Synopsis

Hyperpolarized $$$^{13}\text{C}$$$-imaging techniques a powerful and clinically translatable method to image metabolism. However, owing to the finite and non-renewable magnetisation available to the technique, all proposed imaging sequences necessarily have a comparatively small matrix size compared to conventional anatomical imaging. Typically hyperpolarized images are therefore reconstructed with a large degree of zero-filling. We show here that a modified form of 2D least-squares linear prediction that uses the known analytic properties of trigonometric curves can extrapolate unmeasured Fourier coefficients and hence improve the apparent reconstructed resolution of hyperpolarized images.

Purpose

Hyperpolarized $$$^{13}\text{C}$$$-imaging methods hold enormous promise as a new metabolic imaging technique and are, with [1-$$$^{13}$$$C]pyruvate, translating to the clinic.[1,2] However, the combination of a limited $$$T_2$$$ and the finite and non-renewable magnetisation inherent to all hyperpolarized experiments necessarily limits the achievable matrix size of any imaging acquisition that could be performed, typically to $$$16\times16,\,32\times32$$$ or similar. As comparable anatomical imaging matrices are substantially larger, and such images are subject to severe partial volume effects, it is therefore common to zero-fill the acquired data by a factor of $$$2^{1\rightarrow4}$$$. This corresponds directly to a sinc-shaped interpolation in the image domain.

Several methods have been proposed to recover higher spatial frequency information than that acquired in MR, most famously 1D linear prediction for FIDs,[3,4] and compressed sensing. However, modern compressed sensing methods require sparsity that may be difficult to guarantee with small matrix sizes and often require nonuniform sampling at acquisition time in $$$k$$$-space for acceleration.[5,6] Although compressed sensing approaches have been previously proposed with hyperpolarized MRI, the undersampling regime has typically been one dimensional and limited to a small number of points on the extremities in $$$k$$$-space, in part owing to gradient fidelity limitations and the breakdown of the sparsity assumptions with small matrix sizes.[7]

As an alternative, we propose here the use of a method for analytically extending acquired samples into the Fourier domain that is traditionally viewed as too computationally intensive for use with larger matrix sizes, as commonly found on proton imaging.

Methods

Following the novel approach proposed by Ongie and Jacob,[8] we assume that hyperpolarized images are piecewise-constant apart from a finite number of edges. It has been shown that these edges obey an annihilation relation with a finite subset of trigonometric curves that can be uniquely determined under certain conditions. We use the analytic properties of this fact to perform a least-squares linear prediction of the acquired Fourier samples to new parts of the complex domain. That is, if $$$I(\mathbf{x})$$$ is a (fully-sampled) MR image, and $$$f(\mathbf{x})=\mathcal{FT}(\mathbf{x})$$$ its $$$k$$$-space, then $$$\exists c[\mathbf{k}]$$$ defining a trigonometric curve $$$\mu$$$ such that

\begin{align}\mu(\mathbf{x})\nabla{}f(\mathbf{x}) &=0,\quad\,\text{where for a finite image}\\ \mu(\mathbf{x}) &= \sum_{\mathbf{k}\in\Lambda}c[\mathbf{k}]e^{2\pi i\mathbf{k}\cdot\mathbf{x} },\quad\text{given}\,\,c[\mathbf{k}]\in\mathbb{C},\,\text{and}\,\,\Lambda\subset\mathbb{Z}^2,\,\,|\Lambda|\in\mathbb{N}.\end{align}

It can be shown that if, instead of acquiring a fully-sampled image $$$I$$$, we obtain only a few (low) spatial frequencies of $$$f$$$ given as

\begin{align} \widehat{f[\mathbf{k}]}=\int I(\mathbf{r}) e^{-2\pi i\mathbf{k}\cdot\mathbf{r}} \,\text{d}\mathbf{r},\quad\mathbf{k}\in\Gamma\subset\mathbb{Z}^2\end{align}

then this annihilation condition corresponds to a finite system of linear equations

\begin{align} \sum_{\mathbf{k}\in\Lambda}c[\mathbf{k}]\widehat{\nabla f}[\boldsymbol{\ell}-\mathbf{k}] =0, \quad\forall\boldsymbol{\ell}\in2\Lambda.\end{align}

Furthermore, under a set of assumptions discussed in detail elsewhere,[8] $$$c[\mathbf{k}]$$$ can be recovered from the acquired data through convolution with a filter derived from the data itself, leading to a Toeplitz operator $$$\mathcal{T}\subset \mathbb{C}^2$$$ such that $$$\mathcal{T}c[\mathbf{k}]=0$$$ that is obtained after the data are subject to a Cadzow denoising step that requires the SVD of a large matrix. This gives rise to a Tikunhov-regularised minimisation problem for the unknown Fourier samples beyond the grid of those acquired, namely

\begin{align}\text{argmin}_{\hat{f}[\mathbf{k}]}\left|\left|\mu\nabla\hat{f}[\mathbf{k}] \right|\right|_{2}^{2} + \lambda\left|\left| P_\Gamma\left( \frac{\hat{f}[\mathbf{k}]+\mathcal{T}^*\mathcal{T}}{1+\lambda\text{diag}(\mathcal{T}^*\mathcal{T})} -c[\mathbf{k}] \right) \right|\right|^{2}_{2} \end{align}

where $$$\lambda$$$ is a regularisation parameter and $$$P_\Gamma$$$ a projection operator onto the new resampling space. This process takes several minutes on a modern CPU with $$$32\times32\times16$$$ matrix sizes, owing to the dimensions of $$$\mathcal{T}$$$, and does not scale well with matrix size.

Results

Discussion

Conclusions and Future Work

Acknowledgements

References

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