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

Microscopic fractional anisotropy (μFA) estimation using diffusion MRI is a promising new method for quantifying microstructure, diagnosing pathologies, and measuring brain development. In order for μFA to be a useful metric in clinical and neuroscientific research, its robustness to noise has to be properly understood. In this study, precision of μFA estimation was quantified for different noise levels using propagation of error calculations, simulations, and imaging experiments. We show that μFA is non-linearly sensitive to noise, being the most instable at low values of mean diffusivity (MD) and μFA, and that increasing accuracy by correcting for higher order effects results in decreased precision.

The difference in signal decay between single-dimensional diffusion encoding (SDE)
and multi-dimensional diffusion encoding (MDE) experiments depends on
the anisotropy of microscopic compartments ^{1}. By measuring this difference, it is possible to disentangle
microscopic compartments’ anisotropy and orientation coherence, which can aid diagnosis
of
pathologies and measurements
of changes
in
microstructure
^{2,3}.
Accuracy issues originating from higher
order effects have been recently pointed out ^{4}, but to our knowledge, the error propagation governing
the precision of µFA has not yet been thoroughly studied. Here, we
quantify
the precision of μFA
estimate
using analytical calculations, simulations, and experiments.
Although we
focus on double diffusion encoding (DDE), most
results apply
for q-space trajectory encoding (QTE)
as well.

In the long mixing time regime, signal
deviation between powder averaged data acquired with parallel and
orthogonal gradient pulse pairs depends
on microscopic anisotropy μA,
which
is a measure of compartment anisotropy and
size ^{5}.

$$\varepsilon=\mu A^2=\text{ln}\left(\frac{S_{||}}{S_{\perp}}\right )b ^{-2}\,\,\,\,\,\,\,\,\,\,\,(1)$$

μFA
is a
normalized measure of microscopic anisotropy, that only depends on compartment anisotropy ^{5}.

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

If
data is acquired with multiple
b-values,
a more accurate
estimate of
μFA
can be
calculated by including a third order polynomial in the fit to
correct for higher order terms^{ 4}.

$$\text{ln}\left(\frac{S_{||}^{PA}}{S_{\perp}^{PA}}\right )=\mu A^2b^{2}+P_3 b^3\,\,\,\,\,\,\,\,\,\,\,(3)$$

In
this study, data is powder averaged according to the 5-design, which is a rotationally invariant directional scheme for measuring ε, that holds up to the fifth order of the signal cumulant expansion^{ 5}. We compare the third order fit (equation 3) to the second order fit, which only includes the first term in equation 3.

**Error
propagation calculations:** The
error in
μFA
and ε
was approximated
using:

$$\sigma^2_{f(\textbf{x})}=\sum^N_{i=1}\sigma_i^2\left(\frac{\partial f}{\partial x_i}\right )^2\,\,\,\,\,\,\,\,\,\,\,(4)$$

**Simulations:**
Diffusion
in individual compartments was modelled with
diffusion tensors. Measurements
were calculated for eight
b-values evenly distributed between 500
and 4000
s/mm^{2}.
The noise
properties were studied by adding
Rician
noise to
data. μFA is calculated with equations (2) and (3), and the values of μFA are constrained to be in the interval [0,1].

**MRI
experiments:**
All
experiments were approved by the local competent authority.
A rat brain (N=1) was extracted through standard transcardial perfusion and scanned
in a 9.4 T
Bruker Biospec scanner
harnessing
an 86 mm volume coil for transmission and 4-element array cryocoil
for reception. We used an in-house DDE-EPI
sequence
with four
b-values evenly distributed between 1000
and 4000
s/mm^{2}.
Diffusion-weighting
directions were
given by the 5-design ^{4}.
TE = 69 ms,
TR = 1 s, δ
= 5 ms, ∆ = 15
ms, t_{m}
= 15 ms, voxel size = 0.2 x 0.2 x 0.8 mm, and FOV = 20 x 20 mm.
Data was
denoised using a Marchenko-Pastur-PCA denoising procedure
^{6}.
Gibbs ringing artefacts were reduced using a sub-voxel
shift algorithm
^{7}.
Mean
diffusivity was estimated by fitting a diffusion tensor to the
parallel data at
b = 1000
s/mm^{2}.
Every acquisition was averaged 60 times
to maximize SNR. SNR was quantified as the average of per-voxel
standard deviations divided by the corresponding mean signals over 52 pre-processed
b0-images.

Both simulations and imaging experiments show that μFA’s robustness to noise is highly sensitive to the sizes and shapes of compartments (Figures 3, 4), which is due to μFA’s non-linear dependency on ε (Figure 1). ε is much less sensitive to noise than μFA, and may be the preferred metric in some situations. The error in ε and μFA due to noise can be approximated as

$$\sigma_{\varepsilon}\approx\sqrt{\frac{\sigma^2}{12 S^2_{||}}+\frac{\sigma^2}{60S^2_{\perp}}}b^{-2},\,\,\,\,\,\,\,\,\,\,\,(5)$$

$$ \sigma_{\mu FA}\approx\sigma_{\varepsilon}\sqrt{\frac{27}{200}}\frac{\text{MD}^2}{\varepsilon^{\frac{1}{2}}(\varepsilon+\frac{3}{5}\text{MD}^2)^{\frac{3}{2}}},\,\,\,\,\,\,\,\,\,\,\,(6)$$

where σ is the error in
an individual acquisition. Figure 2 illustrates the magnitude of
σ_{μFA}.

The precision of the
second order and third order fits were compared in simulations
(Figure 3) and imaging
experiments (Figure
4). The
third order fit is more unstable than the second order
fit, and the instability quickly increases as MD and μFA decrease.
Even in the case of no added
noise, the
third order fit results in multiple
negative
values of μA^{2}
which leads to false zero values in the μFA
map (Figure 4).

Our
study shows that in tissues
composing of highly
anisotropic compartments
such as white matter, μFA
is a
precise metric of a well-defined microstructural property. However, in order to use μFA for exploring microstructural changes in voxels with low μFA and/or MD, noise-sensitivity should be taken into account. Second order fit maximizes specificity.

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

4.
Ianuş, Andrada, et al. "Accurate estimation of microscopic
diffusion anisotropy and its time dependence in the mouse brain."
NeuroImage 183 (2018): 934-949.

5. 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. 5.

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

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