PII S0730-725X(99)00083-1

● Original Contribution

FLOW PROPERTIES OF FAST THREE-DIMENSIONAL SEQUENCES FORMR ANGIOGRAPHY

PABLO IRARRAZAVAL ,* JUAN MANUEL SANTOS,* M ARCELO GUARINI,* DWIGHT NISHIMURA†*Departamento de Ingenierı´a Electrica, Pontificia Universidad Cato´lica de Chile, casilla 306, correo 22, Santiago, Chile

and †Department of Electrical Engineering, Stanford University, Stanford, CA 94305-9510, USA

To reduce the scan time of time of flight or phase contrast angiography sequences, fast three-dimensionalk-spacetrajectories can be employed. The best 3D trajectory depends on tolerable scan time, readout time, geometricflexibility, flow/motion properties and others. A formalism for flow/motion sensitivity comparison based on thevelocity k-space behavior is presented. It consists in finding the velocityk-space position as a function of thespatial k-space position. The trajectories are compared graphically by their velocityk-space maps, with simu-lations and with an objective computed index. The flow/motion properties of various 3D trajectories (cones,spiral-pr hybrid, spherical stack of spirals, 3DFT, 3D echo-planar, and shells) were determined. In terms offlow/motion sensitivity the cones trajectory is the best, however, it is difficult to use it for anisotropic resolutionsor fields of view. Tolerating more flow sensitivity, the stack of spirals trajectory offers more geometric flexibility.© 1999 Elsevier Science Inc.

Keywords:Angiography; Velocity k-space; Magnetic resonance angiography.

Magnetic resonance (MR) sequences for angiography aretypically based on the well known effects of Time-of-Flight or flow enhancement,1 and Phase-Contrast2 ant.Almost any imaging sequence can be modified to doangiography based on these effects.

Although angiography can be done as a projection intwo dimensions, most cases require three dimensionalacquisitions. Three dimensional images, beside provid-ing more information at once about the subject, accu-rately register spatial relationships, thus leading to abetter diagnostic and clinical tool. This is especially truefor angiography where vessels are generally not re-stricted to a plane. The drawback of 3D acquisitions is itslonger scan and readout time.

With the development of new fast 3D sequences,3 it ispossible to acquire 3D angiography images in a consid-erable shorter amount of time. This is very important in3D phase contrast angiography where four acquisitionsare needed for collecting full velocity information, in allaxis.

When using 3D trajectories for angiography, their

behavior in the presence of flow and motion is important.This work shows the use of the 3D trajectories forangiography and investigates the properties of these se-quences using the approach proposed by Nishimura etal.4 in which the characteristics of velocityk-space val-ues are analyzed as a function of spatialk-space values.

Time of flight techniques5 exploit the difference intiming between the excitation and the readout to obtainin-flow enhancement (or suppression).

Vessel contrast (flow-related enhancement) is gener-ated by the different excitation patterns seen by thematerial that is flowing into the region of interest and bystationary material. For example, if the repetition time issufficiently short, static material reaches steady statewith spins highly saturated. Spins that are moving intothe region of excitation are relaxed, therefore, producingstronger signal than the ones that have seen previousexcitations.

In phase contrast angiography6–10 the phase of a spinmoving in the presence of a non-uniform magnetic fieldchanges as its resonance frequency varies. Contrast is

RECEIVED 6/24/99; ACCEPTED8/7/99.Address correspondence to: Pablo Irarra´zaval, casilla 306,

correo 22, Santiago, Chile. Tel:156 2 686-4292; Fax:156 2552-2563; E-mail: pim@ing.puc.cl

Magnetic Resonance Imaging, Vol. 17, No. 10, pp. 1469–1479, 1999© 1999 Elsevier Science Inc. All rights reserved.

Printed in the USA.0730-725X/99 $–see front matter

1469

obtained by encoding the spin’s velocity in its phase.Ignoring relaxation and approximating the spin positionr (t) by r 1 vt, the received signal after demodulation is:

s~t! 5 EE m~r , v!e2igSr zE0

tG~t!dt1vzE

0

ttG~t!dtD dr dv,

(1)

wherem(r , v) is the spin density as a function of positionand velocity, andG(t) is the magnetic gradient.

From Eq. (1), the extra phase introduced by the ve-locity is:

Df~t! 5 2pk v~t! z v, (2)

where

k v~t! 5g

2p E0

t

tG~t! dt. (3)

Phase-contrast angiography measuresDf(t) to findthe velocityv. To obtainDf(t) it is needed a referenceand one acquisition with velocity encoding for eachdimension.

To generate the first momentskv bipolar gradients areused so that they do not contribute to the zeroth momentbefore the readout.

3D TrajectoriesNew fast 3D trajectories have been presented by Irar-

razabal et al..3 The trajectories are designed to shortenthe scan time by reducing the number of excitations.They differ from traditional 3D trajectories in that theyemploy time-varying gradients, so the sampling does notoccur in a single straight line. Curved and echo-planartrajectories in general allow longer readouts and in turnreduce the scan time. They are also suited for sampling asphere ink-space to obtain isotropic resolution. Thesetrajectories provide flexibility for trading off SNR forscan time. If SNR is important, it can always be recov-ered by averaging.

The advantages of the 3D trajectories compared withstacking 2D slices are that there is no gap or crosstalkbetween slices;11,12 it is easier to achieve isotropic res-olution;12 it is more flexible in providing trade off be-tween SNR and scan time,13 and it does not need sharpprofiles for the excitation.

Longer readout times and curved trajectories are sen-sitive to off-resonances and motion that are manifestedas blurring and artifacts.14 This, together with the diffi-

culty of interleaving acquisitions, are the main disadvan-tages of the volumetric sequences.

To illustrate the reduction of scan time, Figure 1shows an example of phase contrast angiography ac-quired using the Spherical Stack of Spirals trajectory.The figure shows maximum intensity projections throughthe three orthogonal axes of the magnitude of the veloc-ity. The field-of-view (FOV) is 243 24 3 6 cm with aresolution of 2003 2003 50 pixels (equivalent to 1.231.2 3 1.2 mm). The acquisition was done with 928excitations (tip angle of 30°) at a repetition time (TR) of50 ms for each of the four velocity phase encodes. Thevelocity encoding gradients were designed for a range of650 cm/s.

The total scan time was 3 min 5 s. The same field ofview and resolution 3DFT angiography would have re-quired 10, 000 excitations for each velocity phase en-code, which is equivalent to a total scan time of 33 min20 s if TR is kept in 50 ms.

Fig. 1. Phase contrast angiography acquired with sphericalstack of spirals. Maximum Intensity Projection of the velocitymagnitude in the three orthogonal planes.

1470 Magnetic Resonance Imaging● Volume 17, Number 10, 1999

METHOD AND SIMULATIONS

A major concern when using long readouts in the pres-ence of time-varying gradient is the distortion producedby flowing material. In this section we will compare theflow properties of 3D trajectories using the velocityk-space analysis proposed by Nishimura et al.4

The 3D trajectories to be compared are: cones, Figure2A; spiral-pr hybrid, Figure 2B; spherical stack of spi-rals, Fig. 2C; 3DFT, Fig. 2D; 3D echo-planar, Fig. 2E;and shells, Fig. 2F as described in Reference 3.

Unless the experiment is designed to acquire andreconstruct the six dimensions of Eq. (1), any data ac-quisition for a velocityk-space position different fromzero in the presence of flow will result in distortion of the3D reconstructed data.

Nishimura et al.4 showed the relationship betweenflow and motion distortion and the velocityk-space vec-tor as a function of the spatialk-space position. Their 2Danalysis identified three properties of the velocityk-space vector (kv) that can reduce flow distortion:

● small magnitude near the origin of spatialk-space● continuity (smoothness) over spatialk-space● direction ofkv pointing away from the spatialk-space

origin

These properties are compared for 3D acquisition byinspecting two kinds of representations: one that shows,as gray level, the magnitude ofkv for different slices ofthe 3D vectorial field; and another that depicts the vec-torial fields as arrows corresponding to the projection ofthe field on that particular slice.

We also defined a Flow Sensitivity Index,M, thatrepresents a weighted average of the magnitude ofkv:

M 5 EEE w~r!uk vu dkx dky dkz (4)

wherer 5 =kx2 1 ky

2 1 kz2. For simplicity, the weight-

ing functionw(r) was chosen to be a Gaussian function

Fig. 2. Pictorial representation of 3D trajectories ink-space.

Flow properties of 3D trajectories● P. IRARRAZAVAL ET AL. 1471

Fig. 3. Gray level representation ofku(kx, ky) for each trajectory.

Fig. 4. Gray level representation ofkv (kx, ky) for each trajectory.

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w(r) 5 er2/a so values farther away from the origin areweighted less. Arbitrarily, we useda 5 244, whichmeans that the weight equals a half for pixels a distance13 away from the origin (a diameter approximately aquarter of the size of the data). The value of the index isnot very sensitive toa. It must be noted that this index,as the velocityk-space analysis, does not depend on theflow.

For each 3D trajectory, the velocityk-space vectorkv

was computed as a function of spatialk-space to obtainthe three dimensional vectorial fieldkv 5 [ku(kx, ky, kz),kv(kx, ky, kz), kw(kx, ky, kz)]. For the purpose ofcomparison, the trajectories were designed for a FOV of20 3 20 3 20 cm; a resolution of 1003 100 3 100pixels or 23 2 3 2 mm; a readout duration of 14.336ms; a TR of 50 ms; a maximum gradient amplitude of 10mT/m; and no constraint on the slew rate. 3DFT and 3Decho planar were assumed flow compensated.

Figures 3 and 4 show the gray level representation ofku andkv (components ofkv) for each 3D trajectory, asa function ofkx andky. Figure 5 showskw as a functionof kz andkx. Only the center slice is shown in each case.

To appreciate the direction of thekv vector, Figures 6,7, and 8 show an arrow representing the projection ofkv

onto the spatialk-space plane for each position on theplane.

Table 1 shows the Flow Sensitivity IndexM com-puted for each trajectory along with the number of exci-tations required and scan time, assuming a TR of 50 ms.

To confirm these results two simulations were done.In the first one, a tube of 15 pixels of diameter, 80 pixelsof length and situated in theX–Y plane at an angle of 45°was simulated with a velocity of 1 pixel/ms, for thetrajectories 3DFT, 3D EP, Spherical stack of spirals, andCones. Constant velocity plug flow was considered. Fig-ures 9 and 10 show two orthogonal slices of the 3D datafor each trajectory. The second simulation is a sphere of40 pixels of diameter that oscillates in theZ directionwith a sinusoidal movement of maximum displacementof 2.5 pixels and a period of 10 excitations. Figure 11shows the distortions in the image for each trajectory.The images are all equally windowed to better appreciatethe distortions.

DISCUSSION

The acquisition time of a phase contrast angiography canbe greatly reduced (factor of 10) by using faster 3D

Fig. 5. Gray level representation ofkw (kz, kx) for each trajectory.

Flow properties of 3D trajectories● P. IRARRAZAVAL ET AL. 1473

Fig. 6. Arrows resulting from the projection of [ku (kx, ky), kv (kx, ky)] for each trajectory.

1474 Magnetic Resonance Imaging● Volume 17, Number 10, 1999

Fig. 7. Arrows resulting from the projection of [kv (ky, kz), kw (ky, kz)] for each trajectory.

Flow properties of 3D trajectories● P. IRARRAZAVAL ET AL. 1475

Fig. 8. Arrows resulting from the projection of [ku (kx, kz), kw (kx, kz)] for each trajectory.

1476 Magnetic Resonance Imaging● Volume 17, Number 10, 1999

k-space trajectories. The spherical stack of spirals is ourchosen trajectory because it has good flow and motionproperties; and allows non cubic field of view (or reso-lution).

From Table 1 it is possible to conclude that the conestrajectory presents the best behavior in terms of themagnitude index, which is also confirmed by the simu-lations. It is followed by the spiral-PR hybrid and thespherical stack of spirals that also have relatively smallmagnitude indices. The worst trajectory in terms of flowsensitivity is the shell that presents high values for themagnitude.

For all trajectoriesukvu is smooth, except for the 3Decho planar for which discontinuities are present at theboundaries of interleaving.

As expected, the stacked trajectories (spherical stackof spirals, 3DFT and echo planar) present a smoothbehavior and small values forkw (Figs. 5C–E). Thesetrajectories are less sensitive to flow in thex-y plane.Although the artifacts for flow in that plane are bigger forthe echo planar trajectory due to the discontinuities in theky direction [Figs. 3(E) and 6(E)]. This problem can bediminished by the use of time-shifted gradients, but itstill will be present due to the long readouts. From the

Fig. 9. Flow simulation (x–y slice).

Table 1. Number of excitations, scan time assuming a TR of 50 ms, and the Flow Sensitivity Index for the six studiedtrajectories

Trajectory # ExcitationsScan time(min : s)

Flow SensitivityIndex

Cones 1152 0 : 57 7.3498Spiral-PR Hybrid 1413 1 : 10 9.2339Spherical stack of spirals 628 0 : 31 9.45873DFT flow compensated 10000 8 : 20 10.0368Echo planar flow compensation 2000 1 : 40 14.6688Shells 629 0 : 31 22.2145

Flow properties of 3D trajectories● P. IRARRAZAVAL ET AL. 1477

Fig. 10. Flow simulation (x–z slice).

Fig. 11. Motion simulation.

1478 Magnetic Resonance Imaging● Volume 17, Number 10, 1999

simulated results (Figs. 9–11) for simple flow we canappreciate the distortion for the 3D echo planar trajec-tory. That may not be the case for more complex flow.15

We note that even though the magnitude ofkv for the3DFT trajectory is small in average because of its shortreadouts, the magnitude ofkv is not negligible close tothe origin of k-space, and therefore this trajectory issensitive to flow and motion.

Overall, the cones trajectory is the least sensitive toflow because of its symmetry and small magnitude ofkv

close to the origin. The drawback of this trajectory,besides its longer scan time, is the difficulty of acquiringanisotropic resolution or field of view. The spiral-PRhybrid, next in flow sensitivity, has the same disadvan-tage. The spherical stack of spirals is the best trajectoryin terms of flow (appreciated in the simulations), thatallows some flexibility for the resolution and FOV di-mensions.

CONCLUSION

Using faster 3Dk-space trajectories for doing angiog-raphy allows an important reduction on the total scantime. At the same time it is also possible to reduce theflow and motion artifacts by choosing a trajectory insen-sitive to flow.

The velocity k-space analysis for 3D trajectoriesshows that the cones trajectory has the best properties forreducing flow and motion artifacts. The good character-istics of this trajectory are due to the readouts alwaysstarting at the origin ofk-space, and the symmetry andsmoothness of thekv vectors. The spherical stack ofspirals trajectory is also attractive because it allows moreflexibility for designing anisotropic resolution or field ofview.

An estimate of the flow behavior, such as the magni-tude index, together with the total scan time, minimumecho time, and imaging properties are important factorsto consider when choosing a 3D sequence for angiogra-phy.

Acknowledgments—The authors gratefully acknowledge the supportprovided by FONDECYT 1971310.

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