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تسجيل مستخدم جديدModified-Bloch Equation Based on Fractal Derivative for Analyzing PFG Anomalous Diffusion
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Guoxing Lin
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A modified-Bloch equation based on the fractal derivative is proposed to analyze pulsed field gradient (PFG) anomalous diffusion. Anomalous diffusion exists in many systems such as in polymer or biological systems. PFG anomalous diffusion could be analyzed based on the fractal derivative or the fractional derivative. Compared to the fractional derivative, the fractal derivative is simpler, and it is faster in numerical evaluations. In this paper, the fractal derivative is employed to build the modified-Bloch equation that is a fundamental method to describe the spin magnetization evolution affected by fractional diffusion, Larmor precession, and relaxation. An equivalent form of the fractal derivative is proposed to convert the fractional diffusion equation, which can then be combined with the precession and relaxation equations to get the modified-Bloch equation. This modified-Bloch equation yields a general PFG signal attenuation expression that includes the finite gradient pulse width (FGPW) effect, namely, the signal attenuation during field gradient pulse. The FGPW effect needs to be considered in most clinical MRI applications, and including FGPW effect allows the detecting of slower diffusion that is often encountered in polymer systems. Additionally, the spin-spin relaxation effect can be analyzed, which provides a broad view of the dynamic process in materials. The modified-Bloch equation based on the fractal derivative could provide a fundamental theoretical model for PFG anomalous diffusion.
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Pulsed field gradient (PFG) has been increasingly employed to study anomalous diffusions in Nuclear Magnetic Resonance (NMR) and Magnetic Resonance Imaging (MRI). However, the analysis of PFG anomalous diffusion is complicated. In this paper, a fract
al derivative model based modified Gaussian phase distribution method is proposed to describe PFG anomalous diffusion. By using the phase distribution obtained from the effective phase shift diffusion method based on fractal derivatives, and employing some of the traditional Gaussian phase distribution approximation techniques, a general signal attenuation expression for free fractional diffusion is derived. This expression describes a stretched exponential function based attenuation, which is distinct from both the exponential attenuation for normal diffusion obtained from conventional Gaussian phase distribution approximation, and the Mittag-Leffler function based attenuation for anomalous diffusion obtained from fractional derivative. The obtained signal attenuation expression can analyze the finite gradient pulse width (FGPW) effect. Additionally, it can generally be applied to all three types of PFG fractional diffusions classified based on time derivative order alpha and space derivative order beta. These three types of fractional diffusions include time-fractional diffusion, space-fractional diffusion, and general fractional diffusion. The results in this paper are consistent with reported results based on effective phase shift diffusion equation method and instantaneous signal attenuation method. This method provides a new, convenient approximation formalism for analyzing PFG anomalous diffusion experiments.
Anomalous diffusion exists widely in polymer and biological systems. Pulsed-field gradient (PFG) anomalous diffusion is complicated, especially in the anisotropic case where limited research has been reported. An general PFG signal attenuation expres
sion, including the finite gradient pulse (FGPW) effect for free general anisotropic fractional diffusion based on the fractional derivative, has not been obtained, where {alpha} and b{eta} are time and space derivative orders. It is essential to derive a general PFG signal attenuation expression including the FGPW effect for PFG anisotropic anomalous diffusion research. In this paper, two recently developed modified-Bloch equations, the fractal differential modified-Bloch equation and the fractional integral modified-Bloch equation, were extended to obtain general PFG signal attenuation expressions for anisotropic anomalous diffusion. Various cases of PFG anisotropic anomalous diffusion were investigated, including coupled and uncoupled anisotropic anomalous diffusion. The continuous-time random walk (CTRW) simulation was also carried out to support the theoretical results. There is good agreement between the theory and the CTRW simulation. The obtained signal attenuation expressions and the three-dimensional fractional modified-Bloch equations are important for analyzing PFG anisotropic anomalous diffusion in NMR and MRI.
The studying of anomalous diffusion by pulsed field gradient (PFG) diffusion technique still faces challenges. Two different research groups have proposed modified Bloch equation for anomalous diffusion. However, these equations have different forms
and, therefore, yield inconsistent results. The discrepancy in these reported modified Bloch equations may arise from different ways of combining the fractional diffusion equation with the precession equation where the time derivatives have different derivative orders and forms. Moreover, to the best of my knowledge, the general PFG signal attenuation expression including finite gradient pulse width (FGPW) effect for time-space fractional diffusion based on the fractional derivative has yet to be reported by other methods. Here, based on different combination strategy, two new modified Bloch equations are proposed, which belong to two significantly different types: a differential type based on the fractal derivative and an integral type based on the fractional derivative. The merit of the integral type modified Bloch equation is that the original properties of the contributions from linear or nonlinear processes remain unchanged at the instant of the combination. The general solutions including the FGPW effect were derived from these two equations as well as from two other methods: a method observing the signal intensity at the origin and the recently reported effective phase shift diffusion equation method. The relaxation effect was also considered. It is found that the relaxation behavior influenced by fractional diffusion based on the fractional derivative deviates from that of normal diffusion. The general solution agrees perfectly with continuous-time random walk (CTRW) simulations as well as reported literature results. The new modified Bloch equations is a valuable tool to describe PFG anomalous diffusion in NMR and MRI.
Anomalous diffusion has been investigated in many systems. Pulsed field gradient (PFG) anomalous diffusion is much more complicated than PFG normal diffusion. There have been many theoretical and experimental studies for PFG isotropic anomalous diffu
sion, but there are very few theoretical treatments reported for anisotropic anomalous diffusion. Currently, there is not a general PFG signal attenuation expression, which includes the finite gradient pulse effect and can treat all three types of anisotropic fractional diffusions: general fractional diffusion, time fractional diffusion, and space-fractional diffusion. In this paper, the recently developed instantaneous signal attenuation (ISA) method was applied to obtain PFG signal attenuation expression for free and restricted anisotropic anomalous diffusion with two models: fractal derivative and fractional derivative models. The obtained PFG signal attenuation expression for anisotropic anomalous diffusion can reduce to the reported result for PFG anisotropic normal diffusion. The results can also reduce to reported PFG isotropic anomalous diffusion results obtained by effective phase shift diffusion equation method and instantaneous signal attenuation method. For anisotropic space-fractional diffusion, the obtained result agrees with that obtained by the modified Bloch equation method. Additionally, The PFG signal attenuation expressions for free and restricted anisotropic curvilinear diffusions were derived by the traditional method, the results of which agree with the PFG anisotropic fractional diffusion results based on the fractional derivative model. The powder pattern of PFG anisotropic diffusion was also discussed. The results here improve our understanding of PFG anomalous diffusion, and provide new formalisms for PFG anisotropic anomalous diffusion in NMR and MRI.
Anomalous diffusion exists widely in polymer and biological systems. Pulsed field gradient (PFG) techniques have been increasingly used to study anomalous diffusion in NMR and MRI. However, the interpretation of PFG anomalous diffusion is complicated
. Moreover, there is not an exact signal attenuation expression based on fractional derivatives for PFG anomalous diffusion, which includes the finite gradient pulse width effect. In this paper, a new method, a Mainardi-Luchko-Pagnini (MLP) phase distribution approximation, is proposed to describe PFG fractional diffusion. MLP phase distribution is a non-Gaussian phase distribution. From the fractional diffusion equation based on fractional derivatives in both real space and phase space, the obtained probability distribution function is a MLP distribution. The MLP distribution leads to a Mittag-Leffler function based PFG signal attenuation rather than the exponential or stretched exponential attenuation that is obtained from a Gaussian phase distribution (GPD) under a short gradient pulse approximation. The MLP phase distribution approximation is employed to get a complete signal attenuation expression E{alpha}(-Dfb*{alpha},b{eta}) that includes the finite gradient pulse width effect for all three types of PFG fractional diffusion. The results obtained in this study are in agreement with the results from the literature. These results provide a new, convenient approximation formalism to interpret complex PFG fractional diffusion experiments.
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