To develop a model-based reconstruction technique for real-time phase-contrast flow MRI with improved spatiotemporal accuracy in comparison to methods using phase differences of two separately reconstructed images with differential flow encodings.

The signal model assumes the same magnitude image and the same coil sensitivities for each pair of flow-compensated and flow-encoded datasets, and thus the forward model of the $$$j^{th}$$$ coil in the $$$l^{th}$$$ acquisition is $$$F_{j,l}(x) = P_l \cdot \mathcal{F} \{ \rho \cdot e^{z \cdot S_l} \cdot c_j\}$$$, where $$$\rho$$$ is the magnitude image, $$$z$$$ denotes a complex map which contains the phase differences $$$\Delta \phi$$$ in the imaginary part, while its real part is constrained to be zero, $$$c_j$$$ is the sensitivity map of the $$$j^{th}$$$ coil. The indices $$$S_1 = 0$$$ and $$$S_2 = 1$$$ represent the flow-compensated and the flow-encoded acquisition, respectively. $$$P_l$$$ is the orthogonal projection onto the $$$l^{th}$$$ trajectory¹. The unknowns $$$\rho$$$, $$$z$$$, and $$$c_j$$$ in this nonlinear signal model can be solved by the iteratively regularized Gauss-Newton method as introduced for nonlinear inversion (NLINV)¹. Moreover, a L2-norm regularization on $$$z$$$ is used with the initial guess of zero, and the temporal regularization¹ is adopted on $$$z$$$ as well. The partial derivatives and regularization among unknowns are balanced via an automatic scaling on $$$S_l$$$. The scaling value can be derived via the definition of the complex-difference (CD) image²: $$$CD = |\rho_1 - \rho_2| = M \cdot \sqrt{2[1-cos(\Delta \phi)]}$$$. It can be proven to hold that $$$||\sqrt{2[1-cos(\Delta \phi)]}||_2 \propto ||\Delta \phi||_2$$$, and thus the scaled index is $$$\hat{S}_l = ||M||_2 / ||CD||_2$$$. $$$M$$$ and $$$CD$$$ can be calculated by the mean and complex-difference of the gridded multi-channel k-space data from the flow-compensated and the flow-encoded acquisition, respectively. In fact, based on the linearity of the Fourier transformation, the norm of an image is equivalent to that in k-space.

Real-time flow MRI was based on extremely undersampled radial FLASH (5 or 7 spokes per image) using two sequential acquisitions of a dataset with velocity-compensated gradients in all gradient axes and with velocity-encoding of through-plane flow, respectively. All measurements (TE = 1.70 ms, flip angle 10 degree) 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) or 25.6 ms (5 spokes, TR = 2.56 ms) temporal resolution corresponding to 28 or 39 frames per second, respectively. This work presents real-time flow MRI of the aorta acquired from 10 healthy subjects and 2 patients with combined aortic valve insufficiency and partial stenosis³.

Qualitative comparisons of NLINV and model-based phase-contrast MRI are depicted in Fig. 1 for a normal subject and in Fig. 2 for a patient with aortic valve insufficiency and partial stenosis, respectively. The systolic phase-contrast maps obtained by the model-based reconstruction yield a much better spatial definition in regions with non-zero flow (i.e., vessels). Here, this particularly applies to the descending aorta whose phase-difference presentation is in close agreement with the vessel lumen in the magnitude image. In addition, the implicit a priori knowledge of zero phase in pixels without flowing spins precludes the iterative optimization process to generate residual streaking artifacts in areas around vessels with maximum systolic flow, i.e. for signals with high temporal and spatial frequencies that are most severely affected by k-space undersampling. Moreover, as shown in the phase-contrast map from NLINV in Fig. 2, the pixels in the top-left border of the ascending aorta suffer from partial volume effects. This is unavoidable in the phase-difference reconstruction² when both stationary and flowing magnetizations are present within a single voxel, and usually appears for patients with thicker vessels. However, because of improved spatial acuity, such problems are largely avoided in the model-based reconstruction.

The spatiotemporal improvement achievable by model-based phase-contrast flow MRI may be invested into even faster acquisitions. Fig. 3 advances real-time phase-contrast flow MRI from 7 spokes per image and 35.7 ms total acquisition time to 5 spokes and 25.6 ms temporal resoltuion. This clearly supports the notion that the use of 5 spokes represents an extreme but feasible approach to real-time flow MRI.

Quantitative results are summarized in Tables 1³ and 2 (new healthy volunteers) and found to be in general agreement.Under all conditions, and compared to a previously developed real-time flow MRI method, the proposed method yields quantitatively accurate phase-contrast maps (i.e. flow velocities) with improved spatial acuity, reduced phase noise, reduced partial volume effects, and reduced streaking artifacts. This novel model-based reconstruction technique may become a new tool for clinical flow MRI in real time.