Adam van Niekerk^{1}, Andre van der Kouwe^{2,3}, and Ernesta Meintjes^{1,4}

We explore the possible uses of a 40-kHz switching signal that is proportional to the gradient slew rate. The signal was identified by probing the rate of change of the magnetic flux density in the MRI scanner using a three-dimensional pickup coil. The signal is independent of both the orientation of the pickup coil and location in the scanner bore, and is directed parallel to the static magnetic field. This indicates that it is not caused by current in the gradient coils. These properties make this signal a useful vector reference that could be used for orientation tracking or time frame synchronisation.

The data used comes from a modified gradient echo (GRE) pulse sequence played out on a 3 T Skyra (Siemens, Erlangen, Germany). Before each readout of the GRE sequence, three sinusoidal pulses are played out, one on each gradient axis (lasting a total of 880 μs - Figure 2). Each sinusoid has a frequency of $$$f_n$$$ = 6.25 kHz and a maximum slew rate of $$$s$$$ = 60 T/m/s:

$$\dot{g}=s\cos(2\pi f_n+\theta_n)$$

The WRAD detects the non-selective RF pulse
in the GRE pulse sequence, waits until the start of the sinusoidal playouts,
and then samples each axis of the 3-axis pickup coil using a 12-bit ADC at a
frequency of 200 kHz. The pickup signals are amplified by a factor of 32 and
low-pass filtered using a 3-pole analog *Butterworth*
filter with a cut-off frequency of 32 kHz. This filter was designed before we
had knowledge of the 40-kHz signal; it will therefore be attenuated. In the
ideal case - assuming linear gradients, an axial symmetric z-gradient coil and
negligible rates of change of divergence and curl - the *position encoding* voltage vector induced
in the 3-axis pickup coil sensor can be written as follows^{2}:

$$\vec{v}_p=-a\mathbf{R}\left(\left[\begin{matrix}-\frac{\dot{g_z}}{2}&0&\dot{g_x}\\0&-\frac{\dot{g_z}}{2}&\dot{g_y}\\\dot{g_x}&\dot{g_y}&\dot{g_z}\end{matrix}\right]\vec{p} + B_0\vec{z}\times\vec{\omega}\right)$$

where $$$\mathbf{R}$$$ is a rotation matrix that transforms a vector
from the gradient frame into the WRAD frame, $$$\dot{g_x}$$$, $$$\dot{g_y}$$$ and $$$\dot{g_z}$$$ are the rates of change
of the x, y and z gradients, $$$a$$$ = 32 $$$\times$$$ 0.0052 m^{2} is a geometric scaling
factor relating to the pickup coils, $$$\vec{p}$$$ is the position of the WRAD in the gradient
frame and $$$\vec{\omega}$$$ is the angular rate of change of the WRAD.
We propose that the
high frequency ‘*switching signal*’ superimposed on the sinusoidal pulses in
Figure 3, can be modelled by the following equation:

$$\vec{v}_r=-a\left(\dot{g_x}+\dot{g_y}+\dot{g_z}\right)r\cos(2\pi f_r+\theta_r)\mathbf{R}\vec{z}$$

where $$$f_r$$$ = 40 kHz. Adding the *switching signal* to the *position
encoding* signal and then combining
the result with the pulse sequence waveform shapes results in the
following:

$$\vec{v}=\vec{v}_p+\vec{v}_r=-a\mathbf{R}\left(\mathbf{S}\vec{p}+r\cos(2\pi f_r+\theta_r)\vec{z}\right)s\cos(2\pi f_n+\theta_n)$$

$$~=-a\mathbf{R}\left(s\cos(2\pi{f_n}+\theta_n)\mathbf{S}\vec{p}+\frac{1}{2}rs\left(\cos(2\pi[f_n\pm{f_r}]+\theta_n\pm \theta_r)\right)\vec{z}\right)$$

where
the angular rate term is dropped because it is
perpendicular to the *switching signal* and $$$\mathbf{S}=\left[\begin{smallmatrix}0&0&1\\0&0&0\\1&0&0\end{smallmatrix}\right],~\left[\begin{smallmatrix}0&0&0\\0&0&1\\0&1&0\end{smallmatrix}\right]$$$and$$$~\left[\begin{smallmatrix}-\frac{1}{2}&0&0\\0&-\frac{1}{2}&0\\0&0&1\end{smallmatrix}\right]$$$ for
the RX, RY and RZ readouts (Figure 2).
It follows that the
*switching signal* can be isolated by applying a *Goertzel* filter [single bin Discrete Fourier Transform (DFT)] to $$$\vec{v}$$$ at a frequency of $$$f_n\pm{f_r}$$$.

*Experiment 1) *A normalised
observation from the 3-axis *Hall* effect
magnetometer (MV2, *Metrolab*) can be
interpreted as $$$\mathbf{R}\vec{z}$$$, which is the gradient z-axis observed in the
WRAD coordinate frame. Similarly, if we isolate the *switching signal* and
normalise it, we expect the same result. To evaluate this observation, the WRAD
was placed at a positive z-displacement and then rotated a full
revolution about the centre of the pickup coil assembly. The *switching signal* and
magnetometer observations could then be compared.

*Experiment 2)* The WRAD was
displaced, first along the x, then the y, and finally the z direction in 8 mm
increments. This allowed us to observe how the *switching signal* changes based
on location in the scanner bore.

*Experiment 1)* The *Goertzel* filter effectively separates the *switching signal* from the *position encoding*. The normalised *switching signal* and magnetometer observations closely match and are independent of the WRAD orientation, validating the proposed model (Figure 4).

*Experiment 2)* The *switching signal* is independent of location in the scanner bore and is observable during all three gradient coil playouts (Figure 5). The phase is stable to within 0.6 μs.

1) A. van Niekerk, E. Meintjes, and A. J. W. van der Kouwe, “Hybrid motion sensing with a wireless device, self-synchronized to the imaging pulse sequence.” in Proceedings of the 26th Annual Meeting of the International Society of Magnetic Resonance in Medicine. Paris, France: Wiley, 2018

2) M. A. Bernstein, X. J. Zhou, J. A. Polzin, K. F. King, A. Ganin, N. J. Pelc, and G. H. Glover, “Concomitant gradient terms in phase contrast MR: Analysis and correction,” Magnetic Resonance in Medicine, vol. 39, no. 2, pp. 300–308, feb 1998.