A.M.D. is a founder and holds equity in CorTech Labs, Inc., and also serves on the Scientific Advisory Board. The terms of this arrangement have been reviewed and approved by the University of California, San Diego, in accordance with its conflict of interest policy.

Full Paper

# Real-time rigid body motion correction and shimming using cloverleaf navigators

Article first published online: 6 OCT 2006

DOI: 10.1002/mrm.21038

Copyright © 2006 Wiley-Liss, Inc.

Additional Information

#### How to Cite

van der Kouwe, A. J.W., Benner, T. and Dale, A. M. (2006), Real-time rigid body motion correction and shimming using cloverleaf navigators. Magn Reson Med, 56: 1019–1032. doi: 10.1002/mrm.21038

#### Publication History

- Issue published online: 20 OCT 2006
- Article first published online: 6 OCT 2006
- Manuscript Accepted: 7 JUL 2006
- Manuscript Revised: 27 JUN 2006
- Manuscript Received: 10 FEB 2006

#### Funded by

- National Institutes of Health. Grant Numbers: NIBIB R21EB02530, NCRR P41RR14075, NIBIB R01EB1550, NCRR R01EB00790, NINDS 5P01NS035611, NCRR U24RR021382 (BIRN)
- Mental Illness and Neuroscience Discovery (MIND) Institute
- Siemens Medical Solutions

- Abstract
- Article
- References
- Cited By

### Keywords:

- motion correction;
- shim correction;
- real-time correction;
- brain imaging;
- clover leaf navigator

### Abstract

- Top of page
- Abstract
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- APPENDIX A
- APPENDIX B
- REFERENCES

Subject motion during scanning can greatly reduce MRI image quality and is a major reason for discarding data in both clinical and research scanning. The quality of the high-resolution structural data used for morphometric analysis is especially compromised by subject movement because high-resolution scans are of longer duration. A method is presented that measures and corrects rigid body motion and associated first-order shim changes in real time, using a pulse sequence with embedded cloverleaf navigators and a feedback control mechanism. The procedure requires a 12-s preliminary mapping scan. A single-path, 4.2-ms cloverleaf navigator is inserted every repetition time (TR) after the readout of a 3D fast low-angle shot (FLASH) sequence, requiring no additional RF pulses and minimally impacting scan duration. Every TR, a rigid body motion estimate is made and a correction is fed back to adjust the gradients and shim offsets. Images are corrected and reconstructed on the scanner computer for immediate access. Correction for between-scan motion can be accomplished by using the same reference map for each scan repetition. Human and phantom tests demonstrated a consistent improvement in image quality if motion occurred during the acquisition. Magn Reson Med, 2006. © 2006 Wiley-Liss, Inc.

A substantial amount of clinical and research data are rendered unusable because of subject motion during MRI. Certain subject populations are particularly affected, including children, older subjects, and patients with movement disorders. Longer, high-resolution scans are also more susceptible to motion-induced artifacts. This study presents a method for real-time prospective motion correction using “cloverleaf navigators” (navigators with a trajectory in *k*-space resembling a cloverleaf, as shown in Fig. 1).

Post-hoc methods for registration and averaging images for between-scan motion are well established (1–4). It is common to collect multiple scans of short duration and combine them in postprocessing using these registration algorithms to produce a single high-SNR motion-corrected average, while possibly discarding volumes that are corrupted by motion. However, these procedures do not correct for motion that occurs *during* the scans themselves. Such motion results in ghosting, blurring, and other artifacts that cannot generally be corrected by post-hoc procedures. Methods for detecting the position of an object in the scanner using an external apparatus have also been proposed, including optical (5) and magnetic (6) methods.

The first online prospective real-time methods used straight-line navigators to detect linear motion of organs in the chest (7). These techniques are not applicable for brain scans in which translations and rotations may occur. Thesen et al. (8) developed a system for “prospective motion correction” that corrects for rigid motions of the head during EPI scans in which multiple complete volumes are collected in rapid succession. After each volume in the series is collected and reconstructed, it is registered to the first volume in the series. A correction to the gradients is then made before the next volume is collected, so that the series is prospectively motion-corrected in near real time (there is a lag of one or two TRs between motion and correction). The method described here is intended to correct motion that occurs during the acquisition of a single volume, and operates during acquisition in *k*-space before image reconstruction.

Another approach, proposed by Ward et al. (9), uses so-called orbital and spherical navigators for prospective rigid body motion detection and correction. The orbital (circular) navigator enables the detection of rotation within the plane of the navigator and translations along multiple axes (10). Ward et al. (9) developed a real-time prospective motion-correction scheme in which a set of three circular navigators is used to detect motion in all three planes. An iterative approach is then taken to correct for the motion since the rotations may still be out of the plane of the navigators. The procedure is fairly time consuming and works best for rotations about the cardinal axes. The spherical navigator, which was first described by Irarrazabal and Nishimura (11) and Wong and Roos (12), and implemented in a full 3D rigid body measurement application by Welch et al. (13), addresses the problem of off-axis rotations. A limitation of this approach is the relatively long time required to acquire each spherical navigator, which can greatly affect the scan time required to achieve a certain spatial resolution and signal-to-noise ratio (SNR). Therefore, this approach has been used mainly for between-scan registration (13).

Another approach that has gained popularity is termed “periodically rotated overlapping parallel lines with enhanced reconstruction” (PROPELLER). While the original technique does not attempt to correct for motion prospectively, it has been shown that *T*_{2} acquisitions using PROPELLER in the presence of subject motion are less sensitive to *B*_{0} and motion effects than conventional *T*_{2} FSE (14, 15). PROPELLER is restricted to 2D imaging and rigid motion within the scan plane; however, Pipe and Zwart (16) recently presented an extension of PROPELLER to multislab 3D acquisitions.

The redundant acquisition of the center of *k*-space in variable-density spiral acquisitions has been exploited to provide motion correction for DTI acquisitions in a technique known as self-navigated interleaved spiral (SNAILS) (17). Liu et al. (18) presented a multishot sensitivity encoded (SENSE)-enabled diffusion-weighted SNAILS sequence that additionally uses the oversampling of the center of *k*-space for sensitivity self-calibration. Variable-density spirals have been combined with orbital navigators in a sequence that provides motion compensation and off-resonance correction (19). Ward et al. (20) described a method for real-time first-order shimming during EPI time-course imaging using a “shim NAV.”

The approach presented here was developed to correct motion that occurs during the acquisition of a single volume, and operates during acquisition in *k*-space before reconstruction. Some of the advantages of the previous methods are combined to provide a complete real-time rigid body motion correction scheme with simultaneous first-order shim correction. In this work we used a single-path “cloverleaf” navigator that occupies 4.2 ms of the TR, and thus minimally impacts the total length of the sequence. Compared to spherical and circular trajectories, the cloverleaf trajectory offers the advantages of a shorter duration and simple accurate translation estimates due to the linear segments through the center of *k*-space. Furthermore, preliminary mapping facilitates estimation of off-axis rotations. Corrections to gradient orientation and shim values are made in real time every TR.

### MATERIALS AND METHODS

- Top of page
- Abstract
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- APPENDIX A
- APPENDIX B
- REFERENCES

#### Cloverleaf Navigator Design

It was previously demonstrated that rigid body rotations can be estimated using a single “octant” navigator, for which the trajectory in *k*-space follows the edges of an octant (bounded by adjacent 90° arcs) on the surface of a sphere (21). Several improvements on this design are now presented. To improve estimates of off-axis rotations (i.e., rotations that are not exactly in the plane of any of the three arcs), a rapidly acquired map is used as a reference to correct for out-of-plane errors. Although it is possible in principle to estimate translations using the octant navigator, the estimate is more robust if the navigator has sections that pass through the center of *k*-space. Straight-line sections are added through the center of *k*-space along all three axes in the current “cloverleaf” navigator design. The resulting *k*-space trajectory is illustrated in Fig. 1. A small, straight excursion is made from the center of *k*-space along the readout direction to a little over a third of the radius of the cloverleaf, at which point the traversal reverses direction and reaches constant velocity at a third of the radius before continuing through the center of *k*-space in the readout direction and continuing at constant velocity along the entire trajectory illustrated in Fig. 1. The received signal is sampled during the constant velocity section from the one-third point along the readout direction until it returns to the same point, including the three traversals through the center of *k*-space and the three arcs.

The gradients necessary to implement the cloverleaf navigator are shown in Fig. 2 along with their integrals that equate to the *k*-space trajectory. The figure shows that the sampling of the straight-line sections is asymmetrical in the first and last segments, which ensures that there is a continuous traversal through the center of *k*-space along each axis. The navigator consists of sinusoidal sections for the arcs, constant gradient sections for the traversals through the center of *k*-space, and transition sections (for the loops and smoothed ramps) described by quartic splines. The functions that describe the trajectory are designed to be maximally smooth (vanishing in the fourth or fifth derivative except along the arcs) to reduce slew rate limitations. The slew rate is maximal during the loops (transitions between the straight line sections and the arcs), and this limits the maximum attainable amplitude of the navigator. To simplify the translation calculations, the navigator amplitude is expressed in mm^{–1}. This refers to the radius of the arcs in *k*-space, corresponds to the effective resolution of the navigator, and is proportional to the required gradient amplitudes. Typically, an amplitude of around 0.125 mm^{–1} is used.

The cloverleaf navigator is implemented as a building block with a duration of 4.2 ms. This block is easily incorporated into appropriate sequences, such as 3D fast low-angle shot (FLASH) and 3D echo-planar imaging (EPI). The navigator kernel is not limited by the sample rate or the SNR. The maximum radius in *k*-space is limited by the slew rate at the rounded corners. In principle, any navigator shape that extends in all three dimensions, when used in combination with a preliminary map of navigator rotations (as described below), would suffice to provide rigid body motion estimates. However, computation of the estimates is simplified if the cloverleaf navigator design is used, as described in the following sections.

#### Implementation in a 3D FLASH Sequence

Since sufficient structure in all three dimensions of a reasonably thick slab is required for a good motion estimate (including through-plane translations and out-of-plane rotations), the cloverleaf navigator is best applied in a sequence with a naturally thick slab that is excited every TR, such as 3D FLASH. In our 3D FLASH implementation, the navigator and imaging readout share the same RF excitation pulse. The navigator can be inserted into a sequence kernel after the RF excitation pulse, either before or after the imaging readout, and before the spoiler. Our sequence, as implemented in Siemens IDEA, allows either configuration to be selected.

Four seconds of dummy scans are included at the beginning of the imaging sequence. No data are acquired in the first 3 s while steady state is being attained. In the fourth second, navigators are acquired and corrections are made, but no image lines are acquired. This allows the sequence to correct itself for any positional changes that were made after the subject's map was acquired, and before imaging begins. The required number of dummy TRs is calculated to achieve this fixed timing in seconds (e.g., 200 dummy TRs of 20 ms each).

For every TR of the sequence (i.e., for every navigator), the rotations and translations of the object relative to the initial map are calculated. In the FLASH implementation, we have demonstrated that we can feed back the rotation and translation estimates from the computer receiving the signal samples to the computer controlling the gradients, and make the appropriate correction in real-time at a rate of 71 Hz (TR = 14 ms).

#### Mapping Sequence

Ward et al. (9) described how to estimate rotations within the plane of a circular navigator by measuring the shift in the *k*-space signal along the circle. However, simultaneous out-of-plane rotations can result in features entering or leaving the navigator, which in turn leads to inaccurate in-plane rotation estimates. This is still a problem if a cloverleaf navigator or three perpendicular circular navigators are used. To correct out-of-plane errors, one can collect a map of the *k*-space features in the neighborhood of the cloverleaf navigator by rotating the gradients a few degrees about all three axes and acquiring example navigators at each position. The mapping procedure is completed in a few seconds before the main scan. The subject must remain motionless while the map is acquired. The scanner provides immediate feedback, related to subject motion during the map, after acquisition of the map is complete. This feedback is presented on screen as a line graph (DICOM overlay) together with an overall figure of merit that reflects how much motion occurred during the mapping. This information can be used by the operator to determine whether the map should be reacquired.

When rotations are integrated into the control system, quaternions are used in the software to describe rotations, since this representation is immune to the “gimbal lock” problem of Euler angles (in which two axes align so that a degree of freedom is lost) and does not accumulate errors over successive multiplications (repeated small corrections) the way rigid body rotation matrices do. Because translations and rotations are cleanly separated in the navigator analysis, the disadvantage that quaternions do not represent translations is irrelevant in this application.

The vector part of a quaternion defines the axis of rotation, and the scalar part is given by the cosine of half the angle of rotation about this axis. Therefore, the quaternion representation derives naturally from the “axis-angle” description of rotations. This representation can be used to easily calculate the smallest rotation required to rotate an object from any arbitrary orientation to a desired orientation (22). In one example of the navigator map, navigators are acquired after rotation through every combination of a table of angles (e.g., {–0.5°, –0.375°, –0.25°, –0.125°, 0°, 0.125°, 0.25°, 0.375°, 0.5°} or some multiple of these angles) and rotation axes. The rotation axes are the vectors representing the equally spaced points on a supertessellated icosahedron with 42 vertices as shown in Fig. 3. Note that these are the axes of rotation—not the rotations themselves. In terms of Euler angles, the procedure is equivalent to testing all combinations of small rotations about three orthogonal axes. The map, which consists of 378 navigators, is acquired in 12 s in the FLASH implementation (TR = 20 ms). A small range of angles is sufficient for the map, since during imaging, every TR the fed-back correction ensures that the following navigator realigns to the center of the map, so that the angular range of the map need only exceed the rotational error expected in a single TR. In the Results section, it is shown that a map of rotations in the range of –0.5° to 0.5° is sufficient to correct for absolute rotations of several degrees. In fact, a finer, more local map is preferred because it improves the accuracy of the small corrections from one TR to the next, each of which does not exceed the limits of the map. The null-rotated navigator is sampled redundantly throughout the map. These null-rotated navigators are averaged to calculate a reliable reference navigator for translation estimation. The errors between null-rotated navigators are used to gauge motion during the map and to generate the line graph provided to the operator afterwards. It is also possible to estimate the absolute motion during the map using the constrained method for estimating rotations, as described later in the text.

#### Estimation of Rotations

As a consequence of the shifting property of the Fourier transform, translations in real space correspond to phase rolls in *k*-space:

- (1)

Moreover, rotations in real space correspond to rotations in *k*-space. Therefore, rotations can be estimated from the navigator magnitude information, and once the rotations have been corrected, translations in real space can be estimated from the navigator phase information and corrected by adjusting the phase of the *k*-space data.

Two methods for estimating rotations are described. The first is a very rapid linear method, and the second is a slower but more tightly constrained and potentially more accurate matching method.

The linear method assumes a direct linear relationship between the navigator samples and the rotation angles. This relationship is determined from the navigator map and encoded in the matrix **W**. The method for calculating **W** is described in Appendix A. This matrix is calculated on the scanner in less than 1 s after the map is measured and before any observations of navigators are made during imaging. Once **W** is available, a new estimate of the elemental rotation vector **a** for a new navigator observation **N** (defined in Appendix A) can be made in less than 1 ms on a 3.06 GHz Intel (Santa Clara, CA, USA) Xeon by evaluating the simple expression:

- (2)

Since a correction is made every TR, and this reorients the next acquired navigator to the center of the map, the range of rotations in the map need only be as large as the largest expected rotation over a single TR. The linear algorithm is able to estimate rotations outside the range of the map, but it has the potential to overfit the training data (navigators in the map) and thus reduce the accuracy of new rotation estimates. Although the results showed a robust performance anyway, a second, more tightly constrained algorithm that cannot overfit the data was developed for comparison.

The constrained algorithm for estimating rotations is described in Appendix B. Figure 1 defines the subregions of the navigator used in the algorithm. The navigator samples along the arcs are directly matched to the map by shifting and comparing, after the effects of out-of-plane rotations are removed. In the current implementation, angles in the range of ±10° are tested in steps of 1°. To refine the estimate to fractions of 1°, a quadratic function is fitted to the cost in the local neighborhood of the minimum, and the value for θ_{RP} that minimizes the quadratic function is calculated. While this algorithm cannot overfit the data, it is computationally far more intensive than the linear approach. The entire calculation is achieved for all three angles within a TR of 20 ms, which is fast enough for real-time feedback but substantially slower than the linear algorithm.

#### Estimation of Translations

Translations can be calculated in the frequency or space domain. In the frequency domain, translations are easily calculated from the slope of the phase of the translation sections of the complex navigator samples (T_{R}, T_{P}, and T_{S}) through the center of *k*-space in the readout, phase-encode, and slice directions, respectively. Let T_{R}^{ref}, T_{P}^{ref} and T_{S}^{ref}, represent the translation sections of the reference navigator. Then the phase difference in the readout direction is given by:

- (3)

where the division is scalar. If the *slope* operation is defined as the least-squares fit to the specified elements of a vector, then the translation in the readout direction t̂_{R} is given by

- (4)

where the interval *C* – Δ*K* < *k* < *C* + Δ*K* spans the 11 samples (Δ*K* = 5) acquired around the center sample *C* of *k*-space.

The calculation is the same for the translation t̂_{P} in the phase-encode direction. In the slice direction, however, this approach is confounded by the slab selection, i.e., the phase slope is dominated by the position of the slab as selected by the RF pulse and not by the actual position of the object. In this case, the translation is estimated using a windowed mean square error (MSE) minimization between the fast Fourier transform (FFT) of the straight-line section corresponding to the slice direction within the new navigator and reference navigators that avoids the edges of the slab. Let

- (5)

and

- (6)

where the window operation *win* above zeros all samples in the vector other than those within the indices–W^{−} < *w* < W^{+}. W^{−} and W^{+} are chosen to include all samples of the object within the slab without including the region outside of which the RF slab profile begins to fall off (about 90% of the slab width). The RF pulse duration was increased slightly to allow a sharper profile and a larger window for comparison. The translation estimate is greatly improved if the object's profile within the window contains an edge. Calculate the reference profile:

- (7)

If

- (8)

then the translation estimate t̂_{S} in the slab direction is given by

- (9)

This space domain or “projection” method can also be used to estimate the translations in the readout and phase-encode directions, although the performance of the two methods for these directions appears to be similar. In the readout direction, the translation section of the vector must be padded with zeros before the Fourier transformation because of the asymmetrical design of this particular section of the cloverleaf path in *k*-space.

Translations correspond to phase errors in the *k*-space representation of the image, and as such do not necessarily need to be corrected in real time. They are corrected by adding to every *k*-space sample a phase correction given by the Fourier shifting theorem.

If translations are estimated in the presence of uncorrected rotations, the reference navigator and the test navigator will not be exactly aligned, and there will be an error in the translation estimate. However, because rotation corrections are fed back rapidly to correct the imaging and the navigator gradients, the rotational error between each new navigator and the reference navigator is kept very small (and well within the limits of the map). Furthermore, since the translation estimates primarily use points close to the center of *k*-space, rotational errors are less significant. Potentially the navigator map could be used to construct an appropriately rotated reference navigator, but we assumed that the difference was small enough that it did not warrant such an approach. Conversely, when rotation changes are estimated in the presence of translations, there is no error because translations are manifested as phase changes only, and rotations are estimated from the navigator magnitude data.

#### Estimation of Shim Corrections

Shifts in the position of the object in the *B*_{0} field after shimming may invalidate the shim and result in offsets in the navigator trajectory in *k*-space and artifacts in the image. Moreover, these errors confound the rotations and translations estimated from the navigator, and result in invalid corrections. Linear shim errors appear in the navigator as shifts in the center of *k*-space that can be measured as shifts in the peaks measured during the three traversals through the center of *k*-space. Each shift corresponds to the projection of the X, Y, or Z shim error onto the corresponding imaging axis.

For the traversal along a given axis through the center of *k*-space, let the echo time (TE) represent the expected time from the excitation pulse to the peak. This is obtained from the reference scan derived from the initial map. Let *t*_{E} represent the observed TE. Let *G*(*t*) represent the applied gradient on this axis. Let Δ*G* represent the gradient offset on this axis due to inaccurate shim (assumed constant for the duration of one TR of the sequence).

Then the *k*-space trajectory along this axis in the presence of the shim error is

- (10)

And it is observed that

- (11)

Therefore

- (12)

where ε_{TE} is the observed shift in the peak *t*TE.

Assuming that the peak does not wander out of the constant gradient section of the navigator, i.e.,

- (13)

it follows that the gradient offset Δ*G* is given by

- (14)

Assuming that the signal peak in the center of *k*-space is smooth and spherically symmetrical, the gradient offsets on the three axes may be calculated independently. Linear shim corrections are implemented as offsets on the gradient amplifiers in the Siemens hardware, and therefore can be switched as rapidly as regular gradient pulses in the sequence.

#### Control System

Translation, rotation, and shim error estimates are calculated every TR, and corresponding corrections are made to the gradient orientation, RF pulse center frequency (for slice offset), phase on ADC readout samples, and gradient offsets (first-order shim values). Translations are estimated as absolute offsets from the starting point (head position recorded during map), i.e., the estimates are not integrated to obtain the control signal. Rather, the absolute translation estimates are low-pass-filtered using a third-order Butterworth filter with a cutoff frequency (as a fraction of the sample rate) of 0.05 for the slice-encoding direction, and 0.1 for the phase-encoding and readout directions, and used directly to adjust the RF pulse and ADC phase. Since the translation estimate in the slice-encoding direction is less reliable, the cutoff frequency is lower for this parameter. Rotation estimates are calculated relative to the current gradient orientation, i.e., the relative estimates are integrated to obtain the absolute rotation relative to the starting point. A simple proportional feedback control system is used whereby a fraction of the estimated error in the rotation, calculated using one of the two algorithms described above, is added to the absolute rotation offset every TR. The fraction is given by the “feedback gain” parameter in the sequence protocol. In this case the quaternion representation is useful for two reasons: errors do not accumulate, and the axis/angle components of the quaternion are easily used to calculate a fraction of the rotation offset. The angle in the axis/angle combination is simply multiplied by the feedback gain to yield a correction quaternion, and this is multiplied by the quaternion that describes the current orientation of the object to obtain the updated orientation. The shim control signals are also generated by integrating the estimated errors using proportional feedback, and the shim feedback gain is specified separately in the sequence protocol. The technique benefits from the regularly measured and smoothed estimates of motion and shim offset. While the estimated corrections may be noisy and derived from a limited local map of rotations, when these corrections are made very rapidly (e.g., every 20 ms), they result in smooth, robust tracking of motion over a range of several degrees and centimeters.

### RESULTS

- Top of page
- Abstract
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- APPENDIX A
- APPENDIX B
- REFERENCES

#### Human Tests

The experimental protocol was approved by the Partners Human Research Committee, and the study was in full compliance with Health Insurance Portability and Accountability Act guidelines.

##### Experiment 1

In the first experiment, two subjects were scanned with a Siemens (Erlangen, Germany) 1.5 T Sonata scanner using a single-channel birdcage coil. The purpose of the experiment was to establish whether high-quality averaged 3D volumes could be acquired in the presence of subject motion, and to compare these volumes with uncorrected averages. Six high-resolution scans (3D FLASH, 256 × 192 matrix, 1.3 mm × 1 mm in-plane resolution and 144 × 1.33 mm axial partitions with 11% oversampling, phase encoding left–right, TR = 20 ms, TE = 10 ms, T_{acq} = 7:45, flip angle = 30°, bandwidth = 160 Hz/px, navigator amplitude = 0.125 mm^{–1}, motion feedback gain = 0.4, and shim feedback gain = 0.1) were collected for each subject during a single session. During all six scans the subjects performed deliberate and random head motions at regular intervals (similarly for all scans). Three of the scans were performed with motion correction, and three were performed without motion correction. The quick linear method was used to estimate rotations, and translations were estimated using the phase slope method for the phase- and frequency-encoding directions and the projection method for the through-plane (slice-encoding) direction. The order of scans was randomized and the subjects were not informed of the scan order. The three motion-corrected scans were corrected using the same initial map. In this experiment the navigator was placed immediately after the RF pulse to test the configuration. Figure 4 shows the results for the first subject. Figure 5 shows the average of the scans without motion correction and the average of the scans with motion correction. Because the same map was used for all acquisitions, no coregistration of volumes was required before averaging.

##### Experiment 2

In the second experiment, eight high-resolution brain images of a volunteer were collected using a Siemens (Erlangen, Germany) 1.5 T Avanto scanner. The purpose of the experiment was to compare the motion estimation algorithms and shim correction. The scanning program is described in Table 1. Apart from the algorithm used to estimate rotations, and whether or not shim correction was enabled, the scan parameters were the same for all scans (3D FLASH, 256 × 192 matrix, 1.3 mm × 1 mm in-plane resolution and 112 × 1.33 mm axial partitions with 11% oversampling, phase encoding left–right, TR = 20 ms, TE = 3.2 ms, T_{acq} = 8:16, flip angle = 30°, bandwidth = 698 Hz/px, navigator amplitude = 0.125 mm^{–1}, motion feedback gain = 0.4, and shim feedback gain = 0.1). The navigator was located after the imaging readout.

Scan | Deliberate motion | Motion correction | Rotation (°) | Translation (mm) | Shim | Error | SNR | Reference | RMS difference foreground | RMS difference background |
---|---|---|---|---|---|---|---|---|---|---|

mean ± std | mean ± std | mean ± std | mean ± std | |||||||

- *
Signal-to-noise ratios (SNR) of image volumes, and root mean square differences in the foreground (brain/head) and background (ghosting) relative to reference volumes for various motion estimation algorithms. Constr. = constrained algorithm for estimating rotations, Quick = quick linear algorithm for estimating rotations.
| ||||||||||

1 (ref1) | No | None | – | – | – | – | 6.22 | 5 (ref2) | 28.13 | 13.07 |

2 | Yes | Constr. | 1.31 ± 1.26 | 0.56 ± 1.14 | – | – | 5.30 | 1 (ref1) | 31.88 | 14.86 |

3 | Yes | Constr., shim | 3.26 ± 1.22 | 1.31 ± 1.18 | 0.66 ± 0.27 | 0.00 ± 0.13 | 5.28 | 1 (ref1) | 31.07 | 15.16 |

4 | No | Constr., shim | 3.63 ± 0.15 | 1.88. ± 0.14 | 0.94 ± 0.15 | 0.00 ± 0.11 | 5.84 | 1 (ref1) | 26.77 | 13.65 |

5 (ref2) | No | None | – | – | – | – | 6.21 | 1 (ref1) | 28.13 | 13.07 |

6 | Yes | Quick | 0.80 ± 1.09 | 0.78 ± 1.11 | – | – | 5.54 | 5 (ref2) | 31.77 | 13.83 |

7 | Yes | Quick, shim | 1.74 ± 1.23 | 1.17 ± 1.40 | 0.31 ± 0.30 | 0.00 ± 0.13 | 5.43 | 5 (ref2) | 31.09 | 14.34 |

8 | Yes | None | – | – | – | – | 3.39 | 5 (ref2) | 56.19 | 28.04 |

The volunteer performed deliberate head motions, including rotations and translations about all three axes during scans 2, 3, 6, 7, and 8. The volunteer attempted to perform similar motions during each of these scans. Before runs 1, 5, and 8, an automatic alignment scout was collected and the slices of subsequent scans were automatically positioned (23) to ensure that these references were roughly aligned with one another (to reduce bias). Scans 1 and 5 served as reference scans without motion, and scan 8 served as a reference with motion but without motion correction.

Whenever motion correction was applied, translations were estimated using the phase slope method for the phase- and frequency-encoding directions, and the projection method for the through-plane (slice-encoding) direction. The estimated motion and shim feedback values for scan 3 are shown in Fig. 6 (the results were similar for all scans). Table 1 lists the means and standard deviations (SDs) for the measured and corrected rotations, and translations for each scan that was motion-corrected. The mean values drift away from zero with every subsequent scan after the reference (scans 2–4) because the subject did not return to the exact same starting position after every scan (but the motion was still corrected to the target position in the initial navigator map). For this reason, the second reference (scan 5) and second navigator map were collected after a second automatic alignment.

A binary mask was constructed for the region of signal in each of the two reference volumes using intensity thresholding to detect brain and skull. The inverse of this mask was defined as the background. The SNR for each volume was calculated as the ratio of the mean of the signal in the masked region to the variance of the signal in the background. Table 1 lists these results. Each volume was then compared voxel-by-voxel with the appropriate reference volume to give the square root of the mean square (RMS) difference in the masked region (RMS foreground) and mean square difference in the noise (RMS background).

The results show that when the subject moved, the SNR was higher and RMS differences with respect to the reference were lower with motion correction than without, regardless of the algorithm used. To determine a lower bound on the RMS difference values, the two reference volumes were compared. However, because automatic slice positioning did not exactly align them, there was still a larger difference between the reference volumes than between the motion-corrected volumes and their corresponding references. To minimize this effect, we applied Oxford Centre for Functional Magnetic Resonance Imaging of the Brain's Linear Image Registration Tool (FLIRT) (4) to register the second reference volume to the first before comparing the error and noise differences. For consistency, FLIRT was used to register all volumes to the appropriate reference as listed in Table 1. Table 1 also lists the means and SDs of the shim values for scans 3, 4, and 7. Again, since the subject moved after the map was collected, the mean shim values are not expected to be zero, and the SDs are more relevant.

Figure 7 shows representative slices through the various 3D volumes. Volumes 4 and 5 are omitted because they are very similar to volume 1. It is clear from Table 1 that although the subject attempted to move an equivalent amount during each scan, the differences in the motions, as reflected in the variances, are still substantial. In all cases, motion correction improved the SNR substantially (around 60%) and reduced the RMS difference between the motion-corrected volume and the reference volume collected without motion. The differences in the performance of the rotation estimation algorithms, and the contribution of real-time shim correction, were relatively small. Specifically, Table 1 shows that shim correction improved the SNR by less than 2%, and the quick linear algorithm outperformed the constrained algorithm by around 3–5% in terms of SNR. However, the two algorithms performed almost identically in terms of foreground RMS difference with respect to the reference. These differences in performance may be due to the differences in variance in the amount of motion during each experiment, rather than to true differences in the performance of the algorithms. To disentangle the components of the remaining noise, a more controlled experiment is required. In experiment 3 the algorithms were tested using a phantom on a motorized platform that executed exactly reproducible motions.

#### Phantom Tests

##### Experiment 3

A typical water-filled phantom is inappropriate for testing the navigators because it has insufficient structure in *k*-space along the path of the navigator to provide a unique match to the map, and rotations of phantoms containing liquid are not rigid. In contrast, a pineapple has suitable internal structure and appropriate contrast, and is sufficiently rigid to act as a good model of the head for the purpose of testing navigators. A plastic platform was constructed with distance and angle markings and a motorized lever arm so that the attached pineapple could be exactly manipulated from outside the scanner bore (Fig. 8).

On a Siemens Avanto 1.5 T scanner using a single-channel birdcage coil, we collected a map with the pineapple on the stationary platform, along with a reference scan with no motion and no motion correction, and another reference scan with no motion and motion correction using the constrained algorithm for rotation estimates and shim correction. The platform was then activated to rotate back and forth about the axis that would be anterior–posterior for a supine subject (the Y-axis of the scanner) through a total angle of about 5.5° continuously over a period of 1.8 s. Five additional scans were collected while the phantom was moving. During the first scan, motion correction was disabled. During the remaining four scans, real-time motion correction was applied using the quick linear and the constrained rotation algorithms, each with and without real-time shim correction. In all cases, the phase slope method was used to estimate translations in the phase- and frequency-encoding directions, and the projection method was used to estimate translation in the through-plane (slice-encoding) direction. Figure 9 shows a plot of the estimated motion and shim parameters for a 30-s period of one of these scans (the results for the other scans were similar). The acquired images are shown in coronal section in Fig. 10. Figure 11 shows representative axial and coronal slices with and without motion correction. The scanning protocol was the same as in experiment 2. The periodic nature of the motion results in a few dominant ghosts compared to the human subject's images, in which the ghosts were more randomly distributed in the phase-encoding direction.

To quantify the results, we constructed a mask as described in experiment 2. The SNR was calculated as the mean signal in the masked region divided by the SD of the background noise. For the scans with motion, the RMS differences in signal and noise between each scan and the reference were also calculated.

Table 2 summarizes these results. Since the SNR for the volume with no motion but with motion and shim correction had the highest SNR, it was chosen as the reference. The results show that the constrained algorithm outperformed the quick linear algorithm for estimating rotations in terms of SNR, and that shim correction improved SNR further, although only slightly. The noise estimates also support this conclusion.

Description of volume | SNR | RMS difference foreground | RMS difference background | |
---|---|---|---|---|

Motion | Correction | |||

^{}Constr. = Constrained algorithm for estimating rotations, Quick = quick linear algorithm for estimating rotations.
| ||||

No | Constr. + shim | 15.67 | (Reference) | (Reference) |

No | None | 15.96 | 16.40 | 8.89 |

Yes | None | 5.83 | 73.07 | 28.37 |

Yes | Constr. | 10.09 | 29.33 | 16.15 |

Yes | Constr. + shim | 10.73 | 25.54 | 15.09 |

Yes | Quick | 9.62 | 32.15 | 16.55 |

Yes | Quick + shim | 9.73 | 33.60 | 16.22 |

### DISCUSSION

- Top of page
- Abstract
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- APPENDIX A
- APPENDIX B
- REFERENCES

Our results indicate that a substantial improvement in image quality was achieved with motion correction for a moving object. The phantom results suggest that the use of the constrained algorithm for estimating rotations provided corrected images with a slightly better SNR and reduced signal error and noise (relative to the reference) compared to the quick linear algorithm for estimating rotations. For the phantom, real-time shim correction appeared to marginally improve image quality with both rotation estimation algorithms. Motion correction also substantially improved image quality for the human subject performing deliberate head motions, independently of the applied estimation algorithm.

In experiment 1 the navigator was positioned before the readout and after the RF excitation pulse. This arrangement is expected to result in the best motion estimates, but results in a longer TE for the image. Motion detection worked robustly, but some artifact remained. The artifact was reduced by averaging. The post-readout navigator is slightly less reliable because the signal amplitude is reduced due to the longer TE (and resulting *T*_{2}* signal decay), and because it may be subtly affected by the changing phase-encoding gradients that immediately precede it. However, it was shown in experiment 2 that this configuration is also very effective. Alternatively, a dedicated low-flip-angle RF pulse could be provided for the navigator.

The current mapping sequence runs in 12 s. During this time the subject should not move. The map is redundantly sampled and could be shortened significantly and possibly added at the beginning of the imaging sequence. This would be helpful for subjects who move almost continuously.

While the loops in the cloverleaf *k*-space trajectory may appear to add unnecessary complication and possible loss of fidelity when the path is played out on the gradients, they are actually designed to be maximally smooth and therefore relatively gentle on the gradients. Our results show that the path is reproduced very faithfully from one navigator to the next. The constant gradient sections of the navigator are used to estimate translations and shim offsets. It is assumed that the signal peaks do not wander out of these constant gradient sections, and that their amplitudes do not decrease too much in their projections on the three axes. These assumptions are justified because the corrections on all three axes are rapidly fed back every TR of the sequence along with the translation and rotation corrections. This is also why only a map of the immediate *k*-space neighborhood of the unrotated navigator is necessary even to correct rotations that far exceed the limits of the map. While the currently implemented path is completed in 4.2 ms, this could be shortened depending on gradient system performance.

The slab orientation is restricted to axial for the head so that the non-rigid parts of the head, such as the lower jaw and neck, are not included in the navigator. For translation estimation to work robustly in the slice-encoding direction, a sharp edge is required along this axis; therefore, the imaging volume is selected to include some space above the head. If the navigator volume were separately excited by its own dedicated RF pulse, it could be made larger than the imaging volume, which would limit this restriction; however, in this case a correction would have to be made for the effect of the image volume excitation on the navigator volume.

Since rotations are simulated during the map by manipulating the imaging gradients rather than by physically rotating the object, gradient nonlinearities and shim changes are unaccounted for in the map. This problem should be ameliorated if the range of motion of the object is restricted and it is centered close to the magnet isocenter. The current implementation is limited to single-channel coils. Multichannel implementation requires a correction because the real physical rotations within the sensitivity profiles of the coil elements differ from those mapped by rotating the gradients. The technology has not been tested in other body parts; however, provided that the anatomy of interest moves within the FOV as a single rigid body, the approach should be valid.

The cloverleaf navigator approach is well suited to 3D sequences in which the entire scan volume is excited at every TR. However, a 2D implementation in which the navigator has a separate dedicated excitation pulse is conceivable. In 2D applications, an additional low-energy RF pulse would be required to excite a thicker slab for the navigator between slice excitations.

### CONCLUSIONS

- Top of page
- Abstract
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- APPENDIX A
- APPENDIX B
- REFERENCES

The results presented above suggest that the cloverleaf motion correction technology may substantially improve imaging in subject populations that are traditionally difficult to scan optimally because they move during an MRI acquisition. The current 3D FLASH implementation was motivated by the need for motion correction in the longer, high-resolution scans that are commonly acquired for brain morphometry research. As this type of research becomes more common in pediatric and older subjects, the need for motion correction is expected to increase. Other techniques, such as parallel imaging, can reduce scan times, and a combination of techniques may be optimal. We have demonstrated that cloverleaf navigators can improve the image quality of 3D FLASH scans.

### Acknowledgements

- Top of page
- Abstract
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- APPENDIX A
- APPENDIX B
- REFERENCES

We thank Evelina Busa, Bruce Fischl, Douglas Greve, Franz Hebrank, Berthold Kiefer, John Kirsch, Bernd Kühn, Andreas Potthast, David Salat, Franz Schmitt, Simon Sigalovsky, Alto Stemmer, and Lawrence Wald for valuable insights, technical support, and other help on this project.

### APPENDIX A

- Top of page
- Abstract
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- APPENDIX A
- APPENDIX B
- REFERENCES

#### Linear Algorithm for Estimating Rotations

Let **N** represent a column vector of samples acquired along the cloverleaf navigator path in *k*-space, with all but the samples in the rotation sections set to zero.

Let **N**_{i}, where *i*∈{1,..,*N*}, represent a map of navigators, collected along *N* axis-angle combinations, and represent this map as a matrix **M**, where

A 3 × 3 real skew-symmetric matrix (spin tensor) conventionally represents an infinitesimal rotation of a rigid body in 3D space, i.e., a small rotation about an axis (24). In the limit as the angles approach zero, this representation, the quaternion representation, and the Euler angle representation are equivalent. This “elemental” structure is imposed on a rotation matrix by taking the *n*-th root as follows: Let **R**_{i} represent the rigid body rotation matrix corresponding to the *i*-th axis-angle combination in the map, e.g., in the 378-navigator map that was designed for this method. Then let **a**_{i} represent the vector of elemental angles corresponding with rotation *i*, calculated as follows:

where *n* is the order of the root that imposes the elemental structure on **R**_{i} when it is sufficiently high in value. Theoretically, *n* should be infinite, but in practice a value for *n* of ≥7 was found to be sufficient to ensure that the form of **R**_{i} would be close to skew symmetric.

Let **A** represent the matrix of elemental rotation vectors **a**_{i} where *i* is the index into the navigator map (*i*∈{1,..,*N*}).

Then a weighting matrix **W** relating the navigators to the rotations using the map may be defined as follows:

and calculated using the pseudoinverse **M**^{+} of **M**:

where the regularization factor *r* is a fraction (such as 10^{–6}) of the mean of the diagonal elements of **M**.

### APPENDIX B

- Top of page
- Abstract
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- APPENDIX A
- APPENDIX B
- REFERENCES

#### Constrained Algorithm for Estimating Rotations

In particular **R**_{RP}^{abs}, **R**_{PS}^{abs}, and **R**_{SR}^{abs} are vectors that contain the magnitudes of the samples along the arc (rotation sections) of the navigator in the **R**P, PS, and S**R** planes respectively (**R** = readout, P = phase encoding, S = slice direction).

Let **R**_{RP}^{ref|abs}, **R**_{PS}^{ref|abs}, and **R**_{SR}^{ref|abs} represent the rotation sections of the reference navigator (average unrotated navigator from the map). Let **A** again represent the matrix (order: 3 × *N*) of elemental angle rotations through which the navigators in the map were rotated (for a map of *N* navigators). Let **M**_{RP}^{map|abs}, **M**_{PS}^{map|abs} and **M**_{SR}^{map|abs} represent the matrices of navigator rotation sections in the map.

Define the vector operation resample (res_{x} N) as the result of shifting and resampling (with linear interpolation) the values in the vector by *x* samples:

where *b* = *x* – floor(*x*), *a* = 1 – *b*, and *k* is the sample index in the navigator.

Resample each rotation section (**R**P, PS, and S**R**) of the map by the negative of the rotation angle at which it was acquired (in the same plane). For example, for the **R**P section rotated by Γ_{RP}:

Then the out-of-plane effects are given by

Let **A**_{RP} be that part of the map angle matrix **A** listing the out-of-plane angles PS and S**R** (order 2 × *N*). Then the relationship between the out-of-plane effects on the navigator and map is defined as:

and this relationship is precalculated for all three rotation directions after the map has been acquired.

For every new vector of navigator samples **N** acquired during the imaging sequence, let **R**_{RP}^{N|abs}, **R**_{PS}^{N|abs}, and **R**_{SR}^{N|abs} represent the magnitude of the samples in the rotation sections. For each candidate in-plane rotation θ_{RP}, estimate the out-of-plane effects as follows:

The estimated out-of-plane rotation pair _{RP}=[α_{PS} α_{SR}] is then given by:

And the estimated resampled navigator with in-plane effects removed is:

From which the rotation angle _{RP} in the **R**P plane is chosen from the candidates as follows:

The other two angles _{PS} and _{SR} are calculated similarly. This constitutes the constrained method for estimating rotations.

### REFERENCES

- Top of page
- Abstract
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- APPENDIX A
- APPENDIX B
- REFERENCES

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