Julia Busch^{1}, Valeriy Vishnevskiy^{1}, Maximilian Fuetterer^{1}, Claudio Santelli^{1}, Constantin von Deuster^{1}, Sophie Marie Peereboom^{1}, Mareike Sauer^{2}, Thea Fleischmann^{2}, Nikola Cesarovic^{2}, Christian Torben Stoeck^{1}, and Sebastian Kozerke^{1}

The IDEAL signal model for hyperpolarized metabolic imaging is extended and spatiotemporal regularization and b0-map recalibration is included. The approach is tested on simulated data and in-vivo metabolic imaging data of the heart. Allowing variable b0-fields and including sparsity regularization signal leakage and ghosting can be significantly reduced (average reduction of root-mean-square error (RMSE) by 16% and 30%). Spatial and temporal regularization of the metabolite intensities considerably improved accuracy of the estimate in terms of RMSE with additional reductions by 68% and 20%, respectively. Thus, the metabolic conversion of [1-13C]pyruvate into [1-13C]lactate and 13C-bicarbonate can be measured with improved accuracy.

If a multi-echo acquisition is
combined with an echo-planar imaging readout, the IDEAL signal model^{8,12} has to account
for the chemical shift dependent spatial shift each metabolite undergoes:

$$${u}_{n}\left(\mathbf{k}\right)=\underbrace{\underset{m=1}{\overset{M}{\mathop\sum }}\,{{e}^{i2\pi \Delta{\nu}_{m}{t}_{n}}}{{e}^{-i\mathbf{k}\Delta{\mathbf{r}}_{m}}}\underset{r}{\mathop\sum }\,{{e}^{i\mathbf{kr}}}}_{\mathbf{E}'}\underbrace{{{w}_{m,n}}\left(\mathbf{r}\right)}_{\mathbf{W}\left(\mathbf{w}\right)}\underbrace{{{e}^{i2\pi\tilde{\gamma}{{b}_{0}}\left(\mathbf{r}\right){{t}_{n}}}}}_{\mathbf{W}\left(\mathbf{{b}_{0}}\right)}{{\mathbf{\rho}}_{m}}\left(\mathbf{r}\right)$$$, [1]

with $$$\mathbf{\rho}_m\left(\mathbf{r}\right)$$$: intensities of
the M metabolites at location $$$\mathbf{r}$$$; $$$\Delta\nu_{m}$$$: chemical shift;
$$$u_{n}\left(\mathbf{k}\right)$$$: k-space signal
of the n-th echo; $$$t_{n}=TE+\Delta t_{n}$$$: echo time; $$$b_{0}\left(\mathbf{r}\right)$$$: b_{0}-phase
offsets in Hz; $$$\tilde{\gamma
}=\frac{{{\gamma }_{13C}}}{{{\gamma }_{1H}}}$$$: the scaling ratio between the gyromagnetic ratio of
13C and 1H; $$$\Delta\mathbf{r}_{m}=\Delta\mathbf{r}\left(\Delta\nu_{m}\right)$$$: spatial shift. To
address scaling of the signal magnitude $$$\rho_{m}$$$ between different echoes due to T_{2}*-
dephasing, flip angle dependent signal saturation or inflow effects, a
weighting function $$$w_{m,n}$$$ with $$$\rho_{m,n}=w_{m,n}\rho_{m}$$$ is introduced.
Equation [1] can be written in
matrix notation and formulated as an optimization problem

$$$\arg~\underset{\mathbf{\rho}}{\mathop{\min}}\,\left\| \mathbf{{E}'W}\left(\mathbf{w}\right)\mathbf{W}\left(\mathbf{{b}_{0}}\right)\mathbf{\rho} -\mathbf{u} \right\|_{2}^{2}$$$, [2]

where $$$\left\|\text{ }\right\|_{2}^{2}$$$ denotes the L2-norm.
To account for low SNR of in vivo acquisition
combined with inaccuracies in b_{0}-phase maps and flip angle values, the model is
extended using spatiotemporal regularization and b0-map
recalibration. The regularized model fit quality is defined as

$$$\mathcal{F}\left(\mathbf{\rho},~{\mathbf{b}_{0}}|\mathbf{w},~\mathbf{u} \right)={{\lambda}_{\text{n}}}\left\|\mathbf{{E}'W}\left(\mathbf{w}\right)\mathbf{W}\left({\mathbf{b}_{0}}\right)\mathbf{\rho}-\mathbf{u}\right\|_{2}^{2}+{{\lambda}_{s}}{{\left\|\mathbf{\rho}\right\|}_{1}}+{{\lambda}_{\text{TV}}}\text{TV}\left(\mathbf{\rho}\right)+{{\lambda}_{t}}{{\left\|{{\mathbf{D}}_{2}}\mathbf{\rho}\right\|}_{1}}+{{\lambda}_{{{b}_{0}}}}\text{TV}\left({\mathbf{b}_{0}}\right)$$$. [3]

Here, TV denotes isotropic total variation and $$${{\mathbf{D}}_{2}}$$$ the second order derivative in
temporal direction. The cost function [3]
is iteratively minimized with the optimization algorithm ADAM^{13}.

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Figure1: RMSE of the estimate
computed using different regularization parameters. Each graph shows the effect
of variation in a single regularization parameter. Settings indicated by the pink
dot (λ_{s}=0.13, λ_{TV}=5.0,
λ_{t}=2.5, λ_{b0}=0.01) are used for all experiments.

Figure 2: Columns show results from adding different regularization
terms to the inverse problem according to Equation [3]. RMSEs for each metabolite
over the whole field of view are reported. Standard deviation of noise is 1.Error and ground
truth (first column) for different metabolites using simulated data using the slice
from Figure 1 taken at the time point where corresponding metabolites show
maximum intensity.

Figure 3: Time averaged intensities of metabolite signals
for an examplary in-vivo dataset reconstructed with the unregularized and proposed
reconstruction methods. Please note the improved depiction of lactate and
bicarbonate maps.

Figure 4: Time-intensity curves
of lactate, pyruvate and bicarbonate averaged over the myocardium, left (LV) and right ventricles (RV) obtained from in-vivo data
reconstructed with the unregularized vs. the proposed reconstruction method.