Christopher P Bidinosti^{1}, Pierre-Jean Nacher^{2}, and Geneviève Tastevin^{2}

TRansmit Array Spatial
Encoding (TRASE) MRI uses trains of rf pulses produced by transmit coils which
generate transverse fields of uniform magnitude and spatially varying
directions. These coils also unavoidably generate concomitant rf
fields, which in turn affect magnetisation dynamics during rf flips in low-field NMR. Bloch’s
equation are numerically solved to show that π-pulses imperfectly reverse
transverse magnetisation and that the resulting error in azimuthal angle
linearly increases with B_{1}/B_{0}, with the number of pulses in the TRASE pulse
train, and with distance from the coil axis in the sample. This may induce
significant image distortions or artefacts. Supporting experiments performed at 2 mT will
be reported.

Figure 1
shows a prototype spiral coil and displays computed field maps. The helicoidal
field pattern has a linear z-dependence of its phase φ(z)=k z (with a pitch
coefficient k=0.12 cm^{-1}) and a fairly uniform magnitude of the
transverse component, B_{1t} (aligned with the y-axis, at the coil
centre). The (mainly) x-dependent concomitant rf field component, B_{1z}(x)
= k x B_{1t}, satisfies the rot(**B**_{1})=0
Maxwell’s equation under the assumption of perfectly uniform B_{1t}.
However, this approximation breaks down for distances from the z-axis of order *a*/2 and the actual field maps, which satisfy
both rot(**B**_{1})=0 and div(**B**_{1})=0, are needed
for reliable NMR simulations.
Figure 2 displays magnetisation trajectories (in
the rotating frame) during rf π-pulses computed for sample elements in the z=0
plane, lying either on the coil axis (left: x=0; hence, B_{1z}=0) and
off axis (right: k x =1 = B_{1z}/B_{1t}, chosen for clear effects). They allow for a
comparison between evolution perturbed by the counter-rotating component alone
(see^{3} for further details) or in combination with the concomitant
component B_{1z}. In both cases, the trajectory features depend on
the initial magnetization state. On axis, in spite of deviations
from rotating wave approximation (RWA) trajectories, perfect inversion is
achieved. Off axis, B_{1z} yields deviations with distinct time variation and
maximal amplitudes but, most notably, induces a phase shift α for the East-West
trajectory endpoint. This shift increases with B_{1} magnitude and
distance from the spiral coil axis and is found to amount to α = α_{0}
k y/N_{π}, where α_{0} = 58° and N_{π} = B_{1t}/B_{0}
is the ratio between the π-pulse duration and the rf period. As a result, for a
TRASE phase-encoding sequence comprising tens of pulses, the expected cumulated
added rotation can exceed 90°.

2. Q Deng et al., Magn Reson Imaging 31:891 (2013).

3. CP Bidinosti et al, this conference

4. CP Bidinosti et al., Proc. ISMRM 18 (2010) p. 1509.

5. P-J Nacher et al., Proc ESMRMB15, Magn. Reson. Mater. Phy. 28 Suppl. 1 (2015) p. S64.

6. L Darrasse et al., Magn Reson Imaging 5:559 (1987).

Figure 1: Left: photograph of a spiral
coil used for z-encoding in low-field TRASE experiments^{3} and sketch of the transverse rf field directions it generates. The
spiral coil pitch is k=0.12 cm^{-1}. Right: computed maps of the transverse
and longitudinal field components in the central xy plane, scaled to the
field B_{C} at the coil centre (coil radius: a=6.5 cm). Over the same domain, B_{1x}/B_{C} (not displayed) remains smaller
than 0.02, hence the transverse field has a very uniform direction and amplitude.
As expected for this near-uniform field, B_{1z} only depends on x, with
the expected linear variation (see text) over a wide domain.

Figure 2: Time-evolution of the 3 components of the
magnetisation undergoing a π-rotation driven by a 9-period-long rectangular rf
pulse (N_{π}=9). Top row: N to S; middle row: E to W trajectories. On-axis (kx=0), the trajectories are affected by the
counter-rotating part of the oscillating rf field but the endpoints correspond
to exact π rotations. Off-axis, the added concomitant field component B_{1z} induces
significant deviations of the x-component of the magnetisation. Moreover, the endpoint of a π pulse on initially
transverse magnetisation reveals a significant phase shift. Bottom row: these phase shifts add up for chained TRASE π-pulses.