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A comparison of methods to decrease RF power dissipation and related heating in conductive samples using passive conductors surrounding a sample in a solenoid coil is presented. Full-Maxwell finite difference time domain numerical calculations were performed to evaluate the effect of the passive conductors by calculating conservative and magnetically-induced electric field and magnetic field distributions. To validate the simulation method, experimental measurements of temperature increase were conducted using a solenoidal coil (diameter 3 mm), a saline sample (10 mM NaCl) and passive copper shielding wires (50 μm diameter). The temperature increase was 58% lower with the copper wires present for several different input powers to the coil. This was in good agreement with simulation for the same geometry, which indicated 57% lower power dissipated in the sample with conductors present. Simulations indicate that some designs should be capable of reducing temperature increase by more than 85%.
In MRI and high resolution NMR of small samples, the RF electrical fields within the sample can be a significant factor in temperature increase (ΔT) and SNR . The vast majority of MR microscopy and small-volume NMR studies use solenoidal coils. In this geometry, the conservative electric (Ec) field – mainly caused by the scalar electric potential in the coil winding  – can be a significant component of the total electric field in the sample [1, 3]. This has led to a variety of proposed methods to reduce the Ec field (and associated heating) within the sample while still maintaining a significant B1 field intensity. These methods have included passive [4, 5] or active  conductors between the solenoidal coil and the sample, and use of coil geometries other than the solenoid that have significantly lower Ec fields [7–9].
In this paper we use numerical methods to compare some simple methods to decrease sample power loss and related ΔT using strategically-positioned passive conductors to partially shield the sample from the Ec field of the solenoidal coil. We refer to a set of conductors placed between the coil and the sample to shield conservative E-fields while allowing B1 to penetrate the sample as an “Ec shield.” Full-Maxwell numerical calculations of the electromagnetic fields are performed to evaluate several different arrangements of copper strips oriented along the major axis of the solenoid and arranged about the circumference of the cylinder in a manner similar to that described by Gadian  and applied recently by Wu and Opella , as well as a more recently proposed method, referred to here as the “loop-gap shield” . To validate the numerical calculation method, experiments measuring ΔT with and without a specific Ec shield geometry are performed and results are compared to those from simulations of the same geometry.
The time-rate of change in temperature (dT/dt) within a non-perfused material with an internal heat source, arising from the specific energy absorption rate (SAR) can be written as
where ρ is the mass density, c the heat capacity, and k the thermal conductivity. The SAR can be calculated as
where σ is the local electrical conductivity of the sample and Etotal is the magnitude of the total electric field within the sample. Etotal can be separated into two components, Ec (caused by the spatial distribution of the scalar electric potential ), and Ei, induced by the time-varying magnetic fields. This can be expressed as
where is the vector magnetic potential (Wb/m) defined such that
In order to reduce the absorbed power loss, total should be minimized, but changes in Ei will necessarily require changes in the B1 field distribution also. With a strategic arrangement of passive conductors, however, it is possible to implement a Faraday cage structure that cannot support significant cirumferential RF electrical currents, and thus reduce Ec with minimal effect on the B1 field .
Numerical calculations were used to simulate and compare a variety of designs of Ec shield. To validate the numerical calculation methods, experimental measurement of temperature increase in a phantom with and without an Ec shield was performed and results were compared to simulations.
All MRI experiments were performed on a 14.1 tesla (600 MHz) Direct Drive spectrometer (Varian, Palo Alto, CA) using a custom-made solenoidal coil (3 mm diameter and 4.5 mm length) consisting of 4 turns of 0.35 mm-diameter copper. The sample (10 mM NaCl in a 1.67 mm outer diameter glass tube) was inserted into a polyamide tube (outer diameter: 2.24 mm, length 10.8 mm) which was placed into the solenoidal coil. Five copper wires of 50 μm diameter (California Fine Wire, Grover Beach, CA) oriented parallel to the axis of the coil were glued onto the outside of the polyamide tube. The RF coil was immersed in a perfluorinated liquid (FC43, 3M, Minnesota) for magnetic susceptibility matching and thermal isolation (Figure 1).
Estimation of temperature increases were performed using interlaced periods of RF heating and MRI temperature measurements using the proton reference frequency (PRF) method . Five different time-average input powers (0, 0.15, 0.3, 0.6 and 1.2 Watts) were used for the heating periods. First, a baseline image was acquired using a single slice gradient echo (GE) sequence (TR = 100 ms, TE = 10 ms, field-of-view = 20 × 20 mm, matrix size = 256 × 96, slice thickness = 1 mm, scanning time = 9.6 sec). RF heating used a WALTZ-4 decoupling sequence which was applied for approximately 20 seconds. Imaging and heating procedures were successively repeated sixteen times for each value of the input power. Phase difference images were generated using MATLAB (The MathWorks, Inc., Natick, MA), and phase unwrapping was performed using a previously-published method . Temperature increases (ΔT) with respect to the baseline image were calculated according to 
where Δ is the difference in phase between the particular image and the baseline image acquired before the first heating period.
For comparison of the simulation to experimental results, the modeled coil and Ec shield geometries closely matched those used in experiment. Geometries with no Ec shield and with five copper strips of 50 μm width and 10.86 mm length placed between the solenoid coil and the sample were modeled. These models were defined on a rectilinear grid with 0.012 mm resolution for use with the finite difference time domain (FDTD) method of calculation for electromagnetics.
For numerical comparisons of different Ec shield designs, a model of a 1.6 mm-diameter, 16 mm-long 10 mM saline sample (σ=0.2S/m, εr=78 at 600 MHz) was created within the solenoid coil.. Simulations with no Ec shield, and four different arrangements of evenly-spaced narrow copper strips (0.20 mm width, 10.8 mm length) or wider copper strips (0.60 mm diameter, 10.8 mm length), and a loop-gap shield  were performed with this arrangement. These models were defined on a rectilinear grid with 0.050 mm resolution for use with the FDTD method. These geometries are shown in Figure 2.
All simulation work was performed using commercially-available FDTD software (xFDTD; Remcom, Inc; State College, PA), with the coil driven by a sinusoidally-varying 600 MHz voltage source. Analysis of the results was performed in Matlab (The MathWorks, Inc., Natick, MA). All simulation results of electromagnetic fields were normalized so that Bx = 4 mT at the coil center.
To analyze calculation results, we applied a recently-developed method for separating Ec and Ei from the results of the FDTD calculation . First, i was calculated as
where is the current density in the coil from the FDTD simulation, ω is the Larmor frequency, and μ0 is the permeability of free space. c was then calculated as
where total is the electric field obtained from the FDTD simulation. This method is reasonably accurate when the currents in the coil are much greater than the currents in the sample and when the problem geometry is small enough that wavelength effects are negligible : both conditions are well-satisfied by the experimental setups analyzed here.
Figure 3 shows the experimentally-measured increase in signal phase observed in a rectangular region of interest at the center of the phantom (see dotted line in Figure 4) on gradient echo images acquired between periods of heating when a time-average input power of 1.2W is applied and no Ec shield is present. Most of the phase change (and thus temperature increase) occurs between the first (baseline) and second image, or during the first period of heating. After this, the effects of thermal conduction and mechanisms of heat transfer to the environment surrounding the sample become significant, and the temperature increases at a notably slower rate.
Figure 4 shows maps of the experimentally-determined temperature increase in the phantom between the first (baseline) and second gradient-echo images with and without the presence of the 5-wire Ec shield when a time-average input power of 1.2 W is applied. The dotted rectangle outlines the region used for numerical data in Figures 3 and and55 and in Table 1.
Table 1 and Figure 5 present experimentally-measured temperature increases at the center of the phantom with and without the 5-wire Ec shield for a variety of input powers. The temperature increase is seen to be roughly proportional to the input power level. Values with no applied power during the heating period are close to zero, and indicate the level of error in the measurement. On average the temperature increase is reduced by approximately 58% with the addition of the 5-wire Ec shield in experiment.
Table 2 presents simulated values for the average Ec and B1 fields, and standard deviation of the B1 field with and without the 5-wire Ec shield. It is seen that the presence of the 5-wire Ec shield significantly reduces Ec with minimal effect on the B1 field magnitude or homogeneity. Table 2 also gives the total power dissipated in the sample for each case, showing that the dissipated power is reduced by 57% with addition of the 5-wire Ec shield.
Figure 6 shows the calculated magnitude of the Ec, Ei, and B1 fields within the sample for six different simulation conditions including no Ec shield, four different arrangements of copper strips, and the loop-gap shield (geometries shown in Figure 2). Table 3 gives numerical values for Ec and B1 as well as the power dissipated in the sample for each case. The presence of any of the Ec shields is seen to reduce Ec and the dissipated power throughout the sample significantly. In all but the loop-gap shield this is accomplished with minimal effect on the B1 field distribution. Of the geometries compared here, the 8-strip Ec shield reduces the dissipated power by the greatest amount (88%), with the 6-strip and loop-gap shield designs following closely.
The results of this study confirm that strategically-placed passive conductors as an Ec shield can reduce RF heating of a sample within a solenoid coil. Although this general principle has been noted in, or can be inferred from, previous works [4, 5], here numerical calculations are used to compare a number of designs, showing that the geometry of the Ec shield can largely determine its efficacy. For example, four narrow strips are shown to reduce power dissipation in the sample by 59%, but 8 broad strips can reduce it by 88%.
In general, a number of long, thin conductive strips or wires oriented parallel to the coil axis and placed on the outer surface of the cylindrical sample can reduce sample heating with minimal effect on the B1 field, since the conductors can carry a charge density distribution opposing the Ec field of the coil, which is oriented along the coil axis , but cannot carry any significant current in the circumferential direction and thus cannot significantly affect the B1 field of the solenoid. Another design, based on a single conductor around the surface of the cylinder – continuous except for one slit along its length oriented with the coil axis, the loop-gap shield  – has also been shown to significantly reduce heating, but with a significant effect on the magnitude and uniformity of the B1 field. It was noted previously that the effect on the B1 field is to make it more homogeneous along the whole length of a sample extending well beyond the ends of the solenoid . This is also apparent in Figure 6 of this work. However, in Table 3 we quantified the standard deviation of the B1 field in the sample between the ends of the solenoid only, and found in that region the loop-gap shield did not improve homogeneity, but may actually slightly reduce it. Also, use of the loop-gap shield required approximately 40% greater current in the solenoid to achieve the same B1 field magnitude at the center of the coil compared to the other cases. If the sample volume is significantly longer than the solenoid, this loss of sensitivity per sample volume at the center of the coil may be offset by greater sensitivity to regions of the sample outside the solenoid.
In principle, the use of Ec shields should also affect the signal-to-noise ratio (SNR). Reducing Ec in the sample reduces sample-related noise induced in the coil, but the addition of good conductors to the imaging volume also introduces a potential new source of noise. Careful analysis of this effect would require a different method for numerical calculation (such as a finite element method or a method of moments) which is more capable of accurately representing skin depth effects at the micron level. Work in this area is ongoing.
Funding for this work was provided by the National Institutes of Health (NIH) through R01 EB000454 and by the Pennsylvania Department of Health.
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