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The two-compartment diffusion “standard model” misestimates microscopic anisotropy in-vivo

Rafael Neto Henriques¹, Sune N Jespersen^2,3, and Noam Shemesh¹

¹Champalimaud Neuroscience Programme, Champalimaud Centre for the Unknown, Lisbon, Portugal, ²Center of Functionally Integrative Neuroscience (CFIN) and MINDLab, Clinical Institute, Aarhus University, Aarhus, Denmark, ³Department of Physics and Astronomy, Aarhus University, Aarhus, Denmark

Synopsis

Several microstructural models have been proposed to increase the specificity of diffusion MRI. However, improper model assumptions can compromise the accuracy of model estimates. Here, we compared model-independent metrics extracted from double diffusion encoding (DDE) with the metrics arising from the current (two-compartment) diffusion “standard model” (SM) in in-vivo rat brains. Our results revealed that SM produces overestimated microscopic anisotropy for both white and grey matter. These findings question the validity of SM and calls for future developments of more accurate models.

Introduction

Diffusion MRI (dMRI) is sensitive to microstructural tissue properties that are much smaller than the MRI voxel size¹. Several microstructural models have been proposed to increase the specificity of single-diffusion encoding (SDE) acquisitions^2-4; however, while these can produce very appealing maps, improper model assumptions and constraints can compromise the accuracy of model estimates^4-5. Fortunately, advanced diffusion MRI pulse sequences can be independently employed to extract specific measures decoupled from detrimental mesoscopic effects such as orientation dispersion^5-11, thereby independently mapping selected microstructural features. In turn, these can be used to validate microstructural models^12-14. For instance, Double Diffusion Encoding (DDE) very recently revealed that constrained two-component SM model fitting produces misestimation of microscopic fractional anisotropy (μFA) ex-vivo¹⁴. A fair criticism to this study suggests that perhaps fixation plays a crucial role in those findings. Here, we assess the validity of the more general two-compartment SM^2,4 in living rats.

Theory

DDE μFA measures: At long mixing time, the microscopic anisotropy $$$\left \langle V_\lambda{(\mathbf{D}_i))} \right \rangle$$$ can be estimated from “powder”-averaged DDE signals acquired for parallel and perpendicular pairs of diffusion encoding gradients ($$$S_\parallel$$$ and $$$S_\perp$$$):$$\log{S_\parallel/S_\perp}=\frac{3}{5}\left \langle V_\lambda{(\mathbf{D}_i))} \right \rangle b^2+O(b^3)\,\,\,(1)$$where $$$b$$$ is the total b-value in the two diffusion encodings. $$$\left \langle V_\lambda{(\mathbf{D}_i))} \right \rangle$$$ can be then used to compute8-10 from: $$\mu FA=\sqrt{\frac{3}{2}\frac{\left \langle V_\lambda{(\mathbf{D}_i))} \right \rangle}{\left \langle V_\lambda{(\mathbf{D}_i))} \right \rangle+MD^2}}\,\,\,\,\,(2)$$where $$$MD$$$ is the mean diffusivity.

Two-compartment “standard” model: The two-component SM consists of two Gaussian diffusion components^2,4: $$S_{SM}(b,\hat{n})=\int d\hat{r}ODF(\hat{r})K(b,\hat{n}\cdot\hat{r})\;\;\;\;\left(3\right)$$ and$$K\left(b,\xi\right)=fe^{-bD_i^\parallel\xi^2}+(1-f)e^{-bD_i^\parallel -b\left(D_e^\parallel D_e^\perp\right)\xi^2}\,\,\,\,\,(4)$$where $$$ODF$$$ is the fiber orientation distribution function (here represented up to the sixth order in spherical decomposition), $$$K$$$ is a kernel containing the following rotationally invariant parameters: 1) the intra-axonal volume fraction $$$f$$$; 2) the axial intra-cellular diffusivity $$$D_i^\parallel$$$; 3) the axial extra-cellular diffusivity $$$D_e^\parallel$$$; and 4) the axial intra-cellular diffusivity $$$D_i^\perp$$$ (intra-cellular radial diffusivity is set to 0, according to the narrow diameter assumption). After estimating all parameters from Eqs. 3-4, can be derived from SM (μ$$$FA^{SM}$$$) using Eq.2.

Methods

All animal experiments were pre-approved by the ethics committee operating under EU law (European Directive 2010/63). Data was acquired on N=2 female Long Evans Rats (11 weeks) on a 9.4 T Bruker Biospec MRI scanner equipped with an 86 mm quadrature coil for transmission and 4-element array cryocoil for reception.

SDE: Data were acquired along 60 gradient directions for nine evenly sampled b-values from 0 to 9 ms/μm² (Δ/δ=15/5ms).

DDE: Data were acquired for five b-values (1, 1.5, 2, 3, and 4 ms/μm², Δ=τ/δ=15/5ms). For each DDE b-value, directions are sampled according to the 5-design (12 parallel + orthogonal DDE acquisitions)¹⁰. Additionally, the number of DDE parallel acquisitions was increased by acquiring 45 parallel pairs of diffusion-gradient directions. Other acquisition parameters were as following: TR/TE=1500/65ms, 4 coronal slices, resolution=0.1×0.1×0.8mm, #averages=2.

Data processing: Both SDE and DDE data underwent the same preprocessing steps: 1)Marchenko-Pastur-PCA denoising (10x10 window)¹⁵, 2)sub-pixel image registration for data alignment¹⁶; 3)Gibbs-artefact suppresion¹⁷. SM parameters were fitted from Eqs 3-4 using a non-linear procedure which was repeated for different starting points. The lowest sum of residuals was used to select the final set of parameters, which were then and converted to μFA (Eq.2). μFA estimates from DDE are extracted from Eqs.1-2 (considering the higher order term correction¹¹).

Results

Before pre-processing the SNR of the b=0 images was estimated to be ~40 for both DDE and SDE data. For a representative slice, Fig. 1 presents SM parameter estimates for both rats. µ$$$FA^{SM}$$$ is compared to its DDE counterparts in Fig. 2. To assess the estimates in higher SNR regimes (decreasing noise cofounds), Fig. 3 shows averaged μFA for white and grey-matter ROIs of individual slices. Averaged µ$$$FA^{SM}$$$ estimates were significantly higher than their DDE counterparts (for white and grey-matter p-values =1.8e-2 and 2.5e-6, pairwise t-test).

Discussion

The 2-component SM has been quite extensively employed, but its underpinnings were not thoroughly validated. Comparisons of SM metrics against their counterparts derived using model-free methods such as DDE has previously been presented for ex-vivo data¹⁴, suggesting that SM is incomplete. Here, we extended the analysis to in-vivo experiments not confined to powder-averaged (L=0) SM fitting, and found that µ$$$FA^{SM}$$$ was overestimated, in agreement with the previous ex-vivo studies¹⁴. The DDE acquisitions assume that experiments are in the long mixing time regime, which was indeed observed here independently (data not shown). Therefore, ignoring exchange, DDE can serve as a ground truth. Our results do not point out the “culprit” for the differences between DDE and SM; still, our findings call for more advanced biophysical modelling.

Conclusion

DDE-based measurements revealed that the “standard” two-compartments model overestimates . Since µFA is produced model-free using DDE, better models need to be derived to describe diffusion in the brain.

Acknowledgements

This study was funded by the European Research Council (ERC) (agreement No. 679058).

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Figures

Figure 1. In-vivo SM parametric maps for rat 1 (A) and rat 2 (B): axonal volume fraction (A1 and B1); intra-cellular axonal diffusivity (A2 and B2); extra-cellular axonal diffusivity (A3 and B3); and extra-cellular radial diffusivity (A4 and B4). Maps are plotted for a representative slice.

Figure 2. Comparison between uFA estimates from SM and DDE for rat 1 (A) and rat 2 (B): Maps of uFA estimates from SM (A1 and B1); maps of μFA estimates from DDE and high-order-corrected (A2 and B2). μFA estimates from SM plotted as a function of the μFA estimates from DDE (A3 and B3). Maps are plotted for a representative slice; however, scatter plots are produced for the voxels of all slices. For reference, the identity and regression lines are marked by the red dashed and black solid lines in panels A3 and B3.

Figure 3 – Results of the averaged uFA estimates extracted from white and grey matter maps: A) White matter mask for a representative slice; B) Grey matter mask for a representative slice. C) Average white and grey matter μFA values for both SM and DDE. In the latter panel, individual points correspond to the averages obtained for different slices (all slices of all Rat were considered). Slice μFA averages for SM were significantly higher than the averages for DDE (p-values=1.8e-2 for WM and p-value=2.5e-6 for GM, using pairwise t-test).

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)

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