Andrada Ianus^{1,2}, Sune N. Jespersen^{3,4}, Ivana Drobnjak^{2}, and Noam Shemesh^{1}

Advanced diffusion MRI acquisitions, such as double diffusion encoding (DDE), have been used to provide estimates of microscopic anisotropy, and the majority of DDE studies to date have acquired data at a single b-value. This study shows in simulations and ex-vivo experiments that the D(O)DE derived microscopic anisotropy metric strongly depends on the choice of b-value and proposes a multi shell estimation scheme which provides accurate measurements.

*Diffusion sequences:* To investigate the b-value dependence of the
estimated μA,
double oscillating diffusion encoding (DODE) sequences, illustrated in Figure 1, were
simulated following the 5-design from ^{8}. DODE was
chosen as its mixing time dependence is negligible ^{9}, thus reducing the complexity of the parameter space. Specific sequence parameters
are given in Table 1.

*Microscopic
anisotropy estimation: *For
randomly oriented microdomains with frequency-dependent parallel and
perpendicular diffusivities (D_{‖}(ω) and D_{⊥}(ω)), Figure 1 b) and c), μA can be derived from
the powder averaged DODE signal constructed by using the 5-design. Thus, the cumulant expansion up to second order in b, yields the
following expression ^{10}:

$$\log(\frac{1}{12}\sum S_∥)-\log(\frac{1}{60}\sum S_⊥)= b^2 \frac{2}{15} (D(ω)_∥-D(ω)_⊥ )^2+O(b^3), $$

and the microscopic anisotropy estimated at a given b-value is calculated as

$$\tilde{μA}^2=(\log(\frac{1}{12}\sum S_∥)-\log(\frac{1}{60}\sum S_⊥)) /b^2 =\frac{2}{15} (D(ω)_∥-D(ω)_⊥ )^2.$$

This expression includes only second order terms, which might introduce a bias when higher order terms are not vanishing. To correct for this effect, we can include the next order terms:

$$ \log(S_∥^{(p.a.)} )-\log(S_⊥^{(p.a.)} )= μA^2 b^2+P_3 b^3 $$

where μA^{2 }denotes the corrected microscopic diffusion
anisotropy metric and P_{3} reflects the contribution of 3rd order
terms.

*Simulations:* We use the MISST toolbox^{11,12} to simulate the diffusion signal in
different tissue models as illustrated in Figure 2. Then, for each substrate,
we compare the ground truth value (μA^{2}_{g.t.}) with the
estimated microscopic anisotropy at different b-values as well as the corrected metric.

*Experiments:* All experiments were preapproved by the
Institution’s animal ethics committee and performed on 16.4 T scanner. The
brain was perfused from a healthy mouse, immersed in gadoterate meglumine 2.5mM
for 24h before scanning and placed in a 10mm NMR tube filled with Fluorinert.
The specimen was kept at 37^{o}C during scans. Acquisition parameters are detailed
in Table 1, and the analysis is performed using normalised signals.

*Simulations:* Figure 2 plots the
apparent microscopic anisotropy $$$\tilde{µA}$$$^{2} at a range of b-values (blue curves),
the corrected metric (yellow curve) and the ground truth values
(orange curve) for different models of
microstructure featuring either Gaussian diffusion (Figure 2a, 2b and 2d) or restricted
diffusion (Figure 2c, 2e and 2f). The
results show that metrics estimated from each b-value
independently are biased, and the bias increases with b-value. When µA^{2} is computed using the information from all
b-values to correct for higher order terms, similar values to the ground truth
are obtained.

*Experiments:* Figure 3 illustrates maps measured at each b-value. Indeed, the values decrease with increasing
b-value, with a more pronounced dependence in white matter. For low b-values
(<1000s/mm^{2}) the maps are very noisy, as the difference between
measurements with parallel and perpendicular gradients is very small. Figure 4 presents
the corrected microscopic anisotropy maps, as well as the fitted polynomial
coefficient (P_{3}) corresponding to the third order term in b.

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6. Shemesh, N., et al., Noninvasive bipolar double-pulsed-field-gradient NMR reveals signatures for pore size and shape in polydisperse, randomly oriented, inhomogeneous porous media. J Chem Phys, 2010. 133: p. 044705.

7. Shemesh, N. and Y. Cohen, Microscopic and compartment shape anisotropies in gray and white matter revealed by angular bipolar double-PFG MR. Magn Reson Med, 2011. 65: p. 1216-27.

9. Ianus, A., et al., Double Oscillating Diffusion Encoding and Sensitivity to Microscopic Anisotropy. Magn Reson Med, 2016. 78: p 550–564.

8. Jespersen, S.N., et al., Orientationally invariant metrics of apparent compartment eccentricity from double pulsed field gradient diffusion experiments. NMR Biomed, 2013. 26: p. 1647-62.

10. Ianus, A., et al. Time dependence of microscopic anisotropy in the mouse brain measured with double oscillating diffusion encoding (DODE) MRI. in Annual Meeting of the International Society for Magnetic Resonance in Medicine. 2017. Honolulu, HI, USA.

11. Drobnjak, I., et al., The matrix formalism for generalised gradients with time-varying orientation in diffusion NMR. J Magn Reson, 2011. 210: p. 151-7.

12. Ianus, A., D.C. Alexander, and I. Drobnjak, Microstructure Imaging Sequence Simulation Toolbox, in SASHIMI 2016, S.A. Tsaftaris, Editor. 2016, Springer. p. 34-44.

Figure 1: Schematic
representation of a) DODE diffusion sequences. Schematic representation of the
diffusion model for b) an individual microdomain and c) the ensemble average.

Table 1: Imaging and diffusion
parameters for ex-vivo mouse brain acquisition.

Figure 2: Apparent microscopic
anisotropy ($$$\tilde{µA}$$$^{2}) as
a function of b-value, corrected metric (μA^{2})
and ground truth values μA^{2}_{g.t.} for the following diffusion substrates with isotropically oriented microdomains: a) AstroZeppelins (cylindrically symmetric
tensors with D_{‖}=10^{-3}s/mm^{2}
and D_{⊥}=10^{-4}s/mm2);
b) AstroSticks (sticks with D_{‖} = 2∙10^{-3}s/mm^{2});
c) AstroCylinders (cylinders with D=2∙10^{-3}s/mm^{2} and
Gamma distributed radii of 2μm mean and shape 3); d) AstroZeppelins with a mixture of diffusivities (D_{‖}
= {5,10,10}∙10^{-4}s/mm^{2}, D_{⊥}={1,1,5}∙10^{-4}s/mm^{2} and corresponding f={0.2,0.5,0.3};
e) AstroFiniteSticks (sticks with an equal mixture of
lengths L={5,10,50}μm);
f) AstroFiniteCylinders (finite cylinders with Gamma
distributed radii and an equal
mixture of lengths L={5,10,50}μm).

Figure 3: Apparent microscopic
anisotropy maps ($$$\tilde{µA}$$$^{2})
for DODE sequences with N = 4 (200 Hz) and b values between 500 and 4000 s/mm^{2}.

Figure 4: a) Corrected microscopic
anisotropy maps (μA^{2})
and b) corresponding polynomial coefficient map (P_{3}) for the b^{3}
terms.