Clarissa Wink^{1}, Simon Schmidt^{2}, Jean Pierre Bassenge^{1,3}, Sandy Szermer^{1}, Giulio Ferrazzi^{1}, Bernd Ittermann^{1}, Tobias Schaeffter^{1}, and Sebastian Schmitter^{1}

In this work, we demonstrate the distinction and importance of two virtual time points during excitation for correct flow compensation and quantification: the centre of excitation ($$$t_0^\text{m}$$$) at which spins are excited and thus magnitude is generated, and the isophase time-point ($$$t_0^\text{ph}$$$) at which all excited spins are in phase. A general method to determine $$$t_0^\text{m}$$$ is presented and $$$t_0^\text{ph}$$$ and $$$t_0^\text{m}$$$ are shown to be not necessarily identical. Finally, phantom experiments demonstrate that the knowledge of $$$t_0^\text{m}$$$ is required to remove the displacement artefact in phase-encoding directions to enable correct flow compensation and imaging.

2D-selective excitation has been
employed to accelerate time-resolved flow measurements in 1D^{1-3} and 3D^{4} by
reducing the field-of-view. However, applying conventional flow quantification
and compensation techniques straightforwardly to 2D-selective RF-pulses can lead
to a) wrong velocity quantification and b) displacement^{5} of moving spins
possibly into static tissue.

Here, we identify for the first time
two distinct virtual time-points during excitation: the isophase time-point$$$\;(t_0^\text{ph})\;$$$at which all excited spins are in phase (cf. isodelay^{6}) and the centre of
excitation$$$\;(t_0^\text{m})\;$$$at which spins are effectively excited and thus
magnitude is generated. While those points coincide for standard ‘SINC’-pulses
at nearly half the RF-pulse duration, their distinction is indispensable for
correct flow compensation and quantification using 2D-RF-pulses:

a)$$$\;t_0^\text{ph}\;$$$is needed
for correct velocity quantification and has been identified for spiral
2D-selective excitation at the 2D-RF-pulse end^{7,4}. Here, we investigate how
off-resonances affect phase and velocity quantification.

b)$$$\;t_0^\text{m}\;$$$has not been reported yet but is required to correct the displacement of moving spins caused by the time-delay between excitation and read-out in phase-encode (PE) directions.

The aim of this work is 1.) to present a general method to determine $$$t_0^\text{m}$$$, 2.) to show that$$$\;t_0^\text{ph}\;$$$and$$$\;t_0^\text{m}\;$$$are not necessarily identical, and 3.) to demonstrate in phantom experiments that precise knowledge of$$$\;t_0^\text{m}\;$$$is required to remove the displacement artefact in PE-directions.

**2D-selective excitation**

Two 3.5ms long 2D-selective RF-pulses with spiral k-space trajectory were designed^{7}
as described in^{4} to excite a rectangular bar with 1) a
large field-of-excitation FOX=$$$(\infty\times60\times60)\,\text{mm}\;$$$and
bandwidth-time-product BWT=1.2 (RFL) and 2) a
small FOX=$$$(\infty\times36\times36)\,\text{mm}\;$$$and BWT=0.9 (RFS).

**Simulations**

The velocity-dependent phase of the magnetisation at the 2D-RF-pulse end$$$\;T_\text{end}\;$$$was determined using Bloch-simulations. Additionally, the impact of off-resonance on velocity quantification was investigated by simulating the 2D-RF-pulses once with consecutive flow-encoding and second with flow-compensation gradients. Simulated spins move in a virtual tube oriented in $$$y$$$-direction (PE) with maximum velocities $$$v_y\in[-100:20:100]\,\text{cm}/\text{s}$$$ and are static elsewhere.

The location of $$$t_0^\text{m}$$$ was quantified by evaluating the spatial shifts of the magnitude pattern at $$$T_\text{end}$$$ in Bloch-simulations using moving spins in comparison to stationary spins. The spatial shift of the magnitude signal $$$|S(y,v_y)|$$$ was calculated by $$$\Delta y=\frac{\int|S(y,v_y)|\,y\,\text{d}y}{\int|S(y,v_y)|\text{d}y}$$$ for $$$v_y\in[-100:20:100]\,\text{cm}/\text{s}$$$ and RFS and RFL. The centre of excitation is then calculated by $$$t_0^\text{m}=T_\text{end}-\frac{\Delta y}{v_y}$$$.

**Experiments**

Flow phantoms containing two pipes with constant flow were scanned at 3T
(Magnetom Verio, Siemens) using a 15-channel-knee-coil to verify the simulation
results. RFL- and RFS-excitation pattern shifts were measured for $$$t_\text{exp}=t_0^\text{ph}$$$, $$$t_\text{exp}=t_0^\text{m}$$$, and $$$t_\text{exp}=t_\text{TE}$$$ in the presence and absence of flow. Imaging
parameters of two datasets differing mainly in venc:
dataset1/dataset2: venc(z)=2m/s/4.5m/s, TE=3.35ms/3.25ms, FA=8°/3°,
matrix=64x64x192/96x96x192, resolution=2x2x1mm^{3}/1.5x1.5x1mm^{3}. Receive profiles
were eliminated using separate acquisitions with non-selective excitations.

**Simulations**

Since$$$\;t_0^\text{ph}=T_\text{end}\;$$$for moving spins and zero
off-resonance^{4}, phase at$$$\;T_\text{end}\;$$$is velocity-independent (Fig.1a).
However, for 200 and 500Hz off-resonance, phase is varying non-linearly as a
function of velocity (Fig.1b). Still, the same velocity is quantified
for zero and 200Hz off-resonance (Fig.2a,bottom). Actually, a fit shows that
quantified$$$\;v_\text{qu}\;$$$and actual velocity$$$\;v_\text{act}\;$$$match
for 0 and 200Hz off-resonance (Fig.2b) with fit difference close to zero
(Fig.2c).

The magnitude pattern is smoothened for 200Hz off-resonance and shifted in the moving tissue section (Fig.2a,top). Figure3 depicts the shift of the RFL- and RFS-excitation patterns at $$$T_\text{end}$$$ for different velocities along y-direction. Based on this velocity-dependent shift the magnitude’s centre of excitation was located before$$$\;T_\text{end}$$$ at $$$T_\text{end}-t_0^\text{m}=(0.52\pm0.01)\,\text{ms}$$$ and $$$(0.75\pm0.02)\,\text{ms}$$$ for RFL and RFS respectively, while $$$t_0^\text{ph}$$$ was identified at $$$T_\text{end}$$$.

**Experiments**

Figure4 illustrates that the flow-induced excitation pattern shift of up to $$$12.5\,\text{mm}\,$$$for$$$\;t_\text{exp}=t_\text{TE}\;$$$and$$$\;v=2.9\,\text{m}/\text{s}\;$$$is successfully suppressed choosing$$$\;t_\text{exp}=t_0^\text{m}$$$. Figure5 shows the RFL- and RFS-shift for different velocities and$$$\;t_\text{exp}$$$. The measurements agree with simulations predicting $$$\Delta\boldsymbol{r}=\boldsymbol{v}\cdot(t_\text{exp}-t_0^\text{m})$$$ and thus zero shift for $$$t_\text{exp}=t_0^\text{m}$$$.

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2. Butts K, Hangiandreou NJ, Riederer SJ. Phase velocity mapping with a real time line scan technique. Magnetic Resonance in Medicine. 1993;29(1):134–138.

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Nieters EJ, Roemer PB, Watkins RD, Adams WJ, et al. Real-time acquisition,
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Magnetic Resonance in Medicine. 1993;29(5):667–673.

4. Wink C, Ferrazzi G, Bassenge JP, Flassbeck S, Schmidt S, Schaeffter T, Schmitter S. 4D flow imaging with reduced field-of-excitation. Proceedings of the 27th Annual Meeting ISMRM. 2018.

5. Nishimura DG, Jackson JI, Pauly JM. On the Nature and Reduction of the Displacement Artifact in Flow Images. Magnetic Resonance in Medicine. 1991;22:481-492.

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**Figure 1:**

The transverse
magnetisation's phase as a function of spin velocity for **(a)**
zero and **(b)** 200Hz and 500Hz off-resonance.

**(a)** Since
$$$t_0^{\text{ph}}=T_\text{end}$$$ for moving spins and zero off-resonance^{4},
the phase at $$$T_\text{end}$$$ is nearly independent of velocity and can be
linearly fitted.

**(b)** For 200Hz and 500Hz off-resonance,
the phase varies non-linearly as a function of velocity. However, the effects of off-resonance on
phase are small and flow quantification is not severly impeded as
shown in Fig.2.

**Figure 2:**

Results of Bloch simulations of RFL and consecutive flow-encoding or -compensation gradients for 0Hz and 200Hz off-resonance. Spins move in y-direction with maximal$$$\;v_y=80\,\text{cm}/\text{s}$$$ in the centre of the tube.

**
(a) **Magnitude images **(1st row)
**exhibit the apparent displacement of the moving tissue section. Corresponding flow-encoded**
(2nd row:**$$$~m_1=11.74\,\frac{\text{mT}}{\text{m}}\text{ms}^2$$$**)
**and flow-compensated** (3rd row:**$$$~m_1=0\,\frac{\text{mT}}{\text{m}}\text{ms}^2$$$**)
**phase images, resulting in $$$v_\text{enc}=100\,\text{cm}/\text{s}$$$. The **4th
row **shows the quantified velocity.

**(b) **Quantified versus actual velocity
for off-resonances of 0Hz (blue squares) and 200Hz (red circles) for different
velocities.

**(c)** Difference in quantified velocity
between 0Hz and 200Hz off-resonance (blue 'x') and the respective fit
difference (red line), which is $$$<10^{-12}\,\text{cm}/{\text{s}}$$$.

**Figure 3 (animated):**

**(column1&2)** 2D and 1D cross-section of the excitation pattern at the
end of excitation $$$T_\text{end}$$$ for velocities
$$$v\in[-100,100]\text{cm}/\text{s}$$$ and 2D RF-pulse RFL **(left)** and RFS
**(right)**. The dashed line indicates the respective shift of the excitation
pattern $$$\Delta y$$$.

**(column3)** The **1st row** shows the excitation pattern shift $$$\Delta y$$$
versus velocity $$$v_y$$$ for pulse RFL (filled blue squares) and RFS (red
circles). The **2nd row** shows the resulting time
$$$T_\text{end}-t_0^\text{m}=\Delta y/v_y$$$ as a function of velocity.

**Figure 4:**

**(top)**
Magnitude images obtained using 2D-selective excitation (RFS) in a flow
phantom with two pipes. RO is oriented along the non-selective axis perpendicular to the flow direction.

**(bottom)** The lineplots show the normalized mean
magnitude in the tubes containing flowing (red & yellow) or static water
(blue).

**(a)** Setting the expansion point $$$t_\text{exp}=t_0^\text{ph}=T_\text{end}$$$,
excitation pattern shifts of -1.70±0.32 mm and 1.23±0.03mm are observed.

**(b)** Setting $$$t_\text{exp}=t_0^\text{m}=T_\text{end}-0.75\,\text{ms}$$$, these shifts are well compensated for both flow directions and velocities with residual displacements of 0.26±0.21mm and -0.18±0.03mm.

**(c)** Setting $$$t_\text{exp}=t_\text{TE}=T_\text{end}+3.25\,\text{ms}$$$ even higher shifts of -12.49±0.22mm and 5.70±0.09mm are observed.

**Figure 5:**

The measured, offset-corrected mean shift of the 2D-selective excitation
pattern **(a: **RFL, **b: **RFS**)** as a function of velocity for different choices of
$$$t_\text{exp}$$$. The dashed lines indicate the expected shift $$$\Delta y=v_y\cdot(t_\text{exp}-t_0^\text{m})$$$. Yellow and violet data points were obtained using venc settings of venc1=2m/s and venc2=4.5m/s that led to different echo times of TE1=3.35ms and TE2=3.25ms.
While all other settings of $$$t_\text{exp}$$$ result in displacements scaling linearly with flow velocity and the deviation of $$$t_\text{exp}$$$ from $$$t_0^\text{m}$$$, these artefacts are successfully suppressed for all velocities if $$$t_\text{exp}=t_0^\text{m}$$$ (red symbols) is chosen for calculating bipolar gradients.