Reducing spectral anisotropy in isotropic b-tensor multidimensional diffusion encoding
Henrik Lundell1, Markus Nilsson2, Filip Szczepankiewicz3,4,5, Carl-Fredrik Westin3,6, Daniel Topgaard7, and Samo Lasič8

1Danish Research Centre for Magnetic Resonance, Copenhagen University Hospital Hvidovre, Hvidovre, Denmark, 2Clinical Sciences, Department of Radiology, Lund University, Lund, Denmark, 3Brigham and Women’s Hospital, Boston, MA, United States, 4Harvard Medical School, Boston, Denmark, 5Department of Radiology, Lund University, Lund, Denmark, 6Harvard Medical School, Boston, MA, United States, 7Division of Physical Chemistry, Department of Chemistry, Lund University, Lund, Sweden, 8Random Walk Imaging AB, Lund, Sweden


Multidimensional diffusion encoding isotropic b-tensors eliminate rotational variance from anisotropic Gaussian diffusion, an essential feature for estimating microscopic anisotropy. In anisotropic substrates which exhibit non-Gaussian diffusion on the time scale of the encoding pulse, the variance in temporal characteristics of diffusion encoding across different directions – spectral anisotropy (SA) - may introduce a directional variance in apparent diffusivities. We propose an alternative isotropic encoding with drastically lower SA, which in turn allows accessing intrinsic signatures of non-Gaussian diffusion.


Multidimensional diffusion encoding (MDE) utilizes a range of different gradient waveforms and b-tensor shapes in order to accurately characterize heterogeneous tissues [1–4]. By considering the spectral content of MDE waveforms, we can probe time-dependent diffusion and tune its sensitivity to varying compartment sizes [5]. In waveforms that yield isotropic b-tensors, the spectral content may vary across different directions, which introduces a rotational variance in substrates with time-dependent anisotropic diffusion [6,7]. We refer to this variation as spectral anisotropy (SA), which provides a handle for probing correlations between time dependence and shape of restrictions in heterogeneous samples [8]. Reducing SA is attractive for clean isotropic measurements. A homogeneous spectral content can be fairly well realized over rotations in a 2D plane [8,9], but the 3D case is less straightforward. For instance, the magic angle spinning of the q-vector (qMAS) has high encoding power at relatively low frequencies along the aperture axis of the trajectory compared to its perpendicular plane. In this work we propose an alternative isotropic 3D encoding using two subsequent orthogonal encoding 2D planes and compare its rotational variance to an optimized qMAS trajectory in a Monte Carlo simulation.


Two encoding waveforms were considered and analyzed in terms of the power spectra |F(ω)|2 of their corresponding dephasing trajectories [10]: (1) the original numerically optimized qMAS trajectory [11] and (2) a new alternative scheme based on two orthogonal planar encoding that we call 2DORTHO. A planar encoding was extracted from a projection of the qMAS trajectory with |F(0)|2=0 for all in plane rotations. The full 2DORTHO was realized by repeating the encoding in the xz- and yz-planes around a refocusing pulse. The z-axis gradient strength was scaled to provide an isotropic b-tensor. We performed Monte Carlo simulations for a 1D system with evenly spaced permeable membranes [7,12]. The two waveforms were applied with total encoding time of 45 ms including a refocusing period of 5 ms with 64 uniformly distributed rotations. Membrane permeability was varied in the range p = 0-0.1 µm/ms, membrane spacing was set to a = 1-10 µm and the free diffusivity D0 = 2 µm2/ms. Apparent diffusion coefficients (ADC) and Kurtosis (K) were calculated from the spin phase distributions for: 1) the simulations in individual directions reflecting the intrinsic diffusion for different substrate orientations (ADCi, Ki) and 2) the powder averaged random walks over all directions reflecting the measured signal for a disordered ensemble (ADCPA, KPA) [7,13]. The latter reflect both the distribution of non-Gaussian intrinsic kurtosis (Ki) and the contribution from the multi-Gaussian distribution of ADCi which was calculated as 3) Kv=3·var(ADCi)/⟨ADCi⟩2 [13].

Results and discussion

The proposed 2DORTHO encoding provides a lower degree of spectral anisotropy in the encoding power spectrum compared to the originally proposed qMAS as shown in figure 1. The latter has more encoding power at low frequencies along the z-axis (green spectrum in figure 1 A). In contrast, the encoding power is more evenly distributed across the orthogonal axes in 2DORTHO. This is true for any rotation as shown in high and low frequency portions of the respective power spectra in figure 1 C) and D). As expected, the lower degree of SA yields a more narrow distribution of ADCi in simulations with different rotations both in the case of impermeable and permeable membranes (top histograms in figure 2 A and B). This is also seen in the spread of the signals shown in figure 3. The effect of different SA is reflected also in the kurtosis estimates (lower rows in figure 2), where the lower SA yields smaller Kv. In case of SA~0, KPA for the powder averaged signal is probing Ki without the contribution of Kv. In all cases, we note that the deviations from mono-exponential signal attenuation due to intrinsic kurtosis are in the order of 1% at the highest b-value (highlighted regions in figure 3 B and D).


We propose the alternative isotropic b-tensor encoding 2DORTHO with reduced spectral anisotropy. This encoding provides rotationally invariant isotropic encoding even in anisotropic substrates were non-Gaussian effects are pronounced.


This project has received funding from the European Research Council (ERC) under the European Union’sHorizon 2020 research and innovation programme (grant agreement No 804746).


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A) and B) show dephasing trajectories for the qMAS and 2DORTHO encodings and their respective power spectra |F(ω)|2 for an orientation with large variation in three orthogonal directions (red, blue and green). Black spectra indicate their spherical mean. Gray and black bars under the spectra indicate high- (black) and low- (gray) frequency regions with equal cumulative power in the mean spectra. C and D illustrate rotational variation in cumulative power across the entire frequency range (equivalent to b-value) and across the low and high frequency ranges for the qMAS and 2DORHTO.

Results of Monte Carlo simulations with the isotropic b-tensors by qMAS (left) and 2DORTHO (right). A) Histograms showing the distribution of ADC (top row) and Kurtosis coefficients K (bottom row) over 96 different substrate orientations for a case with impermeable membranes with spacing a = 4 µm. The mean over intrinsic kurtosis values Ki estimated for individual encoding rotations, the kurtosis contribution from variation of apparent diffusivities Kv and the KPA for the powder average signal are indicated with the black, blue and red lines. B) Same as in A) but with membrane permeability of 0.02 µm/ms.

Signals as a function of b-value from the model with membrane spacing a = 4 µm and permeability set to p = 0 (A), B)) and p = 0.02 µm/ms (C), D)) for qMAS and 2DORTHO. The solid black line indicates the powder averaged signals and the thinner red lines the orientations with the highest and lowest degree of signal attenuation. Dotted lines reflect the extrapolated mono-exponential initial slope of the respective curves. Zoomed-in regions of the signals in B) and D) are shown in the far right.

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