Abstract
Pulsed-field gradient (PFG) diffusion experiments can be used to measure anomalous diffusion in many polymer or biological systems. However, it is still complicated to analyze PFG anomalous diffusion, particularly the finite gradient pulse width (FGPW) effect. In practical applications, the FGPW effect may not be neglected, such as in clinical diffusion magnetic resonance imaging (MRI). Here, two significantly different methods are proposed to analyze PFG anomalous diffusion: the effective phase-shift diffusion equation (EPSDE) method and a method based on observing the signal intensity at the origin. The EPSDE method describes the phase evolution in virtual phase space, while the method to observe the signal intensity at the origin describes the magnetization evolution in real space. However, these two approaches give the same general PFG signal attenuation including the FGPW effect, which can be numerically evaluated by a direct integration method. The direct integration method is fast and without overflow. It is a convenient numerical evaluation method for Mittag-Leffler function-type PFG signal attenuation. The methods here provide a clear view of spin evolution under a field gradient, and their results will help the analysis of PFG anomalous diffusion.
Highlights
Anomalous dynamic behavior exists in many polymer or biological systems [1,2,3,4,5,6]
Anomalous diffusion can be detected by pulsed-field gradient (PFG) Nuclear magnetic resonance (NMR) experiments
Much effort has been devoted to studying PFG anomalous diffusion based on fractional calculus, which includes the propagator method [29], the modified Bloch equation method [17,25,30,31], the effective phase-shift diffusion equation (EPSDE)method [18], the instantaneous signal attenuation method [26], the modified-Gaussian or non-Gaussian distribution method [27], etc
Summary
Anomalous dynamic behavior exists in many polymer or biological systems [1,2,3,4,5,6]. Much effort has been devoted to studying PFG anomalous diffusion based on fractional calculus, which includes the propagator method [29], the modified Bloch equation method [17,25,30,31], the effective phase-shift diffusion equation (EPSDE)method [18], the instantaneous signal attenuation method [26], the modified-Gaussian or non-Gaussian distribution method [27], etc. For a homogeneous diffusion spin system, the magnetization amplitude attenuates because of the gradient magnetic field effect, the phase of magnetization keeps constant at the origin of the gradient field Such a specific phase property is employed to derive a PFG signal attenuation equation in this paper. The two methods provide complementary views of PFG anomalous diffusion from both the real space and phase space
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