Synopsis

A model-based iterative reconstruction algorithm for spiral chemical shift imaging (spCSI) of hyperpolarized ¹³C metabolites is implemented and compared to conventional CSI reconstruction. The iterative reconstruction utilizes the prior knowledge of 1) the off-resonance frequencies for each metabolite and 2) the common spatial distribution for pyruvate and pyruvate hydrate. The proposed method improves quantification of bicarbonate and reduces artifacts for pyruvate maps caused by signal from a urea reference phantom.

Introduction

Method

The iterative reconstruction generates the metabolic maps through solving the minimization problem $$$\hat{\rho} = \underset{\rho}{\operatorname{argmin}} \| s - E\cdot \rho\| $$$, where $$$E$$$ represents the encoding matrix relating the metabolic map $$$ \rho $$$ and acquired signal $$$S$$$. For an M-metabolite system, the encoding matrix describes the following forward signal model²:

$$ S\left(t\right)=\int\limits_{V} e^{i\vec{k}\left(t\right)\cdot\vec{r}} \displaystyle\sum_{\text{i}}^{M}\rho_{\text{i}}\left(\vec{r}\right)e^{i2\pi\Delta f_{\text{i}}\cdot t} d\vec{r}$$

$$$\Delta f_{\text{i}}$$$ is the off-resonance frequency for metabolite $$$\text{i}$$$ and $$$\vec{k}\left(t\right)$$$ is the spiral readout trajectory in k-space, both known priorily.

To resolve the overlapped peaks of pyruvate and pyruvate hydrate, we utilize the prior knowledge that they have the same spatial distribution with a fixed ratio $$$k_{p} ={}^{\rho_{pyrh}}/_{\rho_{pyr}}$$$due to their chemical balance. The signal equation can be expanded as:

$$ S\left(t\right)=\int\limits_{V} e^{i\vec{k}\left(t\right)\cdot\vec{r}} \rho_{Pyr}(\vec{r})\left(e^{i2\pi\Delta f_{pyr}\cdot t} + k e^{i2\pi\Delta f_{pyrh}\cdot t}\right)d\vec{r}$$

After contructing the encoding matrix $$$E$$$ properly, the solution $$$\rho$$$ can be derived using conjugate gradient algorithm with fast convergence³.

Results and Discussion

Digital simulations performed in Matlab (R2018a, The MathWorks, Natick, MA) evaluated the effectiveness of iterative reconstruction in removing the chemical shift artifact from spectrally overlapped pyruvate hydrate. Signal acquisition is simulated according to a 2D spiral CSI sequence with FOV = 80mm, matrix size 16×16, 3 spatial interleaves, 32 echoes and 276Hz spectral width (ΔTE=3.62ms). Conventional CSI reconstruction for a 4mm-radius circular disc phantom containing pure [1-¹³C] pyruvate (Δf_pyr=0Hz, relative amplitude (RA)=1.00) is set as ground truth, shown in Figure 1a. Conventional CSI reconstruction for a 4mm-radius circular disc phantom containing pyruvate and pyruvate hydrate mixture (Δf_pyrh=269Hz, RA=0.11) is shown in Figure 1b. Iterative reconstruction using the given prior knowledge for the same mixture phantom is shown in Figure 1c. The result from the iterative reconstruction demonstrates the effect of de-blurring at the side lobes.

Two in vivo experiments were conducted on a 3T MR scanner on healthy rats with intravenous injections of hyperpolarized [1-¹³C] pyruvate. In the first experiment, image data were acquired at a 10mm slice through the kidneys by the same spCSI sequence as in the digital simulation. Metabolic maps for four metabolites were derived: lactate (Δf_lac=195Hz), alanine (Δf_ala=-16Hz), pyruvate (Δf_pyr=-195Hz) and bicarbonate (Δf_bic=-516Hz). Results from both CSI reconstruction and iterative reconstruction are shown in Figure 2. By using the prior knowledge of the spectrum and $$$k_{p}=0.08$$$, the metabolic maps using iterative reconstruction gives better quantification for bicarbonate. Results for iterative reconstruction are prone to error if pyruvate hydrate is treated as an independent metabolite.

In the second experiment, a dynamic 3D spCSI sequence (FOV=80mm×80mm×60mm, matrix size 16×16×12, 4 spatial interleaves, 24 echoes, 274Hz spectral width, 12 time points with 6 seconds temporal resolution) was applied. A time-averaged signal between 48 sec and 72 seconds after injection was used to reconstruct the metabolic maps. Figure 3 shows the results at the slice through the liver, where a urea syringe phantom was placed on top of the rat. Since urea peak (Δf_urea=-242Hz) and pyruvate peak (Δf_pyr=-482Hz) partially overlap in the under-sampled spectrum, severe artifacts exist for the CSI reconstruction, while the iterative reconstruction can resolve the two metabolites successfully.

Conclusion

Acknowledgements

References

1. Mayer, Dirk, et al. "Fast metabolic imaging of systems with sparse spectra: application for hyperpolarized 13C imaging." Magnetic resonance in medicine 56.4 (2006): 932-937.
2. Gordon, J. W., Niles, D. J., Fain, S. B., & Johnson, K. M. Joint spatial‐spectral reconstruction and k‐t spirals for accelerated 2D spatial/1D spectral imaging of 13C dynamics. Magnetic resonance in medicine 71.4 (2014): 1435-1445.
3. Sutton, B. P., Noll, D. C., & Fessler, J. A. Fast, iterative image reconstruction for MRI in the presence of field inhomogeneities. IEEE transactions on medical imaging 22.2 (2003): 178-188.