Ioannis P. Georgakis^{1}, Athanasios G. Polimeridis^{1}, and Riccardo Lattanzi^{2,3,4}

^{1}Center for Computational and Data-Intensive Science and Engineering (CDISE), Skolkovo Institute of Science and Technology, Moscow, Russian Federation, ^{2}Center for Advanced Imaging Innovation and Research (CAI2R), Department of Radiology, New York University School of Medicine, New York, NY, United States, ^{3}Bernard and Irene Schwartz Center for Biomedical Imaging (CBI), Department of Radiology, New York University School of Medicine, New York, NY, United States, ^{4}Sackler Institute of Graduate Biomedical Sciences, New York University School of Medicine, New York, NY, United States

### Synopsis

We introduce a new performance metric for RF shimming, the ultimate intrinsic transmit efficiency (UITXE), which provides an absolute reference independent of any particular coil design. We show in simulation that it represents a performance upper bound, which could be approached with finite transmit arrays with an increasing number of coils. In particular, we demonstrated that a 24-channel array could achieve 70% of the UITXE. UITXE could be employed in a straightforward manner in experiments to assess absolute performance of actual arrays and evaluate RF shimming approaches. The associated ideal current patterns could provide new insight for optimal array design.

*Purpose *

To
introduce the ultimate intrinsic transmit efficiency (UITXE) as a performance
upper bound for any RF shimming experiment, and propose it as a reference to
assess the absolute performance of transmit (Tx) arrays.*Theory and Methods *

The transmit efficiency
metric η, which was proposed for multiple channel transmission^{1,2}, is defined as
$$$η=\frac{\mathbf{w}^{H}\mathbf{\Gamma}\mathbf{w}}{\mathbf{w}^{H}\mathbf{\Phi}\mathbf{w}}$$$ where $$$\mathbf{w}$$$ is a complex-valued vector of RF shimming
coefficients (amplitude and phase modulations) for each Tx channel. The matrix $$$\mathbf{\Gamma=\mathbf{C}^{H}\mathbf{C}}/M$$$ contains the B_{1}+ squared per unit
current for each Tx channel, averaged over the *M* voxels of the region of interest (ROI) for which Tx efficiency is
being maximized. The matrix $$$\mathbf{\Phi}$$$ represents RF power dissipation over the volume
(*V*) of
the sample, and its elements are calculated as $$$\mathbf{\Phi}_{n,m}=\frac{1}{2}\intop_{V}\sigma(\mathbf{r})\mathbf{e}^{(n)}(\mathbf{r})^{*}\cdot\mathbf{e}^{(m)}(\mathbf{r})dV $$$
,
where $$$\mathbf{e}^{(n)}$$$ is the electric field of the n^{th} Tx channel per unit
current. Note that the matrices $$$\mathbf{\Phi},\mathbf{\Gamma}$$$ are Hermitian by definition and
$$$\mathbf{\Phi}$$$ is also a positive definite matrix, in the case
of passive media. Hence the maximum Tx efficiency $$$\tilde{\mathbf{\eta}} $$$ is given by the largest eigenvalue of the
associated generalized eigenvalue problem $$$\mathbf{\Gamma w}=\lambda\mathbf{\Phi w} $$$. The
corresponding eigenvector $$$\tilde{\mathbf{w}} $$$
contains the optimal RF shimming coefficients that
achieve maximum efficiency. In this work, we constructed a complete basis of electromagnetic
fields using vector spherical harmonic modes (4232 in total), generated by
electric currents flowing on a spherical shell (18cm radius), enclosing a
homogeneous sphere (15cm radius) with average brain electrical properties^{3}.
Since the operator that maps currents to fields is compact, $$$\tilde{\mathbf{\eta}} $$$
converges to the UITXE as the number of modes
increases. We investigated UITXE as a function of main magnetic field strength
(B_{0}) and for different target ROIs. We assessed maximum Tx
efficiency with respect to UITXE for various arrays of loop coils at 7T. Coils’ electromagnetic fields were calculated with a volume integral equation solver^{4}
using piecewise linear basis functions. *Results*

**Fig. 1a** shows UITXE for single
voxels across the sphere’s diameter for different B_{0} values, whereas** Fig.
1b** shows UITXE at particular voxel locations for increasing B_{0}. As for the UISNR^{5,6}, UITXE increases as the target voxel approaches the
surface of the sphere. Interestingly, UITXE decreases with B_{0} up to 3T and then starts to slowly
increase for higher B_{0} values. **Fig. 2** shows convergence of the
UITXE calculations vs. number of modes for different voxel positions and B_{0}. The closer the voxel is
to the sphere surface, the more modes are required for convergence. **Fig. 3** illustrates the convergence of UITXE
when maximizing Tx efficiency in 2D and 3D ROIs of different sizes, rather than
at single voxels. UITXE required more modes to converge for 2D than 3D ROIs. **Figs. 4** and **5** show the maximum Tx efficiency achievable at 7T with finite
arrays as a percentage of the UITXE, for a 3D and a 2D ROI, respectively. In
both cases, UITXE was approached more closely by increasing the number of loops.
For the 3D ROI (an inner sphere with radius equal to
30% the object radius), enclosing Tx arrays with 8, 16, and 24 loops yielded 24%, 41%, and
70% of the UITXE, respectively. For the 2D ROI (a circle covering 70% the
central transverse section), belt shaped Tx arrays with 8, 16, and 32 loops yielded
57%, 61%, and 71% of the UITXE, respectively. *Discussion and Conclusions*

UITXE represents
a theoretical upper bound for coil performance, since it depends only on the
size and electrical properties of the sample and on the target ROI, but it’s
independent of any particular coil design. Although
the optimization algorithm does not enforce any particular flip angle
distribution, as in the case of UISAR^{3}, one can expect that
maximizing Tx efficiency would naturally prevent B_{1}+ nulls within
the target ROI^{2}. UITXE could be used as an absolute reference to
assess Tx arrays and RF shimming techniques, since η
is independent of any scale factor in the shimming weights (therefore independent of any overall change in transmit voltage) and, furthermore, can be
practically evaluated in experiments^{2}. Ongoing future work includes
studying UITXE for a realistic human head and combining the current modes using the corresponding optimal shimming coefficients $$$ \tilde{\mathbf{w}} $$$ to
derive ideal current patterns^{7} resulting in UITXE,
which could provide physical insight and a concrete design target to RF
engineers. We also plan to evaluate absolute performance of actual Tx arrays with respect to UITEX in phantom experiments.### Acknowledgements

This work was supported in part by NIH R01 EB024536 and was performed
under the rubric of the Center for Advanced Imaging Innovation and Research
(CAI2R, www.cai2r.net), a NIBIB Biomedical Technology Resource Center (NIH P41
EB017183).### References

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