Oscillating Radial Trajectories for Reduced Undersampling Artifacts

Introduction: In this work, we propose an oscillating radial sampling of k-space which shows a significant improvement in the reconstruction quality over the traditional radial sampling based reconstruction. The incoherency in spatial aliasing artifacts is selected as a criterion for optimizing the k-space trajectory. Adding oscillations reduces the coherency in the k-space trajectory and hence minimizes the aliasing artifacts. In contrast to the other methods for generating randomized trajectories [1], the proposed k-space trajectories are smooth and hence easy to implement on a conventional MRI gradient coil system. We present a systematic way of generating oscillating radial trajectories and show an improvement over the traditional radial sampling using simulations. Theory: Non-Cartesian trajectories (e.g. radial and spiral) have received an increased attention in various applications including cardiac imaging, functional brain imaging, contrast-enhanced MR angiography, and hyperpolarized gas imaging. One major drawback of the non-Cartesian sampling is the enhanced aliasing artifacts which can partially be attributed to the coherence in the k-space sampling pattern, resulting in the point-spread-function (PSF) intensity leaking into the sidelobes [2]. A straight forward approach to reduce the aliasing artifacts is to sharpen the PSF. Method: First we parameterize the radial trajectory by using Eq. 1. In Eq. 1, kx and ky define x and y coordinates of the trajectory, ( ) 0.5, 0.5 τ ∈ − defines pseudo time axis, ( ) 0,1 a ∈ represents the amplitude of sinusoidal oscillations, ( ) 0,10 f ∈ defines the frequency of oscillations, θ describes the orientation of a radial arm, and ( ) 0.5,1 w∈ controls the increase in oscillation magnitude as a function of . τ For 1, w = the oscillation amplitude increases linearly with . τ We have selected incoherency in the k-space sampling as a criterion to optimize the trajectory. An effective method of estimating the incoherency is to measure the intensity E of the PSF that has leaked into the sidelobes. We computed PSF using nonuniform fast Fourier transform [3] of k-space. To compute , E we adopted Eq. 2. In Eq. 2, " " × describes point-by-point multiplication. Now the problem of finding an optimized oscillating radial trajectory ( ), K ψr for a given number of radial arms, translates to finding ( ) , , a w f for which E is minimized, as given by Eq. 3. In this model, since there are only 3 parameters to be adjusted each over a narrow range, we used linear search to find the optimum values of , , a w and . f For a trajectory with 5 radial arms, the optimum values of , , a w and f were found to be 0.39, 0.6, and 4 respectively, while for a trajectory with 10 radial arms, the optimum values of , , a w and f were calculated to be 0.2, 0.6, and 7 respectively. A more comprehensive model, for example the one where each arm of radial acquisition has separate tuning parameters, can also be implemented albeit at a higher computational cost. For reconstruction, we used an iterative method [4] for TV-regularization, with significant acceleration achieved by exploiting the Toeplitz-block-Toeplitz nature of the problem [5]. Results: A 96 96 × numerical phantom, shown in Fig. 2A, was used for the simulation studies. Figures 2B and 2C show the reconstruction from the traditional radial trajectories with 5 and 10 arms respectively, while Figs. 2D and 2E show the reconstruction from the proposed oscillating radial trajectories with 5 and 10 arms respectively. For a 5 arm optimized trajectory, the value of E was reduced by 57% while the mean-square-error (MSE) of reconstruction was reduced by 58% in comparison to the traditional radial sampling. Likewise, for a 10 arm optimized trajectory, the value of E was reduced by 56% while the reconstruction MSE was reduced by 81% in comparison to the traditional radial sampling. Conclusion: We have presented a systematic way of adding oscillations to the traditional radial sampling, resulting in a sharper PSF with reduced aliasing artifacts. The k-space trajectories are smooth and can be implemented with a conventional MRI gradient coil systems. Reference: [1] A. Bilgin, T. P. Trouard, A. F. Gmitro, and M. I. Altbach, ISMRM, Toronto, 2008. [2] M. L. Lauzon and B. K. Rutt, MRM, 36, 1996. [3] S. Kunis and D. Potts, JCAM, 161, 2003. [4] M.A.T. Figueiredo, R.D. Nowak, S.J. Wright, IEEE, 2007. [5] A.H. Delaney and Y. Bresler IEEE 1995. ( ) ( ) ( ) ( ) ( ) ( ) sin