Leevi Kerkelä^{1}, Rafael Neto Henriques^{2}, Matt Hall^{1}, Chris Clark^{1}, and Noam Shemesh^{2}

Double diffusion encoding (DDE) allows the estimation of microscopic fractional anisotropy (μFA), which is a promising metric for studying microstructural properties of brain tissue independent of orientation dispersion. However, a large number of acquisitions is needed to obtain a rotationally invariant measurement of μFA, which poses a problem in clinical settings. A DDE protocol with a reduced number of acquisitions has recently been proposed as a solution. In this study, we assessed the accuracy of this approach and its potential for reporting μFA accurately. Our results show that a reduced number of acquisitions is sufficient for characterizing μFA.

μFA can be calculated as

$$\mu FA=\sqrt{\frac{3}{2}} \sqrt{\frac{\varepsilon}{\varepsilon+\frac{3}{5}\text{MD}^2}},\,\,\,\,\,\,\,\,\,\,(1)$$

where
$$$\varepsilon=\text{ln}(S^{PA}_{||}/S^{PA}_{\perp})b^{-2}$$$ is
quantified from the powder averaged data acquired with parallel and
orthogonal gradient pulse pairs, and
MD stands for mean diffusivity^{ 2}.
The
DDE 5-design holds up to the fifth order of the cumulant expansion and requires 12
parallel and 60 orthogonal gradient pulse pairs for powder-averaging the data (Figure 1A) ^{2}. In case of
Gaussian diffusion, some of the orthogonal gradient pulse pairs measure the same in-plane diffusion tensor trace, and
are thus redundant. We sought to test Yang’s hypothesis that 6
parallel and 6
orthogonal gradient pulse pair directions
can closely approximate $$$\varepsilon$$$ ^{3} (Figure 1B).

All experiments were approved by the local competent authority.

**
Specimen
preparation.** A
rat brain (N=1) was extracted through standard transcardial
perfusion.
The
rat brain was inserted into a fluorinert filled tubed and placed in
the scanner at 23 °
C.

**Data acquisition.**
All experiments were performed on a 9.4 T Bruker BioSpec
scanner harnessing an 86 mm volume coil for transmission and
4-element array cryocoil for reception.
An in-house DDE-EPI pulse sequence was used with
b = 0, 1000, 2000, 3000 s/mm^{2},
δ = 5 ms, ∆ = t_{m}
= 15 ms, TE =
69 ms, TR = 1 s,
voxel
size = 0.2 x 0.2 x 0.8
mm,
FOV
= 20 mm x 20 mm, partial
Fourier =
1.25, and
double sampling.Three slices were acquired with a slice gap of 0.5 mm.

We
compared the
five-design
(A) and the clinical directional scheme
(B), which are shown in figure 1. To maximize SNR and to match
acquisition number in both, acquisitions in A were averaged 30 times and acquisitions in B were averaged 90 times. We
used 12 symmetric acquisitions in B instead of 6 to eliminate possible artifacts arising from cross-terms between the imaging and diffusion gradients.

**Image analysis.** Data was denoised using a Marchenko-Pastur-PCA denoising procedure ^{5},
and Gibbs ringing artifacts were reduced using a sub-voxel shift
algorithm ^{6}.
Voxel-specific
SNR was quantified as the standard deviation divided
by the mean signal over
39 b0-images. Registration was done with single-step DFT algorithm ^{7}. Mean
diffusivity was estimated by fitting a diffusion tensor to the
parallel data at b = 1000 s/mm^{2}. $$$\varepsilon$$$ was fit to multi-shell data. μFA was calculated using equation 1. Synthetic Rician noise was added to
the data to artificially lower SNR.

The high quality of pre-processed data is highlighted in figure 2.

When
comparing the two sets of directions using
multi-shell data, we
find similar µFA values for the studied
b-values (Figure 3). Voxel-wise comparison of the two µFA maps reveals excellent agreement between the two methods: Pearson’s correlation coefficient = 0.91, mean difference = 0.015, std = 0.07 (Figure 4). Similar variance (Pearson's correlation coefficient = 0.92, mean difference = 0.002, std = 0.06) was observed when the
5-design experiment was
repeated. The cost of reducing the number of orthogonal gradient pulse pairs is small.

After confirming that the reduced directions are sufficient for accurately quantifying μFA using multi-shell data, we studied the minimum SNR for obtaining acceptable μFA maps with just 24 acquisitions. Clinically
acceptable protocol duration is
less than ten minutes, which limits SNR. The effect of noise on the
minimal acquisition protocol’s single-shell
μFA
estimates at b = 3000 s/mm^{2} are shown in figure 5. An SNR of 40 in non-diffusion-weighted images appears to be the very minimum for measuring μFA at b = 3000 s/mm^{2}.

We have shown that the accuracy of the clinical directional scheme is comparable to that of the 5-design, and that, given sufficient SNR, μFA can be precisely estimated with just 24 acquisitions rather than the 72 required for using the 5-design. This study thus encourages the use of DDE in the clinic for microstructural imaging.

1. Özarslan, Evren. "Compartment shape anisotropy (CSA) revealed by double pulsed field gradient MR." Journal of Magnetic Resonance 199.1 (2009): 56-67.

2. Jespersen, Sune Nørhøj, et al. "Orientationally invariant metrics of apparent compartment eccentricity from double pulsed field gradient diffusion experiments." NMR in Biomedicine 26.12 (2013): 1647-1662.

3. Yang, Grant, et al. "Double diffusion encoding MRI for the clinic." Magnetic resonance in medicine 80.2 (2018): 507-520.

4.. Lampinen, Björn, et al. "Neurite density imaging versus imaging of microscopic anisotropy in diffusion MRI: a model comparison using spherical tensor encoding." Neuroimage 147 (2017): 517-531.

5. Veraart, Jelle, et al. "Denoising of diffusion MRI using random matrix theory." NeuroImage 142 (2016): 394-406.

6. Kellner, Elias, et al. "Gibbs‐ringing artifact removal based on local subvoxel‐shifts." Magnetic resonance in medicine 76.5 (2016): 1574-1581.

7. Guizar-Sicairos, Manuel, Samuel T. Thurman, and James R. Fienup. "Efficient subpixel image registration algorithms." Optics letters 33.2 (2008): 156-158.

**Figure 5.** A) 1st row:
Examples
of μFA
maps
calculated with single-shell
data acquired with the minimal
protocol at
b = 3000 s/mm^{2}
with synthetic noise added to data to lower SNR. 2nd row: μFA map calculated with multi-shell 5-design data with no additional noise. 3rd row: The
difference between the two (5-design - minimal scheme). B)
The average standard deviation of the voxel-wise μFA difference between single-shell minimal protocol with 24 acquisitions and multi-shell 5-design with 216 acquisitions. 100 repetitions of added noise for each SNR.