Synopsis

Real-time phase-contrast flow MRI is extended from sequential acquisitions of flow-compensated and flow-encoded data with the same set of radial spokes to interleaved acquisitions with different radial spokes oriented by a small Golden angle, thereby improving spatial accuracy and reducing streaking artifacts. To apply model-based reconstructions for this sampling scheme, an automatic scaling of unknowns is proposed, which is capable of balancing partial derivatives and regularizations during the iterative nonlinear inversion.

Purpose

Methods

In current real-time phase-contrast flow MRI, both the flow-compensated and flow-encoded data are sequentially acquired via radial spokes with the same k-space encoding^1,2. In the model-based reconstruction³, the forward model is formulated as $$$F_{j,l} (x) = U_l \mathcal{F} \{ \rho \cdot e^{i \cdot \Delta \phi \cdot S_l} \cdot c_j \}$$$, where the vector $$$x = (\rho, \Delta \phi, c_1, \cdots, c_N)^T$$$ of unknowns comprises $$$\rho$$$ the proton density, $$$\Delta \phi$$$ the phase difference (i.e. velocity), $$$c_j$$$ the $$$j$$$th coil sensitivity map, and $$$U_l$$$ the sampling pattern of the $$$l$$$th acquisition. The velocity-encoding (Venc) indices $$$S_1 = 0$$$ and $$$S_2 = 1$$$ represent the flow-compensated and flow-encoded acquisition, respectively. The unknowns in this nonlinear signal model are solved by the iteratively regularized Gauss-Newton method with Tikhonov regularization on all unknowns. The balance of partial derivatives and regularizations is achieved by scaling of Venc indices: $$$\hat{S_l} = \frac{\left\lVert M \right\rVert_2}{\left\lVert \Delta \phi \right\rVert_2} \cdot S_l$$$, where $$$M$$$ and $$$\Delta \phi$$$ are approximated by the mean and complex difference of the gridded multi-channel k-space data from the flow-compensated and flow-encoded acquisition, respectively.

Here, we modify the pulse sequence to allow for complementary sets of spokes. As shown in Figure 1, each pair of flow-compensated and flow-encoded spokes is sampled in an interleaved manner and separated by a constant Golden angle⁴ of $$$27.19840^o$$$, which is chosen such that the two sets of spokes for one velocity map lead to a nearly uniform coverage of k-space. First, compared to the current sampling scheme^1,2, the proposed scheme samples twice more different spokes, thereby effectively reducing the undersampling factor. Second, this sampling scheme enables an additional sliding window reconstruction, which can further increase the frame rate. The main drawback of this sampling scheme, however, is the incompatibility with the previous scaling³, which requires the calculation of the complex difference. As shown in Figure 2, the complex difference between the gridded flow-compensated and flow-encoded k-space sampled with same spokes emphasizes only the flow region (i.e. aorta), while the complex difference from complementary spokes suffers from severe streaks surrounding not only the flow region but also the “static” region (i.e. chest wall). Consequently, the redundant streaks contribute false-positive information to the scaling calculation.

To allow for complementary sets of radial spokes, the model-based reconstruction requires automatic scaling of unknowns, which seeks for the appropriate scaling during iterative reconstructions⁵. In principle, the scaling of the $$$m$$$th unknown reads as $$$\hat{x}^{(m)} = x^{(m)} \cdot \gamma_m$$$, where $$$\gamma_m$$$ is chosen such that sensitivities of the data with respect to different solution components are balanced, so $$$\gamma_m = \left\lVert DF(x) |_{x^{(m)}} \right\rVert = \left\lVert DF(x) P_m^H \right\rVert$$$ with $$$DF(x)$$$ the derivative of the forward operator and $$$P_m^H$$$ the projection onto the $$$m$$$th unknown. The spectral radius of this matrix norm is determined by its maximal eigenvalue⁶: $$$\gamma_m = \lambda_0 (P_m DF^H (x) DF(x) P_m^H)^{1/2}$$$, approximated by the power method. In the current implementation, the power method with $$$10$$$ iterations is used only for the odd Newton iterations of each frame.

Experimental implementations of real-time phase-contrast flow MRI use undersampled radial FLASH with the Golden-angle interleaved acquisition scheme. All measurements (TE = $$$1.70$$$ ms, flip angle $$$10^o$$$) had $$$1.5$$$ mm in-plane resolution, $$$320$$$ mm field-of-view, $$$6$$$ mm slice thickness, and $$$35.7$$$ ms ($$$7$$$ spokes, TR = $$$2.55$$$ ms) temporal resolution corresponding to $$$28$$$ frames per second.

Results & Discussion

Conclusion

Acknowledgements

References

1. Joseph A, Kowallick JT, Merboldt KD, et al. Real‐time flow MRI of the aorta at a resolution of 40 msec. J Magn Reson Imaging 2014;40:206-213.

2. Untenberger M, Tan Z, Voit D, et al. Advances in real-time phase-contrast flow MRI using asymmetric radial gradient echoes. Magn Reson Med 2016;75:1901-1908.

3. Tan Z, Roeloffs V, Voit D, et al. Model-based reconstruction for real-time phase-contrast flow MRI: Improved spatiotemporal accuracy. Magn Reson Med 2017;77:1082-1093.

4. Wundrak S, Paul J, Ulrici J, et al. A small surrogate for the golden angle in time-resolved radial MRI based on generalized Fibonacci sequences. IEEE Trans Med Imaging 2015;34:1262-1269.

5. Tan Z, Hohage T, Kalentev O, et al. An eigenvalue approach for the automatic scaling of unknowns in model-based reconstructions: Application to real-time phase-contrast flow MRI. NMR Biomed 2017; doi: 10.1002/mrm.26192.

6. Kress R. Numerical analysis. New York:Springer;1998:38-39.