Gradient Non-Linearity Correction for Spherical Mean Diffusion Imaging
Michael Paquette1, Cornelius Eichner1, and Alfred Anwander1

1Neuropsychology, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany


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.


Gradient non-linearity (GNL) in MRI systems induces spatially-varying deviations from the expected gradient direction and intensity1. These deviations are well described for each gradient system and present a real issue, especially for strong gradient systems2. GNL induce geometric distortions and modify the b-values as well as directions of the applied diffusion gradients (DG) at each voxel. Many diffusion models (e.g. DTI3, DKI4, MAPMRI5) can directly correct for this effect by using the voxelwise GNL adjusted b-values and directions in the model fit. Unfortunately, models depending on shells in b-vector space cannot. One category of such models are the ones which rely on the Spherical Mean (SM) of the diffusion signal. These models fit their parameters on the mean signal from a set of uniformly distributed gradient directions at the same b-value to remove sensitivity to the orientation distribution (OD) from the signal. In the presence on GNL, the SM of the signal is no longer independent of the underlying OD (Figure 3). A natural strategy for correction is to model the signal decay and interpolate to the desired b-shell. One could also interpret the signal as a different b-value that better captures the underlying parameters. In this work, we show the severe effects of GNL on SM and explore options for correction thereof.


GNL of the MRI system (3T Connectom System, Siemens Healthineers, Erlangen, Germany) were approximated using solid spherical harmonics (order $$$\le$$$ 15) provided by the vendor. For a given dataset, we computed a GNL tensor (GNLT) $$$T$$$ at every voxel $$$i$$$ such that $$$\tilde{g}(i)=T(i)\cdot g$$$ where $$$g$$$ is the planed DG and $$$\tilde{g}$$$ is the spatially varying DG resulting from GNL. Typically, the effects of GNL are more pronounced at the edges of the volume because the severity of GNL depends mainly on the distance to isocenter1 (figure 2). Figure 1 shows the GNL induced distribution of b-values and direction deviations inside a brain mask of a typical scan on the Connectom system. We explored two modeling strategies for the diffusion signal decay, first-order and second-order. The first-order model fits $$$D_n$$$ at each DG on a single shell such that signal $$$S_n=\exp(-\|\tilde{g}_n\|^2D_n)$$$. The second-order model fits $$$D_n$$$ and $$$K_n$$$ at each DG on a pair of shell such that signal $$$S_n=\exp(-\|\tilde{g}_n\|^2D_n + (\|\tilde{g}_n\|^2D_n)^2K_n/6)$$$. Both models were applied to interpolate a corrected signal for $$$g_n$$$ and spherical averaging was performed. Additionally, we explored a heuristic approach of calculating the mean b-value on the distorted shell and interpreting the signal as an undistorted shell of the mean b-value. To evaluate the accuracy of GNL corrections, we generated a synthetic diffusion signal for multiple microstructural parameters, ODs and GNLTs. A realistic sample of the GNLT space was obtained by clustering the GNLTs inside a representative brain volume. The ODs were generated from mixture of Von-Mises-Fisher distributions for various orientations and concentration parameter $$$k\in [1,16]$$$. We used a microstructure model composed of one stick and one zeppelin compartment modeled by only two free parameters (intra-axonal volume fraction $$$v_{int}$$$ and intra-axonal parallel diffusivity $$$\lambda_{int}$$$). This microstructure model convolved with any OD has a closed form formula for 6 We used $$$\lambda_{int}\in [0.5,2.5]\frac{\mu m^2}{ms}$$$ and $$$v_{int}\in [0.1,0.9]$$$ with b-values ranging from 1 to 20 $$$\frac{ms}{\mu m^2}$$$ in the simulations.


The quality of the first-order and second-order signal approximation is mostly dictated by the product of diffusivity and b-value. Therefore, the first-order recovery (figure 4A) produces spurious results when $$$bD>5$$$. Similarly, the second-order recovery (figure 4B) produces spurious results when using two distant b-values. In practice, it would hard to determine whether or not these models are usable because we do not know the underlying diffusivities. The mean b-value method (figure 5) outperforms the two signal interpolation scheme for all b-values and appear to provide stable error on all configurations. The mean b-value heuristic reduces the GNL bias by 20-fold compared to the uncorrected error.


In conclusion, the effects of GNL cannot be ignored when using high strength gradient systems. The anisotropic nature of GNL threatens the separation of the orientation dispersion and diffusion microstructure parameters assumption of SM models. Interpolating the signal to correct the GNL is possible but depends heavily on the underlying diffusion parameters. The simple heuristic of using a shifted b-value and ignoring the orientation dependence reduce the error greatly.


MP is supported by a NSERC scholarship (PDF-502732-2017) from Canada.


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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.

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