Improved Accuracy of FLASH-based B1+ Mapping by Optimization of the Fourier Encoding Matrix
Omer F. Oran1, L. Martyn Klassen1, and Ravi S. Menon1,2

1Centre for Functional and Metabolic Mapping, University of Western Ontario, London, ON, Canada, 2Department of Medical Biophysics, University of Western Ontario, London, ON, Canada


FLASH-B1 mapping with a Fourier-encoding scheme works only for a limited range of flip-angles due to pronounced saturation effects that occur for short TR. The transmit voltage can be adjusted to satisfy this requirement in problematic Fourier combinations that have high dynamic range. However, this results in an unnecessary reduction of the dynamic range in combinations that are already within the accepted range. This study addresses this problem by optimizing the Fourier encoding matrix such that the flip-angle is reduced only in regions of high flip-angle in the problematic combinations.


Parallel-transmit systems bring in the possibility of mitigating $$$B_1^+$$$ inhomogeneity by RF shimming1 or RF pulse design2. The efficacy of both strategies depends on the accuracy of $$$B_1^+$$$ maps, which is challenged by the high dynamic range of $$$B_1^+$$$ for a single transmit element. This is ameliorated by mapping $$$B_1^+$$$ for linear combinations of transmit channels, i.e. the “channel-$$$B_1^+$$$” information is encoded in the “combination-$$$B_1^+$$$”3,4. The Fourier encoding scheme offers a significant advantage over other schemes since it is less susceptible to RF destructive interferences, an effect that is pronounced at high magnetic fields5.

Regardless of which encoding scheme is used, the conventional $$$B_1^+$$$ mapping techniques are lengthy and often impractical considering that $$$B_1^+$$$ must be mapped for each combination. To combat this, it has been proposed to acquire $$$|B_1^+|$$$ for only one combination (default (CP)-mode) which is complemented by low-flip-angle GRE images for all combinations (see Figure 1 for details)6,7. This technique, referred here to as “FLASH-$$$B_1^+$$$ mapping”, assumes GRE images to be linearly proportional to $$$|B_1^+|$$$ provided that the flip-angle (FA) is sufficiently small to neglect saturation effects. For example, the maximum allowed FA ($$$maxFA$$$) is about 2° in the human brain with a TR of 7 ms (Figure 1). Therefore, in any regions where FA exceeds 2°, combination-$$$B_1^+$$$ is underestimated (Figure 2). The trivial solution to this problem is to adjust the transmit voltage such that $$$FA<maxFA$$$ in all regions and combinations. However, this results in an unnecessary reduction of the dynamic range in combinations that already below the $$$maxFA$$$ and thereby the accuracy is compromised.

In this study, we address this problem by using an iterative method to modify the Fourier encoding matrix. The weights of the combinations for which $$$FA>maxFA$$$ are updated such that only the local regions of high FA are eliminated, while for other combinations, the weights are kept the same.


Iterative Method to Optimize Fourier Encoding Matrix:

The relation between the channel-$$$B_1^+$$$ and the combination-$$$B_1^+$$$ can be written in the matrix form as $$\mathbf{T}_{N_v\times{N_c}}\mathbf{E}_{N_c\times{N_m}}=\mathbf{C}_{N_v\times{N_m}}$$ where $$$\mathbf{T}$$$ contains the complex channel-$$$B_1^+$$$ for each of the $$$N_v$$$ voxels and $$$N_c=8$$$ channels, $$$\mathbf{C}$$$ contains the complex combination-$$$B_1^+$$$ for each of the $$$N_v$$$ voxels and $$$N_m=8$$$ combinations, and $$$\mathbf{E}$$$ is the encoding matrix. In $$$\mathbf{T}$$$ and $$$\mathbf{C}$$$ matrices, the magnitude of $$$B_1^+$$$ was represented by the corresponding flip-angle. The true transmit channel-$$$B_1^+$$$, $$$\mathbf{T}$$$, was obtained using FLASH-$$$B_1^+$$$ mapping with a long TR of 100 ms to reduce saturation effects. The aim of the iterative algorithm is to find $$$\mathbf{E}$$$ such that all elements of $$$\mathbf{E}$$$ are below $$$maxFA$$$ of 2°. This is done once and the optimized encoding matrix ($$$\mathbf{E}_{opti}$$$) is used in subsequent experiments with a short TR of 7 ms. The steps of the algorithm are summarized in Figure 3.

Experimental Methods:

Measurements were performed on a Siemens 7 T head-only MRI scanner (Erlangen, Germany) using an RF coil with 8 transmit and 32 receive channels8. Axial 2D brain images were acquired from a single slice using FLASH (FOV 220x220 mm, image matrix size 128x128, slice thickness 5 mm, TE 1.95 ms, TR 7 or 100 ms). The $$$|B_1^+|$$$-map for the first combination was acquired using Bloch-Siegert-shift-based $$$B_1^+$$$ mapping technique (TE/TR 11/100 ms, FA 25°, Fermi-shaped off-resonance RF pulse (frequency shift 4 kHz, duration 8 ms, nominal peak-$$$|B_1^+|$$$ 5.6 µT)).


When TR=7 ms, $$$B_1^+$$$ was underestimated by up to 25% for some Fourier combinations that have $$$FA>maxFA$$$ (Figure 2). This was also reflected as an underestimation in the channel-$$$B_1^+$$$ (Figure 4). Using the optimized encoding matrix, the maximum FA of the combination-$$$B_1^+$$$ was lowered below $$$maxFA$$$ where needed (Figure 2). Consequently, the corresponding channel-$$$B_1^+$$$ showed no signs of underestimation (Figure 4). Note that the weights were updated only for combinations that have high FAs (Table 1). With a condition number of 3.9, the inverse of $$$\mathbf{E}_{opti}$$$ is still well-defined.

Discussion and Conclusion

This study shows that the FLASH-$$$B_1^+$$$ mapping with Fourier-encoding scheme can fail for short TR values. With the optimization of the encoding matrix, the combination-$$$B_1^+$$$ was maintained within the accepted limits without globally reducing the transmit voltage.

The encoding matrix was optimized using true $$$B_1^+$$$ maps that were obtained from a single subject. However, $$$B_1^+$$$ field distribution in the brain varies for each individual, and $$$\mathbf{E}_{opti}$$$ for one individual might cause errors for others. To overcome this challenge, we plan to acquire true $$$B_1^+$$$ maps from multiple subjects and apply the optimization algorithm by vertically concatenating $$$\mathbf{T}$$$ matrices of each subject.


This work was supported by The Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant to Ravi Menon.


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Figure 1. (a) The summary of the FLASH-$$$B_1^+$$$ mapping technique (b) Illustration of the saturation effects for the FLASH-$$$B_1^+$$$ mapping with typical T1 values of the human brain at 7T. The linear relation between $$$\mathbf{I}_i$$$ and $$$\alpha_i^m$$$ holds only for a limited range of FA, which decreases as the TR decrease. Outside this range, the estimated FA becomes a function of T1, TR, and the actual FA of the first combination (see the relation for $$$\alpha_{i,est}^m$$$ on the left)

Figure 2. Combination-$$$B_1^+$$$ maps obtained with the FLASH-$$$B_1^+$$$ mapping technique for three cases: (i) Fourier encoding matrix was used with TR = 100 ms, (ii) Fourier encoding matrix was used with TR = 7 ms, (iii) Optimized encoding matrix was used with TR = 7 ms. Since the range of the acceptable flip-angles was exceeded in Case (ii), $$$B_1^+$$$ was underestimated by up to 25% in certain regions that are enclosed in white rectangles. For Case (iii), the FA was within the allowed range of 0-2° for TR of 7 ms.

Figure 3. (a) The steps of the proposed iterative algorithm to optimize Fourier encoding matrix. For $$$B_1^+$$$ mapping in the human brain with a TR of 7 ms, the maximum allowed flip-angle ($$$maxFA$$$) was selected as 2° from Figure 1. (b) The norm of the difference between the encoding matrices of consecutive iterations ($$$D(k)$$$) in logarithmic scale. With the relative tolerance $$$TOL=0.001$$$, the number of iterations was 86.

Table 1. The magnitude and phase of the Fourier ($$$\mathbf{E}_0$$$) and the optimized ($$$\mathbf{E}_{opti}$$$) encoding matrices. The magnitude values denote channel weights relative to the transmit voltage that gives mean FA of 3° at the center in the default (CP)-mode (corresponds to the first Fourier combination). Note that the optimization algorithm has not changed the weights for Combinations 2 and 3 since FA was already within the accepted range for these combinations (i.e. 0-2°). The condition number of $$$\mathbf{E}_{opti}$$$ was 3.9. Therefore, the inverse of the matrix is well-defined with sufficiently different distributions of combination-$$$B_1^+$$$ (see Figure 2)

Figure 4. Channel-$$$B_1^+$$$ maps obtained with the FLASH-$$$B_1^+$$$ mapping technique for three cases: (i) Fourier encoding matrix was used with TR = 100 ms, (ii) Fourier encoding matrix was used with TR = 7 ms, (iii) Optimized encoding matrix was used with TR = 7 ms. For Case (ii), the underestimations in the combination-$$$B_1^+$$$ were reflected to the channel-$$$B_1^+$$$ in regions that are enclosed in white rectangles. With the optimized encoding matrix in Case (iii), the channel-$$$B_1^+$$$ was obtained with no errors.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)