Structure-specific magnetic field inhomogeneities and its effect on the correlation time

Exploring electronic, structural and dynamics parameters of phenylbenzothiazole complexes with Mn2+, Cu2+ and Zn2+ for designing new magnetic resonance imaging (MRI) probes: congruence between computation and spectroscopic data

Cancer is one of the leading causes of human death worldwide, being one of the most serious problems faced by mankind. For the diagnosis, Magnetic Resonance Imaging (MRI), through effective contrast agents (Cas), has greatly helped in the diagnosis at the initial stages. However, it is necessary to include new compounds more effective and selective for cancer diagnosis. The complexes with Mn²⁺, Cu²⁺ and Zn²⁺ have received great attention due to their applications as CAs for MRI. Those materials can shorten the T₂ and T₂* transverse relaxation times. Thus, the representative structures for hyperfine coupling constants (HFCCs) were selected from docking results by frequency of occupancy calculations. From the Multivariate Analysis to obtain the PCA graphs in the choice of a representative conformations. it is possible to notice that the variable energy does not present a high correlation with the other variables, and structural factors, such as the spatial positions of the metal atoms, seem to be important in the reactivity of the complexes. Structural factors, such as the spatial positions of the metal atoms, seem to be important in the reactivity of the complexes. Theoretical findings suggest that the compounds are capable of increasing the A_iso values of the water molecules, but the complex [Zn(H₂O)(NNO)] shows a greater influence, being more sensitive to the Electron paramagnetic resonance parameters than the complexes [CuCl(H₂O)NNO] and [MnCl₂(H₂O)(NNO)] with the explicit solvent and the enzyme. MRI contrast agents have generated various problems due to their high toxicity. In this perspective, this compound may be a promising alternative for transporting the CAs into diseased tissue.Communicated by Ramaswamy H. Sarma.

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 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.

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.

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.

Intracellular and extracellular T1 and T2 relaxivities of magneto-optical nanoparticles at experimental high fields

This study reports the T(1) and T(2) relaxation rates of rhodamine-labeled anionic magnetic nanoparticles determined at 7, 11.7, and 17.6 T both in solution and after cellular internalization. Therefore cells were incubated with rhodamine-labeled anionic magnetic nanoparticles and were prepared at decreasing concentrations. Additionally, rhodamine-labeled anionic magnetic nanoparticles in solution were used for extracellular measurements. T(1) and T(2) were determined at 7, 11.7, and 17.6 T. T(1) times were determined with an inversion-recovery snapshot-flash sequence. T(2) times were obtained from a multispin-echo sequence. Inductively coupled plasma-mass spectrometry was used to determine the iron content in all samples, and r(1) and r(2) were subsequently calculated. The results were then compared with cells labeled with AMI-25 and VSOP C-200. In solution, the r(1) and r(2) of rhodamine-labeled anionic magnetic nanoparticles were 4.78/379 (7 T), 3.28/389 (11.7 T), and 2.00/354 (17.6 T). In cells, the r(1) and r(2) were 0.21/56 (7 T), 0.19/37 (11.7 T), and 0.1/23 (17.6 T). This corresponded to an 11- to 23-fold decrease in r(1) and an 8- to 15-fold decrease in r(2) . A decrease in r(1) was observed for AMI-25 and VSOP C-200. AMI-25 and VSOP exhibited a 2- to 8-fold decrease in r(2) . In conclusion, cellular internalization of iron oxide nanoparticles strongly decreased their T(1) and T(2) potency.

Diffusion effects on the CPMG relaxation rate in a dipolar field

The diffusion in the magnetic dipolar field around a sphere is considered. The diffusion is restricted to the space between two concentric spheres, where the inner sphere is the source of the magnetic dipolar field. Analytical expressions for the CPMG transverse relaxation rate as well as the free induction decay and the spin echo time evolution are given in the Gaussian approximation. The influence of the inter-echo time is analyzed. The limiting cases of small and large inter-echo times as well as the short and long time behavior are evaluated.

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

We numerically solve the Bloch-Torrey equation by discretizing the differential operators in real space using finite differences. The differential equation is either solved directly in time domain as initial-value problem or in frequency domain as boundary-value problem. Especially the solution in time domain is highly efficient and suitable for arbitrary domains and dimensions. As examples, we calculate the average magnetization and the frequency distribution for capillaries and cells which are idealized as cylinders and spheres, respectively. The solution is compared with the commonly used Gaussian approximation and the strong-collision approximation. While these approximations become exact in limiting cases (small or large diffusion coefficient), they strongly deviate from the numerical solution for intermediate values of the diffusion coefficient.

Local frequency density of states around field inhomogeneities in magnetic resonance imaging: effects of diffusion

A method describing NMR-signal formation in inhomogeneous tissue is presented which covers all diffusion regimes. For this purpose, the frequency distribution inside the voxel is described. Generalizing the results of the well-known static dephasing regime, we derive a formalism to describe the frequency distribution that is valid over the whole dynamic range. The expressions obtained are in agreement with the results obtained from Kubos line-shape theory. To examine the diffusion effects, we utilize a strong collision approximation, which replaces the original diffusion process by a simpler stochastic dynamics. We provide a generally valid relation between the frequency distribution and the local Larmor frequency inside the voxel. To demonstrate the formalism we give analytical expressions for the frequency distribution and the free induction decay in the case of cylindrical and spherical magnetic inhomogeneities. For experimental verification, we performed measurements using a single-voxel spectroscopy method. The data obtained for the frequency distribution, as well as the magnetization decay, are in good agreement with the analytic results, although experiments were limited by magnetic field gradients caused by an imperfect shim and low signal-to-noise ratio.