Christopher M Collins^{1}, Giuseppe Carluccio^{1}, Bei Zhang^{1}, Gregor Adriany^{2}, Kamil Ugurbil^{2}, and Riccardo Lattanzi^{1}

Experience and general understanding dictate that greater relative permittivity is required to produce a similar effect at lower B0 field strengths and B1 frequencies. Here we use some fundamental explanations and preliminary numerical results for improving receive array performance at different field strengths to propose, more specifically, that permittivity should increase approximately with the inverse of the square of the field strength for an expected effect.

**Introduction**

In Maxwell’s modified Ampere equation, Δ×**B**=μ(**J**+ jωε**E**), **J** is the conduction current and is presumably strongest in the
coil, while jωε**E** is the displacement current, and presumably strongest in a high-permittivity material (HPM) very
near the coil where electric fields **E** are strong. Here, **E** can be seen as having two sources, the conservative electric
field resulting from charge density associated with voltages along the wires of
the coil needed to produce **J**, and
the magnetically-induced electric field associated with the changing magnetic
field via Faraday’s Law, Δ×**E**=-jω**B**. Both Faraday’s Law and Ohm’s Law for conductive segments
of a coil (V=IjωL) indicate that |**E**| will increase proportionally to frequency ω for a given magnetic field **B**, at least in the quasi-static
regime.

If we then assume that for a certain desired
effect of an HPM in a similar configuration of coil, HPM, and sample we would
like for the displacement current in the HPM and conduction current in the coil
to have similar relative contributions (i.e., |**J**|/|jωε**E**| remains
nearly constant with changing frequency), we find that ε should be proportional to 1/ω^{2}.

To determine an optimal permittivity of a thin HPM helmet-shaped shell at
7T, simulations including a numerical model of the human body (Virtual Family’s
“Duke”) in a 16-element stripline array were performed at 300 MHz^{4, 5}. Later
simulations of a similar helmet with an 8-channel receive array both without^{6} and with^{7} tuning and matching of the individual elements showed improvements
in SNR of about 40% on average throughout the whole brain. Further simulations
with a more densely-packed array of 28 elements showed similar improvements
with the HPM helmet at 7T (Figure 1, middle row)^{8}. In all this work,
an optimal ε_{r} of about 110
was found.

To determine an optimal permittivity of a similar helmet-shaped shell at
3T, simulations including the same body model in a body-sized birdcage coil
were performed^{9}. An optimal er
of about 600 was found. Using this permittivity, SNR in the head for the
28-channel receive array were seen to have significant improvements compared to
the array alone (Figure 1, top row).

Noting that between 3T and 7T the optimal permittivity
followed roughly a ε proportional to 1/ω^{2} relationship and considering the theoretical
arguments above, a relative permittivity of 50 was recommended for an ongoing
coil design for head imaging at 10.5T. With no optimization it was seen that
the addition of the same helmet for a very different coil design resulted in
significant improvements in SNR even though addition of the helmet required moving the coil further from the head (Figure 1, bottom row).

**Discussion**

1. Alsop et al., Magn Reson Med 1998;40:49-54

2. Yang et al., J Magn Reson Imag 2006;24:197-202

3. Haines et al., J Magn Reson 2010;203:323-327

4. Collins et al., Proc. 2013 ISMRM, p. 2797

5. Collins et al., Proc. 2014 ISMRM, p. 1340

6. Collins et al., Proc. 2014 ISMRM, p. 404

7. Haemer et al., Proc. 2017 ISMRM p. 4283

8. Carluccio et al., Proc. 2018 ISMRM p. 4405

9. Yu et al., Magn Reson Med 2017;78:383-386