3390
Data-Driven non-Cartesian trajectory correction based on Cartesian reference data1Siemens Healthcare, Erlangen, Germany
Synopsis
Trajectory deformations in non-Cartesian scans can lead to significant image artifacts, while in comparison the effect on Cartesian scans is evidentially minor. We propose a novel approach for trajectory correction relying on the minimization of a suitable measure between a non-Cartesian image and a Cartesian reference image, given a trajectory deformation model. Our proof of concept is applied on simulated as well as measured data. The approach is fast, accurate and easily extendable. Given the right use-case, no extra scan time is required (e.g. mixed spiral-Cartesian MR Fingerprinting).
Introduction
Nominal and realized k-space trajectories in MRI differ due to imperfections of the gradient system. Major effects influencing the realized trajectories are gradient delays and eddy currents1-5.Various approaches for correcting gradient imperfections exist, both on the hardware6-9 and on the acquisition or post processing side. The former focus on shielding gradient coils, thereby reducing eddy current effects. On the acquisition side - in addition to appropriate gradient design24-26 - realized trajectories can be measured2,3,10-18. Other calibration methods determine parameters of a model derived from the gradient impulse response function (GIRF)1,12,19-21. In order to be practical, the latter methods are designed to be calibrated once per scanner in isolated experiments, assuming the determined parameters to be constant over time and consistent for any setup. This assumption is unlikely to fully hold, considering that some effects cannot be covered in the calibration, e.g. the applied gradients and the temperature differ. In contrast, the gradient delay can be determined per measurement22,23.
Evidently, Cartesian scans do not suffer substantially from trajectory deformation: In case of gradient delays, the sampling pattern is shifted uniformly in k-space, resulting in a phase in image space. The effect of eddy currents is minimal, since the gradients are applied in the same direction and are constant during readout.
This does not hold for non-Cartesian scans27, leading to severe image artifacts. As a result of the beforehand described limits of existing calibration techniques, visible effects of trajectory deformation can remain, reducing the robustness and acceptance of non-Cartesian scans in clinical applications.
We propose a novel trajectory correction approach for non-Cartesian scans relying on the fact that the image quality of Cartesian scans is less prone to gradient imperfections.
Methods
Other Trajectory CalibrationsWe compared a Cartesian slice of the NIST phantom39 with spiral images of the same object acquired with a prototype sequence, where different trajectories were employed in the reconstruction: The nominal as well as the corrected ones according to Tan et al.1 and Berzl et al.21, both non-scanner-specific.
Our Prototype Correction Method
A comparison of Cartesian and non-Cartesian scans provides information for trajectory correction. A suitable measure can be defined to quantify their similarity, which is assumed to be maximal if the trajectory is correct. Given a trajectory deformation model, our approach is to optimize the parameters of this model with respect to the measure, which is possible since every operation performed in the reconstruction is differentiable. If the reference scan data can be used apart from the trajectory correction, no extra scan time is required.
Reconstruction of Non-Cartesian Data
Non-Cartesian scans can be reconstructed using the fast non-uniform Fourier transform (NUFFT)28-31. It encompasses gridding the data, prior to applying the FFT, finally correcting for the gridding by dividing by the kernel’s Fourier transform and applying a sampling density compensation32, 33.
Trajectory Deformation Model
We propose to use the GIRF as a physically meaningful model. Hereby it is assumed that the realized gradient $$$g$$$ depends linearly on the nominal gradient $$$\hat{g}$$$:
$$g(t)=\int_{\mathbb{R}}\chi(t-t')\cdot\hat{g}(t')\,dt'$$.
According to Tan et al.1, a simple model regarding gradient delay and eddy currents is given by34:
$$\chi(t)=\theta(t)\cdot\sum_{n}\,a_n\cdot e^{-b_n\cdot(t-t_0)},$$
where $$$\theta$$$ is the step function.
Implementation
We used python3, Tensorflow35 and the ADAM optimizer36.
Simulation
We applied our approach to an artificial slice of the NIST phantom. An Archimedean spiral trajectory was deformed according to the proposed GIRF model (single $$$n$$$). Starting with a different choice of parameters, the optimization was performed by minimizing the mean absolute error between the cartesian and the spiral image.
Application to Real Data
A crucial point is an adequate similarity measure, since the Cartesian and spiral scans exhibit different contrasts (e.g. Fig. 1). We normalized images reconstructed from real measured data to [0, 1] and applied a Sobel edge filter. The optimization was set up to minimize the difference between the edge images.
Results
Other Trajectory CalibrationsFig. 1 and 2 show that both correction methods can fail to remove artifacts in spiral images in this exemplary case.
Simulation
The reconstruction with the nominal trajectory exhibits significant artifacts and rotation (Fig. 3, left), the initial difference measure is visualized in Fig. 4, left side. The optimization converged towards the employed parameters of the GIRF model, resulting in a final reconstruction that matches the ground truth, besides undersampling artifacts (right sides of Fig. 3 and 4).
Application to Real Data
The optimization effectively minimized the difference of the measured Cartesian and Spiral edge images (Fig. 5).
Discussion
Our experiments show that existing calibration methods can fail due to lacking adaptability. We demonstrate that using Cartesian reference scans for a data driven calibration of non-Cartesian k-space trajectories is possible. Future research will concentrate on finding suitable difference measures, quantifying effects of trajectory deformation in images of different contrasts.Conclusion
The proposed trajectory deformation model is extendable, effects beside gradient delay and eddy currents can be introduced. No static calibration is necessary, our method can adapt to temporal deviations of the scanner. It relies on the comparison in image domain, which is the actual subject of interest. Given a suitable use-case, e.g. MR Fingerprinting (MRF)37, 38 with mixed spiral and Cartesian readouts, no extra scan time is required.Acknowledgements
We thank Josef Pfeuffer for inspiring discussions.References
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