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

A g-Factor Metric for k-t-GRAPPA- and PEAK-GRAPPA-Based Parallel Imaging Rebecca Ramb,1* Christian Binter,2 Gerrit Schultz,1 Jakob Assl€ ander,1 Felix Breuer,3 Maxim Zaitsev,1 Sebastian Kozerke,2,4 and Bernd Jung1,5 space. Time-consuming gradient encoding steps are partially omitted and instead, information of spatially varying coil sensitivities is incorporated in the image reconstruction process (1,2). The omission of parts of kspace generally results in a loss of signal-to-noise ratio (SNR) and the reconstruction implies spatially nonuniform noise distribution with spatial correlations. Approaches to capture the varying noise level—avoiding the often not feasible measurement repetitions required for an empirical evaluation—range from: noise statistics based on pseudoreplica (3) to SNR scaled reconstruction (4) to the recently proposed region of interest analysis by (5). For standard nontime-resolved parallel imaging acquisitions, the most prominent formulae to determine the nonuniform variation of noise and SNR—captured by the pixel-by-pixel-based g-factor (2)—have been derived for: sensitivity encoding for fast MRI (SENSE) (2), simultaneous acquisition of spatial harmonics (1), generalized autocalibrating partially parallel acquisitions [GRAPPA; method: (6), g-factor: (7,8)] and parallel magnetic resonance imaging with adaptive radius in k-space (PARS) (9). These methods are applicable only to single slices or volumes in each individual measurement. An extension of the formalism for time-resolved methods is still missing. Time-resolved parallel imaging methods, such as k-tSENSE (10), k-t-GRAPPA (11), PEAK-GRAPPA (12), TSENSE (13), or TGRAPPA (14), use arrangements of kspace undersampling patterns shifted with respect to temporal repetitions and utilize spatial as well as temporal correlations in the image reconstruction process. With these k-t-methods, higher reduction factors relative to frame-by-frame reconstruction have been achieved. However, a previous study (15) has revealed significant differences in the noise treatment with k-t SENSE and k-t GRAPPA: whereas the k-space-based method k-tGRAPPA exhibits smooth noise enhancement over the range of images, the image space-based method k-tSENSE shows greater noise enhancement in the area of motion and less noise in static areas. Nevertheless, the heuristic observations in noise and SNR behavior have not yet been supported by theoretical analyses. The aim of this work is therefore to provide the extension of the g-factor formalism for a noise and temporal analysis of kspace-based time-resolved parallel imaging methods as a step toward the overall goal of obtaining a general intermethod evaluation analysis for k-t-methods. When addressing both k-t-GRAPPA together with its optimized modification PEAK-GRAPPA, the term k-tPEAK-GRAPPA will be used for better readability. The

Purpose: The aim of this work is to derive a theoretical framework for quantitative noise and temporal fidelity analysis of time-resolved k-space-based parallel imaging methods. Theory: An analytical formalism of noise distribution is derived extending the existing g-factor formulation for nontimeresolved generalized autocalibrating partially parallel acquisition (GRAPPA) to time-resolved k-space-based methods. The noise analysis considers temporal noise correlations and is further accompanied by a temporal filtering analysis. Methods: All methods are derived and presented for k-tGRAPPA and PEAK-GRAPPA. A sliding window reconstruction and nontime-resolved GRAPPA are taken as a reference. Statistical validation is based on series of pseudoreplica images. The analysis is demonstrated on a short-axis cardiac CINE dataset. Results: The superior signal-to-noise performance of timeresolved over nontime-resolved parallel imaging methods at the expense of temporal frequency filtering is analytically confirmed. Further, different temporal frequency filter characteristics of k-t-GRAPPA, PEAK-GRAPPA, and sliding window are revealed. Conclusion: The proposed analysis of noise behavior and temporal fidelity establishes a theoretical basis for a quantitative evaluation of time-resolved reconstruction methods. Therefore, the presented theory allows for comparison between time-resolved parallel imaging methods and also nontime-resolved methods. Magn Reson Med 000:000–000, C 2014 Wiley Periodicals, Inc. 2014. V Key words: noise enhancement; g-factor; k-t-GRAPPA; PEAKGRAPPA; pseudoreplica

INTRODUCTION Parallel imaging methods allow for a significant scan time reduction by acquiring only selected parts of k-

1 Department of Diagnostic Radiology, Medical Physics, University Medical Center, University of Freiburg, Freiburg, Germany. 2 Institute for Biomedical Engineering, University and ETH Zurich, Zurich, Switzerland. 3 Research Center Magnetic Resonance Bavaria, Diagnostic Imaging, Wuerzburg, Germany. 4 Imaging Sciences and Biomedical Engineering, King’s College London, London, UK. 5 Institute of Diagnostic, Interventional and Pediatric Radiology, University Hospital Bern, Bern, Switzerland.

*Correspondence to: Rebecca Ramb, Dipl. Math., University Medical Center of Freiburg–Medical Physics, Breisacher Str. 60a, 79106 Freiburg, Germany. E-mail: [email protected] Received 6 April 2014; revised 1 July 2014; accepted 5 July 2014 DOI 10.1002/mrm.25386 Published online 00 Month 2014 in Wiley Online Library (wileyonlinelibrary. com). C 2014 Wiley Periodicals, Inc. V

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FIG. 1. Undersampled acquisition in k-t-space with a reduction factor R ¼ 5. The dimension kx is not displayed. Undersampling pattern for GRAPPA extended to the temporal domain (a), SW reconstruction within the same framework (b), the k-t-undersampling pattern and reconstruction kernels for k-t-GRAPPA (c), and PEAK-GRAPPA (d). For k-t-GRAPPA and SW, the same k-space undersampling shifted for each time frame is used. The PEAK-GRAPPA reconstruction differs in the usage of a permuted pattern to ensure denser sampling in k-t-space. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

article is structured as follows: first, the g-factor formalism for standard GRAPPA is extended by the domain of temporal frequencies. To this end, the k-t-PEAKGRAPPA reconstruction in x-f-space is introduced and kt-PEAK-GRAPPA reconstruction weights in x-f-space are derived. Next, the derived formalism is applied for k-tPEAK-GRAPPA in an in vivo cardiac measurement as well as for a sliding window (SW) method (only timeresolved) (16) and compared to nontime-resolved GRAPPA. The analytical results are validated by statistical evaluations based on series of pseudoreplica images, according to the previously proposed procedure of (3). It is shown that the derived g-factor formalism in x-f-space allows for the derivation of a temporally averaged gfactor map in image space, similar to the known g-factor for nontime-resolved methods, as well as for the investigation of temporal fidelity of k-space-based time-resolved parallel imaging methods. Parts of this work have been published in abstract form (17,18). THEORY For GRAPPA, the standard g-factor formalism builds on the Fourier relation between k-space and image space and is solely based on the reconstruction weights and the receiver noise matrix. Here, the existing g-factor formalism is extended to the dimension of temporal frequencies for k-tPEAK-GRAPPA. The gx -factor referring to the spatial position x :¼ ðx; yÞ hence becomes a pixel-by-pixel-based gxf factor discretized over the range of temporal frequencies. Undersampling in k-t-Space In Cartesian k-t-GRAPPA, the Cartesian k-space undersampling pattern is shifted for each successive time frame by one phase-encoding step. Therefore, each k-space position on the sampling grid is acquired once within R consecutive time frames, where R denotes the reduction factor. The resulting k-t-undersampling pattern, –omitting the k-space dimension kx, –used for k-t-GRAPPA as well as the optimized pattern utilized for PEAK-GRAPPA are depicted for a reduction of R ¼ 5 in Figure 1c,d, respectively. For both,

the acquisition of autocalibration signals (ACS) incorporated in the actual scan is displayed. Further illustrated is the corresponding k-space undersampling of Figure 1a nontime-resolved GRAPPA as well as an example application of the Figure 1b SW approach. The latter method is only time-resolved and, together with the nontime-resolved GRAPPA, isolates the effects of including either k-space or temporal neighbors in the reconstruction.

Reconstruction in k-t-Space Using Time-Resolved GRAPPA Kernels As the resulting noise distribution highly depends on how the image reconstruction process is accomplished, the reconstruction based on GRAPPA kernels is briefly revisited. Reconstruction is accomplished in two steps: calibration of the reconstruction weights and subsequent reconstruction of omitted k-space points on the acquisition grid by linear combination of neighboring data points using the weights. For calibration of the weights, sources and target data points in accordance with the chosen GRAPPA kernel geometry are collected within the densely sampled ACS of each receiver coil and assembled in one major source matrix Ssrc and target matrix Strg. The optimal—here in the least-squares sense—weights in k-t-space x are obtained by solving Strg ¼ v  Ssrc ;

[1]

for the unknown v using the Moore–Penrose pseudoinverse [described in detail in (6,8)]. Whereas for nontime-resolved GRAPPA, only source points within the same time frame are regarded, source points are gathered also from neighboring time frames, hence from three dimensions in total, for two-dimensional imaging k-t-PEAK-GRAPPA. The reconstruction procedure can be mathematically expressed by a convolution—in k-space for GRAPPA and in k-t-space for k-t-PEAK-GRAPPA. A complete Cartesian dataset Sred for the undersampled measurement of a sina gle coil a is obtained by convolving the reconstruction weights vab , corresponding to source coil b, and target

A g-Factor Metric for k-t-GRAPPA and PEAK-GRAPPA

3

coil a, with the undersampled and zero-filled data Sund b for each coil b, i.e, Sred a ¼

Nc X

vab  Sund b ;

where * denotes a convolution. Fourier transform and subsequent combination of all coil images using combination coefficients pa , for the coils a ¼ 1; . . . ; Nc , results in the final reconstructed series of images, denoted by Nc X

   pa F T kx ;ky Sred : a

¼ FT

1 t

( Nc X

) ~I und  Vab : b

[5]

b¼1

[2]

b¼1

I red ¼

Iared

The gxf -Factor for k-t-Reconstruction Methods Using Time-Resolved GRAPPA Kernels The following formulae extend the theory of (8) by the dimension of temporal frequencies. Assuming noise-free und ~ und signal ~I b and additive noise terms n for each coil b, b reconstruction in x-f-space as derived above is described by

[3]

~I red þ n ~ red a a ¼

a¼1

Nc   X ~I und þ n ~ und  Vab ; b b

[6]

b¼1

As SW can also be formulated within the same framework, the difference in formulating the theory between the methods k-t-PEAK-GRAPPA, SW, and nontime-resolved GRAPPA is then solely given by the dimensions used in the reconstruction, i.e., kx, ky, and/ or t. Figure 1 illustrates the different dimensions of the reconstruction kernels for the reduction factor R ¼ 5: (a) nontime-resolved kernels for GRAPPA, (b) the only time-resolved SW kernels, and (c) time-resolved kernels for k-t-GRAPPA, as well as (d) for PEAK-GRAPPA. In addition, explicit treatment of both k-space and temporal borders is required due to finite resolution and finite acquisition time. For instance, to reconstruct missing data points in the first time frame, the source area of time-resolved kernels exceeds the acquired domain as can be deduced from Figure 1. Commonly, either zerofilling or exploiting the circular boundary condition, if given, by copying acquired data points is performed to artificially enlarge the source domain. Reconstruction in x-f-Space Using Time-Resolved GRAPPA Kernels Equivalent to image reconstruction in k-t-space, reconstruction may be performed in x-f-space, which implicitly resolves the treatment of borders assuming the circular boundary condition as mentioned above. Moreover, the reconstruction weights in x-f-space serve as a transfer function from undersampled to reconstructed full datasets in x-fspace, and therefore, provide the basis for analysis of noise and signal transfer in the image reconstruction process. By application of the convolution theorem, Eq. [2] transforms into a point-wise multiplication in x-fspace, i.e., Nc n o   X   und F T kx ;ky ;t Sred ¼ F T S  F T kx ;ky ;t vab ; [4] k ;k ;t x y a b b¼1

where the three-dimensional Fourier transform F T kx ;ky ;t is applied to the undersampled data Sund of each coil b b and the corresponding weights vab . The final series of images in x-t-space is obtained by application of an additional inverse Fourier transform F T 1 along the tempot und ral dimension. Let ~I b denote the transformed undersampled data in x-f-space and Vab be the weights in x-f-space, then the image reconstruction in x-t-space from the undersampled data is given by

the variance of which is calculated by red 2 2 ~ red n red s ð~I a þ n a Þ ¼ s ð~ a Þ ¼s 2

Nc X

! ~ und n b

 Vab

[7]

b¼1

¼

Nc X

jVab j2 s2 ð~ n und b Þþ2

b¼1

Nc X Nc X

~ und jVab Vbg js2 ð~ n und b ;n g Þ:

b¼1 g¼bþ1

[8] Noise distribution hence depends solely on the reconstruction weights in x-f-space as well as noise variances 2 ~ und s2 ð~ n und n und b Þ and covariances s ð~ b ;n g Þ of the multiple receiver coils. Fixing a spatial position x :¼ ðx; yÞ and a temporal frequency f, weights in x-f-space and noise covariances may be expressed as matrices with rows and columns corresponding to the Nc receiver coils, denoted ^ xf and Sf, respectively: by V 1 0 V11 xf    V1Nc xf C B C .. ^ xf :¼ B [9] V C; B ⯗ . ⯗ A @ VNc 1 xf 0 B B Sf :¼ B B @

n und s2xf ð~ 1 Þ ⯗ ~ und n und s2xf ð~ Nc ; n 1 Þ

   VNc Nc xf  ..

.



~ und s2xf ð~ n und 1 ;n Nc Þ ⯗ s2xf ð~ n und Nc Þ

1 C C C: C A

[10]

Computation of the noise variance for each ðx; f Þ then reduces to   H  ^  ^ s2xf ð~ n red : a Þ ¼ V xf  Sf  V xf

[11]

As Sf reflects noise variances and covariances with respect to undersampled data acquisition, noise variances in the fully encoded scenario are obtained by dividing Sf by the reduction factor R for each temporal frequency f. Without addressing the temporal frequency, the formulae are the same as in (8). The relative noise variance in the fully encoded case hence is captured by: 1 s2xf ð~  jSf jaa : n full [12] a Þ¼ R Noise propagation between the reconstructed fully sampled and reconstructed k-t-undersampled data in xf-space for each ðx; f Þ and each coil a is then defined by

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Ramb et al.

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  H ^ xf j ^ xf  Sf  V jV sxf ð~ n red Þ a a qffiffiffiffiffiffiffiffi pffiffiffiffi ¼ ¼: gxf : sxf ð~ n full Þ  R jSf j a

[13]

temporal position and zero otherwise. The reconstructed signal magnitude is captured by ^ mred xf ;a :¼ jV xf  denc ja ;

aa

The coil combined noise distribution with coil combi~ in x-f-space (defined nation coefficient vectors p element-wise as in Eq. [3]) is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    H ^ xf  Sf  p ^ xf j ~T  V ~T  V j p sxf ð~ n Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi ¼ ¼: gxf :  T  T H n full Þ  R sxf ð~ ~ 1 j ~  1  Sf  p j p

^ xf as before. As the signal for each coil a and defining V full magnitude mxf ;a in the fully sampled case is simply the input itself, the magnitude signal transfer between full and reduced scenario is computed by

red

[14] Equations [13] and [14] analytically describe the noise propagation from full to reduced time-resolved image acquisition with k-t-undersampling and when timeresolved GRAPPA kernels are used in the reconstruction.

mred xf ;a mfull xf ;a

mfull xf ;comb

Temporal Average Noise Distribution and Temporal Fidelity

gxavg

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X  2 :¼ gxf : Nt f

^ xf  denc j jV a ¼: tPSFaxf : j1  denc ja

[18]

¼

^ xf  denc j j~ pT  V j~ p T  1  denc j

¼: tPSFxf :

[19]

METHODS k-t-GRAPPA, PEAK-GRAPPA, GRAPPA, and SW The k-t-undersampling pattern and kernel structures were applied as illustrated in Figure 1 for k-t-GRAPPA, PEAK-GRAPPA, nontime-resolved GRAPPA, and SW. Computation of the x-f-Weights

[15]

By the proposed combination of temporal frequencies, a spatial map containing the total noise variances is derived. The temporal frequency combined values are referred to as avg avg gx -factor for two reasons: first, to distinguish gx -factor for time-resolved methods from known gx -factors for nontime-resolved methods and second, to emphasize that these reflect a temporally averaged noise distribution for kt-PEAK-GRAPPA. The latter is substantiated by statistical avg validation as explained in Method Section. The gx -factor now only depends on the regarded position in image space and allows for a comparison with the gx -factor maps of nontime-resolved methods. As a counterpart to the analysis of noise variances in the reconstruction, the temporal average bias is captured by a pixel-wise temporal root mean square error (tRMSE): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N t u1X tRMSE :¼ t jI red  Itfull j2 : Nt t¼1 t

¼

The notation captures the characteristics of a relative temporal point spread function (tPSF). Again, the coil combination is incorporated by applying the coil combination vectors similarly to Eq. 14: mred xf ;comb

With the proposed gxf -factor, the noise behavior in parallel imaging methods using time-resolved GRAPPA kernels can be quantified spatially, i.e., pixel-by-pixel, and discretized over temporal frequencies. To further derive an analysis independent of the temporal frequency domain, the individual contributions of each temporal frequency are combined into one spatial map using Parseval’s theorem of energy preservation:

[17]

[16]

Temporal Point Spread Function Further, the temporal accuracy of k-t-methods is characterized in x-f-space by an analysis of the magnitude signal transfer over temporal frequencies. Therefore, the reconstruction weights are applied to a signal input denc of magnitude unity at all spatial positions for a single

Weights in x-f-space were derived in three steps, the first two of which are illustrated in Figure 2 for k-t-GRAPPA, PEAK-GRAPPA, and SW at a reduction of R ¼ 5. In the first step, the different kernel structures (as depicted in Fig. 1 and repeatedly shown in Fig. 2) were combined into a single convolution kernel. To this end, kernel geometries were shifted and additively overlaid such that their mutual kernel center corresponds to the target point in the reconstruction process. For SW, the convolution kernel consisted of R consecutive entries of unity and was replicated according to the number of coils. In the second step, the convolution kernel was flipped in three dimensions and extended in size by zero-filling to fit dimensions ðNx ; Ny ; Nt Þ. In Figure 2, combination of the kernels and flip of dimensions is further visualized by small arrows indicating the position and orientation of the particular kernel geometries for k-t-GRAPPA. In the third step, the enlarged kernel was Fourier transformed into x-f-space. Computation of gxf -Factors, gavg x -Factors, and tPSF For all time-resolved methods, the same algorithm for the calculation of gxf -factors (Eq. [14]) and the temporal avg average noise distribution, i.e., gx -factors (Eq. [15]), were applied. From the gxf -factor analysis, maps of the spatial noise distribution for each temporal frequency as well as temporal frequency curves for fixed pixels were derived. Moreover, tRMSE was calculated according to Eq. [16], and the temporal correlation was investigated via the tPSF as presented in Eq. [19]. For GRAPPA, the gx -factor algorithm was performed according to (8) using

A g-Factor Metric for k-t-GRAPPA and PEAK-GRAPPA

5

s ^ red xf pffiffiffiffi ¼: g^ xf s ^ full R xf

and

s ^ red xt pffiffiffiffi ¼: g^ xt ; s ^ full R xt

[20]

respectively. The first expression in x-f-space was used for direct comparison with the analytically derived gxf factor maps. From the latter expression in x-t-space, the time average was taken to obtain a temporal mean of the noise distribution map. The temporal mean noise distributions were compared with the temporal frequency avg avg combined gx -factors to show that gx -factors contain temporally averaged noise information for k-t-PEAKavg GRAPPA and SW. The gx -factors, therefore, allow for comparison with gx -factor maps for nontime-resolved methods. In Vivo: Short Axis Scan

FIG. 2. Combination of the kernel geometries for R ¼ 5 into one convolution kernel for k-t-GRAPPA, PEAK-GRAPPA, and SW. The convolution kernel is flipped in all dimensions, enlarged by zero-filling and subsequently, Fourier transformed into x-f-space to provide the reconstruction weights in x-f-space. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

weights derived frame-by-frame. Additionally, GRAPPA weights derived frame-by-frame were averaged over time, on which the same frequency-resolved gxf -factor and tPSF analysis as for the time-resolved methods could be performed. Further, virtual coil arrays reducing the total number of coils were simulated according to (19) and the spatial avg mean of gx -factors and tRMSE were calculated for the reduced coil scenario. The noise covariance matrix of the receiver coils was set to the identity matrix in the avg reduced case. Hence, all gx -factor calculations are based on assuming decorrelated and normalized noise in the virtual receiver channels. Statistical Validation For the validation of the results, statistical evaluation was performed based on generated series of timeresolved pseudoreplica images according to (3). To this end, 256 time-resolved pseudoreplica datasets were iteratively generated for both the undersampled and fully sampled case, with normal distributed noise added in each iteration. With the standard deviation calculated for both scenarios in x-f-space and x-t-space, i.e., ^ full ^ red ^ full s ^ red xf ; s xf ; s xt , and s xt , the statistical noise distribution in x-f-space and x-t-space was calculated by

In an in vivo experiment, a short axis CINE dataset of 42 heart phases was acquired in a healthy volunteer on a Philips Ingenia 3T scanner, using a 28 channel receiver coil, with fully encoded balanced steady-state free precession (bSSFP), during a 20 second breath-hold with retrospective gating and a separate scan of coil sensitivities. Informed consent was obtained prior to the acquisition. The following sequence parameters were used: TE=TR ¼ 1:58=3:15 ms, a ¼ 40 , BW ¼ 1316 Hz/Px, spatial resolution 2  2  8 mm3, and matrix size 148  132. Undersampled acquisition was retrospectively mimicked for reductions of R ¼ 5 and R ¼ 7 and 11 ACS lines according to the patterns introduced for k-t-GRAPPA, PEAK-GRAPPA, GRAPPA, and SW. As the main focus lies on evaluating the effect of applying the kernel, the ACS lines were not subsequently added to the reconstructed datasets. With a heart rate of 66 beats per minute, the duration of one heart cycle was 905 ms, which corresponded to temporal frequencies of 1.1 Hz. For 42 measured time points, the temporal resolution was 21:6 ms and the covered bandwidth of temporal frequencies was approximately 46 Hz. Due to reduction by R ¼ 5 and R ¼ 7, the number of data points for each k-space position in the temporal domain was diminished to 8 and 6 data points, respectively, per time series. Consequently, the sampled bandwidth of temporal frequencies in these cases was approximately 8.8 and 6.6 Hz, respectively. RESULTS Figure 3 shows the analytical and statistical gxf -factor maps derived from the in vivo CINE dataset at reduction factor R ¼ 5. For reference to spatial positions of the moving heart and the surrounding mostly static tissue, the first time frame of each reconstruction is displayed. Each of the noise distribution maps corresponds pixelwise with the spatial positions in the shown magnitude images. However, unlike the single gx -factor map for nontime-resolved methods, gxf -factor maps are obtained as a discrete series of contributions from each temporal frequency. In Figure 3, maps are displayed beginning with the DC component, i.e., 0 Hz, up to 7.7 Hz, and concluding with the highest temporal frequency resolved, i.e., 20.9 Hz. The analytical results are in

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FIG. 3. Analytical gxf -factors and statistical g^ xf -factors (obtained from the time-resolved pseudoreplica) for SW, k-t-GRAPPA and PEAKGRAPPA at reduction factor of R ¼ 5. For spatial reference of moving and static tissue, the first time frame of the series which were reconstructed in x-f-space is displayed on the left. The temporal frequency units results from the measured duration of the heart cycle.

excellent agreement with the in the x-f-domain performed statistical analysis. Although the DC component exhibits values slightly higher than 1, values below 1 occur in maps of higher temporal frequencies indicating a noise filtering effect. This is fundamentally different compared to nontime-resolved methods such as SENSE, where the gx -factor is always greater or equal to 1. Further, the noise reduction exhibits varying spatial appearance over temporal frequencies among the three regarded methods, hence, reveals method-dependent temporal frequency filtering characteristics. Figure 4 depicts analytical and statistical gxf -factors corresponding to a single column of the reconstructed images and over the full range of temporal frequencies. The displayed column is highlighted by a red line in the magnitude image. Whereas no distinction between static and moving tissue is made for SW, the moving tissue remains visible with values around 1 (and slightly higher) at higher temporal frequencies for k-t-GRAPPA and PEAK-GRAPPA. Visually determining a cutoff frequency as the lowest positive temporal frequency where the frequency noise response reduces almost to zero, the cutoff is lower for higher reduction factors in all scenarios due to the increased width of the convolution kernel along the temporal axis. Comparing the three scenarios with each other, the cutoff frequency is the lowest for SW. The performance of k-t-GRAPPA in stationary tissue is similar to SW, however, values are higher in moving tissue. PEAK-GRAPPA generally exhibits a smooth decay with only slight changes with respect to the higher reduction factor. avg Figure 5 displays gx -factors derived according to Eq. [15] and based on the statistical evaluation in the temporal domain (Eq. [20]) for the three time-resolved methods. Within the same figure, however, with separate scaling, the gx -factor calculations for GRAPPA are depicted for a reduction of R ¼ 3. All analytical and statistical results in the x-t-domain/spatial domain are again

FIG. 4. The gxf -factor results of one data column in the reconstruction matrix—as indicated by the red line in the magnitude image—plotted over the range of all temporal frequencies. Analytical and statistical results are depicted next to each other for the three time-resolved methods SW, k-t-GRAPPA, and PEAKGRAPPA at reduction factors R ¼ 5 and R ¼ 7.

A g-Factor Metric for k-t-GRAPPA and PEAK-GRAPPA

FIG. 5. Analytically and statistically derived gavg x -factor maps for SW, k-t-GRAPPA, and PEAK-GRAPPA at reduction factors R ¼ 5 and R ¼ 7, respectively. With separate scaling and with reduction factor R ¼ 3, analytical gx -factors of GRAPPA derived according to (8) with a respective statistically derived map. All maps correspond pixel-wise with the displayed magnitude images of the reconstruction in x-f-space and image space, respectively, displayed on the left side.

in excellent agreement. Although a lower reduction factor was applied, the gx -factor values for GRAPPA already exceed the values obtained in the time-resolved scenarios. For k-t-GRAPPA and PEAK-GRAPPA, values are increased in the area of the moving heart, however, values again only slightly exceed 1. For all three methods, values further decrease for the higher reduction factor in the static tissue due to increased temporal frequency filavg tering effects. The gx -factor maps for the time-resolved methods confirm the overall noise reduction when exploiting temporal correlations in the reconstruction process, where the two time-resolved parallel imaging methods show less noise reduction in areas of moving tissue. While Figure 5 demonstrates the decrease in spatial noise variances for time-resolved methods, the analysis in Figure 6 captures the temporal bias in these methods. Figure 6 presents spatial maps of the tRMSE and reveals that along with the decrease in noise variance (Fig. 5), the temporal bias of the reconstruction increases (Fig. 6), when moving to higher reduction factors. For both reduction factors, the highest tRMSE occurs for SW while the best agreement with the reference is achieved with PEAK-GRAPPA, where there is also only a slight

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increase between the two reduction factors. Single values of the tRMSE are listed for all four methods in Table 1 for two designated pixels, i.e., one pixel within the blood pool, this pixel depicts the myocardium in later time frames, and one pixel within the static area of fat tissue. GRAPPA, despite the lower reduction factor, exhibits the largest error values, whereas PEAK-GRAPPA in both scenarios achieves the lowest error. For k-t-GRAPPA at R ¼ 5, tRMSE values are lower compared to SW, however, higher at R ¼ 7, caused by increasing influence of noise in the reconstruction process. Figures 5 and 6 together analytically demonstrate a spatially varying trade-off between noise reduction and temporal fidelity in time-resolved parallel imaging scenarios. Figure 7 illustrates how the temporal fidelity is affected by blurring and noise, where (a) shows the magnitude time curves of the afore regarded pixels for the three time-resolved methods together with the fully sampled reference (dotted lines) and (b) depicts the same column as in Figure 4 over the range of time frames. For the pixel within the moving tissue in (a), the contrast dip is to some extent smoothed for a reduction of R ¼ 7. The displayal in (b) shows how the GRAPPA reconstruction is strongly affected by noise enhancement. Further, SW exhibits temporal blurring and the reconstruction for k-t-GRAPPA performs slightly better at R ¼ 5, but suffers from noise enhancement in the area of the moving heart at R ¼ 7. PEAK-GRAPPA almost sustains the resolution obtained in the reference reconstruction for R ¼ 5, but appears slightly temporally blurred for R ¼ 7. Figure 8 displays (a) analytical (solid lines) and statistical (dotted lines) gxf -factors and (b) the tPSF, both for the same two pixels and over the range of positive temporal frequencies. The plots allow for a detailed description of signal and noise frequency filtering characteristics, as well as the resulting temporal correlations between time frames. In this way, the characteristic of SW—known from theory, but again evident in the presented framework—can be compared to the here discovered characteristics of the covered time-resolved parallel imaging methods. SW transfer characteristics are independent of the regarded pixel and show a stronger decay of signal and noise transfer with respect to higher reduction factors. Further, SW exhibits zero crossings and side lobes, which increase in number for the higher reduction factor. For kt-GRAPPA and PEAK-GRAPPA, the two example pixels reveal that signal and noise transfer depends on the spatial position. For the pixel which initially depicts the

FIG. 6. tRMSE for SW, k-t-GRAPPA, and PEAK-GRAPPA at reduction factors R ¼ 5 and R ¼ 7, respectively.

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Table 1 Temporal Root Mean Squared Error (tRMSE) of Two Single Pixels: A Single Blood Pool Pixel that in Later Time Frames Contains Myocardium and a Fat Tissue Pixel in static Tissuea Method GRAPPA SW k-t-GRAPPA PEAK-GRAPPA

Reduction factor

Pixel in moving tissue

Pixel in static tissue

R¼3 R¼5 R¼7 R¼5 R¼7 R¼5 R¼7

38.41 22.43 25.51 18.01 33.33 16.71 18.85

9.5 5.73 6.43 5.95 6.72 3.87 4.54

a

The pixels are the same as indicated in the magnitude images of Figures 7 and 8.

blood pool, but later contains the myocardium, the decay for k-t-GRAPPA and PEAK-GRAPPA is evidently smoother. Temporal frequency resolved values were also calculated with the x-f-domain algorithm based on time averaged weights for GRAPPA. In the noise analysis, values around 4 for all temporal frequencies were derived but are not shown in the figure for better visibility. As expected, the tPSF is constant over temporal frequencies for nontime-resolved GRAPPA and is illustrated by dotted lines for R ¼ 3 and R ¼ 5. For the higher reduction factor, values are decreased, indicating an overall decrease in the signal transfer, which however could also be due to the temporally averaged GRAPPA weights. Figure 9 presents spatially averaged values of (a, b) avg gx -factors and (c, d) tRMSE with respect to reducing the number of coils. As SW does not incorporate in vivo coil sensitivities in the reconstruction process, the mean avg gx -factor is independent of the number of coils. The two time-resolved parallel imaging methods, however, exploit in vivo coil sensitivities and hence depend on the number of coils. In the scenario of only a single coil, k-t-GRAPPA and PEAK-GRAPPA become variants of SW and noise reduction mostly depends on the respective window size in the temporal domain which is the widest for k-t-GRAPPA. In the scenario of two or more coils, the

influence of parallel imaging while exploiting temporal correlations becomes evident by a decrease of the mean avg tRMSE while mean gx -factors increase, (a,c) and (b,d), avg respectively. Note that the mean gx -factors yields an average over the whole spatial area, i.e., both low values in static tissue and the for k-t-GRAPPA and PEAKGRAPPA higher values in areas of moving tissue are averaged. This analysis again reveals the trade-off between noise variances versus temporal bias in the time-resolved reconstruction process depending on the combination of exploiting in vivo coil sensitivities and temporal correlations, and also with respect to the reduction factor. When the number of coils is higher than 12, results exhibit only minor changes (a,b,c,d). In all sceavg narios, PEAK-GRAPPA shows the highest mean gx -factor, yet still below 1, but at the same time the lowest mean tRMSE. DISCUSSION The presented analysis comprises the calculation of gxf factors utilized to study the temporal average noise distribution within k-t-methods based on time-resolved kernel reconstructions as well as the temporal fidelity and temporal frequency filter characteristics due to k-treconstruction. All methods are statistically validated based on the pseudoreplica method (3). The gxf -factors were validated in x-f-space by comparison with g^ xf . The frequency comavg bined gx -factor maps were validated in image space via the temporally averaged values of g^ xt . Both g^ xf and g^ xt were derived in Eq. [20]. All analytically calculated values are in excellent agreement with the statistical results (Figs. 3–5 and 8). The pixel-wise gxf -factor maps show the contribution of each temporal frequency separately (Fig. 3). By examining the temporal frequency axis (Figs. 3, 4, and 8), temporal frequency filtering is observed for all time-resolved methods by gxf -factor values below 1, which further decrease when moving to higher temporal frequencies. Moreover, method-dependent temporal frequency filtering characteristics are revealed: First, SW exhibits no spatial variations, as was expected from theory, whereas

FIG. 7. a: Temporal evolution of signal magnitudes for the same pixels as in Table 1 and for all regarded time-resolved reconstruction scenarios as well as for the fully sampled reference (dotted lines). b: The same image column as in Figure 4 displayed along the temporal axis for the fully sampled reference, for GRAPPA at reduction factor R ¼ 3, for k-t-GRAPPA, SW, and PEAK-GRAPPA at reduction factors R ¼ 5 and R ¼ 7, respectively. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

A g-Factor Metric for k-t-GRAPPA and PEAK-GRAPPA

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FIG. 8. a: The gxf -factor results for a fixed pixel plotted over the range of temporal frequencies for SW, k-t-GRAPPA, and PEAKGRAPPA at reduction factors R ¼ 5 and R ¼ 7. Analytical and statistical results are emphasized by solid and dotted lines, respectively. The regarded pixels are displayed in the magnitude image. b: The temporal point spread function for the same pixels. The input for the analysis was a delta signal, i.e., an object of constant value 1 appearing at one distinct time frame, which was reconstructed using the in vivo dataset determined weights. In the plots corresponding to reduction factor R ¼ 5, the results for nontime-resolved GRAPPA with time averaged weights at a reduction of R ¼ 3 and R ¼ 5 are additionally illustrated. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

k-t-GRAPPA and PEAK-GRAPPA exhibit a spatial dependency in temporal frequency filtering, which is a new insight for these methods. Second, differences among temporal frequencies show a sinc-like frequency response for SW, as expected from the box-shaped con-

volution window, whereas k-t-GRAPPA and PEAKGRAPPA reveal their characteristic frequency response due to a nontrivially shaped three-dimensional convolution kernel. Differences among temporal frequency maps between k-t-GRAPPA and PEAK-GRAPPA are directly

FIG. 9. The spatial mean of (a, b) gavg x -factors and (c, d) tRMSE with respect to reduced numbers of coils for R ¼ 5 (a, c) and R ¼ 7 (b, d). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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induced by the differences between reconstruction kernel geometries. For k-t-GRAPPA, the transfer over temporal frequencies is similar as for SW in static tissue, however, in moving tissue gxf -factor values remain higher for higher temporal frequencies. For PEAKGRAPPA, the decay over temporal frequencies, as well as the spatial variation, is much smoother. The smoother decay for PEAK-GRAPPA is explained by the more smoothly varying convolution kernel with smaller temporal extent induced by the optimized kernel structures and denser sampling pattern. By incorporating in vivo coil sensitivities and neighboring time frames, k-tGRAPPA and PEAK-GRAPPA show the ability to differentiate between static and moving tissue in the filtering process. The resulting noise distribution thereby exhibits both spatial and temporal correlations. avg The gx -factor for all time-resolved methods (Fig. 5) demonstrates an overall reduction of noise by values below 1. This confirms the good performance of k-tGRAPPA and PEAK-GRAPPA in terms of noise, which is now supported by a theoretical analysis. Moreover, the avg gx -factor maps are comparable with the g-factor maps known from literature in the nontime-resolved cases, i.e., (2,7,8), and thus facilitate intermethod comparisons. As shown in Figure 5, k-t-GRAPPA and PEAK-GRAPPA possess a superior noise behavior over nontime-resolved GRAPPA reconstructions. However, the reduction of noise variances comes at the expense of temporal frequency filtering which leads to temporal blurring (Fig. 7). Temporal fidelity is reflected in the tRMSE values (Fig. 6, Table 1) and reveals an increased temporal bias as counterpart to the decreased noise variances. The analysis of noise hence cannot be considered as a single predictor of reconstruction quality, but temporal accuracy requires consideration. For SW, where k-space neighbors and in vivo coil sensitivities are disregarded in the reconstruction process, signal and noise transfer is independent of the spatial position, as well as the number of coils, and the same temporal frequency filtering is applied over the whole images. Thereby, tRMSE is high and increases for higher reduction factors. For k-t-GRAPPA and PEAK-GRAPPA, temporal frequency filtering depends on the spatial position and is weaker in areas of moving tissue. Lower tRMSE are achieved. PEAK-GRAPPA exhibits the lowest temporal bias in the comparison of the three methods (Fig. 6, Table 1), but also shows the least noise reduction (Fig. 5). In (20), modulation transfer functions were proposed to evaluate temporal frequency behavior for UNFOLDSENSE (21) and variants of k-t-SENSE, based on the theory of stochastic processes. Here, the signal magnitude transfer is captured by the presented tPSF (Fig. 8) and allows for the application of known filter characteristic measures such as full width at half maximum, main lobe to side lobe amplitude ratio or leakage factor known from signal processing [for instance (22,23)]. All three methods show the characteristics of a low pass filter whose width of the main lobe decreases when moving to higher reduction factors. In SW, zero crossings are visible and indicate the loss of certain temporal frequencies, and temporal frequency leakage due to neighboring lobes. Although the decay of the main lobe is similar for

Ramb et al.

k-t-GRAPPA and SW, it occurs very smoothly with a broader range of temporal frequencies in PEAKGRAPPA. Again, the characteristics for k-t-GRAPPA and PEAK-GRAPPA depend on the pixel considered. The proposed method is applicable to any kernelbased k-t-method which allows for explicit determination of the reconstruction weights prior to reconstruction and considers noise correlations that are invoked between temporal neighboring time frames. Iterative reconstruction methods and methods including a priori information in the reconstruction process, as e.g., done in L1-SPIRiT (24), will be subject of future work. It is further possible to apply the algorithms to nontimeresolved kernel-based methods, however, inducing minor modifications. For nontime-resolved GRAPPA, the frame-by-frame derived GRAPPA weights have to be averaged over time to derive a convolution kernel in k-tspace. This, therefore, allows for reconstruction in x-favg space and calculation of gxf - and gx -factors within the exactly same framework. However, temporal averaging of the frame-by-frame GRAPPA weights prior to computing the g-factor values is not exactly equivalent to temporally averaging the frame-by-frame calculated g-factors. In the present article, the three time-resolved methods were treated within the same framework and compared with the known formalism for nontime-resolved GRAPPA. The observed differences in the results, while applying a similar theory, allow for a classification of parameters and effects. The incorporation of coil sensitivities and/or the temporal domain in the reconstruction process entails a nonuniform noise distribution with spatial correlations and overall noise reduction with temporal smoothing, respectively. The four methods range from using either only coil sensitivites (GRAPPA) or only temporal correlations (SW) to using both (k-tGRAPPA and PEAK-GRAPPA). The observations disclose: (i) Incorporating coil sensitivities, i.e., increasing the influence of parallel imaging, supports the reconstruction fidelity, however, while spatially variably increasing noise. (ii) Employing the temporal domain yields a noise reduction at the expense of temporal inaccuracy in the reconstruction. (iii) The combination of using both, coil sensitivities and temporal correlations, results in a spatially varying noise transfer as well as spatially varying temporal frequency filtering depending on the movement in the regarded areas. Although two-dimensional parallel imaging as presented in the CAIPIRINHA method (25) provides a threedimensional undersampling pattern as in time-resolved two-dimensional parallel imaging methods, the here performed analysis shows how incorporation of the temporal domain is essentially different compared to the dimension of partitions which are again linked to spatially varying coil sensitivities. The analysis of compressed coils scenarios in Figure 9 further substantiates the impact of increasing the influence of parallel imaging onto the reconstruction fidelity. Decreasing the number of coils is feasible, however, removes the parallel imaging part stepby-step such that the temporal domain gains more influence. This results in further noise reduction, however, while introducing artifacts and temporal inaccuracy. Artifacts and temporal inaccuracy are not reflected in the

A g-Factor Metric for k-t-GRAPPA and PEAK-GRAPPA

g-factor, hence, additional investigation of the temporal accuracy is performed here by the tRMSE. The investigation of the reduced coil numbers further shows that no improvement is reached after a number of 12 coils. Please avg note, these are virtual coil arrays and in these cases, gx factors were calculated assuming perfectly decorrelated receiver coils. However, similar observations were also made in other studies as for instance (26). Altogether, the presented method provides a powerful tool in analyzing signal and noise transfer in k-spacebased k-t-methods. One further potential is the application in kernel design with respect to temporal fidelity performance. Thereby, the trade-off between low/high dynamic range and high/low temporal resolution may be balanced with respect to the target application. However, the most important opportunity gained by extending the g-factor formalism to the domain of time-resolved parallel imaging lies in the possibility to evaluate and compare performances: performances between different acquisition strategies and method-dependent parameters like the reconstruction kernel, the amount of ACS data and sampling strategies, performances of time-resolved methods compared to standard nontime-resolved parallel imaging methods, and last but not least, performance comparison with other methodological approaches in time-resolved parallel imaging, such as k-t-SENSE (10) and k-t-PCA (27). In establishing an equivalent framework for both time-resolved methods in k-space and image space, not only intramethod but also intermethod comparisons become tangible. CONCLUSIONS The derived gxf -factor constitutes a general extension of the known g-factor formalism to the domain of timeresolved methods. The presented analysis facilitates quantification of the spatially nonuniform noise distribution in x-f-space and the nonuniform temporal average noise distribution in image space. Further, temporal frequency filter characteristics can be deduced from the presented framework and the noise reduction at the expense of temporal filtering is theoretically confirmed. Formulations were derived and demonstrated for k-tGRAPPA and PEAK-GRAPPA, but are generally applicable to any time-resolved GRAPPA-based parallel imaging reconstruction method. Therefore, the presented framework provides a fast and elegant tool for intermethod and intramethod comparisons of time-resolved and nontime-resolved methods. REFERENCES 1. Sodickson D, Manning W. Simultaneous acquisition of spatial harmonics (SMASH): fast imaging with radiofrequency coil arrays. Magn Reson Med 1997;38:591–603. 2. Pruessmann K, Weiger M, Scheidegger M, Boesinger P. SENSE: sensitivity encoding for fast MRI. Magn Reson Med 1999;42:952–962. 3. Robson PM, Grant AK, Madhuranthakam AJ, Lattanzi R, Sodickson DK, McKenzie CA. Comprehensive quantification of signal-to-noise ratio and g-factor for image-based and k-space-based parallel imaging reconstructions. Magn Reson Med 2008;60:895–907. 4. Kellman P, McVeigh ER. Image reconstruction in SNR units: a general method for SNR measurement. Magn Reson Med 2005;54:1439–1447.

11 5. Hansen MS, Inati SJ, Kellman P. Noise propagation in region of interest measurements. Magn Reson Med 2014. doi: 10.1002/mrm.25194. 6. Griswold M, Jakob P, Heidemann R, Nittka M, Jellus V, Wang J, Kiefer B, Haase A. Generalized autocalibrating partially parallel acquisition (GRAPPA). Magn Reson Med 2002;47:1202–1210. 7. Brau AC, Beatty PJ, Skare S, Bammer R. Comparison of reconstruction accuracy and efficiency among autocalibrating data-driven parallel imaging methods. Magn Reson Med 2008;59:382–395. 8. Breuer F, Kannengiesser S, Blaimer M, Seiberlich N, Jakob P, Griswold M. General formulation for quantitative g-factor calculation in GRAPPA reconstructions. Magn Reson Med 2009;63:1522–2594. 9. Yeh E, McKenzie C, Ohlinger M, Sodickson D. Parallel magnetic resonance imaging with adaptive radius in k-space (PARS): constrained image reconstruction using k-space locality in radiofrequency coil encoded data. Magn Reson Med 2005;53:1383–1392. 10. Tsao J, Boesinger P, Pruessmann K. k-t-BLAST and k-t-SENSE: dynamic MRI with high frame rate exploiting spatiotemporal correlations. Magn Reson Med 2003;50:1031–1042. 11. Huang F, Akao J, Vijayakumar S, Duensing G, Limkeman M. k-t GRAPPA: a k-space implementation for dynamic MRI with high reduction factor. Magn Reson Med 2005;54:1172–1184. 12. Jung B, Ullmann P, Honal M, Bauer S, Hennig J, Markl M. Parallel MRI with extended and averaged GRAPPA kernels (PEAK-GRAPPA): optimized spatiotemporal dynamic imaging. JMRI 2008;28:1226– 1232. 13. Kellman P, Epstein FH, Mcveigh ER. Adaptive sensitivity encoding incorporating temporal filtering (TSENSE). Magn Reson Med 2001;45: 846–852. 14. Breuer FA, Kellman P, Griswold MA, Jakob PM. Dynamic autocalibrated parallel imaging using temporal GRAPPA (TGRAPPA). Magn Reson Med 2005;53:981–985. 15. Jung B, Kozerke S. Comparison of kt-SENSE and kt-GRAPPA applied to cardiac cine and phase contrast imaging. In Proceedings of the 17th Annual Meeting of ISMRM, Honolulu, Hawaii, USA, 2009. p. 4559. 16. d’Arcy JA, Collins DJ, Rowland IJ, Padhani AR, Leach MO. Applications of sliding window reconstruction with Cartesian sampling for dynamic contrast enhanced MRI. NMR Biomed 2002;15:174–183. 17. Ramb R, Binter C, Breuer F, Schultz G, Zaitsev M, Kozerke S, Jung B. Quantitative g-factor calculation in k-t-GRAPPA reconstructions. In Proceedings of the 21st Annual Meeting of ISMRM, Salt Lake City, Utah, USA, 2013. p. 2651. 18. Ramb R, Binter C, Breuer F, Kozerke S, Jung B. A g-factor metric for k-t-GRAPPA reconstructions. Presented at ISMRM workshop on Data Sampling and Image Reconstruction, Sedona, Arizona, USA, 2013. 19. Buehrer M, Pruessmann KP, Boesiger P, Kozerke S. Array compression for MRI with large coil arrays. Magn Reson Med 2007;57:1131– 1139. 20. Chao TC, Chung HW, Hoge WS, Madore B. A 2D MTF approach to evaluate and guide dynamic imaging developments. Magn Reson Med 2010;63:407–418. 21. Madore B. UNFOLD-SENSE: a parallel MRI method with selfcalibration and artifact suppression. Magn Reson Med 2004;52:310– 320. 22. Oppenheim AV, Schafer RW. Digital signal processing. Englewood Cliffs, NJ: Prentice-Hall; 1975. 23. Rabiner LR, Gold B. Theory and application of digital signal processing. Englewood Cliffs, NJ: Prentice-Hall, 1975. 24. Murphy M, Alley M, Demmel J, Keutzer K, Vasanawala S, Lustig M. Fast-SPIRiT compressed sensing parallel imaging MRI: scalable parallel implementation and clinically feasible runtime. IEEE Trans Med Imaging 2012;31:1250–1262. 25. Breuer FA, Blaimer M, Mueller MF, Seiberlich N, Heidemann RM, Griswold MA, Jakob PM. Controlled aliasing in volumetric parallel imaging (2D CAIPIRINHA). Magn Reson Med 2006;55:549–556. 26. Schnell S, Markl M, Entezari P, Mahadewia RJ, Semaan E, Stankovic Z, Collins J, Carr J, Jung B. k-t GRAPPA accelerated four-dimensional flow MRI in the aorta: effect on scan time, image quality, and quantification of flow and wall shear stress. Magn Reson Med 2013. doi: 10.1002/mrm.24925. 27. Pedersen H, Kozerke S, Ringgaard S, Nehrke K, Kim WY. k-t PCA: temporally constrained k-t BLAST reconstruction using principal component analysis. Magn Reson Med 2009;62:706–716.