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    Elena Olariu, M.Sc.

    A thesis submitted to

    the Faculty of Graduate Studies and Research

    in partial fulfillment of

    the requirements for the degree of

    Doctor of Philosophy

    Department of Physics

    Carleton University

    Ottawa, Ontario, Canada

    September, 2009


    2009, Elena Olariu

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  • Abstract

    Traditionally, diffusion in white matter has been interpreted in terms of restricted

    diffusion models with two well defined compartments: the intracellular space and the

    extracellular space. With these models the water molecules are normally considered to diffuse

    as if they were in pure water until they meet a barrier, such as the cell membrane, which

    simply reflects the particle back in a very classical sense. However, these models have not

    been successful in accounting for all of the observed results; in particular, the observed signal

    fractions are inconsistent with the predictions of these models.

    The primary goal of this thesis was to propose a novel explanation for the observed

    diffusion properties of tissue water in human white matter in vivo by considering the

    electrostatic interactions of water molecules with each other and with their surroundings.

    These interactions lead to the formation of hydration layers around macromolecules and along

    membrane surfaces and cause the water in the cell to behave more like a gel than a liquid.

    Incorporation of these considerations into the analysis has allowed developing novel

    explanations for: 1) why the measured fast diffusion component is smaller than the diffusion


  • coefficient of free water, 2) why it is anisotropic, 3) the observed signal fraction values and 4)

    the anisotropy of the signal fractions.

    In this thesis, carefully measured sets of diffusion decays for the splenium and the genu

    of the corpus callosum of two healthy subjects in vivo are presented along with supporting T2

    relaxation time measurements. The diffusion decays were measured as a function of the angle

    between the diffusion direction and the long axis of the axon. All of these data are

    successfully interpreted in terms of the proposed hydration layer model.

    i i i

  • Acknowledgements

    It is a pleasure to thank many people who made this thesis possible.

    Foremost, I would like to thank my supervisor, Dr. Ian Cameron, who shared with me a

    lot of his expertise and research insight. Without his guidance, support, and encouragement

    this thesis would not have been possible. I also thank my committee members, Professors

    Boguslaw Jarosz and Richard Hodgson for their helpful suggestions and comments during the

    entire PhD program.

    It is difficult to overstate my gratitude to Dr. Dave Rogers, Dr. Paul Johns, Dr. Peter

    Raaphorst, Dr. Malcolm McEwen and Dr. David Wilkins for teaching me the clinical aspects

    of medical physics. The road to my graduate degree has been long and winding, so I would

    also like to thank some people from the early days: Dr. Giles Santyr, Dr. Dean Karlen and Dr.

    Hans Mes.

    I am indebted to my lab colleagues for providing a stimulating and fun environment in

    which to learn and grow: Claire Foottit, Marzieh Nezamzadeh, Arturo Cardenas-Bianco and

    Greg Cron. Huge thanks to Sorina Truica and Mihai Gherase for their valuable friendship.


  • Thanks to many other people for Carleton University and the Ottawa General Hospital

    who have helped me in some manner during my studies.

    Thanks to my beloved daughter Karina. Her love always gives me power!

    Lastly, and most importantly, I wish to thank my parents. They have always supported

    and encouraged me to do my best in all matters of life. To them I dedicate this thesis.

    And thanks to you all, who read and criticize my thesis!


  • Contents

    1 Introduction 1

    1.1 Overview of the MR Diffusion Literature 4

    1.1.1 The Apparent Diffusion Coefficient 4

    1.1.2 Biexponential Diffusion 7

    1.1.3 Intracellular and Extracellular Diffusion 8

    1.2 Diffusion in Hydration Layers 12

    1.3 Thesis Summary and Outline 14

    2 Theory 18

    2.1 Principle of Nuclear Magnetic Resonance 18

    2.1.1 Classical Description 18

    2.1.2 Quantum Mechanical Description 23

    2.1.3 Statistical Distribution of Spin States 25

    2.2 Bloch's Equations and Relaxation 26


  • 2.3 Spatial Encoding of the NMR Signal 29

    2.4 Diffusion 29

    2.4.1 Diffusion Theory 30

    2.4.2 Apparent Diffusion in Biological Tissues 32

    2.4.3 Experimental Methods for Diffusion MR 33

    2.5 The Physics ofT2 Relaxation Mechanism 38

    2.5.1 Basics of Dipole-Dipole Interaction 39

    2.5.2 Relaxation and Molecular Motion 41

    2.5.3 NMR Studies of Water in Biological Systems 44

    2.6 Diffusion Tensor Imaging 45

    2.7 Human Brain 48

    2.7.1 Microstructural Organization 48

    2.7.2 Macrostructural Organization 50

    2.8 Tissue Water Model 52

    2.8.1 Hydration Layers 54

    3 Noise Correction 61

    3.1 Introduction 61

    3.2 PDF for MR Magnitude Image Pixel Intensities 63

    3.2.1 The Rician Distribution 63

    3.2.2 The Moments of the Rician Distribution 66

    3.3 The Bias of the Rician Distribution 68

    3.3.1 Review of Rician Bias Correction Techniques 68


  • 3.3.2 Koay and Basser Rician Bias Correction 70

    3.3.3 Improved Rician Bias Correction Technique 72

    3.3.4 Validation of the Linear RB Correction Technique using

    Simulations 76 Technique Validation Using JUR and ag. 76 Convergence Criteria for the Linear RB Correction

    Technique 80 Linear RB Correction Using Nearest Neighbour

    Averaging 85

    3.4 Rician Bias Correction Technique for Magnitude Averaged Signals 88

    3.4.1 The PDF of Magnitude Averaged Signals 89

    3.5 RB Correction Sensitivity to Uncertainties in og 93

    3.6 Validation of the Rician Bias Correction Using a Water Phantom 94

    3.7 Discussion of the RB Correction 98

    3.8 Conclusions 99

    4 Experimental Methods 100

    4.1 MR Experimental Protocol 100

    4.1.1 T2 Measurements 100

    4.1.2 Diffusion Measurements 101

    4.1.3 Subjects and Slice Location 102

    4.2 Decay Curve Analysis 105

    4.3 Water Phantom Measurements 106


  • 5 Results 108

    5.1 T2 Measurements 108

    5.2 Diffusion Measurements 116

    5.2.1 Diffusion Parallel to the Axons in the SCC and the GCC 118

    5.2.2 Diffusion Perpendicular to the Axons in the SCC and the GCC 122

    5.2.3 Diffusion Orientation Dependence in the SCC and the GCC 126

    6 Discussions 133

    6.1 T2 Measurements 137

    6.1.1 T2 Decay Curve Analysis 139

    6.1.2 Interpretation of T2 Components 140 The Short T2 Component 142 The Long T2 Component 144 The Intermediate T2 Component 145

    6.1.3 Hydration Layer and Dis