Introduction

State of the art RF coils are composed of multiple receiving channels. One image per channel is reconstructed and the final image is typically calculated from the root sum of squares of the sub-images (rSoS). A pre-whitening can be done at this stage by inserting the inverse of the noise covariance matrix measured from prescans (covSoS). This combination is quasi-optimal in SNR¹ but in functional MRI, the relevant measure is stability over time, making tSNR the preferred metric. An alternative tSNR optimal coil combination has been proposed by Huber et al.² (STARC) but uses a cumbersome and time-consuming voxel-wise gradient descent algorithm yielding an optimized weighted sum. Here, we propose an analytical solution of the STARC problem and extend the method to propose another coil combination optimal in t-score for the mean³ (TSM) as alternatives to the covSoS. We compare their performance in terms of sensitivity to BOLD signal changes to the covSoS, across task-based fMRI experiments on four volunteers at 7T.

Methods

In this work, we denote N and Nc the number of repetitions and receive channels respectively. For a given voxel, we denote \(I_{\text{STARC}} = AX_{\text{STARC}}\) the \(N × 1\) signal vector resulting from STARC combination with \(A\) the \(N × Nc\) matrix concatenating the magnitude signal time courses from all coils of the voxel and \(X_{\text{STARC}}\) a \(Nc × 1\) weighting vector to be optimized, independently for each voxel. The tSNR of a voxel is typically computed from the ratio of its temporal mean by its standard deviation. Therefore, the tSNR maximization problem can be recast into a minimization of the temporal standard deviation with a constraint on the temporal mean to be equal to a scalar \(b\). It is possible to express the temporal mean and standard deviation of the resulting signal vector with \(X_{\text{STARC}}\). The optimization problem turns out as a quadratic program under linear constraint whose solution satisfies the Karush-Kuhn-Tucker (KKT) conditions. Through the Lagrangian method, the solution is

\(X_{\text{STARC}} = \frac{b\Psi_{t}^{- 1}u}{u'\Psi_{t}^{- 1}u}\) ,

with \(u\) the \(Nc × 1\) column wise (temporal) mean vector of \(A\) and \(\Psi_{t}\) the covariance matrix of \(A\). Since activations can be accounted for signal variations and reducing tSNR, the TSM has been introduced to ignore the effect of neural activity on the measure of signal stability. TSM is the ratio of the temporal mean and the standard deviation of the residual after the General Linear Model (GLM) fit. Starting from its expression and following the same method it is possible to derive a TSM optimal weighting vector

\(X_{\text{STARCtsm}} = \frac{b{\Psi_{\text{TSM}}}^{- 1}u_{\text{TSM}}}{u_{\text{TSM}}'{\Psi_{\text{TSM}}}^{- 1}u_{\text{TSM}}}\) ,

where \(u_{\text{TSM}} = c^{'}D^{+}A\) with \(D^{+}\) the pseudo inverse of the GLM design matrix \(D\), and \(c\) the contrast vector selecting the column of \(D\) estimating the mean of the signal and \({\Psi_{\text{TSM}}}^{- 1}\) the inverse of the covariance matrix of \((Id - DD^{+})A\) with \(Id\) the identity matrix. Since it uses prior information about activations via the design matrix, a regularization of this method is necessary to avoid double-dipping bias. Indeed, under the null hypothesis, the standard deviation of the distribution of t-scores should be equal to one which is not the case here. Regularization was performed by dividing the t-scores by a standard deviation estimation yielded by Monte Carlo simulations.In vivo experiments were performed on four volunteers at 7T (Magnetom, Siemens Healthineers, Erlangen, Germany) with a 3D-EPI sequence⁴ at 1.6mm isotropic resolution (TR/TE=1210/22ms, FOV=208mm, matrix=130×130×88, PF=7/8, GRAPPA 2x4, with CAIPIRINHA sampling, ∆kz=2, N=260) and in pTX mode with Universal pulses⁵ using the 8Tx-32Rx Nova head coil (Nova Medical, Wilmington, MA, USA). The same data were reconstructed with the covSoS, STARC and STARCtsm. The functional runs were performed using a functional localizer⁶ involving visual, auditory and motor tasks. Functional post processing was done on FSL and SPM including motion and distortion correction, normalization and smoothing.

Results

Figure 1 displays tSNR maps and distributions pooled over the 4 volunteers. STARC combination yielded the highest tSNR. Figure 2a displays an example of activation maps yielded by each coil combination. Figure 2b shows the total number of activations across motor, auditory and visual tasks pooled over the four volunteers. covSoS outperforms STARC and the double dipping corrected STARCtsm. Figure 3 shows STARC weights for two voxels, whereby channels with the highest strength (mean) are penalized compared to the weakest ones. In figure 4 we compare for a set of activated voxels (covSoS motor and auditory contrasts) the tSNR ratio STARC/covSoS versus the corresponding gain in t-score and the TSM ratio STARC/TSM versus the corresponding gain in t-score. We see that neither a gain in tSNR nor TSM guarantees a gain in t-score.

Discussion and conclusion

These results suggest that data-driven tSNR optimization approaches might not always improve BOLD signal detection because signal variability also takes into account activation spikes. The TSM also measures signal stability but without taking into account activation spikes by using the GLM residual instead of the standard deviation of the signal. Nevertheless, it leads to double-dipping because the GLM is applied twice. Once the statistical bias is corrected, covSoS remains the optimal coil combination method in terms of t-scores, despite smaller tSNR.

Acknowledgements

ERPT equipment program of the Leducq Foundation and FET-Open AROMA grant agreement n° 885876.

References

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