Fitting the relative B1- maps derived from a B1+ shim acquisition to the Helmholtz equations allows for the calculation of absolute B1- maps. The sensitivity profiles can then be used to optimally combine coils in subsequent acquisitions. In addition, absolute B1- maps allow for the removal of the receive sensitivity from the combined images. This was validated in a GE-EPI sequence and found to produce images with lower phase standard deviation and increased temporal signal-to-noise ratios across the brain. Both of these differences were found to be statistically significant in a Student’s t-test.
The individual coil images used for $$$B_1^+$$$ mapping1 are combined via singular value decomposition which also provides relative $$$B_1^-$$$ maps2. Estimation begins by masking of the relative $$$B_1^-$$$ maps in order to the determine the region for the fit. The masking is done through the ratio of the primary to secondary singular vector as a rough measurement of signal to noise ratio (SNR).
Assuming uniform electrical tissue properties, the $$$B_1^-$$$ should follow the Helmholtz equation. The solution to the Helmholtz equation that is finite at the origin is:
$$\triangledown^2B_1^-=-k^2B_1^-$$
$$B_1^-(r,\theta,\phi)=\sum_{l=0}^{l_{max}}\sum_{m=-l}^{l}c_{lm}j_l(kr)Y_l^m(\theta,\phi)$$
where $$$c_{lm}$$$ are the coefficients to be fitted, $$$Y_l^m$$$ are spherical harmonics and $$$j_l$$$ are spherical Bessel functions of the first order. The $$$B_1^-$$$ profiles are known to be spatially smooth allowing fitting to be completed up to limit order $$$l_{max}$$$.
The voxel-wise relative $$$B_1^-$$$ maps are scaled by the unknown complex scaling composed of the spatial $$$l_2$$$-norm and transmitter phase that is common to all coils. The $$$B_1^-$$$ maps are iteratively fit to the Helmholtz equation solution and a common scaling term. The fit uses a weighted least squares algorithm in order to find the appropriate coefficients. Then, the complex scaling is estimated as the weighted mean of the ratio between the fitted $$$B_1^-$$$ and the measured relative $$$B_1^-$$$. The scaling is then applied to the relative $$$B_1^-$$$ and the next iteration is completed. This is repeated until the fitted $$$B_1^-$$$ converges.
These fitted $$$B_1^-$$$ can then be applied to any scan during reconstruction. Because polynomial fits are inherently poor extrapolators, the fits are only used to interpolate within the convex hull of all fitted points. For points exterior to the convex hull the sensitivity of the closest point on the convex hull is used. The $$$B_1^-$$$ is applied to the complex image values (M) to generate magnetization (m) as follows2 ($$$^H$$$ is hermitian transpose):
$$S=\frac{\mathbf{({B_1^-})^H}}{\mathbf{({B_1^-})^H}\mathbf{{B_1^-}}}\mathbf{M}=\frac{\mathbf{({B_1^-})^H}}{\mathbf{({B_1^-})^H}\mathbf{{B_1^-}}}\left(\mathbf{{B_1^-}}m \right)=m$$
In order to validate this method, accelerated EPI scans were collected and reconstructed using standard system coil combination and receiver sensitivity weighted combination in order to compare temporal signal-to-noise ratio (tSNR) and phase standard deviation. The scan was completed with GRAPPA 3 with 36 reference lines, TR=1250ms, TE=20ms and a 30o flip angle. The fit included terms up to order 5 and took 23 iterations to reach convergence (<10 seconds). The masking threshold used was 15. The images were combined and temporal SNR (tSNR) was calculated after motion correction and linear detrending. The phase images were temporally unwrapped, linearly detrended, underwent motion correction using the magnitude time course parameters, and the standard deviation was taken on a voxel-wise basis. A two sample Student’s t-test was performed on a tight mask of the middle 25 slices in order to compare tSNR and phase standard deviation across the brain.
1 Curtis, A, Gilbert, K, Klassen, L, et al. Slice-by-slice B1+ shimming at 7 T. Magn Reson Med. 2012;68(4):1109–1116
2. Roemer P, Edelstein W, Hayes C, et al. The NMR phased array. Magn Reson Med. 1990;16(2):192-2253.