Simulation of In-Plane FlowImagingIAN MARSHALL

Department of Medical Physics and Medical Engineering, Western General Hospital, University of Edinburgh, EdinburghEH4 2XU, United Kingdom; E-mail: i.marshall@ed.ac.uk

ABSTRACT: Two methods of simulating the imaging process in magnetic resonance( )MR are explained: namely, the k-space and the isochromat summation methods. The lattermethod is more closely related to the actual behavior of magnetic spins, and is used in theremainder of this work. The MR image appearance of in-plane fluid flow is simulated usingthe isochromat summation technique. Flow-related artifacts are demonstrated, together with

( )flow compensation, phase-contrast PC velocity encoding, and cine-PC measurement ofpulsatile flow. The special considerations necessary for simulating pulsatile flow with areverse flow component are discussed. Magnetic field inhomogeneities and susceptibilityeffects are readily incorporated. The computation time required for the simulations isdiscussed in some detail. Simulations of flow imaging have been found useful in thedevelopment and validation of new pulse sequences, and for teaching purposes. Q 1999

John Wiley & Sons, Inc. Concepts Magn Reson 11: 379 ] 392, 1999

KEY WORDS: MRI; simulation; flow; computation; artifacts

INTRODUCTION

Despite the evident usefulness of simulations ofŽ .magnetic resonance imaging MRI , there are rel-

atively few implementations described in the liter-Ž .ature 1]6 and even fewer reports of the simula-

Ž .tion of flow phenomena 7]10 . This may be duein part to the high computational demands re-quired for realistic simulation. However, with theincreasing speed and availability of powerful com-puters, such simulations are within reach of mostlaboratories. With the use of routine clinical MRIexpanding from simple anatomical imaging to theinvestigation of physiological function, there is anincreasing need to develop and evaluate morecomplex pulse sequences. Simulation of such se-quences allows their characteristics to be deter-

Received 7 January 1999; accepted 10 June 1999Ž . Ž .Concepts in Magnetic Resonance, Vol. 11 6 379]392 1999

Q 1999 John Wiley & Sons, Inc. CCC 1043-7347r99r060379-14

mined in advance of their implementation onheavily used scanners, and can also serve as ateaching aid. Simulations are particularly usefulfor the investigation of flow phenomena becauseof the relative difficulty of conducting in ¨itroexperiments. In this article, we consider the simu-lation of in-plane fluid flow in simple and flow-compensated imaging sequences, and in velocity-encoded sequences. The simulation geometry isdiscussed, common artifacts are demonstrated,and methods for reducing those artifacts areshown. The modified simulation geometry neces-sary for pulsatile flow having a periodic reversalof flow direction is outlined. The computationtime is discussed in some detail.

THEORY

Two fundamentally different approaches to simu-Ž .lating magnetic resonance MR imaging se-

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quences have been described in the literature:Ž . Žnamely, the k-space formalism 4 and time-do-

. Ž .main isochromat summation 5 . The two meth-ods will be reviewed briefly before the representa-tion of flow is discussed. For the most part, werestrict the discussion to the formation of two-di-mensional images, i.e., we do not generally con-sider slice selection.

K-Space Method

Ž .This method 4 takes advantage of the convolu-tion theorem, with the simulated image beingformed from the convolution of the ‘‘object’’ and

Ž .the imaging point spread function. The two-di-Ž .mensional fast Fourier transform of the object

is first computed, which yields an idealized k-spacedata set. The data are then multiplied on a point-

Ž .by-point basis by the k-space time domain tra-jectory of the sequence. The effects of limited

Ž .sampling truncation and relaxation are there-Žfore taken into account data points acquired

later are weighted by smaller values owing to T2.decay . The imaging convolution is effectively car-

ried out in k-space by the equivalent multiplica-tion. Finally, the weighted k-space data are in-verse Fourier transformed to yield an image.

The method is very elegant mathematically, inthat MR imaging is represented as an example ofa general imaging process with the usual convolu-tion formulation. It works efficiently for single-shot techniques involving only one radiofre-

Ž . wquency RF excitation such as echo-planar imag-Ž .xing EPI , for which the computational load is

low. However, the k-space method has two short-comings. First, the majority of MR sequences ofinterest use multiple RF pulses, for which thissimple convolution approach breaks down. Oneneeds to consider the transfer of magnetizationbetween different coherence states created by

Ž .every RF pulse 11 and treat them as separateŽ .‘‘partitions’’ 4 . Second, each tissue type with

Ž .differing relaxation time constants T , T and1 2proton density in the simulated object must betreated separately, and the results for all suchtissues summed. This is not a great drawback forsimulating simple objects which may have 10 or sotissue compartments, but becomes problematicalfor more realistic objects based possibly on realmedical images. It is also a serious drawback forsimulations involving complex flow, as there willbe a continuous range of flow velocities, all ofwhich require separate treatment. For a matrixsize of N, the intrinsic N 2 computational load is

thus multiplied by the number of different tissueŽ .types or flow velocities and RF partitions. In the

worst case, each element in the object may havearbitrary parameters, and the k-space methodthen becomes of at least fourth-order complexity.

Isochromat Summation Method

Ž .In the isochromat summation or time-domainŽ .method 5 , the evolution of the magnetization of

each spin is followed explicitly throughout thesequence, using the Bloch equations. The objectis considered as a two-dimensional array of ‘‘spinelements’’ in the x, y plane, with each elementhaving attributes of equilibrium magnetization, T1and T values. RF pulses rotate the magnetiza-2tion of each spin element by a specified flip angleabout a specified axis, whereas evolution in be-tween RF pulses consists of T decay, T recov-2 1ery, motion, and the effects of applied gradients.During ‘‘sampling,’’ the contribution from all theelements is summed at each sample time step, toform a k-space data array. The k-space data arefinally Fourier transformed to yield an image.

Although the isochromat summation methodmay not be as elegant mathematically as thek-space method, it nevertheless warrants studybecause it closely follows the ‘‘real’’ physical pro-cesses involved in spin manipulation and signalacquisition. Furthermore, the method is inher-ently suited to the simulation of arbitrary objects,and it is very straightforward to incorporate theeffects of motion and inhomogeneities. The com-plete time course of magnetization evolution isŽ .in principle available for every spin element.

The drawback of the isochromat summationmethod is its computational complexity. For two-dimensional spin-warp imaging as considered inthis work, the number of time steps increases as

Žthe square of the image matrix size N phase.encodings and readout samples , and since the

number of spin elements also increases as N 2,the entire simulation is intrinsically a fourth-order

Ž .process. Furthermore, Shkarin and Spencer 5explained how it is not sufficient to have a singlespin element per image pixel. In simulations, thecontinuous integral of magnetization across a pixelis replaced by the sum of the contributions from

Ž .the discrete elements see Appendix . This intro-duces an error in reconstruction of the images,which those authors found could be reduced to1.5% by using a ratio of three elements to eachpixel, in each direction. In the Appendix, we alsoconsider the number of spin elements required to

SIMULATION OF IN-PLANE FLOW IMAGING 381

accurately model gradient spoiling and intravoxelphase dispersion.

The computational load of the isochromatmethod thus increases as N 4, whereas that of the

2 Žk-space method ranges from N single-shot se-.quence and only one tissue type to greater than

4 Ž .N multishot sequence and arbitrary object . Inthis work, we used the isochromat summationmethod because it reflects more closely the physi-cal evolution of the elemental spins. The matrixmanipulations for these operations are summa-rized in the Appendix.

Flow

Cyclic object motion and fluid flow have previ-ously been incorporated in imaging simulationsŽ .2, 4, 7]10 , but usually with some simplification.

Ž .Siegel and colleagues 8, 10 , for example, com-bined the output of computational fluid dynamicŽ .CFD calculations with a simple one-dimensionalapproach to MRI in which only the additionalphase shifts caused by the flow were considered.

Ž . Ž .van Tyen et al. 7 and Jou et al. 9 carried outmore complete simulations of two-dimensionalflow in more complex geometry, again couplingCFD with a consideration of the phase evolutionof spins during the imaging sequence. The simu-lation of in-plane flow imaging using the fullisochromat summation method appears, however,to have received little attention, and is the mainthrust of this work.

There are two different ways of representingflow. In the first method, the simulation tracks aset of spins which have a uniform distributioninitially. Associated with each spin is its x, yposition and magnetization M. The position infor-mation is updated according to the flow. In thesecond method, we refer to a fixed grid of evalua-tion points, and associate with each point themagnetization of the nearest spin. As spins move,their magnetization is transferred to an updatedgrid point according to the flow. The two methodsare equivalent, and in fluid dynamics are knownas the Lagrangian and Eulerian views, respec-

Ž .tively 12 . In this work, we used the Eulerianmethod. We restricted the simulations to the sim-ple analytical cases of plug and laminarŽ .Poiseuille flow in straight tubes for two reasons.First, we wanted to compare the simulated imageswith well-known analytical solutions which existonly for these very simple cases. Second, it wasnot our intention at this stage to introduce the

CFD concepts which would be necessary to de-scribe more complex flow patterns.

Arterial blood flow is driven by the heart andis therefore pulsatile in nature. Typically, the flowvelocity rises quickly to a high maximum for arelatively short duration before falling to a lowvalue for most of the cardiac cycle. In manyvessels of interest, the flow in the latter part ofthe cycle may be in the reverse direction, and flowsimulations must be able to handle this situation.

SIMULATIONS

All simulations were based on the isochromatsummation technique as described above and inthe Appendix, and were written in C on a Sun

Ž . ŽUltrasparc 1 170-MHz computer Sun Microsys-.tems, Mountain View, CA . The simple flow ge-

ometry is shown in Fig. 1. Input objects werestored as 512 = 512 data files which were loadedinto the program. Run-time parameters such as

Ž . Ž .field of view FOV , repetition time T , echoRŽ . Žtime T , direction, speed, and type of flow plugE

.or laminar were read in from parameter files.The basic spin-warp sequence is shown in Fig. 2,and consists of RF excitation followed by simulta-neous phase encoding and readout gradient de-

Žphasing, followed by a 1808 refocusing pulse for.spin echo only , and signal acquisition during the

application of the readout gradient. Finally,spoiler gradients S are applied simultaneously inthe x and y directions to minimize residual trans-verse magnetization which would otherwise per-

Ž .sist into the next phase-encoding repetition andlead to stimulated echoes. In the simulations, RFpulses and spin evolution are implemented by

Figure 1 Simple geometry for simulation of in-planeflow. Flow is from left to right. Only spins within theFOV are affected by RF pulses.

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Ž .Figure 2 a Basic gradient echo pulse sequence usedfor the imaging simulations; G s phase encoding gra-pdient; G s readout gradient; S s spoiler pulses; T sr Eecho time. For spin-echo sequences, the polarity of the

Žreadout gradient dephasing lobe is inverted dashed. Ž .line ; b flow compensation pulse to be added to the

Ž .readout gradient; c phase-contrast velocity encodingŽ .pulses to be added to the readout gradient; d bipolar

phase-encoding pulse for reduction of displacementartifact.

means of the matrix manipulations given in theŽ .Appendix. Optional flow compensation 13 and

Ž .phase contrast velocity-encoding gradients 14may be added to the gradient echo version of the

Ž .sequence as shown in Fig. 2 b, c , respectively.Imaging gradient amplitudes were calculated in

Žthe usual way depending on gradient duration,.FOV, and matrix size to give a phase shift of p

per pixel for spins at the edge of the FOV. ForŽ .the phase-contrast PC simulations, it can be

shown that the phase f accumulated by spinsn

moving with velocity n under the influence of thevelocity encoding gradient pulses as shown in Fig.Ž .2 c is

Ž . Ž .f s gnG t 2T y t 1n n

where g is the gyromagnetic ratio, G the veloc-n

ity-encoding gradient amplitude, t the duration ofthe gradient pulses, and T the time between thestart of the first pulse and the end of the secondpulse, and the pulses are placed symmetricallyabout T r2 as shown. G was calculated to pro-E n

duce a phase shift of p for spins moving atvelocity n.

The active region of the coil was taken to beidentical to the selected FOV. In the work pre-sented here, the object consisted of an initiallyuniform field of spins, with a selected portion of

Ž . Ž .them in the vessel in motion flowing across theFOV. The vessel wall was not modeled. Freshspins with equilibrium magnetization enter fromthe left, move across the FOV, and exit from theright, being discarded.

The simulations consisted essentially of fourŽ . Ž . Ž .nested loops Fig. 3 : a phase encoding, b

evolution and sampling of the magnetization fol-Ž .lowing each excitation, c a loop over the x

Ž .direction, and d a loop over the y direction toapply the evolution to every elemental spin. Ateach sampling time point, the contribution of allthe elements was summed and the resultant

Ž .transverse magnetization M and M saved inx yk-space files. Flow was taken into account bycomputing the shift in position of the relevantspins from their velocities and the time incre-ment. In general, the shifts are a nonintegernumber of pixels, and the excess over an integernumber is carried forward to the next time step toavoid cumulative errors. To calculate flow effectscorrectly, spin evolution and shift must be calcu-lated for each time increment whenever a gradi-ent is applied, and not just during the acquisitionperiod. The time increment was set equal to thesample time interval, i.e., the total acquisitionperiod t divided by the matrix size. Simulationacqof images with a matrix of N s 128 typically took8]15 h depending on the gradient durations in aparticular sequence.

Figure 4 shows the revised geometry necessaryfor the simulation of flow which may momentarilybe reversed. With retrograde flow, spins which

SIMULATION OF IN-PLANE FLOW IMAGING 383

Figure 3 Flowchart of the isochromat summationmethod applied to a spin-warp sequence. For an imagematrix of size N, N samples are acquired for each of Nphase-encoding steps. Each sample of the signal con-sists of a two-dimensional summation over all the ele-mental spin magnetizations at the sampling time; thus,the simulation method is inherently fourth order in N.All spin evolutions are calculated over time incrementsof dt, the sample interval. Spin evolution includes theeffects of relaxation times T and T , motion and1 2applied gradients.

Figure 4 Modified geometry for simulation of pul-satile flow with a reverse component. Net flow is fromleft to right. The distance L corresponds to therevintegral of the reverse flow velocity.

Ž .have exited from the right downstream edge ofthe FOV may return, and therefore they cannotsimply be discarded. Similar considerations apply

Ž .at the left upstream edge. It is therefore neces-sary to keep track of spin evolution beyond theFOV, for at least a distance L equal to therevintegral of the reverse velocity of the flow wave-

w Ž .xform see Fig. 8 d .Visualization of the results was achieved by

Ž .using Matlab The Mathworks, Natick, MA . AMatlab script took the k-space data files andperformed a complex two-dimensional Fouriertransform before displaying either the magnitudeor phase of the resulting image. Images were notsmoothed or manipulated in any way.

We studied in-plane flow that had either plugŽ .or laminar Poiseuille characteristics. Spin echo,

and gradient echo sequences with and withoutflow compensation were simulated, together with

Ž .phase-contrast PC velocity measurement se-quences. Pulsatile flow was also investigated, inwhich case the imaging sequence was run in cine-

ŽPC mode, with a series of rapid acquisitions short. Ž .T synchronized to ‘‘triggered’’ by the wave-R

form. In all PC velocity measurement sequences,it is necessary to acquire the images both withand without velocity-encoding gradients, so thatthe phase difference due to motion alone can beextracted. Finally, we looked at a simple exampleof three-dimensional simulation, investigating howthe slice thickness affects the appearance of flow.

RESULTS

Figure 5 shows the result of imaging in-planeŽPoiseuille flow i.e., flow that has a maximum

velocity in the center of the vessel, and with thevelocity falling parabolically with increasing dis-

.tance from the center with a simulated spin-echosequence. The axial velocity was 700 mmrs from

Ž .left to right in the x readout direction, and hasbeen imaged by a gradient-echo sequence. TheFOV is 256 mm and the matrix size is 128, lead-ing to a pixel size of 2 mm. The acquisition timeŽ .t is 7.68 ms, corresponding to a pixel band-acqwidth of 130 Hz, which is typical of commonlyavailable sequences. The sequence repetition timeT is 400 ms, T is 16 ms, and the vessel has aR Eradius of 6 mm. The fluid parameters are T s1

Ž .1000 ms and T s 250 ms to mimic blood 15 ,2and similar fluid surrounds the vessel. The vesselwall is not modeled. With the basic imaging se-quence, only the central thread of the flow is

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Ž .Figure 5 In-plane laminar Poiseuille flow with peak velocity of 700 mmrs in the xŽ .readout direction. Gradient echo sequence with T s 400 ms, T s 16 ms, FOV s 256R E

Ž .mm. The radius of the vessel is 6 mm. a Only the central thread of fluid is visible becauseŽ .of intravoxel phase dispersion; b sequence with flow compensation in the readout direc-

tion. The entire flow is now visible, its brightness being determined by inflow effects.

w Ž .xclearly visible Fig. 5 a . Away from the center,the range of velocities within each pixel leads tointravoxel phase dispersion and partial cancela-

Ž . Ž .tion of signal 13 . Fourier ‘‘ringing’’ artifactsare visible in the background, owing to the trun-

Ž .cated k-space data. In Fig. 5 b , the sequence ismodified to incorporate flow compensation in the

Ž .readout x direction. The flow compensationw Ž .xgradient Fig. 2 b ensures that all spins have

zero phase at the center of the echo, regardless oftheir velocities. This is achieved by setting the

Ž .first moment m of the gradient to zero 13, 14 ,1where

TE Ž .m s Gt dt 2H10

ŽThe corresponding zero-order moment is zero.for all imaging sequences. With flow compensa-

tion in the flow direction, fluid throughout thevessel is now visible. The flowing fluid appearsbright relative to the background because of the

Ž .inflow of fresh spins the ‘‘inflow’’ effect ; those inthe surrounding fluid are partially saturated be-cause of the multiple excitations with a T shorterRthan the fluid’s T . Slower flow away from the1center of the vessel has a progressively lowersignal intensity owing to increased saturation, un-til the signal merges into that of the stationarysurrounding fluid. A parabolic signal void appearsat the extreme left of the flowing fluid, which

corresponds to the distance traveled in the echotime. Spins in this region were outside the activeregion of the coil at the time of the RF excitationpulse, and are thus still in their equilibrium state

Ž .with no observable transverse magnetization.Figure 6 illustrates the well-known displace-

Ž .ment artifact 16 that occurs when imaging in-plane flow that is not aligned along one of theimage axes. Plug flow is used to clarify the behav-ior. It can be seen that the bright flowing fluid isdisplaced to the right relative to its true positionin the vessel. The artifact occurs because of thetime interval between encoding the y position of

Ž .the spins during phase encoding and encodingŽthe x position effectively at the center of the

.echo . During this time, which is approximatelyequal to T , the spins have moved a distanceEwhich depends on T and their velocity. TheEartifact may be minimized by simply reducing T ,Ebut more sophisticated methods are able to al-

Ž .most eliminate the effect 16 . One such methoduses bipolar phase encoding to effectively alignthe time of the phase encoding with the center of

w Ž .xthe readout period Fig. 2 d . Nishimura et al.gave expressions for the timing of the bipolarlobes as

4TE Ž .t s t q 2 y 1 30 (ž /t0

SIMULATION OF IN-PLANE FLOW IMAGING 385

Ž .Figure 6 In-plane plug flow of 700 mmrs in a diagonal direction, with T s 25 ms. aEProminent displacement artifact due to the time interval between phase encoding and

Ž . Ž .readout; b With bipolar phase encoding as shown in Fig. 2 d , the displacement artifact iseliminated. Other parameters as for Fig. 5.

and

t01 Ž .a s 1 y 42 ž /t

where t is the duration of the corresponding0simple phase encoding gradient, T is the echoE

Ž .time, and t and a are as defined in Fig. 2 d .When this modified sequence is run with diagonalplug flow, the displacement artifact is eliminatedw Ž .xFig. 6 b .

Ž .Figure 7 a shows the effect of the displace-ment artifact when imaging laminar flow in adiagonal in-plane direction. Since the displace-

Ž .ment depends on the flow velocity and T , it isEgreatest for the central part of the flow, leadingto a pileup of signal along one wall and distortionin the intensity profile across the vessel. Thiseffect clearly has serious consequences for theinvestigation of flow in ¨itro and in ¨ i o. Use ofthe bipolar phase encoding gradient techniquew Ž .xFig. 7 b effectively removes the artifacts so thatthe flow is imaged correctly, with the brightnessof the flowing spins depending on their velocity.

A phase-contrast velocity encoding sequence isŽ .used in ‘‘cine’’ mode cine-PC with a low flip

Ž . Ž .angle 308 and a short T 50 ms to imageRpulsatile flow in Fig. 8. The pulsatile velocity

w Ž . xwaveform used Fig. 8 d , line represents an arte-

Ž .rial blood velocity waveform 17 . The cycle timeT of 857 ms corresponds to a heart rate of 70cyclebpm. The imaging sequence is ‘‘gated,’’ i.e., trig-gered by the start of the pulsatile waveform, with17 time frames being acquired during each pul-satile cycle. One line of k-space is acquired dur-ing each cycle, so that the acquisition takes a

Ž .simulated 128 cycles. Figure 8 a, b shows themagnitude and phase images corresponding to

Ž .the second time frame at 50 ms . The phase hasbeen ‘‘unwrapped’’ within the object and set tozero outside the object. Note that the brightinflow effect has been removed in the phase dif-

Ž .ference image. Compared with Fig. 5 b , the brightinflow region in the magnitude image is shorterbecause the time-averaged axial flow velocity isonly 78 mmrs rather than 700 mmrs. The inflowsignal void corresponds to the distance traveledbetween spin excitation and signal acquisition for

Ž .this time frame. Figure 8 c shows a vertical pro-file through the phase difference image of Fig.Ž .8 b . The parabolic velocity profile within the

vessel is clearly recovered. The velocity encodingw xgradient was calculated from Eq. 1 to give a

phase shift of p for a maximum velocity of 1000mmrs. Thus, the center of the laminar flow,having a velocity of 950 mmrs, is encoded with a

Ž .phase of 3 0.95p radian. The velocity measuredfrom each frame of the phase images is plotted in

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Figure 7 In-plane Poiseuille flow with peak velocity of 700 mmrs in a diagonal direction.Ž . Ž .Flow compensation in the readout horizontal direction, and T s 25 ms. a The displace-E

ment artifact is worst for the high velocity central flow, leading to an asymmetricalŽ . Ž .appearance of the vessel; b with bipolar phase encoding as shown in Fig. 2 d , the

displacement artifact is eliminated, and the flow appears correctly with its brightness beingdetermined by inflow effects.

Ž .Figure 8 Gated cine phase-contrast cine-PC velocity-encoded sequence applied to pul-Ž . Ž . Žsatile Poiseuille flow. a Magnitude and b phase images taken at t s 50 ms second frame.of the cine sequence , corresponding to maximum velocity flow. The velocity-encoding

w Ž .xgradient pulses Fig. 2 c were selected to give a phase of p for a flow velocity of 1000Ž .mmrs. Flow-compensated sequence with T s 16 ms and T s 50 ms; c vertical profileE R

Ž .through the phase image. The Poiseuille flow had a maximum central velocity of 950Ž . Ž .mmrs, for which the phase is theoretically 3.0 radian; d original velocity waveform line

Ž .and velocity estimates reconstructed from the phase images points .

SIMULATION OF IN-PLANE FLOW IMAGING 387

Ž . Ž .Fig. 8 d points , from which it can be seen thatthe agreement between the original waveformand the extracted velocity is excellent.

Although three-dimensional simulation is notthe main thrust of the present work, Fig. 9 showsa simple example of the effect of slice thicknessand position on the image appearance of in-planeflow. The simulations described so far were re-peated for a number of subplanes at 0.5-mmspacing, with the appropriate flow profile calcu-lated for each position. The resulting k-spacedata sets were then summed and transformed to

Ž .images. For reference, Fig. 9 a is the centralsubplane as presented in earlier examples. It cor-responds to an infinitesimally thin slice thickness.

Ž .In Fig. 9 b , the central subplane data are summedwith the data from the next four subplanes, repre-senting a slice thickness of 5 mm centered on the

Žcenter of the vessel symmetry considerationsmean that we need sum only the subplane data

. Ž .from one side of the central plane . In Fig. 9 c ,the summation has been extended to the first 10planes, representing a central 10-mm-thick slice,or, equivalently, a 5-mm-thick slice offset fromthe central plane by 2.5 mm. Note that the ap-

pearance of the flow is only weakly dependent onthe slice thickness under these conditions.

DISCUSSION

A crucial issue in the usefulness of simulationssuch as these is the time required for their calcu-lation. It is important to optimize the code as

Žmuch as possible for the inner loops i.e., the.matrix computations detailed in the Appendix

where most time is spent. The time incrementused in the calculations must also be carefullyconsidered. During the ‘‘acquisition’’ part of the

Ž .sequence Fig. 2 , we used a time increment equalŽto the sampling interval total acquisition time.divided by matrix size N , so that N steps are

required. Thus, in most of this work, the funda-Žmental time increment was 60 ms 7.68 ms divided

.by 128 . In the absence of flow, it is sufficient toŽcalculate each of the other evolutions during.phase-encoding, spoiling, T filling, etc. as aR

single time increment, since intermediate resultsare not required. However, when simulating

Figure 9 Example of three-dimensional flow simulation showing effect of slice thickness onŽ .imaging of Poiseuille flow in a vessel of radius 6 mm. a Idealized central slice of zero

Ž . Ž .thickness; b 5-mm-thick slice centered on vessel; c 5-mm slice offset 2.5 mm from centerŽ .of vessel. In b, c , the images are calculated by summing data acquired from elemental

planes at 0.5 mm spacing in the slice-selection direction. Flow-compensated gradient echosequence with T s 25 ms.E

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Ž .steady flow, it is necessary to update the spinhistories throughout the periods when gradientsare applied, and we must use a suitably small timeincrement. We used the sample period of 60 msfor convenience. With phase encoding and spoil-ing periods both equal to 3.84 ms, there is thus atotal of 256 time steps per repetition at whichspin evolution must be calculated. This takesabout 8]9 h on the 170-MHz Ultrasparc com-puter used, for the matrix size of 128. When the7.68-ms flow compensation gradient lobe is in-cluded, the simulation time correspondingly in-creases to 12 h. When the flow is pulsatile, it is nolonger possible to treat the T filling time as oneRevolution step. The velocity varies throughout thistime, and the spin evolution must be calculated in

Žsufficiently small increments this also applies to.any delay period when gating the sequence . It is

not necessary to use steps as small as 60 msŽindeed, the pulsatile waveform used here was

.specified at 1-ms intervals , and we chose 10 msas being sufficiently accurate. This adds a further

Ž .80 or so time steps 2 h of computation whenT is 857 ms. In cine-PC simulations, ncycle framestime frames are acquired during each pulsatilecycle, and the computation time rises by a factorof n . It is also necessary to run the simula-framestion twice, once with and once without the veloc-ity-encoding gradients. Consequently, the full

Ž .simulation of a 17-frame T s 50 ms cine-PCRsequence with a matrix size of 128 took morethan a week of computation. For steady flow, thecomputation time is independent of the sequenceT .R

The simulations can, of course, be greatlyspeeded up by reducing the matrix size or byconsidering fewer background spins. For the cine-PC simulation, for example, we used an array of1024 = 128 spin elements, compared with the 512= 512 for the steady flow simulations. As ex-plained in the Theory section, the isochromatsummation method is inherently fourth order withrespect to the matrix size. Thus, all other thingsbeing equal, simulations with a matrix size of 64could be run in - 1 h. If the source object

Ž .resolution of 512 = 512 elements is maintained,the time taken decreases only as the square of thematrix size, and so the simulations would takeapproximately 4 h. However, images with a matrixas coarse as 64 were not considered acceptablefor demonstrating flow effects, and a matrix of128 was judged to be the minimum useful size.Doubling the matrix size to 256 would entail a

Ž16-fold increase in computation time since theobject resolution would also need to be doubled

.to satisfy the criteria discussed in the Appendix ,leading to simulations lasting up to 2 weeks. Thiswas not considered worthwhile with the computercurrently available.

For similar reasons, we have not attempted toŽ .model the third slice selection dimension be-

yond the simple simulation of Fig. 9. To do sowith a realistic spatial resolution would be pro-hibitively slow using the present computer plat-form. However, three-dimensional flow geometryis extremely important, and it would be of greatinterest to perform realistic simulations.

Although we can confidently predict that thespeed of available computers will continue toincrease for some time to come, it is evident thata more drastic solution is necessary if large-scalesimulation is to become a widespread researchtool. Fortunately, the time-consuming matrix op-erations can be carried out largely independentlyfor each spin, and thus the problem of MRIsimulation lends itself to parallel computation

Ž .techniques 6 . In flow simulations, the main re-striction on parallel computation is the need tomove the spins after each time step, which will

Ž .introduce a possibly significant overhead. ForŽstraight flow, each row of spins having the same

.velocity can be treated separately, and the resultscombined. A multiprocessor or distributed work-station platform consisting of a few dozen CPUswould enable high-resolution simulations of two-dimensional flow to be run in a few hours, anduseful three-dimensional flow simulations in afew days.

A simple three-dimensional example of flowsimulation is presented in Fig. 9, in which flowwas in-plane, i.e., with no flow between the sub-planes used in the calculation. A rectangular sliceexcitation profile was assumed for this example,but a more realistic profile could have been simu-lated by appropriately weighting the contributionof the individual subplanes. Another refinementwould be the inclusion of random measurementnoise in the acquired k-space samples, whichwould contribute less to images of thicker slices.The simulation of true, arbitrary three-dimen-

Ž .sional flow i.e., with flow between the subplanesis beyond the scope of the present work.

We have dealt exclusively with simple plug orŽ .laminar Poiseuille flow in straight vessels, for

which analytical solutions are available for com-

SIMULATION OF IN-PLANE FLOW IMAGING 389

Žparison. For more complex geometry such asvessels of varying diameter, curving, and branch-

.ing and for high-velocity flow when turbulencecannot be neglected, the methods of CFD arenecessary to predict the flow patterns. The flowcan then be linked to the MR simulation viasuitable lookup tables for the motion of the spinelements. The complexity of the MR simulationwould be no greater than presented here, but isoutside the scope of the present work. Siegel and

Ž .colleagues 8, 10 applied linked CFD-MR calcu-Ž .lations in a limited one-dimensional way to sim-

ulate MR images of flow in a vessel of varyingŽ .cross-section. Long and colleagues 18 used MRI

and phase-contrast velocity mapping sequences in¨ i o to provide input data for a subsequent CFDcomputation of blood flow at the abdominal bi-furcation of the aorta, but they did not feed backthe calculated flow velocities into imaging simula-tions. The most complete CFD-MR sequencesimulations to date appear to be those of two-di-mensional flow in curved and branching geometry

Ž .carried out by van Tyen et al. 7 and Jou et al.Ž .9 . Again, they coupled CFD calculations of flowstreamlines with a consideration of the phaseevolution of spins during the imaging sequence.They did not model explicitly the evolution ofelemental spins as in the complete isochromatsummation method presented in this work. Theseauthors greatly speeded up their calculations byconsidering only the flowing spins and ignoringthe background. This may be a very useful tech-nique so long as one is aware of potential dis-placement artifacts and is not interested in evalu-ating image contrast. Signal loss due to turbu-

Ž .lence was considered by Gatenby and Gore 19 .In the present work, pulsatile flow was consid-

ered laminar throughout the pulse cycle, althoughthis is a gross simplification. It would be relativelystraightforward to incorporate a time-dependentprofile, based perhaps on CFD or experimentaldata. An approximate analytical solution is avail-able by decomposing the pulsatile waveform intosinusoidal components, for each of which a solu-

Ž .tion in terms of Bessel functions is known 20 .Magnetic field inhomogeneities and magnetic

susceptibility effects are readily incorporated inŽ .the simulations by means of extra gradients 3 .

These gradients are constant with time but maybe position dependent. As discussed above, vary-ing flow in the presence of gradients requires theuse of small time increments to handle the spin

evolution correctly, and therefore these simula-tions take correspondingly longer to run. Presatu-

Žration pulses each consisting of an RF excitation.pulse followed by a spoiler gradient are readily

Ž .modeled, and were studied in previous work 21 .In that work, presaturation bands were simulatedby restricting the effects of the RF pulses to aspatially defined region of the FOV. As a moregeneral case, nonuniformities of the transmittercoil could be simulated by making the excitationflip angle a function of position within the FOV.Similarly, nonuniformities of the receiver coilcould be simulated by suitable spatial weightingof the elemental spin signals that are summed to

Ž .form each k-space sample Fig. 3, inner loop .Although we have simulated conventional spin

warp sequences in this work, there is no reasonwhy other sequences such as fast spin echo, echo

Ž .planar imaging, and BURST 22 could not beŽ .studied 5 . The matrix operations representing

RF excitation and spin evolution are sufficientlygeneral that only a change of loop structure would

Ž .be needed. Similarly, refocused steady-state gra-dient echo sequences can be implemented byomitting the spoiler gradient pulses and addingrefocusing gradient pulses. In such sequences, theapproach to steady-state magnetization is slow,

Ž .and is known to cause additional artifacts 23 .We have considered the action of RF pulses to

be instantaneous. In real sequences, they have afinite duration, and this is especially true whenthey are ‘‘soft’’ pulses used for slice selection. Amore comprehensive simulation would model thepulses more accurately and include spin motion

Ž .during the pulse durations 24 . We have alsoneglected the effects of diffusion in the presentwork. This simplification is generally valid for themodest gradient amplitudes and durations used inmany imaging sequences, but will lead to physi-cally unrealistic results for sequences which in-clude large gradients}for example, in MR mi-croscopy.

In conclusion, we have successfully demon-strated, on a midrange workstation, the simula-tion of MR images of flow, including the appear-ance of artifacts, phase-contrast velocity encod-ing, and gating of pulsatile flow. Although some

weffects of flow can be calculated analytically forexample, the signal attenuation when subjected to

Ž .xa train of RF pulses 25, 26 , many cannot, andin any case such calculations rarely predict imageappearance completely. There is thus a continu-

MARSHALL390

ing need for actual simulations such as thosereported here. Simulations are particularly usefulfor the investigation of flow phenomena becauseof the relative difficulty of conducting in ¨itroexperiments. The full simulation of spin evolutionis, however, time-consuming, taking several hoursper image on commonly available computers. It isa sobering thought that nature carries out thesame operations without approximations, in aslittle time as a few seconds.

APPENDIX

Evolution Matrices

Ž .The magnetization M s M , M , M has initialx y zŽ .value 0, 0, M . In the absence of diffusion, the0

evolution of M can be described by the Blochequations. The instantaneous value of M justbefore an RF pulse is depicted by My, and theeffect of the pulse is to rotate the magnetizationto a new value Mq, where Mqs RMy. R is therotation matrix corresponding to the pulse, andmatrix multiplication is implied. For rotations bya flip angle a about the x and y axes, R takes theforms

1 0 0Ž .R s A10 cos a ysin aa , x ž /0 sin a cos a

and

cos a 0 sin aŽ .R s A20 1 0a , y ž /ysin a 0 cos a

respectively. In between RF pulses, the magneti-zation evolves according to M9 s TM whereaug

w xM is the augmented matrix M M , and T isaug 0the evolution matrix for a time interval of t, givenby

e cos f ye sin f 0 02 2

e sin f e cos f 0 0Ž .T t s 2 2� 0Ž .0 0 e 1 y e1 1

Ž .A3

Ž .where e s exp ytrT describes longitudinal re-1 1Ž .laxation, and e s exp ytrT describes trans-2 2

verse relaxation. The phase f depends on theŽ .applied gradient G and the position r x, y of the

spin according to f s gGr t, where g is the gyro-magnetic ratio.

k-Space Reconstruction and GradientSpoiling

Ž .Shkarin and Spencer 5 pointed out that at leastthree elements per pixel in each direction werenecessary to reduce the k-space reconstructionerrors to - 1.5%. A further requirement thatdoes not appear to have been considered in thecontext of imaging simulations is that gradientspoiling of transverse magnetization must be han-dled very carefully. We have previously shownŽ .21 that the discrete summation of isochromatsin the presence of a spoiler gradient leads to aresidual magnetization that has artifactual max-

Žima as a function of the spoiling strength gradi-.ent amplitude multiplied by time . These false

maxima occur for values of the maximum phasewrap f across a pixel equal to an integer multi-mple of 2 Lp, where L is the number of isochro-

Ž .mats elements along each direction of a pixel.This effect, like the k-space reconstruction error,is a consequence of the difference between theidealized integral

1 ifm Žyif. Žy if t .mŽ . Ž .e df s e y 1 A4Hf f t0m m

and the discrete sum actually used in numericalsimulations

Ly1 Žyif .m1 1 1 y eŽyi jf r L.m Ž .e s A5Ý Žyif r L.mž /L L 1 y ejs0

Ž .True minima occur in both cases when f is amŽ .multiple of 2p but not of 2 Lp . The normal

spin-warp imaging gradients produce phase shiftsof only p across each pixel, and are thereforeimmune to this problem. However, gradient spoil-

Ž .ing is a coherent process 11 , and the use ofspoiler pulses after the acquisition of each line of

Žk-space data to minimize the residual transverse.magnetization may lead to unwanted stimulated

Ž .echoes 21 . The simulation of spoiling demandscareful consideration to avoid the artifactual max-ima, and can lead to very time-consuming simula-tions if L has to be large. In this work, we used a

Žvalue of L s 4 which is also sufficient for the.k-space reconstruction , and we restricted spoiler

SIMULATION OF IN-PLANE FLOW IMAGING 391

pulses to producing a total phase wrap of 4p

across a pixel.

Intravoxel Phase Dispersion

Mention must also be made of how the simula-tions model intravoxel phase dispersion in thepresence of shear flow. The basic gradient echosequence used in this work can be shown to

Ž .produce a phase shift f T of approximatelyE

gGvT t r2 when the echo time T is muche acq E

greater than the duration of the gradient pulses.Here, G is the readout gradient amplitude, v isthe flow velocity in the readout direction and tacq

Ž .is the acquisition sampling time. The velocity inPoiseuille flow is related to the radial position r

r 2Ž . Ž Ž . .in a vessel of radius a by n r s n 1 y ,pk a

where n is the peak velocity occurring in thepk

center of the vessel. For the sequence parametersused in the examples here, and with n s 700pk

w Ž .xmmrs as in Fig. 5 a , the previous two expres-Ž . Žsions can be combined to give f r s 18 1 y

r 2Ž . .. For a pixel spanning radii r and r , the1 2a

total in-plane magnetization signal will be theFresnel integral S s H r2 eif dr, which cannot ber1

evaluated analytically. Table 1 shows how thevalue of S calculated using only four spin ele-

Ž .ments per pixel as in the current simulationscompares with the result obtained for 1000 spin

Ž .elements the correct answer for the three dis-tinct pixels either side of the axis of the vessel in

Ž .Fig. 5 a . It will be seen that agreement is goodunder these conditions. Regions of higher shear,due to higher velocities andror complex flowpatterns, would need more spin elements perpixel to predict intravoxel phase dispersion effectswith comparable accuracy.

Table 1 Modeling of Intravoxel Phase Dispersion

Ž . Ž .Pixel r mm r mm S S1 2 4 1000

Near axis 0 2 0.841 0.830Middle 2 4 0.095 0.108Near Wall 4 6 0.255 0.192

Ž .The normalized signal calculated using four S and 10004Ž .S spin elements per pixel, representing the simulations1000carried out in this work and the ideal result, respectively.Poiseuille flow of peak velocity 700 mmrs in a vessel of 6 mmradius, with simulated 2-mm pixels. Agreement between S4

Žand S is worst for the edge pixel adjacent to the vessel1000.wall where the shear is greatest.

GLOSSARY OF ABBREVIATIONS ANDSYMBOLS

a vessel radiusCFD computational fluid dynamicsCPU central processing unit

Ž .e exp ytrT1 1Ž .e exp ytrT2 2

ECG electrocardiogramEPI echo-planar imagingFOV field of viewG gradient vector

Ž .i imaginary operator square root of y1j summation index

Ž .k k-space time or spatial frequency domaink , k x and y directions of k-spacex yL number of object elements per image pixelL integrated reverse flow velocityrev

Ž .M magnetization M , M , Mx y zM equilibrium magnetization0m first moment of gradient1

Ž . Ž .MR I magnetic resonance imagingN image matrix sizen number of time frames in cine-PC seriesframesPC phase contrastQRS QRS complex of electrocardiogramR rotation matrixr position vectorRF radiofrequencyS signal; spoiler gradientT evolution matrixT echo timeET repetition timeRT T relaxation time constant1 1T T transverse decay time constant2 2t timet acquisition timeacq

t period of pulsatile waveformcycleŽ .x x readout directionŽ .y y phase-encoding direction

a flip angle; timing of bipolarphase-encoding gradient pulse

g gyromagnetic ratiodt sample intervalf phase anglef maximum phase angle across a pixelmt timing of bipolar phase-encoding

gradient pulse¨ , v velocity

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Ian Marshall qualified in Natural Philos-ophy at the University of Oxford, andsubsequently joined the Department ofMedical Physics and Medical Engineer-ing of the University of Edinburgh, wherehe earned a Ph.D. He is currently a Se-nior Lecturer with responsibility for theMRI physics group. His research inter-ests include in ¨ i o proton spectroscopy,

flow imaging, functional imaging, and computer simulation ofMRI.