Synopsis

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.

Introduction

Methods

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 μA² denotes the corrected microscopic diffusion anisotropy metric and P₃ 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 (μA²_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^oC during scans. Acquisition parameters are detailed in Table 1, and the analysis is performed using normalised signals.

Results

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 µA² 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/mm²) 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 (P₃) corresponding to the third order term in b.

Discussion

Conclusions

Acknowledgements

References

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