Lukas Hingerl^{1}, Bernhard Strasser^{1}, Philipp Moser^{1}, Gilbert Hangel^{1}, Siegfried Trattnig^{1,2}, and Wolfgang Bogner^{1}

A density weighted concentrically circular echo-planar trajectories readout scheme is presented for brain MRSI at 7 T. We give an analytic solution for the variable radii distribution in order to intrinsically measure a Hamming weighted k-space. A comparison with post acquisition filtered equidistant concentric circles is done. Invivo metabolic maps and spectra are shown.

Assuming continuous polar sampling, the sample density in k-space is given by the absolute value of the Jacobian determinant $$$J^8$$$ of the transformation

$$k_x=K(k)\text{cos}k_{\phi}$$

$$k_y=K(k)\text{sin}k_{\phi},$$

as already hinted in Lauzon et al.$$$^9$$$. The function $$$K(k)$$$ describes the non-uniform radii distribution and has to fulfill the boundary conditions $$$K(k^{max})=k^{max}$$$ and $$$K(0)=0$$$. Setting $$$J=K'K$$$ proportional to the inverse of the (polar) Hamming filter $$$\mathcal{H}(K(k))=\alpha-\beta \text{cos}(\pi K(k)/k_{max})$$$ we obtain

$$K'(k)K(k)=c/\mathcal{H}(K(k)).$$

Solving the above differential equation by separation of variables gives

$$$k(K)=\left(\alpha\frac{K^2}{2}+\beta \frac{k_{max}^2}{\pi^2}\text{cos}(\pi K/k_{max})+\beta\frac{k_{max} K}{\pi}\text{sin}(\pi K/k_{max})-\frac{\beta k_{max}^2}{\pi^2}\right)/\left(k_{max}\left(\frac{\alpha}{2}-\frac{2\beta}{\pi^2}\right)\right),$$$

where $$$c$$$ and the constant of integration are determined by the boundary conditions. The inverse of the above function, $$$K(k)$$$, is the function of interest which can not be calculated analytically. However we can approximate $$$K(k)$$$ using inverse series expansion. Evaluation of $$$K$$$ at the discrete coordinates $$$k_i=ik_{max}/N$$$ for $$$i=0,...,N$$$ gives the distribution of the $$$N$$$ Radii in order to sample a Hamming weighted k-space.

The authors want to thank Borjan Gagoski for helpful discussions.

This project was funded by the Austrian FFG: #FA771E0801.

1. Furuyama et al, MRM 67, 2012.

2. Kasper et al, NeuroImage 100, 2014.

3. Pohman et al, MRM 45, 2001.

4. Mareci et al, Journal of MR 92, 1991.

5. Strasser et al, NMR Biomed. 26, 2013.

6. Bogner et al, NMR Biomed 25, 2012.

7. Henning et al, NMR Biomed 22, 2009.

8. Hoge et al, MRM 38, 1997.

9. Lauzon et al, MRM 40,1998.

Left: K(k). Equidistant spacing of N coordinates $$$k_i$$$ along the horizontal axis gives the distribution of the radii $$$K(k_i)$$$ in order to measure a Hamming weighted k-space using N circles. Right: k-space trajectories for the radii $$$K(k_i)$$$. The circles are traversed with constant slew rate which corresponds to a constant angular velocity and an equidistant spacing of the samples along each circle. The Radii of the first circle is chosen to match the Nyquist distance 1/FOV.

Localization of an oil phantom with DWA. The 128$$$\times$$$128 matrix corresponds to 1405 circles in k-Space.

The Hamming weighted k-Space for the DWA after gridding. The strongly weighted central peak results from imperfections of the gridding algorithm. However one can compensate that by an additional density compensation which is still SNR efficient.

Metabolic maps for 4 volunteers showing the signal amplitudes of total N-acetylaspartate, total creatine and glutamate plus glutamine for DWA and equidistant circles.

Spectra of the second volunteer for DWA and equidistant circles.