Optimized Quantification of Spin Relaxation Times in the Hybrid State

Purpose: The analysis of optimized spin ensemble trajectories for relaxometry in the hybrid state. Methods: First, we constructed visual representations to elucidate the differential equation that governs spin dynamics in hybrid state. Subsequently, numerical optimizations were performed to find spin ensemble trajectories that minimize the Cram\'er-Rao bound for $T_1$-encoding, $T_2$-encoding, and their weighted sum, respectively, followed by a comparison of the Cram\'er-Rao bounds obtained with our optimized spin-trajectories, as well as Look-Locker and multi-spin-echo methods. Finally, we experimentally tested our optimized spin trajectories with in vivo scans of the human brain. Results: After a nonrecurring inversion segment on the southern hemisphere of the Bloch sphere, all optimized spin trajectories pursue repetitive loops on the northern half of the sphere in which the beginning of the first and the end of the last loop deviate from the others. The numerical results obtained in this work align well with intuitive insights gleaned directly from the governing equation. Our results suggest that hybrid-state sequences outperform traditional methods. Moreover, hybrid-state sequences that balance $T_1$- and $T_2$-encoding still result in near optimal signal-to-noise efficiency. Thus, the second parameter can be encoded at virtually no extra cost. Conclusion: We provide insights regarding the optimal encoding processes of spin relaxation times in order to guide the design of robust and efficient pulse sequences. We find that joint acquisitions of $T_1$ and $T_2$ in the hybrid state are substantially more efficient than sequential encoding techniques.Comment: 10 pages, 5 figure

Cram\'er-Rao Bound Optimized Subspace Reconstruction in Quantitative MRI

We extend the traditional framework for estimating subspace bases that maximize the preserved signal energy to additionally preserve the Cram\'er-Rao bound (CRB) of the biophysical parameters and, ultimately, improve accuracy and precision in the quantitative maps. To this end, we introduce an \textit{approximate compressed CRB} based on orthogonalized versions of the signal's derivatives with respect to the model parameters. This approximation permits singular value decomposition (SVD)-based minimization of both the CRB and signal losses during compression. Compared to the traditional SVD approach, the proposed method better preserves the CRB across all biophysical parameters with negligible cost to the preserved signal energy, leading to reduced bias and variance of the parameter estimates in simulation. In vivo, improved accuracy and precision are observed in two quantitative neuroimaging applications, permitting the use of smaller basis sizes in subspace reconstruction and offering significant computational savings

Hybrid-State Free Precession in Nuclear Magnetic Resonance

The dynamics of large spin-1/2 ensembles in the presence of a varying magnetic field are commonly described by the Bloch equation. Most magnetic field variations result in unintuitive spin dynamics, which are sensitive to small deviations in the driving field. Although simplistic field variations can produce robust dynamics, the captured information content is impoverished. Here, we identify adiabaticity conditions that span a rich experiment design space with tractable dynamics. These adiabaticity conditions trap the spin dynamics in a one-dimensional subspace. Namely, the dynamics is captured by the absolute value of the magnetization, which is in a transient state, while its direction adiabatically follows the steady state. We define the hybrid state as the co-existence of these two states and identify the polar angle as the effective driving force of the spin dynamics. As an example, we optimize this drive for robust and efficient quantification of spin relaxation times and utilize it for magnetic resonance imaging of the human brain

Generalized Bloch model: a theory for pulsed magnetization transfer

Purpose: The paper introduces a classical model to describe the dynamics of large spin-1/2 ensembles associated with nuclei bound in large molecule structures, commonly referred to as the semi-solid spin pool, and their magnetization transfer (MT) to spins of nuclei in Theory and Methods: Like quantum-mechanical descriptions of spin dynamics and like the original Bloch equations, but unlike existing MT models, the proposed model is based on the algebra of angular momentum in the sense that it explicitly models the rotations induced by radio-frequency (RF) pulses. It generalizes the original Bloch model to non-exponential decays, which are, e.g., observed for semi-solid spin pools. The combination of rotations with non-exponential decays is facilitated by describing the latter as Green's functions, comprised in an integro-differential equation. Results: Our model describes the data of an inversion-recovery magnetization-transfer experiment with varying durations of the inversion pulse substantially better than established models. We made this observation for all measured data, but in particular for pulse durations small than 300$\mu$s. Furthermore, we provide a linear approximation of the generalized Bloch model that reduces the simulation time by approximately a factor 15,000, enabling simulation of the spin dynamics caused by a rectangular RF-pulse in roughly 2$\mu$s. Conclusion: The proposed theory unifies the original Bloch model, Henkelman's steady-state theory for magnetization transfer, and the commonly assumed rotation induced by hard pulses (i.e., strong and infinitesimally short applications of RF fields) and describes experimental data better than previous models

Cram\'er-Rao bound-informed training of neural networks for quantitative MRI

Neural networks are increasingly used to estimate parameters in quantitative MRI, in particular in magnetic resonance fingerprinting. Their advantages over the gold standard non-linear least square fitting are their superior speed and their immunity to the non-convexity of many fitting problems. We find, however, that in heterogeneous parameter spaces, i.e. in spaces in which the variance of the estimated parameters varies considerably, good performance is hard to achieve and requires arduous tweaking of the loss function, hyper parameters, and the distribution of the training data in parameter space. Here, we address these issues with a theoretically well-founded loss function: the Cram\'er-Rao bound (CRB) provides a theoretical lower bound for the variance of an unbiased estimator and we propose to normalize the squared error with respective CRB. With this normalization, we balance the contributions of hard-to-estimate and not-so-hard-to-estimate parameters and areas in parameter space, and avoid a dominance of the former in the overall training loss. Further, the CRB-based loss function equals one for a maximally-efficient unbiased estimator, which we consider the ideal estimator. Hence, the proposed CRB-based loss function provides an absolute evaluation metric. We compare a network trained with the CRB-based loss with a network trained with the commonly used means squared error loss and demonstrate the advantages of the former in numerical, phantom, and in vivo experiments.Comment: Xiaoxia Zhang, Quentin Duchemin, and Kangning Liu contributed equally to this wor

On multi-path longitudinal spin relaxation in brain tissue

The purpose of this paper is to confirm previous reports that identified magnetization transfer (MT) as an inherent driver of longitudinal relaxation in brain tissue by asserting a substantial difference between the $T_1$ relaxation times of the free and the semi-solid spin pools. Further, we aim to identify an avenue towards the quantification of these relaxation processes on a voxel-by-voxel basis in a clinical imaging setting, i.e. with a nominal resolution of 1mm isotropic and full brain coverage in 12min. To this end, we optimized a hybrid-state pulse sequence for mapping the parameters of an unconstrained MT model. We scanned 4 people with relapsing-remitting multiple sclerosis (MS) and 4 healthy controls with this pulse sequence and estimated $T_1^f \approx 1.90$s and $T_1^s \approx 0.327$s for the free and semi-solid spin pool of healthy WM, respectively, confirming previous reports and questioning the commonly used assumptions $T_1^s = T_1^f$ or $T_1^s = 1$s. Further, we estimated a fractional size of the semi-solid spin pool of $m_0^s \approx 0.202$, which is larger than previously assumed. An analysis of $T_1^f$ in normal appearing white matter revealed statistically significant differences between individuals with MS and controls. In conclusion, we confirm that longitudinal spin relaxation in brain tissue is dominated by MT and that the hybrid state facilitates a voxel-wise fit of the unconstrained MT model, which enables the analysis of subtle neurodegeneration

Rapid quantitative magnetization transfer imaging: utilizing the hybrid state and the generalized Bloch model

Purpose: To improve spatial resolution and scan time of quantitative magnetization transfer (qMT) imaging without constraints on model parameters. Theory and Methods: We combine two recently-proposed models in a Bloch-McConnell equation: the dynamics of the free spin pool is confined to the hybrid state and the dynamics of the semi-solid spin pool is described by the generalized Bloch model. We numerically optimize the flip angles and durations of a train of radio frequency pulses to enhance the encoding of three marked qMT parameters while accounting for an 8-parameter model. We sparsely sample each time frame along this spin dynamics with a 3D radial koosh-ball trajectory, reconstruct the data with sub-space modeling, and fit the qMT model with a neural network for computational efficiency. Results: We were able to extract qMT parameter maps of the whole brain with a nominal resolution of 1mm isotropic and high SNR from a 12.6 minute scan. In lesions of multiple sclerosis subjects, we observe a decreased size of the semi-solid spin pool and slower relaxation, consistent with previous reports. Conclusion: The encoding power of the hybrid state, combined with regularized image reconstruction, and the accuracy of the generalized Bloch model provide an excellent basis for highly efficient quantitative magnetization transfer imaging

JakobAsslaender/MRIgeneralizedBloch.jl: v0.8.5

<h2>MRIgeneralizedBloch v0.8.5</h2> <p><a href="https://github.com/JakobAsslaender/MRIgeneralizedBloch.jl/compare/v0.8.4...v0.8.5">Diff since v0.8.4</a></p> <p><strong>Merged pull requests:</strong></p> <ul> <li>New version: v0.8.4 (#31) (@JakobAsslaender)</li> <li>CompatHelper: bump compat for LsqFit to 0.14, (keep existing compat) (#32) (@github-actions[bot])</li> <li>CompatHelper: bump compat for LsqFit to 0.15, (keep existing compat) (#33) (@github-actions[bot])</li> </ul&gt

Minimization of Eddy Current Artifacts in Sequences with Periodic Dynamics

Purpose: To minimize eddy-current artifacts in periodic pulse sequences with balanced gradient moments as, e.g., used for quantitative MRI. Theory and Methods: Eddy current artifacts in balanced sequences result from large jumps in k-space. In quantitative MRI, one often samples some spin dynamics repeatedly while acquiring different parts of k-space. We swap individual k-space lines between different repetitions in order to minimize jumps within each repetition. This reordering can be formulated as a travelling salesman problem and we tackle the discrete optimization with a simulated annealing algorithm. Results: The proposed approach is highly effective at minimizing the change of the readout direction of a 3D koosh-ball trajectory. This decreases the eddy-current induced perturbations of the magnetization and reduces artifacts in the derived parameter maps, as demonstrated in the example of a hybrid-state free precession sequence. Conclusion: Beyond this example application, we believe that the approach can be used for most k-space trajectories, other quantitative MRI methods, as well as for cine cardiac imaging.Comment: 7 pages, 5 figure