Synopsis

Quantitative MR (qMRI) at present is clinically unfeasible due to long scan time. Jointly performing image reconstruction and parameters estimation is expected to allow increased acceleration. In this work, we investigate properties of undersampling patterns that are most relevant for parameter estimation using a Cramer-Rao-Lower-Bound (CRLB) based metric for such an approach. We compare key properties of undersampling patterns and conclude that one of these properties, namely low discrepancy, is most relevant for achieving time-efficient qMRI.

Introduction

Method

Let $$$f_j(\boldsymbol\theta_{\boldsymbol{x}})$$$ predict the signal of contrast $$$j\in[1 ... J]$$$ of an acquisition scheme that acquires $$$J$$$ different contrasts with parameter vector $$${\boldsymbol\theta_{\boldsymbol{x}}}$$$. Here $$${\boldsymbol\theta_{\boldsymbol{x}}}= [M_0, T_1, T_2]_\boldsymbol{x}$$$ of voxel $$$\boldsymbol{x}$$$ where $$$\boldsymbol{x}\in\Omega$$$ is the spatial position index in image domain $$$\Omega\subset\mathcal{N}^3$$$, consisting of $$$N$$$ voxels in total, $$$\boldsymbol{k}\in\Omega$$$ is the k-space index and $$$p\in[1...P]$$$ indexes $$${\boldsymbol \theta_{\boldsymbol{x}}}$$$. Let $$$F$$$ be the Fourier transform, and $$$C_{\boldsymbol{x}, c}\in\mathcal{C}$$$ indexed as $$$c\in [1 ... C]$$$ be the coil sensitivity map. Then with an undersampling pattern, $$$U_{j,\boldsymbol{k}}\in\{0, 1\}$$$ the expected value for a k-space sample $$$\mu_{j,\boldsymbol{k},c}(\theta_{\boldsymbol{x}})$$$, is given by

$$\mu_{j,\boldsymbol{k},c}(\boldsymbol{\theta}) =U_{j,\boldsymbol{k}} {\sum_{\boldsymbol{x}\in\Omega}} F_{\boldsymbol{k},\boldsymbol{x}} C_{x,c} f_j(\boldsymbol{\theta}_{\boldsymbol{x}}) .$$

Here, $$$\boldsymbol{\theta}$$$ represents the concatenation of $$$\boldsymbol{\theta_{\boldsymbol{x}}} \forall \boldsymbol{x}\in\Omega$$$. We can derive the CRLB, which quantifies how noise on the collected k-space samples propagates to uncertainty in the estimated parameters maps $$$\boldsymbol{\theta}$$$:

$$\begin{align}\textbf{CRLB}(\boldsymbol \theta) &= I^{-1}(\boldsymbol \theta) \\I(\boldsymbol{\theta})_{(\tilde{p},\tilde{\boldsymbol{x}})(\hat{p},\hat{\boldsymbol{x}})} &= 2(\frac{\Gamma}{| \Gamma|})\sum\limits_{j=1}^{J} \sum\limits_{c=1}^{C} \sum\limits_{\boldsymbol{k}\in\Omega} \frac{\partial \mu_{j,\boldsymbol{k},c} }{\partial \theta_{\tilde{p},\tilde{\boldsymbol{x}}}} \frac{\partial \mu_{j,\boldsymbol{k},c}}{\partial\theta_{\hat{p},\hat{\boldsymbol{x}}}} \end{align}$$

where ($$$\tilde{p}, \tilde{\boldsymbol{x}})$$$ and ($$$\hat{p}$$$,$$$\hat{\boldsymbol{x}}$$$) represent row and column indices corresponding to parameters $$$p$$$ at $$$\boldsymbol{x}$$$, $$$\Gamma$$$ is the complex variance of the noise and $$$I$$$ is Fisher information matrix.

Based on the CRLB and the acquisition time $$$T_{acq}$$$, we compute for each parameter $$$p$$$ the time efficiency $$$\eta_p$$$, which will be used to compare undersampling patterns: $$$ \eta_p= \frac{1}{T_{acq} \textbf{mean}_{\boldsymbol x} \textbf{CRLB}_{(p,\boldsymbol{x})(p,\boldsymbol{x})} (\boldsymbol{\theta}) }$$$

Undersampling patterns

As there are exponentially many $$$U$$$ an exhaustive search is impossible. Hence, we define patterns on the basis of key properties in order to study the features that make an optimal pattern. Key properties are:

1. StdNK: Standard-deviation of the number of times k-space position is sampled
   $$\text{SNK:} = std_k \left(\sum_{j=1}^{J} U_{j,\boldsymbol{k}}\right)$$
2. Discrepancy: Quantifies the deviation from regular sampling and is quantified by the L2 discrepancy.¹

Evaluated undersampling patterns

We study the influence of key properties on time efficiency with following patterns, shown in figure 1.

1. Regular: 2D regular undersampling with acceleration factors $$$R = R_m \times R_n $$$ with $$$R_m$$$ and $$$R_n$$$ ($$$m$$$ $$$\And$$$ $$$n$$$ $$$ \in [1,2,3,4,6,8,12])$$$ the acceleration in the two phase encoding dimensions. For each R with multiple $$$R_m$$$, $$$R_n$$$ pairs, the pair with the best $$$\eta_p$$$ is chosen.
2. Translation: To modify the phase of aliasing (folding) in the echoes, each echo is translated Regularly.
3. TS: To the Translation different sheares are applied to additionally modify the locations of aliasing.
4. Rand: $$$N/R$$$ positions are randomly chosen for each echo independently.
5. RUSK: To ensure each k-space position is sampled equally, a random permutation of the $$$N$$$ k-space positions is repeated $$$J/N$$$ times, selecting $$$N/R$$$ subsequent positions in each echo.
6. RHalton: To ensure quasi-uniform distribution of sampled k-space positions over all echoes, Halton sequence² is used to generate the pseudo-random k-space positions with a low discrepancy.

Acquisition model

A ground truth T₁ and T₂ map shown in figure 2 is used, coil-sensitivity map is computed by the ESPIRIT technique using BART toolbox from actual FSE acquisition with 32-channel head coil.^3,4 Similar to Barral et. al. ⁵, we use an Inversion Recovery prepared 3D Fast Spin Echo to quantify $$$M_0$$$, $$$T_1$$$ and $$$T_2$$$ with inversion times $$$[50, 400, 1100, 2500]$$$ ms, recovery time of $$$2608$$$ ms, echo train length of 18 with $$$6$$$ ms echo spacing and constant flip angle of $$$180^\circ$$$. We model that using $$$f_j(\theta_{\boldsymbol{x}})$$$ and undersampling in phase encoding directions.

Results

Discussion

Most patterns show similar results for lower R but diverge for higher R, indicating sampling patterns are increasingly important for higher R. Comparing StdNK and discrepancy with $$$\eta_p$$$, we observe relevant correlations indicating high influence.

Conclusion

Acknowledgements

References

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