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Prospective motion correction in MR imaging is becoming increasingly popular to prevent the image artefacts that result from subject motion. Navigator information is used to update the position of the imaging volume before every spin excitation so that lines of acquired k-space data are consistent. Errors in the navigator information, however, result in residual errors in each k-space line. This paper presents an analysis linking noise in the tracking system to the power of the resulting image artefacts. An expression is formulated for the required navigator accuracy based on the properties of the imaged object and the desired resolution. Analytical results are compared with computer simulations and experimental data.
Subject motion remains a major problem in MRI. For high resolution brain imaging, measurement times of about 10 to 30 minutes are required to achieve an acceptable signal-to-noise ratio (SNR). During this time, involuntarily movements of the order of millimetres are unavoidable (1). The problem of motion is not limited to high-resolution imaging: in the case of fMRI, motion can produce false-positive activations (2); in routine clinical imaging, motion results in many scans having to be repeated.
Numerous motion-correction methods exist. For abdominal imaging, traditional navigator techniques can be used to selectively acquire data at only a particular point in the breathing cycle (3, 4). Rigid body head motion, however, requires a different approach. PROPELLER (5) has proved popular, although it has disadvantages, such as redundantly oversampling the k-space centre, and being sequence-specific and limited to in-plane motion correction. Other similar techniques (6, 7) suffer from similar drawbacks to PROPELLER, as do approaches that use post-processing correction, such as the autofocus method proposed by Atkinson et al. (8). Image-based navigation and correction methods, such as that proposed by Thesen et al. (9), can only correct for inter-scan motion.
Prospective motion correction is becoming a popular avenue of research, as it has advantages over the above approaches. Navigator data are used to update the scanner gradients, and therefore the position of the imaging volume, before every spin excitation. Although the term ‘prospective’ implies prediction, normally the latest estimate of the pose of the object is used to set the position and orientation of the imaging volume for the next excitation (1). Motion tracking of a rigid body in a full six degrees of freedom (6DOF) is possible and the technique can be applied to any existing pulse sequence to correct for head motion.
There are numerous examples of prospective correction being used during brain imaging in MRI. A variety of navigator systems have been used: reflecting laser beams off a marker attached to the subject (10, 11); optical tracking with multiple cameras (1, 12); optical tracking with a single camera (13, 14); or by tracking using small coils, known as ‘active markers’ (15, 16). The evaluation of these different systems is the subject of current research.
Regardless of the navigation system used, tracking accuracy is limited by noise in the navigator data. When imaging a stationary object, these pose errors generate artefacts in the image.
Despite the popularity of navigators, to our knowledge no analysis of the required navigator accuracy is available in literature (although accuracy has been quantified in the case of navigator echoes (17)). Hence, we develop a formalism to statistically analyse the image artefacts introduced by the prospective motion correction of each k-space line in the presence of tracking noise. It is then possible to predict artefact levels given knowledge of the tracking system parameters and the imaged object. The reverse is also true: one can calculate the required navigator accuracy of a tracking system given a typical target image, scanning parameters and image-quality prerequisites. This will assist the specification and development of such tracking systems.
The k-space signal intensity in traditional spin warp imaging, ignoring saturation, relaxation and dephasing effects can be given as
where ρ (r) is the signal density of the imaged object. Prospective motion correction using a noisy tracking system in the absence of motion adds a small uncertainty, Δ = iΔx +jΔy, into the imaging location relative to the position of the object. We consider in-plane translations only; a justification for the omission of through-plane and rotation errors is left until the Discussion. In gradient echo imaging with line-by-line correction, each realisation of navigator noise is a function of only the phase encoding step, that is, Δ = Δ (ky). The k-space signal intensity is then
where the approximation is valid only for Δ 1 voxel.
Applying the inverse Fourier transform to Eq. (1) to generate an image yields
where ρ (r) is the original signal density and η (r) describes the image-space artefacts that result from the navigator noise.
Eq. (2) can be further analysed statistically to derive the ‘artefact power’, defined here as |η (r)|2, the expected value of the mean-squared artefact intensity in the image. Assuming zero-mean white uncorrelated noise with standard deviation σ and and Δx (ky) Δy (ky) = 0, then
where is the gradient operator.
Expanding Eq. (3) and noting that the artefact power is a function of x, but not y (the direction of phase-encoding), gives
Finally, incorporating the relative size of the FOV gives an expression for the expected value of the mean-squared artefact intensity in terms of the navigator noise in millimetres:
where Ny is the image matrix size in the phase-encoding direction and FOVy is the y-dimension of the field-of-view in millimetres.
Eq. (5) shows that the mean artefact power is proportional to the integral edge power in the image and the navigator noise variance. Additionally, it is clear that the mean artefact power can be separated into the noise generated by the x- and y-components of navigator noise.
To verify the theoretical results presented here, we performed simulations and MR measurements. Both methods produce images corrupted with artefacts caused by navigator noise, which are then compared directly to the predictions made by Eq. (5).
We use the analytical version of the Shepp-Logan phantom for simulation purposes (Fig. 1(a)). The grey levels in the original phantom were chosen to represent the radiographic density variations in the human brain (18). In MRI, radiographic density is not the measured parameter; therefore, the original grey levels are inappropriate and have been adjusted here, as was done by Devaney (19).
Motion simulations are performed by computing each line in k-space individually, where the computed values correspond to a Shepp-Logan phantom in a specified position. This position varies between k-space lines, according to parameters specified by a random number generator. The equation for the Fourier transform of a rotated, shifted ellipse (20) is applied to our situation. The k-space value at coordinates kx and ky for an ellipse with semimajor axis A and semiminor axis B, is computed as
where J1 is a first-order Bessel function of the first kind and xi, yi and θi correspond to the simulated navigator noise (xi, yi) and rotation noise (θi) for line i of k-space (see Fig. 1(b) and (c)). When all lines of k-space are computed, the inverse Fourier transform is applied to generate an image. This image is corrupted by artefacts due to the applied navigator noise. An image of the original artefact-free Shepp-Logan phantom can be then subtracted to show the artefacts alone.
Phantom experiments were carried out on a 3T Magnetom TRIO system (Siemens Healthcare, Erlangen, Germany) using a single-channel birdcage coil. A standard 2D gradient echo sequence was modified to enable real-time motion position input. A stationary phantom was imaged with an in-plane resolution of 0.5 × 0.5 mm2 and slice thickness of 4 mm. Position noise was simulated using a random number generator with a white Gaussian distribution (21). The advantage of using this method, rather than using noise data from an actual tracking system, is that the navigator noise parameters can be tightly specified. Images were reconstructed with no filtering or other post-processing.
An in vivo experiment was performed on a 7T high-field system (Siemens Medical Solutions, Erlangen, Germany). A stereoscopic tracking system (ARTtrack3, Advanced Realtime Tracking GmbH, Weilheim, Germany) was used for prospective motion correction in conjunction with a gradient echo sequence modified for the purpose. Imaging resolution was 0.5 × 0.5 mm with a slice thickness of 4 mm and an axial slice orientation. The field-of-view was deliberately chosen to be much larger than the object, so that any artefacts could be clearly seen outside the support region. The navigator error standard deviations at the centre of the tracking target were σx = 0.08 mm and σy = 0.05 mm in the x- and y-directions, respectively.
Given an Nx × Ny complex-valued reference image, IR(x, y), and an Nx × Ny complex-valued image corrupted by navigator noise, IN (x, y), the ‘measured’ artefact power across the image is computed as
where the vertical bars indicate the complex magnitude. It is also useful to define a signal-to-artefact ratio, given by
Eqs (7) and (8) describe the computation of artefact power for measured data. This can now be compared to the predicted values from Eq. (5), where the theoretical value for AP (x) is equivalent to |η (x)|2. The evaluation of Eq. (5) requires the computation of the partial derivatives in the image, and . This can conveniently be performed by frequency multiplication in the Fourier domain, which avoids errors introduced by a finite difference derivative calculation.
Fig. 2(a) and (b) show images reconstructed from simulations where navigator noise is applied in the x- and y-directions, respectively. The navigator noise standard deviation, σx and σy, was set to 0.1 pixels in each case: this is a realistic value when using our ARTtrack3 tracking system during imaging with 1 × 1 mm resolution. At this noise level artefacts are not easily visible to the eye in printed images. Thus, Fig. 2(c) and (d) show images (a) and (b) with a reference version of the image subtracted, to show only the artefacts. Fig. 2(e) and (f) show the distribution of artefacts when the simulation is run over 100 iterations. As predicted by Eq. (5), it is clear that vertical edges in the image generate artefacts when σx > 0 and horizontal edges in the image generate artefacts when σy > 0.
To explore the relationship of the navigator noise variance to total artefact power, denoted here ΣAP (x) (the denominator in Eq. (8)), the above simulation was carried out for a range of values for σx (Fig. 3). A line of constant slope is plotted for comparison. As predicted by Eq. (5), artefact power increases linearly with the navigator noise variance for small values of . When σx 0.3 pixels, this linear relationship no longer holds, due to the approximation in Eq. (1).
Simulations were also performed for rotation noise with σθ = 0.2°, and the resulting artefacts and artefact power are shown in Fig. 4. Although the extension of the theory to consider rotations is left to future work, it is reasonable to expect that edges lying in a radial direction would be a source of artefacts in this scenario.
The phantom imaging experiments produce similar results to the simulations. Fig. 5(a) shows an artefact-free image. Fig. 5(b)–(d) show sample results with 0.5 × 0.5 mm in-plane spatial resolution and navigator noise with standard deviation of σ = 0.5 mm in (b) both directions, (c) the x-direction only and (d) the y-direction only. Again, vertical edges in the image generate artefacts when navigator noise in the x-direction is present, while horizontal edges in the image generate artefacts when navigator noise in the y-direction is present.
Fig. 6 shows a more quantitative comparison between theory and measurements. The calculated mean artefact power based on the experimental results is plotted against that predicted by the theory based on Eq. (5). Fig. 6(a) and (b) show simulations results; Fig. 6(c) and (d) show results from the imaged phantom. For the simulations in (a) and (b), the total measured signal-to-artefact ratio is almost identical to the predicted value. For the MR data, there is a slight discrepancy: for navigator noise applied in the x-direction, signal-to-artefact ratio is 9.52 × 105 compared to the predicted value of 8.97 × 105; for noise applied in the y-direction, signal-to-artefact ratio is 9.77 × 105 compared to the predicted value of 9.29 × 105. This discrepancy can be explained by system imperfections increasing the difference between the images.
It is now possible to go beyond artefact prediction to specifying the tracking accuracy requirements for prospective motion correction. Assuming that the navigator noise in the x- and y-directions has the same variance, then we define . From Eq. (5),
The procedure to compute the required image accuracy is then
As an example, the above is applied to the brain image in Fig. 7(a). For practical applications it is desirable for the signal-to-artefact ratio to be significantly better than the native image SNR. Here we decide that the artefact level caused by navigator noise is to be 1/4 that of the native image noise, so that signal-to-artefact ratio = 4 SNR. A typical high-resolution image, with 0.5 × 0.5 mm resolution, has an SNR of about 25. Thus, we set signal-to-artefact ratio = 100. Given this, and the edge energy in our target image, ΣAP = 9.0 × 10−6.
Finally, using the FOV size and imaging resolution, we evaluate Eq. (9) and obtain σxy = 0.18 mm. Thus, to meet our target value for signal-to-artefact ratio, tracking noise standard deviation must be 0.18 mm or better in both x- and y-directions.
The navigator noise constraints calculated here are well above the measured parameters from the ARTtrack3 tracking system of σx = 0.08 mm and σy = 0.05 mm for the in vivo example shown in Fig. 7(a). As expected, artefacts due to tracking noise are not easily visible until the image is scaled to emphasise the background (Fig. 7(b)). The predicted artefact power distribution for the image with σx = 0.08 mm and σy = 0.05 mm is shown in Fig. 7(c) together with the image noise floor. It is evident that only the peaks in the distribution will be visible through the noise. To demonstrate that the predictions are reasonable, Fig. 7(d) shows simulated Gaussian-distributed noise, of the same variance as that in Fig. 7(b), combined with the predicted artefacts from Fig. 7(c). A more rigorous analysis of the in vivo data is prevented by the lack of a ground truth image free of motion artefacts and image noise, both of which mask artefacts caused by tracking errors.
We have shown that noise in the tracking data used for prospective motion correction always results in artefacts, which can be quantified in terms of a ‘signal-to-artefact’ ratio. Mean artefact power is proportional to both the integral of the square of the derivatives of the image intensity and to the variance of navigator noise. For small values of navigator noise variance, artefacts are not always visible in the reconstructed image (as is the case in Fig. 2(a) and (b)). Artefact power increases linearly, however, with navigator noise variance. Several tracking systems mentioned in the literature (such as (22), for example) report a level of navigator noise that means artefacts will certainly be visible in the image.
By inverting the derived expression for artefact power, the required navigator accuracy can be computed. This is a function of the object properties and the acceptable level of residual artefacts. In general, however, navigator accuracy should be several times better than the target pixel size for most imaging situations.
Although these requirements seem stringent, some image quality may be recovered in postprocessing, which would relax requirements on the tracking data accuracy to achieve the same final image quality. Techniques for post-processing motion correction, such as that proposed by Bammer et al. (23) or Atkinson and Hill (24), could be applied for reduction of artefacts caused by tracking noise. However, this is only applicable if tracking noise can be estimated after data acquisition. Similarly, approaches such as Kalman filtering, recently used for cardiac navigators (25), are being investigated for head motion tracking (26). This would also relax the accuracy requirements of the tracking system itself.
This analysis only accounts for the effects of in-plane translation errors. Omitting errors in the remaining degrees of freedom is justified if the effect of these errors on artefact power is much smaller than in-plane translation errors. This applies in the case of the ARTtrack3 tracking system used in this study. Provided the target is large relative to the location errors of individual markers, the rotation of the target is accurately estimated. Our simulations show that acquisitions with realistic values for rotation noise do not display visible artefacts. The omission of through-plane translation is justified for 2D imaging, as the slice thickness used is normally around a factor of four greater than the in-plane resolution and is much smaller than navigator noise (Fig. 8). Additionally, the non-ideal slice profile results in a ‘smoothing effect’ if through-plane navigator noise moves the excited plane over a tissue boundary.
The theory presented here makes several assumptions. We assume that such navigator noise is normally distributed: visually, this appears to be the case for noise from the ARTtrack3 tracking system and a Jarque-Bera test (27) applied to the navigator noise data for each DOF was unable to reject the null-hypothesis that the data came from a normal distribution. However, the assumption of Gaussianity is a limitation to this work, as it may not apply to some navigation methods. Additionally, we assume that the imaging properties are independent of the position and orientation of the object. This will not be the case in prospective motion correction with large head movements, due to field distortions and non-uniform coil sensitivity profiles.
Another limitation of the approach described here is that mean noise power does not correlate exactly with human perception of artefacts. If artefacts are evenly distributed in a ‘noise-like’ fashion, then they will be less visible than if they appear concentrated in one area, as occurs in the case of ghosting. Therefore, this work could be usefully extended through the incorporation of a ‘human perception’ factor. This would likely incorporate the maximum value of generated artefacts, rather than their sum over the image.
Here, we compute the image gradient in the x- and y-directions using frequency multiplication in k-space. The result is not identical to the gradient of the imaged object, due to the truncation of k-space. However, we show that it is the edge power in the image after sampling that generates image artefacts, rather than sharp edges present in the object that are lost during imaging. Due to Fourier encoding, the acquired data in MRI is inherently band-limited; thus, errors in frequency components of the object that are not present in the acquired signal, have no effect on the final image. This concept has been recognised in the application of super-resolution to Fourier-encoded MRI data (28) where unsampled spatial components cannot be retrieved using conventional super-resolution techniques involving shifts in the FOV.
It is worth noting that the theory and results presented here can be applied in a more general sense to motion artefacts in MRI. The situation presented is identical to a line-by-line acquisition of k-space, without prospective correction but with Gaussian-distributed position noise caused by patient motion. Although patient motion parameters will not necessarily be Gaussian distributed, a similar expression for artefact generation may apply. It is reasonable to expect that image edges are crucial to artefact generation.
This work derives the effect of navigator noise on image artefact power. Theoretical results closely match simulations and MR data. For small translation errors, artefact power is directly proportional to the variance of the navigator noise in the tracking system. The complete expression for mean artefact power enables the specification of the required accuracy for tracking systems in prospective motion correction. Knowledge of this relationship will aid in the design of tracking systems for prospective motion correction. Tracking system accuracy must be a fraction of a pixel to be effective and to avoid the introduction of extra artefacts when imaging a stationary object.
This work is a part of the INUMAC project supported by the German Federal Ministry of Education and Research (grant 01EQ0605) and the NIH/NIDA project ‘Real-time motion tracking and correction for MR Imaging and Spectroscopy’ (grant 1R01 DA021146). We thank Dr. Valerij Kiselev for helpful discussions.