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SignaltonoiseandsignaltonoiseefficiencyinSMASHimaging

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Signal-to-Noise Ratio and Signal-to-Noise Efficiencyin SMASH Imaging

Daniel K. Sodickson,1* Mark A. Griswold,2 Peter M. Jakob,2 Robert R. Edelman,2

and Warren J. Manning1,2

A general theory of signal-to-noise ratio (SNR) in simultaneousacquisition of spatial harmonics (SMASH) imaging is presented,and the predictions of the theory are verified in imaging experi-ments and in numerical simulations. In a SMASH image, mul-tiple lines of k-space are generated simultaneously throughcombinations of magnetic resonance signals in a radiofre-quency coil array. Here, effects of noise correlations betweenarray elements as well as new correlations introduced by theSMASH reconstruction procedure are assessed. SNR and SNRefficiency in SMASH images are compared with results usingtraditional array combination strategies. Under optimized condi-tions, SMASH achieves the same average SNR efficiency asideal pixel-by-pixel array combinations, while allowing imagingto proceed at otherwise unattainable speeds. The k-spacenature of SMASH reconstructions can lead to oscillatory spatialvariations in noise standard deviation, which can produce localenhancements of SNR in particular regions. Magn Reson Med41:1009–1022, 1999. r 1999 Wiley-Liss, Inc.

Key words: SNR; SMASH imaging; fast imaging; RF coil arrays

Radiofrequency (RF) coil arrays are commonly used toimprove signal-to-noise ratio (SNR) in magnetic resonanceimages. However, spatial information from RF coil arraysmay also be used to accelerate MR image acquisition byshifting some of the burden of spatial encoding away fromthe traditional imaging gradients. Since the initial develop-ment of MR coil arrays, a number of rapid imaging tech-niques have been proposed based on this principle (1–9).Recently, using the simultaneous acquisition of spatialharmonics (SMASH) technique (9), two- to fourfold im-provements in imaging speed have been demonstrated invivo with commercial arrays, and up to eightfold accelera-tions have been obtained in phantom images using special-ized RF hardware (10,11). SMASH operates by substitutinglinear combinations of component coil signals for time-consuming phase-encoding gradient steps, effectively al-lowing the acquisition of multiple lines of k-space inparallel. This approach may be used to multiply theintrinsic speed of MR imaging sequences without increas-ing gradient switching rate or RF power deposition.

While the SNR properties of various reconstructionalgorithms in traditional gradient-encoded acquisitions

with coil arrays have been studied in detail (12,13), nosimilarly detailed studies have documented the effects onimage SNR of accelerated parallel data acquisition inarrays. Carlson and Minemura (8) predicted an approxi-mate Î2 decrease in SNR for a twofold increase in imagingspeed using two nested volume coils, and they also pre-dicted spatial variations in SNR across the image plane, butno explicit SNR measurements were reported. In theproposal by Ra and Rim (5) for an alternative image-domain ‘‘subencoding’’ approach using surface coils, SNRwas not discussed. Kwiat et al (6,7) presented general SNRconsiderations for a proposed many-coil massively parallelimaging apparatus. These general considerations, however,do not indicate what SNR is to be expected for a partiallyparallel imaging technique such as SMASH.

In this report, a detailed description of SNR in SMASHimaging is presented. Analytic expressions for the propaga-tion of signal and noise terms in a SMASH reconstructionare derived. These expressions are used to explore theeffects of noise correlations on image SNR, the dependenceof SNR upon reconstruction parameters, and the scaling ofSNR with acquisition speed. SNR and SNR efficiency inSMASH are also compared with theoretical predictions forconventional gradient-encoded acquisitions and image re-constructions using RF coil arrays (12,13).

These theoretical results are then verified in phantomimaging experiments and in numerical simulations. Forrealistic coil array designs, SMASH images and referenceimages are compared to assess the spatial distribution andacquisition-speed dependence of SNR. Noise variance mea-surements for in vivo images are also compared both withtheoretical predictions and with the results of simulations.

Finally, after a review of the principal determinants ofSNR in SMASH images, implications for the design anduse of RF coil arrays in ultrafast partially parallel imagingapplications are discussed.

THEORY

SMASH Imaging: A Summary

The SMASH procedure is summarized in Fig. 1. Theleft-hand side of the figure shows k-space schematics forthe various stages of data collection and reconstruction,and the right-hand side shows Fourier-transformed imagedata from a water phantom at each of the correspondingstages.

First, MR signal data are acquired simultaneously in thecomponent coils of an RF coil array. A fraction 1/M of theusual number of phase encoding gradient steps are ap-

1Charles A. Dana Research Institute and Harvard-Thorndike Laboratory of theDepartment of Medicine, Cardiovascular Division, Beth Israel DeaconessMedical Center, Boston, Massachusetts.2Department of Radiology, Beth Israel Deaconess Medical Center, Boston,Massachusetts.Grant sponsor: Whitaker Foundation (Biomedical Engineering Grant).*Correspondence to: Daniel K. Sodickson, Cardiovascular Division, BethIsrael Deaconess Medical Center, 330 Brookline Avenue, Boston, MA 02215.E-mail: dsodicks@caregroup.harvard.eduReceived 13 February 1998; revised 10 December 1998; accepted 11December 1998.

Magnetic Resonance in Medicine 41:1009–1022 (1999)

1009r 1999 Wiley-Liss, Inc.

plied, with M times the usual spacing in k-space (M 5 2 forthis schematic example). The component coil signals ac-quired in this way correspond to images with a fraction1/M of the desired field of view (FOV) (Fig. 1a). With Mtimes fewer phase-encoding steps, only a fraction 1/M ofthe time usually required for this FOV is spent on datacollection.

Next, a total of M linear combinations of the componentcoil signals are formed, to produce M shifted compositesignal data sets (Fig. 1b). The linear combinations aredesigned to produce sinusoidal modulations of total RFsensitivity across the image plane. These modulations, or‘‘spatial harmonics,’’ mimic the modulations produced byphase-encoding gradients and may be shown to produce anidentical k-space shift (cf. Ref. 9 and Eq. [4] below). Twosuch spatial harmonic modulations are illustrated schemati-cally in Fig. 1, for a total time savings factor of M 5 2 in thisexample. The maximum achievable time savings factordepends on the total number of spatial harmonics that mayreliably be synthesized through linear combination, andthis number in turn scales up with the number of arrayelements. Identification of the appropriate linear combina-tions for spatial harmonic generation requires independentmeasurement or estimation of the sensitivity profiles ofeach component coil in the array. Several strategies forsensitivity calibration have been described (9,14,15). Fol-

lowing calibration, optimal weighting factors for the com-ponent coils may be derived from a simple numericaloptimization procedure.

After linear combinations have yielded the shifted datasets, these composite signals are interleaved to yield a fullk-space matrix appropriate to the full desired FOV. Finally,this matrix is Fourier transformed to give the reconstructedimage (Fig. 1c). Since the need for multiple phase-encoding gradient steps generally constitutes the temporalbottleneck in MR acquisitions, the SMASH procedurereduces image acquisition times by an amount propor-tional to the number of phase-encoding gradient stepsomitted, namely, by an integer factor of M. Further detailsabout practical implementations may be found in Ref. 9.

The success of a SMASH reconstruction relies upon theexistence of correlations between signals in different com-ponent coils of an RF coil array. These signal correlationsarise from spatial variations in coil sensitivity. The signalfrom each array element may be written as

Sl(kx, ky) 5 ee dx dy r(x, y)Cl(x, y) exp (2ikxx 2 ikyy) [1]

where r(x, y) represents the spin density in a two-dimensional image plane labeled by Cartesian coordinatesx and y, and Cl(x, y) represents the spatial distribution ofRF sensitivity for the lth component coil in the array. kx and

FIG. 1. Schematic representation of the SMASH imaging procedure (k-space cartoon on the left, corresponding phantom images on theright). a: Simultaneous acquisition of data with a reduced number of phase-encoding gradient steps in multiple component coils of an array. b:Formation of shifted data sets using spatial harmonic combinations of component coil signals. c: Interleaving of shifted data sets to generate afull signal matrix, corresponding to an image with full FOV in a reduced total acquisition time.

1010 Sodickson et al.

ky are k-space coordinates corresponding to evolution infrequency-encoding (Gx) and phase-encoding (Gy) gradi-ents, respectively. Here, a two-dimensional imaging experi-ment has been assumed, and this assumption will bemaintained for the purposes of notational simplicity in thesections that follow. The generalization to three-dimen-sional imaging is straightforward, however, and three-dimensional imaging sequences have been successfullycombined with SMASH in practice.

SMASH relies on the fact that, for a suitable array, thecomponent coil sensitivity functions may be combinedwith complex weights 5nl

(m)6 to approximate various spatialharmonic profiles of order m. The m 5 0 harmonicrepresents the composite sensitivity formed by some refer-ence linear combination with weights 5nl

(0)6, i.e.,

C (0)(x, y) 5 ol

nl(0)Cl(x, y). [2]

For the simple case of a linear coil array with elementsdisposed, for example, along the y direction, it is straight-forward to show that higher order harmonics may also beapproximated by appropriate weighted sums:

C (m)(x, y) 5 ol

nl(m)Cl(x, y) < C (0)(x, y) exp (2imDkyy). [3]

Here, Dky 5 2p/FOV, and m is an integer that may bepositive or negative depending on the direction of k-spaceshift that is desired. Using Eqs. [1] to [3], we may derive asimple expression for weighted combinations of compo-nent coil signals:

ol

nl(m)Sl(kx, ky) 5 ee dx dy r(x, y) o

lnl

(m)Cl(x, y)

? exp (2ikxx 2 ikyy)

< ee dx dy r(x, y)C (0)(x, y)

? exp (2ikxx 2 i(ky 1 mDky)y)

< ee dx dy r(x, y) ol

nl(0)Cl(x, y)

? exp (2ikxx 2 i(ky 1 mDky)y)

< ol

nl(0)Sl(kx, ky 1 mDky) [4]

This is the basic relation underlying SMASH imaging. Itillustrates that linear combinations of component coilsignals may in fact be used to produce k-space shiftsanalogous to those produced by magnetic field gradients. Italso indicates that, when imaging is performed in a suit-ably configured RF coil array, the set of MR signal data atone k-space position is linearly related to the signal data atadjacent k-space positions. This is a departure from theusual state of affairs for single-coil imaging, in which such

interdependencies between different k-space signals areprecluded by the orthogonality of the Fourier basis. Anaccurate description of SNR in SMASH must also accountfor any noise correlations that may accompany thesek-space signal correlations.

SNR in SMASH: Signal Correlations and Noise Correlations

In this section, we explore the effects of noise correlations,including both preexisting correlations resulting from thesample and array geometry and new correlations intro-duced by the reconstruction procedure. We have previ-ously reported (9) that when the weights 5nl

(m)6 used forlinear combination of component coil signals form orthogo-nal sets for different spatial harmonics m, no new noisecorrelations are introduced by the SMASH reconstruction,even though a reduced number of measured data points arebeing used to generate the final composite image. To lendmathematical rigor to these general arguments, and tostudy the impact of component coil weighting on SNR, weassess how signal and noise terms propagate through thevarious stages of the SMASH reconstruction procedure. Inthe derivation to follow, we will use a discrete index ratherthan a continuous variable to specify the k-space signal. Inother words, the notation S(kx, ky) from Eqs. [1] to [4] abovewill be replaced by the simpler expression Sk.

The image intensity Ij at any pixel j in an MR image maybe written as a linear combination of N measured signalpoints Sk:

Ij 51

N ok50

N21

WjkSk. [5]

For standard phase warp imaging, Wjk is an appropriatecoefficient from the inverse Fourier transform, i.e.,

Wjk 5 exp 12pijk

N 2 . [6]

To avoid undue complexity in the equations, only onepixel index j has been used here, and the linear combina-tion in Eq. [5] describes a one-dimensional inverse discreteFourier transform (DFT). The generalization to multipledimensions is straightforward, since the non-array-en-coded direction has the standard behavior and may betreated with a separate inverse DFT.

When a coil array is used for image acquisition, multiplesignal points Skl are acquired simultaneously in eachmeasurement step, one data point for each componentcoil l. These component coil signals transform into mul-tiple component coil images Ijl, where l varies from 1 to L,with L representing the total number of array elements.

In a SMASH image reconstruction, linear combinationsof component coil signals are used to generate compositesignals that are shifted in k-space. Suppose that, as isillustrated in Fig. 1, all but a fraction 1/M of the requiredk-space positions have been omitted from the acquisition,to yield component coil images with 1/M of the desired

SNR and SNR Efficiency in SMASH Imaging 1011

FOV. In this case, the k-space index will take values of k 50, M, 2M, . . . , N, and a total of M linear combinations willbe required to fill in the missing data at k-space positionsk 1 m, where m 5 1,2, . . . , M 2 1:

Sk1mSMASH 5 o

l51

L

nl(m)Skl. [7]

To obtain the SMASH-reconstructed full-FOV image, wetake the inverse DFT of the composite signal set (includingthe unshifted m 5 0 composite), i.e.,

IjSMASH 5

1

N ok

om50

M21

Wj,k1mSk1mSMASH

51

N ok

om50

M21

ol51

L

Wj,k1mnl(m)Skl. [8]

The SNR of the SMASH-reconstructed image pixel in Eq.[8] may be expressed in terms of its real and imaginaryparts,

SNR(ReIjSMASH) 5

7ReIjSMASH8

sReIjSMASH

SNR(ImIjSMASH) 5

7ImIjSMASH8

sImIjSMASH

[9]

where the brackets indicate a mean value and s representsthe standard deviation over an ensemble of images sampledfrom the same noise distribution. The numerators in thisexpression are simply the mean of the real or imaginaryparts of Eq. [8], namely,

75Re

Im6IjSMASH8 5

1

N ok

om

ol 75

Re

Im6 (Wj,k1mnl(m)Skl)8 . [10]

The denominators in Eq. [9] take a somewhat morecomplicated form. They may be calculated by taking thevariance of real or imaginary components of Eq. [8],

s5Re Im6I

jSMASH

25 715

Re

Im6IjSMASH2

2

8 2 75Re

Im6 IjSMASH8

2

[11]

and collecting all nonvanishing cross-terms between differ-ent component coils l and spatial harmonics m. Thesecross-terms are evaluated assuming that the distribution ofnoise in the measured signal points Skl is Gaussian, withzero mean and variance s2. Noise voltages for differentk-space sample points are assumed independent, sincenoise is not spatially encoded and is independent ofposition in the image. Correlation of noise voltages be-tween different array elements is expressed in the usualway by a symmetric noise correlation matrix R whoseelements Rll8 express the net noise coupling between

component coils l and l8, whether from residual mutualinductance not eliminated by array design, or from sharedflux of transient current paths in a conductive sample (cf.Refs. 12 and 13). The relevant cross-terms may be ex-pressed as follows:

7ReSklReSk8l88 2 7ReSkl87ReSk8l88

5 7ImSklImSk8l88 2 7ImSkl87ImSk8l88 5 dkk8Rll8s2 [12]

7ReSklImSk8l88 2 7ReSkl87ImSk8l88

5 7ImSklReSk8l88 2 7ImSkl87ReSk8l88 5 0. [13]

Substitution of Eqs. [12] and [13] into an expanded expres-sion for Eq. [11] yields the following result for noisevariance in a SMASH reconstruction:

sReIj

SMASH2

5 sImIj

SMASH2

5s2

N

1

M om

om8

? Re 1Wj,m2m8 ol

ol8

nl(m)Rll8nl8

*(m8)2. [14]

The sums over l and l8 in Eq. [14] express the effects ofpreexisting noise correlations between array elements priorto SMASH reconstruction, as described by the elements Rll8

of the noise resistance matrix. The sums over m and m8, onthe other hand, describe new noise correlations arisingfrom the reconstruction procedure, as a consequence of thefact that different linear combinations of the same acquireddata sets have been used to generate different lines ofk-space.

Combining Eq. [14] with Eqs. [10] and [9] gives the SNRfor a SMASH reconstruction:

SNR(IjSMASH)

5

okom

ol7Wj,k1mnl

(m)Skl8

N 1/2s 11

M om

om8

Re 1Wj,m2m8 olo

l8nl

(m)Rll8n*l8

(m8)221/2

. [15]

SNR in Fully Gradient-Encoded Acquisitions

Simple Linear Combinations

For reference, suppose that a full data set Sklfull is acquired in

the same coil array using the same underlying imagingsequence as in the SMASH acquisition. Now k takes its fullset of N values, k 5 0, 1, 2, . . . , N 2 1. Since M times asmany phase-encoding gradient steps are now required, thetotal acquisition time will be M times as long as for theSMASH image. In the interest of making as direct acomparison as possible, we first use a simple linearcombination of component coil signals to form the compos-ite reference image Ij

simple:

I jsimple 5

1

N ok50

N21

WjkSkfull 5

1

N ok50

N21

ol51

L

Wjknl(0)Skl

full. [16]

1012 Sodickson et al.

Here, nl(0) are the same weights as were used for the m 5 0

combination in the SMASH reconstruction. A simplifiedderivation analogous to the one outlined in the previoussection yields the following expressions for noise varianceand SNR:

sReIj

simple2

5 sImIj

simple2

5s2

N ol

ol8

nl(0)Rll8n*l8

(0) [17]

and

SNR(Ijsimple) 5

ok

ol7Wjknl

(0)Sklfull8

N 1/2s 1ol

ol8

nl(0)Rll8n*l8

(0)21/2

. [18]

Optimal Pixel-by-Pixel Phased Array Image Combinations

The simple reference combination, while a convenientbenchmark for SMASH SNR, does not represent the bestavailable image combination algorithm from the pointof view of SNR. Roemer et al (12) showed that the op-timal SNR for a coil array may be achieved on apixel-by-pixel basis by using appropriate linear combina-tions of corresponding pixels in each of the com-ponent coils. In particular, Roemer et al derived optimalsets of weights 5njl6 for pixel-by-pixel combinations of theform

Ijpixel-by-pixel 5 o

l51

L

njlIjlfull 5

1

N ol

njl ok

WjkSklfull. [19]

For such a combination, the general expression for SNR is

SNR(Ijpixel-by-pixel) 5

7ol

njl ok

WjkSklfull8

N 1/2s 1ol

ol8

njlRll8n*jl821/2

. [20]

The best solution for complex data combination wasshown to involve the noise resistance matrix and thecomplex conjugate of the coil sensitivity functions:

njloptimal 5 lj o

l8(R)ll8

21C*jl8. [21]

Here, lj is a free scaling factor that can vary with each pixelwithout affecting SNR. Ref. 12 also explores a variety ofalternative weighting options that may be used in place ofthe optimal weights of Eq. [21]. The sum of squarescombination procedure, which is now widely used forimage combination in the absence of detailed sensitivityinformation, is not, strictly speaking, a linear technique inthe sense of Eq. [19]. For linear array designs of the sortdescribed in Ref. 12, however, it has been shown toapproximate an optimal linear weighting based on RFsensitivities.

Comparison of SNR in SMASH Versus FullyGradient-Encoded Acquisitions

Comparison With Simple Linear Combinations:Orthonormality Conditions

How does SMASH SNR (Eq. [15]) compare with the SNR ofthe simple linear combination (Eq. [18])? Let us considerwhat happens when the following condition is met:

ol

ol8

nl(m)Rll8n*

l8(m8) 5 dm,m8 o

lo

l8nl

(m)Rll8n*l8

(m) [22]

Subject to this constraint, the m8 sum in Eq. [14] vanishes,and, using Wj0 5 1, the noise variance becomes

sReIj

SMASH2

5 sImIj

SMASH2

5s2

N

1

M om

ol

ol8

nl(m)Rll8n*

l8(m). [23]

(Due to the symmetry of Rll8, the sum Sl Sl8 nl(m)Rll8n*l8(m) is

purely real, and the Re operator may be omitted.) Thisexpression has a familiar form, similar to that derived inRefs. 12 or 13 for the noise variance of traditional arraycombinations. The final noise variance in either case is justa square-weighted sum of the variances s2 of individualsignal points, with mixing dictated by the off-diagonalelements of the noise resistance matrix. The effects ofpreexisting noise correlations remain in the Rll8 term, butno additional noise correlations are introduced by thereconstruction procedure. Comparison of Eq. [23] with Eq.[17] shows that the noise variances sReI

2

jSMASH and sReI

2

jsimple

are identical whenever

1

M om

ol

ol8

nl(m)Rll8n*

l8(m) 5 o

lo

l8nl

(0)Rll8n*l8

(0). [24]

Ideally, the numerator of the SNR will be the same for thesimple reference and the SMASH images, since for aperfect SMASH reconstruction, the composite k-spacematrix is identical to the reference matrix of the simplelinear combination:

Sk1mSMASH 5 o

lnl

(m)Skl 5 ol

nl(0)Sk1m,l

full 5 Sk1msimple. [25]

In other words, the sum over k and m in the SMASH casereduces identically to a sum over the full range of k in thesimple reference image, and

7Ijsimple8 5 71N o

k50

N21

ol51

L

Wjknl(0)Skl

full8

5 71N ok

om50

M21

ol51

L

Wj,k1mnl(m)Skl8 5 7Ij

SMASH8. [26]

Thus, if the SMASH weights allow for an accuratereconstruction and obey both Eqs. [22] and [24], then bothsignal and noise terms will be equivalent to their counter-parts in the simple reference image, and SNR(Ij

SMASH) 5

SNR and SNR Efficiency in SMASH Imaging 1013

SNR(Ijsimple), despite the reduced acquisition time of the

SMASH image.What is the meaning of the conditions in Eqs. [22] and

[24]? For simplicity, we use a vector notation, expressingthe weights 5nl

(m)6 as elements of L-component complexvectors n(m). The noise resistance matrix R is a symmetricL 3 L matrix that operates on these vectors. In this notation,Eq. [15] becomes

SNR(IjSMASH)

5

ok

om

7Wj,k1mn(m)† · Sk8

N 1/2s 11

M om

om8

Re (Wj,m2m8n(m8)† · Rn(m))21/2

[27]

(where the † symbol denotes the Hermitian conjugate, orthe conjugate transpose). Similarly, Eqs. [22] and [24]become

n(m)† · Rn(m8) 5 dm,m8n(m)† · Rn(m) [28]

1

M om

n(m)† · Rn(m) 5 n(0)† · Rn(0). [29]

We may define new effective weight vectors n(m) using thefollowing transformation:

n(m) ; (R1/2)n(m) [30]

which also yields the conjugate relation

n(m)† ; n(m)†(R1/2)† 5 n(m)†(R1/2). [31]

(Here we have used the fact that since R is symmetric, R1/2

is also symmetric, i.e., R1/2 5 (R1/2)†).In this new basis, n(m)† · Rn(m8) = n(m)† · n(m8), and Eq. [28]

takes a simple form:

n(m)† · n(m8) 5 dm,m8 0 n(m) 0 2. [32]

This is an orthogonality relation for the weights used togenerate different harmonics m and m8. Equation [23]describes the noise variance of the SMASH-recon-structed image for the idealized case in which an or-thogonal set of effective weights has been used for thereconstruction and embodies the principle withwhich we began our investigations, namely, that orthogo-nal linear combinations introduce no new noise correla-tions.

Similarly, in the new basis, Eq. [29] becomes

1

M om

0 n(m) 0 2 5 0 n(0) 0 2. [33]

This constitutes a normalization condition for the weightsets 5n(m)6. For the noise variance of the SMASH image to beequal to the variance of the simple reference image, the

various SMASH weight sets must on average have the samenorm as n(0).

In summary, a SMASH image, obtained in a fraction 1/Mof the total acquisition time, will have the same SNR as acomparable full-time reference image formed by simplelinear combination, subject to the orthogonality and normal-ization conditions

ol

ol8

nl(m)Rll8n*l8

(m8)

5 dm,m8 ol

ol8

nl(m)Rll8n*

l8(m) (orthogonality) [34]

1

M om

ol

ol8

nl(m)Rll8n*

l8(m)

5 ol

ol8

nl(0)Rll8n*

l8(0) (normalization). [35]

Whether the SMASH weights approach orthonormalityin practice depends on the constraints of spatial harmonicfitting for any particular image plane and coil array geom-etry. Sample weights are given in a section on experimentalresults to follow. The simplifying conditions of orthonor-mality are by no means a requirement for useful SMASHimaging, but they do serve as a design goal for practicalarray design, and as a helpful conceptual tool for under-standing and predicting SNR behavior. For example, depar-tures from orthonormality will have a number of predict-able consequences for SNR.

Orthogonality: By virtue of the presence of DFT coeffi-cients Wj,m2m8, the expression Eq. [14] for noise variancecontains both constant terms (with m 5 m8) and oscillatingterms (with m Þ m8). The oscillatory terms, which, likeWj,m2m8, have zero mean, characterize spatial variations inthe value of noise variance across the image. These spatialvariations are the result of noise correlations betweendifferent k-space lines in the reconstructed signal matrixand are present whenever the orthogonality condition Eq.[34] is violated. In such cases, the operation of the DFT onthe partially correlated signal data can lead not only tonoise correlations, but also to noise anticorrelations incertain areas of the image, resulting in local reductions ofnoise variance. In the M 5 2 case, all adjacent k-space linesare similarly correlated, resulting in a sinusoidal first-harmonic oscillation of noise variance about its meanvalue. At higher values of the acceleration factor M,contributions from higher harmonics enter into the noisevariance profile, with the relative magnitude of eachcontribution determined by the degree to which the corre-sponding effective weights depart from orthogonality. Thus,orthogonality of the weights controls the spatial distribu-tion of noise in a SMASH image.

Normalization: The value of the constant terms with m 5m8 in Eq. [14] is determined by the norm of the correspond-ing effective weights. Thus, normalization of the weightscontrols the baseline about which noise oscillations occur.If weights for higher harmonics have larger norms than thereference zeroth-harmonic weights, i.e., if 1/M Sm 0 n(m) 0 2 .

1014 Sodickson et al.

0 n(0) 02, then baseline SNR will decrease due to an increasein noise variance out of proportion to signal amplitude. Onthe other hand, if weights for higher harmonics havesmaller norm than the reference weights n(0), average SNRwill actually be improved in the SMASH image comparedwith the reference image formed by simple linear combina-tion.

Comparison With Optimal Pixel-by-Pixel Combinations

The comparison of SMASH SNR with the SNR of optimalpixel-by-pixel combinations is somewhat less straightfor-ward, since the magnitude of SNR gains from pixel-by-pixel techniques depends on the particular structure ofcomponent coil sensitivities, which in turn depends onarray geometry and image plane orientation. In general, nosingle closed form comparison exists independent of aparticular geometry. Nevertheless, the principal effects ofpixel-by-pixel weighting may be estimated using approxi-mate models. The optimal pixel-by-pixel weights havebeen shown to be proportional to the complex conjugate ofthe coil sensitivity functions, C*jl8 (cf. Ref. 12 and Eq. [21]).Such a weighting maintains a maximum additive signal inregions of significant signal amplitude, while discarding,to the greatest extent possible, regions dominated by noise.In other words, the optimal pixel-by-pixel weighting consti-tutes a form of matched filtering. For linear coil arrays, thispixel-by-pixel noise-filtering effect tends to give SNR gainson the order of ÎL compared with simple sums of compo-nent coil images (where L once again represents thenumber of array elements). These arguments indicate that,for the linear arrays used in this work, optimal pixel-by-pixel techniques will yield a factor of approximately ÎLbetter SNR than SMASH reconstructions with well-normalized weights. Departures from the normalizationcondition will affect the comparison by shifting the SMASHnoise baseline up or down, as was outlined in the previoussection. The spatial distribution of SNR in the pixel-by-pixel reconstructions will be determined by the structureof the coil sensitivity functions, whereas in SMASH imagesthe spatial distribution will also be affected by the orthogo-nality of the weights. The results of detailed SNR compari-sons for particular array designs will be presented insections to follow.

Comparison of SNR Efficiency in SMASH Versus FullyGradient-Encoded Acquisitions

One particularly noteworthy feature of the expression forSMASH SNR (Eq. [15]) is that, when the orthonormalityconditions are met, SNR is independent of the accelerationfactor M. In this situation, whether the image is acquired at2 times or at 10 times the standard sequential imagingspeed, the SNR of the SMASH-reconstructed image is thesame. A comparison of how SNR scales with acquisitiontime in partially parallel as opposed to fully gradient-encoded sequential acquisitions is therefore in order.

In traditional sequential imaging, SNR is proportional tothe square root of the acquisition time and thus decreaseswith the square root of imaging speed. This proportionality

has been shown to arise from the integration over acquisi-tion time that effectively takes place during Fourier trans-formation of the MR signal in the course of image genera-tion (16). Inherent in the proportionality, however, is anassumption that total acquisition time scales linearly withmatrix size N for any given imaging sequence, since insequential MRI, each additional line of data requires anadditional gradient-encoding step. This assumption nolonger holds true in a partially parallel acquisition such asSMASH, for which distinct lines of data are sampledsimultaneously, rather than one after the other. To theextent that the various gradient-encoded and coil-encodedlines remain independent of one another, SNR is propor-tional to the total number of data lines, by virtue of the DFToperation. The conditions for independence of gradient-encoded and coil-encoded lines are just the orthonormalityconditions Eqs. [34] and [35]. Thus, for a fixed final matrixsize, SNR in a SMASH reconstruction with orthonormalweights is independent of the total encoding time. Depar-tures from orthogonality change the spatial distribution ofSNR but do not affect its mean value across the image,which does not decrease with increasing imaging speedunless the normalization of the weights is degraded. Bycontrast, for purely sequential techniques, an M-fold in-crease in acquisition speed that preserved matrix size andresolution would either be impossible or else, if possible,would generally be accompanied by a factor of ÎM reduc-tion in SNR.

Figure 2a contains an idealized plot of expected SNR as afunction of acquisition speed for SMASH reconstructionsof reduced data sets compared with simple linear combina-tions or optimal pixel-by-pixel combinations of full datasets in the same coil array. The array in this example iscomprised of four component coils with a linear geometry,and the SNR of pixel-by-pixel combinations is assumed tobe dominated by the noise-filtering effect. Under theseconditions, optimal pixel-by-pixel combinations will havetwice the SNR of simple linear combinations. SNR for bothof these reference combinations falls in proportion to thesquare root of acquisition speed. The SMASH reconstruc-tions, on the other hand, begin at the lower SNR of thesimple combination, but maintain this value as acquisitionspeed is increased by integer factors, as long as orthogonal-ity and normalization conditions continue to be met. (Onceagain, for any set of well-normalized weights, the meanSNR across the image is preserved as acquisition speedincreases, regardless of whether or not the weights areorthogonal.) At the maximum achievable acceleration fac-tor, the SMASH curve intersects the curve for an idealmatched filter combination.

Figure 2b shows the same results expressed in terms ofSNR efficiency, defined here as SNR per square rootacquisition time. Once again, the SMASH curve crossesfrom the lower baseline of the simple combination to meetthe matched filter combination at the maximum accelera-tion factor. This means that, in the same time as wouldnormally be required for a full acquisition, one couldacquire multiple SMASH images, average them together,and obtain the same net SNR as would have been obtained

SNR and SNR Efficiency in SMASH Imaging 1015

for an optimal matched filtered image generated from thefull data set.

One of the most promising applications of SMASH, andof parallel imaging in general, will be to allow increases inspeed even when physical and physiologic constraintswould otherwise prevent further gains. If unit acquisitionspeed in Fig. 2 is taken to lie at the sequential speed limit

(as given by peak gradient performance or by the neuromus-cular stimulation threshold, for example), then the gray-shaded areas in the figure represent a range of speeds thatmay only be achieved using a partially parallel acquisitionstrategy. The dotted continuations of the simple sum or thepixel-by-pixel SNR curves then represent theoreticallypredicted but practically unattainable behavior. An opti-mal SMASH reconstruction has the same SNR as theoptimal pixel-by-pixel reconstruction would have if thelatter could be operated at higher speed.

METHODS

Phantom Imaging Experiments

To verify theoretical predictions, and to explore the behav-ior of SNR under realistic imaging conditions, a resolutionphantom was imaged using a custom-built four-elementlinear array on a Siemens Vision 1.5 T whole-body clinicalMR imager (Siemens Medical Systems, Erlangen, Ger-many). The array, which consisted of four 78 3 230 mmrectangular elements overlapped along their short axis for atotal array extent of 260 3 230 mm, was initially developedfor accelerated SMASH imaging of the heart and of thebrain (17). For the current studies, the array was oriented ina coronal plane, with the elements disposed along thefoot-head B0 field direction. The image plane was alsocoronal, and located approximately 40 mm from the array,with a foot-head phase-encoding direction. A fast low-angle shot (FLASH) imaging sequence was used, with TRof 12 msec, TE 6 msec, matrix size 128 3 128, FOV 300 3

300 mm, flip angle 77, and slice thickness 4 mm. The flipangle and slice thickness were chosen for a target meanreference SNR in the range of 30–40. In this SNR range, theeffects of system instabilities (on the order of 1% variationfrom acquisition to acquisition for our commercial imagingsystem) lie below the noise level and do not significantlycomplicate SNR measurements.

After the phantom had been left in place to stabilize for15 min, 56 separate acquisitions were performed, of whichthe last 48 were used as a measurement ensemble. (Thefirst eight acquisitions were discarded to avoid any non-steady-state effects such as mechanical vibrations in thefluid compartments of the phantom.) Identical reconstruc-tions for reference and for SMASH images were thenperformed on each of the 48 data sets. Reference imagereconstructions were applied to the full acquired data sets,using either a simple sum of component coil signals or atraditional sum of squares combination algorithm (in whichcomponent coil images are combined pixel-by-pixel as thesquare root of the sum of square magnitudes). SMASHreconstructions were performed on reduced data sets,representing accelerated acquisitions with reduced phaseencoding. The weights 5nl

(m)6 for spatial harmonic genera-tion were calculated with a matrix least squares algorithm,using representative signal profiles along the phantomdiameter as internal sensitivity references (cf. Ref. 9). Unitweights (nl

(0) 5 1) were used both for the simple linearcombination reference and for the zeroth spatial harmonic

FIG. 2. Idealized plots of expected SNR (a) and SNR efficiency (i.e.,SNR per unit square root acquisition time; b) as a function ofacquisition speed for SMASH reconstructions of reduced data setscompared with simple linear combinations or optimal pixel-by-pixelcombinations of full data sets. The same four-element array isassumed in all cases, and orthonormal weights are assumed for theSMASH reconstructions. The acquisition speed indicated in thefigure as 1 unit is assumed to be the maximum speed that may safelybe achieved using sequential imaging techniques, so that speeds inthe grayed region are attainable only with partially parallel acquisitions.

1016 Sodickson et al.

FIG. 3. Spatial distribution of signal intensity, noise standard deviation, and SNR for reference and double-speed SMASH images of a resolu-tion phantom using the four-element array. Open circles, experimental data; solid lines, theoretical predictions. Signal, noise, and SNR profiles weretaken along dashed lines in the reconstructed images shown at the top. Image acquisition and reconstruction parameters are reported in the text.

SNR and SNR Efficiency in SMASH Imaging 1017

in SMASH reconstructions. (Such a choice, as opposed to acombination that produces homogeneous signal intensity,for example, is allowed in SMASH reconstructions so longas all higher harmonics are multiplied by the ‘‘zerothharmonic’’ profile; cf. Eqs. [2] and [3]).

SNR values were then calculated on a pixel-by-pixelbasis by taking the mean of each pixel intensity over theensemble of reconstructed images and dividing by thestandard deviation over the same ensemble. Measured SNRvalues were compared with theoretical predictions basedon the observed signal distributions, the calculated SMASHweights, and the measured noise correlation matrix.

In Vivo Measurements

A combination of numerical simulations and empiricalmeasurements was used to assess the spatial variation ofnoise in an in vivo imaging situation. MR signal data,image data, and reconstruction parameters from a previousstudy (9) were used for this analysis. The MR signal datawere obtained in a healthy adult volunteer, using thecommercial Siemens CP spine array with four activeelements on our Siemens Vision 1.5 T system. A centric-reordered single-shot RARE sequence was used, and theFOV was chosen to cover the abdomen and thorax of thevolunteer. Further details on imaging parameters are avail-able in Ref. 9. The original image data included a) areference image acquired in 540 msec and reconstructedusing a sum of squares algorithm; and b) a SMASH imageacquired in 270 msec with half the number of phase-encoding gradient steps and reconstructed using theSMASH procedure with M 5 2.

For the simulations, 16 correlated noise sets were addedto the original in vivo MR signal data sets, to simulate anensemble of repeated experimental measurements. Noisewas generated by calls to a Gaussian random numbergenerator (the function randn in Matlab Version 5.0 (TheMathworks, Natick, MA)), with values of the varianceselected to match empirically estimated SNRs in the sourceimages. Correlation of noise between component coils wassimulated by multiplying component coil noise sets by anoise correlation matrix R derived from experimentalmeasurements in the appropriately loaded array. After thecorrelated noise replicas had been added to the sourcesignal sets, an ensemble of reconstructed images wasgenerated using the sum of squares or the SMASH recon-struction algorithms. Pixel-by-pixel calculations of noisestandard deviation in this ensemble were compared withtheoretical predictions based on the SMASH weights (thesame weights used in the original imaging study) and onthe measured noise correlation matrix.

For comparison with both theory and simulations, noisestandard deviation as a function of position was alsomeasured directly in the original reference and SMASHimages. A small rectangular region of interest (ROI) wasinitially placed in the corner of the in vivo images, and thestandard deviation of image intensities within this regionwas calculated for various ROI positions along the edge ofthe images in the phase encoding direction. Since eachcalculation was performed in an ostensibly signal-free

region, each constituted an approximate measure of noisestandard deviation in that region, and the set of thesemeasurements taken together constituted an estimate of thespatially dependent noise profile.

RESULTS

Phantom Imaging Experiments

Figures 3 and 4 compare experimental results (open circles)with theoretical predictions (solid lines) for SNR as afunction of position in phantom images using the four-element array.

Figure 3 plots pixel-by-pixel signal intensity, noise stan-dard deviation, and SNR profiles for sum of squares,simple sum, and SMASH images. Profiles were taken alonga diameter of the phantom indicated by dashed lines in theimages shown at the top. In signal profile, the double-speed SMASH image is a faithful replica of the full-timesimple sum reference profile. Noise profiles are essentiallyflat for the two reference reconstruction techniques, whilethe standard deviation profile shows a marked oscillationfor the SMASH reconstruction. This oscillation, whichclosely matches the predicted profile, results from nonor-thogonalities of the effective weights used for SMASHreconstruction. Close inspection of the figure shows thatthe oscillation in noise background corresponds to arelative enhancement of SNR at the edges of the SMASHSNR profile compared with the simple sum SNR profile.

Figure 4 plots noise standard deviation profiles as afunction of SMASH acceleration factor M from M 5 2 toM 5 4. The effects of higher harmonic contributions to theoscillatory noise profile are particularly evident at M 5 4.In the SMASH images shown at the right of Fig. 4, slightresidual aliasing artifacts may be seen at the higher SMASHfactors, as has been described in more detail in Ref. 9. Thenoise profiles, however, are unaffected by these residual arti-facts.

Figure 5 shows the scaling of SNR with imaging speedfor phantom images with the four-element array. MeanSNR values were calculated from the reference and SMASHdata sets by averaging the pixel-by-pixel SNR over thecircular region spanned by the phantom. Mean SNR for thesum of squares combination was 33.3, and is representedin the figure as an open triangle. The simple sum combina-tion (open square) had a mean SNR of 14.8, approximatelya factor of 2 lower than for the sum of squares. As in theschematic graph of Fig. 2, dotted lines have been drawn toindicate the theoretically predicted but practically unattain-able square-root scaling of these SNR values for highersequential acquisition speeds. Mean SNR measurementsfor SMASH reconstructions with acceleration factors of 2to 4 are represented as open diamonds, connected by asolid line to guide the eye. The measured SMASH SNRvalues were 14.4 (M 5 2), 14.5 (M 5 3), and 13.8 (M 5 4).(In all cases, estimated measurement error is smaller thanthe marker size.) The dashed line represents the scaling ofSNR with acquisition speed that would be expected forpurely orthonormal SMASH weights. The slight depres-sion of measured SNR at M 5 4 results from a relative

1018 Sodickson et al.

increase in the norm of the weights for the highest spatialharmonic used in the reconstruction.

Indeed, all the general features of the SNR profiles inFigs. 3–5 may be predicted from the values of the weightsused for image reconstruction. For the phantom experi-ments described here, these weights were as follows,

l51 l52 l53 l54m521 0.310.7i 1.020.1i 0.221.0i 20.620.4im50 1 1 1 1m51 0.320.7i 1.010.1i 0.211.0i 20.610.4im52 20.620.7i 1.110.1i 21.110.3i 0.420.8i

[36]

where harmonic number runs from m 5 21 to m 5 2 downeach column, and component coil number runs from l 5 1to l 5 4 along each row. The norms and relative vectorangles of these weights may be calculated from dot prod-ucts of each row in Eq. [36] against itself and against the

other rows:

Norm: m 5 21 m 5 0 m 5 1 m 5 23.2 4.0 3.2 4.2

m 5 21 m 5 0 m 5 1 m 5 2m 5 21 0 71 82 89m 5 0 71 0 71 74m 5 1 82 71 0 70

Angle:

m 5 2 89 74 70 0

[37]

Departures from 907 in the angle matrix of Eq. [37] repre-sent the nonorthogonalities that result in noise varianceoscillations in Figs. 3 and 4. Inspection of the weightvector normalizations confirms that the normalization condi-tion of Eq. [35] is met or exceeded for all but the highestharmonic, since only the m 5 2 harmonic (norm 5 4.2) has alarger norm than the m 5 0 harmonic (norm 5 4.0).

FIG. 4. Noise standard deviation profiles as afunction of SMASH acceleration factor M forphantom images using the four-element array.Left-hand column, noise profiles, with opencircles representing experimental data and solidlines representing theoretical predictions; right-hand column, SMASH-reconstructed images.

SNR and SNR Efficiency in SMASH Imaging 1019

We emphasize that the theoretical curves in Figs. 3–5 areexact results calculated from Eq. [15], making no appeal toorthonormality. The principles of orthogonality and normal-ization serve, however, as useful tools for predicting SNRbehavior using only a small set of component coil weightsas a priori information. For example, the norms andoverlap angles for a sample SMASH reconstruction in theeight-element array used elsewhere to achieve eightfoldaccelerations (10,11) are as follows:

Figure 6 shows the mean SNR as a function of accelerationfactor predicted from these weights. Though residual non-orthogonalities do remain, a large number of the overlapangles in this case approach 907, and noise variance

oscillations are expected to be significantly smaller thanthose shown in Fig. 4 (a fact easily confirmed by simula-tions). Furthermore, the favorable normalization behaviorallows a nearly optimal average SNR efficiency at thehighest SMASH factors.

In Vivo Measurements

Noise oscillations in an in vivo SMASH image are docu-mented in Fig. 7. For purposes of orientation, referencesum of squares and SMASH images, reproduced from Ref.

9, are included in the top panel, with the shifted positionsof the noise measurement region indicated by adjacentboxes with diamonds on the two images. The solid dia-monds in the plots beneath show the corresponding mea-

FIG. 5. Mean pixel-by-pixel SNR as a function of acquisition speedfor reference and SMASH images of a resolution phantom using thefour-element array. Mean SNR was calculated by averaging pixel-by-pixel SNR values in each reference or SMASH image over thecircular region spanned by the phantom. Open triangles, sum ofsquares reference; open squares, simple sum reference; opendiamonds connected by solid line, SMASH; dashed line, SNR thatwould be expected for orthonormal effective SMASH weights. SMASHacceleration factors range from M 5 2 to M 5 4. As in Fig. 2, dottedlines represent reference SNR values that would be expected ifequivalent speed increases were possible using sequential tech-niques alone.

Norm: m 5 23 m 5 22 m 5 21 m 5 0 m 5 1 m 5 2 m 5 3 m 5 45.9 4.5 5.7 12.7 5.7 4.5 5.8 7.8

m 5 23 m 5 22 m 5 21 m 5 0 m 5 1 m 5 2 m 5 3 m 5 4m 5 23 0 86 81 79 88 86 86 82m 5 22 86 0 89 75 82 89 86 89m 5 21 81 89 0 78 77 82 88 89m 5 0 79 75 78 0 78 75 79 81m 5 1 88 82 77 78 0 90 82 83m 5 2 86 89 82 75 90 0 86 77m 5 3 86 86 88 79 82 86 0 77

Angle:

m 5 4 82 89 89 81 83 77 77 0

[38]

FIG. 6. Mean pixel-by-pixel SNR as a function of acquisition speedfor sample SMASH image reconstructions in an eight-element array,as predicted from properties of the weights in Eq. [38]. Opentriangles, sum of squares reference; open squares, simple sumreference; open diamonds connected by solid line, SMASH; dashedline, SNR that would be expected for orthonormal effective SMASHweights. SMASH acceleration factors range from M 5 2 to M 5 8. Asin Fig. 2, dotted lines represent reference SNR values that would beexpected if equivalent speed increases were possible using sequen-tial techniques alone.

1020 Sodickson et al.

sured noise standard deviations as a function of boxposition. Also included in these plots are the simulatedstandard deviation profile (open circles) and the predic-tions of the theory (solid lines). As may be seen, the in vivomeasurements agree well both with simulations and withtheory. Given the substantial spatial variations shown here,it is clear that traditional empirical SNR estimation tech-niques, which measure signal and noise in different re-gions of the image, must be approached with caution inSMASH imaging.

DISCUSSION

As has been demonstrated in the theoretical and experimen-tal investigations presented here, SNR in a SMASH imag-ing study depends in general on the preexisting degree ofnoise correlation in the coil array used for imaging, and onthe weight factors used for SMASH reconstruction. Thesetwo determinants may be combined into a single set ofeffective weights, 5nl

(m)6, as defined in Eq. [30]. Normaliza-tion of the effective weights controls the average noiselevel in the SMASH image, and a departure from thenormalization condition raises or lowers baseline SNR byaltering the noise out of proportion to the signal. On theother hand, orthogonality of the effective weights con-trols the spatial distribution of noise. A departure fromorthogonality introduces correlations between adjacentlines of k-space, which in turn leads to predictable oscilla-tory variations in noise standard deviation across theimage.

Image plane orientation and coil array geometry are theprincipal determinants of SMASH weight factors. In prac-tice, the weights can vary significantly as the position andorientation of the image plane with respect to the coil array

is varied, with large tilt angles away from the plane of thearray generally leading to reductions in SNR as a result ofdegraded normalization behavior. However, the phantomexperiments of Fig. 5 and the predictions of Fig. 6 showthat favorable normalization behavior can be achieved atleast for image planes parallel to realistic arrays. In thefuture, tailored array designs may be developed to opti-mize average SNR for particular anatomical regions orimaging applications.

The presence of noise variance oscillations in SMASHimages raises another interesting possibility for coil arraydesign. Since these noise variance oscillations are notnecessarily accompanied by oscillations in image inten-sity, it may be possible to engineer arrays that havelocalized SNR ‘‘hot spots’’ in particular regions of interest.This would involve a deliberate selection for particularnonorthogonal weights, such that SNR is maximized inanatomic regions of interest, at the expense of SNR in lessinteresting outlying regions.

Additional sequence-dependent effects not explicitlyconsidered in this article can also influence SNR in aSMASH image reconstruction. For example, in single-shotsequences such as echoplanar imaging, SNR can be recov-ered to some extent by virtue of the faster acquisition itself,due to reduced relaxation and hence reduced attenuationof central k-space lines in an accelerated echo train. Thus,the final SNR in any given SMASH image may result from abalance of sequence-specific and reconstruction-specificfactors.

The SMASH technique operates by exploiting correla-tions between the MR signals in different array elements.The presence of such correlations suggests that when dataare acquired in an array of spatially distinct componentcoils, a certain redundancy is introduced into the tradi-

FIG. 7. Noise standard deviation as a functionof position in the phase-encoding direction foran in vivo data set (from a scan of the abdomenand thorax of a healthy adult volunteer using asingle-shot RARE sequence). Top row, sum ofsquares reference image (left) and double-speed SMASH image (right). Boxes with dia-monds indicate positions of the region of interestfor measurements of noise standard deviation.Bottom row, measured, simulated, and theoreti-cally predicted noise standard deviation as afunction of position. Solid diamonds, measurednoise standard deviation in the boxed regionsindicated on the images above; open circles,noise standard deviation profile based on 16simulated replicas; solid lines, theoretically pre-dicted noise standard deviation profiles.

SNR and SNR Efficiency in SMASH Imaging 1021

tional MR spatial encoding procedure using field gradi-ents. Using appropriate reconstruction strategies, some ofthe redundant information may be extracted from the data.Such a line of argument naturally leads to the question ofwhether SMASH, or similar partially parallel imagingschemes, may be used to enhance the total achievable SNRin MR images. If SNR may be preserved in reducedacquisition times, then one would expect that the timesaved could then be applied for SNR enhancement, forexample, via additional signal averaging. When SMASHimages are compared with images from a single RF coil, orwith simple combinations of component coil images in RFcoil arrays, such an averaging procedure would indeedyield an improved SNR. Optimized SMASH reconstruc-tions represent true increases in SNR per unit time overthese reference images. Such references are admittedlynonoptimal, however, since optimal pixel-by-pixel arraycombinations have been shown to have a higher baselineSNR. While SMASH uses the redundancy inherent inmultiple component coil signals to increase imaging speed,the optimal pixel-by-pixel approach abides by the tradi-tional limits on acquisition speed, but resolves the redun-dancy instead in a way that maximizes SNR for a givenacquisition. This dichotomy is inherent in the use of MRcoil arrays: the acquisition and reconstruction strategy maybe designed to optimize either for SNR or for speed, or elsefor some intermediate combination of the two. There is noevidence at present suggesting that the pixel-by-pixeloptimum derived by Roemer et al (12) may be exceeded onthe basis of reconstruction strategy alone. However, se-quence-dependent effects alluded to earlier, such as dimin-ished relaxation in accelerated single-shot images, mayeventually allow net gains in SNR efficiency using par-tially parallel acquisitions.

CONCLUSIONS

SMASH may be used to increase MR imaging speedwithout increasing gradient switching rate or RF powerdeposition, generally with some absolute penalty in SNR,but with the potential for preserved average SNR effi-ciency. The theory, simulations, and experimental resultspresented in this article identify the key parameters thatinfluence SNR in a SMASH acquisition and image recon-struction. An understanding of these parameters may beused to guide coil array design for the optimization of SNRin applications of rapid and ultra-rapid parallel MRI.

ACKNOWLEDGMENTS

The authors have benefited from many lively discussionsof the vicissitudes of signal and noise with a number ofinterested colleagues. D.K.S. acknowledges helpful sugges-tions from Drs. Peter Roemer, Zhi-Pei Liang, and RandyDuensing.

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