MRI using a concentric rings trajectory


  • Hochong H. Wu,

    Corresponding author
    1. Magnetic Resonance Systems Research Laboratory, Department of Electrical Engineering, Stanford University, Stanford, California, USA
    • Packard Electrical Engineering, Room 208, 350 Serra Mall, Stanford, CA 94305-9510
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  • Jin Hyung Lee,

    1. Magnetic Resonance Systems Research Laboratory, Department of Electrical Engineering, Stanford University, Stanford, California, USA
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  • Dwight G. Nishimura

    1. Magnetic Resonance Systems Research Laboratory, Department of Electrical Engineering, Stanford University, Stanford, California, USA
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The concentric rings two-dimensional (2D) k-space trajectory provides an alternative way to sample polar data. By collecting 2D k-space data in a series of rings, many unique properties are observed. The concentric rings are inherently centric-ordered, provide a smooth weighting in k-space, and enable shorter total scan times. Due to these properties, the concentric rings are well-suited as a readout trajectory for magnetization-prepared studies. When non-Cartesian trajectories are used for MRI, off-resonance effects can cause blurring and degrade the image quality. For the concentric rings, off-resonance blur can be corrected by retracing rings near the center of k-space to obtain a field map with no extra excitations, and then employing multifrequency reconstruction. Simulations show that the concentric rings exhibit minimal effects due to Tmath image modulation, enable shorter scan times for a Nyquist-sampled dataset than projection-reconstruction imaging or Cartesian 2D Fourier transform (2DFT) imaging, and have more spatially distributed flow and motion properties than Cartesian sampling. Experimental results show that off-resonance blurring can be successfully corrected to obtain high-resolution images. Results also show that concentric rings effectively capture the intended contrast in a magnetization-prepared sequence. Magn Reson Med, 2007. © 2007 Wiley-Liss, Inc.

The course of development for MRI has continually seen the need to strike a balance between many goals, including scan time, image resolution, and signal-to-noise ratio (SNR). This is especially true for time-sensitive experiments such as dynamic MRI and magnetization-prepared studies. A concentric rings two-dimensional (2D) k-space trajectory can provide new dimensions of flexibility in considering the tradeoff between these goals. Concentric rings are essentially an alternative way to acquire 2D polar data, with many desirable properties that come from its circular geometry and reading out along the angular dimension. Theoretical analysis of the sampling properties and reconstruction effects of polar sampling have been presented by Lauzon et al. (1, 2). Concentric rings as an imaging sequence were first investigated by Matsui and Kohno (3) and used with a fast spin-echo (FSE) sequence by Zhou et al. (4). Liang and Lauterbur (5) also presented the idea of dynamic imaging with concentric rings. Implementations and experimental results were successfully reported, but off-resonance issues and other pertinent properties were not thoroughly investigated. Related imaging trajectories have also been studied, such as spiral-rings by Block et al. (6) and Kerr et al. (7), 3D concentric cylinders by Ruppert et al. (8), and 3D spherical shells by Shu et al. (9). These works all speak about the advantages of using concentric sampling, and it is highly desirable to have more studies that investigate the properties and potential of the concentric rings trajectory.

In this work, we present a study of the concentric rings, including implementation and design procedures, further analysis of its properties and artifacts, and applications of the technique in vivo. We implemented the concentric rings trajectory in a gradient-echo (GRE) sequence. Artifacts due to Tmath image modulation, off-resonance issues, and flow and motion were characterized by simulations. We found that in addition to being inherently centric-ordered and providing a smooth weighting in k-space, this readout trajectory enables shorter scan times and a flexible tradeoff between image resolution and scan time. These properties make the concentric rings well suited as a readout trajectory for magnetization-prepared studies and dynamic MRI. As with other non-Cartesian trajectories, off-resonance blurring in the reconstructed image is a concern. To deal with this issue, we propose a retracing method when acquiring rings near the center of k-space to obtain a field map with no extra excitations. This field map can be used in multifrequency reconstruction to correct for the blur. We present experimental results to show in vivo performance.


There are many ways to realize the idea of circular sampling for MRI. Previously, circular sampling had been incorporated into a spin-echo sequence (3) and an FSE sequence (4). In this work, we consider it in the context of a GRE sequence. We implement a specific concentric rings sequence and analyze its properties. Observations based on this sequence can be extended to more general situations.

Polar Sampling

Polar sampling, as in projection reconstruction (PR) imaging (10), is usually achieved by sampling along radial spokes that cross through the origin of k-space. There are desirable properties associated with radial sampling, such as more distributed flow and motion artifacts.

Radial polar sampling is characterized by the number of full-diameter spokes Nsp, the number of samples along each spoke Nkr, and the sample spacing dkr along the spokes (Fig. 1a). To satisfy the Nyquist criterion at the edges of k-space for a specified field-of-view (FOV), Nsp = (π/2) · Nkr spokes must be acquired. When an insufficient number of spokes are acquired, the reconstruction exhibits “streaking” artifacts (1, 2).

Figure 1.

Polar sampling. The same Nyquist-sampled polar grid in k-space can be acquired using PR spokes (a) or concentric rings (b) with sample spacing of dkr in the radial direction. The resulting 2D PSF (c) has an aliasing-free FOV that is determined by 1/dkr. A radial slice through the PSF is shown in (d).

An alternative way to acquire polar data is to sample data in a collection of concentric rings in k-space. Consider a set of uniformly-spaced rings with radial increments of dkr on a polar grid in k-space (Fig. 1b). The corresponding point spread function (PSF) of the sampling pattern consists of ring lobes at 1/dkr (Fig. 1c and d) (1, 2). Note that since both the PR spokes and the concentric rings cover the same polar grid in this example, they have the same PSF. The FOV is determined by the spacing between these ring lobes, which depends on the choice of dkr. Since the concentric rings have the readout in the angular dimension and phase encoding (PE) in the radial direction, the angular dimension can be oversampled to avoid streaking artifacts.

Trajectory and Gradient Design

A set of N uniformly-spaced concentric rings is used to sample k-space (Fig. 2a). Ring n has radius n · kr,Max/(N − 1), where n = 0,1…N−1. A time-optimal gradient (Fig. 2b and c) is designed for the outermost ring of radius kr,Max = (N − 1)/FOV, which dictates the achievable image resolution. As in Fig. 2b, an initial dephaser gradient is used to first reach the desired radial extent in k-space, stopping at (kr,Max, 0). Sinusoidal gradients are then played out to traverse the readout ring counterclockwise, which is sampled at intervals of Δk = 1/FOV to satisfy the Nyquist criterion. After traversing the ring, a rephaser gradient leads the trajectory back to the k-space origin.

Figure 2.

Gradient design. A set of readout rings is used to cover k-space (a). Gradients are designed for the outermost ring (b) (slew rate shown in (c)) and then scaled down for each ring. We can be more gradient-efficient with the inner readout rings (d) by using a retracing design (e) (slew rate shown in (f)). The inner rings are sampled over two revolutions, corresponding to Set1 and Set2, with a time difference of ΔT in between. Sampling locations are denoted by x for Set1 and o for Set2. The sampling locations for Set1 and Set2 are shown to be evenly interleaved to illustrate the concept, but the rationale is still valid if the relative spacing between x and o is different or even if they coincide. A field map can be computed from the retraced inner rings and used for off-resonance correction.

The first step of the design procedure is to calculate the sinusoidal gradients, which are of the form in Eq. [1] for samples m = 0,1,…ceil(T0/Ts).

equation image(1)

The amplitude Gr and period T0 of the sinusoids can be determined by considering the sampling interval Ts, the maximum gradient amplitude GMax, and the maximum slew rate SMax (see Appendix) (11).

Besides explicitly using Eq. [1] to calculate the sinusoidal gradients, an alternative method is to use the analytically determined Gr and T0 in a time-stepped numerical procedure (see Appendix) that follows from the analysis in Ref.12. This time-stepped procedure monitors the discrete nature of the gradient waveforms more closely, and also has the flexibility to extend to spiral-ring designs (6, 7) and incorporate additional constraints. For example, an asymmetric FOV can be achieved by imposing different FOV constraints when traversing different parts of k-space. We used this time-stepped procedure to generate sinusoidal gradients for all experiments.

The second step of the design is to calculate the dephaser and rephaser gradients. The dephaser gradients are designed to start from the center of k-space at the maximum slew rate and finalize at (kr,Max, 0), transitioning into the gradient-limited regime determined by kr,Max, the sampling interval Ts, and the FOV. An optimization procedure (13) is used to find the time-optimal solution Gdep,x(t) and Gdep,y(t) for the dephasers. To form the rephasers Grep,x(t) and Grep,y(t), the dephasers are mirrored as in Eq. [2]. Tdep denotes the full length of the dephasers.

equation image(2)

Once we have designed this set of gradients for the outermost ring, we scale them down by a factor of n/(N1) for n = 0,1…N−1 to acquire all rings, one ring per TR. We keep the length of TR constant for each repetition and include a gradient spoiler at the end of each repetition. A short readout window is also used to reduce off-resonance and Tmath image effects. This design achieves k-space sampling at a constant angular velocity, which results in a sampling density identical to PR. For the case of n = 0, we sample the constant frequency (DC) point repeatedly throughout the readout window (3, 4). An additional benefit of this design is that gradient delays and timing errors will only manifest as a bulk rotation in the reconstruction.

Retracing Inner Rings for Off-Resonance Correction

For non-Cartesian trajectories, off-resonance effects often cause blurring of the reconstructed image. Even with a high sampling rate, the readout window may not be sufficiently short to avoid off-resonance blurring entirely. As a result, many off-resonance correction (ORC) algorithms have been developed to address the blurring and realize the benefits of non-Cartesian trajectories (14–17). Of the large class of algorithms that rely on information about the field distribution, it is essential to first acquire a field map of reasonable fidelity. However, such a map is often obtained at the expense of extra scan time.

For the concentric rings trajectory, its circular geometry actually allows us to acquire a field map efficiently within the scan itself, without any extra excitations. Off-resonance effects evolve over the length of the readout window, which is constrained by the outermost ring. Although we sample throughout the readout window for each ring, we can be more efficient with gradients when sampling rings in the central region of k-space. That is, instead of collecting all the samples in one revolution around these inner rings, we can retrace them to collect the full set of samples in two or more revolutions within the fixed readout window. Such an approach provides the concentric rings trajectory with the built-in ability to calculate a field map with no additional excitations.

Specifically, for the inner rings n = 0,1…(N/2)−1 (Fig. 2d), we acquire a full set of samples by tracing the rings twice during the fixed readout window to acquire Set1 and Set2, respectively (Fig. 2e and f). For illustrative purposes, the sampling locations “x” for Set1 and “o” for Set2 are shown to be evenly interleaved (Fig. 2d and e), but the rationale and algorithm we propose would still hold if the relative spacing between them were different or even if they coincided. Together, Set1 and Set2 constitute the full set of samples for each inner ring, and as before, the total number of samples acquired for each ring is the same. Since these retraced inner rings are on a twice-oversampled polar grid with this sampling density, both Set1 and Set2 can be used individually to reconstruct a low-resolution full-FOV image, with a time difference of ΔT in between (Fig. 2e). Consequently, a field map spanning the range −1/2ΔT ∼ +1/2ΔT Hz can be calculated from these two images and used for ORC. It is conceivable to use more than two revolutions, but the main goal of retracing in this work will be for ORC. Thus two revolutions are sufficient to demonstrate the technique.

Image Reconstruction

We used gridding (18) to reconstruct our images. Alternatively, one could format the data appropriately and employ filtered-back projection or convolution-back projection, since circular sampling still acquires data on a polar grid. The tradeoff between image resolution and aliasing energy is different for gridding and convolution-back projection (2).

Off-resonance blurring is corrected by first computing a field map

equation image(3)

from the retraced inner rings, and then performing multifrequency reconstruction (14–17). The number of frequency bins needed, L, is determined by the following equation (14):

equation image(4)

where Δfmax denotes the maximum off-resonance and Trdout is the length of the readout. In practice, Δfmax can be set to the maximum off-resonance frequency resolved by fmap, or to a reasonable value by inspecting the histogram of fmap. All acquired samples are used in the final reconstruction by first phase-aligning the two sets Set1 and Set2 (17). The full reconstruction procedure is illustrated in Fig. 3.

Figure 3.

Full reconstruction procedure. Each of the inner rings in k-space is retraced and sampled over two revolutions, which correspond to Set1 (denoted by x) and Set2 (denoted by o). Set1 and Set2 have a time difference of ΔT in between, thus a field map spanning ±1/(2ΔT) Hz can be computed and used for multifrequency reconstruction of the full k-space dataset. The number of frequency bins needed is 8ΔfmaxTrdout. Both Set1 and Set2 are used to reconstruct the final ORC image.

Properties of the Concentric Rings

We investigated various properties of the concentric rings trajectory that may affect the quality of the reconstruction. Important aspects include off-resonance, Tmath image modulation, flow and motion properties, SNR, and scan time efficiency.

The concentric rings trajectory we have implemented starts from the three o'clock position at (kr, 0), revolving counterclockwise around each ring. Off-resonance causes additional phase accrual over the duration of the readout rings, with a common discontinuity along the positive kx-axis. For a point object, the reconstruction becomes an increasingly blurred impulse as the degree of off-resonance increases. To help characterize the off-resonance effects, we simulated a numerical head phantom with a layer of fat circling a disk of gray matter.

Tmath image decay causes amplitude modulation during the course of each ring readout, forming a common discontinuous boundary at the positive kx-axis, just as in the case of off-resonance. Compared to off-resonance, however, the effects of Tmath image modulation are relatively benign. In the current implementation of the retracing design, each revolution of the inner rings is completed over a short readout window of 3.2 ms, for a total readout time of 6.4 ms. Thus a Tmath image of 10 ms would result in a 30% signal reduction over the duration of Set1, causing only minor blurring in the reconstruction. For Tmath image of 100 ms, this is only a 5% decay over the acquisition of Set1, and the blurring is very mild.

Sampling in the angular dimension results in more distributed flow and motion artifacts for the concentric rings than conventional Cartesian sampling, as is the case for the 2DPR trajectory. Furthermore, acquiring one ring per TR has interesting effects. Since each ring contains full-FOV information for a particular frequency band, a quick bandpass snapshot of the object is obtained every TR, albeit at different frequencies as the trajectory progresses through k-space. Through-plane flow and in-plane motion were simulated to allow comparison between 2D Fourier transform (2DFT), 2DPR, and the concentric rings.

There is a penalty in the theoretical SNR efficiency for the current implementation of the concentric rings due to nonuniform sampling. Using the measure of relative SNR efficiency (19), η, the concentric rings trajectory currently implemented has η = 0.87, identical to 2DPR.

When satisfying Nyquist sampling constraints for a prescribed FOV and in-plane resolution, the concentric rings trajectory requires a shorter total scan time compared to 2DPR and 2DFT. This holds over a considerable range of FOVs and resolution, as seen in the simulation results. Although each single TR may be longer for the concentric rings than for 2DPR or 2DFT to accommodate a larger number of samples per excitation, the concentric rings require a smaller number of excitations as each ring covers four quadrants in 2D k-space (4).

Magnetization-Prepared Imaging With Rings

The development of fast GRE imaging sequences has allowed rapid acquisition of a desired imaging section. However, the image contrast obtained with these fast GRE sequences is diminished as compared to conventional T1 or T2-weighted images. To achieve arbitrary image contrast with these fast GRE sequences, the idea of snapshot MRI, or magnetization-prepared imaging, was proposed (20, 21). In such an experiment, the magnetization is first prepared using a preparation RF pulse, and then a fast GRE sequence is used for rapid readout of the intended contrast.

For magnetization-prepared imaging, it is desired to capture as much of the intended contrast as possible in a smooth and time-efficient way. This can be a challenge since the acquisition occurs during signal transition and the prepared contrast disappears quickly. The signal transition creates a weighting of k-space data, which can affect both the final image contrast and the image resolution. Strategies to deal with this weighting effect include centric phase-encode ordering (22, 23) and segmented k-space acquisition (24, 25). Using centric ordering, the intended contrast can be captured quickly by acquiring the low spatial frequencies first. Centric ordering also allows the weighting in k-space to be fairly symmetric with respect to the origin, which can be beneficial in shaping the PSF due to this weighting (22). Segmented acquisitions can help retain the prepared contrast and minimize degradation of image resolution by facilitating a tradeoff between the degree of signal transition that occurs over data acquisition and the total scan time. These two strategies can be used independently or in combination.

From the discussion of strategies to address signal transition, we can appreciate that the concentric rings are well suited as a readout trajectory for magnetization-prepared imaging. Concentric rings are inherently centric-ordered, thus being able to capture the prepared contrast effectively and rapidly. The shorter scan time for concentric rings, as compared to 2DFT or 2DPR, limits the amount of signal transition that occurs even before using k-space segmentation. Furthermore, the concentric rings provide smooth weighting in k-space by distributing the signal transition isotropically in two dimensions. As a result, the image resolution will be isotropic in two dimensions, and the blurring in each individual dimension will be less severe than if the signal transition were concentrated along one centric-ordering dimension (23), as in the case of centric-ordered 2DFT.

To demonstrate the effectiveness of the concentric rings for magnetization-prepared imaging, we specifically consider an inversion-recovery (IR) experiment. The desired set of k-space encodings is acquired as P interleaved segments of Q encodings. For example, if 128 rings are to be acquired at Q = 16 rings per segment, then the P = 8 segments would be rings [0, 8, 16, …, 120], [1, 9, 17, …, 121] up to [7, 15, 23, …, 127]. Interleaved segmentation is more effective at minimizing the discontinuities in k-space weighting due to signal transition than sequential segmentation (24). After each preparatory 180° inversion pulse and evolution over an inversion time (TI), Q encodings are acquired to capture the intended contrast (Fig. 4). Trajectories such as concentric rings, centric-ordered 2DFT, and 2DPR can be used for acquisition. Concentric rings have the advantage that contrast is captured the most rapidly by immediately acquiring the low spatial-frequency information. For centric-ordered 2DFT, the k-space weighting produces no loss of resolution in the readout x direction, but can lead to blurring in the phase-encoding y direction (23). For the rings, k-space weighting is distributed isotropically in two dimensions. Depending on the segmentation parameters P and Q used for acquisition, blurring in the y direction may be reduced in certain cases for the rings.

Figure 4.

Acquisition during IR signal transition. Following an inversion pulse and evolution over a desired inversion time TI, k-space data is acquired as the transitory signal returns to equilibrium. Readout trajectories such as 2DPR, centric-ordered 2DFT, and concentric rings can be used. The rings have the advantage that intended contrast at TI is captured the most rapidly by immediately sampling the low spatial-frequencies and the signal transition creates a smooth weighting in k-space.



Off-Resonance and Tmath image Simulations

As shown in Fig. 5a, a numerical head phantom was used to investigate the effects of off-resonance and Tmath image modulation. The phantom consists of an outer layer and an inner core, and we vary the degree of off-resonance and Tmath image in simulations. A 128-ring trajectory without retracing was used. The readout window for each ring was 3.2 ms.

Figure 5.

Head phantom simulation for Tmath image and off-resonance. A numerical phantom consisting of two layers was used (a). Both layers were first considered to be on-resonance, and Tmath image was varied from 1 s (a) to 10 ms (b) and 1 ms (c). Minimal effects are seen for (a) and (b). Only when Tmath image reaches 1 ms do artifacts become noticeable (arrow in (c)). The Tmath image values were then set to 85 ms for the outer layer and 100 ms for the inner core. The amount of off-resonance for the outer layer was increased from 50 Hz (d) to 100 Hz (e) and 200 Hz (f). Off-resonance effects are minimal for 50 Hz, but become noticeable at 100 Hz. At 200 Hz, off-resonance creates “fringes” in the outer layer (arrow in (f)).

In Fig. 5a, b, and c, off-resonance was fixed at zero, with Tmath image varying from 1 s to 1 ms. Minimal effects are seen for Tmath image values down to an order of 10 ms. Only when Tmath image reaches an order of 1 ms does the modulation create significant degradation (arrow in Fig. 5c).

The set of images in Fig. 5d, e, and f investigated off-resonance. We set the Tmath image of the outer layer and the inner core to 85 ms and 100 ms, respectively, to model fat and gray matter. The degree of off-resonance for the outer layer was set to 50 Hz, 100 Hz, and 200 Hz to appreciate the artifact characteristics. No ORC was performed. For an off-resonance of 50 Hz, 0.16 cycles of extra phase is accumulated and minimal effects are observed (Fig. 5d). For higher off-resonances within 156 Hz, phase accrual over the 3.2-ms window is still less than 0.5 cycles, but artifacts begin to be observed in the reconstruction (Fig. 5e). When the off-resonance reaches 200 Hz, there is wrapping of phase over the [−π, π] range, and artifacts clearly manifest in the reconstruction. Here, we observe an interesting “fringe” artifact, where the fat signal is split into two halves and pushed to the edges of the original structure (arrow in Fig. 5f). Off-resonance blurring can be corrected by using our proposed retracing method. High degrees of off-resonance not characterized by the field map did not cause significant “fringes” in the off-resonance-corrected images (see Experiments).

Flow and Motion Simulations

The effects of flow and motion were considered for 2DFT, 2DPR, and concentric rings (Fig. 6). A 128-ring trajectory without retracing was used in the simulations. The readout window for each ring was 3.2 ms. PE for 2DFT was arranged in the x, or left-right, direction. Full-diameter spokes were used for 2DPR.

Figure 6.

Flow and motion simulations. Simulations were done for 2DFT, 2DPR, and the rings. Column (a) is through-plane pulsatile flow, (b) is in-plane contraction/dilation of a disk, and (c) is in-plane left-right linear movement. Images are windowed up to show the artifacts.

To investigate through-plane pulsatile flow (Fig. 6a), we considered a phantom that was the cross-section of a circular vessel with periodically pulsing blood flow. Acquisitions were simulated over 30 in-flow pulses, since the average scan time of our in vivo experiments was about 30 s or 30 heartbeats. Flow compensation was assumed in the slice direction, and only magnitude effects were simulated. Ghosting is observed for 2DFT in the phase-encoding direction. Very faint streaking occurs for 2DPR. Low-amplitude background halos are seen for the rings, as the pulsatile flow creates aliased datasets.

In-plane beating of an object (Fig. 6b) has characteristics similar to pulsatile flow. Here, a periodically contracting/dilating disk was used for simulation. The radius of the disk was increased by 50% during each contraction/dilation cycle and 30 cycles were simulated. Again, 2DFT shows ghosting in the phase-encoding direction, 2DPR exhibits streaking, and the rings exhibit halos distributed in the FOV.

We also considered in-plane linear movement in the left-right direction (Fig. 6c). Here, the motion is a slow left-right drift from −FOV/32 to +FOV/32 during the scan. For the rings, a particular ring of spatial frequencies is captured each TR over the course of the movement, DC being visible to the left where the object resides initially, with edges progressing over the FOV as the object moves. 2DFT also shows a progression of the edges, but along with faint ghosting across the path of movement. For 2DPR, the incomplete dataset creates a large amount of signal spraying over the extent of the motion.

Scan Time Comparison

The minimum scan time required for a Nyquist-sampled data set was compared for 2DFT, full-diameter 2DPR, and the concentric rings. The minimum TR and number of TRs required were calculated for each sequence over a range of prescribed values: FOV of 18 to 24 cm, isotropic in-plane resolution of 0.4 to 1.2 mm. The same readout bandwidth of ±125 kHz was used for each sequence. Total scan times and the ratios are plotted for comparison (Fig. 7). The concentric rings require the least total scan time for all cases considered.

Figure 7.

Scan time comparison. The minimum total scan time for a Nyquist-sampled dataset was computed over a range of prescribed values for 2DFT, 2DPR, and the rings (a). The ratio of the total scan times is also plotted in (b). The concentric rings enable shorter total scan times than 2DPR or 2DFT.

The concentric rings are faster than 2DPR, with total scan time being approximately half of what 2DPR requires (Fig. 7b). Although the minimum TR is longer for concentric rings than for 2DPR to accommodate more samples per excitation, concentric rings cover k-space in a smaller number of repetitions. For example, a Nyquist-sampled polar grid of 400 full-diameter PR spokes at 256 samples per spoke can be covered by 128 rings at 800 samples per ring. With fewer excitations, the rings sequence omits overhead such as RF excitation and gradient spoiling associated with each repetition. A trend can be seen in Fig. 7b. At low resolution, the rings offer a great reduction in scan time over 2DPR. As the resolution becomes finer, this advantage decreases. This is because higher resolution (for a fixed FOV) for the rings is gained in the current design by expanding the outermost ring and recalculating readout gradients, which increases the number of rings required and also expands the readout window for all rings. On the other hand, higher resolution (for a fixed FOV) can be gained for 2DPR by slightly increasing the readout window for each original spoke.

Off-resonance effects may impose a practical limitation to the length of the readout window for concentric rings. For the imaging experiments we consider, a high readout bandwidth of ±125 kHz allows the readout duration of the outermost ring (hence all rings) to be an acceptable 6.4 ms for a FOV of 24 cm and isotropic in-plane resolution of 0.47 mm. To achieve even finer resolution, we may have to modify the current design and acquire some of the outer rings that exceed a particular readout window length with more than one excitation. However, this modification was not necessary for our imaging experiments.

The rings are also slightly faster than 2DFT. In the current implementation, the rings cover the polar grid with a sampling density resembling 2DPR. Therefore, a greater number of underlying samples is required for a Nyquist-sampled dataset regarding the same prescribed FOV and resolution. This disadvantage is compensated again by the fact that concentric rings require a smaller number of excitations than 2DFT. In the end, the rings enable shorter scan times than 2DFT. The ratio of scan time for 2DFT/rings also diminishes as resolution becomes finer, similar to the previous case for 2DPR/rings (Fig. 7b).


Experiments were conducted on a GE Signa 1.5 T Excite system, with Gmax of 40 mT/m and Smax of 150 mT/m/ms. The readout filter bandwidth was ±125 kHz for all experiments and readout trajectories. Full images were reconstructed with gridding. Written consent was obtained prior to scanning for all subjects.

Phantom Experiments

A sequence of 256 readout rings, with the central 128 retraced, was acquired for a 5-mm slice with a 24-cm FOV, yielding an isotropic in-plane resolution of 0.47 mm. The readout window was 6.4 ms for all rings, with the inner rings acquired over two revolutions of 3.2 ms within the readout window. The time difference of ΔT = 3.2 ms between Set1 and Set2 of the inner rings allowed for a field map spanning ±156 Hz. A 5-bin multifrequency reconstruction was used over the ±156 Hz range. In the uncorrected image (Fig. 8a), blurring is evident at the water/air boundary (arrow). By using the field map (Fig. 8b) for ORC, the blurred signal (Fig. 8c) can be refocused. The final off-resonance–corrected image (Fig. 8d) shows no blurring at the water–air interface (arrow).

Figure 8.

Resolution phantom. A total of 256 rings, with the central 128 retraced, were acquired over a FOV of 24 cm. Plain reconstruction (a) exhibits blurring at the water–air interface (arrow). A field map can be obtained from the retraced inner rings (b) and used for off-resonance correction to refocus the blurred signal (c). The final off-resonance corrected image (d) shows no blurring (arrow).

In Vivo Experiments

We acquired an axial brain image of a normal volunteer near the sinus, using the same sequence as in the phantom experiments. Again, a 5-bin multifrequency reconstruction was used to obtain the final off-resonance–corrected image. Imaging parameters were as follows: slice = 5 mm, FOV = 24 cm, TE = 2 ms, TR = 30 ms, and flip angle = 30°. RF spoiling was used. TE for the rings is defined as the time between the peak of the RF excitation and the first readout sample.

Figure 9 shows this slice taken near the sinus, which exhibits a fair amount of off-resonance at the air–tissue boundaries. Signal loss can be seen near the sinus cavity (arrow in Fig. 9a). A field map (Fig. 9b) is used for ORC to refocus the blurred signal near the sinus (arrow in Fig. 9c). As a note, the flat contrast seen here is due to the imaging parameters and not a result of using the concentric rings for readout.

Figure 9.

Slice near sinus, with off-resonance correction. A total of 256 rings, with the central 128 retraced, were acquired over a FOV of 24 cm. Plain reconstruction (a) exhibits signal loss near the sinus (arrow). By calculating a field map (b) from retraced inner rings and using off-resonance correction, the final image (c) shows refocusing of signal near the sinus (arrow).

Inversion-Recovery Experiments

We acquired an axial brain image of a normal volunteer, comparing 3 readout trajectories: 128 rings at 804 samples/ring with TE/TR = 2.1 ms/8.2 ms, 256-by-256 centric-ordered 2DFT with TE/TR = 3.5 ms/7.6 ms, and 256-by-256 full-diameter 2DPR with TE/TR = 3.5 ms/7.6 ms. No retracing was done for the rings here, as the off-resonance effects did not cause much blurring in this slice. Scan parameters were fixed for all trajectories at TI = 750 ms, slice = 5 mm, flip angle = 30°, FOV = 24 cm, in-plane resolution = 0.93 mm by 0.93 mm, and readout bandwidth = ±125 kHz. The TI = 750 ms was chosen to introduce T1 contrast and suppress fluid signal. TI is defined here as the time between the peak of the IR pulse to the peak of the first excitation pulse. 2DPR is at a relative disadvantage for this kind of experiment, since each spoke acquires data from the low to high spatial frequencies and the final image contrast may not be a clear representation of the intended preparation. But we still include 2DPR here to be consistent in presenting a comparison between 2DFT, 2DPR, and the concentric rings.

First, we considered a single 180° inversion pulse followed by acquisition of all the encodings (Fig. 10a, b, and c). Scan time per image was 1.8 s using the rings (P = 1, Q = 128), and 2.7 s for 2DPR (P = 1, Q = 256) and 2DFT (P = 1, Q = 256). The reconstructed image for 2DPR suffers from various inconsistencies of the dataset and much of the contrast captured is from the steady state, not the preparation (Fig. 10a). Both centric-ordered 2DFT (Fig. 10b) and the concentric rings (Fig. 10c) track the IR-prepared contrast, but the 2DFT image shows horizontal blurring (phase-encoding direction) due to k-space weighting and flow effects (arrows in Fig. 10b). The rings also exhibit better white and gray matter contrast-to-noise ratio than centric-ordered 2DFT. Note that the scan time for the ring images was 33% shorter than that for 2DFT or 2DPR.

Figure 10.

IR experiment with TI = 750 ms, using a single 180° pulse followed by 256-by-256 2DPR (a), 256-by-256 centric-ordered 2DFT (b), and 128 concentric rings at 804 samples/ring with no retracing (c). 2DPR does not capture the intended contrast. Both centric-ordered 2DFT and the rings track the intended contrast, but centric-ordered 2DFT exhibits horizontal blurring (arrows in (b)) due to flow and k-space weighting effects. The experiment was repeated using a segmented acquisition of k-space, with 2DPR at 16 interleaved segments of 16 spokes/segment (d), centric-ordered 2DFT at 16 interleaved segments of 16 lines/segment (e), and rings at eight interleaved segments of 16 rings/segment (f). 2DPR still does not capture the intended contrast. Centric-ordered 2DFT has better top-down (readout direction) resolution than the rings, but the rings require a shorter scan time and show better contrast-to-noise ratio than centric-ordered 2DFT. All images have the same window/level.

We then considered a segmented k-space acquisition of the same slice, using Q = 16 encodings per interleaved segment for each trajectory (Fig. 10d, e, and f). Recovery time was 5 s from the end of readout to the next 180° inversion pulse. Scan time per image was 42 s using the rings (P = 8, Q = 16), and 89 s for 2DPR (P = 16, Q = 16) and 2DFT (P = 16, Q = 16). Both centric-ordered 2DFT (Fig. 10e) and the rings (Fig. 10f) capture the intended contrast, which is now enhanced as compared to the previous case with a single inversion pulse. By acquiring Q = 16 encodings per preparation pulse, k-space weighting effects also become less dramatic than the previous case, as can be seen for centric-ordered 2DFT. Centric-ordered 2DFT now has slightly better top-down (readout direction) resolution than the rings, but the rings still exhibit better white and gray matter contrast-to-noise ratio than centric-ordered 2DFT. 2DPR with interleaved segments still could not capture the intended contrast as effectively as centric-ordered 2DFT or concentric rings (Fig. 10d) (26). Again, note that the ring images required 52% less scan time than that for 2DFT or 2DPR.


The concentric rings trajectory provides an alternative way to sample polar data, with many interesting properties that originate from its circular geometry. When acquiring data with the concentric rings, a full-FOV image always exists, and resolution builds up in an isotropic manner with each new ring. Prepared contrast can be captured smoothly and efficiently by starting from the center of k-space and expanding outwards.

Concentric rings require fewer excitations for a Nyquist-sampled data set when compared to 2DPR and 2DFT. As a result, the rings can enable shorter total scan times in many cases. Time savings up to two times over 2DPR for the same polar data set can be achieved. A Nyquist-sampled data set was considered in this work as the basis for theoretical comparison; however, in practice, 2DPR acquisitions usually collect a smaller number of spokes. Reducing the number of spokes causes streaking artifacts and results in a smaller aliasing-free FOV, which is actually tolerable to quite an extent. Typically, 256 full-diameter spokes with 256 samples along each spoke may be collected when Nyquist constraints warrant a 403-spoke full acquisition. Further reduction to fewer than 256 spokes is also done in practice, which is often the case for reduced-FOV imaging (27–29). We used a 256-by-256 full-diameter 2DPR trajectory for our IR experiments, but even with a reduced acquisition of 256 2DPR spokes, the concentric rings could still cover the original Nyquist-sampled polar grid within a shorter scan time.

The motion and flow properties of the rings are also quite distinct from that of 2DPR or 2DFT. The rings, like 2DPR, have more spatially distributed flow and motion artifacts than 2DFT. Seen from the simulation, 2DPR is more tolerant of pulsatile through-plane flow, while the concentric rings are more robust in the presence of in-plane linear drift. Both 2DPR and the rings have more distributed artifacts than 2DFT for in-plane contraction/dilation. In addition, we observe interesting reconstructions that directly relate to the underlying readout scheme of acquiring a full-FOV bandpass ring each TR. These simulations are by no means exhaustive, but they do provide insight into the type of artifacts to expect when employing the concentric rings trajectory.

From its distinct properties, the concentric rings are well suited for applications in magnetization-prepared studies. The concentric rings are inherently centric-ordered, offer smooth weighting in k-space, and have a shorter total scan time. Prepared contrast is thus captured rapidly and smoothly by the rings. As demonstrated, IR preparation can be combined with the concentric rings trajectory for contrast augmentation and species suppression, with better contrast-to-noise ratio and a shorter scan time than centric-ordered 2DFT. It is possible to extend this concept and use the rings with other types of magnetization-prepared studies (30, 31). These same advantages also make the rings well suited for FSE sequences (4, 6).

For dynamic imaging, the same properties considered for magnetization-prepared imaging also make the rings an attractive choice. Furthermore, since each ring represents a specific band in spatial frequency, we can utilize this feature to flexibly update rings that cover the lower spatial frequencies more often than those that represent the higher spatial frequencies during the course of the scan (5, 32–36).

Off-resonance effects pose a limitation to the application of non-Cartesian trajectories. For the concentric rings, this is exacerbated by the fact that the low spatial frequencies are sampled with minimal gradient amplitudes (4). The retracing method proposed in this work addresses the problem in two ways: higher gradient amplitude in the central region of k-space and the ability to acquire a field map without any extra excitations. This field map is used in multifrequency reconstruction to effectively correct the off-resonance blurring. The current implementation obtains a field map of limited frequency range, thus high degrees of off-resonance or abrupt changes in the field distribution can be potential problems. It is possible to tweak the retracing design to acquire a field map of greater range that suits the imaging experiment.

To further limit the degree of off-resonance artifacts, preparation such as fat-suppression or saturation can be used with the rings, as is often done in spiral imaging (37). We have seen that the concentric rings are effective at capturing prepared contrast. Therefore, fat suppression methods such as IR can be used in combination with the concentric rings trajectory to reduce the effects of off-resonance (4, 6). Smaller degrees of off-resonance can be corrected using the proposed ORC scheme.

Improvements to the present design are possible. Since we designed gradients for the outermost ring and scaled them down to cover k-space, the current design can be readily extended to variable-density k-space coverage. Variable-density sampling strategies offer even more flexibility in balancing SNR, scan time, and resolution (2, 19). The proposed retracing design may also be extended to more than two revolutions for other potential applications, such as retracing the inner rings three or more times to enable multiple-point Dixon reconstruction techniques (38, 39). It is also natural to consider a variable number of rings per excitation, such as spiral-rings (6, 7), or using separately optimized gradient designs for different rings. Asymmetric FOVs can be obtained by incorporating additional constraints in the gradient design procedure. The rings can also be combined with partial k-space approaches and multiple-echo acquisitions. Data can be sampled during the dephaser and rephaser gradients for purposes such as motion correction.

The implementation and analysis of the concentric rings k-space trajectory presented in this work focused on 2D imaging. Extension to concentric 3D coverage is possible by using 2D multislice acquisition schemes or related 3D concentric trajectories such as “stack-of-rings,” “rings-PR-hybrid” (40), concentric cylinders (8), and concentric spheres (9). A 2D multislice acquisition could allow interleaved ordering of slices to accommodate long recovery times and maximize tissue contrast, while a 3D readout trajectory would allow improvements in SNR. Characteristics of a 2D multislice acquisition should follow directly from what we have analyzed in this work, while the behavior of 3D concentric trajectories is a closely related but distinct subject of their own. Many properties of concentric sampling discussed in the context of this work would carry over to the case of 3D concentric imaging, including the advantages for magnetization-prepared imaging, dynamic imaging, and scan-time efficiency.


We implemented and analyzed a concentric rings 2D k-space trajectory which has the built-in ability to correct for off-resonance effects. The concentric rings are inherently centric-ordered, provide a smooth weighting in k-space, and enable shorter total scan times, making it well-suited as a readout trajectory for magnetization-prepared studies. Experimental results show that high-resolution images are obtainable and prepared magnetization can be captured effectively.


H.W. is supported by a Sony Stanford Graduate Fellowship.


Calculation of Sinusoidal Gradients

Sinusoidal gradients to traverse the outermost ring in k-space can be formulated as Eq. [1] (restated below) for samples m = 0,1,…ceil(T0/Ts).

equation image(1)

The amplitude Gr and period T0 of the sinusoids can be determined by following Eq. [A1] through Eq. [A4] below, where Ts is the sampling interval, γ is the gyromagnetic ratio, GMax is the maximum gradient amplitude, and SMax is the maximum slew rate (11). In Eq. [A1], G0 is limited by both the maximum gradient amplitude and the sampling requirements for a prescribed FOV.

equation image(A1)
equation image(A2)
equation image(A3)
equation image(A4)

Besides explicitly using Eq. [1] to calculate the sinusoidal gradients, an alternative method is to use the analytically determined Gr and T0 in a time-stepped numerical procedure. Following the analysis in Ref.12, the desired k-space trajectory for the outermost ring is

equation image(A5)


equation image(A6)
equation image(A7)

The design goal is to find θ(t) while satisfying the following constraints:

equation image(A8)
equation image(A9)

Starting with the initial conditions θ = 0 and equation image is calculated at each time step and then summed discretely to obtain equation image(t) and θ(t). Conditions in Eqs. [A8] and [A9] are imposed at each time step. At the end of the process, G(t) and S(t) are calculated from θ(t) using Eqs. [A6] and [A7].