Patient safety concept for multichannel transmit coils

Authors


Abstract

Purpose

To propose and illustrate a safety concept for multichannel transmit coils in MRI based on finite-differences time-domain (FDTD) simulations and validated by measurements.

Materials and Methods

FDTD simulations of specific absorption rate (SAR) distributions in a cylindrical agarose phantom were carried out for various radio frequency (RF) driving conditions of a four-element coil array. Additionally, maps of transmit amplitude, signal phase, and temperature rise following RF heating were measured by MRI.

Results

Quantitative agreement was achieved between simulated and measured field distributions, thus validating the numerical modeling. When applying the same RF power to each element of the coil array but systematically varying the RF phase between its elements, the maximum of the SAR distribution was found to vary by a factor of about 15.

Conclusion

Our results demonstrate that current RF safety approaches are inadequate to deal with the new challenge of multichannel transmit coils. We propose a new concept based on a systematic investigation of the parameter space for RF phases and amplitudes. In this way the driving conditions generating the highest local SAR values per unit power can be identified and appropriately considered in the RF safety concept of a given MRI system. J. Magn. Reson. Imaging 2007;26:1315–1321. © 2007 Wiley-Liss, Inc.

IN TODAY'S HIGH-FIELD (3–4T) and ultrahigh-field (≥ 7T) whole-body MRI systems the wavelengths of the radio frequency (RF) fields in biological tissue are no longer large compared to the dimensions of the human body (1, 2). The resulting distortions of the RF field distributions, often denoted as “dielectric resonance,” “resonator effects,” or just “wave phenomena,” are serious challenges for the development and application of new hardware and software (3, 4). It is widely believed that parallel transmission (PTX), using arrays of transmit coils with independently adjustable RF amplitude and phase for each element, is the most promising approach to overcome these difficulties (2, 5–9). While the additional degrees of freedom in driving a PTX-coil array are obviously crucial for the enhanced capabilities of these devices, they are simultaneously the source of considerable complications when it comes to RF-safety issues.

The legal requirements for safety of MRI equipment are laid out in the international standard IEC 60601-2-33 (10). To fulfill these requirements it has to be ensured that in head and trunk the locally deposited RF power per unit mass (specific absorption rate, SAR), averaged over 10 g of tissue, nowhere exceeds 10 W/kg. For most practical purposes, however, the global SAR limits of 3.2 W/kg and 4.0 W/kg (values referring to the first level controlled operating mode) for the whole head and the whole body, respectively, are used. For the workhorses of MR transmit coils, whole-body resonator, and volume head coil, such global limits are very convenient, as multiplication by a typical mass for the entire patient or the human head just gives an upper limit for the total RF power Ptot to be delivered to the volume coil. In this way, compliance with safety requirements reduces to controlling and restricting one single parameter, Ptot. This quantity depends only on scanner hardware and patient mass, but not on sequence parameters or on any other individual condition of an adult patient. It can be preset by the manufacturer, therefore, making this a very feasible approach. One has to keep in mind, however, that this practice relies on the concept of a volume coil generating a spatially homogeneous RF magnetic field B1(equation image). In case of inhomogeneous field distributions it can no longer be taken for granted that compliance with global SAR limits will automatically ensure compliance with local limits everywhere in the patient body.

Even for a traditional single-channel transmit coil such inhomogeneities may arise due to the aforementioned wave phenomena at high and ultrahigh field strengths (4, 11). In case of PTX arrays, the global SAR concept definitely breaks down, as one single parameter, Ptot, is no longer adequate to describe the system. In this work we propose a “worst-case analysis” to deal safely with the new complexity of PTX coils. We will demonstrate its feasibility by numerical simulations and phantom measurements for the case of a four-channel PTX array of so-called current sheet antennae (CSA).

MATERIALS AND METHODS

Theoretical Background

In the following sections we report on finite-differences time-domain (FDTD) simulations (12) carried out to determine distributions of the specific absorption rate SAR(equation image) inside a homogeneous agarose phantom. For our description we adopt Hoult's (13) notation, i.e., complex variables are typeset in script font, tildes indicate quantities measured in a rotating coordinate system, and the terms “positive” (superscript (+)) and “negative” (superscript (−)) refer to frames rotating in the same or in the opposite direction, respectively, as the precessing magnetization.

There are two quantities accessible by in vivo MR: | equation imagemath image(equation image)|, the absolute value of the complex amplitude of the positively rotating, and thus transmit-active, component of the RF magnetic field, and ψ(equation image), the phase of the MR signal. In phantom experiments an additional third quantity, namely the temperature increase ΔT(equation image) following RF heating, can also be measured. All this experimental information can and should be used to validate and refine the FDTD calculations.

Exposing a phantom within an MR scanner to continuous wave (CW) RF power for a period ΔtRF will cause the temperature in the phantom to rise above its initial (ambient) value T0. The temperature increase ΔT(equation image) = T(equation image) − T0 can be mapped by MR thermography, e.g., by exploiting the temperature dependence of the proton resonance frequency of water (PRF method) (14, 15). Neglecting thermal diffusion and heat losses during the heating period, the temperature rise can be expressed as ΔT(equation image) = SAR(equation imagetRF/cp, where cp is the specific heat capacity of the phantom material. The specific absorption rate SAR(equation image) inside a homogeneous phantom of electrical conductivity σ and mass density ρ is given by

equation image(1)

where equation image1(equation image) denotes the (real) amplitude of the RF electric field vector.

The positively and negatively rotating components of the complex RF magnetic field are given by

equation image(2a)

and

equation image(2b)

respectively, where ℬ1x, ℬ1y are the complex amplitudes of the x,y components of the equation image1(r→,t) field in the laboratory frame (13). The phase of the complex signal amplitude is given by

equation image(3)

where we have ignored an irrelevant common phase offset as we will always do throughout this work.

Knowing the material parameters cp, ρ, and σ, and taking equation image1(equation image) and equation image1(equation image) from the FDTD simulations, SAR(equation image), |equation imagemath image(equation image)|, and ψ(equation image) can easily be calculated. The latter quantities can then be compared to the experiment, thus allowing to validate the numerical results.

The description given so far applies to a single coil. To extend this to the entire PTX array, the RF driving conditions have to be taken into account. We consider the B1 field generated when the k-th CSA of the array is excited by a reference RF power of P0=1 kW and all other coils are terminated into 50 Ω. We designate the complex amplitude of its positively rotating component as equation imagemath image(P0;equation image). Note that, via RF coupling between elements, the whole array contributes to this quantity; equation imagemath image(P0;equation image) is not identical to the field generated by a single, stand-alone CSA. When coherently exciting all coils of the array at individual RF powers Pk and RF phases ϕk, the complex amplitude of the total B1 field in the rotating coordinate system is obtained from the superposition principle as

equation image(4)

No cross terms are needed in Eq. [4], as all RF coupling is fully included in the definition of the equation imagemath image(P0;equation image). Using the analogous nomenclature we likewise obtain

equation image(5)

In the following, Eqs. [4] and [5] will be used to simulate SAR and equation imagemath image maps for various driving conditions.

MR Scanner Hardware and Calibration of RF Channels

All MR measurements were performed on a 3 tesla whole-body MR scanner (Medspec30/100; Bruker Biospin MRI, Ettlingen, Germany) with four broadband receive channels and four broadband 4-kW RF power amplifiers (LPPA 13040; Dressler Hochfrequenztechnik GmbH, Stolberg, Germany), incorporated into four transmit channels. The output of each amplifier was equipped with a high-power circulator (VAB 1143; Valvo Bauelemente, Hamburg, Germany) and connected to one of the antennae via a separate transmit/receive switch. The third port of each circulator was connected to a 50-Ω high-power dummy load to achieve a source impedance of 50 Ω independently of the loading by the coil array. The scanner electronics allowed to drive all RF power amplifiers coherently, generating four independent RF pulses each with its own sequence timing, pulse shape, frequency, phase, and amplitude.

The output power of each RF power amplifier was calibrated by connecting the corresponding coaxial feeding cable to a calibrated oscilloscope (TH S730A; Tektronix GmbH, Köln, Germany) via a 30-dB high-power attenuator (Model 8322; Bird Electronic Corp., Cleveland, OH, USA), which in turn had been calibrated at 125.3 MHz using a network analyzer (HP4396B; Agilent Technologies Deutschland GmbH, Böblingen, Germany). Based on this procedure, measurement uncertainties of RF powers amount to 15% at most. The relative RF phases of the power amplifiers were measured by sequentially recording free induction decays (FIDs) using different power amplifiers but using the same coil element, loaded by the cylindrical agarose phantom, for transmission.

Four-Channel Transmit/Receive Coil Array and Agarose Phantom

The box-like (length = 16 cm, width = 8 cm, height = 3 cm) TX/RX CSA used in this study were built according to the description given previously (16, 17). Their design is based on a similar construction known from hyperthermia applications (18) and CSA arrays have also been used in the first experimental demonstration (19) of the Transit SENSE technology (7–9). Our four-channel head coil (Fig. 1a) consists of four identical elements mounted symmetrically on a supporting Perspex structure. A detailed assessment of this device was performed but is not subject of this work.

Figure 1.

a: Four-element transmit/receive head coil consisting of four CSA, loaded by cylindrical agarose phantom. b: FDTD model showing dielectric bars on the back sides of the CSA, used for tuning. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com]

The phantom used in all measurements consisted of a cylindrical Perspex vessel (inner diameter = 19 cm, length = 19 cm) filled with a gel containing 20 g/liter agarose, 1.33 g/liter NaCl, and 0.66 g/liter CuSO4 for T1 adjustment. Electrical conductivity and relative permittivity of the gel were determined to be σ = 0.33 S/m and ϵr = 76, respectively, at 25 MHz and 24°C, using a coaxial probe immersed into the gel and connected to a network analyzer.

FDTD Simulations

Today, FDTD simulations (12) are frequently used for quantitative assessment of coil performance and characteristics in MRI. The calculations in the present investigation were performed in effectively the same fashion as in a variety of other studies on these topics (11, 20–22), and only a brief outline is given here.

For our simulations we used the XFDTD software package from REMCOM Inc.(State College, PA, USA). Pre- and postprocessing software to evaluate the steady-state electric and magnetic RF fields inside the phantom was written in IDL (Research Systems Inc., Boulder, CO, USA),. The numerical models were implemented on a 0.5-cm grid consisting of 75 × 75 × 49 cells surrounded by eight perfectly matched layers to achieve free space behavior. The CSA were modeled according to their actual geometry and assumed to consist of planar sheets of perfectly conducting material. Capacitors were treated as a single dielectric bar filling a slot in the top face of the CSA (see Fig. 1b). Its dielectric constant εr was adjusted to tune the antenna to 125.3 MHz. Matching was achieved by a lumped element resistor in parallel to the exciting current source. The phantom was modeled using the measured material parameters and approximating the true geometry on the 0.5-cm grid.

In a first step, several calculations were carried out to tune the coil array and to determine the matching conditions. With all antennae tuned and matched, four final calculations were performed, exciting in each case only one particular coil on resonance. Following excitation, the simulations covered a time period of about 2 μsec to ensure the system reached a steady state. Only the last two oscillation periods of each simulation were then used to extract the desired electric and magnetic field components. For each element these complex quantities were evaluated and normalized to the RF power at the excitation port, which in turn was inferred from the voltage amplitude and the resonance resistance of the antenna.

When driving the entire array at given RF powers and phases for each element, the resulting complex amplitude equation imagemath image(equation image) was calculated by superposition of the four individual, complex steady-state magnetic field amplitudes according to Eq. [4]. Likewise, the total electric field amplitude equation image1(equation image) was obtained using Eq. [5]. Such superpositions do not require new FDTD calculations and, therefore, can be readily computed. In this way, electric and magnetic field distributions can easily be determined for a large number of amplitude and phase settings of the antenna array, which is not feasible to carry out experimentally.

MR Measurements

After applying RF power (125.3 MHz, up to 200 W per channel) from two CW amplifiers (5052F; Ophir RF Inc., Los Angeles, CA, USA) for a period of 15 minutes to two selected antennae of the loaded array inside the scanner, we measured temperature distributions within the cylindrical phantom using the PRF method (14, 15, 23, 24). Starting at ambient temperature, the RF-induced rise was inferred from phase shifts in conventional gradient-echo images (TE = 10 msec), acquired before, during, and after heating. During measurements the RF heating was shut off to avoid interference with the scanner's RF system. These interruptions of just a few seconds did not noticeably affect the temperature distribution within the phantom. In parallel, we monitored the absolute temperature at a selected reference position within the phantom by a fiber-optical sensor (MPM; Luxtron, Mountain View, CA, USA), allowing us to correct the phase images for the dominating errors due to B0 drifts. We estimated the maximum error of the measured temperature changes to be ±1°C and the standard deviation (SD) to be 0.4 °C.

The transmit-active component | equation imagemath image(equation image)| produced by a PTX array can be measured by several flip-angle mapping techniques (25). We applied the simplest but time consuming technique based on a rectangular preparation pulse (τp = 1 ms) followed by spoiler gradients to destroy any transversal magnetization (26). In this way a three-dimensional (3D) distribution of longitudinal magnetization Mz(equation image) was produced depending on the value of |equation imagemath image(equation image)| at the site equation image of each voxel. After preparation, the Mz(equation image)-distribution of a selected slice was mapped by a standard gradient-echo sequence applying an RF pulse delayed by Δt = 12.4 ms. To measure phase maps ψ(equation image) of the complex signal amplitude (see Eq. [3]), spin-echo images with carefully trimmed gradients were recorded in a single axial slice.

RESULTS

Maps of Transmit Amplitude and Signal Phases in Phantom

In this section we compare measured and simulated maps of transmit amplitudes and signal phases intended for validation of the FDTD modelling. Figure 2 shows measured (Fig. 2a) and simulated (Fig. 2b) distributions of | equation imagemath image(equation image)| in the central axial plane of the phantom. Only the antenna at the top (CSA-1, throughout this work the antennae are always numbered as indicated in Fig. 2) was driven and the remaining antennae were terminated into 50 Ω. After applying a global scaling factor of 1.08 to the theoretical results, quantitative agreement with the experimental data was achieved, as illustrated by the cuts along the x-axis (Fig. 2d) and the y-axis (Fig. 2e). The correction factor of 1.08 might reflect uncertainties in modeling the geometry of the loaded array because of the finite resolution (0.5 cm) of the FDTD grid, uncertainties of the measured RF-power, as well as uncertainties in the electric conductivity and relative permittivity of the phantom material. The maps of |equation imagemath image(P0;equation image)| lack mirror symmetry with respect to the y-axis because of RF eddy and displacement currents generated in the sample (see, e.g.,13. Next to CSA-4, there is a circumscribed area of low magnitude |equation imagemath image(P0;equation image)| in Fig. 2a. Due to RF coupling, currents are induced in the other antennae while CSA-1 is driven. The time dependent fields B1(equation image,t) of all coils add up coherently and at the circumscribed area close to CSA-4 mostly |equation imagemath image(P0;equation image)| rather than |equation imagemath image(P0;equation image)| is present (“transmitter blind spot”). The opposite holds true at the mirror site with respect to the y-axis, as is illustrated in Fig. 2c, in which a |equation imagemath image(P0;equation image)| map is depicted. As this quantity is not accessible by experiment, only simulated data can be shown. Since the signal received by CSA-1 is proportional to |equation imagemath image(P0;equation image) × (equation imagemath image(P0;equation image))*|, no signals from the mirror site close to CSA-2 could be recorded (“receiver blind spot”), resulting in a second area of missing data in the experimental field distribution. Besides maps of transmit amplitudes |equation imagemath image(equation image)|, we also compared maps of measured and simulated signal amplitudes |equation imagemath image(equation image)(equation imagemath image(equation image))*| (not shown) and signal phases ψ(equation image) (Fig. 3) in the central axial plane of the phantom. In the depicted situation, also only one antenna (CSA-1) was used for transmitting and receiving. Good agreement between measured and simulated signal amplitudes and phases was achieved in each case. It follows from the results presented in this section that our FDTD modeling of the four CSA coil array and the phantom was validated quantitatively.

Figure 2.

Top row: measured (a) and simulated (b) maps of | equation imagemath image(P0;equation image)| generated by four-element head coil in central transversal slice of cylindrical agarose phantom. c: Simulated map of |equation imagemath image(P0;equation image)|. Antennae indicated by bars. Only CSA-1 was driven at a reference RF power of P0=1 kW, remaining antennae terminated into 50 Ω. Simulated data multiplied by 1.08 for easy comparison with experimental values. Bottom row: simulated amplitudes |equation imagemath image(P0;equation image)| (curves, scaling factor 1.08) and measured data (dots) along horizontal (d, from left to right) and vertical (e, from bottom to top) lines indicated in (a) and (b).

Figure 3.

Measured (left) and simulated (right) signal phases ψ(equation image)=arg{equation imagemath image(equation image)(equation imagemath image(equation image))*} (in degrees) in spin echo images of central axial plane of phantom, with first and second RF pulse delivered by CSA-1. The slice center was arbitrarily assigned the phase ψ(0)=0.

Temperature Distributions in Cylindrical Phantom

Since we are interested in the local SAR, we compared simulated and measured distributions of the temperature rise ΔT(equation image) following CW RF heating of the cylindrical agarose phantom. In each case, 37.5 W of RF power were applied for ΔtRF = 15 min to each of the two antennae CSA-1 and CSA-2, while CSA-3 and CSA-4 were terminated into 50 Ω. Figure 4 shows two repetitions of this experiment, varying only the relative phase Δϕ of the RF currents driving the two active antennae. As can be seen, excellent agreement between simulated (Fig. 4b and d) and measured (Fig. 4a and c) temperature distributions was achieved. For a quantitative comparison, Figs. 4e and f show cross-sections along the (vertical) y-axis and (horizontal) x-axis, respectively, of the Δϕ = 90° temperature maps (Fig. 4c and d). Inside the phantom, measured temperature differences (dots) and theoretical data (solid curves) agree within experimental error limits. Surface temperatures, especially in the vicinity of an active antenna, are expectedly overestimated by the simulation, as heat losses to the environment were neglected. Although the same RF input power was applied, the temperature distributions ΔT(equation image), and hence the local SAR(equation image), for the two driving conditions of Fig. 4 differ conspicuously. They also show features that presumably could not have been easily predicted (like the counter-intuitive hot zone far away from the driving antennae for (Δϕ = 0°).

Figure 4.

ad: Maps of temperature rise ΔT(equation image) = T(equation image)−T0 over initial (ambient) temperature T0 in central axial plane of agarose phantom after RF heating (37.5 W of CW-RF power applied for 15 minutes to each of two CSA). Phase differences between RF currents of 0° (a,b), and 90° (c,d), the remaining two antennae terminated into 50 Ω. Temperature distributions measured (a,c) by PRF method (see MR measurements) and obtained from FDTD simulations (b,d) neglecting thermal diffusion and heat losses to the environment. e,f: Simulated (curves) and measured (dots) temperature profiles along vertical (e, from bottom to top) and horizontal (f, from left to right) lines indicated in (c) and (d).

Worst-Case Analysis of Local SAR in Cylindrical Phantom

It follows from the preceding section that for multichannel transmit/receive arrays the local SAR can vary strongly with RF driving conditions. This means that compliance with global SAR limits alone no longer represents a sufficient safety measure. Applying the same CW-RF power of Pk = 250 W to each of the four CSA of the array, we simulated maps of local SAR(equation image) and |equation imagemath image(equation image)| in the central axial plane of the agarose phantom according to Eqs. [1], [4], and [5]. By systematically varying the RF phases of all other antennae relative to CSA-3 in steps of 15°, a total of 243 different settings were sampled (27). In Fig. 5 we plot the maximum SARmax = max{SAR(equation image)} of the local SAR vs. the maximum |equation imagemath image|max = max{|equation imagemath image(equation image)|} of the transmit amplitude obtained for the same driving conditions, including the results of all 13,824 phase settings. For comparison Fig. 5b shows a plot of SARmax vs. khom = mean{|equation imagemath image(equation image)|}/max{|equation imagemath image(equation image)|}, a dimensionless quantity indicating the homogeneity of each field map. Two results have been singled out in Fig. 5, a “birdcage” mode with phase differences of 90° between neighboring antennae and the “worst case” with the highest local maximum SARmax. For illustration, maps of |equation imagemath image(equation image)| and SAR(equation image) for these two special cases are displayed in the bottom row of the figure. As can be seen from Fig. 5a and b, peak SAR is not obviously correlated to |equation imagemath image| homogeneity, while such a relationship exists with |equation imagemath image|max. Nevertheless, for different driving conditions a variety of SARmax values may correspond to the same maximum |equation imagemath image|max and vice versa. At |equation imagemath image|max=40 μT, for example, SARmax spans the range from about 350–1200 W/kg; i.e., varies by more than a factor of three.

Figure 5.

a: Scatter plot representing maximum values SARmax and | equation imagemath image|max of SAR and transmit amplitude in central axial plane of agarose phantom. Data obtained from SAR(equation image) and |equation imagemath image(equation image)| maps simulated for same RF power of 250 W for each element of coil array. Phase differences between RF currents driving CSA-3 (at bottom) and any of the other antennae were systematically varied in steps of 15°. b: Scatter plot of SARmax vs. dimensionless homogeneity qualifier khom=mean{|equation imagemath image(equation image)|}/|equation imagemath image|max showing no clear correlation between these two quantities. Bottom row: Simulated maps of transmit-active amplitude |equation imagemath image(equation image)| (to the left of color bar) and SAR(equation image) (to the right of color bar) for two particular RF driving conditions, labeled “birdcage” (c, blue circle) and “worst case” mode (d, red triangle, highest value of SARmax); transmit amplitudes are given in μT, SAR values in W/kg.

The maximal values | equation imagemath image|max in the diagram vary by about a factor of 2.7 and SARmax by a factor of more than 15, even though each antenna was always fed by the identical input power. The SARmax value of the “worst case” is about three times higher than that of the “birdcage mode.” As can be seen in Fig. 5d, this “worst case” is characterized by an SAR distribution almost entirely concentrated in the vicinity of CSA-2, located at the right hand side of the array; the corresponding transmit amplitude exhibits an almost linear gradient towards this antenna. In the “birdcage mode” (Fig. 5c), on the other hand, the SAR is enhanced close to each of the antennae and the transmit amplitude is relatively constant over the entire slice.

The SARmax and | equation imagemath image|max values given in the preceding paragraphs always refer to one FDTD voxel containing about 125 mg of tissue. We kept this resolution as we are interested in the most accurate description of coil arrays. Seeking approval from authorities, one would average the SAR data for a given coil assembly over 10 g of tissue, say a sphere of radius 13 mm. The effect would be marginal, however, as all features in the field maps of Figs. 2, 4, and 5 are much larger than this size (the electromagnetic wavelength in water is about 27 cm at 125 MHz).

DISCUSSION

According to the IEC 60601-2-33 standard the limiting value for the local SAR amounts to 10 W/kg and the limit for the global SAR in the head is 3.2 W/kg. In the case of a homogeneous B1 field distribution, compliance with the global SAR limit will ensure that the local SAR will be below the 10-W/kg limiting value (considering only concomitant electric fields generated by the oscillating B1 field and assuming that electric stray fields from lumped elements on the coil may be neglected). Therefore, to our knowledge, the time-averaged RF power transmitted to the coil is monitored and limited in present commercial scanners in order to fulfill both SAR limits and thus comply with the standard. Because of the new capabilities and failure scenarios of PTX arrays this safety concept based on a homogeneous B1 field distribution is no longer adequate and needs to be modified (28).

Based on the results reported above we suggest carrying out the described worst case analysis on a homogeneous phantom by systematically varying relative amplitudes and phases of the RF currents driving the elements of the multicoil array such that the total power delivered to the entire array is always the same. In practice, the total RF power delivered to the array should then be limited to a value ensuring that even for the worst possible case the maximum local SAR falls below the 10-W/kg limit. To validate the simulations for at least a few different driving conditions, this analysis involves MRI experiments, ideally of all field quantities accessible in this way.

Applying the 3.2-W/kg limit for global head SAR to our phantom of 5.4 kg, a typical mass of an adult human head, time-averaged total RF powers of up to Pmath image = 17.3 W would be permissible. Locally, however, we observe for our four-element CSA array a maximal SAR of SARmax≈1430 W/kg (“worst case” point in Fig. 5) at Ptot = P0 = 1 kW, which means a much lower limit Pmath image = 7.0 W is needed, to comply with the 10-W/kg limit for local SAR. This situation is not much changed if we apply 10-g averaging to our SAR data, yielding SARmath image≈1200 W/kg and Pmath image≈8.3 W. In summary, these results emphasize once again the shift from global toward local SAR as the limiting factor for compliance with safety regulations for PTX arrays.

Although not investigated in the present work, we do propose that relative RF amplitudes should also be systematically varied in such a fashion that the total RF power delivered to the array is kept constant. This is a straightforward extension of the phase variations reported in this work and does not require additional FDTD calculations.

Another sincere drawback of traditional approaches toward coil safety in MRI is not resolved by our new proposal for PTX arrays: the specific problems arising from the complicated and nonuniform geometries of the human body and from the inhomogeneities of biological tissue. It is well known that local | equation image1(equation image)| and thus SAR(equation image) can be substantially enhanced at or near tissue interfaces. To address this issue, detailed and reliable numerical dielectric models of the human body in vivo are needed, covering a reasonable range of anatomical variations.

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