Diffusion in porous systems and the influence of pore morphology in pulsed gradient spin‐echo nuclear magnetic resonance studies

Feature attention graph neural network for estimating brain age and identifying important neural connections in mouse models of genetic risk for Alzheimer's disease

Alzheimer's disease (AD) remains one of the most extensively researched neurodegenerative disorders due to its widespread prevalence and complex risk factors. Age is a crucial risk factor for AD, which can be estimated by the disparity between physiological age and estimated brain age. To model AD risk more effectively, integrating biological, genetic, and cognitive markers is essential. Here, we utilized mouse models expressing the major APOE human alleles and human nitric oxide synthase 2 to replicate genetic risk for AD and a humanized innate immune response. We estimated brain age employing a multivariate dataset that includes brain connectomes, APOE genotype, subject traits such as age and sex, and behavioral data. Our methodology used Feature Attention Graph Neural Networks (FAGNN) for integrating different data types. Behavioral data were processed with a 2D Convolutional Neural Network (CNN), subject traits with a 1D CNN, brain connectomes through a Graph Neural Network using quadrant attention module. The model yielded a mean absolute error for age prediction of 31.85 days, with a root mean squared error of 41.84 days, outperforming other, reduced models. In addition, FAGNN identified key brain connections involved in the aging process. The highest weights were assigned to the connections between cingulum and corpus callosum, striatum, hippocampus, thalamus, hypothalamus, cerebellum, and piriform cortex. Our study demonstrates the feasibility of predicting brain age in models of aging and genetic risk for AD. To verify the validity of our findings, we compared Fractional Anisotropy (FA) along the tracts of regions with the highest connectivity, the Return-to-Origin Probability (RTOP), Return-to-Plane Probability (RTPP), and Return-to-Axis Probability (RTAP), which showed significant differences between young, middle-aged, and old age groups. Younger mice exhibited higher FA, RTOP, RTAP, and RTPP compared to older groups in the selected connections, suggesting that degradation of white matter tracts plays a critical role in aging and for FAGNN's selections. Our analysis suggests a potential neuroprotective role of APOE2, relative to APOE3 and APOE4, where APOE2 appears to mitigate age-related changes. Our findings highlighted a complex interplay of genetics and brain aging in the context of AD risk modeling.

Signatures of nanoemulsion jamming and unjamming in stimulated-echo NMR

The unjamming of elastic concentrated nanoemulsions into viscous dilute nanoemulsions, through dilution with the continuous phase, offers interesting opportunities for a pulsed-field gradient (PFG) NMR, particularly if the nanoemulsion is designed to take advantage of the nuclear specificity offered by NMR. Here, we make and study size-fractionated oil-in-water nanoemulsions using a perfluorinated copolymer silicone oil that is highly insoluble in the aqueous continuous phase. By studying these nanoemulsions using ^{19}F stimulated-echo PFG-NMR, we avoid any contribution from the aqueous continuous phase, which contains a nonfluorinated ionic surfactant. We find a dramatic change in the ^{19}F PFG-NMR decays at high field-gradient strengths as the droplet volume fraction, ϕ, is lowered through dilution. At high ϕ, observed decays as a function of field-gradient strength exhibit decay-to-plateau behavior indicating the jamming of nanodroplets, which contain ^{19}F probe molecules, in an elastic material reminiscent of a nanoporous solid. In contrast, at lower ϕ, only a simple decay is observed, indicating that the nanodroplets have unjammed and can diffuse over much larger distances. Through a comparison with bulk mechanical rheometry, we show that this dramatic change coincides with the loss of low-frequency shear elasticity of the nanoemulsion.

Diffusion-relaxation scattered MR signal representation in a multi-parametric sequence

This work focuses on obtaining a more general diffusion magnetic resonance imaging (MRI) signal representation that accounts for a longitudinal T₁ and transverse T₂^⋆ relaxations while at the same time integrating directional diffusion in the context of scattered multi-parametric acquisitions, where only a few diffusion gradient directions and b-values are available for each pair of echo and inversion times. The method is based on the three-dimensional simple harmonic oscillator-based reconstruction and estimation (SHORE) representation of the diffusion signal, which enables the estimation of the orientation distribution function and the retrieval of various quantitative indices such as the generalized fractional anisotropy or the return-to-the-origin probability while simultaneously resolving for T₁ and T₂^⋆ relaxation times. Our technique, the Relax-SHORE, has been tested on both in silico and in vivo diffusion-relaxation scattered MR data. The results show that Relax-SHORE is accurate in the context of scattered acquisitions while guaranteeing flexibility in the diffusion signal representation from multi-parametric sequences.

Transient Anomalous Diffusion MRI Measurement Discriminates Porous Polymeric Matrices Characterized by Different Sub-Microstructures and Fractal Dimension

Considering the current development of new nanostructured and complex materials and gels, it is critical to develop a sub-micro-scale sensitivity tool to quantify experimentally new parameters describing sub-microstructured porous systems. Diffusion NMR, based on the measurement of endogenous water's diffusion displacement, offers unique information on the structural features of materials and tissues. In this paper, we applied anomalous diffusion NMR protocols to quantify the subdiffusion of water and to measure, in an alternative, non-destructive and non-invasive modality, the fractal dimension d_w of systems characterized by micro and sub-micro geometrical structures. To this end, three highly heterogeneous porous-polymeric matrices were studied. All the three matrices composed of glycidylmethacrylate-divynilbenzene porous monoliths obtained through the High Internal Phase Emulsion technique were characterized by pores of approximately spherical symmetry, with diameters in the range of 2-10 μm. Pores were interconnected by a plurality of window holes present on pore walls, which were characterized by size coverings in the range of 0.5-2 μm. The walls were characterized by a different degree of surface roughness. Moreover, complementary techniques, namely Field Emission Scanning Electron Microscopy (FE-SEM) and dielectric spectroscopy, were used to corroborate the NMR results. The experimental results showed that the anomalous diffusion α parameter that quantifies subdiffusion and d_w = 2/α changed in parallel to the specific surface area S (or the surface roughness) of the porous matrices, showing a submicroscopic sensitivity. The results reported here suggest that the anomalous diffusion NMR method tested may be a valid experimental tool to corroborate theoretical and simulation results developed and performed for describing highly heterogeneous and complex systems. On the other hand, non-invasive and non-destructive anomalous subdiffusion NMR may be a useful tool to study the characteristic features of new highly heterogeneous nanostructured and complex functional materials and gels useful in cultural heritage applications, as well as scaffolds useful in tissue engineering.

Measurement of the diffusion of multiple nuclei in restricted spaces by pulsed field gradient NMR

In this study, the restricted diffusion of a solvent, anion, and cation in an electrolyte solution was measured by pulsed field gradient (PFG) NMR for ¹H, ¹⁹F, and ⁷Li nuclei. Further, the time dependences of the diffusion coefficients were measured for a 1 M LiPF₆ electrolyte solution in porous polyethylene, which has pores with sizes of tens of micrometers. The decreasing ratio of the diffusion coefficients of the solvent, cation, and anion based on the diffusion time can be scaled similarly for each diffusion distance. The experimentally obtained time dependences of the diffusion coefficients of the solvent, anion, and cation agreed with the results of the analytical equation with the same structural parameters. Furthermore, the abovementioned experimental results were produced via Monte Carlo simulation in the same model-restricted structure for the solvent, anion, and cation. Based on PFG-NMR, it can be concluded that the solvent, anion, and cation exhibit the same restricted diffusion behavior in polyethylene pores measuring tens of micrometers. It was confirmed that measuring the time dependences of the diffusion coefficients via PFG-NMR with multiple nuclei is effective for studying the diffusion mechanisms of electrolyte solutions in restricted spaces.

Ion Transport in Solid Medium-Evaluation of Ionic Mobility for Design of Ion Transport Pathways in Separator and Gel Electrolyte

Further improvement in the performance of lithium secondary batteries will be an indispensable issue to realize a decarbonized society. Among them, the batteries for electric vehicles still have many issues to be addressed because they are subject to various conditions such as high-power performance, safety, and cost restrictions for widespread use. Those subjects require extensive researches from the improvement of each element material to control the battery system to optimize the total performance. Based on this idea, we have been conducting research focusing on ion movement to elucidate the ion conduction mechanism from the microscopic point of view. It has been recognized that the ionic mobility in the battery, which dominates the power performance of the battery, is affected by the solid environment in which the ions move (separator and electrode materials) and the evaluation of ion movement, including the interaction with the surroundings, is necessary as an essential step for battery design. In this article, I will introduce the evaluation approach of ion dynamics and the evaluation results of mobility and interactive situations of carrier ions in the practical separator membranes and gel electrolytes. Finally, the direction of material design is outlined through this review.

The localization regime in a nutshell

High diffusion-sensitizing magnetic field gradients have been more and more often applied nowadays to achieve a better characterization of the microstructure. As the resulting spin-echo signal significantly deviates from the conventional Gaussian form, various models have been employed to interpret these deviations and to relate them with the microstructural properties of a sample. In this paper, we argue that the non-Gaussian behavior of the signal is a generic universal feature of the Bloch-Torrey equation. We provide a simple yet rigorous description of the localization regime emerging at high extended gradients and identify its origin as a symmetry breaking at the reflecting boundary. We compare the consequent non-Gaussian signal decay to other diffusion NMR regimes such as slow-diffusion, motional-narrowing and diffusion-diffraction regimes. We emphasize limitations of conventional perturbative techniques and advocate for non-perturbative approaches which may pave a way to new imaging modalities in this field.

Localization regime in diffusion NMR: Theory and experiments

In this work we investigate the emergence of the localization regime for diffusion NMR in various geometries: inside slabs, inside cylinders and outside rods arranged on a square array. At high gradients, the transverse magnetization is strongly attenuated in the bulk, whereas the macroscopic signal is formed by the remaining magnetization localized near boundaries of the sample. As a consequence, the signal is particularly sensitive to the microstructure. The theoretical analysis relies on recent mathematical advances on the study of the Bloch-Torrey equation. Experiments were conducted with hyperpolarized xenon-129 gas in 3D-printed phantoms and show an excellent agreement with numerical simulations and theoretical predictions. Our mathematical arguments and experimental evidence indicate that the localization regime with a stretched-exponential decay of the macroscopic signal is a generic feature of diffusion NMR that can be observed at moderately high gradients in most NMR scanners.

INDIANA: An in-cell diffusion method to characterize the size, abundance and permeability of cells

NMR and MRI diffusion experiments contain information describing the shape, size, abundance, and membrane permeability of cells although extracting this information can be challenging. Here we present the INDIANA (IN-cell DIffusion ANAlysis) method to simultaneously and non-invasively measure cell abundance, effective radius, permeability and intrinsic relaxation rates and diffusion coefficients within the inter- and intra-cellular populations. The method couples an experimental dataset comprising stimulated-echo diffusion measurements, varying both the gradient strength and the diffusion delay, together with software to fit a model based on the Kärger equations to robustly extract the relevant parameters. A detailed error analysis is presented by comparing the results from fitting simulated data from Monte Carlo simulations, establishing its effectiveness. We note that for parameters typical of mammalian cells the approach is particularly effective, and the shape of the underlying cells does not unduly affect the results. Finally, we demonstrate the performance of the experiment on systems of suspended yeast and mammalian cells. The extracted parameters describing cell abundance, size, permeability and relaxation are independently validated.

Diffusion tensor distribution imaging

Conventional diffusion MRI yields voxel-averaged parameters that suffer from ambiguities for heterogeneous anisotropic materials such as brain tissue. Using principles from solid-state NMR spectroscopy, we have previously introduced the shape of the diffusion encoding tensor as a separate acquisition dimension that disentangles isotropic and anisotropic contributions to the observed diffusivities, thereby allowing for unconstrained data inversion into diffusion tensor distributions with "size," "shape," and orientation dimensions. Here we combine our recent non-parametric data inversion algorithm and data acquisition protocol with an imaging pulse sequence to demonstrate spatial mapping of diffusion tensor distributions using a previously developed composite phantom with multiple isotropic and anisotropic components. We propose a compact format for visualizing two-dimensional arrays of the distributions, new scalar parameters quantifying intra-voxel heterogeneity, and a binning procedure giving maps of all relevant parameters for each of the components resolved in the multidimensional distribution space.

Spectral Analysis of a Non-Equilibrium Stochastic Dynamics on a General Network

Unravelling underlying complex structures from limited resolution measurements is a known problem arising in many scientific disciplines. We study a stochastic dynamical model with a multiplicative noise. It consists of a stochastic differential equation living on a graph, similar to approaches used in population dynamics or directed polymers in random media. We develop a new tool for approximation of correlation functions based on spectral analysis that does not require translation invariance. This enables us to go beyond lattices and analyse general networks. We show, analytically, that this general model has different phases depending on the topology of the network. One of the main parameters which describe the network topology is the spectral dimension [Formula: see text]. We show that the correlation functions depend on the spectral dimension and that only for [Formula: see text] > 2 a dynamical phase transition occurs. We show by simulation how the system behaves for different network topologies, by defining and calculating the Lyapunov exponents on the graph. We present an application of this model in the context of Magnetic Resonance (MR) measurements of porous structure such as brain tissue. This model can also be interpreted as a KPZ equation on a graph.

Physical characterization using diffusion NMR spectroscopy

NMR diffusion measurements (or dNMR) provide a powerful tool for analysis of solution organization and microgeometry of the environment by probing random molecular motion. Being a very versatile method, dNMR can be applied to a large variety of samples and systems. Here, a brief introduction into dNMR and a summary of recent advances in the field are presented. The research topics include restricted diffusion, anisotropic diffusion, polymer dynamics, solution structuring and dNMR method development. The dNMR studied systems include plants, cells (cell models), liquid crystals, polymer solutions, ionic liquids, supercooled solutions, untreated water, amino acid solutions and more. It is demonstrated how a variety of dNMR methods can be applied to a system to extract the data on particular structures present among, formed by or surrounding the diffusing particles. It is also demonstrated how dNMR methods can be developed to allow probing larger geometries, low sample concentrations and faster processes. Copyright © 2016 John Wiley & Sons, Ltd.

1H NMR Investigations of Activated Carbon Loaded with Volatile Organic Compounds: Quantification, Mechanisms, and Diffusivity Determination

Three volatile organic compounds (VOCs), benzene, cyclohexane, and dichloromethane, were adsorbed onto activated carbon fiber cloth. ¹H (magic-angle spinning (MAS) and pulsed field gradient (PFG)) NMR techniques were carried out, and the signals were analyzed in terms of peak surface areas and shifts. These techniques were shown to be very useful for determining (i) the intrinsic quantification of adsorbed molecules (VOCs and/or water) in the porosity of the materials (the adsorption capacities ranged from 0.2 to 4 mol·kg^-1); (ii) the mechanisms of interactions between adsorbed organic molecules and the carbon walls (illustrations of positions of the molecule inside the pore volume are proposed; the proton-wall distance was less than 0.15 nm); and (iii) the diffusivities (surface diffusion coefficients (D_S) were estimated at ≈4.10^-12 m²·s^-1 for cyclohexane, ≈1.10^-11 m²·s^-1 for benzene, and ≈4.10^-11 m²·s^-1 for dichloromethane).

Stray-field NMR diffusion q-space diffraction imaging of monodisperse coarsening foams

The technique of stray field diffusion NMR is adapted to study the diffusion properties of water in monodisperse wet foams. We show for the first time, that the technique is capable of observing q-space diffusion diffraction peaks in monodisperse aqueous foams with initial bubble sizes in the range of 50-85μm. The position of the peak maximum can be correlated simply to the bubble size in the foam leading to a technique that can investigate the stability of the foam over time. The diffusion technique, together with supplementary spin-spin relaxation analysis of the diffusion data is used to follow the stability and coarsening behaviour of monodisperse foams with a water fraction range between 0.24 and 0.33. The monodisperse foams remain stable for a period of hours in terms of the initial bubble size. The duration of this stable period correlates to the initial size of the bubbles. Eventually the bubbles begin to coarsen and this is observed in changes in the position of the diffusion diffraction maxima.

Direct correlation of diffusion and pore size distributions with low field NMR

The time-dependent diffusion coefficient (D) is a powerful tool to probe microstructure in porous media, and can be obtained by the NMR method. In a real porous sample, molecular diffusion is very complex. Here we present a new method which directly measures the relationship between effective diffusion coefficients and pore size distributions without knowing surface relaxivity. This method is used to extract structural information and explore the relationship between D and a in porous media having broad pore size distributions. The diffusion information is encoded by the Pulsed Field Gradient (PFG) method and the pore size distributions are acquired by the Decay due to Diffusion in the Internal Field (DDIF) method. Two model samples were measured to verify this method. Restricted diffusion was analyzed, and shows that most fluid molecules experience pore wall. The D(a) curves obtained from correlation maps were fitted to the Padé approximant equation and a good agreement was found between the fitting lines and the measured data. Then a sandstone sample with unknown structure was measured. The state of confined fluids was analyzed and structural information, such as pore size distributions, were extracted. The D - T1 correlation maps were also obtained using the same method, which yielded surface relaxivities for different samples. All the experiments were conducted on 2MHz NMR equipment to obtain accurate diffusion information, where internal gradients can be neglected. This method is expected to have useful applications in the oil industry, particularly for NMR logging in the future.

NMR-based diffusion lattice imaging

Nuclear magnetic resonance (NMR) diffusion experiments are widely employed as they yield information about structures hindering the diffusion process, e.g., about cell membranes. While it has been shown in recent articles that these experiments can be used to determine the shape of closed pores averaged over a volume of interest, it is still an open question how much information can be gained in open well-connected systems. In this theoretical work, it is shown that the full structure information of connected periodic systems is accessible. To this end, the so-called "SEquential Rephasing by Pulsed field-gradient Encoding N Time intervals" (SERPENT) sequence is used, which employs several diffusion encoding gradient pulses with different amplitudes. Two two-dimensional solid matrices that are surrounded by an NMR-visible medium are considered: a hexagonal lattice of cylinders and a rectangular lattice of isosceles triangles.

Including diffusion time dependence in the extra-axonal space improves in vivo estimates of axonal diameter and density in human white matter

Axonal density and diameter are two fundamental properties of brain white matter. Recently, advanced diffusion MRI techniques have made these two parameters accessible in vivo. However, the techniques available to estimate such parameters are still under development. For example, current methods to map axonal diameters capture relative trends over different structures, but consistently over-estimate absolute diameters. Axonal density estimates are more accessible experimentally, but different modeling approaches exist and the impact of the experimental parameters has not been thoroughly quantified, potentially leading to incompatibility of results obtained in different studies using different techniques. Here, we characterise the impact of diffusion time on axonal density and diameter estimates using Monte Carlo simulations and STEAM diffusion MRI at 7 T on 9 healthy volunteers. We show that axonal density and diameter estimates strongly depend on diffusion time, with diameters almost invariably overestimated and density both over and underestimated for some commonly used models. Crucially, we also demonstrate that these biases are reduced when the model accounts for diffusion time dependency in the extra-axonal space. For axonal density estimates, both upward and downward bias in different situations are removed by modeling extra-axonal time-dependence, showing increased accuracy in these estimates. For axonal diameter estimates, we report increased accuracy in ground truth simulations and axonal diameter estimates decreased away from high values given by earlier models and towards known values in the human corpus callosum when modeling extra-axonal time-dependence. Axonal diameter feasibility under both advanced and clinical settings is discussed in the light of the proposed advances.

Measuring small compartment dimensions by probing diffusion dynamics via Non-uniform Oscillating-Gradient Spin-Echo (NOGSE) NMR

Noninvasive measurements of microstructure in materials, cells, and in biological tissues, constitute a unique capability of gradient-assisted NMR. Diffusion-diffraction MR approaches pioneered by Callaghan demonstrated this ability; Oscillating-Gradient Spin-Echo (OGSE) methodologies tackle the demanding gradient amplitudes required for observing diffraction patterns by utilizing constant-frequency oscillating gradient pairs that probe the diffusion spectrum, D(ω). Here we present a new class of diffusion MR experiments, termed Non-uniform Oscillating-Gradient Spin-Echo (NOGSE), which dynamically probe multiple frequencies of the diffusion spectral density at once, thus affording direct microstructural information on the compartment's dimension. The NOGSE methodology applies N constant-amplitude gradient oscillations; N-1 of these oscillations are spaced by a characteristic time x, followed by a single gradient oscillation characterized by a time y, such that the diffusion dynamics is probed while keeping (N-1)x+y≡TNOGSE constant. These constant-time, fixed-gradient-amplitude, multi-frequency attributes render NOGSE particularly useful for probing small compartment dimensions with relatively weak gradients - alleviating difficulties associated with probing D(ω) frequency-by-frequency or with varying relaxation weightings, as in other diffusion-monitoring experiments. Analytical descriptions of the NOGSE signal are given, and the sequence's ability to extract small compartment sizes with a sensitivity towards length to the sixth power, is demonstrated using a microstructural phantom. Excellent agreement between theory and experiments was evidenced even upon applying weak gradient amplitudes. An MR imaging version of NOGSE was also implemented in ex vivo pig spinal cords and mouse brains, affording maps based on compartment sizes. The effects of size distributions on NOGSE are also briefly analyzed.

Numerical analysis of NMR diffusion measurements in the short gradient pulse limit

Pulsed gradient spin-echo (PGSE) NMR diffusion measurements provide a powerful technique for probing porous media. 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. Consequently, in general, numerical methods must be employed. Here, a highly flexible method for calculating the results of PGSE NMR experiments in porous systems in the short gradient pulse limit 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 (parallel planes, a cylindrical pore, and a spherical pore) and also to Monte Carlo simulations. The approach is then applied to modelling the more complicated cases of parallel semipermeable planes and a pore hopping model. Finally, the results of a PGSE measurement on a toroidal pore, a geometry for which there is presently no current analytical solution, are presented. This study shows that this approach has great potential for modelling the results of PGSE experiments on real (3D) porous systems. Importantly, the FEM approach provides far greater accuracy in simulating PGSE diffraction data.

How does oil type determine emulsion characteristics in concentrated Na-caseinate emulsions?

Macroscopic properties and ensemble average diffusion of concentrated (dispersed phase 50-60 wt%) Na-caseinate-stabilised emulsions for three different oils (soybean oil, palm olein and tetradecane) were explored. On a volume fraction basis, pulsed gradient stimulated echo (PGSTE)-NMR data show that droplet dynamics for all three systems are similar within a region of the emulsion morphology diagram. The exact limits of the emulsion space depend however on which oil is considered. The reduced solubility of tetradecane in water, and Na-caseinate in tetradecane, result in the stabilisation of flocs during formulation. Floc formation is not observed when soybean oil or palm olein is used under identical emulsion formulation conditions. Linear rheology experiments provide indirect evidence that the local structure and the properties of the thin film interfacial domain of tetradecane emulsions vary from those of soybean oil and palm olein emulsions. Collectively these data indicate that protein/oil interactions within a system dominate over specific oil droplet structure and size distribution, which are similar in the three systems.

Magnetic resonance of porous media (MRPM): a perspective

Porous media are ubiquitous in our environment and their application is extremely broad. The common connection between these diverse materials is the importance of the microstructure (μm to mm scale) in determining the physical, chemical and biological functions and properties. Magnetic resonance and its imaging modality have been essential for noninvasive characterization of these materials, in the development of catalysts, understanding cement hydration, fluid transport in rocks and soil, geological prospecting, and characterization of tissue properties for medical diagnosis. The past two decades have witnessed significant development of MRPM that couples advances in physics, chemistry and engineering with a broad range of applications. This article will summarize key advances in basic physics and methodology, examine their limitations and envision future R&D directions.

Structural disorder and anomalous diffusion in random packing of spheres

Nowadays Nuclear Magnetic Resonance diffusion (dNMR) measurements of water molecules in heterogeneous systems have broad applications in material science, biophysics and medicine. Up to now, microstructural rearrangement in media has been experimentally investigated by studying the diffusion coefficient (D(t)) behavior in the tortuosity limit. However, this method is not able to describe structural disorder and transitions in complex systems. Here we show that, according to the continuous time random walk framework, the dNMR measurable parameter α, quantifying the anomalous regime of D(t), provides a quantitative characterization of structural disorder and structural transition in heterogeneous systems. To demonstrate this, we compare α measurements obtained in random packed monodisperse micro-spheres with Molecular Dynamics simulations of disordered porous media and 3D Monte Carlo simulation of particles diffusion in these kind of systems. Experimental results agree well with simulations that correlate the most used parameters and functions characterizing the disorder in porous media.

Diffusive diffraction phenomenon observed by PGSE NMR technique in a sugar-based low-molecular-mass gel

The paper presents the diffusive diffraction phenomenon observed by the single-pulse-gradient spin-echo (s-PGSE) NMR technique in a real porous material: a gel composed of low-molecular-mass gelator methyl-4,6-O-(p-nitrobenzylidene)-α-D-glucopyranoside and toluene. Thanks to this phenomenon, we can probe the true microstructure (not xerogel) in which the toluene diffuses. To analyze the measured diffusion-diffraction pattern, we employed a composite bicompartmental model that superimposes restricted diffusion in small cavities of the gel matrix within the bundles of crossing fibers, with free diffusion in large and unconfined compartments between the bundles of crossing fibers. For restricted diffusion a pore-hopping formalism was applied. The observation of the diffraction pattern and its analysis leads to the conclusion that the pores, in the slow diffusing compartment of studied gel are ordered, at least locally, and relatively monodisperse with a size of 64 μm. Moreover, the restricting walls formed by the crossing fibers are perpendicular to the direction of the diffusion gradient.

Estimation of pore size in a microstructure phantom using the optimised gradient waveform diffusion weighted NMR sequence

There has been increasing interest in nuclear magnetic resonance (NMR) techniques that are sensitive to diffusion of molecules containing NMR visible nuclei for the estimation of microstructure parameters. A microstructure parameter of particular interest is pore radius distribution. A recent in silico study optimised the shape of the gradient waveform in diffusion weighted spin-echo experiments for estimating pore size. The study demonstrated that optimised gradient waveform (GEN) protocols improve pore radius estimates compared to optimised pulse gradient spin-echo (PGSE) protocols, particularly at shorter length scales. This study assesses the feasibility of implementing GEN protocols on a small bore 9.4 T scanner and verifies their additional sensitivity to pore radius. We implement GEN and PGSE protocols optimised for pore radii of 1, 2.5, 5, 7.5, 10 μm and constrained to maximum gradient strengths of 40, 80, 200 mT m(-1). We construct microstructure phantoms, which have a single pore radius for each phantom, using microcapillary fibres. The measured signal shows good agreement with simulated signal, strongly indicating that the GEN waveforms can be implemented on a 9.4 T system. We also demonstrate that GEN protocols provide improved sensitivity to the smaller pore radii when compared to optimised PGSE protocols, particularly at the lower gradient amplitudes investigated in this study. Our results suggest that this improved sensitivity of GEN protocols would be reflected in clinical scenarios.

Optimising time-varying gradient orientation for microstructure sensitivity in diffusion-weighted MR

Here we investigate whether varying the diffusion-gradient orientation during a general waveform single pulsed-field gradient sequence improves sensitivity to the size of coherently oriented pores over having a fixed orientation. The experiment optimises the shape and the orientation of the gradient waveform in each of a set of measurements to minimise the expected variance of estimates of the parameters of a simple model. A key application motivating the work is measuring the size of axons in white matter. Thus, we use a two compartment white matter model with impermeable, single-radius cylinders, and search for waveforms that maximise the sensitivity to axon radius, intra-cellular volume fraction and diffusion constants. Output of the optimisation suggests the only benefit of allowing the gradient orientation to vary in the plane perpendicular to the cylinders is that we can gain perpendicular gradient strength by maximising two orthogonal gradients simultaneously. This suggests that varying orientation in itself does not increase the sensitivity to model parameters. On the other hand, the variation in a plane containing the parallel direction increases the sensitivity significantly because parallel sensitivity improves the diffusion constant estimates. However, we also find that similar improvement in the estimates can be achieved without optimising the orientation, but by having one measurement in the parallel and the rest in the perpendicular direction. The optimisation searches a very large space where it cannot hope to find the global minimum so we cannot make a categorical conclusion. However, given the consistency of the results in multiple reruns and variations of the experiments reported here, we can suggest that for probing coherently oriented systems, pulse sequences with variable orientation, such as double-wave vector sequences, do not offer more advantage than fixed orientation sequences with optimised shape. The advantage of varying orientation is however likely to emerge for more complex systems with dispersed pore orientation.

A mixed basis approach in the SGP-limit

A perturbation method for computing quick estimates of the echo decay in pulsed spin echo gradient NMR diffusion experiments in the short gradient pulse limit is presented. The perturbation basis involves (relatively few) dipole distributions on the boundaries generating a small perturbation matrix in O(s(2)) time, where s denotes the number of boundary elements. Several approximate eigenvalues and eigenfunctions to the diffusion operator are retrieved. The method is applied to 1D and 2D systems with Neumann boundary conditions.

Mapping of (3) He apparent diffusion coefficient anisotropy at sub-millisecond diffusion times in an elastase-instilled rat model of emphysema

Hyperpolarized (3) He gas can provide detailed anatomical maps of the macroscopic airways in the lungs (i.e., ventilation) as well as insight into the lung microstructure through the apparent diffusion coefficient. In particular, the apparent diffusion coefficient of (3) He in the lung exhibits anisotropic effects that depend on diffusion time (δ), and it has been shown to be extraordinarily sensitive to enlargement in terminal airways and alveoli associated with emphysema. In this study, the anisotropic nature of the (3) He apparent diffusion coefficient is studied in a rat model of emphysema, based on elastase instillation, specifically for δ values less than one millisecond. Longitudinal (D(L) ) and transverse (D(T) ) diffusion coefficients were mapped at δ = 360 μs and δ = 800 μs based on a cylinder model of lung structure and correlated with histological measurement of alveolar damage based on mean linear intercept (L(m) ). Whole-lung mean D(T) measured at δ = 360 μs in the elastase-instilled rat lungs (0.14 ± 0.09 cm(2) /s) demonstrated the most significant increase (p = 0.00195) compared to the sham-instilled cohort (0.06 ± 0.06 cm(2) /s) and had a strong linear correlation with L(m) (Pearson's correlation coefficient of 0.9). These results suggest that measurement of (3) He apparent diffusion coefficient anisotropy, specifically D(T) , can provide a sensitive indicator of emphysema, particularly at very short diffusion times (δ = 360 μs).

Evolution of fat crystal network microstructure followed by NMR

Model systems composed of tristearin in solid state and tricaprin in liquid state with different solid-fat content (SFC) and storage time have been investigated by relaxation NMR and NMR diffusometry. The T(2) relaxation of the tricaprin in the melt exhibited a bimodal distribution as previously observed. The SFC had a major effect on the T(2) relaxation. This effect was explained according to the fast diffusive exchange model in porous media. According to this model the changes in T(2) relaxation as a function of the SFC and storage time were explained by the decrease of the surface-to-volume ratio of the crystal induced by Ostwald ripening. The diffusion coefficient D of the tricaprin in the melt decreased for higher SFC. Since no significant variation of D was observed for different diffusion time, D reflected the long-range connectivity and the tortuosity was calculated. During storage the diffusion coefficient remained constant.

Diffusion and interfacial transport of water in Nafion

Water absorption, membrane swelling, and self-diffusivity of water in 1100 equivalent weight Nafion were measured as functions of temperature and water activity. Free volume per water at 80 °C, determined from water uptake and volume expansion data, decreases with water content in the membrane from 12 cm(3)/mol at λ = 0.5 H(2)O/SO(3) to 1.5 cm(3)/mol at λ = 4. The change in free volume with water content displays a transition at λ = 4. Limiting water self-diffusivity in Nafion was determined by pulsed gradient spin echo NMR at long delay times. The limiting self-diffusivity increases exponentially with water activity; the rate of increase of diffusivity with water content shows a transition at λ = 4. The tortuosity of the hydrophilic domains in Nafion decreased from 20 at low membrane water activity to 3 at λ = 4. It suggested a change in the connectivity of the hydrophilic domains absorbed water occurs at λ ∼ 4. The diffusivity results were employed to separate the contributions of diffusional and interfacial resistance for water transport across Nafion membranes, which enabled the determination of the interfacial mass transport coefficients. A diffusion model was developed which incorporated activity-dependent diffusivity, volume expansion, and the interfacial resistance, and was used to resolve the water activity profiles in the membrane.

Optimizing gradient waveforms for microstructure sensitivity in diffusion-weighted MR

Variations in gradient waveforms can provide different levels of sensitivity to microstructure parameters in diffusion-weighted MR. We present a method that identifies gradient waveforms with maximal sensitivity to parameters of a model relating microstructural features to diffusion MR signals. The method optimizes the shape of the gradient waveform, constrained by hardware limits and fixed orientation, to minimize the expected variance of parameter estimates. The waveform is defined discretely and each point optimized independently. The method is illustrated with a biomedical application in which we maximize the sensitivity to microstructural features of white matter such as axon radius, intra-cellular volume fraction and diffusion constants. Simulation experiments find that optimization of the shape of the gradient waveform improves sensitivity to model parameters for both human and animal MR systems. In particular, the optimized waveforms make axon radii smaller than 5 microm more distinguishable than standard pulsed gradient spin-echo (PGSE). The identified class of optimized gradient waveforms have dominant square-wave components with frequency that increases as the radius size decreases.

Structure, diffusion, and permeability of protein-stabilized monodispersed oil in water emulsions and their gels: a self-diffusion NMR study

Self-diffusion NMR is used to investigate monodispersed oil in water emulsions and the subsequent gel formed by removing the water through evaporation. The radius of the oil droplets in the emulsions is measured using a number of diffusion methods based on the measurement of the mean squared displacement of the oil, water, and tracer molecules. The results are consistent with the known size of the emulsions. Bragg-like reflections due to the restricted diffusion of the water around the oil droplets are observed due to the low polydispersity of the emulsions and the dense packing. The resulting data are fitted to a pore glass model to give the diameter of both the pools of interstitial water and the oil droplets. In the gel, information on the residual three-dimensional structure is obtained using the short time behavior of the effective diffusion coefficient to give the surface to volume ratio of the residual protein network structure. The values for the surface to volume ratio are found to be consistent with the expected increase of the surface area of monodisperse droplets forming a gel network. At long diffusion observation times, the permeability of the network structure is investigated by diffusion NMR to give a complete picture of the colloidal system considered.

Complex geometric models of diffusion and relaxation in healthy and damaged white matter

Which aspects of tissue microstructure affect diffusion weighted MRI signals? Prior models, many of which use Monte-Carlo simulations, have focused on relatively simple models of the cellular microenvironment and have not considered important anatomic details. With the advent of higher-order analysis models for diffusion imaging, such as high angular resolution diffusion imaging (HARDI), more realistic models are necessary. This paper presents and evaluates the reproducibility of simulations of diffusion in complex geometries. Our framework is quantitative, does not require specialized hardware, is easily implemented with little programming experience, and is freely available as open-source software. Models may include compartments with different diffusivities, permeabilities, and T2 time constants using both parametric (e.g. spheres and cylinders) and arbitrary (e.g. mesh-based) geometries. Three-dimensional diffusion displacement probability functions are mapped with high reproducibility, and thus can be readily used to assess reproducibility of diffusion-derived contrasts.

Diffusion processes in homogeneous and phase-separated binary fluid mixtures

Diffusion processes in dynamically asymmetric binary fluid mixtures made of monodisperse polystyrene (PS) and a rodlike nematogen molecule (5CB) are studied by pulsed-field gradient spin echo NMR in the vicinity of the phase-separation/phase-dissolution temperature. The phase-separation process and the loss of mobility of polymer chains at Tg take place simultaneously evidencing the strong effect of elasticity on the sample morphology. Below the instability point of the mixture, two self-diffusion coefficients, named Dfast and Dslow, are observed and assigned to mobile molecules i) dissolved in the polymeric matrix and ii) phase-separated in isolated or interconnected domains, respectively. The temperature dependence of Dfast exhibits an Arrhenius-like behaviour when the mixture is submitted to a slow cooling rate whereas a Vogel-Fulcher-Tamman-Hesse law is obeyed for deep quenches. These results show the dynamic heterogeneities existing in the PS/5CB system below the UCST. Characteristic length scales are estimated from modelling echo attenuation curves exhibiting diffraction-like patterns.

Diffusive diffraction measurements in porous media: effect of structural disorder and internal magnetic field gradients

Pulsed Field Gradient NMR (PFG-NMR) method used to measure the self-diffusion coefficient of liquids can also be exploited to probe the local geometry of porous media. In most practical cases, the measured diffusion attenuation is generally Gaussian and can be interpreted in terms of an apparent diffusion coefficient. Using well chosen experimental conditions, a so called "diffusive diffraction" phenomenon can be observed in the diffusion curve with a specific shape and maxima location characteristic of the system local dimensions. In this paper we investigate this phenomenon by presenting new experimental results obtained on several porous model systems of packed sphere particles. Using different experimental approaches, the diffusion pattern could be finely observed and interpreted in the context of the pore hopping model formalism. Different calibrated systems of polystyrene and glass spheres with known mean diameter and polydispersity were used to investigate specifically the influence of structural heterogeneity and local internal gradients. Structural data obtained in that way were found in close agreement with laser diffraction granulometry measurement and Scanning Emission Microscopy.

Existing Condition and Migration Property of Ions in Lithium Electrolytes with Ionic Liquid Solvent

Ionization conditions of each ionic species in lithium ionic liquid electrolytes, LiTFSI/BMI-TFSI and LiTFSI/BDMI-TFSI, were confirmed based on the diffusion coefficients of the species measured by the pulsed gradient spin-echo (PGSE) NMR technique. We found that the diffusion coefficient ratios of the cation and anion species D(Li)(obs)/D(F)(obs) of the lithium salt and D(H)(obs)/D(F)(obs) of the ionic liquid solvent were effective guides to evaluate the ionization condition responsible for their mobility. Lithium ions were found to be stabilized, forming the solvated species as Li(TFSI)3(2-). TFSI- anion coordination could be relaxed by the dispersion of silica to form a gel electrolyte, LiTFSI/BDMI-TFSI/silica. It is expected that the oxygen sites on the silica directly attract Li+, releasing the TFSI- coordination. The lithium species, loosing TFSI- anions, kept a random walk feature in the gel without the diffusion restriction attributed from the strong chemical and morphological effect as that in the gel with the polymer. We can conclude that the silica dispersion is a significant approach to provide the appropriate lithium ion condition as a charge-transporting species in the ionic liquid electrolytes.

Diffusive diffraction phenomenon in a porous polymer material observed by NMR using radio-frequency field gradients

Pulsed field gradient NMR diffusion experiments can, in principle, lead to the "diffusive diffraction" phenomenon. In practice, its observation by gradients of the static magnetic field is difficult in real systems because they involve internal gradients due to the static magnetic field (necessary for polarizing nuclear spins). This latter drawback can be circumvented by using gradients of the radio-frequency (rf) field (the other magnetic field used in any NMR experiment). For the first time, by means of rf gradients, a so-called diffusive diffraction peak has been observed in a real porous system and its position provides a value of the mean distance between pores; this can be further complemented by the mean pore size determined from the dependence of the apparent diffusion coefficient with respect to the diffusion interval.

Monodisperse emulsions from a microfluidic device, characterised by diffusion NMR

A microfluidic device has been used to create novel monodisperse polymeric oil-in-water emulsions of diameter 15-100 µm, stabilised by surfactants and polymers. The pulse field gradient nuclear magnetic resonance (PFG-NMR) signal attenuation function showed minima that allow a simple calculation of the droplet size and polydispersity. The presence of the minima is due to the restricted diffusion of molecules within the droplets, which can be analysed using standard solutions of the diffusion equation with a spherical boundary condition. However, even when a small population of polymer is unrestricted, the deep troughs in the attenuation function are obscured. The simultaneous water NMR attenuation function gives structural information about the continuous-phase matrix. The NMR size measurements were compared with those obtained by optical and confocal microscopy, and laser diffraction analysis.

Diffusion damping during adiabatic z-rotation pulses for NMR spectroscopy in inhomogeneous magnetic fields

High-resolution nuclear magnetic resonance spectra from samples located in inhomogeneous static and radio frequency magnetic fields can be obtained by applying a train of z-rotation radio frequency pulses to repeatedly refocus the inhomogeneous broadening during signal detection. z-rotation pulses based on an adiabatic double passage are effective over wide bandwidths using a limited amount of radio frequency power at the expense of being time consuming and, consequently, sensitive to motion of the spin bearing molecules. The signal damping resulting from molecular self-diffusion during the pulse was studied experimentally and using Brownian dynamics simulations. The results show that the analytical expression for diffusion damping during a double spin echo is a reasonable approximation for the signal decay during an adiabatic z-rotation pulse. Methods to alleviate the effects of diffusion are discussed.

Temporal diffusion spectroscopy: theory and implementation in restricted systems using oscillating gradients

The theory of temporal diffusion spectra is reviewed. In contrast to q-space spectroscopy, which measures the displacement spectrum of spins in a spatial domain, the spectral density of the velocity correlation function (VCF) in the temporal domain is considered. It is demonstrated that casting diffusion in this domain may facilitate measurements of microscopic geometry and the decomposition of the diffusion signal into components due to disperse flow and restricted diffusion. An oscillating gradient (OG) method of diffusion spectroscopy was developed and implemented. Microscopic pore sizes, surface-to-volume ratios (S/Vs), and diffusion path tortuosities were extracted from model systems using this method. Cases are discussed in which this type of experiment may allow the characterization of pore geometry when spatial domain experiments fail. OGs may be combined with imaging sequences to map complex patterns of diffusion and flow. Moreover, scalar apparent diffusion coefficient (ADC) measurements in complex biological systems may be subtly dependent on specific pulse sequence parameters. Thus, scalar ADC measurements using gradient pulses with different frequency spectra may give different results. Conversely, the frequency dependence of motion-sensitizing gradient pulses may be exploited to deduce the origin of ADC changes.