On the Effective Centre of Excitation and the Point of Gradient Moment Expansion for 2D-Selective Excitation in the Presence of Flow
Clarissa Wink1, Simon Schmidt2, Jean Pierre Bassenge1,3, Sandy Szermer1, Giulio Ferrazzi1, Bernd Ittermann1, Tobias Schaeffter1, and Sebastian Schmitter1

1Physikalisch-Technische Bundesanstalt (PTB), Braunschweig and Berlin, Germany, 2Medical Physics in Radiology, German Cancer Research Center (DKFZ), Heidelberg, Germany, 3Working Group on Cardiovascular Magnetic Resonance, Experimental and Clinical Research Center, a joint cooperation between the Charité Medical Faculty and the Max-Delbrueck Center for Molecular Medicine, Berlin, Germany


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 1D1-3 and 3D4 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) displacement5 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. isodelay6) 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 end7,4. 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.


In MR flow imaging, velocity is quantified by using bipolar gradients$$$\;\boldsymbol{G}(t)\;$$$that impart a velocity dependent phase to the spins$$\phi=\gamma(\boldsymbol{r}\cdot\boldsymbol{m}_0+\boldsymbol{v}\cdot\boldsymbol{m}_1+\cdot\cdot\cdot).$$This equation results from a Taylor expansion around the expansion time-point$$$\;t_\text{exp}$$$8. Here,$$$\;\boldsymbol{r}\;$$$and$$$\;\boldsymbol{v}\;$$$denote the spin’s position and velocity at$$$\;t_\text{exp}$$$, while$$$\;\gamma\;$$$denotes the gyromagnetic ratio. The zeroth and first gradient moment at echo time$$$\;t_\text{TE}$$$ $$\boldsymbol{m}_0=\int_{t_0}^{t_\text{TE}}\boldsymbol{G}(\tau)\text{d}\tau$$ $$\boldsymbol{m}_1=\int_{t_0}^{t_\text{TE}}\boldsymbol{G}(\tau)(\tau-t_\text{exp})\text{d}\tau$$are calculated starting from the pulse's isophase timepoint,$$$\;t_0=t_0^\text{ph}$$$, when all spins are phase aligned. Thus,$$$\;t_0^\text{ph}\;$$$is essential to design velocity encoding gradients. Since position and velocity are encoded at$$$\;t_\text{exp}$$$, a spatial shift (displacement) of the spins $$\Delta\boldsymbol{r}=\boldsymbol{v}\cdot(t_\text{exp}-t_0^\text{m})$$ will be observed, if$$$\;t_\text{exp}\neq\,t_0^\text{m}\;$$$for designing bipolar gradients. While setting$$$\;t_\text{exp}=t_0^\text{m}\;$$$will remove the shift along PE directions, it cannot be compensated along read-out (RO), since$$$\;m_1^\text{RO}\;$$$varies during data acquisition9.


2D-selective excitation

Two 3.5ms long 2D-selective RF-pulses with spiral k-space trajectory were designed7 as described in4 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).


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 $$$t_0^\text{m}=T_\text{end}-\frac{\Delta y}{v_y}$$$.


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=2x2x1mm3/1.5x1.5x1mm3. Receive profiles were eliminated using separate acquisitions with non-selective excitations.



Since$$$\;t_0^\text{ph}=T_\text{end}\;$$$for moving spins and zero off-resonance4, 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 $$$T_\text{end}$$$.


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 $$$t_\text{exp}=t_0^\text{m}$$$.

Discussion and Conclusion

In this work, for the first time, distinct excitation time-points for phase$$$\;t_0^\text{ph}\;$$$and magnitude$$$\;t_0^\text{m}\;$$$are identified for velocity-encoding and compensation. Although off-resonances distort the magnitude pattern and phase varies non-linearly with velocity, flow quantification remains granted. Nevertheless, precise knowledge of$$$~t_0^\text{m}~$$$is essential to suppress the flow-induced displacement artefact in PE-direction. While these findings were derived for 2D-selective RF-pulses, ongoing work suggests them being similarly important for asymmetric 1D-selective excitation pulses, like minimum-phase Shinnar-Le-Roux-pulses10.


No acknowledgement found.


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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-resonance4, 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.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)