Real-Time Free-Breathing Cardiac Imaging with Self-Calibrated Through-Time Radial GRAPPA

Abstract

Purpose

Real-time free-breathing cardiac imaging with highly undersampled radial trajectories has previously been successfully demonstrated using calibrated radial generalized autocalibrating partially parallel acquisition (rGRAPPA). A self-calibrated approach for rGRAPPA is proposed that removes the need for the calibration prescan.

Methods

To investigate the effect of various parameters on image quality, a comprehensive imaging study on one normal swine was performed. Root mean squared errors (RMSEs) were computed with respect to gold standard acquisitions, and several acquisition/reconstruction strategies were compared. Additionally, the method was tested on 13 human subjects, and RMSEs relative to standard through-time radial GRAPPA were computed.

Results

Real-time images with high spatiotemporal resolution were obtained. Image quality was comparable to calibrated through-time rGRAPPA with endocardial and epicardial borders clearly delineated. In the swine, the average RMSE between self-calibrated and gold-standard calibrated images was 5.18±0.84%. In normal human subjects, the average RMSE between self-calibrated and calibrated through-time rGRAPPA was 3.79±0.64%. For lower accelerations rates (R = 6–8) image quality was similar to comparable calibrated scans though RMSE increased for higher degrees of undersampling (R = 12–16).

Conclusion

Highly accelerated real-time imaging with undersampled radial trajectories without additional calibration data is feasible. Image quality was acceptable for real-time cardiac MRI applications demanding high speed.

INTRODUCTION

Cardiac MRI is a strong clinical tool for the noninvasive assessment of cardiac function (1–3) and cardiomyopathy (4–7). Conventional segmented cine imaging is electrocardiogram gated, and the image quality depends on the regularity of cardiac motion. However, images can be nondiagnostic when cardiac arrhythmias are present or when patients cannot hold their breath as required. Although several studies have reported promising results with free-breathing cine imaging employing various methods to help avoid breath-holding—such as motion correction (8,9), respiratory navigation (10), and retrospective self-gating (11)—these methods have not been widely accepted in a clinical setting. Real-time cardiac imaging removes assumptions of periodicity from cardiac motion and obviates the need for breath-holding and is thus an active field of research as a potential alternative to conventional cine imaging (12,13).

To successfully capture wall motion abnormalities and left ventricular function with real-time cardiac MRI, each image should be acquired with sufficient contrast and spatiotemporal resolution to accurately delineate endocardial and epicardial borders (14). If using a standard balanced steady-state free precession sequence and targeting a temporal resolution of approximately 50 ms, fewer than 20 phase encoding lines or radial projections can be acquired given that a typical repetition time (TR) is approximately 2.5 ms. Therefore, parallel imaging techniques that enable image reconstruction from undersampled k-space by incorporating coil sensitivities in either the image domain (15) or k-space (16) are typically used for artifact-free cardiac MRI. Nevertheless, in two-dimensional imaging, image quality can suffer for acceleration factors >4 due to noise amplification inherent in parallel imaging (17). Other techniques that exploit the intrinsic spatiotemporal correlations (i.e., the k-t domain) in cardiac data have demonstrated higher acceleration factors (18,19) but are affected by temporal blurring. Spatiotemporal undersampling has been used in conjunction with the conventional parallel imaging methods as well (19–21), offering higher acceleration factors for real-time cardiac imaging, but it has been shown to be difficult to achieve 50 ms temporal resolution together with adequate spatial resolution using standard Cartesian k-space trajectories.

Non-Cartesian k-space trajectories, such as radial or spiral sampling, can be undersampled with less coherent aliasing artifacts on reconstructed images compared with Cartesian imaging (22–25). For example, up to three-fold undersampling for radial trajectories has been demonstrated to yield images of good quality without any post-processing (24–26). Additionally, non-Cartesian k-space coverage results in reduced motion artifact, which is important for robust cardiac imaging (27). Furthermore, non-Cartesian k-space trajectories permit parallel imaging to achieve high acceleration factors in the form of direct extensions of established parallel imaging techniques such as sensitivity encoding (SENSE) (28,29) or generalized autocalibrating partially parallel acquisition (GRAPPA) (30–34) beyond conventional Cartesian k-space imaging and other more specialized methods (35–39). Among these, the combination of radial sampling with the GRAPPA technique has gained interest due to improved image quality due to new calibration schemes (33,34), which enable robust imaging even at acceleration factors ≥8 (34,40). Furthermore, low-latency reconstructions with radial GRAPPA have been demonstrated (41–43), making true real-time imaging with real-time reconstruction possible.

As originally described, radial GRAPPA (rGRAPPA) requires at least one fully sampled calibration k-space acquired prior to accelerated imaging to compute the GRAPPA weights for every missing point in the undersampled k-space (33). Refinements of the technique use multiple fully sampled frames, which makes reconstruction more robust; however, the calibration time increases, extending scan time (34). Several algorithms for radial GRAPPA without calibration have been proposed (44–48). For example, computation of GRAPPA weights can be achieved by constructing synthetic fully sampled calibration data via multiplication with an image domain mask, which is generated either from coil sensitivity estimates from the densely sampled k-space center (44) or from a post-processing algorithm that attenuates streaking (45). Multiplication with an image domain mask is equivalent to a convolution of the undersampled k-space, which allows the estimation of a fully sampled calibration k-space. The reported image quality is good; however, residual streaking artifacts remain for acceleration factors >4. Another method computes the GRAPPA weights directly from the undersampled data via interpolation of the relative shifts along the azimuthal direction (47). Although this technique yields image quality comparable to that of radial GRAPPA with fully sampled calibration data, reconstructions from highly radially undersampled k-space (with- <50 projections) have not been reported. This is likely because the accuracy of the weight’s interpolation decreases with increasing distance between two adjacent projections, which could be problematic for real-time cardiac imaging where highly undersampled acquisitions are desired to maintain temporal resolution.

In this study, we propose an alternative self-calibration technique for radial GRAPPA to be used in highly accelerated imaging and suitable for true real-time implementation. The technique is validated in one swine and 13 human subjects, and reconstruction error relative to a gold standard is reported.

THEORY

Radial GRAPPA Calibration

To speed up acquisition with GRAPPA sampling, only every 1 out of R adjacent phase encoding lines or projections is acquired, and the missing R-1 phase encoding lines or projections are estimated from the acquired projections in the neighborhood. This estimation of missing spatial frequencies is made possible by exploiting the spatial variations of the coil sensitivities in multichannel receiver arrays that represent a convolution kernel applied during acquisition in the Fourier domain.

For GRAPPA reconstruction, the kernel weight sets that describe the contribution of the source points (acquired data) to the target point (missing data) need to be defined (16). By convention, a B_y×B_x kernel (or block) selects the B_x closest k-space points in the readout direction from the B_y lines closest to the target missing point as source points. The target point is estimated as a weighted combination of N_B = B_y×B_x source points, and this process is performed for every missing k-point.

With Cartesian sampling, the relative geometry of the source points with respect to the target point in k-space are identical throughout the entire undersampled k-space. Hence, the acquisition of a fully sampled k-space provides many occurrences of the kernels to be used in weight set calculation, which requires solving a single linear system (16). However, in azimuthally undersampled radial trajectories, the relative positions of the source points with respect to the target point vary over k-space, resulting in unique kernel geometry per missing point, and requiring solving as many separate linear systems for determination of per-point weight sets. Because the kernel geometries are unique, there exists only one kernel occurrence per each weight set to be calculated, even if an entire k-space is fully sampled. In the original formulation of radial GRAPPA (33), this problem was dealt with by dividing the k-space into small segments spanning a number of projections (N_seg,proj) and readout points (N_seg,read) the size of the segment being N_seg = N_seg,read×N_seg,proj. Each segment was treated as a Cartesian k-space with the assumption that kernel geometries for missing points were similar enough within the segment, and thus, a single set of weights was necessary for the estimation of all target points within the segment.

More recently, an alternative calibration involving a multiplicity of fully sampled k-spaces was proposed (34). With this through-time calibration, required information for determination of the kernel was obtained from multiple repetitions in time instead of occurrences over a segment. Calibration consisting of N_{frames_full} fully sampled frames provided multiple occurrences of the same exact kernel geometry for every target point. This approach drastically increased the accuracy of weight set calculation at the expense of increased calibration scan time.

By integrating both through-time calibration and through–k-space calibration to reduce N_{frames_full} (34), calibration duration was reduced, improving clinical applicability (40). The combination of smaller k-space segments (e.g., N_seg = 4×1), which better preserve kernel geometry assumptions, and through-time calibration allows weights to be computed from kernel occurrences that are more similar in geometry compared with those in the original through–k-space formulation, yielding more accurate reconstructions. This hybrid through-time/through–k-space calibration technique, referred to hereafter as through-time rGRAPPA (TT-rGRAPPA), permits artifact-free high quality real-time imaging from highly undersampled radial data (34).

Self-Calibrated Radial GRAPPA

In through–k-space calibration, fully sampled k-space segments are used to obtain kernel observations, and the geometries of individual kernels used to compute a distinct weight set are similar to one another but not identical. In this study, we propose that weights computation can be accomplished using kernels with approximate geometries, yet we restrict the search for such kernels to the undersampled k-spaces of the actual accelerated scans. As a first step in accomplishing this goal, we reduce the GRAPPA kernel to a one-sided kernel by choosing to include the source points from only the acquired projection that is closer to the missing point. In other words, the number of source points as N_B = 3×2 is reduced to N_B = 3×1, which is hereafter referred to as “half-block” TT-rGRAPPA (HB-TT-rGRAPPA). Then, for any missing point in the k-space, the search for kernels with similar geometries is performed by sliding the N_B = 3×1 source block point by point in the direction of the k-space center and checking whether the geometry is “similar” to original when this block is paired with a point, to serve as the target point, from another acquired projection, as shown in Figure 1c (red arrow). To be more specific, the measure of similarity between two kernels is the inverse difference of each kernel’s distance between the midpoint of the source block and the target point. After the undersampled k-space is searched in this manner, the most similar N_similar kernels are used to calibrate weights for the missing k-space point. Furthermore, the number of kernel occurrences is boosted via repetitions over time from multiple successive accelerated frames (N_{frames _acc}), similar to the TT-rGRAPPA calibration. Thus, the proposed rGRAPPA calibration method is called self-calibrated TT-rGRAPPA (SC-TT-rGRAPPA). Figure 1 depicts the acquisition of the required training data for calibration directly from the undersampled data in detail.

FIG. 1.

Depiction of the through–k-space portion of various rGRAPPA calibration methods. Dashed boxes contain GRAPPA kernels with source (blue) and target points (red). (a, b) Solid blue lines indicate segment size for calibration. (c) The acquisition of training data directly from the undersampled k-space for self-calibration is depicted as two kernels (with green target points) found via searching the k-space toward the center (red arrow) and used as kernel replicates to calibrate the weights for the actual kernel (with red target point). (d) The modified “alternating k-space” acquisition for an R = 4 accelerated scan. The azimuthal positions of the acquired projections are increased by (R/2) · Δk_ϕ (red arrows) every other frame, where Δk_ϕ is the azimuthal angle between consecutive projections in the fully sampled k-space. The metronome-like view ordering to avoid eddy current effects is also depicted (black arrows).

To obtain kernel occurrences with better geometric similarity, one last modification is applied to radial GRAPPA as a variation in the k-space undersampling scheme. Considering a fully sampled radial grid with N_ϕ projections, only N_ϕ,acc= N_ϕ/R projections are acquired per time frame in the accelerated scan. The density of sampling is doubled by alternating azimuthal positions of the acquired projections by (R/2) in the fully sampled grid for every other frame. In other words, the angles of the projections differ by (R/2) · (π/N_ϕ) in successive frames. Using this alternating k-space trajectory results in roughly twice as many k-space samples to be used for weights calibration in SC-TT-rGRAPPA compared with the conventional acquisition scheme. Thus, kernel occurrences with better similarity and/or simply more occurrences with equivalent similarity can be obtained. The sequence modification is depicted in Figure 1 and is referred to hereafter as “alternating k-space.”

METHODS

Data Acquisition

In vivo cardiac data were acquired under free breathing without electrocardiographic gating from one healthy swine and 13 healthy human subjects. Approval for all studies was obtained from our Institutional Review Board and Animal Care and Use Committee, and informed consent was obtained from all human subjects. A modified balanced steady-state free precession sequence with a metronome-like view ordering (34) to prevent eddy current–induced artifacts was used as illustrated in Figure 1. Images were reconstructed with two distinct image matrix sizes at 128×128 and 192×192. The matrix size was determined by the number of samples in the readout, and double oversampling was employed in all scans with 256 or 384 samples per readout. For scans resulting in both image matrix sizes, a variety of settings for N_ϕ (number of projections in a fully sampled calibration scan) and R (acceleration factor) was used in data acquisition. Specifically, 128×128 images were produced from scans that used N_ϕ = [128, 144], and 192×192 images were produced from scans that used N_ϕ = [192, 216, 224, 256]. Scans resulting in 128×128 and 192×192 images are referred to hereafter as standard resolution and high resolution, respectively. Field of view was held constant at 250 mm² for all experiments, yielding spatial resolutions of 1.95×1.95 mm² (standard resolution) and 1.30×1.30 mm² (high resolution).

Animal Studies

For accurate estimation of GRAPPA weights that represent solely the interrelation of the coils’ sensitivity profiles with one another, motion during TT-rGRAPPA calibration is necessary. Computing the weights from temporal replicas of the same image information inevitably copies information from this stationary image into the GRAPPA weights, biasing reconstruction toward the calibration image, hence no stationary phantom studies were performed. Instead, a comprehensive in vivo imaging study was performed on one healthy swine on a 1.5T scanner (Avanto, Siemens Medical Solutions, Erlangen, Germany). Standard chest (2×3 = 6) and spine (3×3 = 9) arrays yielding a total of N_C = 15 receiver coils were used. Other scan parameters were as follows: slice thickness = 7.0 mm, flip angle = 50°, standard resolution/high resolution TR = 2.8/3.2 ms, and echo time (TE) = TR/2. The effect of various acquisition and reconstruction parameters on the performance of the proposed technique was measured. Gold standard reference images were reconstructed with N_{frames_full} = 400 fully sampled calibration frames using N_seg = 1×1 (i.e., no through–k-space calibration) as described by Seiberlich et al. (34). Three short-axis slices and one horizontal long-axis slice were imaged. Defining R_cart = N_PE/N_ϕ,acc as the acceleration rate with regard to a fully sampled Cartesian scan, the following combinations of [N_ϕ; R; R_cart] were used in scanning: standard resolution, [N_ϕ = 128; R = 8; R_cart = 8], [N_ϕ = 144; R = 6; R_cart = 5.3], [N_ϕ = 144; R = 9; R_cart = 8], and [N_ϕ = 144; R = 12; R_cart = 10.7]; and high resolution, [N_ϕ = 192; R = 8; R_cart = 8] and [N_ϕ = 192; R = 12; R_cart = 12]. All accelerated scans were performed using both the conventional k-space and the alternating k-space (Fig. 1) trajectories.

To determine the effects of the self-calibration parameters, several different reconstructions were performed using a variety of N_similar and N_{frames_acc} values for each scan. Specifically, N_similar values were set to [1,2,4,6,8] and N_{frames_acc} was varied between 6 and 120. The evaluation criteria used were chosen to be the normalized root mean squared error (RMSE) with respect to the reference TT-rGRAPPA reconstructions (N_seg = 1×1, N_{frames_full} = 400). TT-rGRAPPA and HB-TT-rGRAPPA reconstructions with N_seg = 4×1 (N_{frames_full} = 80) and N_seg = 16×4 (N_{frames_full} = 6) were also included in the comparison of the images reconstructed by SC-TT-rGRAPPA. The RMSE values were then used to choose the optimal parameters for SC-TT-rGRAPPA.

Normal Human Subject Studies

The human subjects were divided into two sets. For the first set of subjects (n = 8), short-axis cardiac images were acquired on a 1.5T scanner (Avanto, Siemens Medical Solutions, Erlangen, Germany) using the following parameters: N_C = 15 (same coil configuration as in the animal studies), slice thickness = 8.0 mm, flip angle = 45°, standard resolution/high resolution TR = 2.8/3.2 ms, and TE = TR/2. Imaging was performed using the same set of [N_ϕ; R; R_cart] combinations as in the swine study except for [N_ϕ = 128; R = 8; R_cart = 8]. Only the alternating k-space trajectory was employed. For each subject, a stack of 12 adjacent slices were scanned in a sequential manner where each slice was imaged between 5 and 10 seconds to insure coverage through one complete respiratory cycle. In addition, for one midventricular slice, calibration data with 400 fully sampled frames as in the swine study were acquired to enable gold standard TT-rGRAPPA reconstructions and thus computation of RMSEs in the SC-TT-rGRAPPA reconstructions.

The second set of normal subjects (n = 5) was imaged on a 1.5T scanner (Aera, Siemens Medical Solutions, Erlangen, Germany) using combinations of standard chest and spine receiver arrays yielding a larger number of total coils (N_C = 20–30), which permitted imaging at higher acceleration rates and resolution. Other scan parameters were as follows: slice thickness = 6.0 mm, flip angle = 70°, standard resolution/high resolution TR = 2.7/3.1 ms, and TE = TR/2. In addition to the [N_ϕ; R; R_cart] combinations used in the first set of human studies, combinations of [N_ϕ = 216; R = 12; R_cart = 10.7], [N_ϕ = 224; R = 14; R_cart = 12], [N_ϕ = 256; R = 16; R_cart = 12], and [N_ϕ = 192; R = 16; R_cart = 16] were also used for the high resolution scans with a 192×192 final image matrix. For each subject, one short-axis slice and one long-axis slice were imaged using the alternating k-space trajectory.

Image Reconstruction

All image reconstruction was performed in MATLAB (MathWorks, Natick, Massachusetts, USA) and used non-uniform fast Fourier transform (49,50) for gridding. For TT-rGRAPPA and HB-TT-rGRAPPA, GRAPPA weights were computed via the singular value decomposition (SVD)-based pseudo-inverse (i.e. pinv()), whereas Tikhonov regularization with a constant Tikhonov λ per image depending on image resolution (1e-06≤λ≤6e-06]) was used in the calibration of the weights for SC-TT-rGRAPPA. The addition of regularization suppressed noise in the SC-TT-rGRAPPA calibration as further explained in the discussion, and helped improve image quality. SC-TT-rGRAPPA calibrations using N_{frames_acc} = [10, 20, 120] for [N_ϕ = 144; R = 9] and [N_ϕ = 192; R = 8] scans took [130, 148, 238] and [244, 263, 440] seconds on average on our dual-socket six-core Intel Xeon E5-2620 v2 at 2.10 GHz system, respectively. Furthermore, due to the alternating view ordering, residual image artifacts changed every other frame, which presented itself as a high-frequency flicker. Hence, a low-pass temporal filter the following parameters was applied: passband frequency = 0.90 ϖ, stopband frequency = 0.98 ϖ, allowed passband ripple = 1 dB, and stopband attenuation = 60 dB.

RESULTS

Figure 2 shows example images from the swine heart at both end-systole and end-diastole acquired with the alternating k-space trajectory. Images reconstructed from all three methods using fully sampled calibration data look very similar to one another. “Half-block” through-time radial GRAPPA reconstructions are almost identical to those of the standard TT-rGRAPPA, with less apparent grainy noise. This could be attributed to the fact that the linear system in the calibration for HB-TT-rGRAPPA weights is twice as overdetermined as the one in TT-rGRAPPA. Using a larger segment size (i.e., N_seg = 16×4) also does not result in a major difference except for some slight blurring in smaller structures such as the papillary muscles. In the right-most column, images reconstructed with the proposed SC-TT-rGRAPPA method are shown. N_{frames_acc} = 100 was used, which corresponds to 4.4 and 7.7 seconds of real-time imaging for the standard and high resolution scans, respectively. The images reconstructed with the self-calibrated method are comparable to images from the other reconstructions. There is visible contrast-to-noise ratio degradation and some loss of structures near the periphery, but the heart is well visualized with good delineation of the myocardium.

FIG. 2.

(a) Example images from a normal swine reconstructed from nine-fold undersampled data (N_ϕ = 144, N_ϕ,acc = 16) with a spatial resolution of 1.95×1.95 mm² and a temporal resolution of 44.3 ms/frame. For SC-TT-rGRAPPA, a total of 100 undersampled frames (~4.4 s) and N_similar = 6 were used in calibration. (b) The same images with eight-fold undersampling (N_ϕ = 192, N_ϕ,acc = 24) with a spatial resolution of 1.30×1.30 mm² and a temporal resolution of 77.5 ms/frame. An alternating trajectory was used in both scans.

Figure 3 shows plots of the RMSE values in the SC-TT-rGRAPPA reconstructions for each scan of the central short-axis slice acquired with the alternating trajectory from the swine experiment when different self-calibration parameters are used. As expected, RMSE decreases monotonically with the number of undersampled frames used in the weights calibration except for N_similar = 8. Similarly, RMSE decreases as N_similar increases with the exception at N_similar = 8. For small values of N_{frames _acc} (<30), reconstructions with eight similar kernels may outperform those with smaller N_similar values, because the weights calibration system will be only slightly overdetermined with fewer number of kernels; however, N_similar = 8 does not yield the lowest RMSE for higher values of N_{frames _acc}. The unexpected behavior of the RMSE versus N_{frames _acc} when N_similar = 8 is used may suggest that kernel geometries are not sufficiently similar to the original kernel geometry beyond the first six similar kernels that are obtained from the undersampled k-space. With an abundance of the frames used in calibration, N_similar = 6 yields the optimal reconstructions with the lowest RMSE except for N_ϕ/R = 144/6, where N_similar values of 2, 4, and 6 produce indistinguishable results. Hence, N_similar = 6 is chosen as the optimal value for the SC-TT-rGRAPPA reconstructions in the results presented in the remainder of this manuscript.

FIG. 3.

RMSE values of SC-TT-rGRAPPA reconstructions for six different accelerated scans, all of which were acquired with the alternating k-space sequence from a swine short-axis slice. RMSEs are plotted with respect to the number of accelerated frames used in calibration (N_{frames_acc}) with different colors of lines corresponding to various N_similar values. The top horizontal axes indicate the lengths of the accelerated scans used in weights calibration in seconds.

Table 1 displays RMSEs of SC-TT-rGRAPPA reconstructions with N_similar = 6 and N_{frames_acc} = 120 as well as RMSEs of the standard TT-rGRAPPA and HB-TT-rGRAPPA reconstructions for scans in the swine experiment. RMSEs are averaged across slices with the standard deviations averaged in percentage relative their corresponding means prior to averaging. Switching from the regular to the alternating k-space trajectory, there is a noticeable decrease in RMSE for the self-calibrated reconstructions. RMSEs of SC-TT-rGRAPPA reconstructions increase further as the acceleration rate increases as expected. Moreover, RMSEs for N_ϕ = 192 are higher than those for the standard base resolution for all reconstructions. Table 1 also illustrates the feasibility of the proposed half-block rGRAPPA method.

Table 1.

Comparison of RMSEs Obtained After Reconstructing for Diverse Resolution and Acceleration Factors Tested on Normal Swine

Nϕ	R	Standard TT-rGRAPPA		HB-TT-rGRAPPA	SC-TT-rGRAPPA	Pulse Sequence
		Nseg = 4×1	Nseg = 16×4	Nseg = 4×1	Nsimilar = 6
		NB = 3×2	NB = 3×2	NB = 3×1	NB = 3×1
		Nframes_full = 80 (μ±σ, %)	Nframes_full = 6 (μ±σ, %)	Nframes_full = 80 (μ±σ, %)	Nframes_acc = 120 (μ±σ, %)
128 (1.95×1.95 mm2)	8	1.30±0.11	2.19±0.23	1.68±0.24	5.70±0.70	Reg
		1.23±0.29	2.02±0.36	1.52±0.64	5.19±0.92	Alt
144 (1.95×1.95 mm2)	6	1.01±0.27	1.65±0.34	1.21±0.48	7.16±1.15	Reg
		0.94±0.26	1.50±0.29	1.07±0.25	4.55±0.84	Alt
	9	1.24±0.24	2.13±0.34	1.52±0.46	8.69±1.25	Reg
		1.21±0.31	2.02±0.37	1.39±0.31	5.06±0.89	Alt
	12	1.55±0.11	2.68±0.23	1.93±0.32	9.73±0.96	Reg
		1.43±0.29	2.44±0.35	1.70±0.30	5.38±0.90	Alt
192 (1.3×1.3 mm2)	8	1.66±0.13	2.15±0.17	1.68±0.32	7.02±0.73	Reg
		1.68±0.21	2.14±0.23	1.65±0.20	5.25±0.68	Alt
	12	2.11±0.18	2.85±0.22	2.23±0.36	9.02±0.90	Reg
		2.04±0.34	2.72±0.37	2.06±0.33	5.65±0.81	Alt

Abbreviations: Alt, alternating k-space radial trajectory sequence; HB-TT-rGRAPPA, half-block through-time radial GRAPPA; Reg, regular radial acquisition; SC-TT-rGRAPPA, self-calibrated through-time radial GRAPPA; TT-rGRAPPA, through-time radial GRAPPA.

Data are presented as the mean±standard deviation and are relative to gold standard rGRAPPA calibrated with N_{frames_full} = 400 and N_seg = 1×1.

Figure 4 shows short-axis images from a healthy subject in both end-systole and end-diastole for all of the accelerated scans acquired with the alternating trajectory. Images reconstructed with SC-TT-rGRAPPA are qualitatively comparable to those reconstructed with TT-rGRAPPA, except for a subtle loss in blood-to-myocardium contrast that is more visible at R= 12. Figures 5 and 6 show human images of 10 short-axis slices from base to apex at end-diastole and end-systole for N_ϕ/R = 144/9 and N_ϕ/R = 192/12, respectively. The image quality of SC-TT-rGRAPPA reconstructions is consistent across slices, with the loss in contrast-to-noise ratio more apparent in the systolic frames. Figure 7 displays SC-TT-rGRAPPA reconstructions from human data acquired with N_ϕ/R = 144/9 and N_ϕ/R = 192/12 where every other frame from a complete cardiac cycle is shown for each. While the N_ϕ/R = 192/12 reconstruction appears to have noticeably degraded image quality compared with N_ϕ/R = 144/9, the increase in RMSE is not excessive. Figure 8 shows end-diastolic images from a healthy human subject imaged with a larger array of receiver coils and under greater acceleration factors (R≥12). The use of larger receiver arrays permits acceleration factors as high as R = 16 for the standard TT-rGRAPPA technique. Although there is a visible loss of signal-to-noise ratio and slight residual streaking, SC-TT-rGRAPPA reconstructions at these acceleration rates are still feasible.

FIG. 4.

Example images from a human short-axis slice reconstructed with both standard TT-rGRAPPA and self-calibrated TT-rGRAPPA (SC-TT-rGRAPPA) are displayed. Images from data with various undersampling factors are shown. For SC-TT-rGRAPPA, 180 total undersampled frames and N_similar = 6 were used in weights calibration. Δt denotes the temporal footprint of the images. Alternating k-space was employed in all scans.

FIG. 5.

Example end-diastolic and end-systolic images of 10 adjacent short-axis slices (basal to apical) acquired from a human subject. Multiple slices were scanned sequentially with the alternating k-space trajectory in accelerated fashion where data was nine-fold undersampled (N_ϕ = 144, N_ϕ,acc = 16) yielding a spatial resolution of 1.95×1.95 mm² and a temporal resolution of 44.3 ms/frame. Images were reconstructed with both standard TT-rGRAPPA and SC-TT-rGRAPPA.

FIG. 6.

Example end-diastolic and end-systolic images of 10 adjacent short-axis slices (basal to apical) acquired from a human subject. Multiple slices were scanned sequentially with the alternating k-space trajectory in accelerated fashion where data were 12-fold undersampled (N_ϕ = 192, N_ϕ,acc = 16), yielding a spatial resolution of 1.30×1.30 mm² and a temporal resolution of 51.7 ms/frame. Images were reconstructed with both standard TT-rGRAPPA and SC-TT-rGRAPPA.

FIG. 7.

Example images from a human short-axis slice reconstructed with the proposed SC-TT-rGRAPPA method using N_similar = 6 and N_{frames_acc} = 180. Image series where every other frame from successive frames over one cardiac cycle are displayed. RMSEs of images reconstructed with SC-TT-rGRAPPA are computed with regard to the gold standard TT-rGRAPPA reconstructions with N_seg = 1×1 and N_{frames_full} = 400. Alternating k-space was employed in both scans.

FIG. 8.

Example end-systolic and end-diastolic images from a short-axis slice of a healthy human subject reconstructed with both standard TT-rGRAPPA and SC-TT-rGRAPPA are displayed. Images from accelerated scans with various undersampling factors (R≥12) are shown. Δt denotes the temporal footprint of the images. The alternating k-space trajectory was employed in all scans.

A series of videos (Supporting Videos S1–S15) comparing SC-TT-rGRAPPA and standard TT-rGRAPPA at diverse acquisition resolutions is available in the Supporting Information online.

DISCUSSION

The images reconstructed with the proposed self-calibrated radial GRAPPA method presented in this study demonstrate the feasibility of real-time free breathing cardiac imaging without the need for separate training data. In the self-calibrated reconstructions from the animal scans, the mean RMSE averaged over four slices had a maximum of 5.60% with respect to the gold standard TT-rGRAPPA reconstructions. The images reconstructed with the proposed self-calibrated method do exhibit discrepancies when compared visually with their TT-rGRAPPA counterparts, yet image quality is preserved for most cardiac MRI applications. The difference in reconstruction quality is more distinguishable in the short-axis slice from the swine study. This may be due to the greater dynamic range of signal intensity and/or the effect of the coil configuration in that orientation. Figure 4 demonstrates the successful visualization of the myocardium with SC-TT-rGRAPPA for various acceleration rates. At R = 12, blood–myocardium contrast is decreased yet should still be adequate for delineation of the endocardial border. The papillary muscles in the left ventricle are also reconstructed with good definition.

The data in Figure 3 suggest that N_similar = 6 is the optimal number of similar kernels for the extent of this study, and with a N_B = 3×1 kernel, 100 accelerated frames are sufficient for accurate weight estimation in SC-TT-rGRAPPA. For a temporal footprint per image of approximately 44 ms, this corresponds to <5 s per slice. For higher resolution imaging, a maximum of 8 s per slice is needed. Assuming that the left ventricle can be covered with 12 slices, a function study could be completed in 1 to 2 minutes depending on the desired resolution and without any gating or breath-holding. Because real-time scans usually span at least one complete respiratory cycle to ensure acquisition of an end-expiration beat, this is close to the minimum scan duration. Further studies are needed to determine the clinical use of this technique; nevertheless, this is an exciting scenario that could lead to the widespread use of SC-TT-rGRAPPA in situations in which breath-holding is not feasible. Figures 5 and 6 illustrate the feasibility of such time-efficient whole heart imaging with multiple short-axis slices from base to apex reconstructed with both TT-rGRAPPA and SC-TT-rGRAPPA. SC-TT-rGRAPPA-reconstructed images appear noisier compared with the TT-rGRAPPA reconstructions; however, image quality is consistent across all slices.

Based on the RMSEs shown in Table 1, it can be argued that SC-TT-rGRAPPA is suboptimal to TT-rGRAPPA, even when few fully sampled calibration frames are used with a larger segment size (i.e., N_seg = 16×4, N_{frames_full} = 6), hence it may be of lesser value and clinical use compared with TT-rGRAPPA. However, it is important to bear in mind that there is no real ground truth when computing the RMSEs, because we are restricted to a subjective gold standard in the form of TT-rGRAPPA reconstructions using N_seg = 1×1 and N_{frames_full} = 400. The RMSE-based comparison between SC-TT-rGRAPPA and the standard TT-rGRAPPA is therefore biased toward TT-rGRAPPA (with N_seg = 16×4 and N_{frames_full} = 6), because the RMSEs for TT-rGRAPPA is computed with respect to a reconstruction of the same type with the only discrepancy being the larger segment size. Moreover, even though TT-rGRAPPA with N_seg = 16×4 may require a calibration scan of only 2–4 seconds, it is still considerably disruptive given that it needs to be repeated for any change in slice position or orientation during imaging.

The results presented in this study suggest that the acceleration rate (R) is a major factor that hinders the performance of SC-TT-rGRAPPA as in any other parallel imaging technique. Table 1 shows the systematic increase in RMSE as R increases for both standard and high-resolution scans. This is expected, because as R increases, the undersampled k-space from which kernels with similar geometry are obtained gets sparser, thus weakening the assumption that the geometries of the kernels from which weights are calculated are similar enough to the ideal geometry. Considering the reconstructions from scans using the alternating trajectory, the steepest increase in RMSE is at the jump from R = 6 to R = 9, which is consistent with our observations that SC-TT-rGRAPPA on six-fold undersampled data yield images not easily distinguishable from their TT-rGRAPPA counterparts, whereas the images reconstructed by the two methods appear visibly different for acceleration rates of R≥8. Also, even though the increase in RMSE from R = 9 to R = 12 may not seem significant with a mean of 0.32 %, the reference images for R = 12 are already inferior to those of R = 9 in overall image quality, thus the self-calibrated reconstructions at higher accelerations rates may appear with poorer quality than expected.

Due to the alternating view ordering, residual image artifacts changed every other frame, which presented itself as a high-frequency flicker. A low-pass filter successfully removed these artifacts. Although this may have led to a loss of very high frequency cardiac dynamics, its effect on diagnostic quality of imaging was minimal (14). For example, an accelerated scan with image sampling rate of 23 Hz (44 ms/frame) can capture about 10 frequency harmonics on a subject with heart rate of 70 bpm, and the low-pass filter used here would reduce that to nine harmonics. Setser et al. (14) showed that decreasing the captured harmonics from 10 to eight only yields less than 2% and 3% deviation in the computations of left ventricular volume and ejection fraction, respectively. The temporal filter used in this study would lead to smaller deviation in these metrics. The design of a low-pass filter with a higher passband to allow more frequency harmonics can also be considered.

The source and target data points used in any GRAPPA calibration process are assumed to represent samples of k-space of the same object. For standard TT-rGRAPPA, these data are acquired within a narrow temporal window spanning less than one undersampled frame. However, the data used for calibration in SC-TT-rGRAPPA has a broader temporal window, because two contiguous undersampled frames are combined into a single k-space when alternating trajectory is used. The source-target pairing includes points that are acquired more than one image apart, permitting more motion with possible deterioration of the estimation process. More specifically, in TT-rGRAPPA, the sampling times of the source points and the target point within the same kernel vary roughly between TR and (R − 1) · TR, where TR is the time it takes to acquire one projection. On the other hand, this time difference can be as great as 2 · N_ϕ,acc · TR for SC-TT-rGRAPPA calibration. This discrepancy may be partly responsible for the increased RMSEs observed between the two techniques.

In standard TT-rGRAPPA, GRAPPA weights are computed using MATLAB’s implementation of the SVD-based pseudo-inverse [i.e., pinv()]. When GRAPPA weights were computed similarly using pinv() for SC-TT-rGRAPPA, the reconstructed images appeared noisier than images reconstructed with the standard approach. Investigation of the condition numbers of each reconstruction, an indicator for inversion stability and noise amplification, revealed that the condition numbers for the SC-TT-rGRAPPA approach were indeed higher than for standard TT-rGRAPPA, and markedly so for higher acceleration rates. To ameliorate these effects, which were more apparent at higher acceleration rates, Tikhonov regularization was incorporated into the reconstruction at the expense of contrast and blurring. However, the determination of the factors leading to noise enhancement as well as optimizing the choice of Tikhonov λ are subjects of further research. Also, due to the radially decreasing signal-to-noise ratio, it may prove to be beneficial to parametrize λ as a function of radial position to achieve a more optimal regularization.

Having a higher number of coils for reconstruction increased the maximum achievable acceleration rate, further improving temporal resolution as observed in the two subsets of normal subjects. However, the addition of coils increases the number of weights to be determined, and can increase the number of undersampled frames required to achieve a sufficiently overdetermined system for accurate estimation of weights. Nevertheless, to achieve aggressive acceleration rates R≥12, more coils were advantageous and necessary.

Aside from a novel self-calibration method for radial GRAPPA, this study also introduces an alternative for the standard through-time radial GRAPPA—namely, HB-TT-rGRAPPA as depicted in Figure 1. HB-TT-rGRAPPA uses only one acquired projection to reconstruct the nearest missing projections in its neighborhood, whereas rGRAPPA and TT-rGRAPPA use the acquired projections on either side. Note that as the undersampling factor grows, the angular spread between acquired projections in the accelerated scan increases. Thus, when standard radial GRAPPA is employed, missing projections are primarily closer to one of the two acquired projections, the source points on the further projection are relatively far from the target points and may not yield significant contribution in the estimation process. Therefore, the discrepancies between the images reconstructed with standard radial GRAPPA and half-block radial GRAPPA are small with only 0.19% increase in RMSE, on average (Table 1). Because the size of the weights set is reduced in half, HB-TT-GRAPPA is less prone to overfitting when the least-squares problem is not sufficiently overdetermined. The feasibility of half-block radial GRAPPA is demonstrated by comparison with standard radial GRAPPA in Figure 2. In addition, HB-TT-rGRAPPA and standard TT-rGRAPPA exhibit indistinguishable RMSE values and images at higher resolutions. Hence, for these scans, the half-block approach could be used to reduce the precalibration scan duration if necessary. HB-TT-rGRAPPA may also benefit from larger kernels (e.g. N_B = 5×1) to increase reconstruction accuracy, though that also remains a subject for future work.

The proposed self-calibrated radial GRAPPA method has the potential to be translated into the clinical setting with a real-time reconstruction implementation as similarly demonstrated for the standard TT-rGRAPPA (41). Because the weights calibration and image reconstruction can be decoupled and made asynchronous in a parallelized framework (42), real-time display of images can begin rapidly. Figure 3 shows that with N_similar = 6 and 10 undersampled frames (~330–660 ms for standard resolution), images with decent quality can be reconstructed, and with more time to obtain approximately 100–120 undersampled frames, well within the duration of one respiratory cycle, for the more accurate weights calibration, image quality would progressively improve. As a final consideration, the proposed approach for calibration can be directly extended to the three-dimensional TT-rGRAPPA technique (51,52), which could make three-dimensional real-time procedure guidance feasible.

CONCLUSION

The self-calibrated TT-rGRAPPA technique presented in this study demonstrates the feasibility of image reconstructions from aggressively undersampled radial data in the specific application of free breathing nongated cardiac imaging without the need for additional calibration data. Employing undersampling factors ranging from 6 to 16, real-time cardiac images with frame rates as high as 30 (33 ms/image) and 26 (38 ms/image)—at spatial resolutions of 1.95 mm² and 1.30 mm², respectively—are demonstrated without using additional scan time for calibration. The proposed self-calibrated radial GRAPPA method may be applicable in other real-time imaging scenarios wherein a separate calibration scan is not an option.

Supplementary Material

S1. Supporting Video S1.

Real-time images of a human subject’s short-axis slice reconstructed with standard TT-rGRAPPA (left) and self-calibrated TT-rGRAPPA (right). Six-fold undersampling (N_ϕ= 144, N_ϕ,acc= 24) yielded a temporal footprint of 66 ms/frame, and a spatial resolution of 1.95×1.95 mm² was used in data acquisition.

S10. Supporting Video S10.

Real-time images of a human subject’s short-axis slice reconstructed with standard TT-rGRAPPA (left) and self-calibrated TT-rGRAPPA (right). Sixteen-fold undersampling (N_ϕ= 256, N_ϕ,acc= 16) yielded a temporal footprint of 50 ms/frame, and a spatial resolution of 1.30×1.30 mm² was used in data acquisition.

S11. Supporting Video S11.

Real-time images of a normal swine’s short-axis and long-axis slices reconstructed from six-fold undersampled (N_ϕ= 144, N_ϕ,acc= 24) radial data with a temporal footprint of 66 ms/frame and a spatial resolution of 1.95×1.95 mm². Image frames reconstructed with four different methods are shown in four columns in the following order from left to right: standard TT-rGRAPPA using N_seg= 4×1, standard TT-rGRAPPA using N_seg= 16×4, half-block TT-rGRAPPA using N_seg= 4×1, and self-calibrated TT-rGRAPPA using N_{frames_acc}= 100 and N_similar= 6. The second-from-top and the bottom-most rows show absolute difference images between images reconstructed with each of the four methods and the reference standard TT-rGRAPPA reconstruction using N_seg= 1×1 of the short-axis and long-axis slices, respectively. The intensity of the absolute difference images were multiplied by 2 in an effort to present the discrepancies more clearly.

S12. Supporting Video S12.

Real-time images of a normal swine’s short-axis and long-axis slices reconstructed from nine-fold undersampled (N_ϕ= 144, N_ϕ,acc= 16) radial data with a temporal footprint of 44 ms/frame and a spatial resolution of 1.95×1.95 mm². Image frames reconstructed with four different methods are shown in four columns in the following order from left to right: standard TT-rGRAPPA using N_seg= 4×1, standard TT-rGRAPPA using N_seg= 16×4, half-block TT-rGRAPPA using N_seg= 4×1 and self-calibrated TT-rGRAPPA using N_{frames_acc}= 100 and N_similar= 6. The second-from-top and the bottom-most rows show absolute difference images between images reconstructed with each of the four methods and the reference standard TT-rGRAPPA reconstruction using N_seg= 1×1 of the short-axis and long-axis slices, respectively. The intensity of the absolute difference images were multiplied by 2 in an effort to present the discrepancies more clearly.

S13. Supporting Video S13.

Real-time images of a normal swine’s short-axis and long-axis slices reconstructed from 12-fold undersampled (N_ϕ= 144, N_ϕ,acc= 12) radial data with a temporal footprint of 33 ms/frame and a spatial resolution of 1.95×1.95 mm². Image frames reconstructed with four different methods are shown in four columns in the following order from left to right: standard TT-rGRAPPA using N_seg= 4×1, standard TT-rGRAPPA using N_seg= 16×4, half-block TT-rGRAPPA using N_seg= 4×1 and self-calibrated TT-rGRAPPA using N_{frames_acc}= 100 and N_similar= 6. The second-from-top and the bottom-most rows show absolute difference images between images reconstructed with each of the four methods and the reference standard TT-rGRAPPA reconstruction using N_seg= 1×1 of the short-axis and long-axis slices, respectively. The intensity of the absolute difference images were multiplied by 2 in an effort to present the discrepancies more clearly.

S14. Supporting Video S14.

Real-time images of a normal swine’s short-axis and long-axis slices reconstructed from eight-fold undersampled (N_ϕ= 192, N_ϕ,acc= 24) radial data with a temporal footprint of 77 ms/frame and a spatial resolution of 1.95×1.95 mm². Image frames reconstructed with four different methods are shown in four columns in the following order from left to right: standard TT-rGRAPPA using N_seg= 4×1, standard TT-rGRAPPA using N_seg= 16×4, half-block TT-rGRAPPA using N_seg= 4×1 and self-calibrated TT-rGRAPPA using N_{frames_acc}= 100 and N_similar= 6. The second-from-top and the bottom-most rows show absolute difference images between images reconstructed with each of the four methods and the reference standard TT-rGRAPPA reconstruction using N_seg= 1×1 of the short-axis and long-axis slices, respectively. The intensity of the absolute difference images were multiplied by 2 in an effort to present the discrepancies more clearly.

S15. Supporting Video S15.

Real-time images of a normal swine’s short-axis and long-axis slices reconstructed from 12-fold undersampled (N_ϕ= 192, N_ϕ,acc= 16) radial data with a temporal footprint of 52 ms/frame and a spatial resolution of 1.95×1.95 mm². Image frames reconstructed with each of the four different methods are shown in four columns in the following order from left to right: standard TT-rGRAPPA using N_seg= 4×1, standard TT-rGRAPPA using N_seg= 16×4, HB-TT-rGRAPPA using N_seg= 4×1 and self-calibrated TT-rGRAPPA using N_{frames_acc}= 100 and N_similar= 6. The second-from-top and the bottom-most rows show absolute difference images between images reconstructed with each of the four methods and the reference standard TT-rGRAPPA reconstruction using N_seg= 1×1 of the short-axis and long-axis slices, respectively. The intensity of the absolute difference images were multiplied by 2 in an effort to present the discrepancies more clearly.

S2. Supporting Video S2.

Real-time images of a human subject’s short-axis slice reconstructed with standard TT-rGRAPPA (left) and self-calibrated TT-rGRAPPA (right). Nine-fold undersampling (N_ϕ= 144, N_ϕ,acc= 16) yielded a temporal footprint of 44 ms/frame, and a spatial resolution of 1.95×1.95 mm² was used in data acquisition.

S3. Supporting Video S3.

Real-time images of a human subject’s short-axis slice reconstructed with standard TT-rGRAPPA (left) and self-calibrated TT-rGRAPPA (right). Twelve-fold undersampling (N_ϕ= 144, N_ϕ,acc= 12) yielded a temporal footprint of 33 ms/frame, and a spatial resolution of 1.95×1.95 mm² was used in data acquisition.

S4. Supporting Video S4.

Real-time images of a human subject’s short-axis slice reconstructed with standard TT-rGRAPPA (left) and self-calibrated TT-rGRAPPA (right). Eight-fold undersampling (N_ϕ= 192, N_ϕ,acc= 24) yielded a temporal footprint of 77 ms/frame, and a spatial resolution of 1.30×1.30 mm² was used in data acquisition.

S5. Supporting Video S5.

Real-time images of a human subject’s short-axis slice reconstructed with standard TT-rGRAPPA (left) and self-calibrated TT-rGRAPPA (right). Twelve-fold undersampling (N_ϕ= 192, N_ϕ,acc= 16) yielded a temporal footprint of 52 ms/frame, and a spatial resolution of 1.30×1.30 mm² was used in data acquisition.

S6. Supporting Video S6.

Real-time images of a human subject’s whole heart (12 adjacent short-axis slices) reconstructed with standard TT-rGRAPPA (left) and self-calibrated TT-rGRAPPA (right). Nine-fold undersampling (N_ϕ= 144, N_ϕ,acc= 16) yielded a temporal footprint of 44 ms/frame, and a spatial resolution of 1.95×1.95 mm² was used in data acquisition.

S7. Supporting Video S7.

Real-time images of a human subject’s short-axis slice reconstructed with standard TT-rGRAPPA (left) and self-calibrated TT-rGRAPPA (right). Twelve-fold undersampling (N_ϕ= 216, N_ϕ,acc= 18) yielded a temporal footprint of 57 ms/frame, and a spatial resolution of 1.30×1.30 mm² was used in data acquisition.

S8. Supporting Video S8.

Real-time images of a human subject’s short-axis slice reconstructed with standard TT-rGRAPPA (left) and self-calibrated TT-rGRAPPA (right). Fourteen-fold undersampling (N_ϕ= 224, N_ϕ,acc= 16) yielded a temporal footprint of 50 ms/frame, and a spatial resolution of 1.30×1.30 mm² was used in data acquisition.

S9. Supporting Video S9.

Real-time images of a human subject’s short-axis slice reconstructed with standard TT-rGRAPPA (left) and self-calibrated TT-rGRAPPA (right). Sixteen-fold undersampling (N_ϕ= 192, N_ϕ,acc= 12) yielded a temporal footprint of 38 ms/frame, and a spatial resolution of 1.30×1.30 mm² was used in data acquisition.