Extended Signal Modelling and Regularization for Multi-Echo Hyperpolarized Metabolic Image Reconstruction
Julia Busch1, Valeriy Vishnevskiy1, Maximilian Fuetterer1, Claudio Santelli1, Constantin von Deuster1, Sophie Marie Peereboom1, Mareike Sauer2, Thea Fleischmann2, Nikola Cesarovic2, Christian Torben Stoeck1, and Sebastian Kozerke1

1Institute for Biomedical Engineering, University of Zurich and ETH Zurich, Zurich, Switzerland, 2Division of Surgical Research, University Hospital Zurich, Zurich, Switzerland


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.


Dynamic nuclear polarization1,2 enables the assessment of metabolic conversion of 13C substances in-vivo3-6. Multi-echo acquisition schemes in combination with IDEAL reconstruction7,8 have been used for 13C metabolic imaging9,10. Since the IDEAL reconstruction is phase-sensitive, accurate knowledge of the b0 induced phase offsets is of particular importance and more advanced reconstruction techniques have to be applied11. In the present work, the IDEAL signal model is extended and an iterative estimation of the b0 induced phase offsets is included. The approach is tested on simulated data and in-vivo metabolic imaging data of the heart.


If a multi-echo acquisition is combined with an echo-planar imaging readout, the IDEAL signal model8,12 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; $$$b_{0}\left(\mathbf{r}\right)$$$: b0-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 T2*- 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 b0-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 ADAM13.


The algorithm to minimize cost function [3] was implemented in Python using the Tensorflow 1.314 framework and executed on GPU (NVIDIA Titan Xp) with IEEE 754 32-bit floating-point format. Based on the MRXCAT phantom data15 and realistic intensity time curves the metabolic conversion of pyruvate into lactate, alanine and bicarbonate was simulated according to equation [1]. Healthy female pigs (N=6, 30-35kg) were anesthetized using propofol or isoflurane and ventilated with oxygen. Venous catheters for medication and injection of hyperpolarized substances were introduced. Animals were placed in a 3T Philips Ingenia system (Philips Healthcare, Best, The Netherlands) equipped with a custom-built four-channel 13C transmit/receive coil (Clinical MR Solutions, Brookfield WI, USA). A glucose (20%) - insulin (50U/L) solution16-18 was infused with a loading rate of 3ml/kg/h for 60 minute and a maintenance rate (1-3ml/kg/h) thereafter. All experiments were performed in accordance with the Swiss Animal Protection Law and Ordinance. 0.5ml of [1-13C]pyruvic acid was prepared in a commercial SpinLab Hyperpolarizer (GE Healthcare, Waukesha, WI, USA). Upon dissolution, 20ml of the 225mM pyruvate solution was injected into the femoral vein. Immediately following injection, three slices were dynamically acquired using a multiband multi-echo excitation acquisition scheme10 (resolution: 5x5x20mm3, field-of-view: 220x220mm2, 75% partial Fourier, echo time/repetition time: 12ms/32ms, no. echoes: 7, echo spacing: 1.1ms). For excitation, a 1-2-1 binomial excitation pulse of 20ppm bandwidth and 30⁰ flip angle was used. An additional 1H b0-map was acquired using the 2-channel body coil of the scanner. Sequence parameters were: 3 slices, resolution: 2.5x2.5x8mm3, field-of-view: 321x251x48mm3, echo time/repetition time: 1.3ms/3.9ms and flip angle 20°. All regularization weights $$$\lambda=\{\lambda_{S},\lambda_{TV},\lambda_{t},\lambda_{b_0}\}$$$ were tuned on synthetic data by minimizing the mean root mean square error (RMSE) relative to ground truth over the myocardium, left ventricle (LV), right ventricle (RV) and background (Figure 1), yielding $$$\lambda^\star=\{0.13,5.0,2.5,0.01\}$$$.


Without regularization the time-averaged lactate and bicarbonate images show ghosting and signal leakage which can be reduced using isotropic TV regularization on b0 (average reduction of RMSE by 16%). The consecutive introduction of sparsity, temporal and spatial regularization result in further reduction of the RMSE by 30%, 68% and 20%, respectively, with an overall improvement by 85% in RMSE when compared to the unregularized reconstruction (Figure 2). In-vivo, ghosting and signal leakage artifacts are greatly reduced (Figure 3). In particular, lactate and bicarbonate images show reduced artifacts in the myocardium. Temporal signal intensity time curves averaged over cardiac compartments show good separation of the metabolite signals (Figure 4).


Allowing variable b0 fields and including sparsity regularization, signal leakage and ghosting can be significantly reduced. Spatial and temporal regularization of the metabolite intensities considerably improved accuracy of the estimate in terms of RMSE. With the presented sequence and reconstruction algorithm the metabolic conversion of [1-13C]pyruvate into [1-13C]lactate and 13C-bicarbonate can be measured with improved accuracy.


No acknowledgement found.


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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 (λs=0.13, λTV=5.0, λt=2.5, λb0=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.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)