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Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches

Guoxing Lin

2018
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Mathematics
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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
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... -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. displacement is not linearly proportional to diffusion time [2] . These non-Gaussian characteristics make it complicated to analyze PFG anomalous diffusion. Although PFG anomalous diffusion can be approximately analyzed by traditional methods such as the apparent diffusion coefficient method, PFG theories based on the fractional derivative could not only improve the analysis accuracy on the diffusion domain size, diffusion constant, and other variables [17, 18, [25] [26] [27] [28] , but also yield additional information such as the time derivative order α and space derivative order β that are related with diffusion jump time and jump length distributions determined by material properties. 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. Additionally, PFG anomalous diffusions in restricted geometries such as plate, sphere, and cylinder have been investigated [28] . These theoretical methods analyze PFG anomalous diffusion from different angles. Therefore, each of them can have its own advantages in handling certain types of PFG anomalous diffusion. To better apply the PFG technique to studying anomalous diffusion, it is still valuable to develop new theoretical treatments for PFG anomalous diffusion. In this paper, two methods based on the fractional derivative [1,2,6,32-34] are proposed to give general analytical PFG signal attenuation expressions for anomalous diffusion. The first method is the recently developed EPSDE method [18] . This method describes the spin phase evolution by an effective phase diffusion process in virtual phase space [18] . Solving the EPSDE gives valuable information about the phase evolution process such as the phase probability distribution function and the moment of mean phase displacement. Meanwhile, other conventional methods render it difficult to get this phase information, and usually assume an approximate phase distribution such as Gaussian phase distribution [13] . In this paper, it will be shown that a solvable PFG signal attenuation equation can be derived by applying a Fourier transform to the effective phase-shift equation. The second method is to observe the signal intensity at the origin, and is an ultra-simple new method. For a homogeneous diffusion spin system, although 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 above two methods give the same signal attenuation equation, from which the general PFG signal attenuation expression can be derived by the Adomian decomposition method [35] [36] [37] [38] [39] . Besides the Adomian decomposition method, a direct integration method was proposed for the numerical evaluation of the PFG signal attenuation, which is a fast and simple method. The results include the finite gradient pulse width (FGPW) effect [13] [14] [15] , namely, the signal attenuation during each gradient pulse application period. Theoretically, during a short gradient pulse, the PFG signal attenuation can be neglected; nevertheless, the gradient pulse used in a clinical MRI is usually long [16, 40] . Additionally, a longer gradient pulse allows the measuring of slower diffusion under the same gradient maximum intensity, which matters in the study of polymer and biological systems where the molecule diffusion is often slow. Therefore, we need to consider the FGPW effect in many real applications. The two methods here give the same results in terms of the FGPW effect. These results agree with reported results from some other methods [18, 26, 27] and the continuous time random walk simulation [26, 29] . Furthermore, PFG anomalous diffusion of intramolecular multiple quantum coherence (MQC) is also discussed [27] . The MQC effect has the benefit of enhancing the gradient effect on PFG signal attenuation [41] . The two methods provide complementary views of PFG anomalous diffusion from both the real space and phase space.

doi:10.3390/math6020017
fatcat:oopuxwxndbd2zingw7qjoor4ra