Magnetic Resonance Electrical Property Mapping at 21.1 T: A study of conductivity and permittivity in phantoms, ex vivo tissue and in vivo ischemia

Abstract

Objective

Electrical properties (EP), namely conductivity and permittivity, can provide endogenous contrast for tissue characterization. Using electrical property tomography (EPT), maps of EP can be generated from conventional MRI data. This report investigates the feasibility and accuracy of EPT at 21.1 T for multiple RF coils and modes of operation using phantoms. Additionally, it demonstrates the EP of the in vivo rat brain with and without ischemia.

Methods

Helmholtz-based EPT was implemented in its Full-form, which demands the complex B₁⁺ field, and a simplified form requiring either just the B₁⁺ field phase for conductivity or the B₁⁺ field magnitude for permittivity. Experiments were conducted at 21.1 T using birdcage and saddle coils operated in linear or quadrature transceive mode, respectively. EPT approaches were evaluated using a phantom, ex and in vivo Sprague-Dawley rats under naïve conditions and ischemic stroke via transient middle cerebral artery occlusion.

Results

Different conductivity reconstruction approaches applied to the phantom displayed average errors of 12–73% to the target acquired from dielectric probe measurements. Permittivity reconstructions showed higher agreement and an average 3–8% error to the target depending on reconstruction approach. Conductivity and permittivity of ex and in vivo rodent brain were measured. Elevated EP in the ischemia region correlated with the increased sodium content and the influx of water intracellularly following ischemia in the lesion were detected.

Discussion

The Full-form technique generated from the linear birdcage provided the best accuracy for EP of the phantom. Phase-based conductivity and magnitude-based permittivity mapping provided reasonable estimates but also demonstrated the limitations of Helmholtz-based EPT at 21.1 T. Permittivity reconstruction was improved significantly over lower fields, suggesting a novel metric for in vivo brain studies. EPT applied to ischemic rat brain proved sensitivity to physiological changes, motivating the future application of more advanced reconstruction approaches.

Introduction

Electrical properties (EP), i.e. conductivity and permittivity, are fundamental properties of biological tissue that may vary under physiological and pathological conditions (Schepps and Foster 1980, Joines et al 1994, Lazebnik et al 2007, Hancu et al 2015). Therefore, EP of biological tissue potentially could be used as an endogenous quantitative contrast to provide additional information to MRI for diagnostic and therapeutic monitoring purposes (Liu et al 2017b, Katscher and van den Berg 2017, Xiaotong Zhang et al 2014). EP are functions of the frequency of the applied radiofrequency (RF) field and vary relative to tissue biophysical properties, such as water content, ionic concentrations and compartmentalization (e.g., intra- versus extracellular spaces) (Ouwerkerk et al 2007, Pethig 1984, Schwan and Foster 1989, Gabriel et al 1996). Knowledge of EP is also valuable for accurate safety assessments requiring in vivo specific absorption rate (SAR) mapping in high-field MRI (Voigt et al 2012, Zhang et al 2014, Collins et al 2004, Zhang et al 2013b). In recent years, a group of techniques, commonly referred to as MR electrical property tomography (MREPT or EPT), have been proposed to image tissue EP noninvasively at Larmor frequency with minimal alteration of existing MR scanner instrumentation or RF architecture (Katscher et al 2009, Xiaotong Zhang et al 2010, Katscher et al 2013b, Liu et al 2013). Although these approaches have been conducted more commonly at clinical field strengths, the challenges and benefits of applying standard EPT techniques at ultra-high preclinical fields (those above 9.4 T) have not been studied previously. EPT could be affected dramatically at higher frequencies because of constraints placed on certain approximations used in EPT reconstruction as well as RF penetration and skin depth. The current report demonstrates the possibility of implementing EPT at 21.1 T, the highest preclinical field available, for phantoms, ex vivo tissue and in vivo rodent brain. At this field strength, notable accuracy in permittivity reconstructions has been achieved that provided insight into the relative permittivity differences in the ischemic brain lesion.

Determining EP from the MRI radio frequency (RF) field was first proposed in 1991 for a heterogeneous layer model under RF penetration (Haacke et al 1991). Later, experimental and simulation studies showed the strong dependence of the curvature of RF fields to the EP of the biological tissues (Alecci et al 2001, Yang et al 2002, Collins et al 2005, Vaidya et al 2016). Relatively recent studies have shown systematic methods to calculate EP from spatial variation of B₁⁺ (the effective transmit RF field) in phantoms and animal tissue samples (Wen 2003, Katscher et al 2009, Bulumulla et al 2012, Zhang et al 2013b). Subsequently, simplified EPT reconstructions based only on B₁⁺ phase or magnitude were introduced to reduce the imaging and computational time at 1.5 T (Voigt et al 2011) and 7 T (van Lier et al 2012a). In the simplified standard EPT approaches based on the Helmholtz equations, second order derivatives of the B₁⁺ phase or magnitude are used to generate conductivity and permittivity maps, respectively (Van Lier et al 2014). More recently, other reconstruction approaches have been proposed to improve noise robustness of EP mapping (Bevacqua et al 2019, Leijsen et al 2019, Shin et al 2019, Serralles et al 2016, Gurler and Ider 2016, Li et al 2017, Motovilova et al 2015, Wang et al 2019a, Arduino et al 2018), and several studies have demonstrated application of EPT for clinical diagnosis (Katscher and van den Berg 2017, Katscher et al 2013a, Van Lier et al 2011, Huhndorf et al 2013, Gurler et al 2016, van Lier et al 2012b, Wang et al 2019b). Studies have shown novel contrast in the detection of cancer and tumor progression particularly by conductivity mapping (Kim et al 2016, Tha et al 2018, Balidemaj et al 2016, Liu et al 2017a, Shin et al 2015, Kim et al 2018). However, the more routine clinical use of permittivity as well as conductivity mapping is hampered by inaccuracy. As recent studies have shown, the inaccuracy in standard EPT techniques, especially with respect to permittivity mapping (Gavazzi et al 2019) is due to boundary issues (Jin Keun Seo et al 2012, Duan et al 2016), limited signal-to-noise ratios (SNR) and spatial resolution (Shin et al 2015, Mandija et al 2017) at clinical field strengths. More specifically, because numerical approaches to solve the Helmholtz-based EPT equations utilize the Laplacian of the B₁⁺ phase and magnitude maps, the EPT solution is very sensitive to the noise and low SNR of these maps, which can severely degrade the quality of EP maps (Lee et al 2015b, Michel et al 2014). These limitations have motivated the current study as a means to investigate the benefits of implementing Helmholtz-based EPT at 21.1 T. Interestingly, at the Larmor frequency of 900 MHz, most biological tissues (e.g. gray matter with σ = 0.94 S/m and ε_r = 52.72 (Gabriel 1996) ) are between the extreme of ωε₀ε_r ≫ σ for which a magnitude-only based permittivity estimation has increased accuracy and σ>>ωε₀ε_r for which the phase-only conductivity has increased accuracy (Seo et al 2014). As EPT at 21.1 T falls in this intermediate regime, it was necessary to determine if conventional EPT approaches are adequate to produce reasonable and spatially resolved estimates of conductivity and permittivity for preclinical specimen. Although higher field strengths can provide better SNR as well as higher spatial resolution and effects of EP on the B₁⁺ may be more pronounced, the feasibility and accuracy of implementing the Helmholtz-based EPT approach at higher preclinical fields needs to be investigated. Indeed, for clinical MR scanners, a comparison study between fields of 1.5, 3 and 7 T has shown the tradeoff between SNR improvement and validity of B₁⁺ phase approximation (Van Lier et al 2014).

Testing the assumptions intrinsic to Helmholtz-based EPT, this study aimed to compare the precision of Phase-based versus Full-form conductivity mapping at 21.1 T using phantoms, ex vivo and in vivo rat brains. For phantoms, the comparison is extended to two different coils: a linear birdcage and a quadrature saddle coil to investigate the effect of coil structures and operational modes on conductivity estimation at 900 MHz. Additionally, the possibility of generating permittivity maps using Magnitude-based and Full-form reconstructions are investigated in both phantoms and rat brain. As a case study, different EPT approaches were applied to an in vivo rodent model of transient cerebral ischemia to investigate sensitivity to conductivity and permittivity alterations.

Methods

Dielectric Probe Measurements and Phantom Design

Four test solutions containing different sodium chloride (NaCl) concentrations (0, 4.7, 11.7 and 25.6 g/L) dissolved completely in ultra-purified water were evaluated. CuSO₄ (0.15 g/L) was added to each solution to reduce relaxation, yeilding T₁ values between 1.7–1.8 s with T₂ values between 520–675 ms over the full range of the saline solutions used. Conductivity and permittivity were measured for each of the solutions using an HP Dielectric Probe (HP 85070B, Keysight Technologies, Santa Rosa, CA). The dielectric probe had a shield diameter of 1.6 cm and center conductor diameter of 0.9 mm. All measurements were conducted in 50-mL beakers to satisfy the width and depth required by this dielectric probe. According to the probe manufacturer, the accuracy of the permittivity measurement is ±5% (corresponding to a worst case of Δε_r < ±4 for the solutions used), and the accuracy of the loss tangent, tan δ, is ±0.05 (corresponding to a worst case of Δσ < ±0.19 for the ε_r of solutions used). For calibration at 900 MHz, the AC conductivity of ultra-purified water without CuSO₄, found by dielectric probe measurements was 0.14 S/m with relative permittivity of 79, which agrees with the theoretical conductivity (σ_w=0.19) calculated from the Debye dielectric relaxation and Cole-Cole (α=0) models for water at 25 C at this frequency with ε_w(0)=78.36, ε_w(∞)=5.2 and τ=8.27 ps (Kaatze 1997). Note, as 900 MHz enters the microwave regime, the frequency-dependent water conductivity is non-zero. Used as the ground-truth for MR experiments, the conductivities and permittivities measured in this fashion for each solution were tabulated (Tables 1 and 3), with the mean sand standard deviations calculated from five individual measures at 900 MHz.

Table 1.

Conductivity (Mean ± SD) calculated by different EPT reconstruction approaches and dielectric probe measurements. Within a reconstruction approach for a given RF coil, significance (p<0.05) was found between all of the samples using a one-way ANOVA with Tukey’s HSD post-hoc test.

	Dielectric Probe	Linear Birdcage Coil		Quadrature Saddle Coil
Different salinities	Average Conductivity at 900 MHz (S/m)	Full-form EPT Conductivity (S/m)	Phase-based EPT Conductivity (S/m)	Full-form EPT Conductivity (S/m)	Phase-based EPT Conductivity (S/m)
Tube 1 (0 g/L)	0.20 ± 0.003	0.27 ± 0.46	0.42 ± 0.49	0.64 ± 0.32	0.56 ± 0.32
Tube 2 (4.7 g/L)	1.01 ± 0.001	1.32 ± 0.31	1.37 ± 0.36	1.66 ± 0.27	1.54 ± 0.35
Tube 3 (11.7 g/L)	2.04 ± 0.001	2.31 ± 0.35	2.78 ± 0.33	3.09 ± 0.20	2.32 ± 0.21
Tube 4 (25.6 g/L)	4.08 ± 0.002	4.51 ± 0.28	4.61 ± 0.31	4.42 ± 0.21	4.27 ± 0.23

Table 3.

Relative permittivity (Mean ± SD) calculated by Full-form, Magnitude-based reconstruction and dielectric probe measurements.

	Dielectric Probe	Linear Birdcage Coil
Different salinities	Average Permittivity at 900 MHz	Full-form Relative Permittivity	Magnitude-based Relative Permittivity
Tube 1 (0 g/L)	78.74 ± 0.14	86.93 ± 10.03	83.25 ± 09.91^
Tube 2 (4.7 g/L)	77.66 ± 0.01	85.00 ± 13.40	82.81 ± 13.03
Tube 3 (11.7 g/L)	76.13 ± 0.06	81.95 ± 09.12	77.49 ± 10.88^
Tube 4 (25.6 g/L)	73.41 ± 0.05	72.04 ± 13.03*	69.44 ± 12.83*

For EPT reconstructions, significance (*p<0.05) was found between tube 4 and all other samples using a one-way ANOVA with Tukey’s HSD post-hoc test.

The only other significant difference was found between tubes 1 and 3 at the ^p=0.05 level.

For MR experiments, 3 mL of each of the four test solutions were placed in four separate transfer pipettes with diameters of 1.4 cm, filling a length of 4 cm. The pipettes containing the aforementioned NaCl concentrations were labeled as tubes 1–4, respectively. Pipettes were inverted such that the thin-walled pipette bulbs (wall thickness of 0.34 mm) could be placed simultaneously within a 3-cm cylindrical container that was filled with the solution of tube 1. This arrangement permitted multiple solutions to be positioned and scanned simultanouesly within the 33-mm diameter of an in vivo rat head coil and confines of a vertical bore, 900-MHz magnet. Although geometrically different from biological specimens, this phantom covered a range of biologically relevant conductivities and permittivities, and follows the phantom design used widely by others (Zhang et al 2013a, Lee et al 2015c) as an accessible means of testing the EPT reconstruction. Notably, tubes 2 and 3 provide conductivities that cover the range of gray matter (σ = 0.94 S/m) and cerebrospinal fluid (σ = 2.4 S/m) of the human brain at 900 MHz (Gabriel 1996). It also should be noted that at the operating frequency of 900 MHz, the RF penetration through the thin plastic wall is significant, and though an image discontinuity is present, the wall appears transparent to the RF and should not impact the EPT calculations beyond the boundary/image discontinuity. However, DC or conduction currents will be interrupted by even this thin insulating wall thickness (Kwon et al 2014). More biologically relevant assessments were performed on the ex and in vivo rat brain specimens described below for comparison.

Data Acquisition

Data were acquired at the National High Magnetic Field Laboratory using a 21.1-T (¹H frequency of 900 MHz) vertical magnet (Fu et al 2005) equipped with an Avance III spectrometer (Bruker BioSpin, Billerica, MA, USA) and ParaVision software. The 900-MHz system is an ultra-widebore magnet with a diameter of 10.5 cm and gradient bore size of 6.35 cm. The gradient system (Resonance Research Inc., Billerica, MA, USA) can produce magnetic field gradients up to 600 mT/m in all directions, with a rise time of 120 μs. Two home-built RF coils (Figure 1) constructed on a 33-mm diameter former were used for both transmission and reception: 1) a linear sliding ring ¹H/²³Na birdcage coil (Qian et al 2012); and 2) a ¹H saddle coil driven in quadrature mode (Muniz et al 2011, Rosenberg et al 2017). For the ¹H saddle coil, the most homogenous region of the coil was matched to the size of one tube; therefore, conductivities using both EPT approaches were investigated by rotating the phantom inside the coil and acquiring separate data with each tube centered within this homogenous FOV. For phantom acquisitions, the temperature was kept at 25 C in the MR scanner using a regulated water supply.

Figure 1.

RF coils: a) ¹H/²³Na birdcage coil; b) ¹H saddle coil

Two measurements were acquired for EPT: one to map the B₁⁺ magnitude and another to determine the B₁⁺ phase, both using a spin-echo (SE) acquisition from the conventional Bruker pulse sequence library (see supplemental data, Figure 1). The B₁⁺ magnitude map was generated using a double angle method (DAM) (Stollberger and Wach 1997, Cunningham et al 2006) without preparation in which the initial excitation was either α₁=60° or α₂=120° with a 180° refocusing pulse. The ratio of the signal intensities (R=I_α2/2I_α1) was used to calculate the flip map (α = cos⁻¹ R), which was then converted to the |B₁⁺ | map. For the B₁⁺ phase measurement, a SE image was acquired with a 90° initial excitation. As the resultant transceive phase image is the summation of both B₁⁺ and B₁⁻ phases, the transceive phase assumption was utilized to calculate the B₁⁺ phase map as half of the measured transceive phase (Overall et al 2010, van Lier et al 2012a). For both B₁⁺ phase and magnitude measurements, multi-slice 2D spin SE sequences were acquired with a TE/TR=15/2000 ms, resolution of 0.13 × 0.13 mm² for a field of view of 34 × 34 mm² and matrix size of 256 × 256 for phantoms. Nine slices were acquired in an axial direction with a slice thickness of 1 mm using a 6-ms three-lobe sinc excitation pulse. Trapazoidal crushers (10% maximum gradient strength with a 1-ms total duration) were applied before excitation and around the slice selection gradient applied to the 180 pulse to eliminate spurious signals and stimulated echoes. The SE sequence helped to exclude phase contributions from off-resonance effects, including susceptibility distortions. For phantom measurements, no eddy current correction was utilized, although compensated tests were performed subsequently to assess the impacts of eddy currents (see supplemental data, Figure 2). These eddy current correction tests of the phantom demonstrated less than an average 17% and 11% difference in the standard deviation and 13% and 9% in the RMSE of conductivity and permittivity, respectively. These variations were deemed negibile for the phantom, and thus eddy current correction was not persued. However, biological specimen (ex and in vivo) demonstrated potential eddy current impacts, necessitating correction as indicated below. In addition to the SE EPT measurements, a gradient-recalled echo (GRE) sodium image was acquired with TE/TR=1.18/300 and 4 averages using the ¹H/²³Na birdcage to confirm the position and relative salinity of the tubes in the coil.

Ex and in vivo measurements

To show the feasibility of the technique in animal models, data also were acquired from naïve preserved and ischemic stroke rats at 21.1 T.

Ex vivo sample preparation

Under anesthesia, two juvenile (~40-day-old) naïve male Sprague-Dawley rats (~200 g) were transcardially perfused with phosphate buffered saline and then 4% paraformaldehyde. Intact heads were harvested using a guillotine and immersed in 4% paraformaldehyde for at least 24 h. Prior to any imaging session, the preserved rat heads were transferred to phosphate buffered saline and washed for at least 24 h to remove excess fixative from the tissue. Before imaging, the heads were immersed in a non-protonated fluorocarbon (FC-43, 3M Corp, Minneapolis, MN) to provide susceptibility matching during MRI (Webb and Grant 1996).

In vivo stroke model

In accordance with the FSU Animal Care and Use Committee, an ischemic stroke was introduced by a transient middle cerebral artery occlusion (MCAO) in a single juvenile male Sprague-Dawley rat (200 g, ~40 days old) as described previously (Longa et al 1989). Briefly, the internal carotid artery and external carotid artery were exposed, and a rubber-coated filament (Doccol, Sharon, MA) was placed into the external carotid artery and guided 1.9 cm into the ICA until the middle cerebral artery was blocked. Occlusion was sustained for 1.5 h, followed by re-anesthetization and removal of the filament. The MCAO animal was imaged in vivo at 24 h following the occlusion using the ²³Na/¹H double-tuned birdcage RF coil discussed above. It was positioned and secured in the RF coil through the use of a bite bar, which also delivered gaseous anesthesia. Hip and girdle straps restrained the lower body of the animal as MRI experiments at 21.1 T are conducted with the rat in a heads-up, vertical position. The rat was induced with 3% isoflurane in 100% O₂, and maintained on 1–2% isoflurane through experimentation. Respiration was monitored, and acquisitions were triggered using a pneumatic pillow and animal monitoring system (SA Instruments, Inc., Stony Brook, NY).

To achieve ex and in vivo imaging times of 15 min, the acquisition matrix was reduced to 128×128, and a fast spin echo (FSE) with an acceleration RARE factor of four was utilized to generate B₁⁺ magnitude maps from the DAM technique described above as well as the B₁⁺ phase. The temperature was kept at 30 C in the MR scanner using a regulated water supply. Additionally, unlike phantom experiments, there was evidence of eddy current phase artifacts in the rat brain (Figure 2) similar to those identified previously (van Lier et al 2012a) as a decreasing phase superimposed on the overall phase map. Therefore, to eliminate the potential impact of eddy currents, the average phase from two opposite readout directions was used (Bernstein et al 2004, van Lier et al 2012a). All other data acquisition parameters were kept the same as the phantom imaging methods detailed above. In addition to ¹H MRI, ²³Na in vivo images were acquired in 23.5 min using a 3D ²³Na GRE sequence with 36 averages, TE/TR = 1.05/50 ms, FOV of (40 mm)³ and matrix of 40×40×40. The sodium MRI datasets were zero-filled to an image matrix of 64×64×40 to yield an image resolution of 0.625×0.625×1.0 mm³, and partition positions were matched to those used for ¹H EPT acquisitions.

Figure 2.

Phase distribution in x and y directions for phantom and rat brain specimens. a) Unwrapped phase image from the phantom displays a point localized within tube 2 that is subsequently delineated by vertical (black) and horizontal (red) phase lines displayed in b) and c), respectively. d) Unwrapped phase image from an ex vivo sample displaying a point subsequently delineated by vertical (black) and horizontal (red) phase lines displayed in e) and f), respectively. The drop in phase shown in e), which previously has been attributed to eddy currents (van Lier et al 2012a), required the use of eddy current correction for all rat brain specimens. Using reversal of the readout gradient in a subsequent scan and averaging the phase of the two scans, an eddy current corrected phase map was utilized for EP reconstructions. Shown in g) the corrected phase image and h) the corrected phase line correspond to (d) and (e), respectively.

EP reconstruction

A Helmholtz-based EPT technique also called standard EPT (Voigt et al 2011, van Lier et al 2012a, Ammari et al 2015, Van Lier et al 2014) was used to calculate conductivity (σ) and relative permittivity (ε_r). Based on this technique, EP can be calculated for regions with piecewise constant EP from the B₁⁺ field using the homogenous Helmholtz equation:

$${\nabla }^2{B}_1^{+}\approx -\omega {\mu }_0\kappa B_1^{+}$$ [1]

where κ = ε_rε₀ω + iσ, ω is the Larmor frequency, μ₀ and ε₀ are the vacuum permeability and permittivity, respectively. To calculate σ, the imaginary part of Eq. 1 was extracted, and Eqs. 2 and 3 were used for Full-form and Phase-based approach, respectively:

$${\mathrm{\sigma }}_{FF}=\frac{1}{{\mathrm{\mu }}_0\mathrm{\omega }}{\nabla }^2{\phi }^{+}(\text{r})+\frac{2\nabla \left|B_1^{+}\right|\cdot \nabla {\phi }_1^{+}(r)}{\omega {\mu }_0\left|B_1^{+}\right|}$$ [2]

and

$${\mathrm{\sigma }}_{PO}\approx \frac{1}{{\mathrm{\mu }}_0\mathrm{\omega }}{\nabla }^2{\phi }^{+}(\text{r}).$$ [3]

The Full-form approach requires both the amplitude and phase of complex transmit field $B_1^{+}=\left|B_1^{+}\right|\exp \left(-i{\phi }^{+}\right)$. However, if the second term of Eq. 2 is much smaller than the Laplacian of the phase, the first term dominates. In the phase-based approach, only the contribution of this dominant term in the estimated conductivity is considered.

To calculate ε_r, the real part of Eq. 1 was extracted, and Eqs. 4 and 5 were used for the Full-form and Magnitude-based approach, respectively:

$${\epsilon }_r=\left(\frac{{\left(\nabla {\phi }_1^{+}\right)}^2}{{\omega }^2{\mu }_0{\epsilon }_0}-\frac{{\nabla }^2\left(\left|B_1^{+}\right|\right)}{{\omega }^2{\mu }_0{\epsilon }_0\left|B_1^{+}\right|}\right)$$ [4]

and

$${\epsilon }_r\approx \left(-\frac{{\nabla }^2\left(\left|B_1^{+}\right|\right)}{{\omega }^2{\mu }_0{\epsilon }_0\left|B_1^{+}\right|}\right).$$ [5]

In the Magnitude-based permittivity approach, it is assumed that permittivity can be estimated by only considering the dominant term, which is the Laplacian of the magnitude of the B₁⁺. In this study, the accuracy of these simplified methods is compared to the Full-form approaches at the frequency of 900MHz.

In-house MATLAB® (The MathWorks Inc., Natick, MA) scripts were used to calculate conductivity and permittivity from the above equations. Phase unwrapping by Goldstein’s method (Goldstein et al 1988) was included in post-processing of the phase. To compensate for the inherent noise amplification of Laplacian operation in later steps, B₁⁺ magnitude and phase were smoothed with a moving average filter of 6×6 pixels. Additionally, the Laplacian kernels applied to the data incorporate a Gaussian filter as part of the numerical calculation. The Laplacian for both conductivity and permittivity was calculated by convolution with a large noise-robust kernel L of 5×7 presented below:

$$L_x=\left[\begin{array}{ccccccc}1& 2& -1& -4& -1& 2& 1\\ 4& 8& -4& -16& -4& 8& 4\\ 6& 12& -6& -24& -6& 12& 6\\ 4& 8& -4& -16& -4& 8& 4\\ 1& 2& -1& -4& -1& 2& 1\end{array}\right]=L_y^T$$

and

$$L_z=\left[\begin{array}{ccccc}0.5& 0& -1& 0& 0.5\\ 1& 0& -2& 0& 1\\ 0.5& 0& -1& 0& 0.5\end{array}\right]$$

The matrix is extrapolated based on noise-robust kernels published previously (van Lier et al 2012a, Holoborodko 2009). The Laplacian of the B₁⁺ phase or magnitude was calculated in the plane by summing the result of convolving data with kernel L_x to determine the x component and its transpose (L_y) to determine the y component, yielding a symmetric matrix (7×7) in-plane (Mandija et al 2017). Given the reduced number of slices in the z-direction, a small kernel of 3×5 was employed in that direction. Notably, for phantom calculations, the Laplacian of the phase along the direction of the slice selection gradient was close to zero due to symmetry along the tubes, yielding a negligible z component of the Laplacian. Although more noise robust, these larger kernels are susceptible to boundary errors as more pixels in transitional regions are affected.

Imaging Data Analysis

Conductivity and permittivity values were determined for phantom, ex and in vivo measurements based on regions of interest (ROI) analysis for which the means and standard deviations of all pixels within the boundaries of the ROI were calculated. For phantom experiments, EP values (Tables 1 and 3) were calculated from ROI centered in each tube and applied across the five middle slices of the slice package; ROI were chosen to exclude the boundary regions where computational artifacts dominate. In general, ROI placement avoided the 15 pixels nearest the tube periphery to minimize boundary artifacts. Based on this ROI analysis, statistical significance was determined between and within reconstruction and acquisition approaches using a one-way ANOVA with Tukey’s HSD post-hoc test (p<0.05). To compare each reconstruction approach further, the mean differences and root mean square errors (RMSE) were calculated for all pixels within the ROI for conductivity (Table 2) and permittivity (Table 3) separately. The RMSE was calculated using:

$$RMSE=\sqrt{N^{-1}{\sum }_{n\le N}{\left(x_n-\overline{x}\right)}^2},$$ [6]

where $\overline{x}$ is the target conductivity or permittivity measured by the dielectric probe, x_n is the estimated conductivity or permittivity for the n^th pixel, and N represents all pixels from the ROI. In addition, a linear regression (Figure 5) was performed on calculated EP values relative to dielectric probe values for each approach to determine the degree of correlation by means of a Pearson’s coefficient.

Table 2.

Percent Error to Dielectric Probe conductivity and Root Mean Square Error (RMSE) for different conductivity reconstruction approaches.

	Linear Birdcage Coil				Quadrature Saddle Coil
Different salinities	Full-form Conductivity		Phase-based Conductivity		Full-form Conductivity		Phase-based Conductivity
	Percent Error	RMSE (S/m)	Percent Error	RMSE (S/m)	Percent Error	RMSE (S/m)	Percent Error	RMSE (S/m)
Tube 1 (0 g/L)	35%	0.46	110%	0.55	220%	0.55	180%	0.48
Tube 2 (4.7 g/L)	31%	0.42	36%	0.58	64%	0.71	52%	0.64
Tube 3 (11.7 g/L)	13%	0.45	36%	0.88	51%	1.06	14%	0.35
Tube 4 (25.6 g/L)	11%	0.53	13%	0.61	8%	0.41	5%	0.28

Figure 5.

a) Mean conductivities from linear birdcage and quadrature saddle coils for Full-form and Phase-based approaches as compared with dielectric probe values; b) Mean relative permittivities from linear birdcage coil for Full-form and Magnitude-based approaches as compared with dielectric probe values. Lines represent linear regression fits of the dielectric probe measurements for each EPT reconstruction and coil configuration.

For ex and in vivo measurements, ROI were placed on central slices to calculate the pixel mean and standard deviation of conductivity and permittivity for the two reconstruction approaches using only linear birdcage acquisitions. Referenced to the corpus callosum on magnitude MRI, the EP ROI were located in a single slice transecting the striatal region of the ex vivo brain (Figure 6 and Table 5). For the in vivo case modeling ischemic stroke, two slices were analyzed (Figure 7). First, in a slice centered within the ischemic region, the striatum again was analyzed using ipsi- and contralateral placement of EP ROI referenced to the corpus callosum. Second, in a slice distal to the ischemic lesion, left and right hemisphere ROI were placed within the hippocampus. For all in vivo ROI (Table 6), care was taken to avoid the ventricles and major white matter tracks, effectively limiting the EP analysis to gray matter while maintaining central ROI placement to avoid boundary artifacts resulting from the reconstruction.

Figure 6.

Ex vivo rat brain: a) ¹H MR image also displaying the ROI; b) Full-Form conductivity; c) Full-form relative permittivity d) Phase-based conductivity; e) Magnitude-based relative permittivity.

Table 5.

Means and standard deviations for relative permittivity and conductivity measurements in the ex vivo brain.

ROI Left
Conductivity (S/m)		Relative Permittivity
Full-form	Phase-based	Full-form	Mag-based
0.93 ± 0.17	1.01 ± 0.17	57.11 ± 4.43	56.38 ± 4.44
ROI Right
Conductivity (S/m)		Relative Permittivity
Full-form	Phase-based	Full-form	Mag-based
0.71 ± 0.19	0.92 ± 0.17	60.76 ± 5.93	58.86 ± 5.90

Figure 7.

Two slices of in vivo rat brain, the top row is for a slice central to the ischemic lesion, bottom row is a slice distal to the lesion. a) ¹H image through the center of lesion displaying ROI placement b) Central Phase-based conductivity; c) Central Magnitude-based relative permittivity; d) ²³Na image with arrow designating the ischemic hyperintensity; e) Distal ¹H image displaying ROI placement; f) Distal Phase-based conductivity; g) Distal Magnitude-based relative permittivity; h) Distal ²³Na image displaying no sodium abnormalities.

Table 6.

Pixel mean and standard deviations for relative permittivity and conductivity measurements in the in vivo brain.

Slice position to lesion	ROI Left
	Conductivity (S/m)		Relative Permittivity
	Full-form	Phase-based	Full-form	Mag-based
Central	1.23 ± 0.36	1.29 ± 0.38	40.91 ± 5.13	39.66 ± 5.18
Distal	1.19 ± 0.17	1.20 ± 0.16	40.12 ± 7.10	39.02 ± 7.38
Slice position to lesion	ROI Right
	Conductivity (S/m)		Relative Permittivity
	Full-form	Phase-based	Full-form	Mag-based
Central	1.64 ± 0.22	1.91 ± 0.23	64.15 ± 6.67	61.42 ± 6.35
Distal	0.96 ± 0.41	1.21 ± 0.42	40.47 ± 6.70	37.72 ± 6.83

Results

Figure 3 displays the ¹H magnitude MR image, the |B₁⁺| map, the unwrapped transceive phase map, Full-form and Phase-based generated conductivity reconstructions as well as the Full-form and Magnitude-based relative permittivity reconstructions. All data shown are from the middle slice of the slice package for the linear birdcage coil. Figure 3a) acquired using a SE sequence shows loss of signal at the feed points of the linearly excited birdcage, which are adjacent to tubes 2 and 3. Except for these points, RF homogeneity is uniform in the volume birdcage as evident in Figure 3b). For the quadrature saddle coil, Figure 4 displays data acquired from each tube (1–4) positioned within the limited FOV of the saddle coil by rotating the phantom. Quadrature detection provided a 50% increase in SNR over the linear birdcage coil. However, the coil has a less uniform RF field as demonstrated in the line plots of Figure 4, with the |B₁⁺| and transceive phase dropping off as a function of the distance from the face of the saddle coil.

Figure 3.

Phantom with tube 1 = 0.21 S/m, tube 2 = 1.01 S/m, tube 3 = 2.04 S/m and tube 4 = 4.08 S/m in the birdcage coil: a) ¹H MR image displaying tube numbers (arrows point to the RF feed and ground points); b) B₁⁺ magnitude map; c) Phase image; d) Full-form conductivity (S/m); e) Phase-based Conductivity (S/m); f) ²³Na MR image; g) Full-form relative permittivity; h) Magnitude-based relative permittivity; i) ¹H MR image showing positioning of ROI for quantification.

Figure 4.

Phantom in the quadrature saddle coil: columns a through d correspond to positioning of tube 1=0.21 S/m, tube 2 = 1.01 S/m, tube 3 = 2.04 S/m and tube 4 = 4.08 S/m, respectively, in the saddle coil FOV. Row 1 images correspond to the B₁⁺ magnitude map, the blue dot on the upper right image shows the position of the coil. Row 2 images provide the transceive phase; Row 3 plots display the middle lines (element 128 of the matrix) from the B₁⁺ magnitude map (red) and transceive phase (blue) as a function of pixel distance from the coil surface. Rows 4 and 5 are the conductivity maps generated from the Full-form and Phase-based reconstructions, respectively. All the images are masked with 6% threshold.

Table 1 displays mean conductivities and standard deviations for each coil configuration and EPT reconstruction. Served as a ground truth, conductivity and permittivity values for solutions measured with the dielectric probe are also presented in Table 1. Table 2 provides the percent error difference with respect to the dielectric probe and RMSE for each coil configuration and EPT approach. Relative permittivities for both Full-form and Magnitude-based reconstruction of the phantom in the birdcage coil are presented in Table 3 using the same ROI as conductivity measurements. The percent error and RMSE of the relative permittivities are reported in Table 4.

Table 4.

Percent Error to dielectric probe permittivity and Root Mean Square Error (RMSE) for different permittivity reconstruction approaches.

Different salinities	Full-form Relative Permittivity		Magnitude-based Relative Permittivity
	Percent Error	RMSE	Percent Error	RMSE
Tube 1 (0 g/L)	10%	12.59	6%	11.02
Tube 2 (4.7 g/L)	9%	15.53	7%	13.98
Tube 3 (11.7 g/L)	8%	10.66	2%	10.66
Tube 4 (25.6 g/L)	2%	13.21	5%	13.21

Figure 5 (a) shows plots of calculated mean conductivities from any of the approaches versus the ground truth. Regression analysis of this data yielded linear fits with all R² values greater than 0.96 and Pearson’s linear correlation coefficients greater than 0.98 for all reconstruction approaches indicating that measured conductivities from each approach are in good correlation with the ground truth. Similarly, Figure 5b) shows plots of calculated mean permittivities versus the ground truth. Regression analysis of this data yielded linear fits with R² values of 0.97 and Pearson’s linear correlation coefficients of 0.99 for both methods indicating that measured permittivities are in excellent correlation with the ground truth.

Statistical analysis identified a unique and significantly different value for each conductivity as a function of the salinity within a given reconstruction approach and RF coil configuration. On the other hand and independent of the reconstruction method, permittivity values for the highest salinity sample (tube 4) were the only measures found to be consistently and significantly different from the other tubes (1–3). This finding was anticipated for an otherwise high dielectric phantom requiring a large addition of NaCl to induce losses and reduce the permittivity, as reflected in dielectric probe measurements. Additional statistical analysis is provided in the supplemental information (see supplemental data).

Ex vivo conductivity and permittivity maps from both reconstruction techniques are displayed in Figure 6 for a middle slice of a preserved rat brain. Calculated conductivity and permittivity values for the two ROI shown in Figure 6 are presented in Table 5. Figure 7 displays the in vivo result from the MCAO rat for two representative slices of the brain. One slice is selected within the stroke region and another slice distal to the ischemic lesion. Measured conductivity and permittivity for the two ROI, one placed in each hemisphere, for the two slices are shown in Table 6. The ipsilateral ROI was selected in reference to the corpus callosum (namely below this structure) to avoid boundary artifacts but still represent the center of the ischemic lesion and be largely devoid of white matter tracts. Its contralateral compliment and the ROI of the distal slice are matched in size and central location for comparison. ²³Na MRI images are also provided in Figure 7 (d) to illustrate the increase of sodium signal at 24 h post MCAO in the ischemic lesion versus the contralateral side.

Discussion

Higher magnetic fields necessitate application of a higher frequency B₁⁺ field to match the Larmor resonance frequency condition of target nuclei. As a result, sample size becomes non-negligible to the RF wavelength, and sample EP and geometry influence B₁⁺ field curvature. At 21.1 T (900 MHz), for which the B₁ wavelength $\left(\lambda =\frac{{\lambda }_0}{\sqrt{{\epsilon }_r}}\right)$ in pure water (ε_r ~ 80) is 3.7 cm, B₁⁺ fields generated by the RF coils (~3-cm diam.) interact strongly with the sample, and the field uniformity is highly influenced by the geometry and EP of the sample as well as the coil configuration (Amouzandeh et al 2018). On one hand, this coupling helps the detection of EP using Helmholtz-based EPT due to higher RF field curvature. On the other hand, it increases the difference between B₁⁺ and B₁⁻ phases, impacting the validity of the transceive assumption used in EPT reconstructions (Katscher et al 2009, Balidemaj et al 2015). However, this error is minimized if the sample has a cylidrical symmetry. Although not realistic in a biological sense, the use of cylindrical phantoms, which has been used widely by others (Zhang et al 2013a, Lee et al 2015c), provides an accessible means of testing the homogeneous Helmholtz EPT calculations. For the current cylindrical phantoms with diameters less than 3 cm at 21.1 T, the relative agreement of results with dielectric probe measurements demonstrate the potential and limitations of implementing Helmholtz-based EPT with the transceive phase assumption, as error analysis displays the higher susceptibility of conductivity estimates to all sources of phase error in comparison to permittivity estimates. In fact, permittivity measurements displayed good agreement with the dielectric probe and improved accuracy compared to lower fields. Promising results also have been collected for EP measurement of ex and in vivo rat brains using the same method, but point to the limitations of the approach and need for more elegant approaches to preserve resolution and reduce variability.

For the linear birdcage and even with violation of the transceive phase assumption at 900 MHz, an average percent difference of 23% for the Full-form EPT as compared to the dielectric probe measurement across all salinities was calculated (Table 1), with the lowest error of 11% evident in the highest conductivity. This average error is in line with the previous Full-form EPT studies at lower fields (Katscher et al 2009), which showed a mean error range from 3–20%. The Phase-based approach yielded on average a 49% difference from the conductivity values, which is at the lower end of the 44–100% reported at 7 T (van Lier et al 2012a). Table 2 also shows higher percent error and RMSE for the Phase-based approach versus the Full-form for the linear birdcage coil, reinforcing that there is a penalty for ignoring the gradient of B₁⁺ field even with the relatively uniform B₁⁺ profile excited by the birdcage. This penalty differentially impacts the samples evaluated but tends to be greater for samples (tubes 2 and 3) placed near the feed points of the birdcage or samples (tube 1) that have lower conductivities and may be susceptible to dielectric resonance effects. Based on this evaluation, EPT conducted at 21.1 T appears to introduce an average error of 0.47 S/m for the full-form approach; the exclusion of the B₁ magnitude term used in the Phase-only approach increases this error by a factor of 40%.

Outside of phased arrays, few previous studies have investigated implementing EPT with a surface transceive coil configuration, which motivated the evaluation of the quadrature saddle coil at 21.1 T (900 MHz). Based on the standard deviations, increased SNR of the quadrature saddle coil display tangible benefits in improving at least the variability and precision of the EPT estimates. However, as displayed in Figure 4, RF inhomogeneities resulting from B₁ drop-off from the surface of the saddle coil and the generation of standing waves from propagation in a quasi-uniform medium at low conductivities do impact the measured estimates of conductivity. Using the conductivity and relative permittivity of tube 1, the skin depth is δ₁ = 23.6 cm (Bottomley and Andrew 1978) while the wavelength of the B₁ field is λ₁ = 3.67 cm. With a sample diameter of 1.4 cm (>λ₁/4), length of 4 cm (>λ₁) and with no effective attenuation provided by the skin depth, tube 1 experiences standing wave formation similar to dielectric resonance in at least the z-direction. As a result, considerable phase shift and inhomogeneity in the B₁ field unrelated to the conductivity are expected and evident in tube 1, affecting the error percentages for both conductivity reconstruction approaches, though more heavily the Full-form due to its inclusion of the B₁⁺ magnitude term. For tube 4 with significant conductivity that yields skin depth of δ₄ = 1.26 cm, the lossy sample attenuates wave propagation and does not permit standing waves to develop. As a result, higher conductivity dominates the B₁⁺ phase and magnitude profile, resulting in increased accuracy for conductivity reconstructions.

As shown in Tables 1 and 2 for the saddle coil, the Full-form conductivity has an average difference of 86% from the dielectric probe across all salinities while the Phase-based approach yields an average 63% difference, though both averages are weighted heavily by the lowest conductivity that it is impacted by dielectric resonance effects. In comparison to the birdcage coil, the RMSE are more informative, as Full-form EPT of the quadrature coil shows an average of 0.68 S/m, which is 46% higher than the average Full-form RMSE for the birdcage, while Phase-based EPT shows the lowest RMSE of all approaches at an average of 0.44 S/m.

In addition to conductivity, the possibility of reconstructing permittivity was investigated in the phantom. Reconstructed permittivity values are in excellent agreement with the dielectric probe and show an average difference error of 7% and 5% in the Full-form and Magnitude-based reconstructions, respectively, indicating major improvements in the EPT estimations resulting from increased field strength. Permittivity measurements conducted on phantoms at 1.5 T displayed on average 26–90% error from the target; however, studies at 3 and 7 T showed 2–44% and 3–4% errors, respectively (van Lier et al 2012a). A recent study on the impact of B₁ mapping techniques for permittivity reconstructions reported a percent error of 4–54% in phantoms at 3 T depending on the B₁ mapping approach, although standard deviations were found to be greater than 50% of the reported mean permittivity regardless (Gavazzi et al 2019).

Reinforcing the improvement at 21.1 T, the RMSE of measured relative permittivities are consistent across all samples and between the two techniques. Admittedly, the current phantom does not provide a wide range of permittivities; however, it is notable that the lowest permittivity of tube 4 was determined to be significantly different from the other tubes for both reconstructions. This finding underscores the sensitivity of high field EPT to detect even small permittivity changes, even at the level of a 7–10% change in ε_r, with the Magnitude-based as robust as the Full-form reconstruction. Reduced error percentages and RMSE for the measurement of relative permittivity compared to conductivity indicate the relative robustness of permittivity reconstructions with respect to the violation of transceive phase assumption for 21.1 T.

In both conductivity and permittivity reconstructions of the phantom, the violation of the homogeneity assumption in the Helmholtz equations at the sample boundaries and the numerical calculation of the Laplacian have resulted in so-called “boundary artifacts.” At these boundaries, the application of the Laplacian kernel over regions with drastically different EP induces discontinuity that corrupts the EP measurement over a number of interfacial pixels, which are proportional to the width of the Laplacian kernel. As a result, although they offer noise robustness, larger kernel sizes still produce larger boundary artifacts. This limitation is consistent with most EPT approaches that utilize a numerical Laplacian, and high field has no impact on these artifacts but the reduced preclinical sample size (either of phantoms or animal neuroanatomy) does magnify the impact of such artifacts. For phantom experiments, the central regions of each sample tube provided at least an area of 4000 pixel², which was sufficient for EP estimations. However, the limitations of the homogenous Helmholtz EPT approach in this regard constrains the sample regions that can be analyzed while reducing the overall resolution of the EP maps. Additionally, variability in EPT data and relatively high standard deviations result from the implementation of the numerical Laplacian intrinsic to Helmoltz-based EPT. Effecting the conductivity calculation more than the permittivity, the derivative of the B₁⁺ phase (either 2^nd or 1^st order) is impacted by any phase noise. Other EPT reconstruction approaches that better accommodate transitions between boundaries and decrease or eliminate reliance on the numerical Laplacian (Jin Keun Seo et al 2012, Liu et al 2018, Borsic et al 2016, Guo et al 2018) may prove much more applicable at very high field. However, these alternatives often utilize multi-transmit phased array capabilities (Liu et al 2015, Zhang et al 2013a, Liu et al 2017a), which may not be available on preclinical systems (such as the current 21.1-T system employed here), but may motivate instrumentation enhancements to make more complete use of EPT approaches at high field.

In addition to the phantom study, the feasibility of reconstructing EP of the rat brain also was investigated using Helmholtz-based EPT at 21.1 T. For both ex and in vivo cases, the Phase-based conductivity and Magnitude-based permittivity reconstructions were compared with the Full-form calculations. The conductivity and permittivity results do not show a difference between the two forms of reconstructions for biological tissue. As such, the simplified techniques of Phase-based conductivity and Magnitude-based permittivity mapping appear to be as robust as the Full-form at 21.1 T for biological application. Due to the large noise-robust kernels employed in the Laplacian calculation, boundary artifacts are evident at the brain periphery, and there may be tissue boundary artifacts that are present within the more central brain. With non-homogeneous EP compartments, this limitation of Hemholtz-based EPT is significant for the reduced dimensions of preclinical neuroanatomy, and does not permit for the full resolution or sensitivity of high field EPT to be utilized.

Based on the ROI analysis, the mean conductivities and permittivities calculated for rodents at 21.1 T are in good agreement with theoretical EP values based on the Cole-Cole model for excised gray matter of the human brain at 900 MHz, which predicts σ = 0.94 S/m and ε_r = 52.72 (Gabriel 1996). Using Phase-based and Magnitude-based calculations, the percent difference between these theoretical values and the ex vivo EPT conductivity and permittivity are 3% and 9%, respectively. For measures in the non-pathological regions of the in vivo rat brain, the percent differences from the Cole-Cole model are higher, with the conductivity 31% higher and the in vivo permittivity 26% lower. It should be noted that the Cole-Cole model is specific for excised human gray matter, and thus is not impacted by blood flow, motion, temperature and volume averaging across different tissue types (i.e., gray matter, white matter and CSF) and neuroanatomical structures as in the in vivo rat experiments conducted at 21.1 T. Human studies of in vivo brain conductivity at 7 T using Helmholtz-based EPT also yielded higher conductivities than predicted by the Cole-Cole model, namely average EPT conductivity was 26% and 63% higher for gray and white matter, respectively (van Lier et al 2012a). In vivo permittivity assessments in human brain and abdomen up to 7 T did not yield robust permittivity Helmholtz-based EPT estimates that could be deemed reliable or quantitative, limiting their utility in clinically acceptable scans (Gavazzi et al 2019, Van Lier et al 2014). In a preliminary study conducted in a limited ex vivo rat cohort (Jensen-Kondering et al 2017), brain conductivities at 3 T for gray and white matter were 42% and 84% higher, respectively, than the Cole-Cole prediction for 128 MHz. Also, in a study of RF exposure over cellular communication frequencies (Peyman et al 2001), the average dielectric probe measured conductivity and relative permittivity ranges at 881 MHz for intact brains excised 2–4 h after sacrifice were 0.8–0.7 S/m (<1.08% standard deviation) and 49.2–44.3 (<6.15% standard deviation), respectively, for 30–50 day-old Wistar rats, which is in good agreement with the current ex and in vivo EPT measures at 900 MHz. Therefore, in vivo overestimation of conductivity at 21.1 T is at least consistent if not better than published values, with permittivity measures displaying even higher degrees of consistency, which supports the potential of EPT at 21.1 T to provide reliable quantitative estimates in accessible scan times.

More critically, the EPT values evident in the subacute ischemic lesion demonstrate a large increase in Phase-based conductivity (48%) and Magnitude-based permittivity (55%) compared to the contralateral side and more distal non-pathological regions of the in vivo rat brain. As such, these measures indicate significant remodeling of ischemic tissue with a high degree of detectability using Helmholtz-based EPT. Supporting this finding, previous single-patient case studies of ischemic stroke have reported elevated conductivity although with some exceptions (van Lier et al 2012b, Gurler et al 2016). In preclinical evaluations at 3 T, the work of Kim et al. identified a 43.2% increase in conductivity of canine brain 12 h after cerebral ischemia using Helmholtz-based EPT (Kim et al 2015), which is in excellent agreement with the current study’s findings. In another preliminary study conducted at 3 T, an in vivo transient MCAO rat also demonstrated conductivity increases on the order of 2.5 times the contralateral side (Jensen-Kondering et al 2017).

Uniquely, the current study demonstrates a large increase in permittivity for in vivo subacute stroke. This effect has not been evaluated thoroughly in the literature. However, one ex vivo study of edematous tissue (Kao et al 1999) conducted on canine white matter did display a positive correlation between increased water content and higher permittivity over 100–1000 MHz. Although not examining ischemic tissue specifically, the aforementioned RF study (Peyman et al 2001) demonstrated consistently decreasing conductivity (>16%) and permittivity (>10%) as measured by dielectric probe for intact excised rat brain from 0–70 days, which was attributed primarily to the loss of water content and decrease in free-to-bound water fraction with age and structural maturation. Likewise, the recent work of Michel et al. (Michel et al 2017) correlated the anatomically specific water content to conductivity and permittivity, showing that increased EP were evident for structures and tissue types with higher percentages of water. As EP in biological tissue reflect water and ionic content at high frequency (Oh et al 2011, Schwan and Foster 1989), the alteration detected in the EP of the ischemic lesion can be explained by tissue compartment changes following the disruption of cerebral blood flow, which result in the presence of edema as well as increases in sodium concentrations extra- and intracellularly during the subacute phase (Lee et al 2015a, Roussel et al 2018, Hussain et al 2009, Tsang et al 2011, Moseley et al 1990, Boada et al 2005). As such, the increased EP values 24 h post MCAO are consistent with the established increase in tissue water content (displayed by ¹H FSE in Fig 6a) and sodium (evidenced by ²³Na GRE in Fig 6d). The observed increase in the EP of ischemic stroke lesion indicates the importance of further studies and more applicable EPT approaches to track stroke and its dynamic evolution.

Conclusions

In this study, conductivity and permittivity mapping were implemented using Helmholtz-based EPT for ultra-high field (21.1 T). Based on previous studies at fields higher than 3 T, wave interference patterns affecting B₁⁺ phase and magnitude can introduce error in the EPT reconstruction, decreasing the validity of Phase-based approach (Van Lier et al 2014). However, electrical properties and sample/coil geometry also should be considered. For operation at 900 MHz, sample sizes of less than the RF wavelength and increased sample conductivity were shown to minimize the wave interferences such that the calculated conductivity values are in relative agreement with the target using either the Full-form or Phase-based approach. The estimated conductivity values from these two approaches verified that only considering second order variation of transceive phase still approximates the target conductivity, although incorporation of the |B₁⁺| does improve measurement accuracy and precision if the RF field characteristics (such as dielectric resonance or B₁⁺ drop-off as seen in the saddle coil experiments) do not dominate and confound the EPT estimations. Overall, the study shows that the error between Phase-based and Full-form approaches is approximately 25% of the target value at this field strength.

Permittivity mapping at 21.1 T showed a significant improvement in comparison to lower fields. Relative permittivity measured in each tube of the phantom was in good agreement with the dielectric probe value. Additionally, the error between the Full-form or Magnitude-based approach in calculating relative permittivity was less than 5% of the target value, indicating that permittivity distribution is mostly defined by B₁ curvature at higher field.

The feasibility of EP mapping for in vivo, as well as ex vivo rat brains, were investigated in a limited cohort of specimens. Although Helmholtz-based EPT in rat brain significantly suffers from tissue boundaries artifacts and variability, reasonable conductivity and permittivity values were derived from selected ROI. However, limited resolution and contrast were provided solely by homogenous Helmholtz-based EPT. Encouragingly, as a proof of principle, in vivo results showed elevated conductivity and permittivity for an ischemic lesion, reflecting the increased water content and sodium elevation in this region and potentially indicating that permittivity measures may be more robust to ischemia-related alterations. With application of EPT reconstruction approaches that better address heterogeneous compartments over preclinical length scales, conductivity and permittivity as a function of time post occlusion may be able to map the evolution of ischemia over acute and chronic phases.

Although the current findings demonstrate the limited utility of Helmholtz-based EPT approaches at high magnetic fields, these numerical approaches offer the potential for increased precision and reduced error at high field (particularly for permittivity measures). The evident limitations to resolution and variability also serve as motivation for more advanced EPT reconstruction to be pursued, particularly those that avoid the transceive phase assumption (Hafalir et al 2014, Liu et al 2015, Gurler and Ider 2016, Liu et al 2018). As such, this study has ramifications for electrical property tomography at ultra-high field for both materials and biological applications.