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Lin G. Revealing the effect of phase and time coupling on NMR relaxation rate by random walks in phase space. Phys Rev E 2023; 108:034121. [PMID: 37849104 DOI: 10.1103/physreve.108.034121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2023] [Accepted: 09/05/2023] [Indexed: 10/19/2023]
Abstract
Phase-time coupling is a natural process in the phase random walks of a spin system; however, its effect on the nuclear magnetic resonance (NMR) relaxation is a challenge to the established theories such as the second-order quantum perturbation theory. This paper extends the recently developed phase diffusion method to treat the phase-time coupling effect, based on uncoupled phase diffusions, and coupled random walks. The instantaneous projection of the rotating random field is employed to get the accumulated phase of the NMR observable. In the static frame and the rotating frame, the phase diffusion coefficients are derived. The obtained theoretical results show that the phase-time coupling has a significant impact on the NMR relaxation rate: The angular frequency ω in the spectral density is modified to an apparent angular frequency ηω, where η is the phase-time coupling constant. The strongest coupling has η equaling 2, while η equaling 1 corresponds to the traditional results. As an example, the modified relaxation time expressions based on both monoexponential and nonmonoexponential functions can successfully fit the previously reported ^{13}C T_{1} NMR experimental data of polyisobutylene (PIB) in the blend of PIB and head-to-head poly(propylene). In contrast, the traditional relaxation rate expression based on the monoexponential time correlation function cannot fit such experimental data. With the phase-time coupling, the obtained characteristic time of the segmental motion is faster than that from conventional results.
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Affiliation(s)
- Guoxing Lin
- Carlson School of Chemistry and Biochemistry, Clark University, Worcester, Massachusetts 01610, USA
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Liao X, Zheng S, Lin G. Pulsed field gradient signal attenuation of restricted anomalous diffusions in plate, sphere, and cylinder with wall relaxation. Phys Rev E 2020; 101:012128. [PMID: 32069550 DOI: 10.1103/physreve.101.012128] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/06/2019] [Indexed: 11/07/2022]
Abstract
The effect of boundary relaxation on pulsed field gradient (PFG) anomalous restricted diffusion is investigated in this paper. The PFG signal attenuation expressions of anomalous diffusion in plate, sphere, and cylinder are derived based on fractional calculus. In addition, approximate expressions for boundary relaxation induced short time signal attenuation under zero gradient field and boundary relaxation affected short time apparent diffusion coefficients are given in this paper. Unlike the exponential signal attenuation in normal diffusion, the PFG signal attenuation in anomalous diffusion with boundary relaxation is either a Mittag-Leffler-function-based attenuation or a stretched-exponential-function-based attenuation. The stretched exponential attenuations of all three structures clearly show the diffractive pattern. In contrast, only in the plate structure does the Mittag-Leffler-function-based attenuation display an obvious diffractive pattern. Additionally, anomalous diffusion with smaller time derivative order α has a weaker diffractive pattern and less signal attenuation. Moreover, the results demonstrate that boundary relaxation induced signal attenuation is significantly affected by the anomalous diffusion when no gradient field is applied. Meanwhile, the boundary relaxation significantly affects PFG signal attenuation of anomalous diffusion in the following ways: The boundary relaxation results in reduced radius from the minimum of the diffractive patterns, and it results in an increased apparent diffusion coefficient and decreased surfaces to volume ratio in varying the diffusion time experiment; the boundary relaxation also substantially affects the apparent diffusion coefficient of sphere structure in the variation of gradient experiment.
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Affiliation(s)
- Xinli Liao
- Chemistry Department, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, 361005, China
| | - Shaokuan Zheng
- Department of Radiology, UMASS Medical School, Worcester, Massachusetts 01655, USA
| | - Guoxing Lin
- Carlson School of Chemistry and Biochemistry, Clark University, Worcester, Massachusetts 01610, USA
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Guerreri M, Palombo M, Caporale A, Fasano F, Macaluso E, Bozzali M, Capuani S. Age-related microstructural and physiological changes in normal brain measured by MRI γ-metrics derived from anomalous diffusion signal representation. Neuroimage 2018; 188:654-667. [PMID: 30583064 DOI: 10.1016/j.neuroimage.2018.12.044] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2018] [Revised: 12/11/2018] [Accepted: 12/20/2018] [Indexed: 12/29/2022] Open
Abstract
Nowadays, increasing longevity associated with declining cerebral nervous system functions, suggests the need for continued development of new imaging contrast mechanisms to support the differential diagnosis of age-related decline. In our previous papers, we developed a new imaging contrast metrics derived from anomalous diffusion signal representation and obtained from diffusion-weighted (DW) data collected by varying diffusion gradient strengths. Recently, we highlighted that the new metrics, named γ-metrics, depended on the local inhomogeneity due to differences in magnetic susceptibility between tissues and diffusion compartments in young healthy subjects, thus providing information about myelin orientation and iron content within cerebral regions. The major structural modifications occurring in brain aging are myelinated fibers damage in nerve fibers and iron accumulation in gray matter nuclei. Therefore, we investigated the potential of γ-metrics in relation to other conventional diffusion metrics such as DTI, DKI and NODDI in detecting age-related structural changes in white matter (WM) and subcortical gray matter (scGM). DW-images were acquired in 32 healthy subjects, adults and elderly (age range 20-77 years) using 3.0T and 12 b-values up to 5000 s/mm2. Association between diffusion metrics and subjects' age was assessed using linear regression. A decline in mean γ (Mγ) in the scGM and a complementary increase in radial γ (γ⊥) in frontal WM, genu of corpus callosum and anterior corona radiata with advancing age were found. We suggested that the increase in γ⊥ might reflect declined myelin density, and Mγ decrease might mirror iron accumulation. An increase in D// and a decrease in the orientation dispersion index (ODI) were associated with axonal loss in the pyramidal tracts, while their inverted trends within the thalamus were thought to be linked to reduced architectural complexity of nerve fibers. γ-metrics together with conventional diffusion-metrics can more comprehensively characterize the complex mechanisms underlining age-related changes than conventional diffusion techniques alone.
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Affiliation(s)
- Michele Guerreri
- SAIMLAL Department, Sapienza, Piazzale Aldo Moro, 5, 00185, Roma, RM, Italy; Institute for Complex Systems, CNR, Rome, Italy.
| | - Marco Palombo
- Institute for Complex Systems, CNR, Rome, Italy; Department of Computer Science & Centre for Medical Image Computing, University College London, London, United Kingdom
| | - Alessandra Caporale
- Institute for Complex Systems, CNR, Rome, Italy; Laboratory for Structural, Physiologic and Functional Imaging, Perelman School of Medicine University of Pennsylvania, Philadelphia, PA, USA
| | | | | | - Marco Bozzali
- Neuroimaging Laboratory, Santa Lucia Foundation, Rome, Italy
| | - Silvia Capuani
- Institute for Complex Systems, CNR, Rome, Italy; Neuroimaging Laboratory, Santa Lucia Foundation, Rome, Italy
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Yu Q, Reutens D, Vegh V. Can anomalous diffusion models in magnetic resonance imaging be used to characterise white matter tissue microstructure? Neuroimage 2018; 175:122-137. [DOI: 10.1016/j.neuroimage.2018.03.052] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/04/2017] [Revised: 03/13/2018] [Accepted: 03/22/2018] [Indexed: 12/16/2022] Open
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Cai W, Chen W, Xu W. The fractal derivative wave equation: Application to clinical amplitude/velocity reconstruction imaging. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 2018; 143:1559. [PMID: 29604705 DOI: 10.1121/1.5027237] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
This paper proposes a dissipative acoustic wave equation in which the fractal derivative is employed to represent dissipation. The proposed model is derived from the viscoelastic constitutive relationship via the fractal derivative. It is noted that the fractal derivative is a local operator and avoids the expensive computational costs of non-local fractional derivative, which is popular in recent decades to describe frequency-dependent dissipation in acoustic wave propagation in soft materials. The proposed model is tested to simulate the clinical amplitude/velocity reconstruction imaging of breast tumors, where the reflecting plate is imaged as an elevated line in correspondence to tumor. Numerical experiments show that the present model is capable of indicating the size, position and quantity of tumors. The comparative study confirms that the fractal derivative acoustic wave equation has an advantage over the fractional derivative model regarding computational costs.
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Affiliation(s)
- Wei Cai
- College of Mechanical and Electrical Engineering, Hohai University, Changzhou, 213022, China
| | - Wen Chen
- State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, 210098, China
| | - Wenxiang Xu
- State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, 210098, China
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Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches. MATHEMATICS 2018. [DOI: 10.3390/math6020017] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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Lin G. Analyzing signal attenuation in PFG anomalous diffusion via a non-Gaussian phase distribution approximation approach by fractional derivatives. J Chem Phys 2016; 145:194202. [DOI: 10.1063/1.4967403] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
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Lin G, Zheng S, Liao X. Signal attenuation of PFG restricted anomalous diffusions in plate, sphere, and cylinder. JOURNAL OF MAGNETIC RESONANCE (SAN DIEGO, CALIF. : 1997) 2016; 272:25-36. [PMID: 27616657 DOI: 10.1016/j.jmr.2016.08.016] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2016] [Revised: 08/29/2016] [Accepted: 08/30/2016] [Indexed: 06/06/2023]
Abstract
Pulsed field gradient (PFG) NMR is a noninvasive tool to study anomalous diffusion, which exists widely in many systems such as in polymer or biological systems, in porous material, in single file structures and in fractal geometries. In a real system, the diffusion could be a restricted or a tortuous anomalous diffusion, rather than a free diffusion as the domains for fast and slow transport could coexist. Though there are signal attenuation expressions for free anomalous diffusion in literature, the signal attenuation formalisms for restricted anomalous diffusion is very limited, except for a restricted time-fractional diffusion within a plate reported recently. To better understand the PFG restricted fractional diffusion, in this paper, the PFG signal attenuation expressions were derived for three typical structures (plate, sphere, and cylinder) based on two models: fractal derivative model and fractional derivative model. These signal attenuation expressions include two parts, the time part Tn(t) and the space part Xn(r). Unlike normal diffusion, the time part Tn(t) in time-fractional diffusion can be either a Mittag-Leffler function from the fractional derivative model or a stretched exponential function from the fractal derivative model. However, provided the restricted normal diffusion and the restricted time-fractional diffusion are in an identical structure, they will have the same space part Xn(r) as both diffusions have the same space derivative parameter β equaling 2, therefore, they should have similar diffractive patterns. The restricted general fractional diffusion within a plate is also investigated, which indicates that at a long time limit, the diffusion type is insignificant to the diffractive pattern that depends only on the structure and the gradient pulses. The expressions describing the time-dependent behaviors of apparent diffusion coefficient Df,app for restricted anomalous diffusion are also proposed in this paper. Both the short and long time-dependent behaviors of Df,app are distinct from that of normal diffusion. The general expressions for PFG restricted curvilinear diffusion of tube model were derived in a conventional way and its result agree with that obtained from the fractional derivative model with α equaling 1/2. Additionally, continuous-time random walk simulation was performed to give good support to the theoretical results. These theoretical results reported here will be valuable for researchers in analyzing PFG anomalous diffusion.
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Affiliation(s)
- Guoxing Lin
- Carlson School of Chemistry and Biochemistry, Clark University, Worcester, MA 01610, United States.
| | - Shaokuan Zheng
- Department of Radiology, UMASS Medical School, Worcester, MA 01655, United States
| | - Xinli Liao
- Chemistry Department, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, China
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Lin G. Instantaneous signal attenuation method for analysis of PFG fractional diffusions. JOURNAL OF MAGNETIC RESONANCE (SAN DIEGO, CALIF. : 1997) 2016; 269:36-49. [PMID: 27209371 DOI: 10.1016/j.jmr.2016.05.011] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/16/2016] [Revised: 04/15/2016] [Accepted: 05/12/2016] [Indexed: 06/05/2023]
Abstract
An instantaneous signal attenuation (ISA) method for analyzing pulsed field gradient (PFG) fractional diffusion (FD) has been developed, which is modified from the propagator approach developed in 2001 by Lin et al. for analyzing PFG normal diffusion. Both, the current ISA method and the propagator method have the same fundamental basis that the total signal attenuation (SA) is the accumulation of all the ISA, and the ISA is the average SA of the whole diffusion system at each moment. However, the manner of calculating ISA is different. Unlike the use of the instantaneous propagator in the propagator method, the current method directly calculates ISA as A(K(t'),t'+dt')/A(K(t'),t'), where A(K(t'),t'+dt') and A(K(t'),t') are the SA. This modification makes the current method applicable to PFG FD as the instantaneous propagator may not be obtainable in FD. The ISA method was applied to study PFG SA including the effect of finite gradient pulse widths (FGPW) for free FD, restricted FD and the FD affected by a non-homogeneous gradient field. The SA expressions were successfully obtained for all three types of free FDs while other current methods still have difficulty in obtaining all of them. The results from this method agree with reported results such as that obtained by the effective phase shift diffusion equation (EPSDE) method. The M-Wright phase distribution approximation was also used to derive an SA expression for time FD as a comparison, which agrees with ISA method. Additionally, the continuous-time random walk (CTRW) simulation was performed to simulate the SA of PFG FD, and the simulation results agree with the analytical results. Particularly, the CTRW simulation results give good support to the analytical results including FGPW effect for free FD and restricted time FD based on a fractional derivative model where there have been no corresponding theoretical reports to date. The theoretical SA expressions including FGPW obtained here such as [Formula: see text] may be applied to analyze PFG FD in polymer or biological systems with improved accuracy where SGP approximation cannot be satisfied. The method can perhaps provide new insight to FD MRI and hence benefit the development of diffusion biomarkers based on fractional derivative.
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Affiliation(s)
- Guoxing Lin
- Carlson School of Chemistry and Biochemistry, Clark University, Worcester, MA 01610, United States.
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Lin G. Analyzing the special PFG signal attenuation behavior of intermolecular MQC via the effective phase shift diffusion equation method. J Chem Phys 2015; 143:164202. [PMID: 26520505 DOI: 10.1063/1.4934663] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022] Open
Abstract
Inter-molecular multiple quantum coherence (iMQC) has important applications in NMR and MRI. However, the current theoretical methods still have some difficulties in analyzing the behavior of iMQC signal attenuation of pulsed field gradient diffusion experiments. In this paper, the iMQC diffusion experiments were analyzed by an effective phase shift diffusion equation (EPSDE) method, which is based on the idea that the accumulating phase shift (APS) can be viewed as the result of a diffusion process in virtual phase space (VPS) with effective diffusion coefficient K(2)(t) D (rad(2)/s) where K(t)=∫0 (t)γg(t')dt' is a wavenumber and D is the physical diffusion coefficient of the spin carrier in the real space. The term K(t(tot)) z1 needs to be added to the APS when K(t(tot)) is not zero. Most of the time, K(t(tot)) equals zero. However, in iMQC experiments, the condition K(t(tot)) equaling zero or being non-zero for each spin depends on the gradient pulse setting. The signal attenuations of these two types of iMQC, zero or non-zero K(t(tot)), were analyzed in detail for free and restricted diffusions, which shows that there are significant differences between these two types of iMQC. Particularly, if an apparent diffusion coefficient D(app) is used to analyze the signal attenuation, it equals nD for zero K(t(tot)) which agrees with current theoretical and experimental reports, while for non-zero K(t(tot)), it equals (2n - 1) D which agrees with experimental results from the literature; there are no similar theoretical results reported for comparison. The result that D(app) equals (2n - 1) D is important because the higher value of D(app) means that non-zero K(t(tot)) iMQC can potentially provide more contrast and measure slower diffusion rates than zero K(t(tot)) iMQC. The EPSDE method provides a new way to analyze iMQC diffusion experiments.
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Affiliation(s)
- Guoxing Lin
- Gustav H. Carlson School of Chemistry, Clark University, Worcester, Massachusetts 01610, USA
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