A general framework to quantify the effect of restricted diffusion on the NMR signal with applications to double pulsed field gradient NMR experiments

Abstract

Based on a description introduced by Robertson, Grebenkov recently introduced a powerful formalism to represent the diffusion-attenuated NMR signal for simple pore geometries such as slabs, cylinders, and spheres analytically. In this work, we extend this multiple correlation function formalism by allowing for possible variations in the direction of the magnetic field gradient waveform. This extension is necessary, for example, to incorporate the effects of imaging gradients in diffusion-weighted NMR imaging scans and in characterizing anisotropy at different length scales via double pulsed field gradient ͑PFG͒ experiments. In cylindrical and spherical pores, respectively, two-and three-dimensional vector operators are employed whose form is deduced from Grebenkov's results via elementary operator algebra for the case of cylinders and the Wigner-Eckart theorem for the case of spheres. The theory was validated by comparison with known findings and with experimental double-PFG data obtained from water-filled microcapillaries.

Spatio-temporal anomalous diffusion imaging: results in controlled phantoms and in excised human meningiomas

“…In recent years, several approaches have been proposed to quantify non-Gaussian [31][32][33][34][35] and AD processes in heterogeneous systems [8,13,14,[18][19][20][36][37][38][39][40][41]. In this paper, for the first time, we show and compare anomalous subdiffusive Mα maps together with anomalous pseudo-superdiffusion Mγ and conventional MD maps obtained from the same samples (controlled phantom and excised brain tissues).…”

A parametric finite element solution of the generalised Bloch–Torrey equation for arbitrary domains

“…The first was intended to show how the developed method performed in situations where the analytical solution was available [6], whereas the second presented a real application based on experimental data [5]. These examples were chosen between many others just to show the capabilities of the FE formulation in concrete situations.…”

“…To show the versatility of the numerical framework, we simulated the last experiment reported in [5]. It consists in the application of a dPGSE sequence (δ 1 = δ 2 = 4.5 ms, ∆ 1 = ∆ 2 = 40 ms, t m = 0 s, g x = g y = 67.3, 125.3, 202.0, and 375.7 mT /m, g z = 0) perpendicular to impermeable cylindrical microcapillaries of inner diameter 10 ± 1 µm, oriented in the z direction, and centred in the origin.…”

“…These solutions are obtained by proper manipulation of the mathematical expressions describing the physical phenomena. Then, different forms of the solution can be found depending on the mathematical framework used and the approximations made [1,4,5,6,7]. These solutions have been shown to be very important to study the physical basis of experimental results (e.g.…”

“…These solutions have been shown to be very important to study the physical basis of experimental results (e.g. [5]), as well as to perform other mathematical analysis due to their parametric nature [1]. However, since the difficulty of such manipulation increases with the complexity of the domain, there exist solutions only for simple geometries, as multi-layered slabs (1D), cylinders (2D), and spheres (3D).…”