Further development of discrete computational techniques for calculation of restricted diffusion propagators in porous media

Magnetic resonance is a well-established tool for structural characterisation of porous media. Features of pore-space morphology can be inferred from NMR diffusion-diffraction plots or the time-dependence of the apparent diffusion coefficient. Diffusion NMR signal attenuation can be computed from the restricted diffusion propagator, which describes the distribution of diffusing particles for a given starting position and diffusion time. We present two techniques for efficient evaluation of restricted diffusion propagators for use in NMR porous-media characterisation. The first is the Lattice Path Count (LPC). Its physical essence is that the restricted diffusion propagator connecting points A and B in time t is proportional to the number of distinct length-t paths from A to B. By using a discrete lattice, the number of such paths can be counted exactly. The second technique is the Markov transition matrix (MTM). The matrix represents the probabilities of jumps between every pair of lattice nodes within a single timestep. The propagator for an arbitrary diffusion time can be calculated as the appropriate matrix power. For periodic geometries, the transition matrix needs to be defined only for a single unit cell. This makes MTM ideally suited for periodic systems. Both LPC and MTM are closely related to existing computational techniques: LPC, to combinatorial techniques; and MTM, to the Fokker-Planck master equation. The relationship between LPC, MTM and other computational techniques is briefly discussed in the paper. Both LPC and MTM perform favourably compared to Monte Carlo sampling, yielding highly accurate and almost noiseless restricted diffusion propagators. Initial tests indicate that their computational performance is comparable to that of finite element methods. Both LPC and MTM can be applied to complicated pore-space geometries with no analytic solution. We discuss the new methods in the context of diffusion propagator calculation in porous materials and model biological tissues.

A study of water binding, mobility and dehydration in contact lens hydrogels using NMR

Hydrogel polymers are used for the manufacture of soft (or disposable) contact lenses worldwide today, but have a tendency to dehydrate on the eye. In vitro methods that can probe the potential for a given hydrogel polymer to dehydrate in vivo are much sought after. Nuclear magnetic resonance (NMR) has been shown to be effective in characterising water mobility and binding in similar systems (Barbieri, Quaglia et al., 1998, Larsen, Huff et al., 1990, Peschier, Bouwstra et al., 1993), predominantly through measurement of the spin-lattice relaxation time (T1), the spinspin relaxation time (T2) and the water diffusion coefficient (D). The aim of this work was to use NMR to quantify the molecular behaviour of water in a series of commercially available contact lens hydrogels, and relate these measurements to the binding and mobility of the water, and ultimately the potential for the hydrogel to dehydrate. As a preliminary study, in vitro evaporation rates were measured for a set of commercial contact lens hydrogels. Following this, comprehensive measurement of the temperature and water content dependencies of T1, T2 and D was performed for a series of commercial hydrogels that spanned the spectrum of equilibrium water content (EWC) and common compositions of contact lenses that are manufactured today. To quantify material differences, the data were then modelled based on theory that had been used for similar systems in the literature (Walker, Balmer et al., 1989, Hills, Takacs et al., 1989). The differences were related to differences in water binding and mobility. The evaporative results suggested that the EWC of the material was important in determining a material's potential to dehydrate in this way. Similarly, the NMR water self-diffusion coefficient was also found to be largely (if not wholly) determined by the WC. A specific binding model confirmed that the we was the dominant factor in determining the diffusive behaviour, but also suggested that subtle differences existed between the materials used, based on their equilibrium we (EWC). However, an alternative modified free volume model suggested that only the current water content of the material was important in determining the diffusive behaviour, and not the equilibrium water content. It was shown that T2 relaxation was dominated by chemical exchange between water and exchangeable polymer protons for materials that contained exchangeable polymer protons. The data was analysed using a proton exchange model, and the results were again reasonably correlated with EWC. Specifically, it was found that the average water mobility increased with increasing EWe approaching that of free water. The T1 relaxation was also shown to be reasonably well described by the same model. The main conclusion that can be drawn from this work is that the hydrogel EWe is an important parameter, which largely determines the behaviour of water in the gel. Higher EWe results in a hydrogel with water that behaves more like bulk water on average, or is less strongly 'bound' on average, compared with a lower EWe material. Based on the set of materials used, significant differences due to composition (for materials of the same or similar water content) could not be found. Similar studies could be used in the future to highlight hydrogels that deviate significantly from this 'average' behaviour, and may therefore have the least/greatest potential to dehydrate on the eye.

Photo-cross-linked poly(D,L-lactide)-based networks. Structural characterization by HR-MAS NMR spectroscopy and hydrolytic degradation behavior

To date, biodegradable networks and particularly their kinetic chain lengths have been characterized by analysis of their degradation products in solution. We characterize the network itself by NMR analysis in the solvent-swollen state under magic angle spinning conditions. The networks were prepared by photoinitiated cross-linking of poly(dl-lactide)−dimethacrylate macromers (5 kg/mol) in the presence of an unreactive diluent. Using diffusion filtering and 2D correlation spectroscopy techniques, all network components are identified. By quantification of network-bound photoinitiator fragments, an average kinetic chain length of 9 ± 2 methacrylate units is determined. The PDLLA macromer solution was also used with a dye to prepare computer-designed structures by stereolithography. For these networks structures, the average kinetic chain length is 24 ± 4 methacrylate units. In all cases the calculated molecular weights of the polymethacrylate chains after degradation are maximally 8.8 kg/mol, which is far below the threshold for renal clearance. Upon incubation in phosphate buffered saline at 37 °C, the networks show a similar mass loss profile in time as linear high-molecular-weight PDLLA (HMW PDLLA). The mechanical properties are preserved longer for the PDLLA networks than for HMW PDLLA. The initial tensile strength of 47 ± 2 MPa does not decrease significantly for the first 15 weeks, while HMW PDLLA lost 85 ± 5% of its strength within 5 weeks. The physical properties, kinetic chain length, and degradation profile of these photo-cross-linked PDLLA networks make them most suited materials for orthopedic applications and use in (bone) tissue engineering.

NMR relaxation behavior and quadrupole coupling constants of 39K and 23Na ions in glycerol. Comparisons with 39K tissue data

The quadrupole coupling constants (qcc) for39K and23Na ions in glycerol have been calculated from linewidths measured as a function of temperature (which in turn results in changes in solution viscosity). The qcc of39K in glycerol is found to be 1.7 MHz, and that of23Na is 1.6 MHz. The relaxation behavior of39K and23Na ions in glycerol shows magnetic field and temperature dependence consistent with the equations for transverse relaxation more commonly used to describe the reorientation of nuclei in a molecular framework with intramolecular field gradients. It is shown, however, that τc is not simply proportional to the ratio of viscosity/temperature (ηT). The 39K qcc in glycerol and the value of 1.3 MHz estimated for this nucleus in aqueous solution are much greater than values of 0.075 to 0.12 MHz calculated from T2 measurements of39K in freshly excised rat tissues. This indicates that, in biological samples, processes such as exchange of potassium between intracellular compartments or diffusion of ions through locally ordered regions play a significant role in determining the effective quadrupole coupling constant and correlation time governing39K relaxation. T1 and T2 measurements of rat muscle at two magnetic fields also indicate that a more complex correlation function may be required to describe the relaxation of39K in tissue. Similar results and conclusions are found for23Na.

Numerical simulation of anomalous diffusion with application to medical imaging

The first objective of this project is to develop new efficient numerical methods and supporting error and convergence analysis for solving fractional partial differential equations to study anomalous diffusion in biological tissue such as the human brain. The second objective is to develop a new efficient fractional differential-based approach for texture enhancement in image processing. The results of the thesis highlight that the fractional order analysis captured important features of nuclear magnetic resonance (NMR) relaxation and can be used to improve the quality of medical imaging.

Diffusion-sensitive magnetic resonance spectroscopy and imaging in biomedical sciences

We present a mini-review of the development and contemporary applications of diffusion-sensitive nuclear magnetic resonance (NMR) techniques in biomedical sciences. Molecular diffusion is a fundamental physical phenomenon present in all biological systems. Due to the connection between experimentally measured diffusion metrics and the microscopic environment sensed by the diffusing molecules, diffusion measurements can be used for characterisation of molecular size, molecular binding and association, and the morphology of biological tissues. The emergence of magnetic resonance was instrumental to the development of biomedical applications of diffusion. We discuss the fundamental physical principles of diffusion NMR spectroscopy and diffusion MR imaging. The emphasis is placed on conceptual understanding, historical evolution and practical applications rather than complex technical details. Mathematical description of diffusion is presented to the extent that it is required for the basic understanding of the concepts. We present a wide range of spectroscopic and imaging applications of diffusion magnetic resonance, including colloidal drug delivery vehicles; protein association; characterisation of cell morphology; neural fibre tractography; cardiac imaging; and the imaging of load-bearing connective tissues. This paper is intended as an accessible introduction into the exciting and growing field of diffusion magnetic resonance.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

A second order control-volume finite-element least-squares strategy for simulating diffusion in strongly anisotropic media

An unstructured mesh �nite volume discretisation method for simulating di�usion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume �nite-element method and it retains the local continuity of the ux at the control volume faces. A least squares function recon- struction technique together with a new ux decomposition strategy is used to obtain an accurate ux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it signi�cantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes, and appears independent of the mesh quality.