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A Simple Matrix Formalism for Spin Echo Analysis of Restricted Diffusion under Generalized Gradient Waveforms
Abstract
A simple mathematical formalism is presented which allows closed form expressions for the echo attenuation, E(q), in spin echo diffusion experiments, for practically all gradient waveforms and for the case of restricted diffusion in enclosing pores, with or without wall relaxation. The method, which derives from the multiple propagator approach of A. Caprihan et al. (1996, J. Magn. Reson. A 118, 94), depends on the representation of the gradient waveform by a succession of sharp gradient impulses. It leads to E(q) being expressed as a product of matrix operators corresponding quite naturally to the successive sandwich of phase evolution and Brownian migration events. Simple expressions are given for the case of the finite width gradient pulse PGSE experiment, the CPMG pulse train used in frequency-domain modulated gradient spin echo NMR, and the case of a sinusoidal waveform. The finite width gradient pulse PGSE and CPMG pulse trains are evaluated for the case of restricted diffusion between parallel reflecting planes. The former results agree precisely with published computer simulations while the latter calculation provides useful insight regarding the spectral density approach to impeded Brownian motion. Copyright 1997 Academic Press. Copyright 1997Academic Press
... At low gradient strength, perturbation methods are applicable and we shall discuss their limitations when we present bifurcation points in the spectrum. On the numerical side, different computational techniques for dMRI have been developed, including finite difference/finite elements PDE solvers [31][32][33][34][35], Monte-Carlo simulations [36][37][38][39], and spectral methods (matrix formalism) [2,[40][41][42]. However, all of these techniques are numerically challenging at high gradients because of the fine spatial scales involved in the signal formation, as well as the weak signal. ...
... The main idea of the method is to replace the continuous-time gradient profile by a series of infinitely narrow gradient pulses: computing the magnetization is then reduced to solving a series of diffusion problems with different (pseudo-)periodic boundary conditions. Note that the idea of replacing a gradient profile by multiple narrow pulses was introduced and exploited in [40,44,45] to compute the magnetization in bounded domains. One will see that the case of periodic domains is much more subtle. ...
... The gradient direction is arbitrary and may change over time as well. Since the medium is bounded along y and z, the effect of g y and g z can be implemented using standard spectral methods [2,[40][41][42]. Two main schemes were proposed in the literature, in which the gradient is either replaced by (i) a collection of narrow pulses [40,44,45] (similar to our method but without restrictions introduced by periodicity); or (ii) a stepwise function [2,41,42]. ...
The Bloch–Torrey equation governs the evolution of the transverse magnetization in diffusion magnetic resonance imaging, where two mechanisms are at play: diffusion of spins (Laplacian term) and their precession in a magnetic field gradient (imaginary potential term). In this paper, we study this equation in a periodic medium: a unit cell repeated over the nodes of a lattice. Although the gradient term of the equation is not invariant by lattice translations, the equation can be analyzed within a single unit cell by replacing a continuous-time gradient profile by narrow pulses. In this approximation, the effects of precession and diffusion are separated and the problem is reduced to the study of a sequence of diffusion equations with pseudo-periodic boundary conditions. This representation allows for efficient numerical computations as well as new theoretical insights into the formation of the signal in periodic media. In particular, we study the eigenmodes and eigenvalues of the Bloch–Torrey operator. We show how the localization of eigenmodes is related to branching points in the spectrum and we discuss low- and high-gradient asymptotic behaviors. The range of validity of the approximation is discussed; interestingly the method turns out to be more accurate and efficient at high gradient, being thus an important complementary tool to conventional numerical methods that are most accurate at low gradients.
... At low gradient strength, perturbation methods are applicable and we shall discuss their limitations when we present bifurcation points in the spectrum. On the numerical side, different computational techniques for dMRI have been developed, including finite difference/finite elements PDE solvers [31][32][33][34][35], Monte-Carlo simulations [36][37][38][39], and spectral methods (matrix formalism) [2,[40][41][42]. However, all of these techniques are numerically challenging at high gradients because of the fine spatial scales involved in the signal formation, as well as the weak signal. ...
... The main idea of the method is to replace the continuous-time gradient profile by a series of infinitely narrow gradient pulses: computing the magnetization is then reduced to solving a series of diffusion problems with different (pseudo-)periodic boundary conditions. Note that the idea of replacing a gradient profile by multiple narrow pulses was introduced and exploited in [40,44,45] to compute the magnetization in bounded domains. One will see that the case of periodic domains is much more subtle. ...
... The gradient direction is arbitrary and may change over time as well. Since the medium is bounded along y and z, the effect of g y and g z can be implemented using standard spectral methods [2,[40][41][42]. Two main schemes were proposed in the literature, in which the gradient is either replaced by (i) a collection of narrow pulses [40,44,45] (similar to our method but without restrictions introduced by periodicity); or (ii) a stepwise function [2,41,42]. ...
The Bloch-Torrey equation governs the evolution of the transverse magnetization in diffusion magnetic resonance imaging, where two mechanisms are at play: diffusion of spins (Laplacian term) and their precession in a magnetic field gradient (imaginary potential term). In this paper, we study this equation in a periodic medium: a unit cell repeated over the nodes of a lattice. Although the gradient term of the equation is not invariant by lattice translations, the equation can be analyzed within a single unit cell by replacing a continuous-time gradient profile by narrow pulses. In this approximation, the effects of precession and diffusion are separated and the problem is reduced to the study of a sequence of diffusion equations with pseudo-periodic boundary conditions. This representation allows for efficient numerical computations as well as new theoretical insights into the formation of the signal in periodic media. In particular, we study the eigenmodes and eigenvalues of the Bloch-Torrey operator. We show how the localization of eigenmodes is related to branching points in the spectrum and we discuss low- and high-gradient asymptotic behaviors. The range of validity of the approximation is discussed; interestingly the method turns out to be more accurate and efficient at high gradient, being thus an important complementary tool to conventional numerical methods that are most accurate at low gradients.
... A pulsed-gradient spin-echo (PGSE) sequence was applied as depicted in Fig. 3 The numerical computation of the signal in a slab and in a cylinder is effectively reduced to that in an interval and a disk, respectively. For these simple shapes, the most efficient and accurate computation of the signal is realized with the matrix formalism [3,[37][38][39], in which the Bloch-Torrey equation is projected onto the basis of explicitly known Laplacian eigenmodes to represent the signal via matrix products and exponentials (see [40,41] for details). The matrix formalism was also used to compute the transverse magnetization in these two domains (see similar computations in [12]). ...
... Note also that Eq. (24) with only the leading term is not accurate (not shown) so that the correction terms in the exponential are indeed important. For a cylinder of a smaller diameter (2R ¼ 2 mm, see Fig. 7), the signal shows some oscillations that are reminiscent of diffusiondiffraction patterns [14,38,[47][48][49]. This is the case discussed in Section 2.3 where two localization pockets overlap because ' g =ð2RÞ is not small enough. ...
... It is well-known that such domains produce non-Gaussian signals, for example in the limit of narrow pulses (e.g. diffusion-diffraction patterns, see [14,38,[47][48][49]). However, in most former studies the signal from the extracellular space was assumed to be Gaussian, and non-Gaussian effects were attributed to multiple contributing pools. ...
In this work we investigate the emergence of the localization regime for diffusion NMR in various geometries: inside slabs, inside cylinders and outside rods arranged on a square array. At high gradients, the transverse magnetization is strongly attenuated in the bulk, whereas the macroscopic signal is formed by the remaining magnetization localized near boundaries of the sample. As a consequence, the signal is particularly sensitive to the microstructure. The theoretical analysis relies on recent mathematical advances on the study of the Bloch-Torrey equation. Experiments were conducted with hyperpolarized xenon-129 gas in 3D-printed phantoms and show an excellent agreement with numerical simulations and theoretical predictions. Our mathematical arguments and experimental evidence indicate that the localization regime with a stretched-exponential decay of the macroscopic signal is a generic feature of diffusion NMR that can be observed at moderately high gradients in most NMR scanners.
... The numerical computation of the signal in a slab and in a cylinder is effectively reduced to that in an interval and a disk, respectively. For these simple shapes, the most efficient and accurate computation of the signal is realized with the matrix formalism [33,34,35,3], in which the Bloch-Torrey equation is projected onto the basis of explicitly known Laplacian eigenfunctions to represent the signal via matrix products and exponentials (see [36,37] for details). The matrix formalism was also used to compute the transverse magnetization in these two domains (see similar computations in [11]). ...
... One observes the perfect agreement between experiment, matrix formalism computation, and asymptotic formula at high gradients (without any fitting parameter).Note also that Eq. (24) with only the leading term is not accurate here so that the correction terms in the exponential are indeed important. For a cylinder of a smaller diameter (2R = 2 mm, see Fig. 7), the signal shows some oscillations that are reminiscent of diffusion-diffraction patterns [13,42,43,44,34]. This is the case discussed in Sec. ...
... It is well-known that such domains produce non-Gaussian signals, for example in the limit of narrow pulses (e.g. diffusion-diffraction patterns, see [13,42,43,44,34]). However, the signal from extracellular space is almost always assumed to be Gaussian, and non-Gaussian effects are attributed to multiple contributing pools. ...
In this work we investigate the emergence of the localization regime for diffusion in various geometries: inside slabs, inside cylinders and around rods arranged on a square array. At high gradients, the transverse magnetization is strongly attenuated in the bulk, whereas the macroscopic signal is formed by the remaining magnetization localized near boundaries of the sample. As a consequence, the signal is particularly sensitive to the microstructure. Our theoretical analysis relies on recent mathematical advances on the study of the Bloch-Torrey equation. Experiments were conducted with hyperpolarized xenon-129 gas in 3D-printed phantoms and show an excellent agreement with numerical simulations and theoretical predictions. Our mathematical arguments and experimental evidence indicate that the localization regime with a stretched-exponential decay of the macroscopic signal is a generic feature of diffusion NMR that can be observed at moderately high gradients in most NMR scanners.
... An OGSE sequence usually consists of two oscillating pulses of duration σ, each containing n periods, hence the frequency is ω = n 2π σ , separated by a time interval τ − σ (Figure 1.1b). For a cosine OGSE [8,15], the profile f (t) is ...
... When water passes very slowly between the cells and the extra-cellular space as compared to the diffusion time, the cell membranes can be approximated as impermeable to water. In this case, various analytical or semi-analytical expressions have been obtained for the dMRI signal arising from inside the cells [8,19,54,55,67,70]. In general, however, cell membranes are permeable to water, and it is important to study the effect of membrane permeability on the dMRI signal by using a richer numerical model. ...
... 8 shows the convergence slope of the signals of the three-compartment models to the signal of the two-compartment model. Although the L 2 − difference between signals of the three-compartment with anisotropic diffusion in the membrane and the two-compartment model is smaller than that between the threecompartment with isotropic diffusion in the membrane and the two-compartment model, both have the first order convergence versus the thickness. ...
Diffusion magnetic resonance imaging (dMRI) is a non-invasive imaging technique that gives a measure of the diffusion characteristics of water in biological tissues, notably, in the brain. The hindrances that the microscopic cellular structure poses to water diffusion are statistically aggregated into the measurable macroscopic dMRI signal. Inferring the microscopic structure of the tissue from the dMRI signal allows one to detect pathological regions and to monitor functional properties of the brain. For this purpose, one needs a clearer understanding of the relation between the tissue microstructure and the dMRI signal. This requires novel numerical tools capable of simulating the dMRI signal arising from complex microscopic geometrical models of tissues. We propose such a numerical method based on linear finite elements that allows for a more accurate description of complex geometries. The finite elements discretization is coupled to the adaptive Runge-Kutta Chebyshev time stepping method. This method, which leads to the second order convergence in both time and space, is implemented on FeniCS C++ platform. We also use the mesh generator Salome to work efficiently with multiple-compartment and periodic geometries. Four applications of the method for studying the dMRI signal inside multi-compartment models are considered. In the first application, we investigate the long-time asymptotic behavior of the dMRI signal and show the convergence of the apparent diffusion coefficient to the effective diffusion tensor computed by homogenization. The second application aims to numerically verify that a two-compartment model of cells accurately approximates the three-compartment model, in which the interior cellular compartment and the extracellular space are separated by a finite thickness membrane compartment. The third application consists in validating the macroscopic Karger model of dMRI signals that takes into account compartmental exchange. The last application focuses on the dMRI signal arising from isolated neurons. We propose an efficient one-dimensional model for accurately computing the dMRI signal inside neurite networks in which the neurites may have different radii. We also test the validity of a semi-analytical expression for the dMRI signal arising from neurite networks.
... We argue that the GPA is applicable for somas even with very large b-values because the soma signal is already substantially attenuated once it deviates from the GPA at which point the overall signal is dominated by neurites. This was verified by comparing with the exact sphere signal from the multiple correlation function (MCF) approach (Callaghan, 1997;Grebenkov, 2008). ...
... This aligns with the Kärger model's predictions with the diffusivity's minor time dependence possibly attributable to somas as incorporated in the SMEX extensions. We also note that taking higher-order effects into account for the somas using the MCF approach (Callaghan, 1997;Grebenkov, 2008) did not affect the results eliminating the concern for that compartment. This is also supported by the simulations, which intrinsically include higher-order effects, for the somas and also the neurites. ...
Diffusion MRI (dMRI) provides contrast that reflect diffusing spins' interactions with microstructural features of biological systems, but its specificity remains limited due to the ambiguity of its relation to the underlying microstructure. To improve specificity, biophysical models of white matter (WM) typically express dMRI signals according to the Standard Model (SM) and have more recently in gray matter (GM) attempted to incorporate cell soma (the SANDI model). The validity of the assumptions underlying these models, however, remains largely undetermined, especially in GM. Observing the models' unique, functional properties, such as the $b^{-1/2}$ power-law associated with 1d diffusion, has emerged as a fruitful strategy for experimental validation. The absence of this signature in GM has been explained by neurite water exchange, non-linear morphology, and/or obscuring soma signal contributions. Here, we present simulations in realistic neurons demonstrating that curvature and branching does not destroy the stick power-law in impermeable neurites, but that their signal is drowned by the soma under typical experimental conditions: Nevertheless, we identify an attainable experimental regime in which the neurite signal dominates. Furthermore, we find that exchange-driven time dependence produces a behavior opposite to that expected from restricted diffusion, thereby providing a functional signature disambiguating the two effects. We present data from dMRI experiments in ex vivo rat brain at ultrahigh field and observe a time dependence consistent with substantial exchange and a GM stick power-law. The first finding suggests significant water exchange while the second suggests a small sub-population of impermeable neurites. To quantify our observations, we harness the K\"arger exchange model and incorporate the corresponding signal time dependence in SM and SANDI.
... Some methods (split propagation [191], random-walk averaging [95], etc.) do treat spin dynamics at the density matrix level, but introduce diffusion externally in an approximate way [192]. ...
... Additional derivatives appear in the diffusion equation in this case because D is no longer-coordinate independent. The implementation in Spinach uses the full Equation(191) and ...
Many recent magnetic resonance imaging experiments involve spin states other than longitudinal or transverse magnetization. For example, singlet state imaging requires two-spin correlations to be taken into account, hyperpolarised imaging should correctly account for multi-spin dynamics, multiple quantum imaging must consider the dynamics of the corresponding coherences, etc. Simulation of such experiments requires accurate quantum mechanical treatment of spin processes together with an accurate treatment of classical processes, such as diffusion and convection. In this thesis I report the theoretical and software infrastructure for magnetic resonance simulations using the Fokker-Planck formalism, which simultaneously accounts for spatial dynamics and quantum mechanical spin processes. Fokker-Planck equation is superior to the usual Liouville-von Neumann equation formalism in that spatial dynamics processes (diffusion, hydrodynamics, magic angle spinning , off-resonance pulses, etc.) are represented by constant matrices that are more convenient from the programming and numerical efficiency point of view than time-dependent Hamiltonians in the Liouville-von Neumann equation formalism. It is demonstrated below that NMR and MRI experiments with elaborate spatial encoding and complicated spatial dynamics are no longer hard to simulate, even in the presence of spin-spin couplings and exotic relaxation effects, such as singlet state symmetry lockout. Versions 2.3 and later of Spinach library support arbitrary stationary flows and arbitrary distributions of anisotropic diffusion tensors in three dimensions simultaneously with Liouville-space description of spin dynamics, chemical kinetics and relaxation processes. The key simulation design decision that has made this possible is the aban-donment of Bloch-Torrey and Liouville-von Neumann formalisms in favour of the Fokker-Planck equation. The primary factors that have facilitated this transition are the dramatic recent improvement in the speed and capacity of digital computers, the emergence of transparent and convenient sparse matrix manipulation methods in numerical linear algebra, and the recent progress in matrix dimension reduction in magnetic resonance simulations. The principal achievement of this thesis is in programming and software engineering-the reader is encouraged to take a look at the Fokker-Planck module of the Spinach library that the work described in this thesis has made possible, and that was programmed as a part of this work: most of the writing performed within this project was writing code.
... Finally, we come to another important representation of the diffusion MRI signal, derived twenty years ago, that projects the magnetization in the basis of the eigenfunctions of the Laplace operator in the imaged domain. This representation goes under the name of matrix formalism (Callaghan 1997, Barzykin 1999, Grebenkov 2007, Drobnjak et al 2011. There are two advantages to the matrix formalism signal representation . ...
{\bf Objective}\\
The complex-valued transverse magnetization due to diffusion-encoding magnetic field gradients
acting on a permeable medium can be modeled \soutnew{}{by} the Bloch-Torrey partial differential equation.
The diffusion MRI signal has a representation in the basis of the Laplace eigenfunctions of the medium.
However, in order to estimate the permeability coefficient from diffusion MRI data, it is desirable that the forward solution can be calculated efficiently for many values of permeability.
\\{\bf Approach}\\
In this paper we propose a new formulation of the permeable diffusion MRI signal representation in the basis of the Laplace eigenfunctions of the same medium where the interfaces are made impermeable.
\\{\bf Main results}\\
We proved the theoretical equivalence between our new formulation and the original formulation in the case that the full eigendecomposition is used. We validated our method numerically and showed promising numerical results when a partial eigendecomposition is used. Two diffusion MRI sequences were used to illustrate the numerical validity of our new method.
\\{\bf Significance}\\
Our approach means that the same basis (the impermeable set) can be used for all permeability values, which reduces the computational time significantly, enabling the study of the effects of the permeability coefficient on the diffusion MRI signal in the future.
... MISST is a semi-analytical simulation tool based on the matrix formalism approach developed by Callaghan [52]. The experimental parameters were set to values that are achievable on a clinical MR system, matching the settings in the in vivo experiments ( = 10 ms, Δ = 62 ms, b = 2 ⋅ 812 s mm −2 , and m = 10.9 ms). ...
Objective:
To study the origin of compartment size overestimation in double diffusion encoding MRI (DDE) in vivo experiments in the human corticospinal tract. Here, the extracellular space is hypothesized to be the origin of the DDE signal. By exploiting the DDE sensitivity to pore shape, it could be possible to identify the origin of the measured signal. The signal difference between parallel and perpendicular diffusion gradient orientation can indicate if a compartment is regular or eccentric in shape. As extracellular space can be considered an eccentric compartment, a positive difference would mean a high contribution to the compartment size estimates.
Materials and methods:
Computer simulations using MISST and in vivo experiments in eight healthy volunteers were performed. DDE experiments using a double spin-echo preparation with eight perpendicular directions were measured in vivo. The difference between parallel and perpendicular gradient orientations was analyzed using a Wilcoxon signed-rank test and a Mann-Whitney U test.
Results:
Simulations and MR experiments showed a statistically significant difference between parallel and perpendicular diffusion gradient orientation signals ([Formula: see text]).
Conclusion:
The results suggest that the DDE-based size estimate may be considerably influenced by the extra-axonal compartment. However, the experimental results are also consistent with purely intra-axonal contributions in combination with a large fiber orientation dispersion.
... In this appendix, we briefly describe the numerical procedure for computing the spectrum of the Bloch-Torrey operator. Here we rely on the so-called matrix formalism [84][85][86][87][88], in which the differential operator B(g) = −∇ 2 − igx is represented by an infinite-dimensional matrix Λ − igB, where Λ is the diagonal matrix formed by the eigenvalues of the Laplace operator −∇ 2 with Neumann boundary condition, while the matrix B n,n ′ =Ω dr u n (r) x u * n ′ (r) (A.1) ...
We investigate the peculiar feature of non-Hermitian operators, namely, the existence of spectral branch points (also known as exceptional or level crossing points), at which two (or many) eigenmodes collapse onto a single eigenmode and thus loose their completeness. Such branch points are generic and produce non-analyticities in the spectrum of the operator, which, in turn, result in a finite convergence radius of perturbative expansions based on eigenvalues and eigenmodes that can be relevant even for Hermitian operators. We start with a pedagogic introduction to this phenomenon by considering the case of $2\times 2$ matrices and explaining how the analysis of more general differential operators can be reduced to this setting. We propose an efficient numerical algorithm to find spectral branch points in the complex plane. This algorithm is then employed to show the emergence of spectral branch points in the spectrum of the Bloch-Torrey operator $-\nabla^2 - igx$, which governs the time evolution of the nuclear magnetization under diffusion and precession. We discuss their mathematical properties and physical implications for diffusion nuclear magnetic resonance experiments in general bounded domains.
... NMR approaches for the determination of diffusion coefficients are described, for example, in following books and reviews [31,32,[40][41][42][43]. The pulsed field gradientstimulated echo (PFGSTE) method [44] is used in the present study. ...
The ionic liquid propylammonium nitrate (PAN) was elaborated in the present study applying NMR. Temperature dependences of ¹H and ¹³C spin–lattice relaxation rates and diffusion coefficients were measured to describe molecular mobility in PAN. It was discovered that the temperature dependences of the relaxation rates do not have a maximum. An approach to solve the problem of finding a “hypothetical” maximum was applied. Verification of the correctness of the approach on the examples of temperature dependences with a pronounced real maximum was performed. As a result, the correlation times for selected molecular groups of PAN were calculated. Analyzing the data obtained one can conclude that the translational and rotational degrees of freedom in PAN are weakly correlated. The translational motion “slows down” more strongly with the decrease in temperature.
... In this section, we briefly describe the numerical procedure for computing the spectrum of the Bloch-Torrey operator. Here we rely on the so-called matrix formalism [61][62][63][64][65], in which the differential operator ...
We investigate the peculiar feature of non-Hermitian operators, namely, the existence of spectral branch points, at which two (or many) eigenmodes collapse onto a single eigenmode and thus loose their completeness. Such branch points are generic and produce non-analyticities in the spectrum of the operator, which, in turn, result in a finite convergence radius of perturbative expansions based on eigenvalues and eigenmodes that can be relevant even for Hermitian operators. We start with a pedagogic introduction to this phenomenon by considering the case of $2\times 2$ matrices and then explain how the analysis of more general differential operators can be reduced to this setting. We propose an efficient numerical algorithm to find spectral branch points in the complex plane. We employ this algorithm to show the emergence of spectral branch points in the spectrum of the Bloch-Torrey operator $-\nabla^2 - igx$, which governs the time evolution of the nuclear magnetization. We discuss their mathematical properties and physical implications for diffusion nuclear magnetic resonance experiments in general bounded domains. We argue that the example of the Bloch-Torrey operator is representative of some generic properties of non-Hermitian operators.
... This macroscopic approach is outlined in the study by Price 3 and is tractable only for the case of free diffusion. However, modifications to the diffusive attenuation factors of the signals may be made using the short gradient pulse (SGP) approximation 3 or the multiple narrow pulse approximation (MNP) 4,5 to extend the method presented here to systems with restricted/anomalous diffusion. For a fully detailed analysis, boundary conditions for the Torrey-Bloch equations must be included and solutions to these boundary problems may be found using the SGP and MNP approximations. ...
Explicit phenomenological solutions to recurrence relations for the bulk transverse and longitudinal magnetization found using the Torrey–Bloch equations with relaxation effects are used to investigate nuclear magnetic resonance (NMR) diffusion measurements. Of particular interest are steady state NMR (self-)diffusion measurements that reduce experimental time that can extend the techniques to quickly reacting systems. The solutions for bulk transverse and longitudinal magnetization presented here are used to investigate the average behavior of the transverse and longitudinal magnetization in forming a steady state and are used to derive new expressions for the steady state longitudinal magnetization. These solutions can be applied to a noninteracting spin 1/2 ensemble undergoing free diffusion exposed to an arbitrary NMR pulse sequence containing arbitrary magnetic field gradient waveforms. The closed algebraic form method presented here has an advantage over iterative procedures for calculating transverse and longitudinal magnetization for the analysis and development of steady state pulse sequences. Previous theoretical results for steady state diffusion measurements are also reproduced. The Mathematica code for these solutions is provided in the supplementary material.
... Experiments of restricted di usion in simple geometries, such as spheres and cylinders, can be simulated using analytical signal expressions that can be derived for gradient sequences that ful l a set of assumptions, e.g., in nitesimal gradient pulses 48 . Simulation experiments of restricted di usion in simple geometries with arbitrary gradient sequences can be performed using the matrix formalism, which represents the gradient by a succession of sharp gradient impulses, enabling the signal a enuation to be expressed as a computationally inexpensive product of matrix operations 142 ...
Diffusion-weighted magnetic resonance imaging is routinely used for quantifying microstructural properties of brain tissue in both health and disease for its ability to be sensitive to the displacements of water molecules on a microscopic level. Significant effort has been put into the development of methods that provide more information on tissue microstructure than conventional diffusion tensor imaging. Multidimensional diffusion encoding methods render the signal sensitive to the displacements of water molecules that occur along two or three dimensions and can resolve some degeneracies in data acquired with single diffusion encoding methods that measure diffusion along a single dimension. The aim of this thesis is to study the state-of-the-art microstructural imaging methods and to assess their robustness in estimating microscopic diffusion anisotropy, i.e., the average anisotropy of the microscopic diffusion environments irrespective of their orientation dispersion, prior to their adoption in the wider neuroscience research community and possible deployment in clinical studies. First, a massively parallel Monte Carlo random walk simulator is presented. Second, the reproducibility of three commonly used microstructural models is quantified and the shortcomings of such single diffusion encoding methods in estimating microscopic diffusion anisotropy are addressed. Third, the challenges of estimating microscopic diffusion anisotropy in the human brain using double diffusion encoding are addressed using animal imaging experiments and simulations. The results support the feasibility of double diffusion encoding in human neuroimaging but raise hitherto overlooked precision issues when measuring microscopic diffusion anisotropy. Fourth, the accuracy and precision of microscopic diffusion anisotropy estimation using q-space trajectory encoding, a multidimensional diffusion encoding method specifically developed with the limitations of clinical whole-body scanners in mind, are assessed using imaging experiments and simulations. The results suggest that although broken model assumptions and time-dependent diffusion may bias the estimates, the effect of time-dependent diffusion on the estimated microscopic diffusion anisotropy is small in human white matter.
... (41). Further, the accounting for the finite length of the gradient pulse is in general mathematically intractable for cases involving restricted diffusion and various approximations and/or numerical means must be used [15,28,33,69,74]. Due to the complexity of restricting systems a commonly used approach is to analyse the diffusion data assuming free diffusion. ...
Magnetic resonance in the guise of conventional imaging (MRI, also known as k-space imaging) and diffusion imaging (q-space imaging) and combined k- and q-space imaging provides a powerful means of probing porous media and the dynamic processes occurring within such media. This article provides an overview of the imaging process and the accessible length scales of these techniques for probing porous media as well as the type of information that can be obtained. Image reconstruction as well as sources of artefacts are also covered in detail. Some applications of these techniques are also reviewed. Particular problems related to measurements in nanoporous systems are highlighted.
... Another benefit of generalized diffusion waveforms is the ability to integrate intrinsic compensation of gradient imperfections such as eddy currents [64,65] and concomitant fields (Maxwell terms) [66,67] into the encoding waveform and to account for gradient heating [68]. While the b-tensor formalism is applicable only to time-independent diffusion, restriction effects from generalized encoding waveforms can be accounted for simple geometries using the matrix propagator method [69,70]. The matrix propagator formalism discretizes the gradient waveform into well-defined time intervals and assumes that all translational motion occurs at the boundary of those time intervals, allowing analysis using a diffusion propagator approach similar to the analysis for narrow-pulsed gradients. ...
This chapter breaks down acquisition considerations for diffusion MRI into the three key components of diffusion-weighted pulse sequences: (I) diffusion encoding, (II) signal refocusing, and (III) image encoding. The first section describes strategies to sensitize the MR signal to the diffusive motion of water molecules using linear magnetic field gradients. The second section discusses how signal refocusing techniques can be tailored to accommodate the desired diffusion encoding and/or mitigate image artifacts. The third section describes rapid image readout strategies commonly employed for diffusion MRI. Standard single-shot echo-planar imaging is discussed along with advancements in accelerated imaging and encoding strategies which aim to address its shortcomings. The fourth section addresses hardware and system limitations and their effect on diffusion MRI. The final section is devoted to a brief summary of considerations for acquiring diffusion MRI outside of the brain.
... So far, the biophysical models that incorporate permeability rely on assumptions that are either too simplistic (Callaghan, 1997, Codd and Callaghan, 1999, Vangelderen et al., 1994 or do not hold in human tissue (Grebenkov et al., 2014, Kärger et al., 1988). The Kärger model (Kärger et al., 1988) is the most widely used analytical model that incorporates permeability (Nilsson et al., 2010, Stanisz et al., 2005, Lätt et al., 2009). ...
The intra-axonal water exchange time (τi), a parameter associated with axonal permeability, could be an important biomarker for understanding and treating demyelinating pathologies such as Multiple Sclerosis. Diffusion-Weighted MRI (DW-MRI) is sensitive to changes in permeability; however, the parameter has so far remained elusive due to the lack of general biophysical models that incorporate it. Machine learning based computational models can potentially be used to estimate such parameters. Recently, for the first time, a theoretical framework using a random forest (RF) regressor suggests that this is a promising new approach for permeability estimation. In this study, we adopt such an approach and for the first time experimentally investigate it for demyelinating pathologies through direct comparison with histology. We construct a computational model using Monte Carlo simulations and an RF regressor in order to learn a mapping between features derived from DW-MRI signals and ground truth microstructure parameters. We test our model in simulations, and find strong correlations between the predicted and ground truth parameters (intra-axonal volume fraction f: R2 =0.99, τi: R2 =0.84, intrinsic diffusivity d: R2 =0.99). We then apply the model in-vivo, on a controlled cuprizone (CPZ) mouse model of demyelination, comparing the results from two cohorts of mice, CPZ (N=8) and healthy age-matched wild-type (WT, N=8). We find that the RF model estimates sensible microstructure parameters for both groups, matching values found in literature. Furthermore, we perform histology for both groups using electron microscopy (EM), measuring the thickness of the myelin sheath as a surrogate for exchange time. Histology results show that our RF model estimates are very strongly correlated with the EM measurements (ρ = 0.98 for f, ρ = 0.82 for τi). Finally, we find a statistically significant decrease in τi in all three regions of the corpus callosum (splenium/genu/body) of the CPZ cohort (<τi>=310ms/330ms/350ms) compared to the WT group (<τi>=370ms/370ms/380ms). This is in line with our expectations that τi is lower in regions where the myelin sheath is damaged, as axonal membranes become more permeable. Overall, these results demonstrate, for the first time experimentally and in vivo, that a computational model learned from simulations can reliably estimate microstructure parameters, including the axonal permeability .
... Regarding acquisition, both DTI and DKI use single diffusion encoding (SDE), which uses one pair of diffusion-sensitizing gradients. Multidimensional diffusion encoding (Mitra, 1995;Topgaard, 2017), that can be realized by using either double diffusion encoding (DDE) (Cory et al., 1990;Callaghan and Xia, 1991), triple diffusion encoding (Mori and Van Zijl, 1995), or continuous gradient waveforms (Caprihan et al., 1996;Callaghan, 1997;Eriksson et al., 2013), has recently gained attention in clinical studies because these methods provide more specific information about the tissue microstructure. For example, cumulant expansion of the DDE signal up to the fourth-order term of the gradient amplitude (Jespersen, 2012) shows DDE provides unique information that is not contained in the standard diffusion and kurtosis tensors available with SDE. ...
Microstructure imaging by means of multidimensional diffusion encoding is increasingly applied in clinical research, with expectations that it yields a parameter that better correlates with clinical disability than current methods based on single diffusion encoding. Under the assumption that diffusion within a voxel can be well described by a collection of diffusion tensors, several parameters of this diffusion tensor distribution can be derived, including mean size, variance of sizes, orientational dispersion, and microscopic anisotropy. The information provided by multidimensional diffusion encoding also enables us to decompose the sources of the conventional fractional anisotropy and mean kurtosis. In this study, we explored the utility of the diffusion tensor distribution approach for characterizing white-matter degeneration in aging and in Parkinson disease by using double diffusion encoding. Data from 23 healthy older subjects and 27 patients with Parkinson disease were analyzed. Advanced age was associated with greater mean size and size variances, as well as smaller microscopic anisotropy. By analyzing the parameters underlying diffusion kurtosis, we found that the reductions of kurtosis in aging and Parkinson disease reported in the literature are likely driven by the reduction in microscopic anisotropy. Furthermore, microscopic anisotropy correlated with the severity of motor impairment in the patients with Parkinson disease. The present results support the use of multidimensional diffusion encoding in clinical studies and are encouraging for its future clinical implementation.
... Matrix formalism methods involve solving for the eigenmodes of the diffusion propagator [136]. This allows the expected echo attenuation to be calculated mathematically. ...
Cancer is the leading cause of death globally; ahead of heart disease, stroke and chronic obstructive pulmonary disease. There were 8.2 million attributed deaths in 2012 [1], and the number of new cases is expected to increase by 70% in the next two decades [2]. Development of novel new therapies has consequently become a multi-billion dollar industry. However, the development of new cancer therapies is limited by our ability to accurately quantify their effects. This thesis focuses on the development of novel non- invasive biomarkers for assessing changes in tumour microstructure. In chapter 3, the capabilities of an in-house diffusion MRI technique known as Vascular, Extracellular and Restricted Diffusion for Cytometry in Tumours (VERDICT) are inves- tigated. The technique was used to detect changes in tumour microstructure caused by a) tissue fixation and b) administration of temozolomide therapy. In chapter 4, the development of a Monte Carlo tissue diffusion simulation framework is described. The simulation framework is then applied as a tool for validating diffusion MRI models, including VERDICT. Chapter 5 presents an exploration of the potential applications of machine-learning based approaches within the field of diffusion MRI. In a preliminary study, a neural network is trained on synthetic diffusion MRI data, and applied to real-world in-vivo data to try and extract microstructural tissue features without the need for explicit model fitting. The overall aim of this thesis was to assist in the development and validation of advanced diffusion MRI modelling techniques, and explore the future potential of synthetic data and machine-learning models in the extraction of new cancer biomarkers.
... The Monte-Carlo equivalent of the time discretization parameter is the time step size and we simulated with the following choices: To compare with an available analytical solution, we computed the Matrix Formalism [59,60] signal of a rectangle cuboid of size L x × L y × L z using the analytical eigenfunctions and eigenvalues of the rectangle cuboid [61][62][63][64]. We chose L x = 3µm, L y = 100µm, L z = 1µm to be close to the size of the dendrite branches. ...
The diffusion MRI signal arising from neurons can be numerically simulated by solving the Bloch-Torrey partial differential equation. In this paper we present the Neuron Module that we implemented within the Matlab-based diffusion MRI simulation toolbox SpinDoctor. SpinDoctor uses finite element discretization and adaptive time integration to solve the Bloch-Torrey partial differential equation for general diffusion-encoding sequences, at multiple b-values and in multiple diffusion directions. In order to facilitate the diffusion MRI simulation of realistic neurons by the research community, we constructed finite element meshes for a group of 36 pyramidal neurons and a group of 29 spindle neurons whose morphological descriptions were found in the publicly available neuron repository NeuroMorpho.Org. These finite elements meshes range from having 15163 nodes to 622553 nodes. We also broke the neurons into the soma and dendrite branches and created finite elements meshes for these cell components. Through the Neuron Module, these neuron and cell components finite element meshes can be seamlessly coupled with the functionalities of SpinDoctor to provide the diffusion MRI signal attributable to spins inside neurons. We make these meshes and the source code of the Neuron Module available to the public as an open-source package.
To illustrate some potential uses of the Neuron Module, we show numerical examples of the simulated diffusion MRI signals in multiple diffusion directions from whole neurons as well as from the soma and dendrite branches, and include a comparison of the high b-value behavior between dendrite branches and whole neurons. In addition, we demonstrate that the neuron meshes can be used to perform Monte-Carlo diffusion MRI simulations as well. We show that at equivalent accuracy, if only one gradient direction needs to be simulated, SpinDoctor is faster than a GPU implementation of Monte-Carlo, but if many gradient directions need to be simulated, there is a break-even point when the GPU implementation of Monte-Carlo becomes faster than SpinDoctor. Furthermore, we numerically compute the eigenfunctions and the eigenvalues of the Bloch-Torrey and the Laplace operators on the neuron geometries using a finite elements discretization, in order to give guidance in the choice of the space and time discretization parameters for both finite elements and Monte-Carlo approaches. Finally, we perform a statistical study on the set of 65 neurons to test some candidate biomakers that can potentially indicate the soma size. This preliminary study exemplifies the possible research that can be conducted using the Neuron Module.
... So far, mathematical models of white matter ignore or assume simplistic models of permeability [149,89,150], while others explicitly incorporate τ i but rely on assumptions that do not hold in human white matter tissue [90,85]. The Kärger model [85] is the most widely used analytical model that incorporates permeability due to its compatibility with data acquired using clinically available sequences such as SDE and STEAM [6,40,92]. ...
Characterising tissue microstructure is of paramount importance for understanding neurological conditions such as Multiple Sclerosis. Therefore, there is a growing interest in imaging tissue microstructure non-invasively. One way to achieve this is by developing tissue models and fitting them to the diffusion-MRI signal. Nevertheless, some microstructure parameters, such as permeability, remain elusive because analytical models that incorporate them are intractable. Machine learning based computational models offer a promising alternative as they bypass the need for analytical expressions. The aim of this thesis is to develop the first machine learning based computational model for white matter microstructure imaging using two promising approaches: random forests and neural networks. To test the feasibility of this new approach, we provide for the first time a direct comparison of machine learning parameter estimates with histology. In this thesis, we demonstrate the idea by estimating permeability via the intra-axonal exchange time τ_i, a potential imaging biomarker for demyelinating pathologies. We use simulations of the diffusion-MRI signal to construct a mapping between signals and microstructure parameters including τ_i. We show for the first time that clinically viable diffusion-weighted sequences can probe exchange times up to approximately 1000 ms. Using healthy in-vivo human and mouse data, we show that our model's estimates are within the plausible range for white matter tissue and display well known trends such as the high-low-high intra-axonal volume fraction f across the corpus callosum. Using human and mouse data from demyelinated tissue, we show that our model detects trends in line with the expected MS pathology: a significant decrease in f and τ_i. Moreover, we show that our random forest estimates of f and τ_i correlate very strongly with histological measurements of f and myelin thickness. This thesis demonstrates that machine learning based computational models are a feasible approach for white matter microstructure imaging. The continually improving SNR in the clinical scanners and the availability of more realistic simulations open up possibilities of using such models as imaging biomarkers for demyelinating diseases such as Multiple Sclerosis.
... Fig. 8a shows simulated signal decays for the STE protocol used in this study, for spherical compartments with different diffusivities D s and radii r s . The signal resulting from a sphere acquired with arbitrary waveforms was derived by (Codd and Callaghan, 1999) using the matrix formalism (Callaghan, 1997), and we have used the implementation in the MISST toolbox (Drobnjak et al., 2011(Drobnjak et al., , 2010Ianuş et al., 2013) to generate these results. From the signal patterns it becomes apparent that disentangling D s and r s will be challenging. ...
The so-called "dot-compartment" is conjectured in diffusion MRI to represent small spherical spaces, such as cell bodies, in which the diffusion is restricted in all directions. Previous investigations inferred its existence from data acquired with directional diffusion encoding which does not permit a straightforward separation of signals from 'sticks' (axons) and signals from 'dots'. Here we combine isotropic diffusion encoding with ultra-strong diffusion gradients (240 mT/m) to achieve high diffusion-weightings with high signal to noise ratio, while suppressing signal arising from anisotropic water compartments with significant mobility along at least one axis (e.g., axons). A dot-compartment, defined to have apparent diffusion coefficient equal to zero and no exchange, would result in a non-decaying signal at very high b-values (b ≳ 7000 s/mm2). With this unique experimental setup, a residual yet slowly decaying, signal above the noise floor for b-values as high as 15 000 s/mm2 was seen clearly in the cerebellar grey matter (GM), and in several white matter (WM) regions to some extent. Upper limits of the dot-signal-fraction were estimated to be 1.8% in cerebellar GM and 0.2% in WM. By relaxing the assumption of zero diffusivity, the signal at high b-values in cerebellar GM could be represented more accurately by an isotropic water pool with a low apparent diffusivity of 0.12 μm2/ms and a substantial signal fraction of 9.7%. The T2 of this component was estimated to be around 61 m s. This remaining signal at high b-values has potential to serve as a novel and simple marker for isotropically-restricted water compartments in cerebellar GM.
... This approach is often called diffusion-weighted imaging (DWI) [16][17][18][19]. The most promising DWI technique employs Modulated Gradient Spin Echo (MGSE) sequences that enable detailed microstructure characterization [20][21][22][23][24][25][26]. Several works have addressed the estimation of compartment sizes by protocols based on various DWI sequences [23][24][25][26][27][28][29][30]. ...
Characterization of microstructures in live tissues is one of the keys to diagnosing early stages of pathology and understanding disease mechanisms. However, the extraction of reliable information on biomarkers based on microstructure details is still a challenge, as the size of features that can be resolved with non-invasive Magnetic Resonance Imaging (MRI) is orders of magnitude larger than the relevant structures. Here we derive from quantum information theory the ultimate precision limits for obtaining such details by MRI probing of water-molecule diffusion. We show that already available MRI pulse sequences can be optimized to attain the ultimate precision limits by choosing control parameters that are uniquely determined by the expected size, the diffusion coefficient and the spin relaxation time $T_{2}$. By attaining the ultimate precision limit per measurement, the number of measurements and the total acquisition time may be drastically reduced compared to the present state of the art. These results will therefore allow MRI to advance towards unravelling a wealth of diagnostic information.
... In this section, we validate SpinDoctor by comparing SpinDoctor with the Matrix Formalism method (Callaghan, 1997;Barzykin, 1999) in a simple geometry. The Matrix Formalism method is a closed form representation of the dMRI signal based on the eigenfunctions of the Laplace operator subject to homogeneous Neumann boundary conditions. ...
The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium can be modeled by the multiple compartment Bloch-Torrey partial differential equation. Under the assumption of negligible water exchange between compartments, the time-dependent apparent diffusion coefficient can be directly computed from the solution of a diffusion equation subject to a time-dependent Neumann boundary condition. This paper describes a publicly available MATLAB toolbox called SpinDoctor that can be used 1) to solve the Bloch-Torrey partial differential equation in order to simulate the diffusion magnetic resonance imaging signal; 2) to solve a diffusion partial differential equation to obtain directly the apparent diffusion coefficient; 3) to compare the simulated apparent diffusion coefficient with a short-time approximation formula. The partial differential equations are solved by P1 finite elements combined with built-in MATLAB routines for solving ordinary differential equations. The finite element mesh generation is performed using an external package called Tetgen. SpinDoctor provides built-in options of including 1) spherical cells with a nucleus; 2) cylindrical cells with a myelin layer; 3) an extra-cellular space enclosed either a) in a box or b) in a tight wrapping around the cells; 4) deformation of canonical cells by bending and twisting; 5) permeable membranes; Built-in diffusion-encoding pulse sequences include the Pulsed Gradient Spin Echo and the Oscillating Gradient Spin Echo. We describe in detail how to use the SpinDoctor toolbox. We validate SpinDoctor simulations using reference signals computed by the Matrix Formalism method. We compare the accuracy and computational time of SpinDoctor simulations with Monte-Carlo simulations and show significant speed-up of SpinDoctor over Monte-Carlo simulations in complex geometries. We also illustrate several extensions of SpinDoctor functionalities, including the incorporation of T2 relaxation, the simulation of non-standard diffusion-encoding sequences, as well as the use of externally generated geometrical meshes.
... So far, the analytical models that incorporate permeability rely on assumptions that are either too simplistic (Callaghan, 1997, Codd and Callaghan, 1999, Vangelderen et al., 1994 or do not hold in human tissue (Grebenkov et al., 2014(Grebenkov et al., , K ä rger et al., 1988). The Kärger model (K ä rger et al., 1988) is the most widely used analytical model that incorporates permeability (Nilsson et al., 2010, Stanisz et al., 2005, Lätt et al., 2009). ...
The intra-axonal water exchange time {\tau}i, a parameter associated with axonal permeability, could be an important biomarker for understanding demyelinating pathologies such as Multiple Sclerosis. Diffusion-Weighted MRI is sensitive to changes in permeability, however, the parameter has remained elusive due to the intractability of the mathematical models that incorporate it. Machine learning based computational models can potentially be used to estimate such parameters, and recently, a theoretical framework using a random forest (RF) suggests this is a promising approach. In this study, we adopt such an RF approach and experimentally investigate its suitability as a biomarker for demyelinating pathologies through direct comparison with histology. For this, we use an in-vivo cuprizone (CPZ) mouse model of demyelination with available ex-vivo electron microscopy (EM) data. We test our model on noise-free simulations and find very strong correlations between the predicted and ground truth parameters. For realistic noise levels as in our in-vivo data, the performance is affected, however, the parameters are still well estimated. We apply our RF model on in-vivo data from 8 CPZ and 8 wild-type (WT) mice and validate the RF estimates using histology. We find a strong correlation between the in-vivo RF estimates of {\tau}i and the EM measurements of myelin thickness ({\rho_\tau}i = 0.82), and between RF estimates and EM measurements of intra-axonal volume fraction ({\rho_f} = 0.98). When comparing {\tau}i in CPZ and WT mice we find a statistically significant decrease in the corpus callosum of the CPZ compared to the WT mice, in line with our expectations that {\tau}i is lower in regions where the myelin sheath is damaged. Overall, these results demonstrate the suitability of machine learning compartment models with permeability as a potential biomarker for demyelinating pathologies.
... To aid in development of analytical solutions, the short gradient pulse (SGP) approximation is often employed which takes the limit of the gradient time approaching zero (δ → 0) such that the diffusion sensitizing gradient is applied at a single instant. Under this assumption, solutions to the Bloch-Torrey equation solved over simple structures are possible (Tanner and Stejskal, 1968;Szafer et al., 1995;Söderman and Jönsson, 1995), though simple structures can also be solved for more arbitrary pulse profiles (Callaghan, 1997;Codd and Callaghan, 1999), and solutions to multi-layered structures have been presented for a PGSE pulse by Grebenkov (2010). As a first approximation, diffusion is often assumed to be Gaussian which allows the dMRI signal to be modeled as ...
Clinical diffusion MRI (dMRI) is sensitive to micrometer spin displacements, but the image resolution is $\sim$mm, so the biophysical interpretation of the signal relies on establishing appropriate subvoxel tissue models. A class of two-compartment exchange models originally proposed by K\"arger have been used successfully in neural tissue dMRI. Their use to interpret the signal in skeletal muscle dMRI is challenging because myocyte diameters are comparable to the root-mean-square of spin displacement and their membrane permeability is high. A continuum tissue model consisting of the Bloch-Torrey equation integrated by a hybrid lattice Boltzmann scheme is used for comparison. The validity domain of a classical two-compartment tissue model is probed by comparing with the prediction of the continuum model for a 2-D unidirectional composite continuum model of myocytes embedded in a uniform matrix, the validity domain of a classical two-component tissue model is probed. This domain is described in terms of two dimensionless parameters inspired by mass transfer phenomena, the Fourier (F) and Biot (B) numbers. The two-compartment model is valid when B << 1 and F >> 1, or when F << 1 and F ∙ B << 1. The model becomes less appropriate for muscle dMRI as the cell diameter and volume fraction increase, with the primary source of error associated with modeling diffusion in the extracellular matrix.
... Fig. 8a shows simulated signal decays for the STE protocol used in this study, for spherical compartments with different diffusivities D s and radii r s . The signal resulting from a sphere acquired with arbitrary waveforms was derived by (Codd and Callaghan, 1999) using the matrix formalism (Callaghan, 1997), and we have used the implementation in the MISST toolbox (Drobnjak et al., 2011(Drobnjak et al., , 2010Ianuş et al., 2013) to generate these results. From the signal patterns it becomes apparent that disentangling D s and r s will be challenging. ...
The so-called "dot-compartment" is conjectured in diffusion MRI to represent small spherical spaces, such as cell bodies, in which the diffusion is restricted in all directions. Previous investigations inferred its existence from data acquired with directional diffusion encoding which does not permit a straightforward separation of signals from 'sticks' (axons) and signals from 'dots'. Here we combine isotropic diffusion encoding with ultra-strong diffusion gradients (240 mT/m) to achieve high diffusion-weightings with high signal to noise ratio, while suppressing signal arising from anisotropic water compartments with significant mobility along at least one axis (e.g., axons). A dot-compartment, defined to have apparent diffusion coefficient equal to zero and no exchange, would result in a non-decaying signal at very high b-values (b ≥ 7000 s/mm ² ). With this unique experimental setup, a residual yet slowly decaying, signal above the noise floor for b-values as high as 15000 s/mm ² was seen clearly in the cerebellar grey matter (GM), and in several white matter (WM) regions to some extent. Upper limits of the dot-signal-fraction were estimated to be ~2% in cerebellar GM and ~0.2% in WM. By relaxing the assumption of zero diffusivity, the signal at high b-values in cerebellar GM could be represented more accurately by an isotropic water pool with a low apparent diffusivity of 0.11 μm ² /ms and a substantial signal fraction of ~7-16%. This remaining signal at high b-values has potential to serve as a novel and simple marker for isotropically-restricted water compartments in cerebellar GM.
... • OGSE consists of two oscillating pulses of duration σ, each containing n periods, separated by a time interval τ − σ (Fig. 2b). For a cosine OGSE [38,37], the profile f (t) is In this paper, the water exchange between compartments is neglected, yielding the homogeneous Neumann boundary condition: ...
The Bloch-Torrey partial differential equation can be used to describe the evolution of the transverse magnetization of the imaged sample under the influence of diffusion-encoding magnetic field gradients inside the MRI scanner. The integral of the magnetization inside a voxel gives the simulated diffusion MRI signal. This paper proposes a finite element discretization on manifolds in order to efficiently simulate the diffusion MRI signal in domains that have a thin layer or a thin tube geometrical structure. The variable thickness of the three-dimensional domains is included in the weak formulation established on the manifolds. We conducted a numerical study of the proposed approach by simulating the diffusion MRI signals from the extracellular space (a thin layer medium) and from neurons (a thin tube medium), comparing the results with the reference signals obtained using a standard three-dimensional finite element discretization. We show good agreements between the simulated signals using our proposed method and the reference signals for a wide range of diffusion MRI parameters. The approximation becomes better as the diffusion time increases. The method helps to significantly reduce the required simulation time, computational memory, and difficulties associated with mesh generation, thus opening the possibilities to simulating complicated structures at low cost for a better understanding of diffusion MRI in the brain.
... On the measurement time scale the topology is assumed to be fixed. This description goes beyond the traditional models which assume free diffusion of molecules and isolated pores [18][19][20] . Under this description, m i (t) is the magnetization of the ith pore at time t. ...
Unravelling underlying complex structures from limited resolution measurements is a known problem arising in many scientific disciplines. We study a stochastic dynamical model with a multiplicative noise. It consists of a stochastic differential equation living on a graph, similar to approaches used in population dynamics or directed polymers in random media. We develop a new tool for approximation of correlation functions based on spectral analysis that does not require translation invariance. This enables us to go beyond lattices and analyse general networks. We show, analytically, that this general model has different phases depending on the topology of the network. One of the main parameters which describe the network topology is the spectral dimension [Formula: see text]. We show that the correlation functions depend on the spectral dimension and that only for [Formula: see text] > 2 a dynamical phase transition occurs. We show by simulation how the system behaves for different network topologies, by defining and calculating the Lyapunov exponents on the graph. We present an application of this model in the context of Magnetic Resonance (MR) measurements of porous structure such as brain tissue. This model can also be interpreted as a KPZ equation on a graph.
... The simplest generate this signal using a model that as- sumes a particular underlying microstructure and pulse sequence (most often PGSE) [72][73][74]. Approaches such as Callaghan's matrix formulation [75] en- able DW-MR signal to be generated for simple geometries but more complex pulse sequences [76]. The most flexible use a Monte-Carlo approach to simu-late the diffusion of particles within a single voxel, for arbitrary microstructural environments and pulse sequence [77][78][79]. ...
Diffusion-weighted MRI (DW-MRI) is a powerful, non-invasive imaging technique that allows us to infer the structure of biological tissue. It is particularly well suited to the brain, and is used by clinicians and researchers studying its structure in health and disease. High quality data is required to accurately characterise tissue structure with DW-MRI. Obtaining such data requires the careful optimisation of the image acquisition and processing pipeline, in order to maximise image quality and minimise artefacts. This thesis extends an existing MRI simulator to create a simulation system capable of producing realistic DW-MR data, with artefacts, and applies it to improve the acquisition and processing of such data. The simulator is applied in three main ways. Firstly, a novel framework for evaluating post-processing techniques is proposed and applied to assess commonly used strategies for the correction of motion, eddy-current and susceptibility artefacts. Secondly, it is used to explore the often overlooked susceptibility-movement interaction. It is demonstrated that this adversely impacts analysis of DW-MRI data, and a simple modification to the acquisition scheme is suggested to mitigate its impact. Finally, the simulation is applied to develop a new tool to perform automatic quality control. Simulated data is used to train a classifier to detect movement artefacts in data, with performance approaching that of a classifier trained on real data whilst requiring much less manually-labelled training data. It is hoped that both the findings in this thesis and the simulation tool itself will benefit the DW-MRI community. To this end, the tool is made freely available online to aid the development and validation of methods for acquiring and processing DW-MRI data.
We consider the Bloch-Torrey operator, $-\Delta + igx$, that governs the time evolution of the transverse magnetization in diffusion magnetic resonance imaging (dMRI). Using the matrix formalism, we compute numerically the eigenvalues and eigenfunctions of this non-Hermitian operator for two bounded three-dimensional domains: a sphere and a capped cylinder. We study the dependence of its eigenvalues and eigenfunctions on the parameter $g$ and on the shape of the domain (its eventual symmetries and anisotropy). In particular, we show how an eigenfunction drastically changes its shape when the associated eigenvalue crosses a branch (or exceptional) point in the spectrum. Potential implications of this behavior for dMRI are discussed.
Temporal Diffusion Ratio (TDR) is a recently proposed dMRI technique (Dell'Acqua et al., proc. ISMRM 2019) which provides contrast between areas with restricted diffusion and areas either without restricted diffusion or with length scales too small for characterisation. Hence, it has a potential for informing on pore sizes, in particular the presence of large axon diameters or other cellular structures. TDR employs the signal from two dMRI acquisitions obtained with the same, large, b-value but with different diffusion gradient waveforms TDR is advantageous as it employs standard acquisition sequences, does not make any assumptions on the underlying tissue structure and does not require any model fitting, avoiding issues related to model degeneracy. This work for the first time introduces and optimises the TDR method in simulation for a range of different tissues and scanner constraints and validates it in a pre-clinical demonstration. We consider both substrates containing cylinders and spherical structures, representing cell soma in tissue. Our results show that contrasting an acquisition with short gradient duration, short diffusion time and high gradient strength with an acquisition with long gradient duration, long diffusion time and low gradient strength, maximises the TDR contrast for a wide range of pore configurations. Additionally, in the presence of Rician noise, computing TDR from a subset (50% or fewer) of the acquired diffusion gradients rather than the entire shell as proposed originally further improves the contrast. In the last part of the work the results are demonstrated experimentally on rat spinal cord. In line with simulations, the experimental data shows that optimised TDR improves the contrast compared to non-optimised TDR. Furthermore, we find a strong correlation between TDR and histology measurements of axon diameter. In conclusion, we find that TDR has great potential and is a very promising alternative (or potentially complement) to model-based approaches for informing on pore sizes and restricted diffusion in general.
Monitoring time‐dependence with diffusion MRI yields observables sensitive to compartment sizes (restricted diffusion) and membrane permeability (water exchange). However, restricted diffusion and exchange have opposite effects on the diffusion‐weighted signal, which can lead to errors in parameter estimates. In this work, we propose a signal representation that incorporates the effects of both restricted diffusion and exchange up to second order in b‐value and is compatible with gradient waveforms of arbitrary shape. The representation features mappings from a gradient waveform to two scalars that separately control the sensitivity to restriction and exchange. We demonstrate that these scalars span a two‐dimensional space that can be used to choose waveforms that selectively probe restricted diffusion or exchange, eliminating the correlation between the two phenomena. We found that waveforms with specific but unconventional shapes provide an advantage over conventional pulsed and oscillating gradient acquisitions. We also show that parametrisation of waveforms into a two‐dimensional space can be used to understand protocols from other approaches that probe restricted diffusion and exchange. For example, we found that the variation of mixing time in filter‐exchange imaging corresponds to variation of our exchange‐weighting scalar at a fixed value of the restriction‐weighting scalar. The proposed signal representation was evaluated using Monte Carlo simulations in identical parallel cylinders with hexagonal and random packing as well as parallel cylinders with gamma‐distributed radii. Results showed that the approach is sensitive to sizes in the interval 4 – 12 μm and exchange rates in the simulated range of 0 to 20 s‐1, but also that there is a sensitivity to the extracellular geometry. The presented theory constitutes a simple and intuitive description of how restricted diffusion and exchange influence the signal as well as a guide to protocol design capable of separating the two effects.
In quantitative susceptibility mapping, the tissue susceptibility is determined from the magnitude and phase of the gradient echo signal, which is influenced by the interplay of complex susceptibility and diffusion effect. Herein, we analytically analyze the influence of diffusion on magnitude and phase images generated by randomly arranged magnetic spheres as a model of intracerebral iron depositions. We demonstrate that both gradient and spin echo relaxation rate constants have a strong and nonlinear dependence on diffusion strength and give empirical formulas for magnitude and phase. This may be used in the future to improve QSM processing methods. In addition, we show that, in theory, combined acquisitions of gradient and spin echo can be used to determine the dimension of the magnetic spheres and the diffusion strength.
Characterizing neural tissue microstructure is a critical goal for future neuroimaging. Diffusion MRI (dMRI) provides contrasts that reflect diffusing spins’ interactions with myriad microstructural features of biological systems. However, the specificity of dMRI remains limited due to the ambiguity of its signals vis-à-vis the underlying microstructure. To improve specificity, biophysical models of white matter (WM) typically express dMRI signals according to the Standard Model (SM) and have more recently in gray matter (GM) taken spherical compartments into account (the SANDI model) in attempts to represent cell soma. The validity of the assumptions underlying these models, however, remains largely undetermined, especially in GM. To validate these assumptions experimentally, observing their unique, functional properties, such as the b−1/2 power-law associated with one-dimensional diffusion, has emerged as a fruitful strategy. The absence of this signature in GM, in turn, has been explained by neurite water exchange, non-linear morphology, and/or by obscuring soma signal contributions. Here, we present diffusion simulations in realistic neurons demonstrating that curvature and branching does not destroy the stick power-law behavior in impermeable neurites, but also that their signal is drowned by the soma signal under typical experimental conditions. Nevertheless, by studying the GM dMRI signal's behavior as a function of diffusion weighting as well as time, we identify an attainable experimental regime in which the neurite signal dominates. Furthermore, we find that exchange-driven time dependence produces a signal behavior opposite to that which would be expected from restricted diffusion, thereby providing a functional signature that disambiguates the two effects. We present data from dMRI experiments in ex vivo rat brain at ultrahigh field of 16.4T and observe a time dependence that is consistent with substantial exchange but also with a GM stick power-law. The first finding suggests significant water exchange between neurites and the extracellular space while the second suggests a small sub-population of impermeable neurites. To quantify these observations, we harness the Kärger exchange model and incorporate the corresponding signal time dependence in the SM and SANDI models.
Monitoring time-dependence with diffusion MRI yields observables sensitive to compartment sizes (restricted diffusion) and membrane permeability (water exchange). However, restricted diffusion and exchange have opposite effects on the diffusion-weighted signal, which can confound parameter estimates. In this work, we present a signal representation that captures the effects of both restricted diffusion and exchange up to second order in b-value and is compatible with gradient waveforms of arbitrary shape. The representation features mappings from a gradient waveform to two scalars that separately control the sensitivity to restriction and exchange. We demonstrate that these scalars span a two-dimensional space that can be used to choose waveforms that selectively probe restricted diffusion or exchange, in order to eliminate the correlation between the two phenomena. We found that waveforms with specific but unconventional shapes provide an advantage over conventional pulsed and oscillating gradient acquisitions. We also show that parametrisation of waveforms into a two-dimensional space can be used to understand protocols from other approaches that probe restricted diffusion and exchange. For example, we find that the variation of mixing time in filter-exchange imaging corresponds to variation of our exchange-weighting scalar at a fixed value of the restriction-weighting scalar. Numerical evaluation of the proposed signal representation using Monte Carlo simulations on a synthetic substrate showed that the theory is applicable to sizes in the range 2 - 7 micrometres and barrier-limited exchange in the range 0 - 20 s$^{-1}$. The presented theory constitutes a simple and intuitive description of how restricted diffusion and exchange influence the signal as well as how to design a protocol to separate the two effects.
The complex transverse water proton magnetization subject to diffusion‐encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch‐Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold‐standard reference model for the diffusion MRI signal arising from different tissue micro‐structures of interest.
A closed form representation of this reference diffusion MRI signal, called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the tissue geometry and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion‐encoding sequences and b ‐values at negligible additional cost.
In a previous publication, we presented a simulation framework that we implemented inside the MATLAB‐based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for biological cells subject to impermeable membrane boundary conditions. In this work, we extend our simulation framework to include geometries that contain permeable cell membranes. We describe the new computational techniques that allowed this generalization and we analyze the effects of the magnitude of the permeability coefficient on the eigendecomposition of the diffusion and Bloch‐Torrey operators.
This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.
Les maladies neurodégénératives sont caractérisées par des altérations de la microstructure cellulaire (hypertrophie astrocytaire, atrophie neuronale) ainsi que du métabolisme cérébral. Le signal RMN pondéré en diffusion reflète l’environnement exploré localement à l’échelle microscopique par un ensemble de molécules, constituant ainsi un outil de mesure non-invasif de la microstructure cérébrale. L’imagerie de diffusion de l’eau s’est donc imposée comme une approche prometteuse pour mieux comprendre et diagnostiquer les pathologies neurodégénératives. En outre, des mesures de spectroscopie pondérée en diffusion pourraient aider à clarifier la distribution au sein des différents types cellulaires de métabolites impliqués dans le métabolisme cérébral, tels que le lactate. Dans le contexte de l’hypothèse de la navette astrocyte-neurone, cette information est en effet cruciale afin d’éclaircir le rôle métabolique de ces molécules en conditions saine et pathologique. Cependant, l’analyse des données de diffusion de l’eau en termes de microstructure, ou bien des données de diffusion du lactate en termes de distribution cellulaire, suppose d’une part la connaissance précise des propriétés de diffusion au sein des différents compartiments tissulaires, en particulier les compartiments intracellulaires (ICS) et extracellulaire (ECS), et d’autre part que ces propriétés soient suffisamment différentes entre ces compartiments, afin de les discriminer. Par exemple, la diffusion est-elle plus rapide dans l’ECS ou l’ICS ? La diffusion dans l’ECS est-elle libre ou bien est-on au contraire sensible à des phénomènes de restriction et de tortuosité ? Le but de ce travail de thèse est de répondre en partie à ces questions. Dans un premier temps, nous étudions la sensibilité de la spectroscopie RMN à double encodage en diffusion (DDE) aux différents aspects de la microstructure cellulaire, montrant ainsi que le « signal » DDE des métabolites endogènes purement intracellulaire reflète non seulement la diffusion dans des prolongements cellulaire fibreux, mais aussi vraisemblablement la présence d’embranchements de ces prolongements cellulaires. Dans un second temps, nous caractérisons les propriétés de diffusion du sucrose, un marqueur exogène injecté dans l’ECS, grâce à des outils de spectroscopie RMN pondérée en diffusion permettant de collecter des informations complémentaires telles que le coefficient de diffusion apparent, la tortuosité, la restriction et l’anisotropie microscopique. Les mesures DDE se révèlent particulièrement à même de discriminer l’ECS et l’ICS. Enfin, nous explorons le potentiel d’une approche originale utilisant le transfert de saturation par échange chimique (CEST) afin de filtrer la contribution de l’ICS lors des mesures de diffusion. Pour conclure, nous évoquons comment des expériences de ¹³C hyperpolarisé pourraient permettre d’affiner la connaissance des propriétés de diffusion du lactate dans les différents compartiments.
Diffusion Magnetic Resonance Imaging (MRI) plays a very important role in studying biological tissue cellular structure and functioning both in health and disease. Proper interpretation of experimental data requires the development of theoretical models that connect the diffusion MRI signal to salient features of tissue microstructure at the cellular level. In this short review, we present some theoretical approaches to describing diffusion-attenuated magnetic resonance signals. These range from the models based on statistical properties of water molecules diffusing in the tissue- cellular environment, to models allowing exact analytical calculation of the magnetic resonance signal in a specific single-compartment environment. Such theoretical analysis gives important insights into mechanisms contributing to the formation of diffusion magnetic resonance signal and its connection to biological tissue cellular structure.
The complex transverse water proton magnetization subject to diffusion‐encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch‐Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold‐standard reference model for the diffusion MRI signal arising from different tissue micro‐structures of interest. A closed form representation of this reference diffusion MRI signal called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the biological cell and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion‐encoding sequences and b ‐values at negligible additional cost. Up to now, this representation, though mathematically elegant, has not been often used as a practical model of the diffusion MRI signal, due to the difficulties of calculating the Laplace eigendecomposition in complicated geometries. In this paper, we present a simulation framework that we have implemented inside the MATLAB‐based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for realistic neurons using the finite element method. We show that the matrix formalism representation requires a few hundred eigenmodes to match the reference signal computed by solving the Bloch‐Torrey equation when the cell geometry originates from realistic neurons. As expected, the number of eigenmodes required to match the reference signal increases with smaller diffusion time and higher b ‐values. We also convert the eigenvalues to a length scale and illustrate the link between the length scale and the oscillation frequency of the eigenmode in the cell geometry. We give the transformation that links the Laplace eigenfunctions to the eigenfunctions of the Bloch‐Torrey operator and compute the Bloch‐Torrey eigenfunctions and eigenvalues. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.
Axon caliber plays a crucial role in determining conduction velocity and, consequently, in the timing and synchronization of neural activation. Noninvasive measurement of axon radii could have significant impact on the understanding of healthy and diseased neural processes. Until now, accurate axon radius mapping has eluded in vivo neuroimaging, mainly due to a lack of sensitivity of the MRI signal to micron-sized axons. Here, we show how – when confounding factors such as extra-axonal water and axonal orientation dispersion are eliminated – heavily diffusion-weighted MRI signals become sensitive to axon radii. However, diffusion MRI is only capable of estimating a single metric, the effective radius, representing the entire axon radius distribution within a voxel that emphasizes the larger axons. Our findings, both in rodents and humans, enable noninvasive mapping of critical information on axon radii, as well as resolve the long-standing debate on whether axon radii can be quantified.
The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch-Torrey partial differential equation. The spatial integral of the solution of this equation provides a gold-standard reference model for the diffusion MRI signal arising from different tissue micro-structures of interest. A closed form representation of this reference diffusion MRI signal has been derived twenty years ago, called Matrix Formalism that makes explicit the link between the Laplace eigenvalues and eigenfunctions of the biological cell and its diffusion MRI signal. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion-encoding sequences and b-values at negligible additional cost. Up to now, this representation, though mathematically elegant, has not been often used as a practical model of the diffusion MRI signal, due to the difficulties of calculating the Laplace eigendecomposition in complicated geometries. In this paper, we present a simulation framework that we have implemented inside the MATLAB-based diffusion MRI simulator SpinDoctor that efficiently computes the Matrix Formalism representation for realistic neurons using the finite elements method. We show the Matrix Formalism representation requires around 100 eigenmodes to match the reference signal computed by solving the Bloch-Torrey equation when the cell geometry comes from realistic neurons. As expected, the number of required eigenmodes to match the reference signal increases with smaller diffusion time and higher b-values. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.
Nuclear magnetic resonance (NMR) diffusion pore imaging has been proposed to study the shape of arbitrary closed pores filled with an NMR-detectable medium by use of nonclassical diffusion encoding schemes. Potential applications can be found in biomedical imaging and porous media research. When studying non-point-symmetric pores, NMR signals with nonvanishing imaginary parts arise containing the pore shape information, which is lost for classical diffusion encoding schemes. Key limitations are the required high magnetic field gradient amplitudes and T2 relaxation while approaching the diffusion long-time limit. To benefit from the slower T1 decay, we demonstrate the feasibility of diffusion pore imaging with stimulated echoes using Monte Carlo simulations and experiments with hyperpolarized xenon-129 gas in well-defined geometries and show that the necessary complex-valued signals can be acquired. Analytical derivation of the stimulated echo double diffusion encoded signal was performed to investigate the effect of the additionally arising undesired terms on the complex phase information. These terms correspond to signals arising for spin-echo sequences with unbalanced gradients. For most possible applications, the unbalanced terms can be neglected. If non-negligible, selection of the appropriate signal component using a phase cycling scheme was demonstrated experimentally. Using stimulated echoes may be a step towards application of diffusion pore imaging to larger pores with gradient amplitudes available today in preclinical systems.
As the development of the work (J. Phys. Chem. B 2019, 123 (10), 2362–2372) we have investigated the translational mobility in the same set of dried imidazolium-based ionic liquids [bmim]A (A = BF4¯, NO3¯, TfO¯, I¯, Br¯, Cl¯) in a wide temperature range using NMR technique. It is shown that for the [bmim]⁺ cation the temperature dependencies of the product Dη do not follow the Stokes-Einstein relation for most systems studied, i.e. there realized so-called “diffusion-viscosity decoupling”. The correlation between local and translational mobility in pure ionic liquids (IL) of the [bmim][A] type was investigated using the data on NMR-relaxation rates and diffusion coefficients. The most recent hypothesis of "water pockets" in mixtures of IL with water is critically discussed. Considering the totality of data in literature and obtained here, we propose a specific model of the microstructure which may be applied up to water concentrations of 80÷90 mol % (the structure of water-rich solutions is out of our current consideration). To confirm the model molecular dynamics simulations of “IL–water” mixtures were also carried out.
Explosive emulsions are frequently employed in mining operations as they offer a comparatively robust, safe and effective product. These are typically formulated as concentrated ammonium nitrate solutions dispersed in an oil phase in the form of a highly concentrated water‐in‐oil emulsion. Being highly concentrated, determining the droplet size distribution (which is critical to assessing long‐term emulsion stability and explosive effectiveness) of these emulsions is challenging. Here we demonstrate that this is readily achieved using bench‐top Nuclear Magnetic resonance (NMR) Pulsed Field Gradient (PFG) techniques. The initial mean droplet size is shown to decrease with an increase in the concentration of ammonium nitrate or emulsifier or when inert salt was added to the composition, it was found to increase as the concentration of the aqueous (oxidiser) phase was increased. The emulsion droplet size distributions were observed to remain constant for a 3‐month period over the composition range explored.
Ce mémoire rapporte la mise en oeuvre de la technique d'Imagerie par Résonance Magnétique (IRM) de l'hélium-3 hyperpolarisé à 0,1 T puis à 1,5 T, in vivo, chez l'Homme sain. L'utilisation des gaz hyperpolarisés en IRM entraîne deux conséquences majeures : l'absence de régénération de l'aimantation longitudinale et une diffusion très rapide. Prenant en considération ces deux contraintes, différentes stratégies d'acquisition sont comparées à bas champ (0,1 T) : séquences "single-shot" (RARE et EPI) et "multi-shot" (FLASH). On montre ainsi que la séquence RARE permet d'acquérir des images de bonne qualité en moins de 400 ms avec peu de gaz ; la séquence EPI a l'avantage de la rapidité mais entraîne davantage d'artefacts, en particulier en raison du terme de Maxwell ; ces deux séquences sont intrinsèquement limitées en résolution à 5 mm in vivo ; la séquence FLASH, moins efficace en termes de rapport signal sur bruit, permet cependant d'atteindre de meilleures résolutions. Les avantages de l'utilisation d'un bas champ magnétique en régime hyperpolarisé sont discutés : en particulier, le rapport signal sur bruit est, dans une très large gamme, indépendant du champ principal. Des mesures RMN puis IRM de diffusion utilisant la technique CPMG sont présentées et permettent de mettre en évidence la restriction due à la structure alvéolaire des poumons.
The effect of salinity on water-in-oil emulsions was systematically studied using a combination of Nuclear Magnetic Resonance (NMR) pulsed field gradient (PFG) measurements of emulsion droplet size distribution complemented by interfacial tension measurements using the pendant drop method. Long-term emulsion stability over periods of up to five days was found to increase with salinity; this was shown to be independent of whether a monovalent (NaCl) or a divalent (CaCl2) salt was used. The methodology was applied to water-in-oil emulsions formulated with crude oil, paraffin oil, xylene, crude oil with reduced asphaltene content and crude oil with reduced organic acid content as the continuous phase, respectively. In all cases, emulsion stability increased consistently with aqueous phase salinity, with no discernible difference between the continuous oil phases with respect to the extent of this stabilisation. The enhanced stability could thus not be attributed to differences in density, interfacial tension or dielectric permittivity. This leaves a potential increased surface accumulation of stabilising surface active species driven by increasing salinity as the most plausible explanation for the observations reported here.
Phantoms, both numerical (software) and physical (hardware), can serve as a gold standard for the validation of MRI methods probing the brain microstructure. This review aims to provide guidelines on how to build, implement, or choose the right phantom for a particular application, along with an overview of the current state-of-the-art of phantoms dedicated to study brain microstructure with MRI. For physical phantoms, we discuss the essential requirements and relevant characteristics of both the (NMR visible) liquid and (NMR invisible) phantom materials that induce relevant microstructural features detectable via MRI, based on diffusion, intra-voxel incoherent motion, magnetization transfer or magnetic susceptibility weighted contrast. In particular, for diffusion MRI, many useful phantoms have been proposed, ranging from simple liquids to advanced biomimetic phantoms consisting of hollow or plain microfibers and capillaries. For numerical phantoms, the focus is on Monte Carlo simulations of random walk, for which the basic principles, along with useful criteria to check and potential pitfalls are reviewed, in addition to a literature overview highlighting recent advances. While many phantoms exist already, the current review aims to stimulate further research in the field and to address remaining needs.
This paper presents an approach to solving the phase problem in nuclear magnetic resonance (NMR) diffusion pore imaging, a method that allows imaging the shape of arbitrary closed pores filled with an NMR-detectable medium for investigation of the microstructure of biological tissue and porous materials. Classical q-space imaging composed of two short diffusion-encoding gradient pulses yields, analogously to diffraction experiments, the modulus squared of the Fourier transform of the pore image which entails an inversion problem: An unambiguous reconstruction of the pore image requires both magnitude and phase. Here the phase information is recovered from the Fourier modulus by applying a phase retrieval algorithm. This allows omitting experimentally challenging phase measurements using specialized temporal gradient profiles. A combination of the hybrid input-output algorithm and the error reduction algorithm was used with dynamically adapting support (shrinkwrap extension). No a priori knowledge on the pore shape was fed to the algorithm except for a finite pore extent. The phase retrieval approach proved successful for simulated data with and without noise and was validated in phantom experiments with well-defined pores using hyperpolarized xenon gas.
This study is carried out to find relations between the time-dependent molecular self-diffusion and the attenuation of NMR spin-echo. Two cases of diffusion are considered: the Brownian motion in Ornstein's short-time limit and the random walk with memory [13]. The friction and the correlation time describe the mechanism of entrapping interactions between molecules or their bonding to macromolecule chains. The obtained formula for the self-diffusion attenuation is valid at short times and it develops into the well-known Torrey's result in the long-time limit. It fits very efficiently into the NMR data from Refs. [19–20].
The density matrix formalism with the Magnus expansion of the time evolution operator is used to study the nmr response in a pulsed magnetic field gradient (mfg) spin-echo experiment. The results show that the spin-echo cannot only measure the self-diffusion coefficient but can determine the spectrum of the single-particle velocity autocorrelation function as well. The proper combination of rf and mfg pulse sequences are proposed for measuring self-diffusion in spin systems with strong dipolar coupling where the classical method fails.
The PGSE NMR methods relies on the use of two sharp
gradient pulses separated by a well defined time
interval and is therefore naturally suited to time
domain analysis of motion. However it is important to
realise that this particular form of two-pulse
gradient modulation is not unique. In particular a
number of other time-modulation schemes are possible
in which the molecular motion is detected in a
different manner. However, as we shall see, whenever
modulated gradients are used to encode the
spin magnetization for motion rather than position, it is
appropriate to refocus any phase shift due to absolute
spin position by means of a spin echo. In consequence
we refer to this more general type of experiment as
modulated gradient spin echo (MGSE) NMR. A particular
theme of this chapter will be the relationship of the
chosen measurement technique to the type of motion
analysis sought. An important aspect of the discussion
will be the characterisation of motion in molecular
ensembles, and in particular, the effects of deviation
from simple Brownian motion or simple flow. We review
the measurement strategies which may be adopted along
with their associated signal analysis techniques.
These strategies will include time-domain, spatial
frequency-domain and temporal frequency-domain
analyses, spatial localisation of motion, two
dimensional correlation and exchange analyses and
diffraction and scattering analogies.
Nuclear spins (in molecules) are considered to be diffusing in a sphere in a linearly inhomogeneous magnetic field (field gradient) that is imposed during a spin-echo NMR experiment. Relaxation of magnetization both in the bulk medium and on the inner surface of the sphere is assumed to occur. Analytical solutions were obtained for the relevant modified diffusion (partial differential) equation by using separation of variables with a Green's function (propagator) and three different boundary conditions. Neuman [J. Chem. Phys. 60, 4508 (1974)] analyzed the same physical system, but with no relaxation, to obtain an expression that relates the NMR spin-echo signal intensity to the magnitude of the magnetic field gradient, the spin-echo time, and the intrinsic molecular diffusion coefficient. The present analysis was based on that originally used by Neuman and, like the latter, it employed the assumption of a Gaussian distribution of phases of the spin magnetizations. This assumption, while rendering a tractable solution, nevertheless limits its range of applicability; this aspect, and the convergence properties of the series solutions were investigated in conjunction with numerical simulations made with diffusion modeled as a three-dimensional random (Monte Carlo) walk. A novel prediction for spheres with finite surface relaxation and a given radius is the presence of two minima in a graph of the normalized spin-echo signal intensity versus the reciprocal of the diffusion coefficient.
Before 1984 New Zealand was insulated by high levels of protectionism and with a degree of State intervention and regulation unparalleled elsewhere in the western world. Since then New Zealand has experienced one of the most far reaching economic reform programmes of any developed economy. The book describes and analyses the radical economic reform programme undertaken in New Zealand since 1985. These reforms included deregulation of the financial sector, removal of various forms of assistance to producers, particularly in the agricultural sector, increased import liberalisation, radical tax reform, a major overhaul of the public sector and the privatisation of state enterprises. The book seeks to explain why a Labour Government embarked upon the sort of reform programme normally considered the preserve of right-wing administrations elsewhere. It argues that New Zealand's experience provides important lessons for policy-makers elsewhere.
The self-diffusion coefficient of each component in a multicomponent system may be determined by obtaining the Fourier transform of the spin echo using the pulsed gradient, spin echo technique. The pulsed gradient Fourier transform technique also has the individual advantages of Fourier transform NMR and the pulsed gradient technique leading to enhanced signal-to-noise ratio and improved resolution. The method is illustrated for a water(dimethylsulfoxide solution.
Analytical expressions for the NMR signal of an ensemble of spins diffusing in a bounded medium in the presence of a magnetic field gradient exist only in the limit of gradient pulses so short that no diffusion takes place during them. The temporal behavior of any gradient pulse is approximated by a sum of impulses, each being in the narrow-gradient-pulse limit. This approximate, analytical solution for the echo attenuation due to restricted diffusion between two parallel plates with reflecting boundaries in the presence of nonnarrow gradient pulses agrees with that from numerical simulations by Blees [J. Magn. Reson. A109,203 (1994)].
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 pcm(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.
The diffusion of water in a saturated porous medium formed by close-packed monodisperse polystyrene hard spheres has been studied using pulsed gradient spin-echo nuclear magnetic resonance. The echo-attenuation data from three different sphere systems, obtained using a range of diffusive observation times, are analyzed using an analytic function based on a pore-hopping model. We show that diffraction-like coherence peaks, which arise in the echo attenuation, are strongly influenced by sphere size and polydispersity, and that realistic structural parameters can be obtained from the fits to the data.
The NMR pulsed gradient spin echo (PGSE) self-diffusion technique is applied to a system with permeable barriers. The system is a highly concentrated emulsion consisting of 97% water, and its internal structure is one of closely packed water droplets which are separated by thin oil films. These oil films are permeable to water, and the emulsion can therefore be considered as belonging to the general class of porous systems. Recently, Callaghan and co-workers showed both theoretically and experimentally that under certain conditions one may actually detect diffraction-like effects when the ROSE experiment is applied to porous systems. We report here for the first time such diffraction-like effects in an emulsion system, which were observed when examining the water diffusion in the emulsion system mentioned above with the PGSE experiment. From the experiment the mean droplet size may be determined.
Nuclear resonance techniques involving free precession are examined, and, in particular, a convenient variation of Hahn's spin-echo method is described. This variation employs a combination of pulses of different intensity or duration ("90-degree" and "180-degree" pulses). Measurements of the transverse relaxation time ${T}_{2}$ in fluids are often severely compromised by molecular diffusion. Hahn's analysis of the effect of diffusion is reformulated and extended, and a new scheme for measuring ${T}_{2}$ is described which, as predicted by the extended theory, largely circumvents the diffusion effect. On the other hand, the free precession technique, applied in a different way, permits a direct measurement of the molecular self-diffusion constant in suitable fluids. A measurement of the self-diffusion constant of water at 25\ifmmode^\circ\else\textdegree\fi{}C is described which yields $D=2.5(\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}{10}^{$-${}5}$ ${\mathrm{cm}}^{2}$/sec, in good agreement with previous determinations. An analysis of the effect of convection on free precession is also given. A null method for measuring the longitudinal relaxation time ${T}_{1}$, based on the unequal-pulse technique, is described.
Nuclear-magnetic-resonance measurements of the proton-spin relaxation for water in biological cells are known to exhibit a multiexponential decay. A theory, based on the diffusion equation using the bulk diffusivity of water, is developed to explain this phenomenon. It is shown that multiexponential decay arises simply as a consequence of an eigenvalue problem associated with the size and shape of the cell and that this multiexponential decay can only be observed for samples whose size is of the order of a biological cell. As an example, the theory is applied to a previously published data for rat gastronemius cells. Excellent agreement is obtained, and furthermore, the size of the cell is calculated by fitting the theory to the experiment.
We report a combined experimental, theoretical, and simulation study of pulsed gradient spin‐echo (PGSE) nuclear magnetic resonance (NMR) for fluid saturated porous media. A simple pore hopping theory is developed on the basis of the assumption that diffusion within pores is very much faster than diffusion between pores. For suitable periodic media, the theoretical results are found to be in good agreement with random‐walk simulations. The theory for glasslike media is then used to analyze experimental PGSE NMR data for a water‐saturated random loose pack of nearly monodisperse polystyrene spheres. The structural parameters extracted by this method are consistent with the known geometry of such packings. An important observation from the simulations is that the long‐time effective diffusion constant is already accessed at times so short that a single spin will only have diffused across one pore width.
The application of pulsed gradient spin echo nuclear magnetic resonance (NMR) to the case of molecules trapped between two plane parallel boundaries, has been examined theoretically, with computer simulation and by experiment. A new closed‐form analytic expression is obtained for the averaged propagator and the echo attenuation when the walls have finite relaxivity and this expression is verified by computer simulations. It is shown that ‘‘diffraction’’ effects are still strongly apparent when wall relaxation is taken into account and that deviations in the barrier spacing parameter obtained from the position of the echo minimum, are weak. In particular we show that for the pulsed gradient spin echo (PGSE) pulse separation time on the order of a2/2D, the deviation is less than 10% provided that the relaxation is not so severe as to reduce the zero gradient signal amplitude below 10% of its unrelaxed value. We further examine the influence of finite gradient pulse and find, as with wall relaxation, that diffusion during the gradient pulse has the effect of shifting to higher q the position of the first minimum in the echo diffraction pattern. The diffusive diffraction effect is demonstrated experimentally using a stack of pentane‐filled microcapillaries of wall spacing 100 μm, and fits to the data yield realistic values for the known experimental parameters.
THE transport of fluids in porous media is of importance in a wide range of areas, such as oil recovery, heterogeneous catalysis and biological perfusion. The pulsed gradient spin-echo (PGSE) NMR technique has been used for many years to characterize diffusion and flow in such systems1–3. The analogy between NMR measurements in a field gradient and diffraction has been pointed out in the context of NMR imaging4 and, more recently, diffraction-like effects in the PGSE experiment have been discussed for diffusion in both impermeable5 and connected6 structures. The gradient pulse area plays the role of a wavevector, q, which can probe the structure in which the fluid diffuses. Here we report experimental confirmation of these predicted effects from proton NMR studies of a water-saturated, orientationally disordered, loosely packed array of monodisperse polystyrene spheres. The PGSE-NMR experiments may thus be used to provide an indirect, averaged image of the internal structure of porous solids at a resolution higher than that achievable with conventional NMR imaging. This is particularly advantageous for measurements on large samples, as the resolution available with the PGSE method depends only on the available range of gradient pulse amplitude and duration and is unconstrained by the factors determining resolution in conventional NMR imaging.
The applicability of the Fourier relation between the averaged displacement profile and the experimental echo-attenuation curve for the PFG NMR method for studying restricted diffusion is discussed. Formally, this relation is valid only in the limit where the gradient pulse is given in the form of a δ function, the so-called short-gradient-pulse (SGP) limit. Experimentally, this condition can of course never be realized, and usually a block-shaped pulse of finite duration and amplitude is employed. This problem was investigated numerically and the results were compared with the analytic expression valid in the SGP limit. The results presented were calculated for the simplest restricted geometry: the diffusion between two plates.
The Bloch—Torrey equations are modified to include the case of anisotropic, restricted diffusion and flow. The problem of solving these modified equations for the amplitude of a spin echo in a time-dependent magnetic-field gradient subject to restricting boundary conditions is discussed. This problem is solved for a number of selected cases. In particular, it is found that a magnetic-field gradient applied in short, intense pulses is effective in defining the time during which nuclear displacements take place. A simplified equation, suitable for the pulsed-gradient experiment, is presented and solved for two different examples of systems showing restricted diffusion. A procedure for analyzing the data from pulsed-gradient measurements is suggested, and its merits are discussed. Suggestions are made of systems which may well be expected to show restricted, anisotropic diffusion or interesting flow properties.
The phenomenological Bloch equations in nuclear magnetic resonance are generalized by the addition of terms due to the transfer of magnetization by diffusion. The revised equations describe phenomena under conditions of inhomogeneity in magnetic field, relaxation rates, or initial magnetization. As an example the equations are solved in the case of the free precession of magnetic moment in the presence of an inhomogeneous magnetic field following the application of a 90° pulse with subsequent applications of a succession of 180° pulses. The spin-echo amplitudes agree with the results of Carr and Purcell from a random walk theory.
A derivation is given of the effect of a time-dependent magnetic field gradient on the spin-echo experiment, particularly in the presence of spin diffusion. There are several reasons for preferring certain kinds of time-dependent magnetic field gradients to the more usual steady gradient. If the gradient is reduced during the rf pulses, H1 need not be particularly large; if the gradient is small at the time of the echo, the echo will be broad and its amplitude easy to measure. Both of these relaxations of restrictions on the measurement of diffusion coefficients by the spin-echo technique serve to extend its range of applicability. Furthermore, a pulsed gradient can be recommended when it is critical to define the precise time period over which diffusion is being measured.
The theoretical expression derived has been verified experimentally for several choices of time dependent magnetic field gradient. An apparatus is described suitable for the production of pulsed gradients with amplitudes as large as 100 G cm−1. The diffusion coefficient of dry glycerol at 26°±1°C has been found to be (2.5±0.2)×10−8 cm2 sec−1, a value smaller than can ordinarily be measured by the steady gradient method.
The theory of pulsed-gradient spin-echo NMR in the case of molecules trapped within pores has been extended to include wall-relaxation effects. Expressions are obtained for the echo attenuation under the narrow-gradient-pulse condition for rectangular, cylindrical, and spherical pores. The cylindrical result is new, while those for rectangular and spherical pores differ from previously published formulas. The present results are verified in a number of special limits. It is shown that "diffraction" effects are still strongly apparent when wall relaxation is taken into account. Furthermore, deviations in the pore radius, a, obtained from the position of the echo minimum, are weak in practice, provided that the time between the gradient pulses exceeds a2/D, where D is the molecular self-diffusion coefficient.
The so-called short-gradient-pulse approximation, in which the field-gradient pulses are described as δ functions, is most useful in the description of the pulsed held gradient (PFG) experiment when applied to systems showing restricted diffusion. In this contribution, the validity of this approximation is tested by performing Brownian dynamic simulations for molecules entrapped in planar, cylindrical, and spherical geometries, in all cases with reflecting walls. A rather broad variation in the parameters determining the echo decays is covered and, from the data presented, it is possible to chose suitable parameters when applying the PFG experiment to systems showing restricted diffusion. Some limiting behavior of the echo decays are discussed and compared with theoretical results.
A multiple exponential time series expansion is used to analyse the dependence of the NMR pulsed gradient spin-echo amplitude S(q, Delta, tau) on the wavevector q, the pulsed gradient separation Delta and the 90-180° pulse separation tau in multicompartment systems such as cellular tissue. The static structure factor measured in q space microscopy is obtained as the lowest order coefficient in the time expansion at long Delta. Dynamic information is obtained from the wavevector dependence of the higher order coefficients. Spatially inhomogeneous relaxation is shown to distort the apparent static and dynamic structure factors. Changes in compartment morphology alter the coupling of relaxation and diffusion reflected in the wavevector dependent relaxation times obtained when Delta and tau are varied together. By fitting the data for parenchyma tissue of apple with a numerical cell model, a plasmalemma membrane permeability of 1·2 × 10-3 cm s-1 is derived.
A general treatment of diffusion of spin-bearing species in a bounded medium with loss of magnetization at the boundaries is presented. The effects of the loss of magnetization at the boundaries for pulsed-field-gradient (PFG) NMR diffusion measurements and q-space imaging are calculated and demonstrated. If this wall effect is ignored, e.g., by assuming fully reflecting walls, erroneous results for the barrier spacing (a) and interbarrier diffusion coefficient (D) are obtained. Methods for obtaining the correct values of the parameters a, D, and the loss of the magnetization at the boundaries from experimental results are demonstrated. The consequences of loss of magnetization at the walls for the interpretation and application of PFG NMR results are discussed.
We introduce and demonstrate an NMR pulsed gradient stimulated echo method of directly obtaining the molecular translational displacement probability (displacement profile) of a liquid. The temporal development of the displacement profile reflects the presence of diffusion, restrictions to diffusion (e.g., walls, membranes), flow, and spatially dependent relaxation sinks. This approach allows the study of compartments which are too small to be observed by conventional NMR imaging methods. The distribution of spatial properties of compartments can be characterized over a spatial field of about 0.1 to 25 microns, completely independent of the absolute spatial location of the individual compartments.
q-Space imaging (Callaghan, J. Magn. Reson. 88, 493 (1990)) has been used to obtain mouse brain water displacement profiles. These profiles take the form of a unidirectional incoherent-displacement probability density distribution. Two groups of mice were studied, a normal group and one in which surgery had been performed to reduce the supply of blood to the forebrain. In the normal group the incoherent displacement of water was reduced postmortem. Four of the surgically treated mice yielded displacement profiles that resembled those obtained postmortem; the remaining two were near normal. This study demonstrates the feasibility of in vivo q-space imaging. The displacement profile changes that occur subsequent to an interruption of the forebrain blood supply are consistent with the hyperintensity changes seen in diffusion-weighted imaging.
We derive an expression for the magnetization M(k,Delta) in a pulsed-field-gradient experiment for spins diffusing in a confined space with relaxation at the pore walls. Here k=gammadeltag, delta= pulse width, g= gradient strength, gamma= the gyromagnetic ratio, and Delta is the time between gradient pulses. We show that the deviation of -ln[M(k,Delta)/M(0,Delta)] from quadratic behavior in k in experiments in porous media can be a more sensitive probe of the microgeometry (size, connectivity, size distribution, shape, etc.), than either the enhancement of 1/T1 over the bulk water value, or the macroscopic diffusion coefficient, which is derived from the slope of -ln[M(k,Delta)/M(0,Delta)] at small k2, in the limit of large Delta. We propose some simple models of randomly oriented tubes and sheets to interpret the k dependence of the amplitude beyond the leading small-k quadratic behavior. When the macroscopic diffusion coefficient is unobtainable, due to the decay, the present considerations should be useful in extracting geometrical information. The effective diffusion constant derived from NMR exactly equals that derived from electrical conductivity only when the surface relaxivity is zero, but can be close to each other in favorable circumstances even for finite surface relaxivity. Exact solutions with partially absorbing boundary conditions are obtained for a slab and a sphere to infer that the normalized amplitude M(k,Delta,rho)/M(0,Delta,rho) depends only weakly on the surface relaxivity rho for monodisperse convex-shaped pores in the parameter ranges of interest. We also obtain expressions for the mean lifetime of the amplitude in the geometries considered.
We investigate the time-dependent diffusion coefficient, D(t)=/(6t), of random walkers in porous media with piecewise-smooth pore-grain interfaces. D(t) is measured in pulsed-field-gradient spin-echo (PFGSE) experiments on fluid-saturated porous media. For reflecting boundary conditions at the interface we show that for short times D(t)/D0 =1-A0(D0t)1/2+B0D0t+O[(D0t)3/2], where A0=4S/(9 &surd;pi VP) and B0=-HS/(12VP)-tsumi(Li/VP)f(phii). Here D0 is the diffusion constant of the bulk fluid, S/VP is the surface area to pore volume ratio, H is the mean curvature of the smooth portions of the surface, Li is the length of a wedge of angle phii, and the function f(phi) is defined below. More generally, we consider partially absorbing boundary conditions, where the absorption strength is controlled by a surface-relaxivity parameter rho. Here, the density of walkers (i.e., the net magnetization) decays as M(t)=1-rhoSt/VP+..., and D(t) is defined as s/(6t), where s is the mean-square displacement of surviving walkers. When rho!=0 we find that the coefficient A0 of the &surd;D0t term in the above equation is unchanged, while the coefficient of the linear term changes to B0+rhoS/(6VP). Thus, data on D(t) and M(t) at short times may be used simultaneously to determine S/VP and rho. The limiting behavior of D(t) as rho-->∞ is also discussed.
We propose a simple ansatz that relates the diffusion propagator for the molecules of a fluid confined in a porous medium to the pore-space structure factor. Theoretical arguments and numerical simulations show that it works well for both periodic and disordered geometries. The ansatz allows us to deconvolve structural data from momentum dependent pulsed field gradient spin-echo data.