Spin dephasing in the dipole field around capillaries and cells: numerical solution

Influence of flow and susceptibility effects on spin dephasing in lung tissue

PURPOSE

Magnetic resonance imaging (MRI) of the lung can be used for diagnosis and monitoring of interstitial lung disease. Biophysical models of alveolar lung tissue are needed to understand the complex interplay of susceptibility, diffusion, and flow effects, and their influence on magnetic resonance (MR) spin dephasing.

METHODS

In this work, we present a method for modeling the signal decay of lung tissue by utilizing a two-compartment model, which considers the different spin dephasing mechanisms in the alveolar vasculature and interstitial tissue. This allows calculating the magnetization dynamics and the MR lineshape, which can be measured noninvasively using clinical MR scanners.

RESULTS

The accuracy of the method was evaluated using finite element simulations and the experimentally measured lineshapes of a healthy volunteer. In this comparison, the model performs well, indicating that the relevant spin dephasing mechanisms are correctly taken into account.

CONCLUSIONS

The proposed method can be used to estimate the influence of blood flow and alveolar geometry on the MR lineshape of lung tissue.

Dependence of the frequency distribution around a sphere on the voxel orientation

Microscopically small magnetic field inhomogeneities within an external static magnetic field cause a free induction decay in magnetic resonance imaging that generally exhibits two transverse components that are usually summarized to a complex entity. The Fourier transform of the complex-valued free induction decay is the purely real and positive-valued frequency distribution which allows an easy interpretation of the underlying dephasing mechanism. Typically, the frequency distribution inside a cubic voxel as caused by a spherical magnetic field inhomogeneity is determined by a histogram technique in terms of subdivision of the whole voxel into smaller subvoxels. A faster and more accurate computation is achieved by analytical expressions for the frequency distribution that are derived in this work. In contrast to the usually assumed simplified case of a spherical voxel, we also consider the tilt angles of the cubic voxel to the external magnetic field. The typical asymmetric form of the frequency distribution is reproduced and analyzed for the more realistic case of a cubic voxel. We observe a splitting of frequency distribution peaks for increasing tilt of the cubic voxel against the direction of the external magnetic field in analogy to the case for dephasing around cylindrical, vessel-like objects inside cubic voxels. These results are of value, e.g., for the analysis of susceptibility-weighted images or in quantitative susceptibility imaging since the reconstruction of these images is performed in cubic-shaped voxels.

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.

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.

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.

NMR-based diffusion lattice imaging

Nuclear magnetic resonance (NMR) diffusion experiments are widely employed as they yield information about structures hindering the diffusion process, e.g., about cell membranes. While it has been shown in recent articles that these experiments can be used to determine the shape of closed pores averaged over a volume of interest, it is still an open question how much information can be gained in open well-connected systems. In this theoretical work, it is shown that the full structure information of connected periodic systems is accessible. To this end, the so-called "SEquential Rephasing by Pulsed field-gradient Encoding N Time intervals" (SERPENT) sequence is used, which employs several diffusion encoding gradient pulses with different amplitudes. Two two-dimensional solid matrices that are surrounded by an NMR-visible medium are considered: a hexagonal lattice of cylinders and a rectangular lattice of isosceles triangles.

Microstructural Analysis of Peripheral Lung Tissue through CPMG Inter-Echo Time R2 Dispersion

Since changes in lung microstructure are important indicators for (early stage) lung pathology, there is a need for quantifiable information of diagnostically challenging cases in a clinical setting, e.g. to evaluate early emphysematous changes in peripheral lung tissue. Considering alveoli as spherical air-spaces surrounded by a thin film of lung tissue allows deriving an expression for Carr-Purcell-Meiboom-Gill transverse relaxation rates R₂ with a dependence on inter-echo time, local air-tissue volume fraction, diffusion coefficient and alveolar diameter, within a weak field approximation. The model relaxation rate exhibits the same hyperbolic tangent dependency as seen in the Luz-Meiboom model and limiting cases agree with Brooks et al. and Jensen et al. In addition, the model is tested against experimental data for passively deflated rat lungs: the resulting mean alveolar radius of R_A = 31.46 ± 13.15 μm is very close to the literature value (∼34 μm). Also, modeled radii obtained from relaxometer measurements of ageing hydrogel foam (that mimics peripheral lung tissue) are in good agreement with those obtained from μCT images of the same foam (mean relative error: 0.06 ± 0.01). The model’s ability to determine the alveolar radius and/or air volume fraction will be useful in quantifying peripheral lung microstructure.

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.

Numerical solution of a diffusion problem by exponentially fitted finite difference methods

This paper is focused on the accurate and efficient solution of partial differential differential equations modelling a diffusion problem by means of exponentially fitted finite difference numerical methods. After constructing and analysing special purpose finite differences for the approximation of second order partial derivatives, we employed them in the numerical solution of a diffusion equation with mixed boundary conditions. Numerical experiments reveal that a special purpose integration, both in space and in time, is more accurate and efficient than that gained by employing a general purpose solver.

Spin dephasing in a magnetic dipole field

Transverse relaxation by dephasing in an inhomogeneous field is a general mechanism in physics, for example, in semiconductor physics, muon spectroscopy, or nuclear magnetic resonance. In magnetic resonance imaging the transverse relaxation provides information on the properties of several biological tissues. Since the dipole field is the most important part of the multipole expansion of the local inhomogeneous field, dephasing in a dipole field is highly important in relaxation theory. However, there have been no analytical solutions which describe the dephasing in a magnetic dipole field. In this work we give a complete analytical solution for the dephasing in a magnetic dipole field which is valid over the whole dynamic range.