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,2v,22v, …, 2n−1v, 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 T1 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:
Here, I is the image intensity, ρ is the spin density, RX is the receive coil profile (B), TSR is the saturation recovery time, T1 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 T1 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. . 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 v1, v2, …, vmax, where vmax is the voltage corresponding to the first maximum in I(x, y, vi).
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:
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:
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. , the FA profile of each Tx mode, θMode(j)(x,y,V), can be estimated by dividing the low-FA image, IMode(j)(x,y,V), by the density-weighted reception profile along with taking an inverse sine of the resulting image, pixel-by-pixel:
From this FA profile, the quantitative magnitude B profile (nT/V) can then be obtained via Eq. .
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, ϕTXrelative,Mode(j)=ϕTX,Mode(j)+ϕRX. 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, ϕimage=ϕTX+ϕRX, should be equal to the design phase, ϕdesign, with the actual excitation phase being ϕTX = ϕ design − ϕRX.
Figure 2. Flowchart outlining the proposed quantitative B1+ mapping technique, where first a Rx profile of the reception coil is estimated in steps 1–5, after which B1+ maps of the Tx modes/coils can then be obtained via steps 6 and 7.
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Figure 3. In vivo quantitative B1+ mapping of the first gradient Tx mode using a single low-FA acquisition (subject 1). The B1+ map is obtained by dividing the low-FA image with the density-weighted Rx profile estimate along with applying a sine inverse operation.
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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. , 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.
Figure 4. Magnitude (top) and phase (bottom) B1+ maps of the eight optimal modes for (a) the head-shaped phantom, and (b) an axial section in human brain (subject 1).
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