Filip Szczepankiewicz^{1,2} and Markus Nilsson^{1}

Asymmetric gradient waveforms enable efficient diffusion encoding but suffer from signal bias and image artifacts due to Maxwell terms (concomitant fields). We propose a novel method for generating "Maxwell-compensated" waveforms, and we demonstrate that such waveforms retain superior efficiency and exhibit negligible effects due to concomitant fields.

The Maxwell equations dictate
that linear magnetic field gradients, such as those used for diffusion sensitization,
are accompanied by spatially dependent concomitant gradients. In a spin-echo
sequence with the prescribed encoding gradient $$$\mathbf{G}(t)=[G_x(t)~G_y(t)~G_z(t)]^\text{T}$$$, these fields can lead to a non-zero residual moment at readout,
described by a wave vector^{5}

$$\mathbf{k}=\frac{\gamma}{2\pi}\left(\int_{\text{P1}}^{}\mathbf{E}(t)\text{d}t-\int_{\text{P2}}^{}\mathbf{E}(t)\text{d}t\right),\quad\text{Eq.1}$$

where integration is performed
over periods before/after the refocusing pulse (P1/P2), and the true gradient
$$$\mathbf{E}$$$ at location $$$\mathbf{r}=[x~y~z]^\text{T}$$$ is^{5}

$$\mathbf{E}(t,\mathbf{r})=\mathbf{G}(t)+\frac{1}{4B_0}\begin{vmatrix}G^2_z(t)&0&-2G_x(t)G_z(t)\\0&G^2_z(t)&-2G_y(t)G_z(t)\\-2G_x(t)G_z(t)&-2G_y(t)G_z(t)&4G_x^2(t)+4G_y^2(t)\end{vmatrix}\mathbf{r}. \quad\text{Eq.2}$$

Whenever $$$\mathbf{k}\neq0$$$, data may be corrupted by artifacts and signal attenuation
due to through-plane dephasing and T_{2}^{*} decay^{4}. We use the attenuation factor $$$(0\leq\text{AF}\leq1)$$$ as a metric of the signal bias caused by Maxwell terms, according to

$$\text{AF}=\text{AF}_\text{slice}\cdot\text{AF}_\text{phase}=\underbrace{|\text{sinc}(\mathbf{n}_\text{slice}\cdot\mathbf{k}\cdot\text{ST})|}_{\text{AF}_\text{slice}}\cdot\underbrace{\exp(-|\mathbf{n}_\text{phase}\cdot\mathbf{k}|\cdot{\Delta}t/({\Delta}k{\cdot}T_2^*))}_{\text{AF}_\text{phase}},\quad\text{Eq.3}$$

for a rectangular slice profile where $$$\mathbf{n}_\text{slice}$$$ is the normal unit vector, $$$\mathbf{n}_\text{phase}$$$ is a unit vector along the phase encoding direction, $$$\text{ST}$$$ is the slice thickness, $$${\Delta}k$$$ is the distance between k-space lines, and $$${\Delta}t$$$ is the echo spacing. Note that $$${\Delta}k=N/FOV_\text{p}$$$, where $$$N$$$ is the in-plane acceleration factor and $$$FOV_\text{p}$$$ is the field of view along $$$\mathbf{n}_\text{phase}$$$.

We seek a Maxwell-compensated gradient waveform $$$\mathbf{G}(t)$$$ for which ($$$\mathbf{k}=0$$$) so that $$$\text{AF}=1$$$. We propose to do this by minimizing the “Maxwell index”, defined here as the invariant scalar $$$m=(\text{Tr}[\mathbf{MM}])^{1/2}$$$, where

$$\mathbf{M}=\int_{\text{P1}}^{}\mathbf{G}(t)\mathbf{G}(t)^\text{T}\text{d}t-\int_{\text{P2}}^{}\mathbf{G}(t)\mathbf{G}(t)^\text{T}\text{d}t.\quad\text{Eq.4}$$

Importantly, $$$m$$$ is invariant to waveform rotation and when $$$m=0$$$ we know that $$$\mathbf{k}=0$$$ and $$$\text{AF}=1$$$ (unlike minimizing e.g. elements of $$$\mathbf{k}$$$). We include the minimization of $$$m$$$ in the
waveform optimization framework developed by Sjölund et al.^{1} (https://github.com/jsjol/NOW). Note that for sequences with multiple refocusing pulses, Eq.4 is extended with
additional integration periods.

Maxwell-compensated waveforms (Fig.1) that yield spherical tensor encoding (isotropic) were validated in an oil
phantom and in a healthy volunteer. Imaging was performed on a 3T MAGNETOM Prisma (Siemens Healthcare GmbH, Erlangen, Germany) with a prototype spin-echo sequence that enables arbitrary gradient waveforms; using TE=89 ms, TR=5 s, FOV=224×224 mm^{2}, slices=30, resolution=2×2×4 mm^{3},
iPAT=2, ∆t=0.65ms, partial-Fourier=6/8, and ten
equidistant b-values between 0.2–2.0 ms/µm^{2}. The waveforms were rotated according to Fig.1, and $$$\mathbf{n}_\text{phase}$$$ and $$$\mathbf{n}_\text{slice}$$$ were along the $$$y$$$ and $$$z$$$ axes.

Furthermore, we investigated the impact of Maxwell terms on several asymmetric waveforms found in litterature^{1-3,8,9}. The “worst case” within 100 mm of the isocenter was estimated by independently rotating the waveform and phase/slice
directions (kept orthogonal) until the minimal $$$\text{AF}$$$ was found. Analysis was performed in Matlab (The MathWorks, Natick, MA, USA).

The simulations in Fig.2 show that the non-compensated design suffers gros signal error, especially at locations further away from the isocenter. By contrast, Maxwell-compensated waveforms showed negligible effects in all voxels (Fig.2). Measurements in oil (Fig.3) and in brain tissue (Fig.4) confirmed the efficacy of Maxwell-compensation, and visualize the impact of non-compensated waveforms across the FOV. For the current setup, we suggest $$$m<100$$$ (mT/m)^{2}ms as a conservative threshold, for which $$$\text{AF}>0.999$$$ within 250 mm of the isocenter. The
Maxwell-compensated waveform required approximately 3.5 ms longer echo time compared to a similar non-compensated waveform. This cost is small compared to using mirror-symmetric (+18 ms) or repeated waveform designs (+40 ms)^{1}. Finally, Figure 5 shows that Maxwell-compensation can be achieved for
arbitrary b-tensor shapes^{7}, and that several asymmetric designs from literature may be affected by Maxwell terms.

Signal errors and artifacts caused by Maxwell terms
are a critical consideration in diffusion encoding with asymmetric waveforms. We have showed that such effects can be mitigated by minimizing
the Maxwell index, even for arbitrary waveform
rotations (encoding directions) and oblique phase/slice directions. Although prospective and post-hoc
corrections are possible in many scenarios^{4}, a bias-free acquisition may be preferable,
since an accurate correction requires detailed knowledge of the imaging sequence and tissue T_{2}^{*}. Maxwell-compensation does introduce a small penalty to the required encoding time, but it enables asymmetric waveforms with superior efficiency^{1}.

In conclusion, we have proposed a low-cost solution to Maxwell terms in asymmetric waveform design. This approach facilitates efficient diffusion encoding, requires no further correction, and is especially relevant for diffusion imaging at high gradient strength, low main magnetic field, large/oblique/off-isocenter FOV, and/or short T_{2}^{*}.

1. Sjölund et al., Constrained optimization of gradient waveforms for generalized diffusion encoding. J Magn Reson, 2015. 261: p. 157-168.

2. Reese et al., Reduction of eddy-current-induced distortion in diffusion MRI using a twice-refocused spin echo. Magn Reson Med, 2003. 49(1): p. 177-82.

3. Aliotta et al., Convex optimized diffusion encoding (CODE) gradient waveforms for minimum echo time and bulk motion-compensated diffusion-weighted MRI. Magn Reson Med, 2017. 77(2): p. 717-729.

4. Baron et al., The effect of concomitant gradient fields on diffusion tensor imaging. Magn Reson Med, 2012. 68(4): p. 1190-201.

5. Bernstein et al., Concomitant gradient terms in phase contrast MR: analysis and correction. Magn Reson Med, 1998. 39(2): p. 300-8.

6. Meier et al., Concomitant field terms for asymmetric gradient coils: consequences for diffusion, flow, and echo-planar imaging. Magn Reson Med, 2008. 60(1): p. 128-34.

7. Westin et al., Q-space trajectory imaging for multidimensional diffusion MRI of the human brain. Neuroimage, 2016. 135: p. 345-62.

8. Sörland et al., A Pulsed Field Gradient Spin-Echo Method for Diffusion Measurements in the Presence of Internal Gradients. J Magn Reson, 1999. 137(2): p. 397-401.

9. Moffat et al., Diffusion imaging for evaluation of tumor therapies in preclinical animal models. MAGMA, 2004. 17(3-6): p. 249-59.

Fig.1 - Examples of non-compensated and
Maxwell-compensated waveforms that yield spherical tensor encoding (isotropic
encoding). Both render the same b-value for an encoding time of 60 ms, but the
first one has a large residual moment due to concomitant fields (k has large
elements and large Maxwell index), whereas the novel design does not.

Fig.2 - The predicted signal bias in a whole brain FOV,
and worst-case voxel, is shown for the waveforms in Fig.1. This example assumes
in vivo imaging parameters, and no rotation of the waveform. Signal
is simulated assuming mean diffusivity of 1 µm^{2}/ms. The Maxwell-compensated waveform
exhibits negligible signal bias across the entire FOV.

Fig.3 - Signal data from an oil phantom shows that
non-compensated waveforms exhibit a gross signal bias, especially in the peripheral
parts of the FOV, whereas the Maxwell-compensated design exhibits no detectable
bias. Since the bias is relative to the signal value, it is easy to see in oil
where the attenuation due to diffusion is small, i.e. signal is high for all
b-values. Assuming T_{2}^{*} = 20 ms in the selected ROI (black
outline), we can accurately predict the expected attenuation due to Maxwell
terms.

Fig.4 - Signal and predicted error in healthy brain at *b*
= 2 ms/µm^{2} using waveforms depicted in Fig.1. If Maxwell terms exist, the central regions close to the isocenter (e.g. corpus callosum ROI, 34 mm from isocenter) may be negligibly
affected. By contrast, peripheral structures (e.g.
cerebellum ROI, 88 mm from isocenter) show relevant bias. Maxwell-compensated waveforms rendered
no detectable bias. Note that the bias can be imperceptible to the naked eye
(signal from both waveforms looks similar), nevertheless it is relevant for
quantitative methods, especially when multiple diffusion directions and/or oblique
phase/slice directions are considered (Fig.5).

Fig. 5 - Predicted “worst-case” attenuation factors for
asymmetric waveforms found in literature^{1-3,8,9} that yield *b* = 2
ms/µm^{2} at 80 mT/m assuming the imaging parameters from the methods
and tissue T_{2}^{* }= 40 ms. Worst bias due to through-slice
dephasing (AF_{slice}) and T_{2}^{*} decay (AF_{phase})
are found independently by rotating the waveform and phase/slice directions.
The bias due to dephasing is the largest at high b-values, but the T_{2}^{*}
decay may also contribute to gross bias. Maxwell-compensated waveforms showed negligible bias regardless of waveform rotation or slice/phase direction. Maxwell index $$$(m)$$$ is given in (mT/m)^{2}ms.