Introduction

Hyperpolarized ¹³C magnetic resonance imaging has shown potential for imaging metabolism of the in-vivo human heart¹. Multi-echo acquisition² combined with reconstruction techniques that deconvolve signal contributions of individual metabolites based on their known chemical shift³ have been proposed for hyperpolarized ¹³C MRI⁴. However, multi-echo reconstruction is sensitive to B₀ induced phase offsets and therefore more advanced reconstruction techniques are required^5,6. In the present work, a model-based reconstruction jointly estimating image and field map from multi-echo data is proposed and validated using synthetic and in-vivo data.

Methods

An advanced model-based reconstruction of multi-echo, multi-coil hyperpolarized metabolic data is presented (cf. Fig-1). The signal model accounts for chemical shift $$$f(m)$$$ dependent spatial shifts of metabolite $$$m$$$ as well as distortions induced by varying in-plane off-resonances $$$B_0$$$. Sampling along a trajectory $$$\vec{k}$$$ at time points $$$t_s$$$ and considering weighting terms due to coil sensitivities $$$C(\vec{x})$$$ as well as $$$T_1$$$, $$$T_2^*$$$ relaxation, and spectral-spatial excitation $$$\alpha(\vec{x})$$$, yields the signal:
$$\underbrace{s(\vec{k},d,e)}_{\vec{s}}=\underbrace{\sum_{\vec{x}}e^{j\vec{k}\vec{x}}\sum_m{}e^{j2\pi{}f(m)t(e)}\cdot{}C(\vec{x})\cdot{}e^{-\frac{t_{Dyn}(d)}{T_1(m)}}\cdot{}e^{-\frac{t(e)}{T^{*}_2(\vec{x},m)}}\cdot{}\sin\left(\alpha(\vec{x},m)\right)\cdot{}\cos\left(\alpha(\vec{x},m)\right)^{e-1}}_{\mathbf{E}}\cdot{}\underbrace{e^{j2\pi{}B_0(\vec{x})t(e)}}_{\mathbf{\hat{B}_0}}\cdot{}\underbrace{\rho(\vec{x},d,m)}_{\vec{\rho}}+\eta~~~\text{[1]}$$
where $$$t=t_s+TE$$$ with $$$TE$$$ being the echo time of echo $$$e$$$, $$$\rho$$$ denoting the spatiotemporal object of dynamic $$$d$$$ and $$$\eta$$$ Gaussian noise.
Rewriting equation [1] in matrix notation $$$\vec{s}=\left(\mathbf{E}\cdot{}\mathbf{\hat{B}_0}\right)\vec{\rho}$$$, allows to formulate image reconstruction as the following minimization problem optimizing jointly for image and B₀:
$$\underset{\vec{\rho},\mathbf{\hat{B}_0}}{\text{argmin}}{}\left|\left|\left(\mathbf{E}\cdot{}\mathbf{\hat{B}_0}\right){}\vec{\rho}-\vec{s}\right|\right|_2^2+\lambda_{\rho}|\nabla_x\vec{\rho}|_1+\lambda_{\hat{B}_0}|\nabla_x\mathbf{\hat{B}_0}|_1$$
with $$$\lambda_{\rho}$$$, $$$\lambda_{\hat{B_0}}$$$ being regularization parameters and $$$|\cdot|_1$$$, $$$||\cdot||_2$$$ denoting the L1- and L2-norm, respectively.
The mathematical framework was implemented in Python using Tensorflow⁷ and executed on GPUs.

For the simulation study, echo-shift encoded data of metabolic signals were generated using the simulation framework as described previously⁸. An echo-planar imaging (EPI) trajectory (in-plane resolution=5x5mm², FOV=85x85mm²) was used to generate synthetic k-space data for different high-resolution (1x1mm²) B₀ maps and coil sensitivities. In-plane B₀ maps with linear gradients, scaled by varying beta factors (relative phase encoding bandwidth), were used to examine geometric distortion and correction. Scaling by beta>0 stretches the object by a factor of (1+beta) and beta<0 results in object compression with beta=-1 being a projection of the object.

Image reconstruction and off-resonance estimation accuracies w.r.t. noise were analyzed for a single homogenous coil and constant beta=+0.2. The effects of multi-coil encoding were tested for aliasing (beta=+0.5) using simulated coil sensitivities generated using Biot-Savart’s law for one, two and four coils, respectively.

In-vivo applicability was evaluated on synthetic data using six coil sensitivity maps (estimated for reconstruction⁹) and three B₀ map slices obtained from in-vivo data. Data was generated as the average of three sub-slices to account for through-slice B₀ variations. Imaging parameters for the EPI trajectory were set according to in-vivo conditions: in-plane resolution=5x5mm², FOV=220x220mm², slice thickness=20mm, lactate signal-to-noise ratio (SNR) averaged over the myocardium of 15. Both EPI phase-encoding blip directions left/right were analyzed.

Finally, the proposed method was tested on in-vivo data acquired in three healthy pigs at 3T¹⁰ and compared to standard non-linear conjugate gradient-based reconstruction without B₀ correction. Measured field maps (¹H) served as reference for evaluating B₀ estimation.

For the simulation studies, the pixel-wise normalized root-mean-square error (nRMSE) between reconstructed image and ground truth object were compared. For both, synthetic and in-vivo data, the DICE coefficient served as quantification metric for the reconstructed images. B₀ estimation was evaluated based on the pixel-wise RMSE.

Regularization parameters were chosen according to RMSE using grid search.

Results

Fig-2 shows reconstructed images and estimated B₀ maps for different in-plane B₀ gradients beta=[+0.5,-0.5] corresponding to a maximum B₀ range of 236Hz at the carbon resonance frequency at 3T. Reconstruction accuracy in terms of RMSE and DICE decreases linearly with increasing |beta| for reconstruction without B₀ correction, whereas the proposed reconstruction shows improved accuracy for beta=[+0.2,-0.1]. In this range, distortion correction is feasible with less than 20% error.

Fig-3 illustrates the reconstruction for a fixed B₀ gradient as function of SNR and number of coils, respectively. For beta=+0.2, an increase in SNR to 12 significantly improves B₀ estimation accuracy. In case of aliasing (beta=+0.5), increasing the number of coils from one to two/four improves image and B₀ RMSE by 4%/5% and 14%/23%, and DICE by 2%/4%, respectively.

Fig-4 shows reconstruction results for data simulated with in-vivo B₀ gradients for both EPI phase-encoding directions. For in-plane B₀ gradients, the image RMSE of the proposed reconstruction is 14% lower compared to the standard reconstruction. Through-plane B₀ gradients, which are not estimated, deteriorate reconstruction performance.

Fig-5 shows reconstructed in-vivo images with and without B₀ correction. For all three examined in-vivo cases, the average improvement in DICE score accounts for 15% for the proposed method compared to standard reconstruction without B₀ estimation.

Discussion

Advanced multi-echo, multi-coil reconstruction allows for joint estimation of image and field maps thereby addressing off-resonance induced geometrical distortions and resonance separation issues. Given the relatively low spatial resolution and SNR in hyperpolarized MRI, image and field map estimation were found to be limited by SNR and field gradients. Based on accepting 20% reconstruction error, a lactate SNR>=12 and a maximum field gradient covering 94Hz (beta=+0.2) is recommended. Depending on the direction of the field gradients, multi-coil reconstruction was shown to be beneficial in addressing signal folding issues.

In conclusion, joint image and field map estimation holds promise to provide geometrically and chemically-shift consistent metabolic maps in multi-echo hyperpolarized MRI.

Acknowledgements

No acknowledgement found.