3D Undersampled Golden-Radial Phase Encoding forDCE-MRA Using Inherently Regularized Iterative SENSE

Claudia Prieto,1* Sergio Uribe,1,2 Reza Razavi,1 David Atkinson,3 and Tobias Schaeffter1

One of the current limitations of dynamic contrast-enhancedMR angiography is the requirement of both high spatial andhigh temporal resolution. Several undersampling techniqueshave been proposed to overcome this problem. However, inmost of these methods the tradeoff between spatial and tem-poral resolution is constant for all the time frames and needsto be specified prior to data collection. This is not optimal fordynamic contrast-enhanced MR angiography where the dy-namics of the process are difficult to predict and the imagequality requirements are changing during the bolus passage.Here, we propose a new highly undersampled approach thatallows the retrospective adaptation of the spatial and tempo-ral resolution. The method combines a three-dimensional ra-dial phase encoding trajectory with the golden angle profileorder and non-Cartesian Sensitivity Encoding (SENSE) recon-struction. Different regularization images, obtained from thesame acquired data, are used to stabilize the non-CartesianSENSE reconstruction for the different phases of the boluspassage. The feasibility of the proposed method was demon-strated on a numerical phantom and in three-dimensional in-tracranial dynamic contrast-enhanced MR angiography ofhealthy volunteers. The acquired data were reconstructed ret-rospectively with temporal resolutions from 1.2 sec to 8.1 sec,providing a good depiction of small vessels, as well as dis-tinction of different temporal phases. Magn Reson Med64:514–526, 2010. VC 2010 Wiley-Liss, Inc.

Key words: DCE-MRA; undersampling; golden-radial phaseencoding; SENSE; regularization.

The simultaneous high spatial and temporal resolutionrequired in dynamic MRI is limited due to time acqui-sition constraints. This is especially important in appli-cations such as dynamic contrast-enhanced MR angiog-raphy (DCE-MRA), where for example high spatialresolution is required to visualize small vasculaturestructures, whereas high temporal resolution is neededto achieve adequate distinction between arterial and ve-nous vessels (1). To attain this goal, several undersam-pling acquisition and reconstruction techniques havebeen proposed. These techniques speed up the acquisi-

tion of dynamic MRI by reducing the number ofacquired samples in k-t space. The missing data can beestimated by exploiting the high spatial-temporal corre-lation of dynamic process (2–9), by using prior informa-tion (10,11), or by employing spatial sensitivity encod-ing in parallel imaging acquisitions (12–14).

For most of those techniques, the tradeoff between

spatial and temporal resolutions is constant for all the

time frames and must be specified prior to data acquisi-

tion. However, in DCE-MRA a time-varying retrospec-

tive adaptation of this tradeoff would be of great bene-

fit. For instance, a high temporal resolution is desirable

to visualize the arrival of the contrast agent to the ves-

sels of interest, though a lower spatial resolution is

bearable for this part of the dynamic process. Con-

versely, for arterial and venous signal peak phases, a

high spatial resolution is required, while a lower tem-

poral resolution is acceptable. Here, we propose a new

highly undersampled approach that allows the retro-

spective adaptation of this tradeoff between temporal

and spatial resolution to the different stages of the

bolus pass in DCE-MRA. The method combines a three-

dimensional (3D) isotropic radial phase encoding (RPE)

trajectory (15,16), with the golden angle order (17) and

a regularized iterative Sensitivity Encoding (SENSE)

(18) reconstruction.

The proposed method involves new undersampled

acquisition and image reconstruction schemes. The 3D

acquisition combines Cartesian sampling in the readout

direction, with an undersampled radial scheme in the

phase-encoding plane, where the angular step between

two consecutives radial profiles is given by the golden

angle. This trajectory takes advantage of the undersam-

pling properties of the radial trajectory (19,20) and the

retrospective reconstruction flexibility of the golden cut

order (21). Recently, other methods that combine Carte-

sian and projection reconstruction acquisitions have

been proposed (22–24); however, in these cases the

samples are acquired on a Cartesian grid following a

circular phase encoding (PE). In contrast, our method

uses a non-Cartesian trajectory in the PE plane, with

the consequent advantages from the aliasing pattern

point of view.In traditional radial frequency encoding trajectories

(25), undersampling in the radial direction would notmake sense. In contrast, the proposed trajectory allowsthe flexibility of undersampling both the angular and ra-dial directions of the PE plane. This allows great flexibil-ity in designing new trajectories based on their pointspread function (PSF) properties.

Given the non-Cartesian nature of the proposed trajec-tory, iterative SENSE is employed to reconstruct eachframe of the dynamic sequence. A proven approach to

1King’s College London, Division of Imaging Sciences, NIHR BiomedicalResearch Centre at Guy’s & St Thomas’ Foundation Trust London, UnitedKingdom.2Pontificia Universidad Catolica de Chile, Radiology Department, School ofMedicine, Center for Biomedical Imaging, Santiago, Chile.3Centre for Medical Image Computing, University College London, London,United Kingdom.

Grant sponsor: UK Engineering and Physical Sciences Research Council;Grant number: EPSRC E001076/1.

*Correspondence to: Claudia Prieto, Ph.D., Division of Imaging Sciences,The Rayne Institute, 4th Floor, Lambeth Wing, St Thomas’ Hospital,London, SE1 7EH, United Kingdom. E-mail: claudia.prieto@kcl.ac.uk

Received 8 July 2009; revised 24 December 2009; accepted 16 February2010.

DOI 10.1002/mrm.22446Published online in Wiley InterScience (www.interscience.wiley.com).

Magnetic Resonance in Medicine 64:514–526 (2010)

VC 2010 Wiley-Liss, Inc. 514

stabilize the ill conditioning of this kind of reconstruc-tion, and to diminish the residual aliasing and noiseamplification effects, is the use of explicit regularizationmethods (26). The proposed reconstruction scheme takesadvantage of the flexibility and properties of the trajec-tory to estimate a regularization image inherently fromthe same undersampled data. Moreover, different regula-rizations to reconstruct different frames of the dynamicsequence (dynamically adaptive regularization) can beobtained. This is especially suitable for DCE-MRA,where the use of an inappropriate regularization canhighly affect the ability to remove artifacts and the tem-poral resolution by, for instance, contaminating arterialphases with venous signal information.

This paper presents the proposed method and theresults of its application to isotropic 3D DCE-MRA of in-tracranial vessels with undersampling factors up to 52.Simulations and experimental results show the advan-tages of the variable tradeoff between temporal and spa-tial resolution approach in addition to the favorable PSFproperties of the proposed undersampled acquisition.

MATERIALS AND METHODS

Data Acquisition Scheme

The sampling pattern proposed in this work combinesthe recently introduced RPE trajectory (15) with thegolden ratio profile order (17) for dynamic applications.RPE combines a fixed readout direction kx along a Carte-sian grid, while PE is performed in the ky-kz plane on aradial trajectory. The full flexibility to reconstruct imageswith different temporal resolutions is achieved by using

a constant angular profile spacing, given by the goldenratio, yGR ¼ 111.25�. This approach guarantees quasiuni-form profile distribution for an arbitrary temporal resolu-tion. Advantages of this profile order have been shownfor two-dimensional and 3D radial frequency encodingtrajectories (17,27).

In order to speed up the acquisition, the number ofacquired PE steps can be reduced by undersampling theRPE steps in two directions, i.e., along the angular andradial direction, with acceleration factors of Qy and Qr,respectively. Furthermore, undersampling along the RPEdirections can be performed in two ways: (i) noninter-leaved undersampling, where the index of the sampleddata is the same for all angles, or (ii) interleaved under-sampling, where the radial encoding steps are shiftedfrom one angular direction to the next (Fig. 1a-c). Shift-ing encodings between consecutive angles results in tra-jectories with different PSF properties.

The combination of RPE and golden angle profile orderis called here Golden-RPE trajectory (Fig. 1a-c). Golden-RPE allows isotropic spatial resolution in the PE plane,since this is just given by the resolution in the radialdirection, in a similar way to conventional radial fre-quency encoding.

Reconstruction Procedure

As a result of the golden ratio profile order, each frameof the dynamic sequence can be reconstructed retrospec-tively with different temporal resolutions by just select-ing a different number of profiles Ny (also called recon-struction window), as is shown schematically in Fig. 1d.

FIG. 1. Acquisition and reconstruc-tion scheme for Golden-RPE trajec-tory. a: Sampling pattern showing

readout in kx direction and PE planein ky- kz. Sampled data (black dots)in radial-angle plane for Qr ¼ 4 with:

(b) noninterleaved Golden-RPE and(c) interleaved Golden-RPE. d: Top:Different temporal and spatial resolu-tions are achieved by reconstructing,retrospectively, a different number of

angular profiles Ny. Bottom: Regula-rization images (Rs) are used to reg-

ularize different frames of thedynamic sequence, as shown herewith R (1) and R (2).

Undersampled Golden-Radial Phase Encoding for DCE-MRA 515

Full flexibility in the width and starting point ofreconstruction window is given by the golden cut order;however, it has been shown (17) that maximum Signal-to-noise ratio (SNR) and uniformity of the profile distri-bution are reached when Ny is a Fibonacci number. Inthe case of the golden angle order, slightly more thanNr�p/2 profiles are needed in general to assure Nyquistcriterion in the angular direction (17), where Nr is thenumber of samples required to satisfy the Nyquist crite-rion in the radial direction. In Golden-RPE, less thanNr�p/2 profiles are used, in general, to reconstruct eachtime frame, violating Nyquist criterion. Therefore, withthis approach the wider the reconstruction window, thelower the temporal resolution, but also the lower artifactlevel, achieved.

The reconstruction of sensitivity-encoded non-Carte-sian undersampled MRI has been facilitated by the useof iterative techniques. However, the ill conditioning ofthe associated inverse problem produces residual alias-ing and noise amplification. A proven approach to stabi-lize the reconstruction and to diminish these effects isthe use of explicit regularization methods by includingsome sort of prior information. Specifically, non-Carte-sian SENSE reconstruction is given by:

ðEHC�1Eþ aR�1Þ �m ¼ EHC�1d ½1�

where E denotes the Fourier and sensitivity encodingmatrix, C is the noise correlation matrix, a is the regula-rization parameter, R is the explicit regularization, d isthe acquired data, and m is the reconstructed image. In

the case of dynamic images, every frame m(t) is recon-structed separately. If the dynamic information is sig-nificantly different to the static prior knowledgeincorporated in R, the use of the same regularization toreconstruct every frame m(t) could introduce errors. Thisproblem can be overcome by using a dynamically adapt-ive regularization R(t) to reconstruct dynamic images,modifying Eq. 1 to:

ðEHC�1Eþ aRðtÞ�1Þ �mðtÞ ¼ EHC�1dðtÞ ½2�

Taking advantage of the proposed trajectory, low spa-tial and low temporal resolution images reconstructedfrom a high number of profiles (wide reconstruction win-dow) and the center of the k-space are used as R or R(t)to regularize the dynamic reconstruction (Fig. 1d). R(t) isreconstructed in a sliding window fashion using nonre-gularized (a ¼ 0) iterative SENSE.

EXPERIMENTS

Simulations

Sampling Trajectory Properties

The influence of the different undersampling strategiesfor RPE was examined in simulations. Noninterleavedand two-interleaved radial undersampling RPE trajecto-ries were compared against a Cartesian (with circular PE)trajectory (Fig. 2a-d). The interleaved RPE trajectorieswere named according the number of shifted encodingsteps in the radial direction for each angular profile; for

FIG. 2. Trajectories, PSFs, and noise amplification maps for RPE (Qr ¼ 4, Qy ¼ 2) and Cartesian with circular PE (Qy ¼ 4, Qz ¼ 2) cases.

a: Noninterleaved RPE trajectory. b: Interleaved 0-2-1-3 RPE trajectory. c: Interleaved 0-2-0-2 RPE trajectory. d: Cartesian (with circularPE) trajectory. Corresponding PSFs and noise amplification maps are shown in (e-h) and (i-l), respectively.

516 Prieto et al.

instance, the interleaved trajectory 0-2-1-3 for an under-sampling factor of 4 is shown in Fig. 1c. Interleaved RPE0-2-1-3 and 0-2-0-2 were considered here, as is shown inFig. 2b,c. The trajectories were compared according totheir PSF properties (aliasing incoherence), as well astheir performance with parallel imaging reconstructionsusing noise amplification maps. The density compensa-tion functions were numerically estimated using Voronoidiagrams (28), and a weighting scheme to maintain fullresolution (29) was implemented. The PSFs were com-puted using direct Fourier transformation to avoid anyeffect introduced from gridding reconstruction.

Two different approaches were used to quantify alias-ing incoherence: (a) peak-to-side-lobe ratio (PSR) and (b)weighted entropy of the PSF side lobe. The firstapproach is given by AIPSR ¼ max|PSF|/max|PSFSL|,where PSF side lobe (PSFSL) is defined as the PSF outerthe main lobe. Weighted entropy of the PSFSL is givenby AIweighted entropy ¼ Entropy (PSFSL, W), where W is theweighting function that reflects the different contributionof the replicas (aliasing lobes) as a function of their posi-tions (30). The second approach is used here as an indi-cator of the aliasing incoherence. Noise amplificationmaps were calculated using Monte Carlo simulations,and mean and variance were computed for each case.For this purpose, the coil sensitivities of a six-channelcardiac coil were estimated from a cylindrical phantomexperiment (192 � 192 � 192mm3 field of view [FOV],matrix size of 144 � 144 � 144) and the noise amplifica-tion maps were computed following Eggers et al. (31),using 200 independent data sets with uncorrelated noise.

Time-Varying Temporal Resolution Reconstructions

The tradeoff between temporal and spatial resolution wasstudied in a numerical two-dimensional phantom (256 �256 matrix size, 20 time frames). Temporal intensity varia-tions were implemented by simulating the pass of a con-trast agent through different regions. An interleavedundersampling in the radial direction, Qr ¼ 4, and Ny ¼55, 34, and 26 angular profiles were used (total undersam-pling factors of 19, 30, and 39 with respect to the fullysampled Cartesian case, respectively). The acquired spa-tial resolution was employed for Ny ¼ 55, whereas thespatial resolution was reduced using 85% and 70% k-space coverage in the radial direction for Ny ¼ 34 and Ny

¼ 26, respectively (Nyquist not satisfied in all the cases).The acquired spatial resolution can be used for all thereconstructions when we wish to observe the tradeoffbetween temporal resolution and artifact level due toundersampling. The root mean square error was used tocompare quantitatively the different reconstructions.

In Vivo Experiments: Intracranial DCE-MRA

The feasibility of the proposed method was tested in 3DDCE-MR angiography of the intracranial vessels. Experi-ments were performed on a 1.5-T scanner (PhilipsAchieva, Best, The Netherlands) using a six-channelhead coil. Data were obtained on four healthy volunteersand written informed consent was obtained in accord-ance with the ethical rules of our institution.

An isotropic 3D Golden-RPE continuous acquisition ofthe intracranial vessels was obtained using coronal orien-

tation (readout in feet-head direction) in three volunteers.Scanner parameters were as follows: T1-weighted seg-mented Fast Field Echo (FFE) sequence, repetition time/echo time ¼ 3.5 ms/1.91 ms, 25� flip angle, 240 � 210 �210mm3 FOV, 1.25 � 1.25 � 1.25mm3 resolution, Ny ¼375 profiles, interleaved undersampling factor Qr ¼ 4 inthe radial direction. A 0.1 mmol/kg dose of a Gd-basedcontrast agent (Magnevist; Berlex, Wayne, NJ) at a rate of1.5 mL/sec, followed by 20 mL of saline flush at 1.5 mL/sec, was administered with a power injector (SpectrisSolaris, Medrad, Indianola, PA). Contrast injectionstarted simultaneously with the imaging protocol. A ref-erence scan (coil array and body coil, 39.5 sec each) wasperformed, prior to contrast injection, to estimate the coilsensitivity maps.

A sequence of 40 frames (covering the whole acquisi-tion time) with high temporal resolution tr ¼ 1.2s (nr ¼42, Ny ¼ 8, 70% k-space coverage) was reconstructed in asliding window fashion to estimate arterial and venousenhanced phases. From this information two regulariza-tion images (covering arterial and venous enhanced) werereconstructed using Ny ¼ 89 profiles, nr ¼ Nr/Qr ¼ 42samples per profile and 80% k-space coverage (Nyquistnot satisfied). The number of profiles was chosen as themaximum Fibonacci number that avoids strong contami-nation between phases. These images were used asexplicit regularization for the iterative reconstruction ofthe dynamic sequence. Reconstructions to validate (a) thebenefit of retrospective reconstruction with Golden-RPEtrajectory and (b) the influence of dynamically adaptiveregularization for DCE-MRA were performed.

The regularization parameter a was set by ensuringthat there is not excessive influence of the regularizationimage over the acquired data. That was achieved byreconstructing the images corresponding to the first 4 secafter contrast injection with different values of a. Sinceno contrast had arrived to the vessels of interest before 4sec, contrast would be present in the reconstructionsonly when an excessive value for a is employed.

An extra 3D acquisition of the intracranial vessels usingcontrast-enhanced timing-robust angiography (CENTRA)(32) and SENSE was obtained in one volunteer. Scannerparameters were as follows: T1-weighted segmented FFEsequence, coronal orientation, repetition time/echo time ¼2.9 ms/1.85 ms, 25� flip angle, 240 � 210 � 210mm3

FOV, 1.25 � 1.25 � 2.5mm3 resolution, 10 time frames,total undersampling factor of 6. A 0.1 mmol/kg dose of aGd-based contrast agent (Magnevist; Berlex, Wayne, NJ) ata rate of 1.5 mL/sec, followed by 20-mL of saline flush at1.5 mL/sec, was administered with a power injector(Spectris Solaris, Medrad). Contrast injection startedsimultaneously with the imaging protocol. Scanner param-eters and total acquisition time were similar to the onesused for Golden-RPE (with the exception of the resolutionin slice direction).

Benefit of Retrospective Reconstruction WithGolden-RPE Trajectory

The retrospective reconstruction with different temporalresolutions was compared against the classic approach ofconstant predetermined temporal resolution. For thispurpose, the following reconstructions were performed:

Undersampled Golden-Radial Phase Encoding for DCE-MRA 517

• Golden-RPE approach for 10 time frames covering justthe bolus pass period. Undersampling factor of 52 (nr

¼ 42,Ny¼ 13, 90% k-space coverage, tr¼ 1.9 sec).

• Golden-RPE approach for two high-resolutionimages in arterial and venous enhanced phases.Undersampling factor of 20 (nr ¼ 42, Ny ¼ 34, 100%k-space coverage, tr ¼ 5.0 sec).

• Predetermined temporal resolution approach (classicapproach) for 10 time frames, covering the whole ac-quisition period. Undersampling factor of 20 (nr ¼42, Ny ¼ 34, 100% k-space coverage, tr ¼ 5.0 sec).

Influence of Dynamically Adaptive Regularizationfor DCE-MRA

The effect of the dynamic regularization for the non-Car-tesian SENSE reconstruction was studied by reconstruct-ing one specific frame of the sequence with an under-sampling factor of 12 (nr ¼ 42, Ny ¼ 55, 100% k-spacecoverage, tr ¼ 8.1 sec) and two different approaches:

• Without regularization (a ¼ 0).

• With two regularization images R(t) (Ny ¼ 89 pro-files in arterial and venous enhanced phases).

To study the influence of the adaptive regularizedapproach, a sequence of 10 frames (nr ¼ 42, Ny ¼ 13,90% k-space coverage, tr ¼ 1.9 sec) was reconstructedwith the following approaches:

• With a static regularization image R (Ny ¼ 375 cov-ering the whole acquisition period).

• With two regularization images R(t) (Ny ¼ 89 pro-files in arterial and venous enhanced phases).

Non-Cartesian iterative SENSE, using preconditioning(from coil sensitivity maps) and explicit regularization,was used for all the reconstructions. The code wasimplemented on MatLab (The Mathworks) and run on adual 4-core personal computer, 32-GB memory. The four-dimensional reconstruction was transformed into a seriesof two-dimensional reconstructions problems consider-ing each slice in the readout direction and each timeframe of the dynamic sequence independently. Thereconstruction time per slice at a specific time was about11.3 sec, resulting in an average of 36 min per frame ofthe dynamic sequence.

Comparison Between High-Resolution Golden-RPEand Classic Approach

A comparison between the adaptive, single, and nonre-gularized Golden-RPE and the classic (temporal resolu-

tion specified prior to data acquisition) approaches wasperformed for high-resolution reconstructions. In allcases, images were reconstructed with an undersamplingfactor of 20 (nr ¼ 42, Ny ¼ 34, 100% k-space coverage,tr ¼ 5.0 sec). For the adaptive case, two regularizationimages (in arterial and venous enhanced phases) wereconsidered, whereas for the single case one regulariza-tion image (covering the whole acquisition period) wasemployed.

RESULTS

Simulations

Sampling Trajectory Properties

Undersampled trajectories, associated PSFs, and noiseamplification maps are shown in Fig. 2 for (Fig. 2a) non-interleaved RPE Qr ¼ 4, Qy ¼ 2, (Fig. 2b) interleaved 0-2-1-3 RPE, Qr ¼ 4, Qy ¼ 2, (Fig. 2c) interleaved 0-2-0-2RPE, Qr ¼ 4, Qy ¼ 2, and (Fig. 2d) Cartesian (with circu-lar PE) Qy ¼ 2.8, Qz ¼ 2.8. The names 0-2-1-3 and 0-2-0-2 indicate the number of shifted encoding steps in theradial direction for each angular profile (over groups offour angular profiles since Qr ¼ 4) with respect to thenoninterleaved RPE acquisition. An example of the inter-leaved 0-2-1-3 case is shown in Fig. 1c. PSFs were win-dowed for a better depiction of details in Fig. 2. Simi-larly, all noise amplification maps are shown with thesame intensity range. The resolution and number ofacquired samples for all the trajectories were kept con-stant for comparison purposes. Aliasing incoherence(with both peak-to-side-lobe ratio and weighted entropyapproaches) and noise amplification parameters for eachtrajectory are shown in Table 1. Note that the entropy ofthe PSFSL (not its histogram) increases for a more uni-form PSFSL.

A coherent PSF (resulting in strong and coherent spa-tial aliasing artifacts in the reconstruction) was observedfor the Cartesian (with circular PE) case. For all the RPEcases, the aliasing (such as replication, smearing andstreaking artifacts) was distributed over the entire FOV;however, a more coherent PSF (strong and coherent firstside lobe) was observed for the noninterleaved trajectoryin comparison with the interleaved cases. The inter-leaved 0-2-1-3 trajectory also presented a better noiseamplification distribution (mean and standard deviation)across the image with respect to the Cartesian (with cir-cular PE) and noninterleaved RPE trajectories. Consider-ing the aliasing incoherence (33) and the noise amplifi-cation properties as an indicator of the quality of thetrajectory, the interleaved RPE 0-2-1-3 was chosen as the

Table 1Comparison of PSF and Noise Amplification Maps Properties Between RPE and Cartesian (With Circular PE) Trajectories

(AIPSR ¼ Peak-to-Side-Lobe Ratio Aliasing Incoherence and AIweighted entropy ¼ Weighted Entropy Aliasing Incoherence)

Aliasing incoherence of PSF Noise amplification maps

AIPSR AIweighted entropy Mean Standard deviation

Noninterleaved RPE 9.930 2.044 0.239 0.158

Interleaved 0-2-1-3 RPE 8.389 2.156 0.212 0.137Interleaved 0-2-0-2 RPE 7.689 2.081 0.249 0.161Cartesian (with circular PE) 1 0.712 0.258 0.211

518 Prieto et al.

sampling scheme for all the simulations and in vivoexperiments presented in this paper.

Time-Varying Temporal Resolution Reconstructions

The reconstruction results for total undersampling fac-tors of 19, 30, and 39 in comparison with the fullysampled image are shown in Fig. 3. Reconstructionswith undersampling factors of 19 and 30 were in goodagreement with the fully sampled image, with root meansquare errors of 1.17% and 1.49%, respectively, whereasremaining artifacts gave a slight increase of the rootmean square error to 1.82% for an undersampling of 39.

Dynamic signal intensity evolution of three regions ofinterest (ROIs) for all the reconstructions is shown inFig. 3a in comparison with the fully sampled case. Aswas described before, in our scheme, a higher undersam-pling factor involves a higher temporal resolution but atthe same time a lower spatial resolution. We can observethat all ROIs benefit from higher temporal resolution to

recover the correct bolus pass (arrow green in ROI 3),whereas a higher spatial resolution is beneficial to depictsmall structures (arrow red in ROI 2). For ROI 3, we cannotice that an undersampling factor of 19 is inappropri-ate to recover the dynamics of the bolus passage sincethe temporal resolution is not enough, whereas andundersampling factor of 39 is inappropriate because ofremained artifact (arrows in Fig. 3i).

The fully sampled image and the reconstructions fromundersampling factors of 19, 30, and 39 are included fortwo selected time frames in Fig. 3b-i. For the 2nd frame,a good quality of the reconstruction was achieved for allundersampling factors; however, for the 14th framestrong streak artifacts remained for an undersampled fac-tor of 39 (arrows in Fig. 3i). This simple experimentshows the benefit of reconstructing different dynamicframes with different temporal resolutions; for example,in this case, with an undersampling of 39 for the 2ndframe and an undersampling factor of 30 for the 14thframe.

FIG. 3. Comparison of fully sampled image against 19�, 30�, and 39� undersampled reconstructions. a: Dynamic intensity evolutionfor three ROIs (in (b) and (f)), fully sampled (b) frame 2, (f) frame 14; 19� undersampled reconstruction (c) frame 2, (g) frame 14; 30�undersampled reconstruction (d) frame 2, (h) frame 14; and 39� undersampled reconstruction (e) frame 2, (i) frame 14.

Undersampled Golden-Radial Phase Encoding for DCE-MRA 519

In Vivo Experiments: Intracranial DCE-MRA

Benefit of Retrospective Reconstruction WithGolden-RPE Trajectory

A comparison between the proposed Golden-RPE and the

classic (temporal resolution specified prior to data acqui-

sition) approaches is shown in Fig. 4 and Fig. 5. The

dynamic pass of the contrast agent in the intracranial ves-

sels is shown for both cases in coronal and sagittal maxi-

mum intensity projections for 10 time frames. For the

Golden-RPE approach, low-resolution images (frames t1and t2) showed the bolus arrival, whereas the dynamic

pass of the contrast was depicted for the arterial and ve-

nous enhanced phases between t3 ¼ 15.8 sec and t10 ¼26.1 sec (reconstructed from an undersampling factor of

52). In order to obtain the same number of time frames,

an undersampling factor of 20 was fixed for the classicapproach (t1 ¼ 7.5 sec to t10 ¼ 52.5 sec). The retrospec-tive adaptation of Golden-RPE allows reconstructionswith good temporal resolution in the interval of interest,showing a good separation of the different temporalphases. Conversely, despite the good resolution achievedwith the classic approach, the lack of flexibility to choosethe time intervals for the reconstruction showed temporalblurring in some phases of the contrast pass (see arrowsin Fig. 4b and Fig. 5b). Moreover, as can be observed inFig. 4b and Fig. 5b, most of the time frames (t1-t2 and t5-t10) were outside the interval of interest.

Maximum intensity projections (from high-spatial-reso-lution reconstructions) showing arterial and venousenhancement are shown in Fig. 6a,b and Fig. 6c,d,respectively. Good spatial resolution, noticeable in the

FIG. 4. Coronal plane comparison between Golden-RPE and classic approaches. a: Golden-RPE approach for 52� undersampledreconstruction. Ten dynamic frames between bolus arrival and maximum venous intensity are included. b: Classic approach (temporal

resolution specified prior to acquisition) for 20� undersampled reconstruction. Ten dynamic frames covering the complete acquisitiontime are shown. Arrow shows temporal blurring on the arterial phases with the classic approach. Please note the different time resolu-

tion of both reconstructions.

520 Prieto et al.

visualization of the small capillaries, and minimum con-tamination from the opposite phase can be observed.Noisy areas at the bottom of the head were visible insome of the reconstructions; this could be explained duethe low sensitivity of the head coil in this area.

Influence of Dynamically Adaptive Regularizationfor DCE-MRA

The benefit of the regularized reconstruction over thenonregularized case is demonstrated in Fig. 7 for anundersampling factor of 12. Streak artifacts and low sig-nal to noise can be observed for the nonregularizedreconstruction in all planes. Conversely, most of the ali-asing was eliminated with the regularized approach,which is especially clear in the PE plane (Fig. 7f) whereslight streak artifacts remained.

The influence of the adaptive regularized approach isshown in Fig. 8. Three representative time frames (priorto bolus arrival, arterial enhancement, and venous

enhancement) are shown for the adaptive and singleregularized approach. For the adaptive case, two regula-rization images (in arterial and venous enhanced phases)were considered, whereas for the single case one regula-rization image (covering the whole acquisition period)was employed. For comparison purposes, the same regu-larization parameter (a) was used for both cases. As canbe observed, temporal blurring was introduced for thesingle regularized case at t ¼ 12.5 sec and t ¼ 17.3 sec.However, this effect can be diminished by employing adifferent value of the regularization parameter a or byemploying just one regularization image but not coveringthe complete acquisition period.

Comparison Between High-Resolution Golden-RPE andClassic Approach

The benefit of the adaptive and single regularizedGolden-RPE cases over the nonregularized and classic(temporal resolution specified prior to data acquisition)

FIG. 5. Sagittal plane comparison between Golden-RPE and classic approaches. a: Golden-RPE approach for 52� undersampled

reconstruction. Ten dynamic frames between bolus arrival and maximum venous intensity are included. b: Classic approach (temporalresolution specified prior to acquisition) for 20� undersampled reconstruction. Ten dynamic frames covering the complete acquisition

time are shown. Arrow shows temporal blurring on the arterial phases with the classic approach. Please note the different time resolu-tion of both reconstructions.

Undersampled Golden-Radial Phase Encoding for DCE-MRA 521

FIG. 7. Comparison between nonregularized and regularized Golden-RPE 12� undersampled reconstructions. a-c: Nonregularized

reconstruction, (d-f) regularized reconstruction. Residual aliasing is observed in the nonregularized case, especially in the PE plane (c).

FIG. 6. Maximum intensity projections for high-resolution reconstruction. Golden-RPE 20�undersampled reconstruction for: arterial (a,b)and venous (c,d) enhanced phases.

522 Prieto et al.

approaches is shown in Fig. 9 for high-resolution recon-structions. For the adaptive case, two regularizationimages (in arterial and venous enhanced phases) wereconsidered, whereas for the single case one regulariza-tion image (covering the whole acquisition period) wasemployed. Two representative phases (arterial and ve-nous enhanced) and a zoomed ROI for the venousenhanced phase are shown for each case. Temporal blur-ring was introduced for the single regularized Golden-RPE and classic approaches, as is shown in Fig. 9a. Abetter and sharper depiction of small vessels wasobserved with the adaptive and single regularizedGolden-RPE approaches.

The temporal intensity profiles for an arterial and a ve-nous segments are shown in Fig. 9d,e. For the arterialcase, some overestimation was observed for the singleregularized case (1 in Fig. 9d) and for the nonregularizedGolden-RPE and classic approaches (3 in Fig. 9d). Thereduced temporal resolution of the classic approach isobserved in 4 in Fig. 9d. Similar effects were observedfor the venous segment in Fig. 9e.

The adaptive regularized Golden-RPE approach wasalso compared against the standard CENTRA-SENSE ac-quisition, as is shown in Fig. 10. Despite that a head-to-head comparison is not possible in this case since datawere acquired in different voluntaries, similar sharpnessand resolution of the vessels were achieved with both

methods. The temporal resolution, however, was higherwith the Golden-RPE approach than the CENTRA-SENSEacquisition.

DISCUSSION

The Golden-RPE approach allows the reconstruction ofisotropic DCE-MRA, adapting the tradeoff between spa-tial and temporal resolution according to the bolus pas-sage. The combination of the favorable PSF propertiesand flexibility (undersampling in both angular and radialdirections) of the interleaved Golden-RPE trajectory,with the dynamically regularized non-Cartesian SENSEreconstruction, allowed acceleration factors up to 52 (rel-ative to conventional fully sampled Cartesian acquisi-tion). Results show good temporal resolution and qualityof the images in spite of the high undersampling factors.

The feasibility of the proposed approach to achievecorrect timing of the bolus arrival, proper separation ofdifferent temporal phases, and enough spatial resolutionto depict small vessels in specific temporal phases wasshown in MRA of intracranial vessels. The main advant-age of this approach in comparison with conventional3D time-resolved MRA is that specific time intervals canbe chosen retrospectively for the reconstruction. Thisallows, for example, (a) a high-temporal-resolutiondynamic sequence between bolus arrival and maximum

FIG. 8. Adaptive and single regularized approaches for Golden-RPE 52� reconstructions. Coronal planes for three representative time

frames using (a) adaptive and (b) single regularized reconstructions. Axial planes for three representative time frames using (c) adaptiveand (d) single regularized reconstructions. Temporal blurring can be observed with the single regularized approach (arrows).

Undersampled Golden-Radial Phase Encoding for DCE-MRA 523

venous enhancement, and (b) high-spatial-resolutionreconstruction of a selected number of phases (arterialand venous enhancement) of the dynamic sequence.

No previous bolus tracking scan is required with theproposed approach since the acquisition starts simulta-neously with the contrast injection. Moreover, the iso-tropic nonangulated 3D acquisition requires minimalplanning and can be reformatted in any plane afterward.Isotropic resolution is obtained naturally in the PEplane, according to the radial properties.

In spite of the high undersampling factors, subsecondtemporal resolution is still not achievable with the pro-

posed method. To attain this goal, direct extensions ofthe proposed trajectory to (a) half-scan in the radialdirection and (b) anisotropic FOV will be applied infuture volunteers. In the case of the Golden-RPE trajec-tory, two adjacent profiles are, in general, sampled in areversed direction due to the golden ratio profile order;therefore, a half-scan acquisition in the radial directionallows sampling both sides of the k-space, with moresamples concentrated in the center of the PE plane. Ani-sotropic FOV (while maintaining isotropic resolution) isachieved by changing the radial k-space step as a func-tion of the angle, following a similar approach to

FIG. 9. Comparison among adaptive, single, and nonregularized Golden-RPE against classic (temporal resolution specified prior to ac-

quisition) approach for 20� reconstructions. Sagittal planes for (a) arterial enhanced phase, (b) venous enhanced phase, and (c) ROI in(b). Temporal blurring can be observed with the single regularized and classic approaches (arrows in (a)). Sharper and better depiction

of the vessels can be observed in the adaptive and single regularized Golden-RPE reconstructions (arrows in (b) and (c)). Temporal in-tensity profiles for (d) arterial and (e) venous segments; for the arterial case, some overestimation can be observed for the single regu-larized reconstruction (1 in (d)) and for the nonregularized Golden-RPE and classic approaches (3 in (d)); the reduced temporal

resolution of the classic approach is noticeable in 4 in (d).

524 Prieto et al.

anisotropic spiral acquisitions (34). Another possibleapproach to speed up the acquisition would be the useof partial echo.

Aliasing incoherence was measured with twoapproaches to compare RPE and Cartesian (with circularPE) trajectories. However, we considered that AIPSR wasnot a good indicator to differentiate the aliasing incoher-ence between RPE trajectories. This is because this indexdoes not consider the complete distribution of the replicasover the PSFSL but just the maximum value of those repli-cas, which could lead to wrong conclusions. For example,in the studied trajectories the maximum side lobes wereMax(PSFSL) ¼ 0.100 (noninterleaved RPE), Max(PSFSL) ¼0.119 (0-2-1-3 interleaved RPE), Max(PSFSL) ¼ 0.130 (0-2-0-2 interleaved RPE), and Max(PSFSL) ¼ 1 (Cartesian withcircular PE). These values lead to higher AIPSR for thenoninterleaved case in comparison with the interleavescases; however, this conclusion does not correspond withthe visual impression in Fig. 2e-g. Conversely, the entropyapproach correlates well with the level of incoherence, aswas shown in Table 1.

Regularization images, reconstructed from a widereconstruction window, are used in the proposed methodto regularize the dynamic reconstruction from narrowerreconstruction windows. Both single and adaptive regula-rization showed good quality of the reconstructions,

although temporal blurring is introduced with the firstapproach if the regularization parameter a is not chosencarefully. The regularization images are used to stabilizethe iterative reconstruction, rather than used as a directconstraint, which is the case in the noniterative back-pro-jection approach highly constraint back projection(HYPR) (10).

An important consideration in the proposed method isthe choice of the regularization parameter a. For thereported results, the value was found experimentally forthe adaptive regularized case and the same value wasemployed for all the reconstructions. The approachemployed to find a resulted in good-quality images;however, a better and automatic approach for selecting awill be addressed in future studies.

Favorable undersampling properties are preservedbetween frequency and PE radial trajectories; however,motion properties are not. This is explained because k-space center is visited not every pulse repetition timebut every Nr � pulse repetition time. Cardiac triggeringcan be used in cardiac applications to reduce motionartifacts. Undesirable effects of the radial frequencyencoding trajectory, such as eddy currents, off-resonance,and requirement of large FOV in the radial direction, areavoid with the proposed RPE trajectory.

The current implementation of the proposed methodresults in long reconstruction times (36 min per timeframe). However, since each frame and each slice in thereadout direction are reconstructed independently, thealgorithm is highly and easily parallelizable. Therefore,the total reconstruction could be considerably shortenedby using a cluster of workstations or graphical cardsimplementations (35).

Interleaved Golden-RPE trajectories present, in general,high incoherence of spatial aliasing. Therefore, these tra-jectories may be applicable to compressed sensing MRI(11), where aliasing incoherence caused by undersam-pling is desired. Further studies using this reconstruc-tion will be performed and compared against the currentiterative SENSE implementation.

CONCLUSIONS

We have proposed a new highly undersampled approachto reconstruct DCE-MRA. This method allows retrospec-tive reconstructions with different combinations of spa-tial-temporal resolutions according to the pass of thecontrast agent. The method combines a 3D RPE trajectorywith the golden angle profile order and non-CartesianSENSE reconstruction. Moreover, a dynamically adaptiveregularization for the reconstruction can be obtainedfrom the same acquired data. The feasibility of thismethod was demonstrated with in vivo experiments andundersampling factors up to 52. Advantages of the pro-posed approach to achieve correct timing of bolus ar-rival, proper separation among different phases, andenough spatial resolution to depict small vessels in arte-rial and venous enhancement phases were shown.

ACKNOWLEDGMENTS

The authors acknowledge research support from PhilipsHealthcare and financial support from the Department of

FIG. 10. Comparison between adaptive regularized Golden-RPE

20� undersampling and standard CENTRA-SENSE acquisition. a:CENTRA-SENSE, (b) Golden-RPE 20� undersampling. Sets ofdata were acquired in different voluntaries. Similar sharpness and

resolution of the vessels can be observed with both methods;however, a much higher temporal resolution was achieved with

the Golden-RPE approach.

Undersampled Golden-Radial Phase Encoding for DCE-MRA 525

Health via the National Institute for Health ResearchComprehensive Biomedical Research Centre award toGuy’s & St Thomas’ NHS Foundation Trust, in partner-ship with King’s College London and King’s College Hos-pital NHS Foundation Trust.

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