Physical limits to human brain B0 shimming with spherical harmonics, engineering implications thereof

Abstract

Objective

As the MRI main magnetic field rises for improved signal-to-noise ratio, susceptibility-induced B0-inhomogeneity increases proportionally, aggravating related artifacts. Considering only susceptibility disparities between air and biological tissue, we explore the topological conditions for which perfect shimming could be performed in a Region of Interest (ROI) such as the human brain or part thereof.

Materials and methods

After theoretical considerations for perfect shimming, spherical harmonic (SH) shimming simulations of very high degree are performed, based on a 100-subject database of 1.7-mm-resolved brain fieldmaps acquired at 3T . In addition to the whole brain, shimmed ROIs include slabs targeting the prefrontal cortex, both or single temporal lobes, or spheres in the frontal brain above the nasal sinus.

Results and discussion

We show “perfect” SH shimming is possible only if the ROI can be contained in a sphere that does not enclose sources of magnetic field inhomogeneity, which are gathered at the air-tissue interface. We establish a $\sim{13}$Hz inhomogeneity hard shim limit at 7T for whole brain SH shimming, that can only be attained at shimming degree higher than 90. On the other hand, under limited power and SH degree resources, 3D region-specific shimming is shown to greatly improve homogeneity in critical zones such as the prefrontal cortex and around ear canals.

Appendix: Half-sphere and ellipsoidal ROI SH-shimming

Here we intend to demonstrate numerically that a homogeneous spherical ROI is a necessary condition for perfect SH shimming. In practice, we model two examples of non-spherical convex ROIs in the vicinity of an infinitesimal dipole, and try to shim them with Spherical Harmonics as far as possible. The first ROI is a half-sphere, the other one is an ellipsoid.

First consider a 5-cm radius half-sphere ROI, with 1-mm isotropic resolution. Then an infinitesimal magnetic dipole is placed at a distance z below the basis of the half-sphere, z varying between 1 and 10 cm, by steps of 1 cm. At each step, we shim the half-sphere ROI with SH degrees up to 80. We use the same optimization parameters as those explained in Sect. 3.3. The results of such simulations are depicted in Fig. 14. Figure 14a, b show the dipole in red at 4 and 6 cm away from the half-sphere ROI (just inside and outside the bounding sphere). S1c-d depict central slices of the B0 maps in the ROI before shimming, when the dipole is located at 4 and 6 cm, respectively . Note the B0 values are normalized to their standard deviation in the ROI (therefore unitless), and the colormap limits correspond to the B0 extreme values found in the ROI. Figure 14e, f depict the same slices after SH shimming up to 80th-degree. Note that the dipole inside the bounding sphere leaves a non-uniform B0 residual pattern that vanishes when the dipole is placed outside the bounding sphere. In Fig. 14g, we further show the logarithm of the normalized inhomogeneity as a function of the maximum SH degree with which shimming was performed. The 10 colored plots from top to bottom correspond to the dipole located 1–10 cm away from the ROI, respectively. As expected, shimming is improved as the dipole is moved away from the ROI. Yet perfect shimming, achieved when the dipole is outside the bounding sphere (bottom 5 curves), here corresponds to a residual inhomogeneity of less than $10^{-12.5}$, probably due to numerical limitations (note the inhomogeneity curve increases a bit after reaching its minimum, which suggests some numerical instability upon reaching such limitations). For dipoles located inside the bounding sphere, the ROI inhomogeneity increases by at least 2 orders of magnitude, clearly showing perfect SH shimming is not achievable under such conditions. So this tends to demonstrate numerically that a convex ROI (in this example, the half-sphere) cannot be shimmed ‘perfectly’ in presence of a magnetic dipole in its vicinity. By extension, we deduce that a convex ROI with an air-tissue interface in its vicinity does not, in general, fulfill a sufficient condition for perfect SH shimming.

We performed the same simulations as above by replacing the half-sphere by a rugby-ball ellipsoid with a 5-cm large radius along y, and a 3-cm small radius along x and z. We then placed the dipole 1–7 cm away from the ROI, matching the same locations as the last seven configurations used in the half-sphere simulations. Figure 15 then shows the equivalent inhomogeneity log plot as in the half-sphere case. Again, notice how the dipole locations outside the bounding sphere yield a gathering of final B0 inhomogeneities around the same order of magnitude (corresponding to the numerical limit of perfect SH shimming in this ROI configuration). On the other hand, the dipole location in the bounding sphere gives a final inhomogeneity 9 orders of magnitude above. Like for half-spheres, this tends to show that homogeneous ellipsoid ROIs in general cannot be shimmed perfectly when placed in the vicinity of magnetic sources.

In conclusion, homogeneous spherical ROIs seem to be ‘necessary’ for perfect SH shimming. However, any non-spherical ROI contained in such homogeneous spheres can also be shimmed perfectly.

Fig. 14

SH shimming simulation results of a half-sphere ROI located above a magnetic dipole. a, b Location of the dipole (in red) with respect to a 5-cm radius half-sphere: the dipole is away from the ROI by 4 and 6 cm, respectively. c, d Central slices of the B0 maps in the ROI before shimming in the above configurations. B0 values are normalized to their standard deviation in the ROI. e, f Same slices after SH shimming up to 80th-degree. Note that the dipole inside the bounding sphere leaves a non-uniform B0 residual pattern that vanishes when the dipole is placed outside the bounding sphere. g Logarithm of the normalized inhomogeneity as a function of the maximum SH degree with which shimming was performed. Each curve label corresponds to the distance separating the dipole from the ROI. From the 6-cm curve onward, SH shimming reaches its numerical limit (dipole outside the bounding sphere, corresponding to perfect shimming). That limit is not reached when the dipole is inside the bounding sphere

Fig. 15

SH shimming simulation results of an ellipsoidal ROI located above a magnetic dipole. a Different locations of the dipole (in red) with respect to the 3-cm small-radius ellipsoid: the dipole is away from the ROI by 1–7 cm. b Logarithm of the normalized inhomogeneity as a function of the maximum SH degree with which shimming was performed. The 7 curves correspond to the above dipole locations: each curve label is the distance separating the dipole from the ROI. From the 3-cm curve onward, SH shimming reaches its numerical limit (dipole outside the bounding sphere, corresponding to perfect shimming). That limit is far from being reached when the dipole is inside the bounding sphere (1-cm curve)