Advanced numerical modelling of NMR diffusion experiments

Abstract

Pulsed gradient spin-echo nuclear magnetic resonance (PGSE NMR) diffusion measurements provide a powerful technique for measuring the translational motion of molecules. PGSE measurements can be made with a minimum of sample preparation, applied to a large variety of substances, and used to either acquire diffusion coefficients from multiple species in a single sample or isolate a single species from a complex mixture. Consequently, PGSE NMR has tremendous utility in a large variety of disciplines, and is capable of elucidating information on chemical structure, kinetics, and binding in a non-invasive manner. It has particular utility in probing the structure of microscale porous media. By using the diffusion of fluid molecules in the void space of porous materials as a molecular probe, PGSE NMR experiments can reveal detailed information on the structure, size and internal properties of porous systems on a variety of length scales. Such experiments are non-destructive to the porous material, allowing the characterisation of materials in vivo as well as in vitro, meaning they are of benefit to medical diagnostics and treatment as they can safely and accurately probe the properties of biological structures in the human body. Such experiments require mathematical models to describe the relationship between the measured spin-echo signal and the properties of the probe molecule and its interactions with the restricting porous system in which it is contained. However, the derivation of analytical mathematical models for analysing such experiments is only straightforward for ideal restricting geometries and rapidly becomes intractable as the geometrical complexity increases. This is especially true when these models must take into account certain experimental limitations, particularly in clinical NMR spectrometers which do not have hardware as effective as research instruments. Consequently, when attempting to analyse diffusion experiments in complex geometries or including effects such as long magnetic gradient pulses, relaxing boundaries or internal magnetic gradients arising from external magnetic fields, numerical methods become an attractive method for generating these models. In this thesis, a variety of numerical techniques are examined in relation to the diffusion equation, the Bloch equations which describe nuclear magnetism and induction and the Bloch-Torrey equations which combine the two to describe the evolution of nuclear magnetism in a diffusing system. A highly flexible method for calculating the results of PGSE NMR experiments in porous systems based on the finite element method is presented. The efficiency and accuracy of the method is verified by comparison with the known solutions to simple pore geometries in the short gradient pulse limit (parallel planes, a cylindrical pore, and a spherical pore) and outside this limit (parallel planes, a circular pore and a spherical pore). The approach is then applied to modelling the more complicated cases of diffusion in parallel semipermeable planes, an array of connected channels with pore-hopping and a toroidal pore, a geometry for which there is presently no current analytical solution. In addition, experimental data for a model two-plane system are analysed using the technique. Outside the short gradient pulse limit, the method is used to evaluate models of diffusion inside ellipsoidal pores, annular pores and a biconcave ellipsoid used to approximate the shape of human erythrocytes. These models are generated for a variety of diffusion time periods, magnetic gradient pulse lengths, boundary relaxation values and rotations of anisotropic pores against the magnetic gradient, to show the flexibility of the method.

Date of Award	2017
Original language	English