The larmor frequency shift in magnetically heterogeneous media depends on their mesoscopic structure

The Larmor frequency shift of a white matter magnetic microstructure model with multiple sources

Magnetic susceptibility imaging may provide valuable information about chemical composition and microstructural organization of tissue. However, its estimation from the MRI signal phase is particularly difficult as it is sensitive to magnetic tissue properties ranging from the molecular to the macroscopic scale. The MRI Larmor frequency shift measured in white matter (WM) tissue depends on the myelinated axons and other magnetizable sources such as iron-filled ferritin. We have previously derived the Larmor frequency shift arising from a dense medium of cylinders with scalar susceptibility and arbitrary orientation dispersion. Here, we extend our model to include microscopic WM susceptibility anisotropy as well as spherical inclusions with scalar susceptibility to represent subcellular structures, biologically stored iron, and so forth. We validate our analytical results with computer simulations and investigate the feasibility of estimating susceptibility using simple iterative linear least squares without regularization or preconditioning. This is done in a digital brain phantom synthesized from diffusion MRI measurements of an ex vivo mouse brain at ultra-high field.

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.

Incorporating the effect of white matter microstructure in the estimation of magnetic susceptibility in ex vivo mouse brain

PURPOSE

To extend quantitative susceptibility mapping to account for microstructure of white matter (WM) and demonstrate its effect on ex vivo mouse brain at 16.4T.

THEORY AND METHODS

Previous studies have shown that the MRI measured Larmor frequency also depends on local magnetic microstructure at the mesoscopic scale. Here, we include effects from WM microstructure using our previous results for the mesoscopic Larmor frequencyΩ ‾ Meso $$ {\overline{\Omega}}^{\mathrm{Meso}} $$ of cylinders with arbitrary orientations. We scrutinize the validity of our model and QSM in a digital brain phantom includingΩ ‾ Meso $$ {\overline{\Omega}}^{\mathrm{Meso}} $$ from a WM susceptibility tensor and biologically stored iron with scalar susceptibility. We also apply susceptibility tensor imaging to the phantom and investigate how the fitted tensors are biased fromΩ ‾ Meso $$ {\overline{\Omega}}^{\mathrm{Meso}} $$ . Last, we demonstrate how to combine multi-gradient echo and diffusion MRI images of ex vivo mouse brains acquired at 16.4T to estimate an apparent scalar susceptibility without sample rotations.

RESULTS

Our new model improves susceptibility estimation compared to QSM for the brain phantom. Applying susceptibility tensor imaging to the phantom withΩ ‾ Meso $$ {\overline{\Omega}}^{\mathrm{Meso}} $$ from WM axons with scalar susceptibility produces a highly anisotropic susceptibility tensor that mimics results from previous susceptibility tensor imaging studies. For the ex vivo mouse brain we find theΩ ‾ Meso $$ {\overline{\Omega}}^{\mathrm{Meso}} $$ due to WM microstructure to be substantial, changing susceptibility in WM up to 25% root-mean-squared-difference.

CONCLUSION

Ω ‾ Meso $$ {\overline{\Omega}}^{\mathrm{Meso}} $$ impacts susceptibility estimates and biases susceptibility tensor imaging fitting substantially. Hence, it should not be neglected when imaging structurally anisotropic tissue such as brain WM.

Observation of magnetic structural universality and jamming transition with NMR

Nuclear magnetic resonance (NMR) has been instrumental in deciphering the structure of proteins. Here we show that transverse NMR relaxation, through its time-dependent relaxation rate, is distinctly sensitive to the structure of complex materials or biological tissues at the mesoscopic scale, from micrometers to tens of micrometers. Based on the ideas of universality, we show analytically and numerically that the time-dependent transverse relaxation rate approaches its long-time limit in a power-law fashion, with the dynamical exponent reflecting the universality class of mesoscopic magnetic structure. The spectral line shape acquires the corresponding non-analytic power law singularity at zero frequency. We experimentally detect the change in the dynamical exponent as a result of the transition into maximally random jammed state characterized by hyperuniform correlations. The relation between relaxational dynamics and magnetic structure opens the way for noninvasive characterization of porous media, complex materials and biological tissues.

Larmor frequency shift from magnetized cylinders with arbitrary orientation distribution

The magnetic susceptibility of tissue can provide valuable information about its chemical composition and microstructural organization. However, the relation between the magnetic microstructure and the measurable Larmor frequency shift is understood only for a few idealized cases. Here we analyze the microstructure formed by magnetized, NMR-invisible infinite cylinders suspended in an NMR-reporting fluid. Through simulations, we scrutinize various geometries of mesoscopic Lorentz cavities and inclusions, and show that the cavity size should be approximately one order of magnitude larger than the width of the inclusions. We also analytically derive the Larmor frequency shift for a population of cylinders with arbitrary orientation dispersion and show that it is determined by the l = 2 Laplace expansion coefficients p 2 m of the cylinders' orientation distribution function. Our work underscores the need to account for microstructural organization when estimating magnetic tissue properties.

What does FEXI measure?

Filter-exchange imaging (FEXI) has already been utilized in several biomedical studies for evaluating the permeability of cell membranes. The method relies on suppressing the extracellular signal using strong diffusion weighting (the mobility filter causing a reduction in the overall diffusivity) and monitoring the subsequent diffusivity recovery. Using Monte Carlo simulations, we demonstrate that FEXI is sensitive not uniquely to the transcytolemmal exchange but also to the geometry of involved compartments: complex geometry offers locations where spins remain unaffected by the mobility filter; moving to other locations afterwards, such spins contribute to the diffusivity recovery without actually permeating any membrane. This exchange mechanism is a warning for those who aim to use FEXI in complex media such as brain gray matter and opens wide scope for investigation towards crystallizing the genuine membrane permeation and characterizing the compartment geometry.

In-cell NMR: Why and how?

NMR spectroscopy has been applied to cells and tissues analysis since its beginnings, as early as 1950. We have attempted to gather here in a didactic fashion the broad diversity of data and ideas that emerged from NMR investigations on living cells. Covering a large proportion of the periodic table, NMR spectroscopy permits scrutiny of a great variety of atomic nuclei in all living organisms non-invasively. It has thus provided quantitative information on cellular atoms and their chemical environment, dynamics, or interactions. We will show that NMR studies have generated valuable knowledge on a vast array of cellular molecules and events, from water, salts, metabolites, cell walls, proteins, nucleic acids, drugs and drug targets, to pH, redox equilibria and chemical reactions. The characterization of such a multitude of objects at the atomic scale has thus shaped our mental representation of cellular life at multiple levels, together with major techniques like mass-spectrometry or microscopies. NMR studies on cells has accompanied the developments of MRI and metabolomics, and various subfields have flourished, coined with appealing names: fluxomics, foodomics, MRI and MRS (i.e. imaging and localized spectroscopy of living tissues, respectively), whole-cell NMR, on-cell ligand-based NMR, systems NMR, cellular structural biology, in-cell NMR… All these have not grown separately, but rather by reinforcing each other like a braided trunk. Hence, we try here to provide an analytical account of a large ensemble of intricately linked approaches, whose integration has been and will be key to their success. We present extensive overviews, firstly on the various types of information provided by NMR in a cellular environment (the "why", oriented towards a broad readership), and secondly on the employed NMR techniques and setups (the "how", where we discuss the past, current and future methods). Each subsection is constructed as a historical anthology, showing how the intrinsic properties of NMR spectroscopy and its developments structured the accessible knowledge on cellular phenomena. Using this systematic approach, we sought i) to make this review accessible to the broadest audience and ii) to highlight some early techniques that may find renewed interest. Finally, we present a brief discussion on what may be potential and desirable developments in the context of integrative studies in biology.

Influence of diffusion on transverse relaxation rates and phases of an ensemble of magnetic spheres

In quantitative susceptibility mapping, the tissue susceptibility is determined from the magnitude and phase of the gradient echo signal, which is influenced by the interplay of complex susceptibility and diffusion effect. Herein, we analytically analyze the influence of diffusion on magnitude and phase images generated by randomly arranged magnetic spheres as a model of intracerebral iron depositions. We demonstrate that both gradient and spin echo relaxation rate constants have a strong and nonlinear dependence on diffusion strength and give empirical formulas for magnitude and phase. This may be used in the future to improve QSM processing methods. In addition, we show that, in theory, combined acquisitions of gradient and spin echo can be used to determine the dimension of the magnetic spheres and the diffusion strength.

Spin echo formation in muscle tissue

Determination of the spin echo signal evolution and of transverse relaxation rates is of high importance for microstructural modeling of muscle tissue in magnetic resonance imaging. So far, numerically exact solutions for the NMR signal dynamics in muscle tissue models have been reported only for the gradient echo free induction decay, with spin echo problems usually solved by approximate methods. In this work, we modeled the spin echo signal numerically exact by discretizing the radial dimension of the Bloch-Torrey equation and expanding the angular dependency in terms of even Chebyshev polynomials. This allows us to express the time dependence of the local magnetization as a closed-form matrix expression. Using this method, we were able to accurately capture the spin echo local and total magnetization dynamics. The obtained transverse relaxation rates showed a high concordance with random walker and finite-element simulations. We could demonstrate that in cases of smaller diffusion coefficients, the commonly used strong collision approximation significantly underestimates the true value considerably. Instead, the limiting behavior in this regime is correctly described either by the full solution or by the slow diffusion approximation. Experimentally measured transverse relaxation rates of a mouse limb muscle showed an angular dependence in accordance with the theoretical prediction.

Larmor frequency dependence on structural anisotropy of magnetically heterogeneous media

The effect of anisotropic magnetic microstructure on the measurable Larmor frequency offset is investigated in media with heterogeneous magnetic susceptibility using Monte Carlo simulations. The focus is on the transition between the regimes of fast and slow diffusion of NMR-reporting molecules. Simulations demonstrate a perfect agreement with the previously developed analytic theory for fast diffusion. Beyond this regime, the frequency offset shows a pronounced dependence on the medium microarchitecture and the diffusivity of NMR-reporting spins in relation to the magnitude of the susceptibility-induced magnetic field.

Larmor frequency in heterogeneous media

High-precision signal phase measurements ignited an ongoing discussion of the microstructural correlates of the Larmor frequency shift of water in biological tissues. In a broader context, this is the question about the averaged precession frequency in magnetically heterogenous, in particular, porous media. In this study, the Larmor frequency shift is found analytically for water filling connected pore space between NMR-invisible magnetized inclusions with a constant magnetic susceptibility tensor. The magnetic microstructure that encompasses the inclusions' shape and their spatial arrangement is arbitrary as well as the inclusions' magnetic susceptibility tensor. The result is limited to the case of effectively fast diffusion of water molecules. In this limit, the effect of magnetic microstructure on the Larmor frequency shift is represented by only five relevant parameters in the general case and by a single parameter in the case of axially symmetric microstructure. This single parameter enters the dependence of the Larmor frequency on the sample orientation relative to the main magnetic field in the previously performed experiments. The result can help interpreting known experimental data and developing realistic models of biological tissues.

Physical and numerical phantoms for the validation of brain microstructural MRI: A cookbook

Phantoms, both numerical (software) and physical (hardware), can serve as a gold standard for the validation of MRI methods probing the brain microstructure. This review aims to provide guidelines on how to build, implement, or choose the right phantom for a particular application, along with an overview of the current state-of-the-art of phantoms dedicated to study brain microstructure with MRI. For physical phantoms, we discuss the essential requirements and relevant characteristics of both the (NMR visible) liquid and (NMR invisible) phantom materials that induce relevant microstructural features detectable via MRI, based on diffusion, intra-voxel incoherent motion, magnetization transfer or magnetic susceptibility weighted contrast. In particular, for diffusion MRI, many useful phantoms have been proposed, ranging from simple liquids to advanced biomimetic phantoms consisting of hollow or plain microfibers and capillaries. For numerical phantoms, the focus is on Monte Carlo simulations of random walk, for which the basic principles, along with useful criteria to check and potential pitfalls are reviewed, in addition to a literature overview highlighting recent advances. While many phantoms exist already, the current review aims to stimulate further research in the field and to address remaining needs.

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.