a-f BLAST: Non-Iterative Radial k-t BLAST Reconstruction for Real-Time Imaging

Abstract

Non-Cartesian trajectories are advantageous in the collection and reconstruction of dynamic MR images. Like many other fast imaging methods, the reconstruction technique k-t BLAST has been extended to non-Cartesian k-t BLAST through a Conjugate Gradient based approach. However, it is necessary to transform and grid the data to a Cartesian domain as part of the reconstruction, a process which can be time consuming and computationally intensive. In this manuscript, a new non-iterative reconstruction, a-f BLAST, is proposed to extend k-t BLAST to radial sampling. The a-f BLAST reconstruction is performed in a previously unexplored domain termed the angular frequency-temporal frequency (a-f) space and uses similar steps to Cartesian k-t BLAST. The performance of this method was demonstrated on retrospectively undersampled numerical phantoms and compared to k-t BLAST, non-Cartesian k-t BLAST, and the sliding window technique. Additionally, the reconstruction was tested on retrospectively and prospectively undersampled in vivo cardiac data. a-f BLAST is shown to have a similar reconstruction time as k-t BLAST as well as outperform k-t BLAST in terms of RMSE.

I. Introduction

MAGNETIC Resonance Imaging (MRI) uses an external magnetic field and applied radio frequencies to image soft tissues without ionizing radiation. MRI scans often take a significant amount of time, and patient movement, physiological motion, or contrast changes during the collection of data can result in errors in the final images. While some patients can comply with breath-hold commands, physiological motion cannot be controlled during the duration of the scan and long acquisition times make capturing motion, such as the beating of the heart, challenging. Significant research has been focused on accelerating the collection of MRI data to enable rapid dynamic imaging.

The need for faster imaging has led to different reconstruction techniques and a variety of sampling schemes. MRI data are often undersampled to reduce the scan time, but this leads to aliasing artifacts if the Nyquist criterion is not met. In general, there are three approaches to reconstructing artifact-free images by regenerating missing data: exploiting correlations in k-space, correlations in time, and a combination of k-space and temporal correlations [1][2]. Reconstructions that exploit correlations in k-space include parallel imaging techniques, such as GRAPPA [3] or SENSE [4]. In these methods, information is not shared across time, and in a dynamic image series each image frame is reconstructed independently of the other frames.

Techniques that exploit temporal correlations interpolate data in the time dimension. In these methods the temporal dimension is used to share, interpolate, or solve for missing time points without utilizing correlations in the spatial domain. For example, in view-sharing methods, undersampled k-space frames are merged together to yield fully-sampled datasets. More advanced techniques such as MR Fluoroscopy [5], keyhole [6][7], BRISK [8], CURE [9], TRICKS [10], TWIST [11], undersampled radial [12], VIPR [13], and golden-angle radial [12] update the center and periphery of k-space at different rates. However, view-sharing methods sacrifice temporal resolution and are more suited to imaging changes in contrast than changes in anatomical position.

The third category, known as k-t methods, estimates missing k-space points using neighbors in both k-space and time. Because MR images are compressible and contain redundant information, many k-t reconstruction methods [1][14][15][16][17][18][19][20] have been used to exploit combined k-space and temporal correlations. k-t BLAST (Broad-use Linear Acquisition Speed-up Technique) relies on the idea that aliasing artifacts can be offset from one another by interleaving the collection of uniformly undersampled Cartesian k-space data through time, (herein referred to as interleaved Cartesian sampling). Artifact-free images can be reconstructed using knowledge of the point spread function and a separate low-resolution training scan.

The k-t BLAST reconstruction has been used in a variety of applications, typically in imaging situations where periodic motion is expected. In such situations, the temporal information is repeated, and thus the temporal frequency domain is sparse, improving the k-t BLAST reconstruction. For instance, accelerated 2D and 3D cine cardiac data may be reconstructed using k-t BLAST [21][22]. Similarly, real-time free-running and non-gated cardiac scans can also be accelerated using k-t BLAST, thereby enabling rapid imaging of the true motion of the heart [23].

Although k-t BLAST requires interleaved Cartesian sampling, non-Cartesian trajectories (such as the radial trajectory) have many desirable properties, which make them useful for dynamic and real-time imaging. Non-Cartesian trajectories generally have less coherent aliasing artifacts and can be more robust to motion [24][25][26]. Despite these advantages, it is not straightforward to generalize k-t BLAST for radial sampling or other non-Cartesian trajectories. The k-t BLAST reconstruction is dependent on the distinct and predictable offset aliasing in the spatio-temporal frequency space (or x-f space), which is caused by interleaved undersampled Cartesian acquisitions. While Cartesian undersampling produces overlapping replicas of the image, radial aliasing artifacts appear as streaking in the image domain and as noise-like artifacts in the x-f domain. While k-t BLAST has been extended to uniformly undersampled radial data interleaved by timeframe (herein referred to as interleaved radial sampling) using an iterative Conjugate Gradient (CG) based approach, non-Cartesian k-t BLAST [17], this approach requires repeated non-uniform FFT operations and is computationally intensive.

In the present work, a new non-iterative method termed a-f BLAST [27] is proposed to extend k-t BLAST to interleaved radial sampling. The reconstruction steps are similar to k-t BLAST but are performed in a different angular frequency-temporal frequency (a-f) space. Like non-Cartesian k-t BLAST, training data are obtained directly from the undersampled radial scan using the densely sampled center of k-space, eliminating the need for a separate training scan. In this paper, a-f BLAST is tested on numerical phantom simulation data, retrospectively undersampled in vivo cardiac CINE data, and prospectively accelerated real-time in vivo cardiac data. The a-f BLAST reconstructions are compared to both Cartesian and non-Cartesian k-t BLAST reconstructions in terms of root-mean-square error (RMSE), computational speed, and complexity. These comparisons show that a-f BLAST performs similarly or better the other reconstruction methods in terms of RMSE and has a similar reconstruction time to that of Cartesian k-t BLAST, which is significantly faster than non-Cartesian k-t methods. Finally, the performance of a-f BLAST is demonstrated in a real-time in vivo cardiac imaging situation where data are collected over multiple heartbeats, which is the target application for this technology.

II. Theory

A. k-t BLAST

Dynamic images depicting anatomical motion naturally contain redundant information between timeframes. For example, the main movement in a set of dynamic cardiac images is the beating of the heart, while most other organs remain stationary. Therefore, dynamic image frames are correlated over time. Motion over time can be visualized in the spatio-temporal domain (x – t space) or in the related x-f space by performing a 1D Discrete Fourier Transform (1D DFT) along the time dimension (Fig 1a). Anatomical motion is often periodic, so the data are sparse in the x-f domain due to the few dominant frequencies. Because MRI data are collected in k-space, related to the image domain through the 2D Discrete Fourier Transform (2D DFT), uniform undersampling of Cartesian data through time leads to the aliasing in Fig 1b. If the same k-space points are sampled at each timeframe; the same aliasing pattern will appear in every image. In the x-f domain, these aliased signals overlap and are difficult to separate. Alternatively, incrementing the sampling pattern by one phase encoding line for every timeframe yields a lattice sampling pattern (Fig 1c). Due to the Fourier Shift Theorem, the aliasing pattern in the different timeframes have different phase shifts because of the interleaved k-t sampling. When a 1D DFT is performed along the temporal dimension, the resulting aliased replicates will lie on a diagonal in x-f space due to the phase shifts in each timeframe that translate the aliased replicas in the frequency dimension (Fig 1c). The aliased replicas no longer fall directly on top of one another and can be removed more easily due to their separation in both space and temporal frequency [1]. The aliasing in the x-f domain for an acceleration factor of R can be expressed as a mapping of the aliased pixels ρ₁ through ρ_R onto a location d_{ali as} [1]. Mathematically d_{ali as} can be expressed as

$$[\text{1}\hspace{1em}\text{1}\hspace{1em}\cdot \cdot \cdot \hspace{1em}\text{1}]\left[\begin{array}{c}{\rho }_{\text{1}}\\ {\rho }_{\text{2}}\\ ⋮\\ {\rho }_R\end{array}\right]=1\overline{\rho }=d_{alias}$$ (1)

Fig. 1.

Converting time-resolved Cartesian and radial k-spaces of a dynamic image series to the x-f space. a) A series of fully-sampled Cartesian k-space datasets (left) are converted to the image domain (middle) with a 2D DFT. Images are then converted to the x-f domain (right) by applying a 1D DFT along the time dimension. b) An example of a series of Cartesian k-spaces which are uniformly undersamped by a factor of 4. A 2D DFT is applied to produce the aliased images (middle), and then a 1D DFT along time to produce the x-f spaces (right). Because the same k-space lines are sampled in each timeframe, the aliasing in the x-f space appears only in the vertical direction, resulting in significant overlapping. c) An example of a series of Cartesian k-spaces which are uniformly undersamped by a factor of 4 and interleaved by timeframe. This interleaved sampling causes phase shifts in the aliased images (middle), which then results in a diagonally offset aliasing pattern in the x-f domain (right). d) Fully-sampled gridded data collected along a radial trajectory (left) are converted to the image space (middle) using the 2D NUFFT [30]. Images are then transformed to the x-f domain (right). Note that for fully-sampled radial data, the x-f space is nearly identical to that of the fully-sampled Cartesian data. e) Uniformly undersampled and gridded radial k-space datasets interleaved by timeframe with an acceleration factor of 4 (left) are converted to the image domain. The undersampling causes aliasing artifacts, which appear as streaks. The application of a 1D DFT to the time domain results in noise-like aliasing artifacts in the x-f space (right).

In order to reconstruct fully-sampled images from the undersampled data using k-t BLAST [1] (Fig 2), a separate fully-sampled low-resolution training scan must be collected (Fig 2a, bottom). This training dataset has a high temporal resolution, and the information can be used to reconstruct the undersampled data by providing an estimate of the signal magnitudes. Both accelerated and training datasets are transformed to the image domain (x-t space) using a 2D DFT as seen in Fig 2b. A 1D DFT is then performed along the temporal direction, which transforms the data to the x-f domain (Fig 2c). Due to the interleaved Cartesian sampling pattern, the aliasing in the x-f space is diagonally offset and therefore the signal duplicates have little overlap as shown in Fig 2c, top. In the x-f space, the training data is comprised of a low-resolution alias-free version of the expected signal distribution (Fig 2c, bottom), therefore, this training data can be used as a guide to separate the signal duplicates in the higher resolution aliased x-f space. Mathematically, these replicas can be reduced by applying a filter derived from the low-resolution training dataset, m_k,t. As stated in [1], with the additional information about the estimated signal magnitude, a solution to (1) can be found through the weighted-minimum-norm solution [28][29]. Equation (2) describes this process

$$\begin{gathered} \overline{\rho }=\mathbf{M}{(1\mathbf{M})}^{+}{d}_{x,f}={\mathbf{M}}^{\text{2}}{1}^H{\left(1{\mathbf{M}}^{\text{2}}{1}^H\right)}^{-\text{1}}{d}_{x,f} \\ \hspace{1em}=\frac{{[{\left|m_{\text{1}}\right|}^{\text{2}}{\left|m_{\text{2}}\right|}^{\text{2}}\dots {\left|m_R\right|}^{\text{2}}]}^H}{{\left|m_{\text{1}}\right|}^{\text{2}}+{\left|m_{\text{2}}\right|}^{\text{2}}+\dots +{\left|m_R\right|}^{\text{2}}}{d}_{x,f} \\ {\mathbf{d}}_{x,f}=DF{T}_t(DF{T}_{\text{2}D}({\mathbf{d}}_{k,t})) \end{gathered}$$ (2)

where $\overline{\rho }$ is the pixel vector solution of the BLAST reconstruction in the x-f space, d_{x, f} is the undersampled Cartesian data matrix in the x-f domain, d_k,t is the undersampled dataset with undersampled lines zero-filled in k-space (Fig 2a, top), and (.)⁺ represents the Moore-Penrose pseudo-inverse. In order to transform d_k,t to d_{x, f} a 2D DFT is applied, followed by a temporal 1D DFT. M² is a diagonal matrix estimate of the signal covariance and contains the squared estimated signal magnitudes from the densely sampled low-resolution training scan in the x-f domain (Fig 2a–c, bottom). To obtain M, m_k,t is transformed to m_{x, f} using a spatial 2D DFT followed by a 1D DFT along time and its magnitude is taken,|m_{x, f} |. M is a diagonal matrix consisting of |m₁| through |m_R | from |m_{x, f|} corresponding to the locations of pixels ρ₁ through ρ_R aliased on to the location of d_{x, f}. After unfolding every set of aliased pixels (Fig 2d), an inverse 1D DFT is performed on the temporal dimension to recover the reconstructed images (Fig 2e). Due to the fact that the aliased voxels are known for k-t BLAST, the $\overline{\rho }$ and M² in (2) are truncated versions that only contain the data points pertaining to the set of aliased x-f voxels being separated. This simplifies the inverse problem because only small matrices must be inverted.

Fig. 2.

A schematic of the k-t BLAST reconstruction technique and processing of training data. (a, top) Uniformly undersampled Cartesian datasets interleaved by timeframe with high spatial and temporal resolution are collected. (a, bottom) A low-resolution but fully-sampled dataset with a high frame rate is collected in a separate scan and used as training data for the reconstruction. (b, top) High-resolution aliased images result when the 2D DFT is applied to the undersampled accelerated datasets. (c, top) Applying a 1D DFT to the temporal dimension transforms the time-resolved images to the x-f space. Because the undersampled data are collected using an interleaved sampling pattern, the aliased duplicates in the x-f domain are diagonally offset. (a-c, bottom) The low-resolution training data are zero-filled to match the matrix size of the accelerated data, and then converted to the x-f domain in the same way as the accelerated data. (d) Filters derived from the low-resolution training datasets can be used to suppress the aliasing artifacts in the accelerated data via the BLAST reconstruction. (e) The reconstructed images are then generated by applying an inverse 1D DFT to the filtered x-f domain data.

B. Non-Cartesian k-t BLAST

For a fully-sampled radial dataset (Fig 1d) the x-f space looks similar to the Cartesian dataset in Fig 1a. As can be seen in Fig 1e, when interleaved radial data are undersampled, the aliasing that appears in the images, and consequently the x-f space, is different from that in the Cartesian domain (Fig 1c). The aliasing in the image domain from radially undersampled data appears as streaks (Fig 1e, middle) leading to incoherent and noise-like aliasing artifacts in the x-f domain (Fig 1e, right).

The formulation of k-t BLAST described in (2) is useful when the aliasing pattern is simple and known, as in the Cartesian case. However, in general, these matrices can be expanded to describe the entire dataset, and not a subset of aliased pixels. In this case, ${\overline{\rho }}_{x,f}$ is a vectorization of the reconstructed signal and ${\mathbf{M}}_{x,f}^{\text{2}}$ is a diagonal matrix containing all the squared magnitude signal intensities from m_{x, f}. The k-t BLAST reconstruction can thus be generalized using the following formulation [1]

$$\begin{gathered} {\overline{\rho }}_{x,f}={\mathbf{M}}_{x,f}^{\text{2}}{\mathbf{E}}^H{\left(\mathbf{E}{\mathbf{M}}_{x,f}^{\text{2}}{\mathbf{E}}^H+{\mathbf{\Psi }}_{k,t}\right)}^{+}{\overline{\mathbf{d}}}_{k,t} \\ \hspace{1em}\hspace{1em}\mathbf{E}={\mathbf{\Xi }}_{k,t}DF{T}_{x,f\to k,t} \end{gathered}$$ (3)

In (3), ψ_k,t is the noise covariance matrix, E is the encoding matrix which contains the Fourier transforms required to transform data from x-f to k-t domains, and ${\mathbf{\Xi }}_{k,t}$ is a gridding operator [1]. This is the formulation required when the aliased pixels cannot be easily defined, as when using undersampled non-Cartesian trajectories, and is thus the basis for Non-Cartesian k-t BLAST [17].

If an accelerated radial trajectory is employed, no additional training dataset is required; instead, the Nyquist-sampled center from the radial data can serve as the training data. The fully-sampled center for training data is typically chosen to be 1/R of the matrix size, where R is the acceleration factor. In non-Cartesian k-t BLAST, the dynamic data and training data are transformed to the image domain using the 2D Non-Uniform Fourier Transform (NUFFT) [30], and are then converted to the Cartesian x-f space via a temporal 1D DFT as in k-t BLAST. The non-Cartesian k-t BLAST reconstruction can be applied to the rearranged generalized k-t BLAST formulation [17]

$${\overline{\rho }}_{x,f}={\left({\mathbf{E}}^H{\mathbf{\Psi }}_{k,t}^{-\text{1}}\mathbf{E}+{\mathbf{M}}_{x,f}^{-\text{2}}\right)}^{+}{\mathbf{E}}^H{\mathbf{\Psi }}_{k,t}^{-\text{1}}{\overline{\mathbf{d}}}_{k,t}$$ (4)

In (4) ${\overline{\mathbf{d}}}_{k,t}$ is a vector containing the undersampled radial data in k-space. The encoding matrix, E, is defined by $\mathbf{E}=DF{T}_{t\to f}{\mathbf{\Xi }}_{k\to x}=DF{T}_{t\to f}\mathbf{\text{NUFFT}}$ where ${\mathbf{\Xi }}_{k\to x}$ is the gridding operator with Fourier transforms (or NUFFT [30]) to convert the radial k-t data to the x-t domain, and DFT_t→f is the 1D DFT through time to transform the x-t space to the x-f space. Due to the size of $\left({\mathbf{E}}^H{\mathbf{\Psi }}_{k,t}^{-\text{1}}\mathbf{E}+{\mathbf{M}}_{x,f}^{-\text{2}}\right)$ it is unrealistic and computationally intensive to compute the pseudo-inverse directly. Instead, for the non-Cartesian k-t BLAST reconstruction the inverse problem is solved using Conjugate Gradient (CG) descent. This approach is iterative and can be time-consuming, since data must be transformed repeatedly between the x-f and radial k-t domains, which includes gridding and degridding the data. Additionally, the use of multiple gridding steps can introduce errors into the reconstruction. This approach has been accelerated using GPU coding [31], but it is still computationally intensive.

C. a-f BLAST

In this work, a novel reconstruction method termed a-f BLAST is introduced that operates in a radial spatio-temporal frequency domain (Fig 3a). As described above, undersampled radial data does not offer the offset aliasing pattern in the x-f space needed for the Cartesian k-t BLAST reconstruction. The key insight in a-f BLAST is to process the radial data in a matrix with readout points, r, along one dimension and radial projection angle, θ, along the other dimension (Fig 3a, left). This radial k-space is represented by d_r,θ,t. In this form, no gridding has been performed and the radial k-space data resembles Cartesian data.

Fig. 3.

a) Fully-sampled non-gridded radial k-space arranged by read-out and projection angle (left) is transformed to the Radon, or sinogram, domain (left-middle) by applying a 1D DFT along the read-out dimension. A subsequent 1D DFT is performed along the projection angle dimension to obtain the angular frequency-temporal domain, or a-t space (right-middle). A 1D DFT along the time dimension is performed to transform the data into the angular frequency-temporal frequency (a-f) space (right). b) Uniformly undersampled (R=4) non-gridded radial k-space interleaved by timeframe (left). Using a 2D DFT the radial k-space is transformed to the a-t space, which exhibits similar aliasing to Figure 1c, middle. Performing a subsequent 1D DFT along the time dimension results in the radial a-f space (right) with the same diagonally offset aliasing as in Fig 1c, right, without the use of gridding.

a-f BLAST operates on the radial k-space matrix, as seen in Fig 4. First a 1D DFT is performed along the readout direction to convert the radial k-space data into a Radon domain, referred to here as the sinogram space (Fig 3, middle-left) using Equation (5):

$${\mathbf{d}}_{s,\theta ,t}=DF{T}_r({\mathbf{d}}_{r,\theta ,t})$$ (5)

In Equation (5) the 1D DFT along the read-out direction is represented by D F T_r, d_r,θ,t is the radial data matrix, and d_s,θ,t is the data in the sinogram space. Next a subsequent 1D DFT is performed along the projection direction, DFT_θ, to transform d_s,θ,t into a previously undefined space termed “angular frequency-temporal space,” or a-t space (Fig 4b, top), d_s,a,t.

$${\mathbf{d}}_{s,a,t}=DF{T}_{\theta }({\mathbf{d}}_{s,\theta ,t})=DF{T}_{\text{2}D}({\mathbf{d}}_{r,\theta ,t})$$ (6)

Because the undersampled radial data are collected using an interleaved sampling pattern (Fig 4a, top) and lines not sampled are set to zero, the a-t space (Fig 4b, top) has offset aliasing artifacts with similar properties as those seen in the x-t domain of Cartesian k-t BLAST. Finally, a third 1D DFT is performed along the time dimension (Fig 4c, top), as in k-t BLAST, to convert the data into the angular frequency-temporal frequency space (a-f space):

$${\mathbf{d}}_{s,a,t}=DF{T}_t({\mathbf{d}}_{s,a,t})$$ (7)

Although this data is collected along a radial trajectory, it has an offset aliasing pattern in the a-f space, as required for the Cartesian k-t BLAST reconstruction. Thus, the undersampled radial data can be processed using the same steps as those used in k-t BLAST, eliminating the need for multiple gridding steps in the iterative CG k-t BLAST approach.

Fig. 4.

A schematic of the a-f BLAST reconstruction technique and processing of training data. (a) Uniformly undersampled non-gridded radial datasets with high spatial and temporal resolution are collected with an interleaved sampling pattern. (a, bottom) The fully-sampled center of the radially undersampled k-space is cropped and processed to obtain a fully-sampled low-resolution radial k-space with the same matrix dimensions as the accelerated data. (b) Performing a 2D DFT on the accelerated datasets generates the aliased a-t spaces. (c) Applying a 1D DFT along the time domain converts the data to the angular frequency-temporal frequency domain, the a-f space. Because the undersampled data are collected using an interleaved sampling pattern, the aliased duplicates in the a-f domain are diagonally offset. (a-c, bottom) The low-resolution training data are processed identically to the undersampled radial data in order to obtain training information in the a-f domain. (d) Filters derived from the low resolution training datasets can be used to suppress aliasing in the accelerated data using the BLAST reconstruction. (e) The final images are generated by performing inverse DFTs to obtain the reconstructed radial k-space, which is then gridded using the NUFFT to obtain the reconstructed images. With the exception of the final gridding step and manipulation of the training data, both k-t BLAST and a-f BLAST are performed with identical processing steps.

As in non-Cartesian k-t BLAST, the training data for a-f BLAST can be taken from the fully-sampled center of the undersampled radial k-space (Fig 4a). Training data are processed by first extracting the fully-sampled k-space center (radius of (k_r / R) as in [17]) of the rearranged radial k-space. The extracted data, m_r,θ,t, are then interpolated along the angular direction using a cubic spline interpolation to increase the matrix size to match that of the accelerated data in the projection direction; this step is possible as the data have been sampled according to the Nyquist criterion in this part of k-space. The training data are then zero-filled in the readout direction to match the matrix dimensions of the fully-sampled radial k-space data matrix. The radial training k-space is then transformed to the a-f space (Fig 4, bottom), its magnitude is taken and squared, ${\left|\mathbf{m}\right|}_{s,a,f}^{\text{2}}$, such that it contains only the square estimated signal values.

$${\mathbf{m}}_{s,a,f}^{\text{2}}=DF{T}_t{\left(DF{T}_{\text{2}D}\left({\mathbf{m}}_{r,\theta ,t}\right)\right)}^{\text{2}}$$ (8)

Once both the undersampled data and the training data have been converted to the a-f domain, the BLAST portion of the reconstruction can be performed. Similar to (2), Equation (9) describes the BLAST reconstruction in the a-f domain.

$$\overline{\rho }={\mathbf{M}}^{\text{2}}{1}^H{\left(1{\mathbf{M}}^{\text{2}}{1}^H\right)}^{-\text{1}}{d}_{s,a,f}$$ (9)

Here $\overline{\rho }$ is a vector containing the reconstructed a-f space pixels resulting from the BLAST reconstruction in the a-f space (Fig 4d) which are placed in their corresponding locations in the reconstructed matrix ρ_s,a,f. M² is the truncated diagonal matrix containing the magnitude squared estimated signal intensities from m_{s,a, f} corresponding to the locations of pixels aliased together in d_{s,a, f}. Because the locations of the aliased voxels are known as in k-t BLAST, the $\overline{\rho }$ and M² in Equation (9) are similarly truncated to only contain the data points pertaining to the set of aliased a-f pixels, eliminating the need to solve the large inverse problem present in non-Cartesian k-t BLAST. After the BLAST reconstruction in a-f space has been performed for each set of aliased pixels, the reconstructed data are transformed back into radial k-space via Fourier transforms along the temporal and both spatial dimensions. These data are then gridded and an inverse DFT performed using the NUFFT [30] to yield the reconstructed images, (Fig 4e).

$${\rho }_{x,y,t}=DF{T}_{\text{2}D}^{-\text{1}}\left({\mathbf{\Xi }}_{k,t}\left(DF{T}_{\text{2}D}^{-\text{1}}\left(DF{T}_t^{-\text{1}}({\rho }_{a,s,f})\right)\right)\right)$$ (10)

The core steps for the a-f BLAST and k-t BLAST reconstructions are nearly identical, which can be further seen by comparing Fig 2 and Fig 4. The two exceptions are the need for a separate training scan for k-t BLAST, and the necessary training data preparation and final gridding step in a-f BLAST.

III. Materials and Methods

A. Numerical Simulations

In order to test the a-f BLAST reconstruction, simulations were performed using a numerical cardiac phantom with variable and varying periodic motion. The simulated phantom had a matrix size of 144 × 144 and was designed to simulate periodic motion, quasi-periodic motion, and an accelerating heart rate over multiple heartbeats. Note that this numerical phantom simply models motion using shapes with intensities that mimic cardiac images collected using a bSSFP sequence; Bloch equation simulations using appropriate relaxation values were not performed, and fat signal is not included in the simulation. A heartrate of 60 beats per minute (bpm) was used for the regular motion. This numerical phantom was altered to produce quasi-periodic motion by adding small random adjustments to the regular motion of each heartbeat, as shown in Equation (11)

$$H_s=\{\begin{array}{ll}{H}_{\text{0}},\hfill & \text{for}\hspace{0.17em}\text{i}=\text{0}\hfill \\ H_{i-\text{1}}\pm {\mu }_i,\hfill & \text{for}\hspace{0.17em}\text{i}>\text{0}\hfill \end{array}$$ (11)

In Equation (11) H_s is the simulated heartrate, H₀ is the initially specified heartrate of 60 bpm, and µ is a single uniformly distributed random variable in the interval (0,1) which changes every timeframe. For every timeframe the heartrate is randomly increased or decreased by the random variable, adding a slight jitter to the heart movement and small changes to the heartrate so the numerical phantom simulation is not perfectly periodic. Because each acceleration factor was simulated separately, the minimum, maximum, and mean of each simulation varied. As such, the total average of minimum, maximum, and mean bpm were: 54.02 ± 2.03, 61.73 ± 2.62, and 58.04 ± 2.70 respectively. Finally, the motion was altered to accelerate from 60 to 80 bpm over a total scan time (T_c) of 8 seconds to mimic how the heartrate may increase during a breath-hold. The accelerating heartrate was calculated using Equation (12).

$$H_s=\{\begin{array}{ll}{H}_{\text{0}},\hfill & \text{for}\hspace{0.17em}\text{i}=\text{0}\hfill \\ H_{i-\text{1}}+(H_f-H_{\text{0}})/T_c,\hfill & \text{for}\hspace{0.17em}\text{i}>\text{0}\hfill \end{array}$$ (12)

In addition, the performance of a-f BLAST in the presence of continuous-random motion was investigated using a numerical Shepp-Logan phantom. The numerical phantom was modified so that each ellipse moves randomly and continuously between frames.

The reconstruction accuracy of a-f BLAST was tested using these four different motion conditions with various acceleration factors. Data were simulated using both interleaved Cartesian and radial trajectories, where one Cartesian line or radial projection in k-space was collected from each frame of the simulation time series with a simulated TR of 3ms. The total scan time for all simulations was held constant at 8 seconds, and thus the total number of undersampled image frames (N_t) varied according to the acceleration factor. The “fully-sampled” radial k-space data consisted of 144 radial projections collected with a read-out oversampling factor of 2 (matrix size 288 × 144 × N_t). Radial projections were collected linearly with an increment of 1.24^◦ between adjacent projections. Note that while 144 projections was considered to be fully-sampled for the purpose of matching the acquisition time between the radial and Cartesian simulations, the true number of projections needed to fulfill the Nyquist criterion is 144 ∗ π/2 ≈ 226, where 144 represents the “base” image matrix size. In addition to radial data, Cartesian k-space data consisting of 144 projections and 144 readout points (matrix size of 144×144× N_t) was also simulated. Both Cartesian and radial datasets were uniformly undersampled and interleaved by time frame using acceleration factors of 2, 3, 4, 6, and 8. For example, for an acceleration factor of 4, 36 lines of data were collected per time frame. For each time frame the sampling pattern rotated by 1 to achieve the through-time interleaved sampling pattern such that frame 1 contained lines 1, 5, 9, etc., frame 2 contained lines 2, 6, 10, etc., and so on. These interleaved undersampled datasets were then zero-filled to yield matrix sizes of 144 × 144 × N_t and 288 × 144 × N_t for the Cartesian and radial datasets, respectively. The accelerated Cartesian datasets were reconstructed using k-t BLAST, and the accelerated radial datasets were reconstructed using both non-Cartesian k-t BLAST and a-f BLAST. A sliding window reconstruction was also implemented for comparison and to ensure that the a-f BLAST reconstruction was not simply acting as a temporal filter, as in [17]. Training data for non-Cartesian k-t BLAST and a-f BLAST were obtained from the densely sampled center taken from the undersampled radial scan. For Cartesian k-t BLAST, ideal training datasets were simulated by sampling 22 lines at the center of k-space as suggested in [1]. Because the center of k-space of the accelerated data is used as the training data for non-Cartesian k-t BLAST and a-f BLAST, the motion captured in the accelerated data is exactly the same as in the training data. However, this is generally not the case for k-t BLAST. Unless the training data is collected at the same time as the undersampled data these two datasets will typically not have the same motion characteristics. While it is possible to collect the training data for k-t BLAST at the same time as the accelerated scan, in order for the training data to contain the same motion characteristics it would need to be collected for every timeframe in the entire accelerated scan, which would add significantly to the scan time and reduce the achievable acceleration factor. If, instead, the training data is collected for the designated training timeframes at the beginning of the accelerated scan there would still be a mismatch in in motion characteristics between training and undersampled data. Therefore, another simulation was performed show how mismatches in the temporal dynamics of the accelerated and training datasets can affect the k-t BLAST reconstruction. To this end, another more realistic k-t BLAST reconstruction was performed by simulating periodic motion during the accelerated scan and an accelerating heartrate during the training scan. Root mean square error (RMSE) values for each reconstruction were calculated using the corresponding fully-sampled Cartesian or radial datasets as gold-standards.

Additionally, reconstruction times for k-t BLAST and a-f BLAST were recorded and compared for each acceleration factor. Simulations and reconstructions were performed on a 64 bit ASUS N56VZ laptop with an Intel i7–3610QM CPU @ 2.3GHz with 16GB RAM. The non-Cartesian k-t BLAST reconstruction was not timed because it was performed using the Gadgetron framework [32], which ensured an appropriate implementation of the reconstruction. A theoretical analysis to compare differences in computational complexity between the k-t BLAST, a-f BLAST, and Non-Cartesian k-t BLAST reconstructions is provided in the Appendix.

B. Sparsity and Energy Distribution in the x-f and a-f Spaces

Both k-t and a-f BLAST work best when the aliasing artifacts are distinct, and thus separable; if the signal energy in these spaces is diffuse in the fully-sampled cases, there will be significant overlap when the data are undersampled. Thus, by examining the sparsity of these spaces, the relative performance of BLAST in these spaces, and the potential for acceleration, can be better understood.

Two separate analyses were performed to compare these spaces. The first analysis examined the percentage of low signal voxels in the fully-sampled x-f and a-f spaces as a proxy for sparsity. A low signal pixel was defined as having an amplitude of less than 0.1% of the maximum signal amplitude. It is anticipated that the sparser space would indicate more redundancy in the data, and thus the potential for better reconstruction quality. The numerical simulations described above were employed with a total of 288 timeframes, which would be equivalent to approximately 20 seconds worth of real-time data collected with R = 6 and the scanner settings described below for the in vivo data.

In the second analysis, the energy concentrations in the x-f and a-f spaces were examined to understand how the energy is distributed around the center of the domain; concentrated energy would lead to less overlap in the case of acceleration, and therefore enable higher acceleration factors to be reconstructed with higher fidelity. Because Non-Cartesian k-t BLAST is applied in the x-f domain (Fig 1d), the energy concentration would be similar to that of k-t BLAST. In order to quantify the signal distribution, the energy concentration was calculated (including the center DC or 0 Hz line) by summing the magnitude of the signal intensity of the desired section and dividing it by the sum of the signal magnitude of the entire space. Three different measures of energy concentrations were examined: moving outward in the frequency direction from the center 0 frequency line, moving outward in the ±x/a direction, and in concentric rectangles moving outward from the center in both the frequency and x/a direction. A graphical description of the areas used for this analysis can be seen in Fig 5. Equation (13) shows an example of the calculation of the energy along the frequency direction for the x-f and a-f spaces.

$$\begin{gathered} \%Energ{y}_{f_i} \\ \hspace{1em}=\sum _{y/s}\sum _{x/a}\sum _{f_{\text{0}}\to f_i}\left|{\mathbf{d}}_{y/s,x/a,f}\right|/\sum _{y/s}\sum _{x/a}\sum _f\left|{\mathbf{d}}_{y/s,x/a,f}\right| \end{gathered}$$ (13)

Here, f₀ indicates the center 0 Hz line, f_i is how many lines from the center the section includes, and d_y,x,f and d_s,θ,f are the full datasets in the x-f and a-f domains, respectively. Due to the symmetry of the spaces, the analyses were performed using only the right-hand-side of the domain, as seen in Fig 5. Cumulative energy concentrations for the three analyses were plotted against the number of included lines and compared to each other to determine in which space the energy is more centrally concentrated.

Fig. 5.

Graphical representation of the cumulative energy concentration analyses.

C. In vivo Imaging

The relative performance of a-f BLAST, non-Cartesian k-t BLAST, and k-t BLAST was also compared using data collected in vivo. Cardiac data were collected from fourteen volunteers (14 radial scans, 5 Cartesian scans) after obtain-ing informed consent on a 3T Siemens Skyra MRI scanner (Siemens Medical Solution, Erlangen, Germany) with an 18-channel cardiac receiver coil array and a 12-channel built-in spine array in an IRB approved study. Two types of datasets were collected: fully-sampled radial/Cartesian CINE scans acquired with ECG gating, and prospectively accelerated radial/Cartesian real-time scans collected over multiple heart-beats. Both types of radial scans had matrix dimensions of 128 × 128 with 144 projections in the fully-sampled datasets. As in the numerical simulations, 144 projections was considered “fully sampled” for all in vivo radial datasets. Cartesian CINE scans were collected with matrix dimensions of 144 × 144. Prospectively accelerated Cartesian datasets had matrix dimensions of 128 × 128 with 128 phase encoded lines in the fully-sampled Cartesian datasets. A total of 40 timeframes of training data with 20 central k-space lines were collected immediately prior to the prospectively accelerated Cartesian scans for use in k-t BLAST calibration.

The CINE data were collected such that the in vivo performance of the BLAST methods could be compared against a fully-sampled gold-standard. These images were acquired during breathholding at end inspiration using a radial/Cartesian bSSFP sequence with TR = 2.74 – 3.14ms, TE = 1.5ms, BW = 1kHz, FoV = 300mm×300mm, spatial resolution = 2.3mm³ × 2.3mm³ × 8.0mm³, and a flip angle between 37–57 degrees. CINE scans used to test reconstructions had scan durations between 7–45seconds. Scan times are dependent on the volunteer’s heartrate which ranged from 44–96 beats per minute. The temporal resolution for radial and Cartesian scans averaged 29.65ms with a range of 12.56–50.24ms. Because CINE scans are segmented to capture data over the entire heartbeat, these datasets contain only a single heartbeat of 14–108 timeframes. Thus, datasets were extended to mimic multi-heartbeat datasets by concatenating the same CINE dataset multiple times and adding random noise to each concatenation. These fully-sampled but artificial multi-heartbeat CINE scans were retrospectively undersampled (R = 2, 4, 6, and 8, each with 72, 36, 24, and 18 projections, respectively) and reconstructed using a-f BLAST, non-Cartesian k-t BLAST, and k-t BLAST. Just as in the numerical phantom simulations, ideal training datasets were simulated by using 22 central lines of the fully-sampled CINE. Note that no parallel imaging was used—the reconstructions were performed separately on each coil, and the final coil-combined image was produced with the reconstructed single coil images using an adaptive combination method [33]. Finally, the reconstructed images were compared to the fully-sampled images by calculating RMSE values. Additionally, the cumulative energy concentrations for the a-f and x-f spaces were also calculated using the radial and Cartesian CINE datasets.

Fourteen prospectively accelerated real-time radial cardiac datasets and five prospectively accelerated real-time Cartesian cardiac datasets were also collected, each lasting a total of 7–25seconds, in order to acquire images showing multiple heartbeats. These radial and Cartesian datasets used the same scan parameters as the CINE datasets with the following exceptions: Cartesian matrix of 128 × 128; TR = 2.94 – 3.9ms; radial acceleration factors of 2, 4, and 6 (72, 36, and 24 radial projections); Cartesian acceleration factors of 2, 4, and 6 (64, 32, and 21 Cartesian phase encode lines). These datasets were reconstructed using a-f BLAST, non-Cartesian k-t BLAST, and Cartesian k-t BLAST, and results were visually compared to assess performance. All resulting reconstructions have been included in the Supplementary Materials available in the supplementary files/multimedia tab.

IV. Results

The RMSE for the various numerical simulations under different motion conditions are shown in Fig 6 and include the k-t BLAST, sliding window, non-Cartesian k-t BLAST, and a-f BLAST reconstructions. For k-t BLAST, the RMSE values from the regular motion numerical phantom (Fig 6a) and the quasi-periodic motion numerical phantom (Fig 6b) show that k-t BLAST and a-f BLAST result in similar RMSE values over all tested acceleration factors when k-t BLAST is performed with an ideal training dataset (i.e. containing the same motion as the accelerated data). RMSE values increased by approximately four times overall when the training data for k-t BLAST had different motion properties than the accelerated data. For reconstructions of the numerical phantom with an accelerating heartrate (Fig 6c), the a-f BLAST reconstruction resulted in a similar RMSE to k-t BLAST for an acceleration factor of 2. However, a-f BLAST outperforms k-t BLAST for all other acceleration factors tested (R = 3, 4, 6, and 8). Significant differences in performance of k-t BLAST and a-f BLAST arise when the reconstructions are performed using the continuous-random motion numerical Shepp-Logan phantom (Fig 6d). Overall a-f BLAST has a substantially lower RMSE than k-t BLAST. As expected, the sliding window reconstruction resulted in a consistently higher RMSE than a-f BLAST or non-Cartesian k-t BLAST. Plots of RMSE comparisons in Fig 6 show that a-f BLAST has a consistently lower RMSE than non-Cartesian k-t BLAST, with the exception of the continuous-random motion numerical phantom simulation, where a-f BLAST and non-Cartesian k-t BLAST perform similarly for all acceleration factors.

Fig. 6.

RMSE comparison graphs for numerical phantom simulations depicting a heart experiencing a) periodic motion, b) quasi-periodic motion, c) an accelerating heartrate, and d) continuous-random motion. The graphs compare the performance of k-t BLAST with ideal training data, k-t BLAST with separate training data, sliding window, non-Cartesian k-t BLAST, and the a-f BLAST reconstructions.

Example images from the numerical phantom simulation showing diastole and systole are shown in Fig 7. On the left, the fully-sampled images are shown, followed by the R = 4 accelerated Cartesian (left center) and radial (center) images. The corresponding k-t BLAST (right center) and a-f BLAST (right) images from the dataset with a regular heartrate show improved image quality along with some residual aliasing artifacts consistent with their sampling patterns. Overall, the artifacts from both reconstructions are minimal. While both k-t BLAST and a-f BLAST have similar RMSE values for this case (specifically 1.67 and 1.58), the residual Cartesian artifacts appear more prominently.

Fig. 7.

Diastole (top) and systole (bottom) frames from the fully-sampled images, retrospectively undersampled R=4 Cartesian images, retrospectively undersampled R=4 radial images, R=4 k-t BLAST reconstruction from the periodic motion simulation, and R=4 a-f BLAST reconstruction from the periodic motion simulation (from left to right).

A plot comparing the computation times in MATLAB for k-t BLAST and a-f BLAST is shown in Fig 8. For every acceleration factor, k-t BLAST is faster than a-f BLAST due to the added time needed to process the radial training data as well as perform the gridding in a-f BLAST. This additional processing for the radial training data in a-f BLAST required approximately 2.5 seconds longer than for k-t BLAST, and the gridding step for took an average of 0.44 seconds to perform; therefore, reconstruction times differed by approximately 3 seconds for all acceleration factors in this unoptimized code.

Fig. 8.

A graph comparing computation times for the k-t BLAST and a-f BLAST reconstructions. The overall reconstruction was separated into three distinct processing steps: BLAST, processing training data, and, in the case of a-f BLAST, gridding.

The theoretical timing comparison between non-Cartesian k-t BLAST and a-f BLAST provided in the Appendix shows that, when limited to 10 iterations based on [17], non-Cartesian k-t BLAST requires 10 more BLAST calculations, 21 more gridding steps, and 55 more DFTs than a-f BLAST.

The percentage of low signal (< 0.1% of the maximum signal value) pixels for the x-f and a-f space show that the a-f space is sparser for all numerical simulations as well as the artificially extended CINE datasets. In the cardiac numerical phantom, the percentage of low signal pixels in the x-f space is 92.74% under periodic motion conditions, dipping to 87.94% for quasi-periodic motion and 88.75% for the accelerating heartrate. For the continuous-random motion numerical phantom, the percentage of low signal pixels drops drastically to 27.00%. In the artificially extended, and thus perfectly periodic, Cartesian CINE images the x-f space contained 90.16% low signal pixels. Comparatively, the a-f space has markedly more low signal pixels, specifically 99.80% in the periodic numerical phantom, 99.58% in the quasi-periodic phantom, 99.52% in the accelerating heartrate numerical phantom, and 96.71% in the continuous-random motion numerical phantom. Additionally, the percentage of low signal pixels in the a-f space of the artificially extended radial CINE is 98.51%.

While the percentage of low signal pixels is helpful in quantifying the distribution and sparsity of the signal in the x-f and a-f spaces, it is important to look at the spread of these high-signal pixels. This information can be seen in the cumulative energy plots for the frequency dimension in Fig 9a–e, the x /a dimension in Fig 9f–j, and concentric rectangles in Fig 9k–o. In all cases, the energy is more concentrated towards the center of the a-f space than the x-f space.

Fig. 9.

Energy concentration graphs of all phantoms showing the percentage of total energy accumulated from the center line in the frequency dimension (left), x/a dimension (center), and concentric rectangles (right) in the f vs. x/a space. Lines for the x-f space are in dashed blue and lines for the a-f space are in gray.

Example reconstructed images and RMSE values for the retrospectively undersampled in vivo radial and Cartesian CINE scans are shown in Fig 10. For a-f BLAST, the RMSE values of the reconstructed images remained under 1% up to an acceleration factor of 8. The RMSE values for non-Cartesian k-t BLAST are slightly higher than those for the a-f BLAST reconstructions with RMSE values exceeding 1% at an acceleration factor of 6. At the bottom of Fig 10 are corresponding y-t spaces for R = 4 to demonstrate the effects of the different reconstruction techniques on the temporal resolution. All reconstructions show a reduction in the level of aliasing artifacts in the y-t domain, although the temporal resolution of the a-f BLAST images appear to maintain a slightly higher level of fidelity than non-Cartesian k-t BLAST with the CINE images.

Fig. 10.

Example of a-f BLAST, non-Cartesian k-t BLAST, and k-t BLAST reconstructions of retrospectively undersampled radial/Cartesian CINE data and their RMSE values with varying acceleration factors. (bottom) y-t images of a single heartbeat for each image set of R=4 to demonstrate the temporal fidelity of the reconstructions. (right) Diastole and systole images from each image set of R=4 to give a closer view of the heart.

Finally, prospectively accelerated data collected in real-time for an acceleration factor of 4 and 6 were reconstructed with a-f BLAST, non-Cartesian k-t BLAST, and k-t BLAST. Example images can be seen in Fig 11–13, and the complete results are included in Supplementary Materials available in the supplementary files/multimedia tab. Fully-sampled versions of these reconstructions were unavailable due to the real-time nature of the data collection and thus it was not possible to obtain RMSE values. Instead, reconstructions were visually compared to the CINE reconstruction results in order to determine the severity of any remaining artifacts. Image quality for a-f BLAST and non-Cartesian k-t BLAST was determined to be similar to their CINE counterparts up to an acceleration of 6.

Fig. 11.

Example images generated with a-f BLAST and non-Cartesian k-t BLAST performed on prospectively accelerated cardiac images with a temporal resolution of 98.64ms and an acceleration factor of 4 for frames in diastole (top), systole (center), and the y-t space (bottom). Expanded images of the heart in diastole and systole are shown in the colored boxes (right).

Fig. 13.

Example images generated with k-t BLAST, a-f BLAST, and non-Cartesian k-t BLAST performed on prospectively accelerated cardiac images with temporal resolutions of 96.00ms for the Cartesian scan and 98.64ms for the radial scan with acceleration factor of 4 for frames in diastole (top), systole (center), and the y-t space (bottom). Expanded images of the heart in diastole and systole are shown in the colored boxes (right).

V. Discussion

In this study, the feasibility of performing a direct k-t BLAST reconstruction on interleaved radial k-space data, without the iterative gridding operations necessary for non-Cartesian k-t BLAST, is demonstrated. By entering into an angular frequency-temporal frequency space, the a-f domain, the k-t BLAST reconstruction can be used with little modification, eliminating the need for multiple gridding steps and iterative reconstructions. While a comparison of the reconstruction times indicates that the algorithms require a similar computation time for the BLAST part of the reconstruction, the addition of the gridding and extra processing steps for the training data adds approximately three seconds to the MATLAB implementation of the reconstruction. Because these reconstructions are not optimized, it is possible to decrease the processing time of all steps through optimization and parallel computation. While the a-f BLAST reconstruction will always take more time than k-t BLAST due to the need for a gridding step, the a-f BLAST reconstruction is significantly less complex and thus can be performed faster than non-Cartesian k-t BLAST.

In addition to its advantages in complexity and reconstruction time, the performance of a-f BLAST in simulations and in vivo experiments was similar to k-t BLAST and similar or better than non-Cartesian k-t BLAST in terms of RMSE values. The importance of matched training data is evident when comparing RMSE values of the k-t BLAST with identical training data and mismatched training data. Because a-f BLAST uses the Nyquist center for its training data, the motion of the training data is never mismatched to that of the accelerated scan. This advantage is significant in terms of data collection time and the quality of the reconstructed images. Thus, not only does the a-f BLAST require fewer computational steps than non-Cartesian k-t BLAST, and is expected to lead to faster reconstructions, but a-f BLAST also results in lower RMSE values than k-t BLAST with separate training data because the training data for a-f BLAST always contains the same dynamic information.

Energy analysis of the x-f and a-f spaces can be used to show why a-f BLAST has similar and/or lower RMSE values compared to k-t BLAST and non-Cartesian k-t BLAST. Signal intensity concentration and sparsity of these domains is essential to the performance of these k-t reconstructions. As mentioned in [1], k-t BLAST works best in the presence of periodic and pseudo-periodic motion; the x-f space is sparser in these cases and the energy more concentrated in the central portion of the space. In the case of a-f BLAST, the a-f space contains a higher percentage of low signal pixels than the x-f space for all motion conditions and is marginally more concentrated in the frequency dimension (Fig 9a–e). The concentric rectangles analysis showed that the concentration of energy around the center is higher for the a-f space than the x-f space, reaching 90% of the energy concentration for the periodic motion numerical phantom within a centered rectangle of 4 × 8 as compared to the x-f space centered rectangle of 38 × 76.

Reconstruction of retrospectively undersampled in vivo CINE data demonstrated that a-f BLAST could be used to reconstruct in vivo data. The RMSE values of a-f BLAST were similar or lower than non-Cartesian k-t BLAST for all acceleration factors. A visual examination of the image time series and the y-t space for the a-f BLAST reconstruction indicates that temporal information is retained, reducing blur-ring in cases of rapid object motion. The performance of a-f BLAST was also shown using prospectively accelerated cardiac data collected in real-time for an acceleration factor of 4. The results of these in vivo studies coupled with the RMSE values in the numerical phantom simulation studies suggest that it may be preferable in terms of image quality and capturing motion to collect dynamic data along a radial trajectory and use a-f BLAST for the reconstruction as opposed to k-t BLAST, and to use the a-f BLAST formulation over non-Cartesian k-t BLAST due to its decreased computation complexity.

Neither k-t BLAST nor a-f BLAST are able to resolve all aliasing or motion artifacts, as can be seen in the results of the in vivo studies in Fig 11–13 and Supplementary Materials available in the supplementary files/multimedia tab. Note that the pre-dominant failure modes of the two techniques are different; k-t BLAST tends to lead to temporal blurring and a loss in image fidelity in systole at higher acceleration factors, whereas the a-f BLAST reconstruction leaves streak artifacts in highly accelerated images. While a-f BLAST is also affected by some temporal blurring at higher accelerations it is less prominent than in Cartesian and non-Cartesian k-t BLAST. Additionally, at higher accelerations, all reconstructions lead to varying degrees of blurring at the myocardial borders and papillary muscles in systole which can be seen in the R = 6 slides of the Supplementary Materials available in the supplementary files/multimedia tab. In order to improve the temporal fidelity at higher acceleration factors in k-t BLAST, parallel imaging is usually incorporated (k-t SENSE). The inclusion of parallel imaging into the a-f BLAST framework is currently ongoing research.

It is important to note that an expanded regularized residual reconstruction version of (2) was presented in [1] which includes filtering and the incorporation of a baseline estimate subtraction. For the purposes of this work the difference in RMSE between the base reconstruction and the regularized residual reconstruction was minimal and thus the expanded regularized residual reconstruction was not used in order to keep the reconstructions presented as simplified and similar as possible.

While the a-f space has been proposed here to enable an efficient k-t BLAST-type reconstruction, other reconstruction techniques, including non-Cartesian parallel imaging and compressed sensing, may also be feasible in this domain. Compressed sensing reconstructions require incoherent aliasing artifacts and non-linear reconstructions which usually involve iterations between the image and k-space domains. Radial sampling has been used in compressed sensing due its relatively benign streak artifacts, but gridding and degridding steps are required in such iterative reconstructions. The a-f domain may reduce the computational burden in such algorithms, because just as a-f BLAST removes the need for gridding and degridding in non-Cartesian k-t BLAST, the compressed sensing iterations could be performed without the gridding step [34]. While k-t parallel imaging extensions of k-t BLAST have been introduced using the SENSE framework, it is important to note that in its current implementation, a-f BLAST does not incorporate coil sensitivity information. The inclusion of parallel imaging into the a-f BLAST reconstruction could allow for higher acceleration factors and will be the subject of future work.

VI. Conclusion

a-f BLAST is a non-iterative k-t-BLAST-type technique that can be used to reconstruct images from accelerated dynamic radial data. Numerical simulation studies demonstrate that a-f BLAST results in images with similar or lower RMSE values than k-t BLAST or non-Cartesian k-t BLAST under regular and irregular motion conditions. Due to the parallels between these techniques, k-t BLAST and a-f BLAST require similar computational resources and take nearly the same amount of reconstruction time to perform. Additionally, both reconstructions perform best when the motion is regular and periodic; however, with the matched training data a-f BLAST is able to handle more variations in motion characteristics. Retrospectively and prospectively accelerated in vivo data reconstructed using a-f BLAST also showed fewer or similar residual artifacts in comparison to non-Cartesian k-t BLAST. While demonstrated here with the BLAST technique, the new a-f reconstruction space could potentially be used to simplify other reconstruction techniques, including non-Cartesian parallel imaging and compressed sensing.

Fig. 12.

Example images generated from the same scan as Fig 11 with a-f BLAST and non-Cartesian k-t BLAST performed on prospectively accelerated cardiac images with a temporal resolution of 65.76ms and an acceleration factor of 6 for frames in diastole (top), systole (center), and the y-t space (bottom). Expanded images of the heart in diastole and systole are shown in the colored boxes (right).

VII. Appendix

A. Computational Complexity

In order to compare the complexity of the k-t BLAST, a-f BLAST, and non-Cartesian k-t BLAST reconstructions, the number of necessary processing steps was calculated. Overall processing steps required for each reconstruction can be broken into DFTs, processing of the training data, BLAST reconstruction, and gridding steps in the case of non-Cartesian k-t BLAST and a-f BLAST.

The k-t BLAST processing steps include three DFT operations to convert the training data to the x-f space, the DFT operations to convert the undersampled data to the x-f space, a BLAST reconstruction, and a final DFT to get the reconstructed images. The necessary processing steps required for k-t BLAST can be expressed as

$$B+\text{7}F$$ (14)

In (14) B represents the processing of the BLAST reconstruction and F is the processing of a DFT.

a-f BLAST requires similar steps to k-t BLAST with a complex processing of the training data and the gridding transformation. Overall a-f BLAST involves the processing of the training data (extracting Nyquist-sampled center data, cubic spline interpolation, and three DFTs to convert it to the a-f domain), three DFT operations to convert the undersampled data to the a-f domain, a BLAST reconstruction, three DFT operations to transform the results back to the k-t domain, and a final gridding step with two DFTs to obtain the reconstructed images. Therefore, the necessary processing steps for a-f BLAST can be expressed as

$$B+\text{8}F+G+T_r$$ (15)

The same notation is used in (15) as in (14) with the addition of T_T, the processing of the training data, and G which is the gridding operation. Overall, the complexity of a-f BLAST is similar to k-t BLAST.

As previously described, the processing of the training data for non-Cartesian k-t BLAST is the same as a-f BLAST but with the addition of a gridding step. Prior to the CG technique, the undersampled data must be transformed to the x-f domain through gridding and three DFT operations. The inverse problem from (4) is then iteratively solved through the use of the CG technique; each iteration requires two gridding interpolations and six DFTs to move the data from the x-f space to the k-t space and back again. Once converged, the fully reconstructed dataset is converted to the reconstructed images through an inverse DFT. The BLAST reconstruction is defined as the multiplication of the squared training dataset with the undersampled data. It is assumed that this application takes the same amount of time as k-t and a-f BLAST. The processing steps for non-Cartesian k-t BLAST can be expressed as

$$n_i(B+\text{6}F+\text{2}G)+\text{3}F+\text{2}G+T_T$$ (16)

In (16) all notation is the same as in (14) and (15) with the addition of n_i which is the number of iterations required for the CG reconstruction to converge. In [17] n_i was set to 10; therefore, the processing steps can be written as

$$\text{10}B+\text{63}F+\text{22}G+T_T$$ (17)

Note that the non-Cartesian k-t BLAST formulation requires 9 more BLAST operations, 21 more gridding operations, and 55 more DFTs than a-f BLAST, and thus requires significantly more computing power and time.