Michael Paquette^{1}, Chantal M.W. Tax^{2}, Cornelius Eichner^{1}, and Alfred Anwander^{1}

^{1}Neuropsychology, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany, ^{2}CUBRIC, School of Physics, Cardiff University, Cardiff, United Kingdom

We investigate the effect of gradient non-linearities (GNL) on free gradient waveform used for B-tensor diffusion encoding. We show the magnitude of the GNL-bias for strong gradients of $$$300 m\text{T}/\text{m}$$$. We derive a closed-form formula of the voxelwise B-tensor under GNL, independent of the choice of gradient waveform used to encode the B-tensor.

B-tensor encoding is a recent diffusion imaging framework opening new ways to probe the tissues microstructure

From the desired gradient waveform $$$\vec{G}(t)$$$, we define the q-vector as $$$\vec{q}(t) = \gamma \int_0^t \vec{G}(t') \, \text{d}t'$$$. The B-tensor is computed as $$$\mathbf{B} = \int_0^{t_{tot}} \vec{q}(t) \otimes \vec{q}(t) \, \text{d}t$$$ i.e. $$$B_{ij} = \int_0^{t_{tot}} q_i(t)q_j(t) \, \text{d}t$$$ for $$${i,j \in \{x,y,z\}}$$$. The actual gradient waveform under GNL at spatial position $$$\vec{r}$$$ is $$$\vec{G_a}(t) = \mathbf{L}(\vec{r}) \cdot \vec{G}(t)$$$ (fig 1), and the actual q-vector is $$$\vec{q_a}(t) = \mathbf{L} \cdot \vec{q}(t)$$$. Finally, the GNL-distorted actual B-tensor is given by $$$\mathbf{B}_a = [(B_a)_{ij}]$$$ for $$$i,j \in \{x,y,z\}$$$$$\begin{align*}(B_a)_{ij} &= L_{ix} L_{jx} B_{xx} + (L_{ix} L_{jy} + L_{iy} L_{jx}) B_{xy} + (L_{ix} L_{jz} + L_{iz} L_{jx}) B_{xz}\\ & \qquad + L_{iy} L_{jy} B_{yy} + (L_{iy} L_{jz} + L_{iz} L_{jy}) B_{yz} + L_{iz} L_{jz} B_{zz}\end{align*}$$

To showcase the effect of GNL on the desired B-tensor sampling scheme, we generated Maxwell-compensated waveforms

$$\begin{bmatrix}b_{1} & 0 & 0\\0 & b_{2} & 0\\0 & 0 & b_{3}\end{bmatrix}=\frac{b_S}{3}\begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}+\frac{b_P}{2}\begin{bmatrix}0 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}+b_L\begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 1\end{bmatrix}$$

To assess the effect of GNL on estimated diffusion parameters, we simulated a voxel representing fanning white matter pathways. We generated a distribution of diffusion tensors with diffusivities (1.5, 0.3, 0.3) $$$\mu\text{m}^2/m\text{s}$$$ and orientations sampled uniformely in a cone of 30$$$^{\circ}$$$ aperture, centered at direction $$$\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)$$$. We simulated the signal for a uniformly sampled subset of GNL tensors across the entire brain volume. We used the QTI approach

MP is supported by a scholarship (PDF-502732-2017) from the Natural Sciences and Engineering Research Council of Canada (NSERC).

CMWT is supported by a Rubicon grant (680-50-1527) from the Netherlands Organisation for Scientific Research (NWO) and a Sir Henry Wellcome Fellowship (215944/Z/19/Z).

CE is supported by the SPP2041 program "Computational Connectomics" of the German Research Foundation (DFG).

[1] R. Bammer, M. Markl, A. Barnett, B. Acar, M. Alley, N. Pelc, G. Glover, and M. Moseley, “Analysis and generalized correction of the effect of spatial gradient field distortions in diffusion-weighted imaging”, Magnetic Resonance in Medicine, vol. 50, pp. 560–9, 2003.

[2] S. N. Sotiropoulos, S. Jbabdi, J. Xu, J. L. Andersson, S. Moeller, E. J. Auerbach, M. F. Glasser, M. Hernandez, G. Sapiro, M. Jenkinson, D. A. Feinberg, E. Yacoub, C. Lenglet, D. C. V. Essen, K. Ugurbil, and T. E. Behrens, “Advances in diffusion mri acquisition and processing in the human connectome project”, NeuroImage, vol. 80, pp. 125–43, 2013.

[3] H. Y. Mesri, S. David, M. A. Viergever, and A. Leemans, “The adverse effect of gradient nonlinearities on diffusion mri: From voxels to group studies”, NeuroImage, vol. 205, pp. 116–27, 2019.

[4] J. Jovicich, S. Czanner, D. Greve, E. Haley, A. van der Kouwe, R. Gollub, D. Kennedy, F. Schmitt, G. Brown, J. MacFall, B. Fischl, and A. Dale, “Reliability in multi-site structural mri studies: Effects of gradient non-linearity correction on phantom and human data”, NeuroImage, vol. 30, no. 2, pp. 436–43, 2006.

[5] Z. Nagy, N. Weiskopf, D. C. Alexander, and R. Deichmann, “A method for improving the performance of gradient systems for diffusion-weighted mri”, Magnetic Resonance in Medicine, vol. 58, no. 4, pp. 763–8, 2007.

[6] S. Mohammadi, Z. Nagy, H. E. Möller, M. R. Symms, D. W. Carmichael, O. Josephs, and N. Weiskopf, “The effect of local perturbation fields on human dti: Characterisation, measurement and correction”, NeuroImage, vol. 60, no. 1, pp. 562–70, 2012.

[7] M. Paquette, C. Eichner, and A. Anwander, “Gradient non-linearity correction for spherical mean diffusion imaging”, Proc. Int. Soc. Magn. Reson. Med., vol. 27, no. 550, 2019.

[8] D. Topgaard, “Multidimensional diffusion mri”, Journal of Magnetic Resonance, vol. 275, pp. 98–113, 2017.

[9] C.-F. Westin, H. Knutsson, O. Pasternak, F. Szczepankiewicz, E. Özarslan, D. van Westen, C. Mattisson, M. Bogren, L. J. O’Donnell, M. Kubicki, D. Topgaard, and M. Nilsson, “Q-space trajectory imaging for multidimensional diffusion mri of the human brain”, NeuroImage, vol. 135, pp. 345–62, 2016.

[10] J. Sjölund, F. Szczepankiewicz, M. Nilsson, D. Topgaard, C.-F. Westin, and H. Knutsson, “Constrained optimization of gradient waveforms for generalized diffusion encoding”, Journal of Magnetic Resonance, vol. 261, pp. 157–6, 2015.

[11] F. Szczepankiewicz, C.-F. Westin, and M. Nilsson, “Maxwell-compensated design of asymmetric gradient waveforms for tensor-valued diffusion encoding”, Magnetic Resonance in Medicine, vol. 82, pp. 1424–37, 2019.

Figure 1: Gradient waveform and q-vector for a spherical B-tensor ($$$b = 1.97\,m\text{s}/\mu\text{m}^2$$$, $$$\text{G}_{\text{max}}=170\,m\text{T}/\text{m}$$$). The dashed lines show the actual curves under the effect of a "strong" GNL tensor $$$\mathbf{\text{L}}$$$ (approximately the top 10% of voxels for typical brain at Connectom scanner are at least this affected). The actual B-tensor $$$\left(\mathbf{B}_a\right)$$$ has an inflated b-value and is not spherical.

Figure 2: (A - C) Spatial distribution of the gradient non-linearities on a representative brain volume on the Connectom MRI system. We used the Frobenius distance between the gradient non-linearity tensors and the identity matrix as a indicator for GNL severity $$$\left(\text{GNL}_{\text{str}}\right)$$$. (D) Histogram of $$$\text{GNL}_{\text{str}}$$$ over the full brain mask.

Figure 3: Deviations from the desired B-tensor acquisition scheme described in the experiment section for full brain volume incorporating GNL. (A) Max-normalized histogram for each b-value, showing a distribution of b-values up to $$$\pm 15\%$$$ around the desired b-values (vertical red lines). (B - D) For each type B-tensor shape (spherical, planar, linear), we show the spread of their respective normalized shape metrics $$$\left(b_S/b,\,b_P/b,\,b_L/b\right)$$$^{8}.

Figure 4: The GNL tensors for the Connectom, clustered into 100 centroids tensors within a brain mask. Diffusion signal for Fanning WM was simulated using the actual B-tensors (**B**_{a}) at each centroids. The simulated signals were reconstructed using QTI^{9} with GNL corrections (using **B**_{a}) and without (using **B**). (A) The error on the estimated ensemble-averaged diffusion tensor $$$(\langle\mathbf{D}\rangle)$$$. (B) The error on the estimated tensor covariance matrix $$$(\mathbb{C})$$$. The reconstruction errors ignoring GNL show strong correlation with the GNL severity (GNL_{str}).