Introduction

Fluorine-19 (¹⁹F) MRI uses perfluorocarbon (PFC) tracers to enable background-free, “hot-spot” in vivo cell tracking¹. Detection of multiple cell targets requires separating images acquired using different PFC molecules (Fig. 1A-B) with different chemical shifts. The resulting shift artifacts using Cartesian k-space sampling appear as displacements (Fig. 1C) along the readout direction². Moreover, highly sensitive imaging can be achieved using newer, metallo-PFC (MPFC) nanoemulsion probes which have an order of magnitude T₁ reduction³; these tracers are optimally imaged using radially sampled ultrashort echo-time (UTE) pulse sequences⁴ to overcome paramagnetic shortening of T₂, which can also be combined with k-space undersampling and compressed sensing reconstruction for further sensitivity enhancement⁴. A key challenge with radially sampled data is that the frequency offsets lead to non-linear smearing chemical artifacts throughout the image (Fig. 1D).
In this work, we develop a model of the chemical shift effects with radial k-space sampling. Furthermore, we exploit a physics-informed primal-dual unrolling strategy that enables simultaneous artifact removal, as well as low SNR detection.

Theory and Methods

From Fourier slice theorem, radial k-space data $$$s(k, \Phi)$$$ can be obtained by taking 1D Fourier transform $$$F_k$$$ along the $$$t$$$ direction of the projection data $$$R_{\Phi}(t)$$$, where $$$R$$$ denotes Radon transform⁵. We consider two PFC imaging tracers with multiple chemical frequency shifts (Fig. 1E), representing perfluorotert-butyl cyclohexane (PFTBC as $$$\rho_1$$$) and perfluoro-15-crown-5-ether (PFCE as $$$\rho_2$$$). The reference frequency is set at -92.5 ppm for PFCE. By acquiring $$$P\geq2$$$ sets of measurements $$$s=\{s_j\}_{j=1}^P$$$ with varying readout bandwidths and hence varying effective sampling time⁶, distinct harmonic structure can be produced. We model the radial chemical shift as described in Fig. 2. $$$\breve{H}$$$ represents the spectral mixing matrix, where the corresponding convolution kernels can be analytically derived⁷. Given the overall forward sensing operator $$$\mathscr{A}=F_kR\breve{H}$$$ (Fig.2A), the corresponding inverse problem is
$$\min_{x=[\textrm{ρ}_1;\textrm{ρ}_2]}\|s-\mathscr{A}x\|_2+\lambda\mathscr{D}\left(x\right)\quad(1)$$
where the regularizer $$$\mathscr{D}(x)$$$ encodes the structural prior of $$$x$$$. Here we employ the total-variation norm $$$\|\nabla x\|_1$$$, which naturally promotes sparsity. Eq.(1) is solved by a primal-dual hybrid gradient (PDHG) scheme using the explicit proximal operators⁸. To further improve the performance under low SNR and undersampling scenarios, we consider learning an end-to-end data-driven prior from training sample pairs $$$(s^{(1)},x^{(1)}),\dots,(s^{(N)},x^{(N)})$$$. The goal is to learn a non-linear mapping $$$\mathscr{A}_{\theta}^{\dagger}$$$ based on the model in Fig.2A such that $$$\mathscr{A}_{\theta}^{\dagger}(s)\approx x$$$. The objective is to minimize the empirical loss$$\hat{L}(\theta)=\frac{1}{N}\sum_{i=1}^{N}\|\mathscr{A}_{\theta}^{\dagger}(s^{(i)})-x^{(i)}\|_2\quad(2)$$Inspired by the learned CT reconstruction⁸, we replace the proximal operators with convolutional neural networks (CNNs) for learned proximals to unroll the primal-dual (UPD) algorithm for radial multi-spectral deconvolution. We trained UPD with 250 synthetic PFC image pairs (each consisting of two component images of 64² size) to simulate¹⁹F phantoms⁴, along with synthetic UTE data generated using the equation in Fig.2A. This yields input-SNRs of 20 and 5 dB, which corresponds to moderate and high noise, respectively. The number of radial projections were chosen to be 10 and 20 (i.e., 10, 5-fold undersampling). $$$\breve{H}$$$ was derived from the chemical shift spectra displayed on Fig.1E with readout bandwidth (BW) =100 kHz–200 kHz. Test data were generated using a separate set of phantom images with the same $$$\mathscr{A}$$$ and noise levels. We examined the PDHG reconstruction with optimally tuned parameters as a baseline. The network was trained by minimizing Eq.2 using Adam optimizer. All the implementations are based on Python using Operator Discretization Library (ODL)⁹ and Pytorch.

Results

Proper modeling of $$$\mathscr{A}$$$ enabled successful chemical shift deconvolution across all the test cases. For input SNR = 5 dB, some of the weak ¹⁹F signals are not detected using PDHG (Fig. 3B) due to the limitations of structural prior under highly noisy scenarios. In contrast, learned UPD is capable of enhancing the signal detection with much more pronounced noise removal effects, which yields > 10 dB improvement in peak-signal-to-noise-ratio (PSNR,$$$20\log_{10}(M/\|\mathscr{A}_{\theta}^{\dagger}(s^{(i)})-x^{(i)}\|_2)$$$, where $$$M$$$ denotes the maximum possible pixel value) compared to PDHG (Fig. 4). Approximately ~1 dB improvement is achieved using $$$\breve{H}$$$ with BW = 200 kHz, 100 kHz (Fig.4A) against BW = 150 kHz, 100 kHz, respectively (Fig.4B). This implies that joint design of $$$R$$$ and $$$\breve{H}$$$ in the overall sensing operator $$$\mathscr{A}$$$ is necessary to guarantee recovery.

Discussion and Conclusion

The modeling of radial MRI chemical shifts as convolutional effects provides a new avenue to unmix the component images. By leveraging the connection between radial sampling and Radon transform, learned primal-dual through unrolling significantly improves the performance of radial multi-spectral deconvolution of highly noisy and compressed data. In addition to the detection of multiple ¹⁹F tracers in vivo, these methods provide a new avenue to unmix chemical shifts for other X-nuclei applications with short T₂ components.