##### Estimation of the Density-Weighted Reception Profile

The estimation of the density-weighted Rx profile is performed in two steps. First, the mode-1 birdcage transmission profile of the coil array is estimated. This estimate is then used along with the data from an extra acquisition to estimate the Rx profile.

The mode-1 birdcage transmission profile is estimated using a technique similar to the one proposed by Kerr et al. (28). The estimation technique here relies on mode-1 birdcage transmission via slice-selective excitation at a set of *n* voltages, *v*,2*v*,2^{2}*v*, …, 2^{n−1}*v*, and fitting the sinusoidal signal dependence to the Tx voltages. In this method, BIR-4 magnetization saturation pulse (29) is applied after each readout period to allow for a reduced TR and shorter scan times without a detrimental *T*_{1} bias in the FA estimates. Images at a set of transmission voltages are collected and the resulting image intensities are fitted to the following expression, which can be derived from, e.g. Ref.30:

- (1)

Here, *I* is the image intensity, ρ is the spin density, *RX* is the receive coil profile (*B*), *TSR* is the saturation recovery time, *T*_{1} is the longitudinal relaxation time constant, θ is the FA, and *V* is the applied Tx voltage.

With the use of a BIR-4 magnetization saturation pulse, short TR acquisitions can result in high average SAR and reduced SNR (due to reduced *T*_{1} recovery time). To avoid this problem, TR = 1.2s was used in this work. A standard nonlinear search algorithm (Simplex) in MATLAB was employed to perform the fitting of the image intensity to obtain the FA map, (*x,y,V*). We note that the RF pulses for *B* mapping are slice-selective in the current implementation, and thus some care must be taken to minimize the effects of imperfect slice selection at large FA during the fitting of Eq. [1]. To this end, data from very large FAs are excluded from the fitting based on a simple criterion. On a pixel-by-pixel basis, the signal intensity, *I*, was fitted only with measurements corresponding to excitations with *v*_{1}, *v*_{2}, …, *v*_{max}, where *v*_{max} is the voltage corresponding to the first maximum in *I*(*x, y, v*_{i}).

From the FA map, estimation of the transmission profile, (*TX*(*x,y*)), can be obtained by noting the following relationship between excitation FA and transmission profile:

- (2)

where γ is the gyromagnetic ratio, and *RF(t)* is the RF pulse used for the transmission.

With the estimate of the birdcage transmission profile, the density-weighted reception profile, ρ(*x,y*)*RX*(*x,y*), can then be obtained by acquiring an extra slice-selective, low-FA image with mode-1 birdcage transmission at a known voltage (without the magnetization saturation pulse). With the combination of a very low FA (we used a criterion of no more than 8° at maximum, based on the quantitative FA map) and a TR = 1 *s*, this image is approximately proton-density weighted:

- (3)

Again, this equation can be derived from Ref.30. By dividing out sin (*x,y,V*) from the image, the density-weighted birdcage reception profile can be obtained.

##### Estimation of the Individual *B* Transmission Profiles

Once the density-weighted reception profile is estimated, the *B* profile of the individual transmission modes can be obtained though a single slice-selective, low-FA image acquisition of each transmission mode. According to Eq. [8], the FA profile of each Tx mode, θ_{Mode(j)}(*x,y,V*), can be estimated by dividing the low-FA image, *I*_{Mode(j)}(*x,y,V*), by the density-weighted reception profile along with taking an inverse sine of the resulting image, pixel-by-pixel:

- (4)

From this FA profile, the quantitative magnitude *B* profile (nT/V) can then be obtained via Eq. [2].

In addition to the magnitude profile, the accompanying phase profile of *B* is required as well. In this work, we make use of the spatial distribution of *B* phase relative to the Rx profile, ϕ_{TXrelative,Mode(j)}=ϕ_{TX,Mode(j)}+ϕ_{RX}. This phase is obtained from the phase measured in the low-FA acquisition of each mode. By using this phase in the calculation of parallel excitation pulses, the Rx phase variation is taken into account in the excitation design. As a result, the phase of the reconstructed image, which consists of both the excitation and the reception phase, ϕ_{image}=ϕ_{TX}+ϕ_{RX}, should be equal to the design phase, ϕ_{design}, with the actual excitation phase being ϕ_{TX} = ϕ design − ϕ_{RX}.

We also note that once ρ(*x,y*)*RX*(*x,y*)(1 − *e*) was estimated as a by-product of obtaining mode-1 birdcage transmission profile via Eq. [6], we could have used the reset pulse sequence to estimate the *B* profiles of the individual modes without having to estimate ρ(*x,y*)*RX*(*x,y*). Nonetheless, estimating ρ(*x,y*)*RX*(*x,y*) only requires one extra low FA acquisition, and allows us to leave out the use of reset pulse in estimating the *B* profiles of the individual modes. This significantly reduces the SAR requirement for the mapping, which is important for in vivo imaging applications. Furthermore, the estimation of the density-weighted Rx profile is needed in dividing the reconstructed image in the parallel excitation experiments to obtain the excitation profiles.