(3.3) (B) = — f P^logPB (b)db  where pA and pB are, respectively, the marginal probably density function for random variables A and B, and H(A) and H(B) are their respective entropies. Additionally, let H (A, B) =  P AB (a,  b) logp AB (a,b)dadb^(3.4)  where pAB is the joint probability density function and H(A,B) the joint entropy. If random variables A and B are independent then  b) = P A(a)PB(b)^  (3.5)  H(A, B) = (A) + (B) .^  (3.6)  P AB (a,  and  On the other hand, if there are any dependencies then H(A, B) < H(A) + H(B) .^  (3.7)  The difference, I (A, B) = H (A) + H (B) — H (A, B) ,^(3.8)  43  is called mutual information. In the context of image registration, random variables A and B correspond to the fixed image intensity and moving image intensity, respectively. Typically, the marginal and joint probability densities of the image intensity are not available and hence must be estimated from the image data. In this case, Parzen windowing — otherwise known as kernel density estimators — can be used for estimation. In this method, the image intensity is randomly sampled from the image to form set S. A density function is then constructed by super-positioning a kernel function, K(s), centered on the elements of S along the intensity or grey-level axis as shown in Figure 3.5 [38]. Figure removed due to copyright permission. Parzen windowing L. Ng and L. Ibanez, "Mutual Information," in Insight into Images: Principles and Practice for Segmentation, Registration, and Image Analysis, T.S. Yoo, Ed., Wellesley, Massachusetts: A K Peters, 2004, pp. 288 Figure 3.5: In Parzen windowing, the density function of an image can be constructed by super-positioning a kernel function K(s), in this case a Gaussian function, on the elements of the set S, which is randomly sampled from the image [38].  The Mattes mutual information approach is an extension on the mutual information approach with one major difference: Only one spatial sample set is used for the entire registration process instead of using a new sample for each iteration. This results in a much smoother cost function [35].  44  3.3.4 Custom Versor Optimizer The optimizer used in the rigid registration is one that is customized for the registration process. Essentially it uses versor composition for computing the rotational parameters of the parameters, and vector addition for computing the translational components of the parameters [35].  3.4 Non-rigid Image Registration Non-rigid, or sometimes known was deformable, registrations are ones that involve transformations more complicated than those found in the rigid or affine cases; the points in deformable registrations move relatively freely in the image space, movement that can not be described by relatively simple relationships such as rotations, translations, shearing, or scaling. As was described above, deformable registrations are supersets of rigid and affine registrations. While points in the deformable registration case can move relatively freely, certain rules still apply, e.g. conservation of mass, conservation of energy, etc. Thus, deformable registrations aim at finding the corresponding transformation between points of the fixed image, F, and moving image, M, in a physically admissible manner. Note again that even though the transformation is a mapping from the fixed to the moving image, the moving image is the one that will be transformed to look like the fixed image. This idea is illustrated in Figure 3.6.  45  Compute deformation field, T, from  F to M.  Apply reverse of T on M so it esembles F.  Figure 3.6: This diagram illustrates the deformable registration process, where a deformation field that takes the fixed image F to the moving image M is computed, though the moving image is warped to resemble the fixed image at the output of the registration.  Even though a unique solution can be produced from a deformable registration process, it can not be guaranteed that this is the exact one posed in the problem. That is, the 'unique' solution from the algorithm does not necessarily equate the unique solution of what actually took place. In fact, the actual solution may not even have existed in the solution space of the registration in the first place. This means, then, there is not a way to truly verify whether the proposed solution is the exact solution, if it actually exists. Thus, it becomes crucial to verify the accuracy of a deformable registration algorithm, tailored towards the specific application. This is further discussed in Section 3.5. 3.4.1 B spline Deformable Transform -  For the deformation registration process in this work, a B-spline-based transform of order 3 [39] was used as the spatial transform component in the registration framework. Splines are piecewise polynomials where the pieces are smoothly connected together. The joining points of the polynomials are called knots. By considering only those splines with uniform knots and unit spacing, the splines can be uniquely defined by an expansion expression [40]:  46  s(x).  E c(k)f3" (x — k)  kE Z  (3.9)  where c(k) represents the parameters coefficients, or weights, that define the shape of a spline, and fin (x) represent a B-spline of degree n, to be described shortly. In essence, a spline is simply the summation of various degrees and weights of the B-spline shifted from the centre, and can thus be solely represented by the coefficient. B-splines (where the '13' may stand for basis or basic) are symmetrical, bellshaped functions constructed from the (n+1)-fold convolution of a rectangular pulse le:  fi0 x (  1^1 ——<x <— 2^2 1 Ix ' = 2-  )  0  (3.10)  otherwise  fin ( x) = fi0 * fi0 * ...* /30 ( x)^  (3.11)  where * denotes convolution, and the convolution is performed (n+1) times [40]. The Bsplines of degrees 0 to 3 are shown in Figure 3.7.  47  Figure removed due to copyright permission.  Figure removed due to copyright permission.  Figure removed due to copyright permission.  Figure removed due to copyright permission.  B-splines of degree 0  B-splines of degree 1  B-splines of degree 2  B-splines of degree 3  M. Unser, "Splines: A Perfect Fit for Signal and Image Processing," in  M. Unser, "Splines: A Perfect Fit for Signal and Image Processing," in  M. Unser, "Splines: A Perfect Fit for Signal and Image Processing," in  M. Unser, "Splines: A Perfect Fit for Signal and Image Processing," in  IEEE Signal Processing Magazine,  IEEE Signal Processing Magazine,  IEEE Signal Processing Magazine,  IEEE Signal Processing Magazine,  November 1999, pp. 22-38 (a)  November 1999, pp. November 1999, pp. 22-38 22-38 (b) (c)  November 1999, pp. 22-38 (d)  Figure 3.7: B-splines of degrees a) 0, b) 1, c) 2, and d) 3 [40].  In a spline-based deformation, the domain of the image volume can be denoted as = 1(x, y, z) I 0 < x < X, 0 < y < Y, 0 < z < Z1. Let F denote a Vij,k with  mxxmy xm z grid of knots  uniform spacing O. Then, the deformation can be written as the 3D tensor  product of 1D cubic B-splines 3 3 3  T(x,y, z) =ZEE  (u)B „,  1=0 m=0 n=0  (v)B „ ( w)th i+1,j+m,k-Fn^(3.12)  where i=Lx/m„]-1 , ig.x/m 1-1^k =Lx I mj-1 u= x I m,,-Lx I ^v= ylm y - Ly I m y j, y  v= z / m z -Lz / rn, J and where B1 represents  the lth basis function of the cubic B-splines  Bo(u)  = (1 - u) 3 /6  B i (u)  = (3u 3 - 6u 2 + 4) /6  ^  B2(u) = (-3u 3 + 3u 2 + 3u + 1) /6 B3(u) = u 3 /6  48  (3.13)  and T is the optimal transformation T:(x, y, z)^y', z') that maps any point in the image after deformation I(x, y, z, t) at time t into its corresponding point in the image before deformation I(x', y', z', to) [39]. In terms of using B-splines for registration, the input images into the registration framework are subdivided by a grid consisting of kxkxk intersections, i.e. knots, where k is the number of knots per dimension of the image and specified by the user. Via an iterative process, the algorithm uses these knots to transform the moving image incrementally in an attempt to make it resemble the fixed image. And when a certain degree of resemblance is achieved — the satisfactory level of resemblance is determined by the metric component — the iteration process stops and the registration is complete. In other words, the kxkxk knots compose the parameter space of the registration optimization problem. To transform the moving image, the knots undergo displacement independently, and the translational effects are propagated to the voxels around each knot (Figure 3.8). These displacements, in essence, compose the search space of the optimization problem.  ■•■ ■E= 251/11 NMI  ■IR MI■ MUM NMI 11MM 11111111MM. EWA LIMN  Nor NMI NNW EMI MD MIR OW) EMI  PAM! EaRIM NMI 1111111■ !FRE OEM  we  ONIIP MVP MO AIM  EMI bill ■EM WEE  Ilia III till RAI  EMI a m is ii i NMI Figure 3.8: Diagram illustrating the process of transforming an image using a B49  spline-based transformation in order to match a moving image (spinal cord in background) to a fixed image. Thick line represent grid lines, thin lines represent voxels, and red arrows represent the displacement that each knot (red dots) undergoes during an iteration of transformation. The B-spline approach was chosen because it inherently ensures that the deformation is smooth and more physiologically and physically sensible, i.e. more representative of how the spinal cord would deform. It also does not require fiducials or landmarks, which is advantageous in this application as the cord has a homogeneous texture with sparse features. For the rat data, the grid size used was 8 per dimension as it was determined through experimentation to produce highest accuracy. That is, the entire image was deformed using 8x8x8=512 knots. For the VHP data, the grid size used was once again determined through experimentation to optimally be 20 per dimension, or 20x20x20=8000 knots for the entire image. A larger number of knots were needed for the VHP data because it had to be ensured that the detailed FE analysis-based deformation could be fully captured by the registration. 3.4.2 B spline Interpolation -  Unlike the rigid registration case where transformations of rotational and translational natures are involved, the type of transformation each voxel could undergo in the deformable registration case is much more complex and non-linear. Thus, it was important that a robust interpolator was used in this application. The B-spline interpolation scheme [40] was thus chosen. In addition, it offers a good tradeoff between smoothness and computational cost/time. 50  In this scheme, the image to be interpolated — or more specifically, the intensity values at non-grid positions of the image — can be represented as a spline, defined by the B-spline basis functions and coefficients, described above M (x) = Ic(xofin(x—x j ).  (3.14)  Here, c(xi) represents the B-spline coefficients, and fin (x) represents a B-spline of degree n, as defined above in Section 3.4.1. The B-spline coefficients can, in turn, be computed  using a fast recursive filtering technique [41]4421 For B-splines of orders greater than one, both the interpolated image and its derivatives are spatially continuous. The higher the spline order the greater the number of pixels required to compute the interpolated value. The equation of a 1D cubic B-spline kernel is given by 2^ 1  /3(3)(x)  2  — lx1 (2 — lx1) 3^2 1 (2....ixi)3 6^I^I 0  Ox<1 15x<2  (3.15)  2x  3.4.3 Mean Squares Metric Recall that for the rigid registration described in Section 3.3 Rigid Image Registration the Mattes mutual information metric was used, due to its robustness and ability to compensate for the possibility of signal drop-off and motion artifacts between the nondeformed and deformed images. Since the images have already been aligned and accounted for in terms of global changes, a simpler metric — the mean squares metric — was used for the deformable registration case, where the changes are now only local 51  and small, and the distances between corresponding voxels are now closer, which is computationally easier in the registration sense. Its low computational costs have also contributed towards its usage. The mean squares metric [35], is a simple measure of similarity that calculates the mean squared difference over all the pixels in the images, defined by: S(p I F,M ,T)= M (T (x r , p))} 2 N,  ,  (  3.16)  where F is the fixed image intensity function, M is the moving image intensity function, T is the spatial transformation function, xi is the ith pixel of the fixed image region over  which to compute the metric, and N is the total number of pixels in the region. Typically, the value of M(T(xi,p)) needs to be interpolated from the discrete input image due to the transformed intensities falling on non-grid locations (refer to Section 1.6.4 Image Interpolation). Note that for voxels that are mapped to locations outside of the boundaries of the moving images, their contributions are discarded. The mean square metric has a relatively large capture radius with an optimal value of zero, with poor matches between two images resulting in larger values of the metric. 3.4.4 Limited Memory Broyden Fletcher Goldfarb Shannon Minimization with Simple Bounds Optimizer  The optimizer used in the deformable registration is one that is suited for the large, approximately 1536, number of parameters involved — the limited memory Broyden Fletcher Goldfarb Shannon minimization with simple bounds optimizer (LBFGSB) [43].  52  3.5 General Validation Scheme As stated before in Section 3.4 Non-rigid Image Registration, deformable registration, unlike rigid registration, is difficult to validate [44]. Qualitatively speaking, both rigid registration and deformable registration can be validated by finding the image difference between the fixed image and the resultant registered image, where the difference image with intensity values of ideally zero at every voxel corresponding to the case of perfect registration. However, while rigid registration solutions are unique, solutions to deformable registrations are often not unique [45]. This is because, as was stated in Section 3.4 Non-rigid Image Registration, the moving image can be deformed in many different numbers of ways to match the fixed image. Thus, a difference image that shows high correspondence between the original image and the registered result, i.e. low image intensity values in the difference image, does not necessarily mean the registration is accurate and that the produced deformation accurately describes the transformation that took place between objects in the two images. This results in the need to validate the registration results quantitatively. Yet, quantitative validation of deformable registration is much more difficult due to the large degrees of freedom seen in these problems. Whereas in the rigid case only a few parameters need to be calculated — translations, rotations, shearing, and/or scaling — for the entire images, the deformable registration requires the calculation of similar parameters but on the level of voxels in order for the validation to be accurate. It is this difficulty in obtaining the known deformation field, also known as the Gold Standard deformation field, which makes deformable registration validations difficult. One commonly accepted method of quantitatively validating a deformable 53  registration is to apply a deformation to which the deformation field is known a priori, i.e. to establish a ground truth or gold standard [44]. The accuracy can then be established by comparing the computed, registration deformation field against this known deformation field on a voxel-to-voxel basis. However, to assess the accuracy of every single voxel of the registration results is time-consuming and impractical. Thus, the use of a two-step approach to verifying the deformable registration is proposed. First the qualitative accuracy of the registration result is considered by examining the visualization of the computed deformation field. If the deformation field vectors exhibit good correspondence between corresponding voxels in the fixed and moving image, it is an indication that the registration has performed satisfactorily and realistically. Quantitative validation is then performed by finding the difference (i.e. Euclidean distance) between the known deformation field and the computed deformation field for every voxel location where both the known and computed deformation fields exist. As it is impractical to examine the Euclidean distance at every voxel location where it exists due to the shear volume of vectors that need to be studied, the mean and median values across these locations are calculated to produce a measure of registration accuracy. These values are also normalized with respect to the spinal cord width, represented by the mean dorsalventral width. This measure is used because the dorsal-ventral cord direction is where deformation is most likely to take place in an injury event. This two-step approach ensures that the general 'goodness' of registration is considered via the qualitatively analysis while quantitative measures are still provided.  54  Eudidean  Known Deformation Field  Distance .--<Ern3D 1  Figure 3.9: Flow diagram illustrating the general validation process of the proposed deformation algorithm. Note that the 'Analysis' component represents the process illustrated in Figure 3.1.  3.6 Warping and Validation on Synthetic MR Spinal Cord Model Image Having created a synthetic model SC image intended for use in validation, as was explained in Section 2.4, a synthetically-generated, B-spline-based warping technique is then applied to this model. The B-spline technique, the concept of which was discussed in Section 1.6.3, utilizes knots. Furthermore, the warping algorithm takes as input a fixed image and a list of parameters to produce a deformed image and the known deformation field. These input parameters describe the amount of motion or change at each knot, and are organized in the format of: AY1^Az1^Ax2^AY2^Az2^AX kxkxk^kxkxk AZ kxkxk where k is the number of knots per dimension. In this case k = 8, requiring the specification of 8x8x8 = 512 parameters per dimension, or 512x3 = 1536 parameters for the entire image. The advantage of using a B-spline-based technique, again, is that it ensures smoothness in the image to be deformed, as is the case in an actual spinal 55  cord deformation case. The original fixed image was then registered directly to images with increasing amounts of synthetic warping.  Figure 3.10: Flow diagram illustrating the validation process using the synthetic MR SC model image and synthetic warping schemes. Note that the 'Analysis' component represents the process illustrated in Figure 3.1. Two types of warping schemes were applied to the model: 'bending' and `contortion'. Figure 3.11 and Figure 3.12 below illustrate the process of generating these two types of deformations. Cross-sectional View  ^  Top View Lateral (lift)  Dorsal  Lateral^ (right)^ Direction of Applied Warping . Force  Lateral (left)  Caudal  •^  Spinal Cord  Cranial  • Lateral (right)  Ventral  twa H Direction of Applied Warping Force  Figure 3.11: Diagram illustrating the process of synthetically warping (`bending') a spinal cord, in order to be used for validation.  56  Cross-sectional View^ Dorsal •  Top View Direction of Applied Warping Force  (lift)^  If  Lateral Lateral (right)^ I" (left) Spinal  Direction of Applied Lateral Warping Force 44P.'  Caudal  •^  Spinal Cord  Cord  Direction of Applied I::^Ventral Warping Force  Cranial  • Lateral (right) Direction of Applied Warping Force  Figure 3.12: Diagram illustrating the process of synthetically warping (`contortion') a spinal cord, in order to be used for validation.  3.7 Warping and Validation on In Vivo MR Rat Spinal Cord Image Warping schemes applied to the synthetic MR SC model image were also applied to the non-deformed in vivo MR rat SC, described in Section 2.3.1, for validation. Note that for the in vivo rat cord, the mean dorsal-ventral spinal cord width was 1.61 mm (range 1.18— 2.79 mm; standard deviation 0.29 mm), across 64 slices of the data image (cord length of 25.60 mm).  Figure 3.13: Flow diagram illustrating the validation process using the in vivo MR rat spinal cord image and synthetic warping schemes. Note that the 'Analysis' component represents the process illustrated in Figure 3.1. 57  3.8 Warping and Validation on VHP Spinal Cord Model Image From the FE model described in Section 2.5, node points representing the spinal cord only were extracted, and the corresponding node points describing the three types of loading conditions of the cord, illustrated in Figure 2.14, Figure 2.15, and Figure 2.16: distraction, contusion, and dislocation. These nodes points were used as the basis for warping of the 3D VHP SC model image. The FE analysis-based deformations were applied using a thin-plate splines warping algorithm [46]. As was described previously in Section 3.4.1, splines are piecewise polynomials with pieces that are smoothly connected together. In the case of thin-plate splines, for a given set of data points, the interpolation between these points is a weighted combination of the splines, centered on each of these points. The constraint is that the "bending energy" must be minimized. Bending energy is defined here as the integral over 91 3 of the squares of the second derivatives,  I[ f (x, y, z)] = JJJ , (fx2, + f y + fz2z + 2f 2y + 2fz2, + 2fy2z  )dxdydz  (3.17)  The name "thin-plate" is derived from the physical analogy involving the bending of a thing sheet of metal [47]. In terms of the usage, it requires as inputs the fixed image and parameters in the format of Xfl Yfl Zfl Xml Yml Zml  x f2 yf2 Zf2 Xm2 ym2 Zm2 • • • Xfn yfn Zfn x, Ymn Zmn  where f indicates voxel location in the fixed image and m indicates the corresponding voxel location in the moving image (i.e. location specified by  (xfb yfb zfi) is  to be moved  to (xmi, Ymr, zm r), etc.), and n is the number of nodes desired by user for deformation. 58  Note, however, that n cannot exceed —1200 points due to memory constraints for the MATLAB-based algorithm. Furthermore, the nodes obtained from the FE analysis could not be readily used since the measurements/distances were relative to an arbitrary origin in ANSYS and were measured in physical, rather than image/voxel, space. Thus, a program was written in MATLAB to convert the physical locations to voxel locations. The following steps describe the conversion process: 1) Perform arbitrary translation with z mirroring. 2) Register segmented non-deformed SC (fixed) image to node image (moving) using rigid versor to determine translation. 3) Reorient the nodes (in the physical domain) into that of image space by rounding to voxel location closest to physical location, relative to origin in image space. 4) This reorientation step is then viewed as image to ensure the reorientation was done correctly; this adjusts for the z direction padding of zeros discussed above. 5) If process is correct, the locations are converted to landmarks and output to file. 6) This can yet again be viewed to ensure correctness. 7) If correct, use nodes for warping. The advantage of using the thin-plate spline algorithm is that it ensures smoothness in the deformation and is physically realistic, much like the B-spline deformation algorithm, and it accepts specific points/voxels for deformation and deforms the image in a manner that closely resembles the specified points, with little error. But unlike the B-spline case, it ensures that the deformation follows the original FE analysis deformation as closely as possible. Furthermore, the underlying concept of minimum energy is shared between both the FE analysis and the thin-plate splines warping scheme, 59  as such these techniques are complementary on a conceptual level. It should be noted that in this case, the known deformation field used is the field given from the original FE analysis (i.e. the point to point correspondence). This, then, would mean that the accuracy of validation is heavily dependent on how closely the warping can deform according to the given points. Low discrepancy between original nodes and deformed image yields low validation error; high discrepancy between original nodes and deformed image yields high validation error. And since the warping algorithm deforms well, but does not deform perfectly according to the points given, some error is introduced to the validation that does not result from the registration algorithm.  Computed Deformation Field  i^\ (.^ Euclidean Distance —00-CErrorD  Known Deformation Field  \^.1  +  Figure 3.14: Flow diagram illustrating the validation process using the VHP spinal cord model image and FEA-based warping schemes. Note that the 'Analysis' component represents the process illustrated in Figure 3.1.  60  4 Results and Discussion In this chapter the results obtained from the validation processes described in the previous chapter will be examined, both qualitatively and quantitatively. Following this the analyses performed on the rat images acquired using MRI, described in Chapter 2, as well as the limitations present in this work, will be considered.  4.1 Results for Validation on Synthetic MR Spinal Cord Model Image Figure 4.1 below shows the qualitative validation results corresponding to the synthetic B-spline warping of the synthetic MR spinal cord model image described in Section 3.6. Note how the overlaid deformation field arrows (shown in blue) demonstrate good correspondence between locations on the non-deformed cord (shown in yellow) and deformed cord (shown in green).  61  (a)  (b) Figure 4.1: Visualization of the synthetic spinal cord model before (yellow) and after (green) (a) Warp 1 (`bending'), and (b) Warp 2 ('contortion'). Corresponding deformation fields at 3 locations are overlaid (blue arrows). Quantitative results of the same analysis were calculated by finding the Euclidean 62  distance between the computed and known deformation fields and normalized with respect to the dorsal-ventral width of the cord, E% =  1(x — x02^Ylc)2  (Zc Zic) 2  w  where xc and xk represent the x component of the computed and known deformation fields, respectively, w represents the dorsal-ventral width of the cord, and E% represents the error in percentage. These are shown in Figure 4.2. It can be seen that as the amount of deformation increases, so does the amount of error. This is due to increasing difficulty for the algorithm to determine the direction and amount of deformation for each voxel. However, it should be noted that the accuracy of registration would be improved with higher spatial resolution, lower noise, and increased SNR in the image, all of which can be improved through further MRI protocol optmization, as the algorithm would be able to identify corresponding voxels with greater ease.  63  Synthetic Warping of Synthetic Model Data 14 12 10 6 2 Lir 8 c tu  60  4 2-  —40—Warp 1 ('bending') ■• Warp 2 ('contortion')  0 0  20^40^60 43/0  80  100  Max. Deflection (Deformation)  Figure 4.2: Accuracy of measured deformation, represented as median error between the known and calculated deformation fields. Various levels of deformation of the synthetic spinal cord model with Warp 1 (`bending') and Warp 2 (`contortion') are shown. Both the error and deformation extent are measured relative to the cord width.  4.2 Results for Validation on In Vivo MR Rat Spinal Cord Image Figure 4.3 below shows the qualitative validation results corresponding to the synthetic B-spline warping of the in vivo MR rat spinal cord image described in Section 3.7. Note, again, how the overlaid deformation field arrows (shown in red) demonstrate good correspondence between locations on the non-deformed cord (shown in green) and deformed cord (shown in blue).  64  (a)  (b) Figure 4.3: Visualization of MR image of the in vivo rat spinal cord before (green) and after (blue) (a) Warp 1 (`bending'), and (b) Warp 2 (`contortion'). Corresponding deformation fields at 3 locations are overlaid (red arrows). Vertebrae are only shown for the non-deformed position for clarity.  Quantitative results of the same analysis are shown in Figure 4.4. The same trend of increasing error with increasing deformation exhibited in Figure 4.2 is also shown here. It should be noted that at 100% maximum deflection, beyond which deformations are rarely observed clinically, the percentage median error is approximately 6%. Note 65  that the error here is lower compared to those seen in the synthetic model image case because the in vivo image incorporates a lot of shape differences between each slice that is not seen in the synthetic image. This is a demonstration of the effectiveness of the proposed algorithm for real spinal cord images under more realistic loading conditions. Synthetic Warping with In Vivo Rat Data  25 20  a O  5 —0—Warp 1 (bending) '" IN*  0 0  ^  50^100^150  Warp 2 ('contortion)  ^  200^250  % Max. Deflection (Deformation)  Figure 4.4: Accuracy of measured deformation, represented as median error between the known and calculated deformation fields. Various levels of deformation of the in vivo spinal cord data with Warp 1 ('bending') and Warp 2 (`contortion') are shown. Both the error and deformation extent are measured relative to the cord width.  Note that the discrepancy of results between the synthetic data and the in vivo data under the same warping conditions derives from the fact that the synthetic data is quite homogeneous, where the slices are organized as (type 1) (type 2) (type 3) (type 3) (type 2) (type 1) ... (type 3) (type 2) (type 1) and that aside from the noise, all the voxels within the cord are kept same across all 66  slices, with the same intensity values. This causes ambiguities when registering, resulting in higher errors. The in vivo data, on the other hand, has different spinal cord crosssectional shapes throughout the data. So while inner features are not quite distinguishable (e.g. `butterfly'), the boundary of the cord facilitates registration, resulting in lower error as that seen in the synthetic data.  4.3 Results for Validation on VHP Spinal Cord Model Image Qualitatively results corresponding to the FE analysis-based warping of the VHP SC model image, described in Section 3.8, are shown below; Figure 4.5 corresponds to the case of distraction, Figure 4.6 corresponds to the case of contusion, and Figure 4.7 corresponds to the case of dislocation. Note how the overlaid deformation field arrows (shown in red) demonstrate good correspondence between locations on the non-deformed cord (shown in yellow) and deformed cord (shown in green).  (a) 67  (b) Figure 4.5: a) Visualization of the FEA-based distraction deformation node points used to apply deformation to the VHP spinal cord model image for validation, with the spinal cord before (yellow) and after (green) deformation. b) VHP spinal cord model image before (yellow) and after (green) applying the distraction deformation. Corresponding deformation fields at 3 locations are overlaid (red arrows).  (a) 68  (b)  Figure 4.6: a) Visualization of the FEA-based contusion deformation node points used to apply deformation to the VHP spinal cord model image for validation, with the spinal cord before (yellow) and after (green) deformation. b) VHP spinal cord model image before (yellow) and after (green) applying the contusion deformation. Corresponding deformation fields at 3 locations are overlaid (red arrows).  (a)  69  (b) Figure 4.7: a) Visualization of the FEA-based dislocation deformation node points used to apply deformation to the VHP spinal cord model image for validation, with the spinal cord before (yellow) and after (green) deformation. b) VHP spinal cord model image before (yellow) and after (green) applying the dislocation deformation. Corresponding deformation fields at 3 locations are overlaid (red arrows).  Table 1 below shows the quantitative results of validation corresponding to the FE analysis-based warping of the VHP spinal cord model image. The reason for the larger percentage of error values, compared to those expected based on Figure 4.2 and Figure 4.4, is due to the fact that thin-plate splines warping does not generate deformed cord images that perfectly match the original FE analysis-based node points, as was discussed in Section 2.5. And since the registration is based on results of the warping, a larger discrepancy between the deformation field derived from the proposed algorithm and the known deformation field based on the FE analysis-based node points occurs. An improved warping algorithm would thus provide better assessment of the accuracy of this method. 70  Table 1: Accuracy of registration for the case of finite element analysis-based spinal cord data. Percentages are measured relative to the cord width, where 'Distraction' corresponds to Figure 4.5 above, 'Contusion' corresponds to Figure 4.6 above, and `Dislocation' corresponds to Figure 4.7 above.  Max Deflection % Max Deflection Median Error % Median Error  Distraction 1.41 nun 15.1% 0.44 mm  Contusion 2.20 min 23.5% 0.37 mm  Dislocation 8.26 mm 88.3% 1.18 mm  4.7%  3.9%  12.6%  4.4 Analysis of Rat Spinal Cord Deformation Having established the accuracy of the deformation analysis technique, analysis of the deformed spinal cord data described in Section 2.2 was done. While the deformation field to which to compare the computed deformation field against in the MR rat images is not known, the robustness of the method can be gauged by looking at 1) the amount of overlap between the original fixed image and the image resulting from the registration, with the ideal case that the latter resembles and completely overlaps the former; and 2) how plausible and realistic the computed deformation field is in relation to the nondeformed and deformed images. For the in vivo rat spinal cord, as described in Section 2.3.1 where the cord was deformed at the lumbo-sacral region of the vertebra, the analysis results are shown below.  71  (b) Figure 4.8: Visualization of MR image of the in vivo rat spine: a) Visualization of the original, non-deformed spinal cord (yellow) and the cord produced from the registration (green). b) The spinal cord including the vertebrae before (green) and after (blue) a real deformation. Corresponding deformation fields at 3 locations are overlaid (red arrows). Vertebrae are only shown for the non-deformed position for clarity.  Similarly, for the ex vivo rat spinal cord, as described in Section 2.3.2, scans were  72  performed before and after a deformation in the T9-T12 region of the spine. Analysis results are shown below.  (a)  (b) Figure 4.9: Visualization of MR image of the ex vivo rat spinal cord: a) Visualization of the original, non-deformed spinal cord (yellow) and the cord produced from the registration (green). b) The spinal cord before (yellow) and after (green) a real deformation. Corresponding deformation fields at 3 locations are overlaid (red arrows). It can be seen from the figures above that the deformation in the ex vivo case is 73  indeed much larger than that of the in vivo case. As such, one expects that the ex vivo registration would suffer. This is confirmed by the fact that Figure 4.9a), the in vivo case, demonstrates much more overlap between the fixed and moving images than Figure 4.8a), the ex vivo case.  4.5 Limitations of the MRI Approach While the results presented above demonstrate the accuracy of this approach, many technical challenges still exist, especially with regards to imaging. As described above in Section 2.3, one key obstacle is due to the fact that current MRI technology does not allow the imaging of traumatic spinal cord injury. Whereas an injury event takes place in the order milliseconds, minutes to tens of minutes are required for imaging. Thus, the analysis of traumatic spinal cord injury events is not presently possible. Another major issue encountered during this study was the inability to deform the cord as desired. The main reasons for this include 1) confined spacing within the scanner, 2) spinal cord and epicenter of deformation are constrained to be positioned near a surface coil in order to achieve a certain level of image quality, and 3) the kinds of materials allowed in the scanner are restricted (i.e. non-ferromagnetic materials), limiting the possibilities for more effective cord deformation devices. Lastly, while higher image quality is constantly desired in any imaging-related study, the z-axis resolution remains an important issue that warrants attention. The relatively large distances, as compared to the in-plane resolution, between slices of acquired 3D MR images pose some significant problems for any image analysis-related  74  project. This is especially true for the proposed work, involving the study of spinal cords, where the feature to be analyzed, i.e. spinal cord, is shaped longitudinally, and that the smoothness of analysis is of great importance. While the z-axis resolution can be improved, it would be at the cost of the in-plane resolution and other image qualities such as SNR.  75  5 Conclusions and Future Work In this chapter the significance of this work and its contribution to the study of spinal cord deformations will be discussed. This thesis will then be concluded with a discussion on aspects of this study that can be furthered in future studies.  5.1 Thesis Contributions Traditionally, spinal cord-related injury work has been carried out through biomechanical testing where post-injury cords are studied through exposure [1],[2],[3],[4], which alters cord environments [4]. In this thesis, a novel in vivo MRI-based method for non-invasive analysis of spinal cord deformations is proposed. Unlike current techniques, this approach represents a tool for the semi-automated quantitative analysis of cord deformations, specifically myelopathy-related sustained spinal cord compression, in the original biological and anatomical environments. At present, this method is not readily applicable to cases of traumatic spinal cord injury, where the injury event has duration in the order of milliseconds, because present MRI technology requires acquisition time in the order of minutes to tens of minutes. Difficulties were also encountered with regards to imaging rat spinal cord deformations, and indeed rat imaging in general. Yet, this method is well-suited for cases of myelopathy and sustained compression of the spinal cord as might occur in the hours after a traumatic spinal cord injury prior to decompression surgery. The major achievements of this project are as follows: • For the synthetic data set, error levels achieved in this method were consistently below 6% during validation, even for large degrees of deformation up to half of the 76  dorsal-ventral width of the cord (50% deflection). • For the in vivo rat data set, error levels achieved in this method were consistently below 6% during validation, even for large degrees of deformation up to the entire dorsal-ventral width of the cord (100% deflection). • For a more advanced validation approach utilizing finite element analysis, the error levels achieved were as low as 3.9%. • When this method was used to analyze data obtained through imaging of rat spinal cords undergoing actual deformations, the observed deformation fields behaved smoothly and realistically, further confirming the validity of this approach. • This technique is in vivo, non-invasive, and does not require exposure of the cord, ensuring that the physiological and biomechanical properties of the spinal cord and its environment are retained. • Because the analysis is performed on progression of deformation leading to injury in the spinal cord, this approach allows for potential derivation of the causes, mechanisms, and tolerance parameters of spinal cord injuries, as well as the measures used in neuroprotection and regeneration of spinal cord tissue. 5.1.1 List of Publications •  •  Kevin Ming, Rafeef Abugharbieh, Claire F. Jones, Andrew Yung, Piotr Kozlowski,  Wolfram Tetzlaff, and Peter A. Cripton, "Towards In Vivo MRI Based Spinal Cord Deformation Analysis," Spine. (to be submitted)  Kevin Ming, Rafeef Abugharbieh, Claire F. Jones, Andrew Yung, Piotr Kozlowski, Wolfram Tetzlaff, and Peter A.^Cripton, "Computational Analysis of 77  Myelopathy-related Cervical Spinal Cord Compression and Deformation using MRI," Cervical Spine Research Society (CSRS) 36 th Annual Meeting, December 4-6, 2008, Austin, TX, USA. (to be submitted) •  Ming, K., Abugharbieh, R., Jones, C.F., Greaves, C.Y., Yung, A., Tetzlaff, W.,  Kozlowski, P., Cripton, P.A., "Finite Element Model Validation of Spinal Cord Deformation Analysis Based on MR Imaging Data," 26 th Annual National Neurotrauma Symposium, July 27-30, 2008, Orlando, FL, USA. (to be submitted) •  K. Ming, R. Abugharbieh, C.F. Jones, A. Yung, P. Kozlowski, W. Tetzlaff, and P.A.  Cripton, "Computational MR Image Analysis for Spinal Cord Injury Studies,"  International Society for Magnetic Resonance in Medicine (ISMRM) 16 th  Scientific Meeting & Exhibition, May 3-9, 2008, Toronto, ON, Canada.  •  •  Ming, K., Abugharbieh, R., Jones, C.F., Greaves, C.Y., Yung, A., Tetzlaff, W.,  Kozlowski, P., Cripton, P.A., "Finite Element Model Validation of Spinal Cord Deformation Analysis Based on MR Imaging Data," 2008 Northwest Biomechanics Symposium (NWBS), May 9-10, 2007, Boise, ID, USA. Ming, K., Abugharbieh, R., Jones, C.F., Yung, A., Kozlowski, P., Tetzlaff, W.,  Cripton, P.A., "In Vivo MR Image Based Deformation Analysis for Spinal Cord Injury Studies," 2007 Northwest Biomechanics Symposium (NWBS), May 18-19, 2007, Eugene, OR, USA.  5.2 Future Work Spinal cord segmentation has so far been performed manually. In addition to being tedious and time-consuming, these segmentations may not be reproducible or accurate around the boundary of the cord, both of which lead to increased registration error. Applying a semi-automated or automated segmentation technique that can be used to extract the spinal cord from the CSF and surrounding features can reduce pre-processing time and improve accuracy of the analysis. Data acquisition was one of the more difficult aspects in the project, and while most of the issues have been resolved and images were acquired, improvements can still be made: 1) overhead time involving the animal preparation can be reduced by 78  streamlining the process; 2) hardware (e.g. RF coil, deformation apparatus) and procedures used in deforming animal spinal cord for imaging can be enhanced to allow variability and variety in the deformation; and 3) MRI protocol and imaging coil can be further optimized to achieve higher spatial resolutions, reduced noise, and increased SNR, all of which improve accuracy of the proposed algorithm. 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Weidner, "In Vivo High-Resolution MR Imaging of Neuropathologic Changes in the Injured Rat Spinal Cord," American Society of Neuroradiology, vol. 27, pp. 598-604, March 2006. [18] M. Bilgen, B. Al-Hafez, T. Alrefae, Y.Y He, I.V. Smirnova, M.M. Aldur, and B.W. Festoff, "Longitudinal Magnetic Resonance Imaging of SCI in Mouse: Changes in Signal Patterns Associated with the Inflammatory Response," Magnetic Resonance Imaging, vol. 25, no. 5, pp. 657-664, June 2007. [19] Y. Yukawa, F. Kato, H. Yoshihara, M. Yanase, and K. Ito, "MR T2 Image Classification in Cervical Compression Myelopathy: Predictor of Surgical Outcomes," Spine, vol. 32, no. 15, pp. 1675-1678, July 2007. [20] I. Elshafiey, M. Bilgen, R. He, and P.A. Narayana, "In Vivo Diffusion Tensor Imaging of Rat Spinal Cord at 7 T," Magnetic Resonance Imaging, vol. 20, no. 3, 82  pp. 243-247, April 2002. [21] D. Facon, A. Ozanne, P. Fillard, J.F. Lepeintre, C. Tournoux-Facon' and D. Ducreux, "MR Diffusion Tensor Imaging and Fiber Tracking in Spinal Cord Compression," American Society of Neuroradiology, vol. 26, pp. 1587-1594, June 2005. [22] D.N. Loy, J.H. Kim, M. X, R.E. Schimdt, K. Trinkaus, and S.K. Song, "Diffusion Tensor Imaging Predicts Hyperacute SCI Severity," Journal of Neurotrauma, vol. 24, no. 6, pp. 979-990, 2007. [23] L. Ng and L. Ibanez, "Medical Image Registration: Concepts and Implementation," in Insight into Images: Principles and Practice for Segmentation, Registration, and Image Analysis, T.S. Yoo, Ed., Wellesley, Massachusetts: A K Peters, 2004, pp. 239. [24] B.D. Schmit and M.K. Cole, "Quantification of Morphological Changes in the Spinal Cord in Chronic Human Spinal Cord Injury using Magnetic Resonance Imaging," Proceedings of the 26 th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, September 1-5, 2004, San Francisco, CA, USA. [25] O. Coulon, S.J. Hickman, G.J. Parker, G.J. Baker, D.H. 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Unser, "Splines: A Perfect Fit for Signal and Image Processing," in IEEE Signal Processing Magazine, November 1999, pp. 22-38.  [41] M. Unser, A. Aldroubi, and M. Eden, "B-Spline Signal Processing: Part I— Theory," IEEE Transactions on Signal Processing, vol. 41, no. 2, pp. 821-832, February 1993. [42] M. Unser, A. Aldroubi, and M. Eden, "B-Spline Signal Processing: Part II— Efficient Design and Applications," IEEE Transactions on Signal Processing, vol. 41, no. 2, pp. 834-848, February 1993. [43] R.H. Byrd, P. Lu, and J. Nocedal, "A Limited Memory Algorithm for Bound Constrained Optimization," SIAM Journal on Scientific and Statistical Computing, vol. 16, no. 5, pp. 1190-1208, May 1994. [44] J.A. Schnabel, C. Tanner, A.D. Castellano-Smith, A. Degenhard, M.O. Leach, D.R. Hose, and D.L.G. Hill, "Validation of Nonrigid Image Registration Using FiniteElement Methods: Application to Breast MR Images," IEEE Transactions on Medical Imaging, vol. 22, no. 2, pp. 238-247, February 2003. [45] J.M. Fitzpatrick, "Detecting Failure, Assessing Success," Medical Image Registration, J. V. Hajnal, D. L. G. Hill, and D. J. E. Hawkes, Eds: CRC Press,  2001, ch. 1-6, pp. 117-139. [46] M.H. Davis, A. Khotanzad, D.P. Flamig, and S.E. Harms, "A Physics-Based 86  Coordinate Transformation for 3-D Image Matching," IEEE Transactions on Medical Imaging, vol. 16, no. 3, 317-328, June 1997. [47] S. Belongie, "Thin Plate Spline," from Math World--A Wolfram Web Resource, created by Eric W. Weisstein, http://mathworld.wolfram.com/ThinPlateSpline.html.  87  Appendix A — Statement of Co-Authorship All chapters in this thesis were written by Kevin Ming and edited by his supervisors, Dr. Rafeef Abugharbieh and Dr. Peter A. Cripton. The experimental data were collected by Dr. Piotr Kozlowski, Andrew Yung, Claire F. Jones and Kevin Ming. The custome-made inflation device was designed and constructed by Claire F. Jones, while the hyperflexion device was designed and constructed by Kevin Ming. The finite element analysis was developed by Carolyn Y. Greaves. All data analysis methods were designed and developed by Kevin Ming under the supervision of Dr. Rafeef Abugharbieh and Dr. Peter A. Cripton.  88  Appendix B — Imaging Ethics Certificate  89  Page 1 of 1  THE UNIVERSITY OF BRITISH COLUMBIA  ANIMAL CARE CERTIFICATE Application Number: A05-0350 Investigator or Course Director: Piotr Kozlowski Department: Radiology Animals:  Rats Wistar 20 Mice DDS 40  Start Date:  June 1, 2005  Approval Date:  June 19, 2007  Funding Sources: Funding Agency:^UBC Internal Grant Funding Title:^internal  Unfunded title:^N/A The Animal Care Committee has examined and approved the use of animals for the above experimental project. This certificate is valid for one year from the above start or approval date (whichever is later) provided there is no change in the experimental procedures. Annual review is required by the CCAC and some granting agencies. A copy of this certificate must be displayed in your animal facility. Office of Research Services and Administration 102, 6190 Agronomy Road, Vancouver, BC V6T 1Z3 Phone: 604-827-5111 Fax: 604-822-5093  90 https://rise.ubc.ca/rise/Doc/O/FF73U8DKUGL4T2B267COTKD788/fromString.html ^  5/9/2008  

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