INTRODUCTION

Diffusion weighted imaging (DWI) is a prevalent strategy for applications from acute brain ischemia, to tumor detection and monitoring, to tractography¹. A spin-echo DWI sequence consists of large diffusion encoding gradients around the refocusing pulse followed by image readout (Figure 1c). The powerful diffusion gradient lobes can lead to mechanical vibrations in the coils^2,3 caused by the large Lorentz forces acting on them. This vibration is exacerbated when gradient switching occurs at coil mechanical resonant frequencies. Stronger vibrations are more likely to cause signal dropout, trajectory errors, loud acoustic waves, and damage to the gradient system^2–5. The design of time-optimal diffusion lobes has been shown to be convex and has even been extended to minimize induced eddy currents^6,7. Here, a related approach is developed to design time-optimal diffusion lobes with a constrained frequency spectrum to avoid mechanical resonance frequencies.

METHODS

The design of safe time-optimal diffusion gradients is adapted from the CODE method⁶. Here maximizing the objective is equivalent to maximizing the b-value of the finite duration waveform but with a convex form. The allowed duration of the lobe (N) is iterated on using bisection method until the shortest (i.e. time-optimal) feasible solution with b-value greater than equal to the target value is found.$$\begin{aligned}\underset{{\bf{g}}\in{\Bbb{R}}^{N/2}}{maximize}\quad&\sum_{j=0}^M\sum_{i=0}^j{\bf{G}}_i\qquad\qquad\qquad\qquad\qquad\>\mathrm{b-value\>objective}\\subject\>to\quad&\Vert{\bf{g}}\Vert_{2}\leq\mathrm{100mT/m}\qquad\qquad\qquad\quad\>\mathrm{maximum\>amplitude}\\&\Vert{\bf{g}}_{i+1}-{\bf{g}}_{i}\Vert_{2}\leq\mathit{dt}\cdot\mathrm{200T/m}\cdot\mathrm{s}\qquad\mathrm{maximum\>slewrate}\\&{\bf{g}}_{0}=\mathrm{0mT/m}\qquad\qquad\qquad\qquad\quad\>\mathrm{initial\>boundary\>condition}\\&\Vert{}D_{cos}\tilde{\bf{g}}\Vert_{2}\leq\epsilon_{\mathrm{DCT}}\qquad\qquad\qquad\quad\>\>\>\mathrm{DCT\>constraint}\\&|H_{hp}{\bf{g}}|\leq\epsilon_{hp}\qquad\qquad\qquad\qquad\quad\>\>\mathrm{bandlimiting\>constraint}\end{aligned}$$
Our optimization problem includes several extensions. First, the solution is constrained to designing a single gradient lobe, then repeated on either side of a spin-echo pulse of duration T₁₈₀ (Figure 1c). Furthermore, the solution is constrained to be symmetric about the lobe midpoint. Waveform symmetry is enforced by only solving for only N/2 variables and specifying only one boundary condition. This greatly simplifies computational complexity of the problem. The frequency constraint can then be written using the discrete cosine transform ($$$D_{cos}$$$) applied to the full lobe $$$\tilde{\bf{g}}=[{\bf{g}}\>\mathrm{flip}({\bf{g}})]^T$$$. The objective then acts on a concatenation of the full lobe pairs with some intermittent zero padding during the spin-echo pulse $$${\bf{G}}=[\tilde{\bf{g}}\>{\bf{0}}(\mathrm{T_{180}})\>\tilde{\bf{g}}]^T$$$. The solution is also bandlimited to 5kHz by constraining a high-pass filtered signal.

Vibration of the coil following a waveform induces a back-EMF that the current-controlled gradient power amplifier (GPA) bucks in order to maintain zero current. The bridge voltage of the GPA (or back-EMF) then acts as an indicator of vibration magnitude. The EMF is measured for 20ms following each waveform in a set of sinusoids from 50-2000Hz (10Hz spacing), and pairs of safe and unsafe diffusion lobes chosen to maximally excite each gradient axis with b-values close to 1000s/mm². Relative coil response to lobes is estimated by generating trapezoidal lobes of varying b-value and taking the sum of their power spectra weighted by the back-EMF frequency response. Chosen test lobes include 988s/mm², on the X and Y coils, and 1078s/mm² on the Z coil. Additionally, measurements of k-space trajectory over the 20ms period are taken using a field camera (Dynamic Field Camera, Skope, Switzerland).

RESULTS

An example safe and unsafe diffusion lobe pair are shown in Figure 2 along with their power spectrums. Back-EMF measurements following worst-case diffusion lobes at approximately 1000s/mm² are given in Figure 3. De-trended k-space measurements following the diffusion lobes are seen in Figure 4. The power spectra of these measurements are analyzed in Figure 5.

DISCUSSION

Back-EMF frequency response measurements showed two primary mechanical resonance peaks at 1040Hz and 1230Hz in X and Y and two at 1190Hz and 1800Hz in Z. These bands were constrained in safe diffusion designs leading to a great reduction in the post-diffusion back-EMF (Figure 3) indicating significant reduction in vibration. We can see from the Skope camera measurements (Figure 4) that there are also oscillations in the magnetic field. The k_z measurements show that these oscillations were nearly eliminated using the safe diffusion lobe, but some oscillation remains in k_x and k_y. We can see from the power spectrum (Figure 5) that oscillations at the target frequencies were nulled on all axes, but a large amount of energy is deposited at 520Hz in k_x and k_y. The small peak at this frequency in the back-EMF response was ignored, but clearly can be significant for waveforms with high energy at 520Hz. This can easily be addressed by including the band at 520Hz in the frequency constraint.

CONCLUSION

A novel design method for time-optimal frequency constrained “safe” diffusion lobes has been outlined. This method has been shown to avoid gradient mechanical resonance frequencies, reducing vibration of the coils and oscillations in the magnetic field. These methods can be extended to the design of other high-power gradient lobes in sequences such as gradient spoilers and crushers to similarly avoid mechanical resonances.