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Electrical properties tomography (EPT) is a recently introduced technique to reconstruct conductivity and permittivity from measurable radio frequency (RF) magnetic field maps . This quantitative MRI method was already anticipated by Haacke et al. in 1991 . 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 . 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 . Also in breast tumors, an elevated conductivity was reported [7, 8], which was shown earlier for ex vivo measurements . In gastrointestinal research, a fast sequence was used to monitor conductivity of stomach contents during the intake of a fluid .
EPT may also be used to assess RF safety . 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 . 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 , 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 . 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) . 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 . In the latter case, a B1− field is produced in receive that mimics the B1+ field in transmit.
Studies at 1.5T  and 7T  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 . For the phantom, it was therefore investigated at which frequency the accuracy of the assumptions and the SNR are balanced.
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In this work, the homogeneous Helmholtz equation was used for reconstruction of the dielectric properties from the B1+ amplitude and phase :
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:
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−) .
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:
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−:
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 . 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:
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 :
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 . 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 .
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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 . 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) . 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 , 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 , 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.