Module 4. K-Space Symmetry. Review. K-Space Review. K-Space Symmetry. Partial or Fractional Echo. Half or Partial Fourier HASTE

Transcription

1 MRES Fast Imaging Techniques Module 4 K-Space Symmetry Review K-Space Review K-Space Symmetry Partial or Fractional Echo Half or Partial Fourier HASTE Conditions for successful reconstruction Interpolation and Zero Filling k-space Interpolation Elliptical filtering VIBE Summary 1/18

2 Review In 7004, module 2, k-space was introduced, which has some unique properties. By taking advantage of these properties, the acquisition time of a sequence can be reduced, or the echo time, or it can be used to improve image quality in single-shot methods. In 7005, module 1, single shot methods like EPI and SS-FSE were discussed. They are both prone to issues if the echo train length increases in time relative to the T 2 * or T 2 decay curve. 2/18

3 K-Space Review The raw data that is induced in the receiver coil is described as being in "k-space". It contains all the frequency and phase information necessary to reconstruct an image, after inverse Fourier Transform. In Cartesian imaging, the centre line of k-space is acquired where there is no phase encoding gradient (ky=0), and the centre of the echo is the centre of kx (kx=0). Figure 4.1: 3D representation of k-space. The central area holds all of the signal or low frequency information, while the outside of k-space contains the noise plus high frequency or edge information. These areas can be reconstructed separately as in Figure /18

4 Figure 4.2: Reconstruction of only the blue portion of k-space is on the bottom left, and just the outside or red frame of k-space is reconstructed on the bottom right. This is a very important feature when thinking of increasing speed for temporal resolution, which will be covered in more depth in Module 5. 4/18

5 K-space Symmetry During quadrature detection, two signals (real and imaginary) are obtained. These are a cosine and sine signal (Figure 4.3). One portion of the signal is symmetric about the k = 0 point, and the other is anti-symmetric, its amplitude at k=-m is equal and opposite that at k=m. This is also referred to as conjugate symmetry. In quadrature detection, the value at k(m,n) in k-space is A + ib and k(-m,n) is A - ib or the complex conjugate. (A is the real signal and B is the imaginary signal) Figure 4.3: The cosine and sine waves generated by signal detected from two receivers, 90 out of sync. If only the data in the positive portion of k-space is collected, it is a simple task to reconstruct the data for the negative k values. If the object has no phase information, then only half of k-space needs to be acquired, and the other half of k-space is calculated. In practice, the symmetry is not perfect, due to phase fluctuations introduced by B 0 inhomogeneity, motion, or T 2 decay. This means that more than half of k-space needs to be sampled to accurately calculate the missing part of k-space and avoid reconstruction artefacts. 5/18

6 Partial/Fractional Echo Collecting a partial echo means only acquiring part of the k x values (See Figure 4.4 for an example of a half-echo). This is done by reducing the readout dephasing gradient so the centre of the echo starts sooner than normal. For a half echo, there is no dephasing gradient (Figure 4.5a). Figure 4.4: Half of k-space is acquired so that each echo starts immediately, from the centre of k-space. In a conventional sequence, this reduces the echo time. For multiple echo or single shot sequences, the effective echo time is slightly reduced. The impact is not as big as for a sequence like EPI, as each alternate line still starts from the edge of k-space and travels back to the centre (Figure 4.5 b). There is a reduction in echo spacing however, which reduces the geometric distortions in EPI, and reduces blurring in ssfse. The acquisition time is not normally reduced, but might be slightly reduced if reducing the TE allows a shorter TR. Since only half of Nx is actually acquired, the readout gradient is either half as long with the same gradient amplitude, or uses a lower gradient amplitude and the same, or a longer time. The later possibility affects the receiver bandwidth, and can result in a larger water-fat shift. This can be useful when characterising the contribution of fat and water using DIXON or IDEAL sequences. 6/18

7 a) b) Figure 4.5: a) The readout dephasing is removed in conventional imaging and the echo signal starts from the maximum point. b) EPI sampling of k-space with partial echo collection. 7/18

8 Half / Partial Fourier Half or partial Fourier is where part of the k y dimension in k-space is sampled and the rest calculated through conjugate synthesis (an example of half Fourier is given in Figure 4.6). Figure 4.6: Half Fourier sampling of k-space, where only half of k y points are covered. In conventional imaging and FSE this is achieved by using only positive phase encoding gradients, starting with zero phase. In EPI there is only a frequency pre-phase gradient, rather than pre-phase gradients in both frequency and phase (Figure 4.7). Figure 4.7: The effective echo time is reduced in partial Fourier EPI, and the echo train length is halved, resulting in less geometric distortions. In a 3D method, partial Fourier can be applied along either k y or k z directions, as k-space symmetry extends to the k z direction as well. 8/18

9 HASTE HASTE, or half Fourier acquisition single shot turbo spin echo, is a single shot FSE method, which only acquires half of the k y lines of k-space and uses half Fourier method to calculate the other half. It is takes half the acquisition time of a ssfse with the same resolution. The advantage of HASTE over ssfse is that the echo train length is half as long. There is less T 2 decay during sampling, and blurring in the images should be less. Although there can be other artefacts due to the conjugate synthesis reconstruction. a) b) Figure 4.8: a) Pulse sequence diagram of HASTE, and b) the corresponding sampling in k-space. The HASTE sequence can be segmented, so that the desired half of k-space is collected in several repetitions. The images are heavily T 2 weighted, and have a great sensitivity for fluid detection. Motion artefacts are minimized due to the speed of acquisition. HASTE is primarily used for breath-hold imaging. 9/18

10 Figure 4.9: Fat suppressed HASTE image of the liver at 3T. Recommended Reading: Mirrowitz SA, et al. Rapid acquisition spin-echo (RASE) MR imaging: a new technique for reduction of artifacts and acquisition time. Radiology, 175: , Kiefer B, et al. Image acquisition in a second with half-fourier acquisition single shot turbo spin-echo. J Magn Reson Imaging, 4[suppl P]:86-87, /18

11 Conditions for successful reconstruction Conjugate synthesis requires a few conditions to work properly. The user must not be interested in any phase deviations along the image, due to local field inhomogeneities or motion. Secondly, any phase variations must be small, otherwise the reconstruction will result in ghost-like artefacts from locations with large phase shifts. Raw data must be well-centred in k-space for the reflection property to be accurate. This requirement is achieved by a combination of hardware and software engineering. In general, greater than half of k-space is usually sampled as it creates less artefacts, and the centre of k-space holds most of the signal and contrast. The reconstruction of partial k-space images can be improved by using phased constrained reconstruction, which corrects the phase errors. The incidental phase variations that cause problems with the conjugate synthesis calculation can be removed in a number of ways: 1. Homodyne Detection A reference image can be generated from the original image by passing it through a low-pass filter. 2. POCS: projection onto convex set Iteratively generates the missing data and corrects for any k-space offset. 3. Margosian: A low resolution phase map can be generated from a central portion of k-space. 11/18

12 Interpolation and Zero Filling Interpolation of MR image data is possible, such that the final resolution of the image is higher than the acquisition matrix. There are two predominant methods to perform this; the first is zero-filling in k-space. In order to achieve this, the centre of k-space (lower frequency information) is sampled, and the high spatial frequency information is not acquired, but "zero-filled" (Figure 4.10). This means the missing data in the k-space grid is substituted with zeroes before the inverse Fourier Transform or reconstruction. Figure 4.10: Showing zero filling in k-space prior to Fourier transform for a Cartesian sequence. The second method is to perform an image interpolation, where you use the original pixel intensities to determine the value in your new smaller pixels. There are several methods of doing this (e.g. sinc, trilinear, cubic spline). Sinc interpolation in the image gives a similar result to zero-filling the raw k-space data, but can take more time. The sinc interpolation function is sinc z = (sin z)/z where z can be any complex number, including zero. a) 12/18

13 b) Figure 4.11:The result of image interpolation upon a) a low resolution image to create an image with twice the acquired resolution b). 13/18

14 k-space Interpolation Both of these approaches are different from conjugate synthesis discussed above, as the edge of k-space is not acquired on either positive or negative sides, so you can't reflect the data to calculate the missing components. You are either simply replacing with zero in the k-space domain for image frequencies that are not present, or artificially increasing the resolution in the image domain. Too much zero-filling or interpolation will lead to blurred images. You can also interpolate the raw k-space data if you have a non-cartesian sequence. A Fast Fourier Transform (FFT) requires k-space data to be regularly positioned on a grid in k-space. For example, in the case of a RADIAL sequence, the nearby radial points can be used to interpolate what the gridded or Cartesian k-space points should be (See Figure 4.12). Figure 4.12: Showing interpolation of k-space points to create Cartesian grid points from a radial sequence, prior to Fast Fourier Transform (FFT). Brighter areas are where the radial points overlap with the Cartesian Grid, and darker purple is where the data will undergo some type of weighting or interpolation to calculate what the value should be. The black edges of k-space are zero-filled. 14/18

15 Elliptical filtering Elliptical filtering is one way of improving the signal to noise, while potentially decreasing your acquisition time. A central circle of k-space is acquired and the corners are not. Since most of the noise is found in the edges of k-space, this increases the SNR. Some of the high-frequency or edge information is still being acquired, so the image is sharper than where none of the edges are sampled. This type of coverage also saves acquisition time. In the case of single shot methods like EPI, either the gradient amplitude or timing of the readout gradients are modified to achieve the coverage, and it will be 27% faster compared to acquiring full coverage of k-space. The timing between consecutive points in the phase direction changes, so there's no longer a consistent phase bandwidth. For example, the centre still takes a full echo spacing between points, but the edges of k-space there is only a small time difference between points. This means that off-resonance effects will not be a sharp shift in the phase direction, but will appear blurred. Figure 4.13: Example of EPI sequence using a circular filter. The corners of k-space are not acquired 15/18

16 VIBE VIBE or volume interpolated breath-hold examination (also known by other manufacturers as LAVA-XV, THRIVE, or TIGRE), is a 3D spoiled gradient echo method that acquires only the central portion of k z (Figure 4.14). Figure 4.14: Partial acquisition of the z dimension in k-space in VIBE. The acquisition time is greatly reduced as the number of repetitions to complete the second phase encoding direction is reduced. The image is reconstructed to a larger resolution by filling in the missing lines in k y -k z with zeroes.that is, the acquisition size could be 80x40 (Ny x Nz), but the final reconstruction size is 80x80. There is a loss in SNR due to the smaller acquisition size. VIBE provides a fast, 3D, T 1 -weighting method that is usually used for abdominal breath-hold imaging. It normally employs some type of fat saturation. a) b) 16/18

17 Figure 4.15: VIBE images of a) the wrist and b) the abdomen, acquired at 3T. Reference: Rofsky NM, Lee VS, Laub G, Pollack MA, Krinsky GA, Thomasson D, et al. Abdominal MR imaging with a volumetric interpolated breth-hold examination. Radiology 1999;212: /18

18 Summary The symmetry and features of k-space allow for improved acquisition speed and signal to noise. Even an already rapid sequence can be increased by a reduced sampling of k-space. Conjugate synthesis, zero-filling, and interpolation can all be used to increase acquisition speed and reduce some errors. For example, HASTE is a preferred method to single shot FSE as only half the echo train length needs to be acquired and it is not as prone to blurring. Zero-filling and interpolation become very important in 3-dimensional methods like VIBE, in order to allow rapid sampling. In the next module we will continue with our discussion of k-space undersampling, but in order to increase temporal resolution for sequences that are designed to follow an event over time. Review Questions What features of k-space allow us to under sample and image faster? How does the sequence HASTE work? What are its advantages and disadvantages? How does the sequence VIBE work? What are its advantages and disadvantages? What is elliptical filtering? What are its advantages and disadvantages? What is zero filling? What are its advantages and disadvantages? What is interpolation? What are its advantages and disadvantages? When would you use either elliptical filtering, zero filling or interpolation compared to full acquisition? When would you use any one technique compared to the others? 18/18