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J Magn Reson. Author manuscript; available in PMC 2010 August 1.

Published in final edited form as:

Published online 2009 May 21. doi: 10.1016/j.jmr.2009.05.007

PMCID: PMC2722947

NIHMSID: NIHMS133456

Christopher M. Collins, Ph.D., NMR/MRI Building, H066, 500 University Drive, Hershey, PA 17033, Tel: 717-531-7402, Fax: 717-531-8486, Email: ude.usp@snillocmc

The publisher's final edited version of this article is available at J Magn Reson

See other articles in PMC that cite the published article.

This work presents a method to separately analyze the conservative electric fields (Ec, primarily originating with the scalar electric potential in the coil winding), and the magnetically-induced electric fields (Ei, caused by the time-varying magnetic field B1) within samples that are much smaller than one wavelength at the frequency of interest. The method consists of first using a numerical simulation method to calculate the total electric field (Et) and conduction currents (J) in the problem region, then calculating Ei based on J, and finally calculating Ec by subtracting Ei from Et. The method was applied to calculate electric fields for a small cylindrical sample in a solenoid at 600 MHz. When a non-conductive sample was modeled, calculated values of Ei and Ec were at least in rough agreement with very simple analytical approximations. When the sample was given dielectric and/or conductive properties, Ec was seen to decrease, but still remained much larger than Ei. When a recently-published approach to reduce heating by placing a passive conductor in the shape of a slotted cylinder between the coil and sample was modeled, reduced Ec and improved B1 homogeneity within the sample resulted, in agreement with the published results. (196 words)

In high field MR imaging and spectroscopy of small samples, RF energy adsorption in the sample can result in significant sample heating. This has led to a number of designs to produce coils with relatively low electric fields in the sample region [1, 2] and/or to shield the sample from the electric fields produced from the coil [2, 3].

The electric field produced by radiofrequency (RF) coils is often discussed in terms of two components: the conservative electric field (*$\stackrel{\u20d7}{E}$ _{c}*), mainly caused by the scalar electric potential in the coil winding, and the magnetically-induced component of the E-field (

To be able to reduce heating, it is necessary to understand its sources. The magnetically-induced component of the E-field (*$\stackrel{\u20d7}{E}$ _{i}*) cannot be changed without changing the RF magnetic fields while the other (

The method we present for calculating *$\stackrel{\u20d7}{E}$ _{c}* and

The geometry used in this work consists of a cylindrical sample in a solenoidal coil at 600 MHz (14T). The solenoidal coil was based on a published design having 8 turns of 0.15 mm-diameter round copper wire (d), wound into a solenoid with a diameter (d_{coil}) of 1.0 mm, length (*l*_{coil}) of 2 mm, and distance per turn (s) of 0.231 mm (Fig.1). Samples with different relative permittivity (ε_{r}) and electrical conductivity (σ), but with the same diameter (d_{sample} = 0.75 mm) and length (*l*_{sample} = 2.5 mm) were modeled to mimic air (ε_{r}=1, σ=0 S/m), conductor (ε_{r}=1, σ=0.2S/m), dielectric (ε_{r}=78, σ=0 S/m), and weak saline (ε_{r}=78, σ=0.2 S/m). To model a recently-published method for reducing electric fields in the sample [3], we performed an additional calculation with a cylindrical conductor, having a single longitudinal gap, placed between the sample and the coil. Here we refer to this passive conductor as a loop-gap cylinder (LGC). For this application, solenoid and LGC geometries similar to those used in a previously-published work [3] were modeled. The coil had an inner diameter of 3 mm and a length of 3.47 mm, while the LGC had an inner diameter of 2.4 mm and a length of 10.86 mm. The current density *$\stackrel{\u20d7}{J}$* within both the solenoid and LGC were considered In the determination of *$\stackrel{\u20d7}{E}$ _{i}* for this case.

Geometry of the solenoidal coil (blue) and the sample (red). Here d_{coil} is the coil diameter (1.0 mm), s is the distance per turn (s) (0.231 mm), d is the diameter of the round wire (0.15 mm), *l*_{coil} is the coil length (2 mm), d_{sample} is the sample diameter **...**

The calculation of *$\stackrel{\u20d7}{E}$ _{T}* could be performed with any of a variety of field simulation methods. Due to availability of software at our site, we chose to use the Finite Difference Time Domain (FDTD) method [6]. All simulations were performed using commercially available software (xFDTD; Remcom, Inc; State College, PA). In all cases the coil was driven with a constant voltage source (1V) in series with a 50Ω resistor connecting the lead wires. Steady-state values of

Using values for *$\stackrel{\u20d7}{J}$* throughout the coil from the Full-Maxwell calculation, *$\stackrel{\u20d7}{E}$ _{i}* was calculated as

$${\overrightarrow{E}}_{i}=j\omega \overrightarrow{A}$$

(1)

where

$$\overrightarrow{A}(r)=\frac{{\mathrm{\mu}}_{0}}{4\mathrm{\pi}}{\displaystyle \underset{r\text{'}}{\int}\frac{\overrightarrow{J}(r\text{'})}{|r-r\text{'}|}\mathit{\text{dv}}}$$

(2)

and *$\stackrel{\u20d7}{A}$* is the magnetic vector potential, ω is the radial frequency, µ_{0} is the permeability of free space, *r* indicates the location for which *$\stackrel{\u20d7}{A}$* is currently being calculated, and *r*' indicates the location of source current within the solenoid. The integration is performed over the volume of the solenoid wire.

Once both *$\stackrel{\u20d7}{E}$ _{T}* and

$${\overrightarrow{E}}_{c}={\overrightarrow{E}}_{T}-{\overrightarrow{E}}_{i}$$

(3)

In this work, the coil diameter (1 mm) is small enough compared to one wavelength (500 mm in air at 600 MHz) that no significant wavelength effects are expected. Thus the displacement current term is negligible [7].

The power dissipated in the sample can be calculated as [8]

$${P}_{\mathit{\text{abs}}}=\frac{1}{2}{\displaystyle \int \mathrm{\sigma}\left({{E}_{x}}^{2}+{{E}_{y}}^{2}+{{E}_{z}}^{2}\right)\mathit{\text{dv}}}$$

(4)

where σ is the conductivity of the sample, *E*_{x}, *E*_{y}, and *E*_{z} are the amplitude of the electrical field components in the x, y, and z-directions, and the integration is performed over the volume of the sample. After all fields were calculated, they were normalized so that *B _{x}* = 4 µT at the center of the coil.

To ensure our numerical method for calculating *$\stackrel{\u20d7}{E}$ _{c}*,

Using published methods [9], we estimate the inductance of the solenoidal coil to be approximately 24 nH. Thus, the impedance at 600 MHz was approximately j90 Ω. This calculated value was in good agreement with the FDTD numerical simulation result with which had an impedance of 0.05 + j89.38 Ω.

In the numerical calculation, the input was a voltage source with a magnitude of 1V in series with a 50 Ω resistor. Based on the above information,|*$\stackrel{\u20d7}{E}$ _{c}*| can be estimated roughly as

$$\left|{\overrightarrow{E}}_{c}\right|\cong \frac{{V}_{\mathit{\text{coil}}}}{{l}_{\mathit{\text{coil}}}}$$

(5)

where *V _{coil}* is the voltage drop across the solenoidal coil, calculated as

$${V}_{\mathit{\text{coil}}}={V}_{\mathit{\text{source}}}\times \frac{\left|{Z}_{\mathit{\text{coil}}}\right|}{\left|{Z}_{\mathit{\text{total}}}\right|}$$

(6)

where *V _{source}* is the voltage of the input source (1V),

Using Faraday’s law and assuming a homogeneous B_{1} field in the sample near the center of the solenoid, |*$\stackrel{\u20d7}{E}$ _{i}*| within the sample can be calculated as

$$\left|{\overrightarrow{E}}_{i}\right|=\frac{\omega r}{2}\left|\overrightarrow{B}\right|$$

(7)

If |*$\stackrel{\u20d7}{B}$*| is 4 µT, the calculated maximum |*$\stackrel{\u20d7}{E}$ _{i}*| within the sample on the center plane (x=0) is about 2.8 V/m (Table 1).

For a simple analytical estimation of the sample power loss, it was assumed that *$\stackrel{\u20d7}{B}$* was uniform throughout the entire sample. Based on Eq. 3–7 and previous research [8–9],

$${P}_{\mathit{\text{sample}}}\cong \frac{1}{2}{\displaystyle \underset{0}{\overset{L}{\int}}{\displaystyle \underset{0}{\overset{2\pi}{\int}}{\displaystyle \underset{0}{\overset{R}{\int}}\sigma {\left|{\overrightarrow{E}}_{c}+{\overrightarrow{E}}_{i}\right|}^{2}\mathit{\text{rdrd}}\varphi \mathit{\text{dz}}}}}$$

(8-a)

Because *$\stackrel{\u20d7}{E}$ _{c}* is generally in the axial direction and

$${P}_{\mathit{\text{sample}}}\cong \frac{1}{2}{\displaystyle \underset{0}{\overset{L}{\int}}{\displaystyle \underset{0}{\overset{2\pi}{\int}}{\displaystyle \underset{0}{\overset{R}{\int}}\sigma {\left(\sqrt{{\left|{\overrightarrow{E}}_{c}\right|}^{2}+{\left|{\overrightarrow{E}}_{i}\right|}^{2}}\right)}^{2}\mathit{\text{rdrd}}\varphi \mathit{\text{dz}}}}}$$

(8-b)

Where *R* is the sample radius (0.375 mm) and *L* is the sample length (2.5 mm). Using Eq. [7], we can find

$${P}_{\mathit{\text{sample}}}\cong \sigma \pi L\left(\frac{{{E}_{c}}^{2}{R}^{2}}{2}+\frac{{\omega}^{2}{R}^{4}{B}^{2}}{16}\right)$$

(8-c)

If σ is 0.2 S/m, *E _{c}* is 30 V/m,

In all simulation conditions, *$\stackrel{\u20d7}{E}$ _{c}* was much stronger than

Magnitudes of x, y, and z-oriented components of Conservative E-field (*E*_{c}, top) and Magnetically-induced E-field (*E*_{i}, bottom) in the empty solenoidal coil driven at 600 MHz to produce 4 µT at the coil center. On the plane shown, X is axial (up-down **...**

Approximate total magnitude of conservative E-field (*E*_{c}, top), magnetically-induced E-field (*E*_{i}, middle) and magnetic flux density (B, bottom) after normalization when loaded with a cylindrical sample containing various materials. Linear color scale from **...**

Though still remaining significantly larger than *$\stackrel{\u20d7}{E}$ _{i}*, the

*$\stackrel{\u20d7}{E}$ _{c}* is primarily oriented in the × direction in the solenoid because the scalar potential changes along the length of the wire, which is wound along the x-axis (Fig. 1). For

When a cylindrical conductor with a single longitudinal gap is placed between the sample and the solenoid, a reduction in the electric fields and a relative increase in B_{1} field homogeneity are seen, in agreement with previously-published experimental results [3] (Figure 4).

Both the scalar electric potential along the coil wire and the changing vector magnetic potential *$\stackrel{\u20d7}{A}$* produced by the coil current, can create electric fields [7]. In an empty solenoidal coil, *$\stackrel{\u20d7}{E}$ _{c}* can be much bigger than

Based on Faraday’s Law, *$\stackrel{\u20d7}{E}$ _{i}* in the sample is induced by a time-varying magnetic field

As presented in Fig. 2 and Fig 3, the dominant factor of the sample power loss (P = σE^{2}) is *$\stackrel{\u20d7}{E}$ _{c}*, not

The total absorbed power (P_{abs}) in the sample was calculated based on Eq. [4]. Table 2 shows the numerical calculation results of sample power loss. As the relative permittivity (ε_{r}) of the sample is increased from 1 to 78, the power loss of the sample is changed by an order of magnitude. This is mainly caused by the decrease of *$\stackrel{\u20d7}{E}$ _{c}* within the sample due to the polarization field as discussed previously (Fig. 3).

Numerical calculation results of the normalized sample power loss (P_{sample}) caused by the conservative and magnetically-induced electric field components.

The agreement between numerically-calculated values and analytical approximations (Table 1) indicates that our numerical approach yields reasonable results. Comparison to more exact analytical approximations for MR-relevant geometries is, to our knowledge, not feasible at this time. Some differences between numerical and analytical results (Table 1) of the maximum *E _{c}* and sample power loss are likely caused by simplifications and assumptions used in making the analytical approximations, such as assuming negligible displacement current and homogeneous B

For an initial application and demonstration of the method, we simulated a solenoidal coil with and without a passive conductor in the form of a loop-gap cylinder (LGC) inside to shield the interior of the coil from conservative E-fields [3]. As shown in Fig. 4, the addition of the LGC proved to both significantly shield the interior region of the coil from conservative E fields and improve the homogeneity of the B_{1} field along the axis of the coil, in agreement with previously-published experimental results. In our calculations, the improvement in homogeneity along the coil axis is related to both increased sensitivity over a larger volume and lower efficiency at the coil center, as greater coil current is required to maintain the same B_{1} field magnitude there.

In summary, we have presented a new method to calculate conservative E-fields (*$\stackrel{\u20d7}{E}$ _{c}*) and magnetically induced E-fields (

The method of analysis utilized here could be useful as long as no significant wavelength effects are present, and as long as *$\stackrel{\u20d7}{J}$* in the coil and good conductors is much greater than that in the sample. Thus, this method may be useful not only for the evaluation of high field microimaging but also as an alternative method of evaluating fields within loaded gradient coils or larger RF coils at very low-frequencies.

Funding for this work was provided by the National Institutes of Health (NIH) through R01 EB000454 and R01 EB000895, and by the Pennsylvania Department of Health.

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