Accelerating MR Diffusion Tensor Imaging via FilteredReduced-Encoding Projection-Reconstruction

Yi Jiang,1,2* and Edward W. Hsu1,2

MR diffusion tensor imaging (DTI) is a promising tool for char-acterizing the microstructure of ordered tissues. However, itspractical applications are hampered by relatively low signal-to-noise-ratio and spatial and temporal resolution. Reduced-en-coding imaging (REI) via k-space sharing with constrained re-construction has previously been shown to be effective foraccelerating DTI, although the implementation was based onrectilinear k-space sampling. Due to the intrinsic oversamplingof central k-space and allowance for isotropic downsampling,projection-reconstruction (PR) imaging may be better suited forREI. In this study, regularization procedures, including radialfiltering and baseline signal correction to adequately recon-struct reduced encoded PR imaging data, are investigated. Theproposed filtered reduced-encoding projection-reconstruction(FREPR) technique is applied to DTI tissue fiber orientation andfractional anisotropy (FA) measurements. Results show thatFREPR offers improved reconstructions of the reduced en-coded images and on an equal total scan-time basis providesmore accurate fiber orientation and FA measurements com-pared to rectilinear k-space sampling-based REI methods or acontrol experiment consisting of only fully encoded images.These findings suggest a potentially significant role of FREPR inaccelerating repeated imaging and improving the data acquisi-tion-time efficiency of DTI experiments. Magn Reson Med 53:93–102, 2005. © 2004 Wiley-Liss, Inc.

Key works: projection-reconstruction; reduced-encoding imag-ing; diffusion tensor imaging; fiber orientation mapping

INTRODUCTION

MR diffusion tensor imaging (DTI) (1) is a promising non-invasive imaging tool for characterizing the microstruc-tures of ordered tissues such as the brain white matter(2,3), myocardium (4,5), and cartilage (6). In three-dimen-sional space, the generalized diffusion tensor is a symmet-ric, second-order 3 � 3 matrix. A complete solution to thesix independent parameters, plus an extra term for thediffusion-independent magnetization, requires that theDTI data set consist of a minimum of seven images. Thetrade-offs between scan time and the image signal-to-noiseratio (SNR), aggravated by the nature of diffusion sensiti-zation (i.e., via signal attenuation) and the inadvertent T2

weighting during the diffusion encoding gradient pulses,have made the SNR of DTI inherently low. Improvements

in the acquisition methodology, such as optimized diffu-sion encoding gradient directional schemes (7,8) and par-allel (9) or novel rapid imaging techniques (10), have beenhelpful. However, practical applications of DTI continueto be hampered by relatively low spatial resolution, anecessary compromise for faster scan time, improved SNR,or a combination of both. Therefore, improving acquisi-tion-time efficiency (e.g., shortening the scan time withless than the proportional loss in accuracy) is essential forimproving the temporal and spatial resolutions of DTI.

Reduced-encoding imaging (REI) by reconstructing im-age from limited sampling (hence reducing the acquisitiontime) and shared k-space data has been used to accelerateMR imaging. Techniques such as keyhole (11,12) and re-duced-encoding imaging via generalized series reconstruc-tion (RIGR) (13) can be used to accelerate dynamic orrepeated imaging, particularly where the main image-to-image contrast changes, or the underlying tissue struc-tures, are mostly low spatial-frequency in nature, and canhence be sufficiently encoded with limited central k-spacesampling. In general, keyhole and RIGR techniques requirethe acquisition of one or more full k-space reference and aseries of limited central k-space dynamic data. The re-duced encoded dynamic data are then combined with theouter k-space of the reference data and reconstructed togenerate images that are effectively obtained at highertemporal resolution, but with reduced blurring and Gibbsartifacts normally associated with truncated k-space sam-pling.

A DTI experiment is similar to dynamic imaging in thatthe identical pulse sequence (except for diffusion-weight-ing pulses) is repeated over time to acquire images on thesame gross anatomy, but with varying contrast. Conse-quently, diffusion-weighted images may be reasonably re-produced from limited central k-space sampling via key-hole or RIGR technique. A previous study (14) has dem-onstrated that rectilinear sampling-based REI, using thefully encoded nonweighted image to constrain the recon-struction of reduced-encoded diffusion-weighted images,can be used to improve the acquisition-time efficiency ofDTI. Specifically, the application of REI was found toimprove the DTI fiber orientation mapping accuracy on anequal scan-time basis compared to simply reducing thenumber of fully encoded data images.

Although REI studies were based largely on rectilineark-space acquisitions (13–15), radial sampling of k-space orprojection-reconstruction (PR) imaging (16) may be advan-tageous due to the intrinsic oversampling and averaging ofcentral k-space and the possibility to isotropically down-sample the k-space. Moreover, unlike rectilinear k-spacetruncation that reduces the spatial resolution of the entireimage, a proportion of the image center remains artifact-free (e.g., preservation of resolution) in an angularly un-

1Department of Biomedical Engineering, Duke University, Durham, NorthCarolina.2Center for In Vivo Microscopy, Duke University Medical Center, Durham,North Carolina.This study was supported by Whitaker Foundation Research Grant RG-01–0438 and NIH/NCRR P41 RR05959.*Correspondence to: Yi Jiang, Department of Biomedical Engineering, Box90281, Duke University, Durham, NC 27708-0281. E-mail: yi.jiang@duke.eduReceived 5 March 2004; revised 2 August 2004; accepted 2 August 2004DOI 10.1002/mrm.20311Published online in Wiley InterScience (www.interscience.wiley.com).

Magnetic Resonance in Medicine 53:93–102 (2005)

© 2004 Wiley-Liss, Inc. 93

dersampled PR image (17,18). These features of under-sampled PR imaging have been exploited in several stud-ies to accelerate dynamic imaging (19,20). However,because the sampling trajectories converge at the k-spaceorigin, reconstruction by direct substitution of the k-spacedata would lead to an image that is essentially the averagebetween the reference and the target images. Although thismay be adequate for experiments involving small image-to-image variations, due to the dramatic intensity differ-ences especially between the nonweighted and the diffu-sion-weighted images, additional k-space constraints arenecessary to ensure the accuracy of the image reconstruc-tion.

The goals of the current study were to develop appro-priate k-space regularization procedures necessary for ad-equate reconstruction of undersampled PR data and toinvestigate the utility of reduced-encoding PR imaging foraccelerating DTI. The performance of the reduced-encod-ing PR imaging methodology for DTI is evaluated in termsof tissue fiber orientation and fractional anisotropy (FA)(21) measurement errors against comparable rectilinear-sampling REI schemes, including keyhole and RIGR. It isworth noting that the objective here is not to advent PRimaging as a better alternative to rectilinear acquisitionsfor DTI, but to examine the effects of reduced-encodingimaging between the different k-space sampling tech-niques. Results indicate that filtered reduced-encodingprojection-reconstruction (FREPR), which involved radialmask filtering to select for spatial frequency informationand signal baseline correction, can provide not only highlyaccurate reconstructions of reduced encoded images butalso significantly improved acquisition-time efficiency ofDTI.

METHODS

Basic Theory

In the present study, a reduced-encoded DTI data set con-sists of a fully encoded b0 (i.e., b � 0) image and reduced-

encoded diffusion-weighted images. As described in Fig.1, reduced-encoding rectilinear acquisitions are performedby symmetrically sampling the central k-space with re-duced number of phase encoding steps. In contrast, re-duced-encoding PR imaging is achieved by azimuthallydownsampling the k-space. The omitted k-space trajecto-ries of the reduced-encoded data are substituted withthose of the reference data to reconstruct images of thesame matrix size and resolution as the fully encoded im-age.

A key issue in reconstructing reduced-encoded PR im-ages arises from the converging k-space trajectories of theconstituent data. Because image contrast depends heavilyon the signals in the central k-space, relatively moreweighting should be placed on the reduced encoded datathan the reference data to suppress the contribution of thelatter. Because in PR imaging the central k-space is over-sampled, the reduced encoded data alone may providesufficient information in the central k-space for adequateimage reconstruction. A convenient way to achieve thedesired k-space selectivity is to employ radial mask filters.For the reference data, a high-pass radial filter should beapplied to limit its input in the central k-space but pre-serve its contribution in the more sparsely sampled outerk-space. Conversely, to preserve the weighted k-spacesampling density, an opposite low-pass filter needs to beapplied to the diffusion-weighted data. Analytically, asfunctions of the normalized k-space radial distance r (0 �r � 1) and the reduced-encoding factor m (e.g., m � 2corresponds to 50% reduced encoding), the filters for theb0 and diffusion-weighed data, respectively Hb0(r) andHDW(r), should thus satisfy the relationship.

HDW�r� � m � �m � 1�Hb0�r�. [1]

The accuracy of the reconstructed image necessarily de-pends on the radial filters employed. Table 1 shows sev-eral examples of radial filtering schemes that can be usedfor the reconstruction of reduced encoded PR images, in-

FIG. 1. k-space sharing schemesfor rectilinear (top row) and radial(bottom row) k-space acquisitions.The target diffusion-weightedk-space data are reconstructed bycombining reduced-encoded diffu-sion-weighted (darkened trajectoryarrows) and part of the fully en-coded b0 (bright arrows) k-spacedata. To avoid data inconsistencyartifacts, constrained reconstruc-tions (e.g., baseline correction andfiltering) are applied before obtain-ing the final images.

94 Jiang and Hsu

cluding ramp, step, and step-ramp filtering, and the ex-treme cases of zero filtering (i.e., direct substitution ofreference data) and complete filtering (i.e., pure k-spacedownsampling).

Whereas radial filtering applies only to reduced-encod-ing PR imaging, an issue that applies to both rectilinearand PR acquisitions concerns the possible signal disconti-nuity between the reference and reduce encoded k-spacedata. In DTI, direct substitution (i.e., without imposingconstraining conditions) can give rise to signal disconti-nuity (or data inconsistency) artifacts in the reconstructedreduced encoded images. For example, as noted previ-ously for the case of rectilinear sampling (14), combiningthe higher-intensity b0 data in the outer k-space is akin toapplying a step high-pass filter to the diffusion-weighteddata, which causes conspicuous edge-enhancement andringing artifacts in the reconstructed image. Direct substi-

tution is also expected to produce data inconsistency arti-facts in reduced-encoding PR imaging, though the mani-festations may be different due to the radial k-space tra-jectories. To eliminate or alleviate the data inconsistency,the complex (i.e., magnitude and phase) signals across theb0 and diffusion-weighted k-space domains need to bemade continuous. The minimum solution in this regard isa zeroth-order correction to match the baselines of thesignals, which is realized by scaling the replacement b0data by a constant c0 given by (13) the following:

c0 ���d*ref�kx,ky�dRE�kx,ky��� d*ref�kx,ky�dref�kx,ky�

, [2]

where dref and dRE are the reference and reduced encodeddata, respectively, over their shared k-space trajectories

Table 1Radial Mask Filters for Reduced-Encoding PR Imaging

Filter type Hb0 (r) HDW (r)

Zero filtering HZb0 (r) � 1 HZ

DW (r) � 1

Complete filtering HCb0 (r) � 0 HC

DW (r) � m

Ramp filtering HRb0 (r) � r HR

DW (r) � (1 � m)r � m

Step filtering HSb0 (r) � � 0, r � 1/m

1, r � 1/m HSDW (r) � � m, r � 1/m

1, r � 1/m

Step-ramp filtering

HSRb0 (r) � � 0, r �

1m

mr � 1m � 1

, r �1m

HSRDW (r) � � m, r �

1m

m � mr � 1, r �1m

Note. Analytical equations and illustrative plots are shown for a variety of b0 filters, Hb0 (r), and the corresponding filters for thereduced-encoding diffusion-weighted data, HDW (r). The filters are functions of the normalized k-space radial distance r (0 � r � 1) and thereduced-encoding factor m. All filter plots are illustrated for m � 2.

Filtered Reduced-Encoding PR Imaging 95

specified by the variables kx and ky. Although beyond thescope of the current study, it is possible to impose higherorders of signal continuity correction, which combinedwith the use of generalized series (as opposed to Fourierseries) in the reconstruction of rectilinearly sampled k-space data, underlies the basic principle of RIGR (13).

Computer Simulation

All computations and visualizations were performed usingMATLAB 6.5 (The MathWorks Inc., Natick, MA). For clar-ity of identification, the reduced-encoding factor m is an-notated as subscript for each REI scheme (e.g., keyhole2

refers to keyhole reconstruction for m � 2). A nonweighted(i.e., b0) image and a heavily diffusion-weighted image(both 128 � 128 matrix size, with arbitrarily assigneddiffusion constants), with various levels of random noiseadded (average b0 image SNR of 10, 20, 30, 40, 60, or 80)and corresponding PR k-space data (400 half-echo views)were generated for a Shepp–Logan phantom. Reduced en-coded images at m � 2 were reconstructed from retrospec-tively downsampled diffusion-weighted data, with the b0data used as the reference. The effects of radial filtering onreduced-encoding PR imaging were investigated for thecases of zero, complete, ramp, step, and step-ramp filtering(as described in Table 1). For each filtering scheme, FREPRreconstructions were obtained with and without the appli-cation of baseline correction to evaluate the effect of thesignal continuity correction procedure. The accuracy ofeach REI reconstruction was assessed qualitatively bycomputing the absolute difference image and quantita-tively by calculating the cross-correlation coefficient be-tween the fully encoded (i.e., target) and reconstructedreduced encoded images. Results of the computer simula-tions were used to guide the image reconstruction in thesubsequent FREPR DTI experiment.

Experimental Verification

A myocardial fiber orientation mapping experiment wasconducted to verify the utility of FREPR for DTI. An intactfixed sheep heart was placed inside a 10-cm-diameter so-lenoid RF coil and imaged using a 2.0-T MR instrument(Oxford Instruments, Oxford, UK) equipped with shieldedgradients (18 G/cm maximum). Radial sampling data wereobtained via a diffusion-weighted PR sequence (10 cmFOV, 800 half-echo views with 128 sampling points perview reconstructed to 256 � 256 image matrix size,1.0-mm slice thickness, 75° flip angle, 500-ms TR, 19.5-mseffective TE). As a basis of comparison, rectilinearly sam-pled k-space data were also acquired by using a diffusion-weighted gradient-echo pulse sequence (10-cm FOV,256 � 256 matrix size, 1.0-mm slice thickness, 75° flipangle, 500-ms TR, 23.6-ms TE, and 3 averages). Acquisi-tion times for the radial and rectilinear sampling imageswere deliberately made approximately equal (6.7 min and6.4 min, respectively) to control for the likely SNR depen-dence of the performance of the REI schemes. Diffusionencoding for both types of acquisitions was achieved via apair of half-sine gradient pulses (9.0-ms width, 9.0-msseparation, and 18-G/cm gradient amplitude, which corre-sponded to a nominal b value of 514 s/mm2). Each DTI

data set consisted of one b0 and 12 diffusion-weightedimages encoded in each of an optimized set of 12 direc-tions (7), requiring a total scan time of approximately1.4 h. Altogether, 6 separate rectilinear and 6 radial DTIdata sets were acquired.

Reduced-encoding DTI data sets were obtained by retro-spectively applying keyhole2 [with baseline correction(14)], RIGR2, and FREPR2 (with baseline correction andstep-ramp filtering) reconstructions on corresponding rec-tilinear or radial sampling diffusion-weighted k-spacedata, using the b0 data as the reference. Because in re-duced-encoding DTI the scan time savings gained can beotherwise achieved by simply proportionally decreasingthe number of full k-space acquisitions, the performancesof the reduced-encoding DTI schemes were evaluatedagainst a control DTI experiment that consisted of the b0and 6 fully encoded diffusion-weighted scans selectedfrom the original 12-direction data set (so that this subsetof 6 gradient directions yields the minimum fiber angledeviation from the original 12-direction data set). To eval-uate potential gains in using a higher reduced-encodingfactor, a DTI experiment using FREPR4 was also per-formed.

For each DTI data set, diffusion tensors were estimatedusing a previously described formalism (1) on a pixel-by-pixel basis via nonlinear least-squares curve-fitting andthen diagonalized. The eigenvector corresponding to thelargest ranked diffusion tensor eigenvalue was taken to bethe tissue fiber orientation (5,22). The FA (21) value wasalso computed for each pixel to form the FA map. Ideally,a “gold standard” DTI experiment should be used as thebasis to estimate the measurement accuracy. However, toavoid the necessary bias of the “gold standard” toward thepulse sequence used to acquire the data, each reduced-encoded data set was compared to its corresponding fullyencoded data set. For each pixel, the deviation angle was computed between fiber orientations (i.e., principaldiffusion tensor eigenvector) measured by the reduced-encoding (including the control experiment) and the fullDTI datasets, evRE and evfull, respectively, through theirvector inner product according to the following:

� arccos(�evRE � evfull�). [3]

Moreover, for each pixel the normalized FA error FA%was calculated between the FA indices measured by thereduced-encoding (including the control experiment) andthe full DTI data sets, FARE and FAfull, respectively, asfollows:

FA% � 100 ��FARE�FAfull�

FAfull. [4]

The deviation angles and the normalized FA errors wereaveraged over the entire area of the tissue sample. Single-factor repeated-measurement analysis of variance(ANOVA) statistics were performed to determine whetherthe deviation angles or normalized FA errors among theREI schemes and the control experiment (five groups total)were significantly different. The Bonferroni criterion withan overall P � 0.05 significance threshold (i.e., P � 0.005

96 Jiang and Hsu

for each of the 10 possible pairwise comparisons) wasapplied in the intergroup comparisons.

RESULTS

Figure 2 shows the images obtained in the computersimulation experiment for the SNR � 30 case, includingthe fully encoded b0 reference and target diffusion-weighted images, absolute difference images between thetarget diffusion-weighted and FREPR images recon-structed using differing radial filters and difference imageswhen baseline corrections were added. (The intensity ofthe difference images was amplified by a factor of 10 forbetter visualization of small differences.) Without baselinecorrection, image cross-correlation coefficients for zero,complete, ramp, step, and step-ramp filtering were 0.932,0.997, 0.991, 0.991, and 0.998, respectively. On the oneextreme, zero-filtering FREPR, which amounted to directlycombining the b0 reference and reduced-encoded diffu-sion-weighted data, resulted in the largest error because ofthe inclusion of b0 data in the central k-space. On the otherextreme, the complete-filtering FREPR, which representedcomplete exclusion of b0 data and hence pure angulardownsampling of the diffusion-weighted data, demon-strated streaking artifacts from outer k-space undersam-pling (17). Among the radial filters tested for FREPR, thestep-ramp filter yielded the best reconstruction with the

least image difference and highest image cross-correlation.The application of baseline correction resulted in notice-able improvements in the accuracy of all FREPR recon-structions (with respective cross-correlation coefficients of0.970, 0.997, 0.998, 0.998, and 0.999), except for the caseof complete-filtering where correction had no effect be-cause the b0 reference data were essentially excluded. Thesimulation was repeated with different amounts of noiseadded (SNR � 10, 20, 30, 40, 60 or 80). In addition to theexpected SNR dependence of the performance for a givenscheme (getting a little worse as the SNR deceases), amongthe reconstruction schemes a similar trend, that the casewith zero-filtering gave the largest error and the one withstep-ramp filtering yielded the smallest error, was ob-served at each SNR level.

Representative images from the validation DTI experi-ment are shown in Fig. 3, which consists of a fully en-coded reference b0 and target diffusion-weighted imagesof the sheep heart (short-axis view of the midhemisphere)and the absolute difference images obtained from key-hole2, RIGR2, FREPR2, and FREPR4 reconstructions, withcross-correlation coefficients of 0.996, 0.999, 0.999, and0.997, respectively. Between the two rectilinear samplingREI schemes examined, keyhole resulted in relatively largeimage differences, especially at or near the myocardialedges, whereas RIGR shows lower and more dispersivelydistributed deviations. In contrast, the two difference im-

FIG. 2. Effects of radial filtering and baseline correction in FREPR reconstructions. The fully encoded b0 image was used as reference toreconstruct retrospectively reduced encoded data of the target diffusion-weighted image. The absolute difference images (second and thirdrows, image intensity amplified 10� to better visualize small differences) are obtained between the target and FREPR images (atreduced-encoding factor m � 2) reconstructed with various radial-filtering schemes, with and without the application of baseline signalcorrection. The corresponding image cross-correlation coefficients between the target and FREPR reconstructions using zero, complete,ramp, step, and step-ramp filtering are 0.932, 0.997, 0.991, 0.991, and 0.998 in the absence of baseline correction and with correction,0.970, 0.997, 0.998, 0.998, and 0.999, respectively.

Filtered Reduced-Encoding PR Imaging 97

ages obtained from FREPR (at m � 2 and m � 4) bothreveal rather uniformly distributed deviations. As ex-pected, because of the higher degree of reduced encoding,FREPR4 gives higher error than FREPR2.

Figure 4 shows a representative grayscale-coded myo-cardial fiber orientation helix angle map (23), which dem-onstrates the classic epicardial-to-endocardial counter-clockwise rotation of myocardial fiber orientation. Alsoshown are the fiber deviation-angle maps between themyocardial fiber orientations obtained from fully encodedDTI data set and each of the control experiment and re-duced encoded data sets with keyhole2, RIGR2, FREPR2,and FREPR4 reconstructions. Figure 5 displays the FA mapof the same heart demonstrating a relatively uniform FA

distribution throughout the whole myocardium. Alsoshown are the normalized FA error maps resulted from thecontrol experiment or REI schemes including keyhole2,RIGR2, FREPR2, and FREPR4. Qualitatively, the controlexperiment yielded the largest difference angles and FAerrors, whereas FREPR2 produced the smallest deviationsand errors. It is noted that, because the reduced data sets(including the control) are subsets of the “gold standard”data set, the aforementioned errors are likely underesti-mates of the true absolute errors due to the shared randomnoise. However, because the same proportion of the k-space was used in their formation, the impact of sharedrandom noise should be similar among the reduced datasets and does not undermine the relative accuracy advan-

FIG. 3. Representative images and difference images obtained in experimental demonstration of reduced-encoding DTI. The fully encodedb0 image was used in constrained reconstruction of the reduced encoded target diffusion-weighted image, both of which show a short-axisview of the sheep heart midhemisphere. The absolute difference images (intensity scaled 10� for better visualization) correspond to thedifferences between the target and the reduced-encoded diffusion-weighted images reconstructed by keyhole2, RIGR2, FREPR2, andFREPR4. The corresponding image cross-correlation coefficients are 0.996, 0.999, 0.999, and 0.997, respectively. The data showncorrespond to trial 3 in Table 2.

FIG. 4. Target fiber orientationhelix angle map and deviation an-gle maps obtained by reduced-encoding DTI. All angles are gray-scale-coded in degrees. The helixangle map reveals the classic epi-cardium-to-endocardium coun-terclockwise rotation of the myo-cardial fiber orientation. The devi-ation angle maps show theangular differences betweenfiber orientations measured bythe fully encoded DTI data set andthe control or reduced encodeddata sets each reconstructed viakeyhole2, RIGR2, FREPR2 orFREPR4. The data shown corre-spond to trial 3 in Table 2.

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tage of reduced encoding datasets over the control exper-iment. In all cases there are pixels of noticeably higherdeviations located at the borders of the myocardium. Be-cause these points also appear in the control experiment,they may be due to a combination of effects of reducedencoding and systematic factors such as partial volumeaveraging (24).

The sample-mean deviation angles and normalized FAerrors of the control and reduced-encoding DTI experi-ments are tabulated in Table 2 and Table 4, respectively,which is consistent with the qualitative observations inFigs. 3, 4, and 5. The group averages of deviation angles forcontrol, keyhole2, RIGR2, FREPR2, and FREPR4 experi-ments were 10.3° � 0.1°, 7.48° � 0.04°, 7.33° � 0.03°, 4.43°� 0.03°, and 6.89° � 0.03° (mean � SEM, n � 6), respec-tively. The ANOVA results in Table 3 reveal an F-testvalue of 2557 with P ��0.001, suggesting that at least oneof the five groups examined is significantly different. Thepost hoc analysis indicates that all pairwise comparisonsexcept for the keyhole2–RIGR2 pair have statistically sig-nificant group-mean differences. Moreover, the group av-erages of normalized FA errors for control, keyhole2,RIGR2, FREPR2, and FREPR4 experiments were 18.4% �0.1%, 12.8% � 0.1%, 11.1% � 0.1%, 6.66% � 0.06%, and10.6% � 0.1% (mean � SEM, n � 6) and the ANOVAresults in Table 5 reveal an F-test value of 5226 with P��0.001. The post hoc analysis indicates that all pairwisecomparisons have statistically significant group-mean dif-ferences. Among the groups examined, all REI schemes

have lower fiber orientation deviation angles and FA errorsthan the control experiment, which confirms the validityof REI with constrained reconstruction in DTI. AlthoughFREPR4 incurred higher error than FREPR2, both FREPR2

and FREPR4 outperformed either keyhole2 or RIGR2. Thisdemonstrates the superiority of radial acquisition over rec-tilinear sampling in reduced-encoding DTI and that theenhanced acquisition-time efficiency achievable by usingFREPR can be translated into additional savings in acqui-sition time.

DISCUSSION

The minimum DTI data set is comprised of seven imagesof the same gross anatomy but varying diffusion contrast.Moreover, to improve the accuracy, more images in theform of either or a combination of more encoding gradientdirections, gradient steps, or signal averages are added tothe data set. Because the diffusion contrast is relativelylow spatial frequency in nature, not all regions of thek-space sampled contribute equally in improving the DTImeasurement accuracy. A previous study (14) has shownthat, for a given encoding gradient directional scheme,rectilinear sampling reduced-encoding imaging can yieldmore accurate fiber orientation measurements than pro-portionally reducing the number of full-encoding imagesin the DTI data set. The present study shows that thecombination of reduced-encoding imaging and highernumber directional scheme is more advantageous over a

FIG. 5. Target FA map and nor-malized FA error maps (in percent-ages) obtained by reduced-encod-ing DTI. The FA map demonstratesa rather uniform anisotropy distri-bution throughout the myocar-dium. The FA error maps show thenormalized FA error FA% be-tween FA maps measured by thefully encoded DTI data set andthe control or reduced encodeddata sets each reconstructed viakeyhole2, RIGR2, FREPR2, orFREPR4. The data shown corre-spond to trial 3 in Table 4.

Table 2Fiber Orientation Deviation Angles between the Control or Reduced-Encoding DTI Measurementsand the Corresponding Fully Encoded Experiment

Trial Control Keyhole2 RIGR2 FREPR2 FREPR4

1 10.3° 7.52° 7.29° 4.45° 7.03°2 10.1° 7.53° 7.36° 4.45° 6.89°3 10.1° 7.54° 7.38° 4.41° 6.84°4 10.1° 7.32° 7.21° 4.32° 6.88°5 10.6° 7.45° 7.37° 4.49° 6.90°6 10.4° 7.50° 7.40° 4.44° 6.82°

Mean � SEM 10.3° � 0.1° 7.48° � 0.04° 7.33° � 0.03° 4.43° � 0.03° 6.89° � 0.03°

Note. Each entry represents the average over the area of the sample.

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lower number directional scheme with full-encoding im-ages. Moreover, compared to the respective full k-spaceacquisitions, reduced-encoding PR imaging offers moreaccurate DTI measurements than comparable rectilinearimaging. Together, these results indicate that the choice ofk-space sampling (e.g., radial versus rectilinear trajectoriesand preferentially sampling the central k-space versus uni-formly covering the entire k-space) and reconstructionstrategies can make a significant difference in the effi-ciency for capturing the desired DTI information. As such,reduced-encoding imaging with constrained image recon-struction can be a better alternative to simply reducing thenumber of full-encoding images (e.g., number of encodingdirections or signal averages) to accelerate DTI acquisition(or to increasing the number of full-encoding images toimprove the DTI measurement accuracy).

One important concern in REI is that acceptable recon-structions of the target images do not necessarily signifyadequate capturing of the intended dynamic contrastchanges (25). Consequently, the performances of REIschemes for DTI must be evaluated in terms of accuraciesof both the reconstructed diffusion-weighted images andmeasured diffusion tensor quantities (e.g., fiber orienta-tions and FA values). The results shown in Fig. 3 indicatethat the proposed FREPR technique yields more accurate

reconstructions of the target diffusion-weighted imagecompared to baseline-corrected keyhole and RIGR meth-ods. Moreover, in Figs. 4 and 5 and Tables 2–5, FREPRallows significantly more accurate fiber orientation andscalar anisotropy measurements than the rectilinear sam-pling-based methods, even at doubled reduced-encodingfactor. These findings validate the basic premise of thecurrent study that radial k-space acquisitions are inher-ently better than rectilinear sampling for reduced-encod-ing DTI. It should be noted that the present study does notaddress whether PR imaging is a better technique thanrectilinear sampling for DTI. Each is uniquely suited forspecific and likely dissimilar applications. For PR imag-ing, a potential limitation is that the number of viewsneeded to satisfy the Nyquist sampling criterion is higherthan that of rectilinear imaging. For example, a 50% re-duced PR acquisition takes comparable time as full recti-linear sampling acquisition with similar TR and NEX al-though the final SNR may differ. Conversely, PR imagingoffers the advantage of easier motion correction and rela-tive insensitivity to motion artifact such as ghosting (16).Moreover, because REI captures information in the centralk-space which has limited spatial frequency, the loss ofresolution may have in part contributed to the highererrors at the myocardial edge in Figs. 2–5. Therefore, cau-tion is required in extending REI to applications involvinghigh-resolution tissue information, especially for the rec-tilinear sampling case where resolution is directly anduniformly decreased with downsampling.

As shown by the computer simulation results in Fig. 2,beyond direct data substitution of k-space data, properFREPR reconstructions necessitate imposing additionalregularization conditions in the forms of radial filteringand baseline signal correction. The differential (i.e., high-pass versus low-pass) radial filtering of the b0 and reducedencoded data exploits the low spatial-frequency nature ofimage-to-image contrast changes and the radially varyingk-space sampling density in PR imaging. The step-rampfilter, where the b0 reference data are completely excludedin the central k-space but gradually included in outerk-space, provides the most accurate reconstruction. This isconsistent with the fact that, due to the inherent oversam-pling, a proportional area in the central k-space may besufficiently characterized by the downsampled trajectoriesof the reduced-encoded data alone. Consequently, inclu-sion of the b0 reference data in the central k-space, espe-

Table 3Single-Factor Repeated-Measurement ANOVA and Post HocMultiple Comparison Results of Myocardial Fiber OrientationDeviation Data in Table 2

Post hoc pairedcomparison

Student t value P

Control–Keyhole2 48.04 �10�6*Control–RIGR2 50.48 �10�6*Control–FREPR2 100.55 �10�6*Control–FREPR4 58.08 �10�6*Keyhole2–RIGR2 2.44 0.024Keyhole2–FREPR2 52.52 �10�6*Keyhole2–FREPR4 10.04 �10�6*RIGR2–FREPR2 50.08 �10�6*RIGR2–FREPR4 7.61 �10�6*FREPR2–FREPR4 �42.47 �10�6*

Note. ANOVA for five repeated measurements (n � 6); F statistics �2557.42; P � 10�6.*Significant F or Student t statistics, corresponding to P � 0.05 andP � 0.005 (Bonferroni condition for 10 multiple comparisons with anoverall P � 0.05), respectively.

Table 4Normalized FA Errors between the FA Maps of the Control or Reduced-Encoding DTI Measurementsand the Corresponding Fully Encoded Experiment

TrialControl

(%)Keyhole2

(%)RIGR2

(%)FREPR2

(%)FREPR4

(%)

1 18.5 12.9 11.2 6.73 10.72 18.2 12.6 11.0 6.74 10.73 18.1 12.8 11.0 6.81 10.54 18.7 12.9 11.2 6.59 10.65 18.4 12.6 11.1 6.41 10.66 18.7 12.8 11.4 6.69 10.4

Mean � SEM 18.4 � 0.1 12.8 � 0.1 11.1 � 0.1 6.66 � 0.06 10.6 � 0.1

Note. Each entry represents the average over the area of the sample.

100 Jiang and Hsu

cially in the absence of baseline signal correction, hasmore adverse than beneficial effects on the reconstruction.

Despite that radial filtering can achieve much improve-ment in the FREPR reconstruction, further enhancementcan be achieved via k-space baseline correction. The effectof signal correction is to minimize the complex signaldiscontinuities between the reference and reduced-en-coded k-space data, which is especially necessary for ap-plications such as DTI that contain large intensity differ-ences between the reference and reduced encoded data.Mathematically, the baseline correction coefficient (i.e., c0

in Eq. [2]) is equivalent to the magnitude-normalized pro-jection of reference onto the reduced-encoded data in lin-ear algebraic hyperspace. Baseline correction ensures thatthe magnitudes of the reduced encoded and substitutedreference data are continuous. It should be noted that thecurrent study presents only a proof-of-principle investiga-tion of data-regularization techniques for reconstructingreduced encoded PR images and does not preclude furtherimprovements in the FREPR reconstruction that may bepossible by employing higher order of signal continuitycorrection or by applying optimized radial mask filters.

Although in the present study the utility of reduced-encoding PR imaging is demonstrated in the context of DTIof the myocardium, the findings have implications for thebroader reduced-encoding PR imaging, as well as DTIstudies. Previous reduced-encoding PR imaging studieshave used purely angularly downsampled k-space data ofthe target image (equivalent to complete filtering) (18),direct combination of radially sampled reference and re-duced-encoded data (i.e., zero filtering) (19), postacquisi-tion combination of interleaved PR views (similar to stepfiltering) (20), or a sliding window-type reconstruction ofthe PR views with a temporal aperture (26) to improve theeffective time resolution. By applying radial mask filteringand baseline intensity correction, FREPR may providemore accurate reconstructions of the reduced encoded im-ages and be advantageous in these or similar dynamicimaging and interventional MRI applications. In regard toDTI, FREPR can be readily used to image other organ

systems such as the brain, albeit the unique advantagesand challenges of PR imaging [e.g., eddy current (27) orsusceptibility effect (16)] need to be considered for thespecific applications. The methodologies of radial filteringand baseline correction can be directly extended to re-duced-encoding DTI via other FID acquisition techniquessuch as multishot spiral imaging (28). Last, as k-spacesharing (29) has been investigated for echo-planar imaging(EPI) (30), the general principles of constrained imagereconstruction are expected to be applicable to enhancethe reconstruction quality and to improve the acquisition-time efficiency of DTI based on EPI or other rapid acqui-sition methods such as PROPELLER imaging (10).

In summary, the present study investigated regulariza-tion procedures necessary to adequately reconstruct re-duced encoded PR images based on k-space sharing.Marked improvements in the reconstruction accuracywere achieved by employing radial filtering to select fordesired frequency information and baseline correction toassert signal continuity between the reduced encoded andreference data. The application of the proposed FREPRtechnique to DTI tissue fiber orientation and anisotropyindex mapping yielded significantly better experimentaldata acquisition–time efficiency (e.g., higher accuracy us-ing equal scan time) compared to either rectilinear sam-pling-based keyhole and RIGR methods or an equal-scantime control experiment comprising of fully encoded im-ages. These findings demonstrate the inherent advantagesof radial acquisition over rectilinear k-space sampling forreduced-encoding imaging and underscore the utility ofcombining partial k-space imaging and constrained recon-struction in accelerating dynamic imaging, interventionalMRI, DTI, and other repeated imaging experiments.

ACKNOWLEDGMENTS

The authors thank Mr. Gary Cofer and Dr. Anja Brau for thepulse sequence and hardware consultations, and Ms. SallyZimney for editorial assistance.

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