Parsimonious discretization for characterizing multi‐exponential decay in magnetic resonance

Abstract

We address the problem of analyzing noise‐corrupted magnetic resonance transverse decay signals as a superposition of underlying independently decaying monoexponentials of positive amplitude. First, we indicate the manner in which this is an ill‐conditioned inverse problem, rendering the analysis unstable with respect to noise. Second, we define an approach to this analysis, stabilized solely by the nonnegativity constraint without regularization. This is made possible by appropriate discretization, which is coarser than that often used in practice. Thirdly, we indicate further stabilization by inspecting the plateaus of cumulative distributions. We demonstrate our approach through analysis of simulated myelin water fraction measurements, and compare the accuracy with more conventional approaches. Finally, we apply our method to brain imaging data obtained from a human subject, showing that our approach leads to maps of the myelin water fraction which are much more stable with respect to increasing noise than those obtained with conventional approaches.

Investigation of the association between cerebral iron content and myelin content in normative aging using quantitative magnetic resonance neuroimaging

“…Furthermore, MRI mapping of myelin water fraction (MWF), a proxy of myelin content, provides important insights for understanding brain maturation and neurodegeneration ( Bouhrara and Spencer, 2016 ; MacKay and Laule, 2016 ; Whittall et al, 1997 ; Does, 2018 ; Alonso-Ortiz et al, 2015 ; Piredda et al, 2021 ; MacKay et al, 1994 ; Vavasour et al, 2006 ). Advanced analysis methods based on multicomponent relaxometry have been introduced to improve both sensitivity and specificity of MR-based myelin quantification ( Bouhrara and Spencer, 2016 ; MacKay and Laule, 2016 ; Whittall et al, 1997 ; Does, 2018 ; Alonso-Ortiz et al, 2015 ; Piredda et al, 2021 ; Jones et al, 2003 ; Prasloski et al, 2012 ; Bonny et al, 2020 ; M Bouhrara et al, 2021 ; Bouhrara et al, 2015 ); these methods have been extensively applied to characterize cerebral demyelinating diseases and neurodevelopment ( MacKay and Laule, 2016 ; Borich et al, 2013 ; Laule et al, 2006 ; Sirrs et al, 2007 ; Kolind et al, 2012 ; Kolind et al, 2015 ; Dean et al, 2017 ; Dean et al, 2014 ; Dean et al, 2016 ; Deoni et al, 2012 ; M Bouhrara et al, 2020; M Bouhrara et al, 2020; Bouhrara et al, 2018 ; M Bouhrara et al, 2020; Qian et al, 2020 ; Dvorak et al, 2021 ; Papadaki et al, 2019 ). Using MWF and QSM, Yao and colleagues have shown an association between focal iron accumulation and myelin loss in patients with chronic MS lesions ( Yao et al, 2018 ).…”

Circadian Variation of Root Water Status in Three Herbaceous Species Assessed by Portable NMR

“…Firstly, the acquired echo decays were noisier than for the two other species. Consequently, only the first 30 CPMG signal decays (2 days/night periods) satisfied the signal-to-noise ratio (SNR) condition for NNLS analysis (see Figure S1) [34]. Secondly, while these inversions also resulted in two components, the slow relaxing one displayed lower T 2 values, compared to the other two species, of around 50 ms. Thirdly, the two population fractions were closer together than those of the two other species.…”

Span of regularization for solution of inverse problems with application to magnetic resonance relaxometry of the brain

“…While the ground truth for brain MWF in vivo is unknown, we can still implement numerical experiments using this data to compare different reconstruction approaches. First, we defined the MWF as the integral of the DF between abscissa values of ms and ms 15 , 20 for each pixel and created the corresponding map. To compare two different MWF maps and , we define the scaled absolute difference (SAD) by: where is the sum of the absolute values of the elements of a matrix .…”

“…One approach would be to write the DF as the linear combination of a finite set of Gaussian functions , with and representing the unknown mean and standard deviation (SD) of a given element of that set. The discretization of the ’s along the abscissa follows from the discretization of along the abscissa, that is, the choice of the set of abscissa values 15 . With a prior assumption of the number M of Gaussian components required for an adequate description, the determination of can be recast as the non-linear least squares problem: Alternatively, by establishing a dictionary of Gaussian functions of specified and 16 , 17 and incorporating the non-negativity of , we instead have the problem: where , i.e.…”