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Is spherical diffusion encoding rotation invariant? An investigation of diffusion time-dependence in the healthy brain
Filip Szczepankiewicz1,2, Samo Lasic3, Markus Nilsson4, Henrik Lundell5, Carl-Fredrik Westin1,2, and Daniel Topgaard6

1Radiology, Brigham and Women's Hospital, Boston, MA, United States, 2Harvard Medical School, Boston, MA, United States, 3Random Walk Imaging AB, Lund, Sweden, 4Clinical Sciences Lund, Lund University, Lund, Sweden, 55. Danish Research Centre for Magnetic Resonance, Centre for Functional and Diagnostic Imaging and Research, Copenhagen University Hospital Hvidovre, Copenhagen, Denmark, 6Physical Chemistry, Lund University, Lund, Sweden

Synopsis

Recent advances in diffusion weighted MRI have reignited interest in spherical (or isotropic) diffusion encoding. For such encoding to reach high efficiency (minimal echo time), the gradient waveforms have irregular shapes, by design. As such, they lack a well-defined diffusion time and can even be spectrally anisotropic. Most analysis methods based on such encoding assume that diffusion is multi-Gaussian, i.e., that the diffusion is not time-dependent. Since this is a central assumption, we investigate if spherical diffusion encoding is indeed rotation invariant, or if the diffusion time anisotropy has a discernible effect on the diffusion weighted signal in healthy brain.

Introduction

Several recently proposed diffusion MRI techniques employ spherical (or isotropic) diffusion encoding1,2 to probe the microscopic anisotropy and heterogeneity of tissue3-5. To maximize encoding efficiency, the waveforms are often irregular, and therefore lack a straightforward definition of the diffusion time. In 1996, de Swiet and Mitra6 predicted that rotation invariance of spherical encoding could be compromised by time-dependent diffusion7, causing systematic errors in models that ignore time-dependence. Indeed, close inspection of non-conventional encoding waveforms reveals a complex spectral content where diffusion time depends on direction8,9, calling into question the assumption that time-dependence can be ignored. Since ample evidence of time-dependence exists in the brain10,11—albeit at relatively short diffusion times—the reliability of this assumption is imperative to techniques that assume multi-Gaussian diffusion.

In this study, we investigate if spherical diffusion encoding is rotation invariant in a healthy brain by using a waveform that is tuned to exhibit maximal sensitivity to diffusion-time, without impeding its encoding efficiency.

Theory

For axisymmetric diffusion tensors12 with axial $$$(D_\text{A})$$$ and radial eigenvalues $$$(D_\text{R})$$$, the diffusion-weighted signal can be expressed as $$$S=S_0\exp(-bD(\theta))$$$, so that the apparent diffusivity $$$\theta$$$-degrees away from the symmetry axis of the diffusion tensor is$$D(\theta)=\text{MD}+\frac{2}{3}(D_\text{A}-D_\text{R})\cdot{}P_2(\cos\theta)\quad\text{Eq.1}$$where the mean diffusivity $$$\text{MD}=(D_\text{A}+2D_\text{R})/3$$$ and $$$P_2(\cdot)$$$ denotes the second Legendre polynomial. Isotropic diffusion weighting sequences assume a rotationally invariant signal $$$S=S_0\exp(-bD_\text{iso})$$$ where the isotropic diffusivity $$$D_\text{iso}=\text{MD}$$$. However, restrictions may induce time-dependent diffusion and yield a rotation variant inequality between $$$D_\text{iso}$$$ and $$$\text{MD}$$$6,8. The predicted discrepancy can be written as$$\Delta{}D=\frac{2}{3}(D'_\text{A}-D'_\text{R})\cdot{}P_2(\cos\theta)\quad\text{Eq.2}$$where $$$\theta$$$ is now the angle between the low-frequency direction of the spherical encoding and the symmetry axis of the diffusion tensor (Fig.2, right panel), and $$$D'_\text{A}/D'_\text{R}$$$ are the time-dependent apparent axial/radial diffusivities determined by the restriction microstructure and gradient waveform shape. Thus, Eq.2 predicts that $$$\Delta{}D$$$ probed by rotating spherical encoding is proportional to $$$P_2(\cos\theta)$$$ in the presence of anisotropic time-dependence.

Methods

Data was acquired on a 3T (80 mT/m-gradients) in a healthy volunteer, and a liquid crystals phantom with high micro/macro-anisotropy13. Imaging was performed with a prototype spin-echo pulse-sequence where TR/TE=3.2s/91ms, FOV=220x220x60mm3, resolution 2.4mm isotropic, partial-Fourier=7/8, iPAT=2 (Fig.1). Waveforms for spherical encoding were optimized for minimal TE14,15. Importantly, the optimizer was tuned to render maximal diffusion time anisotropy (long diffusion time on x, and short on y and z-axes) without sacrificing the encoding time required to reach the max b-value (Fig.1). Spherical encoding used b=[.1, .7, 1.4, 2] ms/µm2 and ten rotations per shell (rotated so that x-axis pointed along 10 directions). Each series was repeated five times to distinguish variance between and within rotations. Linear encoding used the same b-shells and multiple directions (Fig.1). The number of samples per voxel was 200 for spherical and 82 for linear encoding. Total scan time was 17 min. The fractional anisotropy (FA) and main axis of the diffusion tensor was estimated from conventional diffusion kurtosis analysis16. For each rotation of the spherical encoding, $$$D_\text{iso}$$$ and the isotropic kurtosis $$$V_\text{iso}$$$ (non-normalized5,16) were calculated by fitting the statistical model17$$S_R\approx{}S_0\exp\left(-bD_\text{iso}+\frac{1}{2}b^2V_\text{iso}\right)\quad\text{Eq.3}$$Eq.3 was also used to calculate the mean diffusion and kurtosis (MV) by fitting to signal from all rotations simultaneously. We investigated if the spherical encoding causes rotational variant signal in single voxels of coherent white matter. Furthermore, since single voxel analysis may lack statistical power to detect subtle effects, a comprehensive investigation of rotation dependent diffusivity $$$(\Delta{}D)$$$ and kurtosis $$$(\Delta{}V=V_\text{iso}-MV)$$$ was performed in coherent white matter.

Results

Throughout this study, no evidence was found that spherical diffusion encoding depended on rotation. In single voxels of highly coherent white matter (Fig.2), signal was not dependent on rotation; no stratification of $$$S(b)$$$ and the signal variability between and within rotations was on the same order of magnitude. Fig.2 also visualizes the expected signal if rotation variance was present. A comprehensive search in coherent white matter (Fig.3), also revealed no rotation variance of diffusivity or kurtosis. A similar search, restricted to the corticospinal tract and corpus callosum, also indicated that the spherical encoding is invariant to rotation. For verification purposes, the same analysis was performed in simulated and phantom data where time-dependence is known to be null or negligible (Fig.5).

Discussion and conclusions

We conclude that the current waveform design indeed produces spherical diffusion encoding with negligible rotation variance, at least in the healthy brain. This greatly simplifies the interpretation of techniques based on the multi-gaussian assumption. However, we recognize that experiments are routinely designed to probe much shorter diffusion times18, and that our findings apply only to waveforms designed to prioritize b-value efficiency, yielding relatively long diffusion times. Furthermore, the current results may not generalize to tissues that differ markedly from the healthy brain, e.g., other organs or tissues affected by disease or development. Investigations of such conditions are forthcoming.

Acknowledgements

We thank Siemens Healthcare, Erlangen, Germany, for access to the pulse sequence programming environment. We acknowledge the following research grants NIH P41EB015902, NIH R01MH074794, SSF Framework grant AM13-0090, VR 2016-04482.

References

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Figures

Fig.1 - Acquisition protocol and gradient waveforms used for spherical and linear encoding. The encoding time before/after the refocusing pulse was 35.6/30.8 ms/ms. The power spectrum of the dephasing vector, q(t), shows that the x-axis of the spherical encoding has a peak at low frequencies (long diffusion times) whereas the remaining axes have mean frequencies around 15-20 Hz. The bottom row shows diffusion encoding directions (including antipodal points) used for linear and spherical encoding (spheres were rotated to align the x-axis with the encoding direction). The series were interleaved, and b-values and directions were randomly permuted to alleviate effects of scanner instability.

Fig.2 - No stratification of $$$S_\text{R}(b)$$$ was observed across ten rotations in single-voxels of coherent white matter. The mean signal±SD (across 5 repetitions) overlapped and showed no association between signal intensity and rotation. For visual reference, we calculated the expected effects of time-dependence for an impermeable cylinder for 103 rotations of the currently used gradient waveforms (Fig.1) under the Gaussian approximation of the signal cumulant expansion7. It shows that time-dependence and rotational variance only starts appearing at cylinder radii > 2—3 µm, where higher $$$P_2(\cos\theta)$$$ is associated to lower signal and higher diffusivity.

Fig.3 - An exhaustive search in coherent white matter shows that linear encoding is sensitive diffusion anisotropy, as expected. By contrast, the spherical encoding appears to be rotation invariant; for both diffusivity $$$(\Delta{}D)$$$ and kurtosis $$$(\Delta{}V)$$$, the slope of—and variance explained by—the linear regression based on Eq.2 were both negligible. Thus, no diffusion time dependence was observed in the signal using spherical encoding. Coherent white matter was selected by only including voxels where FA>0.8 (red outlines in axial slice).

Fig.4 - No diffusion time dependence was observed in the corticospinal tract, or the anterior/posterior corpus callosum (red outlines show manual ROIs in an axial slice). As for the exhaustive search in Fig.3, the slope of linear fit, as well as the variance explained by the regression, were both negligible.

Fig.5 - Validation experiments in a liquid-crystal phantom (nano-tubes that exhibit approximately one-dimensional diffusion) and a simulated single-tensor-substrate (DA=2.2 µm2/ms, DA=0.1 µm2/ms). Both examples are free from time-dependent diffusion effects and exhibit the same behavior as the white matter of the brain, i.e., a strong rotation variance for linear encoding (caused by voxel-scale anisotropy), and negligible effects of rotation for the spherical encoding. In the phantom, only voxels where FA>0.8 were included (red ROI outline), to avoid dispersed nano-tubes. The simulations used identical sampling scheme and analysis as for the in vivo case assuming a signal-to-noise ration of 20 at b=0.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)
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