Electrodynamic constraints on homogeneity and radiofrequency power deposition in multiple coil excitations

Abstract

The promise of increased signal-to-noise ratio and spatial/spectral resolution continues to drive MR technology toward higher magnetic field strengths. SAR management and B1 inhomogeneity correction become critical issues at the high frequencies associated with high field MR. In recent years, multiple coil excitation techniques have been recognized as potentially powerful tools for controlling specific absorption rate (SAR) while simultaneously compensating for B1 inhomogeneities. This work explores electrodynamic constraints on transmit homogeneity and SAR, for both fully parallel transmission and its time-independent special case known as radiofrequency shimming. Ultimate intrinsic SAR--the lowest possible SAR consistent with electrodynamics for a particular excitation profile but independent of transmit coil design--is studied for different field strengths, object sizes, and pulse acceleration factors. The approach to the ultimate intrinsic limit with increasing numbers of finite transmit coils is also studied, and the tradeoff between homogeneity and SAR is explored for various excitation strategies. In the case of fully parallel transmission, ultimate intrinsic SAR shows flattening or slight reduction with increasing field strength, in contradiction to the traditionally cited quadratic dependency, but consistent with established electrodynamic principles.

Figure 1

Schematic representation of the sample geometry, the FOV and the shape of the target excitation profiles. Fully homogeneous concentric excitation profiles were modeled with radius equal to 100%, 50%, and 25% of the radius of the sphere. Smoothly varying excitation profiles with the same set of radii were also tested, and are shown in subsequent figures.

Figure 2

Schematic arrangement of two circular surface coils near the surface of a homogeneous sphere. The sphere is centered at the origin of the laboratory reference frame.

Figure 3

Convergence of the ultimate SAR optimization as a function of the number of basis functions used in the calculations. The number of basis functions is equal to 2*(l_max + 1)², where l_max is the order of the multipole expansion, whose range is reported at the top of each plot. Data are reported for three different sphere radii (a = 5 cm, a = 15 cm, a = 25 cm) and two extreme field strengths (B_o = 1T, B_o = 11T). In each of these cases convergence was tested for the three shapes of the target excitation profile shown in Figure 1.

Figure 4

Actual excited profile for various transmit coil configurations and different excitation strategies at 7T magnetic field strength. Leftmost column: coil configuration (with ultimate intrinsic case at bottom). Second column from left (“sum of coils”): excitation profile for simple fixed sums of transmit coil fields. The contributions of the individual coils are summed and scaled by the average of the absolute value of each coil’s transmit sensitivity (see Eq. 42). Third column from left (“forced homogeneity”): excitation profile for the case in which a common tailored gradient and RF excitation are used to correct for sensitivity variations resulting from the simple sum (see Eq. 41). Fourth column from left and rightmost column: these last two columns refer to RF shimming and parallel transmission respectively, showing that the latter enables homogeneous excitations even with a small number of coils. The radius of the modeled homogeneous sphere is 15 cm. The quantities reported are relative measures of SAR and are normalized to the ultimate intrinsic value for parallel transmission (indicated with an asterisk). The SAR values in parentheses for the second and fourth columns represent the case in which excitation is achieved by repeatedly sampling the center of excitation k-space 1024 times with small RF amplitudes, rather than by applying a single high-amplitude spoke in the center during the 32-by-32 EPI trajectory.

Figure 5

Ultimate intrinsic average global SAR for excitation of a transverse FOV through the center of a homogeneous sphere as a function of main magnetic field strength, for SAR-optimized RF shimming (top row) and fully parallel transmission (bottom row). Frequency-dependent average in vivo values of sphere electrical properties were used (see Table 1). The size of the modeled sphere increases from the leftmost to the rightmost column. Results are shown for three different concentric homogeneous target excitation profiles, whose radius is indicated in the legend as a fraction of the sphere radius a.

Figure 6

SAR benefits of relaxing homogeneity constraints. a) Average global SAR and actual excited profile using RF shimming for various transmit coils configurations at 7T magnetic field strength. The case of uniform concentric profiles considered elsewhere in this work is compared against two smoother target excitation profiles: a bi-dimensional Gaussian curve of amplitude one in the center of the FOV and a bi-dimensional quadratic function with amplitude decreasing from one at the edges to zero in the center. b) Average global SAR and actual excited profile using RF shimming for varying degrees of regularization with a 12-element transmit array at 7T. By increasing the tolerance of the SVD-based inversion in Eq. 23, we can loosen the constraint on profile fidelity in order to improve SAR. SVD tolerance (i.e. the threshold for the smallest singular value considered nonzero and included in the inversion) increases from 10⁻¹⁷ to 10⁻¹⁴ to 10⁻¹¹ from left to right, and the resulting SAR values for RF shimming are shown above the resulting excited profiles. The radius of the modeled homogeneous sphere is 15 cm for both (a) and (b). The quantities reported are relative measures of SAR and are normalized to the case with the lowest SAR value in (a), which is indicated with an asterisk in the figure.

Figure 7

Ultimate intrinsic average global SAR for parallel excitations along a transverse FOV through the center of a homogeneous sphere, as a function of the acceleration factor. Each plot refers to a different size of the modeled sphere and shows the behavior for different values of the main magnetic field strength. A uniform, fully homogeneous concentric excitation profile with radius equal to the radius of the sphere was used in all cases.

Figure 8

Normalized distribution (base-10 Log scale) of local SAR within the FOV during unaccelerated parallel excitations with current patterns optimized for global SAR. For each value of the main magnetic field strength, spatial distribution of ultimate intrinsic local SAR is compared for different sizes of the modeled sphere during excitation of the center of k-space.

Figure 9

Normalized distribution (base-10 Log scale) of local SAR within the FOV during accelerated parallel excitations with current patterns optimized for global SAR in a 15 cm sphere. For each value of the main magnetic field strength, spatial distribution of ultimate intrinsic local SAR during excitation of the center of k-space is compared for various acceleration factors.

Figure 10

SAR efficiency of parallel transmission as a function of the number of coil elements in the transmit array. The ratio of SAR resulting from unaccelerated parallel excitations with finite arrays to ultimate intrinsic SAR is plotted for different magnetic field strengths in the case of a 15 cm sphere, using a logarithmic scale.

Figure 11

Behavior, as a function of main magnetic field strength, of the scaling factor that pre-multiplies the SAR per unit flip angle for both the case of circular surface coils and the limit case of a full basis set of spherical harmonics. Permeability of free space was used, whereas conductivity and permittivity were set to frequency-dependent in vivo values as in Ref. (26).