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Estimating microstructural parameters from gradient-echo and spin-echo data: a test of the strong collision approximation
Pippa Storey1,2 and Dmitry S. Novikov1,2
1Bernard and Irene Schwartz Center for Biomedical Imaging, Department of Radiology, New York University Grossman School of Medicine, New York, NY, United States, 2Center for Advanced Imaging Innovation and Research, Department of Radiology, New York University Grossman School of Medicine, New York, NY, United States
1Bernard and Irene Schwartz Center for Biomedical Imaging, Department of Radiology, New York University Grossman School of Medicine, New York, NY, United States, 2Center for Advanced Imaging Innovation and Research, Department of Radiology, New York University Grossman School of Medicine, New York, NY, United States
Synopsis
Keywords: Microstructure, Susceptibility
Motivation: To develop noninvasive methods to interrogate magnetic microstructure.
Goal(s): To test whether the strong collision approximation can accurately characterize microstructure of known geometry and magnetic susceptibility from gradient-echo and spin-echo signals.
Approach: Experimental data were acquired from phantoms containing polystyrene microbeads of 10$$$\mu$$$m, 20$$$\mu$$$m and 40$$$\mu$$$m diameter suspended in gadolinium-doped gelatin. Data were fitted using a published model based on the strong collision approximation and a lookup table prepared from Monte Carlo simulations.
Results: The strong collision approximation overestimated bead size and underestimated magnetic susceptibility from gradient-echo data. For spin-echo data, it yielded poor estimates of susceptibility and was insensitive to bead size.
Impact: The strong collision approximation is a non-perturbative approach for predicting gradient-echo and spin-echo signals in the presence of magnetic microstructure. It employs the Krogh construction and a simplified diffusion propagator. We show how those simplifications affect estimates of microstructural parameters.
Introduction
Many components of tissue microstructure have magnetic susceptibilities that differ from surrounding tissue. This generates magnetic field heterogeneity, which produces characteristic non-exponential signatures in the decay of gradient-echo and spin-echo signals. Diffusion of spins plays a crucial role in determining the effect of magnetic microstructure on signal$$$^1$$$, and exact analytic solutions are not possible except for simple geometries. Elegant perturbative approaches have been pioneered by Jensen, Yablonskiy, Sukstanskii, Kiselev and others$$$^{2-9}$$$. However, their validity is limited to parameter regimes in which the signals are weakly affected by the microstructure.A non-perturbative approach, dubbed the ‘strong collision approximation’, has been proposed by Bauer and Ziener$$$^{10-11}$$$. It approximates diffusion by a stochastic process in which each spin has a probability $$$\exp\left(-\lambda\,\Delta{t}\right)$$$ of remaining at its current position over an interval $$$\Delta{t}$$$, and a complementary probability $$$\left[1-\exp\left(-\lambda\,\Delta{t}\right)\right]$$$ of jumping to a random location anywhere within the system (subject to permeability constraints). In addition, the geometry is simplified using the Krogh construction, which considers a single susceptibility source (e.g. a sphere or cylinder) within a concentric region of the same shape, whose dimensions yield the appropriate volume fraction. Together, these simplifications permit analytic solutions for certain non-trivial geometries. However, their impact on accuracy has not been adequately studied.
The purpose of this work was to investigate the effect of these simplifications by comparing the predictions of the strong collision approximation with Monte Carlo simulations and experimental data from well-characterized phantoms.
Methods
Spherical polystyrene microbeads were suspended in a solution of 2% gelatin doped with 0.07% gadobutrol. Three phantoms, with bead volume fractions of $$$\eta=0.1$$$, were prepared using microbeads of different sizes (Dynoseeds TS-10, TS-20 and TS-40, Microbeads AS, where the number indicates the nominal diameter in microns). A fourth phantom containing only the doped-gelatin medium served as a control. To minimize macroscopic $$$B_0$$$ inhomogeneity, the phantoms were constructed from 25-mL serological pipettes, which approximated an infinite cylinder.Experiments were repeated five times using separate sets of phantoms. 3D gradient-echo data were collected at 3T (Siemens Prisma) with 0.5mm isotropic resolution and 32 monopolar echoes (TE = 2.25ms - 109.82ms). In three experiments, spin-echo data were also collected with quadratically spaced echo times (TE = 6ms - 306ms). To avoid stimulated echoes, a single echo was acquired after each excitation. To minimize effects of temperature drift, data with different TE were collected simultaneously by repurposing the innermost sequence loop to cycle over all echo times between consecutive k-space lines.
Monte Carlo simulations were performed over a 3D volume with periodic boundary conditions containing 4096 spheres, whose distribution was determined by random packing with no overlap$$$^{12}$$$. Spins underwent a random walk without penetrating the spheres. The phase of each spin $$$\phi\left(t\right)$$$ was computed by integrating the local Larmor frequency along the spin’s trajectory. The phase distribution over all spins was used to estimate gradient-echo and spin-echo signals over a range of perturbation strengths using
$$S\left(t\right)=S_0e^{-R_2t}\left\langle{e^{i\phi\left(t\right)}}\right\rangle$$
where $$$R_2$$$ is the molecular relaxation rate.
The perturbation strength was quantified by
$$\alpha=\delta\Omega\,\tau$$
and represents the amount of dephasing over the correlation time. $$$\delta\Omega$$$ denotes the standard deviation in Larmor frequency due to microstructure
$$\delta\Omega=\left\langle\Omega^2\left(\mathbf{r}\right)\right\rangle^{1/2}=\sqrt{\frac{4\eta}{5}}\delta\omega$$
where $$$\delta\omega$$$ is the frequency shift on the equator of a sphere
$$\delta\omega=\frac{1}{3}\gamma\Delta\chi\,B_0$$
We define the correlation time as$$$^{2}$$$
$$\tau=\frac{R^2}{\left(36\pi\right)^{1/3}D}$$
where $$$D$$$ is the diffusion coefficient and $$$R$$$ is the sphere radius.
Experimental data were fitted using a lookup table generated from Monte Carlo simulations and a model based on the strong collision approximation$$$^{11}$$$. Four parameters were estimated, namely bead diameter, susceptibility difference $$$\Delta\chi$$$, molecular relaxation rate $$$R_2$$$, and initial signal $$$S_0$$$.
Results
In the static dephasing regime, predictions of the strong collision approximation deviate from those of Monte Carlo simulations due to use of the Krogh construction (Figure 1).In the presence of diffusion, the models differ most notably for gradient-echo signals at short times (Figure 2) and for spin-echo signals (Figure 3). The discrepancies are attributable to the simplified diffusion propagator used in the strong collision approximation.
Parameter estimation was unstable for the smallest beads, due to inadequate data at short TE, and for gradient-echo data from the largest beads, where the system approaches the static dephasing regime (Figure 4).
The strong collision approximation overestimated bead size from gradient-echo data, and was insensitive to bead size from spin-echo data (Figure 5).
Conclusions
The strong collision approximation replicates the long-time behavior of gradient-echo signals fairly accurately up to moderate perturbation strengths ($$$\alpha\sim{1}$$$). However, it fails at short times and for spin-echo signals. This limits its utility in estimating microstructural parameters from signal decay curves, especially for spin-echo data.Acknowledgements
This work was supported in part by the NYU IT High Performance Computing facility and by NIH grants NS039135 and P41 EB017183.References
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Figures
Figure 1: Left: 2D schematics of the 3D geometries used in Monte Carlo simulations and the strong collision approximation$$$^{11}$$$. Center: Larmor frequency probability distribution $$$P(\Omega)$$$ for both models. Right: Decay of the gradient echo signal (FID) under conditions of static dephasing ($$$D=0$$$) with molecular relaxation rate $$$R_2=0$$$. Note that the FID is the Fourier transform of $$$P(\Omega)$$$. The discrepancies between the models result from use of the Krogh construction in the strong collision approximation, which neglects the disorder in the system.
Figure 2: Predictions of Monte Carlo simulations and the strong collision approximation for gradient echo signals in the presence of diffusion at three values of the perturbation strength $$$\alpha$$$, with $$$R_2=0$$$. For $$$\alpha \gg 1$$$, the system approaches the static dephasing regime (Figure 1). At long times, the strong collision approximation agrees relatively well with Monte Carlo simulations up to $$$\alpha \sim 1$$$. However, at short times it overestimates the curvature of $$$-\log S( t )$$$. This is attributable to its use of a simplified diffusion propagator.
Figure 3: Predictions for spin echo signals with $$$R_2=0$$$. Note that the discrepancies between the two models are larger for spin echo signals (shown here) than gradient echo (Figure 2). This may be due to a greater sensitivity of spin echo signals to the diffusion dynamics, which are modeled in the strong collision approximation by a simple stochastic jumping process. Accurate modeling of diffusion may be more critical for spin echo signals than gradient echo signals since the former do not undergo static dephasing, so their decay due to microstructure depends entirely on diffusion.
Figure 4: Estimates of microstructural parameters from Monte Carlo simulations. Black lines show independently measured reference values. For 10-micron beads (green), parameter estimation was unstable due to the paucity of data at short TE and their sensitivity to transient effects. For 40-micron beads (red) the gradient echo signal approaches the static dephasing regime, where it becomes insensitive to size. For spin echo data, the slight overestimate of diameter for the 40-micron beads may be due to a small fraction of anomalously large beads caused by fusion during manufacture.
Figure 5: Estimates of microstructural parameters from the strong collision approximation. For gradient echo data, the strong collision approximation overestimates bead diameter and underestimates susceptibility difference. This is attributable to its overestimation of the curvature of $$$- \log S( t )$$$ at short times (Figure 2). For spin echo data, it yields poor estimates of $$$\Delta \chi$$$ and is unable to distinguish bead size. This is consistent with the large discrepancies shown in Figure 3 between its predictions for spin echo signals those of Monte Carlo simulations.
DOI: https://doi.org/10.58530/2024/1310