Transverse signal decay under the weak field approximation: Theory and validation

Signatures of microstructure in gradient-echo and spin-echo signals

PURPOSE

To determine whether the spatial scale and magnetic susceptibility of microstructure can be evaluated robustly from the decay of gradient-echo and spin-echo signals.

THEORY AND METHODS

Gradient-echo and spin-echo images were acquired from suspensions of spherical polystyrene microbeads of 10, 20, and 40 μm nominal diameter. The sizes of the beads and their magnetic susceptibility relative to the medium were estimated from the signal decay curves, using a lookup table generated from Monte Carlo simulations and an analytic model based on the Gaussian phase approximation.

RESULTS

Fitting Monte Carlo predictions to spin-echo data yielded acceptable estimates of microstructural parameters for the 20 and 40 μm microbeads. Using gradient-echo data, the Monte Carlo lookup table provided satisfactory parameter estimates for the 20 μm beads but unstable results for the diameter of the largest beads. Neither spin-echo nor gradient-echo data allowed accurate parameter estimation for the smallest beads. The analytic model performed poorly over all bead sizes.

CONCLUSIONS

Microstructural sources of magnetic susceptibility produce distinctive non-exponential signatures in the decay of gradient-echo and spin-echo signals. However, inverting the problem to extract microstructural parameters from the signals is nontrivial and, in certain regimes, ill-conditioned. For microstructure with small characteristic length scales, parameter estimation is hampered by the difficulty of acquiring accurate data at very short echo times. For microstructure with large characteristic lengths, the gradient-echo signal approaches the static-dephasing regime, where it becomes insensitive to size. Applicability of the analytic model was further limited by failure of the Gaussian phase approximation for all but the smallest beads.

Oxygen saturation-dependent effects on blood transverse relaxation at low fields

Objective

Blood oxygenation can be measured using magnetic resonance using the paramagnetic effect of deoxy-haemoglobin, which decreases the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{T}_{2}$$\end{document}T2 relaxation time of blood. This \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{T}_{2}$$\end{document}T2 contrast has been well characterised at the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{B}_{{0}}$$\end{document}B0 fields used in MRI (1.5 T and above). However, few studies have characterised this effect at lower magnetic fields. Here, the feasibility of blood oximetry at low field based on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{T}_{2}$$\end{document}T2 changes that are within a physiological relevant range is explored. This study could be used for specifying requirements for construction of a monitoring device based on low field permanent magnet systems.

Methods

A continuous flow circuit was used to control parameters such as oxygen saturation and temperature in a sample of blood. It flowed through a variable field magnet, where CPMG experiments were performed to measure its \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{T}_{2}$$\end{document}T2. In addition, the oxygen saturation was monitored by an optical sensor for comparison with the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{T}_{2}$$\end{document}T2 changes.

Results

These results show that at low \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{B}_{{0}}$$\end{document}B0 fields, the change in blood \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{T}_{2}$$\end{document}T2 due to oxygenation is small, but still detectable. The data measured at low fields are also in agreement with theoretical models for the oxy-deoxy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{T}_{2}$$\end{document}T2 effect.

Conclusion

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{T}_{2}$$\end{document}T2 changes in blood due to oxygenation were observed at fields as low as 0.1 T. These results suggest that low field NMR relaxometry devices around 0.3 T could be designed to detect changes in blood oxygenation.

Oxygen extraction fraction mapping with multi-parametric quantitative BOLD MRI: Reduced transverse relaxation bias using 3D-GraSE imaging

Magnetic resonance imaging (MRI)-based quantification of the blood-oxygenation-level-dependent (BOLD) effect allows oxygen extraction fraction (OEF) mapping. The multi-parametric quantitative BOLD (mq-BOLD) technique facilitates relative OEF (rOEF) measurements with whole brain coverage in clinically applicable scan times. Mq-BOLD requires three separate scans of cerebral blood volume and transverse relaxation rates measured by gradient-echo (1/T₂*) and spin-echo (1/T₂). Although the current method is of clinical merit in patients with stroke, glioma and internal carotid artery stenosis (ICAS), there are relaxation measurement artefacts that impede the sensitivity of mq-BOLD and artificially elevate reported rOEF values.

We posited that T₂-related biases caused by slice refocusing imperfections during rapid 2D-GraSE (Gradient and Spin Echo) imaging can be reduced by applying 3D-GraSE imaging sequences, because the latter requires no slice selective pulses. The removal of T₂-related biases would decrease overestimated rOEF values measured by mq-BOLD. We characterized effects of T₂-related bias in mq-BOLD by comparing the initially employed 2D-GraSE and two proposed 3D-GraSE sequences to multiple single spin-echo reference measurements, both in vitro and in vivo. A phantom and 25 participants, including young and elderly healthy controls as well as ICAS-patients, were scanned. We additionally proposed a procedure to reliably identify and exclude artefact affected voxels. In the phantom, 3D-GraSE derived T₂ values had 57% lower deviation from the reference. For in vivo scans, the formerly overestimated rOEF was reduced by −27% (p < 0.001). We obtained rOEF = 0.51, which is much closer to literature values from positron emission tomography (PET) measurements. Furthermore, increased sensitivity to a focal rOEF elevation in an ICAS-patient was demonstrated.

In summary, the application of 3D-GraSE improves the mq-BOLD-based rOEF quantification while maintaining clinically feasible scan times. Thus, mq-BOLD with non-slice selective T₂ imaging is highly promising to improve clinical diagnostics of cerebrovascular diseases such as ICAS.

Model-based Bayesian inference of brain oxygenation using quantitative BOLD

Streamlined Quantitative BOLD (sqBOLD) is an MR technique that can non-invasively measure physiological parameters including Oxygen Extraction Fraction (OEF) and deoxygenated blood volume (DBV) in the brain. Current sqBOLD methodology rely on fitting a linear model to log-transformed data acquired using an Asymmetric Spin Echo (ASE) pulse sequence. In this paper, a non-linear model implemented in a Bayesian framework was used to fit physiological parameters to ASE data. This model makes use of the full range of available ASE data, and incorporates the signal contribution from venous blood, which was ignored in previous analyses. Simulated data are used to demonstrate the intrinsic difficulty in estimating OEF and DBV simultaneously, and the benefits of the proposed non-linear model are shown. In vivo data are used to show that this model improves parameter estimation when compared with literature values. The model and analysis framework can be extended in a number of ways, and can incorporate prior information from external sources, so it has the potential to further improve OEF estimation using sqBOLD.

Simulations of the effect of diffusion on asymmetric spin echo based quantitative BOLD: An investigation of the origin of deoxygenated blood volume overestimation

Quantitative BOLD (qBOLD) is a technique for mapping oxygen extraction fraction (OEF) and deoxygenated blood volume (DBV) in the human brain. Recent measurements using an asymmetric spin echo (ASE) based qBOLD approach produced estimates of DBV which were systematically higher than measurements from other techniques. In this study, we investigate two hypotheses for the origin of this DBV overestimation using simulations and consider the implications for experimental measurements. Investigations were performed by combining Monte Carlo simulations of extravascular signal with an analytical model of the intravascular signal. HYPOTHESIS 1: DBV overestimation is due to the presence of intravascular signal which is not accounted for in the analysis model. Intravascular signal was found to have a weak effect on qBOLD parameter estimates. HYPOTHESIS 2: DBV overestimation is due to the effects of diffusion which are not accounted for in the analysis model. The effect of diffusion on the extravascular signal was found to result in a vessel radius dependent variation in qBOLD parameter estimates. In particular, DBV overestimation peaks for vessels with radii from 20 to 30 μm and is OEF dependent. This results in the systematic underestimation of OEF. IMPLICATIONS: The impact on experimental qBOLD measurements was investigated by simulating a more physiologically realistic distribution of vessel sizes with a small number of discrete radii. Overestimation of DBV consistent with previous experiments was observed, which was also found to be OEF dependent. This results in the progressive underestimation of the measured OEF. Furthermore, the relationship between the measured OEF and the true OEF was found to be dependent on echo time and spin echo displacement time. The results of this study demonstrate the limitations of current ASE based qBOLD measurements and provide a foundation for the optimisation of future acquisition approaches.

Spin echoes: full numerical solution and breakdown of approximative solutions

The spin echo signal from vessels in Krogh's capillary model as well in the random distribution vessel model are studied by numerically solving the Bloch-Torrey equation. A comparison is made with the Gaussian local phase approximation, the Gaussian phase approximation and the strong-collision approximation. Differences between the Gaussian local phase approximation and the Gaussian phase approximation are explained. In the intermediate diffusion regime, the full numerical solution shows oscillations which are absent in any of the approximate solutions. In the limit of large diffusion coefficients, where the approximations become exact, the signal shows a linear-exponential decay governed by a single parameter. The features of the exact numerical solution can be explained by an analytically solvable discrete two-level model. There is a one-to-one correspondence between the different diffusion regimes and the three cases of the damped harmonic oscillator.

Transverse NMR relaxation in biological tissues

Transverse NMR relaxation is a fundamental physical phenomenon underpinning a wide range of MRI-based techniques, essential for non-invasive studies in biology, physiology and neuroscience, as well as in diagnostic imaging. Biophysically, transverse relaxation originates from a number of distinct scales - molecular (nanometers), cellular (micrometers), and macroscopic (millimeter-level MRI resolution). Here we review the contributions to the observed relaxation from each of these scales, with the main focus on the cellular level of tissue organization, commensurate with the diffusion length of spin-carrying molecules. We discuss how the interplay between diffusion and spin dephasing in a spatially heterogeneous tissue environment leads to a non-monoexponential time-dependent transverse relaxation signal that contains important biophysical information about tissue microstructure.