Rie B Hansen^{1}, Christian Ø Mariager^{2}, Christoffer Laustsen^{2}, Rolf F Schulte^{3}, Jan H Ardenkjær-Larsen^{1}, and Lars G Hanson^{1}

In this study we demonstrate how reconstruction for IDEAL spiral CSI (spectroscopic imaging scheme developed for hyperpolarized dynamic metabolic MR imaging) can be improved by using regularization with a sparsity constraint. By exploiting sparsity of the spectral domain, IDEAL spiral CSI can achieve chemical shift encoding by acquisition of only few time-shifted echoes. The minimum number of echoes required to avoid noise amplification can be decreased by means of regularization enforcing spectral sparsity, hereby reducing scan time. Improvements achieved by using regularized reconstruction are demonstrated for in vivo data from a hyperpolarized cardiac study of a pig.

Figure 1 shows the spectral encoding scheme of IDEAL spiral CSI. The chemical shift encoding for the IDEAL scheme is:

$$y_{m,n} = \sum_q{E_{m,q}e^{i\omega_q t_n} \xi_q(\mathbf{k}_n)}$$

with $$$E_{m,q} = e^{i\omega_q TE_m}$$$,
where $$$y_{m,n}$$$ is the measured data for the m^{th} echo and the n^{th} k-space sample and $$$\xi_q(\mathbf{k}_n)$$$ is the q^{th} metabolite
distribution in k-space. If $$$\mathbf{E}$$$ is well-conditioned, $$$\xi_q(\mathbf{k}_n)$$$
can be reconstructed by matrix inversion:

$$\xi_q(\mathbf{k}_n) = e^{-i\omega_q t_n} (\mathbf{E}^\dagger \mathbf{y}_n)_q$$

$$$\dagger$$$ denotes the Moore-Penrose pseudo-inverse. Saving scan time by acquiring less echoes, increases the condition number of $$$\mathbf{E}$$$, which leads to noise amplification when it is inverted. To limit this, the reconstruction can be regularized by enforcing a sparsity constraint using the L1-norm and by solving the minimization problem given by the following objective function:

$$\min_x ||\mathbf{E}\mathbf{x}_n - \mathbf{y}_n||_2^2 + \lambda ||\mathbf{x}_n||_1$$

where $$$\mathbf{x}_n = \xi_q(\mathbf{k}_n) e^{i\omega_q t_n}$$$, and $$$\lambda$$$ is a regularization parameter weighting the constraint enforced by the L1-norm. This reconstruction approach was implemented in the pre-existing IDEAL spiral CSI reconstruction pipeline iteratively for each k-space sample using fminsearch in Matlab. The regularization parameter was empirically selected to $$$\lambda$$$ = 100.000.

The method was demonstrated using
in vivo data from a hyperpolarized [1-^{13}C]pyruvate cardiac study of
a healthy female Danish domestic pig weighing 30 kg. For the hyperpolarization
experiment, the pig was first sedated with an intramuscular injection of
Stressnil (2.0 mg/kg bodyweight) and Midazolam (0.1 mg/kg) and subsequently
anaesthetized via continuous intravenous infusion of both Propofol (12 mg
initial bolus dose, 0.4 mg/kg/h thereafter for maintenance) and Fentanyl for
analgesia (8 µg/kg/h). The pig was intubated and mechanically ventilated (60 % O_{2}-air mix) using a respirator system (GE Healthcare,
Broendby, Denmark).
A whole-body clinical 3T GE HDx MR scanner (GE
Healthcare, Milwaukee, WI, USA) was used for imaging together with a
bore-insertable ^{13}C volume resonator (clamp shell design) integrated
into the patient table for excitation (GE Healthcare, Milwaukee, WI, USA). Two arrays
with 8 receive channels each^{2} were placed to cover the heart (Rapid
Biomedical, Rimpar, Germany). The IDEAL spiral CSI was cardiac triggered with 4
excitations per trigger, 11 echoes, echo-time shifting of 0.9 ms, 12
excitations per image, and 8 image repetitions [TE/TR 1.1/100 ms, flip angle 15°,
matrix 60x60, FOV 240x240 mm^{2}, in-plane nominal resolution 4 mm,
slice thickness 50 mm]. Data were averaged over the 8 repetitions before reconstruction.

Using regularization in the IDEAL spiral CSI reconstruction can reduce noise amplification as demonstrated in the lactate images of Figure 2. Signal distribution in the pyruvate-hydrate images in Figure 2(a) was, however, not retrievable using regularization. In this study the regularization parameter $$$\lambda$$$ was empirically selected, but a more thorough investigation of parameter choice could potentially improve the reconstruction. Compared to matrix inversion in IDEAL spiral CSI a longer reconstruction time is expected.

1. Wiesinger F, Weidl E, Menzel MI, et al. IDEAL spiral CSI for dynamic metabolic MR imaging of hyperpolarized [1- 13C]pyruvate. Magn Reson Med. 2012;68(1):8-16.

2. Ringgaard S, Schulte RF, Tropp J, et al. 13C RF coil combination for cardiac and abdominal human and pig studies. In: Proc. Intl. Soc. Mag. Reson. Med. ; 2016:2152.

Figure 1: Acquisition
scheme for IDEAL spiral CSI from^{1}.

Figure
2: Reconstruction results for IDEAL spiral CSI of the short-axis of the
heart for the five metabolites (from top down): lactate, pyruvate-hydrate,
alanine, pyruvate, and bicarbonate. (a) Matrix inversion reconstruction using
11 echoes, (b) matrix inversion reconstruction using 7 echoes, and (c)
regularized reconstruction using 7 echoes.