Evaluations of k-space Trajectories for Fast MR Imaging for project of the course EE591, Fall 2004
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1 Evaluations of k-space Trajectories for Fast MR Imaging for project of the course EE591, Fall 24 1 Alec Chi-Wah Wong Department of Electrical Engineering University of Southern California 374 McClintock Avenue, EEB Los Angeles, CA , USA cwwong@ieee.org Abstract In this paper, various k-space trajectories that are beneficial for fast MR Imaging are explored. First, artifacts of the uniformly under-sampled spiral trajectories are observed. Next, variable-density spiral trajectory is presented to reduce aliasing artifacts. Most energy of the image is concentrated in the low frequency coefficients, which are resided in the center region. By up-sampling the center region and down-sampling the outer region, aliasing can be significantly reduced. On the other hand, frequency can be sampled to a larger extent and hence, the resolution of the image is improved. Second, a 2DFT-based trajectory, the Circular EPI (CEPI), is presented. Some images may have less detail in one direction and more detail in the other direction. By adopting k-space trajectories having different frequency extents in phase encoding and readout directions, the number of readout samples can be reduced without significant degradation of image quality. Third, based on the above observations, a variable-density elliptical spiral k-space trajectory is proposed. It is useful for imaging blood vessels which have long parallel lines when cooperating with selective excitation. I. INTRODUCTION In real-time/fast MR imaging, it is necessary to capture multiple images in short time. As a result, the readout time, or equivalently the number of readout samples per image, is limited. It is desirable to capture the most useful details while minimize artifacts. On the other hand, the shorter the readout time, the more insensitivity to off-resonance the image is. In this article, several kinds of trajectories which are capable of performing fast imaging are discussed. In the first section, spiral trajectory is presented [4], [5]. In the second section, Circular Echo Planar Imaging (CEPI) is discussed [2]. In the third section, elliptical spiral trajectory is proposed. Lastly, a concluding remark is given.
2 2 II. SPIRAL TRAJECTORY Aliasing results when the excited object is larger than the Field Of View (FOV) [4], [5]. In normal 2DFT imaging, aliasing in readout direction can be avoided at no cost by increasing the sampling rate and using an anti-aliasing filter. However, in spiral imaging, aliasing can occur in both directions and cannot be easily avoided. Increasing FOV can solve aliasing problem but the scan time increases accordingly. This is not desirable in fast imaging where the image is captured within breathhold, and in real-time imaging where multiple images are captured within short time. Thus, it is advantageous to reduce aliasing in other means without increasing readout time. In this section, the advantages of using variable-density spiral are explored. Figure 1 shows the original phantom image. Fig. 1. Phantom image of size A. Methods Sometimes the readout or phase encode are not performed on Cartesian planes and hence, non-cartesian image reconstruction is needed [3]. In particular, we adopt 2 gridding reconstruction with deapodization and post-density compensation. A triangular kernel is used. Figure 2 shows the flowchart of the noncartesian reconstruction used. k-space data acquistion 2X gridding reconstruction with post-density compensation and deapodization Extract the middle portion of the reconstructed image Fig. 2. Flow diagram of non-cartesian reconstruction.
3 3 B. Aliasing in spiral trajectory Next, aliasing effects are presented on uniformly under-sampled spiral. In this experiment, four-interleaved spiral is used with each interleave having 28 loops. The total number of data points for each spiral is 5. Figure 3 shows the k-space trajectory normalized to [-.5:.5). It is observed that spiral aliasing interferes with the original image. The most significant one is the additional aliased circle Fig. 3. obtained. Uniformly undersampled spiral. Four interleaved spiral trajectory with 28 loops for each spiral. 5 data points for each spiral is In general, most energy is concentrated in the low frequency region. Thus, by increasing the FOV of low frequency component, aliasing can be reduced. This corresponds to up-sample the low frequency region. As the total number of samples is constant, high frequency components are correspondingly downsampled. As a result, variable-density spiral trajectory is introduced [4], [5]. There are two major ways to achieve variable density spiral. The first method is to use two different spiral densities, one for low frequency region and one for high frequency region. The second method is to use linearly increasing spiral densities from low to high frequency regions. C. Double-density spiral trajectory Figures 4, 5 and 6 show the images reconstructed using double-density spiral. In the center region of the k-space, a dense spiral is used. In the outer region of the k-space, the density is reduced so that in each interleave, the total number of loops and the samples are still 28 and 5, respectively. Two parameters are defined in this experiment α and β. The meaning is such that for each interleave, α loops are within the inner β% of the k-space region. It is observed that the aliasing circle disappear and the image looks cleaner. However, there is an inner loop like spiral noise appears with different sizes by varying α and β. It is noticed that α = 26 and
4 (a) α = 22, β = (b) α = 26, β = 36 Fig. 4. Reduction of aliasing by using double-density spiral trajectories. For each interleave, α loops are within the inner β% of the k-space region. 28 loops are used for each interleave with a total of 5 data points for each interleave. In this figure, β = 36 β = 36 serve the best compromise among the four. It is because it has the least spiral nose and aliasing, while the resolution is not too low. D. Linearly increasing-density spiral Another kind of variable-density spiral is that the density of loop increases from center to outer region of k-space. In this experiment, one parameter is defined α. It is defined such that α inner spiral loops are uniformly up-sampled, with the density of remaining loops increases linearly. The overall spiral trajectory is scaled such that all loops fall within the k-space. Figure 7 shows the effect of aliasing reduction using
5 ky (a) α = 18, β = (b) α = 26, β = 5 Fig. 5. Reduction of aliasing by using double-density spiral trajectories. For each interleave, α loops are within the inner β% of the k-space region. 28 loops are used for each interleave with a total of 5 data points for each interleave. In this figure, β = 5 this method. It is observed that the aliased circle in the original phantom disappears. Similar to the double-density case, spiral noise appears in the reconstructed image. However, it looks more severe than using doubledensity spiral. There are more than one inner loops appear in the reconstructed image.
6 (a) α = 22, β = (b) α = 27, β = 6 Fig. 6. Reduction of aliasing by using double-density spiral trajectories. For each interleave, α loops are within the inner β% of the k-space region. 28 loops are used for each interleave with a total of 5 data points for each interleave. In this figure, β = 6 E. Parameterization of spiral trajectories The equation of Achimedes spiral in radial space can be written as r = awt (1) where r is the radius at time t, and ω is the angular frequency. a is a scaling factor for controlling the spiral density, i.e. the inter-loop distance of spiral. The relationship between k-space and gradients can be summarized as follows: k x (t) = γ 2π k y (t) = γ 2π t G x(τ)dτ t G y(τ)dτ (2)
7 (a) α = (b) α = (c) α = (d) α = 26 Fig. 7. Reduction of aliasing by using linearly increasing density spiral trajectories. For each interleave, α loops uniformly up-sampled and the densities of the remaining loops are increasing. 28 loops are used for each interleave with a total of 5 data points for each interleave.
8 8 The conversion formula from radial space to cartesian coordinates are as follows: Hence, the gradients can be obtained as follows: k x (t) = r cos(ωt) = aωt cos(ωt) k y (t) = r cos(ωt) = aωt sin(ωt) (3) Figure 8 shows an example of the gradients. G x (t) = aω 2 t sin(ωt) + aω cos(ωt) G y (t) = aω 2 t cos(ωt) + aω sin(ωt) (4) 5 G x (t) t 5 G y (t) t Fig. 8. Gradients for spiral trajectory. It is noticed that the tips of the sine functions in the gradient waveforms increase in a approximately linear fashion. If variable-density spiral is used, the tips of the sine functions become exponentially increasing. However, in designing the variable-density spiral or any trajectories, the hardware limitations cannot be exceeded as follows: G 4G/cm dg 16G/cm/ms (5) dt
9 9 III. CIRCULAR EPI Echo Planar Imaging (EPI) is a data acquisition method used in MR imaging which allows rapid readout. Rather than acquiring a single image readout line in k space after the encoding phase of the pulse sequence, the entire MR image is acquired. In general, the method is that multiple rather than single image lines are acquired at each spin-echo. Nayak et. al. in [2] proposed a Circular EPI (CEPI) with different frequency extents in phase encode and readout directions. In some situations, the excited object in one direction is more detailed than the other direction. As a result, by decreasing the resolution in the less detailed direction, it is possible to retain the image quality but can significantly reduce the total scan time, because the total number of samples decreases. Figure 9 displays the CEPI trajectory and the reconstructed image Fig. 9. CEPI trajectory and reconstructed image. To observe the effect of using different frequency extents in phase encode and readout directions, three CEPI variations are presented, which is shown in Figure 1. It is observed that the shape of horizontal lines is preserved. However, the vertical lines are blurred significantly because of the lower resolution in direction. Thus, this technique is useful when the object is oriented in one direction. IV. ELLIPTICAL SPIRAL TRAJECTORY Based on the above two major types of trajectories, a elliptical spiral trajectory is proposed. It is useful in applications where long parallel lines are imaged with different FOVs in x and y directions. Figure 11 shows the phantom used in this experiment. It is noticed that the object allows different FOV in x and y directions, with more details in the direction with a smaller FOV.
10 (a) 1 : 2 CEPI (b) 1 : 4 CEPI (c) 1 : 8 CEPI Fig. 1. CEPI trajectories with different extents in phase encode and readout directions.
11 11 Fig. 11. A phantom image of size With this kind of objects, elliptical spiral trajectories become useful. Figure 12 shows a uniform spiral trajectory. Significant aliasing is observed Fig. 12. Generic spiral trajectory. Two interleaves are used. Each interleave has 1 data points and 38 loops. A. Uniform density elliptical spiral trajectory By using uniform density elliptical spiral, the FOV of the less detailed direction is increased with decreased resolution. In this type of trajectory, trade-off is taken between FOV and resolution. The larger the FOV, the lower the resolution. Figure 13 shows the effect of using different trade-off between FOV and resolution. By increasing the FOV in x direction, the spiral in the direction becomes denser. As the number of loops remain unchanged, the frequency extent in direction is decreased and hence, the
12 12 resolution is reduced. It is observed that aliasing is removed in all elliptical spiral trajectories. In this particular phantom, 1:2 is sufficient to achieve aliasing suppression. B. Variable density elliptical spiral trajectory Now, the total number of samples and the number of loops for each interleave is further reduced such that for each interleave, 28 loops and 6 data points are used. There are still two interleaves in the trajectory. Figure 14 shows the corresponding reconstructed images of the phantom. It is observed that the aliasing is even more severe than the previous experiment, due to the reduced number of loops and samples. In 1:4 elliptical, the aliasing is basically removed. However, the resolution is too low. Therefore, variable density elliptical spiral is proposed so that 1:2 elliptical spiral can be used while aliasing is significantly suppressed. Figure 15 shows the reconstructed image using linearlyincreasing density elliptical spiral. You can see that the aliasing is significantly reduced. As a result, using elliptical spiral with varying density, image quality is retained with significantly reduced number of samples and hence reduced readout time. C. Parameterization of elliptical spiral trajectories For elliptical spiral, the radial coordinate equation is as follows: r = ωt (e cos(ωt))2 + (sin(ωt) 2 ) (6) The gradient waveform can be generated using Equations 2 and 3 and the gradients can be obtained using Matlab using the diff function. An example of the elliptical spiral gradient waveform is shown in Figure 16. (with e =.5) The drawback of using this trajectory is that the waveform looks complicated. Figure 8 shows an example of the gradients. V. CONCLUSION In this article, different types of k-space trajectories that are useful for reducing scan time are mentioned. All of them achieve significant reduction of scan time without much degradation of the image quality. Since the scan time is reduced, off-resonance artifacts are less severe. To further reduce the scan time, partial k-space acquisition and reconstruction can be used. All the trajectories mentioned above can be directly applied with partial k-space reconstruction. For instance, in spiral trajectories, even number of interleaves, e.g. 2n are used. The locations in k-space in interleave number a is the conjugate of the locations in interleave number a + n. Therefore, by appropriately omiting interleaves, partial k-space reconstruction can be used. Since CEPI directly uses 2DFT reconstruction, capability of partial k-space reconstruction is more obvious.
13 (a) 1 : 2 elliptical (b) 1 : 4 elliptical (c) 1 : 8 elliptical Fig. 13. Elliptical spiral trajectories and the reconstructed images. Two interleaves are used. Each interleave has 1 data points and 38 loops.
14 (a) circular (b) 1 : 2 elliptical (c) 1 : 4 elliptical Fig. 14. Elliptical spiral trajectories and the reconstructed images. Two interleaves are used. Each interleave has 6 data points and 28 loops.
15 (a) 1 : 2 elliptical, with 25% of the loops are concentrated in the central region (b) 1 : 2 elliptical, with 5% of the loops are concentrated in the central region, Fig. 15. Variable-density elliptical spiral trajectories and the reconstructed images. Two interleaves are used. Each interleave has 6 data points and 28 loops. REFERENCES [1] Zhi-Pei Liang, Paul C. Lauterbur. Principles of magnetic resonance imaging : a signal processing perspective, Bellingham, Wash. : SPIE Optical Engineering Press ; New York : IEEE Press, 2. [2] Krishna S. Nayak et. al., Real-time interactive coronary MRA, Magnetic Resonance in Medicine, 46:43 435, 21. [3] John M. Pauly, Reconstruction of non-cartesian data, Image Reconstruction Textbook, In progress. [4] Christoph Schroder et. al., Spatial excitation using variable-density spiral trajectories, Journal of Magnetic Resonance Imaging, 18: , 23. [5] Chi-Ming Tsai and Dwight G. Nishimura, Reduced aliasing artifacts using variable-density k-space sampling trajectories, Magnetic Resonance in Medicine, 43: , 2.
16 G x (t) G y (t) t Fig. 16. Gradients for elliptical spiral trajectory.
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