Hamed Y. Mesri^{1}, Martijn Froeling^{2}, Max A. Viergever^{1}, Anneriet M. Heemskerk^{1}, and Alexander A. Leemans^{1}

Nonuniformities of gradient magnetic fields in diffusion-weighted MRI can introduce systematic errors in estimates of diffusion measures. While there are correction methods that can compensate for these errors, as presented in the Human Connectome Project, such non-linear effects are assumed to be negligible for typical applications and, hence, gradient nonuniformities are mostly left uncorrected. In this work, we evaluated the effect of ignoring such diffusion gradient nonuniformities on measures derived from diffusion tensor imaging. In particular, we simulated deviations from the ground-truth in terms of b-value and diffusion gradient orientation and investigated the resulting bias in fractional anisotropy and orientation of the first eigenvector. Our results demonstrate that not including a correction strategy to mitigate diffusion gradient imperfections especially for high quality data may lead to a significant bias for diffusion measure estimates.

**BACKGROUND AND PURPOSE **

**METHODS **

The nonuniformities of gradient magnetic fields result in deviations in the applied gradients [2]. This deviation in the diffusion gradients is decomposed into its angular (α) and magnitude, i.e., b-value, (β) components, i.e.,

$$\alpha=arccos \bigg(\frac{\textbf{g}^T_{nom}(\textbf{r})\textbf{g}_{act}(\textbf{r})}{|\textbf{g}_{nom}(\textbf{r})||\textbf{g}_{act}(\textbf{r})|}\bigg)$$and$$\beta=\big(\frac{|\textbf{g}_{nom}(\textbf{r})|}{|\textbf{g}_{act}(\textbf{r})|} -1\big)×100 $$where $$$\textbf{g}_{act}(\textbf{r})$$$ and $$$\textbf{g}_{nom}(\textbf{r})$$$ represent the actual and nominal gradients at position $$$\textbf{r}$$$, respectively. $$$|\textbf{g}|$$$ denotes the b-value. In this work, we use the methodology presented in [2] to model the nonuniformities in the magnetic field gradient, i.e.,$$\textbf{g}_{act}(\textbf{r})=\textbf{L}(\textbf{r}) \textbf{g}_{nom}(\textbf{r})$$ Here, $$$\textbf{L}(\textbf{r})$$$ denotes the gradient coil tensor at position $$$\textbf{r}$$$, which can be used to calculate the actual gradient vector for each voxel. Figs.1(a-b) show the histograms and spatial variations of the gradient magnitude and angular deviations of the gradient orientation in an example DWI dataset from the Human Connectome Project (HCP) [7]. In order to investigate how the deviations in the gradients propagate into the variations in diffusion estimates, we perform Monte-Carlo simulations from diffusion tensor model [8] with the following parameter settings which are in line with configurations observed in the HCP data set (see Figs.1(a-b)): mean diffusivity (MD)=0.7×10**RESULTS**

**DISCUSSION**** AND ****CONCLUSION **

[1] Conturo, T.E., McKinstry, R.C., Aronovitz, J.A., Neil, J.J.: ‘Diffusion MRI: Precision, accuracy and flow effects’NMR Biomed., 1995, 8, (7), pp. 307–332.

[2] Bammer, R., Markl, M., Barnett, A., et al.: ‘Analysis and generalized correction of the effect of spatial gradient field distortions in diffusion-weighted imaging’Magn. Reson. Med., 2003, 50, (3), pp. 560–569.

[3] Sotiropoulos, S.N., Jbabdi, S., Xu, J., et al.: ‘Advances in diffusion MRI acquisition and processing in the Human Connectome Project’Neuroimage, 2013, 80, pp. 125–143.

[4] Borkowski, K., Klodowski, K., Figiel, H., Krzyzak, A.T.: ‘A theoretical validation of the B-matrix Spatial Distribution approach to Diffusion Tensor Imaging’Magn. Reson. Imaging, 2016.

[5] Janke, A., Zhao, H., Cowin, G.J., Galloway, G.J., Doddrell, D.M.: ‘Use of spherical harmonic deconvolution methods to compensate for nonlinear gradient effects on MRI images’Magn. Reson. Med., 2004, 52, (1), pp. 115–122.

[6] Malyarenko, D.I., Ross, B.D., Chenevert, T.L.: ‘Analysis and correction of gradient nonlinearity bias in apparent diffusion coefficient measurements’Magn. Reson. Med., 2014, 71, (3), pp. 1312–1323.

[7] Van Essen, D.C., Ugurbil, K., Auerbach, E., et al.: ‘The Human Connectome Project: A data acquisition perspective’Neuroimage, 2012, 62, (4), pp. 2222–2231.

[8] Basser, P.J., Mattiello, J., LeBihan, D.: ‘MR diffusion tensor spectroscopy and imaging.’Biophys J, 1994, 66, (1), pp. 259–267.

[9] Jones, D.K., Horsfield, M.A., Simmons, A.: ‘Optimal strategies for measuring diffusion in anisotropic systems by magnetic resonance imaging.’Magn. Reson. Med., 1999, 42, (3), pp. 515–25.

Fig.1:
Histograms of deviations of the magnetic field gradients and the resulting
differences in DWI parameters for a healthy subject from the Human Connectome
Project (HCP); (a) Magnitude deviation (β); (b) Angular deviation (α);
(c) Percentage differences in FA (PD_{F}A) (d) Orientation difference
in the first eigenvector (OD_{FE}) from Diffusion Tensor Imaging (DTI).
The more the distance from the center of the coil, the more the deviations.

Fig.2: Percentage difference for FA (PD_{FA}) as a function of β at three different SNR levels for different combinations of
FA and α. PD_{FA} is higher at lower FA Values and lower SNR levels. The
effect of α and β is swamped by noise at lower SNR levels. Notice that the
offset due to α is more visible at higher
SNR levels or higher FA values. MD
is set at 0.7×10^{-3} mm^{2}/s. The y-axes are
differently scaled.

Fig.3: Coefficient of variation for FA as function of β at three α values for different combinations of FA and SNR. The dispersion (coefficient of variation) in FA is larger for lower FA values and increases as the SNR decreases or α and |β| increase. The effect of α is swamped by noise at lower SNR levels. MD is set at 0.7×10^{-3} mm^{2}/s. Notice that the y-axes are differently scaled.

Fig.4: First eigenvector (FE) orientation difference (OD_{FE}) as a function of α at three different SNR levels for different
combinations of FA and β. The effect of α is lower at lower FA values and lower
SNR levels. Notice the offset due to β at the noise-free case. The effect of β
is dominant over the effect of α at lower FA values. MD is set at 0.7×10^{-3}
mm^{2}/s. The y-axes are differently scaled.