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Stationary States in Magnetic Resonance ImagingWlad T. Sobol and David M. Gauntt

Radiology DepartmentUniversity of Alabama Hospitals and Clinics

University of Alabama at BirminghamBirmingham, Alabama 35233

Contents

I. Introduction

II. The Origins of Ghosting

III. Traditional Vector Models

IV. Kaiser/Hennig Formalism

V. The Balanced Protocol and Interference Artifacts

VI. Unbalanced Protocols

VII. Canonical Stationary States

VIII. Classification of MRI Protocols

IX. Conclusions

X. References

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I. Introduction

In Magnetic Resonance Imaging (MRI), the voxelposition within the imaged slice is encoded using twodifferent mechanisms. First, a magnetic field gradi-ent (MFG) is applied during the readout of the echodata. Thus, different columns of spins, positionedat different distances along the MFG have differentlocal frequencies that are accurately sampled dur-ing the readout time. This scheme is known as fre-quency (or readout) encoding. The second mecha-nism encodes voxel positions within columns; it re-quires that the entire encoding sequence is repeatedmany times with an MFG of varying amplitude ap-plied along the direction within the imaged slicethat is orthogonal to the readout direction. Thisprocedure is known as phase encoding. After allthe data are acquired, they are subject to the 2DFourier Transform (2DFT). The resultant complex2D frequency spectrum forms the final image, but

only the magnitude part of it is usually presented tothe viewer.

Due to the repetitive nature of the data acqui-sition process, stationary states of nuclear magne-tization play an important role in the design andimplementation of the MR imaging protocols. Adetailed understanding of physical mechanisms thatdetermine the properties of these stationary statesis especially important in the design of fast imagingprotocols, when a residual transverse magnetizationexists at the end of the repetition cycle. This reviewpresents a comprehensive theory of stationary statesof nuclear magnetization that focuses on practicalissues related to the design and implementation ofMR imaging protocols. A long list of commercialMR imaging protocols, characterized by differentacronyms, is greatly simplified by using a classifi-cation scheme that is based on physical principlesthat form the foundation of this theory.

194 Bulletin of Magnetic Resonance

II. The Origins of Ghosting

The details of MRI encoding mechanisms arequite complex and could be found in the existingliterature (1). However, it is the repetitive nature ofthe data acquisition process that is a source of mostproblems encountered during the design of data ac-quisition protocols. To better understand the na-ture of this problem, let's ignore the frequency en-coding and focus on phase encoding process only.In the standard MR imaging algorithm the signalS(n) {with n 6 [0, ./V]}, created by the phase en-coding process, defines the so-called k-space in thephase encoding direction. The value of N definesthe number of repetitions required to acquire data ofsufficient spatial resolution (1,2). In the ideal case,the Fourier Transform (FT) of S(n) produces a per-fect, artifact-free image of the investigated object.Assume that due to imperfections in the acquisitionprocess this ideal signal is modulated by some errorfunction f(n). Thus, the resulting k-space data canbe written as:

S(n) = f(n)S(n). (1)

If the original, perfect image, obtained by applyingthe FT to the data set S(n) is denoted I(n), then byapplying the FT to the modulated k-space data oneobtains the following convoluted image:

= F(v)*I(v), (2)where F(v) is an FT of the modulating function f(n).Formally, the f(n) behaves like a point-spread func-tion (PSF). In standard applications the PSF de-scribes image blurring, caused by imperfections ofthe imaging system. In MRI, however, f(n) usu-ally describes ghosting, as will be discussed shortly.The reason for this difference is that in MRI f(n)is always periodic and discrete, since the k-space isalways sampled at some discrete intervals A.

Consider a case when f(n) has a period p; thismeans that its FT spectrum is a series of Dirac deltafunctions (see Figure 1) and:

+P/2i(v) = Fj8 pA) v v = (3)

E

(A)

0 A nA

(B) i

-12A

-1 0 1 1pA w pA 2A

Figure 1: The behavior of the modulation functionf(n), generated by a stationary state of periodicityp (A) and its FT spectrum F(v) (B). For simplic-ity, p is assumed even; the edges of the spectrumF(v) correspond to the borders of the Field of View(FOV) in the regular image.

Thus, a series of ghost objects, centered at positionsj/(pA), will appear in the final image. The intensityof those ghosts is determined by coefficients Fj.

Obviously, ghosts should not appear in MR im-ages. Thus, the function f(n) should have an FTspectrum containing only one component, prefer-ably centered at zero frequency. In such a casethe function f(n) will be independent of n, or willcontain only a DC component. In standard MRIapplications this condition is ensured by using therepetition time, TR, long enough so that the spin-spin relaxation phenomena cause the residual trans-verse magnetization to vanish before the sequenceis repeated again (with a different phase encodingMFG). In practice, this restricts the shortest TRvalues to about 300 ms (most tissues are character-ized by a major T2 component of about 40 - 50 ms).

III. Traditional Vector Models

The situation becomes complicated when TR issufficiently shortened so that a residual transversemagnetization exists at the end of the cycle. Short

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Figure 2: Vector representation of the transversemagnetization evolution within a stationary pulsecycle: (a) magnetization immediately after an rfpulse; (b) immediately before the next rf pulse -notice the precession angle from position a to 6; (c)residual transverse component left in the horizontalplane by the rf pulse; (d) fresh transverse compo-nent generated by the rf pulse from the longitudinalmagnetization. The vector a is a resultant of com-ponents c and d and thus the stationary state isrestored.

TRs are desirable in practice because they reducethe exam time, often from several minutes to sev-eral seconds. Traditionally, fast MR protocol de-signs that produce artifact-free images are based ona vector model of a stationary state (3,4). Withinthis model, a stationary solution is sought for nu-clear magnetization that is subject to a train ofequally spaced rf pulses with equal flip angles a.It is assumed that between rf pulses the magneti-zation relaxes and the phase of. its transverse com-ponent advances by a fixed angle ip. The motion ofsuch a single isochromat is illustrated in Figure 2(the rotating reference frame is used, the B\ vectorof the rf pulses is always pointing in the negative y'direction).

During the evolution period after the rf pulse,the transverse magnetization (black arrow, a) bothshrinks in size (due to T2 processes) and rotatesaround the z-axis (due to local frequency offsets).Thus, at the end of the cycle, immediately beforethe next rf pulse, it is found at some other position(b). The oncoming rf pulse shortens this componentby rotating its x' projection around the y'axis (c).At the same time, the rf pulse rotates the longitudi-nal component of the magnetization, thus creatingfresh horizontal projection (white arrow - d) that re-

stores the stationary transverse vector (black vector(a) is a sum of residual vector (c) and fresh vector(d)). Notice that the tips of vectors a, b, and c areall aligned along a single straight line, parallel to thex' axis. This happens because the rf pulse does notaffect the y' components of transverse magnetiza-tion. Analytical solutions to such a stationary casehave been derived (3,5,6) and successfully used inNMR spectroscopy (7,8), but they have only limitedvalue in MRI applications. Isochromatic solutionsdepend on the phase advance angle ip in a compli-cated way. This is not a problem in NMR spectros-copy, where very uniform magnetic fields are usedand phase distributions across the sample are small.In MRI, wide phase distributions are common dueto the use of MFGs for slice selection and encod-ing of voxel positions; a single pixel value representsan NMR signal from the entire corresponding voxel.Thus, the pixel signal intensities used to create anMR image are calculated as averages over the phasedistributions within the corresponding voxels:

S(n) = J p(<p)M(<p)M(<p,r)dV, (4)V

where p(ip) is a phase distribution function normal-ized over the voxel's volume V and M is the trans-verse component of a stationary magnetization vec-tor. Attempts to calculate these averages using so-lutions derived from the vector model lead to com-plicated equations that cannot be readily analyzed.As a result, computer models were often utilized toexplore the properties of these solutions (6). Sincemany parameters are needed to calculate the sta-tionary solutions (relaxation times Ti and T2, spindensities, rf flip angles a, TR, phase advance (p), nu-merical explorations of vector solutions are tediousand costly.

IV. Kaiser/Hennig FormalismIt turns out that a formalism, first proposed by

Kaiser (9) is much better suited for the studies ofstationary states in MRI. Kaiser used a well knowncomplex notation to represent transverse compo-nents of magnetization, but introduced a new wayof writing a stationary solution as a Fourier series,expanded in terms of the phase advance angle:

+00

Mx + iMy = F = Y, F^lklp (5)fc=—00

196 Bulletin of Magnetic Resonance

-3

Figure 3: A Hennig phase diagram. Horizontalaxis represents time, vertical axis - magnetization'sphase. Vertical lines are spaced by intervals TR;nodes represent different terms of Kaiser expansion.The nodes are labelled with the Kaiser index k.Continuous lines illustrate the evolution of trans-verse components during the TR cycle; dashed linesrepresent the components of the longitudinal mag-netization.

-3

Figure 4: The effect of rf pulse as a "state gen-erator" in the Hennig diagram. A single transversecomponent, marked with an *, enters the white nodefrom the left; rf pulse at that node creates all othercomponents, marked with an * to the right of thisnode line. The effect of the rf pulse on the longitu-dinal component is shown in a similar fashion (un-marked lines). Note the rf pulse mixes longitudinaland transverse components, as expected.

1;i

The longitudinal component of the stationary mag-netization is represented in a similar fashion.

Hennig (10) has shown that this formalism canbe represented graphically in a simple way by usingphase diagrams. To build these diagrams, verticallines are plotted at time intervals TR (Figure 3).A series of nodes appears on each line, representingdifferent expansion terms in eqn. 5. Thus, the verti-cal axis can be viewed as a phase axis and nodes areseparated by the phase advance angle <p. The trans-verse component precesses by an angle ip betweenpulses; this is schematically drawn as a solid linebetween nodes on two consecutive lines. The lon-gitudinal magnetization does not acquire any phasebetween pulses, but "preserves" the existing phasememory among different terms of eqn. 5. This isrepresented by dashed lines joining the nodes. It isimportant to realize that the actual phase evolutionbetween pulses does not matter, as long as the cor-rect phase angle advancement is reached at the endof the cycle, just before the next rf pulse.

The rf pulse "mixes" different components, act-ing as a state generator (see Figure 4). The rf pulsegenerates four new components from every trans-verse component, entering a node: two transverse,and two longitudinal (these components are marked

with * in Figure 4). In contrast, only two compo-nents are generated from a longitudinal trace: onetransverse, one longitudinal (shown as unmarkedcomponents in Figure 4).

The entire history of magnetization evolutionthroughout the pulse train can be plotted usingphase diagrams, constructed using the describedrules. It must be noted that the value of mag-netization at any given node is a superposition ofcontributions arising from magnetization evolutionthrough different phase pathways, as illustrated inFigure 5. If both Ti and T2 were infinite, an infinitenumber of pathways would contribute to each node,which means that the stationary state would neverbe reached. Since in reality both Ti and T2 are fi-nite, any perturbation of magnetization, generatedby an rf pulse at any given time will eventually van-ish due to relaxation phenomena. Thus, a boundarybox can be created around each node: pathways out-side the box do not contribute to the magnetizationvalues at that node. The size of the box is deter-mined by NMR characteristics of the imaged mate-rial (Ti and T2) as well as experimental sequenceparameters (TR, a, <p). Despite this limitation, thenumber of paths contributing to a node can be sub-stantial. For example, Figure 5 shows a few paths

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that originate from the k=2 node and contribute tothe k=0 node positioned six pulses downstream.

V. The Balanced Protocol andInterference Artifacts

Consider now a perfectly balanced MR proto-col (which was introduced in the literature under aname of FISP (11) - see Figure 6). In this protocol,all MFGs used for imaging purposes are shaped insuch a way that they do not contribute to the netphase gain over the repetition interval. The MFCscontribution to the phase gain is equal to:

<p'(r) = -yrTRJ G(t)dt, (6)

where 7 is the gyromagnetic ratio, G represents theMFG, and r identifies a location within the imagedvolume. Thus, for gradient shapes that have a zerointegral over the TR interval, the net phase gain iszero anywhere within the imaged volume. For a per-fectly balanced protocol the only sources of magne-tization precession are local magnetic field inhomo-geneities. In a large, whole-body magnet, the mag-netic field varies slowly within the imaged volume.Thus, within a single voxel the phase advance an-gle is practically constant and the phase distributionover the voxel's volume is a Dirac's delta function,centered at a certain value ipQ. Under such condi-tions, the MR signal intensity for a single voxel canbe calculated from eqns. 4 and 5 and is equal to:

S(n) = j 6(<p - FkeikvdV = • (7)k=~ oo

fc=—00

ikipo

Since ipo varies slowly across the field of view (asthe magnetic field changes), for some voxels ipo—2kirand all components in eqn. 7 will add construc-tively, enhancing the signal intensity. There willbe areas where </?o=(2k+l)7r and the componentswill add destructively, decreasing the signal inten-sity. Thus, the entire image will be composed of al-ternating bright and dark stripes whose shape willreflect the distribution of magnetic field intensitywithin the FOV (12).

(n-6) (n-5) (n-4) (n-3) (n-2) (n-1) nFigure 5: Coherence pathways of different magne-tization components. For clarity's sake, only a fewpaths that originate at a node k—+2 and contributeto the node k=0 six pulses later, are shown.

The MR images created using a true FISP pro-tocol resemble Newton rings that arise in opticalinterference phenomena. In fact, there is an inter-esting analogy between these two phenomena: bothare caused by interference of many components thattravel through different phase pathways and add co-herently. From the practical point of view, trueFISP images are useless because the interferencepatterns severely distort the image features that areneeded for clinical diagnosis.

VI. Unbalanced Protocols

The interference problem can be solved by us-ing unbalanced protocols. In an unbalanced proto-col large phase distributions across voxels are cre-ated using the MFGs and substantial phase gainsare acquired by magnetization components duringthe sequence cycle TR. An example of such a pro-tocol is shown in Figure 7. The unbalanced sliceselect and readout gradients create a uniform phasedistribution across every voxel. Let's assume that,by appropriate adjustment of the MFG shapes, thewidth of this distribution is forced to be a multiple

198 Bulletin of Magnetic Resonance

of 2TT

1!

p(<p) = 2 ^ w h e n -0 otherwise (8)

The imaging gradients impose additional phase evo-lution <j)(t) onto the stationary components of eqn.5 that defines magnetization immediately followingthe rf pulse. Thus, at the center of data acquisitionwindow, identified by the echo time TE, the MRsignal is proportional to

k=-<xS(n,TE) = j p(<p)

V

Fkei{kv+<P(t))~TE/T2dy = p^

where m is an index such that

rrvp + 4>(TE) = 0

(9)

(10)

Under such conditions only one component of theentire Kaiser expansion series contributes to the sig-nal that is acquired to produce an image. All othercomponents, while present, are dispersed in such away that their average value across the voxel is zero.Thus, they cannot be detected by the data acquisi-tion method that samples the MR signals at discretetime increments A. The phase diagrams help to vi-sualize this process. For the protocol shown in Fig-ure 7, the phase path for the node k = 0 is shownin Figure 8. Phase paths for other remaining nodesare parallel to the path shown and are removed fromthe picture for the sake of clarity. It is obvious thatonly path for the node k = 0 crosses the zero phaseline during the TR cycle (at a time TE). Thus, forthis protocol m = 0 and we shall simply label sucha protocol Fo to emphasize the fact that pixel inten-sity values for this protocol are determined by thisexpansion coefficient in eqn. 5.

VII. Canonical Stationary StatesThe above analysis uncovers two fundamen-

tal conditions that are required to produce artifact-free images with fast MRI protocols that use short(<100 ms) repetition times TR. First, carefully de-signed, unbalanced gradients must be used to avoidNewton-ring type artifacts. Second, the componentused to carry the image data, Fm{n), must have adelta-like FT spectrum to avoid ghosts in the final

TR

A\J-\J

TEFigure 6: A balanced MR protocol. All gradientsare shaped in such a way that their net contributionto the magnetization phase, integrated over the timeinterval TR, is zero.

image. We shall use a term cardinal to describea component whose spectral delta function is posi-tioned at the zero frequency, and label canonical acomponent that is characterized by a spectral deltafunction that is positioned at some nonzero offsetfrequency. In other words, a canonical componentwith zero modulation frequency is a cardinal one.

It is desirable to use cardinal components to en-code the MR image data, since it is guaranteed thatsuch an approach will produce artifact-free imageswithout any additional data processing. If a canoni-cal component is used to encode the MR image data,the imaged objects will be shifted off-center. Thisshift will be equal to the frequency offset of the spec-tral delta function for the canonical component usedto encode the data. This offset can be calculatedfrom the analysis of the rf sequence used, and it isa simple matter of implementing an extra process-ing step during the image reconstruction to shift theobjects back to the center of the image.

From the above discussion it follows that it isimportant to identify a class of rf sequences thatwill produce stationary states containing only car-dinal or canonical components. A detailed analysisof this problem (13) uncovered the following con-ditions that are necessary and sufficient to producestationary states with canonical components:• all rf pulses within the sequence must be equallyspaced (by the time interval TR);• all rf pulses must have the same flip angle a;• the phases of rf pulses 6n (n=0,l,2,..) must sat-isfy the following condition (when viewed in a singlerotating reference frame):

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7R

7\V

7TFigure 7: An unbalanced MR protocol, designed toutilize the k=0 node for image encoding. Note thatslice select and readout gradients are unbalanced,but the phase encoding gradient remains balanced.This is because the phase encoding gradient variesamong cycles.

Sn = A + Bn + Cn2 (11)This result is identical to conclusions, derived by Zuret. al. (14) who analyzed ways to "spoil" transversecoherences with phase offsets of the rf pulses usingSekihara's computer simulation method (6). To thebest of our knowledge, it was Zur (15) who firstproposed the use of an rf offset to effectively manip-ulate the stationary state. For a train of pulses withphases that follow this rule, the stationary state hasa form (13):

F(n) = el6n E•+oo

H(n) =

oiCk2 ei(<p-B-2Cn)k

Di{<p-B-2Cn)k

(12)

fc= —oo

This equation is written in a single rotating refer-ence frame; F(n) represents transverse magnetiza-tion as defined by eqn. 5, H(n) represents the logi-tudinal magnetization Mz. If the receiver referencephase follows the phase of the last rf pulse, then theF component in eqn. 12 must be multiplied by anexponential phase factor (TT/2 — 5n) and the signalis free from the transmitter phase:

+ OO

S(n) = i (13)fc=—oo

From eqn. 13 it follows that the value of A willnot affect the stationary state conditions. This is to

Figure 8: Phase path for the node k=0 for the MRprotocol shown in Figure 7. Phase paths for othernodes are parallel to the one shown and are not vi-sualized for simplicity.

be expected, as A represents a constant phase off-set. The value of B describes a frequency offset be-tween the rf carrier frequency and the reference fre-quency of the rotating frame. Under normal imag-ing conditions when unbalanced gradients are used,the value of B does not affect the image contrast, be-cause it simply offsets the center of the distributionfunction p(ip) by B (see eqn. 8). This shift, how-ever, will affect the distribution of the interferencesstripes created with a balanced protocol (see eqn.7). An imaging scheme that takes advantage of thisphenomenon to create true FISP images has beenproposed under the name CISS (16). The selectionof coefficient C will definitely affect the stationarystate. Thus, we shall denote by Fm(C) the proto-col that uses an m-th node of the stationary statecreated with a train of rf pulses determined by thevalue of constant C.

VIII. Classification of MRI Pro-tocols

Using the presented arguments one can classifythe existing MR imaging protocols into categories,according to tissue contrast they produce. Both mand C do affect the image contrast, thus all proto-cols that use Fm(C) component of the Kaiser ex-pansion to encode image data will produce imagesthat are equivalent in terms of image contrast, re-gardless of details of the protocol implementation.This is an important feature since it allows one tosimplify the long list of acronyms that MR equip-

200 Bulletin of Magnetic Resonance

Table 1: Commercial equivalents of FQ(0) protocol.Full Name AcronymFourier Acquired Steady State FASTField Echo FEFast Field Echo FFEFast Field Echo FFEFast Imaging with Steady State Precision FISPFID-based short repetition technique F-SHORTGradient Field Echo with Contrast GFECGradient Recalled Acquisition in the Steady State GRASSSteady State Free Precession SSFP

Note: MS = Medical Systems. Entries are listed in alphabetical order of acronyms.

Manufact urerPicker InternationalToshiba America MSPhilips MSOtsuka ElectronicsSiemens MSElscintHitachi MS AmericaGEMSShimadzu MS

ment vendors introduced in recent years to describethe imaging capabilities of their hardware. Differ-ent hardware platforms and different developmen-tal paths lead to different, proprietary implementa-tion strategies that were described using differentacronyms. However, all protocols must function us-ing the same physics principles and thus, when im-plemented correctly, will produce equivalent resultsdespite hardware-related differences in implementa-tion details.

The protocols with C=0 are easiest to imple-ment, and two are widely used. The first one, Fo(O)is simply an implementation of the method that usesthe m=0 node, as illustrated in Figure 7 and Figure8. This is the most intuitive, and thus most widelyused protocol. The commercial equivalents of thisprotocol are listed in Table 1. Another C~0 protocolis shown in Figure 9. The analysis of phase evolu-tion using the phase diagrams immediately identi-fies this protocol as F-i(0), as seen in Figure 10 (thecommercial equivalents of this protocol are listed inTable 2).

At this stage, the advantages of using Fourier ex-pansion technique with phase diagrams become veryclear. Methods that use traditional vector modelsidentify the Fo(O) protocol as FID, due to its obviousassociation with the signal that is implicitly gener-ated by the rf pulse. Similarly, the JF_I(0) protocolis identified as an "echo" protocol, since it appearsto sample a spin echo signal (the spin echoes werefirst described by Hahn who analyzed a series ofthree arbitrary rf pulses (17)). It is quite obvious

TR

ATE

Figure 9: An F_i MR protocol.

from Figure 5 that such association is misleading,since both signals actually represent superpositionsof many components (that were originally identifiedby Hahn as sources of separate echoes). From thephase diagrams it is also quite obvious that eithersignal can be used to create an MR "echo" any-where within the TR interval by using appropriatelyshaped MFGs (see Figure 8 and Figure 9). If theMFGs are properly designed, no mixing of the twosignals will occur. Furthermore, the vector modeldescribes only two different types of images, associ-ated with either FID or echo signals. It is obviousfrom the phase diagrams that, in principle, an in-finite number of different images can be created byusing different nodes to encode image data. For ex-ample, Figure 11 shows a protocol that encodes them=+l node (see Figure 12). Nobody has used sucha protocol in practice yet, probably because peoplewere unaware of the fact that it could be done.

For the Kaiser expansion to be convergent, thevalues of expansion coefficients Fk must decrease

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Table 2: Commercial equivalents of F_i(0) protocol.Full NameContrast Enhanced Fourier Acquired Steady StateContrast Enhanced Fast Field Echo with T2 weightingEcho-based short repetition techniqueReversed FISPSteady State Free PrecessionSteady State Technique with Refocused FID

AcronymCE-FASTCE-FFE-T2) T2-FFEE-SHORTPSIFSSFPSTERF

ManufacturerPicker InternationalPhilips MSElscintSiemens MSGEMSShimadzu MS

Note: MS = Medical Systems. Entries are listed in alphabetical order of acronyms.

7R

Figure 10: The phase path for node k — — 1 for theMR protocol shown in Figure 9. Note: the defi-nition of the echo time TE for this protocol variesamong different sources and may not correspond tothe value indicated.

quickly with increasing k. This leads to rapid signal-to-noise deterioration when higher order nodes areselected to encode the image data. This signal-to-noise deterioration, as well as the need to use verylarge gradients to encode higher order nodes makestheir use impractical today. But the possibilitiesexist and might be used someday in MRI practice.

The literature describing protocols that usenonzero C values is often confusing, chiefly becausethe vector models do not support such conceptsvery well. It is generally stipulated that such proto-cols lead to "magnetization spoiling" that preventstransverse components from contributing to the sta-tionary state and thus the resulting image contrastdoes not depend on T2 anymore. Existing numericalsimulations (14) and recent analytical analysis (13)of these protocols indicate that this is not the case.Image contrast between two different tissues (char-acterized by different spin densities, Tis and T2s)

7Y rv.TE

Figure 11: An F+i MR protocol.

strongly depends upon the values of C and can beextensively manipulated. It usually exhibits a reso-nant behavior, with resonances of various strengthoccurring for specific values of C that create peri-odic cycles within the sequence train (13). Thus,a continuum of image contrasts is available and animportant question arises how to select the valuesof parameter C to ensure optimal image contrast inclinical applications of these protocols. This issue isnot resolved yet, but it has been significantly facil-itated by recent presentation of analytical solutionsfor resonant values of C (13).

An efficient way to implement the spoiled proto-cols requires digital rf hardware to precisely controlthe phases of rf pulses. In such a case, the basicimaging sequence follows the normal scheme (seeFigures 7, 9, 11), but the phases of rf pulses arevaried according to eqn. 11. An example of such asolution is a commercial SPGR protocol (GE Medi-cal Systems, Milwaukee, WI) that uses C = 46.184°.For systems for which the digital hardware is notyet available, various gradient spoiling techniqueshave been proposed to achieve equivalent results.

202 Bulletin of Magnetic Resonance

Full NameContrast Enhanced Fast Field Echowith Ti weighting

Table 3: Commercial equivalents of FQ(0) protocol.AcronymCE-FFE-Ti, Tx-FFE

Fast Low Angle ShotPartial Saturation

RF spoiled FASTShort repetition techniqueSpoiled GRASSSmall Tip Angle Gradient Echo

FLASHPS

RF-FAST, TVFASTSHORTSPGRSTAGE•

ManufacturerPhilips MS

Siemens MSInstrumentariumImagingPicker InternationalElscintGEMSShimadzu MS

Note: MS = Medical Systems. Entries are listed in alphabetical order of acronyms.

It is much more difficult to avoid image artifactswhen using MFGs to create the required phase off-sets within the TR cycles. The difficulties arise fromthe fact that MFG used in MR imaging protocolsmust often be scaled to allow changes in slice thick-nesses, the size of the field of view, or the data ac-quisition bandwidth (often related to the TE used).The gradient spoiling scheme must produce phaseadvancements that are independent of the above pa-rameters.

To date, all commercial applications utilize the-Fo(C) protocol (see Table 3). Again, this is proba-bly due to the fact that vector models failed to in-dicate that other options are possible. However, forC ^ O the m ^ O components are not cardinal, butcanonical (see eqn. 13). If Cis known, the requiredfrequency shift can be easily calculated, but addi-tional image processing step is required to correctit. It remains to be seen if more complex applica-tions, based on Fm(C) protocols find their way intoclinical practice;

IX. Conclusions

The Kaiser expansion method, combined withthe Hennig phase diagrams provide a powerful toolto investigate stationary magnetization states usefulin generation of artifact-free MR images. Such anapproach opens a way to classify different MR pro-tocols according to their underlying physical princi-ples. In addition to existing commercial protocolsthat all fit into this classification scheme, new ones,so far not implemented in practice, are discovered.Analytical solutions for stationary states responsi-

Figure 12: The phase path for node k = +1 for theMR protocol shown in Figure 11.

ble for tissue contrast in the resulting MR imagesfurther facilitate studies of this exciting, but quiteunwieldy branch of applied NMR physics.

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