Ultimate intrinsic SNR (UISNR) is the theoretically highest SNR for given geometry and electrical properties, independent of the coil design. Here, we introduce an analytic exact expression to calculate the UISNR at the sphere center, enabling to directly analyze the dependence on main magnetic field, sample geometry and electric properties. The analytic expression can approximate the UISNR near the center with < 5% error. This work can enable people without access to the full simulation code to calculate UISNR and use it, for example, as an absolute reference to assess the performance of head coils with spherical phantoms.
All equations are defined in Fig.1. The receive sensitivity Bx(r)−iBy(r) (Eq.(1)) inside a dielectric sphere of radius a<b, where b is the radius of the surface where the current modes are defined, is determined by the vector wave functions Ml,m and Nl,m (Eq.(2)).
At the sphere center (r→0), we found that Ml,m(r→0)=0 ∀(l,m) and Nl,m(r→0)=0 ∀(l,m), except N1,1, N1,0, and N1,−1 (Eq.(3)). Substituting Eq.(3) into Eq.(1), we see that only N1,1 contributes to the receive sensitivity at the center, meaning that the corresponding UISNR (ζr→0) is completely determined by one divergence-free current mode. This explains why the corresponding shape of the ideal current patterns [5] does not change with field strength (Fig.2), and indicates that loops are the optimal coil elements to maximize central SNR.
The analytic expression for ζr→0 is given in Eq.(4), which explicitly shows the dependence of ζr→0 on radius and wave number (electrical properties) of the dielectric sphere.
To validate the analytic solution, we investigated the dependence of ζr→0 on main magnetic field strength B0 and compared the results with corresponding DGF simulation results, with expansion order lmax=55. Average brain tissue was used for the dielectric properties of the sphere, ϵr and σ [4]. We also tested the case with conductivity σ equals to 10−5(Ω−1m−1) to approximate the lossless condition as in Ref. [4].
We used the analytic expression to study the dependence of ζr→0 on the sphere radius (a) and the radius of the surface where the current distribution is defined (b).
Finally, we investigated how the UISNR (ζr) varies with the distance r from the center, either along the x-y plane or z-axis (B0 direction), and determined the range for which the UISNR can be approximated by the value at the center (ζr→0).
Fig.3 shows the double-logarithmic plots of ζr→0vs.B0 for B0=0.5-12 T, replicating the plots in fig.6 of Ref. [4]. The ζr→0 based on the analytic solution (solid lines) was consistent with the results in Ref. [4], which used a multipole expansion to calculate UISNR, and also with values calculated with full DGF simulations (data points). Fig.3 shows that ζr→0 is approximately linear with respect to B0 at low B0 and increase nonlinearly ~B0n with an exponent n>1 at high B0, compatible to published results [4].
The sensitivity of the central UISNR with respect to changes in sphere radius (∂aζr→0) and current radius (∂bζr→0) is shown in Fig.4. The analytic results (solid lines) agree with those from DGF simulations (data points). The UISNR at the center grows monotonically with sphere radius until reaching a plateau (Fig.4a), except at 11 T, where its value oscillates for small sphere radii, likely due to wavelength effects. Intuitively, since there are no conductive losses in air, ζr→0 should not depend on the radius at which the current distribution is defined (b). This is, in fact, confirmed by the analytic expression (Eq.(4)) and by Fig. 4b, where ∂bζr→0=0.
Fig.5 shows that using the UISNR at the center to approximate the UISNR for a voxel at an intermediate position (0<r/a<1) on the x-y plane (Fig. 5a), or z-axis (Fig. 5b) yields an error <5% if r/a<10-20%.
While an approximate solution, with very limited validity, had been proposed for the UISNR [8], our analytic solution of ζr→0 is exact and consistent to full simulations. Calculation speed is 800 times faster than for full simulations.
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