Geometrical setup for the simulations: A void spherical inclusion (inner sphere) with a radius of was placed at the center of an isotropic sample cube (bold box). To avoid edge effects, initial magnetization was attributed only to the central grid points (minus the inclusion) of a cube spanned on a element vector array. Different local dipole fields were modeled using different magnetic susceptibility differences between the inclusion and the surrounding area. Besides the case , local dipole fields of glass or air inclusions with diameter were modeled with diameter voids (outer sphere) with and , respectively. Additionally, the glass sphere was simulated with in a sphere.
iZQC spectroscopy pulse sequence with selective second pulse as used in the simulations. The nonselective 90° excitation pulse (black bar) is followed by the correlation gradient (CG, empty square) and the evolution period. The frequency selective 90° second pulse (sinc shape) acting only on the (solvent) spins creates the DDF, which refocuses transverse magnetization in the detection period . In simulations, both rf pulses and the CG were applied as operators with duration zero.
Simulated iZQC spectra without local dipole field . The presented spectra were summed over the regions indicated by hatched areas in the insets. Gray circle indicates the area where magnetization was set to zero in the simulations. (a) Full spectrum summed over a whole sample cube except the central sphere. Simulation was performed for a homogeneous magnetization over the whole cube (no inclusion). (b) Full spectrum for a simulation where the magnetization in the central sphere with was set to zero. [(c) and (d)] The spectra from the simulation shown in (a), but summed only over the shells indicated in the inset, in (c), and in (d).
Nyquist limit in the simulations. Regions of the sample cube, where the spatial Nyquist frequency was violated in the simulation S6, are shown to the left of the plotted curves as function of the polar coordinates and . Since spatial modulation of the magnetization increases with , the regions grow for longer . Curves are shown for four representative values (labels). Affected regions are almost completely limited to the first analyzed shell as indicated by the vertical lines. Note that only a small fraction of the whole sample cube is shown along the axis.
Simulated iZQC spectra from samples modelling different sources of local dipole fields, summed over the whole sample cube. (a) Glass sphere with and . (b) Sphere with and giving rise to the same local field as in (a). (c) Sphere with and modeling the local field of an air bubble with and . Note the different intensity scales.
Spatial reach of the influence of . Simulated spectra were summed over different shells of a sample that modelled the local dipole field of an air bubble with and . [(a)–(d)] Insets indicate, in a central slice through the sample, over which shell the spectra were summed (hatched area). Gray circles indicate the area where magnetization was set to zero in the simulations. Note the different ratio of and spin peak intensities. Different scales are mostly due to the vastly different number of grid points in the individual shells.
Simulated spectra compared to experimental observations for the case of a glass inclusion. Simulated and experimental spectra from a voxel [(a) and (b)], from a voxel [(c) and (d)], and from a voxel [(e) and (f)], respectively. Arrows indicate the cross peak in the simulations and the water- cross peak in the experiments. Concentration of the spins was threefold higher in the experiments than in the simulations. Scale bars for peak intensities in the simulated spectra and SNR of the cross peak for the measured spectra are indicated.
Minimum sample per inhomogeneity volume ratio giving the detection limit in iZQC spectroscopy in the presence of local dipole fields. was defined as the mean value between the smallest sample volume in which the cross peak was observable and the largest volume in which the peak was not observable, according to the criteria defined in the text. Given errors for “simulation glass inclusion,” “simulation air inclusion,” and “experiment glass inclusion” correspond to the step size in analysis of the data sets S4, S6, and in the experiment, respectively. Note that a threefold higher concentration of the observed spin species was used in the experiments than in the simulations.
Peak intensities and linewidths of the solvent ( spins) and solute ( spins) peaks in the simulated spectra. Values are given for all six simulations (S1–S6, see text) summed over the whole sample cube, and summed over the individual shells (shell 1, ; shell 2, ; shell 3, , and shell 4, ).
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