Filip Szczepankiewicz^{1,2,3}, Cornelius Eichner^{4}, Alfred Anwander^{4}, Carl-Fredrik Westin^{1,2}, and Michael Paquette^{4}

^{1}Harvard Medical School, Boston, MA, United States, ^{2}Radiology, Brigham and Women's Hospital, Boston, MA, United States, ^{3}Clinical Sciences Lund, Lund University, Lund, Sweden, ^{4}Neuropsychology, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany

Gradient non-linearity distorts the shape of the gradient waveform used for diffuison encoding. This distortion also compromises Maxwell compensation in asymmetric gradient waveforms. We show that one of two strategies for Maxwell compensation fails under non-linear gradiets (K-nulling) whereas M-nulling is immune to this effect.

$$$~~~~~$$$The aim of this study was to investigate how GNLs affects Maxwell-compensated asymmetric gradient waveforms. We demonstrate the effects at the Connectom system, where ultra-high gradient amplitudes and severe non-linearities exacerbate the concomitant gradients and their impact on signal accuracy.

$$$~~~~~$$$Gradient waveforms for tensor-valued diffusion encoding can be optimized using two different types of Maxwell compensation, namely socalled M- and K-nulling (2). Briefly, $$$\mathbf{M}$$$ and $$$\mathbf{K}$$$ are matrices that capture the magnitude of the concomitant gradients, calculated from the desired gradient waveform $$$(\mathbf{g}_\mathrm{d}(t)=[g_x(t)~g_y(t)~g_z(t)]^T)$$$, according to (2)

$$\mathbf{K}\propto\int_0^{\tau}~h(t)\begin{bmatrix}g_z^2(t)&0&-2g_x(t)g_z(t)\\0&g_z^2(t)&-2g_y(t)g_z(t)\\-2g_x(t)g_z(t)&-2g_y(t)g_z(t)&4g_x^2(t)+4g_y^2(t)\\\end{bmatrix}\mathrm{d}t$$

and

$$\mathbf{M}\propto\int_0^{\tau}~h(t)\begin{bmatrix}g_x^2(t)&g_x(t)g_y(t)&g_x(t)g_z(t)\\g_y(t)g_x(t)&g_y^2(t)&g_y(t)g_z(t)\\g_z(t)g_x(t)&g_z(t)g_y(t)&g_z^2(t)\\\end{bmatrix}\mathrm{d}t$$

where integration is over the duration of the diffusion encoding, $$$t$$$ is time, and $$$h(t)$$$ is a function that shows the direction of spin dephasing. We note that both matrices are proportional to the square of the gradient waveform, showing that the problem increases with increasing gradient amplitude. If $$$\mathbf{M}$$$ and $$$\mathbf{K}$$$ are zero, the effect of concomitant gradients is negligible. Nulling $$$\mathbf{M}$$$ during waveform optimization ensures negligible effects from concomitant gradients (Maxwell compensation) for arbitrary rotations of the waveform, whereas nulling $$$\mathbf{K}$$$ is more efficient, but does not allow for arbitrary rotation. Since spherical encoding does not require rotations, this restriction is circumvented, facilitating the highest efficiency for K-nulling.

$$$~~~~~$$$We estimate the impact of GNL on the Maxwell compensation by simulating imperfect gradient waveforms using gradient coil tensors from the Connectom system (4). After optimizing the desired gradient waveform $$$(\mathbf{g}_\mathrm{d})$$$ tailored to the system (2,5), the actual gradient waveform $$$(\mathbf{g}_\mathrm{a})$$$ is calculated as $$$\mathbf{g}_\mathrm{a}(\mathbf{r},t)=\mathbf{L}(\mathbf{r})\mathbf{g}_\mathrm{d}(t)$$$ at position $$$\mathbf{r}$$$ (1). The signal error is estimated in the 'concomitant gradient analysis toolbox' (2,6,7) assuming a spin-echo with EPI readout using the parameters detailed in Table 1.

[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.” MRM, 2003.

[2] F. Szczepankiewicz, C.-F. Westin, and M. Nilsson, “Maxwell-compensated design of asymmetric gradient waveforms for tensor-valued diffusion encoding.” MRM, 2019.

[3] C. A. Baron, R. M. Lebel, A. H. Wilman, and C. Beaulieu, “The effect of concomitant gradient fields on diffusion tensor imaging.” MRM, 2012.

[4] M. Paquette and C. M. W. Tax, “https://github.com/mpaquette/gnlcwaveform” GitHub code repository, 2019.

[5] J. Sjölund, F. Szczepankiewicz, M. Nilsson, D. Topgaard, C.-F. Westin, and H. Knutsson, “Constrained optimization of gradient waveforms for generalized diffusion encoding.” JMR, 2015.

[6] M. Nilsson, F. Szczepankiewicz, B. Lampinen, A. Ahlgren, J. D. A. Martins, S. Lasic, C.-F. Westin, and D. Topgaard, “An open-source framework for analysis of multidimensional diffusion MRI data implemented in Matlab.” Proc. Int. Soc. Magn. Reson. Med., 2019.

[7] F. Szczepankiewicz and M. Nilsson, “https://github.com/markus-nilsson/md-dmri/tree/master/tools/cfa” GitHub code repository, 2019.

Figure 1 - Examples of gradient waveforms that yield spherical tensor encoding (top row) with M- and K-nulling optimized for the Siemens Connectom scanner. The bottom row shows the perturbed gradient waveforms, the concomitant gradient waveform, and the k-trajectories of the concomitant gradient waveform (caused by non-linear gradients). In this example, **T**(**r**) is evaluated at position **r** = [0.1, 0.1, 0.1] m). M-nulling yields waveforms that are robust to GNL, whereas K-nulling does not (inset plots show non-zero k-vector at end of the encoding).

Table 1 - Parameters used in the simulation of the signal bias caused by GNLs and concomitant gradients. The sequence is assumed to be a spin-echo with a rectangular slice selection profile and echo-planar imaging readout. More information on how the error is simulated can be found in reference (2).

Figure 2 - When the waveforms are Maxwell-compensated by K-nulling, using a slice thickness of 2 mm, the relative signal errors within the brain mask reach approximately 40%. For M-nulling, the errors were negligible throughout the volume (data not shown). Top to bottom rows show sagittal, coronal and transversal slices, respectively.

Figure 3 - Boxplots show the distribution of relative signal error for M- and K-nulling at slice thicknesses between .1 and 10 mm in a cubic FOV that is 25 cm on each side. The top row shows that M-nulling is expected to exhibit negligible signal loss (worst error is below 0.006%). By contrast, K-nulling shows gross signal errors, reaching complete signal loss at a slice thickness above 1 mm, and as expected from theory (2,3), thicker slices increase the magnitude of the error.