Manuel Stich^{1,2}, Tobias Wech^{1}, Anne Slawig^{1,2}, Gudrun Ruyters^{3}, Andrew Dewdney^{3}, Ralf Ringler^{2}, Thorsten A. Bley^{1}, and Herbert Köstler^{1}

As a linear and
time-invariant (LTI) system, the dynamic gradient system can be described by
the system transfer function. While special measurement equipment like field
cameras can be used to precisely determine even higher orders of the transfer
function, phantom-based approaches were introduced for alternative
determination without additional hardware needed. This study reports on
phantom-based measurements of B_{0}-components, which resulted in
transfer functions with sufficiently high resolution for the characterization
of mechanical resonances.

Experiments were performed with a spherical
phantom placed inside a 3T Siemens MAGNETOM Prisma^{fit} scanner (Siemens
Healthcare, Erlangen, Germany). In a prototype sequence, 12 different input gradients $$$g_{in}(t)$$$ (duration 100
– 320 µs, slew rate = 180 T/m/s) with broad spectral support
were played out. The responding phases $$$\Phi_{1}(t)$$$ and
$$$\Phi_{2}(t)$$$ were measured in two parallel slices at position ±16.5 mm, vertical to the input gradient direction. For each slice, a reference
phase $$$\Phi_{.,ref}(t)$$$ was additionally acquired with no gradient
waveforms played out by the system. The gradient system response
$$$b_{out}(t)$$$ for linear GSTF components are determined as the difference in
phase evolution. In contrast to that, the gradient system response
$$$b_{out}(t)$$$ for the determination of B_{0}-components can then be calculated from the sum of the phase evolution:

$$b_{out}(t)=\frac{1}{2} \cdot \frac{1}{\gamma} \cdot \frac{d}{dt}[(\Phi_{1}(t)-\Phi_{1,ref}(t))+(\Phi_{2}(t)-\Phi_{2,ref}(t))].$$

Finally, the B_{0}-components of the
transfer functions can then be calculated as:

$$H_{B_{0},k}(f)=\frac{\sum_{i=1}^N {G_{in}^{*}}_{k}^{i}(f) \cdot {B_{out}}_{k}^{i}(f)}{\sum_{i=1}^N {{G_{in}}_{k}^{i}(f)}^2},$$

where $$$k$$$ represents the direction in which the input gradient is played out. The index represents a particular triangular waveform, and $$$N$$$ the total number of input gradients. $$$G_{in}(f)$$$ and $$$B_{out}(f)$$$ are the Fourier transforms of $$$g_{in}(t)$$$ and $$$b_{out}(t)$$$.

Relevant
measurement parameters were set to: TR = 5.0 s, slice thickness = 3 mm, slice positions = ±16.5 mm, flip-angle = 90°, bandwidth = 119 kHz, 40 averages. The length of
the acquisition window was ~33 ms such that the GSTF featured a frequency
resolution of 30 Hz. The complete acquisition of one B_{0}-component
takes about 70 min. Fig. 1 visualizes the B_{0}-component measurement
and calculation process schematically.

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