An analytic expression for the ultimate intrinsic SNR in a uniform sphere

The ultimate intrinsic signal-to-noise ratio (UISNR) is normally calculated using electrodynamic simulations with a complete basis of modes. Here, we provide an exact solution for the UISNR at the center of a dielectric sphere and assess how accurately this solution approximates UISNR away from the center.

We performed a mode analysis to determine which modes contribute to central UISNR - ζ(r→0). We then derived an analytic expression to calculate ζ(r→0) and analyzed its dependence on main magnetic field strength, sample geometry, and electrical properties. We validated the proposed solution against an established method based on dyadic Green's function simulations.

Only one divergence-free mode contributes to ζ(r→0). The UISNR given by the exact solution matched the full simulation results for various parameter settings, whereas calculation speed was approximately 1000 times faster. We showed that the analytic expression can approximate the UISNR with <5% error at positions as much as 10-20% of the radius away from the center.

The proposed formula enables rapid and direct calculation of UISNR in the central region of a sphere. The resulting UISNR value may be used, for example, as an absolute reference to assess the performance of head coils with spherical phantoms.