Molecular diffusion is an inherent feature of all fluid systems. The processes and interactions that characterize these systems are in some way dependent upon the mobility of the component molecules. Pulsed field-gradient spin-echo nuclear magnetic resonance (PGSE NMR) is a powerful tool for the study of molecular diffusion; for heterogeneous systems, such as those typically found in biology, this technique is unsurpassed in the diversity of systems that yield to its probing. The aim of the work presented in this thesis was to use an integrated NMR-based approach, in conjunction with computer modeling, for the study of molecular diffusion in compartmentalized and multicomponent biological systems. Erythrocyte suspensions provided an ideal experimental system for the study of compartmentalized diffusion in cells. Water exchanges rapidly between the intra- and extracellular regions and, as the major constituent of the cell, provides a strong and predominant proton NMR signal. In addition, the cells are known to align in the strong static magnetic field of the spectrometer. As a consequence of these two properties, the signal intensity from a suitably designed series of PGSE NMR experiments exhibits a series of maxima and minima when graphed as a function of the magnitude of the spatial wave number vector q. The apparently periodic phenomenon is mathematically analogous to optical diffraction and interference and is referred to here as diffusion-coherence. It is the characterization of this phenomenon, with the aid of computer-based models, which was the focus of a major section of the work described herein. Two quite distinct molecular systems formed the basis of the work in which I investigated diffusion in multicomponent systems. Both systems involved molecules that undergo self-association such that at equilibrium a population distribution of different oligomeric species is present. The first of these was tropoelastin, the monomeric subunit of elastin, which under certain conditions aggregates to form a coacervate. The second system was N-methylacetamide (NMA) which also undergoes extensive self-association. NMA oligomers have previously been studied as peptide analogues due to the presence in the monomer of a peptide linkage. In this work the aim was to use PGSE NMR diffusion measurements, in a manner that is in many ways analogous to analytical ultracentrifugation, to obtain estimates of hydrodynamic and thermodynamic parameters. Computer modeling was also used extensively in this section of work for the interpretation of the experimental data.

Molecular diffusion is an inherent feature of all fluid systems. The processes and interactions that characterize these systems are in some way dependent upon the mobility of the component molecules. Pulsed field-gradient spin-echo nuclear magnetic resonance (PGSE NMR) is a powerful tool for the study of molecular diffusion; for heterogeneous systems, such as those typically found in biology, this technique is unsurpassed in the diversity of systems that yield to its probing. The aim of the work presented in this thesis was to use an integrated NMR-based approach, in conjunction with computer modeling, for the study of molecular diffusion in compartmentalized and multicomponent biological systems. Erythrocyte suspensions provided an ideal experimental system for the study of compartmentalized diffusion in cells. Water exchanges rapidly between the intra- and extracellular regions and, as the major constituent of the cell, provides a strong and predominant proton NMR signal. In addition, the cells are known to align in the strong static magnetic field of the spectrometer. As a consequence of these two properties, the signal intensity from a suitably designed series of PGSE NMR experiments exhibits a series of maxima and minima when graphed as a function of the magnitude of the spatial wave number vector q. The apparently periodic phenomenon is mathematically analogous to optical diffraction and interference and is referred to here as diffusion-coherence. It is the characterization of this phenomenon, with the aid of computer-based models, which was the focus of a major section of the work described herein. Two quite distinct molecular systems formed the basis of the work in which I investigated diffusion in multicomponent systems. Both systems involved molecules that undergo self-association such that at equilibrium a population distribution of different oligomeric species is present. The first of these was tropoelastin, the monomeric subunit of elastin, which under certain conditions aggregates to form a coacervate. The second system was N-methylacetamide (NMA) which also undergoes extensive self-association. NMA oligomers have previously been studied as peptide analogues due to the presence in the monomer of a peptide linkage. In this work the aim was to use PGSE NMR diffusion measurements, in a manner that is in many ways analogous to analytical ultracentrifugation, to obtain estimates of hydrodynamic and thermodynamic parameters. Computer modeling was also used extensively in this section of work for the interpretation of the experimental data.

Doctor of Philosophy / Physics / Christopher M. Sorensen / We present light scattering studies of irregularly shaped particles which significantly affect the climate. We built and calibrated our apparatus which was able to measure all six independent scattering matrix elements. Our apparatus detects light from 0.32° to 157° simultaneously. We studied all six scattering matrix elements of irregularly shaped Arizona Road Dust which behave differently than those of spheres. We strongly focused on the most important scattering matrix element – the phase function, scattered intensity vs. the scattering angle, which we applied Q-space analysis to. Q-space analysis involves plotting the scattering intensity vs. the magnitude of the scattering wave vector q or qR with R the radius of a particle, on a double logarithmic scale. We measured and studied the phase functions of Al₂O₃ abrasives; compared the scattering from the abrasives with the scattering of spheres. To generalize the study, we collected a large amount of experimental and theoretical data from our group and others and applied Q-space analysis. They all displayed a common scattering pattern. The power law exponents showed a quasi-universal functionality with the internal coupling parameter ρ'. In situ studies of the soot fractal aggregates produced from a burner were also conducted. A power law exponent -1.85 is seen to imply the aggregates have fractal dimension of D[subscript f]=1.85. The overall work presented shows Q-space analysis uncovers patterns common to all particles: a q-independent forward scattering regime is followed by a Guinier regime, a power law regime, and sometimes an enhanced back scattering regime. The description of the patterns applies to spheres as well, except the power law regime has more than a single power law. These simple patterns give a unified description for all particle shapes. Moreover, the power law exponents have a quasi-universal functionality with ρ' for non-fractal aggregates. The absolute value of the exponents start from 4 when ρ' is small. As ρ' increases, the exponents decrease until the trend levels off at ρ'≳10 where the exponents reach a constant 1.75±0.25. All the non-fractal particles fall on the same trend regardless of the detail of their structure.

When scattering intensity is plotted against the dimensionless quantity qR, where q is the magnitude of the scattering wave vector and R is the radius of the particle, in log-log scale the scattering curve shows a power-law structure which defines characteristic crossovers. This work reveals some new relationships between the power-law structure and the particle properties. In this work, computer simulation results from T-matrix, Mie theory, and discrete dipole approximation methods are used to study the far field intensity and the internal field of the particles. Scattering by both weakly and strongly refractive particles are studied. For weakly refractive randomly oriented spheroidal particles, how the phasor cancellation-based tip volume method can be applied to predict the scattering envelope is demonstrated. The relationship between backscattering enhancement and the curvature of the weakly refractive particles is explained. In strongly-refractive particles when the phase shift parameter is high, regions with higher field amplitudes start to appear. These regions are recognized as the hot spot regions. In this work, a proper definition is given to the hot spot region. The relationships between the hot spot region and the power-law structure, between the hot spot region and the particle morphology, and between the power-law structure and the particle morphology are extensively studied for scattering by spherical particles. A new semi-quantitative phasor analysis method is introduced, and the new method is used with color-coded phasor plots to identify how different regions of the particle contribute to the scattering pattern to get an insight into the physics behind the scattering. How different regions of the particle contribute to the second crossover (SC) and the backscattering enhancement is presented. Relationships between the SC, particle size, and relative refractive index of the particle are derived. It was identified that the scattering angle at the SC depends only on the relative refractive index of the particle. How the findings of this work can be applied to solve the inverse electromagnetic scattering problem for a single non-absorbing spherical particle is also discussed.

Cette thèse est consacrée à l'élaboration de nouvelles méthodes d'acquisition et de traitement de données en IRM de diffusion (IRMd) afin de caractériser la diffusion des molécules d'eau dans les fibres de matière blanche à l'échelle d'un voxel. Plus particulièrement, nous travaillons sur un moyen de reconstruction précis de l'Ensemble Average Propagator (EAP), qui représente la fonction de probabilité de diffusion des molécules d'eau. Plusieurs modèles de diffusion tels que le tenseur de diffusion ou la fonction de distribution d'orientation sont très utilisés dans la communauté de l'IRMd afin de quantifier la diffusion des molécules d'eau dans le cerveau. Ces modèles sont des représentations partielles de l'EAP et ont été développés en raison du petit nombre de mesures nécessaires à leurs estimations. Cependant, il est important de pouvoir reconstruire précisément l'EAP afin d'acquérir une meilleure compréhension des mécanismes du cerveau et d'améliorer le diagnostique des troubles neurologiques. Une estimation correcte de l'EAP nécessite l'acquisition de nombreuses images de diffusion sensibilisées à des orientations différentes dans le q-space. Ceci rend son estimation trop longue pour être utilisée dans la plupart des scanners cliniques. Dans cette thèse, nous utilisons des techniques de reconstruction parcimonieuses et en particulier la technique connue sous le nom de Compressive Sensing (CS) afin d’accélérer le calcul de l'EAP. Les multiples aspects de la théorie du CS et de son application à l'IRMd sont présentés dans cette thèse. / This thesis is dedicated to the development of new acquisition and processing methods in diffusion MRI (dMRI) to characterize the diffusion of water molecules in white matter fiber bundles at the scale of a voxel. In particular, we focus our attention on the accurate recovery of the Ensemble Average Propagator (EAP), which represents the full 3D displacement of water molecule diffusion. Diffusion models such that the Diffusion Tensor or the Orientation Distribution Function (ODF) are largely used in the dMRI community in order to quantify water molecule diffusion. These models are partial EAP representations and have been developed due to the small number of measurement required for their estimations. It is thus of utmost importance to be able to accurately compute the EAP and order to acquire a better understanding of the brain mechanisms and to improve the diagnosis of neurological disorders. Estimating the full 3D EAP requires the acquisition of many diffusion images sensitized todifferent orientations in the q-space, which render the estimation of the EAP impossible in most of the clinical dMRI scanner. A surge of interest has been seen in order to decrease this time for acquisition. Some works focus on the development of new and efficient acquisition sequences. In this thesis, we use sparse coding techniques, and in particular Compressive Sensing (CS) to accelerate the computation of the EAP. Multiple aspects of the CS theory and its application to dMRI are presented in this thesis.

Cette thèse est dédiée à améliorer la compréhension neuro-scientifique à l'aide d'imagerie par résonance magnétique de diffusion (IRMd). Nous nous concentrons sur la modélisation du signal de diffusion et l'estimation par IRMd des biomarqueurs liés à la microstructure, appelé «Microstructure Imaging». Cette thèse est organisée en trois parties. Dans partie I nous commençons par la base de l'IRMd et un aperçu de l'anisotropie en diffusion. Puis nous examinons la plupart des modèles de microstructure utilisant PGSE, en mettant l'accent sur leurs hypothèses et limites, suivi par une validation par l'histologie de la moelle épinière de leur estimation. La partie II présente nos contributions à l'imagerie en 3D et à l’estimation de microstructure. Nous proposons une régularisation laplacienne de la base fonctionnelle MAP, ce qui nous permet d'estimer de façon robuste les indices d'espace q liés au tissu. Nous appliquons cette approche aux données du Human Connectome Project, où nous l'utilisons comme prétraitement pour d'autres modèles de microstructure. Enfin, nous comparons les biomarqueurs dans une étude ex-vivo de rats Alzheimer à différents âges. La partie III présente nos contributions au représentation de l’espace qt - variant sur l'espace q 3D et le temps de diffusion. Nous présentons une approche initiale qui se concentre sur l'estimation du diamètre de l'axone depuis l'espace qt. Nous terminons avec notre approche finale, où nous proposons une nouvelle base fonctionnelle régularisée pour représenter de façon robuste le signal qt, appelé qt-IRMd. Ce qui permet l'estimation des indices d’espace q dépendants du temps, quantifiant la dépendance temporelle du signal IRMd. / This thesis is dedicated to furthering neuroscientific understanding of the human brain using diffusion-sensitized Magnetic Resonance Imaging (dMRI). Within dMRI, we focus on the estimation and interpretation of microstructure-related markers, often referred to as ``Microstructure Imaging''. This thesis is organized in three parts. Part I focuses on understanding the state-of-the-art in Microstructure Imaging. We start with the basic of diffusion MRI and a brief overview of diffusion anisotropy. We then review and compare most state-of-the-art microstructure models in PGSE-based Microstructure Imaging, emphasizing model assumptions and limitations, as well as validating them using spinal cord data with registered ground truth histology. In Part II we present our contributions to 3D q-space imaging and microstructure recovery. We propose closed-form Laplacian regularization for the recent MAP functional basis, allowing robust estimation of tissue-related q-space indices. We also apply this approach to Human Connectome Project data, where we use it as a preprocessing for other microstructure models. Finally, we compare tissue biomarkers in a ex-vivo study of Alzheimer rats at different ages. In Part III, we present our contributions to representing the qt-space - varying over 3D q-space and diffusion time. We present an initial approach that focuses on 3D axon diameter estimation from the qt-space. We end with our final approach, where we propose a novel, regularized functional basis to represent the qt-signal, which we call qt-dMRI. Our approach allows for the estimation of time-dependent q-space indices, which quantify the time-dependence of the diffusion signal.

Diffusion spectrum imaging is a type of diffusion magnetic resonance imaging, capable of capturing very complex tissue structures, but requiring a very large amount of samples in q-space and therefore time.  The purpose of this project was to create and evaluate a new sampling scheme in q-space for diffusion MRI, trying to recreate the ensemble averaged propagator (EAP) with fewer samples without significant loss of quality. The sampling scheme was created by greedily selecting the measurements contributing with the most mutual information. The EAP was then recreated using the sampling scheme and interpolation. The mutual information was approximated using the kernel from a Gaussian process machine learning model.  The project showed limited but promising results on synthetic data, but was highly restricted by the amount of available computational power. Having to resolve to using a lower resolution mesh when calculating the optimal sampling scheme significantly reduced the overall performance.