Synopsis

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.

Introduction

Theory

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 vector⁵

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

where integration is performed over periods before/after the refocusing pulse (P1/P2), and the true gradient $$$\mathbf{E}$$$ at location $$$\mathbf{r}=[x~y~z]^\text{T}$$$ is⁵

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

Whenever $$$\mathbf{k}\neq0$$$, data may be corrupted by artifacts and signal attenuation due to through-plane dephasing and T₂^* decay⁴. 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 $$$\mathbf{n}_\text{phase}$$$.

Methods

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 mm², slices=30, resolution=2×2×4 mm³, iPAT=2, ∆t=0.65ms, partial-Fourier=6/8, and ten equidistant b-values between 0.2–2.0 ms/µm². 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).

Results

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)²ms 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 shapes⁷, and that several asymmetric designs from literature may be affected by Maxwell terms.

Discussion and conclusion

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 scenarios⁴, a bias-free acquisition may be preferable, since an accurate correction requires detailed knowledge of the imaging sequence and tissue T₂^*. Maxwell-compensation does introduce a small penalty to the required encoding time, but it enables asymmetric waveforms with superior efficiency¹.

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 T₂^*.

Acknowledgements

References

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