Multi-Dimensional Nuclear Magnetic Resonance Methods in the Inhomogeneous Magnetic Field

This thesis introduces new NMR techniques which use the inhomogeneous internal magnetic fields present in the pore space of a porous medium exposed to an external magnetic field to obtain information about the pore size and heterogeneities of the the sample. Typically internal field inhomogeneities are regarded as unwanted due to their effect on various material properties such as relaxation and diffusion. However, in the experiments presented here, we choose samples specifically for their inhomogeneous internal fields and use multi-dimensional NMR methods and simulations to obtain our pore space and heterogeneity information. We first describe software developed to specifically simulate the internal magnetic field and diffusion through the pore space of a simple sphere pack system. This software generates a sphere pack and calculates the internal magnetic field generated by z-aligned magnetic dipoles placed at the center of each sphere. The internal magnetic field gradient is also calculated in the pore space. From there, a random walk method is developed and a realistic reflection off a sphere is introduced. We work through the development of this software and the mathematics behind the algorithms used. This simulation is used in all subsequent experimental chapters. We then use a two-dimensional exchange experiment to separate the susceptibility induced line broadening with the broadening caused by diffusion through the inhomogeneous field. We observe off-diagonal line broadening as the mixing time increases. We attempt to quantify this off-diagonal growth by selecting points on either side of the off-diagonal maximum and plotting their average as a function of mixing time. A biexponential fit to the average intensities with respect to mixing time results in a characteristic time and from that a characteristic length as a fraction of bead diameter. This experiment is simulated and a biexponential growth is also observed in the simulated off-diagonal with characteristic lengths comparable to experiment. To obtain a correlation length directly from experiment and not deduce one from a characteristic time, we add a spatial dimension to our exchange experiment in the form of a propagator dimension. This dimension allows us to select 2D spectra based on their Z-displacement. We observe off-diagonal growth due to both an increase in Z-displacement and an increase in mixing time. We move away from the biexponential fit and move to a relationship based on mixing time, effective diffusion, and Z-displacement to directly calculate a characteristic length. We see these same traits in the simulated data which agrees well with experiment. Lastly, we move away from exchange experiments and move to correlating the transverse relaxation time with the internal field offset. We find that there is correlation at large magnetic field offsets and small T2 times which appear to be indicative of sample heterogeneities. To confirm this we use a highly heterogeneous rock core sample which increases the correlations seen at the previous offsets and times. This experiment is more qualitative than the previous two as we do not have a concrete value for the heterogeneity of our samples. The simulation used throughout the thesis, while showing a definite correlation between field offset and T2 relaxation, is unable to accurately simulate the experiment and requires more development

Orientation-Dispersed Apparent Axon Diameter via Multi-Stage Spherical Mean Optimization

The estimation of the apparent axon diameter (AAD) via diffusion MRI is affected by the incoherent alignment of single axons around its axon bundle direction, also known as orientational dispersion. The simultaneous estimation of AAD and dispersion is challenging and requires the optimization of many parameters at the same time. We propose to reduce the complexity of the estimation with an multi-stage approach, inspired to alternate convex search, that separates the estimation problem into simpler ones, thus avoiding the estimation of all the relevant model parameters at once. The method is composed of three optimization stages that are iterated, where we separately estimate the volume fractions, diffusivities, dispersion, and mean AAD, using a Cylinder and Zeppelin model. First, we use multi-shell data to estimate the undispersed axon micro-environment’s signal fractions and diffusivities using the spherical mean technique; then, to account for dispersion, we use the obtained micro-environment parameters to estimate a Watson axon orientation distribution; finally, we use data acquired perpendicularly to the axon bundle direction to estimate the mean AAD and updated signal fractions, while fixing the previously estimated diffusivity and dispersion parameters. We use the estimated mean AAD to initiate the following iteration. We show that our approach converges to good estimates while being more efficient than optimizing all model parameters at once. We apply our method to ex-vivo spinal cord data, showing that including dispersion effects results in mean apparent axon diameter estimates that are closer to their measured histological values

Multi-Dimensional Nuclear Magnetic Resonance Methods in the Inhomogeneous Magnetic Field

This thesis introduces new NMR techniques which use the inhomogeneous internal magnetic fields present in the pore space of a porous medium exposed to an external magnetic field to obtain information about the pore size and heterogeneities of the the sample. Typically internal field inhomogeneities are regarded as unwanted due to their effect on various material properties such as relaxation and diffusion. However, in the experiments presented here, we choose samples specifically for their inhomogeneous internal fields and use multi-dimensional NMR methods and simulations to obtain our pore space and heterogeneity information. We first describe software developed to specifically simulate the internal magnetic field and diffusion through the pore space of a simple sphere pack system. This software generates a sphere pack and calculates the internal magnetic field generated by z-aligned magnetic dipoles placed at the center of each sphere. The internal magnetic field gradient is also calculated in the pore space. From there, a random walk method is developed and a realistic reflection off a sphere is introduced. We work through the development of this software and the mathematics behind the algorithms used. This simulation is used in all subsequent experimental chapters. We then use a two-dimensional exchange experiment to separate the susceptibility induced line broadening with the broadening caused by diffusion through the inhomogeneous field. We observe off-diagonal line broadening as the mixing time increases. We attempt to quantify this off-diagonal growth by selecting points on either side of the off-diagonal maximum and plotting their average as a function of mixing time. A biexponential fit to the average intensities with respect to mixing time results in a characteristic time and from that a characteristic length as a fraction of bead diameter. This experiment is simulated and a biexponential growth is also observed in the simulated off-diagonal with characteristic lengths comparable to experiment. To obtain a correlation length directly from experiment and not deduce one from a characteristic time, we add a spatial dimension to our exchange experiment in the form of a propagator dimension. This dimension allows us to select 2D spectra based on their Z-displacement. We observe off-diagonal growth due to both an increase in Z-displacement and an increase in mixing time. We move away from the biexponential fit and move to a relationship based on mixing time, effective diffusion, and Z-displacement to directly calculate a characteristic length. We see these same traits in the simulated data which agrees well with experiment. Lastly, we move away from exchange experiments and move to correlating the transverse relaxation time with the internal field offset. We find that there is correlation at large magnetic field offsets and small T2 times which appear to be indicative of sample heterogeneities. To confirm this we use a highly heterogeneous rock core sample which increases the correlations seen at the previous offsets and times. This experiment is more qualitative than the previous two as we do not have a concrete value for the heterogeneity of our samples. The simulation used throughout the thesis, while showing a definite correlation between field offset and T2 relaxation, is unable to accurately simulate the experiment and requires more development.</p