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

TRASE MRI uses rapid π-pulses of phase gradient fields, and
in general requires as many as two distinct phase-gradient coils per encoding
direction. This tends to restrict one to
large amplitude, linear B_{1} fields, which in low B_{0} field
leads to a breakdown of the rotating wave approximation. We have studied this regime both numerically
and experimentally. Our results show a
rich behavior involving a complex interplay of the Bloch-Siegert shift, the B_{1}
start and stop phase, and B_{1} amplitude transients.

**Introduction**

**Methods**

**Results**

**Discussion and conclusion**

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

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4. S Kumaragamage, et al. Proc ESMRMB 16, Magn Reson Mater Phy. 2016:29 Suppl.1;S34.

5. PJ Nacher, et al. This conference.

6. F Bloch and A Siegert. Physical Review. 1940;57:522.

7. D MacLaughlin. Rev Sci Instrum. 1970;41:1202.

8. M Mehring and J Waugh. Rev Sci Instrum. 1972;43:649.

Figure1: Illustration
of the general consequence of the counter-rotating term on the trajectory: rotating
field (blue) versus linear field (yellow). Simulations with N_{π} = 3. Left: Off-resonance rotating field (B =
B_{0} +
δB) and on-resonance linear field (B = B_{0}). Right: On-resonance rotating field (B =
B_{0}) and off-resonance linear
field (B = B_{0} − δB).

Figure 2: Trajectories near the beginning (left) and end
(right) of a π-pulse with N_{π} = 9 and δB=0. With no Bloch-Siegert shift, a perfect π-pulse can only be
achieved with φ = 30^{o}.
The circles indicate time intervals equal to half-periods of the B_{1}
field from the start of the pulse. For φ = 90^{o} these occur at the cusps in the
trajectory, which occur when the net B_{1} field is zero.

Figure 3: Trajectories near the end (left) of a π-pulse with N_{π} = 9 and δB
= β ΔB_{cw}. With
the appropriate de-tuning, a perfect π-pulse can be achieved for any φ. Plot of the analytic form of β for the case where N_{π} = n/2, where n is an
integer. Simulations for N_{π} = 1 and 3
reveal a higher order dependence of β on
the magnitude of B_{1}.

Figure 4: A
π-pulse starting with unit magnetization in the transverse plane: N_{π} = 9 and φ = 0^{o}. Left: On-resonance. The
terminus overshoots or undershoots, depending on the original phase of the
magnetization φ_{M}(0).
Final M_{z} values, shown in the plot, are up
to +/-0.08. Right: Off-resonance (B =
B_{0} −
δB, with δB/B_{0} = -1/(4N_{π})^{2}). Final M_{z} values are reduced to
below 1ppt for the nominal shift (β=-1, points, red dashed line). Using a more
precise value of β to account for higher order effects reduces M_{z} to the ppm
level (points).

Figure 5: Some
experimental results. Left: Bloch-Siegert
shift versus N_{π} for φ = 90^{o} with data (points) and theory
(line). Center: Bloch-Siegert shift versus φ for various N_{π} with data (points) and
theory (lines). Right: Flip angle error (degrees
from 180^{o}) versus de-tuning for CORPSE composite pulse and an N_{π} =12 rectangular pulse. The former remains insensitive to de-tuning despite the departure from the RWA.