Applied Science, Faculty of
Biomedical Engineering, School of
DSpace
UBCV
Sharkey, Liam Rhett
Master of Applied Science - MASc
University of British Columbia
There are many cellular and therapeutic treatments for traumatic spinal cord injury (tSCI) that have shown promise in animal models, however, these treatments have been unsuccessful when applied to humans. A possible reason for this discrepancy is that animal model SCIs are well-controlled, whereas human SCIs are heterogeneous in terms of population, severity and mechanism. It is known that the mechanical injury parameters play an important role in dictating the pathophysiology of SCI. However, to fully describe the mechanics of tSCI, it is necessary to understand the mechanical properties of the spinal cord. In order to investigate the mechanical properties of the rat spinal cord, I created a finite element model to simulate experimental contusion SCIs on rats based on research by Bhatnagar et al., and evaluate morphological similarity between computationally predicted and experimentally deformed spinal cords. The model was used to determine the relative stiffnesses of the grey and white matter by iteratively assigning mechanical properties to each tissue, deforming the spinal cord to match the shape of the experimentally deformed cord, and observing the morphological similarity of the predicted and experimentally deformed grey matter shapes. Using a linear elastic, homogeneous, isotropic material model for both tissues examined, I found that for six of the seven spinal cords examined, the best model agreement occurred when the white matter was modeled as 2-3 times stiffer than the grey matter, while each tissue was held at a Poisson's ratio of 0.45. Furthermore, I found that for contusion injuries inflicted upon the mediolateral center of the spinal cord, the model predicted the deformation well, while for off-center contusions, the model was unable to capture the shape of the grey matter on the side contralateral to the contusion location. I demonstrated that the spinal cord white matter appears to be stiffer than the grey matter and that current strategies of modeling spinal cord injury do not adequately capture the complexity of the deformation of the cord as a whole. My research gives further reason to investigate the mechanical properties of the spinal cord for the purpose of computationally modeling spinal cord injury.
An Inverse Finite Element Approachto Modeling the Rat Cervical SpinalCordFor Use in Determining the Mechanical Properties ofthe Grey and White MatterbyLiam Rhett SharkeyB.Sc.Eng, The University of Manitoba, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Biomedical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)February 2018c© Liam Rhett Sharkey 2018AbstractThere are many cellular and therapeutic treatments for traumatic spinal cord injury (tSCI)that have shown promise in animal models, however, these treatments have been unsuccessfulwhen applied to humans [93]. A possible reason for this discrepancy is that animal model SCIsare well-controlled, whereas human SCIs are heterogeneous in terms of population, severity andmechanism [90]. It is known that the mechanical injury parameters play an important role indictating the pathophysiology of SCI [19, 54, 85]. However, to fully describe the mechanics oftSCI, it is necessary to understand the mechanical properties of the spinal cord.In order to investigate the mechanical properties of the rat spinal cord, I created a finiteelement model to simulate experimental contusion SCIs on rats based on research by Bhatna-gar et al. [3], and evaluate morphological similarity between computationally predicted andexperimentally deformed spinal cords. The model was used to determine the relative stiffnessesof the grey and white matter by iteratively assigning mechanical properties to each tissue, de-forming the spinal cord to match the shape of the experimentally deformed cord, and observingthe morphological similarity of the predicted and experimentally deformed grey matter shapes.Using a linear elastic, homogeneous, isotropic material model for both tissues examined, I foundthat for six of the seven spinal cords examined, the best model agreement occurred when thewhite matter was modeled as 2-3× stiffer than the grey matter, while each tissue was held ata Poisson’s ratio of ν = 0.45. Furthermore, I found that for contusion injuries inflicted uponthe mediolateral center of the spinal cord, the model predicted the deformation well, while foroff-center contusions, the model was unable to capture the shape of the grey matter on the sidecontralateral to the contusion location.I demonstrated that the spinal cord white matter appears to be stiffer than the grey mat-ter and that current strategies of modeling spinal cord injury do not adequately capture thecomplexity of the deformation of the cord as a whole.My research gives further reason to investigate the mechanical properties of the spinal cordfor the purpose of computationally modeling spinal cord injury.iiLay SummaryThere are many treatments for traumatic spinal cord injury (tSCI) that have shown promisein animal models, however, these treatments have been unsuccessful when applied to humans[93]. The failure of these treatments in human trials could be due to the fact that human SCIis highly heterogeneous in terms of the population sustaining the injury and the type of impactto the spinal cord during injury, whereas animal models are homogeneous with respect to theseparameters. In order to understand how the cord deforms during injury, I have created a modelto investigate the stiffness of the two primary materials in the spinal cord; the grey and whitematter.I found that the white matter appears to be stiffer than the grey matter, and that thecurrent strategies of predicting how the cord deforms during injury do not adequately capturethe true cord deformation.iiiPrefaceDr. Antonio Sa´nchez was extremely helpful in guiding me through the process of learning howto use ArtiSynth. Dr. John Lloyd, Dr. Sid Fels and Dr. Tim Bhatnager aided in the ideationof this research project. I performed the majority of the model development, refinement andanalysis under the guidance of my supervisor, Dr. Thomas Oxland.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 FE as an Animal Model Research Tool . . . . . . . . . . . . . . . . . . . 11.1.2 Bench to Bedside . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Project Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Overview of Spinal Cord Injury . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.1 Anatomy of the Human Spine and Spinal Cord . . . . . . . . . . . . . . . 41.3.2 Pathophysiology of Spinal Cord Injury . . . . . . . . . . . . . . . . . . . 131.4 Experimental Modeling of tSCI . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4.1 SCI Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4.2 Experimental Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.5 Determination of Spinal Cord and Brain Tissue Material Properties . . . . . . . 191.6 Computational Modeling of SCI . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.6.1 Strain Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.6.2 Finite Element Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.6.3 Material Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.6.4 Computational Model Findings . . . . . . . . . . . . . . . . . . . . . . . 27v1.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.7.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.7.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.1 Previously Conducted Experimental Data Collection . . . . . . . . . . . . . . . . 302.1.1 Surgical Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.1.2 Imaging Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.1.3 Histological Preservation Methods . . . . . . . . . . . . . . . . . . . . . . 312.1.4 Magnetic Resonance Imaging Post-Processing . . . . . . . . . . . . . . . 322.1.5 Histological Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.1 Image Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.2 Construction of Finite Element Mesh . . . . . . . . . . . . . . . . . . . . 332.2.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2.4 Material Property Investigation . . . . . . . . . . . . . . . . . . . . . . . 362.3 Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.3.1 Simple Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.3.2 True Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.4 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.4.1 Qualitative Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.4.2 Quantitative Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.4.3 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.5 Comparison to Histology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.1 Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.1.1 Mesh Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.1.2 Force Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.1.3 Evaluation of Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1.4 Evaluation of the Modified Hausdorff Distance Metric . . . . . . . . . . . 543.2 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2.1 Qualitative Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2.2 Quantitative Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2.3 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.3 Comparison to Histology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.1 Injury Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.1.1 Gross Mechanical Property Findings . . . . . . . . . . . . . . . . . . . . 78vi4.1.2 Fine Mechanical Property Findings . . . . . . . . . . . . . . . . . . . . . 794.1.3 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.1.4 Comparison to Histology . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2.1 Image Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2.2 Material Model Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . 835 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.3 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 855.4 Concluding Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87viiList of Tables3.1 The number of elements used for each FE test. . . . . . . . . . . . . . . . . . . . 503.2 Best fit relative stiffness value for each cord examined as determined by theminimum Hausdorff distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.3 Best fit relative stiffness value for each cord examined determined by the min-imum Hausdorff distance, using the erroneously segmented grey matter as thetruth condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73viiiList of Figures1.1 An illustration of the regions of the spinal column [28] (reprinted with copyrightpermission). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 An illustration of a typical lumbar vertebra [28] (reprinted with copyright per-mission). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 An illustration of two vertebrae and the adjoining intervertebral disc [23] (reprintedwith copyright permission). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 An illustration of an intervertebral disc attached to a vertebra [28] (reprintedwith copyright permission). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 An illustration of the major features of the spinal cord with the surroundingnerve rootlets and nerve roots [28] (reprinted with copyright permission) . . . . . 81.6 An illustration of the spinal cord, surrounding meninges, and nerve rootlets [28](reprinted with copyright permission) . . . . . . . . . . . . . . . . . . . . . . . . 91.7 An illustration of the basic structure of a myelinated neuron. (Reproducedunder the GNU Free Documentation and the Creative Commons Attribution-Share Alike 3.0 Unported Licenses)(https://commons.wikimedia.org/wiki/File:Neuron.svg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.8 An illustration of the Rexed laminae in the grey matter. (Reproduced under theGNU Free Documentation and the Creative Commons Attribution-Share Alike3.0 Unported Licenses)(https://en.wikipedia.org/wiki/File:Medulla_spinalis_-_Substantia_grisea_-_English.svg) . . . . . . . . . . . . . . . . . . . . . . . 111.9 An illustration of the tracts of neurons in the spinal cord white matter [23].(reprinted with copyright permission) . . . . . . . . . . . . . . . . . . . . . . . . 121.10 An illustration of the ascending and descending tracts in the white matter forthe rat (left) and the human (right) spinal cord [94]. (reprinted with copyrightpermission) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.11 Sagittal view of a burst fracture of the L2 vertebral body in a 30 year old maleresulting from a 20ft. fall [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.12 Sagittal view of a C6-C7 fracture dislocation from a motor vehicle accident in an18 year old male [32]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.13 The deformation of an infinitesimal rectangular element. (Image collected fromthe public domain (https://commons.wikimedia.org/wiki/File:2D_geometric_strain.png)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22ix2.1 Axial and sagittal MR images of the spinal cord before (a,c) and during (b,d) acontusion injury. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 A ‘voxelized’ hexahedral mesh built around the spinal cord of a rat. . . . . . . . 342.3 An illustration of the ICP method used to register a source point cloud (black)to an irregular shaped target point cloud (red). Figures (a), (b) and (c) representthree time points of the registration, from an undeformed state, to a partiallydeformed state to a fully deformed state. . . . . . . . . . . . . . . . . . . . . . . . 352.4 An illustration of the ICP method used to register the undeformed spinal cordsurface mesh to the deformed spinal cord surface mesh. The arrows represent thedirection of the forces around the undeformed ‘source’ mesh (grey), and point inthe direction of the deformed ‘target’ mesh (blue). The FE mesh in the figure isintentionally coarse for illustrative purposes. . . . . . . . . . . . . . . . . . . . . 362.5 An illustration of the calculation of the Hausdorff distance. The Hausdorff dis-tance from curve Y to curve X is shown on the bottom left (black arrow), whilethe Hausdorff distance from curve X to curve Y is shown on the top right (red ar-row). (Reproduced under the GNU Free Documentation and the Creative Com-mons Attribution-Share Alike 3.0 Unported Licenses)(https://en.wikipedia.org/wiki/File:Hausdorff_distance_sample.svg) . . . . . . . . . . . . . . . . 382.6 An illustration of the simple, two-material model exposed to a state of uniaxialcompression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.7 An illustration of the relevant strain parameters in the simple model . . . . . . . 402.8 The nodal normal strains in the vertical direction plotted against the verticaldistance in Figure 2.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.9 Simulation results of the nodal normal strains in the vertical direction plottedagainst the vertical distance in Figure 2.7 for various values of Poisson’s ratio. . . 422.10 The simple model exposed to a uniaxial tensile strain of 70% with the surfacemeshes unrefined (a) and refined into surfaces composed of more faces (b). . . . . 432.11 An illustration of the method used to compute Havg measured from the bluesurface to the red surface. It should be noted that for poorly registered surfaces,like the one shown, distance contributions can be measured between points thatdo not correspond, as is the case for distance d3. . . . . . . . . . . . . . . . . . . 452.12 Illustration of deformed spinal cord (blue), the segmented deformed grey matter(pink), and the segmented deformed grey matter in which the posterior surfaceof the deformed grey matter was extended one pixel in the posterior direction(green). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46x2.13 Sample histological data for NeuN-positive quantification. A - acquired imagesat 20x magnification; B - manually identified ROIs of the ventral horns; C -sample NeuN-positive neurons with the nucleoli presenting as a darkened centerof the cell nucleus. (Figure and caption text reproduced from Bhatnagar et al.under the Creative Commons Attribution Noncommercial License [3].) . . . . . 472.14 An axial MR image of the rat spinal cord with the ventral horns outlined byBhatnagar et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.15 An axial image FE model of the rat spinal cord with the ventral horns illustratedin green. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.1 The changes in Havg with FE mesh refinement for each spinal cord examined inthe study at a relative stiffness value of Ewhite/Egrey = 3.0 and Poisson’s ratioof ν = 0.45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 The changes in maximum principal strain in the grey matter with FE meshrefinement for each spinal cord examined in the study at a relative stiffness valueof Ewhite/Egrey = 3.0 and Poisson’s ratio of ν = 0.45. . . . . . . . . . . . . . . . . 503.3 A cumulative moving average of the force exerted on the FE mesh in the locationof the contusion tip, plotted against the number of data points used in the averagefor the tests conducted. The approximate experimental force of parametricallysimilar experiments conducted by Choo et al. [19] is also shown. . . . . . . . . . 513.4 The segmentation of the grey matter with the MR image overlaid in yellow forthe undeformed and deformed spinal cords 2, 3 and 4. . . . . . . . . . . . . . . . 523.5 The segmentation of the grey matter with the MR image overlaid in yellow forthe undeformed and deformed spinal cords 9, 10, 11 and 12. . . . . . . . . . . . . 533.6 The Havg plotted against the number of sampled points used to calculate theHavg. The sampled points were seeded at the rostrocaudal epicenter. . . . . . . . 543.7 FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord 2compared to the experimental grey matter outline shown in pink. Tested valuesof Ewhite/Egrey = 0.5 and 4.0 had inverted elements and were excluded fromanalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.8 FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord3 compared to the experimental grey matter outline shown in pink. . . . . . . . . 573.9 FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord4 compared to the experimental grey matter outline shown in pink. . . . . . . . . 583.10 FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord9 compared to the experimental grey matter outline shown in pink. Values ofEwhite/Egrey = 3.0 and 4.0 had inverted elements and were excluded from analysis. 593.11 FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord10 compared to the experimental grey matter outline shown in pink. . . . . . . . 60xi3.12 FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord11 compared to the experimental grey matter outline shown in pink. . . . . . . . 613.13 FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord12 compared to the experimental grey matter outline shown in pink. . . . . . . . 623.14 Havg as a function of the relative stiffness of the white matter to the grey matter,Ewhite/Egrey, for each cord at the injury epicenter. Each plot possesses the samespan of 0.03 mm. The red x’s indicate meshes with inverted elements, excludedfrom the study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.15 FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord2 compared to the original experimental grey matter outline (pink) and the erro-neously segmented grey matter outline (green). Tested values of Ewhite/Egrey =0.5 and 4.0 had inverted elements and were excluded from analysis. . . . . . . . . 653.16 FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord3 compared to the original experimental grey matter outline (pink) and theerroneously segmented grey matter outline (green). . . . . . . . . . . . . . . . . . 663.17 FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord4 compared to the original experimental grey matter outline (pink) and theerroneously segmented grey matter outline (green). . . . . . . . . . . . . . . . . . 673.18 FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord9 compared to the original experimental grey matter outline (pink) and theerroneously segmented grey matter outline (green). Values of Ewhite/Egrey = 3.0and 4.0 had inverted elements and were excluded from analysis. . . . . . . . . . . 683.19 FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord10 compared to the original experimental grey matter outline (pink) and theerroneously segmented grey matter outline (green). . . . . . . . . . . . . . . . . . 693.20 FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord11 compared to the original experimental grey matter outline (pink) and theerroneously segmented grey matter outline (green). . . . . . . . . . . . . . . . . . 703.21 FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord12 compared to the original experimental grey matter outline (pink) and theerroneously segmented grey matter outline (green). . . . . . . . . . . . . . . . . . 713.22 Havg as a function of the relative stiffness of the white matter to the grey matter,Ewhite/Egrey, for each cord at the injury epicenter. The red x’s indicate mesheswith inverted elements, excluded from the study. . . . . . . . . . . . . . . . . . . 723.23 The density of NeuN-positive neurons plotted against the minimum principalstrain throughout the ventral horns of the grey matter. . . . . . . . . . . . . . . . 733.24 The density of NeuN-positive neurons plotted against the maximum principalstrain throughout the ventral horns of the grey matter. . . . . . . . . . . . . . . . 74xii3.25 The density of NeuN-positive neurons plotted against the minimum principalstrain in ventral horns of the grey matter for each animal examined. . . . . . . . 753.26 NeuN-positive density data (left axis, red) and minimum principal strain data(right axis, blue) plotted against the distance from the injury epicenter for eachventral horn for spinal cords from animals 2, 3, 4 and 9. . . . . . . . . . . . . . 763.27 NeuN-positive density data (left axis, red) and minimum principal strain data(right axis, blue) plotted against the distance from the injury epicenter for eachventral horn for spinal cords from animals 10, 11 and 12. . . . . . . . . . . . . . 774.1 An illustration of the spinal cord showing the posterior median sulcus [28] (reprintedwith copyright permission) against the absence of an observable sulcus in the MRimages used to conduct the current study. . . . . . . . . . . . . . . . . . . . . . 80xiiiList of AbbreviationsDTI - diffusion tensor imagingFE - finite elementHavg - modified Hausdorff distanceICP - iterative closest pointITK - Insight Tool KitIV - intervertebralMR - magnetic resonanceNeuN - neuronal nucleiRF - radiofrequencytSCI - traumatic spinal cord injuryxivAcknowledgementsI would like to thank my supervisor, Dr. Thomas Oxland, for his support and guidance through-out this project. I would also like to thank Dr. Antonio Sa´nchez, Dr. Tim Bhatnagar, Dr.John Lloyd, Dr. Sid Fels, Dr. Antony Hodgson, Steve Mattucci, Masoud Malakoutian, and allof my lab mates at the Orthopaedic and Injury Biomechanics Group.xvDedicationTo my parents, for their love and support.xviChapter 1Introduction1.1 Background and MotivationTraumatic spinal cord injury (tSCI) is a debilitating condition that can adversely affect theinjured individual’s health and well-being. tSCI occurs when a mechanical insult causes damageto the spinal cord. Currently, there are approximately 44,000 people in Canada living with tSCI,with the annual incidence reported at 1,800 per year [73]. tSCI results in a loss of functionand mobility that depends on a number of factors including level of injury and completenessof injury. While there are a number of pharmacological and cellular therapies that have shownsome success in animal studies, none have been effective in treating spinal cord injuries clinically[73]. It is thought that part of the reason that therapies successful in animal models have failedto produce the same neurological benefits in humans, is due to the inherent heterogeneity ofhuman tSCI. The animal models in which therapies are tested are typically homogeneous withrespect to the injury and population, but human tSCI is highly heterogeneous in terms of thepopulation sustaining injury, as well as the mechanism, severity, and level of the injury [90].It is thought that by better categorizing SCI in terms of the type of damage experienced bythe spinal cord, treatments can be tailored to individual patients [90]. My thesis continues aline of research at the Orthopaedic and Injury Biomechanics Group (OIBG) at the Universityof British Columbia aimed to better understand the link between the mechanics of spinal cordinjury and the resulting tissue damage in the spinal cord. The long term goal of the work is todetermine strategies to tailor treatment of spinal cord injuries based on the mechanical natureof the injuries. Specifically, my thesis focuses on determining the mechanical properties of invivo rat spinal cord tissue in order to improve predictions of how the spinal cord deforms duringinjury.1.1.1 FE as an Animal Model Research ToolWith knowledge of the mechanical properties of the in vivo rat spinal cord, an accurate finiteelement (FE) model can be developed. A validated FE model of the rat spinal cord can be usedfor three main purposes:• To create a more comprehensive understanding of the link between injury mechanics andtissue damage in spinal cord injury;• To establish an injury threshold for individual tissues in the spinal cord;1• To test different treatments of spinal cord injury based on injury severity.In the past, the link between mechanics of SCI and the resulting damage has been studiedby isolating individual injury parameters, such as velocity or compression distance, performingSCIs by varying one parameter at a time, and observing the relevant behavioral and histologicaldifferences in the animals [16, 17, 19, 85]. This method has been useful for determining somerelevant injury parameters, however, it is not feasible to fully categorize SCI by this piece-wise approach. Instead, a validated FE model would enable researchers to categorize injuriesin terms of the stress and strain in the spinal cord, providing a 3D map of the spinal cordthat would facilitate a comprehensive comparison between mechanical stimuli in the spinalcord and the associated pathophysiologic response. Furthermore, a validated FE model usedin conjunction with animal injury trials may lead to the development of an accurate injurythreshold, with which it will be possible to accurately predict the damage in specific tissuesof interest resulting from a spinal cord injury of known parameters. Finally, the developmentof an accurate injury threshold would allow for an investigation of which treatment therapiescan be effectively used for injuries of different severity. This investigation would be done byusing the FE model alongside experimental animal testing, predicting the expected degree oftissue damage, and comparing it to the actual damage seen following the application of varioustherapeutic treatments.1.1.2 Bench to BedsideWhen working in SCI research, it is vital to consider how potential findings can be translatedinto improvement in patient outcomes. Assuming it is true that mechanically different typesof injuries may respond better to different therapies, one must understand how to distinguishthe different types of injuries in humans and how one would decide which treatment to applyas a result. Two potential ways to assess the mechanical nature of human spinal cord injuryare human FE modeling post injury and the use of diffusion tensor imaging (DTI).Human FE ModelOne way to predict the histological damage in human spinal cord injury is to perform patientspecific FE simulations of SCI after the injury in order to determine the likely extent of thedamage to the spinal cord. If it is known that there are specific injury thresholds for spinalcord tissues, then one could recreate the suspected injury with a FE model and observe theexpected deformation of the spinal cord, and make predictions about the injury based on theresults. There are, however, difficulties associated with this idea: it would be time intensive,expensive, it is difficult to know how the cord behaved during the transient injury, and howthe injury has progressed between the time of the injury and the time of the clinical imaging.As such, it may make more sense to use FE modeling only in the animal research phase, andattempt to link relevant mechanical injury parameters with more easily measurable outcomes2in humans.Diffusion Tensor ImagingA more pragmatic approach to predicting the histological damage in human spinal cord injury isto use DTI to quantify histological damage as a result of SCI. DTI is an imaging technique thatuses magnetic resonance (MR) imaging to characterize the diffusion of water in an object. DTIhas been used to characterize microstructural changes with neuropathology and treatment [2].DTI appears to provide insight into the extent of behavioral and histological damage followingSCI. Factors of mean diffusivity, fractional anisotropy, radial diffusivity and longitudinal diffu-sivity have been shown to correlate well with ASIA impairment score in humans and with longterm locomotor recovery in mice [15, 50]. In rat models, myelin water fraction and transversediffusivity correlated well with myelin damage, while longitudinal diffusivity and functionalanisotropy correlated well with axonal damage [52]. Therefore, if we determine that certaintypes of histological damage respond better to different therapeutic treatments using animalmodels in conjunction with FE studies, and we can characterize the type of tissue damage thatexists in human SCI with DTI, we could then apply specific treatments to patients followingspinal cord injury.1.2 Project DefinitionPrevious research has brought forth a need for the development of an FE model to recreateexperimental spinal cord injuries in rats for the purpose of determining the in vivo mechanicalproperties of the primary constituent materials of the rat spinal cord: the grey and whitematter.1.3 Overview of Spinal Cord InjurytSCI occurs when a mechanical insult causes damage to the spinal cord. Most often, the traumais caused by motor vehicle crashes (48%), falls (22%), acts of violence (12%) and sports injuries(10%) [26]. 55% of spinal cord injuries occur in the cervical region (C1-C11), 15% in thethoracic (T1-T11), 15% in the thoracolumbar (T11-L2), and 15% in the lumbosacral (L2-S5)[83]. The annual incidence of spinal cord injury in the developed world varies between 11.5-53.4per million population [92]. Currently, there are approximately 44,000 people in Canada livingwith tSCI, with the annual incidence reported at 1,800 per year [73]. tSCI results in a loss ofmobility and function that depends on a number of factors including the level of injury andcompleteness of injury. In addition to the personal problems associated with loss of sensory andmotor function, the financial burden of tSCI on both the individual and society as a whole arelarge. For individuals with SCI in the US, the average yearly expenses are between $350,000and $1,000,000 in the first year and between $43,000 and $190,000 each subsequent year [72].3Furthermore, the indirect costs and societal impact associated with tSCI are large. One yearafter injury, 81% of those previously employed are unemployed, and 25 years after injury,39.5% remain unemployed. Often, governments or health insurance providers will not cover thenecessary direct and indirect costs associated with tSCI, resulting in a greater financial burdenon the injured individual in addition the the physiologic burden associated with their injury[56].1.3.1 Anatomy of the Human Spine and Spinal CordIn the interest of understanding the fundamentals of human spinal cord injury and investigatingpotential therapeutic treatments in a controlled manner, animal models are often used as asurrogate for human SCI. Rats are the most commonly used SCI animal model for preliminaryresearch due to their relatively low cost, ease of care and similarity to humans in terms ofbiological injury pattern. Although many physiological and functional differences exist betweenrats and humans, there are many similarities between the injury patterns that exist after tSCI,such as the inflammation, neurite regeneration, and spinal motor pattern generators[16]. Assuch, it is necessary to understand the relevant anatomy and physiology associated with humantSCI, as well as the functional differences between the rat model and the human.The Spinal ColumnThe spinal column serves four main purposes: to protect the spinal cord and nerves, to supportthe weight of the body, to provide a semi-flexible axis with which to move the head and toprovide a structure that facilitates posture and locomotion. The human spine is composed of33 vertebrae within five different regions: 7 cervical, 12 thoracic, 5 lumbar, 5 fused in the sacraland 4 fused in the coccygeal region. The spinal column is shown in Figure 1.1.4Figure 1.1: An illustration of the regions of the spinal column [28] (reprinted with copyrightpermission).Between the bony vertebrae are intervertebral (IV) discs and articulating joints that enablethe spinal column to be flexible. Ligaments and muscles attach to the vertebrae to providestability in the column and enable movement. Vertebrae are composed of three primary parts:a vertebral body, a vertebral arch and seven processes, as shown in Figure 1.2.5Figure 1.2: An illustration of a typical lumbar vertebra [28] (reprinted with copyright per-mission).There is some variation in size of features between vertebrae, but they possess the samebasic characteristics. The vertebral body is the anterior portion of the vertebra that providesthe majority of the weight-bearing capability of the vertebra. The vertebral bodies are largestat the base of the spine, where the most weight must be supported. The vertebral arch islocated posterior to the vertebral body and forms the vertebral canal, which surrounds andprotects the spinal cord and meninges. The spinal canal is largest in the cervical spine becauseof the enlarged nature of the spinal cord in the region. The spinous process and two transverseprocesses protrude from the vertebral arch and attach to muscles to facilitate movement. Thefour articular processes function to restrict movement in specific directions. Cervical vertebraefeature two foramen through the transverse processes, which allow vertebral arteries to passthrough.Between the vertebral bodies exist intervertebral discs, Figure 1.3.6Figure 1.3: An illustration of two vertebrae and the adjoining intervertebral disc [23](reprinted with copyright permission).The intervertebral discs function to absorb impacts and allow for some movement of thespine. The intervertebral discs are composed of an anulus fibrosis and a nucleus pulposus. Theanulus fibrosis is composed of concentric lamellae of fibrocartilage that attach to the interver-tebral discs. The nucleus pulposus is a gelatinous material surrounded by the anulus fibrosis(Figure 1.4).Figure 1.4: An illustration of an intervertebral disc attached to a vertebra [28] (reprinted withcopyright permission).The ligaments of the vertebral bodies and processes largely act to limit mobility of the of7the spine and prevent separation of vertebrae, and the articulating joints of the vertebral archesallow for smooth movement between adjacent vertebrae [65].The Spinal CordThe spinal cord is the major reflex center and the conduction pathway between the brain andthe rest of the body. The spinal cord originates in the medulla oblongata and ends betweenvertebrae T12 and L3. The spinal cord branches into 31 pairs of spinal nerves: 8 cervical,12 thoracic, 5 lumbar, 5 sacral and 1 coccygeal. Nerve rootlets leave the dorsal and ventralportions of the spinal cord at these levels, and join together to form nerve roots, as shown inFigure 1.5.Figure 1.5: An illustration of the major features of the spinal cord with the surrounding nerverootlets and nerve roots [28] (reprinted with copyright permission)The dorsal nerve roots relay sensory, or afferent, input from the periphery of the body tothe brain, while the ventral nerve roots relay motor, or efferent, information from the spinalcord to the skin and viscera.Surrounding the spinal cord are three membranes that function to support and protect thespinal cord and spinal nerve roots: the dura mater, arachnoid mater and pia mater, shown inFigure 1.6.8Figure 1.6: An illustration of the spinal cord, surrounding meninges, and nerve rootlets [28](reprinted with copyright permission)The dura mater is the outermost membrane, and is composed of fibrous and elastic tissue.The dura mater is the toughest of the three membranes, and provides protection for the spinalcord. The arachnoid mater is less tough than the dura mater, and is composed of fibrousand elastic tissue. The arachnoid mater is pushed against the dura by the pressure from thecerebrospinal fluid within. The pia mater is the innermost layer of the meninges, and directlycovers the spinal cord. The pia mater attaches to the arachnoid and dura mater via thedenticulate ligament at 21 points, which are partially responsible for holding the cord in place.The subarachnoid space exists between the arachnoid and pia mater. This region is filled withcerebrospinal fluid, a substance which provides cushioning for the spinal cord [65].The spinal cord is responsible for initiating spinal reflex responses, relaying sensory infor-mation to the brain stem and cerebellum, and relaying motor information from the medulla andcerebral cortex to the rest of the body. The spinal cord is composed of nervous tissue, which ismade up of neurons and neuroglia. The neuron, shown in Figure 1.7, is the primary constituentcell of the nervous system and is responsible for sending and receiving electrochemical signals,or action potentials. Neurons have three primary components, the cell body, the dendrite, andthe axon. The cell body synthesizes all neuronal proteins and membranes in the neuron. Thedendrites are branching projections of the cell, which receive electrochemical signals. The signalis then passed to the cell body, which may then send the signal through the axon. The signalis then sent to the axon terminal, which passes the information to the next axon via a synapticcleft. Axons may or may not be surrounded by a myelin sheath composed of fatty tissue, whichenables signals to travel faster through the axon.9Figure 1.7: An illustration of the basic structure of a myelinated neuron. (Reproduced underthe GNU Free Documentation and the Creative Commons Attribution-Share Alike3.0 Unported Licenses)(https://commons.wikimedia.org/wiki/File:Neuron.svg)Neuroglia are nervous tissue cells that act to provide support to the neurons. Neurogliainclude astroglia, oligodendroglia, microglia and ependyma, each of which perform differentfunctions to support neurons. Astroglia provide physical support for other cells in the nervoussystem by their rigidity, and may also be used for exchange of metabolites between neuronsand blood. Oligodendroglia produce and maintain the myelin sheaths of axons in the centralnervous system. Microglia make up about 5% of total neuroglia, but their function is unknown.Ependyma line the ventricular system and may aid in chemical exchange between CSF andneural tissue.The two primary gross components of the spinal cord are the grey and white matter, shownin Figure 1.5. The grey matter is the inner section of the spinal cord that has a butterfly shapewhen viewed in the axial plane. At the center of the grey matter is the central canal, a cavityfilled with CSF. The grey matter consists of three distinct regions: ventral horns, dorsal hornsand an intermediate zone. Broadly, the ventral horn and portions of the intermediate grey areaconduct efferent signals, while the dorsal horn and other portions of the intermediate grey areaconduct afferent signals. The grey matter is composed primarily of cell bodies, neuroglia andcapillaries. The communication network of these nerve fibers forms the neuropil.The white matter is the outer section of the spinal cord surrounding the grey matter, and isdivided into three primary sections: the dorsal funiculus, the lateral funiculus and the ventralfuniculus. The portions of the white matter between the ventral and dorsal horns are calledthe ventral and dorsal commissures, respectively. The ventral commissure is the site at whichpain and temperature axons decuss, or cross the midline of the spinal cord. The white matteris composed mostly of myelinated axons, neuroglia and blood vessels.The spinal cord is composed of designated tracts of neurons, with similar types of neuronsclustered together. In the grey matter, these neuron tracts are called laminae of Rexed, see10Figure 1.8. The different groups of laminae are responsible for modulating different sensoryand motor signals. For example, Lamina I contains large neurons connected to the axons ofthe spinothalamic tract responsible for pain, crude touch and temperature, whereas Lamina IIcontains small, densely packed neurons, which have synaptic contact with cells in laminae I-IV.In the white matter, the tracts typically run along the long axis of the cord, and are namedafter the sites with which the neurons are connected, as seen in Figure 1.9. For example, thecorticospinal tract contains neurons sending motor information from the cortex of the brainthrough the spinal cord, while the spinocerebellar tract sends sensory information from thespinal cord to the cerebellum [23].Figure 1.8: An illustration of the Rexed laminae in the grey matter. (Reproduced underthe GNU Free Documentation and the Creative Commons Attribution-ShareAlike 3.0 Unported Licenses)(https://en.wikipedia.org/wiki/File:Medulla_spinalis_-_Substantia_grisea_-_English.svg)11Figure 1.9: An illustration of the tracts of neurons in the spinal cord white matter [23].(reprinted with copyright permission)Differences Between the Human and Rat Spinal CordsWhen using a rat as a human SCI model, it is essential to understand the anatomical andphysiological differences between the rat and the human.Firstly, the rat spine differs from the human spine in both length and size. The rat spineconsists of 57-60 vertebrae as opposed to 33, with the difference accounted for primarily byvertebrae in the tail. The location of the vertebrae are as follows: 7 in the cervical region, 13in the thoracic region, 6 in the lumbar region, 4 in the sacral region and the rest in the tail[94]. The anterior to posterior vertebral column width of the rat varies from 5.1 mm at the C1level down to 2.6 mm at the T1 level, whereas the human column varies from 18.6 to 13.7 mm[31, 39]. The rat spinal cord itself is 9 cm long and 2-3 mm in anteroposterior diameter, whilethe human cord is 44 cm in length and 10-17 mm in anteroposterior diameter [38, 40, 94].The rat and human spinal cords have approximately the same architecture as defined by dis-tribution of grey and white matter, along with the location of ascending and descending tracts.Additionally, the cellular constitution and structure of nervous tissues are similar between therat and human, as seen in in vitro tissue samples [94]. Some differences exist in terms of thesize and location of specific tracts, as shown in Figure 1.10, which are important for animalmodeling of spinal cord injury if monitoring behavior. There is also some discrepancy betweenthe sizes of rat and human nervous tissue cells. In humans, axons range in diameter from 1-10µm, whereas in rats, axons range from 1.27-5.54 µm [23, 95].12Figure 1.10: An illustration of the ascending and descending tracts in the white matter forthe rat (left) and the human (right) spinal cord [94]. (reprinted with copyrightpermission)The exact degree of similarity between the human and rat spinal cord is difficult to determinegiven the inability to experiment on humans and rats in the same manner, but reasonablesimilarities can be observed that make the rat a useful model for spinal cord injury.1.3.2 Pathophysiology of Spinal Cord InjuryThe pathophysiologic events following spinal cord injury can be subdivided into two phases:primary injury and secondary injury. The primary injury phase consists of cell damage occurringas a direct result of the mechanical insult to the spinal cord. The secondary injury phase consistsof cell damage resulting from a cascade of cellular and molecular events that occur as a resultof a physiologic response to the initial injury [81].Primary InjuryThe primary injury phase consists of mechanical physiologic changes that occur immediatelyfollowing spinal cord trauma. Severing of axons, neural death, glial death and spinal shock im-mediately contribute to the loss of function below the level of injury. The cord becomes swollen,accompanied by hemorrhage in the grey matter and cellular necrosis. Vascular disruption leadsto ischemia and further hemorrhage into the grey matter [81].13Secondary InjuryThe secondary injury phase consists of the cellular damage that occurs in response to the initialinsult of the cord beginning approximately two hours following injury, and lasting for monthsand even years after. Secondary injury can be subdivided into four stages: acute, subacute,intermediate and chronic, with each stage representing a different chronology of cellular events[81].The acute phase of secondary injury occurs between 2 and 48 hours after the initial injury[81]. During this stage, vascular disruption, further hemorrhaging and ischemia occur [91]. Theloss of autoregulatory mechanisms including ionic deregulation and cellular signaling strategieslead to death of oligodendrocytes and demyelination, as well as loss of function of calcium,sodium and potassium channel homeostasis. Each of these events contributes to a disruptionof signal transduction throughout the spinal cord. Damage to, or breakage of, the blood brainbarrier leads to an inflammatory response to the cord, resulting in migration of macrophages andneutrophils to the cord, leading to further damage [71]. Inflammation can also trigger microgliato act in combination with leukocytes to alter normal oxidative metabolism in demyelinatedaxons, causing the formation of a glial scar, which acts as a barrier to axonal regeneration [81].The subacute phase occurs between 2 days and 2 weeks post-injury [81]. During this phase,astrocytes experience cytotoxic edema and necrotic cellular death, followed by astrocytic re-activity causing further progression of the glial scar[76]. The increased phagocytic responseduring this phase may limit the regenerative capabilities of the spinal cord [98].The intermediate phase occurs between 2 weeks and 6 months, and is characterized by glialscar maturation as well as some regenerative sprouting of corticospinal tract axons [81].The final stage of secondary spinal cord injury is the chronic phase, beginning 6 months afterthe injury and lasting until death. During the chronic phase, the glial scar matures, neuroglialcavities form and damaged axons degenerate further [1].1.4 Experimental Modeling of tSCIAnimal injury models are useful tools that enable the study of the effects of different injuryparameters on physiological outcomes. By exposing the spinal cord to different types andseverities of insults, we can gain a better understanding of the biological response to tSCI, andwe can evaluate potential therapies that may be useful in different injury cases. There are fivemain types of injuries used in animal modeling, referred to as injury mechanisms, that aimto replicate different types of clinically relevant injuries: transection, compression, contusion,dislocation and distraction models.141.4.1 SCI MechanismsTransection models of tSCIA common approach to modeling spinal cord injuries is to make a precise surgical incision in thecord. Transection models are often used to study neuronal regeneration after tSCI. The injuryis meant to resemble violent trauma, such as that in a gunshot or stab wound. Transectionmodels use either complete transection, in which the entire cord is cut, or partial transection,in which only a portion of the cord is cut. Complete transection models are useful in that it iseasy to inflict well controlled, reproducible injuries. However, injuries of this type rarely occur.Partial transection injuries offer the benefits of being more clinically relevant and enabling thecomparison between damaged and undamaged sections of the spinal cord, however these modelsare less reproducible [16].Contusion models of tSCIContusion models of tSCI entail directly impacting a laminectomized spinal cord transientlyand acutely. Contusion injuries attempt to replicate relevant injury parameters from a burstfracture, in which axial loading of the spinal cord causes the fracture of a vertebral body, withone or more bony elements impacting the spinal cord. Burst fractures represent roughly 30% ofall spinal cord injuries [83]. A representative example of a burst fracture is shown in Figure 1.11.Burst fractures typically occur from a high energy axial loading event, such as a fall from aheight or a head-first impact.15Figure 1.11: Sagittal view of a burst fracture of the L2 vertebral body in a 30 year old maleresulting from a 20ft. fall [5].The most common methods of inflicting contusion-type injuries are weight drop impactorsand electromagnetic impactors. The Multicenter Animal Spinal Cord Injury Study (MASCIS)impactor, operates by dropping a rod of a specified weight from a known height onto an ex-posed spinal cord to produce injuries of consistent severity while monitoring the parameters ofheight, time of compression, velocity of impact and cord compression. The MASCIS impactoris relatively simple, but may introduce weight bounce, a second impact potentially resulting inadditional uncontrolled injury [97]. The infinite horizon (IH) impactor uses an electromagneticmotor coupled with a metal impounder to induce force-controlled impacts on the exposed spinalcord. The IH device offers the benefits of minimizing error associated with specimen movementdue to its force-controlled nature and eliminates weight bounce, but the clamps introduce vari-ability in firmly holding the spinal column [97]. The Ohio State University (OSU) impactoralso uses an electromagnetic impactor to inflict contusion-type injuries on an exposed cord, butcontrols the impact on the spinal cord based on the compression distance and time of impactrather than force. The primary limitation of the device is the difficulty to establish the exactlocation of the zero compression distance, as the cord moves slightly when first contacted [16].Compression models of tSCICompression models, like contusion models, involve impinging an exposed spinal cord, butdiffer in that the impingement can occur slowly and can easily be held for any specified time16interval. Exposing the cord to a compression that is maintained for some time period is usefulin replicating some clinical injuries, as sustained compressions are common in dislocations andburst fractures.Clip compression uses a modified aneurysm clip to consistently apply a specified force tothe cord over some time period. The only controlled variables in clip compression are force andcompression time, leaving velocity uncontrolled.Calibrated forceps and balloon compression models can apply specified forces for variabletimes, but lack the ability to adequately model the acute nature of tSCI.Spinal cord strapping is a procedure in which a suture is wrapped around the spinal cord andattached to a weighted pulley. The release of the weight initiates a uniform, concentric compres-sion of the cord. This procedure doesn’t require a laminectomy and is relatively non-invasivecompared to other procedures, but only the force of compression can be easily controlled.Dislocation models of tSCIDislocation models of tSCI translate one vertebrae in the dorsoventral or mediolateral directionwith respect to the adjacent vertebrae. This model was designed to replicate fracture-dislocationinjuries, the most common type of SCI observed clinically, representing 40% of all tSCI. Arepresentative example of a fracture-dislocation injury is shown in Figure 1.12.Figure 1.12: Sagittal view of a C6-C7 fracture dislocation from a motor vehicle accident inan 18 year old male [32].The dislocation model was first developed by Fiford et al. in 2004 [30]. This model used alinear actuator to drive a lateral dislocation of vertebrae. This model showed increasing axonal17injury with increasing dislocation distance. Choo et al. developed a dislocation model thatinflicted a dislocation in the dorsal direction to better replicate the types of injuries seen clini-cally. The use of a dislocation model revealed that dislocation injuries yielded more widespreadwhite matter damage compared to that seen in contusion injuries in the rostrocaudal direction.However, high model variability was seen in terms of tissue damage [19]. In 2017, Mattucci etal. developed self-aligning clamps to reduce the relative motion between the clamps and thevertebrae in order to partially address the variability of the model [59].Distraction models of tSCIDistraction models of tSCI create tension in the spinal cord in order to represent flexion-distraction type injury. Flexion-distraction injuries can occur from head-on motor vehicle col-lisions when the upper body is thrown forward while the lower body is constrained.There are several models that have been used for creating distraction injuries in rats includ-ing the Harrington impactor [25], the UBC multimechanism device [19] and the University ofTexas at Arlington (UTA) distractor [82]. Each of these devices attaches to the spine at twodifferent vertebral locations, which are forced apart by a electromagnetic motor, however, eachdevice has drawbacks. The Harrington impactor exhibits high variability within each gradeand uses a velocity of 1 cm/s, which may be too slow to replicate some clinical injuries, theUBC device displays variability due to inconsistent slippage, and the UTA distractor displayedinconsistent behavioral scoring and high variability [12, 16].1.4.2 Experimental FindingsExperiments in animal modeling of spinal cord injury typically observe how different injuryseverity outcomes, such as tissue damage or behavioral deficit, correlate with the injury param-eters of mechanism, injury depth, injury speed and duration of compression.It has been demonstrated that the mechanism of injury plays a role in the salient outcomesof tSCI. Choo et al. found that the different injury mechanisms of contusion, dislocation anddistraction resulted in different histological damage. Compared to contusion injuries, dislo-cation injuries resulted in a larger zone of axonal degeneration in both the mediolateral androstrocaudal directions [17]. Hemorrhage was present in contusion and dislocation, but notdistraction injuries. Furthermore, injurious changes to the nodes of Ranvier were localized inthe contusion injuries compared to the distraction injuries [19]. Clarke et al. compared themechanisms of anterior-posterior fracture dislocation to that of lateral fracture dislocation, andfound that anterior-posterior dislocation produced greater damage as measured by hemorrhagevolume, pathologic accumulation of β-amyloid precursor protein in white matter axons and de-generation of neurons in grey matter [22]. It is well documented that increasing peak spinal corddisplacement correlates with severity of biological damage and negative behavioral outcomesfor both contusion and dislocation type injuries [30, 70, 85].18Sparrey et al. used a rat contusion model to show that increases in injury speed from 3to 300 mm/s result in increases in tissue damage as measured by hemorrhage volume in whitematter, axonal disruption and increase in non-phosphorylated neurofilament staining. Theyalso found that for slow injuries, 83% of the hemorrhage volume was found in the grey matteras opposed to 17% in the white matter [86]. Lam et al. investigated the effects of speed andcompression depth in contusion injuries, and found that at a compression depth of 0.9 mm,varying the injury speed between 8, 80 and 800 mm/s had no effect on behavioral deficit,spared white matter, spared grey matter, and very little effect on percent of damaged nervefibers in the lateral and ventral white matter [54]. However, at compression depths of 1.5 mm,there are large differences in these same injury measures between each tested velocity. Kearneyet al. used a contusion model to vary compression depth and speed independently between25-65% of cord diameter and 1.5-6 m/s, respectively. They found that at lower compressiondepths, functional and anatomic damage is best predicted by compression depth, while at highercompression depths, velocity begins to play an important role in damage prediction [49].Another important mechanical parameter of spinal cord injury is the duration of the com-pression of the spinal cord. Dimar et al used an epidural spacer compression injury in rats toinvestigate the effects of compression depth and time on tissue damage following SCI. Theyfound that neurologic injury existed for rats exposed to a compression of 50% of spinal corddiameter, but not for rats in the 20% or 35% compression groups [27]. They also found thatinjury, as measured by behavior scores and minimum evoked potentials, was more severe withincreasing time of compression for each group tested: 0, 2, 6, 24 and 72 hours. Sjovold et al.used a contusion rat model to investigate importance of residual compression following injury.The injury was performed at 700 mm/s to a depth of 1 mm, with the amount of residual com-pression held at 0%, 40% and 90%. It was found that the rostrocaudal extent of intramedullaryhemorrhage and neuronal nuclei lost in the grey matter region were larger in the 90% residualcompression group compared to the 40% compression group [84].The mechanical parameters of tSCI models vary greatly, and it would be useful to developa more general injury criterion that accounts for the complex interplay between the mechanicalparameters of spinal cord deformation.1.5 Determination of Spinal Cord and Brain Tissue MaterialPropertiesIn order to better understand the mechanical causes of specific aspects of tSCI, there is interestin being able to model injuries computationally [24, 57, 96]. In order to construct biofideliccomputational injury models, one must know how the spinal cord deforms when exposed todifferent loadings. As such, there has been considerable effort in determining the mechanicalproperties of the spinal cord.At the present time, there is no consensus on the mechanical properties of the in vivo human19spinal cord. Most attempts to classify the mechanical properties of the spinal cord have beenperformed on human cadaveric spinal cords or ex vivo animal spinal cords.Oakland et al. performed uniaxial tensile tests on excised spinal cords to investigate thedependence of material properties on time after excision. This study found that the tangentmodulus of the spinal cord tissue increased from approximately 1.2 MPa to 1.9 MPa over thetesting period from 3-72 hours postmortem, indicating the limitations of ex vivo testing ofspinal cord material properties [74].Bilston and Thibault investigated the time dependence of material properties on freshlyexcised human spinal cords at non-injurious strain rates(0.04− 0.24s−1) up to 13% strain andfound that a quasi-linear viscoelastic model adequately described the behavior of the spinal cordin uniaxial tension [6]. Fiford and Bilston found that viscoelastic models accurately describedthe behavior of freshly excised rat spinal cords exposed to uniaxial tension tests at strain ratesbetween 0.002-0.2s−1 for strains of 2-5% [29].It has been shown that the the two primary constituent tissues in the spinal cord, the greyand white matter, may possess different mechanical properties. As such, there has been someeffort to distinguish these tissues separately, both in the spinal cord and the brain. Amongthese studies, some have found that the grey matter is stiffer [20, 37, 45, 51], some have foundthat the white matter is stiffer [9, 53, 60], and one has found no significant difference at all [77].The discrepancy between these studies likely arises from using different methods to test tissuesof different animals at different tissue states.Ichihara et al. performed ex vivo testing of separated bovine grey and white matter underuniaxial tension, and found that the grey matter had a greater tangent modulus at strainvalues between 15% and 35%, while there was no significant difference below 15% or above35%. Between these strain values, the tangent moduli for the grey and white matter rangedfrom 988-1260 kPa and 375-942 kPa, respectively. [45]. Koser et al. performed ex vivo atomicforce microscopy testing of mice within 30 minutes of sacrifice. They found that grey matterwas significantly stiffer than white matter, with reduced elastic modulus values of 130 and 70Pa for grey and white matter, respectively [51]. They found that these results were consistentregardless of directionality or force direction. Christ et al. used scanning force microscopy onex vivo rat cerebellus tissue, and found that the grey matter was significantly stiffer than whitematter, having elastic moduli of 454 and 294 Pa, respectively [20]. Furthermore, Green et al.used magnetic resonance elastography to test in vivo human brain tissue, and found that greymatter (3.1kPa) had a significantly stiffer storage modulus than the white matter (2.7 kPa)[37].Ozawa et al. performed ex vivo tensile testing on the spinal cord of Japanese white rabbitsusing a pipette aspiration technique. They tested the spinal cords in the anteroposterior,mediolateral and rostrocaudal directions, and found that there was no significant differencesbetween the elastic moduli of grey and white matter in any direction [77].Kruse et al. used magnetic resonance elastography on in vivo human brain tissue, and found20that white matter (13.6 kPa) had a significantly greater shear modulus than grey matter (5.22kPa) [53]. McCracken et al. also used magnetic resonance elastography in in vivo brain tissue,and found white matter (12 kPa) to have a greater shear stiffness than grey matter (8 kPa)[60]. Budday et al. used ex vivo bovine brain tissue indentation within 2 hours post mortem,and found that white matter (1.895 kPa) was significantly stiffer than grey matter (1.389 kPa)[9].Due to the contradicting data of brain and spinal cord material properties, there must befurther research into the tissue properties until a consensus is reached.1.6 Computational Modeling of SCI1.6.1 Strain TheoryIn order to classify how the spinal cord, or any material, deforms and experiences damage, onemust define the quantities of stress and strain. Strain is an engineering measure of relativedeformation of an object or material. Stress is an engineering measure of how forces are trans-mitted through an object. In order to understand the deformations of a complex object, it isnecessary to define stress and strain in terms of infinitesimal elements within an object exposedto some external force and resultant deformation.Given a 2D deformation of an infinitesimal rectangular element whose horizontal and verticaldimensions are dx and dy, respectively, that undergoes a deformation shown in Figure 1.13.21Figure 1.13: The deformation of an infinitesimal rectangular element. (Image collectedfrom the public domain (https://commons.wikimedia.org/wiki/File:2D_geometric_strain.png))ab =√(dx+∂ux∂xdx)2+(∂uy∂xdx)2(1.1)Which, for small displacement gradients, reduces to:ab = dx+∂ux∂xdx (1.2)Normal strain, or the relative change in length of a body, in the x-direction, εx, is given by:εx =ab−ABAB(1.3)Given that AB = dx:εx =∂ux∂x(1.4)22In the y and z directions, normal strain is given as:εy =∂uy∂y, εz =∂uz∂z(1.5)Shear strain, or the change in angle between original orthogonal lines, AB and AC is definedas:γxy = α+ β (1.6)Where tan α and tan β can be written as:tan α =∂uy∂x dxdx+ ∂ux∂x dx=∂uy∂x1 + ∂ux∂x, tan β =∂ux∂y dydy +∂uy∂y dy=∂ux∂y1 +∂uy∂y(1.7)For small rotations, equations 1.7 can be written as:tan α = α, tan β = β (1.8)As a result, shear strain can be rewritten as:γxy = α+ β =∂uy∂x+∂ux∂y(1.9)In the y − z and x− z planes:γyz =∂uy∂z+∂uz∂y, γxz =∂ux∂z+∂uz∂x(1.10)The entire state of strain can be rewritten into tensor form, called Cauchy’s strain tensor,as in Equations 1.11-1.13.ε =εxx εxy εxzεxy εyy εyzεxz εyz εzz (1.11)ε = εxx γxy/2 γxz/2γxy/2 εyy γyz/2γxz/2 γyz/2 εzz (1.12)ε =∂ux∂x12(∂ux∂y +∂uy∂x)12(∂ux∂z +∂uz∂x)12(∂ux∂y +∂uy∂x)∂uy∂y12(∂uy∂z +∂uz∂y)12(∂ux∂z +∂uz∂x)12(∂uy∂z +∂uz∂y)∂uz∂z (1.13)Individual elements in the tensor are written as follows:23εij =12(∂ui∂xj+∂uj∂xi)(1.14)Here, i and j represent the rows and columns of the strain tensor, respectively.However, given different orientations, the strain tensor can have different values. As such,it is useful to describe the state of strain in terms of only normal strains, also called principalstrains, which yields a unique solution to the strain tensor. Principal strains represent thelargest tensile, ε1, and largest compressive, ε3, possible normal strains in the element, and areuseful in developing failure criteria for materials.ε =ε1 0 00 ε2 00 0 ε3 (1.15)Although infinitesimal strain theory is typically used only for describing small deformations,the basic strategy of predicting deformation in an object is similar when describing large de-formations, called finite strain theory. Finite strain theory uses a different set of assumptions,and as a result, different strain tensors (e.g. Lagrangian finite strain tensor, Biot strain tensor,etc.) to describe deformations that are large compared to the body being deformed.1.6.2 Finite Element BasicsIn order to use finite element analysis, it is necessary to understand how the finite elementmethod decomposes a continuous material into a set of elements and calculates the resultingdisplacements. The following subsection outlines the standard equations denoting how the finiteelement method calculates nodal displacements of elastic body deformations.For convenience, elastic body deformations are described in terms of mapping undeformedmaterial coordinates, X = (X,Y, Z) to deformed spatial coordinates, x = (x, y, z), with theequation x = φ(X). The derivative of the map with respect to the material coordinates isdefined as the deformation gradient, F, shown in Equation 1.16.F =∂x∂X(1.16)Coordinate displacements u, are described as their deviation, x, from their rest position,X, shown in Equation 1.17.u = (ux, uy, uz) ≡ x−X (1.17)The displacement gradient, ∂u/∂X, is defined in Equation 1.18, and used to define the Cauchystrain tensor, ε, in Equation 1.19.∂u∂X=∂x∂X− I = F− I (1.18)24ε =12(∂u∂X+(∂u∂X)T)=12(F + FT)− I (1.19)Equation 1.19 can be rewritten as a vector, ε, shown in Equation 1.20ε =(∂ux∂x,∂uy∂y,∂uz∂z,∂ux∂y+∂uy∂x,∂ux∂z+∂uz∂x,∂uy∂z+∂uz∂y,)(1.20)The stress vector, σ, for a linear elastic material model is described in Equation 1.21σ = Eε (1.21)E is defined in Equation 1.22.E =λν1− ν ν ν 0 0 0ν 1− ν ν 0 0 0ν ν 1− ν 0 0 00 0 0 12 − ν 0 00 0 0 0 12 − ν 00 0 0 0 0 12 − ν(1.22)The Lame´ constant, λ, is defined in Equation 1.23.λ ≡ Eν(1 + ν)(1− 2ν) (1.23)The elastic modulus, E, is defined as the ratio between the stress and strain for a uniaxialtensile test, as in Equation 1.24.E =σε(1.24)The Poisson’s ratio, ν, is defined in Equation 1.25.ν =−εtransεnormal(1.25)Here, εtrans and εnormal are the transverse and normal strains, respectively.The finite element method decomposes a continuous material into a set of volumetric ele-ments, defined by boundary nodes and shape functions that interpolate quantities at boundarynodes inside elements. x = φ(X) can be approximated by Equation 1.26.x =n∑i=1xiNi(X) (1.26)Here, Ni are the shape functions associated with particular nodes, xi are the deformedcoordinate values at node i, and n is the number of nodes. Displacements, u, are then mapped25in terms of their nodal coordinates, shown in Equation 1.27.u = x−X =n∑i=1(xi −Xi)Ni(X) =n∑i=1uiNi(X) (1.27)ui is the displacement at node i. The displacement gradient is given in Equation 1.28.∂u∂X=n∑i=1ui∂Ni(X)∂X(1.28)Shape functions are defined in terms of their natural coordinates, S, so the displacementgradient can be rewritten in Equation 1.29∂u∂X=n∑i=1ui∂Ni(S)∂S∂S∂X(1.29)Furthermore, ∂S/∂X can be found from the inverse of the element Jacobian, J, defined inEquation 1.30.J ≡ ∂X∂S=(n∑i=1Xi∂Ni(S)∂S)(1.30)The nodal displacements, ui, can be arranged into a vector, ue, as shown in Equation 1.31.u(X) = Hue (1.31)whereH ≡N1(X) 0 0 ... Nn(X) 0 00 N1(X) 0 ... 0 Nn(X) 00 0 N1(X) ... 0 0 Nn(X) (1.32)Strain, ε, can then be calculated, as in Equation 1.33.ε(X) = Bue,B ≡∂∂x 0 00 ∂∂y0 0 ∂∂z∂∂y∂∂x 0∂∂z 0∂∂x0 ∂∂z∂∂yH (1.33)Using the principle of virtual work, BT maps the stress components to nodal forces, nodalforces and stiffness matrix can be determined by Equation 1.34.fe = Keue,Ke =∫VBTEB dV (1.34)26Here, Ke is formed with a volume integral over the element [68].However, the volume integral is not evaluated exactly. It is approximated by contributing aweighted component of the stiffness matrix from selected points, denoted ‘integration points’,to the stiffness matrix associated with a particular node. The stiffness matrix associated witha particular node pair, i, j, is calculated over k integration points, as shown in Equation 1.35.Ke =∑kBTi,kEkBj,kJkwk (1.35)Here, wk is the weight of the integration point and Jk is the local value of the Jacobian atk. After assembling the stiffness matrix, the general equation of motion can be solved, as inEquation 1.36.Mx¨ + Cx˙ + Kx = fx − f0 (1.36)Here, x, x0, and u ≡ x− x0 are the positions, rest positions and deformation of the nodes,while M is the mass matrix, C is the damping matrix, K is the global stiffness matrix, fx arethe external forces, and f0 ≡ −Kx0 [68].Typically, the use of small strain linear elastic models under large deformations leads tounrealistic distortions of the geometry. The use of the technique, ‘stiffness warping’, developedby Muller and Gross resolves unrealistic distortions by removing elemental rotation from thedeformation gradient, F [66]. For this reason, the material model is called a co-rotated linearelastic material model. This technique allows for large deformations and rotations withoutcausing unrealistic volume growth [66]. However, the stress-strain relationship is still dictatedby small deformation assumptions, not taking into account the change in elemental volumewith deformation.1.6.3 Material ModelsThere have been a number of material models used to try to replicate the behavior of thespinal cord and brain tissue during injury, including linear elastic, hyperelastic, viscoelastic,poro-viscoelastic and transversely isotropic, but it has not been determined that one modelbest represents the behavior of the spinal cord [6, 11, 13, 29, 33, 45, 61, 63, 69, 74]. Further-more, the variability of the experiments conducted likely plays an important role in the spinalcord behavior. The variable parameters that have shown to be important include strain rate,preconditioning, time post-mortem, orientation, age of the animal material composition andorientation of the applied loads [6, 14, 21, 29, 34, 35, 44, 45, 62–64, 69, 75, 78].1.6.4 Computational Model FindingsComputational models are often used to complement experimental studies, to give more insightinto how the cord deforms and which areas are exposed to the highest stress or strain, and27therefore most likely to be injured. Computational studies often correlate stress and strainfindings to localized areas of tissue damage to propose an injury criterion. Without havingconclusively established spinal cord material properties, it is not possible to create a perfectstress-based injury criterion, but the models may provide insight into reasonable approximationsabout the deformation of the cord during spinal cord injury.Greaves et al. used a finite element model to simulate contusion, dislocation and distractioninjuries using data from Hung et al. and Maiman et al. for validation [43, 58]. Greaves used alinear elastic material model and modeled the grey and white matter as a single material, andfound that each of the different injury mechanisms resulted in notably different strain patterns[36]. Maikos et al. used a finite element model to recreate weight drop contusion experimentsin rats. They used an Ogden hyperelastic and Prony viscoelastic material model for the spinalcord and found that maximum principal strain correlated well with injury as measured byextravasation. The correlations were seen to be higher in the grey matter ( R2 = 0.84) ascompared to the white matter ( R2 = 0.56) [57]. Sparrey et al. used a finite element modelof a human spinal cord exposed to compression between two plates to show that using variousreported literature values for grey matter, white matter, and pia mater caused high variationin principal stresses and strains [88]. Russell et al. used a finite element model to recreatecontusion and dislocation injuries in rats initially performed by Choo et al. [18]. They useda hyperelastic Ogden model combined with a viscoelastic Prony model, and modeled the greyand white matter as a single material. They found that maximum principal strain correlatedwith tissue damage in white matter for contusion (R2 = 0.86) and dislocation (R2 = 0.52)as measured by axonal membrane permeability. They found that although the white matterhad larger overall strains, more tissue damage was seen in the grey matter [80]. Bhatnager etal. used a deformable image registration approach to quantify the strain field in a contusioninjury by comparing undeformed and deformed MR images of an in vivo rat spinal cord. Theyfound that there was a low correlation between minimum principal strain and tissue damage asmeasured by NeuN staining to measure density of healthy neuronal cell bodies [3].1.7 SummaryGiven that the material properties of the in vivo cord are ill-defined, there exists a need todetermine these properties, and create a biofidelic finite element model with these properties.This work will enable researchers to better understand the relationship between the complexmechanics of spinal cord deformation and the tissue damage resulting from the injury.1.7.1 ObjectivesThe primary objectives of my research project were to:• Create an FE model of the rat cervical spinal cord to recreate previously conductedcontusion-type injuries;28• Use an inverse FE approach to iteratively vary the mechanical properties of the greyand white matter, while qualitatively and quantitatively observing which properties bestmatch the experimental data.The secondary objective of my research project was to:• Compare the strain results from the finite element study to the histological data previouslycollected on the same animals.1.7.2 ScopeThe current study will investigate the mechanical properties of the spinal cord by assessingthe behavior of seven previously conducted contusion-type injuries inflicted upon Sprague-Dawley rats by Bhatnagar et al. [3]. Only the time-independent properties will be investigated,as no transient data were collected in the experiments on which the current study is based.Furthermore, as exact force data was not collected during the experiments, the absolute materialstiffness for the tissues cannot be examined. Only the relative stiffness between the grey andwhite matter can be determined.29Chapter 2MethodsThis section will describe the methods used to build, verify and validate an FE model of spinalcord injury in the rat. The FE model replicates contusion-type injuries to the rat spinal cordthat were performed inside of an MR scanner by a previous doctoral student [3]. The materialproperties that best fit the experimental data are chosen to be the most accurate materialproperties of the spinal cord. This procedure is typically referred to as an inverse FE methoddue to its iterative nature [46, 48]. I provide a brief overview of the experiment on which theFE model is based, followed by an in depth look at the methods used by the current study.2.1 Previously Conducted Experimental Data CollectionBhatnagar et al. performed contusion-type spinal cord injuries on adult male, Sprague-Dawleyrats. They took MR images of the rat spinal cords before the injury, and again while thecontusion was inflicted and held. They then used image registration techniques to estimate thestrain along axial slices of the MR images, and correlated the strains in the ventral horns ofthe grey matter with tissue damage measured using histological techniques. They found thatminimum principal strain correlated with neuron damage as measured by loss of a neuronalnuclei antibody stain across all animals (R2 = 0.19). A more detailed explanation of themethods is given in Sections 2.1.1 to 2.1.5.2.1.1 Surgical PreparationTwelve animals were anesthetized with isofluorane (4% during initiation and maintained at2% administered by nose cone throughout the experiment). The dorsal neck was shaved anddisinfected, and the animals were stabilized in a stereotaxic frame. The soft tissue surroundingthe dorsal aspects of the cervical spine and tissues connecting the scapulae to the dorsal aspect ofthe thorax were resected. Partial laminectomies of the C5 and C6 vertebrae were performed. Aradiofrequency (RF) coil was placed over the cervical spine, with a thin layer of wax separatingthe coil from the animal. Custom clamps were attached to vertebrae C4-C7. The animalswere then inserted into the MR rig. A rectal thermometer, heating pad and respiratory cyclemeasurement transducer were used to maintain body temperature and monitor vital signsduring the experiment.302.1.2 Imaging MethodsThe animal was then positioned in the MR scanner (7T, Bruker BioSpec, Tu¨bingen, Germany)and a T2-weighted sagittal ‘trial’ scan (115 × 150 µm in-plane resolution, 1-mm slice thickness,and 5-min acquisition time) was performed to ensure positioning and orientation). Then, a T2-weighted, high-resolution transverse ‘data acquisition’ scan (140 × 140 µm in-plane resolution,500 µm slice thickness, and 30-min acquisition time) was performed. Finally, a mild (1.1 mmspinal cord compression) or severe (1.8 mm spinal cord compression) contusion injury wasinflicted, and held for 35 minutes while ‘trial’ and ‘data acquisition’ scans of the spinal cordwere taken. A total of 48 axial images were taken over the span of 2.4 cm in the rostrocaudaldirection.Images of the spinal cord before and during a contusion injury are shown in Figure 2.1.(a) (b)(c) (d)Figure 2.1: Axial and sagittal MR images of the spinal cord before (a,c) and during (b,d) acontusion injury.2.1.3 Histological Preservation MethodsFollowing the second ‘data acquisition’ scan, the animal was moved to the fume hood andsacrificed by intracardial perfusion (150 mL of phosphate-buffered saline (PBS) followed by 300mL of 4% paraformaldehyde (PFA)). After perfusion, the spine was laminectomized, the duralsheath was cut from the base of the skull to the C8/T1 level, the dorsal and ventral rootswere cut and the cord was transected at the base of the skull and the C8/T1 level. The spinalcord was placed in a vial of 4% PFA overnight. The cords were then passed through a gradedsucrose system (12%, 18%, and 24% sucrose, in PBS solution) for approximately 24 hours in31each grade. Cords were frozen in a block of embedding medium and cut, transversely, into 20µm sections and mounted on slides. The slides were kept in a -86◦C freezer until staining wasperformed.2.1.4 Magnetic Resonance Imaging Post-ProcessingThe pre-injury and injury image sets were segmented from surrounding tissues. The ventralhorns of the grey matter were manually traced. The segmented images were input into adeformable image registration algorithm, which yields 3D displacement fields mapped onto thepre-injury image. The displacement fields were used to generate transverse-plane Lagrangianfinite strain magnitudes, and then used to calculate maximum and minimum principal strainfields. The strain in the ventral horn in each axial image slice was calculated by averaging thecomputed strains at every voxel contained within the ventral horn.2.1.5 Histological AnalysisA neuronal nuclei (NeuN) antibody was used to identify healthy neurons in the grey matterat the time of perfusion. Disappearance of NeuN staining is used as a marker to identifyneuronal damage. Images were acquired using a Zeiss microscope (Axio Observer A1; CarlZeiss, Thornwood, NY) with a 20× objective (0.65×0.65×5 µm per pixel) and equipped witha Yokogawa spinning disc confocal device and a motorized scanning stage. Images at threedepths were acquired for each tissue section. Exposure settings for image capture were setmanually through Zen software (Blue v1.0.1.0, Carl Zeiss Microscopy GmbH, Germany) toensure adequate contrast between grey and white matter. The injury epicenter was defined asthe section with the largest lesion area. For each transverse image, a region of interest wasmanually drawn around the ventral horns. Within the region of interest outlined, the numberof neurons with an observable nucleolus were counted and the surface area of the region wascomputed, to give a resultant NeuN-positive density, in terms of healthy neurons/mm. Linearregression was then used to compare the average strain in each ventral horn at discrete distancesfrom the injury epicenter to the healthy neuron density at the same locations. They found thatthere was a weak, but statistically significant correlation between minimum principal strainand tissue damage as measured by NeuN staining to measure density of healthy neuronal cellbodies [3].2.2 Model DevelopmentIn order to construct and validate an FE model of the spinal cord, one must discuss the steps ofimage segmentation of the spinal cord, construction of a finite element mesh, model verificationand model validation. Furthermore, the method to evaluate finite element characteristics incomparison to histological data is described here.322.2.1 Image SegmentationThe first step in creating a finite element model based on a specific rat is to segment the spinalcord grey and white matter, separately, from the surrounding tissues captured by the MRimages. This segmentation was done manually using the software, ITK-SNAP (www.itksnap.org/), which enables the construction of 3D solid geometries via the segmentation of individualslices, shown in each anatomical plane.The grey and white matter were segmented in both the pre-injury state and the injury state.These segmented geometries, called surface meshes hereafter, form the basis of the FE modelused in this study.A total of seven injuries were used out of the twelve contusion injuries available, due to theinability to distinguish a clear boundary between the grey and white matter for the remainingfive injuries.Following the manual segmentation of the grey and white matter, the meshes were importedinto MeshLab (http://www.meshlab.net/). Six meshes were then exported for use in each FEmodel: the undeformed and deformed meshes of the grey matter, the undeformed and deformedmeshes of the entire cord, and the undeformed and deformed meshes of the entire cord editedto have the rostral and caudal surfaces removed for later use in the deformation of the FEmesh. The rostral and caudal surfaces were removed for two meshes to control the registration,described in Section 2.2.3.2.2.2 Construction of Finite Element MeshAfter the surface meshes had been constructed, the FE meshes for each specimen were cre-ated in ArtiSynth, a 3D mechanical modeling system that supports the combined simulation ofmultibody and finite element models, developed at UBC [55]. First, the largest possible dimen-sions of the surface mesh of the undeformed spinal cord were calculated. Then, a rectangularcuboid mesh composed of hexahedral elements was created to match the preinjury MR scanof the rat spinal cord. The elements that existed entirely outside of the surface mesh of thespinal cord were then deleted, along with their associated nodes and dependencies, creating a‘voxelized’ FE mesh around the entirety of the spinal cord, as shown in Figure 2.2.33Figure 2.2: A ‘voxelized’ hexahedral mesh built around the spinal cord of a rat.Eight node hexahedral elements were chosen to represent the spinal cord due to the fact thatthey can experience relatively large deformations as compared to tetrahedral elements withoutundergoing volumetric or shear locking. Volumetric, or shear locking is a phenomenon thatcauses artificial stiffening of the element when modeling nearly incompressible materials, suchas biological soft tissue [79]. Hexahedral elements are also more computationally cost effectivethan tetrahedral elements [41].Following the creation of the ‘voxelized’ FE mesh, the surface meshes of the undeformedspinal cord and the undeformed grey matter are then embedded, or skinned into the FE mesh.Mesh skinning is a method for attaching a 3D surface mesh to an FE mesh to achieve realisticdeformations of the surface. Skinning is done by defining each vertex on the surface mesh bythe shape functions of the element containing the vertex. After the element deforms, the shapefunctions are used to calculate the locations of the vertices and update the shape of the surfacemesh. [89]Next the material properties were assigned. For the purposes of explaining the method, I willdescribe the assignment of one set of material properties. For information about the materialproperty investigation, see Section 2.2.4. The primary problem with hexahedral elements isthat it is difficult to use them to create complex shapes [41, 47]. With traditional methods,it is difficult to define and attach separate meshes for the grey and white matter. Howeverthis problem is mitigated in ArtiSynth by carefully defining the material properties within asingle FE mesh for both the grey and white matter. First, a single material model with a set ofproperties is defined for the entire mesh, for example, a linear elastic model with elastic modulusand Poisson’s ratio of 30 kPa and 0.4, respectively. This is the desired material property forthe grey matter of the cord. Then, every integration point within each element in the FEmesh is tested for whether or not it is inside the grey matter surface mesh. If the integrationpoint exists within the grey matter, the point remains unchanged, but if the point exists withinthe white matter, the contribution to the stiffness matrix evaluated at that integration point(as calculated by Equation 1.35) is multiplied by a scaling factor. For example, if the elasticmodulus of the white matter is desired to be twice that of the grey matter, the scaling factor34would be 2.0.2.2.3 Boundary ConditionsThe availability of both undeformed and deformed states of the spinal cord enables the accuratereplication of experimental conditions. In order to perform accurate deformation of the cordfrom the undeformed state to the deformed state, an iterative closest point (ICP) approach wasused. ICP is an algorithm designed to minimize the difference between two point clouds. ICPis performed by defining a point on the point cloud to be registered (the ‘source’ surface), andmoving it to the nearest point cloud on the desired target, (the ‘target’ surface). An illustrationof the method used by a ICP controller is shown in Figure 2.3.(a) (b) (c)Figure 2.3: An illustration of the ICP method used to register a source point cloud (black)to an irregular shaped target point cloud (red). Figures (a), (b) and (c) representthree time points of the registration, from an undeformed state, to a partiallydeformed state to a fully deformed state.In the current study, the ICP approach was implemented by using two surfaces: the unde-formed spinal cord as the source mesh and the deformed spinal cord as the target mesh. Bothmeshes had their rostral and caudal surfaces removed so that no axial registration force wasimparted on the cord from either end, and the surfaces were embedded into the mesh. Foreach vertex on the source mesh, the closest point on the target mesh is found. Then, the 3Ddistance, d, and direction, Fdir, between the two points is calculated. Based on the distancebetween the points and a user specified maximum pressure, Pmax, a force magnitude, Fmag, iscalculated to be applied to the FE mesh, as shown in Equation 2.1.Fmag = Pmax × e−d2/s2 (2.1)Here, s is a user specified force scaling constant. This force scales down to zero at a userspecified distance to the target, and was determined empirically. The force of magnitude Fmagin the direction, Fdir, is then applied to the nodes surrounding the vertex of the surface meshusing a weighting function based on proximity to the surrounding nodes. The equation to runthe ICP controller, (Equation 2.1), is provided in ArtiSynth, and its implementation was foundto work well for the current study. It is of note that the cord is held in place axially due to the35concave shape of the deformed spinal cord at the location of the contusion. A demonstrationof the application of forces on the spinal cord model is shown in Figure 2.4.Figure 2.4: An illustration of the ICP method used to register the undeformed spinal cordsurface mesh to the deformed spinal cord surface mesh. The arrows represent thedirection of the forces around the undeformed ‘source’ mesh (grey), and point inthe direction of the deformed ‘target’ mesh (blue). The FE mesh in the figure isintentionally coarse for illustrative purposes.The forces are updated at every time step until the geometries are sufficiently close to eachother. The ICP approach works well for quasi-static deformations, and can recreate the exactgross deformation pattern of the spinal cord, removing some of the uncertainty associated withthe typical method of estimation of boundary conditions based on the geometry of the vertebraeand impactor. The ICP approach provides a useful starting point for the investigation of thetime-independent properties of the spinal cord. It is of note that the force of gravity was set tozero, as the calculated force of gravity acting on the spinal cord segment Fg = 0.0019 N, wassmall compared to the approximate force reported by previous contusion injuries ( 2 N).2.2.4 Material Property InvestigationAfter defining the FE mesh and boundary conditions applied to the cord, the material model,test conditions, and material property search space had to be defined.Firstly, the material property had to be defined. Previously, the spinal cord has beenmodeled as a linear elastic, Neo-Hookean hyperelastic, Mooney-Rivlin hyperelastic and Ogdenhyperelastic material [7]. The strain energy density, U , is given for each material in Equations2.2-2.5. For a linear elastic material, strain energy density is given in Equation 2.2.U =12Cijklijkl (2.2)Here, Cijkl are the components of the fourth order stiffness tensor defined by the elastic36modulus, E and Poisson’s ratio, ν.The strain energy density of a Neo-Hookean hyperelastic material is given in Equation 2.3.U =µ12(I1 − 3) + K12(J − 1)2 (2.3)Here, µ1 and K1 are the shear and bulk moduli, respectively. The strain energy density of aMooney-Rivlin hyperelastic material is given in Equation 2.4.U =µ12(I1 − 3) + µ22(I2 − 3) + K12(J − 1)2 (2.4)Here, µ1, µ2 and K1 are the material properties.The strain energy density of an Ogden hyperelastic material is given in Equation 2.5.U =N∑i=12µiα2i(λα11 + λα12 + λα13 − 3)K12(J − 1)2 (2.5)Here, µi, αi and K are material properties. [8].Given that two materials were investigated in the study, the grey and white matter, theuse of Mooney-Rivlin and Ogden hyperelastic models was impractical, as each material modelis defined by three parameters. Linear elastic and Neo-Hookean models are each defined bytwo material parameters, so the use of these models significantly reduces the number of testsneeded to examine the overall material search space. A linear elastic representation was chosento be investigated initially, due to the relative ease in interpreting the results from such a test.In order to drive the deformation of the spinal cord iteratively, without user intervention,a Python script was written to interface with ArtiSynth. The script generated the mesh andassigned specific material properties to the grey and white matter, then called the ICP controllerto register the source mesh to the target mesh with a low initial maximum pressure value. Thescript then ran the simulation for a short time and checked to see if the solution was converged.If the solution was not converged, the script then increased the maximum pressure value actingon the FE mesh, again ran for a short time and checked convergence again. This loop wasrepeated until convergence was reached. The purpose of slowly increasing the forces actingon the mesh was to ensure stability of the registration by first allowing translation to occur,followed by a well-controlled deformation process. Convergence was assessed using the Hausdorffdistance, a metric which takes every point on one point cloud, calculates the minimum distanceto a second point cloud, and takes the largest calculated distance, as shown in Figure 2.5.37Figure 2.5: An illustration of the calculation of the Hausdorff distance. The Hausdorffdistance from curve Y to curve X is shown on the bottom left (black ar-row), while the Hausdorff distance from curve X to curve Y is shown on thetop right (red arrow). (Reproduced under the GNU Free Documentation andthe Creative Commons Attribution-Share Alike 3.0 Unported Licenses)(https://en.wikipedia.org/wiki/File:Hausdorff_distance_sample.svg)At a Hausdorff distance of 0.03 mm, about 1% of the diameter of the cord, the registrationwas defined as converged. This value was visually determined to be satisfactory and used forthe remainder of the tests.Given that run times for finite element analyses can be long, and the potential combinationof unique sets of material properties can be large, I first had to define the relevant search spacefor the material properties of the grey and white matter. Most previous literature values for thePoisson’s ratio, ν, of the spinal cord vary between 0.40 to 0.49 [7], while the relative stiffnessvalues between white and grey matter (Ewhite/Egrey) range from 0.54 [51] to 2.6 [53]. It is ofnote that the experiments being replicated by the current FE study did not give explicit forcevalues. As such, an absolute elastic modulus cannot be determined, and the relative stiffnessesbetween the white and grey matter must be evaluated. Initially, I decided to use a startingvalue of ν = 0.45 and vary the relative elastic moduli from Ewhite/Egrey = 0.5 − 4.0. Theserelative stiffness values were chosen based on the limits of the model, as above and below thesevalues, registration could not converge for most spinal cords without inverted elements in themesh. Specifically, the model was run at Ewhite/Egrey values of 0.5, 0.75, 1.0, 1.5, 2.0, 3.0 and4.0 for each spinal cord examined. Each cord was evaluated for similarity to the experimentalvalues at each relative stiffness value, described in Section 2.4.In order to run the dynamic simulations, ArtiSynth uses a Pardiso solver, which chooses38either a direct or iterative solver to calculate the displacements at each time step.2.3 Model VerificationWhen running FE simulations, it is necessary to ensure that the model is behaving as intended.This process is called verification.2.3.1 Simple ModelAs the first step of the verification process, a simple model was constructed to ensure properbehavior of a two-material object with an analytical solution. This process enables the gradualmovement from simple problems with analytical solutions to more complex problems where useof the finite element method is necessary, while maintaining confidence in the model.First, I wished to ensure that the assignment of material properties was correct and thedistribution of nodal strains was correct. In order to do this, a simple two-material block wasconstructed and exposed to a state of uniaxial compression, as shown in Figure 2.6.Figure 2.6: An illustration of the simple, two-material model exposed to a state of uniaxialcompression.The object was exposed to uniaxial compression of 40%. The material on the top andbottom is defined as Mout and the material in the middle is defined as Min. Initially, thematerials were set as linear elastic with the outer material given a elastic modulus twice that ofthe inner material, that is Eout = 2Ein. Both materials were given Poisson’s ratios of ν = 0.0,and is illustrated in Figure 2.7.39Figure 2.7: An illustration of the relevant strain parameters in the simple modelGiven that the sum of the changes in length of each block segment yields the total changein length of the block, and εin = 2εout due to the difference in elastic modulus, we have:∆L1 + ∆L2 + ∆L3 = ∆LtotalεoutL4+ εinL2+ εoutL4= εtotalL = 0.4Lεout2+εin2= 0.4,εout + εin = 0.8εout + 2εout = 0.8εout =0.83= 0.26εin =0.83= 0.53(2.6)Analytically, the strains in the inner material should be twice that of the outer material,and for 40% compression, the strain values should be exactly εin = 0.53 and εout = 0.36. Theanalytical and FE nodal normal strains in the vertical direction of the model as a function ofvertical distance are plotted in Figure 2.8.40−1 −0.5 0 0.5 1−0.6−0.4−0.20y L/2ε yyFEAnalyticalFigure 2.8: The nodal normal strains in the vertical direction plotted against the verticaldistance in Figure 2.7.From Figure 2.8, it can be seen that the model solution agrees well with the analyticalsolution everywhere except at the interface of the two materials, at which point the nodalstrains are intermediate values.Next, I wished to observe how the nodal strain values changed with increasing Poisson’sratio, as the Poisson’s ratio for the spinal cord is typically measured between 0.4 and 0.49. Themodel was again tested at 40% compression, with Eout = 2Ein, and values for Poisson’s ratioset to ν = 0, 0.15, 0.3, 0.45. Again, the nodal normal strains in the vertical direction of themodel are plotted as a function of vertical distance in Figure 2.9.41−1 −0.5 0 0.5 1−0.6−0.4−0.20y L/2ε yyν = 0ν = 0.15ν = 0.3ν = 0.45Figure 2.9: Simulation results of the nodal normal strains in the vertical direction plottedagainst the vertical distance in Figure 2.7 for various values of Poisson’s ratio.Figure 2.9 shows that as Poisson’s ratio is increased, the nodes away from the interfacecompensate by taking on values higher and lower than those seen in the tests with lower valuesfor Poisson’s ratio.Finally, I wished to ensure that the 3D surface meshes were effectively skinned onto theFE mesh. In order to ensure true mesh skinning, the simple model was exposed to 70% axialtensile strain for visual effect. It was found using the initial surface meshes, Figure 2.10a,the surface meshes were not adequately skinned to the FE mesh due to the fact that theylacked an adequate number of faces. Without enough faces, the vertices of a single face on thesurface mesh can be attached to elements that are not adjacent, resulting in surfaces that failto properly conform to the FE mesh. As such, surface mesh refinement is necessary to have asurface mesh that adequately conforms to the FE mesh, as shown in Figure 2.10b.42(a) (b)Figure 2.10: The simple model exposed to a uniaxial tensile strain of 70% with the surfacemeshes unrefined (a) and refined into surfaces composed of more faces (b).With this information, the model of the true spinal cord was ensured to have adequaterefinement.2.3.2 True ModelThe second step in the verification process was to perform verification on the actual model of thespinal cord. The components of this verification process included the mesh independence test,an evaluation of the forces on the model and a qualitative evaluation of the image segmentation.In order to have confidence in the results of an FE study, it is necessary to ensure thatthe solution is mesh independent, i.e. that the results do not depend on the resolution of themesh. In order to achieve mesh independence, the FE test was run first with a very coarsemesh, followed by more refined meshes until the results were deemed converged. The resultswere deemed converged when the Hausdorff distance and the maximum nodal strain inside thegrey matter became constant with additional mesh refinement. Mesh independence tests wereconducted for each spinal cord studied.To ensure that the model was performing as expected, the model forces were evaluatedagainst approximate forces given by previous contusion experiments of similar compressiondistance and speed, conducted by Choo et al. [19]. In order to get a range of possible forcevalues from the model, I set the material properties of the spinal cord to 0.26 MPa and 1.3MPa for two separate tests. These values are the minimum and maximum reported elasticmoduli from previous experimental testing of the spinal cord using traditional material testingmethods [7]. The injury simulations were run, and a cumulative moving average of the forces inthe location of the contusion tip was reported for both tests. A cumulative moving average wasused to remove moment-to-moment force fluctuations, as the simulation is dynamic in nature.43In order to assess the quality of the image segmentation, after importing the surface meshinto ArtiSynth, the MR images of the spinal cord were imported into ArtiSynth and assessedfor similarity. The MR images were imported, and rigidly registered to the spinal cord surfacemesh. The surface meshes of the grey and white matter were then compared qualitatively tothe overlaid MR images. For grey matter surfaces that did not line up with the MR images,the surface was re-segmented and the process was repeated until a satisfactory segmentationwas reached.2.4 Model ValidationFE model validation is a method of testing to ensure that the model accurately represents thesituation being simulated. In the current study, the majority of the validation process wasperformed by checking the results of the grey matter deformation predicted in the FE studyagainst the experimental deformation of the grey matter observed in the animal study. For thepurposes of the current study, the validation process is done concurrently with material modelinvestigation, as an end goal of the study is to develop a biofidelic FE model of spinal cordinjury in the rat.2.4.1 Qualitative MetricsQualitatively, the deformation of the grey matter predicted by the FE model was compared withthe deformation of the grey matter observed in the animal experiments. Overall differences inshapes were observed and points of agreement and notable points of disagreement were noted.Qualitative assessment was a useful approach for determining the functional differences betweenthe deformation behavior of the experimentally observed grey matter and the FE predicted greymatter.2.4.2 Quantitative MetricsIn order to compare the results of the material properties with minimal bias, a quantitativemetric, the modified Hausdorff distance, was used to compare the shapes of the experimentaland FE grey matter shapes. The modified Hausdorff distance, Havg, places n points on thesurface of one mesh, and calculates the distance from each point to the closest point on asecond mesh being compared, and averages the distances, di. The modified Hausdorff distanceeffectively gives an average distance between the meshes, as illustrated in Figure 2.11 andEquation 2.7.44Figure 2.11: An illustration of the method used to compute Havg measured from the bluesurface to the red surface. It should be noted that for poorly registered surfaces,like the one shown, distance contributions can be measured between points thatdo not correspond, as is the case for distance d3.Havg =1n(d1 + d2 + ...+ dn) (2.7)2.4.3 Error AnalysisIn order to evaluate the sensitivity of the results to the segmentation procedure, a segmentationerror analysis was performed in which the posterior surface of the deformed grey matter surfacemesh was extended by one voxel in the posterior direction along the length of the cord. Thefinite element tests were then rerun with the newly segmented deformed grey matter surfaces forthe purpose of observing which material property set best fit the new truth condition. Figure2.12 shows the difference in grey matter shape between the original segmented grey matter andthe erroneously segmented grey matter for a typical specimen.45Figure 2.12: Illustration of deformed spinal cord (blue), the segmented deformed grey matter(pink), and the segmented deformed grey matter in which the posterior surfaceof the deformed grey matter was extended one pixel in the posterior direction(green).2.5 Comparison to HistologyIn order to observe the relationship between the deformation of the spinal cord during injuryand the resulting tissue damage, it was of interest to observe how strain in each model related topreviously collected histological damage of the spinal cords of the rats. NeuN-positive neurondensity, a marker of neuron health in the grey matter, had been collected for the rats used inthe experiment by Bhatnager et al. [3]. Specifically, the NeuN-positive neuron density hadbeen collected at axial slices in the left and right ventral horns of the grey matter at locationsof 3.0, 2.0, 1.2, 0.8, 0.4, and 0 mm from the injury epicenter in the cranial and caudal directions[3]. For each image, a region of interest was manually drawn around the left and right ventralhorns, the number of NeuN-positive neurons were counted, and the area of the region of interestwas calculated to provide an effective NeuN+ neuron density. An example of the histologicaldata collected is shown in Figure 2.13.46Figure 2.13: Sample histological data for NeuN-positive quantification. A - acquired imagesat 20x magnification; B - manually identified ROIs of the ventral horns; C -sample NeuN-positive neurons with the nucleoli presenting as a darkened centerof the cell nucleus. (Figure and caption text reproduced from Bhatnagar et al.under the Creative Commons Attribution Noncommercial License [3].)The matched regions of interest were also outlined directly on the MR images used in thestudy by Bhatnagar et al., as shown in Figure 2.14.Figure 2.14: An axial MR image of the rat spinal cord with the ventral horns outlined byBhatnagar et al.The outlined regions of interest were input into the current FE model so that the nodalstrains in the ventral horns of the grey matter could be collected at the same intervals used in47the histological analysis. The model with the segmented horns input is shown in Figure 2.15.Figure 2.15: An axial image FE model of the rat spinal cord with the ventral horns illustratedin green.In order to evaluate the correlation between strain and neuronal damage in the ventralhorns, the average minimum and maximum principal strain in each ventral horn of the greymatter were collected at the same craniocaudal locations at which the histological analysis wasperformed. Linear regression was used to quantify the relationship between maximum andminimum principal strain and loss of NeuN staining. R2 correlation coefficients were calculatedfor each cord along with corresponding p-values.48Chapter 3Results3.1 Model Verification3.1.1 Mesh IndependenceMesh independence tests were run for each cord modeled as linear elastic at a conservativerelative stiffness value of Ewhite/Egrey = 3.0 and a Poisson’s ratio of ν = 0.45. For the tests,the resolution of the FE mesh was gradually refined until the modified Hausdorff distanceand the maximum strain inside the grey matter no longer changed substantially with meshrefinement. The FE mesh was defined as not changing substantially with mesh refinementwhen an additional refinement of 10,000+ elements resulted in a change in maximum principalstrain of less than 3%. The results of the mesh independence tests illustrating changes in Havgand maximum principal strain in the grey matter for each cord are shown in Figure 3.1 andFigure 3.2, respectively.0 2 4 6 8·1046 · 10−28 · 10−20.10.120.140.16Number of ElementsHavgCord 2Cord 3Cord 4Cord 9Cord 10Cord 11Cord 12Figure 3.1: The changes in Havg with FE mesh refinement for each spinal cord examined inthe study at a relative stiffness value of Ewhite/Egrey = 3.0 and Poisson’s ratio ofν = 0.45.490 2 4 6 8·1040.20.40.60.8Number of ElementsMaximumPrincipalStrainCord 2Cord 3Cord 4Cord 9Cord 10Cord 11Cord 12Figure 3.2: The changes in maximum principal strain in the grey matter with FE mesh re-finement for each spinal cord examined in the study at a relative stiffness valueof Ewhite/Egrey = 3.0 and Poisson’s ratio of ν = 0.45.It can be seen that each FE mesh of the spinal cord converged as measured by the strainin the spinal cord. Havg converged with fewer elements than the maximum principal strain, sothe meshes used were dictated by the point at which the maximum principal strain convergedso that conclusions regarding both Havg and strain in the spinal cord could be made.The number of elements used for each cord is shown in Table 3.1.Table 3.1: The number of elements used for each FE test.Spinal Cord Number of Elements (1000s)Cord 2 46Cord 3 53Cord 4 36Cord 9 67Cord 10 48Cord 11 28Cord 12 583.1.2 Force VerificationThe force at the location of the phantom contusion tip was plotted as a cumulative movingaverage of the number of points used. A cumulative moving average was necessary due to thedynamic nature of the simulation and the resultant fluctuation of the force values. The FEsimulations were run using the smallest and largest values for elastic modulus measured by exvivo testing found in the literature [42],[10],[6]. Egrey and Ewhite were each set to 0.26 MPa forthe first test, and 1.3 MPa for the second test. The resultant force at the contusion tip for both50tests is shown in Figure 3.3. It was found that the possible force values ranged from 0.95 - 4.58N, while the reported experimental force values for contusions of similar compression distanceand speed were approximately 2 N [19]. This finding suggests that the FE model is yielding areasonable result for the force imposed on the cord.0 10 20 30 40 50 6002468Number of pointsForce(N)E = 0.26 MPaE = 1.3 MPaExperimental Force [19]Figure 3.3: A cumulative moving average of the force exerted on the FE mesh in the locationof the contusion tip, plotted against the number of data points used in the averagefor the tests conducted. The approximate experimental force of parametricallysimilar experiments conducted by Choo et al. [19] is also shown.3.1.3 Evaluation of SegmentationThe segmentation was qualitatively evaluated by overlaying the grey matter surface mesh ontop of the rigidly registered axial slices of the MRI. The segmentation was evaluated at theinjury epicenter, the location at which the process of distinguishing grey matter from whitematter was most difficult. In cases where the segmentation was determined to be erroneous, thesegmentation was performed again and reevaluated. The final segmented grey matter meshesare shown against the MR images at the injury epicenter for undeformed and deformed imagesin Figures 3.4 and 3.5.51(a) Cord 2 Undeformed (b) Cord 2 Deformed(c) Cord 3 Undeformed (d) Cord 3 Deformed(e) Cord 4 Undeformed (f) Cord 4 DeformedFigure 3.4: The segmentation of the grey matter with the MR image overlaid in yellow forthe undeformed and deformed spinal cords 2, 3 and 4.52(a) Cord 9 Undeformed (b) Cord 9 Deformed(c) Cord 10 Undeformed (d) Cord 10 Deformed(e) Cord 11 Undeformed (f) Cord 11 Deformed(g) Cord 12 Undeformed (h) Cord 12 DeformedFigure 3.5: The segmentation of the grey matter with the MR image overlaid in yellow forthe undeformed and deformed spinal cords 9, 10, 11 and 12.533.1.4 Evaluation of the Modified Hausdorff Distance MetricIn order to determine the number of points necessary to evaluate Havg, a convergence test wasrun with a sample spinal cord, measuring the Havg as a function of number of points. Thetest points were seeded uniformly throughout the surface mesh of the experimentally deformedgrey matter along an axial slice at the injury epicenter, and the distances to the nearest pointson the undeformed grey matter were recorded. All distances were averaged, and the Havg wasrecorded as a function of the number of points, as shown in Figure 3.6.0 0.5 1 1.5 2 2.5 3·1040.2500.2550.2600.2650.2700.2750.2800.2850.2900.2950.300Number of PointsHavgFigure 3.6: The Havg plotted against the number of sampled points used to calculate theHavg. The sampled points were seeded at the rostrocaudal epicenter.It was found that the Havg converged at roughly 5,000 points, and the difference betweenusing 5,000 points and 30,000 points was 0.00057 mm. A minimum of 5,000 points were usedfor all Havg values for the remainder of the tests conducted.3.2 Model Validation3.2.1 Qualitative MetricsIn order to qualitatively evaluate the predicted set of material properties used for the FE sim-ulations, the shape of the experimentally observed grey matter was overlaid with the deformed54shape of the FE grey matter. Figures 3.7-3.13 depict the grey matter deformations of each FEsimulation and the simulation that best fit the experimentally observed grey matter deforma-tion. In these figures, the deformed shape of the overall spinal cord is shown in blue, the shapeof the experimentally deformed grey matter is shown in pink, and the shapes of the deformedFE grey matter are all shown in grey. For the figures displaying multiple simulations, thegrey matter shapes of simulations with lower relative stiffness values, e.g. Ewhite/Egrey=0.5,consistently lie exterior to those with higher stiffness values, e.g. Ewhite/Egrey=4.0.55(a) Cord 2: Ewhite/Egrey = 0.75, 1.0, 1.5, 2.0, 3.0(b) Cord 2 best fit: Ewhite/Egrey = 1.0Figure 3.7: FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord2 compared to the experimental grey matter outline shown in pink. Tested valuesof Ewhite/Egrey = 0.5 and 4.0 had inverted elements and were excluded fromanalysis.56(a) Cord 3: Ewhite/Egrey = 0.5, 0.75, 1.0, 1.5, 2.0, 3.0, 4.0(b) Cord 3 best fit: Ewhite/Egrey = 3.0Figure 3.8: FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord3 compared to the experimental grey matter outline shown in pink.57(a) Cord 4: Ewhite/Egrey = 0.5, 0.75, 1.0, 1.5, 2.0, 3.0, 4.0(b) Cord 4 best fit: Ewhite/Egrey = 2.0Figure 3.9: FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord4 compared to the experimental grey matter outline shown in pink.58(a) Cord 9: Ewhite/Egrey = 0.5, 0.75, 1.0, 1.5, 2.0(b) Cord 9 best fit: Ewhite/Egrey = 2.0Figure 3.10: FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord9 compared to the experimental grey matter outline shown in pink. Valuesof Ewhite/Egrey = 3.0 and 4.0 had inverted elements and were excluded fromanalysis.59(a) Cord 10: Ewhite/Egrey = 0.5, 0.75, 1.0, 1.5, 2.0, 3.0, 4.0(b) Cord 10 best fit: Ewhite/Egrey = 2.0Figure 3.11: FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord10 compared to the experimental grey matter outline shown in pink.60(a) Cord 11: Ewhite/Egrey = 0.5, 0.75, 1.0, 1.5, 2.0, 3.0, 4.0(b) Cord 11 best fit: Ewhite/Egrey = 3.0Figure 3.12: FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord11 compared to the experimental grey matter outline shown in pink.61(a) Cord 12: Ewhite/Egrey = 0.5, 0.75, 1.0, 1.5, 2.0, 3.0, 4.0(b) Cord 12 best fit: Ewhite/Egrey = 2.0Figure 3.13: FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord12 compared to the experimental grey matter outline shown in pink.3.2.2 Quantitative MetricsHavg plotted against Ewhite/Egrey for each cord analyzed is shown in Figure 3.14. The span ofvalues of Havg is 0.03 mm for all plots.620 1 2 3 40.130.140.15(a) Cord 2Ewhite/EgreyHavg(mm)0 1 2 3 40.130.140.15(b) Cord 3Ewhite/EgreyHavg(mm)0 1 2 3 40.130.140.15(c) Cord 4Ewhite/EgreyHavg(mm)0 1 2 3 40.130.140.15(d) Cord 9Ewhite/EgreyHavg(mm)0 1 2 3 40.130.140.15(e) Cord 10Ewhite/EgreyHavg(mm)0 1 2 3 40.10.110.120.13(f) Cord 11Ewhite/EgreyHavg(mm)0 1 2 3 4789·10−2(g) Cord 12Ewhite/EgreyHavg(mm)Figure 3.14: Havg as a function of the relative stiffness of the white matter to the grey matter,Ewhite/Egrey, for each cord at the injury epicenter. Each plot possesses the samespan of 0.03 mm. The red x’s indicate meshes with inverted elements, excludedfrom the study.63The best fit material properties, as evaluated by choosing the Ewhite/Egrey value that re-sulted in the lowest Havg, are shown in Table 3.2.Table 3.2: Best fit relative stiffness value for each cord examined as determined by the mini-mum Hausdorff distance.Spinal Cord Number Best Fit Relative Stiffness2 1.03 3.04 2.09 2.010 2.011 2.012 3.03.2.3 Error AnalysisAfter determining the qualitative and quantitative best fit for each spinal cord examined, thesame procedure was conducted for the spinal cords with a segmentation error of one voxel inthe posterior direction on the posterior surface of the deformed grey matter. In this analysis,the shape of the experimentally observed grey matter was overlaid with the deformed shape ofthe FE predicted grey matter and the erroneously segmented grey matter shape. Figures 3.15-3.21 depict the grey matter deformations of each FE simulation. In these figures, the deformedshape of the overall spinal cord is shown in blue, the shape of the experimentally deformedgrey matter is shown in pink, the shape of the errant experimentally deformed grey matter isshown in green, and the shapes of the deformed FE grey matter are all shown in grey. Forthe figures displaying multiple simulations, the grey matter shapes of simulations with lowerrelative stiffness values, e.g. Ewhite/Egrey=0.5, consistently lie exterior to those with higherstiffness values, e.g. Ewhite/Egrey=4.0.64(a) Cord 2: Ewhite/Egrey = 0.75, 1.0, 1.5, 2.0, 3.0(b) Cord 2 best fit: Ewhite/Egrey = 0.75Figure 3.15: FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord2 compared to the original experimental grey matter outline (pink) and the erro-neously segmented grey matter outline (green). Tested values of Ewhite/Egrey =0.5 and 4.0 had inverted elements and were excluded from analysis.65(a) Cord 3: Ewhite/Egrey = 0.5, 0.75, 1.0, 1.5, 2.0, 3.0, 4.0(b) Cord 3 best fit: Ewhite/Egrey = 0.5Figure 3.16: FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord3 compared to the original experimental grey matter outline (pink) and theerroneously segmented grey matter outline (green).66(a) Cord 4: Ewhite/Egrey = 0.5, 0.75, 1.0, 1.5, 2.0, 3.0, 4.0(b) Cord 4 best fit: Ewhite/Egrey = 0.75Figure 3.17: FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord4 compared to the original experimental grey matter outline (pink) and theerroneously segmented grey matter outline (green).67(a) Cord 9: Ewhite/Egrey = 0.5, 0.75, 1.0, 1.5, 2.0(b) Cord 9 best fit: Ewhite/Egrey = 1.5Figure 3.18: FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord9 compared to the original experimental grey matter outline (pink) and theerroneously segmented grey matter outline (green). Values of Ewhite/Egrey =3.0 and 4.0 had inverted elements and were excluded from analysis.68(a) Cord 10: Ewhite/Egrey = 0.5, 0.75, 1.0, 1.5, 2.0, 3.0, 4.0(b) Cord 10 best fit: Ewhite/Egrey = 1.0Figure 3.19: FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord10 compared to the original experimental grey matter outline (pink) and theerroneously segmented grey matter outline (green).69(a) Cord 11: Ewhite/Egrey = 0.5, 0.75, 1.0, 1.5, 2.0, 3.0, 4.0(b) Cord 11 best fit: Ewhite/Egrey = 1.5Figure 3.20: FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord11 compared to the original experimental grey matter outline (pink) and theerroneously segmented grey matter outline (green).70(a) Cord 12: Ewhite/Egrey = 0.5, 0.75, 1.0, 1.5, 2.0, 3.0, 4.0(b) Cord 12 best fit: Ewhite/Egrey = 0.75Figure 3.21: FE predicted deformed grey matter for different values of Ewhite/Egrey for Cord12 compared to the original experimental grey matter outline (pink) and theerroneously segmented grey matter outline (green).Havg, evaluated with the erroneous segmentation of the deformed grey matter, plottedagainst Ewhite/Egrey for each cord analyzed is shown in Figure 3.22.710 1 2 3 40.150.160.170.180.19(a) Cord 2Ewhite/EgreyHavg(mm)0 1 2 3 40.110.120.130.14(b) Cord 3Ewhite/EgreyHavg(mm)0 1 2 3 40.150.160.17(c) Cord 4Ewhite/EgreyHavg(mm)0 1 2 3 40.150.160.17(d) Cord 9Ewhite/EgreyHavg(mm)0 1 2 3 40.110.120.13(e) Cord 10Ewhite/EgreyHavg(mm)0 1 2 3 49 · 10−20.10.110.12(f) Cord 11Ewhite/EgreyHavg(mm)0 1 2 3 47 · 10−28 · 10−29 · 10−20.1(g) Cord 12Ewhite/EgreyHavg(mm)Figure 3.22: Havg as a function of the relative stiffness of the white matter to the grey matter,Ewhite/Egrey, for each cord at the injury epicenter. The red x’s indicate mesheswith inverted elements, excluded from the study.72The best fit material properties, as evaluated by choosing the Ewhite/Egrey value that re-sulted in the lowest Havg, are shown in Table 3.3.Table 3.3: Best fit relative stiffness value for each cord examined determined by the minimumHausdorff distance, using the erroneously segmented grey matter as the truth con-dition.Spinal Cord Number Best Fit Relative Stiffness2 0.753 0.54 0.759 1.510 1.011 1.512 0.753.3 Comparison to HistologyThe FE models with the material properties that were determined to best fit the experimentaldata were used to compare average strains in the ventral horns of the grey matter to theneuronal damage, as measured by density of NeuN-positive cells.Figures 3.23 and 3.24 show the linear regression analysis evaluating the dependence of NeuN-positive density values against minimum and maximum principal strain values, respectively.The R-squared values for the dependence of density of NeuN-positive cells on the minimumand maximum principal strain were 0.115 (p=2E-5) and 0.064 (p=0.002), respectively.−0.4 −0.3 −0.2 −0.1 0050100150200Minimum Principal StrainNeuN+Density(#/mm2)R2 = 0.115Figure 3.23: The density of NeuN-positive neurons plotted against the minimum principalstrain throughout the ventral horns of the grey matter.730 0.1 0.2 0.3 0.4050100150200Maximum Principal StrainNeuN+Density(#/mm2)R2 = 0.0644Figure 3.24: The density of NeuN-positive neurons plotted against the maximum principalstrain throughout the ventral horns of the grey matter.Figure 3.25 shows the linear regression analysis evaluating the dependence of NeuN-positivedensity values against minimum principal strain values for each spinal cord tested. The R-squared and accompanying p values for NeuN-positive density dependence on the minimumprincipal strain are listed in the captions of each graph.74−0.4 −0.3 −0.2 −0.1 0050100150200Minimum Principal StrainNeuN+Density(#/mm2)(a) Cord 2: R2=0.197,p=0.0388−0.4 −0.3 −0.2 −0.1 0050100150200Minimum Principal StrainNeuN+Density(#/mm2)(b) Cord 3: R2=0.160,p=0.0651−0.4 −0.3 −0.2 −0.1 0050100150200Minimum Principal StrainNeuN+Density(#/mm2)(c) Cord 4: R2=0.227,p=0.0250−0.4 −0.3 −0.2 −0.1 0050100150200Minimum Principal StrainNeuN+Density(#/mm2)(d) Cord 9: R2=0.309,p=0.00721−0.4 −0.3 −0.2 −0.1 0050100150200Minimum Principal StrainNeuN+Density(#/mm2)(e) Cord 10: R2=0.165,p=0.0603−0.4 −0.3 −0.2 −0.1 0050100150200Minimum Principal StrainNeuN+Density(#/mm2)(f) Cord 11: R2=0.000,p=0.994−0.4 −0.3 −0.2 −0.1 0050100150200Minimum Principal StrainNeuN+Density(#/mm2)(g) Cord 12: R2=0.120,p=0.134Figure 3.25: The density of NeuN-positive neurons plotted against the minimum principalstrain in ventral horns of the grey matter for each animal examined.75The variation in average minimum principal strain and presence of NeuN-positive stainingas a function of rostrocaudal distance from the injury epicenter is shown in Figures 3.26 and3.27.Left Ventral Horn NeuN+ DensityRight Ventral Horn NeuN+ DensityLeft Ventral Horn Average Minimum Principal StrainRight Ventral Horn Average Minimum Principal Strain−3 −2 −1 0 1 2 3050100150Distance from Epicenter (mm)NeuN+Density(#/mm2)−0.5−0.4−0.3−0.2−0.10(a) Cord 2−3 −2 −1 0 1 2 3050100150Distance from Epicenter (mm)−0.5−0.4−0.3−0.2−0.10MinimumPrincipalStrain(b) Cord 3−3 −2 −1 0 1 2 3050100150Distance from Epicenter (mm)NeuN+Density(#/mm2)−0.5−0.4−0.3−0.2−0.10(c) Cord 4−3 −2 −1 0 1 2 3050100150Distance from Epicenter (mm)−0.5−0.4−0.3−0.2−0.10MinimumPrincipalStrain(d) Cord 9Figure 3.26: NeuN-positive density data (left axis, red) and minimum principal strain data(right axis, blue) plotted against the distance from the injury epicenter for eachventral horn for spinal cords from animals 2, 3, 4 and 9.76Left Ventral Horn NeuN+ DensityRight Ventral Horn NeuN+ DensityLeft Ventral Horn Average Minimum Principal StrainRight Ventral Horn Average Minimum Principal Strain−3 −2 −1 0 1 2 3050100150Distance from Epicenter (mm)NeuN+Density(#/mm2)−0.5−0.4−0.3−0.2−0.10(a) Cord 10−3 −2 −1 0 1 2 3050100150Distance from Epicenter (mm)−0.5−0.4−0.3−0.2−0.10MinimumPrincipalStrain(b) Cord 11−3 −2 −1 0 1 2 3050100150Distance from Epicenter (mm)NeuN+Density(#/mm2)−0.5−0.4−0.3−0.2−0.10MinimumPrincipalStrain(c) Cord 12Figure 3.27: NeuN-positive density data (left axis, red) and minimum principal strain data(right axis, blue) plotted against the distance from the injury epicenter for eachventral horn for spinal cords from animals 10, 11 and 12.77Chapter 4DiscussionIn this section, I will discuss the important findings of the current research, as well as thelimitations of the current approach.4.1 Injury SimulationI developed a finite element model of contusion spinal cord injury in the rat cervical spinefor the purpose of investigating the mechanical properties of the grey and white matter. Thisresearch is the first attempt to investigate the material properties of the in vivo spinal cord.The model yields realistic force values, removes the need to model the tissues surrounding thespinal cord, and has the advantage of being compared to in vivo image data with which to assessthe model’s accuracy. This model enables the evaluation of the time independent mechanicalproperties of the spinal cord.4.1.1 Gross Mechanical Property FindingsFor spinal cords 4, 9, 10, 11 and 12, the model best fit the experimental data, both quantita-tively and qualitatively, when the elastic modulus of the white matter was greater than thatof grey matter by a factor of 2-3. For spinal cord 3, model data best fit the experimental dataquantitatively when the white matter was twice as stiff as the grey matter, however there wasrelatively little distinction between the various values of stiffness used. For spinal cord 2, themodel best fit when the data when the elastic modulus of the grey and white matter was thesame.The results of the current study suggest that white matter is stiffer than grey matter bya factor of approximately 2. The current findings agree with those of Kruse et al. [53] andMcCracken et al. [60], who used magnetic resonance elastography of in vivo human brain tissue,and found white matter to have greater shear modulus than grey matter by factors of 2.6 and1.5, respectively. The current findings also agree with those from Budday et al. [9], who used exvivo bovine brain tissue indentation, and found that white matter was stiffer than grey matterby a factor of 1.4. However, the current findings disagree with several results in the literature,including Ichihara et al. [45] and Koser et al. [51], who tested ex vivo bovine and mouse spinalcords, and found that grey matter was stiffer than white matter by factors of roughly 1.3-2.6and 1.9, respectively. Christ et al. tested ex vivo rat cerebellus tissue and found that greymatter was stiffer than white matter by a factor of 1.5 [20]. Finally, Ozawa et al. [77], found78no significant difference between the stiffness of the two tissues.It is difficult to parse the relevant details of the studies that have looked at the properties ofgrey and white matter, as there is variation in terms of the animal model used, the magnitudeof strains imposed on the cord, the methods used for determining the material properties andthe use of in vivo or ex vivo tissues. However, it is of note that all three of the in vivo studiesconducted (including the current one) found white matter to be stiffer than grey matter. Thefact that all in vivo studies found white matter to be stiffer than grey matter may highlight theimportance of time after death in the investigation of neural tissue material properties. Thecurrent study adds in vivo data of the rat spinal cord material properties at injurious strains,which is a useful piece of information for future work in rat models of SCI.4.1.2 Fine Mechanical Property FindingsAlthough there was a trend that modeling the white matter as stiffer than the grey matteryielded better approximations to the experimental data for most of the spinal cords tested,none of the FE deformed cords fit the experimental data perfectly.For tests in which the contusion was inflicted on the spinal cord in the mediolateral center,namely cords 9, 11 and 12, there was good qualitative and quantitative agreement between themodeled grey matter and the experimentally observed grey matter in terms of the comparativeshapes after deformation. Alternatively, for contusions inflicted on the spinal cord lateral tothe mediolateral center, namely cords 2, 3, 4 and 10, the predicted grey matter shapes looknotably different from the experimental shapes, and the quantitative agreement was generallyworse. In these cases, the experimentally observed grey matter shapes look as though they areshifted further from the point of contusion than the FE predicted shapes.The grey matter shape differences between symmetric and asymmetric contusions could bedue to one or more causes, including inadequate modeling of the spinal cord anatomy or thedisregard for potential heterogeneity or anisotropy in the cord. With regards to the potentiallyinadequate modeling of the spinal cord anatomy, one possible limitation was the inability toobserve the posterior median sulcus on the spinal cord. This feature of the spinal cord was notvisible on the MR images, so instead of a clear groove along the posterior portion of the spinalcord, the spinal cord appears to be smooth in this region, as shown in Figure 4.1.79(a) An illustration of the spinal cord (b) MR image used in the current studyFigure 4.1: An illustration of the spinal cord showing the posterior median sulcus [28](reprinted with copyright permission) against the absence of an observable sulcusin the MR images used to conduct the current study.The inability to model this anatomic feature may have resulted in force propagation pri-marily in the anteroposterior direction, while a more realistic representation may yield strainsthat propagate more so along the mediolateral direction across the cord, perhaps producing agrey matter shape shifted further from the location of the contusion point, similar to that seenexperimentally.Another possible reason for the discrepancy between shapes of the modeled and observedgrey matter is spinal cord heterogeneity and anisotropy. In the human spinal cord, axons in thewhite matter vary from less than 1 to 10 µm in diameter. Often, specific regions of the spinalcords contain axons that vary in size and density. For example, fiber density in the lateralfuniculus has been shown to be greater than that in the dorsal funiculus in human cadavers[23]. Furthermore, axons have been shown to vary in orientation in the grey matter. For thespecimens examined by Koser et al., the axons in the ventral horns of the grey matter tendedto have no directional bias, while those in the dorsal horns were preferentially oriented in thecraniocaudal direction. Also, Koser et al demonstrated that the orientation of axons has beenshown to correlate to stiffness as measured by atomic force microscopy. For indentation tests,in locations at which axons were oriented perpendicular to the direction of indentation, thegrey matter tissue had a higher stiffness compared to axons oriented randomly, which in turnhad a higher stiffness compared to axons oriented along the direction of indentation [51]. Thisdata makes it reasonable to suspect that ignoring the potential heterogeneity or anisotropywithin the grey and white matter may be a reason for the difference between the predicted andexperimentally observed grey matter shapes.804.1.3 Error AnalysisComparing the best fit material properties for the spinal cords using the original segmentationto those with the erroneous segmentation yields an understanding of the importance of thesegmentation accuracy. A uniform enlargement of only one pixel along the posterior surfaceof the experimentally deformed grey matter yielded notably different results for the best fitrelative stiffness of the grey and white matter. With the original segmented experimental greymatter shapes, the best fit relative stiffness values, Ewhite/Egrey, varied between 1.0 and 3.0.With the erroneously segmented experimental grey matter shapes, the new range of best fitrelative stiffness values varied between 0.5 and 1.5. The seemingly large difference between therelative stiffness values is likely a result of the low image resolution compared to the size ofthe rat spinal cord. The accuracy of the results of the current study depend on the accuratesegmentation of the MR images. It is important to note that the best fit relative stiffness valuesdecrease with this error analysis, as the process of extending the deformed grey matter in theposterior direction shifts the results toward the case in which grey matter becomes stiffer thanwhite matter. For the results with the initial segmentation, the segmented features have beenverified against MR images overlaid with the segmented surface meshes to reduce the error asmuch as possible.4.1.4 Comparison to HistologyThere was a weak, but statistically significant relationship between neuronal damage in theventral horns of the grey matter as measured by loss of NeuN-positive staining and strain,measured by both minimum and maximum principal strain. Minimum principal strain showeda slightly better relationship to loss of NeuN-positive staining (R2 = 0.115) as compared tomaximum principal strain (R2 = 0.064).The relationship between neuronal damage and strain was examined for each individualanimal in an attempt to better understand the relation between strain and tissue damage. Itwas found that for cords 2, 3, 4, 9 and 10, there was a positive correlation between magnitudeof the principal strain and the loss of NeuN+ staining, with R2 values ranging from 0.16 to0.31, with cords 2, 4 and 9 reaching statistical significance at α = 0.05. However, for cord 11,there was no relation between strain and loss of NeuN+ density, and for cord 12, there was anegative correlation between the strain magnitude and tissue damage.Finally, the minimum principal strain for each ventral horn was compared to the NeuN+density at various discrete distances from the injury epicenter to gain a better understanding ofthe spatial variation in strain and tissue damage. For each animal examined, the average strainin each ventral horn was highest at the injury epicenter, and slowly decreased with distance fromthe injury epicenter. For asymmetric contusions (cords 2, 3, 4 and 10), the magnitude of strainin the right ventral horn was notably greater than the strain magnitude in the left ventral horn.This result is to be expected, as each asymmetric contusion was inflicted to the right of themediolateral center of the cord, and the FE predicted grey matter shape displays low degrees81of deformation in the contralateral side. However, the experimentally observed grey mattershowed a relatively large degree of deformation in the contralateral side, and the histology datashowed similar injury patterns for both the left and right ventral horns. This data indicatethat the current model likely underestimates the strains in the grey matter contralateral tothe contusion site for asymmetric injuries. For two of the contusions in which the cord wasimpacted on the mediolateral center (9 and 12), the strains in the left and right ventral hornswere close in magnitude, as were the neuron density patterns. Cord 11 was unique in that itreceived a contusion that was approximately central on the cord, but displayed different strainmagnitudes between the left and right ventral horns, while maintaining similar injury patternsin both ventral horns.The loss of NeuN+ staining showed the greatest tissue damage occurred at the site of theinjury epicenter for cords 2, 3, 4, 9 and 10, which were each examples that showed a positivecorrelation between strain and damage in the individual animal linear regression analyses.Cords 11 and 12 displayed local minima in tissue damage at the injury epicenter. Furthermore,each graph of tissue damage as a function of distance from the injury epicenter showed largevariations. None of the graphs analyzed showed clear trends of smoothly decreasing tissuedamage with increasing distance from the injury epicenter, as would be expected. These trendssuggest that considerable amount of the variability in the relationship between strain and tissuedamage may be attributable to the method of measuring tissue damage. One possible reason forthe variability in tissue damage reported by NeuN staining is that NeuN expression in the greymatter following acute periods of ischemia after tSCI has not been well-classified [4]. Previousinvestigations, such as that performed by Sjovold et al., reported NeuN-expression 3 hourspost-injury, which may have provided time for more complete expression of tissue damage as aresult of secondary processes [84].4.2 LimitationsThe current study had limitations that could be improved upon by further iterations and futurecorrections.4.2.1 Image QualityThe primary limitation of the current research is the image resolution relative to the size of theanatomical features being imaged. Although the absolute voxel size of the images taken wassmall (140×140×500 µm), the undeformed spinal cord of the rat is only 2-3 mm in diameter.This relatively coarse image resolution may have caused small segmentation inaccuracies whenoutlining the grey and white matter, which may in turn have led to errors in determining therelative stiffness of the grey and white matter. In order to mitigate this effect, the segmentedmesh shapes were verified directly against the MR images, however, this verification was quali-tative in nature, and therefore exhibits some subjectivity. The limitations due to image quality82may have also been responsible for hiding certain anatomic features. The posterior mediansulcus and anterior median fissure were not visible, possibly due to partial volume effects. Thislimitation may have led to mechanical differences between the FE simulation of the injury andthe experimental results.4.2.2 Material Model LimitationsAnother notable limitation of the current methodology is the use of a time independent materialmodel. The experimental data on which the current study is based was collected at two timeintervals: before injury and several minutes after the injury was inflicted and held. The factthat data only exists at these time points means that no information can be gathered from thetransient period of the injury. The analysis is therefore limited in that it ignores the viscoelasticresponse of the cord, and only derives information based on the elastically deformable natureof the cord. As a result, future investigations into the viscoelastic response of the in vivo cordare necessary.A linear elastic model was used, despite the fact that for ex vivo testing of the spinal cord,Ogden hyperelastic material models tend to better capture the spinal cord material properties[87]. The use of a linear elastic material was chosen for a number of reasons. Firstly, only onecontusion test was conducted for each cord, each at approximately the same compression depth.Hyperelastic models are useful for describing the non-linear force-displacement behaviour , butoffer less of an advantage when a static solution is used for a single specimen. Secondly, linearelastic materials deform in an easily understood manner, and as a result it is easier to comparethe deformation of the spinal cord at different values of relative stiffness of the white mattercompared to the grey matter. Furthermore, linear elastic models have fewer material constantsthan complex hyperelastic models. For inverse FE tests, having fewer material constants toadjust significantly reduces the difficulty of navigating the search space of all possible materialconstant combinations, enabling the material property investigation to be completed in a timelyfashion. Finally, the linear elastic material model uses small deformation assumptions to modeldeformations that are reasonably large. The problem of unrealistic volume growth with largerotations is resolved with the use of a co-rotated formulation for linear elasticity [67], however,the stress-strain relationship is still dictated by small deformation assumptions, which do nottake into account the change in element volume with deformation. This assumption may havea slight effect on the best fit values of Ewhite/Egrey.83Chapter 5ConclusionIn this final chapter, I state the degree to which I have achieved my objectives, the contributionsI have made, and my recommendations for future work.5.1 ConclusionsMy research project met my proposed objectives in the following ways:1. Objective: Create an FE model of the rat cervical spinal cord to recreate previouslyconducted contusion-type injuries.I created a novel FE model of rat cervical spinal cord injury based on the geometry ofthe grey and white matter of both undeformed and deformed spinal cord states. Thematerial properties of the grey and white matter are adjustable. For the current study. Imodeled both the grey and white matter as linear elastic, and varied the stiffness of thewhite matter relative to the grey matter. The injury is simulated by adjusting the nodalforce values around the surface of the undeformed cord until the outer external surface ofthe spinal cord lies flush with the surface of the deformed spinal cord.2. Objective: Use an inverse FE approach to iteratively vary the mechanical propertiesof the grey and white matter, while qualitatively and quantitatively observing whichproperties best match the experimental data.I used the FE model created to conduct an inverse FE investigation of the relative stiffnessof the grey and white matter in the rat spinal cord. I varied the relative stiffness of thewhite matter as compared to the grey matter between values of 0.5 to 4.0, as governedby the limits of possible deformations. The best material property combination for eachcord was determined by assessing the shape of the FE predicted grey matter deformationcompared to the deformed grey matter found experimentally. The FE and experimentallydeformed grey matter shapes were assessed for similarity both qualitatively and quantita-tively. For five out of seven spinal cords examined, the best qualitative and quantitativefit was found when the white matter was 2-3× stiffer than the grey matter. For one spinalcord, the best qualitative and quantitative fit was found when there was no difference instiffness, and one cord did not seem to fit the experimental grey matter shape well re-gardless of the material properties used. I also found that contusions inflicted upon themediolateral center of the cord fit well to the experimental data, while contusions inflicted84away from the mediolateral center of the cord were not well predicted by the model, sug-gesting that current modeling conditions are unable to capture the true deformation ofthe spinal cord, perhaps due to heterogeneity, anisotropy, or inaccuracies in anatomicrepresentation.3. Secondary Objective: Compare the strain results from the finite element study to thehistological data previously collected on the same animals.I used the FE model in conjunction with a previously conducted histological analysisto investigate the relationship between strain, as measured by maximum and minimumprincipal strain, and tissue damage, as measured by loss of NeuN-positive neuron den-sity in the grey matter. I found a weak, but statistically significant correlation betweenNeuN-positive neuron density and both maximum and minimum principal strain. Thestrain patterns indicated a need to further refine the FE model, as for the ventral hornscontralateral to the contusion site of asymmetric injuries, the strains appeared to be lowerthan expected to cause the amount of tissue damage observed.5.2 ContributionsThis research is the first investigation of the material properties of the in vivo spinal cord usingexperimental image data to verify the selected material properties.The current study is also the first to use an iterative closest point approach to recreateinjuries in FE models of the spinal cord. This approach is useful for investigating time indepen-dent material properties of the spinal cord, but functions to replicate the deformation boundaryconditions of the injury with great accuracy.My research adds to the conflicting information about the relative stiffnesses of grey andwhite matter. Although there is data that suggests white matter is stiffer than grey matter andvice versa, the only past in vivo studies suggest that white matter is stiffer. My research givesfurther support for white matter being stiffer than grey matter in vivo, and calls into questionprevious data for the mechanical properties of the spinal cord.My research is the first to examine the in vivo material properties of the rat spinal cord,which is of particular use to the researchers at ICORD, who primarily use rat models to inves-tigate tSCI.Finally, this study is the first to concretely illustrate the shortcomings of current FE modelsin their ability to accurately replicate internal deformation of the spinal cord.5.3 Recommendations for Future WorkThe following are recommendations about future work that may be helpful to better modelspinal cord injury in the future.85• The current experiment, that is the spinal cord deformation under MR observation cou-pled with the inverse FE investigation, could be rerun using data from the ex vivo spinalcord of a larger animal to reduce the potential effects of segmentation issues. Performingthe experiment on a large, ex vivo spinal cord would increase the resolution of the MRimages, reveal the entire geometry of the cord and reduce the complexity of the experi-ment. The limitation of this approach is that the ex vivo mechanical properties would bethe ones being measured, which may be different than the in vivo mechanical properties,depending on the amount of time between death and testing.• The current FE experiments could be rerun with different estimates of the geometry ofthe posterior median sulcus to determine the effect of modeling this anatomic feature onthe predicted deformation of the spinal cord. Geometric estimates could be taken fromanatomic literature or direct measurements of rats matched for spinal cord size.• Finally, simple ex vivo material property testing could be performed on the cord to de-termine material properties of different segments. Although the mechanical properties ofthe spinal cord are known to change with time, emphasis could be placed on reducing theamount of time between death and testing or constructing a rig to enable direct materialtesting on anesthetized animals.5.4 Concluding StatementThe current research calls into question the assumption of spinal cord homogeneity made bymost FE studies of the spinal cord. If we want FE modeling to be a useful tool to investigatethe histological damage following spinal cord injury, it is necessary to improve upon our under-standing of the material properties of the spinal cord. Improvement in our understanding ofthe material properties of the spinal cord will enable a better understanding of the link betweenthe complex deformation of the spinal cord and the resulting pathophysiologic changes to thetissues in the spinal cord, and may be useful for the development of an injury threshold forspecific tissues. 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An inverse finite element approach to modeling the rat cervical spinal cord : for use in determining the mechanical properties of the grey and white matter
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