Laplace inversion of low‐resolution NMR relaxometry data using sparse representation methods

Berman, Paula; Levi, Ofer; Parmet, Yisrael; Saunders, Michael A.; Wiesman, Zeev

doi:10.1002/cmr.a.21263

Cited by 105 publications

(99 citation statements)

References 24 publications

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“…[5][6][7][8][9] However, there is at present minimal guidance on the choice of penalty term for regularization in MR relaxometry and related experiments, which is our focus here. [5][6][7][8][9] However, there is at present minimal guidance on the choice of penalty term for regularization in MR relaxometry and related experiments, which is our focus here.…”

mentioning

confidence: 99%

L₁, L_p, L₂, and elastic net penalties for regularization of Gaussian component distributions in magnetic resonance relaxometry

Sabett

Hafftka

Sexton

et al. 2017

Concepts Magnetic Resonance

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Determination of the distribution of magnetic resonance (MR) transverse relaxation times is emerging as an important method for materials characterization, including assessment of tissue pathology in biomedicine. These distributions are obtained from the inverse Laplace transform (ILT) of multiexponential decay data. Stabilization of this classically ill-posed problem is most commonly attempted using Tikhonov regularization with an L 2 penalty term. However, with the availability of convex optimization algorithms and recognition of the importance of sparsity in model reconstruction, there has been increasing interest in alternative penalties. The L 1 penalty enforces a greater degree of sparsity than L 2 , and so may be suitable for highly localized relaxation time distributions. In addition, L p penalties, 1 < p < 2, and the elastic net (EN) penalty, defined as a linear combination of L 1 and L 2 penalties, may be appropriate for distributions consisting of both narrow and broad components. We evaluate the L 1 , L 2 , L p , and EN penalties for model relaxation time distributions consisting of two Gaussian peaks. For distributions with narrow Gaussian peaks, the L 1 penalty works well to maintain sparsity and promote resolution, while the conventional L 2 penalty performs best for distributions with broader peaks. Finally, the L p and EN penalties do in fact outperform the L 1 and L 2 penalties for distributions with components of unequal widths. These findings serve as indicators of appropriate regularization in the typical situation in which the experimentalist has a priori knowledge of the general characteristics of the underlying relaxation time distribution. Our findings can be applied to both the recovery of T 2 distributions from spin echo decay data as well as distributions of other MR parameters, such as apparent diffusion constant, from their multiexponential decay signals. K E Y W O R D Sinverse Laplace transform, inverse problems, NMR relaxometry, non-negative least squares, regularization

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mentioning

confidence: 99%

L₁, L_p, L₂, and elastic net penalties for regularization of Gaussian component distributions in magnetic resonance relaxometry

Sabett

Hafftka

Sexton

et al. 2017

Concepts Magnetic Resonance

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“…Another optimization routine has been proposed which, compared to commercial software using the BRD-routine, claims to provide more reliable distributions [19]. Using either of the methods for inverting the NMR data to distributions through an inverse Laplace transform, it is important to not to forget where the data originates from, knowledge by parsimony!…”

Section: The 1 Dimensional Inverse Laplace Transform (1d-ilt)mentioning

confidence: 99%

Analysis of Dynamic NMR Data

Sørland¹

2014

Dynamic Pulsed-Field-Gradient NMR

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An analysis of one and two-dimensional data sets using the so-called inverse Laplace transforms (ILT) and a discrete approach called Anahess is presented. Actually, the ILT is in mathematical terms an ill posed numerical inversion of Fredholm integrals with smooth kernels, meaning that several and very different solutions may fit the experimental data equally well. This inversion is referred to as an ILT in this book. The ILT and Anahess are fundamentally different methods for analysing the data, and the two methods are evaluated against each other on synthetic data sets. These data sets will represent the NMR data typically found from analysing real systems such as cheese or fluid saturated porous rock core plugs. Singular components and distribution are represented in the synthetic data sets. While the ILT algorithm fits the experimental data to a large set of fixed points in a grid, the Anahess algorithm minimizes the number of points, which are not fixed due to user defined grid as for the ILT algorithm. From the comparison, it will become obvious that both approaches have their benefits and drawbacks. Ultimately, user interpretation is required to decide which particular model fits the system better, either the Anahess algorithm, which fits a limited number of discrete components that are continuously variable, or the ILT algorithm, which assigns a larger set of components from a fixed grid.The analysis of NMR data arising from measurements of dynamic parameters such as diffusion and relaxation in heterogeneous systems is a challenging task. As discussed in Chap. 3 there are several factors besides the dynamic parameters that influence the attenuation of the NMR signal while incrementing the applied magnetic field gradient. The most important ones are the restricting geometries experienced during the prepare and read intervals in the PFG NMR sequence and the possible existence of internal magnetic field gradients. Due to these influences, it is not straightforward to assign a deviation from the mono exponential to a multi exponential behaviour in the measured diffusivity, thus being indistinguishable from experimental artefacts [1][2][3][4][5][6].In the following section a method will be provided to analyse one-dimensional data sets that minimises the afore mentioned artefacts. Afterwards, an analysis of one and two-dimensional data sets using the so-called inverse laplace transforms

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“…NMR relaxation time constant estimation using the exponential curve fitting (ECF) method was used to evaluate the quality of agricultural and food products (Kim and Kim, 2004;Kim and McCarthy, 2010). Inverse Laplace transform (ILT) is also another good alternative for calculating relaxation time constants of heterogeneous materials (Moldovan et al, 2010;Silva et al, 2012;Berman et al, 2013;Xu et al, 2014).…”

Section: Introductionmentioning

confidence: 99%

Detection of Iron Nanoparticles using Nuclear Magnetic Resonance Relaxometry and Inverse Laplace Transform

Kim

2014

Journal of Biosystems Engineering

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Purpose: Rapid detection of bacteria is very important in agricultural and food industries to prevent many foodborne illnesses. The objective of this study was to develop a portable nuclear magnetic resonance (NMR)-based system to detect foodborne pathogens (E. coli). This study was focused on developing a method to detect low concentrations of magnetic nanoparticles using NMR techniques. Methods: NMR relaxometry was performed to examine the NMR properties of iron nanoparticle mixtures with different concentrations by using a 1 T permanent magnet magnetic resonance imaging system. Exponential curve fitting (ECF) and inverse Laplace transform (ILT) methods were used to estimate the NMR relaxation time constants, T1 and T2, of guar gum solutions with different iron nanoparticle concentrations (0, 10 , and 10 -7 M). Results: The ECF and ILT methods did not show much difference in these values. Analysis of the NMR relaxation data showed that the ILT method is comparable to the classical ECF method and is more sensitive to the presence of iron nanoparticles. This study also showed that the spin-spin relaxation time constants acquired by a Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence are more useful for determining the concentration of iron nanoparticle solutions comparwith the spin-lattice relaxation time constants acquired by an inversion recovery pulse sequence. Conclusions: We conclude that NMR relaxometry that utilizes CPMG pulse sequence and ILT analysis is more suitable for detecting foodborne pathogens bound to magnetic nanoparticles in agricultural and food products than using inversion recovery pulse sequence and ECF analysis.

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Laplace inversion of low‐resolution NMR relaxometry data using sparse representation methods

Cited by 105 publications

References 24 publications

L₁, L_p, L₂, and elastic net penalties for regularization of Gaussian component distributions in magnetic resonance relaxometry

L₁, L_p, L₂, and elastic net penalties for regularization of Gaussian component distributions in magnetic resonance relaxometry

Analysis of Dynamic NMR Data

Detection of Iron Nanoparticles using Nuclear Magnetic Resonance Relaxometry and Inverse Laplace Transform

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Laplace inversion of low‐resolution NMR relaxometry data using sparse representation methods

Cited by 105 publications

References 24 publications

L1, Lp, L2, and elastic net penalties for regularization of Gaussian component distributions in magnetic resonance relaxometry

L1, Lp, L2, and elastic net penalties for regularization of Gaussian component distributions in magnetic resonance relaxometry

Analysis of Dynamic NMR Data

Detection of Iron Nanoparticles using Nuclear Magnetic Resonance Relaxometry and Inverse Laplace Transform

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Product

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L₁, L_p, L₂, and elastic net penalties for regularization of Gaussian component distributions in magnetic resonance relaxometry

L₁, L_p, L₂, and elastic net penalties for regularization of Gaussian component distributions in magnetic resonance relaxometry