Spin dephasing in a magnetic dipole field

Brain tumor classification of virtual NMR voxels based on realistic blood vessel-induced spin dephasing using support vector machines

Remodeling of tissue microvasculature commonly promotes neoplastic growth; however, there is no imaging modality in oncology yet that noninvasively quantifies microvascular changes in clinical routine. Although blood capillaries cannot be resolved in typical magnetic resonance imaging (MRI) measurements, their geometry and distribution influence the integral nuclear magnetic resonance (NMR) signal from each macroscopic MRI voxel. We have numerically simulated the expected transverse relaxation in NMR voxels with different dimensions based on the realistic microvasculature in healthy and tumor-bearing mouse brains (U87 and GL261 glioblastoma). The 3D capillary structure in entire, undissected brains was acquired using light sheet fluorescence microscopy to produce large datasets of the highly resolved cerebrovasculature. Using this data, we trained support vector machines to classify virtual NMR voxels with different dimensions based on the simulated spin dephasing accountable to field inhomogeneities caused by the underlying vasculature. In prediction tests with previously blinded virtual voxels from healthy brain tissue and GL261 tumors, stable classification accuracies above 95% were reached. Our results indicate that high classification accuracies can be stably attained with achievable training set sizes and that larger MRI voxels facilitated increasingly successful classifications, even with small training datasets. We were able to prove that, theoretically, the transverse relaxation process can be harnessed to learn endogenous contrasts for single voxel tissue type classifications on tailored MRI acquisitions. If translatable to experimental MRI, this may augment diagnostic imaging in oncology with automated voxel-by-voxel signal interpretation to detect vascular pathologies.

Gibbs point field model quantifies disorder in microvasculature of U87-glioblastoma

Microvascular proliferation in glioblastoma multiforme is a biological key mechanism to facilitate tumor growth and infiltration and a main target for treatment interventions. The vascular architecture can be obtained by Single Plane Illumination Microscopy (SPIM) to evaluate vascular heterogeneity in tumorous tissue. We make use of the Gibbs point field model to quantify the order of regularity in capillary distributions found in the U87 glioblastoma model in a murine model and to compare tumorous and healthy brain tissue. A single model parameter Γ was assigned that is linked to tissue-specific vascular topology through Monte-Carlo simulations. Distributions of the model parameter Γ differ significantly between glioblastoma tissue with mean 〈Γ_G〉=2.1±0.4, as compared to healthy brain tissue with mean 〈Γ_H〉=4.9±0.4, suggesting that the average Γ-value allows for tissue differentiation. These results may be used for diagnostic magnetic resonance imaging, where it has been shown recently that Γ is linked to tissue-inherent relaxation parameters.

Localization regime in diffusion NMR: Theory and experiments

In this work we investigate the emergence of the localization regime for diffusion NMR in various geometries: inside slabs, inside cylinders and outside rods arranged on a square array. At high gradients, the transverse magnetization is strongly attenuated in the bulk, whereas the macroscopic signal is formed by the remaining magnetization localized near boundaries of the sample. As a consequence, the signal is particularly sensitive to the microstructure. The theoretical analysis relies on recent mathematical advances on the study of the Bloch-Torrey equation. Experiments were conducted with hyperpolarized xenon-129 gas in 3D-printed phantoms and show an excellent agreement with numerical simulations and theoretical predictions. Our mathematical arguments and experimental evidence indicate that the localization regime with a stretched-exponential decay of the macroscopic signal is a generic feature of diffusion NMR that can be observed at moderately high gradients in most NMR scanners.

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.

Spin dephasing around randomly distributed vessels

We analyze the gradient echo signal in the presence of blood vessel networks. Both, diffusion and susceptibility effects are analytically emphasized within the Bloch-Torrey equation. Solving this equation, we present the first exact description of the local magnetization around a single vessel. This allows us to deduce the gradient echo signal of parallel vessels randomly distributed in a plane, which is valid for arbitrary mean vessel diameters in the range of physiological relevant blood volume fractions. Thus, the results are potentially relevant for gradient echo measurements of blood vessel networks with arbitrary vessel size.

Pseudo-diffusion effects in lung MRI

Magnetic resonance imaging of lung tissue is strongly influenced by susceptibility effects between spin-bearing water molecules and air-filled alveoli. The measured lineshape, however, also depends on the interplay between susceptibility effects and blood-flow around alveoli that can be approximated as pseudo-diffusion. Both effects are quantitatively described by the Bloch-Torrey-equation, which was so far only solved for dephasing on the alveolar surface. In this work, we extend this model to the whole range of physiological relevant air volume fractions. The results agree very well with in vivo measurements in human lung tissue.

Diffusion effects in myelin sheath free induction decay

Myelin sheath microstructure and composition produce MR signal decay characteristics that can be used to evaluate status and outcome of demyelinating disease. We extend a recently proposed model of neuronal magnetic susceptibility, that accounts for both the structural and inherent anisotropy of the myelin sheath, by including the whole dynamic range of diffusion effects. The respective Bloch-Torrey equation for local spin dephasing is solved with a uniformly convergent perturbation expansion method, and the resulting magnetization decay is validated with a numerical solution based on a finite difference method. We show that a variation of diffusion strengths can lead to substantially different MR signal decay curves. Our results may be used to adjust or control simulations for water diffusion in neuronal structures.

Microscopic interpretation and generalization of the Bloch-Torrey equation for diffusion magnetic resonance

In order to bridge microscopic molecular motion with macroscopic diffusion MR signal in complex structures, we propose a general stochastic model for molecular motion in a magnetic field. The Fokker-Planck equation of this model governs the probability density function describing the diffusion-magnetization propagator. From the propagator we derive a generalized version of the Bloch-Torrey equation and the relation to the random phase approach. This derivation does not require assumptions such as a spatially constant diffusion coefficient, or ad hoc selection of a propagator. In particular, the boundary conditions that implicitly incorporate the microstructure into the diffusion MR signal can now be included explicitly through a spatially varying diffusion coefficient. While our generalization is reduced to the conventional Bloch-Torrey equation for piecewise constant diffusion coefficients, it also predicts scenarios in which an additional term to the equation is required to fully describe the MR signal.

Dephasing and diffusion on the alveolar surface

We propose a surface model of spin dephasing in lung tissue that includes both susceptibility and diffusion effects to provide a closed-form solution of the Bloch-Torrey equation on the alveolar surface. The nonlocal susceptibility effects of the model are validated against numerical simulations of spin dephasing in a realistic lung tissue geometry acquired from synchotron-based μCT data sets of mouse lung tissue, and against simulations in the well-known Wigner-Seitz model geometry. The free induction decay is obtained in dependence on microscopic tissue parameters and agrees very well with in vivo lung measurements at 1.5 Tesla to allow a quantification of the local mean alveolar radius. Our results are therefore potentially relevant for the clinical diagnosis and therapy of pulmonary diseases.

Spin dephasing in a magnetic dipole field around large capillaries: Approximative and exact results

We present an analytical solution of the Bloch-Torrey equation for local spin dephasing in the magnetic dipole field around a capillary and for ensembles of capillaries, and adapt this solution for the study of spin dephasing around large capillaries. In addition, we provide a rigorous mathematical derivation of the slow diffusion approximation for the spin-bearing particles that is used in this regime. We further show that, in analogy to the local magnetization, the transverse magnetization of one MR imaging voxel in the regime of static dephasing (where diffusion effects are not considered) is merely the first term of a series expansion that constitutes the signal in the slow diffusion approximation. Theoretical results are in agreement with experimental data for capillaries in rat muscle at 7T.

W(h)ither human cardiac and body magnetic resonance at ultrahigh fields? technical advances, practical considerations, applications, and clinical opportunities

The objective of this study was to document and review advances and groundbreaking progress in cardiac and body MR at ultrahigh fields (UHF, B0 ≥ 7.0 T) with the goal to attract talent, clinical adopters, collaborations and resources to the biomedical and diagnostic imaging communities. This review surveys traits, advantages and challenges of cardiac and body MR at 7.0 T. The considerations run the gamut from technical advances to clinical opportunities. Key concepts, emerging technologies, practical considerations, frontier applications and future directions of UHF body and cardiac MR are provided. Examples of UHF cardiac and body imaging strategies are demonstrated. Their added value over the kindred counterparts at lower fields is explored along with an outline of research promises. The achievements of cardiac and body UHF-MR are powerful motivators and enablers, since extra speed, signal and imaging capabilities may be invested to overcome the fundamental constraints that continue to hamper traditional cardiac and body MR applications. If practical obstacles, concomitant physics effects and technical impediments can be overcome in equal measure, sophisticated cardiac and body UHF-MR will help to open the door to new MRI and MRS approaches for basic research and clinical science, with the lessons learned at 7.0 T being transferred into broad clinical use including diagnostics and therapy guiding at lower fields. Copyright © 2015 John Wiley & Sons, Ltd.

Contrast-optimized composite image derived from multigradient echo cardiac magnetic resonance imaging improves reproducibility of myocardial contours and T2* measurement

Objectives

Reproducibility of myocardial contour determination in cardiac magnetic resonance imaging is important, especially when determining T2* values per myocardial segment as a prognostic factor of heart failure or thalassemia. A method creating a composite image with contrasts optimized for drawing myocardial contours is introduced and compared with the standard method on a single image.

Materials and methods

A total of 36 short-axis slices from bright-blood multigradient echo (MGE) T2* scans of 21 patients were acquired at eight echo times. Four observers drew free-hand myocardial contours on one manually selected T2* image (method 1) and on one image composed by blending three images acquired at TEs providing optimum contrast-to-noise ratio between the myocardium and its surrounding regions (method 2).

Results

Myocardial contouring by method 2 met higher interobserver reproducibility than method 1 (P < 0.001) with smaller Coefficient of variance (CoV) of T2* values in the presence of myocardial iron accumulation (9.79 vs. 15.91 %) and in both global myocardial and mid-ventricular septum regions (12.29 vs. 16.88 and 5.76 vs. 8.16 %, respectively).

Conclusion

The use of contrast-optimized composite images in MGE data analysis improves reproducibility of myocardial contour determination, leading to increased consistency in the calculated T2* values enhancing the diagnostic impact of this measure of iron overload.

Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions

The cylindrical Bessel differential equation and the spherical Bessel differential equation in the interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R \le r \le \gamma R$$\end{document}R≤r≤γR with Neumann boundary conditions are considered. The eigenfunctions are linear combinations of the Bessel function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{n,\nu }(r)=Y_{\nu }^{\prime }(\lambda _{n,\nu }) J_{\nu }(\lambda _{n,\nu } r/R)-J_{\nu }^{\prime }(\lambda _{n,\nu }) Y_{\nu }(\lambda _{n,\nu } r/R)$$\end{document}Φn,ν(r)=Yν′(λn,ν)Jν(λn,νr/R)-Jν′(λn,ν)Yν(λn,νr/R) or linear combinations of the spherical Bessel functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _{m,\nu }(r)=y_{\nu }^{\prime }(\lambda _{m,\nu }) j_{\nu }(\lambda _{m,\nu } r/R)-j_{\nu }^{\prime }(\lambda _{m,\nu }) y_{\nu }(\lambda _{m,\nu } r/R)$$\end{document}ψm,ν(r)=yν′(λm,ν)jν(λm,νr/R)-jν′(λm,ν)yν(λm,νr/R). The orthogonality relations with analytical expressions for the normalization constant are given. Explicit expressions for the Lommel integrals in terms of Lommel functions are derived. The cross product zeros \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{\nu }^{\prime }(\lambda _{n,\nu }) J_{\nu }^{\prime }(\gamma \lambda _{n,\nu })-J_{\nu }^{\prime }(\lambda _{n,\nu }) Y_{\nu }^{\prime }(\gamma \lambda _{n,\nu }) = 0$$\end{document}Yν′(λn,ν)Jν′(γλn,ν)-Jν′(λn,ν)Yν′(γλn,ν)=0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_{\nu }^{\prime }(\lambda _{m,\nu }) j_{\nu }^{\prime }(\gamma \lambda _{m,\nu })-j_{\nu }^{\prime }(\lambda _{m,\nu }) y_{\nu }^{\prime }(\gamma \lambda _{m,\nu }) = 0$$\end{document}yν′(λm,ν)jν′(γλm,ν)-jν′(λm,ν)yν′(γλm,ν)=0 are considered in the complex plane for real as well as complex values of the index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document}ν and approximations for the exceptional zero \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{1,\nu }$$\end{document}λ1,ν are obtained. A numerical scheme based on the discretization of the two-dimensional and three-dimensional Laplace operator with Neumann boundary conditions is presented. Explicit representations of the radial part of the Laplace operator in form of a tridiagonal matrix allow the simple computation of the cross product zeros.

Free induction decay caused by a dipole field

We analyze the free induction decay of nuclear spins under the influence of restricted diffusion in a magnetic dipole field around cylindrical objects. In contrast to previous publications no restrictions or simplifications concerning the diffusion process are made. By directly solving the Bloch-Torrey equation, analytical expressions for the magnetization are given in terms of an eigenfunction expansion. The field strength-dependent complex nature of the eigenvalue spectrum significantly influences the shape of the free induction decay. As the dipole field is the lowest order of the multipole expansion, the obtained results are important for understanding fundamental mechanisms of spin dephasing in many other applied fields of nuclear magnetic resonance such as biophysics or material science. The analytical methods are applied to interpret the spin dephasing in the free induction decay in cardiac muscle and skeletal muscle. A simple expression for the relevant transverse relaxation time is found in terms of the underlying microscopic parameters of the muscle tissue. The analytical results are in agreement with experimental data. These findings are important for the correct interpretation of magnetic resonance images for clinical diagnosis at all magnetic field strengths and therapy of cardiovascular diseases.