Sebastian Rosenzweig^{1}, Hans Christian Martin Holme^{1,2}, Robin Niklas Wilke^{1,2}, and Martin Uecker^{1,2}

Lately, radial k-space trajectories have become very popular especially for fast MRI. However, radial sampling is prone to streaking artifacts caused by gradient delays. Here, we propose two extensions for a simple but powerful method that compensates gradient delays by estimating the corresponding sample shift using cross-correlation of opposed spokes. First, we show that the opposed spokes don't have to be acquired in calibration scans but can be taken directly from the actual measurements. Second, we show that it is also possible to generate synthetic spoke-pairs for gradient delay estimation.

Introduction and Purpose

In the recent years radial k-space sampling has gained increased interest in the MRI community as it provides significant advantages over conventional Cartesian sampling. However, radial MRI sequences require a very precise gradient timing and therefore are prone to eddy-current induced gradient delays which result in k-space miscentering^{1}.

Amongst others^{2-6}, a very simple and popular method for gradient delay correction was introduced by Block and Uecker^{7}. They suggest to calculate the sample shift by performing a cross-correlation of opposed calibration spokes and compensate for it in the gridding procedure. However, this technique requires the acquisition of diametrically opposed spokes prior to the actual measurement. Hence, modifications to an existing radial MRI sequence must be implemented and moreover, the method cannot be applied to measurements with interactive changes to the imaging plane. Here, we reformulate their approach and propose two extensions that don't rely on calibration scans.

Theory

Eddy-currents induce a k-space sample shift which can by modeled using a quadratic form^{8} (Eq 1)
$$ \delta k(\Theta)=\hat{n}^T\left(\begin{array}{cc}S_x&&S_{xy}\\S_{xy}&&S_{y}\end{array}\right)\hat{n},\;\;\hat{n}^T=(\cos{\Theta},\sin{\Theta})$$
with $$$\Theta$$$ the projection angle of the spoke. $$$S_x$$$ and $$$S_y$$$ correspond to the k-space shifts in x and y direction and $$$S_{xy}$$$ accounts for an additional shift induced by the third physical gradient when measuring oblique slices. We propose three related techniques to estimate these parameters:

**I)** First, we describe a convenient implementation of the method by Block and Uecker^{7}:
**1.** We sequently acquire calibration spokes $$$P_\Theta$$$ and $$$P_{\Theta -180°}$$$ for various projection angles $$$\Theta$$$
**2.** We pick a spoke $$$P_{\Theta}$$$ and its opposed counterpart $$$P_{\Theta-180°}$$$, flip ($$$^f$$$) the latter, Fourier-Transform ($$$\mathcal{F}$$$) both spokes and multiply the $$$\Theta$$$-spoke's transform with the complex conjugate (*) of its flipped counterpart. This yields
$$g(r)=\mathcal{F}P_{\Theta}\cdot(\mathcal{F}{P}^{f}_{\Theta-180°})^*,$$ the Fourier-Transform of the cross-correlation function.
**3.** We calculate the slope of the phase of $$$g(r)$$$ using a scalar product between $$$g(r)$$$ and its duplicate shifted by one pixel $$$g(r+\Delta r)$$$, which magnitude-weights the finite phase-differences. The actual shift is given by
$$\delta k(\Theta)=\frac{1}{2}\;\text{arg}(\langle g(r),g(r+\Delta r)\rangle)\;2\pi/(\text{BaseResolution})$$
**4.** To obtain $$$S_x$$$, $$$S_y$$$ and $$$S_{xy}$$$, we fit the shifts using the quadratic form (Eq. 1).

In practice it is not always possible or desired to perform calibration scans e.g. if e.g. the imaging plane is interactively changed. We therefore propose two methods that don't rely on calibration scans but use the actual measurement data for shift determination:
**II)** We iterate through all or a subset of acquired spokes and for each spoke we search for the one which is approximately opposed to it. With these (almost) opposed spoke-pairs we can continue with step **I.3-4**.

**III)** For highly undersampled radial data there is no guarantee to find sufficiently opposed spokes, thus we generate artificial spoke-pairs by approximating k-space symmetry: $$$P_{\Theta-180°}=P_\Theta^*$$$ and continue with step **I.3**. In practice, the object appears not necessarily real but has a phase in each channel which we account for in a first order approximation using an extended model (Eq. 2):
$$\delta\tilde{k}(\Theta)=\delta{k}(\Theta)+\hat{n}^T\left(\begin{array}{c}D_x\\D_y\end{array}\right)$$
which we use to fit $$$S_x$$$, $$$S_y$$$ and $$$S_{xy}$$$ coil-by-coil and average the results.

Methods

We evaluate the capability of methodResults and Discussion

Fig. 1 shows one frame of a real-time movie (cardiac short-axis) reconstructed with NLINVConclusion and Outlook

Preliminary results suggest that gradient delay correction can be performed using few spokes of the actual measurement without calibration scans. In the future we will evaluate the robustness of the methods.[1] Peters, D. C., Derbyshire, J. A. and McVeigh, E. R. (2003), Centering the projection reconstruction trajectory: Reducing gradient delay errors. Magn. Reson. Med., 50: 1–6. doi:10.1002/mrm.10501 [2] Jang, H. and McMillan, A. B. (2017), A rapid and robust gradient measurement technique using dynamic single-point imaging. Magn. Reson. Med., 78: 950–962. doi:10.1002/mrm.26481 [3] Addy, N. O., Wu, H. H. and Nishimura, D. G. (2012), Simple method for MR gradient system characterization and k-space trajectory estimation. Magn. Reson. Med., 68: 120–129. doi:10.1002/mrm.23217 [4] Vannesjo, S. J., Haeberlin, M., Kasper, L., Pavan, M., Wilm, B. J., Barmet, C. and Pruessmann, K. P. (2013), Gradient system characterization by impulse response measurements with a dynamic field camera. Magn Reson Med, 69: 583–593. [5] Jeff H. Duyn, Yihong Yang, Joseph A. Frank, Jan Willem van der Veen, Simple Correction Method fork-Space Trajectory Deviations in MRI, In Journal of Magnetic Resonance, Volume 132, Issue 1, 1998, Pages 150-153, ISSN 1090-7807. [6] Barmet, C., Zanche, N. D. and Pruessmann, K. P. (2008), Spatiotemporal magnetic field monitoring for MR. Magn. Reson. Med., 60: 187–197. doi:10.1002/mrm.21603 [7] Block, K. T. and Uecker, M. (2011), Simple method for adaptive gradient-delay compensation in radial MRI. In: Proceedings of the 19th Annual Meeting of ISMRM, Montreal, Canada, 2816. [8] Moussavi, A., Untenberger, M., Uecker, M. and Frahm, J. (2014), Correction of gradient-induced phase errors in radial MRI. Magn. Reson. Med, 71: 308–312. doi:10.1002/mrm.24643 [9] Uecker, M., Hohage, T., Block, K. T. and Frahm, J. (2008), Image reconstruction by regularized nonlinear inversion—Joint estimation of coil sensitivities and image content. Magn. Reson. Med., 60: 674–682. doi:10.1002/mrm.21691

Fig. 1: One frame of a real-time movie (cardiac short-axis view, 21 spokes per frame) reconstructed with different gradient delay compensation strategies. a) Reference: No gradient delay compensation. b) Gradient delay compensation using method **II** or **III**. The numbers indicates the amount of spokes used for gradient delay estimation.

Fig. 2: Gradient delay estimation error $$$d$$$ against the number of (evenly distributed) spokes for a simulated 7-channel radial k-space of a shepp-logan phantom.