Steady?state diffusion?weighted imaging: theory

Special Issue Review Article

Received: 28 August 2009, Revised: 23 December 2009, Accepted: 5 January 2010, Published online in Wiley Online Library: 28 June 2010

(wileyonlinelibrary.com) DOI:10.1002/nbm.1509

Steady-state diffusion-weighted imaging:theory, acquisition and analysisy

Jennifer A. McNaba and Karla L. Millera*

Steady-state diffusion-weighted imaging (DWI) has

NMR Biom

long been recognized to offer potential benefits over conven-tional spin-echo methods. This family of pulse sequences is highly efficient and compatible with three-dimensionalacquisitions, which could enable high-resolution, low-distortion images. However, the same properties that lead to itsefficiency make steady-state imaging highly susceptible to motion and create a complicated signal with dependenceon T1, T2 and flip angle. Recent developments in gradient hardware, motion-mitigation techniques and signal analysisoffer potential solutions to these problems, reviving interest in steady-state DWI. This review offers a description ofsteady-state DWI signal formation and provides an overview of the current methods for steady-state DWI acquisitionand analysis. Copyright � 2010 John Wiley & Sons, Ltd.

Keywords: diffusion-weighted imaging; steady-state; SSFP; brain; white matter

* Correspondence to: K. L. Miller, FMRIB Centre, John Radcliffe Hospital, Head-

ington, Oxford, OX3 9DU, UK.

E-mail: [email protected]

a J. A. McNab, K. L. Miller

Centre for Functional MRI of the Brain (FMRIB), University of Oxford,

Oxford, UK

y This article is published in NMR in Biomedicine as a special issue on Progressin Diffusion-Weighted Imaging: Concepts, Techniques, and Applications tothe Central Nervous System, edited by Jens H. Jensen and Joseph A. Helpern,Center for Biomedical Imaging, Department of Radiology, NYU School ofMedicine, New York, NY, USA.

Abbreviations used: ADC, apparent diffusion coefficient; CE-FAST, contrast

enhanced - Fourier acquired steady-state; DWI, diffusion weighted imaging;

DW-PRESTO, diffusion-weighted-principles of echo shifting with a train of

observations; DW-SE, diffusion weighted spi echo; DW-STE, diffusion-

weighted-stimulated echo; DW-SSFP, diffusion-weighted steady-state free

precession; EPI, echo-planar imaging; FID, free induction decay; FLASH, fast

low angle shot; PSIF, reverse fast Imaging with steady-state precession; RARE,

rapid acquisition with relaxation enhancement; RF, radiofrequency; SE, spin

echo; SSFP, steady-state free precession; STE, stimulated echo; TURBINE,

trajectory using radially batched internal navigator echoes. 7

INTRODUCTION

Diffusion-weighted imaging is currently dominated by a singletechnique: diffusion-weighted spin echo (DW-SE) (1–3). Whencombined with a 2D, single-shot echo-planar imaging (EPI)readout (4), DW-SE is a powerful method due to its relatively highefficiency, ability to obtain strong diffusion weighting andinsensitivity to motion. However, this method has a number oflimitations including severe image distortions, limited resolutionand long echo times. Despite its prevalence, many alternativepulse sequences to DW-SE exist; in fact, diffusion contrast can beadded to almost any sequence with inclusion of the appropriategradient pulses. This review considers one set of methods:steady-state diffusion-weighted imaging (DWI).Diffusion-weighted steady-state pulse sequences have long

standing in the NMR and MRI literature. In the context of NMRspectroscopy, the diffusion sensitivity of SSFP pulse sequenceswas first studied by Kaiser in 1974 (5), as a natural extension of theprevious work by Woessner on short RF trains (6). Initialdemonstrations of diffusion-weighted imaging (DWI) usingstimulated- and spin-echo sequences (2,3,7) were quicklyfollowed by reports on DWI with SSFP (8–12).DW-SSFP was recognized early on as a very promising technique

due to its potential to achieve diffusion weighting efficiently (8,10).Like its non-diffusion-weighted cousins, DW-SSFP achieves effi-ciency due to its rapid train of pulse-acquisition cycles. Most MRIpulse sequences sample the magnetization with a TR of severalseconds, while SSFP methods revisit the same magnetization everytens of milliseconds. Although this drives the raw signal levelsdown, the enormous fraction of the time that is spent acquiringdata usuallymore thanmakes up for this, resulting in high signal-to-noise ratio. In addition, as will become clear below, DW-SSFP is ableto achieve strong diffusion weighting with much less gradient areathan conventional methods.Unfortunately, this efficiency has remained elusive in the brain,

primarily due to two issues. First, steady-state DWI data must beacquired using a segmented readout (i.e in multiple shots),making it vulnerable tomotion artifacts.(11,12) Second, the signalin steady-state DWI has relatively complicated dependence

ed. 2010; 23: 781–793 Copyright � 2010

on T1, T2 and flip angle, making analysis (and in particular,quantification) difficult (5,13,14). Following the early interest insteady-state methods for diffusion imaging, the significance ofthese issues led many to conclude that DW-SSFP, whiletheoretically interesting, was unlikely to see any practical use.However, advances in gradient hardware, motion compen-

sation techniques, parallel imaging and signal analysis haverevived interest in steady-state DWI. It is therefore timely toreview the unique idiosyncrasies of steady-state DWI and discussthe potential improvements that steady-state DWI may offer overDW-SE with 2D EPI. This paper aims to offer a clear description ofthe principles behind steady-state DWI signal formation,including a discussion of the range of signal models presentedin the literature. We also give an overview of the current methodsfor steady-state acquisition and analysis. Finally, examples ofapplications of these methods are described.

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Figure 1. Instead of an RF pulse tipping the magnetization by the angle

a, we can think of it acting like a 08 pulse for cos a of the magnetization

and 908 for the remaining sin a. Thus, each pulse splits the magnetizationinto transverse and longitudinal components, which can be treated

independently. A train of pulses creates a spreading tree of magnetization

components, each with a unique history of transverse and longitudinal

periods. The effective RF pulse experienced by a given pathway in each TRis shown above the RF pulse, and the orientation of the magnetization

(longitudinal or transverse) during each TR period is denoted with either

‘L’ or ‘T’. The fraction of magnetization in each pathway is indicated under

the pathway diagram. For example, sin 2a of the total magnetizationexperiences the bottom pathway, with a 908 pulse putting it into the

transverse plane for the first period, and then a second 908 pulse placing it(negatively) longitudinal for the second period.

J. A. MCNAB AND K. L. MILLER

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SIGNAL FORMATION

One obstacle to a full appreciation of DW-SSFP for many casualreaders is its niche in the MRI community. Although there is a vastliterature on the signal dynamics underlying both SSFPsequences and diffusion-weighted imaging, the two are rarelybrought together. Merging these two techniques yields a signalwith considerable complexity, and the intuition held by an experton either method may be insufficient or inappropriate forunderstanding DW-SSFP. This section begins with an overview ofsteady-state signal dynamics, focusing on the conceptualapproach that is most useful for understanding DW-SSFP. Wethen describe signal formation in DW-SSFP and contrast it tomore familiar methods of diffusion weighting.

Steady-state pulse sequences

Steady-state imaging refers to pulse sequences in which themagnetization is not allowed to return to full equilibriumbetween RF pulses. Most MRI sequences (with TR�T1) establish alongitudinal steady state due to incomplete T1 recovery betweenRF pulses. However, ‘steady-state imaging’ typically refers to amore extreme case in which both T1 and T2 relaxation areinterrupted by very rapid RF pulses (TR � T2< T1). Most typically,both TR and flip angle (a) are fixed, as assumed for the remainderof this review. Following an initial equilibration period, bothlongitudinal and transverse magnetization establish a steadystate in which the effect of the flip angle is exactly cancelled by T1and T2 relaxation and precession during the TR.The defining characteristic of steady-state pulse sequences is

that some of the magnetization remains in the transverse plane atthe end of each TR. Depending on the flip angle of the subsequentRF pulse, a fraction of this transverse magnetization is then tippedalong the longitudinal axis, while the rest remains in the transverseplane and persists into the next TR. Similarly, the excitation will tip afraction of the longitudinalmagnetization into the transverse plane,leaving the rest longitudinal. This process, repeated over a train ofrapidly-applied RF pulses, can be visualized as a mixing oflongitudinal and transverse magnetization.

COHERENCE PATHWAYS

The concept of mixing transverse and longitudinal magnetizationis not particularly useful for our purposes. Instead, one canconceive of separating the transverse and longitudinal com-ponents. We can think of each RF pulse as behaving like a 08 pulsefor part of the magnetization and a 908 pulse for the rest. In thisinterpretation, each RF pulse splits the magnetization into twocomponents, so that two pulses result in four distinct components,three pulses result in eight, etc. The train of pulses thus distributesinitial magnetization across a spreading tree, as shown in Figure 1,and the magnetization inhabiting each branch of the tree has aunique history of transverse and longitudinal periods.This process is formally described using ‘coherence pathways’,

and the associated technique of ‘phase graphs’. An excellent(and entertaining) discussion of the theory underlyingcoherence pathways is given by Hennig (15). The basic conceptis to separate the magnetization at every stage into itslongitudinal and transverse components, which are trackedindependently (see Fig. 1). For a train of RF pulses, the totalmagnetization is decomposed into a set of magnetization‘pathways’, each describing a particular history of longitudinaland transverse TR periods. Some fraction of the magnetization

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ends up following each pathway, and one can easily describe themagnetization’s progression based on the pathway history. Themagnetization will experience T1 recovery during longitudinalperiods and T2 decay during transverse periods. Relaxationdictates that older pathways (those excited early in the RF train)eventually fade away. Pathways also undergo loss of phasecoherence (dephasing) during transverse periods due toinhomogeneity and gradient pulses.Signal formation is then described as the summation of the

subset of pathways that form an ‘echo’ in a given TR. That is, apathway contributes signal when its magnetization re-gainsphase coherence, as happens in a spin echo. If we representtransverse and longitudinal periods as T and L, the signal pathwayin a spin echo sequence would be described as:

90��T� 180� � T� Echo

while the signal pathway in a stimulated echo sequence isdescribed as:

90��T� 90��L� 90��T� Echo

Thus far, we describe each RF pulse as generating transverseand longitudinal magnetization. To fully understand echopathways, however, we require the unintuitive result that anyRF pulse (a) acts on the magnetization as if it has a 08, 908 and1808component,(1) as derived in Appendix A. The effect of an RFpulse must therefore be thought of as splitting each existingpathway into three new pathways, rather than the two shown inFigure 1. The third pathway, which corresponds to the1808component, negates longitudinal magnetization and actsas a refocusing pulse on the transverse magnetization. This1808component is important in DW-SSFP.The spin and stimulated echo sequences referred to above

were described as containing a single signal-generating pathway.

ht � 2010 John Wiley & Sons, Ltd. NMR Biomed. 2010; 23: 781–793

STEADY-STATE DIFFUSION-WEIGHTED IMAGING

In steady-state sequences, however, the long train of RF pulsescreate a plurality of pathways, a subset of which will form echoesin any given TR. For example, a steady-state signal would havecontributions from both of the SE and STE pathways alreadydescribed. In these pathways, the first 908 pulse is preceded by animplicit train of longitudinal (L) periods in which the magnetiza-tion is not excited. For example, the full pathway for the primarystimulated echo would be:

L� � � � �L� 90��T� 90��L� 90��T� Echo

In addition, there will be many other pathways contributingechoes, such as:

L� � � � �L� 90��T� 90��L� L� L� 90��T� Echo

Most signal-generating pathways bear this general form, andcan be thought of as a type of stimulated echo, with varyingdurations spent along the longitudinal axis. Nevertheless, thereare also pathways with much more complicated forms, includingmore than two transverse periods, and a given pathway couldform echoes at multiple points in its history. This effectivelyenables the signal from a given magnetization component to berepeatedly harvested on short time frames, creating a highlyefficient signal mechanism.

DIFFUSION CONTRAST IN THE STEADYSTATE

Most sequences (including spin- and stimulated-echo tech-niques) obtain diffusion weighting using a pair of gradient pulseswith identical area, as illustrated in Figures 2a and 2b. The use ofgradient pairs for this purpose was first introduced by Stejskaland Tanner in 1965. (1) At first glance it seems curious, then, thatDW-SSFP uses only a single diffusion-sensitizing gradient pulseper TR, as shown in Figure 2c.However, this sequence makes much more sense in the

context of coherence pathways, where we see that that theequivalent of Stejskal-Tanner gradient pairs occur overmultiple TR periods. Consider the magnetization in a coherence

Figure 2. Diffusion-weighting schemes for (a) spin echo, (b) stimulatedecho and (c) SSFP. The implicit negation of gradients due to 1808 inversionof the magnetization is indicated with gray, dashed lines. For SSFP,

multiple repetition periods are shown to emphasize the similarity ofdiffusion contrast to SE and STE. For illustration purposes, the timescales

of the different sequences are matched, but in general this will not be the

case.

NMR Biomed. 2010; 23: 781–793 Copyright � 2010 John Wiley

pathway with two transverse periods. During the first transverseperiod, the magnetization experiences strong dephasing alongthe direction of the diffusion gradient. If the magnetization in thispathway then undergoes a net rotation of 1808, it will experiencea second, identical gradient and undergo diffusion-attenuatedrephasing exactly like the classic Stejskal-Tanner diffusionpreparation.Provided the net rotation between two transverse periods is

1808, the diffusion gradients are balanced and any combinationof transverse and longitudinal periods can result in an echo. Asshown in Figure 3, this includes echo pathways with multipletransverse periods. However, the amount of signal contributed byeach pathway is limited by its relaxation, and in mostexperimental conditions, signal from pathways with more thantwo transverse periods is small or negligible due to T2 decay.(14)The majority of signal therefore comes from a spin echo pathway(Fig. 3a) and stimulated echo pathways (Figs 3b and 3c).

Factors affecting diffusion sensitivity

It is common convention to describe the diffusion weighting in aDWI experiment using b, (3) which can be broken down into twocomponents. The first factor is the area under each diffusiongradient, which is described by the value q (17) (to be precise, thearea is given by q/g). The second is the time between pairs ofdiffusion gradients, which is often referred to as the ‘diffusion

Figure 3. Five example DW-SSFP echo pathways generated from six TR

intervals. The diffusion encoding gradient is only experienced by spins in

the transverse plane and therefore one axis is used for each echo pathway

to show the ‘effective’ diffusion-encoding gradients for that given path-way. As in Figure 2, diffusion gradients experienced by inverted magne-

tization (i.e. due to a 1808 rotation) are negated to emphasize rephasing.

The pathways shown are: (a) primary spin echo; (b) primary stimulated

echo; (c) stimulated echo with additional longitudinal periods; (d,e)echoes with more than two transverse periods. Note that pathway (e)

forms two echoes during its history.

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time’, and denoted by D. In DW-SSFP, q is well defined as the areaunder the diffusion gradient in a single TR (18) (although the totalgradient area experienced by a pathway depends on how manytransverse periods it contains). However, D is more difficult todefine: each echo pathway has a different number of TR intervalsbetween the transverse pathways forming the echo, which is theprimary time during which the magnetization spins diffuse. Thistranslates into a different D, and therefore a different b, for eachpathway. For an excellent discussion of coherence pathways inDW-SSFP, see Buxton (14).The accumulation of diffusion-sensitization across multiple TR

and via many different echo-pathways is highly efficient, butmore difficult to quantify than with DW-SE or DW-STE. Thefraction of magnetization that populates each pathway dependson the flip angle, as described in Appendix A. At low flip angle, thefraction of magnetization experiencing a 908 pulse in any given TRis relatively small. Intuitively, this means relatively littlemagnetization is excited with each RF pulse. Additionally, it

Figure 4. Signal contributions for echo pathways with two transverse periodsechoes with m longitudinal periods. (a,b) Signal contribution of pathways in

orientations). Longer echoes (largerm) only contribute significant signal at low

to echo number, making it desirable to maximize the contribution from lon

weighting (for q¼ 102 cm�1, achievable with a 40mT/m gradient with 6msbetweenweighted and unweighted occurs at a¼ 248 (arrows). (e) Plots of raw(c) shows that echoes with b> 1000 s/mm2 only contribute significant signa


means any transverse magnetization that gets tipped back alongthe longitudinal axis remains there for longer, increasing theeffective diffusion time. This results in an optimum range ofrelatively low flip angles (Fig. 4).Similarly, as TR increases, the effective diffusion time also

increases, which will tend to increase the diffusion sensitivity.However, the lifetime of each pathway depends on the T1 and T2,placing a limitation on the degree to which diffusion contrast canbe boosted by increasing TR. Hence, the diffusion-sensitization ofthe net signal depends not only on q and TR, but also on the flipangle, T1 and T2.Given the ill-defined b-values associated with DW-SSFP, it

therefore becomes more difficult to provide a readily interpretedmeasure of the diffusion sensitivity of a given acquisition.Provided the area under the diffusion gradient (or q), the TR andflip angle are specified, however, one can readily calculate theexpected signal attenuation for a tissue of interest by assumingsensible T1 and T2 values. It is to be specifically discouraged,

. The spin echom¼ 0, and higher echo numbers correspond to stimulatedthe absence of diffusion weighting (the same surface is shown from two

flip angle, with a peak around 208. (c) The effective b-value is proportionalger echo pathways. (d) The total raw signal with and without diffusion

duration). For the simulated parameters, the greatest signal differencesignal at several flip angles (i.e equal-a profiles from a, b). Comparison with

l at low flip angle.



however, that DW-SSFP studies refer solely to a b-value (or aneffective b-value) due to the inherent ambiguity in such anumber.The concept of combining two gradient pulses across multiple

repetition periods to create a diffusion-weighting pair is alsofundamental to diffusion-weighted principles of echo shiftingwith a train of observations (DW-PRESTO) (10). However, thelatter sequence explicitly combines these pairs of gradients overtwo precisely-defined TR periods, resulting in different propertiescompared to DW-SSFP. DW-PRESTO has a well-characterized b-value and diffusion time, but has lower sensitivity to diffusion pergradient area since it does not create the long diffusion timesprovided by the stimulated echoes.

Figure 5. Pulse sequence diagrams showing two consecutive repetition

periods for (a) the canonical version of DW-SSFP; (b) quasi-isotropicdiffusion-weighting with alternating diffusion direction; and (c) DW-SSFP

with bipolar gradients within each TR.

7

ACQUISITION METHODS

In this review, we consider the steady-state DWI methoddescribed above as the canonical method, which we will referto as DW-SSFP (as opposed to the more general term ‘steady-state DWI’). However, a number of variations on this techniquehave been proposed, as well as DWI methods that use a steady-state acquisition scheme, but which achieve diffusion weightingin a very different way.

Diffusion-weighted SSFP, or DW-SSFP

As described above, the DW-SSFP sequence uses a singlediffusion gradient within each TR; however, we have not yetspecified when the signal is actually acquired. Early workconsidered acquiring the signal either before or after thediffusion gradient (5,8,13). This was motivated by descriptions of(non-diffusion-weighted) SSFP, which considered two distinctsignals during each TR: one immediately after the RF pulse (the‘FID’ component) and a second that peaked just before the RFpulse (the ‘echo’ component).In the context of DW-SSFP, it was soon recognized that,

although the FID component has higher signal, it has lowersensitivity to diffusion (14). This is because the additional signal inthe FID comes from freshly excited magnetization, which is notdiffusion encoded. For this reason, imaging with DW-SSFPtypically acquires data after the diffusion gradient (8,10,14)(Fig. 5a). If one re-casts the diffusion gradient as a simpledephasing gradient, it is essentially a contrast-enhanced Fourier-acquired steady-state technique (CE-FAST, also known as PSIF)(20).By strict definition, CE-FAST/PSIF uses an unbalanced readout,

with a rephasing gradient after the readout, but no dephasinggradient before. More recent implementations of DWSSFP,however, have ensured that all the imaging gradients are fullybalanced within each TR, bearing some resemblance to balancedSSFP methods (Fig. 5a) (18,21,22). This is done to avoid anycontribution to the diffusion weighting from the imaginggradients, since any unbalanced moment will incur somediffusion weighting. This is particularly important if data withdifferent diffusion directions are to be combined, where anunbalanced readout gradient will bias measurements if not takeninto account. In early work, which often only acquired a singlediffusion-encoded direction, this slight bias was likely less of aconcern than minimizing the TR on systems with limited gradientstrength.Another interesting variant of the canonical DW-SSFP

sequence alternates the direction of the diffusion-encoding


gradient between two orthogonal directions every other TR(Fig. 5b) (22). The goal of this work is to create a steady-state withquasi-isotropic diffusion-weighting. However, it is not clear towhat extent the resulting signal is isotropically weighted and towhat extent an oscillating steady-state of directionally-weightedsignal is created.One common characteristic of all SSFP-based imaging

methods is their intensive use of gradients. Overall, SSFP isone of the oldest known pulse sequences, having been described


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by Carr in 1958 (23). The relatively slow uptake of SSFP in theimaging literature can be largely attributed to the need for high-performance gradient technology, attested to by the recentrenaissance in SSFP imaging methods. Due to the need for bothshort TR and a diffusion gradient with area exceeding therequirements of imaging gradients, DW-SSFP is even moregradient-intensive than most other forms of SSFP. DW-SSFPwould almost certainly benefit from improvements in gradientduty cycles and cooling technology.Beyond hardware limits, there are also physiological restric-

tions on achievable protocols, most notably peripheral nervestimulation and RF energy deposition. In our experience, thediffusion-weighting gradients are not the primary source ofperipheral nerve stimulation in DW-SSFP. Rather, versions of DW-SSFP that lead to stimulation tend to be those that use fastreadouts, such as our 3D EPI-based trajectory (24). This is similarto conventional, single-shot diffusion sequences, where the (EPI)readout is the primary contributor to stimulation since thesereadouts include extended periods of near-constant gradientswitching (25,26). RF energy deposition is less of an issue in DW-SSFP due to the use of low flip angles and relatively long TR.Compared to balanced SSFP sequences, which may be limited byenergy deposition at high field strength due to relatively high flipangle (50–708) and short TR (3–4ms) (27), DW-SSFP uses bothlower flip angle (30–408) and longer TR (20–40ms). In ourexperience at 3 Tand below, RF energy deposition is not a limitingfactor in DW-SSFP.

Diffusion-prepared SSFP

A separate set of techniques that are sometimes referred to assteady-state DWI use a fast, steady-state readout to acquiremagnetization that has previously been prepared with diffusionweighting. The first method along these lines was proposed byNorris in 1992 (28). As illustrated in Figure 6, the Stejskal-Tannerscheme (with diffusion gradients on either side of a 1808 pulse) isused to create diffusion weighting, and is immediately followedby a steady-state readout train [e.g. FLASH (29) or low-angle RARE(30)]. An alternate way to prepare the magnetization prior to a

Figure 6. A pulse sequence diagram of a diffusion prepared steady-state

imaging sequence. A diffusion preparation module consisting of a

traditional diffusion-weighted spin echo is followed by a steady-state

readout train. In the driven equilibrium version, a second 908 pulse isadded to the diffusion preparation, placing the prepared magnetization

back along the longitudinal axis.


steady-state read-out train uses what is often referred to as drivenequilibrium. This method uses a Stejskal-Tanner spin-echopreparation, but then applies a 908 pulse to tip the preparedmagnetization back along the longitudinal axis (31) (indicated bygray RF pulse in Fig. 6). The magnetization is then acquired with asteady-state echo train.These methods represent an amalgamation of conventional

diffusion-weighting schemes and steady-state DWI. The keybenefit compared to diffusion-weighted spin echo imaging isthat multiple read-outs can be acquired for each diffusion-weighting period. In theory, the b-value for thesemethods shouldbe based solely on the diffusion preparation, although in practicethe measured signal is contaminated with non-diffusion-weighted signal excited by the echo train. Several improvementson these basic methods have been suggested to reducesensitivity to eddy currents (32), reduce motion sensitivity (33)and incorporate balanced SSFP readouts (34). Although thesediffusion-prepared steady-state techniques are a promising wayto acquire diffusion data rapidly without requiring single-shot EPIacquisitions, we will focus preferentially on the canonical DW-SSFP methods described above in this review.

MITIGATING THE EFFECTS OF MOTION

Motion sensitivity of steady-state DWI

The biggest challenge facing steady-state DWI acquisitions ismotion sensitivity. All diffusion-weighted sequences are sensi-tized to motion on the micron scale, and therefore also to largerscale motion such as patient movement or brain pulsation. Anysource of displacement between diffusion gradients is encodedin the magnetization phase. Diffusive motion results in randomphase for different spins and therefore signal loss, while bulkmotion creates a net phase shift in tissue. If k-space is acquired inmultiple shots, bulk motion causes shot-to-shot inconsistency ofthe object phase, leading to severe image artifacts that corruptthe image magnitude. If k-space can be acquired using single-shot readouts, such as echo-planar imaging (EPI) (4), thesephase artifacts can be neglected, because the image magnitudeis unaffected. For this reason, the vast majority of diffusionimaging uses EPI acquisitions which, when integrated into aDW-SE sequence, result in highly-efficient diffusion imaging. Foran excellent review of motion artifacts in diffusion-weightedimaging, see (35).Unfortunately, single-shot EPI is not readily compatible with

steady-state DWI. In order for the canonical DW-SSFP method tomaintain the magnetization steady-state, the TR must be fairlyshort: typically 20–40ms, but more generally on the order of T2.Given that the TR must also accommodate a 5–10ms diffusiongradient, single-shot readouts are not a viable option. Moreover,single-shot readouts are essentially restricted to 2D multi-sliceacquisitions, which are also incompatible with DW-SSFP. Cyclingbetween 2D slices would increase the TR, while scanning slicessequentially would require repeatedly re-establishing thesteady state for each slice, reducing imaging efficiency. Thus,DW-SSFP requires 3D volumetric imaging acquired over a largenumber of shots. It is worth noting that, although parallelimaging is unlikely to be sufficient to acquire a 3D volume in asingle shot, DW-SSFP is nevertheless likely to benefit fromparallel imaging due to the general compatibility of thesemethods with 3D acquisitions.



7

This restriction of steady-state DWI to multi-shot readoutsmakes motion-induced phase artifacts a major obstacle for in vivoimaging. This motion sensitivity was noted when DW-SSFPimaging was first introduced (11,12), and overcoming thisproblem has been the single greatest area of focus in technicaldevelopment in subsequent years. If motion artifacts can beovercome, the efficiency of DW-SSFP with short, multi-shotreadouts could provide a method with lower distortion and high-resolution capabilities.There are two general types of methods proposed to mitigate

motion sensitivity in DW-SSFP. Ideally, the sequence could bemade intrinsically insensitive to motion, and several methodshave attempted to achieve this. Alternatively, one could attemptto remove motion artifacts in image reconstruction, most likelyusing a measurement of the motion-induced phase, known as a‘navigator’. We discuss both approaches below.

Reducing motion sensitivity

To reducemotion sensitivity, one group has suggested the use of abipolar set of gradients (i.e with the same area and opposing sign)instead of the single diffusion gradient used in canonical DW-SSFP(36), as shown in Fig. 5c. This method is expected to reducemotionsensitivity because the time between the first and secondgradients, during which the sequence can accumulate motion-induced phase errors, is contained within a single TR rather thanpropagating across multiple repetition periods as above. Unfortu-nately, it is exactly this time during which the canonical DW-SSFPmethod accumulates its strong diffusion sensitivity. In other words,the diffusion time, D, is fundamentally tied to the timescale ofmotion sensitivity. As a result, the sequence in Figure 5c hasreduced sensitivity to both bulk motion and diffusion.A generalization of this technique was proposed where a

fraction of the diffusion-encoding gradient was left unbalanced(37), attempting to gain some of the benefits of DW-SSFP whilestill suppressing motion sensitivity. Compared to the completelyunbalanced sequence, obtaining diffusion sensitivity in the brainwith the partially-balanced version requires significantly longertime for the diffusion encoding gradients, increasing the TR andreducing efficiency.This approach has been shown to provide sufficient diffusion

contrast to detect renal diffusion (36), which is 2–3 times fasterthan white matter diffusion. However, even with a 2.5-foldincrease in TR, this method had significantly lower diffusionsensitivity in the brain, insufficient to suppress signal fromcerebrospinal fluid (37).

Correcting for motion artifacts

In order to reduce motion artifacts while maintaining the intrinsicdiffusion sensitivity of the unbalanced DW-SSFP scheme, severalgroups have explored methods of correcting motion-inducedphase errors. These methods all include some form of phasecorrection in image reconstruction using a phase measurement,known as a navigator. However, there is considerable variation inthe navigator measurement and correction schemes.Navigator measurements are based on the assumption that

phase corruption in diffusion imaging can be characterized bylow spatial frequencies. Rigid motions such as shifts and rotationscorrespond to constant and linear phase (38,39), while non-rigid motion of the brain caused by cardiac pulsation createsslowly-varying phase modulations (21). In their most generalform, navigators are additional data acquired at the center of k-


space with the goal of measuring these low-spatial-frequencyphase offsets.The first ‘1D’ navigator methods acquired an additional line in

k-space, which is sufficient to measure constant and linear phaseoffsets along the direction of the navigator line due to rigid bodymotions (38,39). This method was adapted for DW-SSFP (40),improving image quality but not completely removing artifacts.Examining their navigator data, the authors identified respir-atory artifacts, as well as pulsations that appear consistent withcardiac motion (41,42), which are not well-approximated bylinear terms and therefore could not be removed by theircorrection (40).Fortunately, it is relatively straightforward to extend the

navigator concept to two dimensions (43), allowing thecorrection of non-linear phase offsets associated with non-rigidmotion from cardiac pulsations (21,44). Miller and Pauly proposedacquiring a 2D spiral navigator covering only the center of k-space along with each line in a 2DFT acquisition (21) [or,alternatively, acquiring the navigator as part of a variable,interleaved spiral (45)]. The corrected images show a markedimprovement in quality compared to a linear phase correction(Fig. 7). When used in conjunction with cardiac synchronization(where RF pulses continue but no imaging data is acquired duringpeak systole), image quality improves considerably. However, aspresented this method only acquired a single, 2D slice, and thiswork is best considered a proof of concept rather than a viablemethod. A straightforward extension using a 3DFT acquisitionwith a 2D spiral navigator was shown to produce reasonableimages of articular knee cartilage; however, this work suggestedthat a 2D navigator was sufficient largely due to the restriction ofmovement in the leg (46).Several groups have since proposedmethods for acquiring and

correcting 3D navigated DW-SSFP. This has included 3D radialspokes (47), 3D rotated EPI (24) and 3D rotated spirals (48). All ofthese methods are ‘self-navigated’, meaning that these k-spacetrajectories cover a part of central k-space in each imagingreadout, which can be used for navigator correction. In additionto the k-space trajectory, these methods differ in termsof the type of correction that is applied. Radial acquisitionsgenerally only contain sufficient information at the center of k-space for a linear correction. Nevertheless, when used inconjunction with cardiac synchronization, rejection of cor-rupted spokes and a conjugate gradient reconstruction basedon parallel imaging techniques, good image quality can beachieved (47). A different approach was taken with the 3Drotated EPI (or, TURBINE) trajectory (81). Here, cardiacmonitoring is used in the image reconstruction (post-acquisition) to identify EPI blades at a common part of thecardiac cycle and combine them to create a 3D navigator. Thisenables a 3D extension of the non-rigid navigator correctionpresented previously (21).Several important outstanding issues remain regarding

navigator correction of DW-SSFP data. The first is whether 3Dnavigator information is necessary for correcting 3D imagingdata. It is the opinion of the authors of this review that it is,for the reasons illustrated in Figure 8. However, no cleardemonstration of this has been shown. Second, it is unclearwhether any navigator correction will be able to suppressmotion artifacts to a sufficient degree for quantitative or semi-quantitative DW-SSFP (e.g. to compare the signal magnitudefrom different directions of diffusion weighting to estimatediffusion anisotropy).


87

Figure 7. Navigated DW-SSFP images demonstrating improvement in image quality with linear and non-linear (refocusing) phase correction, and

cardiac synchronization (in a separate subject). Motion corruption is muchworse when diffusion weighting is applied along the superior-inferior direction

due to strong motion from cardiac pulsation. These images were acquired using a 2D, slice-selective pulse sequence with a spiral navigator. While

excellent potential for removing artifacts is demonstrated, DW-SSFP images must be acquired in 3D for practical applications. Reproduced withpermission from Miller and Pauly (21).


788

SIGNAL MODELS AND DATA ANALYSIS

The Modified Kaiser model

Kaiser (5) derived the first model for the diffusion attenuation ofSSFP in the presence of a constant magnetic gradient by solvingthe Block-Torrey equations (49). The solution can be representedas a Fourier integral, and simplified to a Fourier series based onthe periodicity of the signal with respect to precession angle. Thiscan be further simplified by assuming symmetry of the Fouriercoefficients, resulting in a fairly complicated, but closed form,expression. For imaging with DW-SSFP, the Kaiser expression wasgeneralized to the case of pulsed gradients by Wu and Buxton(13) using calculations that parallel the Kaiser derivation closely.

Figure 8. Do volumetric (3D) acquisitions require navigation in three dimen

motion-induced phase on 3D k-space data. If the phase profile is perfectly flat

cardiac cycle, the brainstem and spine experience large displacements alongsimulate this kind of motion assuming amaximum velocity difference of 0.3mm

phase corruption when diffusion weighting is along the superior-inferior direc

Multiplication by this three-dimensional phase corruption is equilvalent to co

navigator correction requires 3D phase measurements (i.e 3D coverage of t


We will refer to this as the Modified Kaiser model, presented inAppendix B.Freed and colleagues have more recently noted that the

symmetry assumed by Kaiser to derive the closed-form solutioncanbe very inaccurate (50). They identify specific regimes of signalbehavior by comparing the TR with a characteristic diffusiontime, T�1

D ¼D(g GTR) (2). Greatest diffusion sensitivity in DW-SSFPcorresponds to TR< TD< T2 and aE<a< 1808 (where aE is theErnst angle). The greatest error in the Kaiser model occurs inprecisely this regime. However, Freed and colleagues focus onmuch shorter TR than can be achieved in vivo (1–3ms), suchthat TR TD. By contrast, most in vivo imaging is in theregime TR�TD, where the Kaiser model would appear to be more

sions? Above, we use a T1-weighted structural to simulate the effect of

, the center of k-space has a clear central peak (‘uncorrupted’). During the

the superior-inferior direction, while the cortex is more stable (41,42). We/s between the brainstem and cortex (41), and calculate the image-space

tion (for simplicity, using a Stejskal-Tanner encoding with b¼ 980 s/mm2).

nvolving k-space by a three-dimensional kernel. We therefore argue that

he center of k-space, which is used to de-convolve the corruption).


Figure 9. (a) A 2D apparent diffusion coefficient (ADC) profile for a

diffusion tensor typical in white matter (l1/l1¼ 10�3/10�4mm2/s) and

(b) the resultant DW-SSFP (black dashed) and DW-SE (grey solid) signalprofiles. Theseprofiles representDW-SEwithb¼ 3000 s/mm2andDW-SSFP

with G¼ 40mT/m and d¼ 12ms. Reproduced with permission from (18).


7

accurate. However, the issueof accuracyof theKaisermodel in vivohas not been explored in depth.Although the Modified Kaiser model in its closed form provides

a useful method for calculating the DW-SSFP signal, it is difficultto gain intuition for signal dynamics and dependencies from thisexpression. An alternative calculates the DW-SSFP signal as asummation of diffusion-weighted magnetization pathways. Thisconcept was introduced by Kaiser (5) and shown to agree wellwith the closed form solution to the Bloch-Torrey equationsprovided the TR is not long compared to T1 and T2, which shouldgenerally be true in imaging experiments. This pathway (or‘partition’) analysis and its implications were explored in depth byBuxton (14), and much of the description of DW-SSFP in terms ofpathways given above is drawn directly from this work.

Approximate models

A number of approximate signal models have been proposedthat warrant discussion. Most of these can be considered toapproximate the Modified Kaiser model by neglecting some ofthe echo pathways that contribute signal. The earliest exampleactually preceded and inspired Kaiser’s work. In 1961, Woessnercalculated the signal for the echo pathways that result fromthree- and four-pulse sequences with a constant (non-pulsed)diffusion gradient (6). Similarly, Merboldt presented an approxi-mation that only includes the primary spin and stimulated echoes(10). This approximation is exact for 908 pulses (8–12), but will notbe accurate for lower flip angle.In 1993, Buxton noted that the Modified Kaiser model predicts

increased diffusion sensitivity at low flip angle, and proposed anapproximation that is more accurate in this regime (14). Thisapproximation only considers the echo pathways where twotransverse periods are separated by longitudinal periods whosenet effect is a 1808 rotation, known as the ‘two-transverse periodapproximation’. If T2 � 1.5TR, pathways with more than twotransverse periods will contribute negligible signal, and can beignored without sacrificing much accuracy (14). This approxi-mation has been used for semi-quantitative DW-SSFP in short-T2tissues (46). In situations where this approximation holds, thesignal corresponds to a single q -value (although b is still poorlydefined), providing a useful framework for studying complicatedpatterns of anisotropy and restricted diffusion (18).Oneearlymodelmodified the standardexpressions forbalanced

SSFP (51,52) to include an ‘enhanced T2’ accounting for irreversibledephasing due to diffusion (9). This approximation is not based onisolating certain echo pathways. Although this kind of approxi-mation is useful for single-echo (SE and STE) diffusion-weightedsequences, it does not account for the longer diffusion timeexperienced during longitudinal periods in DW-SSFP and is lessaccurate than the other approximations described above (13).

Quantifying diffusion

Although each coherence pathway has a well-defined b-value, thesignal contributedbyapathwayalsodependsononT1, T2, flipangleandTR. As canbe seen inAppendixB, the signal dependenceon thediffusion coefficient (encapsulated in the A1 and A2 terms) cannotbe easily separated from other terms. Quantification of diffusioncoefficients with DW-SSFP therefore requires knowledge of T1, T2and flip angle. A number of methods for rapid T1 and T2 estimation(which typically estimate the B1 map, as well) have been proposedrecently thatmake this kindofquantificationmore feasible (53–57).QuantificationcouldeitherestimateT1, T2andB1asafirst stage, and


feed thesevalues intoanADCestimation (58), orperformnon-linearfitting on all data simultaneously. Although these parameters canbe measured, they come at a cost of increased scan time andadditional sources of error.In addition to the dependences discussed above DW-SSFP

signal magnitude also exhibits flow sensitivity. This effect that hasbeen neglected by most proposed quantification methods,except inasmuch as the sensitivity to motion is well recognized(11,12). Gudbjartsson and colleagues incorporated the effects offlow velocity into the Kaiser model, and conclude that flow (orother motion approximated as linear velocity) could contributesubstantial error to quantification in DW-SSFP (59).In general, though, ADC quantification with DW-SSFP has

received relatively little attention to date. This is likely in partbecause the DW-SSFP signal is considerably more complicatedthan other diffusion methods.

Diffusion anisotropy and tractography

Detection of diffusion anisotropy with DW-SSFP has beendemonstrated by several groups, both in vivo (21,47) andex vivo (18,24,60). If quantitative ADC measurements can beobtained along a number diffusion-weighting directions, fittingthe diffusion tensor or performing streamline tractography canbe done using identical methods as in DW-SE. However, the needfor additional measurements (T1, T2 and flip angle) incursadditional scan time, as discussed above.If quantitative ADC values are not calculated, there are some

subtleties to the analysis of anisotropy that warrant caution. Asshown in Figure 9, the DW-SSFP signal profile that results from aprolate diffusion tensor typical of whitematter is different from thesignalprofile foraDW-SEexperiment (18). Inparticular, the ‘waist’ ofthe measured signal is thicker (see Fig. 9), a characteristic thatresults from thevariable attenuationof coherencepathwaysdue todiffusion weighting. Nevertheless, the anisotropy and orientationof principal diffusion is obvious, and several groups havedemonstrated qualitatively reasonable anisotropy measures bysimply assuming a b-value and feeding data into a DW-SE analysis(60). However, measures such as mean diffusivity will benonsensical and fractional anisotropy will be biased if the fullmodified Kaiser model is not used (e.g. due to the thicker waist).If tractography is the primary goal, then it is possible to achieve

excellent estimates of the principal diffusion orientation withoutmapping or estimating T1 and T2 (18). One can incorporate theModified Kaiser model into a probabilistic tractography frame-work, which can either estimate these parameters or usemeasured values (18). However, it is important that the full signal


89

Figure 10. (a) In vivo DW-SSFP images obtained with a self-navigated 3D radial trajectory. (b) Ex vivo directionally-encoded colour maps for matched

DW-SE and DW-SSFP acquisitions, demonstrating higher SNR and CNR in DW-SSFP. (c) High resolution (0.125mm3) DW-SSFP images in ex vivo humanbrain showing mean diffusion-weighted image, directionally-encoded colour map, tractography. Data are (a) re-formatted from Jung et al. (47); (b) re-

formatted from McNab et al. (24); (c) reproduced with permission from Benner et al. (60). Acquisition details given in the cited references.


790

model is incorporated for the estimates of the uncertainty inprincipal diffusion direction to be accurate (61).DW-SSFP has been shown to yield sensible tractography results

using both streamline (47,60) and probabilistic (18) approaches(for example, see Fig. 10c). A particularly promising result inex vivo brain demonstrated a higher degree of certainty on theprincipal diffusion direction using DW-SSFP compared to amatched DW-SE method (18) (Fig. 10b).

Non-Gaussian diffusion

One limitation of the full modified Kaiser model is that it implicitlyassumes Gaussian diffusion. The assumption of Gaussiandiffusion has been shown to be inappropriate when complextissue structure is found within a single image voxel. Q-spaceimaging is a model-free analysis method that can represent non-Gaussian diffusion effects such as restrictive barriers and/orcrossing fibres. For DW-SE (and other single echo diffusionimaging methods) there exists a simple Fourier transformrelationship between q-space and the spin displacementprobability density function (provided the diffusion gradientcan be reasonably approximated as a narrow pulse) (17). Due tothe multiple diffusion times of DW-SSFP, the signal reflects the


sum of multiple spin displacement profiles. A forward model forDW-SSFP in terms of an arbitrary spin displacement probabilitydensity function has been presented previously, however, andused to predict the DW-SSFP signal profiles one would expect forcrossing fibres and restrictive barriers (18).

APPLICATIONS

Although steady-state DWI has not been adopted for widespreaduse, at least in part due to difficulties with motion sensitivity andquantification, it has been considered for a few applications. Onegroup in particular has experimented extensively with both DW-SSFP (62–65) and diffusion-prepared steady-state imaging(66,67). Below, we give an overview of the uses of steady-stateDWI that have been explored to date.

In vivo DWI of the brain

Due to the issues of motion sensitivity, DW-SSFP has to date hadlittle application in the brain in vivo. Early studies did present DW-SSFP results for both ischemia (62) and diffusion tensor imagingin amnesia (64). These studies preceded work on navigatorcorrection (21,40) and motion artifacts are clearly visible as



7

regions of intense dropout [e.g. see Figs 1b and 3b in (64)]. Recentadvances in 3D navigated DW-SSFPmay prompt new in vivo brainapplications in the coming years.

Ex vivo DWI of the brain

An area where DW-SSFP has shownmore clear promise is in ex vivoimaging of human brain specimens (24). There has beenconsiderable recent interest in ex vivo diffusion imaging, eitherfor validating the link between in vivo diffusion imaging andhistology (24,68–70) or to study disease (71–74). While the lack ofmotion and ability to scan for long periods may seem to makediffusion imaging of ex vivo tissue a relatively straightforward task,the significant reductions in T1, T2 and diffusion in fixed tissue arein fact highly problematic for DW-SE. In contrast, DW-SSFPperforms quite well under these circumstances, and early resultsindicate that it should outperform DW-SE in this application(18,24). In addition to the first demonstration of the use of DW-SSFP on clinical 3T scanners for scanning whole human brains (24),a recent preliminary report at demonstrated 0.5� 0.5� 0.5mm3

resolution of a human cortical hemisphere (60).

Nerve imaging

Outside the central nervous system, DW-SSFP has beensuggested as a useful alternative for neurography due to itspotential for high resolution and insensitivity to susceptibility-induced field inhomogeneity. An early study demonstratedvisualization of the optic chiasm with DW-SSFP, but image qualityclearly suffered from motion artifacts (no attempts to mitigatemotion were undertaken) (75). A more recent study presentedresults using DW-SSFP of cranial nerves, albeit with a relativelylow level of diffusion weighting (76). DW-SSFP of the sciatic nervehas been reported as a method for determining the locus andcause of sciatic pain (77). More work will be required to determinethe limits of DW-SSFP for neurography, but at present thisappears to be a promising method.

Musculoskeletal imaging

Perhaps the most successful application of steady-state diffusionhas been in musculoskeletal imaging. Many of the tissues ofinterest in the musculoskeletal system have short T2, leading tolow signal in DW-SE sequences due to the need for long TE. DW-SSFP of articular cartilage has been demonstrated by two groups(46,78), which could have application in studying the breakdownof the cartilage matrix in osteoarthritis and other diseases. DW-SSFP has been suggested as an excellent option in the study ofbone fracture (63,65,79). Diffusion-prepared steady-state imaging(e.g. using a RARE readout) has also been considered for imagingof musculoskeletal tumors (66).

Renal imaging

One of the first applications of steady-state DWI was the work byDingandcolleagues, usingapartially-balancedversionofDW-SSFPfor renal imaging (36). As discussed above, this partially-balancedsequence reduces motion sensitivity at the cost of sensitivity todiffusion. However, faster diffusion in the kidney was judged torequire less diffusion sensitivity, and the authors were optimisticabout the level of diffusion contrast obtained. It should be noted,however, that body imaging suffersmuchmore severe cardiac andrespiratory inducedmotion than in thebrainandhenceDW-SSFPofthe kidney still represents a significant challenge.


SUMMARY

Steady-statemethods fordiffusion-weighted imagingpresentbothgreat potential benefit and significant challenges. Although thesemethods were some of the first to be explored for detectingdiffusion in NMR and MRI, the motion sensitivity and complicatedsignal dependencies have prevented them from being taken up inpractical settings. However, recent advances in acquisitiontechniques and signal analysis have made progress in addressingsome of these problems. The potential for efficient diffusionweighting ina low-distortion, 3Dvolumetric acquisitionmeans thatthese methods may find practical applications in coming years.

Note

1. The existence of a 1808component for an RF pulse with flipangle less than 908 is a bewildering concept when firstencountered. However, it is simply a mathematical necessitythat arises when representing a vector rotating in a 3-dimensional space in terms of a complex transverse componentand a scalar longitudinal component. In this representation, 3Drotations appear to behave non-linearly unless one explicitlyrepresents the complex conjugate of the transverse magnetiza-tion. In this representation, the complex conjugate experiencesRF pulses but does not contribute signal unless it is inverted. Forthis reason, it is sometimes suggested that one consider theconjugate as ‘unobservable’ magnetization mixing with the non-conjugated, observable magnetization. Whether or not thisreflects true quantum mechanical behavior, in our experience itcontributes little intuition and tends to encourage fuzzy-mindedobfuscation (16).

Acknowledgements

The authors would like to thank Drs Thomas Benner and Young-kyoo Jung for their kind figure contributions, and Dr Saad Jbabdifor insightful comments on the manuscript.

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7

APPENDIX A

In this section, we derive the result that an arbitrary flip angle canbe decomposed into the net effect of 08, 908 and 1808componentpulses, following the derivation given in x18.3 of (80). The effectof an RF pulse of arbitrary angle (a) applied along the x-axis canbe represented by a rotation matrix about the x-axis applied to a


magnetization vector (MS).

Mþ ¼ RxðaÞM� (A.1)

whereM� represents the magnetization prior to application of theRF pulse, Rx(a) represents a rotation about the x-axis by a degreesand Mþ represents the magnetization after the RF pulse has beenapplied. Neglecting relaxation effects during the application of theRF pulse, the x, y and z components of Mþ can be written as:

Mþx ¼ M�

x

Mþy ¼ M�

y cosaþM�z sina

¼ M�y ½cos2ða=2Þ � sin2ða=2Þ�

þM�z sinðaÞ

Mþz ¼ M�

z cosa�M�y sina

¼ M�z ½cos2ða=2Þ � sin2ða=2Þ�

�M�y sina

(A.2)

If we define the transversemagnetization (Mxy) and its complexconjugate (M�

xy) as:

Mxy ¼ Mx þ iMy

M�xy ¼ Mx � iMy

(A.3)

we can re-write Eq. A.2 as:

Mþxy ¼ M�

x ½sin2ða=2Þ þ cos2ða=2Þ�þiM�

y ½cos2ða=2Þ � sin2ða=2Þ�þiM�

z sinðaÞ¼ M�

xycos2ða=2Þ þM��

xy sin2ða=2Þ

þiMzsinaMþ

z ¼ M�z cos

2ða=2Þ �M�z sin

2ða=2Þ� i

2 ½M��xy �M�

xy �sina

In the context of the transverse (Mxy) and longitudinal (Mz)magnetization, this yields a fairly simple interpretation: cos2ða=2Þexperiences a 08 pulse, sin2ða=2Þ experiences a 1808 pulse andsin(a) experiences a 908 pulse.

APPENDIX B

The equation for the full DW-SSFP signal (S) that is referred to inthis manuscript as the Modified Kaiser Model (13,14) is given by:

S ¼ �M0ð1� E1ÞE2A�2

32 ðF1 � E2A1A

232Þ sina

r � F1s(B.1)

where:

F1 ¼ K �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2 � A2

2

p

K ¼ 1�E1A1 cosa�E22A21A

�23

2 ðE1A1�cosaÞ

E2A1A�43

2 ð1þcosaÞð1�E1A1Þ

r ¼ 1� E1 cos aþ E22A1A132ðcosa� E1Þ

s ¼ E2A1A�4

32 ð1� E1 cosaÞ þ E2A

�13

2 ðcos a� 1ÞA1 ¼ e�b1D

A2 ¼ e�b2D

b1 ¼ ðgGdÞ2TRb2 ¼ ðgGdÞ2dE1 ¼ e�

TRT1

E2 ¼ e�TR

T2

(B.2)

G and d represent the amplitude and duration of the diffusiongradient respectively. Note that we use slightly different notationthan previous work for the b1,2 terms in order to avoid confusionof these terms with the b-value of a given coherence pathway.

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Steady?state diffusion?weighted imaging: theory