Gradient Pulse Width Research Articles

Diffusion Spectrum of Polymer Melt Measured by Varying Magnetic Field Gradient Pulse Width in PGSE NMR.

The translational motion of polymers is a complex process and has a big impact on polymer structure and chemical reactivity. The process can be described by the segment velocity autocorrelation function or its diffusion spectrum, which exhibit several characteristic features depending on the observational time scale—from the Brownian delta function on a large time scale, to complex details in a very short range. Several stepwise, more-complex models of translational dynamics thus exist—from the Rouse regime over reptation motion to a combination of reptation and tube-Rouse motion. Accordingly, different methods of measurement are applicable, from neutron scattering for very short times to optical methods for very long times. In the intermediate regime, nuclear magnetic resonance (NMR) is applicable—for microseconds, relaxometry, and for milliseconds, diffusometry. We used a variation of the established diffusometric method of pulsed gradient spin-echo NMR to measure the diffusion spectrum of a linear polyethylene melt by varying the gradient pulse width. We were able to determine the characteristic relaxation time of the first mode of the tube-Rouse motion. This result is a deviation from a Rouse model of polymer chain displacement at the crossover from a square-root to linear time dependence, indicating a new long-term diffusion regime in which the dynamics of the tube are also described by the Rouse model.

General pulsed-field gradient signal attenuation expression based on a fractional integral modified-Bloch equation

Anomalous diffusion has been investigated in many polymer and biological systems. The analysis of PFG anomalous diffusion relies on the ability to obtain the signal attenuation expression. However, the general analytical PFG signal attenuation expression based on the fractional derivative has not been previously reported. Additionally, the reported modified-Bloch equations for PFG anomalous diffusion in the literature yielded different results due to their different forms. Here, a new integral type modified-Bloch equation based on the fractional derivative for PFG anomalous diffusion is proposed, which is significantly different from the conventional differential type modified-Bloch equation. The merit of the integral type modified-Bloch equation is that the original properties of the contributions from linear or nonlinear processes remain unchanged at the instant of the combination. From the modified-Bloch equation, the general solutions are derived, which includes the finite gradient pulse width (FGPW) effect. The numerical evaluation of these PFG signal attenuation expressions can be obtained either by the Adomian decomposition, or a direct integration method that is fast and practicable. The theoretical results agree with the continuous-time random walk (CTRW) simulations performed in this paper. Additionally, the relaxation effect in PFG anomalous diffusion is found to be different from that in PFG normal diffusion. The new modified-Bloch equations and their solutions provide a fundamental tool to analyze PFG anomalous diffusion in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI).

Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches

Pulsed-field gradient (PFG) diffusion experiments can be used to measure anomalous diffusion in many polymer or biological systems. However, it is still complicated to analyze PFG anomalous diffusion, particularly the finite gradient pulse width (FGPW) effect. In practical applications, the FGPW effect may not be neglected, such as in clinical diffusion magnetic resonance imaging (MRI). Here, two significantly different methods are proposed to analyze PFG anomalous diffusion: the effective phase-shift diffusion equation (EPSDE) method and a method based on observing the signal intensity at the origin. The EPSDE method describes the phase evolution in virtual phase space, while the method to observe the signal intensity at the origin describes the magnetization evolution in real space. However, these two approaches give the same general PFG signal attenuation including the FGPW effect, which can be numerically evaluated by a direct integration method. The direct integration method is fast and without overflow. It is a convenient numerical evaluation method for Mittag-Leffler function-type PFG signal attenuation. The methods here provide a clear view of spin evolution under a field gradient, and their results will help the analysis of PFG anomalous diffusion.

Analyzing signal attenuation in PFG anomalous diffusion via a non-Gaussian phase distribution approximation approach by fractional derivatives.

Anomalous diffusion exists widely in polymer and biological systems. Pulsed-field gradient (PFG) techniques have been increasingly used to study anomalous diffusion in nuclear magnetic resonance and magnetic resonance imaging. However, the interpretation of PFG anomalous diffusion is complicated. Moreover, the exact signal attenuation expression including the finite gradient pulse width effect has not been obtained based on fractional derivatives for PFG anomalous diffusion. In this paper, a new method, a Mainardi-Luchko-Pagnini (MLP) phase distribution approximation, is proposed to describe PFG fractional diffusion. MLP phase distribution is a non-Gaussian phase distribution. From the fractional derivative model, both the probability density function (PDF) of a spin in real space and the PDF of the spin's accumulating phase shift in virtual phase space are MLP distributions. The MLP phase distribution leads to a Mittag-Leffler function based PFG signal attenuation, which differs significantly from the exponential attenuation for normal diffusion and from the stretched exponential attenuation for fractional diffusion based on the fractal derivative model. A complete signal attenuation expression Eα(-Dfbα,β*) including the finite gradient pulse width effect was obtained and it can handle all three types of PFG fractional diffusions. The result was also extended in a straightforward way to give a signal attenuation expression of fractional diffusion in PFG intramolecular multiple quantum coherence experiments, which has an nβ dependence upon the order of coherence which is different from the familiar n2 dependence in normal diffusion. The results obtained in this study are in agreement with the results from the literature. The results in this paper provide a set of new, convenient approximation formalisms to interpret complex PFG fractional diffusion experiments.

Analyzing signal attenuation in PFG anomalous diffusion via a modified Gaussian phase distribution approximation based on fractal derivative model

Pulsed field gradient (PFG) has been increasingly employed to study anomalous diffusions in Nuclear Magnetic Resonance (NMR) and Magnetic Resonance Imaging (MRI). However, the analysis of PFG anomalous diffusion is complicated. In this paper, a fractal derivative model based modified Gaussian phase distribution method is proposed to describe PFG anomalous diffusion. By using the phase distribution obtained from the effective phase shift diffusion method based on fractal derivatives, and employing some of the traditional Gaussian phase distribution approximation techniques, a general signal attenuation expression for free fractional diffusion is derived. This expression describes a stretched exponential function based attenuation, which is distinct from both the exponential attenuation for normal diffusion obtained from conventional Gaussian phase distribution approximation, and the Mittag-Leffler function based attenuation for anomalous diffusion obtained from fractional derivative. The obtained signal attenuation expression can analyze the finite gradient pulse width (FGPW) effect. Additionally, it can generally be applied to all three types of PFG fractional diffusions classified based on time derivative order alpha and space derivative order beta. These three types of fractional diffusions include time-fractional diffusion, space-fractional diffusion, and general fractional diffusion. The results in this paper are consistent with reported results based on effective phase shift diffusion equation method and instantaneous signal attenuation method. This method provides a new, convenient approximation formalism for analyzing PFG anomalous diffusion experiments.

Instantaneous signal attenuation method for analysis of PFG fractional diffusions

An instantaneous signal attenuation (ISA) method for analyzing pulsed field gradient (PFG) fractional diffusion (FD) has been developed, which is modified from the propagator approach developed in 2001 by Lin et al. for analyzing PFG normal diffusion. Both, the current ISA method and the propagator method have the same fundamental basis that the total signal attenuation (SA) is the accumulation of all the ISA, and the ISA is the average SA of the whole diffusion system at each moment. However, the manner of calculating ISA is different. Unlike the use of the instantaneous propagator in the propagator method, the current method directly calculates ISA as A(K(t′),t′+dt′)/A(K(t′),t′), where A(K(t′),t′+dt′) and A(K(t′),t′) are the SA. This modification makes the current method applicable to PFG FD as the instantaneous propagator may not be obtainable in FD. The ISA method was applied to study PFG SA including the effect of finite gradient pulse widths (FGPW) for free FD, restricted FD and the FD affected by a non-homogeneous gradient field. The SA expressions were successfully obtained for all three types of free FDs while other current methods still have difficulty in obtaining all of them. The results from this method agree with reported results such as that obtained by the effective phase shift diffusion equation (EPSDE) method. The M-Wright phase distribution approximation was also used to derive an SA expression for time FD as a comparison, which agrees with ISA method. Additionally, the continuous-time random walk (CTRW) simulation was performed to simulate the SA of PFG FD, and the simulation results agree with the analytical results. Particularly, the CTRW simulation results give good support to the analytical results including FGPW effect for free FD and restricted time FD based on a fractional derivative model where there have been no corresponding theoretical reports to date. The theoretical SA expressions including FGPW obtained here such as Eα,1-Df2∫0tKβ(t′)dt′α may be applied to analyze PFG FD in polymer or biological systems with improved accuracy where SGP approximation cannot be satisfied. The method can perhaps provide new insight to FD MRI and hence benefit the development of diffusion biomarkers based on fractional derivative.

K and q Dedicated to Paul Callaghan

The symbols k and q denote wave numbers in scattering experiments as well as in NMR imaging. Their exploration in NMR is intimately linked to the legacy of Paul Callaghan with his books Magnetic Resonance Microscopy and Translational Dynamics & Magnetic Resonance (Oxford University Press, Oxford 1991 and 2011) placing their focus with their titles on k and q, respectively. Some aspects of k and q have been revisited in the Paul Callaghan lecture of the author at the ISMAR Conference in Shanghai in 2015, which are reviewed here. In particular, there are two definitions of q, one relating to diffusive displacement (q) and the other to coherent flow (qv). Concerning the latter, it turns out, that in the short gradient pulse limit, the common anti-phase pulsed field-gradient scheme can be replaced with schemes employing three and more gradient pulses, which derive from differentiation rules in numerical analysis. Practical gradient modulation schemes with finite gradient pulse widths follow from these to measure velocity with improved accuracy. This approach can be expanded to acceleration and higher order transport coefficients with applications to measurements of flow and potentially also restricted diffusion.

An effective phase shift diffusion equation method for analysis of PFG normal and fractional diffusions

Pulsed field gradient (PFG) diffusion measurement has a lot of applications in NMR and MRI. Its analysis relies on the ability to obtain the signal attenuation expressions, which can be obtained by averaging over the accumulating phase shift distribution (APSD). However, current theoretical models are not robust or require approximations to get the APSD. Here, a new formalism, an effective phase shift diffusion (EPSD) equation method is presented to calculate the APSD directly. This is based on the idea that the gradient pulse effect on the change of the APSD can be viewed as a diffusion process in the virtual phase space (VPS). The EPSD has a diffusion coefficient, Kβ(t)D radβ/sα, where α is time derivative order and β is a space derivative order, respectively. The EPSD equations of VPS are built based on the diffusion equations of real space by replacing the diffusion coefficients and the coordinate system (from real space coordinate to virtual phase coordinate). Two different models, the fractal derivative model and the fractional derivative model from the literature were used to build the EPSD fractional diffusion equations. The APSD obtained from solving these EPSD equations were used to calculate the PFG signal attenuation. From the fractal derivative model the attenuation is exp(−γβgβδβDf1tα), a stretched exponential function (SEF) attenuation, while from the fractional derivative model the attenuation is Eα,1(−γβgβδβDf2tα), a Mittag–Leffler function (MLF) attenuation. The MLF attenuation can be reduced to SEF attenuation when α=1, and can be approximated as a SEF attenuation when the attenuation is small. Additionally, the effect of finite gradient pulse widths (FGPW) is calculated. From the fractal derivative model, the signal attenuation including FGPW effect is exp-Df1∫0τKβ(t)dtα. The results obtained in this study are in good agreement with the results in literature. Several expressions that describe signal attenuation have not been reported and that can be of great importance for the PFG experiments. This EPSD equation method provides a new, simple path to calculate signal attenuation of PFG NMR experiments.

J-modulation effects in DOSY experiments and their suppression: The Oneshot45 experiment

Diffusion-ordered spectroscopy (DOSY) is a powerful NMR method for identifying compounds in mixtures. DOSY experiments are very demanding of spectral quality; even small deviations from expected behaviour in NMR signals can cause significant distortions in the diffusion domain. This is a particular problem when signals overlap, so it is very important to be able to acquire clean data with as little overlap as possible. DOSY experiments all suffer to a greater or lesser extent from multiplet phase distortions caused by J-modulation, requiring a trade-off between such distortions and gradient pulse width. Multiplet distortions increase spectral overlap and may cause unexpected and misleading apparent diffusion coefficients in DOSY spectra. These effects are described here and a simple and effective remedy, the addition of a 45° purging pulse immediately before the onset of acquisition to remove the unwanted anti-phase terms, is demonstrated. As well as affording significantly cleaner results, the new method allows much longer diffusion-encoding pulses to be used without problems from J-modulation, and hence greatly increases the range of molecular sizes that can be studied for coupled spin systems. The sensitivity loss is negligible and the added phase cycling is modest. The new method is illustrated for a widely-used general purpose DOSY pulse sequence, Oneshot.

Towards beyond-1 GHz solution NMR: Internal 2H lock operation in an external current mode

We have commenced a project to develop a beyond-1 GHz solution NMR spectrometer using a HTS coil. Due to a small residual resistance present in the HTS conductor and joint resistance between conductors, a stable persistent current sufficient for NMR measurements is unlikely. Therefore, a current has to be supplied to the HTS coil from an external power supply. The ripple of an external power supply causes a field fluctuation which must be stabilized. In this study we show results of NMR measurements using a 500–600 MHz NMR in such an external current mode: the field fluctuations are stabilized by an internal 2H lock. The field fluctuation from the external power supply comprises a major field fluctuation component at low frequencies, 0.003–0.005 Hz, and superimposed minor field ripples at 2 Hz and 50 Hz. The former limits the time interval of the internal 2H lock, while the latter generates sidebands in the NMR spectrum. Sideband and baseline noise are controlled by appropriate selection of the feedback loop parameters of the lock. The quality of the 1D-solution NMR spectra observed in external current mode is equivalent to that obtained in persistent current mode. However, if the feedback loop time is as short as the gradient pulse width, refocusing of the NMR signal is lost and NMR peaks disappear. The 2D-NOESY and the 2D-HSQC spectra of ubiquitin in an external current mode have been acquired. The quality of the 2D spectra is equivalent to those obtained in persistent current mode; i.e. the internal 2H lock operates stably over an experimental time interval of 40–50 min. To realize a beyond-1 GHz NMR spectrometer, further investigations must be made of (i) the long term stability of a DC power supply, (ii) the enhancement of the compensation field limit for the internal 2H lock, (iii) the extension of the helium refill time interval, and (iv) a method to correct the field homogeneity in the external current mode.

Spin Echo Analysis of Restricted Diffusion under Generalized Gradient Waveforms for Spherical Pores with Relaxivity and Interconnections

AbstractThe problem of restricted diffusion in spherical pores is examined under conditions of finite gradient pulse width in pulsed gradient spin echo (PGSE)‐NMR experiments. Closed form analytic expressions are derived, and the case of interconnected pores is briefly examined. An expression, based on the pore hopping approximation, is presented that predicts the echo attenuation for diffusion between pores under any gradient waveform.

Effects of finite-width pulses in the pulsed-field gradient measurement of the diffusion coefficient in connected porous media

We analytically compute the apparent diffusion coefficient D app for an open restricted geometry, such as an extended porous medium, for the case of a pulsed-field gradient (PFG) experiment with finite-width pulses. In the short- and long-time limits, we give explicit, model-independent expressions that correct for the finite duration of the pulses and can be used to extract the pore surface-to-volume ( S/ V) ratio as well as the tortuosity. For all times, we compute D app using a well-established model form of the actual time-dependent diffusion coefficient D( t) that can be obtained from an ideal narrow-pulse PFG. We compare D app and D( t) and find that, regardless of pulse widths and geometry-dependent parameters, the two quantities deviate by less than 20%. These results are in sharp contrast with the studies on closed geometries [J. Magn. Reson. A 117 (1995) 209], where the effects of finite gradient-pulse widths are large. The analytical results presented here can be easily adapted for different pulse protocols and time sequences.

Validation of diffusion spectrum magnetic resonance imaging with manganese-enhanced rat optic tracts and ex vivo phantoms

Diffusion spectrum imaging (DSI) has been demonstrated to resolve crossing axonal fibers by mapping the probability density function of water molecules diffusion at each voxel. However, the accuracy of DSI in defining individual fiber orientation and the validity of Fourier relation under finite gradient pulse widths are not assessed yet. We developed an ex vivo and an in vivo model to evaluate the error of DSI with gradient pulse widths being relatively short and long, respectively. The ex vivo model was a phantom comprising sheets of parallel capillaries filled with water. Sheets were stacked on each other with capillaries crossed at 45° or 90°. High-resolution T2-weighted images (T2WI) of the phantom served as a reference for the orientation of intersecting capillaries. In the in vivo model, manganese ions were infused into rats’ optic tracts. The optic tracts were enhanced on T1-weighted images (T1WI) and served as a reference for the tract orientation. By comparing DSI with T2WI, the deviation angles between the primary orientation of diffusion spectrum and the 90° and 45° phantoms were 1.19° ± 4.82° and −0.71° ± 4.91°, respectively. By comparing DSI with the T1WI of rat optic tracts, the deviation angle between primary orientation of diffusion spectrum and optic tracts was −0.41° ± 6.18°. In addition, two sequences of DSI using short and long gradient pulses were performed in a rat brain. The bias of the primary orientation between these two sequences was approximately 10°. In conclusion, DSI can resolve crossing fiber orientation accurately. The effect of finite gradient pulse widths on the primary orientation is not critical.

Diffusion measurements with the pulsed gradient nonlinear spin echo method

Spin echo signal attenuation by diffusion is examined for coherence evolution in the course of ordinary pulsed gradient spin echoes and for nonlinear evolution in the presence of a spatially modulated demagnetizing field. It is shown, that, for given field gradient pulse widths (or equivalently for a given gradient strength), echo attenuation by diffusion is much more efficient for nonlinear echoes than for Hahn echoes. Remarkably, in the case of nonlinear echoes the refocusing process is spoiled by diffusion not only during the gradient intervals but also thereafter. The effect of displacements occurring in the gradient intervals is enhanced according to the order of the nonlinear echo the pulse sequence is adjusted for. A second attenuation mechanism takes place after the gradient pulses due to displacements in the presence of the spatially modulated demagnetizing field. This effect even occurs when the gradient intervals are too short to contribute. A complete formalism is presented describing all features of the test experiments. It is shown that nuclear magnetic resonance diffusometry based on nonlinear echo signals permits one to measure small diffusion coefficients with moderate field gradients. Nonlinear echo experiments demonstrate that the coherence pathway dominating by far is of a purely single-quantum nature.

NMR imaging, NMR diffraction and applications of pulsed gradient spin echoes in porous media

Diffraction ideas were first introduced to NMR by Mansfield and Grannell. Here we compare “Mansfield (k-space) diffraction” with the “diffusive (q-space) diffraction,” noting their similarities and differences. The diffusive-diffraction concept is extremely helpful in elucidating the Pulsed Gradient Spin Echo NMR experiment for fluid molecules in porous media and has been applied successfully to the model isolated pore (the “single slit” case) and the orientationally disordered interconnected pore glass (the “powder grating” case). This picture is based on the idea that the PGSE experiment has a scattering analogy in which a wavevector ( q) is conjugate to the displacement of the spins over the time, D, between the gradient pulses. Recently, the theory has been extended to allow for both wall relaxation effects and finite gradient pulse width effects.

Effects of Finite Gradient-Pulse Widths in Pulsed-Field-Gradient Diffusion Measurements

The effects of finite gradient-pulse widths on NMR diffusion measurements for fluids in restricting geometries are studied. It is shown that the echo amplitude is the spatial Fourier transform of a "center-of-mass" propagator, which reduces to the usual diffusion propagator in the limit of zero pulse widths. A finite gradient-pulse width δ effectively changes the pore shape, making isolated pores appear smaller than their actual size. The diffraction analogy still holds for long diffusion times, with the fluid density ρ( r) being replaced by p cm( r, δ). This quantity, the "center-of-mass distribution function," is the spatial probability distribution of the center of mass of Brownian trajectories of duration δ in the pore space. For a periodic pore space, "Bragg" peaks still appear in the amplitude at the reciprocal lattice vectors. The heights of these peaks are enhanced for small δ but reduced for δ. A number of results valid for small δ and piecewise smooth pore surfaces are presented.

Rapid determination of water droplet size distributions by PFG-NMR

In 1972 Packer and Rees described a method for the determination of droplet size distributions in emulsions using the pulsed field gradient (PFG) NMR spin—echo technique. This paper describes an improved method, based on this earlier work, which is directed at the routine determination of water droplet size distributions of water-in-oil emulsions. The improvement is based on the recognition that a set of echo attenuation values ( R) as a function of the field gradient pulse width, obtained under conditions where R is independent of the time allowed for diffusion, contains all the necessary information on the water droplet size distribution. The parameters of a log-normal distribution of water droplet sizes are easily obtained from a small number of measurements which are then compared with a matrix of sets of theoretical echo attenuations, representing different combinations of these parameters, using a simple computer program. The method is described and examples of results are shown. The effect of temperature on the measurements is discussed.