Henrik Lundell^{1}, Markus Nilsson^{2}, Filip Szczepankiewicz^{3,4,5}, Carl-Fredrik Westin^{3,6}, Daniel Topgaard^{7}, and Samo Lasič^{8}

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.

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 ADC[1] D. Topgaard, J. Magn. Reson. 275, 98 (2016).

[2] F. Szczepankiewicz, S. Lasič, D. van Westen, P. C. Sundgren, E. Englund, C. F. Westin, F. Ståhlberg, J. Lätt, D. Topgaard, and M. Nilsson, Neuroimage 104, 241 (2015).

[3] C.-F. Westin, F. Szczepankiewicz, O. Pasternak, E. Özarslan, D. Topgaard, H. Knudsen, and M. Nilsson, MICCAI 14, 209 (2014).

[4] N. Shemesh, S. N. Jespersen, D. C. Alexander, Y. Cohen, I. Drobnjak, T. B. Dyrby, J. Finsterbusch, M. A. Koch, T. Kuder, F. Laun, M. Lawrenz, H. Lundell, P. P. Mitra, M. Nilsson, E. Özarslan, D. Topgaard, and C. F. Westin, Magn. Reson. Med. 75, 82 (2016).

[5] H. Lundell, M. Nilsson, T. B. Dyrby, G. J. M. Parker, H. Cristinacce, F. Zhou, D. Topgaard, and S. Lasic, in In Proc. ISMRM (2017), p. 1086.

[6] T. M. de Swiet and P. P. Mitra, J. Magn. Reson. - Ser. B 111, 15 (1996).

[7] S. N. Jespersen, J. L. Olesen, A. Ianu, and N. Shemesh, ArXiv 1 (2017).

[8] H. Lundell, M. Nilsson, C.-F. Westin, D. Topgaard, and S. Lasic, in In Proc. ISMRM (2018), p. 887.

[9] H. Lundell, C. K. Sønderby, and T. B. Dyrby, Magn. Reson. Med. 73, 1171 (2015).

[10] J. Stepišnik, Phys. B 183, 343 (1993).

[11] D. Topgaard, Microporous Mesoporous Mater. 178, 60 (2013).

[12] J. E. Tanner, J Chem Phys 69, 1748 (1978).

[13] J. H. Jensen, J. A. Helpern, A. Ramani, H. Lu, and K. Kaczynski, Magn. Reson. Med. 53, 1432 (2005).

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
K_{i} estimated for individual encoding rotations, the kurtosis
contribution from variation of apparent diffusivities K_{v} and the K_{PA}
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.