Laplacian eigenfunctions in NMR. II. Theoretical advances

Grebenkov, Denis S.

doi:10.1002/cmr.a.20145

Cited by 46 publications

(70 citation statements)

References 97 publications

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“…Specifying the initial density of particles rðr 0 Þ at time t ¼ 0, one fixes the solution of Eq. [2] and the whole evolution of the system. Among various initial conditions, a point-like source plays a special role.…”

Section: Diffusion Equationmentioning

confidence: 99%

“…Throughout this paper, G t ðr 0 ; rÞ denotes the family of these densities, parametrized by the starting point r 0 , that satisfy Eq. [2] with respect to the arrival point r. The initial condition for all particles concentrated in one point r 0 is mathematically represented by the Dirac distribution: G t¼0 ðr 0 ; rÞ ¼ dðr À r 0 Þ: [4] As it will be shown later, the function G t ðr 0 ; rÞ is symmetric with respect to interchange of r 0 and r: G t ðr 0 ; rÞ ¼ G t ðr; r 0 Þ:…”

Section: Diffusion Equationmentioning

confidence: 99%

“…This tendency is mathematically represented by the phenomenological Fick's law jðr; tÞ ¼ ÀDrcðr; tÞ: [1] Throughout the paper, the free diffusion coefficient D is assumed to be constant (no dependence on time and spatial coordinates). Substituting j(r,t) into the continuity equation, one gets the diffusion (or heat) equation q qt cðr; tÞ À DDcðr; tÞ ¼ 0; [2] where…”

Section: Diffusion Equationmentioning

confidence: 99%

“…The diffusion equation [2] describes the evolution of the density from an initial state. Specifying the initial density of particles rðr 0 Þ at time t ¼ 0, one fixes the solution of Eq.…”

Section: Diffusion Equationmentioning

confidence: 99%

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Laplacian eigenfunctions in NMR. I. A numerical tool

Grebenkov

2008

Concepts Magnetic Resonance

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112

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A geometrical confinement considerably affects the diffusive motion of the nuclei and the consequent signal attenuation under inhomogeneous magnetic fields. In this article, we illustrate the use of Laplacian eigenfunctions to describe this effect. Starting from the classical Bloch-Torrey equation, we obtain the free induction decay (FID) and the spin-echo or gradient-echo signal in a compact matrix form. Each attenuation mechanism (restricted diffusion, gradient dephasing, surface or bulk relaxation) is represented by a matrix which is constructed from the Laplace operator eigenbasis and thus depending only on the geometry of the confinement. In turn, the physical parameters (free diffusion coefficient, gradient intensity, surface or bulk relaxivity) characterize the ' 'strengths' ' of the underlying attenuation mechanisms and naturally appear as coefficients in front of these matrices. Once the Laplacian eigenfunctions for a given confinement are found (analytically or numerically), further computation of the macroscopic signal is more accurate and much faster than by using conventional simulation methods. The matrix technique is actually a simple numerical tool to deal with arbitrary gradient waveforms, including simple or stimulated, single or multiple spin echoes. We illustrate its efficiency by considering restricted diffusion in simple domains: a slab, a cylinder, and a sphere. In a companion paper, we shall focus on theoretical advances achieved by using Laplacian eigenfunctions.

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Section: Diffusion Equationmentioning

confidence: 99%

Section: Diffusion Equationmentioning

confidence: 99%

Section: Diffusion Equationmentioning

confidence: 99%

Section: Diffusion Equationmentioning

confidence: 99%

See 2 more Smart Citations

Laplacian eigenfunctions in NMR. I. A numerical tool

Grebenkov

2008

Concepts Magnetic Resonance

Self Cite

112

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“…In the presented results, the accuracy was verified by increasing N λ and checking whether numerical results remained unchanged. A detailed description of the MCF approach is beyond the scope of this article, but can be found in Grebenkov and Grebenkov [46,47], for example.…”

Section: Methodsmentioning

confidence: 99%

On the Vanishing of the t-term in the Short-Time Expansion of the Diffusion Coefficient for Oscillating Gradients in Diffusion NMR

et al. 2017

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Nuclear magnetic resonance (NMR) diffusion measurements can be used to probe porous structures or biological tissues by means of the random motion of water molecules. The short-time expansion of the diffusion coefficient in powers of t 1/2 , where t is the diffusion time related to the duration of the diffusion-weighting magnetic field gradient profile, is universally connected to structural parameters of the boundaries restricting the diffusive motion. The t 1/2 -term is proportional to the surface to volume ratio. The t-term is related to permeability and curvature. The short time expansion can be measured with two approaches in NMR-based diffusion experiments: First, by the use of diffusion encodings of short total duration and, second, by application of oscillating gradients of long total duration. For oscillating gradients, the inverse of the oscillation frequency becomes the relevant time scale. The purpose of this manuscript is to show that the oscillating gradient approach is blind to the t-term. On the one hand, this prevents fitting of permeability and curvature measures from this term. On the other hand, the t-term does not bias the determination of the t 1/2 -term in experiments.

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A spectral framework forNMRsignal with restricted diffusion

Bar

Sochen

2015

Concepts Magnetic Resonance

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By the multiple correlation function (MCF) formalism, the nuclear magnetic resonance magnetization of diffusing spins can be represented for simple pore geometries. It may be used to infer geometric structure at the scale of microns. This is to be compared to the diffusion tensor imaging, which provides geometric information at the scale of millimeters. The MCF formulation was derived to special cases in which the gradients of the magnetic field are oriented in specific directions. A generalized approach allowing an arbitrary magnetic field direction was introduced by € Ozarslan. In this article, we present a complete account of the generalized MCF mathematical derivation starting from Bloch-Torrey equations. It is aimed to the experts and novices alike. We present two approaches-the indirect derivation is based on € Ozarslan's work, where the standard MCF equations are adopted to arbitrary gradient directions. Our alternative approach is based on direct calculations of the MCF matrices for specified gradient directions. We prove that the two approaches lead to the same equations. Finally, we revise Mitra's microscopic approach and show the relation to the macroscopic MCF approach. We prove that in some limit conditions the two signal equations coincide

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Laplacian eigenfunctions in NMR. II. Theoretical advances

Cited by 46 publications

References 97 publications

Laplacian eigenfunctions in NMR. I. A numerical tool

Laplacian eigenfunctions in NMR. I. A numerical tool

On the Vanishing of the t-term in the Short-Time Expansion of the Diffusion Coefficient for Oscillating Gradients in Diffusion NMR

A spectral framework forNMRsignal with restricted diffusion

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