The gamma distribution model for pulsed-field gradient NMR studies of molecular-weight distributions of polymers

Röding, Magnus; Bernin, Diana; Jonasson, Jenny; Särkkä, Aila; Topgaard, Daniel; Rudemo, Mats; Nydén, Magnus

doi:10.1016/j.jmr.2012.07.005

Cited by 75 publications

(98 citation statements)

References 24 publications

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“…proves to be an efficient and physically plausible model for describing complex polydisperse systems such as polymer solutions [57]. The mean and the dispersion value of the gamma distribution are given by the so-called shape parameter α and the scale parameter β, where D = α · β and μ 2 = α · β 2 , respectively.…”

Section: Estimating Microscopic Fractional Anisotropymentioning

confidence: 99%

Microanisotropy imaging: quantification of microscopic diffusion anisotropy and orientational order parameter by diffusion MRI with magic-angle spinning of the q-vector

et al. 2014

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Diffusion tensor imaging (DTI) is the method of choice for non-invasive investigations of the structure of human brain white matter (WM). The results are conventionally reported as maps of the fractional anisotropy (FA), which is a parameter related to microstructural features such as axon density, diameter, and myelination. The interpretation of FA in terms of microstructure becomes ambiguous when there is a distribution of axon orientations within the image voxel. In this paper, we propose a procedure for resolving this ambiguity by determining a new parameter, the microscopic fractional anisotropy (μFA), which corresponds to the FA without the confounding influence of orientation dispersion. In addition, we suggest a method for measuring the orientational order parameter (OP) for the anisotropic objects. The experimental protocol is capitalizing on a recently developed diffusion nuclear magnetic resonance (NMR) pulse sequence based on magic-angle spinning of the q-vector. Proof-of-principle experiments are carried out on microimaging and clinical MRI equipment using lyotropic liquid crystals and plant tissues as model materials with high μFA and low FA on account of orientation dispersion. We expect the presented method to be especially fruitful in combination with DTI and high angular resolution acquisition protocols for neuroimaging studies of gray and white matter.

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Section: Estimating Microscopic Fractional Anisotropymentioning

confidence: 99%

Microanisotropy imaging: quantification of microscopic diffusion anisotropy and orientational order parameter by diffusion MRI with magic-angle spinning of the q-vector

et al. 2014

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“…Assuming a voxel contains a spectrum of diffusion environments, each exhibiting Gaussian diffusion, the measured signal is given by

\frac{S (b)}{S (0)} = \int^{​} P (D) e^{- b D} d D,

where

P (D)

describes the probability density function (PDF) of diffusivity

D

. Various distributions have been proposed for

P (D)

, including the truncated Gaussian 13, gamma 11, 25, 26, log‐normal 12, 27, and beta 26 distributions. For a given

P (D)

, the mean diffusivity is given by the expectation of

D

E [D]

.…”

Section: Theorymentioning

confidence: 99%

“…The gamma distribution 11, 25, 26 allows more variation in the shape, and in particular the skewness, than the truncated Gaussian model. The PDF is defined as

P (D) = {\begin{matrix} \frac{1}{Γ (k) θ^{k}} D^{k - 1} \exp (- \frac{D}{θ}), & D \geq 0 \\ 0 & D < 0 \end{matrix},

where

Γ

denotes the gamma function (see Eq.…”

Section: Theorymentioning

confidence: 99%

Evaluation of non‐Gaussian diffusion in cardiac MRI

McClymont

Teh

Carruth

et al. 2016

Magnetic Resonance in Med

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PurposeThe diffusion tensor model assumes Gaussian diffusion and is widely applied in cardiac diffusion MRI. However, diffusion in biological tissue deviates from a Gaussian profile as a result of hindrance and restriction from cell and tissue microstructure, and may be quantified better by non‐Gaussian modeling. The aim of this study was to investigate non‐Gaussian diffusion in healthy and hypertrophic hearts.MethodsThirteen rat hearts (five healthy, four sham, four hypertrophic) were imaged ex vivo. Diffusion‐weighted images were acquired at b‐values up to 10,000 s/mm2. Models of diffusion were fit to the data and ranked based on the Akaike information criterion.ResultsThe diffusion tensor was ranked best at b‐values up to 2000 s/mm2 but reflected the signal poorly in the high b‐value regime, in which the best model was a non‐Gaussian “beta distribution” model. Although there was considerable overlap in apparent diffusivities between the healthy, sham, and hypertrophic hearts, diffusion kurtosis and skewness in the hypertrophic hearts were more than 20% higher in the sheetlet and sheetlet‐normal directions.ConclusionNon‐Gaussian diffusion models have a higher sensitivity for the detection of hypertrophy compared with the Gaussian model. In particular, diffusion kurtosis may serve as a useful biomarker for characterization of disease and remodeling in the heart. Magn Reson Med 78:1174–1186, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

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“…However, the scaling parameters of Eq. (1) specific to that polymersolvent system must be found by measuring ⟨D⟩ on fractionated samples of the polymer with known M. Therefore, currently all PGSE NMR-based methods which convert from D to M cannot independently measure the absolute molecular mass distribution [12,13,14,15,16,17,18,19,20,21,22,23,24,25].In this paper we show that ν in Eq.(1) can be directly estimated from a single PGSE experiment in which the extremity (end-group) polymer signal can be spectrally resolved by a chemical shift from the polymer main-chain signal. The scaling exponent, ν, is a measure of the polymer conformation as well as solvent quality [3,26], with bounds of ν = 1/3 for a perfectly coiled, impenetrable, polymer ball and ν = 1 for a perfectly straight polymer rod [17].…”

mentioning

confidence: 99%

“…In this way, M n and M w of each mixture are known. In the following, we introduce PGSE NMR and reproduce equations [24] for the application of the lognormal [29,30,14] and gamma [21,31] distribution models. We then explain sample preparation and outline the procedure for obtaining ν, M w /M n , and the molecular mass distribution.…”

mentioning

confidence: 99%

Scaling exponent and dispersity of polymers in solution by diffusion NMR

Williamson

Röding

Miklavcic

et al. 2017

Journal of Colloid and Interface Science

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Molecular mass distribution measurements by pulsed gradient spin echo nuclear magnetic resonance (PGSE NMR) spectroscopy currently require prior knowledge of scaling parameters to convert from polymer self-diffusion coefficient to molecular mass. Reversing the problem, we utilize the scaling relation as prior knowledge to uncover the scaling exponent from within the PGSE data. Thus, the scaling exponent-a measure of polymer conformation and solvent quality-and the dispersity (M w /M n ) are obtainable from one simple PGSE experiment. The method utilizes constraints and parametric distribution models in a two-step fitting routine involving first the mass-weighted signal and second the number-weighted signal. The method is developed using lognormal and gamma distribution models and tested on experimental PGSE attenuation of the terminal methylene signal and on the sum of all methylene signals of polyethylene glycol in D 2 O. Scaling exponent and dispersity estimates agree with known values in the majority of instances, leading to the potential application of the method to polymers for which characterization is not possible with alternative techniques. Keywords:Pulsed gradient spin echo, pulsed field gradient, Nuclear Magnetic Resonance spectroscopy, Molecular weight distribution, Polymers, DOSY, Polydispersity Index, Self-diffusion, Molar mass, Flory exponent, Lognormal distribution, Gamma distribution, End-group analysis, Scaling law Synthetic polymers have distributions of molecular masses determined by their synthesis [1]. Measuring the molecular mass distribution rather than its average is important because the dispersity can influence polymer properties [2]. Absolute as opposed to relative measurements are needed when using polymer physics to fully realize the potential applications of a polymer [3]. Only a handful of techniques can measure the absolute molecular mass distribution [3]. The gold standard is size exclusion chromatography (SEC) using universal calibration [4], which does not always work [5,6]. New techniques must be developed to aid in the advancement of polymer science.Pulsed gradient spin echo nuclear magnetic resonance (PGSE NMR) [7,8] is a powerful technique for obtaining the distribution of polymer self-diffusion coefficients D [9], from which the distribution of molecular masses M can be obtained by the scaling law [10] ration give PGSE NMR a competitive edge with respect to SEC. Chemical shift information [7], e.g. in a diffusion ordered spectroscopy (DOSY) plot [11], provides the ability to observe chemical heterogeneity and impurity. Sample preparation generally does not require filtration because contaminates from large particles such as dust do not impact the experiment. However, the scaling parameters of Eq. (1) specific to that polymersolvent system must be found by measuring ⟨D⟩ on fractionated samples of the polymer with known M. Therefore, currently all PGSE NMR-based methods which convert from D to M cannot independently measure the absolute molecular mass distribution [...

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The gamma distribution model for pulsed-field gradient NMR studies of molecular-weight distributions of polymers

Cited by 75 publications

References 24 publications

Microanisotropy imaging: quantification of microscopic diffusion anisotropy and orientational order parameter by diffusion MRI with magic-angle spinning of the q-vector

Microanisotropy imaging: quantification of microscopic diffusion anisotropy and orientational order parameter by diffusion MRI with magic-angle spinning of the q-vector

Evaluation of non‐Gaussian diffusion in cardiac MRI

Scaling exponent and dispersity of polymers in solution by diffusion NMR

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