Valery Vishnevskiy^{1}, Jonas Walheim^{1}, Hannes Dillinger^{1}, and Sebastian Kozerke^{1}

Many MR image reconstruction algorithms can be formulated as optimization problems and solved with gradient-based optimization methods of choice. In this work, we present and analyze the performance of the TensorFlow framework for modeling and solving MR image reconstruction problems. We test our approach on undersampled cine cardiac and 4D flow datasets. It is demonstrated that MR image reconstruction is easy to implement in TensorFlow, TensorFlow performs comparably to sophisticated optimization algorithms with theoretical convergence guarantees, and that TensorFlow is as fast as or faster compared to standard MR reconstruction toolboxes.

Cine MRI

Fully sampled short-axis cardiac cine datasets were acquired on 3T system (Philips Healthcare, Best, The Netherlands) with spatial resolution of 1.4x1.4 mm^{3} and $$$N_t=25$$$ cardiac phases. Variable density and temporally incoherent 5-fold undersampling pattern was simulated for each frame. Applying the Fourier transform in the temporal domain to sparsify the signal, the reconstruction problem reads:

To solve problem (1) in TensorFlow, we defined the corresponding computational graph as specified by the cost function. Since gradients of operations are provided by TensorFlow, the backpropagation algorithm calculates $$$\frac{\partial\mathcal{F}}{\partial\,\!X}$$$ via the chain rule. ADAM [6] algorithm is used to minimize $$$\mathcal{F}(X)$$$, with a step length of 0.01.

4D Flow MRI

Fully sampled 4D flow data in the aortic arch of a healthy volunteer were acquired on a 3T Philips Ingenia system (Philips Healthcare, Best, the Netherlands) using a navigated Cartesian four-point phase-contrast gradient-echo sequence with uniform venc of 200 cm/s and a spatial resolution of 1.83x1.83x1.83 mm^{3}. The data were retrospectively undersampled to simulate an accelerated acquisition with factors ranging from 6 to 20. Coil sensitivity maps were calibrated with ESPIRiT [8], k-space data were compressed from 28 to $$$N_b=8$$$ virtual receiver channels [10]. Reconstruction was implemented using temporal total variation [9] regularization:

$$\min_{X_v\in\mathbb{C}^{N_p\times\,\!N_t}}\sum_{j=1,\dots,N_b}\|M_{j,v}\circ(W_{b_j}F_pX_v−Y_{j,v})\|_{2,2}^2+\lambda\|D_tX_v^{T}\|_{1,1},\quad\,v=1,\dots,N_v.\qquad(2)$$

Here $$$X_v$$$ is the estimate for $$$v$$$-th velocity encoding, $$$W_{b_j}$$$ is the diagonal matrix defined by the corresponding coil sensitivity map, $$$Y_{j,v}$$$ are zero-filled k-space samples, $$$M_{j,v}$$$ are the undersampling masks, $$$D_t\in\{−1,1\}^{N_t\times\,\!N_t}$$$ is the temporal finite difference matrix. In BART [2] problem (2) is solved using the ADMM algorithm employing the conjugate gradient method for internal iterations. In TensorFlow, the computational graph for problem (2) was defined in a similar way as described for problem (1) and optimized with the ADAM algorithm using 300 iterations.

Figure 2 shows that both ADMM and TensorFlow reconstructions of the cine MRI data achieve the same performance in terms of optimization cost and estimated residual relative to ground truth. TensorFlow requires more iterations than ADMM to converge (378 for TensorFlow vs. 71 for ADMM). However, when iteration time is taken into account, due to GPU utilization, TensorFlow takes 1.6 seconds to converge, compared to 2.4 seconds using ADMM. Figure 3 illustrates that both reconstruction methods provide visually comparable results.

Figure 4 shows that both methods provide similar reconstruction accuracy for all acceleration factors tested on the 4D flow data. Relative velocity magnitude discrepancy between BART and TensorFlow reconstructions inside the aorta was less than 3%. Reconstruction results are illustrated in Figure 5. Average runtime of BART was 7.5 minutes on a 6-core 3.8 GHz CPU, runtime of TensorFlow was 4.6 minutes on NVIDIA Titan Xp GPU.

Discussion

The authors acknowledge funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 668039.

We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research.

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Figure 1: Comparison of conventional and TensorFlow-based workflows for developing image reconstruction methods. Dashed boxes represent steps that can be carried out automatically using widely available and efficient tools or libraries. Solid outline indicates steps that require manual problem-specific and time-consuming implementation.

Figure 2: Target function cost value (top row) and estimate deviation from the ground truth (bottom row) on every iteration for ADMM and TensorFlow reconstructions on cine MR data. Corresponding values are shown with respect to iteration number (left column) and time in seconds measured from the first iteration (right column).

Figure 3: Illustration of image reconstruction from 5-fold undersampled cine MRI data using ADMM (upper row) and TensorFlow (lower row). Column (a) shows fully sampled ground-truth (upper row) and temporal profile (bottom row) at the green dashed line. (b) and (c) show images reconstructed with ADMM and TensorFlow, and their respective errors in (d) and (e).

Figure 4: (a) Mean absolute value of reconstruction error relative to fully sampled ground truth with ADMM (red) and TensorFlow (blue) for different undersampling factors. (b) Relative velocity magnitude difference between BART and TensorFlow reconstructions with respect to average velocity magnitude of fully sampled dataset inside the aorta as delineated in Figure 5.

Figure 5: Comparison of TensorFlow and BART reconstructions from 8-fold undersampled data relative to the ground truth. Aorta segmentation is delineated with the red dashed line.