Vincent Gras^{1}, Alexandre Vignaud^{1}, Alexis Amadon^{1}, Franck Mauconduit^{2}, Denis Le Bihan^{1}, and Nicolas Boulant^{1}

Small gradient delays with respect to radiofrequency (RF) pulses can have disastrous effects on the performance of bipolar spokes RF pulses employed in parallel transmission (pTx) to mitigate RF field inhomogeneity problems. This work reports a new method to characterize this delay with a precision of ~20 ns, shown to appear necessary for high performance pTx. By the same token, the same physics principles underlying the method suggest a way to correct for it by simply phase-shifting every second spoke RF pulse. The technique is validated with measurements on a water phantom and on an adult volunteer at 7T.

The delay characterization measurement is depicted in Fig.1. The blips shown in the figure correspond to zero-order eddy-currents ($$$\Delta\!B_0(t)$$$ envelopes) inherent to the presence of slice selection gradient ramps and which need to be modelled for an accurate analysis. With spoiled GRE sequences, the first (red blip in Fig.1) and last (blue blip) eddy-currents, the refocusing gradient lobe, the first rising gradient slope before the first RF pulse and the descending gradient slope after the second pulse do not affect the flip angle and thus shall be ignored. During the first gradient lobe and with a gradient delay $$$\Delta\!t$$$, the gradient during $$$T_p-\Delta\!t$$$ induces a rotation $$$R_{z,T_p-\Delta\!t}$$$ on the spins around $$$z$$$ with an angle of rotation proportional to $$$T_p-\Delta\!t$$$. The RF pulse during the first plateau generates the rotation $$$R_{\phi=0}$$$ where, importantly, $$$\phi$$$ corresponds to the global phase of the RF pulse and not the axis of rotation (which may be pointing between the $$$z$$$-axis and the transverse plane). The second pulse, except for the phase change, should be the time-reversed of the first. The rotation induced by the second pulse thus is $$$R_{\phi=\pi+\Delta\!\phi}$$$. The residual eddy-current between the two RF sub-pulses (green envelope in Fig.2) generates $$$R_z(-\phi_{\text{EC}})$$$, i.e. a rotation of angle $$$-\phi_{\text{EC}}$$$ around $$$z$$$. In the absence of a static $$$\Delta\!B_0$$$ offset, the rotation matrix can be shown to be equal to:

$$R_{\text{TOT}} = R_{-z,T_p-\Delta\!t}R_{\phi=\pi+\Delta\!\phi+\phi_{\text{EC}}-2\delta}R_{\phi=0}R_{z,T_p-\Delta\!t},$$

where $$$\delta=\gamma{G_z}\Delta_t$$$, $$$G$$$ being the gradient amplitude and $$$z$$$ the slice position. The eddy-current and gradient delay thus effectively induce a phase-shift on the second RF pulse. If $$$\Delta\!\phi=2\delta-\phi_{\text{EC}}$$$, then the rotation matrix above becomes Identity and there should be no signal. Thus by determining at different slice locations the $$$\Delta\!\phi$$$ value which returns a minimum of signal, the delay $$$\Delta\!t$$$ can be recovered by linear regression of $$$\Delta\!\phi$$$ versus $$$z$$$. Axial measurements were performed at 7T at -25,-20,-10,0,10,20 and 25 mm from the isocenter on a water phantom and $$$\Delta\!\phi$$$ was swept across the interval $$$[-90^{\circ};90^{\circ}]$$$. Once the delay and zero order eddy-current terms $$$\Delta\!t$$$ and $$$\phi_{\text{EC}}$$$ were recovered, their prospective correction was implemented in tailored bipolar 2-spoke configurationsThe research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Program (FP7/2013-2018), ERC Grant Agreement n. 309674.

Bipolar 2-spoke
configuration used for gradient delay and zero order eddy-current
characterization. The RF sub-pulses are padded with zeros before and after
(duration T_{p}). Rotations around the z axis are induced during these
periods which are affected by a gradient delay Δt (R_{z,Tp±Δt} and R_{-z,Tp±Δt}, the sign of the
rotation depending on the sign of the gradient). The RF pulses combined with
the gradient fields generate rotations denoted as R_{φ=0}
and R_{φ=π+Δφ} respectively (global phase of the RF field in
subscript). The middle eddy-currents engender rotations around z, R_{z}(φ_{EC}).

Gradient delay and
zero order eddy-current measurements results. Subplot a) reports the normalized
signal as a function of Δφ in a region of interest where ΔB_{0}~0 and at several slice (z) locations. For each
location, the minimum is achieved when the action of the bipolar 2-spoke pulse
is null at the center of the slice, i.e. when the correct phase correction is
applied. In b), the value of Δφ where the minimum occurs is plotted versus z,
returning after linear regression Δt=-2.69±0.02
µs and φ_{EC}=-8°.

Results of the
phantom experiments (13 out of the 25 axial slices shown, 1 cm separates the
shown adjacent slices). From top to bottom: bipolar 2-spoke pulses with no
correction, bipolar 2-spoke pulses with eddy-current correction only, bipolar
2-spoke pulses with eddy-current and gradient delay compensation, unipolar
2-spoke pulses. The RF spoke-pulses for the bipolar case were tailored for each
slice individually to achieve a target flip angle of 30°. The same RF
coefficients and optimized k-space trajectory were applied to the unipolar
case. The unipolar and corrected bipolar spoke-pulses returned nearly identical
results.

In vivo T_{2}*-weighted axial
images acquired at 7T (13 out of the 25 axial slices shown, 1 cm separates the shown adjacent slices). The same
observations as for the phantom results can be made with very heterogeneous
excitation profiles and signal voids away from isocenter when the gradient
delay is not corrected.