Gaussian Phase Approximation Research Articles

NMR diffusion in restricted environment approached by a fractional Langevin model

The diffusion motion of spin-bearing molecules is considerably affected in confined environments. In Nuclear Magnetic Resonance (NMR) experiments, under the presence of magnetic field gradients, this movement can be encoded in spin phase and the NMR signal attenuation due to diffusion can be evaluated. This paper considered this effect in both normal and anomalous diffusion by means of the Langevin stochastic model within the Gaussian phase approximation (GPA) for a constant gradient temporal profile. The phenomenological approach illustrates the emergence of fractional order exponential decay in the NMR signal and retrieves the classical result from Hanh echo.

Signatures of microstructure in gradient-echo andspin-echo signals.

To determine whether the spatial scale and magnetic susceptibility of microstructure can be evaluated robustly from the decay of gradient-echo and spin-echo signals. Gradient-echo and spin-echo images were acquired from suspensions of spherical polystyrene microbeads of 10, 20, and 40 μm nominal diameter. The sizes of the beads and their magnetic susceptibility relative to the medium were estimated from the signal decay curves, using a lookup table generated from Monte Carlo simulations and an analytic model based on the Gaussian phase approximation. Fitting Monte Carlo predictions to spin-echo data yielded acceptable estimates of microstructural parameters for the 20 and 40 μm microbeads. Using gradient-echo data, the Monte Carlo lookup table provided satisfactory parameter estimates for the 20 μm beads but unstable results for the diameter of the largest beads. Neither spin-echo nor gradient-echo data allowed accurate parameter estimation for the smallest beads. The analytic model performed poorly over all bead sizes. Microstructural sources of magnetic susceptibility produce distinctive non-exponential signatures in the decay of gradient-echo and spin-echo signals. However, inverting the problem to extract microstructural parameters from the signals is nontrivial and, in certain regimes, ill-conditioned. For microstructure with small characteristic length scales, parameter estimation is hampered by the difficulty of acquiring accurate data at very short echo times. For microstructure with large characteristic lengths, the gradient-echo signal approaches the static-dephasing regime, where it becomes insensitive to size. Applicability of the analytic model was further limited by failure of the Gaussian phase approximation for all but the smallest beads.

Diffusion pore imaging in the presence of extraporal water

Diffusion-weighted imaging (DWI) is a powerful, non-invasive tool which is widely used in clinical routine. Mostly, apparent diffusion coefficient maps are acquired, which cannot be related directly to cellular structure. More recently it was shown that DWI is able to reconstruct pore shapes using a specialized magnetic field gradient scheme so that cell size distributions may be obtained. So far, artificial systems have been used for experimental demonstration without extraporal signal components and relatively low gradient amplitudes. The aim of this study was to investigate the feasibility of diffusion pore imaging in the presence of extraporal fluids and to develop correction methods for the effects arising from extraporal signal contributions. Monte Carlo simulations and validation experiments on a 14.1 T NMR spectrometer equipped with a dedicated diffusion probe head were performed. Both by using a filter gradient approach suppressing extraporal signal components as well as by using post-processing methods relying on the Gaussian phase approximation, it was possible to reconstruct pore space functions in the presence of extraporal fluids with little to no deviations from the expectations. These results may be a significant step towards application of diffusion pore imaging to biological samples.

In vivo evaluation of heme and non-heme iron content and neuronal density in human basal ganglia

Non-heme iron is an important element supporting the structure and functioning of biological tissues. Imbalance in non-heme iron can lead to different neurological disorders. Several MRI approaches have been developed for iron quantification relying either on the relaxation properties of MRI signal or measuring tissue magnetic susceptibility. Specific quantification of the non-heme iron can, however, be constrained by the presence of the heme iron in the deoxygenated blood and contribution of cellular composition. The goal of this paper is to introduce theoretical background and experimental MRI method allowing disentangling contributions of heme and non-heme irons simultaneously with evaluation of tissue neuronal density in the iron-rich basal ganglia. Our approach is based on the quantitative Gradient Recalled Echo (qGRE) MRI technique that allows separation of the total R2* metric characterizing decay of GRE signal into tissue-specific (R2t*) and the baseline blood oxygen level-dependent (BOLD) contributions. A combination with the QSM data (also available from the qGRE signal phase) allowed further separation of the tissue-specific R2t* metric in a cell-specific and non-heme-iron-specific contributions. It is shown that the non-heme iron contribution to R2t* relaxation can be described with the previously developed Gaussian Phase Approximation (GPA) approach. qGRE data were obtained from 22 healthy control participants (ages 26–63 years). Results suggest that the ferritin complexes are aggregated in clusters with an average radius about 100nm comprising approximately 2600 individual ferritin units. It is also demonstrated that the concentrations of heme and non-heme iron tend to increase with age. The strongest age effect was seen in the pallidum region, where the highest age-related non-heme iron accumulation was observed.

Spin echoes: full numerical solution and breakdown of approximative solutions

The spin echo signal from vessels in Krogh’s capillary model as well in the random distribution vessel model are studied by numerically solving the Bloch–Torrey equation. A comparison is made with the Gaussian local phase approximation, the Gaussian phase approximation and the strong-collision approximation. Differences between the Gaussian local phase approximation and the Gaussian phase approximation are explained. In the intermediate diffusion regime, the full numerical solution shows oscillations which are absent in any of the approximate solutions. In the limit of large diffusion coefficients, where the approximations become exact, the signal shows a linear-exponential decay governed by a single parameter. The features of the exact numerical solution can be explained by an analytically solvable discrete two-level model. There is a one-to-one correspondence between the different diffusion regimes and the three cases of the damped harmonic oscillator.

Spin dephasing in the Gaussian local phase approximation.

The Brownian motion of spins diffusing in an inhomogeneous magnetic field created by susceptibility effects is considered. Applying the correct form of the Gaussian approximation, the method allows calculating the local magnetization as well as the free induction decay for all diffusion regimes. The phase accumulated during the diffusional motion is treated by an averaging over all possible trajectories in terms of the Gaussian local phase approximation. Predictions of the Gaussian local phase approximation are compared with the Gaussian phase approximation for diffusion in a constant gradient in a slab, a cylinder, and a sphere. The Gaussian local phase approximation, thereby, shows several advantages compared to the Gaussian phase approximation: it is also valid in the static dephasing regime, predicts correctly both transverse components of the magnetization, and yields information about the local magnetization.

Precise Inference and Characterization of Structural Organization (PICASO) of tissue from molecular diffusion

Inferring the microstructure of complex media from the diffusive motion of molecules is a challenging problem in diffusion physics. In this paper, we introduce a novel representation of diffusion MRI (dMRI) signal from tissue with spatially-varying diffusivity using a diffusion disturbance function. This disturbance function contains information about the (intra-voxel) spatial fluctuations in diffusivity due to restrictions, hindrances and tissue heterogeneity of the underlying tissue substrate. We derive the short- and long-range disturbance coefficients from this disturbance function to characterize the tissue structure and organization. Moreover, we provide an exact relation between the disturbance coefficients and the time-varying moments of the diffusion propagator, as well as their relation to specific tissue microstructural information such as the intra-axonal volume fraction and the apparent axon radius. The proposed approach is quite general and can model dMRI signal for any type of gradient sequence (rectangular, oscillating, etc.) without using the Gaussian phase approximation. The relevance of the proposed PICASO model is explored using Monte-Carlo simulations and in-vivo dMRI data. The results show that the estimated disturbance coefficients can distinguish different types of microstructural organization of axons.

New insights about time-varying diffusivity and its estimation from diffusion MRI.

Characterizing the relation between the applied gradient sequences and the measured diffusion MRI signal is important for estimating the time-dependent diffusivity, which provides important information about the microscopic tissue structure. In this article, we extend the classical theory of Stepišnik for measuring time-dependent diffusivity under the Gaussian phase approximation. In particular, we derive three novel expressions which represent the diffusion MRI signal in terms of the mean-squared displacement, the instantaneous diffusivity, and the velocity autocorrelation function. We present the explicit signal expressions for the case of single diffusion encoding and oscillating gradient spin-echo sequences. Additionally, we also propose three different models to represent time-varying diffusivity and test them using Monte-Carlo simulations and in vivo human brain data. The time-varying diffusivities are able to distinguish the synthetic structures in the Monte-Carlo simulations. There is also strong statistical evidence about time-varying diffusivity from the in vivo human data set. The proposed theory provides new insights into our understanding of the time-varying diffusivity using different gradient sequences. The proposed models for representing time-varying diffusivity can be utilized to study time-varying diffusivity using in vivo human brain diffusion MRI data. Magn Reson Med 78:763-774, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

The displacement correlation tensor: Microstructure, ensemble anisotropy and curving fibers

Experiments with multiple diffusion wave vectors are known to carry more information than what is available from standard diffusion experiments. Here we consider a special case of this class of pulse sequences, the double wave vector diffusion experiment, and use the cumulant expansion of the signal to introduce the displacement correlation tensor. We discuss its physical interpretation and properties, noting in particular that its short time behavior allows determination of the surface to volume ratio of the pore space. We present a general expression for the displacement correlation tensor, and provide explicit expressions for a few model geometries. We then show that the scatter matrix characterizing the orientation distribution of an ensemble of cylinders is simply related to the displacement correlation tensor. This result is generalized to ensembles of pores with arbitrary shapes allowing a precise formulation of the influence of microstructural and ensemble anisotropy on the double wave vector diffusion signal in the Gaussian phase approximation. Finally, as a new application of the double wave vector diffusion signal, we analyze its behavior in a curving fiber, and suggest that the displacement correlation tensor may be used to estimate sub-voxel fiber curvature and deflection angle. The theoretical results are corroborated by computer simulations.

Use, misuse, and abuse of apparent diffusion coefficients

AbstractThe notion of effective, time‐dependent or, equivalently, apparent diffusion coefficient (ADC) and its use for characterization of porous media are revisited. It is argued that the dynamic ADC, quantifying the mean‐square displacement of spin‐bearing particles, should not be confused with its counterpart measured by a pulsed gradient spin echo technique. The former is an intrinsic characteristic of the medium, independent of the applied magnetic field. In contrast, the spin‐echo ADC depends on the experimental setup (e.g., gradient intensity and temporal profile), raising potential ambiguities in the interpretation of diffusion‐weighted measurements, which may be strongly misleading when the Gaussian phase approximation (GPA) does not hold. The oversimplified use of a single b‐value is criticized. Several fitting models beyond the GPA are discussed. © 2010 Wiley Periodicals, Inc. Concepts Magn Reson Part A 36A: 24–35, 2010.

Nuclear magnetic resonance restricted diffusion between parallel planes in a cosine magnetic field: An exactly solvable model

We propose a theoretical and numerical analysis of restricted diffusion between parallel planes in a cosine magnetic field. The specific choice of this spatial profile as proportional to an eigenfunction of the Laplace operator in this confining geometry considerably simplifies the underlying mathematics. In particular, exact and explicit relations for several moments of the total phase accumulated by diffusing spins are derived. These relations are shown to provide good approximations for the typical case of a linear magnetic field gradient, for which the theoretical analysis was in general limited to the second moment. We study the structure and the properties of the higher order moments which are responsible for the breakdown of the "Gaussian phase approximation" (GPA) at intense magnetic fields. The limits of applicability of the GPA for nonlinear magnetic fields and the transition to the localization regime are discussed. In particular, a diagram of different restricted diffusion regimes is presented.

Displacement power spectrum measurement by CPMG in constant gradient

The modulation of spin phase produced by Carr–Purcell–Meiboom–Gill (CPMG) sequence in combination with constant magnetic field gradient is appropriate to probe the displacement power spectrum (DPS). The spin-echo attenuation is directly proportional to the DPS value at the applied modulation frequency. Relaxation and selective excitation effects can be factored out while probing the DPS. The modulation frequency is adjusted by varying the pulse separation time while the gradient strength and the time of acquisition are kept constant. In designing the experiment gradient strength limitations, imposed by off-resonance effects, as well as limitations arising from using Gaussian phase approximation must be considered. An effective experimental strategy is presented, supported by experimental results for free and restricted diffusion.

Restricted diffusion within a single pore

The time-dependent diffusion of 3He atoms perpendicular to the axis of a single macroscopically large cylindrical pore is studied using a steady (or constant) gradient-recalled echo sequence. Measurements of the effective 3He diffusion coefficient extending from the free-diffusion regime to the motionally averaged regime are presented, and are well-described by analytic solutions to the Bloch–Torrey equation based on the gaussian phase approximation. Our data yield the value 0.140(6) m 2/s for the self diffusion coefficient of 3He at a temperature of 296 K and a pressure of 1.00 Torr. Adaptations of these methods should enable the study of complex pore geometries as model systems.

Averaged propagator of restricted motion from the Gaussian approximation of spin echo

Apart from the propagator theory of spin-echo diffusion measurement, the average of spin phase fluctuation by the cumulant expansion is developed. The expansion to the second order, the Gaussian phase approximation (GPA), is able to explain the diffusion and flow measurement by the spin echo with the generalized magnetic field gradient sequence, which can be designed in a way to extract information not only about the macroscopic self-diffusion coefficient, but also about the details of the molecular velocity correlation spectrum. By taking into account the distribution of local spin properties resulting from restriction to motional by boundaries, the GPA method covers a broad range of spin echo decay, which throws a new light upon the diffraction-like phenomena, different from that, obtained by the averaged propagator method. The corrections of GPA by the higher order terms of cumulant expansion introduce only small changes to the tail of spin echo decay. The paper demonstrates the advantage of the GPA in this generalized form to deal with the space–time correlations of diffusion and flow in porous medium, closing a fictive gap between NMR diffusometry and q-space NMR.

Nuclear magnetic resonance spin echoes for restricted diffusion in an inhomogeneous field: Methods and asymptotic regimes

We develop systematic formulations for calculating the magnetization of spins diffusing in a bounded region in the presence of the surface relaxation and magnetic field inhomogeneity and compute explicitly the relaxation exponent for the Carr–Purcell–Meiboom–Gill spin echoes. The results depend on the echo number n, and three dimensionless parameters: Lρ/LS, D̃0=(LD/LS)2, the dimensionless diffusion constant, and γ̃=LD2LS/LG3=Δωτ, the dimensionless gyromagnetic ratio, where the restriction is characterized by a size LS, the magnetic field inhomogeneity by a dephasing length, LG, the diffusion length during half-echo time by LD, and a length Lρ characterizes the surface relaxation. Here Δω is the line broadening and 2τ is the echo period. Depending on the length scales, three main regimes of decay have been identified: short-time, localization, and motionally averaging regimes (MAv). The short-time and the MAv regimes are described well by the cumulant expansion in terms of powers of the “small” parameter γ̃. We give simple formulas for decay rates in these two asymptotic regimes. We show that the Gaussian phase approximation (GPA), i.e., the exponent up to the second order in γ̃2 in terms of a full eigenmode expansion interpolates well between these two regimes. In the localization regime, the decay exponent depends on a fractional power, γ̃2/3, denoting a breakdown of the GPA and a breakdown of the cumulant expansion in terms of γ̃.

Relaxation of nuclear magnetization in a nonuniform magnetic field gradient and in a restricted geometry

We study the influence of restriction on Carr-Purcell-Meiboom-Gill spin echo response of magnetization of spins diffusing in a bounded region in the presence of a nonuniform magnetic field gradient. We consider two fields in detail-a parabolic field which, like the uniform-gradient field, scales with the system size, and a cosine field which remains bounded. Corresponding to three main length scales, the pore size, L(S), the dephasing length, L(G), and the diffusion length during half-echo time, L(D), we identify three main regimes of decay of the total magnetization: motionally averaged, localization, and short-time. In the short-time regime (L(D) << L(S), L(G)), we confirm that the leading order behavior is controlled by the average of the square of the gradient, (nablaB(z))(2), and in the motionally averaged regime (MAv), where L(S) << L(D), L(G), by (integral dxB(z))(2). We verify numerically that two different fields for which those two averages are identical result in very similar decay profiles not only in the limits of short and long times but also in the intermediate times, with important practical implications. In the motionally averaged regime we found that previous estimates of the decay exponent for the parabolic field, based on a soft-boundary condition, are significantly altered in the presence of a more realistic, hard wall. We find the scaling of the decay exponent in the MAv regime with pore size to be L(2)(S) for the cosine field and L(6)(S) for the parabolic field, as contrasted with the linear gradient scaling of L(4)(S). In the localization regime, for both the cosine and the parabolic fields, the decay exponent depends on a fractional power of the gradient, implying a breakdown of the second cumulant or the Gaussian phase approximation. We also examined the validity of time-evolving the total magnetization according to a distribution of effective local gradients and found that such approximation works well only in the short-time regime and breaks down strongly for long times. Copyright 2000 Academic Press.

Spin echoes of nuclear magnetization diffusing in a constant magnetic field gradient and in a restricted geometry

We study the influence of restriction on Carr–Purcell–Meiboom–Gill spin echoes response of magnetization of spins diffusing in a bounded region in the presence of a constant magnetic field gradient. Depending on three main length scales: LS pore size, LG dephasing length and LD diffusion length during half-echo time, three main regimes of decay have been identified: free, localization and motionally averaging regime. In localization regime, the decay exponent depends on a fractional power (2/3) of the gradient, denoting a strong breakdown of the second cumulant or the Gaussian phase approximation (GPA). In the other two regimes, the exponent depends on the gradient squared, and the GPA holds. We find that the transition from the localization to the motionally averaging regime happens when the magnetic field gradients approach special values, corresponding to branch points of the eigenvalues. Transition from one regime to another as a function of echo number for a certain range of parameters is discussed. In this transition region, the signal shows large oscillations with echo number. For large n, asymptotic behavior sets in as a function of n for the decay exponent per echo. This is true for all values of the parameters LS, LG, and LD.