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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
Magn Reson Med. Author manuscript; available in PMC 2010 June 15.
Published in final edited form as:
PMCID: PMC2885787

Mapping the Velocity Vector Onto the Spin Vector: Two-Dimensional Velocity-Selective Spin Excitation for MR Flow Imaging


This article presents a two-dimensional velocity-selective spin excitation (2D-VSP) method that enables quantitative imaging of motion in two directions in a single scan, without the need for image subtraction or combination. It is based on the idea of mapping a 2D velocity vector directly onto the transverse magnetization vector, such that the signal intensity reflects the speed of motion, while the signal phase represents the direction of motion. Experimental demonstration is presented in conjunction with an analysis of the accuracy of this method. VSP methods are often limited by inconsistent static signal suppression under variable shim and RF conditions. By using adiabatic RF pulses in a 2D-VSP composite that possesses time-reversal symmetry, consistent background suppression of 30-fold or higher was demonstrated over experimental conditions of ±200 Hz off-resonance and 30% RF field variation.

Keywords: MRI, quantitative flow imaging, velocity-selective, two-dimensional, MR angiography

MRI is inherently sensitive to motion as spins moving along a gradient field accumulate a different phase from static spins (1,2). The development of this concept has been reviewed extensively (3,4). Several classes of flow imaging techniques stem from this basic idea. Velocity-selective spin preparation (VSP) excites or saturates moving spins according to this phase difference, thereby producing image contrasts that reflect the condition of flow (5-15). This article presents a novel 2D-VSP excitation that is simultaneously sensitized to flow in two directions, such that the transverse spin vector forms a representation of the 2D flow vector. Equivalently, the real and imaginary components of the MR signal are made proportional to two independent velocity components in a single image acquisition:


where Mx and My are the orthogonal transverse magnetization components giving rise to the real and imaginary parts of the signal and vx and vy are two independent components of the velocity vector. The velocity components are usually perpendicular to each other, for example, the two orthogonal components within the image slice.

Besides scan-time savings, 2D-VSP improves the contrast-to-noise ratio (CNR) of the flow region over 1D VSP methods. This article also describes refinements to the basic 2D-VSP scheme with adiabatic pulses (14) of time-reversal symmetry to achieve robust background suppression.

The development of differential excitation of spins according to their status of motion begins with saturation-recovery techniques prior to MR imaging (7,16-22). Alternatively, velocity-selective spin preparation or flow excitation relies on motion-induced phase shifts instead of inflow effects. VSP methods have been used as the primary excitation mechanism (5-11,13,14) or as a preparation phase followed by rapid image acquisition (12,15). Static spin suppression was also achieved in steady-state imaging with repeating self-canceling pulse combinations (11). Norris and Schwarzbauer (23) gave a general description of binomial pulses for velocity-selective excitation, inversion, or suppression in the context of solvent suppression techniques in NMR spectroscopy. The opposite approach is suppressing the signal of moving spins with tailored RF and gradient pulses to create dark-blood contrast. Buonocore (24) formulated a class of composite pulses called transparent pulses for saturating moving spins along the through-slice direction. These pulses effectively remove flow artifacts in spin-echo images.

The existing VSP methods are sensitized to flow along one direction per image acquisition. In the following sections, the basic scheme of 2D-VSP as well as several optimization steps are described. The relationship between the MR signal and the velocity vector is also discussed in detail. Experimental demonstration of 2D-VSP is then presented and the robustness of the technique is evaluated under a range of imaging conditions.


Basic 2D-VSP Scheme

The velocity-selective excitation pulse in Fig. 1a (7,8,12) is a typical example of 1D VSP. It results in a transverse magnetization of the form:


where γ is the gyromagnetic ratio and b is the first moment of the gradient lobes. To map two velocity components simultaneously, we look at the effect of the VSP pulse in Fig. 1b. The resulting magnetization vector is:


where bx and by are the first-order moments of the x and y pairs of gradient pulses. To the leading order approximation, the transverse magnetization is approximately proportional to the velocity projection in the xy plane. A single image acquired with this VSP should represent a 2D velocity field.

FIG. 1
Schematics of velocity-selective excitation composites. a: A conventional 1D VSP excitation that is sensitized to motion along the x direction. It is followed by a gradient-recalled echo readout. b: A 2D-VSP excitation that maps a 2D velocity vector onto ...

This VSP composite is essentially a combination of two elements of the form in Fig. 1a. In the first half, the y magnetization is sensitized to the x velocity component and is then stored along the z axis by the 90°(x) flip. What is left in the transverse plane is sensitized to the y velocity component by the subsequent gradient lobes in the y direction. Now the y magnetization primarily encodes the y velocity component. The last 90° flip along the y direction maintains this component, while returning the z magnetization, which was encoded with the x velocity component, back to the x axis. The end result is expressed in Eq. [3]. The nonlinearity of Eq. [3] imposes a limit on the velocity-induced phase shifts. A detailed discussion of this point is presented later in the article.

Improving Background Suppression With Adiabatic RF Pulses Arranged in Time-Reversal Symmetry

VSP techniques rely on precise flip angles. In the presence of B0 and B1 inhomogeneities, adiabatic RF pulses have been proposed for better performance (14). A version of the 2D-VSP using adiabatic 90° and refocusing pulses (25-27) is shown in Fig. 1c. These pulses are not spatially selective, so 2D imaging can be realized with a spin-echo acquisition using a slice-selective 180° refocusing pulse. The magnetization vector at the end of the VSP is still given by Eq. [3].

The adiabatic pulses are still affected by off-resonance effects from imperfect shims and lipid chemical shifts. The result is a residual signal from the static background. Further improvement is achieved with VSP composites that possesses a time-reversal symmetry (28), which is the following: If in a static but not necessarily uniform main field B0, a set of time-dependent RF field B1(t) and magnetization vector M(t) satisfy the Bloch equation, then another set of RF field B1’(t) and magnetization vector M’(t) that is related to M(t) and B1(t) by time reversal and a 180° rotation around the x axis also satisfy the Bloch equation:


Given this symmetry, the VSP excitation shown in Fig. 2a will leave no residual from the static background regardless of the condition of the 90° pulses, if the 180° refocusing pulse is perfect. This is because the second 90° adiabatic pulse is related to the first 90° pulse by Eq. [4]. This idea is straightforward to implement for 2D-VSP, shown in Fig. 2b. The first 90° pulse is a cos/sin half planar rotation pulse (25), with Beff evolving from the y axis to the z axis. The second 90° pulse is related to the first pulse by Eq. [4]. The third 90° pulse follows immediately after the second pulse and is a cos/sin planar rotation pulse with Beff rotating from the x axis to the z axis. The last 90° pulse is related to the third pulse by Eq. [4]. By following through with the rotation matrices of the RF pulses, the resulting magnetization vector is still given by Eq. [3].

FIG. 2
Velocity-selective excitation composites with time-reversal symmetry significantly improves the degree of background suppression. a: A 1D VSP with adiabatic RF pulses and time-reversal symmetry. The trajectories of the effective B-field of the 90° ...

Relationship Between the Velocity Vector and the Complex MR Signal

The relationship expressed in Eq. [3] is approximately linear for a range of speeds, but becomes significantly nonlinear with signal aliasing at high speeds. Figure 3 shows in the (vx, vy) space contours of the percentage difference between the MR signal magnitude and the leading term of Eq. [3] that depends linearly on the speed of motion. The velocity components are measured in the phase shift they induce. Similarly, the difference between the velocity direction and the phase of the signal is shown in Fig. 4. It can be concluded from these two graphs that the gradient first moments should be chosen to be 1.0/(γvmax), where vmax is the maximum speed to be measured. In this range, the underestimation of the flow rate due to nonlinearity is less than 20% and the error in flow direction is less than 10°. The MR signal level at vmax is 80% of the full magnetization and the sensitivity to motion is approximately isotropic for in-plane velocities. Therefore, 2D-VSP generally raises the signal level over 1D methods (7). Note that for the linear region v < vmax, the signal intensity of each image pixel is a relative representation of the volume flow rate, while the velocity distribution in the pixel may not be uniform (29). However, this representation is relative in the sense that the coefficient of proportionality between the signal magnitude and the actual flow rate is unknown without additional calibration measurements.

FIG. 3
This graph is set in the 2D velocity space. The abscissa is the phase shift induced by the velocity component along x, the ordinate is the phase shift induced by the velocity component along y. The dotted contours are the magnitude of the MR signal excited ...
FIG. 4
The coordinates of this graph are the same as Fig. 3. The abscissa is the phase shift induced by the velocity component along x, the ordinate is the phase shift induced by the velocity component along y. The dotted contours are the magnitude of the MR ...


The method of 2D-VSP is demonstrated in a 5.4-cm diameter cylindrical phantom which consisted of flow channels embedded in agarose gel. Cuprite sulfate solution was added to the gel, which shortened the spin relaxation times to T2/T1 ≈ 50 ms / 200 ms. The flow configuration is shown in Fig. 5. All channels were made of polymer tubing of 3.1 mm diameter. The portion of the tubing network forming the planar ring was imaged. The average flow rate in the ring was 0.16 cc/s, average velocity 2.1 cm/s. The experiments were conducted on a 4.7T scanner fitted with a GE Accustar gradient system and a Varian Inova console.

FIG. 5
A diagram of the tubing arrangement in the agarose gel phantom. The in-flow and out-flow tubes are connected to a ring of tubing via T connectors. The direction of water flow is indicated with arrows. An image of the slice containing the tubing ring is ...

The pulse sequence parameters were as follows: In imaging experiments with the asymmetric 2D-VSP (Fig. 1c), the first and third 90° pulses were sin/cos half-passage pulses with peak B1 of 2.7 kHz, frequency sweep of 3.3 kHz, and 1.5 ms duration; the second 90° pulse was a cos/sin half-passage pulse of the same parameters. In experiments using the symmetric 2D-VSP (Fig. 2b), the 90° adiabatic pulses had the same peak B1, duration, and sweep range; however, the first and third 90° pulses were cos/sin half-passage pulses and the second and fourth were sin/cos half-passage pulses. In all experiments the 180° refocusing pulses within the VSP composites were sin/cos(±) adiabatic pulses (26) with peak B1 of 3.0 kHz, frequency sweep of 3.3 kHz, and 3.0 ms duration. The velocity encoding gradients were played in the readout and phase-encode directions, at a sensitivity of 0.247 radians*sec/cm. The image matrix was 256 × 128 and the pixel size was 1.56 × 0.78 mm2 × 5 mm. The B1 field varied by 30% across the cross-section of the phantom, as indicated by the spin-echo image of Fig. 5. The overall length of the asymmetric 2D-VSP pulse (Fig. 1c) was 16.5 ms, that of the symmetric version (Fig. 2b) was 18 ms.


Figure 6 shows a comparison between the images collected with the asymmetric 2D-VSP (Fig. 1c) and without it. The 2D-VSP suppressed the stationary background signal by a factor of 21, on average, and excited moving spins in the ring. The exception was the junctions of the two halves of the ring, which contained opposing flows (Fig. 5), and resulted in local signal cancellation. The symmetric 2D-VSP (Fig. 2b) improved background suppression from 4.8 to 2.2% (Fig. 7). These results also show that spin relaxation of the agarose gel did not prevent its suppression. The reason is that the 2D-VSP composite is essentially a sequence of two spin-echo like elements. The effect of T2 relaxation in each element is a true loss of transverse coherence, which does not affect the nutation of the coherent magnetization back to the z axis, provided the adiabatic 90° pulses are reasonably precise.

FIG. 6
Two images collected with and without 2D-VSP excitation are shown with the same intensity scale. a: A spin-echo image of the slice containing the circular flow channels. b: The same slice acquired with the asymmetric 2D-VSP excitation in front of the ...
FIG. 7
A comparison of the asymmetric 2D-VSP depicted in Fig. 1c and the symmetric 2D-VSP depicted in Fig. 2b. Static background signal was reduced from 4.8–2.2% with the symmetric 2D-VSP. a: This image was acquired with the asymmetric 2D-VSP of Fig. ...

It was shown in the Theory section that the phase values of a 2D-VSP image represent the velocity direction. In practice, the phase map also contains other terms from B0 inhomogeneity, RF field, etc. Figure 8a shows the phase map of the image in Fig. 7b. The arrows point to meeting points of opposite flow directions, marked with abrupt phase jumps. To remove residual phase contributions, a separate image was collected without the 2D-VSP and subtracted from Fig. 8a. With the corrected phase map (Fig. 8b), the direction and relative amplitude of the flow vector at each pixel can be calculated. The result is shown in Fig. 9.

FIG. 8
a: The phase map of the 2D-VSP image of Fig. 7b. The arrows point to the junctions where opposite flow directions meet. They give rise to the abrupt phase jumps. b: The phase map of a contains residuals from B0 inhomogeneity and phase variation of the ...
FIG. 9
This flow vector map is constructed from the 2D-VSP image of Fig. 7b and its corrected phase map of Fig. 8b. The darkness of the arrows is scaled to the relative flow rate as estimated from the magnitude image. The arrows point in the direction of flow, ...

To evaluate the performance of symmetric 2D-VSP under varied shim and chemical shift conditions, a series of images were acquired while the scanner frequency was set off-resonance from −300 to 300 Hz. Figure 10 plots the level of residual background vs. frequency offset. Within ±200 Hz off-resonance, stationary signal was suppressed to below 3.3% of the level without the VSP excitation. The random noise level of these images was 0.35%.

FIG. 10
A graph of the average residual background signal as a percentage of the unsuppressed background vs. off-resonance frequency, measured from a series of symmetric 2D-VSP scans. The random noise level of these images was 0.35% of the unsuppressed background ...


Symmetric 2D-VSP maps the speed and direction of a 2D flow field onto the amplitude and phase of a MR image. It employs adiabatic pulses arranged in time-reversal symmetry to improve the robustness of background suppression. Images collected at 4.7T demonstrated consistent background suppression of 30-fold or higher for the off-resonance frequency range of ±200 Hz.

The relationship between the MR signal amplitude and the speed of motion is ultimately nonlinear. When the motion-induced phase shifts are below 1.0 radian, a near linear relationship is maintained. This allows the signal level to rise to 80% of the full magnetization. In comparison, the CNR in standard phase subtraction methods can reach πM0/noise ideally, where M0 is the full magnetization. However, this difference is offset by the fact that 2D-VSP greatly suppresses the static background. It is therefore possible to reduce the background signal to very low levels by subtracting two images of opposite encoding gradients, thus raising the CNR. Ultimately, the practical value and optimal implementation of 2D-VSP needs to be assessed in vivo, in the context of other flow imaging methods.

The 2D-VSP composites presented in this article are not spatially selective and therefore not sensitive to inflow and outflow effects. It is possible to add inflow sensitivity to the spin preparation with slice-selective adiabatic pulses (30), or by combining it with saturation-recovery techniques. Additionally, many rapid data acquisition strategies have been adapted for MR angiography. Symmetric 2D-VSP is essentially a method for primary excitation. It can be integrated with an image acquisition method that is best suited for the intended application.


I thank Drs. Anthony Aletras and Vinay Pai for critiques on the manuscript.


This article is a US Government work and, as such, is in the public domain in the United States of America.


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