Complex B₁ Mapping and Electrical Properties Imaging of the Human Brain using a 16-channel Transceiver Coil at 7T

Abstract

The electric properties (EPs) of biological tissue provide important diagnostic information within radio and microwave frequencies, and also play an important role in specific absorption rate (SAR) calculation which is a major safety concern at ultrahigh field (UHF). The recently proposed electrical properties tomography (EPT) technique aims to reconstruct EPs in biological tissues based on B₁ measurement. However, for individual coil element in multi-channel transceiver coil which is increasingly utilized at UHF, current B₁-mapping techniques could not provide adequate information (magnitude and absolute phase) of complex transmit and receive B₁ which are essential for EPT, electric field, and quantitative SAR assessment. In this study, using a 16-channel transceiver coil at 7T, based on hybrid B₁-mapping techniques within the human brain, a complex B₁-mapping method has been developed, and in-vivo EPs imaging of the human brain has been demonstrated by applying a logarithm-based inverse algorithm. Computer simulation studies as well as phantom and human experiments have been conducted at 7T. The average bias and standard deviation for reconstructed conductivity in vivo were 28% and 67%, and 10% and 43% for relative permittivity, respectively. The present results suggest the feasibility and reliability of proposed complex B₁-mapping technique and EPs reconstruction method.

INTRODUCTION

Frequency-dependent electric properties (EPs; conductivity σ and permittivity ε_r) of biological tissues have been studied for decades (1–6). A comprehensive survey was conducted (1) which covered a large selection of healthy tissues over a wide range of frequencies (10Hz–20GHz). Several studies have also been conducted on EPs of a variety of cancerous tissue types at radio and microwave frequencies (2–5). These previous results suggested that the contrast of EPs values between malignant and normal tissue (e.g. 10–20% typical differences in permittivity in a variety of tissue types, however ≥300% for breast cancer) is significantly greater than the small (few percent) contrast exploited by conventional imaging modalities like X-ray and ultrasound (6), which suggests a useful imaging contrast in cancer detection and characterization.

Efforts have been made to produce cross-sectional images of EPs in vivo in the past two decades. Electrical Impedance Tomography (EIT) (7) and its variants using magnetic induction (MIT) (8) are cost-effective and can provide dynamic information with regard to tissue EPs, but are limited by their low spatial resolution due to the need to solve an ill-posed nonlinear inverse problem. Magnetoacoustic Tomography with Magnetic Induction (MAT-MI) offers the promise of obtaining high-resolution tissue conductivity (9–11); however, there have been no in vivo experiments reported so far. Based on the Magnetic Resonance Current Density Imaging (MRCDI) technique (12), Magnetic Resonance Electrical Impedance Tomography (MREIT) (13) utilizes the MRI scanner to measure current injection inducted MR phase shifts inside an object. It eliminates the need to solve the ill-posed inverse problem and provides high spatial resolution in conductivity imaging in vivo; however, its requirement of current injection into the body within MR scanner limits its applicability in medical applications.

The idea of extracting EPs from MR images was proposed by Haacke et al. in 1991 (14). In 2003, Wen pointed out the perturbation of the RF field in MRI directly related to the EPs distribution within the sample, and reported phantom experiments at 1.5T and 4.7T with a Helmholtz equation based inverse algorithm (15). Based upon well-developed B₁-mapping theory and technique (16–18), recent efforts have been made on computer simulations, phantom and in-vivo studies using various algorithms to extract EPs from measured B₁ maps (19–29). Differing from other noninvasive imaging techniques, this approach, named as Electric Properties Tomography (EPT) (19), requires no injection of electric currents during MRI scanning. The spatial resolution of MR EPT is essentially determined by MRI data acquisition and B₁-mapping.

Ultrahigh field (UHF) MRI (7T and higher) has been pursued with increasing interest (30). Its advantages include the promise to improve signal-to-noise ratio (SNR) and spatial resolution of MR imaging (31). Compared with traditional birdcage designs which exhibit significant B₁ inhomogeneities in UHF MRI system, the multiple coil element transmit technique has been recognized as a potentially powerful tool for B₁ inhomogeneities compensation at UHF (32). However, management of the specific absorption rate (SAR), which is directly related to RF-induced heating, still remains one of the challenges faced by in-vivo UHF MRI applications. It requires worst-case safety limits on flip angles in pulse sequence designs, which would compromise the underlying increases in SNR and image contrast associated with high field (33–35). Therefore, real-time and patient-specific local SAR estimation is highly desired in UHF MRI applications. Although a conservative global SAR estimation can be obtained by real-time measurement of forward and reflected RF power at the coil ports, a reliable calculation of local SAR hot spots necessitates the knowledge of local tissue EPs.

The EPT technique estimates EPs’ values at the proton Larmor frequency of the operating B₀ field and provides a step toward quantitative SAR assessment. In existing EPT approaches, the transmit B₁ phase, which is needed as an intermediate quantity to derive EPs distribution, is roughly assumed equal to half of the measured transceiver phase within conventional quadrature birdcage coils at 1.5T, 3T, 4.7T or 7T (15,19,21–23,27–29). However, at UHF for example single coil element transceiver, it remains unclear that if such phase assumption would still be feasible since shorter operating wavelength at higher field would cause larger phase variations between transmit and receive B₁ (36). Moreover, complex information of both transmit and receive B₁ is essential for SAR-estimation oriented electric field calculation, whereas complex receive B₁ mapping remains an unsolved problem. Therefore, a new EPT method based on a new complex B₁ mapping technique is highly desired for such multiple element coils at UHF.

In this paper, we demonstrate a new approach for in-vivo EPs imaging of the human brain with a 16-channel transceiver coil at 7T. In addition, based on measured B₁ fields, a direct quantitative estimation of complex transmit and receive B₁ field with the absolute phase for each coil element has been provided. It is anticipated that this work will substantially facilitate local SAR assessment at UHF MRI.

THEORY

Overview of the Proposed EPT Approach

Figure 1 summarizes the proposed EPT approach to image the EPs distributions within the human brain. Using hybrid B₁-mapping techniques and a multi-channel transceiver array coil (with N channels), the initial step consists of measuring, for each coil element, the magnitude of transmit B₁ field ( $\left|{\stackrel{\sim }{B}}_{1,k}^{+}\right|$, “~” denotes complex quantity), the proton density (PD) ratio biased magnitude of receive B₁ field ( $\left|{\stackrel{\sim }{B}}_{1,j}^{-}\right|$), (i.e. ${PD}_{\mathit{ratio}}\times \left|{\stackrel{\sim }{B}}_{1,j}^{-}\right|$), and the relative phase distributions between coil elements, where k ∈ [1,2,…N] and j ∈ [1,2,…N] refer to the transmit and receive components of each coil element, respectively (see MATERIALS AND METHODS). Based on the approximately elliptical symmetry of the human brain in axial views, we extract the PD ratio distribution and estimate the magnitude of receive B₁ field for each coil element. Then, knowing the magnitude and the relative phase of both transmit and receive B₁ field, the absolute transmit and receive phase of each coil element can be calculated by applying Gauss’s Law for magnetism. Finally, based on the resulting full complex B₁ field maps, local conductivity and relative permittivity within the human brain can be reconstructed through a logarithm-based inverse imaging algorithm.

Figure 1.

Schematic diagram of the proposed complex B₁ mapping and EPT approach.

Differing from other EPT algorithms (19,21,22,24–29) which use multi-channel transceiver array technique, our approach provides magnitude of receive B₁ fields as well as absolute transmit and receive phase for each coil element. Therefore, electric fields can be derived by Ampere’s Law, paving the road towards real-time, subject-specific local SAR estimation.

Proton Density Extraction and Receive B₁ Magnitude Mapping

A strong similarity had previously been reported at 7T between, on the one hand, the sum of the magnitude (SOM) of the transmit ${\text{B}}_1({\displaystyle \sum _{k=1}^{16}}\left|{\stackrel{\sim }{B}}_{1,k}^{+}\right|)$ and, on the other hand, the SOM of the receive ${\text{B}}_1({\displaystyle \sum _{j=1}^{16}}\left|{\stackrel{\sim }{B}}_{1,j}^{-}\right|)$, in a 16-channel transceiver array in the context of approximately elliptical symmetry of the human head in an axial plane and of sixteen-fold elliptical symmetry of the coil structure along the major axis Y (37,38). We further empirically observed, both in experiments and computer simulations, a closer match between these two sums when the SOM of the transmit B₁ is flipped (denoted by “↔”) about the Y-axis, i.e. left to right. As a result, the PD ratio, to which the longitudinal magnetization M_Z is proportional, could be extracted by Eq. 1, and receive B₁ magnitudes could then be estimated by Eq. 2:

$${PD}_{\mathit{ratio},\mathit{est}}\approx \sum _{j=1}^{16}({PD}_{\mathit{ratio}}\times \left|{\stackrel{\sim }{B}}_{1,j}^{-}\right|)/\overleftrightarrow{(\sum _{k=1}^{16}\left|{\stackrel{\sim }{B}}_{1,k}^{+}\right|)}$$ [1]

$${\left|{\stackrel{\sim }{B}}_{1,j}^{-}\right|}_{\mathit{est}}\approx {PD}_{\mathit{ratio}}\times \left|{\stackrel{\sim }{B}}_{1,j}^{-}\right|/{PD}_{\mathit{ratio},\mathit{est}}$$ [2]

Absolute Phase Retrieval

Based on Gauss’s Law for magnetism (Eq. 3) and the Principle of Reciprocity (16) described by Eq. 4, we ignore the term of B̃_z gradient and derive Eq. 5, due to much smaller magnitude of the z-component of Cartesian magnetic field B̃_z when compared with that of B̃_x and B̃_y within RF head coil (19,20,39):

$$\partial {\stackrel{\sim }{B}}_x/\partial x+\partial {\stackrel{\sim }{B}}_y/\partial y+\partial {\stackrel{\sim }{B}}_z/\partial z=0$$ [3]

$$\{\begin{array}{l}{\stackrel{\sim }{B}}_1^{+}=({\stackrel{\sim }{B}}_x+i{\stackrel{\sim }{B}}_y)/2\hfill \\ {\stackrel{\sim }{B}}_1^{-}={({\stackrel{\sim }{B}}_x-i{\stackrel{\sim }{B}}_y)}^{\ast }/2\hfill \end{array}$$ [4]

$$\partial {\stackrel{\sim }{B}}_1^{+}/\partial x-i\partial {\stackrel{\sim }{B}}_1^{+}/\partial y+\partial {\stackrel{\sim }{B}}_1^{-\ast }/\partial x+i\partial {\stackrel{\sim }{B}}_1^{-\ast }/\partial y=\partial {\stackrel{\sim }{B}}_x/\partial x+\partial {\stackrel{\sim }{B}}_y/\partial y\approx 0$$ [5]

where * denotes complex conjugation.

On the slice of interest, five adjacent voxels (voxel P and its four adjacent voxels along x and y directions) are chosen as target area for gradient calculation. Taking channel N as the reference, assuming the absolute phase of ${\stackrel{\sim }{B}}_{1,N}^{+}$ in voxel P to be zero, the other nine phase values of ${\stackrel{\sim }{B}}_{1,N}^{+}$ and ${\stackrel{\sim }{B}}_{1,N}^{-}$ in target area are set to be unknowns; with ten known ${\stackrel{\sim }{B}}_{1,N}^{+}$ and ${\stackrel{\sim }{B}}_{1,N}^{-}$ magnitude values in target area, two non-linear equations (one for real part and the other for imaginary part) can be obtained in the form of Eq. 5. Then utilizing measured magnitude and relative phase information of all the other fifteen channels, thirty-two equations with nine unknowns are obtained. By applying certain optimization method, i.e. the large-scale algorithm, the nine unknown phases can be calculated by solving these equations. By moving the target area across the entire imaging slice, we can get axial phase gradients of ${\stackrel{\sim }{B}}_{1,N}^{+}$ and ${\stackrel{\sim }{B}}_{1,N}^{-}$ on the slice, as well as the phase differences between ${\stackrel{\sim }{B}}_{1,N}^{+}$ and ${\stackrel{\sim }{B}}_{1,N}^{-}$ for each voxel. If we choose (arbitrarily) voxel Q and set ${\stackrel{\sim }{B}}_{1,N}^{+}$ phase at this point to be zero, by integration along x and y directions utilizing calculated phase gradients, we finally get the “absolute” phase distribution of ${\stackrel{\sim }{B}}_{1,N}^{+}$ and ${\stackrel{\sim }{B}}_{1,N}^{-}$ (as well as the phases for all the other channels) relative to voxel Q. Note that EPs reconstruction depends on phase gradients (see the following section), thus changing the position of Q will not affect EPs estimation. In this study, we choose the center voxel of slice of interest as the reference point Q to derive the absolute phase distribution across the imaging slice.

Logarithm-based Inverse Algorithm

The general principle of our inverse method is to quantitatively find EPs’ different dependency on the magnitude and phase of complex B₁ distribution. Since logarithm operation over a complex quantity separates its magnitude and phase components into real and imaginary parts, respectively, we would rewrite the wave equation in the form of logarithm operation over complex B₁. We use B̃₁ denoting either complex transmit or receive B₁. Taking gradient over logarithm of complex B₁ gives us Eq. 6 and Eq. 7 as follows,

$$\nabla {\stackrel{\sim }{B}}_1={\stackrel{\sim }{B}}_1\nabla ln{\stackrel{\sim }{B}}_1$$ [6]

$$\nabla ln{\stackrel{\sim }{B}}_1=\nabla [ln\left|{\stackrel{\sim }{B}}_1\right|+iarg({\stackrel{\sim }{B}}_1)]=(\nabla \left|{\stackrel{\sim }{B}}_1\right|)/\left|{\stackrel{\sim }{B}}_1\right|+i\nabla arg({\stackrel{\sim }{B}}_1)$$ [7]

Performing divergence operation on both hand sides of Eq. 6, we get Eq. 8; substituting ∇B̃₁ with B̃₁∇ln B̃₁, we derive Eq. 9 as shown below:

$${\nabla }^2{\stackrel{\sim }{B}}_1=\nabla {\stackrel{\sim }{B}}_1·\nabla ln{\stackrel{\sim }{B}}_1+{\stackrel{\sim }{B}}_1{\nabla }^{2}ln{\stackrel{\sim }{B}}_1$$ [8]

$${\nabla }^2{\stackrel{\sim }{B}}_1/{\stackrel{\sim }{B}}_1=\nabla ln{\stackrel{\sim }{B}}_1·\nabla ln{\stackrel{\sim }{B}}_1+{\nabla }^{2}ln{\stackrel{\sim }{B}}_1$$ [9]

We assume isotropic EPs distribution, and consider the magnetic permeability inside biological tissues to be equal to that in a vacuum. From Helmholtz equation (Eq. 10) which was proposed by Wen (15) and has been further used in (22,24,25,27–29), we find the equivalence between the logarithm of complex B₁ and the EPs which is indicated by Eq. 11:

$${\nabla }^2{\stackrel{\sim }{B}}_1/{\stackrel{\sim }{B}}_1=-{\omega }^2{\mu }_0{\epsilon }_c$$ [10]

$$\nabla ln{\stackrel{\sim }{B}}_1·\nabla ln{\stackrel{\sim }{B}}_1+{\nabla }^{2}ln{\stackrel{\sim }{B}}_1=-{\omega }^2{\mu }_0{\epsilon }_c$$ [11]

where μ₀ is the free space permeability, ω the operating angular frequency, ε₀ the free space permittivity, and ε_c the complex permittivity (ε_c = ε_rε₀ − iσ/ω) with ε_r as the relative permittivity. Combining Eq. 7 with Eq. 10 and separating the real and imaginary parts, we get conductivity and relative permittivity in the expression of magnitude and phase of complex B₁ as Eq. 12:

$$\{\begin{array}{l}\sigma =[2\nabla \left|{\stackrel{\sim }{B}}_1\right|·\nabla arg({\stackrel{\sim }{B}}_1)/\left|{\stackrel{\sim }{B}}_1\right|+\nabla ·\nabla arg({\stackrel{\sim }{B}}_1)]/(\omega {\mu }_0)\hfill \\ {\epsilon }_r=-\{(\nabla \left|{\stackrel{\sim }{B}}_1\right|·\nabla \left|{\stackrel{\sim }{B}}_1\right|)/{\left|{\stackrel{\sim }{B}}_1\right|}^2-\nabla arg({\stackrel{\sim }{B}}_1)·\nabla arg({\stackrel{\sim }{B}}_1)+\nabla ·[(\nabla \left|{\stackrel{\sim }{B}}_1\right|)/\left|{\stackrel{\sim }{B}}_1\right|]\}/({\omega }^2{\mu }_0{\epsilon }_0)\hfill \end{array}$$ [12]

Based on previously measured and estimated complex B₁ distributions, Eq. 12 is our inverse equation for EPs reconstruction in differentiation form.

Ignoring the minor variation of B₁ profiles along Z-axis between adjacent axial slices, assuming homogeneous EPs distribution within integration area S, we perform surface integration on the imaging X–Y plane; applying the Divergence Theorem, the Laplacian operation could be avoided, and Eq. 12 is re-written as follows:

$$\{\begin{array}{l}\sigma =[2{\int }_S\nabla \left|{\stackrel{\sim }{B}}_1\right|·\nabla arg({\stackrel{\sim }{B}}_1)/\left|{\stackrel{\sim }{B}}_1\right|dA+{\int }_{\partial S}\nabla arg({\stackrel{\sim }{B}}_1)·\widehat{n}dl]/(\omega {\mu }_{0}S)\hfill \\ {\epsilon }_r=-\{{\int }_S[(\nabla \left|{\stackrel{\sim }{B}}_1\right|·\nabla \left|{\stackrel{\sim }{B}}_1\right|)/{\left|{\stackrel{\sim }{B}}_1\right|}^2-\nabla arg({\stackrel{\sim }{B}}_1)·\nabla arg({\stackrel{\sim }{B}}_1)]dA+{\int }_{\partial S}[(\nabla \left|{\stackrel{\sim }{B}}_1\right|)/\left|{\stackrel{\sim }{B}}_1\right|]·\widehat{n}dl\}/({\omega }^2{\mu }_0{\epsilon }_{0}S)\hfill \end{array}$$ [13]

Eq. 13 is our final inverse equation using surface-integration.

Eq. 12 illustrates the fact that during the reconstruction process, the conductivity value is more dependent on the phase distribution of complex B₁ field, while the relative permittivity is more dependent on the magnitude distribution. While similar observations have also been demonstrated in (15,21), it should be noted that our surface-integration method only requires single-slice measurement of complex B₁ distribution.

Using either complex ${\stackrel{\sim }{B}}_{1,k}^{+}$ or ${\stackrel{\sim }{B}}_{1,j}^{-}$ (we used sixteen ${\stackrel{\sim }{B}}_{1,j}^{-}$ for inverse problem in this study), by Eq. 13 we could derive sixteen reconstruction results of EPs distributions. However, to avoid including data with low SNR when generating final EPs maps, the coil element contributing the most in ${\stackrel{\sim }{B}}_{1,k}^{+}$ magnitude is identified for each particular voxel (36). The EPs values at this voxel are then picked up from the reconstructed EPs maps which are produced by the most-contributing coil element.

MATERIALS AND METHODS

MR Settings

Imaging experiments were performed with a 7 T magnet (Magnex Scientific, UK) driven with a Siemens console (Erlangen, Germany). A 16-channel transmit/receive head coil, as described in (40), was used for both RF-transmit and signal-receive operations, with the transmit channels powered by 16×1kW amplifiers (CPC, Hauppauge, NY) interfaced with a remotely controlled phase/amplitude gain unit.

In addition to the standard MR console system monitoring the forward and reflected RF power on one channel, a similar 16-channel RF power monitoring system was developed in-house, continuously measuring the RF power delivered on each RF channel, while running averages were constantly updated over two time constants (10s and 600s). Whenever the measured average power exceeds predefined limits (based on International Standard IEC 60601-2-33 2010) on any single RF channel, all RF amplifiers are instantly disabled. More details on the RF monitoring setup can be found in (41). In the meantime, in order to also account for local SAR limits, we utilized numeric simulations with the software XFDTD (Remcom, State College, PA, USA), designing our 16 transceiver coil loaded with a human head. We applied the same RF phase distribution utilized in-vivo, and calculated both global SAR and maximum 10g average local SAR. The maximum global SAR limit of our RF monitoring system was determined at a level where maximum local 10g SAR average is below the maximum allowed in IEC.

A similar imaging protocol was used in both phantom and in-vivo experiments (individual parameters are detailed later) as follows. Scout images were obtained for slice positioning, and B₀ field homogeneity was locally optimized within a box containing 6 contiguous axial slices chosen for the study. A series of sixteen small flip angle 2D GRE images were acquired with only one channel transmitting at a time while receiving on 16 channels separately, so the relative phase maps of ${\stackrel{\sim }{B}}_{1,k}^{+}$ and ${\stackrel{\sim }{B}}_{1,j}^{-}$ between coil elements were calculated as described in (36), respectively. A 3D map of the excitation flip angle was obtained with the Actual Flip Angle (AFI) technique (42), all channels transmitting together, to be merged with previously acquired small flip angle GRE images in order to calculate the magnitude map of ${\stackrel{\sim }{B}}_{1,k}^{+}$ for each of the sixteen coil elements as indicated in (43). (Merging large and small flip angle data for B₁ mapping is referred to as “hybrid B₁ mapping” throughout the context). Finally, a 2D GRE image, was acquired, all channels transmitting together, using a large flip angle (high SNR), a long TR (longitudinal magnetization approximately at equilibrium) and a short TE (negligible T2* relaxation); each of the sixteen images (one per receive channel) of this data set was normalized by the sinus of the excitation flip angle to produce sixteen, proton density biased, ${\stackrel{\sim }{B}}_{1,j}^{-}$ magnitude maps (40).

Phantom Experiments

A multi-compartment phantom was built in order to validate the proposed EPT methods. For this purpose, five pairs of acrylic round tubes, with diameters of 6.99, 3.81, 3.18, 2.22 and 1.27 cm for each pair, were inserted in an acrylic cylinder, 20cm in length and 14.61 cm in inner diameters, following a strict symmetry about the Y-axis as shown in Figure 2. Gels of saline solutions with different concentrations of CuSO₄·5H2O, NaCl, Sucrose (S8501, SIGMA) and Gelatin (G2500, SIGMA) were prepared to fill in different tubes. The concentration of salt, sucrose and copper sulfate influences conductivity, permittivity (44), and relaxation time constants (T1 and T2) within the range of the values of normal human brain tissues, respectively. The phantom was filled three days before scanning. The EPs values were measured with an Agilent 85070D dielectric probe kit and an Agilent E4991A network analyzer at Pennsylvania State University College of Medicine. Structural, solution recipes and measured T1 information are summarized in Table 1.

Figure 2.

The sketch of phantom design using ten numbered tubes. Same gel solution was added in the tubes with same number indicated.

Table 1.

Parameters for Phantom Design

tube No.	tube size (cm)		solution recipes (gram)					measured T1 (ms)
	inner diameter	wall thickness	H2O	NaCl	Sucrose	Cu4SO4·5H2O	gelatin
I	14.61	0.32	100	0	0	0.025	3	621
II	6.99	0.32	100	0.12	0	0.025	3	728
III	3.81	0.32	100	2.50	49.92	0.025	3	343
IV	3.18	0.32	100	2.54	73.97	0.025	3	389
V	2.22	0.16	100	2.74	102.53	0.025	3	459
VI	1.27	0.16	100	0.88	0	0.025	3	682

The phantom was placed in the head coil with its long axis parallel to B₀. Small flip angle GRE images were acquired with TE/TR=2.62/70ms, nominal flip angle (FA) of 5° and 15 averages in 40 minutes; 3D AFI images with TE/TR1/TR2=1.84/20/120ms, FA of 60° and 2 averages in 69 minutes; high SNR GRE images with TE/TR=4.12/8000ms, FA of 70° and 10 averages in 210 minutes. 6-slice images were acquired with resolution of 1.5×1.5×1.5mm³ in FOV of 288×189mm². For inverse reconstruction, surface-integration was performed within 4×4 voxels area.

In-vivo Experiments

In-vivo data were acquired in three healthy volunteers who have signed consent forms approved by the University of Minnesota Institutional Review Board, and lied in supine position with their head centered in the head coil. Small-flip-angle GRE images were acquired with TE/TR=2.92/70ms, FA of 10° and 6 averages in 26.5 minutes; 3D AFI images with TE/TR1/TR2=2.60/20/120ms, FA of 60° and one average in 17 minutes; high SNR GRE images with TE/TR=3.43/8000ms, FA of 70° and one average in 17 minutes. Additionally, a two-contrast, 3D T1-weighted MPRAGE acquisition (45) was obtained to identify brain anatomical structures. 6-slice images were acquired with resolution of 1.5×1.5×1.5mm³ in FOV of 288×189mm². For inverse reconstruction, surface-integration was performed within 4×4 voxels area.

Simulation

Simulation data was utilized to test above methods. XFDTD software version 6.0 was used to perform simulation of B₁ field distributions in the human head at 300 MHz. The coil model design was reproduced from the elliptical 16-channel RF stripline arrays used in the experiments; all RF coil elements were assumed to be completely decoupled. A 120-slice anatomical accurate head model consisting of 20 materials (e.g., cerebrospinal fluid [CSF], white matter [WM] and grey matter [GM]) with 2×2×2.5mm³ resolution was incorporated with proper isotropic EPs values at 300 MHz (46). Software calculations were obtained with single coil element transmitting RF power. Sixteen ${\stackrel{\sim }{B}}_{1,k}^{+}$ and ${\stackrel{\sim }{B}}_{1,j}^{-}$ maps were derived from simulated complex Cartesian magnetic field components and by Eq. 4 for each coil element. For inverse reconstruction, surface-integration was performed within 3×3 voxels area.

All post-processing of B₁ data was completed using MATLAB 2008a (The Mathworks Inc., MA, USA).

RESULTS

Simulations

Figure 3 shows simulated single ${\stackrel{\sim }{B}}_{1,k}^{+}$ and ${\stackrel{\sim }{B}}_{1,j}^{-}$ magnitude distribution for coil element 1 to 16 on one axial slice of interest (Figure 3a and Figure 3b), as well as the sum of all sixteen magnitude maps (Figure 3c and Figure 3d), respectively. Similar twisted patterns, but following opposite rotational directions can be observed in Figure 3a and Figure 3b. When flipped from left to right, the SOM of the ${\stackrel{\sim }{B}}_{1,k}^{+}$ maps exhibits a pattern very similar to the SOM of the ${\stackrel{\sim }{B}}_{1,j}^{-}$ maps, with an overall relative error (RE) of 4.29% and a correlation coefficient (CC) of 0.935. The relative error is defined by Eq. 14.

Figure 3.

Simulated individual magnitude of transmit (a) and receive (b) B₁ for each coil element, and the corresponding sum of magnitude of the sixteen transmit (c) and receive (d) B₁ maps, respectively. A proton density ratio image (e) was applied to produce sixteen individual PD ratio biased receive B₁ magnitude maps (f), and extracted PD ratio image (g), and sixteen estimated of receive B₁ magnitude for each coil element (h) are estimated by Eq. 2. Coil elements positions are denoted in (e).

$$RE=\left|\overleftrightarrow{\sum _{k=1}^{16}\left|{\stackrel{\sim }{B}}_{1,k}^{+}\right|}-\sum _{j=1}^{16}\left|{\stackrel{\sim }{B}}_{1,j}^{-}\right|\right|/\overleftrightarrow{\sum _{k=1}^{16}\left|{\stackrel{\sim }{B}}_{1,k}^{+}\right|}$$ [14]

A PD ratio image (Figure 3e) with proton density ratio (i.e. PD_CSF:PD_WM:PD_GM = 1:0.65:0.8) (47) was used to multiply each individual ${\stackrel{\sim }{B}}_{1,j}^{-}$ magnitude image to mimic PD ratio biased ${\stackrel{\sim }{B}}_{1,j}^{-}$ magnitude images (Figure 3f). Using the method described in Eq. 2, extracted PD ratio image and estimated ${\stackrel{\sim }{B}}_{1,j}^{-}$ magnitude distributions are shown in (Figure 3g) and (Figure 3h), respectively. Compared with pre-assigned PD ratio image, the extracted PD ratio distribution shows an RE of 4.08% and CC of 0.992, while the average RE and CC for sixteen estimated ${\stackrel{\sim }{B}}_{1,j}^{-}$ magnitude distributions are 4.35% and 0.998, respectively. All these statistical values illustrate very desirable accuracy in PD ratio extraction and ${\stackrel{\sim }{B}}_{1,j}^{-}$ magnitude estimation.

Channel 11 was chosen to be the reference channel. Taking the absolute phase of ${\stackrel{\sim }{B}}_{1,11}^{+}$ and ${\stackrel{\sim }{B}}_{1,11}^{-}$ as references and unknowns, with known relative phase and magnitude information, by Eq. 5 we form a set of non-linear equations, and the absolute phase distributions are calculated as shown in Figure 4. The sixteen calculated absolute phase of ${\stackrel{\sim }{B}}_{1,k}^{+}$ (Figure 4c) exhibit a maximum of 0.21 radians difference and 0.993 in CC when compared with simulated target distributions (Figure 4a), while a maximum of 0.27 radians difference and CC of 0.989 for the calculated absolute phases of ${\stackrel{\sim }{B}}_{1,j}^{-}$ (Figure 4d) compared with target ones (Figure 4b), respectively. Since calculated phase errors could be accumulated during integration process, it is necessary to examine the accuracy of calculated phase gradients. By setting the spacing between voxels in each direction as unity, the x and y components of phase gradients of simulated ${\stackrel{\sim }{B}}_{1,11}^{+}$ (of reference channel), as well as simulated ${\stackrel{\sim }{B}}_{1,11}^{-}$ phase gradients, are plotted in Figure 4e–h, and the corresponding gradients of calculated phases are plotted in Figure 4i–l, respectively. Calculated ${\stackrel{\sim }{B}}_{1,11}^{+}$ phase gradients exhibit a maximum of 0.018 radians difference when compared with simulated gradients, while 0.023 radians difference for ${\stackrel{\sim }{B}}_{1,11}^{-}$ phase gradients.

Figure 4.

Simulated absolute phase distributions of transmit B₁ and receive B₁ (a) (b), and calculated absolute phase images (c) (d) for each coil element, respectively. x and y gradients of simulated transmit B₁ and receive B₁ absolute phases for reference channel 11 are shown in (e)–(h), and the corresponding calculated gradients are shown in (i)–(l), respectively. Sagittal views of target conductivity distribution (m), relative permittivity (n), and reconstructed conductivity (o) and relative permittivity (p) distributions; reconstructed conductivity (q) and relative permittivity (r) distributions on five consecutive axial slices (the middle slice was indicated in yellow in (c) and (d)).

Slightly lower accuracy in ${\stackrel{\sim }{B}}_{1,j}^{-}$ phase reconstruction than that of ${\stackrel{\sim }{B}}_{1,k}^{+}$ mainly arose from the errors produced in ${\stackrel{\sim }{B}}_{1,j}^{-}$ magnitude estimation. In addition, several small artifacts can also be observed in the sixteen calculated absolute phase images in Figure 4d, and all these artifacts located within the areas where the ${\stackrel{\sim }{B}}_{1,j}^{-}$ magnitude for coil element 11 (the reference channel) exhibited much lower intensity (Figure 3b) and its absolute phase varied more rapidly in space (indicated in Figure 4b). Note that large absolute phase variation existed between ${\stackrel{\sim }{B}}_{1,k}^{+}$ and ${\stackrel{\sim }{B}}_{1,j}^{-}$ for each coil element, thus the assumption of transmit phase being equal to receive phase (15,19,21–23,27–29) is no longer valid for such coils with multiple transmit elements at UHF.

Applying the logarithm-based inverse algorithm as indicated by Eq. 13, reconstructed EPs distribution results on a sagittal and five consecutive axial slices of interest are shown and summarized in Figure 4m–r and Table 2, respectively. For reconstructed conductivity distributions on the five slices of interest, the REs are within 22–25% while CCs within 0.911–0.918; for reconstructed relative permittivity distributions, the REs are within 16–18% while CCs within 0.944–0.963 for tissues of WM, GM and CSF. Slightly higher reconstruction accuracy is preserved in reconstructed relative permittivity maps.

Table 2.

Target and reconstructed EPs values using simulation data

tissue	conductivity (S/m)		relative permittivity
	target value	reconstructed	target value	reconstructed
CSF	2.30	2.02±0.56	69.21	64.18±11.53
WM	0.49	0.46±0.20	42.81	39.70±10.03
GM	0.83	0.78±0.30	58.31	53.88±20.18

Phantom Data

In phantom experiment, channel 1 was arbitrarily chosen as the phase reference, and using the measured relative phase information, the absolute phase distribution was calculated. Probe-measured and reconstructed EPs results (average values and standard deviations) are shown and summarized in Figure 5 and Table 3. 5.9%~66.7% and 3.1%~12.1% relative errors were observed between measured and calculated values for conductivity and permittivity, respectively. Higher reconstructed conductivity values can be found in the lower part region compared with the upper region inside tube I as Figure 5c indicates, and this difference was caused by accidental leaking. Tube I was contaminated with the gel solution with higher saline concentration from tube II. We also observed such leaking phenomena in measured T1 maps – lower part showing higher T1 than the upper part in tube I.

Figure 5.

Phantom experimental results of probe-measured target conductivity distribution (a), relative permittivity distribution (b), and reconstructed conductivity distribution (c) and relative permittivity distribution (d).

Table 3.

Probe-measured and reconstructed EPs values in phantom experiment

tube No.	conductivity (S/m)		relative permittivity
	probe-measured	reconstructed	probe-measured	reconstructed
I	0.12	0.16±0.13	78	69±21
II	0.34	0.33±0.20	77	71±22
III	1.12	1.35±0.31	65	67±20
IV	0.71	0.91±0.29	61	59±21
V	0.45	0.78±0.31	53	50±21
VI	1.59	2.12±0.37	77	73±19

Human Experiments

Figure 6 shows normalized T1-weighted image (48) of the slice of interest (Figure 6a), measured sixteen individual ${\stackrel{\sim }{B}}_{1,k}^{+}$ magnitude images (Figure 6b), PD ratio biased ${\stackrel{\sim }{B}}_{1,j}^{-}$ magnitude images (Figure 6c), extracted PD ratio image (Figure 6d), and estimated ${\stackrel{\sim }{B}}_{1,j}^{-}$ magnitude distributions (Figure 6e) for each coil element of human subject 1. Similar to simulated distributions as in Figure 3, twisted patterns following opposite rotational directions can be observed in ${\stackrel{\sim }{B}}_{1,k}^{+}$ and ${\stackrel{\sim }{B}}_{1,j}^{-}$ magnitude maps. Extracted PD ratio image exhibits density ratios approximately to be PD_CSF:PD_WM:PD_GM = 1:0.72:0.91. Note that, despite a long TR (8s), residual T1 bias is observed within the CSF in high SNR 2D GRE images: assuming T1=4.2s and using FA=70°, the GRE equation indeed predicts a steady-state longitudinal magnetization at about 88.7% of equilibrium, which explains the lower-than-expected CSF values in the extracted PD ratio image.

Figure 6.

Normalized T1w image of the slice of interest of subject 1(a), measured sixteen individual transmit B₁ magnitude images (b), measured sixteen individual PD ratio biased receive B₁ magnitude images (c), extracted PD ratio image (d), and estimated receive B₁ magnitude (e) for each coil element of human subject 1 (magnitudes are scaled in arbitrary unit). Coil elements positions are indicated in (e). Normalized T1w images of three consecutive axial slices (f–h) of interest of subject 1, and the corresponding reconstructed conductivity (i–k) and relative permittivity (l–n) distribution. The mask area has been denoted in yellow in (f), in which WM, GM and CSF tissues were manually and empirically segmented for statistical evaluations of reconstruction results.

Channel 1 was arbitrarily chosen as the phase reference. Using the measured relative phase information, the absolute phase distribution was calculated, and three consecutive slices of conductivity and relative permittivity reconstruction results of human subject 1 are shown in Figure 6i–n, respectively. The mask area has been indicated on the T1-weighted image in Figure 6f, in which we manually and empirically segmented WM, GM and CSF tissues, and our statistical evaluations are conducted in such segmented areas. Comparisons of measured EPs values of three human subjects with literature values (1) are summarized in Table 4. The CSF in the lateral ventricle can be observed in shape in reconstructed EPs images, while the relative permittivity map provides more structure details compared with the conductivity map. Reconstruction artifacts can be observed to the right of lateral ventricle (i.e. in Figure 6l–n), which was caused by the subject’s movement during scans. Such motion artifacts can also be observed in receive B₁ magnitude maps as in Figure 6e (channel 1–4).

Table 4.

Comparison of literature values with measured EPs values of human subjects

	conductivity (S/m)			relative permittivity
	CSF	WM	GM	CSF	WM	GM
literature	2.22	0.41	0.69	73	44	60
subject 1	2.49±1.20	0.59±0.25	0.86±0.55	79±23	45±18	66±25
subject 2	2.51±1.40	0.62±0.38	0.83±0.32	80±38	52±25	67±29
subject 3	2.43±1.52	0.68±0.31	0.79±0.44	77±26	49±21	64±23

DISCUSSION

Mapping the magnitude and the absolute phase of receive B₁ is highly desirbale but remains a challenging issue, especially in-vivo at high field. In this study, using a multi-channel transceiver array, we have introduced a new method to estimate receive B₁ magnitude and the absolute phase of both transmit and receive B₁ for each coil element. We have demonstrated with an electromagnetic numerical model of a 16 channel transceiver coil loaded with a human head at 7T that this approach provides satisfactory accuracy. Our estimated receive B₁ magnitudes exhibit an average of 4.08% RE and 0.992 CC over all 16 channels, while reconstructed absolute phase for both transmit and receive B₁ shows below 0.27 radians differences with up to 0.023 radians phase gradient differences. We have then demonstrated the feasibility and evaluated the performance of this complex B₁-mapping approach, using the logarithm-based inverse algorithm, in imaging electrical conductivity and relative permittivity distribution in a phantom and in human subjects at 7T. Quantitative reconstruction results suggest the feasibility of the inverse algorithm, as well as the stability of the whole data post-processing methods based on measurable B₁ data by means of the employed hybrid B₁-mapping technique.

In this study, ${\stackrel{\sim }{B}}_1^{-}$ magnitude mapping with PD ratio extraction is the first and a major step towards subsequent absolute phase calculation. To estimate its distribution for each coil element, we utilized both the approximately elliptical symmetry structure of the human head and the sixteen-fold symmetric coil structure along the vertical axis. In the meantime, for our coil system, residual coupling exists between different coil elements, and transmitting through a single channel results in transmitting through all coupled coil elements (see its equation form in APPENDIX of (32)). According to our experience, the level of coupling in our data was sufficiently limited to preserve the overall mirroring symmetry characteristic of the B1 profiles; from extracted PD ratio image of human subject 1 as shown in Figure 6d, PD ratios of PD_CSF:PD_WM:PD_GM = 1:0.72:0.91 are close to literature values (47) while considering the residual T1 bias, which supports our observation of such behavior. Furthermore, one could expect that as long as, either the coupling level itself is sufficiently homogenous through all channels, or the distribution of coupling level between coil elements preserves sufficient symmetry along the dominant axis of symmetry of the head, the mirroring symmetry between coupled transmit and receive B₁ fields along the Y axis would also be preserved; and in the latter case, one can expect that the described properties still apply. Further investigations are needed to verify this behavior in experiments and in simulations, as well as to evaluate the impact on our proposed approach in situations where the symmetry of the human head is significantly altered, such as diseases or anatomical variations, as well as in other body structures that do not exhibit a natural dominant axis of symmetry.

Gauss’s Law for magnetism plays as the fundamental equation in absolute phase calculation. Examining the simulation data indicates that, the magnitude of magnetic field gradients ∂B̃_x/∂x and ∂B̃_y/∂y are usually ten or more times greater than the magnitude of ∂B̃_z/∂z within brain region, which justifies our assumption of negligible z-gradient of B̃_z component and suggests the feasibility of the 2D simplification of Gauss’s Law. In addition, from a theoretical perspective, as few as five transceiver coil elements shall be sufficient to calculate the absolute phase distributions using the proposed method; however, it is anticipated that using a larger number of channels (e.g. 16 vs. 4) should provide more stable solutions, although the spatial distribution of relative SNR in each coil images has to be taken into account. In this study, we utilized data from the most contributing coil elements for any given voxels; more sophisticated SNR optimization could be developed in the future.

Some artifacts were observed in simulation data (see Figure 4d) in the calculated absolute phase maps of receive B₁ for each coil element. The location of these artifacts actually corresponds to areas with extremely low magnitude (Figure 3b) and rapidly varying phase (Figure 4b) in ${\stackrel{\sim }{B}}_{1,11}^{-}$ magnitude and phase maps, i.e. the receive channel eleven utilized as reference for phase computation. We hypothesize that choosing different reference channel for these voxels would address this issue.

The average duration for absolute phase calculation for both simulation and experimental data was 30–60 minutes using a desktop PC with 3GHz CPU and 4GB RAM. We believe, however, that this computational time will be dramatically shortened by using higher end computers, writing a C-language version of the code, and using parallel computing techniques.

In this study, we considered a 2D problem to derive the integral form of inverse equations (Eq. 13) based on negligible variations of B₁ fields along the Z-axis. To verify this assumption, using the simulation data and taking the complex B₁ field of the slice of interest as reference, we find ${\stackrel{\sim }{B}}_{1,k}^{+}$ magnitude on adjacent slices exhibits an overall RE=2.86% with CC=0.999, and ${\stackrel{\sim }{B}}_{1,j}^{-}$ magnitude shows RE of 2.82% with CC of 0.999; for phase distributions on adjacent slices, the absolute phase of ${\stackrel{\sim }{B}}_{1,k}^{+}$ and ${\stackrel{\sim }{B}}_{1,j}^{-}$ both exhibit a maximum of 0.03 radians difference with CC of 0.999, respectively. These statistical values suggest that considering a 2D problem is an acceptable approximation, even though very limited residual biases are naturally to be expected.

EPs reconstruction results both in computer simulations and experimental studies confirm the effectiveness of the proposed inverse algorithm. For simulation results, shown in Table 2, we attribute the slightly lower reconstructed EPs values to the 2D simplification discussed above. For experiment results, while the overall lower SNR of measured B₁ maps affects in general the reconstruction accuracy, the differential EPs dependency on phase and magnitude components of the complex B₁ distribution also translates in a differential impact on EPs map quality. Compared with reconstructed relative permittivity results, higher relative errors of reconstructed conductivity values with larger standard deviations can be found in Table 3 and Table 4. This is in agreement with our previous prediction from the logarithm-based algorithm: the permittivity is more dependent on the B₁ magnitude distribution, while the conductivity value is more dependent on the B₁ phase which is calculated from measured magnitude information, thus the latter is more prone to error propagation.

In phantom experiment, tube walls made by acrylic material with short T2 produce signal void, and very low magnitude pervade within wall areas in measured and calculated B₁ maps. When we perform differentiation (as described in Eq. 13) across such wall areas, larger computation errors would be introduced, and usually higher reconstructed value are observed in tubes periphery region. Such errors are expected to be greater in smaller tube sections. In agreement with this mechanism, for tube VI with smallest inner diameter (0.95mm), applying surface-integration within a 6×6mm² area which is close to the tube cross section, large relative error (40%) in reconstructed conductivity value was observed.

In both simulation and experimental data (Figure 3 and Figure 6), it can be observed that the sixteen ${\stackrel{\sim }{B}}_{1,k}^{+}$ magnitude profiles and the sixteen ${\stackrel{\sim }{B}}_{1,j}^{-}$ magnitude profiles present twisted patterns that are following opposite rotational directions, but are symmetric about the Y-axis (left to right) when comparing pairs such as magnitude maps of ${\stackrel{\sim }{B}}_{1,1}^{+}$ and ${\stackrel{\sim }{B}}_{1,16}^{-},{\stackrel{\sim }{B}}_{1,2}^{+}$ and ${\stackrel{\sim }{B}}_{1,15}^{-}$, etc (36). This phenomenon should be considered together with the high level of symmetry in both the human brain and the considered coil array along the Y-axis. When further examining the simulated absolute phase maps, as Figure 4a and Figure 4b illustrate, similar symmetrical properties can be observed, with opposite phase values, in pairs such as phase maps of ${\stackrel{\sim }{B}}_{1,1}^{+}$ and ${\stackrel{\sim }{B}}_{1,16}^{-},{\stackrel{\sim }{B}}_{1,2}^{+}$ and ${\stackrel{\sim }{B}}_{1,15}^{-}$, etc. It will be of high interest to investigate to which extent these mirroring symmetrical properties could be exploited to further regularize B₁ phase maps derived from experimental in-vivo data; this includes determining variable zero order phase shifts between channels (36).

While equal-phase approximation has been shown to be reliable at low field with suitable polarization reversal through a quadrature hybrid (15,19,21), larger and more noticeable phase distortion has been shown as field strength increases (22). As a result, it is anticipated that EPs results will deteriorate at UHF when using such phase approximation approaches heavily relying on canonical quadrature behavior in volume RF coils. Recently, a new method has been proposed to iteratively calculate the absolute phase distribution of transmit B₁ for single coil element in a multi-transmit system, and EPs imaging and SAR values estimation for each transmit element were demonstrated using a single-compartment cylindrical phantom (49). As presented in this study and in our recent work (23), multi-channel receive B₁ magnitude mapping and absolute phase retrieval have been demonstrated and validated, using a 16 channel transceiver array at 7T, both in simulation and in experimental data obtained in a phantom and in the human brain. With appropriate complex B₁ distributions and reconstructed EPs values, the electric field distribution can be derived by Ampere’s Law, which would substantially benefit us in local SAR calculation for multi-channel transmit at UHF. While we push the envelope to improve the accuracy and stability of both complex B₁ mapping and electrical properties imaging, quantitatively assessing local SAR value will be an important part of our future works.

CONCLUSION

In this study, using a multi-channel transceiver array, we have developed a new method to reconstruct receive B₁ magnitude and absolute phase of transmit and receive B₁ for each coil element. Based on these field maps, we further demonstrate in-vivo imaging of electrical conductivity and relative permittivity distribution in a physical phantom and in the human brain at 7T. The successful development of noninvasive imaging modalities for EPs imaging may facilitate subject-specific local SAR computation in UHF MRI and accurate in-vivo characterization of cancerous tissues, with a significant impact on neuroscience research and, ultimately, on clinical applications.