Recently a method for calculating the ultimate SNR in a realistic body model was presented (1). Thereby the basis of all possible solutions to Maxwell’s equations was computed by randomly exciting many dipoles lying on a conformal surface. In this work, we investigated the ultimate SNR for two configurations with the RF array elements being placed on a cylindrical or spherical holder around the head. The basis set in our approach was created by vector cylindrical harmonics (VCH) and vector spherical harmonics (VSH), which are known to form a complete set of eigenfunctions to Maxwell’s equations in free-space (2).

We arranged a tight fitting spherical and cylindrical surface (infinitely long) around the head of the voxel model “Duke” (3) with isotropic resolution of 5 mm. The radius for both topologies was set to 14 cm and the coordinate center was placed into the midbrain (position 1). Additionally for the VCH a second position was evaluated with the coordinate center shifted slightly above into the ventricle (4). To reduce computational effort of the EM simulations we truncated the voxel model after the neck. We defined the following surface current modes K_cylinder and K_sphere mimicking all possible conductor distributions on the entire cylinder or sphere (5):

$$ \mathbf{K}_{cylinder} = \sum_{n=-N}^{N} \sum_{m=-M}^{M}b^{magn}_{n,m} \mathbf{W}_{n,m}+b^{elec}_{n,m}\mathbf{e}_r\times \mathbf{W}_{n,m}$$

$$ \mathbf{K}_{sphere} = \sum_{l=1}^{L} \sum_{m=-l}^{l}c^{magn}_{l,m} \mathbf{X}_{l,m}+c^{elec}_{l,m}\mathbf{e}_r\times \mathbf{X}_{l,m}$$

Here the vector cylindrical harmonics W_n,m and the vector spherical harmonics X_l,m were used to model divergence-free (loops) current distributions (first term) and curl-free (dipoles) patterns (last term) as indicated in the above equations. The expansion coefficients were denoted as b and c. For the VCH we discretized the continuous spatial frequency m with a resolution of Δm=2 (unit is m^-1) running from -100 to +100 and the discrete index n was truncated at an order of 40 (unit step width). Regarding the VSH we set the expansion order L = 60. Next, we were computing the radiated fields in free-space of both surface current types by applying the framework of dyadic Green’s functions (6). The electric and magnetic fields were afterwards fed into the fast open-source volume-integral solver MARIE (7,8) to compute the total fields inside the human brain. With these steps the basis set was created and the ultimate SNR was calculated according to equation six in (9).

In Fig. 1 we show the convergence of the ultimate SNR for three voxel positions: Except for the peripheral voxels the ultimate SNR reasonably converged for both VSH and VCH. The spatial distribution of the unaccelerated ultimate SNR is visualized in Fig. 2: For both surface current configurations the SNR decayed exponentially from the periphery and had high spatial inhomogeneity. Thus in Fig. 3 we summarized the mean unaccelerated SNR for different brain regions and defined a region of interest (ROI). Within this ROI the average SNR achievable with the VSH was ten times larger than with VCH at position 1 and more than four times than with VCH at position 2. Parallel imaging performance was evaluated with two dimensional acceleration in anterior-posterior and left-right direction. Keeping the maximum value of the g-factor lower than two, with the spherical setup a maximum acceleration of 8x8 was achievable as opposed to 4x4 with the cylindrical one in either position (Fig. 4). We found a large relative drop in highly accelerated SNR (10x10) to the unaccelerated SNR of about 60% for the cylindrical surface at position 1 (73%, position2), where for the spherical geometry this was only 25%. In Fig. 5 we visualized the contribution of the loop-like current patterns to the unaccelerated SNR, which is complemented by dipole-like current patterns to reach the maximum possible SNR. Within the ROI 78% (max 100%, min 58%) of total SNR was reached when using loop-like array elements on a spherical surface and 71% (max 122%, min 38%) when they were placed on a cylinder for position 1. For position 2 these values were 66% (max 100%, min 36%). These results are in accordance with our previous work dealing with a simplified spherical load (10). We calculated the ultimate SNR in a realistic human head model with a complete basis set of cylindrical and spherical vector harmonics. This is of particular interest since RF array holders, coil elements are distributed on, are usually approximating a cylinder or a sphere. When both surfaces had the same radius, the spherical RF array outperformed the cylindrical one (at both positions) regarding receive performance in the human brain. The optimal positioning of the cylinder needs to be determined in future work.