Accelerated 1 H MRSI using randomly undersampled spiral-based k-space trajectories

FULL PAPER

Accelerated 1H MRSI Using Randomly UndersampledSpiral-Based k-Space Trajectories

Itthi Chatnuntawech,1* Borjan Gagoski,2 Berkin Bilgic,3 Stephen F. Cauley,3

Kawin Setsompop,3,4 and Elfar Adalsteinsson1,5,6

Purpose: To develop and evaluate the performance of an

acquisition and reconstruction method for accelerated MRspectroscopic imaging (MRSI) through undersampling of spiral

trajectories.Theory and Methods: A randomly undersampled spiral acqui-sition and sensitivity encoding (SENSE) with total variation (TV)

regularization, random SENSEþTV, is developed and evaluatedon single-slice numerical phantom, in vivo single-slice MRSI,and in vivo three-dimensional (3D)-MRSI at 3 Tesla. Random

SENSEþTV was compared with five alternative methods foraccelerated MRSI.

Results: For the in vivo single-slice MRSI, random SENSEþTVyields up to 2.7 and 2 times reduction in root-mean-square error(RMSE) of reconstructed N-acetyl aspartate (NAA), creatine,

and choline maps, compared with the denoised fully sampledand uniformly undersampled SENSEþTV methods with the

same acquisition time, respectively. For the in vivo 3D-MRSI,random SENSEþTV yields up to 1.6 times reduction in RMSE,compared with uniform SENSEþTV. Furthermore, by using ran-

dom SENSEþTV, we have demonstrated on the in vivo single-slice and 3D-MRSI that acceleration factors of 4.5 and 4 are

achievable with the same quality as the fully sampled data, asmeasured by RMSE of reconstructed NAA map, respectively.Conclusion: With the same scan time, random SENSEþTV

yields lower RMSEs of metabolite maps than other methodsevaluated. Random SENSEþTV achieves up to 4.5-fold accel-

eration with comparable data quality as the fully sampledacquisition. Magn Reson Med 000:000–000, 2014. VC 2014Wiley Periodicals, Inc.

Key words: magnetic resonance spectroscopic imaging

(MRSI); random undersampling; spiral trajectory; compressedsensing; parallel imaging

INTRODUCTION

MR spectroscopic imaging (MRSI) is a technique thatprovides not only the relative intensities of metabolitesignals at each voxel, but also their spatial distributions.Because irregular changes of metabolite concentration inspecific brain regions are precursors to specific physio-logical abnormalities, MRSI plays an important role invarious clinical research applications especially inoncology and neurology (1). 1H-MRSI has been exten-sively used not only for brain, prostate, and breast cancerassessments (2,3), but also to characterize neurologicaland psychiatric diseases such as multiple sclerosis (MS)(4,5), and Alzheimer’s disease (6,7). While MRSI is clini-cally useful, it suffers from fundamental tradeoffs due tothe inherently low signal-to-noise ratio (SNR), longacquisition times, and low spatial resolution. Therefore,reducing the acquisition time of MRSI experiment is awell-motivated problem.

Fast acquisition schemes such as echo planar imaging(EPI) (8–10) and spiral acquisitions (11) have been usedto efficiently collect MRSI data and significantly reducethe data acquisition time compared with the conven-tional Cartesian (phase-encoded) acquisition scheme. Inaddition to these fast acquisition methods, parallel mag-netic resonance imaging techniques such as sensitivityencoding technique (SENSE) (12) and generalized auto-calibrating partially parallel acquisitions (GRAPPA) (13)can be used to further reduce the encoding time even inthe non-Cartesian case (14–19) by exploiting the spatiallyvarying sensitivity profiles of a coil array. For non-Cartesian trajectories, SENSE parallel imaging (16) aswell as its extension to reconstruction and estimation ofcoil sensitivities have been previously proposed for ana-tomical imaging applications (17–19). Parallel imagingalso plays an important role in MRSI (20–25). The com-bination of multiple spin-echo acquisition and SENSEwas shown to reduce the acquisition times of MRSI to afew minutes (20). Proton echo planar spectroscopic imag-ing (PEPSI) was combined with a parallel imaging tech-nique such as SENSE (21,23,24) and GRAPPA (25) tofurther accelerate MRSI acquisition. Spiral MRSI wascombined with iterative SENSE without regularization toreconstruct the data from the uniformly undersampled k-space data (22).

In this work, we combine pseudo-randomly under-sampled spiral-based k-space trajectories along all kx, ky,and kf axes (26) with a total variation (TV) (27) regular-ized SENSE to accelerate MRSI acquisition for the casewhen there is no large dynamic range in amplitudesbetween the desired and undesired signals (e.g., MRSIacquisition with substantially suppressed water and

1Department of Electrical Engineering and Computer Science, Massachu-setts Institute of Technology, Cambridge, Massachusetts, USA.2Fetal-Neonatal Neuroimaging & Developmental Science Center, BostonChildren’s Hospital, Harvard Medical School, Boston, Massachusetts, USA.3A. A. Martinos Center for Biomedical Imaging, Department of Radiology,Massachusetts General Hospital, Charlestown, Massachusetts, USA.4Harvard Medical School, Boston, Massachusetts, USA.5Harvard-MIT Health Sciences and Technology, Massachusetts Institute ofTechnology, Cambridge, Massachusetts, USA.6Institute of Medical Engineering & Science, MIT, Cambridge, Massachu-setts, USA.

Grant sponsor: NIH; Grant number: R01 EB017337.

*Correspondence to: Itthi Chatnuntawech, S.M., Massachusetts Institute ofTechnology, Room 36-776A, 77 Massachusetts Avenue, Cambridge, MA02139. E-mail: [email protected]

Additional Supporting Information may be found in the online version ofthis article.

Received 31 March 2014; revised 10 July 2014; accepted 11 July 2014

DOI 10.1002/mrm.25394Published online 00 Month 2014 in Wiley Online Library (wileyonlinelibrary.com).

Magnetic Resonance in Medicine 00:00–00 (2014)

VC 2014 Wiley Periodicals, Inc. 1

lipid components). Herein, the proposed acquisition-reconstruction method is referred to as randomSENSEþTV. We have demonstrated the feasibility of ran-dom SENSEþTV on three datasets: single-slice numericalMRSI phantom, in vivo single-slice brain MRSI, and invivo brain 3D-MRSI at 3T.

THEORY

Acquisition

In this study, we focus on two types of spiral acquisitionschemes consisting of uniformly and randomly under-sampled spiral MRSI k-space acquisition. On the onehand, the uniformly undersampled acquisition schemeuses spiral trajectories with the same radius (28), andundersampling is achieved by uniformly skipping theinterleaves (22,29). On the other hand, the randomlyundersampled acquisition scheme uses spirals with sev-eral different radii (26). Specifically, in each repetition

time (TR), we select spiral trajectories with randomlychosen radii and play them in succession during thelong readout as shown in Figure 1a to achieve nonuni-form undersampling not only in (kx, ky) space, but alsoalong the kf axis.

Figure 1b shows the projections of possible k-spacetrajectories onto the kx–ky plane. In this specific exam-ple, at R¼ 1, six angular interleaves are used. For theuniformly undersampled acquisition (R¼ 3), two angularinterleaves with their full durations are used. On thecontrary, the randomly undersampled acquisition (R¼ 3)uses all six angular interleaves, but the duration of eachinterleaf is cut down by a random constant. By randomlypicking such a constant whenever a spiral is played, wecould achieve an undersampling pattern that is morerandom along all kx, ky, and kf axes. In addition, weobtain more samples closer to the center of k-space withthis undersampling pattern which results in a better SNRper scan time, compared with the uniformly under-sampled acquisition.

FIG. 1. a: Two types of spiral-based k-space acquisition schemes: Uniformly (top) and randomly (bottom) undersampled spiral k-spaceacquisition schemes. The blue vertical lines represent the time at which a sample is acquired. b: The projections of possible k-space trajec-

tories onto the kx–ky plane. The projection from fully sampled, uniformly undersampled (R¼3), and randomly undersampled (R¼3) acquisi-tion are shown from left to right, respectively. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

2 Chatnuntawech et al.

http://wileyonlinelibrary.com

Given the desired number of distinct radii, Q, andmaximum radius, rmax, we can generate the ith spiral tra-jectory with radius ri ¼ airmax where ai 2 [0,1], andi ¼ 1; 2; . . . ;Q. A simple way to pick ai is to setai ¼ i=Q. Once we have Q different spiral trajectories, wecan determine the number of times that each spiral tra-jectory will be played. Let ni be the number of times thatthe ith spiral is played. A constrained optimization canbe formulated to determine ni’s. The most basic con-straint is directly related to the desired scan time, accel-eration factor, and rmax. Other constraints can beformulated from the desired spatial sampling density aspossibly specified by a probability density function.

To compare the effect of both types of acquisitions, wecan analyze the point spread function (PSF). Let FV bethe undersampled Fourier transform operator, Acart2spiral bethe 3D Cartesian-to-spiral operator, and d be a 3D object withonly one nonzero sample at the location of zero spatial and

temporal frequency. Then, the PSF can be computed as fol-lows (30):

PSF ¼ Acart2spiralFV

� �HAcart2spiralFV

� �d: [1]

Figure 2 shows an example of the PSF of d, which hasa matrix size of (kx, ky, kf)¼ (32 � 32 � 320), obtainedfrom both spiral acquisition schemes (uniform and ran-dom) at R¼ 6. Each image in the figure represents theabsolute value of the PSF at a specific frequency. Twentyimages extracted from the PSF at equally spaced frequen-cies are displayed. The brightest image corresponds tothe PSF at zero frequency. For the randomly under-sampled spiral acquisition, we can see relatively highervalues at all frequencies because it not only randomlyundersamples in (kx, ky) space, but also along the kf axis.Furthermore, for the randomly undersampled spiral k-space acquisition, the energy of the PSF contained at the

FIG. 2. An example of the point spread function (PSF) obtained from the uniform (left) and random (right) spiral k-space acquisitionschemes at R¼6. Each image is the absolute value of the PSF at a specific frequency index, nf 2 f1;2; . . . ; 320g, with the brightest

image corresponding to the PSF at zero frequency (nf ¼161). For the randomly undersampled spiral k-space acquisition, the energy ofthe PSF contained at the zero temporal frequency is equal to 93.6%, compared with 99.9% for the uniformly undersampled k-spaceacquisition. Thus, the interference is more likely to spread out along all three dimensions with the proposed acquisition method. [Color

figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Table 1Single-Slice Numerical Phantom: Acquisition Parameters with the Corresponding RMSEs of the Reconstructed NAA, Creatine, and Chol-ine Maps

Methods No. of averages, Navg Undersampling factor, R

%RMSEs

NAA Cr Cho

Fully sampled NSNR10

3 1 51.6 60.7 64.6Denoised fully sampled NSNR10

3 1 13.3 18.7 20.5Uniform SENSE NSNR10 3 20.8 24.8 30.0

Uniform SENSEþTV NSNR10 3 15.7 18.2 20.9Random SENSE NSNR10 3 15.4 19.4 22.1

Random SENSEþTV NSNR10 3 12.7 16.6 18.3

MRSI by Undersampling of Spiral Trajectories 3


zero temporal frequency is equal to 93.6%, comparedwith 99.9% for the uniformly undersampled k-spaceacquisition. Thus, the interference is more likely tospread out along all three dimensions with the proposedacquisition method.

These observations suggest that the randomly under-sampled spiral acquisition has an advantage over theuniformly undersampled spiral acquisition because itspreads out artifacts along all kx, ky, and kf axes. Theidea of pseudo-randomly undersampling along kx, ky,and kf axes has been proposed for single-coil 3D hyper-polarized 13C MRSI acquired with a fly-back echo-planarspectroscopic imaging readout (31), but not for multi-channel 3D 1H MRSI acquired with spiral readout. Withthe nonuniform undersampling pattern, the proposedacquisition scheme is likely to introduce artifacts thatare less-structured than the uniform acquisition schemeand, hence, works in favor of a regularized reconstruc-tion algorithm (30).

Reconstruction.

Sensitivity encoding with total variation regularization(SENSEþTV) is used to reconstruct the MRSI data. Giventhe undersampled non-Cartesian k-space MRSI data fromall coils, we can reconstruct the underlying Cartesiandata by solving the following optimization problem

minxcart

XNchannel

i¼1

jjAcart2spiralFVCixcart � yspiralijj22 þ ljjGxcart jj1

[2]

where xcart is the reconstructed Cartesian data in theobject domain, yspirali

is the observed non-Cartesiank-space data from the ith coil, Ci is a diagonal matrixthat contains the sensitivity of the ith coil, FV is theundersampled Fourier transform operator, Acart2spiral isthe 3D Cartesian-to-spiral operator, G ¼ ½Gx; Gy; Gf � isthe 3D finite difference in (x, y, f) space, l is a regulari-zation parameter, and Nchannel represents the number ofreceive channels. The first term of the optimization prob-lem ensures that the reconstructed data are consistentwith the observed data from all receive coils. The secondterm, commonly called the total variation term, enforcesthe slowly varying behavior of the data in the objectdomain. The regularization parameter can be interpretedas a relative cost of a constraint violation.

Following the nonuniform fast Fourier transform(NUFFT) algorithm proposed in Fessler and Sutton (32),we implemented a simplified version of the Cartesian-to-spiral operator using a simple interpolation method withoptional oversampling. Specifically, the value of a sam-ple that does not fall on the Cartesian grid, yspiral, is aweighted average of its eight closest neighbors on theCartesian grid (33,34). The contribution (weight) fromeach neighbor to yspiral is determined based on its dis-tance from yspiral. These weights can be assigned inmany ways depending on a type of window functionthat is used. Examples of common window functions aretriangular and Kaiser-Bessel window functions. For sim-plicity, we use a triangular kernel. By using the simpli-fied operator as opposed to the more sophisticated one

(32), we got similar accuracy with faster computationalspeed. Based on numerical phantom and in vivo results(not shown), root-mean-square error (RMSE) of the recon-structed N-acetyl aspartate (NAA), creatine, and cholinemaps obtained from both types of operator were compa-rable, so we used our own implementation for all subse-quent comparisons.

METHODS

Using numerical phantom and experimental data, wetested the performance of six acquisition-reconstructionmethods: (i) Reconstruction without regularization fromfully sampled k-space (fully sampled); (ii) Reconstruc-tion with a TV regularization from fully sampled k-space(denoised fully sampled); (iii) Reconstruction withoutregularization from uniformly undersampled k-space(uniform SENSE); (iv) Reconstruction with a TV regulari-zation from uniformly undersampled k-space (uniformSENSEþTV); (v) Reconstruction without regularizationfrom randomly undersampled k-space (random SENSE);(vi) Reconstruction with a TV regularization from ran-domly undersampled k-space (random SENSEþTV).

In all experiments, constant-density spiral trajectorieswere used to acquire the fully sampled k-space data. Theundersampled k-space data were then simulated by ret-rospectively undersampling the fully sampled data inMATLAB. The reconstruction algorithms were imple-mented in MATLAB using a nonlinear conjugate gradi-ent descent algorithm with backtracking line search (30)and run on a machine with 16 Intel Xeon E5–2670 pro-cessors and 128 GB of memory. For all iterative methodspresented in this manuscript, the following stopping cri-teria were used: the change in values of the cost functionbetween consecutive iterations is less than 0.0001%;change of the solution between consecutive iterations isless than 1%; or maximum number of iterations isreached (500 iterations). With these stopping criteria, thereconstruction time for random SENSEþTV is approxi-mately 10 min per slice. In all experiments, we testedvarious values of a regularization parameter from theinterval [10�7, 102] and only showed the reconstructedMRSI data from the regularization parameter that yieldedthe smallest reconstruction error using RMSE of thereconstructed metabolite maps as a criterion. Each recon-structed metabolite map was obtained by summing theabsolute values of the reconstructed spectra over the fre-quency range that contains the metabolite of interest.

Single-slice Brain MRSI with Matched Acquisition Time:Numerical MRSI Phantom

In this study, we compared the reconstruction quality ofall six methods with the same acquisition time to makefair comparisons. Although there is no undersamplinginvolved in the first two methods, we matched the scantime of the first two methods to that of the rest, byreducing the number of averages.

The numerical phantom was generated in three steps.First, we defined the tissue boundaries of the phantomusing the segmented structural image acquired from ahealthy human subject. The structural image was


segmented using Freesurfer (35). Second, the spectrum ateach voxel was generated using SPINEVOLUTION (36) at3 Tesla (T) with an echo time (TE) of 30 ms. Each spec-trum is a linear combination of various metabolites con-sisting of NAA, creatine (Cr), choline (Cho), glutamate(Glu), and glutamine (Gln). We assigned values reportedin (37,38) to the metabolite peaks in each spectrum.There are no water and lipid components in any spec-trum. Finally, we added complex white Gaussian noiseto the spectra to achieve SNR¼ 10 with respect to theaveraged NAA peak. Here, we assume that the number ofaverages required to achieve this phantom with SNR¼ 10is equal to NSNR10 averages (i.e., Navg¼NSNR10). Theresulting phantom had a matrix size of (kx, ky, kf)¼ (32 �32 � 320).

To generate the observed data, yspirali, we applied the

simulated eight-channel coil sensitivity profile (39),undersampled Fourier transform operator, andCartesian-to-spiral operator to the MRSI phantom withSNR¼10, respectively. The spiral trajectories used inthe Cartesian-to-spiral operator were generated asdescribed in the Acquisition subsection. For the pro-posed acquisition method, the duration of each spiralinterleaf is cut down by a random constant to under-sample not only in (kx, ky) space, but also along the kf

axis. In particular, we used spiral trajectories with eightdistinct radii to achieve an acceleration factor of three.The duration of the spirals was in a range from 0.47 msto 2 ms. The acquisition parameters for different meth-ods are summarized along with the correspondingRMSEs of the reconstructed metabolite maps in Table1. The MRSI phantom without noise (i.e., the resultingphantom from step two) was masked by the fully

sampled spiral trajectory in k-space and then used asground truth.

In Vivo Single-Slice Brain MRSI with Matched AcquisitionTime: Subject A

Fifty average, single-slice MRSI data with a thickness of1 cm were acquired fully sampled in 18 min and thenretrospectively undersampled from a healthy volunteerwith the approval of an Institutional Review Board at 3Tusing the spiral-based trajectories, which were encodedusing six angular and two temporal interleaves, with avoxel size of approximately 0.56 cc, field-of-view in thexy-plane (FOVxy) of 24 cm � 24 cm that resulted in amatrix size of 32 � 32. With 320 kf points acquired andoverall readout duration of 320 ms, the spectral band-width used was 1 kHz. The point resolved spectroscopy(PRESS) excitation (40) was used. The excitation boxhad a size of 9 cm � 9 cm, and it was positioned entirelywithin the head without touching the skull. The watercomponent was suppressed using both chemical shiftselective imaging (CHESS) (41) and additional spectrallyselective water suppression pulses. TR and TE were setto 1.8 s and 87 ms, respectively. The lipid componentwas suppressed using the outer volume suppression(OVS) (42–44). To randomly undersample by a factor ofthree and six, eight spiral trajectories with different radiiwere simulated in MATLAB. The duration of the spiralswas approximately in a range from 0.47 ms to 2 ms.

In addition to the water-suppressed MRSI data, water ref-erence datasets (i.e., without any water suppression) withthe same resolution as that of the MRSI data were acquiredto estimate the coil sensitivity profiles of 32-channel head

Table 3In Vivo Single-Slice MRSI: Acquisition Parameters (Tacq¼42 s) with the Corresponding RMSEs of the Reconstructed NAA, Creatine, andCholine Maps


%RMSEs

NAA Cr Cho

Fully sampled 2 1 52.5 43.9 51.3Denoised fully sampled 2 1 26.5 18.6 19.4Uniform SENSE 12 6 21.0 25.6 31.8

Uniform SENSEþTV 12 6 17.4 22.1 26.1Random SENSE 12 6 9.8 12.7 13.6

Random SENSEþTV 12 6 10.0 12.2 12.9

Table 2In Vivo Single-Slice MRSI: Acquisition Parameters (Tacq¼1.44 min) with the Corresponding RMSEs of the Reconstructed NAA, Creatine,

and Choline Maps


%RMSEs

NAA Cr Cho

Fully sampled 4 1 27.0 24.4 26.7Denoised fully sampled 4 1 17.1 18.8 20.0

Uniform SENSE 12 3 21.6 21.0 22.8Uniform SENSEþTV 12 3 12.8 11.2 12.1Random SENSE 12 3 9.5 12.9 15.3

Random SENSEþTV 12 3 6.8 9.3 9.8


coils. The 50-average data coil combined using the SENSEalgorithm for R¼ 1 were used as ground truth.

The scan time with no acceleration using any of thesix methods was chosen to be 4.32 min which corre-sponds to 12-average MRSI data if acquired fullysampled using the spiral trajectories. The time of 4.32min was chosen so that it is within the range of a typicaland clinically acceptable overall scan time while stillachieving SNR¼ 14. With acceleration, the scan timewas reduced to 1.44 min (R¼3) and 42 s (R¼6), whichcorresponds to four-average and two-average MRSI data,respectively, if acquired fully sampled. The acquisitionparameters for different methods are also summarizedalong with the corresponding RMSEs of the recon-structed metabolite maps in Table 2 and Table 3.

In Vivo Single-Slice Brain MRSI with Matched RMSE:Subject A

Using the already-acquired 50-average data, we per-formed an additional comparison that involved only twoacquisition-reconstruction methods: fully sampled andrandom SENSEþTV. As opposed to the previous com-parison where we matched the scan time, we matchedreconstruction quality based on RMSE without a timeconstraint. The goal of this comparison was to determinethe acquisition time that each method required toachieve the same RMSE with respect to the 50-average,fully sampled data. In particular, we determined thenumber of averages that it would take for the fullysampled data to get the same RMSE of the reconstructedNAA map as that obtained from random SENSEþTVfrom the previous experiment with R¼3.

In Vivo Brain 3D-MRSI with Matched Acquisition Time:Subject B

As opposed to 2D-MRSI, long acquisition time is a majorconstraint in 3D-MRSI. Here, we added the 3D-MRSIdataset to demonstrate the proposed method on thedomain where it is essential to accelerate the acquisitionfor clinical feasibility.

Four-average, 3D-MRSI data were acquired fullysampled from a healthy volunteer at 3T in 20.35 min usinga stack-of-spirals acquisition, which were encoded usingsix angular and two temporal interleaves, with voxelsize¼ 0.5 cc, FOV(x,y,z)¼20 cm � 20 cm � 12.6 cm, num-ber of encoded slices¼ 16, matrix size(x,y,z,f)¼ (25 � 25 �16 � 400), spectral bandwidth¼ 1.25 kHz, TR¼1.6 s, andTE¼ 87 ms. The PRESS-box excitation was used. Theexcitation box had a size in (x,y,z) of 8 cm � 8 cm � 4 cm,and it was positioned entirely within the head withouttouching the skull. The same water and lipid suppressiontechniques as previously applied in the single-slice casewere used. In addition to the water-suppressed MRSIdata, water reference datasets were acquired to estimatethe coil sensitivity profiles of 32-channel head coils.

For this experiment, single-average data (SNR¼ 12 withrespect to the averaged NAA peak) were extracted fromthe four-average data, undersampled at R¼ 3 using bothuniformly and randomly undersampling schemes, andreconstructed using TV-regularized SENSE. The perform-ance of uniform SENSEþTV and random SENSEþTVwere evaluated using the RMSE of reconstructed NAA,creatine, and choline maps. The four-average data coilcombined using the SENSE algorithm for R¼ 1 were usedas ground truth.

FIG. 3. In vivo single-slice MRSI: Reconstructed NAA, creatine, and choline maps obtained from six acquisition-reconstruction methods

with Tacq¼1.44 min. The corresponding RMSE with respect to the ground truth (Navg¼50, R¼1) is displayed under each reconstructedmetabolite map. The acquisition parameters for different methods are summarized in Table 2. [Color figure can be viewed in the onlineissue, which is available at wileyonlinelibrary.com.]



In Vivo Brain 3D-MRSI with Matched RMSE: Subject B

Using the already-acquired four-average data, we per-formed an additional comparison that involved only twoacquisition-reconstruction methods: fully sampled andrandom SENSEþTV. The goal of this comparison was todetermine how much we can accelerate the acquisitiontime using random SENSEþTV while achieving the sameRMSE of the reconstructed NAA map as the single-average, fully sampled data. Again, the four-average datacoil combined using the SENSE algorithm for R¼ 1 wereused as ground truth.

RESULTS

Single-Slice Brain MRSI with Matched Acquisition Time:Numerical MRSI Phantom

Supporting Figure S1, which is available online, showsthe NAA, creatine, and choline maps reconstructed withsix different methods (fully sampled, denoised fullysampled, uniform SENSE, uniform SENSEþTV, randomSENSE, and random SENSEþTV) with the corresponding

RMSEs (Supporting Fig. S1). Random SENSEþTVreduced the RMSE of the reconstructed maps by up to2.2% and 3.0%, compared with the denoised fullysampled method and uniform SENSEþTV, respectively,for the reduction in scan time by a factor of three. Sup-porting Figure S2 shows the spectra reconstructed fromthe fully sampled, denoised fully sampled, uniformSENSEþTV, and random SENSEþTV methods with areduction in scan time by a factor of three.

In Vivo Single-Slice brain MRSI with Matched AcquisitionTime: Subject A

Figure 3 and Supporting Figure S3 show the NAA, crea-tine, and choline maps reconstructed from the fullysampled, denoised fully sampled, uniform SENSE, uni-form SENSEþTV, random SENSE, and randomSENSEþTV methods with Tacq¼ 1.44 min and Tacq¼ 42 s,respectively. Random SENSEþTV yielded up to 2.5 and2.7 times reduction in RMSE, compared with the denoisedfully sampled method for Tacq¼1.44 min and Tacq¼ 42 s,respectively. Random SENSEþTV yielded up to 1.9 and

FIG. 4. In vivo single-slice MRSI: Reconstructed spectra obtained from the fully sampled, denoised fully sampled, uniform SENSEþTV, and

random SENSEþTV methods with Tacq¼1.44 min. The ground truth (red) and reconstructed (blue) spectra were taken from the voxels indi-cated by the red dots in the structural GRE image. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.

com.]




2.0 times reduction in RMSE, compared with uni-form SENSEþTV for Tacq¼ 1.44 min and Tacq¼ 42 s,respectively.

Figure 4 and Supporting Figure S4 shows the spectrareconstructed from the fully sampled, denoised fullysampled, uniform SENSEþTV, and random SENSEþTVmethods with Tacq¼1.44 min and Tacq¼ 42 s, respectively.

In Vivo Single-Slice brain MRSI with Matched RMSE:Subject A

The reconstructed NAA maps (Fig. 5a) and spectra atvarious locations (Fig. 5b) obtained from the 18-averagefully sampled data and random SENSEþTV reconstruc-tion are displayed. The corresponding acquisition timeand RMSE with respect to the ground truth are displayedabove and below each reconstructed NAA map, respec-tively. To achieve approximately the same RMSE, thefully sampled method took 6.48 min (RMSE¼ 6.56%),whereas random SENSEþTV took only 1.44 min

(RMSE¼6.75%). In other words, random SENSEþTVaccelerated the acquisition process by a factor of 4.5,while yielding the equivalent reconstruction quality asthe fully sampled data, as measured by the RMSE ofreconstructed NAA map.

In Vivo Brain 3D-MRSI with Matched Acquisition Time:Subject B

Figure 6a shows the NAA, creatine, and choline maps ofthe representative slice reconstructed from uniformSENSEþTV and random SENSEþTV with the sameacceleration factor of three, which results in an acquisi-tion time of 1.7 min. The corresponding RMSE is dis-played under each reconstructed metabolite map.Uniform SENSEþTV (Navg¼ 1; R¼ 3) and randomSENSEþTV (Navg¼ 1; R¼ 3) methods yielded (16.8, 23.1,18.3) and (10.5, 17.3, 11.9) %RMSE for the reconstructed(NAA, Cr, Cho) maps, respectively. Random SENSEþTVyielded up to 1.6 times reduction in RMSE, comparedwith uniform SENSEþTV.

FIG. 5. a: In vivo single-slice MRSI: Reconstructed NAA maps obtained from the 18-average fully sampled data and random SENSEþTV

reconstruction. The corresponding acquisition time and RMSE with respect to the ground truth (Navg¼50, R¼1) are displayed aboveand below each reconstructed NAA map, respectively. By using random SENSEþTV, an acceleration factor of 4.5 is achievable with the

same quality as the fully sampled data, as measured by the RMSE of reconstructed NAA map. b: The reconstructed spectra of the invivo data obtained from the fully sampled and random SENSEþTV methods. The ground truth (red) and reconstructed (blue) spectrawere taken from the voxels indicated by the red dots in the structural GRE image. [Color figure can be viewed in the online issue, which

is available at wileyonlinelibrary.com.]



In Vivo Brain 3D-MRSI with Matched RMSE: Subject B

Figure 6b shows the reconstructed NAA maps and spec-tra at various locations obtained from the single-average,fully sampled data, and random SENSEþTV (R¼4). Thecorresponding acquisition time and RMSE with respectto the ground truth are displayed above and below eachreconstructed NAA map, respectively. To achieveapproximately the same RMSE, the fully sampled

method took 5.09 min (RMSE¼ 14.3%), whereas randomSENSEþTV took only 1.27 min (RMSE¼12.9%). In otherwords, random SENSEþTV accelerated the acquisitionprocess by a factor of 4, while yielding equivalent recon-struction quality as the fully sampled data, as measuredby the RMSE of reconstructed NAA map.

Figure 7 shows the reconstructed NAA, creatine, andcholine maps of two consecutive slices obtained from

FIG. 6. In vivo 3D-MRSI. a: Reconstructed NAA, creatine, and choline maps obtained from uniform SENSEþTV and random SENSEþTV

with Tacq¼1.70 min. The corresponding RMSE with respect to the ground truth (Navg¼4, R¼1) is displayed under each reconstructedmetabolite map. The ground truth (red) and reconstructed (blue) spectra were taken from the voxels indicated by the red dots in the

structural image. b: Reconstructed NAA maps and spectra obtained from the single-average, fully sampled data, and randomSENSEþTV reconstruction with R¼4. The corresponding acquisition time and RMSE with respect to the ground truth are displayedabove and below each reconstructed NAA map, respectively. By using random SENSEþTV, an acceleration factor of 4 is achievable

with the same quality as the fully sampled data, as measured by the RMSE of reconstructed NAA map.


random SENSEþTV with R¼ 4. Slice 13 has relativelylow concentration of metabolites because the PRESS-boxwas centered at the ventricle of the brain.

DISCUSSION

With the same acquisition time, the reconstructed metabo-lite maps obtained from random SENSEþTV had lowerRMSEs than those from other methods under considera-tion. Moreover, the reconstructed spectra from randomSENSEþTV had a better peak estimation and fitted theground truth better. Compared with the denoised fullysampled method with the same scan time, randomSENSEþTV leads to better reconstruction quality becausethe random acquisition scheme spends unequal amount oftime acquiring samples from different locations in k-space. In particular, the proposed acquisition schemespends more time acquiring samples near the k-space cen-ter where the energy is concentrated and less time acquir-ing samples from the outer k-space region. Compared withuniform SENSEþTV with the same acceleration factor,random SENSEþTV yielded lower RMSEs because the

pseudo-randomly undersampled acquisition not onlysamples the k-space center more frequently, but also intro-duces relatively less-structured artifacts due to increasedrandomness in the sampling pattern.

Regardless of the acquisition method, the results fromthe single-slice datasets indicate that regularizationreduces the RMSEs of the reconstructed metabolite maps.The spectra obtained from the regularized reconstructionsfitted better to the ground truth than those obtained fromthe nonregularized reconstructions because they exploitedthe prior knowledge about the structure of the underlyingdata. TV regularization enforces smoothness of the recon-structed data with preserved edges. TV penalty can bethought of as an l1-norm of the gradient of the data. Whenan l1-norm penalty is imposed on the optimization formu-lation, it seeks a sparse solution while preserving datafidelity. Consequently, if there is an edge at which voxelvalues change dramatically, TV regularization will try topreserve it to avoid a high penalty. As a result, TV regula-rization not only denoises the data, but also preservesmajor edges (i.e., major metabolite peaks in this case)which may result in better peak estimates. The edge-

FIG. 7. In vivo 3D-MRSI: Reconstructed NAA, creatine, and choline maps of two consecutive slices obtained from random SENSEþTVwith R¼4 (Tacq¼1.27 min). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]



preserving property of TV regularization is a reason whythe regularized reconstructions yield lower RMSEs thanthe reconstructions without regularization.

Although regularization term is introduced to the problemformulation to incorporate prior information and reduce ill-posedness of the problem, a proper choice of regularizationparameter is required for successful regularization. In thecurrent work, the optimal regularization parameter of eachmethod was empirically selected to minimize the RMSE ofthe reconstructed metabolite maps. The optimal regulariza-tion parameter results in a reconstruction that has a goodbalance between the data fidelity and regularization term.While the optimally regularized solution is a good compro-mise between the smoothness of data and noise suppres-sion, over-regularized parameter results in a too-smoothsolution with loss of object details, and under-regularizedparameter yields a solution with poor noise suppression.

We found that our method is less sensitive to the choiceof regularization parameter, compared with other methodsunder consideration. For example, %RMSE of the recon-structed NAA map with (

loptimal

10 , loptimal, 10loptimal) obtainedfrom denoised fully sampled, uniform SENSEþTV, andrandom SENSEþTV methods are (42.5, 13.3, 43.7), (16.2,15.7, 33.4), and (14.1, 12.7, 22.1), respectively, for thenumerical phantom with a reduction in scan time by a fac-tor of 3. A similar trend was also observed in the in vivoresults at different acceleration factors and/or with differ-ent metabolites. The smaller deviation of RMSE withunder-regularized or over-regularized parameter from theRMSE obtained with optimally regularized parameter,compared with other methods, suggested that randomSENSEþTV is less sensitive to the regularization parametervalue. Although the optimal regularization parameter wasdetermined empirically in the current implementation, aproper automatic parameter selection such as those thatbased on estimation of the weighted mean squared-error(45–47) should be examined in future research.

Extensions and Limitations

There are several ways to improve the current imple-mentation of random SENSEþTV. First, only a few rep-resentatives of possible randomly undersampled spiralacquisitions were used in the current work. A morecomprehensive way to determine the number of distinctradii and duration of the resulting spirals, number oftimes and ordering that each spiral is played, and opti-mal sampling density should be investigated in futureresearch. Second, the within-brain CHESS excitationand OVS were applied to eliminate strong, undesiredwater and lipid components in the acquisition processbecause the undersampling artifacts from these unde-sired components can contaminate the whole spectrawhich results in highly inaccurate reconstruction.Because random SENSEþTV undersamples the kf axis inaddition to the (kx, ky) space, the strong signals will alsoalias in frequency which further contaminate the desiredmetabolite spectra. Because the dynamic range of waterand lipid signals are at least two orders of magnitudehigher than the metabolite signals, successful suppres-sion of these undesired components are necessary forrecovery of metabolites with undersampling. Therefore,

in its current state, random SENSEþTV is limited to thedomain where there is no large dynamic range betweenthe desired and undesired signals. There are severaltechniques that can be integrated to our method to miti-gate the contamination due to strong lipid signals. Onthe acquisition side, inversion recovery can be appliedto suppress undesired lipid components (48,49). On thereconstruction side, model-based techniques can beincorporated to random SENSEþTV to further mitigatethe undersampling artifacts from the remaining lipidcomponents. For instance, MRSI spectrum, whichincludes lipid components, at each voxel could be para-meterized to improve the condition of the problem andincrease robustness of the reconstruction algorithm(28,29,50,51). Third, the feasibility of randomSENSEþTV was demonstrated only on relatively high-SNR metabolites consisting of NAA, creatine, and chol-ine. The properties of the proposed method in thelower-SNR domain should be investigated in futureresearch to extend the proposed method to other low-SNR metabolites. Finally, the reconstruction algorithmwhich is currently implemented using the conjugate gra-dient descent algorithm could be sped up using othertechniques such as the split Bregman method (52,53) toenable the online reconstruction on a scanner.

CONCLUSIONS

We have proposed the randomly undersampled spiral k-space acquisition-reconstruction method to accelerateMRSI acquisition. The proposed acquisition method usesspirals with several different radii to randomly under-sample the data along the kx, ky, and kf axes, whichleads to less-structured artifacts than the uniform acqui-sition scheme. The proposed reconstruction method usesTV-regularized SENSE to recover the underlying data.We have demonstrated on single-slice numerical phan-tom, in vivo single-slice MRSI, and in vivo 3D-MRSI thatrandom SENSEþTV yields lower RMSEs of the recon-structed metabolite maps, compared with other conven-tional methods with the same scan time. Furthermore,by using random SENSEþTV, we have demonstrated onthe in vivo single-slice and 3D-MRSI that an accelerationfactor of 4.5 and 4 are achievable with the same qualityas the fully sampled data, as measured by the RMSE ofreconstructed NAA map, respectively.

ACKNOWLEDGMENT

The authors thank Audrey Peiwen Fan, Paula Montesi-nos, and Jeffrey Stout for helpful discussions on varioustopics of this work.

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Accelerated 1 H MRSI using randomly undersampled spiral-based k-space trajectories

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