CONtrast Conformed Electrical Properties Tomography (CONCEPT) Based on Multi-channel Transmission and Alternating Direction Method of Multipliers

Abstract

In Magnetic Resonance based Electrical Properties Tomography (EPT), circularly polarized magnetic field B₁ from a transmit radiofrequency (RF) coil is measured and utilized to infer the electrical conductivity and permittivity of biological tissues. Compared to a quadrature RF coil, a multi-channel transmit coil provides a plurality of unique transmit B₁ patterns that help to alleviate the under-determined-ness of EPT reconstruction problem, and it also allows to circumvent the “transceive phase assumption” that fails at ultra-high-field MRI. Here, a new approach, CONtrast Conformed Electrical Properties Tomography or CONCEPT, is proposed based on multi-channel transmission that retrieves electrical properties (EPs) by solving a linear partial differential equation with discriminated L₁ and L₂ norm regularization informed by intermediate EP gradient. The theory of CONCEPT and a fast reconstruction algorithm based on Alternating Direction Method of Multipliers are described and evaluated using numerical simulations, phantom experiment and analysis of in vivo human brain data at 7 T MRI. Compared to multi-channel gradient-based EPT (gEPT) method, this new technology does not require receive B₁ sensitivity profiles and does not rely on symmetry assumption regarding RF coil design and imaged target. Moreover, it is not dependent on external prior information, such as integration seed point or anatomical MRI, which can be sources of bias in reconstructed EP values. By deriving EPs from transmit B₁ profiles only, CONCEPT can be used with RF coils that include receive-only arrays with large channel count which can, in turn, offer substantial gains in Signal to Noise Ratio. It also holds potentials to image unsymmetrical body organs and diseased brain. CONCEPT provides solutions for the practical problems during the implementation of gEPT, thus representing a more generalized framework in the context of multi-channel RF transmission.

I. INTRODUCTION

MAGNETIC Resonance based Electrical Properties Tomography (MREPT or EPT) is a non-invasive approach to retrieve tissue electrical conductivity and permittivity by fitting the measured radiofrequency magnetic field (B₁ field) to time-harmonic Maxwell’s Equations [1]–[4]. The reconstructed electrical properties (EPs) provide fundamentally differential information compared to MR relaxometry, holding promise for additional sensitivity and specificity in clinical diagnoses, such as cancer identification and characterization [5]–[7]. Recent studies also shed light on the associations of EPs with other valuable biomarkers, such as sodium concentration, protein, and water content, that are more challenging for direct quantifications [8], [9]. In addition, subject-specific EP distribution is an essential ingredient for realistic local specific absorption rate (SAR) estimation, especially for ultra-high-field (≥ 7 T) applications where electromagnetic wave behavior becomes prominent [10], [11]. Improvements in the reliability of local SAR estimation would in return unleash the full power of ultra-high-field MRI that is otherwise limited by conservative safety constraints.

The problem of inferring tissue EPs from MRI measurable, circularly-polarized magnetic field B₁ manifests as a nonlinear partial differential equation [12], [13]. In order to decrease the condition number and to obtain numerical stability, conventional EPT methods made the assumption that tissue EPs are piece-wise homogeneous, leading to a simplified reconstruction method known as “Helmholtz equation.” This assumption eliminates the EP spatial variation term, and one B₁ distribution obtained with a quadrature volume coil suffices for a successful EP reconstruction. Actual EPs in the human body, however, are spatially heterogeneous, yielding boundary artifact and noise amplification that significantly thwart proper interpretations of Helmholtz-equation based EP reconstruction.

Recently, efforts have been made to address boundary artifacts coming from the Helmholtz equation. One potential solution is to utilize structural MR images to inform EP reconstruction on piecewise areas of relatively uniform contrast. This can be done by using a spatially weighted reconstruction kernel or adding regularization terms based on edge information [7], [14]. However, structural MRI contrasts and EP contrast do not rely on the same mechanism, therefore, spatial boundaries defined by their respective spatial variations do not necessarily coincide, which may bias the EP reconstruction process. Moreover, signal intensity in structural MRI is highly dependent on which MR sequences and parameters are used, creating additional sources of potential inconsistency that may propagate to EPs.

Other EPT approaches attempted to alleviate the boundary artifact by incorporating the spatial variation term into the reconstruction. Zhang et al. take EPs and their gradient terms as independent unknowns which are simultaneously retrieved using dual excitations of linear-polarization modes in a quadrature coil, demonstrating improved boundary fidelity [4]. However, the algorithm was based on the Cartesian components of the RF magnetic field that are not directly measurable from MRI. “Convection-reaction MREPT (cr-MREPT)” and its conductivity-only variant take the reciprocal of EPs as unknowns to linearize the reconstruction problem, from which 2D slices of EPs can be reconstructed in a single iteration [15], [16]. Nevertheless, the empirical assumptions made about B₁ magnitude and phase, as well as EPs of the imaged target itself may introduce reconstruction errors.

Another category of EP mapping technologies is built upon forward modeling and electromagnetic simulations. In Contrast Source Inversion (CSI), the incident field of an empty coil is simulated, and the contrast source distribution is iteratively updated to minimize the discrepancies between scattering field and the observed field [17]. A similar idea can be implemented in spatial frequency space which retrieves EPs without iterations [18]. In addition, Global Maxwell Tomography (GMT) performs a full-wave simulation with a current guess of EP distribution, which gets updated iteratively to match the measured field [19]. These simulation-based algorithms utilize the integral form of Maxwell equations that naturally incorporates spatial variation of EPs. However, highly accurate electromagnetic simulations need to be performed and carefully registered to experimentally acquired data in order to preserve the delicate EP contrast, which is challenging to implement in realistic scenarios. To the best of our knowledge, there is no experimental validation of these approaches.

Alternatively, a multi-channel RF coil array can be used to effectively address the boundary artifact issue and B₁ field assumptions by leveraging a plurality of B₁ measurements [20]–[22]. Gradient-based EPT (gEPT) solves a pixel-wise inverse problem using both transmit B₁ and receive B₁, or $B_1^{+}$ and $B_1^{-}$, to retrieve the EP gradient, followed by its spatial integration based on a user-assigned seed point [23]. This method successfully retrieves both conductivity and permittivity in high quality with improved robustness against noise. Nevertheless, two caveats need to be taken into consideration which may cause errors in the reconstruction results: i) symmetry assumption about RF coil structure and imaged target; ii) subjective assignment of integration seeds, i.e. tissue voxels with presumptive EP values.

In this study, we report a novel EPT approach, dubbed CONtrast Conformed Electrical Properties Tomography (CONCEPT), based on a linear inverse problem regularized by intermediate EP gradient information, using only $B_1^{+}$ maps from a multi-channel RF coil array. Preliminary ideas about this technology were recently presented in [24]. Compared to gEPT, it does not make symmetry assumptions, nor does it require subjective assignment of integration seed points. Numerical simulations of a digital phantom and a realistic human head model have been performed to illustrate the concept of the proposed approach and to evaluate its performance under various noise levels. A phantom experiment as well as a retrospective in vivo human brain data analysis has been performed at 7 T to demonstrate its applicability in realistic settings. CONCEPT extends the generality of gEPT and holds promise for clinical applications using a generic multi-transmit RF coil.

II. THEORY

In this section, we first derive the central physical equation from the governing electromagnetic theory that relates the measurable $B_1^{+}$ maps to EPs. We then introduce a discriminated regularization strategy using the intermediate EP gradient values, which represents the core of CONCEPT. Lastly, we describe an algorithm based on Alternating Direction Method of Multipliers (ADMM) to rapidly solve the central optimization problem derived from the previous steps. The reconstruction flowchart is shown in Fig. 1.

Fig. 1.

Reconstruction flowchart of CONCEPT.

A. Derivation of Data Fidelity Term

An MRI scan involves electromagnetic radiation from an RF coil at the proton resonance frequency or Larmor frequency. Magnitude and phase distributions of the time-harmonic magnetic field reflect electrical properties of the underlying tissue. Combining Ampère’s law and Faraday’s law, we have

$${\nabla }^2\overrightarrow{H}=-{\omega }^2\mu {\epsilon }_c\overrightarrow{H}-\nabla {\epsilon }_c\times \frac{\nabla \times \overrightarrow{H}}{{\epsilon }_c}$$ (1)

where ⍵ denotes the angular Larmor frequency, μ magnetic permeability, and ε_c := ε − iσ/ω electrical properties comprised of permittivity ε and electrical conductivity σ. By transforming $\overrightarrow{H}$ in the Cartesian system to a rotating frame using $H_1^{+}=\left(H_x+i{H}_y\right)/2$, with assumptions that $B_1^{+}:=\mu H_1^{+}\approx {\mu }_0{H}_1^{+}\text{and}\nabla H_z=\overrightarrow{0}$ for a longitudinal coil element placed in z direction, (1) can be written as

$${\nabla }^2{B}_1^{+}=-{\omega }^2{\mu }_0{\epsilon }_c{B}_1^{+}+\left(\frac{\partial B_1^{+}}{\partial x}-i\frac{\partial B_1^{+}}{\partial y}\right)g_w+\frac{\partial B_1^{+}}{\partial z}{g}_z$$ (2)

where $g_w=\frac{\partial ln{\epsilon }_c}{\partial x}+i\frac{\partial ln{\epsilon }_c}{\partial y}$ is the spatial variation of EPs in the xy plane, and $g_z=\frac{\partial ln{\epsilon }_c}{\partial z}$ is that in the z direction.

Note that $B_1^{+}=\left|B_1^{+}\right|e^{i{\phi }^{+}}$ is a complex quantity with both magnitude and phase components. The magnitude of $B_1^{+}$ can be readily mapped by a variety of well-established “B₁ mapping” methods [25]–[27]. However, the absolute phase of $B_1^{+}$ is not directly measureable, since MRI signal phase has contributions from $B_1^{+}$, $B_1^{-}$, B₀ and chemical shift. Utilizing a multi-channel transmit coil array, signal phase components irrelevant to $B_1^{+}$ can be cancelled out by taking the relative phase with respect to a reference channel r. Liu et al. has shown that the spatial gradient of absolute $B_1^{+}$ phase ∇φ⁺ can be retrieved simultaneously with g_w and g_z by solving (2), given that at least four channels are employed [23], [28]. The retrieved ∇φ⁺ and g_z are then re-introduced back into (2) as known variables. As proposed by Hafalir et al. in the cr-MREPT method [15], (2) can be transform into a linear partial differential equation (PDE) in 2D with respect to the scaled complex admittance ${\gamma }_c=\frac{1}{\omega {\epsilon }_c}$

$$\begin{gathered} ({\nabla }^2{B}_1^{+}-\frac{\partial B_1^{+}}{\partial z}{g}_z){\gamma }_c+\left(\frac{\partial B_1^{+}}{\partial x}-i\frac{\partial B_1^{+}}{\partial y}\right) \\ .\left(\frac{\partial {\gamma }_c}{\partial x}+i\frac{\partial {\gamma }_c}{\partial y}\right)=-\omega {\mu }_0{B}_1^{+} \end{gathered}$$ (3)

with unknown terms written in red. The spatial gradient term of γ_c can be discretized using central difference method

$$\begin{gathered} \frac{\partial {\gamma }_c}{\partial x}+i\frac{\partial {\gamma }_c}{\partial y}=\frac{{\gamma }_c(x+1,y)-{\gamma }_c(x-1,y)}{2dx} \\ +i\frac{{\gamma }_c(x,y+1)-{\gamma }_c(x,y-1)}{2dy} \end{gathered}$$ (4)

where (x, y) is spatial location indices, dx and dy are step size along x and y directions, respectively. As such, (3) can be written as a linear inverse problem for each transmit channel k

$$A_k\mathrm{\Upsilon }=b_k$$ (5)

where ϒ ∈ ℂ^N contains γ_c values of all N pixels in a region of support Ω in a 2D imaging slice, A_k ∈ ℂ^N×N is a sparse matrix with five nonzero elements in each row, and b_k ∈ ℂ^N is a vector of scaled $B_1^{+}$. The inverse problems derived from K different transmit channels can be concatenated as

$$A\mathrm{\Upsilon }=b$$ (6)

where A = [A₁; A₂;...; A_K], b = [b₁; b₂;...; b_K]. In this study, K = 16 sinceB₁ maps from all 16 RF coil elements were used. We utilize L₂-norm of the residue, ${\Vert A\mathrm{\Upsilon }-b\Vert }_2^2$, as a metric of data fidelity to the $B_1^{+}$ measurements, which is optimized by the pseudo-inverse of (6)

$${\mathrm{\Upsilon }}_{PDE}={\left(A^{H}A\right)}^{-1}{A}^{H}b$$ (7)

This solution is referred to as “Multi-channel cr-MREPT” in this paper.

B. Discriminated Regularization

Direct inversion of (6) results in spatially pervasive reconstruction errors due to PDE nature of the problem, intensified by noise effects and inaccuracies in $B_1^{+}$ measurement. Furthermore, numerical errors due to the finite difference model of the problem also deteriorate image quality [29]. Therefore, we propose to use the intermediate variable g_w as prior information to determine the EP transition area and guide image reconstruction. To start, we hard-threshold |g_w| to generate two binary masks

$$\{\begin{array}{l}W=\left(\left|g_w\right|\ge \kappa \right)\cap \mathrm{\Omega }\hfill \\ \overline{W}=\left(\left|g_w\right|<\kappa \right)\cap \mathrm{\Omega }\hfill \end{array}$$ (8)

where κ is an empirical threshold so that approximately 30% of the pixels in Ω are identified as transition area. In the homogeneous area $\overline{W}=1$, the L₂-norm of γ gradient is minimized to simultaneously suppress large spatial variation and error propagation. In the transition area W, the L₁-norm of γ gradient is minimized to promote edge sparsity. Therefore, the central reconstruction is formulated as

$$\underset{\mathrm{\Upsilon }}{min}\frac{1}{2}\Vert A\mathrm{\Upsilon }-b{\Vert }_2^2+{\lambda }_1\Vert W\mathrm{\Psi }\mathrm{\Upsilon }{\Vert }_1+{\lambda }_2\Vert \overline{W}\mathrm{\Psi }\mathrm{\Upsilon }{\Vert }_2^2$$ (9)

where is the finite difference operator calculating ∇γ in the region of support. Since the regularization is informed by intermediate EP contrast instead of structural MRI, this technology is named as “CONtrast Conformed Electrical Properties Tomography”, or CONCEPT.

C. Solving the Central Problem Using Alternating Direction Method of Multipliers

Equation (9) is a large-scale optimization problem involving simultaneous minimization of L₁-norm and L₂-norm. Conventional gradient-based methods, such as gradient descent or conjugate gradient method, are inefficient for solving this type of problem due to the non-smoothness of L₁-norm and the requirement of evaluating function gradient in each iteration. Alternatively, we propose to use Alternating Direction Method of Multipliers (ADMM) to solve (9). ADMM is based on the philosophy of “divide and conquer” in algorithm design: it splits the original complicated problem into a series of simpler sub-problems, which are sequentially addressed until a consensus is reached [30]. ADMM has received increasing attention due to its high efficiency for large-scale image reconstruction problems, especially those with sparsity and low-rank constraints [31]–[33].

In our implementation, we first combine the two L₂-norm terms in (9)

$$\underset{\mathrm{\Upsilon }}{min}\frac{1}{2}\Vert E\mathrm{\Upsilon }-c{\Vert }_2^2+{\lambda }_1\Vert W\mathrm{\Psi }\mathrm{\Upsilon }{\Vert }_1$$ (10)

where $E=[A;{\lambda }_2\overline{W}\mathrm{\Psi }]$ and c = [b; 0]. Introducing an auxiliary variable z so that (10) is equivalent to

$$\underset{\mathrm{\Upsilon }}{min}\frac{1}{2}\Vert E\mathrm{\Upsilon }-c{\Vert }_2^2+{\lambda }_1\Vert z{\Vert }_{1}s.t.W\mathrm{\Psi }\mathrm{\Upsilon }-z=0$$ (11)

The augmented Lagrangian can be formed as

$$\begin{gathered} L_{\rho }(\mathrm{\Upsilon },z,u)=\frac{1}{2}\Vert E\mathrm{\Upsilon }-c{\Vert }_2^2 \\ +{\lambda }_1\Vert z{\Vert }_1+Re\left(\rho u^H(W\mathrm{\Psi }\mathrm{\Upsilon }-z)\right) \\ +\frac{\rho }{2}\Vert W\mathrm{\Psi }\mathrm{\Upsilon }-z{\Vert }_2^2 \end{gathered}$$ (12)

where u is the multiplier and ρ > 0 is the augmented Lagrangian parameter. We can express ADMM updates as

$$\begin{gathered} {{\rm Y}}^{k+1}=arg\underset{\mathrm{\Upsilon }}{min}{L}_{\rho }\left(\mathrm{\Upsilon },z^k,u^k\right) \\ =arg\underset{\mathrm{\Upsilon }}{min}\frac{1}{2}\Vert E\mathrm{\Upsilon }-c{\Vert }_2^2+Re\left(\rho {\left(u^k\right)}^H\left(W\mathrm{\Psi }\mathrm{\Upsilon }-z^k\right)\right) \\ +\frac{\rho }{2}{\Vert W\mathrm{\Psi }\mathrm{\Upsilon }-z^k\Vert }_2^2 \\ =arg\underset{\mathrm{\Upsilon }}{min}\frac{1}{2}\Vert E\mathrm{\Upsilon }-c{\Vert }_2^2+\frac{\rho }{2}{\Vert W\mathrm{\Psi }\mathrm{\Upsilon }-z^k+u^k\Vert }_2^2 \\ ={\left(E^{H}E+\rho {\mathrm{\Psi }}^H{W}^{H}W\mathrm{\Psi }\right)}^{-1}\left(E^{H}c+\rho W\mathrm{\Psi }\left(z^k-u^k\right)\right) \end{gathered}$$

$$\begin{gathered} z^{k+1}=arg\underset{z}{min}{L}_{\rho }\left({\mathrm{\Upsilon }}^{k+1},z,u^k\right) \\ =arg\underset{z}{min}{\lambda }_1\Vert z{\Vert }_1+Re\left(\rho {\left(u^k\right)}^H\left(W\mathrm{\Psi }{\mathrm{\Upsilon }}^{k+1}-z\right)\right) \\ +\frac{\rho }{2}{\Vert W\mathrm{\Psi }{\mathrm{\Upsilon }}^{k+1}-z\Vert }_2^2 \\ =arg\underset{z}{min}{\lambda }_1\Vert z{\Vert }_1+\frac{\rho }{2}{\Vert z-W\mathrm{\Psi }{\mathrm{\Upsilon }}^{k+1}-u^k\Vert }_2^2 \\ =S_{{\lambda }_1/\rho }\left(W\mathrm{\Psi }{\mathrm{\Upsilon }}^{k+1}+u^k\right) \end{gathered}$$

$$u^{k+1}=u^k+W\mathrm{\Psi }{\mathrm{\Upsilon }}^{k+1}-z^{k+1}$$

where S_τ (a) denotes the soft-thresholding function of a with τ. z⁰ and u⁰ are initialized with zero vectors. Note that the algorithm does not require initialization of ϒ, since ϒ¹ is calculated from z⁰ and u⁰ in the first iteration. The stopping criterion is

$$\frac{{\Vert {\mathrm{\Upsilon }}^{k+1}-{\mathrm{\Upsilon }}^k\Vert }_2}{{\Vert {\mathrm{\Upsilon }}^{k+1}\Vert }_2}\le tolerance,ork>k_{max}$$ (13)

We observed that the algorithm typically converges within 15 iterations using tolerance = 10⁻⁴.

III. METHODS

A. Numerical Simulations

Electromagnetic simulations were performed in a Finite-Difference Time-Domain (FDTD) based software (Sim4Life version 3.4, ZMT, Zurich, Switzerland). The 16-channel stripline coil array used in the experiment was numerically modeled and loaded with a cylindrical phantom containing tube anomalies, an extruded (in z-direction) brain slice with a circular anomaly, and a realistic head model (Duke Model, Virtual Family [34]), as shown in Fig. 2. The digital phantom was 160 mm tall, with a long axis of 210 mm and a short axis of 168 mm, and was assigned the mean electrical properties of brain at 298 MHz (0.55 S/m and 52_ε0) [35]. Five sets of tube anomalies with diameters of 5 mm, 10 mm and 20 mm were evenly distributed in azimuthal direction, spanning the physiologically relevant EPs range of 0.2–2.5 S/m and 40_ε0-100_ε0. For the head models, EPs of various tissues were assigned based on Gabriel et al. [35]. The anomaly in Fig. 2b mimics a brain tumor or hemorrhage that disrupts structural symmetry, with EPs of 2 S/m and 76_ε0. The imaged objects were positioned in the center of the coil, mimicking actual experimental setup.

Fig. 2.

Simulation setup. The 16-channel stripline RF coil array is loaded with (a) a numerical phantom, (b) an extruded brain slice with a 50-mm-diameter anomaly in red, and (c) a realistic 3D Duke head model. The last one is shown as voxels after discretization.

Spatially anisotropic discretization of the models was performed by the software based on their geometry and EP distributions to achieve the optimal performance. A maximal spatial step of 1.5 mm was prescribed inside the phantom/head model to ensure high precision. The 16 channels of the coil were excited sequentially with a normalized input power of 1 W. The simulated complex $B_1^{+}$ field was re-gridded to a homogenous 1.5 mm × 1.5 mm × 1.5 mm grid using cubic splines. Relative $B_1^{+}$ phase was calculated taking each channel as reference. Complex white Gaussian noise was added to $B_1^{+}$ to adjust the signal to noise ratio (SNR) to desired levels.

B. Phantom Experiment

An agar phantom was constructed to validate the performance of CONCEPT in realistic settings. The “background” or main body of the phantom contained deionized water, agar, NaCl, NiCl₂· 6H₂O with a mass ratio of 1000:15:1.12:0.5. Eps of the phantom at 298 MHz were measured using a dielectric probe (85070E, Keysight Technologies) as 0.294 S/m and 78.6_ε0. The main body was poured into a cylindrical jar with 120 mm diameter and 150 mm height. The contrast solutions were prepared based on deionized water, with NaCl to adjust conductivity and polyvinylpyrrolidone (PVP) to adjust permittivity [36]. Three solutions with different EPs were prepared and measured using the probe. Solutions were filled into a 25-mm-diameter plastic tube and a 5-mm-diameter straw, and inserted into the phantom background. The whole phantom was allowed to stand for 24 hours in room temperature to solidify. The insulating materials of the anomaly wall do not significantly affect electromagnetic wave propagation and EP reconstruction at Larmor frequency, as investigated in [37].

Experiments were performed on a 16-channel Siemens 7T system equipped with 16 × 1 kW power amplifiers (CPC, Hauppauge, NY) with their transmit power and phase remotely controlled by a phase/gain control unit, interfaced with a locally developed software toolbox. A 16-channel RF coil array, consisting of stripline elements aligned with the main magnetic field direction, was used to obtain multiple $B_1^{+}$ patterns. The coil array was connected to the MRI system through a custom 16-channel T/R switch box. Each coil element was tuned and matched at the operating frequency (297.2 MHz) when loaded with object under detection. $B_1^{+}$ mafgnitude map from each channel was acquired using a hybrid method of multi-slice gradient-echo (GRE) and 3D Actual Flip-angle Imaging (AFI) method [25], [38], [39]. For the former, each coil element was excited sequentially in small-flip-angle regime (< 15°), with TR = 76 ms, TE = 3.3 ms and 8 averages, duration 23.0 min; For the latter, two shimmed $B_1^{+}$ patterns resembling CP+ mode and CP2+ mode, respectively, were imaged with 1 average each, nominal flip angle = 55^◦, TR = 20/120 ms, TE = 3.2 ms, total duration 19.4 min. The entire $B_1^{+}$ mapping session takes approximately 60 min, with additional ~10 min of pilot single-channel $B_1^{+}$ acquisition for shimming purpose, and ~10 min of pilot AFI acquisition for calibration of flip-angle. Hybrid $B_1^{+}$ reconstruction of each transmit channel was initially performed for the two shimmed patterns separately, then combined with weights of the flip angle map, so as to circumvent inaccuracies due to RF nulling spots present in the CP+ and $\text{CP}2+B_1^{+}$ maps. Details are provided in Appendix A. $B_1^{+}$ maps were acquired with in-plane resolution of 1.5 mm × 1.5 mm over a stack of 12 contiguous slices of 3 mm thickness.

C. In Vivo Validation

A retrospective data analysis of human brain was performed. The data was described in [23]. The study was approved by the Institute Review Board at the University of Minnesota. Briefly, single-channel $B_1^{+}$ mapping with relative phase was performed at 7 T in a similar setup to the phantom experiment using the hybrid $B_1^{+}$ mapping method of a single AFI (1 average, 9 min) and a series of small-flip-angle GRE (10 average, 28 min) in a resolution of 1.5 mm × 1.5 mm × 5.0 mm.

D. Image Reconstruction

Simulated noisy data and experimental data were de-noised using a low-pass Gaussian filter with a kernel size of 5×5×3 voxels and standard deviation of 1.2 × 1.2 × 0.8 voxels. The spatial gradient and Laplacian of $B_1^{+}$ were generated based on second order polynomial fitting using a 5 × 5 × 3 Savitzky-Golay filter [40]. Image reconstruction was performed using MATLAB (The MathWorks, MA, USA) codes developed in-house, running on a desktop with a 3.4 GHz GPU and 32 GB RAM.

IV. RESULTS

A. Digital Phantom Simulation

The simulated transmit B₁ field and absolute phase gradient from channel 1 are shown in Fig. 3. The spatial variation of B₁ phase is smooth except for the area corresponding to low B₁. The retrieved phase gradient using (2) is in good agreement with the ground truth.

Fig. 3.

Transmit B₁ field and phase gradient from channel 1 of the phantom simulation. (a) Transmit B₁ magnitude and absolute phase. (b) Phase gradient calculated from the absolute phase juxtaposed with the retrieved phase using equation (2).

EP reconstruction results using Helmholtz equation, Multichannel cr-MREPT, gEPT and CONCEPT are compared in Fig. 4. For Helmholtz equation, accurate reconstruction is limited to large homogenous areas, such as the inside of large tubes as well as the background. Severe oscillations can be observed at the interface of different contrast components. In particular, reconstruction results of small structures (5 mm and 10 mm tubes) are dominated by boundary artifact. For Multi-channel cr-MREPT, boundary artifact is significantly mitigated, thanks to the inclusion of the EP variation term. Nevertheless, chessboard-like artifacts still appear due to the use of finite difference as an approximation of spatial derivative (white arrow). This artifact is more pronounced in the tubes with large EP values because of higher contrast to the background. In addition, erroneously reconstructed values can be identified near the phantom boundary close to individual RF coil elements (green arrows). This artifact arises from inaccurate estimation of local high-order spatial derivatives of $B_1^{+}$ due to its rapid attenuation near each coil element. As shown in the profiles in Fig. 4b, without constraints, such artifacts can bleed into the contrast tubes and bias reconstruction results. For gEPT, one seed point in the center of the phantom was used to implement the algorithm (red asterisk). High-frequency chessboard artifact exists in both EP maps, on top of low-frequency bias induced by violation of the “symmetry assumption”. The EP profiles demonstrate spatially dependent bias in the reconstruction result, which tends to be smaller near the seed point and accumulates going outward radially. For CONCEPT, reconstructed EP values and profiles are close to those of the target. The chessboard artifact and spatial bias are minimized thanks to the informed regularization.

Fig. 4.

Comparison of different reconstruction methods using simulated phantom data. (a) Reconstructed conductivity and permittivity using Helmholtz equation, Multi-channel cr-MREPT, gEPT and CONCEPT are compared with target maps. Red asterisk labels position of the seed point used by gEPT. Both transmit and receive B₁ were employed for gEPT. Other methods do not require seeds or receive B₁. White and green arrows denote chessboard artifact and B₁ peak related artifact, respectively. (b) Reconstructed EP profiles along the red line on the target maps in (a).

Reconstructed EP images using CONCEPT at various SNR levels are shown in Fig. 5a. The obtained EP values inside the large tubes are fitted against the target values in Fig. 5b. Accurate conductivity and permittivity can be obtained for SNR = 200 and 100 within the ranges of interest, with reconstruction errors being ≤ 0.15 S/m for conductivity and ≤ 5.0_ε0 for permittivity. On the conductivity images, the smallest tubes can also be well differentiated from the background and from each other, except for the one with the least conductivity contrast to the background (0.25 S/m). Detection of small structures on the permittivity image is more difficult. Only the two tubes with the largest contrast (≥ 28_ε0) are clearly visible. At SNR = 50, the fitting results show global overestimation in both contrasts. Conductivity retains reasonable imaging quality especially for the large tubes, yet permittivity degrades significantly. Notably, overestimation is pronounced for the tube with the largest EPs. Potential reasons for this overshot are discussed in Appendix B.

Fig. 5.

CONCEPT reconstruction results at SNR = 200, 100 and 50. (a) Reconstructed conductivity and permittivity maps. (b) Linear fitting results of the retrieved values inside the largest tubes against the target values. R² values of fitting for SNR = 200, 100, 50 are 0.99, 0.99, 0.97 for conductivity and 0.99, 0.98, 0.86 for permittivity.

B. Extruded Brain Simulation

Reconstructed conductivity maps of the extruded brain model using gEPT and CONCEPT are compared in Fig. 6. As a consequence of asymmetry in the imaged target, gEPT leads to spatial bias in the result even using the symmetric 16-channel RF coil. Notably the conductivity of the simulated anomaly is underestimated. This bias is alleviated using CONCEPT.

Fig. 6.

Comparison of reconstructed conductivity using gEPT and CONCEPT in the presence of asymmetry. Red rectangular denotes coils from which B₁ is used, and gray denotes coils from which B₁ (both transmit and receive) is discarded.

Fig. 6 also shows results using an asymmetric RF coil array by discarding $B_1^{+}$ and $B_1^{-}$ information from the coils shown in gray, mimicking dead channels due to malfunction of amplifier/TR switch/coil element or a different design of the RF coil. Due to this hardware asymmetry, gEPT reconstruction error is significantly elevated. In contrast, CONCEPT demonstrates robustness with a mild change in the reconstruction results.

C. Realistic 3D Brain Simulation

As an intermediate step of CONCEPT, the determination of the regularization mask from |g_w| is shown in Fig. 7. By solving equation (2), g_w can be retrieved which faithfully reflects the boundary of tissues, even in the noisy case (Fig. 7a). Selecting a threshold for the transition area involves a tradeoff between sensitivity to low-contrast or small structures and robustness against measurement noise. 30% percentile seems to give a good balance between the two in this case, meaning that most transition areas can be correctly identified (true positive rate is high) without including too many noisy pixels (true negative rate is high).

Fig. 7.

Determination of CONCEPT mask in the 3D Duke simulation. (a) Retrieved |g_w| using either noise-less or noisy (SNR = 50) B₁. (b) Effects of the threshold. 20%, 30%, 40% are the percentage of pixels considered as transition pixels.

Three axial slices of the reconstructed brain model with a spatial gap of 12 mm are shown in Fig. 8. Note that the low-pass filter was not applied to the noise-less $B_1^{+}$ data, but only to the noisy $B_1^{+}$ (SNR = 50). For both SNR levels, all three slices show successful reconstruction of the contrasts amongst major brain tissues, e.g. white matter (WM), gray matter (GM) and cerebrospinal fluid (CSF). The reconstructed values of these three tissues are summarized in Table I. Consistent with the digital phantom results, we observe overestimation in the permittivity of CSF, even for the noiseless case. This is potentially due to the increased complexity of the real-3D Duke model compared to the 2D-projected phantom model. The magnitude of this overestimation is smaller for the noisy reconstruction because of the smoothing effects of the low-pass filter.

Fig. 8.

Reconstructed EPs of a simulated 3D Duke head model. Shown are the target maps and reconstruction results using noise-free B₁ data as well as noisy B₁ with SNR = 50. The three axial slices are separated by 12 mm.

TABLE I.

Reconstructed EPs of Duke Head Model

		Target	Noise-free	SNR = 50
WM	σ	0.41	0.58 ± 0.10	0.62 ± 0.12
	ε	43.8	47.2 ± 4.3	52.0 ± 4.8
GM	σ	0.69	0.63 ± 0.30	0.68 ± 0.33
	ε	60.1	56.1 ± 9.5	58.9 ± 9.2
CSF	σ	2.22	1.70 ± 0.34	1.57 ± 0.29
	ε	72.8	105.2 ± 9.6	100.8 ± 7.0

σ in S/m and ε in ε₀

D. Phantom Experiment

A structural image of the phantom and B₁ images from a single channel are shown in Fig. 9. Local spotty artifacts can be seen in the hybrid B₁ map when using a single shim pattern (Fig. 9c white arrows). These artifacts occur at the locations of RF nulls due to complex destructive interference of $B_1^{+}$ between channels [39]. By contrast, B₁ image obtained merging two shim patterns is free of such errors, bearing high resemblance to the B₁ pattern measured by AFI (Fig. 9b).

Fig. 9.

Hybrid transmit B₁ mapping based on dual shims. (a) An axial slice of the agar phantom used in the experiment. Scale bar is 10 mm. (b) Transmit B₁ magnitude from channel 5 (ch5) measured using AFI. (c) Hybrid B₁ mapping results of ch5 based on CP-like shim I and CP2+-like shim II, and their combination. α_I and α_II are measured using AFI. White arrows point to locations with inaccurate calculated B₁ due to RF nulls in the corresponding shimmed B₁ pattern.

Based on the combined hybrid B₁ images, reconstructed EPs of the phantom are shown in Fig. 10, and summarized in Table II. As an intermediate quantity, the spatial variation or gradient strength of EPs across the phantom can be robustly retrieved, enabling a clean separation of EP contrast boundary (Fig. 10a). Importantly, compared to the Helmholtz based reconstruction, CONCEPT gives accurate conductivity reconstruction results not only in homogenous areas but also around contrast boundaries. In addition, all three 5-mm small tubes are visible from the reconstructed map. Similar to the simulations, there is overshot in the reconstructed permittivity of the tubes with high EP values (tube III). Reconstruction errors of the other tubes are relatively smaller.

Fig. 10.

Phantom reconstruction results. (a) Intermediate gradient strength distribution and CONCEPT mask generated by hard-thresholding. (b) Comparison of the reconstruction results using Helmholtz equation and CONCEPT.

TABLE II.

Reconstructed EPs of the Agar Phantom

		Probe Measurement	Reconstructed
Tube I	σ	0.65	0.70 ± 0.05
	ε	50.2	65.0 ± 1.6
Tube II	σ	1.02	0.98 ± 0.02
	ε	62.7	73.6 ± 0.7
Tube III	σ	2.14	2.14 ± 0.07
	ε	78.8	124.0 ± 4.3

σ in S/m and ε in ε₀

E. In Vivo Validation

Reconstructed in vivo images are presented in Fig 11. Large brain compartments, such as the ventricles filled with CSF, and the GM, can be distinguished from white matter with reasonably clear boundaries. However, details of the gyri and sulci are difficult to identify as a result of low-pass filtering. Conductivity in CSF, GM and WM are found to be 1.7±0.58, 0.9 ± 0.42 and 0.5 ± 0.25 in S/m, respectively. Reconstructed permittivity values for the same brain compartments are 93± 25.4, 71 ± 15.2 and 61 ± 3.4 in ε₀.

Fig. 11.

In vivo EPs imaging result using CONCEPT compared to T1-weighted anatomical image.

V. CONCLUSION AND DISCUSSIONS

In this study, we propose a novel approach, dubbed “CONCEPT”, to retrieve EPs from MRI using multiple RF transmission. CONCEPT leverages on the intermediate EP gradient information for edge detection and discriminated regularization. The resultant large-scale, mixed L₁ and L₂ norm image reconstruction problem is solved efficiently using ADMM. A systematic investigation using numerical simulations and phantom experiments at 7 T MRI demonstrate its capability to retrieve accurate EP values in large homogeneous areas (≥ 20 mm), as well as to detect small structures down to 5 mm, and to faithfully reflect the EP transition zone. CONCEPT addresses the symmetry assumptions and seed point requirements of the gEPT method, representing a step forward in clinical applications of the multi-channel EPT approach.

The mere Helmholtz-based approaches of MREPT suffer from boundary artifact, spatial bias, and noise amplification effects, due to a substantial simplification of the governing physical equations and to the use of high-order spatial differentiation operators. These adverse effects can be partially mitigated using numerical regularization with prior information such as contrast edges detected by structural MRI [7], [14]. Nevertheless, spurious EP contrast may be introduced, since it is fundamentally different from magnetic relaxation contrasts and does not necessarily spatially coincide with structural MRI. In comparison, CONCEPT exploits EP gradient information directly derived from $B_1^{+}$, thus it is in principle immune to other sources of contrast such as MR relaxation effects. However, CONCEPT is based on the framework of multi-channel RF transmission, which is not yet widely available in clinical settings. Meanwhile, acquisition of multiple high-resolution B₁ requires lengthy scans which potentially compromise patient comfort and throughput.

For single-channel PDE-based MREPT methods, a Dirichlet boundary condition is usually enforced to produce a stable result, since the central equation is underdetermined and its possible solutions are infinite [15], [16]. However, boundary conditions may be complex and intractable in practical settings, such as imaging of the entire human brain. Erroneously assigned boundary conditions can adversely affect reconstruction result not only in the periphery of the imaged structure, but in the center of it through the spatial differentiation term. Thanks to the plurality of $B_1^{+}$ provided by a multi-transmit RF coil, the central equation of CONCEPT is overdetermined and it does not rely on any boundary condition. Nevertheless, certain forms of relaxed boundary condition, e.g. constrained-value reconstruction in physiologically relavent ranges, may be enforced to further stablize the result.

The global PDE version of MREPT is prone to reconstruction bias in the center of the imaged structure, so-called “low convection field (LCF) artifact” by Hafalir et al. in the cr-MREPT study [15]. It is attributed to small values of $\partial B_1^{+}/\partial x-i\partial B_1^{+}/\partial y$ in the center of a quadrature coil which weights the spatial derivative of EPs. This artifact can also be observed when using multiple transimit coils, since all channels share a $\text{small-}{B}_1^{+}$ region in the center of the imaged structure, making it both low-SNR and low-convection-field [12]. A couple of strategies can be employed to suppress this artifact. Firstly, the LCF can be shifted by padding dielectric material near the subject [15]. For a multi-transmit coil, this can also be potentially done by manipulating B₁ shim patterns. Secondly, regularization and/or a prior information can be employed to correct the bias, e.g. enforcing constrained values in “constrained cr-MREPT” [15] or using regularization in CONCEPT. Last but not the least, spatial propagation of noise should be suppressed by low-pass filering noisy B₁ in preprocessing and utilizing noise-robust differential kernels, such as the Savitzky-Golay filters [40].

In this study, each imaging slice was reconstructed separately, and g_z was not explicitly emploited except as a part of coefficient in (3) weighted by $\partial B_1^{+}/\partial z$ which is a small value in the central part of the longitudinal stripline coil. 3D EPs reconstruction can potentially be performed by incorporating EPs gradient in the z direction into the central equation. However, the following two factors need to be considered. Firstly, the ubiquitous quadrature coil and TEM coils may not be suitable for this purpose. Since EPs gradient in the z direction is weighted by $\partial B_1^{+}/\partial z$ small values of $\partial B_1^{+}/\partial z$ in these coils make z-gradient terms neglegible in the central equation compared to x- and y- gradient terms weighted by $\partial B_1^{+}/\partial x$ and $\partial B_1^{+}/\partial y$ (Fig. 12). Non-TEM coils with a larger $B_1^{+}$ variation in the z direction, such as a small loop array, may better pronounce EPs gradient along z. Secondly, the size of A matrix in (6) scales quadratically with ϒ, and it becomes more complicated internally due to incorporation of z differentiation. These factors increase the computational complexity and size of the problem which may lead to instability of the reconstruction result. 999

Fig. 12.

Comparison of $B_1^{+}$ gradient components in x, y and z directions using the realistic 3D Duke model.

Compared to the multi-channel gEPT method [23], CONCEPT has the following merits. Firstly, the reconstruction process does not require assignment of integration seed, which is unavailable in realistic settings. Secondly, $B_1^{-}$ is no longer needed, leading to a shorter scan time and improved patient compliance. Last but not least, the imaging setup, including both the subject brain and RF coil, does not need to be symmetric, opening the potential to image diseased brain and irregularly shaped body organs using customized RF coil. In fact, since $B_1^{-}$ and $B_1^{+}$ are completely disentangled in CONCEPT, one is no longer limited to transceiver RF coil arrays. Instead, separate transmit and receive coil arrays, that are more ubiquitous than customized transceiver arrays, can be utilized. In addition, such coil type generally has a higher count of receive elements with dedicated high-performance electronics in a close proximity to the imaged target, which significantly improves SNR for MR data acquisition. Additional transmit elements could also be used to obtain a larger number of different $B_1^{+}$ patterns to improve the conditioning of the inverse problem.

In its current form, CONCEPT is based on multiple $B_1^{+}$ maps, yet its counterpart using multiple $B_1^{-}$ maps can be derived in a similar way. The major advantage of using multiple $B_1^{-}$ lies in its high temporal efficiency, since data acquisition can be accomplished in a single acquisition [41]. However, $B_1^{-}$ maps are typically weighted by proton density which cannot be removed in a straightforward way. Computing the ratio of $B_1^{-}$ maps between different channels cancels out the shared proton density weighting, yet the resultant relative $B_1^{-}$ magnitude cannot be directly utilized by CONCEPT. Alternatively, proton density can be taken as unknown variables and reconstructed simultaneously with EPs [28].

The central equation of CONCEPT (9) contains two regularization terms, i.e. L₁-norm and L₂-norm of spatial gradient of ϒ. The effective areas of these two norms are controlled by a binary mask W that is derived from hard-thresholding the intermediate EP gradient. More advanced regularization strategies may be employed to further improve image quality. Firstly, W could be designed as a continuous function of EP gradient, such as linear or logarithm, to better characterize EP distribution of imaged object and gain robustness against measurement noise. Secondly, the L₁-norm of spatial gradient of ϒ, known as total variation (TV), could be replaced by total generalized variation (TGV) containing higher orders of spatial derivative. TGV does not imply the piecewise constant assumption, thus it is capable of alleviating the “staircase” artifact and capturing contrast changes that are naturally continuous [42]. Thirdly, an additional weighting function can be designed and imposed to ϒ to counteract $B_1^{+}$ weighting to the reconstruction results seen in Fig. 8. Lastly, low-rank constraints could be imposed to better represent different types of tissue, as has been shown in other parametric MRI technologies [31]. These features can be added to CONCEPT and solved using ADMM without significant modifications to the current algorithm structure. However, caution needs to be used as a much higher dimension of regularization parameters needs to be explored to obtain the optimal reconstruction outcome.

CONCEPT is based on the 3rd order spatial differentiation of $B_1^{+}$, since it requires the spatial derivative of ∇φ⁺ which is calculated from an equation involving ${\nabla }^2{B}_1^{+}$ [23]. This makes the algorithm more sensitive to noise contaminations, flow effects and motion, which can only be partially mitigated by regularization. Data pre-processing using intensive low-pass filters can alleviate the adverse effects, yet at a price of resolution loss which can be deleterious when targeting small structures. As a result, only large brain compartments are visible in the current in vivo brain imaging result. To visualize complex anatomical structures using CONCEPT, improvements in SNR are warranted. This could be achieved by implementation of recently introduced B₁ mapping methods such as DREAM [27] or Bloch-Siegert shift [26]. In addition, transmit channels could be aggregated to generate a smaller number of “virtual channels” that have better spatial coverage than a single transmit element. The gained SNR can be traded for shorter scan time or higher resolution as desired.

APPENDIX

A. Hybrid Transmit B₁ Mapping based on Dual Shims

At the beginning of the transmit B₁ mapping session, pilot single-channel B₁ is acquired, which is used to determine two static B₁ shim settings resembling CP+ (circularly polarized, $\text{phase}\text{increment}\approx \frac{{360}^{°}}{16}={22.5}^{°}$) and CP2+ $(\text{phase}\text{increment}\approx \frac{{720}^{°}}{16}={45}^{°})$. These two shims are used because they have optimized transmit efficiency and their B₁ patterns are complementary to each other [43]. Flip-angle images α_I and α_II are acquired in these two modes using AFI. Single-channel GRE is acquired once, and combined with α_I and α_II separately using the original hybrid B₁ mapping method [38] to obtain B_1,I and B_1,II (Fig. 9c). Final single-channel B₁ is calculated using weighted sum

$$B_1=\frac{{\alpha }_{\text{I}}{B}_{1,\text{I}}+{\alpha }_{\text{II}}{B}_{1,\text{II}}}{{\alpha }_{\text{I}}+{\alpha }_{\text{II}}}$$ (14)

The Combined B₁ Is Thus Free From Spurious Singularities Due to Small Flip-Angle in Either Shim I or Shim II

B. Analysis of Error Propagation

In this section, we propose and analyze potential reasons why the reconstructed permittivity map has more severe distortion and overshot during CONCEPT reconstruction.

Firstly, as pointed by Marques et al. [41], given that the real and imaginary part of ε_c can be reconstructed at the same SNR level, a change in permittivity of 10ε₀ is commeasurable to a change in conductivity of 0.17 S/m at 7 T. In human brain, the normal physiological range of permittivity is 40_ε0 - 70_ε0, corresponding to a conductivity range of 0.68 S/m - 1.19 S/m, which is much narrower than the actual physiological range of conductivity 0.41 S/m - 2.22 S/m. Therefore, permittivity is relatively more susceptible to disturbances such as noise or B₁ inaccuracy.

Secondly, the overestimation in permittivity may originate from conversion of the reconstructed γ_c to conductivity and permittivity. As shown in Fig. 13, for pixels with larger conductivity values, the same value of the real part of γ_c corresponds to two very different permittivity values. The accuracy of reconstructed permittivity is thus highly dependent on the imaginary part of γ_c, which is unfortunately insensitive to permittivity change in this range. For example, two data points shown in Fig. 13 have the same conductivity value of 2.2 S/m and the same real part of γ_c. Despite their significant difference in permittivity (142ε₀), there is only 21% difference in the imaginary part of γ_c compared to its dynamic range. Therefore, solving for γ_c instead of ε_c can result in unreasonably larger value of permittivity.

Fig. 13.

Projection maps from conductivity and permittivity to complex admittance. Shown are the normalized real (a) and imaginary (b) parts of admittance as functions of conductivity and permittivity, contoured by red lines with a step size of 0.05.

Lastly, reconstruction error on contrast boundary may perfuse to the whole image due to the nature of partial differential equation. Spatially constrained regularization may mitigate this effect but it cannot be eliminated.