2D relaxation/diffusion correlations in porous media

Simple MATLAB and Python scripts for multi-exponential analysis

Multi-exponential decay is prevalent in magnetic resonance spectroscopy, relaxation, and imaging. This paper describes simple MATLAB and Python functions and scripts for regularized multi-exponential analysis methods for 1D and 2D data and example test problems and experiments. Regularized least-squares solutions provide production-quality outputs with robust stopping rules in ~5 and ~20 lines of code for 1D and 2D inversions, respectively. The software provides an open-architecture simple solution for transforming exponential decay data to the distribution of their decay lifetimes. Examples from magnetic resonance relaxation of a complex fluid, a Danish North Sea crude oil, and fluid mixtures in porous materials-brine/crude oil mixture in North Sea reservoir chalk-are presented. Developed codes may be incorporated in other software or directly used by other researchers, in magnetic resonance relaxation, diffusion, and imaging or other physical phenomena that require multi-exponential analysis.

Disentangling the effects of restriction and exchange with diffusion exchange spectroscopy

Diffusion exchange spectroscopy (DEXSY) is a multidimensional NMR technique that can reveal how water molecules exchange between compartments within heterogeneous media, such as biological tissue. Data from DEXSY experiments is typically processed using numerical inverse Laplace transforms (ILTs) to produce a diffusion-diffusion spectrum. A tacit assumption of this ILT approach is that the signal behavior is Gaussian - i.e., the spin echo intensity decays exponentially with the degree of diffusion weighting. The assumptions that underlie Gaussian signal behavior may be violated, however, depending on the gradient strength applied and the sample under study. We argue that non-Gaussian signal behavior due to restrictions is to be expected in the study of biological tissue using diffusion NMR. Further, we argue that this signal behavior can produce confounding features in the diffusion-diffusion spectra obtained from numerical ILTs of DEXSY data - entangling the effects of restriction and exchange. Specifically, restricted signal behavior can result in broadening of peaks and in the appearance of illusory exchanging compartments with distributed diffusivities, which pearl into multiple peaks if not highly regularized. We demonstrate these effects on simulated data. That said, we suggest the use of features in the signal acquisition domain that can be used to rapidly probe exchange without employing an ILT. We also propose a means to characterize the non-Gaussian signal behavior due to restrictions within a sample using DEXSY measurements with a near zero mixing time or storage interval. We propose a combined acquisition scheme to independently characterize restriction and exchange with various DEXSY measurements, which we term Restriction and Exchange from Equally-weighted Double and Single Diffusion Encodings (REEDS-DE). We test this method on ex vivo neonatal mouse spinal cord - a sample consisting primarily of gray matter - using a low-field, static gradient NMR system. In sum, we highlight critical shortcomings of prevailing DEXSY analysis methods that conflate the effects of restriction and exchange, and suggest a viable experimental approach to disentangle them.

Characterization of Pectin-Based Gels: A 1H Nuclear Magnetic Resonance Relaxometry Study

Rare sugars are monosaccharides and their derivatives that are not commonly found in nature. d-Allulose is a rare sugar that is C-3 epimer of fructose and presents an alternative to sucrose with potential health benefits. In this study, different amounts of sucrose, d-allulose, and soy protein isolate (SPI) were used to prepare a set of pectin gels. The effect of these ingredients on the gels was studied at both a molecular level, by ¹H nuclear magnetic resonance (NMR) relaxometry, and a macroscopic level, through the assessment of viscoelastic properties as well as hardness and moisture content measurements. The NMR dispersion profiles were analyzed considering relaxation mechanisms associated with rotational and translational diffusion motions of mono- and disaccharides as well as bound water molecules. Significant variations of the local diffusion coefficient for the studied formulations were evidenced by the model fitting analysis. The viscosity trends observed within each group of samples having the same amount of SPI were mostly in agreement with the diffusion coefficients obtained from the NMR relaxometry. The observed discrepancies could be explained considering hardness and moisture content results, which put into evidence the fact that decreasing the moisture (mainly free water) affects the macroscopic properties of the systems, such as hardness and viscosity, but not the local diffusion processes probed by NMR relaxometry. These findings show the importance of combining both micro- and macroscopic information to analyze the different properties of food products.

MR Study of Water Distribution in a Beech (Fagus sylvatica) Branch Using Relaxometry Methods

Wood is a widely used material because it is environmentally sustainable, renewable and relatively inexpensive. Due to the hygroscopic nature of wood, its physical and mechanical properties as well as the susceptibility to fungal decay are strongly influenced by its moisture content, constantly changing in the course of everyday use. Therefore, the understanding of the water state (free or bound) and its distribution at different moisture contents is of great importance. In this study, changes of the water state and its distribution in a beech sample while drying from the green (fresh cut) to the absolutely dry state were monitored by 1D and 2D ¹H NMR relaxometry as well as by spatial mapping of the relaxation times T₁ and T₂. The relaxometry results are consistent with the model of homogeneously emptying pores in the bioporous system with connected pores. This was also confirmed by the relaxation time mapping results which revealed the moisture transport in the course of drying from an axially oriented early- and latewood system to radial rays through which it evaporates from the branch. The results of this study confirmed that MRI is an efficient tool to study the pathways of water transport in wood in the course of drying and is capable of determining the state of water and its distribution in wood.

High-Resolution Reconstruction for Multidimensional Laplace NMR

As a perfect complement to conventional NMR that aims for chemical structure elucidation, Laplace NMR constitutes a powerful technique to study spin relaxation and diffusion, revealing information on molecular motions and spin interactions. Different from conventional NMR adopting Fourier transform to deal with the acquired data, Laplace NMR relies on specially designed signal processing and reconstruction algorithms resembling the inverse Laplace transform, and it generally faces severe challenges in cases where high spectral resolution and high spectral dimensionality are required. Herein, based on the tensor technique for high-dimensional problems and the sparsity assumption, we propose a general method for high-resolution reconstruction of multidimensional Laplace NMR data. We show that the proposed method can reconstruct multidimensional Laplace NMR spectra in a high-resolution manner for exponentially decaying relaxation and diffusion data acquired by commercial NMR instruments. Therefore, it would broaden the scope of multidimensional Laplace NMR applications.

Effect of the manufacturing process on Fiore Sardo PDO cheese microstructure by multi-frequency NMR relaxometry

The quality of Fiore Sardo cheese, a traditional Italian dairy product, was analyzed by means of Multi-frequency Nuclear Magnetic (NMR) relaxometry. Specifically, ten cheese wheels were purchased from different production chains, either industrial (N = 5) or artisanal (N = 5) samples. The former came from large scale productions and the latter were produced by shepherds in small quantities and in very small dairy factories. A preliminary interlaboratory proficiency testing of Time Domain - NMR (TD-NMR, 20 MHz) relaxometry by five laboratories, consistently showed that product quality is significantly different in terms of molecular mobility according to their production chain (i.e. industrial or artisanal). More detailed information about cheese microstructure was obtained by Multi-frequency Fast Field Cycling NMR (FFC-NMR) at lower magnetic fields (0.01-10 MHz). According to the interpretative model adopted to describe FFC-NMR data, industrially processed cheeses showed a higher para-casein hydration, higher protein protons to water protons ratio and a higher disorder (lower fractal dimension d_f) than artisanal products. It is suggested that differences between artisanal and industrial cheeses generate from the processing steps preceding cheese maturation, and are clearly reflected in the visual appearance of cheeses. This study shows that NMR relaxometry techniques can successfully discriminate Fiore Sardo cheese from different production chains, and paves the way for their implementation in quality control practices of dairy products.

Double diffusion encoding and applications for biomedical imaging

Diffusion Magnetic Resonance Imaging (dMRI) is one of the most important contemporary non-invasive modalities for probing tissue structure at the microscopic scale. The majority of dMRI techniques employ standard single diffusion encoding (SDE) measurements, covering different sequence parameter ranges depending on the complexity of the method. Although many signal representations and biophysical models have been proposed for SDE data, they are intrinsically limited by a lack of specificity. Advanced dMRI methods have been proposed to provide additional microstructural information beyond what can be inferred from SDE. These enhanced contrasts can play important roles in characterizing biological tissues, for instance upon diseases (e.g. neurodegenerative, cancer, stroke), aging, learning, and development. In this review we focus on double diffusion encoding (DDE), which stands out among other advanced acquisitions for its versatility, ability to probe more specific diffusion correlations, and feasibility for preclinical and clinical applications. Various DDE methodologies have been employed to probe compartment sizes (Section 3), decouple the effects of microscopic diffusion anisotropy from orientation dispersion (Section 4), probe displacement correlations, study exchange, or suppress fast diffusing compartments (Section 6). DDE measurements can also be used to improve the robustness of biophysical models (Section 5) and study intra-cellular diffusion via magnetic resonance spectroscopy of metabolites (Section 7). This review discusses all these topics as well as important practical aspects related to the implementation and contrast in preclinical and clinical settings (Section 9) and aims to provide the readers a guide for deciding on the right DDE acquisition for their specific application.

Analysis of three-site T2-T2 exchange NMR

T₂-T₂ exchange NMR is a unique method to investigate the pore space and fluid dynamics in porous media. While two-site relaxation exchange is well understood, three-site exchange is not. We analyze the solutions for three-site T₂-T₂ exchange NMR analytically and by computer simulation. Three main results are obtained. First, the exchange map can be asymmetric in the case of microscale vortex motion in violation of the principle of detailed balance. Second, the apparent longitudinal relaxation times and/or apparent transverse relaxation times can be complex valued. In the case of complex apparent transverse relaxation times, the three-site exchange map coalesces to a two-site exchange map with characteristic oscillations in the time domain. As a result of the oscillations, the shorter relaxation time is less than expected. Third, there can be negative cross-peaks in the exchange map for certain combinations of longitudinal and transverse relaxation times or if the mixing period is shorter than the evolution and detection periods. In view of these results experimental exchange maps may need to be reevaluated.

Magnetic resonance measurements of cellular and sub-cellular membrane structures in live and fixed neural tissue

We develop magnetic resonance (MR) methods for real-time measurement of tissue microstructure and membrane permeability of live and fixed excised neonatal mouse spinal cords. Diffusion and exchange MR measurements are performed using the strong static gradient produced by a single-sided permanent magnet. Using tissue delipidation methods, we show that water diffusion is restricted solely by lipid membranes. Most of the diffusion signal can be assigned to water in tissue which is far from membranes. The remaining 25% can be assigned to water restricted on length scales of roughly a micron or less, near or within membrane structures at the cellular, organelle, and vesicle levels. Diffusion exchange spectroscopy measures water exchanging between membrane structures and free environments at 100 s^-1.

TE dependent Diffusion Imaging (TEdDI) distinguishes between compartmental T2 relaxation times

Biophysical modeling of macroscopic diffusion-weighted MRI signal in terms of microscopic cellular parameters holds the promise of quantifying the integrity of white matter. Unfortunately, even fairly simple multi-compartment models of proton diffusion in the white matter do not provide a unique, biophysically plausible solution. Here we report a nontrivial diffusion MRI signal dependence on echo time (TE) in human white matter in vivo. We demonstrate that such TE dependence originates from compartment-specific T2 values and that it is a promising "orthogonal measure" able to break the degeneracy in parameter estimation, and to yield important relaxation metrics robustly. We thereby enable the precise estimation of the intra- and extra-axonal water T2 relaxation times, which is precluded by a limited signal-to-noise ratio when using multi-echo relaxometry alone.

Identification of Intracellular and Extracellular Metabolites in Cancer Cells Using 13C Hyperpolarized Ultrafast Laplace NMR

Ultrafast Laplace NMR (UF-LNMR), which is based on the spatial encoding of multidimensional data, enables one to carry out 2D relaxation and diffusion measurements in a single scan. Besides reducing the experiment time to a fraction, it significantly facilitates the use of nuclear spin hyperpolarization to boost experimental sensitivity, because the time-consuming polarization step does not need to be repeated. Here we demonstrate the usability of hyperpolarized UF-LNMR in the context of cell metabolism, by investigating the conversion of pyruvate to lactate in the cultures of mouse 4T1 cancer cells. We show that ¹³C ultrafast diffusion–T₂ relaxation correlation measurements, with the sensitivity enhanced by several orders of magnitude by dissolution dynamic nuclear polarization (D-DNP), allows the determination of the extra- vs intracellular location of metabolites because of their significantly different values of diffusion coefficients and T₂ relaxation times. Under the current conditions, pyruvate was located predominantly in the extracellular pool, while lactate remained primarily intracellular. Contrary to the small flip angle diffusion methods reported in the literature, the UF-LNMR method does not require several scans with varying gradient strength, and it provides a combined diffusion and T₂ contrast. Furthermore, the ultrafast concept can be extended to various other multidimensional LNMR experiments, which will provide detailed information about the dynamics and exchange processes of cell metabolites.

Probing molecular dynamics with hyperpolarized ultrafast Laplace NMR using a low-field, single-sided magnet

Laplace NMR (LNMR) offers deep insights on diffusional and rotational motion of molecules. The so-called "ultrafast" approach, based on spatial data encoding, enables one to carry out a multidimensional LNMR experiment in a single scan, providing from 10 to 1000-fold acceleration of the experiment. Here, we demonstrate the feasibility of ultrafast diffusion-T2 relaxation correlation (D-T2) measurements with a mobile, low-field, relatively low-cost, single-sided NMR magnet. We show that the method can probe a broad range of diffusion coefficients (at least from 10-8 to 10-12 m2 s-1) and reveal multiple components of fluids in heterogeneous materials. The single-scan approach is demonstrably compatible with nuclear spin hyperpolarization techniques because the time-consuming hyperpolarization process does not need to be repeated. Using dynamic nuclear polarization (DNP), we improved the NMR sensitivity of water molecules by a factor of 105 relative to non-hyperpolarized NMR in the 0.3 T field of the single-sided magnet. This enabled us to acquire a D-T2 map in a single, 22 ms scan, despite the low field and relatively low mole fraction (0.003) of hyperpolarized water. Consequently, low-field, hyperpolarized ultrafast LNMR offers significant prospects for advanced, mobile, low-cost and high-sensitivity chemical and medical analysis.

Structure and dynamics elucidation of ionic liquids using multidimensional Laplace NMR

We demonstrate the ability of multidimensional Laplace NMR (LNMR), comprising relaxation and diffusion experiments, to reveal essential information about microscopic phase structures and dynamics of ionic liquids that is not observable using conventional NMR spectroscopy or other techniques.

NMR quantification of diffusional exchange in cell suspensions with relaxation rate differences between intra and extracellular compartments

Water transport across cell membranes can be measured non-invasively with diffusion NMR. We present a method to quantify the intracellular lifetime of water in cell suspensions with short transverse relaxation times, T₂, and also circumvent the confounding effect of different T₂ values in the intra- and extracellular compartments. Filter exchange spectroscopy (FEXSY) is specifically sensitive to exchange between compartments with different apparent diffusivities. Our investigation shows that FEXSY could yield significantly biased results if differences in T₂ are not accounted for. To mitigate this problem, we propose combining FEXSY with diffusion-relaxation correlation experiment, which can quantify differences in T₂ values in compartments with different diffusivities. Our analysis uses a joint constrained fitting of the two datasets and considers the effects of diffusion, relaxation and exchange in both experiments. The method is demonstrated on yeast cells with and without human aquaporins.

Sensitivity and resolution of two-dimensional NMR diffusion-relaxation measurements

The performance of 2D NMR diffusion-relaxation measurements for fluid typing applications is analyzed. In particular, we delineate the region in the diffusion - relaxation plane that can be determined with a given gradient strength and homogeneity, and compare the performance of the single and double echo encoding with the stimulated echo diffusion encoding. We show that the diffusion editing based approach is able to determine the diffusion coefficient only if the relaxation time T2 exceeds a cutoff value T2,cutoff, that scales like T2,cutoff∝g(-2/3)D(-1/3). For stimulated echo encoding, the optimal diffusion encoding times (Td and δ), that provide the best diffusion sensitivity, rely only on the T1/T2 ratios and not on the diffusion coefficients of the fluids or the applied gradient strengths. Irrespective of T1, for high enough gradients (i.e. when γ(2)g(2)DT2(3)>10(2)), the Hahn echo based encoding is superior to encoding based on the stimulated echo. For weaker gradients, the stimulated echo is superior only if the T1/T2 ratio is much larger than 1. For single component systems, the diffusion sensitivity is not adversely impacted by the uniformity of the gradients and the diffusion distributions can be well measured. The presence of non-uniform gradients can affect the determination of the diffusion distributions when you have two fluids of comparable T2. In such situations the effective single component diffusion coefficient is always closer to the geometric mean diffusion coefficient of the two fluids.

Obtaining T1-T2 distribution functions from 1-dimensional T1 and T2 measurements: The pseudo 2-D relaxation model

We present the pseudo 2-D relaxation model (P2DRM), a method to estimate multidimensional probability distributions of material parameters from independent 1-D measurements. We illustrate its use on 1-D T1 and T2 relaxation measurements of saturated rock and evaluate it on both simulated and experimental T1-T2 correlation measurement data sets. Results were in excellent agreement with the actual, known 2-D distribution in the case of the simulated data set. In both the simulated and experimental case, the functional relationships between T1 and T2 were in good agreement with the T1-T2 correlation maps from the 2-D inverse Laplace transform of the full 2-D data sets. When a 1-D CPMG experiment is combined with a rapid T1 measurement, the P2DRM provides a double-shot method for obtaining a T1-T2 relationship, with significantly decreased experimental time in comparison to the full T1-T2 correlation measurement.

Ultrafast multidimensional Laplace NMR for a rapid and sensitive chemical analysis

Traditional nuclear magnetic resonance (NMR) spectroscopy relies on the versatile chemical information conveyed by spectra. To complement conventional NMR, Laplace NMR explores diffusion and relaxation phenomena to reveal details on molecular motions. Under a broad concept of ultrafast multidimensional Laplace NMR, here we introduce an ultrafast diffusion-relaxation correlation experiment enhancing the resolution and information content of corresponding 1D experiments as well as reducing the experiment time by one to two orders of magnitude or more as compared with its conventional 2D counterpart. We demonstrate that the method allows one to distinguish identical molecules in different physical environments and provides chemical resolution missing in NMR spectra. Although the sensitivity of the new method is reduced due to spatial encoding, the single-scan approach enables one to use hyperpolarized substances to boost the sensitivity by several orders of magnitude, significantly enhancing the overall sensitivity of multidimensional Laplace NMR.

Laplace NMR provides complementary information to traditional NMR, such as details of molecular motion. Here, the authors report a correlation experiment capable of providing information on the physical environment of molecules while enhancing the chemical resolution and greatly reducing the experiment times.

Ultrafast two-dimensional NMR relaxometry for investigating molecular processes in real time

Nuclear spin-lattice (T1) and spin-spin (T2) relaxation times provide versatile information about the dynamics and structure of substances, such as proteins, polymers, porous media, and so forth. Multidimensional experiments increase the information content and resolution of NMR relaxometry, but they also multiply the measurement time. To overcome this issue, we present an efficient strategy for a single-scan measurement of a 2D T1-T2 correlation map. The method shortens the experimental time by one to three orders of magnitude as compared to the conventional method, offering an unprecedented opportunity to study molecular processes in real-time. We demonstrate that, despite the tremendous speed-up, the T1-T2 correlation maps determined by the single-scan method are in good agreement with the maps measured by the conventional method. The concept of the single-scan T1-T2 correlation experiment is applicable to a broad range of other multidimensional relaxation and diffusion experiments.

Novel 1H low field nuclear magnetic resonance applications for the field of biodiesel

BACKGROUND

Biodiesel production has increased dramatically over the last decade, raising the need for new rapid and non-destructive analytical tools and technologies. 1H Low Field Nuclear Magnetic Resonance (LF-NMR) applications, which offer great potential to the field of biodiesel, have been developed by the Phyto Lipid Biotechnology Lab research team in the last few years.

RESULTS

Supervised and un-supervised chemometric tools are suggested for screening new alternative biodiesel feedstocks according to oil content and viscosity. The tools allowed assignment into viscosity groups of biodiesel-petrodiesel samples whose viscosity is unknown, and uncovered biodiesel samples that have residues of unreacted acylglycerol and/or methanol, and poorly separated and cleaned glycerol and water. In the case of composite materials, relaxation time distribution, and cross-correlation methods were successfully applied to differentiate components. Continuous distributed methods were also applied to calculate the yield of the transesterification reaction, and thus monitor the progress of the common and in-situ transesterification reactions, offering a tool for optimization of reaction parameters.

CONCLUSIONS

Comprehensive applied tools are detailed for the characterization of new alternative biodiesel resources in their whole conformation, monitoring of the biodiesel transesterification reaction, and quality evaluation of the final product, using a non-invasive and non-destructive technology that is new to the biodiesel research area. A new integrated computational-experimental approach for analysis of 1H LF-NMR relaxometry data is also presented, suggesting improved solution stability and peak resolution.

Effect of calcium in brine on salt diffusion and water distribution of Mozzarella cheese during brining

A soft, pasty, high-moisture surface defect occurs with progressive brining of Mozzarella cheese. Addition of calcium is traditionally used to prevent this defect but the underlying mechanism is not clear. Mozzarella cheese was formed into a cylinder inside brine on its plane surface to ensure semi-infinite, unidirectional mass transfer and placed into brine containing 0, 0.1, or 0.25% (wt/wt) calcium chloride. To monitor the effect on cheese composition of calcium in brine, we measured calcium and water contents of the cheese during brining. The extent of calcium loss from the cheese decreased significantly with the addition of calcium. Addition of calcium to a final concentration of 0.25% decreased the loss of calcium from 94.13 to 18.22% from the outside region of the cheese after 30 d, and the water content of the cheese was decreased from 67.8 to 48.8%. To further elucidate the effect of calcium in brine, the Boltzmann method was used to determine the effective diffusion coefficient value, and low-field nuclear magnetic resonance was used to measure the cheese transversal relaxation time. The migration of calcium interfered with salt diffusion. At the end of brining, the amount of water bound to the protein of the cheese significantly increased. Addition of calcium to a final concentration of 0.25% diminished the proportion of bound water by 20.96%. In conclusion, addition of calcium hinders the diffusion of sodium and modifies the distribution of water in Mozzarella cheese during brining.

Relaxation Exchange in Nanoporous Silica by Low-Field NMR

Investigating the 2D-T 2-T 2-relaxation exchange of inter-stitial water in a packing of sedimented Stöber-silicate spheres, we come the conclusion that contrary to its behaviour in macro-pores, water confined in nano-pores of silica exhibits enhanced diffusivity. The 2D-experiments, performed at different temperatures, reveal a temperature-dependent bimodal relaxation distribution and two-site relaxation exchange. Our recently introduced kinetic multi-site exchange model is applied to derive the according exchange rates. The resulting Arrhenius plot produces an exchange activation energy of 7 kJ/mol, which is well below the hydrogen bond energy or the activation energy for self-diffusion of water in the bulk. A possible hopping-mechanism as the source of enhanced proton-diffusion in nanoporous silica is discussed, as well as its significance to mass transfer in porous media.

Exchange relaxometry of flow at small Péclet numbers in a glass bead pack

In this paper we consider low Péclet number flow in bead packs. A series of relaxation exchange experiments has been conducted and evaluated by ILT analysis. In the resulting correlation maps, we observed a collapse of the signal and a translation towards smaller relaxation times with increasing flow rates, as well as a signal tilt with respect to the diagonal. In the discussion of the phenomena we present a mathematical theory for relaxation exchange experiments that considers both diffusive and advective transport. We perform simulations based on this theory and discuss them with respect to the conducted experiments.

Microbial and algal alginate gelation characterized by magnetic resonance

Advanced magnetic resonance (MR) relaxation and diffusion correlation measurements and imaging provide a means to non-invasively monitor gelation for biotechnology applications. In this study, MR is used to characterize physical gelation of three alginates with distinct chemical structures; an algal alginate, which is not O-acetylated but contains poly guluronate (G) blocks, bacterial alginate from Pseudomonas aeruginosa, which does not have poly-G blocks, but is O-acetylated at the C2 and/or C3 of the mannuronate residues, and alginate from a P. aeruginosa mutant that lacks O-acetyl groups. The MR data indicate that diffusion-reaction front gelation with Ca(2+) ions generates gels of different bulk homogeneities dependent on the alginate structure. Shorter spin-spin T(2) magnetic relaxation times in the alginate gels that lack O-acetyl groups indicate stronger molecular interaction between the water and biopolymer. The data characterize gel differences over a hierarchy of scales from molecular to system size.

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.

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.

Structural properties of 2D NMR relaxation spectra of diffusive systems

Much has been learnt and speculated about the form of 2D NMR relaxation spectra of diffusive systems. Herein we show that the eigen-modes formalism can help to establish a number of fundamental structural properties, i.e. symmetries, overall intensities, signs and relative intensities of the diagonal and cross components, of such spectra, on which one can safely rely in analysing experimental data. More specifically, we prove that the correlation T(1)-T(2) spectra will always have negative peaks, thus making questionable the nowadays wide spread strategy in developing inverse Laplace transformation algorithms.

Propagator-resolved 2D exchange in porous media in the inhomogeneous magnetic field

We present a propagator-resolved 2D exchange spectroscopy technique for observing fluid motion in a porous medium. The susceptibility difference between the matrix and the fluid is exploited to produce an inhomogeneous internal magnetic field, causing the Larmor frequency to change as molecules migrate. We test our method using a randomly packed monodisperse 100 microm diameter glass bead matrix saturated with distilled water. Building upon previous 2D exchange spectroscopy work we add a displacement dimension which allows us to obtain 2D exchange spectra that are defined by both mixing time and spatial displacement rather than by mixing time alone. We also simulate our system using a Monte Carlo process in a random nonpenetrating monodisperse bead pack, finding good agreement with experiment. A simple analytic model is used to interpret the NMR data in terms of a characteristic length scale over which molecules must diffuse to sample the inhomogeneous field distribution.

Counterion condensation and effective charge of poly(styrenesulfonate)

The effective charge of poly(styrenesulfonate) has been investigated by diffusion and electrophoresis nuclear magnetic resonance (NMR). While the electrophoretic mobility is determined in the electrophoresis NMR experiment, the hydrodynamic friction is determined from diffusion NMR using Einstein's formula. On the timescale of the NMR experiment a steady state is reached, which results from the force balance between the electric field and the hydrodynamic friction from that the effective charge is calculated without any further model. For the monomer and short polymers the effective charge is equal to the nominal charge, the difference increases with an increasing degree of polymerisation. Increasing the ionic strength of the solution leads to enhanced counterion condensation. If the dielectric constant of the solution is lowered, condensation of counterions is enhanced as well. A lowered effective charge results in reduced repelling forces along the polymer chain and thus in a more compact conformation of the polymer as reflected in the hydrodynamic size. The effective charge of poly(styrenesulfonate) has been studied experimentally as a function of the degree of polymerisation, of the ionic strength and the dielectric constant of the solution.

Observation of molecular migration in porous media using 2D exchange spectroscopy in the inhomogeneous magnetic field

We present a new method for observing fluid diffusion in a porous medium. The method employs 2D exchange spectroscopy for molecules diffusing in the presence of local magnetic field inhomogeneities, in our case distilled water in various sized glass bead packs. Our experiment involves an acquisition and evolution time domain with the two Fourier domains corresponding to the spectral distribution of local fields. We show that exchange in the internal magnetic field can be seen in a 2D spectrum with a characteristic time on the order of that required to diffuse 0.15 sphere diameters with similar behavior found for computer simulations. The method is potentially useful for studying the internal migrations in more complicated systems such as sandstones or other porous media.

Distributions of transverse relaxation times for soft-solids measured in strongly inhomogeneous magnetic fields

The single-sided NMR-MOUSE sensor that operates in highly inhomogeneous magnetic fields is used to record a CPMG (1)H transverse relaxation decay by CPMG echo trains for a series of cross-linked natural rubber samples. Effective transverse relaxation rates 1/T(2,short) and 1/T(2,long) were determined by a bi-exponential fit. A linear dependence of transverse relaxation rates on cross-link density is observed for medium to large values of cross-link density. As an alternative to multi-exponential fits the possibility to analyze the dynamics of soft polymer network in terms of multi-exponential decays via the inverse Laplace transformation was studied. The transient regime and the effect of the T(1)/T(2) ratio in inhomogeneous static and radiofrequency magnetic fields on the CPMG decays were studied numerically using a dedicated C++ program to simulate the temporal and spatial dependence of the CPMG response. A correction factor T(2)/T(2,eff) is derived as a function of the T(1)/T(2) ratio from numerical simulations and compared with earlier results from two different well logging devices. High-resolution T(1)-T(2) correlations maps are obtained by two-dimensional Laplace inversion of CPMG detected saturation recovery curves. The T(1)-T(2) experimental correlations maps were corrected for the T(1)/T(2) effect using the derived T(2)/T(2,eff) correction factor.

Compartment size estimation with double wave vector diffusion-weighted imaging

Double wave vector diffusion weighting uses gradients along two different directions between excitation and acquisition. It has been shown theoretically that for restricted diffusion the signal amplitude in such an experiment can depend on the angle between the two gradient vectors. The highest amplitude is obtained with antiparallel orientation, and the amplitude difference between parallel and antiparallel gradient orientations depends on the compartment size. The validity of this description is experimentally tested for water between polymer beads, for radish, and for porcine spinal cord, using a clinical MR system with limited gradient strength. The results indicate that the phenomenon is observable; however, the size of the signal difference is considerably diminished when compared with theory. This is attributed to violations of the approximating conditions underlying the theoretical description and to free diffusion contributions. It is concluded that the effect could successfully be used as a basis for developing a new noninvasive method for assessing cell size.

Restricted diffusion and exchange of water in porous media: average structure determination and size distribution resolved from the effect of local field gradients on the proton NMR spectrum

NMR Pulsed field gradient measurements of the restrained diffusion of confined fluids constitute an efficient method to probe the local geometry in porous media. In most practical cases, the diffusion decay, when limited to its principal part, can be considered as Gaussian leading to an apparent diffusion coefficient. The evolution of the latter as a function of the diffusion interval yields average information on the surface/volume ratio of porosities and on the tortuosity of the network. In this paper, we investigate porous model systems of packed spheres (polystyrene and glass) with known mean diameter and polydispersity, and, in addition, a real porous polystyrene material. Applying an Inverse Laplace Transformation in the second dimension reveals an evolution of the apparent diffusion coefficient as a function of the resonance frequency. This evolution is related to a similar evolution of the transverse relaxation time T2. These results clearly show that each resonance frequency in the water proton spectrum corresponds to a particular magnetic environment produced by a given pore geometry in the porous media. This is due to the presence of local field gradients induced by magnetic susceptibility differences at the liquid/solid interface and to slow exchange rates between different pores as compared to the frequency differences in the spectrum. This interpretation is nicely confirmed by a series of two-dimensional exchange experiments.

Hydrodynamic Size and Charge of Polyelectrolyte Complexes

Polyelectrolyte complexes have a wide range of applications for surface modification and flocculation and sorption of organic molecules from solutions. As an example, complexes between poly(diallyl dimethyl ammonium chloride) and poly(styrene sulfonate) have been investigated by diffusion and electrophoresis NMR. The formation of primary or soluble complexes is monitored. The hydrodynamic size is characterized by the hydrodynamic radius, calculated from the diffusion coefficient determined by pulsed field gradient NMR. In the combination with electrophoresis NMR, the effective charge of the molecules and complexes is determined. The hydrodynamic size of the primary complex is smaller than that of the pure polyelectrolyte of the larger molecular weight, in the present case poly(styrene sulfonate), in solution, since charges are compensated by the oppositely charged polyelectrolyte and hence the repelling forces diminish. The effective charge of the complexes is drastically reduced.

Propagator resolved transverse relaxation exchange spectroscopy

We present in this communication a novel propagator-resolved transverse relaxation exchange experiment. This experiment enhances the previous technique of transverse relaxation exchange by enabling spatial resolution. Hence, we are able to obtain separate, and remarkably different, T2-T2 exchange plots, corresponding to different spatial displacement of the spin bearing water molecules in a porous sand matrix. This experiment is the first to combine two inverse Laplace dimensions with a Fourier dimension, opening the door to a host of new experiments combining Fourier and inverse Laplace spectroscopy.

Effective charge of polyelectrolytes as a function of the dielectric constant of a solution

The combination of diffusion and electrophoresis NMR is applied to determine the effective charge of poly(styrene sulfonate) in solution. While electrophoresis NMR yields the electrophoretic mobility of the molecules in solution, the hydrodynamic friction is determined from diffusion NMR. From the force balance between electrostatic force and hydrodynamic friction, the effective charge of the molecule is determined free of any model. In the present study poly(styrene sulfonate) has been investigated in mixtures of water and methanol of varying composition. The lower dielectric constant in the mixtures with high methanol content results in a drastically reduced effective charge of the polyelectrolytes. The reduced effective charge along the polymer chain is the reason for a much more compact conformation of the polyelectrolyte, which is seen in a smaller hydrodynamic size of the molecule.

Improved Halbach sensor for NMR scanning of drill cores

A lightweight Halbach magnet system for use in nuclear magnetic resonance (NMR) studies on drill cores was designed and built. It features an improved homogeneous magnetic field with a strength of 0.22 T and a maximum accessible sensitive volume. Additionally, it is furnished with a sliding table for automatic scans of cylindrical samples. This device is optimized for nondestructive online measurements of porosity and pore size distributions of water-saturated full cylindrical and split semicylindrical drill cores of different diameters. The porosity of core plugs with diameters from 20 to 80 mm can be measured routinely using exchangeable radiofrequency coils. Advanced NMR techniques that provide 2D T(1)-T(2) correlations with an average measurement time of 30 min and permeability estimates can be performed with a special insert suitable for small core plugs with diameter and length of 20 mm.

On the use of 2D correlation and exchange NMR spectroscopy in organic porous materials

Two-dimensional (2D) nuclear magnetic resonance (NMR) methods for the investigation of correlation and exchange have been introduced in recent years and have been applied to a range of different systems. Here, we report on the use of 2D NMR diffusion-diffusion correlation spectroscopy for the investigation of diffusion anisotropy in cellular plant tissues and of diffusion-diffusion exchange spectroscopy for the study of the diffusive exchange of dextran in a dispersion of polyelectrolyte multilayer hollow capsules. Furthermore, diffusion-relaxation correlation spectroscopy was applied to both systems.

Observation of exchange of micropore water in cement pastes by two-dimensional T(2)-T(2) nuclear magnetic resonance relaxometry

The first detailed analysis of the two-dimensional (2D) NMR T(2)-T(2) exchange experiment with a period of magnetization storage between the two T(2) relaxation encoding periods (T(2)-store-T(2)) is presented. It is shown that this experiment has certain advantages over the T(1)-T(2) variant for the quantization of chemical exchange. New T(2)-store-T(2) 2D 1H NMR spectra of the pore water within white cement paste are presented. Based on these spectra, the exchange rate of water between the two smallest porosity reservoirs is estimated for the first time. It is found to be of the order of 5 ms{-1}. Further, a careful estimate of the pore sizes of these reservoirs is made. They are found to be of the order of 1.4 nm and 10-30 nm , respectively. A discussion of the results is developed in terms of possible calcium silicate hydrate products. A water diffusion coefficient inferred from the exchange rate and the cement particle size is found to compare favorably with the results of molecular-dynamics simulations to be found in the literature.