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Keywords:

  • EPT;
  • conductivity;
  • permittivity;
  • high-field MRI

Abstract

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. THEORY
  5. METHODS
  6. RESULTS
  7. DISCUSSION
  8. REFERENCES

Purpose

To investigate the effect of magnetic field strength on the validity of two assumptions (namely, the “transceive phase assumption” and the “phase-only reconstruction”) for electrical properties tomography (EPT) at 1.5, 3, and 7T.

Theory

Electrical properties tomography is a method to map the conductivity and permittivity using MRI; the B1+ amplitude and phase is required as input. The B1+ phase, however, cannot be measured and is therefore deduced from the measurable transceive phase using the transceive phase assumption. Also, earlier studies showed that the B1+ amplitude is not always required for a reliable conductivity reconstruction; this is the so-called “phase-only conductivity reconstruction.”

Methods

Electromagnetic simulations and MRI measurements of phantoms and the human head.

Results

Reconstructed conductivity and permittivity maps based on B1+ distributions at 1.5, 3, and 7T were compared to the expected dielectric properties. The noise level of measurements was also determined.

Conclusion

The transceive phase assumption is most accurate for low-field strengths and low permittivity and in symmetric objects. The phase-only conductivity reconstruction is only applicable at 1.5 and 3T for the investigated geometries. The measurement precision was found to benefit from a higher field strength, which is related to increased signal-to-noise ratio (SNR) and increased curvature of the B1+ field. Magn Reson Med 71:354–363, 2014. © 2013 Wiley Periodicals, Inc.


INTRODUCTION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. THEORY
  5. METHODS
  6. RESULTS
  7. DISCUSSION
  8. REFERENCES

Electrical properties tomography (EPT) is a recently introduced technique to reconstruct conductivity and permittivity from measurable radio frequency (RF) magnetic field maps [1]. This quantitative MRI method was already anticipated by Haacke et al. in 1991 [2]. The EPT reconstruction relies on the propagation of the B1+ field (the left-handed rotating component of the transmit RF field) which is affected by the spatial dielectric arrangement. At high field strengths, the interaction between the electromagnetic fields and the dielectric properties leads to well-known B1+ inhomogeneities [3]. This effect may lead to loss of image quality in conventional MRI, but is in fact used for EPT.

Already several applications for EPT were investigated. In oncological research, it was shown that the conductivity in gliomas is higher than in normal tissue [4, 5], which was already observed earlier in an ex vivo study [6]. Also in breast tumors, an elevated conductivity was reported [7, 8], which was shown earlier for ex vivo measurements [9]. In gastrointestinal research, a fast sequence was used to monitor conductivity of stomach contents during the intake of a fluid [10].

EPT may also be used to assess RF safety [1]. In this context, the specific absorption rate is used as a surrogate to assess the risk of patient heating. A model of local dielectric properties is required to calculate the specific absorption rate. The dielectric properties are typically based on ex vivo measurements [e.g., [11, 12], combined with an MRI-based mode of healthy volunteers [13]. In patients, however, the anatomy and dielectric properties can change considerably, e.g., due to tumor growth. For some tumor types the dielectric properties increases with 10–650%, [6, 9]. Nonpatient-specific dielectric properties and models can potentially lead to an underestimation of the specific absorption rate in those patients.

For EPT, several reconstruction methods have been proposed, e.g., methods based directly on the Maxwell Equations [1], or on the Helmholtz wave equation [14-16]. For all methods, reconstruction of the dielectric properties is based on the B1+ amplitude and phase measurements. The B1+ phase cannot directly be obtained from an MRI measurement; instead it is assumed that the transmit and receive phase contribute equally to the measurable transceive phase. This so-called “transceive phase assumption” is based on the reciprocity principle, which states that antenna properties are interchangeable between transmit and receive [17]. In an MRI experiment, however, this relation is complicated by the use of the rotating frame: the left-handed frame corresponds to the excitation of the spins (RF transmission), the right-handed rotating frame is required for optimal receive sensitivity (RF reception) [17]. Therefore, the transceive phase assumption will only hold for linear polarization and under certain conditions for imperfect quadrature excitation with reversal of quadrature combination of the two ports of a birdcage coil in transmit and receive [16]. In the latter case, a B1 field is produced in receive that mimics the B1+ field in transmit.

Studies at 1.5T [15] and 7T [16] showed the possibility of reconstructing conductivity based only on the transceive phase (phase-only conductivity reconstruction). This reconstruction does not require a B1+ amplitude measurement; which shortens the total measurement time considerably. In previous studies, the “phase-only reconstruction” seemingly caused larger errors at 7T than at 1.5T [15, 16]. However, the effect of dielectric properties on the transmit field is more pronounced at higher field strengths, which may lead to a higher precision of reconstructed EPT maps.

The aim of this study is to evaluate the effect of static magnetic field strength on the accuracy and precision of EPT. For this purpose, a systematic evaluation of the transceive phase assumption, the phase-only conductivity reconstruction, and SNR at 1.5, 3, and 7T was conducted. This evaluation is based on simulations and measurements of the human brain and a small phantom. Preliminary results already showed that the transceive phase assumption leads to larger errors at higher field strengths but also that the SNR increases considerably [18]. For the phantom, it was therefore investigated at which frequency the accuracy of the assumptions and the SNR are balanced.

THEORY

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. THEORY
  5. METHODS
  6. RESULTS
  7. DISCUSSION
  8. REFERENCES

In this work, the homogeneous Helmholtz equation was used for reconstruction of the dielectric properties from the B1+ amplitude and phase [16]:

  • display math(1)

where B1+ is the complex transmit field, k is the wave number, µ the magnetic permeability, ɛ0 is the permittivity in free space, ɛr is the relative permittivity, σ is the electrical conductivity, and ω is the angular frequency. Here, the angular frequency is equal to the Larmor frequency, and thus, is field strength dependent. The magnetic permeability µ in the body is assumed to be constant and equal to the permeability in free space.

The relative permittivity and the conductivity can be deduced from Eq. (1), as these parameters affect the real and imaginary part of k2, respectively:

  • display math(2a)
  • display math(2b)

The homogeneous Helmholtz equation is derived from the Maxwell equations under the assumption of homogeneous and isotropic dielectric properties; therefore, the equation is only valid in regions where those properties are piecewise constant.

Transceive Phase Assumption

In Eq. (2a), the complex B1+ is required for reconstruction of the dielectric properties. Several methods are available to map the B1+ amplitude, e.g., [19-22]. The B1+ phase (transmit phase), however, cannot be measured directly, as the RF phase contribution obtained in an MR experiment is affected by both the transmit (φ+) and the receive phases (φ) [17].

Based on the reciprocity principle, one would be tempted to argue that the transmit and receive phase are equal, and can thus easily be disentangled by:

  • display math(3)

In an MRI experiment, however, the reciprocity principle must be applied with care, as the transmit and receive process is performed in a different rotating frame [17, 23]. The transmit field is given by B1+, which is different from the receive field B1:

  • display math(4a)
  • display math(4b)

From Eq. (4a) and (4b), it follows that the transmit and receive phase of a linearly polarized field are equal, thus Eq. (3) can be used to obtain the transmit phase from the transceive phase.

In practice, the RF magnetic field is disturbed by the scanned object. For example, a linearly polarized field incident on a lossy object will induce scatter fields. Those scatter fields interfere with the incident field, perturbing the net polarization in the object. However, the contribution of scatter fields to the B1+ and B1 are canceled, in case two orthogonal linear fields are driven in quadrature mode and are incident on a cylinder. This was shown analytically [16, 24] and numerically [23]. This type of RF excitation and reception is used for a quadrature birdcage coil.

Therefore, Eq. (3) can be used to retrieve the transmit phase even if scatter fields are induced. In this work, the assumption is used that this approach is valid in all objects; we called this the transceive phase assumption. However, for an elliptical geometry, e.g., the head, the scatter fields are not completely canceled. Therefore, it is expected that the validity of the transceive phase assumption is reduced for those geometries. From Faraday's law of induction, it furthermore follows that these scatter fields (or induced currents) increase with the Larmor frequency. Thus, the applicability of the transceive phase assumption is hypothesized to be further reduced at high field strengths.

Phase-Only Conductivity Reconstruction

Recently, it was shown that knowledge of the full complex B1+ field is not always needed to reconstruct the conductivity [5, 15]. With the so-called phase-only conductivity reconstruction, only φ+ is required for reconstruction:

  • display math(5)

An advantage of the phase-only conductivity reconstruction is that it only requires a transceive phase measurement. This reconstruction is only valid under the condition that [16]:

  • display math(6)

Thus, the curvature of the B1+ amplitude should be minimal for Eq. (5) to be valid. Expressed in terms of dielectric properties, it was found that the accuracy of Eq. (5) reduces if ɛ0ɛrω grows larger than σ [15]. These requirements are generally more applicable to the situation at 1.5T than at 7T. Under similar conditions, expressions like Eqs. (5) and (6) to B1 and the conductivity can be found even if the transceive phase assumption is not valid, by simply adding Eq. (5) to its counterpart for φ and dividing by two [15].

METHODS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. THEORY
  5. METHODS
  6. RESULTS
  7. DISCUSSION
  8. REFERENCES

Simulations of a human head placed in a head birdcage coil were performed to assess the effect of field strength on the accuracy of EPT. Also, the effect of object symmetry was investigated. For this purpose, a phantom was used with the approximate dimensions of the human head. In this phantom, an extra compartment was placed which mimicked a lesion. This compartment was placed at two different locations within the phantom. Furthermore, the effect of dielectric properties (conductivity and permittivity) was investigated, as these influence the induced currents. Finally, the simulations were validated by comparing simulations and measurements of the human head and an elliptical phantom with an off-axis inner compartment. The phantom measurements were also used to determine the noise level of the EPT reconstructions at the different field strengths.

Simulations

Finite-difference time-domain (FDTD) simulations were used to assess the transceive phase assumption and the phase-only conductivity reconstruction. Simulations have three advantages over measurements. First, simulations offer the possibility to disentangle the B1+ phase from the transceive phase, which is not possible in measurements. Second, they render a quantitative comparison between the reconstructed and known input dielectric properties. Third, simulations are unaffected by thermal noise, which is field-strength dependent.

A realistic model of a high-pass birdcage coil was used for the simulations (based on the 7T T/R Nova Medical coil, ΦRFshield=0.37 m, Φend ring=0.3 m, # rods=16, type: high-pass birdcage coil; the actual coils at 1.5 and 3T had similar dimensions as the 7T coil). The coil was tuned by adjusting the capacitor values, such that the coil was resonant at 64, 128, or 298 MHz. The coil was fed at four ports in the top end ring. Only the quadrature mode was excited. A model of a human head (Ella, Virtual Family, [13]) or a phantom was placed in the coil (see Fig. 1). The dimensions of the phantom were the same as the measured phantom (see Materials section). Additionally, simulations were performed with different dielectric properties (as listed in Table 1).

image

Figure 1. Overview of simulation set-up. a: Birdcage head coil loaded with a dielectric model of the head. b: Elliptical phantom with two compartments, here the inner compartment is placed off-axis. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Table 1. Dielectric Properties Used for the Comparison Study (Figs. 3 and 4), the Measured Phantom (Figs. 5 and 6), and the Human Head Simulations (Figs. 8 and 9)
  Outer compartmentInner compartmentComments
CaseField strengthσ (S/m)εrσ (S/m)εr 
  1. Cases 1, 2, and 3: Average dielectric properties of the head at 1.5T [12] with average tumor properties at 1.5T in the inner compartment [6]. Case 4: Average dielectric properties of the brain at the individual field strengths [12]. Head: head anatomy (Ella, [13] with tissue and RF-specific dielectric properties based on [12]. Measured phantom: dielectric properties of the phantom were verified by an impedance probe measurement. The dielectric properties did not vary with field strength.

1All0.40830.6288Phantom: inner compartment off-axis
2All0.40830.6288Phantom: inner compartment on long axis
3All0.4083  Phantom: no inner compartment
41.5T0.4083
 3T0.4663  
 7T0.5552
Head    Dielectric properties based on Ref. [11], depend on B0 field strength.
Measured phantomall0.58800.4358Dielectric properties of the phantom were verified with impedance probe.

The magnetic field distributions were simulated with SEMCAD (SEMCAD X, SPEAG, Available at: www.speag.com), stability of all simulations was verified. The irregular grid of the simulations was resampled using spline interpolation to obtain a regular grid with a voxel size of 2.5 × 2.5 × 5 mm3, which was used for further analysis. The simulation grid in the phantom and the head anatomy was slightly irregular due the required higher accuracy for gridding of the capacitors in the RF coil.

Materials

The following scanner types were used for the phantom and in vivo scans: Achieva 1.5T (Philips HealthCare, Best, The Netherlands), Achieva 3.0T (Philips HealthCare, Best, The Netherlands), and Achieva 7.0T (Philips HealthCare, Cleveland, OH). Quadrature birdcage coils were used for transmission and reception (1.5 and 3T: Quadrature Headcoil Philips HealthCare, Best, The Netherlands, 7T: T/R birdcage coil, Nova Medical, Inc., Wilmington, MA).

An elliptical phantom was constructed for the phantom measurements. An elliptical shape was chosen, as this resembles the geometry of the head. Moreover, this shape is hypothesized to create deviations in the transceive phase assumption (see Theory section). The phantom consisted of two compartments; the outer compartment was elliptically shaped (dmajor=19 cm, dminor=16 cm), the inner compartment was circular and placed off-center to break symmetry (d=5.4 cm). To obtain the desired dielectric properties, the following solutions were used: outer compartment: H2O, 3.3 g/L NaCl, inner compartment: 66 vol % H2O, 33 vol % 2-propanol, 6.6g/L NaCl. The dielectric properties were verified using an impedance probe (85070E, Agilent Technologies, Santa Clara, CA): σouter=0.58 S/m, ɛr,outer=80, σinner=0.43 S/m, ɛr,inner=58.

All volunteer experiments were performed with the same person (male, 34 years old, written informed consent obtained). The volunteer's head was fixated in the coil using cushions to minimize movement artifacts.

Measurements

Two measurement types are required for EPT: one to map the B1+ amplitude and another to determine the transceive phase. B1+ mapping was performed using the actual flip-angle imaging method [20] at 3 and 7T (TR1/TR2=50/340 ms, TE=1.64 ms). At 1.5T, the double angle method [19] was used (α1/α2=60°/120°, TR=5000 ms, TE=1.25 ms). For the transceive phase measurement, a spin echo sequence was used to avoid phase contributions from off-resonance effects [1], which was corrected for eddy currents by measuring twice with opposing gradients (TR=1200 ms, TE=5.3 ms [1.5T] or 5.9 ms [3 and 7T]). To enable SNR comparison, the acquired scan resolution and scan time were identical at all field strengths (2.5 × 2.5 × 5 mm, 22 slices).

EPT Reconstruction

The input data for the EPT reconstruction are either complex magnetic field or phase distribution, depending on the reconstruction type, i.e., complex B1+, transceive phase assumption, or phase-only conductivity reconstruction. An overview on the different reconstruction types and the required input data are given in Table 2. For example, a reconstruction with the transceive phase assumption requires the amplitude of the B1+ field and the transceive phase distribution. The transceive phase is constructed using Eq. (3), which is based on the argument of the B1+ and B1 fields.

Table 2. Input data for different EPT reconstructions
 Reconstruction typeEquationsInput data
B1+|B1+|arg(B1+)=φ+arg(B1)=φφ±
  1. For the “transceive phase assumption” and the “phase-only conductivity reconstruction” the phase is based on the transceive phase assumption (Eq. [(3)]), which is a combination of the transmit (φ+) and receive phase (φ). The transceive phase is directly obtained in measurements. For the simulations, the argument of the B1+ and B1 field are used to generate a transceive phase map.

SimulationsComplex B1+[2]+    
Transceive phase assumption[2], (3) +++ 
Phase-only conductivity reconstruction(3), [4]  ++ 
MeasurementsTransceive phase assumption[2], (3) +  +
Phase-only conductivity reconstruction(3), [4]    +

Once the desired dataset is formed, reconstruction is based on the Laplacian (Eqs. (2a) and (5)), sometimes followed by a pointwise matrix division (Eq. (2a)). The Laplacian of a gridded dataset can easily be calculated by convolving the field with an appropriate kernel. The kernel used in this (size 7 × 7 × 5 voxels) is the same as the one as described in Ref. [16]. The EPT reconstruction algorithm is implemented in Matlab (MathWorks, Natick,MA).

Postprocessing

The difference between the simulated φ+ and φ±/2 was mapped to inspect the error in the transceive phase assumption. The effect of this error on the measured dielectric properties was visualized by reconstructing the dielectric properties based on the complex B1+ or |B1+| and φ±/2 according to Eq. (2a) and (2b). The conductivity was also reconstructed using the phase-only reconstruction (Eq. (5)).

To evaluate the effect of field strength on the reconstructed conductivity and permittivity, three parameters were quantified: the systematic reconstruction error, the random reconstruction error, and the noise level.

Systematic Reconstruction Error

The systematic reconstruction error is the difference between the average conductivity (or permittivity) in the outer compartment and the expected conductivity (or permittivity) normalized with the conductivity (or permittivity) as measured with the dielectric probe. This parameter is based on the simulation results of the measured phantom geometry.

Random Reconstruction Error

The random reconstruction error is defined as the standard deviation of the reconstructed conductivity (or permittivity) normalized by the conductivity (or permittivity) as measured with the dielectric probe. This parameter is based on the simulation results of the measured phantom geometry.

Noise Level

The noise level of the conductivity and permittivity map was quantified using a method based on a correlation function [25, 26]. The correlation of a pixel with neighboring pixels is defined by the cross-correlation between the original image and shifted images. The method assumes that the signal intensity is constant over a few voxels, whereas the noise in those voxels is uncorrelated. Therefore, the cross-correlation between neighboring pixels of a noisy and noise-free image is equal. Only the autocorrelation between a pixel and itself (i.e., the zero-offset auto-correlation) is affected by noise. By estimating the difference between the measured and noise-free zero-offset autocorrelation function, the standard deviation of the noise could be calculated. By scaling the standard deviation with the expected conductivity or permittivity the noise level was obtained. The noise level is based on measured EPT maps.

RESULTS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. THEORY
  5. METHODS
  6. RESULTS
  7. DISCUSSION
  8. REFERENCES

In FDTD simulations of the head, it was observed that the B1+ amplitude and the transceive phase distributions are strongly affected by the field strength (Fig. 2). The inhomogeneity of the B1+ amplitude, expressed by the minimum and maximum B1+ amplitude as the percentage of the mean B1+ amplitude, increased from 95–112% at 1.5 T to 63–151% at 7T in a central transverse slice in the head. For the transceive phase, however, the field shape was similar at all field strengths. Only the curvature of this field was stronger for increased field strengths.

image

Figure 2. Simulated B1+ amplitude (|B1+|) and transceive phase (φ±/2) in the head at 1.5, 3, and 7T. The B1+ amplitude was normalized to the maximum B1+ in the slice. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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image

Figure 3. Comparison study conductivity reconstruction (simulation data): field strength, dielectric properties, and asymmetry. For detailed description of the used dielectric properties, see Table 1. Phase error is the difference between φ±/2 and φ+. Equation [2b] is used to test the effect of the “transceive phase assumption” on the conductivity reconstruction, and Eq. (6) is the “phase-only conductivity reconstruction.” For visualization, the same range of conductivity values is shown for all reconstructions. This led to clipping for some 7T results.

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image

Figure 4. Comparison study permittivity reconstruction (simulation data): field strength, dielectric properties, and asymmetry. For detailed description of the used dielectric properties, see Table 1. Equation (2a) is used to test the effect of the phase error on the permittivity reconstruction.

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image

Figure 5. Phantom simulations and measurements. Top row: Conductivity reconstruction based on the simulated complex B1+ (simulations only). Middle rows: The reconstructed conductivity based on the B1+ amplitude and the transceive phase for simulations and measurements. Bottom row: The reconstructed conductivity based on the “phase-only reconstruction” for simulations and measurements. For an overview of the used dielectric properties, see Table 1.

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image

Figure 6. Phantom measurement and simulations. Top row: Reconstructed relative permittivity (εr) based on complex B1+(simulations only). Bottom rows: Reconstructed relative permittivity (εr) based on the “transceive phase assumption.” [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Simulation Study

In Figure 3, an overview is given of the effect of variations in the phantom setup on the transceive phase assumption and the phase-only conductivity reconstruction. In that figure, it can be observed that the transceive phase error depends on the field strength, dielectric properties, and symmetry. For example, in case 3, where the dielectric properties are constant for all field strengths, the standard deviation of the phase error (φ±/2−φ+) was 0.19°, 0.21°, and 1.59° for 1.5, 3, and 7T, respectively. Choosing the correct average dielectric properties of the head per field strength, it can be observed that the error at 7T is tempered (case 4). The standard deviation of the phase error (φ±/2−φ+) in this case was 0.19°, 0.18°, and 0.61° for increasing field strength.

Both these cases (3 and 4) had symmetry axes along the long and short axes of the ellipse; along these axes, an antisymmetric pattern in the phase error was observed. The head and case 2 had only a left–right symmetry. In these cases, the phase error only exhibited an antisymmetric pattern over that symmetry axis. For a completely broken symmetry (case 1), the antisymmetric pattern in the phase error was no longer present.

These phase errors are directly related to the transceive phase assumption (Eqs. (2a) and (3)); i.e., at locations where a large phase error was observed, and therefore, the transceive phase assumption was not valid, also a large error in the conductivity reconstruction was observed. Moreover, the use of field strength dependent dielectric properties (compare cases 3 and 4) also decreased the reconstruction error. Furthermore, the antisymmetric pattern could also be observed in the reconstruction of cases 2, 3, and 4.

Using only the phase for the conductivity reconstruction (Eq. (5)) did not affect the result at 1.5T. However, it slightly degraded results at 3T. Especially at the periphery of the phantom an error emerged. At 7T, this approach led to a large error in all simulated cases. Also at this field strength, the error was most prominent at the periphery of the phantom and the head.

For the permittivity reconstruction, a higher field strength led also to a larger error in the permittivity reconstruction (Fig. 4, case 3). This error was tempered with the use of field strength-specific dielectric properties (case 4). Effects of asymmetry were also observed in the permittivity reconstruction (compare cases 1, 2, and 3).

Phantom Measurements

The simulations were validated with measurements of the head-shaped phantom at 1.5, 3, and 7T (Fig. 5). For this specific case, simulations were performed with the same dielectric properties as measured in the phantom with the probe measurement.

Also for this setup, the transceive phase assumption (Eq. (2b)) led to the largest error at 7T. These errors occurred mainly at the periphery of the phantom. The error pattern in the measurements was similar to the simulations. At 1.5 and 3T, high noise levels dominated the measurements.

The phase-only conductivity reconstruction (Eq. (6)) on simulation data at 1.5T resulted in a correct reconstruction of the conductivity. At 3T, the reconstructed conductivity was higher than the input conductivity, especially at the periphery. However, the contrast between the inner and outer compartment could still be observed. At 7T, the phase-only conductivity reconstruction resulted in an unreliable reconstruction: a large error was observed such that the contrast between the inner and outer compartment could no longer be observed. The error coincided with large gradients in the B1+ amplitude.

Comparison between measurement and simulation data at 1.5T was hindered by the high noise level (Fig. 6). At 3T, also the overestimation of the conductivity at the periphery could not be observed in the measurements due to noise. Only at 7T, the noise level was sufficiently low to enable direct visual comparison between the measurement and the simulation. In both images, a large overestimation at the periphery was observed and the contrast difference between the inner and outer compartment was no longer visible. This indicates that the transceive phase assumption is not valid at this field strength.

To improve the comparison of the measurements with the simulations, and to quantify the effect of the transceive phase assumption and the phase-only conductivity reconstruction, the average conductivity and permittivity as observed in the simulations and measurements were summarized (see Table 3). For the simulations also the standard deviation of the reconstructed dielectric properties is given. These simulation results show that the “transceive phase approximation” led to a minimal error at 1.5 and 3T, as the observed average conductivity value was equal to the input conductivity, and the range of observed conductivity values in the simulations—given by the standard deviation—was limited. Interestingly, the observed average value at 7T deviated only slightly from the input conductivity (0.01 S/m). However, at that field strength, the standard deviation was much larger, meaning that the spread of observed conductivity values was increased.

Table 3. Average Reconstructed Conductivity (σ) and Permittivity (εr) in the Inner and Outer Compartment of the Phantom at 1.5, 3, and 7T Using the “Transceive Phase Assumption” or the “Phase-Only Conductivity Reconstruction” for Simulations (sim.) and Measurements (meas.)
   Probe1.5T3T7T
   σ (S/m)εr (−)σ (S/m)εr (−)σ (S/m)εr (−)σ (S/m)εr (−)
  1. For the simulation data, also the standard deviation of the reconstructed conductivity and permittivity data is given.

 Transceive phase assumptionSim.  0.58±0.00480±0.70.58±0.0180±10.56±0.280±9
Meas.  0.58590.65450.6177
OuterPhase-only reconstructionSim.0.58800.56±0.01 0.64±0.04 1.56±0.7 
 Meas.  0.59 0.71 1.16 
 Transceive phase assumptionSim.  0.43±0.00258±0.40.43±0.00259±0.10.42±0.0158±0.5
 Meas.0.43580.3960.44570.4560
InnerPhase-only reconstructionSim.  0.43±0.004 0.44±0.003 0.71±0.1 
 Meas.  0.38 0.44 0.62 

For the phase-only conductivity reconstruction, the observed mean and standard deviation at 1.5T were only mildly affected. The observed average and standard deviation at 3T was slightly higher. At 7T, the observed mean conductivity in the outer compartment was 2.8 times higher than the input conductivity. Moreover, the standard deviation showed a very large spread in conductivity values.

The observed mean conductivity values in measurements exhibited the same behaviour as the simulated values. At 1.5T, the conductivity reconstructed using EPT was almost equal to the probe measurement, whereas at 3 and 7T, the reconstructed value only slightly overestimated the probe measurement. For the measurements, no standard deviation was given, as this feature is additionally affected by the field strength dependent noise levels as is discussed below.

Permittivity reconstruction of the simulation data showed to be reliable at all field strengths. However, the standard deviation in the simulations was elevated at higher field strengths. The permittivity reconstruction of the outer compartment with measured data deviated considerably from the actual permittivity: 26 and 44%, at 1.5 and 3T, respectively. At 7T, the mean permittivity was more accurate (4% deviation). For the inner compartment, smaller deviations were found 1 and 3% at 3 and 7T, respectively. However, for 1.5 T, a considerably larger deviation was found (90%).

A clear decrease of the noise level with increasing field strength was observed in the measured conductivity and permittivity maps (Figs. 5 and 6, respectively). The noise level was quantified by the standard deviation of the noise in the outer compartment (Fig. 7). The noise level of the conductivity maps was lower than of the permittivity maps at all investigated field strengths. Furthermore, the noise level dropped for increasing field strength. For the conductivity map, the noise level was 3.8 times lower at 7T compared to 1.5T. For the permittivity map, the noise level was 3.1 times lower. Besides the error caused by noise, also the systematic and reconstruction error are summarized in Table 3 and Figure 7.

image

Figure 7. Errors in the EPT reconstruction of the outer compartment of the phantom for different reconstruction methods and field strengths. Three error terms were separated: the systematic reconstruction error, the random reconstruction error, and the noise level. The first two terms are based on simulation results, the latter one on measurements. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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image

Figure 8. In vivo simulations and measurements. Top row: Conductivity reconstruction based on the complex B1+ (simulations only). Middle rows: The reconstructed conductivity based on the B1+ amplitude and the transceive phase for simulations and measurements. Bottom rows: The constructed conductivity based on the “phase-only reconstruction” for simulations and measurements. Because of boundary artifacts, some features of slices below or above the shown slices can appear in the reconstruction. Therefore, e.g., the ventricles appear slightly larger in the reconstruction. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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image

Figure 9. In vivo measurement and simulations. Reconstructed relative permittivity (εr) based on complex B1+ or the transceive phase assumption for simulations and measurements. The complex B1+ results are based on simulations only. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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In Vivo Measurements

In vivo conductivity maps were obtained and compared with simulation results (Fig. 8). A left/right antisymmetry in the transceiver phase error was present at all field strengths. The transceive phase error in the head was minimal at 3T and maximal at 7T (see Fig. 3). It is explained in the Discussion section why the transceive phase error is minimal at 3T. The reconstructed in vivo conductivity was only mildly affected by his error.

For the phase-only conductivity reconstruction, the observed conductivity at 1.5 and 3T was only slightly affected by omitting the B1+ amplitude (see Table 4). Only the conductivity distribution at 7T was altered. At this field strength, the reconstructed conductivity was elevated, although the contrast between white matter, gray matter, and CSF was preserved. The simulation results were confirmed by measurements (Fig. 8).

Table 4. Average and Standard Deviation of the Reconstructed Conductivity (σ) and Permittivity (εr) in the Head (Simulation Data) at 1.5, 3, and 7T Using the “Transceive Phase Assumption” or the “Phase-Only Conductivity” Reconstruction
  1.5T3T7T
  σ (S/m)εr (−)σ (S/m)εr (−)σ (S/m)εr (−)
WMInput0.29680.34530.4144
Transceive phase assumption0.30±0.0768±50.35±0.0853±40.43±0.244±6
Phase-only reconstruction0.31±0.07 0.43±0.09 0.80±0.3 
GMInput0.51970.59740.6960
Transceive phase assumption0.51±0.0997±80.58±0.173±40.71±0.260±8
Phase-only reconstruction0.51±0.09 0.69±0.1 0.92±0.4 
CSFInput2.07972.14842.2273
Transceive phase assumption2.01±0.2596±82.09±0.483±82.07±0.672±9
Phase-only reconstruction2.01±0.26 2.26±0.4 1.98±0.9 

Finally, in vivo permittivity maps were compared to simulations (Fig. 9). Also for the in vivo permittivity maps, high noise levels were observed at low field strengths. These noise levels prevented a reliable reconstruction of the permittivity at 1.5 and 3T. At 7T, however, a good correspondence between the simulated and measured permittivity distribution was shown.

In all resulting conductivity and permittivity maps, errors on the boundaries between the various tissue types were observed. This is caused by the use of the homogeneous Helmholtz equation in combination with a numerical procedure to calculate the Laplacian [1, 16].

DISCUSSION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. THEORY
  5. METHODS
  6. RESULTS
  7. DISCUSSION
  8. REFERENCES

In this study, a systematic comparison between EPT at 1.5, 3, and 7T was performed to investigate the effect of static magnetic field strength on the accuracy and precision of EPT. As EPT has primarily been used in the brain, this comparison was focused on the brain and phantoms with the approximate dimensions of the brain. The EPT reconstruction is not only affected by the SNR of the measurement itself but also by the validity of two assumptions; therefore, an increased accuracy is not necessarily inherent to a higher static magnetic field strength.

In EPT, the B1+ amplitude and phase are required for an exact reconstruction of the local dielectric properties; however, the B1+ phase cannot be measured directly. To estimate the B1+ phase, the transceive phase is divided by two, which implies that B1+ and B1 should be equal. In MRI measurements, the B1+ and B1 field are projected on a clockwise and counter-clockwise rotating frame. Therefore, the fields are only equal if the B1+ is linearly polarized, or under some conditions if there is a quadrature reversal between the transmit and the receive ports.

In earlier work, it was shown the contribution of scatter fields to the B1+ and B1 field are canceled for two orthogonal linear fields incident on a cylinder, in case, there is reversal of the quadrature combination or the two fields between transmit and receive [16]. In that case, the transceive phase assumption holds. In this work, it was verified that, as expected by theory, the transceive phase approximation is not generally valid for elliptical objects. In elliptical objects, the scatter fields are no longer canceled and elliptical polarization will emerge at some specific locations (e.g., most pronounced at the periphery at 7T) [17]. The appearance of elliptical polarization will be more pronounced at higher field strengths as the amplitude of the scatter fields scales with the frequency. This explains the observation that the validity of the transceive phase assumption reduces at an increased frequency.

Scatter fields also explain the difference between the phase error observed in the symmetrical and asymmetrical phantoms. In the symmetrical phantoms, a left/right symmetry was observed, whereas this was not the case in the asymmetrical phantom. Using the approach by Hoult [17], one can show that the asymmetrically placed inner compartment, with different dielectric properties, affects the eddy current paths and its amplitude of the currents. Therefore, the contribution of the scatter fields (i.e., the eddy currents) to the total B1+ and B1 field is no longer canceled.

Besides frequency and phantom symmetry, also the dielectric properties of the object affect the phase error φ±/2−φ+, and thus, the validity of the transceive phase assumption. If one compares cases 3 and 4 at 7T (Fig. 3), it can be observed that decreasing the permittivity tempers the phase error. A lower permittivity results in a lower curvature of the magnetic field and the effect of the scatter fields is reduced. This also explains why the phase error in the head (Fig. 3) is smaller at 3T than at 1.5T.

Phase-only conductivity reconstruction can be a valuable simplification of the EPT algorithm, as it reduces scan time. The requirement for this method is given in Eq. (6), and is related to the spatial fluctuation in B1+ amplitude. As was already observed before [16], this assumption poses problems at 7T due to the large variations in the B1+ amplitude. The error introduced in the phantom at 7T is higher than the error in the head. This is related to the high permittivity in the phantom. A high permittivity shortens the wavelength, and consequently the B1+ amplitude will be more curved. At 3T, the error is reduced and only present at the outer border of the phantom. For 1.5T, the phase-only conductivity reconstruction is a valid simplification of the EPT algorithm, which affects the accuracy of the method only marginally.

Based on the discussion above, it may seem that a low static magnetic field strength would be an ideal candidate for EPT, as the transceive phase assumption and “phase-only conductivity mapping” can be performed best at low radio frequencies. However, the noise level is elevated at low field strengths (Fig. 7), which hampers precise determination of the dielectric properties. This is especially the case for the permittivity, which has a 1.6–2 times higher noise level than the conductivity map.

It would be beneficial to remove the transceive phase assumption, to fully leverage the SNR gain at high field strengths. Recently, strategies that employ transmit arrays were developed to remove this assumption [27-30]. According to the presented study, the use of these strategies is anticipated to be most effective at high field strengths.

A further improvement of EPT can be found in the boundary error. It was shown in earlier work [1, 16], that errors are expected at tissue boundaries, as the reconstruction is only valid for media with piecewise constant dielectric properties. This leads to incorrectly reconstructed dielectric properties close to a tissue boundary, as calculation of the Laplacian requires a kernel with a certain pixel width (7 × 7 × 5 pixels in this work).

The reported results are typical for the used geometry, frequency, and dielectric properties. As was shown in this work, simulations closely match measurements and are therefore a justified tool to investigate the accuracy of the method. A similar approach as given in this article would be required to evaluate EPT for other body parts or geometries, and in case of changed dielectric properties. Additionally, simulations give direct insight in the transceive phase error, which is not possible with measurement data.

REFERENCES

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. THEORY
  5. METHODS
  6. RESULTS
  7. DISCUSSION
  8. REFERENCES
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