Measurements of restricted diffusion using an oscillating gradient spin-echo sequence

An oscillating gradient spin-echo (OGSE) pulse sequence was used to measure the apparent diffusion coefficient (D(app)) of water in the short diffusion time regime in the presence of restrictions. The diffusion coefficients of water in a simple water sample and a water and oil mixture were measured to be the same for different periods of the gradient oscillation, as expected when there are no restriction effects. The D(app) of water in the spaces between closely packed beads was also measured as a function of the gradient oscillation periods in the range 11 to 80 ms. The D(app) of water in restricted systems varies with the period of the gradient oscillation and the dispersion depends on the scale of the restriction. For a sample of packed beads of diameter 9.1 +/- 0.7 microm, the pore surface-to-volume ratio was estimated experimentally by this method to be 1.3 +/- 0.1 microm(-1), corresponding to a mean pore diameter of 6.4 +/- 0.7 microm. A Monte Carlo computer simulation of the NMR OGSE signal from the spins diffusing in a system of compartments was also implemented and the D(app) demonstrated similar behavior with gradient oscillation periods.     

Probing restrictive diffusion dynamics at short time scales

Diffusion imaging gradients serve to spectrally filter the temporally evolving diffusion tensor. In this framework, the design of diffusion sensitizing gradients is reduced to the problem of adequately sampling q-space in the spectral domain. The practical limitations imposed by the requirement for delta-function type diffusion-sensitizing gradients to adequately sample q-space, can be relaxed if these impulse gradients are replaced with chirped oscillatory gradients. It is well known that in many systems of interest, dispersion of velocity will itself produce a peak in the velocity correlation function near w=0, while restricted diffusion will manifest itself in the dispersion spectrum at higher frequencies. In this paper, chirped diffusion-sensitizing gradients are proposed and analytically shown to yield an efficient sampling of q-space in a manner that asymptotically approaches that using delta-function diffusion-sensitizing gradient. The challenge is the consequent reduction in diffusion sensitivity as one probes higher frequency dynamics. This problem is addressed by restricting the gradient power to a spectral bandwidth corresponding to the diffusion spectral range of the underlying restrictive geometry. Simultaneous imaging of diffusion and flow at microscopic resolution and at temporally resolvable diffusion time scales thus becomes possible in vivo. Simulations and experiments validate the proposed approach.     

CPMG relaxation by diffusion with constant magnetic field gradient in a restricted geometry: numerical simulation and application

Carr-Purcell-Meiboom-Gill (CPMG) measurements are the primary nuclear magnetic resonance (NMR) technique used for evaluating formation properties and reservoir fluid properties in the well logging industry and laboratory sample analysis. The estimation of bulk volume irreducible (BVI), permeability, and fluid type relies on the accurate interpretation of the spin-spin relaxation time (T(2)) distribution. The interpretation is complicated when spin's self-diffusion in an inhomogeneous field and restricted geometry becomes dominant. The combined effects of field gradient, diffusion, and a restricted geometry are not easily evaluated analytically. We used a numerical method to evaluate the dependence of the free and restricted diffusion on the system parameters in the absence of surface relaxation, which usually can be neglected for the non-wetting fluids (e.g., oil or gas). The parameter space that defines the relaxation process is reduced to two dimensionless groups: D* and tau*. Three relaxation regimes: free diffusion, localization, and motionally averaging regimes are identified in the (log(10)D*, log(10)tau*) domain. The hypothesis that the normalized magnetization, M*, relaxes as a single exponential with a constant dimensionless relaxation time T*(2) is justified for most regions of the parameter space. The numerical simulation results are compared with the analytical solutions from the contour plots of T*(2). The locations of the boundaries between different relaxation regimes, derived from equalizing length scales, are challenged by observed discrepancies between numerical and analytical solutions. After adjustment of boundaries by equalizing T*(2), numerical simulation result and analytical solution match each other for every relaxation regime. The parameters, fluid diffusivity and pore length, can be estimated from analytical solutions in the free diffusion and motionally averaging regimes, respectively. Estimation of the parameters near the boundaries of the regimes may require numerical simulation.     

Spectral characterization of diffusion in porous media by the modulated gradient spin echo with CPMG sequence

Carr-Purcell-Meiboom-Gill train of radiofrequency pulses applied to spins in the constant magnetic field gradient is an efficient variant of the modulated magnetic field gradient spin echo method, which provides information about molecular diffusion in the frequency-domain instead in the time-domain as with the two-pulse gradient spin echo. The frequency range of novel technique is broad enough to sample the power spectrum of displacement fluctuation in water-saturated pulverized silica (SiO(2)) and provides comprehensive information about the molecular restricted motion as well as about the structure of medium.     

Short-time restricted diffusion in a static gradient and the attenuation of individual coherence pathways

We experimentally explore some of the implications of a recent theoretical study [J. Magn. Reson. 64 (2003) 145] for the measurement of restricted diffusion in connected porous media in a static gradient. In particular, we examine how restriction affects the short-time attenuation of different coherence pathways, all excited with the same sequence of slice-selective radiofrequency (RF) pulses, and how the various pathways make the transition to the long-time or tortuosity regime. We confirm that every pathway contains equivalent diffusional information and, for short times, yields the surface-to-volume ratio (S/V) of the confining space. We find also, in agreement with the theoretical predictions, that different pathways are controlled by different time scales and, thus, exhibit different sensitivity to restriction. This property might be exploited when designing optimal sequences to study restricted motion.     

CAGEBIRD: improving the GBIRD filter with a CPMG sequence

An improvement of the GBIRD-filter is presented. The current approach utilizes Carr-Purcell-Meiboom-Gill type pulse train during the BIRD delay. The method enables recording of purely absorptive 1D spectrum using only one isotope editing element. In the current method, the parent signal leakage due to JHH evolution during the BIRD delay is considerably smaller than in the conventional approach. As a consequence, the t1-noise is smaller also in 2D applications, such as GBIRD-filtered HSQC.     

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.     

Measurement of the spatial backscattering impulse-response at short length scales with polarized enhanced backscattering

In this Letter, we describe an easy to implement technique to measure the spatial backscattering impulse-response at length scales shorter than a transport mean free path with resolution of better than 10 μm using the enhanced backscattering phenomenon. This technique enables spectroscopic measurements throughout the visible range and sensitivity to all polarization channels. Through a combination of Monte Carlo simulations and experimental measurements of latex microspheres, we explore the various sensitivities of our technique to both intrinsic sample properties and extrinsic instrumental properties. We conclude by demonstrating the extraordinary sensitivity of our technique to the shape of the scattering phase function, including higher order shape parameters than the anisotropy factor (or first moment).     

Velocity autocorrelation spectra of fluid in porous media measured by the CPMG sequence and constant magnetic field gradient

Carr-Purcell-Meiboom-Gill (CPMG) train of radiofrequency pulses applied to spins in the constant magnetic field gradient is an efficient variant of the modulated magnetic field gradient spin echo method, which provides information about molecular diffusion in the frequency domain instead of in the time domain as with the two-pulse gradient spin echo. The frequency range of this novel technique is broad enough to sample the power spectrum of displacement fluctuation in water-saturated pulverized silica (SiO2) and provides comprehensive information about the molecular restricted motion as well as about the structure of the medium.     

Dynamic NMR q-space studies of microstructure with the multigrade CPMG sequence

A new approach to q-space studies of microstructure is proposed, which exploits the combined information contained in the water proton transverse relaxation time distribution and the frequency dependence of the apparent water diffusivity in heterogeneous systems. Using an automated two-dimensional multigrade CPMG sequence, both the pulse spacing and the amplitude of the applied field gradient are varied systematically and used to measure the frequency and wave vector dependence of the multiple exponential echo decay constants and amplitudes. Undesirable crossterms in the applied and background field gradients are eliminated by a simple procedure involving a sign reversal in the applied gradient. Nonlinear, local susceptibility-induced field gradients are shown to lead to enhanced, frequency-dependent apparent water diffusivities that are sensitive to the local microstructure.     

Probing lateral diffusion in lipid membranes on nanoscale by PFG NMR with high gradient strength

This work demonstrates the feasibility of noninvasive studies of diffusion on a submicrometer length scale in aligned model lipid membranes using pulsed field gradient nuclear magnetic resonance with ultrahigh (up to 35 T/m) gradient strength. Application of such gradients allows the use of sufficiently small diffusion times under conditions of narrow-pulse approximation. As a result, monitoring anomalous or restricted diffusion in lipid membranes on a length scale in the range of 100 nm becomes possible. The ability to study diffusion in lipid membranes on this length scale is very important because it is comparable with the size of biologically relevant domains (i.e., rafts), which are believed to exist in biomembranes.     

Transition to hexagonal pattern under the variation of intrinsic length scales of a reaction diffusion system

The intrinsic length scales of a reaction diffusion system (Gierer-Meinhardt model) is varied by quasi-statically changing the diffusion constant of the activator and a transition from rolls to hexagon is detected. The transition is hysteretic or first order like. From stability analysis, we also analytically show the possibility of such transitions.Received: 15 June 2004, Published online: 3 August 2004PACS: 87.10. + e General theory and mathematical aspects - 47.70.Fw Chemically reactive flows     

Vacancy-mediated and exchange diffusion in a Pb/Cu(111) surface alloy: concurrent diffusion on two length scales

Pb diffuses in a Pb/Cu(111) surface alloy predominantly by exchange with surface vacancies and, much less frequently, by exchange with thermal Cu adatoms. Because the infrequent adatom exchanges transport Pb atoms much farther, both processes affect observations of Pb transport in the Pb/Cu(111) surface alloy.     

Mapping the intracellular fraction of water by varying the gradient pulse length in q-space diffusion MRI

Finite gradient pulse lengths are traditionally considered a nuisance in q-space diffusion NMR and MRI, since the simple Fourier relation between the acquired signal and the displacement probability is invalidated. Increasing the value of the pulse length leads to an apparently smaller value of the estimated compartment size. We propose that q-space data at different gradient pulse lengths, but with the same effective diffusion time, can be used to identify and quantify components with free or restricted diffusion from multiexponential echo decay curves obtained on cellular systems. The method is demonstrated with experiments on excised human brain white matter and a series of model systems with well-defined free, restricted, and combined free and restricted diffusion behavior. Time-resolved diffusion MRI experiments are used to map the spatial distribution of the intracellular fraction in a yeast cell suspension during sedimentation, and observe the disappearance of this fraction after a heat treatment.     

Background gradient suppression in stimulated echo NMR diffusion studies using magic pulsed field gradient ratios

By evaluating the spin echo attenuation for a generalized 13-interval PFG NMR sequence consisting of pulsed field gradients with four different effective intensities (F(p/r) and G(p/r)), magic pulsed field gradient (MPFG) ratios for the prepare (G(p)/F(p)) and the read (G(r)/F(r)) interval are derived, which suppress the cross term between background field gradients and the pulsed field gradients even in the cases where the background field gradients may change during the z-store interval of the pulse sequence. These MPFG ratios depend only on the timing of the pulsed gradients in the pulse sequence and allow a convenient experimental approach to background gradient suppression in NMR diffusion studies with heterogeneous systems, where the local properties of the (internal) background gradients are often unknown. If the pulsed field gradients are centered in the tau-intervals between the pi and pi/2 rf pulses, these two MPFG ratios coincide to eta=G(p/r)/F(p/r)=1-8/[1+(1/3)(delta/tau)(2)]. Since the width of the pulsed field gradients (delta) is bounded by 0< or =delta< or =tau, eta can only be in the range of 5< or =-eta< or =7. The predicted suppression of the unwanted cross terms is demonstrated experimentally using time-dependent external gradients which are controlled in the NMR experiment as well as spatially dependent internal background gradients generated by the magnetic properties of the sample itself. The theoretical and experimental results confirm and extend the approach of Sun et al. (J. Magn. Reson. 161 (2003) 168), who recently introduced a 13-interval type PFG NMR sequence with two asymmetric pulsed magnetic field gradients suitable to suppress unwanted cross terms with spatially dependent background field gradients.     

Simulations of multiphase flows with multiple length scales using moving mesh interface tracking with adaptive meshing

In multiphase flows, the length scales of thin regions, such as thin films between nearly touching drops and thin threads formed during the interface pinch-off, are usually several orders of magnitude smaller than the size of the drops. In this paper, a number of extra length criteria for adaptive meshes are developed and implemented in the moving mesh interface tracking method to solve these multiple-length-scale problems with high fidelity. A nominal length scale based on the solutions of Laplace’s equations with the unit normal vectors of surfaces as the boundary conditions is proposed for the adaptive mesh refinement in the thin regions. For almost flat interfaces/boundaries which are near to the thin regions, the averaged length of the interior edges sharing the two nodes with the boundary edge is introduced for the mesh adaptation. The averaged length of the interfacial edges is used for the interior elements near the interfaces but outside of the thin regions. For the interior mesh away from the interfaces/boundaries, different averaged length scales based on the initial mesh are employed for the adaptive mesh refining and coarsening. Numerous cases are simulated to demonstrate the capability of the proposed schemes in handling multiple length scales, which include the relaxation and necking of an elongated droplet, droplet–droplet head-on approaching, droplet-wall interactions, and a droplet pair in a shear flow. The smallest length resolved for the thin regions is three orders of magnitude smaller than the largest characteristic length of the problem.     

Heat source reconstruction from noisy temperature fields using a gradient anisotropic diffusion filter

This paper presents a post-processing technique for noisy temperature maps based on a gradient anisotropic diffusion (GAD) filter in the context of heat source reconstruction. The aim is to reconstruct heat source maps from temperature maps measured using infrared (IR) thermography. Synthetic temperature fields corrupted by added noise are first considered. The GAD filter, which relies on a diffusion process, is optimized to retrieve as well as possible a heat source concentration in a two-dimensional plate. The influence of the dimensions and the intensity of the heat source concentration are discussed. The results obtained are also compared with two other types of filters: averaging filter and Gaussian derivative filter. The second part of this study presents an application for experimental temperature maps measured with an IR camera. The results demonstrate the relevancy of the GAD filter in extracting heat sources from noisy temperature fields.     

Determining the static factor in films with a strain gradient by an x-ray diffraction method

Graphical dependences allowing the static-factor parameters to be found from the angular distance between the maxima on the reflection curve are obtained on the basis of the model of a crystal with constant strain gradient in the kinematic approximation of x-ray diffraction theory. X-ray diffraction and theoretical data for the static factor are given for film samples of the solid solution InGaP/(111)GaAs and an autoepitaxial Si/(111) film with boron diffusion.Kabardino-Balkarsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 52–54, June, 1993.