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.
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:
Evaluated undersampling patterns
We study the influence of key properties on time efficiency with following patterns, shown in figure 1.
Acquisition model
A ground truth T_{1} and T_{2} 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.
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.
Figure 3: Time efficiency for all the evaluated patterns for all R. Time efficiency values below 2 are not shown for clarity. Only Regular had values below 2