Michael Paquette^{1}, Cornelius Eichner^{1}, and Alfred Anwander^{1}

Gradient non-linearities are a significant source of errors in MRI systems with strong gradients. In the case of diffusion imaging, they induce spatial deviation of the b-vectors. The spherical mean methods in diffusion relies on the acquisition of spherical b-shell. To recover accurate spherical mean values, it is necessary to undistort the diffusion signal. Therefore, we evaluated three correction methods for gradient non-linearities using the Connectom gradient system as a showcase. We show how a simple heuristic can reduce the spherical mean errors by 20 folds.

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1. Distribution of the b-values and gradient direction deviations of the Connectom gradient system in a representative brain volume. Large portions of voxels are affected by b-value deviations of at least 5%. The orientation deviations are much milder, peaking around 2 degrees. These orientation deviations are mostly negligible when taking into account the spherical mean.

2. Spatial distribution of the gradient non-linearity on a representative brain volume with the Connectom system. The three planes are selected as the center of their respective axis. The graph is the cumulative distribution of the gradient non-linearity (GNL) score over the full brain. The scores are a pseudo distance between GNL tensor and the identity matrix. It is computed as the $$$\ell_2$$$-norm of the singular values of the GNL tensor minus 1 (i.e. $$$\|sv(T) - \vec{1} \|_2$$$). This score penalizes positives and negatives gradient deviations equally, does not prioritize any axis and captures information from the cross-terms.

3. Spherical mean (SM) values for a single microstructure configuration. The x-axis corresponds to different orientations of the model with the same orientation distribution (OD) shape (small subset shown under the axis). The curves correspond to this model computed with different gradient non-linearity (GNL). The undulation in the curves showcase OD dependence of the SM in the presence of GNL. The red horizontal line shows the theoretical SM value in the absence of GNL. The remaining horizontal lines represent the theoretical SM value at the mean b-value for each respective GNL.

4. Error in the spherical mean (SM) value for the first-order correction (A) and second-order correction (B) compared to the uncorrected error. The errors are computed as $$$\frac{|\text{SM after correction} - \text{theoretical SM}|}{|\text{uncorrected SM} - \text{theoretical SM}|}$$$. Errors smaller than 1 (shown in blue) corresponds to configurations were the signal interpolation correction increased the SM accuracy over the uncorrected case.

5. Error of the spherical mean (SM) value for the mean b-value method compared to the uncorrected error. The errors are computed as $$$\frac{|\text{uncorrected SM} - \text{theoretical SM for mean b-value}|}{|\text{uncorrected SM} - \text{theoretical SM for undistorted b-value}|}$$$. Errors smaller than 1 corresponds to configurations were the correction increased the SM accuracy over the uncorrected case. The mean b-value errors are around 20x smaller than the uncorrected errors.