Julia Traechtler^{1}, Valery Vishnevskiy^{1}, Maximilian Fuetterer^{1}, Andreas Dounas^{1}, and Sebastian Kozerke^{1}

^{1}Institute for Biomedical Engineering, University and ETH Zurich, Zurich, Switzerland

An advanced signal model-based reconstruction jointly estimating image and field map from multi-echo, multi-coil acquisition of hyperpolarized metabolic data was developed and validated using synthetic and in-vivo data. Relative to standard multi-echo reconstruction methods, reconstruction accuracy improved by up to 30% for synthetic data considering realistic noise levels and field map gradients. Geometric distortion correction resulted in less than 20% error. For in-vivo data, the average improvement was 15%. Depending on the direction of the field gradients present, multi-coil reconstruction was found to be beneficial for addressing signal folding issues.

$$\underbrace{s(\vec{k},d,e)}_{\vec{s}}=\underbrace{\sum_{\vec{x}}e^{j\vec{k}\vec{x}}\sum_m{}e^{j2\pi{}f(m)t(e)}\cdot{}C(\vec{x})\cdot{}e^{-\frac{t_{Dyn}(d)}{T_1(m)}}\cdot{}e^{-\frac{t(e)}{T^{*}_2(\vec{x},m)}}\cdot{}\sin\left(\alpha(\vec{x},m)\right)\cdot{}\cos\left(\alpha(\vec{x},m)\right)^{e-1}}_{\mathbf{E}}\cdot{}\underbrace{e^{j2\pi{}B_0(\vec{x})t(e)}}_{\mathbf{\hat{B}_0}}\cdot{}\underbrace{\rho(\vec{x},d,m)}_{\vec{\rho}}+\eta~~~\text{[1]}$$

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 B

$$\underset{\vec{\rho},\mathbf{\hat{B}_0}}{\text{argmin}}{}\left|\left|\left(\mathbf{E}\cdot{}\mathbf{\hat{B}_0}\right){}\vec{\rho}-\vec{s}\right|\right|_2^2+\lambda_{\rho}|\nabla_x\vec{\rho}|_1+\lambda_{\hat{B}_0}|\nabla_x\mathbf{\hat{B}_0}|_1$$

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 Tensorflow

For the simulation study, echo-shift encoded data of metabolic signals were generated using the simulation framework as described previously

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 reconstruction

Finally, the proposed method was tested on in-vivo data acquired in three healthy pigs at 3T

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. B

Regularization parameters were chosen according to RMSE using grid search.

Fig-3 illustrates the reconstruction for a fixed B

Fig-4 shows reconstruction results for data simulated with in-vivo B

Fig-5 shows reconstructed in-vivo images with and without B

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

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